  Previous numerical studies on homogeneous Rayleigh-B\'enard convection, which
is Rayleigh-B\'enard convection (RBC) without walls, and therefore without
boundary layers, have revealed a scaling regime that is consistent with
theoretical predictions of bulk-dominated thermal convection. In this so-called
asymptotic regime, previous studies have predicted that the Nusselt number
($Nu$) and the Reynolds number ($Re$) vary with the Rayleigh number ($Ra$)
according to $Nu\sim Ra^{1/2}$ and $Re\sim Ra^{1/2}$ at small Prandtl number
($Pr$). In this study, we consider a flow that is similar to RBC but with the
direction of temperature gradient perpendicular to gravity instead of parallel;
we refer to this configuration as vertical natural convection (VC). Since the
direction of the temperature gradient is different in VC, there is no exact
relation for the average kinetic dissipation rate, which makes it necessary to
explore alternative definitions for $Nu$, $Re$ and $Ra$ and to find physical
arguments for closure, rather than making use of the exact relation between
$Nu$ and the dissipation rates as in RBC. Once we remove the walls from VC to
obtain the homogeneous setup, we find that the aforementioned $1/2$-power-law
scaling is present, similar to the case of homogeneous RBC. When focussing on
the bulk, we find that the Nusselt and Reynolds numbers in the bulk of VC too
exhibit the $1/2$-power-law scaling. These results suggest that the
$1/2$-power-law scaling may even be found at lower Rayleigh numbers if the
appropriate quantities in the turbulent bulk flow are employed for the
definitions of $Ra$, $Re$ and $Nu$. From a stability perspective, at low- to
moderate-$Ra$, we find that the time-evolution of the Nusselt number for
homogenous vertical natural convection is unsteady, which is consistent with
the nature of the elevator modes reported in previous studies on homogeneous
RBC.
