NUMERICAL METHOD FOR CALCULATING THE INTERNAL STRUCTURE OF SHOCK WAVES
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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.Abstract: The internal flow of shock waves is governed by a set of hydrodynamic equations with asymptotic boundary conditions. It is usually converted into an initial value problem and solved iteratively by the shooting method. Yet the flow variables always diverge from their correct values in the test, showing the failure of the shooting method. In this paper, the dynamics of the shock system are analyzed qualitatively by means of the topology of the phase portrait, which suggests the importance of the integral direction. It is found that the downstream point is a saddle point, and any tiny error will be significantly multiplied when the phase point approaches it, leading to divergence. The problem can be solved by a backward marching method, in which an initial value point near the downstream point is first determined using L’Hôpital’s rule and the Euler scheme, and then integrate backward from the downstream to the upstream. This method is unconditionally convergent because the integral curve is always directed to the upstream point. To verify it, calculations were performed for shocks over a wide range of Mach numbers (1.01 ~ 100), and the results show that this procedure can solve the shock structure problem correctly and efficiently.
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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 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
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