Wave trapping by dual porous barriers near a wall in the presence of bottom undulation

Abstract

Trapping of oblique surface gravity waves by dual porous barriers near a wall is studied in the presence of step type varying bottom bed that is connected on both sides by water of uniform depths. The porous barriers are assumed to be fixed at a certain distance in front of a vertical rigid wall. Using linear water wave theory and Darcy's law for flow past porous structure, the physical problem is converted into a boundary value problem. Using eigenfunction expansion in the uniform bottom bed region and modified mild-slope equation in the varying bottom bed region, the mathematical problem is handled for solution. Moreover, certain jump conditions are used to account for mass conservation at slope discontinuities in the bottom bed profile. To understand the effect of dual porous barriers in creating tranquility zone and minimum load on the sea wall, reflection coefficient, wave forces acting on the barrier and the wall, and surface wave elevation are computed and analyzed for different values of depth ratio, porous-effect parameter, incident wave angle, gap between the barriers and wall and slope length of undulated bottom. The study reveals that with moderate porosity and suitable gap between barriers and sea wall, using dual barriers an effective wave trapping system can be developed which will exert less wave force on the barriers and the rigid wall. The proposed wave trapping system is likely to be of immense help for protecting various facilities/ infrastructures in coastal environment.

Keywords

porous barriers / mild-slope equation / reflection coefficient / wave trapping / porous-effect parameter

References

[1] Behera H, Kaligatla RB, Sahoo T. Wave trapping by porous barrier in the presence of step type bottom. Wave Motion, 2015, 57: 219-230

[2] Behera H, Sahoo T, Ng C-O. Wave scattering by a partial flexible porous barrier in the presence of a step-type bottom topography. Coastal Engineering Journal, 2016, 58(3): 1650008

[3] Bennetts LG, Biggs NRT, Porter D. The interaction of flexural-gravity waves with periodic geometries. Wave Motion, 2009, 46(1): 57-73

[4] Berkhoff JCW. Computation of combined refractiondiffraction. Proceedings of 13th International Conference on Coastal Engineering ASCE, 1972, 471-490

[5] Bhattacharjee J G, Soares C. Oblique wave interaction with a floating structure near a wall with stepped bottom. Ocean Engineering, 2011, 38: 1528-1544

[6] Billingham J, King AC. Wave Motion, 2000, Cambridge, United Kingdom: Cambridge University Press

[7] Cerrato A, Gonzalez JA, Rodriguez-Tembleque L. Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries. Engineering Analysis with Boundary Elements, 2016, 62: 22-34

[8] Chamberlain PG, Porter D. The modified mild-slope equation. Journal of Fluid Mechanics, 1995, 291: 393-407

[9] Chwang AT. A porous-wavemaker theory. Journal of Fluid Mechanics, 1983, 132: 395-406

[10] Chwang AT, Chan AT. Interaction between porous media and wave motion. Annual Review of Fluid Mechanics, 1998, 30: 53-84

[11] Das S, Bora SN. Damping of oblique ocean waves by a vertical porous structure placed on a multi-step bottom. Journal of Marine Science and Application, 2014, 13(4): 362-376

[12] Davies AG, Heathershaw AD. Surface-wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics, 1984, 144: 419-443

[13] Dhillon H, Banerjea S, Mandal BN. Water wave scattering by a finite dock over a step-type bottom topography. Ocean Engineering, 2016, 113: 1-10

[14] Huang Z, Li Y, Liu Y. Hydraulic performance and wave loadings of perforated/slotted coastal structures: A review. Ocean Engineering, 2011, 38(10): 1031-1053

[15] Kaligatla RB, Manam SR. Bragg resonance of membranecoupled gravity waves over a porous bottom. International Journal of Advances in Engineering Sciences and Applied Mathematics, 2016, 8: 222-237

[16] Karmakar D, Bhattacharjee J G, Soares C. Scattering of gravity waves by multiple surface-piercing floating membrane. Applied Ocean Research, 2013, 39: 40-52

[17] Karmakar D G, Soares C. Wave transformation due to multiple bottom-standing porous barriers. Ocean Engineering, 2014, 80: 50-63

[18] Koley S, Behera H, Sahoo T. Oblique wave trapping by porous structures near a wall. Journal of Engineering Mechanics, 2014, 141(3): 1-15

[19] Li YC, Liu Y, Teng B. Porous effect parameter of thin permeable plates. Coastal Engineering Journal, 2006, 48(4): 309-336

[20] Liu Y, Li Y. Wave interaction with a wave absorbing double curtain-wall breakwater. Ocean Engineering, 2011, 38(10): 1237-1245

[21] Manam SR, Kaligatla RB. A mild-slope model for membrane-coupled gravity waves. Journal of Fluids and Structures, 2012, 30: 173-187

[22] Mandal S, Behera H, Sahoo T. Oblique wave interaction with porous, flexible barriers in a two-layer fluid. Journal of Engineering Mathematics, 2015, 100(1): 1-31

[23] Massel SR. Extended refraction-diffraction equation for surface waves. Coastal Engineering, 1993, 19(1): 97-126

[24] Michael I, Sundarlingam P, Gang Y. Wave interactions with vertical slotted barrier. Journal of Waterway, Port, Coastal, and Ocean Engineering, 1998, 124(3): 118-126

[25] Mondal A, Gayen R. Wave interaction with dual circular porous plates. Journal of Marine Science and Application, 2015, 14(4): 366-375

[26] Porter D. The mild-slope equations. Journal of Fluid Mechanics, 2003, 494: 51-63

[27] Porter D, Porter R. Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography. Journal Fluid Mechanics, 2004, 509: 145-179

[28] Porter D, Staziker DJ. Extensions of the mild-slope equation. Journal of Fluid Mechanics, 1995, 300: 367-382

[29] Porter R, Porter D. Water wave scattering by a step of arbitrary profile. Journal of Fluid Mechanics, 2000, 411: 131-164

[30] Sahoo T, Chan AT, Chwang AT. Scattering of oblique surface waves by permeable barriers. Journal of Waterway, Port, Coastal, and Ocean Engineering, 2000, 126(4): 196-205

[31] Sahoo T, Lee MM, Chwang AT. Trapping and generation of waves by vertical porous structures. Journal of Engineering Mechanics, 2000, 126(10): 1074-1082

[32] Smith R, Sprinks T. Scattering of surface waves by a conical island. Journal of Fluid Mechanics, 1975, 72(2): 373-384

[33] Suh KD, Kim YW, Ji CH. An empirical formula for friction coefficient of a perforated wall with vertical slits. Coastal Engineering, 2011, 58(1): 85-93

[34] Suh KD, Park WS. Wave reflection from perforated-wall caisson breakwaters. Coastal Engineering, 1995, 26(3): 177-193

[35] Twu SW, Lin DT. Wave reflection by a number of thin porous plates fixed in a semi-infinitely long flume. Coastal Engineering Proceedings, 1990, 1046-1059

[36] Yu X. Diffraction of water waves by porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 1995, 121(6): 275-282