Runup and boundary layers on sloping beaches

Abstract

The present study is devoted to discrepancies between experimental and theoretical runup heights on an inclined plane, which have occasionally been reported in the literature. In a new study on solitary wave-runup on moderately steep slopes, in a wave tank with 20 cm water depth, detailed observations are made for the shoreline motion and velocity profiles during runup. The waves are not breaking during runup, but they do break during the subsequent draw-down. Both capillary effects and viscous boundary layers are detected. In the investigated cases the onshore flow is close to the transitional regime between laminar and turbulent boundary layers. The flow behaviour depends on the amplitude of the incident wave and the location on the beach. Stable laminar flow, fluctuations (Tollmien-Schlichting waves), and formation of vortices are all observed. Comparison with numerical simulations showed that the experimental runup heights were markedly smaller than predictions from inviscid theory. The observed and computed runup heights are discussed in the context of preexisting theory and experiments. Similar deviations are apparent there, but have often been overlooked or given improper physical explanations. Guided by the absence of turbulence and irregular flow features in parts of the experiments we apply laminar boundary layer theory to the inundation flow. Outer flows from potential flow models are inserted in a nonlinear, numerical boundary layer model. Even though the boundary layer model is invalid near the moving the shoreline, the computed velocity profiles are found to compare well with experiments elsewhere, until instabilities are observed in the measurements. Analytical, linear boundary layer solutions are also derived both for an idealized swash zone motion and a polynomial representation of the time dependence of the outer flow. Due to lacking experimental or theoretical descriptions of the contact point dynamics no two-way coupling of the boundary layer model and the inviscid runup models is attempted. Instead, the effect of the boundary layer on the maximum runup is estimated through integrated losses of onshore volume transport and found to be consistent with the differences between inviscid theory and experiments.

Linear stability of boundary layers under solitary waves

“…On the other hand, a flow determined to be stable by one of the above criteria might display instabilities once the initial amplitude is equal or larger than its amplification. In their related study on boundary layers on a 10 • beach Pedersen et al (2013) reported (section IVA) for one case (incident wave with = 0.3) that instability occurred in an up-beach position, immediately after an inflection point in the boundary layer profile was observed. However, this was observed in 3 out of 4 experiments, and never in a location close to equilibrium shoreline, where inflection points in the retardation phase were also present.…”

“…The code was written in MATLAB and we verified the above algorithm by a test problem as well as comparison with the boundary layer model of Pedersen et al (2013), which again was compared to measurements.…”

“…However, for the = 0.5 case, for example, Grimshaw's solution overpredicts the magnitude of the bottom velocity under the crest of the solitary wave by approximately 7%. Hence, we instead use a numerical solution for the full potential Φ by a method derived by Tanaka (1986) combined with a straightforward application of Cauchy's formula Pedersen et al (2013). As mentioned above frictional effects will give rise to a thin viscous boundary layer at the bottom.…”

“…Although very small for solitary waves on relatively large depths, such frictional effects become more important for cases where the layer of water becomes thin. This can be the case for solitary waves on small depths or when the solitary wave is running up a beach which leads to a large discrepancy between theoretical and experimental run up heights Pedersen et al (2013). The bottom boundary layer has been considered to be the more relevant, cf.…”

“…However, a linear stability analysis as presented in Blondeaux et al (2012) cannot predict whether the flow after transition is turbulent or not, contrary to what the title of Blondeaux et al (2012) suggests. In a recent study Pedersen et al (2013) measured boundary layers during runup of solitary waves on a beach. They observed instabilities (undulations and vortices) for Reynolds numbers which were higher than those of Sumer et al (2010), when defined in a corresponding manner.…”