Текст
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2004


y tu< 519.8:533 66K 22.193 11-52 'IeTBepYWICHB 5.H. 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 4YHKUHH pacnpcJlCnCHMJI. 06c}. JlaIOTCJ1 B03MO)l(HOCTH HHTCpnpeTlUHH nOCTpoeHHoR CHCTeMW ypaBHeHHA It IC8'4C. crBC MOACJ1H, onHCWBaIOweR Te'tCHHe BJlJKOrO ra:J8, a T810Ke nOIC83L1Bac. ec cU:SIJ C Tp8l1MU,HOHHWM OnHcaHHeN, onHpaJOWHMCA Ha ypaaHcHHI HUbe-CTOlCca. npMIO.48TC. npMMepw HCnOnb3CB8HHJI npellJ10)KCHHW" anropHntOB nnl MOlleI1H. po.8ItH. COBpeNeHHIX npo6neN ra:JOBoA HH8MMnt. J)OJlbWHHCTBO pe:lYJlltTaroB nOll)"leHO c npHMCHCHHeM BWCOKonpoKJ80D.MTCJ1HWX 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 @ aKynTeT BMHK Mry MM. M.B. ROMoHocoBa.2004 
OrJIasneHHe BBeJl.eHHe rJlasa I. YpaBHeHHe BOJIbaHa H YpaBHeHHJI rasoBoA HHaMHKH  I. OCHOBHbJe nOHSITHSI KHHeTHqecKOA TeOpHB ra30B 1. OHOqacTHqH8,jI CPYHKg pacnpeAeJIeHHSI . . 2. CTOJIKHOBeHHe MOJIeKyn . . . . . . . . . . . . . 3. ,lI.nHH& cBo60oro npo6era.. "lJaCTOTa CTOJIKHoBeHHA  2. Y pasHeHHe BOJ1bMaHa ... . . . . . . . . . . . . . . 1. BbIBO,n ypaBHeHHSI ,lJ.JISI OAHO'lacTH'tIHOA <PYH pacnpe)J.eneHHJI ............ . . . . 2. CBoACTBa HHTerp8J1a CTWIKHoseaHA . 3. H-'reopeMa BOJThMaHa . . . . . . . . 4. PacnpeAeJIeHHe MaxcBeJIJl&. . . . . . 5. YpasHeHHe BOJIhaHa AlIa CMeCH ra30B 6. YpasHeHHe BxaTHar&pa-rpocca-Kpyxa (BrK-Moenh) . . . . . . .. .. . . . 7. Y pasHeHHe 3HCKora . . . . .  3. YpaBHeHHSI ra30BoA HHaMHKH 1. MOMeHTHbJe yp8.BHeHHSI . . . 2. Y paBHeHHSI 9Anepa . . . . . 3. BH3KOCTb H TennonpOBOOCTL .  4. YpaBHeHHSI HaBbe-CTOKCa .... 1. Pa3nO)l(eHHe t.{euMeH&-9HCICOra 2. JIOKaJIbHO HaB&e-CTOKCOBCX&S 4>YHKg pacnpeeJJeHHjI ................. 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 6onhM8.HOBCKHX cxeM 2. CooTHoweHHe 6a.naHca AnSI cPYHKl.\HH pacnpeAeJJeHH .................. 3. Pa3HOCTHble ypaBHellHSI AJlSI ra30HHaMHqecKHX napaMeTpoB . . . . . . . . . . . . . . . . . . . . . . 4. CornacoB8HHoCTb pa3HOCTHhlX annpoKcHMa 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><PepeHH8JIhHO- pa3HOCTHaH <POPM& ypuHeHWI 6anaHca ,l{JlSI cPYHKHH pa.cnpeeJIeHHSI . . . . . . . 2. Ee O)UlH cnoc06 nOJJyeHHg ypasHeHHB 6anaHca . 3. KB33Hra30HHaMHqeCK&Sl CHCTeMa ypaBHeHHA 4. P8.3HOCTHag annpoKcHMHSI . . . . . . . . . . . . . . . . . 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 MoenHpOBaHHB 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.30HHaMH1.{eCKaH CHCTeMa , . . . . . . . .. 118 2. CBSl3b C yp8.BHeHHHMH HaBbe-CTOKca . . . . . . .. 120 3. HccHnaTKBHble qJIeHbJ KBa3Hra30HHaMHqeCKOA CHCTeMbI , . . . . . . . . . . . . . . . . . . . . . 124 4. HeKoTopble pe3ynbTaTbI qJfCJIeHHhlX pacqeTOB 127  3. KHHeTHtfeCKOO OTp8.)KeHMe KBa3Hra30HHaMH\feCKo:A CHCTeMbI ypaBHeHHA . . . .. 131 1. KrY-cHCTeMa H O,ll)lOqacTHaJI <j>YHKlJ)IjI pacnpeeneHHa .................... 131 2. KB83Hra30,l\HHaMH'IeCKaH CHcreMa H B rK - ypaBH eH He . . . . . . . . . . . . . . . . . . .. 133 9 4. 06 HHBapHa.HTHOCTH KBa3Hra30HaMJ[qecKOH CHCTe- MbI ypaBHeHHA . . . . . . . . . . . . . . . . . . . . .. 136  5. 0606ll.\eHHSI Ha ,lq)yrHe 3a'lH MeX8.HHKH cnnOWl10A cpebI . . . . . . . . . . . . . . . . . . . .. 140 1. MoenHpoBaHHe npoecca cPHnbTP81UlH . . . . . .. 140 2. B03MO)KItaJ1 MOB<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-MoenH - LBGK-cxeMbI 151 2. KBa3Hra30AHI-IaMHqecKaft CHCTeMa H LBS . . . . 154  7. MeTo CT86HJIH3aHOHHblx nonpasox H HeKoTopble BapH8HTbI Kr"Y -CHCTeMbi .. . . . . . . . . . . . . 156 1. MeTOA CTa6HJlH3HOHHblX 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 3aq 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 KoppeKHea 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. MHoronpoeccopHbJe BhJ'IHCJIHTeJlbHble CHCTeMbJ .. . 1. HCTOpHfI H nepcneKTHBhI p83BHTHS MHoronpoqEc- COplWX CHCTeM ............. 2. MeTaKOMllblOTHHr H KJlacrepbl . . . . . . . .  5. alrraHjJ KCPC K apXHTeKType MHoronpou.eccopHhlX BLlQHCJ1HTeJIbHbIX CHCTeM . . . . 1. Tpe60B&HHB K BbI'IHCJIHTeJIbHblM anrOpBTM8M . . . 2. MHoronpoeccopHblA BapH8HT KCPC C KoppeKqHea 3. IIOJIYHeHBHaH CXeM& H KCPC H& HecTpYKTypHpo- BaHHhlX ceTKaX - MHOronpOecCOpH8.SI 8J:tanHB . 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)-HTepaHOHHbla MeTOA . . . . . . . . . . . 2. MHoronpoeccopHhlA 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 npoeccopoB 224 224 226 228 rJIasa V. MOelIHpOBaHHe Te'leHHA l'83oBOA cpeAhi 231  1. MOAenHpoSaHHe CTpytbloro Te'leHHH . . . . . . . . 231 
OrJlaBJIeHJle 7  2. MO,lteJIHpOBaHHe HeYCTaHOBHBWHXCH ocu.HJIngOHHLlX TeeHHft. . . . . . . . . . . . . . .. 236 1. t.{HCJJeHHoe MOe.nHpOB8HBe OCItHJI.1UIHOHHbIX 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. MOeJlRpOB8.HHe Typ6YJIeHTHblX HeYC'I'8HOBHBwHXca retJeHHit B OKpecTHOCTH KaSepHbI . . . . . . .. 264  3. reHep8J:lH1I H pa.cnpOCTpa.HeHHe MaJIblX (aKYCTHQeCKHx) B03MyeHHA: . . . . . . . . . . . . .. 272 1. MOtl.eJIHpOBaHHe pacnpOCTp&HeHHSI M8JIbiX B03MY- eHHa . . . . . . . . . . . . . . . . . . . . . . . .. 272 2. MOeJlHpOBalIHe HCTOqHHKa aKYCTlRecKHx B03MY- llI.eHHA .. . . . . . . . . . . . . . . . . . . . . . .. 280 3. MO,D.eJlHpOBaHHe WYM& CBepX3SYKOSOA' CTpYH . 285  4. MoenHpOBaHHe Te'leHHA XHMHQeCKH pearHpylOJ..U.Hx ra30B . . . . . . . . . . . . 287 1. MaTeMaTH'IeCKaB nOCT8.HOBKa 3a'lH . . . . 288 2. Bbl'lHCJJHTenbHhlA 8JIrOpHTM . . . . . 291 3. MOeJlHpOB8.HHe ropeHHSI npHpooro f'83a . . . .. 295  5. MOAeJIHpOBaHHe TeqeHBA YMepeHHO pa3pe>KeHHhlX ra30B. CTPYKTypa yapHoR 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. MOeJ]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>epeHHaJThHOe npH6JJH)KeHHe - KBa3H- ra.30AHHaMH'IecKaSl CHCTeMa YPaBHeHHA (KrY). HHTepeC I( ,D;8HHblM HCCJ1e)\OBaHHSlM B uacTOgee BpeMSI BbJ3Ba.H CJIelOlI(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&lOcA 3alISlTOA 8 CeKYHA}'). B no06HoA CHTYHH B03HHKa.eT TeXHH\feCKa5I B03MO)I(HOCTb npH- crynHTb K lJHCJleUllOMY peWeHHIO HaH60nee CJlO>KHhlX 3aq 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 MOHOCTblO nopJ1):tKa 100 GFLOPS (1011 Onepart: HA C nna.salOIl\eA 3anBToA B CeKYY) H3 p83pSltl-8 YHH- KaJlbHbIX YCTaHOnOK nepexollrSlT B c<pepy 060PYOBaHHHJ ,L\OCTyouoro MHorHM CneHaJlHCTaM. 9TO B npHHHne n03BOJI5IeT WRpOKM <PPOH- TOM BHeAPHTb Bb1COKOnpOH3BOHTeJIbHhle BhlqHCJleHWI B npaKTHKY ra.30AHHaMH'leCKOrO MOemq>OBaHHH. C pyroa CTOpoHbI, nOTeHH8JIbHbIe npeHMYecTBa HCnOJIL- 30BaHHH MHOrOnpOeCCOpHblX 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 nOJIYIHBWHM llaa60J1bWee pacnpocrpaHeHHe MHol'orlpoeccOpHblt.( CHCTeMaM c pacnpeneHHoA naMftTblO. liMe- IOTC.H T8J()Ke 3HaqHTenbHbie npo6neMbI B C03AaBHH CnelUtaJIbHOrO MaTeMaTHQeCKOrO 06eCne'leHHjI, He06xo,nHMoro AlISi npOBeeHHs[ BbI- COKonpOH3BOHTenbHbJX napaJIneJlbHblX BbItfHcneHHA. 
9 OHa.KO HaH60JIbWYIO CJI0JKHOCTb, no MHeHHlO &STopa, BLJ3bJ- BaeT np06JIeMa HAeHTH<I>HJ<.aQHH aHHbJX Bb1COKOnpOH3BOHTenbHbIX BbJt.IHCJleHHA, HCnOnb3YIOHX npOCTpaHCTBeHHbIe cerKH 83 10 9 Y3JIOB H 60J1ee t COnOCTaBJ1eHHH 3TOA HH<PoPMaHH 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 npoeccoB ropeHHSI, B 3a- aqax aapoaxYCTHKH H B PJ>yrHx nOCT&HOBKaX. rAe npHHHnHaJIbHO Sa)KHO npHsneKaTL nopo6Hoe npoCTpaHCTBeHHo-BpeMeHHoe ODHca- HHe cpebl. Pe3YJlbTaTOM MOeJlHpOB8HHS 3,l'(eCb Heo6xo)\HMO HMeTb KapTHHY nOBeeHHH 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- ,4HMOHHO HCnOJIb3yeMoA JVIH OnHcaJlHH Te'leHWI BH3KOro ra3a, - HBJIHeTC OHOA: H3 KJIlOtoIeBhlX B coopeMeHHoA MaTeM&THKe. B CBH3H C BbIWeH3nO)f(eHHblM B HacToHee BpeMSI Bee 60Jlbillee 3Ha'leHHe npH06peTalOT anropHTMbI, KOTopble, C o)J;HoA CTOpoHbl, o- nYCKaJOT npOCTYIO an1O K apXHTeKType MHoronpoeccopHbJX BbJT.fHCJIHTenbHbJX CHCTeM C pacnpeeneBHOa DaMSlThlO, a C .DJ>yro:R - n03B0J111JOT YBepeHHO MO)J.eJlHpoBaTb MHoro06p83HbJe cJ:>H3H'IecKHe HeYCToA'QHBOCTH, B03HHKaI01I(He 8 r830BOA cpe. 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 MaxpocKonHeCKHe ypa.BHeHHB 9AJIepa HJIH HaBbe-CTOKca. Ho ynoMSlHYTble ypaBHe- HHSI, B CBOIO oqepe.D-b. CJ1e)JyIOT H3 6o.nee CJlO>KHOrO ypaBHeHHg BonbMa.Ha AJIH OAHOqacTHQBOA cJ:>YHKlQiH pacnpe,lteneHHSI. npH 3TOM ra.30AHHaMH'-IecKHe napaMeTpbJ: CKOpoCTb, DJIOTHOCTb, aone- HHe H T.. - SlBJISIIOTC MOMeHTaMH cl>YHKHH pacnpeeneHH.R. O,rt- HaKO 3TOT cl>aKT paHee np&KTH'tfecKH He I1PHHHMaJlCjI 80 BHHMaHHe npH C03,AaHHH QHcneHHblx MeTO,ltOB, eCJIH OCTaBHTb B CTopoHe aJ]- 
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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 ."  . "-+00 Il V p B Bblp8,)KeHHH (1.1) CYMMHPYIOTCH MacChI mi BCeX qacTH, OKa- .1aBlliHXCSI BHYTpM 06eMa  V,. CTpeMneHHe DOCJIeOBaTeJIbHOCTH  V p K "YJllO nOHHMa.eTCH B CMblCJ1e "<pH3H'IeCKH 6ecKoHeHO Mana- ro o&beMa" KOP)J.& BHYTPH K&>KAOro H3 06'LeMOB l1 VI' COep)kHTCH AOCTaTOQHO 6onbWOe QHC1IO MOJIeKYJI, He06xo,l\HMoe )l,JlJI npHMeHH- IOCTH npH6J1H>KeHHSI cnnoUIHoA ChI. TeM CaMhlM HeJIBHhlM 06pa- :jOM BBOHTCSJ OrpaHH'IeHHe CHH3Y Ha HaMeTp 06n8CTH . 06pa- TliM BHHMaHHe, qTO HeB, CBSl3aHHasI C OrpaHHQeHHeM CHH3Y - R3 (PW3H'IecKHX npeOJIO)KeHmt - 06'beMa HHTerpHpoBaHHSI, BBnHeT- CH OCHOBOnOJIaraJ()e:A lIpH nocrpoeHHH KCPC H Kry H npH ax HHTepnpeTal.\HH KaK MoenH ng onHcaHHSI ,AHHaMHKH BH3Koro ra3a. rraKoA noo OKa3bJBa.eTaI uone3HLIM H npH paccMOTpeHHH pyrHX :iaatl MexaHHKH CDJlolDHoA cpe (rn. III MOHorpaq,HH). AHaJlorHqHO nJIOTHOCTH OnpeAeJlHM H ,npyrHe napaMeTpbl ra3o- noR cpehl, HanpHMep, CKOpocTb li E :\ mii A 1f 0 u= m , ",",Yp-+ . p....oo p  V p (1.1) 'J,AeCb i - BeKTOp CKOpOCTH i-A MWIeKynhl, !laxOAHeRCSI BflYTPH 061>eMa A V p . O,a)laKO ropa3,n:o 60nee y,n.06Ho HaxoAHTb ra30p)JHaMHQeCKHe Be- Jll1ttHHbI, BBe,ASI nOHSlTHe O,l{HOqacTH'IHoA <l>YHK1I.HH pacnpe,.qeneHHSI f. Onpe,QenHM <PYHKIUfIO paCl1pe,lJ.eneHHSI )l,JlSI oduH4fC06 MOJIeKY n 'lcpe3 Bblpa>KeHHe J(Xt, t)dxdE = dN, (1.2) KOTopoe eCTb O)f(H.AaeMoe qHCJJO MOJIeKYJI B 3JIeMeHTe npocTpaJICTBa dx = dx} dX2dx3 B OKpecTHOCTH TOQKH Xt 06naJOx 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 p8Cnpeene- t1l1S1 f HeSlBHO npe)UIOJlaraeTCSI, q-ro BepowrHOCTH pa3nH'IHbIX MOJIe- I\yn, HaxoASlw.HXCH B TOqKe XJ HMeTb IIpHcye TOnbKO HM CKOpOCTH 
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}fTblBaJOe KOppenRQHJO BepOSITHOCTH otIrHoA H3 MOJIeKyn 06naaTb CKOpOCTblO 1 C BepoHTHO- CTHMH OCTaJlbHbJX (n-l) IOJ]eKYJJ o6JIarrb CKOpOCTjlMH 2'... 'n' pacCM&TpHBaTbCH He 6YJU'T. IIoHHTHB OHOqaCTHtlHOA <l>YHKIJ.HH pac- npeeneHHSI BDWIHe A0CTaTOqHO ,lJ.JJ5I H3J1O>KeHHSI npeACTaBneHHOro B MOHorpa,<J>HH MaTepHana. HwrepeCYIOUJ.He Hac ra30HHaMHqecKHe nap8.Me:rpbl Moryr 6b1Tb nerKO HaAeHb) C nOMOLIO OAHOqacTHHOA <PYHKHH pacnpe,nene- HUH (1.2). IIJlOTHOCTb H CKOpocTb r83a onpe.zx.enalOTcSI Kat p(t,x) = ! mf(x,,t)d, u(t,x) =  / m(f(x,(,t), ( 1.3) ( 1.4) rAe KaJKAaSl MOJIeKYJla HMeerr Macey m. BBeAB nOHJ1THe xaOTHqeCKoi! HJ]H TenJ10BoR CI<0POCTH MoneKYJI c =  - u, 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) 3ecb c 2 = if +  +  . TeMnepaTYPY r8.38 onpeenHM Qepe3 cpeJOro 3HeprHIO rrenJIO- BOro t\BHJKeHHSI MOJIeKYJI 3 k 1 f mc2 2m T= p 2 f(x,,t)d, ( 1. 7) re k - nOCTOmlHasI BOJIbMaHa. ,AaBJIeHHe, Hcnonb3yeMoe B ypaBHeHHHX ra30BOA: AHH3MHKH, no- naraeTCSI paaHblM TpeTH CYMMbI ,[\HarOHanbHblX COCTaBJIHIOIlUIX TeH- 30pa HanpSl>KeHHA: p = 1 (Pu + P22 + P33) · (1.8 ) 
 1. OCHoBHhle nOHSlTHR KllBeTJrlecKOR TeOpHH lB30B 17 YqHTblB8JI (1.5) H (1.7), H3 (1.8) nerKO nOJl)"IHTh: k p = p - T = pR:r  m ( 1.9) pft.e R = k/m - r830B8B nOCTOSlHHaJI. TaKHM 06pa30M, )l.aBneHHe onpe)J.enBeTCfl no KnaccHqecKoA <I>oPMyne KJIaneAposa-MeH,lteneeBa. rOBOpSlT, 'fTO ra30)J.HH&MH'iecKHe napaMeTpbl SlBnS11OTCS1 MOMeH- ',l4MU cpYHKlJ)fH pacnpeeneBHB. 2. CTOJIKHOBeHHe MOJleKYJI npoeccbl CTOJIKHOSeHHJI qacTHIJ. MO>KHO pa3,lI-eJlH1'b Ha Ba THn9. . ynpyrHe H HeynpyrHe. DpH ynpyrHx CTOJIKHOBeHHJlX COXpaHfleTCH nx CYMMapHaH KHHeTH'Iecxas 3HeprHB, a MeHHIOTCSI JlHIW, CKOpOCTH BH)KeHHSI CT8JJKHsalOUJ.KXCSI qacTH. TIpH Heynpyrwc CTOJIKHOOeHH- S1X MeHSeTCH BHYTpeHHHS 9HeprHB qacTHQ. Ca}..f npoecc CTOJIKHOSeHHSI 3aBHCHT OT nOTeHana B3aHMoeA- CTBHH, KOTOpblH onpeeJISleTCB xapaKTepoM KBaHTOBOMexauH'IecKHx H 3JIeKTp0CTaTHQeCKHX npoeccoBJ npOTeK8JOI.qHX npH c6Jm>KeHHH PvIOJIeKyn. npocTeAwHA npoecc 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 HieeT nopSl)J;oK 10- 8 CM. Ana MoenH 6HnbSlPNihlx mapoB fiOTeH- IHan B3aHMOeaCTBHg 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& npoecc B3aHMOeACTBHs MOJIeKYJI. Cpe,D.H ,npyrHx Moenea, 60nee TOQHO OnHCblBa.IOHX Me>KMoneKY- lISlpHoe B3aH M o)\e A CTB He , OTMeTHM nOTeH8JI 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€ 'IacTHb1 1 H 2 HMelOT JJ..O CTOJIKROBeHH3 Maccbl ml , m2 H CKOpocTH l , (2 . XapaKTepHCTHKH 06eHx qacTHD. nocne 38.BepWeHH9 npoecca 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 npOHCXOH'I B OAHOA nJlOCKOCTH, HaTSIHYTOA H3 BeKTOpbl 1 H (2 [9]. B npoecce CTOJIKHO- BeHHB BhlnonHSlIOTC$I 3aKOHbI COXpaHeHHB MacChI, HMDynhca H 31Jep- · AnSI peanbHblx 1'83011 B CTeneHBOM nOTeH8Jle (1.11) nOK83aTeJ1b 8 DpU- HHMaeT 3HaeHHH 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 noeHJUI 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 , mll + m22 - m + E, mId /2 + m2V2 = m2/2 + 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 npHenbHoe paccTOJlHHe: ( 1.13) 191 = 19'1 , b = b ' . (1.14) Ha KaKHX >ICe npHeJlhHblX paccTOSlHHHX bmax ee npoHCXOT 83aHMO;teACTBHe 'facrH? Ha 3TOT BOnpoc nerKO OTBeTHTb, paCCMaT- pHBa MOeJIb 6HJlbSlp;J.HbJX mapoB (1.10). OqeBHAHO, ttTO CTOJIKHo- nCHHe coc'rOHTCH, ecJIR npHI1-enbHOe paccToSlHHe MeHbwe aMeTpa lllapOB d. Cne.nOB8TeJlbHO, )VIR TBepx mapOB b max = d. (1.15) B aHHOM CJJyqae BeJlHqHHY d eCTeCTBeHHO Ha3BaTL 3<P<I>eKTHBHblM CeqeHHeM B3aHMOeaCTBH$1 o. HarJlHOCTb nOHstTHH 3<I><PeKTHBHoro ceqeHHB B3aJ1Mo,neACTBRSI KaK AHaMrpa CTaJIKHBalOIWIxcg wapOB n03BOnSieT nOCTpOHTb ero aHaJlOrH Ii ,lI.J1H ,lWyntX BOB DOTeHqH8Jla CTOJIKHOBeHHB. CTpo- ["0 roBOpSl, yrOJl a OTK.nOHeHWI TpaeKTOpHH qacTHI(bI "PH npOJleTe (pac. 1) OJI)I(eH p8.BHHTbCSl HYmo ecJIH b > u. ,ltrJSI Taro, qTOOLI 1I0CTpOHTb <pH3HQeCKH p83)'MHoe CeqeHHe CTOnKHOBeHHB, 3aMe'I'1Bi, liTO yron a 6blCTpO y6blBaeT C poCTOM b A1ISI nI06oro nOTeHIJ.Hana U, aieKBaTHO ORBCLlBaJOIII-ero pe8JIhHoe B3aHMOeAcTBHe (CM. (1.11), ( 1 .12) ). Y ,lJ.06lIO onpeeJIHTb Sct>cpeKTHBHoe ceqeHue (T NIB KOHKpeT- HOI'O COpTa qacrH c nOMOIIJ.h1O ct>OPMYJlhl [3] 00 u 2 = 21f' 1(1 - coso(b)] bdb. o TaKaH BeJIH'fHH8 q H83bIBaeTCSI 1'p3HcnopTHhlM ceqeHHeM. ,lJ,.njl HeA- TpanbHblX 1"8308 U '" 10- 8 CM. (1.16 ) 
20 rJ1. I. YpasHeHHe BOJlhaHa H ypaBneHHH l3308011 ,ltlfJfaMHKH 3. ,ll..nHHa CB06o,l{Horo llp06era. t.[acTOT& CTOJIKHOBeHHIf )]:JIHHOA cBo60,lUioro npo6era Ha3bIB&eTCSI cpeHee pa,cCT05lHHe, npOXO,l).HMOe qacTHl.{cA Me)l()Q' ,A0YMSI CTMKHoseHHaMH. B 06bl- HblX ra33wX, HaxO,DJI1Jl)IXCSI npH aTMoccpepHOM ,lJ.aBJ1eHHH. 3Ta )VIHHa I "J 10- 5 CM, \lTO B HecKOJIbKO COT p83 60nbWe ,lJ.Ha.MeTpa MOJIeKyn. ,lJJ>yroH BeJIHQHHOA, xapaxTepH3yJOeA cBoAcTBa raJa H TecHO CO3aJfIIOA c )J.JIHHOA cBo6oHoro npo6era. HBJlSleTCSI qaCTOTa CTOJIK- HOBeIlHA: II, paBHaJI KWJUqeCTBY CTOJ1KHOBeHHA. HCDblTLIsaeMbIX MO- JleKynoH B e.LJ.HHHI.(Y speMeHH. Cpe,lJ.Hee BpeMSI Me)K)Jj' CTOJIKHOBeHH- BMH pasHO T = l/v. ( 1.1 7) )l..nHHa cBo6o)J.HOro npo6era H qacTOTa CTOJIKHOSeHMA O[lpeenB- lOT TaKHe BCUKl-lLIe cBoAcTBa ra30B, KaK K03<t>cPHHeHTbI B$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 nOBeeHHH r83osoA CMecH. llYCTb Bce MoneKYJIbl ra3&, KpOMe OHOa np06HoA, HMelOT OAHH&- KOBYIO CKOpOCTb E, a CKOpOCTL np06HoA MOJIeKynbl paBHa 1' 9Ta MOJIeKyna CTOJIKHeTCH B eHIIHUY BpeMeHH co BceNH MOJIeKYJlaMH, neMH BHyrpH HJlHH,lIJ)a C BblCOTOft 9 = I - ll Ii OCHoa&- HHeM (12, paBHLIM K8TY s<p<peKTHBHoro ce"leHHR CTOJlKHOBeHHH. BHYTPH 3TOro IJ)IJlHHtij)& HaxOAHTCSI (72911 MOJIeKyn, r)\e n - qHCJlO MoneKYJI B eHHRne 06'beMa. CnOBaTeJILHO, 1{8CTO'Ia CTOJIKHOse- HHA V H cpeee BpeMSI MeJK)Jy CTOnKHOBeHHSlMH (1.1 7) paBHbI 1 T = 2 · ugn ,llnHHa cBo6oHOro np06era;J.JIs acTHJ o6nalOlI(eR CKOpO- CTbIO 1' paBHa _ 2 v - u gn, (1.18) l(1 = 1111" = Id . (1.19) u 9" ,lJ.nSl TOro 'qTo6hl Ol(eHHTb nopOK 1(1 B QeJlOM AJlSI ra3a, npenOJlo- )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. nOcra.sJISI B (1.18) H (1.19) BMecTO 9 H 111 CKOpoCTb 'JBYKa, nOJIyqHM, '-ITO no nopSlKY BeJIH'IHHbl 'I8.CTOTa CTWlKHOBeHHA " -nnHHa cBo6oHOro npo6era onpeAMslOTcSI C nOMOblO BblpaJKe- tlHA 2 I II  q en) I '" a 2 n · (1.20) HaReHHYIO TaKHM 06pa30M Be.1IH'iHHY l ecrecrBeHHo Ha3BaTb c:pe)J.HeR ,lJ.JlHHoA cB06oHoro npo6era, KOTOPYIO B ,.qanLHeAmeM, ec- JrH cneHanbHO He oroeopeHO. 6yeM npocro HMeHosaTh AJIHHol coo- 60P-IIOro npo6era. H3 (1.18) H (1.20) CJIeYeT npoCTaSl CBH3b Me>KJlj' )I)IHHOH CB06oAHoro npo6era l H CpeM 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 Ill o 1.411 npH fll, CTpeMB- e:ACSI K 6ecKOHe'lHOcTH. YHTbIBM, qTO BenH'iHH& U SlBJIBeTCH ocraTOqHO KOHcepBa- THBHOA, TaK KaK onpetl-eJ1S1eTCH cBoA:CTBaMH OTenbHb1X NOJIeKyn, H3 (1.20) CJJeeT, 'lTO ,D.JJHH& cBo6o,1J)ioro npo6era o6paTHo nponopH- 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 3HaHTeJlbHO 60nbwe pa3MepoB MoneKyn. Eonee CJlO>KIIO npoecc 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 )'feAY MOJIeKYJlaMH. )V1J1 onpeAeneHHSI Y RCnOJJb3YeTCJI p83nOeHHe [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 - yenbHblA 00beM. npoecc neaqH 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(ene3HOOpOHOro COCTa-- sa, Ko['a HMnYJ1bC OT nOKOMOTKBa DOCTeneHHO nepeAaeTCJI OTtTOJllUHW OT Hero BarOHaM.  2. YpaBReHHe BOJIhMaBa KaKHM ypaBHeHHeM OIUlCblBaeTCB O)1;HOQacTIrIHaH $yaKQHH pac- npe):\eJJeHH.H f, SlBJI5I101lI.8SlCSI OCHOBOA AJlB HaxO)I(AeHHSI ra30,AHHS- MH'IeCKHX napaMeTpoB? Ha 3TOT Bonpoc 6blJI a.H OTBe1 1 reHH&nb- HblM aBCTpHAcKHM yqeHblM JIIOBHroM BOJ1bMaHOM (L. Boltzmann) B 1874 ropy. IIpa BblBO,l\e 3TOro ypaBHeHHSi 6YAeM cneAosaTb OpH- rHHaJIbHoMY aBTOpCKOMY H3JlO>KeHHIO [10]. 1. BblBOA ypaBHeHHH AJIB 0AHOIf8CTHqHOR cpYHKlJ)fH pacnpe,l{eJIeHHR BH&\fa.ne pacCMOTpHM npoCTeAmHA C1Iya.A, KorAa Bee 'tI8.CTH.. ) COBepWaJOT )J.BH>KeHHe 6e3 CTOJIKHOBeHHA B fiOJle BHeWHHX CH.n G. COrJl&CHO onpeeneHHIO (1.2) cPYHKRH /, B 3JIeMeHTe cPa30BOro otYbeMa dx d B MOMeHT speueHH t B OlCpecTHOCTH TOKH (x,) COAep>KHTCB f(x(t), (t), t) dx MOJIeKyn. K MOMeHTY BpeMeHH tl = t + t npOH30AeT H3MeHeHHe KOOp)].HHaT H CKopocTeA NO- neKyn x(t) -. X(tl). (t)  E(tl)' TaK KaK ,lI.BH)!(eHHe MOJIeKyn 6eCCTOJJKHOBHTenbHoe, TO BCe MO- neKynbJ, HaxoAgHecft B MOMeHT speMeHH t B OKpecTHOCK TO'iKH (x(t), (t)p nepeA)U'T B OKpecTHOCT& roqKH (X(tl), E(tl)). CJleAOBa- TeJlLHO, ,lVIjf <j>YHKIJ.HH pacnpeAeneHHS 6Y,l\eT CnpaBeHBO paseH- CTBO /(x(t),((t),t)dxd = f(X(tl)I((tl),tl)dx. (2.1) IIonaraa f:1t )J.OCTaTO'UlO MaJJblM, nonytUtY Bblpa>KeHHSi, CB$l3h1- B8JOHe (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 BbJ1 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 ai (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 <PYHKKH f npH HaJlHqHH CTOJIKHG- BeHHA C1IOJKHee. CneA)'eT Y'IecTb, qro B J>e3Y}D)TaTe CTOJIKHOBeHHSI {{acTHbl nOKHaIOT rpynny MoneKYJI co CKOpOCTBMH J Tal( KaK npH -)TOM MeHSleTCJI HX HMnYJlbC. OaKO, TaK>Ke 38, C'IeT CTOJIKHOseHHR, llaCTH[(bl C APyrHMH CKOpOCTSMH J40ryr BORTH B paccMaTpHBaeMYIO rpynny. OnpenHM Monb OTHOCHTeJlbHOA CKOpocTH g Me>K,l\y Mane- KYJIaMH, 06naroMH CKOpocnlMH ( H 1: 'gt = I( - lj · (2.4) Tor)J.a tfHClIO '{acTHl.\1 IlOKHHYBWHX B eHHH BpeMeHH rpyony MO- .'leKyn, HMelO11J,HX CKOpOCTb () 3& CtJeT CT01IKHOBeHHA c 'IacTHaMH CKOpocTH 1' 6YAeT paSHO E- (EI) dx d( = dx  J j(x, (, t) f(x, 1. t) Igi bdbd8. (2.5) F3eCb b - npHe.nbHoe pacCTOJlHHe "PH CTOJIKHOaeHHH MoneKynbJ, ItMelOnt-eA CKOpoCTb t C p83nH'IHblMH WWleKynaMH, 06J1aK>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 ODpeeneHHSI BepoWI'HOCTH CTOJIXHOBeHHB t.faCTML{ co CKOpOCTSlMH  H 1 Heo6xOAHMO BBeCTH nOHSlTHe 6HHapHoA <PYHKl\HH pacnpeeJIeHHSI !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- poTHOCTeA Pa3JlHQHhlX MoneKyn 06n&Tb CKOpoCTSIMH  R l' 9To B&}I(HeAwee ,a)ISI KHHeTHqecKOA TOOpHH r830B onyeHHe 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 OnpeeJlHM 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 npHenbHoe pacCToSlHHe 3TOro CTOJTK- HOBeHHH paSHa b. CTOJIKHOBeHHe npHBe)\eT Knape MOJIeKyn C HeKO- 'fOpblMH (Ollpe,D.eneHHbIMH) CKOpocTSlMH ( H  H npeJ]bHbIM pac- cTOHHeM 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]HHe 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 o6eMa 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 OKOHqaTebHMA BH yp&BHeHHa BOnhnMaHa  OHO8CTHqHOR cPYHK1..(HH pacDpeeJIeIfHjf: 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. OH8J(O OH 06JJap.aeT PSI.u.OM HHTepecHbix CBGACTB, K(}- Topble 6YAYT fiOJIe3HbJ HaM B 8JlLHeAmeM. BHaqane YCTd.HOBHM npeenLJ 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 npHeJlbHhlM pacCTO.BHHSM b onpeenSleTCSI B CJIyqae MOeJIH 6HJlbSIpX wapoB H3 ycnOBHSI 0 < b :5 d, rAe d - )J){&MeTp mapa. ,ltnB ):U)yrHx BHOB nOTeHHaJJa B3aHMoeACTBHg BeJlH'IHHa b max MO>Ke'!' 6b.1Th HaaeHa, c nOMOIIJpIO 3HaeHHSI 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 cPYHKJO cp() H npoHHTerpHpyeM no BceM : Irp(t, x) = I () J(x, E, t)  = = ff({)(E)(f'll-lll)lglbdbd8dlclE. 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 ()YCBHHO) 'iTO BbIpa>I<eHHH (2.13) H (2.14) paSHbJ Benny paslIonpa- BHH CKopoCTeA ( 11 (. YqHTblBaJI (1.14) H TOOpefY JIHYBHJlJlSI (2.9), nOJIYIHr nOJ1YCYMMY (2.13) H (2.14) B BHe Icp =  f f ('P - 'P') u'Jr - Ill) 191 bdbd8 1 . (2.15) 3eCb HcnOnb30B&HbJ ecTeCTBeHHble 0603H&QeHHB t.p = cp(), <p' = = rp(/).  no aHaJIOniH C (2.15) nony'ia.eTCH Bblp8JKeHHe AJUI HHTerpana Il = 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 C1IeIOWYIO OCHOBHYIO <f>opMyny ;:vISI MOMeHTa liHTerpana CTOJJKHOBeHHA= lip =  f / (I() + CPl - cP' - CP) (f'/ - J /1) 191 bdb dO dl d. (2.17) 113 (2.17) BHHO, '1ro ecJIH cPYHKSI 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), YOBJIeTBopJlIOIIJ.He ycnOBHIO (2.18). 1ia3h1BaIOTCSI CYMMaTOpHYMH HHBapHaHTaMH. TaKOBblMH .HBJUlIOTCSI neJ1HQHHbI, YKa3&HHbIe B (1.13): m, mE, m2/2, T.e. Macca, HMoynbc H 3HeprHSI MoneKyn. JIJo60a ,lQ>yroA cYMMaTopHhIA HHBapHaHT npeA- C"'!""aBJ1HeTCSI B BH;lte HX nHHeAHoA KOM6HHaH. 06peHHeM B Hynb MOMeHTa Irp )1nH CYMMaTOpHblX HHBapHaH- rOB Bocnonb3yeMcH B CJlelOeM naparpacj>e npH nOJlyqeHHH YPaB- IleHHa ra30BO:A ,l{HHaMHKH H3 ypaBlleHllSI BOJ]bMaHa. 
28 rJl. I. YpasHeHHe B011bQAlaH8 H ypasneHHH 1330BO#f AHH8MHKH 3. H-TeopeM3 BOJIbl{MaHa BBe,lJ.eM H -ct>YHKHIO 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 BOJIbMaHa (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 bdbd8dl. (2.20) I1pasasl qaCTb (2.20) eCTb MOMeHT I., AJIH <l>YHKQHH «p() == 1 + In f . YqHThlBaJI (2.17), nepenHweM 3T0 Bblp8,)KeHHe B Be [(Hln!) == -1 f in ( : ) (f'/f - Ill) Iglbdbd8d1' (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) ai = oi (flnJ)' ()J a (1 + In f). OXi = {i 8 X i (f In f) · (2.22) HTel'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.ijdE.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 pacnpeMeHHB CTpeMHTC5I K HYJIJO npH ti -)0 :f:oo. BTOpoA HHTerpan B neBoA qacTH (2.23) npeCTaBHM B (2.24) BHAe b div  dx = f * dB , re i = f if In J df. , (2.25) a nOaepXJIOCTHblft HHTerpa.n 6epeTCB no rpaHHe 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), npHXOHM K ypa.a- lIeHHIO dH / dt = 1(1+1n f)dx. D 3aMeTHM TenepL, o MOMenT (2.21), CTOJlHa nOA HHTerp8JJOM n np8.BoA \{acTH (2.26)>> Bcera 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. cPYHKHSI H H3 (2.19) - HeB03paCTaJOIII.aJI <PYHKl\HH BpeMeHH. 13 3TOM 3a.KnlOQaerCSl H -reopeM& BonbMaHa. H -cpYHKIJ.HSI TeCHO CB.H3aH8 C KJlaCCwtecKHM nOHBTHeM 3HTpo- IIHH, KOTOp(UI onpeenHeTCSI KaK ),tepa sepoSiTHocTHoro OCYll.\eCTBJle- IIUSI KaKoro-J1H60 MaKpocKOIlHqeCKOrO COCTOSHHH Bew.ecTsa. He Ba- naSiCb 8 nopo6HOCTH, npHBeAeM CBH3b Me>KM TepMOHa.MH'IeCKOA -HITponHeA H H -cPYHKHea S = -kH, (2.28) 
30 TJl. 1. Yp&BHeHHe BOJlbl{Af8H8. H Yp&BHeHJUI ra3OBOR JVfHaMHKIJ re k - nOCTOBHHaH BonhMaHa. ,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. Cneo8aTe.nbHo, / I(1+1nj)dx = -1!! In ( : ) (f'1 - f Id x D X 'glbdbd8dd1 dx = o. (2.29) PaseUCTBO HynlO Bblpa>KeHHSI (2.29) B03MO>KHO, Kora 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) m2/2. (2.31) «I>YHKIJ.HIO pacnpe,1J.eneHHB yo6.lee Bblpa3HTb epe3 DCHOBHbIe r'd30AHHaMHQecKHe nap8.MeTphl BMeCTO ai, a2i, Q3 . SI 3Toro ):\0- CTaTOqHO nOCTaBHTb 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;rylOee 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)YHKH}1 10 nonYQHJI& Ha3BaHHe J&OmAbHOf pacnpeOeAeHue Ma"c- 8eAAQ. B cnyqae nOK8J1bHo-MaxCBeJIJIOBCKOro pacnpeeneHHSI <l>YHKHH 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- CKOlY. ,LLns nOKaJIbHO-MaKCBeJIJIOBCKOA CPYHKHH HHTerpan CTOJIK- H08eHHA: o6paeTC B uynb, qTO CJleAYeT H3 nOCTaHOBKH nepBoro pascHcTBa (2.30) B 4>opMyny (2.12). B TO >Ke BpeMH cl>YHHB (2.32) 8 06eM cnyqae He YOBneTBOpSleT yp&BHeHHIO BOJIbaaa (2.11) 3& CtteT ero neBoR 'faeTH. B npocTpaHCTBeHHo-oHOPOAHOM cnyqae J10KaJlLHOe pacnpeAe- neHHe nepexoHT B a6conlOTHoe pacnpeeneHHe MaKCBeJlJl& poml/2 { rr (i - UiO)2 } lo() = (211" kT o)3/2 exp - 2kTo · (2.33) I3xOgI.l(He clO)J.a napaMeTpbl Po, Uo, To He 3aBHCSlT OT npOCTp&H- c.rBa H speMeHH. 3BOJIIOU.HSI OT pacnpeeneHHSI (2.32) K pacnpeeneHHJO (2.33) IIpOHCXOHT, KorJJ-a CHCTeMa penaKCHpyerr K a6conIOTHOMY paBHO- HecHIO , Y)Ke HaxOAHCb B rH)JJ>oHHaMHqecKoA CTaJ:(HH, T.e. nOK8Jlh- 110M paBHoBecHH. 3m pen8J<HH onHCblsaeTCa CHCTeMOA ypasHe- flHR ra3o- HJlH rHpO,lJ.HH8.MHKH, 0 qeM noAAeT pe'lb B nOfJIeYIOID-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>YHKHH pacnpe)J.eJIeHHSI oro cOpTa \acrHU. Ii. JIeBaJ1 - ,lJ.HcP<I>epeHH8JlbHaB - '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 B3aHMOeACT8HB '-IaCTH JJ-aHHoro copra. c qaCTHaMH Bcex COpTOB. y t.IHThlBasI BblWecKa38J1HOO, COCTaBHM CHCTeMY ypaBHeHHA ROJIbll.MaHa nH CMeCH ra30B B rJle,nylOeA <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 3eCh G ' tni - COOTBeTCTBeHHO BHelDHSljI cHna, eACTayJOaJI Ha qacTHl"-Y i-ro copra, H Macca nOCJleeA:, 19pil - MOA}'JIL OTHOCHTeJlb- Hoil CKOpOCTH CTaJJKHBaIOHXCB '1aCTH COpTOB pHi. 3aL1eTHMt QTO CKOpOCTH nocne CTOJIKHOSeHHH  H  = ( onpeenSiIOTCSI 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 BOJIhMa.Ha pacCMOTpHM pyr)'10 MO)J;enh AIlS( OnaCaHHjf O,D;HOqacTH\lHOA <l>YHH pacnpeeJIeHBSI t npe,lVlO- )KeHtlylO BxaTHa.rapOM, rpOCCOM H KpYKOM (BrK-MOeJIb; CM. [5J): df at 8j Gi at - = - + i - + - - = v (/0 - f) · dt 8t 8Xi m ai (2.35) 3AeCb /0  nOKaJlbHoe MaKCBeJIJlOBCKoe pacnpeeneHHe (2.32), II(E) - Q8CTOTa CTOJIKHOBeHHA ,LVUJ qaCTHJ 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 DpenonO>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 BOJIbMaHa. QHa 06paeTCH B Hynb, Kor,n;a <l>YHKSI pacnpeeneHHSI 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 nOBeeHH.ft <PYHK pacnpeAeJleHHB I, KOTOpoe BO MHorHX cny'IasIX TPY,D.HO npoBecTH H8 OCHOBe HHTerpo- Hct>cPepeHr(Ha..rlbHoro ypaaHeHHSI BOJILM&Ha. CJleeT 3aMeTH r rb, ttTO ypaBHeHHe BrK, Tal< >Ke K&K H ypaBHe- HHe BonbMaHaJ 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 <PYHKHH f. HeJlHHeAHOCTh BrK-MoAenH, YQHT&JB8JI :iKCnOHeHHaJlhHYIO 3aBHCHMOCTh /0 OT U H T, B KaKOM-TO CMhlC1Ie HBnHeTCB 60nee CHJlbHOR, qeM KBapJ>&THtUlaSi HeJlHHeAHOCTb ypas- HeHIDI BOJIbM8Jfa. 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 pacnpeneHHH J B 3TOA CHTY8lUtH 9HCKorOM npe,D.JIO>KeH& cnWIOll.\aJI MOH<!>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 ) gbdbd8l , 2 {Jx, 8xJ lh: j J" = 2 / lei k i a::X j (f'/ - Ifl)gbdbdDI' 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 MOeJlH 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. OTCIOa B (2.37) TOJILKO nep- 8blA "IJIeH J 1 . HNelOlI{HA HyeBOR nOPJlAOK no a) COBDa.z:t8eT no BII)Ijf C HHTerpanOW CTWlKHOBeHd BOJIbaHa. CJle.ztY1OIIJ.lle 'lJIeHhI p83JlO)l(eBBJI 
no CT HMeJOT 6o.rIee rpoM03KHA 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) cpebI H <PYHAaltJfCHT3J1bHhIX 3aKOHOB COXpaHeHHH MacCbl HM- nynbca H 3HeprHH Bew-eCTBa. 9'1'0 OTHOCHTCSI K ypaBHeHHSlM 9Anepa, onHCbIBalOHM re'leHHSI HeBSl3Koro raaa, H K ypanHeHH.S1M HaBhe- CToKca ,lJ.JISI BSl3KOro TCrInonpoBoHOro ra3a. HanpHMep, ypaSHeHHe Hepa3pbIB- HOCTH  + div(pu) = 0 (3.1) BbIBOHT(JI H3 YCJIOBHSI COXpa.HeHHH MacCbl)VIR KOHe'iHOrO o6'beMa V o , rpa- PHC. 3. K BblBOAY ypaBHenHH HHa 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-OcTporpcKoro, BblpaJKeHHe (3.2) 3aMe- HHM Ha f (+ diV(PU)) dV = o. Va YqHTbIBaSI,  06'heM Vo MO)KeT 61:dTb npOH3BOJIbHblM, nepeM 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, o6eM 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 ee OHO OrpaHH1.feHHe - 6o.nee >KOCTKoe: )J.onYCTHTb K paccMoTpeHHIO Tonb- J«() 06'beMbl Yo, JlHHeAHble pa3Mepbl KOTOpbIX no nopHKY BeJ1HtIHHbl He MeHblllC }J.JIHHbI C8060,L\Horo npo6era I. aHHoe Tpe600aHHe no- ('IY)K"T CPH3HtfOCKOR OCHOBOR .lVfSl IIOCTpoeHHSI HOBOA MoenH) onH- cblBalOw.eA Te'ieJ-IHSI BSl3Koro raaa, no,zq>o6Hoe 06CY>KeHHe 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 CJleCTBHH ypaBHeHHg BOJIbMaHa [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 noo K OnHCaHHfO BH3KHX 'I'eqeHI.r If C03):\a.HHIO KOHKpeTHLIX BbI'iHCJJHTeJlbHbIX 8JlrOpHTMoB. 110 06 3TOM peQb nolteT Y>Ke B CJle.nylOID-HX rJlaBRX. 1. MOMeHTHbIe ypaBHeHHH IIonb3YSlcb onpeAeneHHBMH ra30HaMHqeCKHX napaMeTpOB (1.3)-(1.8) Qepe3 MOMeHThl <}JYHKIJ.HH pacnpeAeneHJIjf) nOnbJTaeMCJI IIUJlyqnTb If a OCHODe YPaBHeHHH BonbMaHa CB$I3b Me)K,lI.Y 3THMH :\tOl\'ICHTaMH. W'IaK. YMHO>KHM ypaBHeHHe BOJIhIJ.MaHR (2.11) Ha CYM- ta'fOpllble MHOapHaHTbI V'(): ffl, m, 2/2, H npoHHTerpHpyeM no nCCM . B pe3ynbTa:re flMeeM / ( 8J 8! Gi Of ) cp() - + i - + - - cl( = 0 · at 8Xi m ai (3.4) IIanoMHHM) t.JTO npaBaJI qaCTb (3.4) pasHa HYJlIO BBH;Jy TOro, '1TO liP" HTerpHpOBaHHH C CYMMaTopHblMH HHBapHaHTaMH HHTerpan ( TOJ1KHOBeHHA 06paaeTCH B HYJlh (2, n.2). PaCCMOTpHM OTe.nbHO HHTerpanbl OT CJlaraeMhlX, BXO,ll.SIHX B ",'uuylO ttacTb ypaBHeHIUI (3.4). nOCKOJIhKY cp() He 3aBHCgT OT npo- 
CTp8.HCTBeHHbIX nepeMeHHhlX H BpeMeHH, cnpaBeAnHBbl BhIpa1KeHHSI f cp()=  ! rp()/, I af a f cp() {i   = a;: rp() {d. Bonee no,a.po6HO OCTaHOBHMCSI Ha HHTerpaJIe OT TpeTberO qneHa H3 nesoA tJacTH (3.4). Bocnonh3yeMcSI <I>oPMynoA HHTerpapOSaHHSI no l.JacTSlM ,lVUI Hanpa.BJIeHHB i CKOpOCTH NOJIeKyn ({J' {Ie - CKOpOCTH MoneKYJI B BYX OCT8JILHhlX HanpasneHHlIX): 00 00 00 J G. of I I / G. 8/ rp«(} rr:  d = k d{; cp() :;;  d{i = -00 -00 -00 00 00 = I d{k / d{j (cp(t)  I) ;: -00 - (3.5) -00 -00 / 00 1 00 / 00 G. 8rp - d{tc d(.j I ;; 8[; d{i · -00 -00 -00 BOCnOJlb3yeMCH ecTeCTBeHHhlM <l>H3H"IecKHM npeonO>KeHHeM 06 y6h1BaHHH <l>YHKHH pacneneHH$I npH f(1  00 6h1CTpee JlI060A creneHHoA <l>YHKH «(). TIpHMepoM TaKO:A CPYHKHH f CJIY>KHT nOKaJIbHOe MaxcBennoBCKoe pacnpeeneHHe (2.32). TaKHM 06P830M, B (3.5) HCQe3aeT qneH, npoHHTerpHpoBaHHblR 110 {i, H OCTaeTCH / rp() Gi 8f  = _0. I f 8cp . (3.6) m 8i m 8{i YqHTblBasi (3.5) H (3.6), nepenHweM (3.4) B Be CHCTeMbl YpaB- HeHHA M.R Ka}f()J.oro CYMMaTOpHoro HHBapH8.HTa:  / mj+ -k I mij = 0, :t I m{j/ + -k f mi{jjdt - Gj f jd( = 0, a f m2 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 onpeeneHH- 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 mijfd( = j + ptJiUj. (3.9) 3To paseHcTBo MO>Ke1' 6blTb nOJJyeHo (JIeA)'IOlD)iM D)"reM. IIpe- CTaBHM CKOpoCTb MOJIeKY nbl B BH)J.e  = c + U H 3anmneM f ijfd( = 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), npeM 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 - TPHHQaTH, c yqeTOM CHM- MeTpHttHOCTH TeH30pa Pij. IIpa 3TOM HMeJIOCb B B, 'ITO fiOJIH8JI 1HeprHH E H3 (3.8) BbIpCSI C nOMoIIJ.hJO npocTblx COOTHOllleHHA ([epe3 JJ.pyrHe MOMeHTbJ (1.4), (1.7)) (1.8): I m2 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 Heo6xoHMO HCnOJJb30BaTb Te HJ1H HHble <l>H3H'ieCKHe rrpnOCbIJl- KH 0 BHe <l>YHKI..\HH pacnpe.n:eneHHHJ n03BOJISlIOHe onpeenHTb o- DOJIHHTeJlbHble CBSI3H Me>KPj' HCKOMhlMH BeJJH'IUHaMH 3TOH CHCTCMbl. BH)J. TaKoA <PYHKQI1H MO)f(eT 6b1Tb HaAeH 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)iHH) Ha3BaTb 3TOT npeeJI 2a300uHaMUeCKUM). ITpH aTOM CaMa cPYHHH pacnpe,lJ.eJIeHHj( 6yeT 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 ra30HaMHqeCKKX BMHHH: Kn = l / L . (3.14) 9TOT napaMeTp Ha3b1BaeTCSI UCAOM KHyoceHa. Kora xapaK'rep- Hbllt p33Mep 3t\ltaqH MHorOKpaTHO npeBblWaeT A1JHHY cB06oHoro npo6era, qHCJIO KHyceHa MaJIO. B 60nbmHHCTse 3aq 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 c806olloro npo6era J TO MO)l(HO rOBopHTb 0 COCTOSlHHH nOKaJ1b- Horo paBHOBeCHH cpebl, B KOTOpOM <l>YHKHSI pacnpeeneHHH SlBn.H- eTCB nOKaJIhHo-MaKCBeJ1J10BCKOR (2.32). BOCn0J1b3yeMc SlBHbIM Bblpax<eHHeM ,LVIB <l>YHKl\HH pacnpee- 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 3aeTC5I, COrJI&CHO (1.5») <I>oPMynoA I{jlij =m f focj=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 HaxoeHHH MOMeHTOB ldaKCBennoBcKoft <PYHKQHH 6yeM 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, . . . OrcfOa 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}KOM H3 COCIaBJIjllOHX 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) nOCTaBnHH 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) re [ - yenbHaJI 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 KnaccHecKHMH ypa6HeHU- JlMU 91l4epa ,QHHaMHKH HeBB3Koro HeTeDnonpoBooro raJa. TeM CaMblM ypaBHeHWI 3Anepa COOTBeTCTBYIOT KOHKpe1'HOMY BH o- HOqacTHQHO:R CPYHKH I, aBae1(oA JlOK8JThHLlM M8.KCBennOBCKHM pacnpeeJIeHHeM (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 npelIonO)KHM, liTO CpeHSI.R qacTOTa CTOJIKHOBeHHR II He 3aBHCHT OT CKOpOCTH MO- neKynbl , a Bblpa>KaeTCa Ka.K II = liT, I'Ae T = llc. BBeeM 6e3pa3- MepHoe BpeMB t ' = t/to, re to - xapaKrepHOe ra30AHHaMH1ieCKOe 
npeMH r H 06e3pa3MepHM ypaBHeHHe (2.35): df 1 dt' = "i (fo - f), re E '" T/to. (3.25 ) B THDUQHbIX ra30BLlX Te'tleHHSlX 6ff3p83MepHhlA napaMeTp E Man H HMeeT nOpBOK 'IHeJIa KHyceHa. 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 BHe OHoro nHmb nepBOro qneHa pa3nO)l(eHHSI (3.26). ,lIpy- I'He 60JIee llOJlHhle ypaBHeHHSI ra3OBOA RHaMHKH MorYT 6bITb no- J1yqeHbJ c YeTOM nOCJIeAYI01QHX 'ineHOB pa3nO>KeHHSI (3.26). TaKHM UYTeM MO>KHO npHRTH K ypaBHeHHSlM BSI3KOro TeWIonpoBOoro ra,- la. 3. BR3KOCTb If TeHJIOnpOBOAHOCTb Cne;o'lOee nOClle nOKaJILHo-MaKCBe1IJlOBCKOro npeCTaBneHHe <1>YHKHH pacnpeeneHH.H, BblTeK8JOee 113 pa..'3JIO)KeHHjI (3.26), 3&- JIHIlJeTCSI B BHe 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 BHe 
HeM H3 (2.32) BblpameflHiI,lVIH 'IacTHblX npoH3BOblX <I>YHK- IJ.HH /0, CTOS1HX 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. OnpeeJIHM HeHarOHaJlbHbIA 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 CYIMe (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 - + - . ] ij V m aXj 8Xi (3.34) LVIHO)KHTeJllO v-I (k / m) T p ecTecTBeHHO npH):t3Tb CMblCJI KJIaC- c'ieCKoro Ko3<l>c.PHHeHTa 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- 'HCK"ft TenJJOnpOBOAHbIA BH BeKTOpa q aT qj = -X-, OXi I',l(e x - K03c1><1>HHeHT TelUIonpoBo)J.HoCTH: (3.36 ) X=  (  )2  . (3.37) 3 aM e q a H He 2. BrK-MOeJIh B eJIOM npasHJIbHO OTp8JKB.eT 3Kcne- pHleHT8J]bHO lIa6nlOaeMb(e <PYHKHoHanbHhle 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, npHcye UH3KOMY renJlOnpOBO,lI)lOMY ra3Y, H3 BrK-uo,D.enH nonY'IeHbI 6bITb He MC>- rYT. HanpHMepJ He cOOTBeTCTDyeT HCTHHe UCAO IIpaHdmAJc, xapaKTepH- 'iYloee OTHoweHHe ,[\Byx K034><1>HHeHTOB: Pr = JlCp/" , (3.38 ) l'iU: c p -- Y,Il.JlbHa.R rennoeMKoCTb npH nOCTORHHOM aBJleHHH. B paMK8.X I)IK-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- MeTpHHYJO 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 npebWw.eM naparpa<j>e npoe)Q'pa nOKa3b1- saeT) TO nOSiBneHHe Bjf3KHX H TennOnpOBO,lJ.HblX qneHOB Henocpe.lJ.- CTBeHHO CBR3aHO C HaJJHQHeU ManoR, npoopqHOH8JlhHOA E t 06a.B- KH K nOK8JIbHOA Ma.KCBeJlJIOBCKOa <PYHKQHH pacnpeeJJeHHSI /0. Ha pa3J10>KelIHH B pft no MaJIOMY napaMeTpY E pemeHHB ypaBHeHHB BonbMaHa OCHOBaHa npoe,zurpa qeoMeHa-9HcKora [2, 3). C n MOhlO 3TOro pa3JIOJKeHHH DWI}"IHM YpaBHeHHjI Hasbe-CTOKca H KOHKpeTHbJR BH K03<tx1>HeHTOB BSI3KOCTH 11 TennonpoBOHOCTH. 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 3ecb J(J, fl)t 38,- SHCHT J1HHeRHO OT K8JKoro B3 060HX aprYMeHTOB. BBeM 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 MeTOa tIeIIMeHa-9HCKOra.. 
1) Ha paccMat'pHBaeMoD rHpOHaMHqeCKOR (ra30HHaA(H'IeCKoA) HH <PYHK paceneHHB 3aBHCHT OT speMeHH TOJ1bKO nOCpe){CTBOM DSiTH napaMeTpOB p, u, T. 2) Bee ra30)J:HHaMH'feCKHe napaMeTpbl MorYT 6b1Tb onpeneHbl c nOMOblO OHOro nepBOro 'L1IeHa p83nO>KeHHSI (4.2). IIpe,L\cTaBHM JleBYIO qacTb ypasHeHHJI (4.1) B Be :;, = ( , + v ) I , of Gi 81 re 1) f = i - + - - · OXi m ai (4.3) 'roral HCIIOJ1b3YSl pa3nO>KeHHe (4.2), ROJIyqHM VI = 'DI(D) + eVJCl) + £2VI(2) + · '. . (4.4) B CBOIO O'tfepebJ HHTerp8JI CTOJIKHOseHH:A MO>KHO ncraBHTb II BH,lJ.e .J(f,/I) = J( E ekl(k), f ePjP») = E f ek+PJ(j(k),lr}) . k=O p=o t=o p=o (4.5) I)BeH 0603HaQeHHe J(n) (JCO),j(l),...,/(n») = E J(!(k),/P)), (4.6) k+p=n nepenHweM BLlpeHHe (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 onpeenglOTCSl qepm npOH3BOHble no Bpe).{e- HH ra30HHaMHqecKHX napaMeTpoB p, u, T, KaK B (4.8). QTo6b1 HaRTH ypaBHelIRS, CBH3bJBaJOHe p83JIHQHhJe cl>YHKU.HH j(p), nocraBHM 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). 11leeM:  [ ( 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><PHHe.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 -... ......... ....... ... .... .................... ........ ... reJ HCXOjI 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 BHe 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 IloCTaBnHH (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-') + qO» ) -  IJUi = 0, m i J  1 m apE +  ( 41 . p _) + q » ) = 0 P - 1 2 at 8Xi LA) l' I , -, ,... "lITo6bl HaATH Bblpa>KeHHSI ra30HH&MHqeCKHX napaMeTpOB, <l>H- I'YpJ.IPYIOIQHX B CHCTeMe (4.15)-(4.17), H8.AO nOCJIeOBaTeJlbHO pe- UlaTL ypaBHeHHH (4.11) OTHOCHTeJIbHO <PYHKHA f(P), P = 0,1, 2,. . . . nepBoe ypaBHeHHe CHCTeMbI (4.11) onpeeJJSleT <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 rlpeblyeM naparpa4>e 6bInO nOK83aHO, \frO ec-JIH B Ka'lecTBe npH6JIH>KeHHSI <PYHKHH pacnpeeJleHHg B3RTb nOKaJIhHhlA MaKCBen- JlRaN, TO H3MeHeHHe ra30HHaMHqeCKHX napaMeTpOB 6YAeT ODHCbI- BaTbCS ypaoHeHH5IMH 3AJIepa. To >Ke caMoe Ha6nlOaeM 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 IIPHCYHM HeBSl3KoMY HeTennOnpoBo,nHOMY ra3Y. TaKHM 06pa3oM, B 3TOM CJlyqae CHCTeMa ypaBHeHmt (4.15)-(4.17) COBnaeT C YpaBHe- HHRMH 9ffJIepa. 2. ROK8JlbHO HaBbe-CTOKCOBCKaH cPYHKHH paclIpe)J.eJleHHH HaA,n:eM Tenepb CJleJU'IOA qneH pa3JJO>KeHHSI Q>YHKHH pac- rrpeeJIeHHR (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>YHKHH «p(1): ( I ) Ci aT B ( s) ( 1 2 ) Ou, in =-A ( s ) --- Cic.-c6'.- T T 8x. 2RT '-> 'J ax.' . , , (4.21) re s = lell V 2RT. CKaJlSlpHhle <I>YHKH A H B onpeAeJlBIOTCSI c nOMO1O peweHHSI HHTerp8JIbHLIX ypaBHeHHA. HanpHMep, ,ZV1H HaxO)KeHHa 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, CJleYIOw.ee 3a nOKaJIbHo-MaxcBeJIJlOBCKHM npH6JIIDKeHHe t lenMeHa-9HCKOra eCTeCTBeHHO, KaK MbI YBH,ltHM HH>Ke, HB3B&Tb AO- aAbHO Ha8be-cmcncco6c"OO CPYHKHeA pacnpe)J.eJIeHHSI. ORa HMeeT nHA INS = 10 [1 - ; ACi ::: - 2T B (CiCj -  c26ij) :; ]. (4.23) 3. ra30AHHaMHQeCKHe ypaBHeHHH HaBbe-CTOKca BbIpaeHHe AJUI CPym<QHH pacnpeeJIeHHH (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: CPYHKH pacnpee- neHHH, BhJpa:>KeHH.H (4.26)-(4.27) nepenmneM B Be <,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>HHeHTaM nponOpD.HOHaJlbHOCTH JJ H X ecTeCTBeBHO npHaTb CMblCJI Koo4><PHQHeHTOB BSI3KOCTH H TeIIJIOnpOBOHOCTH. ORB 3a- 
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 nOMOhlO Bblp8JKeHHA (4.24)-(4.30), nOCT8.BHM COOT- BeTCTBYIOHe HX 3HaqeHHH B MOMeHTHble yp8.BHeHHSI (3.10)-(3.12). TeM caMbiM npHeM 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}) + 8i [P'Ui (f+£) +Uii] - G i 8 8T - ;;; fYlJi = 8Xi x 8 X i ' (4.36) r,lJ.e KOMnOHeHTbl TeH30pa ; onpeenHlOTCSI 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.eI CJlyqae Heo6xoHMO 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 CKJIBaTbCJI H3 ABYX tlacwre:A: KHHeTH(.leCKoA, aHaJlOrH'lHoA p83pe>KeHHhIM ra3&M, B nOTeHanb- HoR, CBjl3aHHOA c nepeHOCOM CTOJIKHOBeHHA, 06ycnoBJIeHHhIM KOHeHhlM pa3MepOM MoneKyn. npH 3TOM C YBenH'feHHeM IIJIOTHOCTH BKn fiOTeH- IlitaJIbHOA COCTaBnHIOeA 6YeT B03pacraTb. B npeenbHOM CJJyqae, KorAa C>KHM8eMOCTLIO cpebl MO)l(HO npeHe6peQL H nnOTHOCTb p DOCTOHHHa, YpaBHeHHB (4.34)-(4.36) ne- peR,nyT B ypaBIeHHSI 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 npoH3BOH&H DO 13peMeHH, F - BHeWHSlSI CHJI&. ,n:eICTBYIOasI Ha eHH 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 onpeeJIe- IIHH K03<p<pHu.HeHToB BSl3KOCTH H TennOnpOBo,nHOCTH. nepBOHaqaJIb- HO OHH HaxO,ll.HJ1HCb 3KcnepHMeHTaJIbHO, KaK Koa<P<PHeHTbI npo- nopHOHanbHOCTH, CBH3bIBaJOHe KOMnOHeHTbI TeH30pa HanpH>Ke- HHA H BeKTopa TenJlOBOrO DOTOKa C COOTBeTCTBYIOID.HMH npOH3BO- (4.41) 
HblMH CKopocTe:A H TeMnepaTyphl. 3ecb >Ke H napaMeTpbl, Ha3bl- BaeMble K03ct><PHQHeHTaMH MOJIeKynSlpHoro nepeHOCa, onpeenHJOTCSI C nOMOblO BblpaJKeHHA (4.31 )-( 4.32), onHpaJOcSl Ha <l>YHAaMeH- TaJIbllhle MoneKYJIHpHble cBo:ACTBa aHHOro ra3a. PacCMOTpHM, IIanpHMep, rrpoCTeAwylO Moenb 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 eIIe OHH 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 BblBOOM C nOMO1O npoe,ll;Y- phI QenMel-la-9HCKora. S1 3TOro pacCMOTpHM npOCTYIO KHHeTH'Ie- CKYIO MOeJlL. ITYCTb I - CPCNfHjI )VIHHa cBo6ooro np06era, a c - cpeHSlSI CKOpOCTb XaoTHeCKoro ,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, HaxoSlDUiecSi B TOqJ(e 1, (Yr- CTOSle OT TOQKH 0 Ha pacCTO- BHHel.CneOBaTenbHO,H3MeHe- HHe HMnYJIbCa PZll 3a Ct.feT Mone- X KYJISpHOrO nepeHoca 6YAeT npo- IIOpOHaJIbHO p33HOCTH CKOpo- PHC. 4. K onpeeJleuHIO Ko3CjJcI>eB- CTeA Uo - Ul B TOqJ(ax" 0 H 1. Ta BB3KOCTH Yq}fTbIB8JI) 'iTO qHcno MOJIeKYJI, nepeceKaJOw.Hx JlHHHIO Y = 0 B eHHUY BpeMeHH, paBHO nc, H Pa3- y 1 1 00 
JlaraH pa3HOCTb UO - Ul B p TeAJlopa, nOJIYHM 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 6naroapjl 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 pacrrpeene- HHH npHBOH'r K ypaBHeHHSlM HaBbe-CTOKca. 9TY npoeAYPY MO)f(- 110 npO,1J.o.n>KaTb aJIbwe. OnHpasIcb Ha TpeTbe ypaBHeHHe CHCTeMbI (4.11), MO)KHO HaRTH CJIeIOHH 'iJIeH p33JIO>KeHHH 1(2) H C era 00- A10blO nOCTpOHTb COOTBeTCTBYI01I\He ypaBHeHHSI )VJg orrpeeneHIDI rd30HaMHqeCKHX napaMeTpOB - ypa8HeHUJI BapHemma [2]. O,.ll,HaKO ypaBlleHHB BapHeTTa He HamJlH CKOJIb-HH6Yb 3aMeT- floro npHMeHeHHH B pemeHHH 38.,L\a'l ra30BoA HHaMHKH. Bo-nepBblX, OHH ropa.3O 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 cnpaBeJlHBbI ypaBHeHHH Hasbe-CTOKca, pe- weHHH 06eHX CHCTeM OqeHb CJla60 OTJlH'laJOTCH APyr OT ,zq>yra. TaM }t(e, re npHMeHeHHe Mo,n:enH BH3Koro TennonpoBoHoro r83a Heo- CTaTOqHO 060CHOBaJIO, HanpHMep) np" YBeJlHQeHHH QHcn8 K.lyceHaJ ypaBHeHHH BapHeTTa AaJOT pe3KOe yxyeHHe peweHHH no cpaBHe- HHIO c ypaBH€HHHMH HaBbe-CToKca. TIO,A06HOe nOBeAeHHe pemeHH.H ypaaHeHHR BapHeTTa He CJIY'IaR- HO.. rrpoQeypa t.{enMeHa-9HCKora KOHCTPYHpOBaJIaCb KaK Pa3JIO)l(e- HHe B pHP- 110 MaJIOMY napaMeTpy e 1 BenHqHHa KOTOpOro TecHO CBH- 181la C 'iHCJIOM KHyceHa. B enOM MeTO,l\HKa nonyqemrSl YP8BHeHHA 3Anepa H HaBbe-CTOKca H3 perneHHH ypaBHeHHSI BOnbM8Jfa no- TBep>K,l\aeT xopOWO H3BecTHblft H3 r830BOH AHHaMHKH cPaKT 06 HX crrpaBenHBOCTH ,lVIH MaJlhlX qHCen KUYAceHa.. 
4. Yp8BHeHHH ){HH8MHKH CMecH r830B no aHa.TlOrKH C 'reM, KaK 83 yp8.BHeHHB BOJlbMaHa c nOMowplO npoQe- AYPbI QenMeHa-9HCKora BblBO)J)ITCjI yp88HeHIUI HaBbe-CTOKCa l onHCbIBa- IOHe O,lJ.HOKOMDOHeHTHblA BJJ3KHA ra3, H3 CHCTeMbI YPaBHeHHH BOnLMatIa AJI)) raaoBOA CMeCH nOJlY'IaIOTCSI YpaBHeHHB HUbe-CTOKCa MHOI'OKOMn(}- HeHTHoA CMeCH (1 J 3J. llepenHweM ee pa3 yp&BHeRHe BOJJhaJt& 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 dfJ ' x, 7ni J p=1 i = 1, . . . ) N i , ( 4.47) rAe i OOH&'IaeT CKOpocTb MoneKynLl copm i. 3aTeM) K8K H B OHOKOM- nOHeHTHOM cnyqae, AJlS <l>YHKR h CTpoHTCR pa3noeHHe 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 msi 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'JOea 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 pOM ,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 - KOHeHT 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) re h, = hi(T, p) - napan&Hble 3HT&J1bnHH. TeH30p HanpJl.>KeHUA Pile, 4>HrYPHPYIO1 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. 
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91. BblBO,l{ KHHeTHlfeCKH COrJl8COBaHHI>lX cxeM 61 MaKCBenJJH3al\HH peweHHB. )J.n.a HaxoeHHH raJOrofHaMH'IecKHX napaMeTpOB -- MOMeHTOB YHKqHH pacnpeAeneHHB - onpeAenB- JIHCb HHTerpanbl B npOCTpaHCTBe CKOpocre:a MOJleKYJJ  OT nOl<aJlbHO- MaKCBeJlJIOBCKOA <PYHKI.J.HH pacnpeeJIeHHSI. ITpH 3TOM C8.Ma OHa onpe,D.eJIH.,1acL C nOMOLIO TOJIbKO Tpex CBOHX 3HaqeHHH no CKOpO- CT5IM MOJTeKyn. I1MeHHo 9TOT <paKT H nocnY)KHJl npHttHJI0A OTHOCH- TeJ1bHO HeBblCOKOA TOtIHOCTH npOBeeHHLIX Ha OCHOBe 3TO cxeMbI pacqeTOB. B pa60Te [16) (Pullin D.L.) nOKaJIbHO-MaKCBeJIJI08CKaH CPYHKH.sI pacnpeeJleHHH npHMeHHJIacb ,lVIH OnpeJIeHHSI nOTOKOB MaCChI, HM- nYJIbCa H 3HeprHH Ha .rl8H>KyeHCg rpaHHl.{e SqeAXH. BnocnecTBHH 3TH nOTOKH HCnOJIb30BaJlHCb ,lVI.H HaxoeHHH 3HaeHHR ra30p-HHa- MHeCKHX napaM€TpOB Ha HOBOM CJloe no BpeMeHH. TaK )Ke KaK H B [15J. B ,QaHHOa pa60Te anropHTM ITpHMeHS1nCJI K MOeJlHpOBaHHlO TetJellH:A ea.TIbHOrO raaa. B pa60Tax [17, 18) (Reitz R.D.; ApHCTOB B.B., LJepeMHcHH cI>.r.) ra30)J.HHaMHQecKHC napaMeTpbl HenocpeCTBeHHO onpeeJIgJlHCb no- c.ne HaxO>Ktl-eHHH <PYHKD-HH pacnpeeJleHHSI H3 perneHH.SI ypaBHeHHR BOJIbMaHa c nOMOwPlO KOHel{HO-pa3HOCTHo:A CXeM:bI. ABTOpbJ pa60Tld (13), Hcnonb3YSl CTyneHQaTO-nOCTOHHHble no- KaJIbliO-MaKCnennOBCKHe <l>YHKIUIH, paccMaTpHBaJIH 6aJIaHC <l>YHK- QHH pacI1peeJleHHa B <l>HKCHpOBaHHOR g'leAKe. KOHeQHO-pa3HOCTHhle ypaBHeHHg CTpOHflHCb nYTeM KHTerpUpOBaHHH 3TOro 6anaacHoro ypaBHeHHSI C CYMMaTopHblMH HHBapHaHTaMH. npH 3TOM TaJ(3<e HC- nOJ1b30BaJIOCb npenOJIO)J(eHHe 0 MrHOBeHllOH MaKCBeJlnH3aIJ)1H raJa tla 110BOJ CJIOO no BpeMeHH. (8TOT noxo)I. nopo6HO pacCMOTpHM B CJleHX nYHKTaX HacTOSIero naparpa<j>a). j!.aHHble cxeMbI no- nY'IHJlH Ha3BaHHe KHHeTH'IecKH cornaCOBaHHblX pa3HOCTHbIX cxeM (KCPC) . CJleeT OTMeTHTL, "liTO B03MO>KHOCTb HaxOJKAeHHSI faaO)J.HHa- MHQecKHX napaMeTpOB HenocpeCTBeHHO H3 KHHeTHQecKHX rrpe- CTaBJIeHHR He HBJISleTCSI peBMIOIJ.HOHHOA. Hao6opoT, OHa BnOJIHe eCTeCTHeHHa, Y\lHTbIBaJI CBIDL Me>K,r:ty KHHeTH'IeCKHM H ra30HHa- MHqeCKHM onHcaHHeM cnnolliHo1t cpeJl:bI. nOOMY 3HaqeHHe nO.HBHB- llIHXC.SI K Ha'lany 80-x roAOH 60JIbMaHOBCKHX (KHIIeTH'fecKHX) cxeu CJJeYeT paccMaTpHBaTb C TOqKH 3peHHSI HX B03MO>KHOCTH COCTaBHTb 
62 TJI. If. K.,HeTH'IeCKH cor.naCOBaHHlJe pa3HOCrHble exeMbl KOHKypeHHIO cYlLI-eCTBOBaBwHM B TO BpeMH TpHOHRbJM a..nro- pHTlaM 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 nOXO,.QaMH H HeMeNIeHHO BbJ3BaJ] cepHlO ,LJ.aJ1bHeAIWiX ny6nHKR asTOpa MOHorpaQ>HH Hera Konner [19)-(26). COBn8AalOIlI.He c KCPC [13] cxeMhl 6b1nH nOJIyIeHbI 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 Moe- JIHpOB8HHSI HeBSI3Koro r83&t IIO;q>o6HO OfiHcaHHbIX B o630pe (29]. 3TOT 0630p TaK}Ke MO)KeT 6bITb peKOMeHAOBRH qwmTeJUlM, 60Jree nop06Ho HHTepecyJOMCS1 p83JIH'IHblMH HanpaBneHHBMH 6anbQ.- MaHOBCKHX cxeM. HaHHa.H c 1985 r. OCHOBHble YCHJ1HB aBTOpa H ero COTpy)].HH- KOB 6b1nH CKOHl.\eHTpHpOSaHbJ }la MoenHpoBaRHH 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 B3rJ1j1bI Ha KCPC C o,LJ.HoA CTOpoHbJ KaK Ha aJIropHTM pacqe-ra, a. c ;q>yroA CTO- poHbl KaK Ha. B03MO>KHyK> Moenb NISI ORHcaJtHB TeqeHHR BSI3KOro ra3a. B 3TOM KnlO'Ie 6YeT paccMOTpeHa CBSI3b KCPC C ypaBHeHH- HMH HaBbe-CTOKca H C'IaBWHMH nonynspHblMH B nOCJIeHee BpeMH Lattice Boltzmann c.xeMaMH (30, 31) (rnaB& III). IIpHCTynHM K BbIBO,llj' KHHeTH'IeCKH cornacoB8HblX cxe11 cne)1.Ya pa60Te (13). 2. COOTHoweHHe 6aJlaHCa ,l{JIJI CPYHKHH 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 <PYHKH pacnpe)\e..neHHB. 1) Ha MOMeHT BpefeHH 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 CPYIIKHjf pacnpeeneHHg, cOBnMa'OIaR 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) r830HHaMHqeCKHe 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 npoHCXOHT MrHOBeHHaJI MaKCBeJt- nH3anHjI cl>YHKHH pacnpE!AeneHHH. IIpH nepexO,lJ.e K MOMeHTY BpeMeHH t j + 2 BCSI npoepa 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 MaxCBeJIJlH3HH, . 1 fl - 3HaqeHHe cl>YHKlt HH pacnpe,AeJJeHHH, onaChlBaeMoR Bbl- pa>KeHHet (1.1), 110CJIe MrHOBeHHoA MaKBeJIJlH3aqua. 06CYHM 060CHOBaHHOCTb 3THX npe)\DonO>KeHHA. OrMeTHM, '1'00 H ra3OAHH8.MHtfeCKHX npoeccoB B AOCTaTOIJHO IIJIOTHblX raaax, HC- cne.n-OB8.HHe KOTOpblX H HBJlHeTC.H OCHOBIIOA l\eJIblO aHHoA MOHorpa- cl>BH, MaKCBeJLnOBCKOe pacnpeeneHHe xopowo onHCblBaeT cl>YHKl\H1O pacnpeAe.neHHB. HMeHHo Ha 6nH30CTH cl>YHKUHH pacnpeAeneHHH K MaKCBennOBCKoA nocTpoeHa npoepa nonyqeHH ypasHeHHA ra- 30BOA p)tHaMHKH H3 ypaBHeHH1I BOJIbQM&Ha (I'Jlasa I). CTOJIKHOBeHHH MoneKYJI 8 r83e npHBOAHT K YCTaHOBneHHIO MaKC- BeJInOBcKoro BH,lJ.& CPYHKHH 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 KHHeTHecKo:A TeOpHH (4.12) (rn. I), cornacHo KOTOpblM no-. 
64 rJl. II. KHHeTHqecKH corJlacOBaHHlJe pa3HOCTHbie cxeMW KaJlbHo-MaKCBennOBCK8.SI cl>YHKQHB onpeenBeTCSI 'fepe3 ra30,ll.kHa- MHqeCKHe napaMeTpbl, KOTOpbJe B C8010 Oqepe,u.b HBIDIIOTCH MOt.!ell- Tal\fH <PYHKHH pacnpeeneHHSI: 1M = /M[J). MrHOBeHH8.H M8.KCBeJInH3au.H HMHTHPYeT npoecc 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 nOMOblO XOpOlllO H3BeCTHoro B BbltUlC1IHTeJIbHOA MaTeMaTHKe npHHna CYM- MapHoA annpOKCHMH J KOra nepeXOA OT OHOrO 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 npeonO>KeHHe BBJlSlercg KnlOtteBhJM ):VIa nOllHMaHWI KCPC K3.K MoenH ,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 pacnpee.neHHH B cTyneH'I&TO-nOCTOSlHIIOM 110 npOCTpaHCTBY BHe 6hlno 06paIl\eHO BHHMaHHe ee B MOHorpa- <l>HH [33) A.A. BnacoB& - 8.BTOpa H3BeCTHoro ypaBHeHHJI, OIIHCblBa- IOrnero nOBeeHHe <l>YHKQHH pacnpe.n-eneHIDI B BhlCOKOTeMnepaTYP- HOlt nn83Me. Ha pHcYHKe 5 CXeMaTHqeCKH H306paJl<eH npoecc 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 qaCTHbl C nOnO>KHTeJlbHblM HanpaeneHHeM MOJleKYJlSlp- HoR CKOpOCTH  > 0 H HMelOHe 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), nponOpHOHaJIbHa MOJIIO 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  pacneneHWI H& OTpe3Ke (i, Zi+l] ,D,JUI ID- reOMeTpHH 3aJ.IHoe t npeAcTaBHM COOTHoweHHe 68JIaHca, onHCbIsalOIII-ero H3Me- HeHHS1 <l>YHKHH pacnpe,l].eneHHSI 8 npoecce 6ecCTOJJKHOBHTeJlbHOrO pa3neTa tJ-nB u.eHTpanbHoA' Hqe:AKH, B CJIetIYlOeM Be: //+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 npenOJIO>KHJIH, '1TO see OTpe3KH HMelOT OAHHaKOBblR p83Mep flx. · 1 3aMeTHM, 1.fTO no OKOHQaHHH pa3ncm <l>YHKHg /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 ra30HHaMHqeCKHX napaMeTpoB, npoHHTerpHPYM 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 onpeeneHHIO c:l>YHKqHH pacnpeeneHHSI (rna- . 1 · 1 . 1 sa I)t COOTBeTCTBeHHO pas}lhI 11+ , (pu)f+ , Ef+ · 3aMeTHM, tfTO "PH 3TOM HeSlBRO HCIIOJIb3YeTCJI CBoAcTBO c:l>YH pacnpeeneHHH, nonO>KeHHoe B OCHOBY npoe)!ypLI QeuMeHa-9HcKora (4.12) (rn. I): I fj+lV'()d = 0, (1.4 ) re () - CYMM&TOPHblR HHBapHaHT, a 6.fj+l OnpeeJUleTCH H3 paseHcTBa I = fM[!J + f. (1.5) HHTerpanbl B npasoA 'facTH (1.3) nerKO 6epyTCSI BB MaKC- sennoBcKoro xap8JCTepa <l>YHKU)fH pacnpeeneHHB npe t = t j . EHH- CTBeHHYIO ,cnO}l(HOCTb npe,nCTaBnHeT onpe)J.eJleHHe HHTerpanOB THna J fI<p()/cl{, KOTOphle, BnpO'leM, TaK)I(e 6e3 oco6oro TpY,Aa BbltlHC- JJS!lOTaI 6naroAapSl Hcnonb30BaHHIO c:l>YHKH 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- qeHHg1H ra30tl-HHa.MHQeCKHX napaMeTpOB pi+l, u j + 1 , T;+1 CTpOHM HOSYIO KYCO'lHo-nOCTOBHHYIO MaKCBMnOBCKYIO <!>YHKII.HJO pacnpe- JleHH.H, H BCSI npoepa nOBTOpBeTCH npH nepexOAe K MOMeHTY Bpe- MeHH t i + 2 . 4. CorJlaCOBaHHOCTb p83HOCTRbIX annpoKcHM1Jl{HR CHCTeMY YP8BHeHHA (1.6)-(1.8) MO)KHO nOJlYHTL, Henocpe,ll.- CTBeHHO OnHpaJICb Ha p83HOCTHYlO cxeMY nepBOro nopKa TOt{HOCTH .AnSI ypaaHeHHSI BonhMaHa 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) re J(li, Ii} - 3HaqeHHe HHTerpan& CTOJIKHOSeBHR, onpeJIHeMoro 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 OTpHaTenhHbIX J cKopocreA MOJIeKYJI He f l _ 1 CJIYQaAHa. OHa onpe- PRC. 6. K BbI(Sopy p&3HOCTHorO wat)noHa .D,1U1l JIHe'l'CB H3 YCJJOBHSI ycrroA- peweHKSI YpaBHelDlJl BOJ1hM&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 BOJIbMaHa. 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. .  11lcp() 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 npeCTaBJIgeTca cosepweHHo eCTeCTseHHblM. CxeMa (1.9)-(1.10) nepooro nop TO'lHOCTH COOTBeTCTBYeT K(- COQHo-nOCTOSlHHoMY IIpeC'IaBneHHIO <PYHKH pacnpeAeneHHSI. Kpo- Me TOro 1 HCnOJIb30BaHHe DpenOJlO}KeHHg 0 MaKCBeJIJIOBCKOM xapak- Tepe <l>YHKH 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 nOJlyeHo 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. OHaJ(O B OT1H'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. 3eCb 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). nO06HbJe npHeMbJ nocTpoeHHSI KCPC MO>KHO npHMeHHTb t HCnOJlL- 3Y1l B KaqeCTBe OCHOBbl H APyrae. OTnHqHble OT (1.11) p83HOCTHbJe CXeMbJ, annpoKcHMHpYIOllJ)fe KHReTHeCKOe ypaBHeHHe. Y1:fHTblBaB CBB3b YPaBBeHHR ra3OBOA AHHaMHKH H ypaBHeHHSI EOnbQMaHa (rJI. I), TPaAHHOHHble 'IHCJleHHble MeTObJ MO>KHO npe- CTaBHTb B BHe cne;l1ylOm.eA enoqKH: KHHeTHQecKoe ypasHeHHe BonbII-MaHa - OCpe,lI.HeHHe ypaBHeHHSI EOJlLMaHa C CYMMaTOpHbJ- t.(H HHBapHaHTaMH H fiOJlyqeHHe ypaBHeHHA ra30BoA )J.HHaMHKH - pa.JHOC'l'tIaJI ,lI.HCKpeTH3HSI ypasHeHHA ra30BoR HHaMHKH. B CBOIO OqepeAb, KCPC MO)KeT 6blTb npeCTaBJleHa B 8He APyroA Docne- OBaTenbJIOCTH eAcTBHA: KHHeTH1.fecKoe ypasHeHHe BOJ1bQMaHa - HcKpeTHaH fO)J.eJIb )).JIB OHO'IacTH'IHOR <PYHKHH pacnpeeneHHSI - OCpe;J.HeHHe HCKperHOA MO,neJIH C CYMMaTOpHblMH HHBapHawra- MH H nonyqeHHe KCPC. TaxHM 06pa30M t no CpaBlieHHIO c Tp- OHHblMH anrOpHTMIH B KHHeTHqeCKH COrJlacoBaHHblx pa3HOCTHbJX cxelt,lax ltJleHSIeTCa nopSlOK npOl.(e)JJ'pbl ocpeAHeHWI H ,lJ.HCKpeTH3a- IUfH B enOqKe, CT8.pTYIOm.eA c ypaBHelIHB BOJIhMaHa. CenaeM ,noa Ba>KH.bIX 3a.Me'laHHB t KOTOpble B anbHeAweM (rJJ. III) 6YT HCnOJIb30BaHbI npH o6cY>KeHHH poJIH KCPC KaK CPH3HttecKOA MOeJIH 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 >KKOCTeA H ra30B. O,IJ.JlaKO npHMeM 80 BHHM8HHe TOT cl>aKT, qro r830AHHaMH'IecKHe na- paMeTpbl KCPC onpeenllJOTCjl 'Iepe3 nOTOKH Ha rpaHHl.\&x KOHe'lHblX lIqe- eK. CaMH ra30,AHHaMH'fecKHe BenH'IHHbI Ha 3THX rp&HHax TepnHT p83pbIB 
70 rJl. II. KHHeTH'leCKH corJl&COBaHHlJe Pa3HOCTHWe CJreMbl H B RBHOM BHe 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 BBeeM nOHftTHe cPYHKIJ.HH pacnpeneHHH li-l/2 H8 rpaHHu.e seaKJ{ (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& rpaHHe 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 1ll 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) II 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 nOqepKHBalOT 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 cBo6oHoro 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 HHTepnpeHH KHHeTHtJecKHX cxeM H KB83Hr830aMH'Ie- CKO.A CHCTeMbl ypasHeHHA KaK MOeJlH NIB OIIHCaHHSI ra30t\HH8MH- I.leCKHX npOL\eCCOB (rn. III). 5. IIpHMepbl pacqeTOB IIpHBeet.-t aHHbIe pac'leTOB BYX npoCTp8JtcrBeHHO-O,D.HOMep- HbiX 3a.u-aQ - 3a)].a'lH 0 pacne ra3OAHHaMH\feCKoro p83pblBa H 3a- P 8 6 4 2 o 4(J so 160 'C PRC. 7. PeweHHe 38J:t&'IH 0 pacDe 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 HCCJIeOBaHHH 8.BTOpa Hero KOJIJJer. Ha pRC. 7 npHBeeHbI 3HaqeHHH fiJlOTHOCTH, B3BThIe H3 pac'leT8, 38.AaQH 0 pacna,ne ra30HHaMHqeCKoro pa.3pbJBa Ha MOMeHT 6e3p83- MepHoro BpeMeHH t = 4. TIpH t = 0 B l\eHTpe paccMaT])Hs&eMoA o6nacTH (x = 80) 3aH ra30HaMHIecKHA pa3pLlB. Cnesa OT pa3- pblBa p = 8, p = pR:r = 480) cnpaoa (J = 1, P = 1. HCnOJlb30oa.rracb pa8HOfepH no npoCTpaHCToy ceTKa (x = 1). 06ee '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 npHBeeHbJ ,AaHHble 'lHcneI-IHOrO pewe- HHSI 3aqH 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 rrOMO1O 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 CJIeeT OTMeTH'rb, LITO npaKTHecKH COBnaJOHe c KCPC pe3YJlbTaTbi npH perneHHH nOCJIeAHe 3aqH paHee 6LlnH nonytteHbI B pa60Te [18). )J.8JlHbIA Q>aKT HeY,l\HBHTeJleH, TaX KaK B 3TOa pa60Te HcnOJIb30B8JlCB a.nropHTM Herrocpe.n;CTBeHHoro onpee- JleHRS <t>YHKHH pacnpeAeJleHHH H3 peUleHHS KHHeTHtleCKOrO ypaB- o o 20 
92. KCPG na npOH3BaJlbHbJX cerK8X 73 HeIlHSI. nOCJJe HaxO}t(,lJ.eHHH cPYHKL\HH pacnpeeneHHH ra30HH8.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- npeeJleHHSI) 'iepe3 onpeeJleHHe ra30HHaMHqecKHX 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 nOMOblO 60J1bMa- HOOCKHX cxeM OTMe'iaJIOCb H B pa60TaX 127, 29, 35]. 38.KJlIO'IaSl )(aHHbIA naparpa<P, CJIe,nyeT OTMeTHTb, 'ITO o6cya- eMbie B HeM, KaK H B OCTaJIbHhlX naparpaQ:>ax AaHHoA rnasbJ, oapH- aHThI KHHeTHqeCKHX, a Ta.K>Ke 6onbM8.}IOBCKHX, cxeN cne;lU'eT pac- cMaTpHBaTb KaK B03MOJKHbre a.nropHTMhl paclleTa 3all ,lI)fHaMH" KH HeaJlbliol'O ra.3a. 9TH anropHTMbl MorYT HMeTb npeHMym;ecTBa (HJI:H HeOCTaTKH) 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 3aq )J.HllaMHKH H)J.ea.nbHoro ra3&. Q)J)iaKO npHHnHnHaJlbHO HOBbIe B03MO>KHOCTH KH- HeTH'IeCKHX cxeM OTKpblBa.K>TC5I npH MOenHpOBaHHH BB3KHX Teqe- "HH.  2. KUHeTHtJecKH COrJl8COBaHHhle cxeMLI U8 npoll3BOJJhHbiX CeTKaX B npeAblAYeM naparpa4>e OCHOBHa..sI HeSl nWlY'IeHHB KCPC 6b1na paccMOTpeHa Ha npHMepe npocreliwero CJJy'tlasr ID reoMeTpHH. PacnpocTpaHHM aHHbJA nOAXO,A Ha UOJIyqeHHe KHHeTHqecKOA Pa3- HOCTHOA annpoKCHMaQHH B 60JJee o6eM C1I}"Iae. BKJlIOQaH HecTPYK- TypHpoBaHHbJe ceTKH. 1. CooTElomeHHe 6aJlaHca AJIB cjJYHKHH pacnpe,l{eJIeHHB na npOH3BOJlbHOlf CeTKe PaccMaTpHBaeMoe B  1 Bblpa>KeHHe (1.3) NtH 6aJJaHca CPYHK- HH pacnpeeneHHg nerKO MO}f(eT 6blTb o606w.eHo Ha npOH3BOJIbHYto ,lI.BytepHYIO HnH TpeXMepHYIO DpocTpaHCTBeHHYIO CeTKY (pHC. 9). 
74 r.n. II. KHHeTH'IecKH corJl8COIJ8.HllWe pa3HOCTHbie CXeMlJ )J.;lS 3TOro aHaJJorHQHO ooMepHoMY CJ1Y'ialO He06xotJ)tMO pacCMOT- peTb npoHHKHoseHHe B D-eHTpanhHYIO H'ieAxy C HOMepoM I M3KCBeJI- JlOBCKHX cl>YHKH:A pacnpeeneHHSI 83 coceHHX SNeeK C HOMepaMH i 8 HanpasneHHH CKopocre:A 1 HMeIOX OTpHaTeJlbHOe CKaJIHpHOe npoH3BeeHHe 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 BHe (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 pacnpeeneHHSI B 1.(eHTpaJJbHoA HqelKe, JiM - 3HaqeHHH WaKC- BeJ1JIOBCKOA <!>YHKIJ.HH pacnpeeJleHHB B coceHHX 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 npHBeeM Bblpa>KeHHSI ,lV]H KHHeTHecKHX 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 o606eHHeM 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. KHHeTHecKH 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 BOJlb 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.3pSly CTPYKTypHpoBaH- HblX CeTOK. O)J.HaKO BO MHorHX 3 a,£taq ax , HMelOIU.HX B nepBYIO oqe- peb 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" annpOKCHMalHH 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 Oqepeb TpeyronbHbte ,lVIH 2D-rOOMeTpHH H TeTpa3- panbHble )J.JlB 3D-reoMeTpHH. IIpH TaKoA annpoKCHMaHH MaTpHl.\bI, BbJpa>KaJOw,He CBB3b ue)f(AY 3HaqeHHflMH ra30,nHHaMHqeCKHX napa- MeTpOB B p83JlHt4HbIX TOtKaX, He HMelOT tJeTKO BLIpa>KeHHYIO JleH- TOt{HYJO CTPYKTYPY. O,ll.H&KO Hap C HX I'JIaBHblM OCTOHHCTBOM - y,n;aqHblM reo- MeTpHQeCKHM npeCTaBneHHeM 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- CHltrlaHIO ypasHeHHA ra30BO.A ,lJ){HaMHKH, OnOJlHHTenbHO o6n8JI-a- 
92. [(CPC H& npOl13BOJll>HblX ceTxax 77 10HX CBOItCTBOM MOHOTOHHOCTH H KOHcepsaTHBHOCTH. BO-BTOpblX, npH HCnOJlb30BatlHH HeSIBHblX cxeM B03HHKaIOT OnOnHHTenbHbJe CJlO>K1IOC1'H, CB3aHHbJe c pemeHHeM CHCTeM CeTO'tJHbIX YPaBHeHHA AJI$I MaTpH, He HMeJOll\HX neHTO'lHYIO crpYKTypy. llpo6neMa He,n;OCTaTOqHOA TOQHOCTH a.Ill1pOKCHM8,QHH )VIR He- CTPYKTypHpOBaHHblX ceTOK B 3Hat.JHTenbHo:A Mepe CHHMaeTCSI CTaB- weA penbHoA B03MO)f(HOCTblO HCnOJIL30BaHHSI MHoronpoecCOpHb1X BbicoKonpoH3BOAHTeJlbHblX BblttHCJIHTeJJbHhlX CHCTeM, KOTOpbJe n03- BOJIHIOT npHMeHSlTb B pactfeTa.X npOCTpaHCTBeHHhte CeTKR, OCTara- IOll(He 10 8 - 10 9 Y3noB. OAHHM H3 B03MO>KHbIX n)'TeA nOBblWeHHft TO'IHOCTH annpOKCHMal\HH J1RJlSleTC$I KOM6HHJI CTPYXTypHpoBaHHbIX H HecTpYKTypUPOB8JlHbIX ce- TOK. npH 3TOM CTPYKTYPHpoBaHHble CeTKH, KaK npaaHno. HCDOJIb3yJOTCH npH annpoKcHMaIUIH nORjI TeeHHjJ 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 BBHY 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 BH- MOCTH, CJIY>KHT TalOKe H OCHOBOA HX XOPOWHX annpOKCHMaHOHHblX CBoRCTB. no CBoeMY nOCTpoeHHIO KCPC HBJISI10TCSI SlBHblMH cxeMaMH. )Lnsr BblnOJlHCHHSI YCJ10BHS YCToAqHBOCTH npH MoenHpoBaHHH 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.HHaMHecKoA CHCTeMbI yaeTCB 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 MoemtpOBalDlJl Te'leHHB OKano WIHH)U>a B nnOCKOY KaHane. 6) pa.rMeHT conpS)J(eHHH TJ>eYI'OJ1LHoA H qeTLlpexyrMHoii CeTOK 
2. [(CPC H& npoH380J1bHhlX CeI'KaX 79 CeTQqHblX ypaBHeHHA. CneYeT 3aMeTHTb, q'fO np06neMa reHepHH ceTOK KNeer CaMo- CTOBTeJ1bHoe cepbe3Hoe HaY'IHOe H npHKnaAHoe 3HaqeHHe H npHBJIe- KaeT K CBoeMY peweHHJO, OCo6eHHO B nOCJ1eee 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 MeTOa 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 onpeeJIHM KOHTponbHblA otYbeM, npe.D-cTa.Bn.SlIOHA 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 oHoA 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.HrYHH ,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 cepeAHHtH rpaHeA TpeyrOJlbHHKoB) HMe- IOHX 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 yoBneTBOp.R- eT npHHHny TpH8.HrYJlHHH 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.HH 06JIacTH TPeyrOJlbHHKH He HMeJOT TY- noro yr J1a, TO B KaqeCTBe cpeAHea TOKH Bbl6HpaeTCH D-eHTp onH- CaJ-IHOH OKPY>KHOCTH. EcJIH :>Ke B TpeyrOJIbHHKe npHCYTCTSYeT TynoA yrOJI, TO I10 cpeHe:R TOKOD nOHHMaeTCH eHTp T.H)((ecTH. BblnHweM 6aJ1aHCHOe COOTHoweHHe (2.1) ,lJ.na KOHTponbHoro o6eMa (pac. 11): k /1+1 = JIM - t L [ShWli(nJi) + SWli(n)] . (2.6) I i=l 3p-eCb CYMMHpOBaHHe npoHcxOHT no sceM k rpaHaM KOHTpORbHO- ro otYbeMa, n, n - BeKTOpbI HopManH K COOTBeTCTBYlOl1I,HM qa- CTHM rpaHHbI, sh, S - ,lVIHHLI COOTBeTCTBYIOw.eA 'lacTH rpa.HBbl KOHTpoJIbHOrO otYbeMa (pRC. 11). B COCTaBnBIOUI,HX (2.6) nOTOKOB HCnO.flb3yeM cneAYIOee 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 il + 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) annpOKCHMHPYIOHX KOHseKTHBHyJO qacrb nOTOKOB, HCnOJlb3YIOTC.H CJIelOru,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 COOTBeTCTBYIOeA 'IacTH rpa.HHbJ KOHTJ)OnLHoro otYbeMa, Un, UT' - KOMJIOHeHTbJ CKOpOCTH B JIOKaJlLHOA eKapTOBOM CHCTeMe KoopHflaT (n, T) : Un = UC08Q + vsina, U r = -usina + 1JCOSO. tLnsr cI>cI>Y3HOHHbJX COCTa.BmDO DOTOKOB Di' 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 paceTaJ Hcnonh3YJOero annpoKcHMaHK) KCPC H& TpeyronbRblx ceTK.8.X, pacCMOTpKM 3aqy 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 rpaHHe 3JlnHnCa njf CKOpOCTH CTaBHJlHCb ycnOBHB HenpOTeK8HHSI. PacqeT npOBo,nHJIca Ha TpeyronbHOA ceTKe, cOCTOSlll(eA H3 5941 Y3JIOB. CeTKa cryaJlacb K nosepXHOCTH 3JInHDca. Ha pUC. 12 nOKa3aHO pacCQ;BTaHHOe pa.cnpeeJIeHHe 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 npeCTaBngeT co- 60a 3a.z:taqy 3HaQHTeJIhHO 60nee C1JO)KHYJO, qeM c03AaHHe IDe BYMep- HblX aHaJloroB. OaKO, ecJIH 3m 3aqa pemeHa, TO IIOCTpoeHHe KCPC annpoKCHM8IJ.HH Ha TaKHX CeTKax, KaK, HanpHMep, TeTpa- 3Apa..nbHbIX, ocyeCTBJlHeTCSI C nOMOU1O npOCTOro o606eHH.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 aHaJlOrHHbI (2.9)-(2.10), XOTS! rop83O 60nee rpoM03H. npHMep paC\ieTa nOTOKa BSl3KOro ra3a Ha TeTpaa,lij)aJIbHblX CeTKaX 6YAeT pac- C10'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, OnHCbJBalOHX Te"leHHe HeaJIbHoro ra3&. IIpH 3TOM KOHBeKTHB- Hide qJ1eHbI J CTOSlIlI.He 8 neBoR qacTH CHCTeMbI (1.6)-(1.8), onHCblBa- IOTCSI eHrrpaJlbHO-pa3HOCTHOA annpOKCHMaHeH, HMelOIII-eA BTOpoA nopH)J.oK TOtJHOCTH, a CTOSlW){e B npaBOa 1IacTH HCCHnaTHBHble qJIe- UbI, aHaJIOrllqHbJe JIeHaM HCKYCCToeHHo:R Bjf3KOCTH, HMelOT nepBbJA nopSlOK 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>KeHHasI elO cxewa (1.6)-(1.8) ):VIR ra30,!tHHaMHecKHX napaMeTpoS. AHaJlOrH'IHLIM nyTeM MO>KHO 110J1YQHTb KHHeTHtJecKHe CXeMhl 60J1ee BblCOKOro nOp,lJ)<a TOqHOCTH, eCJIH nopoaroaSl HX CXeMa ;JISI ypaoHeHHS EOnbMaHa HMeeT 60nee BblCOKHA nOpjfAOK annpoK- CHMaQHH. Cne}1 pa60rre [42], CKOHCTpYHpyeM HeKoTopble B03MO)K- HbIe KHHeTH'IecKHe cxeMhI 60nee BLICOKoro nOpHK8. TOtJHOCTH. 1. CxeMbl I10BblWeHHoro nopRAK3 TOqHOCTB )1.JIH pemeHHH KHHeTH1J€CKOro ypaBHeHHH BbIDHllleM pa3HOCTHY!O cxewY,lJ)IjI ypaBHeHHjf BonblJ).(aHa S cne- )YIOeM 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, nOJ1YHM  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 + II t::.x 8 8x4 i + O(t::.x ). '1'e)1 CaMbJM .lI-H<P<PepeHHaJIbHbJM npU6nH>KeHHeM no rrpOCTpaHCTBY J1.JI pa3HOCTHOA cxeMbI (3.1) HBJISleTC.H CJleroee ypaBHeHHe: Ij+l - Ii +  ( af ) j = -£.t::.x 2 ( ! _ 1 - t/J ) ( lJ3 f ) j _ (3.4) t ax 6 4 ox 3 4 - - 1It::.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'lOeY mlTerpepCJBaJlHH C CYMMaTOpHbJMH HHOapHaHTaMH 06paeTCH B HYJlh. TaKHM 06Pa30M, p83HOCTHaJI cxeM8 (3.4) annpoKcHMHpyer AHcI>- 4>epe}lHanbHYIO 'I&cTb ypaBHeHHH BOJIhMaHa c TOqJfOCThlO O(x3) eCJIH ! _ 1 - 4> = 0 ( 3.5 ) 6 4 ' T.e. "PH tP = 1/3, H C O(x2) B HHbIX C1Iyqx. 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 1'830RRaMHqecKHX nepeMeHHLIX. B CBOJO O'lepe, BeKTOp nOTOKa G onpe,qenHM C nOMOIUp1O CJleylOIIJ.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( + II) li, Gi = f !li( - II) 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 6yeM 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- BeeT K KHHeTHttecKH COrJl8COB8BHOA p83HOCTHOA cxeMe nepBoro no- pSI,D.Ka TOqHOCTH. B C8010 Oqepeb) QT06b1 nO,l\a.BHTh OCnJIBU)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 ODpeenHM KaK min mod[a, b] =  ( sign(a) + sign(b)) min [Ial, Ibl] . BBeeM B npasylO 'IaCTb (3.8) MO,lJ.H4>HpOBa.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 KHHeTHeCKHX cxeM 60nee BblCOKOro nopSl)J.K& n03BOn.HeT IIOBblCHTb TOQHOCTb paceTa 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 3a'IH c DONOJO 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 0606aIOTCB ):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 nOKa 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) Dopa TO'lHOCTH )J.JIJI MenHpOBaJlHa 3cv:ta 0 pacne pa3phIBa. CnnOWUaH KPHBaui - TOHoe pemeHHe CJIeAYlow.eM nYHKTe A8.HHOrO naparpa<pa, CJIeySl 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 allDpOKCHMH 3aKnlO- IJa€TCH 8 TOM, qTO, COrJlacHo [45), ra30AHHaMH'IecKHe napaMeTpbl Ha pacqeTHoR SlQeRKe 3aJOTCH He KaK KycOtJHO-IIOCTOSlHHble, a KaK KYCOlIHO-JlHlleaHble. OH"M H3 B8.)f(HhlX CBoAcTB pa3HOCTHbIX cxeM nH ypaBHeHH nepeHoca HBReTCA 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 rropSlK& an- npoKCHMaJ..{HH 6blJla MOHOTOHHOA, OJJ>KHO BhlDOJIHSlTbCH YCJlOBHe MHHHIYMa 3HaqeHHB rpaAHeHTa npn 3aHHH nHHeAHoro pacnpe- eJIeHHSI cl>Y}JKl\HH Ha pact.teTHoR SlQeAKe (47). B CBOJO Oqepeb, 3TO JlKHeRHoe pacnpe.n.eneHHe M01KHO onpeAeJIHTb nH60 3KCTpaDOJISlIJ.H- eft BBepx no nOToKY, JlH60 HHTepnOJISW)feA. ,.[(sa cnoco6a BbJ60pa pa3HOCTflOro rpHeHTa B ID-CJIyae HJIJlIOCTpHpyer pRC. 14. Ec- JIlt 3KCTpan0J1HHOHHbla rpHelI'.r VQ1 H HHT€pIlOJlSlHOHHbla 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 npoHcxoHT nOK8JlbHoe nepeKnlOtteHHe Ha cxeMY nepBoro nOpHKa. QH-I t.1 i+lf2 i+1 PIlC. 14. HwrepnOJlHHOHH&lAJ 9KCTaDO- JUl1J)IOHIIblA H cpeeB3BemeHIIYA 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 ra30HHaMHqecKHX nepeueHHblX Ha rpaHHe Me>KAY g'tJeRKaMH I H J onpeAeJIHM CJIe,ayIOM 06pa- 30M: Q lJ = QI +  (VQP) llJ , QtJ = QJ - (VQtJ) IlJ · (3.12) 3eCb lr J - BeKTOp, H,lJj'W. HA H3 TOKH I B TOtJKY J. 
90 Tn. II. KHHeTH'IecKIf COrJl8COBaHIlWe p&3HOCTH61e CXeMbl rpHeHT V QP Ollpe,neJISieTCH KaK CpeeB3BemeHHoe MeJKAY HHTepnOJIHOHHblM rp&,l\HeHTOM VQ}O H 9KcrpanOJlS1HOHHblN rpa- HeHTOM VQ'1 ocpeHeHHbIM 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, npHH8JVIeero TaK>Ke TpeyrOJlb- HHKY C BepmHHaMH I, J, K (poc. 15); CYMMHpOBaHHe BeeTCH no BceM TaKHM TpeYroJILHHKaM. 38MeTHM, 'ITO B cnY'Iae Me,n:HaHHbJX - pac'leTHI>IX aqeeK nnow. VI J K pasHa ooA TpeTH WIO8J\H Tpe- yronbHHKa ViJK. Qepe3 (VQ)IJ 0600Ha'ieH rpeHT Q, onpee.nH- eMbJA Ha 3TOM TpeyrOJ1bHHKe: (VQ)/J = [(  ) IJ' () rJ · (3.16) B CBOlO oqepe. COCTa8J1jflOe rpHeHTa aJOTCB BblpaJKeHHSlMH ( fJ Q ) 1 I I ( 8 Q ) 1 f ' ox /J = 2VlJ K S QndS, Oy r J = 2V/J K Qn'JdS, IJK SJJJ( (3.17) I'Ae n = (n, n) - BHeWHSlJI HOpM8JJL K rp8HHe TpeyronbHHKa SIlK. ,nnfl nocTpoeHHB MOHOTOHHOA CXeMbJ H& HeperYJlSlpHOA ceTKe no 8HaJIOrHH c (48) onpeAenHM 3HaqeHHe rpReHTa B nonYeJJoA roQKe j*) JleeA Ha upHMOD, npoxoNleA epe3 TOttKH I H J. 
93. KHHeTU'lecKHe cxeMbl UOBblUleHHoro nofJJJ,lVC8 TOIfHDC1.H 91 OnpeeJIHM 3Haq-eHHSI rp8.,lI.HeHTa B TOqKaX It H J- ) KOTOpbJe BBeeM 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 onpeenBIOTCH CJIe,DJ'IOJ,( 06pa30M: Q IJ = Qr +  L(VQr., QJ - QI), QjJ = QJ - L(VQJ.' QJ - QI). (3.19) 3eCb L(a, b) ecTb Q>YHKSI-OrpaHHqHTenL1 onpeAe.nHIOaJt nODe- J(etlHe nOKaJl1JHblX rpHeHTOB. HCDOJIb30BMOCh Ba Ba 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.MecKHX napaMeTpOB c nOBhlweHHhlM nopHKoM TO'IHOCTH 3&- nHweTCSI B CJJeAYIOeM 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 nOMO1O BLlpa>KeHKA (2.13)- (2.20) . 3aMeTHM, liTO paCCMOTpeHHaJI annpoKCHMHH 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, paccHTaHHOrO 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 yapHoA BMHbl no CpaBHeHHIO C KCPC rrepBoro nopSl,I\Ka. IIpaaa, CJIeeT 38.MeTHTb) 'Iro p'OCTUt{b xopomero OnHC8.HHH I10J1.H 1'e'leHHSI MO>KHO H C nOMOwplO cxeM nepBoro nOpB.l(Ka TOHOCTHJ ec- .nH HCnOJlb30S&Tb AJlSI AHCKpeTH3aU;HB 6onbllloe 'IHCJlO }'3JIOB. Ta.KHe B03MO>KHOCTH npet(OCTaBJI.HeT npHMeHeHHe BbICOKOnpOH3BOHTeJlb- 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 H3BecTHM B KHHeTHt-ieCKOA TeopHH ra30B pa3JIO)KeHHeM qeuMeHa-9HcKora (r.l. I) )I.JlSl npe,ACTaBJIeHHH OHOqacTHHOR Q:>YHKU.HH pacnpeeneHH. ;(nH 3Toro pacC10TpHM BblBO)I. KCPC c nOMOW,b1O npoeAYpbI qeIIMeHa- 9HcKora. oIIHpalOeaCR B HaweM CJIyqae Ha KycOqHo-nocroSlHHoe BHYTPH npocTpauCTBeHHo HeRKH npCTaaneHHe Q:>YHKQHH pac- 
4. I('fHC'I'II'fecKJle CXeMbl H pa3JlO)l(eHHe tfenMeHa-9HclCora 93 npeeneHHjI [37]. BbIIIHweM ypasHeHHe BOnhMaHa 8 CJIe)lJ'lOeM BRAe:  + div(f) = ;JU, J), ( 4.1) r.ll.e £ - M8.1IblA napaMeTp. PemeHHe ypaaHeHHSI (4.1) MO)KeT 6b1Tb npeACTaBJIeHO B CJIeYIOeM HHTerpanbHoM Be: 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 rpaHea 06JIacTH. By.u-CM C"IHTaTb, qTO 8 HCClleAYeMoA: o6nacTH R BBeeHa CeTKa, COCTOSIaR H3 BblnYKnblX MHororpaHHHKOB Vk, k > It C eHTpaMH B TOqK8X Xk H rpa.HHaMH Sk. npHqeM R C U Vk. k 0603HaQHM CHtB0J10M Xk = X(V,,) xapaKTepHCTH'IecKYJO cl>YHK1O MHO)KeCTBa Vie. MMeeM: Vk n V m = </;, k:F m, Xt(X)Xm(X) = 6km, EXt(x) = 1, x E R, Ie re c5km - CHMBon KpoHeKepa. BBet(eM 0600HaQeHHe I(x., t) = Ik«(, t), x E Vk , (4.3) H 6yeM 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) 
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100 TJI. II. KHHeTHilecKH COrJlaCOBaHHble p831focTHhle CXeMbJ 3,n;ecb HHTerpan no HCXOHOA 06JJacTH fipe,nCTaBneH CHaqa.n:& no Ha- npaBJleHMIO x (Bonb 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 TOIKe b rpaHHbl 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 Bxoa 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.IOaSl 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, 3aBaeMbIH 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) npeM K CJlem:eMY 
5. KB83HI'a3Q4HHaMJPleclQ1R CHCreM& YpaBHeHHR 101 )l.KcpcpepeH1J)IaJlbHO-pa3HOCTHOMY ypaBHeHHlO, OnHCbIBaJOeMY 3BO- J1lOI.(HIO <l>YHKHH 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 nOJlyeHHSI KB83Hra- 30,L\HHaMH"IeCKOA CHCTeMbi H no CBoeMY nOCTpoeHBIO 5IBJlHeTCB CJIe,a.- CTBHeM nopo)KalOeA 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, npHBelUHM 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{acTHbl nepeceKaIOT s)'ielKy. BnpOqeM, TaKoe npe)I.IIo- JIO>KeHHe SlBJ1HeTCSI BDonHe eCTeCTSeHHblM NISI ra30B) re HapSl)Jy C ,L1.nHI-toA cBo6oHoro 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 cyecTBe"HbiX OTJlH'IH'I OT ypasHeHH1I BOJ1bMaHa. Bo-nepBwx. OTCYTCTBYeT HHTerp8JI CTWlKHOBeHWI. Bo-BTOPblX t B np8.8oA 'IacTH :noro ypaBHeHHH npHcYTcToyerr ,lJ)I«P4>epeHaJILHbIR onepaTop BTOporo nop- 1<3. 9TOT oneparop  LT LfJ., NO;l\eJIHPYeT npou.ecc KTHBHOro CrJJa.>KH- B8HHJI MaxCBemIOBCKOA <l>YHKlJ.HH pacnpeJ]eHIUI Ha HqeHKe c xapaKTep- HblM pa3MepoM h, C KOTOpblM BpeMjI T CBSl3aHO npOCTbIM COOTHOWeHHel4 (5.3). B CBOIO oepe,.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 BbIBOOM ypaBHeHH5I 6anaHca (5.11) pacCMOTpHM ee 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). rrpenono)J(HMp liTO npH t = t J cl>YHKU)[SI p8CIIpeeneHH.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 npoHCXOT MrHOBeHHaJI MaKCBenJIH- 3H.H f'83a. t;+l f (x, , t j + 1 ) t j IJT JM(X-T,, t j ) PHC. 18. K BWBOAY YpaBHeHHJI 6anaaca <l»yHKIUIH pacnpe.o.eneHH2 3HaqeHHe cl>YHKH pacnpeAeneHHft Ha HOBOM cnoo no BpeMeHH onpeeJIHM c nOMohlO COOTHOmeHH}1 -+1 · f(x, ,tJ ) = f M(X - Tt t tJ),. (5.12) KOTOpoe HBJIHeTC1I cneAcTBHeM CBo6oHOro 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 nopSlKa MaJlOCTH O(U'T]3): '+1 · af I(x,, tJ ) = fM(X, (, t3) - 'T{j 8x' + . 1 1 2 8 2 IIi 3 + 2 {ikT {J (J + O((l{lr) ). Xi XI: flpe06pa3yeM Bblpa>KeHHe (5.13) K CJleeMY BHAY: f(x,,tj+l) - IM(x,(,t i ) +div«(fk> = !: 2 {i  {k ':{IA . (5.14) 7 Vi Xk YpaoHeHHe (5.14) nOXO>Ke Ha ):\H<P4>epeHaJIbHO-p83HOCTHoe ypaBHeHHe (5.11), onHCblBaJOIQee H3MeHeHHe <t>YHKH pacnpeMe- HHB "PH nepexoe 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>YHKg pacnpe- eJleHHSI CJIa60 MeHSleTCB Ha STOM pacCTOHHHH. Ec.nH T eCTb xap8.k- repHoe BpeMSI, 3a KOTopoe qaCTH1 nepeceKaJOT BqeA:KY, a II - HX xapaKTepHaJ1 CKOpocTb, TO B KallecTBe IIT I Ka.K H npH BbJBOe YPaB- HeHH.SJ (5.11), BhlCTynHT pa3Mep SlQeHKH h. TaKHM 06p83ow, speMR T, CTOsee B npaaoA qacrH (5.14), TeCHO CBSI3aHO C xapaKTepHblM pa3MepoM h., Ha KOTOpOM rapaHTHPYeTCSI Crn8JKHB&HHe pemeHHA. OHO MO)Ke'!' OTJIaqSThcsr OT T, CTOgero B nesoA 'IacTH (5.14), onpe- enSlIOero H3MeHeHHe ra30,qHHaMHtteCKHX napaMeTPOB no BpeMeHH H 3aBHCSlero 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&IB8JOeMY H3MeHeHHe <PYHKJJ:HH pacnpeeJIeHHH . (5.13) 3. KBa3Jfr830HHaMHqecK&H CHCTeMa YP8lJHeBJfli ,lLnS1 roro, QTo6bl nonYQHTb ypasHeHHR, onHCblBalOHe H3Me- HetlHe ra30HHaMHqecKHX napaMeTpOB, npoHHTerpupyeN ypaBHeHHe (5.11) C cYMMaTopHblMH HHBapHaHTaMH cp(E) no CKOpocTSIM MOJIe- KYJ1. IIpH 3TOM Ta.K }Ke, KaK H npe fiOJlyqeHHH KCPC. 6yeM CItR- . 1 TaTb) qTO I fi+lcp()dE = f fJ.: cp(). 
IlK 1 JI. 11. J\.HHeTH'tJeCKH COrJIacoSaHlfbie p&3HOCTlfbIe CXeMM BbInHl1Je1 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) - 3aHHbJe ypaBHeHHH COCTOSlHH.R. CHCTela 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<pepeHaJ1LHble '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: MOeJlH) 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 npenonO}f(eHHe 0 CJla6o1 H3Me}leHHH cPYHKHH pacnpe..rIeHHH H8 paccrOHHHH h, CBB- 3aHHbIM C xapaKTepHblM BpeMeHeM 'T. B KCPC nOSlBJIeRHe TaKHX ttneHOB Bb13BaHO KYCOtJHo-nOCTOjlHHhlM npeCTa.BJIeHHeM CPYHK:QHH pacnpe)J.eneHWI B cqeTHoR }I"IeHKe. B Kry -CHCTeMe SlBHblM 06pa3oM npHCyTCTBYIOT a MacIDTa6a speMeHH. IIepBbJA - ra30HHaMHqecKHA MacwTa6, KOTOphIA CBH3a.H  co CKOpOCTblO H3MeReHHB M8.KpoCKOllHtJeCKHX nap&MeTpOB, - oe- AeJlSleT BbI60p illara no BpeMeHH t1t. BTOpoA) T, - KHHeTH'fecKHH MacwTa6, onpeAeJIHeMbI:A Cpe)l:HHM opeMeHeM DpoJIeTa qaCTHqbl qe- pro o6'bet 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 KBa3Hf830HHaMHqeCKOA CHCTeMbI BbJTeKaeT JlHHeAHoe H3MeHeHHe ra3otIrHHaMH'tJecKHX napaweTpOB 3a BpeMSI 6eccronKHo- UH'feJlbHoro PeTa.. 9TO CBoAcTBO noepKHBa.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, CTOBHe B neBoA qacTH YP&BHeHHB, - C I10MOUJ,b1O u,eHTpa.nbHblX pa3HocreA 4;i' .n;HccHnaTHBHhle 'lJIeHbI B npaSbIX qaCTHX CHCTeMbI 3aMeHHIOTC.H p83HOCTBMH Ba (r(ax,))z, ' nH60 (1" (4J)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 6yeT 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 3aq HHaMm<H eanbHoro raaa, nonyqeHHble c nOMOblO Kry H KCPC, npaKTHqecKH COBnaJOT. BMecTe c reM nOAqepK- HeM, TO 60JIbmaH HarnHOCTb KBa3Hra30AHHaMHqecKO:R CHCTeMbJ eJJaeT ee yo6HbJM 06'beKToM K8K JVlH "IHCJleHHOA peanH3aHHt 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 BbJBOe, OItHCa.HHOM B n. 1 8HHOrO naparpa<Pa, B ypasHeHHH (1.11) 3aMeHHM qneH Axl1 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 qaCTHa npOJleTaeT pacqeTHYIO SMeAKYt 38.MeHHM (1.11) IIa cne,n.ylOIl\Y1O pa3- HOCTHYIO annpoKcHMa1O 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>YHKHH pacnpeeneHH.s npeAnonO>KeHHe 0 ee MaKCBenJIOBCKOM xapaKTepe Ha MOMeHT speMeHR t = t;, npHeM K C1Ie,n;YIOea 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>epeHHanbHoe no npOCTpaHCTBY npH6nH>KeHHe NISI CH- CTeMbl (5.22)-(5.24) COBnaeT 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(eHea 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>YHKHH pacnpe- eJ1eHHft B eHTpanb"oA B'JeAKe. KaK 8HHO H3 pac. 19. BnH.SIHHe ;J.Haro- HaJlbHblX jJqeeK 6y,a.eT 3aMeTHO, ec- JlH )I)lHHa cBo60oro 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 oneJJeBHIO 30HhI BJIR- waHHbIX npoH3BQAHbVC ua pe3YJlhTa- JDlJUI ,D.HaroHaJlbHblX gqeeK Ha 6a- TbI paceTOB 6yeT 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 Be, 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 paCneneHJIj( Be muIeTCH 06J13aTeJILHoA. 3aMeqaHHe 5.4. B )J.8HHoA rJIaBe, nOCBeHHoH 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 paccyeHHA, 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) 3HaeHHA <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 ai 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 CTOW.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 BarIILIA: BH)1 KD83Hr83OAHHaMH- qecKoR CHCTeMbl n0380JlBeT HCDOJIb30BaTb ee )1/111 Mo,n.enHpo8&HH;f 3a,L{aQ HeaJ]bHoA ra3OBOA AHHaMHKH B 0606eHHoA 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 Ya.eTCjf 3Haqu- TeJlbHO nOBblCMTb TOqHOCTL pac'leTOB. Bonee no,Iq>06Ho npHMeHeHHe Kry- CHCTeMbl B 0606LU-eHHbIX Koop)J)!HaTaX )l)JB pac'ieT8. 3a'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 KBa3Hra30HaMH'IecKoH CHCTeMbI ,AnH MOe- nHpoB8HHH TeqeuHA HeaJIbHOrO ra3a. Bonee CJlO>KHhIe C MaTeMaTH- QOCKOft TOQKH 3peHHSI H 60nee HHTepecHbie nOCTaHOBKH C TqI(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 3aq, 6y,a;YT paccMoTPeHbI B CJle,L\YIOHx rnasax. 
rJlaBa III IIPHMEHEHHE KIiIHETHqECKH cor JIACOBAHHbIX CXEM H KBA3:urA30HHAMHqECKOit CHCTEMbI JI.H MOEJIHPOBAHH.SI TEqEHHA B.SI3KOrO rA3A PaCCMaTpHoaeMble B npe,lI.bI.D.Ye:A rJlase KCPC H Kry MorYT UblTb HHTepnp€THpOBaHbI KaK anropHTMbI peweHIDI 3a,.a.a'l HeaJIb- HOt\: ra30BoA AHHaMHKH C HCKYCCTseHHblMH HCC.HnaTHBHbJMH tine- lIaMK, nponopHOHa.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{ OHOqaCTH'IHoA (j>YHKQHH pacDpeeJleHBS1 (1 ope- ;-1b1YIQeR 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 MOeJlHpoBaHHH aHHoro KJlacca TeqeHHA A0CTaTOtiHO OqesHHO. Tal( KaK OnHpaeT-  Ha XopoWO H3BeCTHYIO CBB3b Me>K)U' ypaBHeHHeM BOJlbaHa 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- JTHHbIM BapHatlTaM KHHeTHQeCKHX H 60JIbqMaHOBCKHX cxeM npKXO- Jl-HTCH HaxOHTbCB B >KeCTKOA KOHKypeHnHH C J\pyrHMH COBpeMeH- 
Hbl)'-IH aJrrOpHTMaMH pemeHHSI YPaBHeHUA 9A.rIepa. * HllaS{ CHTYHH CKJIblBaeTCjl DPH MoenHpOBaJlHl! TeqeHHH BH3Koro ra3a. 3AeCb aJII'OpHTMbI, nOCTpOeHHble Ha OCH08e KCPC H KBa.3Hra30Jl-HHa.MH"IeCKOA CHCTeMbI, c oHoA 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, npHcYHe ra30BOR cpee. PaCCMoTpeHHe B03MO>KHOCTeA KCPC H Kry HatfHeM C HX 06- cY>KeHHSI KaK <PH3HQecKOH MoenH ,(VIS! OnHCaHHH TeqeHH:B BH3KOro ra3a, o6paTHB oc06oe BHHMaHHe Ha HX CB}13b C KnaCCHqecKMH ypas- HeHHRMH HaBbe-CToKca.  1. KUHeTHqeCKH COrJIaCOBaBHLle cxeMhI KaK MOeJTh n.B ODHCaHRB TeqeHHB BH3KOro I'83a. 1. HeKoTopble 3aMeq&HHH K BblB0JU' YP8BHeHHR r830BOH )1.HHaMRKH IIpe}Ke \{eM paCCMaTpHBaTb, Ha '{eM On>KHO 6LITb OCHOB8HO pa3nHQHe Ii CB.H3b Me>KA}' KCPC H yp8.BHeHHSMH HSBbe-CTOKca t BepHeMC5I ee 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) COcpeOTOqeHHOa B o6'beMe V J OnpeJ]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 BbiCOKonpoH3TeJIhHWX CHCTeM. peWeltHe 3a\( ,D.HH&MHKH HeaJl.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 KOOpH- HaT, a TaK.>Ke TeopeMY raycca-OCTpOrpCKOro, nepenHWeM (1.2) B BHAe ! ( : + div pu ) dV' = O. v aJIee, eJTag 38.KJIIOQeHHe 0 npOH3BOJlhHOM 06eMe V H3 (1.3), 00- JIYQHM H<P4>epeHaJIbHOe ypaBHeHHe Hep83pblBHOCTH (1.1). BepHYBlliHCb K BbIBO,zur ypa.BHeHH (1.1). 3a.,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 cOep)f(aTb onpeeJ1eH}(Oe KOJIH'-IeCTBO MOJIeKYJI. QTo6hI BblnOJIHHJlOCb npH6JIH- )I(€HHe CfiJIOlllHOH cpe,ltbl. aMeTp TaKOrO 06eMa it t XOTb H npe- Bblillaerr HaMeTp MOJIeKYJIbI d , Bce )Ke cyecTBeHHO MeHbIl1e p)IHHbI CB06o.D-Horo npo6era l: ( 1.3) d « 11 « I. ( 1.4) CyecrBYIOT JIll pyrHe OrpaHH'feHHSI HR Bbl60p o6eMa V, Kpo- Me Heo6xoHMOCTH 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- 60Horo 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 nocpeCTBOM 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. MOe.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 nopy BeJlHtUlHhI 6paTL He t.1eHbme JTHHbI cBo6ooro npo6era I. TaKHM 06p8301) eCJIH yqHTbIBaTb 3TO OrpaHH'-IeHHe CHH3Y Ha HatwreTp o&beMa V, TO IIPH nepeXOAe OT HHTerp8J1bHOrO ypauHe- HHH (1.3) K <I><PepeHHaJIbHOMY B noc.ne,QHeM AOJI>KHhI nOSlBHTbCSI OnOJlHHTenb}lble qneHbl nopBKa 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, COCToBaSl H3 ypaoHeaHB THn& (1.6), 06naeT CBoACTBOM KoppeKTHOCTH no KpaAHeR Mepe c <pH3H'IeCKoft TOQKH 3peHHiI, TaK KaK rapaRTHpyeT HaJ1WQHe cna60ro H3MeHeHHSI peweHHSI Ha pacCTOHHH.HX, MeHblIIHX HJlH nopSlKa 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. npoOJI>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 TellJlOnpoBOHOCTH. 9TH ypasHeHH5I npOWJIH HCIILJTa.BWI 60JIee eM BeKoBo:A:, B TOM 'IHCTle H HH,ayCTpHaJIbHOft, npaKTHKoA, nOTBepaJOm;eft TeOpeTB'IecKHe Bb(BOb1, 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 re 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 HccneOB8.Ha CBH3b KBa3H1"8.30,!{HH8.MH'IecKOA' CHCTeMbl H CHCTeMbI ypa.aHeHHA HaSLe-CTOKca. ITone3HOCTh Hcnonh30BaHHB TaKoR CHCTeMbl ypasHeHHA, OTnHqaJOeA:CSI no cBoe:A MaTeM&TH'Ie- CKOH cPopMe OT ypaBHeHHR Hasbe-CTOKca, 6yeT BH)J.Ha H3 npH- Be€HHblX npHMepoo ee HcnOnb30BaRHSI njl MOeJlHpOBaHHg 3a cOBpeMeHHoA a3pOHaMHKH. 2. 3aMeqaHHH 0 AHCCHuaTHBHblX QJleH8X B KCPC MTaK J 6yeM yMeHbmaTh pa.3Mep aqeftKH, H& KOTOpo:a KOHCTpy- HPYeTCH KCPC, o MHHbl cBo6oHoro np06era I. B KBa3Hra30HHa- IHqeCKOA CHCTeMe (5.15)-(5.19) (rn. II) napaMeTp T BLI6epeM pas- HblAf BpeMeHH Me>KAY CTOJlKHOBeHHHMH MOJIeKYJI (T = llc). Tew ca- MblM BbinOJIHSleTC5I OO 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 paccyeHHB CbIrpaJIH pern&IOIQ,y1O JX>1Ih B HCII0JIb30BaHHH KCPC H KB83Hra30laMH'lecKOA: CHCTeMLI ,l].JIH MO)J.eJJHpoBaHID1 TeqeHHA BB3KOro ra3a, KOTopoe 6bL1I0 HaqaTO C cepe.LJ;HHbI 80-x rOOB. 3aMeqaHHe 1.1 I'JI. II nOK83L1BaeT, '11'0 nOTOK MacCLI Ha rpaHH- e paCqeTHOA S1qeKH 8 KCPC paBeH pu, re U - CKOpoCTb BOJIb 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 cB06oHoro npo6ero l, OnOnHHTenbHOe H3MeHeHHe HMnYJlbCa 3a CtfeT AHCCHnaTHBHhlX qneHOB B KCPC 6YeT nponop- 1(HOHaJlbHO pa3I-IOCTHOA npoH3BooA MaKpOCKOnH'IeCKOA CKOpOCTH u C K03c1><PHHeHTOM nponOpHOHaJIbHOCTH JJ, p8.BHblM P, = plc/2. I1MeHHo TaKoR Koa<l>cl>HHeHT 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 pacnpeeneHIDI 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 TenJlOnpOBOHoro qneHa H p33HOCTHOMY aaaJIory pa60TbJ BH3KHX con. COOTBeTCTBeHHO, K03<IxPHHeHTbl 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, BHa8Jle BcnoMHHM p qeM >Ke no cyecrBY 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) OTnHaeTCSI OT (1.7) tI& Ma.nylO aenUqHHY Kn J NS (Kn < 0.001 nJl 3aA&q ra30BoR ,ltHHaMHKH). OAHaKO HyeHHO 3TOT 'IJleH OTBeTCTBeHeH 3a nOBnelIHe 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 MOenHpOBa.HHHt OnHpalOl.U)fXaI H& yp8.B- HellHH 9AJ1epa H ypaBHeHHH HaBbe-CTOKca, npOH8JIHIOTCH TOr,Qa. Kor,n.a 8eJ1HU4HbI (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>HHeHT 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 HenOCpeCTBeHHhlM nepeHOCOM HMnynbca BMecre C MaccoA nOCTynaJOJJJ.eA JKH)U(OCTH. AHanorHHo, nepe- HOC Tenna 3a cQer rennonpOBOAHOCTB He CB1I38J1 C MaKpoCKOnHqecKHM ,l\BH- *eHHeM >KKOCTH HJIH ra38, a ocyeCTBJJHeTCa 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 CBo60Horo np06era) YAOBJIeTBOpHeT Tpe60BaHHSlM. Bbl,nBHHYTblM B n.l aHHOrO naparpaQ>a. C oHoR 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 0606a.a BLJrnecKa3a.HHoe, KCPC MO)fCHO TpaKTOB&Th leaK Ho- BYIO MOeJJb NISI ODHCaHWi TeqeHHSI BH3KOro ra3a. K UKemUeC1roM ompae1iUe.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 pacnpeeneHH:A: no 000 CTOpo- HbI OT rpaHHl.\bI S!'Ie:AKH. llonYMaKCBeJIJIOBCKasI <PYHKR, KOTOPYIO 06pa3YIOT 3TH pacnpeeJJeHHR Ha rp8JfHe, npHBO)J.HT K nOBBJIeHHIO ;J.HCCHnaTMBHbIX COCTaBJISlJOHX B nOTOKaX. IIpHse,LJ.eHHble co06pa>KeHHB SlBWlHCL BaJKHblM MOTHBOM K HC- nonb30BaHHJO KCPC, KOTOpble, H& nepBbIA B3rnH, npeCTaBJIBJOT co6oA cxeMLI C HCKYCCTBeHHoA BB3KOCTblO, OnHChlB&IOe npoeccbl B HeanbHOM raae, AIlS! MOAeJIHPOSaHRSI BB3KHX TeqeHHA. OMf&KO HaH60JJee HHTeHCHBHO STO Moemtp08aRHe, HaqHHaJ1 c pa60T (22), [57]-[59], npOBOHnOCb B pCWK&X KBa3Hra30AHHaMHtJecKoR CHCTeMbl. IIpH 9TOM HaH60nee HaI'JUl,D.HO npocMaTpHBaeTCJI CBB3b Me>KA.}' Kry H ypasHeHH.SIMH HaBbCTOKCa, KOTOPYIO pacCMOTpHM B r.neAYIOeM naparpac}>e.  2. KBa3Hra30HBaMHqeCJaUI 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, npHBeHHbIe B  1, MOHO npeOJJO>KHTb) 'JTO no- cneAHBSI CHCTeM& JJ.&eT npaKTHqeCH 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. OHaKO npe>Ke qeM BblSlCHHTb 3TY CBH3b HenocpeCTBeHHOJ BblnHweM KB83Hr830HHaMHt.(ecK)'1O CI1CTe- MY B 3TOM CJlY'I&e. ):{na 6W1LweA KOMnaKTHOC CHCTeMY ypaBHeHHR 
32. KrY-CHCTeM8 II YpaBHeHHJl HaBCTOKCa 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) 
3ecJ, 'Y - IIOKa3aTeJIh a.D.Ha6aTLI, Pr - q}JCJlO npTJIa, 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(.te (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. no06HaJI npol(eAYpa HJ(eHTH'IHa DOJIyqeHHIO ypaBHeHHA, OOHCblB8IOIII)IX Te'JeHHe B1I3KOro raaa H3 BrK- MO)J.enH (9 3 rJI. I) t re 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) (r1. 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 nOHoro 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 DpeooneHbI C nOMOW»1O pa.3J1H"-IHbIX Bb[qHCJlHTeJILHLIX nOO)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-{aHeM c ypa8HeHH (2.1). YqHTbIB8.H CBb HCCHnaTHBHbIX 'U1e- HOB I1pasoA IJacTH ypasHeHHA (2.2) H (2.3), nepenHweM (2.1) B BHe: 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 nOCTaBHM pa3nO>KeHKH Ba (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), onHChlBaJOMH H3MeHeHHe HMnynbca., ypaBHeHHeM (2.4), OnHCblB8JOllJ.HM H3Me}leHHe 3HeprHH H CQ()TBeT- CTBYlOMH 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 BHe Pik = - 1 "'CiC1c' 8Xi OXi dt HCDOJIb3YS ypasHeHHiI 9AJJepa ARB eanbHOrO ra38., TpaHC<!>OpMH- pyeM npaBYIO 'facTh BbJpaJKeHHB (2.9). Tora 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, CTOSlee no HHTerpaJIOM B npaBoA a.cH (2.9), B Be 8 ( dfo ) 8 8/0 8 a/ o - T-CiCk = -T{pik- + -T{i{k-- 8Xi dt OXi 8z p 8Xi Ot a  a  a  - -TfUk- - -TkUi- + -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 pikTdE = 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, OllHCLlBalOee 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 npoeAYPbl QenMeHa-3HcKora. J fcp()d = J fO'P()d() K TaKOMY )J(e pe3YJIbTaTY npeM. HHTe- rpHpya C secoy p 6aJ1aHCHOe ypaBHeHHe (5.11) H3 rn. II nR oo- qaCTHQHOR cl>YHKHH 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:JOHH3.Ml!qecKOR CRCTe- Mbl. HCDOJtb3YB npOl{e,zurpy, aHaJlOrH"IHYIO (2.9)-(2.13), MO>KHO nOKa- 3aTb, 'ITO H3 6aJIaHCHOrO ypaBHeH1iSl (2.14) 6yeT nonY'IeHO C TOq- HOCTblO o 0(7 2 ) ypaBHeHHe 3HeprHH H3 CHCTeMbI HaBhe-CTOKca, BKJIlOqalOee 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>YHKHOJ-Ia.nbHOM BhIpaJKeHIDI, onHCblBalOel,l 8KJI B ypaBHeHHe 3}IeprHH TennOBoro nOTOKa, qHCJlO npalVl-TnB nOJIaI'aeTCH paBHblM era pea.nbHOMY 3HaIeHHIO ,-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. OaKO OCHOB- HoA 06nacTblO npHMeHeHHjf] re Kry nOKa.38J10 CBOH B03MO>KHOCTH, 8JljleTCH MoenHpoB3.HHe HeHOHapHhIX TeeHH: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, 
OlTHparoeMcH Ha HCnOJIL30BaHHe Bb1COKOnpoH3BOHTenhHblX MHO- ronpoeccopHblX BblQHC1IHTeJlbHhIX CHCTeM. KpOMe TOro. JIpHHnH- aJlbHO 8a*HbJM TIPH nOCTpOeHHH Kry SlSnSleTCS ee HOHapH 3arJHC.b B KOHe'iHo-pa3HOCTHOM Be. BHHMaTeJlbHhlA: qMT&TeJIb 3aMeTHJI, 'ITO npoBe,L\euHoe B ,D:8.HHOM nYHKTe HCCJIe,QOOaHHe CBB3H Kry H CHCTeMbI HaBLe-CTOKCa cye- CTBeHHblM 06p8.30M HCfiOJIb3yeT npenonO)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 nOpgKY BeJlHH}ILI COBrra.IOT c qJleHaMH KOHBeKrHBHbIMM) HanpHMep, B nOrpaHH'IHbIX CJlOSIX. B CJIeA}'IOeM nYHKTe naparpa- <t>a, OnHpagCL Ha pa60TbI (53, 62]) npOBeeM Ol\eHKH HCCHnaTHBHblX qneHOB Kry, CnpaBe,D.nHBble H ,lI,JISI 3TOro CJlyqaH. 3. j{HccHIlaTHBHble CfJleHbl KBa3Hf'830HHaMJfqeCKOR CHCTeMbl OeHKY nopKa ,lJ){CCHllaTHBHbIX QJ]eHOB H81.fHeM c npasoA qa- CTH ypaBHeHHH (2.1), KOTOpylO npeCTaBHM B BRAe CYMMbI {) TOT --A+ --B ax 2 8y 2 (2.15) IAe 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, nponOpOH3JlbHble T, BXOSIe 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 · 3ech L - xapaKTepHhlH pa3Mep 3aqH, 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 ra30HHaMH'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 Oqepeb) 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 ' re qJIel1 T pU / (tgL) CBSI3aH C HecraQ.HOHapHOCThlO KB&3Hr'a30HHa- MHQeCKHX ypasHeHHA: (2.1)-(2.5). TaKHM 06paaoM, 93. (2.19) CJIeTJ tITO OTJIOmeHHe qneHoB) BXO- lJ;SlI.I.J.HX B npaBYIO 1.faCTb (2.1), K qneHaM B neBoA qacrH, HMelOM 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, CTOJIHMH 8 neBoA qacTH 3TOro ypaBHeHIUI. B 3TOM KpoeTai KopeHHoe pa3JIB'IHe we>K.nY ,.[UIC- CHnaTHBHblMH 'IneHaMH 8 ypa8HelIHH (2.1») H qneHaMH, OnHCblB8JOHMH HcrHHHYIO BK3KOCTb B YP8DHeHIDIX HaBb&-CTOKca, KOTOpbJe B DOrpaHH'I- HOM cnoe HldelOT O)J)lH H TOT >Ke nopAOK C KOHBeKTHBHblMH qneHaMH. IIpoBeeM aHaJlOrHQHblH aHaJlH3 .D-HCCHOaTHBHbIX 'IneHOB ypas- HeHHA: (2.2) H (2.3), onHCblBaJOX H3MeHeHHSI HMnYJIbCa B KBa3H- ra30HHaMHqeCKOA 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 3aBHCmIHe 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 Bcera 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 oeHHTb I10pOK QJIeHOB nepBoA rpynUbJ) npecraBHM, HanpHMep, AJISI ypaBHeHHJt (2.20) HX BKJI B CJle.nyJOeM BHe 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 BXOlII)Ie B ypaB- HeHHSI HaBbe-CTOKCa. ,lI;IH OCTaJlbHLIX qJleHOB Bblpa)J(eHHSI (2.15) cnpaseNIHBa oqeHKa no nopHKY BeJIH'IHHbl, pasHaB TIt, [gradp]. TaKHM 06pa30M, MCCHnaTHBHhJe qneHbI B ypaBHeHHJlX (2.2) K (2.3) KBa3Hr330HHaMHqeCKOR CHCTeMLI nH60 no DOPKY Be1lDqH- lib! 8 T / t g p83 MeHbwe KOHBeKTHBHblX, nH60 COBDaIOT C BSl3KHMH . '{JIeHall-I COOTBeTCTBYIOIU.HX ypaoHeHHA HaBbe-CTOKca. YKa3aHliblM Bblme cnoco60M H3 JI.HCCKnaTHBHblX qneHOB ypasHe- HHR (2.2) Ii (2.3) KrY-cHCTeMbI 6blJlH BbJeJIeHhJ 1.IJleHLJ ecTeCTBeH- 110M BSl3KOCTH. H oeHKH OCTd.BWHXCSI qneHOB cyeCTBYIOT ABC B03MO>KHOCTH. B nepBOM cnyqae ORB, KaK 3TO cneeT H3 n.2 aH- Horo naparpaQ:>a, MaJIhI DO CpaBHeHWO C KOHseKTHBHblMH qneH8.MH Ii lJJleHaMH, onHcLiBaroHMH 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 IIpoenaTb H AJlSI ypasHeHHSI 3HeprHH (2.4) KBa3Hra30HHaMHqeCKoA: CHCTeMbI. 4. HeKoTOpble pe3YJlbTaTbI IfHCJ1eHHblX pacqeTOB npoBeeHHbJe B 0.2 H 3 a.HHOrO naparpa<j>a pacCY>KAeHH no- Ka3bJBaJOT, "liTO peweHHSI 3aq B TepMHHax Kry -CHCTeMhI H ypas- HeHHR Hashe-CTOKca AOn)KHbI npaJ<THtfeCKH COBD&AaTb, TO H no- TSep>KAaeTCH MHorotJHCJleH1IbIMH TeCTOSblMH paceT8MH. B KaqecTBe npHMepOB npHBeeM AaHHbIe paceToB 3aqH 0 YMepHOM o6Te- KaHRH rrnacTHHbI no,n: HYJIeBblM yrJloM aT8KH A03BYKOBhlM nOTOKOM B03AYxa (63) H 06TeKaHHB 60nee CJI0>KHOrO Tena HJlHH)ij>HqeckH cHMle'rpHq}fOa <l>OpMLI csepx3BYKOBblM nOTOKOM (64). ,aJIH nepBo:A 38JJ-a'lH pemeHHe KB83HraaOHHaMBqecKOa: CHCTeMbI Cpa.nHHB8JlOCb C TOQHblM aBTOMOenbHblM peweHHeu 3aH 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 CJleYIOeA 3a.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,.o1O 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 OCecHMMeTpHHOro Tena 
j2. KrY-cIICTeM8 I' ypasHeHHR H8.Bbe-CTOKC8 129 KaK BHHO H3 pHC. 20. Me>K ABYMg peweHHHMH 38J\&1.fH, nOJ1Y- qeHHblMH Ha OCHOBe HCnOJIb30BaHHB Pa3JlHQHblX eHcreM ypaBHeHHA) I1paKTlNeCKH OTCYTCTSYIOT pacxoeHHSI. 8 Kat.lecTBe BToporo npH- tepa paccMoTpHr Pe3YJlb- '"raTbl pacqeTOB 06TeKaHWI lHJIHH.APHqecKH cHMMeTpHt.f- HOro TeJla, <l>opMa KOToporo npHBe)J.eHa Ha pRC. 21. Pac- cMaTpHBaJlHCb CJ1eAYIOHe 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): PeHHonbca1 pacCHT8.HHoe EXP - 3KCnepBMeHT, DSMC - pac- 'leT no KHHeTHqecKol MeJIH MeTOOM no OTHoweHHIO K JlHHe MORTe-Kapno, NS - paceT 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.HWH 3KcneplDleHTB H DSMC- MO)l.eJ1HpooaHHH KpOeTCH B B03MO>KIIH 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. PacnpeJleHHe '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 ra3HHawH'IeCKHX napaMeTpoB B y3KOM npHCTeHOHOM 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 npeAbJAyeM npHMepe, DpaKTe- CKH He OTJIHQ8JOTCa OT 8.HHbIX, fiOJIY'IeHHhIX c nOMO1O YPaBHeHHA HaBLe-CTOKca. 9To nOTBepa.eT TeopeTH'IecKHe oeHKHt npoe- JJaHHhle B n.3 H 4 HacTO.S1ero naparpa4>a. CytdMHPYSI pe3ynbTaThI ,qa.HHOro naparpacPa, OTMeTHM CJle)U'1O- ee. K 8Q3U2(J,300uH4MUC"aJi cucme.wa ypa6HUU 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 0Ir036aJOmCR 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 nOAYHH Ha OCH06e uc- 
93. KHHeTHtfeCKDe OrpaHCeHHe KrY-CHCTeMbI 131 nOJt'b306aHUJt ypa6HeHuu Ha8be-Cmcmca. POJ1b 3THX OnOJIHHTenbHbIX qneHOB ):\HC 3aKJI1OaeTCS1 B ra,- panTMpOBaHHOM Crna>KHBaHHH pemeHHB Ha pacCTOSlHHH nopKa )J.JIHHbl cBo60Horo npo6era. KaK 6YeT BHHO 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 nOBeeM HTorH o6cyeHHSI KCPC H Kry -CHCTeM&I KaK <PH3H'IecKOA MoenH SI onHcaHHH Te- t.f€IIHg BSi3KOro ra3a H HX CB3H C BHOM O,[\HO"tfaCTHqHOA Q>YHKQHH paco peeJleHHfl. 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 npeonO)f(eHHe 0 nOKaJlLHO- M8KCBeJInOBCKOM paBHOBecHOM xapaKTepe OHO'facrHqHOR <PYHKn:HH pacnpeeJIeHH.H. YpaBHeHHH HaBbe-CTOKca MorYT 6L1Tb 1I000YQeHbI, eCJIH )VIS npe)J.CTaBneHBB ct>YHKHH pacnpeeJIeHHg HCnOJlb30BaTb a nepBhlX tfJIeHa pa3nO>KeHHB QenMeHa-9HcKora peweHHB KHHe- THeCKoro ypaoHeHHSI BOJIhaHa. KCPC (KrY-cHCTeMa) onHpalOTCH Ha npe.D-D0nO)l(eHHe 0 MaKC- BeJIJlOBCKOM xapaKTepe <PYHKH 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 HanHSI ABYX Pa3J1HQHblX MaKCBennOBCKHX pacneneHHA 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 rpaHHe, HCPcl>epeHanbHbIM OTpa>KeHHeM KOTOphlX BBJISIIOT- 
132 rJl. III. MenHpOB&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 pacnpeeneliHH. KOHKpeTHbIA Blip. oA paBHoBecHOH CPYHKHH BXOtl.HT B cHCTefY (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 pacnpeeneHHSI K MaKCBMJlOBCKOMY p8BHOBec- HO(Y BIf..QY. C ,n;pyroA CTOpOHbI, H3 peweHHB KHHeI'HqeCKOro ypaB- HeHHH CJIeYeT, QTO cPYHKIJ)IH/pacnpeeJleHHSI 6nH3Ka K nOkaJIbHo- p8.BHOBecHOA, eCJIa OHa c.na60 MeHSIeTCH Ha pacCTOSIHHH nopSlKa AnHHhl CBo6oHOro npo6era I. HMeHIIO 3TO CBoRCTBO, HCnOJIb3ye- Moe gBHO, nOJlO}l(eJIO B OCHOBY KHHeTHQecKoro 060cHOBaHHSI KCPC H KrY-CHCTeMbl. JIlo6onblTHa aHaJIOrHB Mey pa3JIO>KeHHHMH B pH no MaJlO- MY napaMeTpy peWeHHjI ypaBHeHH BOJIbl.\MaHa, npeAJIO>KeHHbIMH fHJIb6epToM (21, a 3aTeM 1..JenMeHoM H 9HCKOroM, H npeCTaBJIeHH- eM <PYHKHH pacrrpeAeJ1eHHSI, HcnWlb3yeMblM AJlSI nonyqeHHH KBa- 3Hra30HHaMH'IeCKOH CHCTeMbJ ypaBHeHHA. Pa3JI01KeHHe qenMeHa- 9HCKora BJISIeTCH 60J1ee paHOH8JlbHbIM, 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 pHa ):\JIS <PYHK- D;HH pacnpeAeJIeHHH.. Pau.HOHaJlbHOCTb OCTHraercSl 38. CqeT 51BHOro Y'IeTa, Toro <paKT8. t 'ITO B5JIH3H paSHOBeCHH <PYHKSI pacnpe.n-eJle- HHSI onpeeJl$IeTCSl C nOMOblO nwrH ra30liaMHqecKHx napaMeTpoB p, u, T, KOTopble, B CBOlO oqe, J1S1 K8JKAOrO QneH8. pa3J10JKeHHH HaxOHTCSl H3 CBoe:B CHCTeMbl MaKpocKOllHqecKHX ypaaHeHHA. npH nocTpoellHH KCPC H KB83"ra30HHaMHqecKo: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>YHKHH pacnpeeJIeHHjI B6JIH3H pasHoBeCHlI OT napaMeTpOB p, u, T TaK.>Ke SlDHO Y"IHTblBaeTCR TOT <paKT, tfTO cf>YHK1W pacnpeneHH.H cna60 MeHHeTCH Ha pacCTOs(HHX nOpAKa .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>YHKHR pacnpe,QeJleHHH. 2. KBa3Hra30HHaMHeCKB.H CHCTeMlJ If BrK-ypaBHeHlfe OCHOBOA BblBoa 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-MOMb of 8f at + i OXi = II (f M - f) , (3.2) re 1/-1 HMeeT nOpSl,l(OK BpeMeHH Me.>KAY' CTOJIKHOBeHHHMH T, no- JlYtfHTb KBa3Hra30HHaMHqecKYIO 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>YHKSI pacnpeeJIeHH.H cna60 MeHHeTCSI Ha pacCTOJlHHH no- pRJl.K8. )VIHHbl cBo6oHOro npo6era H 8 Te'tJeHHH BpeMeHH no- pHKa 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-MOeJlb nonY'IeHa H3 ypaBHeUHH BOJIbMa.Ha np" BbInORHeHHH YCJlOBHSI a) (CM. rJl. I). 
BYAeM HCKaTb pemeHHe ypaBHeHHSI (3.2) B Be pa3JJO;)l(eHHH B pSl,l:\ no MaJlOMY napaMeTPY f, no nOpB.AKY BeJIH'lUHbI COIsnaJOe- 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 BXOAe B BbJpa>KeHHe (3.4), 6yeM pacCM&TPH- BaTh Ha MOMeHT speMeHH t = t j + 1 . OHHBM HeKOTOpble qJIeHbJ, BXOHe 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.ildJ:l =! [  {fli{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) C1IeeT, tITO npH HHTerpHpOBaHHH C CYMMaTOp- HblMH HHBapHaHTaMH B BLJpaJKeHHH (3.8) cneAVe'!' yqecTb TOJIbKO 
93. KHHeTHqecxoe OTplUKeSHe KrY-CBCTeJ4bl 135 '-IJIeHbI, crpynnHpoBaHHbJe rJIe)U'IOM 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 npHeM K TOqHO TaKOMY }f(e peaYJlbTaTy, 'ttO H I1PH HHTerpHpOBaHHH 6anaHcHoro ypaBHeHHSI (3.1), TO ecTb K KBa- 3Hra30HHaMHqecKo: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!, onHCblBalOero 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 npoH3BOHble BToporo nopH,nKB. H Bhlwe HMelOT O,lJ;BHaKOBhlA nOpH)J.OK C E. ManwA napa- MeTp  HMeeT OHH 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 nopoK 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- npeeneHHH "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 no06f[oA npOQenypbl C ypaBHeHHeM (3.13) MW BHOBb J K8.K H npH HCIIORb30BaHHH ypaBHeli1Ul (3.1), npeM K KrY-cHCTeMe ypasHeHHA. llpOOn)f(HM H3YQCHHe cBoACTB KrY-cHCTeMhJ, 06CY8 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 orHOCHTeJlbHO cKopoCTeR B ypaBHeHHSX ,!VIR HMIIYJlbCOB (2.3)- (2.3)) I-leJIHHeAHOCTb ee npanblx '1acrea.'" PacCMOTpHM BHaqane npo- 6neMY KHsapHaHTHOCTH KBa3Hra30HHaMHqeCKOa CHCTeMbi OTHOCH- TeJlbHO npe06pa30BaHHR ra.nHJIeH, Koropble onenxlOTCH <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 KB83RI83OAHHaMJflecICOR CnCTeAfW 137 onJ.ICbIBafOee 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>YHHH pacnpeAeJIeHHg B OHHX 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 ) + Ti a + Xi 1   2 {f2 fM(X, (, t j ) O( 3 ) + 2 ikT 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), npHAet K aHanory 6aJJaHCHOrO COOTHOWeHHi1 (3.1)) OTnHqaJOeroCfi 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 npHBOHT K KrY-cHCTeMe. TaKHM 06paaoM J CBoRCTBO HlISapH&HTHOCTH KB83Hra30HHa.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 npoe,L\yphl QeIIMeHa-9HCKora He Y'lHTbIBaJlHCb qneHbJ p83JlO>KeHHH nopSlOB r 2 H Bblwe. B fIpHHHne 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 8I,jlt 3 = 2 rik 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, npHeM 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 HenocpeCTBeHHo nonyqeH B peaYJIbTa:re rpyn- DODoro aH8JIH3a HeKOTophlX qaCTHbIX CJIyqaeB TeqeHH.sI (HanpHMep. npOCTpaHCTBeHHO-YMepHoro H3OTepMHecKOro 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 BOJIh 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 onpeeJ]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 COOTBeTCTBYIOHX ypaBHeHHA: H. 3a1OTCSI CO- OTHoweHHHMH [68] x'  x + at, Q -+ Qt - aQx , y  Qty - aQ%fI , re 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 cyecT8eHHo HCOonb3YeTCH <paKT H8JlH'IHSI HMeHHO KOHe'lHblX pa3HOCTeA no BpeMeHH, KOTOpbJe 3aTeM npe,L\CTaBJJIDOTCa n BHe, 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 BpeHHH, TO 3,[tecb pea.rlH3YeTCSI 60nee npoCToA BapHaHT. HueH- HO, CHCTeta <PcJ>epeH8JIhHblX 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 noxo,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]. nOCTaHOBKa KOOp)J.HIlaT onepaTopa rpynnbJ Bpa- leHHat 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 onpeenSlIOe ypasHeHHSI o6paaeT HX B TOeCT80, qTO H 3aBepmaeT OKa3aTeJlbCTBO. npOBe,lJ.eHHOe paCCA10TpeHHe HHBapHaHTHOCTH KBa3Hra30,l])iHa- MHqecKOA MO,lJ;e.nH, HapSIAY C 060CHOBaHHeM ee BhlBoa C TOQKH 3pe- HK5I KHHrHqeCKOA TeopHH ra30B H 6JJH3OCTH C TOQ}IOCTblO AO O(r 2 ) K KJIaccHqeCKHM ypasHeHHRM H8.Bbe-CTOKca, cnY)KHT ee OHHM nO)I.TBepJKAeHHeM B03MO)l(HOCTH ee HCfiOJlL30BaHHSI KaK cl>H3HQeCKOH MO,ll.eJIH ,D.JUI OnHCaHHSI TeeHlUI BH3KOrO ra3a.  5. 0606eHIUI Ha )q)yrBe 3aqll MexaHIlKH CDJlOWHOI cpehI I1CKJ1lO'-IeHHe H3 MaTeMaTHqeCKOA MoenH paccMOTpeHHSI M&C- WTa60B, MeHbWHX qeM 3TO Heo6xo,n:HMO ,An51 BblnOJlHeHHSI HCXOAHblX cl>H3HQeCKHX npe.n.nOJlo)f(eHHA, nOJlO)KellO B OCHOBY BbIBoa KCPC H Kry. 3TOT I10,!J)CO):{ MO)l(HO pacnpOCTpaHHTb H Ha ,.lIJ>yrHe 3aqH Me- xaHHKH cnJIomHoR CpebI. Bblwe Y>Ke rOBOpHJIOCb 0 nOflS1THH <PU3H- eCKH 6ecKoHeQIlO MaJIOI'O otYbeMa, KOr)\& ero MHHHM8.JJbHbIa pa3Mep OnpeeJISleTCSl 1-13 YCJ10BHH Ha.JIHSI 8 HeM )\OCTa.TOHOro KOJIHtleCTBa MOJ1eKYJI, C TeM QTo6bJ BblnOJlHSlJIOCb npH6nH)I(eHHC CnJIOWHo:A cpe- )I.bl. PaccMoTpHM H pSl JQ>yrHx npHuepoB no,n;o6Horo nooa. 1. MOAeJIHpOBaHlle IJpoecca cPHJlbTpHH KaK H3BeCTHO J KnacCH'leCKHM OD.acaHHeM npoecca cl>HJlhTpa- H }KHKOCTH 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)] , re k - Koacl><PHI.J.HeHT npOIIH1J.aeMoCTH) {3 - K03c1>cI>JIeHT 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(HKOCTH (cl>nIO- Ha) 5IBnSleTCSI 3aKOH cl>opxaAMepa ku + kl 'ufu = - gradp, (5.4) HameAWHA CBoe 3KcnepHMeHTaJIbHOe nOTBepeHHe, 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. rrpeCTaBHM 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). Tora XapaKTepHYJO CKo- KOCTH B nnOCKON KaHane pOCTb TeqeHHH <pnIOHA& U MO}KHO onpeeJJRTD c nOMOIIJ.b1O xnacCH- qeCKOH <POPMYJlbI IIY83e:A:nSl JJU/d2-Vp. (5.5) TeM calwlblM BHHO) 'ITO K03c1><PHeHT 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 nopobI 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'lBHblMH "IJIeHaMH H ero ocpeeHHH Ha xapaKrepHOM Mac- urra6e i (70). Ha npaKTHKe B KaT.lecTBe M8cur:ra6a i ,lI.OCTaTOq}{O B3S1Tb pacCToIIHe nopH;J.Ka COTHH pa3MepOB 3epeH nopo):tDI. 113 npHBeeHHblX pacCY>KeHHa cne,qyeT, qTO paCCMOTpeHHaJI nOCTaJ-IOBKa 3aqH 3KJIeKTHHa no CBoeMY nocrpoeBH1O. B caMOM AeJIe, ypaoHeflHe (5.2) OCblBaeT npoQocchI C MHHHMaJIbHblM xa- paKTepHbIM MacWTa60M I H aanpaBneHO Ha onpeeJIeHHe cpeeR CKOpOCTH <t>HJJbTpal.\HH U (pRC. 26). d ! ArtJl HCC)KHM8.eMOa >KKOCTH CHCTeM8 (5.1)-(5.3) DpHMeT BR,Q div u = O. ku = - gradp. 
B CBOIO Oqepeb, 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 npeoOJIeTb 3TO npoTHBOpeQUe, nonp06yeM BH- OH3MeHHTb ypaBHeHHe (5.1) C yqe- TOM XapaKTepHOrO tacwTa6a ocpet\- HeHHSI l [71). PacCiOTpHM cPYHKHIO pacnpe- eneHHSI f(x,) t) A1IB <!>HJIbTPYJO- w.e:RcSi jf{HAKOCTH, re E - MrHooeH- H8J1 CKOpOCTb MoneKynbl )l{HAKOCTH. . ,L{nSi MOMeHTOB <PYHKIJ)iH f eCTecTBeHHo nOTpe6oBaTb BLlnOJIHeHHSf CJJe,uy101IUIx CBoACTB: PHC. 26. Pa:JJ1HHe Me*M Mt"HO- BeHHOA CXOpOCTblO U' npH o5Texa- HHH 3cpeH DOpo H CKOpocTbIO cl>H.1IhTpaH 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>yHKHH 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 nOHa06HTCSI 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 BOnLMa.Ha (rJI. I,  2). .HBHaJI pa3HOCTHasI annpoKcHMaH 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{Ix '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 BblBoe KBa3Rra.- 30Ha.'4HeCKOH cHcreMLI (rn. II,  5), 3aMeHHM I{I Ha xapaKrepHYIO CKOpOCTb, B KaqecTBe KOTOpoA y06Ho B3RTh CKOpoCTb 3BYxa c. rIpe 3TOM £lOJIYQHM: pf+l - Pi + (pu)f+1 - (PU)Ll = ( X C ) . .6t 2x '. 2 Pi x (5.10) 3aMeTHM, QTO p83HOCTHaJI cxeM8 (5.9) nonyqeHa npH YCJJOBHH KYCO'iHo-nOCTOSlHHoro no npoCTpaHCTBY Ba ct>YHKlJ)fH pacnpee- 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- TpHH U non}"leHO nyreM ocpe,n:HeHHS no npoCTpaacTBY Ha P83- Mep i  10 2 3epeH nopobI CKOpOCTB eACTBHTenbHoro rHAPo,QH- llaMHQeCKoro TetleHHSI u'. U03TOMY _8 K&qeCTBe OrpaHU'IeHUR CHR- 3Y H8 x cneyeT B3S1Tb BenH"qHHY l. YlIHTLlBasI M"OT <PaKT. 38J1H- weM H<P<I>epeHH8JlLHoe IIpH6J1H>KeHHe )J)ISI yp8.BHeHHSI (5.10) Y)Ke B rrpocTpaHCTBeHHo-MHoroMepHoM CJlyqae B Be - op d . d . Ie d at + IV pu = IV"'2 gra p. YpaBHeHHe (5.11) COBMecTHO C 3aKOHOM tI-apca (5.2) 6yeM HC- nOJlb30BaTb B Ka"tfeCTBe MOeJlH NISI onaCaHHSI Te'qeHHj{ JKOCTH. (5.11) 
06CYHM HeKOTOpble cPH3HQeCKHe npeJ1.l10nO)l(eHHSI, nOJlO)«eH- HhIe B OCHOBY BbJBO,l(a ypaBHeHUH (5.11). AnB 60nbIIJHHCTBa nopo,lJ. - HMeeT MeCTO CJleAYIOee OrpaHHt{eHHe no nop5lKY BenUI{HHbl I: 'iTO cyw;eCTBeHHO MeHbme pacC1'OHIDI.A. Ha KOTOphIX npOHCXOHT 3a- MeTHOe H3MeHeHHe napaMeTpOB cl>nJOHa. CneOBaTeJJbHC, H8JJH'iHe OnOJIHHTenhHhIX no CpaBHeHHIO C ypaBHeHHeM Hepa.3pblBHOCTH (5.1) HCCHnaTHBHOCTh qneHa B npasoA 1.facTH yp8.BHeHHB (5.11) He npH- BeeT K CKOJIb-HH6Yb 3aMeTHhIM H3MeHeHHHM no CpaBHeH1l1O C Knac- CH1JeCKOD 3aqeA <l>anhTp8.l.\HH (5.1)-(5.3). TeM He MeHee, HaJI1{qHe 3TOro tI.HCCHnaTHBHoro :JleHa, rapaHTHpYlOrnero CrJl8JKHBaHHe pe- weHHH Ha paCCTOHHH I, MO}KeT ,n:aTb OnOJIHHTenhHble BbI'IHCJIH- TeJlbHble B03..,10)KHOCTH rrpH MoenHpoBaHHH npOQeCCOB cPHnbTpa- QHH. B KaqecTBe npH..,lepa Hcnonb30- BaHHSI MO,a,HcpHlJ)ipOBa.HHOrO ypas- HeHWI Hepa3pbIBHOCTH pacCMOTpHM cnyqaft nJlOCKOpH8JIhHOrO <l>HJIb- TpaQHOHHoro nOTOKa - 1KHKOCTb npHTeKaeT K rH,.DJ)OAHHaMH'JeCKH co- BepmeHHOH Ao6b1BaJOea 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»HlIhToHRoro 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>IIJlbTpR: 1 - TO'DIoe pemeHBe, 2 - pemeBHe n& OCBOBe 38J{&'1H (5.1)-(5.3),3 - pemeHHe 3aqH 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 C1IeYIOll\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 MoenHM: KJlaccHqeCKOa 3aqH <l>HnbTpal.{HH (5.1)-(5.3) H C MOJ(H<I>HU-HpoBaHHbIM ypaBHeHHeM Hepa3pbJBHOCTH (5.11) - npoBOHJIQt MeTOAOM YCTaHOBJIeHHR. npH 3TOM o6e CHCTe- Mbl YPaBHeHHA pelll8JOTCB no SlBHoA creMe C n:eHTpanbHblMH pa3HO- CTBMH OTHOCHTenhHO aBneHHSI. CKOpOCTb 3aTeM HaxOHT H33aKO- Ha )]:apcH, OJIOTHOCTb - H3 ypaBHeHHH COCTOBHHH. HCDOJIL30BaHHe SlBHOA cxeMLI B 3HatfHTMbHOR Mepe o6YCJIOBJIeHO nOCJle.AYlow.HMH nJlaHaMH peweHH 3aq <l>HJlbTpHH Ha MHoronpoeccopHblX BbI- HCJlHTe.nbHblX CHCTeMax. * Ha pHC. 28 H 29 npeCTaBJIeHbI npoQ>W1H aeJIeHHSI H CKOpo- CTH, nOJlyqeHHLIe Ha OCROBe T01.fHOro peweHHSI (5.12) H qMCJIeHHO- ro peweHHSI 3aq (5.1)-(5.3) H C MOH<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 peryJIjlpH3YIOero 'UIeHa 003- BOJlBeT 3Ha'lHffJIhHO YJIyqWHTb Ka'leCTBO pacqeTOB no cpaeHeHHIO c HCJIeHHblM pewellHeM HCXOHOA 3aqH (5.1)-(5.3). npoAOn>KHM paccMOTpeHHe B03MO)l(HOCTH BBe):\eHHSI nO,lJ.06HOrO poa perYJlHpH- 3YIOUUtX ttJleHOB Ha npHMepe ypasHeHHSI, OnHCblBaJOero nOBeeHHe ONlO'lOCTHqHOA <PYH pacnpe,neneHHSI. 2. B03MO)f(HaH MO,l(lfcPHKaIJ.JfH yp&BHeHHH BOJ1bMaHa 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 pacneneHHoA n8MHTH (rn. IV). 
95. 0606Il.{eHHJJ n& ,I{pyrRe 3aJ;tAtfH MexaBHKH ClLIlOUlHOR Cpe.L(bl 147 HOCTSlMH ,nna ypaBlleHHSI BOJIbMaHa HMeeT B 1/+1 - II + { J/+ 1 - 1;-1 = Axl{1 J/+1 - 2// + JLl + llt 2x 2 x2 (5.13) + J(j, I) · 3J1:eCb SI npOCTOTLI H3J101KeHIUI pacCM8.TpHBaeTCH npOCTpaHCTBeHHO- oHoMepHblA CJlyqaA. 3a.nHMCH BonpocoM: OrpaHHeHHe CHH3Y Ha p83Mep jlqeA:KH x CB3aHO TOJIbKO c HMelOMHCSI B H8JIHtJHH BbI'IHCJIHTenbHbIMH B03MO>KHOCT51MH RJIH COep)KHT onpeeJleHHbl1t CPH3HQecKmt CMblCJI? BCnOllIHt, 'qTO no CBoeMY onpeeJJeRHIO / HMeeT BepoHTHoCTHLIA xapaKTep. 11 AJUI TOro, 'lT06bl <l>H3H-qecKH OnpeeJlHTb BepoBTHOCTb pacnpeeJIeH"$J MOJ1eKyn no CKOpocTSlM J Heo6xo,ll.HMO paccMOTpeTb aHCaM6J1b, COCTOSla. 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 6ecKoHeBO MaJIhlM o&b- eMoM Jt B MexaHHKe cnnoUIHoA CbJ - HeT CMbICJla paCCM&TpHBaTb pa3Mepbl, MeHhWHe '(eM 8x. OqellHM no nOpB,lU(Y BeJlHqHHhI STO pacCToHHHe x. B B03e HOpMaJlbHO:A flJIOTHOCTH, c 'IHCJlOM qacTH B lcw3 n "J 10 19 , xapaKTepHblA p83Mep x 6yeT 110pBAJ(a 10- 6 CM. T.e. cy- 1.QecTBelfHo 60nbme xapaKTepHOrO p83Mepa MOJIeKYJlbJ lo  10- 8 CM H JaMeTHO MeHblUe )VIHHbJ CBo6oHoro 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>epeHHaJJbHoe 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, RanHe OnOJIHH- TeJlbHOrO AHCPCPY3HOHHOro qneH& B npaaoft qa.cTH (5.15) He BHe- ceT CKOJIb-HH6Yb 3a.MeTHblX H3MeHeHHft B 3HaqeHHe Tp&HCnOpTHhIX 
K03c1><PHD.HeHTOB (K03<p<pHHeHTOB BSI3KOCTH H TeIIJIOnpOBO)J.HOCTH) J KOTOpbJe npOIlOpHOHanbHbl AJlHHe CBo60,qHoro np06era l. TeM ca- MhI1 Pe3YJlbTaTbJ. 110000'leHHbie Ha OCHOBe ypaBHeHBSI (5.15). 6ytQ'T COOTBeTCTBOBaTb pe3YJlhTaTaM, nOJlyqeHHbIM Ha OCHOBe ypaBHeHHB BOJIbQMaHa. OHaKO ypaBlleHHe (5.15) 06naeT 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 BblBoe KB83Hra30AHHaMHecKoA CHcTeMbI, OT ,L\H<P<t>y3HOHllhlH qJIeH Ha I{I, 3aMeHBSI 3HaqeHHe II 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 HMelOeMY CHMMeTpaqHhI:A BH. 9To ypaBHeHHe, TaK )J(e KaK H (5.15), CrJIa>KHBaeT pemeHHe Ha pacCTOHHHH x, KOTopoe cye- CTBeHHO MeHbwe AJIHHbI cBo60oro np06era I. HCnOJIb30BaHH€ Mo- JI;H<t>HlI.HpOBaHHblx YPaBHeHHA (5.15) H (5.16) M01KeT OKaJaTbCSl no- Jle3HhIM npH AeTaJIbHOM MOAeJlHpOBaHHH nOBeeHHH <t>YHKQHH pac- npe,neneHHH, HcnOJlb3YIOeM 6onbllioe 'fHCJIO npOCTpaHCTBeHHblX )'3- 1108. 3. K Bonpocy 0 napa,n.OKce 6eCKOHetfHO 6blCTporo pacupocTpl1HeHHH TenJla PaccMoTpeHHble B aHHOM naparpae npHMepbl o6DeHHBeT c KCPC H Kry HaJlHqHe HeKOTOporo MHHHMaJlbHOrO MaclllTa6a. MeHbllIe Koroporo aJCbHeAwaR AeTaJIH3g OrmCaHHB Y>Ke He HMe- eT CMbICJIa. B KCPC 3TO NIHH8, CB060HOro npo6era l H BpeMH T Me>KY CTOJlKHOBeHH.SIMH MoneKYJI, B 3aqax <t>HJIbTpaH 9TO pac- CTOSlHHe I10pSlKa COTeH 3epeH nopOAbi. NIB OnHCaHHH O):(HOqacTH'l- HoA <PYHKHH 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 TenJlOIlpOBOHOCTH 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 BbIeJIeHHH B KOHe'IHOa 06JIacTH. 3jI;eCb T - TeMne- paTypa, x - Ko9<l><l>HHeHT 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 Heo- nYCTHMOCTH c Q:>H3H'IeCKOA TO'tfKH 3peHHSI 6ecKoHettHO 60JIbWoA CKo- poCTH pacnpOCTp8.HeHHSi renna. K 6ecKOHeQHO 60JIbWoA CKOpOCTH pacnpoCTpaHeHKH TenJla npeM 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 TenJtonpoBOHOCTH 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:>opMaH.H nepeaeTCg 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 CJJeOBa- 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 nOpHKa BpeMeHH Me>K,lU' CTonKHOBeHH.HMH MOJIeKyn T. Cne,lJ.OBaTenbHO, B RBHoA cxeMe (5.19) 
150 rn. In. MOAenHposlUlJle TeeJlHR BJJ3KOro r&3a He HMeeT CMblCJIa pacCM&TpHBaTb BpeMeHa I1t no nopy SeJlH'IH- HI>I MeHhWHe, '{eM T. 3SMeTHM, '!TO B peaJIbHOA BblqHCJIHTeJIbHOa npaKTHKe 3TO OrpaHH'IeHHe He npH80HT K KaKHM-JIH60 np06neMaM, TaK K8.K B nnOTHbIX raa8.X xapaKTepHoe BpeMfl H3MeHeHHSI a30HHa- MHtfeCKHX napaMeTpos Ra MHOro rropSlKOB 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 6Ye'r OTCyTCTBOBaTb. 3aMeTHM) qTO HMeHHO SlBHag cPopMa 3a- fiaeR C KOHe'IHo-p83HOCTHbJM npeCTaBneHHeM H3MeHeHH9 ra30- HaMHqecKHX napaMeTpOB no BpeMeHH xapaKTepHa A1JH KB3.3Hra3o- )J.HHrH'-IeCKOH CHCTeMbl (2.1)-(2.4). npHBeeHHb1: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 OHH no,.lI)(O - Lattice Boltzmann* cxeMhI J KO'IOpbIA no CBoeMY nOCTpOeHHIO H no B03MO>KHOCTKM OllHcaHHB ra30HHaMHqe- CKHX npoo-eCCOB C nOMO1O MHOronpOecCOpHLIX BhlqHCjJHTeJlbHblX CHCTeM HMeeT MHOro 06HX 'IePT C KHHeTH'IecKH COrJlacoBaHHbIMH cxeMaMH H KB8.3Hra30HHaMHecKoA: 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 6HJ1bBPJ npeCTaBn.HlOm;ero pB3JleT H CTOJIK- HooeHHe >KHKHX 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 HeH 6bUI cen8JI 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 <!>YHKHH pacnpeeneHHSI. B nocnerorHe robJ Ha6nlOaeTCS1 6W1bllloA HHTepec K aHHOMY nOAXO) BbI3B8.H- HbIR B 3HaqHbHOa Mepe npoCToR a;:taHeH nWlyqeHHblX Ha ero OCHOBe anropHTMOB K apXHTeKType MHoronpoQeccopHhlX CHCTeM C pacrlpe,l{eJleHHott naMHTblO. COOTBeTCI'BeHHO Ha6JlIOa.eTCSf H pOCT qHCJI& ny6J1HHA, OKna;:tOB Ha HayqHhlX KOH<PepeHIUtSlX, nOCBH- w.eHHLIX 8.}IHOA TeMe. t{HT8TeJJHM , 3aHHTepeCOB8HHblM 8 nanyqeHHH 60nee nO,l(p06HOA HHQ>opMHH B tJ;&HHOM HanpasneHHH HCCJIeOBa- ""H, MO)K}IO nOpeKOMeHAOSaTb MOHorpa4>ulO S. Succi [30]. npoeMoHcTpHpyeM B03MO:>KHOCTH LBS-MeTOa Ha npHMepe HaR- '-' DOnee npocTOro H liacTO HCfiOJlb3yeMoro BapH&HTa, onHpalOerocSI Ha npH..feHeHHe BrK-MoenH )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), CBSl3b1BaJOaJl3HaqeHHR <!>YHKnHH pacnpeeJIeHHH 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 KJIaccHlecKoA BrK-MOeJlH (rn. I), - B npaBOA '1acTH (6.1) GTOHT f: q , KOTOpas MO>KeT HeCKOJIbKO OTnHaTbca OT 1M. CMLICJI B8eeHHSI ft q 5YAeT nOHSITeH H3 aJILHeamero I13n01KeHHS. Bblpa>KeHHe (6.1) onHCbIS8eT n08eAeHHe cl>YHKHH pacnpee- JJeHHSI qaCTHl.{ co CKOpocTblO k, KOTOpoe CKJla,rtbIS8eTCSI H3 6ec- CTO.,lKHOBHTeJIbHoro paanem B re'leHHe speMeHH At 83 TO'iKH Xi - kt B TOt{KY Xi H CrOJ1KHOBeHHA "IacTHQ, MOeJIHpyeMhlx ne- HOM (f - fa (Xi, (, t j )). .anB paqHOIl8JlbHOA: OpraHH3&qHH BblqHC- flHTeJILHOrO rIpoecca ecTeCTBeHHO nO06paTb 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 - kt cOBna.i1H c Y)Ke CYecTBYIO- e:A IIpOCTp3.HCTBeHHOA ceTKoA. EcTecTBeHHo J qTO B03MO)KHbI paa- JIHmbIe BapHaHTbl nOAo6Horo poa cxeMbI. Tax, Ha pHC. 30,a H306pa- )l(eHa cxeMa DIQ3 ,!VISI npOCTpaHCTBeHHo-O,a.HOMepHhlX 3B,Aatt, CBSI- 3L1BaroaH 3 Y3JIa npoCTpaHCTBeHRoA CeTKH. Ha pHC. 30,6 H306pa- >KeHa cxeMa D2Q9 NISI 2D-3aq) CBa3bIBaIOaJI 9 y3JIOll CeTKH. <PYHKIHJO f i eq )VI.H LBGK-cxeMbI Bbl6HpaeM B CJIelOrn.eM BHe f q = . ( 1 + ioUa + Qiabtlo.Ub ) ( 6.2 )  {XtJ." c 2 c4 · 3ecb i - HH)J.eKC, 03HaQalOIn:HA HYMepaHIO )'3JIOB Ha npebIAY- eM CJ10e no BpeMeHH, Uti, Ub - 3HaqeHHH M8.KpoCKOnH'IeCKHX CKo- pOCTeA B COOTBeTCT8YJOHX HanpaBJIeHHSlX eKaPTOBblX I{OOpHaT, C - CKOpocTh 3BYKa. BecoBble MHO}l(HTenH Wi, Qiab BhI6HpaJOTCSI H3 YCJIOBIUI, '-IT06hl !je q HMHTHpOB8JlO nOKaJlbHO-MaKCBeJlnOBCKYIO <PYHKHIO 1M Ha co- OTBeTCTBYIOW.eA CCTKe. Ee MOMeHTbI, BbIpa)l(eHHb1e 8 BHe CYMMbI rIO COOTBeTCTBYlOl.l\HM Y3JIaM, AOJI>KHhI COBnaTb c ra3OHHaMHqe- CKHMH rrapaMeTpaMH: EWi - 1 , E WiQio.b 0, 1 , E Wii4 - 0, E Wi{itlQibc - 0, i I L WI{i4ib P6o.b , E Wi{io.ibQif)c PUa Ub · t I npOBOASI anhHe:AwylO ananormo Me>K)U' BrK-MOeJlblO H LBGK-cxeMOH] 06paTHM BHHMaHHC Ha TO, qTO B KaqeCTSe 3<P<peK- 
THBHOH ,lI.JIHllbl cBo60Horo 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 cpeeJ ODHCbIBa- eMoA BrK-MOAeJlbJO (6.1). B03MOJKHOCTb ODHcaHWI 5IBJIeHHA ,D)IC- cHnaqHH aeT OCHOBaHHe ,!VUI MOMHpoBaHHSI c nOMOlqblO LBGK- MOAeJIH npOQeCCOB, I1pOHCXOSlII.\HX B BSI3KOit cpee (30). O)J.HaKo, Heo6xoMO yqeCTb,  lset>, KaK npaBHnO, cyeCTBeB- 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, 6yeT 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 >KHKOCTH CYeCTBYeT O,n;HH 6e3pa.3Mep- HbJA napaMeTp - '1HCJlO PeAHOJJbCa, xapaxTepH3YIOee reqeHHe B l\eJIOM: Re = pUL . (6.3) Jl 3ecb p - nnOTHOCTb) U - xapaKTepHaSi CKOpOCTb, L - xapaxTep- HbIA pa.3Mep, JJ - 3cpCPeKTHBHasI BSl3KOCTb, nOJlrIaeM&sI B paMKax LBGK-MoenH. ,l(ns OnHcaHHB peanbHoro npoecca ocraTO'IHO Tal< H3MeHHTL HCXOHhJe napaMeTpbl 3aqJ{, QT06bl nOJIyqaeMOe QHC1JO PeAHOJlb,ll.- ca COBnaJIO c ero ecTeCTBeHHbIM 3HaQeHHeM, HecMOTpSl Ha OTJIH'lHe cP OT 3HaQeHHSI HaTYPaJIbHOA BH3KOCTH. HeB03MO)l(HOCTb HCnWlb30BaHHSI npocTOA nepeHopMHpOBKH ,{HC- JIa PeRHOJIbCa B 60Jlee 06meM CJIyqae nocnY>KHnO npH'tfHHOA Taro, t.{T() OCHOBHbIe pa60TbI B 06JIacTH Lattice Boltzmann cxeM cocpo- TOqeHbl Ha Mo).\enHpOBaHHH TeqeHHA HeC>KHMaeMOH )KH)I.KOCTH H en&- OOC)l(HMaeMoro ra3a. Q)l:HaKO CJIeeT o6paTHTh BHHMaHHe Ha MOTe- pecHYlO pa.6OTY [77], B KOTOpoA paccMaTpHBaeTCSI npHMeHeHHe LBS- MeToa AJlS MoenHpOB8.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. 06paTH1 BHHMaHHe Ha TO, tlro C BLlqHCJIHTeJIbHoA: TOqKH 3pe- HHSI LBGK-a.nropHTM (6.1) qpe3BbJ'IaDHO npoCT. OH IIpeCTaBJIaeT 
co6oa SlBHyID CXeMY, anTaQHB KOTOpo:A Ha apXHTeKTypy MHOrO- npoeccopHbIX BblqHCJIHTeJJbHblX CHCTeM C pacnpeenHHOa DaMSI- TblO He COCT8.BnHeT oco60ro Tpya. 9ToT cPaKT BBnaeTCB OAHO:R H3 OCHOBHblX npHt.fHH B03pacTaIOef'O B nOCJIeHee BpeMSI HH'l'epeCa K LBS-MeT0AY. KopOTKoe 06cY>K,lI.eHHe LBS-cxeM, npOBeeHHoe 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)KeHHIO aH8JlOrHH Me>KY Lattice Boltzmann CXeMaMH, c oHoA CTOPOHbI. KCPC H KB83Hra30HaMHqecKOH CH- CTeloA, C ,LJ.pyroA CTOpOHbl. 2. KBA3Hra30,l1)lHaMHlleCKaH CHCTeM& If LBS rrpOB€eM CpaBIJHTe.nbHbIA aHaJIH3 KBa3HI(t30HHaMHqeCKOA CH- CTelttlbJ, B KOTOpoA HaH60nee HarJlH)UfO npeACTaBJ1S1JOTCH cBoACTBa KCPC KaK cPH3H'qecK0J1 MOeJlHt H CaMoro nonyn.SlpHoro ee BapH- aHTa - LBS, OCHoBaHHoro Ha Hcnonb308aHHH MoenH BxaTHarapa- rpocca-KpYKa. HanoMHHM, qT() B DeHOBe BblBoa KrY-CI-fCTeMLI ne- )KHT npenOJIO>KeHHe 0 6eCCTOJIKHOBHTeJIbHOM paaneTe 1.Ia.cTH B Te- qeHHe BpeMeHH t = t i + 1 - t j H nOCJ1eAYIOIQeA MrHoseBHoA: MaKC- BennH3aHH qaCTHI\ npH t = t j + 1 (rn. II,  5). npeno.nO>KeHHe 0 6eccronKHOBHTeJIbHOCTH pa3neTa npHBoT K CJle,l\YlOeMY COOTIlomeHH1O .n.ng 3Ha'leHHA <l>YHKHM pacnpe,n:eneHHSI }Ia tOMeHTbl BpeMeHH t i H t j + l : "+1 . f{x,E,tJ ) =fM(x-Et,,tJ). I1MHTaHSI npoecca CTOJIKHOBeHmt npOHCXOHT 3a CqeT TOro t liTO cPYIIKU.HH pacnpee.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 nOMO1O BrK-uoeJlM, 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 cHoBeHHA AJI5I t E [t i , t;+l] c nOMOblO HHTerpa.n& CTonKHoBeHHa, 11PHCY- [Qero ypaBHeHHIO BOJlbMaHa. 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. Pa3JlHe 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 npoeyp OCeCTBJIB- 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 ra30HH8.MHqeCKHX napaMeTpOB. AH8J1HTH'Ie- CKHR nepexo D03BOJ1S1eT HCDOJIh30BaTb B Ka:qecTBe MacwTa.60B, OIIH- CblB8.JOlII.HX ,lJ.HCCHnaTHBHble npol\eCCbl, peanblIytO NIHHY cBo60Horo npo6era I H BpeMSI Me)K CTOJIKHOBeHHHMH MOJIeKyn T. fIoMHMo cnoc06a nOCTpoeHHH, Ba..>KHOR o6e:A '1epTOA CXeMbJ LBGK Ii Kry -CHCTeMbI SlBnHeTC5I HaJJH'lHe MHHHMaJIbHOrO Mac- UlTa6a, Ha KOTOpOM rapaHTHpyeTcSI OTCYTCTBHe cenbHbIX BapHaa 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 peaJIH3aHH. 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 COBnaJOT C SlBHblMH anrOpHTMaMH (rJI. IV). KaK H3BeCTHO, HMeHHO BBHble CXeMI:d Ha.H6onee y06HhI K 8.AaUTa,QHH Ha MHOf'OnpOneccopHble BbI'IHCJIHTeJJbHble CHCTeMbI c apxHTeK'rypoii pacnpeeJIeHHOa naMSlTH. BblcoKonpoH3Bo,a.wrenbHbIe MHoronpoeccopHble 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 nOBep)K):\aeTCSI 60JIbWoA BbI'lHCJlH- Te.nbHOH npaKTHKoA HCDOJIb30BaHHB LBGK H KrY). COt.leTaHl1e npHHlJ;HfiHaJIbHOA B03MomHOCTH MOMHpoBaHHSI 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 nOCJIeIOe- ro YCOBepmeHCTBOSaHHjI K8JK)J.Oro H3 nepeQHCJleHHblX nO)J.XO,l).OB.  7. MeTO cT86HJlR3aOHHLIx nonpaBOK H HeKOTophle Bapa8HTbI Kry -CHCTeMhl B 9TOM naparpa4>e, 3a.KJIJOQHTeJIbHOM ,nnsr AaHHOA rnaBhI, pac- CMOTpHM pH,Q MeTOAOB H MoeneA, KOTopbIe npeCTaBnSlIOTCSI OCTa- TOtlHO 6nH3KHMH no cBoeA H,l\eOJlOrHH K KCPC H KB83Hra30HHaMH- qeCKoA: CHCTeMe. 9TH MeTObl H MOAeJlH po,nHHT C npe,1VIO>KeHHbIMH paHee KCPC H KfY HanHIHe MHHHMaJlbHhlX MacmTa60B, Ha KOTO- pblX rapaHTHpYeTcg Crna>KHBaHHe peweHHB, BCJIeCTBHe qero OHH OKa3aJ1aCb Y06HhlMH 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.6HJIH3aHOHHbIX UOnpaBOK 9TOT MeTO,lt y06Ho paCCMOTpeTb Ha npHMepe CTaHOHapHoro ooMepHoro 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><PHeHTbI 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 Be, c Y'leTOM 0603HaQeHHA tl-H<txPY3HOHHOA H a,ltBeKTHBHOA COCTaBJIHIO- lli,HX IIOTOKa: q(x) = [utp](x) - q(x-h) - [uJ(x - h) - 1 (7.2) - 2 [Q(x) + Q(x-h)] h = O. npH BblBoe 3Toro COOTHoweHHSI HHTerpa.n OT HCTOqHHKa Q npe,D;- cmBneH B Be <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[uJ 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) IIocTaBJlSlSI 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 nocneHee Bblpa>KeHHe B 60nee KOMnaKTHOM BHp'e hdr T---=O O<x<l. 2dx' · (7.6) re 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 KOHBeKH-,lI.H<p<PY3HH (7.1). 
C\lHTag <I>ct>Y3HOHHbIA nOTOK H HCTOqHHK MaJJhIMH H OT6pa- CbJBasI I1X COCTa8njllOHe, nponopqHOHaJIbHble h, npH):\eM K CJIe,lO'- lOeMY ypaBHeHHIO nSl OnHCaHHjJ npou.ecca: _TJ d +  ( k dcfJ ) + Q + h  ( U d ) == o. dx dx dx 2 dx dx B CYHOCTH no CpaBHeHHIO C HcxooA 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 KaeCTBO 1.IHCJleHHo- ro pClIIeHHH nOTBep>KaeTCSI MliorO'lIICJIeHHbIMH npHMepaMH TecTO- BblX pac'-teTOB. 06paTHM BHHMaHHe Ha CBH3b TOIIbKO.1fTO paccMOTpeHHoro no- xoa C MOAH<I>HqRpoBaHHoft nOCTaHOBKoR NISI 3a..n.a'IH c:PHnbTpal.{HH (5 aHHoR rJIaBbI). B MOAHcI>HpOBa.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 CTa6HJIH3HPYIOee BJlHSlHHe Ha pe3YJIbTaT HC- neHHblX pac'leTOB. 9Ta CTa6HnH3SI -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 npOHCXOHT 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 cBo6oHoro npo6era l, .RBnSIeTC OCHOBOA )VIS pac- CMOTpeHHH Kry B xatleCTBe MoenH NISI OIlHCaHHH Te'leHHSI BSI3KOro raaa. O,l\HaKO Me>KJ1Y cra6HnH3aOHHOii 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 UeHHH. 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 KBa3Hf830HaMHqeCKOR CHCTeMOA cye- CrrBYIOT ee 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 3TO1 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, onpeenHeMoe 3HaqeHHeM CKOpocTH H warOM CeTKH Bonb 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 pacnpeeneHIDI Ha npOH3BOJIbHOA j1eAKe (rn. II,  5). 3ecb 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. nO06Ha.sI rrpoQeJQ1pa n03BOJIHJla nepeATH K AOCTaTO'lHO yo6- 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 rHpoHHaMHqeCKHX npc>- eCCOB. I1pH aTOM nOJlYQaeTCS1 CHCTeMa ypaBHeHH1t, HMelOaH 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 norocoa, T8J(>Ke pOACTseHHble Kry- CHCTeMe H npHMeHSleMble AnH OIlHCaHHB )J,BlDKeHHSI BR3KOA >KHKO- CTH. 
2. KBa.3HrUAPO,l1.HH3MHtJecKaH CHCTeM& ypaBBeHHR B pa60Te [SOJ IO.B. llIepeToBblM 6bJJIa BBeeHa CHCTeM& ypas- HeHHft, KOTopaa no aHaJIOrHH C KfY-CHCTeMOA 6b1JIa Ha3BaHa KBa- 3HrHlU>OI-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: + ai (Ui(E+p») + div q = pFi(Ui - Wi) - - ()j (UtPik) + ai (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 onpeeJIeHa Qepe3 BH3- KOCTb, AaBJIeHHe H pyrHe MaKpocKORHqeCKHe napaMeTphl cpe (80) 81]. 06paTHMcH K CMblCJIY bBeeHHSI BeKTOpa w. ,l\nH 3Toro nepeOH- meM YPaBHeHHe (7.11) B BHe (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 npeonOiKeHHIO, Ha 
g7. HeKOTOpl:Je SapHatfTbl Kry-CHCTeMbl 161 KOTOpOM CTpOHTC.H KCPC H KBa3HI'830HaMHqeCKaJI CHCTeMa, - CJIa60MY H3MeHeHHIO <l>YHKQHH pacnpeeneHHg H ra30JI-HHaMHeCKHX napaMeTpOB B 06JIacTH, pa3Mepbl KOTOpoA no nopBKY BeJIHqHHbI pasHbI nHHe CBo6oHOro 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- 3HrHpO,[lHllaMHqeCKOa CHCTeMe, K cPaKTOpaM, npKBO.L\M K ero H3feHeHHIO 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 ai (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 CeT ,A06aso'iHoro nepeHoca MaCCbI a:i (Pwi + f1Ui U I; W k) · YqHTbIBaJI npH H3MeHeHHH HMnynbca H nonHoft 3HeprRH H AO- nOJIHHTMbHbJe qneHbJ 1 npHeM K yp8.BHeHHHM (7.12) H (7.13) KBa- 3HrH)J.poHH8.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), onHCbIBalOHX 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 npeCTaBJlgJIOCb 8 BR)].e 8S) + 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])HOH- HblM BnHiI}IHeM Tennonpo80HOCTH H BH3KHX can, ocyecTBnSleTCB 3a CqeT OnOJlHHTeJlbHOro nepeMelIIHBaHHH pw2/T B o6nacTH c xa- paKTepHblM pa3MepOM nopjJKa ATIRHbl CB06oHoro rrpo6era. 3aMeTHM t QTO eKBaTHoe OfiHCaHHe H3MeHeHHH 3HTponHH OT- MeQanOCL ele B nepBblX pa.60TaX, DOCBeHHhIX KCPC, 60JIb- MaHOBCKHM cxeMaM H KBa3Hra30,ll.HHaMOqeCKoit CHCTeMe ypaBHeHHR (24, 28, 35 J · B OTRHQHe OT KrY-cHCTeMbI, KBa3HrpO,l1.HHaMHqeCY..a.H CHCTe- Ma u.enHKOM, BKJIJO'IaJI npOH3BOhle 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 KBa3HrHpoHaMH'IecKoA CHCTeMhl t OTMe- THM, TO OHa XOpOlllO 3apeKOMeOBana COOH "PH paccMoTpeHHH paanHQHhlX Te1.JeHH:A BSl3KoA )KHKOCTB, B nepBYIO oepeh, KOHBeK- THBHblX TeqeHHA [81 J 83]. 3. KBa3Hra30AHHaMHllecKaH MO,l1.e.nb)1.JIR onBC8HHSI TeqeHHH CJIa600KHMaeMblx ffi30B H )l(B).UCOCTeH BapH8HT KBa3Hra30aMHqecKo:A CHCTeMbI ,lVllI Mo,neJIHpOBa- HHjI TeqeHHI cna60C>KHM&eMoro raa& H )KH,a.KOCTeft MO>KeT 6b1Th no- nyqH 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. OaKo B HoHapHo" CJI)''Iae OTJIH&JHe Me*A)' AH<IJ<I>epeH- lJ,H&nbHhlM aHaJlOrOM Kry H ypaBHeBHJIKJI Haabe-CTOKCa ODHCTCJ[ )')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 OHOqacT"q}fOa <PYHKHH paCnpe)];eJIeHHB (84J. nH 3TOro enaeM CJ1eAYIOIInle npeOJIO)KeHHg, CXO>KBe C HcnOJlb3ye- lbIJvtH rIpH BblBOe KCPC H Kry: 1) Ha Bpe(eHHOM CJ10e t = t J <t>YHKg pacrrpeAenesH.Sl J BCIOA}' HMeerr JIOKaJlLHOe paBHOBeCHoe 3HaqeHHe 11'; 2) cPYHKHS pacnpeeHHH CJIa60 MeRjleTCH 3a BpeMSI T H Ha paCCTORHHH " onpe,a.enHeMOM K8J< 1 = rU, re U - xapaKTep- Hoe 3HaqeHHe CKOpOCTH MoneKYJI; 3) B MOMeHT BpeMeKH t j + 1 = t j + T <PYHKHH pacnpe,neneHHH MrHOBeHHO npHHHMaeT paBHOSeCHoe 3HB"IeHHe. 3,n:ech T BbICTynaeT B KalecTBe a.H8JIOra. BpeMeHH Me}l(AY CTOJ1K- HOBeHHHMH , KOTopoe AJl5I UJIOTHLIX f830B onpenHeTCSI c nOMO1O BR3KOCTH I-'t KHHeTHqecKoR COCTaDnmoeA aBneHH PK H KOHCTaH- Tbl CKOpOCTH peaKI:(HH 9A:pHHra K34» CJIM 06p83oM [1]: J1. = PK/K, r = 2/K* = 2JJlpK- (7.18) 3aMeTHM , QTO 3AeCb MbI He npHBH3blSaeMCSI K npeonO>KeHH1O 0 6ecCTOnKHoBwrenbHOM xapaKTepe p83JleTa npH t E [t'. t j + 1 ]) OrpaHH- lJHBaHCb, KaK H npH BblBo,u.e KrY -CBCTeMbI, npHBeeHHOa B 3aMe'fa- IIMH 1  3 ):taHHoA rnasbl HH<I>opMeA 0 TOM, QTO <l>YHH.R pacnpe- AeneHISI npH t == t i H t = t j + 1 npUHHMaeT JIOKallhHo-pasHosecHoe 3HatJeHHe. B C,LJ.ena.HHbIX rrpeOJlO}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), npeCTaBHM 3HaqeHHe <PYHKll)IB fp If HHTerpa.na cTOmcHoBeHHA: J B BHe 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 a2kf"fp T 2 a2kGl/m · fp T 2 ar GI;Glf p + - + - + -  ( 7 20 ) 2 OX/t:OXl 2 aXlta1 2 8ka{f m 2 · T [ '+1 . ] T2 oj 3 + 2 J(x,(,t ' ) + J(x, (, t.1) - 2k aXk + O(r ). qTo6b1 nOJ1Y"lHTb KBa3Hra.30R8.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- = --- mkJdE = 2 OXk OXk 2 a T 2 a r 2 FPPib = _ 8 _ 2 _ 8 (P.)kl = a 2 · Zit Xl X/c nOCJIeHee paBeHCTBO (7.22) cnpase,lVlHBO no npH'IHHe TOro, qTO )VIa paBHoBecHoR CPYHKHH pacnpeeJleRHg 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 npeCTaBJ1BeT 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(HKOCTSlX, 3aK.nlOqaJOIlJ)IACSI B HaJIH'IHH enOqKH nOCJIeOBaTenb- HbJX CTonKHoBeHH:A. 9TOT MexaHH3M ocyw:ecTBn.sreTCH npH HaJlH'lHH 60nee HJ1H MeHee O,nHOpoOa 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 <PYH pacnpeMeHHg, 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) - Tk- 8 -  + Xk m Uk r 2 82klfp (1  ) 1"2lPUf'P T2lPk(G,/m) /1J + - · - Ukl + - + - (. + 2 OXIcOXl 2 8x 2 aXka1 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, onHCbIBaJOHe ecreCTBeHHYIO B- a T 8 KOCl'b H TennonpoBOOCTb. HanpoMep, 'IJleH W::- 2 (PKElp), ax, ax, CTO1I.I,Ha B npaooR qaCTH ypaBHeHHH (7.25), MO>KeT 6b1TL npeCTaB- neH, c yqeTOM (7.18), 8 Be (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 nOTBep)l(AeHHe npHBeeM pe3YJIbTaTbJ conOCTaBneHIDI pacqe- 1'OB Ha OCHOBe YPaBHeHd (7.23)-(7.25) H OnHpalOIQHXCSI Ha HCnOJlb- 30BaHHe ypasHeHHA: Hashe-CTOKca (85] ,LVISI 3a,{H, OnHcOIBarorneR TeqeHHe HeC)I(HMaeMo:A >KKOCTH B KaBepHe c HJlCyeACSI BepxHe:A KpbIillKOA (pHC. 31). Ha pHC. 32 npHBeHbJ AaHe pacqera sepTIt- K8J1bHO:B (Bt);OJIb OCD Y) cocTaBnHIOeR CKOpOCTH B eHTp8JIhHOM ro- pH30HT8JlbHOM Ce'leHHH KaBepHbI, pacCHTaHHble 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, npOBeeHHLlX 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 nOATBep8eT npaSOMOqHOCTL HC- fIOJIb30BaHH.Sl npeO)l(eHHOa CHCTeMhI YPaBHeHHA. BepHeMCSI BHOBh K KBa3HrpoHHaMH'IeCKOa CHCTeMe, pac- cMOTpeHltoA B npebleM 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 cOBneHHe 3THX ,l);MCCHUaTHBHbIx qneHOB J BKJIJOQaB 3aI1HCb )\&BJIeHH5I B oHe 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- MOHHaMHqeCKHX COO6paJKeHHft, a. BO BTOpoM - HenOCpeCTBeHHO 1'13 KHHeTHecKoro ypa.sHeHHSI 9HCKOra. * * * aHHaH rJIasa 3aK8J1qHBaeT 06CY:>KAeHHe KCPC H KBa3Hra30- AHHa.MHqeCKO CHCTeMhl KaJ< <!>H3HQeCKOA MOenH AlI5I ODHcaHHSI Te- 1.{eHH BB3KOro ra.3a. rn8.BHblM B 9ToA MOeJIH 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 CPYliKHH pacnpeAeJ1eHHSI K ee paaHoBecHOMY 3HaQeHHIO. EcTeCTBellHOCTb H <PH3H'tIecKaSI HenpOTHBOpeqHBOCTh 3TOrO JJ.o- nOJII1H'reJIbHOrO Tpe6088.HHg Ha CrJlaJKRBaHHe peIllCHHH npHBOHT K TOMY, tfTO pe3YJlbTaTbl t ItOJlyqeHHhJe Ha 683e HOBOR MOMH, IIpaK- THqecKH He OTnH'iaIOTCSI OT c00l'BeTCTBYIOHX pe3YJlbTaTOB , nOJlY- qeHHblX Ha OCHOBe ypaBHeHHA HaBbe-CTOKca. OAHaKO 3m Moenb o6na.n.aeT DO CBoeMY nOCTpoeHHIO BHYTpeHHeR KOppeKTHOCTblO, qTO n03BOJISleT (rn. V) ycnelllHo HCDOJIb30BaTb ee NISI MO,l\eJIHpOBaHHB CJIO)f(HblX rHAPO- H. ra30HHaMHqecKHX 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 MoenH. IIo cy'rH ,lJ.eJIa. B paMKaX npeA1laraeMoA MoenH BBOAHTCH MH- HHMaJIbHbIA MaclllTa6 (,lJ.JlliHa cBo60Horo npo6era), MeHhWe KOTOpo- ro c <P1i3HQecKOA TOQKH 3peHHS AeT8JIH3al\HB peweHHH OBHTCSI 6eccMblCJJeHHolt. MaTepHaJlbI  5 H nOCJIe,n}'lOx naparpa<poB no- Ka3bIBaIOT, t.JTO npHHl\Hn BbIeJleHHS MHflHMaJlbHblX MacuJ'ra6oB H nOC1'poeHHe MOeJIeA, rapaHTHpyfOX Crna>KHBaHHe peweHHSI Ha HHX, HBJI5JeTCS B OCTaTO'lHOa Mepe yHHBepC8nbHblM. lfMelOTCH H APyrHe MH3KHe not\XO,lJ.ld K nocTpoeHHIO TaKHX MOeJleaJ 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 cpebJ, re npo6neMa KOppeKTHOCTH, CBSl3aHHaJI C BbI- ,l).eJIeHHeM 4>H3HQecKHx H DOAaBJIeHHeM CqeTHbiX HeYCTOAHBOCTea) 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 CJleYlOHX rnaBax IV H V 6y,lQ"r HenocpeCTBeHHO paccMOT- peHbl Bh1QHCJ]HTeJlbHhJe aJIropHTMLI Ha 6a3e KCPC H Kry, HX an- TaJ..\HSI K apXHTeKT)'pe MHoronpou.eCCOpHbJX CHCTeM C pacnpeeJleH- Hott naMSITblO H npHMeHeHHe M'HX a.nropHTMOB NISI MOAMHpOBaHHH Pa3JIHQHbIX 3aq re)Wo- H ra3OBOR HaMHKH. 
rJIaBa IV BbIqHCJIHTEJIbHbIE AJIrOPHTMbI II HCnOJIb30BAHHE MHOrOIIPOQECCOPHbIX CHCTEM JI PEmEHH 3AAq rA30BOa HHAMHKH B aHHOa rnaBe 6y pacCMOTpeHbI HeKOTOpbJe BLlqHCJlHTeJlb- Hble a.nropHTMbI ):VISI MoenHpOBaHIDI 3a)\aq BH3KOrO ra3a H >K- KOCTH, nOCTpoeHHe KOTOpblX cyeCTneHHbJM 06pa30M O:JHpaerca Ha onHcaHHble B 11peb1.l\Y1U)IX Pa3AeJ1ax KCPC H KB83Hra..10HHaM"qe- CKYIO C"CTeIY ypaaHeHHA. Oco6oe BI-IHMaHHe yenHeTCH B03MO>KHO- CTH a,L\an"H 3THX MeTO)J.OB K apxHTeKType MHoronp'JeccOpHblX BbJCJJHTe.nbHLlX CHCTeM c pacnpeeJleHHo:A fiaMBTblO. JIorettecKaH IIpoCTOTa MHorHX aJIropHTMOB, nOCTpoeHHbIX Ha 6a3e KCPC H Kry, ,nenaeT HX YA06HbIM HHCTpYMeHTOM npH peweHHH 3aq rH,lWO- H ra- 30BOft AHHaMHKH Ha COBpeMeHlIblX BblCOKonpoH3BO,ll.HTeJIbHbIX MHO- rOllpOCCCOpHbJX KOMnJleKcax. B CBOIO Oqepe,nb, peaJIbHaJI B03MO)K- HoC'rb MOeJIHpOBaTb c nOM01U.b1O KHHeTHqeCKHX cxeM Ha MHoronpo- QeccopllblX CHCTeMax 3aqH COBpeMeHHoA a3pO,l],MHaMHKH craBHT Ha nOBeCTKY jJ)IH peUIeHHe pSla npo6JJeM napanJIeJIbHoro MaTeM&THqe- CKoro o6ecneQeHWI. 06 3'1'OM TaK>Ke nOeT pe'lb 8 AaHHoA rnaBe. PacCMOTpeHHe MaTepH8Jla Ha'tJHeM C ormcaHHSI KHHeTHqeCKH CO- rnaCOBaHHblX cxeM C KoppeKeA) KOTOpbJe 6blJlH IlepBblM anropHT- MOM 113 KJIacca KCPC, npHMeHeHHblM ,lJ.JIB MOAeJ1HpOBaHI1S1 TeqeHHR B3Koro ra3a. 
 1. KUHeTUqeCKH cornaCOBaUH&Ie cxeMLI C KOppeKIJ.HeA Ha3B3.HHe MeTOa "KCPC c KoppeKHeH' no,a.qepKHBaeT CBSI3b Me KHHeTHqeCKH cornaCOBaHHOH cxeMoft H KB83HraaOHHaMH- recKoA CHcTeMo.R ypaBHeHHA, K KOTOpoA npHMbIKaeT aHHhI.A aJI- ropHTM. npH nOCTpoeHHH 3TOA cxeMbl 6bU1a HCn0J1b30BaHa CBH3h Me>KAY KCPC, Kry H KJlaccHQeCKHMH MOeJljJMH) DonO>KeHHbIMH B OCHOBY OnHCaHHSI BH3KOrO ra3a [57]-(59J, [88]. 1. Pa3HOCTHble cxeMbl fiB, apBMoyrOJIbHblX CeTKaX OCHOBH&g H)J;e51 KHHeTHCofecKHX cxeM C KoppeBeA OCHOBaHa Ha BblAeneHHH 8 KBa3Hr830AHHaMHQeCKOA CHCTeMe 8 BBHOM Be q;re- HOB ecrecTBeHHOa BSl3KOCTH H TerIJIOnpOBO)];HOCTH, npOnOpHOHaJlb- HLIX ,D)IHHe cBo6oHOrO 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. CTOjJHe B npa- BbIX 'IaCTHX p npeCTaBlIeHbJ B <l>opMe CYMlbl ecTeCTBeHHblX H o- nOJIHHTeJIbHblX P.HCCHnaTHBHblX qneHoB. nOCJIeHHe, 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>HHeHT eCTeCTseHHoA TenJlOnpOBOHOCTH, [{)Pil:/8xk] - pa3I-IOCTHaJI annpOKCHMH1I eCTecTBeHHLlX BSl3KHX Ha- npfl>KeHHH, [8/8xk(UiPik)] - pa3HOCTH annpoKCHMaHSI pa60TLJ BSl3KHX CHJ1. CKOpOCTb U onpeeJlgeTCH 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 6YeT 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 nOpJlKOB npeOblwa- eT BpeMB Me>KAY CTOJIKHOBeHHBMH MOJleKYJI T. ,[{aHHM onepHj{ ,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)bMaHOBCKHX CXeM, Heno- cpeCTBeHHO 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, nocBeHHhlX HCnOJIb30BaHHIO KHHeTH'lJeCKHX cxeM NISI MOAeJIHpoBaHHH Te'leHHft BSl3Koro r83&, OT- MeTHM CTaTblO [90]. 2. KCPC C KoppeKlQlei1 H ypSBHeHHH HaBbe-CTOKca npOBeeM oeHKY 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 6yeM 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) nOpSlOK "IJleHOB (1.7), HMelO1IJ)IX CeTO'1Hoe npOHCXO:>KJJ-eHHe, CJleAY- 1OIIJ)f A : [ (pu3);:] <  [pu2] , (1.8) [w (3PU)x] <  [P] · (1.9) 3ecb 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 KOppeKHeJJ 175 H3 ( 1. 7) - ( 1. 9) BbJTeKaeT, qro CeTO'tfHble ,l(06aBKH B n zx MaJlbI no cpaBHeflHJO c COOTBeTCTBYIOMH KOHBeKTHBHbJMH T.fJIeHaUH. AnSI BHeHaroHanbHOa KOMnOHeHThi nOJIHoro TeH30pa HanpSl>Ke- HH:f.t, HanpHMep, II%y = fYUV -  (puv 2 )y - IIUj , (1.11) nOJlyqaeM CJ1e,ll.YIOIDyJO Oel)KY 'iJIeHa CeTOQHOro npoHcxoeHHH: [(puV2)y] <  [puv] . (1.12) OCTaHOBHMCJI Ha nOCJIep)leM qneHe (1.11), OTBeTCTBeHHON 3a ecre- CToeHHYIO BH3KOCTb. IIpH nepexo,ne OT KB83Hra3H"&MH'Iecxoa CHCTeMbl K KCPC C KoppeKIUleA 'Jnet) (r(pu)j)1I TPaHC<l>oPMHPOSaJICft B (Uj)". npH :iTON cyecTBeHHO 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 Kora Ba3KHe 1fJleHbI DO nopKY BeJlH'IHHLI conoCT8- BHMbl C KOHBeKTHBHblNH. B HblX YCJIOBH,.X OTHomeHHe BTOporo cnara- eMoro npaBoA qacTH (1.13) K KOHBeKTHBHbIM '1J1eHaW 6YeT 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)KeHHe, 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 nopa C KOHSeKTHBHhlMH, TO qneHhI CeTO"tJHOrQ ilpOHCXO>K)J;eHBSI MaJIbI no cpaBHeHHIO C TeMH H APyrHMH, T.e. npH P83YMHOA: HCKpeTH3aH (1.10) He HCKa>KaIOT Ka})THHY Te- qeHH.H . B 9TOM KpoeTCSI CYecTBeHHoe OTJIH'tfHe anropHTMa KCPC c KoppeKHeft OT MeTOOB pemeHHSI ypasHeHHA HaBbe-OroKca, B KO- TOpbIX KOHBeKTHBHble qneHhI annpOKCHMHpYIOTCB HanpaBJIeHHblMH (1.13) 
P83HOCTHMH. B nOCJreeM CJIy"tJae B03HHKaIO8$I CXeMHRH B3KOCTb 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 nOpgOK BeJIHtfHHhI CCHnaTHBHbIX qJIe- HOB B YpaBHCHHH Hepa3phlBHOCTH (1.1). Arts 9TOro l(peCTaBHM HX B Be ((pu2 +Ph)x = ((pu2)i)% + (*px)%. KaK cneA)'eT H3 ypaBHeHHH (1.1), nopOK Pi 11860 paBeH nOpSI)J.KY KOHBeKTHBHOrO "IJIeH& (pU) 0 , nH60, eCJIH KOHBeKjI HMeeT OAHHaKo- % BbIA nOpHOK C ecTeCTBeHHOH CCHl1aqHea, paoeH nOpSl,l\KY BjJ3KHX qnefJOB. To ecTb B JI!060M cnyqae HNeeT MecTO oeHKa (1.14) %] < [ 7x 2 ] · (1.15 ) B CBOIO Oqepe)J.b J tVJH pu2J Lx CnpaBe,lVIHBa C1teAYIOag OeHKa: [( (pu2)x) z]  %; [E] · ( 1.16) 113 ( 1.15) H (1.16) BbiTeKaeT, \fTO qJJeHhI, HMelOlIJ)fe CeTOqHOe npoHcxoeHHe, MaJIhJ no cpasHeHHIO C 'IJIeHaMH, CTOSlHMH 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 IJOpKY BeJJH'iHHhI 3H&'lHTeJlbHO npeBbI- WaJOT KOHBeKTHBHbIe. npHcTynHM K ou:eHKe t.fJIeHOB yp&BHelIHH (1.5), onHCLIBalOero H3MeHeJIHe fiOJlHOA 9HeprHH. HaqHeM C oeHKH qneHOB BHa [( u2( E +  p)%) z]  %; [i: (E+p)] · ( 1.1 7) TaKHM 06pa30M, YQHTbIB8J1 MaJIOCTb rrapaMeTpa x/ L:£, MO>KHO YTBep)l(aTb) 'iTO COOTBeTCTBYJOe eHbI ceTO'fHOrO npOHcxoe- HRs( MaJlbl OTHOCHTeJlbHO KOHBeKTBBHblX 'IneHOB. 
.. 91. l\llHeTJfT:JecKH COrJIaCOB&HHble cxeMbl C KoppeKIVIen 177 nepeA,n.eM K OeHKe ,rq>yroro 'IJleHa (1.5), He CBSI3aJ1HOrO c ecTe- CTBeHHblMH BH3KOCThlO H TennonpOBO,lJ)lOCTbIO: [( ; hl)p PX) J < = [CPz] · YttHTblBaH pauee IIpOBeeHHYIO oeHKY (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: CeTOrnhlH qneH npH He CJ1HmKOM MaJlbIX MaKpOCKOllHQeCKHX CKOpOCTBX no nOpH,l\KY BeJIH'fHHbJ OKa.3bIBaeT- SI MeHbllle COOTBeTCTBYIOero KOHB€KTHBHOro t.tJleHa B ypaBHeHHH '-)l-leprHH. 3aMeqaHHe 1.4. H3 oeHKH (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,qBOB HTOI'M npoe.naHHblM OeHKaM) MO)l(HO I'OBopHTb 0 TOM, u YTO npH MOeJ1HpOBaHHH cmOJl038YK06WX U C8epX36Y1C08b£X meeHUU (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 HCCHIJaH nep- Bora nopSl,n:Ka MaJlOCTH, 06blQHO HeraTHBHO BJlHHIOee Ha peweHHe ypaBHeHHA HaBbe-CTOKca, 8 ,It.aHHOM CJlyae He HCKaJKaeT <PH3HQe- CKYIO KapTHHY TeqeHH. Kax 6blno nOKa3aHo) 3TH qneHbI MaJlbl no cpaBHeHHIO c ecTecTBeHHblMH CCHnaTHBHblMH qJ]eHaMH. B nOTBep>K,lJ.eHHe C)J.eJlaHHblX Ol.\eHOK npHBeeM ,lI.Ba npHMepa pacqerOB TpaHC3BYKOBOro H csepX3BYKOBoro TeqeliHA BH3Koro C>KH- 1aeMoro ra33. 9TH pe3YJIbTaTbI, nOJIyqeHHble c nOMOblO KHHeTHe- 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. 3a'tJa paccMaT}>HOanaCb B DpocTpaJICTBeHHo- tJ:BYMepHoA nOCTaHOBKe. IIapaMeTpbJ Ha6eraJoero nOTOKa B03xa 
,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 BOJJb 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 3aaB8JlCSl 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 3a'l{H Ha OCHOBe HcnOnb30BaHHH cxeMhI TpeTberO nopKa TOQaOCTH JUI YPaBHeHH:A Hasbe-CToKca (92) (pHC. 34). B KaqeCTBe BToporo npHMe- pa paccMoTpHM ,n:&HHble pacqe- 1"8, TpaaC3BYKOBoro TeeHHH 803- .L\Yxa 86JIH3H npOCTpaHCTSeHHO- ABYMepHoR KaBepHbI (pHC. 35). TIa.paMeTpbI Ha6eraroef'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, 3aH npocpHJlb CKOpOCTH t COOTBeTCTBYIOHa JlaMHHapHOMY nOrpaHH'IHOMY CJIOIO. Ha pRC. 36 npHBeeHbl )J.aHHble paceTa SepTHK8JJbHOA COCTaBJIB- lOeit 6pa3MepHOro TenJIOBoro DOTOKa, HOpMHpOBaHHOrO Ha ero 3HaqeHHe Ha neooA rpaHHe. ConocTaBnSlIOTCSI pe3ynLTaThl, UOJIY- qeHHble C Hcn0J1b30BaHHeM KCPC c KoppeeR. H 8J{Hble pa60Tbl [93), pacct.JHTaHHhle no ypaBHeHWIM HaBbe,CToKca. Kax H B nepBOM npHMepe, BHHO xopowee COBneHHe ,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 KoppeKeft, 0 0 0 - ,QaHHllle pac'lera (93) KnoHetlHe B pe3YJ1bTaTaX 06'bjJCHSIeTC.R B nepBYIO o'iepeAb TOQHOCThJO IiCnOJIb3yeMoR: pa3HOCTHoA anrIpOKCHMaHH. llpO,ll-eMOHCTpHpOBaHHble pmynbTaTbl nOTBep>K&IOT paHee c;a.e- .naHHblR BbIBO 0 npHMeHHMOCTH KCPC c KoppeKHeA ,lVIH MOe- JllipoBallHSI BSl3KHX TeqeHHR, OfiHCbIB&eMbiX ypaBHeHHSlMH RaBhe- CTOKca. BMeCTe C TeM  KCPC c KoppeKeA Ka.K 1.JUC1IeHHhIA MeTO):t o6J1aeT pSl.LtOM 3aMeTHblX npeHMyeCTB. 1) 9TO oc1Hopolhi'Wu aJl20PUmM, II03BOJIjIIOW.d eHHblM 06pa30M BeCTH pacqeT KaK BSl3KOA, T8.K H HeBSI3KoA: qacTe:A nOTOKa. 2) 9TO Jl8HaJI cxeMa, He 'rpe6YIOID:8JI npHMeHeHHSI cnRanhHblX HTePaQHOHHbIX MeTO,QOB )\nSl peWeHHB CHCTeM CeTOqahIX YP&B- HeHHft. KaK 6Y,lJ;eT BHO 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 MHoronpoeccopHb1X BblqHCJIHTeJIbHhJX CHCTeM C pacnpeeneHHoR naMSITblO. Ee OAHOA CYIl\ecToeHHOH IIOnO)f(HTeJIbHOA qepTOft RBHOA cxe- '1bI (1.1)-(1.5) SlBnHeTOI OCmTO'lHO MgrKOe Orp8.HHeHHe 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 paceTOB OKOJIO- H cBepx3BYKOBblX T€qeHHH BSl3Koro ra3a. TeOpeTHeCKOro HCCJIe,l\o- BaHHSI YCToA1.fHBOCTH HHeHHo:A pa3HOCTHOA CHCTeMbJ KCPC c KoppeKHeH 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 nOpAKa )VIHHbI CBo6oHOro npo6era I AHCCHnaTHBHbIe qJJeHbl CTaHOBSIrcH nponOPQHOIIa.J1bHblMH 1 H OTnHIIHe KCPC C KOp- peKu.HeR OT KrY-CHcTeMhI npHo6peTaeT nOpH,IJ;OK 1 2 , BcnOMHHM, lITO KrY-CHcTeMa BblBOHTCSl H3 MCXO)J)-lO KCPC (rJl. II) 5) KaK P83- IIOCTHafl no BpeMeHH SlBHasI cxeMa, o6J]alOaH orpaHHeHHeM 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 <PYHKH pacnpeeJleHHH, Ha KOTO- poe onHpae'l'CH KrY-cHCTeMa. B CaMOM eneJ H3 pHC. 18 (rJl. n), Hn- nlOCTpHpYJOero H3MeHeHHe <PYHKl\HH pacnpeeJIeHHH, BHOt QTO 3a BpeMjI 6eCCTOJIKHOBHTeJlbHOI'O pa3neTa npH xap8J<TepHblX CKOpO- CTSIX MOJIeKynspHoro AB)KeHHS qacTHa He OJI}l{Ha nponCTaTh pac- CTOHHHe, 60JIbmee pa.3Mepa SI'IeAKH h, Ha KOTOpOM MaKCBennOBCKoe pacnpe)J;e.neHHe C1.fHTaeTC5I cna60 H3MeHjllOMCj!. Yq, HTblB3.SI '1TO · Boo6e, KCPC c KoppeKeA 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 TPoHHoro 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", npeM 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 KoppeKHeR Ha HecTPYKTypHpoB8HHblX CeTKBX ,lLrJs nOCTpoeHHH KHHeTHtJeCKH corJlacoBaHHoR cxeMbl C KOppeK- l\HeA Ha HeCTpYKTypHpOBaHHblX CeTKax o&heHHM reXHHKY KOH- CTpYHpOBaHHH KCPC H& TaKHX CeTKax (rJI. II,  2, n.3) C H,D:eeR HC- nonb30BaHH KHHeTHqecKHX cxeM )I.JlH MOeJ1HpOBaHHSI BR3KHX Te- 'reHHA, paccMoTpeHHoH B fi.l aHHorO naparpaQ>a. npH 3TOM, TaK JKe KaK }if nH npRMoyrOJIbHblX CeTOK, BBIIO BbleJlHM BSl3KHe H Ten- JI0npOBO)J.HbIe q.neHbI. B KaqeCTBe npHMepa npHBe,L\eM p83HOCTHbJe ypaBHeHWI )).JIg ra.- ::lOHHaMHqeCKHX napaMeTpoB Ha TpeyroJlLHblX CeTK8JC. aHaJIOraq- HbJe Bblpa)f{eHHM (2.9)-(2.20) rJl. II. IIpH 3TOM H3MeHeHHe ra30AH- HaMHeCKHX 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, OTHOCSlMHCSl 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 - (punerr(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 3eCbJ 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.roHX ee SqeeK C HOMepoM i. BLlpaJKeHHe )l.nH CCHnaTHBHoro nOTOKa (1.22) 8 ypasHeHHH Hepa3pblBHOCTH 8 TOHOCTH COBnaeT 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 CBaHbl C Bbl)].eJIeHHeM qJIeHOB BHroBOa BH3KOCTH H Terr- JIOnpOBOAHOCTH. B OTJlHqj{e OT ypaBHeHHSI SHeprHH (1.5) B KCPC c KoppeKHeH Ha npHMoyronhHbIX ceTKax, B (1.25) ,lVllI npocToTbI 3arlHCH ClleU)[aJlbHO He BbJengIOTCSI qJIeHhI, CBH3aHHble c pa6oToA B3KHX CHJI. Hx BKJIa,L( B H3MeHeHHe nOJlHoA 3HeprHH nSl OKonO- H CBepX3BYKOBblX TetteHHft HeCYecTBeHeH. B Kaqe('"TBe rrpHMepa npHBe- ,a.eM p€3YJIbTaTbl pacqeTa 3aqH o CBepX3BYKOBOM (Moo = 1.5) 8S13KOM (Re = 104) 06TeKaHHH KpbIJIOBOrO npo<pHJISI NACA0012 nOA HyneBbll yrnoM aTaKH (puc. 37 Ii 38) . TpeyrOJlhHag CeTKa (puc. 37) COep)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 MeTOOM YCTaHOaneHHSI. PacnpeeneHHe Kos4><I>UD.HeHTa aBJIeHHSI (-G p ) BAOJIh OCM cHMMeTpHR H nOBepXHOCTH npo<pHillI nOK83aHO Ha pRC. 38. Pe3YJ1bTaT HCn0JIb30BaHHSI KCPC c KoppeKeA Mano OTnHaeTCH 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.......'fT.'........'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 sij !.'''': ",.: ;.:.;.i....:: :'. '.'; '.'.. .1...,;.:,&1I,...,:........1.'........... "A4" :..:&:.. ;.. !', ;:,/ : ';";';":.';':'; :..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(IpeeneHHe 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_  7v...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  ,.. )II :J r\.    ....-J r:::f'fi oJ L"  II. ['J,j .v  ;'-1"0.1 N f\.l'  r   .... T ..r 1'L1T  ?'r, "    u  LJ't 'V\I  " LofC  An.'IttIl'I ii ,." "''' j '" VVV,;  ,/n,JU , '" V,\  n \I \,j J' ... JL./ V  J- ,",,11 \.i /\/\ J " 111 1\ j,\j  7 -  u "'" '\.I..... 'J   "  \j VV  'J.. r"'t;""'-  VV\l\l\j'(\  J'l7I v  ......" VJ\I\ /\ 1'\  " .J'..... """'IC.._ V -I\. n , looP t1  ..... '"  ".I......... >KD[(K  f-..1I f'   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 0._' - e !! l.tO e . o U D SS Ct' . ... c '.JO 0.21 Ie ... II IIG 'M Time 11. t40 1" "0 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 KB83HnepHOqecKHft 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 HaCTOHee BpeMSI 3TOT no,ltXO ecTeCTBeHHbJM 06pa30M o606lU-eH Ha MO,lteJIKpoBaJlOe npoCTpaHCTBeHHo-TpexMepHblx B3KHX TeqeHHA Ha TeTp83,LqJ&1IbHbIX CeTKax. CneeT Ta!C)f(e CK83aTL. 'ITO HAes KCPC C KoppeKD;HeH MO)Ke'!' COQeTaTbC51 C TeXHOJIOrHeA nOBhIweHHoro nOpHK& annpOKCHMaIJ)UI, OnHCaHHO 8  3 rn. II. T&K, HanpHMep, Anjf BLlqJfClleHHjI ra3H- 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, onpeenSieMbIMH 110 cpopMYJlaM (1.22)-.( 1.25) aHHoro naparpacl>a [44).  2. nOJIYHeSlBHaH cxeMa AnropHTM KCPC C KoppeKIJ;HeA "POCT H yo6eHJ OAHaKO 06- JIacTb ero npHMeHHMOCTH He6e3rpaHHQHa. Kax BHHO H3 npoBeAeH- HLiX BbIwe oeHOK, a T8J()I(e H3 pactfeTHoA npaKTHKH, B CJIyqae, xo- ra xapaKTepHaH MaKpoCKODH'tfecI<aSl CKOpOCTb cyw.eCTBeHHO MeHh- we CKOpOCTH 3BYKa t 3TOT MeTOA He MO>KeT CJlY>KHTb He.>KHblM HH- CTJ>YMeHTOM BblQHCJ1eHHA. ,nng MOeJlHpOB&HHR YMepeHHO 03BYK BbIX TeqeHHA 6h1J1 npe,lVlO>KeR [94, 101] eIQe OH 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 Be 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 TeeHHe KOTOpOro npOHCXO)J)JT CKMh-HH6YJI.h 3aMeTHble H3MeHeHHjI ra30HHaMHqeCKHX napaMeTpOB. BcneACTBHe 3TOro BenH'IRHY T HeYA06Ho HCnOJlb30BaTh B KaqecTBe paceTUoro mara. no BpeMeHH. 
186 ril. IV. l[HCJJeHHwe aJlropHTMhl H MBorOnpOQOCCOpHble CHCTeJlM PaCCMOTpHM OTpe30K BpeMeHH Ilt, KOTOphJH y,u;o6eH )VIB MOe- nHpOBaflHSI H3MeHeHHH r830HaMHqecKHX napaMeTpOB H YAoBne- TBopSlerr COOTHomeHHIO !:it = NT, N» 1. (2.2) Tora 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 - LLfJ.tk '1epe3 \J!j+. CYMMa l CTOBlI.{aH B npaaoA qacTH (2.3), npH6nH)f(eHHO npeCTaBHMa B Be HHTerpana (pHC. 42): ti+N N-l I \II dt  L rWj+A:. t) k=O (2.4) Kax BHO H3 pac. 42. Ha KaJKAOM OTpe3Ke T = t j + 1 - t} BKJI B npH6nHJKeHHOe 3HaqeHHe HHTerpana (2.4) onpeenBeTCB 3HaqeHHeM <t>YHKHH \IIj+k Ha JleBOM Koe 3TOro OTpe3Ka. 
2. IIanYHelIBHa.H cxewa 187 H3 BblpaJKeHHA (2.3) H (2.4) BhITeKaeT COOTHoweHue, CBSJ3bJ- BaJOee 3HaqeHHH <l>YHKqHH pacnpeeJleHHSI Ha CJI0BX t = t i + N H t = t J : fl+ N - fi,M ( ! T ) L f ; (  _ T ) L f j+N = t + 2 + 2t M + 2 2t M = (  + 2t ) L ; Lflr + (  - 2t ) 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 npeCTa- OHM B cne,AylOw;eM BH)J.e: T j T j+l r a j ( A 2 ) 2t Lf M - 2t Lf M = - 2 Ot Lf M + 0 ut · (2.6) OrpaHHt.tHBaSlCb 3aqaUH, B KOTOpbJX xapaKTepHoe BpeMB H3MeHe- HHSf ra30AHHaMJAecKHX napaMeTpOB MHOro 60nhwe T t npeHe6pe)l(eM t.tJleHaMH (2.6) H npeM K cxeMe C BeCOM 1/2, HJIH nO.llYHeJl8HOiJ, )VJst ODHC3HH1I H3MeHeHHa CPYHKU)fH pacnpeJIeHHjJ (101): j1+N Ii I - I,M + ! ( Lfj - L T Lfi ) + t 2 M 2 M +  (Lilt 1 - L;Lfl) = o. (2.7) I1HTerpHpyg KHHeTH'IecKoe ypaBHeHHe (2.7) C cYMMaTOpHblMH HHBapHa.HTaIH, npHxoHM 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 MHoronpoeccopHble CHCTeMbl TpaHc<l:>opMHpyeTCH B CJIeAYIOee: 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 npcraBHTb B onepaTopHOM BHe (2.8) '+1 . Q1  Q1 +  (Aq1 + AQi+l) = O. (2.9) 3eCb Q = (p, pu, pv, pw, E) - oeKTOp r8.30HHaMHqecI(HX nepe- MeHHhIX. A - HeJlHHeHHblft onepaTOPJ BKJlIOlalOa B ce6g Bee npo- CTpaHCTBeHHbJe npOH3BOHbIe 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 MTepHOHHO- 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 npaKTHIecKYIO peanH38WfIO npocTOR, HecMOTpSl Ha BHeWI-IIOJO rpoMo3KOCTb <I>oPMYJI THna (2.8). KaK nOK83b1B&eT onbiT pacqeros, HTepoHHbla npoecc (2.10) CXO,AHTCH IIPH ware D.t, y,n;oBneTBopjJlOeM yenoBHIO KYPaHTa WIH a»(e HeMHoro 60JIb- mel. Ha npaKTHKe )VIR nOJIyqeHHH HCKOMoA TO'IHOCTH CXOHMOCTH 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 CXOHTCH TOJIbKO noCJI€ 6onbllioro 'IHCJJ& HTep8JJ.HA. B 3TOM r.nyqae ;:vIS Haxo)K.ZJ.eFiHsr peweHHSI KfY -CHCTeMbI Ha HOBOM CJloe no BpeMeHH Heo6xoMO HC- nonb30BaTb 60J1ee CJIO}KHbJe HTepaHOHHbIe MeTOAhI. OHa.KO B HeKOTopbIx cnyqaJIX anr9pHTM (2.10) MO>KHO npB- MeHHTb H .AnSI MOeJIHpOBaHWi reqeHHA HeoKHMaeuoro ra3a. npH 3TOM HCfiOJlb3YeTCH npHeM nepeHopMHpoBKH YCJlOBHA 3a'lH no qHC- JlY PeRHOJ1bCa, 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- HOJ1Lca. TIYCTb, HanpHMep, Te'ieHHe o6J1aeT peanhHbIM qHCJIOM Maxa M = 0.01. Tora MO>KHO YBeJlHqJITb BXOYIO CKOpOCTL B 10 pa3, C TeM qT06bI C cj>HKTHBHLJM 'IHCJlOM Maxa "J 0.1 HTepaH- OHHblA npouecc (2.10) HMeJ1 yoBneTBOpHTeJlbHYIO CKOpOCTh CXO- MOC"fH. B CBOIO Oqepeb, o6b1 QHCJ10 PeRHOJIbCa 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 KaeCTBe npHMepa HCllonL30oaHIDI nonYHeHBHoR cxeMbl (2.9) npHBe,n:eM pe- 3YJlbTaTbJ pacqeTa TeqeHHB HeC)I(HMaeMOA >KHKOCTH BHYTpH npHMoyronbHoA K3.BepHbI c ,lJ.BH>KyeAcSi 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 CeqeHHX KaBepHbJ npH t{HC- Jle Pe:AHOJIbCa Re = 100. ,neMoHcTpepyIOTCSI .n-aHHble pacleTOB Ha nocne.n;OBa- TeJIbHOCTH cryalOII.J.HxCSl CeTOK. BHO, 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 3aqH cOBpeMeHHOa rHAJ>O- H ra3oBoA ,nHHaMHKH. B cnelOtneM naparpaQ>e paCCMoTpHM 60nee CJ10>KHbJe anropHl'Mbl pewelfHH KBa3Hra30HHaMHqecKoH 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:hIYw.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 MeTO1 HCITOJIb3YJOT B KaqecToe OCHOBLI QHCTO AHcI>cI>epeHB- aJlbHbJA BapHatIT Kry, C DOCJIeAYIOrneA JIHHeapH3eA no HblOTOHY ypaBHeHHA, BXO-nlll.HX B 3TY CHCTeMY. IIpH paCCMOTpeHHH MaTepH- ana aHHoro naparpaQ:>a 6yeM CJIeOBaTb pa.6oTe (102]. BbJnHWeM K<PcI>epellH8JIbHbJA BapHaHT KBa3Hra30,D-HHaMH'tfe- CKoA CHCTelhI 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 CJleAYlOe 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 MHoronpoecCOpHble CHCTeMlJ aHaJlOra npOH3BO)!.HbIX no BpeMeHa H8 .lJ)I<t><PepeHa.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 annpoKcHMH npoH3BOHOa no opeMeHH BOcnonb3yeMCg napaMe'rpH"ieCl<oA: <I>oPMynoA o6ero BHa [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. BBeeM 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&3Hra30aMHCfeCKOR CHCTeMbl 193 nOCTaBJISISI 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 npoBeeM 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) re A == 8E/8Q - MaTpa HK06H. 3anHCbJB8JI HaqaJIO H KOHe enOqKH (3.6) 8 ,lleJIbTa-<I>oPMe, HMM jE = Aj;Q + O(). (3.7) LfJJyrHe qneHbJ JIHHea.pH3yeM aHaJlOraQHO. qJIeHbI co CMeWaHHblMH npoH3BOMH 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): { 91t ( 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 + 81t [ !.-j-lU + !.-. T Aj-lU ] . (3.9) 1+92 1+92 ax 2 Oy 8y 2 ax 3,necb I - eHHHqHaH MaTpHa, A, B, M, N, A , B - MaTpH 51Ko- 6H) KOHKpeTHhiA B KOTOpblX 6YeT Y IC B KOliQe naparpa<l>a. CxeMa (3.9) )J.J1S1 peweHHH ypasHeHHSI (3.1) 3anHCbJBaeTCS B 60- nee Har JlHHOM 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 MaTpHa 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 npeO>KeHO HCnOJlb30aaTb B pa60Te [104). MCqepnblBalOI.Qee pa3BHTHe 3TOro nO,l),Xot\a 8nOC1IeCTBHH Hamno BbI- paJKeHHe B pa60Te (102). Ha aCROBe cxeMhI (3.10)-(3.13) B03MO)KHO nocrpoeHHe pa3nH'i- flbIX &1JropHTMOB pemeHHH HCXOHOA CHCTeMbI (3.1), BKnlOa.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 nonyqalOHXCB caCTeM JlHHeAHblX CeTOqJJhlX YPaBHe- IIHR MO)KHO npHMeHSlTb Pa3JIHQHble npHMble H HTepOHHhJe 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, OnHp8JOeCSI Ha npeACTaBJIeHHe 8 B (3.12) B BRAe CYMMbI ABYX onepaTopOB H Y'iH- .rbIBaroIIJ.HC cnecJ>HKY KrY-cHCTeMbl. B qacrHOCTH, OKa38JJHCb no- J1e3HbIMH COOTHOmeHH.H Me>K nHHeapH30BaHHbJMH onepaTOpaMH, onHCbIBafOHMH KOHSeKTHBHYIO H cJ><I>Y3HOHHYIO 'IacTb [102]. B 3aeepUJeHHe naparpa4>a p n01IHOTbl H3nO>KeHMJI npHeM 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 b3 "'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 MaTpHbI 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. MHoronpoeccopH&le BhI'lHCJlHTeJlhHhle CHCTeMbi B 3TO1 Ii nOCJle,ll;YIOHX naparp8(j>ax aHHoA: rn8.BbI peQb 6YAeT H,l(TH 0 np06neMax, CBH3aHHblX C IJpHMeHeliHeM BbJCOKOnpOH3BOH- TenbtIbIX MHoronpoeccopHblX BblqHCJIHTeJIbHbJX CHereM ,L\nSl peme- HHB 3aq rHpo- H ra30BOH AHHaMHKH. BKJlroqeHHe STOro MarepH- aJIa B KHHry CB&HO C TpeMB OCHOBHblMR npH'iHHaMH. Bo-nepBbIX) 6ypHblIt pOCT npOH3BOHTeJILHOCTH MHoronpOl..\ec- copHoR BbrqHCJIHTenbHOH TeXHHKH, Ha6nlOaeMbiA B nOC1leHee ea1- THneTHe t OTKpblBaeT wHpoqaRwae B03MO)t(HOCTH B MOCJDIpOBaHHH 33Aaq ra30BoA HHaMaKH H MHOrHX ,LWyrHx npo6neM, OnHCbJBaeMbIX ypaBHefIHSIMH MaTeMaTHQeCKOA <PH3HKH H paHee HeoCTYnHblx ,lI.JIR 'fHCJleHHOI'O peweHHH. B CBH 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 TPYHOCTH a,a.an- HH HMelOllJ.HXCSI Bbl'IHc:JIHTenbHhIX aJlrOpHTMOB K a.pXHTeKTYpe MHoronpOQeCCOpHblX CHCTeM. CecT8YIOT TaK>Ke npo6.neMbI, CBB- 3aHHble co cneqHQ>HKoft napaJIneJIbHOrO MaTeMaTH"IecKoro o6eclle- qeHHH. He MeHee Ba)KHOR np06neMoA, B qeM-TO 8JKe BbIXOlIt-eA B HaC1.'OSlLQCe BpeMSI Ha nepBDe MeCTO, BJlHeTCSI HHTepnpeTaU)fB H Be- PH<I>HKaI.l.HH aHHhlX BbICOKonpOH3BOHTeJIbHblX 8blHCJIeHHA. 9Ta TeMaTHKa "MeeT np.SlMoe OTHOIIleHHe K KnaCCH'ieCKHM np06JIeMaM npHKJIHOH H 06ea 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 BhlCOKOnpOH3BOTeJlbHbIX CHCTeMax. OnHcaHHlO 3THX pe3YJIbTaTOB B 3HaQHTeJThHOH Mepe nOCBeHa rJIasa V. BOJIee TO- ro, peaJIhHasI B03MO>KHOCTb npHMeHeHHR KCPC nSl pemeHHH CTOJIb TPY OeMJ(HX 38Aa "cnpOBOHpo8ana" 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 6YYT paccMCY.rpeHbI B 9 7 rJlaBbJ. OJI.HaKO npe>Ke qeM paccMaTpHBaTb Hcnonb30B8HHe KCPC B napan.nenbHblX BblqHcneHRRx, HM KpaTKYJO HHcPopMaHIO 06 HC- TOpHH pa3BHTHH H nepcneKTHBax MHoronpoeccopHoii BbIIfiICJlH- TenbHoR TeXHHKH. 1. l1cTopHJl H l1epcneKTHBbl pa3BHTHH MHorofIpoecCOpHb1X CHCTeM OrpaHHQeHHOCTb B03MO}K}lOCTeA 9BM C apXHTeKTYPo.A <POI-l HeAMaH8, Kora Bce BbI'IHCJIeHIDI npoBOSlTCH nOCJ1eOBa- 
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94. MHoronpoueccopHble BblqHCJlHTeJlhHbIe CBCTeMbl 201 cneu.HaJlbHble npOl\eCCbl-KOMMYTaTOPbI ,lVIB YBenHqeHHSI CKOpOCTH o6MeHa HHcPopMaHeA. TIpH nOM Ka)l(3JI nOC1JeAYIOaSl MO,ltHtPH- Kal.\H5I BbIQHCJIHTeJ1bHhJX CHCTeM OfiHpaeTCSI KaK Ha OqepeAHYIO 60- nee BbICOKOnpOH3BOTeJlbHYIO MOeJIb 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 BeyIU)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 yrBepaTb, 'tJTO K 2010 r. tJla..KCHMaJIbHbIe I1HKOBbIe MOOCTHt AOCTynHbIe Hcc.neOBa- TeJISIM, 6Y.DYT JJe>KaTb B npeenax 103 -:- 10 4 TFLOPS. YcnexH nOKaJKYTCSI oc06eHHo OmenOMJI.SlIOIIl.HMH, eCJIH conOCTa- 8HTb me c nOJIOeHHeM 25-neTHeR )J.8.BHOCTH, Ko('a OCTynHb1e ,LtJIH 60JIbIDHHCTBa HCCJIeOBaTeneA MOILI.HOCTH HaxOHJlHCb B ana30He OKono 1 MFLOPS. OaKo pOCT DpoH3BOHTenbHOCTH BblqHCJlH- TeJIbHblX CHCTeM 3a qeTBepTb seKa B MHJInHOH Ii 60nee paa He npH- BeJI K aHaJIOrH'-IHOMY nporpeccy B B03MO)KHOCTRX pemeHHH 3aq.". BOJIee TOro, cyrn.ec..."TBYlOID.MA pa3phlB Me B03pacTalOHMH B03- .HanpHMep B Me.OMCTBeHBOM cynepKoMlIhlOTepHOM IJ.eHTpe (MOCKBa) B 2001 r. 6b1J18 BeHa 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 kaeCTBe KOMwyraTOpOB HCnOJlb30Balla CHCTeM8 MHpR- HeT 2000. Ha 3TOH CHCTeMe B nOCJIe,AHHe ro.nbl 8.BTOp Hero KWlnerH npOBOAHJDI paceTbl 1(eJ10rO pSl.lt3 3a'l. OJIHCa.HHe 3THX 38Aa1f H aH8J1H3 pe3YJlbTaTOB me pac'leTOB '-IacTHO HaWJlH OTpa>KeHHe B rJ1. V MOHorpa<pHH. .. QHTaTCJ1H C OCTaT01fH.blM HaY'IHblM CTaJlCeM MorYT BcnOMHHTb 06 ycnem- nOM MOeJ1HpODaHHH MHOrHX cnO)l(HhlX 38,[1.a'l B 70-80 rr. Ha BbI"IHClIHTeJlbHOA TeXHHKe TOro 8petAeHH. 
MO>KHOC'.rS1MH Bbl"lHCJJHTeJIbHOR TexHHKH H pe3ynb'nT8.MH ee Heno- CpeCTBeHHOI'O HcnOnh30BaHHSI, a TaK>Ke HX BJIHSlHHH HR HaY1ffiO- TeXHHQeCKHH nporpecc, SlBJ1j1eTCH CBoero poa Bbl30EOM C()()6ecrBY CneLI.HaJIHCTOB B o6nacTH MaTeM&THqeCKOrO MOeJlHpOBaHHg, npH- KJIHOH MaTeMaTHKH H HHcPopMaTHKH. B KaKoR-TO Mepe J];aHHasI cHTya1I.HSI CBH3aHa c otYheKTHBHhI- MH npHQHIJaMH. HanpHMep, HCfiOJIb30B&Hee BbJCOKonpoH3BoHTeJID- "oR reXHHKH Y>Ke e.n8JIo peanhHbJM pemeHHe HecTaHOHapHbIx npoCTpaHCTBeHHo-TpeXMepHbJX 3aq. 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! KoopaTe) YBeJIH'iHBaeTCSI B nY'lIWeM CJIyqae KaK . OaKO HMelOTCH H APyrHe cPaKTOpbl, ep)KlIBalOe npo- rpecc B npHMeHeHMH MHoronpoeccopHblX BbltJHCJIHTe1IbHbIX CHCTeM. 9To HeOCTaTOqHOCTb p83BHTHS napannbHhlX aJIpHTMOB, na- panneJlbllOrO M8.TeMaTHQecKoro 06ecneQeHHB (parallel software), HH- TepnpeH It oepH<j)HK&QHH ..D:aHHbIX BbiCOKOnpOH3BOHTeJIhHLIX BIdQHCJIeHHA H tij). no06Hb1e npo6neMLI MOryT 6b1TJ. pemeHbI npH COOTBeTCTBYIOHX yCWIBBX cneHanHCTOB. qacTHHO 3TH BOnpo- CLI 6yy'r o6cyaTbCJ1 B nOCJ]eAYIOX naparpa<j)ax rJlaBbI notlt yrJIOPvf 3peHHSI HCn0JIb30SaHHS KCPC )VUI pemeHHH 3aq ra3G- BoD ,lJ.Hf.aMHKH. A aHHblA naparpaQ> 3aBepWHM 0630poM B03MO)K- HOCTeR 06'beHHeHH.SI BLlqHCJIHTeJlbHblX pecypCOB cynepKoMnblOTep- HbIX u.eHTpOB H pa3BHTHH OTHOCHTeJlhHO He6OJ1hWHX MHoronpoec- COpHbIX CHCTeM. 2. MeTaKoMnblOTHHr H KJl8CTepbl nOBJ1eHHe .n;OCTaTO"IHO 6onLworo KonHeCTBa cynepJ4:0MnbJOTepHbIX eHTpoR nOCTaBHnO Ha nOBeCTKY ,lUfSl BOnpoc 0 60nee paHOHanbHOM HC- nOJlb30BaHHH HX pecypCOB. AKTYaJIhHblM CTaHOBHTCR COO,ltaIDle e)J.HHOI Bbl- HCnHTeJIbHOH Cpe)J.bl, 06rbeHHftlOeA 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 38"1H cBo6oAHbOC BbI'IHCJ1HTeJIbHblX 
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1. (O-!3)-HTepaI{JfOHHblH MeTO).{ PacCtOTpHM 3TOT MeTOA Ha npHMepe nSlTHTO"leT.IHoA cxeMbI, 3a- ,n:aHHo:R B np5lMoYroJIbHo:A 06nacTH: BinUi,n-l + KinUi-ln - 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 rpaHHHb1MH 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 · I1peOnO)l(HM, no aH8JJOMfH C xn&CCH'IecKHM MeTO,L\OM nporoH- KH, \{TO peweHHe HCXOOa 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>HHeHTIW. ,L{nsr HaxO.>KeHIDI CHCTeMhI YpaBHeHHA, KOTOpoA YAOBJJeTBOPSl- lOT nporoHOq}{hle Ko3<f>cI>HeHTbJ, npoAenaeM cneAYIOee. C nOMO- IO YCJIOBHA (6.4) nepenHWeM HCXOHoe 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 HeKCbl t, n npH K09<1><1>HeHTaX. cxeMbI (6.1). AHaJlOraqHO t c nOMO1O 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)KeHHg U MO)KHO HCfiOJlb- 30BaTb <l>opMynLl o-nnOMepHoA nporoHKH, KOTOpylO npoBeeM B 'leTbJpex HanpaeneHH.HX B CTOpony YBeJJH1feHHB H YMeHhweHHB HIIAeKCOB i Ii n. npH 9TOM nOJlY1faeTCSI CHCTeMa, COCTOsnu:8.SI BNecTO OHOro ypa.BHeHHH (6.1) H3 BOCbMH ypaBReHH1t ARK onpeeJJeHHSI - - 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. OnpeeJlHB 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 pemeHe HCXOHoro ypasHeHHH (6.1). HanpHMep, 3HaH K03cJ><PHHeHTbI 5, {3, nOCTaBHM 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 3HaeHHa K03<P<1>HtI)leHTOB {3, d, {3, d, B CBOIO Oqe- pe)J.b YOBneTBOpSlIOHX (6.8). ,lJ;JH HaxO>KAeHH5I K03c1>cI>HHeHToB a, 1', a,;:Y H3 CHCTeMbl (6.7) BOCDOJIb3yeMcH, HanpBMept CJleIOIU.HM 
HTepaHOHHbJ M npoeccOM: ai = 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) HaASI KoscP<I>HlI)feHTbJ a, "Y, ii, 1. rIpHCTynHM K onpeeneHH1O - ..... Kos<p<I>HlUteHTOB {3, d, (3, d, pewaH CHCTeMY (6.8) npH nOM01IlwH HTe- pa,wtit l3 k+l) = (/3 k+l) K + Rk+l) B + k)V + F )/ \II 1+ 1 , n In  In In In I i = l,....Ni-1, ,e1) =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 + tlk+l) K + dt+l) E + F )/  Pttn+ 1 In fJn In In , n=l....,Nn-l, +l)=i.I' i=1,...,Ni-1j k+l) = ( k+l)V + l3 k+l) K + dk+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) $IBJIHeTCSI HaH60nee npo- CThlM IeTOOM BbIQHCJ1eHHSI npOrOHOQHbIX K03<pcpHHeHTOB. ,lJpYMfe B03MO}KHhJe BapHaHTbI (a-{3)-HTepaOHHoro MeTOa, o6naJOe 
60nbweA CKOpOCTblO CXOMOCTH, a TaJ()I(e 0606eHHe 8JlropHTMOB Ha CJIY'IaH 3D-reOleTpHH paccMoTpeHbI B MOHorpa$HH (14J. TeoPeTHqecKoe onpeeJIeHHe CKOpoCTH CXO,lI)fMCCTH (a-f3)-HTe- panHOHHOro "'feTOa 3aTpYHeHO BBHAY ero He.rIHHeAlloro xapaxTepa. OeHKH 3eCb npHXOHTaI )1.enaTb, ormpaSCb Ha 3kcnepHMeHTaJlb- Hble aHHble. B C8010 O'lepeb, pac'leThi nOK83bIBaJOr, qTO CKOpoCTb CXO,D.HMOCTH a-npoecca (6.10) HCKJllO'IUTeJIbHO BICOKa. OTHOCH- TeJlbHaH norpelllHOCTb y6bIBaer 60nee 'IeM B 10 paa OT HTepH K HTepHH, BHe 3aBHCHMOCTH OT q}lCJIa y:3JIOB. CKOPOCTb CXOHMOCTH ,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 3aq HHaMHKH H3nY'lalOero 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 MeTOa C1IeyeT OTHeCTH ero "BceH)J.HOCTb" (robustness), n03BOJISIIOllQ'1O - nYCTb H He OqeHb 6bICTpo - pewaTb pa3Hoo6pa3HbIe 3aqHt 8 KOTOpblX BCTpeqaeTCSI CHJIbHaH BapHa.QHH K03<t>4>HeHTOB pa3HOCTHOR cxeMbI. 2. MlloronpoeccopHbIil BapHaHT (a-/3)-MeTOa llpH perneHHH KpaeBblX 3aq H8 MHorOrIpOQeccopHblX CHCTe- Max 06hlqHO Hcnonh3yeTCSI cneAYIO$a nOAxo. KaJK;:\OMY npoec- copy OTBOAHTCH CBO.H no06J1acTb, B KOTOpoA <paKTHtfOCKH CTaBHTCS1 OTAeJlbHast KpaeSaJI 3aa. PacCMOTPHM, KaK npocreawHA: CJlyqaA, p836HeHue HCXOAHOA 06nacTH Ha ,lJ.se noo6JIacTH I H II. B03MO>KHbI Ba cnoco6a p836HeHHSI. nepBhIJt - 6m nepeKpblTHH (pac. 50,a), KO- r,Aa 06W.HMH ;J.JISI no06J]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 CJ1YIae nepeKpblBalOXCH 06J1acTe:A pacqeT HaIHHaeTCH c nOCTaHOBKH HCKYCCTBeHHblX rp8.HHIJHblX YCJlOBHft Ha rp8.HHax no- 06nacTeH ab H cd (puc. 50,6). C STHMH rp8.HH'IHbIMH ycnOBHBMH npOBOAHTCSI pacqeT CeTO'-lHoA <l>YHKHH B nO06nacTgX I H II. no OKOHIaHHH p-aHlloR npoeAYpbl 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. fIpoecc HTepa Me)l( nOA06J1acTH- MH nOUTOpSle1c.H 11.0 CXOHMOCTH. OeBHHO, '{TO "{eM 60J1bIlle nepeKpblTHe no06nacTeA, TeM 6b1- CTpee CXO51TCH HTepaHH Me>K)Jy HHMM. OAHaKo 'tJeM 60nbme 06- HX TQQeK, TeM HH)I<e 3<1><1>eKTHBHocTb napanJIeJIH3Ma (npoeccopbl AY6JIHPYlOT pa60TY ,LJ.pyr pyra). KpoMe TOI'O, 3el><PeKTHBHOCTb Bbl- qHCJIeHHH naeT B pe3YJIhTaTe TOro, qT() YBeJlHQeHHe 'I}lCJJa TOQeK B Ka>K)(oA nO06JIacTH IIpHBOHT, KaK npaBHJIO, K 3&Me)J.JIeHHIO cxo- HMOCTH .HTepaHOHHoro npon:ecca B ee npeeJlax. 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 npoecce HTepao.H:A Me)l(Y no06naCTSlMH. (a-fj)-MrO,lJ. ,n;aeT npoCTYIO B03MO>KHOCTh nonyeHH5I ycnOBHl! Ha BHYTpeHHeR rpaHHe ab B npoecce HTepaQHA Me nOAOMa- CTSlMH. nepeKpblTHe He Tpe6YeTCH, B MeHSleTcg cau npHn napan- JleJ1R3MH anrOpHTMa. CYT!:. B TOM qTO B xoe HTepA BHyrpH Ka.>KOa nO06J1aCTH BblqJfcnjJlOTC5I 3HaQeHHSI He caNol 4>ym<QHH Up a npOrOHO"lHblX K03ct><PHeHTOB. 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 rpaHHe onpe)J.enH- IOTe$! Ko3<P<PHlI.HeHTbl ,.., H d. HaaeHHble 3HaqeHH5I Q H {1 CJIY>KaT HOBblMH rpaHHqHblMH ycnOBHSlMH A1IS1 06nacTH II, a 3HaqeHHH KO- 
220 r.l1. IV. t{HCJlellHble anropHTMbl If MHoronpoeccopHble CHCTeMLl 3<I><t>HeHTOB '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)- HTepOHHOro 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 npoeccopOB < 40, leaK NljI 2D, Tal( H AJIB 3D-3a'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 npoeccopoB B KaqecTBe npHMepa DpHBeeM ,lI;aHHhle pac'feI'OB no Ha.xOe- HHlO peweHHSI 3a'qHJ Hcnonb3ylOeA 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<pHrypaHB OTHOCHnaCb K pac'tJery MeKTpOHIJO-Ab1pOHOA D.na3MbI TpaH3HCTopa. ,lI.n1! pactJeT'd. HCnOJIb30Banacb JJHHeAHaH KOH- <pHrYPIDI CHCTeMbl npoeccopOB (pHC. 44). 06paTHM BHHMaHHe Ha BnOJIHe ecrecTBeHIJoe YMeHbmeHHe 3cj>- <t>eKTHBHocTH napanneJIH3 C poCTOM QHCJla npoeccopOB. O- HaKO CJIe,n.YeT 3aMeTHTb, qrro 3<t><PeKTHBHoCTb napaJIJIeJlbHOrO pac- 1feTa B03pacTaeT C YBeJlHeHHeM o&beMa BblqHCneHHA:] npOH3BOH- MbIX Ka>K)J.bltvf npoeccopoM" 9TOT <PaKT xopowo YKna,nbIBaeTCg B 06I.llJ'1O KOHQenHJO 0 TOM, liTO BbiCOKOnpOH3BO)J.liTenbHbJe MHor npoecCOpHl:de CHCTeMbl OJI}KHhI - B OCHOBHOM - BCn0J1h30aaTbCSI 
96. (0- fJ )-HTepaHOHHb1R MeTO,l{. MHoronpoeCCOpHblR BapllaHT 221 nH o6pa60TKH o6nrnpHblx MacCHBOB HHopMaH. 3. IIpHMepbl paClleTOB B KaqecTBe COAep>KaTMbHblX npHMepoB HCn0JIb30B8.HHH MHO- ronpoeccopHoro BapHaHTa (a-,B)-aJIrOpHTMa pacCMOTpHM HeKO- TOpble pe3YJlbTaTbl MOeJ1HpOBaHHH npoQecca H3BJ1eqeHHSI He<J>TH C nOMOInblO 3aKB.'fKH BO,D.bI B nnaCT, a 3&TeM MetI)JeHHoro Teqe- HHSI MeTaHo-B03IlIHOR CMeCH B IIJIOCKOM KaHane c nO)l.OrpeBOM [ 115 , 127], [132] -[ 133] . IIepsaH 3aqa onHCbIBaeTCSI CHCTeMoA YPaBHeHH:R BaKJIeB- JIeBepeTTa: m a: + div(F(S) U) = Q == qF(S) I q = divU, U = -Kgradp. (6.12) (6.13) 3eCb 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 - HCTOIIHKH H CTOKH >KHKOCTH; K _ _ kwot(S) + koil(S) , (6.14) J..'wat lJoil JJwat,l-toll - Ko<I>HHeHThi BB3KOCTH BObl 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 OHOMY ypaBHeHHJO napa60- J1HQeCKOrO THna, p83HOCTHaH anrrpoKCHMaQ.HH KOTOporo pewanacL HTepa.u.HoHHbIM (a-{3)-MeToAoM. Ha pHC. 52 npHBeeHa OHa H3 'l'HJJH'I HhlX cxeM OpraJm3HH He<l:rreo6blqH C IIOCTOSlHHhlM pacnOJIO>KeHHeM BO.ll:03a.r<a'lHB8JOHX CKBaJKHH. PacqeTHaSl 06J1acTb (Ha pHC. 52 3awrrpHxoBaHa) 06.rIaeT cHMMeTpHeH. Ha pHC. 53 nOKa3aHa XapaKTepHaJI KapTHHa KoeH- TpH He<t>TH B nJlacTe Ha pa3BHTOI CTa.JJ)fH ,q06b1'iH. B pRe BapH- aHTOB paCqeTa 3aMeTHO npOSlBJlHJICSI HeraTHBHblA 3c1><PeKT 06pa.aoBa- HHSI "80HblX llaJIbu.eB", Kora OKOJIO He<l>'reo6bJB8JOllI,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. CeMRTOeqHag cxeNa OPraHH- 38D.HH He4>Te06b1'1H: 6 - BO.n03BK&- "HBaJOHe CKBB)I(HHhI, 0 - Ho- 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. KOHIelrrpau.Rsr He<I>TH 8 nnacre: B&lMH O'M'eHKaNH  3H8eHO ,noneBOe COAep>KaUHe BObI 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 Boa MO>KeT 6blCTpee ,l\OCTHraTb OTKatJHBaJO- W)lX CKBaJKHH. BTopaH 3aqa HenocpeACTBeHHO CBH3aHa c npOH3BOCTBOM 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(. MHoronpoeccopHhlR 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 npOHCXOHT peaxg 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 rrpeBpaeHHIt,. KaK c nOMOblO 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 MeTOOB 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 MeTOOB peweHHH CH- CTeM CeTO"iHhlX ypaBHeHHii, HCnOJlb3yeMhlX aBTOpOM H ero KOJIJlera- MM. PaccMaTpHB8JIHCb aJlropHTMbJ. CBSI3aHHbI€ C p83JIH'iHblMU na- panneJ1bH:bJMH sepCHflMH MeToa HenonHoro pa3JlO}KeHHSI XOJIe- KOro [134, 135] H nonepeMeHHo-TpeyrOJIhHoro MeTOa CaMapcKoro (136) 137]. 3aMeTHM ee 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, HHTepecYIOHMCSI HaqaJlbHblMH CBej!.eHHHMH no aHHOMY Bonpocy, MO>KHO peKoMeHOBaTb MOHorpa<pHIO [138]. · MO,neJIHpOBaHHe npoeCCOB B MHOroSOMDOHeHTHblX XHYlAecKH peantPYIO- UI,IIX raJax Ha OCHOse KrY-CHCTeMbl 6y,neT 60Jlee IIoJq)06HO paccUOTpeRO B cne- YlOlUeA rJ1aBe. 
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B rJlaBe V 8 CBH3H C MOenHpoBaHHeM DpoeCCa ropeHHH MeTaHa. B eJlOM, 3aJ<aHQHBasi AaHHYIO rnaBY, XOTMOCb 6h1 OTMeTHTb, qTO paCCMOTpeHHbie B HeR anfopHTMbI, BKJlIOaB uHoronpoeccop- HYIO peaJ1H3aIHIO, 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 onpeeJJSlnCB B 3Ha'lH" TenbHoA Mepe nH'iHblM OfibJTOM aBTOpa B peweHHH a raaoBoA ,l\HHaMHKH. KOHKpeTHhle pe3ynbTaTbI 9THX pac'leTOB 6YAYT npe- CTaBneHbJ B CJIelOea I'.TIMe. 
rJlaBa v MOEnHPOBAHHETEqEHH rA30BOA CPEbI ,aaHHasI rJIaaa BnHeTCSI 38.BepIIlaJ()ea B MOHorPa4>HH. B HeA Ha OCHOBe paHee npe,lVIO>KeIIHblX <PH3HecKHX MOeJleA H 'IHCJleHHLIX anropHTlvlOB 6YJQ'T pacCMOTpeHbl pe3ynLTaTbi MaTeMaTHQeCKOrO Mo- e.n:HpOBaHHg HeKOTOpblX KnaccoB ra.3OAHHaMHtJ.eCKHX TeqeHHA. ,no 9TOro B OCHOBHOM paccMaTpHB8JIHCb 3aqH 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 npeA1dx fJlasax.* BO.flbWRHCTBO npeCTaBJIeHHbIX B rnaBe pe3ynbTaTOB uonytteao c nOMOll1.blO HcnOnb30BaHHSI BhlCOKonpOH3BO,l:tHTeJIbHbIX MHoronpoec- COpHblX BbIqHCJlHTeJIbHblX CHCTeM.  1. Moen:llpoBaHBe cTpyiaoro Te'leHB.H H Rq}{eM C OTHOCHTeJIbHO npocTOro Knacca HOHapIUdX 3a'l. B AaHHOM naparpa<pe npoeMOHCTpHpyeM HeKOTOpbIe PYJIb- TaTbJ pacqeTa 3aTODJIeHHbIX cTpyA. HapSl)O' C KOHKpeTHbIMH Q>H- 3HQeCKHMH pe-.JYJIbTaTaMH, 06paTHM BHHMaHHe Ha HeKOTopble 8.H- Hble, CBSI3aHHble C HCnOJIb30BaHHeM BblCOKOnpOH3BOHTeJlhHbJX MHO- ronpoeccopHblX CHCTeM. PaCCMaTpHBaeMble pe3ynbTaTbI QHcneHHo- ro MOeJlHpOBaHIDI npeCTa.BneHbJ B pa60Tax [145, 146) H COOTBeT- · B HacTOBII(ee BpeM.9 ae.neTCJI pa6crra no CQaHHIO 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, npoeMOHCTpHpoBaHb1 B rJl. IV. 
CTBYJO'f 3KCnepHMeHTaJlbHbIlvi .AaHHbIM pa60ThI (147). PaccMaTpHoanacb CTPYjI 830m co CJIeIOHMH napaMeTpaMH Ha cpe3e conJla: TeMnepa'I'ypa CTPYH Tn = 249.2 K, a8neHHe Pn = 3.824 · 10 4 rIa, ,n:JIHa cB06ooro npo6era An = 11.73 · 10- 8 M, '-IHCJIO Maxa M n := 1.01, 'IHCJlO KHyceHa, onpeAeJJjleMOe no PHYCY 1. n conna , Kn = An/2rn = 3.747.10- 4 . OTHoIlleHHe 3.BJIeHHH OKPY- }!{8.lOl1.I-eA cpebI Poo K ,lI;aBJIeHHIO Ha cpme canJla AnSI pa3JlHq}fblX aapH3.HTOB 6pa.nOCb CJIe,nYIOee: 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.HJIHHpH'lfeCKOA CHCTeMe KOOp- )I.HHaT {( r, z)- I'eOMeTpH). pacCqHTblBaJI8Cb c nOMow.blO anropHTMa, H3JIO)KeHHOrO B 9 3 npeeM: 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 npeCTaBneHHH. 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 KalOHX np" aTOM CHCTeM TpexTOqe'IHbJX pa3HOCTHbIX ypaaHeHHA ocyecTBnRJlOCb MeTO,l].OM nporoHKH. ).J;rH pac'ieTa Hcnonb30B8JJacb MHoronpoeCCOpHa.H BbJ'IHCJIH- TenbHaS CHCTeMa MBC-IOOO, YCTaHOBJIeHHasI B Me>KBeOMCTBeH- 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- rypaIDI npoeccopoB 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 pH8JIbHOM HaI1p8.BJIeHHH r. Ha PHCYHKax 60 H 61 npeCTaBJleHa scJ><I>eKTHBHOCTb nap8JI- JIeJIbHblX BblQHCJJeHHa B 3aBHCHMOCTH OT KOJIHtfecTBa Hcno.nb3yeMblX IIpoecCOpOB npH Pa3JlllQHblX KOHcJ>HrYPaIJ.U5Ix CHCTeMbl (pHC. 60) 
rlr. JO '0 " G DU 14 . "., 1] . IU 12 ..1,)) - -- '1 ..In) ,. ...u . ....n " ' . ..ua , ...., -- . ...,t  ....1 .- - '\ · -.  -  t · ... )  . - - - ....- ;::Ir -- - - ----   I.IOt _.,.. .,' - ".-  : .:. -.,.  ... .-  , t -; --.. ..-- 2 .... . . " . - -.......= .!I!  ._ - ..:t =-.i '...: :",  --" t. !.. -. 1 ..101 I, -, , ... .. .... _ J' ..".-.. . ".., .0 . 10 100 - 1 2 0 zit. . 20 PUt', 62. ()GHa.H l<apTUtlil ("J py.r: l'f)O/"n = O,U] 1 20 rlr. '5 10 20  z/r. . 10 o PHC, 63. BbIHcmrrenbHu 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>HrypH9 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. BHHLJ 3aMeTHbJe npeHMYecTBa B S<P4>eKTHBHOCTH lIBHO:A cxe- tAbl no CpaBHeHHIO C HeBBHoA (pRC. 61), qTO, Bnpo'tfeM, JJerJ<o nporHO- 3HpOBaJIOCb. .HBHhle CXeMhJ HMelOT ee OHO npeHMyecTBOJ CBSI- 3aHHoe c npOCToA JIOrHqecKoA CTPYKTypolt. OHaKO 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 BblXOOM Ha OHapHblA pe>KHM no HB- 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 YeJIBTh C03,AaHHIO 1lOBbJX napaJIJlhHblX 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. Bo 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 COCTOaag H3 1402 x 1002 Y3nOB. Ha 3TOA DhlqHCJlHTenbHoA ''wmtpeH-<poTorpac:pHH'' xopomo BHAeH ,D.HCK Maxa, B03HHKaroA 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) BOJIb OCB z. CpaBHHBalOTCH aHHble pac;tleTa H 3Kcne- pHleHTa [147]. BH,lI.HO xopomee COBneHHe n xapaKT€pHOM nOBee- 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 onpeenH1'b MHHHMaJlbHOe 3RaqeHHe nnOTHOCT" B o6nacTH ee cMnbHoro H3MeHeHHH. B CJIeyfOln.eM p ropa3O 60JIbllIeM no 06'beMYJ naparpacpe pac- CMOTpHM npHMeHefIHe KHHeTHtJecKHX cxeM MSI MO,lJ.e.nHpoBaHHSI HeycTaHoBHBIIIHXCSI, OCHnJISlQHOHHbIX TeqeHHR B cBepX3cYKOBbIX 1-1 TpaHC3BYKOBbIX nOTOKax BS13KOro raaa.  2. MOeJlHpoBauHe aeYCTaHOBHBillaxC1l u ocnJ]HHOHHLIX TeqeHIIH KJlacc HeyCTaHoBHBWHX TeqelIH:H npeCTaBnileT 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 onpee.neHHoR cPop- Me 06T€KaeMOI"O Tena. ,lJpyrHMH CJlOBaMH, npH onpeeJIeHHblX 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 cPYHaMeHT8JJbHbIMH CBoACTBaMH Ma- TeMaTHQeCKHX MO,ll.eJIeA, OIlHCbIBaJOIlI.HX TeqeHHSI B3KO["O raa&. I1paKTHQeCKaSl Ba>KHOCTb H3YQeHHSI TaKHX SlBneHH:A: OqeBHa. BbICOKHe j"POBHH nynbcaHA aBJ1eHHB "PH TaKHX pe>KHMax SlBJlSI- IOTCSI npHt.IHHOR nOBpeeHH5I MeMeHTOB KOHCTPYKHH H BbJXOa 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 nYJlbCHa BBnseTCSI Ba>KHOH CT)'IIeHhIO npSlMoro 'tHCJleH- 
Horo fO,I:\enHpOBaHIDI Typ6YJIeHTHblX HBJIeHHR. MaTeMaTHQecKoe MOeJlHpOBaHHe HeYCTaHOBHBWHXCR TeqeHHR COnpH>KeHO co 3HaqHTeJIhHbJMH TPYHOCTHMH. B nepoylO Ot.tepeh Hco6xoAHMO. 1.IT06bl B npoecce 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 ra3OHHaMH"IeCKHX nOJIeH B6J1H3H o6Te- KaeMoro Te.lIa. 9TO, B CBOIO o'tfepeb, npHBOHT K Heo6xoHMOCTH HC- nOJIb30BaHH$I BblconpOH3BOJI-HTeJlbHb1X MHoronpoeccopHbIX CRCTeM, {TO TaK>Ke HaKJIbIBaeT orrpeAeJICHHble OrpaHH1.feHH Ha Knacc npH- MeHSIeMbJX anropHTMOB. 113yqeHH€ pe3YJIbTaTOB tJHCTleHHoro MoenHpOBaHHH HeyCTaHo- BHBlIIHXCH pe}KHMOB, nonyqeHHblX c nOMOUtplO KHHeTHqecKHX cxeM) HaqHef C onHcaliH5I TeqemtR 06nH3H npOCTpaHCTBeHHO-BYMepHblX H TpcXepHbJX npHMoyronbHhlx KaBepH. 1. lfHCJlelIHOe MO,l(eJ1HpOBaHHe OCI{HJ1JIHL{HOHHblX pe.IKHMOB a6JlH3H KaBepHbl I1CCJleAOBaHHIO ny nbcaQHOHHblX SlBJIeHHA S6nH3B 2D H 3D Bbl- eMOK nOCnRIneHO MHoro pa6OT. I1HTepec K R3YQeHHIO nOBe,neHHH ra- 30HHaMHqecKHX uOJIeA OnpeM5IeTCH TeM, qTO, c OHOa CTOpOHbl, BbleMKa 51BJI.HeTCSI Oft-HUM H3 HaH6OJ1ee Y06HbIX o&beKTOB CCJIeAO- BaHMSI CPYHAaMeHTaJlbHhlX 3aKOHOMepHOCTeA TeQeHHS1 BH3KHX raaOB. C pyroA CTOpOHbl, npSlMoyronbH8$I BbJeMKa jlBJIHeTC5I peaJIbHOH Mo- AeJlblO ,lVIH MHorHX MeMeHTOB KOHCTPYKl.{HR neTaTeJlbHhlX aJIIIapa- TOB. IIpHMepo1 TaKHX KOHCTPYKIA 6YAYT npa60pHbre OTceKH, OT- ceKH wacCH, TeXIIHqecKHe WBbI, KOHQ>HrYPaQHSI, B03HHKaIOaSl npH OTCYTCTBHM o)J:tloro H3 3JIeMeHTOB Tenn03THoro nOKpbITHB NI5I KOCIHqecKHX Kopa6JIe:A THria ccwaTl'n'" UBypaH". THDHqHast KOIIcpH- rypaHSI 2D H 3D BbJeMOK H306paJKeHa Ha pRC. 65 H 66. B KatfeCTBe an:ropHTMa pacqeTa Hcnonb30BanOCh KCPC C KOp- peKHeR (rJI. IV,  5), BKnlOQaH ee MHoronpoeccopHblA sapH- aUT. JlH 3a,a.at.J B 2D-nOCTaHoBKe pacqeT 0 OCHOBHOM npOBO):tHJICSI Ha MHoronpou.eccopHoA BbIqUCJlHTeJlbHOA CHc'reMe (44 npoeccopa) Parsytec CC o61II-ett npOH3BOHTeJlhHOCTbIO 15 GFLOPS. 3aqH 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'JOHM 06p83oM. Ha neooR rpanHl\€ pa.cqerHoR 06nacTH 3aBanCSI npocPHJJb CKopocrH, COOT- BeTCTBYlOl.1J)I:A ,,'IaMHHapHOMY nOrpaHH'IHoMY CJ1010 [63}: u: =  ( ; ) _  ( ; ) 3 I (2.1) r)J.e 1.£00 = Moo - CKOpOCTh Ha6eralOerO HeB03MYID-eHHoro nOTOKa, 0 - TOJlHa nOrpaHHqHOrO cnO.H. B 2D-reoMeTpHH CKOpOCTb U H U oo 6bL11H HanpaBneHbI SOJIb 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 TBepLIX 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 CB06oHhJX rpaHHQax CTaBHnOCb "MSIrKoe" ycnoBHe CHoca: 8Q = 0, an Q = (p, U, 1J, w, E). (2.5) 
52. MO,.l{eJlllpoBaHHe JleYCTaHOBHBUlHXCR OC!(,HnJlRHOHHblX reqeHHR 239 B paccMaTpHBaeMblx SapHaHTax Ha6eraJOHR 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 OC1UlnIISIHOHHbIX TeqeHRSlX. KaK H3BecTHO, 3a DJlOXO o6TeKa- eMblMH TeJIaMII MorYT 06Pa30BbIBaTbCSI KaK CTaI.\HOH, apHble Tal{ H HeCTHOHapHbIe OTpblBHhle 30Hhl. B CTaHOHapHblX 30Hax B cneK- Tpe nynbcaHa 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 onpeenmoTCSI no xapaKTepHCTHKaM nYJlbca.J.Udt a.BJIe- HH5I. B 3'l'OM CJIyqae cneKTJ)LI nyJIbCa:A HMelOT SpKO Bblpa}KeHHble AHCKpeTHble COCTaBJ1BIOHe. l{a.K nOKa3blBaeT 3KcnepHMeHT, MaKCH- MaJlbHble nynbcau.HH B03HHK8JOT B oKpeCTHoCTSIX TOQeK OTpldBa H npHcoeHHeHHg norpaHHHoro CJlOSI. OCHo8Hbie pe3YJIbTaTbI 'IHCJIeHHOro MoenHpOB8HHjJ 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 MOeJIHpOBaHHe Typ6yneHTHblX Te"le- HHA: C HCnOJIb30BaJIHeM a.nre6paHqecKHX HJlH H<P<I>epeHu.HaJtbHbIX (\.toeJleA Typ6yneHTHoA BH3KOCTH npoe C BbIqHCJlHTeJ'IbHOA TO'q- KH 3peHHSI, qeM npHMoR pac"tJeT YP8.BHeHBit, OIIHCblB8JOHX Te1.felWe B3Koro ra3a. B "QaHHoR rJlaBe B OCHOBHOM npeCTaBJIeHhI pe3ynb- TaTbI 'IHCJle.HHOrO paceTa HecT8HoHapHoro 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:30RaMH'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 npeCTaBJIeHbI qacTOTbI nYJlLcH aBJlel-lHH B TO'tJKe A (IIpaBaB KpoMKa KasepHbl, nJlOCKOCTb CHMMeTpHM) AJlH 2D H 3D nOCTaHOBOK 3a,n.aI. 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 nYJlbcaHH ,L\a.BJIeHRR 8 nOJIoce qaCTOT WHpHHOA 1 r (ak, bJc - K03<1>cPHH- eHTY pa3nOeHH YHKH nynbcalHa 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 ce. 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 HCCJIeOBaHHSI nOKa3bJBaJOT Te >Ke oco6eHHOCTH 3D-Kone6aHHA [150). Ha pRC. 69 H306pa>KeHhl MI'HOBeHHbJe pacnpeeJIeliHH nOJISI CKO- pOCTeA BHy'rpH KaBepHhI .lt1ISI 'ieTbipex Docne,AOoaTenbHbIX MOMeHTOB no OCM a6CHCC OTJlO>Ke- Ha qaCTOTa 8 repax (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'lrPo ) 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-lnJ1HHOHHbIX 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- lOeMY nOTOKY II paCnOnO)f(eno B6JIH3H rrpaBoA KpOMKH KaBepl)W. XOpOUJO npOCJIe)KHBaeTOi lleCTaHOHapHaSl KapTHHa TeqelUfSl, KOTOpaH nepHO,lUlqeCKH nOBTOpeTCSl. Ba>KHOH 'iepTOA 3D-TeqeHH:A HBJ1SeTCSl nOHBJleHHe BHxpeA BOnb 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..  \ -' , \ .... ..04 \'\, " 0 ..as \,I 0.111  ,,- - - ? / '" ...1  - -....  ....  - ..01 2.4 a.I .. 3.0 PHC. 71. noPJJ06HM K2.pTHHa Tee- HRR B neBOM HH>KHeN yrllY BhleMKH o 2 o o , .. _ _ , J . . - . ---- -, ...". .,/' :::;..r..-...... / /' /' ,. , ,I /. - _::: f i     1 r"r'"p_,,- o II..,... ",. , ". . ""'" '- ...  .... ..... .. .. - - .. - -  . , . '" JiAL'" --.-.. -. ..... ...... __., , 2  PHC. 70. BexropHble nons CKOpOCTH H 2D-cnY'l8JI PIlC. 72. CTpYKTYpa Te'leHHR 8 no- rpaHH'IHOM cnoe nepeA OTphIBOU BpeMeHH. B3a>vfOeAcTBHe OTpblBHOrO CJIOB CMemeHHfl C BHemHHM DOTOKOM npHBOHT K nepHOH1.{ecKOMY OTBOY H nOABOY Macchi H3 KaBepHbJ. B TOT MOMeHT BpeMeBH, KOra B OKpeCTHOCTH "paso A KpoMKH BbJ- eMKH TOqKa npHcoeHHeHHSI CJlOSI CMemeHH.H CMelaeTCSI HDJKe KpaJI KpOMKH, ocyeCTBnSleTCSI nOBO 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, CYecTBY1OT ew:e ABa BH- XpH) OHH - B cepeHHe BhleUI<H. pyroA - S6JJH3B npaBoA CTeHKH. 3TH BHXpH nepeteaIOTCB no qacOBOft crpeJ1Ke. B onpeeJIeHHbla MOMeHT H3 3THX yx BHxpeA 06pa3YeTCSI OHH) KOTOpbrlt 3axBaTbl- BaeT 60JlhWYlO 'IacT& nOJlOCTH KaBepHhI. KpHTH"tIecKHA nepen 8.B- neHHjI, Bhl3b1BaJOHft OTphlB no- rpaHHHoro cnOg nepe BeM- KOH, Man. n03TOMY, Kora K JIeBOMY KpalO npHXOttHT BM- Ra CTHg, C03aeTCB nepe- na,A ,l.{aBJIeHHS1, npe8bIWaJOHA: KpHTH'tJeCKHH, BCJIe)J.CTBRe \Ie- ro npOHCXOHT 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: OHH - npH HaJIH-IHH OTpblBHOA 30Hhl t ;WyroA - npH ee BbJpO>KAeHHH. T8.KOA >Ke xapaKTep 06pa30BaHHR Kone6a- HHA Ha6JlIOaeTCH H B 9KcnepHMeH'l'8.JI])HblX HCC1IeOBaHH.HX. npe- CTaBnellHblX B pa60Te (150). BeplIeMCSI K pe3YJlbT8.T8.M qJlCJIeHHoro MO,n;eJIHpOBaJlHJI 3D- KaBepHbJ. Ha pRC. 73 npeCTa.BJ1eBO pacnpeAeJIeHHe Ko<pHeH- 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 . I.. c = 2(1' - Poo) p P u2 · 0000 (2.6) MO)KHO BHeTb, lITO Ha 60JIhweA 'IacTH Ha aBJIeHHe npaKTHQOCKH nOCToBHHO, a 3aTeM pe3KO B03paCTaeT B6JU13H npasoro KpaSl BhleM- KH. TaKoe nOBeeHHe aBJJeHIDI COOTBeTCTBYeT 3KcIIepHMeHT8JThHblM MIHbIM, npecTasJIeHHLlM B pa60Tax [156, 157J. PacCMOTpHM BJIHSlHHe CKOpocTH BHeWHero nOTOKa Ha nYJIh- caHH 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 CTaTl1leCKOI"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'hcal-fA anJleHHH B 1'01IKe A "I)aBoA BeIJxHCH KpOMKH. OTMerHM, t.ITO npH yae.rIHQeHHH Q"rCJI3 laxa RHeilIHeN) nOTOKc1. C Moo == 1.35 AO Moo = 2.1 afnn"TYAa nYJlbcalHj;t YMellblllaeTCH, a npH fv100 = 3. 7 KOJIe6aHH Boo61Ie OTCYTc"rR)O'r. 9TH pe3y.rILTaTbl XOpOITIO corJlaCY'OTC c aBI-IO H3Bec'rHbIMH 3KCnep1tMeH1blfbIMH )aHHLIMH 1158, 1591, H3 KOTOphlX cne,lO'eT, ITO Ma.KCIIMaJlbHaH I-tHTeH- CHBHOCTb KOJTe6aIlHH Ha6JTIOlaeTCiJ 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 Toe A "PH 1oo = 2.1, Re = 3.3 .10 4 J 6/h = O.lJ l/h = 1.05, w/h = 0.5 Kone6aHHH Bo06e He B036yaJOTCSI. Ha pRC. 77 npeCTaBJIeH rpacl>HK nynbcHH aBJIeHHg ,lVISI BbleMKH c pyrH{H reOMeTpWlecKHMH napaMeTpaMH l / h = 1.05, wi h = 0.5. I{o.He6aHHg )VIS 9ToA reOMeTpHH, B OTnH'lHe OT npe- .D.bIycro 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 npeC'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, Kora 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 BOJIb OCM OX H BYX APyrHx HanpaBJIeHH:A. B npoBeeHHOa cepHH paCQe1'OB o5ee HCJIO TOqeK rrpocTpaHCTBeHHo CeTKH co- C'lBJIHJIO 4 . 10 7 Y3JIOB. ECT€CTBeHHO) "11'0 AJlSI peuleHHH 3TOH 3a,natlH HCnCl.lb30Banacb BbICOKOIlpOH3Bo,.QHTeJlbHaH Co 1 TFLOPS nHKoBo:A MOI.1I)iOCTH) MHO- 
2. MO,lCeJlHpOBaHHe neYcmHOBJfBUIHXCH OCHJlJlJllQfOB1lblX Te'leHIIR 247 1M 10  60 u C c» .-  40 a o o 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 ronpoeCCOpH8JI CHCTeMa. Ha pHC. 79 npHBeeHa 3<PcI>eKTHBHOCTh napanneJIbHoro pacqeTa 3D-KaBepHhI B 38.BHCHMOCTH OT 'IJIcn& npo- ecCOpoB. BHHO, 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%. OHaKO, Y"IHThiBaH, tJro HCnOJlb3yeMaJI njJ pacqeTa, KCPC c KoppeKeR 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 paceTHI:dX TOqeK. UpH 3TOM, B CBB3H C OTHOCHTeJIbHblM YMeHhweBHeu 'tJHCJI& nparpaHH'lBhlX TOqeK B noo6nacTIDC, 3<I><j>eKTHBHOCTb napanJIenLHOI 06pa6oTKH B03pac- TaeT. Kax nOK83aJIH pacqeThJ, "PH He6onLmOM OTKnOHeHHH yrna t/J OT HYJIeBOrO 3Ha'leHHH o6aSl 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" Ha6JIJOaerCH OpH t/J = O. 9TOT 
t. - Yron H81U1OH8 cf D Yron H8IC11OH8 2° . Yron H8KI1OH8 4° 200 1. i .........t. -oJ 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, CBHe- TeJlbCTBYWW.HM 06 OCJIa6J1eHHH o6paTHoR CBSl3H Me>KA,Y nepe)\HeR H 3HeA C'I'eHKaMH K8.Bepllbl lIP" "KOCOM)) BAYDe. IIpHBeeHHbla Ha pHC. 80 pe3YJlbTaT BBneTCSI rIOATBepAeHH- eM BbJBOa (rn. III) 0 KOppeKTHOCTH KB33Hra.30.lI)lHaMH'IeCKoA: CHeTe- Mbl H KHHeTHqeCKHX cxeM. BHHO, '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. IIoTBep)KAeHHeM KOppeKTHOCTH npHMeHSIeMoA MoenHJ a TaJ(- )Ke aKTHBHoro HCnOJIb30BaHHH pe3ynbTaTOB, nOJly'tfellHblX B BYMep- HOH nOCTaHOBKe, c:nY>KHT CODOCTaBJIeHHe cneKTpOB nynbit 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 npOOJIbHble 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. rrpeCTaBJUleT(SI BecbMa HH- 
g:.e. MO,lJ.eJlHpOBaHHe neYCTaHOBRBWnXCH OClJ)fJ1nRHOHHb1X 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 (TOKa 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, npHXOSIag K nepeHeA 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 onpeeneHHIO 'IaCTOTHhiX xapaKTepHCTHK nYJ1b- CaIJ.HH. Ha pHC. 82 npeCTaBneHhI cneKTpbl Kone6aHuA ,AaBJIeHH5I B Pa3JIH\{HbIX TOQK8.X npaaoA BepxHeA: KpOMKH. BHO, qT() MHHH- MaJ1bHble YPOBHH 3BYKOBOro a.BJ]eHHjf ,n:OCTHralOTCS1 B WIOCKOCTH CHMMeTpHH. IIpeCTaBJlseTCg TaK>Ke ecreCTBeHHhIM 60nee BbICOKHit ypoBeHb Kone6a.HHR B nOBeTpeHHOM yrny no cpasHeHHIO C HaBeT- peHHblM, Tal( KaK OCHOBHoe BH>KeHHe nOTOKa HanpasneHo B 3TY CTOPOHY. OrMeTIfM, qTO Ha6JIJOaerCH OqeHb xopowee COrJI&COBaJlHe pacqeTHb1X 3HaqemrR OCHOBHbIX qacTOT C aHHblMH 3KcnepHMeHTa (150]. COOTBeTCTBYIOHe pacqeTHLIe H 9KcnepHMeHTaJlLHble 'lHCJJa 
,..  tlO I '" 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 nopeHHOW yrny, 3 - D neBOM HaBeTpeHHOM yrny CTPYxanSl 6b1nH paBHbI (155]: ShF C = 0.307, Shc = 0.616, Shr 'c = 1.22, ShfP = 0.3, Sh;ZP = 0.62, Sh = 1.2. 2. BJlHRHHe cpopMbl BlJeMKB H3 XapaKTepHCTHKH nYJIbCaHoHHoro TeqeHHH. KOJIe6aHHH B6JIH3H HJIHHAPHqeCKHX H cljJepH'CfecKHX TeJ1 B03HHKalOe npH 06TeKaHHH BhJeMOK H ,lij)YnIX TeJI nynhCa- HH B 60nbmHHcTse cnyqaes HOCSIT HeraTHBHhlA xapaKTep no CBO- eMY BnHHHHIO Ha MeMeHT&I KOHCTPYKHH. TI03TOMY peKOMeHaJHH no nOa.BJIeHHIO nYJ1bCHI npeCTaBJISI1OT 3Hat.JHTeJ1bHblA npaKTH- '1ecK"A HHTepec. B npeJJ-bJJl-yeM nYHKTe naparpaQ>a 6h1J1o nOKa3aHO (cpaBHeHHe pe3YJIbTaTOB, npecm.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 nynbcH- OHHOrO TeqeHHSl. a) 6) B Moo a Z ... H. Hz MOl) L B H A ,D PHC. 83. CxeMbI pacqeTHLlX o6nacreA: a) ABYMepH8.SI KaBepHa c BhlCTYD8JOea 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 MoenHpo- BaHH.H [160J CpaBIiHBaJlHCh C aHHbIMH 3KcnepHMeHT8JIhHbIX HCCJle- ,.qOBaHHA. 3aTeM 6YT paccMoTpeHbI pe3YJIhTaTbl MoenHpoBaHHH nYJIb- CHOHHblX TeqeHHA: 86nH3H nHH,Zq)HqecKHx TeJI c BblCTynaJOlMH HrnaMH [89J, [161}-(163). 9TH paCqeThl npOBOHnHCb 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 MHoronpoeccopHl:de TpaHcnbIOTepHble CTaHHH (141). Pe3YJIbTaTbl 9THX pacqeTOB IJ)IJIHH- ,DJ>H'IeCKHX Ten TaK>Ke CpasHHBaJlHCb C aHHbIMH HaTypHbJX 3Kcne- pHMeHTOB [164)-[166J. B KOHe KOpOTKO onHmeM HoseAwHe pe3YJlh'mThl) CBB3aHHhIe c MOeJIHpoBauHeM 06pa30BaHIDI BHXpeBOA Opo>KKH n03 c<pe- phI. 3eCb 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- BOHT K 3aMeTHoMY CHH1KeHHIO MaKCHMaJlbHOrO ypoBH.H )J.HCKpeTHoA rapMoHHKU. ITpH 3TOM '-IaCTOTbl OCHOBI-IbIX H HcKpeTHblx rapMoHHK He3HaHTeJIbHO 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 npeCTaBneHbI 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- 6nlOaeTCSI xopowee COBneHHe 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 cyecTBeHHO CHH:HmeT ypOBeHh nYJl&CHH. npu Z/ Hl = 0.5 nYJIL- CanHH Boo6IJJ.e 3aTYxaJOT H o6TeKaHHe CTaHOBHTCSI CHOHapHblM. TaKH1 06pa3oM) H3 pe3YJIbTaTOB MoenHpOBaHHSI BHH01 '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 IIpHBee}IL1 Ha pac. 87 H 88. HCCJleOBaHHe p83nHtJ}lblX pe)J(HMOB Te- qeHHH ra3a.B6JlH3H TaKHX KOHCTpYKU.HA npeCT8BJIHeT 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 HaJIHlHe 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 TaKe HO- HapHO. IIpH 0.3  (J < 3 Ha6JIIOaIOT01 HeC'J'8.QHOHapHhle nepHOAHQe- CKHe pe)l(HMbI 06TeKaHHH ABYX Tanos. npH KOJIe6a.HHSIX nepBoro poa 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. IIpoecc I1ynLCaQHA BTOporo poa, HMelOIIJ.HA: MeCTO npH 0.3 < 8 < 1.4, 3HaQHTeJILHO c.nO)l(Hee. B nepBoA <l>a3e rrpoecca npo- HCXO)J.HT pa3pyweHHe OTpblBHOA 30KY) BO BTOpoA' - poCT DHOBI:. 06- pB30BaBWeRc OTpbIBHOA 301'lbI) conpoBO)l(alOH:AaI nOTepeR Ko- HHqeCKOA <POpMbl 3a CqeT e<popMHH ee nepeHe:R I.f2,CTH. npH OCTH>KeHHH OTpblBHOR 30HOB MaKCHMaJIhROrO pa3Mepa nepHO,l{ KO- 
92. MOAeJJlfpOBaHHe neYCT&HOBlUJllUfXC1I OC11JlJlR1(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 Boo6e He Ha6nIOA8JOTCiI, a ,lVISI Han830Ha QHCen PeA- HOJIbCa Re = 5 · 10 3 + 10 6 xapaKTep KOJIe6auH1t HOCHT 6nH3KHR xapaKTep. OnHlUcM pe3YJIbTaTbI paceTOB 06TeKaHHSI TeJla C HrJloA: (pHC. 87) co CJIeYlow.HM" napaMeTpaMH BHernHcro nOTOKa H reoMeTpHqeCKH- MH pa.3lepa.MH Tena: Moo = 6; 8 = 1; Re = 8.4 · 10 3 . BLI60p 3TOro OTHOCHTeJIbHO He6o.nbworo lIHCJIa PeAHonbca 06ycnoBJIeH CJle1O- 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 "'10CJlHpOBaHHe B03MO.>KHO H& OTHOCHTeJJbHO rpy6bIX ceT- KaX, 1.fTO 6bJno aKT}'aJIbHO, nOCKOJIbKY pac'tJeTbI rrpOBOWlHCb 8 kOH- e 80-x I'O)).OB, Ha Bbl'lHC'JIHTeJIbHhJX CHCTeM&x He CJIHWKOM BblCOKoA nPOH3BOHTenbHOCTH. 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 yapHa}] BOJIaR OCTH- meT TOpl\a nHHpa (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 yapHbJX BOJIH (pHC. 91), KOTopaH XopOWO Ha6nW,lJ;anaCh B 3KcnepHMeHT8X. 3Ta KOH<pHrypaHSI 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 yapHoA BOJIHOR B03HHKaeT OMacTb nOBbIIDeHHoro aBneHHH. OT- pbJBHaH 30Ha npOOJI)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 TOpla. I10CJIe 3TOro Becb npoecc nOBTOpSleTCSI BROD!>. 
,. (J 1fJ 2D JII IIiI SII . PHC. 89, lb06aphl B nepBOA <pa3e nynhcaQHA ,. III , PRC. 90. H306aphl B Koe 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 COBnaaeT C Ha6mo,l'(aeMbJ M 3KC- neplIMeHTaJIbHO KOJlc6aTeJlbHhIM pe>KHMOM BToporo poa. OTMeTHM, "liTO IlpH pyrHx reOMeTpHqeCKHX pa3Mepax, OTBe1.laJOD1.HX 3HaqeHH- HM rlapleTpa () < 0.3, 8 > 3 Ii 1.4 < (J < 3, B pacqeTa.x, B COrJlaC"H c a.IHblMH 9KcnepHMeHTa, Ha6nlOaJlHCb 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- 6J11OaeMbIR ypOBeHh nYJIbcaQHA. 3TOT <paKT nOT8ep)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 p83eJla OTMCTHM cneYIOee. B I-lacToSlw.ee BpeMg Ha 6a3e KCPC C03aIOTCSl fiaKeTbI MHO- ronpoeCCOpHbJX npHKJIHbIX 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 CJleAYIOIIHe: 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)HHhIe MOMelITbl BpeMeHH 
3pa.nbHOa CeTKe, COCTOge:A H3 70300 y:mOB. CeTKlt cyecTBeHHO crYaJIacb OKOJIO nOBepXHOCTH C<j>ePbI H B CJIeAe ]a neD. AnSI pac- qeTa HcnOnb30BaJIOCb 25 npoecCOpoB CHCTeMhI MBC-IODO. PHC. 92 xapa.KTepH3YeT KapTHHY TeqeHHSI n03H 06TeKaeMO- ro TeJla. llpH qHCJIax PeRHonbca, 60nhWHX 700, TeqeHHe BOKpyr wapa HMh3H ct.JHTaTh perYJISlpRhIM H CTporo nepHo,n;HqecKHM, tTO H nOTBepaeTCSl AaHHbIM pacqeTOM. 06nacTb OTpblBa DOTOKa HMeeT CJIO>KHYIO C'l'PYKTYPY. rpaHHa 3TOa 06naCTH, nOKa3aHttaR Ha pRC. 92 B ABa pa3HblX MOMeHTa opeMeHH, onpeeJI$leTCS1 cpOPMaJIbHO, HCXOAjI H3 nOBeeHH.H perneHHH 3a'lH (167). 3. B3aHMoejCTBHe OCl{HJlJ1HPYIOlI{HX TeIJeHHll rn33 H ynpyrHx IlOBepXHOCTell «I>eHoMeH B3aHMOeaCTBIDI DOTOKa ra3a H ynpyroA nOBepXHOCTH 06TeKaeMoro HM Tena llonyT.JHn Ha3BaHHe np06JIeMbI a3poynpyrocTH. PeweHHe 3TOA npo6J1eMbI Qpe3Bbl'laAHo BHO JlSI coBpeMeHHoA a3poHHaMHKH. He CJlyqaAHO ee HCC1IeOB&HHIO - TeOpeTHqeCKoMY, 3KcnepHMeHT8JIbHOMY H C IIOMOI1t.L1O M&TeM&THqeCKOrO MoenHpoBa- HHB - 3a nOC1le)nlHe 40 neT nOCB.HeHbl MHoroqHCJleHHble ny6JIHKa- HH. YKa}KeM HeKOTOpble H3 HHX: [168]-(173). OTMeTHM, QTO MaTeMaTHQeCKOe MoenHpOB8JlHe 3aq as po- ynPYI'OCTH CTaJlKHBaeTCH C 60J1bWHMH TpYJJ-HOCTMH H MO)l(eT 6blTb OTHeceHO K CllHCKY npo6neM "Grand Challenge". CJJOiKHOCTb npo- 6J1eMbl CSB3aHa C TeM. '1TO Ma.TeMaTH1.feCKaSJ MOJJ-eJ1h, onHCblBalOee 3TO HBJIeHHe, BKJllOqaeT B ce6g KaK ypaBHeHHSI ra30BO:A HaMHKH, TaK H ypaBHeHHSI ynpyroCTH. 9TH ypasHeHHH pewalOTCH COBMeCTHO npH yeTe no,nBHJKHOCTH rpaJmJ TBeporo renal Oc06bJ:R HHTepec C TOqKH 3peHHSI aapoynpyrOCTH Bbl3bIBaJOT HeYCTaHOBHBlliHecs ocu.HnJIHOHHLle nOTOKH. KaK BHO H3 npe- blAYHX MaTepHaJIOB nap arpacpa , B HeYCT8HOBHBmHXCSI TeqeHHHX BB3Koro ra3a Mor)'T B03HHKaTh OCJUll:tHOHJnde pe)KHMbJ c OCTa- TOqftO 60nblIlHMH aMDJIHTYAaMH KOJIe6a.HHA )J.aBJIeHHB. 9TH nepe- nbl aBJIeHHB, 803eACTBYSl Ha ynpyrHe CTeHKH KOHcrpYKHH, 8 CHJ1Y B3aHMHOro BJIHSlHHJI ra30BoR cpeJ H TBepOa 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>oPMHH H a>Ke pa3pymeHBIO KOHCTpYKHH. npoAeMoHCTpHpyeM BO MO>KHOCTH KDHeTlAecKHX M . cxeM "PH MOenHpoBaHHH 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 npoeccoB KOJIe6a.HHA B TBepOM Tene HCDOJIb- 3YeTe WHpoKO RpHMeHReMbIA: MeTO KOHeqHhlX 3JleMeHTOB (176]. B 3TOM CJlyqae CHCTeMa ypasHeHHR, OIDICbJB81OID.mc nepeMeeHHe a, MO>KeT 6hJTb 3anHCaH& B BHe L H 'L JC x=L M8 + D6 + KeY = F, (2.7) r,Ae M - MaTpHa Mace KOHeqaLIX 3JleMeHTOB, D - MaTpHa ewn- cPHPOBaHHB, K - MaTpHa >KecTKOCTH. F - BeKTOp HarpY3KH Ha nOSepXJIOCTb TBeporo 'reJIa, 6 - BeKTOp Y3J10BblX cMew.eHHA Ha. 3HaqeHHH MaTpHQ MacC H >KecTKOCTH CHCTeMl:d onpeeJ1S1IOTCSI reOMeTpHeR KOH<:rpYKHH, a Ta.K;)I(e CBoACTBaMH Ma-repuana (nJlOT- HOCTb, KOHCTaHTbl ynpyroCTH). MaTPa eIdD4>HpOB8HWI oope- )J.eIDIeTC.H 3MHpHqeCKHM nYTeM, no H3MepeHHSlM 3&TyxaHHH eCTe- CTBeHHblX KOJIe6aHHD. YHTbIBaB npH6J1H>KeHHOCTb OnHCaHW1 Mexa- HH3M& eMn<t>HpoB&HHH. qaCTO (176) B KaqecTBe MaTpbJ 0 6epYT JIHlleltHYIO KOM6HHHIO MaTpH K II M: D = o:K + 13M . (2.8) YpaBHeHlfiI, onHCblBaJOe ABlDKeHHe ra3a, H ypasHeHMB, onH- CbI88J01Il.He KOJIOOaHHB nOBepxHoCTH Te1Ia, CBS13aHh1 Me>KA)' c060a 
CJIeIOIqHM 06paaOM. 113 ypaBHeHHA" ra30BoD HHat.tHKH onpeen.Sl- IOTCH KOJ1e6anUB ,lI.a.BneHHSI ra3a B6nH3H nosepXHOC'rH, C nOMOblO KOTOpbJX H Q>OpMHPYeTCSl BeKTOp HarpY3KH F. B CBOIO O'lepe, H3 ypasHeHI1S1 (2. 7) HaxOHTCj1 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 BJIeTCH 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 YeTe nOIJ;BH>KIIOCTH rpaHHQ ... npH pace're ypaBHeIIHH, OI1I-ICblBalOlHX H)I{eHHe ra3a. PacCMOTpHM (pHC. 94) OCTa- TO(HO npoCTYIO nOCTaHoBKY, orpa- HHIHBaSiCb 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 TpaneQHeBHHhlet a OCTaJIbHhle npSlMoyronbHbJe. SI aJlnpOKCH- MaHH ypaBllellHA, onHChlBalOX H)f(eHHe ra3a ria TJ)aneQHe- BH)J.HblX SI'ieAKaX HCnOJIb3YeTCB TOT }ICe caMblR noxo;( nocTpOeHH.H KCPC Ha HOCTPYKTypHpoBaHHblx ceTKaX (rn. II-IV). 06paTHMcSI BHat.f3JIe K pe3YJThTaTaM pacqem 'iHCTO 10HHa.MH- qCCKoA 3a;::{a'-lR C nOBH}KHbIM Kone6JIIOHMCH HOM KaBepHbl. AM- 11JlHTYAY H qacTOTY 3THX KOJIe6aHH 6yeM C'tIHTaTb 3aHIIOH, 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 nOBeeHHe ra30BOro B c hi A o nOTOKa. · B ,AaHHoR MOHorpaHH MY He 6yeM 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 npHBeeHHH ee K AHarOH8JlbHOMY BHAY. 
9 2 . lYIOAeJlHposanHe HeYCT&IlOBJlBUIHXCJl OJlJlJlsmHOH1fWX reeHHR 261 1SO uso - ... - - -- - a SPl ,.0 ----- 120 .P".- .. --- -- 100 0 40000 10000 120000 160000 200000 I (r-.) 180 160 140 SPL 120 100 SO 0 40000 80000 120000 160000 200000 /(riU UIO 160 SPl. 140 120 '00 80 0 40000 80000 120000 16OCIOO 200000 I(r. , +_____._ __ __ 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-1teJIHX KOIIe6ambtx ABa: 6) 38 Kr, B} 54 Krl\, r) 74 J(I' 
HCnOnb30BaJlHCb CJIe)U'IOHe napaueTpbI BHeWHerO DOTOk8. B03- AYxa H reoMeTpHQeCKHX p83MepoB KaBepHhI: 'IRCJlO Maxa Moo = 1.35; \fHCJIO PeaHOJIhca) 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.tHyeHHbIx KOJIe6aHHH He npeSblm8JJ& 10- 3 H. t{acTOTbI 3THX OCIJ.H.J1nSIHa Bb16HpaJIHCb cne)U'IOHe: 11 = 38 Kr, /2 = 54 Krl.\, /3 = 74 IU. Ha pHC. 95 H306paJKeHbI cneKTphl yposlUI nynbcal.\HA aBJIeHHH (B Ael\H6eJIax) 8 TOqKe 86nH3H cepeHHbI Ha KaBepHbl. ,n:nH CJlyqas >KecTKoA CTeHKH (pUC. 95,a) xopowo BHbl )!.Be OCHOBHLJe qacTOTbl nYJlbCal..\HA c COOTB eTCT BYIOMH 3HaQeHHSlMH 38 Kr Ii 76 KrU.. Ha pHC. 95,6 npe,l\CTaBJIeHbI aHaJlOrH'IHhle aHHble, Kor,qa qa- CTOTa Bb1HY>KeHHbIX KOJIe6aHHD /1 = 38 Kr COBnaer 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 Ha6nlOaeTcsr (pHC. 95,r) npH \lacToTe BbJHY- >KeHHblX KOJJe6aHHA 74 Kr. 06paTHM BHHMaHHe H&AYDJleT BTOpoR OCHOBHOR qaCTOTbI. Ero nOSlBJIeHHe xapaKTepHO AJI5I CJIyqaB t Kora qaCTOTa BbIHY)l(eHHhlX KOJIe6auHI 6nH3Ka K c06CTBeHHoa. BecbMa HHTepeceH cnY'IaA, Kora 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 aMnnHTyoA, ;J.ocTHraJOw-eA 150 B. Cne,LU'eT OTMeTHTb, qTO TOT >Ke xapaKTep cneKTpa Ha6.TIlOaeTCB H 8 npaBOM BepXHeM yrny KaBepHbI - TO'IKe, aJIeKO OTCTOHID.eA OT KOJ1e6J11OeroCSI Ha (pRC. 96). TaKHM 06Pa30M, BHlIO) 'iTO ,lVISi HeYCTaHOBHBDIHXCB pe)J(HMOB rrpH HeKOTopblX qaCTOTax BbIHY>KAeHHblX KOJ1e68.HHA aJKe He6OJ1L- woD aMRJlHTYJJ.bI, THn pemeHHj{ MO>KeT cyeCTBeHHo 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 l(ruJ PRC. 96. CneKTp ypoBHH nYJlhcUQd AUJleHHJI B DpaBOY BepXHeM yrJlY Ka8epOhl (qacTOTa BhlHY)KAeHllhiX Kone6awtR I = 54 B:r) 10 h. h, -.-.-- uenOJ\RH*Hoe JU 10 J '10 ..Jta Q.. (/J 100 10 o 4OGQ) I000O taGOOO 160000 200000 I (ru) PHC. 97. CpuHeHHe coeKTpoB nYJlhcaQHA eJlHB AIlS J(asepHbl C HenO,IUSH)I(- HhlM (nYHKTHp) H OC)J.HnJIHPYI01l\HW HON (conowHaJi JlHHHS) 1eo eo .L ----- . 140 -I 120 0. (/) 100 80000 100000 f erR\) PHC. 98. CneKTpbl nYJI))C 3.IJlA aBJIeHHJI AJlJI pa3J1H11J1WX 'lWlII\HH CT&II&HOro ttHa: hI = 2 . 10- 14M (nyHKTHp), 1&2 = 3 .10- 4 MM (cnJlOIDBU JDUDlJI) eo o 2CUX) ..0000 eoooo 
llpe,a.CTaBneHHble Ha pHC. 95 H 96 pe3ynbTaTbi no CYTH eJla nOJlyqeHbJ nYTeM peweHHSI TOJIbKO ypaBHeHHA, onHCblBaJOX Teqe- HH}1 ra3a c nOABHIKHOH rpaHHeD, 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 CJIYaH Bh:HY>K.u.eHHbIX Ko- ne6aHHH. PaCqeTbl npOBO)J.HnHCb JIB pa3nHtJHblX TOJIHH Nla h H P83JlHQHblX MarepHaJIOB (CTaJlb, lOpaJIL). Ha pRC. 97 npeCTaBJIeHO CpaBHeHHe cneKTpOB DYJlbCHH aB- JleHHSI ra3a B TOQKe B6JJH3H cepe,AHHbl AHa )VIH CJlyqas fle[lOBH}I(Ho:A nOBepxHocTH H "PH Y"IeTe B3aHMHOro BJlHjJHH5I Kone6aHHA ra30BOH <Pa3bl H nOBepXHOCTH (CTaJIb TOJImHHoA hI = 2 · 10- 5 MM). BHO, QTO OCHOBHble Kone6aJJH.H OCTaIOTCSI HeH3MeHHbIMH no qacTOTe D 8,M- nnHr)re. O,l\HaKO <POH Kone6aHHA HiUKe B yqae OCJ1HpY1Oero )J;Ha. npOHCXOHT 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$1, KOTopaH DOJIY'feHa nH cnyq8JI HenO,lI.8HmHOH CTeHKH. 4. !vfO,l1.eJIKpOBaHHe Typ6YJIeHTHblX HeYCTaHOBHBUlHXCH 1'eQefIHil B OKpeCTHOCTH K8BepHbl npHBeeHHble B nepBblX nYHK'raJ( naparpa4>a pe3YJIbTaTbJ OT- flOCHJIHCb K MOeJIHpOBaHHIO JIaMMHapHblX HeYCTaHOBHBIIIHXCSI Te- '-JeHUH raJa. BMecT€ C TeM KCPC npeCTaBJIHIOT BC3MO)KHOCTH H ,lI.JISi MOAeJlldpOnaHHSI Te'4eHHA TYP6YJIeHTHblX (178]. HH}Ke, ODHp8.SlCh tIa a.HHb)e pa60TbI [178], 6YAYT npHBeeHbI HeKOTOpbJe pe3YJlbTaTbI MOeJ1HpOBaHHH OCHJJ..1HPYIOIl\HX Typ6yneHTHbIx Te'leHHA o6nH3H npSlMoyrOJlbHoA 20 BbJeMKH. B KaqeCTBe anropHTMa pac'leTa npHMeHSlJlaCb KCPC C KoppeK- eR, 8 KOTOpOft BMecTO ecTeCTBeHHoA BH3KOCTH 1£ HCDOnb30BaJlacb Typ6YJIeHTHa.H BH3KOCTb J.lt. B CBOIO o'{epeb, ,lI)IH HaxO>KeHHjJ JJt 6bIJIa npHBJleqeHa anre6paHQecKaH Moe.nb Ce6e'lH-CMHTa, KOTOpaJI 
92. MOAeJlHpoBaHHe HeYCT3HOBHBUJJfXCSl OCl.{HJIJllIL{HORHblX reqeHH:It 265 onpeeJISlJlaCb CJIe,lU'IOw.HM 06pa30M. ,L(JIH 06J1acTH B6JIH3H TBepoA nOBepxHoCTH npHMeHSlJlaCh <PoP- Myna Pt = pl2 ( : ) 2 + ( :: ) 2 . (2.9) B CBOIO Oqepeb, ,!VIS HaxO)KeHHg ,[I.JIHHbI nYTH CMeWeHHSI l BOC- nonb3yeMcSI BbIp8JKeHHeM 1 = kzDr, (2.10) re k = 0.41 - nOCTOHHHaB KapM8.H&) Z - paCCTOSlHHe OT TBepoA nOBepXHOCTH, Dr - cl>8.KTOp 3aTYxaHHSI BaH-pHcTa Dr = 1 - exp(z$ fA), (2.11 ) z. - 6e3pa3MepHOe paCCTOSlHHe OT CTeHKH, A = 26 - AeMncpHPYIO- 3.)I KOHC1Ta. 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$ - KoopHHaTbl p33en5l1OeA 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 nponopOHanbHa paCCTOBHHIO OT 
CTeHKH B CTeneHH 1/7. ,LVuI TeMnepaTYpbJ Hcnonb3CBanoCb COOTHo- weHHe, H3BecTHoe KaK HHTerpan KpoKKO (63). PaCClvtOTpHM pe3ynbT8Tbl pacqeTa 2D-k8.BepHbl co CJIeIO- 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-£)-MOeJllIMH TYp6yneHTHoft BSI3KOCTH [180J. piP. , II - Ml"'HOKliIlOC ..__.as lit  I .. I  1.0  DI , 'J Ii I 200 . i" '. - 2St) 100 I 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 HecHoHapHbJa xap8.KTep. Ha pHC. 100-101 npe- CTaBneHbI pacnpeeneHIUI cYMMapHoro ypOBHR aKYCTHqecKoro aB- JIeHH Bonb ,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 MOeJlblO TYP- 6yneHTHoR BH3KOCTH BnOJIHe KOHKypeHTocnoco6Ha c 06I1.I-enpH3HaH- HblMH 8JlI'e6paHQecKHMH H H<l>epeHIJ)fanbHblMH MOeJlSlMH 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 MoenH ryp6YJ1eHTHOA B.H3KOCTH 06JIa.IOT 06I1J)fMH 
'80 - ,..... o bperintId --- ... .-. - ..,.. t10 -. ,. ,I.. .-_......... - .- -.-.---.- ... '80 . . i '-' ..... ,4 ....... .' .. , " , '.." ,-- .-' ..' .,' ,,' 1<11'.- _.."-'". '. 'O. ..J a.. U') uo 130 0-0 Q.l 0.4 o,s 0.1 1.0 XJL PHC. 100. Pacnpe.AeneUHe cYMMapHoro ypoBHJI 8KYCTH'IecKOro .o.asneHJIJl B):\WJb AHa Bhlet.1KH 190 - ....... -- ... 1. .__._ CI _I """"' 110  "-'  IdO - -"\ ....----......-  ..---....-----------------.------...--... . .-. .......-...-.- cr- ---- ..-'-..- -. _ ...1\ a 1-'0 I. .1.0 4,1 .." .0.4 00.1 0" YIH PRC. 101. PacnpeAe.neHHe c)'MwapHoro ypoBH8 aKYCTHT.lecKoro ,l1.&BJIeHHS QIIh npasoii CTeHKH. He,AOCTK.aMI: Ka>KP;aJI 83 HHX lL'deeT AOCTaTO'iHO OrpaHH'IeHHYIO o6nacTb npHMeHeHHSI H CYecTBeHHO 3aBHCHT OT BXOAHIIJ.HX B Hee KO- 3<PcPHeHTOB, KOTophle onpeen5llOTCSI nOJI)'3MnHpHQeCKH, npHBjI- 3h1Basrcb HenOCpe,u.CTBeHHO K pewaeMoA 3aqelo 91'0 xopomo BHAHO Ha npHMepe TOJIbKO qTO paCCMOTpeHHOA MOeJ1H (2.9)-(2.13), B KOTO- paa CMeHa MaTeMaTHIJeCKOrO OItHCaHHSI npOHCXOHT B 3aSHCHMOCTH OT TOR HJlH HHOA noo6nacTH TeqeHHSI. IIpH 3TOM B 3Ha'lHTenhHOA 
Mepe HIlcpOpMaHSi onHpaeTCSI Ira Y>Ke H3BeCTHYIO ti3 KOHKpeTHOrO 3KcnepMMeHTa KapTHIIY nOTOKa. BbIXOOM 3eCb SBnHeTCSI 11pBMoe MOeJlJ1pOBaHHe, Kora Ha OCIIOBe HenocpeCTBeHHoro paC1.JeTa KapTHHa TeqeHilH onpeeneT- CSI c t.1aKCHtaJ1bHO 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 MOeJlHpOBaHHSI HeYCTORtlRBOCreii HeCKOnbKHX Ha.H6onee KpynHblx MOClllTa6oB. HIITepecl:IbJM npe,l\CT3BnReTCH KOM6HHHpoBaHHbiR nOOA K MO- eJIHpOBa.IiHIO Typ6YJICHTJIbJX TeqeHI-IA. B ero paMK:)'x C nOMOblO npSll\101"O MO,ll.eJ1HpOBaHIIH Ha BbiCOKOnpOH3BOAHTeJIbHhlX cHCTeMax onpe,QeJI5IeTC5l nOJIe 1'eLIeHHH, BKJIIOt.laIOIee HCYCTOAqH80CTH Kpyn- HblX MacrnwOR. 3aTeM, OnUpa.HCb Ha nOJIyqeHHble TaKHM 06pa.30M xapaK'repHCTHKH TeqelIHH, )VISI HaxO)K)J.eHHH lleYCTOattHBOCTeH 60JIee MeJIKHX laclliTa6oB C He60JIbWHMH xapaKTepHblMH BpeMeHaMH HC- fiOJIb3YlOTCSI pa3JIHt.llIble MOl'eJIH Typ6yneHTHoH B.SI3KOCTH. TIpH 3TOM He.HBHO HCIlOJIb3YeTC.SI TOT <P8.KT, rro qeM MeHbWe MacmT86 HeyCTo:R'- qHHOC1'e, Tetw1 C 60J1bWeA CTenellhlO MO)KHO rOnOpHTb 06 ox 0):\110- pOAHOCI"H H CllpaBeJlHBOCTH npMeHeH:HH JJ..1IH }IX OnHCaHHSI YHHBep- CaJlbHhlX 3aKOIIOR Typ6YJICIITHoc r rH [181]. Ilo,n;o6Hb1A no,lI)(O CTaHO- 8HTC Bce 60JIee norrynspHbIM, B TOM HCJle H B KOJIJleKTHBe HCcnC- ,l\OBaTPJIett PYKOBO):{HMbIX aBTOpoM. npH8CeM pe3YJlbTaTbJ pacqerra CJIOSI CMerneHHSI, nOJlyqeHHbIe c   --.. .\/J = - - - - ---+ - - ----. Ml= - --. - I II F G B . ..-- . . -.-.- PHC. 102. CxeM8 Typ6yneHTHoro CJJO CMeweHH£ 
d 'I f)) PJlC. 10.3. Hh[11c.rIH reJlbHaH lJIJ'HpeH-<!>oTorpacpHH Typ6YJ1CUT"OrO CJI0H CHnIC- HHSI: a) TCf\inCpaT.ypa, 6) Typ6Y.I1CIITIla.H tJJoI:iKoc-rh IIO10LU.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 npYBeAelIa 06uaH 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{YIllfXCH C pa31fOR eKOpOCTblO ('..lID- CB 8(Y3l\yXa. B Ka'leC'rHe tOAPJIH) OnHCbIBaIOe" I)a3BIITHe CpCl\He- If 'leJ1KOMaCllrra6HO 1'yp6y.neHTHOCTH, Ha cpoHC I)acCQHT8.1-IHhIX npH- MblM rly(ref KPYIIHO1cl.(:rJTTa6IrbIX Hej"crrOHQHBOcTel1 HCIIOJlh3()BanaCb (k-E) )H(p(l)cpeH.HTJbHa5I 1fOe.:Ih. Ha pHC. 103 rII)HBeeHLI Bbl- t.tHCJIHTe..rIbHbIe illJIHpeH-cpO'fOrpa(pHH Typ6)'JIell'r1l0rO CJI051 CMcwe- HH5I. ITpH 3'1'01\,t HCIIOJIb30BCi.J1HCb CJIe,uylOIIJ.fe rrapap.,(e'l'IJhi NIH .lI-BH- )K)rllHXC5I 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 )-leB03MYIlCHHoro 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.rIHetaH C IIO1011blO (k-E)- 
MOe.nH. 3 aM e q a H H e 2.1. llpHMoe MoenHpoBaHHe Typ6yneIiTHocTH, TaKJKe KaK no,AXo)J., CO'-leTaJOW)lft npBMoe MOeJIHpOBaHHe H p83JB1Hble MOeJ1H Typ6yneHTHoA BR3KOCTH )J)IH OllHCaHHJI MenKoMacwwHblx HeyCTOA:'IHoo- CTeA, cyecTBeHHo OnHpalOTC1I H& eT&1JbHoe MOeJlHpoBaHHe ra30;J)iHa- MHt.teCKHX DoneR. Q,QHaKO BhlqJtCJIHTeJUlM XopOWO H3BeCT:-IbI TPY,lJ.HOCTH, CBS3aHHble C MO,ll;eJIHpoSaHHeM HeYCTaHOBHBWHXCR TeeHd BB3KOrO ra,.- Ja npa no,cq>o6HoA npocTpaHCTBeHHBpeMeHHoA 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 nop06Hb[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 CBHe- TenbcTByeT 0 B03MO>KHOCTH H36e>KaTb nOA06shIX HeraTHBHbIX eKT08. lfHTepecHo conOCTaBHTb KapTHHhl nOJUl TeeHHSI: no.nyqeHHble np" HCnOJlb30BaHHH pa3JlHtJHhIX paceTHbIX 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 nepexoe K 60nee nO)ij>06HbIM CeTKaM JJ.O (1201 x 1501) j'3- nOB HaqHHaeT npO.HBnHTbCH 60Jlee TOHKaB K8.pTHHa TeqeHHSI. TaK, YAaeTCH Ha6nlOaTbJ Ha"lHH&sI C CeTKH C QHC1IOM y:3JIOB (354 x 451), KaK Ha neBoR KpOMKe BbJeMKH nepHOqeCKH 3apo>KaeTCii napa BH- xpeA, facmTa6 KOTOpblX 3aMeTHO MeHhllle MacmTa6a BHxpeR, cYll.{e- CTBYIOHX BHYTPH BbleMKH. 3aMeTHU, qTO BHXPH BHYTpH BbIeMKH MOeJIHPYIOTCSI Y>Ke Ha rpy6hIx CeTKax. 3Ta rrapa BHxpeR, 3apo)K,n;aJOaHQt Ha KpoMKe (pHC. 104), Bpa- aeTC B pa3Hble CTOpOHbI. OHH H3 BHXpeA )UJH:a<e1:CSI B CTOpoHY HeBo3MyeHHoro nOTOKa, re If HccHnHpyeT. BTOpoB BHXpb BH- >KeTCH no HH>KlleR qacTH CJlOSI CMeweHHH, <POpMHPYSI cBoeo6pa3HblR BHXpeBOA IIOCJ10A. llpH 3TOM KapTHHa nOBeeHHH Kp}'nHOMaCillTa6- HblX BHxpeR BHYTPH BbleMKR HMeeT a6conlOTHO TaKoR )Ke xapaKTep, KaK H B pe3ynbTaTe MoenHpOBaHIDI H& rpy6bIx CeTKaX. 
a -- .- ::::'::: =:   ::::- =-  ::-    -: Tr.  r!   r t 4 1 .' 1! : ! ,  . I I : I . . . . . . . . I . . . . . . . . . . . . . .. . . . . p . .  i' 6 I I . I I f   !  I , , I I , . ... '\ , I I , " " '\ , I I , , " , , , I . , , , " '\ , I I , , " ... , . I I I , " " . , I I , " , " I . . " ... 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, Kora HapsiAY c 60nbWHMH ropoa.MH H OCHOB- HbIMH Ma.rHCTpanaMH nOHsnSlIOTC9 60JIee MenKHe HaceneHHbJe nYHK- TbJ H oporH, HX Coe,l\HHSlIOe. 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 ee OM nO)J.TBep}K)(eHHeM KOppeKTHOCTH KHHeTHqeCKHX cxeM H nepcneKTHB- HOCTH HX npHMeHeHHH B BblCOKOnpoH3BOAHTeJlbHbIX BbI'lHCJIeHHSIX. 
 3. reHepH.SI H pacnpOCTp8.HeHBe MaJIhIX (aKycTHQeCKHx) B03MYIl\eHHA HccneoBaHHe reHepaQRH H pacnpoCTpaHeHHH aKYCTeCKHX B03MYLQeHHH npe,lI.CTaBnfleT co6oA oy H3 HHTepeCHeAmHx H HaH6o- nee TpYbrx 3aq cOBpeMeHHoA BbJ1.lHCJlHTeJJbHO'A rBo:A HHaMH- KH [182]-[186). 9Ta 3a'la HanpgMYlO CBSI38.Ha c pellleHHeM Ba)Kl!blX TeXHHQecKHX nporpaMM p IIanpHMep, c xopomo 3HaKOM01t np06JIeMOR CHH>KeHH ypoBHSI WYMa t HMelOea 6onblUoe COQHaJIbHOe If 3KOJIo- fH\.fecKoe 3HaeHHe. HeYCTalloBHBllIHeCSI H Typ6YJIeH1'Hbie TeqeHHSI npe,nCTaBJI.SIlO'r c060A oco60 HHTepecHblA 06'beKT )l.JISi IICCJIeOBaHHH MaJIblX B03MyeHHR. 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. npolecc reHepa.QHH H pacnpoCTpaHeHHH MaJlbJX B03MYII.\eHHA MO.>1{l)O onHcaTb nYTeM npBMoro pacqeTa TeqeHHR BSl3KOro raaa fIa OCTaT01.JHO nopo6HOR CeTKe. KCPC H KfY B npHHlUine 8JOT Ta- KYIO B03MO)l(HOCTb. O,n;liaKo TaKoA noxo Tpe6ye-r 60JIblliHX BbJ"IHC- JIHTeJIbllblX 3aTpaT H M3JIOpeaJIeH npH cOBpeMeHHoM ypoBHe Pa38H- THH Bbl'lHCJ1HTeJlbHOA TeXHHKH. BoCnOJlb3yeMCiI pyrHM nOOOM K OIIHcaHHIO pacnpocTpaHe- HHg MaJIbIX B03MY'lQeHHR, KOTOpblA .HBJI.HeTai 60nee TpHl\HOHHbIM. B ero paMKaX ypasHeHHH, onHCblBalOll.\He ,ABH)KeHHSI l'a3a, JlHHeapH- 3YIOTC.SI OTHOCHTeJJhHO axYCTHQOCXHX B03MYIIl.eHHA. OR8KO nOMH- MO MO,neJ1HpoB8.HHH pacnpocTpaHeHHSl3BYKOBhIX B03MyeHH:A cye- cTByeT np06JJeMa onHcaHHSI HX reHepH, KOTOpas npeACTaBJlSleTca 6o..nee TPYJJ.HOit. B )J.aHHOM naparpa<Pe, OnHpaBCb H3 KCPC B Ka,I.Je- CTBe anropHTMa, 6YYT npoAeMoHCTpHpoBaHbl B03MO)!(HOCTH pewe- HHSI OOOHX np06J1eM. 1. MOAeJ1HpOBalIHe pacnpocTpaHeHHH M8.JIblX B03MymeHHJt Cne,llJ'H pa60TaM [187]-[189], JIHHeapH3yeM KCPC c KOppeKH- 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 nOpHKOB MaJlOCTH. B CJIyqae 2D-reoMeTpHH llOJlYQHM CJleJOe 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 ) re Q' - BeKTOp MaJlblX B03MyeHHA ra30HHaMH1.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) . CpeIHe BeJIH"tJHHbI r830HHaMHqecKOro 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 - MaTpHhI KOHBeKTHBHblX onepaTOpos, ((BxQ')x) % H (.?f(ByQ')y) y - nHHeapmOBaHHble AODC11JHHTeJIb- HhIe CCHnaTHBHLle qneHbI KCPC C KOppeKll,He:A, nponOpOHaJlb- Hhle x H t::.y ((1.1)-(1.5), rn. IV). S(x, t) - HCTOqHHK aKYCTHqe- CKHX B03Myrn.eHHA (onpeeneHHIO S 6yeT nocBSlIu.eH cneAYIOHH nYHKT naparpaQ:>a): B aHHOM nYHKTe HCTO'tfHHK B03Myem[A CQH- 'meTCH 3aHllbIM. 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, onHpaJOHCB Ha KrY-CHCTeUY (1.1)-(1.5) (rn. IV), KOTophle H8.30BeM .'BapHaRT A", AJlB MO,QeJlHpOBaHHa 38rA8q akYCTHKH RcnOJlb30- Banacb pa3HOBHHOCTb KCPC c KoppeKeA (1.22)-(1.25) (rn. IV) (BapHaBT B). npHUeH1IeMaJI He TOJIbKO sa ftecTPYXTYPHp08a.H11L1X ceTKaX, HO H Ha ceTK&X npa- MoyrolIhHblX. 
MeHeHHSI KB83Hra30HHaMHqeCKOA 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 KOppeKHeR, HCDOJIb30BaHHe SlBHOA: cxeMLI (3.1) npHBOHT K MSlrKOMY OrpanH\feHHIO T ;S h Ha mar no BpeMeHH [187]. 9TO e- naeT CHCTe(\llY (3.1) nerKOocTynHo:R NISI a.AaniiH K apXHTeKTY- pe MHoronpoleccopHblX BhlqHCRHTeJlhHblX CHCTeM C pacnpeeneHHoR natSlTblO, KOTopble HCDOJIb30BaJlHCb [187]-(189] ;:vISi MO,L(eJ1HpOBaHHii 3a.AaQ aKycTHKH. PacCMOTpHM HeKOTO- pble pe3YJlbTaTbl paceTOB pacnpoCTpaHeHHSI 3BYKOBblX KOJIe6aHHA, A.'1H nOJIytteHHH KOTOpblX HCDOJIb30BaJIHCb MHoronpol(eCCOpHble Bbl'lHCJlH- TeJIbHbIe CHCTeMbl Parsytec CC H MBC-IOOO. Ha pHC. 105 npHBeeHa CXeMB. TeqeHHSI BB3- Koro ra3a o6nH3H 2D-KaBepHbl. B TOQKe 0 38.H HCTO'IHHK aKYCTH1.JeCKHX B03MyeIIHa, qacTOTa H aMnJlIiTYa KOTOporo BapbHpOBaJlHCb. MaKCHMaJIbHOO 3HaqeHHe aMnnHT}'):{bl nYJIhA COCTaBJ1S1J10 He 60JIee 1 % OT cpeHero ypoBHSI aaneHHSI. IIapaMeT- phI BHelUHero nOTOKa Moo = 1.35 H reOMeTpHH KasepHLJ L/ H = 2.1 COOTBeTCTBOB8JIH HeYCTaHoBHBmeMYCH OCJIJIHOHHOMY Te'leHHIO, nO,lij)06HO paccMoTpeHHoMY B npeJU'eM naparpa4>e. Ha pRC. 106-109 H306pa>KeHbJ ypoSHH aKycTecKoro aBJIeHHg no cpa.aHeHHIO C ypoBHeM nynhcaHH 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 npeCTaBneH- HblX Pe3YJIbTaTOB. Bo-nepsbIX, BBepx no nOTOKY ypoBeHb aKYCTH'Ie- CKHX nynhcHR cyw.ecTBeHHO HH>Ke. SI Hepe30HaHCHOA qacrOTbI 10 = 32 KrQ npOHCXOT 3HaHTeJIbHOe 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 B03MyeHHR 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 cpebI 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. reHepHR H pacnpoCTpaHe1lBe uanwx B03MyeHHB 277 no cpasHeHH10 C HHTeHCHBflOCTblO HCTOq}fHKa. Ha060pOT, BHM3 no Te1.feHHIO (TOqKa, C) MO}KHO HaMIOaTb yoenHQeHHe ypOBHH aKYCTM- qeCKHX nynbcaHR. CneKTp nYJlbHD 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 nynhcHA 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 IIYJlbCA B roue C (/o = 32 Kr): aepXllBJI KPHBM - nYJlLCaQHH OCUOBHOro nOTOKa., HHXCHBB - &KYCTRQecJ(ue KOJIe6aRHH IloMHMO npeCTaBJleHHbIX Ha pac. 105-110 pmYJIbTaTOB npOBO- ;QHJlHCb paCqeTbI Ii tr.JIn JUJyrHx qacTOT BblHY)KeHHblX Kone6aHHA, B TOM tIHcne H 6JIH3KHX, HO He COBnalOIII,HX C BYMH OCHOBHblMH qa- CTOTaMH c06c r rBeHHblx KOJIe6a.HHA HeYCTaHOBHBllJerOCfl nOTOKa. IIpH 3TOM 8 CneKTpe qacTOT aKYCTH'IecKHX Kone68HHA Ha6J[IO8JIHCb xa- paKTepHbIe ;urnJleTbI. B pacqeTaX co CKOpOCTbIO Ha6er&lOero 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 - ! - 90 E CI) SO 10 0 a) I ! -4.- --1 CA VITY ptlnt C .esII 'Ill' .-T- - ... - I , ......- -t.. - .. - - - . I I I ..- ..... __... ,-- ------. ....L- -.._ . 20 «) 60 10 100 120 140 160 110 200 FREQUENCY (KHz) 130 120 t 10 100 -   -- 90 £ (I) . . '1 . 10 . ---_.. . . . I , I CA Vrry "'1 C . .1.___ -___l.- lDeIh '2x2' . , ._-l-__ --.-... .__-- I - - +. 70 o 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 B0311yeHHlt 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 npeCTaBJIeHbJ 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. BHHOt 'I TO Ha 60- nee nopo6HOR CeTKe cneKTp &BneHHg CY1I"ecTBeHHO CJlO>KHee t qTO npeCTaBJJSleTCjI BnOJlHe eCTeCTBeHHhlM. OHaKO OCHOBHble qaCTOThI KOJ1e6aHHA: Ii HX 8.MnnHTYI npaKTHQecKH MaJIO MeHBIOTCH. 9Ta no- BTOpSleMOCTb pe3ynbTaTOB npH cryw.eHHH CeTKH eMoHCTpHpyeT eIne Pa..1 R03MO)KHOCTM KCPC ;:vIH eKBaTHoro eT8JlbHoro MoenHpo- BaHHSJ m30AHHaMHt.JecKHX Te'ieHHIt. Ha pHC. 112 H306paJKeHa cxeMa pacnpOCTpaHeHWI aKYCTHqe- CKHX B03MyeHHH B npSlMOyroJlbHoA: KaHaBKe. 3m 3aa SlBJlSleTCSI 8YMepHOa B 'IHCTO ra30,lUlHaMlNecKoR nOCTaHOBKe H TpexMepHol M.   ._1l1._.- 130 120 110 LOO 90 ""' 80 ! "-" 70  I - 11.._ rI) 60 so GROOVE poiJIb C 8IId CL ... 'Istxl' - - -..  - «) o   '. ; I .__.- ---. .---.-. --T--- --.....- - . .-- ---,..-- --:---- 20 40 60 80 100 120 1«) 160 1m 2QO FREQUENCY (1Ub) PHC. 113. CneKTp axycTHtleC1CllX 803..yeHldt 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 B03MyeHHA. IIo3ToMY AnSI pacqe- Ta pacnpocTpaHeHHSI MaJIblX B03MyeHHA HcnOJJb30BaJICSI 'IpexMep- Hblll BapHaHT CXeMbJ (3.1) co Cpe,l1.HHMH 3HaqeHHHMH ra30HHaMH'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, oTCToSleA OT TOqKH C Ha pacCT05fHHe L. BH.lIrHOt TO no Mepe YAa- J1eHHSI OT nnOCKOCTH CHMMeTpHH ypoBeHb axYCTHqecKHX B03Mye- HHR na}l:r. TIpH cpaBHeHHH C aH8JlOrHqnbIM cneKTPOM NISI KasepHbl (pRC. 111) 8HHO, "fTO B cnyqae KaHaBKH cneKTp nYJJbcaIJ)IA 6o.nee "MHrKH:R", B HeM MeHee gpKO Bblp8.>KeHbJ OCHOBHble acTOThI KOJIe6a- HHR. 06a aTHX pe3YJJbTaTa npeCT8.BJISIlOlx:B BUOJIlle ecrecTBeHHbJMH. 2. MOAeJlHpoBaHHe HCTOQUHK8. axYCTHqeCKHX B03MyeHHR Ilocne AeMOHCTpaH pe3YJIbTaTOB, CBH3aHHhlX C pacnpocrpa- HeHHeM MaJlblX B03MyeHHA, pacCMOTpHM B03MO)KHOCTH KCPC ,!VIS OfiHcaHHH reHepaHH aKYCTHQeCKHX B03MyeHHD. B H3J10JKeHHH pe- 3ynbr.tClTOB 3'I'oro Ii CJIeAYlOll(ero nYHKTOB 6y)teM CJ]eOBaTb 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 ocpeHeHHSI MeJlKOMac- w11WHblx nynbcaIJ:HH, H npOHHTerpHpyeM no HeMY p83HOCTHYIO cxe- MY. IIpH TaKoD npoe,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 <PYHKHa pacnpe,qeneHIDI (BepOwrHOCTh CKOpOCTH npHHKMaTb TO HJlH HHoe 3t1&tieHHe) MO)KeT 6blTb 3anHcaHa B Be _ ( 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 nynLcaoH- 
93. TeaepaJ.(HH H pa.cnpoCTp&HeHHe M&J1bJX B03Myw.eHHH 281 HoA H ocpeeHHoH COCTaBJUlIOHX CKOpOCTH, P = f I(x, t,' u)du - cpeHHSI 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 re B = o.f1 /0: 2 + 13 2 ' (} = 4 ' 1 fJ = .J2RT ' KpOlwre TOro, BMecrro aBJIeHHH H nOJIHoR 3HepI'HH Ha COOTBeTCTBY- IOHX n03HHX 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 HaxoeHHH KHHeTHtfecKoA sHeprnH T)'p6YJIeHTHo- CTH k H CBH3aHHOA C Hel Typ6YJleHTHoA BH3KOCTH lJe nODOJIHHM NO- AeJIb CHCTeMott (k-£)-ypaaHeHHA ):VIa: HaxO}KeHHH k H CKOpOCTH HCCHnaIHH 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 - 3Pkij ax: - reHepaHH 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 140enb AJ'IB HaxoeHHJI 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 ocpeeHHbIX 3HaqeHHa .RBn.ReTCR HCnOJlb30BaHHe ,ltOCTaTOlfHO nopo6Hb1X CeTOK, C TeM tlTOObJ KpynHoMacwTa6Hb1e HeycToA- '{HBOCTH OnpeJ1BnHCb H3npSiMYIO. Kax Y>Ke OTMe'laIIOCb (n.3  2 JI-&HHoA rnaBbl), aHHbIA no)J)COJi. JlBnJleTCJI O,LUfHN H3 cOBpeNeHHblx HanpaBJIeHHA 'fHCJleHHOrO MoenHpoB&HHJI Typ6y neHTHOCTH. 3aMeaHHe 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"lbHYIO HHct>OpwattHIO, MmKHO A1I nHX eneA npH- MeHHTb H anre6paH'fecKHe ..oenH (191). nOCTIe }JaxoeHWI OCHeHHblx 311a'leHHR ra30;I)UlaMH'IecKHX napaMeTpOB H 3HeprHH Typ6YJleH'I1IOCTH k MO)l(HO npHcTynaT K <POpMHpOB8.HHIO HCTOqHHKa aKYCTH'IecKHx B03MyeHHA S H onHca- HmO we pacnpocTpaHeHHJI C IIOMOIO nHHeapH30saHHblx ypaBHe- HH1t (3.1). OTMeTHM, q'I'() nOCTpoeHHe HCTO'IUHKa B03MyeHHA, a- ItIIJlo.ll...A I'IN.UI..6 . :I I .  r o E . _ ..______...__ .._.._.._.._.____pL. __.._.__. _____.. · i I c. &. Go .J . CI z a JI( m i . I Co. Cpe3 conna .  q o  Bwxc..." 1"..... PRC. 114. CxeM& p8C'lemoi o6nacTII fJII. TeqeHIUI nJlOCXoA C11>YH 
g3. reHepaI.UlSi H pacnpocTpl1lfeHlfe M&1l121X B03MyeRHR 283 >Ke OnHp8.HCb Ha 3HaHHe xapaKTepHCTHK Typ6yneHTHoro Te'leHHSI, npe)J.CTaBJIHeT c06oA: CaMOCTOJlTeJIbHYIO C1IO>KHYIO 3aqy, B 311aqR- TeJ1bHOH Mepe BhIXOHJO 3a paMKH eneR aHHo:R MOHorpa<l>HH. IIpH 3TO[ eo MHO rOM HCDOJIb3yeTCsI Ta HJtH HH8JI 3KcnepHMeHTaJlb- HaSl ftl<POpt..IHSI 06 H3YQaeMOM J1BneUHH. I1poe"'loHcTpHpyeM npoecc nOCTpoeHHjI HCTOIHHKa Ha npHMe- pe 3a,n.aqJ.1 0 cBepx3BYKOBOA nnocKoA: cTpye, cxeMa TeeHHR 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)] . 3ecL u 12 onpegeTCB H3 KHHeTHeCKOA 3HeprHH nYJlbcaHa k B npenOnO)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 aMnJlHTYbJ nYJ1bCA CKOpoCTH u': A = maxlu'l = IP. (3.10) tJ.ng HaxO>KAeUHSJ ttacTOT nYJIbCHft npHBJteeM OnOnHHTeJIb- IfYIO HH<I>opMaalO co CTOpOHbI K8.K HaTYpHoro, 'IaK H BLI'lHCJJHTenb- HOro 3KcnepHMeHTa.-- 'Ha nepBhlA B3rn.R.D. BY3 WBaeT YJ1eHHe BKJIJO'IeHHe B HCTO'IHHX S (3.8) 
H3BecTHO [193]. QTO 'IRena CTPYxaJIlI B CJIoe CMeweHHH CTPYH onpeeJI5IeTCH KaK Sh = fx/u, (3.11 ) re 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 Oqepeb, NIR onpee- neHHR eenHqHHbl k} npHBJICqeM ononHHTenbHblR Bbl'iHCJlHTeJIbHbIR 3KcnepliMeHT. B nOrpaHU"IHOM CJIoe Ha cpe conna 3aBanC}l CTOXaCTHqeCKHR HCTOt.tHHK H3nYleHHSI 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 803MyeHHa Ha aCROBe CHCTeMbI (3.1). Ha pHC. 115 npec'raB- JleMhI xapaKTepHble CneKTpbJ nynbcalJ)lA AaBneHHSI B pSle CeqeHHA Tpy6bl. CneKTpbl t KaK BbISlCJfHnOCb, npaKTH\feCKH He 3aBIICHT OT cno- c06a B036Y)K)eHHjI HCTOQHHKa. 110 BhlBBJleHllLIM nHKaf.{ 8 cneKTpe onpee.nlOrrCjf xapaKTepHble \(acTOTbl H COOTBeTCTBYIOe HM qUC- .na CrpYxaJl5I. 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 npeno- naraeTCSI onpeeJlSlTb qHc.na CTPYxaJIH Ila aCHOBe qaCTOTHbIX MO)). MaKCHMaJlbHOA aMnnHTYbl. 9TH 1fHCJI& OKa3aJ1HCb B AHana30He Sh == O.B -:- 1.1. ,nnH MOAenHpOBaHHSI aKycrrH"IecKoR 3aqH H onpe- AenellHSI IacTOTbl f qHCnO CTPYXaJIjI npHHHMaJlOCb paBHblM Sh = 1. TOnbKO npo)J.OJIbHLlX nYJ1bCA. OAHaKO OIlM. B CJIJIY pemeRHJI ypaBHeHH9 (3.1), MrHOBelWO Bbl3hJBaJOT nOSlBJIeHHe 3KYCTH'IeclOtX B03Myw.eHHA Bcex KOMlIOHeHT. BOJIee nOJIJloe OIIHCaHHe HCTOHHKOB B CHJIy npeNIOJIO)l(eHHB 06 H30TpODHOCTH Typ6yneHTI[OCTH npHBOHT 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.IHA p'aaneHHX n& Pa3JIHqHblX pacCToHHHHX OT cpe3& conna 3. MO,l1JJIHpoBaHHe InYMa CBepX3BYKOBOR CTPYH iloCJIe 3aBepmeHHjI IIOCTpoeHHH MoenH, onHCblBalOeR I'eHepa- If-HIO H pacnpoCTpaHel-lHe WYMOB B cBepx3BYKOBOH cTpye, npHBe.n:eM J-leKOTOpble pe3YJIbTaTbi pacleTOB. PaccMaTpHBaJlaCb IIJIOCKaH CTPY.H (pHC. 114) B03,a.yxa., BLITeKaIOero H3 enH pa3MepOM d = 6 MM npH CJIe,lQlIOHX ra30HH8.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. Tefnepafrypa JJa Bblxoe H3 me.nH Ct:tHTaeTCH paBHoA TeMnepaType OKpY)lffilOllea CpC,l).bJ. TOJIHa 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). BCJ1eCTBHe 3Toro aBJIeHHe Ha OCR CTpYH Me- HHeJ'CSi BOJIHOo6pa3HO, TaK >Ke KaK H B a.H8JIoru1.fHbIX 3KcnepHMeHTaX 
, ,p. . .='"  '::"Lt .11!'" '1t:.. -; .  - . !I" 7- .: . .....;:......:"T ...... : > . - . - ....I. . w:. . ,, . .. _ -.-,,_) ri,q1ir= If -_ .:r.; a) a) 6) 6) PRC. 116. MrHoBeHlloe none nYJlbca- QHA nJIOTHOCTH (a) II ocpHeHHoe 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 npoOJIbHoA CKOpoCTH. Ha pHC. 118 rrpe,l1-CTa.BJIeHbl H30J1HHHH KHReTHtJeCK:)A 3HeprHH Typ6ynefITHocTH. BHO, qTO Typ6yneHTHocTb npaKTH'IeCKH OTCYT- CTBye1' B 06nacTH nOTeHH8JlbHOro 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 CMeaeTCSI K OCM CTPYH. H3MeHeHHe YPOBHB nynbHA aaneHRa BonL n)'qeft 1 H 2 (pHC. 114) npHBeeHbl 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 nyy 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,& BO, ITO 30Ha II&H6onee HHTeHCHBHbIX nynbHA CKOpOCTH neX<HT B OKpeCTHOCTH CJ10H CMemelJHH H <paJ<- THqecKH COBDaeT c 30HOA cyeCTBOBaHHB 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. I1pOOJl>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 JlyeA 1 J.I 2 B Typt5yneHnloM none B MOeJ1HpoBaHHH HaH60J1ee cnO)KHbIX npo6.neM cospeMeHHoA Mf,lQ>O- H ra30BOA HHa.MHKH, nepeA,qeM K paCCM01'peHWO TeeHmt XHMHtJe- CX" pearHpylOx ra3OB.  4. MOeJlHpoB8HHe Teqelllli: XHMII'IeCKII pearHpYIOBX 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&lOea cpe):tbl. BMeCTe C TeM, 3Ta np06neM& BBnSleTCSI o,lJ.HoA H3 HaH60nee C1IO>KHhlX B COBpe- 
MeHHOM MaT€MaTHqeCKOM MOeJ1HpoB8.HKH. 9Ta CJlO)KHOCTb BbJ3BaHa CJIe,ll.Y IOI..I.I.H AH n p HqH Ii 8MB. BO-nepBbIX, nOMHMO anrOpHTMOB, npea3HaqeHHbIX ,l\nSI pac- qeTa QHCTO ra30HlJaMMqeCKHX TeqeHHR, Heo6xOAHMO HCnOnb30BaTb MeTObJ, n03BOJISlI011.I.He MOenHpOBaTb 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 TPYHOCTH B nOC'rpOeHHH MaTeMaTH'IeCKOA MOenH) KOTOpaH Tpe6YeT B Ka>KOM KOIIKpeTHOM CJlY'iae KaK BbIHBJ]eHH OD- THIa.nbfIOR u:enOQKH XHMHqCCKHX peaKH:R, Tax H nO,L\6opa KOHCTaHT CKOpOCTeR nnx peaKHA. OC06bIe CJIO)KHOCTH 3eCb 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, He06xoMMo HCnOJIb30BaTb BhlcOKOnpOH3BOHTenhHble BbIQHCJ1HTeJIbHble CHCTeMbl, \lTO B CBOIO Oqepeb 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- XOHMa KOHeHTpHH YCHJIH9 no BCeM TpeM KOMllOHeHTaM TPH8.,lJ.bl cMOeJlb-anrOpHTM-nporpaMMa" cOBpeueHHoro MaTeMaTH'IecKoro MOAeJ1HpOBaHHH. 1. MaTeMaTHqeCKaH nOCTaHOBKa 3aAatJn g perneHHSI 3TO:tt 3aaqH Bocnonb3yeMcSI KBa.3Hra30AHHaMMe- CKoA CHCTeMoR ypaBHeHH9, KOTOpasI B CJIyqae MHoroKoMDoHeHTHO- 1"0 XHMHQeCKH pearHPYIOero 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 = LPiT, 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 OTeJIhHbJX KOMno- HeHT; Cj - YJ:\eJIhHaJI KOHu:e"lTp5I i-A KOMI10HeHTbI fa3a, P, Pi - CYMMapHOe H napQHaJIbHble 3HaeHHjI 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 - yenb- 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 - 3aa8aeMaSl Ta6J1H'fHO <PYHKIJ)iB; Li, Qi - CKOpoCTH 06pa30BaHHH H norepb i-ro 8eecTBa B XHMH'IecKHX peaxn.H.SIX. CJleYeT OTMeTHTh, TO HCTOqHHKOBbie qneHJ:d Li, Qi, npHcYT- CTBYIOe B ypaBHeHHH (4.1), HHHeaHO 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 rpoMo3KHA BH (194, 195). HaXO>KeHHe STHX 3aBHCHMOCreA npeCTaBngeT co60:9 CaMOCTO.SITeJIbHYIO TpY)J)Iy1O 3aqy reopeTHQe- CKHX H 3KcnepHMeHTaJlbHhlX Hccnet\oBaHuA B 06JIacTH XHMHlleCKOA <PH3HKH. 3 a M e \f a H H e 4.1. BBe,lteHHe <PYHKHH 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. MOeJJHpoBaHHe reqeHd XHMH'lecKH pe&rHPYlOllQfX l'830B 291 3 aM e q a H R e 4.2. 06paTHM BHHMaHHe Ha TO, '{TO KOHeHTpau.HjI no- CJleHeA KOMnOHeHTbl PN. (KaK npaoHno, H& 3TY panh BY6HJ)8.IOT BeU(ecTBO, npeo6na;:t8JOee B CyeCH ra30B) onenJleTCJI oc06hIM 06p83oM. BMOCTO ell(e O,AHOro ;x.HcPcpepeHanbHoro ypaoHeHHH BHA8 (4.1) npHMeHJleTCH 9K- BHBaneHTHoe eMY a.nre6paHqecKoe COOTHoweHHe (4.3). TaKaH npoe,aypa n03BOJI.HeT 3BTOMaTHQeCKH H36e>KaTb 4»HKTHBIILIX HCTO'fHHKOB HJIH CTOKOB MacCbI 3a CqeT BOOMO)KHOro HeaKKypaTHoro ttHClIeHHoro ODHcaBHH XHMH- qeCKHX npe8peHHA KOMnOHeHT. CHCTeMa YPaBHeHHA, 6nH3KaSl K (4.1)-(4.11). 6blna nOJlyqeHa B pa60TaX [196, 197) Ha OCHOBe KHHeTHQOCKoA' MoenH HHaMHKH XH- MHqeCKH pearHPYIOeR ra30BoA CMeCH. B 9TOa MoenH npe,lJ)Ionara- nOCb, '-ITO CPYHKSI pacnpeeneHHSI 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 MoenHpoBaHHH SlsneHHA B XHMH"IeCKH pearHpYIOlL\HX MHOro- KOMlIOHelITl-lbJX raaax, nOMHMO BOOMO)KHOCTH pacqe-ra CJlO>KHhlX ra- 30HHaMHqeCKHX 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 nopKa AnHHbl cBo60oro np06em l. TaxHMH >Ke B03MO)KHo- CTSlMH "PH OnHCaHHH npoecCOB B MHorOKOMDOHeHTHhIX ra3ax o6na- AalOT H Lattice Boltzmann cxeMbI. 2. BhI'tfHCJ11ITeJIbHbIR aJIropHTM ITepBoro B3rmIa 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&>KOa KOUfiOHeH- Thl I'a30BoA CMeCH OJI)KHO pemaTbCSl CBoe ypaBHeHHe (4.1). OAHaKO OCHOBHasI npo6neMa CBSl3aHa C CHJIbHl:dM enH.HHHeM XHMHtJecxHX pe- aKHA Ha H3MeHeHHe nnOTHOCTeA KOMDOHeHT H TeMnepaTyPY CMeCH. 9TO Bbl3BaHO 'reM) qTO XapaKTepHhIe BpeMeHa MHorax XHMHqeCKHX peaKR Ha MHOro nopHOB MeHhwe xapa.KTepHOrO ra30AHHaMH'qe- CKOro BpeMeHH. 9TH CJIO)KHOCTH ycyry6JI.mOTCH pe3KoA HenHReAHOA 38.BHCHMOCTblO KOHCTaHT CKopoCTeA peaKHa 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 IIO3a'IH c nOMOblO MeTOa cYMMapHoA annpOKCHMHH (32, 198). Ha nepBOM 3TaJIe pe- WaeTCH CHCTeMa ypaBllemdt (4.1 )-(4.11). IIpH M B rpaBHeHHH ( 4.1) OTCYTr8YIOT qneflbl Li - PiQi, onHCblBalOHe 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, OllHCbIBaJO8.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'a30HHaMH'ieCKHX ypaB!leHHA IiC- nOJIb3YeTC51 nOJIYHeHBHa cxeMa (2.9) (['n. IV). K KOTOpoA I1pHMeHH- eTCSI HTepHOHHbla 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 pearHpYIOHX 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>epeHanbHblx YPaBHeHHR (O,lI.Y) (4.12)J KaK I1paBHJlO, XBJIeTCSI >KecTKo:A) H60 KOHCTaHThi cKopocrea XHtwIH'tfeCKHX peA 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 norHeCKOa CTpYKTYPe cxo c gBHOA CX€MOH H ero anTag K napaJIJleJIbHblM CHCTeMaM He npeCT8.HJ1HeT TJ>yt\a. CH(."eMa Oy (4.12) no cBoe:A CTPYKTYpe RBnSleTCSI nOKaJlbHOa H Boo611I-e He HYaeTCH ,lVISI Boero peweHHSI B 06MeH&X Mey npoeCCOpaMH. OTMeTHM, q-ro nOKaJIbHOCTH yaeTcSl A06HTbCSl 6J1a- roapH oco6eHHOCTHM Kry -CHCTeMhl, onHCbIB8JOeA npoeccbl ca- MO)l.H<P<I>Y3HH KOMDOHeHT. TaKHM 06p83oM, a.nTaHjJ 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 peaKa BHyrpeHHHft war HIlTerpHpoBaHHH CHCTeMDI (4.12) HeconOCTaBHMO Man no OTHo- weHHIO K r830HaMHqecKoMY wary t. Tax, B paccMaTpHBaeMoA B CJIeAYlOlI.(eM nYHKTe 3aqe 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 3aqe 0 ropeHHH MeTaHa peaKH 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 3a'lH 0 ro- peHHH MeTaHa; TeMHhlA QBeT -- 30Ha XHMH"IeCKHX peaKIUlA, pHMcKHe ltH<PPbl - HOMepa npoeccopoB H COOTBeTCT8YIO no,u.06nacTeA HO, ITO B 60JIbillHHCTBe TOqeK, re 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 HeoHOpOAHbIM. I103TOMY B pacleTe Heo6xoRMO 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 pacnpeeJleHHH npocTpaHCTBeH- HbIX TO'-leK Me)K.lly npoQeccopaMH. npOBO,[(HT 65% OT o6U{ero 06eMa 
94. lVlO,l{eJlHpOBaHRe TeCfeHHR XHI4RIf8CKH pearHpylOupIX 1'&108 295 BLI'fHCJIeHHit. n03TOMY HMeHBO H3 Hero )VIS( 06ecneQeHHSI paBHOMep- MoR 3arpY3KH npoCTpaHCTBeHHhle TOQKH nepealOTCSl pyrHM I1pO- eCCopaM . 3. MOAeJlHpOB3HHe ropeHHH npHpo,l(HOrO ra3a rvIOt'.JIHpOBaHHIO ropeHHSI OpraHH'IecKHX TOUJIHB, BBUAY Sa>KHo- CTH 3TOA npo6JIeMbI, yenHeTCH 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 MoenHpOBaHH.S1 ropeHHSI CTPYH MeTaHa, HCTeKa- lOeA B B03YX. C ropeHHeM npHpoAHorO ra3& cOBpeMeHHoMY 1.JeJlOBeKY npHXO- HTCSl CT8JJKHBaTbCSl nOCTOHHHO. IloMHMo 6b1TOBblX HY, ropeHHe MeTaHa SBJIHeTCSI OCHOBOA MHorHX TeXHOJIOrHtJecKHX npoeccOB. O- HaKO, HapSl)Q' C IIOJIe3HbIM 3<l>ct>eKTOM BbIeJJeHHSI 3HeprHH, npH 1"0- peHHH npHpOHoro ra3a B aTMoc<!>epy Bbl6pacblBaeTCSI 6oJIhwoe Ko- nHQecTBo 'fOKCH'ieCKHX BeeCTB. Cpe;J.H TOKCHqHLIX KOMnOHeHTOB B npoAYK'IaX CropaHHH BbJe- JIHIO'fCH OKC a30Ta NO x , yrapHblA ra3 CO H pa,n:HKaJ1 CH. Ho e- KBarHoe MO,AeJIHpoBaJlHe o6pa30saHHSI OCHOBHblX BpeHbIX npHMe- ceA Tpe6YeT onHcaHHSI npoeccoB, KYAa BOBneqeHO ropa3,n:o 6anbUIe BeecTB. 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 YHTLlB8JlHCb COCT8.BJlSIIOHe HCXOHOR CMecH - MeTaH CH 4 H ra3bJ, 06pa3YIOe B03):ij'X: O 2 , N2; npo,nYKTb1 peaKl\HH ropeHHH H20, CO 2 , CO. 82; pHK8JIhI, HrpaJO- He cyecTBeHHYIO pOJlb B cnO>KHOM npOTeK8.HHH peaKHH ropeHHS1: H, 0, OH, H02, H202, CH3, CHO, CH20, CH 5 0; pa3JluQllLle 830'1'- cOep>KHe BeeCTBa: N, NO x , HN, CN, HNO, HCN; Ha.KOH, RaH60nee Ba)KHbJe H3 BeeCTB H pHK8JIOBJ BXO B enO'IKH 06pa30BaHHH OKCOB a30Ta: CH. CH 2 , H) C 2 H 2, C2 H 3, C2H4) C2 H S, C 2 H 6. BBHY 60nbUIoR TpYAoeMKOCTH pemeHHg 3TOA: 3a'lH B nOJI- 
296 rn. V. MQAenHpoBaHHe Te'leHHIf 18308011 C]>eAbl HoA nOCTaHOBKe A8JKe C HCDOJIb300aHBeM COBpeMeHHhlX BLlCOKOnpo- H3BOHTeJlbHblX CHCTeM, ee HCCJ1eAOBaHHe WJIO ):tBYMSI n}PTIlMH. B IIepBOR nOCTaHOBKe paCCMaTpHBanaCb 1I000H8.S MOAeJIb XHMH'IeCKoit KHHel'HKH (196 peaKU.HH) COBMecTHO C rrpocTOH ra30HaYH'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 peaKHAt 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 3a'JH BO BTOpoA DOCTaJIOBKe HCnOJlb30- Banacb MHoronpoeccopHaB BblqHCJIHTeJJbHaH CHCTeMa MBC-IOOO c nHKoBoA npoH3BOAHTeJIbHOCTblO, )J.ocTHralOe:A 1 TFLOPS. B nepBOa 3aqe H3YQaJJCB npoecc 06pa3oBaJlHH OKCH:)B 830Ta B 3aBHCHMOCTH OT COCTaBa HCXOOa 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 - 6eoA). ,lVIH onpeeneHHH KOHL\eHTpaI.\HA pa3JJH'lHhJX BeeCTB 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. onpee.n:BJlacb H3 3KcnepHMeH- T3J1bHhlX aHHblX [203]. PacCMOTpRM oco6eHHOCTH 06pa30BaHIDI NO - ooA H3 H&H6o- nee TOKCHt{HblX KOMnOHeHT. Ha pOC. 121 H306p8JKeHO pacnpeeneHHe 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 npoecc 06pa30BaHHSI OKCHOB 830Ta npOHCXO)J.HT no-pa3HOMY. B CJIyqae 6eRJfI X cMoce:D HenocpeCTBeH- 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. PacneneHHe KOHeHTpaQHB NO DpH pa3JIH"UIIJIX 3U&'IeHHHX napa- YeTpa Q KOJlHqeCTBO oKcHa a30TS. IIoCJIe npOxO)IC,AeHHH BonHbI ropeHHSI BAa- RH OT Q>POHT'cl. 3a CqeT OCTaT01.JHOR BblCOKOR TeMnepaT)'pbl npoecc 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 KOHeHTpH OKCH,lJ;OB a.3OTa YMeHbWaeT- Cst. 3TH pa60Tbl [202, 204), HecMoTpsr Ha HBHYIO ynpoeHHOCTL n<r CTaHOBKH) TeM He MeHee, n03BOJIHJlH nonY'iHTb P KOHKpeTRDIX Tex- HOJIOrHQeCKHX peKOMeHaI\Idl ,.lI.JlSI CHIDICeHHS1 npoMblD1JIeHHoro 8bJ- 6poca speAHblX npHMeceA (205)-[206]. PacCMOTpHM Teneph peweHHe 3aqH 0 ropeHHH MeTaHa B B03Y- xe Ha OCHOBe HCnOJlb30BaHHH 60- nee DOJIHott CHCTeMbl KBa3Hr830- HHaMH1.JeCK}fX ypaBHeHHA (4.1)- \ J (4.11). 3Ta 3aIa COOTBeTCTBY- '--./ \ '-.-/ e'!' HCTeqeHHIO IIJIOCKoA CTpYH Me-  "--- TaHa B B03,nyX (pHC. 122). MeTaH CMeIIlHBaeTCjI c B03YXOM H ropHT PHC. 122. CxeMa HCTe'leHWl MeTall& B HeM. 9Ta nocmHOBKa SlBnSleTCB B B03A.}'X MOMblO, 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. Pacne.neUHe KOl1lteH- TPattHH MeTana MO,D;eJlHpOBatIHSI 3TOA 3aqH 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 . npenOJla.raJ10Cb, QTO HHHHHpoBa.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 KOHneHTpaHH MeTaHa HS O,ll)lH H3 MOMeHTOB BpeMeHH nepeA nOHroM ra30BoA CMecH. Bo. 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 C03aHHH 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 ropSlero <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 B03Y- xa. B ropfi'leR 30He (o6naCTH HaH- 6onblliero BeneHH9 TerlJIa) npo- HCXOAHT OCHOBHbJe XHMHt.leCKHC pe- HH. Ha pHC. 126,a-r H306pa)l(e- Hbl KOHeHTpaQHH 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 npoecCOpoB, 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'{epebJ CBg3aHHYIO C paccMoTpeHHeM npoec- ca lOpeHHH B JlaMHHapHOA, a He Typ6yneHTHoR nOCTaHoBKe*) au- HbIe pace'J'a afOT none3HYIO HH<PoPM1O 0 xapaKTepe reHepaH TOKCHtJeCKHX KOMllOHeHT npH ropeHHH MeTaHa. .CJIO)J(HOCTb Typ6yneliTHol nOCTaBOBKH 'IH B nepsylO O'lepeb CBJI3aH& C np06neM8MH BcneHHB OCpe;J)ICIIHhlX KORCTaIIT XHMH'lecKHX peal<a.Wt. KOTO- pble OqeHL CWIbHO - 3KcnOHeHanbHlalM 06pa30M - 38.BHCft OT TeMnepaTyphI oKpy*alOw.ero ra3a (194]. 
::suu rn. v. Mo,.qeJIJlpoB8HHe reqeHHH r&3OBOJf Cpe,J1.bI 10000 :. I -..... . ' .. . . ,- " .' I . ::>., .::'. . ,,,," -:.',--' . 'f 1::t)}} ! ':f. : ::".:.- ..,- '",t. . . ,,- ,,.: 1 0000 16000 20000.- ..- i,." 15000 1":..... 't. " . -:. . ... :-,' .f .' !OOO 5000 .'. -:. . , a) 5000 t 00 00 6) °0 6000 t 0000 2«m J- .. 2(looor-' . -----.... ..h.__ . 5000 ;:;i :. r 5000 1 SOOO 1 0000 B) °0 5(100 1 00 DO r) 00 5OGO 10000 Puc 126. XapaK1'epHhre KOHI(CHTPattHH KounOHeHT: a) CO, 6) CO 2 (30HbI HaH- BblcweH KOHleHTpaJ.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 yapHoii SOJIHhI CO;:J.ep>KaHHe 3'!'OrO naparpaQ>a HeCKOJIbKO OTJlHqaerCH 1]0 cBoeD HanpaBJJ'eHHOCTH OT npeblJU'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 ra30AHHaMHeCKOM 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 KHyceHa OrpaHHqHBnCSl MaJlblMH 3HaQeHIUIMH Kn < 0.001, qTO COOTBeTCTBY- eT o6J1aC'rH npHMeHuMocTH ypanHeHHH HaBbe-CToKca. PacCMOTpHM t HacKOJIbKO C rrOMOII.{blO KCPC H 1(883Hra30HHa- MH'-lecKOA CHCTeMbl B03MOJKHO pacWHpHTb npHMeHHMocTb MaKpo- CKOnHQecKoro onHcaHHSI B CTOpoHY 60JIblllHX tlHCen KHYCeHa. npH aHaJlH3e 3THX B03MO)K}IOCTeA KHHeTHecKHX cxeM 6yeM 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 KHyceHa raJOHHaMHqeCKHe napaMeTpLI onpeAenSJIOTCH C nOMO1O pemeHHH KHHeTH'ieCKoro ypaBHeHIDI BOJIbMaHa H nocne)UllOm.ero HHTerpu- pOBaHHH fiOJ1YQeI-IHOH TaKHM 06p33oM <l>YHKHH pacnpe,lI.eJIeHHSI C cYMMaTOpHblMH HHBapaaHTaMH. O}IM H3 OCHOBHDIX HHCTpYMeHToB pemeHHSI ypasHeHH5I BOJIbq- Malia SlBJlHIOTCSI MeTO CTaTHCTHtlecKOro MoenHpOBaHHB, 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 ra308R cpe,ltbl llJeHH ypaBIleHHSI BOJIhMaHa C nOMOblO 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 aTMocepbI, HHTepBaJI IHCeJ1 KHy,n.ceH& OKa3bIOaeTCS TpYHoocTyneH ,Q1H 'iHC- JIeHHOro MOeJIHpOBa.HHH. * llOCMOTPHM, KaKHe B03MO)J(HOCTH OTKpbl- BaeT HCI10nb30oaHHe KCPC )1.1IR MOeJlHpOBaHHH TeqellHA YMepeHHO pa3p€>KeHHWX ra30B. KaK Y)Ke OTMe'laJIOCL (rJl. III), KCPC H Kry 6a3HPYIOTCSI Ha npeACTaBneHHH OOqacTHqHOA YHKQHH pacnpeAeneHHH B BHe MaKCBeJlJIOBCKOA, KOTOpaH CJIa60 MeHSleTCSI H& pacCTOSlHHH ,lVIHHbl cBo6oHoro 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 nOMO1O Q,!{Horo q.neHa pa3JIO)KeHHH ) f=fMa' (5.1 re f Ml - MaKCBMJ10BCKaH OHOqacTHqHaSl <l>YHKlJ. HSI pacnpeeJ1e- HH, nOCTOHH8.SI B SlQeAKe, pa3t.lepbl KOTOpOH paBHbl )VIHHe cBo6o- Horo npo6era. IloCMOTpHM, KaKHe 803- MO)KIIOCTH KCPC, BBnlOIlJ)fX- 01 CJIeACTBHeM pa3nO)f(eHHSI (5.1), cyeCTBYIOT NISI MOAe- JIHpoBaHHH Te'tfeHuR YMepeHHO pa3pe>KeHHblX ra30B [209]. AnSI :fl'oro c nOMO1O pRC. 127 BbI- SlCHHM, Kora YHKHIO Q, SB- nBIDCHooAH3OAHH r I MHtJeCKHX nep€MeHHhlx, MO>K- PHC. 127. npe,ACTaBJIeHHe npo4>KJUI Be- 110 eKBaTHO npeCTa.BHTb B J1HHHhI Q S BHAe Ha60pa cryueH'taTo- BHe Ha60pa CTyneH'faTO-no- nOCTORHU.bIX 3Ha'leHHA CTOSlHHbJX 3HaQeHHA. ITpK 3TOM BeJlHqHHa OTpe3Ka (Xi, Xi+l] paSHa ,!VIHHe CBo- Q  I , I I I I I I I . I ""'-l-fll I , . , , , , , npenOJlO)KHMJ 'i'l'O 6oAHoro np06era l. 113 HarnH){l1LIX co06pa>KeHHii BHOJ QTO rnKYIO KPBYIO 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 MeroawH CT8.THCTHecKoro MOeJIH- pODBHHJI peW&Tb If 3&.r\aqJf co BeC WeHbUIHItOI 'lJJCJIaMH KHyceHa. 
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 npoHcxowr Ha MeHbmeM OTpe3- Ke [xa, Xb] (KpHBaSl II pHC. 127), TO B KaqecTBe L H8AO BbI6HpaTb paCCTOSlHHe L = Xb - X". KOHetlHO, YBeJlHleHHe QHCJ1a HHTepBaJIOB, Ha KOTOpbIe Pa36HT OTpe30K [Xa, XC), AO 10 2 + lQ3 npHBeT K ynyqWeHHJO npe,n:CTaB- J1el-l"H KpHBOA I c nOMOblO CTyneHqaTOA annpOKCHMHH. Ho 3T0 BPSlt( JIM n03BOJIHT nOJ1Y"lHTb Ka,qeCTBeHHO HOOYIO HHcf.>OPMaJJ.HIO OT- 1lOCHTeRbHO nOBe,.a.eHHH <t>YHH Q. MO)J{HO CK33aTb, tJTO BBeeH"e xapaKTepHoro pa3Mepa L HeHBHO nnonaraeT OCTaToq:HOCTb,LVIH KYCOqHO-CTyneHqaTOrO DpeCTaBneHH HecKonbKHX ecRTKOB P83- 6HeHHA. 1l0CJIe 3THX ,ltOCTaTOquO a6CTpaKTHblX pacCY)I(eHHA BepHeMCSI K B03MO)KHOCTH OfiHcaHHB C nOMOblO KCPC TeeHHA YMCpeHHO pa3pe>KeHlfbIX ra:30B. ECJIH cf.>YHK pacnpe,neneHHSI f cryneH"IaTO- nocroflllHaH Ha (X Q , XC], TO Bee ra30H8J.t:HqecKHe napaMeTpbl Q TO- me CTyneHQaTO-nOCTOSlHHbJe. T8.KHM 06paaoM, ):VIR r830HHaMHqe- CKoro OllHCaJIHH C I10MO1O KCPC He06xoo H& OTpe3Ke [Xa, Xc] ,lVIHHOR L HCnOJlb30BaTb He MeHee HeCKOJThKHX ecgTKOB pa36HeHHA [Xi,Xi+l] AJ]HHoA He MeHee NJHHhI cBo6oHoro np06em I. YqHTblBaH onpeAeneHHe q}lCJla KHyCeHa 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 onUC6amb me",eHUJI YMepeH- NO pa3peeHH"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 nOMOblO YPaBHeHHA H8.8OroKca Ha- fi830H AOnyCTHMhlX QHceJJ KHyceHa 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 npeCTaBneHHeM (5.1) H OHO'tfacTHqHOH cPYHKHH pacIIpeeneHIDI f. 3aMe'laHHe 5.1. no NIHHOA CB060HOro 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 cBHoro 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.neT ,.l].HCCHnaTHBHbIe 'JneHbI KBa3Hr830HHaMHqecKoA CHCTeMbl ypaBHe- HHA. Mea ____uu  uu______  -  -- J ----- OCL CHMMeTpHH """"""" PHC. 128. CXeMa TeeHHg ra3a B6nH3H 1JJ18CTHJIY B KaqeCTBe npHMepa pacIJeTa paccMaTpBBaJIaCb 3a.qa. 0 MOe- nHpoBaHHR TeqeHHH B03Aj'xa B6JIH3H nnacTHHhI no HYJIeBblM yrJIOM aTaKH (pHC. 128). napaMeTpbl BXOHOro 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) npHteHHTeJIbHO K rIpoCTpaHCTBeHHo-BYMepHoA reoMeTpHH. IIpH 3TOM, eCJIH mar no KaKOA:-HH6Yb npOCTp8.HCTBeHHOR nepeMeH- HoD npeBbllllaJI 19 (CM. (5.3)), TO CXeM& (2.2)-(2.5) (rn. II) HCfiOJlb- 30BaJlaCb B HeH3MeHHOM BHe. B npoTHBHOM rJIyqae COOTBeTC1.BYKr HH mar npOCTpaHcTBeHBoA nepeMeHHoii nOJlaraJlC51 paBHblM 13<P. Ha pRC. 129 npHBeeHbI rpa4>HKH 3HaqeHHA nnOTHOCl'H B Pa3- JIHqHblX ceqeHHSlX rIO KoopHHaTe 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 KOOpHHaTe %: 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 MOeJlHpOBa.HHjI TOro }l(e TeqeHH {213]. B npe,L\CTaBJ1eHHblX B 3TOa pa60Te AaHHblX TaJOKe, Ha'-lHHasI c ce- qeHHH x = 0.4, nOBBJIeTCSI HeMOHOToHHoe nOBeeHHe nnOTHOCTH C xapaKrrepHbIM MaKCHMYMOM Pmax = 1.6. o AHaKO cy ecTBOBanH H pacxoeHH KOJIH'IeCTBeHHoro xa- paKTepa 8 pe3YJIbT8.TaX pa60' [36] H (213) "PH OllaCa.HHH nOBee- HHSI nJlOTHOCTH. IIpHqHHa pacxoeHMA 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 I1peenhl ,L\HanaaOlla (Kn ;S 0.01), r)J.e npHMeHeHHe KCPC 06oCHOBaHllo. npeCTaBJIeHHbIe B [36) pe3ynhTaThI 6b1nH nepBblMH, n03BOJIHB- 1118MB 06paTHTb BHHMaHHe Ha B03MO>KHOCTH KCPC npH M0tI-enH- pOBaHHH rreqeHHH YMepeHHO pa.3pe)KeHHhlX ra30B. PaCCMOTpHM ee 
vvv 111. v. MQ4enHpOBaHHe reqeHJlII I'83t'JooR CPeAbi ABe B03MO)KHOCfH, n03BOJJHIOHe enaTb npOt\BH>KeHH Ha nyrH HCnOJIb30BaHHH KCPC AJlS1 MOJl-eJlHpoBaHHSI 3THX 3aq. 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) 3eCb U .- 3HaqeHHe CKOpOCTH B HanpaBJIeHHH KaCaTenbHH K no- BepxHoCTH 'rena, n - TeKY3jI KOOpHHaTa 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 rpaHHMH 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 MeTOOB npHMoro CTaTHCTH'IecKO- ro MoenHpOBaH}tlI (DSMC) (214). B paccMaTpHsaeMblx sapHaHTax TeMnepaTypa CTeHKH T w nOJlaranaCb paBHoA TeMoepaType fJeBo3MY- eHHoro nOTOKa. Vl3 pHC. 130, 131 BHHO) '{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. OaKO npH YBeJJHqeHHH qHCJIa Maxa p€3ynbTaTbI, nonY1.feH;Jhle Ha 
95. MOe.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 COBnaIOT C )l.aHHblMH CTaTHCTH'Ie- CKoro Mo,n;enHpOBaIIHH, KOTOpble npHHHMaJlHCb 3B 3T8JIOHHbIe. PacCMOTpl-tM ee OAHH nO,QxOA, CBSl3aHHbIA C Hcnonb30BaHH- eM KCPC H KrY-cHCTeMbI NISI Mo,n;enHpoBaHHSI TeqefIHR YMepeH- 80 pa3pe)KeHHbIX raJOB 1215), KOTOpblH npeCTaBJISiercH OCTaTO"'HO nepcneKTHBllbiM. 3. KCPC H MCTOA domain decomposition B HacTOHee BpeMSi OCTaTOq}fO pacnpoCTpaueH- HblM jlBJISlCTQI CJIeJJYIOHH nOAxOA K MOAeJIHposaHHIO TeqeHHA YMepeHHO pa3pe- .>KeHHbIX f830B t OCHOBaHHblH Ha npHMeHeHHH MeTOOB "domain decomposition u . B o6naCTH B6Jm3H nosepx- PHC. 132. K peweHHIO Ja.n.aH 06reKaHH1I HOCTH Tena, re 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 onpeenSilOTcSI nyreM IIpMOro pelueHHSI ypaBHelIHR BonbMaHa MeTOaMH 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. ra30HHaMHt.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 BonbMaHa, 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 HTepaoH.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. nO06HblR no,Axo):{ ,nOCTaTO'fHO nepcneKTHBeH. OAHa.KO OH CTaJI- KHBaeTCSI C npHHHnH8JlbHblMH TPYHOCT'H. rll)JlqHHoA ax SlBn}f- eTCSI TOT <PaKT, qTO 06nacTL Me)l(A)' KpBBbIMH 1 H 2, re cnpase,ll)lHBO HCIIOJlb30BaHHe ypaBHeHH:A HaBhe-CTOKca (Kn :5 0.001), TpYNJoo- cTynHa An}} npHMeHeHHB MeTOOB CTaTHCTHQeCKOrO MoenHpOB8HH.H. KOHet.JHo, leTO,lJ.aMH CTaTHCTH'IecKoro MoenHpoBaHHH B npH}IHne MO)KHO nOJ1}"iaTb pemeHHSI H "PH M8JIbIX QHCJlax KHyceHa, HO npH M'Otwl pe3KO paCTYT BbI'iHCJlHTeJJbHble 3aTpaTbI Ha pemeHHe KHHeTH- qeCKOrO ypaBHeHHH BOJIbMaHa. OCHOBHaH HeSl pa6oTb1 [215] 3aKJIIOQanacb B onope H 6W1ee WHPOKYJO o6nacTb onHCaHHS, oCTynHYlo KCPC H Kr-y (Kn:5 0.01). CyuecTBYeT Hana.30H tfHCen KHyceHa, pacnOJlO)KeRHblA OKOJIO 3HaqeJIHH (Kn  0.01), rAe, C ooA CTOpOHbI, peaJIbllO HCnOJlb3O- BaTb JeTO,AbI CTaTHCTHQeCKOrO MoenHp083HH ypaBHeHJfil BOJIb- Mal-la, a c pyroR - cnpaBe,nJIHBO ra30HHaMHqecKoe onHcaHHe na OCHOBe KrY-cHCTeMbi. npeCTaBneHHa.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 KHyceHa H n03BOJISleT nOJIYQHTb 3HaQHTeJIbHOe nepeKpbJTHe c pemeHHeM) nony- qeHHblM Ha OCHOBe ypasHeHH BOJIL({MaHa. h- pewellBe ypaBOeHH$I BonhaHa .!.J MeTOOM MOHTe-K&pJlO - YP-HH9 HaBbe-CTOKC3 11 · KCPC H KrY-CCreNa 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 ICOM6BHHH 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 yapHoA BOJIHbI c nOMOmplO YPaBHCHHa: ra30BoR HHaMHKH BBJIjleTCH TP&AHI.J)IOHHldM MSI Teo- pHH ra30B (2]. no BCeA 8HMOCTH, 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 yanocb onHcaTb HeTpHBH8JlbHl:de OCH.rIJ]SlIJ.HH peUleHHH, Ha6JIIOaeMbIe B '3KCnepHMeHTe. AB1'OP 1'aK)I{e XOTen 6bI 06paTHTh BHHMaHHe qHT8Tene Ha pa- 60TY (217J, B KOTOpoR Ha OCHOBe ypaBHeHuii rH,lIJ)OHHaMHKJt nOBbl- llleHHoA TOqHOCTH npoBeAeH pacqeT CTPYKTypld yapHoA BOJIRbl H nOK83aHbI npeHMYIII-ecTBa npe)I.JIaraeMOro OnHCaHHSI no Cp8.BHeHHIO C TpHHOHHbIM ra30Ha.MHqeCKHM. 9TH ypaBHeHHSI r830BOJt HHa- MHKK nOBbilUeHHoA TOqHOCTH 5ISnSlIOTCB CJ1eCTBHeM o606IeHHoro ypanlicHHSI BOJIbMaHa (87) VI V ( VI ) Vt - Vt r Vt = J(f,!), (5.7) re V/Vt  HcJ><j>epeHIUlaJlbHbIH onepaTOp, CTOSHA B JlenoA qa- CTR ypaBHeHHSI BOJIbQMBHa, T - qaCTOTa CTOJJKHOseHHA, J{/, f) - HHTerpan CTOJIKHOBeHHA YpaBHeIlHSI BOJIbQMB.Ha. PacCMOTpHM HeKOTopbIe pe3ynbTaTbI pacqeTOB CTPYKTYPbl YAap- HoA BonHbl, npOBeetI}lbJe Ha aCHOBe Kr'Y -CHCTeMbi [218, 219]. Ha pHC. 134 H306pa>KeHbI proynbTaTbl nOBeeHHH 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 npoxoeHHH Y.Aap- HoA BOJIHbl) 4> - MaKCHMaJlbHblA yron HaKJIOHa npo4>HJJR y,n:apaoa BOJIIJbl. Pe3YJIbTaTbI, nonyqeHflbJe Ha aCROBe pelIIeHHH KBa3Hra30- HHaMHeCKOA 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 OHHaKOBO xopomo onpeenSiIOT TOJJHY yapHO.a BO.lIHbl. OtlHaKO C pOCTOM qHCJla Maxa npeHMyecTBa KrY-cHcreMbI CTaHOBTCH 3aMeTllbIMH. CneyeT 06paTHTb TaK>Ke BHHMa.HHe Ha cJIa60e y6blBaHHe BeJIH'-IHHbl AI6 C pOCTOM "IHCJIa Maxa (M > 4.5) J KaK B aHllblX 3KcnepHMeHTa, Tal< H B pe3YJlbTaTaX pac'leTOB tla OCHo- Be KrY-cHCTeMbl. BOJIee nOtl.p06HO pe3ynbTaTbi pac'leTOB CTPYKTY- pbl YAapHoR BonflbIlla OCHOBe KrY-cHCTeM1:d npeJl-CT8BJ1eHbI B pa6o- Te (218]. IloqeMY *e KBa3Hra30HHaMHeCKag CHCTeMa aer HelIJIOXHC pe3ynb'I"a'1'bJ npH OnHCaHHH CTPYKTypbl YAapHoA BOJIHbI) HecMOTpSl Ha TO, '-ITO XapaKTepHOe tlH('J10 KHY.l{CeHa AJlH 3ToA 3a.a-aH nopKa I? 3'm 3HallHTeJIbHO 60nbwe xapaKTepHOrO 3HaQeHHSI (Kn < 0.01) J npH KOTol>oM B03MO>KHO ODHcaHHe c nOMOIIJ.hIO KCPC H KfY-CHCTeMLI (n.l aHHoro naparpa.<Pa). KaK IIOKa3aHO B pa60Te [221], nH cna6blx yapHbIx BORH HX CrrpYKl'ypa, orrpe)J.eJIeHHM Ha OCHOBe pemeHH9 ypaoHeHHSI BOJIbn- 
312 rJl. V. MD,l{eJIllpOB&HHe reqeHJdt ra:JOBoll cpeAbl MaHa) 6nH3Ka K COOTBeTCTBYIOeMY peWeHHJO ypaBHeHHA Haebe- CTOKCa. C ,l(pyroH CTOpOHbJ t KfY-cHCTeMa, TaM, I'Ae Cnpaae;lVlHDO OnHCaHHe C nOlow.blO ypaBHeHHa HaBhCTOKCa, aeT 6JJH3KHe K HH1 pe3YJIbTa1'b1 (rn. III). 9To H 06"bSlCHSIeT xopomee COBneHHe o60Hx pacqeTHblX pe3YJlb'lTOB "PH MaJlblX qHCJIax Maxa C AaHHbI- MH 3KCnepHMeJ-ITa (pRC. 134). ,AnH 60J1ee CHnbHblX y.n.a.PHbIX BonH xopoWHe pe3YJlbTaTbl npH M0,l.\eJ'"pOBaHHH HX CTPYKrypbl AaeT 6HMOaJlbHOe npH6JJH)KeHHe Mo'r'fa-CMHTa.-TaMMa [2]. CornacHo 3TOfY npH6nH)J(eHHIO, pewe- HHe ypaBlleliJ1B BOJIbl\MRHa HeTC51 KaK JlHHeAHasI KOM6"HHSI BYX MaKCBeJlJIOBCKHX <l>YHHA: pacnpeeneJ[HSI) OTBeqaJOHX paBHoBec- HblM COCTOSlHHSlM raaa no o6e CTOpOHbI OT yapHo:A BOJIHbI. K03H- u.HeHTbl ,lI.aIIHo:R nHfleHHOH KOM6HHal.\HH npe,D.IIOJIaraJOTCH 3&BHCjfH- MH OT npoCTpaHCTBcHHOH KoopHHaTLI. HMeeTCH aH8JIOrHSI Me)KA}' KHHeTHQeCKOR OCHOBOA KCPC H Kry, 6a3HPYloeaCH Ha npeCTaB- neHKH YHKlJ)IH pacnpe,qeJJeHH B BHe MaKCBeJIJlOBCKOA, n':)CTOBIJ- HoR B npoCTpaHCTBeHHo:A SI"-IeRKe) H 6HMOaJlbHhlM npeCTaBJJelIHeM <PYHKIHH pacnpe,lJ.eJIeHHSI. 9Ta CBH3b MO)KeT 06'bHCHHTb OCTaTOq- HO IIel1J10XHe pe3YJIbTaTbl no MOAeJIHpOBaHHlO tPYXTypbl yn.apHoA BOJItlbl, nOJlyqeHHble c nOMOIll-bJO KDa3Hra30HHaMRqeCKOH CilCTeMbJ ypa.BHeHHA. 
3aKJII01:IeHHe ABTOP HeeTCj{) QTO qJITaTenHM 6b1J10 HllTepecHO 03HaKOMHTb- CH c nepCneKrrHBHbIM HanpaBJIeHHCM B BbltfHCJlRTeJJbHOA: ra30BOA )J.H- HaMHKe  KHHeTH"IeCKH COrJlacoBaHllblMH CXeNaJdH. Hx CneL\HcpHKa 3aKJIIO'IaeTCSI 8 TeCHOM nepenneTeHHH KHHeTWlecKOro H ra30)J.HHa- MHQeCKOrO OfiHCaJ-IHSI CpeAhI, KOTOpOe HeIlOCpeCTBeHHO YHacneoBa- HO 8bJQHC.J1HTeJTbllb1M aJ1I'OpHTMM. 51olloe 0TPa)l{euae 8 HCnOJIb3ye- MOt.\: fOenH pH)\a CBoRCTB CnnOWHOH cpehI) )l.OCTHrHyTOe 3a C"IeT npHBJleqeHH.SI KHHeTHqeCKHX npeCTaBJIeHHA, 06eCneQHBa.eT WHpo- KHe 803MO:>KHOCTH KCPC H Kry npH MO)J.eJlHpoBaHHH CJJOJKHbIX ra- 30AHHaMH'-IecKHX .HBJIeHHR:. lID BceA B"HMOCTli, CBoACTBO KCPC H Lattice Boltzmann cxeM, CBH3aHlloe c 60JIee rny60KHM BHe,lij)eHHeM <PH3HqecKBX npe.n:cT8.BJIe- HHA B KOHCTPYKU-HIO anropHTMOB, cmHeT xapaKTepHOR "IepTOA: MHO- rHX BblqHCJlHTeJIbHhlX MeTO,l(OB B 6JlH)K3JtweM 6yyeM. IIpH 3TOM Bbl"lHCJIHTeJIbHaH C'H,l(pO- H ra30BaJI AHHaMHKa 8 3HaQHTenbHoR Mepe cocpet\OTOQHTCH Ha eTaJlbHOM MOeJIHpOBa.aHH cpeAbi - c HCIJOJIb- 30BaHHeM BbiCOKonpoH3BO,[J)lTenbHbJX MHoronpoeccopHblX C14CTeM. IIOTeH'QH8.JI, 38.JIO>Ke8HLJA R KCPC H KfY, aJIeKO He HC'-IepnaH.. B HacTOSlec BpeMH C onopoA Ha Y)I(e HMelOaCH ODblT se.n;yTCSI anbHeHlIlHe pa60TI>1 rIO COBepweHCTBOBaHHlO BbJ'lHCJIHTeJIbHhlX, B TOM 1.(HCJ1e H IlapaJIneJ1bHblX, aJIrOpHTMOB H& 6aae KCPC H kBa3HI'a- 30.l\HHaMHIeCKoA CHCTeMbI. C03,l(aB&eMble na.KeTbI npHKRa,l{HblX npo- rpaMM OTKpblBalOT IUlipOKHe nepcneKTHBbI npHMeHeHHB MeTOoB, OfiHcaHHblX B MOHorpaQ>HH, ,n.nH pemeHHB MHorHX HIWCTPHaJlb}[blX 3aQ . 
COHCOK JlHTepaTYpbI 1. rupwifJeAbiJep A., Kepmuc 'I., Bepd P. MOJJeKynspJla.& TeOpHJI r8.30B H >KHAKOCTeA. M.: :HJI, 1961. 916 c. 2. K024H M.H. J{HHaMHJCa pa3pe>KeHHOI'O no&. M.: HaYK&, 1967. 440 c. 3. t[enMeH C., KaYJlUHZ T. MaTeM&THqecxaa TeopWi BeoAHOpOltblX r830B. M.: I1JI, 1960. 510 c. 4. qefJUHbJlHtJ K. MaTeWaTH"ICCKHe UeTObI B KHHeTH'IeCKOA eopHH ra3OB. M.: MHp 1913. 245 c. 5. JIU60B P. BBeJ(CHlle 8 TeOpHlO KHlleTH'IecKHX ypaollCHHA. M.: MHp, 1974. 371 c. 6. BeoekJtnU'H, B.B. KHHeTHqecKue yp&BHeHHS BOJ1bu8Ha H Bnacooa. M.: <l>H3MaTnwr, 2001. 112 c. 7. 9pH.cm M.X. TO-tHbie peweHWI HeJlJfHeARoro ypaBHeHHB Bwua.Ha H 6J1JnICHX KHHeTH'iOCK.HX ypaBtleHuA. B KH.: HepasHoBocHble BBJIeIlHJI: YpauHeHHe BonbII.MaJfa I nOA p. ,A>K.JI. JIH6oBHQ3, E.Y. MOfiTpol1J1a. M..: MHP, 1986, c.60-131. 8. Bo6b4Ae9 A.B. 06 O"OM KJlacce HHBapRaIn'HhlX peweHHA ypaBHeHHJI BOJIbu.waua / / MH CCCP, 1976, T.231, 1(1 3) c.511-574. 9. .no'Hday JI.,4.} JIUt/Jwuq E.M. Teope-rH'lecKu H3HKa.. T.6. I'HlWOH1la- MHKa. M.: HaYK8. 1988. 736 c. 10. BOJl'bMaH R. neKI\"U no TeopHR ra30B. M.: I'OCTexaT, 1956.554 c. 11. JlOUV.JiHCJCUU .n.r. MexaHHKa )((OCTH H ra3&. M.: Hayn, 198;. 840 c. 12. tJep'UJfibJl.H,U K. 0 MeTO pemeHWI YpaBHeHHR BOJJhaHa. B I(H.: Hepa8HoBeCHhle H,BnefmH: YpauHeHHe BOJIbJJ.M8.H& / n pe,o.. ,LI.>tc.1I. JIH- 6oBHJat E. Y. MOllTpOJlJIa. M. MHP, 1986, c.132-204. 13. BOINUH.C1CaJi, M.H. J TlO8A08 A.H., t[em8epyWft:TH B.H. 06 O)].HA cxeMe HHTer'p"pOR8HHS ypasRelfHft l'a.30BOA .AWl aNH KH: npenpHHT Nt 113. M.: HnM HM. M.B. KeJl,I\bI1ua. All CCCP, 1983. 12 c. 14. flemaepyw1t;Utt B.H. MaTeMaTHqeCKOC MOAeJ1HpoBailHe 3aq JQtHaMHKR H3Jlyt{a.IOero I. M.: HaYKa J 1985. 304 c. 15. nomKu', B.A. KHHeTH'IeCKHA aHaJlU3 pa3HOCTHLIX cxeld AIIH ra30BOA .lI.HHa- MHKH /1 )I{. BbI-tHCJ1. MareN. H ldaTeM. H:J., 1975, 'r.IS, NI 6 J c.1492-1498. 16. PuUln D.I. Direct simulation Dlethods for compressible inviscid ideal-gas flo\\' II J. Comput. Physics, 1980, v.34, No.2. pp.231-244. 17. Rfitz R.D. One-dimensional compressible gas dynamics calculations using the Boltzmann equation /1 J. Comput. Physics. 1981, v.42, No.I t pp.108 123. 
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HaY'4HOe UJdOHlte lfETBEPYlUKHH 60pHC HHKOnaeBH'I KI1HETHQECKHE CXEMbI H KBA311r A30J{HHAMI11JECKAR CWCTEMA YPABHEHHA o P H r H H aJl- M8 KeT E.8. raJIO'4KUHQ, 11.8. t(opodHUllblH KoppekTop: H.8. A6aRaKUH H3,QaTem.cno 000 (<MAKC npeCQ. H:tJl.. JlHIt H.,Q Hi 00510 OT 01.12.99 r. nQAl11tC1HO . nell.,. 29.03.04. cI)opM8T6Ox81 1/16. nC'lanoPncu. 6YMaraotcemu.Ni I. neel. n. 2O,7S. THpac SOO 3D. 3aka3 72 _ OmC'laT8HO . 000 c npccc». 107066. r. Moen.. EnoxcICHA np..... 3. CTp. 2. Ten. 939-38-90. 939-38-91, 928-10-42. Tc.n.ltllCc 939-38-91.