Why does the note played by a flute not change in this case when  the air column is interrupted? I play the flute as a hobby, and I've noticed that when playing middle D or E flat, one can interrupt the air column by releasing a certain key (which is near the middle of the air column), and yet have no effect on the pitch (though the quality changes for the better).
I'll be putting a few diagrams here, since it is hard describing the situation in words. The black portions of the diagrams represent closed holes--basically "air cannot escape from here". The gray represents holes which are closed due to lever action, but need not be. Here's a diagram without coloring (all diagrams are click-to-enlarge):

The mouthpiece is attached on the left, as marked in the diagram.
The second key is just a ghost key connected to the first (and has no hole underneath it), so I'll just remove it from the diagrams:

A few examples
Alright. Normally, when playing consecutive notes, you make the air column shorter by releasing a key. For example, this is F:

This is F#:

And this is G:

One can easily see the physics behind this, an unbroken air column is formed from the mouthpiece.
The weird stuff
Now, let's look at middle D and E flat:
D:

E flat:

Here, the air column is broken in between. I feel that both should play the same note, that is C#:

But they don't. I can close the hole, creating an unbroken air column in both cases, but the sound quality diminishes.
A bit more experimentation (aka "what have you tried?")
Reading this section is strictly optional, but will probably help
I did a lot of experimenting with this key, turning up some interesting results. 
Hereafter, I'm calling the key "the red key", and marking it as such in the diagrams. When the red key is "closed", no air can escape and it forms part of the air column.


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*If I play low D/E flat, I only get a clear note when the red key is closed. With it open, I get a note which has extremely bad quality, as well as being off-pitch. This is markedly opposite with what happens on middle D/E flat (mentioned above), there there is no change in pitch, and the difference in quality is reversed. 

Pictured: Low E flat (for low D extend the RHS of the black portion a bit more). Note that the fingering, save the red key, is the same for middle D/E flat 
Actually, this seems to be happening for all of the low notes--each one is affected drastically when the red key is lifted.

*Going on to the notes immediately after E flat


*

*For E, quality is drastically reduced when the red key is open. The harmonic (second fundamental) of E, which is B, is more prominent than the note itself. One can make the E more prominent by blowing faster, but this reduces quality. Red key closed gives a clear note, as it should.


*For F, a similar thing happens as with E. With the red key closed, it plays normally. With it open, you hear a medium-quality C (first harmonic of F), and no F at all. Blowing faster just gives a high C.



*The notes immediately below D have a fingering starting from no keys pressed (has to happen every octave, obviously). For the first few notes here, lifting the red key gives you C#, as expected. (I'm not explicitly marking the key red here, otherwise it'll get confusing what the correct fingering is)


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*In C#, pressing the red key will obviously change the note


*....to C:

Obviously, lifting the red key here will get you back to C#

*One (half) step lower, we have B, which again goes to C# when the red key is lifted


*It gets interesting again when we play B flat. Lifting the red key here gives a note between C# and C


*And if we go down to A, lifting the red key gives us a C
  


*And a bit of experimentation with the trill keys (the actual holes are on the other side of the flute). Whereas messing with the red key for D and E flat produces no change of pitch, messing with the trill keys (which are the same size as the red key and are furthermore pretty near it) does.


*

*Hitting the second trill key while playing D gives E flat. One should note that this second trill key opens the hole closest to the mouthpiece.

Note the visual similarity between this and the situation in the "the weird stuff" section

*Hitting the first trill key while playing D gives a note between D and E (the two trill keys are close to each other, you may have to see the enlarged version to get the difference)


*Hitting the second trill key while playing E flat gives a note between E and F 

*Hitting the first trill key while playing E flat gives E flat (No diagram here, these two are the same as the last two, except that the far right edge of the black portion is closer)




The Question
Now, the red key(and the trill keys) are about half the diameter of the other keys. I suspect that this is quite significant here, but I can't explain it myself.
My main question is, why does disturbing the air column as shown in the section "the weird stuff" not change the pitch? One has added an escape route for air, the column should then vibrate as if the remaining keys were open--that is C#.
I suspect that the underlying principle is the same, so I have a few other related questions (optional):


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*Why does the red key not change the pitch on D/E flat, but makes it go into the second fundamental/harmonic for E and above?

*Why does the red key change the pitch to notes which are not harmonics, instead close to C#(one of them isn't even part of the chromatic scale--it is between two notes) for B flat and A? 

*The red key is pretty similar to the trill keys with respect to size and general location. Yet, using a trill key on D changes the pitch, whereas using the red key doesn't. Why is this so?

 A: It's an interesting question! I was a flautist some years ago, and I did give it some thoughts, which I will write down below. I do not have any reference to back it up, though, it's just my own understanding of how wind instruments work (and personal experience with the Western concert flute).
My answer is basically: while the flute doesn't have a general, all-purpose octave key (or register key), the left-forefinger key actually acts as one in the particular case of the low and middle D (and D sharp) fingerings.
How does a register key work? It controls a hole in the instrument which, generally smaller than “regular” holes, and located near the beginning of the air column. Because of both its size and position, the hole does control the length of the air column by imposing local equilibrium; otherwise it would change the tune completely, as you noted. However, the register hole disrupts the flow of air, and thus prevents the formation of the lowest vibration mode and helps forming harmonics (octave for the flute, octave and a half for clarinet, …).
In the flute, the harmonics are easier to get that in other woodwind instruments, and you generally don't need a register key to hit the harmonics. However, in the case of D and D sharp, they make your life easier! I do not agree with the statement that it doesn't change the tune, though… I'm sure the two fingerings of middle D (with and without left-forefinger key) don't have the same tune exactly.

In short, the two reasons I see for the left-forefinger key not changing the tune are:


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*its small size: not enough air can exit to impose an open boundary (which is what dictates tune)

*its location: being close to the beginning of the air column, the stable vibration modes that would be consistent with this short wavelength definitely need special effort on the player (blowing high notes requires more pressure and less flow of air, also consistent with the above)

A: I've been doing experiments related to this back in 1994, so it's going to take a bit of recall.
The idea of a flute is that you create standing waves, which have a frequency that depends on the (variable) geometry. The reason they're standing waves is because you fix specific boundary conditions. In particular, p=0 at an open end.
Now, consider that you have a standing wave in a flute, with an wavelength that is a fraction of your flute length. That means that you have several nodes in the middle. If you would open a key at a node, there would be no effect. If you'd open one near a node, the pitch would change slightly.
