RESUMO
We derive an exact equation governing two-particle backwards mean-squared dispersion for both deterministic and stochastic tracer particles in turbulent flows. For the deterministic trajectories, we probe the consequences of our formula for short times and arrive at approximate expressions for the mean-squared dispersion which involve second order structure functions of the velocity and acceleration fields. For the stochastic trajectories, we analytically compute an exact t3 contribution to the squared separation of stochastic paths. We argue that this contribution appears also for deterministic paths at long times and present direct numerical simulation results for incompressible Navier-Stokes flows to support this claim. We also numerically compute the probability distribution of particle separations for the deterministic paths and the stochastic paths and show their strong self-similar nature.
RESUMO
We solve an inverse problem for fluid particle pair statistics: we show that a time sequence of probability density functions (PDFs) of separations can be exactly reproduced by solving the diffusion equation with a suitable time-dependent diffusivity. The diffusivity tensor is given by a time integral of a conditional Lagrangian velocity structure function, weighted by a ratio of PDFs. Physical hypotheses for hydrodynamic turbulence (sweeping, short memory, mean-field) yield simpler integral formulas, including one of Kraichnan and Lundgren (K-L). We evaluate the latter using a space-time database from a numerical Navier-Stokes solution for driven turbulence. The K-L formula reproduces PDFs well at root-mean-square separations, but growth rate of mean-square dispersion is overpredicted due to neglect of memory effects. More general applications of our approach are sketched.
RESUMO
Synthetic models of Eulerian turbulence like so-called kinematic simulations (KS) are often used as computational shortcuts for studying Lagrangian properties of turbulence. These models have been criticized by Thomson and Devenish (2005), who argued on physical grounds that sweeping decorrelation effects suppress pair dispersion in such models. We derive analytical results for Eulerian turbulence modeled by Gaussian random fields, in particular for the case with zero mean velocity. Our starting point is an exact integrodifferential equation for the particle pair separation distribution obtained from the Gaussian integration-by-parts identity. When memory times of particle locations are short, a Markovian approximation leads to a Richardson-type diffusion model. We obtain a time-dependent pair diffusivity tensor of the form K(ij)(r,t)=S(ij)(r)τ(r,t), where S(ij)(r) is the structure-function tensor and τ(r,t) is an effective correlation time of velocity increments. Crucially, this is found to be the minimum value of three times: the intrinsic turnover time τ(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 inertial ranges like the largest achieved in most current KS (about 6 decades long), our model is found to reproduce the t(9/2) power law for pair dispersion predicted by Thomson and Devenish 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 Kraichnan-model-like 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 Thomson and Devenish for KS with large mean velocities, which we argue hold also for KS with zero mean velocity. Our results support the basic conclusion of Thomson and Devenish (2005) that sweeping effects make Lagrangian properties of KS fundamentally differ from those of hydrodynamic turbulence for very extended inertial ranges.
