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Analytical solution to validate code

Hi

I'm trying to validate my code written in spherical coordinates with some small analytical test cases.

I've already tested it with pure diffusion (in spherical coordinates). But now I want to test it with a pure convection.

The equation I'm interested in solving is (spherically symmetric, only radial dependence required)

$\frac{\partial{\phi}}{\partial{t}} + \frac{1}{r^2}\frac{\partial}{\partial{r}}(r^2u_r\phi) = 0$

where

is the radial velocity and $\phi$ is the scalar variable.

This is different to the linear advection equation since I'm not assuming that the velocity is constant.

Does anyone know how I could obtain an analytical solution to the above equation or would it be better to just check it with a linear advection equation?

You should be able to obtain an analytical solution if for example you take $u_{r}$ as being constant.

Or you can try to assume an analytical variation for this, say:

$u_{r}$ =1/ $r^{2}$

Once you've picked an expression in r for $u_{r}$ you can differentiate and try to find solutions of the form:

$\Phi$ = F(r)T(t)

If you are lucky you should be able to find a series solution for your function.

Do

Not to be too nit-picky but I think technically you are verifying your code. One really powerful tool for code verification is the method of manufactured solutions. Two good articles to get started are:

P. J. Roache, Code Verification by the Method of Manufactured Solutions, Journal Fluids Engineering, Vol 124 pp4-10 March 2002

K. Salari and P. Knupp, Code Verification by the Method of Manufactured Solutions, Sandia Report SAND2000-1444, June 2000

Good luck.