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 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Chapter INTRODUCTION _________________________________________________________________________ 1.1. Review of fluid mechanics 1.2. Structure of the course 1.3. Dimensional analysis 1.4. Similarity and models _________________________________________________________________________ Summary This introductory chapter briefly reviews the previous course, in order to remind the students of some basic fluid properties and equations before starting this course on Open Channel Hydraulics. Next, dimensional analysis, similitude and model studies are dealt with and described. Key words Fluid mechanics; open channel flow; dimensional analysis; similitude; Reynolds number; hydraulic model _________________________________________________________________________ 1.1. REVIEW OF FLUID MECHANICS This lecture note is written for undergraduate students who follow the training programs in the fields of Hydraulic, Construction, Transportation and Environmental Engineering. It is assumed that the students have passed a basic course in Fluid Mechanics and are familiar with the basic fluid properties as well as the conservation laws of mass, momentum and energy. However, it may be not unwise to review some important definitions and equations dealt with in the previous course as an aid to memory before starting. 1.1.1. Fluid mechanics Fluid mechanics, which deals with water at rest or motion, may be considered as one of the important courses of the Civil Engineering training program. It is defined as the mechanics of fluids (gas or water). This course will mostly deal with the liquid water. The following properties then are important: (a). Density The density of a liquid is defined as the mass of the substance per unit volume at a standard temperature and pressure. It is also fully called “mass density” and denoted by the Greek symbol  (rho). In the case of water, we generally neglect the variation in mass density and consider it at a temperature of 4C and at atmospheric pressure; then  = 1,000 kg/m3 for all practical purposes. For other specific cases, the densities of common liquids are given in tables in most fluid mechanics books. (b) Specific weight The specific weight of a liquid is the gravitational force per unit volume. It is given by the Greek symbol  (gamma) and sometimes briefly written as sp.wt. In SI units, the specific weight of water at a standard reference temperature of 4C and atmospheric pressure is 9.81 kN/m3. ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 1 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- (c) Specific gravity Specific gravity is defined as the ratio of the specific weight of a given liquid to the specific weight of pure water at a standard reference temperature. Specific gravity, or sp. gr., is presented as: Sp.gr. = Specific weight of liquid Specific weight of pure water Specific gravity is dimensionless, because it is a ratio of specific weights. (d) Compressibility The compressibility of a fluid may be defined as the variation of its volume, with the variation of pressure. All fluids are compressible under the application of an external force, and when the force is removed they expand back to their original volume exhibiting the property that stress is proportional to volumetric strain. In the case of water as well as other liquids, it is found that volumes are varying very little under variations of pressure, so that compressibility can be neglected for all practical purposes. Thus, water may be considered as an incompressible liquid. (e) Surface tension The surface tension of a liquid is its property, which enables it to resist tensile stress in the plane of the surface. It is due to the cohesion between the molecules at the surface of a liquid. Looking at the upper end of a small-diameter tube put into a cup of water, we can easily see the water risen in the tube with an upward concave surface, as shown in Fig. 1a. However, if the tube is dipped into mercury, the mercury drops down in the tube with an upward convex surface as shown in Fig. 1b. If the adhesion between the tube and the liquid molecules is greater than the cohesion between the liquid molecules, we will have an upward concave surface. Otherwise, we get an upward convex surface. The surface tension of water and mercury at 20 ºC is 0.0075 kg/m and 0.0520 kg/m, respectively. Fig. 1.1a. Capillary tube in water Fig. 1.1b. Capillary tube in mercury The phenomenon of rising water in a small-diameter tube is called capillary rise. (f) Viscosity The dynamic or absolute viscosity of a liquid is denoted by the Greek symbol  (mu) and defined physically as the ratio of the shear stress  to the velocity gradient du/dz:  (1-1)  z du dz u  dz du z  where u = velocity in x direction u Fig. 1.2: Velocity distribution ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 2 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Viscosity is its property which controls the rate of flow. In the same tube, the flow of alcohol or water is much easier than the flow of syrup or heavy oil. 1.1.2. Hydrostatics Hydrostatics means study of pressure as exerted by a liquid at rest. Since the fluid is at rest, there are no shear stresses in it. The direction of such a pressure is always at right angles to the surface, on which it acts (Pascal’s law). (a) The total force F on a horizontal, a vertical or an inclined immersed surface is expressed as: F = .A.hgc [kN] (1-2) surface liquid hgc where  = g = specific weight of the liquid [N/m ]; A = area of the immersed surface [m2]; hgc = depth of the gravity center of the horizontal immersed surface from the liquid level [m] (see Fig. 1.3). 3 (b) The pressure center of an immersed surface is the point through which the resultant pressure force acts (see Fig. 1.4):  surface 90 hgc liquid hgc hpc hpc G G P G F = .A.hgc P area A P Fig. 1.4. Vertical and inclined surface (c) The depth of pressure center of an immersed surface from the liquid level, hpc, (see Fig. 1.4) reads: hpc = hpc = where IG =  = IG  h gc A.h gc I G . sin 2   h gc A.h gc [m] (for vertical immersed surface) (1-3) [m] (for inclined immersed surface) (1-4) moment of inertia of the surface about the horizontal axis through its gravity center [m4]; angle of the immersed surface with respect to the horizontal ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 3 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- (d) The pressure center of a composite section is found as follows:  first, by splitting it up into convenient sections;  then, by determining the pressures on these sections;  then, by determining the depths of the respective pressure centers; and  finally, by equating: F.h pc   Fi h pci n (1-5) i 1 where F = n = i = (total) pressure force; number of sections; subscript denoting the ith section. 1.1.3. Continuity equation The continuity principle is based on the conservation of mass as applying to the flow of fluids with invariant, i.e. constant, mass density. The continuity equation of a liquid flow is a fundamental equation stating that, if an incompressible liquid is continuously flowing through a pipe or a channel (the cross-sectional area of which may or may not be constant), the quantity of liquid passing per time unit is the same at all sections as illustrated in Fig. 1.5. Now consider a liquid flowing through a tube. Let Q = flow discharge [m3/s]; V = average velocity of the liquid [ms-1]; A = area of the cross-section [m2]; and i = the number of section. We get: Q1 = Q2 = Q3 = … or V1A1 = V2A2 = V3A3 = … (1-6) (1-7) Q2 Q3 V3A3 Q1 V2A2 V1A1 Fig.1.5. Continuity of a liquid flow 1.1.4. Types of flow     A flow, in which the velocity does not change from point to point along any of the streamlines, is called a uniform flow. Otherwise, the flow is called a non-uniform flow. A flow, in which each liquid particle has a definite path and the paths of individual particles do not cross each other, is called a laminar flow. This flow is void of eddies. But, if each particle does not have a definite path and the paths of individual particles also cross each other, the flow is called turbulent. A flow, in which the quantity of liquid flowing per second, Q, is constant with respect to time, is called a steady flow. But if Q is not constant, it is called an unsteady flow. A flow, in which the volume and thus the density of the fluid changes while flowing, is called a compressible flow. But if the volume does not change while flowing, it is called an incompressible flow. ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 4 OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------   A flow, in which the fluid particles also rotate about their own axes while flowing, is called a rotational flow. But if the particles do not rotate about their own axes while flowing, it is called an irrotational flow. A flow, whose streamlines may be represented by straight lines, is called a onedimensional flow. If the streamlines are represented by curves, the flow is called two-dimensional. A flow, whose streamlines can be decomposed into three mutually perpendicular directions, is called three-dimensional. 1.1.5. Bernoulli’s equation It states: “For a perfect incompressible liquid, flowing in a continuous stream, the total energy of a particle remains the same, while the particle moves along a streamline from one point to another”. This statement is based on the assumption that there are no losses due to friction. Mathematically it reads z where z V2 2g V2 p  = Constant (= energy head) 2g g (1-8) = elevation, i.e. the height of the point in question above the datum; z represents the potential energy; = energy head, representing the kinetic energy, V is the flow velocity along the streamline at the point in question; p g and = pressure head, representing the pressure energy; p is the pressure at the point in question and  is the liquid density. 1.1.6. Euler's equation Euler’s equation for steady flow of an ideal fluid along a streamline is based on Newton’s second law (Force = Mass  Acceleration). It is based on the following assumptions:  The fluid is inviscid, homogeneous and incompressible;  The flow is continuous, steady and along the streamline;  The flow velocity is uniformly distributed over the section; and  No energy or force, except gravity and pressure force, is involved in the flow. Euler's equation in a differential-equation form can be written as: dz  V dV dp  0 g g (1-9) After integrating the above equation, we easily come to Bernoulli's equation in the form of energy per unit weight of the flowing fluid. 1.1.7. Flow through orifices, mouthpieces and pipes  An orifice is an opening (in a vessel) through which the liquid flows out. The discharge through an orifice depends on the energy head, the cross-sectional area of the orifice and the coefficient of discharge. A pipe, the length of which is generally more than two times the diameter of the orifice, and which is fitted externally or internally to the orifice is known as a mouthpiece. When a liquid is flowing through ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 5 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- a mouthpiece, the energy head is declining due to wall friction, change of cross section or obstruction in the flow.  A pipe is a closed conduit used to carry fluid. When the pipe is running full, the flow is under pressure. The friction resistance of a pipe depends on the roughness of the pipe inside. Early experiments on fluid friction were conducted, among others, by Chezy: the frictional resistance varies approximately with: (a) the square of the liquid velocity, and (b) the bed slope.  Frictional resistance per  Frictional resistance =   wetted area  (velocity)2   unit area at unit velocity  Reminder: + Reynolds number: Re  VD  (1-10) where  = kinematic viscosity [m2/s] V = characteristic flow velocity [m/s]; D = characteristic length, e.g. diameter of the pipe [m] L V2 + Darcy–Weisbach’s formula for head loss hf in pipes: h f  f . . [m] D 2g where f = friction coefficient according to Darcy–Weisbach; L = length of the pipe + Chezy's formula for flow velocity V in pipe: V  C Ri [m/s] where C = Chezy's coefficient [m½ s-1]; R (1-11) (1-12) cross  section area A  wetted perimeter P R = hydraulic radius [m] defined as: i = loss of energy head per unit length (= bed slope in uniform flow). 1.1.8. Flow through open channel An open channel is a passage, through which the water flows due to gravity with atmospheric pressure at the free surface. The flow velocity is different at different points in the cross-section of a channel due to the occurrence of a velocity distribution, but in calculations, we use the mean velocity of the flow. In the course on Fluid Mechanics, we have assumed that the rate of discharge Q, the depth of flow h, the mean velocity V, the slope of the bed i and the cross-sectional area A remain constant over a given length L of the channel (see Fig. 1.6). L Q i h VA Q Fig. 1.6. Uniform flow in open channel Discharge through an open channel: Q  VA  AC Ri (1-13) ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 6 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 1.2. STRUCTURE OF THE COURSE 1.2.1. Objectives of the course Open Channel Hydraulics is an advanced course required for all students who follow the field-study of water resources engineering. The subject is rich in variety and of interest to practical problems. The content is focused on the types of problems commonly encountered by hydraulic engineers dealing with the wide fields covered by open channel hydraulics. Due to space and lecturing-time limitations, however, the lecture note does not extend into the specialist fields of mathematical natural flow networks required, for example, for river engineering computations. The course aims to present the principles dealing with water flow in open channels and to guide trainees to solve the applied problems for hydraulic-structure design and water system control. The main objectives of the course are:    To supply the basic principles of fluid mechanics for the formulation of open channel flow problems. To combine theoretical, experimental and numerical techniques as applied to open channel flow in order to provide a synthesis that has become the hallmark of modern fluid mechanics. To provide theoretical formulas and experimental coefficients for designing some hydraulic structures as canals, spillways, transition works and energy dissipators. 1.2.2. Historical note for the course Fluid mechanics and open channel hydraulics began at the need to control water for irrigation purposes and flood protection in Egypt, Mesopotamia, India, China and also Vietnam. Ancient people had to record the river water levels and got some empirical understanding of water movements. They applied basic principles on making some fluid machinery, sailing boats, irrigation canals, water supply systems etc. The Egyptians used dams for water diversion and gravity flow through canals to distribute water from the Nile River, and the Mesopotamians developed canals to transfer water from the Euphrates river to the Tigris river, but there is no recorded evidence of any understanding of the theoretical flow principles involved. The Chinese are known to have devised a system of dikes for protection from flooding several thousand years ago. Over the past 2,000 years, many dikes and canal systems have been built in the Red River delta in the North of Vietnam to contain the delta and drain off its flood water that has always been serious problems. Vietnamese, under Ngo Quyen, have also known to apply the tidal law in Bach Dang river battles in 939 A.D, which has become famous in Vietnamese history. It was not until 250 B.C. that Archimedes discovered and recorded the principles of hydrostatics and flotation. In the 17th and 18th centuries, Isaac Newton, Daniel Bernoulli and Leonhard Euler formulated the greatest principles of hydrodynamics. The work of Chezy on flow resistance began in 1768, originating from an engineering problem of sizing a canal to deliver water from the Yvette River to Paris. The Manning-equation for openchannel-flow resistance has a complex historical development, but was based on field observations. Julius Weisbach extended the sharp-crested weir equation and developed the elements of the modern approach to open channel flow, including both theory and experiment. William Froude, an Engish engineer, collaborated with Brunel in railway construction and in the design of the steamer “Great Eastern”, the largest ship afloat at that time. He contributed to the study of friction between solids and liquids, to wave mechanics and to the interpretation of ship model tests. ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 7 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- The work of Bakhmeteff, a Russian émigré to the United States, had perhaps the most important influence on the development of open channel hydraulics in the early 20th century. Of course, the foundations of modern fluid mechanics were laid by Prandtl and his students, including Blasius and von Kàrmàn, but Bakhmeteff’s contributions dealt specifically with open channel flow. In 1932, his book on the subject was published, based on his earlier 1912 notes developed in Russia. His book concentrated on “varied flow” and introduced the notion of specific energy, still an important tool for the analysis of openchannel flow problems. In Germany at this time, the contributions of Rehbock to weir flow also were proceeding, providing the basis for many further weir experiments and weir formulas. By the mid-20th century, many of the gains in knowledge in open channel flow has been consolidated and extended by Rouse (1950), Chow (1959, 1973) and Henderson (1966), in which books extensive reference can be found. These books set the stage for applications of modern numerical analysis techniques and experimental instrumentation to openchannel flow problems. 1.2.3. Structure of the course The lecture note is divided into three parts of increasing complexity. (a) Part 1 introduces to the basic principles: course introduction (Chapter 1), uniform flow (Chapter 2) and hydraulic jump phenomena (Chapter 3). This part will take 15 teaching hours. (b) Part 2 includes non-uniform flow (Chapter 4) and design application as Spillways (Chapter 5) and Transitions and Energy dissipators (Chapter 6). This part will take 20 teaching hours. (c) Part 3 deals with unsteady flow (Chapter 7). This chapter will take 10 teaching hours. The course approach chart is presented in Fig. 1.7. on the next page. ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 8 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- OPEN CHANNEL HYDRAULICS FOR ENGINEERS + + + Chapter 1: INTRODUCTION 1.1. Review of fluid mechanics. 1.2. Structure of the course. 1.3. Dimensional analysis. 1.4. Similarity and models Chapter 2: UNIFORM FLOW 2.1. Introduction. 2.2.Basic equations in uniform open channel flow. 2.3. Most economical cross-section. 2.4. Channel with compound crosssection. 2.5. Permissible velocity against erosion and sedimentation. Chapter 3: HYDRAULIC JUMP 3.1. Introduction. 3.2. Specific energy. 3.3. Depth of hydraulic jump. 3.4. Types of hydraulic jump. 3.5. Hydraulic jump formulas in terms of Froude-number. 3.6. Submerged hydraulic jump + Chapter 4: NON-UNIFORM FLOW 4.1. Introduction. 4.2. Gradually-varied steady flow. 4.3. Types of water surface profiles. 4.4. Drawing water surface profiles. + Chapter 5: SPILLWAYS 5.1. Introduction. 5.2. General formula. 5.3. Sharp-crested weir 5.4. The overflow spillway. 5.5. Broad-crested weir + Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 6.1. Introduction. 6.2. Expansions and Contractions. 6.3. Drop structures 6.4. Stilling basins. 6.5. Other types of energy dissipators + Chapter 7: UNSTEADY FLOW 7.1. Introduction. 7.2. The equations of motion 7.3. Solutions to the unsteady-flow equations 7.4. Positive surge and negative waves; Surge formation Fig.1.7. Course structure chart ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 9 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 1.3. DIMENSIONAL ANALYSIS Most hydraulic engineering problems are solved by applying a mathematical analysis. In some cases they should be checked by physical experimental means. The approach of such problems is considerably simplified by using mathematical techniques for dimensional analysis. It is based on the assumption that the phenomenon at issue can be expressed by a dimensionally homogeneous equation, with certain variables. 1.3.1. Fundamental dimensions We know that all physical quantities are measured by comparison. This comparison is always made with respect to some arbitrarily fixed value for each independent quantity, called dimension (e.g. length, mass, time, temperature etc). Since there is no direct relationship between these dimensions, they are called fundamental dimensions or fundamental quantities. Some other quantities such as area, volume, velocity, force etc, cannot be expressed in terms of fundamental dimensions and thus may be called derived dimensions, derived quantities or secondary quantities. There are two systems for fundamental dimensions, namely FLT (i.e. force, length, time) and LMT (i.e. length, mass, time). The dimensional form of any quantity is independent of the system of units (i.e. metric or English). In this course, we shall use the LMT-system. The following table gives the dimensions and units for the various physical quantities, which are important form the hydraulics point-of-view. No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. Table 1.1: Dimensions in terms of LMT Quantity Symbol Dimensions in terms of LMT-system Length Area Volume Time Velocity Acceleration Gravitational acceleration Frequency Discharge Force/weight Power Work/Energy Pressure Mass Mass density Specific weight Dynamic viscosity Kinematic viscosity Surface tension Shear stress Bulk modulus L A Vol t V a g N Q F, W P E p m        L L2 L3 T LT-1 LT-2 LT-2 T-1 L3 T-1 LMT-2 ML2T-3 ML2T-2 ML-1 T-2 M ML-3 ML-2 T-2 ML-1 T-1 L2 T1 MT-2 ML-1 T-2 ML-1 T-2 ----------------------------------------------------------------------------------------------------------------------------------Chapter 1: INTRODUCTION 10