IMPROVED EDDY INTERACTION MODELS WITH RANDOM LENGTH AND TIME SCALES DAVID I. GRAHAM School of Mathematics and Statistics, University of Plymouth, Drake Circus, Plymouth, Devon, PL4 8AA, UK (Received 28 August 1996; in revised form 1 September 1997) AbstractÐThe eddy interaction model has been used extensively to model particle dispersion in com- plex turbulent ¯ows. In the model, a particle undergoes a series of interactions with random-velocity eddies. Interaction times, which determine particle dispersion are in¯uenced by the eddy velocity and eddy length and time scales. In general, these can all be random. Recent research has shown some of the shortcomings of the original model, and has suggested improvements be made to ensure that models account for the crossing trajectories, inertia and continuity eects. In this present paper, the performance of variants of the improved model in predicting dispersion of particles in a simple turbu- lent ¯ow one investigated. Each variant is given a dierent combination of eddy length and time distri- butions. Numerical results are compared with previously published analytical results. It is demonstrated that the eects noted above are allowed for in each of the model combinations considered. # 1998 Elsevier Science Ltd. All rights reserved Key Words: particle dispersion, turbulent ¯ow, Lagrangian models, integral scales 1. INTRODUCTION The eddy interaction model (EIM) developed by Gosman and Ioannides (1981) is one of the simplest and most frequently-used methods for simulation of turbulent particle dispersion. In the EIM, individual particles undergo a series of interactions with random-velocity ¯uid eddies. A particle interacts with an individual eddy so long as the particle remains within that eddy and during each interaction the eddy velocity remains constant. In the original model, the particle remains within the eddy until either the eddy ``dies'' when the ``eddy lifetime'' t e is exceeded, or the particle ``crosses'' the eddy, for example, when the separation between the particle and the centre of the eddy exceeds the eddy length l e . Particle motions are determined by evaluating the in¯uence of viscous drag and other forces over the duration of the interaction. On exit from an eddy, the particle immediately enters another eddy with generally dierent characteristics. Eventually, particle phase data are determined by statistical averaging over a large number of trajectories. For particle dispersion in homogeneous isotropic and stationary turbulence (HIST), the orig- inal model of Gosman and Ioannides (1981) would give eddy length and time scales which do not change from eddy to eddy. More recently, however, randomly-sampled scales have been used. Kallio and Reeks (1989) used time scales sampled from an exponential probability distri- bution, while Burnage and Moon (1990) used exponential distributions for both time and length scales. Wang and Stock (1992) used several dierent time scale distributions and developed a general method to ®nd the Lagrangian integral time scale t L for a given distribution. In this paper, we investigate the performance of four dierent eddy interaction models with random length and times scales. We note here that, traditionally, t e has been called the ``eddy lifetime''. Following Graham (1996b), however, and to avoid confusion later, we call t e the ¯uid particle inter- action time or FPIT. Wang and Stock (1992) showed that the probability distribution cho- sen for the FPIT determines the Lagrangian ¯uid velocity auto-correlation. Graham and Int. J. Multiphase Flow Vol. 24, No. 2, pp. 335±345, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0301-9322/98 $19.00 + 0.00 PII: S0301-9322(97)00066-9 335