  Results from direct numerical simulations of vertical natural convection at
Rayleigh numbers $1.0\times 10^5$-$1.0\times 10^9$ and Prandtl number $0.709$
support a generalised applicability of the Grossmann-Lohse (GL) theory, which
was originally developed for horizontal natural (Rayleigh-B{\'e}nard)
convection. In accordance with the GL theory, it is shown that the
boundary-layer thicknesses of the velocity and temperature fields in vertical
natural convection obey laminar-like Prandtl-Blasius-Pohlhausen scaling.
Specifically, the normalised mean boundary-layer thicknesses scale with the
$-1/2$-power of a wind-based Reynolds number, where the "wind" of the GL theory
is interpreted as the maximum mean velocity. Away from the walls, the
dissipation of the turbulent fluctuations, which can be interpreted as the
"bulk" or "background" dissipation of the GL theory, is found to obey the
Kolmogorov-Obukhov-Corrsin scaling for fully developed turbulence. In contrast
to Rayleigh-B{\'e}nard convection, the direction of gravity in vertical natural
convection is parallel to the mean flow. The orientation of this flow presents
an added challenge because there no longer exists an exact relation that links
the normalised global dissipations to the Nusselt, Rayleigh and Prandtl
numbers. Nevertheless, we show that the unclosed term, namely the
global-averaged buoyancy flux that produces the kinetic energy, also exhibits
both laminar and turbulent scaling behaviours, consistent with the GL theory.
The present results suggest that, similar to Rayleigh-B{\'e}nard convection, a
pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers
is not the best description for vertical natural convection and existing
empirical relationships should be recalibrated to better reflect the underlying
physics.
