Sand suction mechanism in artificial beach composed of rubble mound breakwater and reclaimed sand area

ARTICLE IN PRESS

Ocean Engineering 34 (2007) 1104–1119 www.elsevier.com/locate/oceaneng

Sand suction mechanism in artificial beach composed of rubble mound breakwater and reclaimed sand area Dong-Soo Hura,, Tomoaki Nakamurab, Norimi Mizutanib a

Department of Ocean Civil Engineering (Institute of Marine Industry), Gyeongsang National University, Tongyeong 650 160, Korea b Department of Civil Engineering, Nagoya University, Nagoya 464 8603, Japan Received 27 March 2006; accepted 9 August 2006 Available online 3 November 2006

Abstract An artificial beach has been constructed compensating for losing of the natural one caused by the development of coastal area. In this paper, the hydraulic model tests are carried out to investigate the suction phenomenon on the artificial beach constituted of rubble mound breakwater with gravel and the reclaimed sand area. In addition, the numerical model for waves, structures and seabed interaction as well as the numerical method based on the u–p approximation of the Biot equations is developed for investigation of suction mechanism. After verification of the numerical models by comparing numerical results with experimental data, the numerical models are further used to clarify the detailed suction mechanism of the reclaimed sand. The factors that affect the suction phenomenon are examined experimentally and their critical values are presented. Also, it can be pointed out that the vertical discharge velocity as well as the volumetric strain around the still water level of the boundary between the breakwater and the beach gets up to the critical value, the reclaimed sand starts to flow out to the offshore, and it finally leads to caves and cave-ins in the reclaimed zone. r 2006 Elsevier Ltd. All rights reserved. Keywords: Suction mechanism; Rubble mound breakwater; Reclaimed sand area; Newly developed numerical model; Maximum volumetric strain

1. Introduction An artificial beach has been constructed compensating for loss of the natural one caused by the development of coastal area, as well as serving as a location for recreational activities such as sea bathing and fishing. Generally, an artificial beach has two parts: one for the reclaimed material; the other for a protecting facility against washout. It is well known that some structure should be constructed to protect an artificial beach from the outflow due to wave action of the reclaimed sand. Sometimes, the rubble mound breakwater, which is one of the most general seawalls for the protection of coastal zones, is utilized as the structure to protect an reclaimed sand beach. In most cases for artificial beach construction, filter layers and geotextile sheets between the reclaimed material and the Corresponding author. Tel.: +82 55 640 3152; fax: +82 55 640 3150.

E-mail addresses: [email protected] (D.-S. Hur), [email protected] (T. Nakamura), [email protected] (N. Mizutani). 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.08.005

breakwater as a protecting facility have been placed to prevent the suction phenomenon to the artificial beach. However, such a phenomenon may occur if there are holes in the sheet or no filter layer, and it consequently leads to many problems concerning the usefulness of the beaches. Many researchers have investigated analytically and numerically wave-induced seabed response, which affects the suction phenomenon. Yamamoto (1977) and Yamamoto et al. (1978) derived the analytical solutions for the pore water pressure, the soil particle displacements and the effective stresses in a finite and infinite depth seabed based on the consolidation equations of Biot (1941). The solutions indicated that the permeability, the stiffness of porous media and the compressibility of the pore water influenced the seabed response. The consolidation equations, however, considered neither the acceleration of the soil particles nor that of the pore water. Jeng and Lee (2001) obtained analytically the dynamic solution for an isotropic homogeneous seabed of finite thickness based on the Biot equations in the uwp form, which included the acceleration terms. They compared the dynamic solution,

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Nomenclature B CA CD1 CD2 D50 d50 dm E F G g gi Hi Hs h hr hs Kw k L ‘ m mr ms P p pe q Ri1 Ri2 s

breakwater width at the still water level (m) added mass coefficient non-linear drag coefficient linear drag coefficient median grain size of rubble (m) median grain size of sand (m) median grain size of porous media (m) modulus of elasticity (N/m2) VOF function shear modulus of elasticity (N/m2) gravitational acceleration (m/s2) gravitational acceleration vector (m/s2) incident wave height (m) wave height at the front of the reclaimed beach (m) still water depth (m) breakwater height (m) beach height (m) apparent bulk modulus of water (N/m2) hydraulic conductivity (m/s) wavelength (m) beach length at the still water level (m) porosity of porous media porosity of rubbles porosity of sand absolute pressure (N/m2) peak value of dynamic pore water pressure (N/m2) excess pore water pressure (N/m2) source density at the source position (m/s) non-linear drag force vector (m/s2) linear drag force vector (m/s2) breakwater inclination

the quasi-static one (Yamamoto et al., 1978) and the experimental data in a fine sand, and found the importance of the dynamic seabed behavior in a graveled seabed than a sandy bed. All of the aforementioned papers have considered no water particle velocity at the wave-seabed interface, and also assumed an idealized situation, namely, a semi-infinite flat seabed of finite or infinite thickness without any structures. As a result, these analytical approaches cannot be applied to the wave-induced seabed response in the vicinity of coastal structures. In order to investigate waves, breakwaters and seabed dynamic interaction, Mizutani et al. (1998) and Mostafa et al. (1999) developed both a combined BEM–FEM model based on modified Navier– Stokes equations, which took into account the effects of porous media, and a poro-elastic FEM model based on the Biot’s consolidation equations. Although the BEM–FEM model considered continuity of the water particle velocity at the seabed surface, the accelerations of the soil particles

T t Ur Urc u Vs vi vnorm vtan b bij Dxs dij eij ev l m mw v na nw /nS r ra rr rs rw /rS r0

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wave period (s) time (s) Ursell parameter (¼ H i L2 =h3 ) critical Ursell parameter for the sand suction soil displacement vector (m) maximum velocity of water surface fluctuation at the front of the reclaimed beach (m/s) actual velocity vector (m/s) horizontal discharge velocity in the vicinity of the beach inside the breakwater (m/s) vertical discharge velocity in the vicinity of the beach inside the breakwater (m/s) dissipation factor (s1) dissipation factor array (s1) mesh width at the source position (m) Kronecker’s delta strain tensor (positive means expansion) volumetric strain (positive means expansion) Lame´’s constant Lame´’s constant viscosity coefficient of water (kg/(ms)) Poisson’s ratio kinematic viscosity of air (m2/s) kinematic viscosity of water (m2/s) average kinematic viscosity (¼ F nw þ ð1  F Þna ) (m2/s) density of soil (¼ ð1  mÞrs þ mrw ) (kg/m3) density of air (kg/m3) density of rubbles (kg/m3) density of soil particles (kg/m3) density of water (kg/m3) average density (¼ F rw þ ð1  F Þra ) (kg/m3) effective stress tensor (positive means tension) (N/m2)

and the pore water were excluded in the poro-elastic FEM model. Hence, it is possible that the wave-induced effective stresses are inaccurate under non-linear wave conditions (Jeng and Cha, 2003). On the wave–permeable structure interaction, which is another key of the suction phenomenon, van Gent et al. (1994) developed a numerical model based on the volume of fluid (VOF) method (Hirt and Nichols, 1981) with porous flow friction to simulate breaking waves over a permeable low-crested structure. Hur and Mizutani (2003) and Hur et al. (2004) investigated wave forces acting on a three-dimensional body on a permeable submerged breakwater using the three-dimensional numerical method newly developed with the VOF model and the porous body model (Sakakiyama and Kajima, 1992). Their study has revealed that a comparison of the calculated and measured data showed a good agreement. They also reported that their model is extremely useful for the estimation of wave forces on a three-dimensional body. But their model is not

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applicable to analysis for porous flow inside the minute porous materials like a fine sand seabed since a linear drag force term, whose effect is less than that of a nonlinear drag force in case of rubbles, is ignored in the porous body model. In order to cope with this problem, Golshani et al. (2003) expanded the numerical model proposed by Hur and Mizutani (2003) to be able to consider the size effect of the porous media with non-linear and linear drag forces. Although a number of experimental studies on wavedriven seabed behavior such as scour around vertical cylinders exist (e.g., Sumer et al., 2001), there are very few investigations into pore water pressure in a reclaimed zone. Among these investigations, Shigemura et al. (1998) examined experimentally the dynamic pore water pressure in the reclaimed area with sand behind a caisson model, and it was, as a result, revealed that the dynamic pressure attenuated exponentially and its phase shift increased linearly as it propagated in the reclaimed area with sand. However, not all characteristics of pore water pressure and groundwater fluctuation inside the reclaimed zone have been clarified so far. The purpose of this paper is to investigate the suction mechanism of the reclaimed area with sand behind inclined and upright rubble mound breakwaters both experimentally and numerically. In the experiment, we examine effects of incident wave profiles, the relative breakwater width and the median grain sizes of the gravels and the sand, and also measure pore water pressure and groundwater fluctuation around the breakwater. In the numerical study, to begin with, new numerical methods, that is, a numerical wave tank based on the model developed by Golshani et al. (2003) with non-linear and linear drag forces and a finite element method for a reclaimed area with sand based on the up approximation of the Biot equations with continuity of water particle velocity at the wave–sand interface, are developed and validated through a comparison of the experimental data. Next, we explain a suction mechanism with discharge velocities in the breakwater and volumetric strains in the reclaimed area with sand, and show critical values of the velocities and the strains as well. Finally, efficiency of filter layers, which is one of the constructions for the prevention of the suction phenomenon, is demonstrated.

2. Hydraulic model study Hydraulic model experiments were conducted to investigate the suction mechanism of the reclaimed material with the protecting facility against washout in Nagoya University’s wave tank having 30.0 m long, 0.7 m wide and 0.9 m deep. As shown in Fig. 1, a flap-type wave generator was equipped at one side of the tank, and an impermeable bed and wall were installed at the other side. In this paper, two types of breakwater as the protecting facility were adopted: (1) an inclined rubble mound breakwater and (2) an upright rubble mound breakwater. A rubble mound breakwater and reclaimed area were constructed of gravels and sands, respectively, on the impermeable bed. 2.1. Dimensional analyses The suction phenomenon can be described by the following physical parameters: f ðH i ; T; h; g; rw ; mw ; B; hr ; s; rr ; mr , D50 ; ‘; hs ; rs ; ms ; d 50 Þ ¼ 0,

ð1Þ

where Hi is the incident wave height; T is the wave period; h is the still water depth; g is the gravitational acceleration; rw is the water density; mw is the viscosity coefficient of the water; B is the breakwater width at the still water level; hr is the breakwater height; s is the side slope of the breakwater inclination; rr is the density of the rubbles; mr is the porosity of the rubbles; D50 is the median grain size of the gravels; ‘ is the beach length at the still water level; hs is the beach height; rs is the density of the sand; ms is the porosity of the sand and d50 is the median grain size of the sand. Adopting Buckingham’s p-theorem to Eq. (1), the dimensionless parameters can be given by Eq. (2), as follows: pffiffiffiffiffi gh B hr r 0 H i h D50 ; ; f ; ; ; s; r ; mr ; L h L L rw nw  D50 ‘ hs rs d 50 ; ; ; ; ms ; ¼ 0, ð2Þ B L h rw D50 where L is the wavelength and nw ( ¼ mw/rw) is the kinematic viscosity. The dimensionless quantities are Rubble Mound Breakwater

Wave

Reclaimed Sand

1

:7

Wave Generator

33.0 Impermeable Bed

Fig. 1. Wave tank.

Impermeable Wall Unit: cm

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neglected in Eq. (2), since nw, hr, rr, mr, rs and ms are constant in this study. In addition, we adopt the Ursell parameter Ur ¼ H i L2 =h3 instead of the relative water depth h/L. As a result, in this work, the suction mechanism of the reclaimed area with sand behind a rubble mound breakwater is described by the following dimensionless parameters:   B D50 ‘ hs d 50 00 H i f ; Ur; ; s; ; ; ; ¼ 0. (3) L L B L h D50

2.2. Inclined-type rubble mound breakwater model The hydraulic model experiment for an inclined rubble mound breakwater was carried out, referring to the field survey at the Shiroya beach, Japan (see Photo 1). The inclined breakwater (hr ¼ 45.0 cm, s ¼ 1/2) with gravel (D50 ¼ 3.0 cm) and reclaimed beach with sand (d50 ¼ 0.01 cm) were set up on the impermeable bed, as shown in Fig. 2. In this kind of experiment, more attention should be paid to placing gravel for a rubble mound breakwater and sand for a reclaimed area at its onshore side into the wave

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tank. The following procedure was adopted: First, the rubble mound breakwater was placed on the impermeable bed, and the water was poured into the wave tank until the still water depth was about 20 cm. Next, the reclaimed beach was set between the breakwater and the wall, and then the water was poured again. The sand, as a result, flowed into the onshore side in the breakwater, as shown in Photo 2. In this experiment, the incident waves were generated under this initial condition. Five capacitance-type wave gauges were used in order to measure water surface fluctuations at the offshore side of the breakwater. In addition, three groundwater gauges, which were newly developed equipments for measurement of groundwater table fluctuations, were installed inside the breakwater and the reclaimed beach. Here, the groundwater gauge is the capacitance-type wave gauge surrounded by fine wire gauze (mesh size: 75 mm) in order not to touch the capacitance line to rubbles or sand. In preliminary tests, the time variations of water surface profiles obtained by a wave gauge and a new groundwater gauge, which are placed parallel at same distance from wave generator, are compared for regular waves. As a result, it was found that the delay time of the output of the groundwater gauge was 50 ms or less, thus it can be regarded that the gauze around the gauge had little effect on measuring data since the delay time was negligible compared with the incident wave periods. The position of the gauges is shown in Fig. 2. The measurements started with still water conditions, and stored discrete values at sampling frequency of 100 Hz in a personal computer. The suction process of the reclaimed beach has been simultaneously recorded with a video camera (Sony: DCR-PC110) and used in analysis.

Wave

Rubble Mound Breakwater 1 2 3 4 5

Impermeable Wall 6 7 8

45.0

450.0

1:

2

h

B

50.0 50.0 30.0 60.0

25.0

hs

Reclaimed Sand

60.0 30.0 70.0

1-5: Wave Gauge, 6-8: Groundwater Gauge

140.0

Unit: cm

Fig. 2. Hydraulic model set-up for the inclined rubble mound breakwater.

Photo 1. Examples of cave-in observed at the Shiroya beach, Aichi, Japan.

Photo 2. Initial condition for the hydraulic model tests with an inclined rubble mound breakwater.

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Waves generated in this experiment were all regular ones. The incident wave height Hi is changed with the range of 1.8–10.4 cm and the wave period T is varied with the range of 0.7–1.7 s. The still water depth h is set at 30.0 and 35.0 cm and the beach height hs is fixed as 40.0 and 45.0 cm. The experimental conditions are listed in Table 1. In each experiment, the wave generator worked for about 60 min. 2.3. Upright-type rubble mound breakwater model It was revealed that B/L has an effect on the sand suction in case of the inclined rubble mound breakwater (It will be mentioned in Section 3). It is very difficult, however, to discuss simply effects of the wavelength L on the sand suction, since the breakwater width at the still water level B is highly dependent on the still water depth h. In order to investigate effect of B/L more clearly, a hydraulic model experiment for an upright rubble mound breakwater was conducted. In addition, here, to discuss the varying wave field inside the porous media due to the change of experimental condition, the pore water pressures and groundwater fluctuations were also measured. Here, two kinds of experiments were carried out. One was for measurement of pore water pressures and groundwater fluctuations inside the breakwater and the reclaimed area, and the other was for investigation of the suction phenomenon.

Table 1 Experimental conditions of the inclined rubble mound breakwater

2.3.1. Measurement of pore water pressure and groundwater fluctuations To measure pore water pressures and groundwater fluctuations inside the breakwater and the reclaimed beach, the hydraulic experiment was performed. These experimental results will be compared with the numerical results in Section 5. The upright rubble mound breakwater (hr ¼ 45.0 cm and B ¼ 90.0 cm) with gravels of D50 ¼ 3.0 cm and the reclaimed beach (hs ¼ 45.0 cm and ‘ ¼ 150 cm) with sands of d50 ¼ 0.045 cm were installed, as shown in Fig. 3. Also, the fine wire gauze (mesh size: 75 mm) was set up on the offshore side of the beach to prevent the sand from flowing into the breakwater. The pore water pressures and water surface fluctuations were measured with five pressure gauges, three capacitance-type wave gauges and three groundwater gauges, respectively. The waves produced for this experiment were regular, the experimental conditions, including their characteristics, are given in Table 2. In this experiment, the waves were generated for about 1 min. 2.3.2. Investigation of the suction phenomenon As shown in Fig. 4, we installed an upright rubble mound breakwater (hr ¼ 45.0 cm) and reclaimed beach (hs ¼ 45.0 cm and length ‘ ¼ 150:0 cm) in the wave tank with the same procedure in case of the inclined rubble Impermeable Wall Rubble Mound Breakwater G1 G2 G3 G4 G5 G6 G7 G8 G9

Wave

20.0

Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Hi (cm) 3.9 4.0 3.9 6.8 6.9 7.3 6.9 7.1 7.2 7.0 6.8 3.8 3.8 4.0 3.7 4.1 1.9 1.9 1.9 1.8 2.0 10.4 4.0 3.9 4.9 6.5 5.7 7.5

T (s) 0.9 1.3 1.7 0.9 1.3 1.7 0.9 1.3 1.7 1.1 1.5 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 1.0 0.7 0.8 0.8 0.8 0.9 1.0

h (cm)

B (cm)

hs (cm)

D50 (cm)

d50 (cm)

45.0 30.0

30.0

85.0

40.0

2.0

30.0

85.0

40.0

2.0

0.010

30.0

85.0

45.0

2.0

0.010

35.0

35.0

65.0

65.0

65.0

45.0

45.0

45.0

2.0

2.0

2.0

P2 P3 P4 P5 P6 P7 P8 P9 Gauze

0.010 860.0

35.0

P1

0.010

0.010

0.010

P10

Reclaimed Sand

50.0 22.5 22.5 22.5 22.5 15.0 15.0 15.0 15.0 30.0

60.0

G1-G3: Wave Gauge, G4-G9: Groundwater Gauge, P1-P10: Pressure Gauge Unit: cm Fig. 3. Hydraulic model set-up for measurement of pore water pressure and groundwater fluctuations in case of the upright rubble mound breakwater.

Table 2 Experimental conditions for measurement of pore water pressure and groundwater fluctuation inside the porous media in the case of the upright rubble mound breakwater Case

Hi (cm)

T (s)

1 2 3 4 5 6 7 8 9 10

1.9 1.9 1.8 1.8 1.7 4.8 4.7 4.6 4.6 4.5

0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7

h (cm)

B (cm)

hs (cm)

D50 (cm)

d50 (cm)

30.0

90.0

45.0

3.0

0.045

30.0

90.0

45.0

3.0

0.045

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Rubble Mound Breakwater W1

W2

Table 3 Experimental conditions for investigation of suction phenomenon of the reclaimed sand in the case of the upright rubble mound breakwater

Impermeable Wall

W3

Wave

Case

Hi (cm)

T (s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1.8 1.8 1.8 1.8 1.8 4.7 4.6 4.5 4.6 4.6 1.9 1.8 1.8 1.7 1.8 4.8 4.6 4.7 4.5 4.6 1.9 1.8 1.8 1.8 1.7 4.8 4.5 4.4 4.5 4.2 1.9 1.9 1.8 1.8 1.7 4.9 4.5 4.5 4.5 4.6 1.9 1.8 1.8 1.8 1.7 4.7 4.8 4.7 4.6 4.2

0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7

h (cm)

B (cm)

hs (cm)

D50 (cm)

d50 (cm)

45.0

B 30.0

Reclaimed Sand 1000.0

1109

150.0

W1, W2: Wave Gauge, W3: Groundwater Gauge Unit: cm Fig. 4. Hydraulic model set-up for investigation of the suction process in case of the upright rubble mound breakwater.

Photo 3. Initial condition of the hydraulic model tests for investigation of the suction process in case of the upright rubble mound breakwater.

mound breakwater. Hence, the sand flowed into the breakwater (see Photo 3) and this was the initial condition in this experiment. Two capacitance-type wave gauges and a groundwater gauge were set up to examine effects of water surface fluctuations in front of the reclaimed beach, as illustrated in Fig. 4. The measurements were started with still water conditions, and discrete values at sampling frequency of 100 Hz were stored in a personal computer. The suction process of the reclaimed beach has been simultaneously recorded with a video camera (Sony: DCR-PC110) and used in analysis. Incident waves were all regular ones and the still water depth h was set at 30.0 cm. The breakwater width B was set as 60.0, 90.0 and 120.0 cm. Gravels of D50 ¼ 2.0 and 3.0 cm for the breakwater and sands of d50 ¼ 0.01 and 0.045 cm for the reclaimed area were used. These experimental conditions are displayed in Table 3. The waves were generated for 30 min in each experiment, since there was little change of the reclaimed sand shape after 30 min in preliminary tests. 3. Experimental results and discussions First of all, the pore water pressure and the groundwater fluctuations, which were measured in the experimental cases of the upright rubble mound breakwater, are discussed. Next, the suction mechanism is examined in relation to the dimensionless parameters derived in the

30.0

60.0

45.0

3.0

0.045

30.0

90.0

45.0

3.0

0.045

30.0

120.0

45.0

3.0

0.045

30.0

90.0

45.0

2.0

0.045

30.0

90.0

45.0

3.0

0.010

dimensional analyses of the previous section. Finally, effects of the water surface fluctuation in front of the reclaimed beach on the sand suction are investigated. 3.1. Wave field around the upright rubble mound breakwater Here, the damping rates of wave height and pore water pressure against its propagating distance in the rubble

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2.0

1.5

Hi [cm] ≅ 2.0 ≅ 5.0 T = 0.9 [s] T = 1.1 [s] T = 1.3 [s] T = 1.5 [s] T = 1.7 [s]

1.0

Reclaimed Sand

H / Hi

Rubble Mound Breakwater

0.5

0.0 0.0

0.5

(a)

1.0 x/B

1.5

2.0

2.0 Hi [cm] ≅ 2.0 ≅ 5.0 T = 0.9 [s] T = 1.1 [s] T = 1.3 [s] T = 1.5 [s] T = 1.7 [s]

Rubble Mound Breakwater

p /w g Hi

1.5

1.0

As depicted in Fig. 5(b), the range of p/rwgHi in front of the breakwater is wider than that of H/Hi, but its variation tendency is nearly equal to that of H/Hi. To examine the damping rates of wave height and pore water pressure inside the reclaimed beach in details, log(H/ Hi) and log(p/rwgHi) are plotted against x/B in Fig. 6. The upper figure is for the dimensionless wave height H/Hi and the lower one is for the dimensionless peak value of the pore water pressure p/rwgHi. In these Figs. 6(a) and (b), it is shown that there is no connected line between the values at x/B ¼ 1.0 and next ones, because the groundwater gauge of G6 and the pressure gauge of P5 (G6 and P5; see Fig. 3) were fixed inside the gravel side, keeping a small gap from the reclaimed area. Shigemura et al. (1998) revealed in their experimental results on the propagation property of dynamic pressure through the reclaimed sand behind a caisson-type seawall that the dynamic pressure is exponentially decreased with increasing its propagated distance in the reclaimed zone. The similar tendency is also observed for H/Hi and p/rwgHi in this work, as seen from Fig. 6. That is to say, log(H/Hi) as well as log(p/rwgHi) is damped with increasing x/B in

Reclaimed Sand Hi [cm]

1

Reclaimed Sand

0.0 (b)

0.5

1.0 x/B

1.5

2.0

H / Hi

0.5

0.0

Fig. 5. Spatial variations of dimensionless parameters inside the upright rubble mound breakwater and the reclaimed beach. (a) For the dimensionless wave height H/Hi; (b) For the dimensionless peak value of the pore water pressure p/rwgHi.

0.1

0.01

1.0

1.2

1.4

(a)

1.6

1.8

2.0

x/B ≅ 2.0 ≅ 5.0 Hi [cm] T = 0.9 [s] T = 1.1 [s] T = 1.3 [s] T = 1.5 [s] T = 1.7 [s]

1 Reclaimed Sand

p /w g Hi

mound breakwater are discussed. Fig. 5 shows the variation of the dimensionless wave height H/Hi and the dimensionless peak value of the pore water pressure p/rwgHi inside both the upright rubble mound breakwater and the reclaimed beach, with x/B in which x represents the onshore-ward horizontal length from the offshore side of the breakwater. It is shown in Fig. 5(a) that the values of H/Hi in front of the breakwater are more than 1.5 regardless of the wave conditions and then sharply decreased. It is also displayed that the values of H/Hi in front of the beach (0.5ox/Bo1.0) tend to increase or be a stable situation since the standing waves exist at onshore side in the breakwater due to the reflection in front of the beach. This tendency is great effected by increasing wave period T and decreasing incident wave height Hi. They dropped suddenly in the beach again, because the fluid motion is attenuated according to its propagation into the reclaimed area.

≅ 2.0 ≅ 5.0

T = 0.9 [s] T = 1.1 [s] T = 1.3 [s] T = 1.5 [s] T = 1.7 [s]

0.1

0.01

1.0 (b)

1.2

1.4

1.6

1.8

2.0

x/B

Fig. 6. Spatial variations of dimensionless parameters inside the reclaimed beach. (a) For the dimensionless wave height H/Hi; (b) For the dimensionless peak value of the pore water pressure p/rwgHi.

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the reclaimed area, regardless of the incident wave conditions. In addition, it can be pointed out that the damping rate of log(H/Hi) due to its propagating from the breakwater (G6; see Fig. 3) to the beach (G7; see Fig. 3) becomes larger than that of log(p/rwgHi), while the damping rate of them inside the beach is nearly same. 3.2. Suction mechanism of the reclaimed sand 3.2.1. Suction process and its classification In the hydraulic model tests, caves and cave-ins, which are similar to observed ones in field survey, were reproduced. Photo 4 is an example that can explain the suction process of the reclaimed sand. First, a cave occurred around the still water level in the offshore side of the beach due to the incident wave action (Photo 4(b)). And then, if the cave’s size is too big to support the sand weight above it according to successive wave action, a cavein can be observed (Photo 4(c)). Finally successive in coming waves resulted in a cliff-like cave-in as shown in Photo 4(d). In this research, the experimental results for the sand suction are classified into three patters as follows: (1) ‘‘no suction’’ means no cave and cave-in occurred at all; (2) ‘‘cave’’ means caves were formed inside the reclaimed zone due to the sand suction; and (3) ‘‘cave-in’’ means cave-ins occurred on the surface of the reclaimed zone. The typical cave and cave-in are shown in Photos 5(a) and (b), respectively and both ‘‘cave’’ and ‘‘cave-in’’ are called ‘‘suction’’, hereinafter. 3.2.2. Conditions of the sand suction The effect of several parameters on the sand suction is investigated. It is revealed that, among them, the Ursell

Photo 5. Suction classification. (a) Example of ‘‘cave’’ with the condition of Ur ¼ 9.3, B=L ¼ 0:256, D50 =B ¼ 0:050, d 50 =D50 ¼ 0:015; (b) Example of ‘‘cave-in’’ with the condition of Ur ¼ 4.0, B=L ¼ 0:314, D50 =B ¼ 0:031, d 50 =D50 ¼ 0:005.

Photo 4. Suction process of the reclaimed sand behind the inclined rubble mound breakwater with the condition of Ur ¼ 4.0, B=L ¼ 0:314, D50 =B ¼ 0:031, d 50 =D50 ¼ 0:005. (a) Initial condition; (b) 12.5 min later; (c) 15 min later; (d) 30 min later.

ARTICLE IN PRESS D.-S. Hur et al. / Ocean Engineering 34 (2007) 1104–1119

1112

20.0

Suction

0.031

10.0

No Suction

Urc= 1.7 (D50 / B = 0.031)

0.0 0.0

0.2

0.4

(a)

Suction

15.0

B / L = 0.45

20.0

0.6 B/L

0.8

D50 /B 0.022 No Suction Cave Cave-in d50 / D50

1.0

0.025

1.2

0.033 0.050

0.003 0.015 0.023

10.0

No Suction

5.0 Urc = 4.0

0.0 0.0 (b)

the inclined breakwater. This is due to that the still water depth h is only changed in the experiments for the inclined breakwater. Also it is found that there is the weak tendency that the sand suction easily occurs as the rubble/sand diameter ratio d50/D50 becomes smaller, but its detail will be discussed with a numerical simulation in the following section. In this study, since no effects of the wave breaking, the beach length at the still water level ‘ and the beach height hs are found, it is not discussed.

Urc = 4.7 (D50 / B = 0.024)

5.0

Ur

0.024

B / L = 0.60

15.0

Ur

D50 / B No Suction Cave Cave-in

0.2

0.4

0.6 B/L

0.8

1.0

1.2

Fig. 7. Effect of Ur, B/L, D50/B and d50/D50 on the sand suction. (a) In the case for inclined rubble mound breakwater; (b) In the case for upright rubble mound breakwater.

parameter Ur and the relative breakwater width B/L have great effect on it, as depicted in Fig. 7. Figs. 7(a) and (b) are for the inclined and upright rubble mound breakwater, respectively. The sand suction tends to occur with an increase in the Ursell parameter Ur as well as a decrease in the relative breakwater width B/L in the both cases. For the inclined rubble mound breakwater, the sand is sucked out for Ur44.7 in case of D50/B ¼ 0.024 and Ur41.7 in case of D50/B ¼ 0.031 with the range of B/Lo0.60. On the other hand, the sand behind the upright rubble mound breakwater shows the tendency to be sucked out for Ur44.0 or B/Lo0.45, namely, no suction area is defined for Urp4.0 and B/LX0.45. The critical Ursell parameter Urc, which divides the experimental conditions into ‘‘no suction’’ and ‘‘suction’’, here, is proposed. Note that the critical Ursell parameter Urc is dependent on D50/B in the only case of the inclined breakwater because the effect of D50/B is not evident in case of the upright breakwater compared with the case of

3.2.3. Effects of the water surface fluctuation in front of the reclaimed zone As mentioned above, it is revealed that the reclaimed sand around the still water level is sucked out into the rubble mound breakwater. So, in order to examine the mechanism, the water surface fluctuation (W3; see Fig. 4) in front of the reclaimed beach should be focused such as the wave height Hs derived from the fluctuation and the maximum velocity Vs derived from the time derivative of the fluctuation calculated with the Fast Fourier Transform routine. Here we adopt Vs in order to consider the wave non-linearity. pffiffiffiffiffi Figs. 8(a) and (b) show the effects of Hs/h and V s = gh on the sand suction, respectively. Asp shown ffiffiffiffiffi in Fig. 8, it can be pointed out that Hs/h and V s = gh increase with an increase in Ur and a decrease in B/L. That is to say, the sand suction isp disposed to occur easily for larger values of ffiffiffiffiffi Hs/h and V s = gh in front of the reclaimed zone. This tendency is more obviously illustrated in Fig. 9. It is found from Fig. 9 that Hs/h is bigger in case of Ur44.0 and pffiffiffiffiffi B/Lo0.45 even if V s = gh is same in value. Thus, it can be explained again that the reclaimedp sand ffiffiffiffiffi is sucked out due to an increase of Hs/h and V s = gh, in particular, an increase of Hs/h. As a result, in order to protect the p sand ffiffiffiffiffi suction, it is very important to decrease Hs/h and V s = gh, namely, to attenuate the water surface fluctuation in front of the reclaimed area. 4. Numerical simulation In this section, a numerical method for waves, structures and seabed interaction is introduced. And then, another numerical method based on the up approximation of the Biot equations is developed to calculate stress and strain fields inside seabed. Finally how to connect the numerical method for the wave field with that for the seabed is explained in detail. 4.1. Numerical method for wave field Here, the numerical model developed by Golshani et al. (2003) is adopted. But the viscosity term is added to their equations for more precise calculations. Moreover, the governing equations are expanded to be applicable for the air phase flow as well as the fluid phase flow. Therefore, the wave field is governed by modified continuity equation (4) and modified Navier–Stokes equation (5). The volume

ARTICLE IN PRESS D.-S. Hur et al. / Ocean Engineering 34 (2007) 1104–1119

to the whole volume of the cell.

0.10 d50 / D50

0.08

0.003 0.015 0.023

qðmvj Þ ¼ q ; qxj

Ur < 4.0 > 4.0 No Suction Cave Cave-in

Hs / h

0.06

q ¼

qðy; z; tÞ , Dxs

(4)

  1  m qvi qvi 1 þ CA þ vj m qt qxj 1 qP q2 v i  gi þ hni hri qxi qxj qxj  hni qq  Ri2  Ri1 þ  bij vi , 3m qxi

0.04

¼

0.02

0.00 0.0

0.2

0.4

0.6 B/L

(a)

0.8

1.0

1.2

d50 / D50

0.003 0.015 0.023

Ur < 4.0 > 4.0 No Suction Cave Cave-in

0.03

ð5Þ

qðmF Þ qðmvj F Þ þ ¼ Fq , qt qxj

(6)

where vi is the actual velocity, P is the pressure, t is the time, gi is the gravitational acceleration vector, /rS is the fluid density, /nS is the kinematic viscosity of the fluid, m is the porosity, CA is the added mass coefficient, Ri2 , Ri1 are the linear and non-linear drag vectors, respectively, q(y, z; t) is the source density at the source position (x ¼ xs) based on the fifth-order Stokes wave theory (Horikawa, 1988), Dxs is the mesh width at x ¼ xs and bij is the dissipation factor array. Here /rS, /nS, Ri2 , Ri1 , q(y, z; t), gi and bij are as follows:

0.04

Vs / (g h)0.5

1113

0.02

0.01

hri ¼ F rw þ ð1  F Þra ,

(7)

hni ¼ F nw þ ð1  F Þna ,

(8)

0.00 0.0

0.2

0.4

0.6 B/L

(b)

0.8

1.0

1.2

pffiffiffiffiffi Fig. 8. Effect of Hs/h and V s = gh in front of the reclaimed area on the sand suction related to the relative breakwater width B/L. (a) In the case pffiffiffiffiffi for Hs/h; (b) In the case for V s = gh.

Ri2 ¼

12C D2 hnið1  mÞ vi , md 2m

Ri2 ¼

C D1 hnið1  mÞ pffiffiffiffiffiffiffi vi vj vj , 2md m

0.04

qðy; z; tÞ ¼

8   < 1  exp  2t 2U 0 T þh : 2U 0 ZZ0þh s

Vs / (g h)0.5

0.03

(10) Z0 þh Zs þh

if

t T

p3:0

if

t T

43:0

, (11)

gi ¼ 0

0.02

2

0.01

Ur < 4.0 > 4.0 No Suction Cave Cave-in

0.02

0.04

0.06

0.08

0

0 0 60 0 bij ¼ 4 0 0

B / L < 0.45 > 0.45

0.00 0.00

(9)

0.10

Hs / h pffiffiffiffiffi Fig. 9. Effect of Hs/h and V s = gh on the sand suction.

of fluid (VOF) method (Hirt and Nichols, 1981) is used to track the free surface location and the free surface is governed by the VOF function F in Eq. (6), where F represents the rate of volume in a cell occupied by the fluid

g

T

,

(12)

3 0 07 5,

(13)

b

where the subscripts w and a represent values at the fluid phase and the air phase, respectively, CD2 and CD1 are the linear and non-linear drag coefficients, respectively, dm is the median grain size of porous media, T is the wave period, U0 and Z0 are the horizontal velocity and water surface elevation based on the Stokes wave theory, respectively, Zs is the water surface elevation at the source position, g is the gravitational acceleration and b is the dissipation factor which equals zero except for the added dissipation zone (Hinatsu, 1992).

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1114

The numerical calculation for the wave field is carried out using the finite difference method. A staggered mesh is used for the computational discretization. The simplified MAC (SMAC) method (Amsden and Harlow, 1970) is adopted as the time matching scheme and the MICCG (modified incomplete Cholesky conjugate gradient) method is used to solve the Poisson equation. The third-order Adams–Bashforth scheme is used to discretize the time derivative. The convection terms are discretized by the third-order upwind scheme known as the UTOPIA scheme. The central difference method is used to discretize the other terms in Eq. (5). For the advection equation of VOF function F (Eq. (6)), the Euler scheme and the donor– acceptor scheme (Hirt and Nichols, 1981) are adopted to discretize the time derivative and the convective terms, respectively. The cold start, which means all velocities at t ¼ 0 are zero, is utilized as initial conditions. Approximate boundary conditions are adopted on the solid boundary, the lateral boundaries and the free surface. The no-divergence condition is imposed as the pressure boundary condition. 4.2. Numerical method for seabed The up approximation of the Biot equations, in which the relative water particle acceleration is eliminated from the Biot equations in the uwp form, is adopted as the governing equations for seabed. r€u ¼ r  r0  rpe ,

(14)

 qv m k € þ ðrpe  rw uÞ ¼ 0, p_ þ r  rw g qt K w e

(15)

where r ¼ ð1  mÞrs þ mrw is the soil density, rs is the density of the soil particles, rw is the water density, m is the porosity, u is the soil displacement vector, r0 is the effective stress tensor (positive stress means tension), pe is the excess pore water pressure, ev is the volumetric strain (positive strain means expansion), Kw is the apparent water bulk modulus, k is the hydraulic conductivity, g is the gravitational acceleration and the superscript dot represents time derivative. It is assumed that the non-linearity of seabed induced by the wave action is very small, so that the following Hooke law is used as the constitution law. s0ij

¼ lkk dij þ 2mij ,

(16)

l¼

En , ð1 þ nÞð1  2nÞ

(17)

m¼

E ¼ G, 2ð1 þ nÞ

(18)

where l and m are the Lame´ constants, eij is the strain tensor (positive strain means expansion), dij is the Kronecker delta, E is the modulus of elasticity, G is the shear modulus of elasticity and n is the Poisson ratio.

The numerical model test for the seabed is conducted using the finite element method. The finite element equation is solved numerically from the shape function of the two-dimensional isoparametric rectangle elements and the Galerkin method. The Newmark b method and the Crank–Nicolson method is utilized to discretize the time integrals of the soil displacement u and the excess pore water pressure pe, respectively. The system of the linear equations is solved with the band matrix method based on the Gauss elimination. 4.3. Connection method between wave field and seabed The wave field and the seabed are connected with the following process by Mizutani et al. (1998). First, the velocities and pressures are calculated in the whole domain including the seabed with the numerical method for the wave field. Then, new velocities and pressures, which are calculated on the boundary between the wave field and the seabed, are used as the boundary condition for the numerical computation in the seabed. Finally, the stress and strain fields inside seabed are obtained. However, it should be noted that the feedback calculation from the seabed to the wave field is not carried out assuming that the wave-induced soil displacements are adequately small. 5. Numerical results and discussions The numerical model developed in this study, first of all, is verified by comparison with the experimental results obtained in case of the upright rubble mound breakwater. Next, the effects of both velocity and strain fields inside the porous media on the sand suction are investigated in the case where cave-ins occurred on the reclaimed beach in the hydraulic experiments. Finally, the critical values of the velocities and the strains are examined and then, the efficiency of filter layers is discussed for the prevention of sand suction problems. In this section, for simplicity, the upright rubble mound breakwater, which is the simpler form than the inclined one, is adopted for analysis of the sand suction phenomenon. 5.1. Validation of the numerical simulations The numerical wave tank used for verification of the newly developed numerical models is illustrated in Fig. 10. The conditions used in the numerical verification are described below: The gravitational acceleration g is 9.81 m/s2, the densities of water and air are 997 and 1.18 kg/m3, respectively, the kinematic viscosities of water and air are 0.893  106 and 15.4  106 m2/s, respectively, the median grain sizes of the rubbles and the sand are 3.0 and 0.045 cm, respectively, the porosities of the rubbles and the sand are 0.36 and 0.40, respectively, the added mass coefficient CA is 0.15, the non-linear drag coefficient CD1 is 0.5, and the linear drag coefficients CD2 of the rubbles and the sand are 100.0 and 0.1, respectively.

ARTICLE IN PRESS D.-S. Hur et al. / Ocean Engineering 34 (2007) 1104–1119

1115

Impermeable Wall 15.0

Rubble Mound Breakwater

30.0

Open Boundary

Wave

z Reclaimed Sand

x

45.0

h = 30.0

Wave Source

o

90.0

Added Dissipation Zone

150.0

Unit: cm

Fig. 10. Numerical wave tank used for verification of the newly developed numerical models. An added dissipation zone is placed at the offshore side of the tank to avoid wave re-reflection at open boundary that is exposed at the offshore side of the tank.

The newly used conditions in case of the numerical calculation for the seabed are expressed below: The density of the soil particles rs is 2.65  103 kg/m3, the shear modulus of elasticity G is 1.0  108 N/m2, the apparent bulk modulus of water Kw is 1.0  107 N/m2, the Poisson ratio n is 0.33, and the hydraulic conductivity k is 2.20  103 m/s with the following Kozeny–Carman formula: k¼

1 m3 gd 2m , 2 180 ð1  mÞ nw

(19)

where dm represents the median grain size of porous media. In case of the numerical model test for the seabed, it is assumed that the reclaimed area is saturated with water. Moreover, the grid size is set as x ¼ 2.0 cm, y ¼ 1.0 cm and z ¼ 0.5 cm in case of the numerical calculation for the wave field as well as that for the seabed. Fig. 11 displays comparison between the numerical and experimental results for the dynamic pore water pressure pe inside the upright rubble mound breakwater and the reclaimed beach, while Fig. 12 is for the comparison of the water surface fluctuation Z. In these figures, the circles represent the experimental data and the lines are the numerical results in case of numerical simulation for the wave field. In addition, the broken lines, which are only plotted in Fig. 11(b), are the numerical results based on the Biot equations for the seabed. It is shown in Fig. 11(a) that the dynamic pressures pe calculated at P1 and P5 (see Fig. 3) are in good agreement with the experimental data, while there is a small difference in phase between calculated and measured dynamic pressure variations at P2, P3 and P4 inside the breakwater. In case of the time variation of the water surface, the similar tendency is displayed as shown in Fig. 12(a). That is, the small phase difference is also represented at G4 and G5 inside the breakwater like the case of the dynamic pressures. Here, it is not possible to make clear why this phenomenon occurred, but it may be explained by the position of pressure gauges at P2, P3 and P4 and groundwater gauges at G4 and G5 inside the breakwater is slightly shifted as compared with the original planned position. Figs. 11(b) and 12(b) show that the computed dynamic pressure variations and water surface

variations inside the reclaimed area agree well with the experimental results. Incidentally, for the dynamic pressure, VOF-based numerical results (solid line) for the wave field also agree well with the experimental data in the case of this seabed condition. From these results, it is clear, overall, that the numerical calculation is regarded to reproduce fairly the time variation of the dynamic pressure and the water surface inside both the breakwater and the reclaimed zone. Thus, the validity of the newly developed numerical model is verified. 5.2. Suction mechanism of the reclaimed sand Fig. 13 depicts the numerical wave tank used for investigation of suction mechanism of the reclaimed sand. In order to make the similar condition that the reclaimed sand flowed into the onshore side of the breakwater in the hydraulic model test as shown in Photo 3, some part of the breakwater is replaced with the sand as illustrated in Fig. 13. The attention to examine the suction mechanism is focused on the horizontal velocity vnorm and vertical velocity vtan in the vicinity of the beach inside the breakwater. Moreover, the effect of volumetric strain ev around the still water level in the vicinity of the breakwater inside the beach, which is shown with dark gray in Fig. 14, is considered as an object of the examination. For the case that the reclaimed sand behind the upright rubble mound breakwater is sucked out with the cave-in condition (Ur ¼ 12.5, B/L ¼ 0.221, D50/B ¼ 0.050, d50/D50 ¼ 0.015) in the hydraulic experiments, Fig. 15(a) illustrates the spatial variations of the discharge velocities and the volumetric strains ev at the phases when the maximum horizontal velocity vmax norm occurs around the still water level between the breakwater and the reclaimed area, whereas Fig. 15(b) represents those for the phases when the maximum vertical velocity vmax tan occurs. Also, the circled area plotted in Figs. 15(a) and (b) is the position to which more attention should be paid, because the sand suction is started inside the circle as shown in Photos 4(b) and 5(a). It is found in Fig. 15(a) that, although the maximum value of vnorm inside the circle is toward offshore side, it seems that

ARTICLE IN PRESS D.-S. Hur et al. / Ocean Engineering 34 (2007) 1104–1119

1116

0.8 0.4 0.0 -0.4 -0.8

pe /w g Hi

0.8 0.4 0.0 -0.4 -0.8

pe /w g Hi

P1

P2

0.8 0.4 0.0 -0.4 -0.8

P3

0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8

P4

P5 0.0

Exp.,

0.5

(a)

1.0 t/T

1.5

VOF

0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8

P5

P6

P7

P8

P9

P10 0.0

2.0

Exp.,

0.5

(b)

1.0 t/T

VOF,

1.5

Biot 2.0

1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0

G3

G4

G5

G6 0.0

(a)

 / Hi

 / Hi

Fig. 11. Comparison between the experimental data (J J J) and the numerical results (—: by calculation with VOF method,- - -: by calculation with Biot equations) for the dynamic pore water pressure pe under the condition of Ur ¼ 4.6, B/L ¼ 0.332, D50/B ¼ 0.033, d50/D50 ¼ 0.015. (a) Inside the rubble mound breakwater; (b) Inside the reclaimed beach.

Exb.,

0.5

1.0 t/T

1.5

VOF

2.0

1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0

G6

G7

G8

G9 0.0

(b)

Exb.,

0.5

1.0 t/T

1.5

VOF

2.0

Fig. 12. Comparison between the experimental data (J J J) and the numerical results (—) for the water surface fluctuation Z under the condition of Ur ¼ 4.6, B/L ¼ 0.332, D50/B ¼ 0.033, d50/D50 ¼ 0.015. (a) Inside the rubble mound breakwater; (b) inside the reclaimed beach.

ARTICLE IN PRESS D.-S. Hur et al. / Ocean Engineering 34 (2007) 1104–1119

1117

Impermeable Wall 15.0

30.0

B z 4.0

x

26.0

h = 30.0

Wave Source Added Dissipation Zone

Reclaimed Sand

o

45.0

Open Boundary

Wave

Rubble Mound Breakwater

26.0

150.0

Unit: cm

Fig. 13. Numerical wave tank for investigation of suction mechanism. This tank has a small difference from Fig. 10, that is, the rubble inside the rubble mound breakwater is changed for the sand because the reclaimed sand entered into the breakwater before waves were generated in the hydraulic experiment.

5.3. Critical vertical velocity and volumetric strain Wave

Reclaimed Sand

vtan vnorm

4.0 cm

εv

Rubble Mound Breakwater

Impermeable Wall

Fig. 14. Definition of the horizontal velocity vnorm and vertical velocity vtan in the vicinity of the beach inside the breakwater and consideration range of the volumetric strain ev.

the sand suction does not occur because of the negative volumetric strain ev inside the circle, that is, the reclaimed area around the still water level is under compression condition. However, as shown in Fig. 15(b), it may be concluded that the sand suction seems to occur easily in case of the large positive volumetric strain ev inside the circle, which means that the reclaimed area around the still water level is under expansion. In addition, the large vertical velocity, which means upward force on the front surface of the reclaimed zone, should be considered as one of the important effects on the sand suction behind the rubble mound breakwater. On the other hand, the discharge velocities and the volumetric strains ev for the case of no suction condition (Ur ¼ 0.9, B/L ¼ 0.514, D50/B ¼ 0.050, d50/D50 ¼ 0.015) in the hydraulic model tests are also represented in Fig. 16 and this figure is for the phases when the maximum vertical velocity vmax tan occurs around the still water level between the breakwater and the reclaimed area. As compared with Fig. 15(b), Fig. 16 indicates that the vertical velocities vtan and the volumetric strains ev inside the circle are much smaller. This may result in no suction of reclaimed sand. It may be inferred from these results that if the vertical discharge velocity as well as the volumetric strain around the still water level at the boundary between the breakwater and the reclaimed area gets up to the critical value that will be discussed in the following section, the reclaimed sand starts to flow out to the offshore, and it finally leads to caves and cave-ins in the reclaimed zone.

Here, the effect of the maximum value of the vertical velocity vtan and the volumetric strains ev on the sand suction are investigated in detail. Fig. 17 shows the effect of the maximum vertical discharge velocity vmax tan and the maximum volumetric strain max on the suction phenomv max enon. As plotted in Fig. 17, an increase in vmax or  is the tan v cause of the suction of the reclaimed sand. As far as this study is concerned, the suction phenomenon tends to occur pffiffiffiffiffiffiffiffiffiffi 6 max in case of vmax = gD 50 40:022 or/and v 40:15  10 . In tan max addition, v for Ur44.0, whose case represents ‘‘suction’’ (see Fig. 7(b)), is larger than that in case for Uro4.0, so that max significantly affects the sand suction. v pffiffiffiffiffiffiffiffiffiffi max It is also found in Fig. 17 that vmax are tan = gD50 and v in inverse proportion to the median grain size ratio of the sand/rubble, d50/D50. That is topsay, of d50/D50 ffiffiffiffiffiffiffiffiffiffithe increase max leads to the reduction of vmax = gD , which may and  50 tan v protect the sand suction. This result can be physical explanation of the fact that the installation of filter layers is a good countermeasure against the sand suction, which is well known by coastal engineers. 6. Conclusions In the present study, the hydraulic model tests are carried out to clarify the suction phenomenon of an artificial beach that is constituted by the rubble mound breakwater of gravel and the reclaimed area of sand. In addition, the numerical model for waves, structures and seabed interaction as well as the numerical method based on the up approximation of the Biot equations are developed for investigation of suction mechanism. By comparison with experimental results, the validation of newly developed numerical models is well verified. The main conclusions obtained from numerical and experimental results are summarized as follows: 1. Due to wave reflection in front of the beach (reclaimed area), wave height at onshore side of the breakwater is increased, while wave height and dynamic pressure inside the beach attenuated exponentially because of the wave energy dissipation.

ARTICLE IN PRESS D.-S. Hur et al. / Ocean Engineering 34 (2007) 1104–1119

1118

1.0 : 10.0 / (g D50)0.5

t / T = 0.00

0.5

z/h

Rubble Mound

Reclaimed Sand

0.0

v [10-6] 0.3 0.2 0.2 0.1 0.1 0 -0.1 -0.1 -0.2 -0.2 -0.3

-0.5

-1.0 -0.1

0.0

0.1

0.2

(a)

0.3

0.4

0.5

0.6

x/L 1.0 : 10.0 / (gD50)0.5

t / T = 0.34

0.5

z/h

Rubble Mound

Reclaimed Sand

0.0

v [10-6] 0.3 0.2 0.2 0.1 0.1 0 -0.1 -0.1 -0.2 -0.2 -0.3

-0.5

-1.0 -0.1

0.0

0.1

0.2

(b)

0.3

0.4

0.5

0.6

x/L

Fig. 15. Velocity and strain fields around the rubble mound breakwater in the case of ‘‘suction’’ (Ur ¼ 12.5, B/L ¼ 0.221, D50/B ¼ 0.050, d50/ max D50 ¼ 0.015). Occurrence phases of (a) the maximum horizontal velocity vmax norm and (b) the maximum vertical velocity vtan around the still water level between the breakwater and the reclaimed area.

1.0 t / T = 0.30

: 10.0 / (gD50)0.5

0.5 z/h

Rubble Mound

Reclaimed Sand

0.0

v [10-6] 0.3 0.2 0.2 0.1 0.1 0 -0.1 -0.1 -0.2 -0.2 -0.3

-0.5

-1.0 -0.2

0.0

0.2

0.4

0.6 x/L

0.8

1.0

1.2

1.4

Fig. 16. Velocity and strain fields around the rubble mound breakwater at the phase when the maximum vertical velocity vmax tan occurs around the still water level between the breakwater and the reclaimed area in the case of ‘‘no suction’’ (Ur ¼ 0:9, B=L ¼ 0:514, D50 =B ¼ 0:050, d 50 =D50 ¼ 0:015).

2. Caves and cave-ins inside the beach, which are similar to ones observed in field surveys, are reproduced in hydraulic model tests. It is found in the experiments that the sand around the still water level in the offshore side of the beach flows into the onshore side of the breakwater due to incident wave action. 3. Regardless of the breakwater inclination condition, the suction phenomenon of the reclaimed sand is occurs

with an increase in the Ursell parameter and a decrease in the relative breakwater width. 4. The wave height and the maximum velocity around the boundary between the breakwater and the beach have a great effect on the sand suction, so that it is very important to decrease wave energy inside the breakwater to protect it. 5. Numerical simulations based on the volume of fluid (VOF) method and the up approximation of the Biot

ARTICLE IN PRESS D.-S. Hur et al. / Ocean Engineering 34 (2007) 1104–1119

References vmax = 0.15 x 10-6

0.10

0.08

max

vtan / (g D50)0.5

1119

0.06

< 4.0 > 4.0 Ur No Suction Cave Cave-in

0.04

0.02

max

vtan / (g D50)0.5

d50 / D50 0.003 0.015 0.023

= 0.022

0.00 0.0

0.2

0.4 vmax

0.6 [x

0.8

1.0

10-6]

Fig. 17. Effect of the maximum vertical velocity vmax tan and the maximum volumetric strain emax on the suction phenomenon of the reclaimed sand v behind the rubble mound breakwater.

equations are developed and verified with the experimental data. 6. It may be inferred that the vertical discharge velocity as well as the volumetric strain around the still water level of the boundary between the breakwater and the beach gets up to the critical value, the reclaimed sand starts to flow out to the offshore, and it finally leads to caves and cave-ins in the reclaimed zone. 7. An increase in the median grain size ratio, d50/D50, between the sand on the beach and the rubble in the breakwater results in decrease of the vertical velocity and the volumetric strain around the still water level of the boundary between the breakwater and the beach. This result can be physical explanation of the fact that the installation of filter layers is a good countermeasure against the sand suction, which is well known by coastal engineers.

Acknowledgments The authors are grateful to Prof. K. Iwata, the Department of Civil Engineering, Chubu University, for valuable comments and helpful suggestions. Appreciation is also given to Mr. Y. Kuramitsu for his assistance in conducting the hydraulic experiments.

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