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The Structure of Merging Turbulent Jets Beneath a Small Quadrotor
Anoop Kiran, Nora Ayanian, Kenneth Breuer
TL;DR
Quadrotor downwash matters for vehicle performance and formation spacing, yet merged-wake turbulence and its relation to canonical jet scaling remain unresolved. The study applies two-plane PIV to a hovering Crazyflie 2.1 and follows the wake from four rotor sources into a merged column. Mean flow approaches canonical round-jet behavior, while turbulent normal stresses retain a cut-dependent four-rotor signature.
Problem
Prior measurements largely characterized mean flow, leaving higher-order statistics of small-quadrotor merged wakes and their relation to canonical jet scaling unresolved.
Method
Planar PIV measured the hovering Crazyflie 2.1 downwash along diagonal and front-rotor cuts spanning the transition from four sources to a merged wake.
Results
Mean flow and Reynolds shear stress reach canonical self-similarity by z/l ≈ 13, while turbulent normal stresses retain a cut-dependent four-rotor signature.
Takeaways & Limitations
Mean downwash becomes axisymmetric, but turbulence retains structural memory of rotor geometry relevant to multi-UAV proximity flight.
Takeaways & Limitations
The measurements do not resolve out-of-plane velocity fields or time- and phase-resolved structures associated with blade rotation.
Abstract
from arXiv · showhide
The downwash wake of a hovering quadrotor governs both the vehicle's own performance and the safe spacing of multi-rotor formations. Prior measurements have largely characterized the mean flow, using single-point anemometry, volumetric tracking, or planar cuts through part of the rotor system. Higher-order turbulent statistics of the merged wake, and how they relate to canonical jet scaling, have remained unresolved, particularly for small quadrotors at the low-Reynolds-number end of the size range. Here, we present a detailed particle image velocimetry (PIV) study of the downwash of a hovering Crazyflie 2.1 quadrotor (arm length, $l = 46$ mm), sampled along a diagonal cut, passing through rotors along the symmetry axis of the quadrotor, and a front-rotor cut, passing through adjacent rotors. The four rotor jets merge into a single column by $z/l \approx 5$, beyond which the mean velocity profiles progressively approach the canonical round-jet self-similar form, collapsing by $z/l \approx 13$ when scaled by the local centerline velocity and half-width. Centerline decay and half-width growth follow canonical scaling laws with an effective source diameter $D_\text{eff} = 2.29\,l$, effective Reynolds number $Re_{D_\text{eff}} = 3 \times 10^4$, at the low end of the range over which canonical jet scaling has been established, and spreading and decay constants nonetheless within the canonical round-jet range. Resolving both cuts shows that the turbulent normal stresses retain a bimodal, cut-dependent signature of the four-rotor source throughout the measurement domain.
1. Introduction
Quadrotor downwash creates safety-relevant disturbances, but prior work has emphasized mean flow while leaving merged-wake turbulence and canonical-jet behavior unresolved. This study uses two-plane PIV to resolve the transition from four rotor jets toward a single canonical jet.
- Close-formation downwash can reduce control authority, thrust, and stability, motivating quantitative characterization of wake extent and turbulence structure.
- Prior quadrotor studies focused almost exclusively on mean flow using anemometry, visualization, or volumetric PIV for selected configurations.
- Canonical round jets exhibit inverse-linear centerline decay and linear half-width growth, alongside established higher-order turbulence statistics.
- 1.1. Merging jets: Parallel-jet research predicts that merged means approach single-jet self-similarity while Reynolds stresses retain memory of the original sources.
- 1.1. Merging jets: The study resolves mean flow and second-order turbulence along two orthogonal planes to track source-oriented merging and relaxation toward canonical scaling.
2. Methodology
The experiment measured a hovering Crazyflie 2.1 wake with planar PIV while traversing the vehicle to stitch sections across the merged-wake region. Two perpendicular cuts captured opposing-rotor and adjacent-rotor orientations, with velocities referenced to actuator-disk induced speed.
- The Crazyflie 2.1 has mass m = 32 g, motor-arm length l = 46 mm, and rotor diameter D = 45 mm.
- Velocities were normalized by the actuator-disk rotor-induced velocity, with U_i ≈ 4.36 m/s for the Crazyflie 2.1.
- The vehicle remained in steady hover while vertical traversal enabled stitched PIV measurements spanning multiple downstream sections.
- Planar PIV used a diagonal cut through opposing rotors and a front-rotor cut through adjacent rotors to sample different source orientations.
- The analysis used Reynolds decomposition to obtain mean velocities, fluctuations, normal stresses, and shear stress from the PIV fields.
3.1. Mean flow field
The two cuts reveal geometry-dependent near-field merging, which becomes a single spreading and entraining column beyond z/l > 5. Mean-flow topology approaches canonical round-jet behavior while retaining distinct cut-dependent signatures during the transition.
- Near-field merging: By z/l ≈ 2, each cut contains two rotor jets separated by a low-momentum dead zone.
- Near-field merging: The diagonal-cut profile is nearly single-peaked by z/l ≈ 5, whereas the front-cut bimodal profile persists to z/l ≈ 5 and becomes single-peaked by z/l ≈ 7.
- Mean cross-stream velocity: Cross-stream flow changes from divergent rotor-wake spreading to radial entrainment over 2.5 ≲ z/l ≲ 5.
- Mean cross-stream velocity: The front-rotor cut has larger half-widths and a slightly elliptical wake because its closer in-plane rotor pair spreads more strongly.
- Unified schematic of mean flow development: The four sources form disjoint cores at z/l = 1, a clover-shaped region at z/l = 2.5, and a filled single column with faint squareness at z/l = 5.
- Unified schematic of mean flow development: The two cuts assign r = 0 differently: the diagonal cut samples the wake centerline, while the front-rotor cut samples the weak-flow symmetry line between adjacent rotors.
- Unified schematic of mean flow development: For z/l > 5, the wake is a single spreading column whose streamwise and cross-stream profiles collapse onto canonical round-jet self-similarity when scaled locally.
3.2. Centerline decay and jet width scaling
Beyond the merge plane at z/l = 5.0, the diagonal-cut wake follows canonical round-jet centerline decay and half-width growth, with fitted constants within classical ranges. Using the merge-plane half-width defines an effective source diameter of D_eff/l = 2.29, supporting quantitative comparison with canonical jets.
- z/l = 5.0 marks the diagonal-cut wake merge location, identified where peak streamwise velocity coincides with the centerline value.This marks the transition from near-field bimodal structure to a unified merged wake.
- D_eff/l = 2.29 (approximately 105 mm) results from taking twice the wake half-width at the merge plane.
- B = 6.02, S = 0.088, and z0/D_eff = −3.42 describe centerline decay, spreading, and virtual-origin fits over z/l ≥ 5.All three parameters fall within classical canonical round-jet ranges.
- The merged-flow Reynolds number is evaluated from the merge-plane centerline velocity and effective source diameter for comparison with canonical studies.The Reynolds number conventionally corresponds to nozzle-exit states in canonical datasets.
- Beyond the merge plane, inverse centerline velocity decays linearly with normalized distance, while the half-width grows linearly downstream.The canonical forms use decay constant B, spreading rate S, effective source diameter D_eff, and virtual origin z0.
- The self-similarity invariant plateaus near 6 beyond the merge plane, matching canonical round jets and remaining flat across the measured region.Quantitative scaling and decay calculations remain within the experimental field of view, although the downstream extent is constrained by that field of view.
3.3. Self-similarity of mean velocity
The merged wake develops canonical mean-flow self-similarity downstream: normalized streamwise and cross-stream profiles collapse in the far field, while outer profiles depart from constant-transport predictions because of intermittency.
- Similarity construction: S = 0.088 sets the linear half-width growth used to normalize the diagonal-cut profiles by local centerline velocity and half-width.The similarity coordinate is ξ = r/r_1/2(z), with r_1/2(z) = S(z − z_0).
- Far-field collapse: z/l ≳ 13 marks collapse of the normalized streamwise and cross-stream mean-velocity profiles onto canonical round-jet behavior.Stations z/l = 13–17 collapse tightly, matching reference experiment and simulation data.
- Far-field collapse: ⟨v⟩/u_c ≈ 0.018–0.020 is the far-field cross-stream peak amplitude, with the canonical positive peak, zero crossing, and negative entrainment region.The cross-stream profiles reproduce the self-similar free-round-jet form.
- Model departure: ξ ≳ 1 identifies the outer region where both velocity components fall below analytical curves while tracking reference data.Boundary intermittency between turbulent and ambient fluid reduces turbulent transport and steepens the outer gradient relative to constant-transport assumptions.
- Implication: The far-field mean downwash reaches self-similarity, providing a canonical-scaling baseline for evaluating higher-order turbulent moments.This conclusion concerns the mean flow and its use as a reference for turbulent statistics.
3.4. Fluctuating velocity fields
Mean velocity becomes a single quasi-axisymmetric jet before the turbulent normal stresses equilibrate. Both stress components retain source- and cut-dependent structure, with the front-rotor cut relaxing more slowly.
- Streamwise normal stress: The turbulent normal stresses retain a persistent multi-source signature after the mean velocity field has merged into a single quasi-axisymmetric jet.The section compares streamwise and cross-stream normal stresses across diagonal and front-rotor cuts.
- Streamwise normal stress: z/l = 5 shows twin off-axis streamwise-stress peaks and a deeper centerline deficit in the diagonal cut.The peaks broaden and flatten into a central plateau downstream.
- Comparison with multi-source jets: Multi-source stress structure persists beyond mean-flow merging, consistent with the delayed second-moment equilibrium observed in parallel-jet flows.The quadrotor wake follows the established distinction between earlier mean-flow self-similarity and later Reynolds-stress equilibrium.
- Cross-stream normal stress: The front-rotor cut retains twin off-axis cross-stream-stress peaks through z/l = 17, whereas the diagonal-cut centerline depression gradually closes.The two measurement planes therefore exhibit markedly different rates of turbulent-stress evolution.
- Cross-stream normal stress: The cross-stream stress remains multi-rotor-like longer than the streamwise component because strong inner shear layers and lateral velocities localize turbulent kinetic energy.This mechanism is associated with opposing rotor flows in the front-rotor cut.
- Comparison with shear stress: The Reynolds shear stress more closely resembles a canonical turbulent jet than either normal-stress component and mediates radial momentum transport and ambient entrainment.This contrasts the rapid shear-stress adjustment with the persistent bimodal normal-stress structure.
3.5. Reynolds shear stress
Reynolds shear stress follows the merged mean-flow structure more rapidly than the normal stresses, with canonical antisymmetry emerging after near-field rotor-interaction features fade.
- Stress structure: The Reynolds shear stress has a canonical antisymmetric structure linked to the radial mean-velocity gradient and eddy-viscosity closure.Its extrema coincide with the maximum radial gradient of ⟨u⟩.
- Downstream evolution: z/l ≈ 7 marks the disappearance of front-cut secondary peaks near r/l ≈ ±0.5, after which both cuts share the same canonical shear-stress distribution.The secondary peaks arise from inner shear layers between closely spaced rotors.
- Downstream evolution: P = −⟨u′v′⟩∂⟨u⟩/∂r links shear stress to turbulence production and explains its rapid adjustment once the mean flow becomes smooth and single-peaked.The shear stress therefore tracks mean-flow evolution more closely than ⟨u′u′⟩ or ⟨v′v′⟩.
3.6. Self-similarity of turbulent stresses
The turbulent stresses approach self-similarity on distinct timescales: shear stress collapses relatively cleanly, while normal stresses retain rotor-source memory and may remain offset from canonical reference profiles.
- Streamwise normal stress: z/l = 13–17 marks shape collapse of the streamwise normal-stress profiles, although their near-axis amplitude rises from ≈0.02 at z/l = 5 to ≈0.09 at z/l = 17.The far-field profiles remain outboard of the canonical reference, indicating internal self-similarity without full canonical-state matching.
- Streamwise normal stress: z/l ≈ 15 marks the approach of the diagonal-cut streamwise normal stress to the canonical centerline-peaked form.Near-field stations z/l = 5–11 have off-axis maxima at ξ ≈ 0.5–0.7 and an on-axis deficit.
- Reynolds shear stress: ⟨u′v′⟩/u_c^2 collapses cleanly with a peak of ≈0.023 near ξ ≈ 0.8–0.9, matching the canonical reference through the core.Its rapid equilibration follows the gradient transport of mean momentum.
- Reynolds shear stress: ν̂_T = 0.028 fits the measured core shear stress under uniform eddy-viscosity closure, while the closure model closely reproduces the measured core profile.The independent estimate from the measured spreading rate gives ν̂_T = 0.027, consistent with the shear-stress fit.
- Timescale separation: The mean flow merges by z/l ≈ 5 and collapses by z/l ≈ 13, whereas normal stresses retain four-rotor structure and reach canonical centerline-peaked form only by z/l ≈ 15 in the diagonal cut.This comparison demonstrates a separation of timescales between mean flow, shear stress, and normal stresses.
4. Conclusions
The wake transitions from four rotor sources toward a self-similar mean flow, but turbulent normal stresses preserve source- and cut-dependent structure. These findings have practical implications for multi-UAV proximity flight and motivate measurements resolving out-of-plane and phase-dependent rotor structures.
- By z/l≈13, the mean flow and Reynolds shear stress reach canonical self-similarity, while turbulent normal stresses lag behind.The normal stresses begin approaching self-similarity near z/l≈13 and reach it by z/l≈15 in the diagonal cut.
- The front-rotor cut retains twin off-axis peaks through the measurement domain, showing persistent memory of the four-rotor geometry.
- Future measurements should resolve out-of-plane velocities and time- or phase-dependent structures associated with rotor blade rotations.The passage also identifies possible synchronization effects between adjacent rotors as a natural next step.
- Downwash disturbances are driven primarily by sheared, bimodal turbulence rather than the mean velocity deficit.The wake is described as a localized, source-structured disturbance rather than a uniform gust field.
6. Funding & Acknowledgments
The work received support from Brown University and the NSF Graduate Research Fellowship program.
- The work was supported by a Brown University Seed Award from the Office of the Vice President for Research.
- AK received support from an NSF Graduate Research Fellowship, Award 2439559.
- The acknowledgment identifies the Office of the Vice President for Research as the Brown University Seed Award sponsor.