An equation for the probability density, velocity, and temperature of particles in a turbulent flow modelled by a random Gaussian field☆

Abstract The motion and heat exchange of a dispersed admixture of solid particles suspended in turbulent flow are considered on the assumption that the volume concentration of the particles is small, so that the role of collisions between the particles is negligible. The fluctuating motion of the particles is governed by viscous drag from the surrounding turbulent flow and by forces of molecular origin which produce Brownian motion of the particles. Fluctuations in the particle temperature are caused by fluctuations in the heat flux to the particles in the random temperature field of the fluid phase. The turbulent random velocity and temperature fields of the carrier phase are modelled by a Gaussian random process with a given autocorrelation function. In spite of the fact that describing a real turbulent flow by a Gaussian process is a somewhat approximate procedure, this approach, by virtue of its simplicity, is widely used to construct equations for probability density distributions for turbulent flow velocity, and also for studying the turbulent diffusion of passive admixtures, and hence for admixtures of inertial particles /1–8/. Brownian motion of the particles is modelled by a Gaussian process that is δ-correlated with time. A closed equation for the probability density functions (PDFs) of the velocity and temperature of particles in inhomogeneous turbulent flow is constructed using the method of functional differentiation; on the basis of the PDF equations a system of equations for the first and second moments of the fluctuations of the dynamic and thermal characteristics of the solid phase is obtained.