Energy Cascades in Driven Granular Liquids : A new Universality Class? I : Model and Symmetries

TL;DR

This work investigates whether energy cascades in driven granular liquids obey Kolmogorov scaling. It develops a scale-dependent field-theoretic framework that combines Granular Integration Through Transients (GITT) with a stochastic incompressible Navier–Stokes description, treated via the Wetterich nonperturbative RG equation. Qualitative analysis predicts a breakdown of Kolmogorov scaling in granular flows, yielding a spectral exponent $F(k)\propto k^{-3/2}$ (and $E_k\propto k^{-1/2}$ in 3D) due to collisional dissipation, contrasting with Newtonian turbulence that exhibits $F(k)\propto k^{-5/3}$. Symmetry analysis through Ward identities shows that most SNS-like symmetries are preserved in the granular setting, with one key symmetry broken by granular stress contributions that feed into the Kármán-Howarth/Kolmogorov relation, suggesting a new universality class for energy cascades in granular flows. The framework provides a principled, model-independent platform to derive exponents and intermittency effects via controlled RG truncations in future work.

Abstract

This article deals with the existence and scaling of an energy cascade in steady granular liquid flows between the scale at which the system is forced and the scale at which it dissipates energy. In particular, we examine the possible origins of a breaking of the Kolmogorov Universality class that applies to Newtonian liquids under similar conditions. In order to answer these questions, we build a generic field theory of granular liquid flows and, through a study of its symmetries, show that indeed the Kolmogorov scaling can be broken, although most of the symmetries of the Newtonian flows are preserved.