Chapter 1 ppt (2).pptxSCSSMJJUIIkiscscscscscssc

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1. Open Channel Hydraulics 1

2. CHAPTER ONE 1. INTRODUCTION TO OPEN CHANNEL HYDRAULICS 1.1. Definition and Types of Open Channel 1.2. Difference of Open Channel flow and Pipe flow 1.3. Fundamental Equations 1.4. Energy-Depth relationships 2

3. 1.1. Introduction Open channel flow : Any type of flow is either open channel flow or pipe flow . Flow of water (fluid) in which there is an interface between water and air . The surface of flow is open to atmosphere There is only atmospheric pressure on the surface The governing force is the gravitational force component along the channel slope. T he main driving force(source of energy) is gravity Half pipe flow is a type of open channel Example: Water flow in rivers and streams, flow in irrigation canals, sewer systems that flow partially full, storm drains and street gutters. 3

4. 1.2. Difference of Open Channel flow and Pipe flow Open Channel flow Pipe flow have a free surface Has no free surface. exert direct atmospheric pressure There exist no direct atmospheric flows but hydraulic pressure only. Unknown cross-section Have Known, fixed cross-section Flow driven by gravity Flow driven by pressure work Question: Where is the HGL in case of the open channel flow? What is the influence of the change in cross–section of the pipe? Is a half–filled pipe flow open channel flow or pipe flow? 4

5. Energy in pipe flow is expressed as head and is defined by the Bernoulli equation. In order to conceptualize head along the course of flow within a pipe, diagrams often contain a hydraulic grade line. Pipe flow is subject to frictional losses as defined by the Darcy- Weisbach formula . 5

6. Classification of Open Channel  Flows 1. Steady and Unsteady Flows A steady flow occurs when the flow properties, such as the depth or discharge at a section do not change with time. I f the depth or discharge changes with time the flow is termed unsteady . In practical applications, due to the turbulent nature of the flow and also due to the interaction of various forces, such as wind, surface tension, etc., at the surface there will always be some fluctuations of the flow properties with respect to time . Flood flows in rivers and rapidly varying surges in canals are some examples of unsteady flows. 6

7. 2. Uniform and Non-uniform Flows If the flow properties, say the depth of flow , in an open channel remain constant along the length of the channel, the flow is said to be uniform . A flow in which the flow properties vary along the channel is termed as non-uniform flow or varied flow . A prismatic channel carrying a certain discharge with a constant velocity is an example of uniform flow T he free surface will be parallel to the bed. It is easy to see that an unsteady uniform flow is practically impossible, and hence the term uniform flow is used for steady uniform flow . Flow in a non-prismatic channel and flow with varying velocities in a prismatic channel are examples of varied flow . Varied flow can be either steady or unsteady. 7

8. 3. Gradually Varied (GVF) and Rapidly Varied Flows (RVF) If the change of depth in a varied flow is gradual so that the curvature of streamlines is not excessive, such a flow is said to be a gradually varied flow. Frictional resistance plays an important role in these flows. The backing up of water in a stream due to a dam or drooping of the water surface due to a sudden drop in a canal bed are examples of steady GVF. If the curvature in a varied flow is large and the depth changes considerably over short lengths, such a phenomenon is termed as rapidly varied flow (RVF). The frictional resistance is relatively insignificant in such cases and it is usual to regard RVF as a local phenomenon. A hydraulic jump occurring below a spillway or a sluice gate is an example of steady RVF. If some flow is added to or abstracted from the system the resulting varied flow is known as a spatially varied flow (SVF). 8

9. Gradually and Rapidly Varied flow The flow is rapidly varied if the depth changes abruptly over a comparatively short distances; other wise it is gradually varied. Examples of rapidly varied flows are: Hydraulic jump, Hydraulic drop and flow over vweir . 9

10. 4. Depending on the Effect of viscosity The characteristic length (L) commonly used is the hydraulic radius. 10

11. 5 . Effect of gravity : T he effect of gravity upon the state of flow is represented by a ratio of inertial forces to gravity forces. If the value of F r <1, the flow is Subcritical flow If the value of F r >1, the flow is Supercritical flow If the value of Fr = 1, the flow is critical flow 11

12. Geometry of open channel Prismatic   channel :  channel in which the cross sectional shape and size and also  the bottom slope are constant     E.g. Most  of the man made channel (Artificial Channels)     The  rectangle, trapezoid, triangle and circle are some of the commonly used  shapes in made channels.  Non –prismatic: channel having a  varying cross-sections E.g. a ll  natural channels  12

13. Geometric properties necessary for analysis. Depth (y) – the vertical distance from the lowest point of the channel section to the free surface. Stage (z) – the vertical distance from the free surface to an arbitrary datum Area (A) – the cross-sectional area of flow, normal to the direction of flow Wetted perimeter (P) – the length of the wetted surface measured normal to the direction of flow. Surface width(B) – width of the channel section at the free surface or top surface some times denoted by T. Hydraulic radius (R) – the ratio of area to wetted perimeter (A/P) Hydraulic mean depth ( Dm ) – the ratio of area to surface width (A/B) Section factor (Z ) is the product of the water area and the square root of the hydraulic depth. 13

14. Total energy (E) : E= z + y + αv 2 /2g (for small slopes θ with y = d Specific energy (ES) : ES energy in relation to lowest point in a section ES = y + αv 2 /2g Velocity (V) : V= Q/A Velocity head : αV 2 /2g = αQ 2 /2gA 2 Froude number : Fr 2 = αQ 2 B s /gA 3 Friction Slope : S f = Q 2 n 2 /A 2 R 4/3 (manning) Friction Slope : S f = Q 2 /C 2 A 2 R ( chezy ) 14

15. E quations for rectangular, trapezoidal and circular channels. 15

16. 1.3. Fundamental Equations : The equations which describe the flow of fluid are derived from three fundamental laws of physics: Conservation of matter (or mass) Conservation of energy Conservation of momentum Although first developed for solid bodies they are equally applicable to fluids. 16

17. 1. The Continuity Equation (Conservation of mass ) This says that matter cannot be created nor destroyed, but it may be converted (e.g. by a chemical process . For any control volume during the small time interval t, the principle of conservation of mass implies that: The mass of flow entering the control volume - the mass of flow leaving the control volume = the change of mass within the control volume. Application of the continuity principle to unsteady, open channel flow is more difficult. In unsteady open channel flow the water surface will change over a certain distance. 17

18. 2. Conservation of energy This says that energy cannot be created nor destroyed, but may be converted from one type to another (e.g. potential may be converted to kinetic energy). The basic equations can be obtained from the First Law of Thermodynamics. Consider the forms of energy available for the above control volume. If the fluid moves from the upstream face 1, to the downstream face 2 in time dt over the length L . Therefore the energy equation defined as α α = is assumed as unity It is kinetic energy correction factor. Unit weight = mg Mass = Vol*density W=F*S, F= P*A PE= mgh 18

19. 3. Conservation of momentum The law of conservation of momentum says that a moving body cannot gain or lose momentum unless acted upon by an external force . This is a statement of Newton's Second Law of Motion: Force = rate of change of momentum In solid mechanics these laws may be applied to an object which is has a fixed shape and is clearly defined. In fluid mechanics the object is not clearly defined and as it may change shape constantly. To get over this we use the idea of control volumes. To derive the basic equation the above conservation laws are applied by considering the forces applied to the edges of a control volume within the fluid. 19

20. The various forces acting on the control volume in the longitudinal direction are as follows: ( i) Pressure forces acting on the control surfaces, F1 and F2 . ( ii) Tangential force on the bed, F3 , ( iii) Body force, i.e., the component of the weight of the fluid in the longitudinal direction, F4 . By the linear-momentum equation in the longitudinal direction for a steady-flow discharge of Q, Fig.1.17 Definition sketch for the momentum equation 20

21. M1 = β1ρ QV1 = momentum flux entering the control volume , M2 = β2 ρ QV2 = momentum flux leaving the control volume. The momentum equation is a particularly useful tool in analyzing rapidly varied flow (RVF) situations where energy losses are complex and cannot be easily estimated. It is also very helpful in estimating forces on a fluid mass. 21

22. Kinetic energy ( α ) and Momentum ( β ) correction factor The coefficients α and β are both unity in the case of uniform velocity distribution. For any other velocity distribution α > β > 1.0. The higher the non-uniformity of velocity distribution, the greater will be the values of the coefficients . Generally , large and deep channels of regular cross sections and with fairly straight alignments exhibit lower values of the coefficients . Conversely , small channels with irregular cross sections contribute to larger values of α and β. It appears that for straight prismatic channels, α and β are of the order of 1.10 and 1.05 respectively. In compound channels , i.e. channels with one or two flood banks, α and β may, in certain cases reach very high values, of the order of 2.0 22

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24. Velocity Distribution Due to the presence of free surface and to the friction along the channel wall, the velocities in open channel are not uniformly distributed. The measured maximum velocity in ordinary channels usually appear to occur below the free surface at a distance of 0.05 to 0.25 of the depth, the closer the banks the deeper the maximum. As a result of non-uniform distribution of velocities over the channel section, the velocity head of an open channel flow is generally greater than the value computed according to the expression V 2 /2g, where V is the mean velocity. In practice usually average velocity across the flow is taken and correction coefficients are applied. Pressure Distribution The intensity of pressure for a liquid at its free surface is equal to that of the surrounding atmosphere. Since the atmospheric pressure is commonly taken as a reference and of equal to zero, the free surface of the liquid is thus a surface of zero pressure. The pressure distribution in an open channel flow is governed by the acceleration of gravity and other accelerations and is given by the Euler’s equation . 24

25. 1.4. Energy – Depth Relationships 1.4.1. Energy Principle The energy equation and the momentum equation are used in addition to the continuity equation in analyzing fluid-flow situations. They are both derived from Newton’s second law of motion. T he force components on a fluid particle in the direction of its motion are equated to the product of mass of the particle and acceleration along the streamline. F=ma The equation is obtained in differential form and requires the assumption of: A frictionless fluid in order to eliminate all shear in the fluid. Steady flow. Incompressible fluid Prismatic channel Ideal flow Hydrostatic pressure exists 25

26. Figure. Force components on a fluid particle in the direction of streamline. 26

27. let S be a streamline in steady flow, and consider the forces acting on a fluid particle in the direction S of the streamline. On the upstream end the element the force is: PδA , in which P is the pressure intensity at the center of the face. The force on the downstream end of the element is :- 27 acts in the negative direction Any forces acting on the sides of the element are normal to the streamline do not enter the equation. The only other force acting is due to gravity and is γδAδS , acting vertically down ward. The component in the S-direction is:

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34. 1.4.2. Momentum Principle is developed from Newton’s second law of motion. According to Newton's second law of motion : the change of momentum (dmv) per unit time, is equal to the resultant of all external forces acting on a body (body of water flow in a channel in our case). 34

35. The application of Newton's second law, in a one dimensional for to the control volume, i.e. equating the sum of all external forces (F) to the rate of change of momentum (ρ Q V) for any two cross-sections 1 and 2 gives: Or including the momentum coefficient: 35

36. Q Q F1 and F2 are the resultant pressure forces acting on the two sections and w is the weight of the water between the two sections. Ff is the total friction force acting along the surface of the body . The slope is mild assume sin θ ≈ So = 0 ⇒ W sin θ = 0 and for a flow with parallel flow lines the pressure is assumed to be hydrostatic. 36

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38. 38 For a given discharge Q, channel shape and coefficient β the function, M depends only on the water depth y. Plotting M against ay gives a similar figure as for the specific energy Es against depth y. This curve is called specific force curve. In the figure two regions can be determined, namely sub and supercritical flow. For every M > Mmin two water –depths exist, which are called the initial and sequent depth. Together they are the conjugate depths.

39. Q A= by, b=1 thus A=y and q= Q/b 39 Q Q

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41. 41 Q Q

42. Q Q The impulse momentum principle also follows from Newton’s second law. The flow may be: compressible or incompressible, real (with friction) or ideal (frictionless), steady or unsteady moreover, the equation is not only valid along a streamline. 42

43. It has a special advantage for application to problems involving high internal energy changes, such as the problem of the hydraulic jump. If the energy equation is applied to such problems , the unknown internal energy loss represented by hf is indeterminate , and the omission of this term would result in a considerable errors. The advantage of the impulse momentum principle is that only the conditions at the end sections of the control volume govern the analysis. 43

44. If instead the momentum equation is applied to these problems, since it deals only with external forces, T he effects of the internal forces, the effects of the internal forces will be entirely out of consideration and need not be evaluated. The term for frictional losses due to external forces, on the other hand, is unimportant in such problems and can safely be omitted, because the phenomenon takes place in a short reach of the channel and the effect due to external forces is negligible compared with the internal losses. 44

45. The concept of specific energy is first introduced by Bakhmeteft (1932) and has been proven to be very useful in analysis of open channel flow. The “Specific energy” is the average energy per unit weight of water with respect to the channel bottom. Therefore , the specific energy is the sum of the water-depth (y) and the velocity head, if the streamlines are straight and parallel. The total energy of a channel flow referred to datum is given by , H = Z+Y* Cosθ + If the datum coincides with the channel bed at the cross-section, the resulting expression is known as specific energy and is denoted by E. Thus , specific energy is the energy at a cross-section of an open channel flow with respect to the channel bed. E = Y* Cosθ +   45 1.4.3. Specific Energy

46. When Cos θ =1 and the equation of specific energy further simplify as: E= Y + It indicates that the specific energy is equal to the sum of the depth of water and the velocity head. Therefore, we defined as: Specific energy is the energy at a cross-section of an open channel flow with respect to the channel bed. Specific energy is the height of the energy grade line above the channel bottom. In other respect, since V=Q/A, the equation of specific energy may be written as: E= Y + = Y+   46

47. 47 Here, cross-sectional area A depends on water depth y and can be defined as, A = f(y) and also there is a functional relation between the three variables as, f (= E, y, Q) = 0 In order to examine the functional relationship on the plane, two cases are introduced. 1. Constant discharge: Q = Q 1 ⇒ E = f (y, Q). Variation of the specific energy with the water depth at a cross-section for a given discharge Q 1 . 2. Constant Specific Energy E = E 1 ⇒ E = f (y, Q). Variation of the discharge with the water depth at across-section for a given specific energy E.

48. A. Constant Discharge Situation Since the specific energy, E= y+   48 Figure 2.1 Specific Energy Diagram

49. For a channel of known geometry, E =f (y, Q), keeping Q constant it can be seen that, the specific energy in a channel section is a function of the depth of the flow only. The variation E with y is represented by a cubic parabola (Fig 2.1). It is seen that there are two positive roots for the equation of E indicating that any particular discharge Q 1 can be passed in a given channel at two depths and still maintain the same specific energy E. In the Figure 2.1 the ordinate PP’ represents the condition for a specific energy of E 1 . The depth of flow can be either PR=y 1 or PR’=y 1 ’. These two possible depths have the same specific energy are known as alternate depths. In the Figure 2.1, a line OS drawn such that E=y is the asymptote of the upper limb of the specific-energy curve . 49

50. It may be noticed that the intercept P’R’ or P’R represents the velocity head of the two alternate depths, one (PR=y 1 ) is smaller and has a larger velocity head while the other (PR’=y 1 ’) has a larger depth and consequently a smaller velocity head . The condition of minimum specific energy is known as the critical-flow condition and the corresponding depth yc is known as critical depth. Thus, at the critical state the two alternate depths apparently become one. When the depth of flow is greater than the critical depth , the velocity of flow is less than the critical velocity for the given discharge, the flow is subcritical . When the depth of flow is less than the critical depth, the flow is supercritical. Hence , y 1 is the depth of supercritical flow, and y 1 ’ is the depth of subcritical flow. 50

51. At the critical depth, the specific energy is minimum. Thus differentiating Eqn. E= y+ with respect to y (keeping Q constant) and equating to zero,   51 Designating the critical-flow condition by the suffix ‘c’, = 1 which means =   It is easy to see that at the critical flow Y= Y c , then F=F c = 1 If the value of F r <1, the flow is Subcritical flow and If the value of F r >1, the flow is Supercritical flow.

52. 52 B. Variable discharge situation

53. This curve has to limbs AC and BC. The limb AC approaches the horizontal axis asymptotically towards the right. The limb BC approaches the line OD as it extends upward and to the right. Line OD is a line that passes through the origin and has an angle of inclination equal to 45 o . At any point P on this curve, the ordinate represents the depth, and the abscissa represents the specific energy. Which is equal to the sum of the pressure head y and the velocity head. The curve shows that for a certain discharge Q two flow regimes are possible, viz . slow and deep flow or a fast and shallow flow, i.e. for a given specific energy, there are two possible depths, for instance, the low stage y1. and the high stage y2. 53

54. The low stage is called the alternate depth of the high stage, and vice versa . At pint C, the specific energy is minimum. It can be proved that this condition of minimum specific energy corresponds to the critical state of flow. Thus , at the critical state the two alternate depths apparently become one, which is known as the critical depth (YC). When the depth of flow is greater than the critical depth, the velocity of flow is less than the critical velocity for the given discharge, and, hence, the flow is sub critical. When the depth of flow is less than critical depth the flow is supercritical. Hence , Y1, is the depth of a supercritical flow, and Y2 is the depth of a sub critical flow. If the discharge changes, the specific energy will be changed accordingly . The two curves A’B’ and A”B” (Figure 2-3) represent positions of the specific energy curve when the discharge is less and greater, respectively than the discharge used for the construction of the curve AB . 54

55. E=Y+ and Q=A√2g(E-y). The condition for maximum discharge can be obtained by differentiating the above equation with respect to y and equating to zero while keeping E = constant.   55

56. 56 Section Factor See Examples below Q / √g