spiral_exact

spiral_exact, a MATLAB code which computes a 2D velocity vector field that is an exact solution of the continuity equation.

The continuous velocity field (U,V)(X,Y) that is discretely sampled here satisfies the homogeneous continuity equation, that is, it has zero divergence. In other words:

        dU/dX + dV/dY = 0.
      

        U(X,Y) =  10 * d/dY ( PHI(X) * PHI(Y) )
        V(X,Y) = -10 * d/dX ( PHI(X) * PHI(Y) )
      

The underlying function PHI is defined by

        PHI(Z) = ( 1 - cos ( C * pi * Z ) ) * ( 1 - Z )^2
      

The velocity data satisifes the (continuous) continuity equation; this in no way implies that it satisfies the momentum equations associated with Stokes or Navier-Stokes flow! Moreover, a flow solution for those equations would normally also require specifying a value for the scalar pressure field P(X,Y).

Licensing:

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

Languages:

spiral_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:

spiral_exact_test

kdv_exact, a MATLAB code which evaluates an exact solution of the Korteweg-deVries (KdV) partial differential equation (PDE).

gnuplot_test, a MATLAB code which illustrates the use of the gnuplot() graphics program.

navier_stokes_2d_exact, a MATLAB code which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 2D.

navier_stokes_3d_exact, a MATLAB code which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 3D.

pendulum_nonlinear_exact, a MATLAB code which evaluates an exact formula for the solution of the the ordinary differential equations (ODE) that represent the behavior of a nonlinear pendulum of length L under a gravitational force of strength G.

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

Source Code:

• grid_2d.m, returns a regular 2D grid.
• r8vec_uniform_ab.m, returns a scaled pseudorandom R8VEC.
• resid_spiral.m, computes the residual for a spiral velocity vector field.
• spiral_gnuplot.m, writes the velocity vector field to files for GNUPLOT.
• uv_spiral.m, computes a spiral velocity vector field.