Onset of fractional-order thermal convection in porous media

The macroscopic description of buoyancy-driven thermal convection in porous media is governed by advection-diffusion processes, which in the presence of thermophysical heterogeneities fail to predict the onset of thermal convection and the average rate of heat transfer. This work extends the classical model of heat transfer in porous media by including a fractional-order advective-dispersive term to account for the role of thermophysical heterogeneities in shifting the thermal instability point. The proposed fractional-order model overcomes limitations of the common closure approaches for the thermal dispersion term by replacing the diffusive assumption with a fractional-order model. Through a linear stability analysis and Galerkin procedure, we derive an analytical formula for the critical Rayleigh number as a function of the fractional model parameters. The resulting critical Rayleigh number reduces to the classical value in the absence of thermophysical heterogeneities when solid and fluid phases have similar thermal conductivities. Numerical simulations of the coupled flow equation with the fractional-order energy model near the primary bifurcation point confirm our analytical results. Moreover, data from pore-scale simulations are used to examine the potential of the proposed fractional-order model in predicting the amount of heat transfer across the porous enclosure. The linear stability and numerical results show that, unlike the classical thermal advection-dispersion models, the fractional-order model captures the advance and delay in the onset of convection in porous media and provides correct scalings for the average heat transfer in a thermophysically heterogeneous medium.

Figure 1

Schematic showing a saturated porous square enclosure and the choice of thermal boundary conditions. No-slip velocity, i.e., ($u,v=0)$ is applied at all the walls.

Figure 2

Schematic showing the conduction [panel (a)] and convection stable states [panel (b)] for Rayleigh numbers below and above the critical value, respectively.

Figure 3

Comparison of the predicted ${\text{Ra}}_{\text{cr}}$ for different values of $\alpha$ based on single-term (lines) and two-term approximation (symbols) in the Galerkin procedure; (—, $\square$) for $C_{\text{dis}}=1.0$, and ($--,\circ$) for $C_{\text{dis}}=0.8$. The horizontal dotted line indicates the classical value of $4{\pi }^2$ based on integer-order HRL problem.

Figure 4

Map showing the variation of ${\text{Ra}}_{\text{cr}}$ for different values of $\alpha$ and $C_{\text{dis}}.$

Figure 5

Comparison of the predicted ${\text{Ra}}_{\text{cr}}$ for different dispersivity $C_{\text{dis}}$ based on linear stability analysis (lines) and numerical solution of the nonlinear equations (symbols).

Figure 6

(a) The dependency of the observed anomalous behaviors in ${\text{Ra}}_{\text{cr}}$ and Nusselt-Rayleigh curves on the solid-to-fluid thermal conductivity ratio $k_s/k_f$. The solid line in panel (a) is associated with the predictions based on the classical HRL problem with $C_{\text{dis}}=\alpha =1.0$, agreeing with the pore-scale simulations for $k_s/k_f=1$ ($\diamond$ symbols) [25]. The dashed and dotted curves are the predictions of the fractional-order model qualitatively agreeing with the anomalous behaviors observed in pore-scale results [25] for $k_s/k_f<1$ ($\square$ symbols) and $k_s/k_f>1$ ($\circ$ symbols), respectively. (b) Valid ranges of $C_{\text{dis}}$ and $\alpha$ for satisfying both aspects of anomalous behaviors observed in the pore-scale results [25] for different solid-to-fluid thermal conductivity ratio $k_s/k_f$. The gray regions in panel (b) indicate the values of $\alpha$ and $C_{\text{dis}}$ out of the suitable ranges for $k_s/k_f<1$ and $k_s/k_f>1$ cases.