Harmonics are produced by overblowing a lower fundamental pitch. One harmonic can have different fundamental notes from which it is produced. Each of those fingerings influences the tone colour (timbre) of the harmonic. The harmonic series above any fundamental is always: Fundamental, octave, fifth, octave, third, fifth, 1/4-flat seventh, octave, etc. (see diagram below). There are no harmonics possible in the lowest octave.
Composers should note that, on any given fundamental, the higher the partial, the greater the resistance of the flute to produce the pitch. This means that very high partials of a very low fundamental may not be feasible at a quiet dynamic. There are almost always multiple options for an harmonic pitch, each with their own timbre depending on the base fundamental. These fingerings can be used in succession to create sutble microtonal shifts in the pitch. A fast change between two harmonic fingerings is called Bisbigliando or Timbral Trill. The second and fourth harmonics of a fundamental should be in tune with the normal fingering. The third and fifth harmonics of a fundamental tend to be flat compared to the normal fingering with a short air column (left hand fingerings), and mostly in tune with a long air column (both hands). They function the same on piccolo, Alto- and Bass-flute.
To play the Harmonic series, finger the desired fundamental pitch, and without changing fingerings, overblow level by level. The support needed to produce the higher harmonics is the same as that needed for the normal fingering. The character of an Harmonic has a transparent and suspended quality of sound, usually without vibrato. Harmonics are a timbral color effect o the sound. In addition to the actual Harmonic fingerings, you can experiement with using other altered fingerings (trill keys, etc) to create this timbral effect. These are often used as alternative fingerings in fast passages of traditional repertoire. They function the same on piccolo, Alto- and Bass-flute.
Harmonic series on low C
Melodic sample alternating between Harmonics and normal fingerings