Purpose

This study aims to assess the accuracy of degree adaptive strategies in the context of incompressible Navier–Stokes flows using the high-order hybridisable discontinuous Galerkin (HDG) method.

Design/methodology/approach

The work presents a series of numerical examples to show the inability of standard degree adaptive processes to accurately capture aerodynamic quantities of interest, in particular the drag. A new conservative projection is proposed and the results between a standard degree adaptive procedure and the adaptive process enhanced with this correction are compared. The examples involve two transient problems where flow vortices or a gust needs to be accurately propagated over long distances.

Findings

The lack of robustness and accuracy of standard degree adaptive processes is linked to the violation of the free-divergence condition when projecting a solution from a space of polynomials of a given degree to a space of polynomials with a lower degree. Due to the coupling of velocity-pressure in incompressible flows, the violation of the incompressibility constraint leads to inaccurate pressure fields in the wake that have a sizeable effect on the drag. The new conservative projection proposed is found to remove all the numerical artefacts shown by the standard adaptive process.

Originality/value

This work proposes a new conservative projection for the degree adaptive process. The projection does not introduce a significant overhead because it requires to solve an element-by-element problem and only for those elements where the adaptive process lowers the degree of approximation. Numerical results show that, with the proposed projection, non-physical oscillations in the drag disappear and the results are in good agreement with reference solutions.

中文翻译：

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一种用于瞬态不可压缩流的保守度自适应 HDG 方法

 目的

本研究旨在使用高阶可杂交不间 Galerkin （HDG） 方法评估度自适应策略在不可压缩 Navier-Stokes 流背景下的准确性。

设计/方法/方法

这项工作提出了一系列数值示例，以表明标准度自适应过程无法准确捕获感兴趣的空气动力学量，尤其是阻力。提出了一种新的保守投影，并比较了标准程度自适应程序与通过此校正增强的自适应过程之间的结果。这些示例涉及两个瞬态问题，其中流动涡流或阵风需要精确长距离传播。

 发现

标准度自适应过程缺乏稳健性和准确性与将解从给定次数的多项式空间投影到较低次多项式空间时违反自由发散条件有关。由于不可压缩流中速度-压力的耦合，违反不可压缩性约束会导致尾流中的压力场不准确，从而对阻力产生相当大的影响。发现提出的新保守投影可以去除标准自适应过程显示的所有数值伪影。

 原创性/价值

这项工作为度自适应过程提出了一种新的保守投影。该投影不会引入大量开销，因为它需要逐个单元求解问题，并且仅适用于自适应过程降低近似程度的那些单元。数值结果表明，在所提出的投影中，阻力中的非物理振荡消失了，结果与参考解吻合较好。