The nature of triad interactions in homogeneous turbulence

Abstract

Nonlinear interactions in homogeneous turbulence are investigated using a decomposition of the velocity field in terms of helical modes. There are two helical modes per wave vector and thus eight fundamental triad interactions. These eight elementary interactions fit in only two classes, ‘‘R’’ (for ‘‘reverse’’) and ‘‘F’’ (for ‘‘forward’’), depending on whether the small-scale helical modes have helicities of the same or of the opposite sign. In a single-triad interaction, the large scale is unstable when the small-scale helical modes have helicities of opposite signs (class ‘‘F’’), and the medium scale is unstable otherwise (class ‘‘R’’). It is proposed that, on average, the triple correlations in a turbulent flow correspond to these unstable states. In the limit of nonlocal triads, where one leg is much smaller than the other two, the triadic interscale energy transfer is largest for interactions of class ‘‘R.’’ In that case, most of the energy flows locally in wave number, from the medium scale to the smallest, with a comparatively small feedback on the large scale. However, integrating over all scales in an inertial range, the net effect of nonlocal interactions of class ‘‘R’’ is a reverse energy cascade from small to large scales. All other interactions transfer energy upward in wave number. In local triads, this upward energy transfer occurs primarily between modes with helicities of the same sign, through catalytic interactions with a mode whose helicity has the opposite sign. The class ‘‘F’’ interactions transfer energy to the small scales and exist only in three dimensions. The physical processes associated to both classes of interactions are discussed. It is shown that the large local transfers due to nonlocal ‘‘R’’ interactions appear in pairs of opposite signs that nearly cancel each other and the net effect corresponds to an advection in wave space.

Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence

“…In particular, one can always choose h ± =ν ×k ± iν, whereν is an arbitrary unit vector orthogonal to k which satisfies the relationν(k) = −ν(−k), necessary to ensure the reality of the velocity field (Waleffe 1992). Such a requirement is satisfied, for example, by the choiceν = z × k/ z × k , with z an arbitrary vector.…”

“…The starting point of our analysis is the well-known helical Fourier decomposition of the velocity field v(x), expanded in Fourier components, u(k), proposed by Constantin & Majda (1988) and Waleffe (1992). Being divergence-free, k · u(k) = 0, each velocity component in Fourier space has only two degrees of freedom.…”

“…The paper is organized as follows. In § 2 we briefly summarize the exact helical Fourier decomposition of the Navier-Stokes dynamics proposed by Constantin & Majda (1988) and Waleffe (1992), and we discuss the four possible classes of triads which contribute to reconstruction of the whole nonlinear term. In § 3 we summarize some of the results obtained by Biferale et al (2012) by a direct numerical simulation of dNS equations preserving only one class of interactions, namely that one with a well-defined sign of helicity, leading to the split energy-helicity cascades.…”

“…In this paper we propose using an exact decomposition of the Navier-Stokes (NS) equations on a helical Fourier basis proposed originally by Constantin & Majda (1988) (see also Waleffe 1992), in order to highlight some basic properties of helicity common to all turbulent flows. Following Waleffe (1992) we will show that the set of all triadic interactions in NS equations can be divided into four independent classes, as a function of the helical contents of each of the three interacting modes (see below for details).…”

“…Following Waleffe (1992) we will show that the set of all triadic interactions in NS equations can be divided into four independent classes, as a function of the helical contents of each of the three interacting modes (see below for details). Our idea is to study the dynamics of NS equations by preserving only some of these classes, in order to disentangle the energy transfer as a function of the helicity properties of the interacting modes.…”