# Wind Power in a Wind Tunnel

Two questions in one day! What?

Okay, so the power equation for air flow is as follows: $$P = 1/2 \cdot \rho \cdot A \cdot V^3$$ In a wind tunnel due to the Bernoulli principle as the area ($A$) decreases, the velocity ($V$) proportionally increases; but if the flow is assumed to be in-compressible (talking low speed wind tunnels here) then the density does not change. This means that the $V^3$ term dominates in the power equation and the power is increased. So here is my question, what is that actual physical mechanism that adds the power? I mean I get that there's a pressure gradient, but what causes that? I've always imagined it was the shape of the convergent cone that takes the drag force and by Newtons third law it directs some of the drag force back into the air and increasing the pressure, but I want to know for sure.

• -1. Not clear. Are you asking what causes pressure gradients? Or are you asking what causes the drag force? For the latter see Friction in a fluid. And what is the convergent cone? – sammy gerbil Sep 22 '17 at 23:27
• I'm asking what is doing the physical work on the air to add the power to it. If it is just pressure gradients, then yes, what causes the gradients? And maybe cone is the wrong noun, but I'm referring the portion of the wind tunnel that goes from the larger area down to the smaller area in the direction the air is travelling, wherein the air is actually sped up. – SCossano Sep 23 '17 at 0:36
• The reason for the $V^3$ is because power is velocity times pressure, and pressure is proportional to $V^2$ (velocity times momentum). What causes the velocity/pressure change? They both change together; they each cause the other. Just as in $F=ma$, you can't have acceleration without force, and you can't have force without acceleration. Mass is just the ratio of the two. – Mike Dunlavey Sep 26 '17 at 15:02