Rotorcraft Hover Performance

Rotorcraft Hover Performance (flight-mechanics/performance/rotorcraft-hover-performance)

Use when you must compute the hovering performance of a rotorcraft rotor with momentum theory: the ideal induced velocity through the rotor disk, the ideal hover power, the profile power from blade solidity and tip speed, the total hover power through an induced-power factor or through the figure of merit, and the disk loading. This is the first rotorcraft vertical-flight member of the flight-mechanics performance pack: it pairs with flight-mechanics/performance/rotorcraft-forward-flight- performance for the speed-dependent power breakdown and with aerodynamics/ground-effects/ground-effect for hover in ground effect. Momentum theory only: uniform inflow, no ground effect, no recirculation. Geometry (radius, solidity, blade drag coefficient, tip speed) is an input; this leaf does not size the rotor.

Domain quick reference

All quantities are SI. The rotor thrust equals the rotorcraft weight in hover: T = m * g0, g0 = 9.80665 m/s^2.

• Disk area: A = PI * R^2.
• Ideal induced velocity (momentum theory, uniform inflow): v_i = sqrt(T / (2 * rho * A)).
• Ideal hover power: P_ideal = T * v_i.
• Profile power (average section drag model): P_profile = (1/8) * rho * sigma * Cd0 * A * Vtip^3, with sigma the rotor solidity, Cd0 the mean blade drag coefficient and Vtip the tip speed.
• Total hover power with the induced power factor k: P_total = k * T * v_i + P_profile, where k scales the ideal induced power for wake and tip losses (default k = 1.15).
• Total hover power from the figure of merit: P_total = P_ideal / FM, FM in (0, 1].
• Figure of merit: FM = P_ideal / P_total (real rotors sit near 0.5-0.7 with losses).
• Disk loading: DL = T / A in Pa.
• Default density rho = 1.225 kg/m^3 is sea level; pass the density at the chosen density altitude to re-evaluate the hover check there.
• FAR 29 frames the rotorcraft certification context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the operating point: rotor radius R, rotorcraft mass m, density rho at the density altitude, solidity sigma, mean blade drag coefficient Cd0, tip speed Vtip and induced power factor k.
2. Get the disk area with disk_area(radius) and the thrust T = m * G0 (hover thrust equals weight).
3. Compute the ideal induced velocity with induced_velocity(thrust, area, rho).
4. Compute the ideal hover power with ideal_power(thrust, v_i).
5. Compute the profile power with profile_power(rho, area, solidity, drag_coefficient, tip_speed).
6. Combine with total_power(ideal_power, v_i, thrust, profile_power, k) or, if the rotor figure of merit is the input instead, with power_from_figure_of_merit(ideal_power, figure_of_merit).
7. Compute the figure of merit with figure_of_merit(ideal_power, total_power) and the disk loading with disk_loading(thrust, area).
8. For a single-call verdict run hover_performance(weight_kg, radius, rho, solidity, drag_coefficient, tip_speed, k), which returns the full dict and propagates every ValueError from the primitives.
9. Confirm the deterministic checks with the contract test scripts/test_rotorcraft_hover_performance.py.

Worked example

A helicopter at 2200 kg mass with a 5.0 m radius rotor at sea level: rho = 1.225 kg/m^3, solidity 0.08, Cd0 = 0.012, tip speed 220 m/s, k = 1.15. Running hover_performance(2200.0, 5.0, 1.225, 0.08, 0.012, 220.0, 1.15):

• Thrust: 2200 * 9.80665 = 21574.63 N.
• Disk area: PI * 5.0^2 = 78.54 m^2.
• Induced velocity: v_i = 10.59 m/s (hand estimate about 10.6 m/s, within the 9.5-11.5 m/s band).
• Ideal power: P_ideal = 228448 W (within 200000-260000 W).
• Profile power: P_profile = 122935 W (within 100000-150000 W).
• Total power: P_total = 1.15 * 21574.63 * 10.589 + 122935 = 385650 W (within 350000-430000 W).
• Figure of merit: FM = 228448 / 385650 = 0.592 (within 0.50-0.70, the typical rotor FOM band with losses).
• Disk loading: DL = 21574.63 / 78.54 = 274.7 Pa (within 260-290 Pa).

At a density altitude where rho drops to 1.06 kg/m^3 the same rotor needs a higher induced velocity (about 11.4 m/s), so the hover check must be re-run at the chosen density.

Verification

• Confirm hover_performance(2200.0, 5.0) with sea-level defaults returns induced_velocity 10.59 m/s, ideal_power_W 228448, profile_power_W 122935, total_power_W 385650, figure_of_merit 0.592 and disk_loading_Pa 274.7.
• Confirm the momentum-theory closed form: induced_velocity equals sqrt(T / (2 * rho * A)) and total power equals k * T * v_i + P_profile for the same inputs.
• Confirm the figure-of-merit round trip: power_from_figure_of_merit(ideal_power, FM) recovers the total power used to compute FM.
• Confirm every non-positive radius, thrust, area, density, solidity, drag coefficient, tip speed and induced power factor, every negative power, and every figure of merit outside (0, 1] raises ValueError, and that figure_of_merit rejects ideal_power > total_power.
• Confirm determinism: repeated runs return identical floats (no RNG, stdlib only).
• Run the contract test offline: python3 scripts/test_rotorcraft_hover_performance.py (33 tests, deterministic).

Pitfalls

• Forgetting to re-run at the density altitude: the defaults are sea level (rho = 1.225 kg/m3); at a lower density the induced velocity rises (about 11.4 m/s at rho = 1.06) and the hover check must be re-evaluated there.
• Using the figure-of-merit and the k-factor paths together: total power is kTv_i + P_profile with k = 1.15 default, or P_ideal/FM; mixing both conventions double counts the induced losses.
• Reading the ideal power as the total: P_ideal = T*v_i excludes profile power and wake/tip losses, so it is always below the real hover power; the total sits near 385.7 kW on the worked rotor.
• Passing an FM outside (0, 1] or ideal power above total: figure_of_merit rejects ideal_power > total_power and FM outside (0, 1] raises ValueError.
• Disk loading units: DL = T/A comes out in Pa (about 274.7 Pa on the worked rotor), not kgf/m2.

Related leaves

• flight-mechanics/performance/rotorcraft-forward-flight-performance: the sibling rotor leaf for the cruise power split (induced, profile and parasite power versus airspeed) that extends this hover model.
• aerodynamics/ground-effects/ground-effect: hover in ground effect, which this momentum-theory model deliberately excludes.
• flight-mechanics/performance/oei-climb-gradient: the one-engine- inoperative climb verdict for multi-engine rotorcraft, adjacent to the hover power check in a rotorcraft performance pass.
• flight-mechanics/performance/climb-performance and flight-mechanics/performance/glide-performance: fixed-wing vertical performance leaves; this leaf owns the rotorcraft hover case only.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_rotorcraft_hover_performance.py

The test covers the worked example with the spec magnitude bounds (induced velocity 9.5-11.5 m/s, ideal power 200000-260000 W, profile power 100000-150000 W, total power 350000-430000 W, figure of merit 0.50-0.70, disk loading 260-290 Pa), the momentum-theory closed form, the induced power factor model and its default constant, the cubic tip speed scaling of profile power, the figure-of-merit round trip, exact dict keys and primitive consistency of hover_performance, run-to-run determinism, absence of random or external imports, and ValueError rejection of every non-physical input in the validation list.

Compliance

• Standards referenced, not reproduced: FAR 29 (rotorcraft airworthiness, certification context only). The momentum-theory hover relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.