TURBULENCE MODELING FOR BEGINNERS

Transcription

1 TURBULENCE MODELING FOR BEGINNERS by TONY SAAD UNIVERSITY OF TENNESSEE SPACE INSTITUTE hp://ed.nows.

2 The purpose of hs ny gude s o summarze he basc conceps of urbulence modelng and o a comple he fundamenal urbulence models no one smple framewor. Inended for he begnner, no dervaons are ncluded, unless n some smple cases, as he focus s o presen a balance beween he physcal undersandng and he closure equaons. I hope hs maeral wll be helpful.

3 TABLE OF CONTENTS Inroducon... 3 Frs Order Models:... 6 Zero-Equaon Models... 7 One-Equaon Models:... 8 Two-Equaon Models ( ε ): Second Order Models... 1 The Sandard Reynolds Sress Model (RSM) Turbulen Dffuson Modelng Pressure-Sran Correlaon Modelng Modelng of he Turbulen Dsspaon Rae The Algebrac Sress Model Acnowledgmens... 19

4 INTRODUCTION A urbulen flow feld s characerzed by velocy flucuaons n all drecons and has an nfne number of scales (degrees of freedom). Solvng he NS equaons for a urbulen flow s mpossble because he equaons are ellpc, non lnear, coupled (pressure velocy, emperaure velocy). The flow s hree dmensonal, chaoc, dffusve, dsspave, and nermen. The mos mporan characersc of a urbulen flow s he nfne number of scales so ha a full numercal resoluon of he flow requres he consrucon of a grd wh a number of nodes ha s proporonal o Re 9/4. The governng equaons for a Newonan flud are Conservaon of Mass ρ ρu + = 0 x (1) Conservaon of momenum ρu ρuu u p + = μ + ρg + τ s x x x x u () Conservaon of passve scalars (gven a scalar T ) ρct ρ p cut p T + = + s τ x x x (3) So how can we solve he problem? One of he soluons s o reduce he number of scales (from nfny o 1 or ) by usng he Reynolds decomposon. Any propery (wheher a vecor or a scalar) can be wren as he sum of an average and a flucuaon,.e. φ = Φ+ φ where he capal leer denoes he average and he lower case leer denoes he flucuaon of he propery. Of course, hs decomposon wll yeld a se of equaons governng he average flow feld. The new equaons wll be exac for an average flow feld no for he exac urbulen flow feld. By an average flow feld we mean ha any propery becomes consan over 3

5 me. The resul of usng he Reynolds decomposon n he NS equaons s called he RANS or Reynolds Averaged Naver Soes Equaons. Upon subsuon of he Reynolds decomposon (for each varable, we subsue he correspondng decomposon) we oban he followng RANS equaons: Conservaon of Mass ρ ρu + = 0 τ x (4) Conservaon of momenum ρ ρuu U U P + = μ ρuu + τ S u x x x x (5) Conservaon of passve scalars (gven a scalar T ) ρct ρ p cut p T + = ρc τ p u + S (6) x x x Noe: a specal propery of he Reynolds decomposon s ha he average of he flucuang componen s dencally zero, a fac ha s used n he dervaon of he above equaons. However, by usng he Reynolds decomposon, here are new unnowns ha were nroduced such as he urbulen sresses ρ uu and urbulen fluxes (where he overbar denoes an average) and herefore, he RANS equaons descrbe an open se of equaons. The need for addonal equaons o model he new unnowns s called Turbulence Modelng. We now have 9 addonal unnowns (6 Reynolds sresses and 3 urbulen fluxes). In oal, for he smples urbulen flow (ncludng he ranspor of a scalar passve scalar, e.g. emperaure when hea ransfer s nvolved) here 14 unnowns! A sragh forward mehod o model he addonal unnowns s o develop new PDEs for each erm by usng he orgnal se of he NS equaons (mulplyng he momenum equaons o produce he urbulen sresses ). However, he problem wh hs procedure s ha wll nroduce new correlaons for he unnowns 4

6 (rple correlaons) and so on. We hen mgh hn of developng new equaons for he rple correlaons, neverheless, we wll end up wh quadruple correlaons and so on An alernave approach s o use he PDEs for he urbulen sresses and fluxes as a gude o modelng. The urbulen models are as follows, n order of ncreasng complexy: Algebrac (zero equaon) mode ls: mxng lengh (frs order model) One equaon models: model, μ model (frs order model) Two equaon models: ε, l, ω, low Re ε (frs order model) Algebrac sress models: ASM (second order model) Reynolds sress models: RSM (second order model) Zero Equaon Models One Equaon Models Frs Order Models Two Equaon Models Algebrac Sress Models Second Order Models Reynolds Sress Models 5

7 FIRST ORDER MODELS Frs order models are based on he analogy beween lamnar and urbulen flow. They are also called Eddy Vscosy Models (EVM). The dea s ha he average urbulen flow feld s smlar o he correspondng lamnar flow. Ths analogy s llusraed as follows u u u τ = μ + μδ x x 3 x Lamnar FLow: T q = cp x U U τ = ρuu = μ + δρ x x 3 Turbulen Flow: T q = ρu = cp x (7) whch s referred o as he generalzed Boussnesq hypohess. Noe ha: μ = Turbulen Vscosy = Turbulen Knec Energy = Turbulen Conducon Coeffcen The urbulen vscosy and he urbulen conducon coeffcen are flow properes. They are no properes of he flud. They vary from one flow o anoher. So he problem now s he devse means or models o fnd hese unnowns, he urbulen vscosy and he urbulen conducon, because he urbulen sresses and fluxes wll be expressed as funcon of hese new flow properes. 6

8 ZERO EQUATION MODELS In zero equaon models, as he name desgnaes, we have no PDE ha descrbes he ranspor of he urbulen sresses and fluxes. A smple algebrac relaon s used o close he problem. Based on he mxng lengh heory, whch s he lengh over whch here s hgh neracon of vorces n a urbulen flow feld, dmensonal analyss s used o show ha: μ du ν = lu = lm lm (8) ρ dy l m s deermned expermenally. For boundary layers, we have l = κ y for y< δ l m m = δ for y δ (9) and cp = μ (10) Pr Equaons (8) hrough (10) are hen used n he lamnar urbulen analogy and hen bac no he orgnal RANS equaons. 7

9 ONE EQUATION MODELS In one equaon models, a PDE s derved for he urbulen nec energy and he unnowns (urbulen vscosy and conducon coeffcen) are expressed as a funcon of he urbulen nec energy as: ( ) 1 K = u + v + w (11) We also mae use of he fac ha μ lu bu n hs case, he velocy scales s proporonal o he square roo of he nec energy (unle he above case where u was proporonal o he graden of velocy). Therefore, we have: μ = C μ cp = μ Pr Kl m (1) Now ha he urbulen vscosy and urbulen conducon coeffcen are expressed n erms of he urbulen nec energy (herefore he urbulen sresses and urbulen fluxes are funcons of he nec energy), a PDE s developed for he urbulen nec energy. ρk ρuk ρu + = ρuu βgρu...(p + G ) τ x x 1 K ρuu + pu μ...(d ) x x U μ x U x...( ε ) (13) On he oher hand, hs equaon nroduces wo new unnown correlaons; he urbulen and pressure dffusons (D ) and he dsspaon raes (ε ) whch need o be modeled. Fnally, we end up wh he followng: ρk ρuk μ K τ + = μ + + P + G x x σ x ρε (14) 8

10 Where = ρ ρu U μ = U + U P uu x x x x = β T G g c p x (15) (16) And from dmensonal analyss, we oban U U K K 3/ 3/ ε = μ ε = Cd x x l lm (17) 9

11 TWO EQUATION MODELS ( ε ) In he wo equaon models, we develop wo PDEs: one for he urbulen nec energy and one for he urbulen dsspaon rae. The PDE for he urbulen nec energy s already gven by Eq.(13), however, he expresson for he urbulen or eddy vscosy s dfferen. So, he dea s o express he urbulen vscosy as a funcon of K and ε and hen derve PDEs for K and ε. μ 3/ 1/ ul = μ = Cμ ε cp = μ Pr ε (18) The equaon for he urbulen nec energy s repeaed here for convenence ρk ρuk μ K τ + = μ + + P + G x x σ x ρε (19) = ρ ρu U μ = U + U P uu x x x x (0) G = β g c p T x (1) Now nsead of modelng ε, we shall develop an ndependen PDE for s ranspor. We oban ρε ρu ε μ ε μ τ ( P ) + = + + Cε1 + Cε3G Cε x x σε x K ε ε ρ K () = ρ ρu U μ = U + U P uu x x x x (3) 10

12 G = β g c p T x (4) The consans are deermned from smple benchmar expermens. C μ σ σ ε C ε 1 C ε C ε 3 Pr

13 SECOND ORDER MODELS The cenral concep of second order models s o mae drec use of he governng equaons for he second order momens (Reynolds sresses and urbulen fluxes) nsead of he quesonable Boussnesq hypohess. The movaon s o overcome he lmaons of frs order models n dealng wh he soropy of urbulence and he exra srans. The overshoo of hs approach s he large number of PDEs nduced whch nvolve many unnown or mpossble o fnd correlaons. The mos famous models are he Algebrac Sress Model (ASM) and he Reynolds Sress Model (RSM). The second order modelng approach shall be llusraed wh he RSM model only. 1

14 THE STANDARD REYNOLDS STRESS MODEL (RSM) The RSM nvolves he modelng of urbulen dffuson, pressure sran correlaon whch s he mos nvolved par of he RSM, and he urbulen dsspaon rae. The RANS momenum equaon s wren as ρuu ρuuu uu + = μ + D τ x x x + P + G +Φ ε (5) Where U P uu u u U = ρ ρ x x (6) G = ρβ g u ρβ g u (7) And he scalar ranspor equaon s ρu ρuu u + = α + D τ x x x + P + G +Φ ε (8) Where T U P P P uu u = + = 1 ρ ρ x x (9) G = ρβ g (30) The obecve s o fnd models for urbulen dffuson ( D, D ), he pressure sran correlaon ( Φ, Φ ), and he urbulen dsspaon rae ( ε, ε ). TURBULENT DIFFUSION MODELING One way o model he 3 rd order urbulen dffuson ensor s o wre s own ranspor equaon. However, hs becomes very complcaed (handlng ensors by 13

15 self s somemes raher nvolved!). One remedy s o use Generalzed Graden Dffuson Hypohess D C u u uu = s l ε x l (31) and u u D = Cs uu + uu ε x x (3) where he consans are deermned from smple expermens. PRESSURE STRAIN CORRELATION MODELING The role of he pressure sran neracon s o redsrbue he urbulen nec energy over he hree normal sresses. The governng equaon for he evoluon of hs phenomenon aes he form of a Lagrangan negro dfferenal equaon p u 1 u u l u U u l u αg u dv = ρ x 4π I V x x l x x l x x T x x x x S (33) Wh a smlar equaon for pressure scalar correlaon. Whou furher complcaon, he fnal analyss assumes he followng Φ =Φ +Φ +Φ +Φ +Φ +Φ w w w (34) And Φ =Φ +Φ +Φ +Φ +Φ +Φ w w w (35) where Φ, Φ 1 represen he urbulence urbulence neracon,.e. beween flucuaons. 1 They are modeled usng an soropc assumpon, as n he decay of homogeneous urbulence 14

16 ε Φ 1 = C1ρεa = C 1 ρuu δ 3 ρ (36) 1 ε Φ = C ρu = C ρu τϑ (37) Φ, Φ represen he shear effecs. They are also modeled by soropzaon of urbulence 1 Φ = C P δ P 3 (38) Φ = C P (39) Φ Φ 3, 3 shear effecs represen he body force effecs. Ther modelng s smlar o ha of he 1 Φ = C G δ G (40) Φ = CG (41) 3 3 w w w w w w Fnally, Φ, Φ, Φ, Φ, Φ, Φ represen he wall effecs. The basc dea s ha a pressure wave s refleced a a wall and hus affecs he whole flow feld n an ellpc manner ε 3 3 Φ = C ρ uunnδ uunn uunn F w 3 3 Φ = Cw Φl nnlδ Φ nn Φ nn Fn w 3 3 Φ 3 = C3w Φl3nnlδ Φ3nn Φ 3nn Fn w 1 1w l l n (4) and 15

17 ε Φ = C ρ u nn F Φ = C Φ nn F w 1 1w n w w n Φ = C Φ nn F w 3 3w 3 n (43) wh C Fn = (44) κ 34 3 μ K ε y n Where y n represens he dsance from he wall o he concerned pon n he doman. MODELING OF THE TURBULENT DISSIPATION RATE A smple soropc model s used o model ε. ε = δε (45) 3 ε = 0 (46) Wh he followng ranspor equaon for he dsspaon rae ρε ρu ε ε ε 1 + = C ρu u + ( C P C ρε) P = P = ρuu τ x x x ε ε1 ε ε U x (47) Where 1 U P = P = ρuu (48) x The consans appearng n RSM equaons are gven n he followng able 16

18 C s C 1 C C 3 C 1w C w C 3w C ε C ε1 C ε C C 1 C C C 1w C w C 3w

19 THE ALGEBRAIC STRESS MODEL In he algebrac sress model, wo man approaches can be underaen. In he frs, he ranspor of he urbulen sresses s assumed proporonal o he urbulen nec energy; whle n he second, convecve and dffusve effecs are assumed o be neglgble. Algebrac Sress models can only be used where convecve and dffusve fluxes are neglgble,.e. source domnaed flows. 18

20 ACKNOWLEDGMENTS I am ndebed o my undergraduae advsor and uor, Prof. Mchel El Haye a Nore Dame Unversy for nroducng me o urbulence modelng. Mos of hs uoral s based on hs explanaon and presenaons. 19