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A Textbook of Fluid Mechanics and Hydraulic Machine

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The course contents of the book entitled ‘A Text book of fluid mechanics and hydraulic machines’ have been planned in such a way that the book covers the complete course of engineering students of mechanical, civil, electrical aeronautical and chemical.

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Ch_1_(1-33).pdf

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1

CHAPTER

1.1

PROPERTIES OF

FLUIDS

INTRODUCTION

Fluid mechanics is that branch of science which deals with the behaviour of the fluids (liquids or gases) at rest as well as in motion. Thus this branch of science deals with the static, kinematics and dynamic aspects of fluids. The study of fluids at rest is called fluid statics. The study of fluids in motion, where pressure forces are not considered, is called fluid kinematics and if the pressure forces are also considered for the fluids in motion, that branch of science is called fluid dynamics.

1.2 PROPERTIES OF FLUIDS

1.2.1 Density or Mass Density. Density or mass density of a fluid is defined as the ratio of the mass of a fluid to its volume. Thus mass per unit volume of a fluid is called density. It is denoted by the symbol r (rho). The unit of mass density in SI unit is kg per cubic metre, i.e., kg/m3. The density of liquids may be considered as constant while that of gases changes with the variation of pressure and temperature.

Mathematically, mass density is written as r=

 

Ch_2_(34-68).pdf

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2

CHAPTER

2.1

PRESSURE AND ITS

MEASUREMENT

FLUID PRESSURE AT A POINT

Consider a small area dA in large mass of fluid. If the fluid is stationary, then the force exerted by the surrounding fluid on the area dA will always be perpendicular to the surface dA. Let dF is the force dF is known as the intensity of acting on the area dA in the normal direction. Then the ratio of dA pressure or simply pressure and this ratio is represented by p. Hence mathematically the pressure at a point in a fluid at rest is dF

. dA

If the force (F) is uniformly distributed over the area (A), then pressure at any point is given by

p =

F

Force

=

.

A

Area

\ Force or pressure force, F = p ¥ A.

The units of pressure are : (i) kgf/m2 and kgf/cm 2 in MKS units, (ii) Newton/m 2 or N/m 2 and

N/mm2 in SI units. N/m2 is known as Pascal and is represented by Pa. Other commonly used units of pressure are : kPa = kilo pascal = 1000 N/m2 bar = 100 kPa = 105 N/m2.

p =

2.2 PASCAL'S LAW

It states that the pressure or intensity of pressure at a point in a static fluid is equal in all directions. This is proved as :

 

Ch_3_(69-130).pdf

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3

CHAPTER

3.1

HYDROSTATIC FORCES

ON SURFACES

INTRODUCTION

This chapter deals with the fluids (i.e., liquids and gases) at rest. This means that there will be no relative motion between adjacent or neighbouring fluid layers. The velocity gradient, which is equal to the change of velocity between two adjacent fluid layers divided by the distance between the layers, du

∂u

= 0. The shear stress which is equal to m will also be zero. Then the forces acting will be zero or dy

∂y on the fluid particles will be :

1. due to pressure of fluid normal to the surface,

2. due to gravity (or self-weight of fluid particles).

3.2 TOTAL PRESSURE AND CENTRE OF PRESSURE

Total pressure is defined as the force exerted by a static fluid on a surface either plane or curved when the fluid comes in contact with the surfaces. This force always acts normal to the surface.

Centre of pressure is defined as the point of application of the total pressure on the surface. There are four cases of submerged surfaces on which the total pressure force and centre of pressure is to be determined. The submerged surfaces may be :

 

Ch_4_(131-162).pdf

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4

CHAPTER

4.1

BUOYANCY AND

FLOATATION

INTRODUCTION

In this chapter, the equilibrium of the floating and sub-merged bodies will be considered. Thus the chapter will include : 1. Buoyancy, 2. Centre of buoyancy, 3. Metacentre, 4. Metacentric height,

5. Analytical method for determining metacentric height, 6. Conditions of equilibrium of a floating and sub-merged body, and 7. Experimental method for metacentric height.

4.2 BUOYANCY

When a body is immersed in a fluid, an upward force is exerted by the fluid on the body. This upward force is equal to the weight of the fluid displaced by the body and is called the force of buoyancy or simply buoyancy.

4.3

CENTRE OF BUOYANCY

It is defined as the point, through which the force of buoyancy is supposed to act. As the force of buoyancy is a vertical force and is equal to the weight of the fluid displaced by the body, the centre of buoyancy will be the centre of gravity of the fluid displaced.

Problem 4.1 Find the volume of the water displaced and position of centre of buoyancy for a wooden block of width 2.5 m and of depth 1.5 m, when it floats horizontally in water. The density of wooden block is 650 kg/m3 and its length 6.0 m.

 

Ch_5_(163-258).pdf

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5

CHAPTER

KINEMATICS OF FLOW

AND IDEAL FLOW

A. KINEMATICS OF FLOW

5.1

INTRODUCTION

Kinematics is defined as that branch of science which deals with motion of particles without considering the forces causing the motion. The velocity at any point in a flow field at any time is studied in this branch of fluid mechanics. Once the velocity is known, then the pressure distribution and hence forces acting on the fluid can be determined. In this chapter, the methods of determining velocity and acceleration are discussed.

5.2 METHODS OF DESCRIBING FLUID MOTION

The fluid motion is described by two methods. They are —(i) Lagrangian Method, and (ii) Eulerian

Method. In the Lagrangian method, a single fluid particle is followed during its motion and its velocity, acceleration, density, etc., are described. In case of Eulerian method, the velocity, acceleration, pressure, density etc., are described at a point in flow field. The Eulerian method is commonly used in fluid mechanics.

5.3

TYPES OF FLUID FLOW

The fluid flow is classified as :

 

Ch_6_(256-316).pdf

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6

CHAPTER

6.1

DYNAMICS OF FLUID

FLOW

INTRODUCTION

In the previous chapter, we studied the velocity and acceleration at a point in a fluid flow, without taking into consideration the forces causing the flow. This chapter includes the study of forces causing fluid flow. Thus dynamics of fluid flow is the study of fluid motion with the forces causing flow. The dynamic behaviour of the fluid flow is analysed by the Newton’s second law of motion, which relates the acceleration with the forces. The fluid is assumed to be incompressible and non-viscous.

6.2

EQUATIONS OF MOTION

According to Newton’s second law of motion, the net force Fx acting on a fluid element in the direction of x is equal to mass m of the fluid element multiplied by the acceleration ax in the x-direction.

Thus mathematically,

Fx = m.ax

...(6.1)

In the fluid flow, the following forces are present :

(i) Fg, gravity force.

(ii) Fp, the pressure force.

(iii) Fv, force due to viscosity.

(iv) Ft, force due to turbulence.

(v) Fc, force due to compressibility.

 

Ch_7_(317_354).pdf

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7

CHAPTER

7.1

ORIFICES AND

MOUTHPIECES

INTRODUCTION

Orifice is a small opening of any cross-section (such as circular, triangular, rectangular etc.) on the side or at the bottom of a tank, through which a fluid is flowing. A mouthpiece is a short length of a pipe which is two to three times its diameter in length, fitted in a tank or vessel containing the fluid.

Orifices as well as mouthpieces are used for measuring the rate of flow of fluid.

7.2

CLASSIFICATIONS OF ORIFICES

The orifices are classified on the basis of their size, shape, nature of discharge and shape of the upstream edge. The following are the important classifications :

1. The orifices are classified as small orifice or large orifice depending upon the size of orifice and head of liquid from the centre of the orifice. If the head of liquid from the centre of orifice is more than five times the depth of orifice, the orifice is called small orifice. And if the head of liquids is less than five times the depth of orifice, it is known as large orifice.

 

Ch_8_(355-385).pdf

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8

CHAPTER

8.1

NOTCHES AND

WEIRS

INTRODUCTION

A notch is a device used for measuring the rate of flow of a liquid through a small channel or a tank. It may be defined as an opening in the side of a tank or a small channel in such a way that the liquid surface in the tank or channel is below the top edge of the opening.

A weir is a concrete or masonary structure, placed in an open channel over which the flow occurs.

It is generally in the form of vertical wall, with a sharp edge at the top, running all the way across the open channel. The notch is of small size while the weir is of a bigger size. The notch is generally made of metallic plate while weir is made of concrete or masonary structure.

1. Nappe or Vein. The sheet of water flowing through a notch or over a weir is called Nappe or Vein.

2. Crest or Sill. The bottom edge of a notch or a top of a weir over which the water flows, is known as the sill or crest.

8.2

CLASSIFICATION OF NOTCHES AND WEIRS

The notches are classified as :

1. According to the shape of the opening :

 

Ch_9_(387-432).pdf

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9

VISCOUS FLOW

CHAPTER

9.1

INTRODUCTION

This chapter deals with the flow of fluids which are viscous and flowing at very low velocity. At low velocity the fluid moves in layers. Each layer of fluid slides over the adjacent layer. Due to relative du du velocity between two layers the velocity gradient exists and hence a shear stress t = m acts on dy dy the layers.

The following cases will be considered in this chapter :

1. Flow of viscous fluid through circular pipe.

2. Flow of viscous fluid between two parallel plates.

3. Kinetic energy correction and momentum correction factors.

4. Power absorbed in viscous flow through

(a) Journal bearings, (b) Foot-step bearings, and (c) Collar bearings.

9.2

FLOW OF VISCOUS FLUID THROUGH CIRCULAR PIPE

For the flow of viscous fluid through circular pipe, the velocity distribution across a section, the ratio of maximum velocity to average velocity, the shear stress distribution and drop of pressure for a given length is to be determined. The flow through the circular pipe will be viscous or laminar, if the

 

Ch_10_(433-464).pdf

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10

TURBULENT FLOW

CHAPTER

10.1

INTRODUCTION

The laminar flow has been discussed in chapter 9. In laminar flow the fluid particles move along straight parallel path in layers or laminae, such that the paths of individual fluid particles do not cross those of neighbouring particles. Laminar flow is possible only at low velocities and when the fluid is highly viscous. But when the velocity is increased or fluid is less viscous, the fluid particles do not move in straight paths. The fluid particles move in random manner resulting in general mixing of the particles. This type of flow is called turbulent flow.

A laminar flow changes to turbulent flow when (i) velocity is increased or (ii) diameter of a pipe is increased or (iii) the viscosity of fluid is decreased. O. Reynold was first to demonstrate that the rVD

. transition from laminar to turbulent depends not only on the mean velocity but on the quantity m rVD

This quantity is a dimensionless quantity and is called Reynolds number (Re). In case of circular m pipe if Re < 2000 the flow is said to be laminar and if Re > 4000, the flow is said to be turbulent. If

 

Ch_11_(465-558).pdf

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11

CHAPTER

11.1

FLOW THROUGH

PIPES

INTRODUCTION

In chapters 9 and 10, laminar flow and turbulent flow have been discussed. We have seen that when the Reynolds number is less than 2000 for pipe flow, the flow is known as laminar flow whereas when the Reynolds number is more than 4000, the flow is known as turbulent flow. In this chapter, the turbulent flow of fluids through pipes running full will be considered. If the pipes are partially full as in the case of sewer lines, the pressure inside the pipe is same and equal to atmospheric pressure. Then the flow of fluid in the pipe is not under pressure. This case will be taken in the chapter of flow of water through open channels. Here we will consider flow of fluids through pipes under pressure only.

11.2 LOSS OF ENERGY IN PIPES

When a fluid is flowing through a pipe, the fluid experiences some resistance due to which some of the energy of fluid is lost. This loss of energy is classified as :

This is due to friction and it is calculated by the following formulae :

 

Ch_12_(559-610).pdf

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12

CHAPTER

12.1

DIMENSIONAL AND

MODEL ANALYSIS

INTRODUCTION

Dimensional analysis is a method of dimensions. It is a mathematical technique used in research work for design and for conducting model tests. It deals with the dimensions of the physical quantities involved in the phenomenon. All physical quantities are measured by comparison, which is made with respect to an arbitrarily fixed value. Length L, mass M and time T are three fixed dimensions which are of importance in Fluid Mechanics. If in any problem of fluid mechanics, heat is involved then temperature is also taken as fixed dimension. These fixed dimensions are called fundamental dimensions or fundamental quantity.

12.2 SECONDARY OR DERIVED QUANTITIES

Secondary or derived quantities are those quantities which possess more than one fundamental dimension. For example, velocity is denoted by distance per unit time (L/T), density by mass per unit

M volume 3 and acceleration by distance per second square (L/T2). Then velocity, density and accelL

 

Ch_13_(611-656).pdf

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13

CHAPTER

13.1

BOUNDARY LAYER

FLOW

INTRODUCTION

When a real fluid flows past a solid body or a solid wall, the fluid particles adhere to the boundary and condition of no slip occurs. This means that the velocity of fluid close to the boundary will be same as that of the boundary. If the boundary is stationary, the velocity of fluid at the boundary will be zero.

Farther away from the boundary, the velocity will be higher and as a result of this variation of velocity, du the velocity gradient will exist. The velocity of fluid increases from zero velocity on the stationary dy boundary to free-stream velocity (U) of the fluid in the direction normal to the boundary. This variation of velocity from zero to free-stream velocity in the direction normal to the boundary takes place in a narrow region in the vicinity of solid boundary. This narrow region of the fluid is called boundary layer. The theory dealing with boundary layer flows is called boundary layer theory.

According to boundary layer theory, the flow of fluid in the neighbourhood of the solid boundary may be divided into two regions as shown in Fig. 13.1.

 

Ch_14_(657-692).pdf

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14

CHAPTER

14.1

FORCES ON SUB-MERGED

BODIES

INTRODUCTION

When a fluid is flowing over a stationary body, a force is exerted by the fluid on the body.

Similarly, when a body is moving in a stationary fluid, a force is exerted by the fluid on the body.

Also when the body and fluid both are moving at different velocities, a force is exerted by the fluid on the body. Some of the examples of the fluids flowing over stationary bodies or bodies moving in a stationary fluid are :

1. Flow of air over buildings,

2. Flow of water over bridges,

3. Submarines, ships, airplanes and automobiles moving through water or air.

14.2

FORCE EXERTED BY A FLOWING FLUID ON A STATIONARY BODY

Consider a body held stationary in a real fluid, which is flowing at a uniform velocity U as shown in

Fig. 14.1.

FL

FR

FD

U

STATIONARY

BODY

Fig. 14.1 Force on a stationary body.

The fluid will exert a force on the stationary body. The total force (FR) exerted by the fluid on the body is perpendicular to the surface of the body. Thus the total force is inclined to the direction of motion. The total force can be resolved in two components, one in the direction of motion and other perpendicular to the direction of motion.

 

Ch_15_(683-736).pdf

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15

COMPRESSIBLE FLOW

CHAPTER

15.1

INTRODUCTION

Compressible flow is defined as that flow in which the density of the fluid does not remain constant during flow. This means that the density changes from point to point in compressible flow. But in case of incompressible flow, the density of the fluid is assumed to be constant. In the previous chapters, the fluid was assumed incompressible, and the basic equations such as equation of continuity, Bernoulli’s equation and impulse momentum equations were derived on the assumption that fluid is incompressible. This assumption is true for flow of liquids, which are incompressible fluids. But in case of flow of fluids, such as

(i) flow of gases through orifices and nozzles,

(ii) flow of gases in machines such as compressors, and

(iii) projectiles and airplanes flying at high altitude with high velocities, the density of the fluid changes during the flow. The change in density of a fluid is accompanied by the changes in pressure and temperature and hence the thermodynamic behaviour of the fluids will have to be taken into account.

 

Ch_16_(737-802).pdf

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16

CHAPTER

16.1

FLOW IN OPEN

CHANNELS

INTRODUCTION

Flow in open channels is defined as the flow of a liquid with a free surface. A free surface is a surface having constant pressure such as atmospheric pressure. Thus a liquid flowing at atmospheric pressure through a passage is known as flow in open channels. In most of cases, the liquid is taken as water. Hence flow of water through a passage under atmospheric pressure is called flow in open channels. The flow of water through pipes at atmospheric pressure or when the level of water in the pipe is below the top of the pipe, is also classified as open channel flow.

In case of open channel flow, as the pressure is atmospheric, the flow takes place under the force of gravity which means the flow takes place due to the slope of the bed of the channel only. The hydraulic gradient line coincides with the free surface of water.

16.2

CLASSIFICATION OF FLOW IN CHANNELS

The flow in open channel is classified into the following types :

1. Steady flow and unsteady flow,

 

Ch_17_(803-852).pdf

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17

CHAPTER

17.1

IMPACT OF JETS AND

JET PROPULSION

INTRODUCTION

The liquid comes out in the form of a jet from the outlet of a nozzle, which is fitted to a pipe through which the liquid is flowing under pressure. If some plate, which may be fixed or moving, is placed in the path of the jet, a force is exerted by the jet on the plate. This force is obtained from

Newton’s second law of motion or from impulse-momentum equation. Thus impact of jet means the force exerted by the jet on a plate which may be stationary or moving. In this chapter, the following cases of the impact of jet i.e., the force exerted by the jet on a plate, will be considered :

1. Force exerted by the jet on a stationary plate when

(a) Plate is vertical to the jet, (b) Plate is inclined to the jet, and (c) Plate is curved.

2. Force exerted by the jet on a moving plate, when

(a) Plate is vertical to the jet, (b) Plate is inclined to the jet, and (c) Plate is curved.

17.2

FORCE EXERTED BY THE JET ON A STATIONARY VERTICAL PLATE

Consider a jet of water coming out from the nozzle, strikes a flat vertical plate as shown in Fig. 17.1

 

Ch_18_(853-944).pdf

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18

CHAPTER

18.1

HYDRAULIC

MACHINES — TURBINES

INTRODUCTION

Hydraulic machines are defined as those machines which convert either hydraulic energy (energy possessed by water) into mechanical energy (which is further converted into electrical energy ) or mechanical energy into hydraulic energy. The hydraulic machines, which convert the hydraulic energy into mechanical energy, are called turbines while the hydraulic machines which convert the mechanical energy into hydraulic energy are called pumps. Thus the study of hydraulic machines consists of study of turbines and pumps. Turbines consists of mainly study of Pelton turbine, Francis Turbine and

Kaplan Turbine while pumps consist of study of centrifugal pump and reciprocating pumps.

18.2

TURBINES

Turbines are defined as the hydraulic machines which convert hydraulic energy into mechanical energy. This mechanical energy is used in running an electric generator which is directly coupled to the shaft of the turbine. Thus the mechanical energy is converted into electrical energy. The electric power which is obtained from the hydraulic energy (energy of water) is known as Hydroelectric power.

 

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