High-order entropy-stable isothermal wall boundary condition for the compressible Navier--Stokes equations
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 discretely. Provably entropy-stable adiabatic wall boundary conditions were introduced in Parsani 2014~\cite{parsani2014entropy}. These discrete wall boundary conditions were further generalized in Dalcin 2019~\cite{dalcin2019conservative} 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. Currently, no entropy-stable formulation using SBP-SAT operators for isothermal wall boundary conditions is available in the literature. This paper presents new isothermal no-slip wall boundary conditions that: 1) enforce the T|y=0=Twall and v⃗|y=0=v⃗wall 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 are demonstrated and verified on standard benchmark problems for compressible flows.
Related
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- Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form
- Local smooth solutions of the free-boundary compressible primitive equations with physical vacuum
BibTeX
@misc{sayyari2026highorder,
title = {{High-order entropy-stable isothermal wall boundary condition for the compressible Navier--Stokes equations}},
author = {Mohammed Sayyari and Nail K. Yamaleev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09249},
eprint = {2610.09249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09249},
}
RIS
TY - UNPB
TI - High-order entropy-stable isothermal wall boundary condition for the compressible Navier--Stokes equations
AU - Mohammed Sayyari
AU - Nail K. Yamaleev
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.09249
UR - https://arxiv.org/abs/2610.09249
AB - 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 discretely. Provably entropy-stable adiabatic wall boundary conditions were introduced in Parsani 2014~\cite{parsani2014entropy}. These discrete wall boundary conditions were further generalized in Dalcin 2019~\cite{dalcin2019conservative} 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. Currently, no entropy-stable formulation using SBP-SAT operators for isothermal wall boundary conditions is available in the literature. This paper presents new isothermal no-slip wall boundary conditions that: 1) enforce the T|_y=0=T^wall and v⃗|_y=0=v⃗^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 are demonstrated and verified on standard benchmark problems for compressible flows.
ER -