Averaged-equation and diagrammatic approximations to the average concentration of a tracer dispersed by a Gaussian random velocity field

Abstract

Approximate methods for predicting the spread of a passive tracer due to a Gaussian random velocity field are examined. Previous methods such as the direct-interaction approximation [J. Fluid Mech. 11, 257 (1961)] and Phythian’s short-time expansion [J. Fluid Mech. 67, 145 (1975)] make predictions for the second moment of the concentration that are qualitatively consistent with numerical simulations [Phys. Fluids 13, 22 (1970)]. However, it is shown that the higher moments obtained from these approximations are incorrect. The derivations of the direct-interaction approximation using a diagrammatic expansion and the method of averaged equations are reviewed [Phys. Fluids A 1, 47 (1989)]. A higher-order (two-body) approximation is developed which makes predictions for the fourth moment that are qualitatively correct in both the short- and long-time limits.