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The question could be rephrased as simply noting that Bernoulli's
equation is manifestly NOT invariant wrt Galilean transformations
because kinetic energy density is frame-dependent while pressure is
not. To appreciate the apparent strangeness of the result it is
particularly useful to imagine a flowing fluid far from any
boundaries like pipes or airfoils, that is, in a region where there
is a lack of reference *objects* and the frame-dependence of the
fluid velocity becomes more obvious. Since the Bernoulli equation
clearly gives wrong answers unless the correct ("preferred")
reference frame is chosen, the question is, "What determines that
preferred reference frame?"
The answer is that Bernoulli's eqn is derived under the assumption
that the flow is "steady" (i.e., not time-dependent) and is,
therefore, only applicable to steady flows. But, except in boring
cases, steady flows are only steady in *one* reference frame. Of
course, to have a non-boring steady flow, one does also have to have
"stationary" boundaries like pipes or river beds or mountaintops or
airfoils, which then also determine the preferred reference frame,
but that is not the fundamental requirement.
This result is somewhat similar to the equation describing the
classical Doppler shift, which is only valid in the rest frame of the
medium supporting the waves, but I think also slightly more subtle.