ANOMALOUS DISSIPATION IN COMPRESSIBLE FLOWS WITH CONTROLLED ROUGHNESS: A 1D FRACTAL FIELD TEST OF THE DUCHON–ROBERT FRAMEWORK

Abstract

We propose a mathematical test of anomalous dissipation of kinetic energy for compressible flows, using synthetic one-dimensional velocity and density fields with fractal properties. Contrary to existing results based on shock solutions of the compressible Navier-Stokes equations, the present approach focuses on the properties of the fields by assuming velocity and density fields with prescribed Hölder regularity. The Duchon–Robert coarse-graining technique is employed for the evaluation of the energy dissipation rate for the vanishing filter width case. The analytical scaling laws for the energy dissipation rate indicate that the anomalous dissipation rate is finite when the Hölder regularity of the velocity fields is less than or equal to 1/3, extending the Onsager regularity hypothesis for the case of compressible fluids. The numerical evaluation of the Duchon–Robert integrals confirm the analytical results for the threshold behaviour, while an additional dissipation rate is found due to the density field roughness.