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.
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