Local time-stepping algorithm for solving probability density function turbulence model equations

Abstract

A local time-stepping algorithm has been developed to improve the numerical ef ciency of Lagrangian particlebased 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 algorithm has been incorporated into the consistent hybrid nite volume/particle method. The performance of the local time-stepping method is evaluated in terms of numerical ef ciency and accuracy through application to a nonreacting 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 computational domain. 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. NomenclatureC u = Courant number based on particle velocity C Ä = Courant number based on turbulent frequency D b = bluff-body diameter F = mass-density function for uniform time stepping O F = mass-density function for the local time stepping Q f = mass-weighted probability density function M 2 = 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 for U W = isotropic Wiener process X = particle position x = axial distance x = sample space variable for X ± = Dirac delta function ¹ = numerical weight » x = grid-stretching factor in x direction » y = grid-stretching factor in y direction Ä = conditional mean turbulent frequency h i = mean eld 0 = rms uctuating quantity Q = mass-weighted eld

PDF modeling of a bluff-body stabilized turbulent flame

“…It has been shown that the consistent hybrid method is computationally more efficient than the best available alternative solution technique by a factor of an order of magnitude or more [2,6]. The numerical efficiency of the hybrid algorithm is further improved by the recent development of a local time stepping algorithm that has been shown to accelerate the global convergence of the hybrid method as much as by a factor of 10 depending on grid stretching [7]. Combined with the local time stepping algorithm, the consistent hybrid method thus makes the PDF methodology a feasible design tool for the practical applications in engineering or elsewhere.…”

“…In this code, the particle algorithm is largely based on the PDF2DV [25] code and the FV algorithm is a slight modification of the alternating direction implicit (ADI) scheme of Caughey [26]. The local time stepping algorithm that has been recently developed and shown to significantly accelerate the global convergence of the hybrid method [7] is also incorporated into the present hybrid method. Note that a similar local time stepping procedure has been recently presented by Mobus et al [27] for convergence acceleration of Lagrangian PDF methods.…”

“…Combined with the local time stepping algorithm, the consistent hybrid method thus makes the PDF methodology a feasible design tool for the practical applications in engineering or elsewhere. The complete descriptions of the consistent hybrid method and the local time stepping algorithm can be found in the references [5,6] and [7], respectively.…”

Scalar PDF and velocity-scalar PDF modelling of the bluff-body stabilised flame HM1 using ILDM

“…In order to ensure second-order accuracy, the 'midpoint rule' is used [13,14]. A local time-stepping algorithm developed in the framework of statistically stationary problems [15] is applied.…”

Evaluation of a Modified Reynolds Stress Model for Turbulent Dispersed Two-Phase Flows Including Two-Way Coupling

“…The Cartesian grid used for the computations consists of 120 cells in axial direction and 64 cells in the radial direction. A local time-stepping algorithm for the gas phase as well as for the dispersed phase was used to determine the time step used in each cell [15,17]. For the dispersed phase, five particles per class per cell were used and 13 droplet classes were considered.…”