Mach-number-dependent dissipative anomaly in isothermal compressible turbulence

Abstract: Using a comprehensive set of three-dimensional, high-resolution direct numerical simulations, we investigate the existence of a dissipative anomaly in isothermal, homogeneous, isotropic compressible turbulence driven by solenoidal forcing. We find that the total kinetic-energy dissipation rate, as well as its solenoidal and dilatational components, approaches finite asymptotic values with increasing Reynolds number $Re$. The normalized mean dissipation rates collapse onto two distinct branches: one corresponding to the subsonic and transonic regimes, with root-mean-square Mach numbers ($M_{\rm rms}\lesssim 1$), and another to the highly supersonic regime, with ($M_{\rm rms}\ge 3$). This two-branch Mach-number dependence is most pronounced for the total kinetic-energy dissipation. For the solenoidal and dilatational dissipation rate components, the dependence on $Re$ depends in addition on the specific choice of the integral scale and root-mean-square velocity. Despite grid resolutions of up to $2048^3$ points the Reynolds numbers accessible are not sufficiently large to distinguish conclusively between a weak and a strong dissipative anomaly. We substantiate these findings using three complementary approaches: (i) a detailed analysis of the mechanisms responsible for dissipation generation, based on the corresponding dissipation-rate balance equations and their individual production terms; (ii) an investigation of precursors of anomalous dissipation using the Duchon--Robert framework extended to compressible flows; and (iii) a geometrical characterization of regions of intense dissipation. Taken together, these analyses provide consistent evidence for the existence of a dissipative anomaly in isothermal compressible turbulence, while leaving its precise weak or strong character unresolved.