To study turbulent combustion, experiments with expanding, statistically spherical flames ignited by a spark are widely used. The goal of the work is to show that certain trends in the behavior of turbulent flame speed S-t, observed in such experiments, are substantially affected by the curvature of the mean flame brush and by the ignition conditions. For this purpose, simulations of expanding, spherical, premixed flames were performed using the k - epsilon turbulence model and the Turbulent Flame Speed Closure of the balance equation for a progress variable. Three major trends have been observed in the simulations. First, the analysis of various physical mechanisms controlling the increase of S-t has shown that the lime-dependence of the mean heat release rate, invoked by the model, is of substantial importance for small kernels only. For moderately large flames, the development of S-t, is mainly controlled by the relaxation of the reduction effect of the mean flame curvature on the flame speed. The second manifestation of the mean curvature mechanism is the opposite effects of the turbulent length scale L on the speed of asymptotically stationary, planar flames and of moderately large, statistically spherical flames. in the spherical case, a stronger reduction of the flame speed of small kernels is observed in turbulence with a larger scale. As the kernel grows, the reduction effect relaxes and the dependence of S-t on L reverses. Third, when the ignition energy is close to tho critical value igniting the turbulent mixture, a regime of kernel expansion characterized by substantially reduced name speed and burning velocity can occur even in relatively large, statistically spherical turbulent names. The physical cause of this memory effect consists in the formation of a highly dispersed kernel followed by slow after-burning. When the spark energy is kept constant, the increase in turbulent velocity u' increases the critical ignition energy and the transformation to the aforementioned regime occurs. This mechanism can contribute to the decrease of S-t with u', observed in many experiments. Finally, the suppression of counter-gradient diffusion in spherical flames is discussed at the end of the paper.