NAVIER_STOKES_2D_EXACT
Exact solutions to the
2D Incompressible Time-Dependent Navier Stokes Equations

NAVIER_STOKES_2D_EXACT, a FORTRAN90 library which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 2D.

• Lukas: steady flow, pressure is zero everywhere;
• Poiseuille: steady flow, vertical velocity and right hand side are zero everywhere;
• Spiral: velocity is zero on the boundary of the unit square;
• Taylor: source term is zero everywhere.
• Vortex: a steady flow with the same velocity pattern as Taylor.

Licensing:

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

Languages:

NAVIER_STOKES_2D_EXACT is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

NAVIER_STOKES_3D_EXACT, a FORTRAN90 library which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 3D.

NAVIER_STOKES_MESH2D, MATLAB data files which define triangular meshes for several 2D test problems involving the Navier Stokes equations for fluid flow, provided by Leo Rebholz.

SPIRAL_DATA, a FORTRAN90 library which computes a velocity vector field that satisfies the continuity equation, writing the data to a file that can be plotted by gnuplot.

STOKES_2D_EXACT, a FORTRAN90 library which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.

Reference:

1. Maxim Olshanskii, Leo Rebholz,
   Application of barycenter refined meshes in linear elasticity and incompressible fluid dynamics,
   ETNA: Electronic Transactions in Numerical Analysis,
   Volume 38, pages 258-274, 2011.
2. Geoffrey Taylor,
   On the decay of vortices in a viscous fluid,
   Philosophical Magazine,
   Volume 46, 1923, pages 671-674.
3. Geoffrey Taylor, A E Green,
   Mechanism for the production of small eddies from large ones,
   Proceedings of the Royal Society of London,
   Series A, Volume 158, 1937, pages 499-521.

Source Code:

• ns2de.f90, the source code.

Examples and Tests:

• ns2de_test.f90, a sample calling program.
• ns2de_test.txt, the output file.

Lukas Bystricky flow:

• lukas_commands.txt, commands to draw the velocity field.
• lukas_data.txt, velocity field data.
• lukas.png, a PNG plot of the velocity field.

Poiseuille flow:

• poiseuille_commands.txt, commands to draw the velocity field.
• poiseuille_data.txt, velocity field data.
• poiseuille.png, a PNG plot of the velocity field.

Spiral flow:

• spiral_commands.txt, commands to draw the velocity field.
• spiral_data.txt, velocity field data.
• spiral.png, a PNG plot of the velocity field.

Taylor flow:

• taylor_commands.txt, commands to draw the velocity field.
• taylor_data.txt, velocity field data.
• taylor.png, a PNG plot of the velocity field.

Vortex flow:

• vortex_commands.txt, commands to draw the velocity field.
• vortex_data.txt, velocity field data.
• vortex.png, a PNG plot of the velocity field.

List of Routines:

• GET_UNIT returns a free FORTRAN unit number.
• GRID_2D returns a regular 2D grid.
• NS2DE_GNUPLOT writes the Navier-Stokes solution field to files for GNUPLOT.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
• R8VEC_UNIFORM_AB returns a scaled pseudorandom R8VEC.
• RESID_LUKAS returns Lukas Bystricky residuals.
• RESID_POISEUILLE returns Poiseuille flow residuals.
• RESID_SPIRAL evaluates Sprial residuals.
• RESID_TAYLOR returns Taylor residuals.
• RESID_VORTEX returns Vortex residuals.
• RHS_LUKAS evaluates the Lukas Bystricky right hand sides.
• RHS_POISEUILLE returns Poiseuille right hand sides.
• RHS_SPIRAL evaluates Spiral right hand sides.
• RHS_TAYLOR returns Taylor right hand sides.
• RHS_VORTEX returns Vortex right hand sides.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• UVP_LUKAS evaluates Lukas Bystricky's solution.
• UVP_POISEUILLE evaluates Poiseuille solutions.
• UVP_SPIRAL returns Spiral solutions.
• UVP_TAYLOR evaluates Taylor solutions.
• UVP_VORTEX evaluates Vortex solutions.

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