Scaling in Supersonic Turbulence: Energy Spectra and Fluxes using High-Fidelity Direct Numerical Simulations

TL;DR

This study uses high-resolution DNS to explore how energy spectra and transfer mechanisms in supersonic turbulence change with Mach number, revealing a shift from Kolmogorov to Burgers scaling and the dominance of mode interactions.

Contribution

The paper provides detailed insights into energy transfer processes and spectral scaling laws in high Mach number turbulence using advanced numerical simulations.

Findings

Energy spectra steepen from $k^{-5/3}$ to $k^{-2}$ as Mach number increases.

Cross-scale transfer from solenoidal to compressive modes dominates the inertial range.

Pressure dilatation significantly influences energy transfer in supersonic turbulence.

Abstract

Supersonic turbulence is vital to astrophysical and high-speed engineering flows, yet its energy transfer mechanisms remain poorly understood. We present high-resolution ($1024^3$) direct numerical simulations (DNS) of forced compressible turbulence across a range of turbulent Mach numbers ($M_t = 0.2$ to $3.0$). Using the GPU-accelerated solver \texttt{DHARA} with a seventh-order, low-dissipation Targeted Essentially Non-Oscillatory (TENO) scheme, we resolve both fine-scale eddies and sharp shock fronts. Our results reveal a fundamental shift in the energy cascade in the supersonic regime. As $M_t$ increases, the rotational kinetic energy spectrum steepens from a Kolmogorov-like $k^{-5/3}$ scaling toward a Burgers-like $k^{-2}$ scaling. Conversely, the compressive energy spectrum becomes shallower, deviating from Burgers scaling. We show that these spectral modifications are driven by a dominant cross-scale transfer of energy from solenoidal to compressive modes within the inertial range, alongside significant contributions from pressure dilatation. Scaling laws for the root-mean-square compressive velocity ($U_C$) and compressive energy flux ($\Pi_C$) are found to mirror classical Burgers turbulence. Finally, we show that while energy injection rates depend on forcing type rather than Mach number, increased $M_t$ leads to decreased rotational dissipation and increased compressive dissipation and pressure dilatation. These findings elucidate intermodal energy cascade mechanisms, advancing our understanding of energy transfers in supersonic turbulence.