MARKOV CLOSURE FOR THE LUNDGREN-MONIN-NOVIKOV HIERARCHY OF VELOCITY INCREMENTS IN BURGERS TURBULENCE

etc15 Tracking Number 422

A central, yet unsolved issue in the longstanding problem of hydrodynamic turbulence is the closure problem of turbulence, which is due to the nonlinear character of the Navier-Stokes equation. We formulate the closure problem for the many-increment probability distributions (PDF’s) in Burgers turbulence and introduce a new method for closing the hierarchy. To this end, we rely on the experimentally and numerically verified assumption in [1] that the turbulent cascade possesses a Markov property in scale down to the so-called Einstein-Markov length. The hierarchy is closed at the stage of the two-increment PDF corresponding to a three-point closure that allows for a description of intermittency effects, not captured by other closure approximations, i.e. Gaussian closures etc. The proposed closure also opens up a possible way to a perturbative treatment of the Navier-Stokes equation beyond the Einstein-Markov length in successively taking into account a larger and larger scale “history” of the system.