We report on an experimental study of the large-scale flow (LSF) and Reynolds numbers in turbulent convection in a cylindrical sample with height equal to its diameter and heated locally around the center of its bottom plate (locally heated convection). The sample size and shape are the same as those of Narezo Guzman et al. [D. Narezo Guzman, J. Fluid Mech. 787, 331 (2015)JFLSA70022-112010.1017/jfm.2015.701; D. Narezo Guzman, J. Fluid Mech. 795, 60 (2016)JFLSA70022-112010.1017/jfm.2016.178]. Measurements are made at a nearly constant Rayleigh number as a function of the mean temperature, both in the presence of controlled boiling (two-phase flow) and for the superheated fluid (one-phase flow). Superheat values Tb-Ton up to about 11 K (Tb is the bottom-plate temperature and Ton is the lowest Tb at which boiling is observed) are used. The LSF is less organized than it is in (uniformly heated) Rayleigh-Bénard convection (RBC), where it takes the form of a single convection roll. Large-scale-flow-induced sinusoidal azimuthal temperature variations (like those found for RBC) could be detected only in the lower portion of the sample, indicating a less organized flow in the upper portions. Reynolds numbers are determined using the elliptic model (EM) of He and Zhang [G.-W. He and J.-B. Zhang, Phys. Rev. E 73, 055303(R) (2006)PLEEE81539-375510.1103/PhysRevE.73.055303]. We found that for our system the EM is applicable over a wide range of space and time displacements, as long as these displacements are within the inertial range of the temporal and spatial spectrum. At three locations in the sample the results show that the vertical mean-flow velocity component is reduced while the fluctuation velocity is enhanced by the bubbles of the two-phase flow. Enhancements of velocity fluctuations up to about 60% are found at the largest superheat values. Local temperature measurements within the sample reveal temperature oscillations that also used to determine a Reynolds number. These results are generally consistent with the mean-flow EM results and show a two-phase-flow enhancement of up to about 30%.