Library / Dimensional analysis and similarity / DIM-21
Dimensional analysis and similarity
Propeller advance ratio
Ratio of the axial advance speed to the speed of the propeller tip’s rotation, based on diameter.
- Ref
- DIM-21
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- J
J = Va/(nD) compares the axial advance speed with nD, which is proportional to the speed of the blade tips. Here n is the rotational speed in revolutions per second (rpm divided by 60) and D is the diameter. Propeller coefficients for thrust, torque and power are tabulated against J for a given blade geometry and pitch. Allow for changes in blade Reynolds number, tip Mach number and, in water, cavitation when you move to a different size or speed.
Ratio of the axial advance speed to the speed of the propeller tip’s rotation, based on diameter. Propeller thrust, torque and power coefficients are tabulated against it.
Calculator
- Tip speed of rotation, πnD31.42 m/s
- Advance speed at J = 1, nD10 m/s
- Speed in rpm1,200 rpm
- Advance ratio with the speed halved1
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.
- At 5 m/s with 1,200 rpm and a 0.5 m propeller, J = 0.5. Read the thrust and torque coefficients from the propeller data at this advance ratio.
- Two propellers with the same J (and the same blade geometry) are dynamically similar: to keep J fixed when the speed scales by s and the diameter by d, the advance speed scales by s d.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Va Axial advance speed | 0.1 to 300 m/s | 0.01 | 30 | 29.99 | +10% | +10 % |
| n Rotational speed | 1 to 200 rev/s | 0.05 | 10 | 9.95 | +10% | −9.09 % |
| D Propeller diameter | 0.05 to 10 m | 0.025 | 5 | 4.975 | +10% | −9.09 % |
Sweep of Axial advance speed
| Axial advance speed (m/s) | Advance ratio |
|---|---|
| 0.1 | 0.01 |
| 0.272 | 0.0272 |
| 0.7401 | 0.07401 |
| 2.013 | 0.2013 |
| 5.477 | 0.5477 |
| 14.9 | 1.49 |
| 40.54 | 4.054 |
| 110.3 | 11.03 |
| 300 | 30 |
Sweep one variable
Variables and how each one contributes
Advance ratio
What the equation gives you. Everything else feeds it.
Axial advance speed
The axial advance speed of the propeller through the fluid, in m/s: the speed of the vehicle or the inflow. Raises the advance ratio in direct proportion. Raising it by 10 % (to 5.5) moves the result by +10 %, from 0.5 to 0.55.
Rotational speed
The rotational speed, in revolutions per second (rpm divided by 60). Lowers the advance ratio in inverse proportion: a faster-turning propeller at the same advance speed has a smaller J. Raising it by 10 % (to 22) moves the result by −9.09 %, from 0.5 to 0.4545.
Propeller diameter
The propeller diameter, in m. Lowers the advance ratio in inverse proportion. Raising it by 10 % (to 0.55) moves the result by −9.09 %, from 0.5 to 0.4545.
Worked example with the values above
- Rotational term: nD = 20 × 0.5 = 10 m/s.
- Divide the advance speed by it: 5 / 10.
- J = 0.5.
Assumptions and limits
- Steady axial inflow and a propeller of fixed blade geometry and pitch.
- n in revolutions per second, with Va the speed of the fluid approaching the disc.
Common mistakes
- Putting the speed in rpm instead of revolutions per second.
- Using the radius for D.
- Using the vehicle speed without allowing for the wake in the advance speed.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (Va × 1, n × −1, D × −1). The advance speed is the least certain input: the inflow to a propeller is not the vehicle speed, since hulls and fuselages create wakes.
Where it is used
- Finding the operating point on a propeller or fan chart of thrust, torque and efficiency.
- Keeping the operating point fixed when scaling a propeller test to another speed or size.
Maths you need
- Propeller coefficients: thrust, torque and power in dimensionless form
- Axial advance speed: speed of the inflow along the propeller axis