Tips for developing a warm, clear flute tone.
Spend a minute or two each day playing just the head joint. This helps you focus on your air stream and embouchure without worrying about fingers.
Direct your air stream across the embouchure hole, not straight into it. Imagine blowing across the top of a bottle.
Your lips should be relaxed and flexible, not tight or pinched. A small, focused opening works better than a wide one.
As you play, really listen. Is your tone clear or fuzzy? Smooth or wobbly? Adjust your air and lips until it sounds the way you want.
Playing long, sustained notes is the single best exercise for improving your tone. Even 3 minutes a day makes a difference.