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Wave trapping by porous barrier in the presence of step type bottom
Wave Motion 57 (2015) 219–230
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Wave trapping by porous barrier in the presence of step type bottom H. Behera, R.B. Kaligatla 1 , T. Sahoo ∗ Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur-721302, India
highlights • • • • •
The study deals with wave trapping by porous barrier having undulated bottom bed. The physical problem is solved using modified mild-slope approximation. Number of times optimum reflection occurs is more for undulated bed than flat bed. Zero reflection occurs for specific undulated bed profiles. The present study will be useful in the creation of tranquility zone.
article
info
Article history: Received 18 December 2014 Received in revised form 25 March 2015 Accepted 20 April 2015 Available online 8 May 2015 Keywords: Oblique wave trapping Porous barrier Modified mild-slope equation Undulated bed Reflection coefficient
abstract The present study deals with the trapping of oblique wave by porous barrier located near a rigid wall in the presence of a step type bottom bed. The solution of the physical problem is obtained using the eigenfunction expansion method and multi-mode approximation associated with modified mild-slope equation. Assuming that the porous structure is made of materials having fine pores, the mathematical problem is handled for solution by matching the velocity and pressure at interface boundaries. Various numerical results are computed and analyzed to understand the role of bed profiles, structural porosity, depth ratio, oblique angle of incidence, distance between barrier and step edge and, the distance between the porous barrier and rigid wall in optimizing wave reflection and load on the structure/rigid-wall. A comparison of results on wave trapping by porous barriers over flat and undulated bed reveals that for the same distance between the porous barrier and rigid wall, more number of times optimum reflection occurs in case of undulated bed. The present study is likely to be of immense importance in the design of coastal structures for protecting coastal infrastructures. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In recent decades, significant increase in developmental activities along the coast due to the rise in world population is exerting greater pressure on the coastal countries for protection of coastal infrastructures/facilities by various measures. In addition, with an increase in global trading, substantial increase in harbor traffic results in the deterioration of wave conditions in various ports and harbors around the world. Further, the rise in sea level is putting additional pressure on the existing coastal infrastructures. In this context, the study on wave interaction with coastal structures provides useful
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (T. Sahoo).
1 Present address: Department of Applied Mathematics, Indian School of Mines, Dhanbad, India. http://dx.doi.org/10.1016/j.wavemoti.2015.04.005 0165-2125/© 2015 Elsevier B.V. All rights reserved.
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information about various physical processes associated with coastal protection or attenuating wave heights in confined water bodies such as bays/ports/harbors. Most of the coastal structures such as breakwaters, used for shore protection and/or creating a tranquility zone by reducing wave impact near harbors, are vertical and rigid in nature. These vertical rigid structures used to collapse under extreme wave climates like tsunami and storm surges. For reducing wave loads on these type of structures, perforated/wave absorbing structures of various configurations are proposed as alternatives due to their ability to dissipate wave energy. In addition, various study on wave trapping by porous structures reveal that wave loads on existing rigid wall can be minimized by introducing porous structure at a suitable distance from the rigid wall. The minimum wave reflection is often referred as wave trapping. In the context of wave interaction with coastal structures, there are mainly two class of problems namely wave scattering and trapping by porous structures are studied in uniform water depth. However, wave motion over undulated sea bed is of considerable interest in coastal engineering practices including development of infrastructures like ports and harbors. Thus, for protection of marine facilities and coastal infrastructures in coastal zones, there is renewed interest to study wave interaction with coastal structures like breakwaters in the presence of undulated bottom bed. There is an extensive literature on wave trapping by rigid or flexible and/or porous vertical structures in the presence of uniform bottom topography. The most widely used model for wave past thin porous structure is based on the modified boundary condition derived by Yu [1] which was initially proposed by Chwang [2]. However, various study on wave trapping are based on small amplitude water wave theory and will be useful for reducing wave forces on vertical sea/harbor wall and for creating low reflection from harbor walls in various ports (as in McConnell et al. [3]). Sahoo et al. [4] studied trapping and generation of surface gravity waves by submerged vertical permeable barriers. Yip et al. [5] analyzed wave trapping by considering flexibility in porous barrier. Further, the study of Chwang and Chen [6] reveals that a knowledge of wave characteristics and harbor configurations helps in the design of new harbor walls or finding ways to determine wave absorbing sea walls by introducing suitable permeable structures at finite location near the harbor wall. Bhattacharjee and Guedes Soares [7] investigated diffraction of water waves by a finite rigid floating structure near wall, considering vertical step in uniform bottom. Applying Sollitt and Cross model, Koley et al. [8] analyzed oblique wave trapping by porous structures having finite width, placed near a rigid wall. Kaligatla et al. [9] analyzed wave trapping by flexible porous plate by converting the boundary value problem into system of integro-differential equations in both the cases of finite and infinite water depths. These studies on wave trapping are performed in homogeneous fluid of uniform water depth. On the other hand, water wave trapping by porous structures of various configurations in two-layer fluid are studied by Behera and Sahoo [10] whilst, oblique wave trapping by partial flexible and porous barriers of different configurations are studied by Behera et al. [11]. The aforementioned discussions on gravity wave interaction with porous structure are for normalized/obliquely incident waves in uniform water depth. Unlike, these plane waves travel in a fixed direction and the associated standing waves fluctuating vertically in a confined region, short-crested waves are doubly periodic along the horizontal directions, one of which is in the direction of propagation and the other being normal to it (as in Zhu [12] and Tsai et al. [13]). Song and Tao [14] studied the interaction of short-crested waves with a concentric porous cylindrical structure in a quantitative manner. Recently, Mandal et al. [15] studied the hydroelastic response of surface waves by porous and flexible cylinder system. In the presence of varying bottom topography, using the Galerkin method proposed by Massel [16]. Suh and Park [17] developed an analytical model to predict oblique wave reflection by a perforated wall caisson mounted on a rubble mound foundation with sloping step, following the model of Fugazza and Natale [18]. However, the above mentioned Galerkin method for varying bottom beds fails to ensure continuity of mass flow at the interface between two domains. The natural mass conserving jump conditions at the bottom slope discontinuities are derived by Porter and Staziker [19] by means of variational principle. In the last two decades, there is significant progress for developing efficient model equations to deal with wave interaction with varying bottoms. Among those, a typical popular model equation is the modified mild-slope equation (MMSE) derived by Chamberlain and Porter [20]. The MMSE with correct interfacial matching conditions at the locations where the bed slope is discontinuous, has been found to be efficient model for bottom slopes up to 1. Further, MMSE was extended to study scattering of flexural and membrane-coupled gravity waves (as in Porter and Porter [21], Bennetts et al. [22], Bennetts et al. [23], Manam and Kaligatla [24]). Apart from use of MMSE for wave motion in homogeneous fluid medium having bottom undulation, the concept of MMSE is extended to study wave scattering due to bottom undulation in two-layer fluid (see Chamberlain and Porter [25] and Manam et al. [26]). In the present paper, oblique wave trapping by thin porous barrier near a wall in the presence of step of arbitrary bottom bed is studied under the assumption of small amplitude water wave theory. In each of the fluid regions, the velocity potentials are expanded in terms of the eigenfunctions and the full solution of the mathematical problem is obtained by matching the continuity of pressure and velocity at interface boundaries. However, the extended modified mild-slope equation (MMSE) along with suitable jump conditions (as in Porter and Staziker [19]) are used to ensure conservation of mass at the interface boundaries in the fluid region having undulated bottom bed. The mild-slope equation is handled for solution using Runge–Kutta method. The reflection coefficients and wave forces are computed and analyzed for different bottom bed profiles to study the role of the porous barrier in trapping oblique gravity waves in the confined zone between the barrier and the rigid wall. For each bed profile, distance between barrier and rigid wall, distance between barrier and step edge, slope length, depth ratio and oblique angle of incidence on the reflection coefficient, wave forces acting on the barrier and rigid wall are studied. Known results available in the literature are reproduced to check the accuracy of the computational results.
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2. Mathematical formulation In this section, oblique surface wave trapping by thin porous barrier is formulated in the presence of varying bottom topography. The physical problem is studied under the assumption of the linearized water wave theory and the porous barrier is assumed to have fine pores. The problem is considered in the three dimensional Cartesian co-ordinate system with x and y-axis being in the horizontal direction and z-axis being in the vertically upward direction. The fluid domain is divided into four sub-domains as in Fig. 1 depending on the varying bottom bed, position of the porous structure and the rigid wall which can be considered as a simplified profile of a navigational channel/bay. A thin porous barrier, extending from bottom to free surface, is kept vertically at a finite distance L from a rigid wall and at a distance L1 from the edge of a step as in Fig. 1. The varying bottom bed profile z = −h(x) in front of the wall is connected by two unequal uniform bottom levels z = −h1 and z = −h2 . Further, the step of variable depth h(x) span over 0 < x < L2 whilst, the open water region −∞ < x < 0 is of uniform depth h1 . It is assumed that along the y-axis, the fluid domain is horizontally extended over 0 < y < ∞. The fluid is assumed to be incompressible, inviscid with motion being irrotational and simple harmonic in time with angular frequency ω. Further, it is assumed that surface waves propagate by making an angle θ with x-axis which ensures the existence of the velocity potentials Φj such that Φj (x, y, z , t ) = Re{φj (x, z )e−i(βy y+ωt ) } with βy = β0 sin θ and β0 being the wave number of the incident wave in region 1. Subscript j = 1, 2, 3, 4 refers to the fluid domains 1, 2, 3, 4 shown in Fig. 1. Thus, the spatial velocity potentials φj (x, z ) for j = 1, 2, 3, 4 satisfy the Helmholtz equation
∂2 ∂2 2 + − β y φj = 0. ∂ x2 ∂ z2
(2.1)
The boundary condition on the uniform bottoms are given by
∂φj = 0 on z = −hj for j = 1, 3, 4, ∂z
(2.2)
whilst, the bottom boundary condition in the undulated bed region is given by
∂φ2 dh ∂φ2 + = 0 on z = −h(x). ∂z dx ∂ x
(2.3)
The linearized free surface boundary condition is given by
∂φj − K φj = 0 on z = 0, for j = 1, 2, 3, 4, ∂z
(2.4)
where K = ω2 /g and g is the gravitational constant. On the rigid vertical wall, vanishing of horizontal velocity yields
∂φ4 = 0 at x = L2 + L1 + L. ∂x
(2.5)
The boundary conditions on the porous barrier at x = L2 + L1 are given by
∂φ3 ∂φ4 = ∂x ∂x
and
∂φj = iβ0 G(φ3 − φ4 ) for j = 3, 4, ∂x
(2.6)
with G being referred as the complex porous-effect parameter and is given by G=
ϵ , β0 d(f − is)
(2.7)
where ϵ is the porosity of the barrier, f the resistance force coefficient, s the inertial force coefficient, d the thickness of the porous barrier. The real part of the complex porous-effect parameter G represents the resistance effect of the porous material against the flow, while the imaginary part of G represents the inertia effect of the fluid inside the porous material. An increase in either parts or the both of them leads to an increase in the transparency of the porous structure in dissipating and transmitting the wave energy. The situation when the resistance effect of the structure dominates, the imaginary part of G becomes negligible and G reduces to the porous-effect parameter introduced by Chwang [2]. Li et al. [27] studied the physical aspect of the porous-effect parameter associated with a thin permeable structure. 3. Method of solution In order to study wave trapping by porous barrier, the method of solution is described in this section for the mixed boundary value problem formulated in the previous section. It is assumed that the bottom is varying in a finite interval (0, L2 ) and is uniform outside this interval. The bottom profile is assumed to be continuously differentiable function in the
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Fig. 1. Schematic diagram of wave trapping model in the presence of step type bed.
interval (0, L2 ). Further, the bottom is allowed to have slope discontinuities at x = 0 and x = L2 . In region 1, the unknown velocity potential φ1 is written as
φ1 (x, z ) = A0 eiµ0 x f0 (β0 , z ) +
∞
Rn e−iµn x fn (βn , z ),
(3.8)
n =0
where fn (βn , z ) = cosh βn (z + h1 )/ cosh βn h1 with µn =
βn2 − βy2 for n = 0, 1, 2, 3, . . . . The constant A0 is assumed to
be known and is associated with the amplitude of the incident waves and R0 being an unknown constant associated with the amplitude of the reflected wave and Rn s are unknown constants to be determined. Here, β0 is the positive real root and βn for n = 1, 2, 3, . . . are the purely imaginary roots of the dispersion equation β tanh β h1 − K = 0 in β . In region 2, the velocity potential φ2 is written as
φ2 (x, z ) =
∞
ψn (x) Wn (h, z ),
(3.9)
n =0
where ψn (x) are unknown functions and Wn = cosh kn (z + h)/ cosh kn h with µ ˜n =
k2n − βy2 . The wave number k0 is a
positive real root and k1 , k2 , k3 , . . . , are purely imaginary roots of the dispersion equation k tanh kh − K = 0 in k. It may be observed that the roots k0 , k1 , k2 , k3 . . . are functions of bottom profile h(x). The eigenfunctions Wn are borrowed from the flat bottom solution. It may be noted that Eq. (3.9) is an approximation for the velocity potential φ2 (x, z ) which does not satisfy the bottom boundary condition explicitly (as in Berkhoff [28], Porter [29] and the literature cited therein). On the other hand, in regions 3 and 4, the velocity potentials φ3 and φ4 are expressed as
φ3 (x, z ) =
∞ (Bn eiqn x + Cn e−iqn x )gn (γn , z )
(3.10)
n =0
and φ4 (x, z ) =
∞
Dn cos qn (x − x1 )gn (γn , z )
(3.11)
n=0
γn2 − βy2 for n = 0, 1, 2, . . . , x1 = L2 + L1 + L, Bn , Cn , Dn are unknown constants and γ0 is a positive real root and γ1 , γ2 , γ3 , . . . , are purely imaginary roots of the dispersion equation γ tanh γ h2 − K = 0 in γ . Hereafter, the infinite series associated with the evanescent modes for the velocity potentials are truncated after N terms. To obtain the velocity potential φ2 (x, z ) as in Eq. (3.9), using the extended modified respectively, where gn (γn , z ) = cosh γn (z + h2 )/ cosh γn h2 with qn =
mild-slope equation (MMSE) as in Porter and Staziker [19], it can be easily derived that d dx
an
dψn
+
dx
N
bmn − bnm
m=0
dh dψm dx dx
2 d2 h dh + bmn 2 + cmn + dmn − βy2 an ψm = 0, dx
dx
(3.12)
where an (h) =
0
Wn2 dz ,
bmn (h) =
−h
cmn (h) =
0
Wn −h
dbmn dh
0
− −h
∂ Wm dz , ∂h
∂ Wm ∂ Wn dz , ∂h ∂h
∂ 2 Wm dz for n = 0, 1, 2, . . . , N . ∂ z2 −h To determine the functions ψn (x) (n = 0, 1, 2, . . . ., N ) in region 2, the MMSE in Eq. (3.12) can be solved numerically. Assuming that the undulated bed can have slope discontinuities at the interfaces x = 0 and x = L2 and using continuity of dmn (h) =
0
Wn
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pressure across these interfaces, the velocity potentials in Eqs. (3.8)–(3.11) yield
ψ0 (x) = A0 eiµ0 x + R0 e−iµ0 x ψn (x) = Rn e−iµn x
at x = 0 for n = 1, 2, . . . , N ,
(3.13)
at x = L2 for n = 1, 2, . . . , N .
(3.14)
and
ψ0 (x) = B0 eiq0 x + C0 e−iq0 x ψn (x) = Bn eiqn x + Cn e−iqn x
Further, to ensure conservation of mass across the interface boundaries at x = 0 and L2 , using the expressions for ψn (x) as in Eqs. (3.13) and (3.14) and proceeding in a similar manner as in Porter and Staziker [19], the jump conditions are obtained as a0
an
dψ0
+ iµ0 a0 ψ0 + h
dx
′
N m=0
dψn
+ iµn an ψn + h
dx
′
N
bm0 ψm = 2iµ0 a0 A0 bmn ψm = 0
m=0
at x = 0+, n = 1, 2, . . . , N
(3.15)
and a0
an
dψ0
− iq0 a0 ψ0 + h
dx
′
N
bm0 ψm = −2ia0 q0 C0 e
−iq0 x
bmn ψm = −2ian qn Cn e
m=0
dψn
− iqn an ψn + h
dx
′
N
at x = L2 −, n = 1, 2, . . . , N .
(3.16)
−iqn x
m=0
Using continuity of velocity and pressure at x = L1 + L2 as in Eq. (2.6), it is derived that N
Bm e
−iqm x
iqm x
− Cm e
+ iDm sin qm (x − x1 ) Xmn = 0,
m=0 N
m=0
at x = L1 + L2 ,
(3.17)
iβ0 G(Bm eiqm x + Cm e−iqm x ) − Dm {sin qm (x − x1 ) + iβ0 G cos qm (x − x1 )} Xmn = 0,
where 0
gm (γm , z )gn (γn , z )dz ,
Xmn =
for m, n = 1, 2, . . . , N .
−h2
Thus, the system of Eqs. in (3.12) can be solved numerically for ψn for specific bed profile h(x). Afterwards, the numerical solutions of ψn s are used to solve the system of equations in Eqs. (3.13)–(3.17) for the determination of the unknowns. 4. Numerical results and discussion In this section, MATLAB is used for solving the system of algebraic equations to study the effects of wave and structural parameters on the reflection coefficient, horizontal hydrodynamic force on the porous barrier and rigid wall for different bed profiles. Although, the general procedure for multi-mode expansion method is described in the previous section, the various numerical experiments are performed based on single mode approximation. In the numerical computation, the modified mild-slope equation is solved using Runge–Kutta method. For numerical results, various physical parameters are kept fixed as wave length of plane gravity wave in region 1 λ1 = 2π /β0 , time period T = 8 s and acceleration due to gravity g = 9.81 m/s2 unless it is mentioned. The reflection coefficient Kr , magnitude of the horizontal wave forces acting on the porous barrier Cf and on the rigid wall Cw are defined as Kr = |R0 /A0 |, Cf = iρω
0
−h 2
Cw = iρω
(4.18)
{φ4 (x, z ) − φ3 (x, z )}
0
−h 2
φ4 (x, z )
dz .
x=(L1 +L2 )
dz ,
(4.19)
(4.20)
x=x1
Further, the non-dimensional form of the horizontal wave forces on the porous barrier Kf and rigid wall Kw are derived using the formulae Kf =
|Cf | ρ gh21
and
Kw =
|Cw | . ρ gh21
(4.21)
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Fig. 2. Schematic diagrams of undulated bed profiles for Case I.
Fig. 3. Variations of the reflection coefficients Kr versus L2 /λ1 for different values of (a) α with G = 1 + 1i and (b) G with α = 0, and h2 /h1 = 0.25, L/λ1 = 0.4, L1 /λ1 = 0.4, θ = 30°.
In the subsequent discussion, numerical results for various bed profiles (as in Porter and Porter [30]) are analyzed separately in different cases. Case I Here, the bed profiles shown in Fig. 2(a), (b), (c), (d) are considered using the bed function h(x) (0 < x < L2 ) as h(x) = h1 − b{1 − α(1 − x/L2 )2 + (α − 1)(1 − x/L2 )},
(4.22)
with b = (h1 − h2 ) and α is a real constant which will give the different bed profiles to be discussed subsequently. In the bed function h(x) as in Eq. (4.22), α = 0 corresponds to the sloping step type bed as in Fig. 2(b). However, in case of α > 1, there is a protrusion above the depth h2 as shown in Fig. 2(a) and for −1 ≤ α < 0, the bed is concave as in Fig. 2(c) whilst, an increase in depression in bed profile will occur for α < −1 as in Fig. 2(d). In Fig. 3(a) and (b), the reflection coefficients Kr versus non-dimensional slope length L2 /λ1 are plotted for different values of α and porous-effect parameter G respectively. In general, with an increase in the slope length, the reflection coefficient increases in an oscillatory pattern. However, for smaller values of slope length, there is a large variation in the amplitude of the oscillatory pattern of the reflection coefficient. This may be due to the fact that for smaller values of slope length, slope angle becomes large which is leading to higher resonating pattern of the resultant waves in the confined zone. The oscillatory pattern in wave reflection is due to the presence of the vertical rigid wall. Further, the oscillatory pattern in wave reflection decreases with an increase in the slope length which is due to the fact that the sloping bed behaves like a uniform bed. From Fig. 3(a), the reflection coefficient decreases with a decrease in the value of α . This may be due to the protrusion and depression of the bottom bed profile. On the other hand, from Fig. 3(b) it is observed that the reflection coefficient increases with an increase in the absolute values of porous-effect parameter G. This may be due to the transmission of more wave energy through the porous barrier for higher values of G. Further, minimum wave reflection occurs for higher values of slope length with an increase in the absolute value of the porous-effect parameter G. This may be due to the phase shift of the reflected wave with change in complex porous-effect parameter G. In Fig. 4(a) and (b), the reflection coefficients Kr versus the normalized distance between the porous barrier and step L1 /λ1 are plotted for different values of α and porous-effect parameter G respectively. From both the figures, it is observed
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Fig. 4. Variations of the reflection coefficients Kr versus L1 /λ1 for different values of (a) α with G = 1 + 1i and (b) G with α = 0, and h2 /h1 = 0.25, L/λ1 = 0.4, L2 /λ1 = 1, θ = 30°.
Fig. 5. Variations of the reflection coefficients Kr versus L/λ1 for different values of (a) G with h2 /h1 = 0.25 and (b) h2 /h1 with G = 1 + 1i, and α = 0, L1 /λ1 = 0.4, L2 /λ1 = 1, θ = 0°.
that the general patterns of the wave reflection coefficients are periodic in nature with an increase in L1 /λ1 which is due to the uniform depth of the bottom bed beyond the sloping bed. Fig. 4(a) shows that the amplitude in the oscillatory pattern of the reflection coefficient increases with certain phase shift with an increase in α . This may be due to the protrusion and depression of the bottom bed profile as discussed in Fig. 3. Fig. 4(b) shows that the reflection coefficient Kr increases with an increase in the absolute value of the porous-effect parameter G. This is due to the transmission of more wave energy by the porous barrier with an increase in the absolute value of G. In Fig. 5(a) and (b), the reflection coefficients Kr versus the normalized distance between porous barrier and rigid wall L/λ1 are plotted for different values of porous-effect parameter G and depth ratio h2 /h1 respectively. From both figures, it is observed that full reflection occurs periodically. However, a comparison with wave trapping by porous barriers in uniform bed as in Sahoo et al. [4] depicts that due to the presence of bottom undulation more number of times full reflection occurs for the same value of L/λ1 . Fig. 5(a) demonstrates that the reflection coefficient increases with an increase in the absolute value of the porous-effect parameter G as in Figs. 3 and 4. Fig. 5(b) reveals that with an increase in depth ratio h2 /h1 , number of times occurrences of full reflection decreases for the same range of L/λ1 . This may be due to the change in slope of the bottom bed profile. It may be noted that full reflection occurs when the distance between the porous barrier and rigid wall is an integer multiple of half-wavelength of the incident waves for h2 /h1 = 1 as in Sahoo et al. [4]. Further, for h2 /h1 = 0.25, full reflection occurs when the distance between the porous barrier and rigid wall is an integer multiple of one fourth of the incident waves. Thus, in case of a sloping bed, reduction of bed slope increases the occurrences of full reflection. In Fig. 6(a) and (b), the reflection coefficients Kr versus the incident angle θ are plotted for different values of L/λ1 with depth ratio h2 /h1 = 0.25 and h2 /h1 = 1 respectively. It may be noted that the occurrences of full reflection increases with an increase in L/λ1 . Fig. 6(a) reveals that the minimum in the full reflection occurs for higher value of θ with increase in L/λ1 . Further, nearly zero reflection is observed at near θ = 22° for L/λ1 = 1.4. Fig. 6(b) demonstrates that the minimum in wave reflection exponentially increases with an increase in the angle of incidence. For h2 /h1 = 1 with 0 ≤ θ < π /2, i.e., in case of oblique wave trapping by porous barrier having uniform bed, number of times full reflection depend on the ratio λ1 /L and is the greatest integer value of m satisfying the relation λ1 /L ≤ 2/m. It is observed that wave reflection attains a minimum value between two consecutive angle for which full reflection occurs. However, the global minimum in reflection coefficient decreases with an increase in L/λ1 . Similar observation was made by Behera et al. [11] in case of oblique wave trapping by flexible porous barriers in a two-layer fluid in surface mode. A comparison of Fig. 6(a) and (b) demonstrates that with an increase in sloping angle, number of full reflection decreases in case of obliquely incident waves.
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Fig. 6. Variations of the reflection coefficients Kr versus θ for different values of L/λ1 with (a) h2 /h1 = 0.25 and (b) h2 /h1 = 1 with G = 1 + 1i, α = 0, L1 /λ1 = 0.4 and L2 /λ1 = 1.
Fig. 7. Variations of the reflection coefficients Kr versus θ for different values of (a) α with G = 1 + 1i and (b) G with α = 0, and L/λ1 = 0.4, L1 /λ1 = 0.4, L2 /λ1 = 1.
Fig. 8. Variations of the reflection coefficients Kr versus β0 h1 for different values of (a) α with G = 1 + i and (b) G with α = 0, and h2 /h1 = 0.25, L = 10 m, L1 = 10 m, L2 = 4 m, θ = 30°.
In Fig. 7(a) and (b), the reflection coefficients Kr versus the incident angle θ are plotted for different values of α and porous-effect parameter G respectively. From both the figures, it is revealed that for certain angle of incidence, minimum reflection is observed in each case which may be due to the destructive interference of the incident and reflected waves. Fig. 7(a) reveals that in general the reflection coefficient decreases with an increase in α . This is due to the protrusion and depression of the bed profile as discussed in Fig. 3. However, an opposite trend is observed with an increase in the absolute value of porous-effect parameter G as in Fig. 7(b). Unlike the minimum observed in Fig. 7(a) with the change in bed profile, the minimum observed in Fig. 7(b) for a sloping step type bed profile is due to the change in the phase of the complex porous-effect parameter along with the magnitude of the structural porosity. In Fig. 8(a) and (b), the reflection coefficients Kr versus non-dimensional wave number β0 h1 are plotted for various values of α and porous-effect parameter G respectively. From both the Figures, it is observed that the wave reflection is more oscillatory for smaller value of β0 h1 and the oscillatory pattern decreases with increase in β0 h1 . For smaller values of the
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Fig. 9. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus L/λ1 for different values of G with h2 /h1 = 0.25, L1 /λ1 = 0.4, L2 /λ1 = 1,
α = 0 and θ = 30°.
Fig. 10. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus θ for different values of L/λ1 with h2 /h1 = 0.25, L1 /λ1 = 0.4, L2 /λ1 = 1 and α = 0.
wave number β0 h1 , the highly oscillatory pattern in wave reflection may be due to the partial diffraction of long waves by the sloping bed profile whilst, the decrease in the oscillatory pattern of wave reflection for higher values of the wave number β0 h1 may be due to the predominant role of wave energy concentration near the free surface which is reflected by the porous barrier and the rigid wall. Fig. 8(a) reveals that there is negligible change in wave reflection with change in α . However, Fig. 8(b) demonstrates that the reflection coefficient increases with an increase in the absolute value of G which is similar to that in Fig. 3. In Fig. 9(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus L/λ1 are plotted respectively for different values of porous-effect parameter G. From Fig. 9(a), it is observed that the wave force acting on the porous barrier decreases with an increase in absolute value of G which may be due to the dissipation of more wave energy by the porous barrier and an opposite trend is observed for wave force acting on the rigid wall as in Fig. 9(b). It may be noted that the maximum wave force on the rigid wall corresponds to the value of L/λ1 for which wave reflection attends minimum as in Fig. 5(a). Further, it is observed that Kf attends zero minimum for certain values of L/λ1 for which full reflection occurs as in Fig. 5(a). Similar observation was found in case of wave trapping by vertical flexible porous barrier near a wall by Yip et al. [5]. In Fig. 10(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus angle of incidence θ are plotted respectively for different values of L/λ1 . From Fig. 10(a), it is observed that the wave force acting on the porous barrier decreases with an increase in angle of incidence θ . Further, with change in the normalized distance L/λ1 , wave force on the barrier attend zero minimum for certain angle of incidence θ which is related with full reflection as in Fig. 6(a). On the other hand, a comparison of Fig. 10(a) and (b) demonstrates that for oblique angle of incidence, vanishing of wave force on the porous barrier corresponds to maximum wave force on the rigid wall. In Fig. 11(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus β0 h1 are plotted respectively for different values of porous-effect parameter G. From both the figures, it is observed that with an increase in the absolute value of the porous-effect parameter G, forces acting on the porous barrier decrease and on the rigid wall increase. As observed in Figs. 9 and 10, a comparison of Figs. 11(a) and 8(a) reveals that zero wave force acting on the barrier is related with full reflection of the incident waves. Further, with an increase in the wave number β0 h1 , the oscillatory pattern of wave forces acting on the barrier and rigid wall decreases. Further, Fig. 11(b) depicts that wave forces acting on the rigid wall are higher in the absence of porous barrier compared to that of the presence of the porous barrier which is due to the dissipation of wave energy by the porous barrier.
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Fig. 11. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus β0 h1 for different values of G with α = 0, h2 /h1 = 0.25, L = 10 m, L1 = 10 m, L2 = 4 m and θ = 30°.
Fig. 12. Schematic diagrams of the wave trapping model as in Case II.
Fig. 13. Variations of the reflection coefficients Kr versus L2 /λ1 for different values of (a) d/h2 with m = 4 and (b) m with d/h2 = 0.5, and h2 /h1 = 0.25, L/λ1 = 0.4, L1 /λ1 = 0.4 and θ = 30°.
Case II Here, the bed profile as in Fig. 12 is considered using the bed function h(x) (0 < x < L2 ) as h(x) = h2 + b{1 + 2(x/L2 )3 − 3(x/L2 )2 − (d/b) sin(2mπ x/L2 )},
(4.23)
where b = (h1 − h2 ), d is the amplitude of the ripple and m is the number of ripple. In Fig. 13(a) and (b), the reflection coefficients Kr versus non-dimensional slope length L2 /λ1 are plotted for different values of ripple amplitude d/h2 and number of ripples m respectively. From both the figures, it is demonstrated that the pattern of the reflection coefficient is more oscillatory compared to the result observed for bed profiles in Case I as in Fig. 3. Fig. 13(a) depicts that the amplitude of the oscillatory pattern of the reflection coefficient increases with an increase in ripple amplitude d/h2 . On the other hand, Fig. 13(b) depicts that with an increase in the number of ripples in the bed profile, the oscillatory pattern of the reflection coefficient predominates with an increase in the slope length L2 /λ1 . The increase in the amplitude of the oscillatory pattern in the reflection coefficients may be due to the shoaling and refraction of surface waves over the sinusoidal bed profile. In Fig. 14, the reflection coefficients Kr versus (a) non-dimensional slope length L2 /λ1 and (b) non-dimensional wave number β0 h1 are plotted for different values of porous-effect parameter G. The general pattern of the reflection coefficients
H. Behera et al. / Wave Motion 57 (2015) 219–230
229
Fig. 14. Variations of the reflection coefficients Kr versus (a) L2 /λ1 with L/λ1 = 0.4, L1 /λ1 = 0.4 and (b) β0 h1 with L = 10 m, L1 = 10 m, L2 = 4 m, for different values of G along with d/h2 = 0.5, m = 4, h2 /h1 = 0.25 and θ = 30°.
Fig. 15. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus β0 h1 for different values of G with d/h2 = 0.5, m = 4, h2 /h1 = 0.25, L = 10 m, L1 = 10 m, L2 = 4 m and θ = 30°.
in Fig. 14(a) is similar to Fig. 13 whilst, that in Fig. 14(b) is similar to Fig. 8. However, both the figures depict that there are significant increase in the reflection coefficients with the increase in the absolute value of the porous-effect parameter. With an increase in the absolute value of the porous-effect parameter, the porous barrier becomes more transparent to ocean waves leading to reflection of more energy by the rigid wall. Further, Fig. 14(b) depicts that there is an increase in the number of local optima in the reflection coefficient compared to Fig. 8 which may be due to the oscillatory nature of the step type bottom bed. In Fig. 15(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus the non-dimensional wave number β0 h1 are plotted respectively for different values of porous-effect parameter G. The general pattern of the forces Kf and Kw is similar to the result obtained by bottom profile of Case I as in Fig. 11. As has been discussed earlier, full reflection in Fig. 14(b) is related with zero force on the porous barrier. Similar phenomenon was observed while analyzing wave trapping by partial porous barriers in water of uniform depth (Yip et al. [5] and Behera et al. [10]). Further, Fig. 15(b) reveals that force acting on the rigid wall is much higher in the absence of the porous barrier compared to the presence of the barrier which is similar to the observation made in Fig. 11(b). 5. Conclusion In the present manuscript, oblique wave trapping by porous barrier near a rigid wall is studied in the presence of a step of varied configuration. The mathematical problem is handled for solution using the eigenfunction expansion method along the flat bed region and the fourth order Runge–Kutta method to deal with the modified mild-slope equation along the undulated bed region. The emphasis in the present study is in the understanding of the role of various step profiles on wave trapping by porous structure. To understand the role of the porous structure and step of varied configurations in creating an effective wave trapping system, the reflection coefficient, wave load on porous barrier and rigid wall are computed and analyzed for various wave and structural parameters. It is found that more number of times optimum reflection occurs in case of undulated bed compared to flat bed for the same distance between the porous barrier and rigid wall. Further, minimum wave reflection occurs for various angle of incidence in case of undulated bed and nearly zero minimum reflection occurs for certain parametric values. The minimum in wave reflection is often referred as wave trapping. In addition, full reflection occurs for different values of wave number and distance between the porous barrier and rigid wall and for the
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same parametric values, hydrodynamic forces acting on porous barrier become zero. The present study is likely to be of immense importance in the design of coastal structures for protecting coastal infrastructure with less load on vertical rigid wall like sea wall, port and/or harbor walls. Further, the present model can be used in creation of coastal infrastructures for extending coastal activities in the seaward direction in bay/harbor. Acknowledgment HB gratefully acknowledges the financial support received from Ministry of Earth Sciences, Govt. of India, (No. MoES/11MRDF/1/45/P/10) to pursue this research work. References [1] X. Yu, A.T. Chwang, Wave induced oscillation in harbor with porous breakwaters, J. Waterw. Port Coastal Ocean Eng. 120 (2) (1994) 125–144. [2] A.T. Chwang, A porous wave maker theory, J. Fluid Mech. 132 (1983) 395–406. [3] K.J. McConnell, D.M. Ethelston, N.W.H. Allsop, Low reflection walls for harbors: Development of new structures and application in Hong Kong, in: N.W.H. Allsop (Ed.), Coastlines, Structures and Breakwaters, Thomas Telford, London, UK, 1998, pp. 58–69. [4] T. Sahoo, M.M. Lee, A.T. Chwang, Trapping and generation of waves by vertical porous structures, J. Eng. Mech. 126 (2000) 1074–1082. [5] T.L. Yip, T. Sahoo, A.T. Chwang, Trapping of waves by porous and flexible structures, Wave Motion 35 (2002) 41–54. [6] A.T. Chwang, Y. Chen, Field measurement of ship waves in victoria harbor, J. Eng. Mech. 129 (10) (2003) 1138–1148. [7] J. Bhattacharjee, C. Guedes Soares, Oblique wave interaction with a floating structure near a wall with stepped bottom, Ocean Eng. 38 (2011) 1528–1544. [8] S. Koley, H. Behera, T. Sahoo, Oblique wave trapping by porous structures near a wall, J. Eng. Mech. (2014) 04014122. http://dx.doi.org/10.1061/ (ASCE)EM.1943-7889.0000843. 1–15. [9] R.B. Kaligatla, S. Koley, T. Sahoo, Trapping of surface gravity waves by a vertical flexible porous plate near a wall, Z. Angew. Math. Phys. (2015) http:// dx.doi.org/10.1007/s00033-015-0521-2. [10] H. Behera, S. Mandal, T. Sahoo, Oblique wave trapping by porous and flexible structures in a two-layer fluid, Phys. Fluids 25 (2013) 112110. 23. [11] H. Behera, T. Sahoo, Gravity wave interaction with porous structures in two-layer fluid, J. Eng. Math. 87 (2014) 73–97. [12] S. Zhu, Diffraction of short-crested waves around a circular cylinder, Ocean Eng. 20 (4) (1993) 389–407. [13] C.P. Tsai, D.S. Jeng, J.R.C. Hsu, Computations of the almost highest short-crested waves in deep water, Appl. Ocean Res. 16 (6) (1994) 317–326. [14] H. Song, L. Tao, Short-crested wave interaction with a concentric porous cylindrical structure, Appl. Ocean Res. 29 (4) (2007) 199–209. [15] S. Mandal, N. Datta, T. Sahoo, Hydroelastic analysis of surface wave interaction with concentric porous and flexible cylinder systems, J. Fluids Struct. 42 (2013) 437–455. [16] S.R. Massel, Extended refraction–diffraction equation for surface waves, Coastal Eng. 19 (1993) 97–126. [17] K.D. Suh, W.S. Park, Wave reflection from perforated-wall caisson breakwaters, Coastal Eng. 26 (1995) 177–193. [18] M. Fugazza, L. Natale, Hydraulic design of perforated breakwaters, J. Waterw. Port Coastal Ocean Eng. 118 (1992) 1–14. [19] D. Porter, D.J. Staziker, Extensions of the mild-slope equation, J. Fluid Mech. 300 (1995) 367–382. [20] P.G. Chamberlain, D. Porter, The modified mild-slope equation, J. Fluid Mech. 291 (1995) 393–407. [21] D. Porter, R. Porter, Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography, J. Fluid Mech. 509 (2004) 145–179. [22] L.G. Bennetts, N.R.T. Biggs, D. Porter, A multi-mode approximation to wave scattering by ice sheets of varying thickness, J. Fluid Mech. 579 (2007) 413–443. [23] L.G. Bennetts, N.R.T. Biggs, D. Porter, The interaction of flexural-gravity waves with periodic geometries, Wave Motion 46 (1) (2009) 57–73. [24] S.R. Manam, R.B. Kaligatla, A mild-slope model for membrane-coupled gravity waves, J. Fluids Struct. 30 (2012) 173–187. [25] P.G. Chamberlain, D. Porter, Wave scattering in a two-layer fluid of varying depth, J. Fluid Mech. 524 (2005) 207–228. [26] S.R. Manam, Y. Toledo, Y. Agnon, Complementary mild-slope equations in a two-layer fluid, Wave Motion 48 (3) (2011) 223–234. [27] Y.C. Li, Y. Liu, B. Teng, Porous-effect parameter of thin permeable plates, Coastal Eng. J. 48 (4) (2006) 309–336. [28] J.C.W. Berkhoff, Computation of combined refraction-diffraction, in: Proc. of 13th Int’l Conf. on Coastal Eng., Vancouver, Canada, ASCE, 1973, pp. 471–490. [29] D. Porter, The mild-slope equations, J. Fluid Mech. 494 (2003) 51–63. [30] R. Porter, D. Porter, Water wave scattering by a step of arbitrary profile, J. Fluid Mech. 411 (2000) 131–164.
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Wave Motion journal homepage: www.elsevier.com/locate/wavemoti
Wave trapping by porous barrier in the presence of step type bottom H. Behera, R.B. Kaligatla 1 , T. Sahoo ∗ Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur-721302, India
highlights • • • • •
The study deals with wave trapping by porous barrier having undulated bottom bed. The physical problem is solved using modified mild-slope approximation. Number of times optimum reflection occurs is more for undulated bed than flat bed. Zero reflection occurs for specific undulated bed profiles. The present study will be useful in the creation of tranquility zone.
article
info
Article history: Received 18 December 2014 Received in revised form 25 March 2015 Accepted 20 April 2015 Available online 8 May 2015 Keywords: Oblique wave trapping Porous barrier Modified mild-slope equation Undulated bed Reflection coefficient
abstract The present study deals with the trapping of oblique wave by porous barrier located near a rigid wall in the presence of a step type bottom bed. The solution of the physical problem is obtained using the eigenfunction expansion method and multi-mode approximation associated with modified mild-slope equation. Assuming that the porous structure is made of materials having fine pores, the mathematical problem is handled for solution by matching the velocity and pressure at interface boundaries. Various numerical results are computed and analyzed to understand the role of bed profiles, structural porosity, depth ratio, oblique angle of incidence, distance between barrier and step edge and, the distance between the porous barrier and rigid wall in optimizing wave reflection and load on the structure/rigid-wall. A comparison of results on wave trapping by porous barriers over flat and undulated bed reveals that for the same distance between the porous barrier and rigid wall, more number of times optimum reflection occurs in case of undulated bed. The present study is likely to be of immense importance in the design of coastal structures for protecting coastal infrastructures. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In recent decades, significant increase in developmental activities along the coast due to the rise in world population is exerting greater pressure on the coastal countries for protection of coastal infrastructures/facilities by various measures. In addition, with an increase in global trading, substantial increase in harbor traffic results in the deterioration of wave conditions in various ports and harbors around the world. Further, the rise in sea level is putting additional pressure on the existing coastal infrastructures. In this context, the study on wave interaction with coastal structures provides useful
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (T. Sahoo).
1 Present address: Department of Applied Mathematics, Indian School of Mines, Dhanbad, India. http://dx.doi.org/10.1016/j.wavemoti.2015.04.005 0165-2125/© 2015 Elsevier B.V. All rights reserved.
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information about various physical processes associated with coastal protection or attenuating wave heights in confined water bodies such as bays/ports/harbors. Most of the coastal structures such as breakwaters, used for shore protection and/or creating a tranquility zone by reducing wave impact near harbors, are vertical and rigid in nature. These vertical rigid structures used to collapse under extreme wave climates like tsunami and storm surges. For reducing wave loads on these type of structures, perforated/wave absorbing structures of various configurations are proposed as alternatives due to their ability to dissipate wave energy. In addition, various study on wave trapping by porous structures reveal that wave loads on existing rigid wall can be minimized by introducing porous structure at a suitable distance from the rigid wall. The minimum wave reflection is often referred as wave trapping. In the context of wave interaction with coastal structures, there are mainly two class of problems namely wave scattering and trapping by porous structures are studied in uniform water depth. However, wave motion over undulated sea bed is of considerable interest in coastal engineering practices including development of infrastructures like ports and harbors. Thus, for protection of marine facilities and coastal infrastructures in coastal zones, there is renewed interest to study wave interaction with coastal structures like breakwaters in the presence of undulated bottom bed. There is an extensive literature on wave trapping by rigid or flexible and/or porous vertical structures in the presence of uniform bottom topography. The most widely used model for wave past thin porous structure is based on the modified boundary condition derived by Yu [1] which was initially proposed by Chwang [2]. However, various study on wave trapping are based on small amplitude water wave theory and will be useful for reducing wave forces on vertical sea/harbor wall and for creating low reflection from harbor walls in various ports (as in McConnell et al. [3]). Sahoo et al. [4] studied trapping and generation of surface gravity waves by submerged vertical permeable barriers. Yip et al. [5] analyzed wave trapping by considering flexibility in porous barrier. Further, the study of Chwang and Chen [6] reveals that a knowledge of wave characteristics and harbor configurations helps in the design of new harbor walls or finding ways to determine wave absorbing sea walls by introducing suitable permeable structures at finite location near the harbor wall. Bhattacharjee and Guedes Soares [7] investigated diffraction of water waves by a finite rigid floating structure near wall, considering vertical step in uniform bottom. Applying Sollitt and Cross model, Koley et al. [8] analyzed oblique wave trapping by porous structures having finite width, placed near a rigid wall. Kaligatla et al. [9] analyzed wave trapping by flexible porous plate by converting the boundary value problem into system of integro-differential equations in both the cases of finite and infinite water depths. These studies on wave trapping are performed in homogeneous fluid of uniform water depth. On the other hand, water wave trapping by porous structures of various configurations in two-layer fluid are studied by Behera and Sahoo [10] whilst, oblique wave trapping by partial flexible and porous barriers of different configurations are studied by Behera et al. [11]. The aforementioned discussions on gravity wave interaction with porous structure are for normalized/obliquely incident waves in uniform water depth. Unlike, these plane waves travel in a fixed direction and the associated standing waves fluctuating vertically in a confined region, short-crested waves are doubly periodic along the horizontal directions, one of which is in the direction of propagation and the other being normal to it (as in Zhu [12] and Tsai et al. [13]). Song and Tao [14] studied the interaction of short-crested waves with a concentric porous cylindrical structure in a quantitative manner. Recently, Mandal et al. [15] studied the hydroelastic response of surface waves by porous and flexible cylinder system. In the presence of varying bottom topography, using the Galerkin method proposed by Massel [16]. Suh and Park [17] developed an analytical model to predict oblique wave reflection by a perforated wall caisson mounted on a rubble mound foundation with sloping step, following the model of Fugazza and Natale [18]. However, the above mentioned Galerkin method for varying bottom beds fails to ensure continuity of mass flow at the interface between two domains. The natural mass conserving jump conditions at the bottom slope discontinuities are derived by Porter and Staziker [19] by means of variational principle. In the last two decades, there is significant progress for developing efficient model equations to deal with wave interaction with varying bottoms. Among those, a typical popular model equation is the modified mild-slope equation (MMSE) derived by Chamberlain and Porter [20]. The MMSE with correct interfacial matching conditions at the locations where the bed slope is discontinuous, has been found to be efficient model for bottom slopes up to 1. Further, MMSE was extended to study scattering of flexural and membrane-coupled gravity waves (as in Porter and Porter [21], Bennetts et al. [22], Bennetts et al. [23], Manam and Kaligatla [24]). Apart from use of MMSE for wave motion in homogeneous fluid medium having bottom undulation, the concept of MMSE is extended to study wave scattering due to bottom undulation in two-layer fluid (see Chamberlain and Porter [25] and Manam et al. [26]). In the present paper, oblique wave trapping by thin porous barrier near a wall in the presence of step of arbitrary bottom bed is studied under the assumption of small amplitude water wave theory. In each of the fluid regions, the velocity potentials are expanded in terms of the eigenfunctions and the full solution of the mathematical problem is obtained by matching the continuity of pressure and velocity at interface boundaries. However, the extended modified mild-slope equation (MMSE) along with suitable jump conditions (as in Porter and Staziker [19]) are used to ensure conservation of mass at the interface boundaries in the fluid region having undulated bottom bed. The mild-slope equation is handled for solution using Runge–Kutta method. The reflection coefficients and wave forces are computed and analyzed for different bottom bed profiles to study the role of the porous barrier in trapping oblique gravity waves in the confined zone between the barrier and the rigid wall. For each bed profile, distance between barrier and rigid wall, distance between barrier and step edge, slope length, depth ratio and oblique angle of incidence on the reflection coefficient, wave forces acting on the barrier and rigid wall are studied. Known results available in the literature are reproduced to check the accuracy of the computational results.
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2. Mathematical formulation In this section, oblique surface wave trapping by thin porous barrier is formulated in the presence of varying bottom topography. The physical problem is studied under the assumption of the linearized water wave theory and the porous barrier is assumed to have fine pores. The problem is considered in the three dimensional Cartesian co-ordinate system with x and y-axis being in the horizontal direction and z-axis being in the vertically upward direction. The fluid domain is divided into four sub-domains as in Fig. 1 depending on the varying bottom bed, position of the porous structure and the rigid wall which can be considered as a simplified profile of a navigational channel/bay. A thin porous barrier, extending from bottom to free surface, is kept vertically at a finite distance L from a rigid wall and at a distance L1 from the edge of a step as in Fig. 1. The varying bottom bed profile z = −h(x) in front of the wall is connected by two unequal uniform bottom levels z = −h1 and z = −h2 . Further, the step of variable depth h(x) span over 0 < x < L2 whilst, the open water region −∞ < x < 0 is of uniform depth h1 . It is assumed that along the y-axis, the fluid domain is horizontally extended over 0 < y < ∞. The fluid is assumed to be incompressible, inviscid with motion being irrotational and simple harmonic in time with angular frequency ω. Further, it is assumed that surface waves propagate by making an angle θ with x-axis which ensures the existence of the velocity potentials Φj such that Φj (x, y, z , t ) = Re{φj (x, z )e−i(βy y+ωt ) } with βy = β0 sin θ and β0 being the wave number of the incident wave in region 1. Subscript j = 1, 2, 3, 4 refers to the fluid domains 1, 2, 3, 4 shown in Fig. 1. Thus, the spatial velocity potentials φj (x, z ) for j = 1, 2, 3, 4 satisfy the Helmholtz equation
∂2 ∂2 2 + − β y φj = 0. ∂ x2 ∂ z2
(2.1)
The boundary condition on the uniform bottoms are given by
∂φj = 0 on z = −hj for j = 1, 3, 4, ∂z
(2.2)
whilst, the bottom boundary condition in the undulated bed region is given by
∂φ2 dh ∂φ2 + = 0 on z = −h(x). ∂z dx ∂ x
(2.3)
The linearized free surface boundary condition is given by
∂φj − K φj = 0 on z = 0, for j = 1, 2, 3, 4, ∂z
(2.4)
where K = ω2 /g and g is the gravitational constant. On the rigid vertical wall, vanishing of horizontal velocity yields
∂φ4 = 0 at x = L2 + L1 + L. ∂x
(2.5)
The boundary conditions on the porous barrier at x = L2 + L1 are given by
∂φ3 ∂φ4 = ∂x ∂x
and
∂φj = iβ0 G(φ3 − φ4 ) for j = 3, 4, ∂x
(2.6)
with G being referred as the complex porous-effect parameter and is given by G=
ϵ , β0 d(f − is)
(2.7)
where ϵ is the porosity of the barrier, f the resistance force coefficient, s the inertial force coefficient, d the thickness of the porous barrier. The real part of the complex porous-effect parameter G represents the resistance effect of the porous material against the flow, while the imaginary part of G represents the inertia effect of the fluid inside the porous material. An increase in either parts or the both of them leads to an increase in the transparency of the porous structure in dissipating and transmitting the wave energy. The situation when the resistance effect of the structure dominates, the imaginary part of G becomes negligible and G reduces to the porous-effect parameter introduced by Chwang [2]. Li et al. [27] studied the physical aspect of the porous-effect parameter associated with a thin permeable structure. 3. Method of solution In order to study wave trapping by porous barrier, the method of solution is described in this section for the mixed boundary value problem formulated in the previous section. It is assumed that the bottom is varying in a finite interval (0, L2 ) and is uniform outside this interval. The bottom profile is assumed to be continuously differentiable function in the
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Fig. 1. Schematic diagram of wave trapping model in the presence of step type bed.
interval (0, L2 ). Further, the bottom is allowed to have slope discontinuities at x = 0 and x = L2 . In region 1, the unknown velocity potential φ1 is written as
φ1 (x, z ) = A0 eiµ0 x f0 (β0 , z ) +
∞
Rn e−iµn x fn (βn , z ),
(3.8)
n =0
where fn (βn , z ) = cosh βn (z + h1 )/ cosh βn h1 with µn =
βn2 − βy2 for n = 0, 1, 2, 3, . . . . The constant A0 is assumed to
be known and is associated with the amplitude of the incident waves and R0 being an unknown constant associated with the amplitude of the reflected wave and Rn s are unknown constants to be determined. Here, β0 is the positive real root and βn for n = 1, 2, 3, . . . are the purely imaginary roots of the dispersion equation β tanh β h1 − K = 0 in β . In region 2, the velocity potential φ2 is written as
φ2 (x, z ) =
∞
ψn (x) Wn (h, z ),
(3.9)
n =0
where ψn (x) are unknown functions and Wn = cosh kn (z + h)/ cosh kn h with µ ˜n =
k2n − βy2 . The wave number k0 is a
positive real root and k1 , k2 , k3 , . . . , are purely imaginary roots of the dispersion equation k tanh kh − K = 0 in k. It may be observed that the roots k0 , k1 , k2 , k3 . . . are functions of bottom profile h(x). The eigenfunctions Wn are borrowed from the flat bottom solution. It may be noted that Eq. (3.9) is an approximation for the velocity potential φ2 (x, z ) which does not satisfy the bottom boundary condition explicitly (as in Berkhoff [28], Porter [29] and the literature cited therein). On the other hand, in regions 3 and 4, the velocity potentials φ3 and φ4 are expressed as
φ3 (x, z ) =
∞ (Bn eiqn x + Cn e−iqn x )gn (γn , z )
(3.10)
n =0
and φ4 (x, z ) =
∞
Dn cos qn (x − x1 )gn (γn , z )
(3.11)
n=0
γn2 − βy2 for n = 0, 1, 2, . . . , x1 = L2 + L1 + L, Bn , Cn , Dn are unknown constants and γ0 is a positive real root and γ1 , γ2 , γ3 , . . . , are purely imaginary roots of the dispersion equation γ tanh γ h2 − K = 0 in γ . Hereafter, the infinite series associated with the evanescent modes for the velocity potentials are truncated after N terms. To obtain the velocity potential φ2 (x, z ) as in Eq. (3.9), using the extended modified respectively, where gn (γn , z ) = cosh γn (z + h2 )/ cosh γn h2 with qn =
mild-slope equation (MMSE) as in Porter and Staziker [19], it can be easily derived that d dx
an
dψn
+
dx
N
bmn − bnm
m=0
dh dψm dx dx
2 d2 h dh + bmn 2 + cmn + dmn − βy2 an ψm = 0, dx
dx
(3.12)
where an (h) =
0
Wn2 dz ,
bmn (h) =
−h
cmn (h) =
0
Wn −h
dbmn dh
0
− −h
∂ Wm dz , ∂h
∂ Wm ∂ Wn dz , ∂h ∂h
∂ 2 Wm dz for n = 0, 1, 2, . . . , N . ∂ z2 −h To determine the functions ψn (x) (n = 0, 1, 2, . . . ., N ) in region 2, the MMSE in Eq. (3.12) can be solved numerically. Assuming that the undulated bed can have slope discontinuities at the interfaces x = 0 and x = L2 and using continuity of dmn (h) =
0
Wn
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pressure across these interfaces, the velocity potentials in Eqs. (3.8)–(3.11) yield
ψ0 (x) = A0 eiµ0 x + R0 e−iµ0 x ψn (x) = Rn e−iµn x
at x = 0 for n = 1, 2, . . . , N ,
(3.13)
at x = L2 for n = 1, 2, . . . , N .
(3.14)
and
ψ0 (x) = B0 eiq0 x + C0 e−iq0 x ψn (x) = Bn eiqn x + Cn e−iqn x
Further, to ensure conservation of mass across the interface boundaries at x = 0 and L2 , using the expressions for ψn (x) as in Eqs. (3.13) and (3.14) and proceeding in a similar manner as in Porter and Staziker [19], the jump conditions are obtained as a0
an
dψ0
+ iµ0 a0 ψ0 + h
dx
′
N m=0
dψn
+ iµn an ψn + h
dx
′
N
bm0 ψm = 2iµ0 a0 A0 bmn ψm = 0
m=0
at x = 0+, n = 1, 2, . . . , N
(3.15)
and a0
an
dψ0
− iq0 a0 ψ0 + h
dx
′
N
bm0 ψm = −2ia0 q0 C0 e
−iq0 x
bmn ψm = −2ian qn Cn e
m=0
dψn
− iqn an ψn + h
dx
′
N
at x = L2 −, n = 1, 2, . . . , N .
(3.16)
−iqn x
m=0
Using continuity of velocity and pressure at x = L1 + L2 as in Eq. (2.6), it is derived that N
Bm e
−iqm x
iqm x
− Cm e
+ iDm sin qm (x − x1 ) Xmn = 0,
m=0 N
m=0
at x = L1 + L2 ,
(3.17)
iβ0 G(Bm eiqm x + Cm e−iqm x ) − Dm {sin qm (x − x1 ) + iβ0 G cos qm (x − x1 )} Xmn = 0,
where 0
gm (γm , z )gn (γn , z )dz ,
Xmn =
for m, n = 1, 2, . . . , N .
−h2
Thus, the system of Eqs. in (3.12) can be solved numerically for ψn for specific bed profile h(x). Afterwards, the numerical solutions of ψn s are used to solve the system of equations in Eqs. (3.13)–(3.17) for the determination of the unknowns. 4. Numerical results and discussion In this section, MATLAB is used for solving the system of algebraic equations to study the effects of wave and structural parameters on the reflection coefficient, horizontal hydrodynamic force on the porous barrier and rigid wall for different bed profiles. Although, the general procedure for multi-mode expansion method is described in the previous section, the various numerical experiments are performed based on single mode approximation. In the numerical computation, the modified mild-slope equation is solved using Runge–Kutta method. For numerical results, various physical parameters are kept fixed as wave length of plane gravity wave in region 1 λ1 = 2π /β0 , time period T = 8 s and acceleration due to gravity g = 9.81 m/s2 unless it is mentioned. The reflection coefficient Kr , magnitude of the horizontal wave forces acting on the porous barrier Cf and on the rigid wall Cw are defined as Kr = |R0 /A0 |, Cf = iρω
0
−h 2
Cw = iρω
(4.18)
{φ4 (x, z ) − φ3 (x, z )}
0
−h 2
φ4 (x, z )
dz .
x=(L1 +L2 )
dz ,
(4.19)
(4.20)
x=x1
Further, the non-dimensional form of the horizontal wave forces on the porous barrier Kf and rigid wall Kw are derived using the formulae Kf =
|Cf | ρ gh21
and
Kw =
|Cw | . ρ gh21
(4.21)
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Fig. 2. Schematic diagrams of undulated bed profiles for Case I.
Fig. 3. Variations of the reflection coefficients Kr versus L2 /λ1 for different values of (a) α with G = 1 + 1i and (b) G with α = 0, and h2 /h1 = 0.25, L/λ1 = 0.4, L1 /λ1 = 0.4, θ = 30°.
In the subsequent discussion, numerical results for various bed profiles (as in Porter and Porter [30]) are analyzed separately in different cases. Case I Here, the bed profiles shown in Fig. 2(a), (b), (c), (d) are considered using the bed function h(x) (0 < x < L2 ) as h(x) = h1 − b{1 − α(1 − x/L2 )2 + (α − 1)(1 − x/L2 )},
(4.22)
with b = (h1 − h2 ) and α is a real constant which will give the different bed profiles to be discussed subsequently. In the bed function h(x) as in Eq. (4.22), α = 0 corresponds to the sloping step type bed as in Fig. 2(b). However, in case of α > 1, there is a protrusion above the depth h2 as shown in Fig. 2(a) and for −1 ≤ α < 0, the bed is concave as in Fig. 2(c) whilst, an increase in depression in bed profile will occur for α < −1 as in Fig. 2(d). In Fig. 3(a) and (b), the reflection coefficients Kr versus non-dimensional slope length L2 /λ1 are plotted for different values of α and porous-effect parameter G respectively. In general, with an increase in the slope length, the reflection coefficient increases in an oscillatory pattern. However, for smaller values of slope length, there is a large variation in the amplitude of the oscillatory pattern of the reflection coefficient. This may be due to the fact that for smaller values of slope length, slope angle becomes large which is leading to higher resonating pattern of the resultant waves in the confined zone. The oscillatory pattern in wave reflection is due to the presence of the vertical rigid wall. Further, the oscillatory pattern in wave reflection decreases with an increase in the slope length which is due to the fact that the sloping bed behaves like a uniform bed. From Fig. 3(a), the reflection coefficient decreases with a decrease in the value of α . This may be due to the protrusion and depression of the bottom bed profile. On the other hand, from Fig. 3(b) it is observed that the reflection coefficient increases with an increase in the absolute values of porous-effect parameter G. This may be due to the transmission of more wave energy through the porous barrier for higher values of G. Further, minimum wave reflection occurs for higher values of slope length with an increase in the absolute value of the porous-effect parameter G. This may be due to the phase shift of the reflected wave with change in complex porous-effect parameter G. In Fig. 4(a) and (b), the reflection coefficients Kr versus the normalized distance between the porous barrier and step L1 /λ1 are plotted for different values of α and porous-effect parameter G respectively. From both the figures, it is observed
H. Behera et al. / Wave Motion 57 (2015) 219–230
225
Fig. 4. Variations of the reflection coefficients Kr versus L1 /λ1 for different values of (a) α with G = 1 + 1i and (b) G with α = 0, and h2 /h1 = 0.25, L/λ1 = 0.4, L2 /λ1 = 1, θ = 30°.
Fig. 5. Variations of the reflection coefficients Kr versus L/λ1 for different values of (a) G with h2 /h1 = 0.25 and (b) h2 /h1 with G = 1 + 1i, and α = 0, L1 /λ1 = 0.4, L2 /λ1 = 1, θ = 0°.
that the general patterns of the wave reflection coefficients are periodic in nature with an increase in L1 /λ1 which is due to the uniform depth of the bottom bed beyond the sloping bed. Fig. 4(a) shows that the amplitude in the oscillatory pattern of the reflection coefficient increases with certain phase shift with an increase in α . This may be due to the protrusion and depression of the bottom bed profile as discussed in Fig. 3. Fig. 4(b) shows that the reflection coefficient Kr increases with an increase in the absolute value of the porous-effect parameter G. This is due to the transmission of more wave energy by the porous barrier with an increase in the absolute value of G. In Fig. 5(a) and (b), the reflection coefficients Kr versus the normalized distance between porous barrier and rigid wall L/λ1 are plotted for different values of porous-effect parameter G and depth ratio h2 /h1 respectively. From both figures, it is observed that full reflection occurs periodically. However, a comparison with wave trapping by porous barriers in uniform bed as in Sahoo et al. [4] depicts that due to the presence of bottom undulation more number of times full reflection occurs for the same value of L/λ1 . Fig. 5(a) demonstrates that the reflection coefficient increases with an increase in the absolute value of the porous-effect parameter G as in Figs. 3 and 4. Fig. 5(b) reveals that with an increase in depth ratio h2 /h1 , number of times occurrences of full reflection decreases for the same range of L/λ1 . This may be due to the change in slope of the bottom bed profile. It may be noted that full reflection occurs when the distance between the porous barrier and rigid wall is an integer multiple of half-wavelength of the incident waves for h2 /h1 = 1 as in Sahoo et al. [4]. Further, for h2 /h1 = 0.25, full reflection occurs when the distance between the porous barrier and rigid wall is an integer multiple of one fourth of the incident waves. Thus, in case of a sloping bed, reduction of bed slope increases the occurrences of full reflection. In Fig. 6(a) and (b), the reflection coefficients Kr versus the incident angle θ are plotted for different values of L/λ1 with depth ratio h2 /h1 = 0.25 and h2 /h1 = 1 respectively. It may be noted that the occurrences of full reflection increases with an increase in L/λ1 . Fig. 6(a) reveals that the minimum in the full reflection occurs for higher value of θ with increase in L/λ1 . Further, nearly zero reflection is observed at near θ = 22° for L/λ1 = 1.4. Fig. 6(b) demonstrates that the minimum in wave reflection exponentially increases with an increase in the angle of incidence. For h2 /h1 = 1 with 0 ≤ θ < π /2, i.e., in case of oblique wave trapping by porous barrier having uniform bed, number of times full reflection depend on the ratio λ1 /L and is the greatest integer value of m satisfying the relation λ1 /L ≤ 2/m. It is observed that wave reflection attains a minimum value between two consecutive angle for which full reflection occurs. However, the global minimum in reflection coefficient decreases with an increase in L/λ1 . Similar observation was made by Behera et al. [11] in case of oblique wave trapping by flexible porous barriers in a two-layer fluid in surface mode. A comparison of Fig. 6(a) and (b) demonstrates that with an increase in sloping angle, number of full reflection decreases in case of obliquely incident waves.
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Fig. 6. Variations of the reflection coefficients Kr versus θ for different values of L/λ1 with (a) h2 /h1 = 0.25 and (b) h2 /h1 = 1 with G = 1 + 1i, α = 0, L1 /λ1 = 0.4 and L2 /λ1 = 1.
Fig. 7. Variations of the reflection coefficients Kr versus θ for different values of (a) α with G = 1 + 1i and (b) G with α = 0, and L/λ1 = 0.4, L1 /λ1 = 0.4, L2 /λ1 = 1.
Fig. 8. Variations of the reflection coefficients Kr versus β0 h1 for different values of (a) α with G = 1 + i and (b) G with α = 0, and h2 /h1 = 0.25, L = 10 m, L1 = 10 m, L2 = 4 m, θ = 30°.
In Fig. 7(a) and (b), the reflection coefficients Kr versus the incident angle θ are plotted for different values of α and porous-effect parameter G respectively. From both the figures, it is revealed that for certain angle of incidence, minimum reflection is observed in each case which may be due to the destructive interference of the incident and reflected waves. Fig. 7(a) reveals that in general the reflection coefficient decreases with an increase in α . This is due to the protrusion and depression of the bed profile as discussed in Fig. 3. However, an opposite trend is observed with an increase in the absolute value of porous-effect parameter G as in Fig. 7(b). Unlike the minimum observed in Fig. 7(a) with the change in bed profile, the minimum observed in Fig. 7(b) for a sloping step type bed profile is due to the change in the phase of the complex porous-effect parameter along with the magnitude of the structural porosity. In Fig. 8(a) and (b), the reflection coefficients Kr versus non-dimensional wave number β0 h1 are plotted for various values of α and porous-effect parameter G respectively. From both the Figures, it is observed that the wave reflection is more oscillatory for smaller value of β0 h1 and the oscillatory pattern decreases with increase in β0 h1 . For smaller values of the
H. Behera et al. / Wave Motion 57 (2015) 219–230
227
Fig. 9. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus L/λ1 for different values of G with h2 /h1 = 0.25, L1 /λ1 = 0.4, L2 /λ1 = 1,
α = 0 and θ = 30°.
Fig. 10. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus θ for different values of L/λ1 with h2 /h1 = 0.25, L1 /λ1 = 0.4, L2 /λ1 = 1 and α = 0.
wave number β0 h1 , the highly oscillatory pattern in wave reflection may be due to the partial diffraction of long waves by the sloping bed profile whilst, the decrease in the oscillatory pattern of wave reflection for higher values of the wave number β0 h1 may be due to the predominant role of wave energy concentration near the free surface which is reflected by the porous barrier and the rigid wall. Fig. 8(a) reveals that there is negligible change in wave reflection with change in α . However, Fig. 8(b) demonstrates that the reflection coefficient increases with an increase in the absolute value of G which is similar to that in Fig. 3. In Fig. 9(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus L/λ1 are plotted respectively for different values of porous-effect parameter G. From Fig. 9(a), it is observed that the wave force acting on the porous barrier decreases with an increase in absolute value of G which may be due to the dissipation of more wave energy by the porous barrier and an opposite trend is observed for wave force acting on the rigid wall as in Fig. 9(b). It may be noted that the maximum wave force on the rigid wall corresponds to the value of L/λ1 for which wave reflection attends minimum as in Fig. 5(a). Further, it is observed that Kf attends zero minimum for certain values of L/λ1 for which full reflection occurs as in Fig. 5(a). Similar observation was found in case of wave trapping by vertical flexible porous barrier near a wall by Yip et al. [5]. In Fig. 10(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus angle of incidence θ are plotted respectively for different values of L/λ1 . From Fig. 10(a), it is observed that the wave force acting on the porous barrier decreases with an increase in angle of incidence θ . Further, with change in the normalized distance L/λ1 , wave force on the barrier attend zero minimum for certain angle of incidence θ which is related with full reflection as in Fig. 6(a). On the other hand, a comparison of Fig. 10(a) and (b) demonstrates that for oblique angle of incidence, vanishing of wave force on the porous barrier corresponds to maximum wave force on the rigid wall. In Fig. 11(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus β0 h1 are plotted respectively for different values of porous-effect parameter G. From both the figures, it is observed that with an increase in the absolute value of the porous-effect parameter G, forces acting on the porous barrier decrease and on the rigid wall increase. As observed in Figs. 9 and 10, a comparison of Figs. 11(a) and 8(a) reveals that zero wave force acting on the barrier is related with full reflection of the incident waves. Further, with an increase in the wave number β0 h1 , the oscillatory pattern of wave forces acting on the barrier and rigid wall decreases. Further, Fig. 11(b) depicts that wave forces acting on the rigid wall are higher in the absence of porous barrier compared to that of the presence of the porous barrier which is due to the dissipation of wave energy by the porous barrier.
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Fig. 11. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus β0 h1 for different values of G with α = 0, h2 /h1 = 0.25, L = 10 m, L1 = 10 m, L2 = 4 m and θ = 30°.
Fig. 12. Schematic diagrams of the wave trapping model as in Case II.
Fig. 13. Variations of the reflection coefficients Kr versus L2 /λ1 for different values of (a) d/h2 with m = 4 and (b) m with d/h2 = 0.5, and h2 /h1 = 0.25, L/λ1 = 0.4, L1 /λ1 = 0.4 and θ = 30°.
Case II Here, the bed profile as in Fig. 12 is considered using the bed function h(x) (0 < x < L2 ) as h(x) = h2 + b{1 + 2(x/L2 )3 − 3(x/L2 )2 − (d/b) sin(2mπ x/L2 )},
(4.23)
where b = (h1 − h2 ), d is the amplitude of the ripple and m is the number of ripple. In Fig. 13(a) and (b), the reflection coefficients Kr versus non-dimensional slope length L2 /λ1 are plotted for different values of ripple amplitude d/h2 and number of ripples m respectively. From both the figures, it is demonstrated that the pattern of the reflection coefficient is more oscillatory compared to the result observed for bed profiles in Case I as in Fig. 3. Fig. 13(a) depicts that the amplitude of the oscillatory pattern of the reflection coefficient increases with an increase in ripple amplitude d/h2 . On the other hand, Fig. 13(b) depicts that with an increase in the number of ripples in the bed profile, the oscillatory pattern of the reflection coefficient predominates with an increase in the slope length L2 /λ1 . The increase in the amplitude of the oscillatory pattern in the reflection coefficients may be due to the shoaling and refraction of surface waves over the sinusoidal bed profile. In Fig. 14, the reflection coefficients Kr versus (a) non-dimensional slope length L2 /λ1 and (b) non-dimensional wave number β0 h1 are plotted for different values of porous-effect parameter G. The general pattern of the reflection coefficients
H. Behera et al. / Wave Motion 57 (2015) 219–230
229
Fig. 14. Variations of the reflection coefficients Kr versus (a) L2 /λ1 with L/λ1 = 0.4, L1 /λ1 = 0.4 and (b) β0 h1 with L = 10 m, L1 = 10 m, L2 = 4 m, for different values of G along with d/h2 = 0.5, m = 4, h2 /h1 = 0.25 and θ = 30°.
Fig. 15. Variation of non-dimensional horizontal forces (a) Kf and (b) Kw versus β0 h1 for different values of G with d/h2 = 0.5, m = 4, h2 /h1 = 0.25, L = 10 m, L1 = 10 m, L2 = 4 m and θ = 30°.
in Fig. 14(a) is similar to Fig. 13 whilst, that in Fig. 14(b) is similar to Fig. 8. However, both the figures depict that there are significant increase in the reflection coefficients with the increase in the absolute value of the porous-effect parameter. With an increase in the absolute value of the porous-effect parameter, the porous barrier becomes more transparent to ocean waves leading to reflection of more energy by the rigid wall. Further, Fig. 14(b) depicts that there is an increase in the number of local optima in the reflection coefficient compared to Fig. 8 which may be due to the oscillatory nature of the step type bottom bed. In Fig. 15(a) and (b), the non-dimensional horizontal forces acting on the porous barrier Kf and rigid wall Kw versus the non-dimensional wave number β0 h1 are plotted respectively for different values of porous-effect parameter G. The general pattern of the forces Kf and Kw is similar to the result obtained by bottom profile of Case I as in Fig. 11. As has been discussed earlier, full reflection in Fig. 14(b) is related with zero force on the porous barrier. Similar phenomenon was observed while analyzing wave trapping by partial porous barriers in water of uniform depth (Yip et al. [5] and Behera et al. [10]). Further, Fig. 15(b) reveals that force acting on the rigid wall is much higher in the absence of the porous barrier compared to the presence of the barrier which is similar to the observation made in Fig. 11(b). 5. Conclusion In the present manuscript, oblique wave trapping by porous barrier near a rigid wall is studied in the presence of a step of varied configuration. The mathematical problem is handled for solution using the eigenfunction expansion method along the flat bed region and the fourth order Runge–Kutta method to deal with the modified mild-slope equation along the undulated bed region. The emphasis in the present study is in the understanding of the role of various step profiles on wave trapping by porous structure. To understand the role of the porous structure and step of varied configurations in creating an effective wave trapping system, the reflection coefficient, wave load on porous barrier and rigid wall are computed and analyzed for various wave and structural parameters. It is found that more number of times optimum reflection occurs in case of undulated bed compared to flat bed for the same distance between the porous barrier and rigid wall. Further, minimum wave reflection occurs for various angle of incidence in case of undulated bed and nearly zero minimum reflection occurs for certain parametric values. The minimum in wave reflection is often referred as wave trapping. In addition, full reflection occurs for different values of wave number and distance between the porous barrier and rigid wall and for the
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