# spiral_data

spiral_data, a MATLAB code which samples a velocity vector field that satisfies the continuity equation, and writes the nodes and velocities to a file, suitable for display using gnuplot().

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.
```
This is by construction, since we have
```
U(X,Y) =  10 * d/dY ( PHI(X) * PHI(Y) )
V(X,Y) = -10 * d/dX ( PHI(X) * PHI(Y) )
```
which guarantees zero divergence.

The underlying function PHI is defined by

```
PHI(Z) = ( 1 - cos ( C * pi * Z ) ) * ( 1 - Z )^2
```
where C is a parameter.

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).

### Languages:

spiral_data 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:

gnuplot_test, MATLAB codes which illustrate 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.

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

### Source Code:

Last revised on 24 March 2019.