I have a question about non-dimensionalization of the incompressible Navier-Stokes (NS) equations.
My understanding is that the purpose of non-dimensionalization is to
"collapse" solutions onto one curve so that the solution space can
be explored with fewer parameters. It can also give insight into
the physics. Consider the dimensional incompressible NS equation:
$\partial_t u_i + u_j \partial_j u_i = -\rho^{-1}\partial_i p + \nu \partial_{jj} u_i, \qquad \partial_j u_j = 0.$
Non-dimensionalizing using the scales $ [x,y,z] = L, [u] = U, [p] = \rho U^2$. Plus
$[t] = L/U, \qquad \qquad \text{convective time}$
$[t] = L^2/\nu, \qquad \qquad \text{diffusive time}$
Results in two different equations:
$\partial_t u_i + u_j \partial_j u_i = -\partial_i p + Re^{-1} \partial_{jj} u_i, \qquad \qquad \qquad \text{using convective time scale}$
$Re^{-1}\partial_t u_i + u_j \partial_j u_i = – \partial_i p + Re^{-1}\partial_{jj} u_i, \qquad \qquad \text{using diffusive time scale}$
Here $Re = \frac{UL}{\nu}$ is the Reynolds number.
Questions
1) Why is it that these two equations are not the same given the fact that $Re$ is defined exactly the same between the two equations?
2) Can the solutions from each equation be related somehow?
3) Is the span of both parameter spaces the same for both equations?
I'm looking for a methodical view of answering these questions since I'm interested in how understanding the answers extend to a more complicated problem.
I appreciate any help!
Best Answer
The two equations only differ by a trivial rescaling of the time coordinate, and are therefore equivalent. If $u(x,t)$ is a solution of the first equation, then $u(x,{\it Re}\, t)$ is a solution of the second. Re is physically relevant because it governs the relative importance of the advective and dissipative terms. Multiplying $\partial_t u$ by a constant is just a rescaling, and has no physical significance.