Statistical geometry in homogeneous and isotropic turbulence

We consider a phenomenological model, incorporating the main features of hydrodynamic fluid turbulence, aimed at predicting the structure of the velocity gradient tensor, M, coarse-grained at a spatial scale r. This model (M. Chertkov, A. Pumir and B. I. Shraiman, Phys. Fluids, 11, 2394 (1999)) is formulated as a set of stochastic ordinary differential equations depending on three dimensionless parameters. We solve it with two complementary methods. The joint probability distribution functions of the second and third invariants of M, as well as the scaling laws of the average enstrophy, strain and energy transfer, are first computed by using a semiclassical method of resolution of the model. These results are compared with direct numerical simulations (DNS) data. The semiclassical solutions correctly reproduce the DNS data behaviour provided the parameter that controls nonlinearity reduction induced by the pressure is finely tuned. A hybrid Monte Carlo procedure of resolution of the model is then developed. Our approach consists in fixing the vorticity degrees of freedom, treating all the other degrees of freedom by a straightforward Monte Carlo method and then maximising over vorticity. The preliminary results presented here are promising. The comparison of the solutions calculated through both methods enables us to get insight into the model, by evidencing the role of the fluctuations in the different structures of the flow.