On flux terms in volume averaging

• Average fluxes are represented as the sum of global fluxes and local contributions.
• Phases subject to the same large-scale flux times their respective volume fractions.
• Remaining average flux is local: it can be calculated as flow was locally uniform.
• Result does not depend on any specific flow type or regime.
• Disperse flow treated in detail as an example of application of the general theory.

This note examines the modeling of non-convective fluxes (e.g., stress, heat flux and others) as they appear in the general, unclosed form of the volume-averaged equations of multiphase flows. By appealing to the difference between slowly and rapidly varying quantities, it is shown that the natural closure of these terms leads to the use of a single, slowly-varying combined average flux, common to both phases, plus rapidly-varying local contributions for each phase. The result is general and only rests on the hypothesis that the spatial variation of the combined average flux is adequately described by a linear function of position within the averaging volume. No further hypotheses on the nature of the flow (e.g., about specific flow regimes) prove necessary. The result agrees with earlier ones obtained by ensemble averaging, is illustrated with the example of disperse flows and discussed in the light of some earlier and current literature. A very concise derivation of the general averaged balance equation is also given.