Rotorcraft Performance: Method

Rotorcraft Performance

What the tool computes

The power a helicopter or a multirotor needs in hover, in vertical flight and in forward flight, by momentum theory for the induced power and a mean blade drag coefficient for the profile power, and what follows from comparing it with the power or the battery energy on board: ceilings, speeds, rates of climb, endurance, range and the rate of descent in autorotation. The air is the 1976 U.S. Standard Atmosphere at the entered pressure altitude, with an optional temperature offset. The symbols: T thrust, A = πR² disk area, ΩR tip speed, σ solidity, ρ air density.

Hover

Momentum theory gives the induced velocity of a hovering rotor, v_h = √(T/(2ρA)), and the ideal power T v_h. A real rotor needs

P = κ T v_h + ρ A (ΩR)³ σ c_d0/8.

The induced-power factor κ covers nonuniform inflow, tip loss and swirl; the second term is the profile power of blades with a mean drag coefficient c_d0. The figure of merit is the ideal power over the rotor power, FM = T v_h/P. The thrust coefficient is C_T = T/(ρA(ΩR)²); the blade loading C_T/σ and the mean lift coefficient 6 C_T/σ show how hard the blades work, and a mean lift coefficient near the section's maximum means the single drag coefficient no longer holds. The thrust is the weight plus the download, the vertical drag of the airframe in the rotor wake, entered as a fraction of the weight; the tool fades it with the square of v/v_h away from hover (its own simple rule).

Tail rotor, transmission and accessories

The tail rotor of a single-rotor helicopter supplies the thrust T_tr = Q/l that balances the main-rotor torque Q = P/Ω at the arm l. Its power is computed with the same two terms, with its own radius, tip speed, solidity, κ and c_d0, the flight speed lying in its disk plane; fin blockage and the unloading by a fin in cruise are not modelled. A fraction of the main-rotor power can be entered instead, or none for coaxial, tandem and tip-driven rotors. The shaft power at the engine is (main rotors + tail rotor)/η_m plus the accessory power. Several lifting rotors share the weight equally and are treated as isolated: overlap and interference, which raise the induced power of tandem and coaxial rotors, are not included; raise κ to allow for them.

Ground effect

Cheeseman and Bennett represent the ground by an image of the rotor. At constant power the thrust rises by

T/T_∞ = 1 / (1 − (R/4z)² / (1 + (μ/λ)²)),

z the height of the rotor above the ground, μ the advance ratio and λ the inflow ratio; in hover the last bracket is 1. Following NASA/TP-2015-218751 the tool applies the equivalent factor f_g = (T/T_∞)^−3/2 to the induced velocity at constant thrust, uses the relation no lower than z = 0.3 R and applies none above z = 3 R. The hover ceiling is the altitude at which the hover power equals the power available; the tool finds it out of ground effect and at the entered rotor height.

Power available

P_available = P_installed (ρ/ρ₀)^m, not above the transmission limit when one is entered. The exponent m fits the engine: 1 makes the power proportional to the density, 0 holds it constant. The temperature offset acts through the density.

Vertical climb and descent

In axial flight momentum theory gives v (V_c + v) = v_h² in climb and slow descent, so v/v_h = −V_c/2v_h + √((V_c/2v_h)² + 1), and v (V_c + v) = −v_h² in the windmill-brake state, descent faster than 2 v_h. Between them, in the vortex-ring and turbulent-wake states, no slipstream exists and momentum theory fails; the tool uses the approximation of NASA/TP-2015-218751,

(V_c + v)/v_h = (V_c/v_h) [0.373 (V_c/v_h)² − 0.991 + 0.598 (V_x/v_h)²] inside 1.5 (V_x/v_h)² + (2 V_c/v_h + 3)² < 1,

which joins the two momentum branches at V_c = −v_h and −2 v_h. The rotor power is T (V_c + κ v) plus the profile power. The page flags a descent inside the region where Drees found rough, unsteady flight, V_c/v_h between −0.62 and −1.53 at forward speeds below v_h (NASA/TP-2005-213477); the outline the tool draws between those three numbers is a half-ellipse of its own. In that region the thrust fluctuates and the power is only a mean value.

Forward flight

In steady flight at speed V along a path climbing at the angle γ the rotor thrust balances the weight W, the download and the parasite drag D = f ρV²/2, f the equivalent flat-plate area: T is the vector sum, tilted forward by tan⁻¹(D cos γ/(W + D sin γ)). With V_n the component of the flight speed through the disk and V_x the component in its plane, Glauert's momentum equation

v = v_h² / √(V_x² + (V_n + v)²)

is a quartic in v. The tool brackets each of its roots between the turning points of the quartic and takes the one on the branch that continues the hover solution (the windmill-brake branch in steep, fast descents). The power of the rotors is

P = κ T v + T V_n + ρ A (ΩR)³ (σ c_d0/8) F_P, T V_n = D V + W V_c,

induced, parasite, climb and profile power. The profile-power function is the mean cube of the blade-element velocity,

F_P(μ, μ_z) = 4 (1/2π) ∫∫ [(r + μ sin ψ)² + (μ cos ψ)² + μ_z²]^3/2 dr dψ,

integrated over the radius and the azimuth (NASA/TP-2015-218751), μ = V_x/ΩR and μ_z = V_n/ΩR; it starts as 1 + 4.5 μ². The textbook form 1 + K μ² can be chosen instead: K = 3 without the radial flow along the blade, about 4.6 to 4.7 with it and with the reverse-flow region. NACA Report 1266 writes the same power balance as C_P/C_T = C_Pi/C_T + C_Po/C_T + C_Pp/C_T + C_Pc/C_T, with the profile term read from charts computed by blade-element theory with a three-term section drag polar; the page prints these ratios for the entered flight condition.

Speeds, climb and range

The speed of least power is the speed for best endurance and, nearly, for best rate of climb; the speed of least power per unit speed (the tangent from the origin to the power curve) gives the best range in still air. The maximum level speed is where the power required meets the power available. The rate of climb at a speed is found by solving the power balance for V_c, not from the excess power alone, so the change of induced and parasite power with the climb is included. The service ceiling is the altitude at which 100 ft/min remains at the least-power speed. Endurance and range for a fuel weight W_f and a specific fuel consumption c are W_f/(c P) and V W_f/(c P) with the weight held at the entered value; enter a mid-mission weight for a long flight.

Autorotation

With the engine disengaged the rotor is driven by the air: the helicopter descends at the rate for which T (V_n + κ v) + P_o plus the tail-rotor profile power and the accessory power is zero. In vertical descent the solution lies in the turbulent-wake state, where the approximation above applies; measured rotors autorotate near 1.8 v_h. With forward speed the rate of descent is close to the level-flight power divided by the weight, least near the best-endurance speed; the flattest glide is at a higher speed. The flare and the energy stored in the rotor are outside the tool.

Multirotor

Each propeller carries T = √(W² + D²)/N. Its shaft power in hover is T v_h/FM, FM the hover figure of merit of the propeller, which for a fixed-pitch propeller is, to a first approximation, the same at every thrust (its thrust and power coefficients are fixed in static operation). The tool splits that power into the induced part κ T v_h and a profile part (1/FM − κ) T v_h; in forward flight the induced part follows Glauert's equation at the tilt tan⁻¹(D/W), the parasite power D V is added and the profile part keeps its hover form. This split is the tool's own simple model. The electrical power is the shaft power over the motor and controller efficiencies plus the avionics power; endurance is the usable battery energy over that power and range is speed times endurance. Battery voltage sag and the loss of capacity at high discharge rates are not modelled.

Limits

No blade stall, no compressibility drag rise on the advancing blade, no rotor H-force, no rotor-fuselage or rotor-rotor interference, one drag coefficient for all thrusts and speeds. The power curve is cut at μ = 0.5; retreating-blade stall limits most helicopters at lower advance ratios. Results near the vortex-ring region are mean values of an unsteady flow.