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Reflection of oblique ocean water waves by a vertical porous structure placed on a multi-step impermeable bottom
Applied Ocean Research 47 (2014) 373–385
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Reflection of oblique ocean water waves by a vertical porous structure placed on a multi-step impermeable bottom S. Das, S.N. Bora ∗ Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India
a r t i c l e
i n f o
Article history: Received 4 December 2013 Received in revised form 30 March 2014 Accepted 6 July 2014 Keywords: Porous structure Oblique wave Reflection Matching condition Multi-step bottom
a b s t r a c t Based on linear water wave theory, wave reflection by a vertical porous structure placed on an elevated impermeable seabed, assumed to consist of a number of horizontal steps, is considered. The reflection phenomenon is investigated for two forms of the bottom topography: first a 2-step one and then a pstep one. An oblique incident wave propagates through the porous structure and gets reflected by the steps and a vertical solid wall which supports the structure at one end. Boundary value problems are set up in the two different media, the first medium being water and the second medium being the porous structure consisting of p vertical regions – one above each step. By using the matching conditions along the vertical boundaries, a system of linear equations is deduced. Reflection coefficient is obtained by solving this system of equations. The behavior of the reflection coefficient due to different relevant parameters is studied: the effect of various parameters, such as depth, porosity, number of evanescent modes, structure width and angle of incidence is studied graphically for both cases. It is observed that when the porous structure is considered above a 2-step bottom, the number of evanescent modes, porosity and the angle of incidence do not affect the reflection coefficient for relatively long waves. Lower values of friction factor results in oscillation in the reflection coefficient which vanishes with an increase in the values of friction factor. Up to a certain range of the angles of incidence, reflection coefficient is independent of the values of porosity. When a p-step bottom is considered, certain observations remain the same except that the behavior of reflection coefficient against the angle of incidence is different. In the end, both cases are compared by considering the same set of parameters. Justification of our model is presented by matching it with an available one. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The concept of porous media is used in many areas of applied science and engineering: filtration, mechanics (acoustics, geomechanics, soil mechanics, rock mechanics [1]), engineering (petroleum engineering, bioremediation [2], construction engineering), geosciences (hydrogeology, petroleum geology [3], geophysics), biology, biophysics [4], material science, etc. These works amply justify that fluid flow through porous media is a subject of immense common interest and has, hence, emerged as a separate field of study. In coastal areas, porous structures are widely used as breakwaters to protect harbors, inlets and beaches from wave action, and as dissipating sea-walls to attenuate the wave energy in harbors. Because of this enormous significance of interaction of porous structures with ocean waves, we are motivated to
∗ Corresponding author. Tel.: +91 361 258 2604; fax: +91 361 258 2649. E-mail addresses: [email protected] (S. Das), [email protected], [email protected] (S.N. Bora). 0141-1187/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apor.2014.07.001
investigate some specific cases of reflection by a porous structure placed on a step-like topography. Many aspects of interaction between waves and porous media have been studied extensively. Theoretical solutions for reflection and transmission coefficients for certain types of porous structures have been analyzed previously by a number of researchers. The most widely used model of wave-induced flow in porous medium is the one developed by Sollitt and Cross [5]. According to their approach, dissipation of wave energy inside a porous medium is taken into account through a linearized friction term f which is evaluated by an iterative procedure. Madsen [6] derived a simple solution for reflection and transmission from a rectangular porous structure under normal incidence of long waves based on the linearized form of the governing equations and a linearized form of the flow resistance formula. Madsen [7] obtained a theoretical solution for the reflection of linear shallow-water waves from a vertical porous wave absorber placed on a horizontal bottom. The friction term describing the energy loss inside the absorber was linearized and thereafter, by using Lorentz principle of equivalent work, the reflection coefficient was determined as a function of parameters describing the incoming waves and the absorber characteristics.
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Kirby and Dalrymple [8] investigated the diffraction of obliquely incident surface waves by an asymmetric trench in which they developed a numerical solution by matching the particular solution for each subregion of constant depth along the vertical boundaries. An approximate solution based on plane-wave modes was derived and compared with the numerical solution. Sulisz [9] formulated a theory to predict wave reflection and transmission at an infinite rubble-mound breakwater under normal wave incidence. Dalrymple et al. [10] adopted the approach of Sollitt and Cross [5] to analyze the reflection and transmission of oblique incident waves from infinitely long porous structures. Losada et al. [11] extended this study to the case of an infinitely long, homogeneous, vertical structure capped with an impervious element under oblique incident wave. Mallayachari and Sundar [12] took into account the effects of an uneven seabed. The variation of the reflection coefficient with respect to the porosity of the wall, its friction factor and the relative wall width was studied. Zhu [13] used wave induced refractiondiffraction equations for surface waves in the region occupied by a porous structure. He utilized the orthogonality of the depthdependent functions. Liu et al. [14] examined the hydrodynamic performance of a modified two-layer horizontal-plate breakwater consisting of an upper submerged horizontal porous plate and a lower submerged horizontal solid plate. Liu and Li [15] extended this work by considering a double curtain-wall breakwater whose seaward wall was perforated and shoreward wall impermeable. Cho et al. [16] further extended the idea of Liu et al. [14] by considering the lower submerged horizontal plate as a porous one instead of an impermeable one. Das and Bora [17] investigated wave reflection by a rectangular porous structure placed on an elevated bottom and supported by a vertical wall. The variation of reflection with respect to the number of modes, porosity, structure width, etc, was studied. Das and Bora [18] also studied wave damping by a vertical rectangular porous structure placed near and away from a rigid vertical wall. They computed the reflection coefficients for various depths, structure width and porosity. The objective of this work is to solve a water wave scattering problem due to the presence of a vertical porous structure, placed on a 2-step or a p-step horizontal bottom and supported by a rigid vertical wall at one end, and to study the reflection characteristics. The effect of various parameters, such as number of evanescent modes, porosity, friction factor, structure width, angle of incidence on the reflection coefficient is investigated and the results are presented graphically. To the best of the knowledge of the authors, no one has solved the problem of wave propagation through a porous structure that is placed on a step-like impermeable bottom. 2. Mathematical formulation Though the present problem is a specific one, it is fairly important to discuss some general features and equations which usually arise in wave propagation in porous medium. Therefore, in the following subsections a brief description on the specific porous structure under consideration is presented along with these equations ahead of the formulation of the present problem.
Fig. 1. Front view of the structural model.
as cement, ceramics, etc. Depending upon the pore size, porous structures can be categorized into three types, namely, microporous (smaller than 2 nanometers), mesoporous (between 2 and 50 nanometers) and macroporous (larger than 50 nanometers). Metallic foams are good examples of porous materials with higher porosity (ranging from 0.6 to 0.95). The construction of these types of materials can be performed through many means, the main one being the “lost-foam casting”. Porous structures, with high porosity but with considerable stability, to be used as breakwater in ocean and coastal engineering can be constructed from low melting metals and alloys such as copper, aluminum, lead, tin, zinc, etc. The porous structure under consideration in this manuscript is taken as a such type of structure. Fig. 1 presents a rough visual representation of the present structural model. 2.2. General theory for flow inside porous medium Small amplitude wave motion is considered within an undeformable porous medium. It is assumed that the porous structure is homogeneous and isotropic. The fluid motion follows the continuity equation and the equation of motion in terms of the seepage fluid velocity U and dynamic pressure P, which are given by
∇ · U = 0, S
∂U ∇ P + fωU = 0, + ∂t
(2.1b)
where is the density of the fluid, f is the linearized friction factor and ω is the angular frequency of the incident wave. The inertial coefficient S is defined by S =1+
CM (1 − ) ,
(2.2)
where CM is the added mass coefficient and is the porosity of the porous structure. The physical significance and derivation of Eqs. (2.1a,b) and (2.2) are described in Appendix A. A pore velocity potential ˚(x, y, z, t) is introduced to describe the wave-induced fluid motion in the porous medium: U = ∇ ˚.
(2.3)
Integration of equation of motion (2.1b) leads to Bernoulli’s equation
2.1. Porosity and porous structure S A porous medium mostly consists of pores through which fluid (be it gas or liquid such as water) can pass. The skeleton part of the medium is mainly solid (but foam is considered to be porous though). A porous medium may be an aggregate of large number of particles like sand, gravels or a solid containing many capillaries such as porous rock. Many natural substances like sponge, soil, biological materials (bone, lungs, etc.) are few examples of porous material. There are many man-made porous materials such
(2.1a)
∂˚ P + + fω˚ = 0. ∂t
(2.4)
The solution is assumed to be harmonic in time. So the fluid velocity, dynamic pressure and velocity potential can, respectively, be written as U = u(x, y, z) exp(−iωt),
P = p(x, y, z) exp(−iωt),
˚ = (x, y, z) exp(−iωt),
(2.5)
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
where u, p and are spatial functions. Substitution of these values into Eqs. (2.1a), (2.1b) and (2.3) leads to the equations
∇ · u = 0, ∇p
(2.6a)
+ ω˛u = 0,
(2.6b)
u = ∇ ,
(2.6c)
where ˛ = f − iS is the dimensionless impedance of the porous medium. Using Eq. (2.6c) in Eq. (2.6a), we get the spatially dependent potential function which satisfies Laplace’s equation
∇ 2 = 0.
(2.7)
If the free surface is defined by z = (x, y, t), then the linearized dynamic free surface boundary condition can be obtained from Eq. (2.4) with P = g as =−
ω˛ (x, y, 0) exp(−iωt). g
(2.8)
The linearized kinematic free surface boundary condition can be written as
∂ ∂ {(x, y, z) exp(−iωt)} = ∂z ∂t
at z = 0.
(2.9)
Combining Eqs. (2.8) and (2.9), the following single free surface condition is obtained:
∂ − i˛ = 0 at z = 0, ∂z
(2.10)
where = ω2 /g with g as the gravitational constant. The condition to be satisfied at the impermeable bottom is
∂ = 0, ∂n
z = −h(x, y),
(2.11)
where ∂/∂n represents differentiation along the normal to the bottom boundary h(x, y) which depicts a variable bottom topography. For a flat seabed at z = − h, where h is constant, the general solution of the two-dimensional wave motion within a porous medium can be obtained as follows by solving Eqs. (2.7)–(2.11) for the potential in x and z with the help of the technique of separation of variables: =
∞ n=0
˜ n exp(ikn x) + B˜ n exp(−ikn x)]Zn (h, z), [A
(2.12)
375
˜ n and B˜ n are constants and Zn (h, z) is the depth-dependent where A function defined by Zn (h, z) =
cosh kn (h + z) , cosh kn h
n = 0, 1, 2, . . .,
(2.13a)
with the complex wave number kn satisfying the complex dispersion relation i˛ = kn tanh kn h.
(2.13b)
The characteristics of waves in the porous medium can be described by the dimensionless complex wave number kn h. It is known that for a non-dissipative medium (f = 0), k0 is purely real, while kn (n ≥ 1) are purely imaginary. For a dissipative medium, we note that the influence of the friction factor f is to damp the wave motion. This damping can be achieved by adding an imaginary part to k0 and real parts to kn (n ≥ 1). By doing so, the amplitude of both the propagating mode and the evanescent modes will decay. 2.3. Formulation for scattering by a 2-step bottom Let us consider a vertical porous structure of width L + D, a very small portion of which is above the free surface, placed on a 2-step bottom topography above the horizontal ocean bottom, which is at a constant depth h1 from free surface, and resting against a rigid vertical wall. Using rectangular Cartesian coordinate system, the positive x-direction is defined as the direction of the normal wave incident on the porous structure at x = 0, the positive z-direction is considered vertically upwards and the mean free surface as z = 0. Due to the 2-step bottom topography under the porous structure, two different heights of the porous structure exist, namely, h2 when 0 < x < L and h3 when L < x < L + D. The water region (− ∞ < x < 0) is labeled as region I, the porous structure above the first step (0 < x < L) and above the second step (L < x < L + D) as region II and region III, respectively (Fig. 2). The horizontal bottom for region I is considered at z = − h1 . After propagating through the porous structure, the wave gets reflected by the vertical wall at x = L + D. The fluid is assumed to be incompressible, homogeneous and inviscid, and the motion irrotational. According to Sollitt and Cross [5], velocity potentials can be assumed to exist within the porous structure as well as in the fluid region. We define three velocity potentials ˚1 (x, y, z, t) = 1 (x, z) exp(i y − iωt), ˚2 (x, y, z, t) = 2 (x, z) exp(i y − iωt) and ˚3 (x, y, z, t) = 3 (x, z) exp(i y − iωt) for region I, region II and region III, respectively, where = k1,0 sin with k0 and being the
Fig. 2. Schematic diagram of the problem with a 2-step bottom.
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incident wave number and the angle made by the incident wave to the positive x-direction. The governing equation and the boundary conditions in region I (− ∞ < x < 0, −h1 < z < 0) are
where R0 is the complex reflection coefficient, Rn correspond to the decaying modes of reflection; and the depth-dependent function Z1,n (h1 , z) and K1,n are, respectively, given by
∇ 2 1 − 2 1 = 0;
Z1,n (h1 , z) =
∂1 − 1 = 0; ∂z
−∞ < x < 0, −∞ < x < 0,
∂1 = 0; ∂z
−∞ < x < 0,
∂1 = 0; ∂x
x = 0,
−h1 < z < 0, z = 0,
(2.14a) (2.14b)
z = −h1 ,
(2.14c)
−h1 < z < −h2 .
(2.14d)
The governing equation and the boundary conditions in region II (0 < x < L, −h2 < z < 0) are
∇ 2 2 − 2 2 = 0;
0 < x < L,
−h2 < z < 0,
(2.15a)
∂2 − i˛2 = 0; ∂z
0 < x < L,
z = 0,
(2.15b)
∂2 = 0; ∂z
0 < x < L,
∂2 = 0; ∂x
x = L,
z = −h2 ,
2 K1,n = (k1,n − 2 )
1/2
−h2 < z < −h3 .
(2.15d)
L < x < L + D,
−h3 < z < 0,
(2.16a)
∂3 − i˛3 = 0; ∂z
L < x < L + D,
z = 0,
(2.16b)
∂3 = 0; ∂z
L < x < L + D,
∂3 = 0; ∂x
x = L + D,
z = −h3 ,
(2.16c)
1 = i˛2 ,
x = 0,
∂1 ∂2 = , ∂x ∂x
−h3 < z < 0.
−h2 ≤ z ≤ 0,
x = 0,
x = L,
∂2 ∂3 = , ∂x ∂x
(3.19b)
= k1,n tanh k1,n h1 .
(3.19c)
Eq. (3.19c) has one positive real root k1,0 corresponding to the incident and reflected modes of wave propagation, and infinitely many purely imaginary roots k1,n ; n = 1, 2, . . ., which correspond to the evanescent modes. We truncate the infinite sum at n = N so as to consider a finite number of evanescent modes N only: 1 (x, z) = [exp(iK1,0 x) + R0 exp(−iK1,0 x)]Z1,0 (h1 , z) +
N
Rn exp(−iK1,n x)Z1,n (h1 , z).
(3.20)
In region II (0 < x < L), the horizontal impermeable bottom is considered at z = − h2 . Hence, the integrals in this porous region must be from z = − h2 to z = 0 in order to meet the matching conditions and it is required to compute the relevant integrals from z = − h2 to z = 0. By using separation of variables method, the velocity potential 2 (x, z), after truncating the infinite sum at n = N, can be written as
2 (x, z) =
N
{A2,n exp(iK2,n x) + B2,n exp[−iK2,n (x − L)]}Z2,n (h2 , z), (3.21)
(2.16d)
−h2 ≤ z ≤ 0,
−h3 ≤ z ≤ 0,
x = L,
−h3 ≤ z ≤ 0.
(2.17a) (2.17b) (2.17c)
2 − 2 ) where A2,n and B2,n are arbitrary constants; K2,n = (k2,n with k2,n satisfying the dispersion relation
i˛ = k2,n tanh k2,n h2 ,
Z2,n (h2 , z) =
cosh k2,n (h2 + z) , cosh k2,n h2
n = 0, 1, 2, . . ., N.
3 (x, z) =
N
{C3,n cos[K3,n (x − L − D)]}Z3,n (h3 , z),
(3.23)
n=0
i˛ = k3,n tanh k3,n h3 ,
1 (x, z) = [exp(iK1,0 x) + R0 exp(−iK1,0 x)]Z1,0 (h1 , z)
Z3,n (h3 , z) = (3.18)
(3.22b)
In region III (L < x < L + D), the horizontal impermeable bottom is considered at z = − h3 . Hence, the integrals in the porous region must be from z = − h3 to z = 0 in order to meet the matching conditions and it is required to compute the relevant integrals from z = − h3 to z = 0. The velocity potential 3 (x, z), after truncating the infinite sum at n = N, can be written as
The potential 1 (x, z) in region I can be written in the following form:
Rn exp(−iK1,n x)Z1,n (h1 , z),
(3.22a)
2 − 2 ) where C3,n are arbitrary constants; K3,n = (k3,n satisfying the dispersion relation
3. Reflection by the porous structure
1/2
and Z2,n (h2 , z) is the depth-dependent function given by
(2.17d)
Derivation of the matching conditions (2.17a,d) is detailed in Appendix B.
n=1
,
n=0
In addition to the governing equations and different boundary conditions for each region, there exist some other conditions, called matching conditions, along the common boundary of any two successive media. These conditions imply the continuity of pressure and mass flux across the boundary. The matching conditions along the vertical boundaries x = 0 and x = L are given by
+
(3.19a)
n=1
∇ 2 3 − 2 3 = 0;
∞
n = 0, 1, 2, . . .,
with k1,n satisfying the dispersion relation
(2.15c)
The governing equation and the boundary conditions in region III (L < x < L + D, −h3 < z < 0) are
2 = 3 ,
cosh k1,n (h1 + z) , cosh k1,n h1
1/2
with k3,n (3.24a)
and Z3,n (h3 , z) is the depth-dependent function given by cosh k3,n (h3 + z) , cosh k3,n h3
n = 0, 1, 2, . . ., N.
(3.24b)
Using the matching conditions (2.17a–d), the following equations in (4N + 4) unknowns Rn , A2,n , B2,n , C2,n (n = 0, 1, . . ., N) are
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
obtained:
1
N= 0
N
(1 + R0 )Z1,0 +
0.9
N
Rn Z1,n = i˛
n=1
N= 2 0.8
[A2,n + B2,n exp(iK2,n L)]Z2,n ,
|R | 0
N
K1,0 (1 − R0 )Z1,0 −
N= 6
0.6
N= 8
0.5
N= 10
0.4
K1,n Rn Z1,n
0.3
n=1
0.2
N
[A2,n − B2,n exp(iK2,n L)]K2,n Z2,n ,
(3.25b)
0.1 0
n=0
N
0
1
2
3
ν h1
N
[A2,n exp(iK2,n L) + B2,n ]Z2,n =
n=0
C3,n cos(K3,n D)Z3,n ,
N
N
=
5
6
7
Fig. 3. Variation of |R0 | against h1 for different numbers of evanescent modes (N) with f = 1, (L + D)/h1 = 1 and = 0.9.
iK2,n [A2,n exp(iK2,n L) − B2,n ]Z2,n
n=0
4
(3.25c)
n=0
N
N= 4
0.7
n=0
(3.25a)
=
377
iK2,n exp(iK2,n L)B1,n,m A2,n
n=0
K3,n C3,n sin(K3,n D)Z3,n .
(3.25d) +
n=0
N
iK2,n B1,n,m B2,n − K3,m sin (K3,m D)a3,m C3,m = 0,
(3.26d)
n=0
0 2 Zq,m dz =
aq,m = −hq
hq tanh Kq,m hq 2Kq,m hq
0
Bq,n,m =
Zq,n Zq+1,m dz = −hq+1
⎧ ⎪ ⎪ ⎪ ⎨
where 1+
2Kq,m hq sinh 2Kq,m hq
1 2 − K2 K1,n 2,m
(1 − i˛) −
q = 1, 2, K1,n sinh K1,n (h2 − h1 ) cosh K1,n h1 cosh K2,m h2
when q = 1,
⎪ K2,n sinh K2,n (h3 − h2 ) 1 ⎪ ⎪ − ⎩ K2 − K2 cosh K2,n h2 cosh K3,m h3 2,n 3,m
when q = 2.
From Eqs. (3.26a–d) a system of linear equations with (4N + 4) equations in (4N + 4) unknowns can be constructed as follows: MX = b, where M is a square matrix of size (4N + 4), X = [R0 , R1 , . . .RN , A2,0 , A2,1 , . . .A2,N , B2,0 , B2,1 , . . .B2,N , C3,0 , C3,1 , . . .C3,N ]T is the unknown vector and
b = [−B1,0,0 , −B1,0,1 , . . . − B1,0,N , K1,0 B1,0,0 , K1,0 B1,0,1 , . . .K1,0 B1,0,N , 0, . . .. . .. . .0]T .
2N+2 times
By using the orthogonality property of Z2,m (h2 , z) and Z3,m (h3 , z) (for all m = 0, 1, . . ., N) in Eqs. (3.25a–b) and (3.25c–d), respectively, the above equations reduce to N
4. Numerical results B1,n,m Rn − i˛a2,m A2,m − i˛ exp(iK2,m L)a2,m B2,m = −B1,0,m ,
n=0
(3.26a)
N n=0
B1,n,m K1,n Rn + K2,m a2,m A2,m − K2,m exp(iK2,m L)a2,m B2,m
= K1,0 B1,0,m ,
N
(3.26b)
exp(iK2,n L)B1,n,m A2,n
n=0
+
By solving this system, R0 can be evaluated and subsequently the reflection phenomenon within the porous structure can be discussed.
N n=0
B1,n,m B2,n − cos (K3,m D)a3,m C3,m = 0,
(3.26c)
For computational purpose, some specific values of the parameters are considered: L/h1 = 0.5; D/h1 = 0.5; D/L = 1; f = 1; S = 1; = 0.7, 0.8 and 0.9; h1 − h2 = h2 − h3 = h1 /16; = 0◦ and h1 = 2. First we investigate the effect of the number of evanescent modes on the reflection coefficient |R0 | for various wave numbers (h1 ). The ratio (L + D)/h1 = 1 is maintained throughout and the values N = 0, 2, 4, . . ., 10 are considered. From Fig. 3, it is observed that up to approximately h1 < 2.4, reflection can be described by the propagating mode (N = 0) only. As the wave number increases, i.e., for short waves, it is evident that the number of evanescent modes starts affecting |R0 | but it converges to a fixed value for higher numbers of evanescent modes. Porosity of the structure plays an important role in the reflection characteristics. Fig. 4 shows that for relatively long waves (h1 < 1), porosity does not affect reflection, but as the value of the wave number increases, the values of |R0 |, corresponding to the highest value of porosity considered ( = 0.9), are considerably low. In
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S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385 1
1
γ= 0.7
0.9 0.8
0.9
γ= 0.8
γ= 0.8
0.7
0.7
γ= 0.9
0.6
|R0|
γ= 0.7
ν= 0.25
0.8
0.6
|R0|
0.5
γ= 0.9
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0 0
1
2
3
4
5
6
0
7
0.2
0.4
0.6
0.8
νh
1
1.2
1.4
1.6
1.8
2
(L+D)/h1
1
Fig. 4. Variation of |R0 | against h1 for different with (L + D)/h1 = 1, f = 1 and N = 9.
Fig. 7. Variation of |R0 | against dimensionless width of the porous structure ((L + D)/h1 ) for different with f = 1, and N = 9.
1 1
f= 0.25
0.9
(L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1
0.9
0.8
f= 0.5 0.8
0.7 f= 0.75
0.6
|R | 0.5 0
0.7 0.6
f= 1
|R 0|
0.4
0.5 0.4
0.3
0.75 1 1.25 1.5 1.75 2
0.3
0.2
0.2
0.1
0.1
0 0
1
2
3
ν h1
4
5
6
7
0 0
1
2
3
ν h1
Fig. 5. Variation of |R0 | against h1 for different f with (L + D)/h1 = 1, = 0.9 and N = 9.
other words, as the porosity reduces, higher reflection coefficient is encountered. The effect of the friction factor f on the reflection coefficient is studied for various wave numbers. Fig. 5 shows oscillation of |R0 | for lower values of the friction factor considered. As the value of f increases, the oscillation vanishes. Further, Fig. 6 illustrates the effect of friction factor on the reflection coefficient for various dimensionless width ((L + D)/h1 ) of the porous structure. Lower values of f considered result in oscillation in |R0 | which vanishes with an increase in the value of the friction factor (here for f = 1). We also study the effect of the dimensionless width of the porous structure on reflection. |R0 | is plotted against the dimensionless width of the porous structure for different values of porosity in Fig. 7. Reflection coefficient decreases with an increase in (L + D)/h1 before attaining a fixed value for each of the porosity values considered. It is also observed that higher porosity results in lower reflection coefficient.
4
5
6
7
Fig. 8. Variation of |R0 | against h1 for different width of the porous structure ((L + D)/h1 ) with f = 1, = 0.9 and N = 9.
Fig. 8 illustrates the effect of wave number on the reflection coefficient for different values of width (L + D)/h1 . It is observed that oscillation in |R0 | exists for lower values of width and it disappears with an increase in width before converging to each other. Fig. 9 shows the variation of |R0 | against angle of incidence for different values of porosity. It is observed that, for < 45◦ , |R0 | is more or less the same for different porosity values. But for higher values of (>45◦ ), |R0 | attains its minimum value for lower value of porosity (here for = 0.7) and then starts increasing before converging to each other. We further study the effect of wave number on reflection phenomenon for different angles of incidence . Four different values of
, namely = 0◦ , 10◦ , 20◦ and 30◦ , are considered. It is observed from Fig. 10 that for relatively long waves (h1 < 1.5), |R0 | is same for all angles of incidence considered and as the wave number increases further, higher gives rise to lower reflection coefficient.
1
1
0.9
f= 0.25
0.9
0.8
f= 0.5
0.8
0.7
γ= 0.7 γ= 0.8
0.7
f= 0.75
γ= 0.9
0.6
0.6
|R 0|
0.5
f=1
|R0 |
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(L+D)/h1
Fig. 6. Variation of |R0 | against dimensionless width of the porous structure ((L + D)/h1 ) for different f with = 0.9 and N = 9.
0
10
20
30
40
50
60
70
80
90
θ Fig. 9. Variation of |R0 | against incident wave angle ( ) for different with (L + D)/h1 = 1, f = 1 and N = 9.
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
∂j − i˛j = 0 ; ∂z
379
xj−1 < x < xj ,
∂j = 0; ∂z
xj−1 < x < xj ,
∂j = 0; ∂x
x = xj ,
z = 0,
z = −hj ,
−hj < z < −hj+1 .
(5.27b) (5.27c) (5.27d)
The velocity potential p+1 satisfies Eqs. (5.27a–c) along with one extra condition
∂p+1 = 0; ∂x Fig. 10. Variation of |R0 | against h1 for different incident wave angle ( ) with (L + D)/h1 = 1, f = 1, = 0.9 and N = 9.
5. Scattering by a p-step bottom 5.1. Formulation Here we consider p number of horizontal steps, in place of 2 steps, to represent the uneven bottom topography under the porous structure (Fig. 11). hj+1 (j = 2, 3, . . ., p + 1) is considered to be the height of the porous structure above the jth horizontal step and a velocity potential ˚j (x, y, z, t) = j (x, z) exp(i y − iωt) is defined in the region xj−1 < x < xj , −hj < z < 0 for each j where x1 = 0 < x2 < · · · < xp+1 = L
x = xp+1 = L,
−hp+1 < z < 0.
(5.28)
The matching conditions along the boundary x = x1 = 0, −h2 ≤ z ≤ 0, are given by 1 = i˛2 ,
(5.29a)
∂1 ∂2 = . ∂x ∂x
(5.29b)
The matching conditions along the boundary x = xj , −hj+1 ≤ z ≤ 0 (j = 2, 3, . . ., p), are given by j = j+1 ,
(5.30a)
∂j ∂j+1 = . ∂x ∂x
(5.30b)
5.2. Solution
with xj − xj−1 = l.
The boundary value problem in the water region is given by Eqs. (2.14a–d) – exactly the same as in the first case. The governing
j (x, z) =
Depth dependent function Z1,n (h1 , z), K1,n and the potential function 1 in region I satisfy Eqs. (3.19a), (3.19b) and (3.20), respectively. Potential function j (x, z) in the jth porous region is given by
⎧ N ⎪ ⎪ ⎪ {Aj,n exp[iKj,n (x − xj−1 )] + Bj,n exp[−iKj,n (x − xj )]}Zj,n (hj , z), j = 2, 3, . . ., p, ⎪ ⎨ n=0
N ⎪ ⎪ ⎪ ⎪ Cp+1,n cos[Kp+1,n (x − L)]Zp+1,n (hp+1 , z), j = p + 1, ⎩
(5.31)
n=0
equation and the boundary conditions in the jth (j = 2, 3, . . ., p) region are
∇ 2 j − 2 j = 0 ;
xj−1 < x < xj ,
−hj < z < 0,
(5.27a)
where Aj,n , Bj,n and Cp+1,n are arbitrary constants; Kj,n = 2 − 2 ) (kj,n
1/2
with kj,n satisfying the dispersion relation
i˛ = kj,n tanh kj,n hj ,
Fig. 11. Schematic diagram of the problem with a p-step bottom.
(5.32a)
380
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
and Zj,n (hj , z) is the depth-dependent function in region j given by Zj,n (hj , z) =
cosh kj,n (hj + z) cosh kj,n hj
j = 2, 3, . . ., p + 1.
,
(5.32b)
Now using the matching conditions (5.29a–b) and (5.30a–b) along the boundary x = xj , −hj ≤ z ≤ 0 (j = 1, 2, . . ., p − 1), the following equations are obtained:
(1 + R0 )Z1,0 +
N
N
Rn Z1,n = i˛
n=1
[A2,n + B2,n exp(iK2,n l)]Z2,n ,
N
B1,n,m K1,n Rn + K2,m a2,m A2,m
n=0
− K2,m exp(iK2,m l)a2,m B2,m = K1,0 B1,0,m ,
N
exp(iKj,n l)Bj,n,m Aj,n +
n=0
N
Bj,n,m Bj,n − aj+1,m Aj+1,m
n=0
− exp(iKj+1,n l)aj+1,m Bj+1,m = 0,
n=0
(5.34b)
(5.34c)
(5.33a)
K1,0 (1 − R0 )Z1,0 −
N
N
K1,n Rn Z1,n n=0
n=1
=
N
Kj,n exp(iKj,n l)Bj,n,m Aj,n −
N
Kj,n Bj,n,m Bj,n
n=0
− Kj+1,m aj+1,m Aj+1,m − Kj+1,m exp(iKj+1,n l)aj+1,m Bj+1,m = 0, [A2,n − B2,n exp(iK2,n l)]K2,n Z2,n ,
(5.33b)
(5.34d)
n=0 N
N
[Aj,n exp(iKj,n l) + Bj,n ]Zj,n
n=0
=
N
exp(iKp,n l)Bp,n,m Ap,n +
n=0
[Aj+1,n + Bj+1,n exp(iKj+1,n l)]Zj+1,n ,
N
Bp,n,m Bp,n
n=0
− cos(Kp+1,m l)ap+1,m Cp+1,m = 0,
(5.33c)
(5.34e)
n=0
N
N
[Aj,n exp(iKj,n l) − Bj,n ]Kj,n Zj,n
i
n=0
Kp,n exp(iKp,n l)Bp,n,m Ap,n − i
n=0
=
N
[Aj+1,n − Bj+1,n exp(iKj+1,n l)]Kj+1,n Zj+1,n .
N
Kp,n Bp,n,m Bp,n
n=0
− Kp+1,m sin(Kp+1,m l)ap+1,m Cp+1,m = 0.
(5.34f)
(5.33d)
n=0
Eqs. (5.34a–f) reduce to the following system: AX = c, where
A is a square matrix of size [2p(N + 1)], X = [R0 , . . .RN , A2,0 , B2,0 . . ., A2,N , B2,N , . . .. . ., Ap,0 , Bp,0 , . . ., Ap,N , Bp,n , Cp+1,0 , . . ., Cp+1,N ]T is the unknown vector,
and
c = [−B1,0,0 , . . ., B1,0,N , K1,0 B1,0,0 , . . ., K1,0 B1,0,N , 0, . . .. . .. . .. . .0 ]T .
2(p−−1)(N+1)−times
Applying the matching conditions (5.30a–b) along the boundary x = xp separately, the following equations are obtained: N
[Ap,n exp(iKp,n l) + Bp,n ]Zp,n =
n=0
N
5.3. Comparison with existing result Cp+1,n cos(Kp+1,n l)Zp+1,n ,
n=0
(5.33e) N
i
[Ap,n exp(iKp,n l) − Bp,n ]Kp,n Zp,n
n=0
=
N
Cp+1,n sin(Kp+1,n l)Kp+1,n Zp+1,n .
(5.33f)
n=0
Further, using the orthogonality property of Z2,m , Zj+1,m (j = 2, 3, . . ., p − 1) and Zp+1,m , for each m = 0, 1, . . ., N, in Eqs. (5.33a,b), (5.33c,d) and (5.33e,f), respectively, it follows N
By solving the above system of linear equations, R0 can be evaluated.
B1,n,m Rn − i˛a2,m A2,m − i˛ exp(iK2,m l)a2,m B2,m = −B1,0,m ,
n=0
(5.34a)
In order to ascertain that our model is effective, we compare our reflection coefficient plotted against the dimensionless porous structure width with the corresponding result in the work of Madsen [7]. This is accomplished by taking h1 = hp+1 , i.e., by placing the porous structure on the horizontal seabed instead of a p-step bottom. The excellent agreement between our result and Madsen’s result can be observed from Fig. 12. This confirms that our model is valid and hence can be employed effectively to investigate various issues related to scattering of waves by a porous structure placed on a p-step bottom. 5.4. Numerical results: p-step bottom Here the corresponding study of different parameters on reflection coefficient is taken up along with two additional studies on number of steps (p) and height (hp+1 ) of the (p + 1)th region as compared to the 2-step case, with L/h1 = 2; f = 1; S = 1; hp+1 /h1 = 0.5 and = 0.7, 0.8 and 0.9.
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
381
1 Present work ( γ= 0.5) 0.9
f= 1; T= 17.3
Present work ( γ= 0.95)
0.8
Madsen(1983) ( γ= 0.5)
0.7
Madsen(1983) ( γ= 0.95)
0.6 |R 0 |
0.5 0.4 0.3 0.2
Fig. 15. Variation of |R0 | against h1 for different f with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, = 0.9 and N = 9.
0.1 0 0
1
2
3
4 k
1,0
5
6
7
1
L
Fig. 12. Variation of reflection coefficient |R0 | against dimensionless width of porous structure (k1,0 L) for different porosity () with steps=40, h1 = 21, hp+1 = h1 , f = 1, ı = 0.2, T = 17.3, ai = 0.87 and N = 10 (comparison with Madsen [7]).
0.9
f= 0.25
0.8
f= 0.5
0.7
f= 0.75 0.6
|R0 | 0.5
f= 1
0.4 0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
L/h1 Fig. 16. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different f with steps = 8, hp+1 /h1 = 0.5, = 0.9 and N = 9.
Fig. 13. Variation of |R0 | against h1 for different numbers of evanescent modes (N) with steps = 8, hp+1 /h1 = 0.5, f = 1, L/h1 = 2 and = 0.9.
While presenting the reflection coefficient |R0 | by Fig. 13 (against wave number for various number of evanescent modes), Fig. 14 (against wave number for various porosity), Fig. 18 (against width of the porous structure for various porosity), Fig. 19 (against wave number for various width of the porous structure) and Fig. 22 (against wave number for various angles of incidence), it is observed that reflection follows the same pattern as that of the 2-step case except for the fact that for relatively long waves, oscillation is encountered in the reflection coefficient |R0 | before attaining a constant value for relatively short waves.
Figs. 15 and 16 also suggest the same pattern of the reflection coefficient plotted against h1 and L/h1 , respectively, for different friction factors as like that of 2-step case, except for the fact that oscillation vanishes for relatively lower values of the wave number. Fig. 17 shows the effect of wave number on the reflection coefficient for different numbers of steps. Here the number of steps = 4, 6, 8 and 10 are considered. In this case also, oscillation in the reflection coefficient is observed for relatively long waves and it disappears with an increase in the value of the wave number. Higher number of steps gives rise to higher reflection coefficient but converges for sufficiently higher number of steps (here we consider steps = 8 and 10). Further the effect of the angle of incidence on the reflection coefficient is studied for different values of porosity (Fig. 20). It is observed that for values of up to 40◦ (approx.), higher porosity results in lower reflection coefficient, but as the value of increases, the aforesaid effect starts reversing and after = 50◦ (approx.),
1
1
γ= 0.7
0.9 0.8
γ= 0.8
0.7
|R | 0
γ= 0.9
0.6
0.9
steps= 4
0.8
steps= 6
0.7
steps= 8
0.6
|R |
0.5
0
0.4
steps= 10 0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0
1
2
3
ν h1
4
5
6
7
Fig. 14. Variation of |R0 | against h1 for different with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, f = 1 and N = 9.
0 0
1
2
3
νh1
4
5
6
7
Fig. 17. Variation of |R0 | against h1 for different number of steps with hp+1 /h1 = 0.5, L/h1 = 2, f = 1, = 0.9 and N = 9.
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S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385 1
1 γ= 0.7
0.9 0.8
γ= 0.8
/h = 0.75
p+1
0.8
0.7
1
hp+1/h1= 0.625
0.7 γ= 0.9
0.6
|R0|
h
0.9
h
0.6
|R0|
0.5 0.4
/h = 0.5
p+1
1
0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0
0.5
1
1.5
2
2.5
0
3
0
L/h1 Fig. 18. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different with steps = 8, hp+1 /h1 = 0.5, f = 1, and N = 9.
1
10
20
30
40
θ
50
60
70
80
90
Fig. 21. Variation of |R0 | against incident wave angle ( ) for different hp+1 /h1 with steps=8, L/h1 = 16, f = 1 = 0.9, and N = 9.
1
L/h = 0.5 1
0.9
L/h = 0.75
0.9
0.8
L/h = 1
0.8
0.7
L/h = 1.25
0.6
L/h1= 1.5
1
|R | 0
1
0
θ= 20
0.6
|R |
L/h = 1.75 1
0
L/h = 2
0
θ= 30
0.5
1
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0
1
2
3
νh1
4
5
6
0
7
0
Fig. 19. Variation of |R0 | against h1 for different width of the porous structure (L/h1 ) with steps=8, hp+1 /h1 = 0.5, f = 1, = 0.9 and N = 9.
1
2
3
ν h1
4
5
6
7
Fig. 22. Variation of |R0 | against h1 for different incident wave angle ( ) with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, f = 1, = 0.9 and N = 9.
1
1
γ= 0.7
0.9
0.9
0.8
0.6
0.7
γ= 0.9
|R |
steps= 2
0.8
γ= 0.8
0.7
0
θ= 100
0.7
1
0.5
0
θ= 0
|R0|
0.5
steps= 8
0.6 0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0 0
10
20
30
40
θ
50
60
70
80
0
90
Fig. 20. Variation of |R0 | against incident wave angle ( ) for different with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, f = 1 and N = 9.
higher porosity gives rise to higher reflection before converging to each other for very large values of . We also investigate the effect of the angle of incidence on the reflection coefficient for different values of hp+1 /h1 , namely hp+1 /h1 = 0.5, 0.625 and 0.75 (Fig. 21). Up to = 28◦ (approx.), higher hp+1 /h1 results in lower reflection coefficient but as the value of
increases further (in 30◦ < < 50◦ ), higher hp+1 /h1 gives rise to lower reflection coefficient before converging to each other.
1
2
3
νh 1
4
5
6
7
Fig. 23. Variation of |R0 | against h1 for different number of steps (2-step and 8step) for the same set of parameter values (hp+1 /h1 = 0.75, L/h1 = 1, f = 1, = 0.9 and N = 9).
1 0.9
steps= 2
0.8 0.7
steps= 8 0.6
|R |
0 0.5 0.4 0.3
5.5. Comparison between 2-step and p-step bottom
0.2 0.1
To find the difference or similarity between the results for both cases, we move forward to compare graphically the two different solutions for 2-step and p-step cases for the same set of parameter values by comparing |R0 | against the wave number and the dimensionless width of the porous structure for steps = 2 and 8. We consider L/h1 = 1; f = 1; S = 1; hp+1 /h1 = 0.75 and = 0.9.
0 0
0.2
0.4
0.6
0.8
1
L/h1
1.2
1.4
1.6
1.8
2
Fig. 24. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different number of steps (2-step and 8-step) for the same set of parameter values (hp+1 /h1 = 0.75, f = 1, = 0.9 and N = 9).
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
Fig. 25. Variation of |R0 | against h1 for different number of steps (2-step and 8step) for the same set of parameter values (hp+1 /h1 = 0.75, L/h1 = 1, f = 1, = 0.5 and N = 9).
383
but converges for sufficiently higher number of steps. For relatively lower value of the angle of incidence, |R0 | takes higher value for lower porosity but the effect reverses for relatively higher value of the angle of incidence. Same effect mentioned above is observed for different values of hp+1 . Further, comparison between the reflection coefficients of 2-step and p-step bottom is carried out for two different values of porosity against wave number and dimensionless width of the porous structure. It is observed that for relatively lower value of porosity, p-step bottom results in lower reflection coefficient. For the validity of our mathematical model, we compare our work with Madsen [7] by plotting reflection coefficient against dimensionless width of the porous structure for different values of porosity and we find excellent agreement in this regard. This shows that our model will be effective for solving problems of wave scattering by porous structures placed on a p-step bottom. Acknowledgements
1 0.9
The first author is grateful to the Council of Scientific and Industrial Research (CSIR), Govt. of India for providing him senior research fellowship for pursuing Ph.D. at Indian Institute of Technology Guwahati, India. The authors are immensely grateful to the esteemed reviewers and the Editor-in-Chief for their comments and suggestions which have allowed the authors to carry out a much improved version of the manuscript.
steps= 2 0.8 0.7
steps= 8
0.6
|R |
0 0.5 0.4 0.3 0.2 0.1
Appendix A.
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
L/h1 Fig. 26. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different number of steps (2-step and 8-step) for the same set of parameter values (hp+1 /h1 = 0.75, f = 1, = 0.5 and N = 9.
Figs. 23 and 24 show that the values of the reflection coefficient for = 0.9 are more or less the same in both cases. But if lower value of porosity ( = 0.5 in this case) is considered, significant difference occurs (Figs. 25 and 26). Now it is observed that the 2-step approximation results in lower reflection coefficient as compared to the p-step approximation.
Principle of conservation of mass inside a fluid region leads to the following equation known as the equation of continuity:
∂ + ∇ · (U) = 0. ∂t
(A.1)
Now for the steady state flow of fluid having constant density, Eq. (A.1) reduces to
∇ · U = 0.
(A.2)
For a potential flow, i.e., when U = ∇ , the equation of continuity reduces to Laplace’s equation
∇ 2 = 0. 6. Conclusion Based on linear water wave theory, a system of linear equations is derived using matching conditions along the boundary to find the reflection coefficient for oblique water wave scattering by a vertical porous structure placed on an elevated bottom consisting of two steps as well as p steps. It is observed that the propagating mode controls reflection up to a certain wave number. Increment in the height of the porous structure does not affect reflection after certain width of the structure. Higher value of the reflection coefficient is observed for lower values of porosity. It is also observed that lower values of friction factor lead to oscillation in the reflection coefficient which vanishes for higher values of friction factor. For relatively long waves, the values of the angle of incidence do not affect |R0 | but for short waves, higher value of angle of incidence results in lower reflection coefficient. Porosity of the structure does not contribute up to a certain range of values of angle of incidence. Corresponding study of the effect of number of evanescent modes and porosity for p-step impermeable bottom shows the same pattern, but oscillation for relatively long waves appear for a p-step bottom as against a 2-step bottom. Oscillation in |R0 | for higher values of friction factor vanishes for a p-step seabed at a relatively lower value of the wave number as compared to a 2-step sea-bed. Increasing number of steps results in higher reflection coefficient
The incompressible equations of motion inside the porous structure can be written in the following form:
∇ ·U = 0,
(A.3a)
∂U ∇P + resistance forces. =− ∂t
(A.3b)
The resistance forces in Eq. (A.3b) are evaluated by combining known steady and unsteady stress relationships. Under steady state flow conditions the pressure drop through the porous medium is specified by Ward [19] as −
∇P
=
vk Kp
U +
C
f 2 U|U|, Kp
(A.4)
where vk is the kinematic viscosity, Kp is the intrinsic permeability and Cf is a dimensionless turbulent resistance coefficient of the medium. It is hypothesized by Sollitt and Cross [5] that unsteadiness may be accounted for by introducing an additional term which evaluates the added resistance caused by the virtual mass of discrete grains within the medium. The resistance force due to the virtual mass is equal to the product of the displaced fluid mass, the virtual mass coefficient and the acceleration in the approach velocity. The
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S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
resulting force is distributed over the fluid mass within the pore so that the force per unit mass of fluid is simply
Now, inside the porous region
∂U 2 ∇ P2 − fωU 2 , =− ∂t ∂˚2 P2 − fω˚2 , ⇒ S =− ∂t S
1− ∂U CM . ∂t
(A.5)
Combining Eqs. (A.4) and (A.5); and replacing the resistance force in Eq. (A.3b) with them we get 1− ∂U ∇ P vk ∂U U − 2 U|U| − − CM =− , Kp ∂t ∂t Kp Cf ∂U ∇ P vk U − 2 U|U|, − =− ⇒ S Kp ∂t Kp
(A.6)
1− S = 1 + CM .
(A.7)
Kp
fωU · Udt t
=
t+T
Kp
t
V
U +
C
f 2 U|U| Kp
· Udt,
(A.8)
where V is the volume of the flow field and T is the wave period. Now assuming f to be constant throughout the flow field, the following form can be written
f =
1 ω
V
dV
t+T t
V
2 dV
vk U 2 Kp
t+T t
+
Cf |U|3 Kp
U 2 dt
dt .
(A.9)
Appendix B. Let us consider U 1 and U 2 , respectively, to be the velocities of a fluid at any point inside the water and porous region attached to each other. Then the following relation holds true: U i = ∇ ˚i
⇒ −iω1 = −
(B.12)
P1 .
Now, along the vertical boundary between the water and porous regions, continuity of pressure (P1 = P2 ) results in (from Eqs. (B.11) and (B.12)) the following matching condition:
i = 1, 2.
(B.13)
Mass flux per unit volume and unit time inside the porous region is U and the same inside the water region is U. Along the vertical boundary, the continuity of mass flux implies U 1 = U 2 , ⇒
∂1 ∂2 = ∂x ∂x
(B.14) along x-direction.
Moreover, if both the regions consist of the same medium, then Eqs. (B.13) and (B.14) reduce to 1 = 2 ,
(B.15a) along x-direction.
(B.15b)
It is obvious that these matching conditions are valid along the vertical boundary separating any two regions next to each other.
vk
dV
P2 ,
∂U 1 ∇ P1 , =− ∂t ∂˚1 P1 =− , ∂t
∂1 ∂2 = ∂x ∂x
t+T
dV V
ω˛2 = −
(B.11)
P2 ,
1 = i˛2 .
C
f 2 U|U| → fωU.
Combination of Eqs. (A.6) and (A.7) leads to Eq. (2.1b). In order to evaluate f, Lorentz’s principle of equivalent work is applied which says that the average rate of energy dissipation should be identical whether evaluated using the true non-linear resistance law or its linearized equivalent. Since the resistance terms of the above relation represent friction force per unit mass acting at a point in the flow field, the following equality (energy dissipation) holds:
⇒
⇒
It is worth mentioning that S = 1 accounts for two different cases, namely, = 1 or the absence of structure and CM = 0 or the presence of inviscid fluid. Now, linearization of Eq. (A.6) is necessary in order to find an analytical solution and hence, the dissipative stress term is replaced by a linear stress term in U by the following form:
Kp
P2 − fω2 ,
where P1 and P2 are the dynamic pressures of the water and porous regions, respectively. In the water region, Bernoulli’s equation gives
where
U +
−iωS2 = −
⇒ ω(f − iS)2 = −
Cf
vk
⇒
(B.10)
References [1] Xi D, Xu S, Du Y, Yi L. Wave propagation analysis of porous rocks with thermal activated relaxation mechanism. J Appl Geophys 2011;73:289–303. [2] Anglin EJ, Cheng L, Freeman WR, Sailora MJ. Porous silicon in drug delivery devices and materials. Adv Drug Deliv Rev 2008;60:1266–77. [3] Blunt MJ. Physically-based network modeling of multiphase flow in intermediate-wet porous media. J Pet Sci Eng 1998;20:117–25. [4] Clement TP, Hooker BS, Skeen RS. Macroscopic models for predicting changes in saturated porous media properties caused by microbial growth. Groundwater 1996;34:934–42. [5] Sollitt CK, Cross RH. Wave transmissions through permeable breakwaters. In: Proc. 13th Coast. Eng. Conf. Am. Soc. Civ. Eng. 1972. p. 1827–46. [6] Madsen OS. Wave transmission through porous structure. J Waterw Port Coast Ocean Eng Am Soc Civ Eng 1974;100:169–88. [7] Madsen PA. Wave reflection from a vertical permeable wave absorber. Coast Eng 1983;7:381–96. [8] Kirby JT, Dalrymple RA. Propagation of obliquely incident water waves over a trench. J Fluid Mech 1983;133:47–63. [9] Sulisz W. Wave reflection and transmission at permeable breakwaters of arbitrary cross section. Coast Eng 1985;9(4):371–86. [10] Dalrymple RA, Losada IJ, Martin OA. Reflection and transmission from porous structures under oblique wave attack. J Fluid Mech 1991;224:625–44. [11] Losada IJ, Dalrymple RA, Martin OA. Water waves on crown breakwaters. J Waterw Port Coast Eng Am Soc Civ Eng 1993;119(4):367–80.
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[17] Das S, Bora SN. Reflection of oblique ocean water waves by a vertical rectangular porous structure placed on an elevated horizontal bottom. Ocean Eng 2014;82:135–43. [18] Das S, Bora SN. Wave damping by a vertical porous structure placed near and away from a rigid vertical wall. Geophys Astrophys Fluid 2014;108(2): 147–67. [19] Ward JC. Turbulent flow in porous media. Proc J Hydraul Div Am Soc Civ Eng 1964;90(HY5):1–12.
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Reflection of oblique ocean water waves by a vertical porous structure placed on a multi-step impermeable bottom S. Das, S.N. Bora ∗ Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India
a r t i c l e
i n f o
Article history: Received 4 December 2013 Received in revised form 30 March 2014 Accepted 6 July 2014 Keywords: Porous structure Oblique wave Reflection Matching condition Multi-step bottom
a b s t r a c t Based on linear water wave theory, wave reflection by a vertical porous structure placed on an elevated impermeable seabed, assumed to consist of a number of horizontal steps, is considered. The reflection phenomenon is investigated for two forms of the bottom topography: first a 2-step one and then a pstep one. An oblique incident wave propagates through the porous structure and gets reflected by the steps and a vertical solid wall which supports the structure at one end. Boundary value problems are set up in the two different media, the first medium being water and the second medium being the porous structure consisting of p vertical regions – one above each step. By using the matching conditions along the vertical boundaries, a system of linear equations is deduced. Reflection coefficient is obtained by solving this system of equations. The behavior of the reflection coefficient due to different relevant parameters is studied: the effect of various parameters, such as depth, porosity, number of evanescent modes, structure width and angle of incidence is studied graphically for both cases. It is observed that when the porous structure is considered above a 2-step bottom, the number of evanescent modes, porosity and the angle of incidence do not affect the reflection coefficient for relatively long waves. Lower values of friction factor results in oscillation in the reflection coefficient which vanishes with an increase in the values of friction factor. Up to a certain range of the angles of incidence, reflection coefficient is independent of the values of porosity. When a p-step bottom is considered, certain observations remain the same except that the behavior of reflection coefficient against the angle of incidence is different. In the end, both cases are compared by considering the same set of parameters. Justification of our model is presented by matching it with an available one. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The concept of porous media is used in many areas of applied science and engineering: filtration, mechanics (acoustics, geomechanics, soil mechanics, rock mechanics [1]), engineering (petroleum engineering, bioremediation [2], construction engineering), geosciences (hydrogeology, petroleum geology [3], geophysics), biology, biophysics [4], material science, etc. These works amply justify that fluid flow through porous media is a subject of immense common interest and has, hence, emerged as a separate field of study. In coastal areas, porous structures are widely used as breakwaters to protect harbors, inlets and beaches from wave action, and as dissipating sea-walls to attenuate the wave energy in harbors. Because of this enormous significance of interaction of porous structures with ocean waves, we are motivated to
∗ Corresponding author. Tel.: +91 361 258 2604; fax: +91 361 258 2649. E-mail addresses: [email protected] (S. Das), [email protected], [email protected] (S.N. Bora). 0141-1187/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apor.2014.07.001
investigate some specific cases of reflection by a porous structure placed on a step-like topography. Many aspects of interaction between waves and porous media have been studied extensively. Theoretical solutions for reflection and transmission coefficients for certain types of porous structures have been analyzed previously by a number of researchers. The most widely used model of wave-induced flow in porous medium is the one developed by Sollitt and Cross [5]. According to their approach, dissipation of wave energy inside a porous medium is taken into account through a linearized friction term f which is evaluated by an iterative procedure. Madsen [6] derived a simple solution for reflection and transmission from a rectangular porous structure under normal incidence of long waves based on the linearized form of the governing equations and a linearized form of the flow resistance formula. Madsen [7] obtained a theoretical solution for the reflection of linear shallow-water waves from a vertical porous wave absorber placed on a horizontal bottom. The friction term describing the energy loss inside the absorber was linearized and thereafter, by using Lorentz principle of equivalent work, the reflection coefficient was determined as a function of parameters describing the incoming waves and the absorber characteristics.
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Kirby and Dalrymple [8] investigated the diffraction of obliquely incident surface waves by an asymmetric trench in which they developed a numerical solution by matching the particular solution for each subregion of constant depth along the vertical boundaries. An approximate solution based on plane-wave modes was derived and compared with the numerical solution. Sulisz [9] formulated a theory to predict wave reflection and transmission at an infinite rubble-mound breakwater under normal wave incidence. Dalrymple et al. [10] adopted the approach of Sollitt and Cross [5] to analyze the reflection and transmission of oblique incident waves from infinitely long porous structures. Losada et al. [11] extended this study to the case of an infinitely long, homogeneous, vertical structure capped with an impervious element under oblique incident wave. Mallayachari and Sundar [12] took into account the effects of an uneven seabed. The variation of the reflection coefficient with respect to the porosity of the wall, its friction factor and the relative wall width was studied. Zhu [13] used wave induced refractiondiffraction equations for surface waves in the region occupied by a porous structure. He utilized the orthogonality of the depthdependent functions. Liu et al. [14] examined the hydrodynamic performance of a modified two-layer horizontal-plate breakwater consisting of an upper submerged horizontal porous plate and a lower submerged horizontal solid plate. Liu and Li [15] extended this work by considering a double curtain-wall breakwater whose seaward wall was perforated and shoreward wall impermeable. Cho et al. [16] further extended the idea of Liu et al. [14] by considering the lower submerged horizontal plate as a porous one instead of an impermeable one. Das and Bora [17] investigated wave reflection by a rectangular porous structure placed on an elevated bottom and supported by a vertical wall. The variation of reflection with respect to the number of modes, porosity, structure width, etc, was studied. Das and Bora [18] also studied wave damping by a vertical rectangular porous structure placed near and away from a rigid vertical wall. They computed the reflection coefficients for various depths, structure width and porosity. The objective of this work is to solve a water wave scattering problem due to the presence of a vertical porous structure, placed on a 2-step or a p-step horizontal bottom and supported by a rigid vertical wall at one end, and to study the reflection characteristics. The effect of various parameters, such as number of evanescent modes, porosity, friction factor, structure width, angle of incidence on the reflection coefficient is investigated and the results are presented graphically. To the best of the knowledge of the authors, no one has solved the problem of wave propagation through a porous structure that is placed on a step-like impermeable bottom. 2. Mathematical formulation Though the present problem is a specific one, it is fairly important to discuss some general features and equations which usually arise in wave propagation in porous medium. Therefore, in the following subsections a brief description on the specific porous structure under consideration is presented along with these equations ahead of the formulation of the present problem.
Fig. 1. Front view of the structural model.
as cement, ceramics, etc. Depending upon the pore size, porous structures can be categorized into three types, namely, microporous (smaller than 2 nanometers), mesoporous (between 2 and 50 nanometers) and macroporous (larger than 50 nanometers). Metallic foams are good examples of porous materials with higher porosity (ranging from 0.6 to 0.95). The construction of these types of materials can be performed through many means, the main one being the “lost-foam casting”. Porous structures, with high porosity but with considerable stability, to be used as breakwater in ocean and coastal engineering can be constructed from low melting metals and alloys such as copper, aluminum, lead, tin, zinc, etc. The porous structure under consideration in this manuscript is taken as a such type of structure. Fig. 1 presents a rough visual representation of the present structural model. 2.2. General theory for flow inside porous medium Small amplitude wave motion is considered within an undeformable porous medium. It is assumed that the porous structure is homogeneous and isotropic. The fluid motion follows the continuity equation and the equation of motion in terms of the seepage fluid velocity U and dynamic pressure P, which are given by
∇ · U = 0, S
∂U ∇ P + fωU = 0, + ∂t
(2.1b)
where is the density of the fluid, f is the linearized friction factor and ω is the angular frequency of the incident wave. The inertial coefficient S is defined by S =1+
CM (1 − ) ,
(2.2)
where CM is the added mass coefficient and is the porosity of the porous structure. The physical significance and derivation of Eqs. (2.1a,b) and (2.2) are described in Appendix A. A pore velocity potential ˚(x, y, z, t) is introduced to describe the wave-induced fluid motion in the porous medium: U = ∇ ˚.
(2.3)
Integration of equation of motion (2.1b) leads to Bernoulli’s equation
2.1. Porosity and porous structure S A porous medium mostly consists of pores through which fluid (be it gas or liquid such as water) can pass. The skeleton part of the medium is mainly solid (but foam is considered to be porous though). A porous medium may be an aggregate of large number of particles like sand, gravels or a solid containing many capillaries such as porous rock. Many natural substances like sponge, soil, biological materials (bone, lungs, etc.) are few examples of porous material. There are many man-made porous materials such
(2.1a)
∂˚ P + + fω˚ = 0. ∂t
(2.4)
The solution is assumed to be harmonic in time. So the fluid velocity, dynamic pressure and velocity potential can, respectively, be written as U = u(x, y, z) exp(−iωt),
P = p(x, y, z) exp(−iωt),
˚ = (x, y, z) exp(−iωt),
(2.5)
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
where u, p and are spatial functions. Substitution of these values into Eqs. (2.1a), (2.1b) and (2.3) leads to the equations
∇ · u = 0, ∇p
(2.6a)
+ ω˛u = 0,
(2.6b)
u = ∇ ,
(2.6c)
where ˛ = f − iS is the dimensionless impedance of the porous medium. Using Eq. (2.6c) in Eq. (2.6a), we get the spatially dependent potential function which satisfies Laplace’s equation
∇ 2 = 0.
(2.7)
If the free surface is defined by z = (x, y, t), then the linearized dynamic free surface boundary condition can be obtained from Eq. (2.4) with P = g as =−
ω˛ (x, y, 0) exp(−iωt). g
(2.8)
The linearized kinematic free surface boundary condition can be written as
∂ ∂ {(x, y, z) exp(−iωt)} = ∂z ∂t
at z = 0.
(2.9)
Combining Eqs. (2.8) and (2.9), the following single free surface condition is obtained:
∂ − i˛ = 0 at z = 0, ∂z
(2.10)
where = ω2 /g with g as the gravitational constant. The condition to be satisfied at the impermeable bottom is
∂ = 0, ∂n
z = −h(x, y),
(2.11)
where ∂/∂n represents differentiation along the normal to the bottom boundary h(x, y) which depicts a variable bottom topography. For a flat seabed at z = − h, where h is constant, the general solution of the two-dimensional wave motion within a porous medium can be obtained as follows by solving Eqs. (2.7)–(2.11) for the potential in x and z with the help of the technique of separation of variables: =
∞ n=0
˜ n exp(ikn x) + B˜ n exp(−ikn x)]Zn (h, z), [A
(2.12)
375
˜ n and B˜ n are constants and Zn (h, z) is the depth-dependent where A function defined by Zn (h, z) =
cosh kn (h + z) , cosh kn h
n = 0, 1, 2, . . .,
(2.13a)
with the complex wave number kn satisfying the complex dispersion relation i˛ = kn tanh kn h.
(2.13b)
The characteristics of waves in the porous medium can be described by the dimensionless complex wave number kn h. It is known that for a non-dissipative medium (f = 0), k0 is purely real, while kn (n ≥ 1) are purely imaginary. For a dissipative medium, we note that the influence of the friction factor f is to damp the wave motion. This damping can be achieved by adding an imaginary part to k0 and real parts to kn (n ≥ 1). By doing so, the amplitude of both the propagating mode and the evanescent modes will decay. 2.3. Formulation for scattering by a 2-step bottom Let us consider a vertical porous structure of width L + D, a very small portion of which is above the free surface, placed on a 2-step bottom topography above the horizontal ocean bottom, which is at a constant depth h1 from free surface, and resting against a rigid vertical wall. Using rectangular Cartesian coordinate system, the positive x-direction is defined as the direction of the normal wave incident on the porous structure at x = 0, the positive z-direction is considered vertically upwards and the mean free surface as z = 0. Due to the 2-step bottom topography under the porous structure, two different heights of the porous structure exist, namely, h2 when 0 < x < L and h3 when L < x < L + D. The water region (− ∞ < x < 0) is labeled as region I, the porous structure above the first step (0 < x < L) and above the second step (L < x < L + D) as region II and region III, respectively (Fig. 2). The horizontal bottom for region I is considered at z = − h1 . After propagating through the porous structure, the wave gets reflected by the vertical wall at x = L + D. The fluid is assumed to be incompressible, homogeneous and inviscid, and the motion irrotational. According to Sollitt and Cross [5], velocity potentials can be assumed to exist within the porous structure as well as in the fluid region. We define three velocity potentials ˚1 (x, y, z, t) = 1 (x, z) exp(i y − iωt), ˚2 (x, y, z, t) = 2 (x, z) exp(i y − iωt) and ˚3 (x, y, z, t) = 3 (x, z) exp(i y − iωt) for region I, region II and region III, respectively, where = k1,0 sin with k0 and being the
Fig. 2. Schematic diagram of the problem with a 2-step bottom.
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S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
incident wave number and the angle made by the incident wave to the positive x-direction. The governing equation and the boundary conditions in region I (− ∞ < x < 0, −h1 < z < 0) are
where R0 is the complex reflection coefficient, Rn correspond to the decaying modes of reflection; and the depth-dependent function Z1,n (h1 , z) and K1,n are, respectively, given by
∇ 2 1 − 2 1 = 0;
Z1,n (h1 , z) =
∂1 − 1 = 0; ∂z
−∞ < x < 0, −∞ < x < 0,
∂1 = 0; ∂z
−∞ < x < 0,
∂1 = 0; ∂x
x = 0,
−h1 < z < 0, z = 0,
(2.14a) (2.14b)
z = −h1 ,
(2.14c)
−h1 < z < −h2 .
(2.14d)
The governing equation and the boundary conditions in region II (0 < x < L, −h2 < z < 0) are
∇ 2 2 − 2 2 = 0;
0 < x < L,
−h2 < z < 0,
(2.15a)
∂2 − i˛2 = 0; ∂z
0 < x < L,
z = 0,
(2.15b)
∂2 = 0; ∂z
0 < x < L,
∂2 = 0; ∂x
x = L,
z = −h2 ,
2 K1,n = (k1,n − 2 )
1/2
−h2 < z < −h3 .
(2.15d)
L < x < L + D,
−h3 < z < 0,
(2.16a)
∂3 − i˛3 = 0; ∂z
L < x < L + D,
z = 0,
(2.16b)
∂3 = 0; ∂z
L < x < L + D,
∂3 = 0; ∂x
x = L + D,
z = −h3 ,
(2.16c)
1 = i˛2 ,
x = 0,
∂1 ∂2 = , ∂x ∂x
−h3 < z < 0.
−h2 ≤ z ≤ 0,
x = 0,
x = L,
∂2 ∂3 = , ∂x ∂x
(3.19b)
= k1,n tanh k1,n h1 .
(3.19c)
Eq. (3.19c) has one positive real root k1,0 corresponding to the incident and reflected modes of wave propagation, and infinitely many purely imaginary roots k1,n ; n = 1, 2, . . ., which correspond to the evanescent modes. We truncate the infinite sum at n = N so as to consider a finite number of evanescent modes N only: 1 (x, z) = [exp(iK1,0 x) + R0 exp(−iK1,0 x)]Z1,0 (h1 , z) +
N
Rn exp(−iK1,n x)Z1,n (h1 , z).
(3.20)
In region II (0 < x < L), the horizontal impermeable bottom is considered at z = − h2 . Hence, the integrals in this porous region must be from z = − h2 to z = 0 in order to meet the matching conditions and it is required to compute the relevant integrals from z = − h2 to z = 0. By using separation of variables method, the velocity potential 2 (x, z), after truncating the infinite sum at n = N, can be written as
2 (x, z) =
N
{A2,n exp(iK2,n x) + B2,n exp[−iK2,n (x − L)]}Z2,n (h2 , z), (3.21)
(2.16d)
−h2 ≤ z ≤ 0,
−h3 ≤ z ≤ 0,
x = L,
−h3 ≤ z ≤ 0.
(2.17a) (2.17b) (2.17c)
2 − 2 ) where A2,n and B2,n are arbitrary constants; K2,n = (k2,n with k2,n satisfying the dispersion relation
i˛ = k2,n tanh k2,n h2 ,
Z2,n (h2 , z) =
cosh k2,n (h2 + z) , cosh k2,n h2
n = 0, 1, 2, . . ., N.
3 (x, z) =
N
{C3,n cos[K3,n (x − L − D)]}Z3,n (h3 , z),
(3.23)
n=0
i˛ = k3,n tanh k3,n h3 ,
1 (x, z) = [exp(iK1,0 x) + R0 exp(−iK1,0 x)]Z1,0 (h1 , z)
Z3,n (h3 , z) = (3.18)
(3.22b)
In region III (L < x < L + D), the horizontal impermeable bottom is considered at z = − h3 . Hence, the integrals in the porous region must be from z = − h3 to z = 0 in order to meet the matching conditions and it is required to compute the relevant integrals from z = − h3 to z = 0. The velocity potential 3 (x, z), after truncating the infinite sum at n = N, can be written as
The potential 1 (x, z) in region I can be written in the following form:
Rn exp(−iK1,n x)Z1,n (h1 , z),
(3.22a)
2 − 2 ) where C3,n are arbitrary constants; K3,n = (k3,n satisfying the dispersion relation
3. Reflection by the porous structure
1/2
and Z2,n (h2 , z) is the depth-dependent function given by
(2.17d)
Derivation of the matching conditions (2.17a,d) is detailed in Appendix B.
n=1
,
n=0
In addition to the governing equations and different boundary conditions for each region, there exist some other conditions, called matching conditions, along the common boundary of any two successive media. These conditions imply the continuity of pressure and mass flux across the boundary. The matching conditions along the vertical boundaries x = 0 and x = L are given by
+
(3.19a)
n=1
∇ 2 3 − 2 3 = 0;
∞
n = 0, 1, 2, . . .,
with k1,n satisfying the dispersion relation
(2.15c)
The governing equation and the boundary conditions in region III (L < x < L + D, −h3 < z < 0) are
2 = 3 ,
cosh k1,n (h1 + z) , cosh k1,n h1
1/2
with k3,n (3.24a)
and Z3,n (h3 , z) is the depth-dependent function given by cosh k3,n (h3 + z) , cosh k3,n h3
n = 0, 1, 2, . . ., N.
(3.24b)
Using the matching conditions (2.17a–d), the following equations in (4N + 4) unknowns Rn , A2,n , B2,n , C2,n (n = 0, 1, . . ., N) are
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
obtained:
1
N= 0
N
(1 + R0 )Z1,0 +
0.9
N
Rn Z1,n = i˛
n=1
N= 2 0.8
[A2,n + B2,n exp(iK2,n L)]Z2,n ,
|R | 0
N
K1,0 (1 − R0 )Z1,0 −
N= 6
0.6
N= 8
0.5
N= 10
0.4
K1,n Rn Z1,n
0.3
n=1
0.2
N
[A2,n − B2,n exp(iK2,n L)]K2,n Z2,n ,
(3.25b)
0.1 0
n=0
N
0
1
2
3
ν h1
N
[A2,n exp(iK2,n L) + B2,n ]Z2,n =
n=0
C3,n cos(K3,n D)Z3,n ,
N
N
=
5
6
7
Fig. 3. Variation of |R0 | against h1 for different numbers of evanescent modes (N) with f = 1, (L + D)/h1 = 1 and = 0.9.
iK2,n [A2,n exp(iK2,n L) − B2,n ]Z2,n
n=0
4
(3.25c)
n=0
N
N= 4
0.7
n=0
(3.25a)
=
377
iK2,n exp(iK2,n L)B1,n,m A2,n
n=0
K3,n C3,n sin(K3,n D)Z3,n .
(3.25d) +
n=0
N
iK2,n B1,n,m B2,n − K3,m sin (K3,m D)a3,m C3,m = 0,
(3.26d)
n=0
0 2 Zq,m dz =
aq,m = −hq
hq tanh Kq,m hq 2Kq,m hq
0
Bq,n,m =
Zq,n Zq+1,m dz = −hq+1
⎧ ⎪ ⎪ ⎪ ⎨
where 1+
2Kq,m hq sinh 2Kq,m hq
1 2 − K2 K1,n 2,m
(1 − i˛) −
q = 1, 2, K1,n sinh K1,n (h2 − h1 ) cosh K1,n h1 cosh K2,m h2
when q = 1,
⎪ K2,n sinh K2,n (h3 − h2 ) 1 ⎪ ⎪ − ⎩ K2 − K2 cosh K2,n h2 cosh K3,m h3 2,n 3,m
when q = 2.
From Eqs. (3.26a–d) a system of linear equations with (4N + 4) equations in (4N + 4) unknowns can be constructed as follows: MX = b, where M is a square matrix of size (4N + 4), X = [R0 , R1 , . . .RN , A2,0 , A2,1 , . . .A2,N , B2,0 , B2,1 , . . .B2,N , C3,0 , C3,1 , . . .C3,N ]T is the unknown vector and
b = [−B1,0,0 , −B1,0,1 , . . . − B1,0,N , K1,0 B1,0,0 , K1,0 B1,0,1 , . . .K1,0 B1,0,N , 0, . . .. . .. . .0]T .
2N+2 times
By using the orthogonality property of Z2,m (h2 , z) and Z3,m (h3 , z) (for all m = 0, 1, . . ., N) in Eqs. (3.25a–b) and (3.25c–d), respectively, the above equations reduce to N
4. Numerical results B1,n,m Rn − i˛a2,m A2,m − i˛ exp(iK2,m L)a2,m B2,m = −B1,0,m ,
n=0
(3.26a)
N n=0
B1,n,m K1,n Rn + K2,m a2,m A2,m − K2,m exp(iK2,m L)a2,m B2,m
= K1,0 B1,0,m ,
N
(3.26b)
exp(iK2,n L)B1,n,m A2,n
n=0
+
By solving this system, R0 can be evaluated and subsequently the reflection phenomenon within the porous structure can be discussed.
N n=0
B1,n,m B2,n − cos (K3,m D)a3,m C3,m = 0,
(3.26c)
For computational purpose, some specific values of the parameters are considered: L/h1 = 0.5; D/h1 = 0.5; D/L = 1; f = 1; S = 1; = 0.7, 0.8 and 0.9; h1 − h2 = h2 − h3 = h1 /16; = 0◦ and h1 = 2. First we investigate the effect of the number of evanescent modes on the reflection coefficient |R0 | for various wave numbers (h1 ). The ratio (L + D)/h1 = 1 is maintained throughout and the values N = 0, 2, 4, . . ., 10 are considered. From Fig. 3, it is observed that up to approximately h1 < 2.4, reflection can be described by the propagating mode (N = 0) only. As the wave number increases, i.e., for short waves, it is evident that the number of evanescent modes starts affecting |R0 | but it converges to a fixed value for higher numbers of evanescent modes. Porosity of the structure plays an important role in the reflection characteristics. Fig. 4 shows that for relatively long waves (h1 < 1), porosity does not affect reflection, but as the value of the wave number increases, the values of |R0 |, corresponding to the highest value of porosity considered ( = 0.9), are considerably low. In
378
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385 1
1
γ= 0.7
0.9 0.8
0.9
γ= 0.8
γ= 0.8
0.7
0.7
γ= 0.9
0.6
|R0|
γ= 0.7
ν= 0.25
0.8
0.6
|R0|
0.5
γ= 0.9
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0 0
1
2
3
4
5
6
0
7
0.2
0.4
0.6
0.8
νh
1
1.2
1.4
1.6
1.8
2
(L+D)/h1
1
Fig. 4. Variation of |R0 | against h1 for different with (L + D)/h1 = 1, f = 1 and N = 9.
Fig. 7. Variation of |R0 | against dimensionless width of the porous structure ((L + D)/h1 ) for different with f = 1, and N = 9.
1 1
f= 0.25
0.9
(L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1 (L+D)/h = 1
0.9
0.8
f= 0.5 0.8
0.7 f= 0.75
0.6
|R | 0.5 0
0.7 0.6
f= 1
|R 0|
0.4
0.5 0.4
0.3
0.75 1 1.25 1.5 1.75 2
0.3
0.2
0.2
0.1
0.1
0 0
1
2
3
ν h1
4
5
6
7
0 0
1
2
3
ν h1
Fig. 5. Variation of |R0 | against h1 for different f with (L + D)/h1 = 1, = 0.9 and N = 9.
other words, as the porosity reduces, higher reflection coefficient is encountered. The effect of the friction factor f on the reflection coefficient is studied for various wave numbers. Fig. 5 shows oscillation of |R0 | for lower values of the friction factor considered. As the value of f increases, the oscillation vanishes. Further, Fig. 6 illustrates the effect of friction factor on the reflection coefficient for various dimensionless width ((L + D)/h1 ) of the porous structure. Lower values of f considered result in oscillation in |R0 | which vanishes with an increase in the value of the friction factor (here for f = 1). We also study the effect of the dimensionless width of the porous structure on reflection. |R0 | is plotted against the dimensionless width of the porous structure for different values of porosity in Fig. 7. Reflection coefficient decreases with an increase in (L + D)/h1 before attaining a fixed value for each of the porosity values considered. It is also observed that higher porosity results in lower reflection coefficient.
4
5
6
7
Fig. 8. Variation of |R0 | against h1 for different width of the porous structure ((L + D)/h1 ) with f = 1, = 0.9 and N = 9.
Fig. 8 illustrates the effect of wave number on the reflection coefficient for different values of width (L + D)/h1 . It is observed that oscillation in |R0 | exists for lower values of width and it disappears with an increase in width before converging to each other. Fig. 9 shows the variation of |R0 | against angle of incidence for different values of porosity. It is observed that, for < 45◦ , |R0 | is more or less the same for different porosity values. But for higher values of (>45◦ ), |R0 | attains its minimum value for lower value of porosity (here for = 0.7) and then starts increasing before converging to each other. We further study the effect of wave number on reflection phenomenon for different angles of incidence . Four different values of
, namely = 0◦ , 10◦ , 20◦ and 30◦ , are considered. It is observed from Fig. 10 that for relatively long waves (h1 < 1.5), |R0 | is same for all angles of incidence considered and as the wave number increases further, higher gives rise to lower reflection coefficient.
1
1
0.9
f= 0.25
0.9
0.8
f= 0.5
0.8
0.7
γ= 0.7 γ= 0.8
0.7
f= 0.75
γ= 0.9
0.6
0.6
|R 0|
0.5
f=1
|R0 |
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(L+D)/h1
Fig. 6. Variation of |R0 | against dimensionless width of the porous structure ((L + D)/h1 ) for different f with = 0.9 and N = 9.
0
10
20
30
40
50
60
70
80
90
θ Fig. 9. Variation of |R0 | against incident wave angle ( ) for different with (L + D)/h1 = 1, f = 1 and N = 9.
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
∂j − i˛j = 0 ; ∂z
379
xj−1 < x < xj ,
∂j = 0; ∂z
xj−1 < x < xj ,
∂j = 0; ∂x
x = xj ,
z = 0,
z = −hj ,
−hj < z < −hj+1 .
(5.27b) (5.27c) (5.27d)
The velocity potential p+1 satisfies Eqs. (5.27a–c) along with one extra condition
∂p+1 = 0; ∂x Fig. 10. Variation of |R0 | against h1 for different incident wave angle ( ) with (L + D)/h1 = 1, f = 1, = 0.9 and N = 9.
5. Scattering by a p-step bottom 5.1. Formulation Here we consider p number of horizontal steps, in place of 2 steps, to represent the uneven bottom topography under the porous structure (Fig. 11). hj+1 (j = 2, 3, . . ., p + 1) is considered to be the height of the porous structure above the jth horizontal step and a velocity potential ˚j (x, y, z, t) = j (x, z) exp(i y − iωt) is defined in the region xj−1 < x < xj , −hj < z < 0 for each j where x1 = 0 < x2 < · · · < xp+1 = L
x = xp+1 = L,
−hp+1 < z < 0.
(5.28)
The matching conditions along the boundary x = x1 = 0, −h2 ≤ z ≤ 0, are given by 1 = i˛2 ,
(5.29a)
∂1 ∂2 = . ∂x ∂x
(5.29b)
The matching conditions along the boundary x = xj , −hj+1 ≤ z ≤ 0 (j = 2, 3, . . ., p), are given by j = j+1 ,
(5.30a)
∂j ∂j+1 = . ∂x ∂x
(5.30b)
5.2. Solution
with xj − xj−1 = l.
The boundary value problem in the water region is given by Eqs. (2.14a–d) – exactly the same as in the first case. The governing
j (x, z) =
Depth dependent function Z1,n (h1 , z), K1,n and the potential function 1 in region I satisfy Eqs. (3.19a), (3.19b) and (3.20), respectively. Potential function j (x, z) in the jth porous region is given by
⎧ N ⎪ ⎪ ⎪ {Aj,n exp[iKj,n (x − xj−1 )] + Bj,n exp[−iKj,n (x − xj )]}Zj,n (hj , z), j = 2, 3, . . ., p, ⎪ ⎨ n=0
N ⎪ ⎪ ⎪ ⎪ Cp+1,n cos[Kp+1,n (x − L)]Zp+1,n (hp+1 , z), j = p + 1, ⎩
(5.31)
n=0
equation and the boundary conditions in the jth (j = 2, 3, . . ., p) region are
∇ 2 j − 2 j = 0 ;
xj−1 < x < xj ,
−hj < z < 0,
(5.27a)
where Aj,n , Bj,n and Cp+1,n are arbitrary constants; Kj,n = 2 − 2 ) (kj,n
1/2
with kj,n satisfying the dispersion relation
i˛ = kj,n tanh kj,n hj ,
Fig. 11. Schematic diagram of the problem with a p-step bottom.
(5.32a)
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S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
and Zj,n (hj , z) is the depth-dependent function in region j given by Zj,n (hj , z) =
cosh kj,n (hj + z) cosh kj,n hj
j = 2, 3, . . ., p + 1.
,
(5.32b)
Now using the matching conditions (5.29a–b) and (5.30a–b) along the boundary x = xj , −hj ≤ z ≤ 0 (j = 1, 2, . . ., p − 1), the following equations are obtained:
(1 + R0 )Z1,0 +
N
N
Rn Z1,n = i˛
n=1
[A2,n + B2,n exp(iK2,n l)]Z2,n ,
N
B1,n,m K1,n Rn + K2,m a2,m A2,m
n=0
− K2,m exp(iK2,m l)a2,m B2,m = K1,0 B1,0,m ,
N
exp(iKj,n l)Bj,n,m Aj,n +
n=0
N
Bj,n,m Bj,n − aj+1,m Aj+1,m
n=0
− exp(iKj+1,n l)aj+1,m Bj+1,m = 0,
n=0
(5.34b)
(5.34c)
(5.33a)
K1,0 (1 − R0 )Z1,0 −
N
N
K1,n Rn Z1,n n=0
n=1
=
N
Kj,n exp(iKj,n l)Bj,n,m Aj,n −
N
Kj,n Bj,n,m Bj,n
n=0
− Kj+1,m aj+1,m Aj+1,m − Kj+1,m exp(iKj+1,n l)aj+1,m Bj+1,m = 0, [A2,n − B2,n exp(iK2,n l)]K2,n Z2,n ,
(5.33b)
(5.34d)
n=0 N
N
[Aj,n exp(iKj,n l) + Bj,n ]Zj,n
n=0
=
N
exp(iKp,n l)Bp,n,m Ap,n +
n=0
[Aj+1,n + Bj+1,n exp(iKj+1,n l)]Zj+1,n ,
N
Bp,n,m Bp,n
n=0
− cos(Kp+1,m l)ap+1,m Cp+1,m = 0,
(5.33c)
(5.34e)
n=0
N
N
[Aj,n exp(iKj,n l) − Bj,n ]Kj,n Zj,n
i
n=0
Kp,n exp(iKp,n l)Bp,n,m Ap,n − i
n=0
=
N
[Aj+1,n − Bj+1,n exp(iKj+1,n l)]Kj+1,n Zj+1,n .
N
Kp,n Bp,n,m Bp,n
n=0
− Kp+1,m sin(Kp+1,m l)ap+1,m Cp+1,m = 0.
(5.34f)
(5.33d)
n=0
Eqs. (5.34a–f) reduce to the following system: AX = c, where
A is a square matrix of size [2p(N + 1)], X = [R0 , . . .RN , A2,0 , B2,0 . . ., A2,N , B2,N , . . .. . ., Ap,0 , Bp,0 , . . ., Ap,N , Bp,n , Cp+1,0 , . . ., Cp+1,N ]T is the unknown vector,
and
c = [−B1,0,0 , . . ., B1,0,N , K1,0 B1,0,0 , . . ., K1,0 B1,0,N , 0, . . .. . .. . .. . .0 ]T .
2(p−−1)(N+1)−times
Applying the matching conditions (5.30a–b) along the boundary x = xp separately, the following equations are obtained: N
[Ap,n exp(iKp,n l) + Bp,n ]Zp,n =
n=0
N
5.3. Comparison with existing result Cp+1,n cos(Kp+1,n l)Zp+1,n ,
n=0
(5.33e) N
i
[Ap,n exp(iKp,n l) − Bp,n ]Kp,n Zp,n
n=0
=
N
Cp+1,n sin(Kp+1,n l)Kp+1,n Zp+1,n .
(5.33f)
n=0
Further, using the orthogonality property of Z2,m , Zj+1,m (j = 2, 3, . . ., p − 1) and Zp+1,m , for each m = 0, 1, . . ., N, in Eqs. (5.33a,b), (5.33c,d) and (5.33e,f), respectively, it follows N
By solving the above system of linear equations, R0 can be evaluated.
B1,n,m Rn − i˛a2,m A2,m − i˛ exp(iK2,m l)a2,m B2,m = −B1,0,m ,
n=0
(5.34a)
In order to ascertain that our model is effective, we compare our reflection coefficient plotted against the dimensionless porous structure width with the corresponding result in the work of Madsen [7]. This is accomplished by taking h1 = hp+1 , i.e., by placing the porous structure on the horizontal seabed instead of a p-step bottom. The excellent agreement between our result and Madsen’s result can be observed from Fig. 12. This confirms that our model is valid and hence can be employed effectively to investigate various issues related to scattering of waves by a porous structure placed on a p-step bottom. 5.4. Numerical results: p-step bottom Here the corresponding study of different parameters on reflection coefficient is taken up along with two additional studies on number of steps (p) and height (hp+1 ) of the (p + 1)th region as compared to the 2-step case, with L/h1 = 2; f = 1; S = 1; hp+1 /h1 = 0.5 and = 0.7, 0.8 and 0.9.
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
381
1 Present work ( γ= 0.5) 0.9
f= 1; T= 17.3
Present work ( γ= 0.95)
0.8
Madsen(1983) ( γ= 0.5)
0.7
Madsen(1983) ( γ= 0.95)
0.6 |R 0 |
0.5 0.4 0.3 0.2
Fig. 15. Variation of |R0 | against h1 for different f with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, = 0.9 and N = 9.
0.1 0 0
1
2
3
4 k
1,0
5
6
7
1
L
Fig. 12. Variation of reflection coefficient |R0 | against dimensionless width of porous structure (k1,0 L) for different porosity () with steps=40, h1 = 21, hp+1 = h1 , f = 1, ı = 0.2, T = 17.3, ai = 0.87 and N = 10 (comparison with Madsen [7]).
0.9
f= 0.25
0.8
f= 0.5
0.7
f= 0.75 0.6
|R0 | 0.5
f= 1
0.4 0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
L/h1 Fig. 16. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different f with steps = 8, hp+1 /h1 = 0.5, = 0.9 and N = 9.
Fig. 13. Variation of |R0 | against h1 for different numbers of evanescent modes (N) with steps = 8, hp+1 /h1 = 0.5, f = 1, L/h1 = 2 and = 0.9.
While presenting the reflection coefficient |R0 | by Fig. 13 (against wave number for various number of evanescent modes), Fig. 14 (against wave number for various porosity), Fig. 18 (against width of the porous structure for various porosity), Fig. 19 (against wave number for various width of the porous structure) and Fig. 22 (against wave number for various angles of incidence), it is observed that reflection follows the same pattern as that of the 2-step case except for the fact that for relatively long waves, oscillation is encountered in the reflection coefficient |R0 | before attaining a constant value for relatively short waves.
Figs. 15 and 16 also suggest the same pattern of the reflection coefficient plotted against h1 and L/h1 , respectively, for different friction factors as like that of 2-step case, except for the fact that oscillation vanishes for relatively lower values of the wave number. Fig. 17 shows the effect of wave number on the reflection coefficient for different numbers of steps. Here the number of steps = 4, 6, 8 and 10 are considered. In this case also, oscillation in the reflection coefficient is observed for relatively long waves and it disappears with an increase in the value of the wave number. Higher number of steps gives rise to higher reflection coefficient but converges for sufficiently higher number of steps (here we consider steps = 8 and 10). Further the effect of the angle of incidence on the reflection coefficient is studied for different values of porosity (Fig. 20). It is observed that for values of up to 40◦ (approx.), higher porosity results in lower reflection coefficient, but as the value of increases, the aforesaid effect starts reversing and after = 50◦ (approx.),
1
1
γ= 0.7
0.9 0.8
γ= 0.8
0.7
|R | 0
γ= 0.9
0.6
0.9
steps= 4
0.8
steps= 6
0.7
steps= 8
0.6
|R |
0.5
0
0.4
steps= 10 0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0
1
2
3
ν h1
4
5
6
7
Fig. 14. Variation of |R0 | against h1 for different with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, f = 1 and N = 9.
0 0
1
2
3
νh1
4
5
6
7
Fig. 17. Variation of |R0 | against h1 for different number of steps with hp+1 /h1 = 0.5, L/h1 = 2, f = 1, = 0.9 and N = 9.
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S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385 1
1 γ= 0.7
0.9 0.8
γ= 0.8
/h = 0.75
p+1
0.8
0.7
1
hp+1/h1= 0.625
0.7 γ= 0.9
0.6
|R0|
h
0.9
h
0.6
|R0|
0.5 0.4
/h = 0.5
p+1
1
0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0
0.5
1
1.5
2
2.5
0
3
0
L/h1 Fig. 18. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different with steps = 8, hp+1 /h1 = 0.5, f = 1, and N = 9.
1
10
20
30
40
θ
50
60
70
80
90
Fig. 21. Variation of |R0 | against incident wave angle ( ) for different hp+1 /h1 with steps=8, L/h1 = 16, f = 1 = 0.9, and N = 9.
1
L/h = 0.5 1
0.9
L/h = 0.75
0.9
0.8
L/h = 1
0.8
0.7
L/h = 1.25
0.6
L/h1= 1.5
1
|R | 0
1
0
θ= 20
0.6
|R |
L/h = 1.75 1
0
L/h = 2
0
θ= 30
0.5
1
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0
1
2
3
νh1
4
5
6
0
7
0
Fig. 19. Variation of |R0 | against h1 for different width of the porous structure (L/h1 ) with steps=8, hp+1 /h1 = 0.5, f = 1, = 0.9 and N = 9.
1
2
3
ν h1
4
5
6
7
Fig. 22. Variation of |R0 | against h1 for different incident wave angle ( ) with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, f = 1, = 0.9 and N = 9.
1
1
γ= 0.7
0.9
0.9
0.8
0.6
0.7
γ= 0.9
|R |
steps= 2
0.8
γ= 0.8
0.7
0
θ= 100
0.7
1
0.5
0
θ= 0
|R0|
0.5
steps= 8
0.6 0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0 0
10
20
30
40
θ
50
60
70
80
0
90
Fig. 20. Variation of |R0 | against incident wave angle ( ) for different with steps = 8, hp+1 /h1 = 0.5, L/h1 = 2, f = 1 and N = 9.
higher porosity gives rise to higher reflection before converging to each other for very large values of . We also investigate the effect of the angle of incidence on the reflection coefficient for different values of hp+1 /h1 , namely hp+1 /h1 = 0.5, 0.625 and 0.75 (Fig. 21). Up to = 28◦ (approx.), higher hp+1 /h1 results in lower reflection coefficient but as the value of
increases further (in 30◦ < < 50◦ ), higher hp+1 /h1 gives rise to lower reflection coefficient before converging to each other.
1
2
3
νh 1
4
5
6
7
Fig. 23. Variation of |R0 | against h1 for different number of steps (2-step and 8step) for the same set of parameter values (hp+1 /h1 = 0.75, L/h1 = 1, f = 1, = 0.9 and N = 9).
1 0.9
steps= 2
0.8 0.7
steps= 8 0.6
|R |
0 0.5 0.4 0.3
5.5. Comparison between 2-step and p-step bottom
0.2 0.1
To find the difference or similarity between the results for both cases, we move forward to compare graphically the two different solutions for 2-step and p-step cases for the same set of parameter values by comparing |R0 | against the wave number and the dimensionless width of the porous structure for steps = 2 and 8. We consider L/h1 = 1; f = 1; S = 1; hp+1 /h1 = 0.75 and = 0.9.
0 0
0.2
0.4
0.6
0.8
1
L/h1
1.2
1.4
1.6
1.8
2
Fig. 24. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different number of steps (2-step and 8-step) for the same set of parameter values (hp+1 /h1 = 0.75, f = 1, = 0.9 and N = 9).
S. Das, S.N. Bora / Applied Ocean Research 47 (2014) 373–385
Fig. 25. Variation of |R0 | against h1 for different number of steps (2-step and 8step) for the same set of parameter values (hp+1 /h1 = 0.75, L/h1 = 1, f = 1, = 0.5 and N = 9).
383
but converges for sufficiently higher number of steps. For relatively lower value of the angle of incidence, |R0 | takes higher value for lower porosity but the effect reverses for relatively higher value of the angle of incidence. Same effect mentioned above is observed for different values of hp+1 . Further, comparison between the reflection coefficients of 2-step and p-step bottom is carried out for two different values of porosity against wave number and dimensionless width of the porous structure. It is observed that for relatively lower value of porosity, p-step bottom results in lower reflection coefficient. For the validity of our mathematical model, we compare our work with Madsen [7] by plotting reflection coefficient against dimensionless width of the porous structure for different values of porosity and we find excellent agreement in this regard. This shows that our model will be effective for solving problems of wave scattering by porous structures placed on a p-step bottom. Acknowledgements
1 0.9
The first author is grateful to the Council of Scientific and Industrial Research (CSIR), Govt. of India for providing him senior research fellowship for pursuing Ph.D. at Indian Institute of Technology Guwahati, India. The authors are immensely grateful to the esteemed reviewers and the Editor-in-Chief for their comments and suggestions which have allowed the authors to carry out a much improved version of the manuscript.
steps= 2 0.8 0.7
steps= 8
0.6
|R |
0 0.5 0.4 0.3 0.2 0.1
Appendix A.
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
L/h1 Fig. 26. Variation of |R0 | against dimensionless width of the porous structure (L/h1 ) for different number of steps (2-step and 8-step) for the same set of parameter values (hp+1 /h1 = 0.75, f = 1, = 0.5 and N = 9.
Figs. 23 and 24 show that the values of the reflection coefficient for = 0.9 are more or less the same in both cases. But if lower value of porosity ( = 0.5 in this case) is considered, significant difference occurs (Figs. 25 and 26). Now it is observed that the 2-step approximation results in lower reflection coefficient as compared to the p-step approximation.
Principle of conservation of mass inside a fluid region leads to the following equation known as the equation of continuity:
∂ + ∇ · (U) = 0. ∂t
(A.1)
Now for the steady state flow of fluid having constant density, Eq. (A.1) reduces to
∇ · U = 0.
(A.2)
For a potential flow, i.e., when U = ∇ , the equation of continuity reduces to Laplace’s equation
∇ 2 = 0. 6. Conclusion Based on linear water wave theory, a system of linear equations is derived using matching conditions along the boundary to find the reflection coefficient for oblique water wave scattering by a vertical porous structure placed on an elevated bottom consisting of two steps as well as p steps. It is observed that the propagating mode controls reflection up to a certain wave number. Increment in the height of the porous structure does not affect reflection after certain width of the structure. Higher value of the reflection coefficient is observed for lower values of porosity. It is also observed that lower values of friction factor lead to oscillation in the reflection coefficient which vanishes for higher values of friction factor. For relatively long waves, the values of the angle of incidence do not affect |R0 | but for short waves, higher value of angle of incidence results in lower reflection coefficient. Porosity of the structure does not contribute up to a certain range of values of angle of incidence. Corresponding study of the effect of number of evanescent modes and porosity for p-step impermeable bottom shows the same pattern, but oscillation for relatively long waves appear for a p-step bottom as against a 2-step bottom. Oscillation in |R0 | for higher values of friction factor vanishes for a p-step seabed at a relatively lower value of the wave number as compared to a 2-step sea-bed. Increasing number of steps results in higher reflection coefficient
The incompressible equations of motion inside the porous structure can be written in the following form:
∇ ·U = 0,
(A.3a)
∂U ∇P + resistance forces. =− ∂t
(A.3b)
The resistance forces in Eq. (A.3b) are evaluated by combining known steady and unsteady stress relationships. Under steady state flow conditions the pressure drop through the porous medium is specified by Ward [19] as −
∇P
=
vk Kp
U +
C
f 2 U|U|, Kp
(A.4)
where vk is the kinematic viscosity, Kp is the intrinsic permeability and Cf is a dimensionless turbulent resistance coefficient of the medium. It is hypothesized by Sollitt and Cross [5] that unsteadiness may be accounted for by introducing an additional term which evaluates the added resistance caused by the virtual mass of discrete grains within the medium. The resistance force due to the virtual mass is equal to the product of the displaced fluid mass, the virtual mass coefficient and the acceleration in the approach velocity. The
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resulting force is distributed over the fluid mass within the pore so that the force per unit mass of fluid is simply
Now, inside the porous region
∂U 2 ∇ P2 − fωU 2 , =− ∂t ∂˚2 P2 − fω˚2 , ⇒ S =− ∂t S
1− ∂U CM . ∂t
(A.5)
Combining Eqs. (A.4) and (A.5); and replacing the resistance force in Eq. (A.3b) with them we get 1− ∂U ∇ P vk ∂U U − 2 U|U| − − CM =− , Kp ∂t ∂t Kp Cf ∂U ∇ P vk U − 2 U|U|, − =− ⇒ S Kp ∂t Kp
(A.6)
1− S = 1 + CM .
(A.7)
Kp
fωU · Udt t
=
t+T
Kp
t
V
U +
C
f 2 U|U| Kp
· Udt,
(A.8)
where V is the volume of the flow field and T is the wave period. Now assuming f to be constant throughout the flow field, the following form can be written
f =
1 ω
V
dV
t+T t
V
2 dV
vk U 2 Kp
t+T t
+
Cf |U|3 Kp
U 2 dt
dt .
(A.9)
Appendix B. Let us consider U 1 and U 2 , respectively, to be the velocities of a fluid at any point inside the water and porous region attached to each other. Then the following relation holds true: U i = ∇ ˚i
⇒ −iω1 = −
(B.12)
P1 .
Now, along the vertical boundary between the water and porous regions, continuity of pressure (P1 = P2 ) results in (from Eqs. (B.11) and (B.12)) the following matching condition:
i = 1, 2.
(B.13)
Mass flux per unit volume and unit time inside the porous region is U and the same inside the water region is U. Along the vertical boundary, the continuity of mass flux implies U 1 = U 2 , ⇒
∂1 ∂2 = ∂x ∂x
(B.14) along x-direction.
Moreover, if both the regions consist of the same medium, then Eqs. (B.13) and (B.14) reduce to 1 = 2 ,
(B.15a) along x-direction.
(B.15b)
It is obvious that these matching conditions are valid along the vertical boundary separating any two regions next to each other.
vk
dV
P2 ,
∂U 1 ∇ P1 , =− ∂t ∂˚1 P1 =− , ∂t
∂1 ∂2 = ∂x ∂x
t+T
dV V
ω˛2 = −
(B.11)
P2 ,
1 = i˛2 .
C
f 2 U|U| → fωU.
Combination of Eqs. (A.6) and (A.7) leads to Eq. (2.1b). In order to evaluate f, Lorentz’s principle of equivalent work is applied which says that the average rate of energy dissipation should be identical whether evaluated using the true non-linear resistance law or its linearized equivalent. Since the resistance terms of the above relation represent friction force per unit mass acting at a point in the flow field, the following equality (energy dissipation) holds:
⇒
⇒
It is worth mentioning that S = 1 accounts for two different cases, namely, = 1 or the absence of structure and CM = 0 or the presence of inviscid fluid. Now, linearization of Eq. (A.6) is necessary in order to find an analytical solution and hence, the dissipative stress term is replaced by a linear stress term in U by the following form:
Kp
P2 − fω2 ,
where P1 and P2 are the dynamic pressures of the water and porous regions, respectively. In the water region, Bernoulli’s equation gives
where
U +
−iωS2 = −
⇒ ω(f − iS)2 = −
Cf
vk
⇒
(B.10)
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