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Why are people interested in solving the Navier-Stokes equations if people can find a good approximate solution?

Also especially when people have supercomputers?

Joffan
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Victor
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  • Related: http://math.stackexchange.com/questions/717480/why-are-mathematician-so-interested-to-find-theory-for-solving-partial-different – zxq9 Feb 16 '15 at 06:21
  • @mdg: IIRC, Navier-Stokes is one of those equations where we usually can prove a solution exists (certainly when the boundary conditions are taken from physics) even if we can't prove what that solution is. Also, compare it with the root of a second degree poynomial. Because that's solved, we know there are no real solutions when b*b < 4*a*c. – MSalters Feb 16 '15 at 15:13

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The issue is that it's actually exceptionally hard to find a good approximation, even with supercomputers. Furthermore, any solution you compute with sufficient accuracy is typically only relevant for the precise initial conditions you specified. In reality, parameters vary and it is more interesting to solve for broader problems that incorporate this variability.

In addition, most problems where the Navier-Stokes equations apply include other multi-physical phenomena. For example: aeroelastic problems with compressible flow; combustion chemistry in flow fields; magnetohydrodynamics of confined plasmas.

In such a case, solving the Navier-Stokes equations gets you part of a solution. There's a whole lot more that goes on. If we could establish solvability of N-S, we could develop methods that were more efficient in solving these problems. In some cases, we do just this. For example, Large Eddy Simulation uses known properties of the N-S equations to compute fluid profiles. It does this by essentially filtering the N-S equations; to paint with a broad brush, filtering reduces the computational complexity and makes the equations easier to solve with sufficient accuracy.

Emily
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    As a note: there are actually quite a large number of "good approximations" by the definition that they work very very well in a remarkably large number of cases. Unfortunately for those who like approximations, those few cases which they don't work well show up frustratingly often in real life practical situations. As an example, you may be able to design a rotor for which the approximations are good. However, let it age a bit, and pit from random impacts, and suddenly those approximations fall apart. – Cort Ammon Feb 16 '15 at 06:06
  • @CortAmmon Indeed, and in reality we needn't even go that far. For example, compare two brand-new, factory-made fighter jets with consecutive serial numbers. Put them though vibration tests to identify structural modes. It is almost guaranteed that these structural modes will differ, hence they will have different aeroelastic responses. This will only be exacerbated over time. In fact, there is a movement to attempt to create "digital twins" of in-service aircraft, simply because the computational models based on CAD just aren't accurate enough for in-field analysis. – Emily Feb 16 '15 at 16:53