Vortex Methods for Incompressible Flows

不可压缩流的涡旋方法

• 批准号: 432219818
• 负责人: Dr. Matthias Kirchhart
• 金额: --
• 依托单位: Department of Mechanical Engineering
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/432219818?language=en
• 关键词: Vortex; Methods; Incompressible; Flows

Many if not most flow problems occurring in practice feature turbulence. Conventional, mesh-based methods like finite elements face severe problems when applied to such flows: stringent time-step constraints, instabilities, or the introduction of significant amounts of spurious numerical viscosity. While more advanced schemes do exist, such flows remain a significant challenge. Particle methods, on the other hand, are based on an analytical solution of the convective part of the equations and do not suffer from any of these problems. Vortex methods, in particular, feature many desirable conservation properties. Recent progress by the applicant opened new possibilities to apply these methods in the presence of boundaries. The generation of volumetric meshes for complicated geometries has proven to be a labour-intensive, time-consuming task. Almost as a side-product, these same results also created the opportunity of further research into a new class of semi-analytical, mesh-less solvers for the Poisson problem and the Heat equation in bounded domains. These solvers would only require a mesh of the domain's boundary instead of the domain itself, significantly reducing the burden on their users.The specific mathematical structure of vortex methods allows us to consider the non-linear flow equations as a coupling of linear sub-problems. These sub-problems and the methods applied for their solution allow for a mathematically rigorous analysis and convergence properties can be established. For the coupled equations we suggest several numerical test-cases as benchmarks.

在实践中出现的许多（如果不是大多数的话）流动问题都以湍流为特征。传统的，基于网格的方法，如有限元面临着严重的问题时，适用于这样的流动：严格的时间步长约束，不稳定性，或引入大量的虚假数值粘度。虽然确实存在更先进的计划，但这种流动仍然是一个重大挑战。另一方面，粒子方法基于方程的对流部分的解析解，并且不受任何这些问题的影响。特别是涡方法具有许多理想的守恒性质。申请人最近取得的进展为在边界存在的情况下应用这些方法提供了新的可能性。复杂几何体的体积网格生成已被证明是一项劳动密集型，耗时的任务。几乎作为一个副产品，这些相同的结果也创造了机会，进一步研究一类新的半解析，无网格求解泊松问题和热方程在有界域。这些求解器只需要网格的区域的边界，而不是域本身，显着减少了他们的用户的负担。涡方法的特定的数学结构，使我们能够考虑非线性流动方程作为一个耦合的线性子问题。这些子问题和用于解决它们的方法允许数学上严格的分析，并且可以建立收敛特性。对于耦合方程，我们建议几个数值测试的情况下，作为基准。