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Observation and measurement of the bottom boundary layer flow in the prebreaking zone of shoaling waves
Ocean Engineering 29 (2002) 1479–1502 www.elsevier.com/locate/oceaneng
Observation and measurement of the bottom boundary layer flow in the prebreaking zone of shoaling waves Chang Lin a,*, Hwung-Hweng Hwung b a
Department of Civil Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Republic of China b Department of Hydraulic and Ocean Engineering, National Cheng-Kung University, Tainan 600, Taiwan, Republic of China Received 15 May 2001; accepted 8 August 2001
Abstract 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 具U典 at each phase ¯ . The magnitude of U ¯ is non-uniform, which is directly affected by the mean velocity, U increases from zero at the bottom to a local positive maximum at about z / d⬵2.0 苲 2.5 (where z is the height above the sloping bottom and d is the Stokes layer thickness), then decreases gradually to zero at z / d⬵6.0 苲 7.0 approximately, and finally becomes negative as z / d increases further. Moreover, as waves propagate towards shallower water, the rate of increase ˜ ⫹ is greater than that of the offshore in the maximum onshore oscillating velocity component U ⫺ ˜ counterpart U except near the breaking point. The free stream velocities in the profiles of ˜⫺ ˜ ⬁⫹ and U the maximum onshore and offshore oscillating velocity components, U ⬁ are found to appear at z / dⱖ6.0. 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. ˜⫺ ˜ ⬁⫹ is greater than U Although U ⬁ and the asymmetry of the maximum free stream velocities ˜⫺ ˜ ⬁⫹ / U 兩) increases with decrease of water depth, a universal similar profile can be estab(i.e. 兩U ⬁ * Corresponding author. Fax: +886-4-22855182. E-mail address: [email protected] (C. Lin). 0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 1 ) 0 0 0 9 4 - 4
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˜ ⫹ /U ˜ ⬁⫹ 兩) or (兩U ˜ ⫺/U ˜⫺ lished by plotting z/ d versus (兩U ⬁ 兩). 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. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bottom boundary flow; Pre-breaking zone; Shoaling waves; Free stream velocity; Maximum oscillating velocity components; Similar profile
Nomenclature A1 A⬁+ ci d dc Hc Ho/Lo Pi Q 具Q典 ¯ Q ˜ Q Q⬘ →
r Re R∗e T Td Ti Ti t tp tp/T U 具U典 ¯ U ˜ U
Orbital amplitude at the edge of the bottom boundary layer in a linearly (symmetric) oscillatory flow ( ⫽ U1T / 2p) Maximum onshore orbital amplitude at the edge of the bottom ˜ ⬁⫹ T / 2p) boundary layer ( ⫽ U Regression coefficient (i=1, 2,…, 8) Local water depth Water depth measured at the horizontal portion of the wave flume Wave height measured at the horizontal portion of the wave flume Converted deep-water wave steepness ith measuring section (i=1, 2,…, 10) ˜ ⫹ Q⬘) ¯ ⫹Q A physical quantity as function of time ( ⫽ Q ˜) ¯ ⫹Q Phase average of Q as a function of phase ( ⫽ Q Time mean of Q Oscillating component of Q Turbulent fluctuation component of Q Position vector ˜ 1A1 / v) Reynolds number ( ⫽ U ˜ ⬁⫹ A⬁⫹ / v) Reynolds number ( ⫽ U Mean wave period Wave period corresponding to the discarded data set ith wave period Sampling duration Time Phase time varying from 0 to T Phase varying from 0 to 1 ˜ ⫹ U⬘) ¯ ⫹U Horizontal velocity as a function of time ( ⫽ U ˜) ¯ Phase average of U as a function of phase ( ⫽ U ⫹ U Time mean of U (i.e. mean horizontal velocity) Oscillating component of U
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˜1 U ˜ ⫹ U ˜⫺ U ˜⬁ U ˜ ⬁⫹ U ˜⫺ U ⬁ U⬘ z d v h 具h典 h¯ h˜ h⬘
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Maximum free stream velocity at the edge of bottom boundary layer in a linearly (symmetric) oscillatory flow Maximum onshore oscillating velocity component Maximum offshore oscillating velocity component Free stream velocity as a function of phase Maximum onshore free stream velocity (i.e. maximum onshore oscillating velocity component at the edge of bottom boundary layer) Maximum offshore free stream velocity (i.e. maximum offshore oscillating velocity component at the edge of bottom boundary layer) turbulent fluctuation component of U Local height above the sloping bottom Stokes layer thickness ( ⫽ √vT / p) Kinematic viscosity of water Water surface elevation as a function of time ( ⫽ h¯ ⫹ h˜ ⫹ h⬘) Phase average of h as a function of phase ( ⫽ h¯ ⫹ h˜ ) Time mean of h Oscillating component of h Turbulent fluctuation component of h
1. Introduction Gravity waves propagating from deep to shallow water exhibit continuous transformation in wave profile due to shoaling accompanied by nonlinearity, thus leading to steepening of the wave crest and flattening of the wave trough. Eventually, the profile becomes unstable, leading to breaking waves. Since the hydrodynamic characteristics of shoaling waves are closely related to onshore/offshore sediment transport, vertical distribution of suspended loads, and pollutant dispersion in coastal waters; it is of paramount importance to investigate the flow field of shoaling waves, with special emphasis on the velocity distribution within the bottom boundary layer. Although numerous investigations for the wave shoaling have been carried out in the past (e.g. Koh and LeM’ehout’e, 1966; Adeyemo, 1970; Iwagaki et al., 1974; Flick et al., 1981; Hedges and Kirkgo¨ z, 1981; Kirkgo¨ z, 1986; Nadaoka, 1986; Hwung and Lin, 1990; Voropayev et al., 2001), the study for the bottom boundary layer structure of shoaling waves propagating on a sloping bottom is still rather limited. Recently, considerable efforts have been devoted to the phenomenon of wave propagating over horizontal bottom. For example, Horikawa and Watanabe (1968) and Sleath (1970) used hydrogen bubbles and tension wire, respectively, to measure velocity distributions near smooth and rough beds. Jonsson and Carlsen (1976), utilizing a micropropeller flow-meter, measured the velocity profiles of turbulent boundary layer flow over rough beds in an oscillating water tunnel. Employing a laser Doppler
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velocimeter (LDV), Van Doorn (1981), Kemp and Simons (1982), Asano and Iwagaki (1984) and Belorgey et al. (1989) performed non-intrusive measurements of the near-bottom velocities in water waves with or without a current. Teleki and Anderson (1970) may have been the first research to investigate the near-bottom water-particle velocity on a (smooth) slope with a Preston tube. However, their measurements were only made at the instants of wave-crest and wavetrough by alternatively orienting the L-shaped probe upslope and downslope, which interfered with flow field and thus led to measurement error. Recently, Hwung and Lin (1989, 1990) and Kirkgo¨ z (1989) utilized LDV to measure the horizontal velocity profiles in the bottom boundary layer of shoaling waves propagating on 1/15 and 1/12 uniform slopes, respectively. Hwung and Lin (1989, 1990) investigated the vertical distributions of the mean velocity in the pre-breaking zone for different breaking waves. Kirkgo¨ z (1989) examined the vertical distributions of the horizontal velocity under the wave-crest and wave-trough in the transformation zone of plunging breakers. In Kirkgo¨ z (1989), the discrepancy between the measured and predicted velocity distributions was considered to be attributable to the turbulent effect in the near wall region. However, no clear evidence of turbulence was demonstrated, nor was the influence of mass transport on the measured velocity distribution (which might result in such inconsistency) discussed in his investigation. The aim of this study is to elucidate the flow characteristics within the bottom boundary layer induced by nonlinear, asymmetric shoaling waves, propagating on an 1/15 sloping beach. A flow visualization technique with thin-layered fluorescent dye was first used to observe the flow structure of the bottom boundary layer. Then, a one-component LDV was employed to measure the horizontal (water-particle) velocity in the bottom boundary layer across the pre-breaking zone. Finally, some important features of the bottom boundary layer structure are examined and discussed.
2. Experimental set-up and method Experiments were conducted in a glass-walled wave flume located at Tainan Hydraulic Laboratory, National Cheng Kung University. The wave flume is 9.5 m long, 0.3 m wide and 0.7 m deep. The first 3.77 m section had a fixed horizontal smooth bottom starting from the end where a flap-type wave maker was installed. The remaining 5.73 m of the flume had an adjustable sloping smooth bed made of polished stainless plate. A fixed slope of 1/15 was used in this study. Monochromatic waves could be generated with periods ranging from 0.3 to 1.5 s and with wave heights up to 15.0 cm. The experimental conditions are listed in Table 1, in which T is the mean wave period, Hc and dc are the wave height and water depth measured at the horizontal portion of the flume, and Ho/Lo is the converted deep-water wave steepness. Two capacitance-type wave gauges were used to detect the water surface elevations in the experiments. A novel flow visualization technique with thin-layered fluorescent dye was employed to examine the flow characteristics of the bottom boundary layer induced
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Table 1 Experimental wave conditions Case
T (s)
Hc (cm)
dc (cm)
Ho/Lo
Breaker type
1 2 3 4 5 6
1.41 1.41 1.07 0.96 0.74 0.96
5.3 7.5 4.7 5.3 4.3 8.2
33.0 33.0 33.0 33.0 33.0 33.0
0.0186 0.0260 0.0300 0.0405 0.0520 0.0617
Plunging Plunging Plunging Plunging Spilling Spilling
by shoaling waves. The fluorescent dye used was a water solution of either fluorescent sodium (C20H10Na2O5) or rhodamin B (C28H31ClN2O3) with a specific gravity of 1.015 as measured by a precise hydrometer. The dye was injected very slowly and smoothly (through a stainless steel pipe of 2.0 mm in diameter) onto the surface of the sloping bottom near the toe of the 1/15 beach, while the water mass inside the wave flume was kept still. Consequently, the thin-layered fluorescent dye was formed. It was then stretched and transported along the sloping bottom due to the effect of mass transport of shoaling waves after wave maker was started. A floodlight and a 2 W argon-ion laser were used as light sources to illuminate the boundary layer flow structures inside the pre-breaking zone and in the neighbourhood of the breaking point, respectively. In the latter case, a laser beam was emitted from a laser head (Lexel 95-2) and spread into a fan-shaped light sheet (about 1.5 mm thick) using a mirror and a cylindrical lens. In the present experiment, the light sheet was used to illuminate the two-dimensional motion of the dye tracer on a vertical plane along the longitudinal direction of the flume. Photographic equipments, including a camera and a video camera, were also set up in front of a sidewall of the wave flume. Utilizing a LDV system, measurements of the horizontal velocities on the sloping beach were then taken in all or some of the 10 vertical sections (i.e. from P1 to P10) as shown in Fig. 1, depending on the experimental conditions required. The local water depth, d, and the local height above the sloping bottom, z, are also given in Fig. 1. The LDV used in the present study was a one-component, dual-beam optical system, equipped with an acousto-optic Bragg cell driven by a 40 MHz oscillator. The light source was a Lexel 2 W argon-ion laser. All of the transmitting optics were mounted on a three-dimensional translation table. Its movement could be precisely controlled by three optical meters and three digital monitors (with precision of 0.001 mm). A small measuring volume was achieved from the standard TSI optics by using a 2.27× beam expander ahead of a 350 mm focal lens. The dimensions in diameter and length of the measuring volume were estimated to be 0.08 and 1.09 mm for two green beams, respectively. The measuring error of the LDV system was estimated to be ca. 0.14 cm/s. To obtain the velocity measurements at locations near the sloping bottom without beam obstruction due to the edge of the sloping bottom, the transmitting optics were tilted at an angle of 1.2° towards the surface of the sloping bottom (see Fig. 2). The
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Fig. 1.
Schematic diagram of measuring sections and coordinate system.
Fig. 2.
Schematic diagram of the installation of the LDV system.
receiving optics were tilted slightly downwards by ca. 2.5° to reduce the intense light scattered from the surface of the sloping bottom. The zero location of the measuring volume was determined by slightly adjusting the vertical movement of the translation table and observing the digital outputs of optical meter. The positioning error was estimated to be less than ±0.04 mm. Time series of the water surface elevation and the corresponding horizontal velocity were simultaneously measured by a wave gauge and the LDV system at every vertical section. These data were collected using an AD/DA converter. The sampling rate was kept at 100 Hz for all of the experimental cases. To minimize the multireflection problem arising from finite length of the wave-flume, the sampling time for each run was only 8–15 s, depending upon the experimental cases tested. In this way, several runs were completed repeatedly with a total sampling time varying from 45 to 64 s.
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3. Data analysis Hussain and Reynolds (1970) proposed a phase average method that, for a steady turbulent shear flow coexisting with a periodic motion, any physical quantity Q could ˜ , and ¯ , a (periodically) oscillating component Q be decomposed into a time mean Q a turbulent fluctuation component Q⬘. The relation can be expressed as ˜ (→ ¯ (→ r , t) ⫹ Q⬘(→ r , t), Q(r¯ , t) ⫽ Q r) ⫹ Q
(1)
in which → r and t are position vector and time variable, respectively. The time mean of Q is in the form of ¯ (→ Q r) ⫽
冕
1 Ts
Ts
Q(→ r , t)dt,
(2)
0
where Ts is the total sampling time, which is a multiple of the mean wave period T. The phase average of Q is defined as
冘 M
1 ¯ (→ ¯ (→ Q(→ r , tp ⫹ mT) ⫽ Q r) ⫹ Q r , tp), 具Q( r , t p)典 ⫽ M ⫹ 1m ⫽ 0 →
(3)
in which M is the total number of sampling cycle, and tp is the phase time ranging from 0 to T. Hence, at any point in space, the phase average is the average value of Q that is recognized at a special phase time tp in each wave period. Accordingly, the oscillating component of Q is then ˜ (→ ¯ (→ r , tp)典⫺Q r ). Q r , tp) ⫽ 具Q(→
(4)
Therefore, given a reference signal with a specific period (e.g. the free surface elevation h(t)) and a signal Q(t) at a particular location (e.g. the horizontal velocity U(t)), these three components can be determined easily by Eqs. (1)–(4). Note that ¯˜ ⫽ 0, and Q ¯典 ⫽ Q ¯ Q ¯ ⬘ ⫽ 0. 具Q However, the period of each wave cycle in the wave flume cannot be exactly the same due to the precision limit of the wave maker used and the long-term oscillation induced by wave breaking. If the phase average method stated above is used directly, then the problem of phase time shift (drift) will occur frequently during data processing and a false image of oscillating component will be produced. To overcome this problem, the modified phase average method proposed by Hwung and Lin (1989) is adopted in the present study. Some details are briefly illustrated as follows. First of all, the time mean values of the water surface elevation and the horizontal velocity, ¯ are subtracted from h(t) and U(t), respectively. Next, the zero-up crossing h¯ and U points are determined by linear interpolation to the data series of h(t) and U(t) and the data set within an individual wave period Td is discarded when the relative error between Td and T is ⬎2%. These data series are then re-interpolated into the same data points in each qualified wave period, according to the ratio of the ith wave
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period Ti to the mean wave period T. Finally, the modified phase average of Q at a particular phase time, tp, can be rewritten generally as
冘冉 N
冊
1 Ti ˜ → ¯ (→ Q r , tp ⫹Q r) 具Q( r , tp)典 ⫽ Ni ⫽ 1 T →
(5)
in which N is the total sampling cycles qualified.
4. Results and discussions 4.1. Flow visualization results It has been widely recognized that the boundary layer developed over natural sea bottom is a turbulent flow. However, it is not clear if turbulence could occur in the case of laboratory wave. It should be examined in the first stage of this investigation. Figures 3(a, b) show the visualized results of the dye layer flow on the sloping bottom within the pre-breaking zone of case 1 (see Table 1). They were photographed at different times by a camera with different viewing angles when the wave motion had reached a steady state. The direction of wave propagation is from left to right. A seemingly straight fluorescent dye ‘line’ moving parallel along the sloping bottom is clearly observed in Fig. 3(a). Furthermore, a very long ‘strip’ of fluorescent dye with a nearly constant width, partially spanning the sloping bottom, can also be seen
Fig. 3.
Visualized results of the dye layer flow on the bottom within the pre-breaking zone of case 1.
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distinctly in Fig. 3(b). The fluorescent dye layer in both figures is consecutively transported shorewards by Lagrangian mass transport velocity (Longuet-Higgins, 1953; Bijker et al., 1974) before wave breaking. During the process of wave shoaling, neither bursting phenomenon (Kline et al., 1967) nor irregular disturbance could be observed on and near the bottom even by detailed examinations of the photos taken and the films recorded. This holds true for all of the experimental conditions tested. According to Tennekes and Lumley (1972), turbulent flow is characterized by irregularity (or randomness), diffusivity, and high level of fluctuating vorticity, and so on. Hence, we can deduce that the boundary layer structure inside the pre-breaking zone is not of turbulent flow, at least from the tests conducted in our study. But, the flow characteristics of bottom boundary layer at the breaking point and in its close vicinity for case 1 are quite different from those observed within the pre-breaking zone. From the visualized result of Fig. 4(a), it is seen that, when wave phase changes from crest to trough, some of the fluorescent dye layer close to the sloping bottom is disturbed by the turbulent action due to the interaction of mass transport velocity in the pre-breaking zone and the reverse flow inside the surf zone. At the instant of wave trough, the turbulent dye-cloud rises in large flux from the bottom, as seen in Fig. 4(b). Moreover, as shown in Fig. 4(c), a massive cloud of the fluorescent dye rushes upwards as the first plunging action arrives. According to Lin and Hwung (1992), the complex phenomenon is caused by the strong anticlockwise motion of the water particles behind the crest of the breaker, and a counterbalancing motion for those in front of it. 4.2. Characteristics of the phase-averaged velocity and the mean velocity The relationship between the water surface elevation and the horizontal velocity is now introduced. Typical examples of the time histories of the water surface elevation h(t), measured by a wave gauge, and of the corresponding horizontal velocities U(t) obtained by LDV at z=0.20 mm and 0.60 mm in section P9 of case 2, are given in Figs. 5(a, b) and 6(a, b), respectively. It is found that, although extremely tiny fluctuations can be detected in the velocity traces, variations of the water surface elevation and the horizontal velocity are very periodic and highly nonlinear (i.e. non-sinusoidal); and that neither the Tollmien-Schlichting instability of transition phenomenon (Bloor, 1964; Obremski and Fejer, 1967) nor irregularly turbulent fluctuation would be seen in Figs. 5(b) and 6(b). Based on the visualized results delineated in Figs. 3(a, b) and 4(a–c) along with the above discussion, it can thus be confirmed that the boundary layer flow in the pre-breaking zone is laminar in character except in a small region near the breaking point. Hence, the turbulent fluctuation components of the water surface elevation and of the horizontal velocity, h⬘(t) and U⬘(t), may be neglected. Moreover, the results shown in Figs. 5(a, b) and 6(a, b) suggest that the water surface elevation can be used as reference signal, with which the phase relationship and the flow field of velocities (measured by LDV at different times and heights above the sloping bottom) can be closely correlated and reconstructed by the modified phase average method. In the present study, the oscillating component of the
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Fig. 4. Visualized results of the consecutive development of the bottom boundary layer flow in the neighborhood of the breaking point of case 1. (The breaking point is indicated by a white arrow.)
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Fig. 5. Time series of the water surface elevation and the corresponding horizontal velocity U(t) measured at section P9 of case 2: (a) water surface elevation; (b) horizontal velocity obtained at z=0.20 mm.
Fig. 6. Time series of the water surface elevation and the corresponding horizontal velocity U(t) measured at section P9 of case 2: (a) water surface elevation; (b) horizontal velocity obtained at z=0.60 mm.
water surface elevation, h˜ , is selected as a reference signal with phase (tp / T) varying from 0 to 1, where tp / T ⫽ 0 is designated as the phase of zero-up crossing of h˜ . For example, Fig. 7(a) shows the variation of h˜ as a function of phase, obtained at section P6 of case 6. Furthermore, the corresponding changes in the phase-averaged ˜ ), originally measured at six different heights in the same ¯ ⫹U velocity 具U典 (i.e. U section, are given in Figs. 7(b–g), respectively. Some interesting features observed in Figs. 7(a–g) are worthy of mentioning. Firstly, due to the effect of wave shoaling,
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Fig. 7. Temporal variations of the oscillating component of water surface elevation and of the phaseaveraged velocities measured at six different heights at section P6 of case 6.
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the oscillating component of the water surface elevation, h˜ , becomes steepened at the wave crest and flattened near the trough, indicating nonlinear, asymmetric characteristics of wave motion in the pre-breaking zone (Adeyemo, 1970; Flick et al., 1981; Voropayev et al., 2001). The same trend can also be observed from the horizontal velocities shown in Figs. 7(b–g). Secondly, the maximum onshore phase-averaged velocities, 具U典max, measured at different heights of section P6, increase rapidly up to a maximum at a certain height, and then decrease gradually with increasing height z. This infers that a significant velocity gradient and the phenomenon of overshooting actually exist in the bottom boundary layer induced by shoaling waves. Thirdly, the value of tp / T associated with the maximum onshore phase-averaged velocity decreases with decreasing measuring height within the bottom boundary layer, implying the existence of phase difference inside the viscous boundary layer. In reference to the study of monochromatic waves propagating over a horizontal smooth bed, the phase difference is simply caused by the effect of viscous damping, as confirmed by Horikawa and Watanabe (1968), Sleath (1970) and Lin et al. (1995). In the case of shoaling waves over a sloping bed, the phase difference may be affected by both the viscous damping and the higher harmonic components of waves (Flick et al., 1981). However, no attempt was made in this study to differentiate the respective influences which were resulted from these two factors. Similarly, temporal variations in the water surface elevation h˜ and in the vertical distribution of the phase-averaged velocity 具U典 obtained at three different sections of case 1 are depicted in Figs. 8(a, b)–10(a, b), respectively. In these figures, different symbols are used for different phases of the occurrence, which can be identified in the abscissa of Figs. 8(a)–10(a). The local measuring height z, appearing in the ordinate of Figs. 8(b)–10(b), is non-dimensionalized by the characteristic length scale, d ⫽ 冑vT / p (v is the kinematic viscosity of water), which is also known as the Stokes layer thickness. Upon comparing the results presented in Figs. 8(a)–10(a), it is clear that nonlinearity and asymmetry of the water surface elevation h˜ increase with decreasing water depth due to the effect of wave shoaling. The distributions of the phase-averaged velocity 具U典 occurring at some specific phases, taking tp / T ⫽ 0 (marked with ‘䊋’) and the wave-crest phase (with ‘ ’) for examples, also increase rapidly with decreasing water depth for the same dimensionless height z / d. This implies that, very close to the sloping bottom, the onshore shear stress increases considerably in the pre-breaking zone as water depth becomes shallower. However, the phase-averaged velocity distribution taking place at the wave-trough phase (marked with ‘䊎’) increases slightly in magnitude with decreasing water depth. The differences in magnitude and shape of the velocity distributions between the wave crest and trough phases may be attributed to the wave-crest steepening and trough flattening in nonlinear water waves due to wave shoaling (Adeyemo, 1970). It is worth noting that, at every measuring section in the pre-breaking zone for the six different Ho / Lo listed in Table 1, the phase-averaged velocity profile at every phase does not have the shape of time-dependent uniform flow, which is one of the characteristic indicators of oscillatory flow at the edge and outside of the boundary layer. Similar change in the velocity distribution could also be evidenced from the limited
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Fig. 8. Temporal variations of the oscillating component of water surface elevation and of the vertical distributions of the phase-averaged velocity, measured at section P4 of case 1.
results of smaller-scale wave experiments performed by Kirkgo¨ z (1989). However, the exact reason for such a distribution remains unknown. To make clear which mechanism dominating such a velocity profile, the distri¯ corresponding to the phase-averaged velocity 具U典 shown bution of mean velocity U in Figs. 8(b)–10(b), are presented in Figs. 11(a–c), respectively. We can see that the ¯ increases from zero at the bottom to a local positive vertical distribution of U maximum at about z / d⬵2.0 苲 2.5, then decreases gradually to zero at z / d⬵6.0 苲 7.0 approximately, and finally becomes negative as the non-dimensional height increases further. Although only limited data of the mean velocity distribution for case 1 are presented herein, this characteristics is found to be also true for other experimental conditions listed in Table 1. The present results are also in accordance with those measured by Hwung and Lin (1990), in which detailed and unified dis-
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Fig. 9. Temporal variations of the oscillating component of water surface elevation and of the vertical distributions of the phase-averaged velocity, measured at section P8 of case 1.
cussion about the profiles of the mean horizontal velocity within the bottom boundary layer was first reported for various values of deep-water wave steepness.
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Fig. 10. Temporal variations of the oscillating component of water surface elevation and of the vertical distributions of the phase-averaged velocity, measured at section P10 of case 1.
4.3. Vertical distributions of the maximum onshore/offshore oscillating velocity components ˜ at different phases can be obtained, accordThe oscillating velocity components U ing to Eq. (4). As an example, Fig. 12 illustrates some of such results calculated from those shown previously in Fig. 9(b). It is found that, for z / d⭓6.0, the oscillating velocity component occurring at each phase does approach the time-dependent uni˜ ⬁. This indicates that non-uniform distriform velocity, i.e. the free steam velocity U butions of the phase-averaged velocity presented in Figs. 8(b)–10(b) are due to the ¯. effect of mean velocity, U
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Fig. 11. Vertical distributions of the mean velocity at three different sections of case 1: (a) P4; (b) P8; (c) P10.
Fig. 12. Temporal variations of the vertical distribution of the oscillating velocity component corresponding to some of those shown previously in Fig. 9(b). (Symbols used are the same as in Fig. 9(b).)
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Among the vertical distributions of the oscillating velocity component as shown in Fig. 12, the profiles of the maximum velocity in the onshore and offshore directions (i.e. the largest positive and negative) are, undoubtedly, of fundamental importance to the problems associated with wave energy dissipation by bottom friction and nearshore sediment transport. Herein, Figs. 13(a–f) and 14 (a–f) present the collective sketches for the vertical distributions of the maximum onshore and off˜ ⫺, for two shoaling waves with ˜ ⫹ and U shore oscillating velocity components, U different types of breaking — plunging breaker for case 1 and spilling breaker for case 6. ˜ ⫺ across the pre-breaking zone can be ˜ ⫹ and U Some apparent changes of U observed. Firstly, the rate of increase in the maximum onshore oscillating velocity component is greater than that of the offshore counterpart. Secondly, the vertical ˜ ⫺ shown in Figs. 13(f) and 14(f), where the measuring ˜ ⫹ and U distributions of U sections are very close to the breaking points of cases 1 and 6, respectively, seem to be a little scattered. These may be attributed to the interaction between the onshore near-bed mass transport velocity and the reverse flow inside the surf zone, as demonstrated in Figs. 4(a–c) and in the studies of Hwung and Lin (1990) and Lin and Hwung (1992). Thirdly, the rate of increase in the maximum onshore and offshore
Fig. 13. Vertical distributions of the maximum onshore and offshore oscillating velocity components, obtained at six different sections across the pre-breaking zone of case 1 (plunging breaker).
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Fig. 14. Vertical distributions of the maximum onshore and offshore oscillating velocity components, obtained at six different sections across the pre-breaking zone of case 6 (spilling breaker).
oscillating velocity components slows down and may eventually cease as the breaking point is approached. Finally, excluding the velocity distributions measured very close to the breaking points, the dimensionless heights of the maximum onshore and ˜⫺ ˜ ⬁⫹ and U offshore free stream velocities (U ⬁ ) appear at z / d⭓6.0. The value is also found to be independent of the breaker types. Hence, it can be inferred that the nondimensionalized boundary layer thickness (over which the free stream velocity takes place), if the Stokes layer thickness is used as a length scale, remains constant across the pre-breaking zone. Based on the experimental results presented above, it is suggested that both ˜⫺ ˜ ⬁⫹ and U U ⬁ can be determined experimentally at about z / d⬵6.0. However, in order to minimize the measuring error due to finite samplings, these two important velocity scales are obtained by averaging the corresponding data measured for z / d⭓6.0. Accordingly, the non-dimensional forms of the maximum onshore and offshore oscillating velocity components shown in Figs. 13(a–e) and 14(a–e) can be calculated by scaling with respect to the locally maximum onshore and offshore free stream ˜⫺ ˜ ⫹ ˜⫺ ˜ ⬁⫹ and U velocities, U ⬁ , respectively. It is worth mentioning that U⬁ ⬎ 兩U⬁ 兩 for
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nonlinear, asymmetric shoaling waves propagating over a sloping bottom; however, ˜⫺ ˜ ⬁⫹ ⫽ 兩U U ⬁ 兩 for linear, symmetric waves propagating over a horizontal bed. An example of such calculations using data obtained from the six different cases (listed in Table 1) is presented in Fig. 15. It is interesting to note that all data points collapse onto a single distribution curve. Despite the maximum onshore free stream velocity ˜⫺ ˜ ⬁⫹ / U is greater than its offshore counterpart and the asymmetry, 兩U ⬁ 兩, increases as water depth becomes shallower, it is found that similar profiles of the non-dimensionalized maximum oscillating velocity components exist in the onshore and offshore directions; and that, even more excitingly, they are symmetrical with respect to the ordinate z / d. Regression analysis shows that the unique similar profiles of the non-dimensionalized maximum oscillating velocity components may be considered as universal, and that they can be expressed as
冋 冉 冊 册 冋 冉 冊册 再 冋 冉 冊 册 冎
˜⫺ ˜+ U z U ⫽ ⫽ c1exp c2 + ⫺ ˜ ˜ U ⬁ U⬁ d
c3
sin c4
z d
⫹ c5 exp c6
z d
c7
⫺1
(6)
in which
Fig. 15. Vertical distributions of the non-dimensionalized maximum onshore and offshore oscillating velocity components.
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c1 ⫽ 59.8125200, c2 ⫽ ⫺0.7772956, c3 ⫽ 1.1700299, c4 ⫽ 0.0209783, c5 ⫽ ⫺1.0048310, c6 ⫽ ⫺0.2330639 and
(7)
c7 ⫽ 1.9990281. In this figure, solid and dash curves (correspondingly representing the regression equation and the prediction of a linear oscillatory boundary layer theory) agree very well with each other, except for the region of z / d⬵2.3 苲 6.0 where the linear theory underestimates the overshooting. 4.4. Estimation of the Reynolds number Kamphuis (1975) and Jonsson and Carlsen (1976) have defined the maximum wave amplitude Reynolds number, based on Re ⫽
˜ 1A1 U v
(8)
˜ 1T / 2p) is the orbital ˜ 1 is the maximum free stream velocity and A1 ( ⫽ U in which U amplitude at the edge of the bottom boundary layer in a linearly (symmetric) oscillatory flow. These authors indicated that transition occurs at a critical Reynolds number over 105. In the present study, and equivalent maximum Reynolds number, R∗e , for the bottom boundary layer flow induced by nonlinear, asymmetric shoaling waves can be expressed as R∗e ⫽
˜ +⬁A+⬁ U v
(9)
where A⬁⫹ is the maximum onshore orbital amplitude just outside the bottom bound˜ ⬁⫹ T / 2p. From the experimental results presented in Figs. ary layer, i.e. A⬁⫹ ⫽ U ˜ ⬁⫹ is at most about 13(a–f) and 14(a–f), the maximum onshore free stream velocity U ∗ 40.0 cm/s, thus giving the maximum Reynolds number Re about 3.6×104. This further confirms that the boundary layer characteristics across the pre-breaking zone of the six test cases is of laminar flow (except within a small region near the breaking point), which is in agreement with the experimental results presented in Figs. 3(a, b), 5(b) and 6(b).
5. Concluding remarks Experimental work was conducted in a wave flume to investigate the characteristics of the bottom boundary layer flow induced by nonlinear, asymmetric shoaling
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waves, propagating over a smooth bed with an 1/15 uniform slope. A flow visualization technique with thin-layered of fluorescent dye and a laser Doppler velocimeter were both employed. Based on the observation and measurement results depicted above, the following conclusions can be drawn.
1. From the time history of all the horizontal velocities measured and the maximum Reynolds number calculated, it is confirmed that the bottom boundary layer flow across the pre-breaking zone is laminar in character, except very close to the breaking point. This is also strongly supported from the result of flow visualization observations. 2. The non-uniform distributions of the phase-averaged velocity 具U典 at different ¯ , in which its profile phases are to attributable to the effect of the mean velocity, U increases from zero at the bottom to a local positive maximum at about z / d⬵2.0 苲 2.5 then decreases gradually to zero approximately at z / d⬵6.0 苲 7.0, and finally becomes negative as z / d increases further. 3. The rate of increase in the maximum onshore oscillating velocity component, ˜ ⫹ , is greater than that of the maximum offshore oscillating velocity component, U ˜ U⫺, except in a small region near the breaking point. The positions of the ˜ ⬁⫹ and U ˜⫺ maximum onshore and offshore free stream velocities (U ⬁ ) appear at z / d⭓6.0. These properties are independent of breaker types. The non-dimensionalized boundary layer thickness, if the Stokes layer thickness is used as a length scale, remains constant inside the pre-breaking zone. ˜ ⫹ increases considerably with 4. Despite the fact that the vertical distribution of U ˜ ⫺, and that the asymmetry of decreasing water depth as compared with that of U ⫹ ⫺ ˜ ˜ the maximum free stream velocity 兩U⬁ / U⬁ 兩 also increases as water depth becomes shallower, a universal similarity profile can be established by plotting ˜ ⫹ /U ˜ ⬁⫹ ) or (U ˜ ⫺/U ˜⫺ z / d versus (U ⬁ ). 5. It has been demonstrated that the nonlinear, asymmetric distributions of the maximum oscillating velocity components within the bottom boundary layer of shoaling waves prior to breaking can be transformed into a similar profile, which is symmetric and universal across the pre-breaking zone.
It should be noted that, due to the limitations of the wave generator and flume dimensions, the present study only carried out under rather ideal conditions where the bottom boundary layer flow is laminar in the pre-breaking zone. Although experiments on fully turbulent boundary layer flow over horizontal smooth and rough beds can be carried out easily using on oscillating water tunnel, the evolution of the corresponding structure induced by nonlinear, asymmetric shoaling waves has not been successfully simulated using this equipment. Large-scale wave flume experiments will, eventually, be required for study of laminar-transitional-turbulent or fully turbulent boundary layer flow induced by shoaling waves. Moreover, further study associated with the effect of beach slope will be worthwhile in the future.
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Acknowledgement The authors are grateful for the support of the National Science Council, Taiwan, Republic of China, under Grant No. NSC 89-2611-E-005-005.
References Adeyemo, M.D., 1970. Velocity fields in the wave breaker zone. Proc. 12th Int. Conf. Coastal Eng., ASCE 1, 435–460. Asano, T., Iwagaki, Y., 1984. Bottom turbulent boundary layer in wave-current co-existing systems. Proc. 19th Int. Conf. Coastal Eng., ASCE 2, 2397–2413. Belorgey, M., LeBas, J., Grandjean, A., 1989. Application of laser Doppler velocimeter to the study of turbulence generated by swell in the vicinity of walls or obstacles. Coastal Eng. 13, 183–203. Bijker, E.W., Kalkwijk, J.P.T., Pieters, T., 1974. Mass transport in gravity waves on a sloping bottom. Proc. 14th Int. Conf. Coastal Eng., ASCE 1, 447–464. Bloor, M.S., 1964. The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech. 19, 290–304. Flick, R.E., Guza, R.T., Inman, D.L., 1981. Elevation and velocity measurements of laboratory shoaling waves. J. Geophys. Res. 86 (C5), 4149–4160. Hedges, T.S., Kirkgo¨ z, M.S., 1981. An experimental study of the transformation zone of plunging breaker. Coastal Eng. 4, 319–333. Horikawa, K., Watanabe, A., 1968. Laboratory study on oscillatory boundary layer flow. Proc. 11th Int. Conf. Coastal Eng., ASCE 1, 467–486. Hussain, A.K.M.F., Reynolds, W.C., 1970. The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41 (2), 241–258. Hwung, H., Lin, C., 1989. The characteristics of internal flow field and bottom boundary layer of waves propagating on sloping bottoms. Bulletin No. 16. Tainan Hydraulics Laboratory, National Cheng Kung University, Taiwan, R.O.C. (in Chinese). Hwung, H.H., Lin, C., 1990. The mass transport of waves propagating on a sloping bottom. Proc. 22nd Int. Conf. Coastal Eng., ASCE 1, 544–556. Iwagaki, Y., Sakai, T., Tsukioka, K., Sawai, N., 1974. Relationship between vertical distribution of water particle velocity and type of breakers on beaches. Coastal Eng. Jap. 17, 51–58. Jonsson, I.G., Carlsen, N.A., 1976. Experimental and theoretical investigations in an oscillatory turbulent boundary layer. J. Hydraul. Res. 14 (1), 45–60. Kamphuis, J.W., 1975. Friction factor under oscillatory waves. J. Waterwater Harbors Coastal Eng. Div., ASCE 101 (2), 135–144. Kemp, P.H., Simons, R.R., 1982. The interaction between waves and a turbulent current: waves propagating with the current. J. Fluid Mech. 116, 227–250. Kirkgo¨ z, M.S., 1986. Particle velocity prediction at the transformation point of plunging breakers. Coastal Eng. 110, 200–217. Kirkgo¨ z, M.S., 1989. An experimental investigation of plunging breaker boundary layers in the transformation zone. Coastal Eng. 13, 341–356. Kline, S.J., Reynolds, W.C., Schraub, F.A., Runstadler, P.W., 1967. The structure of turbulent boundary layer. J. Fluid Mech. 30, 741–773. Koh, R.C.Y., LeM’ehout’e, B., 1966. Wave shoaling. J. Geophys. Res. 71 (8), 2005–2013. Lin, C., Hwung, H.H., 1992. External and internal flow fields of plunging breakers. Exp. Fluids 12, 229–237. Lin, C., Jeng, C.N., Yen, G.H., Tsai, C.P., 1995. Measurement of velocity and evaluation of bottom shear stress in the laminar boundary layer. Chin. J. Mech. 12 (2), 267–278. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Phil. Trans. R. Soc. A245, 535–581. Nadaoka, K., 1986. A fundamental study on shoaling and velocity field structure of water waves in the
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nearshore zone. Tech. Report No. 36. Department of Civil Engineering, Tokyo Institute of Technology, Tokyo, pp. 33–125. Obremski, H.J., Fejer, A.A., 1967. Transition in oscillating boundary layer flows. J. Fluid Mech. 29, 93–111. Sleath, J.F.A., 1970. Measurements close to the bed in a wave tank. J. Fluid Mech. 42, 111–123. Teleki, P.G., Anderson, M.W., 1970. Bottom boundary layer stresses on a model beach. Proc. 12th Int. Conf. Coastal Eng., ASCE 1, 269–288. Tennekes, H., Lumley, J.L., 1972. A first course in turbulence. The MIT Press, Cambridge, MA. Van Doorn, T., 1981. Experimental investigation of near-bottom velocities in water waves with and without a current. Tech. Rep. M1423. Delft Hydraulics Laboratory, Delft, The Netherlands. Voropayev, S.I., Cense, A.W., McEachern, G.B., Boyer, D.L., Fernando, H.J.S., 2001. Dynamics of cobbles in the shoaling region of a surf zone. Ocean Eng. 28, 763–788.
Observation and measurement of the bottom boundary layer flow in the prebreaking zone of shoaling waves Chang Lin a,*, Hwung-Hweng Hwung b a
Department of Civil Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Republic of China b Department of Hydraulic and Ocean Engineering, National Cheng-Kung University, Tainan 600, Taiwan, Republic of China Received 15 May 2001; accepted 8 August 2001
Abstract 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 具U典 at each phase ¯ . The magnitude of U ¯ is non-uniform, which is directly affected by the mean velocity, U increases from zero at the bottom to a local positive maximum at about z / d⬵2.0 苲 2.5 (where z is the height above the sloping bottom and d is the Stokes layer thickness), then decreases gradually to zero at z / d⬵6.0 苲 7.0 approximately, and finally becomes negative as z / d increases further. Moreover, as waves propagate towards shallower water, the rate of increase ˜ ⫹ is greater than that of the offshore in the maximum onshore oscillating velocity component U ⫺ ˜ counterpart U except near the breaking point. The free stream velocities in the profiles of ˜⫺ ˜ ⬁⫹ and U the maximum onshore and offshore oscillating velocity components, U ⬁ are found to appear at z / dⱖ6.0. 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. ˜⫺ ˜ ⬁⫹ is greater than U Although U ⬁ and the asymmetry of the maximum free stream velocities ˜⫺ ˜ ⬁⫹ / U 兩) increases with decrease of water depth, a universal similar profile can be estab(i.e. 兩U ⬁ * Corresponding author. Fax: +886-4-22855182. E-mail address: [email protected] (C. Lin). 0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 1 ) 0 0 0 9 4 - 4
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˜ ⫹ /U ˜ ⬁⫹ 兩) or (兩U ˜ ⫺/U ˜⫺ lished by plotting z/ d versus (兩U ⬁ 兩). 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. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bottom boundary flow; Pre-breaking zone; Shoaling waves; Free stream velocity; Maximum oscillating velocity components; Similar profile
Nomenclature A1 A⬁+ ci d dc Hc Ho/Lo Pi Q 具Q典 ¯ Q ˜ Q Q⬘ →
r Re R∗e T Td Ti Ti t tp tp/T U 具U典 ¯ U ˜ U
Orbital amplitude at the edge of the bottom boundary layer in a linearly (symmetric) oscillatory flow ( ⫽ U1T / 2p) Maximum onshore orbital amplitude at the edge of the bottom ˜ ⬁⫹ T / 2p) boundary layer ( ⫽ U Regression coefficient (i=1, 2,…, 8) Local water depth Water depth measured at the horizontal portion of the wave flume Wave height measured at the horizontal portion of the wave flume Converted deep-water wave steepness ith measuring section (i=1, 2,…, 10) ˜ ⫹ Q⬘) ¯ ⫹Q A physical quantity as function of time ( ⫽ Q ˜) ¯ ⫹Q Phase average of Q as a function of phase ( ⫽ Q Time mean of Q Oscillating component of Q Turbulent fluctuation component of Q Position vector ˜ 1A1 / v) Reynolds number ( ⫽ U ˜ ⬁⫹ A⬁⫹ / v) Reynolds number ( ⫽ U Mean wave period Wave period corresponding to the discarded data set ith wave period Sampling duration Time Phase time varying from 0 to T Phase varying from 0 to 1 ˜ ⫹ U⬘) ¯ ⫹U Horizontal velocity as a function of time ( ⫽ U ˜) ¯ Phase average of U as a function of phase ( ⫽ U ⫹ U Time mean of U (i.e. mean horizontal velocity) Oscillating component of U
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˜1 U ˜ ⫹ U ˜⫺ U ˜⬁ U ˜ ⬁⫹ U ˜⫺ U ⬁ U⬘ z d v h 具h典 h¯ h˜ h⬘
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Maximum free stream velocity at the edge of bottom boundary layer in a linearly (symmetric) oscillatory flow Maximum onshore oscillating velocity component Maximum offshore oscillating velocity component Free stream velocity as a function of phase Maximum onshore free stream velocity (i.e. maximum onshore oscillating velocity component at the edge of bottom boundary layer) Maximum offshore free stream velocity (i.e. maximum offshore oscillating velocity component at the edge of bottom boundary layer) turbulent fluctuation component of U Local height above the sloping bottom Stokes layer thickness ( ⫽ √vT / p) Kinematic viscosity of water Water surface elevation as a function of time ( ⫽ h¯ ⫹ h˜ ⫹ h⬘) Phase average of h as a function of phase ( ⫽ h¯ ⫹ h˜ ) Time mean of h Oscillating component of h Turbulent fluctuation component of h
1. Introduction Gravity waves propagating from deep to shallow water exhibit continuous transformation in wave profile due to shoaling accompanied by nonlinearity, thus leading to steepening of the wave crest and flattening of the wave trough. Eventually, the profile becomes unstable, leading to breaking waves. Since the hydrodynamic characteristics of shoaling waves are closely related to onshore/offshore sediment transport, vertical distribution of suspended loads, and pollutant dispersion in coastal waters; it is of paramount importance to investigate the flow field of shoaling waves, with special emphasis on the velocity distribution within the bottom boundary layer. Although numerous investigations for the wave shoaling have been carried out in the past (e.g. Koh and LeM’ehout’e, 1966; Adeyemo, 1970; Iwagaki et al., 1974; Flick et al., 1981; Hedges and Kirkgo¨ z, 1981; Kirkgo¨ z, 1986; Nadaoka, 1986; Hwung and Lin, 1990; Voropayev et al., 2001), the study for the bottom boundary layer structure of shoaling waves propagating on a sloping bottom is still rather limited. Recently, considerable efforts have been devoted to the phenomenon of wave propagating over horizontal bottom. For example, Horikawa and Watanabe (1968) and Sleath (1970) used hydrogen bubbles and tension wire, respectively, to measure velocity distributions near smooth and rough beds. Jonsson and Carlsen (1976), utilizing a micropropeller flow-meter, measured the velocity profiles of turbulent boundary layer flow over rough beds in an oscillating water tunnel. Employing a laser Doppler
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velocimeter (LDV), Van Doorn (1981), Kemp and Simons (1982), Asano and Iwagaki (1984) and Belorgey et al. (1989) performed non-intrusive measurements of the near-bottom velocities in water waves with or without a current. Teleki and Anderson (1970) may have been the first research to investigate the near-bottom water-particle velocity on a (smooth) slope with a Preston tube. However, their measurements were only made at the instants of wave-crest and wavetrough by alternatively orienting the L-shaped probe upslope and downslope, which interfered with flow field and thus led to measurement error. Recently, Hwung and Lin (1989, 1990) and Kirkgo¨ z (1989) utilized LDV to measure the horizontal velocity profiles in the bottom boundary layer of shoaling waves propagating on 1/15 and 1/12 uniform slopes, respectively. Hwung and Lin (1989, 1990) investigated the vertical distributions of the mean velocity in the pre-breaking zone for different breaking waves. Kirkgo¨ z (1989) examined the vertical distributions of the horizontal velocity under the wave-crest and wave-trough in the transformation zone of plunging breakers. In Kirkgo¨ z (1989), the discrepancy between the measured and predicted velocity distributions was considered to be attributable to the turbulent effect in the near wall region. However, no clear evidence of turbulence was demonstrated, nor was the influence of mass transport on the measured velocity distribution (which might result in such inconsistency) discussed in his investigation. The aim of this study is to elucidate the flow characteristics within the bottom boundary layer induced by nonlinear, asymmetric shoaling waves, propagating on an 1/15 sloping beach. A flow visualization technique with thin-layered fluorescent dye was first used to observe the flow structure of the bottom boundary layer. Then, a one-component LDV was employed to measure the horizontal (water-particle) velocity in the bottom boundary layer across the pre-breaking zone. Finally, some important features of the bottom boundary layer structure are examined and discussed.
2. Experimental set-up and method Experiments were conducted in a glass-walled wave flume located at Tainan Hydraulic Laboratory, National Cheng Kung University. The wave flume is 9.5 m long, 0.3 m wide and 0.7 m deep. The first 3.77 m section had a fixed horizontal smooth bottom starting from the end where a flap-type wave maker was installed. The remaining 5.73 m of the flume had an adjustable sloping smooth bed made of polished stainless plate. A fixed slope of 1/15 was used in this study. Monochromatic waves could be generated with periods ranging from 0.3 to 1.5 s and with wave heights up to 15.0 cm. The experimental conditions are listed in Table 1, in which T is the mean wave period, Hc and dc are the wave height and water depth measured at the horizontal portion of the flume, and Ho/Lo is the converted deep-water wave steepness. Two capacitance-type wave gauges were used to detect the water surface elevations in the experiments. A novel flow visualization technique with thin-layered fluorescent dye was employed to examine the flow characteristics of the bottom boundary layer induced
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Table 1 Experimental wave conditions Case
T (s)
Hc (cm)
dc (cm)
Ho/Lo
Breaker type
1 2 3 4 5 6
1.41 1.41 1.07 0.96 0.74 0.96
5.3 7.5 4.7 5.3 4.3 8.2
33.0 33.0 33.0 33.0 33.0 33.0
0.0186 0.0260 0.0300 0.0405 0.0520 0.0617
Plunging Plunging Plunging Plunging Spilling Spilling
by shoaling waves. The fluorescent dye used was a water solution of either fluorescent sodium (C20H10Na2O5) or rhodamin B (C28H31ClN2O3) with a specific gravity of 1.015 as measured by a precise hydrometer. The dye was injected very slowly and smoothly (through a stainless steel pipe of 2.0 mm in diameter) onto the surface of the sloping bottom near the toe of the 1/15 beach, while the water mass inside the wave flume was kept still. Consequently, the thin-layered fluorescent dye was formed. It was then stretched and transported along the sloping bottom due to the effect of mass transport of shoaling waves after wave maker was started. A floodlight and a 2 W argon-ion laser were used as light sources to illuminate the boundary layer flow structures inside the pre-breaking zone and in the neighbourhood of the breaking point, respectively. In the latter case, a laser beam was emitted from a laser head (Lexel 95-2) and spread into a fan-shaped light sheet (about 1.5 mm thick) using a mirror and a cylindrical lens. In the present experiment, the light sheet was used to illuminate the two-dimensional motion of the dye tracer on a vertical plane along the longitudinal direction of the flume. Photographic equipments, including a camera and a video camera, were also set up in front of a sidewall of the wave flume. Utilizing a LDV system, measurements of the horizontal velocities on the sloping beach were then taken in all or some of the 10 vertical sections (i.e. from P1 to P10) as shown in Fig. 1, depending on the experimental conditions required. The local water depth, d, and the local height above the sloping bottom, z, are also given in Fig. 1. The LDV used in the present study was a one-component, dual-beam optical system, equipped with an acousto-optic Bragg cell driven by a 40 MHz oscillator. The light source was a Lexel 2 W argon-ion laser. All of the transmitting optics were mounted on a three-dimensional translation table. Its movement could be precisely controlled by three optical meters and three digital monitors (with precision of 0.001 mm). A small measuring volume was achieved from the standard TSI optics by using a 2.27× beam expander ahead of a 350 mm focal lens. The dimensions in diameter and length of the measuring volume were estimated to be 0.08 and 1.09 mm for two green beams, respectively. The measuring error of the LDV system was estimated to be ca. 0.14 cm/s. To obtain the velocity measurements at locations near the sloping bottom without beam obstruction due to the edge of the sloping bottom, the transmitting optics were tilted at an angle of 1.2° towards the surface of the sloping bottom (see Fig. 2). The
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Fig. 1.
Schematic diagram of measuring sections and coordinate system.
Fig. 2.
Schematic diagram of the installation of the LDV system.
receiving optics were tilted slightly downwards by ca. 2.5° to reduce the intense light scattered from the surface of the sloping bottom. The zero location of the measuring volume was determined by slightly adjusting the vertical movement of the translation table and observing the digital outputs of optical meter. The positioning error was estimated to be less than ±0.04 mm. Time series of the water surface elevation and the corresponding horizontal velocity were simultaneously measured by a wave gauge and the LDV system at every vertical section. These data were collected using an AD/DA converter. The sampling rate was kept at 100 Hz for all of the experimental cases. To minimize the multireflection problem arising from finite length of the wave-flume, the sampling time for each run was only 8–15 s, depending upon the experimental cases tested. In this way, several runs were completed repeatedly with a total sampling time varying from 45 to 64 s.
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3. Data analysis Hussain and Reynolds (1970) proposed a phase average method that, for a steady turbulent shear flow coexisting with a periodic motion, any physical quantity Q could ˜ , and ¯ , a (periodically) oscillating component Q be decomposed into a time mean Q a turbulent fluctuation component Q⬘. The relation can be expressed as ˜ (→ ¯ (→ r , t) ⫹ Q⬘(→ r , t), Q(r¯ , t) ⫽ Q r) ⫹ Q
(1)
in which → r and t are position vector and time variable, respectively. The time mean of Q is in the form of ¯ (→ Q r) ⫽
冕
1 Ts
Ts
Q(→ r , t)dt,
(2)
0
where Ts is the total sampling time, which is a multiple of the mean wave period T. The phase average of Q is defined as
冘 M
1 ¯ (→ ¯ (→ Q(→ r , tp ⫹ mT) ⫽ Q r) ⫹ Q r , tp), 具Q( r , t p)典 ⫽ M ⫹ 1m ⫽ 0 →
(3)
in which M is the total number of sampling cycle, and tp is the phase time ranging from 0 to T. Hence, at any point in space, the phase average is the average value of Q that is recognized at a special phase time tp in each wave period. Accordingly, the oscillating component of Q is then ˜ (→ ¯ (→ r , tp)典⫺Q r ). Q r , tp) ⫽ 具Q(→
(4)
Therefore, given a reference signal with a specific period (e.g. the free surface elevation h(t)) and a signal Q(t) at a particular location (e.g. the horizontal velocity U(t)), these three components can be determined easily by Eqs. (1)–(4). Note that ¯˜ ⫽ 0, and Q ¯典 ⫽ Q ¯ Q ¯ ⬘ ⫽ 0. 具Q However, the period of each wave cycle in the wave flume cannot be exactly the same due to the precision limit of the wave maker used and the long-term oscillation induced by wave breaking. If the phase average method stated above is used directly, then the problem of phase time shift (drift) will occur frequently during data processing and a false image of oscillating component will be produced. To overcome this problem, the modified phase average method proposed by Hwung and Lin (1989) is adopted in the present study. Some details are briefly illustrated as follows. First of all, the time mean values of the water surface elevation and the horizontal velocity, ¯ are subtracted from h(t) and U(t), respectively. Next, the zero-up crossing h¯ and U points are determined by linear interpolation to the data series of h(t) and U(t) and the data set within an individual wave period Td is discarded when the relative error between Td and T is ⬎2%. These data series are then re-interpolated into the same data points in each qualified wave period, according to the ratio of the ith wave
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period Ti to the mean wave period T. Finally, the modified phase average of Q at a particular phase time, tp, can be rewritten generally as
冘冉 N
冊
1 Ti ˜ → ¯ (→ Q r , tp ⫹Q r) 具Q( r , tp)典 ⫽ Ni ⫽ 1 T →
(5)
in which N is the total sampling cycles qualified.
4. Results and discussions 4.1. Flow visualization results It has been widely recognized that the boundary layer developed over natural sea bottom is a turbulent flow. However, it is not clear if turbulence could occur in the case of laboratory wave. It should be examined in the first stage of this investigation. Figures 3(a, b) show the visualized results of the dye layer flow on the sloping bottom within the pre-breaking zone of case 1 (see Table 1). They were photographed at different times by a camera with different viewing angles when the wave motion had reached a steady state. The direction of wave propagation is from left to right. A seemingly straight fluorescent dye ‘line’ moving parallel along the sloping bottom is clearly observed in Fig. 3(a). Furthermore, a very long ‘strip’ of fluorescent dye with a nearly constant width, partially spanning the sloping bottom, can also be seen
Fig. 3.
Visualized results of the dye layer flow on the bottom within the pre-breaking zone of case 1.
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distinctly in Fig. 3(b). The fluorescent dye layer in both figures is consecutively transported shorewards by Lagrangian mass transport velocity (Longuet-Higgins, 1953; Bijker et al., 1974) before wave breaking. During the process of wave shoaling, neither bursting phenomenon (Kline et al., 1967) nor irregular disturbance could be observed on and near the bottom even by detailed examinations of the photos taken and the films recorded. This holds true for all of the experimental conditions tested. According to Tennekes and Lumley (1972), turbulent flow is characterized by irregularity (or randomness), diffusivity, and high level of fluctuating vorticity, and so on. Hence, we can deduce that the boundary layer structure inside the pre-breaking zone is not of turbulent flow, at least from the tests conducted in our study. But, the flow characteristics of bottom boundary layer at the breaking point and in its close vicinity for case 1 are quite different from those observed within the pre-breaking zone. From the visualized result of Fig. 4(a), it is seen that, when wave phase changes from crest to trough, some of the fluorescent dye layer close to the sloping bottom is disturbed by the turbulent action due to the interaction of mass transport velocity in the pre-breaking zone and the reverse flow inside the surf zone. At the instant of wave trough, the turbulent dye-cloud rises in large flux from the bottom, as seen in Fig. 4(b). Moreover, as shown in Fig. 4(c), a massive cloud of the fluorescent dye rushes upwards as the first plunging action arrives. According to Lin and Hwung (1992), the complex phenomenon is caused by the strong anticlockwise motion of the water particles behind the crest of the breaker, and a counterbalancing motion for those in front of it. 4.2. Characteristics of the phase-averaged velocity and the mean velocity The relationship between the water surface elevation and the horizontal velocity is now introduced. Typical examples of the time histories of the water surface elevation h(t), measured by a wave gauge, and of the corresponding horizontal velocities U(t) obtained by LDV at z=0.20 mm and 0.60 mm in section P9 of case 2, are given in Figs. 5(a, b) and 6(a, b), respectively. It is found that, although extremely tiny fluctuations can be detected in the velocity traces, variations of the water surface elevation and the horizontal velocity are very periodic and highly nonlinear (i.e. non-sinusoidal); and that neither the Tollmien-Schlichting instability of transition phenomenon (Bloor, 1964; Obremski and Fejer, 1967) nor irregularly turbulent fluctuation would be seen in Figs. 5(b) and 6(b). Based on the visualized results delineated in Figs. 3(a, b) and 4(a–c) along with the above discussion, it can thus be confirmed that the boundary layer flow in the pre-breaking zone is laminar in character except in a small region near the breaking point. Hence, the turbulent fluctuation components of the water surface elevation and of the horizontal velocity, h⬘(t) and U⬘(t), may be neglected. Moreover, the results shown in Figs. 5(a, b) and 6(a, b) suggest that the water surface elevation can be used as reference signal, with which the phase relationship and the flow field of velocities (measured by LDV at different times and heights above the sloping bottom) can be closely correlated and reconstructed by the modified phase average method. In the present study, the oscillating component of the
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Fig. 4. Visualized results of the consecutive development of the bottom boundary layer flow in the neighborhood of the breaking point of case 1. (The breaking point is indicated by a white arrow.)
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Fig. 5. Time series of the water surface elevation and the corresponding horizontal velocity U(t) measured at section P9 of case 2: (a) water surface elevation; (b) horizontal velocity obtained at z=0.20 mm.
Fig. 6. Time series of the water surface elevation and the corresponding horizontal velocity U(t) measured at section P9 of case 2: (a) water surface elevation; (b) horizontal velocity obtained at z=0.60 mm.
water surface elevation, h˜ , is selected as a reference signal with phase (tp / T) varying from 0 to 1, where tp / T ⫽ 0 is designated as the phase of zero-up crossing of h˜ . For example, Fig. 7(a) shows the variation of h˜ as a function of phase, obtained at section P6 of case 6. Furthermore, the corresponding changes in the phase-averaged ˜ ), originally measured at six different heights in the same ¯ ⫹U velocity 具U典 (i.e. U section, are given in Figs. 7(b–g), respectively. Some interesting features observed in Figs. 7(a–g) are worthy of mentioning. Firstly, due to the effect of wave shoaling,
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Fig. 7. Temporal variations of the oscillating component of water surface elevation and of the phaseaveraged velocities measured at six different heights at section P6 of case 6.
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the oscillating component of the water surface elevation, h˜ , becomes steepened at the wave crest and flattened near the trough, indicating nonlinear, asymmetric characteristics of wave motion in the pre-breaking zone (Adeyemo, 1970; Flick et al., 1981; Voropayev et al., 2001). The same trend can also be observed from the horizontal velocities shown in Figs. 7(b–g). Secondly, the maximum onshore phase-averaged velocities, 具U典max, measured at different heights of section P6, increase rapidly up to a maximum at a certain height, and then decrease gradually with increasing height z. This infers that a significant velocity gradient and the phenomenon of overshooting actually exist in the bottom boundary layer induced by shoaling waves. Thirdly, the value of tp / T associated with the maximum onshore phase-averaged velocity decreases with decreasing measuring height within the bottom boundary layer, implying the existence of phase difference inside the viscous boundary layer. In reference to the study of monochromatic waves propagating over a horizontal smooth bed, the phase difference is simply caused by the effect of viscous damping, as confirmed by Horikawa and Watanabe (1968), Sleath (1970) and Lin et al. (1995). In the case of shoaling waves over a sloping bed, the phase difference may be affected by both the viscous damping and the higher harmonic components of waves (Flick et al., 1981). However, no attempt was made in this study to differentiate the respective influences which were resulted from these two factors. Similarly, temporal variations in the water surface elevation h˜ and in the vertical distribution of the phase-averaged velocity 具U典 obtained at three different sections of case 1 are depicted in Figs. 8(a, b)–10(a, b), respectively. In these figures, different symbols are used for different phases of the occurrence, which can be identified in the abscissa of Figs. 8(a)–10(a). The local measuring height z, appearing in the ordinate of Figs. 8(b)–10(b), is non-dimensionalized by the characteristic length scale, d ⫽ 冑vT / p (v is the kinematic viscosity of water), which is also known as the Stokes layer thickness. Upon comparing the results presented in Figs. 8(a)–10(a), it is clear that nonlinearity and asymmetry of the water surface elevation h˜ increase with decreasing water depth due to the effect of wave shoaling. The distributions of the phase-averaged velocity 具U典 occurring at some specific phases, taking tp / T ⫽ 0 (marked with ‘䊋’) and the wave-crest phase (with ‘ ’) for examples, also increase rapidly with decreasing water depth for the same dimensionless height z / d. This implies that, very close to the sloping bottom, the onshore shear stress increases considerably in the pre-breaking zone as water depth becomes shallower. However, the phase-averaged velocity distribution taking place at the wave-trough phase (marked with ‘䊎’) increases slightly in magnitude with decreasing water depth. The differences in magnitude and shape of the velocity distributions between the wave crest and trough phases may be attributed to the wave-crest steepening and trough flattening in nonlinear water waves due to wave shoaling (Adeyemo, 1970). It is worth noting that, at every measuring section in the pre-breaking zone for the six different Ho / Lo listed in Table 1, the phase-averaged velocity profile at every phase does not have the shape of time-dependent uniform flow, which is one of the characteristic indicators of oscillatory flow at the edge and outside of the boundary layer. Similar change in the velocity distribution could also be evidenced from the limited
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Fig. 8. Temporal variations of the oscillating component of water surface elevation and of the vertical distributions of the phase-averaged velocity, measured at section P4 of case 1.
results of smaller-scale wave experiments performed by Kirkgo¨ z (1989). However, the exact reason for such a distribution remains unknown. To make clear which mechanism dominating such a velocity profile, the distri¯ corresponding to the phase-averaged velocity 具U典 shown bution of mean velocity U in Figs. 8(b)–10(b), are presented in Figs. 11(a–c), respectively. We can see that the ¯ increases from zero at the bottom to a local positive vertical distribution of U maximum at about z / d⬵2.0 苲 2.5, then decreases gradually to zero at z / d⬵6.0 苲 7.0 approximately, and finally becomes negative as the non-dimensional height increases further. Although only limited data of the mean velocity distribution for case 1 are presented herein, this characteristics is found to be also true for other experimental conditions listed in Table 1. The present results are also in accordance with those measured by Hwung and Lin (1990), in which detailed and unified dis-
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Fig. 9. Temporal variations of the oscillating component of water surface elevation and of the vertical distributions of the phase-averaged velocity, measured at section P8 of case 1.
cussion about the profiles of the mean horizontal velocity within the bottom boundary layer was first reported for various values of deep-water wave steepness.
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Fig. 10. Temporal variations of the oscillating component of water surface elevation and of the vertical distributions of the phase-averaged velocity, measured at section P10 of case 1.
4.3. Vertical distributions of the maximum onshore/offshore oscillating velocity components ˜ at different phases can be obtained, accordThe oscillating velocity components U ing to Eq. (4). As an example, Fig. 12 illustrates some of such results calculated from those shown previously in Fig. 9(b). It is found that, for z / d⭓6.0, the oscillating velocity component occurring at each phase does approach the time-dependent uni˜ ⬁. This indicates that non-uniform distriform velocity, i.e. the free steam velocity U butions of the phase-averaged velocity presented in Figs. 8(b)–10(b) are due to the ¯. effect of mean velocity, U
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Fig. 11. Vertical distributions of the mean velocity at three different sections of case 1: (a) P4; (b) P8; (c) P10.
Fig. 12. Temporal variations of the vertical distribution of the oscillating velocity component corresponding to some of those shown previously in Fig. 9(b). (Symbols used are the same as in Fig. 9(b).)
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Among the vertical distributions of the oscillating velocity component as shown in Fig. 12, the profiles of the maximum velocity in the onshore and offshore directions (i.e. the largest positive and negative) are, undoubtedly, of fundamental importance to the problems associated with wave energy dissipation by bottom friction and nearshore sediment transport. Herein, Figs. 13(a–f) and 14 (a–f) present the collective sketches for the vertical distributions of the maximum onshore and off˜ ⫺, for two shoaling waves with ˜ ⫹ and U shore oscillating velocity components, U different types of breaking — plunging breaker for case 1 and spilling breaker for case 6. ˜ ⫺ across the pre-breaking zone can be ˜ ⫹ and U Some apparent changes of U observed. Firstly, the rate of increase in the maximum onshore oscillating velocity component is greater than that of the offshore counterpart. Secondly, the vertical ˜ ⫺ shown in Figs. 13(f) and 14(f), where the measuring ˜ ⫹ and U distributions of U sections are very close to the breaking points of cases 1 and 6, respectively, seem to be a little scattered. These may be attributed to the interaction between the onshore near-bed mass transport velocity and the reverse flow inside the surf zone, as demonstrated in Figs. 4(a–c) and in the studies of Hwung and Lin (1990) and Lin and Hwung (1992). Thirdly, the rate of increase in the maximum onshore and offshore
Fig. 13. Vertical distributions of the maximum onshore and offshore oscillating velocity components, obtained at six different sections across the pre-breaking zone of case 1 (plunging breaker).
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Fig. 14. Vertical distributions of the maximum onshore and offshore oscillating velocity components, obtained at six different sections across the pre-breaking zone of case 6 (spilling breaker).
oscillating velocity components slows down and may eventually cease as the breaking point is approached. Finally, excluding the velocity distributions measured very close to the breaking points, the dimensionless heights of the maximum onshore and ˜⫺ ˜ ⬁⫹ and U offshore free stream velocities (U ⬁ ) appear at z / d⭓6.0. The value is also found to be independent of the breaker types. Hence, it can be inferred that the nondimensionalized boundary layer thickness (over which the free stream velocity takes place), if the Stokes layer thickness is used as a length scale, remains constant across the pre-breaking zone. Based on the experimental results presented above, it is suggested that both ˜⫺ ˜ ⬁⫹ and U U ⬁ can be determined experimentally at about z / d⬵6.0. However, in order to minimize the measuring error due to finite samplings, these two important velocity scales are obtained by averaging the corresponding data measured for z / d⭓6.0. Accordingly, the non-dimensional forms of the maximum onshore and offshore oscillating velocity components shown in Figs. 13(a–e) and 14(a–e) can be calculated by scaling with respect to the locally maximum onshore and offshore free stream ˜⫺ ˜ ⫹ ˜⫺ ˜ ⬁⫹ and U velocities, U ⬁ , respectively. It is worth mentioning that U⬁ ⬎ 兩U⬁ 兩 for
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nonlinear, asymmetric shoaling waves propagating over a sloping bottom; however, ˜⫺ ˜ ⬁⫹ ⫽ 兩U U ⬁ 兩 for linear, symmetric waves propagating over a horizontal bed. An example of such calculations using data obtained from the six different cases (listed in Table 1) is presented in Fig. 15. It is interesting to note that all data points collapse onto a single distribution curve. Despite the maximum onshore free stream velocity ˜⫺ ˜ ⬁⫹ / U is greater than its offshore counterpart and the asymmetry, 兩U ⬁ 兩, increases as water depth becomes shallower, it is found that similar profiles of the non-dimensionalized maximum oscillating velocity components exist in the onshore and offshore directions; and that, even more excitingly, they are symmetrical with respect to the ordinate z / d. Regression analysis shows that the unique similar profiles of the non-dimensionalized maximum oscillating velocity components may be considered as universal, and that they can be expressed as
冋 冉 冊 册 冋 冉 冊册 再 冋 冉 冊 册 冎
˜⫺ ˜+ U z U ⫽ ⫽ c1exp c2 + ⫺ ˜ ˜ U ⬁ U⬁ d
c3
sin c4
z d
⫹ c5 exp c6
z d
c7
⫺1
(6)
in which
Fig. 15. Vertical distributions of the non-dimensionalized maximum onshore and offshore oscillating velocity components.
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c1 ⫽ 59.8125200, c2 ⫽ ⫺0.7772956, c3 ⫽ 1.1700299, c4 ⫽ 0.0209783, c5 ⫽ ⫺1.0048310, c6 ⫽ ⫺0.2330639 and
(7)
c7 ⫽ 1.9990281. In this figure, solid and dash curves (correspondingly representing the regression equation and the prediction of a linear oscillatory boundary layer theory) agree very well with each other, except for the region of z / d⬵2.3 苲 6.0 where the linear theory underestimates the overshooting. 4.4. Estimation of the Reynolds number Kamphuis (1975) and Jonsson and Carlsen (1976) have defined the maximum wave amplitude Reynolds number, based on Re ⫽
˜ 1A1 U v
(8)
˜ 1T / 2p) is the orbital ˜ 1 is the maximum free stream velocity and A1 ( ⫽ U in which U amplitude at the edge of the bottom boundary layer in a linearly (symmetric) oscillatory flow. These authors indicated that transition occurs at a critical Reynolds number over 105. In the present study, and equivalent maximum Reynolds number, R∗e , for the bottom boundary layer flow induced by nonlinear, asymmetric shoaling waves can be expressed as R∗e ⫽
˜ +⬁A+⬁ U v
(9)
where A⬁⫹ is the maximum onshore orbital amplitude just outside the bottom bound˜ ⬁⫹ T / 2p. From the experimental results presented in Figs. ary layer, i.e. A⬁⫹ ⫽ U ˜ ⬁⫹ is at most about 13(a–f) and 14(a–f), the maximum onshore free stream velocity U ∗ 40.0 cm/s, thus giving the maximum Reynolds number Re about 3.6×104. This further confirms that the boundary layer characteristics across the pre-breaking zone of the six test cases is of laminar flow (except within a small region near the breaking point), which is in agreement with the experimental results presented in Figs. 3(a, b), 5(b) and 6(b).
5. Concluding remarks Experimental work was conducted in a wave flume to investigate the characteristics of the bottom boundary layer flow induced by nonlinear, asymmetric shoaling
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waves, propagating over a smooth bed with an 1/15 uniform slope. A flow visualization technique with thin-layered of fluorescent dye and a laser Doppler velocimeter were both employed. Based on the observation and measurement results depicted above, the following conclusions can be drawn.
1. From the time history of all the horizontal velocities measured and the maximum Reynolds number calculated, it is confirmed that the bottom boundary layer flow across the pre-breaking zone is laminar in character, except very close to the breaking point. This is also strongly supported from the result of flow visualization observations. 2. The non-uniform distributions of the phase-averaged velocity 具U典 at different ¯ , in which its profile phases are to attributable to the effect of the mean velocity, U increases from zero at the bottom to a local positive maximum at about z / d⬵2.0 苲 2.5 then decreases gradually to zero approximately at z / d⬵6.0 苲 7.0, and finally becomes negative as z / d increases further. 3. The rate of increase in the maximum onshore oscillating velocity component, ˜ ⫹ , is greater than that of the maximum offshore oscillating velocity component, U ˜ U⫺, except in a small region near the breaking point. The positions of the ˜ ⬁⫹ and U ˜⫺ maximum onshore and offshore free stream velocities (U ⬁ ) appear at z / d⭓6.0. These properties are independent of breaker types. The non-dimensionalized boundary layer thickness, if the Stokes layer thickness is used as a length scale, remains constant inside the pre-breaking zone. ˜ ⫹ increases considerably with 4. Despite the fact that the vertical distribution of U ˜ ⫺, and that the asymmetry of decreasing water depth as compared with that of U ⫹ ⫺ ˜ ˜ the maximum free stream velocity 兩U⬁ / U⬁ 兩 also increases as water depth becomes shallower, a universal similarity profile can be established by plotting ˜ ⫹ /U ˜ ⬁⫹ ) or (U ˜ ⫺/U ˜⫺ z / d versus (U ⬁ ). 5. It has been demonstrated that the nonlinear, asymmetric distributions of the maximum oscillating velocity components within the bottom boundary layer of shoaling waves prior to breaking can be transformed into a similar profile, which is symmetric and universal across the pre-breaking zone.
It should be noted that, due to the limitations of the wave generator and flume dimensions, the present study only carried out under rather ideal conditions where the bottom boundary layer flow is laminar in the pre-breaking zone. Although experiments on fully turbulent boundary layer flow over horizontal smooth and rough beds can be carried out easily using on oscillating water tunnel, the evolution of the corresponding structure induced by nonlinear, asymmetric shoaling waves has not been successfully simulated using this equipment. Large-scale wave flume experiments will, eventually, be required for study of laminar-transitional-turbulent or fully turbulent boundary layer flow induced by shoaling waves. Moreover, further study associated with the effect of beach slope will be worthwhile in the future.
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Acknowledgement The authors are grateful for the support of the National Science Council, Taiwan, Republic of China, under Grant No. NSC 89-2611-E-005-005.
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