Lagrangian Tetrad Dynamics and the Phenomenology of Turbulence

Abstract

A new phenomenological model of turbulent fluctuations is constructed by considering the Lagrangian dynamics of 4 points (the tetrad). The closure of the equations of motion is achieved by postulating an anisotropic, i.e. tetrad shape dependent, relation of the local pressure and the velocity gradient defined on the tetrad. The non-local contribution to the pressure and the incoherent small scale fluctuations are modeled as Gaussian white "noise". The resulting stochastic model for the coarse-grained velocity gradient is analyzed approximately, yielding predictions for the probability distribution functions of different 2nd and 3d order invariants. The results are compared with the direct numerical simulation of the Navier-Stokes. The model provides a reasonable representation of the non-linear dynamics involved in energy transfer and vortex stretching and allows to study interesting aspects of the statistical geometry of turbulence, e.g. vorticity/strain alignment. In a state with a constant energy flux (and K41 power spectrum) it exhibits the anomalous scaling of high moments associated with formation of high gradient sheets - events associated with large energy transfer. An approach to the more complete analysis of the stochastic model, properly including the effect of fluctuations, is outlined and will enable further quantitative juxtaposition of the model with the results of the DNS.