Revisiting the scale dependence of the Reynolds number in correlated fluctuating fluids

TL;DR

This paper demonstrates that in spatially correlated fluctuating fluids, the traditional Reynolds number becomes scale-dependent, invalidating the linear approximation at low Reynolds numbers due to the influence of thermal noise correlations.

Contribution

It reveals that spatial correlations in thermal noise cause the breakdown of linearized hydrodynamics at low Reynolds numbers, highlighting the scale dependence of the effective Reynolds number.

Findings

Linearized dynamics relax slower at high wavenumbers in 1D.

Velocity autocorrelation decays more slowly in correlated linear Stokes.

Spatial correlations inhibit small-scale viscous diffusion.

Abstract

For the incompressible Navier--Stokes equation, the Reynolds number ($\mathrm{Re}$) is a dimensionless parameter quantifying the relative importance of inertial over viscous forces. In the low-$\mathrm{Re}$ regime ($\mathrm{Re} \ll 1$), the flow dynamics are commonly approximated by the linear Stokes equation. Here we show that, within the framework of spatially fluctuating hydrodynamics, this linearization breaks down when the thermal noise is spatially correlated, even if $\mathrm{Re} \ll 1$. We perform direct numerical simulations of spatially correlated fluctuating hydrodynamics in both one and two dimensions. In one dimension, the linearized dynamics exhibit significantly slower relaxation of high-wavenumber Fourier modes than the full nonlinear dynamics. In two dimensions, an analogous discrepancy arises in the particle velocity autocorrelation function, which decays more slowly in the correlated linear Stokes case than in the correlated nonlinear Navier--Stokes case. In both settings, spatial correlations inhibit viscous momentum diffusion at small scales, leading to prolonged relaxation under the linear dynamics, whereas nonlinear mode coupling accelerates small-scale relaxation. Thus, the interplay between nonlinear coupling and viscous damping becomes scale dependent, invalidating the use of a single global Reynolds number. Taken together, these findings show that, for spatially correlated fluctuating fluids, the effective Reynolds number must be reinterpreted as a scale-dependent quantity.