Form drag effect and Hopf bifurcation in Dupuit-Darcy thermal convection in a shallow well-packed porous enclosure
Bibliographic record
Abstract
In the present paper, numerical and analytical investigations were performed to assess the effect of the form drag force on Dupuit-Darcy convection in a shallow porous enclosure, subject to a vertical thermal gradient. The effect of the form drag on the threshold of Hopf bifurcation (characterizing the transition from steady to unsteady convection) was also investigated. The governing parameters were the Rayleigh number, RT, the porous layer aspect ratio, A, and the form drag coefficient ($1/P_r^*$1/Pr*), characterized by a modified Prandtl number, $P_r^*$Pr*. In the limit of a shallow enclosure, long wave convective flows were possible where a closed form analytical solution was derived, whereas numerical solutions of the full governing equations were obtained for an arbitrary aspect ratio enclosure. Both the numerical and analytical results were found to be in good agreement as long as the flow remains steady. As known, for a large Rayleigh number, the flow intensity and the Nusselt number were found to vary with RT as $\left| {\Psi _0 } \right| = \sqrt {5/12} R_T^{1/2}$Ψ0=5/12RT1/2 and Nu = 6, for a pure Darcy porous medium, respectively. For a pure Dupuit porous medium, where the Darcy flow is negligible, it was found that $\left| {\Psi _0 } \right| = \left( {2/27} \right)^{1/3} \left( {R_T P_r^* } \right)^{1/3}$Ψ0=2/271/3RTPr*1/3 and Nu = 5. In general, the form drag effect was found to have a significant influence on the flow intensity and the heat transfer rate. The threshold for the onset of stationary convection was unaffected. However, as RT is made very large, transition to unsteady convective flow occurred through a Hopf bifurcation. In infinite or finite aspect ratio enclosure, with single or multiple cells convective flows, the threshold for Hopf bifurcation, $R_{{\rm TC}}^{{\rm Hopf}},$R TC Hopf , was found to increase significantly as the form drag coefficient ($1/P_r^*$1/Pr*) increased. Far from the onset of stationary convection, multiple stable convective solutions were found to exist. In a square enclosure with two-cell convection, broken flow pattern symmetry was observed near and beyond the Hopf bifurcation threshold.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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