  Turbulent shear flows driven by a combination of a pressure gradient and
buoyancy forcing are investigated using direct numerical simulations.
Specifically, we consider the setup of a differentially heated vertical channel
subject to a Poiseuille-like horizontal pressure gradient. We explore the
response of the system to its three control parameters: the Grashof number
$Gr$, the Prandtl number $Pr$, and the Reynolds number $Re$ of the
pressure-driven flow. From these input parameters, the relative strength of
buoyancy driving to the pressure gradient can be quantified by the Richardson
number $Ri=Gr/Re^2$. We compare the response of the mixed vertical convection
configuration to that of mixed Rayleigh-B\'enard convection and find a nearly
identical behaviour, including an increase in wall friction at higher $Gr$ and
a drop in the heat flux relative to natural convection for $Ri=O(1)$. This
closely matched response is despite vastly different flow structures in the
systems. No large-scale organisation is visible in visualisations of mixed
vertical convection - an observation that is quantitatively confirmed by
spectral analysis. This analysis, combined with a statistical description of
the wall heat flux, highlights how moderate shear suppresses the growth of
small-scale plumes and reduces the likelihood of extreme events in the local
wall heat flux. Vice versa, starting from a pure shear flow, the addition of
thermal driving enhances the drag due to the emission of thermal plumes.
