非结构网格体心梯度求解方法的精度分析

赵辉; 张耀冰; 陈江涛; 邓有奇

doi:10.7638/kqdlxxb-2018.0092

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非结构网格体心梯度求解方法的精度分析

研究论文 | 更新时间：2026-05-28

- 非结构网格体心梯度求解方法的精度分析
- The accuracy assessment of gradient computation methods on unstructured grids
- 空气动力学学报 2019年37卷第5期页码：844-854
- 作者机构：
  
  中国空气动力研究与发展中心,绵阳,621000
- 作者简介：
  
  赵辉(1989-), 男, 河南人, 博士, 研究方向:高精度数值计算方法.E-mail:zbhui@mail.ustc.edu.cn
  陈江涛, 研究方向:CFD验证与确认.E-mail:chenjt02@sina.cn
- 基金信息：
- DOI：10.7638/kqdlxxb-2018.0092
  中图分类号： V211.3
- 纸质出版：2019
- 稿件说明：
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赵辉, 张耀冰, 陈江涛, 邓有奇. 非结构网格体心梯度求解方法的精度分析[J]. 空气动力学学报, 2019, 37(5): 844-854.

ZHAO Hui, ZHANG Yaobing, CHEN Jiangtao, et al. The accuracy assessment of gradient computation methods on unstructured grids[J]. 2019, 37(5): 844-854.
赵辉, 张耀冰, 陈江涛, 邓有奇. 非结构网格体心梯度求解方法的精度分析[J]. 空气动力学学报, 2019, 37(5): 844-854. DOI： 10.7638/kqdlxxb-2018.0092.

ZHAO Hui, ZHANG Yaobing, CHEN Jiangtao, et al. The accuracy assessment of gradient computation methods on unstructured grids[J]. 2019, 37(5): 844-854. DOI： 10.7638/kqdlxxb-2018.0092.

摘要

对于空间二阶精度的非结构网格求解器，无黏通量和黏性通量的计算都需要流场变量在单元体心处的梯度值。体心梯度求解的精度很大程度上决定了求解器整体的精度。本文推导了几种常用的梯度求解方法（包括格林高斯法，节点格林高斯法和最小二乘法）求解体心梯度的理论精度，并从数值方面对上述理论精度进行了验证。在数值测试过程中，通过以当前单元体心为基准进行坐标局部缩放的做法，保证了在非结构网格上做精度测试时，网格拓扑能够严格保持不变。理论推导和数值测试一致发现：使用保线性权插值到节点的节点格林高斯法、使用共面单元的最小二乘法和使用共点单元的最小二乘法，不管网格拓扑关系如何，都能保证梯度求解为一阶精度。而格林-高斯法和使用距离倒数权插值到节点的节点格林-高斯法在一般的网格中只有零阶精度，只有在特定的网格和权函数关系下，才能有一阶精度。最后用NACA 0012翼型亚声速绕流模拟进一步验证了梯度求解方法的收敛精度。

Abstract

The precise prediction of the gradient of flow variables at the center of an element is essential to the computation of inviscid and viscous flux for a 2nd order accurate flow solver based on unstructured grids. The overall order of accuracy of a flow solver is largely determined by the order of accuracy of gradient computations. The orders of accuracy for several gradient computation methods

including cell-based Green-Gauss method

node-based Green-Gauss method and least-squares method

are investigated theoretically and verified through numerical experiments. During numerical tests

the local coordinate is scaled down in proportion to guarantee that the grid topology is strictly unaltered during grid refinement. The same conclusions are drawn by theoretical derivation and numerical tests:for cell-based Green-Gauss method or node-based Green-Gauss method cooperated with inverse distance weightings

1st order of accuracy is only achieved on certain grids

but more generally the accuracy is less than 1st order. The 1st order of accuracy for gradient computation is guaranteed independent of the grid topology if the gradient is calculated with least-squares method or node-based Green-Gauss method coupled with interpolation weight which is 2nd order accurate in space. The order of accuracy is further proved by numerical test of subsonic inviscid flow around NACA 0012 airfoil.

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Keywords

references

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