Prandtl number effect on heat transfer and flow structures in Rayleigh–Bénard convection modulated by an oscillatory bottom plate

In this paper, we study the Prandtl number effect on Rayleigh–Bénard convection systems modulated by an oscillatory bottom plate. Direct numerical simulations are carried out in a Prandtl number range of $0.2\le Pr\le 4.6$ and a fixed Rayleigh number of $Ra=10^8$. The initial drop and subsequent rise evolutionary behaviour of the heat transfer efficiency, characterised by the Nusselt number at the bottom plate $N{u}_b$, with respect to the characteristic oscillatory velocity $V_{osc}$ is observed in the whole parameter space under consideration. If the oscillatory bottom plate does not induce boundary layer instabilities but thickens the boundary layer only, then one observes a heat transfer reduction, corresponding to a high $Pr$ and a low $V_{osc}$. If periodic boundary layer instabilities are triggered, then both heat transfer reduction and enhancement are possible. The reduction is generally seen when $Pr\le 1.0$. Under such circumstance, the velocity boundary layer is embedded in the thermal boundary layer, if the instability induced by a certain $V_{osc}$ is not strong enough to compensate the heat resistance of the thermal boundary layer, one still observes a reduction in spite of the boundary layer instabilities. The enhancement is generally seen for a low $Pr$ and/or a high $V_{osc}$, in which case violent boundary layer instabilities will be triggered, leading to a sufficient emission of hot plumes. Furthermore, the critical velocity ${\bar{V}}_c$, characterising the boundary layer instability, is found to be increasing with $Pr$ as $\overline{V_c}\sim P{r}^{0.5}$; and the Reynolds number at the equilibrium state evolves in a similar way as $N{u}_b$. In the end, modal analyses are performed based on standard and extended proper orthogonal decompositions. Energetic contribution of the modes and modal distributions confirm well the modulation of the oscillatory bottom plate and the induced boundary layer instabilities on the heat transfer and flow structures of the convection system.