Derivation of a pdf kinetic equation for the transport of particles in turbulent flows

A transport equation for the particle phase space density (probability density function (pdf) kinetic equation) is derived for the motion of a dilute suspension of particles in a turbulent flow. The underlying particle equation of motion is based upon a Langevin equation but with a non-white noise driving force derived from an Eulerian aerodynamic force field whose statistics are assumed known. Specifically both the particle position and velocity are considered to be functionals of the driving force and an application of a more general form of the Furutsu-Novikov theorem leads to closed expressions for the phase space diffusion current (i.e. the net force due to the turbulence acting on the particles per unit volume of phase space). In the case of a Gaussian random driving force the closed expressions reduce to a simple Boussinesq form in gradients of the pdf with respect to particle velocity and position. As a practical application solutions of the equation are compared with results obtained from particle tracking in a developing simple shear generated by large eddy simulation.