On a kinetic equation for the transport of particles in turbulent flows

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Abstract
The work described is part of a long‐term study in pursuit of a kinetic equation for particles suspended in a turbulent flow. This equation represents the transport of the average particle phase space density and can be used to derive the continuum equations and constitutive relations for a two‐fluid model of a dispersed particle flow and to establish the correct form of boundary conditions. A key role in this study is played by the equation of state of the dispersed particles, which is derived here for a Langevin equation of motion with a random driving force not limited to white noise. A suitable form of kinetic equation is then sought that reproduces this equation of state and predicts Gaussian spatial diffusion in the long‐term limit with the correct form for the diffusion coefficient in homogeneous stationary turbulence. It is shown that both these requirements are automatically satisfied by constructing forms for the phase space diffusion coefficient that preserve invariance to a random Galilean transformation. The most general form is shown to be an expansion in successively higher‐order cumulants of the aerodynamic driving force. When this force is a Gaussian process all terms apart from the first term contract to zero, i.e., the phase space diffusion current is of the form −[μ⋅(∂/∂v) +λ⋅(∂/∂x)]〈W〉 , where 〈W〉 is the average phase space density of particles with velocity v and position x at time t and μ and λ are phase space diffusion tensors whose components are memory integrals involving the correlation of the aerodynamic force along a particle trajectory. With both this form and the more general form of phase space diffusion current, the kinetic equation contracts to the classical Fokker–Planck equation in the white noise limit. Finally with the Gaussian form for the diffusion current, the form of the average phase space density is examined for the important case of a point source diffusing in an unbounded homogeneous stationary flow.