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Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations

Praveen Chandrashekar

arXiv:1209.4994v1math.NAcs.CE

TL;DR

The paper addresses the gap between kinetic-energy-preserving and entropy-conservative finite-volume fluxes for compressible Euler and Navier–Stokes equations. It derives centered fluxes satisfying both properties, then adds scalar, matrix, and blended dissipation for shocks and coarse meshes. The resulting entropy-consistent schemes avoid Roe-type entropy violations, while the hybrid scheme is carbuncle-free for the blunt-body test.

  • Problem

    Existing kinetic-energy-preserving and entropy-conservative fluxes generally do not satisfy both properties simultaneously, while Roe’s scheme can produce entropy-violating shocks.

  • Method

    The paper derives explicit centered fluxes that enforce approximate or exact entropy consistency alongside kinetic-energy preservation, then adds scalar, entropy-variable matrix, and blended dissipation.

  • Results

    The entropy-consistent schemes avoid unphysical sonic-point shocks, and the blended Roe–Rusanov dissipation is shown to be carbuncle-free for the blunt-body problem.

  • Takeaways & Limitations

    The fluxes are attractive for highly resolved direct numerical simulation, while dissipation extends their use to unresolved shocks and coarse meshes.

  • Takeaways & Limitations

    Scalar dissipation requires more added dissipation for strong shocks, which smears contact discontinuities; contact-resolving schemes can also suffer shock instability and carbuncle effects.

Abstract

from arXiv · show

Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form $f^m_\jph = \tp_\jph + \avg{u}_\jph f^ρ_\jph$ where $\avg{u}_\jph = (u_j + u_{j+1})/2$ and $\tp_\jph, f^ρ_\jph$ are {\em any} consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy- variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.

1 Introduction

The paper develops centered finite-volume fluxes that preserve kinetic energy while enforcing approximate or exact entropy consistency. Dissipation is added to stabilize shocks and coarse-mesh computations, including a hybrid approach for carbuncle-free hypersonic blunt-body flows.

  • Motivation: Compressible Euler and Navier–Stokes discretizations must satisfy conservation, entropy, and kinetic-energy requirements, especially across shocks and contact discontinuities.The paper notes that Roe fluxes can produce entropy-violating shocks, while Godunov schemes can excessively damp flow structures through incorrect kinetic-energy dissipation and entropy production.
  • Contributions: The authors construct explicit centered Euler fluxes that preserve kinetic energy and are approximately or exactly entropy consistent.The exact version uses logarithmic averages and the simpler version uses arithmetic averages.
  • Stabilization: Centered fluxes require added scalar or matrix dissipation when shocks are unresolved or meshes are coarse.On very fine meshes for Navier–Stokes equations, physical viscosity can stabilize the centered fluxes.
  • Results: Entropy-consistent schemes avoid the entropy-violating shocks that can occur with the original Roe scheme.The paper evaluates scalar and matrix dissipation strategies for shock and contact-discontinuity problems.
  • Results: A blended dissipation scheme combines Roe-type and Rusanov-type behavior to address carbuncle problems in hypersonic blunt-body flows.The paper presents numerical results for Euler and viscous test cases involving shocks and contact discontinuities.

2 1-D NS equations and finite volume scheme

The paper formulates one-dimensional Navier–Stokes equations in conservation form and advances their finite-volume discretization using numerical inviscid and viscous fluxes. The resulting semi-discrete ordinary differential equations are integrated with a strong-stability-preserving Runge–Kutta method.

  • Governing equations: The one-dimensional Navier–Stokes equations are written in vector conservation form with inviscid and viscous fluxes.The conserved variables include density, momentum, and total energy; viscous terms use Newtonian shear stress and Fourier heat conduction.
  • Finite-volume discretization: Removing viscous fluxes yields the hyperbolic Euler system, while the finite-volume mesh partitions the domain into uniform cells of size Δx.Cell averages represent the solution within each control volume.
  • Finite-volume discretization: The semi-discrete scheme computes numerical inviscid and viscous fluxes at each cell interface.The interface fluxes couple neighboring cell averages in the finite-volume evolution equations.
  • Time integration: The resulting ordinary differential equations are solved numerically with a strong-stability-preserving Runge–Kutta scheme.

3 Kinetic energy preserving scheme

The kinetic-energy-preserving construction targets a discrete balance consistent with the continuous kinetic-energy equation. Its key condition is a momentum flux combining a consistent pressure approximation with arithmetic-average velocity times the mass flux, while entropy considerations later determine the remaining flux terms.

  • Motivation: Kinetic energy is important in turbulent flows because its balance governs cascade dynamics and viscous destruction.Faithful numerical representation is therefore especially relevant to direct numerical simulation.
  • Discrete balance: The desired scheme reproduces the continuous kinetic-energy balance in a discrete finite-volume sense.The balance includes pressure work and irreversible viscous destruction converted into internal energy.
  • Flux condition: Summing the cell equations globally shows that the convective contribution disappears when the momentum flux has the required form.This leaves the pressure-work and viscous terms in the kinetic-energy equation.
  • Flux freedom: Kinetic-energy preservation permits several consistent choices for mass, pressure, velocity, and energy flux components.Jameson’s example uses arithmetic averages, including f^ρ = ρ̄ū and p̃ = p̄.
  • Flux determination: The paper uses entropy considerations to determine the numerical flux uniquely rather than leaving most components unspecified.

4 Entropy condition

The paper defines entropy consistency for finite-volume Euler discretizations and derives fluxes that combine entropy conservation with kinetic-energy preservation. The new exact flux uses logarithmic averages, while dissipation and eigenvalue modifications provide stability and shock resolution.

  • Entropy condition: Entropy conditions select the physically relevant weak solution when conservation and Rankine–Hugoniot jump conditions alone permit non-uniqueness.For discontinuous solutions, the entropy relation becomes a weak inequality.
  • Entropy-stable discretization: Entropy variables and the entropy potential provide the framework for defining entropy-conservative numerical fluxes.Tadmor’s condition yields a semi-discrete entropy conservation equation, while added dissipation generates entropy and produces entropy stability.
  • Euler entropy: For Euler equations, the thermodynamically compatible entropy function is selected from the physical entropy, with corresponding entropy variables and Legendre transform.
  • Accuracy and entropy: The approximate flux satisfies the entropy condition to third-order accuracy, whereas the entropy-conservative flux satisfies it exactly for any mesh size.The exact property is tied to the thermodynamic entropy condition.
  • Kinetic-energy and entropy-conservative flux: The new flux is derived by matching jumps in independent variables so its mass, momentum, and energy components satisfy entropy conservation while retaining the kinetic-energy-preserving momentum form.The momentum flux specifically satisfies the kinetic-energy requirement.
  • Flux comparison: The exact flux has the same structure as the approximate flux but replaces arithmetic averages with logarithmic averages in the mass and energy fluxes.It is also described as computationally less expensive than Roe’s entropy-conservative flux because it does not require a parameter vector.
  • Uniqueness: Entropy consistency and kinetic-energy preservation constrain the linearization choices, supporting uniqueness for two-point fluxes.Alternative linearizations of Δ(βu^2) fail to produce the required momentum-flux form.
  • Flux summary: The centered kinetic-energy-preserving and entropy-consistent fluxes are presented in approximate and exact forms.

5 Scalar artificial dissipation flux

Scalar artificial dissipation augments centered entropy-consistent fluxes for coarse meshes and shocks while preserving kinetic-energy stability and satisfying approximate or exact entropy conditions.

  • Scalar dissipation makes the centered scheme suitable for coarse meshes and inviscid problems, where entropy-consistent fluxes alone are stable only on highly resolved meshes.
  • Kinetic energy stability: The chosen mass and momentum dissipation produces kinetic-energy dissipation rather than spurious kinetic-energy growth.
  • Entropy stability: Approximate entropy dissipation generates entropy in the limit as Δx→0, while an alternative scalar construction is exactly entropy dissipative.
  • Accuracy and shock adaptation: Blending second- and fourth-order dissipation yields second-order accuracy in smooth regions and O(Δx) dissipation near shocks.

6 Matrix dissipation flux

Entropy-variable matrix dissipation is combined with kinetic-energy- and entropy-consistent centered fluxes, with eigenvalue choices controlling stability, contact resolution, and shock smearing.

  • Flux construction: The KEP-ES flux combines entropy-variable matrix dissipation with a kinetic-energy- and entropy-consistent or conservative centered flux.
  • Contact resolution: Exact stationary-contact preservation requires evaluating the interface sound speed using a logarithmic average of β.
  • Kinetic energy stability: Roe-type dissipation fails the kinetic-energy stability condition because its first and third eigenvalues differ, whereas Rusanov dissipation satisfies it.
  • Resolution and robustness: The KES eigenvalue choice resolves stationary contacts exactly, but Rusanov dissipation is highly dissipative and smears shocks, contacts, and shear layers.

7 Monotone resolution of shocks

The shock tests address monotonicity, which entropy stability alone does not guarantee, and show that exact entropy-conservative fluxes improve shock behavior across Mach numbers.

  • Motivation: Monotone discontinuous solutions are desirable because central schemes can oscillate near shocks and produce negative density or pressure.
  • Shock tests: The approximately entropy-consistent flux produces pre-shock oscillations at all tested Mach numbers, while the entropy-conservative flux is monotone in the reported tests.
  • Shock tests: For the entropy-conservative flux, basic Roe-type dissipation gives monotone solutions without increasing eigenvalues at higher Mach numbers.
  • Kinetic-energy-stable variant: The kinetic-energy-stable dissipation is non-oscillatory at Mach 20 but highly diffused in its first-order form.

8 Shock instability, carbuncle and a hybrid scheme

The paper examines shock instability and carbuncle behavior in entropy-stable schemes, then introduces a pressure-jump-based hybrid dissipation that avoids these problems while retaining resolution near contacts and layers.

  • Shock instability and carbuncle: Entropy consistency removes the entropy-violating shock, but entropy stability alone does not guarantee monotone shock solutions.The paper notes that entropy conditions require nonzero dissipation without specifying how much is needed for monotonicity.
  • Shock instability and carbuncle: The KEP-ES schemes avoid the one-dimensional shock instability, whereas added dissipation in KEP-EC1 can reintroduce instability at higher Mach numbers.The instability arises when disturbances reflect from the outflow boundary and repeatedly perturb the shock location.
  • Shock instability and carbuncle: Schemes that exactly resolve grid-aligned stationary contacts tend to suffer carbuncle, while the more dissipative Rusanov scheme avoids it.Increasing dissipation can reduce carbuncle but may degrade boundary-layer resolution.
  • Hybrid scheme: The hybrid scheme switches from Roe-like behavior near contacts to Rusanov-like behavior for strong shocks using a pressure-jump sensor.The switching function satisfies 0≤φ≤1, with φ≈1 for strong shocks and φ≈0 near contact discontinuities.
  • Hybrid scheme: KEP-ES(Hyb) avoids the one-dimensional shock instability and carbuncle effect while retaining good boundary-layer and shear-layer resolution.The dissipation modification preserves entropy stability.

9 2-D NS equations and finite volume method

The two-dimensional Navier–Stokes formulation uses a vertex-based finite volume discretization on unstructured triangular grids, combining compact viscous discretization with entropy-conservative fluxes and dissipation.

  • Governing equations: The equations are written in conservation form with total energy, viscous stresses, and heat fluxes governed by Newtonian and Fourier constitutive laws.The formulation uses u=(u1,u2) and total energy E=p/(γ−1)+ρ|u|^2/2.
  • Finite volume discretization: The finite volume method uses vertex-centered control volumes around triangular-grid vertices, constructed as median-dual or Voronoi cells.Neighboring control volumes share common boundaries, and boundary contributions are included for boundary volumes.
  • Finite volume discretization: Viscous fluxes are discretized with a P1 finite-element approach that produces a compact nearest-neighbor stencil.The method connects only edge-neighboring vertices in the viscous discretization.
  • Numerical flux: The numerical flux combines an entropy-conservative face flux with matrix dissipation based on entropy variables and characteristic eigenvectors.The entropy-conservative flux uses logarithmic averages and a pressure term defined separately in the formulation.
  • Accuracy and implementation: Second-order accuracy uses MUSCL reconstruction of primitive-variable gradients, while time integration uses a three-stage strong-stability-preserving Runge–Kutta method.Triangular grids are generated with GMSH and Delaunay-based tools.

10 Numerical results

Numerical tests assess entropy consistency, stationary-contact resolution, shock and contact sharpness, and viscous-flow accuracy for Euler and Navier–Stokes problems.

  • Test setup: All tests use an ideal gas with γ=1.4 unless stated otherwise.
  • Modified Sod test case: Entropy-consistent schemes avoid the entropy-violating expansion shock produced by the original Roe scheme in the modified Sod problem.The central entropy-conservative flux prevents the violating shock despite vanishing eigenvalues in the expansion fan.
  • Stationary contact: Exact stationary-contact resolution depends on correctly computing enthalpy in the dissipation matrix; logarithmic averaging enables the exact conservative flux to satisfy this property.Arithmetic averaging of β in the approximately entropy-consistent flux does not provide exact contact resolution.
  • Sod test case: The inviscid Sod test resolves the shock within two cells and sharply resolves the contact using entropy-consistent dissipation with MUSCL reconstruction and a minmod limiter.
  • Navier–Stokes test case: In the viscous Sod test, fourth-order dissipation damps density oscillations without reducing accuracy elsewhere, including inside the shock region.The computation uses Navier–Stokes equations at Reynolds number 2000 with 500 cells.

10.4 NS shock structure

The KEP-ES(SD) scheme computes Navier–Stokes shock structures with scalar second- and fourth-order dissipation, remaining non-oscillatory on coarse meshes and accurate on finer meshes. The broader test suite examines the proposed fluxes on supersonic cylinders and transonic airfoil flow.

  • 10.4 NS shock structure: The shock-structure computation uses a kinetic-energy-preserving, entropy-conservative flux with scalar artificial dissipation.The reported dissipation parameters are κ^(2)=1/2 and κ^(4)=1/25.
  • 10.4 NS shock structure: With only fourth-order dissipation on N=200 cells, the scheme produces an accurate solution and damps density and pressure oscillations from the initial discontinuity.The reported fourth-order coefficient is κ^(4)=1/200.
  • 10.5 Supersonic cylinder: For Mach 2 flow over a supersonic cylinder on a Voronoi dual grid, KEP-EC1 resolves density and pressure without a carbuncle effect.The study also compares median and Voronoi dual grids and several dissipation variants at Mach 20.
  • 10.6 Transonic flow past NACA-0012 airfoil: Transonic NACA-0012 calculations compare ROE, KEP-EC1, and KEP-ES(Hyb) using Mach contours at Mach 0.85 and 2 degrees angle of attack.The airfoil computation uses a triangular grid with 6402 vertices and MUSCL reconstruction with a van Albada limiter.

10.7 Laminar flat-plate boundary layer

The paper evaluates the schemes on viscous laminar flat-plate flow and on a Mach 3 forward step with shock reflections. The flat-plate results compare velocity profiles with the Blasius solution, while the forward-step results assess shock, slip-line, and Mach-stem behavior.

  • 10.7 Laminar flat-plate boundary layer: The laminar flat-plate case has Reynolds number 10^5 and incoming Mach number 0.1 on a triangular-grid rectangular domain.Slip, no-slip, and adiabatic boundary conditions are imposed in the inlet and plate regions.
  • 10.7 Laminar flat-plate boundary layer: The computed streamwise and vertical velocities are compared with the Blasius semi-analytical solution in standard nondimensional units.The velocity profiles are reported in Figure 17.
  • Forward step in wind tunnel: The Mach 3 forward-step problem contains reflected shocks, a slip line, and a shock triple point, with spurious entropy at the corner identified as a source of Mach-stem formation.The KEP-EC1 and KEP-ES(Hyb) fluxes are tested using linear reconstruction.
  • Forward step in wind tunnel: At time t=4, both forward-step schemes resolve the main flow features and slip line, while the shock reflection remains free of a spurious Mach stem.The reported contours use 50 equally spaced Mach contours between 0 and 4.8.

11 Summary and conclusions

The paper’s schemes combine kinetic-energy preservation with entropy conservation, then add scalar or matrix dissipation when central fluxes are insufficient. Roe-type eigenvalue augmentation improves shock monotonicity, while hybrid dissipation addresses carbuncle effects and standard tests yield accurate solutions.

  • Contributions: The new Euler flux is entropy conservative and also preserves kinetic energy in the semi-discrete finite volume scheme.The flux is central and therefore requires added dissipation for coarse meshes or poorly resolved shocks.
  • Dissipation and stability: Scalar and matrix dissipation schemes preserve the stability properties, with appropriately chosen matrix eigenvalues producing kinetic-energy dissipation.The matrix schemes use Roe-type dissipation based on entropy variables.
  • Hypersonic and viscous flows: Roe-type eigenvalue augmentation gives monotone shock resolution at hypersonic Mach numbers, while hybrid Roe–Rusanov dissipation is shown to avoid carbuncle effects for blunt-body flow.The hybrid scheme also gives good boundary-layer resolution and is more accurate than Rusanov-type dissipation in the reported comparison.
  • Validation: The schemes are reported to produce accurate solutions across several standard compressible-flow validation cases.The paper also reports that entropy-consistent schemes avoid unphysical shocks at sonic points.
  • Scope and caveat: For shock-free compressible turbulent-flow DNS, kinetic-energy- and entropy-conservative schemes are attractive, but explicit dissipation remains useful for controlling density and pressure oscillations.The paper notes that the KEP scheme has smaller entropy oscillations than entropy-conservative schemes in the Sod test, while not eliminating them completely.
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