Statistical theory of helical turbulence

Abstract

A statistical theory for homogeneous helical turbulence is developed under the condition of strong symmetry. The latter describes reflectional symmetry in planes through and normal to the helical unit vector eξ, which can be achieved by demanding that the mean velocity is zero. The two-point velocity correlation, the pressure–velocity correlation, and the two-point triple correlation are expressed by scalar functions in the helical unit vector system. By introducing the continuity equation for the correlations, the number of unknown functions can be reduced to such an extent that ultimately, a single scalar transport equation remains. Furthermore, a two-point Poisson equation is derived to express the pressure–velocity correlation in terms of the triple correlation. From the two-point version of the transport equation, the single-point limit is derived. Using the single-point equation, it can be shown that all velocity components are generally non-zero. Therefore, it is concluded that the phenomenon of vortex stretching is present in the helical coordinate system. Finally, the theory of axisymmetric turbulence is derived as a limiting case of helical turbulence to show consistency with former work.

Helical fluid and (Hall)-MHD turbulence: a brief review

“…While the fully isotropic fluid case only necessitates one defining function in terms of a field (velocity) correlation tensor, two are needed in the non mirror-symmetric case, corresponding in Fourier space to the energy and helicity spectra. This was re-established recently making use of a helical-vector unit system [23]. It would be of interest to show the equivalence of the secondand third-order tensorial relationships these authors establish with earlier versions of the helical von Kàrmàn equation [24], as well as the helical third-order exact law derived in [15,25] (see §4(b) for more details).…”

“…In fact, the presence of several invariants, beyond global energy and helicity, has also been discussed by a number of authors. For example, the case of helical symmetry, as a generalization of axial symmetry (the axis being now an helix) is discussed in [23]. Such flows are, again, quasi two-dimensional; they have an infinite number of invariants, like for the two-dimensional Euler equations, and are proven as well to be integrable [74].…”

Helical Fluid and (Hall)-MHD Turbulence: a Brief Review

“…When the fully isotropic fluid case only necessitates one defining function in terms of the velocity correlation tensor, two are needed in the non mirror-symmetric case, corresponding in Fourier space to the energy and helicity spectra. This was re-established recently making use of a helical-vector unit system [15]. It would be of interest to show the equivalence of the second-and third-order tensorial relationships these authors establish with earlier versions of the helical von Kàrmàn equation [16], as well as the helical third-order exact law derived in [13,17] (see §IVIV B).…”

“…In fact, the presence of several invariants, beyond global energy and helicity, has also been discussed by a number of authors. For example, the case of helical symmetry, as a generalization of axial symmetry (the axis being now an helix) is discussed in [15]. Such flows are, again, quasi two-dimensional (2D); they have an infinite number of invariants, like for the 2D Euler equations, and are proven as well to be integrable [65].…”