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# Solution of the Navier-Stokes Equations

This example shows how a Navier-Stokes solver for problems on simple domains with periodic boundary conditions can be implemented using the AWFD library. In the /Examples directory there are three source codes
 2DMergingInitial.cc This program generates a file with the initial conditions for velocity and pressure. The model problem is the self-induced movement of three vortices in 2D. Two of the vortices have the same direction of rotation. These vortices merge during the movement, see e.g. this page for further details on the physics. NavierRK4.cc The actual Navier-Stokes solver. When called with the right parameters, a series of files with current velocity/pressure values is written. The numerical scheme is a Chorin-type projection method with a consistent discretization of the pressure Laplacian. The convcection-diffusion problem is discretized by means of a Runge-Kutta(4) / WENO(5) scheme for the convection terms and a Crank-Nicholson scheme for diffusive terms. SparseToOmegaVTK.cc A program to compute the rotation from the above files. The rotation is written in a UDF file.

The solver and the postprocessing program require several parameters. A complete list with explanations is available by calling e.g. ./NavierRK4.exe without any parameters on the command line. For a first try you can call a small shell script (nav.sh) which calls the three programs with meaningfull parameters. Note, however, that the computing times for the solver are relatively large. On my machine (dual PIII 900Mhz) 12 hours. If you dont want to wait so long, edit the nav.sh file and set -ML 8. This should reduce the computing times by a factor of about 4. If you have MATLAB (5.00) you can visualize the computed vorticity and also the adaptive grids by the MATLAB script AWFD/Sources/Examples/nav.m. This produces a slide show. Below are some of the pictures:

Next: Evaluation of Integral Expressions Up: Examples Previous: Adaptive Solution of a
koster 2003-07-29