Minimum scale and spatial resolution requirement for direct numerical simulations of compressible turbulence

In the direct numerical simulation (DNS) of compressible turbulence using Navier-Stokes equations, due to the incomplete resolution of shocklets, the classical grid resolution criterion based on the usual Kolmogorov length scale appears insufficient for high-order statistics. The present study discusses the minimum scale of compressible turbulence under the continuum assumption , and establishes new spatial resolution requirements for DNS. We first define the minimum shock scale for one-dimensional Burgers turbulence, and derive a spatial resolution criterion essential for fully resolving the second- and third-order velocity gradient moments. We demonstrate that this shock scale definition is also applicable to one-dimensional Navier-Stokes turbulence, and validate the spatial resolution requirement through numerical simulations of the Shu-Osher problem. The analysis is then extended to multi-dimensional turbulence. Through theoretical analysis and numerical studies, we conclude that the minimum local Kolmogorov scale , η min , describes the smallest structure in turbulence and is determined by the strongest shocklet. Furthermore, we establish a requirement of η min / Δ x ≳ 1.5 for compressible turbulence, and validate it through DNSs of two-dimensional compressible turbulence with different grid resolutions. The present study provides a reference on spatial resolution requirement for DNS of compressible turbulence.