Library / Aerospace / AER-53
Aerospace
Axial induced speed at an actuator disk
Extra axial speed given to the flow at the disk of an ideal propeller or rotor producing thrust T in a freestream V, in steady incompressible flow.
- Ref
- AER-53
- Rev
- v1
- Field
- Aerospace
- Result
- vi [m/s]
Momentum theory treats the propeller as a disk that adds a speed vi at the disk, and 2vi in the far wake. Thrust equals mass flow through the disk times the wake speed gain: T = ρA(V + vi)(2vi). Solving gives vi = [√(V² + 2T/(ρA)) − V]/2. The page uses the rationalised form (T/(ρA))/(√(V² + 2T/(ρA)) + V), which avoids subtracting two nearly equal numbers at high speed.
Extra axial speed given to the flow at the disk of an ideal propeller or rotor producing thrust T in a freestream V, in steady incompressible flow. The wake speed far downstream is V + 2vi. The page uses a rationalised form of the formula to avoid subtracting two nearly equal numbers. It needs T ≥ 0 and V ≥ 0.
Calculator
- Speed through the disk, V + vi25 m/s
- Far-wake speed, V + 2vi30 m/s
- Disk loading, T/A300 Pa
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The disk speeds the flow up by 5 m/s at the disk and 10 m/s in the wake, so the wake moves at 30 m/s.
- The ideal propulsive efficiency is V/(V + vi) = 0.8, which shows why bigger disks and slower wakes are more efficient.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Thrust | 10 to 100,000 N | 0.1036 | 134.7 | 134.6 | +10% | +8.22 % |
| A Disk area | 0.1 to 100 m² | 0.1242 | 40.99 | 40.87 | +10% | −7.67 % |
| V Freestream speed | 0 to 150 m/s | 0.8288 | 11.18 | 10.35 | +10% | −6.31 % |
| ρ Air density | 0.3 to 1.3 kg/m³ | 4.676 | 14.49 | 9.819 | +10% | −7.67 % |
Sweep of Disk area
| Disk area (m²) | Induced velocity at the disk (m/s) |
|---|---|
| 0.1 | 40.99 |
| 0.2371 | 23.97 |
| 0.5623 | 13.34 |
| 1.334 | 6.955 |
| 3.162 | 3.381 |
| 7.499 | 1.547 |
| 17.78 | 0.6798 |
| 42.17 | 0.2922 |
| 100 | 0.1242 |
Sweep one variable
Variables and how each one contributes
Induced velocity at the disk
What the equation gives you. Everything else feeds it.
Thrust
The thrust, in N. Raises the induced velocity, more slowly than in proportion at high speed. Raising it by 10 % (to 660) moves the result by +8.22 %, from 5 to 5.411 m/s.
Air density
The air density, in kg/m³. A denser flow needs a smaller speed change for the same thrust. Raising it by 10 % (to 1.32) moves the result by −7.67 %, from 5 to 4.616 m/s.
Disk area
The disk area, in m². A larger disk lowers the induced velocity and the losses, which is why propellers are large. Raising it by 10 % (to 2.2) moves the result by −7.67 %, from 5 to 4.616 m/s.
Freestream speed
The freestream speed, in m/s. Lowers the induced velocity as forward speed gives the flow a head start. Raising it by 10 % (to 22) moves the result by −6.31 %, from 5 to 4.684 m/s.
Worked example with the values above
- Thrust per unit disk mass flow: 2T/(ρA) = 500 m²/s².
- Root: √(V² + 2T/(ρA)) = √(400 + 500) = 30 m/s.
- vi = (30 − 20) / 2 = 5 m/s.
Assumptions and limits
- An ideal actuator disk with uniform loading and steady incompressible flow, with no swirl.
- Rotor in hover or propeller in axial flight, with consistent revolutions-per-second definitions.
Common mistakes
- Using revolutions per minute where the formula needs revolutions per second.
- Using the disk area of one rotor for a multi-rotor system.
- Treating the ideal power as the shaft power, which ignores the losses.
Statistics and uncertainty
Momentum theory sets a lower bound on power: real propellers need 10 % to 30 % more than the ideal figure for blade profile drag, swirl and tip losses. Figure of merit of good rotors is 0.7 to 0.8, and propeller efficiency peaks around 0.85 to 0.88.
Where it is used
- Estimating the power needed for propellers, helicopters and drones.
- Reducing test data to coefficients and efficiency.
Maths you need
- Momentum theory: thrust equals mass flow times velocity change
- Advance ratio: J = V/(nD)
- Disk loading: thrust per unit disk area