A New Approach to Treat the RANS-LES interface in PANS [1] Lars - - PowerPoint PPT Presentation

A New Approach to Treat the RANS-LES interface in PANS [1] Lars Davidson

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

PANS Low Reynolds Number Model [4]

∂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 [4] fε = 1.0 LES region: fk = 0.4 RANS region: fk = 1.0

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 2 / 19

PANS Low Reynolds Number Model [4]

∂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 [4] fε = 1.0 LES region: fk = 0.4 RANS region: fk = 1.0 ◮Zonal RANS-LES: fk has a large gradient at the RANS-LES interface

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 2 / 19

PANS: derivation

The PANS k equation is derived by multiplying the RANS k equation by fk. The left hand reads fk Dktot Dt (1) where D/Dt = ∂/∂t + ¯ ui∂/∂xi, ktot = k + kres (modeled plus resolved). If it is assumed that fk constant, Eq. 1 can be re-written as fk Dktot Dt = Dfkktot Dt = Dk Dt , fk = k ktot (2) If fk is not constant, Eq. 2 must be written as (Girimaji & Wallin [2]) fk Dktot Dt = Dfkktot Dt − ktot Dfk Dt = Dk Dt − ktot Dfk Dt This work presents models for the boxed term at RANS-LES interfaces, i.e.

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 3 / 19

PANS: derivation

The PANS k equation is derived by multiplying the RANS k equation by fk. The left hand reads fk Dktot Dt (1) where D/Dt = ∂/∂t + ¯ ui∂/∂xi, ktot = k + kres (modeled plus resolved). If it is assumed that fk constant, Eq. 1 can be re-written as fk Dktot Dt = Dfkktot Dt = Dk Dt , fk = k ktot (2) If fk is not constant, Eq. 2 must be written as (Girimaji & Wallin [2]) fk Dktot Dt = Dfkktot Dt − ktot Dfk Dt = Dk Dt − ktot Dfk Dt This work presents models for the boxed term at RANS-LES interfaces, i.e.

◮ horizontal RANS-LES interface in boundary layer (channel flow) www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 3 / 19

PANS: derivation

The PANS k equation is derived by multiplying the RANS k equation by fk. The left hand reads fk Dktot Dt (1) where D/Dt = ∂/∂t + ¯ ui∂/∂xi, ktot = k + kres (modeled plus resolved). If it is assumed that fk constant, Eq. 1 can be re-written as fk Dktot Dt = Dfkktot Dt = Dk Dt , fk = k ktot (2) If fk is not constant, Eq. 2 must be written as (Girimaji & Wallin [2]) fk Dktot Dt = Dfkktot Dt − ktot Dfk Dt = Dk Dt − ktot Dfk Dt This work presents models for the boxed term at RANS-LES interfaces, i.e.

◮ horizontal RANS-LES interface in boundary layer (channel flow) ◮ vertical RANS-LES interface in embedded LES (channel flow) www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 3 / 19

Interface Model 1

In [2], ktotDfk/Dt is represented by introducing an additional turbulent viscosity, νtr, in the momentum equation ∂ ∂xj (νtr¯ sij) , ¯ sij = 1 2 ∂¯ ui ∂xj + ∂¯ uj ∂xi

• where

νtr = Pktr |¯ s|2 , Pktr = νtr|¯ s|2 = ktot Dfk Dt = k fk Dfk Dt The object of Pktr is to decrease νt and facilitate growth of resolved turbulence on the LES side of an interface Hence, only νtr < 0 is used which corresponds to Dfk/Dt < 0 (from RANS to LES). νt + νtr > 0 in the momentum equation (but not in the k equation)

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 4 / 19

Interface Model 2

This model is identical to Model 1 except that k/fk is replaced by ktot i.e. Pktr = k fk Dfk Dt

Model 1

Pktr = ktot Dfk Dt

Model 2

ktot = k + 1 2¯ u′

i¯

u′

ir.a

where subscript r.a. denotes running average. In PANS, fk is defined as fk = k/ktot In post-processing it is usually found that fk > k/ktot (approx. a factor 4 larger) ⇒ |Pktr |model 2 > |Pktr |model 1

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 5 / 19

Interface Model 3

In Models 1 & 2, νtr = Pktr /|¯ s|2 which may cause numerical problems. Model 3 does not involve νtr. The original term ktotDfk/Dt is used in the k equation Adding the term −0.5¯ u′

i

Dfk Dt − k¯ u′

i

¯ u′

m¯

u′

m

Dfk Dt in the momenum equation corresponds to the time-averaged term − ¯ u′

i¯

u′

i

2 Dfk Dt

• − k¯

u′

i ¯

u′

i

¯ u′

m¯

u′

m

Dfk Dt = −ktotDfk Dt in the kres equation. However, this term causes numerical instability. Hence it is not used. Only the term in the k equation, ktotDfk/Dt < 0, is used.

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 6 / 19

Interface Models: summary

Models 1 & 2: additional turbulent viscosity, νtr = Pktr /|¯ s|2 < 0, in Pk and momentum equations

◮ Limit in momentum equations: νt + νtr > 0 ◮ Model 1: Pktr = k

fk Dfk Dt

◮ Model 2: Pktr = ktotr.a

Dfk Dt

Model 3: additional production term, Pktr , in k equation without use

• f νtr

Pktr = ktotr.a Dfk Dt < 0 Models 1-3 correspond to the non-commutivity in DES beteen filtering and spatial derivative at RANS-LES interfaces (Hamba [3]) ∂f ∂xi = ∂¯ f ∂xi − ∂∆ ∂xi ∂¯ f ∂∆

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 7 / 19

Fully Developed Channel Flow

The URANS and the LES regions. x y wall yint

LES, fk = 0.4 URANS, fk = 1

Reτ = uτδ/ν = 2 000, Re = 4 000 and Re = 8 000 xmax = 3.2, ymax = 2 and zmax = 1.6. 32 × 32 cells in the x − z plane Ny = 80 cells (Reτ = 2 000 and 4 000) or Ny = 96 (Reτ = 8 000)

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 8 / 19

Fully Developed Channel Flow: Results

100 1000 10 20 30

Velocity y + U+

500 1000 50 100 150

Turbulent viscosity y + (νt + νtr)/ν : Model k

fk Dfk Dt

: Model ktot

Dfk Dt .

: no interface model. +: U+ = ln(y +)/0.4 + 5.2

• : location of the computational cell centers.

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 9 / 19

Fully Developed Channel Flow: Results

500 1000 −20 20 40

Production terms in k eq y + Pk + Pktr

0.5 1 −1 −0.5 0.5 1

modeled & resolved stresses y −τ ν

12 − τ12,

−u′v ′ resolved modeled : Model k

fk Dfk Dt

: Model ktot

Dfk Dt .

: no interface model.

• : the location of the computational cell centers.

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 10 / 19

Embedded Channel Flow

x y 2 0.95 5.45

LES, fk = 0.4 RANS fk = 1

Interface Reτ = uτδ/ν = 950 The domain size is 6.4 × 2 × 1.6 (x, y, z) 128 × 80 × 64 cells.

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 11 / 19

Synthetic Fluctuations at the Interface

0.5 1 1.5 2 0.5 1 1.5 2

y

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Bww(ˆ z) ˆ z : u2

rms/u2 τ

: v 2

rms/u2 τ

: w2

rms/u2 τ

: u′v ′/u2

τ

Two-point correlation of synthetic in- let fluctuations

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 12 / 19

Velocity and Skin Friction

10 10

1

10

2

5 10 15 20

Velocity at x = 5.5 y + ¯ u/uτ,in

2 4 6 0.8 0.9 1 1.1

Friction velocity x uτ

• ∗

: Model k

fk Dfk Dt

: Model ktot

Dfk Dt

: no interface model

• : U+ = ln(y +)/0.4 + 5.2

∗: uτ, RANS when x → ∞

• : uτ, PANS when x → ∞

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 13 / 19

Resolved and Modeled Turbulence

2 4 6 5 10 15

Streamwise fluctuation x max

y (u2 rms(x, y)) 2 4 6 20 40 60 80 100

Turbulent viscosities x max

y ((νtot(x, y)/ν

: Model k

fk Dfk Dt

: Model ktot

Dfk Dt

: no interface model νtot = ν + νt + νtr

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 14 / 19

Production at Interface

−1000 −500 500 0.2 0.4 0.6 0.8 1

x = 1 Production of k y : Pk + Pktr , Model k

fk Dfk Dt .

: Pk + Pktr , Model ktot

Dfk Dt .

: Pk, no interface model;

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 15 / 19

Conclusions

Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented:

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19

Conclusions

Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented:

◮

k fk Dfk Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 1

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19

Conclusions

Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented:

◮

k fk Dfk Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 1

◮ ktot Dfk

Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 2

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19

Conclusions

Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented:

◮

k fk Dfk Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 1

◮ ktot Dfk

Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 2

◮ ktot Dfk

Dt added no νttr to the k eq (Model 3)

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19

Conclusions

Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented:

◮

k fk Dfk Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 1

◮ ktot Dfk

Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 2

◮ ktot Dfk

Dt added no νttr to the k eq (Model 3)

Model 2 works very well

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19

Conclusions

Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented:

◮

k fk Dfk Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 1

◮ ktot Dfk

Dt added via νttr to the k eq and mom eq (νttr + νt > 0): Model 2

◮ ktot Dfk

Dt added no νttr to the k eq (Model 3)

Model 2 works very well Model 3 gives identical results to Model 2 (not shown in this presentation)

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19

Three-Day CFD Course at Chalmers

Unsteady Simulations for Industrial Flows: LES, DES, hybrid LES-RANS and URANS 29-31 October 2014 at Chalmers, Gothenburg, Sweden Max 16 participants 50% lectures and 50% workshops in front of a PC Registration deadline: 10 October 2014 For info, see http://www.tfd.chalmers.se/˜lada/cfdkurs/cfdkurs.html

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 17 / 19

References I

[1] Davidson, L. A new approach to treat the RANS-LES interface in PANS. In ETMM10: 10th International ERCOFTAC Symposium on Turbulence Modelling and Measurements (Marbella, Spain, 2014). [2] Girimaji, S. S., and Wallin, S. Closure modeling in bridging regions of variable-resolution (VR) turbulence computations. Journal of Turbulence 14, 1 (2013), 72 – 98. [3] Hamba, F. Analysis of filtered Navier-Stokes equation for hybrid RANS/LES simulation. Physics of Fluids A 23, 015108 (2011).

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 18 / 19

References II

[4] 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.

www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 19 / 19