STATISTICAL-MECHANICAL DESCRIPTIONS OF TURBULENT COAGULATION

A fundamental tenet of statistical mechanics is that the rate of colli sion of two objects is related to the expectation value of their relat ive velocities. In pioneering work by Saffman and Turner [J. Fluid Mec h. 1, 16 (1956)], two different formulations of this tenet are used to calculate the collision kernel Gamma between two arbitrary particle s ize groups in a turbulent flow. The first or spherical formulation is based on the radial component w(r) of the relative velocity w between two particles: Gamma(sph) =2 pi R-2[\w(r)\], where w(r)= w.R/R, R is t he separation vector, and R=\R\. The second or cylindrical formulation is based on the vector velocity itself: Gamma(cyl)= 2 pi R-2[\w\], wh ich is supported by molecular collision statistical mechanics. Saffman and Turner obtained different results from the two formulations and a ttributed the difference to the form of the probability function of w used in their work. A more careful examination reveals that there is a fundamental difference between the two formulations. An underlying as sumption in the second formulation is that the relative velocity at an y instant is locally uniform over a spatial scale on the order of the collision radius R, which is certainly not the case in turbulent flow. Therefore, the second formulation is not expected to be rigorously co rrect. In fact, both our analysis and numerical simulations show that the second formulation leads to a collision kernel about 25% larger th an the first formulation in isotropic turbulence. For a simple uniform shear flow, the second formulation is about 20% too large. The two fo rmulations, however, are equivalent for treating the collision rates a mong random molecules and the gravitational collision rates. (C) 1998 American Institute of Physics.