This flute warmup is one that flutist/conductor/educator Sam Baron (1925-1997) often shared with his students. I have found it particularly useful for finding the low register at the beginning of one’s practice session. Each succeeding measure differs by only one note from the previous one so that it’s quite easy to memorize. Every 5-note grouping should be played initially without vibrato and then with vibrato on the repeat. The low register of the flute is typically the weakest regarding volume and is what most flutists obsess about. However, it serves as the foundation for creating a resonant and well-balanced middle and upper register. Therefore, it needs to be given special attention early in a practice session so that there are no lower register notes that are weaker than any other.
This exercise appears in my woodwind doubling book with my addition of matching the open C# with the first overtone based on the low C# fingering. When these notes are of the same pitch and quality, then the scale of the flute will have the best chance of sounding “in-tune” and the air has a better chance of being properly directed towards the back wall of the head joint. I have found this exercise very helpful when there is only a short amount of time available to warm up.