Optimizing calculation costs of tubulent flows with RANS/LES methods Investigation in separated flows C. Friess, R. Manceau Dpt. Fluid Flow, Heat Transfer, Combustion Institute PPrime, CNRS University of Poitiers ENSMA, Poitiers, France Present address : Lab. Mechanics, Modelling and Clean Processes CNRS Aix-Marseille University, Marseille, France Spring Festival ERCOFTAC, Toulon, 2013/4/9
INTRODUCTION Motivation In recent years: Numerous continuous hybrid RANS-LES methods Friess, Manceau HYBRID RANS/LES METHODS 1
INTRODUCTION Motivation In recent years: Numerous continuous hybrid RANS-LES methods Common feature: ambiguous definition of the resolved variables RANS regions: Reynolds (statistical) averaging LES regions: spatial filter Friess, Manceau HYBRID RANS/LES METHODS 1
INTRODUCTION Motivation In recent years: Numerous continuous hybrid RANS-LES methods Common feature: ambiguous definition of the resolved variables RANS regions: Reynolds (statistical) averaging LES regions: spatial filter Matching in homogeneous turbulence: Spatial filter spatial average when Statistical average = spatial average Friess, Manceau HYBRID RANS/LES METHODS 1
INTRODUCTION Motivation In recent years: Numerous continuous hybrid RANS-LES methods Common feature: ambiguous definition of the resolved variables RANS regions: Reynolds (statistical) averaging LES regions: spatial filter Matching in homogeneous turbulence: Spatial filter spatial average when Statistical average = spatial average Usual configurations: inhomogeneous fundamental inconsistency Friess, Manceau HYBRID RANS/LES METHODS 1
INTRODUCTION Motivation In recent years: Numerous continuous hybrid RANS-LES methods Common feature: ambiguous definition of the resolved variables RANS regions: Reynolds (statistical) averaging LES regions: spatial filter Matching in homogeneous turbulence: Spatial filter spatial average when Statistical average = spatial average Usual configurations: inhomogeneous fundamental inconsistency For inhomogeneous, stationary flows (statistical average independent of t): consistency issue addressed in the frame of temporal filtering Hybrid Temporal LES/RANS (Fadai-Ghotbi, Friess, Manceau, Gatski, and Borée, 2010) Friess, Manceau HYBRID RANS/LES METHODS 1
FORMALISM Temporal filtering approach within the LES formalism: TLES (Pruett, 2000; Pruett, Gatski, Grosch, and Thacker, 2003) ũ(x,t) = 0 G T (τ) u(x,t+τ) dτ ũ i t + ũ iũ j x j = 1 ρ P + 1 2 ũ i x i Re x 2 j τ ijsfs x j 10 0 10-2 E T (ω) Ĝ T Ĝ T E T (ω) 10-4 10-6 10-8 10-10 10-1 10 0 10 1 10 2 10 3 10 4 ω Friess, Manceau HYBRID RANS/LES METHODS 2
FORMALISM Temporal filtering approach within the LES formalism: TLES (Pruett, 2000; Pruett, Gatski, Grosch, and Thacker, 2003) ũ(x,t) = 0 G T (τ) u(x,t+τ) dτ ũ i t + ũ iũ j x j = 1 ρ P + 1 2 ũ i x i Re x 2 j τ ijsfs x j Subfilter stresses τ ijsfs : Dτ ijsfs Dt = P ij +φ ij +D T ij +D ν ij ε ij RANS and TLES equations are form-invariant provides a consistent basis for RANS-TLES hybridization 10 0 10-2 E T (ω) Ĝ T Ĝ T E T (ω) 10-4 10-6 10-8 10-10 10-1 10 0 10 1 10 2 10 3 10 4 ω Friess, Manceau HYBRID RANS/LES METHODS 2
FORMALISM Temporal filtering approach within the LES formalism: TLES (Pruett, 2000; Pruett, Gatski, Grosch, and Thacker, 2003) ũ(x,t) = 0 G T (τ) u(x,t+τ) dτ ũ i t + ũ iũ j x j = 1 ρ P + 1 2 ũ i x i Re x 2 j τ ijsfs x j Subfilter stresses τ ijsfs : Dτ ijsfs Dt = P ij +φ ij +D T ij +D ν ij ε ij RANS and TLES equations are form-invariant provides a consistent basis for RANS-TLES hybridization 10 0 10-2 E T (ω) Ĝ T Ĝ T E T (ω) 10-4 10-6 10-8 ω c 0 10-10 10-1 10 0 10 1 10 2 10 3 10 4 ω Friess, Manceau HYBRID RANS/LES METHODS 2
FORMALISM Temporal filtering approach within the LES formalism: TLES (Pruett, 2000; Pruett, Gatski, Grosch, and Thacker, 2003) ũ(x,t) = 0 G T (τ) u(x,t+τ) dτ ũ i t + ũ iũ j x j = 1 ρ P + 1 2 ũ i x i Re x 2 j τ ijsfs x j Subfilter stresses τ ijsfs : Dτ ijsfs Dt = P ij +φ ij +D T ij +D ν ij ε ij RANS and TLES equations are form-invariant provides a consistent basis for RANS-TLES hybridization 10 0 10-2 E T (ω) Ĝ T Ĝ T E T (ω) ω c 0: SFS stresses tend to Reynolds stresses 10-4 RANS equations recovered 10-6 10-8 ω c 0 10-10 10-1 10 0 10 1 10 2 10 3 10 4 ω Friess, Manceau HYBRID RANS/LES METHODS 2
FORMALISM Temporal filtering approach within the LES formalism: TLES (Pruett, 2000; Pruett, Gatski, Grosch, and Thacker, 2003) ũ(x,t) = 0 G T (τ) u(x,t+τ) dτ ũ i t + ũ iũ j x j = 1 ρ P + 1 2 ũ i x i Re x 2 j τ ijsfs x j Subfilter stresses τ ijsfs : Dτ ijsfs Dt = P ij +φ ij +D T ij +D ν ij ε ij RANS and TLES equations are form-invariant provides a consistent basis for RANS-TLES hybridization 10 0 10-2 E T (ω) Ĝ T Ĝ T E T (ω) ω c 0: SFS stresses tend to Reynolds stresses 10-4 RANS equations recovered 10-6 10-8 ω c 0 ω c 10-10 10-1 10 0 10 1 10 2 10 3 10 4 ω Friess, Manceau HYBRID RANS/LES METHODS 2
FORMALISM Temporal filtering approach within the LES formalism: TLES (Pruett, 2000; Pruett, Gatski, Grosch, and Thacker, 2003) ũ(x,t) = 0 G T (τ) u(x,t+τ) dτ ũ i t + ũ iũ j x j = 1 ρ P + 1 2 ũ i x i Re x 2 j τ ijsfs x j Subfilter stresses τ ijsfs : Dτ ijsfs Dt = P ij +φ ij +D T ij +D ν ij ε ij RANS and TLES equations are form-invariant provides a consistent basis for RANS-TLES hybridization 10 0 10-2 E T (ω) Ĝ T Ĝ T E T (ω) ω c 0: SFS stresses tend to Reynolds stresses 10-4 RANS equations recovered 10-6 10-8 ω c 0 ω c ω c : SFS stresses vanish 10-10 10-1 10 0 10 1 10 2 10 3 10 4 ω Navier-Stokes equations recovered Friess, Manceau HYBRID RANS/LES METHODS 2
CONTROL OF THE RANS LES TRANSITION T-PITM method (Fadai-Ghotbi et al., 2010) Temporal version of the PITM (Chaouat and Schiestel, 2005) Friess, Manceau HYBRID RANS/LES METHODS 3
CONTROL OF THE RANS LES TRANSITION T-PITM method (Fadai-Ghotbi et al., 2010) Temporal version of the PITM (Chaouat and Schiestel, 2005) Three spectral zones [0,ω c ], [ω c,ω d ], [ω d, ] Friess, Manceau HYBRID RANS/LES METHODS 3
CONTROL OF THE RANS LES TRANSITION T-PITM method (Fadai-Ghotbi et al., 2010) Temporal version of the PITM (Chaouat and Schiestel, 2005) Three spectral zones [0,ω c ], [ω c,ω d ], [ω d, ] Partial integration over each zone of the transport equation for E T (x,ω) Dk m Dt Dε Dt = P m ε m +D m [ ] ε = C ε1 P m C ε1 +r(c ε2 C ε1 ) k m } {{ } Cε 2 (ω c ) ε 2 k m +D ε Friess, Manceau HYBRID RANS/LES METHODS 3
CONTROL OF THE RANS LES TRANSITION T-PITM method (Fadai-Ghotbi et al., 2010) Temporal version of the PITM (Chaouat and Schiestel, 2005) Three spectral zones [0,ω c ], [ω c,ω d ], [ω d, ] Partial integration over each zone of the transport equation for E T (x,ω) Dk m Dt Dε Dt = P m ε m +D m [ ] ε = C ε1 P m C ε1 +r(c ε2 C ε1 ) k m } {{ } Cε 2 (ω c ) ε 2 k m +D ε Fundamental parameter: r = k m k m +k r k m = modelled k r = resolved = measure of the energy partition among resolved and subfilter scales Friess, Manceau HYBRID RANS/LES METHODS 3
CONTROL OF THE RANS LES TRANSITION T-PITM method (Fadai-Ghotbi et al., 2010) Temporal version of the PITM (Chaouat and Schiestel, 2005) Three spectral zones [0,ω c ], [ω c,ω d ], [ω d, ] Partial integration over each zone of the transport equation for E T (x,ω) Dk m Dt Dε Dt = P m ε m +D m [ ] ε = C ε1 P m C ε1 +r(c ε2 C ε1 ) k m } {{ } Cε 2 (ω c ) ε 2 k m +D ε Fundamental parameter: r = k m k m +k r k m = modelled k r = resolved = measure of the energy partition among resolved and subfilter scales r = 1 RANS Friess, Manceau HYBRID RANS/LES METHODS 3
CONTROL OF THE RANS LES TRANSITION T-PITM method (Fadai-Ghotbi et al., 2010) Temporal version of the PITM (Chaouat and Schiestel, 2005) Three spectral zones [0,ω c ], [ω c,ω d ], [ω d, ] Partial integration over each zone of the transport equation for E T (x,ω) Dk m Dt Dε Dt = P m ε m +D m [ ] ε = C ε1 P m C ε1 +r(c ε2 C ε1 ) k m } {{ } Cε 2 (ω c ) ε 2 k m +D ε Fundamental parameter: r = k m k m +k r k m = modelled k r = resolved = measure of the energy partition among resolved and subfilter scales r = 1 RANS r = 0 DNS Friess, Manceau HYBRID RANS/LES METHODS 3
CONTROL OF THE RANS LES TRANSITION T-PITM method (Fadai-Ghotbi et al., 2010) Temporal version of the PITM (Chaouat and Schiestel, 2005) Three spectral zones [0,ω c ], [ω c,ω d ], [ω d, ] Partial integration over each zone of the transport equation for E T (x,ω) Dk m Dt Dε Dt = P m ε m +D m [ ] ε = C ε1 P m C ε1 +r(c ε2 C ε1 ) k m } {{ } Cε 2 (ω c ) ε 2 k m +D ε Fundamental parameter: r = k m k m +k r k m = modelled k r = resolved = measure of the energy partition among resolved and subfilter scales r = 1 RANS In between TLES r = 0 DNS Friess, Manceau HYBRID RANS/LES METHODS 3
DERIVATION OF AN EQUIVALENT DES T-PITM: difficulties to reach the expected energy partition Modify the form of the system of equations Friess, Manceau HYBRID RANS/LES METHODS 4
DERIVATION OF AN EQUIVALENT DES T-PITM: difficulties to reach the expected energy partition Modify the form of the system of equations T-PITM Equivalent DES dk m dt = P m ε+d m dk m dt dε dt = C ε ε1 P m C k ε2(ω c ) ε2 D ε m k m = P m ψ(ω c )ε+d m dε dt = C ε ε 2 ε1 P m C ε2 D ε k m k m C ε2(ω c ) = C ε1 +r(ω c )(C ε2 C ε1 ) ψ(ω c ) = k3/2 m /ε L(ω c ) Friess, Manceau HYBRID RANS/LES METHODS 4
DERIVATION OF AN EQUIVALENT DES T-PITM: difficulties to reach the expected energy partition Modify the form of the system of equations T-PITM Equivalent DES dk m dt = P m ε+d m dk m dt = P m ψ(ω c )ε+d m dε dt = C ε ε1 P m C k ε2(ω c ) ε2 D ε m k m dε dt = C ε ε 2 ε1 P m C ε2 D ε k m k m C ε2(ω c ) = C ε1 +r(ω c )(C ε2 C ε1 ) ψ(ω c ) = k3/2 m /ε L(ω c ) Equivalence (Manceau et al., 2010): ψ function that gives the same energy partition as the C ε2 function? Friess, Manceau HYBRID RANS/LES METHODS 4
DERIVATION OF AN EQUIVALENT DES Procedure T-PITM H-TLES dk m dt dε dt = C ε1 = P m ε+d m dk m dt ε k m P m C ε2 ε 2 k m D ε = P m ψε+d m dε dt = C ε ε 2 ε1 P m C ε2 D ε k m k m Friess, Manceau HYBRID RANS/LES METHODS 5
DERIVATION OF AN EQUIVALENT DES Procedure T-PITM H-TLES dk m dt dε dt = C ε1 = P m ε+d m dk m dt ε k m P m C ε2 ε 2 k m D ε = P m ψε+d m dε dt = C ε ε 2 ε1 P m C ε2 D ε k m k m Starting from the RANS system: C ε2 = C ε2 and ψ = 1 same equations same partition (r = 1) Friess, Manceau HYBRID RANS/LES METHODS 5
DERIVATION OF AN EQUIVALENT DES Procedure T-PITM H-TLES dk m dt dε dt = C ε1 = P m ε+d m dk m dt ε k m P m C ε2 ε 2 k m D ε = P m ψε+d m dε dt = C ε ε 2 ε1 P m C ε2 D ε k m k m Starting from the RANS system: C ε2 = C ε2 and ψ = 1 same equations same partition (r = 1) Introducing infinitesimal variations δc ε2 and δψ (perturbation analysis) Friess, Manceau HYBRID RANS/LES METHODS 5
DERIVATION OF AN EQUIVALENT DES Procedure T-PITM H-TLES dk m dt dε dt = C ε1 = P m ε+d m dk m dt ε k m P m C ε2 ε 2 k m D ε = P m ψε+d m dε dt = C ε ε 2 ε1 P m C ε2 D ε k m k m Starting from the RANS system: C ε2 = C ε2 and ψ = 1 same equations same partition (r = 1) Introducing infinitesimal variations δc ε2 and δψ (perturbation analysis) Same variation of the partition of energy if 1 C ε2 C ε1 ψ δψ = 1 ( C C ε2 ε1 C ε1 1 )δc ε2 Friess, Manceau HYBRID RANS/LES METHODS 5
DERIVATION OF AN EQUIVALENT DES Procedure dk m dt dε dt = C ε1 T-PITM = P m ε+d m dk m dt ε k m P m C ε2 ε 2 k m D ε H-TLES = P m ψε+d m dε dt = C ε ε 2 ε1 P m C ε2 D ε k m k m Starting from the RANS system: C ε2 = C ε2 and ψ = 1 same equations same partition (r = 1) Introducing infinitesimal variations δc ε2 and δψ (perturbation analysis) 1 Same variation of the partition of energy if C ε2 C ε1 ψ δψ = 1 ( C C ε2 ε1 C ε1 1 ( ) Cε2 ( ) By integration: Equivalence ψ = 1 + 1 1 r C ε1 C ε2 C ε1 )δc ε2 Friess, Manceau HYBRID RANS/LES METHODS 5
DERIVATION OF AN EQUIVALENT DES Link with the cutoff frequency ψ = k3/2 m /ε L L = 1+ r 3/2 ( )L C ε2 C ε1 1 )(1 r C ε1 int where L int = k3/2 C ε2 ε Friess, Manceau HYBRID RANS/LES METHODS 6
DERIVATION OF AN EQUIVALENT DES Link with the cutoff frequency ψ = k3/2 m /ε L L = 1+ r 3/2 ( )L C ε2 C ε1 1 )(1 r C ε1 int where L int = k3/2 C ε2 ε Evaluation of r as a function of ω c r = k m k = 1 k ω c E T (ω)dω E T (ω) = Eulerian frequency spectrum Friess, Manceau HYBRID RANS/LES METHODS 6
DERIVATION OF AN EQUIVALENT DES Link with the cutoff frequency ψ = k3/2 m /ε L L = 1+ r 3/2 ( )L C ε2 C ε1 1 )(1 r C ε1 int where L int = k3/2 C ε2 ε Evaluation of r as a function of ω c r = k m k = 1 k E T (ω)dω ω c r = 1 β E T (ω) = Eulerian frequency spectrum ( ) 2/3 ( ) 2/3 Us k ω c k ε U s = U 2 +γk is the sweeping velocity (Tennekes, 1975) Friess, Manceau HYBRID RANS/LES METHODS 6
DERIVATION OF AN EQUIVALENT DES Link with the cutoff frequency ψ = k3/2 m /ε L L = 1+ r 3/2 ( )L C ε2 C ε1 1 )(1 r C ε1 int where L int = k3/2 C ε2 ε Evaluation of r as a function of ω c r = k m k = 1 k E T (ω)dω ω c r = 1 β E T (ω) = Eulerian frequency spectrum ( ) 2/3 ( ) 2/3 Us k ω c k ε U s = U 2 +γk is the sweeping velocity (Tennekes, 1975) Cutoff frequency = maximum observable frequency ω c = min ( π dt ; U ) sπ Friess, Manceau HYBRID RANS/LES METHODS 6
DERIVATION OF AN EQUIVALENT DES Subfilter stress model Friess, Manceau HYBRID RANS/LES METHODS 7
DERIVATION OF AN EQUIVALENT DES Subfilter stress model EB RSM (Manceau and Hanjalić, 2002) : a single scalar α accounting for wall effects α+l 2 2 α = 1 φ ij ε ij = α 3 (φ ij ε ij) w +(1 α 3 )(φ ij ε ij) h Friess, Manceau HYBRID RANS/LES METHODS 7
DERIVATION OF AN EQUIVALENT DES Subfilter stress model EB RSM (Manceau and Hanjalić, 2002) : a single scalar α accounting for wall effects α+l 2 2 α = 1 φ ij ε ij = α 3 (φ ij ε ij) w +(1 α 3 )(φ ij ε ij) h Modified EB RSM model Friess, Manceau HYBRID RANS/LES METHODS 7
DERIVATION OF AN EQUIVALENT DES Subfilter stress model EB RSM (Manceau and Hanjalić, 2002) : a single scalar α accounting for wall effects α+l 2 2 α = 1 φ ij ε ij = α 3 (φ ij ε ij) w +(1 α 3 )(φ ij ε ij) h Modified EB RSM model H-TLES : dt dε sfs dt dτ ijsfs = P ijsfs +φ ijsfs +D T ijsfs +D ν ijsfs k3/2 sfs ε ijsfs Lε sfs = standard Friess, Manceau HYBRID RANS/LES METHODS 7
DERIVATION OF AN EQUIVALENT DES Subfilter stress model EB RSM (Manceau and Hanjalić, 2002) : a single scalar α accounting for wall effects α+l 2 2 α = 1 φ ij ε ij = α 3 (φ ij ε ij) w +(1 α 3 )(φ ij ε ij) h Modified EB RSM model H-TLES : dt dε sfs dt dτ ijsfs = P ijsfs +φ ijsfs +D T ijsfs +D ν ijsfs k3/2 sfs ε ijsfs Lε sfs = standard T-PITM : dτ ijsfs dt dε sfs dt = standard P sfs ε sfs = C ε1 C ε 2 sfs ε2 +Diff. k SFS k SFS Friess, Manceau HYBRID RANS/LES METHODS 7
FLOW OVER A PERIODIC HILL Configuration and numerics Periodically constricted channel Friess, Manceau HYBRID RANS/LES METHODS 8
FLOW OVER A PERIODIC HILL Configuration and numerics Periodically constricted channel Re b = U b h/ν = 10595 2 meshes Friess, Manceau HYBRID RANS/LES METHODS 8
FLOW OVER A PERIODIC HILL Configuration and numerics Periodically constricted channel Re b = U b h/ν = 10595 2 meshes Comparison with a 2D RANS & ref LES from (Breuer et al., 2009) Friess, Manceau HYBRID RANS/LES METHODS 8
FLOW OVER A PERIODIC HILL Configuration and numerics Periodically constricted channel Re b = U b h/ν = 10595 2 meshes Comparison with a 2D RANS & ref LES from (Breuer et al., 2009) Numerics Code Saturne: open source finite volume solver developed by EDF CDS convection scheme for resolved momentum + UDS for sfsquantities Friess, Manceau HYBRID RANS/LES METHODS 8
FLOW OVER A PERIODIC HILL Effects of grid refining T-PITM results for a coarse and a fine grid, vs 2D RANS and LES 4 3 LES fine TPITM coarse TPITM 2D RANS 4 3 LES Breuer fine TPITM coarse TPITM 2D RANS y/h 2 y/h 2 1 1 0 0 2 4 6 8 x/h + 2 U/U b 0,03 0,02 0 0 2 4 6 8 LES fine TPITM coarse TPITM 2D RANS x/h + 12 k/u 2 b Cf 0,01 0 0 2 4 6 8 Streamwise velocity, total kinetic energy & friction coefficient x/h Friess, Manceau HYBRID RANS/LES METHODS 9
FLOW OVER A PERIODIC HILL Effects of grid refining Separation / reattachment Cas Maillage N x N y Nz(x/h) sep (x/h) reatt RANS 160 100 1 0.28 5.28 T-PITM 80 100 30 0.23 4.93 T-PITM 160 100 60 0.23 4.56 LES (Breuer et al.) 281 220 200 0.01-0.17 4.93 LES (Fröhlich et al.) 196 128 186 0.22 4.72 Friess, Manceau HYBRID RANS/LES METHODS 10
FLOW OVER A PERIODIC HILL Isocontours of zero-q criterion : coarse & fine grid refining the grid allows resolving more structures Friess, Manceau HYBRID RANS/LES METHODS 11
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Friess, Manceau HYBRID RANS/LES METHODS 12
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Equivalent DES Friess, Manceau HYBRID RANS/LES METHODS 12
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Equivalent DES as for DES, based on a length-scale modification in the subfilter stress equations k3/2 sfs L(r) vs. k 3/2 sfs C DES DES temporal filtering Friess, Manceau HYBRID RANS/LES METHODS 12
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Equivalent DES as for DES, based on a length-scale modification in the subfilter stress equations k3/2 sfs L(r) vs. k 3/2 sfs C DES DES temporal filtering equivalence with T-PITM demonstrated analytically for equilibrium flows (Manceau et al., 2010) Friess, Manceau HYBRID RANS/LES METHODS 12
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Equivalent DES as for DES, based on a length-scale modification in the subfilter stress equations k3/2 sfs L(r) vs. k 3/2 sfs C DES DES temporal filtering equivalence with T-PITM demonstrated analytically for equilibrium flows (Manceau et al., 2010) more convenient than T-PITM Friess, Manceau HYBRID RANS/LES METHODS 12
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Equivalent DES as for DES, based on a length-scale modification in the subfilter stress equations k3/2 sfs L(r) vs. k 3/2 sfs C DES DES temporal filtering equivalence with T-PITM demonstrated analytically for equilibrium flows (Manceau et al., 2010) more convenient than T-PITM main parameter : subfilter to total energy ratio r, can be related to cutoff frequency ω c Friess, Manceau HYBRID RANS/LES METHODS 12
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Equivalent DES as for DES, based on a length-scale modification in the subfilter stress equations k3/2 sfs L(r) vs. k 3/2 sfs C DES DES temporal filtering equivalence with T-PITM demonstrated analytically for equilibrium flows (Manceau et al., 2010) more convenient than T-PITM main parameter : subfilter to total energy ratio r, can be related to cutoff frequency ω c Application to massively separated flows Friess, Manceau HYBRID RANS/LES METHODS 12
CONCLUSIONS Temporal filtering : a consistent formalism for hybrid RANS/(T)LES modelling Equivalent DES as for DES, based on a length-scale modification in the subfilter stress equations k3/2 sfs L(r) vs. k 3/2 sfs C DES DES temporal filtering equivalence with T-PITM demonstrated analytically for equilibrium flows (Manceau et al., 2010) more convenient than T-PITM main parameter : subfilter to total energy ratio r, can be related to cutoff frequency ω c Application to massively separated flows Satisfactory results : RANS << hybrid LES Friess, Manceau HYBRID RANS/LES METHODS 12
REFERENCES Bentaleb, Y., Manceau, R., 2011. A hybrid Temporal LES/RANS formulation based on a two-equation subfilter model. In: Proc. 7th Int. Symp. Turb. Shear Flow Phenomena, Ottawa, Canada. Breuer, M., Peller, N., Rapp, C., Manhart, M., 2009. Flow over periodic hills - Numerical and experimental study in a wide range of Reynolds numbers. Comput. Fluids 38 (2), 433 457. Chaouat, B., Schiestel, R., 2005. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Phys. Fluids 17 (065106), 1 19. Fadai-Ghotbi, A., Friess, C., Manceau, R., Gatski, T., Borée, J., 2010. Temporal filtering: a consistent formalism for seamless hybrid RANS-LES modeling in inhomogeneous turbulence. Int. J. Heat Fluid Fl. 31 (3). Friess, Manceau HYBRID RANS/LES METHODS 13
REFERENCES Manceau, R., Friess, C., Gatski, T., 2010. Of the interpretation of DES as a Hybrid RANS/Temporal LES method. In: Proc. 8th ERCOFTAC Int. Symp. on Eng. Turb. Modelling and Measurements, Marseille, France. Manceau, R., Hanjalić, K., 2002. Elliptic Blending Model: A New Near-Wall Reynolds-Stress Turbulence Closure. Phys. Fluids 14 (2), 744 754. Pruett, C. D., 2000. Eulerian time-domain filtering for spatial large-eddy simulation. AIAA journal 38 (9), 1634 1642. Pruett, C. D., Gatski, T. B., Grosch, C. E., Thacker, W. D., 2003. The temporally filtered Navier-Stokes equations: Properties of the residual stress. Phys. Fluids 15 (8), 2127 2140. Tennekes, H., 1975. Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech. 67, 561 567. Friess, Manceau HYBRID RANS/LES METHODS 14
Temporal filtering and cutoff frequency Contours of U S dt/ : coarse & fine mesh convective time scale governs temporal filtering Friess, Manceau HYBRID RANS/LES METHODS 15