I have been playing the flute for frigging ever… and I STILL can’t play smoothly and easily over the break. In fact, it feels like the scourge of flute playing to me. There’s got to be an answer to this, because I hear people playing and it sure sounds like they know how to do it. Do you??? If so, I’m all ears!
Mmmm. The obvious one word answer is “practice”… but you imply you’ve been doing that.
Are you having trouble altogether with sounding the 2nd 8ve, or just with fingering sequences through the break?
My dim recollection of my early Bohm flute days are of “the break” being built up as some kind of big technical challenge and it being a major achievement to "go over " it. I don’t think that is at all a helpful approach, and if you have a mental block you need to find a way to dump it, or at least to sidestep it.
For starters, in one sense you only have to “go over” the break (on any kind of flute or whistle) in terms of fingering when you do a scale through it, or a scalar passage in a tune… the tricky aspects being the open C# and not dropping the flute (get your hold right!) and the finger swaps from any of B, Cnat or C# to D or E. Those are just an issue of co-ordination, and yes, practice. Slow motion practice of specific tune-contextual sequences are probably the most constructive approach, followed by slow whole scale and stepwise and alternating exercises (“ambulance practice!”).
The other aspect is embouchure, of course. That really ought to be less of a problem - 2nd 8ve notes are just notes, forget the fingerings per se for the minute. Just play a familiar 1st 8ve mid-range note such as A or G or F# and find your way to making it overblow to the harmonic not by blowing harder, but by reducing the size of your embouchure aperture while maintaining the same breath pressure, and also experiment with shifting the aim of your airstream up and down. Work on slurred octaves, note by note, very slowly to start, ever so gently pushing at each low note until it jumps up the octave and then clarifying its tone and making its pitch accurate by further embouchure adjustment, then dropping it back again, all without tongue-attack. Work on this until you can do a steady, full scale of them from bottom D (make sure you practice venting the middle D with L1) up to B at least - beyond that, you need special fingerings for high Cnat, C# and 3rd 8ve D, so don’t go there as yet.
When you have confidence in your support of the flute, about the accuracy and reliability of your embouchure octave shifting in both directions and in your fingering patterns in various note alternations across the break, you’ll wonder what the problem was! Meantime, a bit of self-analysis to define the source of your problem(s) in terms of the three aspects I’ve mentioned is in order. Just doing that may lead you to your own solution, or at least you might then like to tell us what flute you are playing and what method of holding it you have adopted as well as what you perceive to be your problems?
What Jem said.
Plus solid breath support over the break.
Clinton
playing upwards through the break I find a lot easier than playing downwards through it (probably the same for scale runs anywhere on the instrument though). My biggest difficulty is not so much ‘breath’ but consistency of tuning/intonation around the break (e.g. sounding A, B, C or C# rather sharp compared to the d or e above when played in quick succession - making the whole phrase sound unconfident and out of tune). Consciously flattening the left hand notes with lots of embouchure practice seems to be the only answer, like Jem says.
‘The break’, forgive my ignorance, but it’s not a term I have ever heard used. emrys I assume you are refering to the transition from one octave to the next?
The other posters here seem to have some knowledge of what this break is, but I am wondering do you mean playing through the end of a tune and into the repeat?
Or am I on a different wavelength? 
all to no fingers or none to all fingers
not always the octave change (clarinet)
it is a common woodwind term.
When my wife and I separated about 10 years ago, I found it difficult to keep fiddling smoothly over the break.
I had not heard this term used this way before.
However, you just lifts 'em off, or puts 'em down, right? A little woodshedding practicing those couple of notes should do it. As for embouchure, one has to adjust for any interval.
After an interval of 10 years, I am now almost fully adjusted.
I was part way into my first cup when I made it up… I didn’t entirely agree with it at the time and it is not getting any better. However…
Practice would be the answer to most of the pressing issues, innit 
I’m almost with Denny on this one - and it was a term I encountered in this usage (and I don’t think I’ve heard it in another) in my earliest flute-playing days (Tune-a-Day Book 1 stuff). However, to me it has a combination meaning (for flutes, whistles, oboes, even recorders [they’re a bit weird, though] - dunno about clarinets - they’re really weird). Of course, those two things coincide on all the instruments which overblow at the octave. Yes, it means the big finger-swap where you go back to the beginning in the scale sequence of fingerings, but it also means where you start to use the harmonic series (i.e. overblow) - that is, the 2nd 8ve D fingering on all except clarinet and recorder (the thumb-hole of which partially acts as a high D tone-hole, but because it is a speaker-hole too, has to be flattened with L2 - and the first harmonic used and the big fingering swap is to the E).
“The Break” in this sense is, so far as I know, a normal term in British English woodwind usage - I can’t recall ever using the term in conversation with a classical player at least who didn’t know it. Maybe it isn’t current in US wind teaching? It may, though, often not be much thought about once one has conquered it - but beginners can get hung up on it if it is presented as some kind of mountain to climb, which it should not be/is not.
Incidentally (to give an old hobby-horse of mine a gallop), this fact is one of the defining features which I see as indicating that even the Bohm flute is still a D flute with an extended foot. Its mechanism may facilitate a straight-off-in-order fingering sequence of C major, but in the standard fingerings/usage, the overblown C and C# are not used and the first harmonic/overblown note is the 2nd 8ve D.
even recorders [they’re a bit wierd, though]
… but even on one of THE definitive recorder sites http://www.dolmetsch.com/index.htm they use this term when discussing the octave change … can’t quite find the instance at the moment, but I saw it there a couple of days ago. I was quite pleased to see what I had assumed to be the case to be confirmed by an “authoratitive source” 
Talking of “hobby horses”, if you consider a (baroque fingering) recorder as a D whistle with an extended foot, it can become a lot less wierd
They’re the exception to the octave, eh?
Clarinet is an Odd fellow, when they get a Fifth in them.
Quite. What I do know about a one-end-closed cylinder (viz clarinet) is that it overblows to the 12th. What I don’t know is the clarinet fingering scale or how clarinets utilise/overcome their acoustical properties, or even how they are holed - hence I have no idea where the fingering sequences “break” nor how those patterns relate to register changes (using the harmonics).
Thank you everyone, especially “jemtheflute.” You gave me a great deal to chew on, and I really appreciate your expertise and well-written answer! Yes, I do practice a reasonable amount, and I think my main problem is with smoothly fingering over the break. It annoys the heck outta me to have a “bump” in the music, when there shouldn’t really be one there, musically-speaking. I’ve done plenty of slo-mo practicing, but my fingers just don’t naturally want to work together, so it seems.
Seems to me I’ve used the terms “register break” and “octave break” since my earliest days playing winds here in the good ol’ USofA.
Boehm clarinet fingering is straightforward, very similar to Boehm flute. The same pattern in both registers (except at the throat), with the upper note names displaced from the lower (chalumeau) by a 5th. The break is between Bb and B, and the overblow is from bell E to the B a 12th above. Third register is a different story, as usual.
If you play both C and F recorders, clarinet is pretty neat. The lower register fingers (roughly) like an F recorder, and the upper like a C recorder. It’s like 2-in-1.
I have to back up those who know the term “break” - But I also used to play clarinet and other woodwinds in a UK environment.
All Jem the Flute’s advice seems good to me. I would only add one exercise - play a low D and support the air well enough that it will jump into the second octave when you lift your top finger without any other changes to embouchure, breath etc. Not quite sure how to say what this is good for, but it did give me a good feeling for finding the point where the transition happens. And practice the overtones above the first as well.
Not dropping the flute and getting all the fingers on the holes simultaneously really is a matter of practice and making sure that your physical support of the flute is secure. I still have big problems with Middle D followed by a B roll on the wooden flute.
Strangely, although I have no psychological problems with the first/second register transition, I do often have problems with the second/third and with the Low C# and C. These problems are not there if I am playing happily at the top or bottom end for a while and only happen when I play a tune which only just dips briefly into the third octave or only just dips below the low D.