Quantitative analysis of fluctuating hydrodynamics in uniform shear flow

Many theoretical predictions in fluctuating hydrodynamics under uniform shear flow have lacked precise quantitative verification due to analytical approximations whose quantitative impacts are difficult to assess a priori and the limitations of microscopic particle-based simulations. To address this problem, we perform direct numerical simulations (DNS) of the fluctuating Navier-Stokes (NS) equations with shear-periodic boundary conditions. We provide a decisive quantitative validation of two seminal frameworks: the Lutsko-Dufty theory for nonequilibrium long-range correlations, and the dynamic renormalization group (RG) theory by Forster, Nelson, and Stephen (FNS) for anomalous transport. By simulating the linearized fluctuating NS equations, we demonstrate that the predictions of the Lutsko-Dufty theory are quantitatively valid from the viscous-dominated, long-wavelength regime to the shear-dominated, short-wavelength regime, well beyond their originally assumed limits. Moving beyond the linearized equations, we simulate the full nonlinear fluctuating NS equations to test the quantitative predictive capability of the dynamical RG approach by FNS. Our results show that the one-loop RG prediction remains quantitatively accurate up to a strongly nonlinear regime, where conventional perturbation theory fails. Our findings solidify the foundations of these classical theories, paving the way for quantitative analyses using fluctuating hydrodynamics.