In order to test the solver written with Scilab we decided to solve a simple problem which has been used by different authors (see [3], [6]) as a benchmark problem. This benchmark can test different numerical approaches for the solution of the incompressible, steady and unsteady, Navier-Stokes equations. In Figure 4 the problem is drawn, where the geometry and the boundary conditions can be found. The fluid density is set to 1 and the viscosity to 10-3. A parabolic (Poiseulle) velocity field in x direction is imposed on the inlet, as shown in this equation:

Poiseulle_equation

with Um=0.3, a zero pressure condition is imposed on the outlet. The velocity in both directions is imposed to be zero on the other boundaries. The Reynolds number is computed as Re=Umean*D/ν, where the mean velocity at the inlet (Umean=2Um/3), the circle diameter D and the kinematic viscosity ν=μ/ρ have been used.

Geometry and Meshing

benchmark_geometry

the boundary conditions are drawn in blue. The same problem has been solved using different computational strategies in [6]; the interested reader is addressed to this reference for more details.

benchmark_meshing

The two meshes used for the benchmark. On the top the coarse one (3486 unknowns) and on the bottom the finer one (11478 unknowns).

Results with Scilab

benchmark_results

Results with Ansys

benchmark_results_ansys

Starting from top, the x and y components of velocity, the velocity magnitude and the pressure for Reynolds number equal to 20, computed with the finer mesh with Scilab and Gmsh and with the ANSYS Flotran solver (2375 elements, 2523 nodes).

Sources:

[3] J. Donea, A. Huerta, Finite Element Methods for Flow Problems, (2003) Wiley

[6] M. Schafer, S. Turek, Benchmark Computations of laminar Flow Around a Cylinder,
downloaded from http://www.mathematik.unidortmund.de/de/personen/person/Stefan+Turek.html

http://wiki.scilab.org/Tutorials?action=AttachFile&do=get&target=ns_Margonari_enginsoft-fix.pdf