Formula & Calculator
Momentum Theory Induced Velocity (Hover)
Induced (downwash) velocity through a helicopter rotor disk in hover, from simple momentum theory.
Interpretation
Induced velocity in hover: v_i = √(T/(2·ρ·A)), where T is thrust, ρ density, A disk area. It is the downward air velocity through the rotor disk. Example: T=50,000 N, ρ=1.225, A=78.54 m² → v_i = √(50000/(2×1.225×78.54)) = √(50000/192.4) ≈ 16.12 m/s.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v_i | Induced velocity | m/s |
| T | Rotor thrust | N |
| ρ | Air density | kg/m3 |
| A | Rotor disk area | m2 |
What it means
The induced velocity is the velocity of the air as it passes through the rotor disk in hover, from momentum theory. It is a fundamental parameter in rotor analysis, affecting the power required and the flow field. The induced velocity appears in the ideal power equation: P_ideal = T·v_i. In forward flight, the induced velocity decreases. Understanding this velocity is essential for rotor performance calculations and for flow field modelling.
Worked example
Induced Velocity – Two Examples
Real‑World| Parameter | Value |
|---|---|
| T | 20,000 N |
| ρ | 1.225 kg/m³ |
| A | 100 m² |
| Parameter | Value |
|---|---|
| T | 50,000 |
| A | 180 |
Common mistakes
- Momentum theory induced velocity (hover): v_i = √(T / (2·ρ·A)).
- T: Thrust (N).
- ρ: Air density (kg/m³).
- A: Rotor disk area (m²).
- Assumes ideal, uniform inflow.
Applications
Momentum theory induced velocity in hover, v_i = √(T/(2ρA)), is the average downward velocity induced by the rotor to produce thrust. It is used to estimate hover power and to understand the flow physics. Engineers use this to compute ideal power, to size rotors, and to assess ground effect. By understanding induced velocity, aerospace engineers can predict rotor performance and optimise blade design for reduced power consumption, especially important for electric vertical takeoff and landing (eVTOL) aircraft.
- Hover performance and power estimation
- Rotor sizing and induced power calculations
- Ground effect analysis and flight testing
- Blade element theory integration
- Design of low‑power rotors for electric propulsion
Frequently Asked Questions
It gives the induced (downwash) velocity through a helicopter rotor disk in hover, from simple momentum theory.
T = rotor thrust (N)
ρ = air density (kg/m³)
A = rotor disk area (m²)
It determines the induced power and the inflow distribution. It is a fundamental parameter in rotor aerodynamics.
- Applying the simple actuator‑disk (hover) result to forward flight, where induced velocity is significantly lower for the same thrust.
- Using the wrong area (e.g., blade area instead of disk area).
- Ignoring the effect of tip losses.
For T = 50,000 N, ρ = 1.225, A = 201 m², vi = √(50000/(2×1.225×201)) = √(50000/492.45) = √101.5 ≈ 10.07 m/s.
Induced power Pi = T·vi. Higher induced velocity requires more power.
At higher altitude, ρ decreases, so vi increases for the same thrust, increasing power required.
In hover, the downwash velocity far downstream is twice the induced velocity at the disk (v∞ = 2vi).
In ground effect, the induced velocity is reduced, lowering induced power and increasing efficiency.
More advanced models use blade element theory or CFD to compute the induced velocity distribution.