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Runup and boundary layers on sloping beaches
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Image of FIG. 1.
FIG. 1.

Schematic side view of the wave tank.

Image of FIG. 2.
FIG. 2.

Simulated (BIM) swash layers at the time of maximum runup, depicted with true aspect ratio. In the lower panel the two fields of view (FOV) for PIV measurements are shown.

Image of FIG. 3.
FIG. 3.

Upper panel: Definition sketch of wave tank and deep water acoustic gauge. The vertical scale is exaggerated in the sketch. Lower panel: Elevation time series from experiments (exp.), Boussinesq model (Bouss.) and full potential theory (BIM). The model times series start at a finite elevation because the initial conditions in the model are partly situated in front of the gauge location.

Image of FIG. 4.
FIG. 4.

Runup data for θ = 10°.

Image of FIG. 5.
FIG. 5.

Runup data for θ = 12° from Langsholt.

Image of FIG. 6.
FIG. 6.

The shoreline excursion as function of time. The runup height is then ()sin θ, where θ is the beach inclination.

Image of FIG. 7.
FIG. 7.

Image showing part of the shoreline before start of runup, with water to the right. The front of the wave has reached the shoreline, but it has not yet been set it into motion.

Image of FIG. 8.
FIG. 8.

Boundary layers in the runup, with outer flow from full potential theory. The lines normal to the axis show profile locations, while nonlinear and linear profiles are dashed and solid, respectively. Also the free surface is depicted as a bold solid line. Upper panel: / = 0.0985, = 7.40 s. 1 m on the -axis corresponds to 5.84 m/s. Lower panels: / = 0.295, 1 m on the -axis corresponds to 3.89 m/s. The two lower panels correspond to = 6.70 s and = 7.30 s, respectively.

Image of FIG. 9.
FIG. 9.

Measured (dots) and simulated (fully drawn lines) velocity profiles in the boundary layer. Dashes correspond to time simulated profiles with time shifts 0.05 s and 0.02 s, respectively, in the left and right panels. The measurement is made 7 cm in-land and the PIV data are averaged over a distance of approximately 0.5 cm along the beach. Left panel: / = 0.0977 for times 6.90 s, 7.20 s, 7.40 s, 7.60 s, and 7.80 s. Right panel: / = 0.292 for times 5.70 s, 6.00 s, 6.10 s, 6.30 s, and 6.80 s.

Image of FIG. 10.
FIG. 10.

Measured (dots) and simulated (BIM, fully drawn lines) velocity profiles in the boundary layer in the upper field of view for / = 0.292 and times 6.34 s, 6.40 s, 6.58 s, 6.75 s, and 7.00 s.

Image of FIG. 11.
FIG. 11.

Instantaneous streamlines based on PIV velocities approximately 0.8 m from the equilibrium shoreline. Upper left, upper right, lower left, and lower right panels correspond to = 6.40 s, = 6.43 s, = 6.75 s, and = 7.11 s, respectively. The units on the and axes are  m and  mm, respectively.

Image of FIG. 12.
FIG. 12.

A uniform plug retarded by gravity. The shape of the fluid body is irrelevant as long as the boundary layer is thinner than the flow depth.

Image of FIG. 13.
FIG. 13.

Velocity profiles at = 0.4696 s across the viscous boundary layer at various -positions (i.e., τ values) along the beach as indicated on the abscissa. The runup-plug enters the beach with velocity = 1 m/s and the total runup-time is 0.5870s in this case, giving the runup-length 0.2935 m. It is remarked that the profiles are shifted to their locations and that the velocity is zero at = 0 for all profiles.

Image of FIG. 14.
FIG. 14.

Entrainment velocity profile along the outer edge of the viscous boundary layer.

Image of FIG. 15.
FIG. 15.

The wedge-shaped fluid body that surpasses observed inundation. Left panel shows the volume at equilibrium, while the right panel shows the volume at the time, according to the Boussinesq model, at maximum runup. The experimental position for maximum inundation is marked by + in the right panel. The case / = 0.098 has been used for illustration.

Image of FIG. 16.
FIG. 16.

Normalized volume transport defect computed from simulated velocity profiles in the boundary layer.

Image of FIG. 17.
FIG. 17.

The ratio for τ = 0.5 at various locations in the boundary layer.

Image of FIG. 18.
FIG. 18.

The ratio for τ = 0.9 at various locations in the boundary layer is displayed in the figure.


Generic image for table
Table I.

Reynold numbers for solitary waves with = 0.2 m and θ = 10°.

Generic image for table
Table II.

Maximum runup height to amplitude (/). Exp., BIM, and Bouss. correspond to the experiments and the two models, respectively.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Runup and boundary layers on sloping beaches