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Aerospace

Helical propeller tip Mach number

Mach number of the blade tip, from its rotational speed πDn and the axial inflow speed combined at right angles.

Ref: AER-56
Rev: v1
Field: Aerospace
Result: M_tip

M_{tip} \approx \frac{\sqrt{{(π D n)}^{2} + {V_{ax}}^{2}}}{a}

The blade tip moves at the vector sum of its rotational speed πDn and the axial speed Vax, so the helical tip Mach number is Mtip ≈ √[(πDn)² + Vax²]/a. Revolutions per second n is rpm divided by 60. Compressibility losses and noise rise sharply once Mtip exceeds about 0.85, which sets limits on diameter and rotation speed. Swirl, sweep and induction are neglected.

Mach number of the blade tip, from its rotational speed πDn and the axial inflow speed combined at right angles. n is in revolutions per second (divide rpm by 60), and a is the local speed of sound. Swirl, sweep and induction are neglected; if Vax is the freestream only, induction is omitted.

Calculator

Solve for

M_tipTip Mach numberComputed

DPropeller diameter

0.26

nRotational speed

rev/s

5100

V_axAxial inflow speed

m/s

0300

aLocal speed of sound

m/s

290345

Rotational tip speed, πDn188.5 m/s
Helical tip speed195 m/s
Rotational speed in rpm1,800 rpm

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.

Tip Mach number () current value 0.5736

nRotational speedKey driver

0.1741.85

DPropeller diameter

0.1571.67

V_axAxial inflow speed

0.5541.04

aLocal speed of sound

0.5650.672

The blade tips move at Mach 0.5736, below the usual limit of about 0.85 for acceptable efficiency and noise; the rotor runs at 1,800 rpm.
The axial speed adds only 3.46 % to the tip speed, so rotation dominates.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
n Rotational speed	5 to 100 rev/s	0.1737	1.854	1.68	+10%	+9.37 %
D Propeller diameter	0.2 to 6 m	0.1572	1.67	1.513	+10%	+9.37 %
V_ax Axial inflow speed	0 to 300 m/s	0.5544	1.042	0.4877	+10%	+0.688 %
a Local speed of sound	290 to 345 m/s	0.5653	0.6725	0.1072	+10%	−9.09 %

Sweep of Rotational speed

Rotational speed (rev/s)	Tip Mach number
5	0.1737
16.88	0.3448
28.75	0.5513
40.63	0.765
52.5	0.9813
64.38	1.199
76.25	1.417
88.13	1.635
100	1.854

Sweep one variable

Vary

Tip Mach number ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

M_tip

Tip Mach number

Result ·

What the equation gives you. Everything else feeds it.

Propeller diameter

m · typical 0.2 to 6

The propeller diameter, in m. Raises the tip speed in direct proportion. Raising it by 10 % (to 2.2) moves the result by +9.37 %, from 0.5736 to 0.6273.

Rotational speed

rev/s · typical 5 to 100

The rotational speed, in revolutions per second. Raises the tip speed in direct proportion. Raising it by 10 % (to 33) moves the result by +9.37 %, from 0.5736 to 0.6273.

V_ax

Axial inflow speed

m/s · typical 0 to 300

The axial inflow speed at the tip, in m/s. Adds to the tip speed as a vector component, and matters most when it is comparable to πDn. Raising it by 10 % (to 55) moves the result by +0.688 %, from 0.5736 to 0.5775.

Local speed of sound

m/s · typical 290 to 345

The local speed of sound, in m/s. A higher sound speed lowers the Mach number in inverse proportion. Raising it by 10 % (to 374) moves the result by −9.09 %, from 0.5736 to 0.5214.

Worked example with the values above

Rotational speed: πDn = π × 2 × 30 = 188.496 m/s.
Helical speed: √(35,530.6 + 2,500) = 195.014 m/s.
Mtip = 195.014 / 340 = 0.573571.

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

Entering rotational speed in rpm instead of revolutions per second.
Using the freestream sound speed at altitude where the temperature differs.
Leaving out the axial component at high flight speed.

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

Related formulas

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