  We present a systematic study of $\Lambda$ hyperon's polarization observables
using event-by-event (3+1)D relativistic hydrodynamics. The effects of initial
hot spot size and QGP's specific shear viscosity on the polarization
observables are quantified. We examine the effects of the two formulations of
the thermal shear tensor on the polarization observables using the same
hydrodynamic background. With event-by-event simulations, we make predictions
for the Fourier coefficients of $\Lambda$'s longitudinal polarization $P^z$
with respect to the event planes of different orders of anisotropic flow. We
propose new correlations among the Fourier coefficients of $P^z$ and charged
hadron anisotropic flow coefficients to further test the mapping from fluid
velocity gradients to hyperon's polarization. Finally, we present a system size
scan with Au+Au, Ru+Ru, and O+O collisions at $\sqrt{s_\mathrm{NN}} = 200$ GeV
to study the system size dependence of polarization observables at the
Relativistic Heavy-ion Collider.
