Below critical temperature, turbulence prevents the spontaneous phase separation of binary mixtures, resulting in a phase-arrested state of emulsion which is of significant interest for scientific and industrial applications. The current work is an extensive continuation of our initial theoretical investigation into the nature and universality of associated energy cascade [N. Pan and S. Banerjee, Phys. Rev. E 106, 025104 (2022)]. In addition to the previously derived divergence and correlator forms of the exact relations, a Banerjee-Galtier-type [S. Banerjee and S. Galtier, J. Phys. A: Math. Theor. 50, 015501 (2017)] divergence-free form of the exact relation is derived under the explicit assumption of homogeneity. By performing three-dimensional direct numerical simulations with up to ${1024}^3$ grid points, we show that the sum of the kinetic and active energy associates a Kolmogorov-like universal cascade with constant flux rate across inertial scales. Despite term-by-term deviations, the cascade rates computed from all three exact laws show excellent agreement, thereby confirming the equivalence of different exact relations and hence the feasibility of determining the net cascade rate from any of the three formulations. Notably, the two dominant flux rates of the divergence form are found to cross each other roughly at the domain size, which also serves as an infrared cutoff for the inverse cascade of the small-scale active energy. This behavior is phenomenologically justified based on the interplay between the flow and surface dynamics of a phase-arrested binary fluid. Finally, the cascade rate is also calculated from the scale-by-scale energy budget in Fourier space where the inertial range cascade rate becomes constant across scales and matches with that calculated from direct space variables.

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• Complex fluids
• Turbulence