Local Time-Stepping Algorithm for Solving Probability Density Function Turbulence Model Equations

A local time-stepping algorithmhas been developed to improve the numerical ef ciency of Lagrangianparticle-based Monte Carlo methods for obtaining the steady-state solutions of the modeled probability density function equations of turbulent reacting ows. On each step in the pseudo-time-marching algorithm, the properties of each particle are advanced by a time step, the magnitude of which depends on the particle’s spatial location. This algorithmhas been incorporated into the consistent hybrid nite volume/particle method. The performance of the local time-steppingmethod is evaluated in terms of numerical ef ciency andaccuracy throughapplication to a non-reacting bluff-body ow. For this test case, it is found that local time stepping can accelerate the global convergence of the hybrid method by as much as an order of magnitude, depending on the grid stretching. Additionally, local time stepping is found to improve signi cantly the robustness of the hybrid method mainly due to the accelerated convection of error waves out of the computationaldomain.The method is very simple to implement, and the small increase in CPU time per step (typically 3%) is a negligible penalty compared to the substantial reduction in the number of time steps required to reach convergence. Nomenclature Cu = Courant number based on particle velocity CÄ = Courant number based on turbulent frequency Db = bluff-body diameter F = mass-density function for uniform time steppingOF = mass-density function for the local time steppingQf = mass-weighted probability density function M2 = total number of grid cells m = particle mass q = mean particle-mass density R j = jet radius r = radial distance s = a global time variable U = particle velocity V = sample space variable forU W = isotropicWiener process X = particle position x = axial distance x = sample space variable for X ± = Dirac delta function = numerical weight »x = grid-stretchingfactor in x direction »y = grid-stretchingfactor in y direction Ä = conditionalmean turbulent frequency h i = mean eld 0 = rms uctuating quantity Q = mass-weighted eld I.