On the continuum equations for dispersed particles in nonuniform flows

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Abstract
The continuum equations of a dispersed phase of solid, noncolliding particles in a nonuniform turbulent gas flow are derived from a kinetic equation for the transport of the average phase space density 〈W(v,x,t)〉 for particles with velocity v and position x at time t. The crucial feature of this equation is the form given for the phase space diffusion current j representing the net acceleration of a particle from interactions with turbulent eddies. This is based on Kraichnan’s Lagrangian history direct interaction approximation which gives j=−[(∂/∂v)⋅μ+(∂/∂x)⋅λ+γ]〈W(v,x,t)〉, where μ, λ, and γ are dispersion tensors dependent upon displacements in the velocity and position of a particle about v,x in times of order of the time scale of the fluctuating aerodynamic driving force. Most important these tensors are affected by spatial variations in the mean carrier flow velocity and external force as well as inhomogeneities in the carrier phase turbulence; γ is zero for homogeneous turbulence. This form for j is invariant under a random Galilean transformation. The dispersed phase momentum and energy equations deduced from the kinetic equation contain local gradient forms for the net fluctuating interphase force (per unit volume) and its rate of working. The former contains in general an asymmetric stress component (which adds to the particle Reynolds stress) as well as a body force dependent upon inhomogeneities in the turbulence. Conditions are examined under which the mass and momentum equations reduce to a convection–diffusion equation. The convection velocity in this case is the local carrier flow velocity plus a drift velocity proportional to local gradients of the turbulence (zero if the particles follow the flow). The diffusion coefficient is linearly related to the pressure/mean density of the dispersed phase via the particle response time. It is shown to be influenced by the local mean shearing of the carrier flow. Conditions are derived when this contribution can be ignored, the diffusion coefficient reducing to the form for homogeneous turbulence.