  We present high-resolution numerical investigations of heat transport by
two-dimensional (2D) turbulent Rayleigh-B\'{e}nard (RB) convection over the
Rayleigh number range $10^8 \leqslant Ra\leqslant 10^{10}$ and the Prandtl
number range $0.7\leqslant Pr \leqslant10$. We find that there exist strong
counter-gradient local heat flux with magnitude much larger than the global
Nusselt number $Nu$ of the system. Two mechanisms for generating
counter-gradient heat transport are identified: one is due to the bulk dynamics
and the other is due to the competitions between the corner-flow rolls and the
large-scale circulation (LSC). While the magnitude of the former is found to
increase with increasing Prandtl number, that of the latter maximizes at medium
$Pr$. We further reveal that the corner-LSC competitions lead to the anomalous
$Nu$-$Pr$ relation in 2D RB convection, i.e. $Nu(Pr)$ minimizes, rather than
maximizes as in three-dimensional cylindrical case, at $Pr\approx2\sim3$ for
moderate $Ra$.
