OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- i CANTHO UNIVERSITY MHO 5/6 project Civil and Mechanical Engineering OPEN CHANNEL HYDRAULICS FOR ENGINEERS LECTURE NOTE PREPARED BY LE ANH TUAN DELFT, 2003 OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- ii To my wife Hoang Nga, my son Anh Tu and my daughter Hoang Ngan, and all my closed friends . You are lovely rivers flowing in my dreams . LA. Tuan OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- iii PREFACE The subject of Open Channel Hydraulics for Engineers, also called Applied Hydraulics, is a subject required not only for Hydraulic Engineering students but also for other engineering fields involved, such as Construction Engineering, Transportation Engineering and Environmental Engineering. It follows the previous subject named Fluid Mechanics. The knowledge of open-channel hydraulics, which is essential to the design of many hydraulic structures, has made advances by leaps and bounds. The practical importance of this topic in the water resources development together with the challenge posed by the variety and complexity of its problems, has created for a long time the need for more comprehensive and detailed treatment of this subject. As a result, many excellent texts have been written, not only in English, but also in other languages. The best of these texts have imposed a logical and coherent structure in the study of the subject, and created a tradition that present texts consciously follow. This lecture note is prepared as a reference document for the subject. It emphasizes the dynamics of the open-channel flow, by attempting to provide a complete framework of the basic equations of motion of the fluid, which are used as building blocks for the treatment of many practical problems. The structure of the document, with seven chapters totally, follows a logical sequence from a description and classification of Fluid Mechanics and Open Channel Flows, as reviewed in Chapter 1. A development of the basic equation of motion for uniform flow is encountered in Chapter 2. Coming to Chapter 3, the fruitful concepts of specific energy and hydraulic jumps are introduced and developed. Chapter 4 presents a variety of non-uniform flows and applications of drawing water-surface profiles. Spatially-varied flow, found at spillways and weirs is considered in Chapter 5. Transitions and energy dissipators are discussed in Chapter 6. Finally, in Chapter 7, unsteady flow in open channels is introduced generally and an introduction to the method of characteristics is presented. Writing on the subject matter of this lecture note, grateful use has been made of the authoritative texts and treatises on the subject by Ven Te Chow (1973), Henderson (1966), Sergio Montes (1998) and Hubert Chanson (1999), to which frequent references were made. Developing this lecture note is part of the activities within the MHO 5/6 project. This document could not have been completed without the enthusiastic support and advices of Professor Dr. Henri L. Fontijn, Head of the Fluid Mechanics Laboratory of the Faculty of Civil Engineering & Geosciences (CiTG), Delft University of Technology. He spent numerous hours of his time reading and advising on the typed texts. Special acknowledgments are due to the faculty and staff members of CiTG, CICAT, Delft University of Technology and my colleagues at CanTho University, who encouraged me in many ways. Finally, I would like to thank all my family members and friends, whose help in loving support is most gratefully acknowledged. LE ANH TUAN Delft University of Technology, the Netherlands, September 2003 OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- iv CONTENTS Preface . iii Contents . iv List of symbols . vii Chapter 1: INTRODUCTION 1 1.1. Review of fluid mechanics 1 1.1.1. Fluid mechanics 1 1.1.2. Hydrostatics 3 1.1.3. Continuity equation . 4 1.1.4. Types of flow 4 1.1.5. Bernoulli’s equation 5 1.1.6. Euler’s equation 5 1.1.7. Flow through orifices, mouthpieces and pipes 5 1.1.8. Flow through open channel 6 1.2. Structure of the course 7 1.2.1. Objectives of the course . 7 1.2.2. Historical note for the course 7 1.2.3. Structure of the course 8 1.3. Dimensional analysis . 10 1.3.1. Fundamental dimensions 10 1.3.2. Dimensional homogeneity 11 1.3.3. Principles of Dimensional Homogeneity 12 1.3.4. Buckingham’s - theorem 14 1.3.5. Limitations of dimensional analysis . 16 1.4. Similarity and models . 16 1.4.1. Advantages of model analysis 16 1.4.2. Hydraulic similarity . 17 1.4.3. Geometric similarity . 17 1.4.4. Kinematic similarity . 18 1.4.5. Dynamic similarity . 18 1.4.6. Technique of hydraulic modelling 19 1.4.7. Developments in hydraulic model testing . 20 1.4.8. Undistorted models 20 1.4.9. Comparison of an undistorted model and the prototype 21 1.4.10. Distorted models 22 1.4.11. Advantages and disadvantages of distorted models . 23 1.4.12. Comparison of a distorted model and its prototype . 23 Chapter 2: UNIFORM FLOW . 25 2.1. Introduction 25 2.1.1. Definition 25 2.1.2. Momentum analysis 27 2.2. Basic equations in uniform open channel flow 28 2.2.1. Chezy’s formula . 28 2.2.2. Manning’s formula . 30 2.2.3. Discussion of factors affecting f and n 32 2.3. Most economical cross-section . 32 OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- v 2.3.1. Concept . 32 2.3.2. Conditions for maximum discharge . 32 2.3.3. Problems of uniform-flow computation 36 2.4. Channel with compound cross-section 38 2.5. Permissible velocity against erosion and sedimentation . 40 Chapter 3: HYDRAULIC JUMP 46 3.1. Introduction 46 3.2. Specific energy . 47 3.2.1. Specific energy . 47 3.2.2. Critical depth and critical velocity . 48 3.2.3. Types of flows 49 3.3. Depth of hydraulic jump . 51 3.3.1. Concept . 51 3.3.2. Water rise in hydraulic jump . 51 3.3.3. Energy loss due to hydraulic jump . 53 3.3.4. Hydraulic jump features 54 3.4. Types of hydraulic jump . 55 3.4.1. Criterion for a critical state-of-flow . 55 3.4.2. Types of hydraulic jump 58 3.5. Hydraulic jump formulas in terms of Froude-number 61 3.5.1. Momentum-transfer curve 61 3.5.2. Direct hydraulic jump . 62 3.5.3. The initial depth and the sequent depth 62 3.5.4. Energy loss . 64 3.5.5. Efficiency . 66 3.5.6. Height of jump 66 3.5.7. Length of jump 66 3.6. Submerged hydraulic jump . 67 3.6.1. Definition 67 3.6.2. Flow in submerged jump . 68 Chapter 4: NON-UNIFORM FLOW 70 4.1. Introduction 70 4.1.1. General . 70 4.1.2. Accelerated and Retarded flow 71 4.1.3. Equation of non-uniform flow . 73 4.2. Gradually-varied steady flow 75 4.2.1. Backwater calculation concept 75 4.2.2. Equation of gradually-varied flow . 75 4.3. Types of water surface profiles . 77 4.3.1. Classification of flow profiles . 77 4.3.2. Sketching flow profiles . 79 4.3.3. Prismatic channel with a change in slope 82 4.3.4. Composite flow profiles with various controls . 83 4.4. Drawing water surface profiles . 84 4.4.1. Direct-step method 84 4.4.2. Direct numerical integration method 88 OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- vi Chapter 5: SPILLWAYS 90 5.1. Introduction 90 5.2. General formula 91 5.3. Sharp-crested weir 93 5.3.1. Experiments on sharp-crested rectangular weirs 93 5.3.2. Other types of sharp-crested weirs used for flow measurement . 96 5.4. The overflow spillway 98 5.4.1. The spillway crest . 98 5.4.2. The spillway face 100 5.4.3. The spillway toe . 101 5.5. Broad-crested weir 104 5.5.1. Introduction . 104 5.5.2. Broad-crested weir discharge formula . 105 5.5.3. Undular weir flow and discharge coefficients 105 Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS . 107 6.1. Introduction . 107 6.2. Expansions and Contractions 108 6.2.1. The transition problem 108 6.2.2. Expansions and Contractions . 108 6.3. Drop structures . 114 6.3.1. Introduction . 114 6.3.2. Free overfall 114 6.3.3. The head of the overfall . 116 6.3.4. The base of the overfall . 118 6.3.5. The drop structure . 119 6.4. Stilling basins . 121 6.4.1. Concept of stilling basin 121 6.4.2. Simple stilling basin design for canals 121 6.4.3. Specially designed stilling basins 124 6.5. Other types of energy dissipators 128 6.5.1. Stepped spillways . 128 6.5.2. Bucket-type and Ski-Jump . 129 Chapter 7: UNSTEADY FLOW . 130 7.1. Introduction . 130 7.2. The equations of motion . 131 7.2.1. Derivation of Saint-Venant equations 131 7.2.2. The equations of motion 131 7.3. Solutions to the unsteady-flow equations 135 7.3.1. Characteristic differential equations 135 7.3.2. Initial condition 138 7.3.3. The simple-wave problem . 140 7.3.4. Numerical solution of the characteristic differential equations 143 7.4. Positive and negative waves; Surge formation . 145 REFERENCES 147 OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- vii LIST OF SYMBOLS A cross-sectional area [m 2 ] b (average) width of channel [m] B open-channel free surface width [m] C Chezy coefficient [m ½ s -1 ] C c contraction coefficient C d discharge coefficient c natural wave speed [m/s] c f friction coefficient D (circular) pipe diameter [m]; hydraulic depth [m] E mean specific-energy head [m] F force [N]; momentum transfer per unit width [Nm -1 ] Fr Froude-number f friction coefficient according to Darcy-Weisbach g gravity constant [m/s 2 ] H local total energy-head [m] height of water above crest of weir H D design head, (i.e. head over spillway crest) [m] h flow depth, measured perpendicular to channel bed [m] h c critical water depth [m] h e equilibrium flow depth [m] h n normal depth at which flow is uniform [m]; h n = h e h o observed water depth [m] i b slope of channel bed i c slope of critical-depth line i e slope of energy grade line i f friction slope i o observed slope; bed slope e i arithmetic mean slope of the energy grade line L length (of channel, or pipe or weir) [m] L b length of stilling basin [m] L crest crest length in flow direction [m] n resistance coefficient in flow, called Manning's constant [m -1/3 s] P wetted perimeter [m]; weir height [m] P e equilibrium wetted perimeter [m] p pressure [Pa] q discharge per meter width [m 2 /s] Q total volume discharge [m 3 /s] R hydraulic radius [m] Re Reynolds-number s flow direction; coordinate along stream line S (bed) slope; slope of energy gradient line t time [s] V depth-averaged or mean flow velocity [m/s] V o approach velocity to weir [m/s] W channel bottom width [m]; top width of flow area [m]; water weight in channel over a length L [N] x Cartesian coordinate [m] y Cartesian coordinate [m] OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- viii z Cartesian coordinate [m]; altitude or elevation, measured positive upwards [m] z b change of bottom elevation between two cross-sections [m] z crest spillway crest elevation [m] z o reference elevation [m]; bed elevation [m] Greek symbols  velocity distribution coefficient; angle (of slope)  deflection angle h change in flow depth [m] E change in specific-energy head [m] H energy-head loss, i.e. change in total energy-head [m] L length [m] p pressure difference [Pa] s small distance along the flow direction [m] V change in flow velocity [m/s] z o change in bed elevation [m]  specific weight [N/m 3 ]  dynamic viscosity [Pa.s]  kinematic viscosity [m 2 /s]  channel slope; angle of channel bed relative to horizontal  density [kg/m 3 ]  surface tension [N/m]  shear stress [Pa]  o mean longitudinal shear stress acting over perimeter [Pa] Subcripts c critical flow conditions conj conjugate flow property des design flow conditions e equilibrium flow f friction i characteristics of section {i} (in numerical integration process); running index j jump l lateral m model max maximum min minimum p prototype r ratio of prototype to model characteristics; roller x x-component y y-component z z-component 1 upstream flow conditions 2 downstream flow conditions OPEN CHANNEL HYDRAULICS FOR ENGINEERS --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------- ix Abbreviations LHS left-hand side of equation sp.gr. specific gravity sp.wt. specific weight (N/m 3 ) SI International System of Units (Système International d'unités) SAF Saint Anthony Falls Hydraulics Laboratory SIA Swiss Society of Engineers RHS right-hand side of equation USBR United States Bureau of Reclamation . Mechanical Engineering OPEN CHANNEL HYDRAULICS FOR ENGINEERS LECTURE NOTE PREPARED BY LE ANH TUAN DELFT, 2003 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ---------------------------------------------------------------------------------------------------------------------------. of Open Channel Hydraulics for Engineers, also called Applied Hydraulics, is a subject required not only for Hydraulic Engineering students but also for