Axisymmetric cavities in hypersonic flow

This experimental study characterizes the shear layer evolution and flow dynamics of axisymmetric open cavities in Mach 6 flow, revealing how variations in length-to-depth ratio and rear-face height influence the transition from laminar to turbulent states, the dominance of Kelvin-Helmholtz versus flapping modes, and the resulting spectral characteristics.

The Gist

The Big Picture: The "Wind Tunnel" and the "Hole in the Wall"

Imagine you are designing a supersonic jet or a rocket that flies at Mach 6 (six times the speed of sound). At these speeds, the air doesn't just flow smoothly; it behaves like a chaotic, high-pressure fluid that can rip things apart.

The researchers were studying what happens when this super-fast air flows over a cavity (a hole or a recessed area) on the surface of the vehicle. Think of it like a speedboat hitting a wave, but instead of water, it's air, and instead of a boat, it's a rocket nose cone.

These cavities aren't just holes; they are like acoustic traps. When air rushes over the hole, it creates a "feedback loop" that can cause the entire structure to vibrate violently, generate intense heat, or even break apart. The goal of this study was to understand exactly how that air behaves inside the hole so engineers can design better, safer vehicles.

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The Experiment: A Giant Air Cannon

To study this, the team used a Ludwieg Tunnel. Imagine a giant, high-tech air cannon.

• They pressurize a long tube with air.
• They open a valve, and a massive burst of air shoots out at Mach 6.
• They placed a model of a rocket nose cone (with a hole carved into it) in the path of this wind.
• They used high-speed cameras (taking thousands of pictures per second) and lasers to "see" the invisible air currents.

They tested three main variables:

1. How deep and long the hole is (The "Shape").
2. How fast the air is blowing (The "Reynolds Number," which is basically a measure of how "turbulent" the flow is).
3. The shape of the back wall of the hole (Is the back wall higher, lower, or level with the front?).

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The Three Main Discoveries

1. The "Deep Hole" Problem (Length-to-Depth Ratio)

Imagine blowing across the top of a bottle. If the bottle is short, you get a low hum. If it's long, the sound changes.

• Short Holes ([L/D] = 2): The air flows over the hole like a calm river. It stays smooth (laminar). It's quiet and stable.
• Medium Holes ([L/D] = 4): The air starts to get a little restless. It forms little swirls (vortices) that travel across the hole, like eddies in a stream.
• Long Holes ([L/D] = 6): This is where things get wild.
  • At low speeds, the air is calm.
  • At high speeds, the air becomes chaotic. The swirls grow so big and fast that they turn into turbulence (like white-water rapids).
  • The Surprise: In a long hole, the air doesn't just get "noisier"; it actually switches its rhythm. At lower speeds, the whole hole breathes in and out like a lung (a "flapping" motion). At higher speeds, it switches to a rapid, chaotic churning (Kelvin-Helmholtz instability).

Analogy: Think of a long hallway. If you whisper at one end, the sound travels clearly. But if you shout and run down the hallway, the sound bounces off the walls so much that it turns into a chaotic roar. The longer the hallway, the more likely it is to turn into chaos.

2. The "Back Wall" Effect (Rear Face Height)

The researchers changed the height of the back wall of the hole relative to the front.

• The "Sunken" Back Wall (Negative Height): Imagine the back of the hole is lower than the front. The air flows over the hole, creates swirls, and crashes into the ground after the hole. It's like a river flowing over a small dam and splashing down. The air stays mostly "swirly" (vortices) but doesn't shake the whole structure violently.
• The "Raised" Back Wall (Positive Height): Imagine the back wall is higher than the front. This acts like a dam that traps the air.
  • The Result: The entire hole starts flapping up and down like a giant flag in the wind. The air pressure builds up inside, pushes the air layer up, then crashes down. This creates a massive, rhythmic "breathing" motion that is very strong and very dangerous for the structure.
  • The Rhythm: This flapping happens at a specific, slow beat (the 1st Rossiter mode), regardless of how fast the wind is blowing.

Analogy:

• Sunken Back: Like a gutter on a roof. Water flows over it and splashes down.
• Raised Back: Like blowing across the top of a bottle that is partially covered. The air gets trapped, pressure builds, and the whole bottle vibrates.

3. The "Round vs. Square" Difference (Axisymmetric vs. 2D)

Most previous studies looked at flat, 2D cavities (like a square hole in a wall). This study looked at axisymmetric cavities (a round, donut-shaped hole on a cone).

• 2D (Square) Holes: The air behaves predictably. It forms swirls that travel across the hole, and the "noise" frequency stays the same no matter how fast the wind blows.
• Round (Axisymmetric) Holes: The air behaves differently because it can move in a circle (azimuthally).
  • In the round holes, the "flapping" motion (the big breathing motion) is perfectly symmetrical. The whole ring moves up and down together, like a drum skin.
  • In the square holes, the motion is more chaotic and less coordinated.

Analogy:

• 2D Hole: Like a crowd of people walking in a line; they bump into each other and create a messy wave.
• Round Hole: Like a group of people holding hands in a circle; they can move up and down together in perfect unison (the flapping mode).

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Why Does This Matter?

Why do we care about holes in rocket noses?

1. Heat: When the air crashes into the back of the hole (reattachment), it creates a "hotspot." If the air is turbulent, it gets even hotter. This can melt sensors or damage the rocket.
2. Vibration: The "flapping" and "swirling" create intense vibrations. If the rocket vibrates at the wrong frequency, it could shake itself apart.
3. Design: By understanding these rules, engineers can design cavities that either dampen these vibrations (making the rocket quieter and safer) or enhance them (if they want to mix fuel and air quickly in an engine).

The Takeaway

This paper is like a manual for understanding how air behaves when it hits a hole in a super-fast vehicle.

• Short holes are safe and calm.
• Long holes can turn from calm to chaotic depending on speed.
• Raised back walls cause the whole hole to "breathe" violently.
• Round holes behave differently than square ones, often moving in perfect unison.

The researchers used high-speed cameras and lasers to watch these invisible air currents, proving that the shape of the hole and the speed of the wind dictate whether the air will be a gentle breeze or a violent storm.

Technical Summary

1. Problem Statement

Cavities in high-speed flows are critical in aerospace applications (e.g., sensor housings, weapon bays, scramjet combustors) but present complex fluid dynamic challenges. While 2D planar cavity flows are well-studied, axisymmetric cavities (common in sensor housings and trapped combustors) remain less explored, particularly in hypersonic regimes.
Key gaps addressed in this study include:

• The lack of understanding regarding how Reynolds number ($Re_D$) influences shear layer transition and unsteadiness in axisymmetric configurations.
• The effect of geometric asymmetry (specifically the rear face height relative to the front face) on flow modes.
• The distinction between flow dynamics in 2D versus axisymmetric cavities, particularly regarding the propagation of acoustic waves and the dominance of specific instability modes (Rossiter modes vs. Kelvin-Helmholtz instabilities).

2. Methodology

The study employs a comprehensive experimental campaign using the Hypersonic Ludwieg Tunnel (HLT) at the Technion-Israel Institute of Technology.

• Flow Conditions:
  • Freestream Mach number: $M_\infty = 6.0$.
  • Reynolds number range (based on cavity depth $D$): $23,000 \le Re_D \le 74,000$.
  • Test gas: Dry air (with $CO_2$ seeding for PLRS visualization).
• Model Geometry:
  • A conical model with a semi-apex angle of $24^\circ$.
  • Aspect Ratios: Length-to-depth ratios ($L/D$) of 2, 4, and 6.
  • Rear Face Height: Normalized excess height of the rear face ($\Delta h/D$) varied from $-0.5$ to $+0.5$ (specifically focusing on $-0.25, 0, 0.5$ for $L/D=6$).
• Diagnostic Techniques:
  • High-Resolution Schlieren Imaging: Captures density gradients ($\partial\rho/\partial y$) to visualize shock waves and shear layer structures.
  • Planar Laser Rayleigh Scattering (PLRS): Uses $CO_2$ icicles seeded in the flow to visualize density fluctuations and resolve small-scale vortical structures in a streamwise plane.
  • Unsteady Pressure Probes: Kulite sensors mounted at the cavity floor center ($S_1$) and downstream reattachment zone ($S_2$) to capture pressure fluctuations.
  • Data Analysis: Fast Fourier Transform (FFT) for spectral content, Proper Orthogonal Decomposition (POD) for spatial mode extraction, and space-time/frequency ($y-t, y-f$) analysis.

3. Key Contributions

• Systematic Parametric Study: First detailed investigation of axisymmetric cavity flows at $M=6$ across a wide $Re_D$ range, isolating the effects of aspect ratio ($L/D$) and rear face geometry ($\Delta h/D$).
• Mode Switching Mechanism: Identification of a unique Reynolds-number-dependent mode switching phenomenon in long axisymmetric cavities ($L/D=6$), transitioning from a low-frequency flapping mode to a high-frequency Kelvin-Helmholtz (K-H) mode.
• Azimuthal Uniformity Analysis: Demonstration that the "flapping" mode is strictly axisymmetric ($m=0$), whereas K-H vortices exhibit poor azimuthal correlation.
• 2D vs. Axisymmetric Comparison: Direct comparison showing that while 2D cavities maintain a single dominant Rossiter mode regardless of $Re_D$, axisymmetric cavities exhibit complex mode switching and upstream shifting of perturbation origins.

4. Key Results

A. Effect of Aspect Ratio ($L/D$)

• $L/D = 2$: The shear layer remains laminar across all $Re_D$. The flow is stable with minimal unsteadiness, matching the 2nd Rossiter mode.
• $L/D = 4$: K-H vortices appear at higher $Re_D$. The dominant frequency matches the 3rd Rossiter mode and remains invariant of $Re_D$.
• $L/D = 6$ (Critical Case):
  • Low $Re_D$: Dominated by a flapping mode (1st Rossiter mode), characterized by the entire shear layer lifting up and down.
  • Intermediate $Re_D$: A coexistence of the flapping mode and K-H modes (Rossiter 4th mode). The dominant spectral peak depends on the measurement location (sensor vs. shear layer).
  • High $Re_D$ ($>51,300$): The shear layer transitions to turbulence. The dominant mode switches to K-H vortices (Rossiter 4th mode).
  • Perturbation Origin: As $Re_D$ increases, the location where acoustic waves perturb the shear layer moves upstream, providing a longer length scale for instability growth and eventual breakdown.

B. Effect of Rear Face Height ($\Delta h/D$)

• Negative $\Delta h/D$ (e.g., -0.25): The shear layer impinges downstream of the trailing edge. The flow is dominated by K-H vortices (matching the 5th Rossiter mode). The pressure gradient across the cavity is minimal, and the flow is less unsteady.
• Positive $\Delta h/D$ (e.g., +0.5): The shear layer impinges on the rear wall, creating a strong pressure build-up inside the cavity. This drives a strong flapping mode (1st Rossiter mode) where the entire shock/shear system oscillates vertically ("breathing" motion).
  • Mechanism: A cycle of under-expansion (shear layer lift-up, cavity depressurization) followed by over-expansion (strong impingement shock, cavity pressurization).
  • Azimuthal Nature: POD and modal decomposition confirm this flapping mode is axisymmetric ($m=0$), unlike the K-H mode which lacks azimuthal coherence.

C. Comparison with 2D Cavities

• In 2D cavities with $L/D=6$, the dominant frequency remains fixed at the 4th Rossiter mode across all $Re_D$, with no mode switching.
• The perturbation origin in 2D cavities is stationary, whereas in axisymmetric cavities, it shifts upstream with increasing $Re_D$, leading to turbulence transition in the axisymmetric case but not the 2D case under the same conditions.

5. Significance

• Design Implications: The findings are crucial for designing hypersonic vehicles. For instance, sensor housings (small-scale cavities) should avoid geometries that trigger the flapping mode ($\Delta h/D > 0$) to prevent excessive thermal loads and structural vibration. Conversely, scramjet combustors (large-scale) might utilize the mixing enhancement from K-H instabilities.
• Thermal Management: The study highlights that the transition to turbulence in long axisymmetric cavities significantly increases heat loads at the trailing edge, a critical factor for thermal protection systems.
• Fundamental Physics: The research clarifies the role of azimuthal wave propagation in axisymmetric geometries, showing that acoustic feedback loops behave differently than in 2D planar flows, leading to unique instability mechanisms and mode switching behaviors not predicted by standard 2D models.