Advection velocity phase space

The PBE is a simple continuity statement written in terms of the NDE. It can be derived as a balance for particles in some fixed subregion of phase and physical space (Ramkrishna, 2000). Let us consider a finite control volume in physical space O and in phase space with boundaries defined as dO. and dO., respectively. In the PBE, the advection velocity V is assumed to be known (e.g. equal to the local fluid velocity in the continuous phase or directly derivable from this variable). The particle-number-balance equation can be written as... 

In the case of turbulent advection velocity, the transported quantity in the PBE (i.e. the NDF) fluctuates around its mean value. These fluctuations are due to the nonlinear convection term in the momentum equation of the continuous phase. In turbulent flows usually the Reynolds average is introduced (Pope, 2000). It consists of calculating ensemble-averaged quantities of interest (usually lower-order moments). Given a fluctuating property of a turbulent flow f>(t,x), its Reynolds average at a fixed point in time and space can be written as... 

The level set method [1,2] is an interface capturing scheme in which the evolution of an interface is tracked by evolving a level set function (f>(x, t) throughout space but focusing on the location of a specific level surface (or curve in 2D) of (f> to capture the position of the desired interface. In contexts other than two-phase fluid flow, the evolution equation for the level set function can account for advection by a specified external velocity field v(x, t), as well as propagation of the interface in the direction of the local normal vector at a constant speed a, and the motion of the interface in the normal direction at a speed proportional to the local curvature k with proportionality constant —b. The resulting evolution equation for cj) then takes the form... 

For an inflnitesimally small control voltrme, advective transport is the product of the phase velocity with the concentration gradient in space ... 

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