Suppose you have an infinitely-long W-shaped canal....
Here's a similar problem that some people may find easier to analyze:
Imagine two huge airmasses; call them A and B. One is flowing northbound
at XX knots. The other is flowing southbound at YY knots. They meet at a
sharp frontal boundary, a vertical north/south plane. Within each airmass
there is an observer in a balloon, comoving with the airmass, reporting a
pressure of 1013 millibars at low altitude.
All this takes place near the equator so we don't need to worry about
Coriolis effects.
For homework:
1) Tell me it's obvious that if there were a net flow of air from one
airmass to the other, both observers would agree as to which direction
(east or west) the net flow was going.
2) Tell me it's obvious by relativity and symmetry that there is no
tendency for there to be a net flow of air from A to B or vice versa.
3) We overhear observer A, who says in part that
"airmass B is moving, and therefore its ... pressure is lower...."
We also overhear observer B, who says in part that
"airmass A is moving, and therefore its ... pressure is lower...."
Now these are my observers, which means everything they say is true. But
you haven't heard their whole utterances. The assignment is to figure out
what they said and why it is true. In particular:
3a) What did they mean by "... pressure"?
Static pressure?
Dynamic pressure?
Stagnation pressure?
3b) What did they mean by "lower"? What is lower than what?
3c) Write down the static pressure, dynamic pressure, and stagnation
pressure in each of the two airmasses, measured in observer A's frame of
reference.
4) For each item in answer (3c) that is the same for both airmasses,
explain why it _had_ to be the same. For each item that is different,
explain why it changes as you transform from one reference frame to the other.
5) State, roughly speaking, the necessary conditions that must be met
before you can dream of turning the crank on Bernoulli's formula in such a
way that it tells you something about the pressures of two different air
parcels, or the velocities of two different air parcels.