Response of a porous seabed to water waves over permeable submerged breakwaters with Bragg reflection

Ocean Engineering 43 (2012) 1–12

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Response of a porous seabed to water waves over permeable submerged breakwaters with Bragg reflection J.-S. Zhang a,b,c, D.-S. Jeng a,b,n, P.L.-F. Liu d, C. Zhang c, Y. Zhang a a

Center for Marine Geotechnical Engineering Research, Department of Civil Engineering, State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b Division of Civil Engineering, University of Dundee, DD1 4HN, UK c State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University 210098, China d School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA

a r t i c l e i n f o

abstract

Article history: Received 30 March 2011 Accepted 7 January 2012 Editor-in-Chief: A.I. Incecik Available online 31 January 2012

An integrated model is developed for the investigation of wave motion and seabed response around multiple permeable submerged breakwaters subject to different levels of Bragg reflection. In this study, the Volume-Averaged Reynolds-Averaged Navier–Stokes (VARANS) equations are used to describe the non-linear interaction between waves and permeable structures, while Biot’s ‘‘u  p’’ approximation theory is adopted for predicting the wave-induced seabed response. The numerical results show that the reflection coefficient is highly dependent on the wave period and the configuration/number of arrayed breakwaters. Wave motion and its induced seabed response (in terms of pore fluid pressure, vertical effective stress and liquefaction potential) around breakwaters can be largely changed due to Bragg reflection and energy dissipation of permeable structures. With the same incident wave conditions, the maximum liquefaction area decreases in size with an increasing soil permeability or degree of saturation. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Bragg reflection Seabed response Wave motion Permeable submerged breakwaters Mathematical model

1. Introduction Breakwaters have been commonly used for the protection of shorelines worldwide. In recent years, permeable submerged breakwaters have become increasingly attractive to coastal engineers and environmental researchers, as these structures have significant advantages in (i) reflecting efficiently incident wave energy, (ii) dissipating wave energy due to the flow friction within the porous media and (iii) reducing the impact that structures have on water quality, nearby ecosystem and also visual impacts (as the crowns of the structures are under the sea surface). Water waves propagating from offshore to near-shore zones will strongly interact with such marine structures, and part of the incident wave energy is reflected. The reflection by the multiple submerged structures can be amplified when the incident waves are twice as long as the structure spacing, which is called as Bragg reflection. This mechanism is due to the constructive interference of reflected waves from successive structure crests, and it plays a significant role in the evolution of surface waves and in the formulation of offshore ripples (Davies

n Corresponding author at: Center for Marine Geotechnical Engineering Research, Department of Civil Engineering, State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. E-mail address: [email protected] (D.-S. Jeng).

0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2012.01.024

and Heathershaw, 1984; Mei and Liu, 1993; Yu and Mei, 2000). The construction of such kind of marine structures may largely interact with ocean waves and seabed soil. As the seabed response (such as pore fluid pressure, effective stress of soil and liquefaction potential) is mainly dominated by the wave pressure and shear stress at the sea floor, it can be significantly affected by the Bragg reflection of incident waves. Numerous marine structures have been reported in the literature that their failure/ damage may be caused by the wave-induced seabed response, rather than from the construction deficiencies (Smith and Gordon, 1983; Lundgren et al., 1989). Therefore, to accurately estimate the stability of the multiple submerged breakwaters, the wave motion and seabed response due to Bragg reflection of water waves around such marine structures must be understood. Although the Bragg reflection of water waves over impermeable obstacles have been extensively studied (Davies and Heathershaw, 1984; Liu and Cho, 1993; Liu and Yue, 1998; Cho and Lee, 2000; Cho et al., 2001; Hsu and Wen, 2001; Hsu et al., 2007; Tang and Huang, 2008), the knowledge related to permeable structures is limited. Among these, Mase et al. (1995) developed a time-dependent wave equation to describe a wave propagating over permeable ripple beds taking into account of the effects of porous medium, based on two assumptions: (i) the mean water depth and the thickness of porous layer slowly varying compared to the wavelength of surface wave and

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J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

(ii) the spatial scale of ripples being the same as the wavelength of surface waves. Cho et al. (2004) experimentally and Jeon and Cho (2006) numerically investigated the strong reflection of regular waves over a train of submerged breakwaters, and both studies concluded that the reflection coefficients of permeable submerged breakwaters are less than those of impermeable breakwater as wave energy is additionally dissipated due to porous structures. Recently, Lan et al. (2009) developed an analytical solution based on linear wave theory and Biot’s poroelastic theory to predict the Bragg scattering of waves propagating over a series of poro-elastic submerged breakwaters. Their results indicated that the elasticity, permeability and the breakwater height of the series of poro-elastic submerged breakwaters have a significant impact on Bragg reflection. All aforementioned researches mainly have focused on the wave–structure interaction, and did not consider the wave-induced seabed response around the permeable submerged structures. Since the 1990s, numerous studies have been carried out to study the wave-induced seabed response around a permeable submerged breakwater. For example, Mizutani and Mostafa (1998) experimentally and numerically investigated the effects of wave parameters, breakwater dimensions and seabed properties on the dynamic pore pressure and shear stress of the seabed. The laboratory measurements and numerical results of Bierawski and Maeno (2003) showed that the flow motion and seabed response around a permeable breakwater were essentially different from those around the impermeable one. Treating a seabed as a permeable but rigid media, Hur et al. (2008) solved only one set of modified Navier–Stokes (NS) governing equations for the wave–seabed–structure interaction system. Their results indicated that the pore water pressure beneath the permeable submerged breakwater significantly changes when incident wave period varies. By applying an integrated PORO-WSSI II model, Zhang et al. (2011) concluded that the seabed response around the foundation of a permeable submerged breakwater is highly dependent on the process of wave propagation (resulting from the interaction between wave, permeable breakwater and porous seabed). However, all previous studies have investigated a single permeable breakwater. Study on the seabed response and liquefaction potential beneath multiple permeable breakwaters subject to strong Bragg reflection is still not available. In this study, a series of numerical simulations are carried out to study the Bragg reflection and seabed response due to water waves over multiple permeable submerged breakwaters. The PORO-WSSI II model proposed by Zhang et al. (2011) is further enhanced to include Biot’s (1956) dynamic theory. Biot’s dynamic theory known as ‘‘u2p’’ approximation rather than the consolidation theory is adopted for wave-induced seabed response. The influence of dynamic soil behavior on the seabed response has been reported in the literature for idealised 2-D engineering problems (Jeng and Cha, 2003; Ulker et al., 2009). The numerical code for the seabed component was based on DIANA-SWANDYNE II, which was initially designed for 2-D earthquake-induced liquefaction (Chan, 1995), and recently was extended to 3-D wave-induced seabed response (Jeng and Ou, 2010). Detailed information can be found in Chan (1995) and Jeng and Ou (2010). The theoretical formulations are briefly provided in Section 2. The details of numerical experiment are given in Section 3. The effects of number of breakwaters, wave period, soil permeability and degree of saturation on the wave motion and seabed response are discussed in this section. Finally, general conclusions are drawn in Section 4.

enhanced in this study. In this one-way weak coupling model, small poro-elastic deformations of seabed are considered and assumed not to affect the wave transformation above the seabed. Compared with impermeable breakwaters, the permeable breakwaters additionally dissipate wave energy due to the flow friction. To take into account the impact of permeable media on the wave– structure interactions, the Volume-Averaged Reynolds-Averaged Navier–Stokes (VARANS) model developed by Hsu and Liu (2002) is adopted. In the study of Zhang et al. (2011), the Biot’s consolidation equations with poro-elastic theory are solved for the seabed. The main new features of the proposed model are (i) the inclusion of the Biot’s dynamic ‘‘u2p’’ approximation rather than consolidation theory, in which the acceleration due to soil motion is included and (ii) the estimation of liquefaction potential based on the mean effective stress criterion of Ulker et al. (2010). At each time step of simulation, the wave pressure and shear stress calculated from the VARANS model will be imposed as the seabed surface boundary conditions of Biot’s poro-elastic ‘‘u2p’’ model. Only the governing equations and the specification of boundary conditions are provided here. For more details concerning the integrating procedure see Zhang et al. (2011). 2.1. VARANS governing equations The VARANS equations (derived by applying a volume-averaging operator over a representative elementary volume to the NS equations) for the wave motion can be expressed as (Hsu and Liu, 2002) @/u i S ¼ 0, @xi " /u j S @/u i S @/u i S 1 n @/PSf þ ¼  @t 1þ cA nð1 þcA Þ @xj rf @xi # 0 0 @/ui uj S 1 @/t ij S þ þ ng i  @xj rf @xj " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# /ui S að1nÞ2 bð1nÞ /u 1 S2 þ/u 2 S2 ,  þ 2d 1 þcA n2 d2 n 50 50

ð1Þ

ð2Þ

where ui is the flow velocity, xi is the Cartesian coordinate, t is the time, rf is water density, P is pressure, tij is the viscous stress tensor of the mean flow, gi is acceleration due to gravity, and n and d50 are the porosity and the equivalent mean diameter of the porous material. cA denotes the added mass coefficient, calculated as cA ¼ 0:34ðð1nÞ=nÞ. a ¼ 200 and b ¼ 1:1 are empirical coefficients associated with the linear and nonlinear drag force, respectively (Lin and Liu, 1999). The influence of turbulence fluctuation on the mean flow, denoted as /ui 0 uj 0 S, is obtained by solving the modified kE turbulence model where k is the kinetic energy and E is the dissipation rate of kinetic energy (Lin and Liu, 1998). The over-bar represents the ensemble average and the prime denotes turbulent fluctuations with respect to the ensemble mean. The ‘‘/S’’ stands for Darcy’s volume averaging operator and is defined as Z 1 /aS ¼ a dv, ð3Þ V Vf where V is the total averaging volume, and Vf is the portion of V that is occupied by the fluid.

2. Theoretical formulations

2.2. Biot’s u2p approximation theory

The model PORO-WSSI II (Zhang et al., 2011) for predicting wave–porous seabed–permeable structure interaction is further

Considering a homogeneous isotropic seabed, with the same permeability (K) in all directions, the Biot’s ‘‘u2p’’ approximation

J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

3

can be expressed as (Biot, 1956; Zienkiewicz et al., 1980)

r2 p

gw ns bs @p K

@t

þ

rf @2 Es K @t 2

¼

gw @Es K @t

,

ð4Þ

@s0x @txz @p @ 2 us þr 2 , þ ¼ @x @x @z @t

ð5Þ

@txz @s0z @p @2 ws þ þ þ rg ¼ , @z @x @z @t 2

ð6Þ

where p is the pore pressure; gw is the unit weight of pore water; ns is soil porosity; s0x , s0z and txz are the horizontal effective normal stress, vertical effective normal stress and shear stress, respectively. r is the average density of porous seabed ( ¼ rf ns þ rs ð1ns Þ, where rs is the solid density). In (4), the volume strain of soil matrix ðEs Þ and the compressibility of pore fluid ðbs Þ are defined as

Es ¼

@us @ws þ @x @z

and

bs ¼

1 1S þ , Kw Pwo

ð7Þ

in which us and ws are the horizontal and vertical soil displacements, Kw is the true modulus of elasticity of water (taken as 2  109 N=m2 ), Pwo is the absolute water pressure, and S is the degree of saturation. 2.3. Boundary conditions To solve the VARANS model, appropriate boundary conditions are required. For the mean flow field, no-slip boundary condition is imposed on the sea floor surface (ui ¼0), and the zero-stress condition is adopted on the mean free surface by neglecting the effect of air flow ðtij ¼ 0Þ. For the turbulence field, the log-law distribution of mean tangential velocity in the turbulent boundary layer is applied in the grid point next to sea floor, and the zero-gradient boundary conditions are imposed for both the turbulent kinetic energy k and its dissipation rate E on the free surface (@k=@~ n ¼ @E=@~ n ¼ 0, where ~ n is the unit normal on the free surface). Numerical sponge layer is used to damp wave energy and avoid wave reflection at two side boundaries. The evaluation of the wave-induced seabed response also requires appropriate boundary conditions. At the seabed surface, the vertical effective normal stress is assumed to vanish ðsz0 ¼ 0Þ, and pore pressure p and shear stress txz equal the wave pressure and shear stress obtained from the VARANS model. The bottom of porous seabed with finite thickness Hs is treated as impermeable and rigid, and zero displacement and no vertical flow occur in this bottom boundary ðus ¼ ws ¼ @p=@z ¼ 0Þ. As sponge layer method is used at two side boundaries of VARANS model, zero displacement and zerogradient of pore pressure can be adopted in the two vertical sides far away from the concerned region ðus ¼ ws ¼ @p=@x ¼ 0Þ.

3. Results and discussion 3.1. Numerical example configuration The trapezoidal permeable submerged breakwaters are considered in this study, as they are recommended for engineering applications providing a good balance of reflecting capacity and overall performance (Cho et al., 2004; Tang and Huang, 2008). A schematic sketch of trapezoidal shape of permeable submerged breakwaters is shown in Fig. 1. Up to three breakwaters (N ¼1, N ¼2 and N ¼3, N denotes the submerged breakwater number) are considered, and each one has bottom width wb, crown width wt and height h. Porosity n and equivalent mean diameter d50 are the characteristic parameters of permeable material of breakwaters. The distance between two adjacent breakwaters is defined as LS.

Fig. 1. A schematic sketch of trapezoidal shape of permeable submerged breakwaters

The origin of the Cartesian coordinate system is located at the left-bottom corner of the first breakwater (N ¼1). Regular waves with still water depth d, wave period T and wave height Hw are generated at the left-hand-side of breakwaters by using an internal wave-maker (Lin and Liu, 1999). The damping function is applied to absorb the wave energy at the outflow boundary. The seabed has a finite thickness Hs ¼ 10 m, and its soil properties are fixed as follows: soil porosity ns ¼ 0:3, soil permeability K ¼ 103 m=s, the degree of saturation S¼ 1.0, soil shear modulus G ¼ 107 N=m2 , Poisson’s ratio ms ¼ 1=3 and the unit weight of soil gs ¼ 2:65gw . A series of numerical simulations are carried out to model the laboratory experiments of Cho et al. (2004), and the simulated reflection coefficients are compared with the measurements. Then, the wave field and seabed response (in terms of pore fluid pressure and effective stress) due to Bragg reflection of water waves over permeable submerged breakwaters are numerically investigated. 3.2. Comparison with experiments (Cho et al., 2004) Cho et al. (2004) performed a series of laboratory experiments to investigate the strong reflection of regular water waves over a train of permeable submerged breakwaters in a wave flume of 1 m wide, 2 m deep and 56 m long. To numerically model their experiments, the parameters of numerical experiments are defined as follows: still water depth d¼0.8 m, wave height Hw ¼ 0:04 m, structure height h ¼ 0:5d ¼ 0:4 m, bottom width wb ¼ 2d ¼ 1:6 m, crown width wt ¼ 0:5d ¼ 0:4 m, adjacent distance Ls ¼ 2:5d ¼ 2 m, porosity n¼ 0.5 and equivalent mean diameter d50 ¼ 0:076 m. Periods of incident water waves are 1:14 r T r 3:73 s, leading to the relative wave numbers of 0:5 rkd r 2:5. It is noted that the experiments were conducted on an impermeable seabed (i.e., no seabed response is considered, Cho et al., 2004). Therefore, in this comparison, an impermeable seabed is used in the present numerical model. The computational domain covers 50 r x r50 m and 0 r z r 1:2 m. In the vertical direction, Dz ¼ 0:005 m is uniformly distributed. To better fit the shape of the trapezoidal breakwater, a non-uniform grid system is applied in x-direction. Dx ¼ 0:01 m is used in the breakwater region ð0 r x r8:8 mÞ, while Dx ¼ 0:05 m is used in other regions. The internal wave-maker is located at the cross-section x ¼ 20 m. The numerical sponge layer (having a length of twice the wavelength of the surface wave) is applied to reduce the wave reflection at the two side boundaries. When waves propagate into the sponge layer region, their velocities are gradually and artificially reduced by multiplying by an absorption function. To obtain computational stability, the time interval ðDtÞ is automatically adjusted at each time step to satisfy Cournat–Friedrichs–Lewy condition and the diffusive limit condition (Liu et al., 1999). Fig. 2 shows the comparison of the simulated reflection coefficients and experimental measurements. In this example, the reflection coefficient (Kr, a ratio of reflected wave height to incident wave height) is determined by the model of Goda and

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J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

Suzuki (1976). The simultaneous water elevation levels are taken at two adjacent locations placed in front of the submerged breakwaters, and the amplitudes of Fourier components are analyzed by the FFT technique. The amplitudes of incident and reflected wave components are estimated from the Fourier components. Overall, there is a good agreement between simulation and measurement. The results show that the reflection coefficient Kr is highly dependent on wave period (in term of

kd) and it reaches a maximum value when kd ¼0.6 in both cases: two breakwaters (m¼2) and three breakwaters (m¼3). It is noted that m represents the total numbers of breakwaters. The maximum reflection coefficient of three breakwaters (Kr ¼0.52, the average of measurement and simulation) is obviously higher than that of two breakwaters (Kr ¼0.32). This implies that more incident wave energy can be reflected back to the sea with the presence of one more breakwater. In other words, both wave period and the number of breakwaters (m) have a significant impact on the wave motion. As seabed response is mainly dominated by the wave motion, these two parameters may also affect the wave-induced seabed response. 3.3. Wave field The distributions of period-averaged velocity magnitude (unit: m/s) around permeable breakwaters with a wave period of T ¼ 3:16 s are plotted in Fig. 3. The figures clearly show that the wave-induced flow velocity decreases in the vertical direction with an increasing water depth. The flow velocities above the breakwaters are generally higher than those in the adjacent areas, as the existence of structure partly blocks the cross-section space through which the wave flow passes. Due to the Bragg reflection and energy dissipation of permeable breakwaters, the magnitude of velocity decreases largely at the back side of the structures. Compared to the case with two breakwaters (m¼2), the velocities in the case with three breakwaters (m¼ 3) are higher at the upper side of the first arrayed breakwater but smaller at the back of structures. This can be ascribed to the fact that a stronger Bragg reflection takes place and more wave energy is reflected back to the sea in the case with three breakwaters. 3.4. Pore fluid pressures

Fig. 2. Comparisons of computed reflection coefficients with experimental measurements (Cho et al., 2004) with different number of breakwaters: (a) two breakwaters m¼2 and (b) three breakwaters m¼ 3.

The pore fluid pressures in a porous seabed are mainly dominated by the wave pressures along the sea floor. A variation of wave motion consequently results in a change of pore fluid pressure. Figs. 4 and 5 display the distribution of wave-induced

1.2

0.1 0.07

0.06

0.08

04

0.03

0.03

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z/d

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0.

0.02

4

−10

−5

0

5 x/d

10

15

20

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−10

−5

0.06 0.04 0.02

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03 0.

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z/d

4

0.05

0.01

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0.0

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0

0

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1

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0.08

0.04

0.0

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0.07

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0

0.06 0.05

0.6

0.07

0.06

0.8

0.01

0.03

0.05

1

0

5 x/d

10

15

20

0

Fig. 3. Distribution of period-averaged velocity magnitude (unit: m/s) around breakwaters with a wave period of T ¼ 3.16 s: (a) two breakwaters m¼ 2 and (b) three breakwaters m¼3.

J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

pore fluid pressures p=ð0:5gw Hw Þ around breakwaters with a wave period of T¼3.16 s at different time levels. As a result of wave– seabed–structure interaction, the existence of structures has a significant impact on the distribution of wave-induced pore pressure. It is noted that the impact of wave motion on pore fluid pressure is negligible in the region below z=Hs o0:5. Fig. 6 gives the vertical distributions of maximum wave-induced pore pressures 9p9max =ð0:5gw Hw Þ versus z=Hs below the toe of breakwaters where the seabed liquefaction is most likely to take place. The results show that 9p9max =ð0:5gw Hw Þ at the cross-section x=d ¼ 0 (the toe of the first breakwater) is greater than those at other cross-sections. 9p9max =ð0:5gw Hw Þ 4 1 at the point, ðx=d ¼ 0, z=Hs ¼ 0Þ indicates that the height of wave crest is increased in front of the first breakwater due to the Bragg reflection. 9p9max =ð0:5gw Hw Þ o1 at other points, ðx=d ¼ 4:5, z=Hs ¼ 0Þ and ðx=d ¼ 9, z=Hs ¼ 0Þ shows that the transmitted wave energy can be significantly reduced by Bragg

5

reflection and friction dissipation of permeable breakwaters. As shown in Fig. 7, the values of 9p9max =ð0:5gw Hw Þ at points, ðx=d ¼ 0, z=Hs ¼ 0Þ and ðx=d ¼ 4:5, z=Hs ¼ 0Þ in the case with m¼3 are higher than those in the case with m¼ 2, and the difference is about 7% of ð0:5gw Hw Þ. The case with m¼3 is taken as an example to study the impact of the strength of Bragg reflection on wave-induced pore pressure. The source of the Bragg reflection is due to constructive interference of incident and reflected waves and is well known in x-ray diffraction by crystalline materials. When the wavelength of the surface wave becomes closer to the twice of wavelength of bottom undulation, the phenomenon of Bragg reflection becomes stronger (Mei et al., 2005). Three different values of wave period (T ¼3.16, 2.76 and 2.46 s), leading to three different reflection coefficients (Kr ¼0.50, 0.35 and 0.17), are considered here (see Fig. 8). The strength of Bragg reflection has an important impact

0.2 0.2 0.4 0.6 0.8

6 -0.

0

0.2

0

-0

-0.4

0

0

-0.2

0

0

-0.4

0.4

-0.2

0

-0.2

2 0.

2 0. 0

z/Hs

1

.4

-1 -0.8 -0.6 -0.4 -0.2 0

0

0

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0

5

10

15

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0

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0

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0

0

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0

0

0

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0.2

0

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-0.2

4 0.

.2 0

z/Hs

0.2 0.4 0.6 0.8

-0.2

-1 -0.8 -0.6 -0.4 -0.2 0

0

0

5

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15

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.2

-0.6

-0

0

0

.4

0

-0

-0 .4

0

z/Hs

1

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.2

4 .2 0. 0

-0

-0.2

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0.2

-1 -0.8 -0.6 -0.4 -0.2 0

0

0

-0.4

0

0

-5

0

0

5

10

15

x/d Fig. 4. Distribution of the wave-induced pore fluid pressure p=ð0:5gw Hw Þ around two breakwaters (m ¼2) with a wave period of T ¼3.16 s at different time levels: (a) t/T ¼0, (b) t/T ¼ 0.25, (c) t/T ¼ 0.5 and (d) t/T ¼0.75.

6

J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

0.2 -1 -0.8 -0.6 -0.4 -0.2 0

0.2

4 0.

-0

0.2

-0.2

.2

-0.4

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0

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0

0

1

0

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.6 -0

-0 .4

0

0

0

0

0

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0

5

10

15

x/d

0.2 0.4

0

.4

0

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.2

0

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-0

-0.2

-0

0

2 0.

1

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z/Hs

0.2 0.4 0.6 0.8

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0

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0

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.2

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0

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0

0

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2 0. 0

.2 -0

2 0.

-0.2

-0.6

0

0

0

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1

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.2

.2

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-0

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.2 0

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0

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1

-0

4 0. 0.2

4 -0 .

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-0

-1 -0.8 -0.6 -0.4 -0.2 0

0

-5

0

0

-0.4

0

5

10

15

x/d Fig. 5. Distribution of the wave-induced pore fluid pressure p=ð0:5gw Hw Þ around three breakwaters (m¼ 3) with a wave period of T¼3.16 s at different time levels: (a) t/T ¼0, (b) t/T ¼ 0.25, (c) t/T ¼ 0.5 and (d) t/T ¼ 0.75.

on the 9p9max =ð0:5gw Hw Þ in front of first breakwater. For instance, a stronger Bragg reflection (such as the case with wave period of T¼3.16 s) results in a larger value of 9p9max =ð0:5gw Hw Þ, implying more chance for the seabed liquefaction. This is because a higher wave crest takes place in front of structures when the wave reflection becomes stronger. However, the impact of Bragg reflection on 9p9max =ð0:5gw Hw Þ decreases at cross-section x=d ¼ 4:5 and becomes negligible at cross-section x=d ¼ 9. 3.5. Vertical effective stresses Figs. 9 and 10 show the distribution of vertical effective stress

s0z =ð0:5gw Hw Þ around the foundation. The wave period is T¼3.16 s in the example. It can be seen that the value of s0z =ð0:5gw Hw Þ below the breakwaters is different from those away from the structures, as the magnitude of s0z =ð0:5gw Hw Þ is increased due to

the weight of structures. Positive and negative values of

s0z =ð0:5gw Hw Þ within the porous seabed are induced by the wave trough and wave crest, respectively. The magnitudes of s0z =ð0:5gw Hw Þ increase at the toe of breakwaters. Comparison of Figs. 9 and 10 indicates that an additional breakwater (N ¼3) also has an impact on the s0z =ð0:5gw Hw Þ. Fig. 11 shows the vertical distribution of the maximum magnitude of vertical effective stress 9s0z 9max =ð0:5gw Hw Þ versus z=Hs below the toe of breakwaters. In the upper region of seabed ðz=Hs 40:4Þ, the values of s0z =ð0:5gw Hw Þ at cross-sections x=d ¼ 0 and x=d ¼ 4:5 when m¼3 are larger than those when m¼2. The resulted crest height in the case m ¼ 3 is larger than that in the case m ¼ 2, and it will cause higher wave pressure on the structure. Part of the wave pressure on marine structure will be passed into the soil skeleton and increase the vertical effective stress. Generally speaking, a change of configuration/number of arrayed

J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

7

Fig. 6. Vertical distribution of the maximum magnitude of wave-induced pore pressure 9p9max =ð0:5gw Hw Þ versus z=Hs below the toe of breakwaters: (a) two breakwaters m¼ 2 and (b) three breakwaters m¼ 3.

Fig. 8. Effect of strength of Bragg reflection on 9p9max =ð0:5gw Hw Þ at different crosssection in the case m¼ 3: (a) cross-section x/d¼ 0, (b) cross-section x/d ¼4.5 and (c) cross-section x/d ¼9.

vertical effective stress becomes zero (Okusa, 1985). Taking into account the initial vertical effective stress ðs0v0 Þ due to the weight of breakwaters, the liquefaction criterion of Okusa (1985) can be modified as

s0z þ Ds0v0 ¼ 0:

ð8Þ

The liquefaction criterion was further extended for a more general stress state by Tsai (1995) for a free field case and by Ulker et al. (2010) for a breakwater case. The mean effective stress liquefaction criterion modified by Ulker et al. (2010) is Fig. 7. Vertical distribution of the maximum magnitude of wave-induced pore pressure 9p9max =ð0:5gw Hw Þ versus z=Hs at different cross-section: (a) x/d ¼0 and (b) x/d ¼4.5.

breakwaters leads to a variation of the overall interaction of the whole wave–seabed–structure system resulting from the incoming wave, and consequently induces different seabed response. 3.6. Liquefaction potential The liquefaction is an extreme form of seabed instability, which may lead to a vertical movement of sediment. Seabed liquefaction may take place when the wave-induced effective stress is equal to the initial effective stress. To date, several criteria for liquefaction have been suggested. The first is the

1 0 3½ zð1 þ2K 0 Þ þ ð1þ

g

ms Þðs0x þ s0z Þ þ Ds0m0 ¼ 0,

ð9Þ

where K0 is the coefficient of lateral earth pressure at rest, and Ds0m0 is the initial mean effective stress due to the weight of breakwaters. In the previous cases with a laboratory small scale, no seabed liquefaction takes place based on the liquefaction criteria (8) or (9). In this study, a large scale case with a domain being 25 times in dimension of previous laboratory case is adopted. The wave conditions and seabed parameters are given as wave height Hw ¼ 5:0 m, wave period T¼12.5 s, still water depth d¼20.0 m, seabed thickness Hs ¼125 m, soil permeability K ¼ 104 m=s and the degree of saturation S¼0.98. The wave-induced maximum liquefaction areas around three breakwaters estimated by criteria (8) and (9) are given in Fig. 12. The pattern of maximum liquefaction potential estimated by criterion (8) is significantly different from that estimated by criterion (9). When mean

8

J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

0.2 0

0

0 .2 0 .2

.2

0 0

0.

2

0

0

-0.2

-0.2

-0.2

0

-0 .2

0.6 0

-0

0.4

0.2

z/Hs

0.2

0 0.2 0.2

-0.2

0

0

-0.6 -0.4 -0.2

0

0

-0.4 0

-5

0

5

10

15

x/d

0.2

00

-0.2

0. 0

0

0 .2

0

2

- 0 .2

0.4

0

-0.2

- 0 .2

0.6 0. 0 0 2

0

z/Hs

0 .2

0.4

0

2

0.2

0

-0.2

-0 .

0

0.4

-0.6 -0.4 -0.2

0

-0.4

.2

0

-0

0

-5

0

5

10

15

x/d

0.2 0

0.2

0.4

0.6 0

0

0 0

-0 .

0

0 .2

.2

-0

0.2

.2

2

-0

0 .2

0

-0.2

0

2

0 .2

2

-0 .

0 .2

0

0.

z/Hs

00 0

0

-0 .2

-0 .2

-0.6 -0.4 -0.2

0

0 0

0

0

-0.4 -5

0

5

10

15

x/d

0.2

2

0

0

0

-5

0

0

0.

-0 .2

0

-0.2

0.2

-0.2

0 .2

-0.4

0.2

0

-0

000 .2

.4

-0.2

0.6 -0 . 2

0

0.4

0

0.2

0 .2

-0

0.2

0

z/Hs

0

.4

-0.6 -0.4 -0.2

0

0

0

5

10

15

x/d Fig. 9. Distribution of vertical effective stress s0z =ð0:5gw Hw Þ around two breakwaters (m ¼2) with a wave period of T¼ 3.16 s at different time levels: (a) t/T ¼ 0, (b) t/T ¼ 0.25, (c) t/T ¼0.5 and (d) t/T ¼0.75.

effective stress criterion (9) is considered, no obvious liquefaction takes place in the gaps between breakwaters. A maximum liquefaction depth (1.4 m) in front of first breakwater from criterion (9) is less than that (1.5 m) from criterion (8). As discussed by Ulker et al. (2010), the mean effective stress criterion (9) is the most fundamental and appropriate one for the coupled flow and deformation analysis. This criterion is used in studying the effects of the number of breakwater, wave period, soil permeability and degree of saturation on the liquefaction potential. The wave-induced maximum liquefaction areas around different numbers of breakwater are given in Fig. 13. A liquefaction depth (1.52 m) in the case m ¼2 is higher than that (1.4 m) in the case m¼3. The construction of the third breakwater also has a significant impact on the distribution of maximum liquefaction depth within the range of x=d o8:0.

To examine the influence of wave period, the case with three breakwaters is used. In the example, the wave period varies from 12.5 s to 15.0 s. As shown in Fig. 14, the maximum liquefaction area induced by the waves with T¼ 15.0 s significantly differs from that induced by the waves with T¼12.5 s. The soil around the left sub-corner of first breakwater liquefies with a potential depth 1.26 m when the incident wave has a wave period T¼15.0 s, indicating the first breakwater is likely to be damaged due to foundation settlement. Two different types of non-cohesive soils in term of permeability K are examined here: K ¼ 102 m=s for coarse sand, and K ¼ 104 m=s for fine sand. It can be clearly seen from Fig. 15 that the maximum liquefied areas significantly decrease in size with an increasing soil permeability. The degree of saturation is one of the important factors which plays an important role in the

J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

9

0.2 0 .2

0

0.

2

0

0 .2

-0 .2

-0 .2

0.2

-0 .2

0

0.2

0

0

0 .2 .4 00

0

0 0

0.6

0

z/Hs

0.4

0 .2

-0.2 -0.2

-0.2

0.2

0

0

0 0

-0.6 -0.4 -0.2

0

0

0

-0.4 0

0

0

-5

5

10

15

x/d

0.2 0.4

0 0.2.2 0. 2

0

2 0.

0

0

0

-0 .2

0.2

z/Hs

0

.4 0.2 0 0 .2 0 .2

-0.2

0 .4

-0.2

-0.2

0

- 0 .2

0.6 0 0 .2

.2

0.2

-0

0

0

-0.6 -0.4 -0.2

0

- 0 .2

0

0

0.2

-0.4 0

0

-5

0

5

10

15

x/d

0.2

0.4

0.6

-0.2

0

0 0

0

0 .2 0

z/Hs

.2

0 -0. 4

-0.4 -5

0.2

-0

0

-0 .2

0 .2

0

0

0.2

0 .2

0 0 0

0

0.2

-0.2

-0 .4

0

-0 .2

-0.6 -0.4 -0.2

0

0

0.2

0

5

10

15

x/d

.2

2

0.2

0

-0

0

0

0

0 .2

- 0 .2

0

0

-0 -0.4 .6

0 .2

0

0.2

0

-0 .2

0 0

0.2

-0.2

0

0

-0.4

0.6 00

-0 .

0.4

-0 .2

0.2

0

0

-0 .2

z/Hs

0

0. 2

-0.6 -0.4 -0.2

0

-0.6

0.2

-5

0

5

10

15

x/d Fig. 10. Distribution of vertical effective stress s0z =ð0:5gw Hw Þ around three breakwaters (m¼ 3) with a wave period of T¼3.16 s at different time levels: (a) t/T ¼0, (b) t/T ¼ 0.25, (c) t/T ¼ 0.5 and (d) t/T ¼0.75.

wave-induced maximum liquefaction decreases, which can be seen in Fig. 16.

4. Conclusions

Fig. 11. Vertical distribution of the maximum magnitude of vertical effective stress 9s0z 9max =ð0:5gw Hw Þ versus z=Hs below the toe of the breakwaters.

In this paper, based on the VARANS equations and Biot’s ‘‘u2p’’ approximation theory, an integrated model has been developed for the investigation of wave motion and seabed response around multiple permeable submerged breakwaters subjected to different levels of Bragg reflection. Based on the numerical results, the following conclusions can be drawn:

evaluation of the wave-induced liquefaction potential. In the example, the degree of saturation is considered as S ¼0.998, 0.99 and 0.98. As the marine soil gets more saturated,

1. The reflection coefficient Kr is highly dependent on the wave period (in term of kd), and it reaches a maximum value when kd¼0.6. The maximum Kr ¼0.52 of three breakwaters is obviously

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J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

z/d

0.4 0.2 0 -0.2 -5

0

5 x/d

10

15

0

5 x/d

10

15

z/d

0.4 0.2 0 -0.2 -5

Fig. 12. Distribution of the wave-induced maximum liquefaction areas around three breakwaters estimated by: (a) vertical effective stress criterion and (b) mean effective stress criterion.

z/d

0.4 0.2 0 -0.2 -5

0

5 x/d

10

15

0

5 x/d

10

15

z/d

0.4 0.2 0 -0.2 -5

Fig. 13. Effect of the number of breakwaters on wave-induced maximum liquefaction areas: (a) two breakwaters m¼ 2 and (b) three breakwaters m¼ 3.

z/d

0.4 0.2 0 -0.2 -5

0

5 x/d

10

15

0

5 x/d

10

15

z/d

0.4 0.2 0 -0.2 -5

Fig. 14. Effect of wave period on wave-induced maximum liquefaction areas: (a) T¼12.5 s, (b) T¼ 15.0 s.

J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

11

z/d

0.4 0.2 0 -0.2 -5

0

5 x/d

10

15

0

5 x/d

10

15

z/d

0.4 0.2 0 -0.2 -5

Fig. 15. Effect of soil permeability on wave-induced maximum liquefaction areas: (a) K ¼ 104 m=s and (b) K ¼ 102 m=s.

z/d

0.4 0.2 0 -0.2 -5

0

5 x/d

10

15

0

5 x/d

10

15

0

5 x/d

10

15

z/d

0.4 0.2 0 -0.2 -5

z/d

0.4 0.2 0 -0.2 -5

Fig. 16. Effect of degree of saturation on wave-induced maximum liquefaction areas: (a) S ¼0.98, (b) S¼ 0.99, (c) S ¼0.998.

higher than that (Kr ¼ 0.32) of two breakwaters, indicating more incident wave energy can be reflected with the presence of the additional breakwater. 2. Due to the Bragg reflection and the energy dissipation of permeable submerged breakwaters, the magnitude of flow velocity decreases largely at the back side of the structures. Compared to the case with two breakwaters, the velocities in the case with three breakwaters are higher at the upper side of

the first arrayed breakwater but smaller at the back side of structures. 3. At the toe of the first breakwater (cross-section x=d ¼ 0), the maximum wave-induced pore pressure p=ð0:5gw Hw Þ in the case with three breakwaters is higher than that in the case with two breakwaters, and the difference is about 7% of ð0:5gw Hw Þ. A stronger Bragg reflection results in a larger value of p=ð0:5gw Hw Þ at cross-section x=d ¼ 0. However, the impact

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J.-S. Zhang et al. / Ocean Engineering 43 (2012) 1–12

of Bragg reflection on 9p9max =ð0:5gw Hw Þ gets less at the crosssection x=d ¼ 4:5 and becomes negligible at the cross-section x=d ¼ 9. 4. The construction of breakwaters largely affect the distribution of vertical effective stress s0z =ð0:5gw Hw Þ around the foundation. Positive and negative values of s0z =ð0:5gw Hw Þ within the porous seabed are induced by the wave trough and wave crest, respectively. In the upper region of the seabed (z=Hs 40:4), the values of s0z =ð0:5gw Hw Þ at cross-sections x=d ¼ 0 and x=d ¼ 4:5 in case m¼3 are larger than those in case m ¼2. 5. A serial of large scale tests are carried out to investigate the effect of number of breakwater on wave-induced maximum liquefaction potential. The maximum liquefaction area estimated by mean effective stress criterion is smaller than that estimated by vertical effective stress. The results show the maximum liquefaction depths in the case with m¼2 are relatively higher than those in the case with m¼3. 6. The pattern of maximum liquefaction potential in front of first breakwater significantly changes when the incident wave period varies. With the same incident wave conditions, the maximum liquefaction area decreases in size with an increasing soil permeability or degree of saturation.

Acknowledgments The authors are grateful for the support from EPSRC Grant #EP/ G006482/1, State Key Laboratory of Ocean Engineering Self-Development Grant #GKZD010053 (2011), Scotland-China Exchange Program (2010-2011), SciChuan University State Key Laboratory of Hydraulics and Mountain River Engineering Open Fund Scheme #SKLH-OF-1005 (China), NSFC Grant #41176073 (China) and Hohai University State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering Open Fund Scheme #2011491311 (China).

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