The question of the dissipative anomaly, that is, the independence of the mean kinetic energy dissipation rate from viscosity at high Reynolds numbers, has been studied for a long time in incompressible turbulence. For compressible flows, data are scarce and limited to the subsonic regime. Our work stands apart: the closest neighbors in the archive deal either with incompressible turbulence or with magnetic fields and superfluidity, but they do not address the behavior of dissipation across the sound barrier. We are the first to systematically trace how the anomaly changes as the Mach number varies from 0.1 to 10.
We performed a series of direct numerical simulations of three-dimensional isothermal compressible turbulence in a cubic domain with periodic boundary conditions. Turbulence was sustained by a purely solenoidal stochastic force at large scales. The Mach number ranged from 0.1 to 10, and the Reynolds number from 100 to 2400. For the analysis, we used three approaches: balance equations for the dissipation rate with separate contributions from solenoidal and dilatational motions, an extension of the Duchon–Robert formalism to compressible flows for local precursors of anomalous dissipation, and a geometric analysis of intense dissipation regions using box-counting and multifractal formalism.
The normalized mean kinetic energy dissipation rate, as well as its solenoidal and dilatational components, approach finite asymptotic values as the Reynolds number increases. However, these values are not universal: the data split into two branches. One branch corresponds to subsonic and transonic regimes (root-mean-square Mach number less than or about 1), the other to strongly supersonic regimes (Mach number greater than or equal to 3). The difference is most pronounced for the total dissipation. For the solenoidal and dilatational components, the dependence on the Reynolds number is additionally sensitive to the choice of integral scale and root-mean-square velocity. At Mach number 10, the regions of maximum dissipation become two-dimensional sheets, as confirmed by a box-counting dimension of about 2 and multifractal analysis. At Mach number 0.1, intense dissipation is distributed over a network of shear layers with an effective dimension of about 2.5. The generation mechanisms also change: at low Mach numbers, vortex stretching dominates; at high Mach numbers, the interaction of vortical and dilatational motions within shock layers takes over.
Our work shows that the dissipative anomaly in compressible turbulence is not a single phenomenon: it depends on the Mach number. This distinguishes it from the classical picture in incompressible flows, where the anomaly is unique. Unlike neighboring studies that examined either incompressible turbulence or isolated aspects of compressibility, we are the first to link the two branches of the anomaly to a reorganization of the spatial structure of dissipative regions. This result is important for building accurate turbulence models in astrophysics and engineering applications where compressibility is significant.
A natural next step is to include dilatational forcing to test how it affects the branches of the anomaly. One could also go beyond the isothermal approximation, for example by using a polytropic equation of state with an exponent greater than one or by solving the full energy equation. This would help assess how robust the two branches are to changes in thermodynamics. Increasing the Reynolds number in future simulations would help distinguish between weak and strong anomaly.
The results will influence turbulence models in astrophysics, where compressible supersonic flows are common, and engineering calculations of high-speed flows.
Perform simulations with dilatational forcing and at higher Reynolds numbers to refine the nature of the anomaly.
In hydrodynamics, the question of a rigorous proof of the existence of anomalous dissipation for solutions of the Euler and Navier–Stokes equations remains open. Our data provide an empirical basis for mathematical constructions, in particular for convex integration methods, and show that in the compressible case the anomaly may be non-unique.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This study examines the dissipative anomaly in isothermal compressible turbulence driven by solenoidal forcing, using direct numerical simulations with resolutions up to 2048³. It is shown that the normalized kinetic energy dissipation rate approaches finite values as the Reynolds number increases, but these values differ between subsonic/transonic and highly supersonic regimes. The analysis includes balance equations, an extension of the Duchon–Robert formalism, and a geometric analysis of intense dissipation regions.
Where its strength is
The strength of the work lies in its systematic tracking of the Mach number dependence of the dissipative anomaly up to the highly supersonic regime, revealing two branches of the anomaly and linking them to a restructuring of dissipation structures from vortex tubes to shock layers.
Where it could go next
Worth trying
Why this may matter
Understanding the Mach number dependence of the dissipative anomaly is important for astrophysical and engineering applications where turbulence is often supersonic, and it may aid in constructing subgrid dissipation models for numerical simulations.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.51, against a usual 0.64. It looks like few have drilled nearby.
At Mach number 10, the regions of maximum dissipation have a dimension of about 2, meaning they are almost perfectly flat, like sheets of paper, even though the flow itself is three-dimensional.