navier_stokes_2d_exact

navier_stokes_2d_exact, a Python code which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 2D.

• GMS: time dependent, vortices do not decay to zero;
• Lukas: steady, zero pressure;
• Poiseuille: steady, zero vertical velocity, zero source term;
• Spiral: time dependent, zero velocity on the unit square;
• Taylor: time dependent, zero source term, solution decays exponentially.
• Vortex: steady, same velocity pattern as Taylor.

Licensing:

The computer code and data files made available on this web page are distributed under the MIT 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 an Octave version and a Python version.

Related Data and Programs:

navier_stokes_3d_exact, a Python code which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 3D.

stokes_2d_exact, a Python code which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.

Reference:

1. Jean-Luc Guermand, Peter Minev, Jie Shen,
   An overview of projection methods for incompressible flows,
   Computer methods in applied mechanics and engineering,
   Volume 105, pages 6011-6045, 2006.
2. 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.
3. Geoffrey Taylor,
   On the decay of vortices in a viscous fluid,
   Philosophical Magazine,
   Volume 46, 1923, pages 671-674.
4. Geoffrey Taylor, Albert 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:

• navier_stokes_2d_exact.py, the source code.
• navier_stokes_2d_exact.sh, runs all the tests.
• navier_stokes_2d_exact.txt, the output file.

GMS:

• gms_commands.txt, commands to draw the velocity field.
• gms_data.txt, velocity field data.
• gms.png, a PNG plot of the velocity field.
• gms_matplotlib.png, a PNG plot of the velocity field created by MATPLOTLIB.

Lukas Bystricky:

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

Poiseuille:

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

Spiral:

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

Taylor:

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

Vortex:

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