This answer is slightly modified from a forum post I once wrote elsewhere to explain in detail why the flute's "complicated" design is the way it is. It's long as anything, but it does answer the question, so why not? TL;DR included at the bottom.
Let's start with the question about the B flat, and other alternate fingerings. There are some notes with more than one fingering, but in those cases the secondary fingering is there to make certain quick combinations of notes easier to finger. The "thumb B flat" key is there so that if you want to play a scale with a B flat, you don't have to bother using your right hand. Imagine trying to quickly play an A flat to B flat trill without the thumb key... it's a pain; you have to keep flicking your right hand fingers in the opposite way to your left hand fingers. On the other hand, if the B flat was always played with the thumb key, then moving between B and B flat would be really awkward.
Isn't 10 keys + harmonics (in other words overblowing) sufficient to get all the notes of the 3 octave range?
Actually 11, but yes, theoretically. However there are "extra" keys that extend the range in either direction. For example, after C, you are usually expected to then go across the register (overblow), and put all your fingers down. However, in fast passages, this isn't practical, so the trill keys extend the register up a bit further and give you an "extra" D and D# at the top so that it's possible to trill these notes from C and C# (which is essentially impossible for a human to do using the normal fingering).
And, on the bottom end, we have 2 extra keys to extend the register of the flute downwards. This isn't strictly necessary of course, but it's nice to have a couple of extra notes in the bass end without compromising the high end, and composers have seen fit to write music for this full range, (with B and B flat foot joins, this is extended a bit further).
But let's ignore the B flat and the trill keys for a second (which after all, are just alternate fingerings for quick passages, you don't need to use them that often at all). Why are the fingerings the way they are? And why aren't they the same in every octave?
Let's imagine we're designing the flute from scratch. So, how do flutes work? Let's look at the simple, diatonic flute first. Instruments like this are found all over the world, independently invented by different cultures (with some variation but perhaps not as much as you'd expect), except not with keys, but with holes.
For example, for a typical 6 holed flute (could be in any key but in keeping with western tradition I'll write it in D) is fingered like this:
●●● ●●● D
●●● ●●○ E
●●● ●○○ F#
●●● ○○○ G
●●○ ○○○ A
●○○ ○○○ B
○○○ ○○○ C#
●●● ●●● D (overblown, often vented)
(the above could be for example a cheap "Irish flute", a pennywhistle (with a fipple), or indeed an Indian Bansuri. Other similar flutes have more or less holes, the Japanese Shakuhachi has only 5, the carnatic bansuri 7, but the basic principle is the same.)
Over time, in Western Europe, people wanted to be able to play more chromatic notes, but they ran out of fingers to drill more holes. Half holing is always an option (it's what bansuri players do), but it's difficult and imprecise in fast passages. Some "forked" or "veiled" fingerings will sometimes work on these flutes. What this means that instead of just changing the length of the tube, you flatten that note by covering some holes lower down the tube. With an instrument like the french horn, this is comparable to putting your hand in the bell to lower the pitch a little, the tube becoming slightly obstructed.
So if C# is ○○○ ○○○,
You can flatten it to C by covering more holes, so:
○●○ ○○○ , ○●● ○○○, or even ○●● ●○○ (depending on the flute one of these will be pretty much in tune)
Over time flute makers began to understand more and more how to position these 6 holes so these forked/veiled fingerings were in tune. The baroque flute (as seen here
) still only had 6 holes (and 1 key, for E flat), but these holes were tactically placed (and sized) to allow a lot of options for various fingerings.
Note: this is still pretty much just a stick with 6 holes in it (and 1 key at the end to open a 7th hole, but that was it). The specific placement of these holes was perfected to be manipulated (incredibly skillfully) to play all the extra chromatic notes though, by using these "cross-fingerings". This resulted in a flute with many many alternative fingerings for the same notes, with different ones being slightly flatter/sharper (pitch) darker/brighter (timbre). (Note that while to a modern musician this "inconsistency" of tone and pitch may seem like a defect, this was not actually not necessarily undesirable for the music of the time. But going into more detail on that would make this answer even longer...)
e.g.
●●● ●●● D
●●● ●●● □ D#
●●● ●●○ E
●●● ●○● F
●●● ●○○ F#/Gb
●●● ●○○ □ F#/Gb option 2
●●● ○●● F#/Gb option 3
●●● ○○○ G
●●○ ●●● G#
●●○ ○○○ A
●○● ●●○ Bb
●○● ●○● option 2
●○● ●●● □ option 3
●○○ ○○○ B
etc.
The above is for the first octave, but as you get higher, the vented fingerings become more and more complicated, with more and more alternative options that are slightly sharper/flatter darker/brighter louder/quieter. This essentially meant that the further away from the "home key" (D) you got, the more of a pain it was to play. There were no "sonata for flute in Ab"s in the baroque days...
This flute gradually evolved, with more and more keys being added to open extra holes here and there, finally ending up with the concert flute as used at the beginning of the 19th century (now usually called the "simple system" or "old system" flute). So now rather than being a stick with 6 holes on it, it's stick with 6 holes on it and a bunch of other holes covered by metal keys that you can open with a bunch of levers and stuff. (This incidentally is the flute that is still used by Irish flute players to this day, albeit with a different technique than the classical players used to use, and generally the keys are mostly ignored, or even removed).
If you look at the fingering and mechanism of this flute, with all the additional holes and keys here and there, and alternate fingerings for each note, it most certainly does not follow a simple "logical" system. But, it evolved over time, kept all the familiar fingerings for the more simple notes, and allowed the flute to be "more or less, with a lot of finesse", chromatic.
EXAMPLE HERE
The modern flute
So with that out of the way, how would we design a "perfect" chromatic instrument. We want it to play equally as well in all keys, with a strong and even tone that doesn't change throughout the chromatic scale. So that means no veiled fingerings: with the same principle as a diatonic flute, open 1 hole,move up one note, except this time we do it for the chromatic scale not just the diatonic one. Take a big tube, and drill (at least) 11 holes in it (for all 12 notes). And the fingering would be something like this:
●●●●●●●●●●● D
●●●●●●●●●●○ D#
●●●●●●●●●○○ E
●●●●●●●●○○○ F
●●●●●●●○○○○ F#
●●●●●●○○○○○ G
●●●●●○○○○○○ G#
●●●●○○○○○○○ A
●●●○○○○○○○○ A#
●●○○○○○○○○○ B
●○○○○○○○○○○ C
○○○○○○○○○○○ C#
Of course, the biggest problem here is the most obvious one: more notes than fingers.
The modern flute we play now is the Boehm flute, and in fact its whole purpose is to achieve more or less what is shown above, in the simplest way possible: a fully vented flute with 1 large hole for each semitone, each of which is progressively uncovered as you go up each semitone, (effectively shortening the length of the pipe).
Us not having enough fingers though, the genius of Boehm was 1) coming up with this concept itself for a "fully vented" chromatic flute (not just a flute that starts with the assumption of the diatonic flute and then tries to "fix it" to make it chromatic) and 2) coming up with a system of keywork that allows us to control it in a manageable way without needing essentially another hand...
The Principle of the the Boehm system flute's is a "fully vented" system: large, always open tone holes that are opened one by one throughout to go up the chromatic scale, with no veiled or "forked" fingerings. This marked the biggest change the the design of the flute in hundreds of years. Up to this point, changes to the western flute had been incremental: this was a complete step change. Up until this point, the keys added to the flute were closed when not in use. Essentially, when not being used to get the "extra notes", the flute still only had 6 small holes open. The Boehm flute completely scrapped the "6 finger holes in a wooden tube plus some keys" model that had existed for literally hundreds and hundreds of years, and went with a metal tube, with many (far more than fingers on our hands) very large (far bigger than our fingertips) holes on it. This resulted in not only a fully chromatic instrument, but a LOUD one.
The much increased volume, and consistent tone throughout the flutes range and ease of playing ALL keys resulted in its fast adoption through most of Europe (with very minimal modifications) and it has remained almost unchanged for over 150 years at this point (whereas the "old system" flute gradually changed and evolved throughout its life, and always had many competing variations at any given time).
If you look at the fingerings on your flute going chromatically upwards from low C, you will notice that, with the exception of B flat and F sharp, the holes open 1 by one going up the tube, like this
● ●●●●●●●●● ● ● ● C
● ●●●●●●●●● ● ● ○ C#
● ●●●●●●●●● ● ○ ○ D
● ●●●●●●●●● ○ ○ ○ Eb (pinky opens hole at the back)
● ●●●●●●●●○ ○ ○ ○ E
● ●●●●●●●○○ ○ ○ ○ F
● ●●●●●●○○● ○ ○ ○ F# *
● ●●●●●○○○○ ○ ○ ○ G
● ●●●●○○○○○ ○ ○ G#
● ●●●○○○○○○ ○ ○ ○ A
● ●●○○○○●○○ ○ ○ ○ Bb *
● ●○○○○○○○○ ○ ○ ○ B
● ○○○○○○○○○ ○ ○ ○ C
○ ○○○○○○○○○ ○ ○ ○ C#
○ ●●●●●●●●● ● ○ ○ d
If you had an extra hand you wouldn't need the complicated mechanism, and you could just play the flute directly as it is above, but without that, you need a fingering system that triggers these keys.
In fact the earliest (and now pretty much omnipresent) modification to the original Boehm system flute, the closed G#, was a step away from the most "logical" way of triggering these holes. As you can see above: G has one more hole covered than G#, but we finger it the opposite way (using the left hand little finger to open a hole not close it). Originally the "G# key" was just a normal key, pressed down by the little finger of the left hand. Although this was more "logical" chromatically speaking, people disliked having to constantly use the little finger on the left hand to play a G and every single note below it, and so the closed G# became more popular. While this may seem illogical, imagine having to keep the left hand pinkie constantly depressed down to play G, F, F# E, Eb and D. It's a pain... (Note that this is still the situation with the right hand, and the right hand pinkie is to this day one of the more tricky parts of the Boehm flute, causing many students problems).
If you look above this is very nearly a perfect key system: the starred fingerings are the only compromises, and it's easy to see why they were done. Yes, it causes ever so slightly veiled notes, but without these mechanisms for a later hole to close an earlier one, you simply wouldn't have enough fingers to play all the notes!
And how bad are these compromises? Well, with the Bb, it's so far down the tube that it makes almost no difference, and it's possible to play a "perfect" B flat with the thumb key or Bb shake if you want to compare (and I can't hear the difference on my flute at all). With the F sharp it does make a (very small but admittedly audible) difference, and you can tell if you manually close the hole to the left of RH1 and then compare by using RH4 to close the F#, you can hear that a "true" F# is ever so slightly brighter. Also note that the "saxophone" or trill fingering of F# (LH T123 RH 2 4 instead of LH T123 RH 3 4) results in ● ●●●●●●○●○ ○ ○ ○ * instead of * ● ●●●●●●○○● ○ ○ ○ (and is even more veiled). But as a compromise to be able to finger that extra note we need easily, I'd say it's worth it (and you really can barely hear it).
This still leaves another question, why not have all the same fingerings in all octaves?
The answer is "venting". The physics of it is pretty complicated, but basically, yes, if you work hard enough, you CAN use the first octave's "logical" fingering to play up and up in higher octaves. Theoretically of course you could keep going up and up infinitely, playing more and harmonics (like on brass instruments), but the real world physics of overblowing an imperfect pipe starts to get in the way of this, and it becomes pretty much impossible at about B. A "perfect" (physically speaking) flute would be an unobstructed tube, 100% open at both ends, this could play up and up the harmonic series. What a flute actually is is a tube with someone's face at 1 end and holes being opened up at the other end; and while these holes are very large on the Boehm flute, they're not infinitely large.
Luckily, physics also has an answer to this problem: venting: when overblowing (playing upper harmonics of a note) on a flute, you get a clearer, brighter tone by opening up the pipe in certain tactical places. On baroque flutes in order to play the higher octaves, a lot of half holing, minute venting and skill was needed, but the Boehm flute has so many options for holes to open and close independently through the complex key system that it's possible to use these "vented" fingerings to play much higher than would be possible otherwise:
Look at all those holes above: pretty much every one can be opened and closed independently (plus trill keys). This means that you can coax our metal tube to play clearly and in tune up to top C and beyond, by opening up the right holes in the right places to force certain harmonics to resonate properly. Compare a top G with and without the thumb: it's possible to overblow a bottom G up to the 3rd octave, but opening the thumb key to "vent" the note makes it just "pop" into place, and sound much clearer.
TL;DR
The modern flute is more or less logical: keys are opened up 1 by 1 to shorten the length of the pipe. The Boehm flute is designed to follow this principle as close as possible, opening up 1 more key for each semitone, gradually shortening the length of the pipe, with 3 exceptions**
1) There are more notes than we have fingers, so a couple of tricks are needed to allow us to control all the keys. This means that the system of uncovering holes 1 by one going up the flute isn't quite followed for F# (or for one of the B flat fingerings) due to a workaround to let us use the same finger on the same key for multiple functions. However these imperfect "veiled" fingerings on the modern flute are very rare, and barely audible in their difference from a "perfect" note.
2) For high notes, the system of simply opening holes 1 by 1 isn't actually what we want. The non ideal physics of a real flute in the real world means we actually need to open up other (apparently random but not) keys in the middle of the tube to get the highest notes to play clearly and in tune (and to get the very highest notes to even play at all), which is why the fingerings appear to become "illogical" in the highest register. This is called "venting", and is a way of extending the range of a woodwind instrument beyond what can be achieved by simply overblowing the harmonic series.
3) Extra keys are added to allow easier fingering workarounds for certain fast passages. These aren't acoustically necessary, but they give a few extra options to work around our imperfect hands. Oh and there are a couple of keys on the bottom to give a few extra low notes, why not eh?
links:
http://www.oldflutes.com/hybrid.htm
http://robertbigio.com/cleverandrich.htm
http://www.kingmaflutes.com/mySite/ksback.html
http://www.oldflutes.com/articles/reform.htm
