Wavelets & Turbulence - Volume penalization at...

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

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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⊥ ?

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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

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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

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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

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Penalized-Stokes Eigenmodes

!

(S" ) :#u+$p%

1

"&u = 'u

$ (u = 0

)

* +

, +

0 π 2π0

2π

!

" = 0

!

" =1

x1

x2

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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.

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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.

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Behavior when η→0

• Convergence of eigenvalues

39

Behavior when η→0

• Some eigenmodes at η=10-6

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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 »