eVTOL Rotor Wake Interaction and Thrust Loss

Each rotor solves the positive powered-flow branch of Glauert's oblique actuator-disk equations:

The external velocities are induced by the other rotors. Bisection closes the selected fixed-power or held-thrust comparison. A safeguarded Broyden iteration couples those velocities to every rotor's wake unknowns.

The wake is a conservative, circular-section vortex-ring sheet with prescribed Glauert skew. Its pressure jump is $\Delta p=T/A$ and Kelvin shedding gives $\Delta\Gamma=(\Delta p/\rho)\Delta t$. On an interior axial panel, $\Gamma_j(W+\overline{u}_{z,\mathrm{sheet},j})=(\Delta p/\rho)\Delta z_j$; this determines convection time from the calculated sheet velocity, rather than a prescribed acceleration curve. Ring centers convect laterally with the local in-plane ambient flow. The circular section is an approximation to a general deformed wake.

Unknown section radii are solved so that $\int_{A_w}(W+u_z)\,dA=A(W+v_i)$ at every panel boundary, including the disk. The actuator-rim and tail-matching cells use these integral flux constraints to close their circulations; interior cells use the pressure/shedding relation. Four Gauss rings integrate each panel for a pair; a quad uses three to reduce browser cost. No common circulation multiplier is used. The ideal far boundary is $r_\infty/R=\sqrt{(W+v_i)/(W+2v_i)}$. A straight semi-infinite ring cylinder continues the wake beyond 4R; its circulation per axial length is $(\Delta p/\rho)/(W+v_i)$. In hover this is $2v_i$, and the far velocity approaches $2v_i$ while flux stays equal to the disk flux. The tail's axial integral is analytical, with 32-point circumferential quadrature and the exact infinite-cylinder complement in axial flow.

Biot–Savart ring velocity and vector potential use complete elliptic integrals, evaluated by the arithmetic–geometric mean. Finite panels have a regularized core; the ideal far tail is unregularized. Stokes' theorem converts disk flux to a vector-potential boundary integral. Wake-section integrals use symmetry and eight-point Gauss angular quadrature; receiving disks use 32 or 64 boundary azimuths. Equal-area samples provide horizontal means, RMS and harmonics. Quadrature directions rotate with the layout in hover. Animation uses the solved near-wake geometry and convection times.

The pressure/force-free-sheet formulation follows Bontempo and Manna (2019); the reduced finite-volume boundary closure and prescribed-skew extension are implemented here. Leishman (2nd ed., 2006) and NASA NDARC theory supply the momentum limits. This implementation is not a reproduction of Bontempo's high-resolution axisymmetric solver or Guner et al.'s CMTSVT code.

Identical parallel horizontal actuator disks, uniform loading, steady incompressible inviscid flow and no ground or airframe. Positive in-plane airflow is along +x; axial climb flow and rear-row drop are positive downward. Wake sections remain circular and their in-plane convection uses the local ambient flow; wake roll-up and general free-wake deformation are excluded. Rotor speed sets μ = U/(ΩR), axial propeller J = W/(nD) and tip Mach; it does not supply a blade map. Pitch adjustment is implicit. Profile power is constant within each selected comparison. No empirical induced-power correction or measured loss multiplier is applied.

Use for preliminary layout comparisons in hover, climb and low advance ratio (μ ≤ 0.4), with advancing-tip Mach below 0.8. Descent, recirculation, vortex-ring state and windmill operation are excluded. Prescribed skew is an oblique-flow approximation; only the circular-section sheet boundary and circulation are solved. Blade passage, viscous mixing, stall, swirl recovery, acoustics, wing/fuselage download and ground effect require other models. A predicted upwash benefit is not a guaranteed aircraft gain.

For a pair, Standard uses three axial panels/12 finite Gauss rings, 16 area samples and 32 receiving-disk boundary azimuths; Refined uses four panels/16 rings, 64 area samples and 64 boundary azimuths. A quad uses three panels/nine rings: Standard has 16 area samples/32 boundary azimuths, and Refined has 32 area samples/40 boundary azimuths. Refined area sampling uses 32 azimuths in both layouts. This smaller quad grid keeps browser time practical. Panel boundaries cluster cubically near the disk, and a semi-infinite tail replaces wake truncation. The finite-volume rim/tail closure and interpolation between the few section nodes limit near-wake accuracy; flux is enforced at nodes, not at every point between them. For the default pair, Standard and Refined differ by less than 0.1 percentage point of thrust loss. In the default quad geometry, increasing panel quadrature from three to four Gauss rings changed loss by about 1.2 percentage points; the smaller quad grid cannot resolve differences of that size reliably. This is not an accuracy bound for sharper wake edges. Vary core radius and resolution before comparing small benefits.

Sensitivity sweeps run on request. They use the Standard grid, with the accurately solved current point retained at the selected resolution. Standard pair scans use up to five samples; Refined or quad scans use three to limit run time. Pair spacing samples bracket estimated wake ingestion; the advance-ratio scan is local to the current speed. Quad scans are narrower local comparisons (±0.03 in s/D or ±0.01 in μ). Markers are solved samples and connecting segments are linear; narrow peaks between samples can be missed. The isolated thrust input stays fixed across an advance-ratio scan, and isolated power is recalculated at each speed; this is a relative interference comparison, not a constant-motor-power flight sweep. Nonconverged samples are gaps. Inspect a candidate directly with Refined resolution. The animation follows steady convection, not a startup transient.

Coaxial test comparisons allow a 6% retention difference because the tests have torque-balanced fixed-pitch rotors and profile/swirl losses, while this tool adjusts disk loading at fixed power. The NASA far-spacing check tests recovery toward isolated behavior. The near-separated tandem case allows a 4% retention difference without resolving the sign of its small measured loss. The side-by-side case converts fixed-rpm thrust/torque ratios using local hover speed similarity, with a 3% allowance. These are limited benchmarks, not direct fixed-power validation. No coefficient is fitted to the data. Aircraft margins require measured maps, uncertainty allowances and flight-dynamics assessment.