Impact of the formation angle on the drag of bio-inspired $\pmb \vee$-formations

This study uses time-resolved particle image velocimetry (PIV) on axisymmetric cylinders to demonstrate how varying the formation angle of a five-member $\pmb \vee$-formation affects flow dynamics and drag reduction, finding that tighter angles significantly enhance drag reduction for interior members through complex wake-body and wake-wake interactions.

The Gist

The "Drafting" Secret: How Birds and Drones Save Energy

Imagine you are running a marathon. If you run alone, you have to fight the wind every single step of the way. But if you join a group of runners and tuck in closely behind someone, you’ll notice the air feels "thinner" and easier to move through. You are drafting.

This scientific paper explores exactly how that "drafting" magic works, not just for marathon runners, but for birds, fish, and even future swarms of drones.

---

The Experiment: The "Artificial Bird"

Instead of using real birds (which are hard to measure), the researchers built a high-tech simulation. They used five cylinders in a water tunnel to mimic a group of flying objects.

Think of these cylinders as "non-lifting" vehicles—like a group of heavy cargo ships or simple drones that don't use wings to stay up, but just need to push through the water/air. They arranged them in a $\vee$-shape (the classic "V" formation you see when geese fly) and changed the angle of the "V" to see what happened.

The Big Discovery: The "Goldilocks" Angle

The researchers found that the angle of the "V" is the most important factor in how much energy the group saves.

1. The Tight "V" (The Huddle): When the angle is very narrow (like a sharp arrowhead), the members are almost touching. In this setup, the "inner" members of the group get a massive boost. One member saw a drag reduction of 80%! It’s like being in a tight slipstream where the wind is almost completely blocked by your teammates.
2. The Wide "V" (The Social Distancers): As the angle gets wider (more like a flat line), the "magic" disappears. The members are too far apart to help each other. In these wide formations, only the leader gets any benefit, and everyone else has to work just as hard as if they were flying alone.

The Science: "Bleeding Flow" and "Wake-Wrestling"

How does this actually work? The paper explains it using two main concepts:

• The "Bleeding Flow" (The Gap-Filler): Imagine a crowd of people running through a narrow hallway. The air doesn't just sit still; it squirts through the gaps between people. The researchers called this "bleeding flow." In a tight formation, this flow acts like a specialized wind tunnel, directing air in ways that actually "push" the trailing members forward or help them glide more smoothly.
• Wake-Wrestling (The Chaos Behind the Leader): Every object moving through fluid leaves a "wake"—a messy, swirling trail of turbulent water (like the white, bubbly water behind a speedboat). In a $\vee$-formation, the members are essentially "wrestling" with each other's wakes. In a tight formation, the members are positioned so they can "catch" the energy from the leader's wake rather than fighting against it.

Why Does This Matter?

This isn't just about birds. This research provides a "blueprint" for:

• Drone Swarms: Helping a fleet of delivery drones fly longer on a single battery by teaching them the perfect "V" angle.
• Underwater Robots: Designing groups of autonomous subs that can travel vast distances by "huddling" together to save power.
• Engineering: Designing better heat exchangers or marine structures by understanding how groups of objects interact with moving water.

In short: If you want to go far with very little effort, don't just fly together—fly at the perfect angle.

Technical Summary

Technical Summary: Impact of the Formation Angle on the Drag of Bio-inspired $\vee$-formations

1. Problem Statement

The study investigates the aerodynamic/hydrodynamic performance of bio-inspired $\vee$-formations, a strategy used by migratory birds and increasingly by drone swarms to minimize energy expenditure. While previous research has focused on lift-generating bodies (where energy savings are driven by wing-tip vortex upwash), this paper addresses the fundamental mechanics of non-lifting bodies (cylinders). The core research question is how varying the formation angle ($\phi$) affects the drag forces experienced by individual members and the group as a whole, and what complex flow interactions (wake-body, wake-wake, and bleeding flows) drive these changes.

2. Methodology

The researchers employed a high-resolution experimental approach using a water tunnel facility:

• Model Geometry: A 5-member symmetric $\vee$-formation consisting of stationary, non-lifting circular cylinders ($d = 6$ mm). The streamwise distance between rows was kept constant at $2.5d$ to ensure the bodies remained within the "wake interference" region.
• Experimental Variables: The formation angle $\phi$ was varied from $22.6^\circ$ to $67.6^\circ$ in $5^\circ$ increments. The study distinguishes between overlapping formations ($\phi < 43.6^\circ$, where members have frontal overlaps) and non-overlapping formations ($\phi > 43.6^\circ$).
• Flow Visualization & Measurement:
  • Time-resolved, multi-illumination, consecutive-overlapping Particle Image Velocimetry (PIV): This advanced technique was used to overcome shadowing effects caused by multiple bodies, allowing for high-resolution velocity field capture around and between members.
  • Force Calculation: Drag and lift forces were calculated using Reynolds-averaged integral momentum (RAIM) conservation via control volume analysis. Pressure distributions were obtained through line integration of the pressure gradient terms in the RANS equations.
• Flow Characterization: The study analyzed mean and fluctuating velocity fields, turbulent kinetic energy (TKE), streamlines, and instantaneous circulation ($\Gamma$) to characterize vortex shedding patterns.

3. Key Contributions

• Systematic Classification of Flow Regimes: The paper categorizes $\vee$-formations into three distinct physical groups based on flow patterns: Most Overlapping (asymmetric, single recirculating zone), Intermediate Overlap (increasing symmetry, two distinct vortices in the leader's wake), and Non-overlapping (wakes resembling solo cylinders).
• Mechanistic Insight into Drag Reduction: The study identifies the roles of bleeding (gap) flow and pressure recovery in modulating drag. It demonstrates how the interaction between the shear layers of upstream members and the stagnation points of downstream members dictates the force distribution.
• Advanced Imaging Methodology: The use of a multi-light-sheet, consecutive-overlapping PIV approach provides a cost-effective alternative to index-matching, enabling a full-field view of complex multi-body interactions.

4. Key Results

• Drag Reduction Trends:
  • Maximum Benefit: The highest drag reduction ($\sim 80\%$) was observed for interior members in the tightest formations ($\phi = 22.6^\circ$ and $27.6^\circ$).
  • Angle Threshold: For $\phi \leq 50^\circ$, all members experience some level of drag reduction. For $\phi > 50^\circ$, only the leading member (Member 1) benefits significantly.
  • Total Group Drag: The total drag of the formation increases monotonically with the formation angle, but even non-overlapping formations can offer lower total drag than five isolated cylinders.
• Flow Dynamics:
  • Asymmetry: At low angles, the flow is highly asymmetric, causing members in the same row (e.g., 2 vs. 3) to experience different drag forces.
  • Wake Interaction: In overlapping formations, the wake of the leader is "contained" or modified by the bleeding flows between the second-row members, which limits the width of the wake and promotes pressure recovery for downstream members.
  • Turbulence: Overlapping formations suppress the transition to turbulence in the interior of the array, whereas non-overlapping formations allow wakes to develop turbulence similar to solo cylinders.
  • Vortex Shedding: Tight formations can "trap" recirculation regions, leading to non-periodic, flat circulation profiles, whereas wider formations exhibit periodic vortex shedding and higher oscillation amplitudes.

5. Significance

This research provides a critical baseline for the optimization of group motion in both aerial (drones) and aquatic (underwater vehicles) environments. By demonstrating that the formation angle is a powerful "tuning knob" for group performance, the study offers a framework for designing formations that minimize energy consumption. Furthermore, the insights into wake-body and wake-wake interactions contribute to the broader understanding of fluid dynamics in arrays, which is applicable to engineering fields such as heat exchanger design and marine structural engineering.