Progress in Turbulence II

We investigate the decay of freely-evolving, isotropic turbulence whose spectrum takes the form E(k→0)∼Ik

4

, I being Loitsyansky's integral. We report numerical simulations in a periodic domain whose dimensions, l

box

, are much larger than the integral scale of the turbulence, l. We find that, provided l

box

≫l and Re≫1, the turbulence evolves to a state in which Loitsyansky's integral is approximately constant and Kolmogorov's decay law, u

2

∼t

−10/7

, holds true. The approximate conservation of I in fully-developed turbulence implies that the long-range interactions between remote eddies, as measured by the triple correlations, are very weak.

We derive the exact solution for the Burgers equation with a time dependent forcing, which depends linearly on the spatial coordinate. For the case of a stochastic time dependence an exact expression for the joint probability distribution for the velocity fields at multiple spatial points is obtained. We present numerical results for fixed boundary conditions, and analyze the formation of shocks.

Onsager's point vortex model of two dimensional turbulence is extended by the inclusion of time dependent vortex circulations. If the time dependence of the circulations is governed by statistically independent Onstein-Uhlenbeck processes we observe the emergence of scaling regimes for the structure functions of the Eulerian and the Lagrangian velocity increments. Fully developed turbulent flows are flux equilibrium systems leading to selfsimilarity and scaling behaviour of correlation functions. The probability distributions corresponding to the simplest case of idealized, i.e. homogeneous, isotropic, and stationary fully developed flows are unknown although the underlying fluid dynamical equations and its statistical counterparts are well-established [1]. Fluid motions can be treated either from an Eulerian or a Lagrangian point of view. Most analytical theories have been formulated in the Eulerian framework. Point vortex models (see e.g. [2]), which have been extensively investigated especially for the case of two dimensional flows, essentially make use of a Lagrangian description. Since the point vortex equations of an ideal two dimensional fluid, which is not stirred, are of Hamiltonian nature, a statistical treatment based on the microcanonical ensemble can be established. This has been discussed for the first time by Onsager [3]. The purpose of the present Letter is to show that an extension of Onsager's point vortex model, which allows for fluctuating circulations of the point vortices, leads to a state which shows scaling behaviour of the structure functions.