Description 
The goal of this project is to develop
adaptive wavelets solvers for the simulation of turbulence.
Wavelet Representation
of Turbulent Flows
Turbulence in fluid flows is characterized
by localized regions of strong variations in the velocity, pressure of
vorticity fields (coherent structures). It is assumed, that these coherent
structures control the dynamics and statistics of the turbulent flow[1].
The idea is that an adaptive wavelet basis allows for a very sparse representation
of coherent structures. To check this idea we applied wavelet compression
to some instantaneous velocity, pressure and vorticity fields of a turbulent
channel flow (Re=2800,ReTau=178)[2]. The database was kindly provided
by H.J. Kaltenbach (HFI, TU Berlin).
Isosurfaces of spanwise vorticity component
Even at the high compression rate of 60
we obtained a very good preservation of of the statistical quantities (mean,
rms).
This part of work was done in collaboration
with Marie Farge (LMD ENS Paris),
Kai
Schneider (Universitaet Karlsruhe) within the frenchgerman project
on "Numerical Flow Simulation" granted by the DFG(CNRS).
Wavelet Navier Stokes
Solver
On top of our wavelet
sparse grid solver for PDEs we developed a Navier Stokes solver.
The main features are:

Time Discretization:
We employ Chorin's projection method.
Transport step:
Projection step:
In the transport equation K(..) denotes
an explicit (AdamsBashforth or RungeKutta type) discretization of the
convective term.
For the projection step we have to solve
one Poisson equation to obtain the pressure difference p(n+1)p(n).

Pressure Stabilization
To ensure that the velocity is really
discretely divergence free after the projection step, the Laplacian which
appears in the Pressure Poisson equation has to be the nested application
of the discrete pressure gradient and the discrete divergence operator
(consistent pressure Laplacian). Without further measures this operator

leads to spurious pressure oscillations

is not spectrally equivalent to the continuous
Laplace operator, therefore, preconditioning in e.g. a BiCGStab solver
breaks down
To solve this problem we introduced a stabilization
technique which

adds a higher order diffusive term to the
discrete pressure gradient

removes a similar term form the discrete divergence
operator
It can be shown that this removes spurious
oscillations and reestablishes spectral equivalence[3].

Numerical Experiment: Merging of Vortices
The initial configuration are three vortices.
Under the influence of the velocity field they induce, they start to rotate
around each other and then merge.

domain [0,1]x[0,1] , time: 0.0 ... 80.0 units

maximum velocity: 0.07 , Re=55000

time step=5e3 , finest mesh size=1/2048
The left figure shows the vorticity
and the right figure the adaptive sparse grid associated to the current
adaptive basis
Click to see the complete mpeg movie (1.5MB)

Numerical Experiment: 2D Mixing layer
The initial configuration are two flows
in the upper and lower half of [0,1]x[0,1] with opposite sign. The vorticity
in the interfacial layer is randomly perturbated.
KelvinHelmholtz instabilities lead to
the development of vortices which in a later stage rollup and merge. This
is a nice example for the tendency of 2D turbulence to transfer energy
from small to large scales. This leads to a fast decrease of the complexity
of the flow and to a decrease of the number of degrees of freedom required
for an accurate numerical simulation.

domain [0,1]x[0,1] , time: 0.0 ... 80.0 units

minimum/maximum velocity: +0.018 , Re=15000

time step=5e3 , finest mesh size=1/2048
Click to see the complete mpeg movie (0.8MB)

Numerical Experiment: 3D Mixing layer
The initial configuration is the 3D analogue
of the 2D mixing layer. However, due to the 3D character there is an energy
transfer from large to small scales which leads to an increase of the complexity
of the flow.

domain [0,1]x[0,2]x[0,1] , time: 0.0 ... 46.0
units

minimum/maximum velocity: +0.018 , Re=3750

time step=2e2 , finest mesh size=1/512

number of DOF from 1 to 2 million
Click to see the complete mpeg movie (1.1MB);
you can download movies of a longer simulation here(2.5MB)
or here(2.4MB)

Numerical Experiment: 3D Decaying isotropic
turbulence
The initial configuration is a relatively
smooth, periodic flow. Because of the energy transfer from large to small
scales smaller structures (vortices) develop. After a while these decay.

domain [0,1]^3 , time 0.0 ... 0.65 units

maximium velocity (approx.) 3.5 , Re (approx.)
3500

time step=1e4 , finest mesh size= 1/512

number of DOF limited to 2 million
Click to see the complete mpeg movie (0.9MB)

Bibliography 

[1] N. Kevlahan, M. Farge Vorticity
filaments in twodimensional turbulence: creation, stability and effect;
J. Fluid Mech. 346 (1997) pp. 4976

[2] F. Koster, M. Griebel, N. Kevlahan,
M. Farge, K. Schneider Towards an adaptive waveletbased 3D NavierStokes
Solver; Krause E., Hirschel E. (Eds.) Notes on Numerical Fluid Mechanics
(1998)

[3] M. Griebel, F. Koster Adaptive
Wavelet Solvers for the Unsteady Incompressible NavierStokes Equations;
to appear in J. Malek, M. Rokyta (Eds.), Advances in Mathematical Fluid
Mechanics, Springer Verlag, also as Preprint No. 669 (2000), University
of Bonn

[4] F. Koster, K. Schneider, M. Griebel,
M. Farge, Adaptive
Wavelet Methods for the NavierStokes Equations; to appear in E.H.
Hirschel,editor, Notes on Numerical Fluid Mechanics, Vieweg Verlag, Braunschweig.
