两相流松弛模型的物理方面

Z. Bilicki, J. Kestin
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引用次数: 195

摘要

本文探讨了均匀松弛模型(HRM)作为描述绝热、一维、两相流的基础的潜力。为此,一个严格的数学分析突出了这与强调问题的物理和定性方面的均匀平衡模型(HEM)之间的异同。特别注意对分散、特性、窒息和激波的研究。参考合适且方便的HRM相空间Ω发现了最基本的特征,该相空间由压力P、焓h、干燥分数x、速度w和长度坐标z组成。相空间Ω的几何性质使我们能够绘制出模型所有解的拓扑模式。呛阻问题的研究与该模型的一阶微分方程组的奇异点的出现密切相关。非常强大的中心流形定理允许我们将奇异点的研究减少到一个二维平面Π,它与奇异点的解相切,从而证明只有三种奇异点模式可以出现(退化情况除外),即鞍点,节点和螺旋点。分析表明,波的传播存在两个极限速度,即冻结速度af和平衡速度ae。窒息的临界速度是声音的冻结速度。分析明确地证明了从ω < af到w > af的转变只能通过一个奇点发生。这样的条件也可以在信道的末端得到。最后简要讨论了正常、完全分散和部分分散的激波。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Physical aspects of the relaxation model in two-phase flow
The paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P, enthalpy h, dryness fraction x, velocity w, and length coordinate z. The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π, which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity af and the equilibrium velocity ae. The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω < af to w > af can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.
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