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

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.