Numerical Solution of the Incompressible Navier-Stokes Equations
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1 Nmercl Solo of he comressble Ner-Sokes qos The comressble Ner-Sokes eqos descrbe wde rge of roblems fld mechcs. The re comosed of eqo mss cosero d wo momem cosero eqos oe for ech Cres eloc comoe. The deede rbles wll be he ressre d he eloc comoes d he d drecos resecel. The se of mercl mehods o sole he goerg eqos wll follow he sme mehods s sed he reos dscsso o mercl solo of o-ler eqos. The hree goerg eqos o-dmesol cosere form re The rbles he bee o-dmesolzed sg he free srem eloc des ρ scos µ d legh scle show below. The serscr deog o-dmesol form wll be sbseqel droed from here o. Ulke he comressble Ner-Sokes eqos he comressble form does o he me deede erm he mss cosero eqo. he comressble form he me deede erm roded drec referece o des llowg for mercl scheme o sole for des drecl. he comressble for he momem eqos c be sed o sole for d b he co eqo does o referece becse of he bsece o hs erm. rfcl comressbl erm ms be dded o he co eqo o llow for he solo of show below. The erm deoes sedo seed of sod. s he mercl scheme rogresses owrds sed se solo he ew me deede erm eds o zero mgde. Ths mles h he sed se he orgl co eqo s recoered. lc Solo lc solo of he comressble Ner-Sokes s relel srghforwrd ge he reos oes o o-ler erms. Nmeros mehodologes mke be sed forwrd me- cerl sce TCS Dfor rkel ccormck Crk-Ncolso o me ρ µ ρ
2 few. f cerl dfferecg s sed o he coece erms rfcl dsso m be eeded o esre sbl f he olds mber s frl lrge. oher mehod of mrog sbl for comressble flows s wh he se of sggered grd. emle of sggered grd s show below. Ths grd rrgeme rodes sroger colg bewee he ressre d eloc rbles hs mrog sbl. The / P / orgl rmr grd s deoed wh sold les whle he secodr grd s show wh dshed les. The ressre s ssged o he odes o he rmr whle he eloces re defed o he -/ / P -/ / / P / secodr odes. ore secfcll he eloc s defed o he / grd le bewee he odes o he rmr he dreco. Whle he eloc s he / les he dreco s show. elc lgorhm bsed o he sggered grd s show c beg wh he dscrezo of he momem eqos sg frs order dfferece me d cerl scheme sce. The eloc comoes re soled frs sg / / / / / / / / / / / / / / / / / / / / ce he eloc comoes re comed for he me leel he modfed co eresso c be soled for he ressre / / / / Sce he eloces re ol lble o he secodr grd erolo ms be sed o ob eloces os o he rmr grd. The followg romos c be sed / / / / / / / /
3 / / / / / / / / / / / / / / / / / / / / / / lrge mber of cses ressre bodr codo wll o be lble ge bodr. Ths mes h he ressre bodr wll be kow. hs sce he lgorhm s oled ofe clled he rker d Cell roch wll llee he roblem. f he bodres re seleced o cocde wh he secodr grd ressre o he bodres wll o be eeded. or emle cosder flow bodr d wll show below. f he wll s lged wh ½ d he flow wh ½ he ressre does o er o he srfce d s clcled ol he eror. Howeer les of / d / ms be secfed. / flow bodr / / / / / / / / // / wll bodr / / Sce he wll s sold o sl codos rel herefore... / / / / / 5 / / 7 / /... The les sch s / wll be reqred he lss whch c be fod from / / / / whch rodes / /
4 he flow bodr he eloces re secfed drecl from he codos. les sch s / / re kow from roblem defo d eed o secl reme. The comoes of eloc osde of he bodr sch s / 5.. c be fod from erolo oflow bodr oe cold se he sme roch o clcle or ee. oh he me se sze d he le of he sedo seed of sod defe he sbl reqreme for hs mehod. The me se sze s lmed b Here m mes he smlles or o he grd N s he mber of dmesos he dom d s he seed of sod sed he sedo co eqo. Ths le s defed b he mmm eloc Noe h s o ecessr o se sggered grd b does rll eforce cosero he m grd os. lso oe h ll rbles re o-dmesol meg h he comg eloc he flow bodr he boe emle wll be /. mlc Solo The mlc solo of he comressble Ner-Sokes s more esl ccomlshed f he eqos re wre cosere form mlc scheme he o-ler of he fl erms ms be ddressed f ssem of ler lgebrc eqos s o be deeloed. Usg Newo's mehod / / / / / / / [ ] 5 / m N m <
5 5 The erms / d / re Jcob mrces d wll be deoed s d from hs o o. s emle mr s defed s These les c be deermed b re-wrg he orgl fl ecors erms of he comoes of he solo ecor. The fl ecors re wre erms of hese les d dffereed d serg he ler romos o he ecor form of he Ner-Sokes eqos d romg he me dere erm sg Usg rome fcorzo he eqo boe he form c be sered o eqos olg d drecos olg r-dgol mrces... [ ] RHS
6 6 dreco r-dgol eqo dreco hese wo r-dgol eqos secod order rfcl dsso hs bee dded o he lef hd sde HS d forh order o he RHS o esre sbl. The sr ser scr o he dces ermede solo recedg he fl solo of he eqo. sggered grd s o ecessr hs roch de o he colg fforded b he mlc scheme. f cerl dfferece s led o ll sl erms he dreco eqo wold become r-dgol form he eqo s he eqo s RHS [ ] 6 6 RHS
7 7 ferece Hoffm K.. Chg S.T. Comol ld Dmcs ol. geerg dco Ssem Wch Kss 998 SN
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