Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
Title Wavelet-based Navier-Stokes Solver for the Simulation of Turbulence 
Participant Frank Koster , Michael Griebel
Keywords  Turbulence, Coherent Structures, Statistics, Wavelets, Adaptivity, Chorin Projection 
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,Re-Tau=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 french-german 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: 

  1. Time Discretization:

  2. We employ Chorin's projection method. 
    Transport step: 

    Projection step: 

    In the transport equation K(..) denotes an explicit (Adams-Bashforth or Runge-Kutta 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). 

  3. Pressure Stabilization

  4. 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]. 
  5. Numerical Experiment: Merging of Vortices

  6. 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=5e-3 , 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) 

  7. Numerical Experiment: 2D Mixing layer

  8. 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.
    Kelvin-Helmholtz instabilities lead to the development of vortices which in a later stage roll-up 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=5e-3 , finest mesh size=1/2048 

    Click to see the complete mpeg movie (0.8MB) 
  9. Numerical Experiment: 3D Mixing layer

  10. 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=2e-2 , 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)
  11. Numerical Experiment: 3D Decaying isotropic turbulence

  12. 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=1e-4 , finest mesh size= 1/512 
    • number of DOF limited to 2 million 

    Click to see the complete mpeg movie (0.9MB) 
  • [1] N. Kevlahan, M. Farge Vorticity filaments in two-dimensional turbulence: creation, stability and effect; J. Fluid Mech. 346 (1997) pp. 49-76 
  • [2] F. Koster, M. Griebel, N. Kevlahan, M. Farge, K. Schneider Towards an adaptive wavelet-based 3D Navier-Stokes Solver; Krause E., Hirschel E. (Eds.) Notes on Numerical Fluid Mechanics (1998) 
  • [3] M. Griebel, F. Koster Adaptive Wavelet Solvers for the Unsteady Incompressible Navier-Stokes 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 Navier-Stokes Equations; to appear in E.H. Hirschel,editor, Notes on Numerical Fluid Mechanics, Vieweg Verlag, Braunschweig.
Related projects Adaptive PDE Solvers with Wavelets on Sparse Grids
In cooperation with Laboratoire de Météorologie Dynamique / C.N.R.S