Solutions of the Navier-Stokes Equation Through Affine Transformations: The Triad Triplet

Abstract

Considering triads in helical decomposition of three dimensional Navier-Stokes turbulence, three consecutive triads of the same shape and class, where the reference wave-number appears as the smallest, the middle and the largest wave-numbers respectively, constitutes an interesting object called the triad triplet that allows tracing the local transfer in wave-number space due to a given shape and class of triads. It is shown that, evolution equations for the triad triplet can be obtained from the equations of the three legs of a single triad that are put together via affine transformations involving scaling, rotation and reflection. Known power law solutions corresponding to constant flux of energy and helicity appear as exact solutions of a chain of triad triplets, with particular implications for the triadic phases. More interestingly, an exact, time dependent solution is available on an isolated triad triplet, that can be expressed using what appears to be straightforward generalization of Jacobi elliptic functions. These solutions, constitute novel exact nonlinear solutions of the inviscid limit of the Navier-Stokes equations.