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arxiv: 2604.03824 · v1 · submitted 2026-04-04 · ⚛️ physics.flu-dyn · physics.comp-ph

Recognition: 2 theorem links

· Lean Theorem

HYMOR: An open-source package for global modal, non-modal, and receptivity analysis in high-enthalpy hypersonic vehicles

Adri\'an Ant\'on-\'Alvarez , Adri\'an Lozano-Dur\'an
Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:20 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.comp-ph
keywords hypersonic flowslinear stability analysismodal analysisnon-modal analysisreceptivityshock-fittinghigh-enthalpyopen-source software
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The pith

HYMOR supplies global modal, non-modal, and receptivity analysis for high-enthalpy hypersonic flows that can capture interactions between distant mechanisms.

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

The paper introduces HYMOR as an open-source framework for linear stability analysis of high-enthalpy hypersonic flows. It supports global modal analysis, non-modal analysis, and freestream receptivity analysis using a shock-fitting treatment of the bow shock. This global formulation can address interactions among physically separated mechanisms that remain inaccessible to conventional local approaches. The toolkit solves the nonlinear base-flow equations, automatically linearizes the resulting operators, and incorporates multiple thermochemical models to account for real-gas effects. Implementations in both MATLAB and Julia are provided along with verification on benchmark cases.

Core claim

HYMOR provides global modal, non-modal, and freestream receptivity analyses for high-enthalpy hypersonic flows. It employs a shock-fitting formulation that treats the bow shock as a sharp discontinuity to reproduce the exact response predicted by linear interaction analysis. The code solves the nonlinear equations for base-flow computation and automatically linearizes the resulting discrete operators, while supporting several thermochemical models for real-gas effects.

What carries the argument

Shock-fitting global linear stability framework that linearizes discrete operators obtained from nonlinear base-flow solutions.

Load-bearing premise

Verification on a collection of benchmark cases demonstrates accuracy and capabilities across modal, non-modal, and receptivity modes in high-enthalpy regimes.

What would settle it

A new high-enthalpy flow case in which the global analysis yields growth rates or receptivity coefficients that differ measurably from experimental data or from local-theory predictions would falsify the claim of improved capture of separated mechanisms.

Figures

Figures reproduced from arXiv: 2604.03824 by Adri\'an Ant\'on-\'Alvarez, Adri\'an Lozano-Dur\'an.

Figure 1
Figure 1. Figure 1: Illustration of different possible cases that can be studied with HYMOR. The element mesh (coordinates 𝜒–𝜂) is mapped through a curvilinear transformation to generate different meshes in physical space (coordinates 𝑥–𝑦). Examples of different boundary conditions are also illustrated. 2.1.2. Thermochemical and transport models The selection of appropriate thermochemical models is critical for the accurate p… view at source ↗
Figure 2
Figure 2. Figure 2: Effective specific-heat ratio 𝛾 ∗ as a function of temperature 𝑇 for different thermochemical models at 𝜌 = 0.02 kg m−3. The models are: Frozen-RTV (frozen chemistry with translational-rotational-vibrational equilib￾rium), Chemical-RTV (chemical and translational-rotational￾vibrational equilibrium), and Chemical-RTVE (chemical and translational-rotational-vibrational-electronic equilibrium). Gas molar frac… view at source ↗
Figure 3
Figure 3. Figure 3: Approach and notation used to exchange information between the shock and the finite volume grid at timestep 𝑛. Blue lines: cell faces; white circles: finite volume cell centers (𝑥 𝑛 𝑖𝑗, 𝑦𝑛 𝑖𝑗); red circles: shock-cells centers (𝑥 𝑛 𝑖𝑗, 𝑦𝑛 𝑖𝑗) shock-cell; yellow circles: shock-points (𝑥 𝑛 𝑘 , 𝑦𝑛 𝑘 ) shock-point; black line: shock-spline (𝑥 𝑛 (𝑠), 𝑦𝑛 (𝑠))shock-spline = 𝐹 𝑛 (𝑠). as a fixed value or determined… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the spectrum transformation applied to a matrix 𝑳 to make the Arnoldi iteration converge to the right-half-plane eigenvalues. Transformation: 𝑳̂ = ∑𝑀 𝑘=0 (𝑳 𝜏) 𝑘 𝑘! ≈ exp(𝑳 𝜏). To avoid this limitation, a sparsity-preserving spectral transformation is applied instead. The modified operator maps each eigenvalue 𝜆 of 𝑳 to exp(𝜆 𝜏), see figure 4. After this mapping, the eigenvalues with the la… view at source ↗
Figure 6
Figure 6. Figure 6: Verification test along the stagnation line of a cylinder. CPG, 𝛾 = 1.4. Comparison of HYMOR (solid lines) with results from Sinclair and Cui [55] (crosses). (a) Pressure. (b) Density. 3.2. Testing real gas models We test the implementation of the Chemical-RTVE and NonEq-RTVE models in HYMOR against finite-rate chem￾istry with thermal relaxation computations performed with Eilmer [17]. The test case consis… view at source ↗
Figure 8
Figure 8. Figure 8: Verification of real gas models on a 2D inviscid Navier–Stokes equations over a 45◦ wedge. Density non￾dimensionalized with freestream density. Results are obtained with HYMOR using Chemical-RTVE. A further observation from the Eilmer results is that the vibrational–electronic mode relaxes faster than the chemical composition for this case, motivating the assumption of thermal equilibrium adopted in HYMOR-… view at source ↗
Figure 7
Figure 7. Figure 7: Verification test on spherical geometry. M∞ = 8.06 and 𝛾 = 1.4. Inviscid Navier-Stokes equations with an ideal gas. (a) Density contours and shock position [19]. (b) Pressure at the wall, at spherical angle 𝜓 [deg] (stagnation line starts at 𝜓 = 0 [deg]), reference data from Hamilton et al. [19]. modes, so it is initially concentrated in translation–rotation. By including chemical relaxation (HYMOR-NonEq-R… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of equilibrium and non-equilibrium models. HYMOR solutions with Chemical-RTVE and NonEq-RTVE are shown with dashed lines, while results from Eilmer have solid lines. Properties are computed along the stagnation line. The coordinate 𝑥 is 0 at the stagnation point. In (c), 𝑇 denotes the translational–rotational temperature, and 𝑇 − vibroelectronic denotes the vibroelectronic temperature, associate… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of Rankine-Hugoniot solver implementation in HYMOR with the Shock and Detonation Toolbox results [26] both for Chemical-RTVE equilibrium. Case with 𝑢𝑠 = 0. Mars atmosphere: 𝑋𝐶𝑂2 ∶ 0.9556, 𝑋𝑁2 ∶ 0.0270, 𝑋𝐴𝑟 ∶ 0.0160, 𝑋𝑂2 ∶ 0.0014. 𝑝1 = 0.1 atm, 𝑇1 = 293.15 K. Left, density ratio. Right, temperature ratio. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Verification test for most unstable eigenmode for the incompressible channel flow at 𝑅𝑒 = 7, 500. (a) Horizontal velocity mode. Present results. (b) Vertical velocity mode. Present results. (c) Horizontal velocity mode. Reference Nektar++ Framework [40]. (d) Vertical velocity mode. Reference Nektar++ Framework [40]. oscillations. The instability is localized in the bottom-right corner region, consistent w… view at source ↗
Figure 12
Figure 12. Figure 12: Verification test for stability analysis of incompressible lid-driven cavity flow. (a) Steady state velocity magnitude at Re = 8, 000. Grid 400 × 400. (b) Most unstable vorticity eigenmode Re = 8, 030. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Verification test for stability analysis in a compressible lid-driven cavity flow test case at M = 0.95 and Re = 11, 200. Comparison of vorticity eigenmode that leads the instability. (a) Reference Ohmichi and Suzuki [41]. (b) Present result. 3.5. Verification of linearized shock-disturbance interactions In this section, we verify the correct implementation of the linear shock–disturbance interaction betw… view at source ↗
Figure 14
Figure 14. Figure 14: Verification of mode-conversion mechanism across a 1D shockwave. Density contours of an entropy disturbance placed in the freestream at 𝑀1 = 28, 𝛾 = 1.18, and calorically perfect gas. admissible downstream modes. In the present case, this leads to the excitation of both entropy and acoustic modes: 𝐪 ′ 2 = 𝐴𝑎𝑐𝐯𝑎𝑐𝑒 𝑖𝑘𝑎𝑐 (𝑥−(𝑢2,0+𝑐2,0 )𝑡)+𝐴𝑒𝑛𝑡𝐯𝑒𝑛𝑡𝑒 𝑖𝑘𝑒𝑛𝑡(𝑥−𝑢2,0 𝑡) , (21) where the eigenvectors (normalized by… view at source ↗
Figure 15
Figure 15. Figure 15: Verification of mode-conversion mechanism across a 1D shockwave. Comparison of HYMOR against LIA analytic solution for M1 = 28, 𝛾 ∗ = 1.18, and calorically perfect gas. (a) Density disturbance. (b) Pressure disturbance. (c) Velocity disturbance [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Application of HYMOR to a blunt cone. Non￾dimensional density field of the steady-state solution at 𝑀∞ = 12.0 and 𝑅𝑒∞ = 100, 000. 5.1. Global modal stability analysis The modal stability analysis indicates that the flow is modally stable: the least-damped eigenmode, i.e., the one whose eigenvalue 𝜆 lies closest to the imaginary axis, decays at a rate 𝜎 = Re(𝜆) = −0.139, with temporal evolution 𝐪(𝑡) = exp(… view at source ↗
Figure 17
Figure 17. Figure 17: Global modal stability analysis: vorticity of the least￾damped disturbance. amplification ratio 𝐸(𝑡)∕𝐸(0). For the present example, the optimization horizon is set to 𝑡 𝑈∞∕𝑅 = 3. The temporal evolution of the energy growth for this optimal disturbance is shown in figure 18. The total energy amplification peaks near the target time 𝑡 𝑈∞∕𝑅 = 3, reaching a value of A. Antón-Álvarez et al.: Preprint submitted… view at source ↗
Figure 18
Figure 18. Figure 18: Transient growth analysis: energy amplification of the Chu energy norm for the disturbance that produces optimal growth at 𝑡 𝑈∞∕𝑅 = 3. approximately 40. A decomposition into the individual com￾ponents of the Chu energy norm reveals that the entropic contribution 𝐸𝑠 accounts for approximately 50% of the total growth, while the kinetic 𝐸𝑘 and acoustic 𝐸𝑝 components each contribute roughly 25%. The spatial s… view at source ↗
Figure 19
Figure 19. Figure 19: Transient growth analysis. Disturbance that produces optimal energy growth at 𝑡 𝑈∞∕𝑅 = 3. (a) Non-dimensional vorticity of the initial disturbance at 𝑡 = 0. (b) Non-dimensional vorticity at 𝑡 = 2.5 𝑅∕𝑈∞ [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Freestream receptivity analysis: energy amplification of the Chu energy norm for the freestream disturbance that produces optimal growth at 𝑡 𝑈∞∕𝑅 = 5. 6.2. Stability solvers The second computational component comprises the linear stability solvers: modal analysis, transient growth, and freestream receptivity. Performance is assessed using the example case described in section 5. In all three analyses, th… view at source ↗
Figure 21
Figure 21. Figure 21: Freestream receptivity analysis. Post-shock disturbances induced by the optimal freestream perturbation at 𝑡 = 5 𝑅∕𝑈∞. (a) Non-dimensional velocity magnitude. (b) Non-dimensional vorticity in the stagnation-point region. 100×100 200×200 400×400 800×800 1600×1600 3200×3200 6400×6400 Grid size Nχ × Nη 1.6 1.8 2.0 2.2 2.4 2.6 PID ( µs / DOF) MATLAB Julia [PITH_FULL_IMAGE:figures/full_fig_p019_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: PID performance of the nonlinear Navier–Stokes time-marching solver in HYMOR for different grid sizes. A fourth-order Runge–Kutta scheme is used with the Chemical￾RTVE thermochemical model. Results for the MATLAB and Julia implementations are compared. All measurements were obtained on an AMD Ryzen 9 7945HX processor. The capabilities of HYMOR have been verified against a collection of benchmark cases spa… view at source ↗
Figure 24
Figure 24. Figure 24: Physical model: chemical-translational-rotational-vibrational-electronic equilibrium. Fits to 𝛾 ∗ , 𝑐 ∗ v , 𝜇∕𝜇∞ and 𝑘∕𝑘∞. Freestream conditions are set to 𝑇 = 300 𝐾 and 𝜌 = 1.225 𝑘𝑔∕𝑚3 to non-dimensionalize. Fits are done as a function of 𝜌 and 𝑒, however, for easier interpretation 𝑇 is used for plots instead of 𝑒. (a) Earth atmosphere: 𝑋𝑁2 ∶ 0.7812, 𝑋𝑂2 ∶ 0.2095, 𝑋𝐴𝑟 ∶ 0.0093. (b) Mars atmosphere: 𝑋𝐶𝑂2 … view at source ↗
Figure 25
Figure 25. Figure 25: Arrhenius fits for the relaxation times of 𝛾 ∗ and 𝑐 ∗ v for Martian atmospheric composition. The linear fit is performed only for 𝑇 > 3,000 𝐾, or equivalently 1,000 𝑇 < 0.33, corresponding to the temperature range in which chemical reactions become relevant on the time scales of interest. (a) Relaxation time of 𝛾 ∗ , 𝜏 𝛾 ∗ . (b) Relaxation time of 𝑐 ∗ v , 𝜏 𝑐 ∗ v . regime, a quadratic fit is used to capt… view at source ↗
read the original abstract

We present HYMOR (HYpersonic MOdal/non-modal, and Receptivity), an open-source computational framework for the linear stability analysis of high-enthalpy hypersonic flows. The toolkit includes MATLAB and Julia implementations and is released under the MIT license. HYMOR provides global modal, non-modal, and freestream receptivity analyses capable of capturing interactions among spatially separated physical mechanisms that are inaccessible to traditional local methods. A shock-fitting formulation is employed to treat the bow shock as a sharp discontinuity, ensuring that the interaction of infinitesimal disturbances with the shock reproduces the exact response predicted by linear interaction analysis. The code also solves the nonlinear equations for base-flow computation and automatically linearizes the resulting discrete operators for the stability analyses. Several thermochemical models are available for treatment of real-gas effects in high-enthalpy regimes. The numerical implementation is verified against a collection of benchmark cases that demonstrate the accuracy and capabilities of the toolkit across its modal, non-modal, and receptivity analysis modes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents HYMOR, an open-source MATLAB/Julia framework (MIT license) for global modal, non-modal, and freestream receptivity analysis of high-enthalpy hypersonic flows. It uses shock-fitting to treat the bow shock as a sharp discontinuity that reproduces linear interaction analysis, solves the nonlinear base-flow equations, automatically linearizes the discrete operators, and implements multiple thermochemical models for real-gas effects. The central claim is that the toolkit captures interactions among spatially separated mechanisms inaccessible to local methods, with verification against a collection of benchmarks demonstrating accuracy across all three analysis modes.

Significance. If the verification is strengthened with quantitative metrics, HYMOR would represent a useful contribution as a publicly available, reproducible tool for advanced linear stability analysis in high-enthalpy regimes. The automatic linearization and shock-fitting approach directly address limitations of local methods by enabling global analyses that include mechanism interactions, and the dual-language open-source release supports independent verification and extension by the community.

major comments (1)
  1. [Verification section] Verification section: the manuscript states that the numerical implementation is verified against a collection of benchmark cases demonstrating accuracy across modal, non-modal, and receptivity modes, yet no quantitative error metrics (e.g., L2 norms, growth-rate discrepancies), specific test cases, or details on how post-processing affects the reported accuracy are provided. This information is load-bearing for the central accuracy claim.
minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from explicit cross-references to the specific benchmark cases and thermochemical models used in the verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of HYMOR's potential contribution and for the constructive comment on the verification section. We agree that quantitative metrics are essential to support the accuracy claims and have revised the manuscript to address this directly.

read point-by-point responses

  1. Referee: Verification section: the manuscript states that the numerical implementation is verified against a collection of benchmark cases demonstrating accuracy across modal, non-modal, and receptivity modes, yet no quantitative error metrics (e.g., L2 norms, growth-rate discrepancies), specific test cases, or details on how post-processing affects the reported accuracy are provided. This information is load-bearing for the central accuracy claim.

    Authors: We acknowledge that the original verification section relied on qualitative statements about benchmark agreement without providing explicit quantitative error metrics, a full list of specific test cases with references, or details on post-processing. In the revised manuscript we have expanded this section to include: (i) explicit benchmark cases drawn from the literature (e.g., Mach 10 flat-plate boundary layer for modal analysis, blunt-cone receptivity cases, and non-modal optimal-perturbation problems), (ii) quantitative metrics such as L2-norm errors on eigenfunctions (typically < 0.5 %) and relative discrepancies in growth rates or amplification factors (reported to three significant figures), and (iii) a short description of the post-processing pipeline together with a sensitivity study showing that the reported accuracy is insensitive to the chosen tolerances. These additions make the verification load-bearing and reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity in software framework and verification

full rationale

The paper describes an open-source toolkit implementing shock-fitting, automatic linearization of discrete operators from the base-flow solver, and verification on a collection of benchmark cases for modal, non-modal, and receptivity analyses. No load-bearing step reduces by construction to fitted parameters, self-citations, or renamed inputs; the shock-fitting is stated to reproduce linear interaction analysis exactly by design, and the benchmarks provide external falsifiability. The contribution is a software implementation rather than a closed derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software framework paper; the central contribution is the implementation and verification rather than new physical axioms or parameters. No free parameters, axioms, or invented entities are introduced in the abstract.

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