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The Structure of Merging Turbulent Jets Beneath a Small Quadrotor

This study utilizes particle image velocimetry to characterize the merging downwash wake of a hovering Crazyflie 2.1 quadrotor, revealing that while mean velocity profiles transition to canonical round-jet self-similarity beyond five rotor arm lengths, turbulent normal stresses retain a distinct bimodal signature of the four-rotor source throughout the measurement domain.

Original authors: Anoop Kiran, Nora Ayanian, Kenneth Breuer

Published 2026-09-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anoop Kiran, Nora Ayanian, Kenneth Breuer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When a small drone hovers in place, it does not simply sit still; it pushes a powerful column of air downward to stay aloft. This downward rush of air, known as downwash, is the engine of flight for these machines, but it also creates a complex wake that trails beneath them. For engineers and pilots, understanding this wake is a matter of safety and efficiency. If multiple drones fly in close formation, the turbulent air from a leading vehicle can slam into a follower, potentially causing it to lose control or crash. While scientists have long studied the average speed of this air, the hidden, chaotic swirls within it—the turbulence—have remained a mystery, especially for the tiny drones used in modern research and delivery. Knowing how these invisible currents behave is essential for designing safer swarms of robots that can fly close together without colliding.

A team of researchers at Brown University set out to map this invisible world beneath a small, four-rotor drone called the Crazyflie 2.1. To see what was happening, they did not rely on guesswork or simple sensors. Instead, they used a technique called particle image velocimetry, which involves seeding the air with tiny, harmless droplets and illuminating them with a powerful laser sheet. By taking rapid-fire photographs of these droplets as they moved, the team could reconstruct the speed and direction of the air at thousands of points simultaneously. They sliced the wake open in two different ways: one cut ran diagonally through opposite rotors, and the other ran straight through two adjacent rotors. This allowed them to see how the four separate streams of air from the rotors merged into a single column and how the turbulence within that column evolved as it traveled downward.

The researchers found that the four distinct jets of air from the rotors do not merge instantly. Close to the drone, the air behaves like four separate streams, each carrying its own momentum. However, as the air travels downward, these streams begin to interact and blend. By the time the air has traveled a distance equal to about five times the length of the drone's arm, the four streams have coalesced into a single, unified column. Beyond this point, the average speed of the air follows a predictable pattern, slowing down and spreading out in a way that matches the behavior of a standard, single-nozzle jet of air. The researchers calculated that this merged column acts as if it were coming from a single source with an effective diameter of about 105 millimeters, and they confirmed that the turbulence in this far-field region behaves just like the well-understood turbulence of a simple round jet.

However, the story changes when looking at the chaotic, swirling motions within the air, rather than just the average speed. While the main flow of air becomes smooth and uniform, the turbulence retains a memory of its four-rotor origin for a much longer distance. Even after the air has merged into a single column, the intensity of the swirling motion still shows a distinct pattern of two peaks, a signature of the two pairs of rotors that created it. This "turbulence memory" persists much longer than the average flow pattern. In the cut passing through adjacent rotors, this twin-peak structure remained visible all the way to the end of the measurement area, which was about seventeen arm-lengths below the drone. In the diagonal cut, the pattern took longer to fade but still lingered well past where the average flow had already settled.

This discovery reveals a crucial detail about how drone wakes behave: the air can look like a single, smooth column while still hiding a complex, structured turbulence underneath. The researchers noted that the chaotic energy in the air takes longer to redistribute and smooth out than the average speed does. This means that a drone flying in the wake of another might not feel a uniform push or pull, but rather a highly structured, bumpy disturbance that depends on exactly where it is relative to the rotors above. The study confirms that while the overall shape of the wake becomes simple and predictable, the turbulent forces that could destabilize a following drone remain complex and tied to the specific geometry of the four rotors.

These findings have immediate implications for how we think about flying multiple drones together. Current methods for keeping drones stable in windy conditions often assume that the air disturbances are broad and uniform, like a gust of wind. But the wake of a hovering drone is neither broad nor uniform; it is a localized, structured disturbance with hidden peaks of turbulence. To fly safely in close formation, future control systems will need to account for this specific, bumpy nature of the wake. The researchers suggest that simply knowing the average speed of the air is not enough; pilots and algorithms must understand where the turbulent peaks are located to avoid the most dangerous parts of the wake. This work provides the first detailed map of those hidden currents, offering a clearer path toward safer and more efficient drone operations in the future.

Technical Summary: The Structure of Merging Turbulent Jets Beneath a Small Quadrotor

Problem Statement
The downwash wake of a hovering quadrotor dictates both the vehicle's aerodynamic performance and the safe spacing required for multi-rotor formations. While prior research has extensively characterized the mean flow of quadrotor wakes using single-point anemometry, smoke visualization, or planar cuts, higher-order turbulent statistics of the merged wake remain largely unresolved. Specifically, it is unclear how the four discrete rotor jets of small quadrotors (operating at low Reynolds numbers) merge into a single column and whether they adhere to the canonical scaling laws of free turbulent round jets. Furthermore, the relationship between the mean flow evolution and the persistence of multi-source turbulent structures in the second-order statistics has not been fully quantified.

Methodology
The study utilizes Particle Image Velocimetry (PIV) to investigate the downwash of a hovering Crazyflie 2.1 quadrotor (mass $m=32$ g, arm length $l=46$ mm, rotor diameter $D=45$ mm). The effective Reynolds number based on the merged jet is approximately $Re_{D_{eff}} \approx 3 \times 10^4$ .

To capture the three-dimensional transition of the wake, measurements were taken along two orthogonal cutting planes:

Diagonal Cut (Plane A): Passes through opposing rotors along the symmetry axis, intersecting rotors at $r/l = \pm 1$ .
Front-Rotor Cut (Plane B): Passes through adjacent rotors, intersecting them at $r/l = \pm 1/\sqrt{2}$ .

The quadrotor was traversed vertically while the laser sheet and camera remained stationary, allowing for the stitching of five discrete sections to cover a streamwise range of $-1.5 \lesssim z/l \lesssim 17.5$ . Velocities were normalized by the induced velocity derived from actuator-disk theory ( $U_i \approx 4.36$ m/s). The study analyzes time-averaged mean velocities ( $\langle u \rangle, \langle v \rangle$ ) and Reynolds stresses ( $\langle u'u' \rangle, \langle v'v' \rangle, \langle u'v' \rangle$ ).

Key Results

Mean Flow Merging and Scaling:
The four rotor jets merge into a single column by $z/l \approx 5$ . Beyond this point, the mean velocity profiles progressively approach the canonical self-similar form of a round jet.
- Centerline Decay: The centerline velocity decay follows an inverse-linear law ( $u_c \propto 1/(z-z_0)$ ) with a decay constant $B = 6.02$ .
- Spreading Rate: The jet half-width grows linearly with a spreading rate $S = 0.088$ .
- Effective Parameters: The merged wake behaves as a canonical round jet with an effective source diameter $D_{eff} = 2.29 l$ and a virtual origin $|z_0|/D_{eff} \approx 3.42$ . These constants fall within the established ranges for high-Reynolds-number axisymmetric jets, despite the low-Reynolds-number regime of the small quadrotor.
- Self-Similarity: When scaled by local centerline velocity and half-width, the mean velocity profiles collapse onto the theoretical self-similar solution for $z/l \gtrsim 13$ .
Cut-Dependent Near-Field Structures:
The merging process is sensitive to the azimuthal orientation of the measurement plane.
- In the Front-Rotor Cut, the jets merge later ( $z/l \approx 7$ ) due to the closer proximity of the in-plane rotors, resulting in a wider wake cross-section.
- In the Diagonal Cut, the jets merge earlier ( $z/l \approx 5$ ) as off-plane rotors contribute more strongly to the centerline flow.
Turbulent Stresses and Memory of the Source:
A critical finding is the decoupling between the relaxation of the mean flow and the turbulent stresses.
- Reynolds Shear Stress ( $\langle u'v' \rangle$ ): This component collapses rapidly, achieving canonical self-similarity by $z/l \approx 13$ , closely tracking the mean velocity gradient.
- Reynolds Normal Stresses ( $\langle u'u' \rangle, \langle v'v' \rangle$ ): These retain a "bimodal," cut-dependent signature of the four-rotor source throughout the measurement domain. Even after the mean flow has merged into a single column, the normal stresses exhibit twin off-axis peaks and a centerline deficit.
- Persistence: In the diagonal cut, the normal stresses approach a canonical centerline-peaked form only by $z/l \approx 15$ . In the front-rotor cut, the bimodal structure persists to the end of the measurement domain ( $z/l = 17$ ).

Significance and Claims
The paper establishes that while the mean downwash of a small quadrotor obeys canonical round-jet scaling laws in the far-field, the turbulent field retains a distinct "memory" of the four-source geometry for a significantly longer distance. The mean flow achieves axisymmetry and self-similarity much faster than the second-order turbulent statistics.

The authors claim these findings have direct practical implications for multi-UAV proximity flight. They argue that disturbance rejection for follower aircraft cannot rely solely on mean velocity deficits, as the primary disturbances are driven by the sheared, bimodal turbulence structure that persists long after the mean flow has merged. Consequently, control frameworks and trajectory planners for close-formation operations must account for the decoupling between mean-flow axisymmetry and persistent turbulent anisotropy. The study concludes that future work should focus on measuring out-of-plane velocity fields and resolving time- or phase-resolved structures related to blade rotation to fully understand these complex flows.

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