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arXiv · 2609.32249

Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations

Matteo Parsani·Rasha AlJahdali·Lisandro Dalcin
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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 $L_2$-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.

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Matteo Parsani, Rasha AlJahdali, Lisandro Dalcin. 2026-09-26. Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations. https://arxiv.org/abs/2609.32249

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