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

Abstract

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, ${\eta }_{\min }$, describes the smallest structure in turbulence and is determined by the strongest shocklet. Furthermore, we establish a requirement of ${\eta }_{\min }/\Delta x\gtrsim 1.5$ 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.