FDNT-code.m from Analytical and numerical steady states for thermal convection in an inclined, slender, two-dimensional enclosure

The present study considers uni- and multi-cellular natural convection of air in an inclined, slender, two-dimensional, rectangular enclosure, with emphasis on the existence of multiple steady states. The enclosure is subjected to a uniform heat flux along the long sidewalls and adiabatic conditions at the short-end walls. A mathematical model, based on the two-dimensional conservation equations under laminar flow and steady-state conditions, along with the Boussinesq approximation, is first formulated using non-dimensional stream function, vorticity and temperature, and then solved analytically and numerically, for different inclination angles, Rayleigh numbers and aspect ratios. The analytical solution, derived via the parallel-flow approximation, is used to describe the fluid flow and heat transfer in the core region, while detailed flow and temperature distributions are computed by finite difference approximations. By considering the cases of variable inclination angles for a fixed aspect ratio and different aspect ratios in a horizontal enclosure, the results reveal the emergence of multiple steady-state solutions for supercritical Rayleigh numbers around the horizontal configuration. Close agreement between numerical and analytical results confirms the robustness of the approach and provides insights into convective heat transfer mechanisms, including the influence of aspect ratio on the evolution of the steady states in horizontal enclosures.本研究针对倾斜细长二维矩形腔体内空气的单胞与多胞自然对流展开分析，重点关注多稳态解的存在性。该腔体的长侧壁受均匀热流作用，短端壁则满足绝热边界条件。首先基于层流稳态条件下的二维守恒方程，结合布辛涅斯克近似（Boussinesq approximation），采用无量纲流函数、涡量与温度作为变量构建数学模型；随后针对不同倾角、瑞利数（Rayleigh number）及长宽比，分别通过解析与数值方法进行求解。其中，通过平行流近似推导得到的解析解用于描述核心区域的流动与传热特性，而详细的流场与温度分布则通过有限差分近似（finite difference approximations）计算获得。通过开展固定长宽比下倾角可变、以及水平腔体中长宽比可变的两类算例分析，结果显示在近水平构型的超临界瑞利数条件下，可出现多稳态解。数值解与解析解的高度一致性验证了该方法的稳健性，并为对流传热机理研究提供了新的见解，包括长宽比对水平腔体内稳态演化过程的影响。