Backscatter from a scale-similarity model: embedded LES of channel - - PowerPoint PPT Presentation

Backscatter from a scale-similarity model: embedded LES of channel flow, developing boundary layer flow and backstep flow [2] Lars Davidson

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

Embedded LES: Problem Formulation

u′, v ′, w′ x y RANS LES Interface At the interface between RANS and LES, turbulent fluctuations, u′, v ′, w′, are imposed to stimulate growth of resolved fluctuations

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 2 / 22

Embedded LES: Problem Formulation

u′, v ′, w′ x y RANS LES Interface forcing At the interface between RANS and LES, turbulent fluctuations, u′, v ′, w′, are imposed to stimulate growth of resolved fluctuations To promote transition from RANS to LES (reducing the gray area), additional forcing may be used in the LES region

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 2 / 22

Embedded LES: Problem Formulation

u′, v ′, w′ x y RANS LES Interface forcing At the interface between RANS and LES, turbulent fluctuations, u′, v ′, w′, are imposed to stimulate growth of resolved fluctuations To promote transition from RANS to LES (reducing the gray area), additional forcing may be used in the LES region In the present work, forcing is added using a scale-similarity model

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 2 / 22

Momentum Equation

The momentum equations for LES read D¯ ui Dt + 1 ρ ∂¯ p ∂xi = ∂ ∂xk

• (ν + νSGS) ∂¯

ui ∂xk

• − ∂τik

∂xk where D/Dt denotes material derivative. The stress tensor, τik, is

• btained from the scale-similarity model

τik = ¯ ui¯ uk − ¯ ¯ ui ¯ ¯ uk

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 3 / 22

Turbulent Kinetic Energy Eq

Let us take a closer look at the equation for the resolved, turbulent kinetic energy, K = ¯ u′

i¯

u′

i/2, which reads (. denotes averaging in

time) DK Dt + ¯ u′

k¯

u′

i∂¯

ui ∂xk + 1 ρ ∂¯ p′¯ u′

i

∂xi + 1 2 ∂¯ u′

k¯

u′

i¯

u′

i

∂xk = ν ∂2¯ u′

i

∂xk∂xk ¯ u′

i

• −

∂τik ∂xk − ∂τik ∂xk

• ¯

u′

i

• The second line is simply the ¯

u′

i eq. multiplied by ¯

u′

i

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 4 / 22

Turbulent Kinetic Energy Eq (cont’d)

The right side can be re-written as ν ∂2¯ u′

i

∂xk∂xk ¯ u′

i

• εnon

− ∂τik ∂xk ¯ u′

i

• =

ν ∂2K ∂xk∂xk − ν ∂¯ u′

i

∂xk ∂¯ u′

i

∂xk

• ε

− ∂τik ∂xk ¯ u′

i

• εSGS

The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, εnon; this is predominately negative.

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22

Turbulent Kinetic Energy Eq (cont’d)

The right side can be re-written as ν ∂2¯ u′

i

∂xk∂xk ¯ u′

i

• εnon

− ∂τik ∂xk ¯ u′

i

• =

ν ∂2K ∂xk∂xk − ν ∂¯ u′

i

∂xk ∂¯ u′

i

∂xk

• ε

− ∂τik ∂xk ¯ u′

i

• εSGS

The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, εnon; this is predominately negative. The first term on the right side is the viscous diffusion

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22

Turbulent Kinetic Energy Eq (cont’d)

The right side can be re-written as ν ∂2¯ u′

i

∂xk∂xk ¯ u′

i

• εnon

− ∂τik ∂xk ¯ u′

i

• =

ν ∂2K ∂xk∂xk − ν ∂¯ u′

i

∂xk ∂¯ u′

i

∂xk

• ε

− ∂τik ∂xk ¯ u′

i

• εSGS

The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, εnon; this is predominately negative. The first term on the right side is the viscous diffusion the second term, ε, is the (isotropic) dissipation which is positive

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22

Turbulent Kinetic Energy Eq (cont’d)

The right side can be re-written as ν ∂2¯ u′

i

∂xk∂xk ¯ u′

i

• εnon

− ∂τik ∂xk ¯ u′

i

• =

ν ∂2K ∂xk∂xk − ν ∂¯ u′

i

∂xk ∂¯ u′

i

∂xk

• ε

− ∂τik ∂xk ¯ u′

i

• εSGS

The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, εnon; this is predominately negative. The first term on the right side is the viscous diffusion the second term, ε, is the (isotropic) dissipation which is positive The last term, εSGS, can be positive (forward scattering=dissipation)

• r negative (backward scattering=forcing).

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22

Physical Interpretation

The SGS term εSGS = ∂τik ∂xk ¯ u′

i

• consists of a net SGS force vector, T SGS

i

, (per unit mass), multiplied by a velocity fluctuation vector, ¯ u′

i i.e.

εSGS =

• T SGS

i

¯ u′

i

• When the SGS vector, T SGS

i

, opposes the fluctuation, ¯ u′

i, it is

damping the fluctuation, i.e. it is dissipative

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 6 / 22

Select Forward or Backscatter

We want to be able to make the term εSGS dissipative or forcing ν ∂2¯ u′

i

∂xk∂xk ¯ u′

i

• εnon

− ∂τik ∂xk ¯ u′

i

• =

ν ∂2K ∂xk∂xk − ν ∂¯ u′

i

∂xk ∂¯ u′

i

∂xk

• ε

− ∂τik ∂xk ¯ u′

i

• εSGS

The viscous term in the mom. eq., ν ∂2¯ u′

i

∂xk∂xk

• , is dissipative

If −∂τik ∂xk has the same sign as ∂2¯ u′

i

∂xk∂xk , then εSGS is dissipative Otherwise, it is a forcing term (backscatter)

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 7 / 22

Select Backscatter Events

We want the SGS stress tensor to act as backscatter in the K equation. Hence we add −∂τik/∂xk to the momentum equation only when its sign is opposite to that of the viscous diffusion term. i.e. [1] Mik = sign ∂τik ∂xk ∂2¯ u′

i

∂xk∂xk

• ,
• Mik = max(Mik, 0),

∂τik ∂xk − = − Mik ∂τik ∂xk

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 8 / 22

∂2¯ u′

i

∂xk∂xk vs. ∂2¯ ui ∂xk∂xk

Mik = sign ∂τik ∂xk ∂2¯ u′

i

∂xk∂xk

• ,
• Mik = max(Mik, 0),

∂τik ∂xk − = − Mik ∂τik ∂xk ¯ u′

i, is not known at run-time. It could be computed as

¯ u′

i = ¯

ui − ¯ uira, where ¯ uira denotes the running-time average of ¯ ui. It was shown in [1] that, for y + 20 in channel flow, the second derivative of ¯ u′

i is almost 100% correlated with that of ¯

ui Hence, in the present work, the relation at the top-left is replaced by Mik = sign ∂τik ∂xk ∂2¯ ui ∂xk∂xk

• www.tfd.chalmers.se/˜lada

TSFP8, Poitiers, 2013 9 / 22

Stability

The forcing has a positive feedback, i.e. the more the momentum eq is destabilized, the larger the velocity gradients, the larger the forcing Hence, the forcing term has to be limited

• −∂τik

∂xk

• ≤ β(ν + νSGS)
• ∂2¯

ui ∂xk∂xk

• The baseline value is β = 2.

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 10 / 22

PANS Low Reynolds Number Model [3]

∂k ∂t + ∂(kUj) ∂xj = ∂ ∂xj

• ν + νt

σku ∂k ∂xj

• + (P − ε)

∂ε ∂t + ∂(εUj) ∂xj = ∂ ∂xj

• ν + νt

σεu ∂ε ∂xj

• + Cε1P ε

k − C ∗

ε2

ε2 k νt = Cµfµ k2 ε , C ∗

ε2 = Cε1 + fk

fε (Cε2f2 − Cε1), σku ≡ σk f 2

k

fε , σεu ≡ σε f 2

k

fε LRN Damping functions, f2, fµ as in [3] RANS region: fk = 1.0 LES region: i) fk = 0.4 or ii) fk =

1 c1/2

µ

(∆/Lt)2/3 , Lt = (kres + k)3/2/ε Option i and ii give same results. but Option ii unstable in backstep flow with forcing

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 11 / 22

Test Case I: Channel Flow

u′, v ′, w′ x y RANS LES Interface

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 12 / 22

Test Case I: Channel Flow

u′, v ′, w′ forcing x y RANS LES Interface

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 12 / 22

Test Case I: Channel Flow

u′, v ′, w′ forcing x y RANS LES Interface x y 2 0.95 5.45 LES, fk = 0.4 RANS fk = 1.0 Interface Reτ = 950 based on uτ 128 × 80 × 32 (x, y, z) cells zmax = 1.6

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 12 / 22

Test Case II: Boundary Layer Flow

u′, v ′, w′ x y LES

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 13 / 22

Test Case II: Boundary Layer Flow

u′, v ′, w′ forcing x y LES

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 13 / 22

Test Case II: Boundary Layer Flow

u′, v ′, w′ forcing x y LES δin x y H = 15.6 L = 6.4 δin = 1, Zmax = 1.6 Reθ = 3 600 Reδ = Ufreeδin/ν ≃ 28 000

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 13 / 22

Inlet turb. fluctuation, 2-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+2

rms,

: v +2

rms,

: w+2

rms

: u′v ′+

• : inlet;

: x = 3δin

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 14 / 22

Results: Skin Friction

2 4 6 0.8 0.85 0.9 0.95 1 1.05

Channel flow x uτ RANS-LES interface

2 4 6 1 2 3 4x 10

−3Boundary layer flow

x Cf backscatter no backscatter backscatter with β = 3 (limiter) :target value.

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 15 / 22

Results: Resolved Shear Stresses

200 400 600 −1 −0.8 −0.6 −0.4 −0.2

Channel flow x = 3 uv/u2

τ,in

50 100 150 −0.8 −0.6 −0.4 −0.2

Boundary layer flow y + x = 1.25 x = 3 x = 1.25: with markers x = 3: without markers backscatter no backscatter.

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 16 / 22

Backstep flow, computational domain

ReH = 28 000, 336 × 152 × 64 cells (x, y, z), zmax = 1.6 4.6 21 4 H = 1 x y qw

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 17 / 22

Backstep flow, computational domain

ReH = 28 000, 336 × 152 × 64 cells (x, y, z), zmax = 1.6 4.6 21 4 H = 1 x y qw u′, v ′, w′ forcing x y LES

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 17 / 22

Skin friction and St number

−5 5 10 15 20 −2 −1 1 2 3 4 5 6x 10

−3

Skin Friction x Cf

−5 5 10 15 20 0.5 1 1.5 2 2.5 3 3.5 4x 10

−3

Stanton number x St backscatter no backscatter

• : Experiments by Vogel & Eaton [4]

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 18 / 22

Conclusions

The gray area issue at RANS-LES interface has been addressed The stresses, τik, from a scale-similarity model was used for forcing The forcing was achieved be selecting the instants when −∂τik ∂xk corresponds to backscatter It is found that the forcing indeed quickens the transition from RANS mode to LES mode The present approach can also be used for laminar-turbulent transition

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 19 / 22

Three-Day CFD Course at Chalmers

Unsteady Simulations for Industrial Flows: LES, DES, hybrid LES-RANS and URANS 6-8 November 2013 at Chalmers, Gothenburg, Sweden Max 16 participants 50% lectures and 50% workshops in front of a PC Registration deadline: 18 October 2013 For info, see http://www.tfd.chalmers.se/˜lada/cfdkurs/cfdkurs.html

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 20 / 22

References I

[1] Davidson, L. Hybrid LES-RANS: back scatter from a scale-similarity model used as forcing.

• Phil. Trans. of the Royal Society A 367, 1899 (2009), 2905–2915.

[2] Davidson, L. Backscatter from a scale-similarity model: embedded les of channel flow and developing boundary layer flow. In 8th International Symposium on Turbulence and Shear Flow Phenomena (TSFP-8) http://www.tfd.chalmers.se/˜lada/allpaper.html (Poitiers, France, 2013). [3] 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.

www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 21 / 22

References II

[4] Vogel, J., and Eaton, J. 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 TSFP8, Poitiers, 2013 22 / 22