Accommodating LES to high Re numbers: RANS-based, or a new strategy?

Transcription

1 Symposium on Methods July 2005, FOI, Stockholm, Sweden, Accommodating LES to high Re numbers: RANS-based, or a new strategy? K. Hanjalić Delft University of Technology, The Netherlands Guest Professor, Darmstadt University of Technology, Germany Contributions: M. Hadžiabdić, H. Jonker, S. Kenjereš, M. Popovac, M. van Reeuwijk

2 CONTENT: Hybrid RANS/LES (HRL) A priori considerations Zonal matching Seamless approach T-RANS based VLES of R-B convection at extreme Ra s Prediction of ultra-turbulent regime Seamless RANS-LES of R-B convection Possible new strategies Lagrangian partial averaging (Leray-α model) 2

3 RANS Merging Strategies LES CVS - Coherent Vortex Simulations (Farge and Schneider) DES - Detached Eddy Simulations (Spalart et al.) IRLES - Incompletely Resolved LES LANS - Locally Averaged N-S (L-RANS) LEST - Large eddy STimulation (Batten et al., also LNS) PALES - Partially Averaged LES PANS - Partially Averaged N-S (Girimaji) PITM - Partially Integrated Transport Model (Dejoan & Schiestel) PRES - Partially Resolved Eddy Simulations (Pope) PILES - Partially Integrated LES T-RANS - Transient RANS (Hanjalic and Kenjeres)??? 3

4 Zonal (two-layer): Distinct, predefined, interface between RANS (near-wall) and LES (outer region) Different models in two regions: an one-point closure for RANS an ssg model for LES Re τ =2000 Two Approaches: Seamless : A single, continuous model throughout the flow In the limits: y 0 RANS y δ LES (DNS) Re τ =2000 4

5 Key questions Fixed interface (where to locate it?) or a seamless (smooth) blending Which matching conditions are to be used? How does a RANS model react to unsteadiness ( receptivity )? Will the dynamics be rightly returned? What is the impact of RANS layer on the LES region? Will the modelled contribution correctly compensate the reduction in the resolved contribution? Which models are suitable? Problems: detecting the grid and and adjusting the models Zonal modelling: need for extra forcing in the buffer Seamless : how to modify RANS and LES in the buffer region 5

6 Coarse LES problem in attached flows: streaks and super-streaks in a channel, Reτ=2000 Fine LES: 512x128x128 (reference) (Temmerman and Leschziner) Coarse LES: 63x32x64 Contours of instantaneous velocity in plane y + =6 6

7 Channel Flow, Re τ = 2000 Q-isosurfaces in the outer region 400<y + <1600 y + (=Re τ) =2000 y + =100 Fine LES: Vorticity contours in a spanwise cross-section Q-isosurfaces in the inner wall region 0<y + <300 7

8 A zonal (two-layer) scheme: Matching criteria Matching criteria: continuity of total eddy viscosity at the interface ν + ν = ν + ν res res SGS LES t RANS ν res LES with overbar denoting filtered, and <> some local smoothing. Resolved stresses continuous across the interface One-eqn model: C µ = l µ ν = ν k C l t µ µ SGS 0.5 RANS,int = C µ at the interface: k 0.5 ' ' ' ' ( uu i j uu k kδij /3) Sij k-ε model: C µ = f µ Sij Sij ν t ν = SGS C f µ µ 2 ( k / ) 2 µ ( k / ε ) εν ( f ) = ν SGS 2 t 2 k ε 8

9 Adjustment of C µ Function 1 (Leschziner et al.) C ( ) ( y ) 1 exp y = ( C,int 0.09 ) 1 exp int µ µ int Function 2 (Hadziabdic) C µ + y + = 0.09 for y C exp( ( y y( y = 34)) / ) + µ,int + Cµ = y > + 1 exp( yint y( y = 34) / int ) for 27 9

10 Adjustment of C µ (cont.) Variation of C µ across the flow (time invariant) Time histograms of the instantaneous and homogeneously-averaged C µ at the RANS-LES interface 10

11 Interface issues in Two-equation RANS model Interface B.C. for 2-eqn RANS, k int and ε int - Options for k: k a: res int k LES c: Scale similarity k = 2 k = k = 0. 5(Û U ) b: Isotropic spectrum distrib. int = k SGS = 3 2 C C κ 8 / 3 S π 2/ 3 v SGS int 2 SGS where Û i U i i - test-filtered velocity - filtered velocity Interface B.C. for 2-eqn RANS, k int and ε int - Options for ε int : ε int = 3 / 2 k 2. 5y n Or, from least-square error between the total viscosity on both sides of interface. i 11

12 Zonal two-eqn HRL of channel, Reτ =2000: RANS: k-ε with modified Cµ; LES Smagorinski Effect of interface location Modelled eddy viscosity Mean velocity Total and modelled turbulent shear stress 12

13 Zonal two-eqn HRL of channel flow at Re-2000: RANS: k-ε with modified Cµ; LES Smagorinski RANS/LES interface Y + =120 Y + =280 Y + =610 + Fine LES (Temmerman & Leschziner) Contours of streamwise vorticity in planes normal to the flow. (a)- C µ,int =locally averaged, (b),(c)- C µ,int =instantaneous 13

14 Some seamless methods Girimaji: PANS (2003) De Langhe: RNG-based (2003) ( C P C ε ) Dε k-ε model with f Dt τ LLES where f = f f L = RANS L ε1 ε2 = + Dejoan and Schiestel PITM : k-ε model with o Cε2 Cε1 Cε 2 = Cε 1 + 2/ 3 L ( ) 1/3 LES = V LRANS 1+ β L kres + kmod LES LRANS = ε D ε grid detector ( ) 3/2 Problem: providing k res and ε (D&Sch used L RANS =κy ) 14

15 Seamless HRL of channel flow at Re τ =2000: Dejoan and Schiestel k-ε model (Hadziabdic, 2004) 15

16 Energy spectra, Re τ =2000: D&Sch k-ε model (Hadziabdic, 2004) 16

17 A simpler seamless matching (Hadziabdic, 2004) k-ε-ζ-f elliptic relaxation (a version of Durbin s model, ζ=v 2 /k) Dk with k equation Dk Pk Dt = + ξ ε max 1, L 3/2 ξ = max, k ε = ε = Cs V RANS LES Switch from RANS to LES by ν = max( ν, ν ) t t t ( ) 1/3 Re τ =590 Re τ =

18 A simpler seamless matching (Hadziabdic, 2004) Re τ =2000 Re τ = C s =0.8 C s =

19 Some observations ZONAL: Most RANS respond well to LES forcing - Resolved motion in URANS is as strong as in the LES region when forcing from true LES! In the RANS region, both resolved and modelled contributions are large need forf an ad hoc modification (C µ) to reduce the total motion. Fundamental inconsistency in the LES side next to RANS: unrealistic streaks structure, insufficient stress; Needs for extra forcing, artificial backscatter, irrespective of RANS SEAMLESS : Dynamic adjustment of the RANS model to ensure continuity across the interface. No need for extra forcing! Blending still not optimal In general: For identical grids, the HRL results are significantly better than those obtained with LES for the same (coarse) mesh. 19

20 High-Ra number challenge in RB convection Nu Ra 1/3 for Ra<10 12 (Pr O(1)) Nu Ra 1/2 for Ra λ v /H Ra -1/7 λ θ /H Ra -1/3 20

21 High-Ra number challenge in RB convection Nu Ra 1/3 for Ra<10 12 (Pr O(1)) Nu Ra 1/2 for Ra Ra 1/2 Ra- 1/7 Ra- 2/9 Ra- 1/3 Ra 1/3 λ v /H Ra -1/7 λ θ /H Ra -1/3 21

22 High-Ra number challenge in RB convection C f = 2τ W / ρu 2 b Evolution of wall shear stress with Ra τ W = µ U b = 2 U + z βg TH ( ) 1/ 2 V z 2 Evolution of friction coefficient with Re 22

23 T-RANS of R-B: Temperature colored instantaneous trajectories Ra=6.5x10 5 Ra=2x10 14 central horizontal plane (z/d=0.5) inside thermal boundary layer (z/d=0.05): 23

24 : Seamless Approach (Kenjeres & Hanjalic,2004) 3 Equations Reduced T-RANS ASM/AFM model: k ε 2 θ Dejoan and Schiestel (2001) : Present Approach: C ε 2 = C ε L β L RANS LES n C = C 2 ε1 ε + α = max 0.48 α ( 1,L / ) RANS L LES L = κ z RANS W 3 / 2 L f L = ( Vol) 1/ 3 = RANS µ L = ( Vol) 1/ 3 LES LES k ε + simple + interface position self-adjustable -- wall-distance dependence -- additional empirical input (two coefficients) + simple + interface position self-adjustable + no wall-distance dependence + no additional empirical input 24

25 Hybrid RANS/LES: Seamless Approach : blending function C 1.92 ( RANS ) ε 2 C = C 1.44 ( DNS ) ε2 ε1 L RANS /L LES < 1 L RANS /L LES > 1 RANS LES/DNS C ε 2 = C ε L β L RANS LES n C = C 2 ε1 ε α 25

26 HYBRID: between T-RANS and coarse LES Ra=10 9, Pr=0.71 NEW-HYBRID!!! COARSE LES!!! 26

27 Ra=10 9, Pr=0.71 HYBRID: MODELLED <, RESOLVED > -zoom in (0.9< z/h <1) <L RANS /L LES > = 1 27

28 LES (fine grid) 256x256x128 HYBRID 82x82x62 T-RANS 82x82x62 Ra=10 9, Pr=0.71 HYBRID: finer flow structures captured! 28

29 z/h=0.5 LES HYBRID T-RANS z/h=

30 Observation from parallel LES, T-RANS and Hybrid: good agreement between T-RANS with well-resolved LES and with experimental data over a range of Ra Hybrid seamless approach introduced in order to further sensitize T-RANS approach to higher-frequency instabilities and to make predictions less dependent on the subscale model Hybrid approach captures more instabilities and smalles scales compared to T-RANS - whilst still returning accurate predictions of wall heat-transfer and second moments 30

31 Some new old ideas: Lagrangian averaging of the Navier-Stokes equations ( LANS ) U i DNS U ; U U i j j i U i U% i Spatial filtering: LES Ui; U U + ( U U U ju ) j j j j i j i Lagrangian averaging U, U% ; U% U i i j j i - Smoother velocity field - Reduced nonlinearity of NS 31

32 The Leray-α model (1934!) (e.g.cheskidov et. al., Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 2005, 461) U% i Ui Ui 1 p U i + U% j = Fi + ν t xj ρ xi xj xj Filtering operator U% ˆ 2 2 U α % = U x i 2 j Fourier transform i 1 = Uˆ ακ i i f Both fields divergence-free Ui U% x i i = = xi α k 2 0 For filtered field: free slip at the wall! 32 k

33 Opportunities Theoretically well-founded. Simple equations. The filtered velocity is non-local feels walls. Model is dispersive and not diffusive does not dissipate k No tests for most of the standard flow configurations. No tests with scalar transport so far. Our focus: Evaluate the Leray-α model for some generic flows Channel flow, Side-heated and RB convection Opportunity for hybrid models based on this rationale. 33

34 α-filtering in action: R-B at Ra=10 5, Pr=1 α / H = 0 α / H = 0.05 α / H =

35 Expectations Unlike LES, the Leray-α model does not modify the dissipative behavior, but Acts on the production side. Prod. TKE w ~' u ' i u z i log E Leray-α LES Prod. T-var ~' ' w θ z θ log k 35

36 R-B convection with the Leray-α model, Ra=10 5 K-budget θ-budget 36

37 What to make of this? Heat transfer practically unaffected by α. Spectra become more compact coarser grids possible; Production of TKE and temperature variance practically unaffected; P = w% u U +β gwθ k i z i Probably caused by buoyancy production Significantly more variance at large scales. It s harder to generate small scales 37

38 Outlook 1 Getting clarity of what s going on. Increase Ra to 10 7 and perform the same tests. Spectral separation of production and dissipation. Additional tests for channel flow and side-heated convection. Hybrid: Couple α to the mesh-size/turbulent lengthscale and add near wall-modeling. 38

39 Outlook 2: Original Leray-α model U% i Ui Ui 1 p U i + U% j = Fi + ν t xj ρ xi xj xj 2 Ui U% 2 Ui i α % = U = = 0 2 i x x x j Recover N-S: (Note U i is implicitly space-filtered as in LES) Ui Ui Ui τ ij + U% j =... ( U j U% j) = ( UU i j UiU% j) = t x x x x i j j j j τ Model residual stress: τij = 2νt S α ij with ν t = ν sgs 1+ f L L i ij 39

40 Outlook 3: Filter α damps wave no s κ >α -1 ; Decompose Expanding the convection term: U U U U = U % + u x x x i i i j j j j j j U = U% + u i i i Leray-α model neglects the second term which contains some interesting physics! Further expansion: u j Ui U% uu i i j = u j + xj xj xj 123 { back scatter fluctuations Opportunity: recursive or model + ssg or RANS 40

41 A challenge to HRL: Impinging Flows A round impinging jet, Re=20.000, H/D=2-4 Pressure field Hadziabdic and Hanjalić, 2004/5) Q-parameter 41

42 QUESTIONS? 42

43 Motivation: predictions of complex wall-bounded turbulent flows and heat transfer at very high Reynolds and Rayleigh numbers The mainstay of the contemporary industrial CFD are the RANS turbulence closures: affordable, economical, but: too much empiricism, lack of universality, difficulties in predicting complex unsteady and nonequilibrium flows,.. LES: less empirical, captures better the turbulence physics, considered as the future industrial standard, but: expensive and time consuming, especially for high Re and Ra number wall-bounded flows in complex geometries: Solution (compromise) sought in merging two strategies exploiting inherent advantages of each: HYBRID RANS-LES 43

44 A priori study Rationale: Identifying / quantifying the response of the RANS layer to LES Methodology LES provides information to RANS RANS does not provide information to LES LES is solved down to the wall Case Description Periodic channel flow Reb = DNS of Moser, Kim and Mansour (1999) Computational domain: 2πh x 2h x πh Grid: 96 x 64 x 64 with Interface location: SGS model: Smagorinsky 44

45 A-priori test results Instantaneous streamwise velocity profiles for the a-priori RANS and equivalent LES; Time history for the velocity U (y + = 30) and U τ for a-priori RANS and equivalent LES LES RANS 45

46 A-priori test results Time-averaged kinetic energy and eddy viscosity for the reference DNS, the a-priori RANS and the equivalent LES 46

47 Grids issues for LES and RANS for wall-bounded turbulent flows LES of wall-bounded flows require high resolution grid in all directions for resolving near-wall processes ( x + O(50), y + O(1), z + O(20)) For resolving viscous near-wall boundary layer: No of grid cells Re τ 1.8 as compared to Re τ 0.4 for outer layer (Chapman, 1979). For R-B conv.: /H O(Pr 2 /NuRa) 1/4 Total No of grid cells Ra! In contrast, for near-wall RANS N ln Re τ, for R-B N Ra 1/3 Hence, for high Reynolds and Raleigh numbers LES still too expensive Options for very high Re and Ra numbers: Hybrid LES/RANS RANS-based VLES 47

48 Contours of instantaneous velocity in planes parallel to the wall a: y + =20, b: y + =65, c: y + =118, d: y + =230 Dense LES Coarse LES Hybrid k-ε RANS/LES 48

49 MON1(z/D=0.5,x/L=0.5, y/l=0.5) MON2(z/D=0.01,x/L=0.5,y/L=0.5) T-RANS LES Time spectra of <U>, <V>, <W> and <T> signals at characteristic Stockholm, Sweden, monitoring July 14-15, 2005points, Ra=

50 DNS Meeting the high Ra challenges: Transient RANS Comparison of DNS, LES and T-RANS for Ra=6x10 5 and T RANS extrapolation LES (Kenjeres & Hanjalic, Phys Rev. E, 2002) Planform structures and finger-like plumes Ra=6x10 5 Capturing effects of Ra number Ra=10 9 Ra=2x10 14 T-RANS 50