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Feb 22, 2014 (3 years and 1 month ago)

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R
ECONSTRUCTION OF
T
URBULENT
F
LUCTU
A
TIONS FOR
H
YBRID
RANS/LES

S
IMULATIONS
U
SING A
S
YNTHETIC
-
E
DDY
M
ETHOD

N. Jarrin
1
, R. Prosser
1
, J. Uribe
1
, S. Benhamadouch
2

and D. Laurence
1,2


1
School of MACE, the University of Manchester, M601QD, UK

2
EDF R&D, 6 Quai Wat
ier, Chatou, France


dominique.lau
rence@manchester
.ac.uk


Abstract

A

coupling methodology between an upstream
RANS simulation and a LES fu
r
ther downstream is
presented. The focus of this work is on the RANS
-
to
-
LES inte
r
face inside an attached turbulent
boundary layer, where an unsteady LES content has
to be explicitly generated from a steady RANS sol
u-
tion. The performance of the Sy
n
thetic
-
Eddy Met
h-
od (SEM), which generates realistic sy
n
thetic eddies
at the inflow of the LES, is investigated on a wide
var
iety of turbulent flows, from simple cha
n
nel and
duct flows to the flow over an airfoil trailing edge.

The SEM is compared to other existing met
h
ods of
generation of synthetic turbulence, and is shown to
reduce substantially the distance required to d
e
velo
p
realistic turbulence.



1

Introduction

The large number of grid points required to pe
r-
form LES at high Reynolds numbers, complex g
e-
ometries and large domains is the main obstacle to
the application of LES to flows of industrial rel
e-
vance.

In the aeron
autical or automotive industry, eng
i-
neers are interested in LES because it provides an
unsteady turbulent flow field which allows to co
m-
pute the aeroacoustic noise generated by the vehicle
or the airfoil. In practise only a small specific r
e-
gion of interes
t such as the trailing edge of an airfoil
or the rear view mirror of a car is needs to be co
m-
puted with LES. The specification of the upstream
flow conditions for the embedded LES domain r
e-
quires the simulation of the whole g
e
ometry, which
can be achieved
using RANS techniques at a rel
a-
tively cheap computational cost. The challenge is
then
to generate a mature unsteady turbulent LES
solution from a steady RANS solution within as
short a di
s
tance as possible in order to achieve both
a reduction of the total
computational cost of the
simulation by limiting the size of the embedded
LES domain, and a better accuracy of the simulation
by u
s
ing a LES model in the region of interest.


The present investigation thus focuses on the
RANS
-
to
-
LES interface, for what is
often referred
to in the literature as zonal hybrid RANS
-
LES
methods, where LES and RANS regions use sep
a-
rate domains. In this case unsteady turbulent velo
c
i-
ty fluctuations must be explicitly reconstructed and
prescribed at the inflow of the LES region fro
m a
steady upstream RANS solution.

There exists an overwhelming variety of met
h-
ods available to generate inflow boundary cond
i-
tions for LES

(see Keating et al. (2004)

for a r
e-
view
)
.

Although rec
y
cling methods as in Lund et al.
(1998) produce very realisti
c inflow data, they i
n-
crease the cost of the computation and lack gene
r-
a
l
i
ty to be employed in complex industrial applic
a-
tions. Synthetic turbulence generation methods pr
o-
vide an alternative, even though they yield a trans
i-
tion region downstream of the inl
et where the sy
n-
thetic fluctuations imposed at the inlet evolve t
o-
wards real turbulence (Keating et
al., 2004
). In
Keating et al. (2006), synthetic turbulence was su
c-
cessfully used in parallel with a controlled body
force to generate inflow conditions for
hybrid
RANS
-
LES
simulations of non
-
equilibrium boun
d
a-
ry layers.

In this paper, the Synthetic
-
Eddy Method of Ja
r-
rin et al. (2006) is used to generate fluctuations at
the RANS
-
to
-
LES interface of several wall flows.
All of the input parameters of the method

are calc
u-
lated

using only statistical data that is

available
from the upstream RANS simula
tion
.
The SEM is
compared to (and found to perform better than) ot
h-
er existing methods of generation of synthetic tu
r-
b
u
lence. Cases simulated include simple channel
and duct flows, and the more cha
llenging case

of
the turbulent flow over an airfoil trailing edge.



2

Methodology

The governing equations are the incompressible
Navier
-
Stokes equations, filtered (in the LES r
e-
gion) or time averaged (in the RANS region).
In
both regions an eddy viscosity mode is used to close
the equations. The SST model of Menter et al.
(1993) is used in the RANS region, while the sta
n
d-
ard Smagorinsky model with
C
S

= 0.065

and Van
-
Driest damping at the wall is used in the LES r
e-
gion. The
RANS and LES equations are solved with
the collocated finite volume code
Code_Saturne

(Archambeau et al., 2004).



The LES and the RANS simulation are run on
two different domains which overlap so that the
RANS region can provide statistics to the inlet
bou
ndary faces of the LES domain.

The statistics
available from the upstream SST solution are the
mean velocity
U
i
, the Reynolds stress tensor
R
ij
, and
the dissipation rate per unit of kinetic energy
ω
.

The performance of s
everal methods of gener
a-
tion
of infl
ow conditions for LES is investigated.
The simplest

method, referred to as the random
method, generates uncorrelated random
numbers for
each component

of the velocity at each point of the
inlet mesh,
and
at each iteration
. The random nu
m-
bers are then trans
formed using the Cholesky d
e-
composition of the Reynolds stress tensor to create
one
-
point cross
-
correlations between the velocity
components
(
see Appendix B of Lund et al. (1998)
)
.
The method of Batten et al. (2004)
involves the
summation of sines and cosi
nes with random ampl
i-
tudes and phases
.
In all simulations presented here
we

used 20
0
0 random modes. Further details of the
method can be found in Batten
et al.
(2004).

The

main focus of this paper is on

the applic
a-
tion
s

of the
SEM

to RANS
-
LES

coupling. In
Jarrin
et al. (2006)
, the input parameters of the SEM were
derived from a precursor LES or from
ad
-
hoc

fo
r-
mulae. In this paper, all of the input parameters of
the method are calculated

using only
statistics
available from the upstream SST simula
tion
.

The
LES inflow plane
on which we want to ge
n-
erate

synthetic velocity fluctu
a
tions with the SEM

is
a finite set

of points
S
=
{
x
1
,
x
2
,
∙∙∙
,
x
s
}
.

The first step
is to create a box of eddies
B

surrounding
S

which is
going to contain the synthetic eddies. It
s minim
um
and maximum coordinates are

defined by


where
σ

is

a characteristic length scale of the
flow
whose computations from RANS statistics will
be detailed later.

In order to ensure that the density
of eddies i
n
side of the box of eddies is constant, the
number of eddies is set as
N

= max( V
B

/

σ
3

)
, where
V
B

is the volume of the box of eddies
.

The SEM decomposes a turbulent flow field in a
finite sum of eddies.
The

velocity fluctuations ge
n-
erated by
N

eddies have the representation


where the
x
k

are the locations of

the eddies
,

the
ε
k
j

are their respective

intensities and
a
ij

is the Chol
e-
sky decomposition of the Reynolds stress tensor
.

f
σ
(
x

x
k
)

is the velocity distribution of the eddy l
o-
cated at
x
k

. We assume that the differences in the
distributions

between the eddies depend only on the
length scale
σ

and

define
f
σ

by


In all our simulation
f

is a simple tent fun
c
tion,


and
σ

is a parameter that controls the size of the
structures. It is taken as


where

Δ

= max(
Δ
x,
Δ
y,
Δ
z
)
,
ε

= C
μ

k

ω

is the rate of
dissipation, and
δ

is the thickness of the boundary
lay
er considered.
The position
x
k

and the intensity
ε
k
j

o
f each eddy are independent random variables.

At the first iteration,

x
k

is

taken from a uniform di
s-
tribution over the box of eddies
B

and
ε
k
j

=
±
1,
with
equal prob
a
bility to take one value or the other
.


T
he eddies are convected through

the box of e
d-
dies
B

with a constant velocity
U
c

characteristic of
the flow. In our case it is straight forward to co
m-
pute
U
c

as the a
v
eraged mean

RANS

velocity over
the

LES inflow plane
. At each iteration, the new p
o-
siti
on of eddy
k

is given by


where
dt

is the time step of the simulation. If an e
d-
dy

k

is convected out of the box through face
F

of

B
, then it is immediately regenerated randomly on
the inlet face of
B

facing
F

with a new independent
random intensity vecto
r
ε
k
j

still taken from the same
distribution.

The method generates a stochastic signal with

prescribed mean velocity, Reynolds stresses, and
length and time scale distributions.

Although the
SEM involves the summation of a large number of
eddies for each
grid point on the inflow
,

the CPU

time required to generate the inflow data at each i
t-
eration did not exceed 1% of the total CPU time per

iteration of the LES simulation.



3

Results


3.1 Spatially developing channel flow


Hybrid RANS
-
LES simulations of th
e turbul
ent
flow in a plane channel are

pe
r
formed at Re
τ

= 395.
The RANS equations are solved

for

x
/
δ

<

0
.

The
RANS grid
is one
-
dimensional, and
only uses one
cell
with periodic boundary conditions
in the
streamwi
se and spanwise directions.

At
x
/
δ

=
0

the
LES domain, of dimensions
10
πδ

×
2
δ

×

πδ
,

begins.
The g
rid spacing in wall units are
Δx
+



50
,

Δz
+



15
,

Δ
y
+

=
2

at the wall and
Δ
y
=
0
.
1
δ
.

T
he wall
-
normal

grid resolution
is the same as in the RANS
and in the

LES to avoid interpolation of the RANS
data onto the

LES grid.

Several methods of gener
a-
tion of inflo
w conditions for LES are tested and the
simulations pe
r
formed are summarized now.








Figure 1
: Velocity vectors of LES inlet conditions

for hybrid simulations of channel flow. From top to
bottom: precursor LES, SEM, Ba
t
ten's method and
random method
.


A baseline simulation was performed as a co
m-
parison point for all other cases (run P1). Time s
e-
ries of

instantaneous velocity planes were extracted
from a periodic LES (
performed on a shorter d
o-
main but with the same grid refinement and

the
same numer
ical options)
and

imposed at the inlet of
the
present
LES domain. In al
l other simulations,

methods of generation of synthetic turbulence are
used to prescribe

inlet conditions for the LES r
e-
gion. Three hybrid calculations were conducted, u
s-
ing the SEM (ru
n S1),
Batten

s method
(run B1)
and
the random method
(run R1).

Figure 1

shows instantaneous velocity
fluctuations
prescribed at the inlet of the LES domain
.
Although
t
he SEM does not reproduce completely the co
m-
plex structure of the near
-
wall turbulence

o
bserved
in the periodic LES
,

t
he length

scale and the magn
i-
tude of the fluctuations are realistically reproduced
by the SEM.
The velocity fluctuations generated u
s-
ing the method of Batte
n et al. (2004)

exhibit su
r-
prising features. In the near
-
wall region,
the fluctu
a-
tions seem to be un
correlated in

space. In the centre

the fluctuations are correlated in the spanwise dire
c-
tion but seem decorrelated in the wall
-
normal dire
c-
tion.

The reason for these phenomena is the deco
m-
position into Fourier modes used in Ba
tten’s met
h-
od.

The
freq
uencies and wavelengths of the

c
o
sine
and sine functions are allowed to vary in the d
i
re
c-
tion of non
-
homogeneity of the flow (in the pr
e
sent
case the wall
-
normal direction).

The veloc
i
ties at
two points separated even by an infinites
imal di
s-
tance in the wall
-
normal d
i
rection will thus oscillate
at different frequencies, and therefore be completely
decorrelated from each other.

In the direction of
homogeneity of the flow however (the spanwise d
i-
rection in the present case), this proble
m does not
occur since the frequencies and wavelengths are
constant. Thus although the method of Batten et al.
(2004) might appear to be capable of generating
non
-
homogeneous turbulence by allowing the wav
e-
lengths to vary in space, it does so at the expens
e of
destroying the spatial correlations in the non
-
homogeneous directions.


The development of the prescribed fluctuations
downstream of the inlet
will now be studied. Figure
2

shows the downstream development of the coeff
i-
cient of friction.

The horizonta
l dashed line repr
e-
sents the value of the coefficient of friction in the
periodic LES and will be used as a reference point
for the present RANS
-
LES simulations.

Run P1 has
a coefficient of friction in very good agreement with
the periodic LES over the who
le domain.

As e
x-
pected, a
ll three of the other simulations using sy
n-
thetic turbulence exhibit a transient downstream of
the inlet.

When the random method is

used
, the c
o-
efficient of friction drops continuously downstream
of the inlet
, which

indicates that
the flow lam
i
nari
z-
es.

The decay of the coefficient of friction is also
quite important downstream of the inlet when the
method of Batten et al. (2004) is used. However the
coefficient of friction reaches a minimum after
about 8
δ
,

before slowly recovering t
owards its fully
developed value about 25
δ

dow
n
stream of the inlet.
With the SEM, the coefficient of friction decays
downstream of the inlet to reach a minimum about
3
δ

downstream of the inlet (where it has only lost
15% of its initial value), and recovers

its fully d
e-


veloped value only after 10
δ

dow
n
stream of the i
n-
let.



Figure 2
: Coefficient of friction

for hybrid simul
a
tions
of channel flow at

Re
τ

= 395
. Inflow conditions are
generate
d using a precursor LES (
О
), the SEM (___ ),
Batten's method ( _ _ _

) and the ra
n
dom method( .... ).


The performance of the SEM is now tested a
t
two higher Reynolds number (
Re
τ
=
590

and

Re
τ
=
950
)
. Different grids are used for each Re
y
n-
olds number, but the grid refinement in wall units
always satisfies the constraints
Δx
+

≤ 50
,
Δz
+
≤ 15
and

Δ
y
+

=
2

at the wall.

Figure 3

shows that the d
e-
velopment of the coefficient of friction downstream
of the
inlet is the same for the three Reynolds nu
m-
bers
considered

when expressed as a function of
x u
τ

/
υ
.
A
nalysis

of other flow stati
stics not presented in
this paper confirm

that in the near
-
wall region,
the
length of the transition region

scales
approximately
as
x
+
~

3
,
000.


3.2 Spatially developing duct

flow


The SEM is now compared to Batten

s method
and to the random method in th
e case of a turbulent
flow through a square duct at
Re
τ

= 600

(Huser and
Biringen, 1993)
. The computational set
-
up is similar
to the one used in the case of the channel flow. The
RANS domain is positioned upstream of the LES
domain. The upstream SST simulation uses periodic
boundary conditions in the streamwi
se direction.
As
expected the

SST solution does not exhibit any se
c-
ondary motion. The ability of the SEM
, the method
of Batten et al. (2004), and the random method to
yield,

after a short development distance
,

a seco
n
d-
ary motion in the LES region

is invest
igated
.

The
topology of the mean flow is

studied

in a
cross
-
section

at

x/D=15

(roughly x
+

= 9,000) dow
n-
stream of the RANS
-
to
-
LES interface. T
he

simul
a-
tion using the SEM exhibits two mean streamwise
counter
-
rotating vortices in the corner of the

duct,
as sh
own in Figure 4

(a)
.

Their

centre location and
topology are in very good agreement with those
from the

reference fully developed LES.
Due to the

action of the secondary motion, momentum is co
n-
vected from the central region to the w
alls along the
corner bis
ectors, and
the mean streamwise

velocity

distribution (see Figure 5

(a)
) is in excellent agre
e-
ment with the one from the reference LES
.



Figure 3
: Development of the coefficient of fri
c
tion
(normalized by the coefficient of fiction o
b
tained in
the perio
dic LES)

as a function x u
τ

/
υ

for hybrid
simulations of channel flow at

Re
τ

=
395

(

)
,

Re
τ

=
590

(

), and

Re
τ

= 950
(
О
)
.


T
he simulati
on using the random method

does not
exhibit any secondary

motion (see Figure 5 (c)
).
With Batten’s method, two very weak
streamwise
corner vortices can be

observed, but their weak i
n-
tensity does not alter the mean streamwise velocity
distribution in the correct

manner as shown in Fi
g-
ure 5

(b).
We saw that Batten’s method destroys sp
a-
tial velocity correlations in the directio
n of non
-
homogeneity of the flow. In the present case, the u
p-
stream
k

and
Ω

profiles extracted from the SST sol
u-
tion and transmitted to Batten’s method are non h
o-
mogeneous in the two transverse directions. Cons
e-
quently Batten’s method does not generate any two
-
point velocity correlations in the inlet plane. The be
t-
ter results ob
tained than when using the random
method can be explained by the better time correl
a-
tion of the inflow data generated using Batten’s
m
e
thod.





Figure 4
:
Transverse velocity vectors at
x
/
D
=
15
for hybrid simulations of square duct flow with (a)
the SEM,
(b) Batten’s
method, (c) the random met
h-
od and (d) the reference periodic LES.



Figure 5
:

Mean streamwise velocity

distribution
normalized by bulk velocity

at
x
/
D
=
15 for hybrid
simulations
with (a) the SEM, (b) Batten’s method,
(c) the random method an
d (d) the reference per
i
o
d-
ic LES. Contours lines are evenly space between
0
.
3, 0
.
4,
...
, 1
.
2.


3.3 Turbulent flow over an airfoil trailing edge


The airfoil

considered is a two
-
dimensional flat
strut with a circular leading edge and an asymmetric
beveled

trailing

edge with a 25
o

tip angle
.

The g
e-
ometry of the airfoil and the flow conditions are d
e-
scribed
in details
by Blake (1975) and by

Wang and
Moin (2000).


Figure 6
:

Sketch of the hybrid RANS
-
LES simul
a-
tions of the airfoil trailing edge.


As shown i
n Figure 6
, the RANS domain e
n-
closes the entire airfoil and only the rear part of the
trailing edge and the near wake are simulated with
LES (the non
-
equilibrium region of the flow). The
RANS simulation is conducted on a C
-
grid d
o
main
using only 0.1M cells
. The LES domain begins at
x

/

h

=

-
4

and
x

/

h

=

-
2

on the low
-

and high
-
pressure
side of the airfoil, respectively
.

For the LES mesh,
644 cells are uniformly distributed along the upper
su
r
face, and 384 along the

lower surface.
This gives
a grid spacing
in wall units at the inlet of
Δ
x
+
=
41

and
Δ
x
+
=
34

on the upper and lower surfaces, r
e-
spectively. 150 cells are non
-
uniformly distributed
along the wake line. The wall
-
normal grid spacing
increases as the upper and lower walls are a
p-
proached. 64 cells are

used and the near
-
wall grid
spacing is at a min
i
mum at the walls, with
Δ
y
+
~

2
.

In the spanwise direction, 64 cells are uniformly
distributed. The grid spacing in wall units on the
upper surface is around 26 at the inlet of the LES
domain. In total,
the L
ES mesh
has about
3.0
×
10
6

cells.

At the inlet plane of the LES domain,
data are
extracted from the SST

solution, i
n
terpolated onto
the LES grid, and used for the generation of sy
n-
the
t
ic turbulence.

Results on the embedded LES
domain using inflow data gene
rated with the SEM,
Batten

s method and the random method are co
m-
pared

with the finely resolved

LES of Wang and
Moin (2000).

Profiles of mean velocity magnitude

(U
2
+V
2
)
1/2

and streamwise velocity fluctu
a
tions
u’

on the low
-
pressure side of t
he airfoil are
shown in Figure 7
.
The hybrid simulations using the random method
and Batten’s method initially laminarize (as e
x-
pe
cted from previous
computations), and cons
e-
quently show very early separation at
x
/
h
=

2
.
4

and
x
/
h
=

2
.
1
, respectively. When the SEM is use
d
the
mean v
e
locity
profiles are in much better
agreement with the reference LES. The hybrid si
m-
ulation using the SEM detaches at
x
/
h
=

0
.
75
,
slightly after the reference LES (
x
/
h
=

1
.
17
).

The early detachment
observed with the random
method and with B
atten

s method
is caused by the
lack of coherence of the prescribed inlet fluctu
a-
tions
. As a result there is a lack of near
-
wall turb
u-
lent structures close to the inlet, and the flu
c
tuations

are underestimated at the first station. Further
downstream the p
resence of a large separation bu
b-
ble in run B3 and R3 produces larger levels of flu
c-


tuations in the recirculation region (see F
igure 7

at
x
/
h
=

1
.
125

and at
x
/
h
=

0
.
625
).

The effect of the inflow data on the turbulent
structures will now be described. Streamwise velo
c-
ity

fluctuations along the upper surface of the ai
rfoil
are shown in Figure 8
. The simulation using Ba
t-
ten’s

method
-

although le
ading to early separ
a
tion
and weak magnitude fluctuations in the near wall
region

-

still shows features similar to the simul
a-
tion using the SEM: the weak near
-
wall streaks are
elongated

in the streamwise direction (due to the f
a-
vorable pressure gradient e
xperienced by the
boundary layer),

followed by a rapid transition t
o-
wards a more turbulent state (after the removal of
the pressure gradient),

before finally separating
from the wall. With the random method, no turb
u-
lent structures are present

in the near
-
wall region of
boundary layer downstream of the inlet, which also
leads to early separation. Ho
w
ever in this case, the
separation is laminar and leads to the formation of
large scale two
-
dimensional Kelvin
-
Helmotz vo
r
t
i-
ces in the subsequent shear layer.

Fi
nally Figure 9

shows the frequency spectrum
of the
v
-
fluctuations at
x
/
h=4

downstream of the
trailing edge. A strong peak around
f h
/
U
0
=
0
.
6

can
be observed with the random method and Batten’s
method, indicating the presence in the flow of Ke
l-
vin
-
Helmotz
vortices shedding in the wake of the
airfoil. On the contrary the frequency spectrum in
the case of the SEM does not exhibit any peak, in
agreement with observations of instantaneous flu
c-
tuations in the near
-
wake which did not e
x
hibit any
clear vortex shed
ding. This is the physical beha
v-
io
u
r of the flow observed in the reference LES of
Wang and Moin (2000).



Figure 7
:
Profiles of (a) the mean velocity magn
i-
tude and (b) the rms streamwise velocity fluctu
a-
tions
normalized by the edge velocity as a function

of vertical distance from the upper su
r
face, at
x
/
h
=

3
.
125,

2
.
125,

1
.
625,

1
.
125,

0
.
625:

___

, SEM;
_ _ _

, Batten et al. (2004);


...
, random method;

О
, LES Wang and Moin (2000)




Figure 8
:
Streamwise velocity fluctuations in a
plane parallel to th
e wall at
y
+
=
1 fo
r hybrid
sim
u
lations of the trailing edge flow
. From top to
bottom: SEM,

method of

Batten et al. (2004) and

random method.



Figure 9
:
Frequency spectrum of the
v
fluctu
a
tions
in the near wake at
x
/
h
=
4 and
y
/
h
=
0
.
5: ,
SEM; ,
Batten

et al. (2004); and , random method.


3

Concl
u
sions

The SEM was

used to generate i
nlet conditions
for a LES

using only information available

from an
upstream SST

simulation. This hybrid RANS
-
LES
coupling stra
t
egy was

first

tested
in the case of
channel an
d duct flows.
The SEM

was systemat
i
ca
l-
ly compared to other existing methods of gener
a
tion
of synthetic turbulence; the random

method and the
method of Batten et al. (2004). With the SEM, the
develo
p
ment length of the eddies in

the near wall
region was show
n to be approximately 3
,
000 wall
units
in both

cases simulated. This

offers si
g
nificant
promise for the a
p
plication of the method to high
Reynolds number flows of engineering

inte
r
est.


With the random method, the velocity fluctu
a
tions
prescribed at the inl
et were immediately

diss
i
pated
and the flow became laminar. With Batten’s met
h-
od, the use of Fo
u
rier harmonics with spatially

var
y
ing wavelengths leads to a destruction of the
spatial correlations of the signal in the direction of

non
-
homogeneity of the fl
ow.

Finally hybrid simulations

of

the flow
over an
airfoil trailing edge

were pe
r
formed. With the

SEM,
realistic turbulence is generated upstream of the
separation, and thus flow predictions downstream

of
the separation are in good agreement with the refe
r-
ence data.
T
he inlet was positioned only

3 boundary
layer thicknesses u
p
stream of the location where the
boundary layer experiences maximum

acceleration,
this without significant alteration of the results. The

length of th
e LES inlet sections used

did not
allow
either
the random
method

or Batten’s method

to
generate a

reali
s
tic boundary layer upstream of the
regi
on of interest. T
he lack of turbulent stru
c
tures

in
the upstream boundary layer lead to
an early separ
a-
tion

and hence a higher

production of turbul
ent k
i-
netic energy; leading to the growth (after separ
a-
tion) of quasi two
-
dimensional

structures characte
r-
istic of transitional flows.

The

use of small

LES domains (without signif
i-
cant loss of accuracy

compared to the reference d
a-
ta
) in the hybrid simulat
ions using

the SEM led to
substantial sa
v
ings i
n terms of number of cells used.
T
he reduction in terms of CPU

time achieved with
the present

LES

domain is over 40% when co
m-
pared to the domain used in the reference LES of
(Wang and Moin, 2000),

and over 80%

when co
m-
pared to a full domain LES enclosing the entire ai
r-
foil.


References

Archambeau, F., Mehitoua, N., Sakiz M.,
(2004): Code_Saturne: a finite volume code for the
computation of turbulent incompressible flows.
Int.
J. of Finite Volumes
, Vol. 1, No. 1

Batten, P., Goldberg, U., Chakravarthy, S.
(2004): Interfacing Statistical Turb
u
lence Closures
with Large
-
Eddy Simulation.
AIAA Journal
, Vol. 42
No. 3, pp. 485
-
492.

Huser
, A.

and
Biringen
, S. (1993):

Direct n
u-
me
r
ical simulation of turbulent flow in a squa
re
duct.
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, 257:65

95
.

Jarrin, N., Benhamadouche, S., Laurence, D.,
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-
eddy method for
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-
eddy simul
a-
tions,

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585
-
593.

Keat
ing, A., Piomelli, U., Balaras, E., Kalte
n-
bach H.J. (2004):
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and
a po
s
teriori

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-
eddy simulation,
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-
4712.

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a-
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Lund, T.S., Wu X. and Squires D. (1998): Ge
n-
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-
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-
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-
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