Similarities in the free-surface elevations and horizontal velocities of undular bores propagating over a horizontal bed

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.