Abstract In this paper, a spectral element method for the solution of the two-dimensional transient incompressible Navier-Stokes equations is introduced, which combines the segregated velocity-pressure equal-order formulation originated from the SIMPLE (semi-implicit method for pressure linked equations) algorithm. In contrast to previous segregated finite element method based on the SIMPLE algorithm, the pressure equation is derived from the momentum and continuity equations using the element matrices to ensure the convergence of the pressure. High-order element basis functions are adopted to construct the discrete form that presents arbitrary order or accuracy. The validation test with analytical solution verifies the high accuracy of the method. The flow in a lid-driven cavity with different inclination angles and the flow over a backward-facing step are investigated to further illustrate the property of the scheme. The computed results are in excellent agreement with the benchmark solutions, even if fewer grid points are used. The almost periodic solution for the flow of Re=10000 in a lid-driven square cavity is also captured by the present scheme. The numerical results indicate that the proposed high-order method can significantly reduce the number of grid points used in the calculation.

中文翻译：

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二维瞬态不可压缩 Navier-Stokes 方程的分离谱元方法

摘要 本文介绍了一种求解二维瞬态不可压缩 Navier-Stokes 方程的谱元方法，它结合了源自 SIMPLE（压力关联方程的半隐式方法）的分离速度-压力等阶公式。 ） 算法。与之前基于SIMPLE算法的分离式有限元方法相比，压力方程是从动量方程和连续性方程中推导出来的，使用元素矩阵来保证压力的收敛。采用高阶元基函数构造呈现任意阶次或精度的离散形式。分析溶液的验证测试验证了该方法的高准确度。研究了具有不同倾角的盖子驱动腔中的流动和反向台阶上的流动，以进一步说明该方案的性质。即使使用较少的网格点，计算结果也与基准解决方案非常一致。本方案还捕获了 Re=10000 在盖子驱动的方形腔中流动的几乎周期性解决方案。数值结果表明，所提出的高阶方法可以显着减少计算中使用的网格点数。本方案还捕获了 Re=10000 在盖子驱动的方形腔中流动的几乎周期性解决方案。数值结果表明，所提出的高阶方法可以显着减少计算中使用的网格点数。本方案还捕获了 Re=10000 在盖子驱动的方形腔中流动的几乎周期性解决方案。数值结果表明，所提出的高阶方法可以显着减少计算中使用的网格点数。