Playing in tune is a very important part of good flute playing. There are so many variables at play but underpinning everything is the fact that the flute is based on the harmonic series.
The bottom register of the flute is the fundamental note (the lowest note possible with any given fingering), the second register notes are derived from the first harmonic (or partial) of the low register notes and the upper register pitches come from either the 2nd or 3rd harmonic of the low register, fundamental note.
Using Western notation this harmonic series is written as the first harmonic being one octave above the fundamental, the second harmonic a fifth above that and the third harmonic two octaves above the fundamental. The truth of the matter is that the distance between the harmonics is not exactly as written. Western equal temperament has slightly altered the natural pitches series. Basically as you go up further through the harmonic series the pitches become sharper. Since the flute is inextricably linked to the harmonic series this means that as you ascend your pitch will become sharper.
If you are doing nothing to adjust your pitch as you ascend on the flute you will be out of tune.
So the first step in developing good intonation is the realisation that the flute is based on the harmonic series, therefore it naturally pushes sharper as you ascend. You need to be doing something about that if you want to play in tune.
In future posts I will be looking at ways to adjust and control pitch plus some exercises to develop your ability to play with good intonation.
As a first exercise, finger any low register note (low C, C3 or D are good) and without changing your fingers blow through the harmonic series noticing how out of tune these harmonics actually are.