Gregory L. Eyink, Damien Benveniste
Synthetic models of Eulerian turbulence like so called "Kinematic Simulations" (KS) have been criticized by Thomson & Devenish (TD) (2005), who argued that sweeping decorrelation effects suppress pair dispersion in such models. We derive analytical results for Eulerian turbulence modeled by Gaussian random fields for the case with zero mean velocity. Our starting point is an exact integrodifferential equation for the particle pair separation distribution. When memory times of particle locations are short, a Markovian approximation leads to a Richardson-type diffusion model. We obtain a diffusivity tensor of the form $K_{ij}(r,t)=S_{ij}(r)\tau(r,t)$ where $S_{ij}(r)$ is the structure-function tensor and $\tau(r,t)$ is an effective correlation time of velocity increments. This is found to be the minimum value of three times: the intrinsic turnover time $\tau_{eddy}(r)$ at separation $r$, the overall evolution time $t,$ and the sweeping time $r/v_0$ with $v_0$ the rms velocity. We study the diffusion model numerically by a Monte Carlo method. With moderate inertial-ranges like those achieved in current KS, our model is found to reproduce the $t^{9/2}$ power-law for pair dispersion predicted by TD and observed in the KS. However, for much longer ranges, our model exhibits three distinct pair-dispersion laws in the inertial-range: a Batchelor $t^2$-regime, followed by a $t^1$ diffusive regime, and then a $t^6$ regime. Finally, outside the inertial-range, there is another $t^1$ regime with particles undergoing independent Taylor diffusion. These scalings are exactly the same as those predicted by TD for KS with large mean velocities, which we argue hold also for KS with zero mean velocity. Our results support the basic conclusion that sweeping effects make Lagrangian properties of KS fundamentally different from hydrodynamic turbulence for very extended inertial-ranges.
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http://arxiv.org/abs/1208.3497
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