Graphical Abstract Figure
Abstract
The scattering of incident waves by a surface-piercing inverted trapezoidal breakwater (SPTB) encircled by retrofit is numerically examined based on the assumptions of potential flow theory. The dual boundary element method is adopted to evaluate the hydrodynamic performance of the retrofitted SPTB breakwater. The scattering coefficients (i.e., wave transmission, wave reflection, energy loss), and force coefficients acting on the inner SPTB and outer retrofit are reported against relative water depth for various input values of breakwater and incident waves. The SPTB with 10% porosity, spacing , width varied within , and depth is suggested against the incident waves to secure the coastal infrastructure.
References
1.
Liu
, Y.
, Li
, Y. C.
, and Teng
, B.
, 2009
, “Wave Motion Over Two Submerged Layers of Horizontal Thick Plates
,” J. Hydrodyn.
, 21
(4
), pp. 453
–462
. 2.
Dhillon
, H.
, Banerjea
, S.
, and Mandal
, B. N.
, 2016
, “Water Wave Scattering by a Finite Dock Over a Step-Type Bottom Topography
,” Ocean Eng.
, 113
, pp. 1
–10
. 3.
Ning
, D. Z.
, Zhao
, X. L.
, Teng
, B.
, and Johanning
, L.
, 2017
, “Wave Diffraction From a Truncated Cylinder With an Upper Porous Sidewall and an Inner Column
,” Ocean Eng.
, 130
, pp. 471
–481
. 4.
Ding
, W. W.
, Zou
, Z. J.
, Wu
, J. P.
, and Huang
, B. G.
, 2019
, “Investigation of Surface-Piercing Fixed Structures With Different Shapes for Bragg Reflection of Water Waves
,” Int. J. Nav. Archit. Ocean Eng.
, 11
(2
), pp. 819
–827
. 5.
Praveen
, K. M.
, Karmakar
, D.
, and Guedes Soares
, C.
, 2020
, “Wave Interaction With Floating Elastic Plate Based on the Timoshenko–Mindlin Plate Theory
,” ASME J. Offshore Mech. Arct. Eng.
, 142
(1
), p. 011601
. 6.
Mohapatra
, A. K.
, and Sahoo
, T.
, 2024
, “Bragg Scattering of Surface Gravity Waves by a Submerged Composite Wavy Porous Plate
,” ASME J. Offshore Mech. Arct. Eng.
, 146
(3
), p. 031201
. 7.
Vijay
, K. G.
, Venkateswarlu
, V.
, and Karmakar
, D.
, 2020
, “Scattering of Gravity Waves by Multiple Submerged Rubble-Mound Breakwaters
,” Arab. J. Sci. Eng.
, 45
(10
), pp. 8529
–8550
. 8.
Mei
, C. C.
, and Black
, J. L.
, 1969
, “Scattering of Surface Waves by Rectangular Obstacles in Waters of Finite Depth
,” J. Fluid Mech.
, 38
(3
), pp. 499
–511
. 9.
Williams
, A. N.
, and Li
, W.
, 1998
, “Wave Interaction With a Semi-Porous Cylindrical Breakwater Mounted on a Storage Tank
,” Ocean Eng.
, 25
(2–3
), pp. 195
–219
. 10.
Neelamani
, S.
, Uday Bhaskar
, N.
, and Vijayalakshmi
, K.
, 2002
, “Wave Forces on a Seawater Intake Caisson
,” Ocean Eng.
, 29
(10
), pp. 1247
–1263
. 11.
Chandrasekaran
, S.
, and Madhavi
, N.
, 2015
, “Retrofitting of Offshore Cylindrical Structures With Different Geometrical Configuration of Perforated Outer Cover
,” Int. Shipbuild. Prog.
, 62
(1–2
), pp. 43
–56
. 12.
Sahoo
, T.
, Lee
, M. M.
, and Chwang
, A. T.
, 2000
, “Trapping and Generation of Waves by Vertical Porous Structures
,” J. Eng. Mech.
, 126
(10
), pp. 1074
–1082
. 13.
Yip
, T. L.
, Sahoo
, T.
, and Chwang
, A. T.
, 2002
, “Trapping of Surface Waves by Porous and Flexible Structures
,” Wave Motion
, 35
(1
), pp. 41
–54
. 14.
Barman
, K. K.
, and Bora
, S. N.
, 2021
, “Scattering and Trapping of Water Waves by a Composite Breakwater Placed on an Elevated Bottom in a Two-Layer Fluid Flowing Over a Porous Sea-Bed
,” Appl. Ocean Res.
, 113
, p. 102544
. 15.
Venkateswarlu
, V.
, Vijay
, K. G.
, Nishad
, C. S.
, and Behera
, H.
, 2024
, “Oblique Wave Trapping by Sinusoidal Rippled Barrier of Finite Thickness Placed on Closely Spaced Semi-Circular Seabed
,” Phys. Fluids
, 36
(1
), pp. 1
–21
. 16.
Yu
, X.
, 1995
, “Diffraction of Water Waves by Porous Breakwaters
,” J. Waterw. Port Coast. Ocean Eng.
, 121
(6
), pp. 275
–282
. 17.
Kee
, S. T.
, and Kim
, M. H.
, 1997
, “Flexible Membrane Wave Barrier. II: Floating/Submerged Buoy-Membrane System
,” J. Waterw. Port Coast. Ocean Eng.
, 123
(2
), pp. 82
–90
. 18.
Brebbia
, C. A.
, 2017
, “The Birth of the Boundary Element Method From Conception to Application
,” Eng. Anal. Bound. Elem.
, 77
, pp. iii
–x
. 19.
Chen
, J. T.
, Hong
, H. K.
, and Chyuan
, S. W.
, 1994
, “Boundary Element Analysis and Design in Seepage Problems Using Dual Integral Formulation
,” Finite Elem. Anal. Des.
, 17
(1
), pp. 1
–20
. 20.
Chen
, J. T.
, and Chen
, K. H.
, 1998
, “Dual Integral Formulation for Determining the Acoustic Modes of a Two-Dimensional Cavity With a Degenerate Boundary
,” Eng. Anal. Bound. Elem.
, 21
(2
), pp. 105
–116
. 21.
Chen
, K. H.
, Chen
, J. T.
, Chou
, C. R.
, and Yueh
, C. Y.
, 2002
, “Dual Boundary Element Analysis of Oblique Incident Wave Passing a Thin Submerged Breakwater
,” Eng. Anal. Bound. Elem.
, 26
(10
), pp. 917
–928
. 22.
Chen
, K. H.
, Chen
, J. T.
, Lin
, S. Y.
, and Lee
, Y. T.
, 2004
, “Dual Boundary Element Analysis of Normal Incident Wave Passing a Thin Submerged Breakwater With Rigid, Absorbing, and Permeable Boundaries
,” J. Waterw. Port Coast. Ocean Eng.
, 130
(4
), pp. 179
–190
. 23.
Venkateswaralu
, V.
, Vijay
, K. G.
, Nishad
, C. S.
, and Sahoo
, T.
, 2022
, “Gravity Wave Scattering by Retrofitted Circular Breakwaters Using Dual Boundary Integral Formulation
,” Ocean Eng.
, 265
, p. 112259
. 24.
Nishad
, C. S.
, Neelamani
, S.
, Vijay
, K. G.
, and Sahoo
, T.
, 2022
, “Bragg Scattering of Surface Gravity Waves by an Array of Surface-Piercing Variable Porosity Barriers
,” J. Waterw. Port Coast. Ocean Eng.
, 148
(6
), p. 04022021
. 25.
Bird
, H. W. K.
, and Shepherd
, R.
, 1984
, “On the Interactions of Surface Waves With Immersed Structures
,” Int. J. Numer. Methods Fluids
, 4
(8
), pp. 765
–780
. 26.
Hutchison
, B. L.
, 1984
, “Impulse Response Techniques for Floating Bridges and Breakwaters Subject to Short-Crested Seas
,” Mar. Technol.
, 21
(3
), pp. 270
–276
. 27.
Kumar
, P. S.
, Bhattacharjee
, J.
, and Sahoo
, T.
, 2007
, “Scattering of Surface and Internal Waves by Rectangular Dikes
,” ASME J. Offshore Mech. Arct. Eng.
, 129
(4
), pp. 306
–317
. 28.
Qiu
, L. C.
, 2009
, “Modeling and Simulation of Transient Response of a Flexible Beam Floating in Finite Depth Water Under Moving Loads
,” Appl. Math. Model.
, 33
(3
), pp. 1620
–1632
. 29.
Kumaran
, V.
, Neelamani
, S.
, Vijay
, K. G.
, Al-Anjari
, N.
, and Al-Ragum
, A.
, 2022
, “Wave Attenuation by Multiple Slotted Barriers With a Zig-Zag Arrangement-A Physical and Numerical Approach
,” J. Hydro-environ. Res.
, 41
, pp. 25
–37
. 30.
Panduranga
, K.
, and Koley
, S.
, 2022
, “Hydroelastic Analysis of Very Large Rectangular Plate Floating on Shallow Water
,” Z. Angew. Math. Phys. (ZAMP)
, 73
, pp. 1
–22
. 31.
Koley
, S.
, Behera
, H.
, and Sahoo
, T.
, 2015
, “Oblique Wave Trapping by Porous Structures Near a Wall
,” J. Eng. Mech.
, 141
(3
), p. 04014122
. 32.
Behera
, H.
, Gayathri
, R.
, and Selvan
, S. A.
, 2020
, “Wave Attenuation by Multiple Outer Porous Barriers in the Presence of an Inner Rigid Cylinder
,” J. Waterw. Port Coast. Ocean Eng.
, 146
(1
), p. 04019035
. 33.
Athul Krishna
, K. R.
, Abdulla
, K.
, and Karmakar
, D.
, 2023
, “Dissipation of Gravity Waves Due to Submerged Porous Plate Coupled With Porous Structures
,” ASME J. Offshore Mech. Arct. Eng.
, 145
(1
), p. 011201
. 34.
Sahoo
, G.
, Singla
, S.
, and Martha
, S. C.
, 2023
, “Mitigation of Wave Impact on Sea Wall by a Floating Elastic Plate and a Porous Structure
,” ASME J. Offshore Mech. Arct. Eng.
, 145
(5
), p. 051202
. 35.
Keerthi Raaj
, S.
, Vijay
, K. G.
, Neelamani
, S.
, Saha
, N.
, and Sundaravadivelu
, R.
, 2024
, “Gravity Wave Interaction With a Composite Pile–Rock Breakwater
,” ASME J. Offshore Mech. Arct. Eng.
, 146
, p. 0319031
. 36.
Paul
, D.
, Hossain
, M. M.
, and Behera
, H.
, 2024
, “Hydrodynamic Stability Analysis of Shear-Layered Fluid Flow Over a Porous Bed in the Presence of a Floating Elastic Plate
,” Int. J. Non-Lin. Mech.
, 159
, p. 104599
. 37.
Dhanunjaya
, E.
, Sanjeeva Rayudu
, E.
, and Venkateswarlu
, V.
, 2024
, “Hydrodynamic Performance of an Array of Stratified Pile Rock Breakwaters Placed on Elevated Seabed
,” ASME J. Offshore Mech. Arct. Eng.
, 146
(4
), pp. 1
–12
. 38.
Vijay
, K. G.
, Venkateswarlu
, V.
, and Nishad
, C. S.
, 2021
, “Wave Scattering by Inverted Trapezoidal Porous Boxes Using Dual Boundary Element Method
,” Ocean Eng.
, 219
, p. 108149
. 39.
Cho
, I. H.
, and Kim
, M. H.
, 2013
, “Transmission of Oblique Incident Waves by a Submerged Horizontal Porous Plate
,” Ocean Eng.
, 61
, pp. 56
–65
. 40.
Venkateswarlu
, V.
, Praveen
, K. M.
, Vijay
, K. G.
, Anil
, K.
, and Karmakar
, D.
, 2022
, “Oblique Wave Interaction With a T-Layer Pile-Rock Breakwater Placed on Elevated Bottom
,” Ships Offshore Struct.
, 17
(4
), pp. 852
–865
. 41.
Praveen
, K. M.
, Venkateswarlu
, V.
, and Karmakar
, D.
, 2022
, “Hydroelastic Response of Floating Elastic Plate in the Presence of Vertical Porous Barriers
,” Ships Offshore Struct.
, 17
(2
), pp. 457
–471
. 42.
Venkateswarlu
, V.
, Sanjeeva Rayudu
, E.
, Dhanunjaya
, E.
, and Vijay
, K. G.
, 2023
, “Wave Action Analysis of Multiple Bottom Fixed Semi-Circular Breakwaters in the Presence of a Floating Dock
,” ASME J. Offshore Mech. Arct. Eng.
, 145
(6
), p. 061201
. 43.
Lyu
, Z.
, Liu
, Y.
, Li
, H.
, and Mori
, N.
, 2024
, “Multipole Solution With Nonlinear Pressure Loss for Oblique Waves Action on a Submerged Partially Perforated Semi-Circular Breakwater
,” Ocean Eng.
, 291
, p. 116487
. 44.
Molin
, B.
, and Remy
, F.
, 2015
, “Inertia Effects in TLD Sloshing With Perforated Screens
,” J. Fluids Struct.
, 59
, pp. 165
–177
. 45.
Liu
, Y.
, and Li
, H. J.
, 2017
, “Iterative Multi-domain BEM Solution for Water Wave Reflection by Perforated Caisson Breakwaters
,” Eng. Anal. Bound. Elem.
, 77
, pp. 70
–80
. 46.
Lyu
, Z.
, Liu
, Y.
, Li
, H.
, and Mori
, N.
, 2020
, “Iterative Multipole Solution for Wave Interaction With Submerged Partially Perforated Semi-Circular Breakwater
,” Appl. Ocean Res.
, 97
, p. 102103
. 47.
Suh
, K. D.
, Ji
, C. H.
, and Kim
, B. H.
, 2011
, “Closed-Form Solutions for Wave Reflection and Transmission by Vertical Slotted Barrier
,” Coast. Eng.
, 58
(12
), pp. 1089
–1096
. 48.
Huang
, Z.
, Li
, Y.
, and Liu
, Y.
, 2011
, “Hydraulic Performance and Wave Loadings of Perforated/Slotted Coastal Structures: A Review
,” Ocean Eng.
, 38
(10
), pp. 1031
–1053
. 49.
Mackay
, E.
, and Johanning
, L.
, 2020
, “Comparison of Analytical and Numerical Solutions for Wave Interaction With a Vertical Porous Barrier
,” Ocean Eng.
, 199
, p. 107032
. 50.
Yu
, X.
, and Chwang
, A.
, 1994
, “Water Waves Above Submerged Porous Plate
,” J. Eng. Mech. ASCE
, 120
(6
), pp. 1270
–1282
. 51.
Hong
, H. K.
, and Chen
, J. T.
, 1988
, “Derivations of Integral Equations of Elasticity
,” J. Eng. Mech.
, 114
(6
), pp. 1028
–1044
. 52.
Yueh
, C. Y.
, and Chuang
, S. H.
, 2012
, “A Boundary Element Model for a Partially Piston-Type Porous Wave Energy Converter in Gravity Waves
,” Eng. Anal. Bound. Elem.
, 36
(5
), pp. 658
–664
. 53.
Chen
, J. T.
, Yueh
, C. Y.
, Chang
, Y. L.
, and Wen
, C. C.
, 2017
, “Why Dual Boundary Element Method Is Necessary?
,” Eng. Anal. Bound. Elem.
, 76
, pp. 59
–68
. 54.
Patil
, S. B.
, and Karmakar
, D.
, 2023
, “Hydrodynamic Performance of Wave Energy Converter Integrated With Pile Restrained Floating Structure Near a Partially Reflecting Seawall
,” Ocean Eng.
, 285
, p. 115254
. 55.
Ouyang
, H. T.
, Chen
, K. H.
, and Tsai
, C. M.
, 2015
, “Investigation on Bragg Reflection of Surface Water Waves Induced by a Train of Fixed Floating Pontoon Breakwaters
,” Int. J. Nav. Archit. Ocean Eng.
, 7
(6
), pp. 951
–963
. 56.
Zhao
, Y.
, Liu
, Y.
, Li
, H. J.
, and Chang
, A. T.
, 2020
, “Iterative Dual BEM Solution for Water Wave Scattering by Breakwaters Having Perforated Thin Plates
,” Eng. Anal. Bound. Elem.
, 120
, pp. 95
–106
. 
