Exact relations for energy transfer in simple and active binary fluid turbulence

Abstract

Inertial range energy transfer in three-dimensional fully developed binary fluid turbulence is studied under the assumption of statistical homogeneity. Using two-point statistics, exact relations corresponding to the energy cascade are derived in terms of (i) two-point increments and (ii) two-point correlators. Despite having some apparent resemblances, the exact relation in binary fluid turbulence is found to be different from that of the incompressible magnetohydrodynamic turbulence [H. Politano and A. Pouquet, Geophys. Res. Lett. 25, 273 (1998)]0094-827610.1029/97GL03642. Besides the usual direct cascade of energy, under certain situations, an inverse cascade of energy is also speculated depending upon the strength of the activity parameter and the interplay between the two-point increments of the fluid velocity and the composition gradient fields. An alternative form of the exact relation is also derived in terms of the "upsilon" variables and a subsequent phenomenology is proposed predicting a k^{-3/2} law for the turbulent energy spectrum.