We demonstrate that kinematic simulation (KS) of three-dimensional homogeneous turbulence produces fluid element pair statistics in agreement with the predictions of L F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)] even though KS lacks explicit modeling of turbulent sweeping of small eddies by large ones. This scaling is most clearly evident in the turbulent diffusivity's dependence on rms pair separation and, to a lesser extent, on the pair's travel time statistics. It is also shown that kinematic simulation generates a probability density function of pair separation which is in good agreement with recent theory [S. Goto and J. C. Vassilicos, New J. Phys. 6, 65 (2004)] and with the scaling of the rms pair separation predicted by L. F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)]. Finally, the statistical persistence hypothesis (SPH) is formulated mathematically and its validity tested in KS. This formulation introduces the concept of stagnation point velocities and relates these to fluid accelerations. The scaling of accelerations found in kinematic simulation supports the SPH, even though KS does not generate a Kolmogorov scaling for the acceleration variance (except for a specific case and a limited range of outer to inner length-scale ratios). An argument is then presented that suggests that the stagnation points in homogeneous isotropic turbulence are on average long-lived.

We study the statistics of the vertical motion of inertial particles in strongly stratified turbulence. We use Kinematic Simulation (KS) and Rapid Distortion Theory (RDT) to study the mean position and the root mean square (rms) of the position fluctuation in the vertical direction. We vary the strength of the stratification and the particle inertial characteristic time. The stratification is modelled using the Boussinesq equation and solved in the limit of RDT. The validity of the approximations used here requires that L/g < 2π/N < τ η , where τ η is the Kolmogorov time scale, g the gravitational acceleration, L the turbulence integral length scale and N the Brunt-Väisälä frequency. We introduce a drift Froude number F r d = τ p g/N L. When F r d < 1, the rms of the inertial particle displacement fluctuation is the same as for fluid elements, i.e. (ζ 3 − ζ 3 ) 2 1/2 = 1.22 u ′ /N + oscillations. However, whenThat is the level of the fluctuation is controlled by the particle inertia τ p and not by the buoyancy frequency N . In other words it seems possible for inertial particles to retain the vertical capping while loosing the memory of the Brunt-Väisälä frequency.

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