eVTOL Rotor Noise Calculator: BVI and Ground dBA Contours

This is an isolated-rotor compact-strip calculation derived from the monopole/dipole acoustic analogy described by Brentner and Farassat (2003, attached pp. 20–23). Rotating spanwise phase is retained; a full blade-surface FW–H solution is outside its scope.

1. Operating point and coordinates. Ideal-gas density, Sutherland viscosity and sound speed use the supplied atmosphere. Bisection solves $v_i\sqrt{U_d^2+(v_i-W_n)^2}=T/(2\rho A)$. Here $U_d=U\cos\beta-V_d\sin\beta$ and $W_n=U\sin\beta+V_d\cos\beta$. A blade lies at $r(\cos\psi,\sin\psi,0)$ and rotates toward increasing $\psi$. Its tangential airspeed is $U_t=\Omega r-U_d\sin\psi$, consistently with wake convection along $-U_d$.
2. Tip wake and BVI. Each blade sheds a helical filament convected by $(-U_d,0,W_n-v_i)$. Straight-segment Biot–Savart velocities use the finite-core denominator $|d\ell\times r|^2+a^2|d\ell|^2$. Tip circulation follows the mean thrust distribution. Three strips from 0.2R to R receive $dL=\pi\rho cU_t w\,dr$; their cycle means are removed. The outer strip's minimum segment distance is reported. A centred frozen-gust chord average supplies the harmonic sinc factor. The local check reconstructs $\alpha=\alpha_0+\Delta L/(\pi\rho c\,dr\,U_t^2)$ before chord averaging and also checks the mean-removed vortex gust relative to $U_t$.
3. Rotating acoustics. Mean loading varies with $U_t^2$, normalized to thrust and torque. Blade volume follows the integrated NACA four-digit thickness profile, $dV=0.685(t/c)c^2dr$. Eight blade-passing harmonics use phase $\exp[-ihB\psi+ik\hat n\cdot y]$. Loading coefficients are $i\omega B\langle F\cdot\hat n\exp(\cdot)\rangle/(4\pi c_0)$; thickness coefficients are $-\rho\omega^2B\langle dV\exp(\cdot)\rangle/(4\pi)$. BVI adds another loading coefficient. Complex pressures add before taking $2|\hat p|^2$.
4. Broadband. Brooks–Pope–Marcolini NASA RP-1218, attached pp. 67–70, 113–115 and 139–140: pressure-side, suction-side and separation contributions with Reynolds-dependent A/B shapes, K1/K2 amplitudes and displacement thicknesses. Every strip uses its own Mach number. The empirical stalled-flow switch selects $D_l$ from Eq. (B2); other strips use $D_h$ from Eq. (B1). Closed-form azimuth integrals of those directivities are applied before the strip energies are added.
5. Directional cache and propagation. For normal cosine $n_z$ and radial projection $s=\sqrt{1-n_z^2}$, write the pressure as $C(s,\varphi)+n_zA(s,\varphi)$. The cache interpolates $2|C|^2$, $4\Re(CA^*)$ and $2|A|^2$, then reconstructs their quadratic in $n_z$. This preserves coherence and the exact BVI dipole null; a normal-force energy is never interpolated as proportional to $|n_z|$. Observer levels and the observer track use direct source quadrature. For maps, a 17 × 32 table resolves radial projection and azimuth. The eight tone coefficients are absorption-weighted and A-weighted at zero range plus nine logarithmic ranges from 5 to 5,000 m; the 24 broadband bands use a separate 65-range cache. Linear interpolation of these energy coefficients in range avoids repeating every spectral sum at every map node; the analytic normal-force quadratic and strip broadband directivities are retained. Maps use the cached coefficients, spherical energy spreading $N/r^2$, Bass–Sutherland absorption $10^{-\alpha(f)r/10}$ and tone/band A-weighting. Reference pressure is 20 µPa. ISO 9613-1 and IEC 61672 are named for context.

6. Adaptive ground sampling. A 25 × 17 outer mesh includes nine central stations on each axis. A 21 × 21 mesh around the vertiport is solved separately at the final nine approach positions; earlier positions are projected from the outer mesh. Both meshes retain all three source channels. Their maxima are combined before plotting. Four outer display tiles and a 63 × 63 central tile use cached cubic-spline operators, so the dense local picture remains inexpensive during playback.

Rotors are identical, co-located and mutually incoherent; within one rotor, loading, thickness and BVI retain their relative phase. Thrust, torque, speed, disk tilt and descent rate remain constant along the straight segment. The rectangular blades are rigid, with a NACA-like thickness profile, prescribed circulation and a constant vortex core. The wake is neither relaxed nor aeroelastically trimmed.

The airload model uses linear lift and small vortex gusts. Operating points are rejected when a sampled strip exceeds ±10° or the unsteady gust exceeds 10% of local tangential speed. No load clipping or nonlinear stall response is used to make an operating point pass. The BPM separated-flow spectral branch remains implemented with its appropriate low-frequency directivity; it does not supply a nonlinear BVI lift model.

Broadband boundary layers are treated as tripped, with representative strip speed and mean loading-derived angle. Laminar instability, blunt-edge shedding, tip-formation broadband noise and ingested turbulence are excluded. Dry-air thermodynamic properties are used; relative humidity affects absorption. Numerical matrices contain SI-scaled coefficients, loads and pressure energies.

An estimate for subsonic, small-disturbance rotor operation and acoustic far-field receivers. Installation and shielding, rotor–rotor wake coupling, buildings, terrain, background noise, detailed ground reflection, meteorological refraction, translational Doppler and propagation delay are excluded. No blanket ground correction is added.

Resolution is three blade strips, 16B + 16 azimuth samples (64 at the defaults), 32 wake segments per revolution, eight blade-passing harmonics and a 17 × 32 directional coefficient cache. The instantaneous observer and the 33-position observer track retain direct source quadrature and per-frequency absorption. The analytical normal-force factor preserves the BVI dipole null.

The ground calculation uses 25 × 17 outer nodes plus a 21 × 21 local patch. Its half spans are twice the final rotor-to-receiver height, capped at 120 m along track and 80% of map half width across track. The patch is solved directly during the final quarter of the 33-position approach; earlier patch fields are interpolated from the outer grid. This allocation keeps finer angular and near-vertiport resolution within the browser budget. Outer regions and earlier positions have coarser spatial resolution and can miss narrow local lobes. Playback linearly interpolates dBA between the same 33 solved approach positions.

Maps are drawn as five cubic-spline tiles, including a 63 × 63 central picture. Display interpolation is bounded to each solved field's range. It adds no physical samples and does not improve the reported sampled maximum. Observer curves use 129 PCHIP picture points. BVI energy above the eighth harmonic is omitted. Results are emission-position RMS levels and maxima, rather than a sound-level meter's Fast/Slow response.

At 12 default-case probes, six solved nodes and six between-node locations, all three channels are compared against direct quadrature. The final-position picture is checked at all 12 probes; the maximum envelope over all 33 approach positions is checked at two nodal and two between-node probes. Maximum absolute discrepancies in the tested case were 0.221 dB for the directional cache at nodes, 0.220 dB for final pictures at nodes, 0.226 dB for final pictures between nodes and 0.060 dB for the maximum-picture probes. At (0 m, 18 m) at the approach end, the pictured total and BVI errors are 0.090 dB and 0.197 dB. These are numerical comparisons at the specified probes, not a universal error bound or aircraft acoustic validation. To keep the refined angular and spatial calculation near the browser runtime target, tonal absorption is interpolated on nine logarithmic ranges, while broadband retains 65 ranges. This range-grid choice costs up to 0.0203 dB against exact per-frequency absorption at the four default-spectrum validation ranges; spherical spreading remains exact.

Refining the unchanged wake from 64 azimuth samples/48 total segments to 192/144 previously changed default lift RMS by 0.81%, observer total level by 0.56 dB and isolated BVI level by 0.68 dB. Absolute BVI levels remain sensitive to wake position, circulation and core radius; no aircraft-specific measurement calibration is supplied.

Both spectra retain discrete bars. The harmonic display omits components more than 50 dB below the strongest combined tone; calculated pressure energies remain intact. The broadband display retains all 24 bands with six rounded frequency labels. The tool does not provide flight guidance, community-annoyance or certification predictions; check against the standard in force for formal noise assessment.