Scale Invariance

Solutions of the Navier-Stokes equations with slip and vanishingly small viscosity, which can be viewed as regularized solutions of the Euler equations, are scale invariant in the sense that a change of scale in space leaves the solution invariant. This means that the flow around a small wing looks the same as around a big wing. This is not true in the case of no-slip with viscous boundary layers setting an absolute scale.

It follows in particular that the crucial separation pattern at rounded trailing edge of a wing should remain the same as the diameter of the edge tends to zero, and the sharp edge demanded by Kutta-Zhukovsky’s circulation would seem to be unnecessary. This is confirmed by observations showing that diameters smaller than 1% of the chord length give the same lift and drag, see NACA 1938 and NACA 1956.

In this argument we assume that the Reynolds number is so large that viscosity effects do not change the flow pattern, which is to be expected for airplanes but not for insects.

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