A closure theory for the split energy-helicity cascades in homogeneous isotropic homochiral turbulence

We study the energy transfer properties of three dimensional homogeneous and isotropic turbulence where the non-linear transfer is altered in a way that helicity is made sign-definite, say positive. In this framework, known as homochiral turbulence, an adapted eddy-damped quasi-normal Markovian (EDQNM) closure is derived to analyze the dynamics at very large Reynolds numbers, of order $10^5$ based on the Taylor scale. In agreement with previous findings, an inverse cascade of energy with a kinetic energy spectrum like $\propto k^{-5/3}$ is found for scales larger than the forcing one. Conjointly, a forward cascade of helicity towards larger wavenumbers is obtained, where the kinetic energy spectrum scales like $\propto k^{-7/3}$. By following the evolution of the closed spectral equations for a very long time and over a huge extensions of scales, we found the developing of a non monotonic shape for the front of the inverse energy flux. The very long time evolution of the kinetic energy and integral scale in both the forced and unforced cases is analyzed also.Comment: 8 pages, 3 figure

Cascades and transitions in turbulent flows

Turbulence is characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal 3D/2D isotropic cases characterized by the presence of a strictly direct/inverse energy cascade, respectively. We review recent works from a unified point of view and we present a classification of all known transfer mechanisms. Beside the classical cases of direct and inverse cascades, the different scenarios include: split cascades to small and large scales simultaneously, multiple/dual cascades of different quantities, bi-directional cascades where direct and inverse transfers of the same invariant coexist in the same scale-range and finally equilibrium states where no cascades are present, including the case when a condensate is formed. We classify all transitions as the control parameters are changed and we analyse when and why different configurations are observed. Our discussion is based on a set of paradigmatic applications: helical turbulence, rotating and/or stratified flows, MHD and passive/active scalars where the transfer properties are altered as one changes the embedding dimensions, the thickness of the domain or other relevant control parameters, as the Reynolds, Rossby, Froude, Peclet, or Alfven numbers. We discuss the presence of anomalous scaling laws in connection with the intermittent nature of the energy dissipation in configuration space. An overview is also provided concerning cascades in other applications such as bounded flows, quantum, relativistic and compressible turbulence, and active matter, together with implications for turbulent modelling. Finally, we present a series of open problems and challenges that future work needs to address.Comment: accepted for publication on Physics Reports 201

On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations

We study the global regularity, for all time and all initial data in $H^{1/2}$, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the $L^2-$scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the $H^{1/2}-$Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the $H^{1/2}$ and the time average of the square of the $H^{3/2}$ norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato \cite{kato1,kato2}, to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences

Statistical Properties of Turbulence: An Overview

We present an introductory overview of several challenging problems in the statistical characterisation of turbulence. We provide examples from fluid turbulence in three and two dimensions, from the turbulent advection of passive scalars, turbulence in the one-dimensional Burgers equation, and fluid turbulence in the presence of polymer additives.Comment: 34 pages, 31 figure