I'm wondering, will the non dimensionalized NS equations produce the same numerical results with the dimensional NS equations? Technically, it's the same equations, but not really.
Numerical results are approximation of the given PDE. If dimensionless NS equation and dimensioned NS equations produce same results, numerical results would do the same.
https://en.wikibooks.org/wiki/Advanced_Mathematics_for_Engineers_and_Scientists/Nondimensionalization
the dimensional result and non dimensional results are approximately same
Physical phenomena are not dependent on systems of units. Therefore, the results are equivalent (not the same, since the quantities involved in dimensional and in non dimensional equations are different). With the dimensionless results, you can recover quantities like velocity, pressure, etc... in any system of unity. The recovered results coincide with the ones obtained from the original (dimensional) equations.
I agree with everyone. However, to be absolutely precise, it will also be essential to ensure that similar convergence criteria are used when iterating towards a steady state, so that exactly the same relative error is obtained numerically. And when solving for unsteady flows using an implicit scheme, both the timestep-wise convergence criterion and the timestep need to be precisely equivalent between the dimensional and nondimensional forms of the system of equations. This is particularly important when close to a bifurcation point.
I also agree with everyone in principle, in that rendering the equations non-dimensional does not alter the physics of the problem. However, when solving the equations numerically, the order of magnitude difference among the terms will result in different truncation errors between the two forms. This is exactly what Prof. Rees is also saying. The two solutions would be exactly the same as long as the convergence criteria and the truncation errors are the same.
When you nondimensionalize your equation, you 'scale' your variables to have nondimensional variable counterparts. The solution you get is for the new varibales. You have to 'scale back' to the solution in your original variables(if scale is y1=y/D , then scale back is y = y1 D). This will be the same solution as that ou get when solved with the dimensioned equation. Ofcourse, when you 'scale' by a very large or very small dimensioned value D, you may encounter some problems in convergence and truncation error as Profs. Rees and Rokhsaz noted above. But these are rare situations when proper dimension units are not used.(That is, Do not use cm to measure molecule size. Rather use tiny units like micro(?) or nano) Hope this helps.
Non-dimensionalize your dimensional results should get exactly the same results as the dimensionless code. Sometimes to validate my simulations, I purposely run two different sets of dimensional groups of the same dimensionless quantities to check if the results in dimensionless form are exactly the same.
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