I'm sure that this is a frequent question, but when I went to search it up, I only saw one topic that didn't really help me. So, I guess I'll give this a shot.

I have been playing the flute for about five years now, and I am taught by a private instructor. She, however, doesn't seem to find anything wrong with my low notes (mostly low D and down to B), but they really bother me. They're too quiet, and some days the notes won't even come out at all. They don't sound particularly breathy, they're just quiet. They've been like this for a few years now.

I'd like to get my notes stronger, but I'm not sure how. Whenever I try to increase the air speed, I just crack and go up an octave. I've been doing long tones for my problem low notes religiously, but I've come to the conclusion that maybe I'm not even playing them right. Aargh!

What I usually do to play my low notes is direct my airstream downwards just a tad and decrease air speed (because they crack otherwise). Do you guys have any advice or articles that would maybe help me?

It could be a problem with the flute. The low register requires that everything work together just right. Even a tiny leak can make a big difference when you're playing that low. I'd let a tech check out the flute and see what they think. It sounds like you've got a good idea of what you need to be doing to get those pitches. I wouldn't recommend decreasing air speed when you're gunning for the low notes, but rather air pressure. Decreasing the speed will affect your tone negatively. To decrease pressure while maintaining speed, you need a bigger aperture, so try relaxing more when you get to the bottom of the range. One more thing, though. Long tones by themselves don't do a whole lot. You have to go somewhere with them for them to matter. In this case, I would suggest slurring between two pitches chromatically, crescendoing into the second pitch to help expand your dynamic range in the bottom. Also, some flutes are geared towards one end of the range more than the other, which depends largely on how they are built. It could be that your flute is geared more towards the middle/upper registers than the low one. Perhaps experimenting with other headjoints could help you get a bigger low register, or your flute just isn't resistant enough for your playing style.

nothing against your current teacher, but go out and get a second opinion or play in a master class setting or workshop, this could be many problems, but needs to be seen to be addressed, if you do take a lesson with another teacher, be sure to tell your current teacher