# spiral_exact

spiral_exact, an Octave 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.
```
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_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.

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

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

pendulum_nonlinear_exact, an Octave 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, an Octave code which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.

### Source Code:

Last revised on 01 July 2023.