标题： 高阶精度有限体积法求解不可压缩流体流动的旋度-涡量形式的多边形网格任意曲面边界方法 显示英文标题

标题： Very high-order accurate finite volume scheme for the streamfunction-vorticity formulation of incompressible fluid flows with polygonal meshes on arbitrary curved boundaries

摘要： 针对不可压缩流体流动问题的常规数学模型基于以压力和速度表达的纳维-斯托克斯方程。在此背景下，压力与速度的耦合是一个关键问题，并且几十年来已经开发了无数数值技术和方法以高效且精确地求解这些方程。在二维情况下，另一种方法是将纳维-斯托克斯方程重写为两个标量量：流函数和涡度。 与原始变量法相比，这种形式化不需要计算压力，从而避免了与压力-速度耦合相关的固有困难。然而，推导流函数和涡度的边界条件具有挑战性。 本文提出了一种高效的、二阶精度的有限体积离散化方法，用于二维不可压缩纳维-斯托克斯方程的流函数-涡度形式化。详细讨论了如何推导适当的边界条件及其数值处理方法，包括任意曲线边界的情况。 为了避开生成曲线网格所带来的困难（如复杂的网格算法、繁琐的积分规则和复杂的非线性变换），以保持在任意曲线域上的高阶收敛性，采用了离场数据重构方法。该方法通过传统的分段线性逼近来近似任意曲线边界，而边界附近的约束多项式重构满足物理边界的指定条件。 展示了几个非平凡的二维曲线域中的不可压缩流体流动测试案例，并进行了讨论，以证明所提出的方法在实现非常高的收敛阶方面的准确性和有效性。 显示英文摘要

摘要： Conventional mathematical models for simulating incompressible fluid flow problems are based on the Navier-Stokes equations expressed in terms of pressure and velocity. In this context, pressure-velocity coupling is a key issue, and countless numerical techniques and methods have been developed over the decades to solve these equations efficiently and accurately. In two dimensions, an alternative approach is to rewrite the Navier-Stokes equations regarding two scalar quantities: the streamfunction and the vorticity. Compared to the primitive variables approach, this formulation does not require pressure to be computed, thereby avoiding the inherent difficulties associated with the pressure-velocity coupling. However, deriving boundary conditions for the streamfunction and vorticity is challenging. This work proposes an efficient, high-order accurate finite-volume discretisation of the two-dimensional incompressible Navier-Stokes equations in the streamfunction-vorticity formulation. A detailed discussion is devoted to deriving the appropriate boundary conditions and their numerical treatment, including on arbitrary curved boundaries. The reconstruction for off-site data method is employed to avoid the difficulties associated with generating curved meshes to preserve high-orders of convergence in arbitrary curved domains, such as sophisticated meshing algorithms, cumbersome quadrature rules, and intricate non-linear transformations. This method approximates arbitrary curved boundaries with a conventional linear piecewise approximation, while constrained polynomial reconstructions near the boundary fulfil the prescribed conditions at the physical boundary. Several incompressible fluid flow test cases in non-trivial 2D curved domains are presented and discussed to demonstrate the accuracy and effectiveness of the proposed methodology in achieving very high orders of convergence.