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

Abstract: 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 H\"older exponent. Our work is a first step towards a comparison of coherent flow structures in a compressible turbulent flow and related anomalous dissipation.