# How to calculate velocity of air behind a propeller?

Background: I came across this article by Gabriel Staples to calculate static and dynamic thrust of a propeller. In the derivation, it was assumed that the exit velocity is equal to the pitch speed of the propeller, a function of RPM and propeller pitch (inches). However, it was also mentioned that this assumption is far from truth in real life, so the final thrust equation derived was empirically corrected.

Question: I want to know what will be the average airspeed behind the propeller (exit velocity), if the engine is not moving forward (static thrust). Is there a way to calculate it?

(I want to know this speed to calculate the effectiveness of control surfaces submerged in the propwash)

• As engineer: calculate but do not trust results: measure them! Answers below may give you some estimates, but they assume a lot. Commented Jun 17 at 7:12

You want to look into hover momentum theory.

The induced velocity at a hovering rotor $$v_i$$ is:

$$v_i=\sqrt{\frac{T}{A}\frac{1}{2\,\rho}}$$

Where $$T$$ is the thrust and $$A$$ is the area of the disk, so $$T/A$$ is the disk loading. $$\rho$$ is the air density.

After the rotor, the streamtube that passes through the rotor will contract and accelerate. Eventually, it will reach a velocity of twice the velocity at the rotor.

• So what you're saying is exit velocity=2*entry velocity? But then... How do we calculate this, how is entry velocity calculated? How do you get from rotor parameters to exit velocity? Commented Jun 15 at 15:45
• Since the rotor is in hover, far above the rotor, the velocity is zero. At the plane of the rotor, the velocity is v_i. Far after the rotor, the velocity is 2v_i. You calculate v_i (the velocity induced by the rotor) using the equation I gave -- which is in terms of thrust and area. Commented Jun 15 at 17:03

Yes, if you know the propeller dimensions and the thrust this propeller generates.

From this answer we can derive an equation which gives exit speed in relation to thrust:

$$\Delta v = \sqrt{\frac{8\cdot T}{\pi\cdot d_p^2\cdot \rho}}$$

This is only valid for the static case: Entry speed is assumed to be zero.

Nomenclature:
$$\Delta v$$ = exit speed in m/s
$$T\;\,$$ = thrust in N
$$d_p\,$$ = propeller diameter in m
$$\rho\;\;$$ = air density in kg/m³