simulation numérique des écoulements complexes écoulement...
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Simulation numériquedes écoulements complexes
écoulement moyen et viscosité sous-maille
en collaboration avec
- Pierre Borgnat , Hatem Touil (LP, ENSL)- Adrien Cahuzac, Jérôme Boudet, Joëlle Caro
Liang Shao, Jean-Pierre Bertoglio (LMFA, ECL)- Guillaume Balarac (LEGI, Grenoble)- Federico Toschi (TU Eindhoven)
Inhomogénéité - phénoménologie
!
" = #u2
: cisaillement
turbulence homogène et isotropeturbulence cisaillée
écoulement moyen + fluctuations
tourbillon
l 3/23/1)(/ lll!
"" #$uteddy!
"1
sheart
l
!"!
'uL
Kolmogorov’s theory
!"#2'u$
taux de dissipation d’énergie
!
u = u + " u : décomposition de Reynolds
écoulement autour d’un cylindre àReD=47000 (régime turbulent sous-critique)
Simulation numérique desécoulements turbulents
goal= get a physically-sound numerical representation of the flow
-Direct Numerical Simulation (all scales of motion) is usually prohibitive
- alternatives = partial representation of the flow : - Reynolds-Averaged Navier-Stokes : mean flow
- Large-Eddy Simulation : large-scale dynamics of the flow
Does it make sense (from a physical viewpoint) ? What are the governing equations?
subgridgridgridt UU !" div)~(NS
~+=
Ssubgridsubgridsubgrid
~2I)(tr !"" #=#
SGS viscosity = property of the flow, not of the fluid… (very) bad news !
?)etc.,~
,~,( gridgridsubgrid UU !"#
4
shearuniformlocally:j
j
i
iix
x
uuu !+"=
#
#
i
22
2dS),(),(),(
4
1)(T ! "
"
#
$
%%
&
'()+()(=
lB
j
j
iisgs lxul
x
ulxulxu
ll
*
+*
*++
,
!
" # sgs $ S = 2 % sgs(&) $ S 2
' Tsgs(&) & with & = grid spacing
- after some (exact) calculations from the Navier-Stokes equations…
!
" # u ($) % # S & $ and "U($) % S & $ ' ( sgs($) = (Cs$)2& S ) S ( )
- in the context of LES :
Effet du cisaillement en équations
Lévêque et al., J. Fluid Mech. (2007)Shear-Improved Smagorinsky Model
5
If : Smagorinsky model for HI turbulence (Cs =0.17)
If : Shear-dominated SGS viscosity
SS~~
>>!
SS~~!>>
( ) ( ) SCu
C sssgs
~)( 22!"#$
#
#!"#$%
&
( )S
SCssgs ~
~ 2
2!
"#$%
MINUS :PLUS :→ physically-sound, no ad-hoc parameter→ low computational cost→ easy to parallelize→ convenient for boundary conditions
→ requires estimation of mean-flowas the simulation runs!
!
" sgs(#) = (Cs#)2$ S % S ( )
signal processing
Application to complex-geometryunsteady turbulent flows (real-world flows)
How to extract (or reconstruct) the mean flow as the simulation runs ?
- physically sound : smoothing in time mean flow = low-frequency component of the flow
-manageable from computational viewpoint : recursive filter-adaptive : Kalman filtering
Cahuzac et al., in press Phys. Fluids (2010)
grid cut-offsmoothingcut-off (fc)
SGS
very-largescales resolved
scales
SISM
f
Euu(f)in
stab
ilitie
s
turb
ulen
ce cascade
Exponentially-weighted moving averageexponential smoothing
!
cexp =2" fc #t
3$ 3.628 fc #t
conceptually simple local in space low memory storage
MINUS :PLUS :
uniform physically-sound cut-off frequency ? unavoidable phase delay :
!
"t / cexp
)1(
exp
)(
exp
)1()1( ++
!+!"=nnn
ucucu
-cexp : smoothing factor- equivalent tp first-order low pass filter with cut-off frequency fc :
at each grid point
Low-pass Kalman filter adaptive exponential smoothing
)1()()1( +++=
nnn
uuu !
control increment = random noise with fixed variance
!
"# u
=2$ fc %t
3& u
*
)1()1()1( ++++=
nnnuuu !
deviation from the estimated mean = random noise with variance
: observation
: model
K(n+1) optimal Kalman gain (given by theory)!
"#u
(n+1)=max u
*$ u
(n+1)% u
(n+1), 0.1$ u
*2[ ]
periodsstationaryfor/ .expcuu=!!
""
)1()1()()1()1()1( ++++
!+!"=nnnnn
uKuKu
why ?
predict and update :
10!/*ufc =
!
u*
= uwall
3.106 mesh-pointsΔr+=1, RΔθ+=20, Δz+=25
resolution: 49x89x41Δx+=52, Δz+=31, Δy+=0.5…24
!
St = fs "D /U#
!
fc = 2 f s
Strouhal numberfriction velocity
Practical evaluation of the methodTurb’Flow solver (Jérôme Boudet, LMFA):
- Finite volume- Convective fluxes by centered schemes on four points- Diffusive fluxes by centered scheme on two points- SGS model = Shear-Improved Smagorinsky Model (SISM)
plane-channel Flow at Rew = 395 flow past a circular cylinder at ReD = 47,000
Cahuzac et al. Phys. Fluids (2010)
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… spatial average (along x-z)
… exponential smoothing compared tospatial average (along x)
… Kalman filtering
Velocity probes at y+=10 in the plane-channel flow (Rew=395)
12
streamwise velocity profile turbulent intensity profiles
Plane-channel flow
comparisons with DNS data
friction velocity:
1.59.0!
= smuDNS
w
1.exp.54.0!
= smuw
within the 10% level of accuracyreviewed for standard SGS models
→
13
spanwise spectraof streamwise velocity
plane-channel flow
comparisons with DNS data
two-point velocity correlationfor spanwise separation
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plane-channel flow
comparisons with DNS data
two-point correlation functions… promising for aero-acoustics
streamwise
spanwise
!
Cp =p " p#1
2$U#
2
!
Cf ="w
1
2#U$
2
% Re
2
2
1
!
="
U
pC rmsp
#
flow past a circular cylinder
Friction and pressure coefficients
flow past a circular cylinder
mean flow andturbulence intensities
Work in progress...
flow past a circular cylinder
comparisons with experimental dataspectra in frequency
(a)
(a)(b)
(b)
(c)
(c)
1.8 D
… promising for aero-acoustics
fc
« Adaptivity » of the Kalman filter
flow past a circular cylinder
a drawback of the smoothing: phase delay
KalmanKt/!
10 <<KalmanK
… make KKalman maximum inseparation zone
Conclusion
from a physical viewpoint :- take into account mean-flow inhomogeneity + unsteadiness :
SISM (no ad-hoc parameter) + (adaptive) temporal smoothing
from a numerical viewpoint :- inconditionnally stable- low cost algorithms- easy to parallelize + convenient for boundary conditions
To-Do list:- address realistic turbo-machinery flow / aero-acoustic features
Flocon project (LMFA) : Adrien Cahuzac + Jérôme Boudet- reduce phase delay :
- improve Kalman filter- consider Lagrangian filtering (along fluid trajectory)
Plane channel flow (academic flow)Lévêque et al., J. Fluid Mech. (2007)Boudet et al., J. of Therm. Science. (2008)Cahuzac et al. Phys. Fluids (2010)
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flow past a circular cylinder
comparisons with experimental data
flow characteristics
Mesh resolution