The Smallest Scales of Turbulence in Gases Are Not Described by the Navier-Stokes Equations*

We show that the Navier-Stokes equations do not describe turbulent gas flows in the dissipation range because they neglect thermal fluctuations. Energy spectra from Navier-Stokes simulations of turbulent Taylor-Green vortex flow decay exponentially at high wavenumbers, while spectra from corresponding molecular-gas-dynamics simulations grow quadratically due to thermal fluctuations. The quadratic growth begins at length scales much larger than the molecular mean free path, where the Navier-Stokes equations are widely believed to be accurate. Our results show the importance of molecular-level thermal fluctuations and emphasize the need to include them in the Navier-Stokes equations when studying turbulent gas flows.