Unified scaling laws in porous-medium vertical natural convection

This study conducted theoretical analysis and direct numerical simulations (DNS) of vertical natural convection in a two-dimensional cavity filled with porous media, where the imposed temperature gradient is oriented perpendicular to the direction of gravity. Three regimes characterised by distinct flow states and the angle $\theta$ of the isothermal layer are identified. In the steady regime I with $\theta \approx \pi /2$ , the flow is weak and heat transfer is dominated by conduction. In the transitional regime II with rapidly increasing $\theta$ , kinetic and thermal boundary layers gradually develop. In the turbulent regime III with $\theta \approx 0$ , clear boundary layers arise and turbulent thermal convection prevails. Corresponding to these flow states, theoretical analysis is performed to derive the scaling laws of the Nusselt number $\textit{Nu}-1\sim Ra_{D}^{\gamma _1 }\textit{Pr}^{\eta _1}$ and Reynolds number $\textit{Re}\sim Ra_{D}^{\gamma _2 }\textit{Pr}^{\eta _2}$ with respect to Rayleigh–Darcy number $Ra_D$ and Prandtl number $\textit{Pr}$ . We derive $(\gamma _1,\gamma _2,\eta _1,\eta _2)=(2,1,0,-1)$ for the steady regime and $(1/3,4/9,0,-2/3)$ for the turbulent regime. All theoretical scaling exponents in these two regimes are validated by DNS results. Furthermore, we find that the influence of the Darcy number $Da$ becomes almost negligible when it is sufficiently small. Unified models for $\textit{Pr}=1$ are proposed to integrate the three regimes and are applicable across a broad range of $\phi$ and $Ra_D$ , which are satisfactorily verified by DNS results. The unified models provide a predictive framework for heat transport and flow intensity in porous-medium thermal convection, thereby offering practical values for thermal engineering applications.