On the derivation of the Navier–Stokes–alpha equations from Hamilton's principle

Abstract

We investigate the derivation of Euler's equation from Hamilton's variational principle for flows decomposed into their mean and fluctuating parts. Our particular concern is with the flow decomposition used in the derivation of the Navier–Stokes–α equation which expresses the fluctuating velocity in terms of the mean flow and a small fluctuating displacement. In the past the derivation has retained terms up to second order in the Lagrangian which is then averaged. The variation is effected by incrementing the mean velocity, while holding the moments of the products of the displacements fixed. The process leads to a mean Euler equation for the mean velocity. The Navier–Stokes–α equation is only obtained after making a further closure approximation, which is not the concern of this paper. Instead attention is restricted here to the exact analysis of Euler's equation. We show that a proper implementation of Hamilton's principle, which concerns the virtual variation of particle paths, can only be achieved when the fluctuating displacement and mean velocity are varied in concert. This leads to an exact form of Euler's equation. If, on the other hand, the displacement is held fixed under the variation, a term in Euler's equation is lost. Averaging that erroneous form provides the basis of the Navier–Stokes–α equation. We explore the implications of the correct mean equation, particularly with regard to Kelvin's circulation theorem, comparing it with the so called GLM and glm-equations.