Othervise, just for incompressible flows you can assume density to be one and set viscosity to 1/Re; in this manner you have non-dimensional equations.
Andreas has suggested a very good way. I normally do it in a different way. I run the case with dimensional quantities, and presented my results in non-dimensional form. The reason of doing this is that for the same problems, different researcher may choose different definition for their dimensionless parameters. Also, I will purposely tests the two different dimensional problem sets which have exactly the same non-dimensional quantities just to validate my results.
I am also agree with Andreas suggestion....It is the correct way to convert dimensional form into non-dimensional form. In my FVM based code also we do similar procedure.
Non-dimensional equations can be solved using the free open-spurce code OpenFOAM(R). If you look at openfoam.com you can see how the sover is implemented.
Othervise, just for incompressible flows you can assume density to be one and set viscosity to 1/Re; in this manner you have non-dimensional equations.