  We report the results of high resolution direct numerical simulations of
two-dimensional Rayleigh-B\'enard convection for Rayleigh numbers up to
$\Ra=10^{10}$ in order to study the influence of temperature boundary
conditions on turbulent heat transport. Specifically, we considered the extreme
cases of fixed heat flux (where the top and bottom boundaries are poor thermal
conductors) and fixed temperature (perfectly conducting boundaries). Both cases
display identical heat transport at high Rayleigh numbers fitting a power law
$\Nu \approx 0.138 \times \Ra^{.285}$ with a scaling exponent indistinguishable
from $2/7 = .2857...$ above $\Ra = 10^{7}$. The overall flow dynamics for both
scenarios, in particular the time averaged temperature profiles, are also
indistinguishable at the highest Rayleigh numbers. The findings are compared
and contrasted with results of recent three-dimensional simulations.
