  To understand how internal flow structures manifest themselves in the global
heat transfer, we study the correlation between different flow modes and the
instantaneous Nusselt number ($Nu$) in a two-dimensional square
Rayleigh-B\'enard convection cell. High-resolution and long-time direct
numerical simulations are carried out for Rayleigh numbers between $10^{7}$ and
$10^{9}$ and a Prandtl number of 5.3. The investigated Nusselt numbers include
the volume-averaged $Nu_{\text{vol}}$, the wall-averaged $Nu_{\text{wall}}$,
the kinetic energy dissipation based $Nu_{\text{kinetic}}$, and the thermal
energy dissipation based $Nu_{\text{thermal}}$. The Fourier mode decomposition
and proper orthogonal decomposition are adopted to extract the coherent flow
structure. Our results show that the single-roll mode, the horizontally stacked
double-roll mode, and the quadrupolar flow mode are more efficient for heat
transfer on average. In contrast, the vertically stacked double-roll mode is
inefficient for heat transfer on average. The volume-averaged $Nu_{\text{vol}}$
and the kinetic energy dissipation based $Nu_{\text{kinetic}}$ can better
reproduce the correlation of internal flow structures with heat transfer
efficiency than that of the wall-averaged $Nu_{\text{wall}}$ and the thermal
energy dissipation based $Nu_{\text{thermal}}$, even though these four Nusselt
numbers give consistent time-averaged mean values. The ensemble-averaged time
trace of $Nu$ during flow reversal shows that only the volume-averaged
$Nu_{\text{vol}}$ can reproduce the overshoot phenomena that is observed in the
previous experimental study. Our results reveal that the proper choice of $Nu$
is critical to obtain a meaningful interpretation.
