Hybrid LES RANS Method Based on an Explicit Algebraic Reynolds Stress Model

Transcription

1 Hybrid RANS Method Based on an Explicit Algebraic Reynolds Stress Model Benoit Jaffrézic, Michael Breuer and Antonio Delgado Institute of Fluid Mechanics, LSTM University of Nürnberg DESider 7 Corfu, June 7, 7

2 Motivation Hybrid RANS Method Objectives and requirements RANS/ models Linear eddy viscosity model Extension to anisotropy by an explicit algebraic Reynolds stress model Interface definition Numerical Method Test Cases Results Plane Channel Flow Interface Behavior Periodic Hill Flow Conclusions and Outlook Outline

3 Objective: Prediction of complex turbulent flows at high Re using within acceptable simulation times and reasonable accuracy RANS Airfoil at Re = 7 (Spalart) Grid points: ; Time steps: 5 6 Floating Point Operations: = possible in 4 decades (?) Requirements of many aerodynamic applications: I. Computation of attached turbulent thin boundary layers up to separation II. Computation of flows past separation line including Reynolds stresses Idea of RANS coupling: RANS: requirement I fulfilled, but not requirement II : requirement II fulfilled, but requirement I too expensive splitting into attached eddies (RANS) and detached eddies () Hybrid methods with RANS and regions Motivation for Hybrid RANS Methods

4 PSfrag Objectives replacements and Requirements y/h RANS zones zone -RANS Interface Interface: Non Zonal Approach Automatic switching between and RANS Gradual transition between and RANS (no explicit boundaries) No grid influence and no interface predefinition RANS/ Models: Primary Model Requirements for Non Zonal Approach Similar models for both modes to facilitate the blending Cheap models avoiding large computational effort RANS model especially designed for near wall region x/h 4 Hybrid RANS Approach

5 One equation model for both and RANS modes RANS Option : LEVM (Linear Eddy Viscosity Model = L) = Near wall one eq. RANS model by Rodi et al. (99) based on transport eq. for k mod RANS Option : NLEVM (Non Linear Eddy Viscosity Model = nl) = Extension of the LEVM by an Explicit Algebraic Reynolds Stress Model by Wallin and Johansson () = One eq. SGS model based on transport eq. for k sgs by Schumann (975) 5 RANS/ Models

6 Unique one equation model for RANS and k mod t k mod + U j = [ (ν + ν t ) k ] mod x j x j σ k x j (u i u j ) mod U i x j ɛ RANS One eq. model by Rodi et al. (99) (k mod k RANS ) with (v ) / as velocity scale, Durbin (99) Turbulent eddy viscosity: ν t = (v ) / l µ,v v /k RANS = f(y ) for y 6 Dissipation: ɛ = (v ) / k RANS /l ɛ,v Length scale: l µ,v and l ɛ,v analytically determined (l y) SGS model by Schumann (975) (k mod k SGS ) Subgrid scale eddy viscosity: ν t = C µ k / SGS Dissipation: ɛ = C d k / SGS / Length scale: = ( x y z) / defined by filter width 6 Linear Eddy Viscosity Model (LEVM)

7 Motivation: Take Reynolds stress anisotropy into account: a ij = u i u j k δ ij LEVM: u i u j mod = δ ij k mod ν t S ij LEVM + EARSM: u i u jmod = δ ij k mod ν t S ij + a (ex) ij k mod }{{} a ij k mod EARSM of Wallin & Johansson (): (Full RSM weak equilibrium assumption + near wall treatment by Durbin) a (ex) ij = function(s, Ω, S n Ω m, f damp,...) with: τ RANS S = S ij /τ = τ Ω = Ω ij /τ = τ = max ( k ɛ, C τ ( Ui + U ) j x j x ( i Ui U ) j x j x ) i ν ɛ normalized strain tensor normalized rotation tensor time scale [: τ = k ] 7 NLEVM: Extension of LEVM by EARSM ()

8 LEVM by Rodi et al. + EARSM of Wallin & Johansson = RANS k mod t k mod + U j = [ (ν + ν t ) k ] mod x j x j σ k x j (u i u j ) U i mod x }{{} j P k ɛ Production term in NLEVM ( Use extra anisotropy) ( ) P k = ν t S ij + a (ex) ij k Ui U mod x j (in LEVM: P k = ν t S i ij x j ) Turbulent eddy viscosity based on EARSM: Extra anisotropy term from EARSM: Dissipation from LEVM: ɛ = (v ) / k RANS /l ɛ,v ν t = C eff µ k τ RANS a (ex) ij = f(s, Ω, S n Ω m, f damp,...) 8 NLEVM: Extension of LEVM by EARSM ()

9 Objectives Non Zonal Approach Use of a physical (turbulent) parameter in the interface definition Interface dynamically determined PSfrag replacements Applicable for separated flow y/h RANS zones zone -RANS Interface x/h Use of the instantaneous turbulent kinetic energy k (= k mod ) RANS switching criterion y = k / y/ν k mod influence: low k mod = thick RANS region; high k mod = thin RANS region 9 RANS Interface

10 First interface criterion y C switch,y = RANS mode y > C switch,y = mode with y = k / mod y/ν and C switch,y = 6 (Validity region of RANS ) RANS Interface + k mod contour y/h RANS islands x/h PSfrag replacements dynamically computed interface RANS islands in region k ( mod closely ) follows k crit defined as k crit =.5 Cswitch ν representing the RANS interface y RANS Switching Criteria () y/h.5.5 k crit k mod k

11 Second interface criterion Combine the st criterion with a sharp interface y C switch,y = RANS mode y > C switch,y = mode with y = k / mod y/ν and C switch,y = 6 + Sharp Interface Treatment RANS Interface + k mod contour RANS CONVERSION WALL WALL y/h RANS RANS NO CONVERSION RANS RANS WALL WALL x/h RANS islands have no real influence on the statistical results = Computations performed with the st criterion (cheapest formulation) RANS Switching Criteria ()

12 Mod. RANS loge(k) Res. Mod. URANS Mod.: Modeled scales Res.: Resolved scales Res. Mod. k = wave number ag replacements Energy spectrum k c log k RANS mode operates as an Unsteady PSfrag RANS replacements (UR- ANS) loge(k) = Presence of resolved scales in RANS mode k c Total contributions (k tot, (u i u j ) log k tot,...) are the sum of the respective modeled and resolved fields = k tot = k mod + k res = (u i u j ) tot = (u i u j ) mod + (u i u j ) res RANS Method k tot k res k mod DNS. y +

13 OCC (Large Eddy Simulation On Curvilinear Coordinates) Navier Stokes solver (incompressible fluid) D finite volume approach Curvilinear body fitted coordinate system Non staggered (cell centered) grid arrangement Block structured grids Spatial discretization Viscous fluxes: central differences O( x ) Convective fluxes: five different schemes, central diff. O( x ), CDS Temporal discretization Predictor step (moment. eqns.): low storage Runge Kutta scheme, O( t ) Corrector step (pressure correction equation): SIP solver (ILU) Pressure velocity coupling: Momentum interpolation of Rhie & Chow High performance computing techniques Highly vectorized Parallelized by domain decomposition and explicit message passing Vector parallel computers and SMP clusters (Hitachi SR8 F, SGI,...) Numerical Method: OCC

14 Grid A: CVs x+ = O(), z + = O(5) y+stpt =.68 Grid B: CVs x+ = O(6), z + = O() y+stpt =.46 grid resolutions: x/h =. 4 Test Cases 4 x 6 x/h = 6. Lx y z Ly = d x Lz = pi d d y Test Case : Plane Channel Flow Test Case : Periodic Hill Flow Re =,595 Lx = pi d Lz Ly h Reference data provided by Moser et al. (999); DNS at Reτ = 59 Zoom Distribution of k 8

15 Grids Reference Solution: WR Highly Resolved (no wall model); Dynamic Smagorinsky model.4 6 control volumes wall normal resolution ( st CV height): y crest /h =. Present Test Grid designed according to DES requirements control volumes wall normal resolution ( st CV height): y crest /h = Periodic Hill Flow at Re b =, 595

16 Result of the Near Wall LEVM Modification: Lm L Lm DNS k modeled + k resolved = k total L Lm DNS L Lm DNS lacements k mod PSfrag replacements k res PSfrag replacements k tot k res k tot 6. y + k mod k tot. y + k mod k res. Version Lm: modified near wall model using adjusted model const. for high Re Resolved field almost unchanged Modeled field adjusted Better agreement between the hybrid RANS technique and the DNS data Channel Flow Test Case / Adjusted RANS Model y +

17 ements u rms tot v rms tot u v tot ements U + u rms tot u v tot U + v v rms tot L nl DNS Grid A y L nl DNS PSfrag replacements Grid B y + u u rms tot v v rms tot u v tot PSfrag replacements U + u u rms tot v v rms tot U + u v tot L nl DNS Grid B y + L nl DNS Grid B y + Channel Flow at Re τ = 59 L: Linear hybrid method with st interface criterion nl: Non Linear hybrid method with st interface criterion Best predictions by non linear hybrid method

18 RANS Interface + k mod contour.5 RANS islands y/h.5 4 Interface dynamically determined = evolves at each time step as well as spatially x/h Criterion based on inst. k mod = weak RANS island influence on a statistical viewpoint Inst. separation point, k mod = thin RANS region, priority to mode Downstream to separation k mod = thicker RANS region, priority to RANS mode y recognizes the zones of low and high turbulence intensity through k mod, e.g., localization of the inst. separation point Method adapts itself to the turbulent flow features = priority to RANS or 8 Interface Behavior: First criterion

19 Wall resolved (WR ) x sep /h =.9, x reatt /h = Hybrid Version L x sep /h =.54, x reatt /h = 4.75 y y DES Spalart Allmaras model x sep /h =.7, x reatt /h = 5.97 x Hybrid Version nl x sep /h =., x reatt /h = 4.7 x y y x x 9 Results: Streamlines for Hill Flow Test Case

20 ts y/h.5.5 L nl DES WR U/U b Mean Velocity Profiles at x/h =.5 to 8

21 ts S.5 y/h.5.5 L nl WR u u tot/u b u u tot Profiles at x/h =.5 to 8

22 v v tot/u b PSfrag replacements y/h DES τw x/h PSfrag replacements x/h =.5 τ w L nl DES WR x/h y/h L nl WR....4 v v tot/u b Further Examples of Improvements by NLEVM

23 Hybrid RANS Technique: Non Zonal RANS approach Two unique one equation models for the velocity scale LEVM v formulation for the near wall RANS region LEVM + EARSM anisotropy of Reynolds stresses Encouraging results for plane channel and periodic hill flow Use of the EARSM enhances the results More detailed tests of the interface region required Investigations / adjustments of EARSM for more complex flows Future test cases (challenging for RANS) required to evaluate the potential of this hybrid RANS technique Ahmed body, D hill flow, stalled airfoil flow Conclusions and Outlook