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Walsh函数有限体积法的多重网格特征研究

王刚 干源 任炯

王刚, 干源, 任炯. Walsh函数有限体积法的多重网格特征研究. 力学学报, 2022, 54(12): 3418-3429 doi: 10.6052/0459-1879-22-281
引用本文: 王刚, 干源, 任炯. Walsh函数有限体积法的多重网格特征研究. 力学学报, 2022, 54(12): 3418-3429 doi: 10.6052/0459-1879-22-281
Wang Gang, Gan Yuan, Ren Jiong. Investigation on multigrid features of the finite volume method with Walsh basis function. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3418-3429 doi: 10.6052/0459-1879-22-281
Citation: Wang Gang, Gan Yuan, Ren Jiong. Investigation on multigrid features of the finite volume method with Walsh basis function. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3418-3429 doi: 10.6052/0459-1879-22-281

Walsh函数有限体积法的多重网格特征研究

doi: 10.6052/0459-1879-22-281
基金项目: 国家自然科学基金资助项目(92052109)
详细信息
    作者简介:

    王刚, 教授, 主要研究方向: 计算流体力学. E-mail: wanggang@nwpu.edu.cn

  • 中图分类号: V211.3

INVESTIGATION ON MULTIGRID FEATURES OF THE FINITE VOLUME METHOD WITH WALSH BASIS FUNCTION

  • 摘要: Walsh函数有限体积法(FVM-WBF)是一种能够在网格内部捕捉间断的新型数值方法. 持续增加Walsh基函数数目能够稳步提高FVM-WBF方法的求解分辨率, 但计算量暴发式增长和收敛速度下降的问题也会同步出现. 针对Walsh基函数数目增加而引起的计算效率问题, 本文分析了Walsh基函数及其系数所能影响的网格单元局部均值区域尺度, 发现其中隐含类似多重网格的尺度特征, 据此提出一种结合多重网格策略的FVM-WBF方法. 在定常流场计算中根据各级Walsh基函数影响尺度的不同, 对每级Walsh基函数设置满足其稳定性约束的时间步长, 在时间推进求解的过程中快速消除不同波长的数值误差, 实现多重网格的加速收敛效果. 选取NACA0012翼型和二维圆柱的定常无黏绕流问题作为算例, 对引入多重网格策略的FVM-WBF方法和不考虑多重网格策略的FVM-WBF方法进行对比测试. 数值结果证实: 新发展的FVM-WBF方法具备多重网格的关键特征, 在不增加任何特殊处理和计算量的情况下, 只需通过时间步长的调整, 就能够达到多重网格的加速效果, 显著提升计算效率.

     

  • 图  1  单位正方形上二维Walsh基函数示意图

    Figure  1.  Schematic diagram of the two-dimensional Walsh basis function on the unit square

    图  2  FVM-WBF方法对单位正方形区域上的变量$Q(x,y) = - {(x - 0.4)^2} - {(y - 0.6)^2} + 0.1 x\sin (2 y) + 1$拟合数值结果

    Figure  2.  The numerical results of two-dimensional variable $Q(x,y) = - {(x - 0.4)^2} - {(y - 0.6)^2} + 0.1 x\sin (2 y) + 1$ simulated by the FVM-WBF method on unit square area

    图  3  四边形网格的Walsh基函数级数子区域划分示意图

    Figure  3.  Subdivision of a quadrilateral grid according to Walsh basis function series

    图  4  不同级别的基函数所对应的时间步长

    Figure  4.  Correspondence between different levels of basis functions and time steps

    图  5  二维圆柱的计算网格

    Figure  5.  The computational grid of the two-dimensional cylinder

    图  6  圆柱绕流马赫数云图

    Figure  6.  Mach number contours of the low speed flow over cylinder

    图  7  圆柱表面压力系数对比图

    Figure  7.  Comparison of pressure coefficients on cylinder surface

    图  8  FVM-WBF_MG方法在圆柱低速绕流计算中的加速效果

    Figure  8.  Effect of FVM-WBF_MG method in the calculation of the low speed flow over cylinder

    图  9  NACA0012翼型计算网格

    Figure  9.  The computational grid of the NACA0012 airfoil

    图  10  NACA0012翼型低速绕流马赫数云图

    Figure  10.  Mach number contours of the low speed flow over NACA0012 airfoil

    图  11  FVM-WBF_MG方法在NACA0012翼型低速绕流计算中的加速效果

    Figure  11.  Effect of FVM-WBF_MG method in the calculation of the low speed flow over NACA0012 airfoil

    图  12  NACA0012翼型跨声速绕流密度云图

    Figure  12.  Density contours of the transonic flow over NACA0012 airfoil

    图  13  NACA0012翼型表面压力系数对比图

    Figure  13.  Comparison of pressure coefficients on NACA0012 airfoil surface

    图  14  FVM-WBF_MG方法在NACA0012翼型跨声速绕流计算中的加速效果

    Figure  14.  Effect of FVM-WBF_MG method in the calculation of the transonic flow over NACA0012 airfoil

    图  15  NACA0012翼型超声速绕流密度云图

    Figure  15.  Density contours of the supersonic flow over NACA0012 airfoil

    图  16  FVM-WBF_MG方法在NACA0012翼型超声速绕流计算中的加速效果

    Figure  16.  Effect of FVM-WBF_MG in the calculation of the supersonic flow over NACA0012 airfoil

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出版历程
  • 收稿日期:  2022-06-20
  • 录用日期:  2022-10-30
  • 网络出版日期:  2022-10-31
  • 刊出日期:  2022-12-15

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