Shigan Chu, Andrea Prosperetti, On flux terms in volume averaging, International Journal of Multiphase Flow, Volume 80, April 2016, Pages 176-180, ISSN 0301-9322, http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.12.009.
Abstract: 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.
Keywords: Volume averaging; Separation of scales