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

TL;DR

The paper shows that in incompressible fluids, spatially correlated thermal noise invalidates the conventional low-$Re$ linear Stokes limit by making momentum diffusion scale-dependent. By formulating spatially correlated fluctuating hydrodynamics with a nonlocal viscosity $\nu_{eff}(r)=\nu C(r)$ and a Lorentzian correlation $C(r)$ (with Fourier transform $\widehat{C}(k)=\exp(-\ell k)$), the authors derive a scale-dependent Reynolds number $Re_{eff}(k)=u_{eq}/(\nu k \widehat{C}(k))=(u_{eq}/(\nu k))\,e^{\ell k}$ that grows with $k$ for correlated noise. Numerical simulations in 1D and 2D demonstrate that linearized dynamics relax high-$k$ modes much more slowly than nonlinear dynamics when $\ell>0$, and that tracer diffusion in 2D shows large deviations (up to ~90%) between linear and nonlinear predictions at large correlation lengths. These findings collectively establish that inertia-viscosity balance cannot be captured by a single global Reynolds number in spatially correlated fluctuating fluids, and motivate a spectrum of scale-dependent hydrodynamic control parameters for complex or structured fluids.

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.

Figures (4)

• Figure 1: Scale dependence of the Reynolds number in the spatially correlated fluctuating Navier--Stokes equation. (a) For spatially uncorrelated noise, both the noise energy injection and viscous dissipation spectra scale as $k^2$. Therefore, they balance exactly, consistent with the FDR. (b) For spatially correlated noise, the correlation function $\widehat{C}(k)$ modifies both the spectra of energy injection and viscous dissipation, preserving their balance and the FDR. (c) The prefactor of the viscous term remains constant for spatially uncorrelated noise but decreases with increasing $k$ for spatially correlated noise, indicating that the viscous term becomes less important at large wavenumbers. Here, $\hat{\bm{u}}$ is the Fourier mode at wavevector $\bm{k}$, $\bm{N}$ is the nonlinear term, $L$ is the dissipative linear operator, $\ell$ is the characteristic length scale of the correlation function $C(r)$.
• Figure 2: Nonlinear mode coupling in the Burgers equation accelerates velocity relaxation compared with the linear heat equation. (a) For white noise, the autocorrelation contours of the Fourier modes $\hat{u}$ exhibit similar behavior in both the nonlinear (left) and linear (right) dynamics: low-wavenumber modes decay more slowly, whereas high-wavenumber modes relax more rapidly. The VACFs at three representative wavenumbers ($k=1, 13,$ and $126~\unit{\um}^{-1}$) show great agreement between the nonlinear (Burgers) and linear (heat) cases. (b) For correlated noise, however, the nonlinear dynamics yield systematically faster relaxation, most notably at higher wavenumbers ($k\gtrsim10~\unit{\um}^{-1}$), where the linear dynamics exhibit near-frozen high-$k$ modes, illustrating the failure of the linear approximation. The VACFs at the same three wavenumbers in (a) confirm this observation.
• Figure 3: The particles exhibit stronger diffusion in the linear regime than in the nonlinear regime for increasing correlation length. This is manifested by (a) slower long-time decay of the VACFs and (b) larger diffusion coefficients. Results correspond to $\mathrm{Re}\xspace \approx 0.4$, with viscous timescale $t_{\nu} = a^2/\nu$.
• Figure 4: The particles again exhibit stronger diffusion in the linear regime than in the nonlinear regime for increasing correlation length, now at a higher Reynolds number $\mathrm{Re}\xspace \approx 3$ than in \ref{['fig:3']}. The same qualitative behavior observed in \ref{['fig:3']} persists here, namely (a) a slower long-time decay of the VACFs and (b) larger diffusion coefficients with increasing correlation length. The viscous timescale is $t_{\nu} = a^2/\nu$.