A thermodynamics-based turbulence model for isothermal compressible flows
Zhiting Ma, Wen-An Yong, Yi Zhu
TL;DR
Addresses turbulence modeling for isothermal compressible flows by deriving a thermodynamically consistent, hyperbolic closure via Favre averaging and Conservation-Dissipation Formalism (CDF). Introduces entropy-based closures with non-equilibrium variables, yielding a closed first-order system that is compatible with Prandtl's one-equation model in the low Mach regime, and shows that the low-Mach limit recovers an incompressible Reynolds-averaged Navier–Stokes system coupled to Prandtl's equation. A scaling analysis and rigorous justification demonstrate convergence of the rescaled hyperbolic system to the incompressible limit as $\varepsilon \to 0$, with error bounds $\|\mathbf{E}\|_s \le C \varepsilon$. The resulting framework provides a causality-respecting, thermodynamically consistent, hyperbolic closure for compressible turbulence that is amenable to efficient numerical discretization.
Abstract
This study presents a new turbulence model for isothermal compressible flows. The model is derived by combining the Favre averaging and the Conservation-dissipation formalism -- a newly developed thermodynamics theory. The latter provides a systematic methodology to construct closure relations that intrinsically satisfy the first and second laws of thermodynamics. The new model is a hyperbolic system of first-order partial differential equations. It has a number of numerical advantages, and addresses some drawbacks of classical turbulence models by resolving the non-physical infinite information propagation paradox of the parabolic-type models and accurately capturing the interaction between compressibility and turbulence dissipation. Furthermore, we show the compatibility of the proposed model with Prandtl's one-equation model for incompressible flows by deliberately rescaling the model and studying its low Mach number limit.