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激波内部结构的数值求解方法

朱清波 周文元 杨庆春 徐旭

朱清波, 周文元, 杨庆春, 徐旭. 激波内部结构的数值求解方法. 力学学报, 2023, 55(9): 1858-1866 doi: 10.6052/0459-1879-23-093
引用本文: 朱清波, 周文元, 杨庆春, 徐旭. 激波内部结构的数值求解方法. 力学学报, 2023, 55(9): 1858-1866 doi: 10.6052/0459-1879-23-093
Zhu Qingbo, Zhou Wenyuan, Yang Qingchun, Xu Xu. Numerical method for calculating the internal structure of shock waves. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(9): 1858-1866 doi: 10.6052/0459-1879-23-093
Citation: Zhu Qingbo, Zhou Wenyuan, Yang Qingchun, Xu Xu. Numerical method for calculating the internal structure of shock waves. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(9): 1858-1866 doi: 10.6052/0459-1879-23-093

激波内部结构的数值求解方法

doi: 10.6052/0459-1879-23-093
基金项目: 国家自然科学基金资助项目 (51706010)
详细信息
    通讯作者:

    朱清波, 博士研究生, 主要研究方向为激波、超燃冲压发动机、高超声速推进系统. E-mail: qingbozhu@foxmail.com

    杨庆春, 副教授, 主要研究方向为燃烧流动、超燃冲压发动机、高超声速推进系统. E-mail: yangqc@buaa.edu.cn

  • 中图分类号: O354.5, V211.3, V411.3

NUMERICAL METHOD FOR CALCULATING THE INTERNAL STRUCTURE OF SHOCK WAVES

  • 摘要: 激波的内部流动由一组具有渐近边界条件的流体力学方程控制, 这类常微分方程的边值问题一般用打靶法将其转化为初值问题迭代求解. 然而经过验证计算, 打靶法不能有效地求解激波结构, 流动参数总是先趋近波后值, 随后迅速偏离, 直至发散. 文章基于相平面中微分方程相轨迹图的拓扑结构对系统的动力学性质进行了定性分析, 指出波后点是鞍点, 其附近的方向分布导致正向推进计算中任何微小误差都会被放大, 使积分曲线偏离解曲线, 引起发散. 针对该问题, 提出一种逆向推进的数值求解策略及相应的初值确定方法, 先用L’Hôpital法则和Euler格式在波后点附近确定一合理初值点, 然后从该点向上游积分. 由于逆向推进的积分曲线总会被方向场导向波前点, 随着积分的进行误差会不断降低, 计算是无条件收敛的. 为进一步验证该方法的有效性, 对单原子气体中波前马赫数1.01 ~ 100的正激波进行了计算, 结果表明, 逆向推进法能正确而高效地求解激波内部结构.

     

  • 图  1  激波内部结构示意图

    Figure  1.  Schematic diagram of the internal structure of shock waves

    图  2  打靶法计算的单原子气体激波结构 (Ma1 = 2)

    Figure  2.  Profile of a Ma1 = 2 shock in a monoatomic gas calculated with the shooting method

    图  3  V-T平面中的相轨迹

    Figure  3.  Phase portrait in the V-T plane

    图  4  逆向推进法计算的单原子气体激波结构

    Figure  4.  Shock structures in a monatomic gas calculated with the backward marching method

    图  5  打靶法与逆向推进法对激波结构的计算结果的对比

    Figure  5.  Comparison between the shock profiles calculated with the shooting method and the backward marching method

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出版历程
  • 收稿日期:  2023-03-15
  • 录用日期:  2023-07-11
  • 网络出版日期:  2023-07-12
  • 刊出日期:  2023-09-18

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