On roll waves down an open inclined channel
On roll waves down an open inclined channel
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DOI：
10.1098/rspa.1984.0079
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发表时间：
1984-08
期刊：
影响因子：
--
通讯作者：
D. Needham;J. Merkin
中科院分区：
文献类型：
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作者：
D. Needham;J. Merkin
考虑了倾斜明渠的水流。德雷斯勒（Communs pure appl.Math.2，149-194（1949）），利用浅水理论的方程，用Chezy阻力公式扩充，表明当弗劳德数F超过4时，均匀流变得不稳定，在这种情况下，他能够构建一个参数族的不连续周期解的拼接连续部分的波剖面和一系列的水跃。在这里，Dressler的工作被扩展，包括一个进一步的术语，该术语解释了垂直于流动的剪切能量耗散。结果表明，包含这样一个术语并不改变的条件，为稳定的均匀流，当均匀流是不稳定的，一个单参数族的准定常周期解存在（参数化的传播速度U），出现作为一个Hopf分支的临界值Uc = 1 + F-½的均匀流。在证明了周期解的存在性之后，利用Krylov─Bogoliubov─Mitropolski平均方法得到了周期解的一致有效展开式，并通过数值积分将结果推广到更大的振幅.
Flow down an open inclined channel is considered. Dressler (Communs pure appl. Math. 2, 149-194 (1949),) using the equations of the shallow water theory augmented by the Chezy formula for drag, has shown that the uniform flow becomes unstable when the Froude number F exceeds 4, and in this case he was able to construct a one-parameter family of discontinuous periodic solutions by piecing together continuous sections of wave profile and a series of hydraulic jumps. Here the work of Dressler is extended by the inclusion of a further term that accounts for energy dissipation by shearing normal to the flow. It is shown that the inclusion of such a term does not alter the condition for stability of the uniform flow, and that when the uniform flow is unstable, a one-parameter family of quasi-steady periodic solutions exists (parametrized by the propagation speed U), appearing as a Hopf bifurcation out of the uniform flow at the critical value Uc = 1 + F-½. After the existence of these periodic solutions has been shown, uniformly valid expansions for the periodic solutions are obtained by using the Krylov─Bogoliubov─Mitropolski averaging method, and the results are also extended to larger amplitudes by numerical integration.