Solitary Wave Run-up Research Articles

ON THE BOTTOM SHEAR STRESS DURING LONG WAVE RUNUP AND BACKWASH

Laboratory experiments were performed to examine flow characteristics during runup and backwash of a breaking solitary wave. Boundary layer flows were measured by LDV at two locations landward of the still shoreline under different bed conditions. Temporal variations of the bottom shear stress were estimated from nearbed velocity profiles with different methods. The shear stress determined through the logarithmic fitting roughly agreed with that deduced from a depth-integrated momentum balance during turbulent flow phases. The friction factors during the active phases of the runup and backwash were similar in magnitude except in the leading tip where the value sharply increased. A movable bed experiment was also carried out under the same geometry to discuss relations between the bottom shear stress and sediment transport. The result indicated clear dependencies of bed deformations on the magnitude and duration of the exerted shear stress.

Coupling between Shallow Water Equation and k-ω Model for Simulating Breaking Solitary Wave Run-Up

Higher order equation with advanced numerical scheme is able to provide realistic solution for breaking wave case, yet difficult to apply in practical application. Recent development has enhanced SWE, by coupling with k-ω model for better bed stress assessment, and applying FORCE-MUSCL shock capturing scheme to handle breaking wave. In this study, this enhanced SWE model is used to simulate and investigate bed stress accumulation under breaking solitary wave run up (canonical problem). Comparison of bed stress accumulation to the conventional Manning method shows similar general conclusion that bed stress is more significant during run down. Further comparison to non breaking wave case shows that bed stress during run down is greatly emphasize in the breaking wave case.

Efficient simulation of free surface flows with discrete least-squares meshless method using a priori error estimator

In this article, a priori error estimate is employed to improve the efficiency of simulating free surface flows with discrete least-squares meshless (DLSM) method. DLSM is a fully least-squares approach in which both function approximation and the discretisation of the governing differential equations is carried out using a least-squares concept. The meshless shape functions are derived using the moving least-squares (MLS) method of function approximation. The discretised equations are obtained via a discrete least-squares method in which the sum of the squared residuals are minimised with respect to unknown nodal parameters. The governing equations of mass and momentum conservation are solved in a Lagrangian form using a pressure projection method. The proposed simulation strategy is composed of error estimation and a node moving refinement method. Since in free surface problems, the position of the free surface is of primary interest, a priori error estimate is used which automatically associates higher error to the nodes near the free surface. The node moving refinement method is used to construct a nodal configuration with dense nodal arrangement near the free surface. Four test problems namely dam break, evolution of a water bubble, solitary wave propagation and wave run-up on slope are investigated to test the ability and efficiency of the proposed efficient simulation method.

Validation and Verification of a Numerical Model for Tsunami Propagation and Runup

A robust numerical model to simulate propagation and runup of tsunami waves in the framework of non-linear shallow water theory is developed. The numerical code adopts a staggered leapfrog finite-difference scheme to solve the shallow water equations formulated for depth-averaged water fluxes in spherical coordinates. A temporal position of the shoreline is calculated using a free-surface moving boundary algorithm. For large scale problems, the developed algorithm is efficiently parallelized employing a domain decomposition technique. The developed numerical model is benchmarked in an exhaustive series of tests suggested by NOAA. We conducted analytical and laboratory benchmarking for the cases of solitary wave runup on simple beaches, runup of a solitary wave on a conically-shaped island, and the runup in the Monai Valley, Okushiri Island, Japan, during the 1993 Hokkaido-Nansei-Oki tsunami. In all conducted tests the calculated numerical solution is within an accuracy recommended by NOAA standards. We summarize results of numerical benchmarking of the model, its strengths and limits with regards to reproduction of fundamental features of coastal inundation, and also illustrate some possible improvements.

내부조파기법을 활용한 Navier-Stokes 방정식 모형의 고립파 처오름 수치모의

급경사에서의 고립파의 처오름을 예측하기 위해 3차원 수치모형에 내부조파기법을 도입하여 수치모형실험을 수행하였다. 수치모형은 Navier-Stokes 방정식을 유한차분법을 이용하여 계산하는 동수압 모형으로서, 난류의 해석을 위해서 상대적으로 큰 에디(eddy)만을 고려하는 SANS(spatially averaged Navier-Stokes) 방정식을 푸는 LES(large-eddy-simulation) 기반의 수치모형을 사용한다. 엇갈림 격자체계에서 유한차분법을 사용하여 지배방정식을 해석하는 모형으로서 수치기법으로 Two-step projection 기법을 사용하여 SANS 방정식을 풀었으며, Poisson 방정식을Bi-CGSTAB 기법을 이용하여 풀고 압력장을 계산하였다. 또한, 자유수면의 추적을 위하여 2차 정확도의 VOF(volume-of-fluid) 기법을 사용하였다. 먼저 고립파를 3차원 공간의 일정 수심상에서 내부조파하여 해석해와 비교한 후 분산오차에 대해 분석하였다. 그리고 고립파를 내부조파하여 급경사에서의 고립파의 처오름 및 처내림 현상을 예측하고 수리모형 실험결과와 비교 및 분석하였다. A three-dimensional numerical model called NEWTANK is employed to investigate solitary wave run-up with an internal wave-maker on a steep slope. The numerical model solves the spatially averaged Navier-Stokes equations for two-phase flows. The LES (large-eddy-simulation) approach is adopted to model the turbulence effect by using the Smagorinsky SGS (sub-grid scale) closure model. A two-step projection method is adopted in numerical solutions, aided by the Bi-CGSTAB (Bi-Conjugate Gradient Stabilized) method to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate VOF (volume-of-fluid) method is used to track the distorted and broken free surface. A solitary wave is first internally generated and propagated over a constant water depth in the three-dimensional domain. Numerically predicted results are compared with analytical solutions and numerical errors are analyzed in detail. The model is then applied to study solitary wave run-up on a steep slope and the obtained results are compared with available laboratory measurements.

Coupling between Shallow Water Equation and k-ω Model for Simulating Solitary Wave Run-Up

Higher order equation with advanced numerical scheme is able to provide realistic solution for breaking wave case, yet difficult to apply in practical application. Recent development has enhanced SWE, by coupling with k-ω model for better bed stress assessment, and applying FORCE-MUSCL shock capturing scheme to handle breaking wave. In this study, this enhanced SWE model is used to simulate and investigate bed stress accumulation under breaking solitary wave run up (canonical problem). Comparison of bed stress accumulation to the conventional Manning method shows similar general conclusion that bed stress is more significant during run down. Further comparison to non breaking wave case shows that bed stress during run down is greatly emphasize in the breaking wave case.

Hydro- and morpho-dynamic modeling of breaking solitary waves over a fine sand beach. Part II: Numerical simulation

A comprehensive numerical model is developed to predict the transient wave propagation, sediment transport, morphological change, and the elastodynamic responses of seabed due to breaking solitary waves runup and drawdown over a sloping beach. The individual components of the numerical model are first validated against previous analytical, numerical, and experimental results. The validated numerical model is then used to simulate breaking solitary wave runup and drawdown over a fine sand beach, where the experimental results are presented in (Young et al. 2010b. Hydro- and morpho-dynamic modeling of breaking solitary waves over a fine sand beach. Part I: Experimental study). The strengths and weaknesses of the model are assessed through comparisons with the experimental data. Based on the results, sediment transport mechanisms and wave–seabed interactions in the nearshore region are discussed.

Wave runup simulations using Lagrangian blocks on Eulerian mesh

The runup of breaking and non-breaking solitary waves on a plane beach has been simulated using the shallow-water equations and a Lagrangian blocks on Eulerian mesh (LBEM) method. The results obtained for the entire range of wave steepness are compared with the experimental data, the nonlinear theory, and the finite-difference method. The LBEM simulation agrees with the nonlinear theory when the amplitude of the wave is small. On the whole, it produces a better result in agreement with the experimental data when friction is included in the simulation. The simulation also produced better result than the finite-difference simulation.

Nonlinear Internal Wave Run-Up on Impermeable Steep Slopes

Laboratory experiments were conducted to investigate the run-up of internal solitary waves (ISWs) on steep uniform slopes in a two-layered fluid system with a free surface. A 12 m long wave flume, which incorporated a movable vertical gate for generating ISWs, was used in the experiments. A steep uniform slope was modeled at one end of the flume. In the present study we looked at internal waves with small and large amplitudes using a steep uniform slope (from one of θ=30, 50, 60, 90, 120, and 130 deg) much longer than those previously published in the literature. Results collected from a wide range of experimental runs show that the run-up height depends on the planar slope, while the breaking depth is only antecedent to the amplitude of an ISW. The overall quantitative agreement with the linear feature aspects of the incident wave amplitude is encouraging.

Modeling of non-breaking and breaking solitary wave run-up using FORCE-MUSCL scheme

A numerical model based on the non-linear shallow water equations is developed to investigate the propagation and run-up of both non-breaking and breaking solitary waves. The present model utilizes an explicit centered scheme in conjugation with monotonic upstream scheme of conservation laws to estimate interface fluxes along with the surface gradient method for source term treatment. An optimal third-order total variation diminishing Runge-Kutta explicit scheme is used for time integration of the semi-discretized equations. The proposed scheme is second order accurate in space and time and capable of efficiently capturing the breaking induced discontinuities in flow depth. The model accurately represents a breaking wave as a moving bore or a hydraulic jump and shows a remarkable property of mass conservation during the entire run-up and run-down motions. The numerical results satisfactorily agree with available analytical solutions, experimental data as well as with three existing run-up laws.

Modeling of non-breaking and breaking solitary wave run-up using FORCE-MUSCL scheme

Modeling of non-breaking and breaking solitary wave run-up using FORCE-MUSCL scheme

Non‐hydrostatic 3D free surface layer‐structured finite volume model for short wave propagation

AbstractIn this paper a layer‐structured finite volume model for non‐hydrostatic 3D environmental free surface flow is presented and applied to several test cases, which involve the computation of gravity waves. The 3D unsteady momentum and mass conservation equations are solved in a collocated grid made of polyhedrons, which are built from a 2D horizontal unstructured mesh, by just adding several horizontal layers. The mesh built in such a way is unstructured in the horizontal plane, but structured in the vertical direction. This procedure simplifies the mesh generation and at the same time it produces a well‐oriented mesh for stratified flows, which are common in environmental problems. The model reduces to a 2D depth‐averaged shallow water model when one single layer is defined in the mesh. Pressure–velocity coupling is achieved by the Semi‐Implicit Method for Pressure‐Linked Equations algorithm, using Rhie–Chow interpolation to stabilize the pressure field. An attractive property of the model proposed is the ability to compute the propagation of short waves with a rather coarse vertical discretization. Several test cases are solved in order to show the capabilities and numerical stability of the model, including a rectangular free oscillating basin, a radially symmetric wave, short wave propagation over a 1D bar, solitary wave runup on a vertical wall, and short wave refraction over a 2D shoal. In all the cases the numerical results are compared either with analytical or with experimental data. Copyright © 2008 John Wiley & Sons, Ltd.

Numerical simulation on solitary wave propagation and run-up by SPH method

Smoothed Particle Hydrodynamics (SPH) is a meshless method which takes a Lagrangian approach. It is different from the traditional methods using mesh system. A number of numerical simulations of a solitary wave propagation using SPH method on a vertical wall and a sloping wall are carried out. The wave run-ups obtained this simulations are compared with the experimental date and resulted good agreement. The water surface profile on a sloping wall is also compared in detail with the experimental data and a good agreement is obtained. The results of particle configuration and velocity distribution at various times are visualized. It is shown that the method can simulate the sloshing phenomena of a solitary wave propagation, run-up, run-down and backwash with good accuracy.

Two- and three-dimensional computation of solitary wave runup on non-plane beach

Abstract. Solitary wave runup on a non-plane beach is studied analytically and numerically. For the theoretical approach, nonlinear shallow-water theory is applied to obtain the analytical solution for the simplified bottom geometry, such as an inclined channel whose cross-slope shape is parabolic. It generalizes Carrier-Greenspan approach for long wave runup on the inclined plane beach that is currently used now. For the numerical study, the Reynolds Averaged Navier-Stokes (RANS) system is applied to study soliton runup on an inclined beach and the detailed characteristics of the wave processes (water displacement, velocity field, turbulent kinetic energy, energy dissipation) are analyzed. In this study, it is theoretically and numerically proved that the existence of a parabolic cross-slope channel on the plane beach causes runup intensification, which is often observed in post-tsunami field surveys.

Wave overtopping simulations over coastal structures

A large-eddy simulation (LES) model has been developed to investigate nonlinear wave run-up and overtopping. The Navier-Stokes equations are solved by using a finite differential algorithm, and the modified Marker-Density Function (MDF) technique (Miyata and Park, 1995) is employed for solving nonlinear free-surface motions. In order to evaluate the present model, three numerical simulations are presented: solitary wave run-up on a vertical wall, flow over a broad-crested weir, and regular waves overtopping a sloping seawall. The results are compared with data from experiments and other numerical models.

Surf Similarity and Solitary Wave Runup

The notion of surf similarity in the runup of solitary waves is revisited. We show that the surf similarity parameter for solitary waves may be effectively reduced to the beach slope divided by the offshore wave height to depth ratio. This clarifies its physical interpretation relative to a previous parameterization, which was not given in an explicit form. Good coherency with experimental (breaking) runup data is preserved with this simpler parameter. A recasting of analytical (nonbreaking) runup expressions for sinusoidal and solitary waves additionally shows that they contain identical functional dependence on their respective surf similarity parameters. Important equivalencies in the runup of sinusoidal and solitary waves are thus revealed.

Simulation of nonlinear wave run-up with a high-order Boussinesq model

This paper considers the numerical simulation of nonlinear wave run-up within a highly accurate Boussinesq-type model. Moving wet–dry boundary algorithms based on so-called extrapolating boundary techniques are utilized, and a new variant of this approach is proposed in two horizontal dimensions. As validation, computed results involving the nonlinear run-up of periodic as well as transient waves on a sloping beach are considered in a single horizontal dimension, demonstrating excellent agreement with analytical solutions for both the free surface and horizontal velocity. In two horizontal dimensions cases involving long wave resonance in a parabolic basin, solitary wave evolution in a triangular channel, and solitary wave run-up on a circular conical island are considered. In each case the computed results compare well against available analytical solutions or experimental measurements. The ability to accurately simulate a moving wet–dry boundary is of considerable practical importance within coastal engineering, and the extension described in this work significantly improves the nearshore versatility of the present high-order Boussinesq approach.

Run-up of solitary waves on slopes with different profiles

The problem of sea-wave run-up on a beach is discussed within the framework of exact solutions of a nonlinear theory of shallow water. Previously, the run-up of solitary waves with different forms (Gaussian and Lorentzian pulses, a soliton, special-form pulses) has already been considered in the literature within the framework of the same theory. Depending on the form of the incident wave, different formulas were obtained for the height of wave run-up on a beach. A new point of this study is the proof of the universality of the formula for the maximum height of run-up of a solitary wave on a beach for the corresponding physical choice of the determining parameters of the incident wave, so that the effect of difference in form is eliminated. As a result, an analytical formula suitable for applications, in particular, in problems related to tsunamis, has been proposed for the height of run-up of a solitary wave on a beach.

Solitary wave transformation, breaking and run-up at a beach

A validated one-dimensional Boussinesq–non-linear shallow water equations numerical model was used to investigate the interaction of solitary waves with beaches. The numerical model requires two adjustable parameters: the bed friction coefficient and a wave breaking parameter. Excellent agreement was achieved between the numerical predictions of solitary wave transformation and run-up at a plane beach with two sets of high-quality laboratory measurements: one a large number of experiments in a wave flume by Synolakis, the other in the UK Coastal Research Facility. A parameter study investigated the effect of uniform offshore water depth, bed friction and bed slope on solitary wave run-up. A uniform water depth may be associated with a continental shelf region. The non-dimensional run-up was found to be an asymptotic function of non-dimensional wave amplitude at high and low values of initial wave steepness. Both asymptotes scale as (R/ho)∼α(Ao/ho)β where R is run-up (defined as the vertical elevation reached by the wave uprush above still water level), Ao is the offshore wave amplitude and ho is the uniform depth offshore of the beach. The empirical coefficients α and β depend on the beach characteristics. The model is then used to simulate the interaction of a full-scale tsunami event with an idealised beach profile representative of a beach in Eastern Kamchatka.

Interfacial solitary wave run-up in the St. Lawrence Estuary

Density variations show evidence of interfacial solitary waves (ISW) running up the sloping boundary of an island in the St. Lawrence Estuary, confirming inferences based remote sensing. Further detail is suggested by simulations created with a two-dimensional nonhydrostatic numerical model. The simulations confirm theoretical predictions of the location of wave breaking, something that is difficult to observe in the field. Two other results of the simulations match laboratory findings: the creation of turbulent boluses that propagate upslope of the breaking zone, and the creation of an intermediate layer that transports mixed water away from the mixing site. Although our sampling could not resolve the intermediate mixing layer, it did provide evidence of boluses. In addition to ISW breaking the bolus and intrusion effects may also be important in coastal regions.