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Some fellow flutists and I are pretending to be experts at fluid dynamics and reading Patricio Cuadra's thesis on the topic:

On page 17, it says:

The central speed of the jet $U_j$ at the channel exit can be estimated assuming atmospheric pressure outside the channel and using Bernoulli’s equation:

$$U_j = \sqrt{ \displaystyle\frac{2P_f}{\rho_1} }$$

Where $P_f$ = pressure in the cavity before the channel, and $\rho_1$ = gas density

Our question, why does this omit the channel size, how big the lip-hole is, which intuitively seems to have a large affect on speed (e.g. a hose with the nozzle half open vs. open).

What is "central velocity"? Is it what we are thinking of when we say, "jet velocity"?

Or maybe the key phrase is "at the channel exit" (e.g. after the constraints of the channel size no longer apply?)

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Bernoulli's principle states that

$$ \frac{1}{2}U^2+gz+\frac{P}{\rho} = \text{constant} $$

A very important remark here, is that the quantity is only constant a long a streamline. That means that the constant is different from streamline to streamline (e.g., between the center of your flute and the side). Be careful that the theorem does not hold at the side, because viscosity will come into play at some point.

There is a streamline, from the cavity with pressure $P_f$ (pressure on top over atmospheric pressure) and zero velocity, to the center of your flute, with atmospheric pressure (this is an approximation) and velocity $U_j$.

This reduced Bernoulli's equation to

$$ \frac{P_f}{\rho} = \frac{1}{2}U_j^2 $$

which directly reduces to the equation you gave in the question.

As you're looking at streamlines, you don't see how large the system is. The size of the channel is secretly hidden in the measured variable $P_f$. You can imagine, that for a wider channel to blow the same amount of air, you need to blow a lot harder, thus need a higher pressure.

To address jet velocity versus central velocity. In texts like you're referring to, people rarely use two expressions for the same thing, as this is confusing. They are always defined somewhere, but I would assume that the central velocity is the maximum velocity at the centerline, whereas jet velocity is the mean velocity across the whole jet.

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I believe you're missing a square there somewhere... –  Daniel Blay Nov 4 '12 at 12:39
    
@DanielBlay You're right of course, thanks –  Bernhard Nov 4 '12 at 12:40
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