My understanding is that pressure has a square relationship with respect to flow, unless the flow can be made laminar.
This is also my experience with fiddling around with flow circuits.
As you point out with pneumotachs, there is a method of measuring flow that involves measuring differential pressure across a resistance. In other words, you measure pressure immediately prior to the resistance, and immediately following the resistance. You need to take the square root of that differential pressure to get a linear relationship to flow. If you simultaneously make the flow laminar (typically induces some resistance) you don't need to do the square-root correction.
You need to encompass all the flow to take those measurements. In a cannula system, you aren't encompassing all the flow at both pressure
points. But that's not why it has a square-relationship, it's that the flow is not laminar.
That's how I understand it, but I could be wrong.
To confuse the issue a little more, a cannula system is essentially measuring stagnation pressure.
http://www.princeton.edu/~asmits/Bicycl ... oulli.html