  The flow inside a precessing fluid cavity has been given particular attention
since the end of the 19th century in geophysical and industrial contexts. The
present study aims at shedding light on the underlying mechanism by which the
flow inside a precessing cylindrical annulus transitions from laminar to
multiple scale complex structures. We address this problem experimentally using
ultrasonic Doppler velocimetry to diagnose the fluid velocity in a rotating and
precessing cylindrical annulus. When precession is weak, the flow can be
described as a superposition of forced inertial modes. Above a critical value
of the precession rate, the forced flow couples with two free inertial modes
satisfying triadic resonance conditions, leading to the classical growth and
collapse. Using a Bayesian approach, we extract the wavenumber, frequency,
growth rate and amplitude of each mode involved in the instability. In some
cases, we observe for the first time ever experimentally two pairs of free
modes coexisting with the forced flow. At larger precession rates, we do not
observe triadic resonance any more, instead we observe several harmonics whose
frequencies are integer multiples of the rotation frequency.
