Tunable acoustic vortex generation by a compact rotating disk

The study addresses how to generate tunable acoustic vortices (AVs) that carry orbital angular momentum (OAM) in a compact and convenient manner. Unlike light, acoustics are scalar fields and only carry OAM, yet AVs enable applications in manipulation, communication, sensing, and metrology. Existing AV generation approaches split into active (e.g., circular transducer arrays, diffraction gratings) and passive (e.g., spiral phase plates, diffraction gratings, metasurfaces). Active arrays allow flexible tuning but require complex multi-channel control and high power. Passive devices can be compact, but many are single-frequency, have limited flexibility in topological charge, or suffer from low efficiency and thickness constraints around a fraction of wavelength, especially at low frequencies. There is also a need to avoid relying on external incident waves, which complicate applications like rotational information detection due to beam-object size and alignment issues. The authors hypothesize that aerodynamic dipole sources naturally generated by a simple rotating disk at low Mach numbers can be harnessed to form AVs via interference at the rotation frequency and its harmonics, achieving tunable topological charge and working frequency without external incident sources, in a compact configuration.

The paper reviews AV generation methods. Active methods (uniform circular transducer arrays, active diffraction gratings) offer tunable order and frequency with high conversion efficiency but need precise digital control across many elements, increasing complexity and power consumption. Passive methods shape an incident plane wave using structures: spiral phase plates (efficient but single-frequency and thickness near wavelength), spiral diffraction gratings and subwavelength metamaterials (compact and broader bandwidth but may exhibit lower efficiency due to energy blocking and losses). Metasurfaces based on resonance, compound labyrinths, and phase gradients improve wavefront control and efficiency, with some reducing thickness to half or one-third wavelength. However, most passive structures are fixed to a single order and frequency. Reconfigurable metasurfaces introduce tunability, but typically require mechanical adjustment or structural reconfiguration, adding complexity and still constrained by minimum thickness relative to wavelength. Additionally, methods that rely on an external incident beam can hinder applications like rotational Doppler-based sensing due to sensitivity to beam-object size, alignment, and misalignment errors.

Theory and principle: The authors use the Ffowcs Williams–Hawkings (FW-H) equation for sound generation by moving bodies. For a rigid rotating disk at low Mach numbers, the monopole (thickness) and quadrupole (turbulence) terms are neglected; the remaining loading (dipole) term dominates. Under these assumptions, the acoustic pressure field is derived from distributed surface dipole sources (due to surface pressure stresses) on the rotating disk. AVs arise from interference of these dipole sources at the angular rotation frequency Ω and its integer multiples, producing vortices with topological charge l at working frequencies f_working = l Ω / (2π).

Model and parameters: The disk operates in free space, thickness 4 mm, radius 50 mm, made of photo-sensitive resin (Poisson’s ratio 0.42, density 1.12 g/cm³, tensile modulus 2589 MPa). Rotational speed is set to maintain low tip Mach number. The disk surface is treated as a smooth rigid wall to exclude vibrational resonance contributions; only aerodynamic dipoles are considered.

Numerical simulation: A hybrid two-step aeroacoustic approach is used. Step 1: Flow simulation in FLUENT with steady RANS using the SST k–ε turbulence model, followed by unsteady Large Eddy Simulation (LES) with WALE subgrid model and boundary layer mesh (y* ≤ 1). The disk rotation is modeled by a sliding/rotational mesh block around the disk while the outer mesh is stationary. Mesh size totals 4,409,868 elements (1,942,991 in rotational domain; 37,464 on disk surface). Time step is 1e-4 s; after transient stabilization, 4000 time steps of surface pressure are exported. Maximum acoustic solution frequency is 5000 Hz with Δf = 2.5 Hz. Step 2: The unsteady surface pressure is processed (LMS Virtual.Lab) to generate equivalent distributed dipole sources on the disk surface; the radiated sound field is then computed with the finite element method. Simulations focus solely on dipole sources, omitting monopole/quadrupole, and evaluate near- and far-field behaviors.

Experimental setup: Experiments were conducted in a fully anechoic chamber. A brushless DC motor drives the disk with closed-loop speed control and monitoring. Measurements were performed at a plane z = 40 mm in front of the disk. Two microphones are used: Microphone 1 is fixed on a circle of radius 50 mm as phase reference; Microphone 2 scans 145 points (center plus 36 points on each of four circles of radii 10, 20, 30, and 40 mm, with 10° angular spacing). Relative phase at each point is computed as the phase difference between Microphone 2 and the reference Microphone 1, and amplitude is recorded at Microphone 2; linear interpolation visualizes fields. Rotation rates tested include Ω ≈ 139.3, 208.9, and 417.8 rad/s. A roughened disk variant mounts hemispherical bumps (height 1 mm, spacing 7 mm) uniformly on the surface to enhance aeroacoustic dipole energy. Equipment includes B&K 4958-A microphones, a MOTU 16A sound card, and an M+P VibPilot-8 for absolute amplitude calibration when needed. The working frequencies probed are kept below the first disk resonance to avoid vibration-induced sound. Fourier transforms provide amplitude and phase spectra without microphone response compensation or filtering.

Mode purity calculation: The generated AV pressure field p(r, φ, z) is expanded in spiral harmonics exp(i m φ). Coefficients a_m(r, z) are obtained by azimuthal projection; integrating |a_m|^2 over radius yields spiral harmonic power C_m. Mode purity P_m is given by C_m divided by the sum over all modes. Purity within l = −3…+3 is reported.

• AV generation by rotating disk dipole interference: Simulations and experiments show AVs form when the radiation frequency equals l Ω / (2π) for integer l. For Ω ≈ 417.8 rad/s (Ω/2π ≈ 66.5 Hz), AVs with l = 1, 2, 3 are observed at 66.5, 133, and 199.5 Hz, respectively; higher order l = 4 shows weak/blurred features in free-field simulation and is not robust experimentally with a plain disk.
• Field characteristics: Phase wraps by 2π l with a central amplitude null (doughnut profile). As l increases, the near-zero amplitude core expands and peak amplitude decreases, reducing overall plane-integrated radiated energy. First-order AV amplitude decreases with axial distance z, and the core expands (e.g., at z = 10, 25, 50 mm). Sound intensity vectors show a central singularity (zero intensity) and azimuthal power flow.
• Spectral amplitude enhancement: The amplitude-frequency spectrum exhibits distinct peaks only at working frequencies f_working = l Ω / (2π), consistent with AV interference. The peak magnitudes correlate with the fraction of the detection plane having |pressure| near unity in the AV maps and diminish for higher orders.
• Stability–amplitude correlation: AV stability is quantified by the variance of phase differences between adjacent angular measurement points on r = 30 mm. For Ω ≈ 417.8 rad/s, average phase steps are ≈9.8°, 19.7°, 29.4° for l = 1, 2, 3 (reference values 10°, 20°, 30°), with variances 0.79, 2.42, and 8.61 deg^2, respectively. Lower variance (greater stability) coincides with higher field amplitude and stronger spectral peaks. At non-multiples of Ω (e.g., 50/2π, 70/2π Hz), AV phase is absent and no peaks occur.
• Quantitative amplitudes: The first-order AV maximum measured sound pressure amplitude reaches about 0.346 Pa (≈81 dB SPL) at z = 40 mm.
• Tunability across rotation rates: At Ω ≈ 139.3 rad/s, peaks at 22.5, 44.5, 66.5 Hz correspond to l = 1, 2, 3. At Ω ≈ 208.9 rad/s, 33, 66.5, 99.5 Hz correspond to l = 1, 2, 3. At Ω ≈ 417.8 rad/s, 66.5, 133, 199.5 Hz correspond to l = 1, 2, 3. At the same frequency (e.g., 66.5 Hz), different rotation rates yield different l (l = 3, 2, 1 for Ω ≈ 139.3, 208.9, 417.8 rad/s, respectively), demonstrating simultaneous tunability of order and working frequency by adjusting Ω.
• Mode purity: The generated AVs exhibit high mode purity, exceeding 90% for orders within l = −3…+3.
• Roughened disk enhancement: Adding 1 mm-high hemispherical roughness with 7 mm spacing increases aeroacoustic dipole energy, yielding stronger peaks and enabling higher-order AVs up to l = 5 at Ω ≈ 417.8 rad/s. Motor-only rotation noise is much weaker than disk-driven AV peaks, excluding motor noise as the source of the AV features.
• Rotation sensing: Measuring f_working and l allows retrieval of rotation rate via f_working = l Ω / (2π). The sign of l (vortex handedness) encodes rotation direction (e.g., ±Ω ≈ 417.8 rad/s produces opposite-helicity l = ±1 at 66.5 Hz).

The findings validate that distributed aerodynamic dipole sources on a low-Mach rotating disk can interfere to form robust acoustic vortices without any external incident wave or waveguide. This directly addresses the central challenge of achieving both compactness and flexible tunability: the same simple disk source yields multiple AV orders, and by varying Ω, the working frequency and topological charge are jointly and straightforwardly controlled. The amplitude enhancement localized at f_working confirms that AV formation concentrates acoustic energy at specific harmonics of the rotation rate, and the strength of this enhancement links positively to the measured AV stability. Compared with active arrays and passive metasurfaces, the approach removes complex multi-channel control and eliminates fixed-frequency constraints, respectively, while maintaining high mode purity (>90%). The ability to infer rotation rate and direction from AV spectra and chirality offers a practical, contactless sensing modality that avoids misalignment and size-matching issues inherent to methods relying on external incident AVs. Differences between simulation and experiment at higher orders are consistent with turbulence and low-energy vortex susceptibility, as well as modeling approximations in the aeroacoustic pipeline. Overall, the method broadens the toolkit for OAM acoustics with a compact, tunable source and suggests routes to enhanced performance via surface engineering (e.g., roughness).

The work introduces and validates a compact AV generation method based on interference of aerodynamic dipole sources from a rotating disk, achieving tunable topological charge and working frequency governed by f_working = l Ω / (2π). Experiments and simulations show clear AV patterns, spectral amplitude enhancement at working frequencies, high mode purity (>90%), and practical rotation sensing of rate and direction. Surface roughening further increases energy and enables higher-order vortices (up to l = 5). This approach overcomes key trade-offs in existing active and passive methods, offering a simple, incident-free source with flexible tunability. Future research directions include: mitigating free-space diffraction for long-distance applications and multiplexing; strategies to suppress or manage redundant modes; simultaneous multi-order generation at the same frequency; extending operation to underwater environments; and exploiting the link between dipole fields and surface morphology for tailored AV properties and surface identification.

• Higher-order AVs (l ≥ 4) have lower radiated energy and are more susceptible to turbulence and measurement perturbations, reducing robustness in experiments, especially with a plain disk.
• Free-space diffraction limits long-range propagation and may introduce redundant orders in multiplexing scenarios.
• The current method does not achieve simultaneous generation of multiple AV orders at the same frequency.
• Experimental detection uses finite spatial sampling and interpolation, which can introduce phase regression artifacts and limit resolution.
• Numerical modeling involves approximations (e.g., representation as sparse equivalent dipoles, turbulence modeling choices) leading to quantitative differences with measurements.
• Underwater performance remains to be demonstrated, and the influence of different flow regimes (higher Mach) and environmental noise requires further study.