# Fluid flow around pipe, over end of pipe

I am thinking about air flowing around a pipe, from one end to the other, and passing over an open end. Here is a drawing to illustrate what I mean The orange in the middle is a pipe which, as you can see, is closed at one end and open at the other. The dark blue lines represent something around the pipe which is accelerating the air in the direction of the light blue arrows.

As you can see, the air will flow past the open end of the pipe. I am wondering what to expect from the air in the pipe and just beyond the open end of the pipe. I have tried to look up this information, but everything I find is about pressure reduction in fluid flowing through a pipe.

My intuition is telling me that the air just outside the orange pipe opening will be sucked into the lower pressure air that is moving and that effect will continue into the orange pipe causing reduced pressure inside the pipe as well, though possibly not as low pressure as outside the pipe. Also, possibly the pressure would follow a gradient where the pressure increases the farther into the orange pipe (towards the closed end) that you go.

I have other intuitions about reactions that could take place because of this, but first I just want to understand what happens with the air flow and pressure inside the pipe and just outside its opening. I would like to request any explanations of the fluid dynamics and not just the aerodynamic aspects.

How does the air pressure and flow of the pipe react to the air motion described by the arrows in the picture?

The construction and fluid flow is akin to a Pitot tube-in-reverse. Let us label points from right to left as $$1$$, $$2$$ and $$3$$ corresponding to the closed and open ends of the orange tube and far left of the open end of the orange tube respectively. We assume that the structure enveloping the orange pipe does has identical cross-sectional area leftwards starting from point $$2$$.
We now analyze the flow within and without the orange pipe as the fluid initially blows from point $$1$$ to point $$3$$. Indeed, the Bernoulli's equation is a steady-state equation and should not be applied to the time-varying flow we're analyzing, we will apply this approximation. Using Bernoulli's equation $$p_3 + \frac{1}{2}\rho v_3^2 = p_2 + \frac{1}{2}\rho v_2^2$$ for the flow within the structure (outside the orange pipe) while noting that $$v_3 < v_2$$ due to the mass conservation, we have that $$p_2 < p_3$$. Further, for the flow inside the orange pipe $$v_1 = 0$$ so that $$p_2 + \frac{1}{2}\rho v_2^2 = p_1$$. Therefore, $$p_2 < p_1$$. This means that initially, air would flow leftwardswards inside the orange pipe from point $$1$$ to point $$2$$. Subsequently, the pressure within the orange column will equalize $$p_1 = p_2$$ terminating any motivation for the flow to occur.