Navier-Stokes Equations Do Not Describe the Smallest Scales of Turbulence in Gases

In turbulent flows, kinetic energy is transferred from the largest scales to progressively smaller scales, until it is ultimately converted into heat. The Navier-Stokes equations are almost universally used to study this process. Here, by comparing with molecular-gas-dynamics simulations, we show that the NavierStokes equations do not describe turbulent gas flows in the dissipation range because they neglect thermal fluctuations. We investigate decaying turbulence produced by the Taylor-Green vortex and find that in the dissipation range the molecular-gas-dynamics spectra grow quadratically with wave number due to thermal fluctuations, in agreement with previous predictions, while the Navier-Stokes spectra decay exponentially. Furthermore, the transition to quadratic growth occurs at a length scale much larger than the gas molecular mean free path, namely in a regime that the Navier-Stokes equations are widely believed to describe. In fact, our results suggest that the Navier-Stokes equations are not guaranteed to describe the smallest scales of gas turbulence for any positive Knudsen number.

Automated summary

This study compares the Navier-Stokes equations with molecular-gas-dynamics simulations and finds that the equations fail to describe the dissipation range of turbulent gas flows due to the neglect of thermal fluctuations. The research also reveals that the spectra in molecular-gas-dynamics simulations exhibit quadratic growth with wave number in the dissipation range, in contrast to the exponential decay in the Navier-Stokes spectra. Furthermore, the transition to quadratic growth occurs at length scales larger than the gas molecular mean free path.