FLOW6
A Finite Element Code for Fluid Flow

FLOW6 is a FORTRAN90 program whhich uses the finite element method to solve for the steady state velocity and pressure in a two dimensional fluid flow region.

The fluid flow is presumed to be incompressible, and time-independent. The governing equations are the Navier Stokes equations with the continuity equation.

FLOW6 was derived from the FLOW5 code. Alterations and new features include:

• Both 3 and 7 point quadrature rules;
• The no penetration wall condition has been extended to walls that are not along a coordinate direction;
• The elements are now in "standard" form;
• A new free slip boundary condition is being added;
• Boundary conditions handled by BOUNDARY routine.

FLOW6 writes node and element data files that may be read in and plotted by DISPLAY5.

Related Data and Programs:

BUMP is an executable FORTRAN90 program which solves a fluid flow problem in a channel including a bump which obstructs and redirects the flow.

CHANNEL is a program which computes the pressure and velocity for a Navier Stokes flow in a channel. It is available in a FORTRAN77 version and a FORTRAN90 version.

Reference:

1. Max Gunzburger,
   Finite Element Methods for Viscous Incompressible Flows,
   A Guide to Theory, Practice, and Algorithms,
   Academic Press, 1989,
   ISBN: 0-12-307350-2,
   LC: TA357.G86.

Source Code:

• flow6.f90, the source code;
• flow6.csh, commands to compile and load the source code;

Examples and Tests:

• cavity.tec, solution data suitable for input to the graphics program TECPLOT;
• cavity_output.txt, printed output.
• cavity_elements.txt, element information suitable for input to the graphics program DISPLAY5.
• cavity_uvp.txt, node information suitable for input to the graphics program DISPLAY5.
• cavity_u.txt, the finite element coefficient vector.

List of Routines:

• MAIN solves a 2D steady incompressible flow using finite elements.
• BANDWIDTH computes the lower band width of the Jacobian matrix.
• BOUNDARY_INTEGRAL sets data useful for boundary integrals.
• BOUNDARY_INTEGRAL_FREESLIP sets boundary integral data for the freeslip.
• BOUNDARY_INTEGRAL_STEP sets boundary integral data for the step.
• BOUNDARY_INTEGRAL_PRINT prints data useful for boundary integrals.
• BOUNDARY_SHAPE returns the tangent and normal vectors along the boundary.
• BOUNDARY_SHAPE_CAVITY returns the boundary of the cavity problem.
• BOUNDARY_SHAPE_CHANNEL returns the boundary of the channel problem.
• BOUNDARY_SHAPE_FREESLIP returns the boundary of the freeslip problem.
• BOUNDARY_SHAPE_STEP returns the boundary of the step problem.
• BOUNDARY_VALUE evaluates boundary condition functions.
• COEF_READ reads the coefficient data from a file.
• COEF_WRITE writes the coefficient data to a file for possible restart.
• DGB_CHECK checks the dimensions of a general band matrix.
• DGB_FA factors a matrix stored in LINPACK general band storage.
• DGB_PRINT prints nonzero entries of a banded matrix.
• DGB_SL solves a system factored by DGB_FA.
• DVEC_AMAX returns the maximum absolute value in a real vector.
• ELEMENT_PRINT prints out the elements.
• ELEMENT_WRITE writes element information to a file.
• EQUATION_PRINT prints the equation types.
• FP computes the jacobian matrix A of the Navier Stokes residual.
• FP_PRINT prints information about the jacobian.
• FP_DIF estimates the jacobian matrix by finite differences.
• FX computes the residual RES of the Navier Stokes equations.
• GEOMETRY sets up the geometry for any problem.
• GEOMETRY_CAVITY sets up the geometry for the cavity problem.
• GEOMETRY_CHANNEL sets up the geometry for the channel problem.
• GEOMETRY_FREESLIP sets up the geometry for the freeslip problem.
• GEOMETRY_STEP sets up the geometry for the step problem.
• GET_UNIT returns a free FORTRAN unit number.
• HELLO says hello, prints the program name, date and limits.
• INIT zeroes out data used by the program.
• NEWTON applies Newton iteration, seeking a solution of FX(G) = 0.
• NODE_WRITE writes solution information to a file.
• PRESS_INTERP interpolates pressure values at non-pressure nodes.
• REF_BF_L3 evaluates a reference element linear basis function.
• REF_BF_Q6 evaluates shape functions for a 6 node triangle.
• REF_MAP_Q6 returns the interpolation map for data on a 6 node triangle.
• REF_QUAD1 sets abscissas and wquad1s for 1D Gauss-Legendre quadrature.
• REF_QUAD2 sets up the 2D reference triangle quadrature rule.
• SETBAS evaluates the basis functions at each quadrature point.
• STOKES solves the Stokes equations.
• TECPLOT_WRITE writes out solution information for use with TECPLOT.
• TRANS_Q6 calculates the biquadratic reference element transformation.
• UVP_NORM_H1 returns the H1 norm of the solution.
• UVP_NORM_L2 returns the L2 norms of velocity magnitude, and pressure.
• UVP_NORM_LMAX returns the infinity norms of velocity magnitude, and pressure.
• UVP_PRINT prints out the solution.
• UVP_QUAD_VALUE evaluates U, V and P at a quadrature point in a given element.
• UVP_VALUE evaluates U, V and P at any point in a given element.
• X_OF_XSI computes X and Y given XSI and ETA coordinates.
• XY_PRINT prints the X and Y coordinates of the nodes.

You can go up one level to the FORTRAN90 source codes.