Mach reflection of a large-amplitude solitary wave
Mach reflection of a large-amplitude solitary wave
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DOI：
10.1017/s0022112093000941
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发表时间：
1993-03
影响因子：
3.7
通讯作者：
Mitsuhiro Tanaka
中科院分区：
文献类型：
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作者：
Mitsuhiro Tanaka
应用 Dommermuth 和 Yue (1987) 开发的“高阶谱方法”对垂直墙壁对斜入射孤立波的反射进行了数值研究。根据 Miles (1977a, b) 的分析，当 ai [Lt ] 1 时有效，当 ai/(3ai)½ ≤ 1 时，规则反射类型让位于“马赫反射”，其中 ai 是入射波的振幅除以静止水深 d，ψi 是入射角。在马赫反射中，入射波和反射波的顶点以恒定角度（ψ *，例如）远离墙壁，并通过称为“马赫干”的第三孤立波与墙壁连接。迈尔斯模型预测，当 ψi = (3ai)½ 时，马赫干的振幅以及壁上的助跑为 4ai。然而，我们的数值结果表明，大振幅的效应往往会阻止马赫反射的发生。即使马赫反射发生，它也会受到正反射的“污染”，因为表征反射图案的所有重要量，例如茎角ψ*、反射角ψr和反射波ar的幅度，都从迈尔斯理论预测的值转向与正反射相对应的值，即ψ* = 0、ψr = ψi和ar = ai。根据我们对 ai = 0.3 的计算，从马赫反射到规则反射的转变发生在 ψi ≈ 37.8°，这比 (3ai)½ = 54.4° 小得多，并且在 ψi = 35° 处观察到最高马赫干 (ψi/(3ai)½ = 0.644)。尽管对于此处考虑的 ψi 的任何值都没有观察到“四倍放大”，但发现马赫干可以变得高于规定水深的最高二维稳定孤立波。数值结果还与 Johnson (1982) 对一大一小孤立波之间的斜向相互作用的分析进行了比较，当 ψi 足够小时且发生马赫反射时，与 Miles 的分析相比，这与数值结果的一致性要好得多。
Reflection of an obliquely incident solitary wave by a vertical wall is studied numerically by applying the ‘high-order spectral method’ developed by Dommermuth & Yue (1987). According to the analysis by Miles (1977a, b) which is valid when ai [Lt ] 1, the regular type of reflection gives way to ‘Mach reflection’ when ai/(3ai)½ ≤ 1, Where ai is the amplitude of the incident wave divided by the quiescent water depth d and ψi is the angle of incidence. In Mach reflection, the apex of the incident and the reflected waves moves away from the wall at a constant angle (ψ*, say), and is joined to the wall by a third solitary wave called ‘Mach stem’. Miles model predicts that the amplitude of Mach stem, and so the run-up at the wall, is 4ai when ψi = (3ai)½. Our numerical results shows, however, that the effect of large amplitude tends to prevent the Mach reflection to occur. Even when the Mach reflection occurs, it is ‘contaminated’ by regular reflection in the sense that all the important quantities that characterize the reflection pattern, such as the stem angle ψ*, the angle of reflection ψr, and the amplitude of the reflected wave ar, are all shifted from the values predicted by Miles’ theory toward those corresponding to the regular reflection, i.e. ψ* = 0, ψr = ψi, and ar = ai. According to our calculations for ai = 0.3, the changeover from Mach reflection to regular reflection happens at ψi ≈ 37.8°, which is much smaller than (3ai)½ = 54.4°, and the highest Mach stem is observed for ψi = 35° (ψi/(3ai)½ = 0.644). Although the ‘four-fold amplification’ is not observed for any value of ψi considered here, it is found that the Mach stem can become higher than the highest two-dimensional steady solitary wave for the prescribed water depth. The numerical result is also compared with the analysis by Johnson (1982) for the oblique interaction between one large and one small solitary wave, which shows much better agreement with the numerical result than the Miles’ analysis does when ψi is sufficiently small and the Mach reflection occurs.