^{1,a)}, E. Lindstrøm

^{1}, A. F. Bertelsen

^{1}, A. Jensen

^{1}, D. Laskovski

^{1}and G. Sælevik

^{2}

### 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.

The assistance by Arve Kvalheim and Svein Vesterby at the Hydrodynamics Laboratory is gratefully acknowledged. This work is supported by the Norwegian Research Council under the project 205184/F20.

I. INTRODUCTION

II. METHODS AND BACKGROUND

A. Experiments

B. Runup theory

C. Boundary layer equations

D. Stability of boundary layer flow

III. MEASURED AND COMPUTED RUNUP HEIGHTS

IV. BOUNDARY LAYERS

A. The computed and observed boundary layers

B. Analytic boundary layer solutions

C. Estimation of reduced runup height due to the boundary layer

V. DISCUSSION

### Key Topics

- Laminar boundary layers
- 26.0
- Viscosity
- 22.0
- Flow instabilities
- 17.0
- Potential flows
- 15.0
- Experiment design
- 13.0

## Figures

Schematic side view of the wave tank.

Schematic side view of the wave tank.

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.

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.

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.

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.

Runup data for θ = 10°.

Runup data for θ = 10°.

Runup data for θ = 12° from Langsholt. 51

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

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

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 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.

Boundary layers in the runup, with outer flow from full potential theory. The lines normal to the s 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: A/d = 0.0985, t = 7.40 s. 1 m on the s-axis corresponds to 5.84 m/s. Lower panels: A/d = 0.295, 1 m on the s-axis corresponds to 3.89 m/s. The two lower panels correspond to t = 6.70 s and t = 7.30 s, respectively.

Boundary layers in the runup, with outer flow from full potential theory. The lines normal to the s 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: A/d = 0.0985, t = 7.40 s. 1 m on the s-axis corresponds to 5.84 m/s. Lower panels: A/d = 0.295, 1 m on the s-axis corresponds to 3.89 m/s. The two lower panels correspond to t = 6.70 s and t = 7.30 s, respectively.

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: A/d = 0.0977 for times 6.90 s, 7.20 s, 7.40 s, 7.60 s, and 7.80 s. Right panel: A/d = 0.292 for times 5.70 s, 6.00 s, 6.10 s, 6.30 s, and 6.80 s.

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: A/d = 0.0977 for times 6.90 s, 7.20 s, 7.40 s, 7.60 s, and 7.80 s. Right panel: A/d = 0.292 for times 5.70 s, 6.00 s, 6.10 s, 6.30 s, and 6.80 s.

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

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

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 t = 6.40 s, t = 6.43 s, t = 6.75 s, and t = 7.11 s, respectively. The units on the s and z axes are m and mm, respectively.

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 t = 6.40 s, t = 6.43 s, t = 6.75 s, and t = 7.11 s, respectively. The units on the s and z axes are m and mm, respectively.

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.

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.

Velocity profiles at t = 0.4696 s across the viscous boundary layer at various s-positions (i.e., τξ values) along the beach as indicated on the abscissa. The runup-plug enters the beach with velocity U 0 = 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 s locations and that the velocity is zero at z = 0 for all profiles.

Velocity profiles at t = 0.4696 s across the viscous boundary layer at various s-positions (i.e., τξ values) along the beach as indicated on the abscissa. The runup-plug enters the beach with velocity U 0 = 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 s locations and that the velocity is zero at z = 0 for all profiles.

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

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

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 A/d = 0.098 has been used for illustration.

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 A/d = 0.098 has been used for illustration.

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

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

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

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

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

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

## Tables

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

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

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

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

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