A Numerical Investigation of the Reduction of Solitary Wave Runup by A Row of Vertical Slotted Piles

To improve the current understanding of the reduction of tsunami-like solitary wave runup by the pile breakwater on a sloping beach, we developed a 3D numerical wave tank based on the CFD tool OpenFOAM in this study. The Navier Stokes equations were applied to solve the two-phase incompressible flow, combined with an LES model to solve the turbulence and a VOF method to capture the free surface. The adopted model was firstly validated with existing empirical formulas for solitary wave runup on the slope without the pile structure. It is then validated using our new laboratory observations of the free surface elevation, the velocity and the pressure around a row of vertical slotted piles subjected to solitary waves, as well as the wave runup on the slope behind the piles. Subsequently, a set of numerical simulations were implemented to analyze the wave reflection, the wave transmission, and the shoreline runup with various offshore wave heights, offshore water depths, adjacent pile spaces and beach slopes. Finally, an improved empirical equation accounting for the maximum wave runup on the slope was proposed by taking the presence of the pile breakwater into consideration.     

Runup of solitary waves on a straight and a composite beach

Particle Image Velocimetry (PIV) and wave gauges have been used to investigate the runup of solitary waves at two different beaches. The first beach is straight with an inclination of 10°, whereas the second is a composite beach with a change in the 10° inclination to 4° at a vertex point above the equilibrium water level. Comparison with numerical simulations using a Navier–Stokes solver with zero viscosity has been performed for the composite beach. Four different amplitudes of incoming solitary waves are investigated.Measurements of the runup show that the composite beach gives a lower runup compared to the straight beach. Furthermore, the composite beach experiences a longer duration of the rundown compared to the straight beach. This is at least partially assumed to be a result of scaling effects, since the fluid above the vertex creates a relatively thinner runup tongue compared to the straight beach scenario.The appearance of a stagnation point at the beach boundary is clearly visible in both the PIV results and the numerical simulation. This stagnation point is originating at the lowermost part of the beach, and is moving upwards with time. It is found that the stagnation point moves faster upwards for the straight beach than for the composite beach. Further, the stagnation point is moving even faster in the numerical simulation, suggesting that the velocity with which the stagnation point moves is influenced by viscous scaling effects.Finally, the numerical simulation seems to capture the physics of the flow well, despite differences in the phase compared to the PIV results. This applies to both the flow field and the surface elevations.     

Boussinesq modelling of solitary wave and N-wave runup on coast

A total variation diminishing Lax–Wendroff scheme has been applied to numerically solve the Boussinesq-type equations. The runup processes on a vertical wall and on a uniform slope by various waves, including solitary waves, leading-depression N-waves and leading-elevation N-waves, have been investigated using the developed numerical model. The results agree well with the runup laws derived analytically by other researchers for non-breaking waves. The predictions with respect to breaking solitary waves generally follow the empirical runup relationship established from laboratory experiments, although some degree of over-prediction on the runup heights has been manifested. Such an over-prediction can be attributed to the exaggeration of the short waves in the front of the breaking waves. The study revealed that the leading-depression N-wave produced a higher runup than the solitary wave of the same amplitude, whereas the leading-elevation N-wave produced a slightly lower runup than the solitary wave of the same amplitude. For the runup on a vertical wall, this trend becomes prominent when the wave height-to-depth ratio exceeds 0.01. For the runup on a slope, this trend is prominent before the strong wave breaking occurs.     

Scenarios of local tsunamis in the China Seas by Boussinesq model

The Okinawa Trench in the East China Sea and the Manila Trench in the South China Sea are considered to be the regions with high risk of potential tsunamis induced by submarine earthquakes. Tsunami waves will impact the southeast coast of China if tsunamis occur in these areas. In this paper, the horizontal two-dimensional Boussinesq model is used to simulate tsunami generation, propagation, and runnp in a domain with complex geometrical boundaries. The temporary varying bottom boundary condition is adopted to describe the initial tsunami waves motivated by the submarine faults. The Indian Ocean tsunami is simulated by the numerical model as a validation case. The time series of water elevation and runup on the beach are compared with the measured data from field survey. The agreements indicate that the Boussinesq model can be used to simulate tsunamis and predict the waveform and runup. Then, the hypothetical tsunamis in the Okinawa Trench and the Manila Trench are simulated by the numerical model. The arrival time and maximum wave height near coastal cities are predicted by the model. It turns out that the leading depression N-wave occurs when the tsunami propagates in the continental shelf from the Okinawa Trench. The scenarios of the tsunami in the Manila Trench demonstrate significant effects on the coastal area around the South China Sea.     

Statistical characteristics of long waves nearshore

The random long wave runup on a beach of constant slope is studied in the framework of the rigorous solutions of the nonlinear shallow water theory. These solutions are used for calculation of the statistical characteristics of the vertical displacement of the moving shoreline and its horizontal velocity. It is shown that probability characteristics of the runup heights and extreme values of the shoreline velocity coincide in the linear and nonlinear theory. If the incident wave is represented by a narrow-band Gaussian process, the runup height is described by a Rayleigh distribution. The significant runup height can also be found within the linear theory of long wave shoaling and runup. Wave nonlinearity nearshore does not affect the Gaussian probability distribution of the velocity of the moving shoreline. However the vertical displacement of the moving shoreline becomes non-Gaussian due to the wave nonlinearity. Its statistical moments are calculated analytically. It is shown that the mean water level increases (setup), the skewness is always positive and kurtosis is positive for weak amplitude waves and negative for strongly nonlinear waves. The probability of the wave breaking is also calculated and conditions of validity of the analytical theory are discussed. The spectral and statistical characteristics of the moving shoreline are studied in detail. It is shown that the probability of coastal floods grows with an increase in the nonlinearity. Randomness of the wave field nearshore leads to an increase in the wave spectrum width.     

Variations of amplitude, wavelength and phase velocity of plane progressing waves over uniformly sloping beaches

The mysterious variation of the amplitude of waves progressing towards the shore over sloping beaches has been clarified explicitly by linear theory. The phase velocity of the waves increases rapidly as the waves travel across the surf zone towards the shore, as does the wavelength. The phase of the waves near shore is nearly stationary if the beach slope is small. Hence the waves are not sinusoidal near shore but surge towards the shore if the slope is small.     

Run-up of long solitary waves of different polarities on a plane beach

We study the run-up of long solitary waves of different polarities on a beach in the case of composite bottom topography: a plane sloping beach transforms into a region of constant depth. We confirm that nonlinear wave deformation of positive polarity (wave crest) resulting in an increase in the wave steepness leads to a significant increase in the run-up height. It is shown that nonlinear effects are most strongly pronounced for the run-up of a wave with negative polarity (wave trough). In the latter case, the run-up height of such waves increases with their steepness and can exceed the amplitude of the incident wave.     

A two-point method for estimating wave reflection over a sloping beach

This work presents a frequency-domain method for estimating incident and reflected waves when normally incident waves’ propagating over a sloping beach in a wave flume is considered. Linear wave shoaling is applied to determine changes of the wave amplitude and phase due to variations of the bathymetry. The wave reflection coefficient is estimated using wave heights measured at two fixed wave gauges with a distance. The present model demonstrates a high capacity of estimating reflection and shoaling coefficients from synthetic wave-amplitude data. Sensitivity tests for the present model due to measurement errors of wave amplitudes and distance of two probes can more accurately predict the reflection coefficients. The measurement error of wave amplitude affects more significantly than measurement error of distance of two probes on calculating reflection coefficient of waves over a sloping bed.     

海平面变化及其海岸响应

第四纪气温的大幅度冷暖变化，导致全球海平面的变化，引起陆架海侵扣海退。海岸上的各种地貌如海滩、沙坝、三角洲扣陆架沙脊等响应海平面升降而发生新的演化扣变异。东海陆架古岸线、围绕古岸线发育的陆架沙脊、陆架深切河谷扣河谷充填沉积以及冰后期海进型扣海退型沙坝的形成乖演化等沉积事件都是响应海平面升降的结果。近百年来特别是近30年全球海平面普遍上升，引起风暴潮的频度扣幅度的增大。近岸波能增强，越滩浪增多，导致海滩侵蚀，岸线后退。Bruun法则扣其他一些模型能够说明海滩随海平面上升而蚀退的规律，但在预测速率时仍存在很多问题。使用时应注意海平面变化的区域性、海滩发育的滞后性和海滩蚀退因素间的权重关系。     

近岸海域波浪场中污染物运动的实验研究

通过实验模拟,研究了缓坡海岸带(坡度为1:100和1:40)波浪场中污染物的运动.给出了污染物在波浪破碎区和非破碎区运动范围的变化,讨论了在规则波和非规则波作用下波浪破碎区和非破碎区污染物运动的宏观变化趋势.     

Study on Influences of Fringing Reef on Harbor Oscillations Triggered by N-Waves

Influences of topographic variations of the offshore fringing reef on the harbor oscillations excited by incident Nwaves with different amplitudes and waveform types are studied for the first time. Both the propagation of the Nwaves over the reef and the subsequently-induced harbor oscillations are simulated by a Boussinesq-type numerical model, FUNWAVE-TVD. The present study concentrates on revealing the influences of the plane reef-face slope,the reef-face profile shape and the lagoon width on the maximum runup, the wave energy distribution and the total wave energy within the harbor. It shows that both the wave energy distribution uniformity and the total wave energy gradually increase with decreasing reef-face slope. The profile shape of the reef face suffering leading-elevation Nwaves(LEN waves) has a negligible impact on the wave energy distribution uniformity, while for leading-depression N-waves(LDN waves), the latter gradually decreases with the mean water depth over the reef face. The total wave energy always first increases and then decreases with the mean water depth over the reef face. In general, the total wave energy first sharply decreases and then slightly increases with the lagoon width, regardless of the reef-face width and the incident waveform type. The maximum runup subjected to the LEN waves decreases monotonously with the lagoon width. However, for the LDN waves, its changing trend with the lagoon width relies on the incident wave amplitude.     

Characteristics of trapped topographic waves in the northwest part of the Black Sea

We study trapped baroclinic topographic waves in the northwest shelf of the Black Sea for the actual slopes of the bottom and stratification. The time scales of trapped waves are determined. The space scales of the amplitude of oscillations are computed. It is shown that the vertical distribution of the amplitude of oscillations is in qualitative agreement with the experimentally observed distribution. The energy of topographic waves trapped by the sloping bottom is concentrated in the bottom layer, which agrees with the data of measurements. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 44–52, September–October, 2006.     

Runup estimations on a macrotidal sandy beach

This paper presents a methodological approach to calculate runup from the analysis of morphodynamic conditions on a macrotidal sandy beach. The method is based on measurements of the elevation of high-tide deposits and on the analysis of morphological and hydrodynamic changes. A series of measurements has been carried out on the beach of Vougot (Brittany, France) under different wave conditions. This allowed to assess runup formula effectiveness on a macrotidal sandy beach and to determine the best slope parameters to estimate runup. The results suggest that on that macrotidal sandy beach the slope of the active section of the upper beach should be used instead of the entire slope of the foreshore, the latter resulting in an underestimation of runup elevations when used in predictive equations from the literature. Results obtained with widely used equations are relatively well correlated with observed values (r² = 0.63). An analysis of the relationship between observed runup elevations and various variables has enabled the establishment of a runup estimation formula with a relatively good fit to the study site (r² = 0.86).     

On low-frequency waves in the surf and swash

Low-frequency waves in the surf and swash zones on various beach slopes are discussed using numerical simulations. Simulated surface elevations of both primary waves and low-frequency waves across the surf zone were first compared with experimental data and good agreement found. Low-frequency wave characteristics are then discussed in terms of their physical nature and their relationship to the primary wave field on a series of sea bottom slopes. Unlike primary waves, low-frequency wave energy increases towards the shoreline. Low-frequency waves in the surf and swash are a function of incident waves and the sea bottom slope and hence the saturation level of the surf zone. Wave energy on a gently sloping beach is dominated by low-frequency waves while primary waves play a significant role on a steep beach. Low-frequency wave radiation from the surf zone on a given beach depends on primary wave frequency and beach slope. However, a very poor correlation was found between surf similarity parameter and low-frequency wave radiation.     

Edge waves under ice cover at a straight coast with a sloping beach

Edge waves in an ice-covered sea at a straight coast with a sloping beach are analyzed within the linearized theory. Such waves propagate along the coast with an amplitude which exponentially decays offshore. The problem is examined without using the hydrostatic assumption. The seawater is considered to be a homogeneous, inviscid, nonrotating, and incompressible fluid. Ice with a uniform thickness is considered, with constant values of density, cylindrical rigidity, Poisson ratio, and compressive stress in the ice. The normal velocity at the bottom is zero; the linearized kinematic and dynamic boundary conditions are satisfied at the lower surface of the ice. Explicit solutions for the edge flexural-gravity waves and the corresponding dispersion equations are obtained and analyzed.     

A study of radiation stress and set-up in the nearshore region

In this paper the variations of radiation stress and mean water level are studied for the two-dimensional shoaling and breaking of progressive, periodic waves on a plane, gently sloping laboratory beach. Observations of radiation stress and mean water level were made for two initial conditions. The observed variations are compared to the variations as calculated by a linear theory, James' nonlinear theory, and a new nonlinear theory based on Cokelet's work. It is shown that the nonlinear theories are qualitatively and quantitatively superior to the linear theory, thus indicating that the effects of finite amplitude are important for the present case.     

Theoretical and Experimental Study of Breaking Wave on Sloping Bottoms

This paper studies the continuous evolution of breaking wave for the surface water waves propagating on a sloping beach. A Lagrangian asymptotic solution is derived. According to the solution coupled with the wave breaking criteria and the equations of water particles motion, the wave deformation and the continuous wave breaking processes for the progressive water waves propagating on a sloping bottom can be derived. A series of experiments are also conducted to compare with the theoretical solution. The results show that the present solution can reasonably describe the plunging or spilling wave breaking phenomenon.     

Wave runup on a surging vertical cylinder in regular waves

The wave runup caused by a vertical cylinder surging in regular waves is studied both experimentally and numerically. The so-called DualSPHysics Smoothed Particle Hydrodynamics (SPH) code is used for the 3-D numerical modelling. A wide range of cylinder sizes and wave conditions is investigated with results comparing favourably between the experimental and SPH model under both fixed and forced-surge conditions. The experimental and SPH results are further used to predict the maximum runup amplification, in particular the ratio of the runup caused by the surging cylinder to that of the fixed, over the phase difference between the incident wave and surge motion. This maximum runup ratio has been analysed for its dependence on factors such as wave steepness, wave scattering and surge amplitude. An empirical equation is proposed for predicting the maximum runup ratio from known incident wave and surge conditions. Comparison with results from linear solvers suggests that the linear solvers under-predict the full nonlinear runup by a factor of 1.3–1.5.     

Finite volume scheme for the solution of 2D extended Boussinesq equations in the surf zone

In this paper, a hybrid finite volume-finite difference scheme is applied to study surf zone dynamics. The numerical model solves the 2DH extended Boussinesq equations proposed by Madsen and Sørensen (1992) where nonlinear and dispersive effects are both relevant whereas it solves NSWE equations where nonlinearity prevails. The shock-capturing features of the finite volume method allow an intrinsic representation of wave breaking and runup; therefore no empirical (calibration) parameters are necessary. Comparison with laboratory measurements demonstrates that the proposed model can accurately predict wave height decay and mean water level setup, for both regular and solitary wave breaking on a sloping beach. The model is also applied to reproduce two-dimensional wave transformation and breaking over a submerged circular shoal, showing good agreement with experimental data.     

Breaking of a tidal internal wave on a steep shelf as inferred from temperature measurements

The structure, evolution, and breaking of a tidal internal wave on a steep shelf are discussed on the basis of the data of temperature measurements. The bottom slope at the measurement site is close to the critical slope for a tidal wave. The tidal wave and other waves are inclined coastward. The tidal-wave amplitude increases monotonically with increasing horizon depth. The tidal wave is nonlinear in amplitude and turns over on the outer shelf. On the inner shelf, the internal wave is close in shape to rectangular and generates harmonics of its own. The harmonics make the tidal wave steeper and form solitary rises similar to bilateral bores. All these features ensure a more rapid sink for the internal-tide energy.