PANS [4] L ARS D AVIDSON Lars Davidson, www.tfd.chalmers.se/lada - PowerPoint PPT Presentation

L ARGE E DDY S IMULATION OF H EAT T RANSFER IN B OUNDARY LAYER AND B ACKSTEP F LOW U SING PANS [4] L ARS D AVIDSON Lars Davidson, www.tfd.chalmers.se/˜lada

PANS L OW R EYNOLDS N UMBER M ODEL [7] � ∂ k u ∂ t + ∂ ( k u U j ) �� � ∂ k u = ∂ ν + ν u + ( P u − ε u ) ∂ x j ∂ x j σ ku ∂ x j � ∂ε u ε 2 ∂ t + ∂ ( ε u U j ) �� � ∂ε u = ∂ ν + ν u ε u u − C ∗ + C ε 1 P u ε 2 ∂ x j ∂ x j σ ε u ∂ x j k u k u k 2 f 2 f 2 ε 2 = C ε 1 + f k u k k , C ∗ ν u = C µ f µ ( C ε 2 f 2 − C ε 1 ) , σ ku ≡ σ k , σ ε u ≡ σ ε ε u 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 �� 2 � � − y ∗ � � 2 �� � f 2 = 1 − exp 1 − 0 . 3exp − 3 . 1 6 . 5 � R t �� 2 � � 2 �� � − y ∗ 5 � � f µ = 1 − exp 1 + exp − 14 R 3 / 4 200 t Baseline model: f k = 0 . 4. www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 2 / 28

N UMERICAL M ETHOD Incompressible finite volume method Pressure-velocity coupling treated with fractional step Differencing scheme for momentum eqns: ◮ 95 % 2 nd order central and 5 % 2 nd order upwind differencing scheme (baseline) OR ◮ 100 % 2 nd order central differencing Hybrid 1 st order upwind/2 nd order central scheme k & ε eqns. 2 nd -order Crank-Nicholson for time discretization www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 3 / 28

B OUNDARY LAYER FLOW : D OMAIN H y δ inlet x L Inlet: δ inlet = 1 (covered by 45 cells), Re θ = 3 600, U in = ρ = 1. Stretching 1 . 12 up to y /δ ≃ 1. Domain: L /δ in = 3 . 2, H /δ in = 15 . 6, Z max = 1 . 5 δ in Resolution: ∆ z + in ≃ 27, ∆ x + in ≃ 54 Grid: 66 × 96 × 64 ( x , y , z ) www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 4 / 28

A NISOTROPIC S YNTHETIC F LUCTUATIONS : I [3, 2, 5] Prescribe an homogeneous Reynolds tensor, u i u j (here from DNS) www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 5 / 28

A NISOTROPIC S YNTHETIC F LUCTUATIONS : I [3, 2, 5] ( u ′ 2 u ′ 2 ) u ′ 1 ,λ u ′ 2 ,λ = 0 λ ) λ ′ u ′ 1 x 2 ,λ u ( 1 x 1 ,λ Prescribe an homogeneous Reynolds tensor, u i u j (here from DNS) isotropic fluctuations in principal directions, ( u ′ 1 u ′ 1 ) λ = ( u ′ 2 u ′ 2 ) λ , u ′ 1 ,λ u ′ 2 ,λ = 0 www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 5 / 28

A NISOTROPIC S YNTHETIC F LUCTUATIONS : I [3, 2, 5] ( u ′ 2 u ′ 2 � u ′ 1 ,λ u ′ 2 ,λ = 0 λ � λ ′ u 1 ′ x 2 ,λ u ( 1 x 1 ,λ Prescribe an homogeneous Reynolds tensor, u i u j (here from DNS) isotropic fluctuations in principal directions, ( u ′ 1 u ′ 1 ) λ = ( u ′ 2 u ′ 2 ) λ , u ′ 1 ,λ u ′ 2 ,λ = 0 re-scale the normal components, ( u ′ 1 u ′ 1 ) λ > ( u ′ 2 u ′ 2 ) λ , using the eigenvalues u ′ 1 ,λ u ′ 2 ,λ = 0 www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 5 / 28

A NISOTROPIC S YNTHETIC F LUCTUATIONS : II u ′ 2 u ′ 2 x 2 u ′ 1 u ′ 2 � = 0 u ′ 1 u ′ 1 > u ′ 2 u ′ 2 x 1 Transform from ( x 1 ,λ , x 2 ,λ ) to ( x 1 , x 2 ) u ′ 2 = 23, u ′ 2 1 1 = 5 from ( u ′ 1 u ′ 1 ) peak in DNS channel flow, Re τ = 500 u ′ 2 u ′ 2 2 3 www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 6 / 28

I NLET C ONDITIONS FOR k u AND ε u AS IN [6] A pre-cursor RANS simulation using the AKN model (i.e. PANS with f k = 1) is carried out. At Re θ = 3 600, U RANS , V RANS , k RANS are taken. ¯ synt , ¯ synt , ¯ u in = U RANS + u ′ v in = V RANS + v ′ w in = w ′ synt Anisotropic synthetic fluctuations are used. The fluctuations are scaled with k u / k u , max . k u , in = f k k RANS , ε u , in = C 3 / 4 k 3 / 2 u , in /ℓ sgs , ℓ sgs = C s ∆ , ∆ = V 1 / 3 , µ C s = 0 . 05 www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 7 / 28

I NLET TURB . FLUCTUATION , TWO - POINT CORRELATIONS Two-point correlation 1 1000 0.8 800 z ) 0.6 600 B ww (ˆ y / H 0.4 400 0.2 200 0 0 0 0.1 0.2 0.3 0.4 0.5 −2 0 2 4 6 stresses ˆ ˆ z /δ , z / H : u + : v + : w + rms , rms , rms : � u ′ v ′ � + ◦ : inlet; : x = 3 δ in www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 8 / 28

B OUNDARY LAYER : V ELOCITY AND S KIN F RICTION 100 % CDS −3 x 10 25 3.6 3.4 20 3.2 15 U + C f 3 10 2.8 5 2.6 0 1 10 50 1000 0 0.5 1 1.5 2 2.5 3 y + x : 100 % CDS; : 100 % CDS, U in from AKN; : 25 % larger : x = δ in ; : x = 2 δ in ; : inlet fluct.; : 25 % larger in- x = 3 δ in ; � : DNS [8] let fluct., C s = 0 . 07; markers: 0 . 37 ( log 10 Re x ) − 2 . 584 ( + : AKN; ◦ : DNS); www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 9 / 28

R EYNOLDS S TRESSES 1500 1500 1000 1000 y + y + 500 500 0 0 −1 0 1 2 3 −1 0 1 2 3 uv u rms uv v rms , w rms , u rms : x = δ in ; : x = 2 δ in ; : x = 3 δ in ; Markers: DNS [8] x = 3 δ in . www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 10 / 28

B ACKWARD F ACING S TEP : D OMAIN y 4 H q w x H 4 . 05 H 21 H Re H = 28 000 Experiments by Vogel & Eaton [9] Mean inlet profiles from RANS (same as in boundary layer) Grid: 336 × 120 in x × y plane. Z max = 1 . 6 H , N k = 64, ∆ z + in = 31. Anisotropic synthetic fluctuations, u ′ , v ′ , w ′ (same as for boundary layer flow); no fluctuations for t ′ Constant heat flux, q w , on lower wall. www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 11 / 28

B ACKSTEP FLOW . S KIN FRICTION AND S TANTON NUMBER −3 −3 x 10 4x 10 4 3.5 3 3 2 St C f 2.5 1 2 0 −1 1.5 −2 1 −5 0 5 10 15 20 0 5 10 15 x / H x / H : PANS; : PANS, 50 % smaller inlet fluctuations; : WALE; • : : 2D RANS; ◦ , • : experiments [9]. PANS, no inlet fluctuations; www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 12 / 28

B ACKSTEP FLOW : V ELOCITIES . x = − 1 . 13 H x = 3 . 2 H x = 14 . 86 H 2.6 2.5 2.5 2.4 2 2 2.2 2 1.5 1.5 1.8 1 1 1.6 1.4 0.5 0.5 1.2 1 0 0 0 0.2 0.4 0.6 0.8 1 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 � ¯ � ¯ � ¯ u � / U in u � / U in u � / U in : PANS; : PANS, 50 % smaller inlet fluctuations; : WALE; • : PANS, no inlet fluctuations; : 2D RANS; ◦ : experiments [9]. www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 13 / 28

B ACKSTEP FLOW : R ESOLVED S TREAMWISE S TRESS . x = − 1 . 13 H x = 3 . 2 H x = 14 . 86 H 2.6 2.5 2.5 2.4 2 2 2.2 2 y / H 1.5 1.5 1.8 1 1 1.6 1.4 0.5 0.5 1.2 1 0 0 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 u rms / U in u rms / U in u rms / U in : PANS; : PANS, 50 % smaller inlet fluctuations; : WALE; • : PANS, no inlet fluctuations; : 2D RANS; ◦ : experiments [9]. www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 14 / 28

B ACKSTEP FLOW : T URBULENT V ISCOSITIES . x = − 1 . 13 H x = 3 . 2 H x = 14 . 86 H 2.6 2.5 2.5 2.4 2 2 2.2 2 y / H 1.5 1.5 1.8 1 1 1.6 1.4 0.5 0.5 1.2 1 0 0 0 2 4 6 8 0 5 10 15 20 0 5 10 15 ν u /ν ν u /ν ν u /ν : PANS; : PANS, 50 % smaller inlet fluctuations; : WALE; • : PANS, no inlet fluctuations; : 2D RANS/10; www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 15 / 28

F ORWARD /B ACKWARD F LOW Fraction of time, γ , when the flow along the bottom wall is in the downstream direction. 1 0.8 0.6 γ 0.4 0.2 0 0 2 4 6 8 10 12 14 x / H : PANS; : PANS, 50 % smaller inlet fluctuations; : WALE; • : PANS, no inlet fluctuations; ◦ : experiments [9]. www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 16 / 28

S HEAR S TRESSES . x = 3 . 2 H PANS RANS 0.1 0.1 0.08 0.08 0.06 0.06 y / H 0.04 0.04 0.02 0.02 0 0 −5 0 5 10 15 −5 0 5 10 15 −4 −4 x 10 x 10 : ν ∂ � ¯ u � : 2 � ν t ¯ : −� uv � ; ◦ : 2 � ν t ¯ s 12 � ; ∂ y ; s 12 � − � uv � . www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 17 / 28

S HEAR S TRESSES . x = 14 . 86 PANS RANS 0.1 0.1 0.08 0.08 0.06 0.06 y / H 0.04 0.04 0.02 0.02 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 −3 −3 x 10 x 10 : ν ∂ � ¯ u � : 2 � ν t ¯ : −� uv � ; ◦ : 2 � ν t ¯ s 12 � ; ∂ y ; s 12 � − � uv � . www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 18 / 28

T ERMS IN THE � ¯ u � E QUATION . x = 3 . 2 H PANS RANS 0.1 0.1 0.08 0.08 0.06 0.06 y / H 0.04 0.04 0.02 0.02 0 0 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 : ν ∂ 2 � ¯ : − ∂ � ¯ u �� ¯ ; + : − ∂ � ¯ u �� ¯ : ∂ u � u � v � ∂ y ( 2 � ν t ¯ s 12 � ) ; ∂ y 2 ; ; ⋆ : ∂ x ∂ y − ∂ � ¯ ∂ x , △ : − ∂ � uv � p � . ∂ y www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 19 / 28

T ERMS IN THE � ¯ u � E QUATION . x = 14 . 86 H PANS RANS 0.1 0.1 0.08 0.08 0.06 0.06 y / H 0.04 0.04 0.02 0.02 0 0 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 : ν ∂ 2 � ¯ : − ∂ � ¯ u �� ¯ ; + : − ∂ � ¯ u �� ¯ : ∂ u � u � v � ∂ y ( 2 � ν t ¯ s 12 � ) ; ∂ y 2 ; ; ⋆ : ∂ x ∂ y − ∂ � ¯ ∂ x , △ : − ∂ � uv � p � . ∂ y www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 20 / 28