A filtered mass density function approach for modeling separated two-phase flows for LES I: Mathematical formulation

The overall objective of this study is to develop a full velocity-scalar filtered mass density function (FMDF) formulation for large eddy simulation (LES) of a separated two-phase flow. Required in the development of the two-phase FMDF transport equation are the local instantaneous equations of motion for a two-phase flow previously derived by Kataoka. In Kataoka’s development, phase interaction terms are cast in terms of a Dirac delta distribution on the phase interface. For this reason, it is difficult to close these coupling terms in the instantaneous formulation and this difficulty is propagated into the phase-coupling terms in the FMDF transport equation. To address this point a new derivation of the local instantaneous equations for a separated two-phase flow is given. The equations are shown to be consistent with the formulation given by Kataoka, and in the development, a direct link between the conditionally surface-filtered coupling terms, arising in the FMDF formulation, and LES phase-coupling terms is established. Clarification of conditions under which conditionally filtered interphase conversion terms in the marginal FMDF transport equations may be disregarded in a separated continuum-dispersed phase flow is discussed. Modeling approaches and solutions procedures to solve the two-phase FMDF transport equation via Monte-Carlo methods are outlined.