Kolmogorov's exact scaling laws for turbulence — the 4/5 law, the 4/3 law — are among the few rigorous results in a field dominated by approximations. They describe how velocity correlations scale with separation distance in the inertial range, and they hold because energy conservation constrains the cascade regardless of the details of the turbulent fluctuations.
Binary fluid turbulence — two immiscible or partially miscible fluids mixing turbulently — involves an additional degree of freedom: the concentration field, governed by the Cahn-Hilliard equation coupled to Navier-Stokes. Interfaces between the two fluids carry energy and enstrophy. The question is whether exact laws survive.
They do, but with modifications (arXiv:2603.12143). The analogs of the 1/3, 4/3, 2/15, and 4/5 laws exist for Cahn-Hilliard-Navier-Stokes turbulence, derived rigorously through two-point correlators. The new laws incorporate both bulk flow contributions and interface contributions. The 2/15 and 4/5 analogs include additional non-longitudinal terms absent from the classical counterparts — the interfaces break the simplifications that reduce classical turbulence laws to purely longitudinal structure functions.
Direct numerical simulations at 1024³ resolution confirm all derived laws. The transition from divergence form to the isotropic 4/5 form shifts the inertial range toward larger scales, flattening the cascade rate profile. The interface doesn't destroy the cascade; it stretches it.
Exact results are rare in turbulence. Extending them to binary fluids shows that the cascade structure is more robust than the specific fluid: the interfaces add terms but don't break the scaling.