# Turbulence Modelling: Large

Μηχανική

22 Φεβ 2014 (πριν από 4 χρόνια και 3 μήνες)

69 εμφανίσεις

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

RANS

LES

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:

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
)

-
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

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

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.