Oblique wave trapping by a submerged porous ...

INCHOE2014 (Proceedings)

5-7 Feb. 2014

Goa, India

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Oblique wave trapping by a submerged porous structure near a wall S. Koley1, H. Behera2 and T. Sahoo3 Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology Kharagpur.

email address1: [email protected] email address2: [email protected] email address3: [email protected]

ABSTRACT Koley, S., Behera, H. and Sahoo, T., 2014. Oblique wave trapping by a submerged porous structure near a wall, Proceedings of the Fifth Indian National Conference on Harbour and Ocean Engineering (INCHOE2014), 5-7 Feb. 2014, CSIR-NIO, Goa, India. In the present paper, trapping of oblique waves by a bottom-standing submerged porous structure of finite width is analyzed in finite water depth. The problem is analyzed based on the linearized water wave theory in water of uniform depth using the eigenfunction expansion method. The results are compared with the solution obtained via boundary integral equation method. The role of oblique angle in trapping ocean waves is studied. The reflection coefficients and hydrodynamic forces on the rigid wall are computed and analyzed in different cases to understand the role of the porous structure in trapping surface waves. The present study will be of significance importance in the design of various types of coastal structures used in the marine environment for trapping wave energy and protecting sea wall and other marine facilities

ADDITIONAL INDEX WORDS: Submerged porous structure, wave trapping, eigenfunction expansion method, Boundary integral equation method. Sulisz (1985) solved a problem with porous structures extending from bottom to free surface, which comprised of multiple porous regions. Gu and Wang (1992) solved a problem with submerged porous structures, and proposed an efficient method to calculate wave dissipation. In the present paper, oblique wave trapping by a bottomstanding partial porous barrier near a wall is analyzed to find the efficiency of the structure in trapping and reflecting waves. A numerical code is developed using boundary integral equation method and the computational results are validated by comparing the same with the results obtained via eigenfunction method.

1. INTRODUCTION In recent decades, porous structures are introduced into the art of dissipating and reducing the impact of unwanted wave energy on various marine facilities. These structures often alter the phase of the reflected waves which in turn creates a tranquility zone. Further, porous structures are often used as wave absorbers in laboratories for removing unwanted waves during experiments. As a result, study of surface gravity wave interaction with porous structures is becoming increasingly important for design of porous structures like breakwaters for protecting harbors, reducing erosion of the coast due to wave action and protecting coastal platforms from wave action. Recently, to trap ocean waves near a coastal/marine structure, porous structures are in use. With suitable adjustment of the position of the porous structure, ocean waves are being trapped. The process of wave absorption due to the porous structure within the confined region is referred as wave trapping. This phenomenon was initially observed by Chwang and Dong (1984) in case of wave interaction with a porous structure kept near a vertical wall in an infinitely extended open channel of uniform depth. Sahoo et al. (2000) studied the trapping and generation of surface waves by partial porous structures near a channel end wall which was later generalized by Yip et al. (2002) to analyze wave trapping by porous and flexible structures. These works were on wave past thin porous structures. Losada et al. (1996) studied the performance of wave past submerged porous structure of finite width. Recently, Koley et al. (2014) and Behera et al. (2015) studied the wave past porous structures of finite width in single and two-layer fluids. The study has been extended by Koley et al. (2015) to investigate gravity waves interaction with a composite breakwater placed on a sloping sea bed. The use of boundary element method in solving the coupled problem of wave and porous structure interactions does not require the wave numbers within the porous region. Depending on the porous flow mechanism and the structural types, the numerical formulations were somewhat different.

2. MATHEMATICAL FORMULATION It is assumed that a porous structure of finite height a and width b, submerged in a finite water depth h. The structure occupies the region Lb = ( − h ≤ z ≤ − h + a , 0 ≤ x ≤ b) and the assiociated gap region Lg = ( − h + a ≤ z ≤ 0, 0 ≤ x ≤ b) as in Figure 1. Distance between porous structure and the rigid wall is L. Under the assumption that the incident wave is propagating making an angle θ with the x -axis and the motion is simple harmonic in time with frequency ω in the linearized theory of water waves, the velocity potentials are of the form

{

Φ j ( x, y , z, t ) = Re φ j ( x, z )e

− i( k y y −ωt )

}

for

1,2,3,4, (1) j=

where k y = k0 sin θ and k0 is the progressive wave number in region 1. Thus, the spatial velocity potentials φ j ( x, z ) for j = 1, 2,3, 4 satisfy the reduced wave equation

 ∂2 ∂2 2  2 + 2 − k y φ j = 0 z z ∂ ∂   along with the bottom boundary condition given by

Proceedings of the Fifth Indian National Conference on Harbour and Ocean Engineering (INCHOE2014)

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(2)

Koley, S.

∞

φ4 = ∑ Tn cos qn ( x − b − L) I n ( kn , z ),

(13)

n =0

where the eigenfunctions I n (kn , z ) , M n ( pn , z ) and Pn ( pn , z ) for n = 0,1, 2,3... are given by

∂φ j

(3) = 0, on z = − h, for j = 1,3, 4. ∂z The linearized free surface boundary condition is given by ∂φ j (4) − Kφ j = 0, on z = 0, for j = 1, 2, 4, ∂z where K = ω 2 / g , g is the gravitational constant, m is the inertial coefficient and f is the linearized friction coefficient. The condition on the impermeable end-wall is given by ∂φ4 (5) = 0 on x = b + L, − h < z < 0. ∂x The radiation condition is given by

Qn =

(9)

∞

n =0

{

∞

{

− iQn x

φ3 = ∑ An e n =0

− iQn x

+ Bn e

iQn ( x − b )

I n ( kn , z ),

(10)

} M ( p , z),

(11)

+ Bn e

} P ( p , z),

(12)

n

iQn ( x − b )

n

n

n

with

qn = kn2 − k y2

and

pn2 − k y2 . In regions 1 and 4, the eigenvalues kn satisfy

− iq0 x

I 0 ( k0 , z ), with q0 and I 0 ( k0 , z )

being the same as in Eq. (6), the total potential associated with the physical problem is written as φ = φ0 + φ sc , where φ sc is the scattered potential. Thus, in region 1, the scattered velocity potential φ1sc satisfy the following boundary conditions

∂φ1sc = 0 on Γb , Γb , (20) 1 3 ∂n ∂φ1sc − Kφ1sc = 0 on Γ f , (21) ∂n with K is same as defined in Section 2. The condition on the impermeable end wall as in Eq. (5) is rewritten as ∂φ1sc ∂φ0 + = 0 on Γl . (22) ∂n ∂n Further, the radiation boundary condition is given by ∂φ1sc − ik yφ1sc = 0 on Γ c . (23) ∂n

n =0

φ2 = ∑ An e

determined

and is given by φ0 = e

with conditions (3) – (9) associated with the wave trapping by porous structure as in Figure 1 are expressed as iqn x

be

In this Section, a numerical code is developed based on boundary integral equation method (BIEM) to solve the boundary-value problem described in the preceding Section. The developed BIEM can be used for porous breakwater of arbitrary shape and also for arbitrary bottom topography. The fluid domain is decomposed into two regions, as shown in Figure 2. Here, we have taken two regions instead of four regions in Section 2. Assuming that φ0 is the incident potential

In this Section, the eigenfunction expansion method is applied using the matching conditions to derive a system of equations for the determination of the unknowns in the expansion formulae for the velocity potentials. The spatial velocity potentials φ j for j = 1, 2,3, 4 satisfying Eq. (2) along

∞

to

4. BOUNDARY ELEMENT METHOD

3. ANALYTIC METHOD OF SOLUTION

I 0 ( k0 , z ) + ∑Rn e

(17)

Eigenfunctions I n (kn , z ) , M n ( pn , z ) and Pn ( pn , z ) form a complete set of orthogonal functions in their corresponding domains as in Losada (1996). Using the orthogonality to the matching conditions in Eqs. (7) – (9), a system of equations for the determination of the unknowns in the velocity potentials in Eqs. (10) – (13) are obtained.

in the water region 1. The conditions on the submerged interface of the structure at z = − h + a as in Losada et al. (1996) are given by ∂φ2 ∂φ (7) = ε 3 and φ2 = (m − i f )φ3 , for 0 ≤ x ≤ b, ∂z ∂z where ε is the porosity of the permeable material. The continuity of mass flux and pressure at the fluid and porous structure interfaces at x = 0 and x = b are given by z ∈ Lg , φ , φj =  2 (8) z ∈ Lb , (m − if )φ3 ,

− iq0 x

(16)

the dispersion relation (18) K = kn tanh kn h However, in regions 2 and 3, the eigenvalues pn satisfy the dispersion relation (19) K − pn tanh pn h = Fn ( K tanh pn h − pn )

with q0 = k02 − k y2 and I 0 (k0 , z ) is the vertical eigenfunction

φ1 = e

(15)

constants

iq x

 ∂φ2  ∂x , z ∈ Lg , = ∂x  ∂φ3 ε , z ∈ Lb ,  ∂z where j = 1 at x = 0 and j = 4 at x = b.

 ig  cosh pn ( z + h ) − Fn sinh pn ( z + h ) M n ( pn , z ) =   , cosh pn h − Fn sinh pn h ω

where Rn , An , Bn and Tn for n = 0,1, 2,3... are the unknown

(6) φ1 = (e 0 + R0e 0 ) I 0 (k0 , z ), for x < 0, where R0 is the unknown reflection coefficient to be determined

∂φ j

(14)

 ig  (1 − Fn tanh pn a ) cosh pn ( z + h ) Pn ( pn , z ) =   ,  ω  ( s − if )(cosh pn h − Fn sinh pn h ) (1 − G ) tanh pn a ε Fn = , and G = , 1 − G tanh 2 pn a ( s − if )

Figure 1: Schematic diagram of wave trapping

− iq x

 ig  cosh kn ( z + h) , I n (kn , z ) =    ω  cosh kn h


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Wave trapping by a submerged porous structure

Cφ2 + ∫

Γ m 1 +Γ m 2 +Γ m 3

 φ ∂G G ∂φ  −   d Γ  ( m − if ) ∂n ε ∂n 

∂G (31) d Γ = 0, ∂n where C is the solid angle constant and the two integrals covers the entire boundary of each fluid region. To convert the above integral equations to a matrix equation, the entire boundary of regions 1 and 2 is discretized into a finite number ∂φ are of segments. On each segment, the values of φ and ∂n ∂G assumed to be constant, and the singularities G and are ∂n integrated analytically (Au and Brebbia 1982). The integral equations (30) and (31) can then be written in the following discrete form ∑ ( H ij − K G ij )φ1scj + ∑ ( H ij − ik y G ij )φ1scj + ∑ H ij φ1scj + ∫ φ2 Γb 2

Figure 2: Schematic diagram for BEM formulation This radiation condition is introduced in the computational domain as an auxiliary vertical boundary Γ c which is located sufficiently far from the breakwater. The distance of this auxillary boundary from the structure is decided from the computation keeping the convergence of the solution is mind. Further, at the boundary Γ m1 , Γ m 2 and Γ m 3 , the conditions on the porous structure as in Eqs. (7) – (9) are rewritten as ∂φ ∂φ ∂φ1sc ∂φ0 = =ε 2 , + (24) ∂n ∂n ∂n ∂n

φ = φ1sc + φ0 = ( m − if ) φ2 ,

(25)

where the upper bar means the value newly defined at the matching boundary Γ m1 , Γ m 2 and Γ m 3 . Here φ2 , the total velocity potential in region 2, satisfy the bottom boundary condition as in Eq. (3) is rewritten as ∂φ2 = 0 on Γb 2 . (26) ∂n To solve the present boundary value problem, a two-domain boundary integral equation method using simple sources along the entire boundary (as in Kim and Kee 1996) is developed. The fundamental solution (Green function) of the Helmholtz equation (2) is K G = − 0 ( k y r ), (27) 2π where K 0 is the modified zeroth-order Bessel function of the second kind and r is the distance from the source point ( x′, y′) to the field point ( x, y ) . From Eq. (27), the normal derivative of G is obtained as ∂G k y ∂r = K1 ( k y r ) . ∂n 2π ∂n As r → 0, one obtains the asymptotic behavior

(28)

 kyr  (29) K 0 ( k y r ) ≈ −γ − ln  ,  2  where γ = 0.5772 is known as Euler’s constant. By applying Green’s second identity in each of the regions to the unknown potentials φ1sc , φ2 and imposing boundary conditions (20) – (26), the integral equations in each fluid domain can be written as  ∂G   ∂G  − KG  φ1sc d Γ + ∫  − ik y G  φ1sc d Γ Cφ1sc + ∫  Γf Γ c  ∂n  ∂n   ∂G ∂G ∂   − G (φ − φ0 )  d Γ dΓ + ∫ φ − φ0 ) ( Γ m 1 +Γ m 2 +Γ m 3  ∂n ∂ ∂ n n   ∂φ0 sc ∂G (30) +∫ φ1 d Γ = −∫ Gd Γ, Γb1 +Γb 3 Γl ∂n ∂n

+ ∫ φ1sc Γl

Γf

+ ∑H φ ij

= −∑

sc 1 j Γ +Γ b1 b3

∂φ0 j ∂n

∑H

ij

=∑

G ij ∂φ j ε ∂n

φ2 j

Γb 2

G

Γc

+ ∑H

ij Γl

+∑

ij

(φ

j

− φ0 j )

Γ m 1 +Γ m 2 +Γ m 3

∂ + ∑ (φ j − φ0 j ) G ij ∂n

H ij φ ( m − if ) j

Γl

,

(32)

Γ m 1 +Γ m 2 +Γ m 3

Γ m 1 +Γ m 2 +Γ m 3

.

(33)

Γ m 1 +Γ m 2 +Γ m 3

The influence coefficients H and G are defined by ∂G (34) H ij = Cδ ij + ∫ dS , G ij = ∫ GdS . Γ j ∂n Γj We solve (32) and (33) together to get the unknown scattered velocity potential φ1sc and total velocity potential

φ2 on the boundary of the regions 1 and 2 respectively. These calculated scattered velocity potentials will be used further to get velocity potentials at any point inside the regions 1 and 2 respectively.

5. NUMERICAL RESULTS AND DISCUSSION In this Section, a MATLAB program is developed to investigate the effects of different wave and structural parameters on the wave reflection and the hydrodynamic loads on the breakwater and the rigid wall. The analytic solutions described in Section 3 were compared with BEM-based numerical solutions explained in Section 4. The convergence of the BEM method with increase number of uniform segments is shown in Figure 3. The errors based on the the difference between analytic and BEM-based solutions are plotted. The truncation boundary is located three to four times the distance L of Figure 1 away form the structure to ensure that the local wave effect is negligible. It can be seen in Figure 3 that using N=500, the maximum error can be limited to around 1% in the entire frequency range considered. Here N is the total number of boundary elements used in the discretization of the entire domain. In the present study, porosity of the permeable material ε = 0.437, inertial coefficient m = 1.0 are kept fixed through out the analysis unless it is mentioned. The reflection coefficient R, the horizontal wave forces F1 , F2 and F3 on the structure at= x 0,= x b and on the rigid wall respectively

Proceedings of the Fifth Indian National Conference on Harbour and Ocean Engineering (INCHOE2014) 3

Koley, S.

Figure 4. L / λ vs R for different values of a / h.

Figure 3. Convergence test of BEM are computed using the formulae R = R0 , −h+a

−i ρω ∫ φ1 ( 0− , z ) − φ3 ( 0+ , z ) dz, F1 = −h

−h+a

−i ρω ∫ φ3 ( b − , z ) − φ4 ( b + , z ) dz, F2 = −h

0

F3 = −i ρω ∫ φ4 ( b + L, z ) dz, −h

where φ j ( x, z ) for j = 1,3,4 are same as in Figure 1. In all the following figures lines are for analytic solutions and symbol are for BEM solutions. In Figure 4, the reflection coefficient R is plotted versus the normalized position of the porous structure L / λ for different values of structural height a / h with b / h = 0.5, Figure 5. L / λ vs R for different values of b / h.

θ = 150 and f = 0.5. Figure 4 shows that these curves are periodic and each curve repeats itself in every half-wavelength. Similar observation were obtained by Sahoo et al. (2000) in case of wave trapping by thin porous structure. In general, reflection coefficient decreases with an increase of the structural height. This is due to the fact that as the height of the structure increases, less waves can be transmitted and in the process more waves are trapped inside the region between the channel end-wall and the structure. Figure 5 shows the variation of the reflection coefficient R versus the normalized position of the porous structure L / λ for different values of the structural width b / h with

The results reveal that reflection coefficient R decreases with an increase in structural height a / h. However, the reflection coefficient R attends global minimum for the oblique angle θ within the range 850 − 890 for different values of a / h . These global minima are referred as the critical angle of incidence.

= = = a / h 0.9, θ 150 and f 0.5. It is observed that minimum value of the reflection coefficients decreases with an increase of the structural width b / h. which means that less wave energy get reflected. Figure 6 shows the variation of the reflection coefficient R versus the normalized position of the porous structure L / λ for different values of the linearized friction coefficient f = 0.5 with and θ 150. It is observed = a / h 0.9, = b / h 0.5 = that the reflection coefficient decreases with an increase of the linearized friction coefficient f , which means that more wave is being trapped by the structure. In Figure 7, the reflection coefficient R versus angle of incidence θ is plotted for different values of the structural

Figure 6. L / λ vs R for different values of f .

height a / h with width = b / h 0.5, = L / λ 1.0 = and f 0.5.


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Wave trapping by a submerged porous structure

Figure 7. θ ( deg. ) vs R for different values of a / h.

Figure 9. k0 h vs F2 / ρ gh 2 for different values of f .

Figure 8. k0 h vs F1 / ρ gh 2 for different values of a / h.

Figure 10. k0 h vs F3 / ρ gh 2 for different values of b / h.

Figure 8 shows the total wave forces on the rigid wall as function of k0 h with varying structural height a / h. Here

ACKNOWLEDGEMENTS

b / h = 0.5, f = 0.5, θ = 150 , and L / λ = 1. It is observed that the force on the rigid wall decreases as the wave number k0 h

increases. Variation of structural height a / h on wave forces is negligible for higher values of k0 h. In Figure 9, the horizontal wave forces on the front face of the structure are plotted versus non-dimensional wave number k0 h for different values of the linearized friction coefficient f with a / h = 0.6, b / h = 0.5, θ = 150 , and L / λ = 1. It is observed that the wave force initially higher for higher values of f , but for k0 h ≥ 8, variation of f on wave forces is negligible. In Figure 10, horizontal force on the rear face of the structure is plotted versus non-dimensional wave number k0 h

for different values of the structural width b / h with a / h = 0.6, f = 0.5, θ = 150 , and L / λ = 1. It is observed that there is negligible effect of the variation of structural width on the wave forces.

SK acknowledges the support received from CSIR, New Delhi as a Junior Research Fellow. HB gratefully acknowledges the support received from MoES, New Delhi in terms of a Senior Research Fellowship.

CONCLUSIONS In the present study, oblique gravity wave trapping by a bottom-standing porous structure of finite width is analyzed. It is observed that irrespective of barrier length, full reflection by the barrier near the wall occurs when the distance between the structure and rigid wall is approximately an integer multiple of half of the wave length. Further, minimum in wave reflection is observed when the barrier is kept in an intermediate range within half of the wavelength from the rigid wall. Further, it is observed that wave force on the wall and porous structure reduces with an increase in non-dimensional wave number. A minimum in wave reflection is observed for certain critical angle of incidence. The results obtained based on boundary integral equation method is very close to the analytic results derived based on eigenfunction expansion method and thus can be easily generalized to deal with structures of variable cross section and variable water depth.

Proceedings of the Fifth Indian National Conference on Harbour and Ocean Engineering (INCHOE2014) 5

Koley, S.

REFERENCES Chwang, A.T. and Dong, Z. (1984). Wave-trapping due to porous plate. In: Proceedings of the 15th symposium on naval hydrodynamics. National Academy Press, Washington: 407-417. Sahoo, T., Lee, M.M. and Chwang, A.T. (2000). Trapping and generation of waves by vertical porous structures. J. Eng. Mech. 126:1074-1082. Yip, T.L., Sahoo, T. and Chwang, A.T. (2002). Trapping of surface waves by porous flexible structures. Wave motion. 35:41-54. Losada, I.J., Silva, R. and Losada, M.A. (1996). 3-D nonbreaking regular wave interaction with submerged breakwaters. Coastal Eng. 28:229-248. Koley, S., Behera, H. and Sahoo, T. (2014). Oblique wave trapping by porous structures near a wall. J. Eng. Mech. (in press). Behera, H., Koley, S. and Sahoo, T. (2015). Wave transmission by partial porous structures in two-layer fluid. Eng. Anal. Bound Elem. (communicated). Koley, S., Sarkar, A. and Sahoo, T. (2015). Interaction of gravity waves with bottom-standing submerged structures having perforated outer-layer placed on a sloping bed. Appl. Ocean Res. (communicated). Sulisz, W. (1985) Wave reflection and transmission at permeable breakwaters of arbitrary cross-section. Coastal Eng. 9:371-386. Gu, G.Z. and Wang, H. (1992). Numerical modeling for wave energy dissipation within porous submerged breakwaters of irregular cross section. Proc. 23rd ICCE: 1189-1202. Kim, M.H. and Kee, S.T. (1996). Flexible-membrane wave barrier. I: analytic and numerical solutions. J.Wtrwy., Port, Coast. And Oc. Engrg., ASCE 122(1): 46-53. Au, M.C. and Brebbia, C.A. (1982). Numerical prediction of wave forces using the boundary element method. Appl. Math. Modelling. 6(4):218-228.


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