Transitions in a Poiseuille-Rayleigh-Bénard flow in a vertical slender long duct

The flow of air inside a vertical slender long duct with a temperature difference between the horizontal walls characterized by the Rayleigh number $Ra$ and a pressure gradient in the horizontal direction characterized by the Reynolds number $Re$ is studied using the lattice Boltzmann method. In this case, the longitudinal rolls found in ducts with a width larger than the height cannot develop and the flow remains quasi-two-dimensional. Therefore, a two-dimensional approach is used, considering periodic boundary conditions on the vertical walls. For $R{a}_c<Ra\le 2.0\times 10^5$ with $R{a}_c$ the critical Rayleigh number for thermal convection, there are two transitions at $R{e}_0\left(Ra\right)$ and $R{e}_{CP}\left(Ra\right)$, $R{e}_0<R{e}_{CP}$. The first transition at $R{e}_0$ has a sharp decrease in the average Nusselt number and for $0<Re<R{e}_0$ the temperature difference between the horizontal walls is more important than the pressure gradient. The second transition at $R{e}_{CP}$ marks the appearance of a conductive Poiseuille flow with no thermal convection. For $1.0\times 10^5$ ¡ $Ra$ there is a third transition at $R{e}_M$, $R{e}_0<R{e}_M<R{e}_{CP}$. For $Ra<$ $1.0\times 10^5$ and $R{e}_0<Re<R{e}_{CP}$, and for $1.0\times 10^5$ $<Ra$ and $R{e}_M<Re<R{e}_{CP}$, the pressure gradient dominates over the pressure gradient. Both the temperature difference and the pressure gradient are important for $1.0\times 10^5<Ra$ and $R{e}_0<Re<R{e}_M$ with an appreciable decrease of the average Nusselt number at $R{e}_0$ and $R{e}_M$.