Fundamentals of pair diffusion in kinematic simulations of turbulence.

ABSTRACT

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.