Analytical solution to validate code

lost.identity · July 22, 2010, 07:55

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?

DoHander · July 22, 2010, 11:28

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

ptb · July 22, 2010, 20:38

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.

July 22, 2010, 07:55	Analytical solution to validate code	#1
lost.identity Senior Member Join Date: Apr 2009 Posts: 118 Rep Power: 17	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?

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July 22, 2010, 11:28		#2
DoHander Senior Member Join Date: Nov 2009 Posts: 411 Rep Power: 20	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

July 22, 2010, 20:38		#3
ptb New Member Peter Join Date: Apr 2009 Posts: 4 Rep Power: 17	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.