Turbulence Modelling (CFD course)

Transcription

1 Trblence Modellng (CFD corse) Sławomr Kbac Copyrgh 016, Sławomr Kbac

2 Trblence Modellng Sławomr Kbac Conens 1. Reynolds-averaged Naver-Soes eqaons Closre of he modelled erms Eac Reynolds-sress ranspor eqaons Transpor eqaon for rblen nec energy The sandard - model References Append A... 19

3 Trblence Modellng Sławomr Kbac 1. Reynolds-averaged Naver-Soes eqaons The dscsson of he Reynolds or me-averaged Naver-Soes (RANS) eqaons and rblence ranspor eqaons s lmed o he consan-densy (ncompressble) flds. A lnear relaonshp s assmed beween he componens of he sress and deformaon ensors. An eenson o compressble flds s sraghforward and can be fond n many eboos (see for nsance Wlco, 006, 008). In case of compressble fld one has o apply he Favre-averagng (densy-based averagng) nsead of menoned Reynolds-averagng o allow for some smplfcaons n he mass and momenm conservaon eqaons. For ncompressble fld, he nsananeos velocy componen (,) can be wren as he sm of a mean ( ) and a flcang par (,) (fg. 1):,, where he mean velocy s defned as he me-averaged vale (1.1) Fg. 1 Decomposon of he sgnal n mean and flcang componens. T 1 lm,d (1.) T T T s he averagng me nerval. We assme ha T. Ths corresponds o he seady RANS model. For me-accrae RANS (called URANS) hs me nerval has o be sffcenly large wh respec o he me scale T1 of he rblen flcaons (fg. ) and small wh respec o he me scale T of large scale nseadness. So for meaccrae problems Eq. (1.) aes he form: 3

4 Trblence Modellng Sławomr Kbac 1 T T,, d Fg. Tme-averagng wndows. T1 me scale of rblen flcaons, T me scale of nseady moon. The me-averagng wndow T shold be: T1 < T < T Two sefl properes are (seady RANS): - he me-average of a me-averaged vale s agan he same vale (he averaged vale ( ) s no a fncon of me) T T 1 lm d lm d ( ) (1.3) T T T T - and ha he me-averaged vale of he flcang par s zero T 1 lm [(,) ]d ( ) 0 T T (1.4) The oher averagng properes are lsed below for arbrary qanes, and : (me averagng s lnear) (1.5) 4

5 Trblence Modellng Sławomr Kbac (1.6) (1.7) (1.8) 0 (1.9) 0 (1.10) 0 (1.11) The Reynolds-averagng s lnear (1.5) and commes wh he space dervaves (1.6). Relaons (1.7) and (1.8) come from he propery (1.3). Wh (1.9) and (1.10) we ae advanage of he observaon ha he prodc of mean and flcang qany s zero becase he mean of he laer s zero. The qanes and are correlaed. I means he average of her prodc s no zero (Eq. 1.11). For ncompressble fld he conservaon of mass and momenm eqaons (wh he consve relaon nrodced already, see frs lecre) are 0 (1.1) 1 p S (1.13) where s he fld densy, s he dynamc moleclar vscosy, p s he pressre and S s he sran-rae ensor S=1/(/+/). Frs, we average he conny eqaon (1.1). Tang he propery (1.6) we oban:. (1.14) Ne, we average he momenm eqaons (1.13). Usng he propery (1.5) we oban 5 1 p S

6 Trblence Modellng Sławomr Kbac 1 p S (1.15) Here we assme ha he me-averagng commes wh he local me dervave. We can see from fgre 1 ha for T he me-dervave of he mean velocy s zero 0 (1.16) The convecve dervave deserves aenon. We average he prodc of and : (1.17) The second and hrd erm on r.h.s. of eq. (1.17) cancels o (propery 1.9). The las erm s nonzero becase here s some correlaon beween and flcang velocy componens. So sng propery (1.7) we oban: (1.18) Fgre 3 shows he momenm echange process n he shear layers of he plane e a Re=0000. Ths eample s sed o llsrae a correlaon beween and v flcang velocy componens. Le s assme ha he red fld elemen (locaed on lne R) s shfed o he lef. Ths shf s vsble by negave flcang velocy componen (see he coordnae sysem). Snce he red fld elemen s movng from he hgh mean velocy zone VR o he small mean velocy zone VL hs ransfer generaes he posve v flcang velocy (v =VR-VL, v >0). The prodc of hese wo flcaons s negave v <0. The meaveraged prodc of and v s shown n fg. 3 (boom). Ths s he rblen shear sress profle. We clearly see ha he negave flcaon generaes he posve v flcaon. The momenm echange process can also be realzed he oher way arond, so from lef o rgh (see ble fld elemen locaed on lne L). Ths s de o he fac ha formaon of he shear layer s ypcally relaed o evolon (n space and me) of he coheren vore srcres whch sbseqenly breadown no smaller forms. I means ha a one me nsan here s a momenm echange from rgh o lef. B a he oher me nsance here mgh be he momenm echange from lef o rgh. Le s, herefore, assme ha he fld elemen s shfed from lef o rgh (ble conor). Ths shf resls n posve flcang velocy. Snce he ble fld elemen s movng from he low mean velocy zone VL o he hgh mean velocy zone VR hs ransfer generaes he negave v flcang velocy (v =VL-VR, v <0). The prodc of hese wo flcang veloces s, agan, negave v <0. I means ha leads o generaon of he negave shear sress afer me averagng (fg. 3 boom). The Reader can perform a smlar analyss for he rgh par of fg. 3. The resl of hs analyss wll be a formaon of he posve shear sress profle (fg. 3, boom). The sgn 6

7 Trblence Modellng Sławomr Kbac depends on he coordnae sysem. In boh cases he ne effec s he momenm ransfer from he mean flow o he flcang flow (reslng n ncreased rblen shear sress). B he mos mporan remar s ha n rblen flow he momenm echange n one flow drecon cases he momenm echange n he oher flow drecons (flow s hree dmensonal). So here s some correlaon beween boh and flcang componens. Ths resls n he shear sress ensor (rghmos erm n Eq. 1.18). Gong bac o he me-averagng. The erms on r.h.s. of eqaon (1.15) can be easly averaged: S S 1 p 1 p (1.19) Fnally, we oban he Reynolds-Averaged Naver-Soes (RANS) eqaons (eq. 1.14, 1.18, 1.19): 0 (1.0) ( ) 1 p S (1.1) where 1 S (1.) The las erm on r.h.s of Eq. (1.1) s he dvergence of he Reynolds-sress ensor. The componens of he Reynolds sress ensor are denoed by.ths erm reqres closre model (see below). 7

8 Trblence Modellng Sławomr Kbac L <0 R v >0 >0 v <0 Y X Fg. 3. Momenm echange n he shear layers of he plane e. Mean V velocy componen (op) and he shear sress profle (boom). 8

9 Trblence Modellng Sławomr Kbac 9. Closre of he modelled erms.1. Eac Reynolds-sress ranspor eqaons If we denoe by N() he Naver-Soes operaor: 0 p 1 ) N( (.1) We can oban he eac Reynolds-sress ranspor eqaon, by mlplyng he momenm eqaon (.1) by flcang qany and addng o hs erm a smlar erm wh he ndees nerchanged and averagng. The followng ranspor eqaon s obaned (Wlco, 006):. p 1 p 1 p 1 (.) Dervaon of Eq. (.) s show n Append A. The physcal meanng of he erms on r.h.s. of Eq. (.) can be descrbed as follows. - Prodcon (frs and second erm). Trblen sresses are generaed a he epense of mean flow energy by mean flow deformaon. Ths erm does no need any closre. - The hrd erm represens he sress dsspaon whch manly occrs a he smalles scales ( shold be modelled). - Pressre flcaons (forh erm) redsrbe he rblen sress among componens o mae rblence more soropc. - Transpor erm (las erm n Eq..). Ths erm consss of several pars (n sqare brace): Frs erm s he ranspor erm and s called he moleclar dffson. Second, here s he rblen dffson erm (ranspor hrogh velocy flcaons). A closre model s necessary for hs erm. A hrd and forh par of he ranspor erm s he pressre ranspor. Ths erm also needs closre model.

10 Trblence Modellng Sławomr Kbac. Transpor eqaon for rblen nec energy An eac eqaon for he rblen nec energy follows from he eqaons for he Reynolds sress componens (.) by conracon rogh png = and mang he sm over = 1,,3: (.3) We ae advanage of he conny eqaon for he flcang velocy componens. (.4) whch means ha for an ncompressble fld sded here he pressre sran-rae erm cancels o n Eq. (.3). We defne he rblen nec energy as: (.5) Ne, by nserng Eq. (.5) o Eq. (.) we oban an eac eqaon for he rblen nec energy (.6) In Eq. (.6) he frs erm on r.h.s descrbes prodcon of he rblen nec energy. Ths erm s modeled sng he Bossnesq hypohess: S (.7) 3 We laer wre he prodcon erm as P S / SS. n Eq. (.7) denoes he rblen vscosy (see dscsson below). The second erm on r.h.s. of eq. (.6) represens dsspaon of he rblen nec 10

11 Trblence Modellng Sławomr Kbac energy (.8) In he frame of he wo-eqaon models dscssed here he erm (.8) s obaned by solvng an addonal ranspor eqaon (dscsson below). The second and hrd erm n braces descrbe he rblen ranspor process by velocy flcaons and he flcang pressre velocy ndced dffson. Boh erms are modeled sng he graden hypohess: (.9) where s a ceran consan. Fnally, he ranspor eqaon for he rblen nec energy reads: (.10).3. The sandard - model The model whch s called he sandard - rblence model was developed by Jones and Lander (197). In hs model, he RANS-eqaons are sed ogeher wh he - eqaon (.10), he -eqaon (nrodced below) and he eddy-vscosy epresson C. The eddy vscosy epresson C reqres he nowledge of he rblen nec energy, whch we can derve from eqaon (.10). The second ngreden s he dsspaon rae. Remar ha we are no resrced o an eqaon of. Any qany whch s a combnaon of and may be sed. The mos obvos s o se an eqaon for. The epresson for s nown, Eq. (.11). So, we cold se an approach smlar o he one as sed for he rblen nec energy o derve a ranspor eqaon. Techncally, hs s possble, b he eqaon s mch more complcaed han he eqaon for and conans many more erms ha need modellng. So, loong a he -eqaon (.10) and ang no accon ha he parameer ha we can form wh and s he me scale =/, he -eqaon can be ransformed no an -eqaon by c 1 P c (.11) where we nrodce he consans cε1 and cε n he ransformaon of he prodcon 11

12 Trblence Modellng Sławomr Kbac erm and he dsspaon erm. We frher nrodce he dffson coeffcen (see dscsson laer). The sandard vales for he model parameers are: c 0.09, 1.0, 1.3, c 1.44, c The vale for c comes from he observaon of hn shear flows wh appromae balance beween prodcon and dsspaon: D free e mng layers and he so-called neral regon or logarhmc layer n a bondary layer flow. Fgre 4 shows he erms n he rblen nec energy eqaon (Eq..6) along normal o he wall drecon n he rblen bondary layer a Re =1410. The resls have been obaned sng DNS by Laren e al. (01). The moleclar dffson s mporan a he wall. Boh prodcon and dsspaon become domnan a y + >5. Noe a sgn change on he profles of he rblen and moleclar dffson nsde he bondary layer (dffson parly acs as a sn and parly as a sorce of he rblen nec energy). Ths maes he prodcon and dsspaon far more mporan han he dffson processes a sffcenly large dsance from he wall. Moreover, we can assme ha prodcon almos eqals dsspaon for y + >10: P=ε. Wh y as cross-sream coordnae hs resls n: ( ) y 'v' c ( ) For hn shear flows, he epermenal observaon s 'v' ( ) 0.3, reslng n c = prodcon dsspaon Fgre 4. Terms n he rblen nec energy eqaon nsde he rblen bondary layer a Re =1410. DNS resls by Laren e al., (01). 1

13 Trblence Modellng Sławomr Kbac The consan c s deermned by consderng decayng homogeneos rblence. Fgre 5 shows he vore srcres obaned for DNS smlaon of decayng soropc and homogeneos rblence n he perodc bo (Dbef Delcayre, 000). In hs neresng flow he sascs of all qanes n braces on r.h.s. of Eq. (.6) are consan. Ths means ha here s no dffson. Moreover, he sascs of mean velocy dervaves are zero. I means ha here s no prodcon eher. The -eqaon redces o D D, D c (.1) D These eqaons are sasfed for an observer movng wh he flow by n, m 1, dla n, m n 1 (.13) c 1 Epermens of Come-Bello and Corrsn (1966) ndcae vales n = 1. o 1.3 whch mples c = 1.83 o The vale c = 1.9, seleced by Jones and Lander dffers somewha becase hey dd some nmercal opmsaon of hs consan over a range of flows. Fgre 5. Homogeneos and soropc rblence n a perodc bo. (Dbef Delcayre, 000) 13

14 Trblence Modellng Sławomr Kbac The consan c 1s deermned from a homogeneos shear flow epermen. Fgre 6 shows a sech of sch flow. Homogeneos shear flow appears o reach an eqlbrm sae wh and growng n sch a manner ha he rblen me-scale / approaches an appromaely consan vale. The governng eqaons are D D c P c P D, 1 D / (.14) whch can be combned wh he assmpon of a consan rblen me-scale o yeld he followng relaons. Assmng D/D=0 we arrve a: D( / ) D 1 D D D D P c 1P c 1 ( ) P c 1 (c11) 0 (.15) whch leads o c 1 c 1 1 P / Usng c =1.83 and he shear flow daa of Tavolars and Corrsn (1981), where P / 1.8 and /consan, he vale c resls. Agan, Jones and Lander ae he somewha dfferen vale c by nmercal opmsaon. Fgre 6. Flow n D mng shear layer 14

15 Trblence Modellng Sławomr Kbac The dffson coeffcen σ s a pror aen o ny, snce governs he rblen dffson of he rblen nec energy by he rblen moon self. The second dffson coeffcen σε comes from consderaon of he dffson of ε n he logarhmc zone of he bondary layer. The followng eqaons can be wren n a zero pressre graden bondary layer flow: 0 ( ) cons an w y y y 0 ( ) ( ) y y y 1 0 c ( ) c ( ) y y y (.16) (.17) (.18) In he log-layer, d (.19) dy y From Eq. (.16) and (.19) we oban y (.0) If we assme P= and he consan -profle n he logarhmc par of he rblen bondary layer (no dffson of ) Eq. (.10) redces o: P 'v' ( ) y y ( ) y( ) y y 3 (.1) y Tang he defnon of he dynamc rblen vscosy no Eq. (.16) we oban: C and nrodcng 15

16 Trblence Modellng Sławomr Kbac w 'v' c y c y 3 y c c y c c 1 ( y) ( y) y 4 1 ( y) y y (.) c Inrodcng Eq. (.19), (.1), (.) and C no Eq. (.18) we oban: c c y 0 c 1 y c ( ) y y ( y) ( y) y Ths resls n c c ( ) y c 1 1 (c c ) (c c ) c 1 (.3) In sandard -, he vales 1 c = 1.44, c = 1.9, c = 0.09 and = 1.3 are sed, whch followng (.3) mples = 0.43, whch s a somewha larger vale han wha s sally assmed for he Von Karman consan ( = ). Agan, hs dfference comes from he nmercal opmsaon by Jones and Lander. One of he shorcomng of he sandard - model s s nably o reprodce a correc level of he rblen shear sress n he near-wall regon. Fgre 7 shows he predced by he - model (dashed lne) and comped sng DNS (sold lne) profles of he rblen vscosy nsde he bondary layer for smlaon of he channel flow. As shown he - model overesmaes he rblen vscosy n he near-wall regon. In order o lm hs shorcomng some dampng erms are employed n fron of he eddyvscosy formla. The oher approach consss n sng separae rblence model, whch shows beer performance han he classc - model n he near-wall regon, and s blendng wh sandard - frher away from walls. Ths obvosly complcaes he modelng approach sng he sandard - model. 16

17 Trblence Modellng Sławomr Kbac Fgre 7. Predced sng he - model (dashed lne) and comped sng DNS (sold lne) profles of he rblen vscosy nsde he bondary layer for smlaon of he channel flow a Re =590. Drbn and Peersson Ref, (003). References J.B. Cazalbo, P.R. Spalar and P. Bradshaw. On he behavor of wo-eqaon models a he edge of a rblen regon. Physcs of flds, 6(5): , G. Come-Bello and S. Corrsn. The se of a conracon o mprove he soropy of grdgeneraed rblence. J. Fld. Mech., 5:657-68, Y. Dbef and F. Delcayre, On coheren-vore denfcaon n rblence, Jornal of Trblence, 1, N11, 000. P.A. Drbn and B.A. Peersson Ref. Sascal Theory and Modelng for Trblen Flows. John Wley, 001. W.P. Jones and B.E. Lander. The predcon of lamnarzaon wh a wo-eqaon model of rblence. AIAA J., 15: ,197. B.E. Lander and B.I. Sharma. Applcaon of he Energy Dsspaon Model of Trblence o he Calclaon of Flow Near a Spnnng Dsc. Leers n Hea and Mass Transfer, 1(): ,

18 Trblence Modellng Sławomr Kbac C. Laren, I. Mary, V. Gleze, A. Lera, D. Arnal, DNS daabase of a ransonal separaon bbble on a fla plae and applcaon o RANS modelng valdaon, Compers & Flds 61: 1 30, 01. Tavolars and Corrsn., Epermens n Nearly Homogeneos Trblen Shear Flow wh a Unform Mean Temperare Graden. J. of Fld Mechancs, 104: , D.C. Wlco. Reassessmen of he Scale Deermnng Eqaon for Advanced Trblence Models. AIAA J., 6(11): , D.C. Wlco. Comparson of Two-Eqaon Trblence Models for Bondary Layers wh Pressre Graden. AIAA J., 31(8): , D.C. Wlco. Trblence Modelng for CFD. Grffn Prnng, Glendale, Calforna, D.C. Wlco. Trblence Modelng for CFD. DCW Indsres, 006. D.C. Wlco, Formlaon of he - rblence model revsed. AIAA J., 46: ,

19 Trblence Modellng Sławomr Kbac Append A Mlplyng Eq. (.1) by he flcang velocy and addng hs erm o a smlar erm wh he ndees nerchanged and averagng we oban: (A.1) Afer some algebra we oban: (A.) Assmng he Reynolds decomposon (mean pls flcaon): (A.3) and png Eq. (A.3) o (A.) we oban several erms whch are dscssed below: 1. Local me dervave (A.4) 19

20 Trblence Modellng Sławomr Kbac The propery (1.10) was sed n order o smplfy Eq. (A.4) (second lne). The followng relaon was sed o derve he las erm n Eq. (A.4) (A.5). Convecve erm (A.6) 3. Pressre graden erm (A.7) 4. Vscos erm (A.8) Fnally, png Eq. (A.4), (A.6), (A.7) and (A.8) o Eq. (A.) we oban he eac form of he Reynolds-sress eqaon: 0

21 Trblence Modellng Sławomr Kbac (A.9) Noe ha Eq. (A.9) can be rewren n he form gven by Eq. (19) sng he followng relaons: (A.10) 1