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Helicopter Rotor

Helicopter Rotor Induced Velocity Simulator

Use momentum theory to compute the induced velocity that a helicopter rotor drives downward in hover, and estimate disk loading, Figure of Merit and required power. Change the helicopter type, rotor diameter, tip speed, take-off weight, forward speed and altitude to see in real time how each parameter shapes rotor performance.

Parameters
Helicopter type
Representative helicopter / eVTOL presets
Rotor diameter
m
Number of blades
Blade chord
m
Tip speed
m/s
Shock losses rise sharply above Mach 0.8
Take-off weight
kg
Forward speed
km/h
0 means hover; cruise around 200 km/h
Flight altitude
m
Higher altitude lowers air density
Results
—
Disk area A (m²)
—
Air density ρ (kg/m³)
—
Hover v_i (m/s)
—
Hover P_i (kW)
—
Disk loading W/A (kg/m²)
—
Figure of Merit
—
Helicopter side view — induced velocity field animation

The rotor drives air downward and supports the airframe weight. The arrows below the disk represent the induced velocity v_i, while the cyan band at the ground marks the In-Ground-Effect (IGE) region.

Required power vs altitude
Disk loading and power by helicopter type
Theory & Key Formulas

$$v_{i} = \sqrt{\dfrac{W}{2\,\rho\,A}}, \qquad A = \pi R^{2}$$

Hover induced velocity v_i from momentum theory. W: airframe weight [N], ρ: air density [kg/m³], A: rotor disk area [m²], R: rotor radius [m].

$$P_{i} = W\,v_{i}, \qquad P_{0} = \tfrac{1}{8}\,\sigma\,A\,\rho\,V_{tip}^{3}\,C_{d0}$$

Induced power P_i and profile (drag) power P_0. σ: solidity ratio = N·c/(πR); V_tip: blade tip speed; C_d0: mean drag coefficient (this tool uses 0.01).

$$FM = \dfrac{P_{i}}{P_{i}+P_{0}}, \qquad \rho(h) = 1.225\,e^{-h/8400}$$

Figure of Merit and altitude-corrected density. Well-designed main rotors reach FM 0.7–0.8. ρ(h) is a simplified ISA exponential model with altitude h in metres.

Helicopter Rotor Induced Velocity & FM — Momentum Theory

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Helicopters just spin their rotor and lift off the ground. Where does that lift physically come from?
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Great starting point. In plain words the rotor is a "fan that throws air downward and gets lifted by the reaction". Momentum theory looks at the mass flow ṁ = ρ·A·v_i passing through the disk area A. Multiplying by an exit speed of 2v_i gives the thrust T = ṁ·2v_i, and in hover T = W, so v_i = sqrt(W/(2ρA)). That is exactly the formula this tool evaluates.
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When I switch from "Lightweight (R44)" to "Heavy (CH-47)" the disk loading shoots up. Is that a bad thing?
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More like a trade-off between efficiency and airframe size. The larger the disk loading W/A, the larger v_i, so the more energy you spend to hold the same weight. That is why lightweight helicopters such as the R44 sit around 12 kg/m² and stay fuel-efficient. The CH-47 trades efficiency for payload by running near 50 kg/m². Many eVTOLs also push the disk loading high through ducted fans to keep the vehicle compact.
🙋
Why do helicopters always stick around a tip speed of 220 m/s? Wouldn't spinning faster give more lift?
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There is both an upper and a lower limit. The upper one is the speed of sound: above a tip Mach number of about 0.85 (≈290 m/s at sea level) shock-wave drag explodes. The lower one is stall: if the rotor turns too slowly the relative flow on the blade drops below the stall angle. Real machines settle into 200–230 m/s. In this tool you can see the profile power P_0 climb with the cube of tip speed.
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The FM came out as 0.52. Is that a poor number?
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This tool uses a conservative profile-power coefficient, so absolute values come out a bit low. Real main rotors more often sit at 0.70–0.80. What matters here is the relative trend: "bigger rotor, lower tip speed, more blades, optimised chord" — which of these moves FM up. If you grow the diameter from 11 m to 14 m, you will see FM rise as disk loading drops and induced power dominates.
🙋
When I lift the altitude from 0 to 4000 m the required power climbs. Is this why mountain rescue with heavy loads is so hard?
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Exactly — that is the heart of the "altitude–temperature–weight" chart that pilots use. As altitude rises ρ falls, so v_i = sqrt(W/(2ρA)) grows and induced power P_i = W·v_i grows with it. The tool's exponential model gives about 70% density at 3000 m and 50% at 6000 m. Engine power also degrades. That is why helicopters operating in the Himalayas or the Alps fly with take-off weights roughly halved from sea-level limits.

Frequently Asked Questions

In momentum theory the rotor sustains the weight W by pushing air downward. The induced velocity is v_i = sqrt(W / (2·rho·A)), where A is the rotor disk area and rho is the air density. For an R44 (W=10.8 kN, A=95 m², rho=1.09 kg/m³ at 1000 m), v_i is about 7.2 m/s. The larger the disk loading W/A, the larger v_i and the more induced power the rotor needs.
FM is the ratio of ideal induced power to actual power and is the standard yardstick of rotor aerodynamic efficiency. This tool uses the simplified form FM ≈ P_i / (P_i + P_0); the smaller the profile (drag) power P_0 is relative to P_i, the closer FM gets to 1. A well-designed main rotor typically reaches 0.70–0.80, while small low-Reynolds rotors stay around 0.50.
At low forward speeds hover demands the most power; induced power minimises around 40–80 km/h because the inflow lets the rotor "grab" air more efficiently. The power-required curve therefore has a U shape. At high speeds the fuselage parasite power P_p grows with V cubed and the curve climbs again. In this tool the parasite-power term increases visibly as you raise the forward speed, letting you check the margin against maximum continuous power.
Air density rho drops with altitude. This tool uses the exponential model rho = 1.225·exp(−h/8400), giving about 70% at 3000 m and 50% at 6000 m. To sustain the same weight v_i ∝ 1/sqrt(rho) grows, so induced power P_i = W·v_i grows too. Combined with the engine power loss that comes from lower air mass flow, mountain operations must reduce take-off weight or use a larger rotor.

Real-World Applications

Initial sizing of light helicopters: Momentum theory drives the early sizing decisions for two-to-five seat singles like the R44 or Bell 206. Keeping disk loading at 12–18 kg/m² and tip Mach around 0.65 lets a piston or turbine maintain steady hover within continuous power. Picking the R44 preset gives roughly 78 kW of induced power and 214 kW of total required power — a bit higher than the real machine's hover power of 130–150 kW because this tool's simplified profile-power model is conservative.

External-load limits for heavy transport rotors: On tandem-rotor heavies like the CH-47 Chinook, disk loading runs near 50 kg/m² and induced power dominates. Disaster-relief or military missions in mountainous terrain require HOGE (Hover Out of Ground Effect) calculations at the destination altitude. Sweeping weight and altitude in this tool gives a tactile feel for how density couples into required power, making it useful as a training aid.

Power budgets for eVTOL and drones: Urban-air-mobility eVTOLs and delivery drones live or die by the battery-capacity-to-hover-power ratio. Multirotor layouts keep disk loading at 20–35 kg/m², while tip speeds of 150–200 m/s help meet residential noise limits below 65 dB(A). The hover induced-power number from this tool, divided into battery capacity (kWh), gives an order-of-magnitude estimate of hover endurance.

Pre-CFD checks in rotor education: Momentum theory is usually the first lecture in any university course on rotary-wing aerodynamics. Before launching detailed CFD (OVERFLOW, CFD++) or BEMT analysis, engineers use momentum theory to bound v_i and FM. Demonstrating "weight up — v_i up" with this tool first, then moving on to a CFD mesh, accelerates how quickly students grasp design space.

Common Misconceptions and Pitfalls

First, momentum theory ignores ground effect (IGE) and viscous losses. The v_i = sqrt(W/(2ρA)) formula assumes uniform inflow, no viscosity and an infinite atmosphere. In reality, hovering within one rotor diameter of the ground reduces induced power by 10–20%, while gusts or sloped terrain raise it. This tool applies a 1.15× factor at altitude 0 m as a rough hint, but detailed work needs an actuator-disk CFD or full-vehicle CFD analysis. Final design decisions must always be backed by flight-test data.

Second, Figure of Merit is highly model-dependent. The simplified FM ≈ P_i / (P_i + P_0) form used here assumes a constant C_d0 = 0.01. In reality the airfoil, blade thickness, twist and tip shape (BERP, OGE tips) move FM significantly: F-35B lift-fan designs exceed 0.85, while micro-drones at low Reynolds may drop to 0.4. Look at relative gains from parameter changes rather than the absolute number this tool reports.

Third, momentum theory does not apply directly to forward flight. This tool models forward speed crudely as a parasite drag P_p ∝ V³. In real forward flight, once the advance ratio μ = V/V_tip exceeds about 0.15 the retreating blade's reverse-flow region and the advancing tip's shock losses dominate, and the simple v_i formula breaks down. Production design relies on Glauert's forward-flight model, BEMT and CFD. Treat the forward-speed dependence here as a "slightly different from hover" hint and reserve serious sizing for the hover and take-off/landing conditions.

How to Use

  1. Enter rotor diameter in meters (typical range: 8–16 m for utility helicopters)
  2. Input number of blades (2–5 for most designs) and blade chord length in meters
  3. Set rotor tip speed in m/s (typically 200–250 m/s to avoid compressibility effects)
  4. Select air density ρ in kg/m³ (1.225 at sea level, 1.0 at 2000 m altitude)
  5. Review calculated disk area A, induced velocity v_i, induced power P_i, disk loading W/A, and figure of merit FM

Worked Example

A Sikorsky UH-60 Black Hawk with rotor diameter D = 16.76 m, 4 blades, chord c = 0.53 m, and tip speed Ω = 225 m/s in sea-level hover (ρ = 1.225 kg/m³). Disk area A = π(8.38)² ≈ 220.6 m². Using momentum theory, for a hover weight of 22 kN (5000 lbf), disk loading W/A ≈ 100 kg/m². Induced velocity v_i ≈ 6.2 m/s. Induced power P_i = W·v_i/2 ≈ 68 kW. Figure of merit FM = 0.62 indicates rotor efficiency relative to ideal actuator disk.

Practical Notes

  1. Disk loading W/A directly affects induced velocity; reducing it by 20% decreases v_i by ~10%, improving hover efficiency in rescue/transport missions
  2. Tip speed limits: subsonic designs stay below 240 m/s to minimize compressibility losses and blade vibration on CH-47 Chinooks
  3. Figure of merit typically ranges 0.55–0.68 for conventional rotors; values below 0.50 suggest blade twist losses or excessive blade solidity (number of blades × chord / disk area)
  4. Altitude density corrections: at 3000 m elevation, ρ ≈ 0.91 kg/m³, increasing v_i and P_i by ~15% for same hover weight

🎬 Watch it in motion

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