Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations
Abstract
We develop entropy-stable isothermal-wall conditions for the compressible Navier--Stokes equations at continuous and semi-discrete levels. For a stationary, impermeable, no-slip wall at constant positive temperature, adding total-energy density divided by wall temperature to the canonical convex entropy cancels the wall contribution without suppressing heat transfer. This affine modification preserves the entropy Hessian, relative entropy, and volume entropy production. It is proportional to total ballistic-energy density and yields a conditional L2-type bound based on the initial field under the stated assumptions. Two wall formulations use a common discontinuous Galerkin discretization with the summation-by-parts property and simultaneous approximation terms. Both give zero adapted wall contribution without a penalty and a nonpositive contribution with it. The complete-residual formulation penalizes the full wall-data residual and balances fluid energy through its complete numerical wall flux. The Fourier-energy-exact formulation combines an entropy-neutral skew correction with a momentum-projected penalty so that the complete numerical outward energy flux equals the one-sided numerical Fourier heat flux at every wall node and spatial resolution. Pointwise tests verify the local identities. Manufactured-solution studies with entropy-stable interior coupling demonstrate convergence rates consistent with order ps+1 in the discrete L2 norm, where ps is the solution polynomial degree. Thermal-relaxation calculations verify the semi-discrete energy and adapted-entropy balances. Three-dimensional calculations demonstrate balance closure in a closed domain, convergence of curved-wall traces, and applicability to a complex supersonic separated flow.
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BibTeX
@misc{parsani2026isothermal,
title = {{Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations}},
author = {Matteo Parsani and Rasha AlJahdali and Lisandro Dalcin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32249},
eprint = {2609.32249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32249},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations
AU - Matteo Parsani
AU - Rasha AlJahdali
AU - Lisandro Dalcin
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32249
UR - https://arxiv.org/abs/2609.32249
AB - We develop entropy-stable isothermal-wall conditions for the compressible Navier--Stokes equations at continuous and semi-discrete levels. For a stationary, impermeable, no-slip wall at constant positive temperature, adding total-energy density divided by wall temperature to the canonical convex entropy cancels the wall contribution without suppressing heat transfer. This affine modification preserves the entropy Hessian, relative entropy, and volume entropy production. It is proportional to total ballistic-energy density and yields a conditional L_2-type bound based on the initial field under the stated assumptions. Two wall formulations use a common discontinuous Galerkin discretization with the summation-by-parts property and simultaneous approximation terms. Both give zero adapted wall contribution without a penalty and a nonpositive contribution with it. The complete-residual formulation penalizes the full wall-data residual and balances fluid energy through its complete numerical wall flux. The Fourier-energy-exact formulation combines an entropy-neutral skew correction with a momentum-projected penalty so that the complete numerical outward energy flux equals the one-sided numerical Fourier heat flux at every wall node and spatial resolution. Pointwise tests verify the local identities. Manufactured-solution studies with entropy-stable interior coupling demonstrate convergence rates consistent with order p_s+1 in the discrete L_2 norm, where p_s is the solution polynomial degree. Thermal-relaxation calculations verify the semi-discrete energy and adapted-entropy balances. Three-dimensional calculations demonstrate balance closure in a closed domain, convergence of curved-wall traces, and applicability to a complex supersonic separated flow.
ER -