A N EW A PPROACH OF Z ONAL H YBRID RANS-LES B ASED ON A T WO - EQUATION k − ε M ODEL [2] L ARS D AVIDSON ETMM9, Thessaloniki, 7-9 June 2012 Lars Davidson, www.tfd.chalmers.se/˜lada Financed by the EU project ATAAC (Advanced Turbulence Simulation for Aerodynamic Application Challenges) DLR, Airbus UK, Alenia, ANSYS, Beijing Tsinghua University, CFS Engineering, Chalmers, Dassault Aviation, EADS, Eurocopter Deutschland, FOI, Imperial College, IMFT, LFK, NLR, NTS, Numeca, ONERA, Rolls-Royce Deutschland, TU Berlin, TU Darmstadt, UniMAN
PANS L OW R EYNOLDS N UMBER M ODEL [3] � ∂ k ∂ t + ∂ ( kU j ) ∂ k = ∂ �� ν + ν t � + ( P k − ε ) ∂ x j ∂ x j ∂ x j σ ku � ∂ε ∂ t + ∂ ( ε U j ) ε 2 �� � ∂ε = ∂ ν + ν t ε + C ε 1 P k k − C ∗ ∂ x j ∂ x j ∂ x j k ε 2 σ ε u f 2 f 2 k 2 ε 2 = C ε 1 + f k k k ν t = C µ f µ ε , C ∗ ( C ε 2 f 2 − C ε 1 ) , σ ku ≡ σ k , σ ε u ≡ σ ε f ε f ε f ε C ε 1 , C ε 2 , σ k , σ ε and C µ same values as [1]. f ε = 1. f 2 and f µ read � R t − y ∗ �� 2 � � � � 2 �� f 2 = � 1 − exp 1 − 0 . 3exp − 3 . 1 6 . 5 � R t �� 2 � � 2 �� − y ∗ � 5 � � f µ = 1 − exp 1 + exp − R 3 / 4 14 200 t Baseline model: f k = 0 . 4. Range of 0 . 2 < f k < 0 . 6 is evaluated www.tfd.chalmers.se/˜lada Zonal PANS 2 / 18
PANS L OW R EYNOLDS N UMBER M ODEL [3] � ∂ k ∂ t + ∂ ( kU j ) ∂ k = ∂ �� ν + ν t � + ( P k − ε ) ∂ x j ∂ x j ∂ x j σ ku � ∂ε ∂ t + ∂ ( ε U j ) ε 2 �� � ∂ε = ∂ ν + ν t ε + C ε 1 P k k − C ∗ ∂ x j ∂ x j ∂ x j k ε 2 σ ε u f 2 f 2 k 2 ε 2 = C ε 1 + f k k k ν t = C µ f µ ε , C ∗ ( C ε 2 f 2 − C ε 1 ) , σ ku ≡ σ k , σ ε u ≡ σ ε f ε f ε f ε C ε 1 , C ε 2 , σ k , σ ε and C µ same values as [1]. f ε = 1. f 2 and f µ read � R t − y ∗ �� 2 � � � � 2 �� f 2 = � 1 − exp 1 − 0 . 3exp − 3 . 1 6 . 5 � R t �� 2 � � 2 �� − y ∗ � 5 � � f µ = 1 − exp 1 + exp − R 3 / 4 14 200 t Baseline model: f k = 0 . 4. Range of 0 . 2 < f k < 0 . 6 is evaluated www.tfd.chalmers.se/˜lada Zonal PANS 2 / 18
PANS L OW R EYNOLDS N UMBER M ODEL [3] � ∂ k ∂ t + ∂ ( kU j ) ∂ k = ∂ �� ν + ν t � + ( P k − ε ) ∂ x j ∂ x j ∂ x j σ ku � ∂ε ∂ t + ∂ ( ε U j ) ε 2 �� � ∂ε = ∂ ν + ν t ε + C ε 1 P k k − C ∗ ∂ x j ∂ x j ∂ x j k ε 2 σ ε u f 2 f 2 k 2 ε 2 = 1 . 5 + f k k k ν t = C µ f µ ε , C ∗ ( 1 . 9 − 1 . 5 ) , σ ku ≡ σ k , σ ε u ≡ σ ε f ε f ε f ε C ε 1 , C ε 2 , σ k , σ ε and C µ same values as [1]. f ε = 1. f 2 and f µ read � R t − y ∗ �� 2 � � � � 2 �� f 2 = � 1 − exp 1 − 0 . 3exp − 3 . 1 6 . 5 � R t �� 2 � � 2 �� − y ∗ � 5 � � f µ = 1 − exp 1 + exp − R 3 / 4 14 200 t Baseline model: f k = 0 . 4. Range of 0 . 2 < f k < 0 . 6 is evaluated www.tfd.chalmers.se/˜lada Zonal PANS 2 / 18
C HANNEL F LOW : Z ONAL RANS-LES LES, f k < 1 k u , int , ε u , int RANS, f k = 1 . 0 y int y wall x Interface: how to treat k and ε over the interface? They should be reduced from their RANS values to suitable LES values The usual convection and diffusion across the interface is cut off, and new “interface boundary” conditions are prescribed k u , int = f k k RANS Nothing is done for ε x max = 3 . 2 (64 cells), z max = 1 . 6 (64 cells), y dir: 80 − 128 cells CDS in entire region www.tfd.chalmers.se/˜lada Zonal PANS 3 / 18
( N x × N z ) = ( 64 × 64 ) . y + int = 500 0 30 −0.2 25 20 −0.4 uv + U + 15 −0.6 10 −0.8 5 0 −1 1 100 1000 30000 0 0.05 0.1 0.15 0.2 y y + Re τ = 4 000 Re τ = 8 000 Re τ = 16 000; Re τ = 32 000. www.tfd.chalmers.se/˜lada Zonal PANS 4 / 18
I NTERFACE LOCATION . Re τ = 8 000 . 0 30 25 −0.2 20 −0.4 uv + U + 15 −0.6 10 −0.8 5 0 −1 1 100 1000 8000 0 500 1000 1500 2000 y + y + y + = 130 y + = 500 y + = 980 www.tfd.chalmers.se/˜lada Zonal PANS 5 / 18
E FFECT OF f k . Re τ = 16 000 . y + int = 500 0 30 25 −0.2 20 −0.4 uv + U + 15 −0.6 10 −0.8 5 0 −1 1 100 1000 10000 0 0.05 0.1 0.15 0.2 y y + f k = 0 . 2 f k = 0 . 3 f k = 0 . 5 f k = 0 . 6 www.tfd.chalmers.se/˜lada Zonal PANS 6 / 18
E FFECT OF R ESOLUTION : V ELOCITY ( N x × N z ) = ( 32 × 32 ) ( N x × N z ) = ( 128 × 128 ) 30 30 25 25 20 20 U + U + 15 15 10 10 5 5 0 0 1 100 1000 30000 1 100 1000 30000 y + y + Re τ = 4 000 Re τ = 8 000 Re τ = 16 000; Re τ = 32 000. www.tfd.chalmers.se/˜lada Zonal PANS 7 / 18
E FFECT OF R ESOLUTION : R ESOLVED S HEAR S TRESS ( N x × N z ) = ( 32 × 32 ) ( N x × N z ) = ( 128 × 128 ) 0 0 −0.2 −0.2 −0.4 −0.4 uv + uv + −0.6 −0.6 −0.8 −0.8 −1 −1 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 y y Re τ = 4 000 Re τ = 8 000 Re τ = 16 000; Re τ = 32 000. www.tfd.chalmers.se/˜lada Zonal PANS 8 / 18
E FFECT OF R ESOLUTION : T URBULENT V ISCOSITY Re τ = 4000 Re τ = 8000 0.015 0.015 ν t / ( u τ δ ) 0.01 0.01 0.005 0.005 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 Re τ = 16 000 Re τ = 32 000 0.015 0.015 ν t / ( u τ δ ) 0.01 0.01 0.005 0.005 0 0 0 0.05 0.1 0.15 y 0.2 0.25 0.3 0 0.05 0.1 0.15 y 0.2 0.25 0.3 ( N x × N z ) = 64 × 64 32 × 32 128 × 128 www.tfd.chalmers.se/˜lada Zonal PANS 9 / 18
E FFECT OF R ESOLUTION : T URBULENT V ISCOSITY Re τ = 4000 Re τ = 8000 ! 0.015 0.015 y t i s o c ν t / ( u τ δ ) s 0.01 0.01 i v t n e l 0.005 0.005 u b r u t t 0 0 n 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 e Re τ = 16 000 Re τ = 32 000 d n 0.015 0.015 e p e d n ν t / ( u τ δ ) 0.01 0.01 i d i r G 0.005 0.005 0 0 0 0.05 0.1 0.15 y 0.2 0.25 0.3 0 0.05 0.1 0.15 y 0.2 0.25 0.3 ( N x × N z ) = 64 × 64 32 × 32 128 × 128 www.tfd.chalmers.se/˜lada Zonal PANS 9 / 18
SGS M ODELS B ASED ON G RID S IZE When the grid is refined, ν t gets smaller www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
SGS M ODELS B ASED ON G RID S IZE When the grid is refined, ν t gets smaller E P k , res ε sgs κ κ c www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
SGS M ODELS B ASED ON G RID S IZE When the grid is refined, ν t gets smaller E P k , res ε sgs , ∆ ε sgs , 0 . 5 ∆ κ κ c 2 κ c www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
SGS M ODELS B ASED ON G RID S IZE When the grid is refined, ν t gets smaller ε sgs , ∆ = ε sgs , 0 . 5 ∆ u � s ij � − � τ 12 , t � ∂ � ¯ s ij ¯ ε sgs = 2 � ν t ¯ ∂ y E P k , res Grid refinement ⇒ must be s ij ¯ s ij accompanied with larger ¯ s ij ¯ s ij must take place at ⇒ ¯ ε sgs , ∆ higher wavenumbers ε sgs , 0 . 5 ∆ if not ⇒ grid dependent κ κ c 2 κ c www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
w ′ ∂ ¯ P OWER D ENSITY S PECTRA OF ν 0 . 5 t ∂ z One -eq k sgs model Zonal PANS 0.02 0.03 w ′ /∂ z ) 2 � 0.025 0.015 0.02 ν t ( ∂ ¯ 0.01 0.015 0.01 0.005 � 0.005 E 0 0 0 20 40 60 80 100 0 20 40 60 80 100 κ z κ z ( N x × N z ) = 64 × 64 32 × 32 128 × 128 www.tfd.chalmers.se/˜lada Zonal PANS 11 / 18
SGS DISSIPATION VS . W AVENUMBER Energy spectra of the SGS dissipation show that the peak takes place at surprisingly low wavenumber (length scale corresponding to 10 cells or more). E ( κ ) ε sgs κ κ c www.tfd.chalmers.se/˜lada Zonal PANS 12 / 18
SGS DISSIPATION VS . W AVENUMBER Energy spectra of the SGS dissipation show that the peak takes place at surprisingly low wavenumber (length scale corresponding to 10 cells or more). ε sgs ,κ E ( κ ) ε sgs κ κ c www.tfd.chalmers.se/˜lada Zonal PANS 12 / 18
SGS DISSIPATION , Re τ = 8000 u � s ij � − � τ 12 , t � ∂ � ¯ u ′ u ′ s ij ¯ SGS dissipation in the ¯ i ¯ i / 2 eq, ε sgs = 2 � ν t ¯ ∂ y One -eq k sgs model Zonal PANS 15 15 10 10 ε sgs ε sgs 5 5 0 0 0.1 0.15 0.2 0.25 0.3 0.1 0.15 0.2 0.25 0.3 y y ( N x × N z ) = 64 × 64 32 × 32 128 × 128 www.tfd.chalmers.se/˜lada Zonal PANS 13 / 18
L OCAL E QUILIBRIUM . Re τ = 4000 , N x × N z = 64 × 64 . k equation ε equation 0.4 4E−3 4E−3 0.03 4E−4 4E−4 0.3 0.02 0.2 2E−3 2E−3 2E−4 2E−4 0.01 0.1 0 0 0 0 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0 0 0 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 y y � P k � + � C ε 1 P k /ε � + � C ε 2 ε 2 / k � + � ε � + Left vertical axes: URANS region; right vertical axes: LES region. www.tfd.chalmers.se/˜lada Zonal PANS 14 / 18
L OCAL E QUILIBRIUM IN ε E QUATION . How can both the k eq. and ε be in local equilibrium?? www.tfd.chalmers.se/˜lada Zonal PANS 15 / 18
L OCAL E QUILIBRIUM IN ε E QUATION . How can both the k eq. and ε be in local equilibrium?? If � P k � = � ε � www.tfd.chalmers.se/˜lada Zonal PANS 15 / 18
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