Particle transport in a random velocity field with Lagrangian statistics

The transport properties of a random velocity field with Kolmogorov spectrum and time correlations defined along Lagrangian trajectories are analyzed. The analysis is carried out in the limit of short correlation times, as a perturbation theory in the ratio, scale by scale, of the eddy decay and turnover time. Various quantities such as the Batchelor constant and the dimensionless constants entering the expression for particle relative and self-diffusion are given in terms of this ratio and of the Kolmogorov constant. Particular attention is paid to particles with finite inertia. The self-diffusion properties of a particle with Stokes time longer than the Kolmogorov time are determined, verifying on an analytical example the dimensional results of Olla [Phys. Fluids 14, 4266 (2002)]. Expressions for the fluid velocity Lagrangian correlations and correlation times along a solid particle trajectory are provided in several parameter regimes, including the infinite Stokes time limit corresponding to Eulerian correlations. The concentration fluctuation spectrum and the nonergodic properties of a suspension of heavy particles in a turbulent flow, in the same regime, are analyzed. The concentration spectrum is predicted to obey, above the scale of eddies with lifetime equal to the Stokes time, a power law with universal -4/3 exponent, and to be otherwise independent of the nature of the turbulent flow. A preference of the solid particle to lie in less energetic regions of the flow is observed.