CAVITY
A Sample 2D Navier-Stokes Problem

CAVITY is a square region that is 1 unit wide and 1 unit high. The tangential velocity is specified to be 1 along the top boundary, with a zero normal component. On all other parts of the boundary, the velocity is specified to be zero.

To run the problem directly, you only need the user-supplied routines in dirichlet_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

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

Some of the files needed to run the problem include:

• constants.m, defines flow constants; for now, just the dynamic viscosity.
• dirichlet_condition.m, the user-supplied routine to evaluate the Dirichlet boundary conditions.
• rhs.m, the user-supplied routine to evaluate the right hand side.
• cavity_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 linear 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.