Langevin Fokker-Planck Research Articles

Some issues of turbulence statistics

The issue of dropping the random force f_i and the arbitrariness ofchoosing the basic variable in the variational approach to turbulence closure prob-lem, pointed out recently by the Russian scientists Bazdenkov and Kukharkin, are dis-cussed. According to the mean-square estimation method, the random force f_i shouldbe dropped in the error expression of the LFP (Langevin-Fokker-Planck) model. How-ever, f_i is not neglected, its effect has been taken into account by the variationalapproach. In order to optimize the perturbation solution of the Liouville equation,the LFP model requires that the basic variable is as near to Gaussian as possible.Hence, the velocity, instead of the vorticity, should be chosen as the basic variablein the three-dimensional turbulence. Although the LFP model and the zero-orderGaussian term of PDF (probability density function) imply whiteness assumption(zero correlation time of f_i), the higher-order non-Gaussian terms of PDF correspondto the nonwhiteness of turbulence dynamics, the variational approach does calculatethe nonwhiteness effect properly.

Variational approach to the closure problem of turbulence theory

A new method is proposed to solve the closure problem of turbulence theory and to drive the Kolmogorov law in an Eulerian framework. Instead of using complex Fourier components of velocity field as modal parameters, a complete set of independent real parameters and dynamic equations are worked out to describe the dynamic states of a turbulence. Classical statistical mechanics is used to study the statistical behavior of the turbulence. An approximate stationary solution of the Liouville equation is obtained by a perturbation method based on a Langevin–Fokker–Planck (LFP) model. The dynamic damping coefficient η of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution; this leads to a convergent integral equation for η to replace the divergent response equation of Kraichnan’s direct-interaction (DI) approximation, thereby solving the closure problem without appealing to a Lagrangian formulation. The Kolmogorov constant Ko is evaluated numerically, obtaining Ko=1.2, which is compatible with the experimental data given by Gibson and Schwartz.