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But is there a surface geometry of scale so small that Reynold's number ceases to have meaning? I want to give an example where the "scale" doesn't necessarily mean physical scale. To do this, I will consider increasing altitude in Earth's atmosphere, therefore decreasing density. Consider increasing altitude at which a plane (or a rocket) is moving. ...

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To add to Lalylulelo's excellent answer, the Reynolds number only has meaning with respect to a particular flow geometry. That is, it's only useful in comparing two flows of the same configuration. A Reynolds number corresponding to laminar flow in a pipe geometry might correspond to unstable or turbulent flow in some other geometry (never mind the fact ...

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The answer of user3823992 is correct: Reynolds number will cease to have meaning when the hypothesis of continuum mechanics will cease to be verified. To complete his answer, if the characteristic length scale of the studied geometry is close to the free mean path (i.e. Knudsen number close to 1), you can no longer consider the Navier-Stokes equations to ...

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As far as I understood this distinction in plasma physics, you refer to weak/strong in terms of your Rayleigh decomposition of fields: Any field $X =$, $\vec v$,$\rho$,$T$,$\vec B$ can be decomposed into a background part $\bar X$ and a fluctuating part $X'$, while constructing $X'$ thus that the average $\bar{X'}=0$. The usual scheme for wave-analysis is ...

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I'd say part of the answer must be that whatever dynamic variable you use, like Enstrophy, Vorticity, their potential analogues, etc. those are always 'filtered' fields. Filtered in the sense, that you start with the velocity field $\vec v = \sum u_i \vec e_i$ that has full information over the dynamics and then apply some operators (integration and ...

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Under the definition you use for a Fourier transform, which is the discrete Fourier transform, then we have: $$\mathbf{u}(\mathbf{x}) = \sum_{i} \hat{\mathbf{u}}(\mathbf{k}_{i}) e^{i \ \mathbf{k}_{i} \cdot \mathbf{x}}$$ which has the inverse given by:  \hat{\mathbf{u}}(\mathbf{k}) = \sum_{j} \mathbf{u}(\mathbf{x}_{j}) e^{- i \ \mathbf{k} \cdot ...

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