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Thesis Submitted to the University of Nottingham for the Degree of Doctor of Philosophy, October, 1982

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Thesis Submitted to the University of Nottingham for the Degree of Doctor of Philosophy, October, 1982 UNIVERSITY OF NOTTINGHAM DEPARTMENT OF CIVIL ENGINEERING OSCILLATORY BOUNDARY LAYER OVER FIXED ROUGH BEDS by S. M. BORGHEI, B. Sc. Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy, October, 1982. I i. ABSTRACT The accurate study of characteristics of the flow under gravity waves has become of prime importance due to the growing demand for structural engineering designs in the coastal environment. Although many investigations have been carried out, the progress of fundamental research was slow due to the lack of an adequate velocity measuring instrument. However in recent years, the development of the Laser Doppler Velocimeter has made it possible to obs- erve the orbital velocity very close to a bed without dis- turbing the flow. This technique was used in this invest- igation, in which observations of the oscillatory flow under gravity waves were carried out above smooth, two- dimensional and three-dimensional rough beds. For the smooth bed case it was found that the velocity profile throughout the depth was well presented by the Stokes second order shear wave equation, except that the theoretical predictions underestimated the observed results, and a linear relationship was obtained for the velocity co- efficients between the two sets of values. As for mean velocity the profile was in close agreement with the Longuet- Higgins conduction solution, and it was found to have a negative value (in opposite direction to wave progression) in the bulk of fluid and always positive values within the boundary layer. ii. The rough beds made little change to the flow in the bulk of fluid. As for inside the boundary layer, the laminar boundary layer was eliminated due to the large size of the rough bed, but for a small size rough bed the flow became laminar at the edge of the boundary layer, and a. perturbed laminar boundary layer velocity profile was traceable. How- ever, the two rough beds had similar influences on the flow except for the roughness size and Reynolds number values. Inside the roughness elements of the rough beds vortex form- ation was clearly observed and the comprehensive range of measurements of these formations are analysed and discussed. iii. ACKNOWLEDGEMENTS I am indebted to many people for their invaluable assistance and support in the preparation of this thesis. I acknowledge this debt of gratitude to the following people: - To Dr. G. B. Parr for his stimulating influence and helpful discussion and suggestions whilst supervising my progress, and for his continual help and direction through- out my work on the project. To Professor R. C. Coates for allowing me the use of the resources of the department and to Dr. F. Rieband for supervising the work. I would also like to express my gratitude to the following: - Dr. Williams of the Department of Mechanical Engineering for his assistance and guidance in using the Analogue to Digital Converter of Fourier Analyser Computer. Mr. J. G. Redfern, the Chief Technican, Mr. J. Moody, Mr. N. Hardy, Mr. J. Leonardi and Mrs. E. Nickeas, staff members of the Civil Engineering laboratories for their assistance. The staff at the University of Nottingham Computing Centre for their guidance. The Applied Science Faculty workshops for manufacturing and repairing the electronic equipment. iv. The Science Library for making available the references. Miss S. Close for her skill and limitless patience in typing the manuscript. Mehrar, my brother, for his unselfish help and expertise in the compilation and drawing of figures and diagrams for my thesis. My colleagues in research especially Dr. N. W. Beech for his discussion. And, finally, special thanks to my parents for their endless support throughout my education. V. Page ABSTRACT ACKNOWLEDGEMENT NOTATIONS xii. ABBREVIATIONS xv. INTRODUCTION 1. CHAPTER 1: A REVIEW OF WAVE THEORIES 1.1 Introduction 4. 1.2 The Classification of Waves 4. 1.2.1 Deep water waves 5. 1.2.2 Shallow water waves 8. 1.2.3 Intermediate depth water waves 8. 1.3 Wave Theories Classifications 8. 1.4 Linear Theory 9. 1.4.1 Progressive waves 10. 1.4.2 Standing waves 18. 1.5 Finite Amplitude Waves 2O. 1.5.1 Stokes second order theory 20. 1.5.2 Stokes third and higher order theories 24. 1.5.3 Introduction and significance of Ursell parameter 25. 1.6 Cnoidal Waves 27. 1.7 Solitary Waves 27. 1.8 Trochoidal Wave Theory 28. 1.9 Comparison of the Wave Theories 29. vi. Page CHAPTER 2: REVIEW OF THE OSCILLATORY BOUNDARY LAYER 2.1 Introduction 32. 2.2 Stokes Shear Wave Equation 32. 2.3 Boundary Layer Thickness 34. 2.4 Boundary Layer Equations under Laminar and Turbulent Conditions 39. 2.4.1 Laminar and turbulent flow in oscillatory flow 39. 2.4.2 Critical Reynolds number and the effect of roughness 41. 2.5 The Boundary Layer Velocity Equation 51. 2.5.1 Laminar flow 51. 2.5.2 Turbulent flow 51. 2.6 Mass Transport 55. 2.7 Roughness'Effect to Mass Transport Velocity 58. 2.8 Flow Separation within the Boundary Layer 59. 2.9 Boundary Shear Stresses 62. CHAPTER 3: INSTRUMENTATION 3.1 Introduction 65. 3.2 Wave Channel and Wave Measuring Instruments 65. 3.2.1 General 65. 3.2.2 The Wave Channel 66. 66. 3.2.2.1 The channel 66. 3.2.2.2 Wave generator 3.2.2.3 Wave breaker 66. 69. 3.2.3 Wave celerity probe 3.2.4 Wave surface probe 72. 75. 3.2.5 Tests on the channel vii. Page 3.3 The Flow Measurement Equipment 75. 3.3.1 General 75. 3.3.2 The Laser Doppler System 76. 3.3.2.1 Laser 76. 3.3.2.2 Doppler effect 77. 3.3.2.3 Principle of Laser Doppler velocimeter 77. 3.3.3 Some remarks about L. D. V. 81. 3.3.3.1 Measurements in liquid flow 81. 3.3.3.2 Measurements of flow direction 83. 3.3.3.3 The advantages and disadvantages of L. D. V. 85. 3.3.4 The equipment 86. 3.3.5 Calibration of the velocimeter 90. 3.3.6 Velocity measurements 92. 3.3.6.1 Horizontal velocity measurements 94. 3.3.6.2 Vertical velocity measurements 94. CHAPTER 4: EXPERIMENTAL METHODS, DATA COLLECTION AND ANALYSIS 4.1 Introduction 97- 4.2 Procedure and Data Collection 97" 97= 4.2.1 Water depth and wave period 98. " 4.2.2 Smooth bed 98. 4.2.2.1 Surface wave profile 4.2.2.2 Flow measurements in the bulk of fluid 99" 4.2.2.3 Velocity at the edge of the viscous boundary layer 99. viii. 4.2.2.4 Velocity in the viscous boundary layer 4.2.3 Rough beds 4.2.3.1 The geometry of the rough beds 4.2.3.2 Velocity inside the roughness elements 4.2.3.3 The sampling location 4.2.3.4 The viscous boundary layer velocity 4.2.3.5 Velocity outside the boundary layer 4.3 Data Analysis, Methods and Units 4.3.1 Data tape 4.3.2 The timing disc pulse generator 4.3.3 The analogue to digital convertor (A. D. C. ) 4.3.4 The 1906A computer software 4.3.4.1 The curve fitting routine 4.3.5 Data shifting 4.3.6 Boundary layer velocity 4.3.7 Velocity outside boundary layer 4.3.8 Velocity in the bulk of fluid 4.3.9 Velocity inside the roughness elements 4.3.10 Analysis of the data from sampling locations CHAPTER 5: RESULTS AND ANALYSIS OF WAVE PROFILE AND ORBITAL VELOCITIES 5.1 Introduction 5.2 The Classification of Generated Waves ix. 5.2.1 Relative depth 5.2.2 Ursell parameter 5.2.3 Limitations of wave theories 5.3 Data of Surface Wave Profile 5.4 Orbital Velocity in the Bulk of Fluid 5.4.1 Velocity Maxima and Minima 5.4.2. Drift velocity 5.4.3 Velocity-outside the viscous boundary layer 5.4.3.1 Horizontal velocity 5.4.3.2 Vertical velocity 5.5 Results and Stokes second order predictions 5.6 Influence of Roughness the orbital Motions . on in the Bulk of Fluid 5.6.1 Two-dimensional rough bed 5.6.1.1 Horizontal velocity 5.6.1.2 Vertical velocity 5.6.2 Three-dimensional rough bed 5.6.2.1 Horizontal velocity 5.6.2.2 Vertical velocity 5.6.3 Results over rough beds 5.7 Concluding Remarks CHAPTER 6: BOUNDARY LAYER VELOCITY OVER SMOOTH AND ROUGH BEDS 6.1 Introduction 6.2 Smooth bed 6.2.1 Boundary layer velocity profile 6.2.2 Mean velocity X. Page 6.3 Rough Beds and the Boundary Layer. Velocity Profile 190. 6.3.1 Two dimensional rough bed 190. 6.3.2 Three dimensional rough bed 225. 6.4 The Influence of Roughness on Boundary Layer Velocity 244. 6.5 Roughness Effect on Mass Transport Velocity 264. 6.6 Concluding Remarks 265. CHAPTER 7: VELOCITY RESULTS AND ANALYSIS OF FLOW INSIDE THE ROUGHNESS ELEMENTS 7.1 Introduction 267. 7.2 The Results 267. 7.2.1 Two dimensional rough bed 268. 7.2.2 Three dimensional rough bed 283. 7.3 The Significant 't' Tests for the 3 and 5 Cycles Sample Population 297. 7.4 Concluding Remarks 300. CHAPTER 8: CONCLUSIONS 8.1 Introduction 302. 8.2 The Performance of the Wave Breaker 302. 8.3 The Measuring Instrument 303. 303. 8.3.1 The wave surface probe 303. 8.3.2 Wave celerity probes 8.4 The Method of Data Analysis 304.
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