FEM2D_NAVIER_STOKES_CHANNEL
A Sample 2D Navier-Stokes Problem

FEM2D_NAVIER_STOKES_CHANNEL is a MATLAB library which supplies information defining a Navier-Stokes flow problem in a channel. The channel is a rectangular region that is 3 units wide and 1 unit high. This problem is used as a test case for the Navier-Stokes solver.

Usage:

To run the problem directly, you only need the user-supplied routines in boundary_type.m, constants.m, dirichlet_condition.m, neumann_condition.m and rhs.m, the node data in nodes6.txt, and the element data in triangles6.txt.

You can run the program with the MATLAB command

        fem2d_navier_stokes ( 'nodes6.txt', 'triangles6.txt' )
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

FEM2D_NAVIER_STOKES_CHANNEL is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM2D_NAVIER_STOKES, a MATLAB program which solves the 2D incompressible Navier Stokes equations in an arbitrary triangulated region. In order to run, it requires user-supplied routines that define problem data.

Some of the files needed to run the problem include:

• boundary_type.m, determines the type of boundary conditions imposed.
• constants.m, defines flow constants, for now just the dynamic viscosity NU.
• dirichlet_condition.m, evaluates the right hand sides of the Dirichlet boundary conditions.
• neumann_condition.m, evaluates the right hand sides of the Neumann boundary conditions.
• rhs.m, evaluates the right hand sides (source terms) of the Stokes equations.
• channel_output.txt, output from a run of the program;

The geometry is defined by sets of nodes and triangles. The velocities use the full set of nodes, and quadratic (6 node) triangles.

• nodes6.txt, a text file containing the velocity nodes;
• nodes6.png, a PNG image of the velocity nodes;
• triangles6.txt, the quadratic triangulation;
• triangles6.png, a PNG image of the order 6 triangulation;

The pressures are associated with a subset of the nodes called "pressure nodes", and linear (3 node) triangles. Note that, in the order 3 triangulation, the nodes are renumbered, and do NOT inherit the labels used in the order 6 triangulation.

• nodes3.txt, a text file containing the pressure nodes;
• nodes3.png, a PNG image of the pressure nodes;
• triangles3.txt, the linear triangulation;
• triangles3.png, a PNG image of the linear triangulation;

The Stokes equations are solved first, providing the solution of a linear system that can be used as a good estimate of the solution, especially for high values of the viscosity.

• stokes_pressure3.txt, a text file containing the Stokes pressure P at each pressure node;
• stokes_pressure3.png, a PNG image of a contour plot of the Stokes pressure, produced by TRIANGULATION_ORDER3_CONTOUR.
• stokes_velocity6.txt, a text file containing the Stokes velocity (U,V) at each velocity node;
• stokes_velocity6_dir.png, a PNG image of the Stokes velocity direction field, created by VECTOR_PLOT.
• stokes_velocity6_vec.png, a PNG image of the Stokes velocity field, created by VECTOR_PLOT.
• stokes_velocity6_arrows.png, a PNG image of the Stokes velocity field, created by VELOCITY_ARROWS.

The nonlinear Navier Stokes equations are solved, using the Stokes solution as a starting point.

• navier_stokes_pressure3.txt, a text file containing the Navier Stokes pressure P at each pressure node;
• navier_stokes_pressure3.png, a PNG image of a contour plot of the Navier Stokes pressure, produced by TRIANGULATION_ORDER3_CONTOUR.
• navier_stokes_velocity6.txt, a text file containing the Navier Stokes velocity (U,V) at each velocity node;
• navier_stokes_velocity6_dir.png, a PNG image of the Navier Stokes velocity direction field, created by VECTOR_PLOT.
• navier_stokes_velocity6_vec.png, a PNG image of the Navier Stokes velocity field, created by VECTOR_PLOT.
• navier_stokes_velocity6_arrows.png, a PNG image of the Navier Stokes velocity field, created by VELOCITY_ARROWS.