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

NAVIER_STOKES_2D_EXACT, a Python 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 Python library which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 3D.

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

STOKES_2D_EXACT, a Python 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:

• grid_2d.py, returns a regular 2D grid.
• ns2de_gnuplot.py, writes files to create a GNUPLOT image of the flow.
• ns2de_matplotlib.py, writes files to create a MATPLOTLIB image of the flow.
• parameter_poiseuille_test.py, monitors the Poiseuille solution norm for various values of the parameters.
• parameter_spiral_test.py, monitors the Spiral solution norm for various values of the parameters.
• parameter_taylor_test.py, monitors the Taylor solution norm for various values of the parameters.
• parameter_vortex_test.py, monitors the Vortex solution norm for various values of the parameters.
• r8vec_amax.py, returns the maximum absolute value entry in an R8VEC.
• r8vec_amin.py, returns the minimum absolute value entry in an R8VEC.
• r8vec_max.py, returns the maximum entry in an R8VEC.
• r8vec_min.py, returns the minimum entry in an R8VEC.
• r8vec_norm_l2.py, returns the L2 norm of an R8VEC.
• r8vec_print.py, prints an R8VEC.
• r8vec_uniform_ab.py, returns a scaled pseudorandom R8VEC.
• resid_lukas.py, evaluates the Lukas Bystricky residual at any point (x,y) and time t.
• resid_poiseuille.py, evaluates the Poiseuille residual at any point (x,y) and time t.
• resid_spiral.py, evaluates the Spiral residual at any point (x,y) and time t.
• resid_taylor.py, evaluates the Taylor residual at any point (x,y) and time t.
• resid_vortex.py, evaluates the Vortex residual at any point (x,y) and time t.
• rhs_lukas.py, evaluates the Lukas Bystricky right hand sides at any point (x,y) and time t.
• rhs_poiseuille.py, evaluates the Poiseuille right hand sides at any point (x,y) and time t.
• rhs_spiral.py, evaluates the Spiral right hand sides at any point (x,y) and time t.
• rhs_taylor.py, evaluates the Taylor right hand sides at any point (x,y) and time t.
• rhs_vortex.py, evaluates the Vortex right hand sides at any point (x,y) and time t.
• timestamp.py prints the YMDHMS date as a timestamp.
• uvp_lukas.py, evaluates the Lukas Bystricky velocity and pressure field at any point (x,y) and time t.
• uvp_poiseuille.py, evaluates the Poiseuille velocity and pressure field at any point (x,y) and time t.
• uvp_spiral.py, evaluates the Spiral velocity and pressure field at any point (x,y) and time t.
• uvp_taylor.py, evaluates the Taylor velocity and pressure field at any point (x,y) and time t.
• uvp_vortex.py, evaluates the Vortex velocity and pressure field at any point (x,y) and time t.

Examples and Tests:

• navier_stokes_2d_exact_test.py, calls all the tests.
• navier_stokes_2d_exact_test.sh, runs all the tests.
• navier_stokes_2d_exact_test.txt, the output file.

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.

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