Modelling of turbulent flows: RANS and LES

Transcription

1 Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der Strömungsmechanik: RANS und LES Markus Uhlmann Institut für Hydromechanik Karlsruher Institut für Technologie SS 2012 Lecture 7 1 / 16

2 LECTURE 7 2 / 16

3 Questions to be answered in the present lecture How can the linear eddy viscosity assumption be avoided without the need for solving transport equations? algebraic stress models nonlinear eddy viscosity models 3 / 16

4 Intermediate between RSM and Boussinesq approximation Reynolds stress transport models naturally incorporate transport effects describe stress production exactly BUT: high computational cost (equations for 6 components) Standard eddy-viscosity models (Boussinesq approximation) (A) local relation between Reynolds stress and mean strain (B) linear relation between Reynolds stress and mean strain (A) is inevitable (B) can be changed non-linear Reynolds stress/mean strain relationships 4 / 16

5 Reynolds stress equations for transport model: D u i u j + (T k ) Dt,k = P + R 2 }{{} 3 εδ D (transport) Basic idea of algebraic stress models (ASM): approximating transport terms D by local expressions resulting model is free from derivatives: 6 algebraic equations relating u i u j, k, ε, u i,j approach benefits from known models for pressure-strain R 5 / 16

6 equilibrium assumption Reynolds stress equations for transport model: D u i u j + (T k ) Dt,k = P + R 2 }{{} 3 εδ D (transport) Simplest local equilibrium assumption: neglect the transport term altogether: D = 0 implies for the turbulent energy: 1 2 D ll = P ε = 0 problem: equality P = ε not verified in general! 6 / 16

7 weak equilibrium assumption Reynolds stress equations for transport model: D u i u j + (T k ) Dt,k = P + R 2 }{{} 3 εδ D (transport) Weak equilibrium assumption (Rodi 1972) rewriting Reynolds stress in terms of anisotropy and TKE: u i u j = 2k b k δ neglecting transport of anisotropy: D u i u j ( Dk Dt + u i u j ) Dt = u i u j k k D Dt k u i u j k applying the approximation to the entire transport term: D u i u j k (transport of k) = u i u j 1 k 2 D ll = u i u j k (P ε) final model: u i u j k Dk Dt (P ε) = P + R 2 3 εδ 7 / 16

8 CHAPTER 11: REYNOLDS-STRESS AND RELATED MODELS ASM predictions for homogeneous shear flow Turbulent Flows Stephen B. Pope Cambridge University Press, 2000 LRR-IP pressure-strain model R = C R 2 εb C 2 (P 2 3 Pδ ) corresponding ASM: b = 1 2 (1 C 2) C R 1+P/ ε P 2 3 δ P ε in homogeneos shear flow: b has finite limit for P ε b b b stress remains realizable 0.3 b c Stephen B. Pope 2000 b 11 b 22 =b 33 b b 12 (k-ε) P/ε P/ ε, Figure 11.20: Reynolds-stress ASM predictions; anisotropies as, functions k-ε model of P/ε according to the LRR-IP algebraic stress model. The dashed line shows b12 according (from to the Pope k-ε Turbulent model. Flows, 2000) 8 / 16

9 Stress/mean strain relation implied by ASM CHAPTER 11: REYNOLDS-STRESS AND RELATED MODELS Turbulent Flows Stephen B. Pope Cambridge University Press, 2000 c Stephen B. Pope 2000 LRR-IP pressure-strain model define: u v = C µ k 2 ε u,y with unknown function C µ substituting ASM: C µ = 2 3 (1 C 2)(C R 1+C 2 P/ ε) (C R 1+P/ ε) 2 C µ decreases with P/ ε 0.30 C µ P/ ε P/ε Figure 11.21: The value, of Cµ as ASM a function predictions of P/ε given by the LRR-IP algebraic stress model (Eq ). (from Pope Turbulent Flows, 2000) 9 / 16

10 Assessing the ASM approach Achievements of algebraic stress models partial differential equations reduced to algebraic equations physics of pressure-strain model is carried over Problems of the ASM approach implicit system of equations dependence is in general non-linear system can have multiple solutions numerical stiffness 10 / 16

11 Performance in free shear flow Performance in flow with system rotation Explicit ASM or non-linear eddy viscosity models Explicit ASM (EASM) explicit expressions for the stress components are numerically desirable there are two routes (viewpoints) to achieve this: 1. construct an implicit ASM (as above): b = f i (b, k ε u i,j ) then derive equivalent explicit form analytically b = f e ( k ε u i,j ) 2. construct an explicit expression for the Reynolds stresses: b = f e ( k ε u i,j ) both approaches have been realized results also known as non-linear eddy viscosity models 11 / 16

12 Deriving explicit algebraic stress models Performance in free shear flow Performance in flow with system rotation ansatz: ) b = B (Ŝ, Ω where normalized mean rate of strain/rotation are defined: Ŝ k 2 ε ( u i,j + u j,i ), Ω k 2 ε ( u i,j u j,i ) most general consistent expression (Pope 1975): ) B (Ŝ, Ω = 10 n=1 G(n) (n) T with independent, symmetric, deviatoric functions: T (1) = Ŝ (2) T T (4) = Ω trace( Ω 2 (5) )I T T (7) = ΩŜ Ω 2 Ω 2 Ŝ Ω (8) T T (10) = ΩŜ2 Ω 2 Ω 2 Ŝ 2 Ω = Ŝ Ω ΩŜ T (3) = ΩŜ2 Ŝ2 Ω T (6) = Ŝ ΩŜ2 Ŝ2 ΩŜ T (9) and undetermined scalar coefficients G (n) = Ŝ2 1 3 trace(ŝ2 )I = Ω 2 Ŝ + Ŝ Ω trace(ŝ Ω 2 )I = Ω 2 Ŝ 2 + Ŝ2 Ω trace(ŝ2 Ω 2 )I 12 / 16

13 Examples of EASM Performance in free shear flow Performance in flow with system rotation Linear case Boussinesq hypothesis G (1) = C µ ; G (n) = 0 for n 2 b = C µ Ŝ Statistically two-dimensional flow (Pope, 1975) sum contains only three terms (G (n) = 0 for n 4) General three-dimensional flow all 10 terms are non-zero 1. ASM approach: Gatski & Speziale (1993), based on linear pressure-strain 2. direct approach: Shih, Zhu & Lumley (1995), based on realizability 13 / 16

14 7 4 0 " ",, I O I O O S O O O IIIIIIIIIIIIIIIIIII I I I I IO O O O S S " ', ", O OOOOOOOO S ',", ' ", ',", ", ", ","","","", ", ", ", "", ", ", ", ", ", """" S S S S SSSSSSSSSSSS ''''''''''''''''''' ' Performance in free shear flow Performance in flow with system rotation Performance of EASM in mixing layer flow Self-similar mixing layer spreading rate dδ/dx: exp. EASM (SZL) k-ε EASM by Shih et al. not well calibrated for free shear flows anisotropy well predicted (normal stresses) y δ / 41 2 /30 /0 1 2,,,,,,,,,,,, ' ' '!#" $%$& ()+*-, $. " " " " " " " "" ' ' ' ' ' ' ' ' ''' II I I I I I I O O O S S S OOOOO O O O O I I OO S S S S S S S S SSS :;<= :>? :@A: BDCEF FHGJI KLNMPO K%LNM /RQTS 4 6 4%1 42 4% %1 52 4%1 b 11 Fig. 4.10: Profils transversaux de la composante axialeb11 de l anisotropie. (experiment of Bell & Mehta, 1990) 14 / 16

15 Performance in free shear flow Performance in flow with system rotation Performance of EASM for rotating pipe flow 104 S. Wallin and A. V. Johansson Rotation in axial direction rotation has stabilizing effect (production term) EASM predictions are reasonable linear eddy-viscosity fails U z U mean 0.8 u r/r Figure 7. Axial velocity in rotating lines: pipe EASM flowof for Wallin different & Johansson rotation(2000) ratios. Computatio current EARSM (EARSM 0) based on K ω (lines) compared to experiment by Imao (symbols) for Z =1( )( ), Z =0.5 symbols: ( experiment )( ) and Z of =0( Imao et al. )( ). (1996) 1.0 no rotation; medium; strong / 16

16 Performance in free shear flow Performance in flow with system rotation Performance of EASM for rotating channel flow u Rotation in spanwise direction rotation causes non-symmetric profiles EASM predictions are comparable to full transport model y/h, EASM of Gatski & Speziale (1993) experiment of Johnston et al. (1972) 16 / 16

17 Conclusion Further reading Summary of today s lecture How can the linear eddy viscosity assumption be avoided without the need for solving transport equations? transport terms eliminated by weak equilibrium assumption 1. algebraic stress models (ASM) inherit properties of pressure-strain model often numerical difficulties 2. nonlinear eddy viscosity models (EASM) provide general explicit expressions for Reynolds stresses allow for prediction of complex straining fields 1 / 3

18 Conclusion Further reading Summary (2): A hierarchy of RANS models (a) elliptic relaxation RSM (b) standard RSM (c) ASM (with k-ε equations) (d) nonlinear eddy-viscosity model (with k-ε) (e) standard (isotropic eddy-viscosity) k-ε model (f) one-equation k-model (with l m ) (g) mixing-length model Which assumption is added when stepping from (a) to (g)? 2 / 3

19 Conclusion Further reading Further reading S. Pope, Turbulent flows, 2000 chapter 11 D.C. Wilcox, Turbulence modeling for CFD, 2006 chapter 6 3 / 3