Analysis of the clustering of inertial particles in turbulent flows

An asymptotic solution is derived for the motion of inertial particles exposed to Stokes drag in an unsteady random flow. This solution provides an estimate for the sum of Lyapunov exponents as a function of the Stokes number and Lagrangian strain-and rotation-rate autocovariance functions. The sum of exponents in a Lagrangian framework is the rate of contraction of clouds of particles, and in an Eulerian framework, it is the concentration-weighted divergence of the particle velocity field. Previous literature offers an estimate of the divergence of the particle velocity field, which is applicable only in the limit of small Stokes numbers [Robinson, Comm. Pure Appl. Math. 9, 69 (1956) and Maxey, J. Fluid Mech. 174, 441 (1987)] (R-M). In addition to reproducing R-M at this limit, our analysis provides a first-order correction toR-M at larger Stokes numbers. Our analysis is validated by a directly computed rate of contraction of clouds of particles from simulations of particles in homogeneous isotropic turbulence over a broad range of Stokes numbers. Our analysis and R-M predictions agree well with the direct computations at the limit of small Stokes numbers. At large Stokes numbers, in contrast to R-M, our model predictions remain bounded. In spite of an improvement over R-M, our analysis fails to predict the expansion of high Stokes clouds observed in the direct computations. Consistent with the general trend of particle segregation versus Stokes number, our analysis shows a maximum rate of contraction at an intermediate Stokes number of O(1) and minimal rates of contraction at small and large Stokes numbers.