Numerical modeling of wave runup and forces on an idealized beachfront house

doi:10.1016/j.oceaneng.2007.07.009

Ocean Engineering

Volume 35, Issue 1, January 2008, Pages 106-116

https://doi.org/10.1016/j.oceaneng.2007.07.009 Get rights and content

Abstract

In this paper, a numerical wave model based on the incompressible Reynolds-averaged Navier–Stokes (RANS) and k–ε equations is used to estimate the impact of a solitary wave on an idealized beachfront house located at different elevations on a plane beach. The locations of the free surface are reconstructed by volume of fluid (VOF) method. The model is satisfactorily tested against the experimental data of wave runup, and the analytical solution of wave forces on vertical walls. The time histories of wave profiles, forces, and overturning moments on the idealized house are demonstrated and analyzed. The variations of wave forces and overturning moments with the elevation of the idealized beachfront house are also investigated.

Introduction

Many houses have been built along the coastlines of United States in the past several decades because of the attractive and beautiful ocean view. Some beachfront houses are located within the wave breaking and runup zones or coastal high hazard areas (FEMA, 2005) where wave action and/or high-velocity water can cause serious structural damage during hurricane or tsunami events. Waves on high water levels during storm surge can cause extreme erosion and severe damage to beachfront buildings. The most severe damage is caused by breaking waves. The force created by waves breaking against a vertical surface is often 10 or more times higher than the force created by high winds during a storm event (FEMA, 2007). An example is given in Fig. 1 that shows multiple family beachfront dwelling damaged by waves and storm surge during Hurricane Ivan in 2004. More examples of structural damages of beachfront houses and structures during Hurricane Ivan can be found in the assessment report by BBCS (2004).

The leading waves of tsunamis have been modeled as solitary waves (Liu et al., 1991; Lin et al., 1999). Most of the analytical approaches to study solitary wave runup are based on the nonlinear shallow water equations, disregarding dispersion and other higher order effects. For example, Synolakis (1987) solved both the linear and nonlinear initial value problems of a long wave propagating first over a flat ocean bottom and then over a sloping beach, and presented an asymptotic formula for solitary wave runup on a plane beach. Other notable analytical solutions include Tuck and Hwang (1972) for wave runup generated by a rigid block sliding down a slope, and Thacker (1981) for wave resonance in a circular parabolic basin. A comprehensive review of tsunami hydrodynamics can be found in Synolakis and Bernard (2006).

In addition to theoretical solution, researchers have developed various numerical models based on linear shallow water wave (LSW) equation, nonlinear shallow water wave (NSW) equations, Boussinesq equations and Reynolds-averaged Navier–Stokes (RANS) equations. Among them, the applications of RANS model to estimate wave runup and force on coastal structures have become increasingly popular. Volume of fluid (VOF) method is usually adopted in RANS models to reconstruct the free surface. Petit et al., (1995) developed the numerical model SKYLLA based on RANS equations and FLAIR method for simulation of breaking waves on coastal structures. The breaking wave profiles are obtained and analyzed on a 1:20 plane slope of a submerged structure. Sabeur et al., (1997) proposed numerical techniques based on VOF method for the simulation of waves with highly distorted water–air interfaces at a plane slope. The velocity fields during the process of wave runup on a 1:3 plane slope were investigated by the numerical techniques. Lin and Liu, (1998) described the development of a numerical model based on RANS and k–ε equations to study the evolution of a wave train, shoaling and breaking in the surf zone. Good agreement between numerical results and experimental data was observed for shoaling and breaking cnoidal waves on a sloping beach. Kawasaki (1999) proposed a numerical wave model for two-dimensional wave field in the vertical plane. The model combines the VOF method with a non-reflective wave generator in addition to the open boundary treatment to achieve stable computations. Both the wave breaking due to a submerged breakwater and post breaking wave deformation are numerically investigated using the proposed model. Lo and Shao (2002) simulated the near shore solitary wave runup using an incompressible smoothed particle hydrodynamics (SPH) method together with large eddy simulation (LES) approach. The incompressible Navier–Stokes equations in Lagrangian form are solved using a two-step fractional method. Hur et al. (2004) developed a direct numerical model which combines the VOF method and the porous body model to accurately simulate the nonlinear interaction between water waves and a porous structure. The transverse, uplift and inline wave forces on an asymmetric structure are accurately predicted by the model.

In this study, numerical computations are carried out to study the wave impact on an idealized beachfront house located at different elevation on a plane beach. A model based on RANS and k–ε equations is employed. Two test problems, one for a breaking solitary wave runup on a mild slope beach, the other for wave forces on vertical walls are used to validate the model. Comparisons of free surface profiles are made among numerical, experimental, and analytical results. The validated model is applied to simulate solitary wave runup on a plane beach, as well as wave forces acting on the idealized beachfront house. The variations of wave forces and overturning moments with the elevation of the idealized beachfront house are also investigated based on time histories of wave profiles, forces and overturning moments calculated by the numerical model.

Section snippets

Governing equations

For wave–structure-interaction problems, water can be viewed as incompressible viscous material. The RANS equations (Eqs. (1), (2)) are used to solve the mean flow velocity and k–ε turbulent model (Eqs. (3), (4)) is used for the closure of RANS equations.

RANS equations $\frac{\partial u_{i}}{\partial x_{i}} = 0,$ $\frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} = - \frac{1}{ρ} \frac{\partial p}{\partial x_{j}} + g_{i} + \frac{\partial τ_{ij}}{\partial x_{j}} .$ k–ε model $\frac{\partial k}{\partial t} + u_{j} \frac{\partial k}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(\frac{ν_{t}}{σ_{k} +} + ν) \frac{\partial k}{\partial x_{j}}] + G - ε,$ $\frac{\partial ε}{\partial t} + u_{j} \frac{\partial ε}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(\frac{ν_{t}}{σ_{ε}} + ν) \frac{\partial ε}{\partial x_{j}}] + C_{1 ε} \frac{ε}{k} G - C_{2 ε} \frac{ε^{2}}{k},$ in which $τ_{ij} = 2 (ν + C_{d} \frac{k^{2}}{ε}) σ_{ij} - \frac{2}{3} k δ_{ij}, σ_{ij} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}), ν_{t} = C_{d} \frac{k^{2}}{ε},$ $G = 2 ν_{τ} σ_{ij} \frac{\partial u_{i}}{\partial x_{j}}, C_{d} = 0.09, C_{1 ε} =$

Projection method

In the numerical model, the computation domain is discretized using staggered grid. All scalar quantities, i.e., p, k, ε, ν_t, and the VOF function F are defined at the center of the cells, and the vector quantities are defined at the center of the cell faces.

The two-step projection method (Chorin, 1968) has been used to solve the RANS equations. The first step is to introduce an intermediate velocity ${\hat{u}}_{i}^{n + 1}$ , which carries the correct vorticity, as $\frac{{\hat{u}}_{i}^{n + 1} - u_{i}^{n}}{Δ t} = - u_{j}^{n} \frac{\partial u_{j}^{n}}{\partial x_{j}} + g_{i} + \frac{\partial τ_{ij}^{n}}{\partial x_{j}},$ where the

Comparison of wave profile in breaking and runup experiment

The experimental data of breaking solitary wave runup and down (Synolakis, 1986) is used to validate the model's ability to capture wave profile in breaking case. The beach has the angle of 2.88° and the slope is about σ=tan(2.88°)≈1.20. The still water depth (d) varies from 0.21 to 0.29 m. A solitary wave with the ratio of wave height (H) to still wave depth (d), $H / d = 0.28$ , was generated.

The computational domain is 6.5 m×0.32 m. Model grid sensitivity studies have been conducted under different

Wave runup and forces acting on an idealized beachfront house at different elevation

In this section, the wave model is applied to simulate the process of a solitary wave runup a mild plane slope and its impact on an idealized house standing on the beach. Firstly, the maximum vertical runup height (R) on the plane beach is calculated by the model and compared with empirical equation. Then, the idealized beachfront house is placed at five different elevations on the beach related to the maximum wave runup height, that is −1/4, 0, 1/2 and 3/4 of total runup height, and the

Conclusion

The goal of this paper is to evaluate the variation of wave forces and overturning moment (at the hell) with the elevation of an idealized beachfront house. The interactions of a solitary wave (H=2 m) and the idealized house located at five typical elevations ((h−d)/R=−1/4, 0, 1/4, 1/2, 3/4) on a plane slope are studied using a VOF-based RANS numerical model. The time histories of wave force and overturning moment on the house are obtained and analyzed. The results show that the wave force,

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Numerical modeling of wave runup and forces on an idealized beachfront house

Abstract

Introduction

Section snippets

Governing equations

Projection method

Comparison of wave profile in breaking and runup experiment

Wave runup and forces acting on an idealized beachfront house at different elevation

Conclusion

Coastal Engineering

Journal of Computational Physics

Coastal Engineering

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The numerical computation of turbulent flows, Computer Methhods in Applied Mechanics and Engineering

Numerical solution of the Navier–Stokes equations

Mathematics of Computation

A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions

Journal of Fluid Mechanics

Design of seawalls and breakwaters

Journal of the Waterways and Harbors Division, Proceedings of the American Society of Civil Engineers

Numerical simulation of breaking and post-breaking wave deformation process around a submerged breakwater

Coastal Engineering