PARTICLE METHOD FOR TURBULENT FLOWS - INTEGRATION OF STOCHASTIC-MODELEQUATIONS

A numerical method is developed to integrate the stochastic differenti al equations that arise in a particle method for modelling turbulent f lows. These equations present several challenges, the foremost being t he presence of multiple time scales, the smallest of wh ich can be sig nificantly less than an acceptable time-step size, Delta t. The essenc e of the approach adopted is to transform and decompose the equations so that the stochastic components (which contain the small time scales ) appear as strictly linear stochastic differential equations. Analyti c solutions to these equations (with frozen coefficients) are then exp loited to produce a stable and accurate scheme. When the method is use d to advance the properties of N particles, the resulting numerical er ror can be decomposed into three contributions: statistical error, bia s, and time-stepping error. Comprehensive tests to study these errors are reported for two test cases. A novel variance-reduction technique is described that significantly reduces the statistical error, which s cales as N--1/2. In general, the bias is smaller, and scales as N-1 (i n accord with a simple analysis). The time-stepping error is less than 1% for a nondimensional time step of 1/32-which may be several times larger than the smallest time scale. Over the range of time-step size investigated, the dominant time-stepping error varies as Delta t(3/2). The method has the requisite stability, accuracy, and efficiency for incorporation in multi-dimensional particle methods. (C) 1995 Academic Press, Inc.