Scale-resolving simulations and data-driven modal analysis of turbulent transonic buffet cells on infinite swept wings

This study employs scale-resolving simulations and modal analysis on infinite swept wings to demonstrate that transonic buffet arises from the superposition of quasi-2D shock oscillations and separation-driven 3D instabilities, with the latter emerging as dominant spanwise-travelling modes only when mean flow separation at the shock is sufficiently pronounced.

The Gist

Imagine you are flying a commercial airplane. Suddenly, the plane starts to shudder violently, like a car hitting a patch of ice at high speed. This isn't just a bump; it's a rhythmic, self-sustaining shaking called transonic buffet.

This paper is like a high-tech detective story that tries to figure out why this shaking happens and how to stop it. The researchers used supercomputers to create a "virtual wind tunnel" to watch the air behave around a wing in slow motion.

Here is the breakdown of their discovery, explained simply:

1. The Problem: The "Shaking Wing"

When a plane flies fast (near the speed of sound), a shockwave forms on the wing. Think of this shockwave like a traffic jam in the air.

• The 2D Problem (The Traffic Jam): Usually, this shockwave moves back and forth along the wing, like a car braking and accelerating. This is a 2D problem (moving left and right).
• The 3D Problem (The "Buffet Cells"): But sometimes, the air doesn't just move left and right; it starts to swirl in giant, rolling bubbles along the length of the wing. The researchers call these "buffet cells." Imagine a long, skinny snake slithering across the wing. These snakes are the 3D instability that makes the shaking much worse.

2. The Challenge: Why Wasn't Anyone Seeing the Snakes?

For years, scientists tried to simulate this on computers, but their "virtual wings" were too short (like a tiny model airplane).

• The Analogy: Imagine trying to study how a long river flows by looking at a 5-foot section of a creek. You might see the water moving, but you'll never see the big, rolling eddies that form in a wide river.
• Because the computer models were too narrow, they missed the "snakes" (the 3D buffet cells). They only saw the "traffic jam" (the 2D shockwave).

3. The Experiment: Building a "Super-Wing"

The researchers in this paper built a massive virtual wing.

• The Scale: They made the wing 3 times wider than the distance from the front to the back of the wing. This was huge for a computer simulation, requiring the power of a supercomputer (using thousands of graphics cards running for months).
• The Sweep: They tested wings that were straight and wings that were swept back (like the wings on a Boeing 787 or Airbus A350).

4. The Big Discovery: It's All About the "Separation"

They ran two main scenarios:

• Scenario A (The Calm Wing): The air flowed smoothly over the wing, with only a tiny bit of separation (detachment) near the back.
  • Result: Even with the huge wing, the "snakes" didn't show up. The wing just shook back and forth (2D). The shockwave was in control.
• Scenario B (The Rough Wing): They increased the angle of the wing, causing the air to separate (detach) significantly right where the shockwave hits.
  • Result: Boom! The "snakes" appeared. Giant, rolling bubbles of separated air formed along the wing.

The Key Insight: The 3D "buffet cells" (the snakes) only appear when the air is rough and separated at the shockwave. If the air is smooth, the 3D chaos doesn't happen.

5. The Twist: How Sweep Changes the Game

When they swept the wing back (angled it), something fascinating happened to the "snakes":

• On a straight wing: The separation bubbles just grew and shrank in place (like a breathing lung).
• On a swept wing: The bubbles started traveling sideways along the wing, like a wave moving across a stadium crowd.
• The Speed: As they increased the sweep angle, these traveling waves moved faster and shook the wing at a higher frequency.

6. The Conclusion: Why This Matters

The paper solves a long-standing mystery: Why do real airplanes (which have swept wings) seem to only have the 3D "snake" problem, while simple computer models only show the 2D "traffic jam"?

• The Answer: Real wings are swept, and real wings often have enough air separation to trigger the 3D cells. Once the wing is swept, the 3D "snakes" take over and become the dominant force, drowning out the simple back-and-forth shaking.
• The Takeaway: To stop the shaking, engineers can't just look at the shockwave. They have to manage the separation of the air right where the shock hits. If they can keep the air attached (smooth) at that spot, the dangerous 3D snakes won't form.

Summary Analogy

Think of the wing as a trampoline.

• The 2D mode is someone jumping up and down in the middle.
• The 3D buffet cells are a ripple traveling from one side of the trampoline to the other.
• This paper found that the ripple only happens if the trampoline is wet and sticky (separated flow) and if you tilt the trampoline (sweep). If you keep the trampoline dry and flat, you only get the up-and-down jump.

By understanding this, engineers can design wings that avoid the "sticky" conditions, making future flights smoother and safer.

Technical Summary

1. Problem Statement

Transonic buffet is a self-sustained shock-wave/boundary-layer interaction (SBLI) phenomenon that causes severe structural vibrations in aircraft. It is characterized by two distinct instability modes:

• 2D Mode: Low-frequency chordwise shock oscillations ($St \sim 0.05 - 0.1$).
• 3D Mode: Spanwise-modulated flow separation bubbles known as "buffet cells" ($St \sim 0.2 - 0.4$).

Key Challenges:

• Computational Limitations: Previous high-fidelity simulations (LES/DNS) were restricted to narrow aspect ratios ($AR = L_z/c \sim 0.05 - 0.25$), insufficient to capture the full 3D buffet cell structures which require $AR \ge 1$.
• Modeling Uncertainties: Low-fidelity methods (RANS/URANS) often struggle to accurately predict unsteady separated flows and the interplay between small-scale turbulence and large-scale 3D structures.
• Mechanism Ambiguity: The relationship between the 2D shock oscillation and the 3D buffet cells, particularly how they coexist and how sweep angle influences the transition from quasi-stationary 3D modes to travelling waves, remains unclear.

2. Methodology

The authors employed Implicit Large Eddy Simulations (ILES) combined with Spectral Proper Orthogonal Decomposition (SPOD) to investigate these phenomena.

• Numerical Setup:

  • Solver: OpenSBLI, a high-order finite-difference compressible flow solver running on GPU-accelerated HPC clusters.
  • Geometry: Infinite swept wings (span-periodic) based on the NASA Common Research Model (CRM) airfoil.
  • Domain: Aspect ratios up to $AR = 3$($L_z = 3c$) to accommodate multiple wavelengths of 3D structures.
  • Mesh: Approximately $8 \times 10^9$ grid points per case.
  • Conditions: $Re_c = 0.5 \times 10^6$, $M_\infty = 0.72$. Sweep angles ($\lambda$) ranged from $0^\circ$ to $35^\circ$.
  • Cases: Two primary angles of attack ($\alpha$) were studied:
    • $\alpha = 5^\circ$: Minimally separated mean flow at the shock.
    • $\alpha = 6^\circ$: Largely separated mean flow at the shock.

• Analysis Technique:

  • SPOD: Applied to time-resolved flow snapshots (pressure, streamwise velocity $u$, spanwise velocity $w$) to extract coherent structures and their energy content. This allowed for the separation of overlapping frequencies and the identification of travelling vs. stationary modes.

3. Key Contributions

• Scale: Performed the largest scale-resolving simulations of transonic buffet to date, bridging the gap between narrow-span LES and full-3D configurations.
• Mechanism Link: Established a direct link between the quasi-stationary 3D separation mode observed on unswept wings and the spanwise travelling buffet cell mode observed on swept wings.
• Sweep Sensitivity: Quantified the specific effects of sweep angle on the frequency, energy, and convection velocity of 3D buffet cells.
• Separation Dependency: Demonstrated that significant mean flow separation at the shock location is a necessary condition for the emergence of dominant 3D buffet dynamics.

4. Key Results

A. Minimally Separated Case ($\alpha = 5^\circ$)

• Flow Behavior: The flow remains essentially quasi-2D. The shock oscillation is uniform across the span ($St \approx 0.08$).
• 3D Effects: Even with $AR=3$ and sweep up to $25^\circ$, no large-scale buffet cells form at the shock. Weak, intermittent separation patches appear only near the trailing edge but do not interact significantly with the shock.
• Conclusion: At low angles of attack, the 2D shock mode dominates regardless of sweep or aspect ratio.

B. Largely Separated Case ($\alpha = 6^\circ$)

• Flow Behavior: Significant mean separation at the shock triggers the emergence of pronounced 3D buffet cells with a characteristic spanwise wavelength $\lambda_z \approx 1 - 1.5c$.
• Effect of Sweep:
  • Unswept ($\lambda=0^\circ$): A quasi-stationary 3D mode exists at very low frequency ($St \approx 0.02$).
  • Swept ($\lambda > 0^\circ$): The mode transforms into a spanwise travelling wave. As sweep increases, the frequency shifts monotonically to intermediate values ($St = 0.06 - 0.35$).
  • Frequency vs. Sweep: The 2D shock mode frequency ($St \approx 0.08$) is insensitive to sweep. In contrast, the 3D cell frequency increases linearly with the tangent of the sweep angle.
• Energy Distribution: At high sweep angles (e.g., $\lambda = 35^\circ$), the energy content of the 3D mode becomes comparable to or exceeds that of the 2D shock mode.

C. Modal Analysis Insights

• SPOD Utility: SPOD based on the spanwise velocity ($w$) was crucial for distinguishing the 3D travelling mode from the 2D shock mode, even when their frequencies overlapped.
• Convection Velocity: The convection velocity of buffet cells ($V_c$) follows a scaling law $V_c \propto \tan(\lambda)$. The results align well with previous URANS and Global Stability Analysis (GSA) models (e.g., $V_c \approx 0.78 \tan(\lambda)$), confirming the physical mechanism is separation-driven.
• Wavelength Independence: The spanwise wavelength of the 3D mode is determined by the domain aspect ratio ($\lambda_z = L_z/2$) and remains independent of the sweep angle.

5. Significance and Implications

• Resolution of the 2D vs. 3D Debate: The study clarifies that transonic buffet on realistic swept wings is a superposition of distinct 2D shock motion and separation-driven 3D instabilities. The apparent "absence" of 2D modes on finite wings in previous experiments is likely due to the dominance of the 3D mode at high sweep angles.
• Design Relevance: For modern commercial aircraft (typically $\lambda \sim 35^\circ$), the 3D buffet cell dynamics are the dominant instability. This explains why 2D airfoil studies often fail to predict buffet behavior on full wings.
• Validation of High-Fidelity Methods: The results validate the use of ILES/DNS for capturing complex SBLI physics where RANS models often fail, providing a benchmark for future turbulence modeling.
• Physical Mechanism: The work confirms that mean flow separation at the shock is the critical trigger for 3D buffet cells. Without sufficient separation, the flow remains quasi-2D even in the presence of sweep.

In summary, this paper provides a definitive high-fidelity characterization of how sweep angle and flow separation interact to drive the transition from 2D shock oscillations to complex 3D buffet cell instabilities, offering critical insights for the design and control of transonic aircraft.