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Darcy flow in a two dimensional rectangular domain heated at the bottom and cooled at the top with perfectly insulated sidewalls is the topic of interest for this research. For Rayleigh numbers less than the critical value, $Ra_{cr}$, any disturbances will decay to a motionless solution and heat transfer will occur via conduction only. Above $Ra_{cr}$, natural convection develops in the domain. At some second critical Rayleigh number, $Ra_t$, the steady convection cells lose stability and the solution transitions to a weakly turbulent state. An approximate analytical solution was derived, using weak nonlinear analysis, by asymptotically expanding the solution about $Ra_{cr}$, resulting in a nonlinear system of equations that is equivalent to the Lorenz system. This solution is able to more accurately predict the transition from steady convection to weak turbulence because the initial conditions are taken into account in the analysis. However, the approximate analytical solution is only valid for Rayleigh numbers sufficiently close to $Ra_{cr}$. This research investigated the validity domain of the Lorenz system as a model for natural convection in porous media. The temperature and velocity fields given by the Lorenz system were compared to a numerical solution for the temperature and velocity fields for increasing Rayleigh numbers. Near $Ra = 100$, the number of convection cells predicted by the numerical solution increase from two to three resulting in a significant increase in difference between the Lorenz system and the numerical solution. To provide a comparison between the Lorenz solution and the numerical solution that is global in scale relative to the problem domain, the Nusselt numbers resulting from each solution are compared to each other and also to experimental data.
References
[1] Y. Katto and T. Masuoka, “Criterion for the onset of convective flow in a fluid in a porous medium,” Int. J. Heat Mass Transf., vol. 10, no. 3, pp. 297–309, 1967.
[2] P. Vadasz, “Analytical prediction of the transition to chaos in Lorenz equations,” Appl. Math. Lett., vol. 23, no. 5, pp. 503–507, 2010.
[3] P. Vadasz, “Capturing analytically the transition to weak turbulence and its control in porous media convection,” J. Porous Media, vol. 118, no. 11, pp. 1075–1089, 2015.
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