Abstract Natural convection (NC) in high permeable porous media is usually investigated using the Darcy-Lapwood-Brinkman model (DLB). The problem of the porous squared cavity is widely used as a common benchmark case for NC in porous media. The solutions to this problem with the DLB model are limited to steady-state conditions. In this paper, we developed a time-dependent high accurate solution based on the Fourier-Galerkin method (FG). The solution is derived considering two configurations dealing with unsteady and transient modes. The governing equations are reformulated using the stream function. The Temperature and the stream functions are expended as unknowns in space using Fourier series which are appropriately substituted in the equations. The equations are then projected to the spectral space using Fourier trigonometric trial functions. The obtained developed equations form a nonlinear differential algebraic system of equations. An appropriate technique is used to integrate the spectral system in time and to ensure high accuracy. The results of the FG method are compared to a finite element solution for different Rayleigh and Darcy numbers values. The transient and unsteady solutions are obtained with a feasible and low computational cost. The paper provides high accurate time-dependent solutions useful for benchmarking numerical models dealing with NC in porous media. The results of the developed solutions are efficient to gain physical insight into the time-dependent NC processes.

中文翻译：

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使用 Darcy-Lapwood-Brinkman 模型求解多孔外壳中自然对流的瞬态解

摘要 通常使用 Darcy-Lapwood-Brinkman 模型 (DLB) 研究高渗透多孔介质中的自然对流 (NC)。多孔方腔问题被广泛用作多孔介质中 NC 的常见基准案例。DLB 模型解决此问题的方法仅限于稳态条件。在本文中，我们开发了一种基于傅立叶伽辽金法 (FG) 的时间相关的高精度解决方案。该解决方案是考虑到处理非稳态和瞬态模式的两种配置而得出的。使用流函数重新制定控制方程。使用傅立叶级数将温度和流函数扩展为空间中的未知数，在方程中适当地替换。然后使用傅立叶三角试验函数将方程投影到谱空间。得到的方程构成了一个非线性微分代数方程组。使用适当的技术及时积分光谱系统并确保高精度。将 FG 方法的结果与不同瑞利数和达西数值的有限元解决方案进行比较。以可行且低计算成本获得瞬态和非稳态解。本文提供了高精度的时间相关解决方案，可用于对多孔介质中处理 NC 的数值模型进行基准测试。开发的解决方案的结果可以有效地获得对时间相关的 NC 过程的物理洞察。将 FG 方法的结果与不同瑞利数和达西数值的有限元解决方案进行比较。以可行且低计算成本获得瞬态和非稳态解。本文提供了高精度的时间相关解决方案，可用于对多孔介质中处理 NC 的数值模型进行基准测试。开发的解决方案的结果可以有效地获得对时间相关的 NC 过程的物理洞察。将 FG 方法的结果与不同瑞利数和达西数值的有限元解决方案进行比较。以可行且低计算成本获得瞬态和非稳态解。本文提供了高精度的时间相关解决方案，可用于对多孔介质中处理 NC 的数值模型进行基准测试。开发的解决方案的结果可以有效地获得对时间相关的 NC 过程的物理洞察。