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Текст
nPHKJIAJ{HMI MA TEMA THKA
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KHHETHqECKHE CXEMbI
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2004
y tu< 519.8:533
66K 22.193
11-52
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1I-52 KHHeTR..ecKRe cxeMW B KSa3HI"'830j!RHaMH'IeCKU
CHCTeMa ypaBHeHHI. - M.: MAKC npecc, 2004. - 332 c.
ISBN 5-317 -00974-X
B MOHOrp8( HH H3naraeTCi H081..A I1OAXO;t Ie nOCTpoeHHIO BbI'iHCTKTenb-
"wx MeTOllOB B nuo80R lIHH8MHKC" HCnOCpcllCTBCHHOR OCHOBOA nnl nOJI/"ICHHJI
K010pIJIX C11Y*8T Monelnt Ani 0l1HO'f8CTHIfHOA 4 YHKUHH pacnpcJlCnCHMJI. 06c} .
JlaIOTCJ1 B03MO)l(HOCTH HHTCpnpeTlUHH nOCTpoeHHoR CHCTeMW ypaBHeHHA It IC8'4C.
crBC MOACJ1H, onHCWBaIOweR Te'tCHHe BJlJKOrO ra:J8, a T810Ke nOIC83L1Ba c. ec
cU:SIJ C Tp8l1MU,HOHHWM OnHcaHHeN, onHpaJOWHMCA Ha ypaaHcHHI HUbe-CTOlCca.
npMIO.48TC. npMMepw HCnOnb3CB8HHJI npellJ10)KCHHW" anropHntOB nnl MOlleI1H.
po.8ItH. COBpeNeHH IX npo6neN ra:JOBoA HH8MMnt. J)OJlbWHHCTBO pe:lYJlltTaroB
nOll)"leHO c npHMCHCHHeM BWCOKonpoKJ80D.MTCJ1 HWX MHOronpoueccopHWX 8W-
'fHCnHTenIaHY" CMCTeN. OcH08HU 'IacTL HCc.nCJIOBaHMA BhlnOJlHCH8 npH n'HaHco-
BOA nOJlll\:p*lCe PoccHAclCoro +oH.aa 4»YHJ18NeHTaJlbHbIX HCCnClIOBaHHA no npoelC"
TY K 03-01-00606.
.nn. cneQHaJ1HCT08 no lTeN811I'ICCICOMY MOJleJ1MpoBBHHIO. r830BoR llHtl8-
MHKC H npHK.naLlHOA MlTeM8THKC acnHpaHTOB H CTYAeHTOB COOTBeTCTB)'JOutHX
cneUHMbHocTCM.
Y JJ.K S 19.8:533
66K 22.193
Pe.aaKT'Op: fl.8. ,/l0p()(}HullbrH
ISBN S-317-00974-X
@ lfeTBepywkHH 6.H., 2004
@ aKyn TeT BMHK
Mry MM. M.B. ROMoHocoBa.2004
OrJIasneHHe
BBeJl.eHHe
rJlasa I. YpaBHeHHe BOJIb aHa H YpaBHeHHJI rasoBoA
HHaMHKH
I. OCHOBHbJe nOHSITHSI KHHeTHqecKOA TeOpHB ra30B
1. O HOqacTHqH8,jI CPYHK g pacnpeAeJIeHHSI . .
2. CTOJIKHOBeHHe MOJIeKyn . . . . . . . . . . . . .
3. ,lI.nHH& cBo60 oro npo6era.. "lJaCTOTa CTOJIKHoBeHHA
2. Y pasHeHHe BOJ1b MaHa ... . . . . . . . . . . . . . .
1. BbIBO,n ypaBHeHHSI ,lJ.JISI OAHO'lacTH'tIHOA <PYH
pacnpe)J.eneHHJI ............ . . . .
2. CBoACTBa HHTerp8J1a CTWIKHoseaHA .
3. H-'reopeMa BOJTh MaHa . . . . . . . .
4. PacnpeAeJIeHHe MaxcBeJIJl&. . . . . .
5. YpasHeHHe BOJIh aHa AlIa CMeCH ra30B
6. YpasHeHHe BxaTHar&pa-rpocca-Kpyxa
(BrK-Mo enh) . . . . . . .. .. . . .
7. Y pasHeHHe 3HCKora . . . . .
3. YpaBHeHHSI ra30BoA HHaMHKH
1. MOMeHTHbJe yp8.BHeHHSI . . .
2. Y paBHeHHSI 9Anepa . . . . .
3. BH3KOCTb H TennonpOBO OCTL .
4. YpaBHeHHSI HaBbe-CTOKCa ....
1. Pa3nO)l(eHHe t.{euMeH&-9HCICOra
2. JIOKaJIbHO HaB&e-CTOKCOBCX&S 4>YHK g
pacnpe eJJeHHjI .................
3. ra30,l1.HHaMHQeCKHe ypaBHeHHSI HaBbe-CTOKca
4. Yp8.BHeHHH HHaMHKH CMecO ra30B
5. YpaBHeHHH fps,l(a . . . . . . . . . . . . . . . . .
8
14
14
14
17
20
22
22
26
28
30
31
32
33
34
35
38
41
44
44
48
49
54
56
4
OrllaBJIeHHe
rnaBa II. KUHeTH1.feCKH COrJlaCOBaHHhle pa3HOCTHble
cxeMbI
9 1. BblBo KHHeTHQeCKH COrJlacoB&HHbJX cxeM .
1. KpaTKHA o63op 6onh M8.HOBCKHX cxeM
2. CooTHoweHHe 6a.naHca AnSI cPYHKl.\HH
pacnpeAeJJeHH ..................
3. Pa3HOCTHble ypaBHellHSI AJlSI ra30 HHaMHqecKHX
napaMeTpoB . . . . . . . . . . . . . . . . . . . . . .
4. CornacoB8HHoCTb pa3HOCTHhlX annpoKcHM a
5. IIpHMepb! paCqeTOB . . . . . . . . . . . . . . . .
2. KHHeTHqecKH cornaCOBaHHhle CXeMI:d
H& npoH3BOJILHblX CeTKaX . . . . . . . . . . . . . .
1. CooTHoweHHe 6anaHca nB cPYHK
pacnpe,neneHHSI Ra npOB3BOJIbHoA: CeTKe . 73
2. KCPC Ha npSlMoyroJIbHblX CeTKaX . . . . 75
3. KCPC Ha TJ>eyronbHblx CeTKax. . . . . . 76
3. KHH. HqecKHe cxeMbi nOBblmeHHoro nopB,ll,Ka TO'IHOCTH 84
1. CxeMbI nOBblweKfloro nops)J)<a TOqHOCTH NISI peme-
HHB KHHeTHqecKOro ypaBHeHHSI. . . . . . . . . . . .
2. CxeMbl nOBhlweHHoro nop TO'lHOCTH
Ha TpeyrOJIbHblX CeTKax. . . . . . . . . . . . . . . .
4. KHHeTHQecKHe CXeMLJ H pa3JIO>KeHHe
QenMeHa-9HcKora . . . . . . . . . . . . . . . . . . . .
5. KBa3Hra30t\HH8MHqeCKaJI CHCTeM& ypaaHeHuA . . . . .
1. ,n:H<t><PepeH H8JIhHO- pa3HOCTHaH <POPM& ypuHeHWI
6anaHca ,l{JlSI cPYHK HH pa.cnpe eJIeHHSI . . . . . . .
2. E e O)UlH cnoc06 nOJJy eHHg ypasHeHHB 6anaHca .
3. KB33Hra30 HHaMHqeCK&Sl CHCTeMa ypaBHeHHA
4. P8.3HOCTHag annpoKcHM HSI . . . . . . . . . . . . .
. . . .
59
59
59
62
65
67
71
73
. . . .
85
88
92
97
97
102
103
107
rnaBa III. IIpHMeHeHHe KHHeTU'IeCKH COrJlaCOB8HHhlX
cxeM H KBa3Hra30,t\HH8MHQeCKOA CHCTeMbl ,lJ.JIB
Mo enHpOBaHHB TeqeHHA BJl3KOrO ra3a 111
1. KHHeTHQeCKH cornacoBaHHble cxeMbi Ka.K
Mo.n-enb NISI OnHCaJlHSI TeqeHHA: BSI3Koro raJa . . . .. 112
OrJ1aBJleHHe 5
1. HeKOTOpbJe 3a.Me'laHHH K BbIB0A}' ypaBHeHHA raao-
BOA: HHaMHKH . . . . . . . . . . . . . . . . . . . .. 112
2. 3aMeQaHWi 0 AHCCUnaTHBHLIX qJIeHax B KCPC . .. 115
2. KBa.3Hra30,lJ.HllaMHQeCKaH CHCTeMa ypaBHeHHa H ypau-
HeHHSI HaBbe-CTOKca ..... . . , . . . . . .. 118
1. KBa3Hra.30 HHaMH1.{eCKaH CHCTeMa , . . . . . . . .. 118
2. CBSl3b C yp8.BHeHHHMH HaBbe-CTOKca . . . . . . .. 120
3. HccHnaTKBHble qJIeHbJ KBa3Hra30 HHaMHqeCKOA
CHCTeMbI , . . . . . . . . . . . . . . . . . . . . . 124
4. HeKoTopble pe3ynbTaTbI qJfCJIeHHhlX pacqeTOB 127
3. KHHeTHtfeCKOO OTp8.)KeHMe
KBa3Hra30 HHaMH\feCKo:A CHCTeMbI ypaBHeHHA . . . .. 131
1. KrY-cHCTeMa H O,ll)lOqacT HaJI <j>YHKlJ)IjI
pacnpe eneHHa .................... 131
2. KB83Hra30,l\HHaMH'IeCKaH CHcreMa H
B rK - ypaBH eH He . . . . . . . . . . . . . . . . . . .. 133
9 4. 06 HHBapHa.HTHOCTH KBa3Hra30 HaMJ[qecKOH CHCTe-
MbI ypaBHeHHA . . . . . . . . . . . . . . . . . . . . .. 136
5. 0606ll.\eHHSI Ha ,lq)yrHe 3 a'lH MeX8.HHKH
cnnOWl10A cpe bI . . . . . . . . . . . . . . . . . . . .. 140
1. Mo enHpoBaHHe npo ecca cPHnbTP81UlH . . . . . .. 140
2. B03MO)KItaJ1 MO B<j>HI<aD;HSI ypaBHeHHSI BonhltMaHa. 146
3. K Bonpocy 0 nap8J\oKce 6ecKOHe'IHO 6wCTporo
pacnpocTp8JfeHHH TeIIJJa .. . . . . . . . . . . . 148
9 6. Lattice Boltzmann H KHHeTHqecKHe cxeMbI. . . . 150
1. LBS Ha OCHOBe BrK-Mo enH - LBGK-cxeMbI 151
2. KBa3Hra30AHI-IaMHqecKaft CHCTeMa H LBS . . . . 154
7. MeTo CT86HJIH3a HOHHblx nonpasox H HeKoTopble
BapH8HTbI Kr"Y -CHCTeMbi .. . . . . . . . . . . . . 156
1. MeTOA CTa6HJlH3 HOHHblX nOnpaBOK . . . . . . 156
2. KB33HrH;J;pO,ll)tHaMHqecKasJ CHCTeMa ypaBHeHHA 160
3. KB83Hra30AHtlaMH'IeCKaB MOAeJlb AJlB OIlHCaHHSI Te-
qeHHH CJla60C>KHMaeMbiX neOB H >KH.ltKocTeA . . .. 162
6
OrnaBJJeHlfe
rna.Ba IV. BbI'IHCJlHTeJlbHble aJIrOpHTM&I H
HCDOJJb30B8HHe MHoronpoqeccopHhIX CHCTeM nSl
pemeHIUI 3 aq 1'830801 HaMHKH 110
1. KSHeTKqecKH cornacOB&HHhle cxeMbI C KOppe K 1I.BeA .. 171
1. Pa3HOCTHble cx:eMhI Ra npSlMoyrOJIbHblX CeTKaX . .. 171
2. KCPC c KoppeK Hea H ypaaHeHHSI H&BLe-CTOKCa.. 174
3. KCPC C KOppeKu,leA H& HeCTpYKTYPHpOBaHHhlX
Ce'I'Kax .. . . . . . . . . . . .
2. nonYHesBH8S1 cxeMa . . . . . . . . . . . . . .
9 3. HegSHble CXeM&1 S1 pemeHHB
KBa3Hra30)J;HHaMHQeCKOR CHCTeMbi YPaBHeHHA . . . . .
4. MHoronpo eccopHbJe BhJ'IHCJIHTeJlbHble CHCTeMbJ .. .
1. HCTOpHfI H nepcneKTHBhI p83BHTHS MHoronpoqEc-
COplWX CHCTeM .............
2. MeTaKOMllblOTHHr H KJlacrepbl . . . . . . . .
5. alrra HjJ KCPC K apXHTeKType
MHoronpou.eccopHhlX BLlQHCJ1HTeJIbHbIX CHCTeM . . . .
1. Tpe60B&HHB K BbI'IHCJIHTeJIbHblM anrOpBTM8M . . .
2. MHoronpo eccopHblA BapH8HT KCPC C KoppeKqHea
3. IIOJIYHeHBHaH CXeM& H KCPC H& HecTpYKTypHpo-
BaHHhlX ceTKaX - MHOronpO ecCOpH8.SI 8J:tan HB .
96. (a-,8)-HTepaII.HoHHhIA MeTOA. MHoronpoQeCCOpHbJA
181
185
191
196
197
. 202
204
205
207
212
'. 213
214
218
221
BapHailT . . . . . . . . . . . . . . . . . . . . . . . .
1. (a-,8)-HTepa HOHHbla MeTOA . . . . . . . . . . .
2. MHoronpo eccopHhlA BapHaHT (a-{J)-MeTo;J.&
3. n pHMepb! paCqeroB . . . . . . . . . . . . . . . .
7. HeKoTOpble AOfiOJlHHTenbHble BonpOChl
napaJIJIeJlLHblX BhI'IHCJIeHHA . . . . . . . . . . . . . . .
1. BH3YaJIH3&11.HSI auHblX MHoronpo:qeccopHLJX
Bb! lJHCJ1eHHA . . . . . . . . . . . . . . . . .
2. Pan:HoHa.nbHoe pa36HeHHe Ha nO)J.o6JIaCTH .
3. ,LI.HHaMHQeCK8J1 nepe3arpY3Ka npo eccopoB
224
224
226
228
rJIasa V. MO elIHpOBaHHe Te'leHHA l'83oBOA cpeAhi 231
1. MOAenHpoSaHHe CTpytbloro Te'leHHH . . . . . . . . 231
OrJlaBJIeHJle 7
2. MO,lteJIHpOBaHHe HeYCTaHOBHBWHXCH
ocu.HJIng OHHLlX Te eHHft. . . . . . . . . . . . . . .. 236
1. t.{HCJJeHHoe MO e.nHpOB8HBe OCItHJI.1UI HOHHbIX pe}KH-
IOB B6nH3H KaBepHbJ . . . . . . . . . . . . . . 237
2. BJlHSlHHe CPOpMbI BbleMKH H& xapaKTepHCTHKH
nYJJbC8.QHOHHOro Te'leHHSI. Kone6aHHB 06JIH3H
HnHHAPH'IeCKHX H c<}>epa'iecKHX Ten . . . . . 250
3. B3aHMOAeACTBHe OCl.\HJlnHpylO TeqeHHI ra3a
H ynpymx nOBepXHoCTeA . . . . . . . . . . . . . .. 258
4. MO eJlRpOB8.HHe Typ6YJIeHTHblX HeYC'I'8HOBHBwHXca
retJeHHit B OKpecTHOCTH KaSepHbI . . . . . . .. 264
3. reHep8J:lH1I H pa.cnpOCTpa.HeHHe MaJIblX
(aKYCTHQeCKHx) B03My eHHA: . . . . . . . . . . . . .. 272
1. MOtl.eJIHpOBaHHe pacnpOCTp&HeHHSI M8JIbiX B03MY-
eHHa . . . . . . . . . . . . . . . . . . . . . . . .. 272
2. MO eJlHpOBalIHe HCTOqHHKa aKYCTlRecKHx B03MY-
llI.eHHA .. . . . . . . . . . . . . . . . . . . . . . .. 280
3. MO,D.eJlHpOBaHHe WYM& CBepX3SYKOSOA' CTpYH . 285
4. Mo enHpOBaHHe Te'leHHA XHMHQeCKH
pearHpylOJ..U.Hx ra30B . . . . . . . . . . . . 287
1. MaTeMaTH'IeCKaB nOCT8.HOBKa 3 a'lH . . . . 288
2. Bbl'lHCJJHTenbHhlA 8JIrOpHTM . . . . . 291
3. MO eJlHpOB8.HHe ropeHHSI npHpo oro f'83a . . . .. 295
5. MOAeJIHpOBaHHe TeqeHBA YMepeHHO pa3pe>KeHHhlX
ra30B. CTPYKTypa y apHoR BOJIHbI . . . . . . . . . 301
1. B03MO>KHOCTb npHMeHeBHa KCPC A1U1 ODHC&HHB
re'ieHHA YMepeHHO pa3peJKeHRLJX ra30B . . . . . .. 301
2. Kry -CHCTeMa C rpaHH'lHhIMH ycnOBHSlMH
CKOJIb>KeHHSI . . . . . . . . . . . . . . . . 306
3. KCPC H MeTO domain decomposition . 308
4. MO eJ]HpOBaHHe CTPYKTYpbJ y,n:apHoA BOJIHLI . . .. 310
3aKJIIO"IeHHe 313
CnHcoK nHTepaTyphi 314
BBeAeHHe
,LI.aHHasI MOHOrpa<l>H9 nOCBBl.I.\eHa oco6oMY THny anrOpHTMOB,
npetl.H33HattellHhlX )J.J1S1 OnHCaHHH WHpOKOro KJIacca aq rH;JJ>O- H
ra30BOA )].HHaMHKH. 9ro KHHeTHtJeCKH cOrnaCOBaHHhle p83HOCTHbIe
CXeMbi (KCPC) Ii HX AH<I>4>epeH HaJThHOe npH6JJH)KeHHe - KBa3H-
ra.30AHHaMH'IecKaSl CHCTeMa YPaBHeHHA (KrY). HHTepeC I( ,D;8HHblM
HCCJ1e)\OBaHHSlM B uacTOg ee BpeMSI BbJ3Ba.H CJIe lOlI(HMH o6cros-
TenbCTBaMH.
C O)l.HOR CrrOpoHbl, Ha6nlO,AaeTCSI 6b1CTpb1A nporpecc 8bJttHCJIH-
TenbHOA TeXIIHKH, nHKOBbJe MOw.aOCTH KOTOpoR Y>Ke npeBbJWaJOT
10 TFLOPS (10 13 Onepal.\HA c nJJae&lO cA 3alISlTOA 8 CeKYHA}').
B no 06HoA CHTY HH B03HHKa.eT TeXHH\feCKa5I B03MO)I(HOCTb npH-
crynHTb K lJHCJleUllOMY peWeHHIO HaH60nee CJlO>KHhlX 3 aq rHJ.ij)o-
H r830BOA ,L\HHa.\fHKH. ,L(onoJlHHTMbHbIA HMnynbc BbiCOKOnpOH3BO-
AHTeJ1bHbJM BbJtJHCJlelIH.ftM TaK)J(e npHll.aeT TO 06CTOBTeJlb<:TBO, "{TO
MHOrOnpo11-eCCOpHbIe CHCTeMbl MO HOCTblO nopJ1):tKa 100 GFLOPS
(1011 Onepart: HA C nna.salOIl\eA 3anBToA B CeKY Y) H3 p83pSltl-8 YHH-
KaJlbHbIX YCTaHOnOK nepexollrSlT B c<pepy 060PY OBaHHHJ ,L\OCTyouoro
MHorHM Cne HaJlHCTaM. 9TO B npHH Hne n03BOJI5IeT WRpOK M <PPOH-
TOM BHeAPHTb Bb1COKOnpOH3BO HTeJIbHhle BhlqHCJleHWI B npaKTHKY
ra.30AHHaMH'leCKOrO MO emq>OBaHHH.
C pyroa CTOpoHbI, nOTeH H8JIbHbIe npeHMY ecTBa HCnOJIL-
30BaHHH MHOrOnpO eCCOpHblX BbICOKOnpOH3BOJ:(HTeJlbHbJX CHCTeM B
3Ha.1.IHTeJlbHOH Mepe 6J10KHPYIOTCSI TPY,lJ)IOCTHMH 8..l\anTal.\HH K ax ap-
XHTeKType Bbl'lHCJlHTenbHblX anropHTMOB. B nepoylO Oqepe,Ab CKa-
3aHHOe OTHOCMTCH K nOJIY IHBWHM llaa60J1bWee pacnpocrpaHeHHe
MHol'orlpo eccOpHblt.( CHCTeMaM c pacnp eneHHoA naMftTblO. liMe-
IOTC.H T8J()Ke 3HaqHTenbHbie npo6neMbI B C03AaBHH CnelUtaJIbHOrO
MaTeMaTHQeCKOrO 06eCne'leHHjI, He06xo,nHMoro AlISi npOBe eHHs[ BbI-
COKonpOH3BO HTenbHbJX napaJIneJlbHblX BbItfHcneHHA.
9
O Ha.KO HaH60JIbWYIO CJI0JKHOCTb, no MHeHHlO &STopa, BLJ3bJ-
BaeT np06JIeMa HAeHTH<I>HJ<.aQHH aHHbJX Bb1COKOnpOH3BO HTenbHbIX
BbJt.IHCJleHHA, HCnOnb3YIO HX npOCTpaHCTBeHHbIe cerKH 83 10 9 Y3JIOB
H 60J1ee t COnOCTaBJ1eHHH 3TOA HH<PoPMa HH C ,AaHHbJMH HaTypHoro
3KcnepHMeHTa. OC06etIHO OCTpo YKa.3a.HHaJI npo6JJeMa BCTaeT npH
pacqeTe HeYCTaHOBHBIlIHXCB TeqeHHA BB3KOrO ra3&. npH npSlMOM Mo-
enHpOB8JfHH TYP6yneHTHblX Te'lelIHA H npo eccoB ropeHHSI, B 3a-
aqax aapoaxYCTHKH H B PJ>yrHx nOCT&HOBKaX. rAe npHH HnHaJIbHO
Sa)KHO npHsneKaTL no po6Hoe npoCTpaHCTBeHHo-BpeMeHHoe ODHca-
HHe cpe bl. Pe3YJlbTaTOM MO eJlHpOB8HHS 3,l'(eCb Heo6xo)\HMO HMeTb
KapTHHY nOBe eHHH eCTecTBeHHhlX HeycroA:"IHBocreA, Bh!3B8JlijblX
cj>H3HQeCKHMH npH'IHH3MH, a He HCkYCCTseHHblX HeYCTOAqHBOCTeA
BLlqHCJlHTenbHOro xapaKTepa. PemeHHe 3TOit np06J1eMhl HanpBMYJO
CB938HO C BblSlCHeHHeM KOppeKTHOCTH Hcnonh3yeMLIx aJIropHTMOB
Ii M&TeMaTHqecKHX MO,.lJ.eneA. Hccne..a-OBaHHjI B aHHOM Hallpa.BJle-
HHH npe,L\CTaBJlHIOTCH HenpocTOA 3a.u.a1JeA. OTMeTHM, sanpHMep, qrro
npo6neMa KoppeKTHOCTH ypaBHeHHR HaBbe-CTOKca - MO,l\eJJH, Tpa-
,4H MOHHO HCnOJIb3yeMoA JVIH OnHcaJlHH Te'leHWI BH3KOro ra3a, -
HBJIHeTC O HOA: H3 KJIlOtoIeBhlX B coopeMeHHoA MaTeM&THKe.
B CBH3H C BbIWeH3nO)f(eHHblM B HacToH ee BpeMSI Bee 60Jlbillee
3Ha'leHHe npH06peTalOT anropHTMbI, KOTopble, C o)J;HoA CTOpoHbl, o-
nYCKaJOT npOCTYIO an 1O K apXHTeKType MHoronpo eccopHbJX
BbJT.fHCJIHTenbHbJX CHCTeM C pacnpe eneBHOa DaMSlThlO, a C .DJ>yro:R
- n03B0J111JOT YBepeHHO MO)J.eJlHpoBaTb MHoro06p83HbJe cJ:>H3H'IecKHe
HeYCToA'QHBOCTH, B03HHKaI01I(He 8 r830BOA cp e. KHHeTH'leCKH co-
rnacoBaJlHLle pa3HOCTHble cxeMhl CTOBT B PBJIY T8.KHX a.nropHTMOB.
B qeM OTnHqHe KHHeTRqeCKHX cxeM 01' tU>ynvc MeTOAoB pe-
weHHSI 33.AaQ ra30BoA HHaMHKH? 06b1QHO B Ka'leCTBe nepBOOCHO-
BbI )VIR HanHca.HHSI pa3HOCTHoA cxeMbI cnY)l(8T MaxpocKonH eCKHe
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10
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11
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12
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13
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rep H HMeeT D:eJlblO 06ner'fHTb ,lJ.8JIbHeatuee 'ITeHUe KHKPH. BOJIee
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1. OAHOqacTHtIHaR cPYHK JrJl pacIIpeJ{eJleHHB
KaK H3BeCTHO, ra3 COCTOHT H3 orpoMHoro 'IHCJla Mo,leKyn. Hx
c806o Hoe nepeMem.eHHe H CTOJIKHOBeHHa ,z:u>yr C pyroM onpe,lI.e-
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9 1. OCHoBHble nOHBTHH lCHHeTH'fecKOif reopHH ra3DB
15
onpeAeJIeHa KaK
NJI
I e L- i= l ffli A I' 0
P = 1m 1 u ." .
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.1aBlliHXCSI BHYTpM 06 eMa V,. CTpeMneHHe DOCJIe OBaTeJIbHOCTH
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ro o&beMa" KOP)J.& BHYTPH K&>KAOro H3 06'LeMOB l1 VI' CO ep)kHTCH
AOCTaTOQHO 6onbWOe QHC1IO MOJIeKYJI, He06xo,l\HMoe )l,JlJI npHMeHH-
IOCTH npH6J1H>KeHHSI cnnoUIHoA C hI. TeM CaMhlM HeJIBHhlM 06pa-
:jOM BBO HTCSJ OrpaHH'IeHHe CHH3Y Ha HaMeTp 06n8CTH . 06pa-
TliM BHHMaHHe, qTO H eB, CBSl3aHHasI C OrpaHHQeHHeM CHH3Y - R3
(PW3H'IecKHX npe OJIO)KeHmt - 06'beMa HHTerpHpoBaHHSI, BBnHeT-
CH OCHOBOnOJIaraJ() e:A lIpH nocrpoeHHH KCPC H Kry H npH ax
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li E :\ mi i A 1f 0
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(1.1)
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061>eMa A V p .
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Jll1ttHHbI, BBe,ASI nOHSlTHe O,l{HOqacTH'IHoA <l>YHK1I.HH pacnpe,.qeneHHSI
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'lcpe3 Bblpa>KeHHe
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(1.2)
KOTopoe eCTb O)f(H.AaeMoe qHCJJO MOJIeKYJI B 3JIeMeHTe npocTpaJICTBa
dx = dx} dX2dx3 B OKpecTHOCTH TOQKH Xt 06n aJO x CKOpOCTHMH
u 3..rIeMeHTe npOCTp8HCTB8 cKopocreA d = ld{2d(3 B OKpecTHOCTH
I'OQKH (, B MOMeHT BpeMeHH t.
IIpH BBet\eHHH nOHHTHH O)J.HO'IaCTH'IHOR <l>YHKD.HH p8Cnpe ene-
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16 r.n. 1. YpaBHeHHe BOJJLL(Jlana H yp8B'feHlIH ItJ3DBOR HHaMHKH
He CBH3aHbl pyr C .lI.pyrOM. Bonee CJJO)f(Hbie n- acTH'{lIble <j>YHK-
nHH pacnpeAeJleHHH f n (x, l' · · · t n' t), Yq}fTblBaJO e KOppenRQHJO
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pacCM&TpHBaTbCH He 6YJU'T. IIoHHTHB O HOqaCTHtlHOA <l>YHKIJ.HH pac-
npe eneHHSI BDWIHe A0CTaTOqHO ,lJ.JJ5I H3J1O>KeHHSI npeACTaBneHHOro
B MOHorpa,<J>HH MaTepHana.
HwrepeCYIOUJ.He Hac ra30 HHaMHqecKHe nap8.Me:rpbl Moryr 6b1Tb
nerKO HaA eHb) C nOMO LIO OAHOqacTH HOA <PYHK HH pacnpe,nene-
HUH (1.2). IIJlOTHOCTb H CKOpocTb r83a onpe.zx.enalOTcSI Kat
p(t,x) = ! mf(x, ,t)d ,
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( 1.3)
( 1.4)
rAe KaJKAaSl MOJIeKYJla HMeerr Macey m. BBeAB nOHJ1THe xaOTHqeCKoi!
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onpe.a-enHM TeH30p HanpHJKeHHA Pt.;' H oeKTop TenJlOBOrO nOTOKa q:
1>;.j = m I CiCj f(x, , t) d , i,j = 1,2,3, (1.5)
qi= /c2Gf(x, ,t)d , i=l,2,3. (1.6)
3 ecb c 2 = if + + .
TeMnepaTYPY r8.38 onpe enHM Qepe3 cpe JOro 3HeprHIO rrenJIO-
BOro t\BHJKeHHSI MOJIeKYJI
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( 1. 7)
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30pa HanpSl>KeHHA:
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(1.8 )
1. OCHoBHhle nOHSlTHR KllBeTJrlecKOR TeOpHH lB30B
17
YqHTblB8JI (1.5) H (1.7), H3 (1.8) nerKO nOJl)"IHTh:
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BH)KeHHSI CT8JJKHsalOUJ.KXCSI qacTH . TIpH Heynpyrwc CTOJIKHOOeHH-
S1X MeHSeTCH BHYTpeHHHS 9HeprHB qacTHQ.
Ca}..f npo ecc CTOJIKHOSeHHSI 3aBHCHT OT nOTeH ana B3aHMo eA-
CTBHH, KOTOpblH onpe eJISleTCB xapaKTepoM KBaHTOBOMexauH'IecKHx
H 3JIeKTp0CTaTHQeCKHX npo eccoBJ npOTeK8JOI.qHX npH c6Jm>KeHHH
PvIOJIeKyn. npocTeAwHA npo ecc CTOJIKHOBeHHA onHCblBaeTCSI MO,n;e-
JiblO 6HJ1bHp,l\Hb1X rnapOB. MoneKynLl 3)J:ecb paccMaTJ)HBaIOTCSI K8J(
>KeCTKHe ccl>epbl HaMeTpa d. B peaJJLHLIX ra3ax Ha.MeTp MoneKyn
H ieeT nopSl)J;oK 10- 8 CM. Ana Mo enH 6HnbSlPNihlx mapoB fiOTeH-
I Han B3aHMO eaCTBHg MO>KeT 6bITh 3a)taH B Blltte
U(r) = { +00, r < d,
o t r > d,
(1.10)
IJl.e r - pacCToRHHe Me lteHTp&MH MOJIeKyn.
MO)l;eJlb 6HnhBp,ll;HLIX mapoB, HeCMOTpJl Ha CBOIO npoCToTY, n03-
BOnSIeT 3<Pcl>eKTHBHO ODHChlBaT& npo ecc B3aHMO eACTBHs MOJIeKYJI.
C pe,D.H ,npyrHx Mo enea, 60nee TOQHO OnHCblBa.IO HX Me>KMoneKY-
lISlpHoe B3aH M o)\e A CTB He , OTMeTHM nOTeH 8JI CTeneHHoro BH):\&
A
U(r ) = 1
r'-
(1.11)
11 nOTeHlJ)faJI JIeHHapp-a-,lVKoHca
rov.no [
", .: :.c,::: .u£1i) = (d,.o/r)n - (ro/r)m] ,
lIIII..cnfY"
ttOC .1 pc, .,M ..w':' ,......pc..,.,).
61P16nMOTEKA
(1.12)
18 rJf. I. YpaBHeHHe BOJlblQtlU8 H ypasHeHHR ra30soit ,l1.HHaAfHKH
rAe At 8 , B, dr O I ro, n H m - nOJIOJKHTeJlbHble KOHCTaHTbl..
z
Jg
b
z'
PHC. 1. CxeM& CTOJIKHOBelDlJl ABYX q&CTHlt
npHBeAeM KOpoTKO HeKOTophie CBe;r.eHRg, K&CaJOUJ,HecB CTOJIK-
HOBeHHft BYX Ten C nOTeHu.HaJlOM 83a.HMO,l\eBCTBH.B U(r), 3aBHCB-
M OT pacCTORHHB Me>K)Q' eHTp&MH ax Macc. IIyCTh AB€ 'IacTH b1
1 H 2 HMelOT JJ..O CTOJIKROBeHH3 Maccbl ml , m2 H CKOpocTH l , (2 .
XapaKTepHCTHKH 06eHx qacTHD. nocne 38.BepWeHH9 npo ecca CTOJIK-
HOBeHH)I 6Y,L\eM 0603HaQaTb W1'pHx&MH. BBe,ll.eM BeKTop g = 1-E2
OTHOCHTeJlbHOA CKOpOCTH ,{acTHJJ. 1 H 2. HanpaBHM (pHC. 1) OCb Z H3
l(eHTpa qacTHDPI 1 DpoTHB BeKTopa g H 0603H8.'IHM '1epe3 b npH-
l\eJJLHoe pacCTOsHHe Me>KA)' TpaeKTOpHe:A BTOpOft 'i8.CTHQ.hJ r(t) H
DeLIO Z.
npH TaKOM CTOJIKHOBeRHH Bee ABH>KeHHe npOHCXO H'I B OAHOA
nJlOCKOCTH, HaTSIHYTOA H3 BeKTOpbl 1 H (2 [9]. B npo ecce CTOJIKHO-
BeHHB BhlnonHSlIOTC$I 3aKOHbI COXpaHeHHB MacChI, HMDynhca H 31Jep-
· AnSI peanbHblx 1'83011 B CTeneHBOM nOTeH 8Jle (1.11) nOK83aTeJ1b 8 DpU-
HHMaeT 3Ha eHHH Me 10 B 16. npH S = 5 HMeeN T8JC H83YBae14L1e NUC-
BeJlnOBCKHe MoneKYJIW, ,AJlR ICOTOphIX DOJIY'IeH PJlA HHTepecHYX TeOpe-nt'lecKHX
pe3ynbT&TOB OTHOCHTe1JI,HO no eHJUI CPYHK p8CrtpeAeneRH8(7, 8). B (1.12)
u8H6onee qaCTO HCD0JIh3YlOTC8 nOKa3aTeJUI creneHeA n = 12. m = 6.
. 1. OCHOBHlJe nOH.RTIISI KHH8TJfVecKoA TeOpHll r&30B
19
I'U":
ml + m2
, ,
- ml + m2 ,
ml l + m2 2 - m + E ,
mId /2 + m2 V2 = m 2/2 + 2/2.
BenH'-IHHbl <p( ): m, m(, m{2/2 B KHHeTH'IecKOA TeopHH ra30B "0-
.iJyqHJIH lla3BaHee cyMMamopH'b£Z UH6apuaHmo6. KpoMe CYMMaTOp-
IIblX HHsapHaHTOB, npH CTOnKHOBeHHH COXP8HHIOTCSI Mo,zurnb OTHo-
CHrreJIbHOA CKOpOCTH H npH enbHoe paccTOJlHHe:
( 1.13)
191 = 19'1 ,
b = b ' .
(1.14)
Ha KaKHX >ICe npH eJlhHblX paccTOSlHHHX bmax e e npoHCXO T
83aHMO;teACTBHe 'facrH ? Ha 3TOT BOnpoc nerKO OTBeTHTb, paCCMaT-
pHBa MO eJIb 6HJlbSlp;J.HbJX mapoB (1.10). OqeBHAHO, ttTO CTOJIKHo-
nCHHe coc'rOHTCH, ecJIR npHI1-enbHOe paccToSlHHe MeHbwe aMeTpa
lllapOB d. Cne.nOB8TeJlbHO, )VIR TBep x mapOB
b max = d.
(1.15)
B aHHOM CJJyqae BeJlHqHHY d eCTeCTBeHHO Ha3BaTL 3<P<I>eKTHBHblM
CeqeHHeM B3aHMO eaCTBH$1 o.
HarJl HOCTb nOHstTHH 3<I><PeKTHBHoro ceqeHHB B3aJ1Mo,neACTBRSI
KaK AHaM rpa CTaJIKHBalOIWIxcg wapOB n03BOnSieT nOCTpOHTb ero
aHaJlOrH Ii ,lI.J1H ,lWyntX B OB DOTeHqH8Jla CTOJIKHOBeHHB. CTpo-
["0 roBOpSl, yrOJl a OTK.nOHeHWI TpaeKTOpHH qacTHI(bI "PH npOJleTe
(pac. 1) OJI)I(eH p8.BHHTbCSl HYmo ecJIH b > u. ,ltrJSI Taro, qTOOLI
1I0CTpOHTb <pH3HQeCKH p83)'MHoe CeqeHHe CTOnKHOBeHHB, 3aMe'I'1Bi,
liTO yron a 6blCTpO y6blBaeT C poCTOM b A1ISI nI06oro nOTeHIJ.Hana U,
ai eKBaTHO ORBCLlBaJOIII-ero pe8JIhHoe B3aHMO eAcTBHe (CM. (1.11),
( 1 .12) ). Y ,lJ.06lIO onpe eJIHTb Sct>cpeKTHBHoe ceqeHue (T NIB KOHKpeT-
HOI'O COpTa qacrH c nOMOIIJ.h1O ct>OPMYJlhl [3]
00
u 2 = 21f' 1(1 - coso(b)] bdb.
o
TaKaH BeJIH'fHH8 q H83bIBaeTCSI 1'p3HcnopTHhlM ceqeHHeM. ,lJ,.njl HeA-
TpanbHblX 1"8308 U '" 10- 8 CM.
(1.16 )
20 rJ1. I. YpasHeHHe BOJlh aHa H ypaBneHHH l3308011 ,ltlfJfaMHKH
3. ,ll..nHHa CB06o,l{Horo llp06era. t.[acTOT& CTOJIKHOBeHHIf
)]:JIHHOA cBo60,lUioro npo6era Ha3bIB&eTCSI cpe Hee pa,cCT05lHHe,
npOXO,l).HMOe qacTHl.{cA Me)l()Q' ,A0YMSI CTMKHoseHHaMH. B 06bl -
HblX ra33wX, HaxO,DJI1Jl)IXCSI npH aTMoccpepHOM ,lJ.aBJ1eHHH. 3Ta )VIHHa
I "J 10- 5 CM, \lTO B HecKOJIbKO COT p83 60nbWe ,lJ.Ha.MeTpa MOJIeKyn.
,lJJ>yroH BeJIHQHHOA, xapaxTepH3yJO eA cBoAcTBa raJa H TecHO
CO 3aJfIIOA c )J.JIHHOA cBo6o Horo npo6era. HBJlSleTCSI qaCTOTa CTOJIK-
HOBeIlHA: II, paBHaJI KWJUqeCTBY CTOJ1KHOBeHHA. HCDblTLIsaeMbIX MO-
JleKynoH B e.LJ.HHHI.(Y speMeHH. Cpe,lJ.Hee BpeMSI Me)K)Jj' CTOJIKHOBeHH-
BMH pasHO
T = l/v.
( 1.1 7)
)l..nHHa cBo6o)J.HOro npo6era H qacTOTa CTOJIKHOSeHMA O[lpe enB-
lOT TaKHe BCUKl-lLIe cBoAcTBa ra30B, KaK K03<t>cPH HeHTbI B$I3KOCTH,
TenJlOnpOBO)\HOCTH H B3aHMHOR )J.H<I>CPY3HH P83JlHt{HbiX J<OMnOHeHT
ra30BoA CMeCH. C APyroA CTOpOHbl, OHI1 Henocpe,QCTBellHO CBH3aHbI
co 3Ha'leHHSIMH CeqeHHH CTOJIKHOSeHHB, nnOTHOCTH ra3& H xaoTHqe-
CKoA (TenJ1oBoA) CKOpOCTH ABH)KeHHR MOJIeKyn.
PacCMOTpHM ynpow.eHHYIO KapTHHY nOBe eHHH r83osoA CMecH.
llYCTb Bce MoneKYJIbl ra3&, KpOMe O HOa np06HoA, HMelOT OAHH&-
KOBYIO CKOpOCTb E, a CKOpOCTL np06HoA MOJIeKynbl paBHa 1' 9Ta
MOJIeKyna CTOJIKHeTCH B e HIIHUY BpeMeHH co BceNH MOJIeKYJlaMH,
ne MH BHyrpH HJlHH,lIJ)a C BblCOTOft 9 = I - ll Ii OCHoa&-
HHeM (12, paBHLIM K 8TY s<p<peKTHBHoro ce"leHHR CTOJlKHOBeHHH.
BHYTPH 3TOro IJ)IJlHHtij)& HaxOAHTCSI (72911 MOJIeKyn, r)\e n - qHCJlO
MoneKYJI B e HHRne 06'beMa. Cn OBaTeJILHO, 1{8CTO'Ia CTOJIKHOse-
HHA V H cpe ee BpeMSI MeJK)Jy CTOnKHOBeHHSlMH (1.1 7) paBHbI
1
T = 2 ·
ugn
,llnHHa cBo6o HOro np06era;J.JIs acTH J o6n alOlI(eR CKOpO-
CTbIO 1' paBHa
_ 2
v - u gn,
(1.18)
l(1 = 1 111" = I d . (1.19)
u 9"
,lJ.nSl TOro 'qTo6hl Ol(eHHTb nop OK 1(1 B QeJlOM AJlSI ra3a, npe nOJlo-
)ICBM t qTO CKOpoCTH 1 H g COH3MepHMbI C O,lUfoA xapaKTepHoA nS1
.fl. OCHOBHble nOILRTHSllCHHerH'lecl<oR TeOpHH r8308
21
Bcero ra3a BeJlH1JHHOA, B Ka'JecTBe KOTOpOft ecrecTBeHHO Bbl6paTb
CKOpOCTb 3SYKa C, KOTOpasr CJIy)f(HT MepoA xaoTH'leCKoro TennoBoro
;.{BH>KelIWl. nO cra.sJI SI B (1.18) H (1.19) BMecTO 9 H 1 11 CKOpoCTb
'JBYKa, nOJIyqHM, '-ITO no nopSl KY BeJIH'IHHbl 'I8.CTOTa CTWlKHOBeHHA
" -nnHHa cBo6o HOro npo6era onpeAMslOTcSI C nOMO blO BblpaJKe-
tlHA
2 I
II q en) I '" a 2 n · (1.20)
HaR eHHYIO TaKHM 06pa30M Be.1IH'iHHY l ecrecrBeHHo Ha3BaTb
c:pe)J.HeR ,lJ.JlHHoA cB06o Horo npo6era, KOTOPYIO B ,.qanLHeAmeM, ec-
JrH cne HanbHO He oroeopeHO. 6y eM npocro HMeHosaTh AJIHHol coo-
60P-IIOro npo6era. H3 (1.18) H (1.20) CJIe YeT npoCTaSl CBH3b Me>KJlj'
)I)IHHOH CB06oAHoro npo6era l H Cpe M speMeHeM T Me>KA}' CTOJIK-
HooeHHBMH MoneKyn:
T lIe.
(1.21 )
3aMeTHM, qTO pmpoc l() OTHOCHTenhHO BeJJHqm{LJ I ,lVIHHhI
cBo6o)J)foro npo6era. HeBeJIHK: OH NeHfiercg OT 0 DPH MaJlblX OTHo-
CHTeJ1hHO CKOpOCTH 3BYKa 3HaQeHWIX I ll o 1.411 npH f ll, CTpeMB-
e:ACSI K 6ecKOHe'lHOcTH.
Y HTbIBM, qTO BenH'iHH& U SlBJIBeTCH ocraTOqHO KOHcepBa-
THBHOA, TaK KaK onpetl-eJ1S1eTCH cBoA:CTBaMH OT enbHb1X NOJIeKyn, H3
(1.20) CJJe eT, 'lTO ,D.JJHH& cBo6o,1J)ioro npo6era o6paTHo nponop H-
OHaJIbH& IIJIOTHOCTH ra3a.
BblweJ.t3nO>KeHHOe OTHOCHnOCb K AocmroqHO pa.3pe)l(eHHbIM raa&M)
uanpHr.,lep, H&xO,lVlInHMCJI "PH aTwoccpepHOM AaBneHHH. B HHX cpe.n.Hee
pacCTOBHHe Me)K,IUI MoneKynaMH 3Ha HTeJlbHO 60nbwe pa3MepoB MoneKyn.
Eonee CJlO>KIIO npo ecc CTOJIKHOBeHHA npOXQAHT B I1JIOTHhlX ra:3&x H )KH.lt-
KOCTRX. B HHX HMeeT Mecro S<l>cPeKTHBHoe YBenH'IeHHe 'I8C1'OTbi CTOJIKHO-
nCHHA B Y p33, TaK KaK ce'leHHe tT ConOCTaBBMO co Cpe)J)lHM paCCTOJlHHeM
)'fe AY MOJIeKYJlaMH. )V1J1 onpeAeneHHSI Y RCnOJJb3YeTCJI p83nO eHHe [1]
Y =: 1 + 0.801 /V + 0.625 (botvf + 0.2869 (bo/Vr +... I (1.22)
rn.e bo = i 7rN A0 3 - nOCTWlHH8J1 Baa-Aep-BaanLca, NA - 'CHCJlO ABO-
ltW>O, V = 1/ P - y enbHblA 00beM.
npo ecc ne aqH HunynLca H 3HeprHH COCTOn H3 enO\{KH nocne-
,..'\OBaTeJlbHblX CTOIIKHoBeHHIt ManeKyn. K8JK;lJ.8JI H3 KOTOpbIX npoxOAHT OT-
HOCHTenbHO He60nbwoe p&CCTOBHHe, Cpa8HHMoe c (1. B K&KOM-TO CMblC1le
22 rJJ. I. YpaBHeH1le BOJII>U.MIUf8 H YPaBHeHJJB f830BoR ;CtIIHaMHKH
3,lteCh YMeCTHa aJlMOrHjf C HaqUOM ,D.BH)KeHHa )l(ene3HO OpO HOro COCTa--
sa, Ko[' a HMnYJ1bC OT nOKOMOTKBa DOCTeneHHO nepeAaeTCJI OTtTOJllUHW OT
Hero BarOHaM.
2. YpaBReHHe BOJIh MaBa
KaKHM ypaBHeHHeM OIUlCblBaeTCB O)1;HOQacTIrIHaH $yaKQHH pac-
npe):\eJJeHH.H f, SlBJI5I101lI.8SlCSI OCHOBOA AJlB HaxO)I(AeHHSI ra30,AHHS-
MH'IeCKHX napaMeTpoB? Ha 3TOT Bonpoc 6blJI a.H OTBe1 1 reHH&nb-
HblM aBCTpHAcKHM yqeHblM JIIO BHroM BOJ1b MaHOM (L. Boltzmann)
B 1874 ropy. IIpa BblBO,l\e 3TOro ypaBHeHHSi 6YAeM cneAosaTb OpH-
rHHaJIbHoMY aBTOpCKOMY H3JlO>KeHHIO [10].
1. BblBOA ypaBHeHHH AJIB 0AHOIf8CTHqHOR cpYHKlJ)fH
pacnpe,l{eJIeHHR
BH&\fa.ne pacCMOTpHM npoCTeAmHA C1Iy a.A, KorAa Bee 'tI8.CTH..
) COBepWaJOT )J.BH>KeHHe 6e3 CTOJIKHOBeHHA B fiOJle BHeWHHX CH.n
G. COrJl&CHO onpe eneHHIO (1.2) cPYHK RH /, B 3JIeMeHTe cPa30BOro
otYbeMa dx d B MOMeHT speueHH t B OlCpecTHOCTH TO KH (x, )
COAep>KHTCB f(x(t), (t), t) dx MOJIeKyn. K MOMeHTY BpeMeHH
tl = t + t npOH30A eT H3MeHeHHe KOOp)].HHaT H CKopocTeA NO-
neKyn x(t) -. X(tl). (t) E(tl)'
TaK KaK ,lI.BH)!(eHHe MOJIeKyn 6eCCTOJJKHOBHTenbHoe, TO BCe MO-
neKynbJ, HaxoAg Hecft B MOMeHT speMeHH t B OKpecTHOC K TO'iKH
(x(t), (t)p nepeA)U'T B OKpecTHOCT& roqKH (X(tl), E(tl)). CJleAOBa-
TeJlLHO, ,lVIjf <j>YHKIJ.HH pacnpeAeneHHS 6Y,l\eT CnpaBe HBO paseH-
CTBO
/(x(t),((t),t)dxd = f(X(tl)I((tl),tl)dx . (2.1)
IIonaraa f:1t )J.OCTaTO'UlO MaJJblM, nonytUtY Bblpa>KeHHSi, CB$l3h1-
B8JO He (X(tl), ((tl») c (x(t), E(t»:
X(tl) = x(t) + ((t) 8t,
G
((tl) = ((t) + - t.
m
9'2. YpasHeHHe BOJ1bl{UaHA
23
nepenHllJeM <popMyny (2.1) B BH):\e
/(x(t), (t), t) dx d =
= J(x(t) + ((t) ft.t, ((t) + G ft.t, t + ft.t) dx . (2.2)
m
IJ a3naraJI npaoylO '{acTb (2.2) B p TeAnopa H OrpaHHqHB&HCh ne
BbJ 1 ttneHOM pa3J10)f(eHHH, nonYQHM YpaBHeHHe ):VIH <}>YHKlJ.HH pac-
npeAeneHHH B CJIyqae 6eCCTOJIKHOBHTeJlbHOrO pa3JleTa
a f + i a f + a f = 0 .
at 8Xi m a i
(2.3)
3,!J.eCb, KaK H 8 ,lJ.aJlbHeAlIIeM) no nOBTOpHlOll\HMCSI HHAeKC&M npOH3-
nO)J.HTai CYMMHpOBaHHe. HanpHMep,
81 3 8J
i- = L{i-'
ax;, i= 1 8Xi
IIonytlHTb ypasHeHHe ):VIa <PYHK KH f npH HaJlHqHH CTOJIKHG-
BeHHA C1IOJKHee. CneA)'eT Y'IecTb, qro B J>e3Y}D)TaTe CTOJIKHOBeHHSI
{{acTH bl nOKH aIOT rpynny MoneKYJI co CKOpOCTBMH J Tal( KaK npH
-)TOM MeHSleTCJI HX HMnYJlbC. O aKO, TaK>Ke 38, C'IeT CTOJIKHOseHHR,
llaCTH[(bl C APyrHMH CKOpOCTSMH J40ryr BORTH B paccMaTpHBaeMYIO
rpynny.
Onp enHM Mo nb OTHOCHTeJlbHOA CKOpocTH g Me>K,l\y Mane-
KYJIaMH, 06n aro MH CKOpocnlMH ( H 1:
'gt = I( - lj ·
(2.4)
Tor)J.a tfHClIO '{acTHl.\1 IlOKHHYBWHX B e HHH BpeMeHH rpyony MO-
.'leKyn, HMelO11J,HX CKOpOCTb () 3& CtJeT CT01IKHOBeHHA c 'IacTH aMH
CKOpocTH 1' 6YAeT paSHO
E- (EI) dx d( = dx J j(x, (, t) f(x, 1. t) Igi bdbd8. (2.5)
F3 eCb b - npH e.nbHoe pacCTOJlHHe "PH CTOJIKHOaeHHH MoneKynbJ,
ItMelOnt-eA CKOpoCTb t C p83nH'IHblMH WWleKynaMH, 06J1 aK>IqHMH
24 rJl. I. YpasHeHHe BOJlbqM8JI& 1-1 YP8BHeJlJlR r&10BOI1 AHH&AUUCJI
z
g
PRC. 2. K HaxO)I(AeUHIO 6an8Hca 1f8CT 3& CqeT CTOJIlCHoBettHA
CKOpOC'.NIMH 1' 6 - yron 8 I1JIOCKOCTH, nepneH):\HKY.1IBpHOA: BeKTOpy
OTHOCH're11bHOA CKOpoCTH g (pac. 2).
"tJTo6bJ HaiTH fiOJlHOO qHCJlO q&CTH ) nOKHHYSWHX paccMaTpH-
BaeMYIO r'pynny 3a C'IeT CTOJIKHoseHd, He06xoAHMo npoHUTerpHpo-
BaTh BblpaJKeHHe (2.5) no BeeM CKOpOCTSlM 1:
!;- dx = dx d( j(x, , t) J f(x, (1' t) 191 bdbdO 1 · (2.6)
3HaqeHHe f(x, t t) MO)KHO BblHeCTH 3& 3Ha.K HHTerpHpoBaJlHB, TaX
KaK CKOpoCTH Ii 1 HsnBIOTCSI H e38..B HCHNhlMH ,lij>yr OT APyra.
CTporo rOBOpjI, A/lSi ODpe eneHHSI BepoWI'HOCTH CTOJIXHOBeHHB
t.faCTML{ co CKOpOCTSlMH H 1 Heo6xOAHMO BBeCTH nOHSlTHe 6HHapHoA
<PYHKl\HH pacnpe eJIeHHSI !2(X, , 1' t). B Bblpa>KeHHSlX (2.5) H (2.6)
<PaxTHqecKH HCfi0JJb30B8JIOCb paseHCTBo
f2(X, , (1) t) = f(x, (, t) · f(x, (1' t) ·
(2.7)
aTO 03HaqaeT, \!TO npHHWI'O npep.nOJlO>KeHHe 0 He38.BHCHNOCTH Be-
po THOCTeA Pa3JlHQHhlX MoneKyn 06n &Tb CKOpoCTSIMH R l' 9To
B&}I(HeAwee ,a)ISI KHHeTHqecKOA TOOpHH r830B ony eHHe Ha3hJSaeT-
CSI rHnOTe3oA MOAeKyJLR.pH020 XGOCQ.
92. YpaBHeHHe l>o.nbL(M8If&
25
3aMeTHM T8.K>Ke) 'iTO npH nOJlyqeHHH BeJ1HqHHbI E- nO):ij>83YMe-
Banach, \fTO BepOSITHOCTb TpoAHhIX CTOJIKHOSeRHA H CTOnKHOBeHHA
60JIbllleA KpaTHOCTH npeHe6pe)KHMO Mana no CpaBHeHHIO C BepOST-
HOCTblQ CTOJIKHOBeHWI ABYX qa.cTH .
Tenepb Onpe eJlHM BMB'IHHY E+ - KOJIHtfecTBO MOJIeXYJI, npH-
06peTaJOlIJ)fX 3HaqellHe CKOpOCTH B pmYJIbTaTe CTOJIKHOBeHHA. ,l1.na
= TOro Hatl-O HaaTH CKOpOCTH ( H 1 TaKHe, 'ITO nocne CTOJIKHOBeHHSI
BYX o6na,L\alOUJ,Hx HMH MoneKyn ax CKOpocTH CTaHYT paBHbIMH H
l' CII0c06 onpeAeJJeHHSI HCXOAHbJX ( H BbJTeKaeT H3 06paTHMO-
CTH 3axOHOB KnaCCHqeCKOR MexaHHKH.
PacCMOTpHM CTOJIKHOSeHHe MOJIeKynbI co CKOpOCTblO H MQJle-
KynbI co CKOpOCTblO l. llYCTL npH enbHoe pacCToSlHHe 3TOro CTOJTK-
HOBeHHH paSHa b. CTOJIKHOBeHHe npHBe)\eT Knape MOJIeKyn C HeKO-
'fOpblMH (Ollpe,D.eneHHbIMH) CKOpocTSlMH ( H H np eJ]bHbIM pac-
cTO HHeM nocne CTOJIKHOSeHHH II. Ecnll }l(e CTonKHyTOI MOJIeKynbl co
CKOpOCTHMH (, H npHQeJ1bHblM paccTOSlHHeM b', TO pe3ynbTaTOM
6YAeT na.pa MOJIeKyn c napaMeTpaMH (, 1' b. nOM-OMY Bblpa.>KeHHe
;(.,1H E+ aHaJIOrHq}{O (2.6) Ana E-:
L+ dxd = dxd( f I(x, (, t) f(x, , t) 19'1 b' db' dB . (2.8)
3,necb) B OTJ]H He OT <PoPMYJIbJ (2.6), HeJlb3B BblHecTH f(x,(,t) 3a
'lHaK HHTerp8JIa, nOCKonbKY CKOpOCTH ( H Y>Ke He SlBJISlJOTCSI
ue3aOHCHMbIMH: I.fX 3Ha'leHHB CBSl3aHbI Qepe3 3HaqeliHjf HCXOt{HI:dX
CKOpoCTeR H l.
B npaBOA \iacTH (2.8) MO>KHO 3aMeHHTb, CJle,lQ'SI ycnOBHJO (1.14),
19'1 Ha 191 H b ' Ha b. KpoMe TOro, H3 TeopeMLJ nH}'BHnna 0 coxpaHe-
IIHH cl>a30BOro o6 eMa HMeeM
d( = d(l.
(2.9)
'TaKHM 06P330M, 06'be,lJ.HHjlSI (2.6) H (2.8), nOJIY'IHM BhJpa)KeHHe AJlSI
H3MeHeHH5I QHcna "IaCTH 3a C'IeT CTonKHOBeHHA:
(E+ - E-) dx t = dx dE t f(f'f{ -1/1) 191 bdbd9 d(l. (2.10)
13 npaBoA "IacTH (2.10) HCnOJlb30B&HbI 06w.enpHHSlTble B KHHeTH'Ie-
CKoA TeOpHH ra30B 0603Ha'feBHSl
f = f(x, , t), /1 = f(x, El,t), j' = f(x,E', t), /{ = /(x) , t).
26 Tn. I. YpaBHeHHe BOJIb }.faHa H YpalJlfeRWl ra30BoJf PJlHaMHKH
O&be)).HHIDI nepeHOC (2.3) H CTOnKHOBeHHH (2.10), nOJlYQHM
OKOHqaTe bHMA BH yp&BHeHHa BOnhnMaHa O HO 8CTHqHOR
cPYHK1..(HH pacDpe eJIeIfHjf:
af of OJ 8J J( " f ) I I () At
at + i 8 X i + -;; = I'll - I 1 9 bdbd l'
YpaBHeHHe BQJU, MaHa BBJlSleTCH HenHHeAuhlM HHTerpo- <pcPepeH-
IJ.HaJlbHblM ypaBHeHHeM. HHTerpan, L'TOSlJJ..(HA B ero npaBoR qaCTH,
nonYQHJI H33BaHHe UHme2paAa cmOJItCHOOeHUU.
(2.11 )
2. CBoRCTBa HHTerpaJl& CTOJlKHOBeHHif
0630P CBoAcTB peWCHH5I ypaBHeHHSI BOJJbQM8.H& H&q}leM c aHa-
J1H3a HHTerpa.na CTWlKHoseHHR, CTOSIw.ero B ero npaBoA \JacTH. 9TOT
HHTerpaJI 6epeTCH no ID1TH CK8JJHpHblM nepeMeHHhlM, H CTpYKTypa
ero HenpoCTa. O H8J(O OH 06JJap.aeT PSI.u.OM HHTepecHbix CBGACTB, K(}-
Topble 6YAYT fiOJIe3HbJ HaM B 8JlLHeAmeM.
BHaqane YCTd.HOBHM npe enLJ HHTerpHpOBaHHH no BeeM nepe-
MeHHbIM. no CKOpOCTflM 1 H yrnaM f) MH'rerpHpoSaHHe 8 (2.11) BhI-
UOJIIJHeTCSf 80 BeeM ,AHana30He HX H3MeHeHHSI -00 < l,i < 00,
i = 1,2, 3j 0 < () 211". 06J1RCTb HHTerpHpOBaHHSI no npH eJlbHhlM
pacCTO.BHHSM b onpe enSleTCSI B CJIyqae MO eJIH 6HJlbSIp X wapoB
H3 ycnOBHSI 0 < b :5 d, rAe d - )J){&MeTp mapa. ,ltnB ):U)yrHx BH OB
nOTeH HaJJa B3aHMo eACTBHg BeJlH'IHHa b max MO>Ke'!' 6b.1Th Haa eHa,
c nOMOIIJpIO 3Ha eHHSI TpaHcnopTHoro ceqeHHB (1.16).
0603HaQHM Hwrerpan C'I'OJIKHOOeIIHA Qepe3
J(x, , t) = J(/'/1 - f /1) Igi b dbd8 dEl ·
(2.12)
YMHO)l{HM J(x, (, t) Ha cPYHK JO cp( ) H npoHHTerpHpyeM no BceM
:
Irp(t, x) = I ( ) J(x, E, t) =
= ff({)(E)(f'll-lll)lglbdbd8d lclE.
HapH,rty c lcp pacCMOTpHM HHTerpa..n [vi :
(2.13)
lrpl(t,x) = J cp( /) J(x,(', t) d( =
= If ( ') (f /1 - f'/ l ) 19'1 b' dJ/ d8 dE d( ·
(2.14)
j2. YpasHeHHe Bonbl{Mall8
27
()YCBH HO) 'iTO BbIpa>I<eHHH (2.13) H (2.14) paSHbJ Benny paslIonpa-
BHH CKopoCTeA ( 11 (. YqHTblBaJI (1.14) H TOOpe fY JIHYBHJlJlSI (2.9),
nOJIY IH r nOJ1YCYMMY (2.13) H (2.14) B BH e
Icp = f f ('P - 'P') u'Jr - Ill) 191 bdbd8 1 .
(2.15)
3 eCb HcnOnb30B&HbJ ecTeCTBeHHble 0603H&QeHHB t.p = cp( ), <p' =
= rp( /).
no aHaJIOniH C (2.15) nony'ia.eTCH Bblp8JKeHHe AJUI HHTerpana
I l = J<P( l)J(X' l,t) l:
Icp, = l f f ('PI - 'I'D (f,! - //1) 191 bdbd8 d(ll
(2.16)
rn.e <PI = CP((l), ({J = cp( ). ,n:anee 1 MOMeHTbJ If/) H Irpl paBHhI BBH-
; v paaHOnpaBHH cKopocreA ( H l' II03TOMY nOJIYCYMMa paBeHCTB
(2.15) H (2.16) aeT C1Ie IOWYIO OCHOBHYIO <f>opMyny ;:vISI MOMeHTa
liHTerpana CTOJJKHOBeHHA=
lip = f / (I() + CPl - cP' - CP ) (f'/ - J /1) 191 bdb dO d l d . (2.17)
113 (2.17) BH HO, '1ro ecJIH cPYHK SI cp(.f) TaK8Jl, qro
, , 0
cP + V'l - <P - Cf'l = ,
(2.18)
TO COOTBeTCTBYIOW)dt MOMeHT HHTerp8JI& CTOnKHOBeHHA IV' o6pa.-
THTcg B HYJIb. tPYHKQHH cp(E), Y OBJIeTBopJlIOIIJ.He ycnOBHIO (2.18).
1ia3h1BaIOTCSI CYMMaTOpHYMH HHBapHaHTaMH. TaKOBblMH .HBJUlIOTCSI
neJ1HQHHbI, YKa3&HHbIe B (1.13): m, mE, m 2/2, T.e. Macca, HMoynbc
H 3HeprHSI MoneKyn. JIJo60a ,lQ>yroA cYMMaTopHhIA HHBapHaHT npeA-
C"'!""aBJ1HeTCSI B BH;lte HX nHHeAHoA KOM6HHa H.
06p eHHeM B Hynb MOMeHTa Irp )1nH CYMMaTOpHblX HHBapHaH-
rOB Bocnonb3yeMcH B CJle lO eM naparpacj>e npH nOJlyqeHHH YPaB-
IleHHa ra30BO:A ,l{HHaMHKH H3 ypaBlleHllSI BOJ]b MaHa.
28 rJl. I. YpasHeHHe B011bQAlaH8 H ypasneHHH 1330BO#f AHH8MHKH
3. H-TeopeM3 BOJIbl{MaHa
BBe,lJ.eM H -ct>YHK HIO c nOMOwplO cne.L\YJOw. ero BbIpa)KeHHS:
H(f} = JdxJ flnf .
D
(2.19)
HHTerpHpOBaHHe npoHCXOAHT no HCClIe,rureMo.A npOCTp8fICTBeHHOA
06naCTH D H no seeM CKOpOCTHM . PacCMOTpHM 3BOJ1IOIJ.HIO H-
ct>YHKIJ.HH 80 BpeMeHH.
YpaBHeHHe BOJIb MaHa (2.11) YMHO>KHM Ha (1 + In f) H DPOHH-
TerpHpyeM DO Deeu :
I ( 1 + In f) ( + {i : + ; ) d =
= f (1 + In!) (f 1 l1- f 11) 'gi bdbd8d l. (2.20)
I1pasasl qaCTb (2.20) eCTb MOMeHT I., AJIH <l>YHKQHH «p( ) == 1 + In f .
YqHThlBaJI (2.17), nepenHweM 3T0 Bblp8,)KeHHe B B e
[(Hln!) == -1 f in ( : ) (f'/f - Ill) Iglbdbd8d 1' (2.21)
,AnSI HHTerpanoB. CTO.HLU.HX B neBoA tlacTH (2.20), BOCnCnb3yeMcSI
paaeHcTBaMH
8j 8 8f a
(l+lnJ) &t = ot (flnf)t (l+lnf) a i = o i (flnJ)'
()J a
(1 + In f) . OXi = {i 8 X i (f In f) · (2.22)
H Tel'pHPYSl (2.20) no npocrpaHCTBeHHoH o6nacTH D H Y'IHTbJBaSi
(2.22), nOJIYqHM:
I dx f fln/ + f dx I i -k(flnf)d +
D D
+ f dx f -k (f In f) dE = f I(1+ln f)dx · (2.23)
D D
92. YpaBHeHHe Banbl.{MaH8
29
nOC.l1e,cufHA HHTerp8JI B JleBOA 'I&CTH (2.23) COCTOHT H3 CJI&raeMhlX
UliAa
f dX;; f i (f In f) df.i jdE.t: =
D
I G. f 00
= dx -2 jdE.k (/ In!) = o.
m =-oo
D
PaBeHCTBO HynlO (2.24) 06'bSlCHSleTCB TeM, 'ITO mo6aa <PR3H'IecKH
OCMbJCJleHHaSl <l>YHKlVfSl pacnpe MeHHB CTpeMHTC5I K HYJIJO npH
ti -)0 :f:oo. BTOpoA HHTerpan B neBoA qacTH (2.23) npe CTaBHM B
(2.24)
BHAe
b div dx = f * dB , r e i = f if In J df. ,
(2.25)
a nOaepXJIOCTHblft HHTerpa.n 6epeTCB no rpaHH e o6nacTH D. Ec.riH
CHCTeMa 3aMKHYTa, TO nOTOK 'Iepe3 rpaHHl.\Y paBeH H}'JlIO. nOACT88-
)IHH B (2.23) BblpaJKeHIDI (2.19), (2.24) H (2.25), npHXO HM K ypa.a-
lIeHHIO
dH /
dt = 1(1+1n f)dx.
D
3aMeTHM TenepL, o MOMenT (2.21), CTOJl Ha nOA HHTerp8JJOM
n np8.BoA \{acTH (2.26)>> Bcer a HenonO)KHTeJJeH, Ta.K KaK OeJIH'IHHbI
In(/'!{) - In(f Ii) H (f'f{ - f 11) BCerA& o):\Horo 3HaKa. H3 (2.26)
(2.26 )
CJre,AYeT, '1TO
dH / dt < 0 ,
(2.27)
T.e. cPYHK HSI H H3 (2.19) - HeB03paCTaJOIII.aJI <PYHKl\HH BpeMeHH.
13 3TOM 3a.KnlOQaerCSl H -reopeM& Bonb MaHa.
H -cpYHKIJ.HSI TeCHO CB.H3aH8 C KJlaCCwtecKHM nOHBTHeM 3HTpo-
IIHH, KOTOp(UI onpe enHeTCSI KaK ),tepa sepoSiTHocTHoro OCYll.\eCTBJle-
IIUSI KaKoro-J1H60 MaKpocKOIlHqeCKOrO COCTOSHHH Bew.ecTsa. He B a-
naSiCb 8 no po6HOCTH, npHBeAeM CBH3b Me>KM TepMO Ha.MH'IeCKOA
-HITponHeA H H -cPYHK Hea
S = -kH,
(2.28)
30 TJl. 1. Yp&BHeHHe BOJlbl{Af8H8. H Yp&BHeHJUI ra3OBOR JVfHaMHKIJ
r e k - nOCTOBHHaH Bonh MaHa. ,Ly>YrHMH C1JOBaMH t H3 H-reopeMbi
BOJlbl.\MaHa CJle.nYeT xopowo H3BeCTHbIA B repMO,ll;HI18MHKe ct>aKT: 8
.,
3aM7CHymou CUCme.Me 3Hmponua 803paCmaem.
4. Pacupe/l.eJleHHe MaKCBeJlJIa
TepMO,lJ.HH3.MH'IeCKOMY p8.BHOBeCWO a,lt.Ha6aTH'IeCKOJf CHCTeMbI
OTBeqaeT COCTOSlHHe C MaKCHMYMOM 3HTponHH. B 3TOM cnY'Iae dH / dt
O. Cne o8aTe.nbHo,
/ I(1+1nj)dx = -1!! In ( : ) (f'1 - f Id x
D
X 'glbdbd8d d 1 dx = o.
(2.29)
PaseUCTBO HynlO Bblpa>KeHHSI (2.29) B03MO>KHO, Kor a
III = f'f{ HJ1H Inf + In/I = inj' + Infr.
(2.30)
HHb))'1H C'J10BaMH t In f HsnseTCSI CYMM&TOPHblM HHB&PU&HTOM H MO-
JKeT 6brrb npe,QCTaBJIeH B BHAe
in f = al (t, x) m + a2i(t, x) i + a3(t, x) m 2/2.
(2.31)
«I>YHKIJ.HIO pacnpe,1J.eneHHB y o6.lee Bblpa3HTb epe3 DCHOBHbIe
r'd30AHHaMHQecKHe nap8.MeTphl BMeCTO ai, a2i, Q3 . SI 3Toro ):\0-
CTaTOqHO nO CTaBHTb f H3 (2.31) B onpeAeneHWI BeJ1HqHH (1.3), (1.4)
H (1.7). aJlee nYTeM HecJIOX<HhJX, HO OBonblfO rpoM03,AKHX Bhlxna-
OK nonYQHM CJle;rylO ee Bhlpa>KefIHe AJUI f:
...
p(t.x)m 1 / 2 { m({i-Ui(t,X))2 }
lo(xt , t} = (21f'kT(t,x»3j2 exp - 2kT(tt x ) ·
(2.32)
fl)YHK H}1 10 nonYQHJI& Ha3BaHHe J&OmAbHOf pacnpeOeAeHue Ma"c-
8eAAQ.
B cnyqae nOK8J1bHo-MaxCBeJIJIOBCKOro pacnpe eneHHSI <l>YHK HH
I 3aBHCHT OT npocTpalfCTBeHHblx nepeMeHHblx x H BpeMeHH t 'Iepe3
nHTb napa"-leTpOB p, u, T . T8J<HM nyTeM oCyll.teCTBJlSleTaI nepexoA
92. YpasHeHJfe BanhL(MAH8
31
OT KHHeTH'IeCKoro ODHC3JlHH aHcaM6nSl 'IacTH K rH,lQ)otJ.HHaMHqe-
CKO lY. ,LLns nOKaJIbHO-MaKCBeJIJIOBCKOA CPYHK HH HHTerpan CTOJIK-
H08eHHA: o6p aeTC B uynb, qTO CJleAYeT H3 nO CTaHOBKH nepBoro
pascHcTBa (2.30) B 4>opMyny (2.12). B TO >Ke BpeMH cl>YH HB (2.32)
8 06 eM cnyqae He Y OBneTBOpSleT yp&BHeHHIO BOJIb aaa (2.11) 3&
CtteT ero neBoR 'faeTH.
B npocTpaHCTBeHHo-o HOPOAHOM cnyqae J10KaJlLHOe pacnpeAe-
neHHe nepexo HT B a6conlOTHoe pacnpe eneHHe MaKCBeJlJl&
poml/2 { rr ( i - UiO)2 }
lo( ) = (211" kT o)3/2 exp - 2kTo ·
(2.33)
I3xO gI.l(He clO)J.a napaMeTpbl Po, Uo, To He 3aBHCSlT OT npOCTp&H-
c.rBa H speMeHH.
3BOJIIOU.HSI OT pacnpe eneHHSI (2.32) K pacnpe eneHHJO (2.33)
IIpOHCXO HT, KorJJ-a CHCTeMa penaKCHpyerr K a6conIOTHOMY paBHO-
HecHIO , Y)Ke HaxOAHCb B rH)JJ>o HHaMHqecKoA CTaJ:(HH, T.e. nOK8Jlh-
110M paBHoBecHH. 3m pen8J< HH onHCblsaeTCa CHCTeMOA ypasHe-
flHR ra3o- HJlH rH pO,lJ.HH8.MHKH, 0 qeM noAAeT pe'lb B nOfJIe YIOID-HX
lIaparpaQ>ax rJ1aBbJ I.
5. Yp8BlleHHe BOJIbIVdaHa p,JIH CMecH ra30B
PaHee BbJDHCaHHoe KHHeTHtJOCKoe ypasHeHHe (2.11) 6L1J1o Dony-
'-(euo )J.JISI ra38) COCTOBl1.\erO H3 'IacT O,lJ)loro COpTa. JlIur CMecH, B
KOTOpoA npHCYTCTBYIOT qaCTHl(bJ Hi p83JlH'qHhlX TROOS J Heo6xoP;HMO
BwnHcaTb ypasHeHHe ,l].JlR <l>YHK HH pacnpe)J.eJIeHHSI oro cOpTa
\ acrHU. Ii. JIeBaJ1 - ,lJ.HcP<I>epeH H8JlbHaB - 'iacTb 6YAeT COBna.na.Tb
( ,l{H<P<I>epeHlI.HaJIb}IOA qaCTbIO KJI&CCH\feCKoro ypaBHeHHSI BOJJbnM&-
lIa. I1HTerpan CTOJIKHOBeHHA ,.QQ1I}f(eli OUHcblBaTb B3aHMO eACT8HB
'-IaCTH JJ-aHHoro copra. c qaCTH aMH Bcex COpTOB.
y t.IHThlBasI BblWecKa38J1HOO, COCTaBHM CHCTeMY ypaBHeHHA
ROJIbll.MaHa nH CMeCH ra30B B rJle,nylO eA <J>opwe:
[) Ii a Ii G} {J Ii J ( I' f )
at + j ax, + 'mi {Je. = L.J Iii" - Ii , Ig,il bdbd8d(" ( 2.34 )
J '-, p= 1
i = 1, · · . 1 Ni ·
32 rn. I. Yp8BHeHHe Banhl(AlaB8 H yp&BHeHHJI r&3OBOJ1 Jl.HHaMHKH
3 eCh G ' tni - COOTBeTCTBeHHO BHelDHSljI cHna, eACTayJO aJI Ha
qacTHl"-Y i-ro copra, H Macca nOCJle eA:, 19pil - MOA}'JIL OTHOCHTeJlb-
Hoil CKOpOCTH CTaJJKHBaIO HXCB '1aCTH COpTOB pHi. 3aL1eTHMt QTO
CKOpOCTH nocne CTOJIKHOSeHHH H = ( onpe enSiIOTCSI H3 nOJI-
HOa CHCTeMbl YPaBHeHHA: (1.13) c Y'ieroM B03MO)KHOrO HepaBeHCTB8
MacC MoneKYJI m" H mi.
6. YpaBHeHHe BX&THarapa-rpocca-KpYK8 (BrK-MO,J1.eJIb)
HapB C ypaaHeHHeM BOJIh Ma.Ha pacCMOTpHM pyr)'10 MO)J;enh
AIlS( OnaCaHHjf O,D;HOqacTH\lHOA <l>Y HH pacnpe eJIeHBSI t npe,lVlO-
)KeHtlylO BxaTHa.rapOM, rpOCCOM H KpYKOM (BrK-MO eJIb; CM. [5J):
df at 8j Gi at
- = - + i - + - - = v (/0 - f) ·
dt 8t 8Xi m a i
(2.35)
3AeCb /0 nOKaJlbHoe MaKCBeJIJlOBCKoe pacnpe eneHHe (2.32), II(E)
- Q8CTOTa CTOJIKHOBeHHA ,LVUJ qaCTH J HMelOUWX CKOpOCTL (:
v(E) = I fO( I) 191 bdbd8d(l.
srK-YPaBHeHHC (2.35) MO)KeT 6LITb nonyqeHO H3 ypasHeHHH
BOnb1J.MaHa (2.11) B cnyqae COCTOBHHH ra3a, 6nH3Koro K pasKoBecHO-
MY. npH 3TOM CYID-ecTseHHo Hcnon:b3yeTCJI Dpe nonO>KeHHe: tITO no-
CAe cmOAXH06eHtLJI pacnpede.AeHUe cmu cmaHOBUmCJI AO"aAf)tw-
MaW;C6eJ1J1OBCKUM.
npaBasi qacrb ypasHeoHSI (2.35) nepeHHMaeT PsrA cBOlcrs Y HH-
Terpana CTOJIKHOBeHHA H3 yp8.8HeHMSI BOJIb MaHa. QHa 06p aeTCH
B Hynb, Kor,n;a <l>YHK SI pacnpe eneHHSI nOKaJIbHo-MaKCBeJ1JlOBCKaH.
npH HHTerpHpOSaHHH C CYMM8TOPHblMH HHBapHaHTaMH B Kaqe-
CTBe BeCOBoro MHO>KHTeJl5I npaBaJI Q8CTb (2.35) TaXx<e 06paTHTCB
B Hynb. 9TH cBoAcTBa BrK-YPaBHeHHSI ,l\enaroT ero YAo6HOA NO-
eJlblO Anfl HCCJleJJ:oB8.HHjI nOBe eHH.ft <PYHK pacnpeAeJleHHB I,
KOTOpoe BO MHorHX cny'IasIX TPY,D.HO npoBecTH H8 OCHOBe HHTerpo-
Hct>cPepeHr(Ha..rlbHoro ypaaHeHHSI BOJIL M&Ha.
CJle eT 3aMeTH r rb, ttTO ypaBHeHHe BrK, Tal< >Ke K&K H ypaBHe-
HHe Bonb MaHaJ HeJlHHeAHO. flpaBaH 't{&CTb 3TOro YPaBHeHHJI 'fepe3
2. YpaBHeHHe BDJlbUMaH8
33
10 H II 3aBHCHT OT p, U H T, KOTOpble, B CBOIO 01lepe)J.b, BSnBIOT-
CSt MOMeHTaMH <PYHK HH f. HeJlHHeAHOCTh BrK-MoAenH, YQHT&JB8JI
:iKCnOHeH HaJlhHYIO 3aBHCHMOCTh /0 OT U H T, B KaKOM-TO CMhlC1Ie
HBnHeTCB 60nee CHJlbHOR, qeM KBapJ>&THtUlaSi HeJlHHeAHOCTb ypas-
HeHIDI BOJIb M8Jfa.
7. YpaBHeHHe 9HcKora
,ana MOTHbIX ra30B H )l(H.D.KoCTeA xapaKTepHO TOt 'lTO AJlHH& cso6o,lt-
HOro npo6era B HHX conOCIaBHM8 C JJ.HaMeTpOM wOJIeKynbl u. ,lVI.JI onHe&-
IIHH <!>YHKQHH pacnp eneHHH J B 3TOA CHTY8lUtH 9HCKorOM npe,D.JIO>KeH&
cnWIOll.\aJI MO H<!>HK3IU1J1 ypaBHeHIUI BOJIbw.t8Ha (I):
8/ 8f G i lJl
+ {i _ 8 + - 8 & = J 1 + J2 + J 3 + J4 + J 6 + J8 ,
ut Xi m '..i
(2.36)
rJl-e
J 1 = Y j(!'!{ - J J1) gbdbd8d{l .
/ ( ,l)/ fJ/ 1 )
J 2 = uY lei I 8x. + / 8Xi gbdbd8d{11
J 3 = ; f ; (/'1; + Ifl)gbdbdIJd{l. (2.37)
(72 f ( , 82 ft IJ2 11 )
J. = _ 2 Y leiki / 8 8 - f 8 8 gbdbd8d{l ·
Zj %j Z, %j
J[, = f le i 8Y lei ( 1' 8 fI - 1 81t ) gbdbd8 l ,
2 {Jx, 8xJ lh: j
J" = 2 / lei k i a::X j (f'/ - Ifl)gbdbdD I'
3.necb Y - 34>cl>eKTHBHoe YBeJlH'IeHHe qacronl CTOJIKHOBeHHA, onpet\e-
JUleMoe Bhlpa>KeHHeM (1.22). k - eAHHH'IHYA BeKTOp, HanpaoneHHbfA OT
l(eHTpa o,lJ.HoA MOIIeKynhl K l\eHTPY APYroA.
B MO eJlH 3HcKora npHHHM&eTCR 80 BHHUaHHe TOT <i>aJ<T, "ITO eH-
TPbl ABYX CTa11KH8&IOUJ.HXCSI MoneKYJI H&XopJITCH B pa3HbDC roqxax npo-
C1'paHCTB& x. 9TO B8JKHO, nOCKOJIbKY (CM. g 1, 0.3) HMnynbc B 3HeprHR
nepe;r.aJO'rCI MeJK,Ay 'I&CTHIJ.8MH no [\enO'lKe. OTCIO a B (2.37) TOJILKO nep-
8blA "IJIeH J 1 . HNelOlI{HA Hy eBOR nOPJlAOK no a) COBDa.z:t8eT no BII)Ijf C
HHTerpanOW CTWlKHOBeHd BOJIb aHa. CJle.ztY1OIIJ.lle 'lJIeHhI p83JlO)l(eBBJI
no CT HMeJOT 6o.rIee rpoM03 KHA BH,A. Buec-re C Teu, B CCRone Mo,neJlH no-
npe)KHeMY Jle)KHT rHnOTe3a 0 Ue3&BHCHMOM oepWiTHOCTHOM pacDpeAeJ1e-
HHO BYX qacTHIl..
3. Y paBHeHHB ra3oBoA HaMHKH
YpaBHeHHSI ra30BoA HHaMHKH MorYT 6bITb nOJIY'IeHLI <peaoMe-
HOJIOrlll.JeCKHM nYTeM fll) Ha aCROBe nOHilTHg HenpepblBHoA (cnnolU-
HOlt) cpe bI H <PYHAaltJfCHT3J1bHhIX 3aKOHOB COXpaHeHHH MacCbl HM-
nynbca H 3HeprHH Bew-eCTBa. 9'1'0 OTHOCHTCSI K ypaBHeHHSlM 9Anepa,
onHCbIBalO HM re'leHHSI HeBSl3Koro raaa, H K ypanHeHH.S1M HaBhe-
CToKca ,lJ.JISI BSl3KOro TCrInonpoBo HOro ra3a.
HanpHMep, ypaSHeHHe Hepa3pbIB-
HOCTH
+ div(pu) = 0 (3.1)
BbIBO HT(JI H3 YCJIOBHSI COXpa.HeHHH
MacCbl)VIR KOHe'iHOrO o6'beMa V o , rpa-
PHC. 3. K BblBOAY ypaBHenHH HH a KOTOporo He MeaHeTCSI BO Bpe-
Hep&3pblBHOCTH MeHH. B CaMOM )J.eJIe) 3MeHeHHe Mac-
Cbl B 06'beMe Vo pasHO ee KMe'lecTBY, nepeHOCHMOMY Qepe3 I10BepX-
HOCTL S, OKPY}l(3J()UJyIO P-aHHbIA: o&beM (pRC. 3):
f p dV = - f (pu · 0) dB ·
Vo 5
(3.2)
CJIeA}'51 <popMyne raycca-OcTporp cKoro, BblpaJKeHHe (3.2) 3aMe-
HHM Ha
f ( + diV(PU)) dV = o.
Va
YqHTbIBaSI, 06'heM Vo MO)KeT 61:dTb npOH3BOJIbHblM, nep eM OT
TO*,neCTBa (3.3) K ypaBHeHHIO Hepa3phlBHOCTH (3.1).
A Ha..nOruqHO, ODHpasICb Ha ycnOBHSI coxpaHeHHSI HMOYJlbCa H
3HeprHH SI 06'beMa Vo, nonYQHM OCT8J1bHhle ypasHeHHSI raaOBOH
HHaMHKH.
(3.3)
BHHMaTeJlbHhlA 'iHTa:reJIb cpaay 3aMeTHT, qTO, CTpOrO rOBOpSl,
o6 eM Vo He MO)f(e'!' 6blTb npOH3BonhHhlM. Ha Hero no MeHbweA
Iepe HaKJIa,lI.bJBalOTC9 OrpaHH'IeHHH t CBSI3aHHbJe C He06xOAHMOCTblO
( )rm:eCTDOBaHHH 6ecKoHeQHO MaJIOrO eMa (91, 11.1). 3a6eras BUe-
pe , YKa>KeM Ha TO, t{TQ K 9TOMY QeJlecOo6p83HO Ao6asHTb e e O HO
OrpaHH1.feHHe - 6o.nee >KOCTKoe: )J.onYCTHTb K paccMoTpeHHIO Tonb-
J«() 06'beMbl Yo, JlHHeAHble pa3Mepbl KOTOpbIX no nopH KY BeJ1HtIHHbl
He MeHblllC }J.JIHHbI C8060,L\Horo npo6era I. aHHoe Tpe600aHHe no-
( 'IY)K"T CPH3HtfOCKOR OCHOBOR .lVfSl IIOCTpoeHHSI HOBOA Mo enH) onH-
cblBalOw.eA Te'ieJ-IHSI BSl3Koro raaa, no,zq>o6Hoe 06CY>K eHHe KOTOpoD
IIpOBO,lI.HTCSI B rJ1ase III.
B 3'TOM H CJleA,YlO1QeM naparpa4>ax ypaBHeHHSI 9A.1Iepa H Hasbe-
C;ToKca 6YlU'T nonytfeHbl KaK CJle CTBHH ypaBHeHHg BOJIb MaHa [IJ-
IS]. TaxOH flYTb n0380JIHeT He TOJIbKO nyqwe nOH1ITh o6nacTb npHMe-
IIHMOCTJ.I ypaBHeHH:R raJOBO.R AHHa.MHKH J HX CBB3L C KHHeTHQecKoD
TeopHeA, HO H CKOHCTpYHpoBaTb HOBblA no o K OnHCaHHfO BH3KHX
'I'eqeHI.r If C03):\a.HHIO KOHKpeTHLIX BbI'iHCJJHTeJlbHbIX 8JlrOpHTMoB.
110 06 3TOM peQb nolt eT Y>Ke B CJle.nylOID-HX rJlaBRX.
1. MOMeHTHbIe ypaBHeHHH
IIonb3YSlcb onpeAeneHHBMH ra30 HaMHqeCKHX napaMeTpOB
(1.3)-(1.8) Qepe3 MOMeHThl <}JYHKIJ.HH pacnpeAeneHJIjf) nOnbJTaeMCJI
IIUJlyqnTb If a OCHODe YPaBHeHHH Bonb MaHa CB$I3b Me)K,lI.Y 3THMH
:\tOl\'ICHTaMH. W'IaK. YMHO>KHM ypaBHeHHe BOJIhIJ.MaHR (2.11) Ha CYM-
ta'fOpllble MHOapHaHTbI V'( ): ffl, m , 2/2, H npoHHTerpHpyeM no
nCCM . B pe3ynbTa:re flMeeM
/ ( 8J 8! Gi Of )
cp( ) - + i - + - - cl( = 0 ·
at 8Xi m a i
(3.4)
IIanoMHHM) t.JTO npaBaJI qaCTb (3.4) pasHa HYJlIO BBH;Jy TOro, '1TO
liP" HTerpHpOBaHHH C CYMMaTopHblMH HHBapHaHTaMH HHTerpan
( TOJ1KHOBeHHA 06pa aeTCH B HYJlh ( 2, n.2).
PaCCMOTpHM OT e.nbHO HHTerpanbl OT CJlaraeMhlX, BXO,ll.SI HX B
",'u uylO ttacTb ypaBHeHIUI (3.4). nOCKOJIhKY cp( ) He 3aBHCgT OT npo-
CTp8.HCTBeHHbIX nepeMeHHhlX H BpeMeHH, cnpaBeAnHBbl BhIpa1KeHHSI
f cp( ) = ! rp( )/ ,
I af a f
cp( ) {i = a;: rp( ) {d .
Bonee no,a.po6HO OCTaHOBHMCSI Ha HHTerpaJIe OT TpeTberO qneHa
H3 nesoA tJacTH (3.4). Bocnonh3yeMcSI <I>oPMynoA HHTerpapOSaHHSI
no l.JacTSlM ,lVUI Hanpa.BJIeHHB i CKOpOCTH NOJIeKyn ({J' {Ie - CKOpOCTH
MoneKYJI B BYX OCT8JILHhlX HanpasneHHlIX):
00 00 00
J G. of I I / G. 8/
rp«(} rr: d = k d{; cp( ) :;; d{i =
-00 -00 -00
00 00
= I d{k / d{j (cp(t) I) ;: -00 - (3.5)
-00 -00
/ 00 1 00 / 00 G. 8rp
- d{tc d(.j I ;; 8[; d{i ·
-00 -00 -00
BOCnOJlb3yeMCH ecTeCTBeHHhlM <l>H3H"IecKHM npe onO>KeHHeM 06
y6h1BaHHH <l>YHK HH pacn eneHH$I npH f(1 00 6h1CTpee JlI060A
creneHHoA <l>YHK H «(). TIpHMepoM TaKO:A CPYHK HH f CJIY>KHT
nOKaJIbHOe MaxcBennoBCKoe pacnpe eneHHe (2.32). TaKHM 06P830M,
B (3.5) HCQe3aeT qneH, npoHHTerpHpoBaHHblR 110 {i, H OCTaeTCH
/ rp( ) Gi 8f = _0. I f 8cp . (3.6)
m 8 i m 8{i
YqHTblBasi (3.5) H (3.6), nepenHweM (3.4) B B e CHCTeMbl YpaB-
HeHHA M.R Ka}f()J.oro CYMMaTOpHoro HHBapH8.HTa:
/ mj + -k I m ij = 0,
:t I m{j/ + -k f m i{jjdt - Gj f jd( = 0,
a f m 2 8 J 2 f
at 2 Idt+ 8Xi tTl.{i?:fdt-Gi {ifd =O.
j = 1, 2, 3,
(3.7)
llpeo6p33yeM ypaBHeHHH (3.7), BOCn0Jlb30BaBWHCh onpe eneHH-
HMH ra30,1J.HHaMHllecKHX nap8.MeTpOB (1.3)-(1.9), Bhlp&>KeHHeM ):VIS!
nonHoA 3HeprHH
E= l i 2 I
(3.8)
J.f paBeHCTBOM
f m i jfd( = j + ptJiUj.
(3.9)
3To paseHcTBo MO>Ke1' 6blTb nOJJy eHo (JIeA)'IOlD)iM D)"reM. IIpe -
CTaBHM CKOpoCTb MOJIeKY nbl B BH)J.e = c + U H 3anmneM
f i jfd( = m f Cicjld + mUiUj J fd +
+ mUj I Ci/d( + mu, J cjld!..
rloCJIej\HHe ):\Ba qJIeHa B npaaoA 'IacTH paBHhI HYJlIO B CHJIy (1.4)
Ii onpeAe.nelJHS c. YqHTbJOaH Bblpa>KeHHSI ,lJ,11S1 r8.30,D:HHaMHQeCKHX
Ue.nH1.fHH (1.3)-{1.5), np eM K cpopMyne (3.9).
HTaX, nepenHmeM CHCTeMY ypasHeHHR (3.7) B <I>oPMe
op 8
at + (pud = 0 1 (3.10)
a 8 BP i; G j
lit (ptJj ) + 8 X i (Pf.'iUj) = - 8Xi +;;- p, j = 1,2,3, (3.11)
BE a Gi
+ _ 8 (pu,E + Uj ; + qi) - - fJU1 = O. (3.12)
u, Xi m
CHCTeMa MOMeHTHblX ypaBHeHHA (3.10)-(3.12) He BBJI$leTCSI 3a-
fKHYTOa. B HeR q}fC1l0 ypaBHeHHA paDRO DSlTH, a 'fHCJlO He3a.BHCH-
MbIX HCKOMblX BeJIUqHH p, 1./,]' P;.j, qj - TPHH QaTH, c yqeTOM CHM-
MeTpHttHOCTH TeH30pa Pij. IIpa 3TOM HMeJIOCb B B , 'ITO fiOJIH8JI
1HeprHH E H3 (3.8) BbIp CSI C nOMoIIJ.hJO npocTblx COOTHOllleHHA
([epe3 JJ.pyrHe MOMeHTbJ (1.4), (1.7)) (1.8):
I m 2 f u2 / c2
E= 2 I =m 2f +m 2f =
u 2 3 pkT
=P"2+ 2 m ·
(3.13)
KaK >Ke 3aMKHYTb CHCTeMY YPaBHeHHA: (3.10)-(3.12)? ,nJIS 3TO-
ro Heo6xo HMO HCnOJJb30BaTb Te HJ1H HHble <l>H3H'ieCKHe rrp nOCbIJl-
KH 0 BH e <l>YHKI..\HH pacnpe.n:eneHHHJ n03BOJISlIO He onpe enHTb o-
DOJIHHTeJlbHble CBSI3H Me>KPj' HCKOMhlMH BeJJH'IUHaMH 3TOH CHCTCMbl.
BH)J. TaKoA <PYHKQI1H MO)f(eT 6b1Tb HaA eH no npH6JIH}I(eHHH K 2Udpo-
auHaMU4t(eC"oMY npeaeAY (B COOTBeTCTBHH C npHtlSlTOA: B a.HHOa IO-
HorpaQ>HH TepMHHOJIOrHeA 6anee ecTeCTBeHHO, BonpeKH Tpa,D)i HH)
Ha3BaTb 3TOT npe eJI 2a300uHaMU eCKUM). ITpH aTOM CaMa cPYH HH
pacnpe,lJ.eJIeHHj( 6y eT 3aBHCeTh OT npoCTpaHCTBeHHbIX rrepeMeHHbIX
H BpeMeHH nocpeACTBOM nBTH napaMe'rpOB p, Ui, Tt qHCJIO KOTOphlX
COBfia.,IJ.aeT C t.JHCJIOM MOMeHTHblX ypaBHeHHA (3.10)-(3.12).
2. 'paBlleHHR 3R.nepa
Bae,.neM 6e3pa3MepHbIA napaMeTp, paBHbIA OTHoweHHIO ):VJHHbI
CB06o)I)ioro npo6era l K xapaKTepHOMY paCCTOHHHIO L, Ha KOTOpOM
npoHCXO,ll.HT 3aMeTHoe H3MeHeHHe ra30 HaMHqeCKKX BMH HH:
Kn = l / L .
(3.14)
9TOT napaMeTp Ha3b1BaeTCSI UCAOM KHyoceHa. Kor a xapaK'rep-
Hbllt p33Mep 3t\ltaqH MHorOKpaTHO npeBblWaeT A1JHHY cB06o Horo
npo6era, qHCJIO KHy ceHa MaJIO. B 60nbmHHCTse 3 aq ra30BoA )J.H-
H&MHKH, onHCblBaeMblX ypasHeHHjlMH 9bepa HnH HaBLe-CTOKCa,
Kn < 10- 3 .
ECJlH napaMeTpbl raJa cna60 MeHHIOTCH Ha pacCTO.SIHHH MHorax
NJI-iH c806o lloro npo6era J TO MO)l(HO rOBopHTb 0 COCTOSlHHH nOKaJ1b-
Horo paBHOBeCHH cpe bl, B KOTOpOM <l>YHK HSI pacnpe eneHHH SlBn.H-
eTCB nOKaJIhHo-MaKCBeJ1J10BCKOR (2.32).
BOCn0J1b3yeMc SlBHbIM Bblpax<eHHeM ,LVIB <l>YHKl\HH pacnpe e-
JIeHHH (2.32), '-IT06LI onpeAeJ1HTb TeH30p HanpH>KeHHA: P ij Ii BeKTOp
TennOBoro nOTOKa q H reM caMhIM 3aMKHYTb CHCTeMY MOMeHTHblX
ypaBHeHHA (3.10)-(3.12).
9JIeUeHT j "PH i:/: j 3 aeTC5I, COrJI&CHO (1.5») <I>oPMynoA
I{jli j =m f fo cj =m j (2: :/2 exp{- ; } x
-00
f oo { m } / 00 { m4 }
x Ci exp - 2kT de; Cj exp - 2kT de;. (3.15)
-00 -00
H Haxo eHHH MOMeHTOB ldaKCBennoBcKoft <PYHKQHH 6y eM fiOJlb-
30BaTbCSI 3HaQeHHSlMH Ta6nH'IHblX HHTerpaJIOB
1 x2n-le-Il:z;2dx = 0, 1 e- :z;2 dx = ..j1rlfJ ,
-00 -00
00
f 2n -{jx 2 _ (2n-l)(2n-3)... 3.1
x e dx - 2R V 7J2n+l '
-00
(3.16)
n = 1 t 2, . . .
OrcfO a B npOH3se,tteHHH HHTerpaJIOB (3.15) sTOpoA H TpeTHA COMHo-
IKHTeJIH paBHbl HYJlIO. CJIe,lJ.OaaTeJlbHO, ,lVIji nOK8JlbHO-MaKcsennOB-
CKoA <l>YHK pacnpe,n:eneHHB
j = 0 , i j .
(3.17)
,lI,HarOHaJlbH}'1O KOMnOHeHTY TeH30pa Hanpg)J(eHHR Pjj nony-
tfHM, HCDOJIb3YSl BbtpameHRSI ARSI HHTerpanOB (3.16):
/ 00 {Yml/2 { me; } f oo { }
p'j = m (21rkT)3/ 2 exp - 2kT dCI exp - 2kT d x
-00 -
1 00 2 { mc3 }
x Cj exp - 2k; dc; = pR:1',
-00
j=1,2,3.
(3.18)
)J;ISI BeKTopa TeIIJIOBOrO nOTOKa (1.6)
m f ( 2 ) pml/2
qj = 2" c1 + q + c, Cj (211'kT)3/2 X
{ m(cl+ 4 +c1) }
x exp - 2kT de
(3.19)
B Ka}K OM H3 COCIaBJIjllO HX B&lp8JKeHHe (3.19) IIOBTOpHhlX HHTe-
rpa.nOB npUCYTCTBYIOT MOMeH'I'bl OT He'leI'HblX CTeneHeA Cj. ncaTO-
MY, 8 CHJIY (3.16), BeKTOp rennoBoro nOTOKa q paseH HynlO:
q = O.
(3.20)
nO CTaBnHH Bblpa)KeHHS1 (3.17), (3.18), (3.20) B CHCTeMY MO-
MeHTHbiX ypaBHeHHA (3.10)-(3.12), nOJIY'IHM:
8p a.
lit + aXi (pud = 0,
8 (} 8p Gj
- (pUj ) + - (f'UiUj ) = - - + - P t j = 1, 2, 3,
at OXi OXj m
8E a G i
at + fJx i (pui E + UiP) - -;;; PUi = 0,
p = p(T, p), E = pu2/2 + e(T, p) p J
(3.21)
(3.22)
(3.23)
(3.24)
r e [ - y enbHaJI BHYTpeHHSlR 9HeprHR oeIr(ecTB&.
3 a M e 'I a H H e 1. Ami O)UlOKOMnOHeHTHoro O)l)foaTOMHoro raJa cnpa-
Be;vIHBbl 60nee npocTble Bhlpa>KeHH.R P.J1B ,QaaneHHB H BHYT])eHHeA 3HeprHH
k 3 k
p == p m T, E = m T ·
HcnOJlb3yeM&ft 8 (3.24) AOCTaTOqHO 06 tPoPMa 3aDHCH YP&8ReHHA co-
CTO.RHHR MO>KeT 6blTb KOHKpeTH3HpOBaHa )J.JIa K&>KAoro cnY"l8JI r&3OBOA
CMecH.
YpaBHeHHB (3.21)-(3.24) SlBJlHIOTCH KnaccH ecKHMH ypa6HeHU-
JlMU 91l4epa ,QHHaMHKH HeBB3Koro HeTeDnonpoBo oro raJa. TeM
CaMblM ypaBHeHWI 3Anepa COOTBeTCTBYIOT KOHKpe1'HOMY BH o -
HOqacTHQHO:R CPYHK H I, aBae1(oA JlOK8JThHLlM M8.KCBennOBCKHM
pacnpe eJIeHHeM (2.32).
,[VIS! TOro, tJTo6b1 HaMeTHTh D}'Tb DonyqeHH.H 60nee fiWIHLlX no
cpaBHeHHIO C ypaBHeHHBMH 9AJIepa ypaBHeHHA ra30BO:A: )\HHaMHKH,
BocnOJ1b3yeMcS1 BrK-Mo,a.eJ1blO (2.35). H npocTOTbl npe lIonO)KHM,
liTO Cpe HSI.R qacTOTa CTOJIKHOBeHHR II He 3aBHCHT OT CKOpOCTH MO-
neKynbl , a Bblpa>KaeTCa Ka.K II = liT, I'Ae T = llc. BBe eM 6e3pa3-
MepHoe BpeMB t ' = t/to, r e to - xapaKrepHOe ra30AHHaMH1ieCKOe
npeMH r H 06e3pa3MepHM ypaBHeHHe (2.35):
df 1
dt' = "i (fo - f), r e E '" T/to.
(3.25 )
B THDUQHbIX ra30BLlX Te'tleHHSlX 6ff3p83MepHhlA napaMeTp E Man
H HMeeT nOpB OK 'IHeJIa KHy ceHa. Yq}fThlBaJI 3TO, CT8.BHM pe-
IueHAe ypaBHeHHSI (3.25) B BHAe p83JlO)KeHH.H B PBA no MaJlOM)' na-
paMeTpy t::
dfo 2 tPfo
f = fo - E dt! + E dt'2 +...
(3.26)
YpaOHeHHSlM 9Rnepa OTBeqaeT rrpe):tCTaBJIeHHe <PYHKIUlH pacnpeAe-
JleHHSI B BH e O Horo nHmb nepBOro qneHa pa3nO)l(eHHSI (3.26). ,lIpy-
I'He 60JIee llOJlHhle ypaBHeHHSI ra3OBOA RHaMHKH MorYT 6bITb no-
J1yqeHbJ c Y eTOM nOCJIeAYI01QHX 'ineHOB pa3nO>KeHHSI (3.26). TaKHM
UYTeM MO>KHO npHRTH K ypaBHeHHSlM BSI3KOro TeWIonpoBO oro ra,-
la.
3. BR3KOCTb If TeHJIOnpOBOAHOCTb
Cne;o'lO ee nOClle nOKaJILHo-MaKCBe1IJlOBCKOro npe CTaBneHHe
<1>YHK HH pacnpe eneHH.H, BblTeK8JO ee 113 pa..'3JIO)KeHHjI (3.26), 3&-
JIHIlJeTCSI B BH e
dlo
1 = 10 - E dt' ·
(3.27)
113MeHeHHSI 8 TeH30pe BH3KHX HanpSl>KeHHA j H B BeKTOpe TenJlo-
80ro nOTOKa q no cpasHeHHIO C BbIpa>KeOHBMH (3.17), (3.18), (3.20)
6y,nyT onpe,l\enstTbCH BTOpbIM JIeHOM B npaBoA qacTH (3.27), KOTO-
pwA: nepenHweM 8 BH e
H eM H3 (2.32) BblpameflHiI,lVIH 'IacTHblX npoH3BO blX <I>YHK-
IJ.HH /0, CTOS1 HX 8 npaBoR '-IacTH (3.28): .
8/ 0 _ ml/2 { _ m ( i-Uj)2 } _ l!!
8p - (211'kT)3/ 2 exp 2kT - p ,
8/ 0 = pml/2 ex { _ m( t-Ui)2 } m({j-Uj) =
aUj (21rkT)3/2 p 2kT kT
m
= kT( j - Uj)/o,
8fo [ 3 pml/2 pml/2 m( I-Ul)2 J
aT = - 2 (211"k)3/2 T5/2 + (21rkT)3/2 2kT2 x
{ m ( i-Ui)2 } ( 3m ( i - 1£t)2 )
X exp - 2kT = -2T + 2kT2 fOl
8/ 0 _ _ pml/2 { _ m ( i-Ui)2 } m ( i - Ui) _
8f.i - (21rkT)3/2 exp 2kT kT-
m
= -kf({i - Ui)/O.
Onpe eJIHM He HarOHaJlbHbIA 3JIeMeHT TeU30pa HanpH)KeHHH
(3.29)
(3.30)
(3.31 )
(3.32)
R. . 1 = f ( t o - E ) ml"--c' ,l = -€ f dfo m,.:c. de ( 3.33 )
1.} i#;'j JI dt' 'J dt' .. 1 ,
nOnb3YBCb cpopMynaMH (3.28)-(3.32). AHaJlH3 BXOAH B (3.28)-
(3.32) BbIpa.>KeHIdt no Ka3h1BaeT, QTO HCTBeHHblA qneH. KOTOpbJjt
BHOCHT BKn B HHTerpan (3.33)] ecTb
a/o i Oui. .
au j OXi
Cne,noBaTeJ1bHO, ):VIS j MO>KHO HaDHCaTb
P.. . 1 = -E Ou, J 8fo ""c' de =
'1 ".J." _.. '-5 1
Ir1 uX s vul
m OUl J
= -£ kT8x, c, (U,+c s ) CiCj!odc,
I1pHQeM B CY IMe (US+C,,) HaJIHqHe U 8 He SnH.SIeT Ha pe3ynbTaT. B
) HOHlIOH CYMMe rlo l H S He nOHBJlHlOTai He'ieTHble MOMeHTbJ TOJIbKO
111)11 I = i) s = j JlH60 I = j) s = i. Bocnonb30BaswHcb Bblpa-
iKeFIHSlMH tvlSI 1.feTHbiX HHTerpa.noB (3.16), a 3aTeM Bep"YBuIHCb K
I>a3l\'lepHblM BeJlHqYHaM, nOJly"tJHM
I lk ( OOi au; )
Pi. = -- -Tp - + - .
] i j V m aXj 8Xi
(3.34)
LVIHO)KHTeJllO v-I (k / m) T p ecTecTBeHHO npH):t3Tb CMblCJI KJIaC-
c'i eCKoro Ko3<l>c.PH HeHTa BSI3KOCTH IJ:
1 k
p, = --Tp.
11m
(3.35)
AHaJlOrHtJHblM 06paaoM MO)KHO nonY'tJHTb BB3KHe 068.BKH K
HarOH8JIbHblM KOMllOHeHTaM TeH30pa HanpH>KeHHA i H KnaccH-
'H CK"ft TenJJOnpOBOAHbIA BH BeKTOpa q
aT
qj = -X-,
OXi
I',l(e x - K03c1><1>H HeHT TelUIonpoBo)J.HoCTH:
(3.36 )
X= ( )2 .
(3.37)
3 aM e q a H He 2. BrK-MO eJIh B eJIOM npasHJIbHO OTp8JKB.eT 3Kcne-
pH leHT8J]bHO lIa6nlO aeMb(e <PYHK HoHanbHhle 3aBHCHMOCTH OT ra3O)J.HHa-
M H qeCKHX napaMeTpoB TeH30pa Ba3KHX HanpS>KeOHA H BeKTOpa TeDJI080ro
IIOTOKa.. Q)J.HaKO HeKOTOpbIe KonH'IecTBeHHble 38.KOIfOMepHoCTH, npHcy e
UH3KOMY renJlOnpOBO,lI)lOMY ra3Y, H3 BrK-uo,D.enH nonY'IeHbI 6bITb He MC>-
rYT. HanpHMepJ He cOOTBeTCTDyeT HCTHHe UCAO IIpaHdmAJc, xapaKTepH-
'iYlo ee OTHoweHHe ,[\Byx K034><1>H HeHTOB:
Pr = JlCp/" ,
(3.38 )
l'iU: c p -- Y,Il. JlbHa.R rennoeMKoCTb npH nOCTORHHOM aBJleHHH. B paMK8.X
I)I K-Mo,LJ.eJlJ1, C Y\feTOM Bblpa>KeHHA (3.35) AIlS! IJ H (3.37) ;lJ.JIB x, B cnyqae
o,J'lHoaTOMHoro raJa (c p = k/m) BcerJta Pr = I, TOr,lJ.a KaK B peanbHOCTH
Pr=2/3.
1.fT06bl npeOtl-OneTL ,lJ.8.HHYIO CHTY , B BrK-MO,l!-enb 880AJ1T aCHM-
MeTpH HYJO N8.KCBeJJnOBCKYJO cPYHKl(HIO pacnpe,neJIeHHR (12):
d/
- = 1/ (+ - f) ,
dt
(3.39)
r,ne
= p1r- 3 / 2 (detA)1/2exp{-AiiCiCj},
-1 2RT l-Pr
(A )i; = Pr 6 ij - 2 Pr Pij/P'
(3.40)
(6 ij - CHMBDn KpoHeKepa.)
4. Y pasHeHHB HaBbe-CToKca
IIpHBeAeHI-rasr B npe bWw.eM naparpa<j>e npo e)Q'pa nOKa3b1-
saeT) TO nOSiBneHHe Bjf3KHX H TennOnpOBO,lJ.HblX qneHOB Henocpe.lJ.-
CTBeHHO CBR3aHO C HaJJHQHeU ManoR, npo opqHOH8JlhHOA E t 06a.B-
KH K nOK8JIbHOA Ma.KCBeJlJIOBCKOa <PYHKQHH pacnpe eJJeHHSI /0. Ha
pa3J10>KelIHH B pft no MaJIOMY napaMeTpY E pemeHHB ypaBHeHHB
Bonb MaHa OCHOBaHa npo e,zurpa qeoMeHa-9HcKora [2, 3). C n
MO hlO 3TOro pa3JIOJKeHHH DWI}"IHM YpaBHeHHjI Hasbe-CTOKca H
KOHKpeTHbJR BH K03<tx1> HeHTOB BSI3KOCTH 11 TennonpoBO HOCTH.
1. Pa3J10)f(eHHe t.fenMeHs-9HcKora
3anmneM ypaBH€HHe Bon]) M8.Ha B 6e3pa3MepHOA <PoPMe
df 1
dii = E: J(f, It) ·
(4.1 )
HHTerpaJI CTOJIKHOBeHHA (2.12), 0603H&1feHHblR 3 ecb J(J, fl)t 38,-
SHCHT J1HHeRHO OT K8JK oro B3 060HX aprYMeHTOB. BB eM pa3Jl
>KeHHe cPYHKQHH f B p no CTeneHS1M MaJlOro napaMeTpa E
/ = /(0) + E: j(l) + E 2 J<2) + ... .
(4.2)
C<popMynHpyeM nOCTynaTbl, nonO)KeHHbIe 8 OCHOBY MeTO a
tIeIIMeHa-9HCKOra..
1) Ha paccMat'pHBaeMoD rH pO HaMHqeCKOR (ra30 HHaA(H'IeCKoA)
HH <PYHK pac eneHHB 3aBHCHT OT speMeHH TOJ1bKO
nOCpe){CTBOM DSiTH napaMeTpOB p, u, T.
2) Bee ra30)J:HHaMH'feCKHe napaMeTpbl MorYT 6b1Tb onp eneHbl
c nOMO blO O HOro nepBOro 'L1IeHa p83nO>KeHHSI (4.2).
IIpe,L\cTaBHM JleBYIO qacTb ypasHeHHJI (4.1) B B e
:;, = ( , + v ) I ,
of Gi 81
r e 1) f = i - + - - ·
OXi m a i
(4.3)
'ror al HCIIOJ1b3YSl pa3nO>KeHHe (4.2), ROJIyqHM
VI = 'DI(D) + eVJCl) + £2VI(2) + · '. . (4.4)
B CBOIO O'tfepe bJ HHTerp8JI CTOJIKHOseHH:A MO>KHO n craBHTb
II BH,lJ.e
.J(f,/I) = J( E ekl(k), f ePj P») = E f ek+PJ(j(k),lr}) .
k=O p=o t=o p=o
(4.5)
I)Be H 0603HaQeHHe
J(n) (JCO),j(l),...,/(n») = E J(!(k),/ P)), (4.6)
k+p=n
nepenHweM BLlp eHHe (4.5) B BH)\e PHAa
J(f,/d = JCO) (/(0)) +eJ(1) (/(O),j(1») +
+ e 2 .1<2) (/(0),/(1),/(2)) + · · · . (4.7)
<I>YHKl.\HH / 3aBaCHT OT BpeMeHH Qepe3 napaMeTpbJ P. u, T.
Cne,1J.OB, aTeJlbHO
{}j = oj + + ?1 (4.8)
at 8p at QUj Ot aT 8t ·
Onpe,lI- HM onepaTOp {J / at CJleA}'IQ1II)iM 06pa30M:
a 80 2
-=-+e-+e -+...
at at at EJt '
(4.9)
r,a.e OnepaTOpbl 8 p / 8t onpe englOTCSl qepm npOH3BO Hble no Bpe).{e-
HH ra30 HHaMHqecKHX napaMeTpoB p, u, T, KaK B (4.8).
QTo6b1 HaRTH ypaBHelIRS, CBH3bJBaJO He p83JIHQHhJe cl>YHKU.HH
j(p), no craBHM pa3J10>KeHHC (4.2), a T8lOKe BbJpa,)KeHHS1 AnSI HH-
TerpaJIa crrOJIKHOBeHHIt (4.7) H onepaTOpOB V (4.4) H a/at (4.9) B
ypaBHeHHe BOJlbl_MaHa (4.1). 11 leeM:
[ ( au + E 01 + c: 2 lh + . . . ) ( /(0) + € j(l) + E,2 ,(2) + . . . ) +
&t at at J
+ (viO) + EVfCl) + E 2 VjC2) +...) J = J(O) (/0») + (4.10)
+ E J(l) (f(O) I f(1») + E 2 J(2) (f(O) , f(1) I J(2») + ... .
npHpaBHHBasr K03cl><PH He.ITbl npH O,l\HHaKOBhlX CTeneHHX c, npH-
,a.eM K 6eCKOlleQHOH CHCTeMe ( enO'iKe) YP8BHeHHA: NIB onpeAeJIeHHSI
f (p) :
0 - J(O} (f(O}) ,
( + V)J(O) - J(l) (f(O), J(l}) I
(1.11 )
( 80 +V)f(l)+ f(O} - J(2) (f(O), f(l) I 1(2») ,
at 8t
-... ......... ....... ... .... .................... ........ ...
r eJ HCXO jI H3 (4.8),
Olel(p) 8f(p) 8kP 8f(p) a"Ui 8f(p) B"T
= -+ +
at op 8t au. 8t 8T at ·
YqHTblBaSl) qTO p, U, T onpe,lJ.eJISUOTCH JlHIUL nepBblM qJJeHOM
p8.3JIO.>KeHHSI (4.2), nOJIYQMM
f f{O) d =
mc 2 /2
f J{p) = 0 ,
mc 2 /2
p
pu
p T
2 m
p = 1,2,. . .
( 4.12)
BeKTOp 'l'ennOBOro nOTOKa H TeH30p BH3KHX Hanpa.>KeIWR, co-
('naCHO pa3JIO>KeHHIO (4.2), 38JlHWeM B BH e
Pi) = ! f mCiCj = E £P / j(p)me;.cj = E r) I
,=0 p=O
( 4.13)
qj = J f mc 2 cj d( = f e: P J /(p)m Cj d( = E qf>. (4.14)
p=o p:O
Ilo CTaBnHH (4.12)-(4.14) B paHee nocTpoeHllble MOMeHTHhle
) paBHeHHH (3.10)-(3.12) H Y1.JHTblBasI, 'ITO p, u, T CYTb BeJIHqHHbl
0(1), a TaK>Ke Bblpa>KeHHe (4.9) ,lVIg onepaTOpa alOt, IIOJIYQHM:
fJop a
{Jt + 8Xi (PUi) = 0,
{JpP = 0 1 2
at ' P = , ,...
(4.15 )
80 a {JP ) G.
- ( pu. ) + - ( ptJi U . ) = - 'J + J P ,
at J aXi 3 OXi m
8 8P.( )
p ( ) IJ
at Pu j = - {JXi '
(4.16)
p = 1, 2, . . .
fJoE + ( fJUi E + u..P-' ) + q O» ) - IJUi = 0,
m i J 1 m
apE + ( 41 . p _ ) + q » ) = 0 P - 1 2
at 8Xi LA) l' I , -, ,...
"lITo6bl HaATH Bblpa>KeHHSI ra30 HH&MHqeCKHX napaMeTpOB, <l>H-
I'YpJ.IPYIOIQHX B CHCTeMe (4.15)-(4.17), H8.AO nOCJIe OBaTeJlbHO pe-
UlaTL ypaBHeHHH (4.11) OTHOCHTeJIbHO <PYHK HA f(P), P = 0,1, 2,. . . .
nepBoe ypaBHeHHe CHCTeMbI (4.11) onpe eJJSleT <l>YHKU)f1O pacnpe-
lI:e.JIeHHSI ,(0), AJl5I KOTOpoR HHTerpan CTOJIKHOBeHHA pa.BeH HynlO.
COrJlacHo (2.29), TaKaS! Q>YHKQHSI OJI>KHa 6blTL nOK8JlhHO-MaKCBeJI-
."lOBCKO :
( 4.17)
B rlpe bl y eM naparpa4>e 6bInO nOK83aHO, \frO ec-JIH B Ka'lecTBe
npH6JIH>KeHHSI <PYHK HH pacnpe eJleHHg B3RTb nOKaJIhHhlA MaKCBen-
JlRaN, TO H3MeHeHHe ra30 HHaMHqeCKHX napaMeTpOB 6YAeT ODHCbI-
BaTbCS ypaoHeHH5IMH 3AJIepa. To >Ke caMoe Ha6nlO aeM H 3Aecb.
KaK cneA)'eT H3 (4.13)-(4.14), 06pbIB pa3JJO>KeHHR (4.2) Ha. HYJIeSOM
'fJIeHe /(0) npHBo)\HT K 3Ha'feHHHM p.; = p (!r./m) T6ijt qj = 0 t
IIPHCY HM HeBSl3KoMY HeTennOnpoBo,nHOMY ra3Y. TaKHM 06pa3oM, B
3TOM CJlyqae CHCTeMa ypaBHeHmt (4.15)-(4.17) COBn aeT C YpaBHe-
HHRMH 9ffJIepa.
2. ROK8JlbHO HaBbe-CTOKCOBCKaH cPYHK HH paclIpe)J.eJleHHH
HaA,n:eM Tenepb CJleJU'IO A qneH pa3JJO>KeHHSI Q>YHK HH pac-
rrpe eJIeHHR (4.2), BHOBb OlIHp8JICL H& cHCTeMY ypaBHeHHA (4.11).
)I.nH 3TOro, 3HasI (4.18), He06xo):\HMo pemHTb YPaBHeHOe
( : + V) 10 = ;(1) (/Oti 1 » .
(4.19)
BYAeM HCKaTb peweHHe /(l) 31'01'0 ypasHeHHSI B BRAe
I(i) = 10 cp(l).
(4.20)
He OCT8.HaBJIHB8JlCb B AeT8JUIX H8 OCTaTOqHO I'pOM03,AKoA: npo-
e,lQ'pe pemeHHB HHTerpanbHoro ypaBHeHHB (4.19), BblDHIIleM 8bl-
pa>KeHHe JV1S1 HCKOMOA: <I>YHK HH «p(1):
( I ) Ci aT B ( s) ( 1 2 ) Ou,
in =-A ( s ) --- Cic.- c6'.-
T T 8x. 2RT '-> 'J ax.'
. , ,
(4.21)
r e s = lell V 2RT. CKaJlSlpHhle <I>YHK H A H B onpeAeJlBIOTCSI
c nOMO 1O peweHHSI HHTerp8JIbHLIX ypaBHeHHA. HanpHMep, ,ZV1H
HaxO)K eHHa B HcnGnh3yeM paeeHCTBO
* j ((c;Cj - c26ij) B) = -2 ( c;Cj -lC26ij ) ,
r-ne HHTerpanbHblA OIlepaTOp J(<p) aeTCH Bhlpa>KeHHeM
(4.22)
J(<p) = J 10( 1) (<p( ) + rp«() - CP«(l) - rp(E)) Igi bdb dO l ·
I1TaK, CJle YIOw.ee 3a nOKaJIbHo-MaxcBeJIJlOBCKHM npH6JIIDKeHHe
t lenMeHa-9HCKOra eCTeCTBeHHO, KaK MbI YBH,ltHM HH>Ke, HB3B&Tb AO-
aAbHO Ha8be-cmcncco6c"OO CPYHK HeA pacnpe)J.eJIeHHSI. ORa HMeeT
nHA
INS = 10 [1 - ; ACi ::: - 2 T B (CiCj - c26ij) :; ]. (4.23)
3. ra30AHHaMHQeCKHe ypaBHeHHH HaBbe-CTOKca
BbIpa eHHe AJUI CPym<QHH pacnpe eJIeHHH (4.23) n03BOJUIeT
onpeAeJIHTL TeH30p HanpSl>KeHHA Pij 11 BeKTOp TennoBoro IIOTOKa
qJ :
j = m J Cicjl NS = 6ij pR:I' + l) I
m J 2 (I)
qj ="2 CjC f NS d = qj I
( 4.24)
(4.25 )
r )1.e
l) = - 2 :; f B(s) CiCj ( Ck C ' - ! c26 kl ) 10 d{,
(1) m 8T f 2
qj = - 2T aXi A(s) CiCjC 10 df.,
(4.26)
( 4.27)
MCfiOJlb3YSl BH)J. nOKaJlbHo-MaKCBeJJnOBcKoA: CPYHK H pacnpe e-
neHHH, BhJpa:>KeHH.H (4.26)-(4.27) nepenmneM B B e
<,l) = _ JJ ( + Ouj ) 1 i 1= j ,
, ax; OXi
p.<.l) = -IL ( 2 Ou; _ 2 diVU )
11 r- 8 X i 3 '
q(1) = -:J( aT .
1 aXj
( 4.28)
(4.29)
(4.30)
Ko3<P<t>H HeHTaM nponOpD.HOHaJlbHOCTH JJ H X ecTeCTBeBHO npH aTb
CMblCJI Koo4><PHQHeHTOB BSI3KOCTH H TeIIJIOnpOBO HOCTH. ORB 3 a-
IOTCH <l><>PMynaMH
00
8pKI'e f _,2 6
#J = 15y'7r e B(8) 8 ds,
o
2 00
- 8pR T€ f -,2 A( ) 6 ds
X-...ji e 88.
o
(4.31 )
( 4.32)
HanOMHHM, tfTO K BRAY (4.28) NISI KOMnOHel(T TeH30pa Hanpsr-
>KeHH:A MbJ Y)I(e npHlllna B (3.34), pa3na.ras B no MaJlOMY napa-
MeTpy peweoue MOAeJlbHOI'O BrK-YP8BHeHHSI (3.25). Me Bhlpa-
)l(eHHSlMH (4.28) H (3.34) MO>KHO YCTaHOBHTb IIOJIHoe COOTBeTCTBHe,
ecJJH B KaqeCTBe BenH'IHHhI T, 06paTHoA 3HSQeHHIO 'IacTOTbI CTOJIK-
HOBeHHA B BfK-MoAenH, HCnOJIb30B8Tb HHTerpan
00
8 e f -. B( ) 6 d
r=i5 yr,r e 888.
o
(4.33)
Onpe):\enHB TeH30p BSI3KHX Hanpsl>KeHHA j H BeKTOp TeWlODO-
ro UOTOKa q C nOMO hlO Bblp8JKeHHA (4.24)-(4.30), nO CT8.BHM COOT-
BeTCTBYIO He HX 3HaqeHHH B MOMeHTHble yp8.BHeHHSI (3.10)-(3.12).
TeM caMbiM npH eM K CHCTeMe yptJ8HeHut1 Ha8be-Cmo"ca, onHCbI-
88.1OIlI.HX Te"leHHe BB3KOro TeIIJIonpoBoAHoro raaa:
op a
at + 8Xi (pud = 0,(4.34)
a a {JR. · G ·
- {J (puj) + (puiUj) = - {) 'J + p, j = 1,2,3, (4.35)
t V'" i :ri m
(p +PE(T,P}) + 8 i [P'Ui (f+£) +Ui i] -
G i 8 8T
- ;;; fYlJi = 8Xi x 8 X i ' (4.36)
r,lJ.e KOMnOHeHTbl TeH30pa ; onpe enHlOTCSI c nOMOI.U.blO BbIpa>Ke-
BRa
Pi; = p(T,p} 6 ij + 1).
( 4.37)
B CBOIO oqepe , pH) orrpe,l\eJUllOTCSI H3 (4.28)-(4.29). ,n8.BJIeHHe p
II BHYTpellHHSI 3HcprHB e tVISI OAHoaToMHoro rB3a, npH Y\fe-re TOJIbKO
ynpyrHx CTOJIKHOBeHHA, 3a.n.alOTCSl cPoPMYJlaMH
p = pRTt
3
e = 2 RT,
R = k/m.
( 4.38)
L-3 o6w.e I CJlyqae Heo6xo HMO nonb30B8.ThCH 60JIee C1I0)l(HhlJdH 3a-
BHCHl¥10CTSlMH p(T, p) H e(T, p), DOJIyqaeMhlMH TeOpeTH'tJecKH JIH60
:) Kcnep H M eH1ClJI bRO.
3 aM e 'I a H He 1. YpaBHeKBfl Haabe-CTOKca MorYT 6b1Th nOJIY'IeHbI H
113 ypaBHeHHH 9HcKora (2.36),l])IB nJlOTHLIX r830B H KOCTeA. "pH STON
AaBneHHe p H BeKTOp TennoBOro fiOTOK8 q 6YIl:!T CKJI BaTbCJI H3 ABYX
tlacwre:A: KHHeTH(.leCKoA, aHaJlOrH'lHoA p83pe>KeHHhIM ra3&M, B nOTeH anb-
HoR, CBjl3aHHOA c nepeHOCOM CTOJIKHOBeHHA, 06ycnoBJIeHHhIM KOHe HhlM
pa3MepOM MoneKyn. npH 3TOM C YBenH'feHHeM IIJIOTHOCTH BKn fiOTeH-
IlitaJIbHOA COCTaBnHIO eA 6Y eT B03pacraTb.
B npe enbHOM CJJyqae, KorAa C>KHM8eMOCTLIO cpe bl MO)l(HO
npeHe6peQL H nnOTHOCTb p DOCTOHHHa, YpaBHeHHB (4.34)-(4.36) ne-
peR,nyT B ypaBI eHHSI Hasbe-CTOKca NIB HecX<:HMaeMOA B$J3KOA: )l(H)J.-
KOCTH:
p ; = - gradp + div IJgradu + F,
( 4.39)
( 4.40)
div u = 0 .
dE QUj au. {) err
p - = J.l - ':..::2 + - x - + UiFi ,
dt ax j OXi OXi OXi
rAe d/dt = fJ/8t+UiO/8xi - cy6CTaHQHORaJIhHaH npoH3BO H&H DO
13peMeHH, F - BHeWHSlSI CHJI&. ,n:eICTBYIO asI Ha e HH o61,eMa.
I1CTopH'lecKH ypaBHeHHjJ HaBbe-CTOKca (4.34)-(4.36), TaX >Ke
KaK Ii ypa.BHeHHB 9Anepa. , 6h1J1H nWIY'lfeHhl cPeHoMeHOJIOraqecKHM
nYTeM - Ha OCROBe HCfiOJIb30BaHWI BHTerp8J1bHhlX 3&1<OH08 coxpa-
HeHHii. OJJ;HaKo CHCTeMa yp8.BHeHHA (4.34)-(4.36) OTnHqaerCB OT
opHrHHaJIbHoro ct>eHoMeHOJIOrHqeCKOrO nOAXOA& MeTO,n:OM onpe eJIe-
IIHH K03<p<pHu.HeHToB BSl3KOCTH H TennOnpOBo,nHOCTH. nepBOHaqaJIb-
HO OHH HaxO,ll.HJ1HCb 3KcnepHMeHTaJIbHO, KaK Koa<P<PH eHTbI npo-
nop HOHanbHOCTH, CBH3bIBaJO He KOMnOHeHTbI TeH30pa HanpH>Ke-
HHA H BeKTopa TenJlOBOrO DOTOKa C COOTBeTCTBYIOID.HMH npOH3BO -
(4.41)
HblMH CKopocTe:A H TeMnepaTyphl. 3 ecb >Ke H napaMeTpbl, Ha3bl-
BaeMble K03ct><PHQHeHTaMH MOJIeKynSlpHoro nepeHOCa, onpe enHJOTCSI
C nOMO blO BblpaJKeHHA (4.31 )-( 4.32), onHpaJO cSl Ha <l>YHAaMeH-
TaJIbllhle MoneKYJIHpHble cBo:ACTBa aHHOro ra3a.
PacCMOTpHM, IIanpHMep, rrpoCTeAwylO Mo enb 6HJIbBP;J;IIblX ma-
POD HaMeTpa d. ,lVIsr Hee
5 vkmT
JJ = 16 .jidl 1
( 4.42)
3
x = '2 Cpl-t
( 4.43)
HJlH (CM. (3.38)) x Pr = jJc p npH Pr = 2/3 .
B03MO'l<eH eII e O HH nyrb nOJlyqeHJi BH3KHX qJIeHOB B YPaB-
HeHIDlX ['a3OBoR j\HHaMHKH, 3af1HMaJOIII.HA B KaKOM-TO CMbICJJe npo-
Me>KYTO'fHoe nOJIO)l(eHHe Me)K)ly <l>eHoMeHOJIOrH'leCKHM nOCTpoeHH-
eM ypaBHeH.HA Haobe-CTOKca H HX BblBO OM C nOMO 1O npo e,ll;Y-
phI QenMel-la-9HCKora. S1 3TOro pacCMOTpHM npOCTYIO KHHeTH'Ie-
CKYIO MO eJlL. ITYCTb I - CPCNfHjI )VIHHa cBo6o oro np06era, a c
- cpe HSlSI CKOpOCTb XaoTH eCKoro ,lI;BH>KeHHH. COrJIacHo onpe,AeJIe-
HHIO TCMnepaTypbl,
c = vR:I' .
( 4.44)
06paTHMCSI K PHCYHKY 4. TIe-
peHOCHTh CBOR HMny JlbC Qepe3
JIBHHIO Y = 0 MorYT MOJIeKY-
nhl, Haxo SlDUiecSi B TOqJ(e 1, (Yr-
CTOSl e OT TOQKH 0 Ha pacCTO-
BHHel.Cne OBaTenbHO,H3MeHe-
HHe HMnYJIbCa PZll 3a Ct.feT Mone-
X KYJISpHOrO nepeHoca 6YAeT npo-
IIOp OHaJIbHO p33HOCTH CKOpo-
PHC. 4. K onpe eJleuHIO Ko3CjJcI> eB- CTeA Uo - Ul B TOqJ(ax" 0 H 1.
Ta BB3KOCTH Yq}fTbIB8JI) 'iTO qHcno MOJIeKYJI,
nepeceKaJOw.Hx JlHHHIO Y = 0 B e HHUY BpeMeHH, paBHO nc, H Pa3-
y
1
1
00
JlaraH pa3HOCTb UO - Ul B p TeAJlopa, nOJIY HM
au
Pz;y = -IJ 8ii '
1 1
p, = 2" mnc = 2" pel ·
( 4.45)
(4.46)
IHO)KHTeJIb 1/2 B BbIpa>KeHHH (4.46) o6paayeTCSI 6naro apjl yqeTY
BH)KeIIH MOJleKyn noJI. pa3nHq}JbIMH yrnaMH K OCH y.
06pblB pa3nO)l(eHH5I QenMeHa-9HcKora Ha HyneBoM "IJleHe npH-
BO,LJ.HT K ypaBHeHHSlM 9AJIepa JIS Ma.KpOCKOnHqecKHX napaMeT-
pOB. I1cnOJIb30BaHHe nepBoro npH6nH)KeHHB <j>YHKl.{HH pacrrpe ene-
HHH npHBO H'r K ypaBHeHHSlM HaBbe-CTOKca. 9TY npo eAYPY MO)f(-
110 npO,1J.o.n>KaTb aJIbwe. OnHpasIcb Ha TpeTbe ypaBHeHHe CHCTeMbI
(4.11), MO)KHO HaRTH CJIe IO HH 'iJIeH p33JIO>KeHHH 1(2) H C era 00-
A10 blO nOCTpOHTb COOTBeTCTBYI01I\He ypaBHeHHSI )VJg orrpe eneHIDI
rd30 HaMHqeCKHX napaMeTpOB - ypa8HeHUJI BapHemma [2].
O,.ll,HaKO ypaBlleHHB BapHeTTa He HamJlH CKOJIb-HH6Y b 3aMeT-
floro npHMeHeHHH B pemeHHH 38.,L\a'l ra30BoA HHaMHKH. Bo-nepBblX,
OHH ropa.3 O CJIO>KHee YPaBHeHHR HaBbe-CTOKca.. Bo-BTOphlX, HaA-
eHHbIe C HX rIOMOIII.bIO peweHHH He YJIyttWalOT Ka'tfeCTBO c:lJH3HQeCKO-
ro onHcaHHS1. TaM, r,a.e cnpaBe JlHBbI ypaBHeHHH Hasbe-CTOKca, pe-
weHHH 06eHX CHCTeM OqeHb CJla60 OTJlH'laJOTCH APyr OT ,zq>yra. TaM
}t(e, r e npHMeHeHHe Mo,n:enH BH3Koro TennonpoBo Horo r83a He o-
CTaTOqHO 060CHOBaJIO, HanpHMep) np" YBeJlHQeHHH QHcn8 K.ly ceHaJ
ypaBHeHHH BapHeTTa AaJOT pe3KOe yxy eHHe peweHHH no cpaBHe-
HHIO c ypaBH€HHHMH HaBbe-CToKca.
TIO,A06HOe nOBeAeHHe pemeHH.H ypaaHeHHR BapHeTTa He CJIY'IaR-
HO.. rrpoQe ypa t.{enMeHa-9HCKora KOHCTPYHpOBaJIaCb KaK Pa3JIO)l(e-
HHe B pHP- 110 MaJIOMY napaMeTpy e 1 BenHqHHa KOTOpOro TecHO CBH-
181la C 'iHCJIOM KHy ceHa. B enOM MeTO,l\HKa nonyqemrSl YP8BHeHHA
3Anepa H HaBbe-CTOKca H3 perneHHH ypaBHeHHSI BOnb M8Jfa no -
TBep>K,l\aeT xopOWO H3BecTHblft H3 r830BOH AHHaMHKH cPaKT 06 HX
crrpaBe nHBOCTH ,lVIH MaJlhlX qHCen KUYAceHa..
4. Yp8BHeHHH ){HH8MHKH CMecH r830B
no aHa.TlOrKH C 'reM, KaK 83 yp8.BHeHHB BOJlb MaHa c nOMowplO npoQe-
AYPbI QenMeHa-9HCKora BblBO)J)ITCjI yp88HeHIUI HaBbe-CTOKCa l onHCbIBa-
IO He O,lJ.HOKOMDOHeHTHblA BJJ3KHA ra3, H3 CHCTeMbI YPaBHeHHH BOnL MatIa
AJI)) raaoBOA CMeCH nOJlY'IaIOTCSI YpaBHeHHB HUbe-CTOKCa MHOI'OKOMn(}-
HeHTHoA CMeCH (1 J 3J.
llepenHweM e e pa3 yp&BHeRHe BOJJh aJt& AIDI <l>YHKll,HH pa.cnpeAe-
neHH.A i-ro COpTa 'IacTHD. (2.35) ( 2, 0.5):
i Ni
81. i {} Ii G; {J Ii - f< I' J ) I I b .
at + j a-: + ---:- a& - L..i Iii l' - Ii p 9pi db dB d fJ '
x, 7ni J p=1
i = 1, . . . ) N i ,
( 4.47)
rAe i OOH&'IaeT CKOpocTb MoneKynLl copm i. 3aTeM) K8K H B O HOKOM-
nOHeHTHOM cnyqae, AJlS <l>YHK R h CTpoHTCR pa3no eHHe qenMeH8-
3HcKora, B KOTOpoM 6ePYTCJI ,ltBa nepBhIX 'lneH&. C Y'IerOM DOJIY'feHHb!X
Bhlpa>KeHHA, ypaBHeHHe (4.47) HHrerpHpyeTCJI C CYMM&'l'OpHbIMH HHBapH-
aHTaMH
H, Hi
Ip({) = ffl1, L ms i J L ml: 2 /2 ·
1:=1 i= l
(3,necb 3HaK I: nO)q>83YNeBaeT YMHIDKeHHe oro ypaBHeHHB (4.47)
H& COOTBeTCTBYJOUU'1O cl>YHKU.HIO, HHTerpHpoSaHHe no (.i H CYMMHpoBalIHe
no seeM KONnOHeHTaM.) 9TO npHBo)1wHT K CJIe)U'JO ea CHCTeMe YPaBHellHA
HaBbe-CToKCa:
i + diV(Pi U ) + div ji ::: 0, i ::: 1,... ,N i ,
Hi
8 ( ) () BP j .. Pi i
Fi PUll + _ 8 . (pujUt) = - fh + L- ---:- G Ie ,
z, J i=l m.
(p +Pt:) + ; [PUi( +t) +U le P jlc ] -
N,
Pi G i d .
- L..i ---:- jUJ = - IV q.
. m.
1=1
( 4.48)
k = 1, 2, 3,
( 4.49)
( 4.50)
Yp8.BHeHH1I Haa&e-CTOKca ,l),J1B MHoroKoMnoHesTHoro r83a (4.48)-
(4.50) OTJIH'I8JOTCJ1 OT 06blQHoA CHCTeMbI (4.35)-(4.37) ypaBHeHHHMH nnOT-
HocTeA' OT,l\eJlbHhIX KOMnOHeHT Pi H eD.\e p OM ,neT8JleA. B (4.48) npHcYT-
...rnYeT ji - ,l\H<P<PY3HOHHbIA nOTOK i-I KOMnOHeHTbI. IIpocTeAwHM opa-
f)J1J }f(eHHeM JI.JU1 Hero CTlY>KHT 3&KOH HK&
ji = -pDt gradCi t
( 4.51)
I'}(e P -- cYMMapHaH nnOTllOCTb:
Hi
P = LPi i
i=1
l)i - KO H eHT R4>4>y3HH, Ci = p,/p - YAeJlbll8Jl KOHl(esTp&U.HJI
I-A KOMOOHeHTbI r83&. npH 9TOM CYMM& AH<I>4>Y3ROHHblX nO'lOKOB Bcer,Q&
paBH& HYJlIO:
Hi
Eji = 0 .
i=1
B ypaBHeHHH (4.50) BHyrpeHIUIJI 3HeprHB £ npe,ll.CT&BJJJllOTaI B BHAe
N,
E = L CiEi(T, p},
-1= 1
r,n.e E i - napLl.H8J1hHbIe 3H&'1eHHJI BHyrpeHHeA 3HeprHH. TennoBOA nOTOK
q 3afiHChIBaeTCJI K&K
N,
q = -HgradT + Lj i l4,
i= 1
( 4.52)
r e h, = hi(T, p) - nap an&Hble 3HT&J1bnHH.
TeH30p HanpJl.>KeHUA Pile, 4>HrYPHPYIO 1 B ypuHeRHJlX (4.49)-
(4.50), HMeeT TOT )Ke BH.D., qTO H B o,l\BOKOMDOHeHTBOW raae:
Pjlc = 'i -p ( au + 8uJ; - di"diVU )
J lJz. 8z j 3 1
r e nOJIBoe AABneHHe p ecT& CyMM8 napQ.HaJ1hHhIX A&BJIeHHA:
( 4.53)
Ni
P = LPi(T,p).
i=-1
K034><I>HQHeHTbI Jl-a<P<l>Y3HH D i t rennonpoBO,lUlOCTH '" H BJI3KOCTB IJ B BhI-
pa>KeHHHX (4.51)-(4.53) 3&8HCBT He TOJIhKO OT pMOAKHa"HqecKHX napa-
MeTpoB, 80 H OT COCT8B8 CMecH.
KaK Y)Ke ynoMHHaJlOCh Bb1We, B 6onbWHHC7Be 3a.n.a ra3000A ,l\HHa-
MHKH MHOrOKOMnOHeUTHYIO CMeCb MO>KHO C ycnexoM 38.MeHH1'b 3<P<PeKTHB-
HbIM O)l.HOKOMnOfleHTHbIM ra.30M, AJUI KOTOporO cnpaoeAJIHBbI KJlaccHqe-
CKHe ypa8ueHHSI Hasbe-CTOKCa H nO,lJ.6HpaJOTCSI COOTBeTCTBYIO He ypas-
aeHHJI COCTOJIHIDI. r,n:e eACTBHTenbHO Ba>KHO rfHTbIBaTb UHoroKoMn(}-
HeHTHYIO DPHpO,ay CpeAbI - TaK 3TO npK MOAeJIHpOBaHBH re'ieKHA XH-
MH'lecKH pearHpYIOlI(HX raJOB. H8JIl{qHe XHMH eCKHX npeBpa1l\eHHR onpe-
eneHHbIM 06Pa30M BH,nOH3MeHReT ypaoHeHH'I (4.48)-(4.50). 0 np06JIewax
H pe3ynbTaTax Mo enHpOBaHIDI TeqeHHA c peaKlJ.HHMH noitD.eT peqb B rna-
Be V.
5. YpaBHeHIlH rp3Aa
YpaBHeHHH Ha.Bhe-CTOKca He 98JJBlOTQI HCTBeHHoA MaTeMaTH-
'IecKOH Mo,qenblO, onHCblBaIOlQeA TeqeHHe 8S13KOro rB3a. B 1949 r. rp A
(H. Grad) n O>KHJI CBol nOAXOA [1), npHBe)J.IIIHA K nocTpoeHHIO OPHrH-
H8.JIbHOA CHCTeMbl YPaBHeHHii, BBnftJOIIOtXCB 8 KaKOM-TO CMb:CJJe aJ1bTep-
Ha1"HBoA ypaBHeHHlIM Ha.Bbe-CTOKca.
QT06bI nonY HTb 3TY cHcreMY, YMHO>KHM ypaBHeHHe BOJIbl(MaH& Ba
D011HHOMbl 3pMHTa BeKTOpHoro aprYMeHTa Hn(c/ vRT ) H npOHHTerpHPY-
eM no BeeM CKOpocT.ftM MOJIeKYJI. OAHOBpeMeHHO npe CTaBHW <PYHKU)f1O
pacnpe,.qeJIeHBR B B e npoH3B eHHJI nOKaJJbHoro MaKCBeJUIHBHa Ila CYM-
MY KOHeqHoro qacna nonHHoMoB 9pWHT8. UpH 3TOM BOOHHKHeT CHCTeMa
AHQ><I>epeHlI,HaJIbHbIX ypaoHeHHA OTHOCHTeJlhHO IIpOH3BOJlbHbIX BeJIH'fHH p,
Ui, T I Pij, qi H MOMeHTOB 60JIee BhlCOKOI"O nopJJ.ltK8. IIpHewneMblM qHcnOM
MOMeHTOB 6Y eT N = 13, KOr)l.B HeH3BecTHblMH 6y;u'T P. 'Ui, T, Pij H
9i, r)l.e Pi; = Pa; - v6ij - TeH30p BJl3KHX Hanpj()KeHHA. CHCTCMa 3'rHX
ypasHeHHR nOJIyqHJI& Ha3B&HHe 13-woMeuTHoro npH6ntDKeHIUI rp3)1.a.
B yK83aHHO.R Mo enH TeH30p BJI3KHX Hanp1l>KeHHA H BeKTOp nOTOKa
Ten.na HBJIHIOTCJI He3&8HCHNhWH <l>YHKD.H.RWH, KOTOpble ODPe,I(eJljflOTCJl H3
COOTBeTCTSYlO1I(HX YPaBHesHR
Bpi.j {) [ 8u i aUj 2 aUk ]
- + - ( UIcPij ) + P - + - - - 6 ij - +
8t OXic 8xj ax, 3 8x"
2 ( 8qi 8q; 2 OQ1c ) au j P
+ - - + - - - c5 i . - + (Pit + plci) - + - Pi' = 0 ,
:; ax, lJXi 3 J OXIt aXle JJ J
( 4.54)
oq, a 5 fJT OPik
_ a + a - ( Uj:qi ) + 2 Rp 8 - + KI' a +
t XI: Xi X/c
7 fJT Pij ( 8p (Jpjk ) 9 au,
+ - RPit - - - - + + - qA; - +
2 eXt P ax, OXic 5 aXle
qA: ( lJui, aUk ) au, 5Rp
+- --- +q,-+-qi=O.
2 Ox Ie 8Xi aXle 2"
(4.55)
B <PopuYJ1e (4.54) o6paTHM BHHMaHHe Ha Bblpa>KeHHe B KBa,.rJJ>aTHblx cKo6-
KaX H Ha nocneJ])lHA qneH neBOA 'I8CTH. C HeKO'I'OpbIMH Or080pKaMH MO>K-
HO YTBep>K aTb, tfTO Bj(3KHe HanpH>KeHHjI (4.54) H TennOBoA nOTOK (4.55)
peJta"c'UPYJOm K CBOHM KnaCCH'qecKHM 3Ha'leHHRM (4.28)-(4.30) 83 ypaB-
HeHHA HaObe-CTOKca.
BnpoqeM, ypaBReHHB rp3Aa He HawnH 3aMeTHOro npHMeHeHH1I B
np&KTHKe Mo enHpoB&HHjJ Te'leHHA Bjf3KOro raaa. B<rnepBblX, QHCJ1eH-
Hoe pellleHHe 13-MoMeHTHbIx YPaBHeHHR COnpH>KeHO C ropaJAo 6onbWH-
MH TPY,lUIOCTRMH, qeM peweHHe DWI'II YPaBHeHHA HaBbe-CTOKca. BropbIM
Het(OCTaTKON (CM. (4)) 3TOA CHCTeMbI HBJlHeTCH TO, 'lTO npe.ll.CTa.8JleHHe
ct>YHKnHH pacnpe)\eneHHjI B BHAe M8.KCBeJlJlOBCKOA, YMHO>KeHHOA ua KO-
He'lHYIO CYMMY nonHHOMOB 9PMHT8, cnpaBeA1I HB o A1UI BeupepblBHblX no
CKOpOCTBM MOJIeKYJI <j>YHKl(HA. OAHa.KO Ha rpaJIH e paccMaTpHBaeuLIX 06-
JlacTeA, r)J.e THDHtJHO <I>oPMHpoBaJlHe DorpallWiHbIX CJlOeB, onpe.nenSIO HX
Bee TeqellHe BJl3KOrO raJa, <PYHK SI pacnpeJJ-eneUHH MO)Ke'!' HcnblTbIBaTb
pa3pbIB.
.
*
*
B 3aKnJOQeHHe aHHOH rnaSbJ 8BTOp XOTeJI 6bI em.e paJ OCTa-
HOBHThal Ha Ba>KIIOCTH KHHeTHqeCKOA TeOpHH nSl HccneAoBaHHH
ra30 HHa)'fHqeCKHX npoqeccOB.
Bo-nepBbJX, 6naro apSi TaKOMY no,l).X0AY B03HHKaeT nOHHMa-
HHe rny6HHHbJX CBH3eft Me)K):ty ra30 HBaMH'IeCKHM (rH po HHaMH-
qecKHM, MaKpOCKOnH'IeCKHM) OllaCaHHeM CllJIOWHOR cpe,lI.bI B ee
MOJIeKY JI pHO- KHBeTuqecKHM OnHC&HHeM.
Bo-BTOpblX, 60nee 'ieTKO Onpe eJlHeTCSl o6nacTL napaMeTpOB,
rtl-e )J.JIH OfiHcaHHH )I.BH>KeHHB ra3a OCTaTOqHO OrpaHH'IHTbCH YPaB-
HeHHH IH 3A:nepa HJlH HaBbe-CTOKCa, a r,l1.e He06xo HM nepexoA K
ypaBHeHHIO BOJIbD-MaHa HnH ,lij)yrHM KHHeTHqeCKHM MOAeJIHM.
5B TJJ. 1. YpaBHeHJle Banb MaHa H yp&BHeHHll ra30BOR AHH&MHKH
B-TpeTbHX, K03ct>ct>HlI,HeHTbl BB3KOCTH H TennOnpOBO HOCTH Mo-
rYT 6b1Tb Orrpe,lJ.MeHbl He TOJIbKO nocpe,a.CTBOM HaTypHoro 3KcnepH-
MeHTa, XOTS B pSl)J.e CJ1}"1aeB HMeHHO TaKOft nYTb npHBO HT K 60-
nee TOqHbI 1 pe3YJIbTaTaM H SlBJlSleTCSI OCTaTOttHO y.n06HbIM. ,LVJ$!
HaxO)K)J.eHHH 3THX K03c1><PHQHeHTOB c nOMOIUpIO KHHeTH'IeCKOR Teo-
pHH C03 aHa Cne HaJlbHaji npOl\eAYpa Bblq}JcneHHA, HCnOJTb3YJO aH
<J>YHAaMeHT8J1bHbJe MoneKYJISlpHble cBoftCTBa r830B.
Ha nepBOM MeCTe He cnY'IaAHo BblAeneHO nOHHM3HHe CBSl3H C
KHHeTHqecxUM onHcaHHeM. K CO>K8JIeHHIO BblQHCJIHTenbH8JI r po-
H ra30sas HHaMHKa Mano OnHp8JI8Cb Ha KHHeTH'IecKYJO TEOpHJO.
llpaolta, B DOCJIe,AHee BpeMjI STOT np06en H&'Ia.n HHTeHCHBHO BOC-
nOJ1HSlTbCH.
B aHHOR: pa60Te paccN3TpHBalOTCSI BhlqHCJTHTeJIbHbIe anropHT-
MbI, llenocpe,nCTBeHHO CBSI38.HHhle c nOIDITHeM ct>YHK H pacnpeAe-
JleHH$I H KHHeTH ecKHMH MO enJlMH, onHCbIBalOll.\HMH ee n08e elfHe.
KpOMe TOro t npSMaSi CB93b Me Bblq}fCJIHTeJlbHhlMH acneKTaMH
H cPR3H\fecKHMH npe CTa.BJleHHSI)AH 0 MOJleKYJIHpHblX cBoACTBax ra-
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onHcaHHSI Te'ieHHR r83a. HMeHHO B H3BecTHOM CMhlC1Ie HeTp H-
OHHbla B3rJlB,q Ha KOHCTPYK IO ra30 HHaMHqecKHX MO eJIeft H nOJlY-
qeHHe HOBbIX Bblt.lHcnHTenbHblX anropHTMoB nOCJlY)l(Hn no6y H'I'eJIb-
HbIM MOTHBOM K npOBe eHHIO HCCJIe OBaHHAJ DOJIO>KeHHbJX B OCHOBY
HacTOSIlIl-eA MOHorpa<t>UH.
rJlaBa II
KHHETUqECKH cor JIACOBAHHbIE
PA3HOCTHbIE CXEMbI
aHHaSi rnaB8 nocBsw.eHa BblBO.D-Y KHlieTHqecKR COrJlacoBaHJIh1X
pa3HOCTHbiX cxeM H KB83Hra30 HHaMHqeCKo1t CHCTeMbI yp8.BHeHHA.
B Ka;qeCTBe OCHOBbJ RX nOJIyqeHH HCnOJlb3YIOTOi ,lJ.HCKpeTHl:de MO-
eJIH ,lI.JIS1 OnHCaHHSI OAHO acrHqHO.A cpYHKIJ.HH pacnpe,a:eneHHS. Pac-
CMaTpHBaeTCjI npHMeHeHHe KHHeTHqeCKH cornacoBaH.HLJX cxeM tl-nSI
MO e.nHpOBaHHSI TeqeHHA eaJI.bHoro ra.3a.
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1. KpaTKHIf 0630p 60Jlh MaHOBcKIL" cxeM
K HacToBw.eMY BpeMeHH, 3a 60JJee qeM nonYBeKOBYIO HCTOpHIO
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pa HJIH HaBbe-CToKca. ,L(ocTlfrHYTble ycnexH B C03AaHHH a.nropHT-
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nOCJIe,D)lHe rO bI, CTaB.HT H3 nOBecTKY HH pemeHHe Bce 60J1ee cnO>K-
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perneHHfl TaKHX 3 atI BbI3blBae'f nOSlDJleHHe HOBhIX a.nropHTMOB t Xo-
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TeXHHKH. O HHM H3 T8.KHX anropl1TMOB CT&1'lH KHHe'rHqeCKH corna-
COBaHHble p83HOCTHble CXeMbJ (KCPC).
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pa3pa60TKe aJJropHTMOB pemeHHB ypaBHeHHA 9RJIepa H HaBbe-
60
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..list SlBJJSlIOTCR CJIe CTBHeM 60J1ee CJIO)KHOrO KHHeTKqeCKOrO YJ>aBHe-
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B ).J.OCTaTO\.fHO nnOrrHbJX ra3ax. )J.nSl paJpe)KeHBblX ra308 HCnOJlb30Ba-
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CSI ecTeCTBeHtlhlM H HMeeT Jl.OCT&ro'lJHO OJIryJO npeAbICTOpHfO, XOT.SI
6bl DOTOMY, 'ITO Henocpe CTBeHHoe npHMeHeHHe )I.JISI ex OllHCa.HHSI
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B KCPC 6a30BOa Sl8JIReTCH HCKpeTHaSl MO eJIb .LVISI onH HHH
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MOJIeKYJI ypaBHeHHe, onHCbIBaJOw:ee 3TY )J;HCKpeTHYJO MO,n.eJIb, C CYM-
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THqeCKH cornacoBaHHble pa3HOCTHble cxeMbI AJISI ra30AHHaMH"iOCKHX
napa feTpoB (13J. Hcc.neAosaHIDI B 06JlacTH KHHeTH'feCKHX cxeM J1)IH
aBTOpa SlBHJIHCb eCTeCTBeHl{b1M npo on)f(eHHeM ero t{ejITenbHOCTH
B 06J1a.cTH MO,lI-CJIHpOBaHIDI 38.,l{a'I )J.HHaMHKH H3nyqalO ero ra3a J B
KOTOpblX TeCHO nepenneJIHCb KHHeT eCKHe npOl\eCCbI nepeHoca 83-
Jly eH}IH Ii ra30 HHaMHqeCKHe npo eccbl (14].
KCPC MO>KHO BKJllO'-IHTh B 60JIe€ IlIHpOKHA KJIacC a.nropJi:TMOB
- 60J1bIJ.MaHOBCKHe cxe lbI, B KOTOphlX TaK>Ke B KaqeCTBe nepBO-
OCHOBbJ Henocpe CTBeHHO HCIIOJIb3YlOTCR KHHeTHt.feCKHe Mo,[(e.nH )J.JIB
O-nHOqacTHqHOH cpYHKL\HH pacnpeAeJIeHHSI. B )J.aHHoA MOHorpacpHH
He CTaBHROCb 3a)J.aQeA no p06Hoe H3JIO)KeHHe B03MO>KHhJX BapHaJ-I-
TaB 60J1b MaHOBCKHX cxeM. Orp8JJH'-IHMC.H CIIHCKOM nHTepaTYI)bl no
60Jlb ialIOBCKHM cxeMaM H KpaTKHM 06cY)KAeHHeM 'reX H3 HHX, KO-
TopbIe HaH60J1ee 6J1H3K" K KCPC.
B pa60Te (15)) KOTOpaJI) no BeeR BH MOCTH) 6blna nepBoA B
KJIacce 6onb aJIOBCKHX cxe IJ Ha OCHoae ,nHCKpeTHoA MO,LI.enJi ,lVJH
cPYHK HH pacnpe,AeneHHSI 6bJna nOCTpoeHa p83HOCTHasI CXeM& pe-
llleH fI 3a.n.a ra30BoA: HHaMHKH 8 narpaH»ceBoA CHCTeMe KOOpttH-
HaT." TIpH 3TO f B Y3JIax pa.3HOCTHoA ce-rKH Ha npeJJ:bI ym.eM CJlae
no BpeMeHH t ' cPYHKl.\H.R pacnpe eneHHH rrpe)\llOJlaranacb JIOKaJIbHO-
lvlaKCBeJ1nOBCKoR. Ha HOBO 1 CJIoe t j + 1 npoHcxoAW13 MI'HOBeHHaJI
.9rn PaOOT8 6b1lta C OPMYnHpOBaHa aBTopoM MOHorpa4>HH H BbU1OJ1HHJI8Cb
no,IJ. ere PYKOBOACTBOM B Ka'leCTBe .LJ)tI1JlOMHOA pa.60Tbl. K rny60KoMY COa<a1le-
HHJO) BCKope noc., e ee 3aeepmeHHH B.A.nOTKHJI Tpa.rH'ieCKH norH6.
91. BblBO,l{ KHHeTHlfeCKH COrJl8COBaHHI>lX cxeM
61
MaKCBenJJH3al\HH peweHHB. )J.n.a Haxo eHHH raJOrofHaMH'IecKHX
napaMeTpOB -- MOMeHTOB YHKqHH pacnpeAeneHHB - onpeAenB-
JIHCb HHTerpanbl B npOCTpaHCTBe CKOpocre:a MOJleKYJJ OT nOl<aJlbHO-
MaKCBeJlJIOBCKOA <PYHKI.J.HH pacnpe eJIeHHSI. ITpH 3TOM C8.Ma OHa
onpe,D.eJIH.,1acL C nOMO LIO TOJIbKO Tpex CBOHX 3HaqeHHH no CKOpO-
CT5IM MOJTeKyn. I1MeHHo 9TOT <paKT H nocnY)KHJl npHttHJI0A OTHOCH-
TeJ1bHO HeBblCOKOA TOtIHOCTH npOBe eHHLIX Ha OCHOBe 3TO cxeMbI
pacqeTOB.
B pa60Te [16) (Pullin D.L.) nOKaJIbHO-MaKCBeJIJI08CKaH CPYHK H.sI
pacnpe eJleHHH npHMeHHJIacb ,lVIH Onp eJIeHHSI nOTOKOB MaCChI, HM-
nYJIbCa H 3HeprHH Ha .rl8H>Ky eHCg rpaHHl.{e SqeAXH. Bnocne cTBHH
3TH nOTOKH HCnOJIb30BaJlHCb ,lVI.H Haxo eHHH 3Ha eHHR ra30p-HHa-
MH eCKHX napaM€TpOB Ha HOBOM CJloe no BpeMeHH. TaK )Ke KaK H
B [15J. B ,QaHHOa pa60Te anropHTM ITpHMeHS1nCJI K MO eJlHpOBaHHlO
TetJellH:A ea.TIbHOrO raaa.
B pa60Tax [17, 18) (Reitz R.D.; ApHCTOB B.B., LJepeMHcHH cI>.r.)
ra30)J.HHaMHQecKHC napaMeTpbl Henocpe CTBeHHO onpe eJIgJlHCb no-
c.ne HaxO>Ktl-eHHH <PYHKD-HH pacnpe eJleHHSI H3 perneHH.SI ypaBHeHHR
BOJIb MaHa c nOMOwPlO KOHel{HO-pa3HOCTHo:A CXeM:bI.
ABTOpbJ pa60Tld (13), Hcnonb3YSl CTyneHQaTO-nOCTOHHHble no-
KaJIbliO-MaKCnennOBCKHe <l>YHKIUIH, paccMaTpHBaJIH 6aJIaHC <l>YHK-
QHH pacI1pe eJleHHa B <l>HKCHpOBaHHOR g'leAKe. KOHeQHO-pa3HOCTHhle
ypaBHeHHg CTpOHflHCb nYTeM KHTerpUpOBaHHH 3TOro 6anaacHoro
ypaBHeHHSI C CYMMaTopHblMH HHBapHaHTaMH. npH 3TOM TaJ(3<e HC-
nOJ1b30BaJIOCb npe nOJIO)J(eHHe 0 MrHOBeHllOH MaKCBeJlnH3aIJ)1H raJa
tla 110BO J CJIOO no BpeMeHH. (8TOT no xo)I. no po6HO pacCMOTpHM B
CJle HX nYHKTaX HacTOSI ero naparpa<j>a). j!.aHHble cxeMbI no-
nY'IHJlH Ha3BaHHe KHHeTH'IecKH cornaCOBaHHblX pa3HOCTHbIX cxeM
(KCPC) .
CJle eT OTMeTHTL, "liTO B03MO>KHOCTb HaxOJKAeHHSI faaO)J.HHa-
MHQecKHX napaMeTpOB Henocpe CTBeHHO H3 KHHeTHQecKHX rrpe -
CTaBJIeHHR He HBJISleTCSI peBMIOIJ.HOHHOA. Hao6opoT, OHa BnOJIHe
eCTeCTHeHHa, Y\lHTbIBaJI CBIDL Me>K,r:ty KHHeTH'IeCKHM H ra30 HHa-
MHqeCKHM onHcaHHeM cnnolliHo1t cpeJl:bI. nO OMY 3HaqeHHe nO.HBHB-
llIHXC.SI K Ha'lany 80-x roAOH 60JIb MaHOBCKHX (KHIIeTH'fecKHX) cxeu
CJJe YeT paccMaTpHBaTb C TOqKH 3peHHSI HX B03MO>KHOCTH COCTaBHTb
62
TJI. If. K.,HeTH'IeCKH cor.naCOBaHHlJe pa3HOCrHble exeMbl
KOHKypeH HIO cYlLI-eCTBOBaBwHM B TO BpeMH Tp HOHRbJM a..nro-
pHT laM pellleHHB 3a.AaQ ra3oBoA HHaMHKH. 3aMeTHM, TO npe):\no-
)l(eHHbIA B pa60Te [13] anropHTM, - MO)l(eT 6bJTb, no CJIyqaAJiOMY
CTet.IeUHIO 06CTOHTeJIbcTB - Cpa3Y )Ke nOK838JJ C Y1.£eTOM BbJlIHClIH-
TMbHblX 3aTpaT CBOIO KOHKYPCHTOCDOc06HOCTb C Tpa,qHIJ)iOIlHhIMH
nO XO,.QaMH H HeMeNIeHHO BbJ3BaJ] cepHlO ,LJ.aJ1bHeAIWiX ny6nHK R
asTOpa MOHorpaQ>HH Hera Konner [19)-(26).
COBn8AalOIlI.He c KCPC [13] cxeMhl 6b1nH nOJIy IeHbI B 1986 r.
B pa60TaX [27, 28] (Deshpande S.M.) H TaK>I<e nOCJ1Y)KHnH OCHO-
BOa: aJ1LHeAmHX HHTeHCHBHblX HccneAOB8.HHA, KaCaJO UJ.HXCH Mo e-
JIHpOB8HHSI HeBSI3Koro r83&t IIO;q>o6HO OfiHcaHHbIX B o630pe (29].
3TOT 0630p TaK}Ke MO)KeT 6bITb peKOMeHAOBRH qwmTeJUlM, 60Jree
no p06Ho HHTepecyJO MCS1 p83JIH'IHblMH HanpaBneHHBMH 6anbQ.-
MaHOBCKHX cxeM.
Ha HHa.H c 1985 r. OCHOBHble YCHJ1HB aBTOpa H ero COTpy)].HH-
KOB 6b1nH CKOHl.\eHTpHpOSaHbJ }la Mo enHpoBaRHH c nOMorn.blO KH-
HeTHqeCI<:HX cxeM CJIO>KHblX TeqeHHR BSI3KOI'O TeDJIOnpOBO)I.Horo f'a-
3a (rnaBa III). npH 3TOM Bee 60JlbWe nepenJIeTaJlHCb B3rJ1j1 bI Ha
KCPC C o,LJ.HoA CTOpoHbJ KaK Ha aJIropHTM pacqe-ra, a. c ;q>yroA CTO-
poHbl KaK Ha. B03MO>KHyK> Mo enb NISI ORHcaJtHB TeqeHHR BSI3KOro
ra3a. B 3TOM KnlO'Ie 6Y eT paccMOTpeHa CBSI3b KCPC C ypaBHeHH-
HMH HaBbe-CTOKca H C'IaBWHMH nonynspHblMH B nOCJIe Hee BpeMH
Lattice Boltzmann c.xeMaMH (30, 31) (rnaB& III).
IIpHCTynHM K BbIBO,llj' KHHeTH'IeCKH cornacoB8HblX cxe 11 cne)1.Ya
pa60Te (13).
2. COOTHoweHHe 6aJlaHCa ,l{JIJI CPYHK HH pacrzpe,l{eJIeHHH
06cY>KAeHHe cnoco6oB HaxO.>KAelIH5I KCPC Ha'iHeM C H8H6W1ee
npoc'roro npOCTpaHC'l'BeIIHO-OAHoMepHoro CJlyqMl (lD-reoueTpHB).
np"MeM CJ1e)1.YIOll.Q'1O MOAeJlb H3MeHeHHSI <PYHK H pacnpe)\e..neHHB.
1) Ha MOMeHT Bpe feHH t j Ha Ka)f(,nOM OTpe3Ke [Xi, Xi+ 1] pactteT-
HoA 06nacTH, paCI10J10>KeHHOH Ha OCH x (puc. 5), CYll(CCTBYeT
CBO nOCTOHliHaH 110 npOCTpaHCTBY (H8 01'pe3Ke x E [Xi, Xi+l))
O,I(HOqacTf.{q}lail CPYIIK Hjf pacnpe eneHHg, cOBnMa'OI aR c
jl. BIJBOA KHHeTH&leCKH OOrJl&COBaHHblX cxeM
63
M8.KCBenJlOBCKoA:
Pi { ( i - Ui)2 }
Ii = JiM = (21rR11)3/2 exp - 2RT i ·
(1.1)
r830 HHaMHqeCKHe napaMeTpbJ Pi, U 1 , 11 - ITOCTOSlHHbl Ha OT-
pe3Ke X E [Xi, Xi+lJ.
2) 3a OTpe30K BpeMeHH 6.t = t j + 1 - t' ra3 cosepwaeT 6eCCTOJIK-
HOBHTeJlbHblA pa3JleT.
3) Ha MOMenT BpeMeHH t j + 1 npoHCXO HT MrHOBeHHaJI MaKCBeJt-
nH3anHjI cl>YHK HH pacnpE!AeneHHH. IIpH nepexO,lJ.e K MOMeHTY
BpeMeHH t j + 2 BCSI npo e pa nOBTOpHeTCH SHOSh. B8JKHhlM
06CTOSlTeJ1bCTBOM 3,necb SlBJlHeTCSI BblIlOJJHeHHe npH t = tJ"+l
COOTHorneHHSI
I J!+1cp«()d( = J J/ 1V'«()d(,
(1.2)
.+1
r,!te 'P( ) -. CYMMaTOpHbIe HHBapHaHTbI, It 3HaqeHHe
cl>Y}I pacnpe,JXeneHH.SI IIpH t = t i + 1 11..0 MaxCBeJIJlH3 HH,
. 1
fl - 3HaqeHHe cl>YHKlt HH pacnpe,AeJJeHHH, onaChlBaeMoR Bbl-
pa>KeHHe t (1.1), 110CJIe MrHOBeHHoA MaKBeJIJlH3aqua.
06CY HM 060CHOBaHHOCTb 3THX npe)\DonO>KeHHA. OrMeTHM, '1'00
H ra3OAHH8.MHtfeCKHX npo eccoB B AOCTaTOIJHO IIJIOTHblX raaax, HC-
cne.n-OB8.HHe KOTOpblX H HBJlHeTC.H OCHOBIIOA l\eJIblO aHHoA MOHorpa-
cl>BH, MaKCBeJLnOBCKOe pacnpe eneHHe xopowo onHCblBaeT cl>YHKl\H1O
pacnpeAe.neHHB. HMeHHo Ha 6nH30CTH cl>YHKUHH pacnpeAeneHHH K
MaKCBennOBCKoA nocTpoeHa npo e pa nonyqeHH ypasHeHHA ra-
30BOA p)tHaMHKH H3 ypaBHeHH1I BOJIbQM&Ha (I'Jlasa I).
CTOJIKHOBeHHH MoneKYJI 8 r83e npHBOAHT K YCTaHOBneHHIO MaKC-
BeJInOBcKoro BH,lJ.& CPYHK HH pacnpe,L\eneHWl. HanOMHKM, qTO npe.n--
nOJIO>KeHHe 06 ycraHoBneHHH Cpa3Y nOCJIe CTOJ1KHOBeHHH MaKBeJI-
nOBcKoro pacnpe):\e.neHHH llonO)l(eHO B OCHOBY BbIBo.n;a BfK-MO,L\eJlH
AJISI KHHeTHtJeCKoro ypasHeHHg (2.35) (rn. I).
COoTllollleHHe (1.2) T3J(}I(e cooTB e'rcr nyerr OCHOBHblM nonox<e-
HHHM KHHeTH ecKo:A TeOpHH (4.12) (rn. I), cornacHo KOTOpblM no-.
64
rJl. II. KHHeTHqecKH corJlacOBaHHlJe pa3HOCTHbie cxeMW
KaJlbHo-MaKCBennOBCK8.SI cl>YHKQHB onpe enBeTCSI 'fepe3 ra30,ll.kHa-
MHqeCKHe napaMeTpbl, KOTOpbJe B C8010 Oqepe,u.b HBIDIIOTCH MOt.!ell-
Tal\fH <PYHK HH pacnpe eneHHSI: 1M = /M[J).
MrHOBeHH8.H M8.KCBeJInH3au.H HMHTHPYeT npo ecc CTOJIKtIOBe-
HHSI MoneKyn, KOTOpblA 8 ypaBHeHHH BOJILU).(aHa OnHCbIBa.eTCSI HenH-
HeAIlblM HHTerpa.nOM CTOJIKHooeHHD. [(en01.fKa "6eCCTOnKHOBHTeJ1h-
HblA pa3JIeT - MrHOBeHHaH MaKCBeJlJIH3&qHB" no CYTH ena HEJIH-
eTOi peaJIH3aJJ.HeA peweHHSI KHHeTH'IecKoro ypasHeHHSI C nOMO blO
XOpOlllO H3BeCTHoro B BbltUlC1IHTeJIbHOA MaTeMaTHKe npH Hna CYM-
MapHoA annpOKCHM H J KOr a nepeXOA OT O HOrO speMeHHoro Cnos
K pyroMY onHCblBaeTcg C nOMOIIJ.b1O nOCJle)J.OBaTeJILHOCTH pa3nH1.f-
HbJX cl>H3H1.JeCKHX npou.eCCOB (32] (CaMapCKHA A.A.). t.JTO >ICe Ka.Ca-
eTC5I cTyneHQaTO-DOCTOHHHoro xapaxTepa cI>)'HKl\HH pacnpeAeneHHSI.
TO cl>R3HQecKHA CMblC1I 9ToA MO)l;MH 6Y)I.eT nO)q)06HO o6cY)K)J.aTbCSI
B CJIe)O'lOm;eA rJIaBe. 3aMeTHM, TO 9TO npe onO>KeHHe BBJlSlercg
KnlOtteBhJM ):VIa nOllHMaHWI KCPC K3.K Mo enH ,IVI5I OnHCaHH5I rretJe-
HHA B.SI3KOrO ra3a. OAHaKO Cpa3Y 06paTHM BHHMaHHe, qTO Ha HCK_llO-
QHTeJlbHYIO nepcneKTHBHOCTb HCn0Jlb30BaHHSI B <pH3HKe p8.BHOBeC-
HblX 3Ha\.leHHA: <l>YHKnHH pacnpe e.neHHH B cTyneH'I&TO-nOCTOSlHIIOM
110 npOCTpaHCTBY BH e 6hlno 06paIl\eHO BHHMaHHe e e B MOHorpa-
<l>HH [33) A.A. BnacoB& - 8.BTOpa H3BeCTHoro ypaBHeHHJI, OIIHCblBa-
IOrnero nOBe eHHe <l>YHKQHH pacnpe.n-eneHIDI B BhlCOKOTeMnepaTYP-
HOlt nn83Me.
Ha pHcYHKe 5 CXeMaTHqeCKH H306paJl<eH npo ecc H3MeHeHWI OA-
HO'IacTHqHOA cl>YHKIJ:UH pacnpeAeneHHB Ha OTpe3Ke x E [xI' X 1 +1] 3a
BpeMSI t = t j + 1 - t i . CJIeBa 8 eHTpaJIbHYIO B'IeAKY x E (Xi,Xi+l]
nona.p.aJOT qaCTH bl C nOnO>KHTeJlbHblM HanpaeneHHeM MOJleKYJlSlp-
HoR CKOpOCTH > 0 H HMelO He MaKCBeJIJIOBCKOe pacnpe,l1.eneBHe
It-I,M. 91'0 MaJ<CBeJlnOBCKOe pacnpeAeJIeHHe, COrJIaCHO (1.1), 3a.BH-
CHT OT ra30AHHaMHQeCKHX rrapaMeTpOB Pi-I' u1- 1 , TL l' Cnpa.aa B
}1QeAKY nona,o.YT "4aCTHQbI., HMelO1ll.He CKOpoCTb < 0 H MaKCBP..-n-
JtOBCKoe pacnpeAeneHHe fl+ 1 ,M.
IIJlon.-a,nh B ",eHTpaJIbHOR SlqeAKe, 3aHHTaH BHORb npHwettWH-
MH 1.JaCTHu-aMH (pHC. 5), nponOp HOHaJIbHa MO JIIO HX CKOpOCTII Ii
BpeMeHH 6eccTOJIKHOBHTeJlbHOrO p&3neTa t. YqHThlBaJI BblweCKa-
* 1. BblBO,l{ I<HHeTH'IeCKlf COrJ18COBalfllhlX cxeM
65
li-l,M h M Ji+l.M
I,
x' 1 x. Xi+l X
J - I
PHC. s. H3NeneH1te pacn eneHWI H& OTpe3Ke ( i, Zi+l] ,D,JUI ID-
reOMeTpHH
3aJ.IHoe t npeAcTaBHM COOTHoweHHe 68JIaHca, onHCbIsalOIII-ero H3Me-
HeHHS1 <l>YHK HH pacnpe,l].eneHHSI 8 npo ecce 6ecCTOJJKHOBHTeJlbHOrO
pa3neTa tJ-nB u.eHTpanbHoA' Hqe:AKH, B CJIetIYlO eM B e:
//+1 ({) = Ji,M({) + Llt J/+1'M({ :Ll'M( ) _
. . .
_ A t fl+l,u( ) - 211. u ({) + fl-I,M({)
u 2 x ·
( 1.3)
3,u.eCb MbI )VIS} npOCTOTbJ npe nOJIO>KHJIH, '1TO see OTpe3KH HMelOT
OAHHaKOBblR p83Mep flx.
· 1
3aMeTHM, 1.fTO no OKOHQaHHH pa3ncm <l>YHK Hg /1+ ( ), CTOSl-
a.sr B JIeBoA qacTH (1.3), Y)I(e He SlBJlHeTCB MaKCaennOBCKoA. TaKO-
BOa OHa CTaHOBHTaI TOJIbKO nocne BblnonHeHHB npH t j + 1 9TalIa 3, Mo-
t(MHpylOw.ero nepexo K paBHOBeCHOMY COCTOSHHIO 3a cqer CTOJIK-
HOBeHHR.
3. Pa3HOCTHbIe ypasHeHHH AJIR ra30ftHH3MH'lecKHX
napaMeTpoB
QT06bJ nOJ1YQHTb p83HOCTHble ypa.nHeHHSI, OnHCblB3.I01.IJ)ie H3Me-
HeHHe ra30 HHaMHqeCKHX napaMeTpoB, npoHHTerpHPY M 6aJ1aHCHOe
COOTHomeHHe (1.3) C cYMMaTOpHbJMH HHBapHa.HTdMH. npH aTOM B ne-
· I "+1 . I
RoR qacTH nOSlBHTCSI MOMeHTbI f fl+ , J II { , f fl+ ( 2/2) d ,
66
rJJ. II. KHHeTJf'lecICH COrJJ&COBalIHble pa3HOCTHble cxeWbl
KOTOpble, cornaCHO onpe eneHHIO c:l>YHKqHH pacnpe eneHHSI (rna-
. 1 · 1 . 1
sa I)t COOTBeTCTBeHHO pas}lhI 11+ , (pu)f+ , Ef+ · 3aMeTHM, tfTO
"PH 3TOM HeSlBRO HCIIOJIb3YeTCJI CBoAcTBO c:l>YH pacnpe eneHHH,
nonO>KeHHoe B OCHOBY npo e)!ypLI QeuMeHa-9HcKora (4.12) (rn. I):
I fj+lV'( )d = 0,
(1.4 )
r e ( ) - CYMM&TOPHblR HHBapHaHT, a 6.fj+l Onpe eJUleTCH H3
paseHcTBa
I = fM[!J + f.
(1.5)
HHTerpanbl B npasoA 'facTH (1.3) nerKO 6epyTCSI BB MaKC-
sennoBcKoro xap8JCTepa <l>YHKU)fH pacnpe eneHHB npe t = t j . E HH-
CTBeHHYIO ,cnO}l(HOCTb npe,nCTaBnHeT onpe)J.eJleHHe HHTerpanOB THna
J f I<p( )/ cl{, KOTOphle, BnpO'leM, TaK)I(e 6e3 oco6oro TpY,Aa BbltlHC-
JJS!lOTaI 6naroAapSl Hcnonb30BaHHIO c:l>YHK H oWH6oK
2 %
erf(x) = I: J exp( -z2)dz.
v 1r o
HTaK, DOJIy'lHM CHCTeMY p83HOCTHbIX ypaBHeHHA ,lJ.J]B r830AHH&-
MHqeCKHX napaMeTpoB
'+1 ·
P[ A; PI + (pu) = x [puerf(s) + exp(-s2)]=; (1.6)
;+1 j
(PU)i - (PU)i + (pu2 + p)o =
t1t %
= T [(pu2 + p) erf(s) + M exp( -s2)] b ;
E!+l - E!
16 & + (u(E+p))o=
t %
= x [U(E + p) erf(s) + p (E + ) ex p (-s2>t:r:.
3,Aecb HCnOJlb30BaHbJ CJIe)U'lOIII.He 0603Ha'leRHJI: p = pRT - aB-
neHHe ra3a, E = pE + pu2/2 - DOJIHaSi 3HeprWl. E = RT/('Y- 1 )
(1.7)
(1.8 )
51. BblBOA KHHeTHlfeCKJI COrn8COBaHH61X cxe",
67
- BHYTpeHHjlH 3HeprHa, 'Y .- nOK83aTeJlb 8,D.Ha6aTbl, {3 = 1/ V 2R:r,
s = {3u,
ao ==
%
(1.;+1 - B,;-l
2 :z;
,
4i+l - 2ai + ai+l
az:r: = x2
COOTBeTCTBeHHO) l.\eHTp8J1bH8.SI pa3HOCTb JVlH annpoKCHManHH
nepBOa npocTpancTBeHHoR npoH3BOABoA: H p83HOCTHaJI BTOplUl npo-
H3 BO,ll.H a.5I.
Ha MOMeHT BpeMeHH t = t j + 1 B COOTBeTCTBHH C HOBblMH 3Ha-
qeHHg 1H ra30tl-HHa.MHQeCKHX napaMeTpOB pi+l, u j + 1 , T;+1 CTpOHM
HOSYIO KYCO'lHo-nOCTOBHHYIO MaKCBMnOBCKYIO <!>YHKII.HJO pacnp e-
JleHH.H, H BCSI npo e pa nOBTOpBeTCH npH nepexOAe K MOMeHTY Bpe-
MeHH t i + 2 .
4. CorJlaCOBaHHOCTb p83HOCTRbIX annpoKcHM1Jl{HR
CHCTeMY YP8BHeHHA (1.6)-(1.8) MO)KHO nOJlY HTL, Henocpe,ll.-
CTBeHHO OnHpaJICb Ha p83HOCTHYlO cxeMY nepBOro nop Ka TOt{HOCTH
.AnSI ypaaHeHHSI Bonh MaHa
I j + 1 ,j I i ,;
i - J i + i - 1 ';-1 = J(fi,jj))
l:1t x
,.;+1 f ' f J I;
J i - i + i+l - I i = J(/j, I;),
t
{ > 0,
(1.9)
< 0,
(1.10)
r e J(li, Ii} - 3HaqeHHe HHTerpan& CTOJIKHOSeBHR, onp eJIHeMoro
c nOMOID.blO YHK pacnpeAeneHHfi, KOTOpaa B MOMeHT speMeHH
t = t i DOCTOSIHH& H& OTpe3Ke x E [Xi, Xi+l].
P83HaH <I>opW& 3a- Jt1
rIMCM ypaBHeHHA (1.9) H 9 0 I.
(1.10) M nOJIO>KHTMb- Sl-1II
HblX H OTpH aTenhHbIX J
cKopocreA MOJIeKYJI He f l _ 1
CJIYQaAHa. OHa onp e- PRC. 6. K BbI(Sopy p&3HOCTHorO wat)noHa .D,1U1l
JIHe'l'CB H3 YCJJOBHSI ycrroA- peweHKSI YpaBHelDlJl BOJ1h M&Ha
QHBOCTH (pHC. 6) SBHOR
ft'
<O
f
Ii+Jn II_In
--* *---
/j
I
:E1+1J2
f:+ 1
68
rn. II. KHHeTHtJeclCH corJlBCOBaHHble p83HOCTHble cxeJ,fbJ
pa3HOCTHOR cxeMbI A'lH ypaBHeHHg nepeHoca (32]. qaCTHbJM CJJy-
qaeM Taxoro ypaBHeHHSI HBJISleTCSI JleBaJI ,ltH<I><t>epeHD;HaJIbHaJI qacTL
ypaBHeHHS1 BOJIb MaHa.
Pa3HOCTHLle cxeMbI (1.9) H (1.10) MO}KHO 3&IIHCaTb e)J.MHblM 06-
pa30M C nOMO(QLlO Bblpa.>KeHHjI
I j + 1 1 1 I ' / '
i - i + 1+1 - i-I
6t 2Lix
= 6% I{I 1/+1 - 21/2 + It-I + J(f, f).
2 x
(1.11)
IIpoHHTerpHpyeM p83HOCTHoe Bblpa)f(eHHe (1.11) no CKOpOCTH MO-
neKyn C CYMMaTOpHbIMH HHBapHaHTaMH cp({) = 1, {, 2/2. HcnOJIb-
3YH 06pall..\eHHe B HYJIb COOTBeTCTBYJ01U)iX MOMeHTOB .QJIH HHTerpaJIa
CTW1KHOBeHHA (rn. I)t a TaK>Ke npeADOJIarasI npH HaxO>K,l\eHHH HH-
TerpanOB
f rJ ,1 ! f j 2- 2 ck de ,
Ji.CIcCp . .
11 lcp( ) 1/ d ,
q'ro <PYHKL\HB pacnpeAe.neHH B MOMeHT BpeMeHH t = t j KYCOqHo-
MaKCBeJInOBCKaSI no npoCTpaJlCTBY BHOBb npHAeM K pa3HOCTHbIM
ypasHeHHHM (1.6)-(1.8).
3TO COBoa,.neHHe npe CTaBJIgeTca cosepweHHo eCTeCTseHHblM.
CxeMa (1.9)-(1.10) nepooro nop TO'lHOCTH COOTBeTCTBYeT K (-
COQHo-nOCTOSlHHoMY IIpe C'IaBneHHIO <PYHK H pacnpeAeneHHSI. Kpo-
Me TOro 1 HCnOJIb30BaHHe Dpe nOJlO}KeHHg 0 MaKCBeJIJIOBCKOM xapak-
Tepe <l>YHK H pacnpe,QeneHHg npH t = t J "PH nonyqeHHH yp8BHe-
HHA (1.6)-(1.8) H3 pa3HOCTHOA cxeMhI (1.11) rrpHBO)J.HT K TeM >Ke
YCJlOBHftM, c nOMOI.U:bIO KOropblX nOJly eHo COOTHoweHHe 68J1aHCa
(1.3) .
CHCTeMY pa3HOCTHbJX ypasHeHHA (1.6)-(1.8) MO}l(HO Tp&KTO-
BaTb KaK cxeMY peWeHHjI YPaBHeHHA ra3oBoA AHH8MHKH C HCKYC-
CTBeHHbIMH HCCHIIaTHBHYMH tUleHaMH B npaso" qacTH. O HaJ(O
B OT 1H'IHe OT APyrHx anropHTMOB pacqeTa, HCDOnb3YlOl1I.Hx HCKYC-
CTBeHHYIO BSl3KOCTb, 3TH HCCHnaTHBHbIe 'IJIeHhl nonY'IeHbJ He <t>e-
HOMeHOJlOrUqeCKHM O}'TeM, KOTOpbJA OnHpaeTCjl Ha Te HnH HHble
1. BblBOA KHlle'rl,",eCKH cornacoS8IfHblX CXeM
69
4>H3HQecKHe npe.n.nonO}KeHHji 0 XapaKTepe r830AHRaMHt.tecKoro 're-
tfeHHSl. 3 eCb BH ,l\HCCanaTHBHblX 'IJIeHOB COrJlacoBaH C Pa3HOCT-
HoR annpoKcHMaqHeA (1.11) KHHeTHtiecKOro ypaBHeHHSI. n03TOMY
eCTecTBeHHo nOJIyqeHHOA Td.KHM 06p830M cxeMe (1.6)-(1.8) ,1J.aTb Ha-
3HaJIHe KUHemU"4ec"" c02AacoeaHnoU pa3HocmtWlI cxeMW (KCPC).
nO 06HbJe npHeMbJ nocTpoeHHSI KCPC MO>KHO npHMeHHTb t HCnOJlL-
3Y1l B KaqeCTBe OCHOBbl H APyrae. OTnHqHble OT (1.11) p83HOCTHbJe
CXeMbJ, annpoKcHMHpYIOllJ)fe KHReTH eCKOe ypaBHeHHe.
Y1:fHTblBaB CBB3b YPaBBeHHR ra3OBOA AHHaMHKH H ypaBHeHHSI
EOnbQMaHa (rJI. I), TPaAH HOHHble 'IHCJleHHble MeTO bJ MO>KHO npe -
CTaBHTb B BH e cne;l1ylOm.eA enoqKH: KHHeTHQecKoe ypasHeHHe
BonbII-MaHa - OCpe,lI.HeHHe ypaBHeHHSI EOJlL MaHa C CYMMaTOpHbJ-
t.(H HHBapHaHTaMH H fiOJlyqeHHe ypaBHeHHA ra30BoA )J.HHaMHKH -
pa.JHOC'l'tIaJI ,lI.HCKpeTH3 HSI ypasHeHHA ra30BoR HHaMHKH. B CBOIO
OqepeAb, KCPC MO)KeT 6blTb npe CTaBJleHa B 8H e APyroA Docne-
OBaTenbJIOCTH eAcTBHA: KHHeTH1.fecKoe ypasHeHHe BOJ1bQMaHa -
HcKpeTHaH fO)J.eJIb )).JIB O HO'IacTH'IHOR <PYHK HH pacnpe eneHHSI
- OCpe;J.HeHHe HCKperHOA MO,neJIH C CYMMaTOpHblMH HHBapHawra-
MH H nonyqeHHe KCPC. TaxHM 06pa30M t no CpaBlieHHIO c Tp -
OHHblMH anrOpHTM IH B KHHeTHqeCKH COrJlacoBaHHblx pa3HOCTHbJX
cxelt,lax ltJleHSIeTCa nopSl OK npOl.(e)JJ'pbl ocpeAHeHWI H ,lJ.HCKpeTH3a-
IUfH B enOqKe, CT8.pTYIOm.eA c ypaBHelIHB BOJIh MaHa.
C enaeM ,noa Ba>KH.bIX 3a.Me'laHHB t KOTOpble B anbHeAweM
(rJJ. III) 6Y T HCnOJIb30BaHbI npH o6cY>K eHHH poJIH KCPC KaK
CPH3HttecKOA MO eJIH JVlH OnHCaHHS Te'leHHH ra3a.
3 aM c q a H H e 1.1. B npaBOA 'facTH YpaBHeHHB (1.6) npHcYTCTBY-
lOT ,lJ)tCCHnaTHBllble t.tJIeHbI, ,z:u.t<pc:PepeHIUlaJ1bHblM npHMH>KeHHeM KOTo-
J>blX 6Y)J.eT Bblpa)l(eHBe
ilx 8 ( P :2 )
AW p = 2 Ox puerf(s) + fJ.fi exp( -8) ·
TeM caMblM K nOTOKY M3CCbl pu npH6aBJljleTCH ,nonOJlHHTenbHblH nOTOK
(- Wp)t I.fTO Ha nepBbIA B3rJlBA BCTynaeT B npOTHBOpe"fHe C KJIacCH'Ie-
CKOH TeopHeA >K KOCTeA H ra30B.
O,IJ.JlaKO npHMeM 80 BHHM8HHe TOT cl>aKT, qro r830AHHaMH'IecKHe na-
paMeTpbl KCPC onpe enllJOTCjl 'Iepe3 nOTOKH Ha rpaHHl.\&x KOHe'lHblX lIqe-
eK. CaMH ra30,AHHaMH'fecKHe BenH'IHHbI Ha 3THX rp&HH ax TepnHT p83pbIB
70
rJl. II. KHHeTH'leCKH corJl&COBaHHlJe Pa3HOCTHWe CJreMbl
H B RBHOM BH e He onpe.n.eneHbl. BblDHweM, HanpHMep, narox MacChI W p
Ha neBOA rp8JIH1l.e SI'qeAKH (pHC. 5):
Wp _1/2 = J {!i-l,Md{ + f fi.Md .
>o «0
BBe eM nOHftTHe cPYHKIJ.HH pacnp eneHHH li-l/2 H8 rpaHHu.e s eaKJ{
(1.12)
f . - { Ii-I,M, > 0,
'-1/2 - !i,M, { < o.
Tor,tta BLlpa>KeHHe (1.12) nepenHweTCH B BRAe
00
W p '_1/2 = f fi-1/2d{ = (PU)i-l/2'
-00
(1.13)
"fTO 3KBHBaneHTHO KJI&CCH'feCKOWY Bblp&)KeHHIO ,D.J1B nOTOKa MacCbI npM
x = Xi-l/2'
TaKHM 06pa30M, 4>YHAaMeHTaJIbHbIA 3&KOH COXpaHeHHR MacChi. CBB-
3bIB81OUUlA ee H3MeBeUHe no BpeMeHH C COOTBeTCTBYIOII(HMH DOTOKaMIf 11&
rpaHH e 06'beMa, AIlS ypaaHeHHJI (1.6) BblnWllIBeTCB. nOBoneHHe )Ke ;J.O-
nOJlHHTeJJbHbIX. no cpaaHeHHIO C ypaBHeHHeM Hep83pblBHOCTM, AHCCllna-
THBHhlX 'tneHOB CBj(38HO C HCnOJ1h30BaHHeM KOHettHoro 06'beMa H'leAKH, B
KOTOpoM pa8HOBecHaR cPYHKIJ.WI pacn}>eAeneHHJI (8 A8.HHOM CJlY'l8e MaKC-
BenJIOBCKaJI) C'IHTaeTCJI DOCTOJlHHOA no npocrpaHCTBY.
3aNe'laHHe 1.2. HcCHnaTHBHhle '1neHbI B npaBOA 'IaCTH CH:Te-
Mbl ypaBHeHHA (1.6)-(1.8) 3&BHCJlT OT p83wepa S'IeAKH z. YNeHbI:JHM
pa3Mep JI'IeAXH IJ.O AJlHHW CB06oAHOro npo6era I ( x = l). npeACTa8HM
nocneAHHA linea COOTHOmeHJIR 6anaHC& (1.3) B BH)J.e
1 ll 1.+1, M - 2/i,M + f.-I.M +i+l/2 - 4-i-l/2
T 6,z2 = :z'
(1.14)
rAe
. _ h+1,M - JiM
,+ 1/2 - 2 dx .
PaccMoTpHM cnY'IaA, KorAa MaKpocxonHqeCKaS CKOpoCTb ra3a cY11te-
CTBeHHO MeHbwe CKOpocTH 3BYKa
lul <e:: c.
(1.15)
Torp.8 "PH KHTerpHpOBaHHB (1.3) C cYMMaTOpHblM HHBapH8HTOM 3a.Me-
HHM B (1.14) I I Ha CK0J!OCTb 3BYI<&, Tal( }l(e, KB.K M'o,n:enaeTCSI B 3J1eMeHTap-
HoA KHHeTHqeCKOI TeOpHH (rn. I). npH TaKoA 3aMeHe nacne HHTerpHpooa-
"WI no CKOpOCTJlM MOJIeKyn nOHBHTCH KJI&CCHqeCKHR Bft3KHA qneH (P,Ui)% C
* 1. BblBO,lt KI-lHeTH'IecKH COrJIacoB8HlfbIX crew
71
Koo4><I>HQ.HeHTOM BB3KOCTH /.J, OTBeQarol1\HM 3J1eWeHT8pHoA KHHeTH'IecKOA
reopHH (4.46) (rn. I): J.t:: pcl/2.
3aMetfaHHH 1.1 H 1.2 nO qepKHBalOT B&>KHOCTb orpaHUqeHHSI
CHH3Y KOHeQHblX 06beMOS, ne>KaIIJ.HX B OCROBe nocrpoeHHH KHHeTH-
qecKHX cxeM, a T8.K>Ke B03MO>KHOCTL ODHC8HHSI C nOMOIIJ,b1O KCPC
ecTeCTBeHHblX HCCHnaTHBHhlX npO:QeccoB, "PH YCJlOBHH paoeHCTBa
no nopSAKY BeJJH'IHHbI xap8.KTepHblX p83MepoB gqeAKH z NIBBe
cBo6o Horo npo6era l. 3TH <l>aKTLI DOCJIY>KHnH OCHODoR HCDonb3O-
BaHHR, Hat.tHHaJI C 1985 ('., KCPC ,lVIS! pacqeTa, TeqeHHA BS3KOro raaa,
a 'I'aK)J(e HHTepnpe HH KHHeTHtJecKHX cxeM H KB83Hr830 aMH'Ie-
CKO.A CHCTeMbl ypasHeHHA KaK MO eJlH NIB OIIHCaHHSI ra30t\HH8MH-
I.leCKHX npOL\eCCOB (rn. III).
5. IIpHMepbl pacqeTOB
IIpHBe et.-t aHHbIe pac'leTOB BYX npoCTp8JtcrBeHHO-O,D.HOMep-
HbiX 3a.u-aQ - 3a)].a'lH 0 pacn e ra3OAHHaMH\feCKoro p83pblBa H 3a-
P
8
6
4
2
o
4(J
so
160 'C
PRC. 7. PeweHHe 38J:t&'IH 0 pacD e raJOAHH8WH'IecKoro pa3pbl8a: 1 - fta OCHOBe
KCPC, 2 - no MeTOJU' C.K. I'oA,YHOBa,3 - DO cxeMe SHASTA; TOHICOA JIHHHeA
noJCa33HO TOqJfoe peWeHze
72
T.l1. II. KHHeTJf'lecKH cornacOBaHHMe p&3HOCTHble cxeMbl
aqH 0 BH)f(eHHH nOpWHJI (13, 24J. I1pHBeAeHHhle pe3YJ1hTaTbI He
npereH)Q'JOT Ha IlaYQHYIO H npaKTHQeCKYIO HOBH3HY, a CKopee Ae-
MOHCTPHPYIOT KOHKypeHTocnoc06HOCTb cxeMbi (1.6)-(1.9) no cpas-
HeHHIO C Tpa..ltHI1.HOHHbJMH Cyru-ecTBOB&BWHMH Ha Ha'laJIO 80-x rO,l).OB
aJlrOpHTM&MH, 'liTO H DOCJIy>KHJIO OCHOBOA anbHeawHX HHTeHCHB-
HbIX HCCJIe OBaHHH 8.BTOpa Hero KOJIJJer.
Ha pRC. 7 npHBe eHbI 3HaqeHHH fiJlOTHOCTH, B3BThIe H3 pac'leT8,
38.AaQH 0 pacna,ne ra30 HHaMHqeCKoro pa.3pbJBa Ha MOMeHT 6e3p83-
MepHoro BpeMeHH t = 4. TIpH t = 0 B l\eHTpe paccMaT])Hs&eMoA
o6nacTH (x = 80) 3 aH ra30 HaMH IecKHA pa3pLlB. Cnesa OT pa3-
pblBa p = 8, p = pR:r = 480) cnpaoa (J = 1, P = 1. HCnOJlb30oa.rracb
pa8HO fepH no npoCTpaHCToy ceTKa ( x = 1). 06 ee 'IHCJ10 TOt.JeK
no npocrpaHcTBY paoHo 160. mar no BpeMeHH paBeH 0.02.
Ha TOM :>Ke pHCYHKe npHBe)\eHbl auHble H3 pacqeTOB no CXeMaM
C.K. rOAYHoSa H SHASTA, B3HTbJe H3 pa60Tbl [34J.
T Ha pRC. 8 npHBe eHbJ
,AaHHble 'lHcneI-IHOrO pewe-
HHSI 3 aqH 0 ,l\BHJKeHHH
nopWHSI, KOTOpbJe cpasHHBa-
IOTCSI c aHaJIOrH'IHLJMH pe-
3ynLTaTaMH, nonyqeHHblMH C
110MOmDIO CXeMbJ C.K. rOAY-
ROBa. Pe3YJJbTaTbJ, nOJl)"ieH-
Hble c rrOMO 1O KCPC, Ma-
no OTJ1HtlalOTaI OT aHaJ1I1TH-
4.0
I
,
I
I
,
I
I 2
V
I 1
2.0
qeCKoro peweHHSI npH qHCJle
Maxa M = 8. Oc06oe BHH-
40 x 6
MaHHe cne)JyeT 0 paTHTb Ha
nOqrnf nOJIHoe OTcyrcTBHe 3H-
PIIC. 8. PewellHe 3a.lI:a"tfH 0 nopwHe: 1 - c
HCfiOJIL30DalfHeM KCPC, 2 - no MeTOA)' TpoIIHftHoro cneAa.
C.K. rOA}'HOBa CJIe eT OTMeTH'rb, LITO
npaKTH ecKH COBn aJO He c
KCPC pe3YJlbTaTbi npH perneHHH nOCJIeAHe 3 aqH paHee 6LlnH
nonytteHbI B pa60Te [18). )J.8JlHbIA Q>aKT HeY,l\HBHTeJleH, TaX KaK B
3TOa pa60Te HcnOJIb30B8JlCB a.nropHTM Herrocpe.n;CTBeHHoro onpe e-
JleHRS <t>YHK HH pacnpeAeJleHHH H3 peUleHHS KHHeTHtleCKOrO ypaB-
o
o
20
92. KCPG na npOH3BaJlbHbJX cerK8X
73
HeIlHSI. nOCJJe HaxO}t(,lJ.eHHH cPYHKL\HH pacnpe eneHHH ra30 HH8.MH-
qecKHe napaMeTpbl onpe,lJ.eJUlJJHCb KaK ee MOMeHTbJ. 3aMeTHM, 'fTO
B KCPC pemeHHe KHHerHqecKoro ypaBHeHHlI npOHCXOft.HT HeSlBHO
(BB npeAI10nO>KeHHSI 0 CYID.ecTBOB8.HHH MaKCBeJJnOBCKoro pac-
npe eJleHHSI) 'iepe3 onpe eJleHHe ra30 HHaMHqecKHX uapaMeTpOB.
n03TOMY eCTeCTBeHHO, QTO pe3YJILTaTbl pacq:eTOB no 3THM BYM aJI-
ropHTM8.M OKa3aJIHCb 6nH3KH. 06paTHM TaJ()Ke BHHM8HHe H8 TO, liTO
aKKypaTHoe OIIHcaHHe H3MeHeHHB 311TpOnHH C nOMO blO 60J1b Ma-
HOOCKHX cxeM OTMe'iaJIOCb H B pa60TaX 127, 29, 35].
38.KJlIO'IaSl )(aHHbIA naparpa<P, CJIe,nyeT OTMeTHTb, 'ITO o6cy a-
eMbie B HeM, KaK H B OCTaJIbHhlX naparpaQ:>ax AaHHoA rnasbJ, oapH-
aHThI KHHeTHqeCKHX, a Ta.K>Ke 6onb M8.}IOBCKHX, cxeN cne;lU'eT pac-
cMaTpHBaTb KaK B03MOJKHbre a.nropHTMhl paclleTa 3 all ,lI)fHaMH"
KH H eaJlbliol'O ra.3a. 9TH anropHTMbl MorYT HMeTb npeHMym;ecTBa
(HJI:H He OCTaTKH) no COOTHomellHIO Ka1iecTBa pacIJerOB H Bbl'lHC-
nHTeJ1bHblX 3aTpaT, y.L\06CTBa pe8JIH3B.l.\HH B CpaBHeHHH C APyrHMH
nOCTOHHHO Pa.3BHBatOIlf.HMHCB MeTOAaMH pemeHHB 3 aq )J.HllaMHKH
H)J.ea.nbHoro ra3&. Q)J)iaKO npHHnHnHaJlbHO HOBbIe B03MO>KHOCTH KH-
HeTH'IeCKHX cxeM OTKpblBa.K>TC5I npH MO enHpOBaHHH BB3KHX Teqe-
"HH.
2. KUHeTHtJecKH COrJl8COBaHHhle cxeMLI
U8 npoll3BOJJhHbiX CeTKaX
B npeAblAY eM naparpa4>e OCHOBHa..sI H eSl nWlY'IeHHB KCPC
6b1na paccMOTpeHa Ha npHMepe npocreliwero CJJy'tlasr ID reoMeTpHH.
PacnpocTpaHHM aHHbJA nOAXO,A Ha UOJIyqeHHe KHHeTHqecKOA Pa3-
HOCTHOA annpoKCHMaQHH B 60JJee o6 eM C1I}"Iae. BKJlIOQaH HecTPYK-
TypHpoBaHHbJe ceTKH.
1. CooTElomeHHe 6aJlaHca AJIB cjJYHK HH pacnpe,l{eJIeHHB
na npOH3BOJlbHOlf CeTKe
PaccMaTpHBaeMoe B 1 Bblpa>KeHHe (1.3) NtH 6aJJaHca CPYHK-
HH pacnpe eneHHg nerKO MO}f(eT 6blTb o606w.eHo Ha npOH3BOJIbHYto
,lI.By tepHYIO HnH TpeXMepHYIO DpocTpaHCTBeHHYIO CeTKY (pHC. 9).
74
r.n. II. KHHeTH'IecKH corJl8COIJ8.HllWe pa3HOCTHbie CXeMlJ
)J.;lS 3TOro aHaJJorHQHO o oMepHoMY CJ1Y'ialO He06xotJ)tMO pacCMOT-
peTb npoHHKHoseHHe B D-eHTpanhHYIO H'ieAxy C HOMepoM I M3KCBeJI-
JlOBCKHX cl>YHK H:A pacnpe eneHHSI 83 coce HHX SNeeK C HOMepaMH
i 8 HanpasneHHH CKopocre:A 1 HMeIO X OTpH aTeJlbHOe CKaJIHpHOe
npoH3Be eHHe C BeKTOpoM HOpMaJJH COOTBeTCTByrorneA rpaHH (pe-
6pa) (CM. pHC. 9).
e JI M
I,
VI
.
/f,M
/!+l,M
e
PHC. 9. nonY'IeHHe KCPC "8 npoH3BMbHOA ceTXe
YqHThIB8.B 3'1'0, BblnHweM 6anaHCHoe COOTHoweHHe B BH e (361-
[38) :
/ 3+ 1 I i Ie ,j+l,3
1 - J,M + 1 L / ( 'D.il)J i,M + J i,M do =
.6t mes VI '-1 2
1- Sil
Ie
= 2 1 Vi '" / 1( ,nil)1 ul M - fi M)do-. (2.1)
mes 1 ,
I-IS"
3,a.ecb 11t M H II - 3HaqeHHB MaKCBeJlJIOBCKOit H npoCTO cl>YHK-
IJ.HH pacnpe eneHHSI B 1.(eHTpaJJbHoA HqelKe, JiM - 3HaqeHHH WaKC-
BeJ1JIOBCKOA <!>YHKIJ.HH pacnpe eJleHHB B coce HHX jlqeAKaX, k - qHC-
JlO rpaueA (pe6ep ,lVISi 2D-reoMeTpRH) eHTp8JIbHOA S'IeAKH, mes \Ii
- 06'beM (nn01I.1-3Ab) eHTPa.nbHoA H'IeAKH, n,J - HOpMaJIb K rpaHa
(pe6py) srQeAKH. HHTerpaJIbJ B Bblp8JKeHHH (2.1) 6epYTCSlI1o rpaHSM
Sir (pe6paM) L\eHTpanLHoA gqeAKH.
f2. KCPC H3 npOH3BOJIbHhIX ceTK&X
75
t.{To6b1 nOJlYttHTb KCPC Ha npOH3BOJIbHhlX CeTKax J AOCTaTOt.!-
HO npOHHTerpHpOBaTb BblpIDKeHHe (2.1) no CKOpOCTJ1M MoneKyn c
cYMMaTopHblMH HHBapHaHmJAH. B noCJIe.L\YIOIU.HX nYHKT8X I1aparpa-
cPa npHBe eM Bblpa>KeHHSI ,lV]H KHHeTH ecKHX cxeM B cnY'la.e 2D-
reoMeTpHH ):tnR npSlMoyronbHblx H TpeyronLHblx CeTOK.
2. KCPC na rrpRMoyrOJ1bHblX CeTxax
Ha npjlMOYI'OJIbHJ:dX ceTKaX KHHeTHQeCKH cornacOB8HHble CXeMbJ
SlBJlSlIOTCH npOCTblM o606 eHHeM Bblpa>KeHHA (1.6)-(1.8):
'+1 -
pfn - pfn + (pu)o + (ptJ)o =
Ilt % II
= x [pu erf(su) + p exp( -s ) t% +
6.y [ P 2 ]
+"2 pv erf(sv) + p.Ji exp( -sv) !ill'
(2.2)
j+l j
(PU)in - (PU)in + (pu2 + p)o + (pu1J)o =
t % r
= x [(pu 2 +p)erf(s ) + M exP(-s )t% +
6.y [ Pu 2 ]
+"2 pti1J erf(s.,) + .8v'i exp( -8 v ) y I
(2.3)
';+1 j
(PV)in - (PV)in + (puv) 0 + (ptJ2 + p)o =
t % u
x [ rm 2 ]
== 2 (puv)erf(su) + pV; exp(-su) z% +
+ f [(pv2 +p)erf(s.,) + M exP(-s )]jtl'
(2.4)
76
rn. II. KHHeTH ecKH COI'JI&COBaJlHb.e paJlloC'rffble cxeMhl
E!+l - E!
In In + (u ( E + P ) )o + (v(E + p))o =
t % !I
6.x [ E + p/2 2 ]
= T u(E + p) erf(su) + f3../ii exp( -Su) xx +
!I [ E + p/2 2 ]
+ 2 v(E + p) erf(sv) + fJVi exp(-sv) j;y ·
3Aecb, B AOnOJlHeHHe K o603HaQeHIDIM CHCTeMbI (1.6)-(1.8), Hcnonb-
.
3YIOTCS: I, n H u, V - COOTBeTCTSeHHO, HOMepa CeTO'\fHbIX Y3JIOB H
CKOpOCTH B OJlb HanpaaneHHH oceA ox H OY; 8" = {3u, Sv = {3V.
CXeMbI Ha npHMoyronbHblX CeTK8.X )J)IjI TpeXMepHOA reoMeTpHH
TpHBHanhHO nOJlY'IaJOTCSI H3 (2.2)-(2.5) 06aBJIeHHeM pa3HOCTHOA
annpOKCHM8J.{HH eID:e no oAlloa KOOpNfHare OZ.
(2.5 )
3. KCPC H8. TpeyrOJlbllblX ceTKaX
IIpSiMoyronbHble Ce1.'KH OTHOCSlTCSI K pa.3pSl y CTPYKTypHpoBaH-
HblX CeTOK. O)J.HaKO BO MHorHX 3 a,£taq ax , HMelOIU.HX B nepBYIO oqe-
pe b HH)JjICTpHaJlbHOe npHMeHeHHe, HCnOJlb3yeMbIe paCt.feTHble 06na.-
CTH HMelOT tt-0CTaTOqHO ClIO>KHYIO reoMeTpH'IecKYIO crpYKTYPY. Hx
pa.36HeHHe Ha CqeTHbJe )1qeRKH C J..(eJ1hlO llOCJleAYJOw-eA p83HOCTHO"
annpOKCHMal HH 3aQaCTYIO TPY)J.HO npoBecTH c nOMOUT,b1O ce'fOK, HME-
IOIIUlX ynopSl)I-O"leHHYIO TOnOJlOrHIO CTPYKTypUPOBaHHblX CeTOK. no-
3TOMY 80 MHorHX CJIyqamc npOCTpaHCTBeHHyro HCKpeTH3au.HIO pac-
\leTHoA 06naCTH npoBO)UlT, HCfi0JIb3Ysr HecrPYKTypHpoB8.HHble ceT-
KH , 8 nep8YlO Oqepe b TpeyronbHbte ,lVIH 2D-rOOMeTpHH H TeTpa3 -
panbHble )J.JlB 3D-reoMeTpHH. IIpH TaKoA annpoKCHMa HH MaTpHl.\bI,
BbJpa>KaJOw,He CBB3b ue)f(AY 3HaqeHHflMH ra30,nHHaMHqeCKHX napa-
MeTpOB B p83JlHt4HbIX TO tKaX, He HMelOT tJeTKO BLIpa>KeHHYIO JleH-
TOt{HYJO CTPYKTYPY.
O,ll.H&KO Hap C HX I'JIaBHblM OCTOHHCTBOM - y,n;aqHblM reo-
MeTpHQeCKHM npe CTaBneHHeM Dong Te1.JeHHB - npHMeHeHHe He-
CTPYKTypHpOBaHHhlX CeTOK CDSI38J10 C l.\eJJblM pSl,QOM npo6neM. Bo-
nepBblX, Ha TaKHX CeTKRX) B OTnHt.JHe OT CeTOK CTPYKTYPHpOBaHHblX,
OC'1'aTOqHO CJ10)t(HO nOCTpoHTb 8KKypaTHYlO p83HOCTHYIO annpOK-
CHltrla HIO ypasHeHHA ra30BO.A ,lJ){HaMHKH, OnOJlHHTenbHO o6n8JI-a-
92. [(CPC H& npOl13BOJll>HblX ceTxax
77
10 HX CBOItCTBOM MOHOTOHHOCTH H KOHcepsaTHBHOCTH. BO-BTOpblX,
npH HCnOJlb30BatlHH HeSIBHblX cxeM B03HHKaIOT OnOnHHTenbHbJe
CJlO>K1IOC1'H, CB 3aHHbJe c pemeHHeM CHCTeM CeTO'tJHbIX YPaBHeHHA
AJI$I MaTpH , He HMeJOll\HX neHTO'lHYIO crpYKTypy.
llpo6neMa He,n;OCTaTOqHOA TOQHOCTH a.Ill1pOKCHM8,QHH )VIR He-
CTPYKTypHpOBaHHblX ceTOK B 3Hat.JHTenbHo:A Mepe CHHMaeTCSI CTaB-
weA penbHoA B03MO)f(HOCTblO HCnOJIL30BaHHSI MHoronpo ecCOpHb1X
BbicoKonpoH3BOAHTeJlbHblX BblttHCJIHTeJJbHhlX CHCTeM, KOTOpbJe n03-
BOJIHIOT npHMeHSlTb B pactfeTa.X npOCTpaHCTBeHHhte CeTKR, OCTara-
IOll(He 10 8 - 10 9 Y3noB.
OAHHM H3 B03MO>KHbIX n)'TeA nOBblWeHHft TO'IHOCTH annpOKCHMal\HH
J1RJlSleTC$I KOM6HH JI CTPYXTypHpoBaHHbIX H HecTpYKTypUPOB8JlHbIX ce-
TOK. npH 3TOM CTPYKTYPHpoBaHHble CeTKH, KaK npaaHno. HCDOJIb3yJOTCH
npH annpoKcHMaIUIH nORjI Te eHHjJ B6nH3H nOBepxHocTH Te1I8 (pHC. 10,a),
r;J.e Tpe6YeTCB t HanpHMep, H3-3a HaJlH'fHJt nOl"pallll'lHbIX cnoeB, 6o.nee Bbl-
COKaJl TO'lHOCTb. OTMeTHM, '11'0 B paMKaX KCPC H3 OCHOBe 6an8.HCHoro
COOTHoweHHH (2.1) nerKO OpraHH30BaTb conpIDKeHHe pa3J1H'IHbIX THDOB
Ce'l"OK, HanpHMep, TpeyronbHblX H qeTblpexyrOJlbHbIX, H306p8JKeHHbIX Ha
pHC. 10 , 6.
Bonpochl MOHOTOHHOCTH H KOHcepBaTHBHOCTH ,LV1SJ KCPC nerKO
CHHMalOTCSI BBH Y xapaKTepa ax BbIBO;(& 83 COOTHOWeHH.H 6aJ18.Hca
(2.1) ,LI.JIH OAliOqacTH'-IHOA cl>YHKQHH pacnpeAeJIeHHa. HarJIH,lI.HblA a
P33YMHblA C cl>H3HQeCKOA TO\{}(H 3peHHH BbIBO KCPC, no BeeR B H-
MOCTH, CJIY>KHT TalOKe H OCHOBOA HX XOPOWHX annpOKCHMa HOHHblX
CBoRCTB.
no CBoeMY nOCTpoeHHIO KCPC HBJISI10TCSI SlBHblMH cxeMaMH.
)Lnsr BblnOJlHCHHSI YCJ10BHS YCToAqHBOCTH npH Mo enHpoBaHHH Te'le-
HHA H,n.ea.nbHoro Pa3a OCTaTOttHO orpaHH'feHHSI Ha war no BpeMeHH
onpe,l{enSle""oro H3 KPHTePHSI KypaHTa. BOJ1ee TOro, MB ,AOCTaTO'\fHO
WHpOKoro Knacca. TeqeHHA BH3Koro ra3a B paMKaX KCPC H KBa3H-
ra3O)J.HHaMH ecKoA CHCTeMbI y aeTCB nOCTpOHTL HBHhle a.nropHTMbl,
YCTOAqHBOCTb KOropblX T8J<>Ke onpe)J.engeTCSI H3 YCJlOBHSI KYPaHTa
(rn. III). Cne,rureT 38.MeTHTh J 1.fro BblnOJlHeHHe 3TOro YC1l0BHSI YCToA-
tJHBOCTH enaeT npHMeHeHHe SlBHblX cxeM BnOJlHe KOHKypeHTOCno-
C0611bIM )J.JISI pemeHHR MHorHX 38Aa . C )JJ)yroA CTOpOHbl, 'ITO HBJlH-
eTCSI KpaRHe B8JKHbIM t HCnOJ1b30BaHHe BBHbIX cxeM Ha HecTpYKTypH-
pOBaHHblX ceTKax nOJlHOCTblO CHHMaeT npo6neMY peweHHR CHCTeM
18
rJl. II. KHHe1'HqecKJI COrnaCOB8JfHble p&3HOCTHble cxeMbl
a)
6)
PUC. 10. KOM6HHau.HH CTPYKTYPHpoB8HHoA H HCCTPYKTYPHpOBaHHOif ceTOK:
a) npHMep Mo emtpOBalDlJl Te'leHHB OKano WIHH)U>a B nnOCKOY KaHane.
6) pa.rMeHT conpS)J(eHHH TJ>eYI'OJ1LHoA H qeTLlpexyrM Hoii CeTOK
2. [(CPC H& npoH380J1bHhlX CeI'KaX
79
CeTQqHblX ypaBHeHHA.
Cne YeT 3aMeTHTb, q'fO np06neMa reHep HH ceTOK KNeer CaMo-
CTOBTeJ1bHoe cepbe3Hoe HaY'IHOe H npHKnaAHoe 3HaqeHHe H npHBJIe-
KaeT K CBoeMY peweHHJO, OCo6eHHO B nOCJ1e ee BpeMg, BHHMaHHe
MHorHX Cne(J.HaJIHCTOB. O,ll.HaKO B aHHoA MOHorpa,Q>HH 3TH BonpoChi
cneQHaJlbllO He pacCM8TpHBaIOTCSI. ,!VIa TpHaHl'ynHD-HH pac'feTHoA
06nacTH HCnOJIb3YlOTCB H3seCTHhle aJIrOpHTMbI H tl-OCTynHhle npo-
rpaMMbJ J 8 '-IacTHOC'rH, UMGAFD-2D {reHep .SI H aHaJlH3 ABYMep-
HblX HecTpYXTypHpooaHHblX ceTOK Ha OCHOBe MeTO a npOABHHYToro
<l>poHrm (39]).
PHC. J1. KOHTpo1lhtWA o&beu )J.JJJI annpOKCHMaqmt R;' TpeyronbHoA ceTXe
IIpHCTyoHM K nOCTpoeHHJO KHHeTH'IeCKH cornacoBaHHbIX cxeM
Ha TpeyrOJIbHblX ceTKaX. BHaqane onpe eJIHM KOHTponbHblA otYbeM,
npe.D-cTa.Bn.SlIO HA co6oA MHoroyronLHYIO SNeAKY (pRC. 11). B CBOIO
OqepeAb, npH nOCTpoeHHH MHoroyrOJ1bHoR gqeAKH BOKpyr Y3JIa Tpe-
YroJ1bHOA CeTKH 3aMeTHM, qTO B Ka>KAOM TpeYI'O.1lhHHKe B o HoA
TOQKe t);OJIJI<HbI DepeceKaTbc 11'H rpaHH HOBO:A CeTKH aeperYJlSlp-
HoR CTPYKTypLI. 06m;aH TOqKa 06S13&TenhHO OJI.>KHa ne)K8,Tb BHY-
TPH TpeyronbHHKa. HanpHMep, 3TO MO>KeT 6h1TL eHTp OnHCaHHOA
OKPY>KHOCTH AJlH TpHa.HrY HH ,L(enoHe, KOTOPD[ 38.Be,ltOMO ne>KHT
BHYTPH TpeyrOJlbHHKa. B HameM C1Iyqae CoeNIHHM OTpe3KaMH npSJ-
80
rn. II. K,fHeTHqeClCH COrJ1aCOBlJJlllhIe pa3HOCTHbIe cxeMhI
MoA eHTpbJ TH)KecTH c cepeAHH tH rpaHeA TpeyrOJlbHHKoB) HMe-
IO HX 06I.lYJ'IO BepWHHY I. llpH 3TOM nonYQUM pacqeTHYIO s:rqeAxy,
npeACT8.BJlHlOl.lYJ'fO co60a MHoroyronbHHK. BBJISlIOw.aAca KOHTponb-
HbIM 06'beMOM (pac. 11).
3aMeTHM, 'lTO ecnH HCXO)\HaSi TpeyroJILHM CeTKa y oBneTBOp.R-
eT npHH Hny TpH8.HrYJlH HH t(enOHe, TO nOCTpOeHH3J1 H& ee OCHose
HeperYJljlpHa.R CeTKa 6Y,lI.eT COCTOSlTb H3 SI'IeeK HpHX1Ie. B 06ID-eM
cnY"Iae, B OTJlH'tHe OT srqeeK ,l(HpHxne, KOHTpoJIblILJA o&beM - He
06SI3aTeJ1bHO Bbmyt<JIblR) HO C 60nbWHM t{HCJ]Ot.l rpaHeA: MHOroyronb-
HHK, tITO n03BOJ1BeT cI>eKTHBHO Y'JeCTb 60JThIDOe 'IHCJ!O HanpaBJIe-
HH MOJ1eKYJIHpHblX CKopoCTeA.
ECJIH npH TpHaHrYJI.H H 06JIacTH TPeyrOJlbHHKH He HMeJOT TY-
noro yr J1a, TO B KaqeCTBe cpeAHea TO KH Bbl6HpaeTCH D-eHTp onH-
CaJ-IHOH OKPY>KHOCTH. EcJIH :>Ke B TpeyrOJIbHHKe npHCYTCTSYeT TynoA
yrOJI, TO I10 cpe He:R TO KOD nOHHMaeTCH eHTp T.H)((ecTH.
BblnHweM 6aJ1aHCHOe COOTHoweHHe (2.1) ,lJ.na KOHTponbHoro
o6 eMa (pac. 11):
k
/1+1 = JIM - t L [ShWli(nJi) + S Wli(n )] . (2.6)
I i=l
3p-eCb CYMMHpOBaHHe npoHcxO HT no sceM k rpaHaM KOHTpORbHO-
ro otYbeMa, n , n - BeKTOpbI HopManH K COOTBeTCTBYlOl1I,HM qa-
CTHM rpaHH bI, sh, S - ,lVIHHLI COOTBeTCTBYIOw.eA 'lacTH rpa.HB bl
KOHTpoJIbHOrO otYbeMa (pRC. 11). B COCTaBnBIOUI,HX (2.6) nOTOKOB
HCnO.flb3yeM cneAYIO ee Bhlpa>KeHHe
Wli(n) = Fli(n) - Dli(n),
(2.7)
rAe
1
FJi(n) = 2 n(fl + Ii),
1
Dli(n) = I(nl(fi - !r),
(2.8)
n - 3HaqeHHe MoneKYJlSlpHo:A CKOpOCTH B HanpaBJIeHHH, nepneH.l\H-
KYJIHpHOM qacTH rpaHHlJ.hl KOHTponbHoro OOheMa, no KOropOA' np
H3BO.l\HTCSI CyuMHpoBaHHe.
92. KCPC H8 npoH3BOJ1bHblX CeTlC8X
81
HHTerpHPYS COOTHoweHue 6aJIaHca (2.6) nH Q>YHK1I.HH pacnpe-
AeneHHSI C cYMMaTopHLIMH HHBapHaHTaMH 1, {3:t Y' 2/2, nOJlYQHM
CHCTeMY pa3HOCTHblX ypaBHeHHA Ha TpeYroJIbHOA ceTKe ;J.JIH ra30-
H}IaMHqeCKHX napaMeTpoB:
. Ie
(puH+ 1 = (pu)1- ' L[sfaF]j(nfi) - DYi(nri» +
I i=l
+ S ( (n ) - DYi(n »],
. It
(pv) +1 = (pv) _ J L[S (Ffi(nti) - DYi(nfi» +
I i l
+ S (Fli(n ) - Dji(n »] J
. Ie
.+1 . tJ " [ L E L E L
Ej = E: - V L.J S[i(F li (n/i) - D/i(nli» +
I i==l
+ S (Fl(n ) - D (n »],
nSl F , FJi, Ffi, Fl ) annpOKCHMHPYIO HX KOHseKTHBHyJO qacrb
nOTOKOB, HCnOJlb3YIOTC.H CJIe lOru,re Bbtp&>KeHHH:
. Ie
_j + 1 6.t' [ L P L p L
IFI = PI - 11: L.J Sli(Fli(nli) - D/i(n/i)) +
I i=l
+ S (F (n ) - D:i(n »],
F = [(Pfinh + (PUn)i] J
FYi = [(pu + ph + ( + P)i] cosali +
+ [(pu,.u,o)/ + (PUnUT)i] sinari,
Fii = [(pu + ph + (pu + P)i] sinari -
- [(PUnu T )/ + (J)UnUT)i] casali J
FR = [(E + p)un]r + [(E + P)un]i'
(2.9)
(2.10)
(2.11 )
(2.12)
(2.13)
(2.14)
(2.15 )
(2.16)
J h. U. l\HHeTHqecKJf COrJlaCOBaHHble p&3HOCTHble CXeJ4bl
3.n.eCb cos a, sin a - KOOp.LI-HHaTbi e)J.HHH'IHoro BeKTopa HopMa-
nH n K COOTBeTCTBYIO eA 'IacTH rpa.HH bJ KOHTJ)OnLHoro otYbeMa,
Un, UT' - KOMJIOHeHTbJ CKOpOCTH B JIOKaJlLHOA eKapTOBOM CHCTeMe
Koop HflaT (n, T) :
Un = UC08Q + vsina, U r = -usina + 1JCOSO.
tLnsr cI>cI>Y3HOHHbJX COCTa.BmDO DOTOKOB D i' Dr,t DYi' D
HCIIOJIb3YIOTCH cne;QYIOIIJ.He Bblpa>KeHHSI:
p 1 [ ( ( exp( - s ) )
Dfi = '2 fYUn erf sn) + P fJ..(i i
( exp(-s:) ) ]
- pu,. erf(sn) + P fJ..jir /'
u 1 [( 2 eXP(-8 ) )
D 1i = 2 (pun + p) erf(Sn) + PUn fJ..(i i
( 2 exp( -8;) ) ]
- (PUn + p) erf(sn) + fJIJn (3 I cos ali +
1 [( exP(-S » )
+ 2 P'UntJr erf(sn) + PUT (3..jir i -
( f( ) exp( -8;» ] ·
- fJUn'U T er Sn + f1Ur f3 1 sma",
1J 1 [( 2 exp(-s;) )
D ri = 2 (PUn + p) erf(sn) + J'Un 13.Ji i
(( 2 ) f( ) exp( -S ) ) ] ·
- PUn + p er Sn + PUn fJ / smali-
1 [( exP(-S ) )
- 2 P'UnUT erf(Sn) + f1Ur fJ..jir i -
( exp( -S ) ) ]
- PtJnUr erf(sn) + rm-r (3..fi I COS Qri ,
(2.17)
(2.18)
(2.19)
92. KCPC H8 npoH3BOJIbHblX ceTlC&X
83
D = [ (E + p)Un erf(sn) + ( E + !p) eX ';; ) ) i
- (E + p)Unerf(sn) + (E +!p) ';; ) ) 1] I
1
/3 = v'2RT ' 8n = PUn.
,L{atlHblA a.nropHTM 6b1JI nOCTpOeH B pa60Te (40).
B KaqeCTBe npHMepa pac eTaJ Hcnonh3YJO ero annpoKcHMa HK)
KCPC H& TpeyronbRblx ceTK.8.X, pacCMOTpKM 3 aqy 0 CBepX3SYKo-
80M 06TeKaHHH 3JInHnCa no HYJIeBhIM yrnoM aTaKH. B pacCM&TpR-
naeMOM BapH8.HTe t.fHCJIO Ma.xa (OTHomeHHe CKOpocTH K CKOpOCTH
3BYKa) Ha 6ecKOHeQHOCTH Moo = 2. OTHomeHHe nonyoce:A: 3JlnHnCa
b/a = 1/2. nOKa3aTenL a6aTbI 'Y = 1.4. Ha rpaHH e 3JlnHnCa
njf CKOpOCTH CTaBHJlHCb ycnOBHB HenpOTeK8HHSI.
PacqeT npOBo,nHJIca Ha TpeyronbHOA ceTKe, cOCTOSlll(eA H3 5941
Y3JIOB. CeTKa cry aJlacb K nosepXHOCTH 3JInHDca.
Ha pUC. 12 nOKa3aHO pacCQ;BTaHHOe pa.cnpe eJIeHHe a.BneHHSI Ha
MOMeH'r YCTaHOBJIeHHSl. Kax BHAHO, pe3YJlhTaTbI 6nH3KH K XOPOWO
H3BeCTHblM B BhIQHCJlHTM&HOft r830BoA HHaMHKe aHHbJM pa60TbI
[41].
ITocTpoeHMe 3D HeCTpYKTYPHpOBaJlHldX ceTOK npe CTaBngeT co-
60a 3a.z:taqy 3HaQHTeJIhHO 60nee C1JO)KHYJO, qeM c03AaHHe IDe BYMep-
HblX aHaJloroB. O aKO, ecJIH 3m 3 aqa pemeHa, TO IIOCTpoeHHe
KCPC annpoKCHM8IJ.HH Ha TaKHX CeTKax, KaK, HanpHMep, TeTpa-
3Apa..nbHbIX, ocy eCTBJlHeTCSI C nOMOU 1O npOCTOro o606 eHH.H pe-
3YJIhTaTOB aHHoro naparpacl>a. llpH 9TOM B xac&TeJIhHOA K KaJK-
oR rpaHH IL1IOCKOCTH paCCMaTpHBalOTCSI ARe OpTOrOHaJIhHble KOM-
nOHeHTbJ CKOpOCTH U-rl H U r 2. I1OJI)"IaIOI1\HecH "PH 3TOM BbIpa)KeHHH
aHaJlOrH HbI (2.9)-(2.10), XOTS! rop83 O 60nee rpoM03 H. npHMep
paC\ieTa nOTOKa BSl3KOro ra3a Ha TeTpaa,lij)aJIbHblX CeTKaX 6YAeT pac-
C 10'rpeH B rJI. V.
(2.20)
lS4 I. II. 1J.
6
5
4
AHBeTHtJeCKH COrn8COSaHHore p83HDCTHble CKeMDl
u 3
=
=
; 2
1
o
o 10 20 30 40 50 60 70 80 90
Yrort cp
PHC. J 2. Pa.cnpe.z:\eneHHe ABBneHHH H& ronoBHoA nOBepXHOCTH nHnca; cnnow-
naH KpHBaJI nOJlY'feu& Ha OCHOBe BCDOJlL30B A-HAg KCPC, ... - ;lJ.aHHble pa60Thi
(41 ]
3. KUHeTHqeCKHe cxeMbi nOB))IWeHHOro DOp
TOQHOCTH
PaCCMOTpeHHble B 9 1 Ii 2 KCPC no CYTH eJla HBnSlIOTCSI cxe-
MaMH nepBoro nop9.AKa TO"IHOCTH no npocTpaHCTBeHHblM nepeMeH-
Hbl)"1. B caMOM AeJJe, cxeMY (1.6)-(1.8) MO)l(HO HHTepnpeTHpOB8Tb
KaK cxeMY nepBoro nopH,lt.Ka AJIR annpOKCHMal1.HH YPaBHeHRH 9AJle-
pa, OnHCbJBalO HX Te"leHHe H eaJIbHoro ra3&. IIpH 3TOM KOHBeKTHB-
Hide qJ1eHbI J CTOSlIlI.He 8 neBoR qacTH CHCTeMbI (1.6)-(1.8), onHCblBa-
IOTCSI eHrrpaJlbHO-pa3HOCTHOA annpOKCHMa HeH, HMelOIII-eA BTOpoA
nopH)J.oK TOtJHOCTH, a CTOSlW){e B npaBOa 1IacTH HCCHnaTHBHble qJIe-
UbI, aHaJIOrllqHbJe JIeHaM HCKYCCToeHHo:R Bjf3KOCTH, HMelOT nepBbJA
nopSl OK no npOCTpaHCTBY 0(&£).
KaK cne,nyeT H3 1, KHHeTHQOCK8JI cxeMa (1.6)-(1.8) TaK1Ke Mo-
>KeT 6blTb nOJIyqeHa H3 pa3HOCTHOR cxeMbI (1.9)-(1.10) HnH 3KBHBa-
neHTHo:A (1.11) cxeMbl J annpoKcHMHpYI() e:A ypaaaeHHe BOJIbI.\MaHa.
Cxet..ta (1.9)-(1.10) C HanpaBJIeHHbIMH pa3HOCTSlMH SJBJISIeTCR cxeMoA
'3. KHHeTHtieCKHe CXeldhl nOBblWeHIfOrO nOpH,l{Ka ro'lHOCTH
85
nepBoro nOpg TO'lHOCTH. n09TOMY HeYAHBHTeJIbHO, qTO TOT )Ke
fIOpH):tOK TO'IHOCTH HMeeT nOpO>K eHHasI elO cxewa (1.6)-(1.8) ):VIR
ra30,!tHHaMH ecKHX napaMeTpoS.
AHaJlOrH'IHLIM nyTeM MO>KHO 110J1YQHTb KHHeTHtJecKHe CXeMhl
60J1ee BblCOKOro nOp ,lJ)<a TOqHOCTH, eCJIH nopo aro aSl HX CXeMa
; JISI ypaoHeHHS EOnb MaHa HMeeT 60nee BblCOKHA nOpjfAOK annpoK-
CHMaQHH. Cne }1 pa60rre [42], CKOHCTpYHpyeM HeKoTopble B03MO)K-
HbIe KHHeTH'IecKHe cxeMhI 60nee BLICOKoro nOpH K8. TOtJHOCTH.
1. CxeMbl I10BblWeHHoro nopRAK3 TOqHOCTB )1.JIH pemeHHH
KHHeTH1J€CKOro ypaBHeHHH
BbIDHllleM pa3HOCTHY!O cxewY,lJ)IjI ypaBHeHHjf BonblJ).(aHa S cne-
) YIO eM BH)J.e
Ij+ - Ii + £. 1i+1/2;X h - 1 / 2 = J(J, f), (3.1)
..Ae
fi+l + Ii 1 - <P
2 + 4 (DIi-I/2- D Ji+l/2), {>O, (3.2)
fi+l/2 = fi+l + Ii 1 - 4J
2 + 4 (Dfi+I/2 - Dli+3/2), £. < 0,
D fi+l/2 == /i+l - fie (3.3)
I1CllOJlb3Ysr pa3nO}KeHHe 4>YHK / B p TeAnopa no creneWlM
x, nOJ1Y HM
li+l/2 -1i-l/2 = ( 8/ ) + 6.z 2 ( ! _ 1 - tP ) ( lJ3 f ) +
t1.z ax i 6 4 ox 3 i
3 1 - 4> ( 04 ' ) 4
+ I I t::.x 8 8x4 i + O(t::.x ).
'1'e)1 CaMbJM .lI-H<P<PepeH HaJIbHbJM npU6nH>KeHHeM no rrpOCTpaHCTBY
J1.JI pa3HOCTHOA cxeMbI (3.1) HBJISleTC.H CJle ro ee ypaBHeHHe:
Ij+l - Ii + ( af ) j = -£.t::.x 2 ( ! _ 1 - t/J ) ( lJ3 f ) j _ (3.4)
t ax 6 4 ox 3
4 -
- 1 It::.x3 1 t/J ( :x{ ) J + J(J, f) + O( x4).
o
1 Jl. J1.. AHHeTHlfecKII COrJlacoBaHHble p&3HOCTHMe cxeMbI
3,l\eCb He o6cYJK,lJ;aIOTCa cnoco6w annpOKCHM&I.\HH HHTerpaJta CTOJIK-
HoseHHH J(f, f}, KOTOphift IIPH nocne)Q'lO eY mlTerpepCJBaJlHH C
CYMMaTOpHbJMH HHOapHaHTaMH 06p aeTCH B HYJlh.
TaKHM 06Pa30M, p83HOCTHaJI cxeM8 (3.4) annpoKcHMHpyer AHcI>-
4>epe}l HanbHYIO 'I&cTb ypaBHeHHH BOJIh MaHa c TOqJfOCThlO O( x3)
eCJIH
! _ 1 - 4> = 0 ( 3.5 )
6 4 '
T.e. "PH tP = 1/3, H C O( x2) B HHbIX C1Iyq x. llpH 4> = -1 "PH-
eM K aH8JIory cxeMhI BTOporo nopBAJ(& TO'iHOCTH C Hanp8.BJ1eHHLIMH
pa3HOCTSIMU:
/j+l - Ii {+ 1{11.5/1- 21!-1 + 0.5/!-2
l:1t + 2 x - (3.6)
_ { 1{ll.511- 2/t + 0.5/1+2 = J(j,J).
ITpoHHTerpHpyeM ypaBReHHe (3.6) no CKOpoCTSM MOJIeKYJI C
CYMMaTopHblMH HHBapH8JlTaMH = (1, , {2/2). npH 3TOM Dony-
\fHM KHHeTyqecKYIO cxeMY, KOTOpylO 3anHweM B 06D1-eM BHAe
( +l - Q1 ) + G i +1/2 - G i - 1 / 2 = 0 (3.7)
t. x '
&
r)J;e Q = (p, pu, E) - BeKTOp 1'830 RRaMHqecKHX nepeMeHHLIX. B
CBOJO O'lepe , BeKTOp nOTOKa G onpe,qenHM C nOMOIUp1O CJle ylOIIJ.Hx
Bblpa}KeHHR:
G a + 1 / 2 = (Gt + !3i+l) + 1 : 4> (DGi+l/2 - DG;_1/2) +
I-t/J ( )
+ 4 DGtH/2 - DG i+ 3/2 ,
(3.8)
Gt = J w( + I I) li , Gi = f !li( - I I) li ,
% I {:f: I{I Gf+l - Gr
DGi+l/2 = If' 2 (fi+l - Ii) d{ = 2 ·
(3.9)
(3.10)
93. KHHeTHqecKHe cxeMhl IlOBhlilleHHoro nop1fAlC& 7'OrwOCTIl
87
'LnH HaXO>K.l\eHHH DOTOKOB G C I10MOIU.blO BLlpaJKeHH1I (3.9) TaM J
['A e 3'ro He06xo,l\HMO, sweCTO Ii 6y eM HCnOJIh308aTb nOKaJ1bHo-
MaKCSeJIJlOBCKYIO 4>YHKlJ)f1O JiM.
CJle)Q'eT 06paTHTb BHHMaHHe H& TO. liTO Hcrronb30B&HKe nHWb
nepoblx "IJJeHOB Gt H G i + 1 B npasoA 1:IacTH BbJp&>KeHH.H (3.8) npH-
Be eT K KHHeTHttecKH COrJl8COB8BHOA p83HOCTHOA cxeMe nepBoro no-
pSI,D.Ka TOqHOCTH.
B C8010 Oqepe b) QT06b1 nO,l\a.BHTh OC nJIBU)fH HCKYCCTBeHHoro
xapaKrrepa, B03HH.KaJ() He B KHHeTHqecKHX cxeM&x 6WIee BLiCOKoro
nopSI)J.Ka TOqHOCTH, BBe.n-eM Orp8JIH"IeHHH pa3HOCTH nOTOKOB DG%,
BXO)J.HUJ.HX B npaayJO acTb Bblpa>KeHHSI (3.8). )Lnsr 3ToA QeJlH HC-
IlOJIb3yeM MO <I>H:QHpOBauHYJO paaHOCTb nOTOKOS, npHMeHSlSI one-
l>aTOp MHHHMYMa MOA}'JIB (min mod), KOTOpblA ODpe enHM KaK
min mod[a, b] = ( sign(a) + sign(b)) min [Ial, Ibl] .
BBe eM B npasylO 'IaCTb (3.8) MO,lJ.H4>H pOBa.HHble p83HOCTH BeK-
TOpaS nOTOKOB (3.10):
- :I: · [ % :i: ]
Gi+1/2 = mID mod DGi+112' R "DG i - 1 / 2 I
-- :l: · [ :!: :t ]
DG i + 1/2 = mtnmod DG i + 1 / 2 , R. DG j + 3 / 2 '
r',l\e R - qHCJlOBOA <paKTOp:
(3.11 )
3-tfJ
O<R< I-tJl "
Hcnonb30BaHHe KHHeTH eCKHX cxeM 60nee BblCOKOro nopSl)J.K&
n03BOn.HeT IIOBblCHTb TOQHOCTb pac eTa no CpaBHeHHIO c npHMeHe-
llHeM cxeM nepBoro nops;:\Ka. B KaqeCTBe npHMepa Ha poco 13 npH-
Re,neHbl )J.8.HHble. B3S1Tble H3 pa60TbJ [29], xap8JCTepH3YIOw,He pacqeT
TeCTOBoR 3 a'IH c DONO JO KHHeTH'IeCKHX cxeM nepBOro H 60nee
BblCOKHX nopS.lI.KOB TO'IHOCTH.
PaccMoTpeHHble KHHeTHqecKHe cxeMbl nonY'IeHhI AJlH cnY'lasl
ID-reoMeTpHH. OAHaKO OHH J1erKO 0606 aIOTCB ):VIS DpBMoyronb-
HblX CeTOK B cnyqae 2D H 3D reOMeTpHA. HecKOJIbKO 60J1ee CJlO>K-
HO, XOTjI Ha 683e ToA )Ke caMoA HJ(eonOrHH, CTpOSITCB KCPC nOB&!-
weHHoro nO Ka TO'lHOCTH Ha HecTpYKTYPHpoBaHHblX CerKaX. B
88
rJ1. II. KJlHeTHf.JeCICH COrJI8COBlUIHlJe p&3HOCTHhle cxeMLI
1.0
0.1
x
o
o 0.1 0.2 0.3 0.4 0.6 0.8 0.7 0.8 ..8 1.Q
PUC. 13. HCfiOJJL3OBaHHe KHHerlNecJCmc cxeM nepsoro ( )J BTOporo (x) H Tpe-
Thero (0) Dop a TO'lHOCTH )J.JIJI M enHpOBaJlHa 3cv:ta 0 pacn e pa3phIBa.
CnnOWUaH KPHBaui - TO Hoe pemeHHe
CJIeAYlow.eM nYHKTe A8.HHOrO naparpa<pa, CJIe ySl pa60TaM [43. 44),
pacCMOTpHM HX nOCTpoeHHe Ha TpeYroJIbHblX CeTKax.
2. CxeMbl IIOBLIweHHoro nopH,l1)(a TOqHOCTH
H8 TpeyrartbHblX ceTKaX
OCHOBHaH H,nesr nOBbIweHHH nopSl,lI)<a allDpOKCHM H 3aKnlO-
IJa€TCH 8 TOM, qTO, COrJlacHo [45), ra30AHHaMH'IecKHe napaMeTpbl
Ha pacqeTHoR SlQeRKe 3 aJOTCH He KaK KycOtJHO-IIOCTOSlHHble, a KaK
KYCOlIHO-JlHlleaHble. O H"M H3 B8.)f(HhlX CBoAcTB pa3HOCTHbIX cxeM
nH ypaBHeHH nepeHoca HB ReTCA CBoAcTBO MOHOTOHHOCTH, HnH
YCJIOBHe TVD (Total Variation Diminishing), J1H HMHHeHHoro f'H-
nep60J1HQeCKOrO ypaBHeHHSI [46]. 9TO CBo1tCTBO n03BOJISIeT H36e>KaTb
HecpH3HQeCKKX oClJ)fJUlSIQHA pa3HOCTHOro peweHHB 8 oKpecrHocTH
93. KHHeTlftJecKJfe CJceMbl IlOBblweHHoro nop1fAK& TO"lBOCTH
89
pa3pblBHblX reqeHH:ft (YAapHblx BOJ1K).
J1B TOro, q'f()6bl p83HOCTHaH cxeMa nOBLlweHHoro rropSl K& an-
npoKCHMaJ..{HH 6blJla MOHOTOHHOA, OJJ>KHO BhlDOJIHSlTbCH YCJlOBHe
MHHH IYMa 3HaqeHHB rpaAHeHTa npn 3 aHHH nHHeAHoro pacnpe-
eJIeHHSI cl>Y}JKl\HH Ha pact.teTHoR SlQeAKe (47). B CBOJO Oqepe b, 3TO
JlKHeRHoe pacnpe.n.eneHHe M01KHO onpeAeJIHTb nH60 3KCTpaDOJISlIJ.H-
eft BBepx no nOToKY, JlH60 HHTepnOJISW)feA. ,.[(sa cnoco6a BbJ60pa
pa3HOCTflOro rp HeHTa B ID-CJIy ae HJIJlIOCTpHpyer pRC. 14. Ec-
JIlt 3KCTpan0J1H HOHHbla rp HelI'.r VQ1 H HHT€pIlOJlSl HOHHbla VQ}n
HMelOT O,LJ.HHaKOBblR 3HaK, TO H3 HHX Bhl6HpaeTCSI HaHMellbWHA 110
a6COJIlOTHO:A BMHqHHe. EcJIR >Ke 3Ha.KH pa3Hb1e (TOQKa nOKaJlh"OrQ
3KCTpeMYMa), TO H.X 3Hat.teHHB npHpaBHHBalOTCSI HynIO. TaxHM o6pa-
30M, 8 OKpeCTHOCTSIX pa3pblBOB H 60nbmHX rpa)I.HeHTOB npoHcxo HT
nOK8JlbHoe nepeKnlOtteHHe Ha cxeMY nepBoro nOpH Ka.
QH-I
t.1
i+lf2 i+1
PIlC. 14. HwrepnOJlH HOHH&lAJ 9KCTaDO-
JUl1J)IOHIIblA H cpe eB3BemeHIIYA rpCVJ,H-
eHTbI (lD.reoueTpHs)
PHC. 15. KOlITpollhHhle 06'beMbJ
A1UI cxeM DOBwweRRoro nOPiIAK&
TO'lHOCTH
B paCCMaTpHBaeMoM CJIyqae TPeyrOJlLlI1:aIx CeTOK npeAJIOJlO)KHM,
o HCKOMaSI cl>YHKQHH SlBJlSleTCH JIHHeaHOA B Ka)l(AoM TpeyrOJILHHKe
C BepmHHoA I (pRC. 15). 3HaQeHWI ra30 HHaMHqecKHX nepeueHHblX
Ha rpaHH e Me>KAY g'tJeRKaMH I H J onpeAeJIHM CJIe,ayIO M 06pa-
30M:
Q lJ = QI + (VQP) llJ , QtJ = QJ - (VQtJ) IlJ ·
(3.12)
3 eCb lr J - BeKTOp, H,lJj'W. HA H3 TO KH I B TOtJKY J.
90
Tn. II. KHHeTH'IecKIf COrJl8COBaHIlWe p&3HOCTH61e CXeMbl
rp HeHT V QP Ollpe,neJISieTCH KaK Cpe eB3BemeHHoe MeJKAY
HHTepnOJIH OHHblM rp&,l\HeHTOM VQ}O H 9KcrpanOJlS1 HOHHblN rpa-
HeHTOM VQ'1 ocpe HeHHbIM no KOHTponbHOMY o6'beuy B ,.o;aHHOM
Y3JIe (CM. pHC. 14):
vqj = PVQ1 + (l-,B)VQ}n,
(3.13 )
[',l\ e
VQ}n = QJ - QI,
(3.14)
" Q e - fVl VQdV -..!.. J V QdV =
v /- r dV -1/jL-,
JV/ 1_
VIJK
1 " -
= VI L-, VlJK(VQ)lJ ·
(3.15)
-
3,n.ecb VI - KOHTponbHbIA o&beM C eHTpoM B TO'lKe I, VI J K -
qaCTb KOHTponbHoro o&beMa VI, npHH8JVIe ero TaK>Ke TpeyrOJlb-
HHKY C BepmHHaMH I, J, K (poc. 15); CYMMHpOBaHHe Be eTCH no
BceM TaKHM TpeYroJILHHKaM. 38MeTHM, 'ITO B cnY'Iae Me,n:HaHHbJX
-
pac'leTHI>IX aqeeK nnow. VI J K pasHa o oA TpeTH WIO 8J\H Tpe-
yronbHHKa ViJK. Qepe3 (VQ)IJ 0600Ha'ieH rp eHT Q, onpe e.nH-
eMbJA Ha 3TOM TpeyrOJ1bHHKe:
(VQ)/J = [( ) IJ' ( ) rJ ·
(3.16)
B CBOlO oqepe . COCTa8J1jflO e rp HeHTa aJOTCB BblpaJKeHHSlMH
( fJ Q ) 1 I I ( 8 Q ) 1 f '
ox /J = 2VlJ K S Qn dS, Oy r J = 2V/J K Qn'JdS,
IJK SJJJ(
(3.17)
I'Ae n = (n , n ) - BHeWHSlJI HOpM8JJL K rp8HH e TpeyronbHHKa
SIlK.
,nnfl nocTpoeHHB MOHOTOHHOA CXeMbJ H& HeperYJlSlpHOA ceTKe
no 8HaJIOrHH c (48) onpeAenHM 3HaqeHHe rp ReHTa B nonY eJJoA
roQKe j*) Jle eA Ha upHMOD, npoxoNl eA epe3 TOttKH I H J.
93. KHHeTU'lecKHe cxeMbl UOBblUleHHoro nofJJJ,lVC8 TOIfHDC1.H 91
Onpe eJIHM 3Haq-eHHSI rp8.,lI.HeHTa B TOqKaX It H J- ) KOTOpbJe BBe eM
aHaJlOrH'IHblM 06Pa30W:
VQ/. = 2(VQ ) - (QJ - QJ),
VQJ. = 2('\7 ) - (QJ - Qr).
(3.18)
3Ha'ieHHB QiJ H QtJ H8. rpaHHI.(e Me>KAY YMg paCqeTHLJMH '1ea-
KaMH onpe enBIOTCH CJIe,DJ'IO J,( 06pa30M:
Q IJ = Qr + L(VQr., QJ - QI),
QjJ = QJ - L(VQJ.' QJ - QI).
(3.19)
3 eCb L(a, b) ecTb Q>YHK SI-OrpaHHqHTenL1 onpeAe.nHIO aJt nODe-
J(etlHe nOKaJl1JHblX rp HeHTOB. HCDOJIb30BMOCh Ba B a T8.kHX
<PYHKIJ.HR [45, 49]:
L(a, b) = 8(a, b) · min(Jal, Ibf),
21a( Ibl
L(a, b) = Sea, b) lal + Ibl '
l",l\e S(a, b) = (sign(a) + sign(b»)/2.
TaKHM 06pa30M) CHCTeMa YPaBHeHHH ,lI.1UI onpeAeneHHSI ra30.Q)l-
Ha.M ecKHX napaMeTpOB c nOBhlweHHhlM nopH KoM TO'IHOCTH 3&-
nHweTCSI B CJJeAYIO eM BHAe:
min mod:
(3.20 )
van Leer:
"+1 . t' +
= - VI L SIJ (FIJ (QIJ, QIJ) - D/J(QIJ, QtJ»' (3.21)
rAe SfJ - qaCTb rpaHHD;bJ KOHTpOnbHoro o&beMa (pec. 12), FIJ H
D I J - COOTBeTCTBeHHO , KOHBeKTHBHble H AHQ>Q>Y3HOHHhle COCTaB-
JllIlOIqHe n01'OKOB 1 onpe).\eJISleMbJe c nOMO 1O BLlpa>KeHKA (2.13)-
(2.20) .
3aMeTHM, liTO paCCMOTpeHHaJI annpoKCHM HH OTHOCHTC5I K
KJIacCY TaK H83b1BaeMhlX MUSCL (Monotonic Upstream Schemes
for Conservation Laws) cxeM [45].
Ha pllC. 16 npHBe.neHbI nons a.sneRHB (30 J1HHHA YPOBHSI), pac-
C'lHTaHHbJ€ no cxeuaM nepsoro H IIOBblllIeHHOro nops TOqHOCTH
92
TJI. II. KBneTHqeacH COrn3COBaHHble p83HDCTHble cxeW1:d
03
D25
0.2
0.15
0.1
o JJ!:»
0
a) 0
OJ
025
02
o J.!
0.1
OD!
0
6) 0
1)25
0.5
035
1
0.25
os
0.75
1
PHC. 16. Pacnpe.lleneHHe AaBJIeHIIB, pacc HTaHHOrO DO KCPC (a) H no KCPC c
Hcnonb30BaHHeW MUSCL-annpoKCHMaJ.J)UI (6) n& TpeYl'OJIbHoA ceTKe
({3 = 1/2). Xopomo BHAHO, 'Iro pactJeT no cxeMe, HcnOJlb3ylC' eA
MUSCL-annpoKcHMannlO, )J.aeT 60nee tieTKO BLJpa>KeHHYIO CTPYK-
TYPY y apHoA BMHbl no CpaBHeHHIO C KCPC rrepBoro nopSl,I\Ka.
IIpaa a, CJIe eT 38.MeTHTb) 'Iro p'OCTUt{b xopomero OnHC8.HHH I10J1.H
1'e'leHHSI MO>KHO H C nOMOwplO cxeM nepBoro nOpB.l(Ka TO HOCTHJ ec-
.nH HCnOJlb30S&Tb AJlSI AHCKpeTH3aU;HB 6onbllloe 'IHCJlO }'3JIOB. Ta.KHe
B03MO>KHOCTH npet(OCTaBJI.HeT npHMeHeHHe BbICOKOnpOH3BO HTeJlb-
HhlX MHororlpOlteCCOpHblX BbI'IHCJlHTMhHbIX CHCTeM.
9 4. KHBeTH'IeCKHe cxeMhI B pa3J10:>KeBHe
1.JeIlMeHa-9aCKora
B3rJIHHeM Ha KHHeTHqecKH COrJlaCOBaHHble cxeMbI ew.e c o,lJHoR
CTOpOHbI. BblHBHM aHaJIOrHIO Me)KAY' HHMH H XOpowo H3BecTH M B
KHHeTHt-ieCKOA TeopHH ra30B pa3JIO)KeHHeM qeuMeHa-9HcKora (r.l. I)
)I.JlSl npe,ACTaBJIeHHH O HOqacTH HOR Q:>YHKU.HH pacnpe eneHH . ;(nH
3Toro pacC 10TpHM BblBO)I. KCPC c nOMOW,b1O npo eAYpbI qeIIMeHa-
9HcKora. oIIHpalO eaCR B HaweM CJIyqae Ha KycOqHo-nocroSlHHoe
BHYTPH npocTpauCTBeHHo H eRKH np CTaaneHHe Q:>YHKQHH pac-
4. I('fHC'I'II'fecKJle CXeMbl H pa3JlO)l(eHHe tfenMeHa-9HclCora 93
npe eneHHjI [37].
BbIIIHweM ypasHeHHe BOnh MaHa 8 CJIe)lJ'lO eM BRAe:
+ div( f) = ;JU, J),
( 4.1)
r.ll.e £ - M8.1IblA napaMeTp. PemeHHe ypaaHeHHSI (4.1) MO)KeT 6b1Tb
npeACTaBJIeHO B CJIe YIO eM HHTerpanbHoM B e:
1 ,
f(x, , t + /},t) = /(XAtl (, t) + € [ JU, J)(x" (, t + r)dr, (4.2)
j).t > 0, x" = x,(x, ) = x + s , 0 < s < t.
fIpH tIanHcaHHH Bblpa>KeHHfI (4.2) He Y'lHThISaeTCg TOT WIH HHOA
3aKOH B3aHMOtr-eRCTBH 'facTHl.\ c rpaH ea 06JIacTH.
By.u-CM C"IHTaTb, qTO 8 HCClleAYeMoA: o6nacTH R BBe eHa CeTKa,
COCTOSI aR H3 BblnYKnblX MHororpaHHHKOB Vk, k > It C eHTpaMH
B TOqK8X Xk H rpa.HH aMH Sk. npHqeM
R C U Vk.
k
0603HaQHM CH tB0J10M Xk = X(V,,) xapaKTepHCTH'IecKYJO cl>YHK 1O
MHO)KeCTBa Vie. MMeeM:
Vk n V m = </;, k:F m,
Xt(X)Xm(X) = 6km,
EXt(x) = 1, x E R,
Ie
r e c5km - CHMBon KpoHeKepa.
BBet(eM 0600HaQeHHe
I(x. , t) = Ik«(, t), x E Vk ,
(4.3)
H 6y eM HCKaTL peweHHe ypaBHeHHH (4.2) B BH)J;e
l(x, , t) = L h;((, t)Xk(x),
k
npH x E Sic onpe,a:eJIeHbI
f+(x, ,t)= lim f(x+an,E,t),
Q-+O+
f-(x,E,t) = lim J(x + an,E,t),
0-.0-
x E U (VA: \ Sk)'
Ie
(4.4)
(0, Dk) > 0,
(n, Dk) < 0,
(4.5)
94
rJl. II. KlllfeTH'lecKH COrJlacoBaHHble pa::JHOCTHble c>:eMbI
r e n - HeKoTopblD e HHHqHhl:R oeKTop, Dk - BHemllSlB OTHOCH-
TeJIhHO VA: HOpMaJIb K nOBepxHocTH Sic B T01.{Ke x (pe6pa BhlOYKJIblX
MHororpaHHRKoB HCKnlOt.fHM H3 06JIacTH onpeAeneHHjI cPYHKlJ.H1( f))
(0, nk) - CKaJlS1pIIOe npoH3Be)J.eHBe BeKTOpOB D H nt. OTMeTHM 1aK-
>Ke, ({TO
!k((, t) = 1 Vi J f(x, , t) dx.
mes Ie
V,
AnllpoKCHMHPYSl HHTerpa.n 8 npaaoll 'iacTH (4.2) no cPopMyne
Tpaneu.wA, nOJlY"IHM cne,n;YlOrn.ee KHHeTH'fecKoe ypaBHeHKe, KOTopoe
6YAeT CJlY>KHTb OCHOBOa 8JIbHeRmHx paccY>K)J.eHHA,
f(x, ) t + t) = /(x_ t, E, t) +
+ [J{f,J)(XI ,t + .!It) + J{f,J)(X-Ah(,t)].
(4.6)
OCTaTO Hble qJIeHbl B (4.6) HMelOT nopSl OK O( t3)..
0603Ha'lHM 'Iepe3 J ) MaKCBeJL1IOBCKyro <l>YHK HJ() pacllpe eJIe-
HHSI c raaot\HHaMlRecKHMH napaMeTpaMH, flBJIHIODJ.HMHCSI MOMeHT"d.-
MH <l>YHK HH fk( ) t), T8K tfTO
p,ui:) - ftc) = 0, k > 1, 0 < p < 4, P p ! = J ""f , (4.7)
1/;" - cYMMaTopHblR HHBapeaHT,
(k)
1 M ( , t) = !M(/k((, t)).
Yq'reM B (4.6) npe)I.CT8.BneHHe (4.4). npH 3TOM nOJIY'IHM
L fk( , t + t) Xt(x) = L Jk( , t) Xk(X- ,) + (4.8)
k k
+ [ J(fkl Jk)(E, t + L\t)XA:(x) + J(fAa fAJ(E, t) XA;(X-At)] .
.,[(.nx TOro, ttT06b1 OCTaTO'IHh1e "IJIeHbl (4.6) 6hUIH uanw no CpaBHeWlIO C
OCTaJIbHWNH, He06xo o, 'IT06h1 t '" £. CMWCJI 3TOro orp&llH etlHB CT&HeT
6o.nee nOR"THnIM H3 M&TepH&110B rJl&8h1 ill, r,[\e KCPC 06cY &eTc. xa.K <pR3H-
-feClCas ).(o enb .ang onHCaHHSI TeqeHRA BS3KOro ra.3a.
f4. KHHeTH'lecICHe cxeMbl II pa3JJC»lCelUle QenMeHa-3HCKora 95
3.a.eCb HCDOJ1b30B8JlOCb COOTHoweHHe
J(f, J)(x, , t) = E J(flc, fk)( , t) Xk(X).
k
CneJ\YH crra apTHLJM paccY)J(AeHHSlM MaTeMaTHqecKoA TeopHH
ypasHeHHH BOJIL Mana H CBOA:CTS8M 4>YHK H pacnp eneHIDI, no-
JIO>KeHllbIM B OCHOBY Dpo e bl QenMeHa-3HCKora (rJl. I), BBeAeM
Q:>YHK HIO Fit;, TaKYIO, 'Iro
fle{(, t) = Jl:>{(, t) + Fk«(, t),
P,Ft = 0, k > 1, 0 p < 4.
Y'CfHTblB3.SI (4.9) H HCDOJIb3YSl 0603Ha'leHHe j,,(€, t+dt) = fl+1,
nonYQHM H3 (4.8)
(4.9)
L {[f;{tl(k)xk(X) - fltXtc(X-lU) - t(LkFi+1)Xk(X) -
k
- b.t(L k Fl)Xt(x-6£)] + t: [Fl+ 1 xk(x} - F1Xk(X-6t) - (4.10)
1 ( '+1 "+1 .. )] }
- 2 t J{FI ,Fi )XA:(x)+J(FI,Fl)xk(x-6£) =0.
3AeCb CHMBOJIOM L k 0603HaQeH mlHeapH30B&HHblA OTHOCHTeJlbHO
MaKCOennOBCKoR Q:>YH HH pacnpe e.neHHSI C ra30,lJ.HBaMH1IecKHMH
napaMeTp fH P p / onepaTOp CTOJIKHOBeHHB
Lie J = J(fJ:> , J).
(4.11 )
H3 COOTHomeHWI (4.10) uueeM CJIeAYJO ee npH6nH>KeHHe nep-
Bora nopS.D;Ka no c;
L [{ 1J:)j+1 - tLkFi+1} Xk(X) -
Ie
- {JD Ie ) - tLkF1} Xk(X- 6 £)] = o.
( 4.12)
cI>HKcHpyeM HeKOTOpylO HqeAKY VI c COCeAHMH \Ii H OCpe,l\HHM
ypaBHeHHe (4.12) no VI:
/{;/ - tLI Fl+1 = L (J )j + tLk F1) C:( t), (4.13)
Ie
rAe
C:( s) = 1 \1, j XI(X)Xt(X - s)dx, (4.14)
mes I
R
npeqeM C;(O) = dkl H npH nlO6oM <PHKCHpoSaHHOM k t$YHKU;JfSl
C7(Es) cpHHHTHa no E.
B cHny cBoAcTBa Ek Xk(X) = 1 HMeeM
LC;( S) = 1 \1, j X1dx = 1
k mes 1
R
( 4.15)
JI)I JIJ06oro .
C001'HOllleHHe (4.13)t yqUTblBasJ {4.15)1 MO>KHO nepenBC&Tb B
BH)l.e
ftt/ - tLI Fl+ 1 = 11M +
+ L (/1:)j - JiM + tLkFl) C;( t). (4.16)
Ie
Y'-IHTLIBaH CBoACTBO (4.9) <PYHK HH Fie. Ha OCHOBe BbIpa>KeHHH
(4.16) npH.D-eM K CJIe)JJ'lOm.eMY COOTHomeHHIO:
-+1 .
PpflM = Pp!lM +
+ LP, [u1: b - 11M + tLkF1)C;(( t)]. (4.17)
Ie
BbIpax<eHHe (4.17) npe,LI.CTaBJISIeT B 06 eR <popMe Knacc KCPC
H, KpoMe TOro, OHO HBJlSIeTCSI YCJlOBHeM pa3peWHMOCTH ypaBHeHHSI
(4.16) Ha HOBOM CJIoe no BpeueHu:
j j+t I j [/ (k); l ' ]
L[ Fj+l = 1M - 1M _" M - 1M + Lit: F/ Ck ( t t ) .
I 6.t L...J 8t Ie 1 \t
k
95. KSa3Hra30AlI1f8MHtleCK&R CHCTeua ypaBH8HHJt
97
CJle YeT OTMeTHTb, QTO BceB03MO>KHbJe BapHaHTbJ KCPC (4.17) pa3-
JlH'IaIOTCSI cnoco6auH arrnpoKCHM al\A BeJJHqHH C: ( .6.t).
B03Bpa.J.QaHcb K npol\eAYpe llOJ1rteHHSI: BhIpa>KeHHR (4.17), MO)K-
110 C)\eJlaTL BLlBO t '1TO paccMoTpeHHLle B 2 3TOA rnaBbl KCPC
coomeemcmeywm nepBoMY 4&.IIeHY pa3J1,O eHWI lJen.MeHa-9tic1'02a
aJU(, pa3HocmHoeo aHaA02a ypa6H.eHUJI BOAbqMaoo.
OnHCaHHaJI npoQe,aypa TaK)Ke ,lI.onycKaeT nOCTpoeHHe BapHaH-
TOB KCPC Ha OCHOBe HCn0Jlh30BaHHSI cneAYIOw.ux qJ]eHOB pa3no>Ke-
HHH no napaMeTpy c: YHK H fk( , t). Q,l\HaKO B paMKax JJ.8JiHoA
MOllorpa<pHH nop-06Kble cxeMbI 8 ,n8JIhHeAweM He paccMaTpHBaJOTCSI.
5. KBa3Hra30,L\HHaMI£IIecKaB CHCTeM8 YPaBHeHHA
B aHHOM naparpa<pe 6y eT paccMOTpeH BbIBO)\ KEa3Hra30,nH-
HaMH"IecKoA CHCTeMbl ypaaHeHIdt (KrY). 9Ta CHCTeMa caMblM rec-
HbiM 06Pa30M CB 3aHa C KHHeTHqeCKH CQrnacoBaHHbIMH cxeMaMH H
SlBRSleTCH o HoA H3 OpM HX <p<PepeHLI.HaJlbHOR 3anHCH. B CWty
60nee HarJJSlAHOrO MaTeMaTINeCKoro 0T06pa)KeHM Kry qaCTO HC-
UOJJb3yeTcH B 'KaqeCTSe OCHOBbJ BbI HCJ]HTeJJbHbIX anropHTMOB. Ha
ee 683e 6blJlH npOBe eHbJ OCHOBHhle HCCJIe OBa.HHSlJ pacKpLlBaro He
B03MO)J(}IYIO pOJIb KCPC H KfY KaK MO eJIH AJISI OIlHCaHRjI Te'ieHH.H
BSl3Koro ra3a (rn. III).
1. ,l{HljJljJepeHI{HaJlbHo-pa3HOCTHaH cPOpM8 ypaBHeHBll
6aJl&HCa ,IJ.JIH ljJYHK HJl pacnpe"QeJleHHR
BepHeMC.H BHOBh K COOTHomeHHIO 6anaHca )VI.SI <PYHK HH pac-
Dpe eJleHHSI) 3&I1HCa.HHOMY 8 06lU-ea opMe (2.1). PacCMOTpHM npa-
BYIO acrb (2.1)
Ie
DIS = 2 1 L ! I«(,nil}/ (JIM - J/M)du, (5.1)
mes 1. 1
1= S
il
KOTopasr OTBeTCTBeHH& 3a reHepaJJ.H1O ,lJ,HCCHoaTHBHblX qneHOB
KCPC.
J.Jl. 11. l\.JfHeTlI'leCKH COrJl8C08l1lfflble pa3HOC' Hble CXeMbl
I
Tp&HCcJ>OPMKpyeM BLlpa>KeHHe (5.1). ,n.,m 9TOro YUHO}KHM H
pa3 enHM Ka>KJJPljt 'IneH CYMMbI H& J ( , Oil) I = I iln t H 11,,1.
1 Ie / C2 J... j ; f '
DIS = L iln ' ''il iM - 1M dU J (5.2)
2 mes VI i=l S l ilnJ 1
it
rAe 11n - CKOpOCTb MOJIeKynhl B HanpasneHHH t nepnell,l1.HKYJ1S1pHOM
COOTBeTCTBYIO eH rpaHH eHTpaJIbHOa SqeAKH (pHC. 9), h.;t - pac-
CTOSlHHe Me y eHTpaMH SI'IeeK.
BeJlu'tHHa hi! /I iln I HMeeT <l>H3HQecKHft CMhlCJI speMeHH. 3a Ko-
TOpoe qacTH aJ o6n aIOIl(aSi CKOpOCThlO {iln, npoxo HT pacCToSlHHe,
coe HHHIO ee eHTpbJ HqeeK. C,n:enaeM O HO B3JKHOe Dp6).\nOJIO.>Ke-
HHe. By eM C'IHTaTb, t{T() 3TO BpeMSl He 3aBHCHT 8M OT a6COJIIOTHOR
BeJlH'IHHbI CKOpOCTK (,8M OT HanpaBJIeHHSI nOJIeTa MOJI€KYJIbI. To
eCTb cy eCTByeT xapaKTepHoe BpeMSI
T = htl /I ilnl,
(5.3)
3a KOTopoe MOJIeKYJIbi r83a nepeceK3JOT pac'leTHYIO SNeAK)'. no CpR
)J.ena, BBe,AeHHe BpeMeHH T aH&110I"'H"IHO BBe,lteHHIO opeMelfH Me)J( Y
CTOJIKHOBeHHSlMH acTH , onpeJJ;eJIaeMOMY COOTHoweHHe ..,. = lie
(CM. (1.21), rn. I). B aJlbHeAweM 3Ta aHaJlOrHH 6Y eT aKTHBHO HC-
nOJlb30S3TbCSl I1PH o6cY>K eHHH B03MO>KHOcreA Kry )Uta fJnHCaHHSI
Te'ieHHA BB3Koro raaa.
B C+\eJlaHHblX npeADOnOJKeHHSIX HCXO Hoe COOTHOllieHHe 68JIaHCa
(2.1) ,lVISi <l>YHK HH pacnp eneHHB 3aITHllieTCH B BH e
Jl+ 1 - JIM + 1 t / (EJ DiI) J1M + JIM du =
t mes VI i=l S 2
II
k j j
= 1 L / lln'r .fiM - h..M.. du.
2 mes VI . 1 hil
1= S
il
(5.4)
COOTHoweHHe (5.4) lIBnSleTCSI HHTerpanbHoit <I>oPMoA, H3 KOTO-
paR BnoC1I cTBHH 6YAeT nOJlyqeHa <P<I>epeHlJ)laJIhHO-pa3HOCTHa.H
KB83Hra30AHHaMH'IecKaSi CHCTeMa ypasHeHHR. nOSlBJIeHHe B np&BoA
95. KB831fr&3OAHHaMHQeCKaSl CHCTeMIJ, YP&BHeHHR
99
qacTH K83JI.paTa MOJIeKYJISlpHOA CKOpOCTH BMeCTO ee MOAYn.H 003-
BOnSleT nOJlY HTb 60nee npOCTbIe H y o6Hble ):VllI aJlhHeAmHX HC-
CJle OB3HHA (no cpaBHeHHJO c paaee pacCMOTpeHHhIMH sapHa.HTaMH
KCPC) Bblp8jKeHHB AJlSI HaxO)l( eHHSI ra30j\HHaMHqecKHX nap8.MeT-
pORe Ew:.e pa3 OTMeTHM, 'tfTO Tp8JfCcl>oPM g HCXO HOro BhIpa)KeHHR
(2.1) B (5.4) npoH30wna 6naro apSi BBe eHHlO cpe)J.Hero BpeMeHH T,
3a. Korropoe qacTH LI nepeceKaJOT paC'IeTHYIO B eRKY.
IIpe)J.noJlomHM, q-r() cPYHK HH / M H f lJIa60 MeHSUOTcg Ha pac-
CTOflHHH h. HBnHJOlll,HMCB XapaKTepHhlM HaMeTpoM cqeTHoR )1qeA-
KH, Ii HMelOT OCTaTO Hb(A nOpH)J.OK rn KOCTH. 3'00 ,n;aer B03MO)K-
HOCTb 3aMeHHTb (5.4) BbJpa)KeHHeM
f ( f:+1 iJM ) dV + / {nfM(r, )cJq = ! e T : roo, (5.5)
s s
I1{C S - nOBepXHOCTb, OrpaHHqHBalOmasr KOHTpOJIhHhlA o&beM, r -
TeKY 3.H TOqKa nosepXHOCTH, (dJ Jdn)fr - llpOH3BO,lUlag B TO'Ixe r
8 HanpasneHHH BHemHeA HOpMan" K nOSepXHOCTH S.
TIoBepxHOCTHblA HHTerpaJI B JIe-
BoA 'facTH (5.5) nen<o npe06p83yeT-
CSI B 06DeMHblA c nOMO blO <PoPMY-
nbl raycca-OCTporp CKOro
I nJM(r, )da = J div((fM)dV.
s V
nSl npe06pa3oBaHHSI nOBepXHOCTHo- PJJC. 17. K nony eRHJO AH«I>-
1'0 HHTerpana B npasoA qaCTH (5.5) <p e peHIJ.H8.II&Hol cI>oPI4hl AlIJI 6a-
Bocnonb3yeMcH npHeMOM, npe,lVIO>KeH- naacHoro yp&BHettHS
HbIM B MOHorpa4>HH B.C. Bn HMH-
pOBa [50] np" HccneAOBaHHH ca.MocOnpSl>KeHHOrO YPaBHeHBSI, onH-
CblB8JOw.ero nepeHoc 'IacTH .
npeo6p83yeM HHTerpan E no BblnyKnoA 06JIacTH t orpaHuqeuHoA
flOBepXHOCTblO S B B e nOBTopHoro KHTerpana
1 d 2 d j / l bd2df
E = - 1"-dV = dx(J. - T-dx .
VI dx dx( A dz(
( 5.6)
100
TJI. II. KHHeTHilecKH COrJlaCOBaHHble p831focTHhle CXeMbJ
3,n;ecb HHTerpan no HCXO HOA 06JJacTH fipe,nCTaBneH CHaqa.n:& no Ha-
npaBJleHMIO x (B onb HanpaBJIeHHSlIIOneTa MoneKYJIbI co CKOpOCTblO
), 3aTeM - no HanpaBneHHIO x .J. (nepneH,qIiKYJISlpHO Hanpa.sneHHIO
CKOpoCTH ).
BHYTpeHHHA HHTerpa.n B npaBoA tfacTH (5.6) MO}i(HO B3HTb TOQ-
HO
/ dx .l. {" d d eT d df dx{ = / ( {2T d d! r(b» ) dx{J. =
Jo. x{ x{ n I'(a)
- J dx 2T [ d f _ df ]
- .1. dn r(b) dn r(a) ,
PAe (df /dn)lr(b) - 3HaqeHHe npoH3BO,IJ.Hoft BAOJIb BHeWHeA HopMa-
JIM B TO IKe b rpaHH bl r (TOqKa Bblxo,n:a J1YQa C HanpanJleHHeM
H3 KOHTponbHoro 06'beMa), (df /dn)lr(a) - 3H8qeHHe npOH3BOAHOA
B)J.onb BHYTpe}IHeH HOpMaJIH B TOqKe a rp8.HHl{.b.1 r (TOt.JKa Bxo a
nyqa C Hanpa.BJIeHHeM B KOHTpoJIbHblH 06'beM).
MeHHH HanpaBJIeHHe nOJ1eTa MOJIeKYJILI B TOtfKe a Ha npoTHBO-
UOJIO)KHOe (- ), nepenHW€M npaBYlO qaCTb (5.7) BRAe
(5.7)
J dx 2r [ df - !! ] = J eT df duo (5.8)
dn r(b) dn r(a) dn r
s
TaKHM 06paaOM , npaBasi qaCTb BblpaJKeHHH (5.5), COBna,a.a.IO aSl
c npaBoA qaCTbIO (5.8), MO>Ke1' 6b1Tb Bblpa>KeHa 'Iepe3 o&beMHLIA
KH'rerpan 1: 83 (5.6). 9To aeT B03MO>KHOCTb npeo6pa300aTb Bblpa-
>KeHHe (5.5) K BH,n;y
J [ ( f/+1 JiM ) + div( fk) - £7" Lfi,] dV =.0, (5.9)
V,
rAe L .- OnepaTOp, 3 aBaeMbIH Bblpa>KeHHeM
LJ = ( ,grad!).
(5.10)
IIpe,LI.IIOJ1araH npOH3BQnbHOCTb otYbeMa VI B (5.9), C yqeTOM CJla6o-
ro H3MeHeHHjI f M H f Ha pacCTOSlHHH h) np eM K CJle m:eMY
5. KB83HI'a3Q4HHaMJPleclQ1R CHCreM& YpaBHeHHR
101
)l.KcpcpepeH1J)IaJlbHO-pa3HOCTHOMY ypaBHeHHlO, OnHCbIBaJO eMY 3BO-
J1lOI.(HIO <l>YHK HH pacnpeAeJJeHHB
j+1 j
11 - JIM + div ( Jj ) = !LrLP
t M 2 M.
,LJ:H<p<pepeHQHaJlbHO-pa3HOCTHoe ypaBHeHHe (5.11) 6bIJlO DOJJY-
-rreHO B pa60Te [24] HeCKOJIbKO HHbIM cnoco6oM J KOTOp.bla H3JIO>KHM
n03)J{e. 3TO ypaBHeHHe 9BJ1SleTCjf OCHOBOA ,({JIg nOJly eHHSI KB83Hra-
30,L\HHaMH"IeCKOA CHCTeMbi H no CBoeMY nOCTpoeHBIO 5IBJlHeTCB CJIe,a.-
CTBHeM nopo)K alO eA ero KCPC. IIpH ero BbIBO)J.e HCnOJlb300aJIHCb
Te >Ke CaMbJe OCHOBHble npe,lJ.nOJIO)KeHIDI) Ha aCROBe KOTOphlX ( l
aHHoA rnaBLI) 6blJIH nOJIyqeHbI KCPC:
(5.11 )
a) Q>YHKQHH pacnpeAeneHHH CJIa60 MeHSeTCSI BHYTP" gqeAKH C
xapaKTepHbIM pa3MepOM h;
6) nocne CTOJIKHOBeHHH MoneKYJI Ha MOMeHT opeMeHH t j + 1 cpYHK-
HH pacnpeAeJleHHH BROSh CTaHOBHTCSl MaKCBeJ1JIOBCKOR.
E)J)ftICTBeHHbIM OnOJlHHTeJlbHbIM npeAJ10nO)KeHHeM, npHBe lUHM K
60nee npoc-l'oR Hcp<l>epeHI..\HanbHo-pa3HOCTHo:R cJ>oPMe (5.11) 6a-
JIaHCHOrO COOTHomeHH , HBJIHeTCH sse)J.eHHe cpe)J)lero BpeMeHH r t
3a KOTopoe 't{acTH bl nepeceKaIOT s)'ielKy. BnpOqeM, TaKoe npe)I.IIo-
JIO>KeHHe SlBJ1HeTCSI BDonHe eCTeCTSeHHblM NISI ra30B) r e HapSl)Jy C
,L1.nHI-toA cBo6o Horo npo6era l paccMaTpHB8JOT CBjf3aHHOe C HHM xa-
paKTeplioe BpeNS Me>K,qy CTOnKHOBeHHgMH ". = lie (rJI. I).
3 a M e q aH H e 5.1. ,lI.HcI>4>epeHl\HaJIbHaB qacTb ypaBHeHH.SI (5.11)
KMeeT .D.Ba cy ecTBe"HbiX OTJlH'IH'I OT ypasHeHH1I BOJ1b MaHa. Bo-nepBwx.
OTCYTCTBYeT HHTerp8JI CTWlKHOBeHWI. Bo-BTOPblX t B np8.8oA 'IacTH :noro
ypaBHeHHH npHcYTcToyerr ,lJ)I«P4>epeH aJILHbIR onepaTop BTOporo nop -
1<3. 9TOT oneparop LT LfJ., NO;l\eJIHPYeT npou.ecc KTHBHOro CrJJa.>KH-
B8HHJI MaxCBemIOBCKOA <l>YHKlJ.HH pacnp eJ]eHIUI Ha HqeHKe c xapaKTep-
HblM pa3MepoM h, C KOTOpblM BpeMjI T CBSl3aHO npOCTbIM COOTHOWeHHel4
(5.3). B CBOIO o epe,.ll,b) CTO.RKHOBeHHe MOJIeKyn Mo,nenHpyeTCJI MrHOBeRHoft
MaKCBeJ1nH3aU)1eA Ha NoueRT BpeMeHH t :::: t,;+I.
2. EIl.{e Ot(HH cnoc06 no.nyqeBBH YPaBHeHHH 6a.n8Hca
HapS1 c paCCMOTpeHHblM B n.1 ,l\8.HHOro naparpa4>a BbIBO OM
ypaBHeHH5I 6anaHca (5.11) pacCMOTpHM e e 0AHH cnoc06 ero nony-
qeHH [24). EcTecTseHHo, q-ro 3TOT BbIBO onHpaeTCSI Ha Te >Ke caMbre
nOCTYJlaTbJ, KOTOpbIe 6bInH nOJlO)KeHbI B OCHOBY BblBOAa ypaoHeHHH
(5.11).
rrpe nono)J(HMp liTO npH t = t J cl>YHKU)[SI p8CIIpe eneHH.9 onpe-
enSieTCH ;J.OCTaTOqJfO rn8.AKoA nOKaJlbHo-MaKCBennOBCKoR cl>yHKl\H-
eft fM(X,E,t). llpeADOIlO,)l(HM T8KJlCe, q'I'() B TeqeHHe BpeMeHH T =
t j + 1 - t j qacTHl.\hl raa& COBepnralOT 6eCCTOJIKHOBHTenbHhIA pa3neT,
a D MOMeHT speMeHH t = t j + 1 npoHCXO T MrHOBeHHaJI MaKCBenJIH-
3 H.H f'83a.
t;+l
f (x, , t j + 1 )
t j
I JT
JM(X- T, , t j )
PHC. 18. K BWBOAY YpaBHeHHJI 6anaaca <l»yHKIUIH pacnpe.o.eneHH2
3HaqeHHe cl>YHK H pacnpeAeneHHft Ha HOBOM cnoo no BpeMeHH
onpe eJIHM c nOMo hlO COOTHOmeHH}1
-+1 ·
f(x, ,tJ ) = f M(X - Tt t tJ),.
(5.12)
KOTOpoe HBJIHeTC1I cneAcTBHeM CBo6o HOro p33nera ra3a B TetIeHHe
BpeMeHH T (pac. 18).
P83J10)KHM npaBYIO Q8CTb Bblp8,)KeHHSI (5.12) B PH)); TeAnopa no
npoCTpaacTBeHHblM nepeMeHHblM C TO'lHOCTblO AO 'lneHOB TpeTberO
95. KS83Hr&30AHHBWHtleCKaSl CHCTeM& yp8BHeHHR
103
nopSl Ka MaJlOCTH O(U 'T]3):
'+1 · af
I(x, , tJ ) = fM(X, (, t3) - 'T{j 8x' +
. 1
1 2 8 2 IIi 3
+ 2 {i kT {J (J + O((l{lr) ).
Xi XI:
flpe06pa3yeM Bblpa>KeHHe (5.13) K CJle eMY BHAY:
f(x, ,tj+l) - IM(x,(,t i ) +div«(fk> = !: 2 {i {k ':{IA . (5.14)
7 V i Xk
YpaoHeHHe (5.14) nOXO>Ke Ha ):\H<P4>epeH aJIbHO-p83HOCTHoe
ypaBHeHHe (5.11), onHCblBaJOIQee H3MeHeHHe <t>YHK H pacnpe Me-
HHB "PH nepexo e OT MOMeHTa BpeMeHH t i K MOMeHTY SpeMeHH t i + 1 .
YT01.lHHM CBSl3b Me>K.n}' 3THMH ypaBHeHHSlMH.
Pa3JIO>KeHHe (5.13) B no MaJlOMY napaMeTPY, HMeJ() eMY
pa3MepellocTb AJlHHbJ, HeSlBHO npeAn on araeT, liTO 4>YHK g pacnpe-
eJleHHSI CJIa60 MeHSleTCB Ha STOM pacCTOHHHH. Ec.nH T eCTb xap8.k-
repHoe BpeMSI, 3a KOTopoe qaCTH 1 nepeceKaJOT BqeA:KY, a I I - HX
xapaKTepHaJ1 CKOpocTb, TO B KallecTBe I IT I Ka.K H npH BbJBO e YPaB-
HeHH.SJ (5.11), BhlCTynHT pa3Mep SlQeHKH h. TaKHM 06p83ow, speMR
T, CTOs ee B npaaoA qacrH (5.14), TeCHO CBSI3aHO C xapaKTepHblM
pa3MepoM h., Ha KOTOpOM rapaHTHPYeTCSI Crn8JKHB&HHe pemeHHA.
OHO MO)Ke'!' OTJIaqSThcsr OT T, CTOg ero B nesoA 'IacTH (5.14), onpe-
enSlIO ero H3MeHeHHe ra30,qHHaMHtteCKHX napaMeTPOB no BpeMeHH
H 3aBHCSl ero npH 'iHCJIeHHOA pe8JIH38.l\HH B TOM C1Ie OT ycno-
BHSI YC'I'OH'iHBOCTH HCnOJIb3yeMbIx p83HOCTHblX cxeM. Y'iHTblBaSI 3TO
11 3aMeHSISI B neBoA qaCTH (5.14) T Ha t, npH,D.eM K paHee DOJIY-
qeHHOMY ypaBHeHHIO (5.11), OnHC&IB8JO eMY H3MeHeHHe <PYHKJJ:HH
pacnpe eJIeHHH .
(5.13)
3. KBa3Jfr830 HHaMHqecK&H CHCTeMa YP8lJHeBJfli
,lLnS1 roro, QTo6bl nonYQHTb ypasHeHHR, onHCblBalO He H3Me-
HetlHe ra30 HHaMHqecKHX napaMeTpOB, npoHHTerpupyeN ypaBHeHHe
(5.11) C cYMMaTopHblMH HHBapHaHTaMH cp(E) no CKOpocTSIM MOJIe-
KYJ1. IIpH 3TOM Ta.K }Ke, KaK H npe fiOJlyqeHHH KCPC. 6y eM CItR-
. 1
TaTb) qTO I fi+lcp( )dE = f fJ.: cp( ) .
IlK
1 JI. 11. J\.HHeTH'tJeCKH COrJIacoSaHlfbie p&3HOCTlfbIe CXeMM
BbInHl1Je 1 nOnY'IeHHYIO TaKHM 06pa30M CHCTeMY ypasHeHHft
Cpa.."3Y B npOCTpaflCTBeHHo-TpexMepHoM CJlY'Iae.
p1+1 - pi 8 a 8
t + ax (pu) + By ( (TV ) + & (pw) =
8 ( T {} 2 ) a ( " a )
= - --(pu + p) + - --(pu1J) +
ax 2 ax ox 2 8y
a ( T 8 ) f) ( T {) )
+ - --(puv) + - --(puw) +
ay 2 ax ax 2 oz
a ( T a ) a ( T a 2 )
+ - --(puw) + - --(pv +p) +
OZ 2 ax {}y 2 8y
a ( T a ) a ( T a )
+ - --(pvw) + - --(pvw) +
oy 2 oz OZ 2 8y
{} ( T lJ 2 )
+ iJz 2 az (pw + p) ,
(5.15)
( ).i+l - (pu)j 8 a a
pu. Ilt + ih(pu 2 + p) + iJy (puv) + a;(puw) =
8 ( T 8 3 ) {) ( 7" {} 2 )
= - - - (pu + 3pu) + - - - (pu v + pv) +
ax 2 ax ax 2 8y
a ( T a 2 ) a ( T a 2 )
+ - - - (pu v + pv) + - - - (pu w + pw) +
oy 2 ax ax 2 8%
a ( T () 2 ) 8 ( T () 2 )
+ - - - (pu w + pw) + - - - (puv + pu) +
& 2 8y 2ay
a ( T 8 ) {J ( 7 a )
+ - --(puvw) + - --(puvw) +
{)y 2 8z 8z 2 Oy
a ( 7 a 2 )
+ - --(puw + pu) ,
8z 2 8z
(5.16)
g5. KBa3Hra3QAHHaMHflecKaSl CJlCTeM8 YP3BHeH}fjf
105
(pv)j+l - (pv)j a lJ a
6t + ax (puv) + ay(pv 2 + p) + a;(pvw) =
a ( T a 2 ) a ( T {) 2 )
= - --(pu tJ + pv) + - --(pv u + pu) +
ax 2 ax ax 2 Oy
a ( 1 a 2 ) a ( 7 8 )
+ - --(pv u + pu) + - --(pu11W) +
ay 2 ax ox 2 8z
a ( T a ) fJ ( " 8 3 )
+ - --(puvw) + - --(pv + 3pv) +
8z 2 ax By 2 8y
a ( T a 2 ) a ( T () 2 )
+ - - - (pv w + pw) + - - - (pv w + pw) +
2& & 2
a ( T a 2 )
+ 8z 2 oz (pw 11 + ptJ) ,
( )j+ 1 - (pw)j a a a
pw t + OX (puw) + ay(pvw) + 8z (pw2 + p) =
lJ ( T () 2 ) a ( T a )
= - --(pu w+pw) + - --(puvw) +
OX 2 ax ox 2 By
a ( T a ) a ( T a 2 )
+ - --(puvw) + - --(pw u +pu) +
By 2 ax ax 2 oz
a ( 1"" {} 2 ) {} ( ' a 2 )
+ - - - (pw u + pu) + - - - (pv w + pw) +
& 2 2
a ( 7 a 2 ) {} ( 7" a 2 )
+ - --(pw tJ + pv) + - --(pw tJ + pv) +
8y 2 oz oz 2 8y
8 ( T 8 3 )
+ oz 2 8z (pw + 3pw) ,
(5.17)
(5.18)
100
IJJ. JJ. llJllleTRtfecKH corJlactJB&IIlIWe pa3HOCTflble CXeMhI
Ej+l - 19 8 8 a
Ilt + 0; [u(E + p)) + ay[t1(E + p)] + 8z [weE + p)) =
Or8 [ 2 ] OTO [ ]
= - - - u (E + 2p) + - - - uv( E + p) +
AX 2 ox ax 2 8y
8T8 [ ] OTO [ ]
+ - - - uv(E+p) + - - - uw(E+p) +
&y 2 ax ax 2 8z
8ro [ ] 8rO [ 2 ]
+ - - - uw(E+p) + - - - v (E+ 2p) +
8z 2 8x iJy 2 8y
OTO [ ] 8TO [ ]
+ - - - vw(E + p) + - - - tlw(E + p) +
av 2 8z 8% 2 Oy
{} T a [ 2 ] lJ T {} [ p ]
+ - - - w (E + 2p) + - - - -(E + p) +
oz 2 oz ax 2 ox p
{} T {} [ p ] a T fJ [ p ]
+ - - - -(E + p) + - - - -(E + p) ,
8y 2 &y p OZ 2 oz p
(5.19)
u 2 + tJ2 + w 2
P = p(T,p), E = p 2 + pt(T,p). (5.20)
3,a.ecb W - CKOpOCTb ra3a 8 HanpaBJ1eHHH OCR 0 Z) p(T, p) If e (T, p)
- 3 aHHbJe ypaBHeHHH COCTOSlHH.R.
CHCTe la YPaBHeHHA (5.15)-(5.20) ITOJ1ytfHna Ha3BaHHe K6a3uza-
300uHaMu",ec7Cou cucmeMW ypa8HeHtJu (24] (KfY). 06paTHM BHHMa-
HHe H8 HeKOTOpble OC06eHHOCTH Kf'Y-cHCTeMbi.
npOCTpaHCTBeHHble H<p<pepeH aJ1LHble 'IneHhI) KaK KOHBeK-
THBHbIe, TaK H )J.HCCHOaTHBHhJe. BbI'ifUcnSlIOTCH SlBHblM o6paaoM Ha
ope..,leHHoM CJJoe t = t j . no CyTH Aena B <p<>pMynHpOBKe KB83Hra-
30AHHaMHQeCKOR CHCTeMbl CO)J.ep}l(HTCSI H KOHKpeTHbiA cnoco6 onpe-
e.neHHS ra.3OAHHaMHtleCKHX napaMeTpoB npH H3MeHeHHH BpeMeHH
Ha OTpe3Ke t E (t j , t j + 1 ). TaKaS! <l>opMa 3&nHCH Benocpe,nCToeHHO
BblTeKaeT H3 q,H3INeCKoA: MO eJlH) nOnO)KeHHOA B OCHOBY KCPC H
Kry -CHCTeMbl.
Bo Bcex ypaaHeHBRX (5.15)-(5.20) B DpaBhlX 'I acTS X npHCYT-
CTBYIOT CHJlbHO HenHHeAHhie no CKOpOCTH )l;HCCHII&THBHbJe qneHLI.
Ha.npHMep, B ypaBHeHHSlX (5.16)-(5.18), onHChlBaJOI1J.HX H3!.feHeHHe
j5. KBa3Hra30AHll&Mll&fec«aR CHCTeMa YPaBHellHR
107
HMOYJJbCa, HCCHnaTHBHbJe t{lIeHId 3aBHCHT OT CKOpocTH B TpeTheA
CTeneHH, a B ypaaHeHHH 3HeprHH (5.20) OT CKOpoCTH B 1JeTBepToR
cTeneHH.. ITOftBJIeHHe 3THX HCCHI1aTHBHbIX qneHOB CB.H3aHO C TeM,
qTO no cnoeMY noc'rpoeHHIO Kr'Y onHpaeTai Ha npe nonO}f(eHHe 0
CJla6o 1 H3Me}leHHH cPYHK HH pacnp e..rIeHHH H8 paccrOHHHH h, CBB-
3aHHbIM C xapaKTepHblM BpeMeHeM 'T. B KCPC nOSlBJIeRHe TaKHX
ttneHOB Bb13BaHO KYCOtJHo-nOCTOjlHHhlM npe CTa.BJIeHHeM CPYHK:QHH
pacnpe)J.eneHWI B cqeTHoR }I"IeHKe.
B Kry -CHCTeMe SlBHblM 06pa3oM npHCyTCTBYIOT a MacIDTa6a
speMeHH. IIepBbJA - ra30 HHaMHqecKHA MacwTa6, KOTOphIA CBH3a.H
co CKOpOCTblO H3MeReHHB M8.KpoCKOllHtJeCKHX nap&MeTpOB, - o e-
AeJlSleT BbI60p illara no BpeMeHH t1t. BTOpoA) T, - KHHeTH'fecKHH
MacwTa6, onpeAeJIHeMbI:A Cpe)l:HHM opeMeHeM DpoJIeTa qaCTHqbl qe-
pro o6'be t c xapaKTepHbIM ,lJ)IaMeTpoM h. BnpOqeM, BpeMeHa 6.t H
r He SIBJlDIOTCSI He3aOHCHMblMH BeJ1HttHHaMH. HanpHMep, H3 SlBHo-
ro xapaK'repa cHcreMbi (5.15)-(5.20) BblTeKaeT OrpaHHQelJHe Ha mar
t, CBst3aHHOe c yCJJOBHeM YCToAq}fBOCTH. KpoMe TOro, H3 caMOro
nOCTpoeHHH KBa3Hf830 HHaMHqeCKOA CHCTeMbI BbJTeKaeT JlHHeAHoe
H3MeHeHHe ra3otIrHHaMH'tJecKHX napaweTpOB 3a BpeMSI 6eccronKHo-
UH'feJlbHoro P eTa.. 9TO CBoAcTBO no epKHBa.eTC9 p&3HOCTHLlM
XapaKTepOM H3MeHeHH no BpeMeHH BeJlHqHH p, pu, E.
4. Pa3HOCTHaH annpOKCJlMaIJ.HB
KOpoTKO 06Cyt\HM aeKOTopbIe BOnpOCbl, CBSI3aHHb1e C 1.fHCJleH-
HhlM perneHHeM CHCTeMbI (5.15)-(5.20).
llpOCTpaHCTBeHHbIe npoH3BOAHhIe B KB83Hra.30,l\HHaMB'lfeCKOa CH-
creMe annpOKCHMHPYIOTCH H3 npHMOyronbHoA ceTKe 06bI'IHblM 06-
pa30M. KOHBeKTHBHhle qneHbl, CTOB He B neBoA qacTH YP&BHeHHB,
- C I10MOUJ,b1O u,eHTpa.nbHblX pa3HocreA 4;i' .n;HccHnaTHBHhle 'lJIeHbI
B npaSbIX qaCTHX CHCTeMbI 3aMeHHIOTC.H p83HOCTBMH B a
(r(ax,))z, '
nH60
(1" (4 J) k' ilk.
· CHJIbHaJI HenHHeAHOCTL IS DpaBbiX qacTSX KB83H r " aMHqecl(oA CHCTeMLI
Ae.rraeT aKTYaJIbHbIl4 06cy>KlteHHe npo6neNY ee HHBapHaHTHOCTH OTHOCHTe.llbHO
KJIaCCH'IOCKOrO npeOOp830BaJlHS rMBJIeJl. STOT BOrtp()C 6y eT paccKOTpeH B CJ1e-
JJj'IOIQeA rnaBe.
ItRS
J:n. 11. l\HHeTXqecKH COrJI8COBaHHble p&3HOCTHlJe CXeJ.fhl
9TH cxeMbI HeCKORbKO npow.e, qeu HCXo,nHble annpOI<CHMa "H
KCPC. OJl.HaKO CJle.LU'eT OTMeTHTb, pe3YJ1bTaTbl ttHCJIeHHbIX pac-
'feTOS 3 aq HHaMm<H eanbHoro raaa, nonyqeHHble c nOMO blO
Kry H KCPC, npaKTHqecKH COBn aJOT. BMecTe c reM nOAqepK-
HeM, TO 60JIbmaH Harn HOCTb KBa3Hra30AHHaMHqecKO:R CHCTeMbJ
eJJaeT ee y o6HbJM 06'beKToM K8K JVlH "IHCJleHHOA peanH3a HHt Tax
H AJIS1 TeOpeTH'-leCKHX HCCJ1e,n.OB8JtHA.
KBa3Hra30AH:naMHQecKYIO pa3HOCTHYIO armpoKCHMaII.mo MO)l(HO
Henocpe.L(CTBeHHO nOJlyqHTb H3 pa3HOCTHOH CXeMbJ (1.11) .LUISI YPaB-
HeHHH BOJIbl1.MaHa. TaK >Ke KaK H "PH BbJBO e, OItHCa.HHOM B n. 1
8HHOrO naparpa<Pa, B ypasHeHHH (1.11) 3aMeHHM qneH
Axl 1 11+. - 21! + 1/-1
2 x2
Ax e /1+1 - 2Jl + /L.
a8 m-'2 Ax2
B AaJIbHeArneM 1 BBO xapaKTepHOe BpeMH T, 3a KOTopoe qaCTH a
npOJleTaeT pacqeTHYIO SMeAKYt 38.MeHHM (1.11) IIa cne,n.ylOIl\Y1O pa3-
HOCTHYIO annpoKcHMa 1O KHHeTHQeCKoro ypaoHeHHH:
1 )+1 I i j J' - I j t:2 I i -2 f J + f j
i - i + t' i+l i-I = T i+l i i-I + J (f I) ( 5 21 )
l:1t 26.x 2 x2 ' · ·
¥fHTerpHPY.H Bblpa>KeHHe (5.21) no CKOpOCTSIM MOJIeKYJI C CYM-
MaTopHblMH HHBapHaHTaMH H HcnOJlL3YSl ,lVIH HaxO>K,[J.eHHR MOMeH-
TOB cl>YHK HH pacnpe eneHH.s npeAnonO>KeHHe 0 ee MaKCBenJIOBCKOM
xapaKTepe Ha MOMeHT speMeHR t = t;, npH eM K C1Ie,n;YIO ea CH-
CTeMe paaHocTHbIx ypaoHemdt (24), KOTOpylO BblIlHmeM Ha npHMepe
npOCTpaHCTBeHHO-OAHOMepRoro cnY\f8.H:
+1 - pi ( 7 ( 2 ) )
At + (pu) = 2 pu + p % :z:'
( )j+l _ (pu)J' ( T )
rm At + (pu2 + P ); = 2 (pu3 + 3puh: :z:'
Ej + I - EJ ( T )
t + ('U(E + p» = 2 (u 2 (E + 2p»)% :I: +
+ ( ; ( : (E+P»)f)%.
(5.22)
(5.23)
(5.24)
5. KBa3Hra3OJIHHaMHCfecK&H CHCTeAl3 ypaBHeHHR
109
).bI<pct>epeH HanbHoe no npOCTpaHCTBY npH6nH>KeHHe NISI CH-
CTeMbl (5.22)-(5.24) COBn aeT C BapHaHTOM CHCTeMbI (5.15)-(5.19)
)I.JIB npOCTpaHCTBeHHO-OAHOMepHoro cnY'IaJI.
3 a M e q a H He 5.2. KaK JlerKO 3aMeTHTh, B npocrpaaCTBeUHo-)(8Y-
MepnoM H TpexMepHoM cnyqamc KB83Hf830)J.HHaMHQecKaJI annpOKcHMau,HS,
nonyqeUH&a Henocpe;J.CTBeHHO 83 p33HOCTHOro aHaJIora ypanUeHHJ1 BOnbl(-
MaHa H3 npsMoyronbHLIX CeTKaX, 6YA OTJJHqalfbCJI OT p83HOCTHOA ao-
npoKCHMaI.l)fH CHCTeMbl (5.15)-(5.19). B nOCJ(e Hea 6YAYT npHcYTCTBOBaTb
q.neHbl, COO'l'BeTCTBYIOIlUIe CMernaHHblM npOH3BO,IJ)1blM no npocTpaHCTBY.
CMelIIaHHble npOH3BO,lJ)fble onH-
CbIBalOT 8JIHgHHe ;J)IaroHaJIbHbfX g-qe-
eK (112M, 114M, f 16M 1 f ISM Ha pHC. 19)
Ha pMHpoBaHHe Q>YHK HH pacnpe-
eJ1eHHft B eHTpanb"oA B'JeAKe. KaK
8H HO H3 pac. 19. BnH.SIHHe ;J.Haro-
HaJlbHblX jJqeeK 6y,a.eT 3aMeTHO, ec-
JlH )I)lHHa cBo60 oro npo6era I 6y-
eT COnOCTaBHMa C JIHlleAHblM pa3Me-
poM SI'Ie:etKH ax. B npoTMBHOM cnyqae,
'-{TO H 1I0)J;I'Bep>K-naeTCSI 8 b('{HCJIRTeJIb-
HblM 3KCnepHMeHTOM, BJlWI!me CMe- PHC. 19. K on eJJeBHIO 30HhI BJIR-
waHHbIX npoH3BQAHbVC ua pe3YJlhTa- JDlJUI ,D.HaroHaJlbHblX gqeeK Ha 6a-
TbI pac eTOB 6y eT Hecym.eCTBeHHbrM. nauc «J>YHKLUlH pacnpe,.QeneHHB
3TOT cnyqaif xopowo COO'I'BeTCToyeT 38J1.8'1aM eaJlbHOrO raa&, B KOTO-
pbIX, teaK npaBHJlO, 6.x (D, COOTB eTCTBe HHO) T) BhroHpaJOTCH H3 YCJlOBH.9
I < A2:.
-/:
UJI
- .t;IM
. "'"
-J:v,
-i.
'I71iI
-//111
-1
'14M
-
-I..
'14M
3aMe-qaHHe 5.3. 06paTHw BBHMaHHe R& ypaBHeJlHe (5.20) KB&-
3Hr&"JOAHHaMH'IeCKOH CHCTeMbi (5.15)-(5.20). 3anHcb ypaBHeHHH COCTOB-
HHJI He B B e, CJle)]j'IOID.ew H3 Ma.KCBennOBCKoA 4>YHKII,HH pacnpe,neJleHHR
1) = pRT, a 8 60J1ee 06w.eA 4>oPMe He BBJljleTCJI <PoPMaJ1bHblM axTON.
KaK 6ytJ;eT BH)JJIO H3 paccMaTpHBaeMoro B cneA.)llOlQeA rJIaBe cPH3HQeCKo-
('0 cMbIcna KCPC, npHBJl3Ka nenocpeACTBeHHO K MaKCBennOBCKOMY paBHO-
BeCHOMY onHcaHHIO 4>ym<nHH paCn eneHJIj( Be muI eTCH 06J13aTeJILHoA.
3aMeqaHHe 5.4. B )J.8HHoA rJIaBe, nOCB eHHoH BblB KCPC H
K883Hra.30AHHaMH'IecKoA CHCTeMhl, H3 coo6paJKeHHH KOMn&KTHOCTB H3JIO-
>KeHHH pe3ynbTaTOB He paccM8TpHBancs cnyqaH BJlHBHHH nons BHeWHHX
CM)]. O,ll.HaKO nOJlYq}!Tb CHCTeMY KB83Hra30,ll.HHaMH'IecKHX ypaBHeHHA He
COCTaBnS1eT Tpy,a.a, cPaKTH'IeCKH nOBTOpJll1 D-enO'IKY paccy eHHA, npH-
HeJUllHX K Bblpa>KeHHJlM (5.12)-(5.14).
110
rJI. II. KHHeTH'leCICH COrJlaCOBa.JfHble pa:3HOCTHbCe CXeMbl
B caMOM ene, H3 npe,lUlOJlO>KeHWI 6ecCTOJIKHOBHTenbHOro p83JIeTa
COH3b (5.12) 3Ha eHHA <PYHKJJ)III pacnpeAeneHHJI npH t == t j + 1 H t = t; C
Y'leT01vt nOJIlI 8HeWHHX CHn F npHMeT BH)I.
J(x,(., t-i+ 1 ) = fM(X - (-r, - Fr, t i ).
P33.1IaI'a$! 3TO Bblp&)l(eHHe B PBA TeAJlopa C TO'fHOCThlO JJ.O 'IJIeHOB 1peTbero
nops,l1Ka lrlaJIOCTH, npHAeM K BbIp&>KeHHJO, aHaJlOrH'fHOMY (5.14):
'+1 . 8 j
f(x, ,tJ ); fM(X, ,tJ) + div((/lt) + Fi 7 =
1 () af 1 a 8/1.t 1 8 8/1.,
= 2 f..t {JXi f..k T {JXIe + 2 f..i {JXi F" (JE..lc + 2 Fi a i FItT a " ·
(5.25)
MHTerpHpyg (5.25) C cYMMaTOpHbIMH HHBapHaHTaMH, npH):t.eN K CH-
creMe K8a3Hra30AHHaMH'IecKHX ypaoHeHu:A, aHaJlorHqnoA (5.15)-(5.20)
npH yqeTe nW111 BHeWHHX CHn. He npHBO)J.ft 9TY CHCTer.ty 88HA)' ee rpo-
M03,[I.KOCTH) OTMeTHM, tJTO yqeT qJJeHOB F i o/M/8E.iJ CTO W.HX B neBOA qa-
CTH (5.25), npHoetI.eT K nOBBneHHJO B TOttHOCTH TaKHX .>Ke QJleHOB, Ka.K H B
ypaBHeHHHX 3Hnepa., onHCbIB3101QHX H3MeHeHHe Hwnynbca H 3I1eprHH raJa
3a CtteT non1l BHeWIIHX CHn.
3 a Me 'f a H He 5.5. OCTaTOqHO Bar IILIA: BH)1 KD83Hr83OAHHaMH-
qecKoR CHCTeMbl n0380JlBeT HCDOJIb30BaTb ee )1/111 Mo,n.enHpo8&HH;f 3a,L{aQ
H eaJ]bHoA ra3OBOA AHHaMHKH B 0606 eHHoA CHCTeMe KOOp,ll,llHaT (51, 52).
BSe.D:eHHe o606w:eHHLIx KOOp,D)fH3T )J;aeT B03ldWKHOCTb aDTHpoBaTb pac-
\leTHYIO CeTKY K nOBepXHOCTH o&reKaeMOrO Tena, rlJ.e, Kale npaSHJJO, npo-
HCXO;J.BT Hau60nee HHTeHCHBHbJe H3MeHeHH.H ra3OAHBaNl£ClecKHX noneA. C
nOMO[UbJO TaKHX CeTOK "PH HKCHpoBaHHOM '1HCJIe Y3nOB Y a.eTCjf 3Haqu-
TeJlbHO nOBblCMTb TOqHOCTL pac'leTOB. Bonee no,Iq>06Ho npHMeHeHHe Kry-
CHCTeMbl B 0606LU-eHHbIX Koop)J)!HaTaX )l)JB pac'ieT8. 3 a'l HAeaJJbHoro ra3a
paccMOTpeHO B paaee BblweAweA MOHorPa4>HH aBTOpa, nOCBRIUeHHoA KH-
HeTH"IeCKHM CXeMaM (53].
a.HHbIR naparpacP 3aKaH'lHBaer paccMoTpeHHe HCDOJIL30BaKHR
KHHeTHqecKHX cxeM H KBa3Hra30 HaMH'IecKoH CHCTeMbI ,AnH MO e-
nHpoB8HHH TeqeuHA H eaJIbHOrO ra3a. Bonee CJlO>KHhIe C MaTeMaTH-
QOCKOft TOQKH 3peHHSI H 60nee HHTepecHbie nOCTaHOBKH C T qI(H 3pe-
HHSI npHJlO>KeHHR OTHOCBTCR K Hccne,nOBa.HHIO BSI3Koro TennOnpOBo -
Horo rB.3a. BonpocbI, CBSl3aHHbJe C HCTIOJIb30B8HHeM I<HHeTWiecKHX
cxeM )l.JISi pemeHHB 3THX 3 aq, 6y,a;YT paccMoTPeHbI B CJle,L\YIO Hx
rnasax.
rJlaBa III
IIPHMEHEHHE KIiIHETHqECKH
cor JIACOBAHHbIX CXEM
H KBA3:urA30 HHAMHqECKOit
CHCTEMbI JI.H MO EJIHPOBAHH.SI
TEqEHHA B.SI3KOrO rA3A
PaCCMaTpHoaeMble B npe,lI.bI.D.Y e:A rJlase KCPC H Kry MorYT
UblTb HHTepnp€THpOBaHbI KaK anropHTMbI peweHIDI 3a,.a.a'l H eaJIb-
HOt\: ra30BoA AHHaMHKH C HCKYCCTseHHblMH HCC.HnaTHBHbJMH tine-
lIaMK, nponop HOHa.nbHbJMH p33uepy pac'IeTHoA tleAKH h. 9TH
)\HCCHnaTHBHbIe qneHbJ 6LLnH nOJIY'IeHhI He <l>eHoMeHOnOrH'IecKHM
nYTeM, a cornacoBaHbl C p83HOCTHoA annpOKCHMau.ueA ypaBHeHHH
BOJlbLtMaHa tIlls{ O HOqaCTH'IHoA (j>YHKQHH pacDpe eJleHBS1 ( 1 ope-
;-1b1 YIQeR rnaBbI). 8m cornacOBaHHOCTb H ftBJIH eTCSI npHqHHOA xo-
pomero KaqeCTBa KaK aJ1I'OpHTMOB, nOCTpOeHHhlX H& 6aae KCPC,
TaK H 6oJIbqMaHOBCKHX cxeM APyrHX a.BTOpOB )V(.H MOAeJ1HpOBaHHR
TeqeHuA HeBH3Koro ra3a.
BMecTe C TeM cneAYeT OTMeTHTb, tlTO C TeOpeTH'iecKOA TOq-
KH 3peHHSI HCDOJIb30BaHHe KHHeTHtfecKHX cxeM ,lJ,1UI MO eJlHpoBaHHH
aHHoro KJlacca TeqeHHA A0CTaTOtiHO OqesH HO. Tal( KaK OnHpaeT-
Ha XopoWO H3BeCTHYIO CBB3b Me>K)U' ypaBHeHHeM BOJlb aHa H
ypaBHeHHHMH 3Anepa. He06xo)UlMO TaK)f(e HMeTb B BH.DJ1, tJTO B no-
CJ1e).J.HHe )J.eCHTHneTHSI nOHBHnOCb MHOro anropHTMoB, rapaHTHpylO-
II.\HX BblCOKYIO "rOttuOCTb pacqe-ra H,l(eaJIbHOrO ra3a npH OTHOCHTenh-
liO l-1e60J1bUIHX Bbl'iHCJIHTe!IbHbIX 3aTpaT8X. TaKHM 06pa.JOM 1 P83-
JTH HbIM BapHatlTaM KHHeTHQeCKHX H 60JIbqMaHOBCKHX cxeM npKXO-
Jl-HTCH HaxO HTbCB B >KeCTKOA KOHKypeHnHH C J\pyrHMH COBpeMeH-
Hbl)'-IH aJrrOpHTMaMH pemeHHSI YPaBHeHUA 9A.rIepa. *
HllaS{ CHTY HH CKJI blBaeTCjl DPH Mo enHpOBaJlHl! TeqeHHH
BH3Koro ra3a. 3AeCb aJII'OpHTMbI, nOCTpOeHHble Ha OCH08e KCPC
H KBa.3Hra30Jl-HHa.MH"IeCKOA CHCTeMbI, c o HoA CTOpOHLI, OTHOCBTCSI
K TeM Hep.,IHorHM t KOTopble xopowo a.,n;anTHpyIOTCI K apXHTeKType
MHoronpOQeccopHbIX CHCTeM c pacnpe,ZJ;eneHHOH "&MBThlO, a C APY-
reA - n03BOJISlIOT YBepeHHo OITHChlBaTb pa3JIHq}loro po,na <PH3Htte-
CKHe HeycTOAqHBOCTH, npHcY He ra30BOR cpe e.
PaCCMoTpeHHe B03MO>KHOCTeA KCPC H Kry HatfHeM C HX 06-
cY>K eHHSI KaK <PH3HQecKOH Mo enH ,(VIS! OnHCaHHH TeqeHH:B BH3KOro
ra3a, o6paTHB oc06oe BHHMaHHe Ha HX CB}13b C KnaCCHqecK MH ypas-
HeHHRMH HaBbe-CToKca.
1. KUHeTHqeCKH COrJIaCOBaBHLle cxeMhI KaK
MO eJTh n.B ODHCaHRB TeqeHHB BH3KOro I'83a.
1. HeKoTopble 3aMeq&HHH K BblB0JU' YP8BHeHHR r830BOH
)1.HHaMRKH
IIpe}K e \{eM paCCMaTpHBaTb, Ha '{eM On>KHO 6LITb OCHOB8HO
pa3nHQHe Ii CB.H3b Me>KA}' KCPC H yp8.BHeHHSMH HSBbe-CTOKca t
BepHeMC5I e e pa3 K <PeHOMeHOJIOrH'IeCKOMY BbJBOAY ypaaHeHHA m-
30BOA ,lJ}ItJ8.MHKH. PacCMOTpHM, HanpHMep, BbIBO)J. ypaBHeHIiSi Hep8.3-
pblBHOCTH (rn. It 3, pHC. 3):
::: + div(pu) = 0, (1.1)
Ko'ropoe BbITeKaeT H3 ycnOBIDI COXpa.HeHHS MacCY NISI KOHeqHoro
061>eMa V (pHC. 3). 9TO YCJlOB1iIe coxpaHeHIUI 03H8'1aeT, '1TO CKOpOCTb
H3MeHeHHSI MacChI) COcpe OTOqeHHOa B o6'beMe V J Onp eJ]5IeTCSI ee
nOTOKOM Qepe3 nOBepXHOCTb S, orpaHHttHBalOIIIJ'IO a.HHLJa 06'beu:
; / pdV' = - f pu,.du.
v s
(1.2)
.nO MHeHHIO aaTOpa, - 803'(0)1(80 , CnOpHO&.lY, - B CBR3H C n09BneHIIeM KaK
anropHMOB nOBblWeHHoA TO'lHOCTH, Tax H BbiCOKonpoH3 TeJIhHWX CHCTeM.
peWeltHe 3 a\( ,D.HH&MHKH H eaJl.bHoro raaa. B 3H&'lHTenbHOA uepe npeBpam:aeTOl
K3 HaY1.lHoA B HH)I(eHepHYJO npo6.neMY.
l. KrfIle'ff,qeCKH cornaCOB4IIHl:de CXeMhI Ka.K MO.o.eJIb BR3KOro r83a 113
Yq}f'rbIBaJI He3aBHCHMOCTb BpeMeHH H npoCTpaHCTSeHHbJX KOOp H-
HaT, a TaK.>Ke TeopeMY raycca-OCTpOrp CKOro, nepenHWeM (1.2) B
BHAe
! ( : + div pu ) dV' = O.
v
aJIee, eJTag 38.KJIIOQeHHe 0 npOH3BOJlhHOM 06 eMe V H3 (1.3), 00-
JIYQHM H<P4>epeH aJIbHOe ypaBHeHHe Hep83pblBHOCTH (1.1).
BepHYBlliHCb K BbIBO,zur ypa.BHeHH (1.1). 3 a.,a){MCSI BonpOCOM:
He'!' nH KaKHX-J1H60 OrpaHH'IeHBA Ha Dbl60p <>&LeMa V? 113 KJIacCH-
lJecKoA MeXaHHKH cnnowHoR Cpe)\bl XopOWO H3BeCTHO nOHHTHe c:l>H-
3Ht.JecKH 6eCKOHetfHO MaJlOro 06'beMa [11 J. B COO'fBeTCTBHH C 3THM
nOHHTHeM, 061>eM. no KOTOPOUY npOH3BOAHTCH HHTerpHpOBaIIHe, He
MO)KeT 6b1Tb 6ecKOHe1lHO MaJlblM. 9TOT o&beM OJI)f(eH cO ep)f(aTb
onpe eJ1eH}(Oe KOJIH'-IeCTBO MOJIeKYJI. QTo6hI BblnOJIHHJlOCb npH6JIH-
)I(€HHe CfiJIOlllHOH cpe,ltbl. aMeTp TaKOrO 06 eMa it t XOTb H npe-
Bblillaerr HaMeTp MOJIeKYJIbI d , Bce )Ke cy ecTBeHHO MeHbIl1e p)IHHbI
CB06o.D-Horo npo6era l:
( 1.3)
d « 11 « I.
( 1.4)
Cy ecrBYIOT JIll pyrHe OrpaHH'feHHSI HR Bbl60p o6 eMa V, Kpo-
Me Heo6xo HMOCTH BblnOJlHeHHB YCJlOBH$I <l>H3HQeCKH 6OCKOHeQHO
1vlaJIOrO 06'beMa? IIpocTbJe CO06pa>KeHHSI nOK83b1BaJOT, 'ITO no rIO-
PSIAKY BeJIHqHHbJ HaMeTp V OJt}KeH 6b1Tb He MeHLwe ,LVIHHbI CBo-
60 Horo np06era I.
B caMOM Aene, po- HJIH ra30AHH8.MH'IeCKOe npH6JDOKeHHe no-
CTpoeHo Ha 6JTH3OCTH <PYHKQRH pacnpe,L\eneHHSI K ee paBHOOecHOMY
(8 KOHKpeTHOM CJIyqae MaKCBennOBCKOMY) 3HaQeHHIO II' (rJI. I):
f /1' = fOe
(1.5)
B CBOIO Oqepe,nb) 113 ycnOBHSI (1.5) CJIe,lfjleT, "ITO cPYHKIJ:HSI pacnpe-
;\e..nCHHS1 cna60 MeHHeTCS1 Ha pacCTOHHHH l. YqHTbIBaSl, 'iTO S6JTH3H
paBHOBecHH c:l>YHKI\HH pacnpeAeneflHH 3aSHCHT OT npocrpaHCTBa H
BpeMeHH nocpe CTBOM nHTH HHaMHqeCKHX napaMeTpOB, npa-
tl-eM K BbIBO,l\}', QTO p, u, T OJ1>KHhI CJta60 MeHBTbCR H& paccTOSI-
UHH l. ABTOMaTHqecKH rapaHTHpOBaTb BblnOJIHeHHe 3TOro YCJlOBIrIH
114
TJI. III. MO e.nHpo88Jflle Te'leHHR 81f3KOro r83&
MO>KHO, ecJIH B HHTerpanhHhlx COOTHOWeHY1IX THna (1.2), OnHCblsa-
IOW)iX 3aKOHhI COXpaHeHHS MacChI, HMnYJIl>Ca H 3HeprHH, p83MepbI
KOHTpOJIbHOrO 06'heMa V no nop y BeJlHtUlHhI 6paTL He t.1eHbme
JTHHbI cBo6o oro npo6era I.
TaKHM 06p830 1) eCJIH yqHTbIBaTb 3TO OrpaHH'-IeHHe CHH3Y Ha
HatwreTp o&beMa V, TO IIPH nepeXOAe OT HHTerp8J1bHOrO ypauHe-
HHH (1.3) K <I><PepeH HaJIbHOMY B noc.ne,QHeM AOJI>KHhI nOSlBHTbCSI
OnOJlHHTenb}lble qneHbl nopB Ka l:
+ div pu = O{l).
( 1.6)
AHaJlOf'HqHblM 06paaoM, AononHHTeJJbHbIe qneHbl 0(1) OnJKHbI no-
iIBHTbCH lot B ypaBHeHHSlX, onHCblBalOUJ.HX H3MeHeHHe HMoYJlbCa H
3HeprHH.
IIOJIyqeHHaB TaKBM 06pa3oM CHCTeMa ypaBHeHd, COCToB aSl
H3 ypaoHeaHB THn& (1.6), 06n aeT CBoACTBOM KoppeKTHOCTH no
KpaAHeR Mepe c <pH3H'IeCKoft TOQKH 3peHHiI, TaK KaK rapaRTHpyeT
HaJ1WQHe cna60ro H3MeHeHHSI peweHHSI Ha pacCTOHHH.HX, MeHblIIHX
HJlH nopSl Ka l. 3a.MeTHM, {{TO JI-OKa3aTenbCTBO TeopeMbi CYm.eCTBO-
BaflHB H e)J.HHCTseHHOCTH NISI CHCTeMbI ypaaHeHHA Haobe-CTOKca
CTaJlkHSae'rca co 3HaqHTeJ1bHbJMH TPY,l].HOCTHMH, HanpHMep, 113110-
>KeHHbIMH B xopomo H3BecTHbJX MOHOrpa<PH1IX O.A. JI3AbJ>KeHCKOR
[54), A.M. KOWeJIeB& [55], P. TeMaMa (56]. ASTOp He 6epe:r H8 coos
CMeJIOCTb enaTh YTBep>KAeHHe 0 BOOMO>KHOCTH enH HeB03MOXOlOCTH
TaKora OKa3aTeJ1bCTBa. OTMeTHM nHWb J 1fTO Ha 8acTOBUJ.KA MOMeHT
STOT sonpoc no-npe)KlfeMY OCTaeTCSI OTKpIdTbJM.
npo OJI>KaS1 HaqaTbIe paCCY)K)J;eHHSI, 3aMeTHM, 'ITO B CHCTeMe
Hasbe-CTOKca TaK>Ke npHC)'TCTBYIOT ,QHCCHnaTHBHble qneHbl O(l) 8
ypaBHeHHHX HMnynbca H 3HeprHH, KOTOpblMH B AaHHOM CJlY'Iae gB-
JI.HlOTcg 'iJ1€HbI eCTeCTBeHHoA BR3KOCTH H TellJlOnpoBO HOCTH. 9TH
ypasHeHH5I npOWJIH HCIILJTa.BWI 60JIee eM BeKoBo:A:, B TOM 'IHCTle
H HH,ayCTpHaJIbHOft, npaKTHKoA, nO TBep aJOm;eft TeOpeTB'IecKHe
Bb(BO b1, nOJJyqeHHhle Ha HX OCROBe. n09TOMY BHOBb nOJlyqeHHbIe
ypaBHeHHjI THIta (1.6), Y"IHTblB&IOID.He orp8HHQeHHe CHH3Y H& KOH-
TponbHbIH o&beM V, AOJI>KHbl ,l].8B&Th pe3YJIhT&Tbl J npaKTHtJeCKH He
OTJlHQaJOW)ieCB OT pe3ynbTaTOB. llOJIyqeHHhlX Ha OCHoae ypaBHeHHA
91. KIIHeTUtlecKH COrJlBCOB&IIHblB cxeMW ICalC MOAeJI'b BH3KOro r&3a 115
HaBbe-CTOKca. B nepBYIO Otiepe 9TO OTHOCHTCSI K cT&QHoHapHbJM
l-1.J1H OTHOCHTenbHO CJIa.60 MeHHIOIIJ1ICS no npOCTpaHCTBY H speMeHH
Tet.teHHSlM 8H3Koro ra3a p r e npHMeHeHHe ypaBHeHHA Hasbe-CTOKca
He Bbl3bISaeT cOMHeHIdt.
06paTHM BHHMaHHe Ha TO, qTO KCPC H KBa3Hra30AHHaMHtfe-
CKaH CHCTeMa ypasH€HHA no CBoeMY nOCTpoeHHIO HCR01Ib3ye-r ol'pa-
mrqeHHe CHH3Y aa pa3Mephl KOHTpOJlhHOro o&beM& V, TO ecTL OT-
HOCHTCH K ypaBHeHH}1U THna (1.6). B 1I0CJIeAYIO naparpa<j>ax
r JIaBbl 6y eT Hccne OB8.Ha CBH3b KBa3H1"8.30,!{HH8.MH'IecKOA' CHCTeMbl
H CHCTeMbI ypa.aHeHHA HaSLe-CTOKca. ITone3HOCTh Hcnonh30BaHHB
TaKoR CHCTeMbl ypasHeHHA, OTnHqaJO eA:CSI no cBoe:A MaTeM&TH'Ie-
CKOH cPopMe OT ypaBHeHHR Hasbe-CTOKca, 6y eT BH)J.Ha H3 npH-
Be €HHblX npHMepoo ee HcnOnb30BaRHSI njl MO eJlHpOBaHHg 3 a
cOBpeMeHHoA a3pO HaMHKH.
2. 3aMeqaHHH 0 AHCCHuaTHBHblX QJleH8X B KCPC
MTaK J 6y eM yMeHbmaTh pa.3Mep aqeftKH, H& KOTOpo:a KOHCTpy-
HPYeTCH KCPC, o MHHbl cBo6o Horo np06era I. B KBa3Hra30 HHa-
IHqeCKOA CHCTeMe (5.15)-(5.19) (rn. II) napaMeTp T BLI6epeM pas-
HblAf BpeMeHH Me>KAY CTOJlKHOBeHHHMH MOJIeKYJI (T = llc). Tew ca-
MblM BbinOJIHSleTC5I O O H3 Tpe60BaHHA K ypaBHeHH}1).1, onHCblBaJO-
HM ,rt;BH)j(eHHe BSl3KOro ra3a (n.1 aHHOro naparpa4>a), - mpaH-
THSI CYll\eCTBOBaHHg paaHOBeCHoro COCTOSlHHSI NtH OAHOqacTH1{HOA:
<PYHKQHH pacnpeAeneHIDI B KOHTpWJbHOM o&beMe.
JI.SI roro, 'IT06bI npOKOH'rponHpoBaTb BhlnOJlHeHHe BTOpOro
rrpe60BaHHH, rapaHTHpYIOm;ero eKBaTHOCTh pe3YJlbTaTOB p nony-
qeflflblX c nOMOI1J.b1O KCPC H C nOMOwplO ypaBHeHHA HaBhe-CTOKca,
BHOBb 06paTHMCSI K 3aMeQaHUSIM 1.1 H 1.2 (rJl. II). CJleAYeT OTMe-
THTb , {.ITO npHB€,AeHlible HH}Ke paccy eHHB CbIrpaJIH pern&IOIQ,y1O
JX>1Ih B HCII0JIb30BaHHH KCPC H KB83Hra30 laMH'lecKOA: CHCTeMLI
,l].JIH MO)J.eJJHpoBaHID1 TeqeHHA BB3KOro ra3a, KOTopoe 6bL1I0 HaqaTO C
cepe.LJ;HHbI 80-x rO OB.
3aMeqaHHe 1.1 I'JI. II nOK83L1BaeT, '11'0 nOTOK MacCLI Ha rpaHH-
e paCqeTHOA S1qe KH 8 KCPC paBeH pu, r e U - CKOpoCTb B OJIb
BHemlieH HOpMaJIH K nOBepXHOCTH H1.feRKH, TO ecTb 3TOT nOTOK eH-
116
rJ1. III. MO,lJ.enllpoB&HHe Tetlemdl BH3KOro ra38
THqeH nOTOKY MacChI Qepe3 nosepxHoCTh KOHTpOJlbHOro 06'beMa, no-
nOJKeHHOrO B OCHOBY BbIBO.n:a KnaccH'IecKoro YPaBHeHHB HepaapblB-
HOCTH.
B03Bpa.r.naSlCb K 3&Me"laHHIO 1.2 (rJI. II), BROBb o6paTHM BHH-
MaHHe Ha TO, qT() B OTCYTCTBHe MaxpoCKOnHqeCKOA CKOpoCTH, "PH
xapaK'repHOM 3HaqeHHH a6cOJlIOTHOA BeJIHtJHHbI MOJ1eKynjlpHoA CKO-
poC'rH, paBHoA" CKOpocTH 3BYKa (1(1 = c) H npH pa3Mepe BqeAKH ,
paBHOM )].JIHHe cB06o Horo npo6ero l, OnOnHHTenbHOe H3MeHeHHe
HMnYJlbCa 3a CtfeT AHCCHnaTHBHhlX qneHOB B KCPC 6Y eT nponop-
1(HOHaJlbHO pa3I-IOCTHOA npoH3Bo oA MaKpOCKOnH'IeCKOA CKOpOCTH
u C K03c1><PH HeHTOM nponOp HOHaJIbHOCTH JJ, p8.BHblM P, = plc/2.
I1MeHHo TaKoR Koa<l>cl>H HeHT BB3KOCTH BbiTeKaeT H3 9neMeHmpHoR
KHHeTHt.JeCKoA TeopHH (rJl. I).
AHaJIOrHQHO MOJKHO nOK33aTb, ({TO npH HllTerpHpoBaHHH C()OT-
HoweHHSI 6aJ1a1iCa (2.1) (['n. II) .LJ)ISI <pYHKnHH pacnpe eneHIDI C oe-
COM 2 /2 H npH xa.paxrrepHoM pa3Mepe SI'IeRKU, pasHOM AnHHe CBo-
6oAHoro npo6era l, 38.MeHSIH B npaBol qaCTH (2.1) B BhlpaJKeKHH
Ie
1 L J I( 'Ili.l)I{f/M - fl M )dl1
mes 1 i=l
CKOpOCTb Ha CKOpoCTb XaOTH"IeCKOro ABH)I(eHHSI NoneKYJI, llOJIY'IHM
8 npaBoA ttacTH KHHeTH'IeCKH cornacoBaJIHOA pa3HOCTHOA cxeMLI
HccHnaTHBHbJe qneHbI, onHCbIBaJOIU)Ie H3MeHeHHe nonnoA 3HeprHH.
8TH HCCHnaTHBHble 'IneHbI 6Y,lIj'T COOTBeTCTBOBaTb pa3HOCTHOMY
aJlaJ10ry TenJlOnpOBO Horo qneHa H p33HOCTHOMY aaaJIory pa60TbJ
BH3KHX con. COOTBeTCTBeHHO, K03<IxPH HeHTbl BB3KOCTH H TeIJJIo-
npOBO)VfOCTH B HHX 6y paBHbJ Kos<l><PHII-HeHTaM, onpeJJ-eJISleMhlM
Ha OCHOBe 3JJeMeHTapHoA KHHeTH'fecKoA TeopHH.
aHHble BblBO;J.bI DOJ1Y'feHbI npH ycnOBHH 1 QTO CKOJ>OCTb 3BYKa 3H3'1H-
TeJlbHO npeBblWaeT MaxpocKonH'tocKYIO CKOpoCTb ra3a. t{T06bJ OneHHTh
BnHHHHe 3TOro o rpaoHQeOH.ft, BHa 8Jle BcnoMHHM p qeM >Ke no cy ecrBY
ypaaHeHHH HaBbe-CTOKca ornH'IaJOTCH OT ypaBHeHHA 9ltJIepa. BhIBO,lJ.
ypaoueHHA 9iinepa OCHOBaH H3 npe,ltCTaBJIenHH Q:>YHKl(HH pacnpe,QeneHHJI
B JlOKaJlbHo-UaKCBenJlOBCKOM BH,lJ.e
f == /0 ,
( 1. 7)
ql. KUHe'rfl'leCKH COrJl&COBaHHble CXeMbl K.aJ( MOAe.nb B1I31(0ro raJa 117
a BbiBotI- ypaBHeHHH HaBbe-CTOKca - Ha ee npe)J.CT88J1eHHH B BH.Ae )l)JYX
'fJIeHOB p83JIOJICeHH$I t{enMeHa-9HCKora:
f = /0 + Kn/NS.
( 1.8 )
Pa3nO>KeHHe (1.8) OTnH aeTCSI OT (1.7) tI& Ma.nylO aenUqHHY Kn J NS
(Kn < 0.001 nJl 3aA&q ra30BoR ,ltHHaMHKH). OAHaKO HyeHHO 3TOT 'IJleH
OTBeTCTBeHeH 3a nO BnelIHe BeKTopa TennOBoro nOTOK& qi H H3MeHeHHH
BLlp8.)f(eHHSI M TeH30pa HanpH>KeHHA Ptj = 6 ij p + Pij:
J ( 1 2 ) ! Cic2
1'1, = Kn Cac; - c 6 i i fNSd , qi = Kn T fNSd(. (1.9)
P83JIHQHJI B pe3ynbT8:ra.x MO enHpOBa.HHHt OnHpalOl.U)fXaI H& yp8.B-
HellHH 9AJ1epa H ypaBHeHHH HaBbe-CTOKca, npOH8JIHIOTCH TOr,Qa. Kor,n.a
8eJ1H U4HbI (1.9), I1eCMOTpSl H& MaJIOCTb qHcna KHYJl.CeH&, CT3JfOBSlTCSI co-
nOC'raBHMbl'-fH C cOOTBeTCTBYJOI1l.HMH MOMeHTaMH OT CYMM8TOPHbiX HHBa-
pHaHTOB H 2/2 <PYHK1X.HH pacnpe)J.eneHHJI (1.7). 83 (1.9) BHt\HO, 'fTO ponL
Pij t qi B03pacTaeT, KOl'A8 CKOpocTb 38Yxa C (Cpe.lJ.HH1I reMOB8JI CKOpOCTb
ABH)KeHHfl MoneKYJ1) npeBOCX MaKpoCKOnH'fecKYIO CKOpOCTb B KaKOM-
TO HanpaB.neHHH. 3aMeTHW, 'ITO npe,u;enbHblH BapHaHT TaKoA cMTyanHH,
KOr,ll.a ABa CJ10H )I(H,lUCOCTH HJIH 1'a38 ABHra.lOTCJl nap8JIneJlbHO APyr APyry,
HCTOpHqeCKH nocnY)KHJ) OCHOBOA BBe,l{eUHH nOHSlTHJI BRJKOCTH. BS3KOCTb
TpaKTOBaJIacb KaK Ko3<j>cI>H HeHT nponopUHOH8JIhHOCTH Me HanpH>Ke-
"HeM "rpeHIDI H npOH3BO,ll)foA CKOpOCTH B HanpaoneHHH, nepneBAHKynap-
HOM )1.8 H)I(eHHJO cnoeB [11).
TaKHM 06p830M, TeH30p Bft:)KHX HanpJl>KeHHA ofipeAeJiReT Ty qacrb
nOTOKa HMnYJ1bC8, KOTOpaH He CBJl3aH8 C HenOCpe CTBeHHhlM nepeHOCOM
HMnynbca BMecre C MaccoA nOCTynaJOJJJ.eA JKH)U(OCTH. AHanorH Ho, nepe-
HOC Tenna 3a cQer rennonpOBOAHOCTB He CB1I38J1 C MaKpoCKOnHqecKHM ,l\BH-
*eHHeM >K KOCTH HJIH ra38, a ocy eCTBJJHeTCa 38 C'feT MoneKynSipHoro
nepeHoca 3HeprHH. IIon'OMY ou.eHKH )J.HCCHnaTHBHblX 'fReHOB KCPC npH
HcnOJJb30oaHHH YCJIOBHH C > U BnOllue 060CHOBaBbI C cl>H3H'IeCKOA TO'lKH
3peHHJI.
TaKHM 06Pa30M, KHHeTHtJecKH cOrJlaCOBaHHaJI pa3HOCTHaH cxe-
Ma, paCCMOTpeHHa H& 06naCTH C pa3MepOM SlqeAKH, paBHbIM ARHHe
CBo60 Horo np06era) YAOBJIeTBOpHeT Tpe60BaHHSlM. Bbl,nBHHYTblM B
n.l aHHOrO naparpaQ>a. C o HoR CTOpoHbJ, rapaHTHpYeTCB HaJIHqHe
paBHoBecHoro COCTOHHHSI B SI'Ie:HKe) c pyroa CTOpOHhI, pe3ynbTaTLI
pacqeTOB, nOJlyqeHHbIe Ha OCROBe HX npHMeHeHHSI, OJI>KHhI COOT-
oeTCTBOBaTb pe3YJlbTaTaM pac'leTOB C HCn0JIb30BaHHeM ypaBHeHHA
HaBbe-CTOKca.
118
r.n. m. MOAeJ1HpoB3HHe TeCfeHHIl BJl3KOC'O 1&38
0606 a.a BLJrnecKa3a.HHoe, KCPC MO)fCHO TpaKTOB&Th leaK Ho-
BYIO MO eJJb NISI ODHCaHWi TeqeHHSI BH3KOro ra3a. K UKemU eC1roM
ompa e1iUe.M 3m MOdeAu Jl6AJ1emCJI npeOcma8AeHt4e oOH01U1CmU"'.
Hm1 Y"" ttU pacnpedeAeHtJJI 8 BUOe Ma"CBeAA06C"oU. 9mo .Ma"C-
eeNt08C"Oe pacnpeaueHue He AOJCaA'bHOe. OHO nOCmOJlHHO no npo-
cmpaHcm6Y 8 o6Aacmu C pa3MepaMU, paSH'MMU dAUHe ce06oOHo2o
npo6eza I.
llOSIsnelIHe AHCCHnaTHBHblX qneHOB B KCPC CB.SI3a.HO C HaJlH"IH-
eM P.Byx pa3JlHt.fHhJX Id&KCBenJlOBCKHX pacnpe eneHH:A: no 000 CTOpo-
HbI OT rpaHHl.\bI S!'Ie:AKH. llonYMaKCBeJIJIOBCKasI <PYHK R, KOTOPYIO
06pa3YIOT 3TH pacnpe eJJeHHR Ha rp8JfH e, npHBO)J.HT K nOBBJIeHHIO
;J.HCCHnaTMBHbIX COCTaBJISlJO HX B nOTOKaX.
IIpHse,LJ.eHHble co06pa>KeHHB SlBWlHCL BaJKHblM MOTHBOM K HC-
nonb30BaHHJO KCPC, KOTOpble, H& nepBbIA B3rnH , npe CTaBJIBJOT
co6oA cxeMLI C HCKYCCTBeHHoA BB3KOCTblO, OnHChlB&IO e npo eccbl
B H eanbHOM raae, AIlS! MOAeJIHPOSaHRSI BB3KHX TeqeHHA. OMf&KO
HaH60JJee HHTeHCHBHO STO Mo emtp08aRHe, HaqHHaJ1 c pa60T (22),
[57]-[59], npOBO HnOCb B pCWK&X KBa3Hra30AHHaMHtJecKoR CHCTeMbl.
IIpH 9TOM HaH60nee HaI'JUl,D.HO npocMaTpHBaeTCJI CBB3b Me>KA.}' Kry
H ypasHeHH.SIMH HaBb CTOKCa, KOTOPYIO pacCMOTpHM B r.neAYIO eM
naparpac}>e.
2. KBa3Hra30 HBaMHqeCJaUI CIICTeMa YPaBHeHHH B
ypaBHeBBB HaBbe-CToKca
1. KBa3Hra30,l{HHaMJfQeCK&Sl CHCTeMa
YtlHTblBaJI CBH3b Me>KA)' KCPC H Kry (rn. II) H OnHpaJICh Ha
pacCY)I(.n;eHIDI, npHB eHHbIe B 1, MO HO npe OJJO>KHTb) 'JTO no-
cneAHBSI CHCTeM& JJ.&eT npaKTHqeC H TO )l(e OnHC8.HRe BJI3KHX re'le-
HHR t 'iTO H ypaBHeHHSI H&Bbe-CTOKca npH T, p8.BHOM BpeMeHH Me>K-
l1Y CTOJIKHOBeHHSlMH MOJIeKYJI. O HaKO npe>K e qeM BblSlCHHTb 3TY
CBH3b Henocpe CTBeHHOJ BblnHweM KB83Hr830 HHaMHt.(ecK)'1O CI1CTe-
MY B 3TOM CJlY'I&e. ):{na 6W1LweA KOMnaKTHOC CHCTeMY ypaBHeHHR
32. KrY-CHCTeM8 II YpaBHeHHJl HaB CTOKCa
119
IJblnHWeM B npOCTp&HCTBeHHo- BYMepHoM CJIyqae:
pi+l - r>' I) fJ a T 8 ( 2 )
+ - (pu) + - (pv) = - - - pu + P +
T fJx 8y ax 2 8x
{}TO ( 2 ) ora orB
+ fJy 2 Oy pv + P + ax 2 ay(PUtJ) + 8y 2 a; (puv);
( U ) j+l ( ) j a {)
p - flU + _ ( pu2 + P ) + - (puv) =
r Ox 8y
OT{) ( 3 ) or8 ( 2 )
== --- flU + 3pu + --- pu1J + pu +
ax 2 8x 8y 2 8y
8ro ( 2 ) oro ( 2 )
+ --- flU 1J+PV + --- pu v+pu ;
ox ?, 8y Oy 2 ax
( ) 1+1 ( ) j 8 8
pv - pv + - (puv) + - ( pv2 + P ) =
; ax 8y
8TO ( 2 ) 8TO ( 3 )
= --- pu V+p1J +--- pv +3pv +
ox 2 ax fJy 2 Oy
8rO ( 2 ) 8r8 ( 2 )
+ --- pv u+pt& + --- pv u+pu ;
ax 2 8y 8y 2 ax
E1+ 1 - Ej a a
+ (u(E + p)) + (1J(E + p)) =
T uZ uy
a T {) ( 2 ( 5 ) 1 2 ) {} r a ( )
= --- u E+-p + -p1J + --- uv(E+p) +
ax 2 ax 2 2 ax 2 fJy
8T8 ( 2 ( 5 ) 12 ) 8T8 ( )
+ --- tJ E+-p + -pu + --- uv(E+p) +
2Oy 2 2 8y28x
'Y [ a T p fJp fJ T P 8 ( P )]
+ ')'-1 8X 2 p 7h + ax "2 ax p +
'Y [ 8TP8p {JTP 8 ( P )]
+ ')'-1 ay 2 p a; + 8y 2 P; BY p ·
(2.1)
(2.2)
(2.3)
(2.4)
3 ecJ, 'Y - IIOKa3aTeJIh a.D.Ha6aTLI, Pr - q}JCJlO np TJIa,
2J.l
r = -, p = p(T,p),
p
u 2 + v 2
E = P 2 + E(T,p).
(2.5 )
3 a M e q a H H e 2.1. HCnOJJb3YjI npoueAYPY BblBOJ].a Kry, OI:HcaHHYIO
B 95 (rJI. II), npH,l\eM K ypaBHeHHIO 3HeprHH (2.4) C t.IJIcnOM np8.H;trJ1S1,
paoHbIM eAHH(.t e (Pr = 1), QTO He COO1'BeTCTByeT CHTYaI.l)fH B peaJIb-
HbIX ra3a.X. n03TOMY ,.lI)I1I OnHC8.HHH npou.ecca -renJJOnpOBQlJ.Hocr'H B paM-
KaX Kry 8MecTO 1 BOJIeBblM 06Pa.30M CTaBHTCg "IHCJlO Pr, COOTOeTCTBYJO-
w.ee ero 3Ha'feHHIO ,Ana .QaHHOro r838. no 06HaJI npol(eAYpa HJ(eHTH'IHa
DOJIyqeHHIO ypaBHeHHA, OOHCblB8IOIII)IX Te'JeHHe B1I3KOro raaa H3 BrK-
MO)J.enH (9 3 rJI. I) t r e BMecTO JIOKaJIhKo-MaKCBeJJJJOBCKOro pacnpe,lteneOHJI
,lI,JI}{ nonyqeHBJI peanbfioro 'IHcn& Pr HCnOJJb3YeTCH [12) HCKYCCTBeHHO BOe-
)\eHHoe aHH30TponHoe nOKaJIbHO-MaKCBeJlnOBCKoe pacnpeJl-eneHHe (3.39)-
(3.40) (r 1. I).
AHaJIOrHqHM CHTyaQHH C BMHqHHaMH Ko3<f> HlJ)IeHTa BmKOCTH 11,
YPaBHelJHA COCTOHHHH p(T,p) H e(T, p) H nOK83&TeJlB a;J)ta6aTbJ 1, KOTt>-
pble B ra3ax He Bcer)J.a COBna.zr.8JOT C HX 3H8'1eHH.HMH, nOJlyqeHBblMH H&
OCHOBe 3JleMeHmpHOH KHHeTH'IecKOA TeopHH. B 3TOM CJlyqae, UK H 06YQ-
HO B ra30BOH )J.HHaMHKe, B KrY-cHCTeWe HCnOJlb3yJOTCH pe&1ILHble, no -
TBep)l(JI-elIHble 3KcnepHMeHT3J)bHb1MH H TeopeTH'IecKHUH ,D.8BHbIMH 3Haqe-
HI4S1 Ko3<l><J>HI1.HeHT8 BH3KOCTH J.I H YPaBHeHHA COCTOHHJ.f1l.
3 aM e q a Hit e 2.2. npo6neM& H3J1HWHe nO Horo mara no BpeMe-
HM H3-3a ManOCTH T no CpaBHeHHIO C xapaKTepHhlM ra30)J.HH8MH'feCKHW
BpeMeHeM t0 1 TaK >Ke Ka.K H HeY,.Q.o6cTBa. CBB3aHHhle C B03MO>KHbIM H3Me-
HeHHeM T DO npocTp8JfCTBeHHblM nepeMeHHblM, MorYT 6blTb Dpeo oneHbI
C nOMOW»1O pa.3J1H"-IHbIX Bb[qHCJlHTeJILHLIX nO O)J;OB (53). HeKOTOpbIe H3
B03MO)f(HbiX anropHTMOB pemeHHSI CHCTe.w (2.1)-(2.5) 6YAYT paccMoTpe-
Hbi B rnane IV.
3 aM e q a H If e 2.3. HeKOTOpoe BHeWHee OTJlHilHe ypaeHeHHH (2.4),
ODHCbIBaJOw.ero H3MeHeHHe nonoo8 3ueprHH, OT yp&8HenHJI (5.19) (rn. II)
'Y lJ IT {J
CB"3aHO C .HBHblM BblAeJleHHeM qJleHOB BH)I.a -L- I ft::XPl:C"""Kr. COOT-
')'- aXi rr UXi
BeTCTBYIO rennOnp080,lI)IhlM 'J.neHaM.
2. CBH3b c ypaBHeHHHMH HaBbe-GTOKca
PacCMOTpHM rrenepb Henocpe)J.CTseHHO COOTHoweHIDI Me*JJY Koa-
3Hra30AHHaMH\fecKOA CHCTeMOA H ypaoHeHHHMH HaBbe-CTOK':a, [60].
2. KrY-CHCTeMa H yp88HenHH H8Bbe-Croxc&
121
l-{a HeM c ypa8HeHH (2.1). YqHTbIB8.H CB b HCCHnaTHBHbIX 'U1e-
HOB I1pasoA IJacTH ypasHeHHA (2.2) H (2.3), nepenHweM (2.1) B BH e:
pi+l _ P' + apu'; + apvi = _ ( puj+l - puJ ) T _
T ax Oy Ox T 2
. 1 .
_ ( JW'+ .,.- pvJ ) ; + 0(1"2).
(2.6)
Pa3JIO}l(HM puj, pui H pi B PSA TeAnopa, HanpHMep:
( 8 ) j+l T2 ( &2 ) ;+1
pui = puj+1 -1' ;; +"2 ;; + 0(.,-3),
(2.7)
H nO CTaBHM pa3nO>KeHKH B a (2.7) B (2.6). npH 3TOM nOJlY1fHM
8pi+l T &2';+1 opuj 8pui T a apui+l
8t - 2 8t 2 + 8x + 8y = - 2 ax iJt
-+1
_ T arm' + 0(.,.2)
2 oy at
HJlM
8pi+l 8puj 8pui a ( 8puj+l 8pvi+ 1 )
at + ax + 8y +r 8t ax + fJy -
_ ( opu.j+l 8ptJj+l ) _ T 8 2 pi +1 _ 2
at 2 ax + 8y 2 lJt2 - O( T ).
(2.8)
CYMMa llOCJIeAHHX ABYX qJJeaOB JlesoA qacTH (2.8) pasHa HYmo
C TOqllOCTblO ,QO 0(,2). C APyroA CTOpOHbI:
opui opvi ( 8PU (Jpv ) ;+l _ Opuj+l 8pui+l 2
ax + 8y +.,. at ax + 8y - ax + 8y + 0(.,- ).
To eCTb ypasHeHHe (2.1) C TOqHOCTbJO l1.0 "tIJIeHOB 0(7 2 ) nepeH,1\eT B
KJlaccH\ieCKOe ypasHeHHe HepaapbJBHOCTH
+ 8pu + apv ::; 0
at & By ·
122
rn. III. MD,lte.JlHpoBAHlle Te'lelfldl BR3KOro ra3a
HeCKOJ1bKO 60J1ee CJlO)l(HblM 06pa3oM Hccne,nyeM CBB3b Me}K)Jy
ypaBHeHHSlMH (2.2) H (2.3), onHChlBaJO MH H3MeHeHHe HMnynbca.,
ypaBHeHHeM (2.4), OnHCblB8JOllJ.HM H3Me}leHHe 3HeprHH H CQ()TBeT-
CTBYlO MH ypasHeHHRMH HaBbe-CToKca.
BH3KHe qneHbI B ypaoHeHHSlx Hasbe-CToKca Moryr 6b1T Heno-
cpe,nCT8eHHO uonyqeHbl H3 BrK-MO.lI.eJIH. npH 3TOM BKJI TeH30pa
BSl3KHX Hanpa)KeHHA Pile 3aI1HWeTCSI B BH e
Pik = - 1 "' CiC1c '
8Xi OXi dt
HCDOJIb3YS ypasHeHHiI 9AJJepa ARB eanbHOrO ra38., TpaHC<!>OpMH-
pyeM npaBYIO 'facTh BbJpaJKeHHB (2.9). Tor a NISI BB3KHX Hanpg>Ke-
HH:R DOJIyqHM:
Ci = {i - tJi.
(2.9)
( OOi &u t 2 au l )
Pile: = P, - + - - - Oilc - ·
aXle OXi 3 OXl
(2.10)
npeACTaBHM SblpaJKeHHe, CTOSl ee no HHTerpaJIOM B npaBoA
a.c H (2.9), B B e
8 ( dfo ) 8 8/0 8 a/ o
- T-CiCk = -T{p i k- + -T{i{k--
8Xi dt OXi 8z p 8Xi Ot
a a a
- -T fUk- - -T kUi- + -TUjUt-.
8Xi dt 8Xi dt 8Xi dt
(2.11)
nOCJIe.AHHe TpH qneHa B npasoA tfacTH (2.11) 06paTSlTCSI B aYJIb C
TO'IHOCTbJO IJ.O O(r 2 ) np" HHTerpHpOSaHHH no CKOpOCTSlM (. B CBOJO
Oqepe)J;b, B pe3YJIbTaTe TaKOro HHTerpHpoBaHHB AJ]a nepBoro 'lJIeH&
B npasoA qacTH (2.11) nOJlyqHM
f () 8/0 1 f 8 {}fo
p i kT dE = 2 _ 8 . T{p(i{k +
V%1 uX p X, uX p
1 f 82 fo
+ 2 T{p 8t 2 d(.
(2.12)
TaKHM 06Pa30M, yp8.BHeHHe, OllHCLlBalO ee H3MeHeHHe HMnynb-
ca B CHCTeMe HaBbe-CTOKca, MO)l(eT 6h1Tb nonyqeHO npH HHTerpe-
poSaHHH C cYMMaTOpHblM HHBapHaHTOM {" c roqHOCTbIQ AO O(T 2 )
92. KrY-ClfCTeAlB H ypaBHeHIIJI HUbe-OroKca
123
H3 BbJpaJKeHHH
f & ( _! lP 10 a 10 _
c"p at 2 T &t2 + c"k ax"
lJ 810 a T 810 )
- T- /c- - - i{k-- d(.
at ax" 8Xi 2 8x,=
(2.13)
Y"IHTblB8$i OCHOBlloe IIOJJO>KeHHe npo eAYPbl QenMeHa-3HcKora.
J fcp( )d = J fO'P( )d() K TaKOMY )J(e pe3YJIbTaTY np eM. HHTe-
rpHpya C secoy p 6aJ1aHCHOe ypaBHeHHe (5.11) H3 rn. II nR o o-
qaCTHQHOR cl>YHK HH pacnpe,a.eneHH9
'+1 j
J1 r- 10 + div( /oP = L ; LI6 J
(2.14)
TO eCTb K ypasHeHHBM (2.2) H (2.3) KBa3Hra:JO HH3.Ml!qecKOR CRCTe-
Mbl.
HCDOJtb3YB npOl{e,zurpy, aHaJlOrH"IHYIO (2.9)-(2.13), MO>KHO nOKa-
3aTb, 'ITO H3 6aJIaHCHOrO ypaBHeH1iSl (2.14) 6y eT nonY'IeHO C TOq-
HOCTblO o 0(7 2 ) ypaBHeHHe 3HeprHH H3 CHCTeMbI HaBhe-CTOKca,
BKJIlOqalO ee B Ka'4ecTBe CCHnaTHBHbIX 'IneHOB pa60Ty Ba3KHX
CHn 8(UkPilc)/8xi H divq, r,ne q - BeKTOp TennoBoro nOTOKa.
IIpH 3TOM ew.e pa3 3aueTHM, 'fT() 8 paMKaX KrY-MO,neJIH B
cl>YHK HOJ-Ia.nbHOM BhIpaJKeHIDI, onHCblBalO el,l 8KJI B ypaBHeHHe
3}IeprHH TennOBoro nOTOKa, qHCJlO npalVl-TnB nOJIaI'aeTCH paBHblM
era pea.nbHOMY 3Ha IeHHIO ,-Qn51 COOTBeTC1."BYI01LI.HX ra30B H cMeceA.
Taft:tJM 06pa30M, Kea3U2a3oduHl1.Wu4tectcaJi CUCmeMa (2.1)-(2.4)
c mo"tHocmb70 do O(r 2 ) nepe.xooom 8 cucme.wy yptI6HeHu(J Ha6'be-
CmOKca.
Pe3ynbTaTLI, pacCMOTpeHHhle B aHHOM nYHKTe, COOTBeTCTBylOT
pa60Te [60). PaJlee B pa60rre [61J paCCMaTPHBa.nacb CBH3b OHap-
HbIX sapHaHToB Kry H ypaBHeHHft Hasbe-CTOKca. O aKO OCHOB-
HoA 06nacTblO npHMeHeHHjf] r e Kry nOKa.38J10 CBOH B03MO>KHOCTH,
8JljleTCH Mo enHpoB3.HHe He HOHapHhIX Te eHH:A BH3KOro r33a,
r.Dte CYW.eCTB)'IOT HaH60nbillHe npo6neMbl, CBSl3aJIHbJe C HCnOJlb30Ba-
HHa.l YPaBHeHHA HaBbe-CTOKca. 9TH TPYAHOCTH HaH60JIee 3a.MeT-
HbI npH eTaJlbHOM npocTpaHCTBeHHO-BpeMeHHOM ODecamm cpe ,
OlTHparo eMcH Ha HCnOJIL30BaHHe Bb1COKOnpoH3BO HTenhHblX MHO-
ronpo eccopHblX BblQHC1IHTeJlbHhIX CHCTeM. KpOMe TOro. JIpH HnH-
aJlbHO 8a*HbJM TIPH nOCTpOeHHH Kry SlSnSleTCS ee H OHapH
3arJHC.b B KOHe'iHo-pa3HOCTHOM B e.
BHHMaTeJlbHhlA: qMT&TeJIb 3aMeTHJI, 'ITO npoBe,L\euHoe B ,D:8.HHOM
nYHKTe HCCJIe,QOOaHHe CBB3H Kry H CHCTeMbI HaBLe-CTOKCa cy e-
CTBeHHblM 06p8.30M HCfiOJIb3yeT npe nonO)KeHHe 0 MaJIOCTH ,lJ){CCHna-
THBHblX qJleHOB no CpaBHeHHlO C q.neH8.MH KOHBeKTHBHhIMH. OAHaKO
B TeqeHHHX ra3a Hepe.AJ(O BCTpeqalOTCSI cHTyaqHH, KorAa Bfl3KHe qne-
Hbi no nOpg KY BeJlH H}ILI COBrr a.IOT c qJleHaMH KOHBeKrHBHbIMM)
HanpHMep, B nOrpaHH'IHbIX CJlOSIX. B CJIeA}'IO eM nYHKTe naparpa-
<t>a, OnHpagCL Ha pa60TbI (53, 62]) npOBe eM Ol\eHKH HCCHnaTHBHblX
qneHOB Kry, CnpaBe,D.nHBble H ,lI,JISI 3TOro CJlyqaH.
3. j{HccHIlaTHBHble CfJleHbl KBa3Hf'830 HHaMJfqeCKOR
CHCTeMbl
O eHKY nop Ka ,lJ){CCHllaTHBHbIX QJ]eHOB H81.fHeM c npasoA qa-
CTH ypaBHeHHH (2.1), KOTOpylO npe CTaBHM B BRAe CYMMbI
{) TOT
--A+ --B
ax 2 8y 2
(2.15)
I Ae
a 2 a {) a 2
A = ax (pu + p) + ay(puv), B = ax (puv) + ay(pv + p).
113 YPaBHeHHA (2.2) H (2.3) cneA}'eT, TfTO
(pu)j+l_(pu)' (pv)j+l_(pv)i
A = - + EA, B = - + E B1 (2.16)
T T
r)];e EA H EB - t{J]eHbl, nponOp OH3JlbHble T, BXO SI e B np8.BbJe
qaCTH ypaBHeHHA (2.2) H (2.3). nOpSlAOK Be.nIRHHbI 3THX 'CI.'1eHOB
{ pU3 U }
[EA, E8] < max T £2 ,T £2 ·
3 ech L - xapaKTepHhlH pa3Mep 3 aqH, C H U - Xap8.KTepHble
CKOpOCTH 3BYKa H MaKpocKonH'IeCKOrO TeqeHHH raaa.
(2.17)
.92. I<rY-CJICTeMa H YPaBHeHHH .H8JJMrOroKca
125
YqHTLlBM, '1TO U 2 / £2 = lit; H c 2 / L 2 = 1 It - BeJUI'lHHhI,
o6paTHble KB3Jij>aTaM ra30 HHaMH'IeCKOrO BpeMeHH t g H BpeMeHH
pacnpOCTpaHeHHB 3BYKOBbJX B03MYW.eHHA t c , nepenHlUeM (2.17) B
BH)J.e
{ T pU T pU }
(EA,EBJ:$ max 2" t ' 2 t ·
B CBOIO Oqepe b) 83 (2.16)-(2.18) CJle,n; YeT , QTO
[ 8 'T a T ] { T pU T 2 pU T 2 pU }
ax 2 A + 8y 2 B max: t g L' t; L' t L '
r e qJIel1 T pU / (tgL) CBSI3aH C HecraQ.HOHapHOCThlO KB&3Hr'a30 HHa-
MHQeCKHX ypasHeHHA: (2.1)-(2.5).
TaKHM 06paaoM, 93. (2.19) CJI eTJ tITO OTJIOmeHHe qneHoB) BXO-
lJ;SlI.I.J.HX B npaBYIO 1.faCTb (2.1), K qneHaM B neBoA qacrH, HMelO M
nopH):t.OK pU I £, eCTb nH60 T ltg, JUl60 T 2 /t;, JIH60 T 2 /t . Yqm'b(B8.H,
t{'ro J,\nH nnOTHblX ra30B BpeMSI Me>K,Ay CTOJIKHOBeHHjlMH MWIeKYJI
T MaJlO no cpa8HeHHIO C raJO,D.HHaMH'9:OCKHM BpeMeHeM t c , MO>KHO
c,aenaTb BbIBOA, 1.fTO HCCHDaTHBHbJe qJleHbl J CTOBlI\He B npasoA qa-
CTH (2.1), MaJIbl no cpasaeHHIO C Qnelf8.MH H3 JIeBo:A 'IaCTH 3Toro
ypaBHellHSI.
3 aM e q au He 2.4. r83QlX,H HaMH'IeCKOe H 3BYKOBoe BpeM ,LI.a>Ke AJIjI
1'OHKHX nOrpaH""IHbIX CJ10eB B 6onLwHHcToe 3lV1.&'I MHOro 6011bwe BpeweHH
Me)t(JQ' CTOJIKHOBeHHSlNH MoneKYJI'T. n03TOMY qneHbl J CTO.RInHe B npaeoA
qaCTH (2.1), BCer,Aa MaJIbI no cpa8HeHMIO C QneH8MH, CTOJI HMH 8 neBoA
qacTH 3TOro ypaBHeHIUI. B 3TOM KpoeTai KopeHHoe pa3JIB'IHe we>K.nY ,.[UIC-
CHnaTHBHblMH 'IneHaMH 8 ypa8HelIHH (2.1») H qneHaMH, OnHCblB8JO HMH
HcrHHHYIO BK3KOCTb B YP8DHeHIDIX HaBb&-CTOKca, KOTOpbJe B DOrpaHH'I-
HOM cnoe HldelOT O)J)lH H TOT >Ke nop AOK C KOHBeKTHBHblMH qneHaMH.
IIpoBe eM aHaJlOrHQHblH aHaJlH3 .D-HCCHOaTHBHbIX 'IneHOB ypas-
HeHHA: (2.2) H (2.3), onHCblBaJO X H3MeHeHHSI HMnYJIbCa B KBa3H-
ra30 HHaMHqeCKOA CH(''TeMe. 3anHUleM 3TH ypaBHeHH5I B epreHT-
HOA'I BH.n.e:
(2.18)
(2.19)
(pu)1+1 - (pu)j on%Z aII%JI _ 0
+ a + 8 -,
T X Y
(pv)1+1 - (pv)j 8IIyz all w _ 0
T +ax+{)y-'
(2.20)
(2.21 )
126
rJI. III. MQlteJlHpOB8JfJle TeYfenHR BJl3KOro m.Ja
KOMIIOHeHTbI DOJIROrO TeH30pa Hanpa>KeHHA Onpe,n:eJlSlIOTCSJ CJJe -
IOI1I.HM 06pa30M:
2 TO 3 r8 2
fin = pu + P - 2 a;(ptc + 3pu) - 2 &y (pu V + pv), (2.22)
; a 2 i 8 2
fix" = fI yz = fYUv - 2 ax (pu v + pv) - "2lii/(PV u + pu), (2.23)
II = pv2 + P _ T (pv2u + pu) _ T (pv3 + 3pv) (2.24)
yy 2 8:& 2 8y ·
,LLHccHoaTHBHbJe T.fJleHLJ, nponOpUHOHaJlLHble T, CTORIl\He B npa-
BblX 4a.cTSiX (2.22)-(2.24). pa3)J.enHM Ha ,lU)e rpynIIW. B nepsyIO
rpynrry BKJIlOqHM qJIeHbI, JIHHeAHO 3aBHCmI He OT CKOpoCTH, 80 BTO-
PYIO - qneHbI, nponOpIJ,HOHanbHble TpeTbe:R CTenelIH MaKpOCKORH-
qecKoA CKOpOCTH.
OrHorneHHe ttJIeHOB 8TOpoR rpynnbJ K KOHBeKTHBHblM t{JIeHaM
pu2, puv, pv2 Bcer a MaJTO K onpe)I.CJlHeTCH JIH60 KaK r(ul Lz), nH60
KaK r(v/ Ly). ,l(pyrHMH CJlOBaMH, ,lVIB HHX, KaK H AnB ,lI.HCCHnaTHB-
HbIX t.fJleHOB (2.1), Crrpa.BeA1IHB& 0l\eHKa no BMH1.fHHe MaJlOCTH OTHO-
CHTeJIbHO qJleHOB 83 neBoR 8CTH YPaBHeHHA (2.2) H (2.3), paBHaH
rJt g .
QTo6bJ o eHHTb I10p OK QJIeHOB nepBoA rpynUbJ) npe craBHM,
HanpHMep, AJISI ypaBHeHHJt (2.20) HX BKJI B CJle.nyJO eM BH e
fJ T au 8 T 8u a ( [ 00 lJ1J ] )
D = a;P 2 {h + 8y P"2 8y + ax pT & + 8y +
+ ( 8TP . + 8Tp . &v ) + ( T u + v 8p ) + (2.25)
2 ax ax By ax ox 2 ax 2 8y
a r 8p
+ ay'2 u a;'
YqHTbJ8aSl Bblpa>KeHHe (2.5) nj! onpeAeJIeHH9 T, nepenHweM
nepBble )J.BR \{JJeHa B npaBoA 'facTa (2.25) B BHAe
a8u aau
Dt = -P- + -IJ-,
ax ax {}y 8y
(2.26)
j2. KrY-CHCTeM8. H yp&BHeHHll HUbe-CTOKC&
127
KOTopble npeft-CTaB.n1lJOT c060A BSI3KHe 1{JIeHbI J BXO lII)Ie B ypaB-
HeHHSI HaBbe-CTOKCa. ,lI;IH OCTaJlbHLIX qJleHOB Bblpa)J(eHHSI (2.15)
cnpaseNIHBa oqeHKa no nopH KY BeJIH'IHHbl, pasHaB TIt, [gradp].
TaKHM 06pa30M, MCCHnaTHBHhJe qneHbI B ypaBHeHHJlX (2.2) K
(2.3) KBa3Hr330 HHaMHqeCKOR CHCTeMLI nH60 no DOP KY Be1lDqH-
lib! 8 T / t g p83 MeHbwe KOHBeKTHBHblX, nH60 COBD aIOT C BSl3KHMH
.
'{JIeHa ll-I COOTBeTCTBYIOIU.HX ypaoHeHHA HaBbe-CTOKca.
YKa3aHliblM Bblme cnoco60M H3 JI.HCCKnaTHBHblX qneHOB ypasHe-
HHR (2.2) Ii (2.3) KrY-cHCTeMbI 6blJlH BbJ eJIeHhJ 1.IJleHLJ ecTeCTBeH-
110M BSl3KOCTH. H o eHKH OCTd.BWHXCSI qneHOB cy eCTBYIOT ABC
B03MO>KHOCTH. B nepBOM cnyqae ORB, KaK 3TO cne eT H3 n.2 aH-
Horo naparpaQ:>a, MaJIhI DO CpaBHeHWO C KOHseKTHBHblMH qneH8.MH
Ii lJJleHaMH, onHcLiBaro HMH eCTeCTBeHH}'1O BSI3KOCTh. Bo BTOpON J
eCJIH He npHHHMaTL DO BHHMaflHe ycnoBHe MaJlOCTH eCTeCTBeHHbIX
CCHnaTHBHbIX 'UleHOB no CpaBHeHHIO C KOHBeKTHBHhlMH. OUB MaJIhI
no CpaBHeHHIO C KOHBeKTHBHhIMH. AHaJIOrUqHYJO OQeHKY HCCHna-
THBHblX qJleHOB J C BbiAeJIeHHeM ttJJel10S ecTeCTBeHHOA TennonpOBOA-
HOCTH If pa60ThI BSl3KHX CHJI, MO)KHO IIpo enaTb H AJlSI ypasHeHHSI
3HeprHH (2.4) KBa3Hra30 HHaMHqeCKoA: CHCTeMbI.
4. HeKoTOpble pe3YJlbTaTbI IfHCJ1eHHblX pacqeTOB
npoBe eHHbJe B 0.2 H 3 a.HHOrO naparpa<j>a pacCY>KAeHH no-
Ka3bJBaJOT, "liTO peweHHSI 3 aq B TepMHHax Kry -CHCTeMhI H ypas-
HeHHR Hashe-CTOKca AOn)KHbI npaJ<THtfeCKH COBD&AaTb, TO H no -
TSep>KAaeTCH MHorotJHCJleH1IbIMH TeCTOSblMH pac eT8MH. B KaqecTBe
npHMepOB npHBe eM AaHHbIe pac eToB 3 aqH 0 YMepHOM o6Te-
KaHRH rrnacTHHbI no,n: HYJIeBblM yrJloM aT8KH A03BYKOBhlM nOTOKOM
B03AYxa (63) H 06TeKaHHB 60nee CJI0>KHOrO Tena HJlHH)ij>HqeckH
cHM le'rpHq}fOa <l>OpMLI csepx3BYKOBblM nOTOKOM (64).
,aJIH nepBo:A 38JJ-a'lH pemeHHe KB83HraaO HHaMBqecKOa: CHCTeMbI
Cpa.nHHB8JlOCb C TOQHblM aBTOMO enbHblM peweHHeu 3 a H BJla3H-
yea, llOJIyqeHHblM Ha OCHOBe ypaBHeHHa Ha.Bbe-CTOKCa "PH CKOpo-
CTH Ha6eralOInero nOTOKa H& 6ecKOHeQHOCTH Moo = 0.08. t.{HCJIeH-
HbIe MeTOtI-bJ peWCHHSI KrY-cHCTeMbJ, KOTOphlM 6YAeT nOCBHIII-eHa.
rnaBa IV, .lI.JI.H 3TO:A H CJle YIO eA 3 a.qH 3,l\eCb He 06cY>KAa.IOTca.
1215
rJl. Ill. MOAeJ1HpoBaHHe TelfeHH# BS3KOro ra:Ja
1.0012
T
1.0010
1 0004
1.0008
1.0006
1 .0002
T
1.00DO ..----.....---.....----...----
a) 0 2 4 e
0.10
U
0.08
D.De
C.04
0.02
000
6) 0 2 8
8
9 10
8
V 10
PIIC. 20. Pac&feT 38tt8'IH 06 06TexaHlIH DJJ8CTHHW C no,.o 1O KfY-atCTeMhI
(x x x) H peWeJfHe ypaaueKHA Hube"-CT()J(ca (CDJIOIDH8JI JIHBHB): a) npo(j>HJD.
TeNnepaTyphi B OAHOM IC cpe30B. nepOeR;ulxyngpHWX DJ1&CTHHe. 6) upocpHJlb
CKOpoCTH
L 0.1011 m
0.0433
Q.<W
15°
0.0575
PHC. 21. KOHcPHryp81l;JlJl OCecHMMeTpH HOro Tena
j2. KrY-cIICTeM8 I' ypasHeHHR H8.Bbe-CTOKC8
129
KaK BH HO H3 pHC. 20. Me>K ABYMg peweHHHMH 38J\&1.fH, nOJ1Y-
qeHHblMH Ha OCHOBe HCnOJIb30BaHHB Pa3JlHQHblX eHcreM ypaBHeHHA)
I1paKTlNeCKH OTCYTCTSYIOT pacxo eHHSI.
8 Kat.lecTBe BToporo npH-
tepa paccMoTpH r Pe3YJlb-
'"raTbl pacqeTOB 06TeKaHWI
l HJIHH.APHqecKH cHMMeTpHt.f-
HOro TeJla, <l>opMa KOToporo
npHBe)J.eHa Ha pRC. 21. Pac-
cMaTpHBaJlHCb CJ1eAYIO He
napaMeTpbI Ha6erarorne-
1"0 nOTOKa: Poo = 6.3 na,
Too = 51 K, Moo = 9.91.
TeMncpaTypa nOBepXHOCTH
Tena npe,n.nOJlaraJ1acb nocTO- PHC. 22. PacnpeAeJleHHe )J.aBJIeHHH
HHHOA: Tw = 293 K. '-IHCJIO B;J:Mb OCH OX (ycnoBHe npHJllmauHB):
PeHHonb ca1 pacC HT8.HHoe EXP - 3KCnepBMeHT, DSMC - pac-
'leT no KHHeTHqecKol M eJIH MeTO OM
no OTHoweHHIO K JlHHe MORTe-Kapno, NS - pac eT H& OCROBe
L = 0.1017 M (pRC. 21), ypauHeHKA HaDbe-CTOKC&. QGD - pac-
6b1J1o paBHO Re = 18916. 'leT IJ3 OCHORe KrY-CHcTeWLJ
Ha pHC. 22-24 npeJtCTaB1IeHbI npotj)HJlH -naBJIeHHJI, I1JIOTHOC11f H TeM-
neparypbl, nonyqeHHhle H8 OCHOBe 3KcnepHMeHT8J1bHb1X ,Z:taHHbIX, paclfe-
T8 nYTeM CT8THCTH'IeCKoro MO,IJ.enHpOBaHHH C npHMCHeHHeM MeTOJt8
MOHTe-Kapno H Jl.8HHhlX paCIJeT8, nonyqeHHhlx C nOM01IJ.blO YpaBHeHHH
HaBbe-CTOKca H KrY-CHCTeMbI*. Ha pHC. 23 H 24 npe,aCTaBneHhl npo-
<PunK nJlOTHOCTH H TeMnep81Yphl, AJ1J1 paClfeT8 KOTOpblX B ypaBHeHHJI
HaSbe-CTOKca H KrY -CHCTeMbl BKJllOqanHCh kaK ycnOBHI npHflHnaHHJI,
TalC H ycnOBHSI CKOnb)I(CHHJI.
rpaHHQHble yCJ10BHJf CKOnb)KeHWI
.jip.8u
u= -,
P v 2RT an
l"lI-e a 18n - npoH3BOAHaJI no HOpMaJIH K nOBepXHOCTH Te.na t H TeMnepa-
TypHoro CKa'lK8
c
10 ·
DSIIC
. EXP
____lIS
-.-.--- aGD
.
0.5
1
XlL 1.5
(2.27)
Vi ( 1 HaT )
Tl - Tw:; 2R ";2[([' p an I
(2.28 )
· npHt.tHHa ueKOTOphiX pacxO>K.JJ-eHHA c A8.H WH 3KcneplDleHTB H DSMC-
MO)l.eJ1HpooaHHH KpOeTCH B B03MO>KII H raJOAHHaMH'lecKOro OnHCaHHR npH
QHCJ1aX KHY!J.ceHa Kn > 0.01. OHa 6y,neT o6cY1K.n eJJa 60Jlee IIoAP06HO B 15 rJla-
BhI V.
0.,
YI\.
.
.1
y
U!I
DSIIC
----. ..
....,...-.
am
.
- DIMe
- - :r--
-.- -. QGG
u..o.1
o
I
I
pip""
.
I TIT""
PHC. 23. PaCnpe.lI.e.neHHe OJ1OTHOCTH
B,Ltonb OCH OY AIlS! rpaHH'fHhlx ycno-
8HH npWIHI1aHHH (nonslip) H CKOJIb-
)l(eHHS (slip) npH xl L = 0.3
PHC. 24. Pacnp eJleHHe 'teMoepaTy-
phi B)!OJIIt OCM OY npH %IL = 0.3
rJJ-e T w - TeMnepaTypa rena, a T 1 - TeMnepaTypa r838 B OKpeCTHOCTH
TeJI&, xopowo H3BeCTHbI 8 TeOpHH p83pe>KeHBhlX r830B [65]. 9TH ycnOBHB
n03BOMIIOT He paccuaTpHB&Th B ,D.eTaJIBX nOBeAeHHe ra3 HHawH'IeCKHX
napaMeTpoB B y3KOM npHCTeHO HOM CJloe nopH,AKa HeCKOJIbKHX A1IHH CB()-
6oJUIoro npo6era.
Pe3YJlbTaTbl, npep-CTaBJIeHHble Ha pHC. 22-24. no K83.bISa.1OT) QTO
aHHble, fiOJlY\feHHhle Ha OCHOBe HCUOJlb30BaHHH KBa3Hra30AHHaMH-
lfecKoA CHCTeMbJ, TaX )ICe KaK H B npeAbJAy eM npHMepe, DpaKT e-
CKH He OTJIHQ8JOTCa OT 8.HHbIX, fiOJIY'IeHHhIX c nOMO 1O YPaBHeHHA
HaBLe-CTOKca. 9To nO TBep a.eT TeopeTH'IecKHe o eHKHt npo e-
JJaHHhle B n.3 H 4 HacTO.S1 ero naparpa4>a.
CytdMHPYSI pe3ynbTaThI ,qa.HHOro naparpacPa, OTMeTHM CJle)U'1O-
ee. K 8Q3U2(J,300uH4MU C"aJi cucme.wa ypa6 HUU MO:>ICem 6t.lmb
npeacmaeJr.eHa 6 800e ypatmeHuU Ha6'be-CmOKOO, " 1Wmo ao-
6a8/&eH1M dOnOAHUme.JlbH&£e duccunamU6HtJle eH1»C:
Kry =H-C + ,[(HC.
9mu HW ,l1.HC 0Ir03 6aJOmCR MaJ& U no Cpa8HeHUto c ecme-
cm6eHti'bLMU duCC!JnamUSHt»&MU eHl1MU. Hx Rp1Jcymcmaue He
8Hocum C1COAb1CO-'liu6ycJb 3aACemHtIU U3MeHeHul1 8 peweHue KFY-
cucmeMW no Cpa8HeH!J1O C peweHtJeM J nOAY HH Ha OCH06e uc-
93. KHHeTHtfeCKDe OrpaHCeHHe KrY-CHCTeMbI
131
nOJt'b306aHUJt ypa6HeHuu Ha8be-Cmcmca.
POJ1b 3THX OnOJIHHTenbHbIX qneHOB ):\HC 3aKJI1O aeTCS1 B ra,-
panTMpOBaHHOM Crna>KHBaHHH pemeHHB Ha pacCTOSlHHH nop Ka
)J.JIHHbl cBo60 Horo npo6era. KaK 6Y eT BH HO H3 AaJlLHeftmero, 3'1'0
CBoRCTBO OKa3bJBaeTCSI seCLMa nOJle3HblM npH MO):teJIHpOBaHHH MHO-
rHX CJIO)KHblX TeqeHHR Ba3Koro ra3a.
3. KHHeTHqeCKOe OTpa>KeOHe
KBa3Hr830,ll;HHaMH'qecKol CHCTeMLI ypaBHeBHH
B 3TOM naparpa<t>e OKOHQaTeJlbHO nO Be eM HTorH o6cy eHHSI
KCPC H Kry -CHCTeM&I KaK <PH3H'IecKOA Mo enH SI onHcaHHH Te-
t.f€IIHg BSi3KOro ra3a H HX CB 3H C BH OM O,[\HO"tfaCTHqHOA Q>YHKQHH
paco pe eJleHHfl.
1. KrY-cHCTeM& H OAHOqacTHqHaH cPYHKl.{HSl
pacnpe)1.eJleHHH
CHcTeuy ypaBHeHHA 9Anepa)VI.H HeBSl3Koro, ReTenJlOnpoBO,L\HO-
ro r83a MO}KHO nOJlyqHTb, ecnH enaTb npe onO)f(eHHe 0 nOKaJlLHO-
M8KCBeJInOBCKOM paBHOBecHOM xapaKTepe O HO'facrHqHOR <PYHKn:HH
pacnpe eJIeHH.H. YpaBHeHHH HaBbe-CTOKca MorYT 6L1Tb 1I000YQeHbI,
eCJIH )VIS npe)J.CTaBneHBB ct>YHK HH pacnpe eJIeHHg HCnOJlb30BaTb
a nepBhlX tfJIeHa pa3nO>KeHHB QenMeHa-9HcKora peweHHB KHHe-
TH eCKoro ypaoHeHHSI BOJIh aHa.
KCPC (KrY-cHCTeMa) onHpalOTCH Ha npe.D-D0nO)l(eHHe 0 MaKC-
BeJIJlOBCKOM xapaKTepe <PYHK H pacnpe.a-eneHHR. 9'1'8, O)I.HOqacTH'I-
t18J1 cPYHKI.{HSI pacnpe,r(eneHHH nOCT05lHHa B 06n8CTH p JIHHeftHLle Pa3-
Mephl KOTOpoR paBHbl )J.JIHHe cB06oAHoro npo6era I. (,lI:nsr Kry-
CHCTeMbI paBHOBecHaJI <pYHKnHH pacnpe,a.eJIeHH.R on>KHa cna60 Me-
HS1TbCH Ha paCCTOSlHHH nOp.SlAKa MHHbI l).
)])tCCHDaTHBHbre qneHhI KCPC nOmuISDOTCB 3a Ct{eT Han HSI
ABYX Pa3J1HQHblX MaKCBennOBCKHX pacn eneHHA no o6e CTOpOHhI
OT rpaHHQbJ HqeAKH. ,l(Ba paaJIHIfHblX paBHOBeCHhlX pacnpe.a-eneHHSI
H npHBO,lJ.SlT K nOSlBJIeHHIO AH(j><PY3HOHHblX MOJIeKyngpHLIX nOTOKOB
Ha rpaHH e, HCPcl>epeH anbHbIM OTpa>KeHHeM KOTOphlX BBJISIIOT-
132
rJl. III. M enHpOB&Hlle recteHH# BH3KOro raJa
CH npaBhle qaCTH B KBa3Hra30.AHH \(HqecKOA CHCTeMe. ilOA'IepKHeM,
\lTO JIH KrY-cHcTeMhI Ba>KHO JlHlIIb cYlI(eCTBOBaHHe cna60 MeHHIO-
eetcR 8 npOCTpaHCTB€HHOA 05J1&CTH paJMepa l paBHOSOCHOH <l>YHK-
QHH pacnpe eneliHH. KOHKpeTHbIA Blip. oA paBHoBecHOH CPYHK HH
BXOtl.HT B cHCTe fY (2.1)-(2.5) HeHBHblM 06pa30M Qepe3 K03cP<PHU.H-
eH'ThI nepeHoca Ii ypaBHeHHSI COCTORHHSI.
OTMeTHM cw.e pa3, qTO HHKaKOro KapAHHaJIbHOrO nepecMoTJ)a
KIUJeTHT.feCKOro o60cHOBaHHS ypaBHeHHA, ODHCbIBaJO r'a3Q,n;HHa-
MHqeCKHe npoll.eCCbl, He COBepWaeTCSI. B CaMOM ,n.ene, BbJBO ypasHe-
HHIt ra. oBoR ,lX.HHaMHKH) B TOM qHCJle H HaBhe-CTOKca, OnHpaeTCSI Ha
6nH30CTb <PYIIKQHH pacnpe eneHHSI K MaKCBMJlOBCKOMY p8BHOBec-
HO (Y BIf..QY. C ,n;pyroA CTOpOHbI, H3 peweHHB KHHeI'HqeCKOro ypaB-
HeHHH CJIe YeT, QTO cPYHKIJ)IH/pacnpe eJleHHSI 6nH3Ka K nOkaJIbHo-
p8.BHOBecHOA, eCJIa OHa c.na60 MeHSIeTCH Ha pacCTOSIHHH nopSl Ka
AnHHhl CBo6o HOro npo6era I. HMeHIIO 3TO CBoRCTBO, HCnOJIb3ye-
Moe gBHO, nOJlO}l(eJIO B OCHOBY KHHeTHQecKoro 060cHOBaHHSI KCPC
H KrY-CHCTeMbl.
JIlo6onblTHa aHaJIOrHB Me y pa3JIO>KeHHHMH B pH no MaJlO-
MY napaMeTpy peWeHHjI ypaBHeHH BOJIbl.\MaHa, npeAJIO>KeHHbIMH
fHJIb6epToM (21, a 3aTeM 1..JenMeHoM H 9HCKOroM, H npe CTaBJIeHH-
eM <PYHK HH pacrrpeAeJ1eHHSI, HcnWlb3yeMblM AJlSI nonyqeHHH KBa-
3Hra30 HHaMH'IeCKOH CHCTeMbJ ypaBHeHHA. Pa3JI01KeHHe qenMeHa-
9HCKora BJISIeTCH 60J1ee pa HOH8JlbHbIM, qeM p83JlO>KeHHe rHJJb-
6epTa, T8.K KaK onHcaHHe BSl3Koro ra3a B03MO>KHO C IIOMOUJ.b1O TOJIb-
KO JlH11Jh nepBbJX ,n.nyx t.tJIeHOB acHMllTOTHqecKOro pH a ):\JIS <PYHK-
D;HH pacnpeAeJIeHHH.. Pau.HOHaJlbHOCTb OCTHraercSl 38. CqeT 51BHOro
Y'IeTa, Toro <paKT8. t 'ITO B5JIH3H paSHOBeCHH <PYHK SI pacnpe.n-eJle-
HHSI onpe eJl$IeTCSl C nOMO blO nwrH ra30 liaMHqecKHx napaMeTpoB
p, u, T, KOTopble, B CBOlO oqe , J1S1 K8JKAOrO QneH8. pa3J10JKeHHH
HaxO HTCSl H3 CBoe:B CHCTeMbl MaKpocKOllHqecKHX ypaaHeHHA.
npH nocTpoellHH KCPC H KB83"ra30 HHaMHqecKo:A CHCTeMbI
nSl OnHCaHJ.1i! BSI3KOro TeqeHHS, no CYTH AeJla, HCllOJJb3YeTCSI Bcero
· Bonee Toro, oc-raeTCR OTXPWTOA C8M& npHH1.UlDHBJlbH8S B03M01KHOCTb OIDl-
. C8JlHH C nOMOUJ.b1O pa3J1o)f(eHHJI rBJlb6epTa Te-JeHHJI BH3KOro ra:Ja npH HaJIH-
'4HH o6nacTeii CHJJbHOrO H3MeHeHH napaweTpoD, Hanpm.cep, norp&llJlq)fLlx CJlo-
e8 (29].
93. KHHeTlf'lecKoe OTJ)l1)KeHse KrY"CHCI'eMIJ
133
OAHH 'IneH B pa3JJO>KeHHH. ,D,anhHeAmee YMeHbmeHHe qHCJIa tJJJeHOB 8
pa3JlO}KeJ-IHH o&bsrcHSIeTCH 'reM, 'ITO nOMHMO 38BHCHMOCTH <l>YHK HH
pacnpe eJIeHHjI B6JIH3H pasHoBeCHlI OT napaMeTpOB p, u, T TaK.>Ke
SlDHO Y"IHTblBaeTCR TOT <paKT, tfTO cf>YHK 1W pacnp eneHH.H cna60
MeHHeTCH Ha pacCTOs(HH X nOp AKa .lV1HIILl CBo6o,n;Horo npo6era I.
B nJIaHe anbHeAwero o6cY>KAeHH KHHeTH\feCKOI'O o60cHoBa-
HHH KrY-CHcTeMbl paccMOTpHM ee BblBO,l\, HCnOJlb3YS1 HenOCpeA-
CTBellHO ft,IO,lJ.eJlb EXaTHarapa-rpocca-KpYKa (rn. I) .nnSi ODHcallHSI
3BonlO4HH cf>YHK HR pacnpe,QeJleHHH.
2. KBa3Hra30 HHaMH eCKB.H CHCTeMlJ If BrK-ypaBHeHlfe
OCHOBOA BblBo a Kry -CHCTeMbI SlBJIBeTC$l 6anaHcHoe ypaBlleHHe
(rn. II):
/ ' + 1 / j
- M + Lf1..t = L T Lfl"
T 2
{}f
I' e Lf = {i-.
{JXi
(3.1 )
PacCMOTpH Tenepb, KaJ(, OnHpaHCb Ha BrK-MO Mb
of 8f
at + i OXi = II (f M - f) ,
(3.2)
r e 1/-1 HMeeT nOpSl,l(OK BpeMeHH Me.>KAY' CTOJIKHOBeHHHMH T, no-
JlYtfHTb KBa3Hra30 HHaMHqecKYIO CHCTeMY ypaBHeHHA [53, 66). Ha-
nOMHHM OCIIOBHhle ycnOBHB, "PH KOTOpblX 6b1J1& nWJyqeHa Kry-
CHCTeMa:
a) nOCJIe CTOJIKltOBeHHSI <J:>YHKl.\HH pacnpeAeJIeHHSl CTaHOBHTCSI MaKC-
BennoBcKoH;
6) cf>YHK SI pacnpe eJIeHH.H cna60 MeHHeTCSI Ha pacCTOJlHHH no-
pRJl.K8. )VIHHbl cBo6o HOro npo6era H 8 Te'tJeHHH BpeMeHH no-
pH Ka BpeMeHH Me>K.LU' CTOJIKHOSeHHHMH MOJIeKYJI;
8) ,LJ,JUI CYMMa.TOpHbIX HHBapaaHT08 cp(,) = 1,£,,{2/2 HMeeT MeCTO
COOTHOllleHHe I fj+1f{)( )d = f If: Ip(£,) .
B CBOIO O'iepe , BrK-MO eJlb nonY'IeHa H3 ypaBHeUHH BOJIb Ma.Ha
np" BbInORHeHHH YCJlOBHSI a) (CM. rJl. I).
BYAeM HCKaTb pemeHHe ypaBHeHHSI (3.2) B B e pa3JJO;)l(eHHH B
pSl,l:\ no MaJlOMY napaMeTPY f, no nOpB.AKY BeJIH'lUHbI COIsn aJO e-
MY C OTtJOWeHHeM BpeMeHH Me)J( Y CTOJIKHOBeHH.SIMH MOJIeKYJI T Ii
xapaK'repHbJM ra30,lJ.HHaMHQeCKMM speMeHeM t g .
1
£ I'J T It g rev -,
vt g
(3.3)
1 = 1M - aiM _ ! 81M + !. lP 1M + (i 81M +
1/ 8t v 8 X i 11 2 &t 2 1J2 at iJXi,
+ a 2 f.if.kf M + O(c 3 ).
£12 OXi(}Xk
Bee 1.JneHbl t BXOA e B BbJpa>KeHHe (3.4), 6y eM pacCM&TPH-
BaTh Ha MOMeHT speMeHH t = t j + 1 . O HHBM HeKOTOpble qJIeHbJ,
BXO He B npaBYIO qacTb (3.4), npH 9TOM DOJIarasI T = 2/11:
(3.4)
1 [ 0 / i+1 82 / j+l ] 1 [l j + 1 - fj ]
_ _ M + M = _ _ M M + O(E 3 },
v at EJt2 V T
(3.5)
! [ - i 811;1 + 2 f.i 8 1 1: 1 ] = _ 8 1lt. + O(c 3 ). (3.6)
v 8Xi 1/ at 8Xi II 8Xi
OCTaBWHACSI 'IneH npaBo:A 'faCTO BbIp8JKeHHH (3.4) npe,n:CTaBHM 8
BH)J.e
82f.i ldJ:l =! [ {fl i{kJl, ] + 0(£3). (3.7)
11 2 8XiOXit v 2 8XiOXIc
IIo,ACTaBJlSlR (3.5)-(3.7) B p83JI01KeHHe (3.4), OKOH'IaTeJlbHO nony-
'lHM:
J1+1 = 11;1 -
_ ! [ !1;1 - IL + {i 8Jt., - if.A: ] + O(!3).
II T OXi 2 8Xi 8xk,
(3.8)
H3 cBoAcTBa B) C1Ie eT, tITO npH HHTerpHpOBaHHH C CYMMaTOp-
HblMH HHBapHaHTaMH B BLJpaJKeHHH (3.8) cneAVe'!' yqecTb TOJIbKO
93. KHHeTHqecxoe OTplUKeSHe KrY-CBCTeJ4bl
135
'-IJIeHbI, crpynnHpoBaHHbJe rJIe)U'IO M 06Pa30N:
.1+1 j
1 M r- 1 M + LfL = LiLf .
(3.9)
B ypaoHeHHH (3.7), B cHny CBOACTB& B), npB HHTerpHpOBaHHH
. 1
C CYMMaTOpHblMH HHBapHaHTaMH qneH fJ: MO)f(HO 3aMeHHTb Ha
fi+ 1 . B 3TOM cnyqae npH eM K TOqHO TaKOMY }f(e peaYJlbTaTy, 'ttO H
I1PH HHTerpHpOBaHHH 6anaHcHoro ypaBHeHHSI (3.1), TO ecTb K KBa-
3Hra30 HHaMHqecKo:A CHCTeMe.
3 aM e q aH He 3.1. npH Hcnonb30BaHHH npe.a.nonO>KeHHA 0 TOM, 'fro B
MOMeHT BpeMeHH t :;; t; YHKlJ1Ig pacnpe-ZJ.eJIeHHJI SBnJleTaJ M&KCBennOB-
CKoA fJ = f H B MOMeHT BpeMeHH t j + 1 OHa BHOBb CTaHOBHTaI M8KCBeJ1-
JIOBCKOA, KrY-CHCTeM& MO>KeT 6b!Tb fiOJ1}"1eHa H3 60nee cnO)l(HOro YP&BRe-
IIHS!, onHCblBalO ero H3MeHeHHe y pacnpe.L\eneHWI H8 BpeMeBHOM
nonYHHTepBaJIe t E [t j J t; + T'):
8/ 8f 1
+ i 0- :;; -J(/, I).
v, U%i E
(3.10)
3;J.ecb J(f'J f) - HHTerpa.n CTOJlKHOBeHHI YpaBHeHHJI BOJIhl(MaH&, Ha
KOTOpbrA HaJIO>l<eHO OnOJIHHTenbHoe -rpe600aHHe, 'frO ern npoH3BO Hble
BToporo nopH,nKB. H Bhlwe HMelOT O,lJ;BHaKOBhlA nOpH)J.OK C E. ManwA napa-
MeTp HMeeT O HH nOpjlAOK C ,,-J - OTHOmeHHeM BpeMeHH we>K;Jy CTOJIK-
HOBeHHHMH ,. H 133O)J.HHaMH\JecKOro BpeMeHH t g .
H3 c.neJIaHHblX npeAfiOnO)KeHHft cne,AyeT, 'fTO npH t = t i + 1 (,Qo M8XC-
oennH3a1J)lH) peweHHe ypaBHeHHSI (3.10) MOJKeT 6WTb npe,ACTaBneHO B BH-
e
l(x, (, t j + 1 ) = I M(X - f.r', , t i ) +
,
+ ! 'j JU, I)(x - t', I t.i+l - t')dt'.
Eo
AnnpOKcHMHpyeM HHTerpan p CT()jJIltHA B np8.80A 'IacTH Bhlpa>KeHHH
(3.11), no <popMYJle rpanelUiA:
(3.11)
I
f(x, ,tj+l) = /M(X - T', ,ti) + [J(J,/)(x, ,ti+1) +
+ JU, I)(x - r', , t i )] + 0(£3).
(3.12)
TpeTl1A nop oK annpoKcHwau.HH cne;urer H3 en&HBb[X npe)l.l1ono)l(eHHA
OTHOCHTeJIbHO npoH3BOAHblX J{/) f).
136
Tn. III. MOAeJlllpOBl1Iflle TetleRH:II BR3KOro ra:Ja
YqHTblBaB, qTO B cHny MaKCBeJlJJOBCKOro xapaKTepa <lJYHKlUfH pac-
npe eneHHH "PH t = t J
J(j,j)(x - {T',(,t j ) = 0,
rIpe06p83yeM BbIpaJKe"He (3.12) K BHAY
/1+ 1 (x) ) - JI.,(x, ) afit ".' 8 2 fi,
I + t a = _ 2 ' Ii; a 8 +
T Xi Xi %k
+ 2 JU, 1)1+1 + 0(£2).
(3.13)
HHTerpa.n CTOnlOIOBeHHH J{f, ip+l npH HHTerpIIpOB8HHH C CYMwa-
TOpHblt.fH HtiBapHaHTaMH 06paTHTclI B IIYJlb (I'JI. I). n03TOMY npH npose-
,.qeHHH no 06f[oA npOQenypbl C ypaBHeHHeM (3.13) MW BHOBb J K8.K H npH
HCIIORb30BaHHH ypaBHeli1Ul (3.1), np eM K KrY-cHCTeMe ypasHeHHA.
llpO On)f(HM H3YQCHHe cBoACTB KrY-cHCTeMhJ, 06CY 8 Bonpoc
o ee HHBapHaHTHoCTH OTHOCHTcnLHO npe06paaOBaHHS rananeg H
rpynnbI oparn.eHHR (67].
4. 06 HHBapHaHTHOCTH KB83Br830,ll)UlaMH'IeCKOA
CHCTeMhI ypaoHeHHB
Bonpoc 06 HHBapHaHTHOCTH Kry -CHCTeMLI npe,lI.CTaBJIBeTCB JJ.O-
CTaTOt.fHO HeTpHBHaJlbHbIM) YQHTblB3.SI cHnllHYJO) HanpHMep, Ky6HQ-
HYIO o rHOCHTeJlbHO cKopoCTeR B ypaBHeHHSX ,!VIR HMIIYJlbCOB (2.3)-
(2.3)) I-leJIHHeAHOCTb ee npanblx '1acrea.'" PacCMOTpHM BHaqane npo-
6neMY KHsapHaHTHOCTH KBa3Hra30 HHaMHqeCKOa CHCTeMbi OTHOCH-
TeJlbHO npe06pa30BaHHR ra.nHJIeH, Koropble on enxlOTCH <POPMY-
JlaMB
{G}: x'-+x+at) t'-+t, (-+ +a. (4.1)
OCHOBO:A ,lVIg nonyqeHHfl KrY-cHcTeMbi BJIjleTCSI COOTHoweHHe
(rn. II, 5):
f(x, E, t j + 1 ) = fM(X - T, , t'),
(4.2)
· CneroreT OTMeTHTb, 'lTO C npaKTH'fecKOA TO'lKH 3peRR8 pe3YJlbT&TN pacqe-
T8. c 6o.nbWHMil CKOpOCTHAUI (,n.o HecXMbKHX AecRTXOB CeJI Maxa) H8 OCBOBe
HCDOJIb30BaHHK Kr'Y He Bhl3h1B&JJH cOMHeHHA, 'fTO He CHH>AaeT BaJlCHOCTH 'reO-
peTH"IecKoro HCcne;J.OB8HHSI HHBapHaHTHOCI'H 3THX ypaBHeHHA.
94. 06 J.IIIBapH&H7'HOCTH KB83RI83OAHHaMJf lecICOR CnCTeAfW 137
onJ.ICbIBafO ee 6eccTOJJKHOBHTeJIhHbIR p83JIeT qacTH , HMelOIIJ.HX B
MOMeHT BpeMeHH t J MaKCBeJInOBCKOe pacnpe,l1.eneHHe.
B H08blX WTpHxoBaHHbix nepeMeHHblx BLIpa>KeHHe (4.2) npHMeT
B
f (x' , E', t j + 1 ) = f M (X' - (T, ( , t' ) ·
( 4.3)
3,I1.eCb yqTeHO, '1TO npn npe06pa30B8JlHH fa.rIuJleSI (4.1) BpeMSI He
H3MeH5IeTCSI.
113 HHBapHaHTHOCTH Be",iHqHHbJ x - (r CJIe)U'eT, \.ITO BbJpa)KeHHe
(4.2) TaK}f{e HHBapHaHTHO t TO ecTb He3aBHCHMO OT CHCTeMbI KOOp-
,1];HHaT. OHO CBB3bmae-r cl>YH HH pacnpeAeJIeHHg B O HHX H Tex >Ke
TO '{KaX <pH3U'IeCKOrO npoCTpaHCTBa.
Pa3JIO:>KHM o6e qacTH (4.3) B p TeAnopa H BHOBb BOCTlOJIb3y-
eMC.sI RHBapH8.HTHOCTb1O x - Er:
J(x',(,t j + 1 ) = J(x,(,t j + 1 ) +afT O!(X'a::tj+l) +
2 '+1
1 2 8f(x) ttJ) 0( 3 )
+ 2 C1iakT fJ + T,
vXi Xk
, · I. 8JM(X,(,t;}
JM(X - T,C;, t J ) = IM(X,f. ,t J ) + T i a +
Xi
1 2 {f2 fM(X, (, t j ) O( 3 )
+ 2 i kT a f) + 7,
Xi XA:
nO,L\CTaBnaSi 3TH p83JIO)KeHHa B (4.3) H YQHTbJBaH, 'ITO
( 4.4)
(4.5)
( I , fl . ) ( , ) ,OfM(X,(, t j ) 2
f M x - r.; T, ( , = f M x, r.; , t - r i lJxi + D( T ),
"+1 , j
f(x, (, tJ ) = fM(X, E , t ) + O(r),
npHAe t K aHanory 6aJJaHCHOrO COOTHOWeHHi1 (3.1)) OTnHqaJO eroCfi
OT Hero 3aueHoA Ha ' = + a:
J(x,f.',t j + 1 ) ;fM{X,(',t j ) +div( '!M(X, ',tj») =
= T &I 8fM(X,(, t j ) O( 2 )
2 & a k 8 + T.
Xi Xk
(4.6)
138
Tn. Ul. MQAeJlHpOBaJUle TeqeRllii BB3ICOro ra3a
I1HrerpHpOBaHHe no nepeMeHHoA ( Bblpa>KeHH11 (4.6) C cYMMa-
TOpHbJ:"fH HHBapHaHTaMH npHBO HT K KrY-cHCTeMe.
TaKHM 06paaoM J CBoRCTBO HlISapH&HTHOCTH KB83Hra30 HHa.MH-
qecKoR CHCTeMbI OTHOCHTe1IbHO npeo6pa30SaHH.H r8JIHJIe BblfiOJI-
HHeTCSI C TO'IHOCTLJO )1.0 q.neHOB nopJlAKa MaJIOCTH ,2. 8TOT Q>8J<T
npe"qcraBJISl€TCS HeYAHBHTeJIhHLlM, ecJIH BCnOMHHTb t TO C8Ma Kry-
CHCTeMa nonyqeH& H& OCHOBe p83JlO>KeHHB HHsapH8HTHoro COOTHo-
weHHfl (4.2) C TOQHOCTblO 11.0 qneHOB O(T 2 ) (rJI. II). C pyroA CTO-
POHbl. npH BbJBO,ll-e ypaBHeHHH HaBbe-CTOKca H3 YPaBHeHHj( BOJIhl\-
MaHa c nOMOllU»1O npo e,L\yphl QeIIMeHa-9HCKora He Y'lHTbIBaJlHCb
qneHbJ p83JlO>KeHHH nopSl OB r 2 H Bblwe.
B fIpHH Hne 803140>KHO nonyqeHHe aHaJ10ra KrY-CHCTeMb! C 60-
nee BbJCOKOA TOqHOCTblO BblIIOJIHeHH9 cBo:ACTBa HHsapHaHTHoCTH,
HanpuMep, C TOqHOCTblO ):\0 'iJIeHOB nOpS1;1:Ka O( r 3 ). )]:JIS 3TorO IJ.O-
CTaTO'iHO npaBYIO qaCTb BhJpa)l(eHHa (4.2) p83nO)f(HTb B pH TeA.rIo-
pa C TO'IHOCTblO IJ.O O(T 4 ). npH 3TOM BMeCTO 6anaHCHoro COOTHome-
HH (3.1) UOJIYQHM
/(x,(,t j + l ) - fM{X,(,t i ) + div(E/L) =
T
1 8 a/It r 2 a a 8 I,jlt 3
= 2 r i k a + _ 6 _ 0 i- {J {k a + O(r ).
VXi Xk Xi XI: X p
(4.7)
HHrerpHpys Bhlpa>KeHHe (4.7) C cYWMaTOpHhIMH HHBapHaHTaMH,
npH eM K pa3HOBH,lI)IOCTH KBa3Hra30;ltHHaMH"IecKOIt CHCTeMhJ. HBJISI-
lOw.eaCSI aHanoroM ypaBHeHHR BapHeTTa (rJI. I, 4).
TOT >Ke caMbIA pe3ynbTaT 06 HHBapHaHTHOCTH C TOtJHOcrblO 11-0
0(.,.2) MO)Ke'.(' 6b1Tb Henocpe CTBeHHo nonyqeH B peaYJIbTa:re rpyn-
DODoro aH8JIH3a HeKOTophlX qaCTHbIX CJIyqaeB TeqeHH.sI (HanpHMep.
npOCTpaHCTBeHHO- YMepHoro H3OTepMH ecKOro TeqeHHSI C llOCTO-
HHHbJM T), onaCbIBaeMoro KrY-cHCTeMol. npoBepHM HHBapHaHT-
HOCTb KrY-CHCTeMbi OTHOCHTeJIbHO rpynnhl npe06pa30BaHHst IanH-
neH 1 tlanpHMep, KOr)\a CKOpOCTb a = {a, 0, O} HanpaBneHa B OJIh
OCH OX (HHBapHaHTHOCTh OTHOCHTeJlbHO rpynnbl npe06p83oBaHaA
co CKOpOCTblO a B)).onb oceD OY H OZ npoBepjleTCil aHaJIOrH'IHbIt.1
06p830M).
94. 06 HHlJapHaJlTIIOCTH KB&3Hra3QL{HHaMsl',ecKoH CHcreMbI
139
HH<PHHHTe3HM8JIbHbIA onepaTOp, 38AalO 'rdXyro TOqeqjfYIO
rpynny rlpe06pa3oBaHIDt, HMeeT BH
x = ta:;: + au.
(4.8)
KOHe\.lllWC npe06pa3oBaHHR rpynrIbl t onpe eJ]HIOw.He 3aKOH H3MeHe-
IIHH He TOJIbKO He3aBHCHMblX H 3&BHCHMblX nepeMeHHhlX, HO H HX
npoH3Bo,a.HbIX AO BTOporo nop BeJIHqHHbI BKJlIO'IHTenbHO, no-
nytlaJO'l'CS1 peWeIJHeM COOTBeTCTBYIO HX ypaBHeHHA: H. 3 a1OTCSI CO-
OTHoweHHHMH [68]
x' x + at,
Q -+ Qt - aQx ,
y Qty - aQ%fI ,
r e Q = {p, u, tJ } .
3aJIHCbIB KB83Hra30,lJ.HHaMHQecKYIO CHCTeMY 'Iepe3 lUTpHXO-
BaHHbJe nepeMeHHbIe H HcnOJIb3YSl COOTHoweHHSI (4.9), J1erKO npoBe-
pHTb, 'l{'ro ypaBHeHHjJ COXpauBIOT CBoR BH npH rpynnoBoM npe06-
pa30BaIIHH (4.8) C TOqHOCTbJO AO qneHOB O(T 2 ). npH OKa3aTenb-
CTBe cy ecT8eHHo HCOonb3YeTCH <paKT H8JlH'IHSI HMeHHO KOHe'lHblX
pa3HOCTeA no BpeMeHH, KOTOpbJe 3aTeM npe,L\CTaBJJIDOTCa n BH e, Ha-
npHMep1
'11' u + a,
Q Qu - aQ%% ,
Qu --. Qtt - 2aQtz + a 2 QZ% ,
(4.9)
"+1. 2
pJ - P' _ fJp T O( 2 ) (4.10)
T - at + 2 lJt2 + T.
l{TO )l(e KaCaeTCSI HHBapHaJlTHOCTH Kry OTHOCHTenbHO rpynnbI
Bp eHHH, TO 3,[tecb pea.rlH3YeTCSI 60nee npoCToA BapHaHT. HueH-
HO, CHCTe ta <PcJ>epeH 8JIhHblX YpaBHeHd, KOTOp8JI 1I000yqaeTCSI
H3 KOHetJIJO-pa3HOCTHhIX Kry npeo6pa.30BaHHeM (4.10) c nocneAY-
IOll\HM oropacbJB8JlHeM 'IneHOB O(T 2 ), SlBJlSleTCB HHsapHaHTHOA OT-
HOCHTeJ1bHO rpynlIhl BPa.w.eHHft. npH AOKa3aTeJlbCTBe HCnOJlL3YeTCB
CTaHAapTHbI:A t OCllOBaHHbIA Ha HH<PHHHTe3HM8JIbHOM no xo,n:e, aJ)-
ropH'J'M rpynrroBoro aHaJIH3a, fiPH KOTOpOM ,lVIB HaxO)l(jJ.eHHH KOOp-
)].HHaT oncpa1'Opa ,L\OnYCKaeMOH rpynnbJ pewalQTCj) onpe,qenBIOIIJ)fe
ypaBHellHH [68]. nO CTaHOBKa KOOp)J.HIlaT onepaTopa rpynnbJ Bpa-
l eHHat HanpHMep, B nJ10CKOCTSIX {x,y} H {u,v} OAHOBpeMeHHO:
Z = x8y - y{Jz + uO v - 1JO u ,
(4.11 )
140
Ln. ill. MO,lteJIlfpoSaHHe TetJeHHR BJf3KOro l&J3
- B 9TH onpe enSlIO e ypasHeHHSI o6pa aeT HX B TO eCT80, qTO
H 3aBepmaeT OKa3aTeJlbCTBO.
npOBe,lJ.eHHOe paCCA10TpeHHe HHBapHaHTHOCTH KBa3Hra30,l])iHa-
MHqecKOA MO,lJ;e.nH, HapSIAY C 060CHOBaHHeM ee BhlBo a C TOQKH 3pe-
HK5I KHH rHqeCKOA TeopHH ra30B H 6JJH3OCTH C TOQ}IOCTblO AO O(r 2 )
K KJIaccHqeCKHM ypasHeHHRM H8.Bbe-CTOKca, cnY)KHT e e O HHM
nO)I.TBepJKAeHHeM B03MO)l(HOCTH ee HCfiOJlL30BaHHSI KaK cl>H3HQeCKOH
MO,ll.eJIH ,D.JUI OnHCaHHSI Te eHlUI BH3KOrO ra3a.
5. 0606 eHIUI Ha )q)yrBe 3 aqll MexaHIlKH
CDJlOWHOI cpe hI
I1CKJ1lO'-IeHHe H3 MaTeMaTHqeCKOA Mo enH paccMOTpeHHSI M&C-
WTa60B, MeHbWHX qeM 3TO Heo6xo,n:HMO ,An51 BblnOJlHeHHSI HCXOAHblX
cl>H3HQeCKHX npe.n.nOJlo)f(eHHA, nOJlO)KellO B OCHOBY BbIBo a KCPC H
Kry. 3TOT I10,!J)CO):{ MO)l(HO pacnpOCTpaHHTb H Ha ,.lIJ>yrHe 3 aqH Me-
xaHHKH cnJIomHoR Cpe bI. Bblwe Y>Ke rOBOpHJIOCb 0 nOflS1THH <PU3H-
eCKH 6ecKoHeQIlO MaJIOI'O otYbeMa, KOr)\& ero MHHHM8.JJbHbIa pa3Mep
Onpe eJISleTCSl 1-13 YCJ10BHH Ha.JIH SI 8 HeM )\OCTa.TO HOro KOJIHtleCTBa
MOJ1eKYJI, C TeM QTo6bJ BblnOJlHSlJIOCb npH6nH)I(eHHC CnJIOWHo:A cpe-
)I.bl. PaccMoTpHM H pSl JQ>yrHx npHuepoB no,n;o6Horo no o a.
1. MOAeJIHpOBaHlle IJpo ecca cPHJlbTp HH
KaK H3BeCTHO J KnacCH'leCKHM OD.acaHHeM npo ecca cl>HJlhTpa-
H }KH KOCTH CJlY)KHT CHCTeMa ypasaeHHD (69):
op d . 0
at + IV PU = ,
ku = - gradp,
(5.1)
(5.2)
(5.3)
t
p = Po [1 + PCp-Po)] ,
r e k - Koacl><PHI.J.HeHT npOIIH1J.aeMoCTH) {3 - K03c1>cI>JI eHT OKHMa-
eMOCTH,* Po H PO - IIOCTOBHHh1e 3Ha'leHHSI aBneHHSI H nnOTHOCTH.
· 3.n.ecb paCCMOTpeH oouud CJlY.faA J )"IHThI»aJOUJ.HA B eBOeA nOCTBHOBKe H3-
MeHeHHe MOTHOCTH >KHAKOCTH H BJIKJlHHe nnOTHOCTH Ha H3J.teHeHHe ...Q8.BJ1euHB.
5. 0606U(eHHH H3 JWyrne 3MaqH MexaHHKH cnJlowHoR Cpe,I{lJ 141
3aKOH (5.2), CBH3bIBaJOW){A 3Ha'feHHe CKOpOCTH CPHJlbTPaI.J.HH C
rp8AHeHTOM aaneHHH 6hJn o6HapY>KeH 3KcnepHMeHTaJlbHO ,[(apcH
OKono 150 neT TONY Ha3 . 0606ll\eHHeM 38KOH& ,[(apcH H& CJIyqa,D
OCTaTOt.fHO 60nblUHX cKopocreA HnbTpYlOl1.\eRcSl }l(H KOCTH (cl>nIO-
H a) 5IBnSleTCSI 3aKOH cl>opxaAMepa
ku + kl 'ufu = - gradp, (5.4)
HameAWHA CBoe 3KcnepHMeHTaJIbHOe nO TBep eHHe, KaK H 38.KOH
,L(apcH, OCTaTOqHO asHO (OKOJIO 100 neT TOMY Ha3 ) [69].
3aKoH ,L(apcH MO>KeT 6h1Tb TaK-
}l(e nonY'IeH H3 npOCTblX pa3MepHOCT-
Hb!X co06pa>KeHHH. rrpe CTaBHM H-
>KeHHe <PnIOH)J.a Qepe3 nopHcT}'1O cpe-
B BH,l(e AOH)KeHIDI BSI3KOR )f{H.L\-
KOCTH B nJIOCKOM KaHane 'c pacC'IO-
5lHHeM MC>KAY CTeHKaMH, paBHbJM d PIIC. 25. ;:J.aH>I<eHHe BH3KOA )I( -
(pRC. 25). Tor a XapaKTepHYJO CKo- KOCTH B nnOCKON KaHane
pOCTb TeqeHHH <pnIOHA& U MO}KHO onpe eJJRTD c nOMOIIJ.b1O xnacCH-
qeCKOH <POPMYJlbI IIY83e:A:nSl
JJU/d2 -Vp. (5.5)
TeM calwlblM BH HO) 'ITO K03c1><PH eHT k B 3aKOHe apCH (5.2) onpe-
e.nHeTCg KaK Qepe3 BB3KOCTb cpnlOHt{a 1', TaK H Qepe3 CBOACTS& DO-
pO;J:bl (xapaKTepHble pa3Mephl 3epeH nopo bI H xapaxrepHoe pacCTO-
BRHe Jt,fe>KA)' HHMH).
OTMeTHM, QTO 3a.KOH apcH BblTeKaeT HenOCpe,qCTBeHHO H3
ypaBHeHHSI BHx<eHHSI CHCTeMbJ H8.Bbe-CTOKCa B npeHe6pe>KeHHH
KOHBeK'l BHblMH "IJIeHaMH H ero ocpe eHHH Ha xapaKrepHOM Mac-
urra6e i (70). Ha npaKTHKe B KaT.lecTBe M8cur:ra6a i ,lI.OCTaTOq}{O B3S1Tb
pacCTo IIHe nopH;J.Ka COTHH pa3MepOB 3epeH nopo):tDI.
113 npHBe eHHblX pacCY>K eHHa cne,qyeT, qTO paCCMOTpeHHaJI
nOCTaJ-IOBKa 3 aqH 3KJIeKTH Ha no CBoeMY nocrpoeBH1O. B caMOM
AeJIe, ypaoHeflHe (5.2) O CblBaeT npoQocchI C MHHHMaJIbHblM xa-
paKTepHbIM MacWTa60M I H aanpaBneHO Ha onpe eJIeHHe cpe eR
CKOpOCTH <t>HJJbTpal.\HH U (pRC. 26).
d
!
ArtJl HCC)KHM8.eMOa >K KOCTH CHCTeM8 (5.1)-(5.3) DpHMeT BR,Q
div u = O. ku = - gradp.
B CBOIO Oqepe b, ypasHeHHe
(5.1) fiOOBOnjleT oneCblBaTL cnO)f(HOe
MHKpOCKOnHqecKoe BR>KeHHe cj>JllO-
BAa co CKOpocThlO U' (PIIC. 26) Ha
=> u paCCTOmlHH pa3Mepa 3epHa nopo-
hI H MeHbweM. QT06bJ npeo OJIeTb
3TO npoTHBOpeQUe, nonp06yeM BH-
OH3MeHHTb ypaBHeHHe (5.1) C yqe-
TOM XapaKTepHOrO tacwTa6a ocpet\-
HeHHSI l [71).
PacC iOTpHM cPYHK HIO pacnpe-
eneHHSI f(x, ) t) A1IB <!>HJIbTPYJO-
w.e:RcSi jf{HAKOCTH, r e E - MrHooeH-
H8J1 CKOpOCTb MoneKynbl )l{HAKOCTH.
.
,L{nSi MOMeHTOB <PYHKIJ)iH f eCTecTBeHHo nOTpe6oBaTb BLlnOJIHeHHSf
CJJe,uy101IUIx CBoACTB:
PHC. 26. Pa:JJ1H He Me*M Mt"HO-
BeHHOA CXOpOCTblO U' npH o5Texa-
HHH 3cpeH DOpo H CKOpocTbIO
cl>H.1IhTpa H u
f f(x, E, t) = pet, x), f f(x, , t)E' = p(t, x)u'(t, x), (5.6)
rAe u'(t,x) - MrHOBeHHaJI CKOpoCTb 1HKpOCKOnH'IeCKOrO H>KeHBSI
(pHC. 26).
AnB OllHCaHHSI H3MeHeHHa <l>yHK HH pacnpe;:t.eJIeHHH pacCMOT-
pHM KHHeTJNecKoe ypasHeHHe 9HcKora (rJl. It 2):
81 lJl 6
- + i- = 2: J k o
at OXi k=l
(5.7)
HaM nOHa 06HTCSI TOJ[bKO 3aKOH COXPaHeJfHH MacChi. Heo6xo;\HMbIA:
NISI era nOCTpoeHHH MOMenT OT HHTerpana CTOJIKHOBeHHA YPaBHe-
HHSI 9HCKora paBeH HYJlIO:
f t J" dE == 0,
k=l
(5.8)
TdK >Ke KaK H ATIH ypaoHeHHH BOnL Ma.Ha (rJI. I, 2).
.HBHaJI pa3HOCTHasI annpoKcHMa H ypasHeHH1I (5.7)t KOTOPYIO
AJI.H npOCTOTbI Bhlnmn€M B npoCTpaHCTBeHHo-O,lJ;HOMepHOM CJIY'Iae,
g5. 0606w,eHHH H8 APyrHe 34\l(&'lH Mex&HlfKH cnJlOUIHOR cpe,l(&I 143
npHArfeT BH
. .
f/+l({) = il( ) + fl+l( ;l-l({) tlt-
j j j 6
_ 1{I x 'Hi ({) - 2f i ({) + f i - 1 ({) llt + J llt
2 A2 L.-,k.
X k=l
(5.9)
I1poHHTerpHpyeM ypasHeHHe (5.9) c seCOM m (M&CCa MOJIeKynhl)
no CKOpoCTSIM , HcnOJlb3YSl CBOHCTBO (5.8) HHTerpana CTOJIKHOBeHHA
ypaBHeHHH 3HcKora. npH 3TOMt Tal( >Ke KaK H "PH BblBo e KBa3Rra.-
30 Ha.'4H eCKOH cHcreMLI (rn. II, 5), 3aMeHHM I{I Ha xapaKrepHYIO
CKOpOCTb, B KaqecTBe KOTOpoA y 06Ho B3RTh CKOpoCTb 3BYxa c. rIpe
3TOM £lOJIYQHM:
pf+l - Pi + (pu)f+1 - (PU)Ll = ( X C ) .
.6t 2 x '. 2 Pi
x
(5.10)
3aMeTHM, QTO p83HOCTHaJI cxeM8 (5.9) nonyqeHa npH YCJJOBHH
KYCO'iHo-nOCTOSlHHoro no npoCTpaHCTBY B a ct>YHKlJ)fH pacnpe e-
JIeHHR f:
f(x, ,t) = Ji( ,t), x E [Xi,Xi+l].
CBH3aHO J1H OrpaHH'IeHHe Ha x CHH3Y TOJ1bKO C B03MO>KHO-
CTHMH BbltJHCJJHTeJlbHO:A TeXHHKH HJIH HMeeT 60nee rny60KBA <!>H-
3HQeCKHA CMbICJI? BepHeuDI K pHC. 26. 3HaqeHHe CKOpOCl'H <!>BJIb-
Tp HH U non}"leHO nyreM ocpe,n:HeHHS no npoCTpaacTBY Ha P83-
Mep i 10 2 3epeH nopo bI CKOpOCTB eACTBHTenbHoro rHAPo,QH-
llaMHQeCKoro TetleHHSI u'. U03TOMY _8 K&qeCTBe OrpaHU'IeHUR CHR-
3Y H8 x cne yeT B3S1Tb BenH"qHHY l. YlIHTLlBasI M"OT <PaKT. 38J1H-
weM H<P<I>epeH H8JlLHoe IIpH6J1H>KeHHe )J)ISI yp8.BHeHHSI (5.10) Y)Ke
B rrpocTpaHCTBeHHo-MHoroMepHoM CJlyqae B B e
-
op d . d . Ie d
at + IV pu = IV"'2 gra p.
YpaBHeHHe (5.11) COBMecTHO C 3aKOHOM tI-apca (5.2) 6y eM HC-
nOJlb30BaTb B Ka"tfeCTBe MO eJlH NISI onaCaHHSI Te'qeHHj{ JK OCTH.
(5.11)
06CY HM HeKOTOpble cPH3HQeCKHe npeJ1.l10nO)l(eHHSI, nOJlO)«eH-
HhIe B OCHOBY BbJBO,l(a ypaBHeHUH (5.11). AnB 60nbIIJHHCTBa nopo,lJ.
-
HMeeT MeCTO CJleAYIO ee OrpaHHt{eHHe no nop5l KY BenUI{HHbl I:
'iTO cyw;eCTBeHHO MeHbme pacC1'OHIDI.A. Ha KOTOphIX npOHCXO HT 3a-
MeTHOe H3MeHeHHe napaMeTpOB cl>nJOH a. Cne OBaTeJJbHC, H8JJH'iHe
OnOJIHHTenhHhIX no CpaBHeHHIO C ypaBHeHHeM Hepa.3pblBHOCTH (5.1)
HCCHnaTHBHOCTh qneHa B npasoA 1.facTH yp8.BHeHHB (5.11) He npH-
Be eT K CKOJIb-HH6Y b 3aMeTHhIM H3MeHeHHHM no CpaBHeH1l1O C Knac-
CH1JeCKOD 3 aqeA <l>anhTp8.l.\HH (5.1)-(5.3). TeM He MeHee, HaJI1{qHe
3TOro tI.HCCHnaTHBHoro :JleHa, rapaHTHpYlOrnero CrJl8JKHBaHHe pe-
weHHH Ha paCCTO HHH I, MO}KeT ,n:aTb OnOJIHHTenhHble BbI'IHCJIH-
TeJlbHble B03..,10)KHOCTH rrpH Mo enHpoBaHHH npOQeCCOB cPHnbTpa-
QHH.
B KaqecTBe npH..,lepa Hcnonb30-
BaHHSI MO,a,HcpHlJ)ipOBa.HHOrO ypas-
HeHWI Hepa3pbIBHOCTH pacCMOTpHM
cnyqaft nJlOCKOp H8JIhHOrO <l>HJIb-
TpaQHOHHoro nOTOKa - 1KH KOCTb
npHTeKaeT K rH,.DJ)OAHHaMH'JeCKH co-
BepmeHHOH Ao6b1BaJO ea CKBa)f(HHe
(69] (pec. 27). IJ.eHTp CKB8.)KHHbI
pacnOJlOJKeH B Hat{8J]e KOO PtI.HH aT,
rc - P8.D.HYC CKB8JKHHbI, Rk - pa-
HYC KOHTypa IIHTaHBSI (KIT) CKB8-
>KHHbj. JIHHHH TOKa Pa.z\H8JJbHhle H PHc. 27. CxeMa IUIOCKOPB.lnIaJlb-
HaITpaBneHbl K CKBaJKHHe. 3a):{a'tf& HOro 4»HlIhT oHRoro DOTOxa K
BBnHeI'CR OAlloMepHoA: lJ)iJIHH)J;pH- CKB&>ICHHe
ttecKH CHMMeTpH\fHOa.
rpaHH'llfble YCJlOBHH Ha 3a60e CKBa>KHHbI H KOHType nHTaHH.H
nOCTOSlHHbl. B 3THX YCJlOBHBX cPOPMYJIbl .lVISi YCTaHOBHBweroca nnoc-
KOP8AHaJIbHOrO TetteHHSI R3BecTHbJ:
I ;5 0.01 M,
Pk - Pc r
p = Pc + I (R / ) In-,
n Ie rc Tc
y
o
R K X
Co
PI: - Pc 1
lul = k In(R.t/rc) r ·
(5.12)
J5. 0606U{eHJlH aa JtPYf'He a Mex&lfHKH cnJ1omnoR cpe.tVJ 145
1x10'
8x10.
8)c10'
P ---- -1
-2
4x10. · . · -3
2x10'
o
20
40
80
80
100
r
PRC. 28. aBJIeHHe: 1 - TO'IHoe pemeHHe (5.12), 2 - peweHHe H& 0CII0Be 3&,l).8qJf
(5.1)-(5.3)t 3 - peweHHe 'IB c ypaBHeHReM (5.11)
2.Ox1 o I
I
I
1.5x10'"
J u r
1.0Jr10 ----1
-2
· · · -3
5.Ox10"
o
20
80
80
100
r
PHC. 29. CKOpoCTb cJ>IIJlbTp R: 1 - TO'DIoe pemeHBe, 2 - pemeBHe n& OCBOBe
38J{&'1H (5.1)-(5.3),3 - pemeHHe 3 aqH c ypaBHeHHeM (5.11)
3,L1.eCb HH,n:eKcaMH C H k 0603Ha'leHbI 3HaQeHHSI COOTBeTCTBYK>IUHX
BeJIH'-IHH Ha CKBa>KHHe H KOHType nHTaHHSI.
HCnOJlb30BaJIHCb C1Ie YIOll\He 3HaqeHHg napaMeTpOB:
6 r 7 r r
Pc = 10 2 ' Pic = 10 2 ' P == 1 3 ,
CM · C CM · C CM
. 2
k = 10 7 ' fJ = lO-g CM C , Rk = lO"CM.
CM .e r
Pact-leT no 06eI-fM Mo enHM: KJlaccHqeCKOa 3 aqH <l>HnbTpal.{HH
(5.1)-(5.3) H C MOJ(H<I>HU-HpoBaHHbIM ypaBHeHHeM Hepa3pbJBHOCTH
(5.11) - npoBO HJIQt MeTOAOM YCTaHOBJIeHHR. npH 3TOM o6e CHCTe-
Mbl YPaBHeHHA pelll8JOTCB no SlBHoA creMe C n:eHTpanbHblMH pa3HO-
CTBMH OTHOCHTenhHO aBneHHSI. CKOpOCTb 3aTeM HaxO HT H33aKO-
Ha )]:apcH, OJIOTHOCTb - H3 ypaBHeHHH COCTOBHHH. HCDOJIL30BaHHe
SlBHOA cxeMLI B 3HatfHTMbHOR Mepe o6YCJIOBJIeHO nOCJle.AYlow.HMH
nJlaHaMH peweHH 3 aq <l>HJlbTp HH Ha MHoronpo eccopHblX BbI-
HCJlHTe.nbHblX CHCTeMax. *
Ha pHC. 28 H 29 npe CTaBJIeHbI npoQ>W1H aeJIeHHSI H CKOpo-
CTH, nOJlyqeHHLIe Ha OCROBe T01.fHOro peweHHSI (5.12) H qMCJIeHHO-
ro peweHHSI 3 aq (5.1)-(5.3) H C MO H<I>Bn:HpoBaHHhJM ypaBHeHH-
eM Hepa3pbIBHOCTH (5.11). KaK BH.l(HO, HCn0JIh30BaHHe ypaBHeHHS[
(5.11) npHBO,AHT K '!'eM )Ke proYJlbTaTaM, '11'0 Moryr 6h1Tb uonY'IeHhI
Ha OCHOBe TOQHOrO pemeHHB CHCTeNbl (5.1)-(5.3). HaJIHqHe B npa-
BoA qacTH (5.11) OnOJlHHTenbHoro peryJIjlpH3YIO ero 'UIeHa 003-
BOJlBeT 3Ha'lHffJIhHO YJIyqWHTb Ka'leCTBO pacqeTOB no cpaeHeHHIO
c HCJIeHHblM pewellHeM HCXO HOA 3 aqH (5.1)-(5.3). npoAOn>KHM
paccMOTpeHHe B03MO)l(HOCTH BBe):\eHHSI nO,lJ.06HOrO po a perYJlHpH-
3YIOUUtX ttJleHOB Ha npHMepe ypasHeHHSI, OnHCblBaJO ero nOBe eHHe
ONlO'lOCTHqHOA <PY H pacnpe,neneHHSI.
2. B03MO)f(HaH MO,l(lfcPHKaIJ.JfH yp&BHeHHH BOJ1b MaHa
BepHeMCSI K BblBO,qy KHHeTHqeCKH cornacoBaHHOA pa3HOCTIIOA
cxeMbl (rJl. 11 1 1). CxeMa nepBoro nOpH,D.Ka C aanpaBJleHHhlMH pa3-
· Kax H3BecTHO. aBHhle cxeMbi lIaH60nee YJJ.06HY AJlB 8A&rr.ra.nHH K napa.n-
JleJ1bHblb4 Bh['lBCJIHTenhHhlM CHCTeMaM C apXBTeICTypoA pacn eneHHoA n8MHTH
(rn. IV).
95. 0606Il.{eHHJJ n& ,I{pyrRe 3aJ;tAtfH MexaBHKH ClLIlOUlHOR Cpe.L(bl 147
HOCTSlMH ,nna ypaBlleHHSI BOJIb MaHa HMeeT B
1/+1 - II + { J/+ 1 - 1;-1 = Axl{1 J/+1 - 2// + JLl +
llt 2 x 2 x2 (5.13)
+ J(j, I) ·
3J1:eCb SI npOCTOTLI H3J101KeHIUI pacCM8.TpHBaeTCH npOCTpaHCTBeHHO-
o HoMepHblA CJlyqaA.
3 a.nHMCH BonpocoM: OrpaHH eHHe CHH3Y Ha p83Mep jlqeA:KH
x CB 3aHO TOJIbKO c HMelO MHCSI B H8JIHtJHH BbI'IHCJIHTenbHbIMH
B03MO>KHOCT51MH RJIH CO ep)KHT onpe eJleHHbl1t CPH3HQecKmt CMblCJI?
BCnO llIH t, 'qTO no CBoeMY onpe eJJeRHIO / HMeeT BepoHTHoCTHLIA
xapaKTep. 11 AJUI TOro, 'lT06bl <l>H3H-qecKH Onpe eJlHTb BepoBTHOCTb
pacnpe eJIeH"$J MOJ1eKyn no CKOpocTSlM J Heo6xo,ll.HMO paccMOTpeTb
aHCaM6J1b, COCTOSl a. no MeHhweA Mepe H3 HecKonbKHX AecsrrKOB
qacTH .
9TO KOJIHqeCTBO qaCTHI\ He nOK8JJH30BaHO B roQKe, a pacno-
nO>KeHO B KOHeQHOM 06'heue C xapaKTepHLIM JIHHeAHblM p83MepoM
6x. To ecTb - no aHaJlOI'HH C "cpH3HQeCKH 6ecKoHe BO MaJIhlM o&b-
eMoM Jt B MexaHHKe cnnoUIHoA C bJ - HeT CMbICJla paCCM&TpHBaTb
pa3Mepbl, MeHhWHe '(eM 8x. OqellHM no nOpB,lU(Y BeJlHqHHhI STO
pacCToHHHe x.
B B03 e HOpMaJlbHO:A flJIOTHOCTH, c 'IHCJlOM qacTH B lcw3
n "J 10 19 , xapaKTepHblA p83Mep x 6y eT 110pBAJ(a 10- 6 CM. T.e. cy-
1.QecTBelfHo 60nbme xapaKTepHOrO p83Mepa MOJIeKYJlbJ lo 10- 8 CM
H JaMeTHO MeHblUe )VIHHbJ CBo6o Horo npo6era l 10- 4 CM:
10 « /).X « t.
(5.14)
HanOMHHM, "ITO pa3HOCTHaH CXeM& (5.13) nOJIY'ieHa H3 ycnOBIDI
nOCTOHHCTsa f Ha OTpe3Ke .X E [Xi) XI+1J, H ee AH<PcI>epeH HaJJbHoe
npH6JlH)J(eHHe
+ { f!1 = 1 1{16X 8J + J(f, f) (5.15)
at ax 2 ax ox
rapaHTHpyeT Crna)l(HSaHHe peweHHB ypaBHeHHSI (5.15) Ha pacCTO-
SIHHH nOpSiAKa. !!"x. BBHA}' TOro, 1:fTO dx « I, RanH e OnOJIHH-
TeJlbHOrO AHCPCPY3HOHHOro qneH& B npaaoft qa.cTH (5.15) He BHe-
ceT CKOJIb-HH6Y b 3a.MeTHblX H3MeHeHHft B 3HaqeHHe Tp&HCnOpTHhIX
K03c1><PHD.HeHTOB (K03<p<pH HeHTOB BSI3KOCTH H TeIIJIOnpOBO)J.HOCTH) J
KOTOpbJe npOIlOp HOHanbHbl AJlHHe CBo60,qHoro np06era l. TeM ca-
MhI 1 Pe3YJlbTaTbJ. 110000'leHHbie Ha OCHOBe ypaBHeHBSI (5.15). 6ytQ'T
COOTBeTCTBOBaTb pe3YJlhTaTaM, nOJlyqeHHbIM Ha OCHOBe ypaBHeHHB
BOJIbQMaHa. O HaKO ypaBlleHHe (5.15) 06n aeT KoppeKTHOCTblO
(no KpaRHeA Mepe c <pH3H'ieCKo.A TOQKH 3peHHB), rapaHTHpya crna-
>KHSaHHe pemeflHH Ha paCCTOSlHH" nop5JAJ(a x.
POJIb ):\onOJIHHTeRbHOrO AH<t>cPY3HOHHOro qneHa B npa.BoA qacTH
ypaBHeHH.SI (5.15) COCTOHT HMeHHO B Crn8.)KHBaJlHH pemeBHH Ha pac-
CTOllHHH x. Ero KOHKpeTHblA BH,.lJ., npH BhlnOJIHeHHH 3TOro YCJlO-
BHH, lIe CTOJIh Y>K H 88..>KeH. n03TOMY YMHO}KHM H pa3AenHM, KaK
3TO AenanOCb IIPH BblBo e KB83Hra30AHHaMH ecKoA CHcTeMbI, OT
,L\H<P<t>y3HOHllhlH qJIeH Ha I{I, 3aMeHBSI 3HaqeHHe I I 8 3H8.MeH&Tene
Ha CKOpoCTb 3BYKa. TeM CaMblM BMecTO (5.15) npBAeM K ypaBHeHHIO
[72, 73]:
of of () x Of
t + i- {) = - i{k +J{f,f), (5.16)
v Xi uXi C uX/c
HMelO eMY CHMMeTpaqHhI:A BH . 9To ypaBHeHHe, TaK )J(e KaK H
(5.15), CrJIa>KHBaeT pemeHHe Ha pacCTOHHHH x, KOTopoe cy e-
CTBeHHO MeHbwe AJIHHbI cBo60 oro np06era I. HCnOJIb30BaHH€ Mo-
JI;H<t>HlI.HpOBaHHblx YPaBHeHHA (5.15) H (5.16) M01KeT OKaJaTbCSl no-
Jle3HhIM npH AeTaJIbHOM MOAeJlHpOBaHHH nOBe eHHH <t>YHKQHH pac-
npe,neneHHH, HcnOJlb3YIO eM 6onbllioe 'fHCJIO npOCTpaHCTBeHHblX )'3-
1108.
3. K Bonpocy 0 napa,n.OKce 6eCKOHetfHO 6blCTporo
pacupocTpl1HeHHH TenJla
PaccMoTpeHHble B aHHOM naparpa e npHMepbl o6De HHBeT
c KCPC H Kry HaJlHqHe HeKOTOporo MHHHMaJlbHOrO MaclllTa6a.
MeHbllIe Koroporo aJCbHeAwaR AeTaJIH3 g OrmCaHHB Y>Ke He HMe-
eT CMbICJIa. B KCPC 3TO NIHH8, CB060 HOro npo6era l H BpeMH T
Me>K Y CTOJlKHOBeHH.SIMH MoneKYJI, B 3 aqax <t>HJIbTpa H 9TO pac-
CTOSlHHe I10pSl Ka COTeH 3epeH nopOAbi. NIB OnHCaHHH O):(HOqacTH'l-
HoA <PYHK HH STO JIHHe:AHblii p83Mep 06'beMa l1x, BKn1OQ8JOw.ero B
ce6S1 HeCKOJIbKO eCstTKoB MOJIeKyn. PacCMOTpHM Ha I1pHMepe KJJac-
95. 0606U(eHHR H8 ,apyrJle 3N(a&JH A1exaHHKJ.I CUJIOlUHOif Cpe,l{LI 149
CH'IeCKoft 38AatJH 0 paCnpOCTpaHeHHH Tenna, K8.K HaJIHqae TaKHX MH-
HHM8JIbHbJX M8CWTa60B fi03BOJUIeT HCKnlO'IHTb HeKOTOpble 3c:Pcl>eK:-
TbI He<pu3H1;feCKOrO cBoAcTBa.
Kax H3BeCTHO [74], H3 pemeHHSJ yp8.BHeHWI TenJlOIlpOBO HOCTH
aT a 8T
at = ax x ax + Q,
(5.17)
KOTOpoe Mbl pacCMOTpHM Ha npHMepe npOCTpaHCTBeHHo-O,u.HOMepHo-
ro CTlyqaJ), CJIe,ayeT 3Q><PeKT 6ecKOHe'lHO 6bJCTpOrO pacnpOCTpaHeHH
TeIL1Ja, npH ero BbI eJIeHHH B KOHe'IHOa 06JIacTH. 3jI;eCb T - TeMne-
paTypa, x - Ko9<l><l>H HeHT TeIIJIOnpOBO,lJ.HOCTH, Q - Orp8.HHQeHHhIA
B npOCTpaHCTBe HCTO'IHHK TelIJla.
TOT cJ>aKT, qTO BJ1HHHHe HCTO'IHHKa Q Ha OCmTOqHO YAaneHHblx
OT Hero pacCTOSlHHSlX HHqTO)I(HO MaJIO) He CHHMaeT OCTpOTY He o-
nYCTHMOCTH c Q:>H3H'IeCKOA TO'tfKH 3peHHSI 6ecKoHettHO 60JIbWoA CKo-
poCTH pacnpOCTp8.HeHHSi renna. K 6ecKOHeQHO 60JIbWoA CKOpOCTH
pacnpoCTpaHeHKH TenJla np eM TaK3(e , pemaSi ypaBHeHHe (5.17)
no a6ConfOTIIO YCTOH'IHBOA HeBBHoA CXCMe
Tj+l - T1 .
= (xTi) +l + Q.
At
(5.18)
Ha060p01't npH perneHHH ypaoHeHHH TenJtonpoBO HOCTH c nOMa-
W.blO HBHOH cxeMbI
Ti+ 1 - Tj .
= ( xT- ) J + Q
t % :1:
(5.19)
CKOpOCTb pacnpocTpaHeHH.H Tel1JI& KOne'lHa: 3a o war no speMe-
HH HHQ:>opMa H.H nepe aeTCg JIHillb COCeAHHM }'3JIaM. OAH8.KO "PH
crpeMneHHH warn t K HYJlIO, CBH33.HHOM C nOBbImeHHeM TO"IHo-
CTH pacqeTa, CKOpOCTb pacnpoCTpaHeHHH Tenna TaKx<e crpeMHTCH K
6eCKOHeQHOCTH.
Y'ITeM, qTO COCTOHHHe, MH3Koe K p8.BHOBeCHOMY, a CJJe OBa-
TenbHO - H OnHCaHHe nepeHoca renna c nOMOIl\hIO MexaHH3Ma TeD-
JlOnpOBO)\HOCTH - B03MO)l(HO) eCJIH qacrHIJPI HaxOAHTCH B ycnOBHSIX,
CJla60 MeHmow.HXca B TeqeHHe BpeMeHH nOpH Ka BpeMeHH Me>K,lU'
CTonKHOBeHH.HMH MOJIeKyn T. Cne,lJ.OBaTenbHO, B RBHoA cxeMe (5.19)
150
rn. In. MOAenHposlUlJle Te eJlHR BJJ3KOro r&3a
He HMeeT CMblCJIa pacCM&TpHBaTb BpeMeHa I1t no nop y SeJlH'IH-
HI>I MeHhWHe, '{eM T. 3SMeTHM, '!TO B peaJIbHOA BblqHCJIHTeJIbHOa
npaKTHKe 3TO OrpaHH'IeHHe He npH80 HT K KaKHM-JIH60 np06neMaM,
TaK K8.K B nnOTHbIX raa8.X xapaKTepHoe BpeMfl H3MeHeHHSI a30 HHa-
MHtfeCKHX napaMeTpos Ra MHOro rropSl KOB 60J1hwe T.
TaKHM 06pa30M, npH HcnOnh30BaHHH BBHOR: cxeMbI (5.19), c yqe-
TOM OrpaHH"IeHHSI Ha war l1t
6t > 'T )
(5.20)
HecpH3H"IeCKHR 3cP<PeKT 6eCKOHeT.JHO 6b1CTpOI"O paCnpoCTpaHelDlB Ten-
na 6Y e'r OTCyTCTBOBaTb. 3aMeTHM) qTO HMeHHO SlBHag cPopMa 3a-
fiaeR C KOHe'IHo-p83HOCTHbJM npe CTaBneHHeM H3MeHeHH9 ra30 -
HaMHqecKHX napaMeTpOB no BpeMeHH xapaKTepHa A1JH KB3.3Hra3o-
)J.HH rH'-IeCKOH CHCTeMbl (2.1)-(2.4).
npHBe eHHb1:A npHMep, MO>KeT 6b1Tb H He CTOJIb B(UKHLJ:A e roq-
KH 3peHHB BbI'tfHCJ1HTeJIbHOA npaKTHKH, TeM lIe MeHee rrOK83blBaeT
B03MO)f(HOCTH MOt\eneA: c nO,A06Horo pO;D.a perynapH3aTOpaMH WlH
KoppeKTHoro OnHC&HHSI cnnomHo:A cpe . B 3TOM IIJIaHe pacCMOT-
pHM ew.e O HH no,.lI)(O - Lattice Boltzmann* cxeMhI J KO'IOpbIA no
CBoeMY nOCTpOeHHIO H no B03MO>KHOCTKM OllHcaHHB ra30 HHaMHqe-
CKHX npoo-eCCOB C nOMO 1O MHOronpO ecCOpHLIX BhlqHCjJHTeJlbHblX
CHCTeM HMeeT MHOro 06 HX 'IePT C KHHeTH'IecKH COrJlacoBaHHbIMH
cxeMaMH H KB8.3Hra30 HHaMH ecKoA: CHCTeMoA ypaBHeHHA.
6. Lattice BoltzlIl8.IlD B KHHeTOqeCKHe cxeMhl
OCHOBOA )VIS! pa3BHTHSI Lattice Boltzmann cxeM (LBS) nOCJ1y-
}Kana pa60Ta [75]J B KOTOpoA 6hIJlO nOKa3aHO, tfrO Ha OCHOBe HeKOTO-
pora aHaJlOra Hrpbl B 6HJ1bBP J npe CTaBn.HlOm;ero pB3JleT H CTOJIK-
HooeHHe >KH KHX qacTHL'(, MO)KHO npmtTH npH HeKOTOphIX OrpaHH-
qeHHHX K onHcaHHIO cpe):{bl, 6nH3KoMY K nonyqeHHOMY rtpH HCnOJIb-
30BaHHH ypaoHeHHA HaBbe-CToKca. t):aJlLHeAmHA Ba>KHbIA: war B
· ABTOp He B3BJ1 H& cOOs CMeJ10CTh ,qaTb nepeBOA Ha PYCCKHA il3b1lC ,l{&HHOrO
TepMHHa. BOMHe B03MO>ICBO J &frO B 6nwKaAweM 6y.nyw.ey eKBaTHLU1 pyccKHA
&Manor aHHOrO TePMHU& nOBBHTCK C&MOCTOSTeJIhHO Ha OCHOBe npaKTHKH HC-
DWJb30D&UBH )l.aHHOro IIO .
*6. Lattice Boltzmann If KHHeTJfqeCKlfe CXeMhi
151
pa3BHTHH STott H eH 6bUI c en8JI B pa60rre (76], B KOTopOft B Ka,-
qecToe OCHOBbi ,lVIH non}"leRHSI m30,J].HHaMH'IecKtiX napaMeTpOB HC-
fiOJIb30BaJIHCb HCKpeTHhle MOAeJIH KHHeTHqecKHX ypaBHeHHA )VIB
onHcallHH otI-HOqaCTHqHOA <!>YHK HH pacnpe eneHHSI. B nocnerorHe
ro bJ Ha6nlO aeTCS1 6W1bllloA HHTepec K aHHOMY nOAXO ) BbI3B8.H-
HbIR B 3HaqH bHOa Mepe npoCToR a;:ta HeH nWlyqeHHblX Ha ero
OCHOBe anropHTMOB K apXHTeKType MHoronpoQeccopHhlX CHCTeM C
pacrlpe,l{eJleHHott naMHTblO. COOTBeTCI'BeHHO Ha6JlIO a.eTCSf H pOCT
qHCJI& ny6J1H HA, OKna;:tOB Ha HayqHhlX KOH<PepeHIUtSlX, nOCBH-
w.eHHLIX 8.}IHOA TeMe. t{HT8TeJJHM , 3aHHTepeCOB8HHblM 8 nanyqeHHH
60nee nO,l(p06HOA HHQ>opM HH B tJ;&HHOM HanpasneHHH HCCJIe OBa-
""H, MO)K}IO nOpeKOMeHAOSaTb MOHorpa4>ulO S. Succi [30].
npo eMoHcTpHpyeM B03MO:>KHOCTH LBS-MeTO a Ha npHMepe HaR-
'-'
DOnee npocTOro H liacTO HCfiOJlb3yeMoro BapH&HTa, onHpalO erocSI
Ha npH..feHeHHe BrK-Mo enH )JftB O)I;HO'tlacTHqHOA <!>YHKI.\HH pacnpe-
e!IeHHSI.
1. LBS na OCHOBe BrK-MOAeJlH - LBGK-cxeMI1l
npOCTeRmaH p83HOCTHaJI cxeMa LBS,l1)1B BrK-MO,l:teJIH (LBGK-
cxeMa), CBSl3b1BaJO aJl3HaqeHHR <!>YHKnHH pacnpe eJIeHHH B MOMenT
BpeMeHH t 11 t + t, HMeeT B
. 1 ( . )
J(Xi' k,t+/}.t)=J(Xi- k/}.t'(k,t1)+ T J:q-/i(X, ,t') . (6.1)
BMeCTO / M -- JIOKaJlbHO-MaKCSeJIJlOBCKOA <PYHKIJ)IH, npHcyrcTBYIO-
meR B KJIaccH lecKoA BrK-MO eJlH (rn. I), - B npaBOA '1acTH (6.1)
GTOHT f: q , KOTOpas MO>KeT HeCKOJIbKO OTnH aTbca OT 1M. CMLICJI
B8e eHHSI ft q 5YAeT nOHSITeH H3 aJILHeamero I13n01KeHHS.
Bblpa>KeHHe (6.1) onHCbIS8eT n08eAeHHe cl>YHK HH pacnpe e-
JJeHHSI qaCTHl.{ co CKOpocTblO k, KOTOpoe CKJla,rtbIS8eTCSI H3 6ec-
CTO.,lKHOBHTeJIbHoro paanem B re'leHHe speMeHH At 83 TO'iKH
Xi - k t B TOt{KY Xi H C rOJ1KHOBeHHA "IacTHQ, MO eJIHpyeMhlx ne-
HOM (f - fa (Xi, (, t j )). .anB paqHOIl8JlbHOA: OpraHH3&qHH BblqHC-
flHTeJILHOrO rIpo ecca ecTeCTBeHHO nO 06paTb KOOp,ll.HHaTbl npo-
CTpaHCTBeHHblX Y3nOB Xi H 3HaqeHHH CKOpOCTeA k TaK, QT06b1 B
a)
6)
Ij
e
1,+1
e
fj
e
PHC. 30. llla6JTOHbI LBGK-cxeM: a) cxewa DIQ3 1 6) CXeMa D2Q9
MOtvreHT BpeMeHH t j TO'lKH Xi - k t cOBn a.i1H c Y)Ke CY ecTBYIO-
e:A IIpOCTp3.HCTBeHHOA ceTKoA. EcTecTBeHHo J qTO B03MO)KHbI paa-
JIH mbIe BapHaHTbl nOAo6Horo po a cxeMbI. Tax, Ha pHC. 30,a H306pa-
)l(eHa cxeMa DIQ3 ,!VISI npOCTpaHCTBeHHo-O,a.HOMepHhlX 3B,Aatt, CBSI-
3L1Baro aH 3 Y3JIa npoCTpaHCTBeHRoA CeTKH. Ha pHC. 30,6 H306pa-
>KeHa cxeMa D2Q9 NISI 2D-3 aq) CBa3bIBaIO aJI 9 y3JIOll CeTKH.
<PYHKI HJO f i eq )VI.H LBGK-cxeMbI Bbl6HpaeM B CJIe lOrn.eM BH e
f q = . ( 1 + ioUa + Qiabtlo.Ub ) ( 6.2 )
{XtJ." c 2 c4 ·
3 ecb i - HH)J.eKC, 03HaQalOIn:HA HYMepa HIO )'3JIOB Ha npe bIAY-
eM CJ10e no BpeMeHH, Uti, Ub - 3HaqeHHH M8.KpoCKOnH'IeCKHX CKo-
pOCTeA B COOTBeTCT8YJO HX HanpaBJIeHHSlX eKaPTOBblX I{OOp HaT,
C - CKOpocTh 3BYKa.
BecoBble MHO}l(HTenH Wi, Qiab BhI6HpaJOTCSI H3 YCJIOBIUI, '-IT06hl
!je q HMHTHpOB8JlO nOKaJlbHO-MaKCBeJlnOBCKYIO <PYHK HIO 1M Ha co-
OTBeTCTBYIOW.eA CCTKe. Ee MOMeHTbI, BbIpa)l(eHHb1e 8 BH e CYMMbI
rIO COOTBeTCTBYlOl.l\HM Y3JIaM, AOJI>KHhI COBn aTb c ra3O HHaMHqe-
CKHMH rrapaMeTpaMH:
EWi - 1 , E WiQio.b 0,
1 ,
E Wi i4 - 0, E Wi{itlQibc - 0,
i I
L WI{i4 ib P6o.b , E Wi{io. ibQif)c PUa Ub ·
t I
npOBOASI anhHe:AwylO ananormo Me>K)U' BrK-MO eJlblO H
LBGK-cxeMOH] 06paTHM BHHMaHHC Ha TO, qTO B KaqeCTSe 3<P<peK-
THBHOH ,lI.JIHllbl cBo60 Horo npo6era l B cxeMe (6.1) BbJCTynaeT xa-
paKTepflbIA pa3Mep C1.JeTHolt }1'{eAKH. HanHq]fe lscP o6ecne'tfHBaeT cy-
eCTBOBa.HHe peaJIbHbIX HCCHnaTHBHbIX CBoRCTB B cpe eJ ODHCbIBa-
eMoA BrK-MOAeJlbJO (6.1). B03MOJKHOCTb ODHcaHWI 5IBJIeHHA ,D)IC-
cHnaqHH aeT OCHOBaHHe ,!VUI MO MHpoBaHHSI c nOMOlqblO LBGK-
MOAeJIH npOQeCCOB, I1pOHCXO SlII.\HX B BSI3KOit cpe e (30).
O)J.HaKo, Heo6xo MO yqeCTb, lset>, KaK npaBHnO, cy eCTBeB-
HO 60nhwe pe8JIbHOA ,lVIHHhl cB06o.L\Horo npo6era I. ECTecTBeHHo,
'iTO 3<PCPeKTHBHbIA Kos<PcI> HeHT BB3KOCTH J peaJIH3yeMblH B paM-
K&x LBGK-Mo,n.eJlH, 6y eT 3HaQHTeJlbHO npeBblmaTb ero HaTypanh-
HblR' aHaJIor. ,lJ;JSI HeC)KHMaeMOA )I( KOCTH H.1IH CJIa6OC>KHMaeMOro
fa:3a 1 8 KOTOpoM ABH)KeHHe npoHCXOtl-HT CO CKOpOCTSlMH, 3Ra'IHTe1Ib-
HO leHbUJMMH, {leM CKOpOCTb 3BYKa, 3TY TPYAHOCTL nerKO o6oATH.
B 3TOM cnyqae lVISi BSl3KO:A >KH KOCTH CY eCTBYeT O,n;HH 6e3pa.3Mep-
HbJA napaMeTp - '1HCJlO PeAHOJJb Ca, xapaxTepH3YIO ee reqeHHe B
l\eJIOM:
Re = pUL . (6.3)
Jl
3 ecb p - nnOTHOCTb) U - xapaKTepHaSi CKOpOCTb, L - xapaxTep-
HbIA pa.3Mep, JJ - 3cpCPeKTHBHasI BSl3KOCTb, nOJlrIaeM&sI B paMKax
LBGK-Mo enH.
,l(ns OnHcaHHB peanbHoro npo ecca ocraTO'IHO Tal< H3MeHHTL
HCXO HhJe napaMeTpbl 3 aqJ{, QT06bl nOJIyqaeMOe QHC1JO PeAHOJlb,ll.-
ca COBn aJIO c ero ecTeCTBeHHbIM 3HaQeHHeM, HecMOTpSl Ha OTJIH'lHe
cP OT 3HaQeHHSI HaTYPaJIbHOA BH3KOCTH.
HeB03MO)l(HOCTb HCnWlb30BaHHSI npocTOA nepeHopMHpOBKH ,{HC-
JIa PeRHOJIb Ca B 60Jlee 06meM CJIyqae nocnY>KHnO npH'tfHHOA Taro,
t.{T() OCHOBHbIe pa60TbI B 06JIacTH Lattice Boltzmann cxeM cocp o-
TOqeHbl Ha Mo).\enHpOBaHHH TeqeHHA HeC>KHMaeMOH )KH)I.KOCTH H en&-
OOC)l(HMaeMoro ra3a. Q)l:HaKO CJIe eT o6paTHTh BHHMaHHe Ha MOTe-
pecHYlO pa.6OTY [77], B KOTOpoA paccMaTpHBaeTCSI npHMeHeHHe LBS-
MeTo a AJlS Mo enHpOB8.HHK OKHMaeMoro ra3a. no BeeR BH,L\HMOCTH,
AaJlbHeftwHe HHTeHCHBHbIe Hccne)I.OBa.HHB B 3TOM H&npasneHHH npH-
Be,n;y'r K pacWHpeHHIO 06nacTH npHMeHHMoCTH LBS-MeTOAa.
06paTH 1 BHHMaHHe Ha TO, tlro C BLlqHCJIHTeJIbHoA: TOqKH 3pe-
HHSI LBGK-a.nropHTM (6.1) qpe3BbJ'IaDHO npoCT. OH IIpe CTaBJIaeT
co6oa SlBHyID CXeMY, anTaQHB KOTOpo:A Ha apXHTeKTypy MHOrO-
npo eccopHbIX BblqHCJIHTeJJbHblX CHCTeM C pacnpe en HHOa DaMSI-
TblO He COCT8.BnHeT oco60ro Tpy a. 9ToT cPaKT BBnaeTCB OAHO:R H3
OCHOBHblX npHt.fHH B03pacTaIO ef'O B nOCJIe Hee BpeMSI HH'l'epeCa K
LBS-MeT0AY.
KopOTKoe 06cY>K,lI.eHHe LBS-cxeM, npOBe eHHoe B aHHOM na-
parpa<f>e) JIHIlIb B MaJloA CTeneHH pacKpLiBaeT 803MO)KHOCTH 9TOro
nepcneK'rHBHoro MeTO,lI.a. OHO CJIY>KHT ,nns nO)l.rOTOBKH MaTepHana
K nOCJIe,rorIOIUeMY 06cY)K eHHIO aH8JlOrHH Me>K Y Lattice Boltzmann
CXeMaMH, c o HoA CTOPOHbI. KCPC H KB83Hra30 HaMHqecKOH CH-
CTe loA, C ,LJ.pyroA CTOpOHbl.
2. KBA3Hra30,l1)lHaMHlleCKaH CHCTeM& If LBS
rrpOB€ eM CpaBIJHTe.nbHbIA aHaJIH3 KBa3HI(t30 HHaMHqeCKOA CH-
CTelttlbJ, B KOTOpoA HaH60nee HarJlH)UfO npeACTaBJ1S1JOTCH cBoACTBa
KCPC KaK cPH3H'qecK0J1 MO eJlHt H CaMoro nonyn.SlpHoro ee BapH-
aHTa - LBS, OCHoBaHHoro Ha Hcnonb308aHHH Mo enH BxaTHarapa-
rpocca-KpYKa. HanoMHHM, qT() B DeHOBe BblBo a KrY-CI-fCTeMLI ne-
)KHT npe nOJIO>KeHHe 0 6eCCTOJIKHOBHTeJIbHOM paaneTe 1.Ia.cTH B Te-
qeHHe BpeMeHH t = t i + 1 - t j H nOCJ1eAYIOIQeA MrHoseBHoA: MaKC-
BennH3a HH qaCTHI\ npH t = t j + 1 (rn. II, 5).
npe no.nO>KeHHe 0 6eccronKHOBHTeJIbHOCTH pa3neTa npHBo T K
CJle,l\YlO eMY COOTIlomeHH1O .n.ng 3Ha'leHHA <l>YHK HM pacnpe,n:eneHHSI
}Ia tOMeHTbl BpeMeHH t i H t j + l :
"+1 .
f{x,E,tJ ) =fM(x-E t, ,tJ).
I1MHTa HSI npo ecca CTOJIKHOBeHmt npOHCXO HT 3a CqeT TOro t liTO
cPYIIKU.HH pacnpe e.neHHSI rap8.HTHposaHHO CTaHOBHTCiI nOKaJIbHO-
MaKCBe..rmOBcKoit npH t ;: t j , t j + 1 , t 1 + 2 , . . . .
KaK 6bIno nOKa3aHO (rJl. III, 3). KfY-cHCTeMa MO)l(eT 6bJTb no-
ny'tfeHa c nOMO 1O BrK-uo eJlM, HBnSlIOm.eACH OCHOBOA )\JISI LBGK-
aJIropHTMa (6.1). HanOMHHM TaK>Ke (rn. III, 3), 'ITO KrY-cHCTeMa
MO>KeT 6blTb nOJIyqeH& H npH OnHCaHHH npoQecca c HoBeHHA
AJI5I t E [t i , t;+l] c nOMO blO HHTerpa.n& CTonKHoBeHHa, 11PHCY-
[Qero ypaBHeHHIO BOJlb MaHa. B8,)KHO nHIDh, '{TO B MOMeHThI Bpe-
96. Lattice Boltzmann H ICIIHeTHt.feCKHe cxeMbI
_ 155
t. eHH t = t j , t J + 1 , t i + 2 , . .. Q>YHKL\HR pacnpetteneHHH HBJUlJIaCb 6hJ
nOKanbHO-MaKCBennoBcKoft.
Pa3JlH e Me.>KAY CHCTeMaMH LBGK H Kry npOgBJISleTCH B TOM,
qTO ,LBGK-a.rlropHTM coxpaHHeT HCKpeTHI:dA xapaKTep. B paMKaX
>Ke KrY-cHCTeMbl 3a CtleT aHaJIHTUqOCKHX npo e yp OC eCTBJIB-
eTC5I nepexo K 68JIaHCHOMY YPaBHeHHIO (rn. II, 5), a 3aTeM H K
P-H<p<PepeHnHa.nbHo-pa3HOCTHhlM (no BpeueHH) ypaBHeHHHM, onHCIiI-
BalOIlI)IM H3MeHeHHe ra30 HH8.MHqeCKHX napaMeTpOB. AH8J1HTH'Ie-
CKHR nepexo D03BOJ1S1eT HCDOJIh30BaTb B Ka:qecTBe MacwTa.60B, OIIH-
CblB8.JOlII.HX ,lJ.HCCHnaTHBHble npol\eCCbl, peanblIytO NIHHY cBo60 Horo
npo6era I H BpeMSI Me)K CTOJIKHOBeHHHMH MOJIeKyn T.
fIoMHMo cnoc06a nOCTpoeHHH, Ba..>KHOR o6 e:A '1epTOA CXeMbJ
LBGK Ii Kry -CHCTeMbI SlBnHeTC5I HaJJH'lHe MHHHMaJIbHOrO Mac-
UlTa6a, Ha KOTOpOM rapaHTHpyeTcSI OTCYTCTBHe cenbHbIX BapHa a
pewelItlSl. HarroMHHM, "ITO AJlSI LBGK-cxeMbl B Ka'leCTSe TaKora
MacmTa6a Bblc1'ynaeT XapaKTepHbIA mar npocTpaHCTBeHHoA ceTKH.
3TOT Q>aKT, no KpaRHe:A Mepe c cJ>H3HtJecKOA TOq](H 3peHHH) rapau-
THpye-r KOppeKTHOCTb peweHHSI, 11onyqaeMoro H8 aCROBe 060HX 3THX
IIO)J.XOAOB.
,lIpyroR Ba)KHOA tfepTOA LBG K H Kry SlBJlSleTC1I npocToTa HX
qHCneHfIoA peaJIH3a HH. LBGK-MeTO no CBoeMY nOCTpoeHHIO SlBJISI-
eTCSI )iBHOH cxeMoR C OTHOCHTeJIbHO M.HrKHM orpaHH'leHHeM H8 war
no BpeMeHH, BbITeKalOll\HM H3 YCJlOBHH YCToA'IHBOCTH
J).t ,$ h.
KfY-cHCTeMa (2.1)-(2.4) no cBoeA cpopMe 3aJ1HCH TO>Ke HOCHT
HBHbJ:A xapaKTep. KOHKpeTHble BhlqHCJlHTeJIbHble aJlrOpHTMbI, HC-
nOJIb3ycMble NISI pacqerra KBa3Hra30;I.HHaMH'lecKoH CHCTeMLI, B 6onb-
IIIlfHCTBe CBoeM nH60 SlonHlOTCSI SlBHbJMH CXeMaMH. JlH60 no cBoeA no-
rHQeCKOA CTpYKTYpe COBn aJOT C SlBHblMH anrOpHTMaMH (rJI. IV).
KaK H3BeCTHO, HMeHHO BBHble CXeMI:d Ha.H6onee y 06HhI K 8.AaUTa,QHH
Ha MHOf'OnpOneccopHble BbI'IHCJIHTeJJbHble CHCTeMbI c apxHTeK'rypoii
pacnpe eJIeHHOa naMSlTH.
BblcoKonpoH3Bo,a.wrenbHbIe MHoronpo eccopHble CHCTeMbl n03-
BOJIHlO'f HCII0J1.b30BaTb )I)lSI QUCJleHHOro MO,n;eJ1HpOBaHHS p83Horo po-
a HeycTo:AtfHBOCTeA CeTKH B 60..1b1DHM q}{CJlOM npoCTpaHCTBeHHblX H
BpeMeHHbIX yanOB. BHYTpeHHSlSI KOppeKTHOCTb CXeM n03BonHeT Mo-
eJlHpOBaTb HMeHHO ecTeCTBeHHble HeYCToftq}fBOCTH (:¥rOT 4>aKT He
OKa.3aH MaTeMaTHttOCKH, HO nO Bep)K):\aeTCSI 60JIbWoA BbI'lHCJlH-
Te.nbHOH npaKTHKoA HCDOJIb30BaHHB LBGK H KrY).
COt.leTaHl1e npHHlJ;HfiHaJIbHOA B03MomHOCTH MO MHpoBaHHSI
Heyc'roA'lHBbIX TeqeHHA C B03MO}KHOCTSlMH npocroR anTaU:HH K ap-
XHTeKType napaJIJIe.nhHbIX CHCTeM AeJlaIOT KCPC, KB83Hra30,lI.HHa-
MHQeCKYIO CHCTeMY H LBS-cxeMhI O)tHHM H3 nepcneKTHBHblX Hanpas-
JIeHHR conpeMeHHo:A BblqJiCJ1HTeJ1&HOA rH)ij)O- H ra3OBO:A HaMHKH.
I1x aJ1bHeAIUee COBMeCTHOe paaBHTHe, ,[\ononHeHHoe HeH36e}KHbIMH
3J1eMeHT8.MH KOHKypeHIJ)IH, n03BonHT HaRTH nyTu AJIS nOCJIe IO e-
ro YCOBepmeHCTBOSaHHjI K8JK)J.Oro H3 nepeQHCJleHHblX nO)J.XO,l).OB.
7. MeTO cT86HJlR3a OHHLIx nonpaBOK H
HeKOTophle Bapa8HTbI Kry -CHCTeMhl
B 9TOM naparpa4>e, 3a.KJIJOQHTeJIbHOM ,nnsr AaHHOA rnaBhI, pac-
CMOTpHM pH,Q MeTOAOB H Mo eneA, KOTopbIe npe CTaBnSlIOTCSI OCTa-
TOtlHO 6nH3KHMH no cBoeA H,l\eOJlOrHH K KCPC H KB83Hra30 HHaMH-
qeCKoA: CHCTeMe. 9TH MeTO bl H MOAeJlH po,nHHT C npe,1VIO>KeHHbIMH
paHee KCPC H KfY HanH IHe MHHHMaJlbHhlX MacmTa60B, Ha KOTO-
pblX rapaHTHpYeTcg Crna>KHBaHHe peweHHB, BCJIe CTBHe qero OHH
OKa3aJ1aCb Y 06HhlMH nH ODHCaHH.H pa3nH"IHOrO poAa 4>H3HQeCKHX
HeYCTOAqHBOCTea. PaCCMOTpeHHe Haq}feM C MeTo,[{a cra6HJJH3a.QHOH-
HblX nonpaaOK, npe;::vIO>KeHIloro E. Onate [78, 79].
1. MeTOA CTd.6HJIH3a HOHHbIX UOnpaBOK
9TOT MeTO,lt y 06Ho paCCMOTpeTb Ha npHMepe CTa HOHapHoro
o oMepHoro ypaBHeHHSI KOHBeK - Hct><I>Y3HH, KOTOpoe npHHHMa-
eT BH,lI;
d lfJ.
-u dx + k cJx2 + Q = O. (7.1)
llpe,l\nOJlO>KHM TaK>Ke J+fI.H npOCTOThI, qTO Kos4><P HeHTbI U H k
nOCTOHHHbI, U > O. 0603Ha1.fHM ,lJ.H<I>4>Y3HOHHYIO acTL nOTOKa Qepe3
q = -d /dx, a B.ABeKTHBHYIO ero cOCTaBnHIOUIyIO - epe3 u<P.
Y paBHeHHe 6aJ1aHca A1UI 0Tpe3Ka AB J Ha OCROBe KOToporo nOJIY-
4eHO Bhlpa)l(eHHe (7.1), MO>KeT 6h1Tb 3aJTHCaHO B CJle)Q'lOm.eM B e,
c Y'leTOM 0603HaQeHHA tl-H<txPY3HOHHOA H a,ltBeKTHBHOA COCTaBJIHIO-
lli,HX IIOTOKa:
q(x) = [utp](x) - q(x-h) - [u J(x - h) -
1 (7.2)
- 2 [Q(x) + Q(x-h)] h = O.
npH BblBo e 3Toro COOTHoweHHSI HHTerpa.n OT HCTOqHHKa Q npe,D;-
cmBneH B B e <POPMYJlbl TpanelUiR
j Qdx ' [Q(x) + Q(x - h)]h.
x-h
BblpaJKeHHSI AnSI nOTOKOB H HCTO'IHHKa B TO'qJ(e XB - h npe -
CTaBHM Qepe3 pa3JIO)KeHHe B pSiA TeAJlopa
d[u J h 2 Jl(ucp]
[uc)](xB-h) = [Uc)j(XB) - h dx B + 2" dx 2 B + O(h 3 ), (7.3)
q(xB-h} = q(xn) - h ;;: + ; + O(h 3 ), (7.4)
dQ
Q(xB-h) = Q(XB) - h dx + O(h 2 ). (7.5)
IIo cTaBJlSlSI Bblpa,)KeHHB (7.3)-(7.5) B (7.2), nOJlYIIHM
de) d ( d ) h d [ d d ( dip ) J
-u- + - k- + Q - - - -u- + - k- + Q = O.
dx dx dx 2 dx dx dx dx
nepeTlHWeM nocne Hee Bblpa>KeHHe B 60nee KOMnaKTHOM BHp'e
hdr
T---=O O<x<l.
2dx' ·
(7.6)
r e
difl cP
r = -1J dx + k dx 2 + Q. (7. 7)
3aMeTHM, ttTO, K3.K c.ne)U'eT H3 (7.6)-(7.7) 1 6anaHCHoe ypasHeHHe
(7.2) npH h = 0 npHBetI-eT K ypaBHeHHIO KOHBeK H-,lI.H<p<PY3HH (7.1).
C\lHTag <I>ct>Y3HOHHbIA nOTOK H HCTOqHHK MaJJhIMH H OT6pa-
CbJBasI I1X COCTa8njllO He, nponopqHOHaJIbHble h, npH):\eM K CJIe,lO'-
lO eMY ypaBHeHHIO nSl OnHCaHHjJ npou.ecca:
_TJ d + ( k dcfJ ) + Q + h ( U d ) == o.
dx dx dx 2 dx dx
B CY HOCTH no CpaBHeHHIO C Hcxo oA q,opMynHpOtlKoA (7.1) B
ypaBHeHHH (7.8) rrOSBHJlCSI AOnOJlHHTeJlLllbJ:A H<PCPY3HOHHbIA qneH
h d dtb
2llX u dx ' KOTOpblA, Ka.K nOKa3a.HO B (79], 06ecneQHBaeT CTa.6HJIb-
HOCTb pelIleHH ) B qacTHOCTH, CBH3a.HHYIO C OTCYTCTBHeM fle<l>H3H-
qeCKHX OTpHl.\aTeJIbHbJX 3HaqellHH 4). Xopollee Ka eCTBO 1.IHCJleHHo-
ro pClIIeHHH nO TBep>K aeTCSI MliorO'lIICJIeHHbIMH npHMepaMH TecTO-
BblX pac'-teTOB.
06paTHM BHHMaHHe Ha CBH3b TOIIbKO.1fTO paccMOTpeHHoro no -
xo a C MOAH<I>HqRpoBaHHoft nOCTaHOBKoR NISI 3a..n.a'IH c:PHnbTpal.{HH
( 5 aHHoR rJIaBbI). B MOAHcI>H pOBa.HHOM ypaBHeHHH HepaapblB-
HOCTH (5.11) B npasoA QaC1'H TaK>Ke nOSlBnHIOTCH ononHHTenbHbIe
,lJ.H<PQ:>y3HoHHble IJleHbI, KOTOpble) KaK BHAHO H3 npHBe,n.eHHblx npH-
MepOB, OKa3b1BaIOT CTa6HJIH3HPYIO ee BJlHSlHHe Ha pe3YJIbTaT HC-
neHHblX pac'leTOB. 9Ta CTa6HnH3 SI -AnH ypaBHellHSI (7.8) CBH3a-
Ha c rapaHTHpOBaHHeM CrJlaJKHBaHHSI pel1!eHHSI Ha pacCTOHHHH h, a
)J.IDI YP8.BHeHHS (5.11) - Ha paccTORHHH " paBHbIM npU6JIH3HTMb-
HO COrl'He Ha.MeTpOB 3epeH nopoAhl. To ecTb B TOM " ,lJJ>yroM CJIyqae
BBO,ll.HTCSI MHHHMaJlbHhlA MacmTa6, Ha KOTOpOM npOHCXO HT crna-
>KHBaHHe pellleHHH. O.lJ.HaKO) ecJIH B CJlyqae ypasHeHHft (7.8) 3TOT
MacwTa6 HOCHl' CeTOT.JHblA xap8.KTep, TO AJlH ypaBHeHH.R (5.11) OH
u.eJlllKOM orrpe,l{eJ1j1eTC.H H3 <PH3HQecKHX co06pa>KeHd..
BneAeHHe MHHHMaJIbHOro MaculTa6a, B KaqeCTBe KO'fOPOro BbI-
cTynaeT )J.JIHHa cBo6o Horo npo6era l, .RBnSIeTC OCHOBOA )VIS pac-
CMOTpeHHH Kry B xatleCTBe Mo enH NISI OIlHCaHHH Te'leHHSI BSI3KOro
raaa. O,l\HaKO Me>KJ1Y cra6HnH3a OHHOii TeXHBKO:A, paccMOTpeHHOR
(7.8)
· CJ1CA}'eT OTMeTHTb, qro -repMHH "cerOq)fYA" He nO,lQ)a3YUeBaeT nOJIBoro OT-
CYTCTBHjJ peanbuoro cl>wH"IecKoro CQAep>KaHHS B ero U eHHH. B C8.MOM Aene)
paJMep wara CeTKH h He B noCJIe):tH1OJO oqepe.D.b CBS3aH C BemAHHoA Macnna6a
<bH3H'teCKHX HeotJ;JtopOJJ.HOCTeA , xorophfe >KenaTe.1JhHO OUHcaTb B npoBO,D)IMOM
BW'lfHCJlHTeJ1hHOM 3KcnepHMeHTe.
7. HeKOTOpLle BapJlaHTbl KrY-CHCTeJ.fb/
159
B )J.aHHOM naparpa4>e, H KBa3Hf830 HaMHqeCKOR CHCTeMOA cy e-
CrrBYIOT e e H .AOnOJlHHTeJlbHble RH8JJOrHH.
TIoJlaraJI U > 0, YMHO)KHM H DO)\MHM OnOJIHHTen&HblA ,L\H<P<Py-
h d d
3HOHHbItt qJJeH 2 Cfiiu([X B BblpaJKeHHH (7.8) Ha BeJIH'iHHY lul. npH
3TO 1 nOJIY1.lHM:
d d ( d ) d 2d
-u- + - k- + Q + -TU - = 0,
dx dx dx dx dx
(7.9)
r)J.e
h
T = 21ul
(7.10)
SlBJ1Sle'rcSI, COrJIaCHO TepMHHOJIOrHH [79], BHYTp€HHHM BpeMeHeM.
npH pacCMOTpeHHH IlpOCTpa.HCTBeHHo- BYMepHbIx H TpexMep-
HblX 3CV\aQ ,LVIH Ka)KAOrO HanpaBJIeHHSI HCIIWlb3YeTCSI CBoe 3Ha'leHHe
T, onpe enHeMoe 3HaqeHHeM CKOpocTH H warOM CeTKH B onb COOT-
BeTcTBYlOlQeii KOOPAHHaTbI.
nepexo OT ypaeHeHHSI (7.3) K (7.9)-(7.10) HMeerr npswylO aHa-
J10rHIO C BbJBO,lJ.OM KBa.3Mra3o,L\HHaMH'ieCKOA CHCTeMbI H3 paCCMOTpe-
HWI cl>YHKQHH pacnpe eneHIDI Ha npOH3BOJIbHOA j1 eAKe (rn. II, 5).
3 ecb TaK>Ke paCCMaTpHB aeTCSI xapaKTepHoe BpeMB T = h;r II{ilnf
((5.3) rJl. II), 3a KOTopoe qacTHQa nepeceKaJla paCCMaTpHsaeMYJO
S(QeAKY. BnocneJ1-CTBHH BeJlHt!HHa ' ilnl 3aMeHHJI8Cb Ha ee xapaK-
TepHoe 3Ha'-l€HH€, B KaqecTBe KOTOporo BbJ6Hpa.n:acb CKOpOCTb 3SYKa
c. nO 06Ha.sI rrpoQeJQ1pa n03BOJIHJla nepeATH K AOCTaTO'lHO y o6-
HOMY 6aJ1aHCHoMY ypaBtleHHIO ((5.9) rn. II), H3 KOTOporo BfiOCJleA-
CTBRH HHTerpHposaHHeM C CYMMaTOpHblMH HHSapHaHTaMH nOJ1YQa-
eTOi KrY-cHCTeMa.
Cne)JyeT TaK>Ke O'I'MeTHTb, qrrO CTa6HJIH3aI\HOHHaH TexHHKa
ycnemHo HCnOJlb30BaJlacb JI.S1 OnHCaHHSI rH po HHaMHqeCKHX npc>-
eCCOB. I1pH aTOM nOJlYQaeTCS1 CHCTeMa ypaBHeHH1t, HMelO aH Il.O-
nonHHTenbHbJe AHCCHnaTHBHble qJleHbI IIO cpasHeHHIO C YPaBHeHHSI-
MH HaBbe--CTOKca. 9TH OnOnHHTeJIbHble 1.f:JleHbI no cBoeR CTpYKTY-
pe noxomH Ha aH8JIOrHQHble )J.HCCHnaTHBHbJe qJleHbI KrY-cHCTeMbJ.
PaccMOTpHM e Ba noroco a, T8J(>Ke pOACTseHHble Kry-
CHCTeMe H npHMeHSleMble AnH OIlHCaHHB )J,BlDKeHHSI BR3KOA >KH KO-
CTH.
2. KBa.3HrUAPO,l1.HH3MHtJecKaH CHCTeM& ypaBBeHHR
B pa60Te [SOJ IO.B. llIepeToBblM 6bJJIa BBe eHa CHCTeM& ypas-
HeHHft, KOTopaa no aHaJIOrHH C KfY-CHCTeMOA 6b1JIa Ha3BaHa KBa-
3HrHlU>O I-IaMl-rqeCKo:A. CHCTeMoA ypaBHeHd:
'::: + div(pu) = div(pw) ,
a () fJp 8piJ:
-(pUk) + -(pUiUk) + - = pFk - +
at 8Xi ax k 8Xi
{)
+ _ 8 ( PWi'Uk + PWkUi ) ,
Xi
a: + a i (Ui(E+p») + div q = pFi(Ui - Wi) -
- () j (UtPik) + a i (wi(E+p) + f1Ui U /cw/c) ·
3)I.eCb E = p (u 2 /2 + £) - nOJlHaSi 3HeprnSi p e - BHYTpeHHSlB 3Hep-
rHH, Pik - TeH30p ec'recTBeHHLJX BSl3KHX HanpS)KeHHA, q - BeK-
TOp eCTec1"BeHHoro 1'enJIOBoro nOTOKa (CM. BblpaJKeHHSI (4.28)-(4.30)
rn. I) I
(7.11)
(7.12)
k = 1. 2, 3,
(7.13)
T [ Ouk op ]
Wk = - {J'Ui - + - - pFk ·
P OZj Ox"
B CBOIO Oqepe,ll.b, BeJmqHlla T, HMelOI.1(aH cpH3HtleCKHA CMblCJI spe-
MeHH Me*AY C'l'OnKHOBeHHBMH, MO>KeT 6blTh onpe eJIeHa Qepe3 BH3-
KOCTb, AaBJIeHHe H pyrHe MaKpocKORHqeCKHe napaMeTphl cpe
(80) 81].
06paTHMcH K CMblCJIY bBe eHHSI BeKTOpa w. ,l\nH 3Toro nepeOH-
meM YPaBHeHHe (7.11) B BH e
(7.14)
8p d .. 0
lit + IV J = ,
(7.15)
r e j = p( u - w).
HTaK. BeKTop nOTOKa MaCCbl onJ>eAengeTC.H ABYMH KOMnOHeH-
TaMH. IIepBas (pu) CBH3a.Ha C KOHBeKTHBHhJM nepeHOCOM. BTOpag
(pw) o6B3aHa CBOHM nOSBneHHeM OCHOBHOMY npe onOiKeHHIO, Ha
g7. HeKOTOpl:Je SapHatfTbl Kry-CHCTeMbl
161
KOTOpOM CTpOHTC.H KCPC H KBa3HI'830 HaMHqeCKaJI CHCTeMa, -
CJIa60MY H3MeHeHHIO <l>YHKQHH pacnpe eneHHg H ra30JI-HHaMH eCKHX
napaMeTpOB B 06JIacTH, pa3Mepbl KOTOpoA no nopB KY BeJIHqHHbI
pasHbI nHHe CBo6o HOro npo6era. 110 CYTH eJla, :na OnOJlHHTeJIL-
H8.SI KOMnOHeHTa pw 06eCne'IHBaeT nepeMeWHB8JlHe H BhIpaBHHBaHHe
MaCCbI BHyrpH 3TOA o6nacTH.
QTo6b1 nonYQHTb ypaBIIcHHSI AJlH H3Mel-JeHIDI HMnYJIbCa B KBa-
3HrH pO,[lHllaMHqeCKOa CHCTeMe, K cPaKTOpaM, npKBO.L\ M K ero
H3 feHeHHIO B CHCTeMe ypaBHeHHR Ha.Bbe-CTOKca - KOHBeKTHBHOMY
nepeHOCY HMnYJlbCa pU,Ut, CHJI8.M ):t&SneHHSI Pt BHeWHHM (F) H Bjf3-
KHM (Pile) CHnaM, - o6aBHM nepeHOC HMnYJlbCa 3a CqeT OnOJIHH-
TeJ1bHOrO nOTOKa MaCChi (- pw). 9Ta OnOnHHTeJILHaH KOMnOHeHTa
onpeAeJlHeTCSI KaK
a i (PWiUA; + PWkUi) ·
A HaJlOrHQHO, B ypaBHeHHH 3HeprHH HOBbie no OTHOllleHHIO K
CHCTeMe ypa.BHeHHA: HaBbe-CTOKCa qneHbI o6YCJIOBJIeHbl OnOJIHH-
TenbHblM nepeHocoM 3HeprnH wE H pa6oToa 3a C eT ,A06aso'iHoro
nepeHoca MaCCbI
a:i (Pwi + f1Ui U I; W k) ·
YqHTbIBaJI npH H3MeHeHHH HMnynbca H nonHoft 3HeprRH H AO-
nOJIHHTMbHbJe qneHbJ 1 npH eM K yp8.BHeHHHM (7.12) H (7.13) KBa-
3HrH)J.po HH8.MHqecKoR CHCTeMbI.
B03BPaIU.aJlCb K opHrHH8.JIbHOMY cnoc06y nOJlY'IeHHH KBa3HrHA-
P0,I1.HHaMH"lecKOA: CHCTeMbI [80], OTMeTHM t QTO OH 6b1n OCHOBaH Ha
YCJlOBHH corJIaCOSaHHB ypaBHeHH.SI 6aJ1aHca 3HTponHH H ypaBHeHHR
(7.11)-(7.14), onHCbIBalO HX 6a.naHcbI MacChl, HMnynbca H 3HeprHH.
Yc.noBHeM T8.Koro COrJ1acoBaHHSI SlBWlOCb HCUOJIb30B8.HHe TOm,L\eCTBa
rH66ca
TdS = dE + pd(l/p).
(7.16)
IIpH 9TOM ypaBHeHHe 6an&Hca 3HTpOllHH S npe CTaBJlgJIOCb 8 BR)].e
8 S) + div(puS) = div(pwS) + div (X ) +
( V T ) 2 PikPti pw2
+ x T + 2p.T + T ·
(7.17)
91:0 03HaQaeT, '11'0 pOCT 3HTpOUHH B CHCTeMe, HapSl c T]) H OH-
HblM BnHiI}IHeM Tennonpo80 HOCTH H BH3KHX can, ocy ecTBnSleTCB
3a CqeT OnOJlHHTeJlbHOro nepeMelIIHBaHHH pw2/T B o6nacTH c xa-
paKTepHblM pa3MepOM nopjJ Ka ATIRHbl CB06o Horo rrpo6era.
3aMeTHM t QTO eKBaTHoe OfiHCaHHe H3MeHeHHH 3HTponHH OT-
MeQanOCL el e B nepBblX pa.60TaX, DOCB eHHhIX KCPC, 60JIb -
MaHOBCKHM cxeMaM H KBa3Hra30,ll.HHaMOqeCKoit CHCTeMe ypaBHeHHR
(24, 28, 35 J ·
B OTRHQHe OT KrY-cHCTeMbI, KBa3Hr pO,l1.HHaMHqeCY..a.H CHCTe-
Ma u.enHKOM, BKJIJO'IaJI npOH3BO hle no BpeMeHH, 3anHcaaa 8 H<t>-
cpepeHQH8JIbHOR cl>opMe. 9TO n03BOJISleT OKa.3aTb ee 6J1H3OCTb K
yp8J3HeHHRM Hashe-CTOKca C TO'IHOCThfO o O( r 2 ) TOJ]bKO nB CTa,-
I{HOHapHoro cnyqasr..
3aBepWaH OnHCallHe KBa3HrH po HaMH'IecKoA CHCTeMhl t OTMe-
THM, TO OHa XOpOlllO 3apeKOMe OBana COOH "PH paccMoTpeHHH
paanHQHhlX Te1.JeHH:A BSl3KoA )KH KOCTB, B nepBYIO o epe h, KOHBeK-
THBHblX TeqeHHA [81 J 83].
3. KBa3Hra30AHHaMHllecKaH MO,l1.e.nb)1.JIR onBC8HHSI TeqeHHH
CJIa600KHMaeMblx ffi30B H )l(B).UCOCTeH
BapH8HT KBa3Hra30 aMHqecKo:A CHCTeMbI ,lVllI Mo,neJIHpOBa-
HHjI TeqeHHI cna60C>KHM&eMoro raa& H )KH,a.KOCTeft MO>KeT 6b1Th no-
nyq H Ha 6a- e Hcnonb30B8.HHH ypaBHeHIDI 9HcKora (2.36) (rn. I, 9 2)
.CnelXYeT OTldeTHTb, QTO attaJIOm'lHblA pe3YnbT8T B OHapHOU cny-
'Ia.e NO)KeT 6blTh nOJ1YQen H AJUI CTO AH <l>epeHI(H&1IhROro aH8JlOra. Kry-
CHCTeMbI [821. O aKo B H oHapHo" CJI)''Iae OTJIH&JHe Me*A)' AH<IJ<I>epeH-
lJ,H&nbHhlM aHaJlOrOM Kry H ypaBHeBHJIKJI Haabe-CTOKCa ODHC TCJ[ )')ICe
'IJIeHaMH nepaol'O nopSl;:tKa ManOCTH DO T. 9TOT alCTt )"IHThlBaJl pe3YJlbmTbI
2 .lt3HHOA rnashl, yX83h1BaeT H& npHH &1IhRYIO TeOpeTH'iecKYIO B8JKHOCTb
AH<I><l>epeHII.HaJlbHo-pa3HOCTHOA 4>oPWhl 3aIJHCH KrY-cHCTeM .
N7. HelCOTOpbIe sapHaHTbl KrY-CHCTeMhl
163
,WIg ODHCaHH.SI O HOqacT"q}fOa <PYHK HH paCnpe)];eJIeHHB (84J. nH
3TOro enaeM CJ1eAYIOIInle npe OJIO)KeHHg, CXO>KBe C HcnOJlb3ye-
lbIJvtH rIpH BblBO e KCPC H Kry:
1) Ha Bpe (eHHOM CJ10e t = t J <t>YHK g pacrrpeAenesH.Sl J BCIOA}'
HMeerr JIOKaJlLHOe paBHOBeCHoe 3HaqeHHe 11';
2) cPYHK HS pacnpe eHHH CJIa60 MeRjleTCH 3a BpeMSI T H Ha
paCCTORHHH " onpe,a.enHeMOM K8J< 1 = rU, r e U - xapaKTep-
Hoe 3HaqeHHe CKOpOCTH MoneKYJI;
3) B MOMeHT BpeMeKH t j + 1 = t j + T <PYHK HH pacnpe,neneHHH
MrHOBeHHO npHHHMaeT paBHOSeCHoe 3HB"IeHHe.
3,n:ech T BbICTynaeT B Ka lecTBe a.H8JIOra. BpeMeHH Me}l(AY CTOJ1K-
HOBeHHHMH , KOTopoe AJl5I UJIOTHLIX f830B onp enHeTCSI c nOMO 1O
BR3KOCTH I-'t KHHeTHqecKoR COCTaDnmo eA aBneHH PK H KOHCTaH-
Tbl CKOpOCTH peaKI:(HH 9A:pHHra K34» CJI M 06p83oM [1]:
J1. = PK/K ,
r = 2/K* = 2JJlpK-
(7.18)
3aMeTHM , QTO 3AeCb MbI He npHBH3blSaeMCSI K npe onO>KeHH1O 0
6ecCTOnKHoBwrenbHOM xapaKTepe p83JleTa npH t E [t'. t j + 1 ]) OrpaHH-
lJHBaHCb, KaK H npH BblBo,u.e KrY -CBCTeMbI, npHBe eHHOa B 3aMe'fa-
IIMH 1 3 ):taHHoA rnasbl HH<I>opM eA 0 TOM, QTO <l>YH H.R pacnpe-
AeneH ISI npH t == t i H t = t j + 1 npUHHMaeT JIOKallhHo-pasHosecHoe
3HatJeHHe.
B C,LJ.ena.HHbIX rrpe OJlO}KeHHgX H3 ypaBHeHHSJ 9HcKora (2.36)
(PJI. I) CJJe)Q'eT:
f(x, . t j + 1 ) = f,(x - T , E - TG/m, t j ) +
+ [J(x) ,tj+l)+J(x-rEI -rG/m,tj)] +O(r 3 ).
(7.19)
OnHpaHCb Ha CBoACTBO 2), npe CTaBHM 3HaqeHHe <PYHKll)IB fp
If HHTerpa.na cTOmcHoBeHHA: J B BH e p83nO)KeHHJI B pH TeftJIopa B
OKpecTHOCTH TDqKH (x, , t j ). IIpH 3TOM COOTHoweHHe (7.19) nepeA-
eT B
j+1 _ i ) _ all!. _ rGIc a,p
f(x, , t ) - fp(x, , t T{k a a ' +
Xl: m k
r 2 a2 kf"fp T 2 a2 kGl/m · fp T 2 ar GI;Glf p
+ - + - + - ( 7 20 )
2 OX/t:OXl 2 aXlta 1 2 8 ka{f m 2 ·
T [ '+1 . ] T2 oj 3
+ 2 J(x,(,t ' ) + J(x, (, t.1) - 2 k aXk + O(r ).
qTo6b1 nOJ1Y"lHTb KBa3Hra.30 R8.MH'IecKoe ypaBHelrne Hep8.3-
pblBHOCTH, npOHHTerpBpyeM (7.20) c seCOM m. llpH 3TOM Y'fTeM, 1.fTO
B C.Hny CBoACTB HHTerpana CTOnKHoseHH:A ,AJIB ypaBHeHHSI 9HCKora
HMeeM
I m [J(x,EI ti+1) + J(x,E,t-i)] = 0,
J T2 aJ 8 T 2 f
- m- k- = --- m kJdE =
2 OXk OXk 2
a T 2 a r 2 FPPib
= _ 8 _ 2 _ 8 (P.)kl = a 2 ·
Zit Xl X/c
nOCJIe Hee paBeHCTBO (7.22) cnpase,lVlHBO no npH'IHHe TOro, qTO
)VIa paBHoBecHoR CPYHK HH pacnpe eJleRHg fiOTeHIJ.HaJlhHCUi \facTb
Jl-aBJIeHHSI Pill eCTb (P )/d = P.dkl.
TaKHM 06pa.30M, B pe3YJ1bT&Te HHTerpHpOBaHBR (7.20) ( CYMMa-
TOpllbJM HHBapHaHTOM m, DOJIY'1HM:
(7.21)
(7.22)
pi+l - ,J 8 . 8 T a .
+ _(pUt)1 = ---(pukUI)J +
T aXle 8Xt 2 OXl
a T 8pi {J ( T pGk ) '
+ 8XA; 2 aXle - ax,- '2 m I
(7.23 )
rAe )J.aBJIeHHe p npe CTaBJ1BeT 00608 CYMMY KHHeTHqecKOA u nOTeH-
anbHOa COCTaBJIHI01.1J)iX: p = P K + p ·
.g7. HeKorophle BapHI111TbJ KrY-CHCTeMbJ
165
Y paBHeHHH ,lVISi OnUCaHRjI H3MeHeHIDI HMnYJIhCa H nanaoA 3Hep-
rUB nonyqeHbI aHaJlOrH'C.fHO:
(p U k)j+l - (puk)i ( lJP'UlUk ) j 8pi ( pGle ) '
+ = -- + +
T OXI ax/( m
8T8 . ora -
+ ---(PK · Uk)' + ---(pukUrU,)J - (7.24)
ox I 2 8Xl {Jx r 2 8Xl
_ ( T Gl pUk(l + Ok'» ' I
aXI 2 m
Ej+l - Ej a . ( pG1ttl ) j
+ -((E+p)u,)J - =
T &t m
8 T a ( 2 E ) ' () T a ( ) '
= --- (PUl+PK)- + --- (E+2PK)Uk U l +
{)Xl 2 aXl P /C#l lJxA; 2 Ox,
( 3T pG l ) ; a T ( Gz Gk 2 ) '
+ - - -- -PUtUI + -(put + 3PK) ·
2 m 2 aXle 2 m 2m
llpH BblBO)J.e ypaBHeHHA (7.24) H (7.25) Yq}[TblBaJlcsr 60JIee CJlO)K-
HbJA MeXRHH3M nepeAaQH HMnynbca H 3HeprHH B nJlOTHblX raaax H
)f(H KOCTSlX, 3aK.nlOqaJOIlJ)IACSI B HaJIH'IHH enOqKH nOCJIe OBaTenb-
HbJX CTonKHoBeHH:A. 9TOT MexaHH3M ocyw:ecTBn.sreTCH npH HaJlH'lHH
60nee HJ1H MeHee O,nHOpo Oa CKOpocTH nepe})oca HMnynhca H 3Hep-
rHH ,lVI5I BCex qacTH[J.. B Ka1lecrBe TaKoR CKOpOCTH BbJ6epeM no Ka)K-
OMY HanpaaneHHIO Xk BMH'lHHY
(7.25)
JUA:I = (u + c 2 )1/2.
(7.26)
B HTOre 6aJ1aHCHoe ypaBHeRHe NIB <PY H pacnpe MeHHg, H3
KOTOporo 6ytU'T nOJIyqeHhI ypaBHeHHSI, ODaCLlBaJOll\He H3MeHeHHH
HMnynbCa. (7.24) H fiOJlHOA 9HeprHH (7.25), npHMeT BHA
j+l . alp rG. alp
f(x, , t ) = fp(x, , tJ) - T k- 8 - +
Xk m U k
r 2 82 k lfp (1 ) 1"2lPU f'P T2lP k(G,/m) /1J
+ - · - Ukl + - + - (. +
2 OXIcOXl 2 8x 2 aXka 1
T 2 dlGIcGl I p r [ j+l ( i )]
+ "2 8{k a {, ;;;2 + 2 J(x, (, t ) + J x, (, t ·
113 AHCCHuaTHBHbIX qneHOB CHCTeMbJ (7.23)-(7.25), 'raK >Ke KaK
H3 npaablx qaCTeR OpHrHH8JIbHOA KBa3Hra30AHBaMHqecKOA CHCTe-
MLlJ fO}l(I-IO BbI.nenHTb 'VIeHbJ, onHCbIBaJO He ecreCTBeHHYIO B -
a T 8
KOCl'b H TennonpoBO OCTb. HanpoMep, 'IJleH W::- 2 (PKElp),
ax, ax,
CTO 1I.I,Ha B npaooR qaCTH ypaBHeHHH (7.25), MO>KeT 6b1TL npe CTaB-
neH, c yqeTOM (7.18), 8 B e
(7.27)
8 T 8 8 T 8 2
8 X l 2 ax, (PKE/p) = ax, 2 ax, PK(U,,/2 + E) =
a T 8 2 a ( " 8pK ) 8 or
= 8Xl 2 ax, {PK u k/ 2 l + ax, 2 £ aXl + ox, :K ax, '
TaK )Re, KaK H AJlSI KrY-cHCTeMhI, MO>KHO nOK83aTb, TO OCTaJIb-
H.bJe t);HCCHnal'HBHbIe 'IJIeHbI OKa3b1BaJOTCH MaJlbJMH no cpaOHeHHIO C
KOHseKTHBHblMH H ecTeCTBeHHhlMH AHCCHnaTHBHblMH 'tJJIebaMH.
B nO TBep)l(AeHHe npHBe eM pe3YJIbTaTbJ conOCTaBneHIDI pacqe-
1'OB Ha OCHOBe YPaBHeHd (7.23)-(7.25) H OnHpalOIQHXCSI Ha HCnOJlb-
30BaHHe ypasHeHHA: Hashe-CTOKca (85] ,LVISI 3 a,{H, OnHcOIBarorneR
TeqeHHe HeC)I(HMaeMo:A >K KOCTH B KaBepHe c HJlCy eACSI BepxHe:A
KpbIillKOA (pHC. 31). Ha pHC. 32 npHB eHbJ AaH e pacqera sepTIt-
K8J1bHO:B (Bt);OJIb OCD Y) cocTaBnHIO eR CKOpOCTH B eHTp8JIhHOM ro-
pH30HT8JlbHOM Ce'leHHH KaBepHbI, pacC HTaHHble no 06eHM MeTOJ{H-
KaM. PaccMaTpHBaJICR KOHKpeTHblA sapHaHT npH tJHCJIe PemOnb,ll.Ca
Re = 400. Pac1.JeT CHCTeMhl (7.23)-(7.25) BenQ1 H& paBHoMepHoA
npHMoyrOJlbHoA CeTKe c tJHCJIOM )'3nOB 161 X 161. KaK BD,ltHO. pe-
3YJlbTaTbI pacqerOB, npOBe eHHLlX Ha OCHOBe CHCTeMbl (7.23)-(7.25),
npa.Kl'HQeCKH He OTnlNaJOTCSI OT a RH X, nonY'IeHHhIX c nOMOU(hlO
(7.28)
g7. HeKoropble BapHaHTbI KrY-CHCTeMbl
161
y 0.-:
o
PHC. 31. K nOCTaHODKe 3M&1{U 0
KaBepHe C ,l1BH1Kyw.eltcR KpWW-
KoA
Y.W£LAEI"
PHC. 32. npo<pHnb BepTHK8J1hBoH CKo-
pOCTH B eHTpanbHOM CeqeHHH KaBepHW:
CnnOIllHaB KpHB8JI - pe3YJIhT&T p&C'IeTOB
H& OCHORe CRCTeMlII (7.23)-(7.25),
X x x - ,..8HHhle pa60Tw (85J
YPaBHeHHA Hasbe-CTOKca, qTO nOATBep 8eT npaSOMOqHOCTL HC-
fIOJIb30BaHH.Sl npe O)l(eHHOa CHCTeMhI YPaBHeHHA.
BepHeMCSI BHOBh K KBa3Hr po HHaMH'IeCKOa CHCTeMe, pac-
cMOTpeHltoA B npe bl eM nYHKTe 8HHOrO naparpac:Pa. HaJIH'IHe
rotccHnaTHBHblx qJIeHOB B npaBO:H 'facTH ypaBHeHIDI Hepa3pblBHo-
CTH SIBnHeTCSI rnaBHblM, '1TO OTJ1HIJaeT 3TY CHCTeMY OT YPaBHeHHa
Haobe-CTOKca. 06paTHM BHHMaHHe Ha llOJIHOe cOBn eHHe 3THX
,l);MCCHUaTHBHbIx qneHOB J BKJIJOQaB 3aI1HCb )\&BJIeHH5I B oH e CYM-
MbI KHHeTw-IeCKOA H DOTeHIJ.H8JIbHOR cOCT8BnHIO , B KBa3Hru,lij)o-
HHaMH'feCKOa CHCTeMe H caCTeMe (7.23)-(7.25). HanOMHHM, '1TO B
nepBOM CJIyqae 3TH HCCHnaTHBHbIe qJ'(eHhl 6b1nH nOJlyqeHbJ H3 rep-
MO HHaMHqeCKHX COO6paJKeHHft, a. BO BTOpoM - HenOCpe CTBeHHO
1'13 KHHeTH ecKoro ypa.sHeHHSI 9HCKOra.
*
*
*
aHHaH rJIasa 3aK8J1qHBaeT 06CY:>KAeHHe KCPC H KBa3Hra30-
AHHa.MHqeCKO CHCTeMhl KaJ< <!>H3HQeCKOA MO enH AlI5I ODHcaHHSI Te-
1.{eHH BB3KOro ra.3a. rn8.BHblM B 9ToA MO eJIH BBnseTCSI Crn8JKHB&-
HHe pellleHHg Ha pacCTOSlHHH nop ,lVIHHbI CBo60,a.Horo np06era
l. Tpe60BaHHe Crn8JK;IofB8JlHS pemeHH5I Ha 3TOM pacCToaHHH eCTe-
CTBeHHO. OHO BbiTeKaeT H3 KRHeTHqeCKoro 06ocHOSaHHSI ypaBHeHHH
rH)U>o- H ra30BOA AHllaMHKH, 8 CBOIO Oqepe.n;b OnHp8JOl1\erOCSl Ha
6nH3OCTb CPYliK HH pacnpeAeJ1eHHSI K ee paaHoBecHOMY 3HaQeHHIO.
EcTeCTBellHOCTb H <PH3H'tIecKaSI HenpOTHBOpeqHBOCTh 3TOrO JJ.o-
nOJII1H'reJIbHOrO Tpe6088.HHg Ha CrJlaJKRBaHHe peIllCHHH npHBO HT K
TOMY, tfTO pe3YJlbTaTbl t ItOJlyqeHHhJe Ha 683e HOBOR MO MH, IIpaK-
THqecKH He OTnH'iaIOTCSI OT c00l'BeTCTBYIO HX pe3YJlbTaTOB , nOJlY-
qeHHblX Ha OCHOBe ypaBHeHHA HaBbe-CTOKca. OAHaKO 3m Mo enb
o6na.n.aeT DO CBoeMY nOCTpoeHHIO BHYTpeHHeR KOppeKTHOCTblO, qTO
n03BOJISleT (rn. V) ycnelllHo HCDOJIb30BaTb ee NISI MO,l\eJIHpOBaHHB
CJIO)f(HblX rHAPO- H. ra30 HHaMHqecKHX reqeHHA, BKmoqasr HeYCToA-
HBble TeqeHHSI. KpoMe Toro, B paMKaX KCPC H Kry JlerKO CTpOHTb
napa.nJ1eJ1bHhle 8JJrOpHTMbI (rn. IV), 1:fTO T8J()f(e SlBJ1S1eTCSI KOCBeH-
HbIM OTp8JKeHHeM ecTeCToeHHOCTH &HHOH Mo enH.
IIo cy'rH ,lJ.eJIa. B paMKaX npeA1laraeMoA Mo enH BBOAHTCH MH-
HHMaJIbHbIA MaclllTa6 (,lJ.JlliHa cBo60 Horo npo6era), MeHhWe KOTOpo-
ro c <P1i3HQecKOA TOQKH 3peHHS AeT8JIH3al\HB peweHHH OBHTCSI
6eccMblCJJeHHolt. MaTepHaJlbI 5 H nOCJIe,n}'lO x naparpa<poB no-
Ka3bIBaIOT, t.JTO npHHl\Hn BbI eJleHHS MHflHMaJlbHblX MacuJ'ra6oB H
nOC1'poeHHe MO eJIeA, rapaHTHpyfO X Crna>KHBaHHe peweHHSI Ha
HHX, HBJI5JeTCS B OCTaTO'lHOa Mepe yHHBepC8nbHblM. lfMelOTCH H
APyrHe MH3KHe not\XO,lJ.ld K nocTpoeHHIO TaKHX MO eJleaJ Be Hawe -
ruRe OTpa)l{eHH.B B AaHHoA: MOHorpa.c.pHH, B 'iacTHOCTH, H3J!O)KeHHble
B pa60TaX [86, 87].
B03MO)KHOCTb nOCTpoeHIDI TaKHX MOAeJIeR BHOCHT 3aMeTHYIO
OJIIO onTHMH3Ma B rrepcneKTHBDI HCDOJIb30BaHH.Sl BblcOKOnpOH3BO-
HTenbHbJX BbI'fHCJlHTMhHblX CHCTeM P.JISI pemeHHa 3CVJ-aQ Me xaHH -
KH cnJlomHoA cpe bJ, r e npo6neMa KOppeKTHOCTH, CBSl3aHHaJI C BbI-
,l).eJIeHHeM 4>H3HQecKHx H DOAaBJIeHHeM CqeTHbiX HeYCTOA HBOCTea)
npHo6peTaeT oco6oe 3HaqeHHe npH eT8JIbHOM npOCTpaHCTBeHHOM
OnHCaHHH npou:ecca. OnOJlHHTeJlbHbJe no cpaoHeHHlO C I<na.cCH'Ie-
CKHMH Mo.n-eJI)1MH (YPaBHeHHH HaBbe-CToKca) HCCHIIaTHBHbIe qne-
Rbi Ha npaKTHKe cnY>KaT 3cpcpeKTHBHblMH perynSlpH3aTOpa.,\lH, rac.sI-
HMH C"IeTHbJe HeYCTOAlIHBOCTH.
N7. HeKcrropble Bapll8HTbl KrY-CJfCTeJlbl
169
B CJle YlO HX rnaBax IV H V 6y,lQ"r Henocpe CTBeHHO paccMOT-
peHbl Bh1QHCJ]HTeJlbHhJe aJIropHTMLI Ha 6a3e KCPC H Kry, HX an-
TaJ..\HSI K apXHTeKT)'pe MHoronpou.eCCOpHbJX CHCTeM C pacnpe eJleH-
Hott naMSITblO H npHMeHeHHe M'HX a.nropHTMOB NISI MOAMHpOBaHHH
Pa3JIHQHbIX 3 aq re)Wo- H ra3OBOR HaMHKH.
rJIaBa IV
BbIqHCJIHTEJIbHbIE AJIrOPHTMbI II
HCnOJIb30BAHHE
MHOrOIIPOQECCOPHbIX CHCTEM
JI PEmEHH 3A Aq rA30BOa
HHAMHKH
B aHHOa rnaBe 6y pacCMOTpeHbI HeKOTOpbJe BLlqHCJlHTeJlb-
Hble a.nropHTMbI ):VISI Mo enHpOBaHIDI 3a)\aq BH3KOrO ra3a H >K -
KOCTH, nOCTpoeHHe KOTOpblX cy eCTneHHbJM 06pa30M O:JHpaerca Ha
onHcaHHble B 11pe b1.l\Y1U)IX Pa3AeJ1ax KCPC H KB83Hra..10 HHaM"qe-
CKYIO C"CTe IY ypaaHeHHA. Oco6oe BI-IHMaHHe y enHeTCH B03MO>KHO-
CTH a,L\an "H 3THX MeTO)J.OB K apxHTeKType MHoronp'J eccOpHblX
BbJ CJJHTe.nbHLlX CHCTeM c pacnpe eJleHHo:A fiaMBTblO. JIorettecKaH
IIpoCTOTa MHorHX aJIropHTMOB, nOCTpoeHHbIX Ha 6a3e KCPC H Kry,
,nenaeT HX YA06HbIM HHCTpYMeHTOM npH peweHHH 3 aq rH,lWO- H ra-
30BOft AHHaMHKH Ha COBpeMeHlIblX BblCOKonpoH3BO,ll.HTeJIbHbIX MHO-
rOllpO CCCOpHbJX KOMnJleKcax. B CBOIO Oqepe,nb, peaJIbHaJI B03MO)K-
HoC'rb MO eJIHpOBaTb c nOM01U.b1O KHHeTHqeCKHX cxeM Ha MHoronpo-
QeccopllblX CHCTeMax 3 aqH COBpeMeHHoA a3pO,l],MHaMHKH craBHT Ha
nOBeCTKY jJ)IH peUIeHHe pSl a npo6JJeM napanJIeJIbHoro MaTeM&THqe-
CKoro o6ecneQeHWI. 06 3'1'OM TaK>Ke nO eT pe'lb 8 AaHHoA rnaBe.
PacCMOTpeHHe MaTepH8Jla Ha'tJHeM C ormcaHHSI KHHeTHqeCKH CO-
rnaCOBaHHblX cxeM C KoppeK eA) KOTOpbJe 6blJlH IlepBblM anropHT-
MOM 113 KJIacca KCPC, npHMeHeHHblM ,lJ.JIB MOAeJ1HpOBaHI1S1 TeqeHHR
B 3Koro ra3a.
1. KUHeTUqeCKH cornaCOBaUH&Ie cxeMLI
C KOppeKIJ.HeA
Ha3B3.HHe MeTO a "KCPC c KoppeK HeH ' no,a.qepKHBaeT CBSI3b
Me KHHeTHqeCKH cornaCOBaHHOH cxeMoft H KB83HraaO HHaMH-
recKoA CHcTeMo.R ypaBHeHHA, K KOTOpoA npHMbIKaeT aHHhI.A aJI-
ropHTM. npH nOCTpoeHHH 3TOA cxeMbl 6bU1a HCn0J1b30BaHa CBH3h
Me>KAY KCPC, Kry H KJlaccHQeCKHMH MO eJljJMH) DonO>KeHHbIMH B
OCHOBY OnHCaHHSI BH3KOrO ra3a [57]-(59J, [88].
1. Pa3HOCTHble cxeMbl fiB, apBMoyrOJIbHblX CeTKaX
OCHOBH&g H)J;e51 KHHeTHCofecKHX cxeM C Koppe BeA OCHOBaHa Ha
BblAeneHHH 8 KBa3Hr830AHHaMHQeCKOA CHCTeMe 8 BBHOM B e q;re-
HOB ecrecTBeHHOa BSl3KOCTH H TerIJIOnpOBO)];HOCTH, npOnOp HOHaJlb-
HLIX ,D)IHHe cBo6o HOrO npo6era. B rJlaBe III 6bIno nOKa3aHO, QTO )1.0-
nOJlHHTeJIbHble ,l1.HCCHIIaTHBHble qneHbI Kry AaJOT MaJILlA BKJI B
pemeHHe no CpaBHeHHIO C ecTeCTBeHHblMH HCCHDaTHBHhIMH qJJeHa-
MH. II09TOMY peilleHHB Kry -CHCTeMhI H ypaBHeHHH HaBbe-CTOKCa
npaKTH"ieCKH He OTJIH'I8JOTCH yr OT ,npyra.
B KCPC C KOppeKJJ.He:R HCCHnaTHBHLle qneHhJ. CTOjJ He B npa-
BbIX 'IaCTHX p npe CTaBlIeHbJ B <l>opMe CYM lbl ecTeCTBeHHblX H o-
nOJIHHTeJIbHblX P.HCCHnaTHBHblX qneHoB. nOCJIe HHe, no aH8JJOf'HH
C OpHrHHaJlbHo.R KrY-cHCTeMoA (rJl. II, 95), 6epYTcSI nponopD.ao-
HaJIbHhIMH mary npocTpaHcTBeHHoA ce'l'KK h. BbInHmeM 3TY cxeMY
Ha npS1MoyrOJIbHOA CeTKe ,[VUI CJlyqa.st 3D-reoMeTpHH:
. 1 . ·
p1+ - p1 , · · ( dX 2 ) ,
6.t + (pu) + (ptJ) + (pw): = 2U (pu + ph % +
( y 2 ) ' ( Z 2 ) i
+ 2U (pv + p)ji II + 2U (pw + ph z'
(1.1)
(pu)j+ ; (puV + (pu2 + p )! + (fYU V ) + (puW ) =
x II :
( X 3 ) j ( y 2 ) ;
= 2U (pu + 3PU>t x + 2fj(pv U), +
( 4% 2 ) ; [ a ] i
+ 2U (pm u}z z + Oxic (P%Ic) t
. 1 ·
(pv)1+ (pu)1 + (ptJv) + (pv2 + p) + (pvw) =
z z
( 6.x 2 ) j ( Lly 3 ) j
= 2U (pu v}z :r: + 2U (pv + 3pv)j y +
( Z 2 ) j [ () ] j
+ 2U (pw v}z z + 8Xk (PyA:) ,
. 1 .
(pw)J+ tJ. (pw)J + (puw)! + (pvw) + (pw2 + p)! =
% Y z
( X 2 ) J ( fly 2 ) j
= 2U (pu w)x :r: + w(pt) W)g 11 +
( z 3 ) j [ a ] j
+ 2U (pw + 3pw}z z + aXle (Pzk) t
(1.2)
(1.3)
(1.4)
91. KHHerHqecKJI COrJl8COBtlHHlde CXeMbi C KOppeKlJ}IeH
173
Ej + 1 Ej . . .
+ (u (E+p») + (V (E+P») + (w (E+p») =
- [ (u 2 (E + p)) ZJ: + [ (v 2 (E + p)) y]: +
. .
+ [ (W2(E+ P))J: + [ ('y!l)P P%]: + (1.5)
[ y p ] i [ Z P ] i
+ W {')'-I)pPy 11 + 2U (-y_I)pPi % +
+ (xT%) + (xTy) + (XTi) + [ a:k {UiPik>r.
3,l\eCb x - K03<1><I>H HeHT eCTeCTseHHoA TenJlOnpOBO HOCTH,
[{)Pil:/8xk] - pa3I-IOCTHaJI annpOKCHM H1I eCTecTBeHHLlX BSl3KHX Ha-
npfl>KeHHH, [8/8xk(UiPik)] - pa3HOCTH annpoKCHMa HSI pa60TLJ
BSl3KHX CHJ1. CKOpOCTb U onpe eJlgeTCH Bhlp8.>KeHHeM
u = v u 2 + c2,
(1.6)
T.e. OHa HBJlSeTCSI xapaKTepHoA CKOpOCThlO, C KOTOpoA: l{acTH ra3a
nepeceKalOT pacqeTHYJO gqeaKY.
3 a Melt a H He 1.1. 06psTHM BHHM&HHe Ha OTCyTcTBHe 'fneHOB co cwe-
lllattHbfMK npoH3BO,ll.HbIMH, nporroplUfOHarIbHhlNH p8.3Mepy "qelICH, KOTO-
pble OTHOCHTCR K p33pH):\)' OnWlHHTeJlbHbDC no OTHomeHH1O K CCHn&-
H YPaBHeHHA Haobe-CTOKca. Kax 6Y eT nOK83&HO B CJle,lU'lOlQeM nYHKTe
naparpa4>a, onOnHHTeJIbHble ,D;HCCHnaTHBHhIe 'IneHbI BrpaJOT pOJlb pery-
J1gpH33TOpOB, cra6HJ1H3HPYJOIII-HX QHCJIeHHOe pemeHHe. )J..nH 3THX ue.neA
anOJIHe ,l(OCTaTO'tJHO OrpaHM'IHTbCH Ha60poM AHCCHOaTH8HbiX "fneHOS, HMe-
fQlI(HXCH B BLlpaJKeHHjlX (1.1)-(1.5).
3 a M e q a H He 1.2. B OTnH'IHe OT opHrHHMbHoA KBa3Hra30JU{H8MH'Ie-
CKoA CHCTeMbI (rn. III), B BLlpa.>KeHH1IX (1.1)-(1.5) H3MeBeHHe DO BpeMeHH
BeKTOpa ra30,J].HHaMHQecKHX nepeMeHHbIx Q = (p, pu, E) OnHC&1BaeTC
'1neHOM
Q,+1 _ Qi
t
BMecTO
Qi+l _ Qf
T
rAe t - war no BpeMeHH) HCIIOJIb3yeMblR B p8C'IeTe. fiQAo6HaJI 3&MeH&
BhI3BaH8 TeM, '1TO B 60nbmHHCTBe p&C'IeTHhiX 3a,a.a.Q xapaKTepHOe spews
H3MeHeHHSI rc130)J.HHaMHtteCKHX napaMeTpOB H& WHOro nOpJl KOB npeOblwa-
eT BpeMB Me>KAY CTOJIKHOBeHHBMH MOJleKYJI T. ,[{aHHM onep Hj{ ,ltOnYCTH"
M8, Tal( KaK B paMKaX pacCM&TpBBaeMoro anrOpHTM& perYJUlpH3HPYIOIDJ'1O
pOJlb HrpalOT qneHbI) nponoplUIOHMbHLJe t, &HaJIOrHtfHO 'IJIeHaM. npo-
nOplJ)iOHaJILlibIM h.
CJleAYeT nO,L\ttepKHYTb) qTO KCPC C KoppeKQHeA 6b1nH nep-
BhlMH H3 KJIacca KHlieTHQeCKHX HJlH 6()J)b MaHOBCKHX CXeM, Heno-
cpe CTBeHHO HCnOJlb3yeMbIx, HaqHHaJI C 1986 r. [88. 89], jJ pacqe-
Ta 3aAa1.l AHHaMHKH BS3Koro ra3a , B TOM 'IRene HeYCTaHOBHBWHXCSI
Te"leHHR. B 3HaQUTeJILHOM 06'beMe 3TH pacqeTbI 6YPJ'T OlIHC8JlbI B
rJIaoe V. 113 pa60T pyrHx aBTopos, nocB eHHhlX HCnOJIb30BaHHIO
KHHeTH'lJeCKHX cxeM NISI MOAeJIHpoBaHHH Te'leHHft BSl3Koro r83&, OT-
MeTHM CTaTblO [90].
2. KCPC C KoppeKlQlei1 H ypSBHeHHH HaBbe-CTOKca
npOBe eM o eHKY nopH,LU<a QJIeHOB CHCTeMbJ pa3HOCTIIb1X ypaa-
HennA (1.1)-(1.5), 6b1 BLIHBHTb ee CBSl3b c pemeHHeM ypasHeHHA
HaHbe-CTOKca [91). llpH OM B 3Ha'lHTeJlbHoft Mepe 6y eM HCnOJIb-
30BaTb T€XHHKY AOKa3aTeJlbCTBa H3 rn. III, 2, n.3.
Ha'iHeM c npasoA qaCTH ypaBHeHIDI (1.2). )I.HaroHaJJbHaSi KOM-
rIOHeHTa pa3HOCTHOro aHaJIora nonHoro TeH30pa HanpS)I(eHHA npH-
HHMaeT BH
n%S = pu2 + P - W (pu3 + 3pu).i .
( 1.7)
nOpSl OK "IJleHOB (1.7), HMelO1IJ)IX CeTO'1Hoe npOHCXO:>KJJ-eHHe, CJleAY-
1OIIJ)f A :
[ (pu3);:] < [pu2] , (1.8)
[w (3PU)x] < [P] · (1.9)
3 ecb H AaJIee Lz, Ly - xapaxTepHbJe pa3MepbI pacqeTHOA 06nacTH
B HanpaBJ1eHHH oceD OX H OY.
Y'IHTbIBaSl, TO JlI06aB ecTeCTBeHH8.H pa3HOCTHaH annpOKCHMa-
HSI npe,lJ)1OJ1araeT COOTHOWeHHJI
x / Lz « 1 ,
Day / Ly « 1 ,
( 1.10)
gl. KHIleTfllfecKH COrJl&COBaHHble CXeMhl C KOppeK HeJJ
175
H3 ( 1. 7) - ( 1. 9) BbJTeKaeT, qro CeTO'tfHble ,l(06aBKH B n zx MaJlbI no
cpaBHeflHJO c COOTBeTCTBYIO MH KOHBeKTHBHbJMH T.fJIeHaUH.
AnSI BHe HaroHanbHOa KOMnOHeHThi nOJIHoro TeH30pa HanpSl>Ke-
HH:f.t, HanpHMep,
II%y = fYUV - (puv 2 )y - IIUj ,
(1.11)
nOJlyqaeM CJ1e,ll.YIOIDyJO O el)KY 'iJIeHa CeTOQHOro npoHcxo eHHH:
[ (puV2)y] < [puv] .
(1.12)
OCTaHOBHMCJI Ha nOCJIep)leM qneHe (1.11), OTBeTCTBeHHON 3a ecre-
CToeHHYIO BH3KOCTb. IIpH nepexo,ne OT KB83Hra3 H"&MH'Iecxoa CHCTeMbl
K KCPC C KoppeKIUleA 'Jnet) (r(pu)j)1I TPaHC<l>oPMHPOSaJICft B ( Uj)".
npH :iTON cy ecTBeHHO HCIT01Ib30BaJIOI TOT <l>axT t 'ITO nocne nO'lJleHHoro
:x.H<I> epel)l.\HpoBa.HH H
(T(PU)-) = ( JJ pu- ) + ( IJ u p- )
I'll P 51 P II
U 11
BTOpoA qJleH 8 npaBoA 'f8CTH cYUJ.OCTBeHHo MeHbme nepBoro, DO KpaAHeii
Mepe, B TOM CJIyqae J Kor a Ba3KHe 1fJleHbI DO nop KY BeJlH'IHHLI conoCT8-
BHMbl C KOHBeKTHBHblNH. B HblX YCJIOBH,.X OTHomeHHe BTOporo cnara-
eMoro npaBoA qacTH (1.13) K KOHBeKTHBHbIM '1J1eHaW 6Y eT ODpe,l\eJlJlTbCB
(a.nbIM napaueTpoM TIt, t r)J.e t 9 - xapaKTepHoe ra30,lJ.HHaa.tH'iecKoe Bpe-
Mg.
TaKHM 06pa30U, B ypaoHeHHH )J.BH>KeHHB (1.2) qneHbI, HMelOII.\He
CeTO'IHoe npOHCXO)K eHHe, MaJlbJ no cpaBHeHHIO C KOHBeKTHBHblMH
t.fJIeHaMH. EcnH HCTHHHble BB3KHe qJJeHbI TaK>Ke MaJIbI OTHOCHTe1IbHO
KOHBeKTHBHbJX, TO ,lJ.OnOJIHHTeJJbHble qneHbI ,ll.HCKpeTHOro npoHCXO-
)I(,geHHH HI'palOT pOJIh HCKYCCTBeHHoD BH3KOCTH, Hcnonb3yeMo:A npn
pemeHHH ypaaHeHHI 9AJIepa AJlH H,[I.eaJlbHOro r83a. EcnH >Ke HCTHH-
HbJe BSI3KHe qneHld o,D)loro nop a C KOHSeKTHBHhlMH, TO qneHhI
CeTO"tJHOrQ ilpOHCXO>K)J;eHBSI MaJIbI no cpaBHeHHIO C TeMH H APyrHMH,
T.e. npH P83YMHOA: HCKpeTH3a H (1.10) He HCKa>KaIOT Ka})THHY Te-
qeHH.H .
B 9TOM KpoeTCSI CY ecTBeHHoe OTJIH'tfHe anropHTMa KCPC c
KoppeK Heft OT MeTO OB pemeHHSI ypasHeHHA HaBbe-OroKca, B KO-
TOpbIX KOHBeKTHBHble qneHhI annpOKCHMHpYIOTCB HanpaBJIeHHblMH
(1.13)
P83HOCTHMH. B nOCJre eM CJIy"tJae B03HHKaIO 8$I CXeMHRH B 3KOCTb
MO:>KeT 6brrb COnOCTaBHMa C HCTHHHOR HnH p-aJKe 60JILwe ee TaM, rAe
ecTeCTBeHHble HCCHnaTHBHble qneHbl Onpe.neJIHIOT xapaKTep Teqe-
HHBt HanpHMep, B nOrpaHHqHOM cnoe.
PaccMOTpHM renepb nOpg OK BeJIHtfHHhI CCHnaTHBHbIX qJIe-
HOB B YpaBHCHHH Hepa3phlBHOCTH (1.1). Arts 9TOro l(pe CTaBHM HX
B B e
( (pu2 +Ph)x = ( (pu2)i)% + (*px)%.
KaK cneA)'eT H3 ypaBHeHHH (1.1), nop OK Pi 11860 paBeH nOpSI)J.KY
KOHBeKTHBHOrO "IJIeH& (pU) 0 , nH60, eCJIH KOHBeK jI HMeeT OAHHaKo-
%
BbIA nOpH OK C ecTeCTBeHHOH CCHl1aqHea, paoeH nOpSl,l\KY BjJ3KHX
qnefJOB. To ecTb B JI!060M cnyqae HNeeT MecTO o eHKa
(1.14)
%] < [ 7x 2 ] ·
(1.15 )
B CBOIO Oqepe)J.b J tVJH pu2J Lx CnpaBe,lVIHBa C1teAYIO ag O eHKa:
[( (pu2)x) z] %; [E] ·
( 1.16)
113 ( 1.15) H (1.16) BbiTeKaeT, \fTO qJJeHhI, HMelOlIJ)fe CeTOqHOe
npoHcxo eHHe, MaJIhJ no cpasHeHHIO C 'IJIeHaMH, CTOSl HMH B Jle-
BOa qacTH ypaBHeHHS1 (1.1) H COOTseTCTByrow.HMH KnacCH'IecKOMY
ypasHeHHIO HepaapbJBHoCTH.
3 a M e q aH He 1.3. TIOJIY'IeHHWe OneHKH He KaC81OTCS1 Te'leHHA CTOK-
ca) B KOTOpblX BJl3KHe 'IJIeHhI no IJOp KY BeJJH'iHHhI 3H&'lHTeJlbHO npeBbI-
WaJOT KOHBeKTHBHbIe.
npHcTynHM K ou:eHKe t.fJIeHOB yp&BHelIHH (1.5), onHCLIBalO ero
H3MeHeJIHe fiOJlHOA 9HeprHH. HaqHeM C o eHKH qneHOB BH a
[( u2( E + p)%) z] %; [i: (E+p)] ·
( 1.1 7)
TaKHM 06pa30M, YQHTbIB8J1 MaJIOCTb rrapaMeTpa x/ L:£, MO>KHO
YTBep)l( aTb) 'iTO COOTBeTCTBYJO e eHbI ceTO'fHOrO npOHcxo e-
HRs( MaJlbl OTHOCHTeJlbHO KOHBeKTBBHblX 'IneHOB.
..
91. l\llHeTJfT:JecKH COrJIaCOB&HHble cxeMbl C KoppeKIVIen
177
nepeA,n.eM K O eHKe ,rq>yroro 'IJleHa (1.5), He CBSI3aJ1HOrO c ecTe-
CTBeHHblMH BH3KOCThlO H TennonpOBO,lJ)lOCTbIO:
[( ; h l)p PX) J < = [CPz] ·
YttHTblBaH pauee IIpOBe eHHYIO o eHKY (1.15) )JftH rpAAHeHTa ,a;as-
jleHHjI, nepenHIlIeM (1.18) B BH,Ae
[( 6X p ) ] l1x [ pu2 ] 6.x [ pu2 ]
2U (-y-l)p Pi :r: < Lx C Lx Lx U Lx ·
( 1.18)
(1.19)
:>TO 03HaQaeT, QTO aHHbIH: CeTO rnhlH qneH npH He CJ1HmKOM MaJlbIX
MaKpOCKOllHQeCKHX CKOpOCTBX no nOpH,l\KY BeJIH'fHHbJ OKa.3bIBaeT-
SI MeHbllle COOTBeTCTBYIO ero KOHB€KTHBHOro t.tJleHa B ypaBHeHHH
'-)l-leprHH.
3aMeqaHHe 1.4. H3 o eHKH (1.19), Tal< >Ke KaK H3 3a.MeQaHHH 1..3,
fJ1e;IyeT, QTO CXeM& (1.1)-(1.5) npHMeHHM8,[(J1S1 OI1HCa.HHjI BB3KHX Teqetudl:
C He CJIHmKOM HH3KHMH MaKpOCKODHtfeCKHMH CKOpocTfiMH.
IIo,qBO B HTOI'M npo e.naHHblM O eHKaM) MO)l(HO I'OBopHTb 0 TOM,
u
YTO npH MO eJ1HpOBaHHH cmOJl038YK06WX U C8epX36Y1C08b£X me eHUU
(a 1'aK}I(e ,lI.030YKOBbIX co CKOpOCTSlMH nOTOKa, conOCTaBHMblMH co
CKOpOCTblO 3BYKa) KCPC C KoppeKqHeA aJOT pe3ynbTaTbI, 6nH3-
KHe K peIlleHHSlM YPaBHeHHA 9Anepa HJlH HaBbe-CTOKca. CKa3aH-
Hoe cnpaBe,n:nHBO H AJIjI HeBg3KHX, H JlSI BH3KHX reqeHHii. Ew:e p8.3
OTMeTHM, QTO IIOSlBJIeHHe QJIeHOB HCKYCCTBeHHoft HCCHIJa H nep-
Bora nopSl,n:Ka MaJlOCTH, 06blQHO HeraTHBHO BJlHHIO ee Ha peweHHe
ypaBHeHHA HaBbe-CTOKca, 8 ,It.aHHOM CJly ae He HCKaJKaeT <PH3HQe-
CKYIO KapTHHY TeqeHH . Kax 6blno nOKa3aHo) 3TH qneHbI MaJlbl no
cpaBHeHHIO c ecTecTBeHHblMH CCHnaTHBHblMH qJ]eHaMH.
B nO TBep>K,lJ.eHHe C)J.eJlaHHblX Ol.\eHOK npHBe eM ,lI.Ba npHMepa
pacqerOB TpaHC3BYKOBOro H csepX3BYKOBoro TeqeliHA BH3Koro C>KH-
1aeMoro ra33. 9TH pe3YJIbTaTbI, nOJIyqeHHble c nOMO blO KHHeTH e-
CKH COrJlacoBaHHoit cxeMbl C KOppeKn:ueH) CpaBJmBalOTCH C AaHHbJMH
pyrHx aBTopOB) nOJ1YQeHHblMH Ha OCHOBe HCnOJIb30BaHHSI YP8.BHe-
HHA HaBbe-CToKca.
Ha pHC. 33-34 H306p8.)KeHbI pe3ynbTaTbl pacqerra aBJIeHHS 3a
o6paTHblM YCTynOM. 3 a'tJa paccMaT}>HOanaCb B DpocTpaJICTBeHHo-
tJ:BYMepHoA nOCTaHOBKe. IIapaMeTpbJ Ha6eraJo ero nOTOKa B03 xa
,y
....-..
x
PHC. 33. reOMeTpH'IecKHe napaMeTphi
o6paTHoro ycryna
D.g
0.0
0.7
G.'
IIC
-1.0 1.1 2.0 1.1 5.' 1.0 10.0
Pac. 34. Be3pa.3Mepfioe aaneHHe
B OJJb 0CJi :t: AaHHble, DWlY"IeH-
Bble H3 OCHOBe KCPC C KoppeKlUleA,
- - - Hhle pa60TLI(92)
CJleAYlOlI.{He: Moo = 2.9; Re oo = 3.4.10 3 ; Pr = 0.72; Too = 300 K.
Ha neBo:R rpaHHIJ.e 3a aB8JlCSl npo<l>HJlh CKOpOCTH t MeHHIOIlJ)IAC1I OT 0
/1,0 Moo no KJTacCH'iecKOMY 3aKoHY ,LVISI naMHHapHoro OrpaHHqHOrO
CJlO5I TOJIlU.HHbI 0 (pHC. 33). BbI'IHCJleHHble pe3ynbTaTbi cpaOHHBa-
IOTCS C )J.aHHbIMH pacqeTa, ToR >Ke 3 a'l{H Ha OCHOBe HcnOnb30BaHHH
cxeMhI TpeTberO nop Ka TOQaOCTH JUI YPaBHeHH:A Hasbe-CToKca
(92) (pHC. 34).
B KaqeCTBe BToporo npHMe-
pa paccMoTpHM ,n:&HHble pacqe-
1"8, TpaaC3BYKOBoro Te eHHH 803-
.L\Yxa 86JIH3H npOCTpaHCTSeHHO-
ABYMepHoR KaBepHbI (pHC. 35).
TIa.paMeTpbI Ha6eraro ef'o noTO-
PRC. 35. reoNeTpH'IecKHe napaMeTpbl Ka CJle,lJJ'IOIlI)ie: Moo = 1.05 ;
K&BepHbI Re oo = 103; Pr = 0.72. Ha JIeBOA
rpaHHQe 'r8.K )l{e, KaK H B npe)J.bI)Q' eM npHMepe, 3 aH npocpHJlb
CKOpOCTH t COOTBeTCTBYIO Ha JlaMHHapHOMY nOrpaHH'IHOMY CJIOIO.
Ha pRC. 36 npHBe eHbl )J.aHHble pac eTa SepTHK8JJbHOA COCTaBJIB-
lO eit 6 pa3MepHOro TenJIOBoro DOTOKa, HOpMHpOBaHHOrO Ha ero
3HaqeHHe Ha neooA rpaHH e. ConocTaBnSlIOTCSI pe3ynLTaThl, UOJIY-
qeHHble C Hcn0J1b30BaHHeM KCPC c Koppe eR. H 8J{Hble pa60Tbl
[93), pacct.JHTaHHhle no ypaBHeHWIM HaBbe,CToKca. Kax H B nepBOM
npHMepe, BH HO xopowee COBn eHHe ,L\8.HHblX ABYX pa.cqeTOB. OT-
)'L
x
q_" aT
an
00 0000
1.
0.5
PHC. 36. TenJloBoA nOTOK B Hanpasnemm OCH OY Ha nOBepxBOCTH KaBepllY:
- KCPC c KoppeK eft, 0 0 0 - ,QaHHllle pac'lera (93)
KnoHetlHe B pe3YJ1bTaTaX 06'bjJCHSIeTC.R B nepBYIO o'iepeAb TOQHOCThJO
IiCnOJIb3yeMoR: pa3HOCTHoA anrIpOKCHMa HH.
llpO,ll-eMOHCTpHpOBaHHble pmynbTaTbl nO TBep>K &IOT paHee c;a.e-
.naHHblR BbIBO 0 npHMeHHMOCTH KCPC c KoppeK HeA ,lVIH MO e-
JllipoBallHSI BSl3KHX TeqeHHR, OfiHCbIB&eMbiX ypaBHeHHSlMH RaBhe-
CTOKca. BMeCTe C TeM KCPC c KoppeK eA Ka.K 1.JUC1IeHHhIA MeTO):t
o6J1 aeT pSl.LtOM 3aMeTHblX npeHMy eCTB.
1) 9TO oc1Hopolhi'Wu aJl20PUmM, II03BOJIjIIOW.d e HHblM 06pa30M
BeCTH pacqeT KaK BSl3KOA, T8.K H HeBSI3KoA: qacTe:A nOTOKa.
2) 9TO Jl8HaJI cxeMa, He 'rpe6YIOID:8JI npHMeHeHHSI cn RanhHblX
HTePaQHOHHbIX MeTO,QOB )\nSl peWeHHB CHCTeM CeTOqahIX YP&B-
HeHHft.
KaK 6Y,lJ;eT BH O H3 anbHeRmero M&TepHaJIa rJI&BbI, CBolCTBO O,lJ;-
HOpO)J.HOCTH H SlBHOCTH B8.)KHO )VIS! ycnewHoA a,a.aII'I'aQHH anropHT-
Ma K apXHTeKType MHoronpo eccopHb1X BblqHCJIHTeJIbHhJX CHCTeM C
pacnpe eneHHoR naMSITblO.
E e OAHOA CYIl\ecToeHHOH IIOnO)f(HTeJIbHOA qepTOft RBHOA cxe-
'1bI (1.1)-(1.5) SlBnHeTOI OCmTO'lHO MgrKOe Orp8.HH eHHe Ha mar
no BpeMeHH - THna YCJlOBHH Kypa.H'm:
Dat < oh, a = 0(1).
( 1.20 )
YCTOJ:tqHBOCTb QHCJ1eHHOrO pemeHHSI npH ycnOBHH (1.20) eCTb 3Kcrre-
pHA.leHTaJ1hHbIA <PaKT) notl.TBep)K,AeHHblR npaKTHKoA pac eTOB OKOJIO-
H cBepx3BYKOBblX T€qeHHH BSl3Koro ra3a. TeOpeTH eCKOro HCCJIe,l\o-
BaHHSI YCToA1.fHBOCTH H HeHHo:A pa3HOCTHOA CHCTeMbJ KCPC c
KoppeK HeH He npOBOAHJIOCb. O)\HaKO OTMeTHM p CO06pa.>KeI-IHft
8 nOJIb3Y ,AOCTaTOqHO MjlrKoro YCJIOBH YCTOAq}fBOCTH (1.20).
Bo-nepBblx, B npaBblX qacTSIx CHCTeMbl (1.1)-(1.5) npHCYTCTBY-
lOT )J.HCCHnaTHBHble {JIeHbI, nponOPQHOHaJlbHLIe wary CeTKH h. 9'fH
qJIellbl MaJJbl no CpaBHeHHIO C IJIeHaMH ecTeCTBeHHoA: BSl3KOCTH H
TeWlonpoBOAHOCTH) QAKaKO OHH c.nY.>KaT perynHpH33TopaMH QHCJ1eH-
Horo peweHHSI KBa3Hra30AHHaMHqecKOR CHCTeMLI [94J.*
BO-BTOPhIX, npH rHnOTeTHqeCKOM YMeHbWeHHI{ mara. ceTKH o
pa3MepOB nOp AKa )VIHHbI CBo6o HOro npo6era I AHCCHnaTHBHbIe
qJJeHbl CTaHOBSI rcH nponOPQHOIIa.J1bHblMH 1 H OTnHIIHe KCPC C KOp-
peKu.HeR OT KrY-CHcTeMhI npHo6peTaeT nOpH,IJ;OK 1 2 , BcnOMHHM, lITO
KrY-CHcTeMa BblBO HTCSl H3 MCXO)J)-lO KCPC (rJl. II) 5) KaK P83-
IIOCTHafl no BpeMeHH SlBHasI cxeMa, o6J] alO aH orpaHH eHHeM Ha
BpeMeHHoA war..
t 'f.
(1.21 )
3TOT cPaKT CJIe)JJ'eT Y)f(e H3 cnoco6a nOCTpoeHHSI 6anaHCHoro
ypa.aHeH )VIS1 o,l{l)OQaCTHQHOA <PYHK H pacnpe eJleHHH, Ha KOTO-
poe onHpae'l'CH KrY-cHCTeMa. B CaMOM eneJ H3 pHC. 18 (rJl. n), Hn-
nlOCTpHpYJO ero H3MeHeHHe <PYHKl\HH pacnpe eJIeHHH, B HOt QTO
3a BpeMjI 6eCCTOJIKHOBHTeJlbHOI'O pa3neTa npH xap8J<TepHblX CKOpO-
CTSIX MOJIeKynspHoro AB )KeHHS qacTH a He OJI}l{Ha nponCTaTh pac-
CTOHHHe, 60JIbmee pa.3Mepa SI'IeAKH h, Ha KOTOpOM MaKCBennOBCKoe
pacnpe)J;e.neHHe C1.fHTaeTC5I cna60 H3MeHjllO MCj!. Yq, HTblB3.SI '1TO
· Boo6 e, KCPC c KoppeK eA YO)ICHO TpaKTOBaTb H Kax BblqHCJIHTeJlhHhIA
aJlr0PU'fM )tJIJI peweHHH ypaBHeHHA HULe-CTOKC&.. npH TaKOM B3rJ1Jl)te AHCCR-
naTHBHble "IneHhI, npOnoplI,HOHaJlbHbIe h, 6Y.lU'T HHTCpOpeTHpOBaKbl KaK onon-
HHTeJlbHble perYJlHpH3aropbJ HMeHHO 3THX ypaBHeHHA: (94). IIpaB,D.3, B paMKaX
CHCTeMbJ HaDbe-CToKca nOJlyqHTb nOAo6Hble 068.BoqHhle qneHbI MO)f(HO 6hIJ1o
6h1 TOJIhKO 6JIacoAapB CJIy'laHHoA 3MDHpH'IeCKOA HaxoAKe.
..YcnoBHe (1.21) OTnH'laeTaI OT TP oHHoro ycnOBHJI YC':'OA'IHBOCTH DHAa
t :s T'. 3TO OTJlHQRe CBB3aHO C Te!d, 'lTO 8 Kry mar no BpeweHH He uox<eT
6h1Tb cyw.ecTBeHHO MeHbwe BpeMeHH Me>IC,Ay CTOJIKHOBelDUlMH MOJIeKYJI T t 'IMhI
rapaHTHpoBaTb HaJlH'lHe p8.8HOBOCHJI B BileAKe.
l1.JI.R KfY 3TO paCCTOSlHHe paBHO 0(1), H l1pHHHMaH 80 BHHMaHHe
CBH3b Me)l{ I If 7", np eM K ycnOBHIO (1.21).
OrMeTHM, 1JTO YClIOBHe YCToRqHBOCTH (1.20) 6blJIO T8.K)f(e nOJlY-
'leHO TeOpeTH"IeCKH (95] npH aaaJIH3e nHHeapH30BalIHblX KHHeTH'-le-
CKH cor JlacosaHHbIX cxeM.
3. KCPC c KoppeK HeR Ha HecTPYKTypHpoB8HHblX CeTKBX
,lLrJs nOCTpoeHHH KHHeTHtJeCKH corJlacoBaHHoR cxeMbl C KOppeK-
l\HeA Ha HeCTpYKTypHpOBaHHblX CeTKax o&he HHM reXHHKY KOH-
CTpYHpOBaHHH KCPC H& TaKHX CeTKax (rJI. II, 2, n.3) C H,D:eeR HC-
nonb30BaHH KHHeTHqecKHX cxeM )I.JlH MO eJ1HpOBaHHSI BR3KHX Te-
'reHHA, paccMoTpeHHoH B fi.l aHHorO naparpaQ>a. npH 3TOM, TaK
JKe KaK }if nH npRMoyrOJIbHblX CeTOK, BBIIO Bbl eJlHM BSl3KHe H Ten-
JI0npOBO)J.HbIe q.neHbI.
B KaqeCTBe npHMepa npHBe,L\eM p83HOCTHbJe ypaBHeHWI )).JIg ra.-
::lO HHaMHqeCKHX napaMeTpoB Ha TpeyroJlLHblX CeTK8JC. aHaJIOraq-
HbJe Bblpa)f{eHH M (2.9)-(2.20) rJl. II. IIpH 3TOM H3MeHeHHe ra30AH-
HaMH eCKHX napaMeTpOB PI, (pu)l, (pv)r H EI 6Y,lJ.eT nO-npe)KHeMY
OIIHCblBaTbCH ypaBHeHHSlMH (2.9)-(2.12) (rJI. II), HO C H3MeHeHHblMH
no cpaBHeHUIO C (2.17)-(2.20) (rJI. II) 'IJIeH&MH, OTHOCSl MHCSl K
HccHnaTHBHOa qaCTH nOTOKOB. BblnmneM 3TH 'IJ1eHbJ:
p 1 [( ( ) exp( -8;) )
Dli = 2' fYUn erf 8n + P fJ-li i
( exp( -S » ) ]
- PUn erf(sn) + P fJVi I I
( 1. 22)
Ii 1 [( 2 eXP(-S ) )
D/i = 2 (PUn + p) erf(sn) + PUn 13.../i & -
( 2 exp( -s ) ) ]
- (PUn + p) erf(sn) + flUn 13.Ji I COSO/i +
+ [(/1UnU r erf(Sn)}i - (JlUnU T erf(Sn» I] sin 01& +
p,{3"," [ ( exp( -8;) ) ( exp( -S ) ) ] .
+ h . PUr R_r::. - PUT R sInet/i,
p II fJV7r i Py1r 1
v 1 [ ( 2 ) ( ) exp( -S ) )
D Ii = 2 ( + P erf 8n + PUn {3';:; i -
(( 2 ) ( ) exp( -8 ) ) ] .
- PUn + P erf Sn + PUn 13..fii I SIn Qli -
- [(pUnUrerf(Sn»i - (pun err(sn»l] COSOIi-
f3Vi [( exP(-s ) ) ( exP(-s » ) ]
- h . fYUr II t= - pu.,. fJ r; cos ali ,
p 11 /Jy1f i v 1r 1
( 1.23)
( 1.24)
E 1 [( + u exp{ -S ) )
Du = 2" (E + p)u n erf(sn) + P 2 fJ../i & -
( u2 + u 2 exp ( 82 )) ]
- (E + p)Un erf(sn) + P n T - n +
2 I3Ji 1
+ x{3../ff [( pe eXP(-8 » ) _ ( pe exp(-8 » ) ] .
fJCp h Ii f3 .;:; i 13';:; 1
3 eCbJ Tal( >Ke KaK H 8 OPMYJIax (2.9)-(2.20) rJIaBbJ II, fJ =
= 1/ vl2 RT J Sn = PUn, hli - pacCTOSlHHe Me)l(,[Iy l.\eHTpaMH BHY-
TpeHHe.A HqeRKH I H oJnloA H3 OKPY>Ka.ro HX ee SqeeK C HOMepoM i.
BLlpaJKeHHe )l.nH CCHnaTHBHoro nOTOKa (1.22) 8 ypasHeHHH
Hepa3pblBHOCTH 8 TO HOCTH COBn aeT c nepSOaa'l8JIbHI:aJM BapHaH-
TOM (2.17) (rn. II). H3MeHeIWR B Bblpa>KeHHSIX AIlS! n01'OKOB D'fi'
( 1.25 )
1. KHHeTllqeCKH COrJI8COBaHHlJe cxeMbl C KoppeKl{HeR
183
D'li 1 Dfi CB aHbl C Bbl)].eJIeHHeM qJIeHOB BHroBOa BH3KOCTH H Terr-
JIOnpOBOAHOCTH. B OTJlHqj{e OT ypaBHeHHSI SHeprHH (1.5) B KCPC
c KoppeK HeH Ha npHMoyronhHbIX ceTKax, B (1.25) ,lVllI npocToTbI
3arlHCH ClleU)[aJlbHO He BbJ engIOTCSI qJIeHhI, CBH3aHHble c pa6oToA
B 3KHX CHJI. Hx BKJIa,L( B H3MeHeHHe nOJlHoA 3HeprHH nSl OKonO- H
CBepX3BYKOBblX TetteHHft HeCY ecTBeHeH.
B Kaqe('"TBe rrpHMepa npHBe-
,a.eM p€3YJIbTaTbl pacqeTa 3 aqH
o CBepX3BYKOBOM (Moo = 1.5)
8S13KOM (Re = 104) 06TeKaHHH
KpbIJIOBOrO npo<pHJISI NACA0012
nOA HyneBbl l yrnoM aTaKH
(puc. 37 Ii 38) . TpeyrOJlhHag
CeTKa (puc. 37) CO ep)KHT 5877
Y3JJOB, crYUl-alOIU-HXCjI K nOBepx-
4 PIIC. 37. q.parWeHT pac'feTHoA Tpe-
HOCTH TeJIa (h min = 8.53 · 10-). YI'OJu,HoA Ce-rKH (5877 }'3J10B)
Pac"leT npoBO)l.HnCH MeTO OM
YCTaHOaneHHSI. Pacnpe eneHHe Kos4><I>UD.HeHTa aBJIeHHSI (-G p )
BAOJIh OCM cHMMeTpHR H nOBepXHOCTH npo<pHillI nOK83aHO Ha pRC. 38.
Pe3YJ1bTaT HCn0JIb30BaHHSI KCPC c KoppeK eA Mano OTnH aeTCH
OT BbICOKOTOqHOrO pacqerra no ypasHeHHSlM HaBbe-CTOKca [96].
Ha pHC. 39-41 npeACTa.BneHbl J>e3YJIb'IaThi pacqeTOB 06TeK8.HHSI
KpyroBOro HnHHApa 03BYKOBbIM (M = 0.54) llOTOKOM Bjl3KOro
1.0 -c
p
0.5
0-0
'0.5
-1.0
.1.5
-2.0
0.0
0.2
0.4
, & . ". "&'&. &.,a'r..f.. ,....,..,, : ". ..( "4...1.9-
.... -:- ....'.....I... .... 'f T.'........'fa.,..,, ;:: t: ...f'.... A.4
........... . :1 .....,........ .",,"",,""'.' ,..,"',,'." !. " .. .....t. .... .':t. · 't
... . : '...,.....,.....:.........,...,r.i.I",'.t: .r ........ . . ...,..t.-t.t ;.., I..
r; .... ":, ., .,...'...........'....,....,. "r.'..,; I,' ...". .A.Jo''': .......Y'..'P
... 4t..." .. ..",... ..'.'....1.........,. ",' I.' '..' .....".. '..I1..".'.: 'iI." .. ...:.t :1... 4 '.....
@ ; r(:; \ F ;fi :;:/ : ,:,'} .': '. ::. ::\\::: ;;:::;":: ::::' .' :} :::::.? ;?m: ,:
j1! iik..:: \':i: :?:;;:(: : :::.:: ;, :,,' .: :. ::.:.. :'.X ;Ml s ij
!.'''' : ",.: ;.:.;.i.. ..:: :' . '.'; '.'.. .1...,; .:,&1I,...,:........1 . ' . ..........
"A 4" :..:&:.. ;. . !', ;:,/ : ';";';":.';':'; :..4 '& , 'f..,;1':1 ..( .... .....A.
." I': 'Ii:!...y..... Y. _ · ..'.." . · .. AI .. .".. .... ... ,-.... a.
" . . 4 '..r. ';"A.4";' ",..' .'''.'fA.... ,...."..
"A '4" J:. ... , , I>. to -
0.6
0..8
1.0
PHC. 38. Pac(Ipe eneHHe Ko3<l»«PHU;HeHTa AAWJeHHJI no OCH CHNMeTpHH Z H no
nosepXHOCTH npa<PHn1l: KCPC C KOppeXI{HeH) - - - aHJlhle pa60Thl (96)
184 rJl. IV. q'lClleHHWe anrOpllTMlJ H MHoronpOneccopHble CHCTeMlJ
-4.JJVU_ 7 v... rvvrvl L..", I\.
\. "Cr )t1\I'u ,f' V" Il
"\ "":I ....1t1'N '" r.. - U\,,'V\) .... r. ,"
.., "-.:v 7r..... ';f" ,, I-tt'" r"\ \J
I'J v r ,(\... -'\J"!' \ , 11"\
IlL __ 1\ "J J J "" \. \.
... ..."L . 1
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....-J r:::f'fi oJ L"
II. ['J,j .v ;'-1"0.1 N f\.l'
r
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......" VJ\I\ /\ 1'\ " .J'..... """'IC.._ V -I\.
n
, looP
t1
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> KD[(K
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PRC. 39. cl»parMeHT TpeYroJIbHOA CeTKH B6nH3H "KpyroBOro L\HnllH.lQ)a
PHC. 40. none OTUOCHTeJ1bHOA CKOpocTH ful/c Ha. OAKH H3 MOMeHTOB 8peMeJJe
I
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PHC. 41. Koscl>4>HQHeuT COnpOTHBJ1CIUUI
.
92. nonYlfeHBHaJI cxeM8
185
(Re = 500) r83a. PacqeTHaH CeTKa. (pRC. 39) COCTOHT H3 21517 Tpe-
YroJIbHHKOB. Xopowo BHAHa 06Pa30Baswasca BHXpeBaH AOpO>KKa
(pRC. 40). TeqeHHe HOCHT KB83HnepHO qecKHft xapaKTep (pHC. 41).
PaccMoTpeHHbIA B )J.aHHOM nyHKTe aJlropHTM KCPC C KOppeK-
1J.He'A: na TpeyrOJIbHblX CeTKax 6bIn paaee OfiHCaH B pa60Tax (40 52),
[97]--(100]. B HaCTOH ee BpeMSI 3TOT no,ltXO ecTeCTBeHHbJM 06pa30M
o606lU-eH Ha MO,lteJIKpoBaJlOe npoCTpaHCTBeHHo-TpexMepHblx B 3KHX
TeqeHHA Ha TeTp83,LqJ&1IbHbIX CeTKax.
Cne eT Ta!C)f(e CK83aTL. 'ITO HAes KCPC C KoppeKD;HeH MO)Ke'!'
COQeTaTbC51 C TeXHOJIOrHeA nOBhIweHHoro nOpH K& annpOKCHMaIJ)UI,
OnHCaHHO 8 3 rn. II. T&K, HanpHMep, Anjf BLlqJfClleHHjI ra3 H-
HaMH'IeCKHX napaMeTpOB Ha TpeYroJIbHblX CeTKaX MO)KeT 6b1Tb HC-
nonb30oaHO BbJpa>KeHHe (3.21) (rn. II) c HCCHnaTHBHblMH COCTaBJISI-
fOUJ.HMH nOTOKOB D Ii, onpe enSieMbIMH 110 cpopMYJlaM (1.22)-.( 1.25)
aHHoro naparpacl>a [44).
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AnropHTM KCPC C KoppeKIJ;HeA "POCT H y o6eHJ OAHaKO 06-
JIacTb ero npHMeHHMOCTH He6e3rpaHHQHa. Kax BH HO H3 npoBeAeH-
HLiX BbIwe o eHOK, a T8J()I(e H3 pactfeTHoA npaKTHKH, B CJIyqae, xo-
r a xapaKTepHaH MaKpoCKODH'tfecI<aSl CKOpOCTb cyw.eCTBeHHO MeHh-
we CKOpOCTH 3BYKa t 3TOT MeTOA He MO>KeT CJlY>KHTb H e.>KHblM HH-
CTJ>YMeHTOM BblQHCJ1eHHA. ,nng MO eJlHpOB&HHR YMepeHHO 03BYK
BbIX TeqeHHA 6h1J1 npe,lVlO>KeR [94, 101] eIQe O H aJlropHTM, Keno-
cpe,nCTBeHHO BbiTeKalOlI.\HA H3 yp8.BHeHHR (5.11) (rn. II), onHCblBa-
lOm.ero H3MeHeHHe cpYHKJ.J.IUI pacnpe)J.eJleHH.H OT BpeMeHH6ro CJIOg t j
K cnOIO t j + 1 . IIepenHweM 3TO ypaaHeHHe B B e
j+l _ j j T j
II - f r . M - TLf M + 'TL 2 LfM'
(2.1 )
rAe OIlepaTOp L onpeAeJISleTCSI BbIpa.>KeHHeM Lf = {i Of fOXi.
B ra3e HOpMaJlbHOA nJlOTHOCTH BpeMSI Me)KAj' CTOJIKHOBeHHBMH
MOJIeKYJI T 3HaQHTeJlbHO MeHblIle BpeMeHH D.t, B Te eHHe KOTOpOro
npOHCXO)J)JT CKMh-HH6YJI.h 3aMeTHble H3MeHeHHjI ra30 HHaMHqeCKHX
napaMeTpOB. BcneACTBHe 3TOro BenH'IRHY T HeYA06Ho HCnOJlb30BaTh
B KaqecTBe pac eTUoro mara. no BpeMeHH.
186 ril. IV. l[HCJJeHHwe aJlropHTMhl H MBorOnpOQOCCOpHble CHCTeJlM
PaCCMOTpHM OTpe30K BpeMeHH Ilt, KOTOphJH y,u;o6eH )VIB MO e-
nHpOBaflHSI H3MeHeHHH r830 HaMHqecKHX napaMeTpOB H YAoBne-
TBopSlerr COOTHomeHHIO
!:it = NT, N» 1.
(2.2)
Tor a H3 (2.1) CJle,lQ'eT. \lTO nepexoA OT speMeHH6ro CJIOst t j K CJ1010
t j + N MO}l(eT 6bITb OnHCaH C nOMOllJpJO BbIp3JKeHH
N-l [ ]
J1+N = fl,M - E". LfJ.tk - L ; Lft+ k ·
s=o
(2.3)
'¥
.
I
.
.
.
.
.
I
I
:
/1 I
t j + 1 t j + k {J+k+ 1 t j + N
PHC. 42. K nocTpOeBHIO nonYHeBBHoH cxeww
0603Ha'lHM p33HOCTb LfJ.:k - L LfJ.tk '1epe3 \J!j+ . CYMMa l
CTOBlI.{aH B npaaoA qacTH (2.3), npH6nH)f(eHHO npe CTaBHMa B B e
HHTerpana (pHC. 42):
ti+N N-l
I \II dt L rWj+A:.
t) k=O
(2.4)
Kax BH O H3 pac. 42. Ha KaJKAOM OTpe3Ke T = t j + 1 - t} BKJI B
npH6nHJKeHHOe 3HaqeHHe HHTerpana (2.4) onpe enBeTCB 3HaqeHHeM
<t>YHK HH \IIj+k Ha JleBOM Ko e 3TOro OTpe3Ka.
2. IIanYHelIBHa.H cxewa
187
H3 BblpaJKeHHA (2.3) H (2.4) BhITeKaeT COOTHoweHue, CBSJ3bJ-
BaJO ee 3HaqeHHH <l>YHKqHH pacnpe eJleHHSI Ha CJI0BX t = t i + N H
t = t J :
fl+ N - fi,M ( ! T ) L f ; ( _ T ) L f j+N =
t + 2 + 2 t M + 2 2 t M
= ( + 2 t ) L ; Lflr + ( - 2 t ) L ; LfJjN.
(2.5)
B BbJpa>KetlHH (2.5) YQacmy10T TOJIbKO CJIOH j H (j+N), B CBB3H C
qeM nepeo603Ha'tfHM jj+N KaK 1 , + 1 H nepeA,n.eM K pacqeTHOMY mary
no Bpe}"IeHH 6.t. ·
Bbl6epeM B neooft t.tacTH (2.5) tU1eHbl, CYWMY KOTOpblX npe CTa-
OHM B cne,AylOw;eM BH)J.e:
T j T j+l r a j ( A 2 )
2 t Lf M - 2 t Lf M = - 2 Ot Lf M + 0 ut ·
(2.6)
OrpaHHt.tHBaSlCb 3 aqaUH, B KOTOpbJX xapaKTepHoe BpeMB H3MeHe-
HHSf ra30AHHaMJAecKHX napaMeTpOB MHOro 60nhwe T t npeHe6pe)l(eM
t.tJleHaMH (2.6) H np eM K cxeMe C BeCOM 1/2, HJIH nO.llYHeJl8HOiJ, )VJst
ODHC3HH1I H3MeHeHHa CPYHKU)fH pacnp eJIeHHjJ (101):
j1+N Ii
I - I,M + ! ( Lfj - L T Lfi ) +
t 2 M 2 M
+ (Lilt 1 - L;Lf l) = o.
(2.7)
I1HTerpHpyg KHHeTH'IecKoe ypaBHeHHe (2.7) C cYMMaTOpHblMH
HHBapHa.HTa IH, npHxo HM K p83HOBHAHOCTH KBa3Hra30,1J)lHaMH\fe-
CKOH CHCTCMhl (5.15)-{5.19) (rn. II), 3anHCaHHOR B DOJIYHeSBHOM
BH.L\e [62. 101J. HanpHMep, ypasHeHHe AJUI nnOTHOCTH (5.15) (rn. II)
188 rJl. IV. t{HClIeHHbJe anropHTMEJ H MHoronpo eccopHble CHCTeMbl
TpaHc<l:>opMHpyeTCH B CJIeAYIO ee:
p1+1 - p1 1 [ {) a fJ ] j
t + 2 ox (ph) + Oy (pu) + oz (pw) +
1 [ 8 a a ] j+l
+ 2 iJx (pu) + ay(pv) + a;(pw) =
1 [ 8T8 2 ora {)T8
= 2 ax 2 ax (pu +p) + ax 2 8y (puv) + 8y 2 ax (puv) +
OT{J ()ro OTO 2
+ a;, 2 8z (puw) + & '2 a; (puw) + 8y 2ay(PV +p) +
(} r {} {} T a [} T 8 2 ] j
+ ---(pvw) + ---(pvUJ) + ---(pw +p) +
8y 2 8z 8z 2 8y oz 2 8z
1 [ 8"-8 2 878 8ro
+ 2 ax 2 iJx (pu +p) + ax 2lJy (puv) + ay'2 ax (puv) +
oTB ora 8r8 2
+ ax 2 az (puw) + ih 2 ax (puw) + iJy 2 8y (pu +p) +
a r a a r IJ {} ". () 2 ] 1+ 1
+ ay2&(PVW) + &2"ay(PVW) + 8z 2 OZ (pw +p) ·
nOJlYHeBBHYIO KrY-cHCTeMY B eJIOM MO>KHO np craBHTb B
onepaTopHOM BH e
(2.8)
'+1 .
Q1 Q1 + (Aq1 + AQi+l) = O. (2.9)
3 eCb Q = (p, pu, pv, pw, E) - oeKTOp r8.30 HHaMHqecI(HX nepe-
MeHHhIX. A - HeJlHHeHHblft onepaTOPJ BKJlIO lalO a B ce6g Bee npo-
CTpaHCTBeHHbJe npOH3BO HbIe KrY-cHCTeMbl - Kal< KOHBekTHBHble,
TaK H HCCHnaTHBHbIe.
IIpH t{HCJIeHHOM peweHHH CHCTeMbI {2. onepaTOp A 3a.MeH.SleT-
CD HeJIHHeAHhlM pa3IIOCTHhIM onepaTopOM A, B KOTOpOM KOHBeKTHB-
HhIe qneHbI annpoKCHMHPYJO'I'CSI eHTpanbHblMH paaHOCTBMH THna
(a)o J a HCCHnaTHBHbIe - BTOpb1MH pa3HOCTHhIMH npOH3BOAHblMH
x
92. UonyneHBHaR cxeMa
189
( (b)ii)x flH60 ( (b);)y . nOJlyqeHHaSl TaKHM 06pa30M CHCTeMa Pa3-
lIQC'fHblX ypaBHeHHA pewaeTOl C nOMOIJJ.L1O npocTOI'O MTep HOHHO-
10 npOn;ecca
'+1 (k+l) ·
(Q1 ) t - QJ + (Aq1 + A{q1+1){k») = 0,
o < k < Nk; (Qj+l)(O) = q1.
(2.10)
IIo cBoeR CrrpYKType anropUTM (2.10) aHaJlOrHt.feH HBHOA cxe-
Me. 3TO .u-eJlaeT ero npaKTH IecKYIO peanH38WfIO npocTOR, HecMOTpSl
Ha BHeWI-IIOJO rpoMo3 KOCTb <I>oPMYJI THna (2.8). KaK nOK83b1B&eT
onbiT pacqeros, HTep oHHbla npo ecc (2.10) CXO,AHTCH IIPH ware
D.t, y,n;oBneTBopjJlO eM yenoBHIO KYPaHTa WIH a»(e HeMHoro 60JIb-
me l. Ha npaKTHKe )VIR nOJIyqeHHH HCKOMoA TO'IHOCTH CXO HMOCTH
06b1QHO OKa3bIBaeTCSI ,AOCTaTOQHO BhlfiOJlHHTb 3-4 HTeP8.Q.HH npH He
CJJHlliKOM MaJIbJX qHCJJ&x Maxa (M 0.1). npH 60nee HH3KHX QHC-
JIax Maxa MeTO,n npocro:B HTepau.HH (2.10) CTaHOBHTCH Hes<I>cj>eK-
THBHbJM nH peweHIDI CHCTeMbJ (2.9)t TaJ( KaK OH CXO HTCH TOJIbKO
noCJI€ 6onbllioro 'IHCJJ& HTep8JJ.HA. B 3TOM r.nyqae ;:vIS Haxo)K.ZJ.eFiHsr
peweHHSI KfY -CHCTeMbI Ha HOBOM CJloe no BpeMeHH Heo6xo MO HC-
nonb30BaTb 60J1ee CJIO}KHbJe HTepa HOHHbIe MeTOAhI.
O Ha.KO B HeKOTopbIx cnyqaJIX anr9pHTM (2.10) MO>KHO npB-
MeHHTb H .AnSI MO eJIHpOBaHWi reqeHHA HeoKHMaeuoro ra3a. npH
3TOM HCfiOJlb3YeTCH npHeM nepeHopMHpoBKH YCJlOBHA 3 a'lH no qHC-
JlY PeRHOJ1b Ca, aHaJlOm'lHhiA npHMeHSIeMOMY B npaKTHKe Lattice
Boltzmann cxeM (rn. III) 6) .
B caMOM ene, "PH xapaKTepHblX t{HCJIax Maxa M < 0.2 o6b1Q-
HblH ra3 MO}f(HO CqHTaTb np3.KTHQeCKH HeC>KHMaeMbIM H )J.lliI Hero
e,l],HHCTB€HHbIM 6e3pa3MepHblM napaMeTpOM SlBnBeTCSI tfHCJIO PeD-
HOJ1L ca. TIYCTb, HanpHMep, Te'ieHHe o6J1 aeT peanhHbIM qHCJIOM
Maxa M = 0.01. Tor a MO>KHO YBeJlHqJITb BXO YIO CKOpOCTL B
10 pa3, C TeM qT06bI C cj>HKTHBHLJM 'IHCJlOM Maxa "J 0.1 HTepa H-
OHHblA npouecc (2.10) HMeJ1 y oBneTBOpHTeJlbHYIO CKOpOCTh CXO -
MOC"fH. B CBOIO Oqepe b, o6b1 QHCJ10 PeRHOJIb Ca
Re = McpL
Jj
190 r.n. IV. tlJICJleHHble anroplJTMbJ H MHoronpOl.(eccopH61e CHC1'eMbl
COOTBeTCTBOBaJIO cBoeMY HCTHHHOMY 3HatJeHHIO) MeHSIIOTCSI pyrHe
napaMeTpbl 3a,r(aQH, HanpHMep, JIHHeAHbJe pa3Mephl YMeHLmalOTCSI
B 10 paJ.
B Ka eCTBe npHMepa
HCllonL30oaHIDI nonYHeHBHoR
cxeMbl (2.9) npHBe,n:eM pe-
3YJlbTaTbJ pacqeTa TeqeHHB
HeC)I(HMaeMOA >KH KOCTH
BHYTpH npHMoyronbHoA
K3.BepHbI c ,lJ.BH>Ky eAcSi
KpbllUKOR (rn. III, 8) H
HX cpaSHeHHe c aHHbJMH
pa60TLI (85). Ha pHC. 43
nOKa3aHbl npo<pHnH ropH-
3OHTanbHOA H BepTHk&nbHOR
CKOpOCTH B o.eHTpanbHblx
CeqeHH X KaBepHbJ npH t{HC-
Jle Pe:AHOJIb Ca Re = 100.
,neMoHcTpepyIOTCSI .n-aHHble
pac leTOB Ha nocne.n;OBa-
TeJIbHOCTH cry alOII.J.HxCSl
CeTOK. B HO, q'fO npH ):U)o6neHHH mara ceTKH qHcneHHoe pemeHHe
CTpeMHTCSI K 3TaJ10HHOMY pemeHHIO H3 pa60TbJ (85).
0.00
1.0
0.25
x
0.&1
100
0.2
O.7!5
Q8
.(lot
01
0.0
0.6
-01
v
.0.2
,
04
-03
0.2
00
.(t.
.0.5
1.0
0.0
.
0.5
PUC. 43. npo4>HnH ropH30HTaJlbHOA
(Kp"naJl 1) H OCpTHKaJlhHOA (2) CKOpo-
cm B eHTpanbHhlX ceqeHHaX JC8BepHbI:
Re = 100
B 3a.KJIIOt.teHHe OTMeTHM, qTO paccMoTpeHHble B nepBbIX I1..BYX
naparpa<Pax rJIaBbl MeTO,L\bI - KCPC c KOppeK "eR H nOJlYHeaBH8.SI
cxeMa - .SISnHIOTCH npocrblMH no cBoeA J10THQeCKOA CTpYKType 8Jl-
ropHTMaMH. OIIH nerKO aJlTHpyIOTCH K apXHTeKTYpe MHoronpo-
eccopHbIX BblqHCJIHTeJIbHblX CHCTeM C pacnpetteneHHoA naMfiTblO
H aloT B03MO>KHOcrb MO)J.eJIHpOB&Tb Ha BbIcoKonpoH3Bo,lJ.HTeJIhHbJX
KOMDbIOTepax pa3J1Ht.JHbIe 3 aqH cOBpeMeHHOa rHAJ>O- H ra3oBoA
,nHHaMHKH. B cne lOtneM naparpaQ>e paCCMoTpHM 60nee CJ10>KHbJe
anropHl'Mbl pewelfHH KBa3Hra30 HHaMHqecKoH CHCTeMbl ypaBHeHHR.
3. HeJlBHble CXeMW J1,JJ1l peWeHHJl KB83Hra3OAJlH8NHlfecKoll CHCTeMbl 191
3. HeSlBHhle cxeMbI SI pemeBHB
KB8.3Hra30,ll.HHaMHQeCKOI CHCTeMId ypaBHeRBA
.l\.nropHTMb!J OnHCaHl-lb!e B npe)J:hI Yw.w< naparpa4>ax rnaBLJ, no
cBoeJ:i CTPYKTYpe aHaJ10rHQHbl SlBHblM CXeMaM H 683HPYJOTCB H8 OPH-
rHHaJlhHolt KrY-cHcTeMe (2.1)-(2.4) (rJI. III), p83HOCTHOA no ope-
MeHH, c npHMeHeHHeM AH<I>Q:>epeH1.\HaJlhHbIX onepaTopOB H8 HH>KHeM
cnoe. B npoTHBOnOJlO>KHOCTb 9TOMY - H3JJaraeMble B AaHHOM napa-
rpa4>e MeTO 1 HCITOJIb3YJOT B KaqecToe OCHOBLI QHCTO AHcI>cI>epeH B-
aJlbHbJA BapHatIT Kry, C DOCJIeAYIOrneA JIHHeapH3 eA no HblOTOHY
ypaBHeHHA, BXO-n lll.HX B 3TY CHCTeMY. IIpH paCCMOTpeHHH MaTepH-
ana aHHoro naparpaQ:>a 6y eM CJIe OBaTb pa.6oTe (102].
BbJnHWeM K<PcI>epell H8JIbHbJA BapHaHT KBa3Hra30,D-HHaMH'tfe-
CKoA CHCTe lhI Ha IlpHMepe npoCTpa.HCTBeHHo-ABYMepHoro CJJyqast:
BQ = _ BE _ of + T ( BV +p 8H + aG + au ) +
at ax Oy ax 2 8x ox p ax Oy
{J T ( OW au pOG au )
+ 8y 2 8y + P7iij + p {}y +"& ·
(3.1 )
3,4eCb HcnOJlb30BaHbi CJleAYlO e o603HaQeHHH:
p pu pv
Q= pu E= ptt2 + P F= p1J1J
pv , PU1J pv2 + P ,
E u (E + p) v{E+p)
pu2 + P pv2 + P
v= pu3 + 3pu w= pv 2 u + pu
pu 2 v + pv pv3 + 3pv
u 2 (E+2.5p) + pv 2 /2 v 2 (E+2.Sp) + pu2/2
0 0 puv
H= 1 0 G= 'Y 0 u= pu 2 v + pv
(1- 1 )Pr 0 , 1'-1 0 , pv 2 u + pu
pIp p uv(E+2p)
3 a)d e q &H He 3.1. CHcTewa (3.1) OTJllrlaeTOI OT OpHrHHaJlhHOA KBa-
3HI'aJoAHHaMHqeCKOA CHCTeMbI (2.1)-(2.4) (rn. III) 3aMeHoA pa3HOCTHOro
J J]I. 1 V . 1fCJleHHble anrOpllTMY H MHoronpo ecCOpHble CHCTeMlJ
aHaJlOra npOH3BO)!.HbIX no BpeMeHa H8 .lJ)I<t><PepeH a.nhHble Bblpa>KeRHfl
H OTKa30M 01' .SIBHOA <I>oPMbl 3&nHCH (no Hb(M c npeAbl,aym;ero CJlOB
no BpeMeHH) KOHBeKTHBHblX H ;J.HCCHnaTHBHbIX qJIeHOB. npoCToA &HaJJH3
nOKa3blBaeT t "Iro 3TH ;:tBe CHCTeMbI OTJJHqalOTCJI H8 BeJlH'IHHbI
T {)2Q
2 8t 2
H
() T 8
at 2 8 X i (AQ) I
(3.2)
r,l(e 8(AQ)j8xi - KOHBeKTHBHble qneBLI. 3aMeTHM, QTO 3THIdH >Ke'lneH3-
MH nepBoro nOPHAKR no T 6Y,lI.eT ODpe)\enSTbCB paanH'IHe Me)K)Ql CHCTeMOA
(3.1) H ypaBHeHHJlMH Haow-CTOKca. IIopB,l(oK BeJJH'fHHbI 3TBX 'IJIeHOB Ta-
KoA )ICe, KaK y eCTOCTBeUHblX HCCHnaTHBHblX tt11eHOB.
O)],HaKO eCJIH paccMaTpHsaeMbrA KJlacc Te'leHHH 6nH30K K CTau;lOHap-
HhiM HnH no KpaHHeH MCpe He npe)J.nonaraeT HHTeHcHBHoro H3MeHeHHSI
npocTpa.HCTBeHHo-BpeMeHHhlX xapaKTepHCTHK, TO npHMeneHHe CHCTeMN
(3.1) ,lVIH HX OnHcaHH5I HBJlSeTCB Onpan,lIPHHbIM. KaK nOK83blBaeT npax-
THKa paCqeTOB J ,a.aHHble. DOJIY'IeHHble Ha OCHOse HcnOnb30Ba1IH.ft CHCTeMbI
(3.1), xopomo COOTBeTCTOyroT pe3YJlbTa:raM, OTHOCHlIJ)WCSI K yp8.BHeHHBM
HaBbe-CTOKca. Kpoue TOro, <l><>PM& 3aI1HCH CHCTeMbI (3.1) aeT AononHH-
Te.1IbHble BOOMO)KHOCTH AJlB nOCTpoeHHH BLPDICJIHTenbllblX a.nropHTMOB.
,lLn annpoKcHM H npoH3BO HOa no opeMeHH BOcnonb3yeMCg
napaMe'rpH"ieCl<oA: <I>oPMynoA o6 ero BH a [103]:
Qj+l _ Qj _ 1 ( aQj+l _ a Qj )
t - 1 +8 2 81 at + ( 1 (h) 8t +
. . 1
82 Q1 - Q1- ( 1 J2 3 )
+ 1+(J2 t + 0 (8. - - (J2) r + t ·
(3.3)
aHHaH cl>opMYJla 3a. CqeT BbI60pa napaMeTpOB 6) H ()2 n03B01IHeT
nOJ1YQaTb MHorHe H38eCTHbJe pa3HOCTHble cxeMhI, BKJlJO'IaH BBHYIO
npH 8 1 = ()2 = O.
BBe eM 0603HaQeHHe jQ = q1+1 - Qj H nepenHweM (3.3) B
TaK Ha3b1BaeMOA eJIbTa-<I>opMe (103):
JQ = Llt8. ( jQ) + tJ..t aQ; + 1J 2 j-lQ +
1+82 at 1+92 at 1+
+ 0 ((8. - - (J2) t3 + t4). (3.4)
j3. HeJlBHhle cxeMbJ p'JlH peWeHHR KB&3Hra30 aMHCfeCKOR CHCTeMbl 193
nO CTaBJISISI B (3.4) Bblpa>KeHHe 8Q/8t H3 (3.1), UOJ1Y'IHM:
. t (Jl [ () ( . T (). T (J .
lQ = - -6 J E + - -/),.1V + -p- JH +
1 +02 AX 2 ax 2 ax
rp a. 'T iJ I ) a ( . T lJ .
+--- JG+-- JU +- -AJF+-- JW+
2 p ax 2 lJy 8y 2 8y
T (). T P 8. T 8 . )]
+ -p- lH + ---L\JG + -- JU +
2 lJy 2 p Oy 2 ax
t [ f:) ( j 'T BV; T 8Hi ; poGi T aUi )
+ - - +- +-p +-- +- +
1 +82 OX 2 ox 2 ax 2 p ax 2 8y
a ( . Taw; r 8Hj T poGj raUj )]
+ - -F' + - + -p + - - + - +
{}y 2 {}y 2 8y 2 p 8y 2 ax
+ (h j-l Q + O (( () _ 1 _ 9 ) l:1t 3 + t4 )
1 +82 1 2 ·
(3.5)
JIHHeapH3yeM qneHbJ E, AF, il V, D. W , LlG, H B ypaSHe-
HUH (3.5) "YTeM pa3JIO>KeHHB B pH TeiiRopa, Koropoe npoBe eM Ha
npHMepe E.
E'+l _ Ei = ( )1 D.t + O(D.t 2 ) =
= ( ; ) j At + O(D.t 2 ) = Aj (q;+1 - Qi) + O(At 2 ),
(3.6)
r e A == 8E/8Q - MaTp a HK06H. 3anHCbJB8JI HaqaJIO H KOHe
enOqKH (3.6) 8 ,lleJIbTa-<I>oPMe, HM M
jE = Aj ;Q + O( ).
(3.7)
LfJJyrHe qneHbJ JIHHea.pH3yeM aHaJlOraQHO. qJIeHbI co CMeWaHHblMH
npoH3BO MH 3a.MeHHM C IIOMOwplO COOTHomeHHSI
ju = j-lU + O( t2).
(3.8)
B pe3YJlbTaTe npeo6pa30BaHHA (3.7)-(3.8) npHXOAHM K JIHHea-
pH30BaHHOMY YpaBHeHHIO (3.5):
{ 91 t ( a ( . T. a · T . a . ". a - , )
1+ -;- AJ - -11' -MJ - -(p/p)l-N' - _-A' +
1+02 ax 2 ax 2 ax 2 ax
a ( . T. a . T . 8 T a - . ) ] } .
+ 8y B' - 2 P' ay M1- 2 (Pjp):J 8y NJ - 2 8y BJ JQ =
_ Llt [ !.- ( _ . T 8V i T -8Hi T aGi ::. 8U j )
- 1+82 8x E1 + 2 ax + 2P' ax + 2 pj 8x + 2 8y +
a ( . roWi T' aRi T 8GJ TaUj )]
+ 8y -F:J + 2 8y + 2 Y 8y + 2 tft 8y + 2 ax +
+ 8 2 j-lQ + 81 t [ !.- j-lU + !.-. T Aj-lU ] . (3.9)
1+92 1+92 ax 2 Oy 8y 2 ax
3,necb I - e HHHqHaH MaTpH a, A, B, M, N, A , B - MaTpH 51Ko-
6H) KOHKpeTHhiA B KOTOpblX 6Y eT Y IC B KOliQe naparpa<l>a.
CxeMa (3.9) )J.J1S1 peweHHH ypasHeHHSI (3.1) 3anHCbJBaeTCS B 60-
nee Har JlH HOM BHJJ-e:
( I + t 81 Bi ) Ll;Q =
1+82
1 · (J 2 · 1 (J l '
= t A' + 6. ' - Q + 6t CJ .
1+92 1+92 1+82
B CBOIO OqepeAb, BeKTOpbl Aj J cj H MaTpH a Hi 8JOTc.H BIdpa>Ke-
HHSlMH
(3.10)
, 8 ( . T BV" T. oBi T p1 aw T au i )
AJ = - - E' + - + -Y + - --: + - +
ax 2 ax 2 ax 2 P' ax 2 J
( _ j :. 8W j :'. 8Hj :. p1 8W T aU' ) ' ( 3.11 )
+ 8y F + 2 8y + 21" 8y + 2 pi By + 2 ax '
. a ( . T. a . T . 8 · T {} - . )
8 J = ax A' - 2 1" ax M' - "2(pjp)J ax N1- 2 {hAJ +
a ( . T' 8 .".. . 8 . , {} - " )
+ By BJ - 2 P' By M:J - 2 (p / p)J ()y NJ - 2 By BJ =
= B + 8 )
(3.12)
g3. HeJlBHble cxeMJ:J ,l1.JIH pelUeHJlR KBa3Hr83DAHH8M1fQeCKOR CHCTeMhl 195
c j = 'T j-lU + :!:. j-lU . (3.13)
ox 2 By 8y 2 ax
OT (eTHM) 'I'O BllepBble nHHeapH3aI.J.HIO no HblOTOHY OHap-
BoA KfY-cHcTeMbl 6hlno npe O>KeHO HCnOJlb30aaTb B pa60Te [104).
MCqepnblBalOI.Qee pa3BHTHe 3TOro nO,l),Xot\a 8nOC1Ie CTBHH Hamno BbI-
paJKeHHe B pa60Te (102).
Ha aCROBe cxeMhI (3.10)-(3.13) B03MO)KHO nocrpoeHHe pa3nH'i-
flbIX &1JropHTMOB pemeHHH HCXO HOA CHCTeMbI (3.1), BKnlO a.H HBHble
cxeMhI. Cpe,n.H paccMoTpeHHblx B [102] aJIrOpHTMOB HMelOTCH aHa-
norD cxeM BHMa-YopMBHra H CTerepa-YopMHHra, OnHCaHHe KJI&C-
CH"IeCKHX BapHaHTOB KOTOpLIX MO)l(HO HaaTH B MOHOrpa4>HH (103).
,LVJsr perneHHH nonyqalO HXCB caCTeM JlHHeAHblX CeTOqJJhlX YPaBHe-
IIHR MO)KHO npHMeHSlTb Pa3JIHQHble npHMble H HTep OHHhJe Me-
rrO,l).bl (105), 8KJJfO'iaB MeTO M&TpH'lHoA npOroHKH. O,lJ.H8J<O HaH6o-
nee 3QxPeKTHBHbI MeTO)\hI paCll(eOJleHH}1 DOTOKOB, OnHp8JO eCSI Ha
npeACTaBJIeHHe 8 B (3.12) B BRAe CYMMbI ABYX onepaTopOB H Y'iH-
.rbIBaroIIJ.HC cne cJ>HKY KrY-cHCTeMbl. B qacrHOCTH, OKa38JJHCb no-
J1e3HbIMH COOTHOmeHH.H Me>K nHHeapH30BaHHbJMH onepaTOpaMH,
onHCbIBafO HMH KOHSeKTHBHYIO H cJ><I>Y3HOHHYIO 'IacTb [102].
B 3aeepUJeHHe naparpa4>a p n01IHOTbl H3nO>KeHMJI npH eM BH.Q
MaTpHU 511<061-1 ,D)JJI nHHea.pH30BaHHb1X BeKTOpoB.
0 1 0 0
8E O.5«,,-3)u 2 + ( -1)v2) (3-ry)u -(-y-l)v ')'-1
A---
- 8Q- -Ul1 11 U 0
441 a42 -(1'-l)uv 11£
r JJ-e
CLtl = -u ('YE/p-{')'-I)(u 2 +v 2 »,
o
B _ of _
- 8Q -
-U1J
O.5«(1'-3)v 2 + (i- 1 )u 2 )
b41
a42 = "Y E lp-O.5(')'-1) (3u 2 +v 2 ) ;
010
1) u 0
-(,-l)u (3-7)V --y-l
-(1'-l)uv b 3 "'IV
r ;I.e
b 41 = -v (7 E /P-(,,-1)(u 2 +v 2 » J
all (3-,,)u
- av l a22
A---
- 8Q - 031 (3-,,)uv
a41 B42
b 43 = ')'E/p-O.5(1'-1) (3v 2 +u 2 ) ;
-(,,-l)v ,.,-1
-.3('Y- 1 )uv 3(1'-1)u
a33 ( 'Y -1 )11
3 a.w
,
r)J.e
all = O.5(,-3)u 2 + (1'-1)v 2 ),
(121 = -1.1, (3(')'-1) E/ p - (3,-5) u 2 - 3(1'-1) v 2 ),
Q22 = 3(1'-1) EJp - 1.5(3,-5) u 2 + (')'-1) v 2 ) 1
Q31 = -v (,-1) E/ p + (3-1') u 2 - (,,-1) 1J2) ,
a33 = ('}'-1) E/p - 0.5«'1-3) u 2 + 3(1-1) v 2 ) J
a41 = «3-51')u 2 - (,¥-1)v 2 ) Ejp + (1-1) (3.75u 4 + O.75v 4 + 4.5 U 2 V 2 ))
ii.t2 = U ((5"(-3) Elp - ('Y-1)(5u2 + 3v 2 )))
B.t3 = 1) (1'-1) (E I p - 3(U 2 +V 2 ) , 4 = 0.5 «5'y-3) ,,2 + (ny-I) v 2 ) .
A1S1 IlonyqCHWI 3neMeHTOB MaTpH bI B = 8W /8Q HY)t(fIO nOMefl.RTL
MecTat.IH BTOpoA H TpCTHH CT0JJ6QbI, BTOpyJO H TJ)eTblO CTpokH, a TaK)I(e
80 Bcex Bblp3>KeHMBX nOMeH.RTb MeCTaMH KOMnOHeUTbl CKOpOCTH U H v.
0 0 0 0
M = DB = -.:L 0 0 0 0
8Q Pr p 0 0 0 0
1.1 2 + v 2 - E/p u tJ 1
0 0 0 0
8G 0 0 0 0
N=-=, 0 0 0 0
8Q
O.5(U 2 +V 2 ) -u -v 1
4. MHoronpo eccopH&le BhI'lHCJlHTeJlhHhle
CHCTeMbi
B 3TO 1 Ii nOCJle,ll;YIO HX naparp8(j>ax aHHoA: rn8.BbI peQb 6YAeT
H,l(TH 0 np06neMax, CBH3aHHblX C IJpHMeHeliHeM BbJCOKOnpOH3BO H-
TenbtIbIX MHoronpo eccopHblX BblqHCJIHTeJIbHbJX CHereM ,L\nSl peme-
HHB 3 aq rH po- H ra30BOH AHHaMHKH. BKJlroqeHHe STOro MarepH-
aJIa B KHHry CB &HO C TpeMB OCHOBHblMR npH'iHHaMH.
Bo-nepBbIX) 6ypHblIt pOCT npOH3BO HTeJILHOCTH MHoronpOl..\ec-
copHoR BbrqHCJIHTenbHOH TeXHHKH, Ha6nlO aeMbiA B nOC1le Hee ea1-
THneTHe t OTKpblBaeT wHpoqaRwae B03MO)t(HOCTH B MO CJDIpOBaHHH
33Aaq ra30BoA HHaMaKH H MHOrHX ,LWyrHx npo6neM, OnHCbJBaeMbIX
ypaBHefIHSIMH MaTeMaTHQeCKOA <PH3HKH H paHee He oCTYnHblx ,lI.JIR
'fHCJleHHOI'O peweHHH. B CB H C 3THM XOTenOCb 6bl 06paTHTb BHH-
MaHRe IHTaTeJIeA Ha pa60TY O.M. BenOQepKoBcKoro [106], HanHcaH-
HYIO OKQJIO 20 neT TOMY Ha3 , B KOTOpOH nepcneKTHBbI peme}IHH
4. Mlforonpou.eccopHble BbI'IHCJlHTeJlf)HWe CHCTeMbI
197
CJIOiKHblX 3a,.naQ r830BOA IIaMHKH HanpSMYIO CBSl3bIB8JIHCb C 6y,ay-
I.qHM nporpeCCOM B npOH3BO,QHTeJIhHOCTH BhI'IHCJIHTeJIhH:bIX CHCTeM.
C ,z:r.pyroA CTOpoHbl, HM€IOTCR 3HaqHTeJIhHLle TPY HOCTH a,a.an-
HH HMelOllJ.HXCSI Bbl'IHc:JIHTenbHhIX aJlrOpHTMOB K a.pXHTeKTYpe
MHoronpOQeCCOpHblX CHCTeM. C ecT8YIOT TaK>Ke npo6.neMbI, CBB-
3aHHble co cneqHQ>HKoft napaJIneJIbHOrO MaTeMaTH"IecKoro o6eclle-
qeHHH. He MeHee Ba)KHOR np06neMoA, B qeM-TO 8JKe BbIXO lIt-eA B
HaC1.'OSlLQCe BpeMSI Ha nepBDe MeCTO, BJlHeTCSI HHTepnpeTaU)fB H Be-
PH<I>HKaI.l.HH aHHhlX BbICOKonpOH3BO HTeJIbHblX 8bl HCJIeHHA. 9Ta
TeMaTHKa "MeeT np.SlMoe OTHOIIleHHe K KnaCCH'ieCKHM np06JIeMaM
npHKJI HOH H 06 ea MaTeM3THKH, a BMeHHO, K BOnpOCa.M KoppeKT-
HOCTH HCllOJlh3yeMbIx 8JIrOpHTMOB H HCXOAHhlX MaTeMaTlAeCKHX M
,L\eJIeR.
HaJ<OHeQ, OCHOBHbIe anropHTMbI, coo,n:aHHble Ha 6a3e KCPC If
KrY-CHcTeMbI, OKa38J1HCb H)I.eaJlbHO npHcnoco6J1eHHblMH K apXHTeK-
Type COBpeb,leHHbJX MHoronpOQeccopHblX KOMDLIOTepoB. BblnH no-
CTpOeHbJ napaJIJIMbHbJe BapHaHThl 3THX a.nropHTMOB, ttTO n03BOJIH-
JIO npOBeCTH MOAeJIHpOBaHHe IJ.enoro pgAa CJlO)KHblX 3a,AaQ ra3OBOA
HHaMHKH Ha BhlCOKOnpOH3BO TeJlbHbIX CHCTeMax. OnHcaHHlO 3THX
pe3YJIbTaTOB B 3HaQHTeJThHOH Mepe nOCB eHa rJIasa V. BOJIee TO-
ro, peaJIhHasI B03MO>KHOCTb npHMeHeHHR KCPC nSl pemeHHH CTOJIb
TPY OeMJ(HX 38Aa "cnpOBO Hpo8ana" asTopa H ero KOJIJJer Ha HC-
CJIe,l(OBaHHe BonpOCOB, K838JIOCb 6L1, CTOmIJ)iX B CTopOHe OT OCHOBHo-
ro HanpaOJIeHMH eHTenbHOCTH. CIOAa OTHoarrCH np06JIeMbI napaJ1-
neJIbHOrO MaTeMaTHqecKoro o6ecneQeHHg [73 , l07}, KOTOpble 6Y YT
paccMCY.rpeHbI B 9 7 rJlaBbJ.
OJI.HaKO npe>K e qeM paccMaTpHBaTb Hcnonb30B8HHe KCPC B
napan.nenbHblX BblqHcneHRRx, HM KpaTKYJO HHcPopMa HIO 06 HC-
TOpHH pa3BHTHH H nepcneKTHBax MHoronpo eccopHoii BbIIfiICJlH-
TenbHoR TeXHHKH.
1. l1cTopHJl H l1epcneKTHBbl pa3BHTHH MHorofIpo ecCOpHb1X
CHCTeM
OrpaHHQeHHOCTb B03MO}K}lOCTeA 9BM C apXHTeKTYPo.A
<POI-l HeAMaH8, Kor a Bce BbI'IHCJIeHIDI npoBO SlTCH nOCJ1e OBa-
TeJ1bHO, Y)Ke 6blna H3BeCTHa Ha 3ape KOMfibIOTepHOA 3phJ: B KOH e
40-x - Haqa.ne 50-x rr. XX BeKa. OKOJIO DSlTH)J.eCSITH JIeT TOMY
H83 MaTeMaTHKH HaqanH p83pa6aTblBaTb TeOpmo napanneJlLHhJX
BbIqHCJIeHHA, KOTOpas npeBpaTHJlaCb HbIHe B caWOCTOHTeJlbHOe
HanpaaJIeHHe BbI'IHCJIHTeJlbHOA MaTeMaTHKH [108]. O,n;HaKo napa.n-
JIeJIbHble BbIqHCJrHTeJlbHhle CHCTeMhI, H& KOTOpbIX MO>KHO 6hlJlO
61d peaJI.H30BaTh OCTH)I(eHWI TeOpHH t ,l\onroe BpeM5I lIpaKTHQeCKH
OTCYTCTBOB8JIH. OT enbHble nODblTKH C(3)1.a.H1UI MHOroIIpO ec-
COpHOH TeXHHKH, XOT.H H OK83aJ1HCb nWle3HbIMH npH peweHKH
pH a KOflKpeTHblX aq, B enOM He MelUlnH o6m:eA KapTHHbI
HCnOJ1b30BaHHst BbI'-IHCJlHTeJlbHbIX MOIIJ.HOCTeA, r e 6e3pa3 eJlbHO
apCTBOBanH o onpo eccop.lble KOMIThJOTepbJ.
IIono>KeHHe Pa.AHKaJlbHO H3MeHHnOCb B cepe,n;HHe 80-x rr., Kor a
cne anHc -aMH cpVlpMbI INMOS (BeJIHK06pHTaHH}1) 6bInH CKOHCTPY-
HpoBaHbI mpaKCn'bwmepbt. TpaHClIhlOTePLI OTHOCBTCSI K ceMeACTBY
RISC-npol.\eccopoB (Reduced Instruction Set Computer) - npo ec-
COpoB C orpaHUqeHHblM Ha6opoM KOM8H)J:.
BamHo:A: OTnH'IHTeJ1bHOA oco6eHHOCTblO TpaJICnblOTepOB 110
cpasHeHKIO C OCTaJlbHbJMH RlSC-npO ecCOp8.MH Ha TO speMSI SlBJIB-
nOCb H8JIH'iHe t.JeTblpex KaH8JJOB CBSl3H (links). 9TH k8.HaJILI o6ec-
ne"lHBaJIH 6b1C1'pbJA 06MeH HHcPopM8 He:A KaK Me OT bHhlMH
npo eccopaMH"TpaHCnbK>Tepa.MH, TaX H Me>KA)' TpaucnbJOTepaMH H
,DJ>yrHMH BbI'fHCJIHTeJlbHblMH cpe CTBaMH - nepcOHaJ:LHblMH KOM-
nblOTepauH Ii pa60QHMH CTaHQWlMH (109).
IIepBble Tp8.HCllblOTepbl T-414, cepHAHN:A BblnYCK KOTOpblX 6hIJI
HaqaT 8 1985 r., HMeJIH (IpOH3BO HTeJlhHOCTb 0.1 H 0.025 MFLOPS
MSI 32- H 64-pa3PSl.lJ.HbIX BOHqHbIX QHCeJI COOTBeTCTBEHHO. Cne -
IOInaJI MO H<PH HH TpaHCIlhIOTepoB T -800 (cepHAHbJA BhInYCK -
1988 1".) o6JI a.na npOH3BOAHTeJJbHOCTh1O 11.0 2 MFLOPS Ha O H
rrpo eccop, a MaKCHManbH&H O HOCTOpOHHfi CKOpOCTb nepe aqH
aHHLIX no KaHaJJY CBB3H COCTaBnHJIa 1.2 M6aAT/C.
Bnaro ap apXHTeKTYpe TpaHcnbIDTepoB, OHH o6he HHnHcb
B MHOrOnpOl.\eCCOpHble pa60Que CTaHU,HH. I1pHMepbl KOHcpHrypa-
l\HA npo eccopoB nOK83aHbJ Ha pHC. 44 H 45. B KOH e 80-x I'I'.
cYMMapHaH npoH3BoJ).HTMbHOCTb TaKoA: CTaH HH Morna oCTHraTb
100 MFLOPS, H 3TH MO HOCTH HCnOJIh30BaJIHCb B TOM 1fHC1Ie H nH
PC
Host
rsoo
Root
noo
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PHC. 44. llpIDdep JlHHeAHoH KOH H-
rypattHB Dpo ecCOpOB
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QeccopHoA CBCTeMhI B e peWeTKH
paCQeT8. 3 aq, onHCbIBaeMhlX YpaBHeHH.HMH M&TeMaTH'tfeCKOI <PH-
3HKH.
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TepOB T -9000 - c npOH3BO TeJIbHOCTbIO 11.0 25 MFLOPS H& npo-
eccop - <l>aKTH'feCKH He COCTOSlnc . Ho paapa60TQHKH MHoronpo-
l..(eccOpHblX BbI'IHCJlHTeJIhHLIX CHCTeM HaInJIB HOBoe peweHHe. B CH-
CTeMax CJ1e,.QYIO ero nOKOJIeHHSI hla npo eccop- npe CTaBJlHJI
co6oA: Y>Ke cBoeo6p83HhIA' Knacrep, B KOTOpOM B Ka1.leCTBe apH4>-
MeTH1.IeCKoro YCTpoiicTBa BhlCTYDaJI B:bICOKOnpOH3BO Te.nhHhlA: (H&
HaqanO 90-x rr.) npo eccop, HanpHMep, INTEL-860, Power PC-601
Ii MJ. npH 3TOM TpaHcnblOTep T-800 (HJ1H ero MO H<I>HKaI.\HH T-805),
6ypy\.IH qacTblO KJIaCTepa, Hrpan poJIb CBSl3YIOw.ero 3BeHa, o&be -
HSIIO ero OTAeJIbHble npo eccopbI B napaJIJIenbHYIO BLICiHCJJHTMb-
HYIO CHCTeMY. IIpoH3Bo TeJlbHOCTb npo eccopOB, 06P83ytOID.HX o -
llO apH<pMeTH'IeCKOe YCTpoAcTBO, 6h1na MH3Ka K 100 MFLOPS, a
napanneJIbHble CHCTeMbI Ha me OCHOBe OCTHranH IIHKOBOR npoH3so-
)J.HTenbHOCTH nopSlAKa 10 GFLOPS..
,l].anbHeAwHA 6blCTpblR pOCT npoH3Bo):{HTeTlbHOCTR Ha O)J;HH
rrpo eccop, KOTOpaH Ha KOHe 90-x rr. YJKe COCTaBR5IJI& nopH a
1 GFLOPS, HMen H HeraTHBHbIe nocne,L\CTBHSI. B MHoronpo eccop-
· MHoronpon:eccopHhle CHCTeJ4b1, OCHOBalUlhle H3 TpanCDhJOTepax) Y>lCe 60nee
10 neT He onpe eJImoT TeH.JI-e B TeXHOJIOMtRX na.pa.nneJu,HYX BLlCUlcneJOd.
Q)J)IaKO OCTaTO't"O nOJij)06Hoe ODHcaHHe HCTOpHB HX p83BHTWI en.eTc. B 113-
BeCTHOH Mepe aJlbIO 6naro apHocTH WRorHX Cne aJlHCTOB, BKJI.I)'I8JI 8BTOpa ,
)THM npe-nCTaBH 'enRN 3J1eMeHTHoA 683b1, B03HHKnOBeHHe KOTOphiX B SO-x ro,ltax
nOCTaBHJ10 B peaJJbnylO nnOCKOCTb napannenbHhle 8h1'1HCJleHHg.
HblX CHCTeMax t r e Ka>K,ltbIA npOD-eccop pa60TaeT DOttTH KaK &BTO-
J-IOMHbIH KOMnblOTep) o6n asJ C06CTBeHHOR OnepaTMBHfJA naMjlTbIO,
y3KHM MecTOM CTaJJa CKOpOCTb 06MeHa HH<l:>opMa HeA, 06ecneQHBa-
eMag TpaaCnblOTepHblMH KaHaJIaMH CBR3H.
B CaMOM AeJIe, pOCT MOUI,HOCTH OT MbHOrO nponeccopa npH-
BO HT K ecTeCTSeHHOMY CTpeMJIeHHIO n0Jlb30BaTeJlH pelUaTb 3 aqH
co Bee 66JIbmHM o&beMoM HH<POpM HH. Ii KaK 6bI H3 CTapaJIHCb
MaTe"'1aTHKH-BbI HCJlHTeJlH 3a CqeT Bhl60pa cneQHaJlbHLlX a.nropHT-
MOB YMeHLWHTb 06MeHLJ, KaKaB-TO qacTb HHcI:>OpMa.QHH B 6onhWHH-
CToe 3a.n:a OJ])KHa nepeAaBaTbCB OT o oro npou:eccopa. K ,lIJ)yrOMY.
TaKHM 06Pa30M, B a6cOJlIOTHOM Bblpa)KeHHM KOJIH'IeCTBO 06MeHHBa-
eMoR H}[cPOPMau.HH, KaK npasHJlO, AOJI}f(HO paeTH O HCBpeMeHHO c
npOH3BO;J.HTenbHOCTblO npo eccopOB. HMeHHO 3'rO YCHJJHBaeT Tpe6o-
BaHHH K 6bICTpo eAcTBHJO cneQHaJlbHblX YCTpoRcTB, o6ecneQUBaJO-
n HX HHcP<>PM HOHHblH 06MeH.
Bbl'lHCJ1HTeJJbHble CHCTeMbl c pacnpet\eneHHoA naMBTblO B H&-
CTOB ee BpeMB ,AOMHHHPYIOT B npaKTHKe npHMeHCHHB BbICOKonpo-
H3BOAH r reJIbHOA napanneJIbHOR TeXHHKH. Q,lJ.HaKO HapSl C HHMH
TaK1Ke HCnOJlb3YlOTCB Mlloronpo eCCOpHbJe KOMI1bJOTepbI C apXHTeK-
T)'poA, 6a3HpYIOIneRCH H& o6xneA aKTHBHOA naMHTH. 3TO 03HaQaeT,
1.ITO Ka>K bIA 83 npOQeccopoB HMeeT B03MO)KHOCTb 06p aTbCH K
o HoA naMBTH, 06 eA JlB Bcex npol\eccopoB.
BSH,lQ' Toro, '{TO JIH'lHbJA: BhlQHC1IUTeJlbHbIR OIIhlT CLBTopa (53,
73, 98) 100, 107], [110)-[121] B OCHOBHOM 6blJI cocpe OTO eH Ha HC-
nOJIb30BaI-IHH MHoronpo eccopHblx CHCTeM c pacnpe eneHHOa naMSI-
TblO J OTHOCHT€nhHO CHCTeM C o6w.eA Da.MHTblO OrpaHHQHMCB 3 eCL
JIHWb Kpa:rKMM nepeqHCJleHHeM me AOCTOHHCTB M He Ot'TaTKOB. K
qJ{cny npaCYInHX HM OCTOHHCTB cne,.Qyer OTHeCTH TO, tfTO 'IHC-
neHHble aJ1rOpHTMbl anTHPYIOTal K HX apXHTeKType 3HaqHTeJlh-
HO npow;e) qeM K apxHTeKType caCTeM C pacnpe.n;eJIeHHO naMHThlO
(06 3TOM 60JIee no p06Ho CK3JKeM B CJle lO eM naparpa4>e). 113
HeAOCTaTKOB cne)tYeT OTMeTHTb cy ecTBeHHO 66JJbmylO CTOHMoC'rb
CHCTeM c o6 eR naM5ITblO no CpaBHeHHIO C MHoronpoQeccopHhlMH
CHCTeuaMH, 6a3HPYIO HMHCSI Ha pacnpe)J.eJIeHHOA R8.MHTH.
C CepeJI,HHbl 9O-x ro OB pa3BHBaeTCH HOBoe nOKOJIeHHe MHOro-
npou.eccopHblX CHCTeM c pacnpe,.neneHHoA na.MSlTblO, HCnOJlh3yIDlIJ)Ix
94. MHoronpoueccopHble BblqHCJlHTeJlhHbIe CBCTeMbl
201
cneu.HaJlbHble npOl\eCCbl-KOMMYTaTOPbI ,lVIB YBenHqeHHSI CKOpOCTH
o6MeHa HHcPopMa HeA. TIpH nOM Ka)l( 3JI nOC1JeAYIO aSl MO,ltHtPH-
Kal.\H5I BbIQHCJIHTeJ1bHhJX CHCTeM OfiHpaeTCSI KaK Ha OqepeAHYIO 60-
nee BbICOKOnpOH3BO TeJlbHYIO MO eJIb npOD-eccopa t TaK H Ha HOBbie
KOMMYTaTOpbJ) o6ecneQysalOIIJ;He BCe 60JIee BbICOKYIO CKOpoCTb 06-
MeHOB M€.>KAY npon;eccopaUH.
Ha 3TOM nYTH AOCTHrHYT BneqaTnHIOll\HA nporpecc B pOCTe npo-
H3BO,!l.HTenbHOCTH BbI1.£HCJIHTeJJbHblX CHCTeM. TaK, eCJIH B cepeAHHe
90-x I'O,u.OB nHKOBas npOH3BO,D:HTeJIbHOCTb B 100 GFLOPS K&3aJIaCb
He,nOCTH)t(HMO BbJCOKOA, TO B CaMOM Haqane XXI Bexa. )J.JIH Be yIU)iX
MHpOBblX HaY'iHblX eHTpoB peanhHOCTblO CTaJlH BhlqHCJIHTenbHble
cnCTeUbI C llHKOBO:tt npoH3BONITenbHOCTblO no 1 TFLOPS.*
TeMllbl pa3BHTHjf BbIqHCJIHTeJlbHblX CHCTeM He CHH)K8J()TCH H B Ha-
liK )J.HH. TaK, KOMnaHHSI IBM 06'bSlBHJIa 0 CBoeN npoeKTe C03A8JfHR
K 2005 r. BblQHCJ1HTenbHoA CHCTeMbI C fiHKOBOA rrpOH3BOJl-HTenbHo-
CTblO B 360 TFLOPS. CKOpee Bcem, K MOMeJITY BhIXOAa ,lI;aHHoA
KHHrH 31'o'r npoeKT CTaHeT peanbHOCTblO. ilporHo3HPYsr HCCKOHhKO
aJ1bwe) MO)KHO C onpeAeneHHoA YBepeHHOCTblO yrBep aTb, 'tJTO K
2010 r. tJla..KCHMaJIbHbIe I1HKOBbIe MO OCTHt AOCTynHbIe Hcc.ne OBa-
TeJISIM, 6Y.DYT JJe>KaTb B npe enax 103 -:- 10 4 TFLOPS.
YcnexH nOKaJKYTCSI oc06eHHo OmenOMJI.SlIOIIl.HMH, eCJIH conOCTa-
8HTb me c nOJIO eHHeM 25-neTHeR )J.8.BHOCTH, Ko(' a OCTynHb1e ,LtJIH
60JIbIDHHCTBa HCCJIe OBaTeneA MOILI.HOCTH HaxO HJlHCb B ana30He
OKono 1 MFLOPS. O aKo pOCT DpoH3BO HTenbHOCTH BblqHCJlH-
TeJIbHblX CHCTeM 3a qeTBepTb seKa B MHJInHOH Ii 60nee paa He npH-
BeJI K aHaJIOrH'-IHOMY nporpeccy B B03MO)KHOCTRX pemeHHH 3 aq.".
BOJIee TOro, cyrn.ec..."TBYlOID.MA pa3phlB Me B03pacTalO HMH B03-
.HanpHMep B Me. OMCTBeHBOM cynepKoMlIhlOTepHOM IJ.eHTpe (MOCKBa)
B 2001 r. 6b1J18 B eHa B 3KCnny8T81..\HJO CHCTeMa MBK-lOOO C DHXOBOA npo-
H380,AHTe1IbHOCTblO 1 TFLOPS. R1J h[A KOMnJ1eXC COCTOHT H3 768 RpOQecco-
pOD -\LPHA 21264. B ka eCTBe KOMwyraTOpOB HCnOJlb30Balla CHCTeM8 MHpR-
HeT 2000. Ha 3TOH CHCTeMe B nOCJIe,AHHe ro.nbl 8.BTOp Hero KWlnerH npOBOAHJDI
pac eTbl 1(eJ10rO pSl.lt3 3 a'l. OJIHCa.HHe 3THX 38Aa1f H aH8J1H3 pe3YJlbTaTOB me
pac'leTOB '-IacTH O HaWJlH OTpa>KeHHe B rJ1. V MOHorpa<pHH.
.. QHTaTCJ1H C OCTaT01fH.blM HaY'IHblM CTaJlCeM MorYT BcnOMHHTb 06 ycnem-
nOM MO eJ1HpODaHHH MHOrHX cnO)l(HhlX 38,[1.a'l B 70-80 rr. Ha BbI"IHClIHTeJlbHOA
TeXHHKe TOro 8petAeHH.
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TeXHHQeCKHH nporpecc, SlBJ1j1eTCH CBoero po a Bbl30EOM C()()6 ecrBY
CneLI.HaJIHCTOB B o6nacTH MaTeM&THqeCKOrO MO eJlHpOBaHHg, npH-
KJI HOH MaTeMaTHKH H HHcPopMaTHKH.
B KaKoR-TO Mepe J];aHHasI cHTya1I.HSI CBH3aHa c otYheKTHBHhI-
MH npHQHIJaMH. HanpHMep, HCfiOJIb30B&Hee BbJCOKonpoH3Bo HTeJID-
"oR reXHHKH Y>Ke e.n8JIo peanhHbJM pemeHHe HecTa HOHapHbIx
npoCTpaHCTBeHHo-TpeXMepHbJX 3 aq. Ho KaKOBhI )!(e B03MO)KHOCTH
pewaTb Te >Ke 3a.)J.&QH Ha 60nee nOAP06Hblx CeTKaX no npoCTpaH-
CTBY H BpeMeHH? npH paCTe npOH3BOAHTeJIbHOCTH Q CHCTeMbl CTe-
neHb eTaJlhHOCTH OnHCaHHSI (QHCJlO Y3JJOB no OAHO! Koop aTe)
YBeJIH'iHBaeTCSI B nY'lIWeM CJIyqae KaK .
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rpecc B npHMeHeHMH MHoronpo eccopHblX BbltJHCJIHTe1IbHbIX CHCTeM.
9To He OCTaTOqHOCTb p83BHTHS napann bHhlX aJI pHTMOB, na-
panneJlbllOrO M8.TeMaTHQecKoro 06ecneQeHHB (parallel software), HH-
Tepnpe H It oepH<j)HK&QHH ..D:aHHbIX BbiCOKOnpOH3BO HTeJIhHLIX
BIdQHCJIeHHA H tij). no 06Hb1e npo6neMLI MOryT 6b1TJ. pemeHbI npH
COOTBeTCTBYIO HX yCWIBBX cne HanHCTOB. qacTH HO 3TH BOnpo-
CLI 6y y'r o6cy aTbCJ1 B nOCJ]eAYIO X naparpa<j)ax rJlaBbI notlt
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HOCTeR 06'be HHeHH.SI BLlqHCJIHTeJlbHblX pecypCOB cynepKoMnblOTep-
HbIX u.eHTpOB H pa3BHTHH OTHOCHTeJlhHO He6OJ1hWHX MHoronpo ec-
COpHbIX CHCTeM.
2. MeTaKoMnblOTHHr H KJl8CTepbl
nO BJ1eHHe .n;OCTaTO"IHO 6onLworo KonH eCTBa cynepJ4:0MnbJOTepHbIX
eHTpoR nOCTaBHnO Ha nOBeCTKY ,lUfSl BOnpoc 0 60nee pa HOHanbHOM HC-
nOJlb30BaHHH HX pecypCOB. AKTYaJIhHblM CTaHOBHTCR COO,ltaIDle e)J.HHOI Bbl-
HCnHTeJIbHOH Cpe)J.bl, 06rbe HHftlO eA pecYPCLl KaK 8 PaMYAX OTAMbHblX
CTpaH, TaK H B 60nee WHpOKOM Macnrra6e - BnJIOTb Il.O MHpoBOro.
TaKaR <:XYbe,D.HHeHHaJI BhlqJfCJIHTeJlbHasi CeTb ("Grid":' npeAOCTaBHT
cneA}'lOlJJ){e B03MOX<HOCTH.
1) Hcnonb30BaHHe,LI,JIJl pemeHHH 3 8"1H cBo6oAHbOC BbI'IHCJ1HTeJIbHblX
94. MHoroupo eccopHbIe Bbl'IHC/IHT8JlbHbIe CHCTeMbl
203
MOl11.HOCTeA ){PY1"'HX BLJ IRCJJHTenhHb1X ueHTJ>OB. HaxO,n)1 HXCR, 6h1Tb
MO>KeT , 8 pyrHx CTpaHax.
2) CneJJ.Ha.iIH3alUfjl TOrn HJlH "HOrD BblqHCJlHTeJJbHOrO QellTpa 8 Ka e-
CTBe XpaHHflHw.a 6a.3b1 aHHbIX 6oJlbWOro otYbeMa no onpe eJIeHHO-
IY nanpaBneHHfO HCCJle OBaTenbCKOa Hnu X03j1HCTBeHHoA Ae1lTMb-
HOCTH. TaKoe p8.3,l1.eneHHe <PYHKIJ)fH n03BOJJHT CKon eHTpHpoBaTb
BOKpyr AaHHoro n.eJIT})a YCHJlHH cneU)laJIHCTOB B KOHKpeTHOR o6J1a-
CTH H 60JIee 3<P<PeKTHBHO 3KcnnyaTHPOBaTb TeXHHtJecKHe peCypCbI
u,eHTpa B BbJ6paHHOM HanpaBJIeHHH. 9TH 6a3b1 ,lI.aHHbIX CTaHOBJlTCSI
AocTynHblMH BeeM yqacTHHKaM npOeKTa, pa60T8JO M B pa3JIH'IHbIX
L\eHTpax.
3) HCfiOJlb30BaHHe HeCKOJIbKHX BblCOKonpoH3BOJ1)ITenLHbIX CHCTeM, BXo-
,lI.ftlU)fX 8 061UJ'1O ceTb, R pemeHHB o.lJrHoA CJIo>KHoA 3a.l\3'1H, Hide-
lOll(eA 6onbWyJO HaYQHYIO HnH fipaKTHtfeCKYIO u.eHHOCTL. llpH 3TOM
3 aqa MO>KCT pa36HBaTbCB Ha OT,nenbHble qacTH, CJIa60 CBH3aHHbIe
MeJK,lIJ' co6oA, KOTopble pacnpe,l\eJlJlIOTC5I Me>K)]j' cynepKOMnLJOTep-
HblMH CHCTeMaMH. yroR pa3HOBH,D.HOCTbfO CTpaTerHH HBJlBeTCB
peweHHe ua pa.3JIH\lHbIX Bbl'lHCJ1HTEUIbHblX cHcTeMax p83Hoo6paa-
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1l11JOInHXC}1 napaMeTpax. K CJlOBY CKa3aTb J TaKoA npHeM He jiBnJleTCSI
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6wJIO 3aAeHCTBOBaHO CP83Y HeCKOJIbKO 9BM. Ha oA H3 KOTO-
pblX peWaJ1CJ1 CBoA co6cTBeHHblA BapHaHT.
4) 11, 1'18KoHeu., HaH60llee CJIO>KHCUI C M&TeMaTH'feCKOA TO'JKH 3peHiUI
np06JIeMa CBR38H& C pacqeTOM O OH 3 a"lH o oBpeMenHo Ha
IlecKonbKHX MHoronpo eccopHblX BblQHCJlHTenLHhlX cHcTeMax. Pe-
menHe 3TOR npo6ne 1bl n CTaBJUleT oco6b1R HHTepec AIDI HaY'lH
HCCJIe,[\OBaTeJlbCKHX HHCTHTYTOB, B KOTOpbIX -- no pa311H'fHblM npH-
CfHHaM - OKa3aJIHCL B HaJlHqHH MHoronponeccopHhle CHCTeMLI co
cpe Hea npoH3Bo,nHTeJlbHOCTbJO H HOO,lI,HH&KOBblX TUDOB.
npo6neMbI MeTaKOMnblOTHHra - 3<M>eKTHBHoro HCnOJ1b30BaHHJI ce-
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paBHOMepHoft 3a.rPY3KH. ae06xo,D;DMo HCnMh30B&n. CDeQHaJIbHhIe nporpaM-
MY ,lI)IH&MH ecJ(oA nepe3arpY3KH npo ecCOpoB. IIpJnlepy 3&A&'1, HnnIOCTpHPY-
lO e npHMeneHHe 'l'axHX nporpaMM t 6ypyr paccMoTpeHhl 8 rn. V.
)l{ecTKoe OrpaHHtJeHHe (5.3) H& mar no BpeMeHH nOJtHOCTblO nHKBHAHPyeT
npeHMytneCTBa B CKOpOM peweHJtH Ja.D.a'iH Ha BblCOKODp0H380 HTeJlbHhIX
CHCTeMax npH crYll(eHHH RpocrpauCTBeHHOA ceTKH. A HMeHHO H& ,D.eranb-
Hoe OnHC&HHe raJO HaMHqecKOro nOTOKa nocpe CTBOM nO,lq>06HblX CeTOK
opHeHTHpOBaHO npHMeHeHHe T8KHX BbNHCJ1H1'MbHhlX CHc..--reM.
i\nropHTM (5.1) B cany CBoero HBHoro XapaKTepa H Op)fOpO HO-
CTH npocT J1S1 nporpaMMHpOBaHHSI. Pa3HOCTHble cxeMbi (1.1)-(1.5))
HeCl\IlOTpR Ha OBOJIbHO rpOM03/l.KHA BH ) C TOQKH 3peHHSI J1oruqe-
CKoA CTPYKTypbI OqeHb npocTbI) QTO 06JIerQaeT nepeBo nporpaMM
C o,l.\HoA MHoronpo eccopHoA CHCTeMbl HR ro>yrylO. HanpHMep, nepe-
HeceHHe npOrpaMMbI pacQeT8 BYMepHhJX neYCTaHoBHBlllHXCB Teqe-
HHA BSI3Koro ra3a c MHoroDpo eccopHoa TpallCnLIOTepHoA CTdJ-lQHH
APS48 (npoH3Bo CTBO BOJIrapHH) Ha BbI'IHCJlHTenbHblA: KOMnneKC
PowerXplorer (repMaHHH) B 1994 r. 3&HSlJIO He 60J1ee )\BYX He e.nL.
Bcero, Ha HHaJI c 1988 r., nporpaMMbI H3 OCROBe cxeMbi (5.1) HCnOJIb-
30BaJlHCb t{JlSI paC'ieTOB Ha eBgTH Pa3JIHQHblX MHoronpo eccopHblX
CHCTeMax.
OCTaJlOCb 06cY)J.HTb nocne Hee Tpe60BaHHe K anropHTMY - KOp-
peKTHoCTb. KaK Heo HoKpaTHo roBopHnoCb Bblwe (rJl. III), KBa3HIQ-
30 HHaMHqecKa.S1 CHCTeMa ypaBHeHHA H KCPC rapaHTHPYIOT crna-
>KHBa.HHe peweHHH Ha pacCTOBHHH nopSl Ka JIHHbI cBo60AHoro npo-
6era l. TeM CaMbIM OHH SlBJliIIOTCH KoppeKTHbIMH C cl>H3HQecKOD TOq-
KH 3peHHH. BbJ'IHC1IHTeJIbHblA aJlrOpHTM KCPC C KOppeKl.\HeA npe -
CTaBJIHeT c06oA. 06Pa3HO I'OBOpSl, rH6p Kry H KCPC. Crna.>KHBasi
perneHHe c nOMO 1O ecTeCTBeHHblX CCHnaTHBHbIX qneHOB, npo-
nOpIJ.HOHaJIbHblX l, H OCTaJIbHblX )J.HCcunaTHBHhIx eHOB, rrponop-
QHOHaJIbHblX wary CeTKH h.
CTporo roBOp5l, aHBoe pacCYQeHHe He SlBmIeTCB OKa3aTenb-
CTaOM KOppeKTHOCTH KCPC c KoppeK eH H MO)l(eT paccMaTpHBaTb-
CSI JlHmb KaK HaBOAH ee C006p8.)KeHHe B D01Ih3Y npHMeHeHHB MeTo a
"QIDI pemeHHSI 3 aq ra30Boft HHaMHKH Ha BbICOKonpOH3BO HTeJIb-
HbIX CRCTeMax. CnpaBeA1lHBOCTb T8.Koro MHeHHH nO,lJ.TBep)l( aeTCSI
MHOrOneTHeA npaKTHKoA HCnOJIb30BaJm1l anropHTMa ,lVIS1 MO eJIH-
poBaHHSI HeYCTOAqHBbIX TeqeHHtt BSI3Koro ra3a.
TaKHM 06pa30M, Bce OCHOBHhle Tpe60BaHHH, npe,lfbSBnBeMbIe K
ttHCJleHHblM anrOpHTMaM NIH HX ycnewHoA a HH K apXHTeK-
5. .4A'a.nTal{HJJ KCPC K MHoronpOI{eCCOpHoJl apXHTeKType
211
Type MHoronpOQeccopHblX BbJqHCJIHTeJlhHbIX CHCTeM C pacnpe eneH-
HOU naMHTbIO,)I.J1H cxeMhl (5.1) BblnOJlHeHbl. BBe eM Tenep& nOHgTHe
:JfjxfJe'ICmUt3HOCmU napoAJleAf)HOU peaJlU3GttUU BhNHCJleHHR.
nYCTb SJ pemeHHSI 3 aqH Ha O HOM upo eccope 3aTp&QeHO
npeMSI tl, a Ha CHCTeMe, cOCToSl eA H3 N TOt.JHO TaKHX )f(e npo ec-
COPOB, BpeMSI paCtleTa paBHO tN. Tor a 9<j>Q>eKTHBHOCThlO napan-
JIeJlbHOH peanH3aL\HH H830BeM BeJlHqHHY
tl
Pen = N · 100%.
· tN
(5.4)
3 a M e q a H He 5.2. 9To onpe eJleHHe 3<1>cI>eKTHBROCTH OCT8.TO'IHO
OqeBH,lUiO H B TOft HJ1H HHOA <I>oPMe HcnOJJ:b3YeTCR Cne &1IHcraMH B 06J1a-
CTH napaJ1nenbHbIX BbI'lHcneHHA. XOTB cnpase,a,JlHBOCTH p B O 38Me-
THTb, q-r(),NIR Hax eHH$I Pel/Bblp8JkeHRe (5.4) He Bcer)J.& npRMeHHMO H&
npaK'rHKe. Kor 3 a'fa '1'pe6YeT Cuepx6wIbWOro 06'beMa BhI'IHCJ1eHHA H
pemaeTCSI H& npe.Ll.eJIe B03UO>KHOCTeA MHoronpoo.eccopHoA CHCTeMbI, BPRA
JlH pea..rIbHO H3MepHT& HanpJlMYIO BpeMB ee paC'IeTa 8 O,lJ.HonpOneCCOpHOY
pe}l(HMe tl. B M cnY'l8e npHBJIeK&lOTCS KOCBeBHhle MeTO,l(bI olIpeAene-
HHjf 34><PeKTRBHoCTH: HanpHMep, ,lJ.JUI 0AHO POAJlYX CJCeM HCnOnb3YeTC.R CJI
)tYIOI1\HA npHew. 3a.u.a,'Ia peW&eTCJI H8 O):tHOM nponeccope co 3Ha'lHTeJIbHO
MeHbWHM - B N' pa3 - 'IHCJlON pac'leTHblX )'3JIOB, qeM B MHoronpo ec-
COpHOM BapHaH're. 3aTeM SpeMJI tl Haxo,qHTCJI no <l>opMyne
tl :::: t; · N'.
(5.5)
MHOl"OJIeTHSlj1 npaKTHKa pac'leTOB no Ka3an a , qTO ,LVIH cxeMhI
(5.1) pe8JIbHaJI 9cP<i>eKTHBHOCTb nap8JIJlMbHhlX BbIqJfCJIeHHA: npH
pemeHHH aq Ha OCTaTOquO nO,lij>o6HL1x npoCTpaHCTBeHHblX CeT-
KaX npH6nH>KaeTCSI K 100%. ::hOT 4>8.KT 6hIn BnOJIHe npe CK83yeM,
H60 npH 6anbWOM '1Rcne C'IeTHbIX Y3JIOB H gBHhlX cxeM KOJIH-
qecTBO 06MeHHBaeMoft Me>KAY npo eCCOp&MH HH<PoPM&qHH OTHOCH-
TeJIbHO HeBenHKO.
3 aM e q a H H e 5.3. B HaY HoA JlHTeparype HHor):\& npHBOAHTOI pe-
3YJ1bTaTbI) nOK&3LIBaJO He HH3KYIO S<i><l>eKTHBHOCTh napan.neJIbHhIX BhI-
'IHcneHHA A8JKe AJlJI JlBHbIX cxeM. IIpH 3TOM Bcnonb3yeMoe 'IHCJIO CqeT-
HhlX y3JIOB 06bI'IHO HeBeJIHKO, TaK 'ITO 3 &qa 6m TPYJ\a pewaercg H B
o.nHonpOl.{eCCOpHOM pe>KHMe. C n 06BoA CHTY8ltHeA npHXO,l1HnOCh CT8JI-
KHBaTbCB R aBTopy. 9<1>cPeKTHBBOCTb IJSpannenbHbIX pac'leTOB no cxeue
J. 1 JI. l.". '1HCJleHHble anropHTMhI H MHOrOnp01(ecoopHbIe CHCTeMhl
(5.1) Ha rpy6bIx ceTKaX HaqHH8JI8 pe3KO n Tb. H BCe )ICe 3TH pe3ynb-
T8TbI HMeKYI' CKOpee nCHXOnOrH'IecKoe 3HaQeHHe, qeM npaJcr'HqecKoe. Be)l,b
B peaJ1bHOCTH - nooropHM ew.e pa3 - BbIOOKonpoH3BO,ltH bHhle MHOro-
npOnecCOpHbJe CHCTeMbi npe a3HaqeHbI ,LVlB eTaJIbHOro MOAe.rmpOB8HHB
}13 nOJJJ)()6HbIX ceTKaX.
3. IIonYHeHBHaH cxeMa H KCPC lfa HecTpYKTYPHPOB3HHbiX
ceTKax - MHoronpo eccopHaR anTa HR
OCTaHOBHMCSI BKpaT e Ha pyrHX anropHTMax, paccMOTpeHHhIX
B Ha4a.ne al[HOH rnaBhI.
KaK H KCPC C KOppeKD.UeR t nonYHeSBHasi cxeMa, (2.9) JlerKO
a,LJ,an"rHpYeTCH K apXHTeKType MHoronpo eccopHblX CHCTeM c pac-
npe,((eJ1eJ[HOH IlaMHTblO. 3anHIlIeM e e pa3 MeTO npocTbJX HTepa-
RR (2.10))VIR BbJQHCTIeHHH HOBOro CJIOSI B 3TO:A cxeMe:
'+l(k+l) .
Ql dt - Q1 + (AQi + AQ1+1(k») == 0, 0 s: k S Nk, (5.6)
Ql+1(O) =Qi.
Ha pHC. 48 H306pa>KeH ma-
6nOH cxeMhl (5,6), AJlH HarnSI.LJ.-
n.i+l(k) nOCTH, 8 ID-cnY'-Iae. AnropHTM
i+l (5.6) 06J] aeT ToR )Ke JIOrHt.fe-
CKoA CTPYKTypoA) 'ITO H cxeMa
(5.1). OH O Hopo eB. BBH,QY TO-
ro, 'ITO (5.6) MO)KHO HHTepnpe-
. Q1:: (1) THpOBaTb KaK 5IBHYlO cxeMY, 06-
MeH mI<PoPM ea MC>KftY npo-
u.eCCOpaMH rrpOH3BOAHTCH TOJIb-
KO Qepe3 3HaqeHHSI ra30)J.HHa-
MH\feCKHX napaMeTpOB B Y3JIax,
PHC. 48. Pa3HOCTHNA wa.6noH )US!
DOJIYHeftBHOA Hrepa HoHHoA CXeMhI: npHJIeraJO X K rpaHHQaM no-
ID-reoMeTpHs Ao6nacreR.
HeCKOJIbKO 60nee C1IO}l(Ha
a.Aan HH K apxHTeKType MHoronpo eccopHLlX CHCTeM KHHeTHtJe-
CKH COI'naCOBalIHbIX cxeM C KOppeKQHe:A, nOCTpoeHHblX Ha Hepery-
.
Q j+l(k+l)
i .
Ql+ 1 (k)
.
Q1
Q1+1
g6. (U-P)-HTePa.u.HOHHlJIlf MeTO,.l{. MHoronpo eccopHMlt SapH8lfT 213
JJ£pHbIX CeTKaX ( 1, n.3). 9TH cxeMhI TaK.>Ke O opo,l\HbJe, SlBHble H
npOCTbJe C "rO'tlKH 3peHHH JlOrHqOCKOA CTPYKTypbl. 3 ecb CBoACTBO
HBHOCTH npe CTaBJI5IeTCH oco6eHHo Ba.>KHbIM, TaK KaK n03BOJISIeT 83-
6e.>KaTb perneHHg CHCTeM CeTOqHblX ypaBHeHHA npHMblMH HJIH HTe-
p HOHHbIMH MeTO aMH, KOTopbIe B CJlyqae HecTPYKTYPHpOBaHHbJX
CeTOK CHnbHO YCJlO>KHj{IOTCR. O HaKo npH HCn0Jlh30BaHHH npHH H-
na reOMeTpHqCCKOro napaJlJleJIH3Ma 06eCneQeHHe MHHHMH3a.I.1.HH 06-
MeHOB Me)K)ly npOnecCOpaMH He CTOnb TpHBHaJJbHO, KaK )VIS{ ceTOK
perYJISlpHbIX. BOJIee nOAP06Ho np06J1eMa 6Y eT paccMoTpeHa B
9 7 a.HHOa rnaBbI.
AnH qHC'JIeHHOro pemeHHSI nOJlYHeaBHoR CHCTeMbI ypaBHeHHH
MO}KHO npHMeHHTb pyrHe HTepa Ol18ble MeTO bJ - 60nee a<p<peK-
TRBHble (HO H ropa3 O 60nee CJlO}KHbIe) no cpaBHeHHIO c anropHTMoM
(5.6). 9TH )Ke MeTO,QbI npHMeHHMLI H AfIB HaxO>K eHHSI pemeHWI nH-
HeapH30BaHllbiX cxeM, pacCMOTpeHHblX B 3 aHHOa rnaBbl. Cne eT
OTMeTHTb, QTO C03AaHHe 9Q:>cI>eKTHBHblx nap8JIJIeJIbHbJX HTep HOH-
HblX MeTOAOB npe CTaBJlgeTCS1 aJIeKO He TpHBHaJIbHoA aqeR.
6. (a-.B)-HTep HoHHbIA MeTO .
MHoronpo eccopHhlii: BapHaHT
B aTOM naparpaQ>e 6Y eT H3JlO)KeH OpHrHHaJlbHhlA (a-fj)-HTe-
pau.HOHHbIA MeTO [14, 126], BKRlOqasJ ero MHoronpo eCCOpHbJa Ba-
PHaH'l'. MCTO npHMeHHJlC8 aBTOpOM Hero KonneraMH ,LVliI Mo e-
JIHpOBaHH)1 npo eccoB 06hIqH 0PraHHqecKHX TOnnHB, aMHKH
3JIeKTpOHHO-AhlpoQHOR IIJI83MhI DOJIynpoBoAHHKOBLIX npH60poB, ro-
peHH.H rrpHpo HOro ra3a (lIlt 115, 127, 128]. COAep}K3.HHe 3TOro H
nocne,l\YIOU1,ero naparp B Henocpe CTBeHHO He CBft38HO C KOHe-
THQeCKHM nOAxO)J.OM B ra30BO:A )\HHaMHKe. ORO HeCHT AOCTaTO'lHO
06W.HH XapaKTep H HanpasneHo Ha 60nee nOJlHoe HH<I>OpMHpOBaHHe
t.tHTaTeneA 0 B03MO)l(HOCTIDC napaJUleJlhHblX BbI'IHCJIeHHR.
1. (O-!3)-HTepaI{JfOHHblH MeTO).{
PacC tOTpHM 3TOT MeTOA Ha npHMepe nSlTHTO"leT.IHoA cxeMbI, 3a-
,n:aHHo:R B np5lMoYroJIbHo:A 06nacTH:
BinUi,n-l + KinUi-l n - CinUin + Ean U i+l l n + VinUi,n+l +
+ n=O, i=1,...,Ni- 1 , n=l,...,N n -l.
(6.1 )
CHCTeMY (6.1) p-OnOJlHHM rpaHH Hb1MH ycnOBHSlMH
UO,n = <l'l,n U l,R + {i.n ,
Ui.O = i,l Ui,l + i,l ,
UNi,n = «PNi-l,nUNi-l.n + N,-l.n'
(6.2)
Ui,N n = C(Ji)N. -1 Ui.N" -I + {i,N" -1 ·
I1pe OnO)l(HM, no aH8JJOMfH C xn&CCH'IecKHM MeTO,L\OM nporoH-
KH, \{TO peweHHe HCXO Oa CaCTeMLJ (6.1) YAOBJIeTBOpgeT COOTHO-
meHHSlM
Uin = Qi+l.nU i + 1 ,n + {3i+l.n ,
-
Uin = ai,n+l Ui,n+l + {3"n+l ,
Uin = 'Yi-l.n U i-l,n + -l,n ,
-
Uin = .:yi,n-1Ui,n-l + .n-l t
(6.3)
(6.4)
-., -
I'Ae a,"Y,a,;Y,{3,d,j3,d - HeH3SeCTHble noxa KCY3<1><I>H HeHTIW.
,L{nsr HaxO.>K eHIDI CHCTeMhI YpaBHeHHA, KOTOpoA YAOBJJeTBOPSl-
lOT nporoHOq}{hle Ko3<f>cI>H eHTbJ, npoAenaeM cneAYIO ee. C nOMO-
IO YCJIOBHA (6.4) nepenHWeM HCXO Hoe ypaBHeHHe (6.1) B BHAe
KUi-l.n - (C- a in B --1in V) Uin + EUi+l.n + F +
- -
+ Pin B + c4n V = 0 · (6.5)
3,D.eCb H anee onYCKaeM H eKCbl t, n npH K09<1><1>H eHTaX. cxeMbI
(6.1). AHaJlOraqHO t c nOMO 1O YCJlOBHA: (6.3), nOJIY'lHM:
BUi,n-l - (C-Qin. K -'Yin E ) Uin + VUi.n+l + F +
+ PinK + c4n E = O. (6.6)
CXeMaTH'IeCKH nepexo
K ypaBHeHHHM (6.5) PI (6.6)
H306pa}KeH Ha pHC. 49. 9TH
ypaBHeHH5I HBJIglOTCH TpexTO-
qe'fHblMH ypaBHeHHSlMH OTHo-
CHTeJlbl-lO 3HaqeHHH CeTO q}fO It
<PYHKIJ.HH {Ui-1,n, Uin, Ui+l,n} P K
HC. 49. BblBo.z:u' ypaaueHBA AllJI ICO-
H {U i ,n-l, Uin, Ui,n+l}' ,!VIs RItHeHTOB nporOHKH
HaXO)K eHHg U MO)KHO HCfiOJlb-
30BaTb <l>opMynLl o-nnOMepHoA nporoHKH, KOTOpylO npoBe eM B
'leTbJpex HanpaeneHH.HX B CTOpony YBeJJH1feHHB H YMeHhweHHB
HIIAeKCOB i Ii n. npH 9TOM nOJlY1faeTCSI CHCTeMa, COCTOsnu:8.SI BNecTO
O HOro ypa.BHeHHH (6.1) H3 BOCbMH ypaBReHH1t ARK onpe eJJeHHSI
- -
nporOHO'ffihIX Ko9<l><I>HQHeHTOB a, 1,0, 1, fJ, d, /3, d:
V
K C E
B
v
-
K C E
B
0i+l.ft = Elwin, \II = C - OinK - Oin B -7in V,
al,n=CPl,n, i=1,...t N i- 1 t n=l,...,N n -l;
/'i-l,n = K/'llln , \IIl n = C - 'Yin E - Qin B -7in V t
'YN;-l,n = <PNi-iln, i = Ni- 1 )..., 1, n = 1,.. · t N n -l;
(6.7)
- -
a i,n+l = V/wfn, \II?n = C - ii;n B - oinK - 'Yin E J
aill = c,oi, 1 , i = 1,. · · , N i -1 , n = 1, · · · , N n -1 ;
-" -
'Il in = C -/in V - G'in K - 'Yin E ,
i= 1,...,Ni-1, n=Nn-l,...,l;
1i,n-l = B /q,ln ,
.-..
'Yi,Nn. -1 = <Pi,N" -I ,
- - 7
t4-1 t R = (dinE + f3in B + din V + F)/\I!in ,
dNi-1Jn= Ni-l.nt i=Ni-l,...,l, n=l,...,Nn-l;
Pi,n+l = (Pin B + PinK + nE + F)/ :',
-
Pi 11 = i) 1 , i = 1,.. · , Ni -1, n = 1, · · · , N n - 1 ;
(6.8)
di,n-l = ( n V + l3in K + nE + F)/\J!in'
-
,Nn-l = i,Nn-l, i= 1,...,Ni- 1 , n=Nn--l,...,l.
Onpe eJlHB napy Ha60pOB nporOHO\{UbIX K03<PcI> HeHToB
- -
(o:,P), (I,d), (a,fj) J1H6o (;Y)d) H3 pemeHHSI CHCTeMbI (6.7)-(6.8),
nerKO HaATH pemeH e HCXO Horo ypasHeHHH (6.1). HanpHMep, 3HaH
K03cJ><PH HeHTbI 5, {3, nO CTaBHM HX Bhlpa>KeHHe (6.4) B COOTBeT-
CTBYIOIU.ee rpaHHqHOe YCJIOBHe (6.2) H DOJIY'IHM:
..-
Vin = Oi,n+1Ui,n+l + {3i.n+l, n = Nn-l,... ,0,
....,
Ui l N n = (C{Ji J Nn-ll3i I N" + {i,Nn-l)/(l - C{)i.Nn-la'.N,,) , (6.9)
i = 1, . . . , Ni -1.
IIoCJle BblnOJlHeHHH npOQe;urpbI (6.9) 3Ha-qeHH cJ>}.HKLtHH U "PH
i = 0 H i = Ni nerKO HakrH H3 rpaIIH'IHbJX YCJlOBH.A (6.2).
HeJIUlleHHaJI CHCTeMa o-ypaBHeHHA (6.7) aBTOHOMHa. Ee peIlle-
- -
HHe He 3aBHCHT OT 3Ha eHHa K03<P<1>HtI)leHTOB {3, d, {3, d, B CBOIO Oqe-
pe)J.b Y OBneTBOpSlIO HX (6.8). ,lJ;JH HaxO>KAeHH5I K03c1>cI>H HeHToB
a, 1', a,;:Y H3 CHCTeMbl (6.7) BOCDOJIb3yeMcH, HanpBMept CJle IOIU.HM
HTepa HOHHbJ M npo eccOM:
a i = E / (C - a +l) K - a +1) B - ;Y!:) V) ,
"" (8+1) _ _ 1 AT I -
i = 1,. . · , "i-I, a l,n - l.n , n - ,..., J" n - t
1! 8t. = K / (C -1 +1) E - a +l) B -;yi:)V),
N (s+l) - T
1 == i-I,..., 1, 'YNi-l,n -I.(JN,-l,n 1 n = 1,..., 1Yn-1;
a +;l = V / (C - a +l) B - a :+l) K -1'! +1) E),
-(-,+1)
n = 1,. . · 1 N n - 1 , i t 1 = c,oi,l t i = 1, · . · , Ni -1 ;
- $+l) = B/ (c _ - J+l) V _ &+l) K _ s+l) E)
'Y I ,n-l 'r ln a ln 1.n ,
AT -(8+1) - . · - l AT. 1
n = J Y n -1, · · . , 1, 'Y i, N n -1 - <p....N fa -1 , ,- t..., 1 Y I - ·
(6.10)
HaA SI KoscP<I>HlI)feHTbJ a, "Y, ii, 1. rIpHCTynHM K onpe eneHH1O
- .....
Kos<p<I>HlUteHTOB {3, d, (3, d, pewaH CHCTeMY (6.8) npH nOM01IlwH HTe-
pa,wtit
l3 k+l) = (/3 k+l) K + R k+l) B + k)V + F )/ \II
1+ 1 , n In In In In I
i = l,....Ni-1, ,e 1) = l.n, n= 1,...,N n -l;
Jk+l) = ( Jk+l) E + k+l) B + d1 k )V + F )/ \I17
I-l.n In fJzn in In ,
i = Ni- I ,. · . ,I, J; ,+I2, n = N,-I.n, n = 1,..., Nn-lj
(k+l) = (fJ - k+l) B + tl k+l) K + d t+l) E + F )/
Pttn+ 1 In fJ n In In ,
n=l....,Nn-l, +l)= i.I' i=1,...,Ni-1j
k+l) = ( k+l)V + l3 k+l) K + d k+l) E + F )/ \)7
1,n-l In In In tn'
k+l) c ·
n = N n -1, · · · ,1) i,Nn-l = iJNn-l , I = 1, · · · ,Nj-l.
(6.11 )
HTepaI.\HOtIHbIA rrpOl\ecc (6.10)-(6.11) $IBJIHeTCSI HaH60nee npo-
CThlM IeTO OM BbIQHCJ1eHHSI npOrOHOQHbIX K03<pcpH HeHTOB. ,lJpYMfe
B03MO}KHhJe BapHaHTbI (a-{3)-HTepa OHHoro MeTO a, o6n aJO e
60nbweA CKOpOCTblO CXO MOCTH, a TaJ()I(e 0606 eHHe 8JlropHTMOB
Ha CJIY'IaH 3D-reO leTpHH paccMoTpeHbI B MOHorpa$HH (14J.
TeoPeTHqecKoe onpe eJIeHHe CKOpoCTH CXO,lI)fMCCTH (a-f3)-HTe-
panHOHHOro "'feTO a 3aTpY HeHO BBHAY ero He.rIHHeAlloro xapaxTepa.
O eHKH 3 eCb npHXO HTaI )1.enaTb, ormpaSCb Ha 3kcnepHMeHTaJlb-
Hble aHHble. B C8010 O'lepe b, pac'leThi nOK83bIBaJOr, qTO CKOpoCTb
CXO,D.HMOCTH a-npo ecca (6.10) HCKJllO'IUTeJIbHO B ICOKa. OTHOCH-
TeJlbHaH norpelllHOCTb y6bIBaer 60nee 'IeM B 10 paa OT HTep H K
HTep HH, BHe 3aBHCHMOCTH OT q}lCJIa y:3JIOB. CKOPOCTb CXO HMOCTH
,8-HTepaQH (6.11) CYlQecTBeHHO HH>Ke. HMeHHo OHK H onpe)J.enSlJOT
B KOHet.tlIOM HTore CKOpOCTb Bcero (Q-p)-npou.ecca.
OnbIT paCqeTOB 3 aq HHaMHKH H3nY'lalO ero ra3a, nony-
npoBo .IHKOBOH 3JleKTpoHHo-AblpO'tJHOR nJl83MbI, 3atJ.aQ HOHOC<Pep-
HoR <PW3HKH (14), [129)-[131} nOKa3bIBaeT, qTO (a-p)-MeTO)J. SlBJlSI-
eTC Ha,L\e>I(HbIM HHCTPYMeHTOM peweHHSI CHCTeM CeTOqHbIX ypaa-
HeHHR. K AOCTOHHCTOY MeTO a C1Ie yeT OTHeCTH ero "BceH)J.HOCTb"
(robustness), n03BOJISIIOllQ'1O - nYCTb H He OqeHb 6bICTpo - pewaTb
pa3Hoo6pa3HbIe 3 aqHt 8 KOTOpblX BCTpeqaeTCSI CHJIbHaH BapHa.QHH
K03<t>4>H eHTOB pa3HOCTHOR cxeMbI.
2. Mlloronpo eccopHbIil BapHaHT (a-/3)-MeTO a
llpH perneHHH KpaeBblX 3 aq H8 MHorOrIpOQeccopHblX CHCTe-
Max 06hlqHO Hcnonh3yeTCSI cneAYIO$a nOAxo . KaJK;:\OMY npo ec-
copy OTBOAHTCH CBO.H no 06J1acTb, B KOTOpoA <paKTHtfOCKH CTaBHTCS1
OTAeJlbHast KpaeSaJI 3 a a. PacCMOTPHM, KaK npocreawHA: CJlyqaA,
p836HeHue HCXOAHOA 06nacTH Ha ,lJ.se no o6JIacTH I H II. B03MO>KHbI
Ba cnoco6a p836HeHHSI. nepBhIJt - 6m nepeKpblTHH (pac. 50,a), KO-
r,Aa 06W.HMH ;J.JISI no 06J]acTeA BmIlOTCg nHWb TOqKH BHYTpeHHeA
rpaHHLI-bl. llpH BTOpOM cnoc06e (overlapping) o6naCTH nepeKpblBa-
tOTeR I HMeR 061IJj'1O qacTb: Ha pHC. 50,6 OHa 3anJTpHXOBaHR.
B CJ1Y Iae nepeKpblBalO XCH 06J1acTe:A pacqeT Ha IHHaeTCH c
nOCTaHOBKH HCKYCCTBeHHblX rp8.HHIJHblX YCJlOBHft Ha rp8.HH ax no-
06nacTeH ab H cd (puc. 50,6). C STHMH rp8.HH'IHbIMH ycnOBHBMH
npOBOAHTCSI pacqeT CeTO'-lHoA <l>YHK HH B nO 06nacTgX I H II. no
OKOH IaHHH p-aHlloR npo eAYpbl Ha'lHHaeTCH CJIetU'1011I)Ia 3TaU: pac-
56. (0- {3 )-wrep8.l.(HOHHbIR MeTO,Lt. MHoronpOl.{eccopHblJf BapHaHT 219
.
a
CD <4 , U... CID
'"
y,d
a)
b
b
d
PRC. 50. a cnoco6a pa36HeHHB 06nacm: a) 6e3 nepeKpLlTH8 nO,D.ooJlacTeA;
6) c nepeKpblTHeM nOJJ:06nacTcR
qeT B 06J1acTH I C HOShIMH rpaHHqHhlMH 3HaQeHWlMH Ha nHHHH 'cd,
BbJqHCJIeHlfblMH nSI o6nacTH II, H 8H8JIOrHQHO - B 06nacTH II co
3HaqeHHHMH ria JlHHHH ab. fIpo ecc HTep a Me)l( nOA06J1acTH-
MH nOUTOpSle1 c.H 11.0 CXO HMOCTH.
O eBH HO, '{TO "{eM 60J1bIlle nepeKpblTHe no 06nacTeA, TeM 6b1-
CTpee CXO 51TCH HTepa HH Me>K)Jy HHMM. OAHaKo 'tJeM 60nbme 06-
HX TQQeK, TeM HH)I<e 3<1><1>eKTHBHocTb napanJIeJIH3Ma (npo eccopbl
AY6JIHPYlOT pa60TY ,LJ.pyr pyra). KpoMe TOI'O, 3el><PeKTHBHOCTb Bbl-
qHCJIeHHH n aeT B pe3YJIhTaTe TOro, qT() YBeJlHQeHHe 'I}lCJJa TOQeK
B Ka>K)(oA nO 06JIacTH IIpHBO HT, KaK npaBHJIO, K 3&Me)J.JIeHHIO cxo-
HMOCTH .HTepa HOHHoro npon:ecca B ee npe eJlax.
BapHaHT p836HeHHjI 003 nepeKpblTHH (pRC. 50,a) cB06o.n;eH OT
YKa3aHHbIX HeAOCTaTKOB. O,ll.HaKO B 9TOM CJIyqae OCTaeTC1l HenOIDIT-
HbIM MeXaHH3M OpMHpOBaHHH rp3.HH'IHOrO ycnOBHSI H& JIHHHH ab
B npo ecce HTepao.H:A Me)l( Y no 06naCTSlMH.
(a-fj)-M rO,lJ. ,n;aeT npoCTYIO B03MO>KHOCTh nony eHH5I ycnOBHl!
Ha BHYTpeHHeR rpaHH e ab B npo ecce HTepaQHA Me nOAOMa-
CTSlMH. nepeKpblTHe He Tpe6YeTCH, B MeHSleTcg cau npH n napan-
JleJ1R3 MH anrOpHTMa. CYT!:. B TOM qTO B xo e HTep A BHyrpH
Ka.>K Oa nO 06J1aCTH BblqJfcnjJlOTC5I 3HaQeHHSI He caNol 4>ym<QHH
Up a npOrOHO"lHblX K03ct><PH eHTOB. IIocne OKOHQaHHB pactJe'J'a, B
06JIaCTH I Ha JIHHHH ab (pHC. 50,a) DOJIYtlaIOTCSI HOBble 3Ha'leHHH
Koa<p<pHIJ.HeH'fOB a H {J, a B 06JI8CTH II Ha 9TOR rpaHH e onpe)J.enH-
IOTe$! Ko3<P<PHlI.HeHTbl ,.., H d. Haa eHHble 3HaqeHH5I Q H {1 CJIY>KaT
HOBblMH rpaHHqHblMH ycnOBHSlMH A1IS1 06nacTH II, a 3HaqeHHH KO-
220 r.l1. IV. t{HCJlellHble anropHTMbl If MHoronpo eccopHble CHCTeMLl
3<I><t>H eHTOB '1 H d - rpa.HHqHbJMH YCJlOBIDIMH NUl 06JIaCTH I. no-
CTpOeHHbJA TaKHM 06pa30M anropHTM HaXO)f( eHHjI 8HYTpeHliHX rpa-
HHQHblX YCJIOBHA CJ1Y>KHT OCHOBOA nap8JIJleJlbHOrO BapHaHTa (Q- fJ)-
HTep OHHOro npOl.\ecca.
Kax nOK838.JIH npOBe,.qeHHble pac'tJeTbI, a<l><l>eKTRBHoCTb napa.n-
JIeJlH3Ma ,LVISI (a-,B)-MeToAa cOCTasnaeT 60nee 50% 8 cHcTeMax c
'iHCJlOM npo eccopOB < 40, leaK NljI 2D, Tal( H AJIB 3D-3 a'l.
30
25
- Tl!opMA
-6- Ce1Q481-IO
Ce1u481J120
--- CI"TU 48111:30
5
.. 20
s
%
1 15
u
10
0-
o
5
10
1
20
25
30
4 np0LtlCcopOll
P"c.51. 3aBHCHMOCTb eKTHBnOCTH napanJ1e.nIt3 OT 1:fHcna npo eccopoB
B KaqecTBe npHMepa DpHBe eM ,lI;aHHhle pac'feI'OB no Ha.xO e-
HHlO peweHHSI 3 a'qHJ Hcnonb3ylO eA pa3HOCTHYIO annpOKCIDAaQHIO
onepaTOpa JIannaca (pHC. 51). PaccMaTpHsanacb 06JJacTh, BblTSHY-
TaJI B,lJ.Mb OCM X H C 60nblllHM t{HCJlOM }'3nOB B STOM HanpasneHHH.
TaKas KOH<pHrypa HB OTHOCHnaCb K pac'tJery MeKTpOHIJO-Ab1pO HOA
D.na3MbI TpaH3HCTopa. ,lI.n1! pactJeT'd. HCnOJIb30Banacb JJHHeAHaH KOH-
<pHrYP IDI CHCTeMbl npo eccopOB (pHC. 44).
06paTHM BHHMaHHe Ha BnOJIHe ecrecTBeHIJoe YMeHbmeHHe 3cj>-
<t>eKTHBHocTH napanneJIH3 C poCTOM QHCJla npo eccopOB. O -
HaKO CJIe,n.YeT 3aMeTHTb, qrro 3<t><PeKTHBHoCTb napaJIJIeJlbHOrO pac-
1feTa B03pacTaeT C YBeJlH eHHeM o&beMa BblqHCneHHA:] npOH3BO H-
MbIX Ka>K)J.bltvf npo eccopoM" 9TOT <PaKT xopowo YKna,nbIBaeTCg B
06I.llJ'1O KOHQen HJO 0 TOM, liTO BbiCOKOnpOH3BO)J.liTenbHbJe MHor
npo ecCOpHl:de CHCTeMbl OJI}KHhI - B OCHOBHOM - BCn0J1h30aaTbCSI
96. (0- fJ )-HTepa HOHHb1R MeTO,l{. MHoronpo eCCOpHblR BapllaHT 221
nH o6pa60TKH o6nrnpHblx MacCHBOB HH opMa H.
3. IIpHMepbl paClleTOB
B KaqecTBe COAep>KaTMbHblX npHMepoB HCn0JIb30B8.HHH MHO-
ronpo eccopHoro BapHaHTa (a-,B)-aJIrOpHTMa pacCMOTpHM HeKO-
TOpble pe3YJlbTaTbl MO eJ1HpOBaHHH npoQecca H3BJ1eqeHHSI He<J>TH
C nOMOInblO 3aKB.'fKH BO,D.bI B nnaCT, a 3&TeM MetI)JeHHoro Teqe-
HHSI MeTaHo-B03 IlIHOR CMeCH B IIJIOCKOM KaHane c nO)l.OrpeBOM
[ 115 , 127], [132] -[ 133] .
IIepsaH 3 aqa onHCbIBaeTCSI CHCTeMoA YPaBHeHH:R BaKJIeB-
JIeBepeTTa:
m a: + div(F(S) U) = Q == qF(S) I
q = divU, U = -Kgradp.
(6.12)
(6.13)
3 eCb S -- OTHOCHTenbHoe cO)J.ep>KaHHe BO)J.hl B )KHAKoR <l>aae) U
- CKopoc'rb Te'ieHHH Byx<l>a3HOa >KHAKOCTH, p - ,n;aOJleHHe, Q -
HCTO IIHKH H CTOKH >KH KOCTH;
K _ _ kwot(S) + koil(S) ,
(6.14)
J..'wat lJoil
JJwat,l-toll - Ko <I>H HeHThi BB3KOCTH BO bl H He<P'rH; F(S) - HeJle-
HeAHa5I <l>YHKQHj1 B8.KJleSI-JIesepeTra:
F ( S ) = kwo.,(S)/IJ'Wo.t . ( 6 15 )
kwat ( S) I fLwat + koil (S) / JJoil ·
CHc'reMa (6.12)-(6.13) CBeJlacL K O HOMY ypaBHeHHJO napa60-
J1HQeCKOrO THna, p83HOCTHaH anrrpoKCHMaQ.HH KOTOporo pewanacL
HTepa.u.HoHHbIM (a-{3)-MeToAoM.
Ha pHC. 52 npHBe eHa O Ha H3 'l'HJJH'I HhlX cxeM OpraJm3 HH
He<l:rre o6blqH C IIOCTOSlHHhlM pacnOJIO>KeHHeM BO.ll:03a.r<a'lHB8JO HX
CKBaJKHH. PacqeTHaSl 06J1acTb (Ha pHC. 52 3awrrpHxoBaHa) 06.rI aeT
cHMMeTpHeH. Ha pHC. 53 nOKa3aHa XapaKTepHaJI KapTHHa Ko eH-
Tp H He<t>TH B nJlacTe Ha pa3BHTOI CTa.JJ)fH ,q06b1'iH. B pR e BapH-
aHTOB paCqeTa 3aMeTHO npOSlBJlHJICSI HeraTHBHblA 3c1><PeKT 06pa.aoBa-
HHSI "80 HblX llaJIbu.eB", Kor a OKOJIO He<l>'re o6bJB8JOllI,HX CKBaJKHH
222 rJl. IV. T.[HCJleHHLle a.nropJlTAlbl H MHoronpOl.{eccopHble CJICTeMbl
o
6
eo. .!
o
"
..
.
'1-..
..... .. t
o
6
0'--'-":'- - 0
0- -
o
PHC. 52. CeMRTO eqHag cxeNa OPraHH-
38D.HH He4>Te 06b1'1H: 6 - BO.n03BK&-
"HBaJO He CKBB)I(HHhI, 0 - H o-
6h1B8.IOIr(He CKB8.>KHHbJ
II G.1
_ 0.2
....: II () 1
II . .
"R ; . . I
. .: II 0 4
.,. .... .. ..,.
.. II 0.,5
11 0.6
D (J. 7
. ...: 0 0.8
.. '.
... . '.
PHC. 53. KOHI elrrpau.Rsr He<I>TH 8
nnacre: B&lMH O'M'eHKaNH
3H8 eHO ,noneBOe COAep>KaUHe BO bI
B )l(H,D;KoA cPa3e
KOHL1.eHTpa1I-H5I HecpTH OKa3b1BaeTCSI HH.>Ke) qeM B COCe)!.HHX o6.rIaCTSIX.
9To BL13BaHO TeM, o y BO,.D.I>I BSI3KOCTb cym.ecrBeHHO MelIbIlle, qeM
y He<t>TH, 3a Cq€T qero Bo a MO>KeT 6blCTpee ,l\OCTHraTb OTKatJHBaJO-
W)lX CKBaJKHH.
BTopaH 3 aqa HenocpeACTBeHHO CBH3aHa c npOH3BO CTBOM
tCCHHTe3-ra3a", KOTOpbJA 06p83YeTCB B npoQecce Ka:raJlHTINeCKOrO PO-
peflH MeT8Jia CH 4 H CXCMaTHqeCKH K306Pa>KaeTCH cI>oPMYJloA
1
CH4 + 2 02 -) 2H2 + CO.
CxeMa npoD-ecca H306p8.}Kena Ha pHC. 54. raJOBasi CMOCb, co-
Pt TK=1200K
Ar,CH 4 ,02 -+: CO, H 2
. .
Pt TK=1200K
PHC. 54. Cxewa X8.Ta.1JHnrqecXOro peaICTopa: Pt - KaTaJlHTB'ICCKaB nnaTRtlOBa1l
rp8.HHQ8, H&rpeT8.illJ,O TeMnepaTyphl TK = 1200 K
96. (0-{3 )-Hrep&UHoHIlbIJI AleTO,l(. MHoronpo eccopHhlR BSpJlaHT 223
CTORllJ.aH H3 MeTana, aprOHa H B03,lQ'xa, HarpeTaJI o TeMnepaTYphI
800 K, BXO)J)IT C He6OJ1bwoA CKOpOCTblO ( 1 M/C) B y3KHA KaHan.
B6JIH3H HarpeTblX ( o 1300 K) CTeHOK KaH8Jla B npHcYTCTBHH DJJa-
THHOBOro KaTaJIH3aTOpa npOHCXO HT peax g 06paaoB8.HHB CHHTe3-
ra3a.
Ha pHC. 55 nOKa3aH <ppar-
MeHT nOJlSl TeMnepaTYp. ,aaJIHhIe
fiOJlY'feHbJ B q]iCTO ra.3OABHa-
MHQecKOA: nOCTaHOBKe, 6e3 Y'fe-
T8. XHMHQeCKHX rrpeBpa eHHIt,.
KaK c nOMO blO ROJIYHeSlBHoA
CXeMbJ (92 ,n;aIIHoR r.naBbI)) TaK
II C fIOM01J..I.btO paaHOBH):\HOCTH
KB33Hfa30,L\HHaMH"IeCKOR CHCTe- PHC, 55. none TCMnepaTYP B KaHane
MbI (7.23) (7.25) (rn. III). npu-
MeHeHHe 060HX MeTO OB 8JlO npaKTH'IecKH COBna,naJOIIJ.He pe3YJlbTa-
TbJ. ,lLrI.H HaJCO)l(tl-eHHa peweHHB CHCTeM CeTOqtlblX YPaBHeHBI, B03-
JIHKaIOI.U.HX npH HeHBHo:A: allnpOKCHMaJJ.HH, HCnOnL30BaJICH (a- (3)-
HTepa[J.HoHHblA aJlrOpHTM.
Q'rMCTHM HeCHMMeTpHJO nOJIH TeMnepaTYP Ha pHC. 55. Ee 803-
HHKHOBeHHe CBSl3aHO C yqeTOM CHnbJ TSI:>KeCTH, KOTOpaJI "PH He6onb-
WHX CKOpOCTSIX Te'leHHR MOJKeT OK33bIBaTb 3aMeTHOe BJIHSlUHe Ha
06Il(YJO KapTHHY npol\ecca.
B 3aKJ11OtJeHHe naparpa4>a CKa>Keu, qTO (a-fj)-anropHTMoM
He HC'tJepnblBaeTCB Ha60p napaJIJIenhHblX MeTO OB peweHHH CH-
CTeM CeTO"iHhlX ypaBHeHHii, HCnOJlb3yeMhlX aBTOpOM H ero KOJIJlera-
MM. PaccMaTpHB8JIHCb aJlropHTMbJ. CBSI3aHHbI€ C p83JIH'iHblMU na-
panneJ1bH:bJMH sepCHflMH MeTo a HenonHoro pa3JlO}KeHHSI XOJIe -
KOro [134, 135] H nonepeMeHHo-TpeyrOJIhHoro MeTO a CaMapcKoro
(136) 137]. 3aMeTHM e e pa3, QTO HaJIHqHe 118.A e >KHbIX H eKTHB-
HblX napannenbHblX a.nropHTMoB peweHHH CHCTeM CeTOqHblX ypaB-
HeHHIit HBJlSleTCH BaJKtlhIM MeMeHTOM MHoronpou:eccopHbIX BhI'IHC-
neHHH. t.{HTaTeJI.HM, HHTepecYIO HMCSI HaqaJlbHblMH CBej!.eHHHMH no
aHHOMY Bonpocy, MO>KHO peKoMeH OBaTb MOHorpa<pHIO [138].
· MO,neJIHpOBaHHe npo eCCOB B MHOroSOMDOHeHTHblX XHYlAecKH peantPYIO-
UI,IIX raJax Ha OCHOse KrY-CHCTeMbl 6y,neT 60Jlee IIoJq)06HO paccUOTpeRO B cne-
YlOlUeA rJ1aBe.
7. HeKoTopble ,L\OnOJlHHTeJJbHhle BonpoChI
nap8.JIJleJIbHhJX BhIqHCJIeHH:I
8 3TOM naparpa4>e 6Y T KpaTKO paccMoTpeHbI cn Hct>HqecKHe
Bonpocbl, CBSl3aJIHhIe C BH3yanH3aJJ:HeA ,o;aHHbIX BbJCOKf)npOH3Bo H-
TenbHblX MHoronpo eccopHblX BbIqHCJIeHHA, pa HOHanLI-lbiM p836a-
eHHeM Ha no,!(o6nacTH lIP" HCnOJlb30BaHHH HeCTpYKTypupoBaHHbix
CeTOK, a TaK)Ke AI-iHaMUqecKoA nepe3arpY3KoA npo eccopOB. B npaK-
THKe peweHHH 3CV1-aQ ra30BoR HH3.MHKH Ha MHoronpo eccopHblX CH-
cTeMax nepeqHCJIeHHble np06neMbJ OKa3aJIHCL BeCbMa axrYaJIbHbIMH.
IIpR HX pa3pellleHHH aKTHBHO HCfiOJlb30B8JlaCb HH<j)opMa HjI 0 xa-
paKTepe r(30)J;HHaMHQeCKoro reqeHHSI H 6b1JI nepepa6OTa.H onbIT BbJ-
'IHCJIHTeJIhHOA MaTeMaTHKH (73, 119).
1. BH3YaJIH3aIl.HH AaHHblX MHoronpo eccopHbIX
Bbl'lHCJlelIU#
ITpo6neMa BH3YaJIH3 H ,n;aHHblX MHoronpOQeccopHblX BblttHC-
neHHH B nepBYIO o lepe,ltb CBS13aHa C orpoUHblM o&beMoM HHcpop-
M HH, nonyqaeMOH 8 pe3ynLTa:re 3THX pacqeTOB. Ha OCHODe B
KOHCqHOM CqeTe c-rpoHTCSI BH3Y8JIbHaJI KapTHHa TeqeHHS I'a3a. 3aqa-
CTYIO 3apaHee lleH30eCTHbJ KaK MOMeHTbI BpeMeHH, Tal( H JIOAo6JIa-
CTH npOCTpaHCTBa, NlH KOTOphIX >KMaTenbHO nOJlyqHTb BH3yanH-
3a H}O r830 HHaMHlJecKHx noneA. B 3TOM CJIyttae Bbl60p aHHblX
)VISi nOCTpoeHHii rpacPWleCKoro H306p8JKeHHB MO>KeT nOTpe60BaTb
HeonpaB,L\aHI-IO 60JlbUIHX 3aTpaT BpeMeHH.
BbIXO,nOM 3 eCb npeACTaBJIfleTCH HCnOJlb30BaJlHe TeXHMorHH
"KnHelfT-cepBepn [139, 140). B paMKaX 3ToA Mo,n:enH, cepBep, 3any-
eHHblA Ha MHOrOrIpo eccopHoii CHCTeMe, npoH3Bo HT 06pa6OTKY H
Bbl60PKY aHHbIX 1-13 cBepx6onbworo o&beM& HHcI>OpMa.u.H , a KJIHeHT
nOAAep>KHBaeT ,QHaJIOr C fiOJlb30BaTeJleM H Bblno.nHfteT OKOII'Iarrenb-
Hoe <POpMHpOBaHHe H BblBOA H306paJKeHHSI Ha 3KPaH. nOCJIe,ll,HHM
qacro cnY>KHT HcnJleA 06YttHOro nepCOH8JlbHOrO KOMnbIOTepa. OT-
MeTHM) 'iTO CTPYKTYPH3aQHB AaHHblX, 06ner aro ajl MHDronpo ec-
COPHYIO Bbl60PKYJ MO}Ke'!' 6h1TL ocym:eCTBnelIa c nOMOllJ,blC npHH na
reOMeTpHqeCKoro napaJIJIeJIH3Ma Ha OCBOBe pacqeTHbIX no,[{o6.naCTeH
g7. j{OfIOJlHHTeJlbHble BOnpocbl naplJJlnenhHblX BW'lJfCJleHHR
225
oro npoueccopa.
Ba>KHbIM npeHMy ecTBOM 3TOH TeXHOJIOrHH OCTaeTCH npeeM-
CTBeHIIOCTb npOrpaMMHoro o6ec1Ie1.JeHHH IIpH nepexO,L\e K MHoronpo-
eccopHolt CHCTeMe HOBoro Tuna. IIpa 3TOM BCSI rpa<PHqOCKaSl qaCTb
OCTaC'rc.R Y KJlHeHTa H 6y eT 6e3 H3MeHeHHA B3aHMOAeHCTBOBaTb C
HOBhIM cepBepOM, pa60TaJOIU.HM Ha HOBOA CHCTeMe.
IIoMHMO 06W.HX npo6neM, cy e-
CTBYJO'l' oco6eIIHoc r rH BH3YaJIH3aI HH
aJillbIX 60.nbUlHX ra30AHHa)'tH eCKHX
BblqHCJICHHR, HanpHMep, nOCTpoeHHe
TpaeK'ropHR aCTH . )];BIDKeHHe qa-
CTHlJ.bI, HaqaBWHCb B nOAo6J1aCTH I
(pHC. 56), KOTopa paCCqHTbJBaeTCH CQ-
OTBeTCTBYlOlWfM npo eccopOM, tO>KeT
nepcMecTMTbcSI B no 06JIacTH II HIlI.
BH3YaJIH3a.Il)-1j.] 'raKHJ( TpaeKTopHA Tpe-
6yeJ.' nO,l\XO,ll.S1rn.ero o6ecneQelJHH, Y'IH-
Tb)BalO ero nepexo,LI. qaCTHqbI H3 pa6o-
qero nOJIH oAHoro npo eccopa B nMe
APyroro.
HCnOJIb30BaHHe BbICOKOnpoH3BO,l\H-
TenbHblX CHCTeM npHBoAHT K TOR CH-
T)t HH, Kor,n.a qHcnO TOlieK pacqeTHoA ceTKH MO>K€T 3Ha HTeJlbHO
npeobllllaTb qI-lCJ10 rpaQ:>HttecKHx TOtfeK B OKHe BH3yanH38J.J.HH. B pe-
3}t",1h'TaTe cyw.ec'rBeHHasr acTb TOHKHX ,D:eTaJleR TeqeHHH. p onH-
CaHHH KOTOpbJX H npeNIPHHHMaJIHCb MHoronpo eccopHble BbitUlC-
..TIeHHH, YCKOJ1b3aeT OT BHHMaHHSI HCCJIe,.a;OBaTE'.J1H. KOMneHcHpoBaTb
3TOT Hera rHBHbltt 3CP<j>eKT MO)KHO C nOMOIl\blO BBe,L\eHHH Cne HaJ1b-
HblX HH HKaTOpOB, YK83bJBaJOIIJ.HX Ha I10jlBneHHe HUTepecHblx oco-
6eHItOCTeA perneHIDI. Bhl60p TaKoro H KaTOpa 3aBHCHT OT xaPaK-
Tepa H3Y'-laeMoro <l>H3HtlecKoro npo ecca. HanpHMep, )VIS! BH3YaJIH-
3aUHH Me.1lKOH BKxpeBoA CTpYKTypbl B xaqeCTBe HH HKaTopa MO)KHO
HCnOJIb30Ba 1 rb 3Ha1:IeHHe CPYHK HH rot u.
I
ii
-
) -o-
'0 J
'OA
N
ill
PHC. 56. KapTHH8 HeCTa-
OHa.pHOro Te'leHHSI ra3a:
o - TpaeKTopHB qacTIJ ;
I, D. III, IV no..no6nacTH,
npHJIa.u.ne>Ka e OT enbHhlM
rIpOQecoopaM
2. PaI:(HOHaJIbHOe pa36HeHHe n8 nOAo6JlacTH
KaK Y)Ke OTMeqaJIOCb ( 5, n.3 aHHo:A rJIaBbJ), "PH HCDOJIb30Ba-
HHH HeCTpYKTypHpOaaHHhlX CeTOK paa6HeHHe Ha no 06J1aCTH npe,lJ.-
CTa8J1 eTCH Henpoc'ro.R 3 aqea. 9'1'0 CBSl3aHO) npe)K e Bcero, co cno-
co6otvr HYMepa HH y3J10B HeCTPYKTYPHpoaaHHoR CeTKIf. B perYJIHp-
HbIX CeTKaX )/3JIbI HMelOT CTOJIbKO HH):\eKCOB, KaKoaa pa.3MepHOCTb
npoCTpaHCTBa C Ba HJ1H TpH), TOr a KaK ,lVI IIeC1"pYKTypHpoBaH-
HbJX ceTOK B nlO6oM CJI Iae BBO HTCSI OAHOHH eKCHaJI HYMepa Hs[
yanOB. KaK npaBHJIO, HOMepa Y3J10B CB.H3aHbJ C anropHTMOM) C no-
MOW-bID KOTOporo 6bIna creHepHpOBaHa ceTKa.
PacCMOTpHM TpHBHaJIhHOe pa36HeHHe Ha nOAo6"lacTH, KOr;ll.a,
HanpHMep, TOqKH C HOMepaMH OT 1 11-0 1000 npHH(lJJ.ne)f(aT pa6oQe-
MY nonlO npoQcccopa I, C 1001 no 2000 - lIpo eccopy II H T. . 3TO
npHBO HT K pa3 eny Ha nO 06JIacTH, KOTopble HMelOT BeCbMa npOTH-
>KeHHbJe rpaHHQbI t Tal< '1TO 3HaQHTenbflasr AOJISI Y3JIOB OK83blBaJOTCH
npHrpaHHQHblMH. npHMepoM SlBJlSleTCH pactfeT 3 at.jH 06 06TeKa-
HHH 3JJnHnCa (pHC. 57). CeTKa COCTOHT H3 11 537 TpeYroJIbHHKOB,
Ha KOTOpblX annpoKcHMHPYIOTCSI ra30 Ha.MH'{eCKHe ypaBlleUHSI. 1'13
piIC. 57,6 BH,lI.HO, qTO no 06nacTHJ K asI H3 KOTOphIX a eHTHcPH H-
pyeTcH cBoeA TOHaJlblolOCTblO, HMeIOT CJIHlliKOM YNlHHeHHYIO ct>oPMY,
np:Rt.leM HI-JOr a 06p83YIOTCSI no ofinacTH HecSH3Hble .1H60 Heo)J)lo-
CBSI3Hble. Pa:3YMeeTCSI, KOnHqeCTBO o6MeHoB Me>KA)' npo eCCOpaMH
npH TaKOM pa36HeHHH 6YAeT BeJlHKO, H scPcPeKTHBHocTb napa..rrnenb-
HblX Bbl'tfHCJleHHA CTaHOBHTCJI HH3KOft.
TIepecPopMHpyeM no 06naCTH TaK, QTo6bI YMeHbillHTb KOJlH"fe-
CTBO CBH3e:A Me>KAY H8MH. AnSI 9TOR l enH npHMeHSIeTcr MeTO CrJeK-
Tp8J1bHOrO pa36HeHHH [141). B OCHOBe MeTo a ne)f(HT HCDOJIb30-
BaHHe co6cTBeHHblx BeKTOpOB MaTp bl JIanJIaca A1ISI 3 aHHoro
rpacPa. BOJIbrnaH qaCTb TeopeTH'ieCKoA: pa60Tbi no HCCJIe,l\OBaHHIO
CBOA:CTB 3THX BeKTOpOB 6blJla npoBe eHa B [142, 143). H3nO)KeHHble
TaM OCHOBHble MAeH 6blJlH 38TeM Hcnonb30BaHbl B (144) "PH peweHHH
npo6J1eMbI 6aJIaHCHpOBKH 38rpY3KH npol.\eccopOB.
C onopoA Ha 3TH HCCJIet\OBaHHH, B paMKaX KCPC 6b1JIa C03 a-
Ha no,nnporpaMMa, BbmOJlHjllOIU.aH p836HeHHe ceTKH Ha nO;l\o6JIa-
CTH nepeA Ha anOM ra30,lUtHaMHtteCKHX pa.cqeTOB [100, 140]. npH-
,97. 1.0f1OJlHIf1'eJIbllh1e BOnpOC&l nap8JIJleJlbHblX BblllHCJleHJliI
227
. - 4 :0.043
',....' .:;/, 0.603
:.1. ::'--', . 1 ...
.-' t]h' ' ''f I 1
· ,:,:..: -' - , . . . .. 1 . 122
.' ... - '"'': . I .
'I,).if. r. .. . >'.. _2.282
. .:. - f. . -2.142
- . : : . . .3.402
.: ' . .3.911
.04 . .4.521
.5.081
".5.611
it)
. : '< !:' ., ;
J . .J; '
. J:; ' :-;=..
- I '..'t'J_:: ,.,
-. - -.l '- .J.j
6)
n)
PHC. 57. 06TeKaJIHe 3J1nHnca a) pac-
upe-a.eJlelll1e )\a.BnelllUl; 6) TpHBH8J1h-
Hoe paJ6HeHHe CCTKH; 8) pa.u.uOHaJ1b-
floe paa6HeHMe CeTKlI
a)
6)
"'J
I
,
I
,
n)
" . ;I: :
I,. .'
. I
I _t.
, .... ..
---'-
Pue. 58. 06TeKaliHe ItBYX3JIeMeHT-
HorO npo4>HJJs: a) cl>parMeHT ceT-
KH; 6) pa36HeHHe)JJ1J1 8 npou.eccopo8;
D) pa36HeHHe AID! 32 npOl.(eCCopOB
Mel) pa llOIIanbHoro pa36HeHHH AJIH 3 aqH 06 06TeKaHHH 3JIJIHnCa
npHBe,ll.eH Ha pHC. 57,8. B OTRHQHe OT nepBOHa'-laJIbH:Jro pa36HeHHH
(pHC. 57,6) t 06JIacTH CTaJlH AOCTaTO IHO KOMnaKTHblMH. KOJIH IeCTBo
CBSi3ett t.1e>K;J:Y npou.eccopaMH .H c.nyqas 8 npOl.\eCCOpOB YMeHLIIIH-
JlOCh no cpaBHeHHlO C TpHBHaJIbHblM p836HeHHeM npH6JIH3u'fenbHO
8 5 p33, a JVlH 12 npol eccopoB - B 6.3 pa.3a.
,lWyroH npHMep pa OHanblloro pa.36ueHHSI OTHOCHTC K 3 aqe
06 06TeKaHHH ABYKPbIJ10Boro npo<l>HJIH (pHC. 58). B 3TOM CJlY'lae
CeTKa 6blna nOCTpoeHa 113 75 790 TpeyrOJlbHHKOB.
,lLnH 3all.aQH, rrpHBeAeHHoH Ha pUC. 57, 3<Pct>eKTHBHOCTh napaJI-
JIenH3a MH 12-npoQeccopHoro BapHaH'fR B03pocna OT 56% ([PH TPH-
BHaJIbHOM paa6HeHHH AO 72% npH CneKTpaJIbHOM pa36HeHHH. Pac-
qeTbl npOBO HJ]HCb Ha 12-npo eccopHoA: BbIQUCJlHTenbHoD CHCTeM€
PARSYTEC CC.
TIpH YBeJIH'IeHHH QHCJIa npo eccopOB, H COOTBeTCTSeHHO, 'lHCJ1a
Y3JIOB CeTKH, poCT 3<l><PeKTHBHOCTH npH nepeXOAe K ?aqHOHaJIbHO-
MY pa36HeIJHIO 6Y eT rOpa3AO 60JIee 3aMeTeH. 0):\113.1<0 npH 6oJIb-
WeM 'tIHC1Ie npOCTpaHCTBeHHblX Y3JIOB 803HHKaeT HOBasI TPY HOCTb.
CaM aJ11 pHTM p HOHaJIbHOrO p836HeHIDl, BbioonHSleNblH nepe)J. Ha-
'IaJIOM pac ler.ra, Tpe6yer )J;JIH cBoelt peaJIH3aQ,HH BbICOKOnpoH3BO-
HTenbHOH MHoronpou.eccopHOH cHcrreMhI. K CO>KaJIeHHIO, Sa)l{HbJe
BOnpOCbl napannMbHbIX BbI tlHCJICU Hit, KaK TO: napa.:1JJeJlbHaJI pe-
aJ1H3 HSI aJ]J:'OpHTM3 pa36aeHHSI Ha no o6nacTH, p&36HeHHe Allil
npOCTpaHCTBeHHo-TpeXMepHbJX aq - BLJXO wr 33 paMKH OCHOB-
HOlt TeMaTHKH MOHOrpaQ>HH H B HeR paccMaTpHBaTbCSl He 6Y)Q'T.
3. 4HI-laMH£lecKaH nepe3arpY3K3 npou.eccopOB
KaK Y>Ke OTMeqaJIOCb B 4 aHHoA rJIaBbI, npo6.neMa pa.BHOMep-
HOlt 3ar'PY3KH npo eccopOB B cnyqae Hcnonb30BaHHSI O HOpO HbIX
a.nropH'TMOB peillaeTCH npoCTO. I10 H TaK >Ke npOCTO 3Ta npo6ne-
Ma pelliaeTCH, Kor a Heo opo)J;HOCTb aJlrOpHTMa HOCHT ccJlHHeAHblA"
XapaKTep, T.e. CMCHa THna 8JlropHTMa He 3aBHCHT OT peWel-lHB 3 a-
qH, a npOHcxo,n;HT 8 3apaHee H3BeCTllblX nOAo6na.cTHx npocTpaHcTBa
HnH }fa cPHKcHpoBaHHbix HHTepBanax BpeueHH.
t..JT06bl o6ecnet:(HTb paBHOMepnylO 3arpY3KY npo occopOB, o e-
-7. )1.onOJlHllTeJJbllbie BOrrpOCbl IIapan.ne.nbHblX Bbl'lHC1leHHH
229
HUM - TeOpeTH qecKH HnH nYTeM u npo 6HOrO" pac eTa - BpeMSI BbI-
QHCJ1eH}I.A, 3aTpaQHBaeMOe Ha O RH y3en. IIpH pa36HeHHH Ha no-
06J1acTH, B Ka>K oA H3 KOTOpblX pacqeT AOJI>KeH npOBO,ltHTbaJ co6-
crneHHbJM npoQeccopOM, qHCJlO Y3JIOB B HHX 6y eM Bbl6HpaTb o6paT-
HO nponOPQHOHaJIbHbIM 3TOMY pacqeTHOMY BpeMCHH.
BOJIee CJTO)KIIO OprauH3YeTCSI pasHoMepH8JI 3arpY3Ka npo ecco-
pOS npH tre11HHeRHOM THoe HeONlOpO,ll;HOCTH, KOr)1.a xapaKTep aJI-
ropHTMa MeHSIeTCSI B 3aBHCHMOCTH OT nOBe eHHH pemeHIDI 3 aq}f.
B 3TO f cnyqae npOHCXO HT nOCTOSlHHblA MOHHTOpHHI' BpeueHH, 3&-
TpaQHBaeMOro Ka)K bIM npo eccOpoM H& BbInOJIHeHHe KOHKpeTHoro
3TaJ1a Bbl HcneHHa, H onpe,n:enslOTCSl HaH60nee 3arpY.>KeHHhle npo-
ecCOpbl (Ha pHC. 59 3TO npo eccop IV) [140].
I
II
III IV
v
VI VII
PIIC. 59. Cxewa Ha.MHqecKOA nepe3arPY3KH npou.eccopoB: I.... 1 VII - HOMepa
npoQ.occopoB; KO.nllQeCTBO CTpenoK npODOPQHOHaJIbHO '{Heny y3J10B, nep aD
MblX OT OJI,Horo npoQeccopa. K ApyrOMY
3aTeM qacTb T01.IeK B HenpepblBHoM pe)KHMe rrepe,[\aeTCH OT 3TO-
1'0 npoQeccopa K coce HHM (pHC. 59: npo eccopbI ill H V). B CBOIO
o epe b) ttT06bI 3TH npo eccopbI He TOpM03HJ1H BbI'fHtJIeHHSI, qacTb
TOqeK OT HHX, HO Y}Ke B MeHbmeM o&beMe, nepe,qaeTCR 60J1ee 8JIb-
HUM npo cccopaM II H VI (pHC. 59). npo e,lQ'pa nepe3arpY3KH B
KaKOM-TO CMb1CJ1e aHaJIOrHQHa npou.eccy paCnpOCTpaHeHH.H TellJIa H3
o6nacTH. r,ll.e npOH30mno 9HeproBbI eneHHe.
)J.HllaMWIecKYIO nepe3arpY3KY npoqeccopoB 6LIBaeT MO>KHO ocy-
w.eCTBJlHTb lot 60JIee npoCTO, onHpascb Ha HH<I>OpM 1O 0 xapaxrre-
pe pewaeMoR 3 a ". TaK, HanpHMep) CMeHa THlla anropHTMa MO-
}KeT npOHCXO HTL npH OCTH)KeHHH HeKoTopoH ra30,AHHaMH'ieCKOR
BeJlHtt}IHOA !<pHTHtieCKOrO 3Ha'teHHH. B 3TOM CJJyqae ,1],JISI BKnIOQe-
HHfI MeXaHI.f3Ma nepe,ll.aqH )'3JIOB OT o,LtHoro npo eccopa K pyrHM
AOCTaTOIJIIO CJle HTb 3a H3MeHeHHeM r830 HaMHqecKoro napaMeT-
pa, a tIe npoH3BoJ]jtTb rIOCTOSIHHblA MOHHTopHHr 3arpy3KH npo ec-
COpoB. IIpHMep HCnOJIb30BaHHH TaKoro I1o o)1.a 6Y,!!.eT paccMOTpeH
B rJlaBe V 8 CBH3H C MO enHpoBaHHeM Dpo eCCa ropeHHH MeTaHa.
B eJlOM, 3aJ<aHQHBasi AaHHYIO rnaBY, XOTMOCb 6h1 OTMeTHTb,
qTO paCCMOTpeHHbie B HeR anfopHTMbI, BKJlIO aB uHoronpo eccop-
HYIO peaJ1H3aI HIO, HonHIOTCH RHIlIb qaCTblO Tex qJtCJIeBHbJX MeTO;J.OB,
KOTopble 6b1JlH nocrpoeHbJ Ha 683e KCPC H Kry HJ1H MorYT 6blTb
C03)].aHbJ B nepcneKTHBe. BbJ60p MaTepHana onpe eJJSlnCB B 3Ha'lH"
TenbHoA Mepe nH'iHblM OfibJTOM aBTOpa B peweHHH a raaoBoA
,l\HHaMHKH. KOHKpeTHhle pe3ynbTaTbI 9THX pac'leTOB 6YAYT npe -
CTaBneHbJ B CJIe lO ea I'.TIMe.
rJlaBa v
MO EnHPOBAHHETEqEHH
rA30BOA CPE bI
,aaHHasI rJIaaa BnHeTCSI 38.BepIIlaJ() ea B MOHorPa4>HH. B HeA
Ha OCHOBe paHee npe,lVIO>KeIIHblX <PH3H ecKHX MO eJleA H 'IHCJleHHLIX
anropHTlvlOB 6YJQ'T pacCMOTpeHbl pe3ynLTaTbi MaTeMaTHQeCKOrO Mo-
e.n:HpOBaHHg HeKOTOpblX KnaccoB ra.3OAHHaMHtJ.eCKHX TeqeHHA. ,no
9TOro B OCHOBHOM paccMaTpHB8JIHCb 3 aqH B OTHOCHTeJIbHO npo-
CTblX TecTOBblX nOCTaHOBK8X, C IlaMepeHHeM 60JIee anellanpaBneHHO
npO,QeMOHCTpHpOBaTb B03MO)l(HOCTH KCPC H Kry. KOTOpbIe c TeO-
peTHqeCKOH TOt{KH 3pellHSI o6cY>KAaJIHCb B npeA1d x fJlasax.*
BO.flbWRHCTBO npe CTaBJIeHHbIX B rnaBe pe3ynbTaTOB uonytteao c
nOMOll1.blO HcnOnb30BaHHSI BhlCOKonpOH3BO,l:tHTeJIbHbIX MHoronpo ec-
COpHblX BbIqHCJlHTeJIbHblX CHCTeM.
1. Mo en:llpoBaHBe cTpyiaoro Te'leHB.H
H Rq}{eM C OTHOCHTeJIbHO npocTOro Knacca HOHapIUdX 3 a'l.
B AaHHOM naparpa<pe npo eMOHCTpHpyeM HeKOTOpbIe P YJIb-
TaTbJ pacqeTa 3aTODJIeHHbIX cTpyA. HapSl)O' C KOHKpeTHbIMH Q>H-
3HQeCKHMH pe-.JYJIbTaTaMH, 06paTHM BHHMaHHe Ha HeKOTopble 8.H-
Hble, CBSI3aHHble C HCnOJIb30BaHHeM BblCOKOnpOH3BO HTeJlhHbJX MHO-
ronpo eccopHblX CHCTeM. PaCCMaTpHBaeMble pe3ynbTaTbI QHcneHHo-
ro MO eJlHpOBaHIDI npe CTa.BneHbJ B pa60Tax [145, 146) H COOTBeT-
· B HacTOBII(ee BpeM.9 ae.neTCJI pa6crra no CQ aHHIO KHoronpoo.eccopHbIX os.-
KeTOB nporpaMM )VIR Mo.z.eJlHpOBaHHJI ewm BH3Koro ra3a B 2D H 3D reOMeT-
pHH. B KaqeCTBe OCHOBhI ,lJJIS annpOKCHM Hcnonb3YIOTCJ1 KecrpYKTypnpo-
BaHHblC ceTKH. 't.JacTH"IHo B03MO)KHOCTB KCPC, oOHpaJOUUiXcB H& HCnOJILJOB&-
HHe TpeyrOJIbl1hlX Ii TeTpa3;ij)aJIbHblX CeTOK, npo eMOHCTpHpoBaHb1 B rJl. IV.
CTBYJO'f 3KCnepHMeHTaJlbHbIlvi .AaHHbIM pa60ThI (147).
PaccMaTpHoanacb CTPYjI 830m co CJIe IO HMH napaMeTpaMH
Ha cpe3e conJla: TeMnepa'I'ypa CTPYH Tn = 249.2 K, a8neHHe
Pn = 3.824 · 10 4 rIa, ,n:JI Ha cB06o oro npo6era An = 11.73 · 10- 8 M,
'-IHCJIO Maxa M n := 1.01, 'IHCJlO KHy ceHa, onpeAeJJjleMOe no P HYCY
1. n conna , Kn = An/2rn = 3.747.10- 4 . OTHoIlleHHe 3.BJIeHHH OKPY-
}!{8.lOl1.I-eA cpe bI Poo K ,lI;aBJIeHHIO Ha cpme canJla AnSI pa3JlHq}fblX
aapH3.HTOB 6pa.nOCb CJIe,nYIO ee: Poo/Pn = 0.011, 0.00262, 0.00131 ,
0.000732.
configuration "grid", 561-92
100
90
80
70
10
..
Q" 50
40
30
20
10
0
2
configuration "grid"
50 100 150 200 260 300 350 400 450 500
Proces8or number
PHC. 60. 9<t><I>exTHBHoCTb llap8JIJIeJIH3a1J.HII ,l1)1B paJnH'lHblX KOHcperypa.u.Wi
npol. eCCOpOB (HeHBH3S1 cxewa): 1 - JlH"eAHaJI KOH4>Hrypa , 2 - KOHcI>Hry-
PaI(JlSI "peWeTK3."
KfY -cHCTeMa, 3anHcaHHaH s- U.HJIHH pH'lfeCKOA CHCTeMe KOOp-
)I.HHaT {( r, z)- I'eOMeTpH ). pacCqHTblBaJI8Cb c nOMow.blO anropHTMa,
H3JIO)KeHHOrO B 9 3 npe eM: rnaBbI. lIpH 3TOM paccMaTpHBaJICSI
KaK HBHbIA) TaK H HeHBHblA SapHaHT JIHHeapH30BaHHoA cxeMbI. ,IljIsr
HeSlBHoR cxeMbi Hcnonb30B8J1CH 6e3h1TepaLI.HOHHbIA anrcpHTM, OCHO-
BaHHblR Ha nOKaJIbHO-O,AHOMepHOM npe CTaBneHHH. PeweHHe B03HH-
configuration "grid", 561x92
100 -----.-..--------.-.----------------------...---------..--.
80
1
&0
..
'=:
Q..
40
20
o
50
100 150 200
Proc...or number
300
250
PHC. 61. 94><I>eKTHBHoCTb napaJIJIeJUf3811.HH .D)IH pa3JIH'IHblX 8.JIrOpHTMOB
1 - ftBH(UI cxeMa J 2 - HeW3UaJI cxeM8
KalO HX np" aTOM CHCTeM TpexTOqe'IHbJX pa3HOCTHbIX ypaaHeHHA
ocy ecTBnRJlOCb MeTO,l].OM nporoHKH.
).J;rH pac'ieTa Hcnonb30B8JJacb MHoronpo eCCOpHa.H BbJ'IHCJIH-
TenbHaS CHCTeMa MBC-IOOO, YCTaHOBJIeHHasI B Me>KBe OMCTBeH-
HOM cynepKoMnbfOTepHOM eHTpe (MCQ, r. MOCKBa), c nHKO-
BoA npOH3Bo)J.HTenbHoCTblO 1 TFLOPS. 06w.ee tJHCJlO npoQeccopOB
(ALPHA 21264) CHCTeMbI ,n:OCTHraJlO 768. npH pac'iere aHHoR 3a-
aqH npHMeHS1JIOCb ,lI.O 600 npoqeccopOB. HCnOJlb30B8JJOCb KOH<I>H-
rypa IDI npo eccopoB KaK J1HHeAH&s1 (pHC. 44, rJl. IV, 4), TaK H B
BH.A€ pellleTKH (pRC. 45).
KOHKpeTHble BapHaHTbl pacqeTHbIX CeTOK COCTOSlJIH H3 281 x 91,
561 x 91 1 1402 x 1002 y3JIOB COOTBeTCTBeHHO BAOJIb OCR Z (HanpaB-
neHHC ,nBH}KeHHjI CTPYH) H B p H8JIbHOM HaI1p8.BJIeHHH r.
Ha PHCYHKax 60 H 61 npe CTaBJleHa scJ><I>eKTHBHOCTb nap8JI-
JIeJIbHblX BblQHCJJeHHa B 3aBHCHMOCTH OT KOJIHtfecTBa Hcno.nb3yeMblX
IIpo ecCOpOB npH Pa3JlllQHblX KOHcJ>HrYPaIJ.U5Ix CHCTeMbl (pHC. 60)
rlr.
JO
'0
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14 . ".,
1] . IU
12 ..1,))
- -- '1 ..In)
,. ...u
. ....n
" ' . ..ua
, ....,
-- . ...,t
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.- -
'\ · -. - t · ...
) . - - - ....- ;::Ir -- - - ---- I.IOt
_.,.. .,' - ".- : .:. -.,. ... .- , t -; --.. ..-- 2 ....
. . " . - -.......= . !I! ._ - ..:t =-.i '...: :", --" t. !.. -. 1 ..101
I, -, , . .. .. .... _ J' ..".-.. . "..,
.0 . 10 100 - 1 2 0 zit.
.
20
PUt', 62. ()GH a.H l<apTUtlil ("J py .r: l'f)O/"n = O,U] 1
20
rlr.
'5
10
20
z/r.
.
10
o
PHC, 63. BbI HcmrrenbHu mJlHpeH-cIMnorpa4»BB CTpYH B6nH3H conna
H pa3nHl.IHblX a.nropHTMax (pHC. 61). B KaqecTBe OCHOBbl pacqe-
Ta HCDOJlb30Banacb CeTKa c qHCJIOM y3J10B 561 X 91. BD,lI.Ho. '{TO
,lVIj( HeHBHol cxeMbI JlHHeAH8JI KOHcI>Hryp H9 3aMeTHO npoerpbl-
BaeT KOHcpHryp "peweTKa" B s<I><PeKTHBHOCTB. 9TO CBft3aHO C
60J1bWHM KOnHqecTBoM CBSI3eR H Hepau.aoH8JlbHOA HX OpraHH3 -
eft AJlH J1HHeHHoA Kou<l>HrypanHH. B CJIyqae HCnOJlb30B8.HHR SBHOA
cxeMbl JlHHeAH&H KOHcpHrYPaI.\HjI TaKx<e npoHrphlB8err Koa<l>Hrypa-
(\HH "peweTKa", XOTSI H p83J1H'lHe He CTOJIh SlpKO Bblpa>KeHO, KaK ,lJ.JISI
HeBBHoA cxeMbI.
BH HLJ 3aMeTHbJe npeHMY ecTBa B S<P4>eKTHBHOCTH lIBHO:A cxe-
tAbl no CpaBHeHHIO C HeBBHoA (pRC. 61), qTO, Bnpo'tfeM, JJerJ<o nporHO-
3HpOBaJIOCb. .HBHhle CXeMhJ HMelOT e e O HO npeHMy ecTBOJ CBSI-
3aHHoe c npOCToA JIOrHqecKoA CTPYKTypolt. O HaKO B eJI0M Bpe-
p/P mln B
,
5
4
3
2
10
..
.
.
. .
. .
. axpedl t
- - - - h.-O.1a;.. h,-O.s .2 J
-.-.-.- h..O.1 .b..O.2St; "JI
It,. 0 J. . h..O.1.. 11M t
5
15
10
20
Z (rM'I)
Puc. 64. OTJIOCHTeJ1hH8.B WIOTHOCTb: cDJlowaaa KpHBU R nyHKTHpbl - pacqeT
H& pa3nH'IHblX ceTKax. . - 3KcnepHMeHT (147)
MH TOqHOrO paCqeTd, C BblXO OM Ha OHapHblA pe>KHM no H B-
Holf cxeMe 6bIJIO MeHbme, qeM 110 HBHOA, BBH,[U' B03MO>KHOCTH HC-
nOJlb30BaHHSI 3Ha'lHTeJ1bHOro 60JIhwero BpeMeHH6ro marn. YtfHTbJ-
Basi npIlBe,n.eHHb1e pe3YJILTaTbI, MO}KHO CAeJI&Tb BbJBO)\ 0 He06xo -
MOCTH HCIIOnb30BaHHH H pa.3BHTH.H K8.K BBHhlX, T8J( H HeSBHblX 8JI-
ropHTMOB. npH 3TOM 3Ha1.JHTMLHWe YCHHHjI Heo6xo,lJ)lMO Y eJIBTh
C03,AaHHIO 1lOBbJX napaJIJl hHblX anropHTMOB pemeHHSI CHCI'eM ce-
TOqtlblX ypaBHeHHA.
Ha pHC. 62 H306pa>KeHa o6w.a,g KapTHHa CTPYH tVliI BapHaHTa
BHewHero AaBJJeHHB Poo/Pn = 0.011. B o 06pa30BaHHe 60QK006-
pa3HLIX HtJeeK, xapaKTepHbIX ,lVIH csepX3BYKOBblX crpy8. Ha pHC. 63
nOKa3aHO 60JIee no,rq>06Ho fiOJIe TeqeHHSI B6nH3H conna. a pac-
Qel'a Hcnonb30Banacb CeTKa J COCTOa ag H3 1402 x 1002 Y3nOB. Ha
3TOA DhlqHCJlHTenbHoA ''wmtpeH-<poTorpac:pHH'' xopomo BHAeH ,D.HCK
Maxa, B03HHKaro A Cpa3Y nocne 06nacTH nepSOHaQ8JIhHOrO pac-
IllHpeHHH CTPYH.
Ha pRC. 64 H306pa,)KeH rpa4>HK OTHOCHTeJlhlIOR fiJlOTHOCTH (HOp-
MHpoBaHa Ha ee MHHllMaJlbHOe 3HaqeHHe nOCJIe nep80HaQaJIbHOrO
paCIllHp€UHSI) B OJIb OCB z. CpaBHHBalOTCH aHHble pac;tleTa H 3Kcne-
pH leHTa [147]. BH,lI.HO xopomee COBn eHHe n xapaKT€pHOM nOBe e-
HIDI fIJ10THOCTH, nOJ1Y'ieHHOM B pacqeTe H 3KcnepHMeHTe. IIp" 3TOM
KD.nH\.feCTBeHHbIA CJ1.BIIT 06CHX KpHBblX BnOJlHe MO>Ke'I' 6blTb o6'bSlc-
HeH YCJIOBHSlMH HOpMHpOBKH .AnSI 3KclIepHMeHT3JIbHhlX aHHbIX, B
KlYfOpblX TPY,QHO onpe enH1'b MHHHMaJlbHOe 3RaqeHHe nnOTHOCT" B
o6nacTH ee cMnbHoro H3MeHeHHH.
B CJIe yfOln.eM p ropa3 O 60JIbllIeM no 06'beMYJ naparpacpe pac-
CMOTpHM npHMeHefIHe KHHeTHtJecKHX cxeM MSI MO,lJ.e.nHpoBaHHSI
HeycTaHoBHBIIIHXCSI, OC HnJISlQHOHHbIX TeqeHHR B cBepX3cYKOBbIX 1-1
TpaHC3BYKOBbIX nOTOKax BS13KOro raaa.
2. MO eJlHpoBauHe aeYCTaHOBHBillaxC1l
u
oc nJ]H HOHHLIX TeqeHIIH
KJlacc HeyCTaHoBHBWHX TeqelIH:H npe CTaBnileT 60JIbIII0:9 TeO-
peTHQeCKHR H npaKTHqecKHH HHTepec. TeopeTHtleCKHH HHTepeC BbI-
3BaH B nepBYIO Oqepe,[l.L 1'€M, QTO 3TH Te'feHHB Mor)'T ocyw.eCTBnSlTb-
Cst npH flOCTOSIHHOM 8HeUIHeM IIOTOKe ra.3a. Kone6aTenbHblA pe1KHM
B 3TOM Y'iae peaJIH3YeTCH He BcerA8, a J1Hlllb npH: I-leKOTOpLJX xa-
paKTepHhlX napaMeTpax Ha6eraJOI1(ero ra3a H onpe e.neHHoR cPop-
Me 06T€KaeMOI"O Tena. ,lJpyrHMH CJlOBaMH, npH onpe eJIeHHblX na-
paMeTpax) xapaKTepH3YlOlIJ.HX BHeWHHA nOTOK H cl>OPMY rena, ocy-
eCTBJ1HeTCSl He IIOCTOHHHbl:A:, KaK Ka3aJ10Cb lla nepBbIH B3rnSl , a
HeycraHoBHBllIHACSI pc>KHM Te"leHHH. 110 CYTH AeJIa, 3TH nOTOKH
HerIOCpe,lJ;CTBeIIHO CBH3aHbI C cPYH aMeHT8JJbHbIMH CBoACTBaMH Ma-
TeMaTHQeCKHX MO,ll.eJIeA, OIlHCbIBaJOIlI.HX TeqeHHSI B 3KO["O raa&.
I1paKTHQeCKaSl Ba>KHOCTb H3YQeHHSI TaKHX SlBneHH:A: OqeB Ha.
BbICOKHe j"POBHH nynbca HA aBJ1eHHB "PH TaKHX pe>KHMax SlBJlSI-
IOTCSI npHt.IHHOR nOBpe eHH5I MeMeHTOB KOHCTPYK HH H BbJXO a
H3 CTpoH rrpH6opOB JlemrenbHblX annapaTOB. Bonee HH3KHe ypOBHH
nynbcan.HA cnY)I(aT HCTOtIHHKOM 3BYKOBblX B03Myw.eHHR. KpOMe T0-
w
rO J B03MO>KHOCTb pacqeTa HeYCTaHOBHSWHXCSI TeqeHHH C p83BHTbIM
cneKTpOM nYJlbC Ha BBnseTCSI Ba>KHOH CT)'IIeHhIO npSlMoro 'tHCJleH-
Horo fO,I:\enHpOBaHIDI Typ6YJIeHTHblX HBJIeHHR.
MaTeMaTHQecKoe MO eJlHpOBaHHe HeYCTaHOBHBWHXCR TeqeHHR
COnpH>KeHO co 3HaqHTeJIhHbJMH TPY HOCTHMH. B nepoylO Ot.tepe h
Hco6xoAHMO. 1.IT06bl B npo ecce pac'leTa Bbl1.J:HCJIHTenbHbJD aJIrOpHTM
peanH30BbIB3J1 KOJIe6aTeJJbHYIO, a He nOCTOSIHHYro BeTBb peweHHH.
KpOMe rroro, onHcaHHe nOAo6HbiX pe}KHMOB) KaK npaSHJIO, Tpe6YeT
3HaHHH ,neTa.lIbHOA KapTHHbI ra3O HHaMH"IeCKHX nOJIeH B6J1H3H o6Te-
KaeMoro Te.lIa. 9TO, B CBOIO o'tfepe b, npHBO HT K Heo6xo HMOCTH HC-
nOJIb30BaHH$I BblconpOH3BOJI-HTeJlbHb1X MHoronpo eccopHbIX CRCTeM,
{TO TaK>Ke HaKJI bIBaeT orrpeAeJICHHble OrpaHH1.feHH Ha Knacc npH-
MeHSIeMbJX anropHTMOB.
113yqeHH€ pe3YJIbTaTOB tJHCTleHHoro Mo enHpOBaHHH HeyCTaHo-
BHBlIIHXCH pe}KHMOB, nonyqeHHblX c nOMOUtplO KHHeTHqecKHX cxeM)
HaqHe f C onHcaliH5I TeqemtR 06nH3H npOCTpaHCTBeHHO- BYMepHblX
H TpcX epHbJX npHMoyronbHhlx KaBepH.
1. lfHCJlelIHOe MO,l(eJ1HpOBaHHe OCI{HJ1JIHL{HOHHblX pe.IKHMOB
a6JlH3H KaBepHbl
I1CCJleAOBaHHIO ny nbcaQHOHHblX SlBJIeHHA S6nH3B 2D H 3D Bbl-
eMOK nOCnRIneHO MHoro pa6OT. I1HTepec K R3YQeHHIO nOBe,neHHH ra-
30 HHaMHqecKHX uOJIeA Onpe M5IeTCH TeM, qTO, c O HOa CTOpOHbl,
BbleMKa 51BJI.HeTCSI Oft-HUM H3 HaH6OJ1ee Y 06HbIX o&beKTOB CCJIeAO-
BaHMSI CPYHAaMeHTaJlbHhlX 3aKOHOMepHOCTeA TeQeHHS1 BH3KHX raaOB.
C pyroA CTOpOHbl, npSlMoyronbH8$I BbJeMKa jlBJIHeTC5I peaJIbHOH Mo-
AeJlblO ,lVIH MHorHX MeMeHTOB KOHCTPYKl.{HR neTaTeJlbHhlX aJIIIapa-
TOB. IIpHMepo 1 TaKHX KOHCTPYKI A 6YAYT npa60pHbre OTceKH, OT-
ceKH wacCH, TeXIIHqecKHe WBbI, KOHQ>HrYPaQHSI, B03HHKaIO aSl npH
OTCYTCTBHM o)J:tloro H3 3JIeMeHTOB Tenn03 THoro nOKpbITHB NI5I
KOC IHqecKHX Kopa6JIe:A THria ccwaTl'n'" UBypaH". THDHqHast KOIIcpH-
rypa HSI 2D H 3D BbJeMOK H306paJKeHa Ha pRC. 65 H 66.
B KatfeCTBe an:ropHTMa pacqeTa Hcnonb30BanOCh KCPC C KOp-
peK HeR (rJI. IV, 5), BKnlOQaH ee MHoronpo eccopHblA sapH-
aUT. JlH 3a,a.at.J B 2D-nOCTaHoBKe pacqeT 0 OCHOBHOM npOBO):tHJICSI
Ha MHoronpou.eccopHoA BbIqUCJlHTeJlbHOA CHc'reMe (44 npo eccopa)
Parsytec CC o61II-ett npOH3BO HTeJlhHOCTbIO 15 GFLOPS. 3 aqH B
PHC. 65. reOMeTpH'IeCKqe xaPaK-
TepHCTHXH 2D-Bblew1CH
PHC. 66. reoMeTpa'IecKHt xapa.KTepHCTH-
KH 3D-8h1eMKH
3D-noCTaHOBKe peW8JlHCb DB. BbI'IHCJlHTeJ1bHOA CHCTeMe MBC-IOOO
c nHKOBOA npOH3BOAHTeJ1bHOCTblO 1 TFLOPS.
rpaHH'C.fHbJe YCJJOBHB CTaBHJIHCb CJIe,lQ'JO HM 06p83oM. Ha neooR
rpanHl\€ pa.cqerHoR 06nacTH 3 aBanCSI npocPHJJb CKopocrH, COOT-
BeTCTBYlOl.1J)I:A ,,'IaMHHapHOMY nOrpaHH'IHoMY CJ1010 [63}:
u: = ( ; ) _ ( ; ) 3 I
(2.1)
r)J.e 1.£00 = Moo - CKOpOCTh Ha6eralO erO HeB03MYID-eHHoro nOTOKa, 0
- TOJl Ha nOrpaHHqHOrO cnO.H. B 2D-reoMeTpHH CKOpOCTb U H U oo
6bL11H HanpaBneHbI S OJIb OCR OX (pHC. 65). )VIH 3D-3bleMKH 3TH
CKOpOCTH IlanpaSneHbl nOtl; yrJloM 1/J (pHC. 66) K DJlOCKOCTU (x, z).
Pacnpe,n:eneHHe TeMnepaTypbl B nOl"paJlHtUlOM c.noe TaK>Ke 3a-
aBanOCb B COOTBeTCTBHH C (63]. Ha TBep LIX CI'eIlK8JC CTaBHJlOCb
YCJlOBHe npHJIHn8BH.H. KOr,Aa Bee KOMnOHeHTbI CKOpocTH paBHbJ HY-
JIJO :
u = v = tv = 0,
a TaK>Ke YCJlOBHe ,LVI.sI a.BJIeHHH:
(2.2)
(}Plan = o.
(2.3)
nOBepXHOCTH CTeHOK BHyrpH H SHe BbIeMKH C'iBTaJlHCb a;J:Ha6aTH'Ie-
CKHMH:
fJT/On = O.
(2.4 )
Ha CB06o HhJX rpaHHQax CTaBHnOCb "MSIrKoe" ycnoBHe CHoca:
8Q = 0,
an
Q = (p, U, 1J, w, E).
(2.5)
52. MO,.l{eJlllpoBaHHe JleYCTaHOBHBUlHXCR OC!(,HnJlR HOHHblX reqeHHR 239
B paccMaTpHBaeMblx SapHaHTax Ha6eraJO HR DOTOK 6bIJ1 cBepx-
3BYKOBbIM (Moo > 1). B 03BYKOBOA o6nacTH nOrpaHH'tfHOI'O C1IOSI
CTaBHJ1HCb HeOTpa>KalOIIJ.He rpaHH'iHble YCJIOBHSI, aIOI1UIe B03MO)K-
HOC1'b 6ecnpenHTcTBeHHoro npOXO)f( eHHSI BBepx no nOToKY MaJIblX
B03Myw.elllIA. B 3Ha"lHTenbHoA "'Iepe nocTpoeHHe rpaHMqHblX YCJlO-
BHA (2.2)-(2.5) onHpa.nocb Ha aHHble pa60T [148, 149J.
Kpa'l'KO onHweM HMelOIllYIOCSI HH4>oPMaI.\H1O 0 HeYCTaHOBHBWHX-
CH OC1UlnIISI HOHHbIX TeqeHRSlX. KaK H3BecTHO, 3a DJlOXO o6TeKa-
eMblMH TeJIaMII MorYT 06Pa30BbIBaTbCSI KaK CTaI.\HOH, apHble Tal{ H
HeCT HOHapHbIe OTpblBHhle 30Hhl. B CTa HOHapHblX 30Hax B cneK-
Tpe nynbca Ha anJleHHSI BBHO BblpaJKeHHble HCKpeTHbie COCTaBJlSI-
lOIJJ.He He B03HHKaIOT. 9'rOT cneKTp e AJl5I Typ6YJleHTHoro Teqe-
HHSI XapaKTepn.YeTCH ' enbIM WYMOM". B036Y>K,lJ;aeMbIM Typ6yneHT-
HOCTblO, H OTCYTCTBHeM nlO6bJx nynhca.u.uA nSl n&MHHapHOro nOTO-
Ka [150, 151].
HeCTaIJ.HOHapHble 30Hbl OTpblBa B 3KCnepHMeHTaJIbHblX HCCJIe-
OBaHH.sIX onpe enmoTCSI no xapaKTepHCTHKaM nYJlbca.J.Udt a.BJIe-
HH5I. B 3'l'OM CJIyqae cneKTJ)LI nyJIbCa :A HMelOT SpKO Bblpa}KeHHble
AHCKpeTHble COCTaBJ1BIO He. l{a.K nOKa3blBaeT 3KcnepHMeHT, MaKCH-
MaJlbHble nynbcau.HH B03HHK8JOT B oKpeCTHoCTSIX TOQeK OTpldBa H
npHcoe HHeHHg norpaHH Horo CJlOSI.
OCHo8Hbie pe3YJIbTaTbI 'IHCJIeHHOro Mo enHpOB8HHjJ HeYCTaHO-
BHBllIHXCH 'reqeHHH CBB3aHbI C HCCJIeAOBaHHeM TYP6yneHTKblx noTO-
KOB. 9TOT cpaKT 06ycnoBJIeH He TOJIbKO BaJKHOCTblO H3YQeHHSI nO,lJ.o6-
HbIX otYbeKToB, HO H TeM, {{TO MO eJIHpOBaHHe Typ6yneHTHblX Te"le-
HHA: C HCnOJIb30BaJIHeM a.nre6paHqecKHX HJlH H<P<I>epeHu.HaJtbHbIX
(\.to eJleA Typ6yneHTHoA BH3KOCTH npo e C BbIqHCJlHTeJ'IbHOA TO'q-
KH 3peHHSI, qeM npHMoR pac"tJeT YP8.BHeHBit, OIIHCblB8JO HX Te1.felWe
B 3Koro ra3a. B "QaHHoR rJlaBe B OCHOBHOM npe CTaBJIeHhI pe3ynb-
TaTbI 'IHCJle.HHOrO pac eTa HecT8 HoHapHoro OTpblBHOro Te"leHIUI B
npsMoyrOJlbHoH TpeXMepHoA H YMepHOa BhIeMKax, 06TeKaeMbJX
CBepX3BYKOBbIM JlaMHHapHblM DOTOKOM [117], [152J-[155].
PacCldOTpHM HeKOTopLle pe3ynbTaTbI MOAeJ1HPOBaHHSI Te'leHHB
B6nH3H 2D Ii 3D KaBepH "PH CJle)Q'lOIU.HX ra:30 RaMH'IecKHX H reo-
MeTpHQeCKHX napaMeTpax: Moo = 1.35; ReA = 3.3 · 104; 6/ H = 0.1;
L / H = 2.1; W / H = 1 (NIH TpexMepHoro cnY'las). 3;J;ech QHcnO
Pe:AIIORb,IX.Ca onpe,lJ;eJl5IeTC5I no rny6HHe BbleMKH. ,lI;1S1 3D-reOMCTpHH
yrOJ1 'f/; (pHC. 66) nOJlaranCH paBHbIM HYJlIO.
Ha pHC. 67 H 68 npe CTaBJIeHbI qacTOTbI nYJlLc H aBJlel-lHH B
TO'tJKe A (IIpaBaB KpoMKa KasepHbl, nJlOCKOCTb CHMMeTpHM) AJlH 2D
H 3D nOCTaHOBOK 3a,n.a I.
r,ll.e 0"0 - 2. 10- 5 na -
onopHoe axYCTH\lecKoe AaB-
J1eHH€ (nopor CJ1bIWHMOCTH),
Ulru. = (ai + b ) - cPe.D.He-
KBa.lQ)aTH\lHOe 3Ha'leHHe aM-
..
DJIHTYA nYJlbca HH ,L\a.BJIeHRR
8 nOJIoce qaCTOT WHpHHOA
1 r (ak, bJc - K03<1>cPH H-
eHTY pa3nO eHH YHK H
nynbcal Ha aBneHHSI B TO'tf-
Ke A B pS.D. cl>YPbe); Po = PRC. 68. Q8CTOTHWA cneKTp NUl 3D-
= 101.325 Kna - CT8HJ1-apT- reoMeTpHH
Hoe aTMocQ>epHoe ,l(8BneHHe,
Poo - AaBJleHHe 8 OKPY)I(a!()r.n eA c e.
113 COnOCTaOJIeHWI pRC. 67 H 68 BHMfO, '11'0 B TpeXMepHOM CJIY-
qae B03pacTaeT YPOBeHb ,l\HCKpeTHbiX COcraBJIHIOUJ)iX cneKTpa KOJIe-
6aHHH. TaKme YBeJlH"lJHBaeTCSJ ypoBeHb WYM3 BAWlb BCero cneKTpa
KOJJe6aHJ.IH. 3KcnepUMeHTaJlbHbIe HCCJIe OBaHHSI nOKa3bJBaJOT Te >Ke
oco6eHHOCTH 3D-Kone6aHHA [150).
Ha pRC. 69 H306pa>KeHhl MI'HOBeHHbJe pacnpe eJIeliHH nOJISI CKO-
pOCTeA BHy'rpH KaBepHhI .lt1ISI 'ieTbipex Docne,AOoaTenbHbIX MOMeHTOB
no OCM a6C HCC OTJlO>Ke-
Ha qaCTOTa 8 rep ax (rQ) ,
no OCH Op;:lHHa.T - aMIIJIHTY-
Ahl rapMOllHK, npe"nCTaBJleH-
HhIe B clx>PMe npHoe,l.(€HHOrO
cneKTpanbHoro ypoBHH nynb-
cau.uif a8JIeHHst L 1 rn' BbIpa-
)KeHHoro 8 ,lI.eIJ,H6enax ()J:6)
cornaCHO <PoPMyne
( O'lr Po )
L1fu = 20log 10 ,
C70 Poo
LI'" (AI))
..
a.
UG
t.
..
.
j'x lo'ru
PKC. 67. qacTOTllhlA cneJ<Tp A1UI 2D-
reOMeTpHH
III r I' (,.,';)
J.
PI
-
-
.
f)( 10' ru
92. MOAeJlHpOBllHHe l1CYCTanOBJIBlUJIXCH OCl.(J-lnJ1H HOHHbIX Te'leHHIt 241
. , , , ", - .._.. ....,."..
, i i :::::: ::::
, , P ': 4-,,, ;
. .
. .. .
,. "".,
{ " \ . L
, I ' · · ,
' . . . ,
\\\\""
,\ \ \\"'..'
IIa -. -. ......
1
. , ., '.1'", .....
. , ,\ I.'........
11
'" '-'
"
- --. " "" _'\ -
.. -"" ,\
" , - " \
I .. ..\\"
J , .,\"
'I' .., ,'
\ \ \ ....",
. . ..
, , t1
, I I
, I I
, , I
I , l
, ,
1
'\
, , ... .... " " '" ,.
.. .... " "'
.. "... ,,"\
, , , .. , , \ ,
I ' I I -- -
I i ;::::::
JIll".."
t I , \ , ' " . . · .'
PHC. 69. lIMe CKOpocTeH B CeqeHHH xl L = 0.95
BpeMeUH B Ce"feHHH xl L = 0.95, KOTOpoe IIepneH,n:HKYlliIPHO Ha6era-
lO eMY nOTOKY II paCnOnO)f(eno B6JIH3H rrpaBoA KpOMKH KaBepl)W.
XOpOUJO npOCJIe)KHBaeTOi lleCTa HOHapHaSl KapTHHa TeqelUfSl,
KOTOpaH nepHO,lUlqeCKH nOBTOp eTCSl. Ba>KHOH 'iepTOA 3D-TeqeHH:A
HBJ1SeTCSl nOHBJleHHe BHxpeA B Onb 60KOBOA CTeHKH BbleMKH, qTO
XOpollJO BH,.[\HO H3 pRC. 69. I1MeHHo 3TH BHXPH, HaJIHqHe KOTOpblX
cpHKcHpyeTCSI B 3KcnepHMeHTe p Ii nOBbllUaJOT ypOBeHb aKYCTH'ieCKHX
B03MYLl.I.eHHA .
BOJIee nOlI-p06HO pacCMOTpHM KapTIDIY TeqeHHH S6JIH3H 2D-
KaBepHbl, KOTopaH HMeeT MHoro 06w.Hx qepT C Te'tfeHHeM B nJlOCKO-
CTH cHMMeTpHH (y = z = 0) NtH 3D-CJIYQas. Ha pllC. 70 nOKa3aHbI
UOJI5I CKopocTeR ,LVISI 2D-K8.BepHbl Ha 'tIeTbipe Pa3JlHqHblX MOMeHTa
0.. \ -' , \
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HRR B neBOM HH>KHeN yrllY BhleMKH
o
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PHC. 70. BexropHble nons CKOpOCTH
H 2D-cnY'l8JI
PIlC. 72. CTpYKTYpa Te'leHHR 8 no-
rpaHH'IHOM cnoe nepeA OTphIBOU
BpeMeHH.
B3a>vfO eAcTBHe OTpblBHOrO CJIOB CMemeHHfl C BHemHHM DOTOKOM
npHBO HT K nepHO H1.{ecKOMY OTBO Y H nOABO Y Macchi H3 KaBepHbJ.
B TOT MOMeHT BpeMeBH, KOr a B OKpeCTHOCTH "paso A KpoMKH BbJ-
eMKH TOqKa npHcoe HHeHHSI CJlOSI CMemeHH.H CMel aeTCSI HDJKe KpaJI
KpOMKH, ocy eCTBnSleTCSI nO BO MacChi BHYTpb BbleMKH H 803HHKa-
eT BOJIHa aBJJeHHB, KOTOpaH paCnpoc1'paHileTCSI BBepx no nOTOKY. B
JleBOM yrny BbIeMKH B6nH3H ee ,lI.Ha nOCTOSlHHO paCDQ1IaraeTCH BHXpb
He60nbworo pa3Mepa (pHC. 71), HHTeHCHBHOCTb KOToporo MeHSleTCH
2. kloAennpoBaHHe neyCTaHOBHBUlHXCR HQJlOHHhlX Tetlemdt 243
B pa3Hble MOMeH1'bI BpeMeHH. KpoMe Toro, CY ecTBY1OT ew:e ABa BH-
XpH) O HH - B cepe HHe BhleUI<H. pyroA - S6JJH3B npaBoA CTeHKH.
3TH BHXpH nepe te aIOTCB no qacOBOft crpeJ1Ke. B onpe eJIeHHbla
MOMeHT H3 3THX yx BHxpeA 06pa3YeTCSI O HH) KOTOpbrlt 3axBaTbl-
BaeT 60JlhWYlO 'IacT& nOJlOCTH KaBepHhI.
KpHTH"tIecKHA nepen 8.B-
neHHjI, Bhl3b1BaJO Hft OTphlB no-
rpaHH Horo cnOg nepe B eM-
KOH, Man. n03TOMY, Kor a K
JIeBOMY KpalO npHXOttHT BM-
Ra C THg, C03 aeTCB nepe-
na,A ,l.{aBJIeHHS1, npe8bIWaJO HA:
KpHTH'tJeCKHH, BCJIe)J.CTBRe \Ie-
ro npOHCXO HT MHKpOOTpLIB no-
rpaHHqHOrO CJIOS. Ha pec. 72 PIlC. 73. Pacope.AeJleHHe KO*4>HInt-
npe,n:CTaBneHI:d nOJI CKOpocTeft eHTa cpeAHero CT&TlAeckoro ,It&BJ1eBHB
Ha ,lJ.Ba apaKTepHbJX MOMewra AHa 3D-BbleyXH
BpeueHH B 3TOA 30He: O HH - npH HaJIH-IHH OTpblBHOA 30Hhl t ;WyroA
- npH ee BbJpO>KAeHHH. T8.KOA >Ke xapaKTep 06pa30BaHHR Kone6a-
HHA Ha6JlIO aeTCH H B 9KcnepHMeH'l'8.JI])HblX HCC1Ie OBaHH.HX. npe -
CTaBnellHblX B pa60Te (150).
BeplIeMCSI K pe3YJlbT8.T8.M qJlCJIeHHoro MO,n;eJIHpOBaJlHJI 3D-
KaBepHbJ. Ha pRC. 73 npe CTa.BJ1eBO pacnpeAeJIeHHe Ko <pH eH-
Ta cpeAHero CTaTHCTH'IecKoro aBneHHg. B nJIOCKOCTH CHMMeTpHH
(y = z = 0) H& He BbleMKH. Cpe)].Hee asneHHe p H KOSqxs,OII)l-
eHT CTJ onpe,n:MSlIOTCH C nOM01l\hlO Bblp8JKeoH1t
'I
P = 1 J P dt )
t I - ti
t;
Cp
o
Q
a
o
I
o
I
o
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c = 2(1' - Poo)
p P u2 ·
0000
(2.6)
MO)KHO BH eTb, lITO Ha 60JIhweA 'IacTH Ha aBJIeHHe npaKTHQOCKH
nOCToBHHO, a 3aTeM pe3KO B03paCTaeT B6JU13H npasoro KpaSl BhleM-
KH. TaKoe nOBe eHHe aBJJeHIDI COOTBeTCTBYeT 3KcIIepHMeHT8JThHblM
MIHbIM, npe cTasJIeHHLlM B pa60Tax [156, 157J.
PacCMOTpHM BJIHSlHHe CKOpocTH BHeWHero nOTOKa Ha nYJIh-
ca HH tr.aBneJIH5I 8 K&BepHe. Ha pac. 74-76 npeACT&.8JIeHbl rpa-
p!jJ 7I
"I
ltf'7o = J.35
0.
..
L
Puc. 74. PacnpC,Ae.;Ip.Ha.te 110 npeMellH ftrrrHODCIJHOro CTaTl1 leCKOI"O ,l{anJleHHSI V/Poc
ua npaRoA Kp01t.IKe R TO.JK A 3D-BbleJ.fKH JIB Moo := 1.35
IJ/Pa;
1\f f70 = 2. 1
1.
Pile. 75. Pa.cnpe/l.elleKJle no npCMeHH MrHo8CHHoro c'raT}f1:JecKoro ,n,anneHHSI p/Poo
U TO'IJ<e ..4 ;UI toc == 2.1
(PHKH n}r 'hca l-fA anJleHHH B 1'01IKe A "I)aBoA BeIJxHCH KpOMKH.
OTMe rHM, t.ITO npH yae.rIHQeHHH Q"rCJI3 laxa RHeilIHeN) nOTOKc1. C
Moo == 1.35 AO Moo = 2.1 a fnn"TYAa nYJlbcal Hj;t YMellblllaeTCH, a
npH fv100 = 3. 7 KOJIe6aHH Boo61I e OTCYTc"rR) O'r. 9TH pe3y.rILTaTbl
XOpOITIO corJlaCY'OTC c aBI-IO H3Bec'rHbIMH 3KCnep1tMeH1 blfbIMH
) aHHLIMH 1158, 1591, H3 KOTOphlX cne,lO'eT, ITO Ma.KCIIMaJlbHaH I-tHTeH-
CHBHOCTb KOJTe6aIlHH Ha6JTIOl aeTCiJ rTPH Moo 1.0, a IIPH Moo > 2.5
92. MO,JJ.eJlHpOBaHHe HeYCTaIfOBHBIDHXCS OCl(llJ1JlHl(HOHHbJX Te'leHHR 245
p/p.
I.
1.5
M(JD = 3.7
I.
1.
I.
I.
PIlC. 76. Pacnpe)J.eneHHe no BpeMeHH MrHonellHoro (:T3THQeCKoro aane"HJI p/Poo
8 TOQJ(e A )\J'Si Moo = 3.7
pip,;
1.2
1.
I.
0.
I.',
PHC. 77. Pacnpe,neneHHe no BpeMeHH MmOBeBHOrO CTaTHqecxoro -A&sneHBB p/Poo
B To e A "PH 1oo = 2.1, Re = 3.3 .10 4 J 6/h = O.lJ l/h = 1.05, w/h = 0.5
Kone6aHHH Bo06 e He B036y aJOTCSI.
Ha pRC. 77 npe CTaBJIeH rpacl>HK nynbc HH aBJIeHHg ,lVISI
BbleMKH c pyrH {H reOMeTpWlecKHMH napaMeTpaMH l / h = 1.05,
wi h = 0.5. I{o.H e6aHHg )VIS 9ToA reOMeTpHH, B OTnH'lHe OT npe-
.D.bI y cro CJIyqa51 (pHC. 75), 3aTYxaJOT. Cne,u.YeT OTMeTHTb, tlTO H3-
MeHel-lHe llIHpHHbl BbleMKH OT wlh = 1 )1.0 w/h = 0.5 npaKTHqeCKH
He MeHHeT KapTHHbl TeqeHHSI B IIJIOCKOCTH CHMMeTpHH.
pi Pa..
1.4
I. !
1.2
1.
0.8
o
240 I
PRC. 78. Pacnpe,l(eneHHe no BpeMeHH MrHOBeHHOro CTa.TH'ICCKOrO a.aneHH.H p/Poo
B npa80M BepXHCM yrny TpexMepHoA KaBepUbl)J.JJSI Moo = 2.1, 6/h = 0.1, l/h =
2.1 J w/h = 0.5
Bonee CJIO)KHaH Kap'rHHa Kone6aHHa fla6JIIOAaeTCSI npH Y"QaJIe-
111-111 OT WlOCKOCTH CHMMeTpHH. Tax, Ha pHC. 78 npe C'IaBJIeHO pac-
npet\eJIeHHe AaBJ1eHHSI B roqKe, pacnOJIo>KeHHoR Ha nepece'tleHHH
npaBoM Ii 60KOBOB CTeHOK BbleMKH. B cneKTpe tJacTOT ):pOBeHb wy-
Mel 3Ha"llHTenbHO BbInIe. IIo A o6HMI KapTHHa CBD3aHa co CJlO)KHoR HH-
rep<pepeH'-"HcH aKYCTH qeCKHX B03MYW.eHHR, KOTopble reHepHpYI01:
BHXPHMH, o6pa3YlOrnHMH OKOJIO 60KOBOA CTeHKH KaBepHhl.
npe,lXbIAYl[ He pe3YJlbTaTbi no MO,D.eJJHpOBaHHIO 3D-Kaaepl-lbl OT-
HOCHnHCb K CJIyqaro, Kor a yron 1/J Ha6eralOLQero 110TOKa 6bIJI pa-
BeH HYJIlO (pHC. 66). JJ:Jl pacqeTa BapHaHTOB C yrJIOM 1/;, OTJIHQHblM
OT HYJI)f, Tpe6yeTcSI 3HaQHTenbHO 60Jlbme npOCTpaHcToeHHblx Tot.leK,
TaK KaK, BO-llepBbIX, OTCYTCTBYeT CHMMeTpHH OTHOCHTeJlbHO nJlOC-
KOC'l'H (y, z), a BO-BTOpbIX, ):VIH OfiHCaHH.H B03HBK8JO < npH 3TOM
oco6eHHocreA TeqeHH lole06xoAHMa 60JIee nOJijJ05uaH )IHCKpeTH3a-
HSl B OJIb OCM OX H BYX APyrHx HanpaBJIeHH:A. B npoBe eHHOa
cepHH paCQe1'OB o5 ee HCJIO TOqeK rrpocTpaHCTBeHHo CeTKH co-
C'l BJIHJIO 4 . 10 7 Y3JIOB.
ECT€CTBeHHO) "11'0 AJlSI peuleHHH 3TOH 3a,natlH HCnCl.lb30Banacb
BbICOKOIlpOH3Bo,.QHTeJlbHaH C o 1 TFLOPS nHKoBo:A MOI.1I)iOCTH) MHO-
2. MO,lCeJlHpOBaHHe neYcmHOBJfBUIHXCH OC HJlJlJllQfOB1lblX Te'leHIIR 247
1M
10
60
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40
a
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100 1GI 3M 401 SIt
Number 01 processors
600
PHC. 79. 3<1>q,eKTHBHOCTh napan.nenbHhIx BY'IIICJleHHA (3D-K8.BepHa) B 3aBHCR-
NOCTH OT 'iHC1Ia npoqeccopoB
ronpo eCCOpH8JI CHCTeMa. Ha pHC. 79 npHBe eHa 3<PcI>eKTHBHOCTh
napanneJIbHoro pacqeTa 3D-KaBepHhI B 38.BHCHMOCTH OT 'IJIcn& npo-
ecCOpoB. BH HO, qTO npH q}fCJIe HCD0JIh3yeMhlx npOQeccopos 60nee
500 3<P<t>eKTH8HoCTb HeCKOJIbKO D8AaeT, XOTSJ H He onycKaeTCSI HH:>Ke
OnOJlHe npHeMJIeMOrO ,LVIa napaJIJIeJlbHDIX BblQHCJ1eHHA nOKa3aTeJJSI
B 60%. O HaKO, Y"IHThiBaH, tJro HCnOJlb3yeMaJI njJ pacqeTa, KCPC
c KoppeK eR HOCHT BBHbJA xap8.KTep, 3<1><l>eKTHBHOCTb napanneJIb-
IILIX BblqHCJleHHA MO>KHO 3Hat.fHTeJILHO nOBhlCHTh. J(nSi 3Toro Ao-
CTaTO'iHO YBeJIH'IHTh o6II(ee trilcno pac eTHI:dX TOqeK. UpH 3TOM, B
CBB3H C OTHOCHTeJIbHblM YMeHhweBHeu 'tJHCJI& nparpaHH'lBhlX TOqeK
B no o6nacTIDC, 3<I><j>eKTHBHOCTb napanJIenLHOI 06pa6oTKH B03pac-
TaeT.
Kax nOK83aJIH pacqeThJ, "PH He6onLmOM OTKnOHeHHH yrna t/J
OT HYJIeBOrO 3Ha'leHHH o6 aSl xapTHHa retJeHHSI 8 enOM COxp8.HSIeT
CHOB qePTbI. Ha pHC. 80 npe,L\CTaBneHLI crreKTpbl nynLcan:HA AaBne-
HHH B TOqKe A B 38.BHCHMOCTH OT yrJla,p. BH,D.Ho, 'lTO npH H3MeHeRHH
yrna B Han830He 0° < 1/J 2° OCHOBHLIe MOAbI Kone6aHHA HMeIOT
np3KTHQecKH TY >Ke qacrOTY H 8.MDJJRTY,LQ', He 38.BHCSI OT 1/1. M8.k-
CHMaJIbHhIH ypOBeHL u6enoro WYMa" Ha6JIJO aerCH OpH t/J = O. 9TOT
t.
- Yron H81U1OH8 cf
D Yron H8IC11OH8 2°
. Yron H8KI1OH8 4°
200
1.
i
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0-
m
121
tlO
.
o .. 4I0OI .. ... 11I0OI t2llOlO t....
4 acroJ8 (r
PHC. 80. CneKTpY nynhCaQHA AaaneHHSI B TO'IKe A AJlH pamH-uIhIX yrnoB 1/J
4>aKT XOpOIIIO COO'l'BeTCTsyer 3KcnepHMeOTaJIhHbIM AaHHblM, CBH e-
TeJlbCTBYWW.HM 06 OCJIa6J1eHHH o6paTHoR CBSl3H Me>KA,Y nepe)\HeR H
3 HeA C'I'eHKaMH K8.Bepllbl lIP" "KOCOM)) BAYDe.
IIpHBe eHHbla Ha pHC. 80 pe3YJlbTaT BBn eTCSI rIOATBep AeHH-
eM BbJBO a (rn. III) 0 KOppeKTHOCTH KB33Hra.30.lI)lHaMH'IeCKoA: CHeTe-
Mbl H KHHeTHqeCKHX cxeM. BH HO, '1TO MaJIOe, BIIOJIHe onYCTHMoe
C 3KcnepHMeI-ITaJIbHOA TOQKH 3peHHSI, H3MeHeHHe na.paMeTpa 1/J He
npHBO,lJ,HT K 3HaqHTeJIbHblM H3MeHeHHSlM peweIDIH )1a>Ke B 06nacTH
B03H}IKHOBeHHSI HeYCTOA'IHBOCTH TetfeHHSI.
IIo TBep)KAeHHeM KOppeKTHOCTH npHMeHSIeMoA Mo enHJ a TaJ(-
)Ke aKTHBHoro HCnOJIb30BaHHH pe3ynbTaTOB, nOJly'tfellHblX B BYMep-
HOH nOCTaHOBKe, c:nY>KHT CODOCTaBJIeHHe cneKTpOB nynb it aB-
neHHH, nOJIyqeHHbIX B 2D H 3D paCqeTaX B TO"lJKe A, 1/J ="0 (pIIC. 81).
B OTJIH'IHe OT 2D-cnYQ8JI, B TpexMepHo:A nOCTaHOBKe npH 1/1 = 0
Ha6J1IO,AaJIOCb nepHOAHqecKoe HCTeqeHUe H nOCTynneHHe ra3a Qepe3
npO OJIbHble KpOMKH. ITpH 1/J =/; 0 3TOT 3<P4>eKT TOJIbKO YCHJIHBaeT-
CSI. ITpJ.t 3'l'OM B03HHKaeT nOnepeQHoe BHXpeBoe H>I<eHHe II BHXpH
B IIJIOCKOCTH (x, y) BHYTPH BbleMKH. rrpe CTaBJUleT(SI BecbMa HH-
g:.e. MO,lJ.eJlHpOBaHHe neYCTaHOBRBWnXCH OClJ)fJ1nR HOHHb1X Te'leHHiI 249
20D
- TpexMepwul
· · AiI)' l4epuuq pac"'"
tlO
1..
140
120
100
o 20000 4000G I000O toot» 1 00000 t2fXHJO 1 40000
1l8amm (rQ)
PHC. 81. CneKTphl nyJlbC3.ltH A ,AaBneHWI, nonyqeHBJale np" pacqeTe B 2D R 3D
nOCT8HOBKe (TO Ka A, t/J = 0)
TepecHbIM aKT HCQe3HOBeHHSI OTpbIBa norpaHHqHoro CJlOSI OT ne-
BOa KpOMKH. aHHb1A aQ><PeKT MO}l(eT 6LITb 06'bHCHeH OCJIa6J1eHHeM
06paTHOH CBB3H Me>KA)' JIeBoft H npasoa KpOMKOA. 1b-3a 6oKOBO-
ro BHm BOJIHa C)KaTHH, npHXO SI ag K nepe HeA KpOMKe, cna6ee,
({eM np" 1/J == O. pa3IIH:Qa aBJIeHHA He ,AOCTHraeT KpHTH1.JecKOH Be-
nH'IHHbI H He MO)KeT OTTeCHHTb Ha6eI'alOllI-H:R nOTOK.
B03BpaTHMCH K onpe eneHHIO 'IaCTOTHhiX xapaKTepHCTHK nYJ1b-
CaIJ.HH. Ha pHC. 82 npe CTaBneHhI cneKTpbl Kone6aHuA ,AaBJIeHH5I B
Pa3JIH\{HbIX TOQK8.X npaaoA BepxHeA: KpOMKH. BH O, qT() MHHH-
MaJ1bHble YPOBHH 3BYKOBOro a.BJ]eHHjf ,n:OCTHralOTCS1 B WIOCKOCTH
CHMMeTpHH. IIpe CTaBJlseTCg TaK>Ke ecreCTBeHHhIM 60nee BbICOKHit
ypoBeHb Kone6a.HHR B nO BeTpeHHOM yrny no cpasHeHHIO C HaBeT-
peHHblM, Tal( KaK OCHOBHoe BH>KeHHe nOTOKa HanpasneHo B 3TY
CTOPOHY. OrMeTIfM, qTO Ha6JIJO aerCH OqeHb xopowee COrJI&COBaJlHe
pacqeTHb1X 3HaqemrR OCHOBHbIX qacTOT C aHHblMH 3KcnepHMeHTa
(150]. COOTBeTCTBYIO He pacqeTHLIe H 9KcnepHMeHTaJlLHble 'lHCJJa
,..
tlO
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I ::
fit
it:
a As ...... B1OIIC8 1
· 81'0'418 2
.Q8BneMte 8 1'0'«8 3
. II'" ..... ..... ..... 1..110 110.. 1411'0
,. cr
PlfC. 82. CneKTpY KOJIOO8JIHA aBJIeHHH B pa3HhIX TO'lKaX npaBOa KpOMKR BbI-
eMKH npH ..p = 2°: 1 - B ewrpe KpOMKH , 2 - B IlpaBOM no peHHOW yrny,
3 - D neBOM HaBeTpeHHOM yrny
CTPYxanSl 6b1nH paBHbI (155]:
ShF C = 0.307, Sh c = 0.616, Shr 'c = 1.22,
ShfP = 0.3, Sh;ZP = 0.62, Sh = 1.2.
2. BJlHRHHe cpopMbl BlJeMKB H3 XapaKTepHCTHKH
nYJIbCa HoHHoro TeqeHHH. KOJIe6aHHH B6JIH3H
HJIHHAPHqeCKHX H cljJepH'CfecKHX TeJ1
B03HHKalO e npH 06TeKaHHH BhJeMOK H ,lij)YnIX TeJI nynhCa-
HH B 60nbmHHcTse cnyqaes HOCSIT HeraTHBHhlA xapaKTep no CBO-
eMY BnHHHHIO Ha MeMeHT&I KOHCTPYK HH. TI03TOMY peKOMeH aJ HH
no nO a.BJIeHHIO nYJ1bC HI npe CTaBJISI1OT 3Hat.JHTeJ1bHblA npaKTH-
'1ecK"A HHTepec. B npeJJ-bJJl-y eM nYHKTe naparpaQ>a 6h1J1o nOKa3aHO
(cpaBHeHHe pe3YJIbTaTOB, npe cm.BJleHHb1X Ha pHC. 75 H 77), QTO H3-
MeHeHHe q,OpMbl Tena npH 'reX )l(e napaMeTpax BHewHero nOTOKa
MO)KeT BHeCTH 3aMeTHhle KOppeKTHBbl B xapaKTepHCTHKH nynbc H-
OHHOrO TeqeHHSl.
a)
6)
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PHC. 83. CxeMbI pacqeTHLlX o6nacreA: a) ABYMepH8.SI KaBepHa c BhlCTYD8JO ea
KpblJUKOA; 6) ywepH&JI xaBepHa C HaKnOHHOA CTef[J(oA
BHaqane HccnepyeM BJIHSIHHe CPOpMbI BbleMKB (xaaepHa C KpblW-
KoA H HaK.,1JOHHoA: CTeHKoft, pRC. 83,a,6) Ha xapaKTepHCTHKH nYJIhCa-
QHOHHOro ,!J.BYMepHoro TeqeHHSI. Pe3ynhTaTbl 1.fHCJIeHHOrO Mo enHpo-
BaHH.H [160J CpaBIiHBaJlHCh C aHHbIMH 3KcnepHMeHT8JIhHbIX HCCJle-
,.qOBaHHA.
3aTeM 6Y T paccMoTpeHbI pe3YJIhTaTbl Mo enHpoBaHHH nYJIb-
C HOHHblX TeqeHHA: 86nH3H nHH,Zq)HqecKHx TeJI c BblCTynaJOl MH
HrnaMH [89J, [161}-(163). 9TH paCqeThl npOBO HnHCb Ha OCHOBe cxeM
K3 KJlaCCa KCPC C KoppeKnHeA) allTHpOB8.HHbIX K HH.Lij)Hqe-
cKoA CHCTeMe KOOPAHHaT. ,lJ.ng MO,neJIUPOBaHHH, nOMHMO O,nHonpo-
eccopHbIX cHCTeld, Hcnonb30BaJIHCb H nepBble MHoronpo eccopHl:de
TpaHcnbIOTepHble CTaH HH (141). Pe3YJIbTaTbl 9THX pacqeTOB IJ)IJIHH-
,DJ>H'IeCKHX Ten TaK>Ke CpasHHBaJlHCb C aHHbIMH HaTypHbJX 3Kcne-
pHMeHTOB [164)-[166J.
B KOH e KOpOTKO onHmeM HoseAwHe pe3YJlh'mThl) CBB3aHHhIe
c MO eJIHpoBauHeM 06pa30BaHIDI BHXpeBOA Opo>KKH n03 c<pe-
phI. 3 eCb HCn0JIb30BaJIHCb HecTPYKTypUpOBaHHble TeTpa9AP8JlbHble
CeTKH, nOCTpoeHHble AJlH TpeXMepHoA eKapToBoA reoMeTpHH. Pac-
'C.{eTbl BeJ1HCb Ha cospeMeHHbIX MHoronponeccopHblX cRCTeMax.
Ha pHC. 84 npHBeAeHbI rpa<PHKH 3aBHCHMOCTH Ma.KCHManhHO-
10 YPOBHR OCHOBHOa HCKpeTHOA rapMOHHKH OT yrna HaKnOH& a
252
.
Lz:.L
180
17S
170
16'
160
1"
150 0
rJl. V. MOAeJlHpOBaHHe Te'leuHit r&30BOB Cp
I ---;;-- , ., """-'
. " .
....
. . '
", .,'
-.. ,
........ ,.
...... . '
...........-.
0.2
0.4
0,6
0.8 .xii. a.
PHC. 84. Pacnpe,!.te.ueHHe ypoBHJI aKYCTH'IecKOro ,lUUJJ1eHHH DO ,lJ.HY BhleMKH CO
I.ltHTKOM (a) (%/ Hl = 0.25) H 6e3 l1tHTKa (6); 3&BHCHMOCTb MaKCHMaJlbHoro ypOB-
HJI OCH08HOA CKpeTHOA rapMoHHKH L- OT yrJ1a HaKnOHa CTeHXH (8)
38.,l.J)1eH CTeHKH_ IIpH 3TOM tvIHH& L2 OCTaBaJIacb cj)HKCHPOBaHHOA, a
H3MeHeHHe yrJla Q onpe.n.emInOCb BeJlH'UlHoA Lt. napaMe'rpbI BHew-
Hero nOTOKa H reoMeTpHH KaBepHhl (pHC. 83,6) 6p8JIHCb CJle)Q'lOJ.U.He:
Moo == 1.35 t Reh = 3.3-10 4 , SIH = 0.04, L21H = 2.1, a = 32°+90°.
H3 PHCYHKa 848 BH,L{HO, liTO YMeHbweHHe yrJla HaKJIOHa CTeHKH npH-
BO HT K 3aMeTHoMY CHH1KeHHIO MaKCHMaJlbHOrO ypoBH.H )J.HCKpeTHoA
rapMoHHKU. ITpH 3TOM '-IaCTOTbl OCHOBI-IbIX H HcKpeTHblx rapMoHHK
He3Ha HTeJIbHO MeHSIIOTCS C H3MeHeHHeM yrna Q.
L
180
160
a)
...... . ..,. 6)
140
.
120
0 2 4 6 8 /.10.
PHC. 85. \.{a.cTOTHble cneKTphi npB p&3J11f'1BL1X yrnax HaxnoHa CTeHKR:
a) (} = 32°, 6) Q = 48° (L JJ.6, I rLt)
2. WIO,l{eJlHpon8111fe HeYCTaHOBIlBUJHXCH OClfHJlJIJll.{HOHHbIX reqemdt 253
B pac\.{eTaX, eMoHcTpHpyeMblx Ha pHC. 85, npetI.CTaBJleHbI qa-
CTOT}IbIe cneKTpbl Ha npasoA CTeHKe B 3aBHCHMOCTH OT yrna a
(L2/ H == 3.7). ,Ang 0 = 32° cneKTp HWHpoKononocHbl:R". qacTOTHblA
cneKTp TaKOra BH.l(a B03HHKaeT, HanpHMep) Kor):\a B cnoe CMeweHHH
npHcYTCTBYIOT MacIllwHbIe BHxpeBhle CTpYKTYPbJ. 9TO SlBJIeHHe HC-
t{e3aCT npH a = 48° .
J.2 ..
Sh 0,8
. --- 5
---0-" 1 I
__ _ . ._.__ -II- 6_
-- -0- -- 2
. 9' 7
· . .6- -- 3 I
---. ..v--- 4 -.- ---4. 8
I x 9
0,6 - t-_ :. I
--- r- ----
I .IJ
0,4 -
0.2
40
so
M
70
RO
90 a
PHC. 86. CpasHeHMe .naHHhlX pac'teT8 H 3 KcuepHMewra: 1 2 - 3KCUepHMeHT H
3,4 - pac'leT "PH Moo = 2; 5-7 - 3KCOepHMeHT H 8,9 - pac'leT npR Mao = 1.5
Ha pRC. 86 npe CTaBneHbI rpaQ>HKH 3aBHCHMOCTH qucen CTPY-
xaJISi Sh = J L2/U oo MB nepBoA H BTOpOH MO Kone6&HHA ( aJI-
Hbl€ 3KcnepHMeHTa 5-7 nj{ TpeX MO)J. KWle6aamt OT yrna HaKnOHa
Ct. nonyqeHHbIMH no .AaHHLIM 3KcnepHMeHTa {156] H pacqeTa). Ha-
6nlO aeTCSI xopowee COBn eHHe SKcnepHMeHTaJlbHblX II paC'IeTHhIX
aHHbIX.
BepHeMcB K pliC. 84,a,6, Ha KOTOpOM H306pa>KeHbJ ypOB11H
CTaTHQeCKOrO aBJleHHSI AlISI KaBepHhI co IIJ)fTKOM (KpHBaJI a)
(ZI Ht = 0.25) H 6e3 lIJ)fTKa (KpHB8JI 6). BHJJ;HO, 'tfTO H8JlHq}fe HTKa
cy ecTBeHHO CHH:HmeT ypOBeHh nYJl&C HH. npu Z/ Hl = 0.5 nYJIL-
CanHH Boo6IJJ.e 3aTYxaJOT H o6TeKaHHe CTaHOBHTCSI C HOHapHblM.
TaKH 1 06pa3oM) H3 pe3YJIbTaTOB Mo enHpOBaHHSI BH H01 'ITO ny-
Tef\.1 o'rHOCH'l'enbHO He6oRbworo H3MeHeHHSI cpOpMbJ 06TeKaeMoro Te-
na MO:>KHO CYI.J..I-eCTBeHHO nOHH3HTb HHTeHCHBHOCTb KOJle6a.HHA HJIH
aJKe nOJlHOCTbIO HX HCKJIJO"IHTb.
tJ't
l:n. V. MOJteRlfpoBaHHe TeQeHJflll33OBoR cPeAbl
L
D
)
>
Mco
M.
L
-l
D
PlfC. 87. l(Hn.HHAPH'lecKoe Teno C Hr-
noA
PHC. 88. KOHH'IecKoe TeJlO C HrnoA
06paTHMc.H K pe3ynbTaTaM 1fHCJleHHOrO MOAenHpoBaHH.R OCecHM-
MeTPHqHbIX TeJI C BbICTynaJOIl\HMD HrnaMH. KOH<j'JHrypaIJ)lH TeJI,
oOKpyr KOTOphlX npOH3BO,lJ.HTCSI pac'leT ra3O)J.HHaMHI.fOCKHX nOJle t
IIpHBe e}IL1 Ha pac. 87 H 88. HCCJle OBaHHe p83nHtJ}lblX pe)J(HMOB Te-
qeHHH ra3a.B6JlH3H TaKHX KOHCTpYKU.HA npe CT8BJIHeT HHTepec MSI
cBepx3BYKOBOA aBualUlH H paKeTHoA TexHHKH.
OnHmeM Bllaqa.ne KpaTKO pe3YJIbTaTbI 3KcnepHMeUTOB [163]-
(165) "PH Moo = 6 H Re = 10 6 .
npH () - L/ D < 0.3 HaJIH lHe HrnbJ He BJlHSleT Ha KapTHHY Te-
qeHHH, Tal< KaK OTOwe.QwaSl y,n.apHas BOnH& He B3aHMO,ll,eACToyeT C
HrJlolt. TeqeHHe CTa.QHOHapHO. IIpa 8 > 3 TeqeHHe TaK e HO-
HapHO.
IIpH 0.3 (J < 3 Ha6JIIO aIOT01 HeC'J'8.QHOHapHhle nepHOAHQe-
CKHe pe)l(HMbI 06TeKaHHH ABYX Tanos. npH KOJIe6a.HHSIX nepBoro
po a 1.4 < 8 < 3 OTpLIBH8.SI 06JIacTb HMeeT KOHlNeCK}'IO <j'JOPMY,
TO'l{Ka OTpbIBa pacnonO>KeHa Y KOHQa HrnhI. HeCT8J.J.HoHapaocTb npo-
BBJlSleTCSI B H3MeHeHHH H8JUlOH& CKaqKa YWloTHeoHjJ. CPopMa 30Hbl
OTpbIBa H3MeHBeTCSI OTHOCHTenbHO CJla60 H so Bee MOMeHTbJ opeMe-
fiR ocraeTCSI 6J1H3KOA K KOHHqecKOA.
IIpo ecc I1ynLCaQHA BTOporo po a, HMelOIIJ.HA: MeCTO npH
0.3 < 8 < 1.4, 3HaQHTeJILHO c.nO)l(Hee. B nepBoA <l>a3e rrpo ecca npo-
HCXO)J.HT pa3pyweHHe OTpblBHOA 30KY) BO BTOpoA' - poCT DHOBI:. 06-
pB30BaBWeRc OTpbIBHOA 301'lbI) conpoBO)l( alO H:AaI nOTepeR Ko-
HHqeCKOA <POpMbl 3a CqeT e<popM HH ee nepe He:R I.f2,CTH. npH
OCTH>KeHHH OTpblBHOR 30HOB MaKCHMaJIhROrO pa3Mepa nepHO,l{ KO-
92. MOAeJJlfpOBaHHe neYCT&HOBlUJllUfXC1I OC 11JlJlR1(HOHHMX Te'leHHR 255
JIe6aHH:A 3aKaHQHBaeTCH H npQqecc nOBTOpSeTCB BHOBb. Ba}KHblM
pe:JynbTaTOM 3KcnepHMeHTa BBJlHeTCSI TOT <l>aKT, I.fTQ npH Re < 10 3
KOJIe6aHHi1 Boo6 e He Ha6nIOA8JOTCiI, a ,lVISI Han830Ha QHCen PeA-
HOJIb Ca Re = 5 · 10 3 + 10 6 xapaKTep KOJIe6auH1t HOCHT 6nH3KHR
xapaKTep.
OnHlUcM pe3YJIbTaTbI pac eTOB 06TeKaHHSI TeJla C HrJloA: (pHC. 87)
co CJIe Ylow.HM" napaMeTpaMH BHernHcro nOTOKa H reoMeTpHqeCKH-
MH pa.3 lepa.MH Tena: Moo = 6; 8 = 1; Re = 8.4 · 10 3 . BLI60p 3TOro
OTHOCHTeJIbHO He6o.nbworo lIHCJIa PeAHonb ca 06ycnoBJIeH CJle 1O-
1IJ.HMH 06CTOBTeJIbCTBaMH. Bo-nepBbIX, 3KcnepHMeHTbJ nOKa3blBalOT
CJla60e H3MeHeHHe KapTHHbI TetJeHHSI B lIIHpOKOM an83oHe QHCeJI
PeAHOJIbtI:Ca. Bo-BTOphlX, qTO He MeHee Ba)KHO, ,a.nH Re = 8.4 · 10 3
qHCJIeHHOe "'10 CJlHpOBaHHe B03MO.>KHO H& OTHOCHTeJJbHO rpy6bIX ceT-
KaX, 1.fTO 6bJno aKT}'aJIbHO, nOCKOJIbKY pac'tJeTbI rrpOBO WlHCb 8 kOH-
e 80-x I'O)).OB, Ha Bbl'lHC'JIHTeJIbHhJX CHCTeM&x He CJIHWKOM BblCOKoA
nPOH3BO HTenbHOCTH.
Ha pHC. 89-91 npe)J;CTaBneHbI pac"leTHbJe H306apbl ,LVISl p33JIHQ-
HbIX CTa.D)iA nYJIbCaI.\Hft. K MOMeHTY speMeHH. Kor):\a TOttKa oTpblBa
nOTOKa HaqHH&eT BHraTbCSl K TOpJJ.Y, A8.BJIeHHe Ba HeM OCTHraeT
MHHHMYMa (pHC. 89). K KOHJJ.Y nepBoA <l>a3hl y apHa}] BOJIaR OCTH-
meT TOpl\a nHH pa (pHC. 89). B 91'0 BpeMSI aBJIeHHe IIa TOPQe y
KpOMKH OCTHraeT cBoero MaKCHM8JI))HOrO 3HaQeHHSI. IIepBasi <l>a3a
3a.KaH'IHBaeT'CSI.
Bo BTOpoR cJ>a3e B03HHK&eT -KoHcPHrYPaQHH y apHbJX BOJIH
(pHC. 91), KOTopaH XopOWO Ha6nW,lJ;anaCh B 3KcnepHMeHT8X. 3Ta
KOH<pHrypa HSI o6pa3YeTcSJ KOCOA:, ronOBHoA H npSlMblMH y,n:apHhl-
MH BOJIHaMH. llOCJle npoXO)K,qeHH.SI qepm npSlMYJO YAapRYIO BOJIHY
,LJ;aBJIeHHe ra3a nOBbIWaeTCSI H OH YCTpeMnseTC51 B o6nacT!> 60nee
HH3KOrO aBJIeHWI - OTpbIBHYIO 30HY. nO,nIIHThiBaeMaJI TaKHM 06-
pa30M oTpbIBHaSl 30Ha 6b1CTpO PacTeT.
,Aanee B OKpecTHOCTH HrJIbI 3a npHcoeJI)fHeHHOA K ee KOH
y apHoA BOJIHOR B03HHKaeT OMacTb nOBbIIDeHHoro aBneHHH. OT-
pbJBHaH 30Ha npO OJI)K8.eT YSeJIIAHBaTbCS, HO TepSJeT CB010 KOHH-
qecKYIO CPOPMY. QHa OCTHraeT cBOero MaKCHM8.JIbHOro pa3Mepa, H
H36b1'roQRoe KOJ1HtfeCTBO ra3a nOKH,[I.aerr OTphlSHyJO 30HY, npoxo
H KpOMKOA TOpl a. I10CJIe 3TOro Becb npo ecc nOBTOpSleTCSI BROD!>.
,.
(J
1fJ
2D
JII
IIiI
SII
.
PHC. 89, lb06aphl B nepBOA <pa3e nynhcaQHA
,.
III
,
PRC. 90. H306aphl B Ko e nepnoA <I>&3b1
tJ
16
III
t$
1 1.29
2 5.00
J 1.71
A 12.42
5 1'.13
. 1).14
7 2 ..18
. 2":J.7
. 30.98
10 34.1'
z
1 1.2'
2 11.31
3 21.51
4 31.11
5 41.72
. 11.82
7 61.93
. 72.04
I 12.14
1012.21
a
1 1.2'
2 11.1'
3 21.0.1
. 30.92
, 40.10
. 50.17
7 10.11
. 7O.A3
. 10.31
10 H. ...
.. .
PIiC. 91. H306apbl B MOMenT nOBBJleHHJI -06paJHoA KOHcpHrypaQHH
2. MO,l{eJIHpOBa.nne neYCTBHOBHBJUHXCH OCl{HJlJlJll(HOHHlUX reqeHHll 257
Xo"q npou.ecca Ot.feHb xopomo COBna aeT C Ha6mo,l'(aeMbJ M 3KC-
neplIMeHTaJIbHO KOJlc6aTeJlbHhIM pe>KHMOM BToporo po a. OTMeTHM,
"liTO IlpH pyrHx reOMeTpHqeCKHX pa3Mepax, OTBe1.laJOD1.HX 3HaqeHH-
HM rlap leTpa () < 0.3, 8 > 3 Ii 1.4 < (J < 3, B pacqeTa.x, B COrJlaC"H
c a.IHblMH 9KcnepHMeHTa, Ha6nlO aJlHCb COOTBeTCTBeHHO CTaQHO-
HapHbJe pe)l(HMbl H KOJIe6aTeJlhHble pe>KHMhl nepBoro po,a.a.
KaK nOKa3bJOalOT ,a.aHHbIe 9KcnepHMeHTa, YCTaHOBKa aa HrJIe Ko-
HHtfCCKOro HTKa no yrnoM 0- K OCR HrJIbl (pHC. 88) CHH>KaeT Ha-
6J11O aeMbIR ypOBeHh nYJIbcaQHA. 3TOT <paKT nO T8ep)K.a-aeTcg pac-
teTaM". Cor naCHO 3THM t{aHHblM H ,n.aRHblM qHCJleHHOro MOAeJJHpO-
BaHHH [166], Kone6aHHSI Boo6w.e 3aTyxaroT, ecrm yrOJI a < 48°.
8 3aKJIJO'IeHHe AaHHOrO p83 eJla OTMCTHM cne YIO ee.
B I-lacToSlw.ee BpeMg Ha 6a3e KCPC C03 aIOTCSl fiaKeTbI MHO-
ronpo eCCOpHbJX npHKJI HbIX DporpaMM ,IVI MO,D.eJIHpoBaJlHH npo-
cTpaHcTBeHHo-TpexMepllbIx BH3KHX Te'ieHHA, B TOM '1HC1Ie H Ha
HecTpYKTypupOBaHHblX ceTKaX. B KaqeCTBe npHMepa npHBe)J.eM pe-
3ynbTaTbI BblnonHeHHoro H.B. A6anaxHHhlM H C.A. CYKOBbIM pac-
qeTa Bjf3KOrO HeYCTaHOBHBllleroCH 06TeKaHHH c<pepbl. napaMeTpbJ
Ha6eralOlD-ero nOTOka r83a CJleAYIOII He: qHCJlO Maxa M = 0.15 , qHC-
no PeAHOnb,lJ.Ca Re = 1250. AnnpoKcHM CTpoH.1IaCb Ha TeTpa-
1
(l,
PHC. 92. ITooepxHocTH Scixt>eKTHBHOH rpaHHlJ,b1 BHXpeBoA JOHhI B p83J)H HhIe
MOMelITbl BpeMeHH
3 pa.nbHOa CeTKe, COCTOg e:A H3 70300 y:mOB. CeTKlt cy ecTBeHHO
crY aJIacb OKOJIO nOBepXHOCTH C<j>ePbI H B CJIeAe ]a neD. AnSI pac-
qeTa HcnOnb30BaJIOCb 25 npo ecCOpoB CHCTeMhI MBC-IODO.
PHC. 92 xapa.KTepH3YeT KapTHHY TeqeHHSI n03 H 06TeKaeMO-
ro TeJla. llpH qHCJIax PeRHonb ca, 60nhWHX 700, TeqeHHe BOKpyr
wapa HMh3H ct.JHTaTh perYJISlpRhIM H CTporo nepHo,n;HqecKHM, tTO H
nO TBep aeTCSl AaHHbIM pacqeTOM. 06nacTb OTpblBa DOTOKa HMeeT
CJIO>KHYIO C'l'PYKTYPY. rpaHH a 3TOa 06naCTH, nOKa3aHttaR Ha pRC. 92
B ABa pa3HblX MOMeHTa opeMeHH, onpe eJI$leTCS1 cpOPMaJIbHO, HCXOAjI
H3 nOBe eHH.H perneHHH 3 a'lH (167).
3. B3aHMo ejCTBHe OCl{HJlJ1HPYIOlI{HX TeIJeHHll rn33
H ynpyrHx IlOBepXHOCTell
«I>eHoMeH B3aHMO eaCTBIDI DOTOKa ra3a H ynpyroA nOBepXHOCTH
06TeKaeMoro HM Tena llonyT.JHn Ha3BaHHe np06JIeMbI a3poynpyrocTH.
PeweHHe 3TOA npo6J1eMbI Qpe3Bbl'laAHo B HO JlSI coBpeMeHHoA
a3po HHaMHKH. He CJlyqaAHO ee HCC1Ie OB&HHIO - TeOpeTHqeCKoMY,
3KcnepHMeHT8JIbHOMY H C IIOMOI1t.L1O M&TeM&THqeCKOrO Mo enHpoBa-
HHB - 3a nOC1le)nlHe 40 neT nOCB.H eHbl MHoroqHCJleHHble ny6JIHKa-
HH. YKa}KeM HeKOTOpble H3 HHX: [168]-(173).
OTMeTHM, QTO MaTeMaTHQeCKOe Mo enHpOB8JlHe 3 aq as po-
ynPYI'OCTH CTaJlKHBaeTCH C 60J1bWHMH TpYJJ-HOCT MH H MO)l(eT 6blTb
OTHeceHO K CllHCKY npo6neM "Grand Challenge". CJJOiKHOCTb npo-
6J1eMbl CSB3aHa C TeM. '1TO Ma.TeMaTH1.feCKaSJ MOJJ-eJ1h, onHCblBalO ee
3TO HBJIeHHe, BKJllOqaeT B ce6g KaK ypaBHeHHSI ra30BO:A HaMHKH,
TaK H ypaBHeHHSI ynpyroCTH. 9TH ypasHeHHH pewalOTCH COBMeCTHO
npH y eTe no,nBHJKHOCTH rpaJm J TBep oro renal
Oc06bJ:R HHTepec C TOqKH 3peHHSI aapoynpyrOCTH Bbl3bIBaJOT
HeYCTaHOBHBlliHecs ocu.HnJI HOHHLle nOTOKH. KaK BH O H3 npe-
blAY HX MaTepHaJIOB nap arpacpa , B HeYCT8HOBHBmHXCSI TeqeHHHX
BB3Koro ra3a Mor)'T B03HHKaTh OC JUll:tHOHJnde pe)KHMbJ c OCTa-
TOqftO 60nblIlHMH aMDJIHTYAaMH KOJIe6a.HHA )J.aBJIeHHB. 9TH nepe-
n bl aBJIeHHB, 803 eACTBYSl Ha ynpyrHe CTeHKH KOHcrpYK HH, 8
CHJ1Y B3aHMHOro BJIHSlHHJI ra30BoR cpe J H TBep Oa ynpyroA no-
BepxHOCTH MorYT npHBecTH K8J( K Ka'leCTBeHHOMY H3MeHeHHJO ra3o-
2. l\t!OAeJJJ,pOB&HJle neYCTaIIOBHBmRXCR OCIVfJlJlJUl,JlOHHblX TelleHHJI 259
HHaMHqecKoro nOTOKa, T8J( H K Aecl>oPM HH H a>Ke pa3pymeHBIO
KOHCTpYK HH.
npoAeMoHCTpHpyeM BO
MO>KHOCTH KDHeTlAecKHX
M
.
cxeM "PH MO enHpoBaHHH
np06JIeMbl aspoynpyroCTH. B
KaQecT8e o6TeKaeMoro TMa
pacCMOTpHM 2D-KaBepHY. y
KOTOpOa CTeHKH )l(ecTKHe, 3a
HCKJlIOqeHHeM Ha, KOTOpoe
rl0 anHS1HHeM KOJIe6aHHA ra3a
MO>KeT HCnblTblBaTb OC1(HJI- PRC. 93. CPopwa kaBepHhl: 1 - JKeCTKoe
J1HJ..UI¥I. CXeMaTHQecKH 3'Ia HO, 2 - ynpyroe o
KaBepHa H306pa)f(eHa Ha pHC. 93. IIpe,L\nonara.eTCiI, 'ITO BHeWHn
Ha6erarolQHH nOT OK HMeeT CKOpocTb Moo = 1.35. npeACTaBJIeHHbJe
HH>Ke pe3ynbTaThi COOTBeTCTBYIOT aHHblM pa60T [174, 175J.
,JJ:n.H ODHcaHIDI npo eccoB KOJIe6a.HHA B TBep OM Tene HCDOJIb-
3YeTe WHpoKO RpHMeHReMbIA: MeTO KOHeqHhlX 3JleMeHTOB (176]. B
3TOM CJlyqae CHCTeMa ypasHeHHR, OIDICbJB81OID.mc nepeMe eHHe a,
MO>KeT 6hJTb 3anHCaH& B BH e
L
H
'L
JC
x=L
M8 + D6 + KeY = F,
(2.7)
r,Ae M - MaTpH a Mace KOHeqaLIX 3JleMeHTOB, D - MaTpH a ewn-
cPHPOBaHHB, K - MaTpH a >KecTKOCTH. F - BeKTOp HarpY3KH Ha
nOSepXJIOCTb TBep oro 'reJIa, 6 - BeKTOp Y3J10BblX cMew.eHHA Ha.
3HaqeHHH MaTpHQ MacC H >KecTKOCTH CHCTeMl:d onpe eJ1S1IOTCSI
reOMeTpHeR KOH<:rpYK HH, a Ta.K;)I(e CBoACTBaMH Ma-repuana (nJlOT-
HOCTb, KOHCTaHTbl ynpyroCTH). MaTP a eIdD4>HpOB8HWI oope-
)J.eIDIeTC.H 3M HpHqeCKHM nYTeM, no H3MepeHHSlM 3&TyxaHHH eCTe-
CTBeHHblX KOJIe6aHHD. Y HTbIBaB npH6J1H>KeHHOCTb OnHCaHW1 Mexa-
HH3M& eMn<t>HpoB&HHH. qaCTO (176) B KaqecTBe MaTp bJ 0 6epYT
JIHlleltHYIO KOM6HH HIO MaTpH K II M:
D = o:K + 13M .
(2.8)
YpaBHeHlfiI, onHCblBaJO e ABlDKeHHe ra3a, H ypasHeHMB, onH-
CbI88J01Il.He KOJIOOaHHB nOBepxHoCTH Te1Ia, CBS13aHh1 Me>KA)' c060a
CJIe IOIqHM 06paaOM. 113 ypaBHeHHA" ra30BoD HHat.tHKH onpe en.Sl-
IOTCH KOJ1e6anUB ,lI.a.BneHHSI ra3a B6nH3H nosepXHOC'rH, C nOMO blO
KOTOpbJX H Q>OpMHPYeTCSl BeKTOp HarpY3KH F. B CBOIO O'lepe , H3
ypasHeHI1S1 (2. 7) HaxO HTCj1 CKOpocTb CMem.eHHH nOBepxllocTH , KO-
TopaH HCfiOJIb3YeTCH B KaqeCTBe rpaHHqHLIX ycnOBH:A: ,lVIB ypaBHe-
HUH, onHCblBaJOI.QHX H)KeIIHe ra3a.
PacqeT KOJIe6aHHA nJIacTHHhI C 3aKpenJleHHbIMH KpasIMH, Koett
BJI eTCH AHO KaBepHbI, OCTaTOq}fO rrpOCT.. JIjI pacqeTa Te"leHHSI
ra3a HcnOJlb30BaJlacb paccM(YrpeHHaB B 1 rJl. IV KCPC C KoppeK-
D:HeA. OC'mHOBHMCH 6OJlOO nOAP06Ho Ha Y eTe nOIJ;BH>KIIOCTH rpaHHQ
...
npH pac e're ypaBHeIIHH, OI1I-ICblBalOl HX H)I{eHHe ra3a.
PacCMOTpHM (pHC. 94) OCTa-
TO (HO npoCTYIO nOCTaHoBKY, orpa-
HH IHBaSiCb 3.MDJlHrry ,[{oA KOJI e6 a-
HHA. KOTopbIe He npeBbIllIaJOT Be-
nH'lHHY h BepTHKaJIbHOrO pa3Me-
pa nepBoro CJIOSI npHMOYroJIbHbIX
CqeTHblX .HtteeK. IIpH 9TOM 8 nep-
PHC. 94. CTPYKTypa paC'ieTHOA HOM CJloe gqeRKH (pHC. 94) HqeR-
ceTKH OKono ,lUIa KaBepHhl KH TpaneQHeBH Hhlet a OCTaJIbHhle
npSlMoyronbHbJe. SI aJlnpOKCH-
Ma HH ypaBllellHA, onHChlBalO X H)f(eHHe ra3a ria TJ)aneQHe-
BH)J.HblX SI'ieAKaX HCnOJIb3YeTCB TOT }ICe caMblR no xo;( nocTpOeHH.H
KCPC Ha HOCTPYKTypHpoBaHHblx ceTKaX (rn. II-IV).
06paTHMcSI BHat.f3JIe K pe3YJThTaTaM pacqem 'iHCTO 10 HHa.MH-
qCCKoA 3a;::{a'-lR C nO BH}KHbIM Kone6JIIO HMCH HOM KaBepHbl. AM-
11JlHTYAY H qacTOTY 3THX KOJIe6aHH 6y eM C'tIHTaTb 3 aHIIOH, He
CBg3aHHo:A C TeqeHHeM ra3a. TaKaSI nOCTaHOBKa 3a/1.a'iH MO>KeT Mo-
)l.eJIHpOBaTb BnHHHHe BH6PaD.HH, r€HepHpyeMoH pa3JIHttHbIMH Mexa-
llH3MaMH BHYTPH JleTaTenbHoro annapaTa, Ha nOBe eHHe ra30BOro
B
c
hi
A
o
nOTOKa.
· B ,AaHHoR MOHorpa HH MY He 6y eM K8C&ThCH 06Il(eH38ecTHLIX MeTOJ1.0B pe-
weHHS CHCTeMbi (2.7) CPeAH KOTOpyX MO)l(HO YKa3aTb WeTO):tLI HBTerpHpODaHHS
c nOMOD.lb1O HeRBHblX H SBHblX cxew) a Ta.K>Ke MeTO , OCHOBaHHhle Ha peweHHH
0606weHHoA np06JJeMhl c.o6cTBemtblX 3HaqeHHA CHCTeMhl (2.7) H npHBe eHHH ee
K AHarOH8JlbHOMY BHAY.
9 2 . lYIOAeJlHposanHe HeYCT&IlOBJlBUIHXCJl O JlJlJlsmHOH1fWX re eHHR 261
1SO
uso - ... - - -- -
a
SPl ,.0 -----
120 .P".- .. --- --
100
0 40000 10000 120000 160000 200000
I (r-.)
180
160
140
SPL
120
100
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0 40000 80000 120000 160000 200000
/(riU
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SPl. 140
120
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0 40000 80000 120000 16OCIOO 200000
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, +_____._ __ __ o-
r
'40 - ---- ...
SPL
120 - ---
100
o
40000
10000
120000
leoooo 200000
I (lit)
PIlC. 95. CneKTphi KOJIe6aHHA AAB1Iemur ra38 Y AHa xaaepHhl B TO'IKe z = L/2:
a) >KeCTKaR CTeHKa, 6), B), r) - DpH BhlHy >K-1teJIH X KOIIe6ambtx ABa: 6) 38 Kr ,
B} 54 Krl\, r) 74 J(I'
HCnOnb30BaJlHCb CJIe)U'IO He napaueTpbI BHeWHerO DOTOk8. B03-
AYxa H reoMeTpHQeCKHX p83MepoB KaBepHhI: 'IRCJlO Maxa Moo = 1.35;
\fHCJIO PeaHOJIh ca) onpe,n:ensreMoe no napaMeTpaM BHellillero fiOTOKa
H rny6uHe BbleMKH, Reh = 3.3 · 10 4 . TOJIID.HHa norpaHlt'iHOrO CJlOSl
hi H = 0.041; LI H = 2.1 (L = 6.2 MM, H = 3 MM). AMnnHTY-
a S.b.tHy eHHbIx KOJIe6aHHH He npeSblm8JJ& 10- 3 H. t{acTOTbI 3THX
OCIJ.H.J1nSI Ha Bb16HpaJIHCb cne)U'IO He: 11 = 38 Kr , /2 = 54 Krl.\,
/3 = 74 IU .
Ha pHC. 95 H306paJKeHbI cneKTphl yposlUI nynbcal.\HA aBJIeHHH
(B Ael\H6eJIax) 8 TOqKe 86nH3H cepe HHbI Ha KaBepHbl. ,n:nH CJlyqas
>KecTKoA CTeHKH (pUC. 95,a) xopowo B Hbl )!.Be OCHOBHLJe qacTOTbl
nYJlbCal..\HA c COOTB eTCT BYIO MH 3HaQeHHSlMH 38 Kr Ii 76 KrU..
Ha pHC. 95,6 npe,l\CTaBJIeHbI aHaJlOrH'IHhle aHHble, Kor,qa qa-
CTOTa Bb1HY>K eHHbIX KOJIe6aHHD /1 = 38 Kr COBn aer c nepBOa
OCHOBHoR qacTOTOA c06cTBeHHhlx KOJIe6aHHA ra3a. B QenOM nOJlY-
'1eHHble cneKTphl BeCbMa 6nH3KH, a He6OJ1hWOe YCHneHHe aMnnHTY-
)J;bl OCHOBHO:R qacTOTbl BnOJlHe O&DBCHHMO.
BnH3KaH KapTHHa Ha6nlO aeTcsr (pHC. 95,r) npH \lacToTe BbJHY-
>K eHHblX KOJJe6aHHA 74 Kr . 06paTHM BHHMaHHe H&AYDJleT BTOpoR
OCHOBHOR qaCTOTbI. Ero nOSlBJIeHHe xapaKTepHO AJI5I CJIyqaB t Kor a
qaCTOTa BbIHY)l( eHHhlX KOJIe6auHI 6nH3Ka K c06CTBeHHoa.
BecbMa HHTepeceH cnY'IaA, Kor a qacrom BLlny>KAeHHbIX KO-
ne5aHHA /2 = 54 Kr oaxOABTCSI nocpeAHHe Me>KPY ,lUJYMH c06-
CTBeHHbIMH qacrOTaMH (puc. 95,B). CneKTP nYJlbC&lJ)lR CHJIhHO OT-
JlH1iaeTCB OT CJ1yqas )KeCTKOA CTeHKH (pRC. 95ta). OC06blA HHTepec
Bbl3bIOalO'!' paHee OTCYTCTBOBaBWHe KOJIe6aHHSI HH3KOA 'IaCTOThi -
npH6JIH3HTenbHO 18 Kr - c aMnnHTy oA, ;J.ocTHraJOw-eA 150 B.
Cne,LU'eT OTMeTHTb, qTO TOT >Ke xapaKTep cneKTpa Ha6.TIlO aeTCB H
8 npaBOM BepXHeM yrny KaBepHbI - TO'IKe, aJIeKO OTCTOHID.eA OT
KOJ1e6J11O eroCSI Ha (pRC. 96).
TaKHM 06Pa30M, BH lIO) 'iTO ,lVISi HeYCTaHOBHBDIHXCB pe)J(HMOB
rrpH HeKOTopblX qaCTOTax BbIHY>KAeHHblX KOJ1e68.HHA aJKe He6OJ1L-
woD aMRJlHTYJJ.bI, THn pemeHHj{ MO>KeT cy eCTBeHHo MeHHrLCS1. IIo-
06Hbla 9<P<peKT, CBil3aHHbIA C nOSlBJIeHHeM HOBbJX qaCTOT HHTeH-
CHBHbIX KOJIe6aHHR B ra3e) MO>KeT OJCa3&TbCH BaJKHbIM AJUI npaxTH-
qeCKHX npHJlO)l(eHHA.
N2. MOAe.JlIlpOB8JfJle HeYCTalfOBHBWHXCR OCl(HJIJlBl.{HOHHbIX Tetlemdl 263
180
U30
SPt
140
'20
100
o
40000
80000
120000
160000
200000
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PRC. 96. CneKTp ypoBHH nYJlhcUQd AUJleHHJI B DpaBOY BepXHeM yrJlY Ka8epOhl
(qacTOTa BhlHY)KAeHllhiX Kone6awtR I = 54 B:r )
10
h. h,
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PHC. 97. CpuHeHHe coeKTpoB nYJlhcaQHA eJlHB AIlS J(asepHbl C HenO,IUSH)I(-
HhlM (nYHKTHp) H OC)J.HnJIHPYI01l\HW HON (conowHaJi JlHHHS)
1eo
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140
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100
80000 100000
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PHC. 98. CneKTpbl nYJI))C 3.I JlA aBJIeHHJI AJlJI pa3J1H11J1WX 'lWlII\HH CT&II&HOro
ttHa: hI = 2 . 10- 14M (nyHKTHp), 1&2 = 3 .10- 4 MM (cnJlOIDBU JDUDlJI)
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llpe,a.CTaBneHHble Ha pHC. 95 H 96 pe3ynbTaTbi no CYTH eJla
nOJlyqeHbJ nYTeM peweHHSI TOJIbKO ypaBHeHHA, onHCblBaJO X Teqe-
HH}1 ra3a c nOABHIKHOH rpaHH eD, H3MeHeHHe no BpeMeHH KOTOpoA
3a.naHO. PHCYHKH 97 H 98 CMOHCTpHPYJOrr pe3YJIbTaTbl COBMecTHoro
peweHRH ypaBHeHHR raaoBoA HI-laMHKH H ypaBHeHHA (2.7)) onH-
CbIBaJOUI.HX ynpyrHe 3<PcI>eKTbl B KOJIe6.nIOll\eMCSI He KaBepHbI. reo-
MeTpHQeCKHe napaMeTpbJ KaBepHbI H napaMeTpLI BHeWHero nOTOKa
TaKHe, KaK " )J.JUJ panee"paCCMOTPeHHoro CJIY aH Bh:HY>K.u.eHHbIX Ko-
ne6aHHH. PaCqeTbl npOBO)J.HnHCb JIB pa3nHtJHblX TOJI HH Nla h H
P83JlHQHblX MarepHaJIOB (CTaJlb, lOpaJIL).
Ha pRC. 97 npe CTaBJIeHO CpaBHeHHe cneKTpOB DYJlbC HH aB-
JleHHSI ra3a B TOQKe B6JJH3H cepe,AHHbl AHa )VIH CJlyqas fle[lO BH}I(Ho:A
nOBepxHocTH H "PH Y"IeTe B3aHMHOro BJlHjJHH5I Kone6aHHA ra30BOH
<Pa3bl H nOBepXHOCTH (CTaJIb TOJImHHoA hI = 2 · 10- 5 MM). B HO,
QTO OCHOBHble Kone6aJJH.H OCTaIOTCSI HeH3MeHHbIMH no qacTOTe D 8,M-
nnH r)r e. O,l\HaKO <POH Kone6aHHA HiUKe B yqae OC J1HpY1O ero
)J;Ha. npOHCXO HT cBoeo6pa3118.S1 nepeKa'lKa 3HeprHH KOJJe6aHHH &C6e-
J1oro mYMa" B 3HeprHIO nYJlbcaQHH nnacTHHbI. PDC. 98 AeMOHCTpH-
pyerr eCTeCTBeHHblJt pe3YJlbTaT, 'iTO ,lI)ISI 6o.nhweA TOJ: Hb1 DJIaCTH-
Hbl KapTHHa Te'leHHH 6nH)Ke K 1'0$1, KOTopaH DOJIY'feHa nH cnyq8JI
HenO,lI.8HmHOH CTeHKH.
4. !vfO,l1.eJIKpOBaHHe Typ6YJIeHTHblX HeYCTaHOBHBUlHXCH
1'eQefIHil B OKpeCTHOCTH K8BepHbl
npHBe eHHble B nepBblX nYHK'raJ( naparpa4>a pe3YJIbTaTbJ OT-
flOCHJIHCb K MO eJIHpOBaHHIO JIaMMHapHblX HeYCTaHOBHBIIIHXCSI Te-
'-JeHUH raJa. BMecT€ C TeM KCPC npe CTaBJIHIOT BC3MO)KHOCTH H
,lI.JISi MOAeJlldpOnaHHSI Te'4eHHA TYP6YJIeHTHblX (178]. HH}Ke, ODHp8.SlCh
tIa a.HHb)e pa60TbI [178], 6YAYT npHBe eHbI HeKOTOpbJe pe3YJlbTaTbI
MO eJ1HpOBaHHH OC HJJ..1HPYIOIl\HX Typ6yneHTHbIx Te'leHHA o6nH3H
npSlMoyrOJlbHoA 20 BbJeMKH.
B KaqeCTBe anropHTMa pac'leTa npHMeHSlJlaCb KCPC C KoppeK-
eR, 8 KOTOpOft BMecTO ecTeCTBeHHoA BH3KOCTH 1£ HCDOnb30BaJlacb
Typ6YJIeHTHa.H BH3KOCTb J.lt. B CBOIO o'{epe b, ,lI)IH HaxO>K eHHjJ JJt
6bIJIa npHBJleqeHa anre6paHQecKaH Mo e.nb Ce6e'lH-CMHTa, KOTOpaJI
92. MOAeJlHpoBaHHe HeYCT3HOBHBUJJfXCSl OCl.{HJIJllIL{HORHblX reqeHH:It 265
onpe eJISlJlaCb CJIe,lU'IOw.HM 06pa30M.
,L(JIH 06J1acTH B6JIH3H TBep oA nOBepxHoCTH npHMeHSlJlaCh <PoP-
Myna
Pt = pl2 ( : ) 2 + ( :: ) 2 .
(2.9)
B CBOIO Oqepe b, ,!VIS HaxO)K eHHg ,[I.JIHHbI nYTH CMeWeHHSI l BOC-
nonb3yeMcSI BbIp8JKeHHeM
1 = kzDr,
(2.10)
r e k = 0.41 - nOCTOHHHaB KapM8.H&) Z - paCCTOSlHHe OT TBep oA
nOBepXHOCTH, Dr - cl>8.KTOp 3aTYxaHHSI BaH- pHcTa
Dr = 1 - exp(z$ fA),
(2.11 )
z. - 6e3pa3MepHOe paCCTOSlHHe OT CTeHKH, A = 26 - AeMncpHPYIO-
3.)I KOHC1 Ta.
80 BHemHeH qaCTH norpaHH'IHOrO MOR Typ6YJIeHTHasi Bg3KOCTb
onHCblBaeTCS cl>opMYJloii KnaY3epa
JJt = O.0168puI6$ I,
(2.12)
I' e Ul - CKOpOCTb B HanpaoneHHH nOTOKa Ha BHeWHeii rpaHHn:e BSI3-
KOH o6nacTH, o. - TOJI1l\HHa BblTeCHeHH.SI, I - Kos4>cI>Hl.\HeHT nepe-
MeX<aeMOCTH.
B o'rpblBHoH 301le BHYTPH BbleMKH Typ6YJleHTHaH BSI3KOCTb BhI-
QHCJISeTCH no cPopMYJle f179]
J.lt = O.168pul6* ( JL ) Dr,
Yd.
(2.13)
rAe Yd$ - Koop HHaTbl p33 en5l1O eA nHHHH TOKa Me HpKynSl-
HOHHhJM H BHeIIlHHM TeqeHHeM.
BHe nOrpaHHQHOrO cnOH Typ6yneHTH&H BB3KOCTb nonaranacb
pa.BHoA HynlO. B Ka'leCTBe rpamrqHblX ycnOBHH B )J;ODOJIHeHHe K
(2.2)-(2.5) npeAnonara.nOCb) 'iTO sa Bxo,qHoR rpaHHqe B Typ6YJIeHT-
HOM nOrpaHHqHOM CJJoe CKOpoCTh nponop OHanbHa paCCTOBHHIO OT
CTeHKH B CTeneHH 1/7. ,LVuI TeMnepaTYpbJ Hcnonb3CBanoCb COOTHo-
weHHe, H3BecTHoe KaK HHTerpan KpoKKO (63).
PaCClvtOTpHM pe3ynbT8Tbl pacqeTa 2D-k8.BepHbl co CJIe IO -
MH reOMeTpH(.leCKHMH nap8.MeTpaMH H lIapaMeTpaMH nOTOKa. tJHCJlO
Maxa Moo = 1.35, qHCJlO PeAHOJIb,l\Ca Re = 2.15.10 5 , HHa K8BepHbl
L = 0.121; L/ H = 5.07. Pe3ynbTa'I:bI pacqem CpaBHHBaJIHCb C aH-
HbJMH 3KcnepHMeHTa (151] H AaHHbIMH pacqeTOB H& OCHOBe HCnOJlb-
30BaHHg YPaBHeHHA' HaBbe-CToKca C MOp-eJlblO BOJIA}'HHa-JIoMaKCa
[157] H (k-£)-MO eJllIMH TYp6yneHTHoft BSI3KOCTH [180J.
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PHC. 99. OTHOCHTeJIbHOe aBJJeHHe p/Poo B TO'IKe %/ H = -0.4 HlDKe BepxHero
npaaoro Kp8JI BhleUKH
Ha pRC. 99 npe,ACTaBneHa 3aBHCHMOCTb a.BJIetlHSI OT BpeMeHH
B o,ltHoR H3 TOqeK npaBoA KpoMKH BhleMKH. PeWeHHe HOCHT HpKO
Bblpa)l(eHHblH Hec HoHapHbJa xap8.KTep. Ha pHC. 100-101 npe -
CTaBneHbI pacnpe eneHIUI cYMMapHoro ypOBHR aKYCTHqecKoro aB-
JIeHH B onb ,lJ).la H npasoA CTeHKH BbleMKH. TaM )ICe npHBeAeHhI
,n;aHHble 3KcnepHMeHTa [151] H pe3YJIbTaTOB pac'ieTOB, B3S1TblX H3 pa-
60T [157, 180]. KaK DRAliO H3 pHC. 100-101. KCPC c MO eJlblO TYP-
6yneHTHoR BH3KOCTH BnOJIHe KOHKypeHTocnoco6Ha c 06I1.I-enpH3HaH-
HblMH 8JlI'e6paHQecKHMH H H<l> epeHIJ)fanbHblMH MO eJlSlMH Typ6y-
neHTHOCTH, npHMeHHeMhlMH B paMKaX YPa.BHeHHft HaBbe-CToKca.
TeM He MeHee, CJIeAYeT 38.MeTHTb, '1TO Bce CYI.I.\ecT8YIOIl\He B Ha-
CTOHll(ee BpeMSI Mo enH ryp6YJ1eHTHOA B.H3KOCTH 06JI a.IOT 06I1J)fMH
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AHa Bhlet.1KH
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npasoii CTeHKH.
He,AOC TK.aM I: Ka>KP;aJI 83 HHX lL'deeT AOCTaTO'iHO OrpaHH'IeHHYIO
o6nacTb npHMeHeHHSI H CY ecTBeHHO 3aBHCHT OT BXOAHIIJ.HX B Hee KO-
3<PcP HeHTOB, KOTophle onpe en5llOTCSI nOJI)'3MnHpHQeCKH, npHBjI-
3h1Basrcb HenOCpe,u.CTBeHHO K pewaeMoA 3 aqelo 91'0 xopomo BHAHO
Ha npHMepe TOJIbKO qTO paCCMOTpeHHOA MO eJ1H (2.9)-(2.13), B KOTO-
paa CMeHa MaTeMaTHIJeCKOrO OItHCaHHSI npOHCXO HT B 3aSHCHMOCTH
OT TOR HJlH HHOA no o6nacTH TeqeHHSI. IIpH 3TOM B 3Ha'lHTenhHOA
Mepe HIlcpOpMa HSi onHpaeTCSI Ira Y>Ke H3BeCTHYIO ti3 KOHKpeTHOrO
3KcnepMMeHTa KapTHIIY nOTOKa.
BbIXO OM 3 eCb SBnHeTCSI 11pBMoe MO eJlJ1pOBaHHe, Kor a Ha
OCIIOBe Henocpe CTBeHHoro paC1.JeTa KapTHHa TeqeHilH onpe en eT-
CSI c t.1aKCH taJ1bHO B03MO)KHOH AeTaJ1bHOCTblO. ECTeCTBelIHO J q'l'() B
3TOM CJIY1.lae aKTYaJIbHblM CTaHOBHTCfl HCnOJlb30aaHHe BbICOKOnpOH3-
BO,lIJ1 r reJIbHb1X MHoronpOQeccopHbIX BblqHCJIHTeJlbHblX CHCTeM, MO -
HOCTH K01'OpbIX Y>KC B HacTOSlmee HpeMH aIOT B03MO)KHOCTD )J)IH
npHMOrO MO eJlHpOBaHHSI HeYCTORtlRBOCreii HeCKOnbKHX Ha.H6onee
KpynHblx MOClllTa6oB.
HIITepecl:IbJM npe,l\CT3BnReTCH KOM6HHHpoBaHHbiR nO OA K MO-
eJIHpOBa.IiHIO Typ6YJICHTJIbJX TeqeHI-IA. B ero paMK:)'x C nOMO blO
npSll\101"O MO,ll.eJ1HpOBaHIIH Ha BbiCOKOnpOH3BOAHTeJIbHhlX cHCTeMax
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fiOJIb3YlOTCSI pa3JIHt.llIble MOl'eJIH Typ6yneHTHoH B.SI3KOCTH. TIpH 3TOM
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qHHOC1'e , Tetw1 C 60J1bWeA CTenellhlO MO)KHO rOnOpHTb 06 ox 0):\110-
pOAHOCI"H H CllpaBe JlHBOCTH npMeHeH:HH JJ..1IH }IX OnHCaHHSI YHHBep-
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8HTC Bce 60JIee norrynspHbIM, B TOM HCJle H B KOJIJleKTHBe HCcnC-
,l\OBaTPJIett PYKOBO):{HMbIX aBTOpoM.
npH8C eM pe3YJlbTaTbJ pacqerra CJIOSI CMerneHHSI, nOJlyqeHHbIe c
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HHSI: a) TCf\inCpaT.ypa, 6) Typ6Y.I1CIITIla.H tJJoI:iKoc-rh
IIO 10LU.hJ{) IIO; ()6}IOrO IIO) XOlI.a (; aHHhJe rlpe,[J.ocraBJlelIbl I1.B. l\6a-
.,1aKHllhJM VI rr.I<. K03)'6cKOa). Ha pl-IC. 102 np YBeAelIa 06u aH cxeMa
pa...'BJ.1rrH rl'.Y )()j'JIeHTHOl'O CJIOH C {( IJleHI.ffl, B0311}U\c1.I{)Ju.ero Ha JIH-
lIHH COIII>HI(OCHOBeHH5I ,l\B}'X ABH)I{YIl lfXCH C pa31fOR eKOpOCTblO ('..lID-
CB 8(Y3l\yXa. B Ka'leC'rHe tOAPJIH) OnHCbIBaIO e" I)a3BIITHe CpCl\He-
If 'leJ1KOMaCllrra6HO 1'yp6y.neHTHOCTH, Ha cpoHC I)acCQHT8.1-IHhIX npH-
MblM rly(re f KPYIIHO 1cl.(:rJTTa6IrbIX Hej"crrOHQHBOcTel1 HCIIOJlh3()BanaCb
(k-E) ) H(p(l)cpeH .H TJbHa5I 1fO e.:Ih. Ha pHC. 103 rII)HBe eHLI Bbl-
t.tHCJIHTe..rIbHbIe illJIHpeH-cpO'fOrpa(pHH Typ6)'JIell'r1l0rO CJI051 CMcwe-
HH5I. ITpH 3'1'01\,t HCIIOJIb30BCi.J1HCb CJIe,uylOII J.fe rrapap.,(e'l'IJhi NIH .lI-BH-
)K)rll HXC5I CJIO€R: Ma == 0.7; Mb 0.3; 1c = 0.19 (3ACCb
Me := (lI a - Ub)/(C a + Cb), f' e C - CKOpOCTh 3BYK8. COOTBeTCTnylO-
rn;ero )-leB03MYIl CHHoro CJIOH); 6 00 = 0.107 MM, Re oeo = U a 6 00/ JJaJ
l'.1I.e j(J, - rr)'I)6)fJleHTH(\'H BSl3T{OCTh) Ol1pe/ e.rIHe taH C IIO 1011 blO (k-E)-
MO e.nH.
3 aM e q a H H e 2.1. llpHMoe Mo enHpoBaHHe Typ6yneIiTHocTH, TaKJKe
KaK no,AXo)J., CO'-leTaJOW)lft npBMoe MO eJIHpOBaHHe H p83JB1 Hble MO eJ1H
Typ6yneHTHoA BR3KOCTH )J)IH OllHCaHHJI MenKoMacwwHblx HeyCTOA:'IHoo-
CTeA, cy ecTBeHHo OnHpalOTC1I H& eT&1JbHoe MO eJlHpoBaHHe ra30;J)iHa-
MHt.teCKHX DoneR. Q,QHaKO BhlqJtCJIHTeJUlM XopOWO H3BeCT:-IbI TPY,lJ.HOCTH,
CBS3aHHble C MO,ll;eJIHpoSaHHeM HeYCTaHOBHBWHXCR Te eHd BB3KOrO ra,.-
Ja npa no,cq>o6HoA npocTpaHCTBeHH BpeMeHHoA AHCKpeTlf38U)fH ypasHe-
HHii, onHCblBalOlIIwHx DOBe,LJ.eHHe ra3O,lJ.HH8MH'IecKHX DOJIeA. DPH'IUHOA 3THX
TPY,lJ.HOC'reA HBJlBeTCB TOT <!>aKT. qTO CXeMHaH Bjl3KOCTb H3 nO.LI.p06Hh1x
CeTKaX OKa3LIBaeTCJI MaJloA H Y>Ke He MO>KeT OKa3hIBaTb cra6HJ1H3HPYIO-
Inero anHS1HHH Ha pe3YJlbTaTbI QHcneHHoro peweHHB. KOCBe:lHbD4 nOATBep-
>K)J.eHHeM 3Toro Te3HC3 1IBnfteTal TO , QTO pe3ynbTaTbi MO)J.eJlHp003HH$I Ha
no p06Hb[x CeTKaX nepecTaJOT KOppeJlHpOBaTb C aHaJlOrH'IHbIMH ,lI.aHHbl-
MH, nOJIyqeHllbIMH npH HCnOJIb30BaHHH I'py6hIX CeTOK. npH 3TOM ,.ltOCTO-
BepHocTb MO,LI.eJlHpOBaHIDI, 8 nepBYJO oqepe,lJ.b HeycroAqHBOCTeA MeJ1KHX
MacwTa60B, BeCbMa np06J1eM&TH'tIHa. 3aMeTHM, 'C.J1'O MHOroneTHUA onblT
Hcnonb30BaHHB KCPC H Kry,D.JUI MOAMHpOB8HIUI T&KHX ,:,eqeHHA CBH e-
TenbcTByeT 0 B03MO>KHOCTH H36e>KaTb nOA06shIX HeraTHBHbIX eKT08.
lfHTepecHo conOCTaBHTb KapTHHhl nOJUl Te eHHSI: no.nyqeHHble
np" HCnOJlb30BaHHH pa3JlHtJHhIX pac eTHbIX ceTOK. B Kaqe<:rBe npH-
Mepa paCCMOTpHM BapH3flT 2D-Ka.oepHbI, pe3ynbT8Tbl ?ac"leTa KOTo-
paR Ha OTHOClfTeJ1bHO rpy60A: CeTKe (115 x 151) Y3noB npe,n:CTaBJIeHbI
Ha pHC. 70-72 a.HHOro naparpa<l>a.
IIpH nepexo e K 60nee nO)ij>06HbIM CeTKaM JJ.O (1201 x 1501) j'3-
nOB HaqHHaeT npO.HBnHTbCH 60Jlee TOHKaB K8.pTHHa TeqeHHSI. TaK,
YAaeTCH Ha6nlO aTbJ Ha"lHH&sI C CeTKH C QHC1IOM y:3JIOB (354 x 451),
KaK Ha neBoR KpOMKe BbJeMKH nepHO qeCKH 3apo>K aeTCii napa BH-
xpeA, facmTa6 KOTOpblX 3aMeTHO MeHhllle MacmTa6a BHxpeR, cYll.{e-
CTBYIO HX BHYTPH BbleMKH. 3aMeTHU, qTO BHXPH BHYTpH BbIeMKH
MO eJIHPYIOTCSI Y>Ke Ha rpy6hIx CeTKax.
3Ta rrapa BHxpeR, 3apo)K,n;aJO aHQt Ha KpoMKe (pHC. 104), Bpa-
aeTC B pa3Hble CTOpOHbI. O HH H3 BHXpeA )UJH:a<e1:CSI B CTOpoHY
HeBo3My eHHoro nOTOKa, r e If HccHnHpyeT. BTOpoB BHXpb BH-
>KeTCH no HH>KlleR qacTH CJlOSI CMeweHHH, <POpMHPYSI cBoeo6pa3HblR
BHXpeBOA IIO CJ10A. llpH 3TOM KapTHHa nOBe eHHH Kp}'nHOMaCillTa6-
HblX BHxpeR BHYTPH BbleMKR HMeeT a6conlOTHO TaKoR )Ke xapaKTep,
KaK H B pe3ynbTaTe Mo enHpOBaHIDI H& rpy6bIx CeTKaX.
a
--
.- ::::'::: =:
::::- =- ::- -:
Tr . r! r
t 4 1 .' 1! :
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Puc. 104. nponecc 06pa30BaHHa OTpbIBHOA 30H&I H8 neuoA KpoMKe BhleMKH
ITo CYTH ena B ,l).aJlHOM pacqeTe HMeerr MeCTO CHTYa.IJ.HB, a.HaJIO-
rH1JHaJI nepexoAY OT KpyrmoMacmTa6HoR K MenKOM8CWTa6HOI reo-
rpaQ>HqeCKOA KapTe, Kor a HapsiAY c 60nbWHMH ropo a.MH H OCHOB-
HbIMH Ma.rHCTpanaMH nOHsnSlIOTC9 60JIee MenKHe HaceneHHbJe nYHK-
TbJ H oporH, HX Coe,l\HHSlIO e. rIpe 3TOM MeCTopacnOJlO)KeHHe
60nblliHX ropo,L\OB H MarHCTpaJIe:B OCTaeTCSI Ha TOM )Ke Mecre, qTO H
AJISI KPynHoMacnrra6HoA K3PTbl.
aHHblA pe3YJlbTaT, HapBAY C OnHCaHHeM cneKTpa uynbCa.IJ.HR
3D-Ka.BepllDI "PH p83nH'IUhlX yrJIax t/J (pRC. 80) t CJlY)KHT e e O M
nO)J.TBep}K)(eHHeM KOppeKTHOCTH KHHeTHqeCKHX cxeM H nepcneKTHB-
HOCTH HX npHMeHeHHH B BblCOKOnpoH3BOAHTeJlbHbIX BbI'lHCJIeHHSIX.
3. reHep H.SI H pacnpOCTp8.HeHBe MaJIhIX
(aKycTHQeCKHx) B03MYIl\eHHA
Hccne oBaHHe reHepaQRH H pacnpoCTpaHeHHH aKYCT eCKHX
B03MYLQeHHH npe,lI.CTaBnfleT co6oA o y H3 HHTepeCHeAmHx H HaH6o-
nee TpY brx 3 aq cOBpeMeHHoA BbJ1.lHCJlHTeJJbHO'A r Bo:A HHaMH-
KH [182]-[186). 9Ta 3 a'la HanpgMYlO CBSI38.Ha c pellleHHeM Ba)Kl!blX
TeXHHQecKHX nporpaMM p IIanpHMep, c xopomo 3HaKOM01t np06JIeMOR
CHH>KeHH ypoBHSI WYMa t HMelO ea 6onblUoe COQHaJIbHOe If 3KOJIo-
fH\.fecKoe 3Ha eHHe. HeYCTalloBHBllIHeCSI H Typ6YJIeH1'Hbie TeqeHHSI
npe,nCTaBJI.SIlO'r c060A oco60 HHTepecHblA 06'beKT )l.JISi IICCJIe OBaHHH
MaJIblX B03My eHHR. B caMOM eneJ xapaKTepHoft HX qeproA SlBJIB-
eTCH B03HI1KIIOOeHHe H HCt.{e3IIOOeHHe BHxpeA pa3HOrO tacmTa6a ( 2
aHHoH I'JlaBbJ), Ka)l{,[\bIA H3 KOTOpblX SHOCHT CBoA BK.1AA B cneKTp
nYJIbCalUtH ,AaBneHHH. HeKoTopbIe H3 3THX nynbcaQHA no cBoeMY
ypOBHIO MO>KfIO xapaK 'epH30BaTb KaK axYCTH1iecKHe.
npol ecc reHepa.QHH H pacnpoCTpaHeHHH MaJlbJX B03MYII.\eHHA
MO.>1{l)O onHcaTb nYTeM npBMoro pacqeTa TeqeHHR BSl3KOro raaa fIa
OCTaT01.JHO no po6HOR CeTKe. KCPC H KfY B npHHlUine 8JOT Ta-
KYIO B03MO)l(HOCTb. O,n;liaKo TaKoA no xo Tpe6ye-r 60JIblliHX BbJ"IHC-
JIHTeJIbllblX 3aTpaT H M3JIOpeaJIeH npH cOBpeMeHHoM ypoBHe Pa38H-
THH Bbl'lHCJ1HTeJlbHOA TeXHHKH.
BoCnOJlb3yeMCiI pyrHM nO O OM K OIIHcaHHIO pacnpocTpaHe-
HHg MaJIbIX B03MY'lQeHHR, KOTOpblA .HBJI.HeTai 60nee Tp Hl\HOHHbIM.
B ero paMKaX ypasHeHHH, onHCblBalOll.\He ,ABH)KeHHSI l'a3a, JlHHeapH-
3YIOTC.SI OTHOCHTeJJhHO axYCTHQOCXHX B03MYIIl.eHHA. O R8KO nOMH-
MO MO,neJ1HpoB8.HHH pacnpocTpaHeHHSl3BYKOBhIX B03My eHH:A cy e-
cTByeT np06JJeMa onHcaHHSI HX reHep H, KOTOpas npeACTaBJlSleTca
6o..nee TPYJJ.HOit. B )J.aHHOM naparpa<Pe, OnHpaBCb H3 KCPC B Ka,I.Je-
CTBe anropHTMa, 6Y YT npoAeMoHCTpHpoBaHbl B03MO)!(HOCTH pewe-
HHSI OOOHX np06J1eM.
1. MOAeJ1HpOBalIHe pacnpocTpaHeHHH M8.JIblX B03MymeHHJt
Cne,llJ'H pa60TaM [187]-[189], JIHHeapH3yeM KCPC c KOppeK H-
eA, KOTOpag B 2 HCfiOJlh30B8JIacb n.H pac'lera, HeYCTaHoBHBillHXCH
J3. reHepaJ.{HJl H paCnpocrpaHeHHe MaJ1b1X BOOMYU{eHHR
273
TeqeHHR.. rIpH 3TOM, KaK 06blQHO, HHTeHCHBHOCTb aKYCTHqecKHX Ko-
ne6aliuA CLJHTaeTOI OCTaTOqHO M&1IOa, C TeM T06hJ MO>KHO 6bJno
npeHe6pet.{b 1.fJIeH8.MH BTOpOrO H 60nee BhlCOKHX nOpH KOB MaJlOCTH.
B CJIyqae 2D-reoMeTpHH llOJlYQHM CJle JO e Bblpa>KeHH.H:
Q'j+l _ Q'j I ,
f}..t + (AxQ ) + (A 7I Q )y =
= ( (BxQ'}z) x + ( (ByQ')j)]I + Hivs + 8,
(3.1 )
r e Q' - BeKTOp MaJlblX B03My eHHA ra30 HHaMH1.JeCKHX napaMeT-
pOB:
Q' = (p', (pu)', (pv)', E 1 ) ,
(pu)' = Upl + pu', (pv)' = Upl + fJv', (3.2)
C = p'(u 2 + fJ2)/2 + pu u' + jJfJv' + p' /(,.,-1) .
Cpe IHe BeJIH"tJHHbI r830 HHaMHqecKOro nOTOKa Q onpe,ll.CJI1I-
IOT nYTeM ero MOAeJIHpOaaHHB H& YMepeHHO nO)ij>06HoH CeTKe,
H NS - JIHHeapH30BaHHble p83HOCTHble eCTeCTBelJHble CCHlIaTHB-
HhIe qJleHhI, KOTOpble MorYT 6blTb BhIAeJleHbI H3 KB83Hm30AHHaMH-
qecKoA CHCTeMbl, Ax, A'll - MaTpH hI KOHBeKTHBHblX onepaTOpos,
( (BxQ')x) % H (.?f(ByQ')y) y - nHHeapmOBaHHble AODC11JHHTeJIb-
HhIe CCHnaTHBHLle qneHbI KCPC C KOppeKll,He:A, nponOp OHaJlb-
Hhle x H t::.y ((1.1)-(1.5), rn. IV). S(x, t) - HCTOqHHK aKYCTHqe-
CKHX B03Myrn.eHHA (onpe eneHHIO S 6y eT nocBSlIu.eH cneAYIO HH
nYHKT naparpaQ:>a): B aHHOM nYHKTe HCTO'tfHHK B03My em[A CQH-
'meTCH 3 aHllbIM.
B03MO}f(HOCTb HCn0JIb30BaHWI ypaBHeHHa (3.1) MSI OnHC8.HIDI
pacnpoCTpaHeHHSI fa..nbJX B03MYtQeHHA onHpaeTCH Ha pa.Hee BblSlB-
neHHYIO CBH3b (rn. III H IV) pemeHHH, llOJIyqeHHblX Ha DCHose npH-
· HapRAY co CXeMaMH, onHpaJO HCB Ha KrY-CHCTeUY (1.1)-(1.5) (rn. IV),
KOTophle H8.30BeM .'BapHaRT A", AJlB MO,QeJlHpOBaHHa 38rA8q akYCTHKH RcnOJlb30-
Banacb pa3HOBH HOCTb KCPC c KoppeK eA (1.22)-(1.25) (rn. IV) (BapHaBT B).
npHUeH1IeMaJI He TOJIbKO sa ftecTPYXTYPHp08a.H11L1X ceTKaX, HO H Ha ceTK&X npa-
MoyrolIhHblX.
MeHeHHSI KB83Hra30 HHaMHqeCKOA CHCTeMbI H KCPC C KOppeKQH-
el, C pemeHHeM, nonyqeHHLIM H& OCHOBe HCnOJIb3OBaHHSI ypasHeHHA
HaSbe-CTOKCa. OqeHh Ba,)KHO OTMeTHTb, qTO TaK >Ke, KaK If AIlS!
KCPC c KOppeK HeR, HCDOJIb30BaHHe SlBHOA: cxeMLI (3.1) npHBO HT
K MSlrKOMY OrpanH\feHHIO T ;S h Ha mar no BpeMeHH [187]. 9TO e-
naeT CHCTe(\llY (3.1) nerKO ocTynHo:R NISI a.Aan iiH K apXHTeKTY-
pe MHoronpol eccopHblX BhlqHCRHTeJlhHblX CHCTeM C pacnpe eneHHoR
na tSlTblO, KOTopble HCDOJIb30BaJlHCb [187]-(189] ;:vISi MO,L(eJ1HpOBaHHii
3a.AaQ aKycTHKH.
PacCMOTpHM HeKOTO-
pble pe3YJlbTaTbl pac eTOB
pacnpoCTpaHeHHSI 3BYKOBblX
KOJIe6aHHA, A.'1H nOJIytteHHH
KOTOpblX HCDOJIb30BaJIHCb
MHoronpol(eCCOpHble Bbl'lHCJlH-
TeJIbHbIe CHCTeMbl Parsytec CC
H MBC-IOOO. Ha pHC. 105
npHBe eHa CXeMB. TeqeHHSI BB3-
Koro ra3a o6nH3H 2D-KaBepHbl.
B TOQKe 0 3 8.H HCTO'IHHK
aKYCTH1.JeCKHX B03My eIIHa,
qacTOTa H aMnJlIiTY a KOTOporo
BapbHpOBaJlHCb. MaKCHMaJIbHOO 3HaqeHHe aMnnHT}'):{bl nYJIh A
COCTaBJ1S1J10 He 60JIee 1 % OT cpe Hero ypoBHSI aaneHHSI. IIapaMeT-
phI BHelUHero nOTOKa Moo = 1.35 H reOMeTpHH KasepHLJ L/ H = 2.1
COOTBeTCTBOB8JIH HeYCTaHoBHBmeMYCH OC JIJIH OHHOMY Te'leHHIO,
nO,lij)06HO paccMoTpeHHoMY B npe JU' eM naparpa4>e.
Ha pRC. 106-109 H306pa>KeHbJ ypoSHH aKycT ecKoro aBJIeHHg
no cpa.aHeHHIO C ypoBHeM nynhca HH aBneHHH }VIS pa3JlHq}{bJX qa-
CTOT /0 = 32 Kr H /0 = 38.35 KfQ B TOqKe O. OTMeTHM, tfTO
qaCTOTa fo = 38.35 Kr COOTBeTCTBYeT nepBoA c06CT8eHHoA qacro-
Te KOJIe6aHH.R )VIS! HeYCTaHOBHBWeroCSI nOTOKa.
06paTHM BHHMaHHe H& HeKOTOpLie oc06eHHocTH npe CTaBneH-
HblX Pe3YJIbTaTOB. Bo-nepsbIX, BBepx no nOTOKY ypoBeHb aKYCTH'Ie-
CKHX nynhc HR cyw.ecTBeHHO HH>Ke. SI Hepe30HaHCHOA qacrOTbI
10 = 32 KrQ npOHCXO T 3Ha HTeJIbHOe YMeHbweHHe ax YPOBHjI
M.
Puc. 105. CXeM& pacnpocTpalleHHB
K01Ie6aHHA 8 KaBepHe c ,It'BIDEYW;HMCJI
ra30M: 0 - TO&fKa reHepaltHR Kone-
6aHHii, A H C - TO'IKR H3MepeHHSI
ypoBRJl KOJ1e6aHHft
g3. 1 'enepal.{HR H pacnpocTp&HeHlfe J4aJlbIX B03My eHHR
275
point A
pojnt 0
0.001
0.004
.. . .- ."
" ". . -
.. ., . '. . . .0
.. . .
. . , .
. , . . . . .
. .
.
w
....
0.000
::E
<
. .
.
. .
. . .
. . ,
". o'
. .' . . . . . .
. . . . .
". " . '.
-O.OCM
1%0 130 140 ISO 160 170 180 190 200
TIME
PHC. 106. YpoBHH nynhC A&8JIeHHJI B TOqKax 0 H A AIlg qacTOTW BYHY-
acAeHllhiX KOJIe6aHHA /0 = 32 B
0.008 -
polntC
. . .. , ,.. point 0
0.00"
w
CI
;J
..
0.000
-0.004 -
i' :
:. , :, '. :.
'. : 0 . : . :
. .
.'
..
.', ::t :: ,': . ; f\/ :':
: . \ : .
:: : :: : 0
. . r1 . ·
... e.= I. :
,. . .
.' . .
. 0
.. "
I. . .
. . . I
. . .. .
, .
. J .' -.
.. . .' .
. ".
: II' "
. .
. .
.
j :
. : : : :
: ., 1 0 "I ." . .
'. : . .. :. ,
'.
: ' .
: . " 0 '
:: :. :
... :,,"
.
. I . . . . - I . I
. .. I . I
120 IJO 140 150 160 170 110 190 200
TIM!
PRe. 107. YpOBHH nyJIbCaJ..tHA eHJI. B TO'IK&X 0 H C .AJUI 'I&cTOTbl
/0 = 32 Kr
276
rH. v. MO,J{enHpoB3JfHe TetleHHlI Ta3oBoll cpe bI
0.04
point A
point 0
0.02
r&3
0.00
.....
A..
-0.02
-<
-0.04
60
80
100
120
TIME
J40
160
180
PRC. 108. YponnH nynhcaJt,JdI MeHHg B TOqx&x 0 H A ):VIB '1ac1'OTY
/0 = 38.35 Kr
0.04
point C
.. . . . . .. point 0
0.02
0.00
....
...:a
D..
-0.02
-<
-0.04
60
80
100
110
TIM!
140
160
180
PHC. 109. YpOBItH rrYJlbCau.aA A8BJleIIHH B TOqx&X 0 H C AJlH 'IacTOThI
/0 = 38.35 KJ:'q
N3. reHep HR H pacnpoCTpaHe1lBe uanwx B03My eHHB
277
no cpasHeHH10 C HHTeHCHBflOCTblO HCTOq}fHKa. Ha060pOT, BHM3 no
Te1.feHHIO (TOqKa, C) MO}KHO HaMIO aTb yoenHQeHHe ypOBHH aKYCTM-
qeCKHX nynbca HR. CneKTp nYJlb HD OCTaTOqHO CJI0)KeH, B HeM
MO>KHQ Bb).Ae.tlHTb HecKonLKO xapaKTepHbJX \{acTOT (pHC. 110). Bo-
BTOpblX, npR lacTOTe BblHY)KAeHHblX KOJIe6aHHA: 1 paolloA pe30HaHC-
HOft 10 = 38.35 Krl.\J YPoBeHb aKYCTHT.JeCKHX nynhc HA 3Haq}{TeJlb-
80 Bblllle (pHC. 108 H 109).
180
point C
160
140
..-.-
.ca
e. 120
-J
=--
100
80
60
o 10 20 30 40 50 60 70 80 90 100
FREQUENCY (KHz)
PHC. 110. CneKTpY IIYJlbC A B roue C (/o = 32 Kr ): aepXllBJI KPHBM -
nYJlLCaQHH OCUOBHOro nOTOKa., HHXCHBB - &KYCTRQecJ(ue KOJIe6aRHH
IloMHMO npe CTaBJleHHbIX Ha pac. 105-110 pmYJIbTaTOB npOBO-
;QHJlHCb paCqeTbI Ii tr.JIn JUJyrHx qacTOT BblHY)K eHHblX Kone6aHHA, B
TOM tIHcne H 6JIH3KHX, HO He COBn alOIII,HX C BYMH OCHOBHblMH qa-
CTOTaMH c06c r rBeHHblx KOJIe6a.HHA HeYCTaHOBHBllJerOCfl nOTOKa. IIpH
3TOM 8 CneKTpe qacTOT aKYCTH'IecKHX Kone68HHA Ha6J[IO 8JIHCb xa-
paKTepHbIe ;urnJleTbI. B pacqeTaX co CKOpOCTbIO Ha6er&lO ero no-
TOKa Moo = 3.7 (TetteHHe B 3TOM cnyqae CTaU-HOHaplIO) He Ha6nlO-
,lI.aJlOCb proOHaHCHOro YCHneHHSI ypoSHB aKYCTHqecKHX kOJIe6aHHA.
.H aapHaHTa c Moo paccMaTpHBaJIcsr T8J()Ke HCTO'IHHK co cny-qaA-
HoA qaCTOTOA
A{t) = ao sin(21rfra.ftd t ),
(3.3)
130
120
110
100
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10
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20 «) 60 10 100 120 140 160 110 200
FREQUENCY (KHz)
130
120
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100
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£
(I) . . '1 .
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70
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20 40 60 80 100 120 140 160 110 200
FJUtQUENCY (KHz)
6)
PRC. 1J1. CneKTp nYJ1bCau,HA AAMeHWI B TOliKe C: a) ceTXa 1 x 1; 6) CeTKa 2 x 2
93. feHepa.I{JlSI H pacnpocrpaseHHe M8JlblX B0311y eHHlt
279
KOTOpas MeHBJIacb 8 Han830He OT 1 o 200 Q.
HHTepOCHO 06paTHTb BHHM&-
RHe Ha pe3ynbTaTbI, oOJlyqeHHble
;:vIH eapHaHTa Moo 3.7 Ha
p83JIH'IHblX paCqeTHblX CeTKaX. Ha
pHC. 111 npe CTaBJIeHbJ cneKTpLi Pac. 112. Cxeua upsMoyrom.aol
nYJIbCaI.\HA ,n;a.BneHHSI B TOqKe C, K8HaBKH: . - 30H& HCTOtlHKK8 3By-
nonytteHHble c nOMOIII.b1O paCCfeTa K&
Ha 06bl'-IHOR CeTKe (pHC. lll,a) H ceTKe, B KOTOpoA 'IHCJlO Y3JIOB
no Ka)f(,n:oMY HanpaBJIeHHIO B Ba pa3a 60J1bwe. BH HOt 'I TO Ha 60-
nee no po6HOR CeTKe cneKTp &BneHHg CY1I"ecTBeHHO CJlO>KHee t qTO
npe CTaBJJSleTCjI BnOJlHe eCTeCTBeHHhlM. O HaKO OCHOBHble qaCTOThI
KOJ1e6aHHA: Ii HX 8.MnnHTY I npaKTHQecKH MaJIO MeHBIOTCH. 9Ta no-
BTOpSleMOCTb pe3ynbTaTOB npH cryw.eHHH CeTKH eMoHCTpHpyeT eIne
Pa..1 R03MO)KHOCTM KCPC ;:vIH eKBaTHoro eT8JlbHoro Mo enHpo-
BaHHSJ m30AHHaMHt.JecKHX Te'ieHHIt.
Ha pHC. 112 H306paJKeHa cxeMa pacnpOCTpaHeHWI aKYCTHqe-
CKHX B03My eHHH B npSlMOyroJlbHoA: KaHaBKe. 3m 3 a a SlBJlSleTCSI
8YMepHOa B 'IHCTO ra30,lUlHaMlNecKoR nOCTaHOBKe H TpexMepHol
M.
._ 1l1._.-
130
120
110
LOO
90
""' 80
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"-" 70
I
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rI) 60
so
GROOVE poiJIb C 8IId CL
... 'Istxl'
- - -.. -
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._ _.- -- -. .---.-. --T--- --.....- - . .-- ---,..-- --:----
20
40 60 80 100 120 1«) 160 1m 2QO
FREQUENCY (1Ub)
PHC. 113. CneKTp axycTHtleC1CllX 803..y eHldt B Toqse C (aepXB.. KpHBU) B CL
(HH)KHHJI KpHB8JI)
u
1 B. V. MOJ{eJlHpoB3IlBe reqeHHII ra3osolt cpe,L{bl
BBH,DJ' 6nH3KoH K TOqeQHOR 30Hb! HCTOtJHHK8. 3BYKa nB OrmCaHHH
pacnpoCTpaneHHSI axycTHqeCKHX B03My eHHA. IIo3ToMY AnSI pacqe-
Ta pacnpocTpaHeHHSI MaJIblX B03My eHHA HcnOJJb30BaJICSI 'IpexMep-
Hblll BapHaHT CXeMbJ (3.1) co Cpe,l1.HHMH 3HaqeHHHMH ra30 HHaMH'qe-
CKHX napaMeTpOB, DOJIyqeHHbIMH np" pemeHHH 3aAattH B ,n:BYMepHoA
I10CTaHOBKe.
Ha pHC. 113 H306p3.)KeHbI ( nH BapH&HTa Moo = 3.7) CneKTpbJ
axycrrHqecKtlX B03MYIl\eHHR B TOqKe C H )J.pyroA TO'-lKH KPO}AKH CL,
oTCToSl eA OT TOqKH C Ha pacCT05fHHe L. BH.lIrHOt TO no Mepe YAa-
J1eHHSI OT nnOCKOCTH CHMMeTpHH ypoBeHb axYCTHqecKHX B03My e-
HHR na}l: r. TIpH cpaBHeHHH C aH8JlOrHqnbIM cneKTPOM NISI KasepHbl
(pRC. 111) 8H HO, "fTO B cnyqae KaHaBKH cneKTp nYJJbcaIJ)IA 6o.nee
"MHrKH:R", B HeM MeHee gpKO Bblp8.>KeHbJ OCHOBHble acTOThI KOJIe6a-
HHR. 06a aTHX pe3YJJbTaTa npe CT8.BJISIlOlx:B BUOJIlle ecrecTBeHHbJMH.
2. MOAeJlHpoBaHHe HCTOQUHK8. axYCTHqeCKHX B03My eHHR
Ilocne AeMOHCTpa H pe3YJIbTaTOB, CBH3aHHhlX C pacnpocrpa-
HeHHeM MaJlblX B03My eHHA, pacCMOTpHM B03MO)KHOCTH KCPC ,!VIS
OfiHcaHHH reHepa HH aKYCTHQeCKHX B03My eHHD. B H3J10JKeHHH pe-
3ynbr.tClTOB 3'I'oro Ii CJIeAYlOll(ero nYHKTOB 6y)teM CJ]e OBaTb pa60Te
[190J.
Pa3HOCTHYfO cxeMY THIla KCPC C KOppeKU)'IeA (sapHaHT B, BbI-
paJKeHHB (1.22)-(1.25), rn. IV) HCnORb3yeM ,lI.JISi onHcaHHSI nOBeAe-
HH.SI MrHoneHHbIX 3Ha (eIIHH raaOAHHaMH1.IeCKHX napaMeTpoB. Bbl6e-
peM MacwTa6 BpeMeHH) ;\OCTaTOq}JLlA )VISI ocpe HeHHSI MeJlKOMac-
w11WHblx nynbcaIJ:HH, H npOHHTerpHpyeM no HeMY p83HOCTHYIO cxe-
MY. IIpH TaKoD npo e,LCYpe HeYCTOA'WBoCTH KpynHbIx Macurra60B 6y-
AYT BbJqHCJlSI'fbCjl HenOCpe,lI-CTBeHHO "3 ocpe)UieHHoA cxeMbI.
BbJ t lHCJ1eHHH MOMeHTOB ocpe.n;HeHHoA cxeMbl npe)J.nonO>KHM,
'ITO <PYHK Ha pacnpe,qeneHIDI (BepOwrHOCTh CKOpOCTH npHHKMaTb
TO HJlH HHoe 3t1&tieHHe) MO)KeT 6blTb 3anHcaHa B B e
_ ( 4 ) 3/2 { 3( u - U)2 }
f(x, t, u) = P 31rk exp - 4k '
(3.4)
r)J.e u, u', U - cOOTBecTBeHHo, 3H8'1eHHH MrHOBeHHOH t nynLca oH-
93. TeaepaJ.(HH H pa.cnpoCTp&HeHHe M&J1bJX B03Myw.eHHH
281
HoA H ocpe eHHoH COCTaBJUlIO HX CKOpOCTH, P = f I(x, t,' u)du -
cpe HHSI nnOTHOCTb, u = (lIp) I uJ(x, t, u)du - Cpe)I)lSlH CKOPOCT&,
u' = U - u .
1 J t2
k = u J(x,t,u)du
- KHHeTH'iecKaJI 3HeprHB MenKoMacWTa6HhJX nYJIbc .
B pe3YJIbTaTe TaKOro ocpeAHeHIDI B KHHeTHqeCKH cornacoBaH-
HYIO cxeMY B Ka'leCTBe aprYMeHToB <l>YHK erf S H exp( -8 2 ) 6Y!lYT
BXO,lI.HTb SenHQHHbl SUi = BUiJ r e
B = o.f1
/0: 2 + 13 2 '
(} = 4 '
1
fJ = .J2RT '
KpOlwre TOro, BMecrro aBJIeHHH H nOJIHoR 3HepI'HH Ha COOTBeTCTBY-
IO HX n03H H X 6Y,l1J'T CTOSlTb p" H E-:
p. = p + (2/3){Jk l E* = p(ii 2 + fj2)/2 + pk + fJ/("(-l). (3.5)
rrOJlY"leliHa TaKHM 06paaoM CHCTeMa ypaBHeKHA .HBnHeTCR He3a-
MKllyrroA. R Haxo eHHH KHHeTHtfecKoA sHeprnH T)'p6YJIeHTHo-
CTH k H CBH3aHHOA C Hel Typ6YJleHTHoA BH3KOCTH lJe nODOJIHHM NO-
AeJIb CHCTeMott (k-£)-ypaaHeHHA ):VIa: HaxO}K eHHH k H CKOpOCTH
HCCHnaI HH Typ6YJleHTHO 9HeprHH E:
opk op Uf, k a [ ( /At ) Ok ] -7") _
- + = - p + - - + r - petJ
at OXi. OXi (fk 8 X i
ape 8PUiC lJ [( JJ t ) 8E ] C _£2
at + 8 X i = 8Xi IJ + (h aXi + GEl k 'P - GE2P"k'
rAe
(3.6)
(3.7)
pk 2
IJtt = Cp.-
E
- Typ6YJIeHTHaH BH3KOCTb,
( 2 ) au.
P = IJtSij - 3Pk ij ax:
- reHepa HH KHHeTH\feCKOA SHeprHH Typ6yneHTROCTH,
8Ui au . 2 &uk
Si' = - + -L - - - 6i · .
, OXj 8Xi 38xIe 1
3 a M e 'f a H H e 3.1. nOCTpOeIIH&H 140 enb AJ'IB Haxo eHHJI ocpep.-
HeHHbIX 3H&'IeHHA ra30;nlHaWHQeCKHX napaMeTpOB H SHeprHH TYP6yneHT-
HOCTH CYll\eCTBeHHO ODHp&eTCSI Ha npeA11OJIO>KeHHe 0 MenKoM&cwTa6nocTH
nYJlbcanWt. B C&MOM Aene, BYpa>KeHHe (3.4) A1IJI ct>YHKnmt pacnpeAeneHHH
CnpaBe,lVlHBO .nHWb AJlR IDOTpOnHoA Typ6yneHTHOCTH, npHcYll{eR HeYCToA-
qBBOCTHM MeJlKHX Macnrra60B. a H caMO npHMeHeHHe (k-E)-MOJteJIH,
CTporo rOBOp.R, onpM,QaHO npH BLIDOJIHeHHH OCHOBHhlX 3aK3HOB TYp6y-
neHTHOCTH, CnpaBe,nJ1HBblX A1IR MeJlKOMBCWT86HLIX O,lUfOPO){HblX HeycroA-
ttHBOcreA. n03TOMY Heo6xo;I.HMb1M YCJlOBHeM npHMeHeHHB nOAo6HoR npo-
u.e,aypbl HaxO)K)J.eHHB ocpe eHHbIX 3HaqeHHa .RBn.ReTCR HCnOJlb30BaHHe
,ltOCTaTOlfHO no po6Hb1X CeTOK, C TeM tlTOObJ KpynHoMacwTa6Hb1e HeycToA-
'{HBOCTH Onp eJ1BnHCb H3npSiMYIO. Kax Y>Ke OTMe'laIIOCb (n.3 2 JI-&HHoA
rnaBbl), aHHbIA no)J)COJi. JlBnJleTCJI O,LUfHN H3 cOBpeNeHHblx HanpaBJIeHHA
'fHCJleHHOrO Mo enHpoB&HHJI Typ6y neHTHOCTH.
3aMe aHHe 3.2. (k-e)-wo,r\eJlb, KOHe'lHO, He HBJlSeTai eAHHCTBeH-
HO B03MO>KHblU cnoco6oM Ja.MblKaHHB CHCTeMbI OCpe.D)leHHLIX ypaoHeHHA.
HCnOJ1b3YJl ,AOnOJ1HH"l bHYIO HHct>OpwattHIO, MmKHO A1I nHX eneA npH-
MeHHTb H anre6paH'fecKHe ..o enH (191).
nOCTIe }Jaxo eHWI OC HeHHblx 311a'leHHR ra30;I)UlaMH'IecKHX
napaMeTpOB H 3HeprHH Typ6YJleH'I1IOCTH k MO)l(HO npHcTynaT K
<POpMHpOB8.HHIO HCTOqHHKa aKYCTH'IecKHx B03My eHHA S H onHca-
HmO we pacnpocTpaHeHHJI C IIOMO IO nHHeapH30saHHblx ypaBHe-
HH1t (3.1). OTMeTHM, q'I'() nOCTpoeHHe HCTO'IUHKa B03My eHHA, a-
ItIIJlo.ll...A I'IN.UI..6
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PRC. 114. CxeM& p8C'lemoi o6nacTII fJII. TeqeHIUI nJlOCXoA C11>YH
g3. reHepaI.UlSi H pacnpocTpl1lfeHlfe M&1l121X B03My eRHR
283
>Ke OnHp8.HCb Ha 3HaHHe xapaKTepHCTHK Typ6yneHTHoro Te'leHHSI,
npe)J.CTaBJIHeT c06oA: CaMOCTOJlTeJIbHYIO C1IO>KHYIO 3 aqy, B 311aqR-
TeJ1bHOH Mepe BhIXO H JO 3a paMKH eneR aHHo:R MOHorpa<l>HH.
IIpH 3TO [ eo MHO rOM HCDOJIb3yeTCsI Ta HJtH HH8JI 3KcnepHMeHTaJlb-
HaSl ftl<POpt..I HSI 06 H3YQaeMOM J1BneUHH.
I1po e"'loHcTpHpyeM npo ecc nOCTpoeHHjI HCTO IHHKa Ha npHMe-
pe 3a,n.aqJ.1 0 cBepx3BYKOBOA nnocKoA: cTpye, cxeMa Te eHHR KOTO-
poA H306pa>Kella Ita pec. 114. Cor nacHo pa60Ta.M [190 1 192], HC-
TO'iHHK S B CHCTeMe (3.1) MO>KeT 6hJTb 32.AaH B BHAe BeKTOpa
S = (81182, S3, 84) C KOMnOHeHTaMH
81 = 0, 82 = -!; [p{ u 12 - u 12 )] , S3 = 0,
84 = [p( u,2 - u'2)] + :x [P u ( u f2 - u (2 )] + (3.8)
+ [pu(U'2 - u'2)] .
3 ecL u 12 onpe geTCB H3 KHHeTH eCKOA 3HeprHH nYJlbca Ha k B
npe nOnO)l(eHHH, QTO Typ6yneHTHoCTb S1BJIJleTcg H3OTponHoA:
U12=jk.
(3.9)
113 ou.eHO\fHoro COOTHoweHHSI
..; u 12 0.7max lu'l
H Bblpa>KeHHB (3.9) cne)Q'eT npu6nIDKeHHasI cPOPMYJI& AnB BbJtfHC-
JIeHHjI aMnJlHTY bJ nYJ1bC A CKOpoCTH u':
A = maxlu'l = IP.
(3.10)
tJ.ng HaxO>KAeUHSJ ttacTOT nYJIbC Hft npHBJte eM OnOnHHTeJIb-
IfYIO HH<I>opMa alO co CTOpOHbI K8.K HaTYpHoro, 'IaK H BLI'lHCJJHTenb-
HOro 3KcnepHMeHTa.--
'Ha nepBhlA B3rn.R.D. BY3 WBaeT Y J1eHHe BKJIJO'IeHHe B HCTO'IHHX S (3.8)
H3BecTHO [193]. QTO 'IRena CTPYxaJIlI B CJIoe CMeweHHH CTPYH
onpe eJI5IeTCH KaK
Sh = fx/u,
(3.11 )
r e x - paCCTO.SIHHe OT cpe3a conna, 1.1, - XapaKTepHoe 31la'ieHHe
CKOpOCTH B CeqeHHH crpYH. I103TOMY xapaKTepHaJI qaCTOTa f HC-
TO"IHHKOB B Ka)K,l\OM CeqeHHH 6Y,lI.eT
f = k 1 u/x.
(3.12)
KaJKAOe Ce'lel-[He CTPY" H3nYQaeT mYM Ha COOTBeTCTBylOlIJ.HX qa-
CTOTax: Ha\f8J10 CTPYH ,n:aeT BblCOKOqaCTOTHble Kone6aHHSI, ,.qanbHHe
yqacrrKH CTPYH - HH3KOqacTOTHble. B CBOIO Oqepe b, NIR onpe e-
neHHR eenHqHHbl k} npHBJICqeM ononHHTenbHblR Bbl'iHCJlHTeJIbHbIR
3KcnepliMeHT.
B nOrpaHU"IHOM CJIoe Ha cp e conna 3 aBanC}l CTOXaCTHqeCKHR
HCTOt.tHHK H3nY leHHSI He60JIbWoA aMn.nHTY . C 3THM HCTOqHHKOM
H C nonyqeHHblMH pal-lee 3HaqeHHHMH ocpe,lVleHHblX ra3OAHHaMHQe-
CKHX napaMeTpOB npOBO)J.HnCSl pacqeT pacnpOCTpaHelfUH aKYCTH'ie-
CKHX 803My eHHa Ha aCROBe CHCTeMbI (3.1). Ha pHC. 115 npe c'raB-
JleMhI xapaKTepHble CneKTpbJ nynbcalJ)lA AaBneHHSI B pSl e CeqeHHA
Tpy6bl. CneKTpbl t KaK BbISlCJfHnOCb, npaKTH\feCKH He 3aBIICHT OT cno-
c06a B036Y)K) eHHjI HCTOQHHKa. 110 BhlBBJleHllLIM nHKaf.{ 8 cneKTpe
onpe e.n lOrrCjf xapaKTepHble \(acTOTbl H COOTBeTCTBYIO e HM qUC-
.na C rpYxaJl5I.
PeanbH3.jI crPYSI "PH B03AeHCTBHH Ha Bee cTOxacTHqeCKOrO my-
Ma H3JIYQaCT 3BYK Ha xapaKTepnCTHqeCKHX qacTO'IaX, COOTBeTCTBY-
IOIIJ.HX llHKaM 8 CneKTpaJILHbIX HarpaMMax. II03TOMY H npe no-
naraeTCSI onpe eJlSlTb qHc.na CTPYxaJIH Ila aCHOBe qaCTOTHbIX MO)).
MaKCHMaJlbHOA aMnnHTY bl. 9TH 1fHCJI& OKa3aJ1HCb B AHana30He
Sh == O.B -:- 1.1. ,nnH MOAenHpOBaHHSI aKycrrH"IecKoR 3 aqH H onpe-
AenellHSI IacTOTbl f qHCnO CTPYXaJIjI npHHHMaJlOCb paBHblM Sh = 1.
TOnbKO npo)J.OJIbHLlX nYJ1bC A. OAHaKO OIlM. B CJIJIY pemeRHJI ypaBHeHH9 (3.1),
MrHOBelWO Bbl3hJBaJOT nOSlBJIeHHe 3KYCTH'IeclOtX B03Myw.eHHA Bcex KOMlIOHeHT.
BOJIee nOJIJloe OIIHCaHHe HCTO HHKOB B CHJIy npeNIOJIO)l(eHHB 06 H30TpODHOCTH
Typ6yneHTI[OCTH npHBO HT K aHaJIorllq)fol KapTHHe pacnpocTpaueDBH 3BYKa,
KOTOpas B ropa3):{O 60J1bWeA CTeneHB 3&BHCHT OT aMIIJIH'I'y)l.bl A H acTOT nyJlL-
cau.HH u / .
3. IeHepaIJ,HH H pacnpOCTpa..neHHe M&1IbIX B03MymBHHR
285
110
100
90
80
70
-C
60
50
40
30
20
0
....-...-. x 18 mm
......... J( == fT1fT1
X = 95 mm
.. x = 206 mm
IJ:--: ...... ... -"
20
40
60
80
100
f(kHz)
PHC. 115. CnexTpLi (IYJ1bC 3.I HA p'aaneHHX n& Pa3JIHqHblX pacCToHHHHX OT cpe3&
conna
3. MO,l1J JIHpoBaHHe InYMa CBepX3BYKOBOR CTPYH
iloCJIe 3aBepmeHHjI IIOCTpoeHHH Mo enH, onHCblBalO eR I'eHepa-
If-HIO H pacnpoCTpaHel-lHe WYMOB B cBepx3BYKOBOH cTpye, npHBe.n:eM
J-leKOTOpble pe3YJIbTaTbi pac leTOB. PaccMaTpHBaJlaCb IIJIOCKaH CTPY.H
(pHC. 114) B03,a.yxa., BLITeKaIO ero H3 enH pa3MepOM d = 6 MM npH
CJIe,lQlIO HX ra30 HH8.MHqecKHX napaMeTpax: lfHCJIO Maxa Ha BbIXO-
e H3 w.eJlH Ma = 1.01; 'lfHC1IO PeA:HOJIbACa, pac'tJHTaHHOe no napa-
MeTpaM IIl.eJIH Red = 105; creneHb HepacqeTHOCTH n = Pa/Poo = 2.1.
Te fnepafrypa JJa Bblxo e H3 me.nH Ct:tHTaeTCH paBHoA TeMnepaType
OKpY)lffilOll ea CpC,l).bJ. TOJI Ha nOrpaHH'tJHoro c.nOH nepe,n. TOqKo:A
OTphlBa Ct.JHTaeTCSI paDRoA 0.6 MM.
B pe3ynbTaTe pacqeTa, Cpe,AHHX 3HaqeHHIt ra30AHHaMH'lecKHX
rrapaMeTpOB qeTKO BH)J.H& 6otfKoo6p83H8S CTPYKTypa CKaqKOB ynnOT-
HeHH.H, xapaKTepHaJI NJH HepaC'IeTHOrO pe>KHMa HCTe'leHHH CTPYH
(pRC. 116)6 H 117 J6). BCJ1e CTBHe 3Toro aBJIeHHe Ha OCR CTpYH Me-
HHeJ'CSi BOJIHOo6pa3HO, TaK >Ke KaK H B a.H8JIoru1.fHbIX 3KcnepHMeHTaX
, ,p. . .='" '::"Lt .11!' " '1t:.. -; .
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a)
6)
6)
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QHA nJIOTHOCTH (a) II ocp HeHHoe no-
ne nnOTHOCTH (6)
PJlC. 117. MraOBeHHoe none nynb"
carntA npOAOJ1bHOA CKDpOCTH (a) H
OCpe,lUleHHOe none npc;:t.OJILHoA CKQ..
poem (6)
(150). TaM >Ke (pHC. 116,a H 117,a) npHBeAeHbI MrHOBeHHble nonS!
nynb "A IIJIOTHOCTH H npo OJIbHoA CKOpoCTH.
Ha pHC. 118 rrpe,l1-CTa.BJIeHbl H30J1HHHH KHReTHtJeCK:)A 3HeprHH
Typ6ynefITHocTH. BH O, qTO Typ6yneHTHocTb npaKTH'IeCKH OTCYT-
CTBye1' B 06nacTH nOTeH H8JlbHOro SI)ij)& CTPYU. BblCOKHe ypOBHH
Typ6yneHTIIOCTH Ha6J1IO,AaJOTcg H& KpOMKe III-eJIH. no Mepe p83BHTHSI
rJlOB CMemeHHH C pOCTOM paCCTOHHHSI OT eJlHJ MaKCHMYM ypOBItH
Typ6y..neHTHOCTH CMe aeTCSI K OCM CTPYH.
H3MeHeHHe YPOBHB nynb HA aaneHRa B onL n)'qeft 1 H 2
(pHC. 114) npHBe eHbl Ha pHC. 119. MaKCHMaJIbHhlA nme COOTBeT-
CTByerr JlY"lJY 2 B MeCTe ero rrepecetfeHHH nepnoA c'60QKH" CTpYH. B
TO >Ke BpeMH pacnpeJ1:MeHHe aKYCTHtJecKoro ,ttaBneHHSI no ny y 1
MHHHMaJlbHO Ha T8.KOM .>Ke paCCI'OHHHH a.nOJlh JlytIa. 9'00 CBSI3aHO C
TeM, t.tTO ,n3.HHaH TOttKa nyqa 1 ne)KHT 3a npeAeJlaMH ''60QKH u .
113 pRC. 116,a H 117,& B O, ITO 30Ha II&H6onee HHTeHCHBHbIX
nynb HA CKOpOCTH neX<HT B OKpeCTHOCTH CJ10H CMemelJHH H <paJ<-
THqecKH COBD aeT c 30HOA cy eCTBOBaHHB M&KCHMaJ1bHO:R KOBe-
THQecKOft 3HeprHH Typ6yneHTHOCTH. B TO >Ke BpeMSI 308h1 Ma.KCH-
MaJ1bHbIX 3Ha'leHHA nYJlbca R.nOTHOCTH ( aBJ1eHHB) <P&KTHTIecKH
OrpaHHQeHbI ' OqKaMHn CTpYH.
I1pO OJl>KasI pa3roBOp 0 B03MO)KHOCTHX KCPC H KrY-CHCTeMbl
J4. MO,lteJlBpOBaHHe TeCfeHH;; XJfMH'feclUf pearHPYIO ra30B 287
10
10
20
30
40
50
o 0
PHC. 118. none ICHHeTlI'IocxoA 3HeprHH Typ6yneHTHOCTH
175
170
165
160
155
150
... 145
140
135
130
125
I
. ! - - .- --- -- -r-
I
. :: . I
--,.. ... . ..r.. . ..._- .---r"- ------ ----:. . r--.
.
. j
--..---....T... . ... -.--.. ---....----..- -- --....-... -..-..--
- -. . - ..- . t . - - .. .. - - .. - - ... to' . .. . . -.. . \ .-..
o 20 40 80 80 100 120 140 180 180
Ll (MY)
PHC. 119. OCpeAHeHHblA ypOBeHh nynb p;aaneDBJI B p83nH'IHhlX TOqxax
Jly eA 1 J.I 2 B Typt5yneHnloM none
B MO eJ1HpoBaHHH HaH60J1ee cnO)KHbIX npo6.neM cospeMeHHoA Mf,lQ>O-
H ra30BOA HHa.MHKH, nepeA,qeM K paCCM01'peHWO Te eHmt XHMHtJe-
CX" pearHpylO x ra3OB.
4. MO eJlHpoB8HHe Teqelllli: XHMII'IeCKII
pearHpYIO BX r830B
MOAeJlHpoB8.HHe MHOrOKOMIlOHeHTBhlX TeqeHd raa& C XHMIl'le-
CKHMH peaKl.\HBMH npeACTaBJlSleT 60J1bWoA npaKTH'IeCKHA HHTepec
B CBSl3H C MIIOrO"lHCJIeHHblMH TeXHOJIOrHqeCKHMH npHJIO>KeHHHMH H
npo6neMaMH, CBSI3aHHhlMH C oxpaaoR oKPY>K&lO ea cpe):tbl. BMeCTe
C TeM, 3Ta np06neM& BBnSleTCSI o,lJ.HoA H3 HaH60nee C1IO>KHhlX B COBpe-
MeHHOM MaT€MaTHqeCKOM MO eJ1HpoB8.HKH. 9Ta CJlO)KHOCTb BbJ3BaHa
CJIe,ll.Y IOI..I.I.H A H n p HqH Ii 8MB.
BO-nepBbIX, nOMHMO anrOpHTMOB, npe a3HaqeHHbIX ,l\nSI pac-
qeTa QHCTO ra30 HlJaMMqeCKHX TeqeHHR, Heo6xOAHMO HCnOnb30BaTb
MeTO bJ, n03BOJISlI011.I.He MO enHpOBaTb XHMH'IecKHe peaKII-HH, H npH-
TOM OnYCKalOw.He COBMecTHoe pellIeHHe ypaaHeHHA ra30BOA: HHa-
MHKH H ypaoHellHH XHMH'IeCKOa KHHeTHKH. Bo-BTOPLIX, HMelOTCSI
3Ha'iHTeJIbHbJe TPY HOCTH B nOC'rpOeHHH MaTeMaTH'IeCKOA MO enH)
KOTOpaH Tpe6YeT B Ka>K OM KOIIKpeTHOM CJlY'iae KaK BbIHBJ]eHH OD-
TH Ia.nbfIOR u:enOQKH XHMHqCCKHX peaK H:R, Tax H nO,L\6opa KOHCTaHT
CKOpOCTeR nnx peaK HA. OC06bIe CJIO)KHOCTH 3 eCb B03HHKalOT npH
nOCTpOeH}lH M&TeMaTH'tJecKHX MO,L\eneii Typ6yneHTHbJX XHMH'IecKH
pearHpyrorn;HX ra30B (194]. B TpeTbHX, BBBA)' 6oJIbmoro 06'beMa
Bblt.IHCJJeHHlt, He06xo MMo HCnOJIb30BaTb BhlcOKOnpOH3BO HTenhHble
BbIQHCJ1HTeJIbHble CHCTeMbl, \lTO B CBOIO Oqepe b Tpe6yer npuMeHe-
llliH cnCQHaJJbHblX MeTO)J.OB J onycKa1OI.lJ)iX pacnapanneJ1HBaHHe) H
,
COOTBeTCT8YI01U-ero napaJlJleJlbHOrO MaTeM&THtJecKoro ooecneqelIHH
(73) 119]. ,lIpyrHMH CJIOBaMH, )J:nSl peweHHSI 3TOA npo6J1eMbJ He06-
XO HMa KOH eHTp HH YCHJIH9 no BCeM TpeM KOMllOHeHTaM TPH8.,lJ.bl
cMO eJlb-anrOpHTM-nporpaMMa" cOBpeueHHoro MaTeMaTH'IecKoro
MOAeJ1HpOBaHHH.
1. MaTeMaTHqeCKaH nOCTaHOBKa 3aAatJn
g perneHHSI 3TO:tt 3a aqH Bocnonb3yeMcSI KBa.3Hra30AHHaMM e-
CKoA CHCTeMoR ypaBHeHH9, KOTOpasI B CJIyqae MHoroKoMDoHeHTHO-
1"0 XHMHQeCKH pearHPYIO ero ra3a (B npocTpaHCTBeHHo- BYMepHoH
IJOCTaHOBKe) npHMeT BHA
- 1 .
pf+ - p! () a () ". 8 ( 2 )
At t + ax (Pi U ) + ax (Pi'v) = ax 2 ax PiU + Pi +
or8 ( 2 ) ora Bra
+ ay 2 8y Pi'V + Pi + ax 2 By (Piuv) + a:y 2 ax (Piuv) +
+ L i - PiQi,
i = It. . · , Ni- 1 ;
(4.1 )
J4. MOAeJUlpoBaHHe TeqeHJfR XJlMJrfeclCH pearHpylOlIVfX l'830B 289
+l - pi a iJ 8 T a ( 2 )
Llt + ax (pu) + ax (pt]) = ax 2 ax pu + p +
( 4.2)
8rO ( 2 ) (Jr{) a"o
+ 7iii 2 ay f:N + P + ax 2 By (puv) + BY 2 ax (puv) j
Ni- 1
PHi = P - L Pi j (4.3)
i=l
( )i+l ( ) j 8 a
pu - pu + _ ( pu2 + P ) + - (puv) =
t ax By
8 T 8 ( 3 ) a T 8 ( 2 )
= --- p1l. + 3pu + --- (mV + pu + (4.4)
ax 2 ax 8y 2 By
OT8 ( 2 ) 8r8 ( 2 )
+--- pu v+pv +--- pu v+pv j
Gx28y Oy2oz
(pv)j+l - (pv)j +!-. (puv) + ( pv2 + P ) =
t ax IJy
= 1" ( pu 2 v + pv ) + 1" ( pv3 + 3pl1 ) + (4.5)
ax 2 ax 8y 2 8y
+ !...!.. ( pv 2 u + pu ) + '!.. ( pv 2 u + pu ) ;
ax 2 8y Oy 2 ax
EJ+l - Ej 8 a
Llt + ax (u( E + p)) + 8Y (v( E + p») =
= ( u 2 (E + 2p» ) + !.. ( v 2 (E + 2p) ) +
ax 2 ax 8y 2 IJy
a a a a ( 4.6)
+ _ T _ ( uv(E + p) ) + _ T - ( u1J(E + p» ) +
ax 2 8y By 2 aX
N.
+ !..- '[ + !. - t act (£i + hoi)i
8x28x 8y2Oy i=lOt
N. ( U2 + V2 )
e = LPi + 2 ;
1=1
(4.7)
N. N.
. , ,
P = LPi = LPi T,
i=l i=l
= Ro/mi;
(4.8)
N,
[; = LCi£il
i=l
N..
h=Lcihil
i=l
Ci = Pi / P;
(4.9)
T 214
= Je".dT, T =-;
o ' P
CVi = c Pi - , hQi = CPi. (4.11)
B CHCTeMe ypasHeHHA (4.1)-(4.11) HCllOJlb30BaJlHCh cne,LQ'lOUJ.He
0603JlaQeHHH: Ni - qHCJlO KOMnOHeHT CMecH; P, Pi - COOTBeTC'rBeH-
HO J CYMMapHaSl nnOTHOCTb CMeCH H nnOTHOCTH OT eJIhHbJX KOMno-
HeHT; Cj - YJ:\eJIhHaJI KOHu:e"lTp 5I i-A KOMI10HeHTbI fa3a, P, Pi -
CYMMapHOe H napQHaJIbHble 3Ha eHHjI A8.BJIeHHSl; T - TeMnepaTypa;
E -- nonHaSi 3HeprHSI; £, &j - CYMMapHOe H napqHaJIblIbJe 3HaQeHHSI
BHYTpeHHeA 9HepI"HH; U, V - KOMfiOHelITbI BeKTOpa CKOpocTH; h, II;, -
CYMMapHOe H napl\HaJJ.hHbIe 3HaQeHHSI 3HTaJlbnHH; c"i 1 C\'i - y enb-
Hble TenJlOOMKOCTH i-A KOMDOHeHTbl raaa npH nOCTOSIJHOM 8.BneHHH
H nOCTOSlHHOM 06'beMe; mi - MOJ1HpHbIA Bee i-a KOMnOHeHThI; Ro -
YHHBepCaJlbHMI ra30BaH IIOCTOHHHasI; tpi - 3a a8aeMaSl Ta6J1H'fHO
<PYHKIJ)iB; Li, Qi - CKOpoCTH 06pa30BaHHH H norepb i-ro 8e ecTBa
B XHMH'IecKHX peaxn.H.SIX.
CJle YeT OTMeTHTh, TO HCTOqHHKOBbie qneHJ:d Li, Qi, npHcYT-
CTBYIO e B ypaBHeHHH (4.1), H HHeaHO 3aBHCSlT OT KOlW.eHTpaQHA
HeCKOJlbKHX KOMnOHeHT H TeMnepaTypbl, a. 1'8K}I(e CBSI3aHbI co CKo-
pOCTHMH np5lMbiX H o6paTHhix peaKl.\HR, 'l'eM caMblM npBo6peT8$1 JJ..O-
BDnbHO rpoMo3 KHA BH (194, 195). HaXO>K eHHe STHX 3aBHCHMOCreA
npe CTaBngeT co60:9 CaMOCTO.SITeJIbHYIO TpY)J)Iy1O 3 aqy reopeTHQe-
CKHX H 3KcnepHMeHTaJlbHhlX Hccnet\oBaHuA B 06JIacTH XHMHlleCKOA
<PH3HKH.
3 a M e \f a H H e 4.1. BBe,lteHHe <PYHK HH e H3 (4.7) B ypaBHeHRe )J.JIR
nOJlHoA 3HeprHH (4.6) n03BOJIBeT Y'IecTb nepeMeHHblA COCTaB ra3OBOA CMe-
CM.
T
e;, = f cviclI',
o
( 4.10)
j4. MO eJJHpoBaHHe reqeHd XHMH'lecKH pe&rHPYlOllQfX l'830B 291
3 aM e q a H R e 4.2. 06paTHM BHHMaHHe Ha TO, '{TO KOH eHTpau.HjI no-
CJle HeA KOMnOHeHTbl PN. (KaK npaoHno, H& 3TY panh BY6HJ)8.IOT BeU(ecTBO,
npeo6na;:t8JO ee B CyeCH ra30B) on enJleTCJI oc06hIM 06p83oM. BMOCTO
ell(e O,AHOro ;x.HcPcpepeH anbHoro ypaoHeHHH BHA8 (4.1) npHMeHJleTCH 9K-
BHBaneHTHoe eMY a.nre6paHqecKoe COOTHoweHHe (4.3). TaKaH npo e,aypa
n03BOJI.HeT 3BTOMaTHQeCKH H36e>KaTb 4»HKTHBIILIX HCTO'fHHKOB HJIH CTOKOB
MacCbI 3a CqeT BOOMO)KHOro HeaKKypaTHoro ttHClIeHHoro ODHcaBHH XHMH-
qeCKHX npe8p eHHA KOMnOHeHT.
CHCTeMa YPaBHeHHA, 6nH3KaSl K (4.1)-(4.11). 6blna nOJlyqeHa B
pa60TaX [196, 197) Ha OCHOBe KHHeTHQOCKoA' Mo enH HHaMHKH XH-
MHqeCKH pearHPYIO eR ra30BoA CMeCH. B 9TOa Mo enH npe,lJ)Ionara-
nOCb, '-ITO CPYHK SI pacnpe eneHHSI KaJK,n:oro COpT&, MOJIeKyn npHHH-
MaeT IIa Oqepe,nHOM BpeMeHHOM cnoe MaKCBeJIJIOBCKHR B , O,tt.HaKO
XHMHqeCKOrO paBHoBecuSi Me)K,[Iy KOMnOHeHTaMH He HacrynaeT.
CJIeAYeT OTMeTHTb t QTO O,lI)lHM H3 OCTOHHCTB KrY -CHCTeMhI
npH Mo enHpoBaHHH SlsneHHA B XHMH"IeCKH pearHpYIOlL\HX MHOro-
KOMlIOHelITl-lbJX raaax, nOMHMO BOOMO)KHOCTH pacqe-ra CJlO>KHhlX ra-
30 HHaMHqeCKHX TeqeHHA, HBnSleTCSI eCTeCTBeHHOCTb ODHCaHHH npo-
ecca )J.H4>cPy3HH pa3JIH'iHbIX KOMnOHeHT. 9To 06eCne'lHBaeTCSI Ha-
JIHqHeM )I.HCCanaTHBHbIX qJIeHOB B npaBo:A 'I&eTO ypaBHeHHSI (4.1) t
nOHBneHHe KOTOpblX BbI3BaHO Crna)l(HBaHHeM pemeHHH Ha paccrOSl-
KKK nop Ka AnHHbl cBo60 oro np06em l. TaxHMH >Ke B03MO)KHo-
CTSlMH "PH OnHCaHHH npo ecCOB B MHorOKOMDOHeHTHhIX ra3ax o6na-
AalOT H Lattice Boltzmann cxeMbI.
2. BhI'tfHCJ11ITeJIbHbIR aJIropHTM
ITepBoro B3rmI a Ha CHCTeMY YPaBHelJHA (4.1)-(4.11) OCTaTOq-
80, QTo6bJ nOHSITL ee nOBblWeHHYIO CJIO>KHOCTb ,LVrn qRcneHHOro pe-
meHHB. I1pHMetJaTeJlbHO TO, qTO AIlS WlOTHOCTH K&>K Oa KOUfiOHeH-
Thl I'a30BoA CMeCH OJI)KHO pemaTbCSl CBoe ypaBHeHHe (4.1). OAHaKO
OCHOBHasI npo6neMa CBSl3aHa C CHJIbHl:dM enH.HHHeM XHMHtJecxHX pe-
aK HA Ha H3MeHeHHe nnOTHOCTeA KOMDOHeHT H TeMnepaTyPY CMeCH.
9TO Bbl3BaHO 'reM) qTO XapaKTepHhIe BpeMeHa MHorax XHMHqeCKHX
peaK R Ha MHOro nopH OB MeHhwe xapa.KTepHOrO ra30AHHaMH'qe-
CKOro BpeMeHH. 9TH CJIO)KHOCTH ycyry6JI.mOTCH pe3KoA HenHReAHOA
38.BHCHMOCTblO KOHCTaHT CKopoCTeA peaK Ha OT TeMnepaTypbl.
292
TJI. V. MO,l{eJUlpOSaHHe TeT.feHHR ra30soJi Cpe):tbl
B Ka,'leCTBe nepBoro aTaIIa Bblt.JRCJ'IHTeJlbHOrO aJlropliTMa pa3o-
6heM pemaeMYIO cHcTeMY (4.1)-(4.11) Ha)l;Be IIO 3 a'IH c nOMO blO
MeTO a cYMMapHoA annpOKCHM HH (32, 198). Ha nepBOM 3TaJIe pe-
WaeTCH CHCTeMa ypaBllemdt (4.1 )-(4.11). IIpH M B rpaBHeHHH
( 4.1) OTCYT r8YIOT qneflbl Li - PiQi, onHCblBalO He XBMHQecKHe
npeBpaw.eHHR KOMTIOHeHT. llo 3aBepmeHMH 3Toro 3TaI1a <POpMaJlb-
HO 6y,n:eM C"IHTaTb, qTO nOJl}"leHHble 3HaQeHHSI napaMeTpOB COOTBeT-
CTOYIOT wary no BpeMeHH (j + 1/2).
Ha BTOpoM 3Tane. CTapTYSI C nonY1.feHHblX Ha nepBo:A CTa)I.HH 3Ha-
qeHaA pi + 1/2 I pf +1/2, Tj+ 1/2 , uj +1 /2 I vi + 1/2 I B TeqeHHe D}JeMeHH [j.t
pellla.eTCSI CHc'reMa ypaBHeHHA, OllHCbIBaJO 8.Si XHMHQeCKHe peaKl(HH
8p'
at' = L i - PiQi. (4.12)
TIOJIY"IeHHhle B pe3YJlbT8.Te pemellHSI CHCTeMbl (4.12) 3H&Qe:-JHSI nnOT-
HOCTeA XHMHqeCKHX KOMnOHeHT CqHTaJOTCH OKOH\fRTenbHbIMH ax
3HaqeHHHMH Ha MOMeUT speMeHH t = t)'+l.
.lLn pemeHHH CHCTeMbI KBa3HI'a30 HHaMH'ieCKHX ypaB!leHHA IiC-
nOJIb3YeTC51 nOJIYHeHBHa cxeMa (2.9) (['n. IV). K KOTOpoA I1pHMeHH-
eTCSI HTep HOHHbla npoQ€cc (2.10) (rn. IV). HTOI'OBhlR B ,lJ. cxeMbI
npOHnJlIOCTpupyeM Ha npHMepe ypaBHeHHjf ,LVIH napU;H8JI bH hIX DJlOT-
HOCTeA (4.1):
p{+Hs+1) _ PI 1 [ a () ] i
[j.t + 2 ox (Pi U ) + {)y (Pit}) +
1 [ a a ] j+ (S)
+ 2 ox (pi U ) + ay (Pi V ) =
= [ ;:x (Pi u2 + Pi) + ; (Pi v2 + Pi) +
{} T 8 {} T a ] ;
+ --- (Pi UV ) + --- (PfUV) +
ax 2 By 8y 2 ax
+ [ :x i a: (Piu 2 + Pi) + ; (Pi 1J2 + Pi) +
8 T a 8 T a ] j+ (8)
+ ox 2 fJy (pi UtJ ) + (}y 2 ax (p;uv) , s = 0, · · · , N s ·
( 4.13)
g4. MOAennpoBaHHe TeqeBsR XHMH1fecKH pearHpYIO HX 18308 293
3Ha'4CHH5I ra30jI.HHaMH1fecKHX napaMeTpOB Ha CJloe (j + 1/2) npH
s = 0 nonaraloTCSI paBHl:dMH HX 3tla"leHHHM npH t = t'.
CHcTeMa 06bIKHOBeHHbJX H<t><t>epe HanbHblx YPaBHeHHR
(O,lI.Y) (4.12)J KaK I1paBHJlO, XBJI eTCSI >KecTKo:A) H60 KOHCTaHThi
cKopocrea XHtwIH'tfeCKHX pe A pa3nHQaJOTCH Ha MHorHe nopH,lI.KH.
,A;JSI perneH"H 3THX CHCTeM cym.ecTBYeT MHO)f(eCTBO aJlI'OpHTMOB
H roTOBbIX nporpaMM qHCJIeHHOrO HHTerpHpoBaHHH. B HaInHX
pactleTax NISI peweHHSI CHCTeMbI (4.12) HCfiOJlh30BaJIC}1 naKeT
VODE [199) c aBTOMaTHqecKHM BbJ60poM wara no BpeueHH H
a<l><I>eKTHBHbIM anropHTMoM ero YBeJIHqeHHjJ.
OCTaHOBHMCI Ha BOnpocax, CBB3aHHblX C peanH3aQHeA anrOpHT-
Ma Ha MHoronpoQeccopHbIX BhlQHC1IHTeJIbHblX CHCTeMax c pacnpe-
eJ)eHHoA naUHTblO. BbllJHCJlHTenbHblA aJlrOpHTM (4.13) no cBoeii
norH eCKOa CTpYKTYPe cxo c gBHOA CX€MOH H ero anTa g K
napaJIJleJIbHblM CHCTeMaM He npe CT8.HJ1HeT TJ>yt\a.
CH(."eMa O y (4.12) no cBoe:A CTPYKTYpe RBnSleTCSI nOKaJlbHOa
H Boo611I-e He HY aeTCH ,lVISI Boero peweHHSI B 06MeH&X Me y
npo eCCOpaMH. OTMeTHM, q-ro nOKaJIbHOCTH y aeTcSl A06HTbCSl 6J1a-
ro apH oco6eHHOCTHM Kry -CHCTeMhl, onHCbIB8JO eA npo eccbl ca-
MO)l.H<P<I>Y3HH KOMDOHeHT. TaKHM 06p83oM, a.nTa HjJ paccMaTpH-
8aeMoro anropHTMa peweHHSI CHCTeMbI (4.1)-(4.11) K apxHTeKType
MHoronpoqeccopliLlX CHCTeM C pacnpe,a:eneHHoH n3MBTblO npe):\CTaB-
JIHeTc.sI .LI-0CTaTOQHO npoCToR.
06paTHM BIIHMaHHe H& O,AHY OCo6eHHOCTh MOAenHpoBaHHSI Te-
qeHHA XHMHqecKH pearHpyJOI.U.Hx ra30B. KaK npaBHno, HecMOTpH Ha
BHeWHlO1O rrpocTOTY CHCTeMbI (4.12)) ee peweHHe 38.HHMaeT 66J1LlliYIO
qacTb Bcem BpeMeHH pacqeTa. CBSI3aHO 3TO C 'reM, tJTO H3-3a O'leHh
BbICOKO:A CKOpOCTH HeKOTopblX XHMHqeCKHX peaK a BHyrpeHHHft
war HIlTerpHpoBaHHH CHCTeMDI (4.12) HeconOCTaBHMO Man no OTHo-
weHHIO K r830 HaMHqecKoMY wary t. Tax, B paccMaTpHBaeMoA
B CJIeAYlOlI.(eM nYHKTe 3 aqe 0 ropeHHH MeTaHa pacqeT CHCTeMhI
(4.12) Tpe60BaJI 90-95 npOneHTOB MamHHHOrO BpeMeHH.
C pyroH CTOpoHbI. XHMH'IeCKHe peaKIJ.HH, KaK npaBHnO) npOTe-
K8JOT B OTHOCHTeJ1bHO He60nbWHX no pa3Mepy npOCTpanCTBeHHLIX
06nacTSIx. HanpHMep, B 3 aqe 0 ropeHHH MeTaHa peaK H YT B
nOKRnbHbJX 30Hax nOBbirneRHoA TeMoepaTYphl (pHC. 120). OqeBH -
2500 .
.
II. aaJ7
O.II1W7
11.IIHeI
d.d3!l!
.. r....
.. ISIR
II. IItti:!II
a.171'
It. ...43
cr......
2000
o
1500
1000
o
500 1000
PHC. 120. CxeMa nepepacnpeAeneHHSI 3al'PY3KH npou.eccopoB NUl 3 a'lH 0 ro-
peHHH MeTaHa; TeMHhlA QBeT -- 30Ha XHMH"IeCKHX peaKIUlA, pHMcKHe ltH<PPbl -
HOMepa npo eccopoB H COOTBeTCT8YIO no,u.06nacTeA
HO, ITO B 60JIbillHHCTBe TOqeK, r e TeMnepaTypa He CTOJIb DblCOKa,
BeCTH pacqCT XHMHqeCKHX peaKQHA He HMeeT CMbICJIa.
BMecTe C TeM, BKJllO'IeHHe WIH BhlKJIIOqeHHe H3 o6IU.ero anrQ-
pHTMa CHCTeMbI (4.12) B 3aBHCHMOCTH OT llOJIyqaeMOA B paCqeTe
TeMIIepaTypbl AeJIaeT tJHCJleHHhJR MeTO Heo HOpOAHbIM. I103TOMY
B pac leTe Heo6xo RMO HCnOJIh30BaTb MeTOAbi t{HHaMHt.tecKoA nepe-
3arpY3KH npoQeccopoB (rJI. IV, 97). Ha pHC. 120 H306pa>KeHa 301la
HHTeHCHBHoro ropeHHSI MeTaHOBoro cpaKena. BHP-HO, qTO QeHTpaJIb-
HblA npoQeccop, npH paBHOMepHOM pacnpe eJleHHH npocTpaHCTBeH-
HbIX TO'-leK Me)K.lly npoQeccopaMH. npOBO,[(HT 65% OT o6U{ero 06 eMa
94. lVlO,l{eJlHpOBaHRe TeCfeHHR XHI4RIf8CKH pearHpylOupIX 1'&108 295
BLI'fHCJIeHHit. n03TOMY HMeHBO H3 Hero )VIS( 06ecneQeHHSI paBHOMep-
MoR 3arpY3KH npoCTpaHCTBeHHhle TOQKH nepe alOTCSl pyrHM I1pO-
eCCopaM .
3. MOAeJlHpOB3HHe ropeHHH npHpo,l(HOrO ra3a
rvIO t'.JIHpOBaHHIO ropeHHSI OpraHH'IecKHX TOUJIHB, BBUAY Sa>KHo-
CTH 3TOA npo6JIeMbI, y enHeTCH OCTaTOqHO MHoro BHHM8HH.H. HeKO-
Topble BonpoCbl, CBSl3aHHbIe C 3TOa np06neMo:A, xopOWO H3nO)l{eHbI
8 0630pe [200). B )I-aHHOD MOHorpaq.HH 6YAYT npe,QCTasneHbi HeKO-
Topb1e pe3ynLTaTbi Mo enHpOBaHH.S1 ropeHHSI CTPYH MeTaHa, HCTeKa-
lO eA B B03 YX.
C ropeHHeM npHpoAHorO ra3& cOBpeMeHHoMY 1.JeJlOBeKY npHXO-
HTCSl CT8JJKHBaTbCSl nOCTOHHHO. IloMHMo 6b1TOBblX HY , ropeHHe
MeTaHa SBJIHeTCSI OCHOBOA MHorHX TeXHOJIOrHtJecKHX npo eccOB. O -
HaKO, HapSl)Q' C IIOJIe3HbIM 3<l>ct>eKTOM BbI eJJeHHSI 3HeprHH, npH 1"0-
peHHH npHpO Horo ra3a B aTMoc<!>epy Bbl6pacblBaeTCSI 6oJIhwoe Ko-
nHQecTBo 'fOKCH'ieCKHX Be eCTB.
Cpe;J.H TOKCHqHLIX KOMnOHeHTOB B npoAYK'IaX CropaHHH BbJ e-
JIHIO'fCH OKC a30Ta NO x , yrapHblA ra3 CO H pa,n:HKaJ1 CH. Ho e-
KBa rHoe MO,AeJIHpoBaJlHe o6pa30saHHSI OCHOBHblX Bpe HbIX npHMe-
ceA Tpe6YeT onHcaHHSI npo eccoB, KYAa BOBneqeHO ropa3,n:o 6anbUIe
Be ecTB. Tax, B pa60Te {201J B nopSlAKe HccneAOB8.lJHSI 06Pa30BaHlUI
NO x BblnHcaHa enoqKa, COCTOH H3 196 npslMblX H o6paTHbIX
(196+196) 3JIeMeHTapHhlX XHMHqecKHX peaK , B KOTOphlX 3&AeR-
CTBOBaHbI 32 KOMDOHeHTbl. CpeAH HHX Y HTLlB8JlHCb COCT8.BJlSIIO He
HCXO HOR CMecH - MeTaH CH 4 H ra3bJ, 06pa3YIO e B03):ij'X: O 2 , N2;
npo,nYKTb1 peaKl\HH ropeHHH H20, CO 2 , CO. 82; p HK8JIhI, HrpaJO-
He cy ecTBeHHYIO pOJlb B cnO>KHOM npOTeK8.HHH peaK HH ropeHHS1:
H, 0, OH, H02, H202, CH3, CHO, CH20, CH 5 0; pa3JluQllLle 830'1'-
cO ep>K He Be eCTBa: N, NO x , HN, CN, HNO, HCN; Ha.KOH ,
RaH60nee Ba)KHbJe H3 Be eCTB H p HK8JIOBJ BXO B enO'IKH
06pa30BaHHH OKC OB a30Ta: CH. CH 2 , H) C 2 H 2, C2 H 3, C2H4)
C2 H S, C 2 H 6.
BBH Y 60nbUIoR TpYAoeMKOCTH pemeHHg 3TOA: 3 a'lH B nOJI-
296
rn. V. MQAenHpoBaHHe Te'leHHIf 18308011 C]>eAbl
HoA nOCTaHOBKe A8JKe C HCDOJIb300aHBeM COBpeMeHHhlX BLlCOKOnpo-
H3BO HTeJlbHblX CHCTeM, ee HCCJ1eAOBaHHe WJIO ):tBYMSI n}PTIlMH. B
IIepBOR nOCTaHOBKe paCCMaTpHBanaCb 1I000H8.S MOAeJIb XHMH'IeCKoit
KHHel'HKH (196 peaKU.HH) COBMecTHO C rrpocTOH ra30 HaYH'lecKoH
MOAeJIblO) He CBH3aHHoA C HCrJleJ(yeMbIMH B ,AaIIHoR MOHorpa4>HH.
Bo BTopoA - KfY-cHCTeMa (4.1)-(4.11) pemaJI8Cb A1IH 18 KOMno-
HeHT: ClLt, C H 3J CH 2 0 t CH 3 0, CHO, CO, C02, H2, O 2 : H, OH,
H022 112 0 , N2, N, NO, N02, N20, KOTOpble yqacTSYIOT 8 64 nps-
MblX H o6paTHblx peaKIIrHHX. Bbl60p BelI\eCTB Ii peaK HAt HaH60nee
Ba)KHblX np" 06pa30BaJIHH TOKCH'iecKHX KOMnOHeHT, B 3Ha'lHTeJJb-
BoA Mepe OfiHpanCB Ha pe3yn:bTaTbi t RonY'IeHHbIe H8 OCHOse nepBOH
nOCTaHOBKH. ,D.nH paC'IeTa 3 a'JH BO BTOpoA DOCTaJIOBKe HCnOJlb30-
Banacb MHoronpo eccopHaB BblqHCJIHTeJJbHaH CHCTeMa MBC-IOOO c
nHKoBoA npoH3BOAHTeJIbHOCTblO, )J.ocTHralO e:A 1 TFLOPS.
B nepBOa 3 aqe H3YQaJJCB npo ecc 06pa3oBaJlHH OKCH :)B 830Ta
B 3aBHCHMOCTH OT COCTaBa HCXO Oa CMeCH [202]:
CH 4 + a(2 02 + 7.569 N 2 ),
( 4.14)
rt(e 0 - Kos<p<l>HQaeHT H36b1TKa. B03A;rxa, KOTOpblA sapbH;>OBaJIC5I
(0 = 1 COOTBeTCTBYeT CreXHOMeTpH'IeCKOA CMecH, npH a < 1 CMecb
Ha.3LISaeTCSI 6oraToH t npH Q > 1 - 6e oA).
,lVIH onpe eneHHH KOHL\eHTpaI.\HA pa3JJH'lHhJX Be eCTB RCnOJIb-
30BaJIOCb ypaBHeHHe (4.1) C IIOCTOBRHOA CKOpoCTb1O u. H TeMoepa-
Typoa T*. B KaqeCTBe T* 6p8JJOCb 3HaQeHHe, paBHoe 3HaQeHHIO PaB-
HOBecHo:A TeMnepaTypbl a6aTH'IeCKOrO Bldropa.HHSI CMecH MeTaHa
H B03A.}'xa npH HaqaJIbHOA TeMnepaTYpe T = 300 K H nOCTommOM
AaBJIeHHH p = 1 aTM. CKOpocTh u. onpe e.n:BJlacb H3 3KcnepHMeH-
T3J1bHhlX aHHblX [203].
PacCMOTpRM oco6eHHOCTH 06pa30BaHIDI NO - o oA H3 H&H6o-
nee TOKCHt{HblX KOMnOHeHT. Ha pOC. 121 H306p8JKeHO pacnpe eneHHe
KOHu.eHTpaQHH NO B ppm (lMOJIb/(MOJIL CMeCH) = 10 6 PPIT ) B 3a-
BHCRMOCTH OT COCTaBa MeTaHOB03pylIIHOA CMecB. KaK BH.AHO. .lVIH
6e)J.HbIX H )J.nSl 60raTbIx cMeceA npo ecc 06pa30BaHHSI OKCH OB 830Ta
npOHCXO)J.HT no-pa3HOMY. B CJIyqae 6eRJf I X cMoce:D Henocpe CTBeH-
BO Ha <PPOHTe BDnHbI ropeHHH (z = 0) 06pa3yercH nHWb He6onLllIoe
j4. MO,l(e.J1HpOB&HHe Te'leHHR XHWH'IeCKH pe&rHPYlOuqIX ra30B 297
2 ppm
a=O.9
o
-,..
PHC. 121. Pacn eneHHe KOH eHTpaQHB NO DpH pa3JIH"UIIJIX 3U&'IeHHHX napa-
YeTpa Q
KOJlHqeCTBO oKcH a a30TS. IIoCJIe npOxO)IC,AeHHH BonHbI ropeHHSI BAa-
RH OT Q>POHT'cl. 3a CqeT OCTaT01.JHOR BblCOKOR TeMnepaT)'pbl npo ecc
06pa30BaHHSI NO npop-OJ])KaeTCSI. AnSI 60raTbIX cMeceR Ha ct>POHTe
OMHbJ ropeHHH Cpa3Y 06p83YeTCH 60J1hwoe KonHt.JeCTBO NO. O,n:Ha-
KO nOCJIe IIpOXO>K,lI.eHHB BOJIHhI ropeHHH H3-3a CHJlbHhIX BOCCTaHOBH-
TeJlbHblX CBOaCTB cpeAbl KOH eHTp H OKCH,lJ;OB a.3OTa YMeHbWaeT-
Cst.
3TH pa60Tbl [202, 204), HecMoTpsr Ha HBHYIO ynpo eHHOCTL n<r
CTaHOBKH) TeM He MeHee, n03BOJIHJlH nonY'iHTb P KOHKpeTRDIX Tex-
HOJIOrHQeCKHX peKOMeH aI\Idl ,.lI.JlSI CHIDICeHHS1 npoMblD1JIeHHoro 8bJ-
6poca speAHblX npHMeceA (205)-[206].
PacCMOTpHM Teneph peweHHe
3 aqH 0 ropeHHH MeTaHa B B03 Y-
xe Ha OCHOBe HCnOJlb30BaHHH 60-
nee DOJIHott CHCTeMbl KBa3Hr830-
HHaMH1.JeCK}fX ypaBHeHHA (4.1)- \ J
(4.11). 3Ta 3 a Ia COOTBeTCTBY- '--./ \ '-.-/
e'!' HCTeqeHHIO IIJIOCKoA CTpYH Me- "---
TaHa B B03,nyX (pHC. 122). MeTaH
CMeIIlHBaeTCjI c B03 YXOM H ropHT PHC. 122. CxeMa HCTe'leHWl MeTall&
B HeM. 9Ta nocmHOBKa SlBnSleTCB B B03A.}'X
MO MblO, HMHTHpYK) eH asapHHHblH Bbl6poc MeT8H& Ha npOMblCJIax
c nOCJIe,nylO1II,HM nO .>KHra.HHeM ra.30BOrO ct>axena [207]. Pe3ynhTaTbI
2QOOi) f
5000
- ---y-- -......-----,---
5000 10000
1 OOO
1000D
.
o
o
PRC. 123. none CKOpocTeA B CMeCH
MeTaH+Bo.. AYX
PHC. 124. Pacn e.neUHe KOl1lteH-
TPattHH MeTana
MO,D;eJlHpOBatIHSI 3TOA 3 aqH H3JJO)KeHbI B pa60Tax IIIB, 121, 127,
154, 208J.
Pa3Mep m.enH, H3 KOTOpoA BLl6pachlBaJlCB MeTaH, COCTaBJIHn
d = 10 CM. CKOpoCTb Bbl6poca npHpO)J.HOrO ra3a H3 w.enH Vo =
= 1.2 · l04 cM / c . npe nOJla.raJ10Cb, QTO HHH HHpoBa.HHe nO )KHra
coOepWaJIOCb nYTeM oHeceHHR 8 tfacTb 06naCTH, 3aHgTO:A CMecblO
MeTaH+003AYX C Q 1, nepBOHaqa...rThHO BhICOKOA TeMnepaTypbl
T = 1800 K.
Ha pHC. 123 H 124 H306p8JKeHhI nonS! cKopocreR B CMecH
MeTaH+Bo3AYX H pacnpe,..qeJIeHHe KOHneHTpa HH MeTaHa HS O,ll)lH H3
MOMeHTOB BpeMeHH nepeA nO HroM ra30BoA CMecH. B o. QTO
CMeweHHe ra30B rrpOHCXOp)iT 3& cqeT BTJlrHBaHH.SI B03A}'xa B CTPYIO
MeTaHa H3 unpH3eMHbJX CJIOeB" (B6Jm3H III-eJ1U). B )I.&1lhHeAmeM aJ(-
THBHYIO pOJIb 8 C03 aHHH CMecB ra30B Hrpae-r BHXpeBoe BIDKeHHet
06pa3oBaBweecSi B 30He aKTHBHOrO TOpMO)KeHWI CTPYH OKPY>K8JO-
j4. MOAeJlHpoB&HHe TeqeHHlt XlfMHCfecKH pearllpYIOlI{HX r&.1OB 299
Ha pRC. 125 npHBeAeHo xa-
paKTepJ-loe npOCTpaHCTBeHHoe pac-
npeAe.rICHHe TeMnepaTypbI ropSl ero
<paKe.na. BH,l1-HO, 'ITO 06J1acTb nOBbI-
meHHoA 'reMuepaTYpbl (H& pac. 125
OHa 3aKpameHa) pacnOnO)KeHa Ha
nepH<l>epHH $aKeJIa. B 3ToA 30He
HaxOAHTCH Ofi'l'HMaJIbHOe (a 6JIH3KO
K 1) COOTHomeHHe Meralla H B03 Y-
xa. B ropfi'leR 30He (o6naCTH HaH-
6onblliero B eneHH9 TerlJIa) npo-
HCXOAHT OCHOBHbJe XHMHt.leCKHC pe-
HH. Ha pHC. 126,a-r H306pa)l(e-
Hbl KOH eHTpaQHH HeKOTOpblx XHMH-
qeCKHX KOMnOHeHT CO, NO, CH3,
PRe. 125. XapalCTepHbIA npocj)HJIh
CO 2 . 9TH BeID-ecTBa TaK)Ke COCpe.rto- TeMJIepaTYpw <tJaKena
TOqeHbl B OCHOBlfOM 8 ropSlQeA 30He,
QTO em.e pa3 onpaBAbJnaeT CTpaTerHIO neproarpY3KH npo ecCOpoB,
paCCMO'rpeHlIYJO B n.2 aHHoro naparpaQ>a.
ea ee aTMoc<l>epo:A.
.11)0'
, L)O:)
, 0000
5000
00
6000
t 0000
CneAYeT OTMeTHTb, 'lT0 HeCMOTpB Ha HeKOTOpylO ynpow.eHHOCTh
OnHCaHHB (B nepBYIO o'{epe bJ CBg3aHHYIO C paccMoTpeHHeM npo ec-
ca lOpeHHH B JlaMHHapHOA, a He Typ6yneHTHoR nOCTaHoBKe*) au-
HbIe pac e'J'a afOT none3HYIO HH<PoPM 1O 0 xapaKTepe reHepa H
TOKCHtJeCKHX KOMllOHeHT npH ropeHHH MeTaHa.
.CJIO)J(HOCTb Typ6yneliTHol nOCTaBOBKH 'IH B nepsylO O'lepe b CBJI3aH& C
np06neM8MH B cneHHB OCpe;J)ICIIHhlX KORCTaIIT XHMH'lecKHX peal<a.Wt. KOTO-
pble OqeHL CWIbHO - 3KcnOHeH anbHlalM 06pa30M - 38.BHCft OT TeMnepaTyphI
oKpy*alOw.ero ra3a (194].
::suu
rn. v. Mo,.qeJIJlpoB8HHe reqeHHH r&3OBOJf Cpe,J1.bI
10000
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16000
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5000
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B)
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1 00 DO
r)
00
5OGO
10000
Puc 126. XapaK1'epHhre KOHI(CHTPattHH KounOHeHT: a) CO, 6) CO 2 (30HbI HaH-
BblcweH KOHl eHTpaJ.J.H" 3aKpa1UCHhI); D) NO, r) CH3
j5. MOJ{enHpoBtufHe TeqeHHR YJ,lepeBHO pa3pe)f{eHHbIX l&10B 301
5. Mo,n:eJIHpOBaHHe TeqeBHA YMepeHBo
pa3pe)l(eHHhIX ra30B.
CTPYKTYPa y apHoii SOJIHhI
CO;:J.ep>KaHHe 3'!'OrO naparpaQ>a HeCKOJIbKO OTJlHqaerCH 1]0 cBoeD
HanpaBJJ'eHHOCTH OT npe blJU'DJ.HX M&rrepHaJIOB. B 1- 4 pe1.lb mJIa
o MO,[(eJ1HpOBaHHH Te1.feHHA ,L\OCTaTOqHO I1J1OTHOrO raJa - np06neMbI
B npHliQHne ODHCblBaeMoA B rH.lij>o- HJlH ra30AHHaMH eCKOM npH-
6nH>KeHHH. OCTOHIICTBa KCPC H KrY-cHcTeMhl npOSlonHIOTCSI B
TOM, t{TO OI1HpasCb Ha CBOHCTBO KOppeKTHOCTH, OCTaTOqHO OqeBH -
Hoe c cPH3HQeCKOH TOQKH 3peHHS, YAaeTCB ycnewHo Mo,n;enapoBaTb
Ha BbICOKonpOH3BO)J.HTenbHhJX Bbl1.fHCJIHTeJlbHblX cHCTeMax CJlO)KHLIe
raJO)J.HHaMHqeCKHe TeqeHHjf. UpH 3TOM Hana30H ,(HC KHy ceHa
OrpaHHqHB nCSl MaJlblMH 3HaQeHIUIMH Kn < 0.001, qTO COOTBeTCTBY-
eT o6J1aC'rH npHMeHuMocTH ypanHeHHH HaBbe-CToKca.
PacCMOTpHM t HacKOJIbKO C rrOMOII.{blO KCPC H 1(883Hra30 HHa-
MH'-lecKOA CHCTeMbl B03MOJKHO pacWHpHTb npHMeHHMocTb MaKpo-
CKOnHQecKoro onHcaHHSI B CTOpoHY 60JIblllHX tlHCen KHY CeHa. npH
aHaJlH3e 3THX B03MO)K}IOCTeA KHHeTH ecKHX cxeM 6y eM ormpaTbCB
Ha pe3YJlbTaTbJ pa60T [36, 38) 97, 209, 210).
1. B03MO)f(HOCTb npHMeIlelfHH KCPC AJIB OfiHC&1UfH
Te[lelll.,JI YMepellHo pa3pe>KeHHblX ra30B
OnHCaHHe TeqeHHA raaa c nOMOlI1,b1O ypasHeHHA 3Anepa H
HaBbe-CToKca B03MO)l(HO JlHWb ,LVIii He6OJ1blllHX 3HaqeHH:B queen
KIIYAceHa Kn ;S 0.001. nSl 60nbIDHX 3HaqeHHA qucen KHy ceHa
raJO HHaMHqeCKHe napaMeTpLI onpeAenSJIOTCH C nOMO 1O pemeHHH
KHHeTH'ieCKoro ypaBHeHIDI BOJIb MaHa H nocne)UllOm.ero HHTerpu-
pOBaHHH fiOJ1YQeI-IHOH TaKHM 06p33oM <l>YHK HH pacnpe,lI.eJIeHHSI C
cYMMaTOpHblMH HHBapaaHTaMH.
O }IM H3 OCHOBHDIX HHCTpYMeHToB pemeHHSI ypasHeHH5I BOJIbq-
Malia SlBJlHIOTCSI MeTO CTaTHCTHtlecKOro Mo enHpOBaHHB, H3JIO>Ke-
Hue KOTOpblX MO>KHO Ha.9:TH, HanpHMep, B MOHorpa<j>IfHx [211, 212].
9TH MeTOAbl 3<t><PeKTHBHbI ATlH A0CTaTOqHO 60J1billHX qacen KHY));-
ceHa (Kn > 0.1). I1pH YMeHbllleHHH tlHCJla KHY,QceHa TPY)J.HOCTH pe-
302
Tn. V. MOAenllpOBaJlJfe TeqeHHH ra308 R cpe,ltbl
llJeHH ypaBIleHHSI BOJIh MaHa C nOMO blO MeTO,n:OB CTaTHc:-IrIecKo-
ro tO,n;eJJHpoBaHHSI pe3KO B03pacT8.IOT. T8.KHM o6pa3ou, npaKTHQe-
CKM Ba)KHbla ,!VIR npH.nO)l(eIIHH, HanpHMep, CBH38HHblA C onJrICaJlHeM
BXO>l< eHHSl JJeTaTenbHblX annapaTOB B nnOTHble CJlOH aTMoc epbI,
HHTepBaJI IHCeJ1 KHy,n.ceH& OKa3bIOaeTCS TpY Ho ocTyneH ,Q1H 'iHC-
JIeHHOro MO eJIHpOBa.HHH. * llOCMOTPHM, KaKHe B03MO)J(HOCTH OTKpbl-
BaeT HCI10nb30oaHHe KCPC )1.1IR MO eJlHpOBaHHH TeqellHA YMepeHHO
pa3p€>KeHHWX ra30B.
KaK Y)Ke OTMe'laJIOCL (rJl. III), KCPC H Kry 6a3HPYIOTCSI Ha
npeACTaBneHHH O OqacTHqHOA YHKQHH pacnpeAeneHHH B BH e
MaKCBeJlJIOBCKOA, KOTOpaH CJIa60 MeHSleTCSI H& pacCTOSlHHH ,lVIHHbl
cBo6o Horo npo6era. l. 31'0 )J;aeT B03MO:>KHOCTb, B OTnH'lHe OT pa3J1o-
>KeHH YenMeHa-9HcKora, onHcaTb 9<1><I>eKTbl MOJIeKynHpHoro nepe-
Hoca c nOMO 1O Q,!{Horo q.neHa pa3JIO)KeHHH )
f=fMa' (5.1
r e f Ml - MaKCBMJ10BCKaH O HOqacTHqHaSl <l>YHKlJ. HSI pacnpe eJ1e-
HH , nOCTO HH8.SI B SlQeAKe, pa3t.lepbl KOTOpOH paBHbl )VIHHe cBo6o -
Horo npo6era.
IloCMOTpHM, KaKHe 803-
MO)KIIOCTH KCPC, BBn lOIlJ)fX-
01 CJIeACTBHeM pa3nO)f(eHHSI
(5.1), cy eCTBYIOT NISI MOAe-
JIHpoBaHHH Te'tfeHuR YMepeHHO
pa3pe>KeHHblX ra30B [209]. AnSI
:fl'oro c nOMO 1O pRC. 127 BbI-
SlCHHM, Kor a YHK HIO Q, SB-
nB IDCHo oAH3 OAHH
r I
MHtJeCKHX nep€MeHHhlx, MO>K-
PHC. 127. npe,ACTaBJIeHHe npo4>KJUI Be- 110 eKBaTHO npe CTa.BHTb B
J1H HHhI Q S BHAe Ha60pa cryueH'taTo- BH e Ha60pa CTyneH'faTO-no-
nOCTORHU.bIX 3Ha'leHHA CTOSlHHbJX 3HaQeHHA. ITpK 3TOM
BeJlHqHHa OTpe3Ka (Xi, Xi+l] paSHa ,!VIHHe CBo-
Q
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6oAHoro np06era l.
113 HarnH){l1LIX co06pa>KeHHii BH OJ QTO rn KYIO KP BYIO I
Ha OTpe3Ke [xc, Xc] (pRC. 127) MO}KHO xopomo npe)\CTaBHTb B BHAe
.OTMeTHM) 'lTO nporpecc B pa3BHTHH BbiCOKOnpOH3BO,D;BTeJJbHWX MHoronpo-
ecCOpllbI.X CHCTeM n03BWtgeT nocreneHHO Mero awH CT8.THCTH ecKoro MO eJIH-
pODBHHJI peW&Tb If 3&.r\aqJf co BeC WeHbUIHItOI 'lJJCJIaMH KHy ceHa.
95. MOAeJlJlpOBl1IfHe TelfeHlfR YMepeHHO pa:JpeRCeHHbJX r&30B 303
cTynefIqaTOA annpoKcHMaI.{HH, ecnH 'IHCJIO HHTepBaJIOB, Ha KOTOphle
p836HT HCXO,D.HbIA OTpe30K, COCTaSnBeT HecKOJJbKO eCSlTKOB. IIpH
3TOM HeHBDO npeAnOJlaraeTC.H) qTO 3aUeTHble H3MeHeHHB <PYHKu;HH Q
npOHCXO,l(SIT HMeHHO Ha 3TOM xapaKTepHOM p8.CCTOSlHHH L = Xc - XC.
EC1JH )Ke 3aMeTHbie H3MeHeHHH Q npoHcxo wr Ha MeHbmeM OTpe3-
Ke [xa, Xb] (KpHBaSl II pHC. 127), TO B KaqecTBe L H8AO BbI6HpaTb
paCCTOSlHHe L = Xb - X".
KOHetlHO, YBeJlH leHHe QHCJ1a HHTepBaJIOB, Ha KOTOpbIe Pa36HT
OTpe30K [Xa, XC), AO 10 2 + lQ3 npHB eT K ynyqWeHHJO npe,n:CTaB-
J1el-l"H KpHBOA I c nOMO blO CTyneHqaTOA annpOKCHM HH. Ho 3T0
BPSlt( JIM n03BOJIHT nOJ1Y"lHTb Ka,qeCTBeHHO HOOYIO HHcf.>OPMaJJ.HIO OT-
1lOCHTeRbHO nOBe,.a.eHHH <t>YH H Q. MO)J{HO CK33aTb, tJTO BBe eH"e
xapaKTepHoro pa3Mepa L HeHBHO n nonaraeT OCTaToq:HOCTb,LVIH
KYCOqHO-CTyneHqaTOrO Dpe CTaBneHH HecKonbKHX ecRTKOB P83-
6HeHHA.
1l0CJIe 3THX ,ltOCTaTOquO a6CTpaKTHblX pacCY)I( eHHA BepHeMCSI
K B03MO)KHOCTH OfiHcaHHB C nOMO blO KCPC Te eHHA YMCpeHHO
pa3pe>KeHlfbIX ra:30B. ECJIH cf.>YHK pacnpe,neneHHSI f cryneH"IaTO-
nocroflllHaH Ha (X Q , XC], TO Bee ra30 H8J.t:HqecKHe napaMeTpbl Q TO-
me CTyneHQaTO-nOCTOSlHHbJe. T8.KHM 06paaoM, ):VIR r830 HHaMHqe-
CKoro OllHCaJIHH C I10MO 1O KCPC He06xo o H& OTpe3Ke [Xa, Xc]
,lVIHHOR L HCnOJlb30BaTb He MeHee HeCKOJThKHX ecgTKOB pa36HeHHA
[Xi,Xi+l] AJ]HHoA He MeHee NJHHhI cBo6o Horo np06em I.
YqHTblBaH onpeAeneHHe q}lCJla KHy CeHa
Kn = i/L,
(5.2)
nepect>oPMYJlHpyeM ere KaK OTHoweHBe BMH'fHHbI OTpe3Ka paa6ne-
HIUl [Xi, Xi+l] K xapaKTepHOMY pa3Mepy L. BcnOMHHasI, ttTO TaKHX
MHHHMaJlbHblX OTpe3KOB OJI)I(HO 6'bJTh He MeHbwe UeCKOJIbKHX 1J.e-
C)1TKOB J llOJIyttUM KCPC AWzym onUC 6amb me",eHUJI YMepeH-
NO pa3pe eHH"blX 2a306 om uceA K HyOcetta, pa6H&LW HeCKOAbfroM
com U HU3fCe: Kn 0.01 [97, 209). BcnoMHHM, '1TO AIlS! r830)J.H-
HaMH'iecKoro OfiHcaHHSI C nOMO blO YPaBHeHHA H8.8 OroKca Ha-
fi830H AOnyCTHMhlX QHceJJ KHy ceHa Ha IIOpSAOK MeHbme. aTO pac-
lllHpeHHe AHaI1830Ha CT8JIO B03MO>KHhlM H& OCHOBe 60JIee npOCTOro
->U4
1 'n. V. MO,D.eJ1HpOBaHHe TelfeHHR ra3OBOR Cpe,J{bl
no cpaSHeHHIO C p83nOJKeHHeM t{enMeHa-9HCKora npe CTaBneHHeM
(5.1) H O HO'tfacTHqHOH cPYHK HH pacIIpe eneHIDI f.
3aMe'laHHe 5.1. no NIHHOA CB060 HOro npo6era I I10HHM&eTCJI
TO xapaKTepHoe paCCTOSHHe, KOTOpoe q ac-l.t1u. bI ra3& "}X>neT8JOT, He HcnbI-
TaB CTOJlKHOBeHHR. 9'1'0 paCCT01IHBe npH 60nbWHX csepX3BYJ<OBbIX CKOpo-
CTHX ra3a MOJKeT 3Ha'fHTeJIbHO YBeJlH'IHBaTbCJI. HanpHMep, B HanpaBJIeHHH
BH)f(eHHS r33a co CKOpOCTblO U 3<jxpeKTHBHasI )VIHH8 cB Horo npo6era
[36, 97) eCTb
J u 2 + c2
cI> = I · c2'
(5.3)
BOJIee He3a8HCDMOA B 3TOM CMblCJle enBeI'CJ1 .D.PYraJI BeJIH'IHH& - qacToTa
CTOJIKHOBeHHA
T = l / c.
OHa 8bIQHCJIJ1eTCH C nOMOUJ.b1O TeMnepaTypbl H nnOTHOCTH raJ08 Ii onpe-
e.n eT ,.l].HCCHnaTHBHbIe 'JneHbI KBa3Hr830 HHaMHqecKoA CHCTeMbl ypaBHe-
HHA.
Mea
____uu uu______ - -- J -----
OCL CHMMeTpHH """""""
PHC. 128. CXeMa Te eHHg ra3a B6nH3H 1JJ18CTHJIY
B KaqeCTBe npHMepa pacIJeTa paccMaTpBBaJIaCb 3 a.qa. 0 MO e-
nHpoBaHHR TeqeHHH B03Aj'xa B6JIH3H nnacTHHhI no HYJIeBblM yrJIOM
aTaKH (pHC. 128). napaMeTpbl BXO HOro nOTOKa Bh16HpaJIHCb CJIe-
)1y1OI.IJ.He: Moo = 5.5; Kn = 0.04. Be3p83MepHaJI )J)IHHa DJlacTHHbl
L = 1. TeMnepaTypa Ha IIOBepXHocTH nnacrHHbJ paBHa TeMnepaTY-
pe Ha6eralOll\erO nOTOKa Too (B 6e3p83MepHo:A BMH'IHHe Too = 48).
,LVIH pacqeTa (36) HCnOJlb30BaJIaCb cxeMa KCPC (2.2)-(2.5)
(rJl. II) npH teHHTeJIbHO K rIpoCTpaHCTBeHHo- BYMepHoA reoMeTpHH.
IIpH 3TOM, eCJIH mar no KaKOA:-HH6Y b npOCTp8.HCTBeHHOR nepeMeH-
HoD npeBbllllaJI 19 (CM. (5.3)), TO CXeM& (2.2)-(2.5) (rn. II) HCfiOJlb-
30BaJlaCb B HeH3MeHHOM BH e. B npoTHBHOM rJIyqae COOTBeTC1.BYKr
HH mar npOCTpaHcTBeHBoA nepeMeHHoii nOJlaraJlC51 paBHblM 13<P.
Ha pRC. 129 npHBe eHbI rpa4>HKH 3HaqeHHA nnOTHOCl'H B Pa3-
JIHqHblX ceqeHHSlX rIO Koop HHaTe x. 06paTHM BHHMaHHe Ha TO, 'ITO
5. IOAeJllfpOBaHHe Te'leHHii YMepeHHO pa3peJKeHHblX r830B 305
1.0
0.0 0.1 0.2 0_3 0.4 0.5
1.6
1.4
c
1.2
PRC. 129. TIpo$W11I nnO'I'HOC'l'1l B CeqeHHRX no KOOp HHaTe %: X = 0,
- - - x = 0.2, - . - . - x = 0.4, ... x == 0.6, DO 0 x = 0.8, + + + x = 1
B6JIH3H nepeAHeA KpOMKH (x == 0) nnOTHOCTb MeIDIeTC51 npaKTMt.fe-
CKH MOHOTOHHO. tI-anee, no Mepe y,L\aJ1eHHH OT HOCHKa UJlaCTHHbl,
IIP<><l>HJlH CTaHOBSlTCSI Bce 60nee HeMoHOTOHHbJMH, HMeSl xapaKTep-
UblH MaKcHMYM. 3TH pe3YJIbTaTbI xopowo cornacYIOTCSI C AaHHhlMH
npSlMoro CTaTHCTI1'iecKoro MO eJlHpOBa.HHjI TOro }l(e TeqeHH {213].
B npe,L\CTaBJ1eHHblX B 3TOa pa60Te AaHHblX TaJOKe, Ha'-lHHasI c ce-
qeHHH x = 0.4, nOBBJI eTCSI HeMOHOToHHoe nOBe eHHe nnOTHOCTH C
xapaKrrepHbIM MaKCHMYMOM Pmax = 1.6.
o AHaKO cy ecTBOBanH H pacxo eHH KOJIH'IeCTBeHHoro xa-
paKTepa 8 pe3YJIbT8.TaX pa60 ' [36] H (213) "PH OllaCa.HHH nOBe e-
HHSI nJlOTHOCTH. IIpHqHHa pacxo eHMA 3aKnIOttaeTCH B CJleAYIO-
eM. HaH60J1ee CHJIbHble H3MeHeHHSI napaMeTpOB B Ha'laJIe nnacTH-
libl npOHcxo,.q.HT B Y3KOM CJJoe H nnacTHHoA: (pHC. 129). qHCJlO
KHYAceHa, paCCqHTaHHOe no TOJ]I.IJ.HHe 3TOro CJlOSl) OCTaTOqHO Be-
JIHKO (Kn ::::: 0.5) H BbIXO)J.HT 3a I1pe enhl ,L\HanaaOlla (Kn ;S 0.01),
r)J.e npHMeHeHHe KCPC 06oCHOBaHllo.
npe CTaBJIeHHbIe B [36) pe3ynhTaThI 6b1nH nepBblMH, n03BOJIHB-
1118MB 06paTHTb BHHMaHHe Ha B03MO>KHOCTH KCPC npH M0tI-enH-
pOBaHHH rreqeHHH YMepeHHO pa.3pe)KeHHhlX ra30B. PaCCMOTpHM e e
vvv
111. v. MQ4enHpOBaHHe reqeHJlII I'83t'JooR CPeAbi
ABe B03MO)KHOCfH, n03BOJJHIO He enaTb npOt\BH>KeHH Ha nyrH
HCnOJIb30BaHHH KCPC AJlS1 MOJl-eJlHpoBaHHSI 3THX 3 aq. HaqHeM C
OfiHCaHHii pe3YJ1bTaTOB HCOonb30BaHHH KrY-CHcTeMbl C rpaJIHqHbI-
MH YCJIOBHflMH CKOnb)l(eHHSI (214J.
2. KrY-cHCTeM8 c rpaHHTfHblMH YClIOBHRMH CKOJlb)f(eHHH
OnOJIHHM KB83Hra30AHHaMH"IeCKYIO CHCTeMY ypaBlleftHA (2.1)-
(2.4) (rn. III) rpaHH'lHblMH YC1l0BHHMH Ha nOBepXHOCTH ofueKaeMoro
Tena
t!!!. - 0
- ,
on
ViJJoo
u= -.
p V2 RT an
( 1 x liT )
Tl - Tw = 2R -.12m' pan '
(5.4)
(5.5 )
(5.6)
3 eCb U .- 3HaqeHHe CKOpOCTH B HanpaBJIeHHH KaCaTenbH H K no-
BepxHoCTH 'rena, n - TeKY 3jI KOOp HHaTa B HanpaaneHHH, neprreH-
,aHKynSipHoM K nosepxHocTH.
rpaJIH'iHOe YCJlOBHe (5.4), HCnOnb3yeMoe B KrY-cHCTeMe, CDSI-
3aHO C YCJIOBHeM HenpOTeK8.HHSI raa& Qepe3 nOBepXHOCTb Tena (24).
rp8.HHttHbIe YCJ10BHSI CKa1.£Ka 6b1nH paHee pacCMOTpeUbi 8 rJlaBe III
(Bblpa>KeHHH (2.27)-(2.28» H xopowo H3BecTHLJ B TeopHH nL30B.
Pe3YJlbTaTbI pacqeTOB Kry -CHCTeMbI C rpaHH MH ycnOBHsr-
MH (5.4)-(5.6) (Ha pHC. 130, 131 OHH 0603HaQeHbJ KaK QGS) cpaB-
HHBaJ1HCb C aHaJlOrUQHblMH paC'fe'IaMH 118 aCROBe Hcoonb30BaHHSI
ypaBIlcHHB HaBbe-C1.'OKCa (NS) H MeTO OB npHMoro CTaTHCTH'IecKO-
ro Mo enHpOBaH}tlI (DSMC) (214). B paccMaTpHsaeMblx sapHaHTax
TeMnepaTypa CTeHKH T w nOJlaranaCb paBHoA TeMoepaType fJeBo3MY-
eHHoro nOTOKa.
Vl3 pHC. 130, 131 BH HO) '{TO npH He6OJIb11JHX QHC1Iax Maxa Mo-
enbt OIIHpaf01I\&HCiI Ha HCnOJIb30BaHHe ypaBHeHHA HaBhe-CTOKca c
rpaHH1.IHblMH ycnOBHSlMH CK01Ih1KeHHSI, H KrY-CHCTeMa C TeMH )Ke
rpallH1.fHl:dl\lfH YCJIOBHSlMH alOT 6JIH3KHe no Ka'leCTBY Pe3YJJbTaTbI.
O aKO npH YBeJJHqeHHH qHCJIa Maxa p€3ynbTaTbI, nonY1.feH;Jhle Ha
95. MO e.JIHpoBaHHe reqeHHlJ YMepeHHD pa3peJICeffHhlX ra30B
307
100
eo
40
2ID
a)
.,
1.
.
10
.to
20
6) ,
DIMe
QCI)
HI
DStC
-.--.... QCK)
..-.. HI
PHC. 130, npo4>HJ1H nJIOTHOCTH (a) H aBJIelDUl (6) B Ce'lCUHBX no KOOp,lJ.HllaTe
x: Moo = 2
'I
3D
DSMC .
QGD
NS
I
.
to
:..'-.-
--.---
- - ._- ..
- - ,. - . ---'-";;a
- ..- - .
JD
a)
1.0
p
10
r .
3D \
,
,
,
-. .-
--. :-.........-.-...
. ...,.
.... /
.' .
--- .-. .' , 11..&0
""7---.-. _
.. ,. ..... "J
;.."
,
---. ...._..... _....!:-:. a
,. -
D8NC
aGD
HI
'10
p
6)
40
PHC. 131. llPOcpHJIH nnOTHOCTH (a) H aBJIeHHH (6) B CeqeHHJrX no Koop,n;HHare
x: .foo = 20
OCROBe Kry -CHCTeMbJ, Jlyqwe COBn aIOT C )l.aHHblMH CTaTHCTH'Ie-
CKoro Mo,n;enHpOBaIIHH, KOTOpble npHHHMaJlHCb 3B 3T8JIOHHbIe.
PacCMOTpl-tM e e OAHH nO,QxOA, CBSl3aHHbIA C Hcnonb30BaHH-
eM KCPC H KrY-cHCTeMbI NISI Mo,n;enHpoBaHHSI TeqefIHR YMepeH-
80 pa3pe)KeHHbIX raJOB 1215), KOTOpblH npe CTaBJISiercH OCTaTO"'HO
nepcneKTHBllbiM.
3. KCPC H MCTOA domain decomposition
B HacTOH ee BpeMSi
OCTaTOq}fO pacnpoCTpaueH-
HblM jlBJISlCTQI CJIeJJYIO HH
nOAxOA K MOAeJIHposaHHIO
TeqeHHA YMepeHHO pa3pe-
.>KeHHbIX f830B t OCHOBaHHblH
Ha npHMeHeHHH MeTO OB
"domain decomposition u . B
o6naCTH B6Jm3H nosepx-
PHC. 132. K peweHHIO Ja.n.a H 06reKaHH1I HOCTH Tena, r e pemeHHe
MeTOD-OM "domain decomposit.ion"
HcnblTbIBaeT pe3KIJe rpa-
,
l\HeHTbl }Ia pacCTOSlHHH nopSIAK8 HecKOJIbKHX A1IHH CBO'JOAHOI'O
Ilp06era, ra30,.[tHHaMU"IeCKKe na.paMeTpbI onpe enSilOTcSI nyreM
IIp MOro pelueHHSI ypaBHelIHR Bonb MaHa MeTO aMH craTHCTH-
"IecKQf'O Mo,nenHpOBa.1IH5I. Ha pRC. 132 9'Ia 06nacTb orpaHH'-IeHa
nOBepXHOCTblO TeJla (KpHBaSi 0) H KpHBOA 1. B qacTH npocTpaacTBa,
Jle}f{a.J.l(eA SHe 06J1aCTH, OrpaHHQeHHOA nYHKTHpHOA KpHBOA: 2.
ra30 HHaMHt.leCKHe napaMeTpbl onpe.u-eJIHIO'rCH Ha OCHoae pemeHHA
ypaBHeflHA HaBbe-CToKca. IIpa 3TOM 3HaQeHHSI napaMe'IpOB Ha
KpUBOA 2, HaA,lJ.eHHbJe H3 pemeHHS1 ypasHeHHSI Bonb MaHa, CJIYJKaT
rpaHHQHblMH YCJIOBHSlMH ,lJ.J1S1 ypa.BHeHHA HaBbe--CTOKca. H Hao60-
poT) pemeHHe YPaBHeHHA Hasbe-CTOKca H& KpHBOA 1 CJIY)f(HT nSl
cl>opMHpoaaflHH rpaHIRHblX YCJJOBHR nH ypasHeHHSI B01IhqMaHa.
3TH rpaHHqHble ycnOBHH cl>OPMHPYJOTCjI HTepa oH.lhIM nYTeM
Ha OCHoae cepHH nocne,ll.OBarenbHbIx peweHmt yp8.BHeHHH BOJIbl.{Ma-
Ha H ypasHeHHR HaBbe-CToKca. 3a.MeTHM. liTO, KaK npaBHJlO t 3TOT
HTepaQHOHHblA npol.\ecc CXOAHTCH 'reM 6blCTpee, qeM 60J1bWe c6nacTb
.-......---.
- ')"p-tl_ HUIlbC-<....lmlCU. .-_... 2
,-
,,,,
"..
"
...' ..' ..-
.
I
I
l
.
I
I
o
-:5
.
.
.
.
.
.
r
I
-. .....-.. . :
. p-ttMC: r,(J_I ._
5. MQAeJlHpoBaHHe TeIJe,mR yuepeHHO pa3peHCeHHblX r830B 309
''nepeKpbJTHB'') ne)l( aH Me>K)U' KpHBblMH 1 H 2.
nO 06HblR no,Axo):{ ,nOCTaTO'fHO nepcneKTHBeH. OAHa.KO OH CTaJI-
KHBaeTCSI C npHH HnH8JlbHblMH TPY HOCT 'H. rll)JlqHHoA ax SlBn}f-
eTCSI TOT <PaKT, qTO 06nacTL Me)l(A)' KpBBbIMH 1 H 2, r e cnpase,ll)lHBO
HCIIOJlb30BaHHe ypaBHeHH:A HaBhe-CTOKca (Kn :5 0.001), TpYNJo o-
cTynHa An}} npHMeHeHHB MeTO OB CTaTHCTHQeCKOrO Mo enHpOB8HH.H.
KOHet.JHo, leTO,lJ.aMH CTaTHCTH'IecKoro Mo enHpoBaHHH B npH}I Hne
MO)KHO nOJ1}"iaTb pemeHHSI H "PH M8JIbIX QHCJlax KHy ceHa, HO npH
M'Otwl pe3KO paCTYT BbI'iHCJlHTeJJbHble 3aTpaTbI Ha pemeHHe KHHeTH-
qeCKOrO ypaBHeHHH BOJIb MaHa.
OCHOBHaH H eSl pa6oTb1 [215] 3aKJIIOQanacb B onope H 6W1ee
WHPOKYJO o6nacTb onHCaHHS, oCTynHYlo KCPC H Kr-y (Kn:5 0.01).
Cyu ecTBYeT Hana.30H tfHCen KHy ceHa, pacnOJlO)KeRHblA OKOJIO
3HaqeJIHH (Kn 0.01), rAe, C o oA CTOpOHbI, peaJIbllO HCnOJlb3O-
BaTb JeTO,AbI CTaTHCTHQeCKOrO Mo enHp083HH ypaBHeHJfil BOJIb -
Mal-la, a c pyroR - cnpaBe,nJIHBO ra30 HHaMHqecKoe onHcaHHe na
OCHOBe KrY-cHCTeMbi. npe CTaBneHHa.g Ha pHC. 133 cxeMa nOKa-
3bJBaeT "peHMYIIJ.ecTBa KB33Hra30AHHaMHQeCKo:A CHCTeMbI, KOTOpLJe
npoHBJISlIOTaI B "rOM, qTO OHa no cpaOHeilHIO c ypaBHeHHSlMH HaBbe-
CTOKca onYCKaeT npHMelleHHe AJlH 60J1bWHX QHCen KHy ceHa H
n03BOJISleT nOJIYQHTb 3HaQHTeJIbHOe nepeKpbJTHe c pemeHHeM) nony-
qeHHblM Ha OCHOBe ypasHeHH BOJIL({MaHa.
h- pewellBe ypaBOeHH$I Bonh aHa
.!.J MeTO OM MOHTe-K&pJlO
- YP-HH9 HaBbe-CTOKC3 11
· KCPC H KrY-C CreNa 1
J ,
10- 4 10- 3 0.01 0.1
.
[)
1 Kn
PHC. 133. K 06OCHOBaHR1O MOAenapo RA.lIV jI Te'leHIUI ra:Ja H& OCHOBe ICOM6BH HH
peweHHB KrY-cHCTeMb[ H ypaBHeHH8 BOJIblJ.Maa8.
310
rJl. v. MOAeJ1IlpOMlfHe Te1.JeHHR ra30BOH Cpe,I(
4. MO,l{eJlHpOBaHHe CTPYKTypbl YAapnoR BOJIHbI
Bonpoc 0 B03MO)KHOCTH OlIHcaHHSI y apHoA BOJIHbI c nOMOmplO
YPaBHCHHa: ra30BoR HHaMHKH BBJIjleTCH TP&AHI.J)IOHHldM MSI Teo-
pHH ra30B (2]. no BCeA 8 HMOCTH, nepBoA: ny6JIHKa[ Hea, B kOTopoA
6L1J1H nOKa3aHbl B03MO>KHOCTH KCPC H Kry -CHCTeMbI B 3ToR 06-
JlacTR, 6bJna pa60Ta [216]. B HeR KB83Hra30P.HHaMH'IeCKag CHCTeMa
HCnOnb30Banacb ,lVIH Mo,.qeJlHposaHHB y,napHblx BQ1IH B HORoc<j>epe
3eMJ1H. TIpH 3TOM y anocb onHcaTb HeTpHBH8JlbHl:de OC H.rIJ]SlIJ.HH
peUleHHH, Ha6JIIO aeMbIe B '3KCnepHMeHTe.
AB1'OP 1'aK)I{e XOTen 6bI 06paTHTh BHHMaHHe qHT8Tene Ha pa-
60TY (217J, B KOTOpoR Ha OCHOBe ypaBHeHuii rH,lIJ)O HHaMHKJt nOBbl-
llleHHoA TOqHOCTH npoBeAeH pacqeT CTPYKTypld y apHoA BOJIRbl H
nOK83aHbI npeHMYIII-ecTBa npe)I.JIaraeMOro OnHCaHHSI no Cp8.BHeHHIO C
Tp H HOHHbIM ra30 Ha.MHqeCKHM. 9TH ypaBHeHHSI r830BOJt HHa-
MHKK nOBbilUeHHoA TOqHOCTH 5ISnSlIOTCB CJ1e CTBHeM o606I eHHoro
ypanlicHHSI BOJIb MaHa (87)
VI V ( VI )
Vt - Vt r Vt = J(f,!),
(5.7)
r e V/Vt HcJ><j>epeHIUlaJlbHbIH onepaTOp, CTOS HA B JlenoA qa-
CTR ypaBHeHHSI BOJIbQMBHa, T - qaCTOTa CTOJJKHOseHHA, J{/, f) -
HHTerpan CTOJIKHOBeHHA YpaBHeIlHSI BOJIbQMB.Ha.
PacCMOTpHM HeKOTopbIe pe3ynbTaTbI pacqeTOB CTPYKTYPbl YAap-
HoA BonHbl, npOBe etI}lbJe Ha aCHOBe Kr'Y -CHCTeMbi [218, 219].
Ha pHC. 134 H306pa>KeHbI proynbTaTbl nOBe eHHH BeJlHqHHbI AI is
B 3alllfCHMOCTH OT "IRena Maxa, rAe A - AJlHlla cBo6o,lJ.HOrO np06era. B
Ha6eraJOIneM nOTOKe t 6 - 'rOJ) Ha YAapHoA BOJlHbI, Onpe)l.eJ1HeMa..sI
H3 COOTHorneHHSI
5 = (P2 - PI) / tg 4J,
(5.8)
r)J.e PI H P2 - 3HaqellHH nnOTHOCTH o It nOCJ1e npoxo eHHH Y.Aap-
HoA BOJIHbl) 4> - MaKCHMaJlbHblA yron HaKJIOHa npo4>HJJR y,n:apaoa
BOJIIJbl. Pe3YJIbTaTbI, nonyqeHflbJe Ha aCROBe pelIIeHHH KBa3Hra30-
HHaMH eCKOA CHCTeUbJ I Cp8.BHHBaJ1HCh C aHHblMH 3KcnepHMeHT8.
(220).
55. MOAeJ1HpOB8Ifne TeqeHHiI YJ4epelfHO pa3peJKeHHblX Ia30B 311
0.1a
/
;/
J ...--- .
P r-'" .
.. .1-
, -u -'1iiI
r. I -, . . I .- .
,
J II .1 -.
II' I .
.
3 3 . , - . - ., - I '" . .... I ,. .1
O.M
D.51
0.411
It. o.
8
0.32
D.
0.11
0..
D..
I
M
PHC. 134. PaC'ieT TOJIIlJ)fHhl y,a.apaoii BOJIHhI B 3a.BHCHMOCTH OT 'IHena Maxa.:
· - ypaBHeHHS Ha8be-CTolCca J 0 - KrY-CHCTeUa.. - SKcnepUMeHT
TIpH He60JIbmHX 1.iHCJlaX Maxa ypaBHeSHB H8.Bbe-CTOKca H
KrY-CHCTeMa O HHaKOBO xopomo onpe enSiIOT TOJJ HY y apHO.a
BO.lIHbl. OtlHaKO C pOCTOM qHCJla Maxa npeHMy ecTBa KrY-cHcreMbI
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