Numerical solution for the second-order wave interaction with porous structures

Int. J. Numer. Meth. Fluids 29: 265 – 288 (1999)

NUMERICAL SOLUTION FOR THE SECOND-ORDER

WAVE INTERACTION WITH POROUS STRUCTURES

HU-HSIAO HSU AND YUNG-CHAO WU*

Department of Ci6il Engineering, National Chiao-Tung Uni6ersity, Hsinchu, Taiwan, Pro6ince of China

SUMMARY

This study mathematically formulates the fluid field of a water-wave interaction with a porous structure as a two-dimensional, non-linear boundary value problem (bvp) in terms of a generalized velocity potential. The non-linear bvp is reformulated into an infinite set of linear bvps of ascending order by Stokes perturbation technique, with wave steepness as the perturbation parameter. Only the first- and second-order linear bvps are retained in this study. Each linear bvp is transformed into a boundary integral equation. In addition, the boundary element method (BEM) with linear elements is developed and applied to solve the first- and second-order integral equations. The first- and second-order wave profiles, reflection and transmission coefficients, and the amplitude ratio of the second-order components are computed as well. The numerical results correlate well with previous analytical and experimental results. Numerical results demonstrate that the second-order component can be neglected for a deep water-wave and may become significant for an intermediate depth wave. Copyright © 1999 John Wiley & Sons, Ltd.

KEY WORDS: boundary element method; porous structure; reflection coefficient; transmission coefficient

1. INTRODUCTION

In ocean engineering, porous structures such as rubble-mound breakwaters have been widely constructed to protect harbors, inlets and beaches from wave action. Porous structures provide shelter from wave attack by reflecting and dissipating incident wave energy. These structures are also frequently used as absorbers in laboratories to remove unwanted waves during experiments. In addition, the functional efficiency of these structures can be evaluated by calculating the reflection and transmission of waves. In permeable breakwaters, a portion of the incident wave energy is transmitted through the porous structure. Water depth, as well as wave properties such as wave period and wave height, and structural properties markedly influence the distribution of reflected, transmitted and dissipated wave energies. The major structural properties are geometry, porosity, permeability, size distribution and shape function of the components of the porous structures.

The interaction among progressing waves and a porous structure can be addressed in several ways. Previous investigations have derived theoretical solutions for the reflection and transmis-sion coefficients by using eigenfunction expantransmis-sions in the fluid and in the porous structure [1 – 4]. However, their solutions are valid only for structures with rectangular cross-sections

* Correspondence to: Department of Civil Engineering, National Chiao-Tung University, Hsinchu, Taiwan, Province of China.

CCC 0271 – 2091/99/030265 – 24$17.50 Copyright © 1999 John Wiley & Sons, Ltd.

Recei6ed 6 September 1996 Re6ised 4 September 1997

under normally incident linear waves. Dalrymple et al. [5] extended the solutions to structures subjected to an oblique wave attack. Recently, Lee and Lan [6] analyzed the porous structural problems up to the second-order. These investigators also analyzed trapezoidal breakwaters by considering an equivalent breakwater of rectangular cross-section [1,4] or by boundary element models [7].

In addition to the above efforts, other scholars have closely examined the porous problems using various methods. Chwang [8] applied Taylor’s [9] concept of a porous screen to avert the complicated porous structure flow and develop the so-called porous wavemaker theory. Huang and Chao [10] proposed a solution involving the porous Reynolds number, thereby avoiding the complex bvp. Owing to the problem’s complexity, experimental studies are considered indispensable tools in providing reliable information for engineering applications, Iwasaki and Numata [11], Dattatri et al. [12], Aminti and Franco [13] and Oumeraci and Partensucky [14]. However, such results are restricted only to specific types of porous structures.

In this study, a generalized potential theory is applied to describe both water and porous flow regions. The original non-linear boundary value problem (bvp) is linearized by the perturbation method. The numerical solutions are derived up to the second-order. The boundary element method (BEM) is used to solve the first- and second-order bvps. An attempt is also made to increase the numerical solution’s accuracy by using the linear element to perform computations. Moreover, the numerical solution’s accuracy is demonstrated by comparing the analytical solution and the experimental data.

2. THEORETICAL FORMULATION OF THE PROBLEM

Herein, the interaction of a gravity wave train with a single homogeneous, isotropic, porous structure of width b, between two semi-infinite fluid regions of constant depth h, as shown in Figure 1, is considered. A Cartesian co-ordinate is chosen with the origin located at the still water surface. The porous structure is specified with given porous features and is considered to be rectangular with a horizontal bottom. The incident wave is specified propagating in the + x-direction, with a wave height H and a period T. The incident wave train encountering the porous structure face is partially reflected and partially transmitted. The wave motion inside the porous structure decays as it propagates through the pores. Then, as it encounters the leeward structural face, it is partially reflected back into the structure and partially transmitted into the leeward semi-infinite fluid region. Inside the porous structure, the transmitted and reflected waves are subsequently reflected and transmitted back and forth between the two outside faces.

To resolve the above problem, the domain is divided into three regions, i.e. a porous structure region and two water regions in front of and behind the porous structure. In this study, the mathematical model used to describe the flow regions for the porous structure and water regions is generalized into one theory. Only the flow mechanism in the porous structure is mathematically described.

By making the usually assumptions of classical hydrodynamics, the wave field outside the structure can be specified by velocity potentials [1],F₁ in the seaward region (denoted region 1) andF₃in the leeward region of the porous structure (region 3). In the porous structure, the incompressible fluid motion for the discharge velocity is also describable by a potential (F₂), and a modified free-surface boundary condition. The velocity potentials Fj(x, z, t) can be expressed as:

Fj(x, z, t) = Real[fj(x, z) e− ivt], j = 1, 2, 3, (1) where i =−1, and the velocity potentials fj(x, z) must satisfy the Laplace equation

92_f

j= 0. (2)

The unsteady Bernoulli equation for the flow in the porous structure can be expressed as [1,6]:

Figure 1. Schematic diagram of waves propagating over a porous structure; (a) physical domain; (b) computational domain.

HU-HSIAO HSU AND YUNG-CHAO WU 268 − Sj (Fj (t + Sj 2

n

where the subscript j denotes the jth flow region, Sj represents the virtual mass coefficient,r is the fluid density, g denotes the gravity acceleration, and v represents the wave frequency. The final term of the left-hand side of Equation (3) represents the flow resistance and fj is referred to as the friction coefficient. In addition, the friction coefficient fjis defined by using the Lorentz condition of equivalent work, and can be written as [15]:

fj= 1 vÍ Ã Ã Á Ä ojn Kpj +oj 2_C fj Kpj

&

&

&

&

where Vb jdenotes the real part of the seepage velocity,n represents the kinematic fluid viscosity,

Cfjdenotes the turbulence drag coefficient,ojis the porosity of the medium, and Kpjrepresents the intrinsic permeability of the porous medium. For Sj= 1 and fj= 0, Equation (3) is reduced to the well-known Bernoulli equation for the irrotational flow of an ideal fluid.

Equations (3) and (4) indicate that the non-linearity of the flow resistance is maintained in an implicit form. In the solution, the problem is iteratively solved by assuming a friction fj value, which is then recalculated from the solved flow velocity. In the first-order solution, the linear velocity is used to calculate the friction coefficient. However, in the second-order solution, the linear velocity and the second-order component of the velocity are used to calculate the friction coefficient. Furthermore, the use of Equation (3) implies irrotational flow and the generalized velocity potential function exists. A theoretical proof was given by Lee [4]. Figure 1(a) depicts the physical domain of the present problem. The porous structure is subjected to incident waves with frequency v and wave height H. The first region includes incident wavesF1Iand reflected wavesF1R, and the third region contains transmitted wavesF3T.

Based on the above theoretical framework, the boundary conditions for the jth region (Figure 1) are given below.

1. The kinematic free-surface boundary condition: (hj (t+ (Fj (z− (hj (x (Fj (x= 0 on z =hj; j = 1, 2, 3. (5) 2. The dynamic free-surface boundary condition:

− Sj (Fj (t + Sj 2

n

3. The boundary condition at the water bottom: (Fj

(n
= 0 on z = − h, (7)

indicates that the bottom is impermeable. The vector n
is the unit normal vector pointing out of the computation domain.

4. The radiation conditions: These conditions model the behavior of an outgoing wave at a distance far away from the structure (where the water depth remains constant).

5. The matching boundary conditions at the interface of the (j − 1)th and jth regions (j = 2, 3):

o(j − 1) (F(j − 1) (x =oj (Fj (x, (8) − S(j − 1) (F(j − 1) (t + S(j − 1) 2

n

n

These equations can be applied to pure fluid regions, j = 1 and j = 3, by specifying the virtual mass coefficient to be unity (Sj= 1) and the friction coefficient to be zero (fj= 0) in the equations.

The resulting non-linear bvp can be reformulated using a perturbation method. With the perturbation approach, it is assumed that the solution relies on the presumed small quantitye, the wave steepness. The series form of the velocity potential can be written as [16]:

Fj(x, z, t) = %

n = 1

Fjn(x, z, t), (10)

where a perturbation parameter, e, is implicitly included in the expansion. Other physical quantities such as dynamic pressure, surface elevation and velocities can also be similarly expanded. The fact that the location of the water surface is a priori unknown accounts for why the free-surface boundary conditions must also be expanded in the Taylor series around the still water level.

By applying the perturbation procedures, the bvps of the first- and second-order are formulated. The first-order bvp can be expressed as:

92_F j1= 0, j = 1, 2, 3, (11) Sj (2_F j1 (t2 + g (Fj1 (z +vfj (Fj1 (t = 0 on z = 0, (12) (Fj1 (z = 0 on z = − h. (13)

The matching boundary conditions on the interface of the (j − 1)th and jth regions (j = 2, 3), are o(j − 1) (F(j − 1)1 (x =oj (Fj1 (x, (14) S_{(j − 1)}(F(j − 1)1 (t +vf(j − 1)F(j − 1)1= Sj (Fj1 (t +vfjFj1 (15)

and radiation conditions, i.e. an outgoing wave at far-field.

The free-surface elevation can be computed according to the linearized Bernoulli equation evaluated at the free-surface:

hj1= 1

g

!

(Fj1

(t +vfjFj1" on z = 0, (16)

which is identical to the linear problem given by Sollitt and Cross [1]. The second-order bvp can be expressed as:

92_F

Sj (2_F j2 (t2 + g (Fj2 (z +vfj (Fj2 (t = − Sjhj1 (3_F j1 (z (t2−vfjhj1 (2_F j1 (z (t− Sj (hj1 (t (2_F j1 (z (t −vfj (hj1 (t (Fj1 (z + Sj (Fj1 (x (2F j1 (x (t+ Sj (Fj1 (z (2F j1 (z (t + g(hj1 (x (Fj1 (x − ghj1 (2_F j1 (z2 on z = 0, (18) (Fj2 (z = 0 on z = − h. (19)

The matching boundary conditions at the interface of the (j − 1)th and jth regions (j = 2, 3), are o(j − 1) (F(j − 1)2 (x =oj (Fj2 (x, (20) S(j − 1) (F(j − 1)2 (t +vf(j − 1)F(j − 1)2− S(j − 1) 2

n

n

The free-surface elevation at z = 0 can be computed from the second-order Bernoulli equation: hj2= 1 g

!

n"

The non-homogeneous terms in Equations (18) and (21) involve the products of the first-order solutions. Although complex variables are used in the expression of the linear solution, only the real part represents physical quantities. Therefore, redundant results attributed to multiplication of the imaginary parts should be excluded. By using Vantorre’s expression [17], Equations (18), (21) and (22) can be reformulated into a time-dependent term proportional to e− i2vt and a time-independent term. The second-order velocity potential and surface elevation can be written in the form

Fj2(x, z, t) =fj2(x, z) e− i2vt+8j2(x, z), (23)

hj2(x, t) =hˆj2(x) e− i2vt+h˜j2(x). (24)

Equations (17) – (22), can be decomposed into two bvps: a time-dependent problem,fj2(x, z), and a time-independent problem,8j2(x, z). In addition, the bvp for the time-dependent part of the second-order solution can be written as:

92_f j2= 0, (25) − 2v2_(2S j+ ifj)fj2+ g (fj2 (z = Dj(x) on z = 0, (26) (fj2 (z = 0 on z = − h, (27) where Dj(x) denotes

Dj(x) = v5_{( − i3S} j+ 2fj)(Sj+ ifj)2 2g2 (fj1)2+ v(−i2Sj+ fj) 2

The matching boundary conditions at the interface of the (j − 1)th and jth regions (j = 2, 3), are o(j − 1) (f(j − 1)2 (x =oj (fj2 (x, (29) v[−i2S(j − 1)+ f(j − 1)]f(j − 1)2− S(j − 1) 4

n

n

In region 1, the time-dependent potential function contains the second-order incident wavef12I

and a priori unknown reflected wave f12R. The second-order time-dependent free-surface

elevation at z = 0 can be expressed as hˆj2= 1 g

!

n

"

The bvp, Equations (25) – (30), in terms of fj2 contains non-homogeneous boundary

conditions, i.e. Equations (26), (29) and (30). To facilitate the solution of the problem, it is further decomposed into two parts: a Stokes wave problem,fj2s, and a free-wave problem,fj2f.

fj2=fj2s +fj2f , (32)

wherefj2s must satisfy Equations (25) – (27). Thefj2f must satisfy Equations (25) and (27), the radiation conditions and a homogeneous free-surface boundary condition [18].

The bvp for the time-independent part of the second-order solution can be expressed as: 92₈ j2= 0, (33) g(8j2 (z = Ej(x) on z = 0, (34) (8j2 (z = 0 on z = − h, (35)

in which Ej(x) can be expressed as

Ej(x) = iv5_S j(Sj+ ifj)(Sj− ifj) 2g2 fj1f(j1+ vfj 2 (fj1 (x (f(j1 (x + v(−iSj+ fj) 2 fj1 (2f( j1 (x2 , (36)

wheref(j1, (f(j1/(x and (2f(j1/(x2represent the complex conjugate offj1, (fj1/(x and (2fj1/ (x2 _{respectively. The matching boundary conditions between the (j − 1)th and jth regions}

(j = 2, 3) are o(j − 1) (8(j − 1)2 (x =oj (8j2 (x, (37)

HU-HSIAO HSU AND YUNG-CHAO WU 272 −vf(j − 1)8(j − 1)2+ S(j − 1) 4

n

n

where (f(j1/(z denotes the complex conjugate of (fj1/(z. The time-independent part of the corresponding free-surface elevation at z = 0 can be expressed as

h˜j2= 1 g

!

n

"

The procedures to solve the time-dependent bvp can be applied herein to resolve the time-independent bvp8j2. Moreover, the solution can be written as the sum of a solution of a bvp, 8j2s, and a solution of a bvp, 8j2f , i.e.

8j2=8j2s +8j2f , (40)

where8j2s must satisfy Equations (33) – (35). Also,8j2f must satisfy Equations (33) and (35), the radiation conditions and a homogeneous free-surface boundary condition [19].

3. OPEN BOUNDARY TREATMENT

At far-field, x =9(W(+(b/2)), the radiation conditions stipulate that both the reflection wave and the transmission wave propagate away from the porous structure. Therefore, the following discussion encompasses the velocity potentials, the first- and second-order, which satisfy governing equation and radiation conditions in the region of constant depth, h. The first-order velocity potential at the reflection side that satisfies Equations (11) – (13) and the radiation condition, can be expressed as [18]:

F1r(x, z, t) =f1r(x, z) e− ivt, (41) f1r= gH 2v cosh[k(h + z)] cosh kh exp[ik(x + W( +(b/2))] +gHr 2v cosh[k(h + z)]

cosh kh exp[ − ik(x + W( +(b/2))]+evanescent modes. (42)

The subscript represents the order of magnitude. The first term is the velocity potential of the first-order incident wave; the second term is the velocity potential of the first-order reflection wave, where Hrdenotes the wave height of the reflection wave. The last term decays away to

zero at far-field, W( =6h and therefore, is neglected herein. In addition, k and v must satisfy the dispersion relation,

v2_{= gk tanh kh. (43)}

At interface AB (Figure 1(b)), x = − (W( +(b/2)), the matching conditions provide continu-ity of pressures and horizontal velocities normal to the vertical interface. The relation between velocity potential,f11, and normal velocity,(f11/(n, on the vertical interface AB is derived in

Appendix A (Equation (69)). In the present context, it can be written as f11= H g v cosh k(h + z) cosh kh + cosh k(h + z) ikQ0

&

Similarly, on the vertical interface CD (x = (W( +(b/2))), the relation between f31and(f31/(n

can be established as (Appendix A) f31= coshk(h + z) ikQ0

&

The time-dependent velocity potential,F2rs=f2rse− i2vt, at the reflection side of the

second-order Stokes wave can be analytically solved as [18]: f2rs= i

n

sinh4_(kh) Hr2exp[ − i2k(x + W( +(b/2))]+evanescent modes.

(46) The first term is due to the interaction of the first-order incident wave and the first-order reflection wave; the second term is the velocity potential of the second-order Stokes incident wave, the third term is attributed to the interaction of the first-order reflection wave. The final term decays away to zero at far-field and is therefore, neglected herein.

The time-dependent velocity potentialF₂rf_=f 2

rf_e− i2vt_{at the reflection side of the}

second-or-der free-wave can be expressed as [18]:

f2rf= A( cosh b(h+z) exp[−ib(x+W(+(b/2))]+evanescent modes, (47)

where A( is an unknown coefficient. Evanescent modes decay away to zero at far-field. b and v must satisfy the dispersion relation

(2v)2_{= gb tanh bh. (48)}

Similarly, at interface AB, using the matching conditions provides continuity of pressures and horizontal velocities normal to the vertical interface. The open boundary condition can be expressed as (Appendix B): f12= R1(z) + − i coshb(h+z) bW0

&

On the transmission side, f2ts=

− i3v

32 sinh4_khHt2cosh 2k(h + z) exp[i2k(x − W( −(b/2))]+evanescent modes (50)

and

f2tf= B( cosh b(h+z) exp[ib(x−W(−(b/2))]+evanescent modes, (51)

where B( denotes an unknown coefficient. The relation between f32 and f32n on the interface

CD (x = (W( +(b/2))) can be established as (Appendix B): f32= R2(z) + − i coshb(h+z) bW0

&

The time-independent velocity potential,8₂rs_{, at the reflection side of the second-order wave}

HU-HSIAO HSU AND YUNG-CHAO WU 274 82rs= iv 32

cosh2_kh

+iv 32

3 sinh2_kh+

1

cosh2_kh

The time-independent velocity potential,82rf, at the reflection side of the second-order wave

can be expressed as [19]: 82rf= C( x+ %  n = 1 A.n cosmn(h + z) exp[mn(x + W( +(b/2))], mn= np h, n = 1, 2, 3, . . . , (54) based on the assumption of no-uniform current, i.e. C( =0, and the series summation term decays away to zero at far-field. Therefore, at interface AB, x = − (W( +(b/2))

812=82rs+82rf= iv 32

Similarly, on the transmission side,

82ts= 0 (56) and 82tf= K( x+ %  n = 1 B.ncosmn(h + z) exp[ −mn(x − W( −(b/2))], (57)

assuming no-uniform current (K( =0) exists and the series summation term decays away to zero at far-field. Therefore, at interface CD, x = (W( +(b/2))

832=82ts+82tf= 0. (58)

Equations (44), (45), (49), (52), (55) and (58) are the open boundary conditions on AB and CD of Figure 1(b), which correspond to the first-order, the time-dependent part of the second-order and the time-independent part of the second-order problems respectively.

4. BEM FORMULATION

The BEM has been used to solve a variety of problems in theoretical hydrodynamics and elasticity theory [20]. For a bvp in which the free-space Green’s function is known, the BEM can be used to perform computations only on the domain’s boundary.

Utilizing the BEM initially involves converting the bvps into an integral equation represen-tation. Using Green’s second identity

&

G

(n

&

(f.92_{q − q9}2_{f.) dV, (59)}

where q denotes the fundamental solution of the governing equation, G represents the

boundary of the solution domain, V is the solution domain, and f. denotes the velocity potential at a selected point of the boundary.

The fact that the governing equation of the fluid domain is the Laplace equation accounts for why the fundamental solution is [21]

q =1

2pln

r

where r represents the distance from the source point to the field point. From Equation (59) any velocity potential f.j of the boundary is given by

−b 2pf.j=

&

where j denotes the source point and b represents the internal angle of the source point j. The numerical procedure of the BEM entails dividing the boundary into N segments or elements. To increase the numerical method’s accuracy, the linear element is used to perform computations on the domain’s boundary. Next, Equation (61) is numerically integrated using Gaussian quadrature. Finally, a system of N equations is established. The system of N equations can be written as

[H]{f.}=[G]

!

(n

"

in which [H] and [G] are known and rely only on the geometry. Therefore, instead of attempting to obtain the unknown distribution of the boundary values over the discretized surface, the problem is reduced to finding the unknownf. and (f./(n at the nodal points.

In the previous section, the relation between f. and (f./(n was established. Herein, the discretized forms of the open boundaries are established on the basis of linear element. By combining Equation (62) with the discretized form of the open boundary conditions, a sufficient number of equations exist to solve unknown quantities. After rearranging the equations in such a manner that all unknowns are taken to the left-hand side and all the knowns are moved to the right-hand side, Equation (62) can be written as:

[A][X] = [B], (63)

where [X] denotes the vector of unknown fjm and (fjm/(n (m=1, 2), [B] represents the known vector and [A] is the matrix of coefficients. Equation (63) can be solved by employing the Gauss elimination method.

5. NUMERICAL RESULTS AND DISCUSSION

This work has examined the second-order problem of the interaction of progressive wave and porous structure in water of constant depth by the BEM. To ensure the accuracy of the computation, the numerical solutions of the BEM are compared with available analytical solutions of Lee and Lan [6] and experimental results of Sollitt and Cross [1]. The friction coefficient used to describe the flow mechanism in the porous region is calculated by the Lorentz’s condition [15] of equivalent work, i.e. Equation (4). In addition, the flow velocities of the first- and second-order are defined using the perturbation definition:

The first order: Vb1=Real{ux21}2+ Real{uz21}2 (64)

where the first subscript of u represents velocity direction, the second subscript denotes flow region and the third subscript is the order of magnitude.

The reflection and transmission coefficients of the first- and second-order waves are defined as K1r=

'

'

'

'

where E1i, E1r and E1tare the first-order energies of incident, reflected and transmitted waves

respectively. Moreover, E2i, E2r and E2t are the complete second-order total energies of

incident, reflected and transmitted waves respectively.

Figure 2 presents the numerical solutions for wave steepness H/L = 0.01, dimensionless width of the porous structure b/h = 1.0, porosity o2= 0.439, turbulence drag coefficient

Cf2= 0.295, intrinsic permeability Kp2= 0.1138 × 10− 5(ft2) and kinematic viscosityn=1.09× 10− 5_(ft2_s− 1_{). This figure also compares numerical results in this study with points taken from}

analytical results of Lee and Lan [6]. The results of this study closely correlate with those of Lee and Lan [6], with the only discrepancy found in the linear reflection coefficient. Next, the proposed numerical model’s accuracy is confirmed by reanalyzing the linear problem by the method of matched eigenfunction expansion, as used by Lee and Lan [6]. The analytical solutions, plotted in Figure 2, closely correlate with the BEM solutions. Therefore, the proposed numerical model’s accuracy is verified. Reasons for the discrepancy between this study and Lee and Lan [6] still remain unknown.

This study also considers two different wave steepnesses, H/L = 0.01 and 0.02, and two different porous structural properties (Table I). The first-order problems are solved by both the BEM and the analytical method; the second-order problems are solved by the BEM only. Figures 3 – 5 plot the numerical results.

Figures 3 – 5 present the reflection and transmission coefficients for a permeable breakwater of rectangular cross-section, as a function of relative water depth, kh. Notably, more wave energy is dissipated in the porous structure for a deeper water-wave. The effect of second-order

is important in the region of intermediate depth wave, p/10BkhBp. In addition, the

importance of the second-order effect increases with an increasing wave steepness.

The surface elevation can be computed according to Equations (16) and (22). For kh = 0.4 and 1.0, Figures 6 and 7 display the temporal waveforms at four different locations. Each figure includes the temporal waveformsh1_, h1+h2s _andh1₊h2s+h2f_{, where}h1 _{denotes the}

first-order wave profile, h2s _{represents the second-order Stokes’ wave profile, and h}2f _{is the}

second-harmonic free-wave profile. These figures reveal that the surface elevations differ at different position. This phenomenon is attributed to the fact that the second-order free-wave moving with phase velocity is smaller than the Stokes’ wave, (Equations (43) and (48)).

Figure 6, kh = 0.4, indicates that the second-order free-surface elevation, h1_+h2_{, has an}

obvious deformation, implying that the second-order component is important. On the other hand, Figure 7, kh = 1.0, indicates that the deformation of the second-order free-surface is insignificant, i.e. the second-order component can be neglected. Therefore, it can be concluded that the effect of second-order wave component gradually decreases with an increasing relative water depth. Herein, an attempt is also made to further understand the relative importance of

Figure 2. Friction (a), reflection (b) and transmission (c) coefficients (H/L = 0.01,o2= 0.439, Kp2= 0.1138 × 10− 5, Cf2= 0.295).

Figure 2 (Continued)

the second-harmonic free-wave and the second-order Stokes’ wave. Figure 8 plots the ratio of free-wave amplitude and Stokes’ wave amplitude,h2f_/h2s _{for different relative water depths.}

According to this figure, the second-harmonic free-wave is overwhelmingly larger than the Stokes’ wave.

6. CONCLUSIONS

This study establishes the BEM with linear elements to examine second-order wave interaction with a porous structure in water of finite depth. The accuracy of the proposed BEM is verified by comparing the present numerical results with previous analytical solutions and experimental data. For a low value of relative water depth, the numerical results indicate that second-order components may be important in the region of intermediate depth wave. The second-order components consist of a second-harmonic free-wave and a second-order Stokes’ wave. In addition, the wave height of a free-wave is markedly higher than that of a Stokes’ wave. Moreover, the fact that the free-wave travels with lower speed down the leeside of the porous structure accounts for why the water surface profile meanders, i.e. surface profiles vary at different positions.

Table I. Media physical properties

Kp2(ft2) Cf2 o2(%) 0.439 0.295 0.1138×10−5 0.434 0.4820×10−5 0.282

Figure 3. Friction (a), reflection and transmission (b) coefficients (H/L = 0.01, o2= 0.434, Kp2= 0.4820 × 10− 5, Cf2= 0.282).

Figure 4. Friction (a), reflection and transmission (b) coefficients (H/L = 0.02, o2= 0.439, Kp2= 0.1138 × 10− 5, Cf2= 0.295).

Figure 5. Friction (a), reflection and transmission (b) coefficients (H/L = 0.02, o2= 0.434, Kp2= 0.4820 × 10− 5, Cf2= 0.282).

Figure 6. Time variation of the surface elevation at different positions (H/L = 0.01,o2= 0.439, Kp2= 0.1138 × 10− 5, Cf2= 0.295, kh = 0.4).

Figure 7. Time variation of the surface elevation at different positions (H/L = 0.01,o2= 0.439, Kp2= 0.1138 × 10− 5, Cf2= 0.295, kh = 1.0).

Figure 8. The ratio of second-order wave amplitudes,h2f_/h2s _{(H/L=0.01, o}

2= 0.439, Kp2= 0.1138 × 10− 5, Cf2= 0.295).

ACKNOWLEDGMENTS

The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC-85-2611-E-009-001.

APPENDIX A

According to the match conditions of continuity of pressure and horizontal velocity normal to the vertical interface AB and Equation (42), one gets

f11=f1r= g 2v cosh k(h + z) cosh kh (H + Hr), (67) f11n= −f1xr = − i gk 2v cosh k(h + z) cosh kh (H − Hr). (68)

The relationship between f11 and f11n on the interface AB can be established as [18]:

f11= H g v cosh k(h + z) cosh kh + cosh k(h + z) ikQ0

&

&

WAVE INTERACTION WITH POROUS STRUCTURES 287

Q0=

&

cosh2_{k(h + z) dz. (71)}

Similarly, on the vertical interface CD, f31=f1t= gHt 2v cosh k(h + z) cosh kh , (72) f31n=f1xt = i gkHt 2v cosh k(h + z) cosh kh . (73)

The relationship between f31 and f31n on the interface CD can be established as:

f31= cosh k(h + z) ikQ0

&

&

According to the match conditions of continuity of pressure and horizontal velocity normal to the vertical interface AB and Equations (46) and (47), one gets

f12=f2rs+f2rf= i

n

2_{)cosh 2k(h + z) + ibA( cosh b(h+z). (77)}

The relationship between f12 and f12n on the interface AB can be established as [18]:

f12= R1(z) + − i coshb(h+z) bW0

&

&

&

&

f32=f2ts+f2tf=

− i3v

32 sinh4_khHt2cosh 2k(h + z) + B( cosh b(h+z), (83)

f32n=f2xts+f2xtf =

3kv

16 sinh4_khHt2cosh 2k(h + z) + ibB( cosh b(h+z). (84)

The relationship between f32 and f32n on the interface CD can be established as:

f32= R2(z) + − i coshb(h+z) bW0

&

&

&

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