Pitot tube question

Question

I think i get the concept of the Bernoulli equation pretty well, but I have a hard time understanding the Pitot tube.

We have the Bernoulli equation for the Pitot tube: $P_1=P_2+(\rho V_2^2)/2$, where $P_1$ is pressure in the stagnation point $1$ and $P_2$ is the pressure in the point $2$.

Now, let's assume that the aircraft is not moving. Let's blow a stream of air over the point $2$, like in the theoretical example, but we don't blow any air towards point $1$, like we did in the theoretical example. The velocity at point $1$ will still be zero, and the Bernoulli equation will have the same form: $P_1=P_2+(\rho V_2^2)/2$, Which should not be correct.

Where is the mistake in my reasoning?

Answer 1

Bernoulli only says that $P + \frac{\rho v^2}{2}$ remains constant along a streamline in inviscid, steady flow (neglecting gravity). In some cases, such as what I think you are showing in the first diagram, the constant is the same for all streamlines. But in the second case, you have air flowing in some layer at the top, and sitting still in a layer underneath, and the value of the constant is different in these two regions.