Pith. sign in

REVIEW 4 major objections 6 minor 1 cited by

An Entropy-Stable/Double-Flux scheme for the multi-component compressible Navier-Stokes equations

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims a new interface flux that joins Entropy-Stable and Double-Flux methods satisfies a semi-discrete entropy inequality, preserves kinetic energy, and suppresses pressure oscillations at species interfaces with variable…

desk verdict The ES/DF combination is genuinely new and the numerics are serious, but Theorem 4.5's entropy-stability proof rests on two lemmas that are false for variable-r mixtures, so the paper's central claim is unsupported as written. read the letter →

arxiv 2506.13231 v1 pith:BLPOVPO2 submitted 2025-06-16 cs.CE

Vahid Badrkhani , T. Jeremy P. Karpowsk , Christian Hasse This is my paper · ORCID
classification cs.CE MSC 65M0876N1565M12
keywords Entropy-StableschemesDouble-Fluxmethodmulti-componentcompressibleflowfinite-volumemethodskineticenergypreservationpressureoscillationsadaptivemeshrefinementsupersonicsimulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that a numerical flux can combine two previously incompatible ideas: the Entropy-Stable flux, which guarantees the second law of thermodynamics in discrete form, and the Double-Flux method, which freezes each cell's effective specific-heat ratio and reference energy during a timestep to stop spurious pressure oscillations at species interfaces. The proposed flux is said to satisfy a semi-discrete entropy inequality, to preserve kinetic energy, and to remain low-dissipation while keeping material interfaces oscillation-free. A sympathetic reader would care because uniting these properties in one flux would remove a known trade-off in multi-component compressible flow simulation: entropy stability without interface noise, or oscillation-free Double-Flux without thermodynamic consistency. The paper supports the claim with a theorem, a hybrid dissipation strategy, and benchmark tests ranging from moving interfaces through shock-bubble interaction to an under-expanded hydrogen jet.

What carries the argument

The load-bearing object is the interface flux of Equation (42), assembled from logarithmic and arithmetic means of the reconstructed left and right states, with $\gamma^*$ and $e_0^*$ frozen per cell as in Double-Flux. Entropy stability is certified through the shuffle condition of entropy-conservative flux theory, which relates the jump of entropy variables to the jump of the entropy potential $\psi = r(Y)\rho v$. The dissipation layer uses the eigenvector scaling relation $A_0 = R T R^T$ and the hybrid eigenvalue matrix $|\Lambda| = (1-\theta)|\lambda| + \theta|\lambda_{\max}|I$, blending Roe-type and Lax-Friedrichs-type dissipation by a local pressure-jump indicator. A key identity used in the proof is Lemma 4.1, $[\![r(Y)\rho]\!] = \bar{r}[\![\rho]\!]$, which assumes the mixture gas constant does not jump across the face.

What would settle it

Evaluate the identity $[\![r(Y)\rho]\!] = \bar{r}[\![\rho]\!]$ at a face separating pure hydrogen from pure nitrogen at equal pressure and temperature: the jump in $r$ is large, and the equality omits the $\bar{\rho}[\![r]\!]$ term, so the identity is false exactly at the interfaces the method targets. A numerical corollary would be to run the one-dimensional moving-interface test and record the discrete entropy residual each stage; a scheme delivering the claimed inequality should never show entropy decreasing across the species jump.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a new inviscid numerical flux, $F^{\mathrm{es}^*}$ in Equation (42), that splices the Entropy-Stable flux for multi-component Euler equations together with the Double-Flux freezing of $\gamma^*$ and $e_0^*$ per cell. Contracting the semi-discrete scheme with entropy variables reduces the entropy production to the jump expression $\sum_i r_i Y_i [\![\ln Y_i]\!] \rho^{\ln}$, which Lemma 4.2 shows is non-positive; hence the flux is claimed to satisfy the entropy inequality and be Entropy-Stable. The same flux is shown to meet the structural condition for kinetic-energy preservation, and a hybrid dissipation term built from eigenvector scaling is added in Equation (54) without breaking the entropy inequality. If these claims hold, the method delivers a thermodynamically consistent, oscillation-free, low-dissipation discretization for multi-component compressible flows with variable specific-heat ratios.

Load-bearing premise

The proof of entropy stability assumes the mixture gas constant does not change across a cell face, but at the species interfaces the method is designed for — hydrogen against nitrogen, helium against air — that constant jumps sharply, so the key identity used in the theorem does not hold there.

Editorial extensions

If this is right

  • Moving species interfaces with different specific-heat ratios are captured without the pressure and velocity oscillations that plain Entropy-Stable fluxes produce.
  • The semi-discrete scheme obeys the second law in an integral sense, so entropy production is non-negative even near shocks.
  • Kinetic-energy preservation keeps numerical dissipation low, which should improve under-resolved turbulent and mixing simulations.
  • The hybrid dissipation adds Lax-Friedrichs-type damping near strong shocks and Roe-type low dissipation elsewhere, preserving sharp oblique shocks.
  • Benchmark results reproduce reference shock-bubble trajectories and Mach-disk positions for the under-expanded hydrogen jet, matching previous simulations and experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable repair of the proof would add dissipation or a correction proportional to $[\![r]\!]$ at species interfaces, restoring the jump identity and letting the discrete entropy inequality be checked directly.
  • Because Double-Flux freezes $\gamma^*$ and $e_0^*$ per cell over each stage, the scheme is not fully conservative; measuring accumulated mass and energy errors on long interface advection would quantify the trade-off between oscillation-free capture and conservation.
  • The same flux structure could be ported to real-gas or transcritical mixtures by replacing the frozen perfect-gas closure with tabulated thermodynamics, an extension suggested by the paper's own real-gas references.
  • If the entropy-stability claim survives the proof repair, the scheme is a natural candidate for large-eddy simulation of reacting high-speed jets, where low dissipation and thermodynamic consistency both matter.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a numerical flux for multi-component compressible Navier-Stokes equations that combines an entropy-stable formulation with the double-flux model. The central claim, stated in the abstract and in Section 4.2, is that the proposed flux satisfies a semi-discrete entropy inequality and is entropy-stable, while preserving kinetic energy and suppressing pressure oscillations at material interfaces. The method is implemented in OpenFOAM with adaptive mesh refinement and is tested on a moving interface, a two-dimensional shock-bubble interaction, and a three-dimensional under-expanded hydrogen jet. The paper also introduces a hybrid dissipation strategy intended to blend low and high dissipation mechanisms while maintaining entropy stability.

Significance. If the theoretical claim were valid, the scheme would be a valuable contribution to multi-component compressible flow simulation: it promises thermodynamic consistency, kinetic energy preservation, and oscillation-free behavior at species interfaces, which are typically conflicting goals. The numerical benchmark suite is reasonably broad, includes quantitative comparisons (Mach disk dimensions and jet penetration in Section 6.4), and reports an OpenFOAM implementation that could be useful to practitioners. However, the central proof of entropy stability rests on two lemmas that are false for the intended applications, so the paper's main theoretical conclusion is unsupported. The numerical experiments do not monitor a discrete entropy budget and therefore cannot compensate for the faulty proof.

major comments (4)
  1. [Section 4.1, Lemma 4.1 (Eq. 34)] Lemma 4.1 assumes [[r(Y)]] = 0, but r(Y) = Σ_i Y_i r_i is discontinuous at species interfaces, which are exactly the cases targeted by the paper (H2/N2 in Section 6.2, He/air in Section 6.3). The correct jump identity is [[rρ]] = \bar r [[ρ]] + \bar ρ [[r]]. Since Eq. (43) of Theorem 4.5 uses Lemma 4.1 directly, the proof of the semi-discrete entropy inequality is invalid for the intended applications.
  2. [Section 4.1, Lemma 4.2 (Eq. 35)] The claimed inequality Σ_i r_i \bar Y_i [[ln Y_i]] ρ^ln ≤ 0 is false when the species gas constants differ. For two species with r1=2, r2=1 and Y1: 0.1→0.9, Y2=1−Y1, the sum equals 0.5 ln 9 ≈ 1.099, which is positive. The proof in Eq. (37) passes from Σ_i r_i Y_i [[ln Y_i]] to Σ_i [[ln Y_i]] r ρ^ln without a valid inequality, and the subsequent bound is therefore a non-sequitur. Since Eq. (46) in Theorem 4.5 uses this lemma as the final dissipation step, the entropy-stability conclusion is unsupported.
  3. [Section 4.2, Theorem 4.5] Because Theorem 4.5 relies on Lemmas 4.1 and 4.2, both of which fail for variable-r mixtures, the proof that the flux (42) is entropy-stable collapses. The numerical experiments do not monitor the discrete entropy residual, so they cannot remedy this. The claim in the abstract and Section 4.2 that the flux 'satisfies a semi-discrete entropy inequality' is therefore not established.
  4. [Section 4.3, Eq. (55)] The statement that the dissipation term −½ (vR−vL)^T R |Λ| T R^T (vR−vL) is a negative quadratic form requires proving that R |Λ| T R^T is symmetric positive semi-definite. The manuscript does not provide this proof for the modified scaling based on γ*, and the derivation in Eq. (53) introduces R^{-1} in a way that is not the standard entropy-variable dissipation structure, so the entropy-stability claim for the hybrid flux (54) is not justified.
minor comments (6)
  1. [Introduction] The word 'challange' in the second paragraph should be 'challenge'.
  2. [Section 3.3, Eq. (32)] The definition of c*_{pi} is unclear: the integration variable appears as T in both the integrand and the denominator, and the reference state T0 is not specified. Please clarify the notation.
  3. [Section 4.1, proof of Lemma 4.2] The notation ρ^ln and Y_i^ln in Eq. (37) is not defined, and the inequality steps are not explained; this makes the already problematic proof even harder to follow.
  4. [Section 2.1, Eq. (11)] The entropy inequality is written for the inviscid entropy flux only, while the Navier-Stokes equations include viscous terms that contribute to entropy production; the statement should clarify the role of the viscous fluxes.
  5. [Section 6.1, Figure 3] The reported convergence order Δt^{2.91} is close but not equal to the formal third order of SSP-RK3; the text should explain how the measured order is obtained and why it differs from 3.
  6. [Section 6.4] The boundary condition at the inlet is described only as Dirichlet, but the velocity profile and turbulence specification are not given, which limits reproducibility of the three-dimensional jet simulation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy-stability claim rests on invalid lemmas, not on a circular reduction or fitted input.

full rationale

The paper's central derivation is anchored in Tadmor's entropy-stability framework: the entropy-stable property of flux (42) is checked by contracting with entropy-variable jumps and comparing with the entropy potential jump, Eqs. (43)-(46). This is a standard external criterion, not a quantity fitted from the benchmark data. No parameter in the scheme is calibrated to the test cases, and the numerical benchmarks are presented as validation rather than as inputs to the proof. The self-citations in the paper ([4], [42], [49]) concern solver infrastructure, matrix-free methods, and other numerical techniques; they are not used to justify the entropy-stability theorem or to exclude alternative formulations. The serious defect in Section 4 is a mathematical invalidity, not circularity: Lemma 4.1 assumes [[r]]=0 (Eq. 34) with proof '[[r]]=0', which is false exactly at the species interfaces the method is designed to treat (H2/N2, He/air), and Lemma 4.2's inequality (35) is false when the species gas constants r_i differ. These false lemmas make Theorem 4.5's conclusion unsupported, and the final dissipation argument in Section 4.3 inherits that failure. However, an unsupported or false proof is not the same as a derivation whose conclusion is equivalent to its assumptions by construction, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. Therefore, under the stated rules, no circular step can be exhibited, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard entropy-variable framework from Tadmor and prior multicomponent entropy-conservative fluxes, plus new assumptions in Lemmas 4.1 and 4.2. The main added axiom is the false assumption [[r]] = 0, which is load-bearing. No constants are fitted to benchmark data; the only user-chosen simulator settings are CFL and AMR threshold, which are not part of the central claim.

assumptions (5)
  • ad hoc to paper The jump of the mixture gas constant across an interface is zero: [[r(Y)]] = 0.
    Used in Lemma 4.1 (Eq. 34) to prove Theorem 4.5. False for species interfaces, which is exactly the target application.
  • domain assumption The entropy pair uses s = sum Y_i (c_vi ln T - (R/m_i) ln(rho Y_i)), with constant c_vi.
    Eqs. 9 and 10. This assumes calorically perfect species entropies, which is inconsistent with the general polynomial c_p(T) form mentioned in the paper.
  • standard math Barth's eigenvector scaling theorem gives A0 = R T R^T for the gamma*-modified scaling.
    Invoked in Eq. 52 from Refs. [36, 19]. Requires the average states and eigenvector scaling to be consistent with the Double-Flux modification.
  • domain assumption The Double-Flux freezing of gamma* and e0* is compatible with a conservative entropy-stability analysis.
    Section 3.3 states Double-Flux is inherently non-conservative, but Theorem 4.5 analyzes the flux in Eq. 42 as a conservative numerical flux without treating non-conservative correction terms.
  • ad hoc to paper Lemma 4.2: sum r_i Y_i [[ln Y_i]] rho_ln <= 0.
    The proof chain in Eq. 37 does not establish this inequality for unequal r_i, and counterexamples exist for valid species mass fractions and gas constants.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Entropy-Stable/Double-Flux scheme for the multi-component compressible Navier-Stokes equations." pith.science (2026). https://pith.science/paper/BLPOVPO2

@misc{pith2026250613231,
  author       = {Pith},
  title        = {Pith review of: An Entropy-Stable/Double-Flux scheme for the multi-component compressible Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLPOVPO2}},
  note         = {Machine review of arXiv:2506.13231}
}
read the original abstract

We present a novel combination of numerical techniques to improve the efficiency, accuracy, and robustness of multi-component compressible flow simulations. At the core of our approach is an Entropy-Stable formulation that preserves kinetic energy and integrates a Double-Flux scheme tailored for multi-component flows with variable specific heat ratios. This formulation yields low-dissipation, oscillation-free solutions and enhances stability compared to standard fully conservative methods. To further improve robustness, we introduce a new hybrid dissipation strategy that blends the Entropy-Stable/Double-Flux approach with conventional dissipation mechanisms. We provide a rigorous proof that the resulting numerical flux satisfies a semi-discrete entropy inequality, ensuring consistency with the second law of thermodynamics. For time integration, we employ an explicit Runge-Kutta scheme in combination with adaptive mesh refinement to capture local flow features dynamically. The method is implemented within an existing compressible Navier-Stokes solver based on OpenFOAM. Benchmark cases, including multi-dimensional interface and shock-interface interactions, demonstrate the effectiveness of the proposed framework. The results confirm its favorable stability and robustness, validating the approach as a promising advancement for high-fidelity simulations of supersonic flows.

Figures

Figures reproduced from arXiv: 2506.13231 by the authors.

Figure 1
Figure 1. Simplified flowchart illustrating the coupling of the density-based solver in OpenFOAM with [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. A grid structure with colored density gradients for the shock-wave and reacting helium bubble [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Change in entropy s(t) for both Entropy-Stable/Double-Flux and conservative method using a Lax-Friedrichs (LF) fluxes. The convergence of the change in entropy s(t) at the final time t = 2s converges to zero as O [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Pressure and velocity profiles for the one-dimensional moving interface problem, [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Schematic of the two-dimensional Shock-bubble interaction problem. [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Evolution of helium mass fraction (left) and density gradient magnitude (right) for the interaction [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Showing spatiotemporal evolution of characteristics points along the bubble interface from simu [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Sequence of local refinements for the two-dimensional Shock-bubble interaction problem. Time [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: Computational domain for hydrogen injection. The right side presents the setup dimensions, while [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: Comparison between the density gradient |∇ρ| of the Double-Flux scheme (left) and the Entropy￾Stable/Double-Flux method (right). Time stamps from top to bottom: t = 25 µs and t = 55 µs. of the intercepting shock. The high turbulence fluctuations at the nozzle exit of …
Figure 11
Figure 11. Figure 11: Two-dimension slice of the velocity field for under-expanded hydrogen injection into air at [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An entropy-stable and kinetic energy-preserving macro-element HDG method for compressible flows

    cs.CE 2025-07 conditional novelty 6.0 of 10

    A macro-element HDG method with entropy-stable, kinetic energy-preserving fluxes satisfies a discrete entropy inequality and achieves about 2.7x speedup over standard HDG on vortex benchmarks.

Reference graph

Works this paper leans on

60 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [1]

    Latini, O

    M. Latini, O. Schilling, W. S. Don, Effects of weno flux reconstruction order and spatial resolution on reshocked two-dimensional richtmyer–meshkov instability, Journal of Computational Physics 221 (2) (2007) 805–836

    work page 2007

  2. [2]

    Kawai, H

    S. Kawai, H. Terashima, A high-resolution scheme for compressible multicomponent flows with shock waves, International journal for numerical methods in fluids 66 (10) (2011) 1207–1225

    2011

  3. [3]

    Movahed, E

    P. Movahed, E. Johnsen, A solution-adaptive method for efficient compressible multifluid simulations, with application to the richtmyer–meshkov instability, Journal of Computational Physics 239 (2013) 166–186

    work page 2013

  4. [4]

    Vakilipour, M

    S. Vakilipour, M. Mohammadi, V. Badrkhani, S. Ormiston, Developing a physical influence upwind scheme for pressure-based cell-centered finite volume methods, International Journal for Numerical Methods in Fluids 89 (1-2) (2019) 43–70

    work page 2019

  5. [5]

    R. W. Houim, K. K. Kuo, A low-dissipation and time-accurate method for compressible multi- component flow with variable specific heat ratios, Journal of Computational Physics 230 (23) (2011) 8527–8553

    work page 2011

  6. [6]

    Johnsen, T

    E. Johnsen, T. Colonius, Implementation of weno schemes in compressible multicomponent flow prob- lems, Journal of Computational Physics 219 (2) (2006) 715–732

    2006

  7. [7]

    Xiong, C.-W

    T. Xiong, C.-W. Shu, M. Zhang, Weno scheme with subcell resolution for computing nonconserva- tive euler equations with applications to one-dimensional compressible two-medium flows, Journal of Scientific Computing 53 (1) (2012) 222–247

    work page 2012

  8. [8]

    Renac, Entropy stable, robust and high-order dgsem for the compressible multicomponent euler equations, Journal of Computational Physics 445 (2021) 110584

    F. Renac, Entropy stable, robust and high-order dgsem for the compressible multicomponent euler equations, Journal of Computational Physics 445 (2021) 110584

    work page 2021

Show all 60 references
  1. [9]

    Y. Lv, M. Ihme, Discontinuous galerkin method for multicomponent chemically reacting flows and combustion, Journal of Computational Physics 270 (2014) 105–137

    2014

  2. [10]

    M. T. H. de Frahan, S. Varadan, E. Johnsen, A new limiting procedure for discontinuous galerkin methods applied to compressible multiphase flows with shocks and interfaces, Journal of Computational Physics 280 (2015) 489–509

    2015

  3. [11]

    Gaburro, W

    E. Gaburro, W. Boscheri, S. Chiocchetti, M. Ricchiuto, Discontinuous galerkin schemes for hyper- bolic systems in non-conservative variables: quasi-conservative formulation with subcell finite volume corrections, Computer Methods in Applied Mechanics and Engineering 431 (2024) 117311

    2024

  4. [12]

    Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws

    E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. i, Mathematics of Computation 49 (179) (1987) 91–103

    1987

  5. [13]

    Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numerica 12 (2003) 451–512

    E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numerica 12 (2003) 451–512

    2003

  6. [14]

    P. Chandrashekar, Kinetic energy preserving and entropy stable finite volume schemes for compressible euler and navier-stokes equations, Communications in Computational Physics 14 (5) (2013) 1252–1286

    2013

  7. [15]

    A. Jameson, Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes, Journal of Scientific Computing 34 (2008...

    2008

  8. [16]

    P. K. Subbareddy, G. V. Candler, A fully discrete, kinetic energy consistent finite-volume scheme for compressible flows, Journal of Computational Physics 228 (5) (2009) 1347–1364

    2009

  9. [17]

    Coppola, F

    G. Coppola, F. Capuano, S. Pirozzoli, L. de Luca, Numerically stable formulations of convective terms for turbulent compressible flows, Journal of Computational Physics 382 (2019) 86–104

    2019

  10. [18]

    Y. Kuya, K. Totani, S. Kawai, Kinetic energy and entropy preserving schemes for compressible flows by split convective forms, Journal of Computational Physics 375 (2018) 823–853

    2018

  11. [19]

    Gouasmi, K

    A. Gouasmi, K. Duraisamy, S. M. Murman, Formulation of entropy-stable schemes for the multicom- ponent compressible euler equations, Computer Methods in Applied Mechanics and Engineering 363 (2020) 112912

    2020

  12. [20]

    Abgrall, S

    R. Abgrall, S. Karni, Computations of compressible multifluids, Journal of computational physics 169 (2) (2001) 594–623. 30

    2001

  13. [21]

    Karni, Multicomponent flow calculations by a consistent primitive algorithm, Journal of Computa- tional Physics 112 (1) (1994) 31–43

    S. Karni, Multicomponent flow calculations by a consistent primitive algorithm, Journal of Computa- tional Physics 112 (1) (1994) 31–43

    1994

  14. [22]

    M. J. Castro, U. S. Fjordholm, S. Mishra, C. Par´ es, Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems, SIAM Journal on Numerical Analysis 51 (3) (2013) 1371–1391

    2013

  15. [23]

    T. Y. Hou, P. G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Mathematics of computation 62 (206) (1994) 497–530

    1994

  16. [24]

    Abgrall, S

    R. Abgrall, S. Karni, A comment on the computation of non-conservative products, Journal of Com- putational Physics 229 (8) (2010) 2759–2763

    2010

  17. [25]

    Matheis, S

    J. Matheis, S. Hickel, Multi-component vapor-liquid equilibrium model for les of high-pressure fuel injection and application to ecn spray a, International Journal of Multiphase Flow 99 (2018) 294–311

    2018

  18. [26]

    Schmitt, L

    T. Schmitt, L. Selle, A. Ruiz, B. Cuenot, Large-eddy simulation of supercritical-pressure round jets, AIAA journal 48 (9) (2010) 2133–2144

    2010

  19. [27]

    Schmitt, Large-eddy simulations of the mascotte test cases operating at supercritical pressure, Flow, Turbulence and Combustion 105 (1) (2020) 159–189

    T. Schmitt, Large-eddy simulations of the mascotte test cases operating at supercritical pressure, Flow, Turbulence and Combustion 105 (1) (2020) 159–189

    2020

  20. [28]

    Terashima, M

    H. Terashima, M. Koshi, Approach for simulating gas–liquid-like flows under supercritical pressures using a high-order central differencing scheme, Journal of Computational Physics 231 (20) (2012) 6907–6923

    2012

  21. [29]

    Billet, R

    G. Billet, R. Abgrall, An adaptive shock-capturing algorithm for solving unsteady reactive flows, Com- puters & fluids 32 (10) (2003) 1473–1495

    2003

  22. [30]

    P. C. Ma, Y. Lv, M. Ihme, An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows, Journal of Computational Physics 340 (2017) 330–357

    2017

  23. [31]

    P. C. Ma, H. Wu, D. T. Banuti, M. Ihme, On the numerical behavior of diffuse-interface methods for transcritical real-fluids simulations, International Journal of Multiphase Flow 113 (2019) 231–249

    2019

  24. [32]

    Tadmor, A minimum entropy principle in the gas dynamics equations, Applied Numerical Mathe- matics 2 (3-5) (1986) 211–219

    E. Tadmor, A minimum entropy principle in the gas dynamics equations, Applied Numerical Mathe- matics 2 (3-5) (1986) 211–219

    1986

  25. [33]

    Zhang, C.-W

    X. Zhang, C.-W. Shu, A minimum entropy principle of high order schemes for gas dynamics equations, Numerische Mathematik 121 (3) (2012) 545–563

    2012

  26. [34]

    B. J. McBride, NASA Glenn coefficients for calculating thermodynamic properties of individual species, National Aeronautics and Space Administration, John H. Glenn Research Center . . . , 2002

    2002

  27. [35]

    S. K. Godunov, An interesting class of quasilinear systems, in: Dokl. Akad. Nauk SSSR, Vol. 139, 1961, pp. 521–523

    1961

  28. [36]

    T. J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, in: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws: Proceedings of the International School on Theory and Numerics for Conservation Laws, Freiburg/Littenweiler, O...

    1997

  29. [37]

    Gottlieb, C.-W

    S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM review 43 (1) (2001) 89–112

    2001

  30. [38]

    Ismail, P

    F. Ismail, P. L. Roe, Affordable, entropy-consistent euler flux functions ii: Entropy production at shocks, Journal of Computational Physics 228 (15) (2009) 5410–5436

    2009

  31. [39]

    U. S. Fjordholm, S. Mishra, E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM Journal on Numerical Analysis 50 (2) (2012) 544–573

    2012

  32. [40]

    Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: a quasi conser- vative approach, Journal of Computational Physics 125 (1) (1996) 150–160

    R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: a quasi conser- vative approach, Journal of Computational Physics 125 (1) (1996) 150–160

    1996

  33. [41]

    Fleischmann, S

    N. Fleischmann, S. Adami, X. Y. Hu, N. A. Adams, A low dissipation method to cure the grid-aligned shock instability, Journal of Computational Physics 401 (2020) 109004

    2020

  34. [42]

    Badrkhani, R

    V. Badrkhani, R. R. Hiemstra, M. Mika, D. Schillinger, A matrix-free macro-element variant of the hybridized discontinuous galerkin method, International Journal for Numerical Methods in Engineering 124 (20) (2023) 4427–4452

    2023

  35. [43]

    P. L. Roe, Approximate riemann solvers, parameter vectors, and difference schemes, Journal of com- putational physics 43 (2) (1981) 357–372. 31

    1981

  36. [44]

    H. G. Weller, G. Tabor, H. Jasak, C. Fureby, A tensorial approach to computational continuum me- chanics using object orientated techniques, Computers in Physics 12 (1998) 620–631

    1998

  37. [45]

    D. G. Goodwin, Cantera c++ user’s guide, California Institute of Technology 32 (2002) 358

    2002

  38. [46]

    Heylmun, M

    J. Heylmun, M. E. Fadeli, G. Macpherson, J. Hoehler, Openfoam-Com, Luanmingyi, STFS- TUDa/blastAMR: Load-balanced AMR library release, Zenodo (Oct. 2023). doi:10.5281/ZENODO. 8427733

    2023 doi

  39. [47]

    Heylmun, P

    J. Heylmun, P. Vonk, T. Brewer, blastFoam version 6.0 User Guide (Aug. 2022). URL https://github.com/synthetik-technologies/blastfoam

    2022

  40. [48]

    M. Sun, K. Takayama, Conservative smoothing on an adaptive quadrilateral grid, Journal of Compu- tational Physics 150 (1) (1999) 143–180

    1999

  41. [49]

    Badrkhani, M

    V. Badrkhani, M. F. ten Eikelder, R. R. Hiemstra, D. Schillinger, The matrix-free macro-element hybridized discontinuous galerkin method for steady and unsteady compressible flows, International Journal for Numerical Methods in Fluids 97 (4) (2025) 462–483

    2025

  42. [50]

    G. J. Gassner, A. R. Winters, D. A. Kopriva, A well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations, Applied Mathematics and Compu- tation 272 (2016) 291–308

    2016

  43. [51]

    Y. Wang, R. Deiterding, J. Liang, An adaptive solver for accurate simulation of multicomponent shock- interface problems for thermally perfect species, Computers & Fluids 291 (2025) 106587

    2025

  44. [52]

    J.-F. Haas, B. Sturtevant, Interaction of weak shock waves with cylindrical and spherical gas inhomo- geneities, Journal of Fluid Mechanics 181 (1987) 41–76

    1987

  45. [53]

    Marquina, P

    A. Marquina, P. Mulet, A flux-split algorithm applied to conservative models for multicomponent compressible flows, Journal of Computational Physics 185 (1) (2003) 120–138

    2003

  46. [54]

    J. J. Quirk, S. Karni, On the dynamics of a shock–bubble interaction, Journal of Fluid Mechanics 318 (1996) 129–163

    1996

  47. [55]

    Terashima, G

    H. Terashima, G. Tryggvason, A front-tracking/ghost-fluid method for fluid interfaces in compressible flows, Journal of Computational Physics 228 (11) (2009) 4012–4037

    2009

  48. [56]

    Crist, D

    S. Crist, D. Glass, P. Sherman, Study of the highly underexpanded sonic jet., AIAA journal 4 (1) (1966) 68–71

    1966

  49. [57]

    R. S. Snedeker, et al., A study of free jet impingement. part 1. mean properties of free and impinging jets, Journal of fluid Mechanics 45 (2) (1971) 281–319

    1971

  50. [58]

    Vuorinen, J

    V. Vuorinen, J. Yu, S. Tirunagari, O. Kaario, M. Larmi, C. Duwig, B. Boersma, Large-eddy simulation of highly underexpanded transient gas jets, Physics of Fluids 25 (1) (2013)

    2013

  51. [59]

    Hamzehloo, P

    A. Hamzehloo, P. Aleiferis, Large eddy simulation of highly turbulent under-expanded hydrogen and methane jets for gaseous-fuelled internal combustion engines, International Journal of Hydrogen Energy 39 (36) (2014) 21275–21296

    2014

  52. [60]

    A. J. Ruggles, I. W. Ekoto, Ignitability and mixing of underexpanded hydrogen jets, International Journal of Hydrogen Energy 37 (22) (2012) 17549–17560. 32

    2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.