Large Eddy Simulation

Turbulence Modellingk-ε Models- Short computational time- Simple, robust- Limited range of validity

Reynolds Stress Models, RSM- More general, still not universal- More complex, seven PDE:s- Extensive modeling- Longer computational time

Large Eddy Simulation, LES- Resolving large scales- Modeling small scale effects- Long computational time

Direct Numerical Simulation, DNS- No Model, resolves all scales - Very long computational time- Limited to low Re

Approximations

The conceptual steps

• Filtering• Govering equations including Sub-Grid

Scale (residual) stress tensor• Closure by modelling SGS stress

tensor• Numerical solution

∫= ')'()',( dxxuxxGu

Computational Grid

u u

• Spatial filtering rather than ensemble average

Large-Eddy Simulation for Turbulent Flows

Turbulent kinetic energy spectrum

(Kolmogorov theory for isotropic & homogenous turbulence)

log(

E(k)

)

log(k)

Dissipationsubrange

Inertialsubrange

Largescales

Universal range

-5/3

log(∆)

Filter cutoff

LES-spectrum

−∆

∆= rHrG

211)(

Box filter

∆

∆

=κ

κκ

21

21sin

)(Ĝ

Gaussian filter

∆

−∆

= 22

2/12

6exp)6()( rrGπ

∆−=

24exp)(ˆ

22κrG

Sharp spectral filter

( )rrrGππ ∆

=/sin)(

−∆∆

= κπκ HG 1)(ˆ

Filtering in 3D

{2/02//1

´)(3

∆>∆≤∆

=−i

i

xforxfor

zzG”Box” filter

−

∆−

∆=− 22

2/32 ´

6exp)6(´)( zzzzGπ

”Gaussian” filter

Filtered governing equations

0=i

i

xu∂∂

jj

i

ij

jii

xxu

xp

xuu

tu

∂∂∂ν

∂∂

ρ∂∂

∂∂

+−=+1

Mass:

Momentum

Now consider the non-linear term

Filtered governing equations

Momentum ( )j

ij

jj

i

iji

j

i

xxxu

xpuu

xtu

∂∂τ

∂∂∂ν

∂∂

ρ∂∂

∂∂

−+−=+1

jijiij uuuu −=τSub-grid stresses:

( ) ( )jijij

jijj

ji uuuux

uuxx

uu−+=

∂∂

∂∂

∂∂

Leonards decomposition

jijiij uuuu −=τTurbulent stresses

Leonard’s decomposition ( )( ) jiijjijijjiiji uuuuuuuuuuuuuu ′′+′+′+=′+′+=

ijijijjijiij RCLuuuu ++=−=τ

jijiij uuuuL −=

ijjiij uuuuC ′−′=

jiij uuR ′′=

Leonard tensor

Cross-term tensor

Reynolds stresses

Filtered governing equations

Momentum

j

ij

jj

i

ij

ij

i

xxxu

xp

xuu

tu

∂∂τ

∂∂∂ν

∂∂

ρ∂∂

∂∂

−+−=+1

Leonard stresses:

( ) ( ) ( )jijij

jij

jij

uuuux

uux

uux

−+=∂∂

∂∂

∂∂

jijiij uuuuL −=

( )j

ji

j

ijji

j xuu

xuuuu

x ∂∂

∂∂

∂∂

+=

jijiij uuuu −=τSub-grid stresses:

Germano’s decomposition

jijioij uuuuL −=

ijjiijjioij uuuuuuuuC ′−′−′−′=

jijioij uuuuR ′′−′′=

Leonard tensor

Cross-term tensor

Reynolds stresses

Interaction among resolved scales

Interaction between resolved and non-resolved scales

Interaction among non-resolved scales

Governing Equations

LES modelling of non-linearities:

“Convective”: a. Gradient diffusion type (Smagorinsky)b. Scale similarity typec. Dynamic typesd. Others

Examples for incompressible SGS models

Governing Equations - LES

The Space Filtered Governing Equations

Transport Equation for an Inert Additive

j

j

jjj

j

xxxcD

xcu

tc

∂∂

−∂∂

∂=∂

∂+∂∂ ϕρρ 2

Subgrid Scale Mixing )( cucu jjj −= ρϕ

A note on kinetic energy

rfijijijjij

fj

f PpSuxx

Eu

tE

−−=

−−

∂∂

−∂∂

+∂∂

εδρ

τν2

ijijf SSνε 2=

ijijr SP τ−=

iiuuE 21

=

rf kEE +=

iif uuE 21

=

Filtered kinetic energy

Decomposition

Kinetic energy of the resolved scales

Small if η>>∆

Can be negative

Subgrid Scale Models

What do they do?

1. Account for the effects of small scales on the resolved ones

1. Dissipate the energy transferred to the smallest scales

Subgrid Scale Models• Subgrid Scale Stress Tensor

- No explicit model- Smagorinsky (“Eddy diffusivity”)- Stress Similarity Model- Dynamic Divergence Model- Exact Differential Model

• Subgrid Scale Mixing- No Explicit Model- Eddy Diffusivity type- Scale Similarity Model - Exact Differential Model

Subgrid Scale ModelsSubgrid Scale Stress Tensor

Smagorinsky

SSM

DDM

EDM

)( jijiij uuuu −= ρτ

( )jijiLkkijij uuuuC −−= ρτδτ 31

jijm

jkkij SSSC ,2

1

klkl2)(

,jij, ) 2(23

1 ∆−

= ρτδτ

)(qfq =

Liu, S., Meneveau, C. And Katz, J., J. Fluid Mech., 1994.

Held, J and Fuchs, L., AIAA paper, 97-1931, 1997.

Fuchs, L., Kluwer Academic Publishers, 1996.

)(21;)(

31 2/1

i

j

j

iijijklklLkkijij x

uxuSSSSC

∂∂

+∂∂

=−= ρτδτ

SGS Model

4

43

3

1

||41

j

ij

j

jxuxu

∂∂

∆− ∑=

ρ

No explicit model: an example

The largest term in the truncation error of the third order upwind scheme proposed by Kawamura and Kuwahara

Kawamura, T. and Kuwahara, K., “Computation of High Reynolds Number Flow around a Circular Cylinder with Surface Roughness”, AIAA-84-0340, 1984.

IMM

Advantages

+ Simple and fast, (no explicit SGS Model used) + Correct asymptotic behavior (h->0, LES->DNS)

Disadvantages

- Not physically related

SGS Model

)(21;)(

31 2/1

i

j

j

iijijklklLkkijij x

uxuSSSSC

∂∂

+∂∂

=−= ρτδτ

Smagorinsky

Momentum:

Scalar

jcj x

CA∂∂

= ρϕ

)( cucu jjj −= ρϕ

)( jijiij uuuu −= ρτ

Smagorinsky

Advantages

+ Simple + Dissipative

Disadvantages

- A “free” model parameter: Not universal- Does not account for “back-scatter”- Near wall treatment

SGS Model

( )131

=

−−=

L

jijiLkkijij

C

uuuuCρτδτ

The Stress Similarity Model (SSM)

The Liu et al. type of model

(SSMC)

1)(

=

−=

c

jjcj

CcucuCρϕ

SSM

Advantages

+ Simple + Correct asymptotic behavior in the inertial subrange+ Accounts for some “back-scatter”

Disadvantages

- A “free” model parameter - Requires an additional filter- Not absolutely dissipative

SGS ModelThe Dynamic Smagorinsky Model (Germano’s model)

) 2(2 31 2

1

klkl2

ijij ijijkkij SSSC ∆−=+

= ραατδτ

S)2(2 31

ij21

2ij klklijkkijij SSCTT ∆−=+

= ρββδ

mnmnklkl

ijijg

jijiijijij

SSSL

C

uuuuTL

αβ

τ

−=

−=−= )(

Sub-grid scale stresses:

Sub-test scale stresses:

Germano’s identity

ijklklgkkijij SSSC2/12 )(

31

∆−= ρτδτ

StressTurbulent Resolved = Lij

SGS ModelThe Dynamic Smagorinsky Model (Germano’s model)

mnmnklkl

ijijg

jijiijijij

SSSL

C

uuuuTL

αβ

τ

−=

−=−= )(Germano’s identity

StressTurbulent Resolved = Lij

Gives 6 values of the model parameter, choose by minimizing the error:

( )( )2ijijij CCLQ αβ −−=

( )( )( )mnmnmnmn

ijijijg

LC

αβαβαβ

−−−

=

Germano’s model

Advantages

+ No “free” model parameter+ Correct asymptotic behavior (h->0, LES->DNS) Disadvantages

- Requires an additional filter- May require limiting of the model parameters- Isotropic coefficient

SGS ModelThe Dynamic Divergence Model (DDM)

jijj

kkij C ,,

jij, 31 ατδτ +

=

31

,,

, jijj

kkijjij CTT βδ +

=

( )

lilkik

jiji

jjijijijjijjij

LC

uuuuTL

,,

,

,,,, )(

αβ

τ

−=

−=−=

Sub-grid scale stresses:

Sub-test scale stresses:

One model parameter for each direction

Model directly the divergence of the SGS stresses:j

ijjij x∂

∂=

ττ ,

SGS ModelThe Dynamic Divergence Model (DDM)

Bounding of the model parameter

Total dissipation must be positive 0≥++ numSGSvisc εεε

( )

∂∂

+

∆

=≥

j

iijklkl

i

xuS

h

SSCC

3

21

2min

)2(2

1 ν

( )

−

∆∆

=≤ νt

h

SSCC

klkl

i2

21

2max

)2(2

1

DDM

Advantages

+ No “free” model parameter+ Correct asymptotic behavior (h->0, LES->DNS) + Non-isotropic coefficients

Disadvantages

- Requires an additional filter- May require limiting of the model parameters

SGS ModelKinetic energy SGS model

Subgrid kinetic energy: ( )kkkksgs uuuuk −= 21

The SGS-stresses: ijsgskijsgsij SkCk

T

ν

δτ 2/1)(232

∆−=−

∂∂

∂∂

+∆

−∂∂

−=∂∂

+∂

∂

j

sgs

k

T

j

sgs

j

iij

j

sgsj

sgs

xk

xk

Cxu

xk

ut

kσντ ε

2/3

Accounts for the transport of SGS turbulent kinetic energy

SGS Model

The Exact Differential Model (EDM)

∫ −= ξξξ dqxGq )()(

We look for filters (G) such that the “error” of filtering becomes:

qDPqq )(−=−

Where P(D) is a polynomial of differentials.

(1)

(2)

SGS Model

Fourier transform ( ) relations (1) and (2):

qikPqqandqGq ˆ)(ˆˆˆˆˆ −=−=

We look for filters (G) such that the “error” of filtering becomes:

*̂

)(11ˆ

ikPG

−=

For realizability require that P is an even polynomial.

To lowest order (P(D)= ∆) one gets:

SGS Model

~

~

~~

~

~ tu~

=q~ where ~

2

22i2

j,ij

2

22222

−+∆+++∆−=

=∇∆−=∇∆−=

j

i

ij

ji

j

ji

j

ji

j

xu

xp

xuu

xuu

xuu

xqqqqqqq

∂∂

µ∂∂

∂ρ∂

∂ρ∂

∂ρ∂

∂ρ∂

τ

∂∂

The Exact Differential Model (EDM)

An additional term enters into the continuity equation

0~

2 =∂∂

∆−∂∂

k

k

i

i

xu

xu

EDMAdvantages

+ Exact expression for the SGS term + No model parameter+ Enhanced understanding of SGS roles + Can estimate the contribution of each term + Correct asymptotic behavior (h/D ->0 for all flows!

h = cell size and D = filter width)

Disadvantages

- Additional (model dependent) boundary conditions- Boundary conditions have to be computed implicitly - Direct use may be numerically unstable

Issues: Inflow/outflow boundary conditions

Handling walls and near wall effects

Accuracy

Modelling of Turbulent Flows

Model comparisonsImpinging jet

Impinging Jet

Impinging Jet

Impinging Jet

EXAMPLES

LES of turbulent flows

Results

Concentration

Velocity

Area shown in the animation

Results

Instantaneous velocity field in the wall jet.

Instantaneous concentration field.

Impinging jets

Impinging Jets: Scalar Mode 0,1,2

Largest variance (instead of energy)

Asymmetric

Impinging Jets

Mode 0,1,2 (the most energetic)

Coherent vortices

T

vu

vu

R

=

Vector valued u:

49

Unsteady Impinging Jet

Basic case H/D=2, S=0, Re=20000, LES

x

y

z

Free jet region

Wall jet region

Stagnation region

50

Centreline (x/D=0) frequency spectra u-vel.

Unsteady Impinging Jet

(-): y/D=0.004

(--): y/D=1

St2

St1

• Vortex shedding at x/D=0.5

• St1: fundamental freq.

• St2: sub-harmonic freq

• Spectra normalized with local RMS and energy levels

• Initially inviscid instabilities (inflection point, laminar flow + perturbations)

• Rolling up of vortices, growing downstream

• Vortex shedding, vortex pairing

LES Case1 (H/D=2, Re=20000, S=0)

Space/frequency relation

St2 St1 St2

St1

Energy levels of St1 and St2Dominant frequencies at x/D=0.5

y / D

St E

Thomas Hällqvist

LES (H/D=2, Re=20000, S=0,1)

Space/frequency relation

St2 St1

Dominant frequencies at x/D=0.5

y / D

Thomas Hällqvist

Bildnummer 1Turbulence Modelling The conceptual stepsBildnummer 4Bildnummer 5Bildnummer 6Bildnummer 7Bildnummer 8Bildnummer 9Bildnummer 10Bildnummer 11Bildnummer 12Bildnummer 13Bildnummer 14Bildnummer 15Bildnummer 16Bildnummer 17Bildnummer 18Bildnummer 19Bildnummer 20Bildnummer 21Bildnummer 22Bildnummer 23Bildnummer 24Bildnummer 25Bildnummer 26Bildnummer 27Bildnummer 28Bildnummer 29Bildnummer 30Bildnummer 31Bildnummer 32Bildnummer 33Bildnummer 34Bildnummer 35Bildnummer 36Bildnummer 37Modelling of Turbulent FlowsModel comparisonsImpinging JetImpinging JetImpinging JetBildnummer 43ResultsResultsBildnummer 46Bildnummer 47Bildnummer 48Bildnummer 49Bildnummer 50Bildnummer 51Bildnummer 52