Volume penalizationat vanishing viscosity
GDR turbulence/IFS - ESPCI - 3.11.2010
Romain Nguyen van yen1, Marie Farge1
Dmitry Kolomenskiy2, Kai Schneider2
1 Laboratoire de météorologie dynamique-CNRS, ENS Paris, France 2 Laboratoire de mécanique, modélisation et procédés propres-CNRS,
CMI-Université d’Aix-Marseille, France
2
wall
3
Outline
Vanishing viscosity limit with walls
Volume penalization
Scaling issues
Dipole-wall collision
4
Walls at vanishing viscosity(some open questions)
5
Mathematical formulation
• incompressible fluid with constant densitycontained in a 2D torus (3D case briefly mentioned),
• the torus may either– be filled with fluid only => wall-less,– contain solid obstacle(s) => wall-bounded.
• The main question we are trying to address is: how to deal with the wall-bounded case?
6
Mathematical formulation
!
uRe(t,x)
!
u(t,x)
the Reynolds number Re = ULν-1 appears when non dimensional quantities are introduced.
solution
solution
Navier-Stokes equations withno-slip boundary conditions:
Incompressible Euler equations: ?
?!
Re"#
(NS)
(E)
7
Known convergence results
• Without walls, for smooth initial data, we have the strongconvergence result (Golovkin 1966, Swann 1971, Kato 1972) :
in all Sobolev spaces for all time in 2D, and as long as the smooth Eulersolution exists in 3D (the constant in the O may blow up very fast in time).
• With walls,– what is the limit of uRe ?– is there energy dissipation in this limit ?– what equation is satisfied by this limit ?
!
uRe"u =
Re#$O(Re
"1)
8
Remarks on numerical approximation
1. There exists exponentially accurate schemes for the wall-less Navier-Stokes equations (i.e. the error decreasesexponentially with computing time),
2. In the 2D wall-less case, the numerical discretization sizeshould satisfy:
(remark : the proof of that is not yet complete)
• 1+2 imply that, in the 2D wall-less case, solving NSprovides an order 2 scheme to approach the vanishingviscosity limit (i.e. solving Euler), (at least in the energy norm)
what about the wall-bounded case ??
!
"x #Re$ 12
9
What does this have to do with turbulence ?
• focus on fully deterministic initial value problem,
• many steps away from statistical theories of turbulence !(like molecular dynamics compares to the kinetic theory of gases)
• look for new “microscopic hypotheses” that could be usedto improve current statistical theories,
• cf. recent discoveries by Tran & Dritschel*, who showedthat one of the basic “microscopic hypotheses” of theKraichnan-Batchelor 2D turbulence theory is slightlyincorrect.
*JFM 559 (2006), JFM 591 (2007)
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Volume penalization
11
Volume penalization method
• For efficiency and simplicity, we would like to stick to aspectral solver in periodic, Cartesian coordinates.
• as a counterpart, we need to add an additional term in theequations to approximate the effect of the boundaries,
• this method was introduced by Arquis & Caltagirone (1984),and its spectral discretization by Farge & Schneider (2005),
• it has now become classical for solving various PDEs.
!
"tu+ (u # $)u = %$p +1
Re&u%
1
'(0u
$ #u = 0, u(0,x) = v
)
* +
, + (PNS)
solution
!
uRe,"
12
Behavior at vanishing viscosity
• No theorem guarantees convergence.• Scaling estimates:
– penalization error :
– discretization error :
!
"#( )1/ 2
k$
!
k"K#1
!
k"
!
K
is the maximal wall-normal wavenumber of the solution,
is the maximal discretization wavenumber.
!
K > C "#( )1/ 2
!
K > Ck" and
13
What is the scaling of k⊥ ?
14
Re1/2laminar + attachedboundary layer
1904Prandtl-Blasius
Re3/4 (?) 3D homogeneous +isotropic + similarity
1941Kolmogorov
Obukhov
Re1uτ (?) similarity + stationarity19301932
von KármánPrandtl-Nikuradse
≥ Re1energy dissipation19231984
BurgersKato
(?) 2D homogeneous +isotropic + similarity
Condition
Re1/2
Scaling
19671969
YearAuthors
KraichnanBatchelor
Proposed scalings for k⊥
15
Vanishing viscosity limitof a 2D dipole-wall collision
16
Choice of geometry
• We consider a channel, periodic in the y direction
where U is the RMS velocity and L is the half-width.
solid fluid
!
Re =UL
"
17
Choice of initial conditions
vorticity dipoleVorticity decays exponentiallyand the circulation is zero.Velocity also decaysexponentially.
Pressure compatibility condition notsatisfied exactly but we havechecked that it does not creat toomuch problems at t=0.
18
Choice of parameters
• To resolve the Burgers-Kato layer, we impose:
•We take for η the minimum value allowed by the CFLcondition, which implies:
!
N "Re#1
!
"#Re$1
Parameters of all reported numerical experiments
19
Results
vorticity movie for Re = 3940 zoom on collision for Re = 7980
!
L
Re
20
Boundary conditions
• A posteriori, check what were the boundary conditions seenby the flow.
• Define the boundary as the isoline χ = 0.02, where theviscous term approximately balances the penalization termin the penalized equations.
• To avoid grid effects interpolate the fields along this isoline.
• The normal velocity is smaller than 10-3 (to be compared with the initial RMS velocity 0.443),
• but the tangential velocity reaches values of order 0.1 !!
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Boundary conditions
• We plot the tangential velocity as a function of thetangential stress:
22
Boundary conditions
• A linear relationship with correlation coefficient above 0.98appears:
• The flow hence sees Navier boundary conditions with aslip length α satisfying:
!
uy +"(Re,#,N)$xuy = 0
!
" #Re$1# (%& )1/ 2
23
Subregions
We define two subregions of interest in the flow :• region A : a vertical slab of width 10N-1 along the wall,• region B : a square box of side 0.025 around the center of
the main structure.
Now we consider the local energy disspation rate:
(sometimes called pseudo-dissipation rate)
and we integrate it over A and B respectively.
!
" = # $u2
24
Subregions
25
Conclusion
• energy dissipation in the vanishing viscosity limit of2D flows– can be observed using volume penalization method at
high resolution K ~ Re1,– is difficult to observe with other methods due to lack of
resolution (adaptivity needed),– is compatible with Burgers-Kato scaling k⊥ ~ Re1,– is incompatible with all other scalings, except VK law of
the wall if uτ does not vanish with Re,
• residual slip with α ~ Re-1 may be an issue.What about no-slip ? more dissipation ?
26
Thank you !
References :• J.M. Burgers, « On the resistance experienced by a fluid in
turbulent motion », Proc. KNAW 26, p. 582 (1923)• T. Kato, « Remarks on zero viscosity limit for nonstationary
Navier-Stokes flows with boundary », Sem. nonlin. partialdiff. eq., MSRI Berkeley (1984), p. 85
• C. Bardos & E. Titi, « Euler equations for incompressibleideal fluids », Russ. Math. J. (2007)
• I. Marusic et al., Phys. Fluids 22, #065103• preprints of our papers available on request.
Codes available under GNU GPL license athttp://justpmf.com/romain
http://wavelets.ens.fr
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Lagrangian point of view
28
Spectral analysisof the penalized Stokes operator
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Stokes Eigenmodes
!
(S0) :
"u+#p = $u
# %u = 0
u&' = 0
(
) *
+ *
0 π0
2π
x1
x2
30
Penalized-Stokes Eigenmodes
!
(S" ) :#u+$p%
1
"&u = 'u
$ (u = 0
)
* +
, +
0 π 2π0
2π
!
" = 0
!
" =1
x1
x2
31
Mathematical properties
• Self-adjoint, positive operators with compact inverses.• Sequence of positive eigenvalues:
and corresponding eigenfunctions:
!
(S0) : "
0
(0)# "
0
(1)# ...# "
0
(i)# ...
!
(S" ) : #"(0) $ #"
(1) $ ...$ #"(i) $ ...
!
(S0) : v
0
(0),v
0
(1),...,v
0
(i),...
!
(S" ) : v"(0),v"(1),...,v"
(i),...
32
A quick remark: Schrödinger analogy
• Note that (Sη) is analog to the eigenvalue problem for theSchrödinger operator with a « hard well » potential:
• η going to zero corresponds to semi-classical limit.• there is a classical theory of such problems, which studies
in particular the exponential decay of solutions (related tothe quantum tunnel effect, mathematical ref.: S. Agmon),
• however the effect of pressure term is specific to Stokes,
this analogy remains to be exploited
!
V =1
"# η-1
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Semi-analytical solution
• Fourier series in the x2 (wall-parallel) direction,• eliminate pressure and û2,• write 4th order linear ODE on û1(x1,k2), with
constant coefficients on ]0,π[ and ]π,2 π[,• solve by complex exponentials on these intervals,• reconnect at x1=0 and x1=π, yielding 8 by 8 linear
system, determined using symbolic computationsoftware,
• vanishing determinant yields (complicated)eigenvalue equation, which can be solvednumerically.
34
Semi-analytical solution
• Assuming that:
!
" < k2
+1
#
no oscillations inside the wall
Modes with
!
" > k2
+1
#are spurious.
(they do not converge to any Stokes eigenmodes)
35
Semi-analytical solution
• Assuming that:
!
" < k2
+1
#
!
ˆ u 1+(x1,k2)=
Acos q0 x "#
2
$
% &
'
( )
$
% &
'
( ) + Bcosh k x "
#
2
$
% &
'
( )
$
% &
'
( ) for 0 < x < #
C exp q1 x " 2#( )( ) + exp "q1 x "#( )( )( ) + Dcosh k x "3#
2
$
% &
'
( )
$
% &
'
( ) for 0 < x < #
*
+
, ,
-
, ,
symmetric modes
36
Semi-analytical solution
• Assuming that:
!
ˆ u 1"(x1,k2)=
Asin q0 x "#
2
$
% &
'
( )
$
% &
'
( ) + Bsinh k x "
#
2
$
% &
'
( )
$
% &
'
( ) for 0 < x < #
C exp q1 x " 2#( )( ) " exp "q1 x "#( )( )( ) + Dsinh k x "3#
2
$
% &
'
( )
$
% &
'
( ) for 0 < x < #
*
+
, ,
-
, ,
!
" < k2
+1
#
!
ˆ u 1+(x1,k2)=
Acos q0 x "#
2
$
% &
'
( )
$
% &
'
( ) + Bcosh k x "
#
2
$
% &
'
( )
$
% &
'
( ) for 0 < x < #
C exp q1 x " 2#( )( ) + exp "q1 x "#( )( )( ) + Dcosh k x "3#
2
$
% &
'
( )
$
% &
'
( ) for 0 < x < #
*
+
, ,
-
, ,
symmetric modes
antisymmetric modes
37
Preliminary analysis of analytical results
• The most interesting feature is the behavior insidethe boundary, which is like
η1/2 exp(-kx)and not
η1/2 exp(-x/η1/2)as for the Laplace operator !
• This is due to the nonlocal effect of pressure.• We now present some preliminary « raw » results
for k=3.
38
Behavior when η→0
• Convergence of eigenvalues
39
Behavior when η→0
• Some eigenmodes at η=10-6
40
Behavior when η→0
• Qualtitative convergence of eigenmode
41
Behavior when η→0
• Convergence of wall values to zero
42
Behavior when η→0
• Plot of wall value rescaled by
!
" # k 2( )1/ 2
$1/ 2
43
Behavior when η→0
• « Slip length »