高精度间断Galerkin方法及其在叶轮机械内部流动与传热模拟中的应用研究

Discontinuous Galerkin is a discretization method which combines the ideologies of both finite volume and finite element methods. It uses the variational principle to construct the higher dimensional space of the local approximate solution inside each element, and accounts for the physics of wave propagation by the Riemann discontinuity decomposition at each element interface. Thus the method guarantees the realizability of higher order accuracy, meanwhile, retains the ability of capturing discontinuities. The development of DG also provides a more advanced numerical simulating technique than conventional ones for the complex flow and heat transfer phenomena in modern high performance turbomachinery. In allusion to the simulations of real flow and heat transfer features in modern compressors and turbines, this proposal plans an intensive investigation for some of the critical problems of the high order DG method in theory and application. The involved theoretical problems include the base functions and limiters on mixed grids, discretization of diffusion term, unified algorithms of compressible-incompressible flow, pre-conditioning techniques, implicit time schemes, and the new PnPm method deriving from DG. The applied problems include the DG treatments for the modeling equations of non-linear eddy viscosity models based on ARSM and localized transition models, the effects of DG method combined with advanced turbulence models LES/DES, and the numerical theory of fluid-solid conjugate heat transfer based on DG. The achievements of this proposal will be of great significance for the precise predictions of aerodynamic and cooling performance in turbomachinery.

间断Galerkin方法是由有限体积与有限元各自的思想结合而成的离散方法，它在单元内部借助变分原理来构造高维局部近似解空间，而在界面处则由黎曼间断分解来反映波动传播特性，从而使其兼具高阶精度的可实现性和良好的间断捕获能力；它的发展同时也为现代高性能叶轮机械内部复杂的流动及传热现象提供了比常规方法更为先进的模拟手段。本申请拟以现代压气机和透平内部真实流动与传热特征的模拟为导向，针对高精度DG方法在理论及应用上尚存的部分关键问题展开深入的研究。所涉理论问题包括混合网格上的基函数和限制器、扩散项处理、高低速流动统一算法、预处理技术、隐式时间格式等，以及由DG衍生的PnPm方法；应用问题包括基于ARSM的非线性涡粘模型与局部化转捩模型等模化方程的DG方法实现、DG结合LES/DES先进湍流模式及其效果，以及基于DG的气固耦合传热计算理论等。研究成果对叶轮机械气动与冷却性能的准确预测具有重要的意义。

间断Galerkin 方法是由有限体积与有限元各自的思想结合而成的离散方法，它在单元内部借助变分原理来构造高维局部近似解空间，而在界面处则由黎曼间断分解来反映波动传播特性，从而使其兼具高阶精度的可实现性和良好的间断捕获能力；它的发展同时也为现代高性能叶轮机械内部复杂的流动及传热现象提供了比常规方法更为先进的模拟手段。本项目以现代压气机和透平内部真实流动与传热特征的模拟为导向，针对高精度DG 方法在理论及应用上尚存的部分关键问题进行了深入的研究。所涉理论问题包括结构-非结构混合网格上的基函数和限制器、扩散项处理、高低速流动统一算法、预处理技术、隐式时间格式等，以及由DG 衍生的PnPm 方法；应用问题包括基于ARSM 的非线性涡粘模型与局部化转捩模型等模化方程的DG 方法实现、DG 结合LES/DES 先进湍流模式及其效果，以及基于DG 的气固耦合传热计算理论、基于DG方法的热弹性力学数值模拟方法、基于DG方法的热弹性力学数值模拟方法以及时空DG方法在叶轮机械结构动力学中的应用等。研究成果对叶轮机械的设计与分析有着重要意义。