NAVIER_STOKES_3D_EXACT
Exact solutions to the
3D Incompressible Time-Dependent Navier Stokes Equations

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.

The given velocity and pressure fields are exact solutions for the 3D incompressible time-dependent Navier Stokes equations over any region.

To define a typical problem, one chooses a bounded spatial region and a starting time, and then imposes boundary and initial conditions by referencing the exact solution appropriately.

In the Ethier reference, a calculation is made for the cube centered at (0,0,0) with a "radius" of 1 unit, and over the time interval from t = 0 to t = 0.1, with parameters a = PI/4 and d = PI/2, and with Dirichlet boundary conditions on all faces of the cube.

Licensing:

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

Languages:

NAVIER_STOKES_3D_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_2D_EXACT, a Python library which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 2D.

NAVIER_STOKES_MESH3D, MATLAB data files defining meshes for several 3D test problems involving the Navier Stokes equations for fluid flow, provided by Leo Rebholz.

Reference:

1. Martin Bazant, Henry Moffatt,
   Exact solutions of the Navier-Stokes equations having steady vortex structures,
   Journal of Fluid Mechanics,
   Volume 541, pages 55-64, 2005.
2. Johannes Burgers,
   A mathematical model illustrating the theory of turbulence,
   Advances in Applied Mechanics,
   Volume 1, pages 171-199, 1948.
3. C Ross Ethier, David Steinman,
   Exact fully 3D Navier-Stokes solutions for benchmarking,
   International Journal for Numerical Methods in Fluids,
   Volume 19, Number 5, March 1994, pages 369-375.

Source Code:

• burgers.py, evaluates the Burgers solution or its residual at any point (x,y,z) and time t.
• ethier.py, evaluates the Ethier solution or its residual at any point (x,y,z) and time t.
• r8_erf.py, evaluates the error function.
• 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_print.py, prints an R8VEC.
• r8vec_uniform_ab.py, returns a scaled pseudorandom R8VEC.
• timestamp.py prints the YMDHMS date as a timestamp.

Examples and Tests:

• navier_stokes_3d_exact_test.py, calls all the tests.
• navier_stokes_3d_exact_test.sh, runs all the tests.
• navier_stokes_3d_exact_test.txt, the output file.

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