This paper presents a comprehensive theoretical study on the dynamical behavior of natural convection flow in the confined region between horizontal half-cylinders. For this purpose, a low-order spectral model with three modes will produce for the fluid flow system using the Galerkin technique. It proved that the generated model is physically meaningful, as it conserves energy in the dissipationless limit and has bounded solutions in the phase space. Analytical procedures indicate that the system has three stationary points and the onset of instability in the flow is when the Rayleigh number reaches a critical value. With an appropriate Lyapunov function is proved that for the Rayleigh numbers below the critical value, the flow is globally stable. As the Rayleigh number gets higher and reaches a fixed value, a Hopf bifurcation occurs, and chaotic motion appears in the system. The critical and Hopf Rayleigh numbers relation are derived parametrically based on dynamical system theories. Also, numerical simulations will carry on the presented low-order model. Different dynamical behaviors of this flow and its transition from regular to chaotic motion are explained, with phase portraits and velocity-temperature diagrams obtained by numerical solutions. This parametric study can pave the way for future researchers to determine at around values of critical parameters should an experiment or direct numerical simulation be performed to have more accurate data without resorting to tests at all operating conditions.

本文对水平半圆柱体之间受限区域内自然对流的动力学行为进行了全面的理论研究。为此，将使用 Galerkin 技术为流体流动系统生成具有三种模式的低阶谱模型。证明了生成的模型在物理上是有意义的，因为它在无耗散极限中保存能量并且在相空间中具有有界解。分析程序表明，系统具有三个驻点，并且当瑞利数达到临界值时，流动开始不稳定。通过适当的李雅普诺夫函数证明，对于低于临界值的瑞利数，流动是全局稳定的。随着瑞利数变高并达到固定值，出现 Hopf 分岔，系统中出现混沌运动。临界数和 Hopf Rayleigh 数关系是基于动力系统理论参数化推导的。此外，数值模拟将继续提出的低阶模型。解释了这种流动的不同动力学行为及其从规则运动到混沌运动的转变，并通过数值解获得了相图和速度-温度图。如果执行实验或直接数值模拟以获得更准确的数据，而不需要在所有操作条件下进行测试，这项参数研究可以为未来的研究人员确定关键参数的值铺平道路。数值模拟将继续提出的低阶模型。解释了这种流动的不同动力学行为及其从规则运动到混沌运动的转变，并通过数值解获得了相图和速度-温度图。如果执行实验或直接数值模拟以获得更准确的数据，而不需要在所有操作条件下进行测试，这项参数研究可以为未来的研究人员确定关键参数的值铺平道路。数值模拟将继续提出的低阶模型。解释了这种流动的不同动力学行为及其从规则运动到混沌运动的转变，并通过数值解获得了相图和速度-温度图。如果执行实验或直接数值模拟以获得更准确的数据，而不需要在所有操作条件下进行测试，这项参数研究可以为未来的研究人员确定关键参数的值铺平道路。