Analytical and Numerical Solution of the Equation for the Probability Density Function of the Particle Velocity in a Tur
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 5, September, 2020
GENERAL PROBLEMS OF TRANSPORT THEORY ANALYTICAL AND NUMERICAL SOLUTION OF THE EQUATION FOR THE PROBABILITY DENSITY FUNCTION OF THE PARTICLE VELOCITY IN A TURBULENT FLOW I. V. Derevich and A. K. Klochkov
UDC 532.517.45
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 finitedifference 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. Keywords: probability density function, stochastic ordinary differential equation, two-phase turbulence, difference scheme, autocorrelation function, Green′s function, direct numerical modeling. Introduction. Two-phase turbulent flows are widespread in nature and have numerous engineering applications. Up to now, there have been no complete procedures for calculating two-phase flows. There are two alternative approaches, which are based on Lagrange and Euler concepts. In the Lagrange approach, calculation of the dynamics of a dispersed phase is based on the modeling of an enormous number of random particle paths (of the order of 104–108) followed by averaging to obtain practical information. In the Euler method, a system of dynamic equations of a dispersed phase is written in a continuous approximation, where a collective of particles is interpreted as a continuous medium. In the literature, ill-founded semiempirical assumptions are sometimes used to pass to a continuous description. A rigorous approach to the passage from the dynamics of individual particles to continuum equations is possible when the probability density function (PDF) of distribution of random parameters of the dispersed phase is used. Numerical implementation of the Lagrange approach requires much smaller intellectual resources than the Euler approach. However, processor resources necessary for calculating and storing a large array of random parameters of particles in Lagrange variables exceed substantially resources required for calculating averaged parameters of the dispersed phase in the Euler method. Advantages of the Lagrange and Euler approaches depend on a concrete type of problems. For example, if the characteristic residence time of a dispersed turbulent flow in the apparatus is comparable with the characteristi
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