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 θ of the isothermal layer are identified. In the steady regime I with θ ≈ π /2 , the flow is weak and heat transfer is dominated by conduction. In the transitional regime II with rapidly increasing θ , kinetic and thermal boundary layers gradually develop. In the turbulent regime III with θ ≈ 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 Nu-1~ RaDγ ₁Prη ₁ and Reynolds number Re~ RaDγ ₂Prη ₂ with respect to Rayleigh–Darcy number RaD and Prandtl number Pr . We derive (γ ₁,γ ₂,η ₁,η 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 Pr=1 are proposed to integrate the three regimes and are applicable across a broad range of φ and RaD , 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.