  We present results on the effect of dispersed droplets in vertical natural
convection (VC) using direct numerical simulations based on a two-way fully
coupled Euler-Lagrange approach with a liquid phase and a dispersed droplets
phase. For increasing thermal driving, characterised by the Rayleigh number,
$Ra$, of the two analysed droplet volume fractions, $\alpha = 5\times10^{-3}$
and $\alpha = 2\times 10^{-2}$, we find non-monotonic responses to the overall
heat fluxes, characterised by the Nusselt number, $Nu$. The $Nu$ number is
larger when the droplets are thermally coupled to the liquid. However, $Nu$
retains the effective scaling exponents that are close to the ${1/4}$-laminar
VC scaling, suggesting that the heat transport is still modulated by thermal
boundary layers. Local analyses reveal the non-monotonic trends of local heat
fluxes and wall-shear stresses: Whilst regions of high heat fluxes are
correlated to increased wall-shear stresses, the spatio-temporal distribution
and magnitude of the increase is non-universal, implying that the overall heat
transport is obscured by competing mechanisms. Most crucially, we find that the
transport mechanisms inherently depend on the dominance of droplet driving to
thermal driving that can quantified by (i) the bubblance parameter $b$, which
measures the ratio of energy produced by the dispersed phase and the energy of
the background turbulence, and (ii) $Ra_d/Ra$, where $Ra_d$ is the droplet
Rayleigh number, which we introduce in this paper. When $b \lesssim O(10^{-1})$
and $Ra_d/Ra \lesssim O(100)$, the $Nu$ scaling is expected to recover to the
VC scaling without droplets, and comparison with $b$ and $Ra_d/Ra$ from our
data supports this notion.
