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Numerical simulation of wave damping over porous seabeds
Coastal Engineering 53 (2006) 845 – 855 www.elsevier.com/locate/coastaleng
Numerical simulation of wave damping over porous seabeds S.A.S.A. Karunarathna a , Pengzhi Lin a,b,⁎ a
b
Department of Civil Engineering, National University of Singapore, 117576, Singapore State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, 610065, China Received 14 October 2005; received in revised form 23 February 2006; accepted 1 May 2006 Available online 21 June 2006
Abstract This paper presents a study of wave damping over porous seabeds by using a two-dimensional numerical model. In this model, the flow outside of porous media is described by the Reynolds Averaged Navier–Stokes equations. The spatially averaged Navier–Stokes equations, in which the presence of porous media is considered by including additional inertia and nonlinear friction forces, is derived and implemented for the porous flow. Unlike the earlier models, the present model explicitly represents the flow resistance dependency on Reynolds number in order to cover wider ranges of porous flows. The numerical model is validated against available theories and experimental data. The comparison between the numerical results and the theoretical results indicates that the omission or linearization of the nonlinear resistance terms in porous flow models, which is the common practice in most of analytical models, can lead to significant errors in estimating wave damping rate. The present numerical model is used to simulate nonlinear wave interaction with porous seabeds and it is found that the numerical results compare well with the experimental data for different wave nonlinearity. The additional numerical tests are also conducted to study the effects of wavelength, seabed thickness and Reynolds number on wave damping. © 2006 Elsevier B.V. All rights reserved. Keywords: Numerical modeling; Wave damping; Porous seabeds; Nonlinear frictional force; Nonlinear wave; Reynolds number
1. Introduction Natural seabeds are usually composed of permeable beds that enable mass and momentum transfer across the interface. Rigid porous seabeds allow the fluid to transfer through the body without deforming the soil structure. In non-rigid seabeds soil skeletons are deformed as the waves interact with the beds. When waves propagate over a porous seabed the water–seabed interface will experience a positive dynamic pressure under a wave crest and negative dynamic pressure under a wave trough. The pressure gradient within the interface results in wave induced flow through the pores inside the porous seabed. As the flow propagates through the porous media, resistances will act on the flow fluid due to fluid viscosity and the interaction of pore fluid with soil particles. Furthermore, the inertia force due to the unsteady motion of the pore fluid also becomes significant under certain combination of wave and seabed conditions. Wave energy dissipation due to flow resis⁎ Corresponding author. Department of Civil Engineering, National University of Singapore, 117576, Singapore. Tel.: +65 65161314; fax: +65 67791635. E-mail address: [email protected] (P. Lin). 0378-3839/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2006.05.003
tances inside the porous bed results in the decrease of wave heights as the waves propagate over. This phenomenon is known as wave damping. Numerous experimental, theoretical and numerical studies have been performed to study wave damping over porous seabeds. Wave damping over rigid porous seabeds has been studied experimentally by several researchers (e.g., Savage, 1953; Sawaragi and Deguchi, 1992). Savage (1953) conducted laboratory experiments to find out wave energy losses and wave height attenuation by bottom friction and percolation. His experiment revealed that the energy losses by percolation are insignificant for sands of low permeability and they are significant for coarse sands. An experimental study on solitary wave propagating over smooth and rough bottoms was conducted by Özhan and Shi-Igai (1977). They found that for both smooth and rough regimes, the friction factor depends on the instantaneous wave height. Gu and Wang (1991) conducted laboratory experiments for standing waves oscillating over a porous bottom and the experiments revealed that the boundary layer at the interface is an important issue in wave damping prediction. On the other hand, Sawaragi and Deguchi's (1992) experiment of periodic wave damping over porous seabeds
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showed that the effects of boundary shear on wave damping are negligible when compared to the effects of permeability. Besides experimental studies, a lot of theoretical studies have been performed to study wave damping over rigid porous seabeds. In the earliest theoretical investigations, theoretical expressions for wave energy dissipation due to individual effects of bottom friction (e.g., Putnam and Johson, 1949) and percolation (e.g., Putnam, 1949) were derived by considering that the energy dissipation is caused due to the wave induced pressure fluctuation at the interface. Putnam (1949) in his derivation assumed that the flow within the permeable seabed satisfies the unsteady Darcy's law. Wave damping over porous seabeds of infinite depth has been studied by solving the boundary value problem by a number of researchers (e.g., Hunt, 1959; Murray, 1965; Liu, 1973). These studies were limited by the fact that they were based on small amplitude wave theories with viscous laminar flow in porous media. For porous seabeds with finite thickness, Liu and Dalrymple (1984) obtained a complex wave dispersion relationship by applying Dagan's (1979) porous flow model to describe the flow through the porous media. In their wave dispersion relationship, wave number was a complex number in which the real part represents the change of wavelength whereas the imaginary part represents the damping of wave height. They found that wave damping reaches the maximum at a certain permeability of the soil. The same problem was investigated by Gu and Wang (1991) but including the effect of the nonlinear frictional resistance to the porous flow. However, the linearization of the nonlinear terms was made to simplify the mathematics. Thus, the contribution of the nonlinear term on wave energy dissipation was not completely included in their model, which will be further discussed later. Direct numerical modeling of wave interaction with seabeds has the advantage of including irregular seabed geometry, inhomogeneous porous media, nonlinear waves and nonlinear frictional force. However, probably due to the difficulty of coupling turbulent flow models inside and outside of the porous beds, only limited number of such study was reported before. Several numerical models based on mild-slope equation (e.g., Rojanakamthorn et al., 1990), Boussinesq equations (e.g., Cruz et al., 1997) and Navier–Stokes equations (e.g., Huang et al., 2003) have been developed to study wave interaction with porous submerged breakwaters. Cruz et al. (1997) derived a set of Boussinesq-type equations to model wave propagation on a porous bed. The weak dispersive nature of the equations was corrected by adding dispersion terms and the equations were solved numerically. Recently,
Chang (2004) studied propagation of periodic and solitary waves over porous seabeds by solving two-dimensional Navier–Stokes equations for the flow outside the porous media together with adapted Navier–Stokes equations for the flow inside the porous media. In his study, the flow outside the porous seabed was assumed to be laminar. The laminar flow assumption neglects the turbulent shear stress and thus it may cause significant errors, especially when the waves are highly nonlinear. In this paper, wave damping over rigid finite porous seabeds (Fig. 1) is studied by using a numerical model extended from an earlier model developed by Liu et al. (1999). In the numerical model, the mean flow outside the porous media is described by Reynolds Averaged Navier–Stokes (RANS) equations in which the turbulence field is modeled by k–ε equations. For the flow motion inside the porous media, a new porous flow model will be derived by spatially averaging Navier–Stokes equations while the effect of the porous media is treated by the additional inertia and drag forces. These forces are modeled according to Morison's equation assuming porous media is composed of equivalent spherical particles. The derived porous flow model is incorporated into the numerical model. This numerical model will be validated against various theories and experimental data for wave damping over different porous seabeds. The model is used to study the effects of wave nonlinearity, wavelength, seabed thickness and Reynolds number on wave damping. 2. Governing equations in the numerical model In Liu et al.'s (1999) numerical model, for the domain outside the porous media, the mean flow is described by the RANS equations in which the effects of turbulence field are modeled by the k–ε model. The flow inside the porous media is described by spatially averaged Navier–Stokes equations while the resistance from the porous media is modeled by a combination of linear and nonlinear frictional forces, which are expressed by empirical formulae suggested by Van Gent (1995). The details of governing equations and numerical implementations can be found in Liu and Lin (1997), Lin and Liu (1998) and Liu et al. (1999) and they will not be reiterated here. Based on a series of numerical tests conducted for different wave and porous media interactions, it is found that the coefficients associated with the empirical formulae for frictional forces need to be adjusted according to Reynolds number (Re) of the flow in order to reproduce correctly the experimental results for a wide
Fig. 1. Definition sketch of wave damping over a porous seabed.
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range of porous materials. Therefore, in this study, a new porous flow model is proposed to include Re effects into the frictional forces. For this purpose, the spatially averaged Navier–Stokes equations are re-derived. Taking the spatial average of the original Navier–Stokes equation in the fluid domain, which can be subsequently transformed in the domain of entire porous media (e.g., see Su and Lin, 2005), we have: Aui ¼0 Axi
ð1Þ
1 Aui uj Aui 1 Ap0 m A2 ui 1 Aui Wuj W þ 2 ¼− þ − fi − 2 n At n Axj q Axi n Axj Axj n Axj
ð2Þ
In Eqs. (1) and (2), t = time, ρ and ν = density and kinematic viscosity of fluid, and n = porosity. i, j = 1, 2 represent the horizontal and vertical direction, respectively and the over-bar denotes the spatially averaged quantities and the double prime denotes the spatially fluctuated quantities. The variables x, u, p0 and f are direction, velocity, effective pressure (= p + ρgixi, where p is the pressure and g is the gravitational acceleration) and resistance force, respectively. In Eq. (2), the force fIi is caused by the presence of porous material in flow and it includes inertia and drag forces i.e., fIi ¼ fIi þ fDi . The inertia force represents the additional momentum required to accelerate water in the porous medium that is known as added mass phenomenon. By assuming that the porous medium is composed of uniform spherical particles and summing up the contribution from each spherical particle and taking the average, the inertia force term can be written as: fIi ¼
gp ð1−nÞ Aui At n2
ð3Þ
where γp is the virtual mass coefficient and the suggested value by Van Gent (1995) is γp = 0.34. The drag force fDi can be derived in the similar way as: 3 ð1−nÞ 1 fDi ¼ CD 3 uc ui 4 n d50
ð4Þ
where d50 is the mean diameter of sediments, uc p is ffiffiffiffiffiffiffiffi theffi characteristic velocity which can be estimated by uc ¼ ui ui and CD is the drag force coefficient. It is well known that for a single spherical particle, the drag coefficient CDS is the function of Reynolds number Re ¼ uc md50, which can be approximated by the formula (Fair et al., 1968): CDS ¼
24:0 3:0 þ pffiffiffiffiffiffi þ 0:34 Re Re
ð5Þ
In this study, CD is expressed similarly to CDS in terms of Reynolds number. However, two coefficients c1 and c2 are introduced in order to account for the effects of porosity, roughness, shape factor, and etc. on the change of drag coefficient. The final form of CD reads: 24:0 3:0 7:5 þ c2 pffiffiffiffiffi þ 0:34 1 þ CD ¼ c1 Re KC Re
ð6Þ
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In Eq. (6), KC ¼ Undmax50T is the Keulegan–Carpenter number where Umax = maximum water particle velocity and T = wave period and it is included to account for the effects of oscillatory flows on the change of nonlinear frictional force inside the porous medium. The derived porous flow model is implemented into Liu et al.'s (1999) numerical model neglecting the last term on the right hand side of Eq. (2) with the assumption of negligible mean turbulence shear effect inside the porous media. The coefficients in Eq. (6) are determined to be c1 = 7.0 and c2 = 2.0 from a series of numerical calibrations against available experimental data for wave interaction with porous structures and porous seabeds. The above coefficient values will be used throughout this study except in Section 3 where comparisons are made to different theoretical works, in which the respective coefficient values suggested in the original theoretical papers will be used in the numerical modeling for consistency. 3. Model validation against theories The numerical model will be first validated against the theoretical models derived by Liu and Dalrymple (1984) and Gu and Wang (1991) for wave damping over porous seabeds. In Liu and Dalrymple's model, the nonlinear resistance term is not included for the porous flow model whereas in Gu and Wang's model the nonlinear resistance term is approximated by an “equivalent” linear term. In contrast, in the numerical model the fully nonlinear porous flow model is included. In Sections 3.1 and 3.2, we shall briefly discuss the formulations of Liu and Dalrymple's and Gu and Wang's theoretical models, respectively. In order to validate the numerical model, these theoretical models are incorporated into the numerical model and the numerical results are compared to the corresponding theoretical results. For this comparison, empirical expressions for linear, nonlinear and inertial resistance terms and associated coefficients in each model are set to be consistent with those used by Liu and Dalrymple (1984) and Gu and Wang (1991). 3.1. Liu and Dalrymple's theory (1984) Theoretical expression derived by Liu and Dalrymple (1984) was based on Dagan's (1979) porous flow model, which is a generalized form of Darcy's law that can be applied for non-uniform flows. In addition, an acceleration term was also introduced into the porous flow model in order to represent the oscillatory flow motion inside the porous bed. However, the effect due to added mass was not considered. In the derivation, free surface profile was assumed to vary with distance and time as η(x,t) = aeiðk w x−σtÞ where a is the wave amplitude and σ is the angular frequency. Here, the wave number kw is a complex variable, which can be written as kw = kr + iki, in which kr represents the change of wavelength and ki represents the damping of wave height. With these conditions the homogeneous solution of the boundary value problem has led to a complex wave dispersion relationship as: r2 −gkw tanh kw h i tanh kw dðgkw −r2 tanh kw hÞ ¼− ð1=R−ibÞ
ð7Þ
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Fig. 2. Validation of the numerical model against Liu and Dalrymple's (1984) theory and Gu and Wang's (1991) theory.
where h is the water depth, d is the seabed thickness, R ¼ rK m is the k m permeability parameter with K ¼ gp ¼ specific permeability in which kp = permeability of porous material, and β = 1 / n. The first and second terms in the denominator of the right hand side of Eq. (7) represent the linear and inertia frictional resistances, respectively. In order to calculate wave damping rate ki, expression (7) needs to be solved iteratively. To simulate Liu and Dalrymple's theory, we set c1 ¼ a0 nð1−nÞ2 in which the coefficient a0 = 570, c2 = 0 and γp = 0. 18 The corresponding wave and porous media parameters are: wave period T = 5 s, initial wave height Hi = 0.5 m, water depth h = 5 m, porous bed thickness d = 5 m and porosity n = 0.4. For the numerical tests, wave damping for a distance of x = 300 m over different permeable seabeds with different permeability parameters R are numerically simulated by changing diameter of sediments. The corresponding diameter for each permeability parameter is obtained by using the expression for specific permeability defined by Gu and Wang 2 2 (1991) i.e., K ¼ n d 50 3. a0 ð1−nÞ According to Fig. 2, numerical results and analytical results of wave decay coefficients agree well. This ensures that the numerical model can correctly represent the theoretical model by Liu and Dalrymple (1984) when the nonlinear term and the added mass effect in the inertial term are neglected in the numerical model. In order to view the effect of added mass in the inertial term, we also plot the theoretical results from Liu and Dalrymple (1984) by including added mass effect, which will change the coefficient in the inertia resistance to be b ¼ ba ¼ nþgpnð1−nÞ . According to the 2 figure, when the term that represents the added mass is included into the Liu and Dalrymple's model (1984), the damping rates are reduced compared to the damping rates from their original model. The permeability parameter that gives the peak-damping rate is also slightly changed.
3.2. Gu and Wang's (1991) theory The complex wave dispersion relationship derived by Gu and Wang (1991) includes the linear and nonlinear resistance terms for the unsteady porous flow model. In order to include energy dissipation due to nonlinear frictional resistance in their theoretical model, the energy dissipation within a true nonlinear system was equated to an equivalent linearized system. Then, the complex wave dispersion relationship was obtained as: r2 −gkw tanh kw h ¼ −
i tanh kw dðgkw −r2 tanh kw hÞ f0
ð8Þ
Here, both wave number kw and the linearized friction coefficient f0 are complex variables. The relationship for f0 is given by: f0 ¼
1 Cd jkw Dðkw ; f0 Þsinh kw dj −iba þ R rjrf0 j
ð9Þ
in which Dðkw ; f0 Þ ¼
ga cosh kw h cosh kw d ½1−i=f0 tanh kw h tanh kw d ð10Þ
where ā is the average of ae−ki x over one wavelength. In Eq. (9), the terms on the right hand side represent the linear, inertia and nonlinear resistances, respectively. The parameter Cd relates to 1 porosity n and sediment mean diameter d50 by Cd r ¼ b0 1−n , ffiffiffi n3 d50 C which has been approximated as Cd ¼pCffiffiKf ffi ¼ pffiffif ffi rm where R b0 Cf ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. The coefficients a0, b0 and γp takes the values 2 n
a0 ð1−nÞ
of 570, 3.0 and 0.46, respectively. In order to calculate wave damping rate ki, expressions (8) and (9) need to be solved iteratively.
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In order to solve the theoretical model by Gu and Wang (1991), the porous flow equation ( Eq. (8)) is implemented into the numerical model. When Eq. (8) was derived, the pore pressure equation has been linearized while keeping the inertia resistance term together with the nonlinear resistance term. In order to implement it into the numerical model, we have changed its form by replacing f0 with fl − iβa where fl is an unknown complex coefficient which is given by:
fl ¼
1 Cd jkw Dðkw ; fl Þsinh kw dj þ R rjrðfl −iba Þj
ð11Þ
where
Dðkw ; fl Þ ¼
ga cosh kw h cosh kw d ½1−i=ðfl −iba Þtanh kw h tanh kw d ð12Þ
This wave damping model can be equivalently implemented into the numerical model when the coefficients in the numerical rfl n3 2 model are given by c1 ¼ 18m 1−n d 50 and c2 = 0 and γp = 0.46. In Fig. 2, wave damping results from Gu and Wang's (1991) theory, the numerical model with linearized pore pressure equation, and the fully nonlinear numerical model are shown. In the case of fully nonlinear model simulation, in order that the coefficients are consistent with the corresponding coefficients in Gu and Wang's model, the coefficients in the numerical model are 2 ffiffiffiffi þ 1:02 , neglecting KC effect and set to c1 ¼a0 nð1−nÞ ; c2 ¼ 4b0 = p9:0 Re 18 γp = 0.46. According to Fig. 2, when the linearized pore pressure equation is implemented into the numerical model, a good agreement between the numerical results and Gu and Wang's (1991) results is obtained. However, the wave damping results computed by using the fully nonlinear numerical model are different. It is clear that the reason for this difference is caused by the linearization of the nonlinear friction term in Gu and Wang's treatment. In other words, the linearization will not provide the exactly equivalent energy dissipation for a full range of permeability of porous bed.
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4. Model verification by experimental data In this section, the numerical model with the coefficient values of c1 = 7.0, c2 = 2.0 and γp = 0.34 will be further verified against the experimental data by Savage (1953) and Sawaragi and Deguchi (1992) for wave damping over porous seabeds. It is noted that Savage's (1953) experimental data represent small permeability cases whereas Sawaragi and Deguchi's (1992) experimental data represent large permeability cases. 4.1. Savage's (1953) experimental data Savage (1953) conducted an experiment in a laboratory flume of length 29.3 m, width 0.46 m and depth 0.61 m. The porous bed was 18.3 m long and 0.30 m thick. Water depths above the seabed were 0.229 m, 0.152 m and 0.102 m. He divided the flume into two parts and one side was used as a smooth side in which bottom friction and percolation would not exist. The other side was used to conduct wave damping tests in which the bottom consisted with sediment particles. Waves were generated by a wave generator, which produced waves with the period ranged from 0.5 s to 5 s. Wave heights were measured using wave gauges located at 0.61 m or 1.22 m intervals. Several runs were performed to measure wave height attenuation due to bottom friction and percolation. In order to eliminate the energy loss due to side wall friction, the energy loss in the sand side was corrected according to the loss in the smooth side. For the purpose of further verification of the numerical model, decay of wave heights along the bed in Run No. 1 of Savage's (1953) experiment is selected (Fig. 6 in Savage, 1953). For this run wave period T = 1.27 s, wave height Hi = 0.054 m, water depth h = 0.229 m, seabed thickness d = 0.30 m and the porous media had a mean diameter d50 = 3.82 mm and specific permeability K = 4.49 × 10− 9 m2. Since the porosity was not given in the paper we decided to use n = 0.3, which has been used by both Liu and Dalrymple (1984) and Gu and Wang (1991) for the same experiment. Numerically simulated free surface profile at t = 25 T is shown in Fig. 3. Corresponding dimensionless wave heights along the direction of wave propagation that has been non-dimensionalized with wavelength λ is compared to the experimental data in Fig. 4. Savage (1953) corrected his experimental wave damping
Fig. 3. Simulation results of free surface profile at t = 25 T for the Run No. 1 of Savage's (1953) experiments.
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Fig. 4. Comparison of numerical results for wave damping along the porous bed with the corresponding experimental results (Run No. 1 of Savage, 1953) and theoretical results (Liu and Dalrymple, 1984; Gu and Wang, 1991).
results by eliminating the side wall effects. For this experimental condition, wave damping results from the theoretical models by Liu and Dalrymple (1984) and Gu and Wang (1991) are also shown. According to the figure, the numerical result has very good agreement with the corrected experimental results. However, both theoretical models have underestimated wave damping slightly. According to Savage's (1953) experimental results, waves having different instantaneous wave heights provide different wave damping rates when the other parameters i.e., wave period, water depth, seabed thickness and sediment diameters are the same (Table 4 in Savage, 1953). Some of these experimental runs are numerically simulated to study the effects of wave nonlinearity on wave damping. Table 1 provides the comparison of the numerical simulation results to the experimental (Savage, 1953) and the theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) results.
k −k
In Table 1, the relative error is defined by D% ¼ i im 100% where kim kim is the measured wave damping rate. It is shown that the numerical results are in better overall agreement with the experimental results than the theoretical models. The differences between the numerical results and theoretical results, however, are not very significant because all these cases have small permeability (e.g., log R ≈ −2) where the wave decay rates predicted by various methods are close (refer to Fig. 2). Fig. 5 shows the variation of dimensionless wave damping rate with the increase of wave nonlinearity in different water depths. It is observed that with the increase of wave nonlinearity wave damping remains almost a constant for large water depth. However, when water depth becomes smaller, wave nonlinearity plays an important role such that wave damping is increased with the increase of wave nonlinearity, which is
Table 1 Comparison of the numerical wave damping results to the experimental and theoretical wave damping results for waves with different wave heights Run no.
3 4 12 14 21 22 23 5 7 15 16 17 24
T (s)
1.27
h (m)
0.229
0.152
1.00
0.229
0.152
Hi/h
0.299 0.111 0.388 0.114 0.418 0.167 0.401 0.388 0.133 0.333 0.277 0.117 0.167
ki/(σ2/g) Experimental results (Savage, 1953)
Present numerical results (ΔP%)
Gu and Wang's results (1991) (ΔGW%)
Liu and Dalrymple's results (1984) (ΔLD%)
0.0131 0.0143 0.0148 0.0164 0.0255 0.0190 0.0263 0.0122 0.0119 0.0106 0.0117 0.0112 0.0135
0.0133 (+1.5) 0.0139 (− 2.8) 0.0138 (− 6.8) 0.0132 (− 19.5) 0.0228 (− 10.6) 0.0206 (− 8.4) 0.0226 (− 14.1) 0.0116 (− 4.9) 0.0090 (− 24.4) 0.0092 (− 13.2) 0.0118 (+0.8) 0.0092 (− 17.9) 0.0160 (− 18.5)
0.0131 (+0.0) 0.0148 (+3.5) 0.0125 (− 15.5) 0.0148 (− 9.8) 0.0230 (− 9.8) 0.0256 (+34.7) 0.0232 (− 11.8) 0.0087 (− 28.7) 0.0087 (− 26.9) 0.0089 (− 16.0) 0.0093 (− 20.5) 0.0103 (− 8.0) 0.0191 (+41.5)
0.0135 (+3.0) 0.0135 (− 5.6) 0.0135 (− 8.8) 0.0135 (− 17.7) 0.0240 (− 5.9) 0.0240 (+26.3) 0.0240 (− 8.7) 0.0097 (− 20.5) 0.0097 (− 18.5) 0.0097 (− 8.5) 0.0097 (− 17.1) 0.0097 (− 13.4) 0.0182 (+34.8)
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Fig. 5. Variation of wave damping rates with wave nonlinearity for T = 1.27 s in different water depths.
correctly predicted by the numerical model when compared to the experimental data. This finding also agrees with the results by Chang (2004), who found by a numerical analysis that the increase of wave nonlinearity from 0.05 to 0.2 results in an increase of wave damping rate. Gu and Wang's theoretical model, however, predicts the opposite trend for the nonlinear wave effect, whereas Liu and Dalrymple's linear model gives the same decay coefficient regardless of wave height. 4.2. Sawaragi and Deguchi's (1992) experimental data In this section, we shall examine the behavior of the numerical model for a larger permeability of the bed (Sawaragi and Deguchi, 1992). The experiments for wave damping over porous seabeds were performed in a two-dimensional wave tank of length 30 m, width 0.7 m and depth 0.9 m. The permeable bed was made on the horizontal bottom of the wave tank and the length and the thickness were 3.5 m and 0.15 m, respectively. The water depths
above the permeable bed were either 0.15 m or 0.25 m for different experimental cases. Rubble stones of mean diameters either 1.80 cm or 3.07 cm were used to construct the permeable bed. A series of experiments was performed for different water depths, sediment diameters, wave periods and wave heights. In order to further verify the numerical model for the case of larger permeability when turbulence inside porous media becomes strong and nonlinear friction force predominates, we simulate two experimental conditions i.e., J-2 and J-6 from a series of experiments performed by Sawaragi and Deguchi (1992) to determine wave attenuation over permeable beds. The wave periods for the two cases were T = 1.50 s and T = 1.00 s with wave height Hi = 0.0358 m. Both experimental cases were conducted in water depth h = 0.15 m and seabed thickness d = 0.15 m. The mean diameter of the sediments in the permeable bed was 3.07 cm. Since the porosity was not provided in the paper, we use the value of n = 0.4, which was estimated by Chang (2004) for same experiments based on some other similar experiments by the
Fig. 6. Comparison of numerical results for the wave damping along the porous seabed to the experimental (Sawaragi and Deguchi, 1992), theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) and numerical (Chang, 2004) results for case J-2.
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Fig. 7. Comparison of numerical results for the wave damping along the porous seabed to the experimental (Sawaragi and Deguchi, 1992) and theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) results for case J-6.
same authors (Deguchi et al., 1988). Figs. 6 and 7 present the comparison of the numerical results to the experimental and theoretical results (Liu and Dalrymple, 1984; Gu and Wang, 1991) for wave heights along the permeable bed. In Fig. 6, the numerical results according to Chang (2004) are also provided. According to Fig. 6, the present numerical results as well as numerical results by Chang (2004) are in good agreement with experimental data. The results from Gu and Wang's theoretical model also provides good agreement with experimental data. However, Liu and Dalrymple's theoretical model has significantly underestimated wave damping. In Sawaragi and Deguchi's experiments, permeability was much larger compared to permeability in Savage's experiments and the flow inside the porous media could become turbulent in contrast to laminar flow in Savage's experiments. The good agreement between numerical results and
experimental data for both experiments indicates that the numerical model is accurate for a wide range of permeability, especially when porous flow becomes turbulent. 5. Further discussion of other flow parameters In this section, the verified numerical model will be used for the parameterization study of wave damping under different wave and flow conditions. 5.1. Influence of the wavelength on wave damping Fig. 8 presents the variation of wave damping rate with the non-dimensional wave parameter kwh for different permeable seabeds. For the numerical tests, fixed values of Hi = 0.5 m,
Fig. 8. Variation of (a) dimensional and (b) non-dimensional wave damping rate with kwh for different porous beds.
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Fig. 9. Variation of wave damping rate with the seabed thickness for different porous beds.
h = 5 m, d = 5 m and n = 0.4 are chosen while T is decreased from T = 22 s to T = 2.6 s in order to increase kwh from kwh = 0.2 to kwh = 3.0. According to Fig. 8(a), for a particular d50 the maximum wave damping occurs at a certain kwh while for both smaller and larger kwh, wave damping rates decrease. The reason is that the energy dissipation depends on the product of the pressure on the bed and the velocity of flow into and out of the bed. For short waves, fluctuation of velocity on the bed is less and hence energy dissipation is less. On the other hand, for long waves, pressure variation during a wave cycle occurs in a longer distance, consequently reducing velocity on the bed and wave induced flow inside the porous bed. Therefore, energy dissipation is less for long waves. It is also found that for the
same kwh, the maximum wave decay rate occurs in certain middle range of sediment size. According to Fig. 8(b), it is observed that the non-dimensional wave damping rate always decreases with the decrease of wavelength. This implies that the relative energy loss with one wavelength will always decrease as the decrease of wavelength. 5.2. Influence of seabed thickness on wave damping Fig. 9 shows the variation of wave damping rate with the increase of seabed thickness for the fixed values of Hi = 0.5 m, h = 5 m, T = 5 s and n = 0.4. It is seen that the wave damping rate increases with the increase of the seabed thickness up to a certain
Fig. 10. Variation of wave damping rate with Reynolds number.
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thickness of the bed. The further increase of the bed will not influence on wave damping. For smaller seabed thickness, since the wave induced flow penetrates into a smaller volume of porous media wave energy dissipation is less. With the increase of the seabed thickness, energy dissipation inside the porous bed will increase. However, for a particular wave, the wave-induced porous flow will be concentrated into a certain depth of the porous bed. Therefore, wave damping attains a constant for thick beds. 5.3. Influence of Reynolds number on wave damping Numerical tests are conducted for the case of fixed values of Hi = 0.5 m, h = 5 m, d = 5 m, T = 5 s and n = 0.4 while d50 varies from d50 = 0.001 m to d50 = 0.4 m in order to obtain wave damping rates for a wide range of Reynolds number. Fig. 10 displays the variation of non-dimensional wave damping rates with Reynolds number. Here, Reynolds number is defined as umax;c d50 H gk 1 Re ¼ where umax;c ¼ i w is the maximum m 2r cosh kw h characteristic velocity that occurs at the seabed under a wave crest. It is seen that the maximum wave damping occurs at a certain Reynolds number. Smaller or larger Reynolds numbers result in smaller wave damping rates. The results indicate that Reynolds number is an important parameter in wave damping over porous seabeds. 6. Conclusions Wave damping over porous seabeds is studied by using a two-dimensional numerical model. The model is extended from an earlier model developed by Liu et al. (1999) by including a new porous flow model in which the Reynolds number Re is explicitly formulated. The new model is able to simulate porous flows in a wide range of permeability. The model is first validated against the theories by Liu and Dalrymple (1984) and Gu and Wang (1991). A significant difference is found for the wave damping rate between the present numerical results and the theoretical results by Liu and Dalrymple (1984), who assumed the linear friction in the porous media. On the other hand, the difference between the numerical results and the theoretical results from Gu and Wang (1991), who linearized the nonlinear friction in the derivation, is less significant, with the peak damping rate shifting to the lower permeability for the fully nonlinear numerical model. The results confirm that the theoretical models that have been derived by neglecting the nonlinear resistance term can only be used at low Re flows. On the other hand, the linearization of nonlinear porous flows may also result in the underestimation or overestimation of wave damping in difference range of soil permeability. The model is further validated against experimental data by Savage (1953) and Sawaragi and Deguchi (1992). It is found that the numerical model gives better overall comparisons to the experimental data than the theoretical models for a wide range of soil permeability and wave nonlinearity. It is also found that wave nonlinearity plays an important role in shallow water depth, whereas in deep water wave damping is little affected by wave nonlinearity. Compared to the experimental data, the present numerical results predict the correct trend of the increasing wave
damping with the increase of wave nonlinearity. This is contrary to Gu and Wang's theory that predicts the opposite trend of the decreasing wave damping rate with the increase of wave nonlinearity. The effects of wavelength, seabed thickness and Reynolds number on wave damping are also investigated by using the numerical model. The wave damping rate for both short and long waves is smaller than that for waves in intermediate depth. The results also reveal that the wave damping rate increases with the increase of the seabed thickness up to a critical value, beyond which the damping rate remains the same because the wave energy dissipation only occurs within a finite depth of the top soils. The numerical results also show that under a given wave condition the maximum wave damping occurs in the middle range of flow Reynolds number, which is related to soil permeability and flow condition. Notations a wave amplitude a0 coefficient associated with linear friction ā averaged wave amplitude b0 coefficient associated with nonlinear friction CD drag force coefficient for porous media CDS drag force coefficient for a single sphere Cd, Cf parameters in Gu and Wang's model c1, c2 coefficients for linear and nonlinear resistances d seabed thickness d50 mean sediment diameter fl a complex variable f0 linearized resistance coefficient fDi spatially averaged drag resistance force in i-direction fIi spatially averaged inertia resistance force in i-direction fIi spatially averaged resistance force in i-direction g gravitational acceleration Hi initial wave height h water depth K specific permeability k kinetic energy ki imaginary part of complex wave number kim measured wave damping rate kp permeability kr real part of complex wave number kw wave number KC Keulegan–Carpenter number n porosity p pressure p0 spatially averaged effective pressure R permeability parameter Re Reynolds number T wave period t time Umax maximum particle velocity uc characteristic velocity umax;c maximum characteristic velocity ui ; uj spatially averaged velocity in i-, j-direction ui W; uj W fluctuation of velocity in i-, j- direction x horizontal distance
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xi , xj y
distance in i-, j-directions vertical distance
Greek symbols β parameter representing inertial resistance term without added mass effect βa parameter representing inertial resistance term including added mass γp virtual mass coefficient Δ relative error ε kinetic energy dissipation rate η free surface displacement λ wavelength ν kinematics viscosity ρ density σ angular frequency Acknowledgements The study was supported, in part, by the research grants provided by Defence Science and Technology Agency (DSTA), Singapore (R-347-000-021-422) and China National Science Foundation (50525926). References Chang, H.-H., 2004. Interaction of water waves and submerged permeable offshore structures. Ph. D. thesis, The National Cheng Kung University, Taiwan. Cruz, E.C., Isobe, M., Watanabe, A., 1997. Boussinesq equations for wave transformation on porous beds. Coast. Eng. 30 (2), 125–156. Dagan, G., 1979. The generalization of Darcy's law for nonuniform flows. Water Resour. Res. 15 (1), 1–7. Deguchi, I., Sawaragi, T., Shiratani, K., 1988. Applicability of nonlinear unsteady Darcy's law to the waves on permeable layer. Proc. 35th Japanese Conf. on Coast. Eng. JSCE, pp. 487–491 (in Japanese). Fair, G.M., Geyer, J.C., Okum, D.A., 1968. Waste water engineering Vol. 2: Water purification and wastewater treatment and disposal. New York, Wiley.
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Gu, Z., Wang, H., 1991. Gravity waves over porous bottoms. Coast. Eng. 15 (5), 497–524. Huang, C.-J., Chang, H.-H., Hwung, H.-H., 2003. Structural permeability effects on the interaction of a solitary wave and a submerged breakwater. Coast. Eng. 49 (1), 1–24. Hunt, J.N., 1959. On the damping of gravity waves propagated over a permeable surface. J. Geophys. Res. 64 (4), 437–442. Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. J. Fluid Mech. 359, 239–264. Liu, P.L.-F., 1973. Damping of water waves over porous bed. J. Hydraul. Div. 99 (12), 2263–2271. Liu, P.L.-F., Dalrymple, R.A., 1984. The damping of gravity waves due to percolation. Coast. Eng. 8 (1), 33–49. Liu, P.L.-F., Lin, P., 1997. A Numerical Model for Breaking Wave: The Volume of Fluid Method. Res. Rep. No. CACR-97-02. Ctr. for Appl. Coast. Res., Oc. Eng. Lab., University of Delaware, Newark. Liu, P.L.-F., Lin, P., Chang, K.-A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. J. Waterw. Port Coast. Ocean Eng. 125 (6), 322–330. Murray, J.D., 1965. Viscous damping of gravity waves over a permeable bed. J. Geophys. Res. 70 (10), 2325–2331. Özhan, E., Shi-Igai, H., 1977. On the development of solitary waves on a horizontal bed with friction. Coast. Eng. 1, 167–184. Putnam, J.A., 1949. Loss of wave energy due to percolation in a permeable sea bottom. Trans., Am. Geol. Union 30 (3), 349–356. Putnam, J.A., Johson, J.W., 1949. The dissipation of wave energy by bottom friction. Trans., Am. Geol. Union 30 (1), 67–74. Rojanakamthorn, S., Isobe, M., Watanabe, A., 1990. Modeling of wave transformation on submerged breakwater. Proc. 22nd Coast., Eng. Conf., ASCE, Delft, Netherlands, pp. 1060–1073. Savage, R.P., 1953. Laboratory study of energy losses by bottom friction and percolation. Beach Erosion Board, Corps of Engineers, Technical Memorandum, No. 31. Sawaragi, T., Deguchi, I., 1992. Waves on permeable layers. Proc. 23rd Coast. Eng. Conf., ASCE, Venice, pp. 1531–1544. Su, X., Lin, P., 2005. A hydrodynamic study on flow motion with vegetation. Mod. Phys. Lett. B 19 (28–29), 1659–1662. Van Gent, M.R.A., (1995). Wave interaction of permeable coastal structures. Ph. D. thesis, Delft University, The Netherlands.
Numerical simulation of wave damping over porous seabeds S.A.S.A. Karunarathna a , Pengzhi Lin a,b,⁎ a
b
Department of Civil Engineering, National University of Singapore, 117576, Singapore State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, 610065, China Received 14 October 2005; received in revised form 23 February 2006; accepted 1 May 2006 Available online 21 June 2006
Abstract This paper presents a study of wave damping over porous seabeds by using a two-dimensional numerical model. In this model, the flow outside of porous media is described by the Reynolds Averaged Navier–Stokes equations. The spatially averaged Navier–Stokes equations, in which the presence of porous media is considered by including additional inertia and nonlinear friction forces, is derived and implemented for the porous flow. Unlike the earlier models, the present model explicitly represents the flow resistance dependency on Reynolds number in order to cover wider ranges of porous flows. The numerical model is validated against available theories and experimental data. The comparison between the numerical results and the theoretical results indicates that the omission or linearization of the nonlinear resistance terms in porous flow models, which is the common practice in most of analytical models, can lead to significant errors in estimating wave damping rate. The present numerical model is used to simulate nonlinear wave interaction with porous seabeds and it is found that the numerical results compare well with the experimental data for different wave nonlinearity. The additional numerical tests are also conducted to study the effects of wavelength, seabed thickness and Reynolds number on wave damping. © 2006 Elsevier B.V. All rights reserved. Keywords: Numerical modeling; Wave damping; Porous seabeds; Nonlinear frictional force; Nonlinear wave; Reynolds number
1. Introduction Natural seabeds are usually composed of permeable beds that enable mass and momentum transfer across the interface. Rigid porous seabeds allow the fluid to transfer through the body without deforming the soil structure. In non-rigid seabeds soil skeletons are deformed as the waves interact with the beds. When waves propagate over a porous seabed the water–seabed interface will experience a positive dynamic pressure under a wave crest and negative dynamic pressure under a wave trough. The pressure gradient within the interface results in wave induced flow through the pores inside the porous seabed. As the flow propagates through the porous media, resistances will act on the flow fluid due to fluid viscosity and the interaction of pore fluid with soil particles. Furthermore, the inertia force due to the unsteady motion of the pore fluid also becomes significant under certain combination of wave and seabed conditions. Wave energy dissipation due to flow resis⁎ Corresponding author. Department of Civil Engineering, National University of Singapore, 117576, Singapore. Tel.: +65 65161314; fax: +65 67791635. E-mail address: [email protected] (P. Lin). 0378-3839/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2006.05.003
tances inside the porous bed results in the decrease of wave heights as the waves propagate over. This phenomenon is known as wave damping. Numerous experimental, theoretical and numerical studies have been performed to study wave damping over porous seabeds. Wave damping over rigid porous seabeds has been studied experimentally by several researchers (e.g., Savage, 1953; Sawaragi and Deguchi, 1992). Savage (1953) conducted laboratory experiments to find out wave energy losses and wave height attenuation by bottom friction and percolation. His experiment revealed that the energy losses by percolation are insignificant for sands of low permeability and they are significant for coarse sands. An experimental study on solitary wave propagating over smooth and rough bottoms was conducted by Özhan and Shi-Igai (1977). They found that for both smooth and rough regimes, the friction factor depends on the instantaneous wave height. Gu and Wang (1991) conducted laboratory experiments for standing waves oscillating over a porous bottom and the experiments revealed that the boundary layer at the interface is an important issue in wave damping prediction. On the other hand, Sawaragi and Deguchi's (1992) experiment of periodic wave damping over porous seabeds
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showed that the effects of boundary shear on wave damping are negligible when compared to the effects of permeability. Besides experimental studies, a lot of theoretical studies have been performed to study wave damping over rigid porous seabeds. In the earliest theoretical investigations, theoretical expressions for wave energy dissipation due to individual effects of bottom friction (e.g., Putnam and Johson, 1949) and percolation (e.g., Putnam, 1949) were derived by considering that the energy dissipation is caused due to the wave induced pressure fluctuation at the interface. Putnam (1949) in his derivation assumed that the flow within the permeable seabed satisfies the unsteady Darcy's law. Wave damping over porous seabeds of infinite depth has been studied by solving the boundary value problem by a number of researchers (e.g., Hunt, 1959; Murray, 1965; Liu, 1973). These studies were limited by the fact that they were based on small amplitude wave theories with viscous laminar flow in porous media. For porous seabeds with finite thickness, Liu and Dalrymple (1984) obtained a complex wave dispersion relationship by applying Dagan's (1979) porous flow model to describe the flow through the porous media. In their wave dispersion relationship, wave number was a complex number in which the real part represents the change of wavelength whereas the imaginary part represents the damping of wave height. They found that wave damping reaches the maximum at a certain permeability of the soil. The same problem was investigated by Gu and Wang (1991) but including the effect of the nonlinear frictional resistance to the porous flow. However, the linearization of the nonlinear terms was made to simplify the mathematics. Thus, the contribution of the nonlinear term on wave energy dissipation was not completely included in their model, which will be further discussed later. Direct numerical modeling of wave interaction with seabeds has the advantage of including irregular seabed geometry, inhomogeneous porous media, nonlinear waves and nonlinear frictional force. However, probably due to the difficulty of coupling turbulent flow models inside and outside of the porous beds, only limited number of such study was reported before. Several numerical models based on mild-slope equation (e.g., Rojanakamthorn et al., 1990), Boussinesq equations (e.g., Cruz et al., 1997) and Navier–Stokes equations (e.g., Huang et al., 2003) have been developed to study wave interaction with porous submerged breakwaters. Cruz et al. (1997) derived a set of Boussinesq-type equations to model wave propagation on a porous bed. The weak dispersive nature of the equations was corrected by adding dispersion terms and the equations were solved numerically. Recently,
Chang (2004) studied propagation of periodic and solitary waves over porous seabeds by solving two-dimensional Navier–Stokes equations for the flow outside the porous media together with adapted Navier–Stokes equations for the flow inside the porous media. In his study, the flow outside the porous seabed was assumed to be laminar. The laminar flow assumption neglects the turbulent shear stress and thus it may cause significant errors, especially when the waves are highly nonlinear. In this paper, wave damping over rigid finite porous seabeds (Fig. 1) is studied by using a numerical model extended from an earlier model developed by Liu et al. (1999). In the numerical model, the mean flow outside the porous media is described by Reynolds Averaged Navier–Stokes (RANS) equations in which the turbulence field is modeled by k–ε equations. For the flow motion inside the porous media, a new porous flow model will be derived by spatially averaging Navier–Stokes equations while the effect of the porous media is treated by the additional inertia and drag forces. These forces are modeled according to Morison's equation assuming porous media is composed of equivalent spherical particles. The derived porous flow model is incorporated into the numerical model. This numerical model will be validated against various theories and experimental data for wave damping over different porous seabeds. The model is used to study the effects of wave nonlinearity, wavelength, seabed thickness and Reynolds number on wave damping. 2. Governing equations in the numerical model In Liu et al.'s (1999) numerical model, for the domain outside the porous media, the mean flow is described by the RANS equations in which the effects of turbulence field are modeled by the k–ε model. The flow inside the porous media is described by spatially averaged Navier–Stokes equations while the resistance from the porous media is modeled by a combination of linear and nonlinear frictional forces, which are expressed by empirical formulae suggested by Van Gent (1995). The details of governing equations and numerical implementations can be found in Liu and Lin (1997), Lin and Liu (1998) and Liu et al. (1999) and they will not be reiterated here. Based on a series of numerical tests conducted for different wave and porous media interactions, it is found that the coefficients associated with the empirical formulae for frictional forces need to be adjusted according to Reynolds number (Re) of the flow in order to reproduce correctly the experimental results for a wide
Fig. 1. Definition sketch of wave damping over a porous seabed.
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range of porous materials. Therefore, in this study, a new porous flow model is proposed to include Re effects into the frictional forces. For this purpose, the spatially averaged Navier–Stokes equations are re-derived. Taking the spatial average of the original Navier–Stokes equation in the fluid domain, which can be subsequently transformed in the domain of entire porous media (e.g., see Su and Lin, 2005), we have: Aui ¼0 Axi
ð1Þ
1 Aui uj Aui 1 Ap0 m A2 ui 1 Aui Wuj W þ 2 ¼− þ − fi − 2 n At n Axj q Axi n Axj Axj n Axj
ð2Þ
In Eqs. (1) and (2), t = time, ρ and ν = density and kinematic viscosity of fluid, and n = porosity. i, j = 1, 2 represent the horizontal and vertical direction, respectively and the over-bar denotes the spatially averaged quantities and the double prime denotes the spatially fluctuated quantities. The variables x, u, p0 and f are direction, velocity, effective pressure (= p + ρgixi, where p is the pressure and g is the gravitational acceleration) and resistance force, respectively. In Eq. (2), the force fIi is caused by the presence of porous material in flow and it includes inertia and drag forces i.e., fIi ¼ fIi þ fDi . The inertia force represents the additional momentum required to accelerate water in the porous medium that is known as added mass phenomenon. By assuming that the porous medium is composed of uniform spherical particles and summing up the contribution from each spherical particle and taking the average, the inertia force term can be written as: fIi ¼
gp ð1−nÞ Aui At n2
ð3Þ
where γp is the virtual mass coefficient and the suggested value by Van Gent (1995) is γp = 0.34. The drag force fDi can be derived in the similar way as: 3 ð1−nÞ 1 fDi ¼ CD 3 uc ui 4 n d50
ð4Þ
where d50 is the mean diameter of sediments, uc p is ffiffiffiffiffiffiffiffi theffi characteristic velocity which can be estimated by uc ¼ ui ui and CD is the drag force coefficient. It is well known that for a single spherical particle, the drag coefficient CDS is the function of Reynolds number Re ¼ uc md50, which can be approximated by the formula (Fair et al., 1968): CDS ¼
24:0 3:0 þ pffiffiffiffiffiffi þ 0:34 Re Re
ð5Þ
In this study, CD is expressed similarly to CDS in terms of Reynolds number. However, two coefficients c1 and c2 are introduced in order to account for the effects of porosity, roughness, shape factor, and etc. on the change of drag coefficient. The final form of CD reads: 24:0 3:0 7:5 þ c2 pffiffiffiffiffi þ 0:34 1 þ CD ¼ c1 Re KC Re
ð6Þ
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In Eq. (6), KC ¼ Undmax50T is the Keulegan–Carpenter number where Umax = maximum water particle velocity and T = wave period and it is included to account for the effects of oscillatory flows on the change of nonlinear frictional force inside the porous medium. The derived porous flow model is implemented into Liu et al.'s (1999) numerical model neglecting the last term on the right hand side of Eq. (2) with the assumption of negligible mean turbulence shear effect inside the porous media. The coefficients in Eq. (6) are determined to be c1 = 7.0 and c2 = 2.0 from a series of numerical calibrations against available experimental data for wave interaction with porous structures and porous seabeds. The above coefficient values will be used throughout this study except in Section 3 where comparisons are made to different theoretical works, in which the respective coefficient values suggested in the original theoretical papers will be used in the numerical modeling for consistency. 3. Model validation against theories The numerical model will be first validated against the theoretical models derived by Liu and Dalrymple (1984) and Gu and Wang (1991) for wave damping over porous seabeds. In Liu and Dalrymple's model, the nonlinear resistance term is not included for the porous flow model whereas in Gu and Wang's model the nonlinear resistance term is approximated by an “equivalent” linear term. In contrast, in the numerical model the fully nonlinear porous flow model is included. In Sections 3.1 and 3.2, we shall briefly discuss the formulations of Liu and Dalrymple's and Gu and Wang's theoretical models, respectively. In order to validate the numerical model, these theoretical models are incorporated into the numerical model and the numerical results are compared to the corresponding theoretical results. For this comparison, empirical expressions for linear, nonlinear and inertial resistance terms and associated coefficients in each model are set to be consistent with those used by Liu and Dalrymple (1984) and Gu and Wang (1991). 3.1. Liu and Dalrymple's theory (1984) Theoretical expression derived by Liu and Dalrymple (1984) was based on Dagan's (1979) porous flow model, which is a generalized form of Darcy's law that can be applied for non-uniform flows. In addition, an acceleration term was also introduced into the porous flow model in order to represent the oscillatory flow motion inside the porous bed. However, the effect due to added mass was not considered. In the derivation, free surface profile was assumed to vary with distance and time as η(x,t) = aeiðk w x−σtÞ where a is the wave amplitude and σ is the angular frequency. Here, the wave number kw is a complex variable, which can be written as kw = kr + iki, in which kr represents the change of wavelength and ki represents the damping of wave height. With these conditions the homogeneous solution of the boundary value problem has led to a complex wave dispersion relationship as: r2 −gkw tanh kw h i tanh kw dðgkw −r2 tanh kw hÞ ¼− ð1=R−ibÞ
ð7Þ
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Fig. 2. Validation of the numerical model against Liu and Dalrymple's (1984) theory and Gu and Wang's (1991) theory.
where h is the water depth, d is the seabed thickness, R ¼ rK m is the k m permeability parameter with K ¼ gp ¼ specific permeability in which kp = permeability of porous material, and β = 1 / n. The first and second terms in the denominator of the right hand side of Eq. (7) represent the linear and inertia frictional resistances, respectively. In order to calculate wave damping rate ki, expression (7) needs to be solved iteratively. To simulate Liu and Dalrymple's theory, we set c1 ¼ a0 nð1−nÞ2 in which the coefficient a0 = 570, c2 = 0 and γp = 0. 18 The corresponding wave and porous media parameters are: wave period T = 5 s, initial wave height Hi = 0.5 m, water depth h = 5 m, porous bed thickness d = 5 m and porosity n = 0.4. For the numerical tests, wave damping for a distance of x = 300 m over different permeable seabeds with different permeability parameters R are numerically simulated by changing diameter of sediments. The corresponding diameter for each permeability parameter is obtained by using the expression for specific permeability defined by Gu and Wang 2 2 (1991) i.e., K ¼ n d 50 3. a0 ð1−nÞ According to Fig. 2, numerical results and analytical results of wave decay coefficients agree well. This ensures that the numerical model can correctly represent the theoretical model by Liu and Dalrymple (1984) when the nonlinear term and the added mass effect in the inertial term are neglected in the numerical model. In order to view the effect of added mass in the inertial term, we also plot the theoretical results from Liu and Dalrymple (1984) by including added mass effect, which will change the coefficient in the inertia resistance to be b ¼ ba ¼ nþgpnð1−nÞ . According to the 2 figure, when the term that represents the added mass is included into the Liu and Dalrymple's model (1984), the damping rates are reduced compared to the damping rates from their original model. The permeability parameter that gives the peak-damping rate is also slightly changed.
3.2. Gu and Wang's (1991) theory The complex wave dispersion relationship derived by Gu and Wang (1991) includes the linear and nonlinear resistance terms for the unsteady porous flow model. In order to include energy dissipation due to nonlinear frictional resistance in their theoretical model, the energy dissipation within a true nonlinear system was equated to an equivalent linearized system. Then, the complex wave dispersion relationship was obtained as: r2 −gkw tanh kw h ¼ −
i tanh kw dðgkw −r2 tanh kw hÞ f0
ð8Þ
Here, both wave number kw and the linearized friction coefficient f0 are complex variables. The relationship for f0 is given by: f0 ¼
1 Cd jkw Dðkw ; f0 Þsinh kw dj −iba þ R rjrf0 j
ð9Þ
in which Dðkw ; f0 Þ ¼
ga cosh kw h cosh kw d ½1−i=f0 tanh kw h tanh kw d ð10Þ
where ā is the average of ae−ki x over one wavelength. In Eq. (9), the terms on the right hand side represent the linear, inertia and nonlinear resistances, respectively. The parameter Cd relates to 1 porosity n and sediment mean diameter d50 by Cd r ¼ b0 1−n , ffiffiffi n3 d50 C which has been approximated as Cd ¼pCffiffiKf ffi ¼ pffiffif ffi rm where R b0 Cf ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. The coefficients a0, b0 and γp takes the values 2 n
a0 ð1−nÞ
of 570, 3.0 and 0.46, respectively. In order to calculate wave damping rate ki, expressions (8) and (9) need to be solved iteratively.
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In order to solve the theoretical model by Gu and Wang (1991), the porous flow equation ( Eq. (8)) is implemented into the numerical model. When Eq. (8) was derived, the pore pressure equation has been linearized while keeping the inertia resistance term together with the nonlinear resistance term. In order to implement it into the numerical model, we have changed its form by replacing f0 with fl − iβa where fl is an unknown complex coefficient which is given by:
fl ¼
1 Cd jkw Dðkw ; fl Þsinh kw dj þ R rjrðfl −iba Þj
ð11Þ
where
Dðkw ; fl Þ ¼
ga cosh kw h cosh kw d ½1−i=ðfl −iba Þtanh kw h tanh kw d ð12Þ
This wave damping model can be equivalently implemented into the numerical model when the coefficients in the numerical rfl n3 2 model are given by c1 ¼ 18m 1−n d 50 and c2 = 0 and γp = 0.46. In Fig. 2, wave damping results from Gu and Wang's (1991) theory, the numerical model with linearized pore pressure equation, and the fully nonlinear numerical model are shown. In the case of fully nonlinear model simulation, in order that the coefficients are consistent with the corresponding coefficients in Gu and Wang's model, the coefficients in the numerical model are 2 ffiffiffiffi þ 1:02 , neglecting KC effect and set to c1 ¼a0 nð1−nÞ ; c2 ¼ 4b0 = p9:0 Re 18 γp = 0.46. According to Fig. 2, when the linearized pore pressure equation is implemented into the numerical model, a good agreement between the numerical results and Gu and Wang's (1991) results is obtained. However, the wave damping results computed by using the fully nonlinear numerical model are different. It is clear that the reason for this difference is caused by the linearization of the nonlinear friction term in Gu and Wang's treatment. In other words, the linearization will not provide the exactly equivalent energy dissipation for a full range of permeability of porous bed.
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4. Model verification by experimental data In this section, the numerical model with the coefficient values of c1 = 7.0, c2 = 2.0 and γp = 0.34 will be further verified against the experimental data by Savage (1953) and Sawaragi and Deguchi (1992) for wave damping over porous seabeds. It is noted that Savage's (1953) experimental data represent small permeability cases whereas Sawaragi and Deguchi's (1992) experimental data represent large permeability cases. 4.1. Savage's (1953) experimental data Savage (1953) conducted an experiment in a laboratory flume of length 29.3 m, width 0.46 m and depth 0.61 m. The porous bed was 18.3 m long and 0.30 m thick. Water depths above the seabed were 0.229 m, 0.152 m and 0.102 m. He divided the flume into two parts and one side was used as a smooth side in which bottom friction and percolation would not exist. The other side was used to conduct wave damping tests in which the bottom consisted with sediment particles. Waves were generated by a wave generator, which produced waves with the period ranged from 0.5 s to 5 s. Wave heights were measured using wave gauges located at 0.61 m or 1.22 m intervals. Several runs were performed to measure wave height attenuation due to bottom friction and percolation. In order to eliminate the energy loss due to side wall friction, the energy loss in the sand side was corrected according to the loss in the smooth side. For the purpose of further verification of the numerical model, decay of wave heights along the bed in Run No. 1 of Savage's (1953) experiment is selected (Fig. 6 in Savage, 1953). For this run wave period T = 1.27 s, wave height Hi = 0.054 m, water depth h = 0.229 m, seabed thickness d = 0.30 m and the porous media had a mean diameter d50 = 3.82 mm and specific permeability K = 4.49 × 10− 9 m2. Since the porosity was not given in the paper we decided to use n = 0.3, which has been used by both Liu and Dalrymple (1984) and Gu and Wang (1991) for the same experiment. Numerically simulated free surface profile at t = 25 T is shown in Fig. 3. Corresponding dimensionless wave heights along the direction of wave propagation that has been non-dimensionalized with wavelength λ is compared to the experimental data in Fig. 4. Savage (1953) corrected his experimental wave damping
Fig. 3. Simulation results of free surface profile at t = 25 T for the Run No. 1 of Savage's (1953) experiments.
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Fig. 4. Comparison of numerical results for wave damping along the porous bed with the corresponding experimental results (Run No. 1 of Savage, 1953) and theoretical results (Liu and Dalrymple, 1984; Gu and Wang, 1991).
results by eliminating the side wall effects. For this experimental condition, wave damping results from the theoretical models by Liu and Dalrymple (1984) and Gu and Wang (1991) are also shown. According to the figure, the numerical result has very good agreement with the corrected experimental results. However, both theoretical models have underestimated wave damping slightly. According to Savage's (1953) experimental results, waves having different instantaneous wave heights provide different wave damping rates when the other parameters i.e., wave period, water depth, seabed thickness and sediment diameters are the same (Table 4 in Savage, 1953). Some of these experimental runs are numerically simulated to study the effects of wave nonlinearity on wave damping. Table 1 provides the comparison of the numerical simulation results to the experimental (Savage, 1953) and the theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) results.
k −k
In Table 1, the relative error is defined by D% ¼ i im 100% where kim kim is the measured wave damping rate. It is shown that the numerical results are in better overall agreement with the experimental results than the theoretical models. The differences between the numerical results and theoretical results, however, are not very significant because all these cases have small permeability (e.g., log R ≈ −2) where the wave decay rates predicted by various methods are close (refer to Fig. 2). Fig. 5 shows the variation of dimensionless wave damping rate with the increase of wave nonlinearity in different water depths. It is observed that with the increase of wave nonlinearity wave damping remains almost a constant for large water depth. However, when water depth becomes smaller, wave nonlinearity plays an important role such that wave damping is increased with the increase of wave nonlinearity, which is
Table 1 Comparison of the numerical wave damping results to the experimental and theoretical wave damping results for waves with different wave heights Run no.
3 4 12 14 21 22 23 5 7 15 16 17 24
T (s)
1.27
h (m)
0.229
0.152
1.00
0.229
0.152
Hi/h
0.299 0.111 0.388 0.114 0.418 0.167 0.401 0.388 0.133 0.333 0.277 0.117 0.167
ki/(σ2/g) Experimental results (Savage, 1953)
Present numerical results (ΔP%)
Gu and Wang's results (1991) (ΔGW%)
Liu and Dalrymple's results (1984) (ΔLD%)
0.0131 0.0143 0.0148 0.0164 0.0255 0.0190 0.0263 0.0122 0.0119 0.0106 0.0117 0.0112 0.0135
0.0133 (+1.5) 0.0139 (− 2.8) 0.0138 (− 6.8) 0.0132 (− 19.5) 0.0228 (− 10.6) 0.0206 (− 8.4) 0.0226 (− 14.1) 0.0116 (− 4.9) 0.0090 (− 24.4) 0.0092 (− 13.2) 0.0118 (+0.8) 0.0092 (− 17.9) 0.0160 (− 18.5)
0.0131 (+0.0) 0.0148 (+3.5) 0.0125 (− 15.5) 0.0148 (− 9.8) 0.0230 (− 9.8) 0.0256 (+34.7) 0.0232 (− 11.8) 0.0087 (− 28.7) 0.0087 (− 26.9) 0.0089 (− 16.0) 0.0093 (− 20.5) 0.0103 (− 8.0) 0.0191 (+41.5)
0.0135 (+3.0) 0.0135 (− 5.6) 0.0135 (− 8.8) 0.0135 (− 17.7) 0.0240 (− 5.9) 0.0240 (+26.3) 0.0240 (− 8.7) 0.0097 (− 20.5) 0.0097 (− 18.5) 0.0097 (− 8.5) 0.0097 (− 17.1) 0.0097 (− 13.4) 0.0182 (+34.8)
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Fig. 5. Variation of wave damping rates with wave nonlinearity for T = 1.27 s in different water depths.
correctly predicted by the numerical model when compared to the experimental data. This finding also agrees with the results by Chang (2004), who found by a numerical analysis that the increase of wave nonlinearity from 0.05 to 0.2 results in an increase of wave damping rate. Gu and Wang's theoretical model, however, predicts the opposite trend for the nonlinear wave effect, whereas Liu and Dalrymple's linear model gives the same decay coefficient regardless of wave height. 4.2. Sawaragi and Deguchi's (1992) experimental data In this section, we shall examine the behavior of the numerical model for a larger permeability of the bed (Sawaragi and Deguchi, 1992). The experiments for wave damping over porous seabeds were performed in a two-dimensional wave tank of length 30 m, width 0.7 m and depth 0.9 m. The permeable bed was made on the horizontal bottom of the wave tank and the length and the thickness were 3.5 m and 0.15 m, respectively. The water depths
above the permeable bed were either 0.15 m or 0.25 m for different experimental cases. Rubble stones of mean diameters either 1.80 cm or 3.07 cm were used to construct the permeable bed. A series of experiments was performed for different water depths, sediment diameters, wave periods and wave heights. In order to further verify the numerical model for the case of larger permeability when turbulence inside porous media becomes strong and nonlinear friction force predominates, we simulate two experimental conditions i.e., J-2 and J-6 from a series of experiments performed by Sawaragi and Deguchi (1992) to determine wave attenuation over permeable beds. The wave periods for the two cases were T = 1.50 s and T = 1.00 s with wave height Hi = 0.0358 m. Both experimental cases were conducted in water depth h = 0.15 m and seabed thickness d = 0.15 m. The mean diameter of the sediments in the permeable bed was 3.07 cm. Since the porosity was not provided in the paper, we use the value of n = 0.4, which was estimated by Chang (2004) for same experiments based on some other similar experiments by the
Fig. 6. Comparison of numerical results for the wave damping along the porous seabed to the experimental (Sawaragi and Deguchi, 1992), theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) and numerical (Chang, 2004) results for case J-2.
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Fig. 7. Comparison of numerical results for the wave damping along the porous seabed to the experimental (Sawaragi and Deguchi, 1992) and theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) results for case J-6.
same authors (Deguchi et al., 1988). Figs. 6 and 7 present the comparison of the numerical results to the experimental and theoretical results (Liu and Dalrymple, 1984; Gu and Wang, 1991) for wave heights along the permeable bed. In Fig. 6, the numerical results according to Chang (2004) are also provided. According to Fig. 6, the present numerical results as well as numerical results by Chang (2004) are in good agreement with experimental data. The results from Gu and Wang's theoretical model also provides good agreement with experimental data. However, Liu and Dalrymple's theoretical model has significantly underestimated wave damping. In Sawaragi and Deguchi's experiments, permeability was much larger compared to permeability in Savage's experiments and the flow inside the porous media could become turbulent in contrast to laminar flow in Savage's experiments. The good agreement between numerical results and
experimental data for both experiments indicates that the numerical model is accurate for a wide range of permeability, especially when porous flow becomes turbulent. 5. Further discussion of other flow parameters In this section, the verified numerical model will be used for the parameterization study of wave damping under different wave and flow conditions. 5.1. Influence of the wavelength on wave damping Fig. 8 presents the variation of wave damping rate with the non-dimensional wave parameter kwh for different permeable seabeds. For the numerical tests, fixed values of Hi = 0.5 m,
Fig. 8. Variation of (a) dimensional and (b) non-dimensional wave damping rate with kwh for different porous beds.
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Fig. 9. Variation of wave damping rate with the seabed thickness for different porous beds.
h = 5 m, d = 5 m and n = 0.4 are chosen while T is decreased from T = 22 s to T = 2.6 s in order to increase kwh from kwh = 0.2 to kwh = 3.0. According to Fig. 8(a), for a particular d50 the maximum wave damping occurs at a certain kwh while for both smaller and larger kwh, wave damping rates decrease. The reason is that the energy dissipation depends on the product of the pressure on the bed and the velocity of flow into and out of the bed. For short waves, fluctuation of velocity on the bed is less and hence energy dissipation is less. On the other hand, for long waves, pressure variation during a wave cycle occurs in a longer distance, consequently reducing velocity on the bed and wave induced flow inside the porous bed. Therefore, energy dissipation is less for long waves. It is also found that for the
same kwh, the maximum wave decay rate occurs in certain middle range of sediment size. According to Fig. 8(b), it is observed that the non-dimensional wave damping rate always decreases with the decrease of wavelength. This implies that the relative energy loss with one wavelength will always decrease as the decrease of wavelength. 5.2. Influence of seabed thickness on wave damping Fig. 9 shows the variation of wave damping rate with the increase of seabed thickness for the fixed values of Hi = 0.5 m, h = 5 m, T = 5 s and n = 0.4. It is seen that the wave damping rate increases with the increase of the seabed thickness up to a certain
Fig. 10. Variation of wave damping rate with Reynolds number.
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thickness of the bed. The further increase of the bed will not influence on wave damping. For smaller seabed thickness, since the wave induced flow penetrates into a smaller volume of porous media wave energy dissipation is less. With the increase of the seabed thickness, energy dissipation inside the porous bed will increase. However, for a particular wave, the wave-induced porous flow will be concentrated into a certain depth of the porous bed. Therefore, wave damping attains a constant for thick beds. 5.3. Influence of Reynolds number on wave damping Numerical tests are conducted for the case of fixed values of Hi = 0.5 m, h = 5 m, d = 5 m, T = 5 s and n = 0.4 while d50 varies from d50 = 0.001 m to d50 = 0.4 m in order to obtain wave damping rates for a wide range of Reynolds number. Fig. 10 displays the variation of non-dimensional wave damping rates with Reynolds number. Here, Reynolds number is defined as umax;c d50 H gk 1 Re ¼ where umax;c ¼ i w is the maximum m 2r cosh kw h characteristic velocity that occurs at the seabed under a wave crest. It is seen that the maximum wave damping occurs at a certain Reynolds number. Smaller or larger Reynolds numbers result in smaller wave damping rates. The results indicate that Reynolds number is an important parameter in wave damping over porous seabeds. 6. Conclusions Wave damping over porous seabeds is studied by using a two-dimensional numerical model. The model is extended from an earlier model developed by Liu et al. (1999) by including a new porous flow model in which the Reynolds number Re is explicitly formulated. The new model is able to simulate porous flows in a wide range of permeability. The model is first validated against the theories by Liu and Dalrymple (1984) and Gu and Wang (1991). A significant difference is found for the wave damping rate between the present numerical results and the theoretical results by Liu and Dalrymple (1984), who assumed the linear friction in the porous media. On the other hand, the difference between the numerical results and the theoretical results from Gu and Wang (1991), who linearized the nonlinear friction in the derivation, is less significant, with the peak damping rate shifting to the lower permeability for the fully nonlinear numerical model. The results confirm that the theoretical models that have been derived by neglecting the nonlinear resistance term can only be used at low Re flows. On the other hand, the linearization of nonlinear porous flows may also result in the underestimation or overestimation of wave damping in difference range of soil permeability. The model is further validated against experimental data by Savage (1953) and Sawaragi and Deguchi (1992). It is found that the numerical model gives better overall comparisons to the experimental data than the theoretical models for a wide range of soil permeability and wave nonlinearity. It is also found that wave nonlinearity plays an important role in shallow water depth, whereas in deep water wave damping is little affected by wave nonlinearity. Compared to the experimental data, the present numerical results predict the correct trend of the increasing wave
damping with the increase of wave nonlinearity. This is contrary to Gu and Wang's theory that predicts the opposite trend of the decreasing wave damping rate with the increase of wave nonlinearity. The effects of wavelength, seabed thickness and Reynolds number on wave damping are also investigated by using the numerical model. The wave damping rate for both short and long waves is smaller than that for waves in intermediate depth. The results also reveal that the wave damping rate increases with the increase of the seabed thickness up to a critical value, beyond which the damping rate remains the same because the wave energy dissipation only occurs within a finite depth of the top soils. The numerical results also show that under a given wave condition the maximum wave damping occurs in the middle range of flow Reynolds number, which is related to soil permeability and flow condition. Notations a wave amplitude a0 coefficient associated with linear friction ā averaged wave amplitude b0 coefficient associated with nonlinear friction CD drag force coefficient for porous media CDS drag force coefficient for a single sphere Cd, Cf parameters in Gu and Wang's model c1, c2 coefficients for linear and nonlinear resistances d seabed thickness d50 mean sediment diameter fl a complex variable f0 linearized resistance coefficient fDi spatially averaged drag resistance force in i-direction fIi spatially averaged inertia resistance force in i-direction fIi spatially averaged resistance force in i-direction g gravitational acceleration Hi initial wave height h water depth K specific permeability k kinetic energy ki imaginary part of complex wave number kim measured wave damping rate kp permeability kr real part of complex wave number kw wave number KC Keulegan–Carpenter number n porosity p pressure p0 spatially averaged effective pressure R permeability parameter Re Reynolds number T wave period t time Umax maximum particle velocity uc characteristic velocity umax;c maximum characteristic velocity ui ; uj spatially averaged velocity in i-, j-direction ui W; uj W fluctuation of velocity in i-, j- direction x horizontal distance
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xi , xj y
distance in i-, j-directions vertical distance
Greek symbols β parameter representing inertial resistance term without added mass effect βa parameter representing inertial resistance term including added mass γp virtual mass coefficient Δ relative error ε kinetic energy dissipation rate η free surface displacement λ wavelength ν kinematics viscosity ρ density σ angular frequency Acknowledgements The study was supported, in part, by the research grants provided by Defence Science and Technology Agency (DSTA), Singapore (R-347-000-021-422) and China National Science Foundation (50525926). References Chang, H.-H., 2004. Interaction of water waves and submerged permeable offshore structures. Ph. D. thesis, The National Cheng Kung University, Taiwan. Cruz, E.C., Isobe, M., Watanabe, A., 1997. Boussinesq equations for wave transformation on porous beds. Coast. Eng. 30 (2), 125–156. Dagan, G., 1979. The generalization of Darcy's law for nonuniform flows. Water Resour. Res. 15 (1), 1–7. Deguchi, I., Sawaragi, T., Shiratani, K., 1988. Applicability of nonlinear unsteady Darcy's law to the waves on permeable layer. Proc. 35th Japanese Conf. on Coast. Eng. JSCE, pp. 487–491 (in Japanese). Fair, G.M., Geyer, J.C., Okum, D.A., 1968. Waste water engineering Vol. 2: Water purification and wastewater treatment and disposal. New York, Wiley.
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