Purpose

The boundary integral method (BIM) provides unparalleled computational efficiency for solving problems wherever it is applicable. For Stokes flows, the BIM in its current form can only be applied to a limited class of problems that generally comprises boundaries with either specified velocity or stress. This study aims to radically extend the applicability by developing a general method within the BIM framework that can handle periodic, symmetry, zero normal-velocity gradient and the specified pressure boundary conditions. This study is limited in scope to steady-state flows.

Design/methodology/approach

The proposed method introduces a set of points near the boundary for the symmetry, zero normal-velocity gradient and specified pressure boundary conditions. The formulation for the first two boundary conditions use a spatial discretization procedure within the BIM framework to arrive at a set of equations for the unknowns. The specified pressure boundary condition warrants the decomposition of the unknown traction term into simpler components before the discretization procedure can be executed. Though the new methodology is illustrated in detail for two-dimensional rectangular domains, it can be generalized to more complex three-dimensional cases. This will be the subject for future investigations.

Findings

The current endeavor has successfully demonstrated the incorporation of the above boundary conditions through simple Stokes flow problems like plane channel flow, flow through ribbed duct and plane wall jet. The predicted results matched adequately with either analytical solutions or with available literature data.

Originality/value

To the best of the author’s knowledge, this is the first time that the exit boundary conditions like zero normal-velocity gradient and specified pressure have been formulated within the BIM for Stokes flows. These boundary conditions are extremely powerful and the current research initiative has the potential to dramatically increase the range of applicability of the BIM for Stokes flow simulations.

中文翻译：

---

稳态 Stokes 流中不同边界条件下边界积分法的扩展

目的

边界积分法 (BIM) 为解决任何适用的问题提供了无与伦比的计算效率。对于 Stokes 流，当前形式的 BIM 只能应用于有限类别的问题，这些问题通常包含具有指定速度或应力的边界。本研究旨在通过在 BIM 框架内开发一种通用方法来从根本上扩展适用性，该方法可以处理周期性、对称性、零法向速度梯度和指定压力边界条件。本研究仅限于稳态流动的范围。

设计/方法/途径

所提出的方法在对称性、零法向速度梯度和指定压力边界条件的边界附近引入了一组点。前两个边界条件的公式使用 BIM 框架内的空间离散化程序来得出一组未知数方程。指定的压力边界条件保证在执行离散化程序之前将未知牵引项分解为更简单的分量。虽然新方法针对二维矩形域进行了详细说明，但它可以推广到更复杂的三维情况。这将是未来调查的主题。

发现

目前的努力已经通过简单的 Stokes 流动问题成功地证明了上述边界条件的结合，例如平面通道流动、通过肋管的流动和平面壁射流。预测结果与分析解决方案或现有文献数据充分匹配。

原创性/价值

据作者所知，这是第一次在 Stokes 流的 BIM 中制定出口边界条件，如零法向速度梯度和指定压力。这些边界条件非常强大，当前的研究计划有可能显着增加 BIM 在斯托克斯流模拟中的适用范围。