Reply to the comments of McMullen et al. (arXiv:2510.04828)

TL;DR

The paper defends the reliability of clean numerical simulations (CNS) for studying Navier–Stokes turbulence by showing that numerical noise can be suppressed up to a critical predictable time $T_c$, and argues that DNS noise may obscure or misrepresent turbulent statistics. It challenges the expectation that simulations with explicit random velocity fluctuations must display a thermal-fluctuation-dominated range, pointing to evidence where DNS noise does not yield such a range and where statistical results can remain robust, albeit with potential disturbances. The authors propose a deterministic CNS-based model that introduces a discrete Gaussian velocity perturbation of strength $σ$ to emulate environmental and thermal disturbances, and discuss the potential for solving LLNS equations within CNS in future work. Overall, the paper advocates thorough verification of numerical noise effects across approaches, introduces a simple noise-influence framework, and calls for deeper integration of stochastic disturbance effects with mathematical turbulence models to better connect simulations with real turbulent flows.

Abstract

McMullen et al. [1] comment that the numerical simulations that explicitly include random velocity fluctuations ``should exhibit a thermal-fluctuation-dominated range'' consistent with the literature, so that our results (J. Fluid Mech. 1008, R2, 2025) [2] ``contradict other results in the literature''. First of all, we would give an opposite example against this viewpoint: DNS results (that are badly polluted by numerical noises quickly, as mention in Section 2) implicitly include random numerical noises, but they also DO NOT exhibit a thermal-fluctuation-dominated range. In other words, DNS results in the literature qualitatively agree with ours at this point. In addition, we highly suggest that influences of numerical noises on statistics of turbulent flows given by ALL numerical approaches should be carefully checked, since numerical noises might have huge influences on statistics of chaotic systems (including turbulence), as pointed by Lorenz [3] in 2006. Detailed replies are given below.