Coaxial Rotor Calculator: Spacing, Interference and Efficiency

The hover calculation follows the axial-inflow BEMT of Leishman and Ananthan (2008), with a finite-spacing prescribed wake. Each rotor is divided into radial annuli. With all radii and velocities normalized by the upper radius and upper tip speed, the annular balance is

Here $\lambda_c$ is incoming flow from the other rotor, $\lambda$ is total axial velocity and $F$ is the iterated Prandtl tip-loss factor. The positive-flow quadratic is solved in every annulus; negative local lift is retained only while total axial throughflow remains positive. A real quadratic root alone does not establish that the hover branch is valid. Profile drag is $C_d=C_{d0}+k_d C_l^2$. Shaft power integrates $\lambda\,dC_T$ and profile drag separately.

Wake and mutual inflow: Thiele, Obster and Hornung (2019), equations 19 and 23–30, supply the upstream axial influence and Landgrebe wake-age/contraction law. The upper wake approaches $R_u/\sqrt2$. Continuity accelerates the mapped upper induced velocity by $1/(R_w/R_u)^2$. Integration on the lower rotor is split exactly at the wake edge; its outer annuli ingest ambient air. The lower rotor's area-averaged self-induced velocity feeds back to the upper rotor. This uses axial coupling; the 2019 paper's additional swirl model is not included.

A damped finite-difference Newton iteration adjusts collective pitch to hold the requested total thrust, and, for balanced operation, makes upper and lower aerodynamic torque magnitudes equal. Both rotors run at the same speed. The design search repeats that trim at each spacing/diameter pair and excludes stalled, reversed-throughflow or unconverged solutions. Sign changes of the combined constraint margin trigger additional boundary solves in diameter and at the spacing/diameter domain intersections; the map interpolates the continuous margin instead of requiring neighboring grid nodes to be feasible.

Reported metrics: footprint FM is $T^{3/2}/[\sqrt{2\rho A_u}P]$. Thrust-sharing merit uses $[T_u^{3/2}/\sqrt{2\rho A_u}+T_l^{3/2}/\sqrt{2\rho A_l}]/P$. The interference factor divides coaxial induced power by two isolated BEMT rotors at the same individual thrusts and blade geometries. It includes mutual induction but excludes profile power. The separate ideal-disk factor also includes nonuniform inflow losses. Syal's 2008 thesis, Table 2.1, gives independent ideal momentum reference values; their normalization is stated in each check.

Merit normalization: footprint FM is a single-disk reference applied to a two-rotor system. It can exceed 1 and is not a thermodynamic efficiency. The extra effective rotor area changes that comparison, as discussed by Giovanetti (2015), printed p. 52, equations 3.21–3.22. Thrust-sharing merit instead uses each rotor's own thrust and area; its isolated-disk ideal-power datum is the one used for the efficiency-bound check.

• Steady, out-of-ground-effect hover in still, incompressible air; two concentric, counter-rotating rotors at one common speed.
• Fixed upper diameter fixes the footprint. The lower diameter is at most the upper diameter; changing its diameter preserves the entered absolute chord ratio, blade count, taper and normalized root cutout.
• Collective is measured at 75% of each rotor's own radius. Linear twist is the pitch change over one full radius, extrapolated to the shaft; positive offset means more collective on the lower rotor.
• One linear lift slope and quadratic drag polar represent both rotors. The lift limit is an applicability limit, not a post-stall correction. Generic polar defaults are engineering starting values, not measured properties of a named airfoil.
• The lower rotor influences the entire upper disk through a uniform area-averaged axial inflow. Wake contraction is prescribed, not solved as a free-vortex wake. No empirical induced-power multiplier is fitted to Harrington's data.

Use for comparative hover sizing with attached flow and subsonic tips. The small-inflow-angle, independent-annulus BEMT omits explicit swirl recovery, radial mixing, blade-vortex encounter loads, compressibility corrections, Reynolds-number-dependent polars, flapping, elastic deflection and installation blockage. It does not predict forward flight, descent/vortex-ring state, acoustic levels, motor/ESC losses, yaw-control reserves or blade-clearance margins. Validate the actual propellers on a thrust/torque stand before flight sizing.

The selected and winning sampled designs use 128 annuli per rotor. To keep browser values near three seconds, the search uses 7 × 7 geometry points and 64 annuli per rotor, with the lower grid split at the wake edge. Candidates are ranked at search resolution, then checked at 128 annuli; if a candidate fails, the next ranked candidate is checked before a winner is reported. Adjacent diameter stations with opposite constraint-margin signs receive additional safeguarded secant solves. The same refinement locates crossings at the minimum and maximum diameter boundaries and adds solved spacing columns there. Refinement stops at a normalized margin of 0.0001 or a geometry-ratio bracket of 0.0001, with at most five iterations. The margin combines section lift, total throughflow, real quadratic roots, thrust/power signs and trim convergence. This is a bounded sampled search, not a continuous or global blade-shape optimum; small interior optima can fall between stations. The 51 × 37 map and 121-point curves interpolate both merit and signed constraint margins with PCHIP, then mask the negative-margin side. Thus the displayed feasible envelope is an interpolation of refined boundary solves, not a guarantee at every unsolved point; confirm a design near that envelope with its direct 128-annulus solution. A gray contour marks the interpolated feasibility boundary. The shared color range and contour levels adapt to the finite displayed merits, including values above 1. The open circle marks the selected geometry when it lies within the search bounds, even though the map always shows torque-balanced pitch.

Harrington HR1 is an approximate performance comparison: the reconstructed straight taper and generic drag-rise coefficient do not reproduce its spanwise thickness-dependent polar. Its check permits an absolute FM difference of 0.08 (about 13% of the measured 0.635); it is not a calibration or a claim of test-level accuracy. Ideal momentum checks are much tighter. Numerical residuals verify trim and annular momentum consistency, not the physical accuracy of the prescribed wake.