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In probability density function (PDF) methods a transport equation is solvednumerically to compute the time and space dependent probability distribution ofseveral flow variables in a turbulent flow. The joint PDF of the velocitycomponents contains information on all one-point one-time statistics of theturbulent velocity field, including the mean, the Reynolds stresses andhigher-order statistics. We developed a series of numerical algorithms to modelthe joint PDF of turbulent velocity, frequency and scalar compositions forhigh-Reynolds-number incompressible flows in complex geometries usingunstructured grids. Advection, viscous diffusion and chemical reaction appearin closed form in the PDF formulation, thus require no closure hypotheses. Thegeneralized Langevin model (GLM) is combined with an elliptic relaxationtechnique to represent the non-local effect of walls on the pressureredistribution and anisotropic dissipation of turbulent kinetic energy. Thegoverning system of equations is solved fully in the Lagrangian frameworkemploying a large number of particles representing a finite sample of all fluidparticles. Eulerian statistics are extracted at gridpoints of the unstructuredmesh. Compared to other particle-in-cell approaches for the PDF equations, thismethodology is non-hybrid, thus the computed fields remain fully consistentwithout requiring any specific treatment. Two testcases demonstrate theapplicability of the algorithm: a fully developed turbulent channel flow andthe classical cavity flow both with scalars released from concentrated sources.

Joint probability density function modeling of velocity and scalar in turbulence with unstructured grids