On Markov modelling of turbulence

We consider Lagrangian stochastic modelling of the relative motion of two fluid particles in the inertial range of a turbulent flow. Eulerian analysis of such modelling corresponds to an equation for the Eulerian probability distribution of velocity-vector increments which introduces a hierarchy of constraints for making the model consistent with results from the theory of locally isotropic turbulence. A nonlinear Markov process is presented, which is able to satisfy exactly, in the statistical sense, incompressibility, the exact results on the third-order structure function, and the experimental second-order statistics. The corresponding equation for the Eulerian probability density of velocity-vector increments is solved numerically. Numerical results show non-Gaussian statistics of the one-dimensional Lagrangian probability distributions, and a complex shape of the three-dimensional Eulerian probability density function. The latter is then compared with existing experimental data.