Analytical and Numerical Solution of the Equation for the Probability Density Function of the Particle Velocity in a Turbulent Flow

A study has been made of the random motion of inertial particles in a homogeneous isotropic turbulent gas flow. Fluctuations of the gas velocity along the particle path were modeled by the Gaussian random process with a finite time of degeneracy of the autocorrelation function. A closed equation has been obtained for the probability density function of the particle velocity, for which two methods of numerical solution have been proposed: using the finite-difference scheme and using one based on direct numerical modeling of an empirical probability density function. The empirical probability density function was obtained as a result of the averaging of random particle paths, which are a solution of a system of ordinary stochastic differential equations. The results of numerical calculation have been compared with the analytical solution describing the dynamics of the probability density function of the particle-velocity distribution.