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

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.