The multi-flow framework and the generalized Lagrangian mean theory

A mathematical framework for describing different flows of the same fluid and their relationships is introduced. This theory, named the Multi-flow Framework (MfF), considers the referential, spatial, and relative descriptions of fields 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 flow is carried out using the particle or position distributions from another flow. The basic kinematics of the mixed distributions is developed with emphasis on those fields giving the circulation and flux in one flow per unit of length and area of another flow. These fields are called here the relative circulation and the relative flux fields, 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-flow density, and balance of linear momentum given in the Generalized Lagrangian Mean theory. The MfF puts the GLM mean flow density into an exact correspondence with the reference mass density, of every flow at the average position, per unit of spatial volume of the average flow 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 flow in the material average flow, expressed in terms of the rate of change of the relative circulation velocity. (C) 2003 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.