The wave boundary layer is a thin layer near the bottom generated by progressive waves. The flow structures are complex and the sediment motions are intense in the wave boundary layer A better understanding of the flow structure of the wave boundary layer is necessary for the prediction of sediment transport and morphological evolution in coastal areas. To obtain flow velocity in the boundary layer, previous studies usually solved the Navier-Stokes type momentum equations numerically, thus a huge computational effort is needed. In this study, the explicit expression of the vertical velocity is derived directly from the mass-balance equation taking the bottom slope and free surface into consideration. The explicit expression of the vertical velocity corresponds well with the measured data collected from the large-scale wave flume. Finally, the sensitivity analysis of bottom slopes and wave characteristics on flow structures are provided. The implication of the study is that this type of explicit expression of the vertical velocity can be embedded in numerical models to reduce the computational cost significantly.
In shallow water, the wave particles under the progressive wave oscillate at the bed surface, which leads to a time-varying boundary layer (Nielsen 1992). A thorough understanding of the wave boundary layer is a prerequisite for predicting sediment movement, coastal morphological evolution, and surface wave dissipation, which is of great significance for coastal protection, beach evolution, port construction. (Grant and Madsen 1986). The frequent action of flow, the large gradient of velocity, the shear stress in the layer, and the turbulence effect under natural conditions, make the field measurement difficult. In the past decades, studies of the wave boundary layer are mainly based on indoor experiment. Many scholars have carried out relevant experiments, so they have accumulated much experimental data and constantly deepened their understanding of the wave boundary layer. Sumer et al. (1993) studied the velocity and shear stress of turbulent boundary layers on three different slopes using the oscillating tunnel, and found that there is a constant, non-zero, periodic average flow in the offshore direction during the convergent half cycle, and the flow velocity increases with the increase of slope. Yuan et al. (2018) studied the wave nonlinearity and bottom slope effects in an inclined oscillating tunnel and found that the boundary layer flow due to velocity skewness and wave asymmetry greatly enhanced the net sediment transport rate. Fuhrman et al. (2009) simulated the dynamic changes of the wave boundary layer caused by slope in an oscillating tunnel based on the Navier-Stokes equation and two-equation turbulence model. It was found that the slope of the bed surface would cause time-averaged shore flow and shear stress in the boundary layer. However, their studies focused on the wave boundary layer in oscillating tunnels, ignoring the free surface of waves, which is quite different from the wave boundary layer on the real-worldcoasts. In the real-worldcoast, the free surface can cause the vertical movement of wave particles, and the Reynolds stress generated affects the velocity distribution in the wave boundary layer. Horikawa and Watanabe (1968), and Slash (1970) conducted experiments on smooth and rough bed surfaces in wave flumes respectively and proposed that turbulence would affect the thickness of the boundary layer and the phase advance of free flow velocity. Dixen et al. (2008) discussed the boundary layer flow of a bed composed of ripples and table tennis balls in a wave flume. Mirfenderesk and Young (2003) directly measured the bottom bed shear stress in a small wave flume and analyzed the bottom friction under the corrugated bottom bed. Their researches focus on flat bottom, however, the real-world coast usually presents a certain slope. On the slope, morphological changes will generate additional horizontal velocity gradient and vertical velocity, which often have different directions in the first and second half cycles, bring additional vertical momentum exchange and wave-induced Reynolds stress, and change the vertical distribution of time-averaged velocity. Therefore, the dynamic structure of the boundary layer under slope and free surface conditions still requires more in-depth and comprehensive study.