粘性流动计算的高品质混合网格生成方法研究

The hybrid mesh with prismatic elements in viscous regions could achieve the best trade-off between ease of use and solution accuracy in the computations of viscous flows. In this proposal, we plan to solve three fundamental issues as below to support the development of the next generation of high-quality hybrid mesh generation method for both complicated geometric models and complicated flows. Firstly, Poisson equations are used as the global control equations for the advancing of marching directions, considering the anisotropic flow features. At the same time, the source function of the equation will be determined and a fast multipole boundary element method will be developed to efficiently solve the equation. Secondly, to incorporate with multiple marching directions, a numerical solution method based on high order boundary elements will be developed to solve the proposed control Poisson equation, aimed at improving the boundary layer mesh quality in complicated geometric regions. Thirdly, to considering the problem of sizing transition of both boundary layer mesh and unstructured mesh, an optimization model will be formulated and solved efficiently, which would provide a smooth transition between the anisotropic sizing of boundary layer mesh and the isotropic sizing of unstructured mesh. It is highly hopeful that the above techniques, once developed successfully as expected, would enable the establishment of the theory of hybrid mesh generation and the development of a new generation of high-quality hybrid mesh generator and their applications in viscous flow simulations for complicated aircraft model.

含边界层网格单元的混合网格在粘性流动计算中可取得易用性和计算精度的最佳平衡，针对复杂几何外形和复杂流动，本项目以开发新一代高品质混合网格生成方法为目标，研究解决以下科学难题：1）利用泊松方程建立边界层单元法向推进的全局等价模型，以考虑流场内部含各向异性特征的边界层网格生成问题。相应地，发展泊松方程源项的计算方法和泊松方程的快速多级边界元解法。 2）发展泊松方程的高阶边界元数值解法，以在几何模型的凸边和凸点处集成多法向推进技术，进一步提高复杂几何区域的边界层单元质量。3）建立统一的、同时考虑边界层网格单元和非结构网格单元尺度过渡问题的数学优化模型，开发相应的快速求解算法，实现边界层单元尺寸和非结构单元尺寸的光滑过渡，以提高后续数值计算的精度和效率。上述研究有助于建立成体系的混合网格生成理论及开发新一代高品质混合网格生成软件，并促进其在复杂飞行器粘性流动计算中的应用，有重要的理论和实践意义。

针对复杂几何外形和复杂粘性流动计算，通过在物面附布置边界层网格而在远物面区域布置非结构单元，既可以较少网格单元获得较高粘性计算精度又可提升网格对复杂问题的适应能力。在本项目的研究中，我们提出了基于物理模型的边界层网格生成方法，并设计了该物理模型的快速边界元解法和高阶解法，以适应大规模边界层网格生成问题和边界层网格多法向增长问题，在边界层网格生成效果和效率上都可跟商业软件Pointwise相媲美；针对不同类型网格单元的尺寸过渡问题，提出了具有全局最优解的各向异性网格单元尺寸场光滑化数学模型，实现了包含各向同性网格和以及用于捕捉各向异性流场特征的各向异性网格间的网格单元尺寸光滑过渡；此外，针对边界层单元为六面体的应用需求，提出了基于矢量场的分块结构化四边形网格生成算法。上述研究成果为新一代高品质混合网格生成软件提供了算法基础，部分算法已集成于国家数值风洞一期网格生成软件中，有望为复杂粘性流动计算应用提供软件支持。