Rotorcraft Turn Performance

Rotorcraft Turn Performance (flight-mechanics/performance/rotorcraft-turn-performance)

Use when the task is the banked-turn performance of a helicopter rotor at load factors above one: the uniform-inflow momentum theory of a steady coordinated level turn, where the disk banks so the thrust axis carries the resultant n times the weight, the turning induced velocity at that raised thrust, the turn power breakdown into induced, profile and parasite terms, and the sustained-load-factor an available power supports at the turn speed. This leaf implements the momentum-theory solve in pure Python, stdlib only, with the disk-plane free-stream convention of the level-flight Glauert model. It pairs with flight-mechanics/performance/rotorcraft-forward-flight-performance, which supplies the n = 1 power conventions and worked rotor this leaf reuses, and with flight-mechanics/performance/rotorcraft-hover- performance, whose v_h reference is the V = 0 identity of the turning inflow.

Domain quick reference

• Turn thrust: the rotor thrust in the banked level turn is the load factor times the weight, T = n * W with W = m * g0. At n = 1 the disk carries the level-flight weight; above one the rotor carries more than the aircraft weight.
• Turning inflow (momentum theory, uniform inflow): the generalized induced velocity v_i of the turn solves n * W = 2 * rho * A * v_i * sqrt(V2 + v_i2), v_i >= 0, with A = pi * R**2 the disk area. The left side is strictly increasing on v_i >= 0, so the root is unique; a FIXED-COUNT bisection on [0.0, sqrt(n * W / (2 * rho * A))] (BISECT_ITER = 120 iterations, no tolerance early exit) returns it. At n = 1 the value reproduces the level-flight Glauert inflow; at V = 0 it is sqrt(n * W / (2 * rho * A)) = sqrt(n) * v_h.
• Induced power: P_i = k * n * W * v_i with the induced power factor k = K_DEFAULT = 1.15, the hover and forward-flight convention.
• Profile power: P_prof = (1/8) * rho * sigma * Cd0 * A * V_tip**3, the average-section-drag model shared with the hover and forward- flight leaves; the turn keeps the rotor speed fixed, so the profile power of the turn equals the level-flight value.
• Parasite power: P_par = 0.5 * rho * V3 * f with f the equivalent flat-plate drag area; the V3 term makes the parasite power grow fast with the turn speed.
• Total turn power: P_total = k * n * W * v_i + P_prof + P_par, strictly increasing in the load factor because both n * W and v_i rise with n.
• Sustained maneuver: the power-sustained-load-factor n_s is the largest n whose turn power the available power covers, found by a FIXED-COUNT bisection on [1.0, N_CEILING]; bank angle phi = acos(1 / n_s), turn rate omega = G0 * sqrt(n2 - 1) / V and turn radius R_t = V2 / (G0 * sqrt(n**2 - 1)) close the maneuver with G0 = 9.80665 m/s2.
• Units are SI throughout: N, W, m/s, m2, kg/m3, rad. Angles are in rad; radians convert to degrees by 180 / pi.
• 14 CFR Part 29 (far-29) frames the rotorcraft context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the turn state and the rotor: load factor n >= 1, true airspeed V, rotor radius R (disk area A = pi * R**2), rotorcraft mass m (weight W = m * g0), density rho, solidity sigma, blade drag coefficient Cd0, tip speed, flat-plate drag area f and induced power factor k. Record the hover reference v_h = sqrt(W / (2 * rho
   • A)).
2. Raise the thrust to the load factor: thrust_for_turn returns T = n * W and rejects load factors below 1 with ValueError.
3. Solve the turning-flight-inflow: generalized_induced_velocity bisects on [0.0, sqrt(n * W / (2 * rho * A))] for v_i at the n-times-weight thrust. Check the identities: speed zero returns sqrt(n) * v_h, and at fixed n the inflow falls as the speed grows.
4. Break down the banked-turn-power: turn_power returns the dict {load_factor, thrust, induced_velocity, induced_power, profile_power, parasite_power, total_power} through induced_power, profile_power and parasite_power in the fixed order above. Confirm the n = 1 total equals the level-flight total of the forward-flight leaf at the same speed.
5. Invert for the power-sustained-load-factor: sustained_load_factor bisects h(n) = total power at n minus available power on [1.0, N_CEILING] and returns the sustained-load-factor with its bank_angle, the turning induced velocity, the power terms and a note of "power-limited" or "power-excess above ceiling". Get the same bank angle in one solve with max_bank_from_power.
6. Close the maneuver kinematics: bank_from_load_factor gives the power-limited-bank-angle acos(1 / n); turn_rate and turn_radius give omega and R_t at the sustained point. Verify omega * R_t = V and cos(bank) = 1 / n.
7. Confirm determinism: repeated calls return bit-identical values and the note strings are fixed, then run the contract test scripts/test_rotorcraft_turn_performance.py.

Worked example

Shared worked rotor (identical to the hover and forward-flight worked rotors): R = 5.0 m (A = 78.5398 m2), m = 2200 kg so W = m * g0 = 21574.63 N, rho = 1.225 kg/m3, solidity sigma = 0.08, Cd0 = 0.012, tip speed 220 m/s, f = 2.2 m2, k = 1.15. Sea level.

• Hover reference at V = 0: v_h = sqrt(W / (2 * rho * A)) = 10.5887 m/s at n = 1; at n = 2 the turn inflow is sqrt(2) * v_h = 14.9747 m/s.
• Banked turn at n = 2.0, V = 60 m/s (module outputs): thrust T = 43149.3 N, turning induced velocity 3.73017 m/s, induced power 185097 W, profile power 122935 W, parasite power 291060 W, total turn power 599092 W; bank angle acos(1/2) = 1.0472 rad (60 deg), turn rate 0.283094 rad/s, turn radius 211.944 m. The induced velocity at n = 2 is about twice the level-flight value 1.86778 m/s at the same speed, and the level-flight total power of 460336 W (the n = 1 identity, matching the forward-flight leaf worked example) grows by about 138 kW to sustain the 2 g turn at 60 m/s.
• Second state, n = 1.5 at V = 40 m/s: induced velocity 4.18175 m/s, induced power 155629 W, total 364804 W. Third inflow probe, n = 3.0 at V = 60 m/s: induced velocity 5.58195 m/s.
• Power-sustained maneuver: available power 600000 W at V = 60 m/s gives sustained-load-factor 2.00491, bank angle 1.04861 rad (60.081 deg), turning induced velocity 3.73929 m/s, induced power 186005 W, total power at the sustained point 600000 W, turn rate 0.28402 rad/s and turn radius 211.253 m; max_bank_from_power returns 1.04861 rad.
• With available power 450000 W at V = 40 m/s the sustained-load- factor falls to 1.86867 (rate 0.387015 rad/s, radius 103.355 m), and at the same power and V = 50 m/s it falls further to 1.69094 (rate 0.267439 rad/s, radius 186.959 m): the parasite V**3 growth cuts the sustained-load-factor as the speed rises at fixed power.

Verification

• Confirm generalized_induced_velocity(2.0, 21574.63, 78.5398, 1.225, 60.0) returns 3.73017 m/s and that the n = 1, V = 0 and n = 2, V = 0 calls return 10.5887 and 14.9747 m/s (sqrt(2) times v_h).
• Confirm turn_power at n = 2, V = 60 m/s totals 599092 W and that the n = 1 total at 60 m/s is 460336 W, equal to the forward-flight leaf worked total at that speed.
• Confirm sustained_load_factor(600000.0, 21574.63, 78.5398, 1.225, 60.0, 0.08, 0.012, 220.0, 2.2) returns load_factor 2.00491 with note "power-limited" and that turn_power at the returned load factor reproduces the available power (round trip).
• Confirm every non-physical input raises ValueError: load factor below 1, non-positive weight, area, rho, solidity, drag coefficient, tip speed, k or available power, negative speed, a ceiling at or below 1, and available power below the n = 1 total at the speed (level flight not sustainable).
• Run the contract test offline: python3 scripts/test_rotorcraft_turn_performance.py (34 tests, deterministic).

Related leaves

• flight-mechanics/performance/rotorcraft-forward-flight-performance: the level-flight Glauert inflow and power conventions this leaf reuses at n = 1; its worked rotor and total power are identical.
• flight-mechanics/performance/rotorcraft-hover-performance: the hover v_h reference and V = 0 limit of the turning inflow.
• flight-mechanics/performance/turn-performance: the fixed-wing level-turn kinematics (omega, R_t, load factor from bank) for aircraft without a rotor inflow model.
• flight-mechanics/performance/rotorcraft-vertical-climb-performance: the bracket convention of the "power-excess above ceiling" note and its climb power solves.
• flight-mechanics/performance/rotorcraft-range-endurance: fuel closure on the level power curve, downstream of the turn power estimate.

Pitfalls

• Reusing the level-flight induced velocity in the turn: at n = 2 the turning inflow is 3.73017 m/s against 1.86778 m/s level at 60 m/s, so the power must be built on the n-times-weight thrust or the turn is understated by a factor near n in the induced term.
• Forgetting the induced power factor: the induced term is k * n * W * v_i with k = 1.15, not the ideal n * W * v_i; dropping k understates P_i by 15 percent (185097 W against 160954 W at n = 2, 60 m/s).
• Feeding n into the profile or parasite power: the turn keeps the rotor speed fixed and the free stream in the disk plane, so profile_power and parasite_power do not depend on n; only the induced term carries the load factor.
• Reading the excess case as a bisection result: when the total power at n = N_CEILING stays below the available power, the sustained solve returns load_factor = N_CEILING with note "power-excess above ceiling"; the bracket convention does not extrapolate beyond the ceiling.
• Ignoring the level-flight floor: an available power below the n = 1 total at the turn speed (460336 W at 60 m/s on the worked rotor) raises ValueError, because no banked turn exists at a speed whose level flight the power cannot sustain.
• Writing the compound terms with spaces: sustained-load-factor and power-limited-bank-angle are the leaf's routing tokens; keep them hyphenated in the description so they do not tokenize into generic words owned by the fixed-wing turn leaf.

Contract test

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

python3 scripts/test_rotorcraft_turn_performance.py

It covers the turning-flight-inflow anchors at n = 1, 1.5, 2 and 3 against the level-flight Glauert and sqrt(n) * v_h identities, the banked-turn-power breakdown anchors (induced 185097 W, profile 122935 W, parasite 291060 W, total 599092 W at n = 2, 60 m/s), the n = 1 total power identity with the forward-flight leaf, the monotone power and inflow behaviors, the sustained-load-factor solves with their power round trips and fixed note strings, the kinematics anchors with the omega * R_t = V and cos(bank) = 1 / n identities, determinism of repeated calls, and ValueError rejection of every non-physical input class in the spec validation list.

Behavior contract (gate 3)

The leaf is deterministic: pure stdlib math, FIXED-COUNT bisection schedules (BISECT_ITER = 120), no RNG, no network, no external processes. The contract test above must pass in full before the leaf is rated; its 34 test methods exercise every workflow step of this SKILL.md and every validation item of the wave-41 spec.

Compliance

• Standards referenced, not reproduced: 14 CFR Part 29 (far-29) is US government work in the public domain; the momentum-theory relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.