Novel porous media formulation for multiphase flow conservation equations

摘要

Multiphase flows consist of interacting phases that are dispersed randomly in space and in time. An additional complication arises from the fact that the flow region of interest often contains irregularly shaped structures. While, in principle, the intraphase conservation equations for mass, momentum, and energy, and their initial and boundary conditions can be written, the cost of detailed fluid flow and heat transfer analysis with explicit treatment of these internal structures with complex geometry and irregular shape often is prohibitive, if not impossible. In most engineering applications, all that is required is to capture the essential features of the system and to express the flow and temperature field in terms of local volume-averaged quantities while sacrificing some of the details. The present study is an attempt to achieve this goal by applying time averaging after local volume averaging. Local volume averaging of conservation equations of mass, momentum, and energy for a multiphase system yields equations in terms of local volume-averaged products of density, velocity, energy, stresses, and field forces, together with interface transfer integrals. These averaging relations are subject to the following length scale restrictions: d l L, where d is a characteristic length of the pores or dispersed phases, l a characteristic length of the averaging volume, and L is a characteristic length of the physical system. Solutions of local volume-averaged conservation equations call for expressing these local volume-averaged products in terms of products of averages. In nonturbulent flows, this can be achieved by expressing the "point" variable as the sum of its intrinsic volume average and a spatial deviation. In turbulent flows, the same can be achieved via subsequent time averaging over a duration T such that τ_(HF) T τ_(LF), where τ_(HF) is a characteristic time of high-frequency fluctuation and τ_(LF) is a characteristic time of low-frequency fluctuation. In this case, and instantaneous "point" variable ψ_k of phase k is decomposed into a low-frequency component ψ_(kLF) and a high-frequency component ψ′_k, similar to Reynolds analysis of turbulent flow. The low-frequency component consists of the sum of the local intrinsic volume average ~(3i) < ψ_k >_ (LF) and its local spatial deviation ψ_(kLF) Time averaging then reduces the volume-averaged products to products of averages plus terms representing eddy and dispersive diffusivities of mass, Reynolds and dispersive stresses, and eddy and dispersive conductivities of heat, etc.