navier_stokes_3d_exact

navier_stokes_3d_exact, an Octave code 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.

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.

For the Poiseuille flow, a typical region is the infinite cylinder along the x axis, with radius 1, for which the velocity is zero on the boundary.

Licensing:

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

Related Data and Programs:

navier_stokes_3d_exact_test

navier_stokes_2d_exact, an Octave code 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 flow 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:

• resid_burgers.m, evaluates the Burgers residual in the velocity and pressure equations at any point (x,y,z) and time t.
• resid_ethier.m, evaluates the Ethier residual in the velocity and pressure equations at any point (x,y,z) and time t.
• resid_poiseuille.m, evaluates the Poiseuille residual in the velocity and pressure equations at any point (x,y,z) and time t.
• uvwp_burgers.m, evaluates the Burgers velocity and pressure field at any point (x,y,z) and time t.
• uvwp_ethier.m, evaluates the Ethier velocity and pressure field at any point (x,y,z) and time t.
• uvwp_poiseuille.m, evaluates the Poiseuille velocity and pressure field at any point (x,y,z) and time t.