# Open channel hydraulics for engineers. Chapter 1 introduction

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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.......

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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 4C 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 4C 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: Specific weight of liquid Sp.gr. = 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 u dz z where u = velocity in x direction dz du 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 surface an inclined immersed surface is expressed as: F = .A.hgc [kN] (1-2) liquid hgc where = g = specific weight of the liquid [N/m ]; 3 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). (b) The pressure center of an immersed surface is the point through which the resultant pressure force acts (see Fig. 1.4): 90 surface hgc liquid hpc hgc hpc G G P G F = .A.hgc P P area A 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: h gc IG hpc = [m] (for vertical immersed surface) (1-3) A.h gc I G . sin 2 hpc = h gc [m] (for inclined immersed surface) (1-4) A.h gc where IG = 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 = (total) pressure force; n = number of sections; i = 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. Q2 Q3 Let Q = flow discharge [m3/s]; V3A3 V = average velocity of the liquid [ms-1]; Q1 V2A2 A = area of the cross-section [m2]; and i = the number of section. V1A1 We get: Q1 = Q2 = Q3 = … (1-6) Fig.1.5. Continuity of a liquid flow or V1A1 = V2A2 = V3A3 = … (1-7) 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 one- dimensional 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 V2 z p = Constant (= energy head) (1-8) 2g g where z = elevation, i.e. the height of the point in question above the datum; z represents the potential energy; V2 = energy head, representing the kinetic energy, V is the flow velocity along 2g the streamline at the point in question; p and = pressure head, representing the pressure energy; p is the pressure at the g 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 0 dV dp (1-9) g g 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: Re VD + Reynolds number: (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] (1-11) 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] (1-12) where C = Chezy's coefficient [m½ s-1]; cross section area A R = hydraulic radius [m] defined as: R wetted perimeter P 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 h Q i VA 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 open- channel-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 open- channel 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 open- channel 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 cross- section. 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. Table 1.1: Dimensions in terms of LMT No. Quantity Symbol Dimensions in terms of LMT-system 1. Length L L 2. Area A L2 3. Volume Vol L3 4. Time t T 5. Velocity V LT-1 6. Acceleration a LT-2 7. Gravitational acceleration g LT-2 8. Frequency N T-1 9. Discharge Q L3 T-1 10. Force/weight F, W LMT-2 11. Power P ML2T-3 12. Work/Energy E ML2T-2 13. Pressure p ML-1 T-2 14. Mass m M 15. Mass density ML-3 16. Specific weight ML-2 T-2 17. Dynamic viscosity ML-1 T-1 18. Kinematic viscosity L2 T1 19. Surface tension MT-2 20. Shear stress ML-1 T-2 21. Bulk modulus ML-1 T-2 ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 10
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- All variables used in science or engineering are expressed in terms of a limited number of basic dimensions. For example, we can designate the dimensions of velocity as: LT 1 Dis tan ce L [V] = (1-14) Time T Here the brackets [x] mean "dimension of". So, equation (1-14) reads as follows: "the dimension of velocity V equals the ratio of the distance to the time". In this case, L represents the dimension of distance and T that of time. Note: There are four systems of units, which are commonly used and universally adopted. These are known as: SI Units (International System of Units or Système International d'unités in French): a unified and systematically constituted system of fundamental and derived units for international use have been recommended by the 11th General Conference of Weights and Measures (CGPM). SI is widely used in Vietnam as an official unit system. The fundamental units of LMT (length, mass and time) are meter, kilogram and second, respectively. CGS Units: the fundamental units of LMT are centimeter, gram and second, respectively. MKS Units: the fundamental units of LMT are meter, kilogram and second, respectively. English Units: the fundamental units of LMT are foot, pound and second, respectively 1.3.2 Dimensional homogeneity Let us consider the common equation of hydraulics. Q =A.V We can write: L T = L2 x LT-1 = L3T-1 3 -1 (1-15) Above example goes without saying that all equations must balance in magnitude. However, all rational equations (those developed from basic laws of physics) must also be dimensionally homogeneous. An equation is called dimensionally homogeneous, if the fundamental dimensions have identical powers of LMT on both sides. That is, the left-hand side (LHS) of the equation must have the same dimensions as the right-hand side (RHS). Moreover, every term in the equation must have the same dimensions. Such an equation would essentially be independent of the system of measurement (i.e. English or SI). Note: Two dimensionally homogeneous equations can be multiplied or divided without affecting the homogeneity. But the two dimensionally homogeneous equations cannot be added or subtracted, as the resulting equation may not be dimensionally homogeneous. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 11
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 1.3.3. Principles of Dimensional Homogeneity The principle of dimensional homogeneity has a number of applications. The following issues are important from the point of view of the subject. (a) Determining the dimension of a physical quantity The dimensions of any physical quantity may be easily determined with this principle, e.g. the dimension of energy: Energy = Work = Force Distance (1-16) = [LMT ] [L] -2 (Force, [F ]= [LMT-2]) = [ML2T-2] Example 1.1: Determine the dimension of the following quantities in the LMT-system: (i) Force (ii) Pressure, (iii) Power, (iv) Specific weight, and (v) Surface tension Solution: We know the dimension of ‘force‘ in the LMT-system: (i) Force = Mass x Acceleration = [M] 2 [MLT 2 ] Length [ML] 2 Ans. Time [T ] Force [MLT 2 ] Similarly, (ii) Pressure = 2 [ML1T 2 ] Ans. Area [L ] Work done Force Distance (iii) Power = Time Time [MLT -2 ][L] = [ML2T -3 ] Ans. [T] Weight Force (iv) Specific weight = (Weight = Force) Volume Volume [MLT 2 ] = 3 [ML2 T 2 ] Ans. [L ] Force [MLT -2 ] and (v) Surface tension = = [MT-2] Ans. Length [L] Example 1.2. Determine the dimension of the following quantities in the LMT-system (i) Discharge, (ii) Torque and (iii) Momentum. Solution: We know the dimension of ‘discharge’ in the LMT-system: Volume [L3 ] (i) Discharge = [L3T 1 ] Ans. Time [T] Similarly (ii) Torque = Force Distance = [LMT-2][L] = [ML2T-2] Ans. and (iii) Momentum = Mass Velocity = [M] [LT-1] = [LMT-1] Ans. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 12
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- (b) Checking the dimensional homogeneity of an equation The dimensional homogeneity of an equation may be easily checked with this principle; e.g. let us consider Darcy - Weisbach’s formula for loss of energy head in pipes: L V2 hf f . . (1-17) D 2g The dimension of 2 in the denominator is not considered. The dimension of f, being constant, is taken as 1. Now substituting the dimensions on the LHS and RHS of the equation, we get: [1] x [L] x [LT -1 ] 2 [L] = = [L] (1-18) [LT 2 ][ L] Example 1.3. Check the dimensional homogeneity of the following common equations in the field of hydraulics: (i) Q C d A 2gH and (ii) V C Ri Solution (i) Given equation, Q = C d A 2gH Substituting the dimensions on the LHS and RHS of the equation (the dimension of Cd, being a discharge coefficient, is taken as 1): [L3T-1] = [1] [L2] [[1] ½ [LT-2 x L]½ = [L3T-1] Since the dimensions on both sides of the equation are the same, the equation is dimensionally homogeneous. Ans. (ii) Given equation, V = C Ri Substituting the dimensions on the LHS and RHS of the equation (the dimension of i, being dimensionless is taken as 1): LT-1 = C [L 1]1/2 = C [L]1/2 Since the dimensions on both sides of the equation are not the same, the equation is not dimensionally homogeneous. Ans. From the above equation, we find that: [ LT 1 ] C [ L1 / 2 T 1 ] [L]1 / 2 (c) Changing the coefficient of an equation while using an other system of units The coefficient of an equation may be easily changed, while using the same equation in an other system of units, for example from English to MKS or vice versa. Let us consider Manning’s formula for the velocity 1 2 3 12 V= .R .i (1-19) n ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 13
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- where n is a resistance coefficient, called Manning's constant. Now substituting the dimensions on the LHS and RHS of the equation, we get: [L]2/3 [1]1/2 = [L]2/3 1 1 [LT-1] = n n (The dimension of the bed slope i is 1). Since the dimensions of both sides are not the same, the equation is dimensionally non- homogeneous. From the above equation, we find that: 1 [LT ] 13 1 L .T 1 [L] 3 = 2 n Now, in order to make the above equation applicable to English units, the coefficient of M has to be changed. We know that 1m = 3.281 ft, and the unit of time is the same in both the systems. Therefore the new constant is: L1/3 = 3.2811/3 = 1.486 1.486 2 3 1 2 It is obvious, that the equation for English units will be V= .R .i (1-20) n (d) Using the dimensional analysis methods There are several methods that may be used to carry out the process of dimension analysis, such as the Step-by-Step method, the Exponent method, .... Students can find them in reference books. In this course, Buckingham's -theorem will be introduced shortly in the next section. 1.3.4. Buckingham’s - theorem Buckingham’s -theorem states, “If there are n variables in a dimensionally homogeneous equation and if these variables contain m fundamental dimensions such as (L, M, T), they may be grouped into (n-m) non-dimensional independent -terms.” Mathematically, if a variable X1 depends on the independent variables X2, X3, X4, ..., Xn the function may be written as: X1 = k (X2, X3, X4, ..., Xn) The equation may be written in its general form as: f(X1, X2, X3, X4, ..., Xn) = C where C is a constant, and f represents the functional relationship. In this equation, there are n variables. If there are m fundamental dimensions, then according to Buckingham’s - theorem f1 (1,2,3,...,n-m) = Constant where is a dimensionless term. Students can read for understanding details and how to apply in reference hydraulics books. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 14
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Example 1.4: Flow through a closed conduit with rectangular cross-section. Let us determine the wall friction as a dependent quantity. Solution: Let be the average wall shear stress over the full perimeter depends on: b = internal breadth [m] h = internal height [m] k = dimension of wall roughness [m] = specific mass density of fluid [kgm-3] = dynamic viscosity of fluid [kgm-1s-1] V = fluid velocity, averaged over cross-section [ms-1] = f1(b, h, k, , , V) [] = ML-1T-2 [b] = L [h] = L [k] = L [] = ML-3 [] = ML-1T-1 [V] = LT-1 We have totally 7 quantities and 3 basic dimensions, viz. M. L and T There are 4 independent dimensionless parameters. b k f2 , , V 2 h h Vh = cfV2 cf V2 Therefore, cf is a friction factor, i.e. dimensionless b k Vh c f f 3 , , Re h , with Re h h h h k Vb c f f 4 , , Re b , with Re b or b b ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 15
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 1.3.5. Limitations of dimensional analysis Some problems may be met when applying dimensional analysis: In order to use dimensional analysis, we must first decide which variables are significant. If we do not understand the problem well enough to make a good initial choice of variables, dimensional analysis seldom provides clarification. One error might be the inclusion of variables whose influence is already accounted for. For example, one might tend to include two or three length variables in a scale- model test, where only one may be sufficient. Another serious error might be the omission of a significant variable. If this is done, one of the significant dimensionless parameters will likewise be missing. How do we know whether a variable is significant for a given problem? Probably the proper answer is from experience. After working in the field of fluid mechanics and open- channel hydraulics for several years, one develops a feeling for the significance of variables to certain kinds of application. 1.4. SIMILARITY AND MODELS Since the beginning of the twentieth century, the engineers engaged on the creation or design of hydraulic structures (such as dams, spillways or large hydraulic machines) have developed a new and scientific method to predict the performance of their structures and machines. This is done by preparing physical scale models and testing them in a laboratory; so as to form some opinion, about the working and behaviour of the proposed hydraulic structures, after their completion or actual installation. The structure, of which the model is prepared, is known as prototype and the model is known as scale model or simply physical model. 1.4.1 Advantages of model analysis Though there are numerous advantages of model testing, yet the following is to be mentioned: 1. The behaviour and working details of a hydraulic structure or a machine can be easily predicted from its physical model. The smooth and reliable working of a hydraulic structure or a machine can be ascertained by spending a relatively small sum of money, which is a negligible fraction of the total cost to be spent on the prototype. 2. If the hydraulic structure or machine is made directly, then in case of its failure, it is very difficult to change its design. Moreover, it is very costly. Laboratory tests can result in saving human labour and material. 3. With the help of model testing, a number of alternative designs can be studied. Finally, the most economical, accurate and safe design may be selected. 4. When the existing hydraulic structure is not functioning properly, then model testing can help us in detecting and rectifying the defects. 5. Sometimes, it is difficult to design a particular portion of a complex hydraulic structure or machine. In such a case, model testing is very essential in order to ascertain the safety and reliability of that particular portion of the prototype. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 16
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 1.4.2. Hydraulic similarity If we look at a photograph of a man, very carefully, we can have an idea of the proportion of various parts of his body. The photograph will also give an idea of the features of each part of the man. Similarly, to know the complete working and behaviour of the prototype from its model, there should be a complete similarity between the prototype and its scale model. This similarity is known as hydraulic similitude or Hydraulic Similarity. Three types of hydraulic similarity are important, viz.: 1. Geometric similarity, 2. Kinematic similarity, and 3. Dynamic similarity 1.4.3. Geometric similarity Geometric similarity is said to exist between the model and the prototype, if both of them are identical in shape, but differ only in size. Or in other words, geometric similarity is said to exist between the model and the prototype, if the ratios of all corresponding linear, geometrical dimensions are equal (see Fig. 1.8). Lp (a) Bp Lm (b) Bm Fig.1.8. Geometric similarity: (a) Prototype and (b) Model Let Lp = length of the prototype Bp = breadth of the prototype, Dp = depth of the prototype, and Lm, Bm and Dm = corresponding values for the model. Now, if geometric similarity exists between the prototype and the model, then the linear ratio of the prototype and the model (also called scale ratio) reads as: L B D Lr = p = p = p L m Bm D m Similarly, it hold for the area ratio of the prototype and the model: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 17
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- L B D 2 2 2 Ar = p = p = p L m Bm D m And for the volume ratio: L B D 3 3 3 Vr = p = p = p L m Bm D m 1.4.4. Kinematic similarity Kinematic similarity is said to exist between the model and the prototype, if both of them have corresponding motions or velocities. Or in other words, kinematic similarity is said to exist between the model and the prototype, if the ratio of the corresponding velocities at corresponding points are equal. Let V1p = velocity of liquid in prototype at point 1, V2p = velocity of liquid in prototype at point 2, V1m, V2m = corresponding values for the model. Now, if kinematic similarity exists between the prototype and the model, then the velocity ratio of the prototype and the model reads as: V1p V2 p V3 p Vr = ... V1m V2 m V3 m 1.4.5. Dynamic similarity Dynamic similarity is said to exist between the model and the prototype, if both of them have corresponding forces. Or in other words, dynamic similarity is said to exist between the model and the prototype, if the ratios of the corresponding forces acting at corresponding points are equal: Let F1p and F1m = force acting in prototype and model at point 1; F2p and F2m = force acting in prototype and model at point 2. Now, if dynamic similarity exists between the prototype and the model, then the force ratio of the prototype and the model reads as: F F F Fr = 1p = 2p = 3p =... F1m F2m F3m Consider the flow over the spillway shown in Fig. 1.9. Here corresponding masses of fluid in the model and the prototype are acted on by corresponding forces. These forces are the force of gravity Fg, the pressure force Fp, and the viscous resistance force Fv. These forces add vectorially in Fig. 1.9 to yield a resultant force FR, which will in turn produce an acceleration of the volume of fluid in accordance with Newton’s second law. FRm M m a m FRp Mpa p ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 18
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- forces on fluid element prototype force polygon FPp FPp Fgp Fvp FRp = Mp.ap (a) Fvp Fgp model force polygon FPm FPm Fgm Fvm FRm = Mm.am (b) Fvm Fgm Fig. 1.9: Dynamic similarity (a) Prototype and (b) Model 1.4.6. Technique of hydraulic modelling The technique of hydraulic modelling involves the following steps: a. Selection of suitable scale, b. Operation of the hydraulic model, and c. Correct prediction a. Selection of suitable scale This depends on many factors. But the following is important concerning this issue: Availability of funds; Availability of time; Availability or space for accommodating the model; and Availability of employees. The usual practice is to make the model geometrically similar to the prototype. But in some cases distorted models are also employed. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 19
- OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- b. Operation of the hydraulic model After selecting the type, the scale and the materials of the model, the next step is to construct the model accurately according to the plan. High-tech instruments can be essential for precisely measuring the hydraulic quantities in the experiments. Great care and patience are required for correctly interpreting the model results. c. Correct prediction After obtaining the precise measurements of the required hydraulic quantities in an experiment, the next step is to predict the correct working of the prototype. We shall study the correct prediction of prototypes in the following pages. 1.4.7. Developments in hydraulic model testing The model testing is the most scientific and common feature of the design and successful working of hydraulic structures and machines. Two types of facility are important and need to be dealt with: 1) the wind tunnel, and 2) the water tunnel. 1. Wind tunnel A wind tunnel is a standard equipment for aircraft design. It provides a steady flow of air around the model which is suspended in the stream. Though the walls of the tunnel will interfere, to some extent, with the stream of air, yet its effect is generally neglected. In a wind tunnel, the air is set in motion by means of a compressor. The model under investigation is mounted in the path of the wind stream. Sometimes, the compression of air in the wind tunnel produces an appreciable rise in temperature, which must be dissipated by a cooling device. 2. Water tunnel A water tunnel is a standard equipment for the design of turbines, pumps and ships. In water tunnels, a uniform stream of water is produced and the model under investigation is mounted in the path of the water. The size of the water tunnel is, usually, expressed as the diameter of its best section. The existing water tunnels range as size from 10 cm to 150 cm. 1.4.8. Undistorted models All the hydraulic models may be broadly classified into the following two types: 1. Undistorted models, and 2. Distorted models. A model, which is geometrically similar to its prototype is known as an undistorted model. The prediction from an undistorted model is comparatively easy and the results obtained from the model, can be easily transferred to the prototype, if the basis condition (of geometric similarity) is satisfied. A distorted model will be discussed in Section 1.4.10. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 1: INTRODUCTION 20

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