We study the stability and dynamic transitions of thermal convection in a fluid layer overlying a saturated porous media based on the Navier–Stokes–Darcy–Boussinesq model. We take a hybrid approach that combines analysis with numerical computation. The center manifold reduction theory is applied to reduce the infinite dynamical system to a finite dimensional one together with a non-dimensional transition number that determines the types of dynamical transitions. Careful numerical computations are performed to determine the transition number as well as related temporal and flow patterns etc. Our result indicates that the system favors a continuous transition in which the steady state solution bifurcates to a local attractor at the critical Rayleigh number. Unlike the one layer case, jump transitions can occur at certain parameter regime. We also discover that the transition between shallow and deep convection associated with the variation of the ratio of free-flow to porous media depth is accompanied by the change of the most unstable mode from the lowest possible horizontal wave number to higher wave numbers which could occur with variation of the height ratio as well as the Darcy number and the ratio of thermal diffusivity among others.

我们基于Navier–Stokes–Darcy–Boussinesq模型研究了覆盖饱和多孔介质的流体层中热对流的稳定性和动态转变。我们采用一种将分析与数值计算相结合的混合方法。中心流形归约理论用于将无限动力学系统与确定动态过渡类型的无维过渡数一起简化为有限维系统。进行了仔细的数值计算以确定跃迁数以及相关的时间和流型等。我们的结果表明，该系统支持连续跃迁，其中稳态解在临界瑞利数处分叉到局部吸引子。与单层情况不同，跳变可以在某些参数范围内发生。