This paper presents experimental results regarding the existence of similarities and/or Froude number similitudes in the time series of the free-surface elevation (FSE) and horizontal velocity of undular bores (UBs) propagating over a horizontal bed. Two wave gauges and high-speed particle image velocimetry are employed to measure the FSEs and velocity fields. A complete evolution of the FSE (or horizontal velocity) of UBs is divided into four temporal stages, named stages I–IV. The surge-wave amplitudes, constant characteristic horizontal velocities, and specified time differences are then identified as the characteristic length scale, velocity scale, and timescale of UBs. Data for the dimensionless FSE and free-stream velocity vs the dimensionless shifted time are found to collapse onto two unique similarity trends. This demonstrates the existence of Froude number similitudes for the entire evolution of UBs with geometric similarity. Under the same water-depth ratio and length ratio, but distinct values of the relative basin and travel length for different UBs, larger values of the relative basin length produce greater end times for the final occurrence of the constant FSE and characteristic horizontal velocity. This demonstrates that the end time is not influenced by the relative travel length. Under identical values of the water-depth ratio and relative basin length, a smaller relative travel length results in a greater rate of decrease in the unit discharge during stage IV. Correspondingly, this results in earlier bifurcation of the time series of FSE (or free-stream velocity), which has a smaller (or larger) rate of decrease, from their counterparts with larger relative travel lengths.

中文翻译：

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在水平床上传播的波状钻孔的自由表面高程和水平速度的相似性

本文介绍了关于自由表面高程 (FSE) 和在水平床上传播的波状钻孔 (UB) 的水平速度的时间序列中存在相似性和/或弗劳德数相似性的实验结果。采用两个波规和高速粒子图像测速仪来测量 FSE 和速度场。UBs 的 FSE（或水平速度）的完整演变分为四个时间阶段，称为阶段 I-IV。然后将浪涌波振幅、恒定特征水平速度和指定的时间差确定为 UB 的特征长度尺度、速度尺度和时间尺度。发现无量纲 FSE 和自由流速度与无量纲偏移时间的数据折叠成两个独特的相似趋势。这证明了具有几何相似性的 UB 的整个演化过程中存在 Froude 数相似性。在相同的水深比和长度比下，但不同 UB 的相对流域和行进长度值不同，相对流域长度值越大，最终出现恒定 FSE 和特征水平速度的结束时间越长。这表明结束时间不受相对行程长度的影响。在水深比和相对流域长度相同的情况下，相对行进长度越小，第四阶段单位流量下降率越大。相应地，这导致 FSE（或自由流速度）的时间序列的更早分叉，其具有较小（或较大）的下降率，