  It is shown that the acoustic scaling patterns of anisotropic flow for
different event shapes at a fixed collision centrality (shape-engineered
events), provide robust constraints for the event-by-event fluctuations in the
initial-state density distribution from ultrarelativistic heavy ion collisions.
The empirical scaling parameters also provide a dual-path method for extracting
the specific shear viscosity $(\eta/s)_\mathrm{QGP}$ of the quark-gluon plasma
(QGP) produced in these collisions. A calibration of these scaling parameters
via detailed viscous hydrodynamical model calculations, gives
$(\eta/s)_\mathrm{QGP}$ estimates for the plasma produced in collisions of
Au+Au ($\sqrt{s_{NN}}= 0.2$ TeV) and Pb+Pb ($\sqrt{s_{NN}}= 2.76$ TeV). The
estimates are insensitive to the initial-state geometry models considered.
