  It has recently been proved that, in the presence of vortex flows, the
fluctuation dynamics of a rotating photon-fluid model is governed by the
Klein-Gordon equation of an effective massive scalar field in a
$(2+1)$-dimensional acoustic black-hole spacetime. Interestingly, it has been
demonstrated numerically that the rotating acoustic black hole, like the
familiar Kerr black-hole spacetime, may support spatially regular stationary
density fluctuations (linearized acoustic scalar `clouds') in its exterior
regions. In particular, it has been shown that the composed
rotating-acoustic-black-hole-stationary-scalar-field configurations of the
photon-fluid model exist in the narrow dimensionless regime
$\alpha\equiv\Omega_0/m\Omega_{\text{H}}\in(1,\alpha_{\text{max}})$ with
$\alpha_{\text{max}}\simeq 1.08$ [here $\Omega_{\text{H}}$ is the angular
velocity of the black-hole horizon and $\{\Omega_0,m\}$ are respectively the
effective proper mass and the azimuthal harmonic index of the acoustic scalar
field]. In the present paper we use analytical techniques in order to explore
the physical and mathematical properties of the acoustic scalar clouds of the
photon-fluid model in the regime $\Omega_{\text{H}}r_{\text{H}}\gg1$ of
rapidly-spinning central supporting acoustic black holes. In particular, we
derive a remarkably compact analytical formula for the discrete resonance
spectrum $\{\Omega_0(\Omega_{\text{H}},m;n)\}$ which characterizes the
stationary bound-state acoustic scalar clouds of the photon-fluid model.
Interestingly, it is proved that the critical (maximal) mass parameter
$\alpha_{\text{max}}$, which determines the regime of existence of the composed
acoustic-black-hole-stationary-bound-state-massive-scalar-field configurations,
is given by the exact dimensionless relation
$\alpha_{\text{max}}=\sqrt{{{32}\over{27}}}$.
