  We report high-quality H${\alpha}$/CO, imaging spectroscopy of nine massive
(log median stellar mass = 10.65 $M_{\odot}$), disk galaxies on the
star-forming, main sequence (henceforth `SFGs'), near the peak of cosmic galaxy
evolution ($z\sim$1.1-2.5), taken with the ESO-Very Large Telescope, IRAM-NOEMA
and Atacama Large Millimeter/submillimeter Array. We fit the major axis
position-velocity cuts with beam-convolved, forward models with a bulge, a
turbulent rotating disk, and a dark matter (DM) halo. We include priors for
stellar and molecular gas masses, optical light effective radii and
inclinations, and DM masses from our previous rotation curve analyses of these
galaxies. We then subtract the inferred 2D model-galaxy velocity and velocity
dispersion maps from those of the observed galaxies. We investigate whether the
residual velocity and velocity dispersion maps show indications for radial
flows. We also carry out kinemetry, a model-independent tool for detecting
radial flows. We find that all nine galaxies exhibit significant non-tangential
flows. In six SFGs, the inflow velocities ($v_r\sim$30-90 km s$^{-1}$, 10%-30%
of the rotational component) are along the minor axis of these galaxies. In two
cases the inflow appears to be off the minor axis. The magnitudes of the radial
motions are in broad agreement with the expectations from analytic models of
gravitationally unstable, gas-rich disks. Gravitational torques due to clump
and bar formation, or spiral arms, drive gas rapidly inward and result in the
formation of central disks and large bulges. If this interpretation is correct,
our observations imply that gas is transported into the central regions on ~10
dynamical time scales.
