A numerical study of wave interaction with porous media

A NUMERICAL STUDY OF WAVE INTERACTION WITH POROUS MEDIA S. A. SUVINI ANUJA KARUNARATHNA NATIONAL UNIVERSITY OF SINGAPORE 2005 A NUMERICAL STUDY OF WAVE INTERACTION WITH POROUS MEDIA S. A. SUVINI ANUJA KARUNARATHNA (B.Sc.Eng. (Hons.), University of Peradeniya, Sri Lanka) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Dedicated with love to my parents ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my supervisor, Assoc. Prof. Lin Pengzhi for assisting me in my Ph.D. study at NUS. His valuable advice, constructive feedback, suggestions and guidance was very much helpful for my research work. The financial support during the final semester of my graduate study is also greatly acknowledged. Without his assistance completion of this thesis would not have been possible. I would also like to thank the chairman of my thesis advisory committee, Prof. Cheong Hin Fatt for valuable advice and suggestions on my research and thesis. I am also thankful for the professors who examined my thesis. Their supportive comments and suggestions were very much useful for the further improvement of the thesis. Support from my former co-supervisor, Dr. Guo Junke John is also appreciated. Thanks are also due to all the teaching faculty members from whom I have taken courses during the graduate study. The financial assistance from the National University of Singapore through the NUS research scholarship is deeply appreciated. The support provided by Mr. Krishna Sanmugam and Mrs. Norela Bte Buang in the use of computer facilities is also appreciated. Thanks are given to all the friends from the hydraulics group: Dr. Liu, Dr. Gu, ChuanJiang, DongChao, Zhang Dan, XioHui, Didi, DongMing, WenYu, HaoLiang, QuangHong and others for the support and friendship. Thanks are also extended to Dulakshi, XinYing, Doan and Atique from the hydro-informatics group for sharing enjoyable times. Warm appreciation goes to Pradeep and Sita, who were my constant i companions throughout my university life for the friendship, endless support and encouragement. I also thank Dammi, Dumindu, Buddi, OG, and other Sri Lankan friends for the support. The friendship shared with all these friends made my stay in Singapore a pleasant period of time. Special thanks are due to my friends in Sri Lanka, Dil and Prigie for loving friendship, encouragement and endless support and care extended to my family during my stay in Singapore. I would like to express my deepest gratitude and love to my parents, Mr. S.A. Karunarathna and Mrs. R.A.S. Ranasinghe for love, affection, endless support in my life and continuous encouragement for my studies. I also like to thank my sister, Neranja and brother-in-law, Ajith for love and support. I would like to remind my grandmother, who always wished my success, with gratitude and love. I also like to thank my mother-in-law, Mrs. E.A. Seelawathie for care and affection. Finally, I would like to thank my best friend and husband, Dammika for love, care, understanding and constant encouragement for my studies. ii TABLE OF CONTENTS Page Acknowledgements i Table of contents iii Summary viii List of tables x List of figures xi Nomenclature xv Chapter Introduction 1.1 Wave interaction with porous media 1.2 Wave damping over porous seabeds 1.3 Wave transformation by porous breakwaters 1.4 Modeling of wave and porous media interaction 1.5 Objectives of the study 1.6 Outline of the thesis 11 Literature review 13 2.1 Fundamental governing equations 13 2.2 Turbulent boundary layer 15 2.2.1 Turbulent boundary layer over an impermeable wall 15 2.2.2 Influence of injection and suction on turbulent Chapter boundary layer 2.2.3 Effects of transpiration on velocity profiles of turbulent boundary layers 2.3 21 24 Porous flow models 29 2.3.1 Stationary flows 30 2.3.2 Non-stationary flows 34 iii Page 2.4 2.5 2.6 Chapter Wave damping over porous seabeds 38 2.4.1 Wave damping over rigid porous seabeds 38 2.4.2 Wave damping over non-rigid porous seabeds 39 Wave interaction with porous breakwaters 41 2.5.1 Interaction of periodic waves with porous breakwaters 41 2.5.2 Interaction of solitary waves with porous breakwaters 42 Conclusions 43 Turbulent boundary layer flows above a porous surface subject to flow injection 45 3.1 Introduction 45 3.2 Derivation of analytical expressions 46 3.2.1 Mathematical formulation 46 3.2.1.1 Governing equations 46 3.2.1.2 Mathematical simplification 47 3.2.1.3 Analytical solution 49 3.2.2 Determination of u b + and y b 54 3.2.3 Interpretation of analytical solution 56 3.2.3.1 Final solution 56 3.2.3.2 Asymptotic solution for low injection rate 58 Comparison with experimental data 60 3.2.4.1 Review on experiments 60 3.2.4.2 Methodology 63 3.2.4.3 Velocity profiles 64 3.2.4.4 Bed shear stress 64 3.2.4 3.3 Conclusions 65 iv Page Chapter Description of the numerical model 74 4.1 Introduction 74 4.2 Governing equations 75 4.2.1 Flow motion outside of porous media 75 4.2.2 Flow motion inside of porous media 77 4.2.2.1 Spatially averaging of Navier-Stokes equations 77 4.2.2.2 Derivation of forces 79 4.2.2.3 Final governing equations 82 4.3 4.4 Initial and boundary conditions 83 4.3.1 Boundary conditions on the free surface 84 4.3.2 Boundary conditions on the rigid boundary 85 4.3.3 Boundary conditions on the permeable boundary 86 Numerical computation 88 4.4.1 Computational domain 88 4.4.2 Two-step projection method 90 4.4.3 Spatial discretization 91 4.4.4 Discretization of k − ε equations 94 4.4.5 Volume of fluid method 97 4.4.6 Computational cycle 99 4.4.7 Stability and error analysis 100 4.5 Model calibration 101 4.6 Conclusions 102 Wave damping over rigid porous seabeds 103 5.1 Literature review 103 5.2 Motivation 107 Chapter v Page 5.3 5.4 5.5 5.6 Chapter 6.1 Model comparison with theories 109 5.3.1 Liu and Dalrymple’s theory (1984) 110 5.3.2 Gu and Wang’s theory (1991) 111 5.3.3 Numerical simulations 113 5.3.3.1 Setup of numerical experiments 113 5.3.3.2 Wave generation 115 5.3.3.3 Boundary conditions 116 5.3.3.4 Results and discussion 117 Model verification by experimental data 120 5.4.1 Savage’s (1953) experimental data 120 5.4.2 Sawaragi and Deguchi’s (1992) experimental data 126 Further discussion of other wave and porous bed parameters 128 5.5.1 Influence of the wavelength on wave damping 129 5.5.2 Influence of seabed thickness on wave damping 131 Conclusions 132 Solitary wave interaction with porous breakwaters 134 Literature review 134 6.1.1 Interaction of periodic waves with porous breakwaters 134 6.1.2 Interaction of solitary waves with impermeable 6.1.3 structures 139 Interaction of solitary waves with porous breakwaters 141 6.2 Motivation 6.3 Model testing for solitary wave propagation on constant 6.4 143 water depth 145 Model validation 147 6.4.1 148 Model validation against various theories vi Page 6.4.2 6.5 6.4.1.1 Madsen’s theory (1974) 148 6.4.1.2 Vidal et al.’s theory (1988) 151 Model validation against experiments 155 6.4.2.1 Vidal et al.’s (1988) experimental data 155 6.4.2.2 Lynett et al.’s (2000) experimental data 157 Numerical results and discussions 159 6.5.1 Setup of numerical experiments 160 6.5.2 Tabulated results of RTD coefficients for different combinations of a b h and d 50 6.5.3 6.6 h ratios 161 Analysis of solitary wave transformation 166 6.5.3.1 Wave reflection and transmission 166 6.5.3.2 Wave energy dissipation 169 6.5.4 Effects of depth of submergence 171 6.5.5 Effects of porosity 172 6.5.6 Scale effects 174 Conclusions 177 Conclusions and recommendations 180 7.1 Conclusions 180 7.2 Recommendations for future research 184 Chapter References 186 Appendix A Calibration of the roughness function 202 Appendix B Velocity profile in a turbulent boundary layer above a porous surface subject to flow suction 204 Appendix C Integral energy equation 211 List of publications 215 vii Twu, S. -W., Liu, C. -C. and Twu, C. -W. Wave damping characteristics of vertically stratified porous structures under oblique wave action. Ocean Engrg., 29, pp. 12951311. 2002. Van der Meer, J. W., Petit, H. A. H., Van den Bosch, P., Klopman, G. and Broekens, R. D. Numerical simulation of wave motion on and in coastal structures. Proc. 22nd Int. Conf. Coast. Engrg., ASCE, Venice, Italy, pp. 1772-1784. 1992. Van Gent, M. R. A. Formulae to describe porous flow. Communications on Hydr. and Geotech. Engrg., No. 92-2. ISSN 0169-6548. Delft University, The Netherlands. 1992. Van Gent, M. R. A. Modelling of wave motion on and in coastal structures. Coast. Engrg., 22, pp. 311-339. 1994. Van Gent, M. R. A. Wave interaction of permeable coastal structures. Ph. D. thesis, Delft University, The Netherlands. 1995. Van Gent, M. R. A., Tönjes, P., Petit, H. A. H. and Van den Bosch, P. Wave action on and in permeable structures. Proc. 24th Int. Conf. Coast. Engrg., ASCE, Kobe, pp. 1739-1753. 1994. Vidal, C., Losada, M. A., Medina, R. and Rubio, J. Solitary wave transmission through porous breakwaters. Proc. 21st Int. Conf. Coast. Engrg., ASCE, Costa del SolMalaga, Spain, pp. 1073-1083. 1988. Wang, K. -H. Diffraction of solitary waves by breakwaters. J. Wtrwy, Port, Coast., and Oc. Engrg., 119(1), pp. 49-69. 1993. Ward, J. C. Turbulent flow in porous media. J. Hydr. Div., HY5, pp. 1-12. 1964. Wilson, K. W. and Cross, R. H. Scale effects in rubble-mound breakwaters. Proc. 13th Int. Conf. Coast. Engrg., ASCE, Vancouver, B.C., Canada, pp. 1873-1884. 1972. Wu, S. and Rajaratnam, N. A simple method for measuring shear stress in rough boundaries. J. Hydr. Res., 38(5), pp. 399-400. 2000. 200 Wurjanto, A. and Kobayashi, N. Irregular wave reflection and runup on permeable slopes. J. Wtrwy, Port, Coast., and Oc. Engrg., 119(5), pp. 537-557. 1993. Yalin M. S. Mechanics of sediment transport. 1st Ed., Pergamon Press, New York. 1972. Yalin, M. S. River Mechanics. Pergamon Press, Inc., Oxford, U.K. 1992. Yamamoto, T., Koning, H. L., Sellmeijer, H. and Van Hijum E. On the response of a poro-elastic bed to water waves. J. Fluid Mech., 87(1), pp. 193-206. 1978. Yasuda, T. and Hara, M. Breaking and reflection of a steep solitary wave caused by a submerged obstacle. Proc. 22nd Int. Conf. Coast. Engrg., ASCE, Delft, Netherlands, pp. 923-934. 1990. Yu, X. and Chwang, A. T. Wave motion through porous structures. J. Engrg. Mech., 120(5), pp. 989-1008. 1994. Zhu, S. and Chwang, A. T. Investigations on the reflection behaviour of a slotted seawall. Coast. Engrg., 43, pp. 93-204. 2001. Zhuang, F. and Lee, J. -J. A viscous rotational model for wave overtopping over marine structure. Proc. 25th Int. Conf. Coast. Engrg., ASCE, Orlando, Florida, pp. 2178-2191. 1996. 201 APPENDIX A CALIBRATION OF THE ROUGHNESS FUNCTION The calibration of the roughness function involves the determination of the coefficients c1 , c2 , c3 and c4 in (3.23). Based on (3.22), we can recast it into: ( ) ( ) B1 ks+ = Bs − B2 vs+ = Bs + vs+ − 2a vs+a (A.1) In this study, we have made use of large amount of experimental data for flows subject to vertical injection. These experiments include air flows (McQuaid, 1967, Simpson, 1967) and water flow (Cheng and Chiew, 1998b). Firstly, for each experimental run, yb is determined by the least square method with the use of (3.25), (3.26), (3.20) and the experimentally determined u* in order to give the best fit of velocity profile between the theory and laboratory measurements. Secondly, the corresponding value of Bs is obtained by using the expression yb = k s exp(−κBs ) , and the information of ks . Thirdly, B1(ks+ ) is obtained from (A.1) with the corresponding value of vs+ . After we plot B1 against ln ks+ for all runs, we have identified its similar behavior to the roughness function for impermeable walls (Nikuradse, 1933) (Fig. A.1). Finally, the coefficients c1 , c2 , c3 and c4 in (3.23) are determined with the use of least square method again. The fitted curve is compared to experimental data in Fig. A.1. 202 B1 −2 −4 −6 B =(c *lnk+)/[1+c *exp[c *sinh(c *lnk+)]] −10 −2 s s Data from Simpson(1967) Data from McQuaid(1967) Data from Cheng & Chiew(1998b) −8 −1 ln(k+s ) Fig. A.1. Least square fit of (3.23) with the data determined from the experiments 203 APPENDIX B VELOCITY PROFILE IN A TURBULENT BOUNDARY LAYER ABOVE A POROUS SURFACE SUBJECT TO FLOW SUCTION B.1 Introduction In Chapter 3, expressions for the velocity profile in the fully turbulent region of a turbulent boundary layer above a porous surface subject to flow injection were derived by solving momentum equations and turbulent kinetic energy equation. In the derivation the advection of turbulent kinetic energy is retained whereas the earlier studies have neglected it. A deflection point is defined within the velocity profile and the horizontal coordinate of this deflection point increases from negative infinity to positive values with the increase of injection rate from zero. Furthermore, the vertical coordinate of the deflection point increases with the increase of flow injection. However, in turbulent boundary layers subject to flow suction, the deflection point will not really exist since the streamlines are pulled towards the surface by flow suction. Instead, a slip velocity occurs at the bed while the origin of the velocity profile is located inside the porous surface (Mendoza and Zhou, 1992; Chen and Chiew, 2004) as shown in Fig. B.1. In this appendix, we shall present a derivation of an analytical expression for the velocity profile in turbulent boundary layers subject to flow suction by solving momentum equations and turbulent kinetic energy equation. Similar to the derivation of analytical expressions for the case of flow injection, here also the advection of turbulent kinetic energy will be retained. However, instead of defining the velocity and 204 the coordinate of the deflection point, in this derivation we shall define the slip velocity at the bed u s and the displacement of the origin of the velocity profile y . y u(y) y0 O x vs us Fig. B.1. Illustration of velocity profile of a turbulent boundary layer above a horizontal permeable bed subject to flow suction B.2 Governing equations Similarly in Chapter 3, here also the governing equations are the continuity equation, momentum equations and turbulent kinetic energy equation [equations (2.5), (2.6), (2.7) and (3.1)]: ∂u ∂v =0 + ∂x ∂y (B.1) u ∂p ∂τ xy ∂τ xx ∂u ∂u +v =− + + ρ ∂x ∂x ∂y ρ ∂ x ρ ∂y (B.2) u ∂p ∂τ yy ∂τ yx ∂v ∂v +v =− + + −g ∂x ∂y ρ ∂ y ρ ∂y ρ ∂x (B.3) u ∂k ∂k τ xx ∂u τ yy ∂v τ yx ∂v τ xy ∂u ∂ ⎛ ν t ∂k ⎞ ⎟ −ε +v = + + + + ⎜ ∂x ∂y ρ ∂x ρ ∂y ρ ∂x ρ ∂y ∂y ⎜⎝ σ k ∂y ⎟⎠ (B.4) 205 B.3 Mathematical simplification The momentum equations and turbulent kinetic energy equation are simplified similar to the simplification made in the derivation of analytical expressions for flow injection. Thus, within the fully turbulent region the continuity and momentum equations reduce to: vs ∂u ∂τ xy = ∂y ρ ∂y (B.5) Integrating (B.5) and applying the boundary conditions of y = , u = u s and τ = τ b we get: τ xy = τ b + ρuv s − ρu s v s (B.6) The simplified kinetic energy equation reads: vs ∂k τ xy ∂u = −ε ∂y ρ ∂y (B.7) Turbulent kinetic energy and kinetic energy dissipation rate in (B.7) can be related to Reynolds shear stress by using the same expressions that are used for injection: k= τ xy (B.8) aρ ⎛τ xy ⎞ ⎜ ρ ⎟⎠ ⎝ ε= L (B.9) Since the boundary condition is different for turbulent boundary layers subject to flow suction compared to that for turbulent boundary layers above impermeable walls, it is reasonable to assume that the mixing length will be modified when 206 turbulent boundary layers are subjected to flow suction. Thus, for suction conditions, the mixing length L can be expressed as (Chen and Chiew, 2004): L = κ ( y + y0 ) (B.10) By substituting (B.8), (B.9) and (B.10) into (B.7), we get: ⎛τ xy ⎞ ⎜ ⎟ ∂ ⎛ τ xy ⎞ τ xy ∂u ⎝ ρ ⎠ ⎜ ⎟− + vs =0 ∂y ⎜⎝ 2a ρ ⎟⎠ ρ ∂y κ ( y + y ) B.4 (B.11) Analytical solution After substituting the simplified momentum equation (B.6) into the simplified kinetic energy equation (B.11) and integrating with respect to y the implicit solution for velocity distribution is obtained as: + ⎞ ⎛ y+ y ⎞ ⎛ vs ⎛⎜ + + + + ⎞ ⎟⎟ −1⎟ = ln⎜⎜ + u vs − us vs −1⎟ + + ⎜ + ⎜ ⎟ + + + ⎝ ⎠ a y κ vs + − u v u v ⎝ ⎠ s s s ⎠ ⎝ (B.12) The explicit solutions for (B.12) can be obtained as: u+ = u+ = u+ = 8a κ vs + 8a κ vs + 8a κ vs + ⎡ ⎛ ⎛ y + y0 ⎞ ⎞ + ⎛ y + y0 ⎞ + ⎤ ⎛ y + y0 ⎞ + ⎟ ⎜ ⎢− 4a 2κ + 4a 2κ ln⎜ ⎟⎟ vs + ⎥ ⎟ ⎜ ⎟ v a ln vs + 2aκ ln⎜⎜ + s ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎢ ⎥ ⎝ y0 ⎠ ⎝ y0 ⎠ ⎝ ⎝ y0 ⎠ ⎠ ⎢ ⎥ ⎛ ⎛ y + y0 ⎞ + ⎢ +4 ⎥ +2 ⎞ 2 + + ⎟ ⎜ ⎢κ vs + 8a κ vs us + ⎜ 2aκ + a ln⎜⎜ y ⎟⎟vs + κvs ⎟ × As1 ⎥ ⎝ ⎠ ⎠ ⎝ ⎣ ⎦ ⎡ ⎛ ⎛ y + y0 ⎞ ⎞ + ⎛ y + y0 ⎞ + ⎤ ⎛ y + y0 ⎞ + ⎟ ⎜ ⎢− 4a 2κ + 4a 2κ ln ⎜ ⎟⎟ vs + ⎥ ⎟ ⎜ ⎟ v a ln vs + 2aκ ln⎜⎜ + s ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎢ ⎥ ⎝ y0 ⎠ ⎝ y0 ⎠ ⎝ ⎝ y0 ⎠ ⎠ ⎢ ⎥ ⎛ ⎛ y + y0 ⎞ + ⎢ +4 ⎥ +2 ⎞ 2 + + ⎟ ⎜ ⎟ ⎜ ⎢κ vs + 8a κ vs us − ⎜ 2aκ + a ln⎜ y ⎟vs + κvs ⎟ × As1 ⎥ ⎝ ⎠ ⎠ ⎝ ⎣ ⎦ ⎡ ⎛ ⎛ y + y0 ⎞ ⎞ + ⎛ y + y0 ⎞ + ⎤ ⎛ y + y0 ⎞ + ⎟ ⎜ ⎢− 4a 2κ − 4a 2κ ln⎜ ⎟⎟ vs + ⎥ ⎟ ⎜ ⎟ v a ln vs − 2aκ ln⎜⎜ + s ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎢ ⎥ ⎝ y0 ⎠ ⎝ y0 ⎠ ⎝ ⎝ y0 ⎠ ⎠ ⎢ ⎥ ⎛ ⎛ y + y0 ⎞ + ⎢ +4 ⎥ +2 ⎞ 2 + + ⎟ ⎜ ⎟ ⎜ ⎢κ vs + 8a κ vs us + ⎜ 2aκ − a ln⎜ y ⎟vs + κvs ⎟ × As ⎥ ⎝ ⎠ ⎠ ⎝ ⎣ ⎦ (B.13) (B.14) (B.15) 207 u+ = 8a κ vs + ⎡ ⎛ ⎛ y + y0 ⎞ ⎞ + ⎛ y + y0 ⎞ + ⎤ ⎛ y + y0 ⎞ + ⎟ ⎜ ⎢− 4a 2κ − 4a 2κ ln⎜ ⎟⎟ vs + ⎥ ⎜⎜ ⎟ ⎜ ⎟ v a ln v a κ ln − + s s ⎜ y ⎟ ⎜ ⎜ y ⎟⎟ ⎢ ⎥ 0 ⎝ y0 ⎠ ⎠⎠ ⎠ ⎝ ⎝ ⎝ ⎢ ⎥ ⎛ ⎛ y + y0 ⎞ + ⎢ +4 ⎥ +2 ⎞ 2 + + ⎢κ vs + 8a κ vs us − ⎜⎜ 2aκ − a ln⎜⎜ y ⎟⎟vs + κvs ⎟⎟ × As ⎥ ⎠ ⎝ ⎠ ⎝ ⎣ ⎦ (B.16) where ⎛ y + y0 ⎞ + ⎛ y + y0 ⎞ + ⎛ y + y0 ⎞ + ⎟⎟vs − 4aκ 2vs + + a ln⎜⎜ ⎟⎟ vs + 2aκ ln⎜⎜ ⎟⎟vs + κ 2vs + As1 = 4a 2κ + 4a 2κ ln⎜⎜ y y y 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ and ⎛ y + y0 ⎞ + ⎛ y + y0 ⎞ + ⎛ y + y0 ⎞ + ⎟⎟vs − 4aκ 2vs + + a ln⎜⎜ ⎟⎟ vs − 2aκ ln⎜⎜ ⎟⎟vs + κ 2vs + As = 4a 2κ − 4a 2κ ln⎜⎜ ⎝ y0 ⎠ ⎝ y0 ⎠ ⎝ y0 ⎠ The graphical representation of these four explicit solutions is presented in Fig. B.2. 0.08 solution (B.13) solution (B.14) solution (B.15) solution (B.16) 0.07 0.06 y+ 0.05 0.04 0.03 0.02 0.01 −10 −5 10 + u Fig. B.2. Graphical representation of four explicit solutions for horizontal velocity 208 According to Fig. B.2, among the four explicit solutions only one solution i.e., (B.13) provides realistic velocity distribution. Thus, the final solution for the velocity profile is: u+ = 8a κ vs + ⎡ ⎛ ⎛ y + y0 ⎞ ⎞ + ⎛ y + y0 ⎞ + ⎤ ⎛ y + y0 ⎞ + ⎟ ⎜ ⎢− 4a 2κ + 4a 2κ ln⎜ ⎟⎟ vs + ⎥ ⎜⎜ ⎟ ⎜ ⎟ v a ln v a κ ln + + s s ⎜ y ⎟ ⎜ ⎜ y ⎟⎟ y ⎢ ⎥ 0 ⎠ ⎝ ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ ⎢ ⎥ ⎛ ⎛ y + y0 ⎞ + ⎢ +4 ⎥ +2 ⎞ 2 + + ⎢κ vs + 8a κ vs us + ⎜⎜ 2aκ + a ln⎜⎜ y ⎟⎟vs + κvs ⎟⎟ × As1 ⎥ ⎝ ⎠ ⎠ ⎝ ⎣ ⎦ (B.17) where ⎛ y + y0 ⎞ + ⎛ y + y0 ⎞ + ⎛ y + y0 ⎞ + ⎟⎟vs − 4aκ 2vs + + a ln⎜⎜ ⎟⎟ vs + 2aκ ln⎜⎜ ⎟⎟vs + κ 2vs + As1 = 4a 2κ + 4a 2κ ln⎜⎜ y y y 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ For low injection rates the solution can be expanded into series form by using Taylor series expansion. Thus, (B.17) in series form can be expressed as: ⎡ ⎛ y + y0 u + = us+ + ⎢ ln ⎜ ⎣ κ ⎝ y0 ⎞ ⎤ vs+ ⎟⎥ + ⎠⎦ ⎡ ⎛ y + y0 ⎢ ln ⎜ ⎣ κ ⎝ y0 ⎞⎤ vs+ ⎡ ⎛ y + y0 ⎢ ln ⎜ ⎟⎥ + a ⎣ κ ⎝ y0 ⎠⎦ ⎞ ⎤ vs+ ⎟⎥ + ⎠ ⎦ 4a ⎡ ⎛ y + y0 ⎢ ln ⎜ ⎣ κ ⎝ y0 ⎞⎤ ⎟ ⎥ + . ⎠⎦ (B.18) It can be seen that (B.18) reduces to the universal logarithmic law when there is no flow suction. Furthermore, the leading order terms in (B.18) are the same as modified logarithmic law for flow suction (e.g., Chen and Chiew, 2004). B.5 Conclusions An analytical solution for velocity profile above a porous surface subject to flow suction is derived by solving momentum equations and turbulent kinetic energy equation. In the derivation a slip velocity at the bed and the displacement of the origin of the velocity profile are defined. The solution reduces to universal logarithmic law when there is no flow suction and the leading order terms are same as the modified logarithmic law. Due to time constraint the derived expression is not validated with experimental data. A complete analysis needs to be carried out in the future to validate 209 the derived expressions. Furthermore, an appropriate roughness function and an expression for slip velocity as the function of suction velocity and bed roughness need to be determined. 210 APPENDIX C INTEGRAL ENERGY EQUATION The numerical results for RTD coefficients that are presented in Section 6.4 is based on the integral energy equation presented by Lin (2004). In this appendix, we shall briefly describe the derivation of the integral energy equation and its application to calculate RTD coefficients for the numerical results of wave structure interaction simulations. C.1 Derivation of integral energy equation The integral energy equation is derived for general three-dimensional turbulent flows, which can be described by RANS equations [equations (2.3) and (2.4)]. In equation (2.4), the mean pressure is split into dynamic and static pressure components i.e., p = p D + p S = p D + ρx k g k . Substituting this relation into (2.4) and with the use of (2.3) one can obtain: ⎡1 2⎤ ∂ ⎢ ρu i ⎥ ⎣ ⎦ + ∂t 2⎤ ⎡ ∂ ⎢u j ρ u i ⎥ ∂( p D u i ) ∂ρ ⎣ ⎦ =− − xk g k ui ∂x j ∂x i ∂x i ⎧ ∂ ⎪⎪ + ⎨μ ∂x j ⎪ ⎪⎩ ⎫ ⎡1 ⎤ ∂⎢ ui ⎥ ⎪ ⎡ ∂u ∂u ∂u ⎤ ⎪ ⎣2 ⎦ i − ρu i ' u j ' i ⎥ − ρu i u i ' u j '⎬ − ⎢ μ i ∂x j ⎥⎦ ∂x j ⎪ ⎢⎣ ∂x j ∂x j ⎪⎭ (C.1) With the definition of kinetic energy per unit fluid volume as E kin = ρu i / and potential energy as E pot = − ρx k g k we have: 211 ∂ ( E kin + E pot ) ∂t ⎤ ⎡ ∂E kin − ρu i u i ' u j '⎥ ⎢− p D u j − ρu j E kin + μ ∂x j ⎥⎦ ⎢⎣ ⎡ ∂u ∂u i ∂u ⎤ − ⎢μ i − ρu i ' u j ' i ⎥ ∂x j ⎥⎦ ⎢⎣ ∂x j ∂x j = ∂ ∂x j (C.2) Integrating (C.2) within a control volume V which is fixed in time and space and applying Gauss divergence theorem, we get: ∂ EdV = FE | As − ∫ DdV ∂t V∫ V (C.3) where E = E kin + E pot is the total mechanical energy, FE As is the rate of total energy flux through control surface As and D is the rate of local energy dissipation within V . They are defined as follows, respectively: FE | As = Fp + Fc + Fm + Ft = ∫ − pDun dAs + ∫ − ρun Ekin dAs + As As ∂Ekin ∫A μ ∂xn dAs + A∫ − ρuiui ' un ' dAs s s D = Dm + Dt = μ ∂u i ∂u i ∂u − ρu i ' u j ' i ∂x j ∂x j ∂x j (C.4) (C.5) where un is the velocity normal to the control surface. The rate of total energy flux is composed of pressure-induced energy flux (Fp), convective energy flux (Fc), and energy fluxes induced by molecular and turbulence stress (Fm and Ft). The rate of local energy dissipation is contributed from molecular and turbulence effects (Dm and Dt). For two-dimensional free surface flows, we consider a rectangular control volume that may include various structures. In numerical experiments, the control volume can be considered as a numerical wave tank where zero energy fluxes through bottom and upper sides can be assumed. Then, the effective control surface reduces to left and right boundaries, S1 and S2 respectively. If we select these boundaries away 212 from structures inside the computational domain energy fluxes induced by molecular and turbulence stress (Fm and Ft) can be neglected and equation (C.3) reduces to: [ ] ∂ Edxdy = F p ( S1 ) − F p ( S ) + [Fc ( S1 ) − Fc ( S )] − ∫ Ddxdy ∂t ∫ η ( S2 ) η ( S2 ) ⎤ ⎤ ⎡ η ( S1 ) ⎡ η ( S1 ) = ⎢ ∫ p D udy − ∫ p D udy ⎥ + ⎢ ∫ ρuE kin dy − ∫ ρuE kin dy ⎥ − ∫ Ddxdy ⎥⎦ ⎥⎦ ⎢⎣− h ( S1 ) ⎢⎣− h ( S1 ) −h( S2 ) −h( S2 ) C.2 (C.6) Application of the integral energy equation Integrating (C.6) between t1 and t2, we have: ∫ Edxdz | t =t − ∫ Edxdz | t =t1 t2 ⎞ ⎛ t2 ⎜ + ∫ dt ∫ Ddxdz = ∫ F p ( S1 )dt + ∫ Fc ( S1 )dt ⎟ ⎟ ⎜t t1 t1 ⎠ ⎝1 t2 ⎞ ⎛ t2 − ⎜ ∫ F p ( S ) dt + ∫ Fc ( S )dt ⎟ ⎟ ⎜t t1 ⎠ ⎝1 t2 (C.7) In a numerical simulation, we can always choose t1 and t2 such that ∫ Edxdz |t =t = ∫ Edxdz |t =t . Equation (C.7) then reduces to: t2 t2 ⎛ t2 ⎞ ⎛ t2 ⎞ ⎜ F p ( S ) dt + Fc ( S ) dt ⎟ − ⎜ F p ( S ) dt + Fc ( S ) dt ⎟ dt Ddxdz = 1 2 ∫t ∫ ∫ ∫ ∫ ∫ ⎜t ⎟ ⎜t ⎟ t1 t1 ⎝1 ⎠ ⎝1 ⎠ ⇔ TD = (EF p1 + EFc1 ) − (EF p + EFc ) t2 (C.8) where EFp1 represents the integration of pressure induced energy flux through section between t1 and t2 and EFc1 the integration of convective energy flux through section between t1 and t2. EFp2 and EFp2 are similarly defined. TD represents the total energy dissipation within the control volume between t1 and t2. Realizing the fact that both the incident wave and reflected wave pass section and only transmitted wave passed section 2, we can rewrite equation (C.8) as follows: 213 TD = (EF p1 + EFc1 ) − (EF p + EFc ) ⇔ TD = (EFinc + EFref ) − (EFtrans ) (C.9) The information in (C.9) can then be easily used to calculate the wave reflection, transmission, and dissipation coefficients (KR, KT, and KD) as follows: KR = − EFref EFinc KT = EFtrans EFinc KD = TD = − K R2 − K T2 EFinc (C.10) It is noted that EFref is always negative because the reflected wave propagates against the incident wave direction. 214 LIST OF PUBLICATIONS INTERNATIONAL JOURNAL PAPERS Lin, P. and Karunarathna, S. A. S. A. A numerical study of solitary wave interaction with porous breakwaters. Submitted for possible publication in the J. Wtrwy, Port, Coast., and Oc. Engrg., ASCE, 2005. Karunarathna, S. A. S. A. and Lin, P. Numerical simulation of wave damping over porous seabeds. To be appear in the Coast. Engrg., 2006. Lin, P. and Karunarathna, S. A. S. A. Turbulent boundary layer flows above a porous surface subject to flow injection. To be appear in the J. Engrg. Mech., ASCE, 2006. INTERNATIONAL CONFERENCE PAPERS Karunarathna, S. A. S. A. and Lin, P. Numerical simulation of solitary wave interaction with porous breakwaters. Proc. 5th Int. Symp. WAVES 2005, ASCE. Madrid, Spain, Paper No. 109. 2005. Karunarathna, S. A. S. A. and Lin, P. Numerical simulation of wave attenuation over porous seabeds. Proc. 6th Int. Conf. on Hydr., Perth, Australia, pp. 205-211. 2004. Karunarathna, S. A. S. A. and Lin, P. Wave attenuation over a porous seabed. Proc. 6th Int. Conf. on Coast. and Port Engrg. in Developing Countries, Colombo, Sri Lanka, Paper No. 081. 2003. 215 [...]... whereas in deep water depths wave damping is less affected by wave nonlinearity The results also reveal that both short waves and long waves have smaller wave damping when compared to damping of waves in intermediate water depths The present numerical model is validated for both long wave and solitary wave interaction with porous breakwaters The model is employed to study solitary wave interaction with. .. defined as a physical constitution of solid particles and associated voids that allows the fluid to pass through In the field of coastal engineering, wave interaction with porous media is an important phenomenon and its applications include wave interaction with porous seabeds and wave interaction with porous structures Porous seabeds are usually of two types i.e., rigid porous seabeds and non-rigid porous. .. protection of harbors and beaches from ocean waves Wave reflection, transmission and energy dissipation are the primary features of breakwaters The most common existing breakwaters are permeable that are made of quarry stones or artificial blocks Permeable breakwaters have several advantages over impermeable breakwaters They allow the waves to penetrate into the breakwater thus reducing wave reflection This... porous beds 131 Fig 6.1 Definition sketch of solitary wave interaction with a porous breakwater 144 Fig 6.2 Comparison of numerical and analytical free surface profiles for solitary wave propagation in a constant water depth at different 12 times t (g h ) =13.3 (A) , 44.6 (B), 76.0 (C): solid line – numerical free surface profile, dotted line – analytical free surface profile 147 Fig 6.3 Standing wave. .. (Liu and Dalrymple, 1984; Gu and Wang, 1991) 122 Fig 5.9 Variation of wave damping rates with wave nonlinearity for waves with T = 1.27 s in different water depths 125 Fig 5.10 Comparison of numerical results for the wave damping along the porous seabed to the experimental (Sawaragi and Deguchi, 1992), theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) and numerical (Chang, 2004) results for case... infiltration occurs under a wave crest and exfiltration occurs under a wave trough However, the net seepage flux is nearly zero (Lee et al., 200 2a) This seepage flux results in various modifications in characteristics of coastal water such as wave characteristics (Lee et al., 200 2a) , characteristics of boundary layer adjacent to a porous seabed (Mendoza and Zhou, 1992; Conley and Inman, 1994) and hydrodynamic... Fig 5.11 Comparison of numerical results for the wave damping along the porous seabed to the experimental (Sawaragi and Deguchi, 1992) and theoretical (Liu and Dalrymple, 1984; Gu and Wang, 1991) results for case J-6 128 Fig 5.12 Variation of (a) dimensional and (b) non-dimensional wave damping rate with k w h for different porous beds 130 Fig 5.13 Variation of wave damping rate with the seabed thickness... dissipation rate The porous flow model is used to describe the flow through porous media The revised numerical model is first validated for the study of wave damping over porous seabeds The effects of wave nonlinearity, wavelength and seabed thickness on wave damping are studied by using the numerical model It is found that in shallow water depths, the wave nonlinearity has a distinct effect on wave damping... linearized porous flow model The shallow-water equations models, on the other hand, can only describe the weakly nonlinear and dispersive waves The limitations of mild-slope equation models and depth averaged models can be avoided by the models based on Navier-Stokes equations 1.4 Modeling of wave and porous media interaction The modeling of wave and porous media interaction is based on coupling of two... solitary wave interaction with porous breakwater with ab h = 2.0 and d 50 h = 0.05 153 Fig 6.6 Comparison of the variation of reflection and transmission coefficients ( K R and K T ) with the increase of k w a between the present numerical results and the theoretical results (Vidal et al., 1988); (a) small porous size and (b) large porous size: dashed dotted line – numerical K R , dotted line – numerical . A NUMERICAL STUDY OF WAVE INTERACTION WITH POROUS MEDIA S. A. SUVINI ANUJA KARUNARATHNA (B.Sc.Eng. (Hons.), University of Peradeniya, Sri Lanka) A THESIS. effect on wave damping whereas in deep water depths wave damping is less affected by wave nonlinearity. The results also reveal that both short waves and long waves have smaller wave damping when. damping over rigid porous seabeds 38 2.4.2 Wave damping over non-rigid porous seabeds 39 2.5 Wave interaction with porous breakwaters 41 2.5.1 Interaction of periodic waves with porous breakwaters