Nonlinear helical interactions in Navier-Stokes and shell models for turbulence

The turbulent dynamics can be thought as the cumulative result of a large network of Fourier triadic interactions between wavevectors. Our aim is to study the role played by helicity in the non-linear evolution of Navier-Stokes (NS) equations. In order to disentangle the contributions coming from different interactions, we use the decomposition in complex helical waves. The results here presented are based on direct numerical simulations of the NS equations, or, when that it is not feasible, on numerical simulations of shell models: a simplified set of equations that show a phenomenology very close to real turbulence, for a subset of observables. We first study the influence of helicity on turbulent dynamics. In particular, we look at the restoring of the parity symmetry at small scale, broken by a large scale helical forcing, by analyzing the scaling properties of a new set of structure functions which are sensitive to parity symmetry-breaking, both in direct numerical simulations of NS equations and in shell models. We then focus on the transfer properties of the different helical triadic interactions, at varying the geometry of the triads. Following [Waleffe, F., 1992. The nature of triad interactions in homogeneous turbulence. Physics of Fluids A: Fluid Dynamics (1989-1993)], one can make a series of predictions on the direction of the energy cascade in the NS dynamics. In some cases, an inversion in the energy cascade direction is expected. We test these predictions in helical shell models of turbulence. We also look at soliton-like solutions of the inviscid shell model equations. These solutions are intimately connected to the finite time blowup of the equations. The signature of such a solution in the turbulent velocity field is the development of a coherent structure that travels from small to large wavenumbers. It has been argued [Mailybaev, A.A., 2013. Blowup as a driving mechanism of turbulence in shell models. Physical Review E] that such structures may have a significant impact on the statistics of the energy transfers. We study the instantons developed by the different helical triadic interactions, by means of helical shell model simulations, and show that for some specific interactions the instanton becomes chaotic. We further observe a correlation between the chaoticity of the instanton and the presence or not of intermittency in the stationary dynamics of the same triadic interaction. Finally, we address the issue of time-irreversibility in the lagrangian statistics of a turbulent flow, again using shell models. After defining the shell model analogous of the lagrangian velocity field in the NS equations, we study the third moment of the lagrangian energy derivative, which is a quantity sensitive to the breaking of time-reversibility symmetry. Both in the NS and shell models equations the viscous term explicitly breaks time-reversal symmetry. To highlight the independence of the lagrangian time-irreversibility from the viscous term, we study modified shell model equations with a time-dependent viscosity, which makes the equations formally time-reversible, but with a dynamics which is equivalent to the original ones.