Extending the Duchon-Robert framework for anomalous dissipation to compressible fluid flows

Anomalous dissipation, the persistence of a finite mean kinetic energy dissipation as the Reynolds number tends to infinity, occurs in flows with sufficiently spatially rough velocity fields. Compressible turbulence adds further anomalous dissipation mechanisms, which we investigate in this work. To this end, the Duchon-Robert framework (DR) for anomalous dissipation is extended from the incompressible to the compressible Navier-Stokes flow case. We obtain three integral dissipation terms, two anomalous and a viscous one, which arise from the pressure-dilatation and density variations, differently from the incompressible case. Subsequently, fully compressible one-dimensional flows with traveling and mutually crossing shock waves are analysed in detail. In such flows, DR reveals a local maximum of anomalous dissipation at the shock front. Furthermore, DR is compared with a coarse-grain cascade theory of compressible turbulence due to Aluie (AL) and the relevant dissipation flux terms of both frameworks are identified and compared with each other. The comparison shows that each contribution related to the compressibility effects in DR has its analogue in AL. Finally, a piecewise linear shock-type velocity profile, which approximates the crossing of two shock waves from the simulations, is used for an analytical analysis of the anomalous dissipation terms of DR to analyse the dependence of the terms on the local Hölder exponent. Our work is a first step towards a comparison of coherent flow structures in a compressible turbulent flow and related anomalous dissipation.

💡 Research Summary

The paper extends the Duchon‑Robert (DR) framework, originally devised for incompressible turbulence, to the full compressible Navier–Stokes equations and investigates the resulting anomalous (or “dissipative”) energy dissipation mechanisms. By introducing the density‑weighted velocity w = √ρ u, the authors rewrite the momentum equation in a form that isolates pressure, density, and viscous stress contributions. Applying a smooth spatial filter φε at scale ε, they regularize the fields (e.g., wε = φε * w) and combine the original and filtered equations to obtain a local kinetic‑energy balance. In the limit ε → 0 three extra terms appear in addition to the usual viscous dissipation:

1. D₍wwu₎ – the classic DR term arising from velocity‑velocity increments, representing the non‑viscous cascade of kinetic energy.
2. D₍wχp₎ – a new term involving increments of the density‑weight factor χ = 1/√ρ and pressure, which captures a pressure‑work channel specific to compressible flows.
3. D₍wχτ₎ – a term coupling density‑weight increments with the viscous stress tensor, representing a mixed viscous‑compressibility contribution.

These three terms survive in the ε → 0 limit only if the corresponding fields are sufficiently rough, i.e., their local Hölder exponents satisfy h < 1/3 (or analogous thresholds for density and pressure). The authors then present the Aluie (AL) framework, which uses Favre (density‑weighted) filtering to separate large‑scale and sub‑scale dynamics. In AL the kinetic‑energy budget contains a pressure‑work flux Πℓ, a viscous‑dissipation flux, and a compressibility‑transfer term. By explicit comparison the paper shows a one‑to‑one correspondence: D₍wwu₎ ↔ the inertial‑cascade flux, D₍wχp₎ ↔ Πℓ, and D₍wχτ₎ ↔ the compressibility‑viscous transfer term. This structural equivalence validates both approaches and clarifies the physical meaning of the new DR contributions.

To illustrate the theory, the authors conduct one‑dimensional numerical experiments of compressible gas dynamics featuring (i) a single traveling shock and (ii) two shocks that intersect. Using high‑resolution shock‑capturing schemes, they compute the filtered fields and evaluate the three DR dissipation densities. Near the shock front the velocity and density gradients become extremely steep, the local Hölder exponent drops below the critical value, and both D₍wwu₎ and D₍wχp₎ exhibit sharp peaks. In the crossing‑shock configuration the pressure‑work term is amplified by the superposition of the two compression fronts, leading to a pronounced maximum of total anomalous dissipation precisely at the interaction zone. These results confirm the DR prediction that anomalous dissipation is localized where the flow is most singular.

Finally, the paper provides an analytical model of the crossing‑shock region using a piecewise‑linear velocity profile that mimics the numerical data. By inserting this profile into the DR expressions, the authors obtain closed‑form scaling laws for each dissipation term as a function of the local Hölder exponent h. For example, D₍wwu₎ ∝ ε^{3h‑1}, showing that when h < 1/3 the term remains finite as ε → 0. The pressure‑density term D₍wχp₎ acquires additional dependence on the Hölder exponents of density and pressure, leading to a more intricate ε‑scaling. This analytical treatment demonstrates that shock‑like structures, characterized by very low Hölder exponents, act as precursors and primary contributors to anomalous dissipation in compressible turbulence.

Overall, the study achieves three major advances: (1) a rigorous extension of the DR anomalous‑dissipation framework to compressible flows, (2) a clear mapping between DR and Aluie’s coarse‑graining approach, and (3) concrete numerical and analytical evidence that shock fronts—and especially their interactions—are the dominant sites of anomalous energy loss. The work opens the path for future investigations of three‑dimensional compressible turbulence, where coherent structures such as vortex‑shock complexes could be examined with the combined DR/AL toolbox to quantify anomalous dissipation in high‑Mach-number astrophysical and aerospace applications.