Comment on "Physical significance of artificial numerical noise in direct numerical simulation of turbulence"

TL;DR

The paper scrutinizes the claim that numerical noise in DNS of the deterministic Navier-Stokes equations can mimic thermal fluctuations. It anchors the critique in the equilibrium spectra of thermal fluctuations, with $E_\mathrm{th}^\mathrm{2D}(k) = \frac{k_B T}{2 \pi \rho} k$ and $E_\mathrm{th}^\mathrm{3D}(k) = \frac{3 k_B T}{4 \pi^2 \rho} k^2$, and cites a dissipation-range crossover at $k_c \eta \sim 3$ (and $k_c \eta_\Omega \sim 3$ for enstrophy-scale) observed in DSMC and fluctuating hydrodynamics simulations. The authors show that the claimed equivalence would require spectra matching the thermal form over a decade of wavenumbers, which is not observed in Liao & Qin's results. They further demonstrate that Liao & Qin's CNS implementation is inconsistent with fluctuating hydrodynamics because fluctuations should enter via a random stress tensor obeying the fluctuation-dissipation relation, not as an additive random velocity field, and because their temporal integrator does not converge for $O(\sqrt{\Delta t})$ Wiener increments. Consequently, numerical noise cannot be treated as a surrogate for physical thermal fluctuations, and the CNS approach, given realistic noise amplitudes, would not reproduce the correct thermal-dominated dissipation-range behavior.

Abstract

Recently, Liao and Qin [J. Fluid Mech. 1008, R2 (2025)] claimed that numerical noise in direct numerical simulation of turbulence using the deterministic Navier-Stokes equations is "approximately equivalent" to the physical noise arising from random molecular motion (thermal fluctuations). We show here that it this claim not supported by their results and that it contradicts other results in the literature. Furthermore, we demonstrate that the numerical implementation of thermal fluctuations in their so-called "clean numerical simulations" is incorrect.