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High-order entropy stable isothermal wall boundary condition for the compressible Navier--Stokes equations

Mohammed Sayyari, Nail K. Yamaleev

arXiv:2610.09249v1math.NAcs.NA

TL;DR

Existing isothermal wall treatments in SBP-SAT schemes lack a provably entropy-stable formulation, despite the need to control wall entropy production. This paper develops a high-order temperature-gradient penalty within an SBP-SAT scheme. The penalty is design-order accurate, and convergence of the ballistic-energy bound shows the scheme is entropy-stable.

  • Problem

    Prior isothermal wall treatments have unresolved entropy-production or accuracy issues, and an existing formulation cannot be adapted to SBP-SAT schemes.

    Source: §1 · pp. 1–2

  • Method

    The paper derives a high-order isothermal wall penalty as a temperature correction within an SBP-SAT discretization.

    Source: §3 · pp. 3, 6, §6 · p. 18

  • Results

    The penalty is design-order accurate, and convergence of the ballistic-energy bound shows that the scheme is entropy-stable.

    Source: §6 · p. 19

  • Takeaways & Limitations

    A relaxed parameter α = Δx_avg^-q, q < p, is sufficient to enforce the isothermal condition even for the Ma = 10 heated-cylinder flow with an approximately 4000 K stagnation temperature difference.

    Source: §6 · p. 19

    The design-order choice α = Δx^-p produces very stiff problems that require implicit time-steppers.

    Source: §6 · p. 19

Abstract

from arXiv · show

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 a new isothermal no-slip wall boundary conditions that: 1) enforces the $T|_{y=0}=T^{wall}$ and $\vec{v}|_{y=0}=\vec{v}^{wall}$ conditions with the design order of accuracy while maintaining stability, 2) provides the correct sign of entropy production at the wall, and 3) mimics 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.

1. Introduction.

Earlier isothermal wall treatments had unresolved entropy-production or accuracy issues, and the existing formulation could not be adapted to SBP-SAT schemes. This paper develops and verifies a new isothermal wall condition with design-order accuracy, stable entropy production, and discrete entropy balance.

1. Introduction.

  • The formulation in mimics the continuous entropy inequality but is not provably entropy-stable and can produce positive wall entropy production; its accuracy and order were not verified.The treatments in likewise reported positive wall entropy production when imposing the isothermal condition through the viscous state.Source: §1 · pp. 1–2
  • The proposed condition enforces wall temperature and velocity at design order, gives the correct sign of wall entropy production, and mimics the discrete Navier–Stokes entropy balance.Its entropy boundedness and order properties are verified on internal and external compressible-flow cases.Source: §1 · p. 2
  • The isothermal condition in cannot be adapted to SBP-SAT schemes because it requires T_wall > T.The paper therefore develops its condition from the work in instead.Source: §1 · p. 2
  • The reported tests include channel and cylinder convergence studies, subsonic flow around a square cylinder, hypersonic cylinder flow, and Couette flow.The paper identifies the convergence studies as tests of design order and the hypersonic case as demonstrating versatility.Source: §1 · p. 2

2. The compressible Navier–Stokes equations and the entropy inequal-

This section presents the compressible Navier–Stokes equations and their entropy inequality, including how wall conditions affect entropy production. For isothermal walls, the physical wall entropy term makes stability depend on the heat-entropy flux.

2. The compressible Navier–Stokes equations and the entropy inequality

  • The compressible Navier–Stokes equations are written in curvilinear coordinates using a conservative-variable vector containing density, momentum, and total energy.The supplied passage identifies U as [ρ, ρv1, ρv2, ρv3, ρE]⊤.Source: §2 · p. 2
  • The inviscid and viscous fluxes are stated using Einstein summation and Cartesian coordinates.The passage notes that the summation convention applies to the equations and that the fluxes are given in this formulation.Source: §2 · pp. 2–3
  • Pressure is given by P = ρRgT, κ denotes heat conductivity, and μ denotes dynamic viscosity.The stress tensor is also introduced in this formulation.Source: §2 · p. 3
  • The entropy inequality uses the Godunov entropy pair and applies when the boundary conditions are entropy-stable.An adiabatic wall has zero entropy production, whereas an isothermal wall contributes physical wall entropy production.Source: §2 · p. 3
  • An isothermal wall is entropy-stable only if its heat flux does not exceed the physical heat-entropy flux; entropy production during wall cooling violates the inequality.Source: §2 · p. 3
  • The discrete penalty is introduced as a correction to the temperature gradient.Source: §2 · p. 3

3. Entropy bounded isothermal boundary condition.

The section introduces a temperature-jump penalty as a wall heat-entropy flux correction and analyzes its entropy and ballistic-energy bounds. Under a bounded-boundary-term assumption, the bound yields an L2 solution bound; the high-order discretization uses SBP-SAT operators.

3. Entropy bounded isothermal boundary condition

  • The wall correction g(t) is defined from the temperature jump and applied as a heat-entropy flux source; α is a penalty parameter and κ comes from the Fourier-term coefficient.The analysis adds the penalty to the continuous equations to study transition to an isothermal wall from an initial temperature different from T_wall.Source: §3 · p. 3
  • The entropy-balance argument bounds the entropy time derivative when the heat-entropy flux is bounded, but the temperature-gradient term has no a priori estimate.The entropy production at the wall is considered in deriving this bound.Source: §3 · p. 4
  • The ballistic-energy estimate uses the reciprocal penalty parameter to assess convergence numerically despite the lack of an a priori temperature-gradient estimate, assuming the boundary term is bounded.The estimate is motivated by the need for a controllable bound.Source: §3 · p. 5
  • If the ballistic energy and the boundary term are bounded, the solution is bounded in the L2 norm by initial and boundary data.The section states this result as Theorem 3.1 for the isothermal wall condition.Source: §3 · p. 6
  • The high-order entropy-stable scheme uses SBP operators and SAT penalties, with discontinuously collocated Legendre–Gauss–Lobatto nodes and weak enforcement at interfaces and boundaries.The SBP operators mimic integration by parts discretely; entropy-stable boundary conditions are required for the scheme's provable entropy stability.Source: §3 · p. 6

4. Entropy-stable high-order scheme.

The scheme combines SBP-based spatial discretization and weak wall penalties with a heat-flux correction for isothermal walls. Its entropy analysis derives a discrete wall balance and a bound on the discrete solution.

4. Entropy-stable high-order scheme.

  • SBP operators provide a telescoping form of the derivative and extend to multiple dimensions by tensor products.The operators satisfy polynomial accuracy and a discrete integration-by-parts structure.Source: §4 · pp. 6–7

4.1. Semi-discrete Navier–Stokes equations.

  • The high-order scheme is augmented with entropy-residual-based artificial dissipation and a first-order hybrid when thermodynamic variables become negative.The resulting hybrid scheme is described as conservative, entropy-stable, GCL-satisfying, and design-order.Source: §4.1 · pp. 7–8

4.2. Entropy-stable wall boundary conditions.

  • Wall conditions are imposed weakly with SAT penalties, including a dissipative matrix penalty and a heat-entropy flux source term.The heat-flux correction is defined as an L2 function and can represent a prescribed wall heat flux.Source: §4.1 · p. 8, §4.2 · p. 9

4.3. The isothermal wall boundary condition.

  • The isothermal condition uses a heat-flux-consistent penalty, with α tuned not to exceed the physical wall heat-entropy flux.For design-order properties, α scales with inverse grid spacing to the polynomial degree; stiffness may motivate a smaller capped power.Source: §4.2 · p. 9, §4.3 · p. 10

4.3.1. Contribution to the discrete entropy balance.

  • The wall entropy balance is a discrete analogue of the continuous balance, with a nonzero isothermal contribution when the temperature gradient does not vanish.The analysis also identifies positive-definite viscous dissipation and an entropy-gradient penalty.Source: §4.3.1 · pp. 10–11

4.3.2. Contribution to the semi-discrete ballistic energy.

  • The ballistic-energy analysis derives a bound on the discrete solution; benchmark cases include channel, square-cylinder, supersonic-cylinder, and thermal-Couette flows.The cases assess design-order accuracy, extreme wall heat fluxes, and comparison with a reference case.Source: §4.3.2 · p. 13, §5 · p. 13

5. Numerical results.

The numerical results assess wall-condition accuracy across channel and cylinder flows, test stability under extreme wall heat fluxes, and compare thermal Couette flow with a reference solution.

5.1. 2D Channel flow.

  • For p = 3 and p = 4, wall-temperature convergence reaches at least O(h^p+1), while p = 2 is slightly suboptimal, possibly due to initial-temperature oscillations.The authors conclude that the higher-order cases demonstrate design-order wall-temperature accuracy.Source: §5.1 · p. 14, Table 1 · p. 15
  • Wall-velocity convergence follows the expected O(h^p+1) rate for every polynomial order tested, and the new isothermal condition does not affect its convergence.The reported velocity rates are consistent with.Source: §5.1 · pp. 14–15, Table 2 · p. 15
  • The channel’s ballistic-energy bound converges to zero with mesh refinement, although its convergence rate need not be design order.The bound is assessed even though the wall temperature gradient is not known a priori.Source: §5.1 · p. 15, Table 3 · p. 16

5.2. Subsonic flow around a cooling square cylinder.

  • For the cooling square cylinder, scaling the penalty parameter as α = Δx^-p yields design-order errors in both wall-temperature and wall-velocity values.This setup substantially increases stiffness; the reported simulations use implicit backward Euler to alleviate it.Source: §5.2 · pp. 15–16, Table 4 · p. 17

5.3. Bow shock ahead of a heated cylinder.

  • In the Ma = 10 cylinder case, the scheme enforces wall temperature despite an approximately 4000 K difference between the stagnation bubble and wall, and remains stable under extreme wall heat fluxes.The figure shows the temperature field and temperatures along the stagnation and side lines at t = 10.Source: §5.3 · p. 17, Figure 4 · p. 17

5.4. Thermal Couette flow.

  • In thermal Couette flow, the wall condition leaves the velocity profile unaffected, and the solution matches the Navier–Stokes reference.The proposed high-order isothermal penalty is derived as a temperature correction to the Fourier term.Source: §5.4 · p. 18, §6 · p. 18, Figure 5 · p. 18

6. Conclusions.

The penalty is design-order accurate, and convergence of the ballistic energy bound demonstrates entropy stability; relaxed scaling can still enforce the isothermal condition in an extreme heated-cylinder case.

6. Conclusions.

  • The penalty is design-order accurate, and convergence of the ballistic energy bound demonstrates that the scheme is entropy-stable.The bound estimates the solution's L2 norm continuously and discretely through SBP-SAT operators.Source: §6 · p. 19
  • Design-order scaling α = Δx^-p produces stiff problems that require implicit time-steppers, a limitation of the scheme.Source: §6 · p. 19
  • Relaxed scaling with q < p still enforces the isothermal condition in a Ma = 10 heated-cylinder flow with an approximately 4000 K stagnation temperature difference.The relaxed parameter uses the average grid spacing raised to −q.Source: §6 · p. 19
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