Turbulence Modelling: Large

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Turbulence Modelling: Large
Eddy Simulation


Turbulence Modeling:


Large Eddy Simulation and Hybrid RANS/LES


Introduction



LES Sub
-
grid Models



Numerical aspect and Mesh



Boundary conditions



Hybrid approaches



Sample Results

Turbulence Structures

Introduction: LES / other Prediction Methods


Different approaches to make turbulence computationally tractable:


DNS: Direct Simulation.


RANS: Reynolds average (or time or ensemble)


LES: Spatially average (or filter)


DNS

3D, unsteady

RANS

Steady / unsteady


LES


3D, unsteady

RANS vs LES

RANS model1

RANS

model 2

LES

Why LES?


Some applications need explicit computation of accurate unsteady fields.



Bluff body aerodynamics


Aerodynamically generated noise (sound)


Fluid
-
structure interaction


Mixing


Combustion





LES : difficulties


Cpu expensive for atmospheric modelling:


Unsteady simulation


Turn around time is weeks/months (rans is hours or days)


Cannot afford grid independence testing



Still open research issues


combustion, acoustics, high Prandtl, shmidt mixing problem, uns.


Wall bounded flows


Need expertise to reduce cpu, human cost


Mesh


models


strategy


Analysis of the instantaneous flow


Knowledge about turbulent instabilities, turbulent structures




Energy Spectrum



Large eddies:


responsible for the transports of
momentum, energy, and other
scalars.



anisotropic, subjected to history
effects, are strongly dependent on
boundary conditions, which
makes their modeling difficult.





k,f


Small eddies


tend to be more isotropic and less
flow
-
dependent (universal), mainly
dissipative scales, which makes their
modeling easier.





E
u
,E
T

e

LES: Filtering
-

Decomposition



E
Energy spectrum against
the length scale













scale
subgrid
scale
resolved
t
t
t
,
,
,
x
u
x
u
x
u





t
,
x
u
f


2


t
,
x
u

Anisotropic

Flow dependent

Isotropic

Homogeneous

Universal

Filtering




The large or resolved scale field is a local average of the complete field.



e.g., in 1
-
D,






Where G(x,x’) is the filter kernel.



Exemple: “box filter” G(x,x’) = 1/D if |x
-
x’|<D/2, 0 otherwise

( ) (,') (')'
i
i
u x G x x u x dx


Filtered Navier
-
Stokes Equations

Filtering the original Navier
-
Stokes equations gives filtered Navier
-
Stokes equations
that are the governing equations in LES.
























j
i
j
i
j
j
i
i
x
u
x
x
p
x
u
u
t
u


1
Filter

j
ij
j
i
j
i
j
j
i
i
x
x
u
x
x
p
x
u
u
t
u




























1
j
i
j
i
ij
u
u
u
u



Needs modeling

Sub
-
grid scale (SGS) stress

N
-
S
equation

Filtered N
-
S
equation

Available SGS model


Subgrid stress : turbulent viscosity




Smagorinsky model (
Smagorinsky,
1963
)


Need ad
-
hoc near wall damping



Dynamic model (
Germano et al., 1991
)


Local adaptation of the
Smagorinsky constant





Dynamic subgrid kinetic energy
transport model (
Kim & Menon 2001
)


Robust constant calculation
procedure


Physical limitation of backscatter






2
t s
v C S
 


2
/
1
sgs
k
t
k
C
v
























j
sgs
k
sgs
j
sgs
j
i
ij
j
sgs
j
sgs
x
k
x
k
C
x
u
x
k
u
t
k



e
2
/
3
1
2
3
ij kk ij t ij
v S
  
  


2
t D
v C S
 
Smagorinsky’s Model


Hypothesis: local equilibrium of sub
-
grid scales


Simple algebraic (0
-
equation) model (similar to Prandtle Mixing
length model in RANS)








Cs
= 0.065 ~ 0.25


The major shortcoming is that there is no
Cs

universally applicable
to different types of flow.


Difficulty with transitional (laminar) flows.


An
ad hoc

damping is needed in near
-
wall region.


Turbulent viscosity is always positive so no possibility of
backscatter.



ij
ij
S
S
S
2
,
3
/
1




with



2
t s
v C S
 
Dynamic

Smagorinsky’s

Model


Based on the similarity concept and Germano’s identity (Germano et al.,
1991; Lilly, 1992)


Introduce a second filter, called the test filter with scale larger than the
grid filter scale


The model parameter (
Cs

) is automatically adjusted using the resolved
velocity field.


Overcomes the shortcomings of the Smagorinsky’s model.


Can handle transitional flows


The near
-
wall (damping) effects are accounted for.


Potential Instability

of the constant



Basic Idea : consider the same smagorinsky model at two different scales
, and adjust the
constant accordingly


Constant value = error minimization using least square method and Germano’s Identity

T
ij


ij

L
ij


Error minimization





Test Filter

Grid Filter


ij i j i j
L u u u u
 
ij
ij ij
E L T

  
Dynamic Smagorinsky’s Model

Dynamic Subgrid KE Transport Model


Kim and Menon (1997)


One
-
equation (for SGS kinetic energy) model










Like the dynamic Smagorinsky’s model, the model constants (
Ck
,
C
e
) are automatically adjusted on
-
the
-
fly using the resolved velocity
field.


Backscatter better accounted for


ij
sgs
k
ij
kk
ij
S
k
C




2
/
1
2
3
1



























j
sgs
k
sgs
j
sgs
j
i
ij
j
sgs
j
sgs
x
k
x
k
C
x
u
x
k
u
t
k



e
2
/
3
Mesh



Grid
resolution

:



Constraint Based on LES hypothesis:


Explicit Resolution of production mechanism (whereas production
is modeled with RANS)


Resolution of anisotropic and energetic large scales


Cell size must be included inside the inertial range, in between
the integral scale (L) and the Taylor micro
-
scale (l).


Integral scale L


Energy peaks at the integral scale. These scales must be resolved
(with several grid points).


Crude Estimation of L :


Use correlation (mixing layer, jet) for L


Perform RANS calculation and compute L = k3/2 / e


Energie E(k)

Dissipation D(k)

1/


1/L

1/


Mesh



Grid resolution:



Taylor micro
-
scale l:


Dissipation rate peaks at l.


Not necessary to resolve l but useful to define a lower bound for the cell
size.


Estimation of l ~ L ReL
-
1/2



( for an homogeneous and isotropic turbulence = 151/2 L ReL
-
1/2)




Temporal resolution: resolve characteristic time scale associated to the cell size
(ie CFL=U Dt / Dx <1).



As for the numerical scheme (minimization of numerical errors) use of hexa and
high
quality of mesh

(very small deformations) is recommended


Numerics: time step




Dt must be of the order (or even less for acoustic purpose) of the
characteristic time scale t corresponding to the smallest resolved scales.


As t~Dx/U , it correspond to approx CFL = 1 (Courant Dreidrich Levy
Number) (where U is the velocity scale of the flow)





Numerics: discretization scheme



Discretization scheme in space should minimize numerical dissipation


LES is much more sensitive to numerical diffusion than RANS


2nd Order Central Difference Scheme (CD or BCD) perform much better than high order
upwind scheme for momentum


A commonly used remedy is to blend CD and FOU.


With a fixed weight (
G

= 0.8), this blending scheme has been found to still introduce
considerable numerical diffusion


Bad idea!


Most ideally, we need a smart, solution
-
adaptive scheme that detects the wiggles on
-
the
-
fly
and suppress them selectively.

Boundary Conditions (LES)



Near
-
wall resolving



Near
-
wall modelling



Inlet Boundary Conditions





Inlet Boundary Conditions

Inlet Boundary conditions :





Laminar case:


Random noise is sufficient for transition


Turbulent case:









mean velcocity field
turbulent fluctuations
,,
i i i
u t U u t

 
x x x
Precursor domain


Realistic inlet turbulence


Cpu cost


not universal

Vortex Method


Coherent structures


preserving turbulence


Need to use realistic profiles (U, k,
e
)


otherwise risk to force the flow

Spectral synthesizer


No spatial coherence


Flexibility for inlet (profiles of full reynolds stress or k

e,

constant values,
correlation)


Boundary Conditions



Near Wall treatment



1/ Near Wall Resolving


All the near

wall turbulent structures are explicitly computed
down to the viscous sub
-
layer.



2/ Near Wall Modeling



All the near
-
wall turbulent structures are explicitly computed
down to a given y+ >1



3/ DES:


No turbulent structures are computed at all inside the entire
boundary layer (all B.L. is modeled with RANS).




Boundary Conditions


1/ Near wall resolving : explicit resolution of the boundary layer.



Motivation: separated flows, complex physics (turbulence control)


High resolution requirement due to the presence of (anisotropic) wall turbulent
structures: so called
streaks.



Necessary to resolved correctly these near wall production mechanisms





Boundary layer grid resolution :


y+<2


Dx+ ~ 50
-
150, Dz+ ~ 15
-
40





Not only wall normal constraints, but also
Span
-
wise and stream
-
wise
constraints

due to the streaky structures






Boundary Conditions


Near wall modeling :



Wall function:


Schumann/Grozbach:


Instantaneous wall shear stress at walls and instantaneous
tangential velocity in the wall adjacent cells are assumed to be in
phase.


Log law apply for mean velocity (necessary to perform acquisition)



Werner & Wengle (6.2):


Instantaneous wall shear stress at walls and instantaneous
tangential velocity in the wall adjacent cells are assumed to be in
phase.


Filtered Log law (power 1/7) applied to instantaneous quantities

Hybrid LES
-
URANS


Near Walls: URANS 1
-
equation model


Core region: LES 1
-
equation SGS model


Hybrid LES
-
URANS


Navier Stokes time averaged in the near
wall and filtered in the core region reads:


LES
-
URANS hybrid


Use 1
-
equation model in both LES and
URANS regions


LES region

RANS region

LES
-
URANS hybrid


Problems:


LES region is supplied with bad BC from the URANS
regions


The flow going from URANS to LES region has no
proper time or length scale of turbulence


Solution:


Add synthesized isotropic fluctuations as the source
term of the momentum equations at the LES
-
URANS
interface.

Inlet BC and forcing

35 M

65 M

Side Box (max)

8 mm

6 mm

Rear Box (max)

8 mm

6 mm

Nb Prism layer

5

5


Side box



Rear box


Volume mesh: Gambit & «

Sizing
Functions

» to control both growth
rate and cell size in specified box

LES of a realistic Car model exposed to Crosswind

Results:


Model

Exp

SST
k
-
w

LES
WALE
(35 M
cells)

LES
WALE
(65 M
cells)

RS
M

v
2
-
f









Drag

(SCx)

0,70

0,66

0,69

0,68

0,71

0,73

Side

(SCy)

2,22

2,00

2,19

2,18

2,30

2,10

Forces

Lift

(SCz)

1,40

1,66

1,27

1,30

1,82

1,77

Yawing

(SCn)

-
0,64

-
0,60

-
0,57

-
0,59

-
0,47

-
0,47

Rolling

(SCl)

-
0,42

-
0,36

-
0,49

-
0,49

-
0,46

-
0,41

Moments

Pitching

(SCm)

0,12

0,10

0,21

0,23

0,03

0,07










Simulation of flow over a 3D mountain

Comparision between RANS and LES
-
RANS hybrid model


RANS using SST model






Hybrid RANS
-
LES model



Conclusion


RANS/URANS is not always reliable.


LES is closer to reality than RANS/URANS.


LES is computationally very expensive.


In the absence of enough experimental data one is left
with no choice but to use LES wherever feasible.