Storm Wave Forces on Selected Prototype Coastal Bridges on the Island of Oahu

Submitted to: Hawaii Department of Transportation Coastal Bridge and Port Vulnerability to Tsunami and Storm Surge Project Project No: DOT-08-004, TA 2009-1RHydrodynamic study of storm wave loads on four selected coastal bridges (prototype scale) around the Island of Oahu is presented here. These include NewMakaha Stream bridge, New South Punaluu Stream bridge, Maili Stream (Maipalaoa) bridge and Kahaluu Stream bridge on the Island of Oahu. Maximum water level at the location of the selected bridges is determined under extreme conditions of a Category 5 Hurricane making landfall on the island. The maximum wave height and wave period are estimated theoretically based on the highest water level. Several different scenarios are considered for each of the selected bridges. The wave loads on the bridges are calculated by use of several theoretical methods. One is based on Euler’s equations coupled with the Volume of Fluid method, for which OpenFOAM, an open access computational fluid dynamics (CFD) package is used to perform the computations, and another one is based on the Green-Naghdi (Level I) nonlinear shallow water wave equations, and is applied to the cases in which the bridge is fully submerged. Existing theoretical and empirical relations, including the Long-Wave Approximation for a fully submerged bridge, developed based on the linear potential theory, and the empirical relations for an elevated bridge deck are also used. Re- sults are compared with each other. The condition that results in the maximum wave forces for each of the bridges is summarized at the end of the report.This work is partially based on funding from State of Hawaii’s Department of Transportation (HDOT) and the Federal Highway Administration (FHWA), grant numbers DOT-08-004, TA 2009-1R. Any findings and opinions contained in this paper are those of the authors and do not necessarily reflect the opinions of the funding agency

On bore dynamics and pressure:RANS, GN, and SV Equations

Propagation and impact of two and three dimensional bores generated by breaking of a water reservoir is studies by use of three theoretical models. These include the Reynolds Averaged Navier-Stokes (RANS) equations, the Level I Green-Naghdi (GN) equations and the Saint-Venant (SV) equations. Two types of bore generations are considered, namely (i) bore generated by dam-break, where the reservoir water depth is substantially larger than the downstream water depth, and (ii) bore generated by an initial mound of water, where the reservoir water depth is larger but comparable to the downstream water depth. Each of these conditions correspond to different natural phenomena. This study show that the relative water depth play a significant role on the bore shape, stability and impact. Particular attention is given to the bore pressure on horizontal and vertical surfaces. Effect of fluid viscosity is studied by use of different turbulence closure models. Both two and three dimensional computations are performed to study their effect on bore dynamics. Results of the theoretical models are compared with each other, and with availably laboratory experiments. Information is provided on bore kinematics and dynamics predicted by each of these models. Discussion is given on the assumptions made by each model and differences in their results. In summary, SV equations have substantially simplified the physics of the problem, while results of the GN equations compare well with the RANS equations, with incomparable computational cost. RANS equations provide further details about the physics of the problem

Bore pressure on horizontal and vertical surfaces

Bores generated by dam-break and initial mound of water and their propagation over horizontal and inclined surfaces are studied by use of theoretical approaches. Calculations are carried out in two and three dimensions and particular attention is given to the bore impact on horizontal and vertical surfaces. Downstream of the initial mound of water may be wet or dry. Discussion is provided on the influence of the downstream water on the bore behaviour and impact. Three methods are used in this study, namely the Reynolds-Averaged Navier-Stokes equations (RANS), the Green-Naghdi (GN) equations and Saint Venant equations (SV). The governing equations subject to appropriate boundary conditions are solved with various numerical techniques. Results of these models are compared with each other, and with laboratory experiments when available. Discussion is given on the limitations and applicability of these models to study the bore generation, propagation and pressure on horizontal and vertical surfaces. It is found that the GN equations compare well with the RANS equations, while the SV equations have substantially simplified the solution.</p

Diffraction of Cnoidal Waves by Vertical Cylinders in Shallow Water

Diffraction of nonlinear waves by single or multiple in-line vertical cylinders in shallow water is studied by use of different nonlinear, shallow-water wave theories. The fixed, in-line, vertical circular cylinders extend from the free surface to the seafloor and are located in a row parallel to the incident wave direction. The wave–structure interaction problem is studied by use of the nonlinear generalized Boussinesq equations, the Green–Naghdi shallow-water wave equations, and the linearized version of the shallow-water wave equations. The wave-induced force and moment of the Green–Naghdi and the Boussinesq equations are presented when the incoming waves are cnoidal, and the forces are compared with the experimental data when available. Results of the linearized equations are compared with the nonlinear results. It is observed that nonlinearity is very important in the calculation of the wave loads on circular cylinders in shallow water. The variation of wave loads with wave height, wavelength and the spacing between cylinders is studied. Effect of the neighboring cylinders, and the shielding effect of upwave cylinders on the wave-induced loads on downwave cylinders are discussed.</p

Diffraction and refraction of nonlinear waves by the Green-Naghdi equations

Abstract Diffraction and refraction of nonlinear shallow water waves due to uneven bathymetry are studied by use of the Green–Naghdi (GN) equations in three dimensions. A numerical wave tank consisting of deep, transitional, and shallow regions is created. Various forms of three-dimensional bathymetry, consisting of ramps with nonuniform profiles and large slopes, are used to connect the deep-water side of the tank to the shallow water shelf. A wavemaker is placed at the upwave side of the domain, capable of generating solitary and cnoidal waves of the GN equations. A numerical wave absorber is located downwave of the domain to minimize the wave reflection back into the domain. The system of equations is solved numerically in time domain by use of a second-order finite-difference approach for spatial discretization, and in a boundary-fitted coordinate system, and by use of the modified Euler method for time marching. Results include solitary and cnoidal wave surface elevation and particle velocities and are compared with the existing solutions where possible. Overall, very good agreement is observed. Discussion is provided on the nonlinearity and dispersion effects on the wave diffraction and refraction by the various forms of the ramps, as well as on the performance of the GN equations in solving these problems.</jats:p

A comparative study on generation and propagation of nonlinear waves in shallow waters

This study is concerned with the generation and propagation of strongly nonlinear waves in shallow water. A numerical wave flume is developed where nonlinear waves of solitary and cnoidal types are generated by use of the Level I Green-Naghdi (GN) equations by a piston-type wavemaker. Waves generated by the GN theory enter the domain where the fluid motion is governed by the Navier&ndash;Stokes equations to achieve the highest accuracy for wave propagation. The computations are performed in two dimensions, and by an open source computational fluid dynamics package, namely OpenFoam. Comparisons are made between the characteristics of the waves generated in this wave tank and by use of the GN equations and the waves generated by Boussinesq equations, Laitone&rsquo;s 1st and 2nd order equations, and KdV equations. We also consider a numerical wave tank where waves generated by the GN equations enter a domain in which the fluid motion is governed by the GN equations. Discussion is provided on the limitations and applicability of the GN equations in generating accurate, nonlinear, shallow-water waves. The results, including surface elevation, velocity field, and wave celerity, are compared with laboratory experiments and other theories. It is found that the nonlinear waves generated by the GN equations are highly stable and in close agreement with laboratory measurements

Moored elastic sheets under the action of nonlinear waves and current

This study is concerned with the interaction between nonlinear water waves and uniform current with moored, floating elastic sheets, resembling floating solar panels, floating airports, tunnels and bridges, and floating energy systems. The Green-Naghdi theory is applied for the nonlinear wave-current motion, the thin plate theory is used to determine the deformations of the elastic sheet and Hooke’s law defines the effect of the mooring lines. The horizontal displacement of the floating sheet is determined by substituting the forces induced by the fluid flow and the tensions generated in the mooring lines into the equations of motion of the floating body. The resulting governing equations, boundary and matching conditions are solved in two dimensions with a finite-difference technique. The results are compared with the available numerical data. Overall, very good agreement is observed. In the model developed here, the sheet is allowed to drift due to the wave-current impact, and hence the mooring lines partially restrict both deformation and the horizontal motions of the sheet. The influence of the mooring lines on the dynamics of the floating sheet is assessed in terms of wave- and current-induced elastic deformations and surge movements of the sheet. It is demonstrated that the mooring lines attached to the leading and trailing edges of the sheet can be highly effective in mitigating the horizontal oscillations and vertical elastic deformations of the floating sheet subjected to the wave and current actions. Special attention is given to the horizontal periodic motions of the sheet, which are analysed by use of a Fourier transform technique. It is shown that the moored elastic sheet can oscillate at a frequency different from its exciting frequency as a result of restoring forces from the mooring lines, exciting resonance when both frequencies meet. Extensive study in a broad range of sheet parameters, mooring stiffnesses and wave-current conditions established the location of resonant regimes of different configurations of the moored systems. Analysis of wave reflection and transmission coefficient revealed that mooring lines of increasing stiffness intensify the wave reflection and, consequently, result in smaller energy transformation downwave

Bore impact on decks of coastal structures

Bore impact on horizontal fixed decks of coastal structures is studied by use of the Level I Green–Naghdi (GN) equations and the Navier–Stokes (NS) equations. The bore is generated by the breaking of a water reservoir, and may represent the propagation of a tsunami on land or broken storm waves. The bore-induced horizontal and vertical forces are determined and their variation with the bore and deck conditions is studied in this work. Various conditions of deck location with respect to the water level are considered, including cases with the deck under or above the still-water level. Two types of bore are considered, namely (i) a bore generated by a dam break, where the reservoir water depth is substantially larger than the downstream depth, and (ii) a bore generated by an initial mound of water, where the reservoir water depth is comparable to the downstream depth. It is shown that these mechanisms result in the formation of significantly different bore shapes. It is also shown that the relative height of the reservoir and the downstream water depth play a significant role in the bore generation and its impact on coastal structures. It is also found that the bore-induced forces vary almost linearly with the change in amplitude of the reservoir, while a change in the length of the reservoir has little effect on the loads. The horizontal force on submerged decks is shown to be independent of the submergence depth of the deck; this is due to the uniform velocity distribution over the water column of the bore. Results of the GN and NS models are compared with each other for submerged cases and the limitations, accuracy, and efficiency of these models in studying this problem are discussed. Results of the GN equations are in close agreement with the NS equations, making them a computationally efficient alternative for the study of this problem.<br/