The analytical theory on Darcy–Bénard convection is dominated by normal-mode approaches, which essentially reduce the spatial order from four to two. This paper goes beyond the normal-mode paradigm of convection onset in a porous rectangle. A handpicked case where all four corners of the rectangle are non-analytical is therefore investigated. The marginal state is oscillatory with one-way horizontal wave propagation. The time-periodic convection pattern has no spatial periodicity and requires heavy numerical computation by the finite element method. The critical Rayleigh number at convection onset is computed, with its associated frequency of oscillation. Snapshots of the 2D eigenfunctions for the flow field and temperature field are plotted. Detailed local gradient analyses near two corners indicate that they hide logarithmic singularities, where the displayed eigenfunctions may represent outer solutions in matched asymptotic expansions. The results are validated with respect to the asymptotic limit of Nield (Water Resour Res 11:553–560, 1968).

中文翻译：

---

具有非解析角的多孔矩形中的振荡对流开始

Darcy-Bénard 对流的分析理论以常模方法为主，从本质上将空间阶数从 4 级减少到 2 级。本文超越了多孔矩形中对流开始的正常模式范式。因此，研究了一个精心挑选的情况，其中矩形的所有四个角都是非解析的。边缘状态随着单向水平波传播而振荡。时间周期对流模式没有空间周期性，需要通过有限元方法进行大量数值计算。计算对流开始时的临界瑞利数及其相关的振荡频率。绘制了流场和温度场的二维特征函数的快照。两个角附近的详细局部梯度分析表明它们隐藏了对数奇点，其中显示的特征函数可能表示匹配渐近展开式中的外解。结果根据 Nield 的渐近极限得到验证（Water Resour Res 11:553–560, 1968）。