Vol.4 : Computational Turbulent Incompressible Flow

Dear Claes,

Here are the promised comments on you paper with J. Hoffman on "Finally: Resolution of d'Alembert's Paradox". After some discussion with Paolo and John, I think that JMFM could publish your paper as a note for stimulating discussion of an important problem if you agreed to the following suggestions:

1. The overall attitude of the paper should be less provocative and its tone less pompous. I should look like a note pointing out some observations which are hoped to shed new light at an old problem. This should initiate new discussion and investigation of the problem. Please avoid formulations which could be understood as calling classical Fluid Mechanics ignorant and misled (see the paragraph at the end of section 3).

2. JMFM is a mathematical paper and the terms "proof", "show", etc. have a certain meaning of mathematical rigor. Your "Resolution" is not mathematically rigorous, neither is Prandl's. You should make clear what you mean by "Resolution" and its possible consequence to numerical computation.

3) Your paper offers a rigorous proof of the instability of classical Euler solutions, which may actually not be new, but the rest of the argument is heuristic based on computational observation which needs discussion. Please make clear which claim is mathematically "proved" and which is only suggested by numerical observation. Also the term "turbulent behavior" is used without explanation which may confuse readers with a background in turbulence theory.

4. Much of the citicism of the first reviewers concern your trusting in computational results which may be blured by numerical diffusion. At this point a bit more detail on the numerical scheme and the precautions taken to limit the diffusive effects would be helpful. You could also add some remarks on the validation of the computer code used particularly the studies of the effects of numerical viscosity.

5. The title should sound less emphatic, for example: "A new approach to resolve d'Alembert's paradox" or simply (what I would prefer) "A note on d'Alembert's paradox".

6. Some statements could be a bit weakened, e.g., "We propose a new ..." rather than "We present a new ...". I actually find the tone of the Outline of the paper on page 2 very appropriate.

7. From another experts opinion on the paper, I freely recall a statement concerning your argument on vorticity creation in irrotational flow. Please oberve this comment.

"The mathematical "proof" about the possible generation of vorticity in a perfect fluid which is initially irrotational and is in contact with a rigid boundary given in section 7 is questionable. Helmholtz theorem, which states that in an Euler fluid if a particle carries no vorticity initially then it carries no vorticity at all times, does not require the boundedness of the gradient of velocity, but only that the Jacobian (from Eulerian to Lagrangean coordinates) is not zero. However, if the Jacobian vanishes, everything can happen and whatever happens I would hardly describe as physically reasonable."

I hope that you will agree on these, I think, minor suggestions which would give the paper a somewhat milder and more acceptable attitute without changing its intended impact. If this is the case please send me a revised version. I will be happy to explain my comments in more detail if necessary on the phone.

Best regards,


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