ZHANG Zhen-wei, ZOU Zhi-li

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: zwzhang@live.cn

(Received February 15, 2012, Revised May 10, 2012)

VERTICAL DISTRIBUTION OF LONGSHORE CURRENTS OVER PLANE AND BARRED BEACHES*

ZHANG Zhen-wei, ZOU Zhi-li

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: zwzhang@live.cn

(Received February 15, 2012, Revised May 10, 2012)

Experiments were conducted to determine the vertical profile of the longshore currents over plane and barred beaches. The logarithmic law is applied to fit the data for the region below the wave trough and an adjusted logarithmic profile without the mass transport velocity is applied to the region above the wave trough. The results indicate that the logarithmic law fits the data well for both plane and barred beaches. The friction velocity and the relative roughness obtained by the data fitting are compared with relevant calculated results.

waves, coast, logarithmic law, surf zone

Introduction

When the waves propagate obliquely to the shoreline, they will break in a shallow water with reduced wave height. This change in the wave height will lead to a momentum transfer from the breaking waves to the surrounding fluid and induce the longshore current and wave set-up. The present paper studies the vertical distribution of the longshore currents by fitting the laboratory measurements using a logarithmic law.

Visser[1]used the dye release method to observe the vertical structure of the longshore currents on a plane beach. Hamilton and Ebersole[2]measured the velocity by placing an Acoustic Doppler Velocimeter (ADV) at different vertical locations on a plane beach. Wang et al.[3]used the similar method to observe the vertical distribution of the longshore currents over a sand beach. A field observation on the longshore current vertical structure was conducted during theDUCK94[4]experiment over a barred beach. An active external recirculation system driven by pumps was developed by Visser[1]to maximize the longshore uniformity. A similar system was also employed by Hamilton and Ebersole[2]and Wang et al.[3].

The present paper investigates experimentally the vertical distribution of the longshore currents by measurements using ADVs over plane and barred beaches. The logarithmic law was applied to fit the measured velocity vertical profile. The paper is organized as follows. In section 1, the experiment setup is presented and the longshore uniformity of the measured longshore currents is shown. Section 2 gives the fitting of the measured velocity vertical profile by the logarithmic law. In section 3, are the concluding remarks.

1. Experiment set-up

Fig,1 Experimental set-up and measurement transects

Fig.2 Photo of the experiment

The experiment was performed in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The wave basin is 55 m long, 34 m wide and 0.7 m deep. Two concrete beaches were constructed, a plane beach with the slope of 1:40 and a barred beach formed by superimposing a bar on the plane beach. Figures 1 and 2 show the experimental set-up for the bared beach. For the plane beach, the experimental set-up is the same as that for the barred beach but with the bar removed. The beach was rotated at 30owith respect to the wavemaker paddles in order to increase the beach length. The still water depth was 0.45 m in the horizontal bottom part of the basin. Two wave guides were built from the wave-maker paddle to the toe of the beach. The wave absorbing layers formed by net boxes filled with plastic scraps were placed on the inner side of the wave guide wall to prevent the wave reflection against the walls.

The coordinate system (x,y) with the origin at the upstream end of the still water shoreline was adopted with x-axis pointing offshore and y-axis downstream. A Gaussian bar profile with a crest height of 0.08 m and width of 2.0 m and located atx=5.0m was adopted for the bared beach, from which, the corresponding still water depth can be calculated as

Unidirectional obliquely incident waves, both regular and irregular, were generated by a multipaddle type wave maker. Table 1 lists the wave conditions in the experiment, where T is the wave period, H is the incidence significant wave height andξ0is the surf similarity parameter, ξ0=tanβ(H0/L0)-1/2, where the subscript 0 denotes the deepwater conditions[5]and tanβ is the beach slope. The breaker is a spilling wave for the plane beach, but for the barred beach, the slope is approximately 1:10 on the seaward side of the bar, resulting in a plunging breaker over the bar crest.

The wave surface elevations were measured by 57 capacitance wave gauges at three sections normal to the shore line, Sections I through III, with the interval 5.0 m between them. The distance from the upper end of the beach to Section I is 7.0 m. The vertical distribution of the longshore currents was only measured at 3-9 locations with interval 0.5 m at Section IV (located at y=14.0 m), which cover the cross shore distance fromx=2.0 m to the offshore toe of the sand bar (x=6.0 m). The depth averaged longshore current was measured by 18 ADVs along Section V, located at y =14.5 m, which were set at 1/3 of the still water depth from the bottom.

Table 1 Wave conditions

In order to check the longshore uniformity of the longshore current, a velocity meter array of 11 ADVs running in the longshore direction was set up at =x 2.5 m or 4.0 m for the plane beach, and at =x3.0 m, 4.0 m for the barred beach, as shown in Fig.1.

The ADV used is produced by the Nortek Company, Norway, with accuracy of 0.5% of the measured value plus ±0.001 m/s. A set of five ADVs was used to measure the vertical variation of the longshore current, which were positioned at five different vertical levels and mounted together on a movable shelf (the gaps between ADVs are from 0.07 m to 0.13 m), as shown in Fig.2. Two sets of such velocity meter array were used simultaneously, located in the shallow and deep water regions, respectively. For the plane beach, the distances from the bottom of the sensors of the five ADV’s are 0.005 m, 0.021 m, 0.040 m, 0.065 m, 0.086 m for measurements in the shallow water region, and 0.005 m, 0.026 m, 0.051 m, 0.073 m, 0.099 m for measurements in the deep water region. For the barred beach, the distances from the bottom of the sensors of the five ADV’s are 0.005 m, 0.017 m, 0.037 m, 0.055 m, 0.072 m for measurements in the shallow water region (including the measurements over the bar crest), and 0.005 m, 0.016 m, 0.041 m, 0.059 m, 0.080 m used for measurements in the deep water region.

All velocity measurements were sampled at 20 Hz and the total sampling duration is 500 s for the regular waves and 620 s for the irregular waves. Data collection began several seconds before the wave is generated in order to record the still water level for the measurements of the wave set-up. The steady velocity data from the sampled data of about 200 s in the time domain were used to determine the mean longshore currents.

To form the circulation system caused by the longshore currents, a passive recirculation system was employed, which was formed by the channel at each end of the beach and behind the beach model, as shown in Fig.1 by arrows. The channel width is 4.4 m at the two ends of the beach and in a range from 4.0 m to 8.0 m behind the beach. The channel water depth is the same as that in the horizontal bottom in front of the beach. The flow in the downstream channel will be driven by the longshore currents and goes through the channel behind the beach model to reach the upstream channel. Then, it feeds into the upstream end to form a closed fluid circulation. This form of a passive circulation system was discussed theoretically and experimentally by Dalrymple et al.[6]. Compared with the active circulation system used by other researchers (Visser 1991), this system has an advantage that it is not required to have a man-made flow discharge fed in the beach upstream and drained from the beach downstream end, as is adopted in a pump driving circulation system and this discharge is difficult to be determined accurately. Although this passive circulationsystem only has a uniformed longshore current in the middle part of the beach, the length of the uniformed longshore current is long enough for the measurement of the vertical velocity distribution.

Fig.3 Longshore distributions of longshore current over plane and barred beaches

Figure 3 shows the velocity longshore distributions for the plane and barred beaches (Tests AR1, AR2, AI1, AI3 for the plane beach and Tests BR1, BR2, BI1, BI4 for the barred beach), along two measuring lines, x=2.5 m, 4.0 m for the plane beach and x=3.0 m, 4.0 m for the barred beach. It can be seen in the figure that for the plane beach the longshore currents is uniform from y=8.0 m to 18.0 m for test AR1, y=8.0 m to 16.0 m for Test AR2 (regular wave cases) and from y=10.0 m to 18.0 m for Test AI1, AI2 (irregular wave cases). For the barred beach, the longshore currents is uniform from y=6.0 m to 15.0 m for Test BR1, y=6.0 m to 16.0 m for Test BR2 (regular wave cases) and Test BI1, BI2 (irregular wave cases). So the length of the region with a uniform longshore current comprises about one third of the total beach length. The vertical velocity profile is chosen to be measured in this region, at y=14.0 m from the beach upstream end.

Figure 4 shows the longshore uniformity of the wave height and the set-up for AI1 (irregular wave) and BI2 (irregular wave) over plane and barred beaches, respectively. As shown in the figure, the wave height and the set-up are uniform in the basin. The wave field is approximately homogeneous for other waves.

Three runs of tests for each condition were conducted. The relative error of the mean longshore current measurements (averaged over all cross-shore measurements) for the three runs was 5.68% for the plane beach and 8.86% for the barred beach, respectively. This shows that the repeatability of ADV measurements is acceptable.

2. Logarithmic fitting for vertical velocity distribution

Fig.4 Longshore uniformity of wave height and set-up for plane and barred beaches (the beach scaled by 0.5)

The vertical profile of the measured longshore currents is usually fitted by a logarithmic law. Here we also follow this practice to see if this law is applicable for the data obtained in the present experiments. Since the oscillation of the surface elevation affects the flow above the wave trough level, the logarithmic law should be applied below the wave trough level. So the whole water column is divided into two layers, the upper part above the wave trough levelz=MWLHrms=-Hrms(the surface layer) and the lower part below the wave trough level (the bottom layer), where Hrmsis the root mean square (rms) wave height and MWL stands for the mean water level. In the surface layer, the improved logarithmic law proposed by Faria et al.[4]is applied. In the bottom layer, the following logarithmic law is applied where z is the vertical coordinate pointing upward and with its origin at the still water level, κ the Von Karman constant, generally taking the value of 0.4,h the local still water depth, v*the friction velocity, zathe apparent roughness[7-9]. zais different from thebot tomrough ness z0(z0=ks/30,ksthe Nikuradse roughness), which involves the impact of waves on the bottom boundary layer. The interaction between the waves and the currents within the bottom boundary layer increases the bottom boundary layer height, thus zais greater than z0. A linear regression least squares method is used to determine the constants in Eq.(2), and we have

with

where a and bare obtained from data fitting.

In the surface layer, one or two ADVs may move in and out of water, which leads to noises in the ADVs readings. The data acquired during this measuring period needs a special treatment in the data processing. It is desired to decide whether the ADVs are in or out of water. In the present study, the correlation coefficient between the velocities at different moments was used for this purpose, as the value of the correlation coefficient will have a sudden drop to 0.2 or lower if an ADV is out of water[10]. Therefore, the threshold value taken as 0.2 for the analysis; if the correlation coefficient of the corresponding measured data is less than or equal to 0.2, the measured velocity will be set to zero. This treatment only removes the noise in the data time series, and will not influence the results of the mean longshore currents.

After the above treatmentfor the velocity data in the surface layer, the following modified mean current within the surface layer is applied where V＇(z)is the logarithmic profile as Eq.(2),η the mean water level, η the surface elevation, P(η) the cumulative surface elevation probability density function (pdf) following the Gaussian distribution pdf. The percentage of time when the current meter is in water is given by 1-P(η). Hrmsis calculated by, where σ is the standard dev i ation calculated from the surface elevation time series. P(η) is computed by

Fig.5 Comparison of vertical profiles with and without the mass transport (Test AR3). Upper: without mass transport velocity, lower: with mass transport velocity

Expression (5) is different from the equation used by Faria et al.[4], the former excludes the mass transport velocity associated but the latter does not, which reads

in which the last term is the mass transport velocity,＜＞the ensemble average,θ the mean wave incident angle, ＜U(z)＞ the ensemble mass transport velocity defined by

where U(A) is the Stokes wave mass transport velocity[11]with wave amplitudeA, and p(A) the Rayleigh probability density function. To show the effect of the mass transport velocity, Figure 5 gives the vertical profile of the longshore currents given by Expressions (5) and (7), and it is shown that the inclusion of the mass transport (using Expression (7)) leads to a deviation of the velocity profile near the breaking point (x=5.5 m) from the measured data between the trough level and the mean water level. But without the mass transport velocity, the vertical profile (given by Expression (5)) agrees well with the measured data.

Figure 6 shows the measured and the fitting velocity profiles at different locations along the cross shore direction for the plane beach. The logarithmic profiles in the bottom layer are indicated by solid lines while the adjusted logarithmic profiles in the surface layer by dash lines. It is seen that the logarithmic profiles fit the measured data well, whereas the adjusted logarithmic profiles sees some deviation from the measured data at the most upper points (for example, the location =x3.0 m of Test AR1). It may be caused by the difficulty in measuring the velocity above the mean water level.

Figure 7 shows the measured and fitting velocity profiles at different locations along the cross shore direction for the barred beach. The agreement is similar to those for the plane beach discussed above. Even for the location at the bar crest, the agreement is also acceptable, where there are only two set of measured data and only the adjusted profile can be applied (at water depth of 0.045 m).

2.1 Friction velocity

From the above data fitting with a logarithmic law, the values of the apparent roughness zaand the friction velocity v*can be obtained. For v*, a comparison is made between the logarithmic fitted values and the predictions using the following shear stress formula[12]

in which θ is the angle between the longshore current and the wave propagation direction. Generally, the waves tend to approach the shoreline normally due to the refr action, thus θ= π/2 in th e present study.isthedepth-averagedlongshorecurrent, ρ the

Fig.6 Vertical profiles of longshore currents for regular and irregular waves over plane beach

water density,|u~| the average of the fluid particle velocity in a period on the bottom,|u~|=2um/π,umthe maximum horizontal velocity of the water particle on the bottom, um=Aω/sinh(kh ),A the wave amplitude, =A/2hγ in the surf zone and γ the wave breaking index, =γ0.6, 0.7[13,14]for irregular and regular waves, ω the wave frequency,cfthe wave friction coefficient, with the value of 0.00225. From the relation=ρv2, the friction velocity can be wri-* tten as

For the plane beach, Fig.8 shows the friction velocities obtained from the logarithmic fitted values and those from formula (10) for regular and irregular waves, respectively. It can be seen that the logarithmic fitted results agree well with the theoretical calculations in magnitude, both of which have a decrease trend with the increase of the distance from the shoreline. Figure 9 shows the corresponding results for the barred beach and the good agreement is found.

2.2 Apparent roughness height

For za, its value can be obtained from values of za/z0by the data fitting with z0=ks/30 and with the Nikuradse roughness taking the value asks= 0.001 m for the smooth concrete bottom used in the present experiment. The results obt ained in such a waywerecomparedwiththetwowidelyappliedempirical relationships for za/z0

Fig.8 Friction velocities for regular and irregular waves over plane beach

[15,16].

The comparison indicates that none of these empirical relations can predict the apparent roughness reasonably, maybe because the present measurement can not give an accurate value of za/z0, since the obtained results of za/z0depend on the velocity measurement accuracy and within the thin wave boundary layer, the velocity measurement can not be performed. calculations. But the obtained relative roughnessza/ z0can not be predicted by the two empirical relations, maybe because the relative roughness is associated with t he bottom roughness and the velocity within the waveboundary layer, which are dependent on the tur-

3. Conclusions

A series of laboratory tests were performed to determine the vertical profile of the longshore currents over plane and barred beaches. The vertical profiles of the longshore currents fall into two parts: the surface layer and the bottom layer. From the test data, a logarithmic profile can be found for the bottom layer. Due to the presence of waves, the velocity profile for the surface layer is described by an adjusted logarithmic profile without the mass transport velocity.

The friction velocity and the apparent roughness given by data fitting were analyzed. The magnitude of the obtained friction velocity agrees with the theoretical formula commonly used for the longshore currentbulence condition and difficult to be determined accurately in the experiment.

Fig.9 Friction velocity results for regular and irregular waves over barred beach

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10.1016/S1001-6058(11)60296-5

* Project supported by the National Natural Science Foundation of China (Grant No. 51079024), the Funds for Creative Research Groups of China (Grant No. 50921001).

Biography: ZHANG Zhen-wei (1979-), Male,

Ph. D. Candidate

ZOU Zhi-li,

E-mail: zlzou@dlut.edu.cn