E MBEDDED LES U SING PANS [2] L ARS D AVIDSON 1 AND S HIA -H UI P ENG - PowerPoint PPT Presentation

E MBEDDED LES U SING PANS [2] L ARS D AVIDSON 1 AND S HIA -H UI P ENG 1 , 2 1 Department of Applied Mechanics Chalmers University of Technology, SE-412 96 Gothenburg, SWEDEN 2 FOI, Swedish Defence Research Agency, SE-164 90, Stockholm, SWEDEN

PANS L OW R EYNOLDS N UMBER M ODEL [3] � ∂ k u ∂ t + ∂ ( k u U j ) ∂ k u = ∂ �� ν + ν u � + ( P u − ε u ) ∂ x j ∂ x j ∂ x j σ ku � ∂ε u ∂ t + ∂ ( ε u U j ) ε 2 �� � ∂ε u = ∂ ν + ν u ε u u + C ε 1 P u − C ∗ ∂ x j ∂ x j ∂ x j k u k u ε 2 σ ε u f 2 f 2 k 2 ε 2 = C ε 1 + f k u k k ν u = C µ f µ , C ∗ ( C ε 2 f 2 − C ε 1 ) , σ ku ≡ σ k , σ ε u ≡ σ ε f ε f ε f ε ε u 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 − y ∗ �� 2 � � 2 �� � 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 Davidson& Peng AIAA, Hawaii, 27-30 June 2011 2 / 30

C HANNEL FLOW : D OMAIN Interface LES, f k < 1 RANS, f k = 1 . 0 d y x δ 2 . 2 δ Interface: Synthetic turbulent fluctuations are introduced as additional convective fluxes in the momentum equations and the continuity equation f k = 0 . 4 is the baseline value for LES [3] Davidson& Peng AIAA, Hawaii, 27-30 June 2011 3 / 30

I NLET FLUCTUATIONS 2 1 w ′ w ′ two-point corr 0.8 1.5 0.6 y 1 0.4 0.2 0.5 0 0 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 z � u ′ v ′ � , v 2 rms , w 2 rms , u 2 rms / u 2 ˆ τ Anisotropic synthetic fluctuations, u ′ , v ′ , w ′ , Integral length scale L ≃ 0 . 13 (see 2-p point correlation) Asymmetric time filter ( U ′ ) m = a ( U ′ ) m − 1 + b ( u ′ ) m with a = 0 . 954 , b = ( 1 − a 2 ) 1 / 2 gives a time integral scale T = 0 . 015 ( ∆ t = 0 . 00063) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 4 / 30

I NTERFACE C ONDITIONS FOR k u AND ε u For k u & ε u we prescribe “inlet” boundary conditions at the interface. First, the usual convective and diffusive fluxes at the interface are set to zero Next, new convective fluxes are added. Which “inlet” values should be used at the interface? ◮ k u , int = f k k RANS ( x = 0 . 5 δ ) , ε u , int = C 3 / 4 k 3 / 2 u , int /ℓ sgs , ℓ sgs = C s ∆ , µ ∆ = V 1 / 3 ◮ Davidson& Peng AIAA, Hawaii, 27-30 June 2011 5 / 30

I NTERFACE C ONDITIONS FOR k u AND ε u For k u & ε u we prescribe “inlet” boundary conditions at the interface. First, the usual convective and diffusive fluxes at the interface are set to zero Next, new convective fluxes are added. Which “inlet” values should be used at the interface? ◮ k u , int = f k k RANS ( x = 0 . 5 δ ) , ε u , int = C 3 / 4 k 3 / 2 u , int /ℓ sgs , ℓ sgs = C s ∆ , µ ∆ = V 1 / 3 ◮ Baseline C s = 0 . 07; different C s values are tested Davidson& Peng AIAA, Hawaii, 27-30 June 2011 5 / 30

C HANNEL F LOW : V ELOCITY AND S HEAR S TRESSES 30 1 25 0.5 20 � u ′ v ′ � + U + 0 15 10 −0.5 5 −1 0 0 1 2 0 0.5 1 1.5 2 10 10 10 y + y + x /δ = 0 . 19 x /δ = 1 . 25 x /δ = 3 Davidson& Peng AIAA, Hawaii, 27-30 June 2011 6 / 30

C HANNEL F LOW : STRESSES AND PEAK VALUES VS . x x /δ = 3 4 100 3.5 resolved stresses 3 � ν u /ν � max max 2.5 � u ′ u ′ � + 2 2 50 1.5 1 0.5 0 0 0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3 3.5 3.5 x 0 0 200 400 600 800 y /δ � u ′ u ′ � + � u ′ u ′ � + max (left) � v ′ v ′ � + � ν u � + max (right) � w ′ w ′ � + Davidson& Peng AIAA, Hawaii, 27-30 June 2011 7 / 30

C HANNEL F LOW : DIFFERENT C s VALUE FOR ε interface k u , int = f k k RANS ε u , int = C 3 / 4 k 3 / 2 u , int /ℓ sgs , ℓ sgs = C s ∆ µ x /δ = 3 30 1 25 0.5 20 � u ′ v ′ � + U + 0 15 10 −0.5 5 −1 0 0 1 2 0 0.5 1 1.5 2 10 10 10 y + y + C s = 0 . 07 C s = 0 . 1 C s = 0 . 2 Davidson& Peng AIAA, Hawaii, 27-30 June 2011 8 / 30

C HANNEL F LOW : DIFFERENT C s VALUE FOR ε interface x /δ = 3 6 1.05 5 1 4 � ν u � /ν u τ 3 0.95 2 0.9 1 0 0.85 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 y + x /δ C s = 0 . 07 C s = 0 . 1 C s = 0 . 2 Davidson& Peng AIAA, Hawaii, 27-30 June 2011 9 / 30

C HANNEL F LOW : DIFFERENT f k VALUES x /δ = 3 1 20 0.5 � u ′ v ′ � + 15 U + 0 10 −0.5 5 −1 0 0 1 2 0 0.5 1 1.5 2 10 10 10 y + y + f k = 0 . 4 f k = 0 . 2 f k = 0 . 6 Davidson& Peng AIAA, Hawaii, 27-30 June 2011 10 / 30

C HANNEL F LOW : DIFFERENT f k VALUES x /δ = 3 4 1.05 3.5 3 1 � ν u � /ν 2.5 u τ 2 0.95 1.5 1 0.9 0.5 0 0.85 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 x /δ y + f k = 0 . 4 f k = 0 . 2 f k = 0 . 6 Davidson& Peng AIAA, Hawaii, 27-30 June 2011 11 / 30

H UMP F LOW x I / c = 0 . 6 S R NTS 2D RANS PANS Inlet, Separation x S / c = 0 . 65; reattachment x R / c = 1 . 1 Re c = 936 000 U ij c ( U in = c = ρ = 1, ν = 1 / Re c ν H / c = 0 . 91, h / c = 0 . 128, x / c = [ 0 . 6 , 4 . 2 ] Mesh: 312 × 120 × 64, Z max = 0 . 2 c (baseline) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 12 / 30

B ASELINE I NLET FLUCTUATIONS 0.4 1 w ′ w ′ two-point corr 0.35 0.8 0.3 0.6 y / c 0.25 0.4 0.2 0.2 0.15 0 −1 0 1 2 3 4 5 6 0 0.02 0.04 0.06 0.08 0.1 z � u ′ v ′ � , v 2 rms , w 2 rms , u 2 rms / u 2 ˆ τ Integral length scale L ≃ 0 . 04 (see 2-p point correlation) Asymmetric time filter ( U ′ ) m = a ( U ′ ) m − 1 + b ( u ′ ) m with a = 0 . 954 , b = ( 1 − a 2 ) 1 / 2 gives a time integral scale T = 0 . 038 ∆ t = 0 . 002. 7500 + 7500 time steps (100 hours one core) Fluctuations multiplied by f bl = max { 0 . 5 [ 1 − tanh ( y − y bl − y wall ) / b ] , 0 . 02 } , y bl = 0 . 2, b = 0 . 01. Davidson& Peng AIAA, Hawaii, 27-30 June 2011 13 / 30

P RESSURE : A MPLITUDES O F I NLET F LUCT 0.8 0.7 0.6 0.5 − C p 0.4 0.3 0.2 0.1 0 0.5 1 1.5 2 x / c 1 . 5 × (baseline inlet fluct) baseline inlet fluct 0 . 5 × (baseline inlet fluct) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 14 / 30

S KIN F RICTION : A MPLITUDES O F I NLET F LUCT zoom −3 3x 10 −3 10x 10 2 8 6 1 4 C f 0 2 −1 0 −2 −2 0 0.5 1 1.5 0.6 0.8 1 1.2 1.4 1.6 x / c x / c baseline inlet fluct 1 . 5 × (baseline inlet fluct) 0 . 5 × (baseline inlet fluct) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 15 / 30

V ELOCITIES : A MPLITUDES O F I NLET F LUCT x / c = 65 x / c = 80 0.25 0.25 0.2 y 0.2 0.15 0.1 0.15 0.05 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 x / c = 100 x / c = 110 0.25 0.25 0.2 0.2 y 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0 0.5 1 0 0.5 1 U / U b U / U b baseline 1 . 5 × (baseline) 0 . 5 × (baseline) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 16 / 30

V ELOCITIES : A MPLITUDES O F I NLET F LUCT x / c = 120 x / c = 130 0.25 0.25 0.2 0.2 y 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 U / U b U / U b baseline 1 . 5 × (baseline) 0 . 5 × (baseline) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 17 / 30

R ESOLVED AND M ODELLED ( < 0 ) S HEAR STRESSES x / c = 0 . 65 x / c = 0 . 80 0.25 0.25 0.2 y 0.2 0.15 0.1 0.15 0.05 0.1 −5 0 5 10 15 0 0 0.01 0.02 0.03 −3 x / c = 1 . 10 x 10 x / c = 1 . 00 0.2 0.2 0.15 0.15 0.1 y 0.1 0.05 0.05 0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 � τ 12 , u � , �− u ′ v ′ � / U 2 � τ 12 , u � , �− u ′ v ′ � / U 2 b b baseline 1 . 5 × (baseline) 0 . 5 × (baseline) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 18 / 30

S HEAR STRESSES : A MPLITUDES O F I NLET F LUCT Resolved and Modelled ( < 0) Shear stresses x / c = 1 . 20 x / c = 1 . 30 0.2 0.2 0.15 0.15 y 0.1 0.1 0.05 0.05 0 0 0 0.01 0.02 0.03 0 0.01 0.02 0.03 � τ 12 , u � , �− u ′ v ′ � / U 2 � τ 12 , u � , �− u ′ v ′ � / U 2 b b baseline inlet fluct 1 . 5 × (baseline inlet fluct) 0 . 5 × (baseline inlet fluct) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 19 / 30

T URB V ISCOSITY : A MPLITUDES O F I NLET F LUCT x / c = 0 . 80 x / c = 0 . 65 0.2 0.2 0.18 0.15 0.16 0.1 y 0.14 0.05 0.12 0 0.1 0 5 10 15 20 0 20 40 60 80 100 120 x / c = 1 . 00 x / c = 1 . 10 0.2 0.2 0.15 0.15 y 0.1 0.1 0.05 0.05 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 � ν t � /ν � ν t � /ν baseline 1 . 5 × (baseline) 0 . 5 × (baseline) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 20 / 30

T URB V ISCOSITY : A MPLITUDES O F I NLET F LUCT x / c = 1 . 20 x / c = 1 . 30 0.2 0.2 0.15 0.15 y 0.1 0.1 0.05 0.05 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 � ν t � /ν � ν t � /ν baseline 1 . 5 × (baseline) 0 . 5 × (baseline) Davidson& Peng AIAA, Hawaii, 27-30 June 2011 21 / 30

P RESSURE : f k = 0 . 5 ; NO INLET FLUCT ; N k = 128 0.8 0.7 0.6 0.5 − C p 0.4 0.3 0.2 0.1 0 0.5 1 1.5 2 x / c N k = 128 f k = 0 . 5 no inlet fluct Davidson& Peng AIAA, Hawaii, 27-30 June 2011 22 / 30