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

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

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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 )

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SLIDE 1

LARGE EDDY SIMULATION OF HEAT TRANSFER IN BOUNDARY LAYER AND BACKSTEP FLOW USING PANS [4] LARS DAVIDSON

Lars Davidson, www.tfd.chalmers.se/˜lada

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SLIDE 2

PANS LOW REYNOLDS NUMBER MODEL [7]

∂ku ∂t + ∂(kuUj) ∂xj = ∂ ∂xj

  • ν + νu

σku ∂ku ∂xj

  • + (Pu − εu)

∂εu ∂t + ∂(εuUj) ∂xj = ∂ ∂xj

  • ν + νu

σεu ∂εu ∂xj

  • + Cε1Pu

εu ku − C∗

ε2

ε2

u

ku νu = Cµfµ k2

u

εu , C∗

ε2 = Cε1 + fk

fε (Cε2f2 − Cε1), σku ≡ σk f 2

k

fε , σεu ≡ σε f 2

k

fε Cε1, Cε2, σk, σε and Cµ same values as [1]. fε = 1. f2 and fµ read f2 =

  • 1 − exp
  • − y∗

3.1 2 1 − 0.3exp

  • −

Rt 6.5 2 fµ =

  • 1 − exp
  • − y∗

14 2 1 + 5 R3/4

t

exp

  • −

Rt 200 2 Baseline model: fk = 0.4.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 2 / 28

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SLIDE 3

NUMERICAL METHOD

Incompressible finite volume method Pressure-velocity coupling treated with fractional step Differencing scheme for momentum eqns:

◮ 95% 2nd order central and 5% 2nd order upwind differencing

scheme (baseline) OR

◮ 100% 2nd order central differencing

Hybrid 1st order upwind/2nd order central scheme k & ε eqns. 2nd-order Crank-Nicholson for time discretization

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 3 / 28

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SLIDE 4

BOUNDARY LAYER FLOW: DOMAIN

x y L H δinlet Inlet: δinlet = 1 (covered by 45 cells), Reθ = 3 600, Uin = ρ = 1. Stretching 1.12 up to y/δ ≃ 1. Domain: L/δin = 3.2, H/δin = 15.6, Zmax = 1.5δin 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

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SLIDE 5

ANISOTROPIC SYNTHETIC FLUCTUATIONS: I [3, 2, 5]

Prescribe an homogeneous Reynolds tensor, uiuj (here from DNS)

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 5 / 28

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SLIDE 6

ANISOTROPIC SYNTHETIC FLUCTUATIONS: I [3, 2, 5]

( u

′ 1

u

′ 1

)

λ

x1,λ ( u

′ 2

u

′ 2

)

λ

x2,λ u′

1,λu′ 2,λ = 0

Prescribe an homogeneous Reynolds tensor, uiuj (here from DNS) isotropic fluctuations in principal directions, (u′

1u′ 1)λ = (u′ 2u′ 2)λ,

u′

1,λu′ 2,λ = 0

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 5 / 28

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SLIDE 7

ANISOTROPIC SYNTHETIC FLUCTUATIONS: I [3, 2, 5]

( u

′ 1

u

′ 1

  • λ

x1,λ ( u

′ 2

u

′ 2

  • λ

x2,λ u′

1,λu′ 2,λ = 0

Prescribe an homogeneous Reynolds tensor, uiuj (here from DNS) isotropic fluctuations in principal directions, (u′

1u′ 1)λ = (u′ 2u′ 2)λ,

u′

1,λu′ 2,λ = 0

re-scale the normal components, (u′

1u′ 1)λ > (u′ 2u′ 2)λ, using the

eigenvalues u′

1,λu′ 2,λ = 0

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 5 / 28

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SLIDE 8

ANISOTROPIC SYNTHETIC FLUCTUATIONS: II

u′

1u′ 1 > u′ 2u′ 2

x1 u′

2u′ 2

x2 u′

1u′ 2 = 0

Transform from (x1,λ, x2,λ) to (x1, x2) u′2

1

u′2

2

= 23, u′2

1

u′2

3

= 5 from (u′

1u′ 1)peak in DNS channel flow, Reτ = 500

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 6 / 28

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SLIDE 9

INLET CONDITIONS FOR ku AND εu AS IN [6]

A pre-cursor RANS simulation using the AKN model (i.e. PANS with fk = 1) is carried out. At Reθ = 3 600, URANS, VRANS, kRANS are taken. ¯ uin = URANS + u′

synt, ¯

vin = VRANS + v′

synt, ¯

win = w′

synt

Anisotropic synthetic fluctuations are used. The fluctuations are scaled with ku/ku,max. ku,in = fkkRANS, εu,in = C3/4

µ

k3/2

u,in/ℓsgs, ℓsgs = Cs∆, ∆ = V 1/3,

Cs = 0.05

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 7 / 28

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SLIDE 10

INLET TURB. FLUCTUATION, TWO-POINT CORRELATIONS

−2 2 4 6 200 400 600 800 1000

stresses y/H

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

ˆ z/δ, ˆ z/H Bww(ˆ z) Two-point correlation : u+

rms,

: v+

rms,

: w+

rms

: u′v′+

  • : inlet;

: x = 3δin

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 8 / 28

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SLIDE 11

BOUNDARY LAYER: VELOCITY AND SKIN FRICTION

1 10 50 1000 5 10 15 20 25

y+ U+ 100%CDS

0.5 1 1.5 2 2.5 3 2.6 2.8 3 3.2 3.4 3.6 x 10

−3

x Cf : x = δin; : x = 2δin; : x = 3δin; : DNS [8] : 100% CDS; : 100% CDS, Uin from AKN; : 25% larger inlet fluct.; : 25% larger in- let fluct., Cs = 0.07; markers: 0.37 (log10Rex)−2.584 (+: AKN; ◦: DNS);

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 9 / 28

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SLIDE 12

REYNOLDS STRESSES

−1 1 2 3 500 1000 1500

y+ uv urms

−1 1 2 3 500 1000 1500

y+ uv vrms, wrms, urms : x = δin; : x = 2δin; : x = 3δin. x = 3δin; Markers: DNS [8]

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 10 / 28

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SLIDE 13

BACKWARD FACING STEP: DOMAIN

4.05H 21H 4H H x y qw ReH = 28 000 Experiments by Vogel & Eaton [9] Mean inlet profiles from RANS (same as in boundary layer) Grid: 336 × 120 in x × y plane. Zmax = 1.6H, Nk = 64, ∆z+

in = 31.

Anisotropic synthetic fluctuations, u′, v′, w′ (same as for boundary layer flow); no fluctuations for t′ Constant heat flux, qw, on lower wall.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 11 / 28

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SLIDE 14

BACKSTEP FLOW. SKIN FRICTION AND STANTON

NUMBER

−5 5 10 15 20 −2 −1 1 2 3 4 x 10

−3

x/H Cf

5 10 15 1 1.5 2 2.5 3 3.5 4x 10

−3

x/H St : 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 12 / 28

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SLIDE 15

BACKSTEP FLOW: VELOCITIES.

0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

¯ u/Uin x = −1.13H

−0.2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5

¯ u/Uin x = 3.2H

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5

¯ u/Uin x = 14.86H : 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

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SLIDE 16

BACKSTEP FLOW: RESOLVED STREAMWISE STRESS.

0.05 0.1 0.15 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

urms/Uin y/H x = −1.13H

0.05 0.1 0.15 0.2 0.5 1 1.5 2 2.5

urms/Uin x = 3.2H

0.05 0.1 0.15 0.5 1 1.5 2 2.5

urms/Uin x = 14.86H : 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

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SLIDE 17

BACKSTEP FLOW: TURBULENT VISCOSITIES.

2 4 6 8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

νu/ν y/H x = −1.13H

5 10 15 20 0.5 1 1.5 2 2.5

νu/ν x = 3.2H

5 10 15 0.5 1 1.5 2 2.5

νu/ν x = 14.86H : 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

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SLIDE 18

FORWARD/BACKWARD FLOW

Fraction of time, γ, when the flow along the bottom wall is in the downstream direction.

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 1

x/H γ : 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

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SLIDE 19

SHEAR STRESSES. x = 3.2H

−5 5 10 15 x 10

−4

0.02 0.04 0.06 0.08 0.1

y/H PANS

−5 5 10 15 x 10

−4

0.02 0.04 0.06 0.08 0.1

RANS : 2νt¯ s12; : ν ∂¯ u ∂y ; : −uv; ◦: 2νt¯ s12 − uv.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 17 / 28

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SLIDE 20

SHEAR STRESSES. x = 14.86

0.2 0.4 0.6 0.8 1 x 10

−3

0.02 0.04 0.06 0.08 0.1

y/H PANS

0.2 0.4 0.6 0.8 1 x 10

−3

0.02 0.04 0.06 0.08 0.1

RANS : 2νt¯ s12; : ν ∂¯ u ∂y ; : −uv; ◦: 2νt¯ s12 − uv.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 18 / 28

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SLIDE 21

TERMS IN THE ¯ u EQUATION. x = 3.2H

−0.06 −0.04 −0.02 0.02 0.04 0.06 0.02 0.04 0.06 0.08 0.1

y/H PANS

−0.06 −0.04 −0.02 0.02 0.04 0.06 0.02 0.04 0.06 0.08 0.1

RANS : ∂ ∂y (2νt¯ s12); : ν ∂2¯ u ∂y2 ; : −∂¯ u¯ u ∂x ; +: −∂¯ u¯ v ∂y ; ⋆: −∂¯ p ∂x , △: −∂uv ∂y .

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 19 / 28

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SLIDE 22

TERMS IN THE ¯ u EQUATION. x = 14.86H

−0.06 −0.04 −0.02 0.02 0.04 0.06 0.02 0.04 0.06 0.08 0.1

y/H PANS

−0.06 −0.04 −0.02 0.02 0.04 0.06 0.02 0.04 0.06 0.08 0.1

RANS : ∂ ∂y (2νt¯ s12); : ν ∂2¯ u ∂y2 ; : −∂¯ u¯ u ∂x ; +: −∂¯ u¯ v ∂y ; ⋆: −∂¯ p ∂x , △: −∂uv ∂y .

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 20 / 28

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SLIDE 23

HEAT FLUXES. x = 3.2H

−1 −0.8 −0.6 −0.4 −0.2 0.02 0.04 0.06 0.08 0.1

y/H PANS

−1 −0.8 −0.6 −0.4 −0.2 0.02 0.04 0.06 0.08 0.1

RANS : νt σt ∂¯ t ∂y

  • ;

: ν σℓ ∂¯ t ∂y ; : −v′t′. ◦: νt σt ∂¯ t ∂y

  • − v′t′.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 21 / 28

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SLIDE 24

HEAT FLUXES. x = 14.86H

−1 −0.8 −0.6 −0.4 −0.2 0.02 0.04 0.06 0.08 0.1

y/H PANS

−1 −0.8 −0.6 −0.4 −0.2 0.02 0.04 0.06 0.08 0.1

RANS : νt σt ∂¯ t ∂y

  • ;

: ν σℓ ∂¯ t ∂y ; : −v′t′. ◦: νt σt ∂¯ t ∂y

  • − v′t′.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 22 / 28

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SLIDE 25

TERMS IN THE ¯ t EQUATION. x = 3.2H

−100 −50 50 100 0.02 0.04 0.06 0.08 0.1

y

PANS

−100 −50 50 100 0.02 0.04 0.06 0.08 0.1

RANS : ∂ ∂y νt σt ∂¯ t ∂y

  • ;

: ν σℓ ∂2¯ t ∂y2 ; : −∂¯ u¯ t ∂x ; +: −∂¯ v¯ t ∂y ; △: −∂v′t′ ∂y .

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 23 / 28

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SLIDE 26

TERMS IN THE ¯ t EQUATION. x = 14.86H

−50 50 0.02 0.04 0.06 0.08 0.1

y/H PANS

−50 50 0.02 0.04 0.06 0.08 0.1

RANS : ∂ ∂y νt σt ∂¯ t ∂y

  • ;

: ν σℓ ∂2¯ t ∂y2 ; : −∂¯ u¯ t ∂x ; +: −∂¯ v¯ t ∂y ; △: −∂v′t′ ∂y .

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 24 / 28

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SLIDE 27

CONCLUDING REMARKS

Developing boundary layer

◮ Synthetic fluctuations give fully developed conditions after a couple

  • f boundary layer thicknesses

◮ 5% upwinding dampens resolved fluctuations; can be compensated

by 25% larger inlet fluctuations

Backstep flow

◮ Very good agreement with experiments ◮ 2D RANS predicts turbulent diffusion surprisingly well ◮ Synthetic inlet fluctuations give an improved Stanton number;

  • therwise small effect in the reciculation region

◮ LRN PANS and WALE equally good ◮ 5% upwinding has a negligble effect in the recirculation region www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 25 / 28

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SLIDE 28

REFERENCES I

[1] ABE, K., KONDOH, T., AND NAGANO, Y. A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows - 1. Flow field calculations.

  • Int. J. Heat Mass Transfer 37, 1 (1994), 139–151.

[2] BILLSON, M. Computational Techniques for Turbulence Generated Noise. PhD thesis, Dept. of Thermo and Fluid Dynamics, Chalmers University of Technology, G¨

  • teborg, Sweden, 2004.

[3] BILLSON, M., ERIKSSON, L.-E., AND DAVIDSON, L. Modeling of synthetic anisotropic turbulence and its sound emission. The 10th AIAA/CEAS Aeroacoustics Conference, AIAA 2004-2857, Manchester, United Kindom, 2004.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 26 / 28

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SLIDE 29

REFERENCES II

[4] DAVIDSON, L. Large eddy simulation of heat transfer in boundary layer and backstep flow using PANS (corrected version can be downloaded at http://www.tfd.chalmers.se/˜lada/). In Turbulence, Heat and Mass Transfer, THMT-12 (Palermo, Sicily/Italy, 2012). [5] DAVIDSON, L., AND BILLSON, M. Hybrid LES/RANS using synthesized turbulent fluctuations for forcing in the interface region. International Journal of Heat and Fluid Flow 27, 6 (2006), 1028–1042. [6] DAVIDSON, L., AND PENG, S.-H. Embedded LES with PANS. In 6th AIAA Theoretical Fluid Mechanics Conference, AIAA paper 2011-3108 (27-30 June, Honolulu, Hawaii, 2011).

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 27 / 28

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SLIDE 30

REFERENCES III

[7] MA, J., PENG, S.-H., DAVIDSON, L., AND WANG, F. A low Reynolds number variant of Partially-Averaged Navier-Stokes model for turbulence. International Journal of Heat and Fluid Flow 32, 3 (2011), 652–669. 10.1016/j.ijheatfluidflow.2011.02.001. [8] SCHLATTER, P., AND ORLU, R. Assessment of direct numerical simulation data of turbulent boundary layers. Journal of Fluid Mechanics 659 (2010), 116–126. [9] VOGEL, J. C., AND EATON, J. K. Combined heat transfer and fluid dynamic measurements downstream a backward-facing step. Journal of Heat Transfer 107 (1985), 922–929.

www.tfd.chalmers.se/˜lada THMT-12, Palermo, Sept 2012 28 / 28