基于群论和微扰论的射流气动声场问题的Hamilton方法及应用

This project is to one kind of low Mach jet pneumatic sound field as a physical research object. In this project, the Hamilton system (includes kinetic energy, potential energy and energy )which is in the framework of the universal analytical mechanics is used to replace Newton mechanics system and two types of frameworks in the research of the pneumatic sound field which are CAA and Lighthill Theory at present, and Group theory and Perturbation theory is applied in the research of the pneumatic sound field. Under the Hamilton system, try to combine theory and application, explore mathematics model and method which is used to describe one kind of typical low Mach jet pneumatic sound field. Using the combination of Group theory and Perturbation theory as the research program, Group theory of modern mathematics is applied in the symmetry analysis and the level research of the jet pneumatic sound field.The Hamilton quantity of system and the perturbation caused by the harmonic oscillator are used to calculate the energy conservation characteristics of this typical kind of low Mach jet pneumatic sound field and others ; The calculation results and the experimental data are compared with others. Based on this calculation and evaluation, mathematics model and method are presented to calculate and analyse the jet pneumatic sound field in the Hamilton system, and it also provides the preliminary theory foundation and other application assistance for the development of new methods and new ways of one kind of low Mach jet pneumatic sound field.

本项目以一类低马赫射流气动声场作为研究的物理对象，采用普适的分析力学框架下的Hamilton体系(动能，势能及能量)，替代Newton 力学体系下的气动声场研究的CAA及Lighthill 拟声源理论，将群理论与微扰论应用于气动声场问题的研究，尝试从理论与应用的结合上，在Hamilton体系下，探索一种描述典型低马赫射流气动声场的数学模型及方法. 采用群理论与微扰论相结合的研究方案，将现代数学中的群理论应用于射流气动声场的对称性分析及能级研究；用体系的Hamilton量及谐振子所构造的微扰，计算这类典型低马赫数射流气动声场及其它的能量守恒特征，并将结果与实验数据及其它计算结果相比较. 通过以此为基础的计算与评估，提出一种Hamilton体系下的射流气动声场计算及分析的数学模型和方法，研究并为发展一类射流气动声场问题研究的新方法和新途径，提供初步的理论基础及其它的应用支持.

随着计算机技术的发展，采用数值模拟方法求解射流气动声场及噪声问题已成为国内外相关研究的主流趋势。我们针对求解射流气动声学问题的数值算法问题进行了深入研究。首先，我们设计并搭建了亚音速射流噪声测量试验平台，对冲击射流的噪声特性及声源机理进行了试验与数值研究，补充并完善了宏观射流气动声场的相关试验和仿真计算数据，获得了具有系统性和可靠性的研究射流流场与声场的数值计算方法。将数值方法应用于暖通空调、室内通风等工程领域的产品结构优化和设计中，获得了一系列研究成果。引入并拓展了气动声学数值研究的不同途径和方法：采用气体动理学格式（GKS）对稀薄气体亚音速射流模型进行了数值研究，获得了不同稀薄条件下的亚音速射流的流场特征，为求解稀薄条件下射流气动声学问题奠定了基础；引入基于Hamilton体系的辛算法对气动声学中声波传播问题进行了理论研究，与传统有限差分算法相比，辛算法在抑制数值频散和计算效率等方面具有优越性；提出了一种基于群论和微扰论相结合的Hamilton方法来求解气动声学中的声传播问题，在Hamilton谐振子数学模型基础上，引入Morse势能函数替代原来的简谐振子势模型，初步完成了一维气体分子链中声传播问题的研究。