# Why does the pressure drop behind a windmill actuator disk?

The canonical derivation of Betz's law for wind turbines discusses pressure before and after a windmill actuator disk.

It's obvious that the wind resistance of the disk would cause a pressure increase before the windmill. But why is there a similar pressure drop behind the windmill (negative overshoot below ambient pressure)?

Note that this question is not about turbulence which is fully ignored in actuator disk momentum theory. This also does not depend on any aerodynamics like Bernoulli force on the wing of the turbine as again, all momentum actuator disk theory is independent of the implementation of the wind machine.

Indeed this question was controversial between 1865 and 1920 with many feeling a pressure drop was inconceivable despite Froude's assertion that it was there (Parsons' model was rejected).

But I can't find an intuitive or simple explanation in terms of momentum, energy or incompressible flow as to what causes the pressure drop.

Help?

Addendum: I note that there is a circular answer to this question that isn't helpful in explaining the phenomena. The circular answer is that if the wind speed through the actuator disk is higher than the final wake wind speed, then there has to be a pressure change. That answer is circular logic since the calculation of the higher wind speed presupposes a pressure drop!

As a counter proposition, suppose the wind speed through the actuator disk were the same as the wake wind speed. There still would be a pressure build up in front of the actuator disk to power the windmill. And you could then have ambient pressure behind the windmill. Why is this the wrong answer? I can't see what I'm missing.

In the canonical derivation, I don't believe that it is postulated a priori that the pressure after the disk is lower than the atmospheric pressure. It is the result of the calculation. So I don't see why the argument would be circular?

The main result of the model is $$v=(v_1+v_2)/2$$ and it is incompatible with your counter proposition $$v=v_2$$ since it would lead to $$v=v_1=v_2$$ ?

Hmm.. perhaps we're burying ourselves too deep in the equations. Since you need only an intuitive answer:

How could the pressure NOT drop?

If the pressure on the leeward side (immediately downstream from the turbine) equalled or exceeded the pressure on the windward side (immediately upstream from the turbine), wouldn't the air flow be reversed? [By "immediately" before and after, what I mean is: so close to the rotor disc that the diameter of the flow "tube" passing across the rotor disc is virtually unchanged.]

The only way for air to be driven through the rotor disc, and

• to be slowed down by it, and
• (in the non-ideal case) to compensate for the additional frictional/viscosity losses ...

... would be for the pressure to drop across that disc, right?