**spiral_exact**,
a Fortran90 code which
defines a 2D velocity vector field that satisfies the
continuity equation, and writes the nodes and velocities to a file,
suitable for analysis or plotting.

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:

This is by construction, since we havedU/dX + dV/dY = 0.

which guarantees zero divergence.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

where C is a parameter.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)**.

The information on this web page is distributed under the MIT license.

**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.

GNUPLOT, Fortran90 programs which illustrate the use of the gnuplot graphics program.

NAVIER_STOKES_2D_EXACT, a Fortran90 library which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 2D.

STOKES_2D_EXACT, a Fortran90 library which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.

- spiral_exact.f90, the source code.
- spiral_exact.sh, compiles the source code.