Entropy stable high-order schemes for the compressible Navier--Stokes equations with isothermal no-slip wall boundary conditions

Abstract

Entropy stable boundary conditions are critical for ensuring that the corresponding numerical scheme satisfies the discrete entropy inequality, thus, mimicking the second law of thermodynamics for the compressible Navier-Stokes equations. Provably entropy-stable adiabatic wall boundary conditions were introduced in Parsani 2014. These discrete wall boundary conditions were further generalized in Dalcin 2019 to a moving adiabatic solid wall or a wall with a prescribed heat flux for the compressible Navier–Stokes equations discretized by using summation-by-parts (SBP) and simultaneous-approximation-term (SAT) operators. Also, a similar approach was used in Chan 2022 to impose adiabatic and isothermal no-slip wall boundary conditions for the modal Discontinuous Galerkin method for the compressible Navier-Stokes equations. Although the adiabatic wall boundary conditions developed in Chan 2022 are provably stable in the entropy sense, the authors report that while the proposed discrete isothermal no-slip wall boundary conditions mimic the continuous entropy inequality, they are not entropy-stable and invariably lead to positive entropy production. Moreover, to the best of our knowledge, no entropy-stable formulation for isothermal wall boundary conditions is currently available in the literature. This talk presents a new isothermal no-slip wall boundary conditions that: 1) enforce the $T|{y=0}=T^\text{wall}$ and $v|{y=0}=v^\text{wall}$ conditions with the design order of accuracy while maintaining stability, 2) provide the correct sign of entropy production at the wall, and 3) mimic the entropy balance of the Navier-Stokes equations at the discrete level. The structure-preserving and design-order properties of the proposed methodology will be demonstrated and verified on standard benchmark problems for compressible flows. References: - M. Parsani, M. H. Carpenter, and E. J. Nielsen, “Entropy stable wall boundary conditions for the compressible Navier–Stokes equations,” tech. rep., 2014. - L. Dalcin, D. Rojas, S. Zampini, D. C. D. R. Fernández, M. H. Carpenter, and M. Parsani, “Conservative and entropy stable solid wall boundary conditions for the compressible Navier–Stokes equations: Adiabatic wall and heat entropy transfer,” JCP, vol. 397, p. 108775, 2019. - J. Chan, Y. Lin, and T. Warburton, “Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier–Stokes equations,” JCP, vol. 448, p. 110723, 2022.

Mohammed Sayyari

Postdoctoral Research Associate, Adjunct Assistant Professor

Mohammed is a systems thinker. His Mathematical interests are in Numerical Analysis and Modeling of Evolutionary PDEs. His Scientific interests are in the advancement of accurate, efficient, economic and enviromentally friendly technologies.