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

Abstract

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 ∝ 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 ∝ 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. * thomas.gomez@univ-lille1.fr 2

Three-dimensional turbulence without vortex stretching

“…The forcing term in (3.3) is defined by (Briard et al. 2017), with determined such that the integral of is unity. This ensures an energy input which is fixed at and an enstrophy input close to unity ().…”

“…Typical examples of such modifications are the removal of the pressure terms from the governing equations as first investigated by Burgers (1950), and later by Polyakov (1995) and Boldyrev (1999), the removal of a certain class of modes upon which the flow-field is projected to focus on certain triadic interactions (Biferale, Musacchio & Toschi 2012;Alexakis 2017;Briard, Biferale & Gomez 2017;Qu, Naso & Bos 2018), or the decimation of Fourier space to change the fractal dimension of space (Frisch, Lesieur & Sulem 1976;Frisch et al 2012;Lanotte et al 2015). The change of the dimension of space can also be directly investigated by reformulating turbulence in more than three dimensions (Gotoh et al 2007;Yamamoto et al 2012;Berera, Ho & Clark 2020), or by considering intermediate systems such as axisymmetric turbulence, with properties of both two-and three-dimensional systems (Leprovost, Dubrulle & Chavanis 2006;Naso et al 2010;Qu, Bos & Naso 2017;Qin et al 2020), or thin-layer turbulence (Celani, Musacchio & Vincenzi 2010;Benavides & Alexakis 2017;Favier, Guervilly & Knobloch 2019).…”

“…Typical examples of such modifications are the removal of the pressure terms from the governing equations as first investigated by Burgers (1950), and later by Polyakov (1995) and Boldyrev (1999), the removal of a certain class of modes upon which the flow-field is projected to focus on certain triadic interactions (Biferale, Musacchio & Toschi 2012; Alexakis 2017; Briard, Biferale & Gomez 2017; Qu, Naso & Bos 2018), or the decimation of Fourier space to change the fractal dimension of space (Frisch, Lesieur & Sulem 1976; Frisch et al. 2012; Lanotte et al.…”

Cascades and transitions in turbulent flows

“…From this perspective, being homochiral is more fragile than being two-dimensional three-component (2D3C), u(x, t) = (u x (x, y, t), u y (x, y, t), u z (x, y, t)), or of being fully anti-symmetric, u(−x, t) = −u(x, t), which are two symmetries exactly preserved by the time evolution of NSE. Before ending this section, we mention that similar conclusions concerning the existence of inverse energy cascades for homochiral turbulence can also be obtained by studying absolute equilibrium spectra for the H-NSE with α = ν = 0 and f = 0, if restricted to a finite number of Fourier modes [103] or by using second-order closures based on the so-called Eddy Damped Quasi Normal Markovian Approximation [230]. Recently it was also found that the dynamics of triads belonging to class-II have an extra quadratic invariant whose dimension depends on the triad's geometry [20,231].…”

Staircase scaling of short-time energy transfer in turbulence

“…We remark that Proposition 1 is a general result for the scaling of short-time energy transfer. However, the growth rate can be affected by phase correlations and the helical properties of the velocity modes [44][45][46] constituting the initial field. For instance, we can imagine a single-scale initial velocity field constructed by homochiral modes (i.e., s p =s q ), and u s k (k) is the same for all wave vectors k with the same length k. It is easy to show that this (perhaps somewhat pathological) case is a fixed point of the Euler equation corresponding to zero energy transfer.…”