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I am looking for a nice way to motivate the Strouhal number definition. Let me illustrate what I mean on the Reynolds number. (As ususal, $\mathbf{u}$, $p$, $\rho$, $\nu$ denote the flow velocity, pressure, density and kinematic viscosity respectively.)

Sure, there are multiple good ways to show the importance and properties of the Reynolds number. I particularly like the one based on the momentum equation (the Navier-Stokes equation) scaling. The equation reads:

$$ \frac{\partial \mathbf{u}}{\partial t} + \left( \mathbf{u} \cdot \nabla \right)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u} $$

then by introducing $X_i = x_i/L$, $U_i = u_i/V$, $P = p/(\rho V^2)$, $\tau = \nu t/L^2$ we obtain:

$$ \frac{\partial U_i}{\partial \tau} + \text{Re}\left( U_k \frac{\partial U_i}{\partial X_k} + \frac{\partial P}{\partial X_i} \right) = \frac{\partial U_k}{\partial X_k \partial X_k} $$

i.e. the Reynolds number $\text{Re} = \frac{UL}{V}$ appears naturally as the only control parameter of the scaled system.

And now, is there a way to obtain the Strouhal number by a similar procedure?

Notes and notions:

  • The particular beauty of the aforementioned procedure is that "it is autonomous". I presume that for the Strouhal number there should be an assumption such as "let the flow instability be described by a time-harmonic function".
  • It must be based on Euler equations rather then Navier-Stokes equations.
  • Would recasting the momentum equation in Crocco's form be of any help?
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In your definition of the dimensionless time you have assumed that the characteristic scale for time is $\frac{L^2}{\nu}$. If you instead assume that the characteristic scale is the inverse of the vortex-shedding frequency $f^{-1}$ and redo the analysis you will retrieve the Strouhal number. You will need to rescale the characteristic scale for the pressure accordingly.

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