Some spatially integrated time-averaged identities for fluids

Abstract

This paper deals with the derivation of a series of seemingly simple but ignored relations, and presents several identities for fluids never obtained before (relations of integrals over different sets of independent variables and connected by an invertible mapping). These identities are based on mass conservation and a mathematical transform with no restriction on dynamics. They are, however, crucial to some fundamental concepts and the interpretation of results from dynamics. The identity for momentum is of most importance and, when averaged over time, yields a relationship between the spatially integrated and time-averaged Lagrangian momentum and the spatially integrated and time-averaged Eulerian momentum. For a constant density fluid, the averaged identity reduces to a relation between the integrated mean displacement of the particles and the integrated mean Eulerian velocity. For an exactly oscillatory flow in an Eulerian description this identity yields a zero integrated mean displacement of particles. In the case of a progressive surface gravity wave, which is periodic but not exactly oscillatory in an Eulerian description such that there are no particles above the trough during part of the period, the mean momentum identity ensures that the integrated mean Eulerian momentum is equal to the integrated mean momentum of the particles. Therefore, there is essentially no superiority as to which description gives a better estimate of the total momentum or total transport at the same order of approximation. The widely used relation, that the Lagrangian velocity equals the Eulerian velocity plus the Stokes velocity, is not based on a 1 to 1 invertible mapping and is therefore ambiguous. This relation is not valid where there is a discontinuity, particularly above the wave trough. It can therefore give an incorrect result on the Lagrangian velocity. The Generalized Lagrangian Mean (GLM) theory uses a mapping between the mean positions of particles and the particles themselves and apparently avoids the non-invertibility. However, this mapping is actually not invertible in general because different particles may have the same mean position.