Free convection in an undulating saturated porous layer: resonant wavelength excitation

Abstract

Thermal convection in a saturated porous medium contained between two undulating fixed boundaries of mean horizontal disposition is considered when the layer is heated from below. In an analytic study, the amplitudes of the two-dimensional undulations are assumed to be small compared with the mean depth, and the wavelength is taken to be close to the critical wavelength for the onset of Lapwood convection. For values of the mean Darcy-Rayleigh number Ra below the Lapwood critical value Rac an analytical formula is found for the mean Nusselt number. As Ra→Rac, convection driven by baroclinic effects induced by boundary variations is greatly amplified by convective instabilities. The natures of the resultant bifurcations are examined when the configuration is varicose and also non-varicose. Consideration is given to both longitudinal and transverse modes and to the effects of detuning. The effects of finite amplitude and larger Rayleigh number are examined, for the varicose configuration, in a numerical study of two-dimensional convection. Periodic solutions are found and the existence of the flows delimited in the parameter space of Ra and the boundary amplitude a.