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Entropy-stable moving-wall boundary conditions for the ALE formulation of the compressible Navier-Stokes equations

Luca Galimberti, Roberto Nuca, Lisandro Dalcin, Alberto Guardone, Matteo Parsani

arXiv:2608.25146v1math.NAmath.AP

TL;DR

Moving and deforming ALE meshes make entropy-stable wall treatment difficult because physical fluxes, grid velocity, and metric terms interact in the entropy balance. The paper derives moving-wall conditions and extends their entropy properties to a semi-discrete SBP framework, with numerical tests demonstrating accuracy, robustness, scalability, and applicability across moving-boundary flow regimes.

  • Problem

    Entropy-stable wall treatments for compressible Navier–Stokes equations do not directly extend from static grids to general ALE boundaries, where grid velocity and metric terms enter the entropy balance.

  • Method

    The paper derives continuous moving-wall entropy conditions and proves their semi-discrete conservation or stability within a DG-SBP-SAT framework designed to extend across diagonal-norm SBP methods.

  • Results

    Moving-wall conditions are entropy conservative for the compressible Euler equations and entropy stable for the compressible Navier–Stokes equations, while numerical tests validate accuracy, robustness, and scalability.

  • Takeaways & Limitations

    High-order entropy-stable schemes can be applied to complex moving-motion cases across varied flow regimes and multiphysics interactions.

  • Takeaways & Limitations

    The stability analysis applies to spatial semi-discretization; fully discrete entropy behavior also depends on the temporal approximation.

Abstract

from arXiv · show

We present a high-order entropy-stable framework for the compressible Euler and Navier-Stokes equations on moving domains. The space-time mapping describing the domain motion is recast in an arbitrary Lagrangian Eulerian (ALE) formulation, in which the physical inviscid fluxes and the contributions induced by mesh motion are treated in a unified manner. At the continuous level, we prove that the proposed moving-wall boundary conditions are entropy conservative for the Euler equations and entropy stable for the Navier-Stokes equations. The no-slip condition is formulated in terms of the velocity relative to the moving wall, yielding a bounded inviscid contribution to the entropy balance, while the viscous terms contribute only entropy dissipation. Using diagonal norm summation-by-parts (SBP) operators together with appropriate numerical fluxes, these properties are extended to the semi-discrete formulation, resulting in nonlinear stability in the $L^2$ sense. The accuracy, robustness, and scalability of the proposed method are demonstrated in practice through an extensive set of numerical experiments, ranging from canonical two-dimensional verification cases to large-scale turbulent and supersonic simulations involving moving boundaries and fluid-structure interaction. The results confirm the suitability of high-order entropy-stable schemes for complex moving-domain problems across a broad range of flow regimes and multiphysics applications. Because the analysis relies on the SBP property rather than on a particular discretization, the framework naturally extends to a broad class of methods based on diagonal-norm SBP operators, including finite volume, finite element, and flux reconstruction schemes.

1. Introduction

ALE methods support simulations on moving and deforming domains while preserving body-fitted resolution, but entropy-stable wall treatments must account for mesh motion and metric terms. This work develops and analyzes moving-wall conditions that extend entropy stability to general ALE boundaries and discretizations.

  • Motivation: ALE formulations support moving-domain simulations, body-fitted discretizations, and redistribution of resolution near solid walls and boundary layers.Mesh motion can improve resolution where it matters most while retaining a body-fitted mesh.
  • Motivation: High-order methods can achieve a given accuracy with fewer degrees of freedom and improved dispersion and dissipation properties, but require provably stable formulations.The stability challenge is especially important for under-resolved flows, discontinuities, sharp gradients, and complex boundaries.
  • Motivation: Entropy stability provides a nonlinear stability mechanism consistent with the second law when positivity is satisfied, making boundary treatment central to compressible-flow robustness.Boundary contributions decisively affect the entropy balance.
  • Research gap: Moving and deforming ALE meshes couple physical fluxes, grid velocity, and metric terms, so static-grid entropy-stable wall treatments do not directly extend to general ALE boundaries.Existing analyses addressed periodic domains or did not provide a moving-wall analysis and proof.
  • Contribution: The work derives continuous and semi-discrete entropy conditions for moving-wall boundaries, including no-slip adiabatic walls and prescribed entropy heat flux.The construction is presented in a DG-SBP-SAT setting but is formulated for broader discretization families, including finite-volume, finite-difference, finite-element, and flux-reconstruction methods.
  • Validation: Numerical experiments assess convergence, accuracy, and realistic moving-boundary applications across canonical and challenging configurations.The paper includes convergence and accuracy tests as well as more realistic applications.

2. Continuous entropy stability theory

The paper derives a continuous ALE entropy framework for moving and deforming domains, showing how metric identities, entropy variables, and moving-wall conditions control the entropy balance. No-slip walls eliminate the ALE-inviscid entropy flux, while viscous contributions are dissipative under the stated positivity and boundary assumptions.

  • ALE formulation: The ALE formulation maps the compressible Navier–Stokes equations from reference to physical space and combines inviscid fluxes with grid-motion contributions.The mapping is assumed invertible, with grid velocity and Jacobian entering the transformed equations through metric identities.
  • Entropy framework: Under positive density and temperature, strict convexity makes the entropy estimate a nonlinear stability measure related to an L2-type bound.The conservative-to-entropy-variable mapping remains one-to-one when density and temperature stay bounded away from zero.
  • Scope: The continuous analysis applies to spatial semi-discretization only; time integration can introduce additional entropy production or dissipation.Implicit schemes may add numerical entropy dissipation, whereas general explicit methods can produce entropy even with entropy-conservative spatial discretization.
  • Entropy balance: Satisfied time and space metric identities remove the corresponding metric terms from the differential entropy equation and yield the global entropy statement.The transformed entropy balance is integrated over the physical domain to obtain a global conservation relation.
  • Entropy balance: Viscous dissipation contributes a non-positive term to mathematical entropy, so entropy increases can arise only through boundary contributions.The dissipation quantity satisfies DT ≥ 0; for smooth flows, the entropy inequality becomes an equality.
  • Moving-wall conditions: The ALE no-slip condition sets fluid velocity equal to wall velocity and makes the ALE-inviscid entropy flux vanish, while weaker no-penetration suffices for the inviscid contribution.The viscous wall contribution is controlled separately by prescribing the heat entropy flow through the wall.

3. Semi-discrete equations

The semi-discrete ALE formulation combines diagonal-norm SBP operators, flux differencing, numerical fluxes, and weak SAT boundary treatments for moving and deforming elements. Its entropy analysis uses SBP identities, geometric conservation laws, entropy-conservative two-point ALE fluxes, and optional interface dissipation.

  • SBP operators: The diagonal-norm SBP framework maps hexahedral elements to a fixed reference cube with Legendre–Gauss–Lobatto volume nodes.Each element uses N^3 volume nodes and polynomial degree p = N − 1.
  • SBP operators: SBP differentiation, quadrature, boundary, and restriction operators are constructed through tensor products in the three computational directions.The multidimensional operators extend scalar constructions componentwise to the five-component Navier–Stokes system.
  • SBP operators: The multidimensional SBP identity provides a discrete integration-by-parts relation that underpins the semi-discrete entropy analysis.It converts volume contributions into boundary terms under appropriate contractions.
  • Semi-discrete formulation: Discrete space- and time-geometric conservation laws support freestream preservation and entropy conservation on moving and deforming elements.The space-GCL uses the same SBP operators as the derivative computation, while the time-GCL advances the deforming-cell mapping equation with the system.
  • Semi-discrete formulation: The ALE convective operator uses nonlinear flux differencing assembled from symmetric, consistent, entropy-conservative two-point fluxes, while viscous divergence uses standard SBP derivatives.The two-point flux may incorporate convective relative velocity due to grid motion; the viscous contribution separates into surface flux and non-negative volume dissipation.
  • Interface treatment: Entropy-conservative interface fluxes can be augmented with interface dissipation, producing entropy stability for discontinuous elements.The dissipation is intended to maintain stability near steep gradients while relaxing entropy conservation to entropy stability.

4. Entropy-conservative and entropy-stable moving-wall boundary conditions

The moving-wall treatment separates ALE inviscid and viscous boundary contributions. Relative-velocity no-slip conditions make the inviscid wall contribution entropy preserving, while the viscous treatment is entropy conservative for adiabatic walls and entropy stable with bounded prescribed heat-entropy flow.

  • Boundary entropy balance: The semi-discrete entropy analysis reduces the moving-wall balance to physical boundary terms after volume terms are reduced and interior-face contributions cancel.The ALE boundary flux explicitly contains normal grid velocity, unlike the stationary-wall case.
  • Inviscid moving-wall treatment: The inviscid contribution induced by grid motion is entropy preserving for the proposed moving-wall treatment.The proof establishes the desired entropy balance for general wall motion while mesh motion enters through the ALE advective flux.
  • Inviscid moving-wall treatment: Theorem 4.1 identifies a primitive-variable boundary state that makes the penalty ALE flux contribution entropy conservative.The wall velocity is assumed to coincide with the mesh velocity, and the inviscid state reflects fluid velocity relative to the wall in the normal direction.
  • Viscous moving-wall treatment: The viscous wall treatment is entropy-conservative for an adiabatic wall and entropy-stable for a prescribed, bounded heat-entropy flow.The analysis retains the stationary-grid viscous entropy structure because ALE terms do not modify viscous fluxes or dissipation.
  • Viscous moving-wall treatment: The additional viscous wall penalty is entropy dissipative for an arbitrarily moving wall.Its dissipativity follows from the symmetric negative-semidefinite penalty structure; grid velocity appears through the relative no-slip condition.

5. Numerical tests

The numerical tests verify entropy conservation, convergence, accuracy, robustness, and stability of the ALE formulation across moving-grid, wall-bounded, aeroelastic, and supersonic flows.

  • Isentropic vortex: Entropy conservation is achieved up to machine precision for isentropic-vortex simulations with two prescribed grid motions.With added upwind dissipation, the computations achieve the expected p + 1 convergence rate; for p = 5, convergence plateaus near machine precision and the time-integration tolerance.
  • Isentropic vortex: The isentropic-vortex convergence study achieves the expected p + 1 rate for all tested polynomial degrees.For p = 5, convergence plateaus very close to machine precision and the time-integration tolerance.
  • Annular Poiseuille flow: The annular Hagen–Poiseuille study preserves accuracy with moving grids, attaining computed orders close to the formal value of approximately p + 1 for non-periodic boundaries.The study compares the default static-grid solver with the ALE solver across polynomial degrees p = 2, 3, 4, 5.
  • Heaving-pitching NACA-0012: The heaving-pitching airfoil simulation reproduces the reference aerodynamic-force magnitudes and temporal evolution, while its resolved flow structures remain consistent with reference computations.Agreement in aerodynamic loads and spanwise vorticity fields demonstrates accurate reproduction of the strongly unsteady moving-boundary problem.
  • Instability of a 2-DOF airfoil: The coupled two-degree-of-freedom airfoil develops a bounded large-amplitude response, with pitch spanning approximately α ∈[−60°, 60°] and nondimensional heave reaching h/c ∈[−0.2, 0.2].The nonlinear hardening spring bounds the rotational response, while large excursions produce repeated separation, reattachment, coherent vortices, and pronounced lift and drag fluctuations.
  • Supersonic bluff body: The supersonic bluff-body computation remains stable throughout simultaneous three-dimensional rotations, including strong compressibility effects, bow shocks, and unsteady wake dynamics.The test uses a rotating box at Ma = 1.5 and Re = 1 × 10^4.

6. Conclusions

The framework proves entropy-compatible moving-wall treatments for Euler and Navier–Stokes equations and extends them to nonlinear semi-discrete stability. Numerical tests demonstrate expected accuracy, robustness, scalability, and applicability across moving-boundary flow regimes.

  • Moving-wall boundary conditions are entropy conservative for compressible Euler and entropy stable for compressible Navier–Stokes equations.Relative-velocity no-slip conditions bound the inviscid entropy contribution, while viscous terms contribute only entropy dissipation.
  • The ALE nodal discontinuous Galerkin framework uses SBP operators and penalty-type interface and boundary operators to extend the entropy proof semi-discretely.The construction yields nonlinear stability in the L2 sense for continuous and semi-discrete equations.
  • The isentropic vortex and annular pipe tests achieve the expected convergence order (p + 1).These tests assess canonical periodic and wall-bounded flows, respectively.
  • Moving-airfoil and bluff-body simulations validate forces, fluid-structure interaction robustness, turbulent scalability, and extended-time supersonic computation.The tests include 2D heaving-pitching flow, 3D turbulent subsonic flow with dynamic instabilities, and supersonic large-scale rotation.
  • The results support applying high-order entropy-stable schemes to complex moving-motion cases across varied flow regimes and multiphysics interactions.

CRediT authorship contribution statement

The authors contributed across conceptualization, methodology, software, validation, analysis, investigation, writing, supervision, and project administration, with responsibilities distributed among five contributors.

  • Luca Galimberti contributed to nearly every project stage, including conceptualization, methodology, software, validation, formal analysis, investigation, data curation, writing, and visualization.
  • Roberto Nuca contributed methodology, investigation, formal analysis, writing review, and visualization.
  • Lisandro Dalcin contributed software, validation, writing review, supervision, and project administration.
  • Alberto Guardone contributed methodology, writing review, supervision, project administration, and funding acquisition.
  • Matteo Parsani contributed conceptualization, methodology, formal analysis, resources, writing, supervision, project administration, and funding acquisition.

AppendixA. Symbolic proof: moving-wall boundary condition

The appendix symbolically verifies the moving-wall boundary-condition analysis, including ALE flux consistency, entropy-flux identities, viscous dissipation, and boundary-state constructions for arbitrary wall motion.

  • The symbolic proof checks entropy conservation for the moving-wall boundary condition on arbitrary moving grids.It uses a pointwise relation between ghost and boundary nodes and matches the paper’s physical quantities and flux definitions.
  • Manufactured boundary states impose moving-wall conditions through relative normal velocity and velocity reflection about the wall velocity.The inviscid penalty term is formed from the physical ALE flux and the entropy-consistent boundary flux.
  • Symbolic checks verify equal-state consistency of the ALE flux and independently recover the physical ALE entropy flux.The expected entropy flux is expressed using density, entropy, and velocity relative to the wall.

AppendixB. Symbolic proof: volume contribution

The appendix verifies the semi-discrete volume contribution using diagonal-norm tensor-product SBP operators, geometric conservation identities, entropy-conservative fluxes, and nonlinear telescoping relations.

  • The symbolic volume proof retains spatial and temporal geometric conservation-law residuals while checking ALE entropy reduction and interior entropy-conservative coupling.
  • The construction generates LGL nodes, differentiation matrices, quadrature weights, and tensor-product diagonal-norm SBP operators.The polynomial degree p determines p + 1 LGL nodes in each coordinate direction.
  • The appendix verifies the one-dimensional and tensor-product SBP identities, exact differentiation of constants, and discrete telescoping relations.
  • A generic nonlinear telescoping identity is verified for symmetric two-point fluxes by balancing direct, boundary, and shuffle terms.
  • The ALE volume implementation combines physical entropy-conservative flux identities with grid-velocity terms and checks consistency, symmetry, and Tadmor shuffle conditions.The appendix explicitly defines the ALE inviscid flux and entropy-conservative flux using the metric normal and grid velocity.
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