Transonic flow past the complex cavity-sub-cavity configurations

This study employs Detached Eddy Simulation and Spectral Proper Orthogonal Decomposition to analyze unsteady transonic flow in a complex cavity-sub-cavity system derived from a scramjet-integrated launch vehicle, revealing feedback-driven pressure oscillations and demonstrating that a ventilated sub-cavity is the most effective passive control strategy for suppressing these loads.

The Gist

The Big Picture: The "Whistling Hole" Problem

Imagine you are driving a car with the windows down, and you stick your hand out. If you cup your hand just right, you might hear a loud whistle. That's because the air rushing past your hand creates a vibration.

Now, imagine that "hand" is actually a giant, deep hole (a cavity) on the side of a supersonic rocket or a scramjet engine. When air flies over this hole at high speeds, it doesn't just whistle; it screams. It creates massive, rhythmic pressure waves that shake the vehicle, create noise, and can even damage the engine.

This paper is about a specific, complicated version of this problem: a cavity inside a cavity. Think of it like a Russian nesting doll, but instead of wood, it's made of air and metal, and it's moving at the speed of sound.

The Setup: A Rocket's "Open Mouth"

The researchers are studying a specific scenario involving a Scramjet engine (a super-fast jet engine) attached to a launch vehicle (a rocket).

• The Main Cavity: This is the engine's nozzle (the back part where hot gas shoots out).
• The Sub-Cavity: This is a deep, hidden tunnel inside the engine (the isolator) that sits right in front of the nozzle.

When the rocket launches, the protective cover flies off, and the air rushes into this complex "mouth." The air creates a swirling, unstable dance that causes the engine to vibrate violently.

The Experiment: Watching the Invisible Dance

Since you can't easily put a high-speed camera inside a rocket engine flying at Mach 1.2 (1.2 times the speed of sound), the scientists used a computer simulation.

• The Method: They used a technique called Detached Eddy Simulation (DES). Think of this as a high-tech weather forecast for a tiny, specific patch of sky. It's smart enough to see the big storms (large air swirls) but also detailed enough to see the raindrops (small turbulence).
• The Validation: Before trusting the computer, they built a small plastic model of the engine and tested it in a wind tunnel at the Indian Institute of Technology. The computer results matched the real-world wind tunnel results perfectly, proving their "digital twin" was accurate.

What They Found: The "Feedback Loop"

The study revealed that the air inside this double-cavity system is stuck in a feedback loop, like a microphone too close to a speaker.

1. The Swirl: Air flows over the top of the hole and creates a giant, rolling vortex (a swirling tube of air), similar to a smoke ring.
2. The Crash: This swirl crashes into the back wall of the cavity.
3. The Echo: That crash creates a sound wave (a pressure pulse) that travels backwards upstream, hitting the front of the hole.
4. The Trigger: That backward wave hits the incoming air and tells it to roll up into a new swirl.
5. Repeat: This happens over and over, thousands of times a second, creating a self-sustaining, violent vibration.

Key Finding 1: Speed Matters. As the rocket goes faster (from Mach 0.9 to 1.2), the pressure hitting the back wall of the sub-cavity gets significantly stronger. It's like the "whistle" getting louder and more painful the faster you drive.

Key Finding 2: Shape Matters. The shape of the cavity changes everything.

• They tested a rectangular box shape.
• They tested a ramp shape (like a slide).
• They tested an inverted ramp (like a bowl).
• Result: The "bowl" shape (Inverted SERN) actually helped calm the flow down a bit, while the ramp shape made the vibrations worse. The shape changes how the air "rolls" and crashes.

The Solution: How to Stop the Screaming

The researchers tried two "passive" ways to stop the vibration. "Passive" means no moving parts, no batteries, and no complex electronics—just changing the shape of the metal.

Strategy A: The "Chamfer" (Smoothing the Corner)

• What they did: They cut the sharp back corner of the cavity at an angle (like beveling the edge of a picture frame).
• The Result: It helped a little. It reduced the vibration by about 60%. It's like putting a piece of foam on a speaker; it dampens the sound, but the speaker is still trying to scream.

Strategy B: The "Vent" (The Magic Hole)

• What they did: They drilled small slots (vents) in the floor of the inner sub-cavity.
• The Result: This was a game-changer. It reduced the vibration by 96%.
• Why it works: Imagine the cavity is a drum. When the air hits the back, it builds up pressure. The vents act like a "pressure release valve." Instead of the pressure building up and crashing back, some of it escapes through the holes. This breaks the feedback loop. It's like putting a hole in the bottom of a bucket so the water can't build up enough pressure to splash back out.

The "SPOD" Analysis: Listening to the Music

The researchers used a fancy math tool called Spectral Proper Orthogonal Decomposition (SPOD).

• The Analogy: Imagine a chaotic orchestra playing a messy song. SPOD is like a super-smart conductor who can listen to the noise and say, "Okay, the violins are playing the main melody, the drums are playing the rhythm, and the flutes are just making noise."
• The Finding: In the uncontrolled cavity, the "drums" (the big, loud pressure waves) were dominating the song. In the "vented" cavity, the vent changed the song entirely. The loud drums stopped, and the music became much quieter and more organized.

The Conclusion

This paper tells us that when designing high-speed rockets and engines, we can't just look at the main shape; we have to look at the little "pockets" inside them.

1. Speed increases the pain: Faster flight means harder shaking.
2. Shape is key: The geometry of the engine dictates how the air vibrates.
3. Simple fixes work best: You don't need complex machines to stop the shaking. A simple hole (a vent) in the right place can silence the screaming engine almost completely.

This research helps engineers build safer, quieter, and more stable supersonic vehicles for the future.

Technical Summary

1. Problem Statement

The study addresses the unsteady flow physics and high-amplitude pressure oscillations occurring in complex cavity-sub-cavity geometries typical of aerospace applications, specifically the integration of a Scramjet engine with a launch vehicle.

• Context: During the "throttling stage" of a launch vehicle, the scramjet engine is exposed to the freestream while the bay door is closed, creating a complex flow environment.
• Geometry: The configuration consists of a primary cavity (representing the Single Expansion Ramp Nozzle, SERN) and an embedded deep sub-cavity (representing the scramjet isolator).
• Challenge: Unlike simple open or closed cavities, this complex topology creates intricate hydrodynamic-acoustic feedback loops. These loops lead to self-sustained oscillations, severe pressure fluctuations, and aeroacoustic noise, which can compromise vehicle stability and engine operability.
• Gap: While extensive research exists on simple rectangular cavities, there is a lack of comprehensive understanding regarding complex, realistic cavity-sub-cavity systems in the transonic regime ($0.8 \le M_\infty \le 1.2$), where compressibility effects and shock-shear layer interactions are highly complex.

2. Methodology

The research employs a multi-faceted approach combining numerical simulation, experimental validation, and advanced spectral analysis.

• Numerical Approach:
  • Solver: 2D Detached Eddy Simulation (DES) using ANSYS Fluent. DES was chosen to balance the computational cost of Large Eddy Simulation (LES) with the limitations of Unsteady RANS (URANS) in capturing global instabilities.
  • Turbulence Model: $k-\omega$ SST with Delayed DES (DDES) shielding.
  • Domain: A 2D computational domain representing the scramjet-launch vehicle interface, extending upstream and downstream to prevent boundary reflections.
  • Conditions: Simulations were conducted at freestream Mach numbers $M_\infty = 0.9, 1.0, 1.1,$ and $1.2$ at an altitude of 25 km.
• Experimental Validation:
  • Experiments were conducted in a perforated transonic induction wind tunnel at IIT Madras ($M_\infty = 0.9$).
  • Time-resolved pressure measurements were taken using Endevco transducers to validate the numerical solver's ability to capture fluctuating pressures and spectral peaks.
• Analysis Techniques:
  • Statistical Analysis: Mean pressure, standard deviation, and Power Spectral Density (PSD) via Fast Fourier Transform (FFT).
  • Modal Analysis: Spectral Proper Orthogonal Decomposition (SPOD) was used to extract coherent structures and dominant flow modes, separating hydrodynamic and acoustic mechanisms.
  • Control Strategies: Two passive control methods were investigated:
    1. C1: Chamfering the trailing-edge wall.
    2. C2: Introducing ventilation slots (slotted sub-cavity).

3. Key Contributions

• Characterization of Complex Topology: The study moves beyond simplified rectangular models to analyze a realistic, integrated SERN-isolator geometry, revealing how the coupling between the primary cavity and deep sub-cavity alters flow dynamics.
• Transonic Regime Insights: It elucidates the unique non-monotonic pressure behavior in the transonic regime, where localized subsonic and supersonic regions coexist, creating competing mechanisms that differ from purely subsonic or supersonic flows.
• Mechanism Elucidation: The paper details the specific feedback loop in complex geometries, identifying how mass ingestion/ejection cycles, shock reflections, and vortex impingement sustain global unsteadiness.
• Effective Passive Control: It demonstrates that ventilated sub-cavities are significantly more effective than geometric chamfering for suppressing pressure loads in this specific configuration.

4. Key Results

A. Flow Physics and Feedback Loops

• Self-Sustained Oscillations: A feedback loop was identified where the shear layer rolls up (Kelvin-Helmholtz instability), impinges on the trailing edge, generates compression waves, and propagates upstream to perturb the leading edge.
• Mass Flux Coupling: There is a strong coupling between the primary cavity and the sub-cavity. The sub-cavity experiences periodic mass inflow and outflow, with the sub-cavity end-wall acting as a critical pressure load point.
• Mach Number Dependence:
  • Mean pressure loading on the sub-cavity end-wall increases monotonically with Mach number (rising ~15-30% from $M=0.9$ to $M=1.2$).
  • Peak spectral power increases by 31.5% at $M=1.2$ compared to $M=0.9$.
  • The dominant frequency (Rossiter mode) remains relatively insensitive to Mach number changes.

B. Influence of Cavity Topology

Three geometries were compared: Baseline (Rectangular), SERN (Ramp), and Inverted-SERN.

• SERN (SG): Exhibited the highest mean pressure and pressure fluctuations due to larger mass entrainment.
• Inverted-SERN (IG): Showed reduced mass exchange and lower pressure fluctuations compared to the baseline, as the geometry restricted the expansion fan and low-pressure zone formation.
• Mass Flux: The Baseline geometry had the highest mean mass flux, while IG had the lowest, confirming that topology directly controls the intensity of the feedback loop.

C. Passive Control Effectiveness

• Chamfered Trailing Edge (C1): Reduced peak spectral power by 60% at the sub-cavity end-wall. However, it slightly increased inflow and did not significantly alter the shear-layer dynamics.
• Ventilated Sub-Cavity (C2): Proved to be the most effective strategy.
  • Achieved a 96% suppression of pressure fluctuations at the sub-cavity end-wall.
  • Reduced mass inflow by 38% compared to the baseline.
  • Diverted ~6% of the inflow through the slots, breaking the periodicity of the feedback loop.
  • SPOD analysis showed that C2 restructured the flow, shifting the dominant dynamics from low-frequency acoustic resonance to higher-frequency, localized wavepacket structures.

D. SPOD Analysis Findings

• Baseline Cases: Dominated by low-frequency ($St \approx 0.07$) large-scale Orr-type structures (acoustic dominance) and high-frequency Kelvin-Helmholtz (KH) instabilities.
• Control Cases:
  • C1: Showed multiple peaks at higher frequencies, indicating a shift toward KH-type instability dominance.
  • C2: Exhibited a single dominant peak at a much higher frequency ($St = 8.7$) confined near the slot, indicating that the ventilation successfully disrupted the global acoustic feedback loop and localized the unsteadiness.

5. Significance

This research provides critical insights for the design of hypersonic and scramjet-powered launch vehicles.

1. Design Guidance: It establishes that simple geometric modifications (like chamfering) may be insufficient for complex integrated systems, whereas ventilation strategies offer a robust solution for mitigating destructive pressure oscillations.
2. Operational Safety: By identifying the mechanisms driving pressure loads in the transonic regime, the study aids in predicting and preventing structural fatigue and control surface failures during the critical launch and separation phases.
3. Methodological Value: The successful application of DES and SPOD to complex cavity-sub-cavity systems validates these tools for future high-fidelity analysis of aerospace flow problems where 3D effects are secondary to dominant 2D unsteady mechanisms.

In conclusion, the paper demonstrates that topological modifications, specifically ventilated sub-cavities, are highly effective in decoupling the hydrodynamic-acoustic feedback loop in complex scramjet configurations, offering a viable passive control strategy for next-generation aerospace vehicles.