Noncontinuum effects at the smallest scales of turbulence

Currently accepted turbulence theory assumes that the flow is continuum at all length scales, including the smallest length scale of turbulence, known as the Kolmogorov scale. Kolmogorov in his celebrated 1941 theory [A. N. Kolmogorov, C. R. Acad. Sci. URSS 30, 301-305 (1941)] asserted that the fine-scale turbulent structures in the energy cascade are universal. According to Kolmogorov’s theory, the energy dissipation rate and kinematic viscosity alone describe this universal behavior in terms of the Kolmogorov length, time, and velocity scales. However, it has been suggested [R. Betchov, J. Fluid Mech. 3, 205-216 (1957); D. Bandak, N. Goldenfeld, A. A. Mailybaev, and G. Eyink, Phys. Rev. E 105, 065113 (2022)] that thermal fluctuations, absent from the continuum description of gases, can terminate the energy cascade at a length scale larger than mean-free-path considerations alone would suggest. Additionally, for high-Mach-number turbulent flows, the Kolmogorov length scale can be comparable to the gas-molecule mean free path, which could induce noncontinuum molecular-level effects in the turbulent energy cascade. To investigate these two issues, compressible Taylor-Green vortex flow is simulated using the direct simulation Monte Carlo (DSMC) method and direct numerical simulations (DNS) of the Navier-Stokes equations. It is found that the molecular-gas-dynamics spectra grow quadratically with wavenumber in the dissipation range due to thermal fluctuations instead of decreasing exponentially as the continuum description predicts. Macroscopically, thermal fluctuations appear to break the flow symmetries and thereby produce different but statistically similar routes from the initial non-turbulent flow to the long-time turbulent flow.