Observation and measurement of the bottom boundary layer flow in the prebreaking zone of shoaling waves

In this paper, the characteristics of the bottom boundary layer flow induced by nonlinear, asymmetric shoaling waves, propagating over a smooth bed of 1/15 uniform slope, is experimentally investigated. Flow visualization technique with thin-layered fluorescent dye was first used to observe the variation of the flow structure, and a laser Doppler velocimeter was then employed to measure the horizontal velocity, U.The bottom boundary layer flow is found to be laminar except within a small region near the breaking point. The vertical distribution of the phase-averaged velocity U at each phase is non-uniform, which is directly affected by the mean velocity, . The magnitude of increases from zero at the bottom to a local positive maximum at about z/δ2.02.5 (where z is the height above the sloping bottom and δ is the Stokes layer thickness), then decreases gradually to zero at z/δ6.07.0 approximately, and finally becomes negative as z/δ increases further. Moreover, as waves propagate towards shallower water, the rate of increase in the maximum onshore oscillating velocity component is greater than that of the offshore counterpart except near the breaking point. The free stream velocities in the profiles of the maximum onshore and offshore oscillating velocity components, and are found to appear at z/δ≥6.0. This implies that, if the Stokes layer thickness is used as a length scale, the non-dimensionalized boundary layer thickness remains constant in the pre-breaking zone. Although is greater than and the asymmetry of the maximum free stream velocities (i.e. ) increases with decrease of water depth, a universal similar profile can be established by plotting z/δ versus ( ) or ( ). The final non-dimensional profile is symmetric and unique for the distributions of the maximum onshore and offshore oscillating velocity components within the bottom boundary layer, which are induced by nonlinear, asymmetric shoaling waves crossing the pre-breaking zone.     

Prediction of wave pressures on smooth impermeable seawalls

Simple prediction methods are proposed to estimate the wave induced pressures on smooth impermeable seawalls. Based on the physics of the wave structure interaction, the sloped seawall is divided into a total of five zones (zones 1, 2 and 3 during run-up (corresponding pressures are called as positive pressures) and zones 4 and 5 during run-down (corresponding pressures are called negative pressures)) (Fig. 1). Zone 1 (0<z<d−H_i/2), where the wave pressure is governed by the partial reflection and phase shift; Zone 2 (d−H_i/2<z<d), where the effect of wave breaking and turbulence is significant; Zone 3 (d<z<Run-up height), where the pressure is induced by the run-up water; Zone 4 (Run-down<z<d), where the wave pressure is caused by the run-down effect and Zone 5 (0<z<d-Run down), where the negative wave pressures are due to partial reflection and phase shift effects. Here d is the water depth at the toe of the seawall, H_i is the incident wave height and z is the vertical elevation with toe of the seawall as origin and z is positive upward. For wave pressure prediction in zones 1 and 5, the empirical formula proposed by Ahrens et al. (1993) to estimate the wave reflection and Sutherland and Donoghue's recommendations (1998) for the estimation of phase shift of the waves caused by the sloped structures are used. Multiple regression analysis is carried out on the measured pressure data and empirical formulas are proposed for zones 2, 3 and 4. The recommendations of Van der Meer and Breteler (1990) and Schüttrumpf et al. (1994) for the prediction of wave run-down are used for pressure prediction at zone 4. Comparison of the proposed prediction formulas with the experimental results reveal that the prediction methods are good enough for practical purposes. The present study also shows a strong relation between wave reflection, wave run-up, wave run-down and phase shift of waves on wave pressures on the seawalls.     

Comparison of turbulence schemes for prediction of wave-induced near-bed sediment suspension above a plane bed

Based on a wave bottom boundary layer model and a sediment advection-diffusion model, seven turbulence schemes are compared regarding their performances in prediction of near-bed sediment suspension beneath waves above a plane bed. These turbulence algorithms include six empirical eddy viscosity schemes and one standard two-equation k-ε model. In particular, different combinations of typical empirical formulas for the eddy viscosity profile and for the wave friction factor are examined. Numerical results are compared with four laboratory data sets, consisting of one wave boundary layer hydrodynamics experiment and three sediment suspension experiments under linear waves and the Stokes second-order waves. It is shown that predictions of near-bed sediment suspension are very sensitive to the choices of the empirical formulas in turbulence schemes. Simple empirical turbulence schemes are possible to perform equally well as the two-equation k-ε model. Among the empirical schemes, the turbulence scheme, combining the exponential formula for eddy viscosity and Swart formula for wave friction factor, is the most accurate. It maintains the simplicity and yields identically good predictions as the k-ε model does in terms of the wave-averaged sediment concentration.     

Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport

A large number of studies have been done dealing with sinusoidal wave boundary layers in the past. However, ocean waves often have a strong asymmetric shape especially in shallow water, and net of sediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviors influenced by the effect of asymmetry are different from those in sinusoidal waves. Characteristics of the turbulent boundary layer under breaking waves (saw-tooth) are investigated and described through both laboratory and numerical experiments. A new calculation method for bottom shear stress based on velocity and acceleration terms, theoretical phase difference, φ and the acceleration coefficient, a_c expressing the wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined empirically from both experimental and baseline k–ω model results. The new calculation has shown better agreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by other existing methods. It was further applied into sediment transport rate calculation induced by skew waves. Sediment transport rate was formulated by using the existing sheet flow sediment transport rate data under skew waves by Watanabe and Sato [Watanabe, A. and Sato, S., 2004. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. Proc. of 29th ICCE, ASCE, pp. 1703–1714.]. Moreover, the characteristics of the net sediment transport were also examined and a good agreement between the proposed method and experimental data has been found.     

Does the Ulleung eddy owe its existence to β and nonlinearities?

A nonlinear theory for the generation of the Ulleung Warm Eddy (UWE) is proposed. Using the nonlinear reduced gravity (shallow water) equations, it is shown analytically that the eddy is established in order to balance the northward momentum flux (i.e., the flow force) exerted by the separating western boundary current (WBC). In this scenario, the presence of β produces a southward (eddy) force balancing the northward momentum flux imparted by the separating East Korean Warm Current (EKWC).It is found that, for a high Rossby number EKWC (i.e., highly nonlinear current), the eddy radius is roughly 2R_d/ε^1/6 (here ε≡βR_d/f₀, where R_d is the Rossby radius), implying that the UWE has a scale larger than that of most eddies (R_d). This solution suggests that, in contrast to the familiar idea attributing the formation of eddies to instabilities (i.e., the breakdown of a known steady solution), the UWE is an integral part of the steady stable solution. The solution also suggests that a weak WBC does not produce an eddy (due to the absence of nonlinearity).A reduced gravity numerical model is used to further analyze the relationship between β, nonlinearity and the eddy formation. First, we show that a high Rossby number WBC which is forced to separate from the wall on an f plane does not produce an eddy near the separation. To balance the northward momentum force imparted by the nonlinear boundary current, the f plane system moves constantly offshore, producing a southward Coriolis force. We then show that, as β is introduced to the problem, an anticyclonic eddy is formed. The numerical balance of forces shows that, as suggested by the analytical reasoning, the southward force produced by the eddy balances the northward flow force imparted by the boundary current. We also found that the observed eddy scale in the Japan/East Sea agrees with the analytical estimate for a nonlinear current.     

On the importance of resolving the western boundary layer in wind-driven ocean general circulation models

The western boundary layer (WBL) plays a fundamental role in basin-scale wind-driven ocean circulations. In idealized ocean models with flat bottom topography, this layer is required not only to balance the interior Sverdrup transport to close the gyre circulation, but also to dissipate the vorticity imposed by the wind-stress curl. The width of the WBL in Munk-type models is estimated to be δ_M(A_H/β)^1/3, where A_H and β are horizontal eddy viscosity and the meridional derivative of the Coriolis parameter respectively. For commonly used values of A_H, the boundary-layer width δ_M ranges from 30 to less than 200 km in the mid-latitude ocean. This scale is often poorly resolved in large-scale climate models.This paper intends to demonstrate some consequences when the western boundary layer is not adequately resolved. It is found that coarse resolution models reach equilibrium states by distorting some important dynamics in order to dissipate wind-imposed vorticity. In three-dimensional models, for instance, very strong spurious upwelling and downwelling can occur along the WBL. In models of two-dimensional flow, however, spurious recirculations may develop near the boundary. These false features can be removed when the boundary layer is better resolved. We propose a method in which a spatially varying A_H is used to broaden the WBL without affecting mixing in the interior. The method improves the model results considerably.     

Seabed shear stress and bedload transport due to asymmetric and skewed waves

A simple conceptual formulation to compute seabed shear stress due to asymmetric and skewed waves is presented. This formulation generalizes the sinusoidal wave case and uses a variable friction factor to describe the physics of the boundary layer and to parameterize the effects of wave shape. Predictions of bed shear stresses agree with numerical computations using a standard boundary layer model with a k–ε turbulence closure. The bed shear stress formulation is combined with a Meyer-Peter and Müller-type formula to predict sheet flow bedload transport under asymmetric and skewed waves for a horizontal or sloping bed. The predictions agree with oscillatory water tunnel measurements from the literature.     

The vertical structure of combined wave–current flow

A simple numerical model, based on the Reynolds stress equations and k–ε turbulence closure scheme, is developed for the coastal wave and current bottom boundary layer. The current friction velocity is introduced to account for the effect of currents on waves. The implicit Crank–Nicolson finite difference method discretizes the governing equations. Vertical changing step grids with the constant ratio for two adjacent spatial steps are used together with the equal time steps in the modeling. Vertical profiles of mean current velocity and wave velocity amplitude are obtained. These modeled results are compared with the laboratory experimental data of Van Doorn [1981. Experimental investigation of near bottom velocities in water waves with and without a current. Report M1423, Delft Hydraulics Laboratory, Delft, The Netherlands; 1982. Experimenteel onderzoek naar het snelheidsveld in de turbulente bodemgrenslaag in een oscillerende stroming in een golftunnel. Report M1562, Delft Hydraulics Laboratory, Delft, The Netherlands]. It has been shown that modeled and observed (Van Doorn, T., 1981. Experimental investigation of near bottom velocities in water waves with and without a current. Report M1423, Delft Hydraulics Laboratory, Delft, The Netherlands; 1982. Experimenteel onderzoek naar het snelheidsveld in de turbulente bodemgrenslaag in een oscillerende stroming in een golftunnel. Report M1562, Delft Hydraulics Laboratory, Delft, The Netherlands) mean velocity profiles within the wave and current bottom boundary layer are in better agreement than outside. Modeled and observed (Van Doorn, T., 1981. Experimental investigation of near bottom velocities in water waves with and without a current. Report M1423, Delft Hydraulics Laboratory, Delft, The Netherlands) wave velocity amplitude profiles within the wave and current bottom boundary layer are in better agreement than outside. Modeled wave velocity amplitudes are in good agreement with the laboratory experimental data of Van Doorn [1982. Experimenteel onderzoek naar het snelheidsveld in de turbulente bodemgrenslaag in een oscillerende stroming in een golftunnel. Report M1562, Delft Hydraulics Laboratory, Delft, The Netherlands].     

A simple analytical model for surface water waves on a depth-varying current

A perturbation analysis is presented in which a series of small amplitude regular waves co-exist with an arbitrarily sheared current, U(z). Assuming that the current velocity is weak, i.e. U(z)/c=O(ε), the solution is extended to O(ε²), where c is the phase velocity and ε=ak the wave steepness. This provides a first approximation to the non-linear wave-current interaction, and allows simple explicit solutions for both the modified dispersion relation and the water-particle kinematics to be derived. These solutions differ from the existing irrotational models commonly used in design and, in particular, highlight the importance of the near-surface vorticity distribution. These results are shown to be in good agreement with laboratory data provided by Swan et. al. [J. Fluid Mech (2001, in press)]. Perhaps more surprisingly, good agreement is also achieved in a number of strongly non-linear wave-current combinations, where the results of the present analytical solution are compared with a fully non-linear numerical wave-current model.     

An analytic solution to the wave bottom boundary layer governing equation under arbitrary wave forcing

Existing models of the wave bottom boundary layer have focused on the vertical and temporal dynamics associated with monochromatic forcing. While these models have made significant advances, they do not address the more complicated dynamics of random wave forcing, commonly found in natural environments such as the surf zone. In the closed form solution presented here, the eddy viscosity is assumed to vary temporally with the bed shear velocity and linearly with depth, however, the solution technique is valid for any eddy viscosity which is separable in time and space. A transformation of the cross-shore velocity to a distorted spatial domain leads to time-independent boundary conditions, allowing for the derivation of an analytic expression for the temporal and vertical structure of the cross-shore velocity under an arbitrary wave field. The model is compared with two independent laboratory observations. Model calculations of the bed shear velocity are in good agreement with laboratory measurements made by Jonsson and Carlsen (1976, J. Hydraul. Res., 14, 45–60). A variety of monochromatic, skewed, and asymmetric wave forcing conditions, characteristic of those found in the surf zone, are used to evaluate the relative effects on the bed shear. Because the temporal variation of the eddy viscosity is assumed proportional to the bottom shear, a weakly nonlinear interaction is created, and a fraction of the input monochromatic wave energy is transferred to the odd harmonics. For a monochromatic input wave, the ratio of the third harmonic of velocity at the bed to the first is <10%. However, for a skewed and asymmetric input wave, this ratio can be as large as 30% and is shown to increase with increasing root-mean-square input wave acceleration. The work done by the fluid on the bed is shown to be a maximum under purely skewed waves and is directed onshore. Under purely asymmetric waves, the work done is significantly smaller and directed offshore.     

Bed friction and dissipation in a combined current and wave motion

Two simple two-layer eddy viscosity models, which facilitate analytical solutions, are presented in order to describe the velocity field and associated shear stress in a combined current wave motion. The models, which have the same eddy viscosity in the current boundary layer, but different eddy viscosities in the wave boundary layer, cover together the whole rough turbulent regime. Straightforward definitions are made for the wave friction factor and the current friction factor for the combined motion, which are in accordance with the results for pure waves and pure currents. In this way one avoids the fictitious reference velocities and elliptic integrals which e.g. Grant and Madsen (1978, 1979) experienced. The two friction factors turn out to be functions of four dimensionless parameters. A detailed calculation procedure is presented. Comparison with laboratory experiments yields promising results. A new relation connecting dissipation and bed shear stress is also developed.     

On a theoretical model of rough turbulent wave boundary layers

An analytical theory which describes the motion in a turbulent wave boundary layer near a rough sea bottom by using a two-layer time invariant eddy viscosity model is presented. The eddy viscosity in the inner layer increases quadratically with the height above the sea bottom. In the outer layer the eddy viscosity is taken as a constant. The mean velocity and shear stress profiles, the bottom shear stress and the bottom friction coefficient are presented, and comparisons are made with experimental results.     

Investigating the impact of surface wave breaking on modeling the trajectories of drifters in the northern Adriatic Sea during a wind-storm event

An accurate numerical prediction of the oceanic upper layer velocity is a demanding requirement for many applications at sea and is a function of several near-surface processes that need to be incorporated in a numerical model. Among them, we assess the effects of vertical resolution, different vertical mixing parameterization (the so-called Generic Length Scale –GLS– set of k–ε, k–ω, gen, and the Mellor–Yamada), and surface roughness values on turbulent kinetic energy (k) injection from breaking waves.First, we modified the GLS turbulence closure formulation in the Regional Ocean Modeling System (ROMS) to incorporate the surface flux of turbulent kinetic energy due to wave breaking. Then, we applied the model to idealized test cases, exploring the sensitivity to the above mentioned factors. Last, the model was applied to a realistic situation in the Adriatic Sea driven by numerical meteorological forcings and river discharges. In this case, numerical drifters were released during an intense episode of Bora winds that occurred in mid-February 2003, and their trajectories compared to the displacement of satellite-tracked drifters deployed during the ADRIA02-03 sea-truth campaign.Results indicted that the inclusion of the wave breaking process helps improve the accuracy of the numerical simulations, subject to an increase in the typical value of the surface roughness z₀. Specifically, the best performance was obtained using α_CH = 56,000 in the Charnok formula, the wave breaking parameterization activated, k–ε as the turbulence closure model. With these options, the relative error with respect to the average distance of the drifter was about 25% (5.5 km/day). The most sensitive factors in the model were found to be the value of α_CH enhanced with respect to a standard value, followed by the adoption of wave breaking parameterization and the particular turbulence closure model selected.     

Wave reflection characteristics of plane, dentated and serrated seawalls

The hydrodynamic performance of vertical and sloped plane, dentated and serrated seawalls were investigated using physical model studies. Regular and random waves of wide range of heights and periods were used. Tests were carried out for different inclinations of the seawall (i.e. θ=30, 40, 50, 60 and 90°) and for a constant water depth of 0.7 m. The wave reflection was measured to assess the dissipation character of the seawalls. It was observed that the serrated seawall was superior to the plane and dentated seawall in reducing the wave reflection. Even for the vertical case, the coefficient of reflection due to regular waves for dentated seawall ranged from 0.6–0.99 and for the vertical serrated seawall it was 0.45–0.98, whereas for the vertical plane wall, it was almost 1.0. It was found that the characteristic dimension of the seawall (i.e. L/W) and the relative water depth (i.e. d/L) were better influencing parameters compared to the conventionally used surf similarity parameter ‘ξ’ (ξ=tan θ/(H_i/L)^0.5) in predicting the reflection from the dentated and serrated seawall, where L is the local wave length, W the width of the dent along the length of the seawall slope, d the water depth at the toe and H_i is the incident wave height. A similar trend was observed for the random waves too. The reduction in the wave reflection due to random waves for the dentated seawall as compared to the plane seawall was about 18% and for the serrated seawall, it was 20%. It was observed that the reflection due to random waves was lesser for all the three different walls than the regular waves, due to the mutual interaction of random waves. Multiple regression analysis on the measured data points was carried out and predictive equations for the reflection coefficient were obtained for both regular and random waves. This study will be useful in the design of energy dissipating type vertical quay walls in ports and harbours, sloped seawalls for shore protection from erosion and sloped caisson as breakwaters. Comparison of predictive formulae with the experimental results revealed that the prediction methods were good enough for practical purposes.     

Wave friction factor rediscovered

The wave friction factor is commonly expressed as a function of the horizontal water particle semi-excursion (A _wb) at the top of the boundary layer. A _wb, in turn, is normally derived from linear wave theory by \fracU_\textwbT_\textw2p \frac{{{U_{\text{wb}}}{T_{\text{w}}}}}{{2\pi }} , where U _wb is the maximum water particle velocity measured at the top of the boundary layer and T _w is the wave period. However, it is shown here that A _wb determined in this way deviates drastically from its real value under both linear and non-linear waves. Three equations for smooth, transitional and rough boundary conditions, respectively, are proposed to solve this problem, all three being a function of U _wb, T _w, and δ, the thickness of the boundary layer. Because these variables can be determined theoretically for any bottom slope and water depth using the deepwater wave conditions, there is no need to physically measure them. Although differing substantially from many modern attempts to define the wave friction factor, the results coincide with equations proposed in the 1960s for either smooth or rough boundary conditions. The findings also confirm that the long-held notion of circular water particle motion down to the bottom in deepwater conditions is erroneous, the motion in fact being circular at the surface and elliptical at depth in both deep and shallow water conditions, with only horizontal motion at the top of the boundary layer. The new equations are incorporated in an updated version (WAVECALC II) of the Excel program published earlier in this journal by Le Roux et al. Geo-Mar Lett 30(5): 549–560, (2010).     

Eigensensitivity analysis of moisture-related buckling of marine composite panels

In this paper, an approximate closed-form solution is presented to compute the moisture-related buckling of symmetric angle-ply laminates. The environment corresponds to a steady state condition, which provides a uniform moisture distribution for the laminate. The laminate consists of four layers [θ/−θ]_s constructed of low, moderate and high stiffness ratio materials. Comparative results using the Rayleigh–Ritz method provides a means of assessing the accuracy of the expression. For certain laminate architectures, several modes must be computed to ascertain the lowest buckling mode, and once identified, provides an excellent approximation for the mode computed using the Rayleigh–Ritz method.     

An experimental study of stratified mixing caused by internal solitary waves in a two-layered fluid system over variable seabed topography

Stratified mixing is observed in a wave flume on an internal solitary wave (ISW) of depression or elevation type propagating over a submarine ridge. The submarine ridges, which comprise the seabed topography, are either semicircular or triangular. Tests are performed in a series of combinations of submarine ridges with different heights and ISW in different amplitudes within a two-layer fluid system. When the thickness of the top layer is less than that of the lower layer (i.e., H₁<H₂), a depression-type ISW may produce a strong hydraulic jump with downwards motion and continuous eddy diffusion. During diffusion, the leading profile of the ISW transforms a wrapped vortex on the front face of the ridge, and a vortex separation at the apex of the ridge. Meanwhile, an elevation-type ISW causes a vortex in the lee of a submarine ridge, which resembles a surface solitary wave in terms of wave transmission process. The degree of wave-obstacle interaction is determined by energy loss, which is induced by submarine ridge blockage. The experiment results suggest that degree of blocking can be applied to classify various degrees of ISW-obstacle encounter in the stratified two-layer fluid system.     

An approach for eliminating re-reflected waves

Theoretical and experimental studies were conducted to eliminate the re-reflected waves in a wave channel by installing a wavefilter in front of the wavemaker. A thin porous mesh is installed in front of the wavemaker.to serve as a wavefilter. A porous-effect parameter [Chwang, A. T. and Li, W. (1983), A piston-type porous wavemaker theory. J. Engng Math. 17, 301–313], G₀ = bω/μL₀, is employed to characterize the transmissability of the wavefilter. Theoretical relationships are established between the amplitude of progressive wave and G₀, and the distances between the wavefilter, the wavemaker and the test structure. The proper location for the wavefilter to eliminate re-reflected waves can be determined. Several experimental tests were conducted to verify the theory.Both theoretical and experimental studies show that re-reflected waves can be effectively eliminated by placing a wavefilter at a proper position between the wavemaker and the test structure, provided G₀ ≤ 1. For G₀ > 1 however, the wavefilter would fail.     

The experimental study on infra-gravity wave

A series of experiments were conducted in a super-wave flume (300 m×5 m×5.2 m) to examine the low-frequency motion induced by waves with different incident steepness, sloping gradients and normalized frequency (sideband space). Two kinds of waves including initial uniform wave train and modulated wave train (one carrier with a pair of sidebands) are utilized for incident wave conditions. From the experimental results, it is found that for a given slope gradient the infra-gravity wave component decreases as wave nonlinearity increases and frequency downshift is a predominant factor. Furthermore, the magnitude of low-frequency component decreases with slope gradients for a given initial wave condition. In addition, the maximum value of low-frequency motion is found to be close to the normalized frequency, δ=1.0.     

The stability of the antifer units used on breakwaters in case of irregular placement

The primary aim of the study is to experimentally investigate the stability performance of antifer units on the trunk section of breakwaters under the effect of regular and irregular waves in case of irregular placement. The stability performance tests were conducted for different slopes, i.e. cot α=1.25, 1.5, 2.0, 2.5, under irregular waves and for cot α=2.5 under regular waves. Hudson’s formula was employed in order to characterize the stability performance of antifer units for the irregular placement technique. Different representative wave height parameters, i.e. H_s, H_1/10 and H_max, were examined to determine the one best characterizing breakwater stability. Furthermore, the effects of wave period and wave steepness on the stability of the breakwater were explored.