The multi-flow framework and the generalized Lagrangian mean theory

Abstract

A mathematical framework for describing di erent ows of the same uid and their relationships is introduced. This theory, named the Multi-ow Framework (MfF), considers the referential, spatial, and relative descriptions of ÿelds of both pure and mixed distributions. A mixed distribution, as opposed to a pure distribution, results when the transformation between descriptions of some quantity in one ow is carried out using the particle or position distributions from another ow. The basic kinematics of the mixed distributions is developed with emphasis on those ÿelds giving the circulation and ux in one ow per unit of length and area of another ow. These ÿelds are called here the relative circulation and the relative ux ÿelds, respectively.The MfF is applied to both the material and the spatial ensemble average theories. The results in the material average framework are then applied to obtain a new formulation and interpretation of the conservation of mass, in terms of the mean-ow density, and balance of linear momentum given in the Generalized Lagrangian Mean theory. The MfF puts the GLM mean-ow density into an exact correspondence with the reference mass density, of every ow at the average position, per unit of spatial volume of the average ow relative to the reference (initial) volume. The MfF formulates the GLM momentum equation as the spatial ensemble average of the relative circulation balance of momentum relative to the actual ow in the material average ow, expressed in terms of the rate of change of the relative circulation velocity.