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

This paper introduces HYMOR, an open-source MATLAB and Julia framework that enables global modal, non-modal, and receptivity analyses of high-enthalpy hypersonic flows by employing shock-fitting techniques and real-gas thermochemical models to capture complex physical interactions inaccessible to traditional local methods.

The Gist

Imagine you are trying to predict when a smooth, glassy river of air flowing over a supersonic rocket will suddenly turn into a chaotic, churning white-water rapid. This moment, called transition, is the difference between a rocket that flies efficiently and one that burns up or loses control.

The paper introduces HYMOR, a new, free software tool designed to be the ultimate "weather forecaster" for these extreme high-speed flights. Here is a breakdown of what it does, using everyday analogies.

1. The Problem: The "Glassy River" vs. The "White Water"

When a hypersonic vehicle (like a rocket or spacecraft) flies at 12 times the speed of sound, the air around it gets incredibly hot—hot enough to break apart the air molecules themselves. This is called high-enthalpy flow.

Scientists want to know: When and where will the smooth air (laminar) turn into turbulent chaos?

• Why it matters: Turbulent air creates massive friction and heat. If you don't predict this, your thermal protection system (the heat shield) might be too thin, and the vehicle could melt.

2. The Old Way vs. The New Way (HYMOR)

The Old Way (Local Methods):
Imagine trying to understand a river by looking at just one single drop of water. You might see it spinning, but you miss the fact that a rock 100 feet upstream is causing a ripple that eventually hits that drop. Traditional tools looked at small, isolated sections of the flow and missed how disturbances travel and interact over long distances.

The New Way (HYMOR - Global Analysis):
HYMOR looks at the entire river at once. It sees how a tiny ripple at the nose of the rocket interacts with the shockwave, travels down the side, and eventually triggers a storm of turbulence. It connects the dots between distant parts of the flow that older tools couldn't see.

3. The Three Superpowers of HYMOR

HYMOR doesn't just look at the flow; it asks three specific questions:

• A. Modal Analysis (The "Tuning Fork" Test):

  • The Analogy: If you tap a wine glass, it rings at a specific note. If the note is too loud, the glass shatters.
  • The Science: HYMOR checks if the air flow has a natural "ring" (instability) that will grow louder and louder on its own. If it finds a "shattering note," it knows the flow will become turbulent.

• B. Non-Modal Analysis (The "Push" Test):

  • The Analogy: Even if a swing is stable, a sudden, hard push from the side can make it swing wildly for a moment before settling down.
  • The Science: Sometimes, the flow is stable on its own, but a specific, temporary gust of wind (a disturbance) can cause a massive, temporary spike in energy that triggers turbulence. HYMOR calculates the "perfect push" that would cause the most chaos.

• C. Receptivity Analysis (The "Front Door" Test):

  • The Analogy: How does a noise outside the house get inside? Does it come through the window, the door, or the chimney?
  • The Science: The rocket is flying through a messy sky full of dust and wind. HYMOR simulates how these outside disturbances crash into the rocket's bow shock (the invisible wall of compressed air in front of it) and how they "convert" into trouble inside the boundary layer.

4. The Secret Sauce: The "Shock-Fitting" Trick

This is the most clever part of the paper.

• The Problem: In computer simulations, shockwaves (the invisible walls of air) are usually treated like a blurry, thick fog. This "blur" creates fake errors that mess up the math, especially when tiny disturbances hit the shock.
• The HYMOR Solution: HYMOR treats the shockwave like a razor-sharp knife edge. It uses a mathematical technique called "shock-fitting" to make the shock a perfect, infinitely thin line.
• Why it matters: It's the difference between trying to measure a ripple on a foggy mirror versus a ripple on a perfect sheet of glass. This ensures the results are mathematically exact, not just an approximation.

5. Real-Gas Effects: The "Melting Ice Cube"

At normal speeds, air acts like a simple gas. But at hypersonic speeds, the air gets so hot that oxygen and nitrogen molecules start to vibrate, break apart, and even ionize (turn into plasma).

• The Analogy: Imagine air isn't just a gas, but a crowd of people holding hands. As they run faster (get hotter), they start letting go of hands (breaking bonds) and running wild (ionizing).
• HYMOR's Job: It has built-in models to track this "melting ice cube" behavior, ensuring the physics are accurate even when the air is chemically changing.

6. Who Made It and How to Use It?

• Open Source: It's free! Anyone can download it, look at the code, and improve it.
• Two Languages: It's written in both MATLAB (popular in engineering) and Julia (a modern, fast coding language), so everyone can use it.
• GPU Power: It's designed to run on powerful graphics cards (GPUs), making the heavy math calculations happen much faster than on a standard computer.

Summary

HYMOR is a free, high-tech toolkit that helps engineers predict exactly when a hypersonic rocket will go from smooth flying to chaotic turbulence. By looking at the whole picture, treating shockwaves with razor-sharp precision, and accounting for the fact that hot air changes its chemistry, it gives scientists a much clearer map to design safer, faster spacecraft.

Technical Summary

1. Problem Statement

Predicting the laminar-to-turbulent transition in high-speed hypersonic flows remains a critical challenge in aerodynamics. Transition drastically increases skin friction and aerodynamic heating, necessitating accurate prediction for thermal protection system design.

• Limitations of Existing Methods: Traditional Linear Stability Theory (LST) and Parabolized Stability Equations (PSE) rely on local or weakly non-parallel assumptions. They fail to capture interactions between spatially separated physical mechanisms (e.g., shock-boundary layer interactions) and often struggle with global instability modes.
• Shock Treatment Issues: In high-enthalpy flows, strong bow shocks are present. Standard "shock-capturing" numerical methods introduce artificial shock thickness, which corrupts the linear interaction analysis (LIA) of infinitesimal disturbances. This leads to inaccurate receptivity predictions.
• Software Gaps: Existing stability tools (e.g., LASTRAC, STABL) often lack integrated global analysis capabilities, do not uniformly support high-enthalpy real-gas effects, or are not open-source. There is a lack of a unified framework for global modal, non-modal (transient growth), and freestream receptivity analysis in high-enthalpy regimes.

2. Methodology

The authors present HYMOR (HYpersonic MOdal/non-modal, and Receptivity), an open-source computational framework implemented in both MATLAB and Julia.

A. Governing Equations and Physics

• Flow Solver: Solves the compressible Navier-Stokes equations in non-dimensional form.
• Thermochemical Models: Supports five distinct models to handle real-gas effects in high-enthalpy regimes:
  1. Calorically Perfect Gas (CPG).
  2. Frozen Chemistry with Translational-Rotational-Vibrational (RTV) equilibrium.
  3. Chemical and RTV equilibrium.
  4. Chemical and Translational-Rotational-Vibrational-Electronic (RTVE) equilibrium.
  5. Chemical Non-Equilibrium with RTVE equilibrium (using a reduced-order Landau-Teller approach).
• Transport Properties: Uses Sutherland's law or collision-integral-based models (via Cantera) for viscosity and thermal conductivity.

B. Numerical Implementation

• Spatial Discretization: Second-order finite-volume scheme on structured curvilinear grids.
• Shock-Fitting Approach: A key feature where the bow shock is treated as a sharp discontinuity (immersed interface) using Rankine-Hugoniot jump conditions rather than a smeared shock-capturing method. This ensures exact linear interaction analysis (LIA) behavior for infinitesimal disturbances.
  • The shock position is parametrized by cubic splines that evolve independently of the grid, allowing subgrid accuracy.
  • The solver automatically linearizes the discrete operators around the base flow, including the shock perturbation vector.

C. Stability Analysis Capabilities

HYMOR implements three distinct linear stability approaches:

1. Global Modal Analysis: Solves the eigenvalue problem $\lambda \mathbf{q}' = \mathbf{L} \mathbf{q}'$ to find unstable eigenmodes.
   • Optimization: Uses a sparsity-preserving spectral transformation ($\hat{\mathbf{L}} \approx \exp(\mathbf{L}\tau)$) combined with the Implicitly Restarted Arnoldi Method (ARPACK). This avoids the memory explosion associated with "shift-and-invert" methods for large grids.
2. Global Transient Growth (Non-Modal): Analyzes the maximum energy amplification of disturbances over a finite time using Chu's energy norm.
   • Formulated as a generalized Rayleigh quotient solved via the Lanczos iteration.
   • Identifies optimal initial disturbances that maximize energy growth, crucial for bypassing transition scenarios.
3. Freestream Receptivity: Models the interaction between freestream disturbances and the post-shock flow.
   • Treats the freestream as a frozen turbulence field convected toward the vehicle.
   • Solves an input-output system where the input is periodic freestream forcing and the output is the post-shock response, accounting for mode conversion across the shock.

D. Performance

• Hardware Acceleration: The inner loops of the stability solvers (matrix-vector multiplications for matrix exponentials) are offloaded to GPUs (NVIDIA CUDA), significantly reducing memory bandwidth bottlenecks.
• Scalability: The code demonstrates near-size-independent scaling for time-marching and $O(N^{1.5})$ scaling for stability solvers, where $N$ is the grid size.

3. Key Contributions

1. Unified Open-Source Framework: HYMOR is the first open-source package to integrate global modal, non-modal, and freestream receptivity analyses specifically for high-enthalpy hypersonic flows.
2. Shock-Fitting for Stability: The rigorous implementation of shock-fitting ensures that the linear response of infinitesimal disturbances to the shock is physically accurate, eliminating artifacts common in shock-capturing methods.
3. Real-Gas Capability: The inclusion of multiple thermochemical models (including chemical non-equilibrium) allows for accurate stability analysis in high-enthalpy regimes where real-gas effects are dominant.
4. Efficient Spectral Transformation: The use of an exponential spectral transformation preserves matrix sparsity, enabling the analysis of large grids (up to $800 \times 800$) that would be intractable with traditional shift-and-invert methods due to memory constraints.
5. Dual-Language Implementation: Providing identical functionality in both MATLAB and Julia facilitates accessibility and high-performance computing integration.

4. Results and Verification

The authors validated HYMOR against a comprehensive set of benchmark cases:

• Shock-Fitting Verification: Density and pressure contours for Mach 2.5 flow over a cylinder and Mach 8.06 flow over a sphere matched reference solutions (Carpenter & Casper; Hamilton et al.) and experimental data.
• Real-Gas Models: Comparisons with the Eilmer code for flow over a wedge showed that HYMOR correctly captures translational-rotational temperature spikes and chemical non-equilibrium effects behind the shock.
• Linear Interaction Analysis (LIA): The code successfully reproduced the mode-conversion mechanism (entropy to acoustic waves) across a 1D shock, matching analytical LIA solutions.
• Stability Benchmarks:
  • Incompressible Channel Flow: Converged eigenvalues matched reference data (Canuto) with high precision.
  • Lid-Driven Cavity: Correctly predicted the critical Reynolds number for Hopf bifurcation ($Re_c \approx 8030$) and captured complex unstable modes in compressible regimes.
• Application Case: A hypersonic blunt cone ($M_\infty = 12$) was analyzed.
  • Modal: The flow was found to be modally stable, with the least-damped mode associated with the boundary layer.
  • Transient Growth: Identified optimal disturbances (vortical streaks) producing an energy amplification of ~40, driven by the Orr mechanism.
  • Receptivity: Showed massive energy amplification (~30,000) due to shock transmission and shear-layer interaction, highlighting the critical role of freestream disturbances.

5. Significance

HYMOR addresses a critical gap in hypersonic research by providing a tool that bridges the gap between local stability theories and the complex, global physics of high-enthalpy flows.

• Scientific Impact: It enables researchers to study transition mechanisms driven by non-modal growth and freestream receptivity, which are often the dominant factors in hypersonic transition but are inaccessible to traditional LST/PSE.
• Engineering Impact: By accurately modeling shock-disturbance interactions and real-gas effects, the toolkit aids in the design of more robust thermal protection systems and vehicle margins.
• Community Impact: Being open-source (MIT license) and GPU-accelerated, it lowers the barrier to entry for high-fidelity stability analysis and promotes reproducibility in the hypersonic community.