Two-particle dispersion in turbulentlike flows

Lagrangian calculations of average concentrations require knowledge of one-particle statistics. However, if Lagrangian calculations of concentration fluctuations and concentration covariances are to account for turbulent mixing associated with relative dispersion, then such calculations must incorporate some features and properties of two-particle statistics @1#. The calculation of concentration covariances is important in the prediction of reaction rates in chemical reactors and in the atmosphere because chemical reaction rates depend on concentration covariances and not on average concentrations. The calculation of concentration fluctuations is also important for air-quality control, combustion, and pollutant dispersal in geophysical flows. Perhaps the most important statistic of two-particle dispersion ~certainly the most frequently studied! is the mean square distance between two fluid elements ~also referred to as particles in this paper!, D(t), which is of course a function of time t . In certain circumstances, such as downstream of a linear concentration gradient @1#, D(t) is the only twoparticle statistic needed to calculate concentration fluctuations. In general, D(t) is one of the fundamental quantities of interest in the theory of turbulent dispersion. In a series of papers starting in 1926, Richardson @2# studied the turbulent diffusivity (d/dt)D(t) as a function of the distance D between two particles advected by atmospheric turbulence. Richardson’s empirical finding, (d/dt)D;(D), implies D;t ~neglecting the initial distance D0 between pairs of particles under the assumption that D0 !D2 at a time t that is sufficiently large!. Obukhov @3# and Batchelor @4# derived Richardson’s dispersion law theoretically by applying Kolmogorov’s similarity arguments to D(t) and obtained D(t);et in an intermediate inertial range of times t ~e is the average rate of dissipation per unit mass of fluid!. When the time t is much larger than correlation integral time scales, D(t);t because the two particles move apart independently