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Fluid Mechanics and Fire Hydraulics

This book is a part of the course by Jaipur National University, Jaipur.This book contains the course content for Fluid Mechanics and Fire Hydraulics.

JNU, JaipurFirst Edition 2013

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Index

ContentI. ...................................................................... II

List of FiguresII. ........................................................... V

List of TablesIII. ......................................................... VII

AbbreviationsIV. ......................................................VIII

Case StudyV. .............................................................. 114

BibliographyVI. ......................................................... 118

Self Assessment AnswersVII. ................................... 120

Book at a Glance

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Contents

Chapter I ....................................................................................................................................................... 1Fluid Properties ............................................................................................................................................ 1Aim ................................................................................................................................................................ 1Objectives ...................................................................................................................................................... 1Learning outcome .......................................................................................................................................... 11.1 Introduction .............................................................................................................................................. 21.2 Matters-Solid, Liquid and Gases .............................................................................................................. 21.3 Fluid Properties ........................................................................................................................................ 2 1.3.1 Mass Density (ρ) ...................................................................................................................... 2 1.3.2 Specific Weight (Γ) .................................................................................................................. 3 1.3.3 Relative Density (RD) or Specific Gravity (S) ........................................................................ 3 1.3.4 Viscosity ................................................................................................................................... 3 1.3.5 Surface Tension – (σ) = force per unit length .......................................................................... 4 1.3.6 Capillarity ................................................................................................................................ 71.4 Types of Fluids ....................................................................................................................................... 101.5 Properties of Gases ................................................................................................................................ 12Summary ..................................................................................................................................................... 13References ................................................................................................................................................... 13Recommended Reading ............................................................................................................................. 13Self Assessment ........................................................................................................................................... 14

Chapter II ................................................................................................................................................... 16Dimensional Analysis ................................................................................................................................. 16Aim .............................................................................................................................................................. 16Objectives .................................................................................................................................................... 16Learning outcome ........................................................................................................................................ 162.1 Introduction ............................................................................................................................................ 172.2 Dimension .............................................................................................................................................. 172.3 Dimensional Homogeneity .................................................................................................................... 192.4 Dimensional Analysis ............................................................................................................................ 20 2.4.1 Rayleigh Method .................................................................................................................... 20 2.4.2 Buckingham’s π Theorem ...................................................................................................... 212.5 Similitude ............................................................................................................................................... 222.6 Significance of Dimensionless Number ................................................................................................. 232.7 Types of Models ..................................................................................................................................... 252.8 Geometric Distortion in River Models .................................................................................................. 26Summary ..................................................................................................................................................... 27References ................................................................................................................................................... 27Recommended Reading ............................................................................................................................. 27Self Assessment ........................................................................................................................................... 28

Chapter III .................................................................................................................................................. 30Fluid Statics ................................................................................................................................................ 30Aim .............................................................................................................................................................. 30Objectives .................................................................................................................................................... 30Learning outcome ........................................................................................................................................ 303.1 Fluid Pressure......................................................................................................................................... 313.2 Variation of Static Pressure .................................................................................................................... 323.3 Absolute and Gauge Pressure ................................................................................................................. 343.4 Pressure Measurement ........................................................................................................................... 35 3.4.1 Simple Manometers ............................................................................................................... 35 3.4.2 Differential Manometer ......................................................................................................... 373.5 Pressure on Plane Surfaces .................................................................................................................... 39

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3.5.1 Total Pressure on Horizontal Plane Surface ........................................................................... 39 3.5.2 Total Pressure on Vertical Plane Surface ............................................................................... 40 3.5.3 Total Pressure on Inclined Plane Surface ............................................................................... 413.6 Total Pressure on Curved Surface .......................................................................................................... 423.7 Pressure Diagram ................................................................................................................................... 43Summary ..................................................................................................................................................... 44References ................................................................................................................................................... 44Recommended Reading ............................................................................................................................. 44Self Assessment ........................................................................................................................................... 45

Chapter IV .................................................................................................................................................. 47Fluid Kinematics ........................................................................................................................................ 47Aim .............................................................................................................................................................. 47Objectives .................................................................................................................................................... 47Learning outcome ........................................................................................................................................ 474.1 Introduction ............................................................................................................................................ 484.2 Description of Fluid Motion .................................................................................................................. 48 4.2.1 Langrangian Method .............................................................................................................. 48 4.2.2 Eulerian Method .................................................................................................................... 484.3 Velocity .................................................................................................................................................. 48 4.3.1 Steady and Unsteady Flows ................................................................................................... 49 4.3.2 Uniform and Non-Uniform Flows ......................................................................................... 49 4.3.3 Laminar and Turbulent Flows ................................................................................................ 49 4.3.4 Compressible and Incompressible Flows .............................................................................. 50 4.3.5 Rotational and Irrotational Flows .......................................................................................... 50 4.3.6 One, Two and Three-Dimensional Flows .............................................................................. 504.4 Flow Patterns ......................................................................................................................................... 51 4.4.1 Stream Line ............................................................................................................................ 51 4.4.2 Path Line ................................................................................................................................ 52 4.4.3 Streak Line ............................................................................................................................. 52 4.4.4 Streamtube ............................................................................................................................. 534.5 Continuity Equation for Three Dimensional Flow in Cartesian Coordinate ......................................... 534.6 Continuity Equation for One Dimensional Flow ................................................................................... 554.7 Velocity and Acceleration ...................................................................................................................... 564.8 Rotational and Irrotational Flows .......................................................................................................... 58 4.8.1 Components of Rotation ........................................................................................................ 59 4.8.2 Circulation ............................................................................................................................. 604.9 Velocity Potential (Ø)............................................................................................................................. 614.10 Stream Function(ψ) .............................................................................................................................. 624.11 Relation between Stream Function and Velocity Potential .................................................................. 634.12 Flow Nets ............................................................................................................................................. 63Summary ..................................................................................................................................................... 65References ................................................................................................................................................... 65Recommended Reading ............................................................................................................................. 65Self Assessment ........................................................................................................................................... 66

Chapter V .................................................................................................................................................... 68Dynamics of Fluid Flow ............................................................................................................................. 68Aim .............................................................................................................................................................. 68Objectives .................................................................................................................................................... 68Learning outcome ........................................................................................................................................ 685.1 Introduction ............................................................................................................................................ 695.2 Forces Acting on Fluid in Motion .......................................................................................................... 695.3 Eulers Equation of Motion for Three Dimensional Flow (Cartesian Coordinates) ............................... 715.4 Euler’s Equation of Motion along a Stream Line .................................................................................. 74

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5.5 Bernoulli’s Equation from Euler’s Equation for (Incompressible Fluid) .............................................. 755.6 Bernoulli’s Equation for Steady Compressible Flow ............................................................................. 755.7 Significance of Various Terms in Bernoulli’s Theorem ......................................................................... 765.8 Limitations of Bernoulli’s Equation ....................................................................................................... 775.9 Modifications of Bernoulli’s Equations ................................................................................................. 775.10 Hydraulic Gradient Line and Total Energy Line .................................................................................. 775.11 Energy Correction Factor ..................................................................................................................... 785.12 Applications of Bernoulli’s Equation ................................................................................................... 78Summary ..................................................................................................................................................... 84References ................................................................................................................................................... 84Recommended Reading ............................................................................................................................. 84Self Assessment ........................................................................................................................................... 85

Chapter VI .................................................................................................................................................. 87Flow Through Orifices ............................................................................................................................... 87Aim .............................................................................................................................................................. 87Objectives .................................................................................................................................................... 87Learning outcome ........................................................................................................................................ 876.1 Introduction ............................................................................................................................................ 886.2 Classification of Orifices ....................................................................................................................... 886.3 Experimental Determination of Hydraulic Coefficients ........................................................................ 896.4 Flow through Submerged Orifice .......................................................................................................... 916.5 Time of Emptying a Tank ....................................................................................................................... 946.6 Flow over Notches ................................................................................................................................. 956.7 Elbow Meter or Bend Meter .................................................................................................................. 956.8 Rotameter ............................................................................................................................................... 96Summary ..................................................................................................................................................... 97References ................................................................................................................................................... 97Recommended Reading ............................................................................................................................. 97Self Assessment ........................................................................................................................................... 98

Chapter VII .............................................................................................................................................. 100The Boundary Layer Theory .................................................................................................................. 100Aim ............................................................................................................................................................ 100Objectives .................................................................................................................................................. 100Learning outcome ...................................................................................................................................... 1007.1 Introduction .......................................................................................................................................... 1017.2 Boundary Layer Thickness (δ) ............................................................................................................. 1017.3 Boundary Layer Profile over a Flat Plate ............................................................................................ 1047.4 Laminar Boundary Layer ..................................................................................................................... 1057.5 Turbulent Boundary Layer ................................................................................................................... 1077.6 Laminar Sub Layer .............................................................................................................................. 1077.7 Separation of Boundary Layer ............................................................................................................. 1087.8 Methods of Controlling Boundary Layer ............................................................................................. 1097.9 Hydro-dynamically Smooth and Rough Boundaries ............................................................................110Summary ....................................................................................................................................................111References ..................................................................................................................................................111Recommended Reading ............................................................................................................................111Self Assessment ..........................................................................................................................................112

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List of Figures

Fig. 1.1 Three forms of matter- solid, liquid and gases ................................................................................. 2Fig. 1.2 Viscosity of fluid............................................................................................................................... 3Fig. 1.3 Various forces on fluid molecule – surface tension .......................................................................... 5Fig. 1.4 Forces on the water droplet .............................................................................................................. 6Fig. 1.5 Pressure ............................................................................................................................................. 6Fig. 1.6 Pressure inside a liquid jet ................................................................................................................ 7Fig. 1.7 Capillary rise in water ....................................................................................................................... 7Fig. 1.8 Capillarity in water and mercury ...................................................................................................... 8Fig. 1.9 Stress vs. Volumetric strain ............................................................................................................... 9Fig. 1.10 Graphical representation of velocity vs. Shear stress- types of fluid ............................................11Fig. 3.1 Forces on the element in fluid at rest .............................................................................................. 32Fig. 3.2 Forces on static fluid element ......................................................................................................... 32Fig. 3.3 Pressure at a point ........................................................................................................................... 33Fig. 3.4 Pressure at the base of container of different shapes ...................................................................... 34Fig. 3.5 Absolute and gauge pressure .......................................................................................................... 34Fig. 3.6 Piezometer ...................................................................................................................................... 35Fig. 3.7 U tube manometer to measure small pressure ................................................................................ 36Fig. 3.8 U tube manometer to measure large pressure ................................................................................. 36Fig. 3.9 U tube differential manometer ........................................................................................................ 37Fig. 3.10 Micromanometer .......................................................................................................................... 38Fig. 3.11 Inclined manometer ...................................................................................................................... 39Fig. 3.12 Pressure on horizontal plane ......................................................................................................... 39Fig. 3.13 Pressure on vertical plane ............................................................................................................. 40Fig. 3.14 Pressure on inclined plane ............................................................................................................ 41Fig. 3.15 Pressure on curved surface ........................................................................................................... 42Fig. 3.16 Pressure diagram ........................................................................................................................... 43Fig. 4.1 Velocity of a fluid particle .............................................................................................................. 48Fig. 4.2 Streamline ....................................................................................................................................... 51Fig. 4.3 Path line .......................................................................................................................................... 52Fig. 4.4 Streakline ........................................................................................................................................ 52Fig. 4.5 Streamtube ...................................................................................................................................... 53Fig. 4.6 Continuity equation for three dimensional flow ............................................................................. 53Fig. 4.7 Continuity equation for one dimensional flow ............................................................................... 55Fig. 4.8 Fluid displacement .......................................................................................................................... 58Fig. 4.9 Rotation .......................................................................................................................................... 59Fig. 4.10 Force acting on body immersed in fluid ....................................................................................... 60Fig. 4.11 Limitation of flow net ................................................................................................................... 64Fig. 5.1 Types of forces that influence the fluid motion .............................................................................. 70Fig. 5.2 Forces acting as the parallelepiped ................................................................................................. 71Fig. 5.3 Flow in a stream tube ...................................................................................................................... 74Fig. 5.4 Hydraulic gradient line ................................................................................................................... 77Fig. 5.5 Simple pitot tube ............................................................................................................................. 78Fig. 5.6 Venturimeter ................................................................................................................................... 80Fig. 5.7 Orifice meter ................................................................................................................................... 82Fig. 6.1 Types of orifices ............................................................................................................................. 88Fig. 6.2 Hydraulic coefficients with a tank filled with water ...................................................................... 90Fig. 6.3 Totally submerged orifice ............................................................................................................... 91Fig. 6.4 Partially submerged orifice ............................................................................................................. 92Fig. 6.5 Large rectangular orifice................................................................................................................. 93Fig. 6.6 Discharge time of water through orifice ......................................................................................... 94Fig. 6.7 Elbow meter .................................................................................................................................... 95Fig. 7.1 Types of boundary layer ............................................................................................................... 101Fig. 7.2 Boundary layer thickness .............................................................................................................. 102

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Fig. 7.3 Boundary layer thickness .............................................................................................................. 104Fig. 7.4 Graphical representation of zones ................................................................................................ 107Fig. 7.5 Velocity distributed for flow over a curved surface...................................................................... 108Fig. 7.6 Separation of flow region for energy ............................................................................................ 109Fig. 7.7 Development of boundary layer flow in pipe ................................................................................110

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List of Tables

Table 1.1 Fluid characteristics ..................................................................................................................... 12Table 2.1 Units and dimensions of different quantities ............................................................................... 18

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Abbreviations

SI - Système International d’unités (International System of Units)Kg - Kilogramm - MeterK - KelvinN - NewtonRD - Relative densityS - Specific gravityLHS - Left hand sideRHS - Right hand sideRe - Reynold’s numberFr - Froude NumberMa - Mach NumberEu - Euler NumberWe - Weber Number+ve - positiveCP - Centre of PressureH.G.L - Hydraulic Gradient LineT.E.L - Total Energy LineL.B.L - Laminar Boundary layerT.B.L - Turbulent Boundary LayerCf - local drag coefficient

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Chapter I

Fluid Properties

Aim

The aim of this chapter is to:

explain the nature of fluid•

analyse the properties of fluid•

categorise fluid into different types•

Objectives

The objectives of this chapter are to:

highlight viscosity and explain Newtonian and non-Newtonian fluids•

describe different types of matter•

discuss surface tension•

Learning outcome

At the end of this chapter, you will be able to:

explain the appropriate physical properties•

recognise the properties of gases•

describe the concept of compressibility and bulk modulus of elasticity•


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1.1 IntroductionFluid mechanics is a science which deals with the behaviour of fluids when subjected to a system of forces. It involves the study of statics, kinematics and dynamics of fluids (both, liquids and gases.)

Kinematics: • It deals with space time relationship of fluid flow, i.e., velocity acceleration and rotation of fluid mass.Dynamics:• It deals with certain aspects of fluid flow concerning with force, energy losses and their variation.Statics:• It refers to the study of fluid behavior when it is at rest.

1.2 Matters-Solid, Liquid and GasesMatter can be distinguished into three states: solid, liquid and gaseous. Liquid and gaseous states are combined and are known as fluids.

Collision

CollisionMean

free pathSolid GasLiquid

Fig. 1.1 Three forms of matter- solid, liquid and gases

The three differ from each after due to their molecular structure- spacing between the molecules and the ease with which they move. In solids, molecules are closely spaced and therefore, they have very strong intermolecular attractive forces. They have fixed size and shape. Liquid molecules have comparatively larger spacing and therefore also have weaker intermolecular attractive forces. They occupy a certain volume of the container they don’t have fixed size while in gases, the spacing is very large, therefore, intermolecular attractive forces are extremely small. They don’t have fixed size or shape. They occupy entire volume of the container.

The fluid is a substance having particles which readily change their positions with respect to the body. The fluid deforms continuously under the action of shear force. Fluids involve both, liquids and gases. Fluids lack the ability to resist deformation. It flows under the action of the force. Its shape will change continuously as long as the force is applied. While in solids, when a shear force is applied, it may cause some displacement but the solid does not continue to move indefinitely.

1.3 Fluid PropertiesFollowing are the different properties of fluid:

1.3.1MassDensity(ρ)It is defined as mass per unit volume.Dimension- [M1L-3T0]SI unit- Kg/m3

ρ water = 1000 Kg/m3

ρ Mercury = 13546 Kg/m3

ρ Air = 1.23 Kg/m3

(at pressure = 1.013 X 10-5N/m2, temperature =288.15K )

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1.3.2SpecificWeight(Γ)It is defined as the weight per unit volume.Dimension- [M1L-2T-2]SI unit- N/m3

Г water = 9787 N/m3

Г Mercury = 132943 N/m3

Г air = 12.07 N/m 3

1.3.3RelativeDensity(RD)orSpecificGravity(S)It is defined as the ratio of mass density of a substance to some standard mass density.Dimension – [M0L0T0]SI Unit – No unitFor solids and liquids, this standard mass density is the maximum mass density for water (which occurs at 40c) at atmospheric pressure.S water = 1S mercury =13.5

1.3.4 ViscosityIt is the property of fluid by which it offers resistance to deformation under the action of shear force. As long as shear force remains in existence, fluid undergoes continuous deformation. The rate of deformation depends upon magnitude of shear force. Consider a fluid contained between two parallel plates, the bottom one is stationary while the upper one is moving with a uniform velocity under the action of shear force F as shown in figure below.

Upper LayerLower Layer

dy

y

u+du

du

u

u

Solid boundary

y=b, u=U

y=0, u=00

1 2 3 F

Fig.1.2Viscosityoffluid

Initially, all the fluid particles are at rest. When the top plate starts moving, the particles sticking to it start moving with the same velocity U, while particles adhering to lower plate are at rest. The velocity of particles vary from u = o (at y = 0) to u = U (at y = b). If the gap between the plates is very small, then the velocity distribution will be linear. The fluid particles originally lying on line 01 after a certain time interval t = ∆t, will occupy the positions indicated by the line 02. After t = 2∆t, the fluid particles will be along 03 and so on. The maximum deformation will occur at y=b distance where it is equal to U/b while deformation is zero at y=0

Let θ - angle of deformationu - point velocity at a distance y from the lower plateF - shear force required to move the surface at a constant velocity URate of deformation = Shear stress τ= F/A


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= =

But, the rate of deformation depends upon magnitude of shear stress. Therefore,

τ = ∝

τ = µ

τ = µ

where, µ is constant of proportionality and known as absolute or dynamic viscosity. The equation is known as Newton’s law of viscosity. The term is the change in velocity with distance y and this is known as velocity gradient and may be written in the differential form as . The fluid which obeys this equation is known as Newtonian fluid.

Relation between viscosity and temperatureViscosity µ is the property of a fluid due to cohesion and interaction between molecules, which offers resistance to shear deformation. Different fluids deform at different rates under the same shear stress. In gases, the molecules are widely spaced therefore very negligible cohesion, hence, viscosity is due to the transfer of molecular momentum which increases with increase in temperature. Viscosity of gases increases with increase in temperature, in liquids the viscosity is due to the cohesive forces which decreases with increase in temperature, therefore, viscosity of liquid decreases with increase in temperature.

Absolute or Dynamic Viscosity (µ)

µ=

Dimension –M1L-1T-1

SI unit – N.S/m2

Unit – Poise1Poise – 0.1 N.S/m2

Kinematic Viscosity (v)v = µ/ρDimension – L2T-1

SI Unit – m2/sUnit - stokes1 stokes = 10-4m2/sv water = 0.01 x 10-4 m2 /sv air = 0.146 x 10-4 m2 /sv Hg = 1.145 x 10-4 m2 /s

1.3.5 Surface Tension – (σ)=forceperunitlengthDimension – [MT-2]SI unit - N/mCohesion means intermolecular attraction between molecules of the same liquid. Adhesion means attraction between molecules of a liquid and the molecules of a solid boundary surface in contact with the liquid. Surface tension is caused by the cohesive force at the free surface.

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Fig.1.3Variousforcesonfluidmolecule–surfacetension

Consider a molecule C that is below the liquid surface. Neighboring molecules will attract it equally from all sides; therefore, it will be stable. However, the molecule Y at the free surface will have a resultant downward force perpendicular to the surface, as there are no liquid molecules above the surface to balance the force of the molecules below it. As a result, the surface will be pulled down, causing curvature to the surface, which develops tension in the surface known as ‘surface tension’. It is a force per unit length or surface energy per unit area. It will be same everywhere on the surface. Its magnitude depends upon the relative magnitude of adhesive and cohesive forces. As cohesive force decreases with rise in temperature, the surface tension will also decreases with rise in temperature.

Surface tension of water and mercury in contact with air:Water – air σ= 0.073 N/m at 20oCMercury – air σ = 0.1 N/m

Some important examples of phenomenon of surface tension are as follows:Capillary rise• Water droplet• Break up of liquid jet•

Pressure inside a water droplet: A falling rain drop becomes spherical due to cohesion and surface tension.Let p – Pressure inside the droplet, above the outside pressure d – Diameter of the droplet σ- Surface tension of the liquid

In case of a liquid or water droplet, the action of surface tension is to increase the pressure inside, than that on the outside surface, so that pressure difference is created.


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Fig. 1.4 Forces on the water droplet

Fig. 1.5 Pressure

From free body diagrams (fig.1.5), there are two forces, namely, pressure force and surface tension force, acting along the circumference. Under equilibrium conditions, these two forces will be equal and opposite.

p d2 = σπd p = 4

Liquid Jet: Consider a cylindrical liquid jet of diameter (d), of length (l). Following figure shows a semi-jet, where there are two forces.

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Fig. 1.6 Pressure inside a liquid jet

Pressure force = p x l x dSurface tension force = 2σl

Under equilibrium condition,P x l x d = 2σlp = 2

1.3.6 CapillarityIt is a phenomenon by which a liquid (depending upon its specific gravity) rises into a thin glass tube above or below its general level. This phenomenon is due to the combined effect of cohesion and adhesion of liquid particles. Figure given below shows the phenomenon of rising water in the tube of smaller diameter.

Fig. 1.7 Capillary rise in water


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Let d = diameter of the capillary tube, θ = angle of contact of the water surface h = height of capillary rise σ = surface tension force for unit length and Γ = specific weight

Now upward surface tension force (lifting force) = weight of the water column in the tube (gravity force)

πd σ Cos θ = d2hΓ

∴ h =

This shows that h∝ , i.e, if a tube of smaller diameter is used, capillary rise will be more. Therefore, whenever a glass tube is to be used for pressure measurement in order to give correct pressure, it is necessary that the rise of water or any other liquid should not be influenced by capillary action. To ensure this, the diameter should be large enough so that capillary rise / fall is negligible. Therefore, diameter of tube should not be less than 1 cm.

For water and glass θ > 0Hence, the capillary rise of water in the glass tube

h = 4

In case of mercury, there is a capillary depression (Refer to figure below) and the angle of depression is θ ≈ 140o. [It may be noted that here cos θ = cos140o = cos (180 – 40o) = - cos 40o, therefore, ‘h’ is negative capillary depression].

Fig. 1.8 Capillarity in water and mercury

Following points are important:Smaller the diameter of the capillary tube, greater is the capillary rise or depression.• For the measurement of liquid level in laboratory capillary (glass) tubes should not be smaller than 8 mm.• Capillary effects are negligible for tubes longer than 12 mm.• For wetting liquid (water): θ < • For water: θ = 0 when pure water is in contact with clean glass but as θ increases as high as 25o when water is • slightly contaminated.For non-wetting liquid (mercury): θ > • (For mercury: θ varies between 130o to 150o)

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The effects of surface tension are negligible in many fluid flow problems except those involving.• Capillary rise Formation of drops and bubbles The break up of liquid jets and Hydraulic model studies where the model or flow depth is small.

Compressibility and bulk modulus of elasticityThe property by virtue of which fluids undergo a change in volume under the action of external pressure is known as compressibility. It decreases with the increase in pressure of fluid as the volume modulus increases with the increase of pressure.

The variation in volume of water, with variation of pressure, is so small that for all practical purposes it is neglected. Thus, the water is considered to be an incompressible liquid. However in case of water flowing through pipes when sudden or large changes in pressure (e.g., water hammer) take place, the compressibility cannot be neglected. The compressibility in fluid mechanics is considered mainly when the velocity of flow is high enough reaching 20 percent of speed of sound in the medium.

Fig. 1.9 Stress vs. Volumetric strain

Elasticity of fluids is measured in terms of bulk modulus of elasticity (K) which is defined as the ratio of compressive stress to volumetric strain. Compressibility is the reciprocal of bulk modulus of elasticity.

Consider a cylinder fitted with a piston as shown in the above figure.Let V = volume of gas enclosed in the cylinder and p = pressure of gas when volume is V

= where, A is the area of cross-section of the cylinder.

Let the pressure is increased to p + dp, the volume of gas decreases from V to V - dV Then increase in pressure = dp Decrease in volume dv ∴Volumetric strain =


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(Negative sign indicates decrease in volume with increase of pressure) ∴Bulk modulus of elasticity E or K = Compressibility =

A fluid can be compressed by the application of pressure thereby reducing its volume and giving rise to volumetric strain. This compressed fluid will expand to its original volume when applied pressure is removed. This properly of compressibility of a fluid is expressed by E.

E =

Negative sign indicates decrease in volume with increase in pressure. As most of liquids have very high value of E, therefore b is nearer to zero. Liquids are practically incompressible.

The bulk modulus of elasticity (K) of a fluid is not constant, but it increases with increase in pressure. This is so because when a fluid mass is compressed its molecules become close together and its resistance to further compression increases, i.e., K increases (e.g. the value of K roughly doubles as the pressure is raised from 1 atmosphere to 3500 atmosphere).

The bulk modulus of elasticity (K) of the fluid is affected by the temperature of the fluid. In the case of liquids there is a decrease of K with increase of temperature. However, for gases since pressure and temperature are inter-related and as temperature increases, pressure also increases, and increase in temperature results in an increase in the value of K.

At NTP (normal temperature and pressure)Kwater = 2.07 ´106kN /m2

Kair = 101.3kN/m 2

1.4 Types of FluidsFluids, like most other forms of matter, are made up of tiny particles [molecules], which are separated by large spaces. A fluid is anything that would spill or float away if it weren’t in a container. There are five different types of fluid. They are:

Newtonian Fluidsτ = µ

Fluids which obey Newton’s law of viscosity where value of m is constant are known as Newtonian fluids.e.g. air, water. It is true for most common fluids.

Non Newtonian Fluids

τ = µ

Fluids in which the value of µ is not constant are known as non-Newtonian fluids. In non-Newtonian fluids the relationship between shear stress and the velocity gradient is nonlinear. This relationship can be seen in the graph.

Pseudo plastics τ = µ n<1e.g. Milk, clay, cement, paper pulp

Dilant τ = µ n>1e.g. Quicksand, butter, concentrated sugar solution, printing ink

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Fig.1.10Graphicalrepresentationofvelocityvs.Shearstress-typesoffluid

Each of these lines can be represented by the equation

τ = A + B where A, B and n are constants

PlasticShear stress must reach a minimum value before flow commences. Ideal/Bingham Plastics – They are characterized by the fact that they have a definite yield stress beyond which the shear stress is linearly related with the velocity gradient.

τ = τy + µ

e.g. Sewage sludge, tooth paste

Thixotropic SubstancesThe relationship is nonlinear beyond a certain yield stress.

τ = τy + µ

e.g. Printers ink, paint , enamels

Ideal Fluid τ=0Fluid which does not possess viscosity, compressibility and surface tension is known as ideal fluid. Shear stress is always zero regardless of the motion of fluid. It is represented by the horizontal axis.


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1.5 Properties of GasesGases are easily expandable and compressible unlike solids and liquids. Gases have a measurement of pressure. Pressure is defined as force exerted per unit area of surface. It can be measured in several units such as kilopascals (kPa), atmospheres (atm), and millimeters of Mercury (mmHg). Gas has a low density because its molecules are spread apart over a large volume.

Vapour pressureAll liquid have a tendency to vaporize when expressed to air or atmosphere. This vaporization rate depends upon the molecular energy of the liquid and atmospheric condition.

Consider a liquid contained in a sealed container, inside which a constant temperature is maintained. Some molecules have sufficient energy to leave the liquid surface and enter the air space. After a certain time, the air will contain enough liquid molecules to exert a partial pressure of air on some molecules forcing them to re-enter the liquid surface. When equilibrium is established, the rate at which molecules leave the surface is same as the rate at which the molecules re-enter the surface. In this condition, the air is saturated with liquid vapour molecules and the pressure exerted by air on the liquid surface is vapour pressure. The boiling of a liquid is closely related with vapour pressure. When pressure impressed on the liquid surface is below the vapour pressure, liquid starts boiling. Thus boiling can be achieved either by raising the temperature of the liquid so that its vapour pressure rises or by lowering the pressure of the overlying air below the vapour pressure of the liquid.

PV=nRTfor a perfect gasIsothermal process – PV = ConstantP1V1 = P2V2 = P3V3 = …… ………. = PnVn

Adiabatic Isentropic: If the process is such that no heat is added or withdrawn from the gas and no friction is involvedPVk = ConstantWhere, k= Cp/Cv

Fluid characteristics are shown in the following table:

Sl.No. Characteristics Symbol units

1 Mass Density ρ Kg/m3

2 Specific Weight Γ N/m3

3 Specific Volume m3/N

4 Specific Gravity, Relative Density S No Unit

5 Dynamic Viscosity µ N.s/m2

6 Kinematic Viscosity v m2/s

7 Bulk Modulus of Elasticity K, E N/m2

8 Surface Tension σ N/m

Table 1.1 Fluid characteristics

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SummaryFluid mechanics involves the study of statics, kinematics and dynamics of fluids.• Matter can be distinguished into three states: solid, liquid and gaseous.• The characteristics of fluid are mass density, relative density, specific weight and specific gravity.• Viscosity is the property of fluid by which it offers resistance to deformation under the action of shear force.• Surface tension is caused by the cohesive force at the free surface.• The fluid undergoes deformation under the action of shear force. The shear stress • τ is related with the rate of deformation, which is expressed by Newton’s law of viscosityτ = µ

Fluids, like most other forms of matter, are made up of tiny particles [molecules], which are separated by large • spaces. A fluid is anything that would spill or float away if it weren’t in a container. There are five different types of fluids; Newtonian fluids, Non-Newtonian fluids, plastic, thixotropic substances and ideal fluid. When the pressure is applied, fluid may be compressed. The bulk modulus of elasticity is given as,•

K or E = -

ReferencesProperties of gases. • [Online] Available at: <http://library.thinkquest.org/19957/gaslaws/propertiesgasbody.html>. [Accessed on 8 April 2011.]Types of Fluid Motion.• [Online] Available at: <http://www.adl.gatech.edu/classes/lowspdaero/lospd2/lospd2.html>. [Accessed on 8 April 2011.]Fluids Mechanics and Fluid Properties.• [Online] Available at: <http://www.efm.leeds.ac.uk/CIVE/CIVE1400/PDF/Notes/section1.pdf>. [Accessed on 8 April 2011.]

Recommended ReadingMenzies, J. Kay and Nedderman, R. M, 1985. • Fluid mechanics and transfer processes, 2nd ed., CUP Archive. Som and Biswas, K, 2008. • Intro to Fluid Mechanics, 2nd ed., Tata McGraw-Hill Education.Streeter, V. Lyle, 1958. Fluid mechanics, 2• nd ed., Tata McGraw-Hill Education.


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Self AssessmentThe fluid undergoes ___________ under the action of shear force.1.

viscosity a. compressibilityb. surface tensionc. deformationd.

Fluid which does not possess viscosity, compressibility and surface tension is known as ____________.2. plastica. Non Newtonian fluidb. ideal fluidc. Newtonian fluidd.

Real fluids are __________.3. compressiblea. non-compressibleb. viscousc. non-viscousd.

Formula of Kinematic viscosity is __________________.4. v = ρa. µ

v =b.

v = c.

v =d.

What is the unit of dynamic viscosity?5. N/ma. 2

N/mb. 2sNs/mc. 2

Nmd. 2/s

Match the following:6.

Newtonian Fluids1. A. τ = τy + µ

Non Newtonian Fluids2. B. τ = τy + µ

Plastic3. C. τ = µ

Thixotropic substances4. D. τ = µ

1-d, 2-c, 3-b, 4-aa. 1-c, 2-a, 3-d, 4-bb. 1-b, 2-a, 3-d, 4-cc. 1-d, 2-a, 3-b, 4-cd.

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___________ is easily expandable and compressible.7. Solida. Liquidb. Gasc. Vapourd.

In which fluids the relationship between sheer stress and the velocity gradient is nonlinear?8. Newtoniana. Non-Newtonianb. Dilatentc. Ideald.

The viscosity of liquids ______________ with increase in temperature.9. decreasesa. increasesb. first decreases and then increasesc. first increases and then decreasesd.

________________ is a phenomenon where a liquid (depending upon its specific gravity) rises into a thin glass 10. tube above or below its general level.

Surface tensiona. Capillarityb. Cohesionc. Adhesiond.


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Chapter II

Dimensional Analysis

Aim


describe the concept of dimensional analysis•

discuss the concept of dimension•

introduce the concept of dimensional homogeneity•

Objectives


analyse the concept of similitude•

highlight Rayleigh method•

explain Buckingham’s II Theorem•

Learning outcome


describe dimension less parameter•

analyse the significance of the dimensionless numbers•

esxplain the types of models•

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2.1 IntroductionIn engineering, the application of fluid mechanics in designs makes much of the use of empirical results sought from a lot of experiments. This data is often difficult to present in readable form. Even from graphs, it may be difficult to interpret. Dimensional analysis provides a strategy for choosing relevant data and for deciding how it should be presented.

This is useful technique in all experimentally based areas of engineering. If it is possible to identify the factors involved in a physical situation, dimensional analysis can form a relationship between them. The resulting expressions may not at first sight appear rigorous, but these qualitative results converted to quantitative forms can be used to obtain any unknown factors from experimental analysis.

With rapid development of various branches of engineering, scientists these days come across various complex problems in fluid mechanics, structural analysis etc. These problems can easily be solved by carrying out dimensional analysis based upon similitude technique. It is carried out in three stages:

Identification of more predominant variables for that fluid problems and application of dimensional analysis • for grouping the variables to form dimension less number.Achieving complete similarity by realising relative importance of dimensionless groups.• Actual execution of the similarity criteria in making a geometric model, deciding the kinematical condition • and carrying out tests.

This technique is used to:test dimensional homogeneity of any equation• derive equation expressed in terms of dimensionless parameter• construct model and carry out model test• analyse complex flow problems•

2.2 DimensionIn all branches of science, we come across various terms like acceleration, force, pressure etc. These can be expressed in terms of only three quantities, which are known as primary quantities. These are Mass (M), length (L) and time (T) which are independent variables. The expression is called dimension of that quantity.

In dimensional analysis, we are only concerned with the nature of the dimension, i.e., its quality not its quantity. The following common abbreviations are used.Length = LMass = MTime = T


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Quantity SI Unit Dimension

Area L2 M0L2T0

Volume m3 M0L3T0

Slope - -

Velocity m/s M0L1T-1

Angular Velocity rad/s M0L0T-1

Acceleration m/s2 M0L1T-2

Angular Acceleration rad/s2 T-2

Force N M1L1T-2

Energy Nm M1L2T-2

Power Nm/s M1L2T-3

Pressure N/m2 M1L-1T-2

Torque Nm M1L2T-2

Weight N M1L1T-2

Mass density kg/m3 M1L-3T0

Specific Weight N/m3 M1L-2T-2

Specific Gravity - -

Dynamic Viscosity Ns/m2 M1L-1T-1

Kinematic Viscosity m2/s M0L2T-1

Momentum Kg. m/s M1L1T-1

Discharge m3/s M0L3T-1

Relative density - -

Surface tension N/m M1L0T-2

Modulus of Elasticity N/m2 M1L-1T-2

Table 2.1 Units and dimensions of different quantities

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2.3 Dimensional HomogeneityFourier’s principle states that any equation which represents some fluid flow problem must be algebraically correct and dimensionally homogenous. Dimensionally homogenous equation is that equation in which the dimension of LHS terms is same as the dimension of RHS in terms of the equation. Some of the examples are as follows:

For the period of oscillation of simple pendulum,•

t = 2π where, t is time, λ is length and g is gravitational acceleration.Dimension of LHS = [M0L0T1]

Dimension of RHS 2π = [M0L0T0] [L/M0L1T-2] (1/2) = [M0L0T1]

Dimension of LHS = Dimension of LHSTherefore, it is dimensionally a homogenous equation.

For laminar flow through a circular pipe,• ∆P = where, µ is dynamic viscosity, V is a volume of the liquid poured and D is the diameterDimension of LHS ∆P = M1L-1T-2

Dimension of RHS 32 μ V L/D2 = [M1L-1T-1] [M0L1T-1] [L]/ [L2]= [M1L-1T-2]Dimension of LHS = Dimension of LHSTherefore, it is dimensionally homogenous equation.

This property of dimensional homogeneity can be useful for:checking units of equations• conversion between two sets of units• defining dimensionless relationships•

These equations are useful, if an equation is dimensionally homogenous, then the value of constant is same in all systems of units.

For example,

t = 2π

wheret is timeλ is length g is gravitational acceleration

= 2π = C

whereC = Constant

All dimensionally homogenous equations can be expressed in the form of equation involving only dimensionless parameter.

∆P = 32 µ VL/D2 where, ∆P is the pressure drop, µ is dynamic viscosity, V is a volume, L is the length and D is the diameter.


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= 32

M0L0T0 = M0 L0T0

The number of dimensionless groups that can be formed, is equal to the number of variables involved in the equation minus the number of primary units.

For flow over a weir,Q = Cd. H3/2 where, Q = flow rate (m3/s), H = head on the weir (m), g = 9.81 (m/s2) - gravity,

Cd = discharge constant for the weir - must be determined

Variables involved = 4

Q- L3T -1, g- LT -2

No. of Primary units: L, T, = 2

Dimensionless groups: Cd, = 2

2.4 Dimensional AnalysisDimensional Analysis is a tool to find or check relations among physical quantities, by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass, length, time, electric charge, and temperature) which describes it; for example, speed has the dimension length/time, and may be measured in meters per second, miles per hour, or in other units. Dimensional analysis is based on the fact, that a physical law must be independent of the units used to measure the physical variables. Dimensional analysis has two theorems. They are as follows:

2.4.1 Rayleigh MethodThis method was first proposed by Rayleigh for determining the effect of temperature on the viscosity. In this method, a relationship of some variables is expressed in the form of an exponential equation which is dimensionally homogenous equation.

Suppose we want to write a functional relation between variables x1, x 2 …...xn

Then x1 = f n(x2x3….. xn)

∴ x1= C. …..

Where C is constant whose value can be determined from experiment and the value of exponent a, b…n is determined on the basis that equation is dimensionally homogenous is equation.

For laminar flow in a pipe, the drop in pressure • ∆p is a function of pipe length (l), diameter (d), mean velocity (u) of flow and dynamic viscosity of fluid (µ ).Use Rayleigh method to obtain an expression for ∆p∆p = fn (1, d, u,µ)∆p = C λadbucµd

ML -1T -2 = [M0L0T0] [L]a [L]b [LT-1]c[M1L-1T-1]d

ML-1T-2=MD La+b+c-d T-c-d

∴ d = 1, a = -1-b, c=1

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∆p = C. l (-1-b) u1 db μ1

∆p = C (u μ/l) (d/l) b

∆p = (u μ/l) Ø (d/l)Find an expression for the drag force on sphere of diameter D, moving with velocity V in a fluid of density r • and dynamic viscosity (µ).F = f n (D, V, ρ, μ)F = C. Da Vbρc μd

MLT -2 = [Mo LoTo] [La] [LT -1] b [ML-3] c [M1L-1T-1] d

MLT -2 = Mc+d L a+b-3c-d T -b-d

c=1-d, a=2-d, b=2-d∴ F=C. D2-d V (2-d) ρ (1-d) m d

F= C. ρV2 D2 (μ/ρ V D) d

F = ρV2 D2 Ø (μ / ρ V D)

But this method is useful only when the number of variables is up to 3 or 4, otherwise it is very difficult to find an expression for dependent variable. Therefore, Buckingham’s π theorem method is used.

2.4.2 Buckingham’s π TheoremIt states that if there are ‘n’ dimensional variables involved in a phenomenon which can be completely described by ‘m’ primary quantities or dimensions and are related by a dimensionally homogenous equation, then the relationship among these ‘n’ variables can always be expressed by exactly (nm) dimensionless and independent π terms.

Mathematically, if Q1 is the variable which depends upon number of variables like Q 2, Q3……Qn thenQ1 = fn (Q2, Q3……Qn)fn (Q1, Q2 , Q3……Qn) = constant

That is, if there are ‘n’ variables Q1, Q2,Q3……Qn and they can be expressed in terms of ’m’ variables Q1,Q2,Q3…. .Q m, then in accordance with π theorem, a dimensionally homogenous equation is obtained as,fn, (P1 P2.... Pn-m) = Constant

Each π term is obtained by combining ‘m’ variables out of ‘n’ variables with remaining (n-m) variables, i.e.,

π1 = ( ……… )

π2 = ( ……… )

πn-m = ( ……… )

So, these variables which occur repeatedly in each of these equations are called repeating variables. Each equation is dimensionally homogenous equation. The exponents a, b are determined by considering that the each equation is dimensionally homogenous equation. The final expression can be obtained by expressing any one π term in terms of the remaining one.

Steps involved in Buckingham’s π theorem:List all the variables in the given phenomenon. Note the dimension of each variable. Therefore, number of • primary units, m is known so that there will be (n-m) π terms.Select ‘m’ variables as repeating variables. These variables should be such that:•

they should not be dimensionless no two variables have the same dimension They themselves don’t form dimension less parameter.


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Guidelines to select there variablesSelect the first variable representing geometry of flow; length, width and depth Select the second variable representing fluid property; mass density, viscosity and surface tension Select the second variable representing fluid motion; velocity, discharge, pressure and force.

Write the general equation for each • π term. The π terms are expressed as a product of the repeating variables, each raised to some unknown power and one of the remaining variables, taken in turn, raised to known power, usually one.Compute the values of exponent by equating the exponents of the respective functional dimension on the both • sides of each of the dimensionally homogeneous equation and write different π termsWrite the final expression in term of • π terms.

Manipulation of the π groupsIdentified manipulation of the π groups is permitted. These manipulated do not change the number of groups involved, but may change their appearance drastically.Taking the defining equation as: Ø (π1 π2 π3 ……πm-n) = 0

Then the following manipulations are permitted:Any number of groups can be combined by multiplication or division to form a new group which replaces one • of the existing. For example, π1 and π2 may be combined to form π1a= π1 x π2 so the defining equation becomes Ø (π1a, π2, π3 ……πm-n) = 0The reciprocal of any dimensionless group is valid.• So, Ø (• π1, , π3, ….. )=0 is valid.Any dimensionless group may be raised to any power.• So, Ø • , , …….. = 0 is valid. Any dimensionless group may be multiplied by a constant.• Any group may be expressed as function of the other groups.• For example, • π2 = Ø (π1, π3, πm-n)In general, the defining equation could look like: • Ø (• π1, , π2, ……….. 0.5 πn-m) = 0

2.5 SimilitudeBefore undertaking any expensive heavy engineering project or structure, scientists are always interested to know how the structure will behave after it is constructed. In order to get such information, generally models are constructed and tests are carried on them. Results obtained from these tests give the desired information which can be used to modify the structure if it is necessary. It also helps to check the validity of assumptions made in the design of certain hydraulic structure. It is always necessary to find out whether the assumptions made are on the safer side or not and also to ascertain whether the structure so designed will serve the desired purpose.

Since it is not always possible to have same fluid flow as it is in the prototype or to have the model of same size as that by the prototype, (to transfer, the test results) it is necessary to have similarity between the two systems. For complete similarity to exist between model and prototype, the two systems must be geometrically, kinenamatically and dynamically similar.

Geometric similarityIt is obtained when the solid boundaries that control the fluid flow are geometrically similar and the ratios of corresponding length dimension in model and in prototype are same.

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Length scale ratio

= = lr

Area scale ratio

= = Ar =

Volume scale ratio

(Lm X bm X dm) / (Lm X bm X d m) = Vr = lr3

Kinematic similarityIt signifies the similarity of fluid motion. Such similarity exists if

The paths of homogenous moving particles are geometrically similar.• The ratio of velocity and acceleration of the homogenous moving particles are equal. Such similarity can be • obtained if flow nets for the two systems are geometrically similar. Then the velocity and acceleration vectors at the corresponding points must point in the same direction .

Velocity scale ratio

v1 = = = x =

Acceleration scale ratio

ar = = = =

Dynamic similarityIt is the similarity of forces. For flows to be dynamically similar, identical types of forces must be parallel and must bear the same ratio at all the corresponding points. For flows to be dynamically similar, the corresponding sides of the two force polygon must be parallel and equal.

2.6SignificanceofDimensionlessNumberIf the numbers of variables affecting a flow problem are known, then they can be arranged into some dimensionless parameter by dimension analysis and then by experiment the factors which are less important can be dropped out. We get the parameter which has greater influence on that phenomenon. Then the dynamic or complete similarity between model and prototype can be obtained by making these parameters same for both the systems.

Reynold’sNumber(Re)Re = = Fi = mass x acceleration = ma = ρL2V2

Fv = µ VLRe = Re = where, Re = Reynolds Number (non-dimensional), ρ = density, μ = dynamic viscosity, L = characteristic length, ν = kinematic viscosity.

In all problem where viscous resistance to flow occurs, Reynolds number is important. For example, flow through pipe, resistance offered by submarines, airplane and so on.


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Depending upon Reynolds’s no., the flow is classified as:Laminar Re < 2000, transitional Re – 2000 to 4000 and turbulent Re >4000.

Reynolds Model Law

(Re)m = (Re)p

(ρm Vm Lm/µm) / (ρm Vm Lm/µm)

(ρr Vr Lr/µr) =1

∴Vr = =

tr = = ρr

ar = = x =

Qr = arear Vr = =

FroudeNumber(Fr)Fg = ρgL3Fi = mass x acceleration = ma = ρL2V2

Fr = Depending upon Froude’s number, the flow is classified as subcritical Fr<1, critical Fr=1, supercritical Fr>1For example, flow through an open channel.

Froude Model Law

=

= 1

Vr =

Vr =

Tr =

Qr = x = gr

Fr = ar. mr = ρr gr

Qr = x =

MachNumber(Ma)

Ma = =

Ma =

Cauchy No. =

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Depending upon Mach number, the flow is classified as subsonic flow Ma<1, sonic flow Ma=1 and supersonic flow Ma>4.

For example, Phenomenon where velocity of flow is greater then velocity of sound: water Hammer

EulerNo(Eu)

Eu =

WeberNo(We)

We = = =

We =

2.7 Types of ModelsFollowing are the types of models:Undistorted model: Undistorted model is that which is geometrically similar to its prototype, i.e., scale ratios for corresponding linear dimension of the model and its prototype are same. Because of this, the results obtained from model tests can be directly transferred to its prototype.

Distorted model: In these models, one or more terms of the model are not identical with their corresponding counterparts in the prototype. Since the basic condition of perfect similitude is not satisfied, the results obtained from the distorted model are liable to distortion. It may have geometrical, material or distortion of hydraulic qualities or a combination of these.

In geometric distortion, distortion is of dimension or that of configuration. Many times, different scale ratios are adopted for the longitudinal, vertical transverse dimension , then it is geometrical distortion. Mainly it is used in river models where a different scale ratio for depth is adopted in vertically exaggerated models. Vertical scales are greater than horizontal scales.

Material distortion takes place when the physical properties of the corresponding materials in the models and its prototype don’t satisfy the similitude condition. Material distortion is also used in river models constructed for the sediment transport study.

Distorted models are adopted to:maintain accuracy in vertical measurement.• maintain turbulent flow• obtain suitable bed material & its adequate movements.• obtain suitable roughness condition• accommodate the available facilities such as space, money, time.•

Models with free surfaceRivers, estuaries etc. are the models with free surface. When modeling rivers and other fluid with free surfaces, the effect of gravity becomes important and the major governing non-dimensional number becomes the Froude (Fr) number. The resistance to motion formula would then be derived with gas an extra dependent variables to give an extra p group. So the defining equation is:Ø(R, ρ, u, l, µ, g) = 0


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From the dimensional analysis gives:

R = ρu2l2Ø ( )

R = ρu2l2Ø (Re, Fn)

Generally the prototype will have a very large Reynolds number, in which case slight variation in Re causes little effects on the behaviour of the problem. Unfortunately, models are sometimes so small and the Reynolds numbers are large and the viscous effects take effect. This situation should be avoided to achieve correct results. Solutions to this problem would be to increase the size of the model – or more difficult – to change the fluid (i.e. change the viscosity of the fluid) to reduce the Reynolds number.

2.8 Geometric Distortion in River ModelsWhen river models are to be built, it is very difficult to choose a suitable scale for the model and to keep geometric similarity. A model which has a suitable depth of flow will often vary. Reducing the size and retaining geometric similarity can give tiny depth where viscous force comes into play. These results in the following problems:

To measure accurate depths and depth changes become very difficult to measure;• The bed roughness of the channel becomes impracticably small;• Sometimes laminar flow may result – (turbulent flow is normal in river hydraulics.)•

The solution often adopted to overcome these problems is to abandon strict geometric similarity by having different scales in the horizontal and the vertical.

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SummaryFourier’s principle states that any equation which represents some fluid flow problem must be algebraically • correct and dimensionally homogenous. Dimensionally homogenous equation is that equation in which the dimension of LHS terms is same as the • dimension of RHS terms of the equation.Dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. • The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass, length, • time, electric charge, and temperature) which describe it; for example, speed has the dimension length / time, and may be measured in meters per second, miles per hour, or other units. Dimensional analysis is based on the fact that a physical law must be independent of the units used to measure • the physical variables. Dimensional analysis has two theorems; Rayleigh method, Buckingham’s π theoremIf the numbers of variables affecting a flow problem are known, then they can be arranged into some dimensionless • parameter by dimension analysis and then by experiment the factors which are less important can be dropped out. We get the parameter which has greater influence on that phenomenon. Then the dynamic or complete similarity between model and prototype can be obtained by making these parameters same for both the systems.

ReferencesRayleigh’s method of dimensional analysis• [Online] (Updated on 15 March 2011) Available at: http://en.wikipedia.org/wiki/Rayleigh%27s_method_of_dimensional_analysis. [Accessed on 12 April 2011.]2004. • Buckingham’s pi-theorem [Online] Available at: http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf. [Accessed on 12 April 2011.]Reynolds number• [Online] (Updated on 11 April 2011) Available at: <http://en.wikipedia.org/wiki/Reynolds_number >. [Accessed on 12 April 2011.]

Recommended ReadingGibbings, J.C., 2011. • Dimensional Analysis, 1st ed., Springer.Szirtes, T., 2006. • Applied Dimensional Analysis and Modeling, 2 ed., Butterworth-Heinemann.Bridgman, P. W., 2001.• Dimensional Analysis, General Books LLC.


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Self AssessmentA dimensionally homogenous equation is applicable to ______________.1.

only C.G.S systema. only F.P.S systemb. M.K.S and SI systemc. all system of unitsd.

Dimensional analysis is a tool to find or check relations among __________ by using their dimensions.2. repeating variablesa. geometric variablesb. physical quantitiesc. N dimensionless parametersd.

What is dynamic similarity? 3. Similarity of motiona. Similarity of lengthsb. Similarity of forcesc. Similarity of heightd.

What does kinematic similarity signifies?4. Similarity of dischargea. Similarity of shapeb. Similarity of fluid motionc. Similarity of sized.

Match the following.5.

1. Geometric Similarity A. Geometrically similar to its prototype.

2. Kinematic Similarity B. For this, the identical types of forces must be parallel and must bear the same ratio at all the corresponding points.

3. Dynamic Similarity C. The paths of homogenous moving particles are geometrically similar.

4. Undistorted modelD. It is obtained when the solid boundaries that control the fluid flow are

geometrically similar and the ratios of corresponding length dimension in model and in prototype are same.

1-D, 2-C, 3-B, 4-Aa. 1-C, 2-D, 3-A, 4-Bb. 1-B, 2-A, 3-D, 4-Cc. 1-D, 2-A, 3-B, 4-Cd.

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In all problem where viscous resistance to flow occurs, __________ is important.6. Froude’s numbera. Weber’s numberb. Reynold’s numberc. Mach’s numberd.

In which of the following, one or more terms of the model are not identical with their corresponding counterparts 7. in the prototype?

Undistorted modela. Material distortionb. Distorted modelc. Froude’s numberd.

When modeling rivers and other fluid with free surfaces, the effect of gravity becomes important and the major 8. governing non-dimensional number becomes the _______________.

gravity forcesa. elastic forcesb. Mach’s numberc. Froude (Fr) numberd.

Whic of the following is an example of 'Models with free surface'?9. Estuariesa. Bridgeb. Gravityc. Viscous forcesd.

Which of the following uses Material distortion?10. River modelsa. Dams b. Harboursc. Bridgesd.


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Chapter III

Fluid Statics

Aim


describe pressure at a point•

introduce the concept of fluid pressure•

discuss variation of static pressure•

Objectives


highlight manometers•

analyse the concept of absolute and gauge pressure•

explain measurements of pressure•

Learning outcome


explain the concept of pressure on plane surfaces•

recognise total pressure on curved surface•

illustrate pressure with the help of diagram•

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3.1 Fluid PressurePrinciples of fluid statics find application in engineering, related to forces acting on submerged bodies such as gates, submarines, dams etc. and in the analysis of stability of floating bodies such as ships.

Consider a small area ∆A in a large mass of fluid. The fluid surrounding this area exerts a force on that area. If the fluid is stationary, this force is always perpendicular to the surface. If ∆F is the elemental force acting on area ∆A in the normal direction, then

p =

where p is known as intensity of pressure or simple pressure. The unit of pressure is N/m2. If the force ∆F is not perpendicular to ∆A, it will have a component parallel to area ∆A- a shear stress. But a shear stress cannot exist unless there is a relative motion between different fluid layers. Therefore, p acts normal to the surface.

Pressure at a pointPressure is used to specify the usual force per unit area at a given point, acting on a specified plane within the fluid mass of interest. The question is how this pressure varies as we change the orientation of the plane around this point. We can answer this question by studying the following.

Pascal’s LawIt states that the pressure at a point in a fluid at rest is same in all the directions, i.e., when a pressure is applied at any point in a fluid at rest. The pressure is equally transmitted in all the directions and to every other point in the fluid. To prove this, consider an infinitesimal wedge shaped fluid element at rest as a free body.

Since the fluid is at rest and τ=μdu/dy, whatever may be the viscosity, τ= 0. Therefore, only two forces are acting. Pressure acts normal to the surface and gravitational force acts vertically downward. As the element is in equilibrium, the sum of forces in any direction is equal to zero.

= M ax =0; = M ay =0; = M az =0

i. = px δy δz - ps δy δz Sin θ = 0 δz = δs Sin θ = px δy δz - ps δy δz = 0

(px - ps) δy δz = =0

px = ps

ii. = 0 ∴-↓Ps.Cosθ.δsδy + Pz.δxδy - γ But δs Cosθ = δx

Pz. δxδy - Ps. δxδy - γ = 0


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Fig.3.1Forcesontheelementinfluidatrest

Since the choice of wedge was completely arbitrary it can be seen that pressure at any point is the same in all the directions.

But δxδyδz →0 to find pressure at a point,∴Pz. δxδy - Ps. δxδy = 0∴(Pz – Ps) δxδy = 0∴Pz – Ps = 0∴ Pz = Ps =Px Px=Py=Pz=Ps

3.2 Variation of Static Pressure

Fig.3.2Forcesonstaticfluidelement

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Consider a fluid element of size dx, dz and of unit length. Let p be static pressure at the center of the element 0(x, z). The fluid is in equilibrium under the action of pressure and gravity force.

Pressure on AB = i. dz

CD = dz

BC = dx

AD = dx

Weight of the element = ii. γ.dx.dz.1For equilibrium, the sum of force in x and z direction must be zero.iii. iv. = 0

∴ dz - dz = 0

∴ dxdz = 0

as dx and dz can’t be zero; = 0 i.e., there is no pressure variation in X direction.

v. = 0

dx - dx - γ.dx.dz.1 = 0

dxdz - γ.dx.dz = =0

- = γ

= γ

p1 -p2 =Г ( - )

+ z1 = + z2 p = -γ z + constant

Fig. 3.3 Pressure at a point


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At z=H p=0 gauge pressure0 = -γ z + constantPressure at a point∴Constant = γH∴P=-γ Z+γ H∴P=γ (H-Z) =γh

P=γhwhere h is measured from the free surface. The equation is known as hydrostatic law of pressure variation. P = γ h shows that pressure varies linearly with the distance of point from the free surface and it doesn’t depend upon the shape of the bounding container.

Fig. 3.4 Pressure at the base of container of different shapes

The above figure shows four containers of different shapes having same base area and are interconnected and have the same height of liquid. The pressure of the base of container is rh.P = Pa + γ hP-Pa = γ hF = (P-Pa) AF = γ hA

Asγ, h and A are same, the hydrostatic force on the bottom is same regardless of their shape. (Ah) representing volume of prism and γ Ah weight of liquid prism. Hydrostatic force on the bottom is equal to the weight of the liquid prism over the bottom.

3.3 Absolute and Gauge Pressure

Fig. 3.5 Absolute and gauge pressure

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Pressure can be expressed in two different systems. The pressure measured above the absolute zero is absolute pressure and the pressure measured when expressed as difference between its value and local atmospheric pressure, is gauge pressure. If the pressure is below local atmospheric pressure, then it is Vacuum or negative gauge or suction pressure.

Gauge pressure = Absolute pressure – Local atmospheric pressureGauge pressure or vacuum = Local atmospheric pressure – Absolute pressure

The standard atmospheric pressure is the mean pressure at sea level and it is 10.3 m of water or 76 cm of mercury.

3.4 Pressure MeasurementThe principle of measurement of pressure of a liquid is to balance the pressure by means of column of another liquid. The apparatus used for pressure measurement is known as manometer. The device in which pressure is balanced by the force of spring or dead weight is known as mechanical pressure gauge.

ManometersThere are two types of manometers; simple manometer and differential manometer.

3.4.1 Simple ManometersIt consists of glass tube of specific diameter, with one end connected to a point where pressure is to be measured and other end open to the atmosphere.

Piezometer

Fig. 3.6 Piezometer

It is used for measuring moderate pressure. It consists of a glass tube inserted in the wall of pipe or a vessel containing a liquid whose pressure is to be measured. The pressure at a depth h below the free surface of a liquid of sp wt Г is Гh. The tube is installed at a point or near the point where pressure is to be measured. The height of liquid raised in the glass tube measured above the point is a measure of pressure at that point. PA = γ h. These tubes are known as Piezometers. They measure only gauge pressure as they are open to atmosphere. Piezometer may be inserted either in the top or the side or the bottom of the container even then rise of liquid level in the tube will be same.

Piezometer is not used for measuring small pressures, as it does not give accurate results. Also, it is not used for measuring high pressures, since height of the tube required will be more and will not be easy to handle. The internal diameter of the tube should not be less than 12 mm. Otherwise, capillary rise will affect the height of liquid column. Also diameter should not be more than 25 mm.


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U Tube ManometerThe U tube manometer is used to measure small, moderate and high pressure at point in a fluid at rest or in motion. It is used to measure both gauge and vacuum pressures. It consists of a glass tube of U shape with one of its end connected to the point where pressure is to be measured and other open to the atmosphere.

If the same liquid is used to measure small positive or negative pressure, then the rise or fall of the liquid in the tube above or before the reference point gives the positive or negative pressure as shown in figure below.

Fig. 3.7 U tube manometer to measure small pressure

To measure greater positive or negative pressure, the U portion of the tube is filled with heavier liquid like mercury immiscible with the liquid of which pressure is to be measured as shown in figure below.

M M

Fig. 3.8 U tube manometer to measure large pressure

For positive pressure (as shown in figure above)Pm + Г w S1 h1 - Г w S2 h2 = 0Pm = Г w S2 h2 - Г w S1 h1 = S2 h2 - S1 h1 m of water

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For negative pressure (as shown in fig. above)Pm + Г w S1 h1 + Г w S2 h2 = 0Pm = -Г w S2 h2 - Г w S1 h1 = -S2 h2 - S1 h1 m of water

3.4.2 Differential ManometerIt is used to measure the difference in pressure between two points in a pipe or in two different pipes. The choice of manometric liquid depends upon the range of pressures to be measured. For large pressure differences, heavy liquid like mercury is used while for small pressure difference carbon tetrachloride is generally used. For determination of unknown pressure or difference of pressure, the following procedure can be adopted.

To write gauge equation:Start from either A or from free surface in the open end of the manometer.• The pressure at the free surface in the open end is atmospheric pressure and can be taken as zero gauge • pressure.Then find the change in pressure as proceeding from one level to another adjacent level of contact of liquids of • different specific gravity. Use positive sign if the next level is lower than first and vice versa.Continue process until the other end of the gauge is reached and equate the expression to the pressure at other • end.

U tube differential manometer

Fig. 3.9 U tube differential manometer

PA + γw.S(y + x) - γw.Sm x - γw .Sy = PB

= -xS-yS+x.Sm+yS

= x

= x m of liquid of specific gravity S

Inverted ∩ tube differential monometerFor inverted ∩ tube differential manometer an immiscible lighter fluid, namely, oil or air is used as manometric fluid. As the manometric fluid is lighter, a large deflection can be obtained even for small pressure difference.


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PA - γw.S(y + x) + γw.Sm x + γw .Sy = PB

= yS+xS+x.Sm-yS

= x

= x m of liquid of specific gravity S.

Micromanometer

Fig. 3.10 Micromanometer

For measuring very small pressure difference or for the measurement of pressure difference with very high precision, micro manometers are used. It consists of U shaped manometer with two tanks. The heavier liquid S1>S2 fills lower portion and lighter liquid upper portion of the tube. These two liquids of different specific gravity are immiscible with each other and with the fluid for which pressure difference is to be measured. Before the manometer is connected to pressure points at A and B the manometric liquids of Specific gravity S1 and S2 are at DD’ and CC’ respectively. When the manometer is connected to pressure points (if PA > PB), then manometric liquid S2 < S1 will fall in the left tank and rise in right tank by the same amount ∆Z. Similarly, manometric liquid S1 will also fall in left and rise in right tank by the same amount.

A (∆z) = a 2∆z =

A and a are c/s area of the tank and tube respectively. Then, the volume of the fluid displaced in the tank is equal to volume of the fluid displaced in the tube.

The quantities S1, S2, a, A, S3 are constant for a particular manometer. Therefore, by measuring x the pressure difference between any two points can be found out.

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Inclined manometerIt is used for precise measurement of small pressures in low velocity gas flow. When the opening of tank C is exposed to the atmosphere, the inclined glass tube has the liquid level at O. When the small pressure is applied to the tank liquid, the level of the liquid in the inclined tube rises.

Fig. 3.11 Inclined manometer

Let l is inclined distancep Sγ hw Sγw l Sinθ γ lSinθγ–sp is wt of manometric liquid. For a small pressure, it is not possible to measure h with reasonable precision but as l is comparatively larger, it can be observed.

3.5 Pressure on Plane SurfacesWhen a static mass of fluid comes in contact with a surface either plane/curved, a force is exerted by the fluid on the surface, which is known as total pressure. When fluid is at rest there are no shear forces. Therefore total pressure acts perpendicular to the surface and the point of application of total pressure on the surface is known as center of pressure.

3.5.1 Total Pressure on Horizontal Plane Surface

Fig. 3.12 Pressure on horizontal plane

Consider a plane surface held in a horizontal position at a depth h below free surface of the liquid. Since every point on the surface is at the depth of h below the free surface, the pressure intensity p = rh is constant over the entire plane surface. If A is total area of the surface then the total pressure on the horizontal surface P = rhA. The total pressure passes through the CG of the area. Therefore CG coincide CP, where CG – centre of gravity, CP – centre of pressure.


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3.5.2 Total Pressure on Vertical Plane SurfaceConsider a plane surface of arbitrary shape, total area A wholly submerged in a static fluid of sp wtγ. Let CG is at depth of below free surface of the liquid.

Fig. 3.13 Pressure on vertical plane

Since depth of liquid varies from point to point on the surface, the pressure intensity is not uniform. Therefore total pressure on the surface can be determined by dividing the entire surface into a number of small parallel strips of width b and computing total pressures on each of these strips. A summation of all these total pressures on the strips gives the total pressure on the entire surface.

Consider a horizontal strip of thickness dx and width b lying at a depth of x below the free surface. Therefore p = γ x (assuming uniform pressure intensity)Total pressure on strip dP = p.dA = γ x.bdxP = = γ = γ

-first moments of the areas of the strip at axis OX= AP = γA

Since Ax is the pressure intensity at the CG of the surface area, it can be stated that the total pressure on a plane surface is equal to the product of the surface area and pressure intensity at CG of the area.

Centre of PressureLet vertical depth of the CP below the free surface.dp = γ .x(b.dx) and its moment at OX = dp.x =γ x2 (bdx)

Considering a no. of strips and summing the moments of the total pressure on these strips at OX=γ .bdxbut the moments of the total pressure on the entire plane surface at OX=P

Therefore, according to moment principle, the moment of the resultant of a system of forces at an axis is equal to the sum of moments of its components at the same axis,∴P = γ .bdxbut ò x2 .bdx … 2nd moment of area of the strip at axis OX =IoIo= ò x2 .bdx ……. M I of the plane surface at OX\ P = γ Io

but according to parallel axis theoremIo = IG=Ax

-2

IG - M I of the area at an axis passing through CG & parallel to ‘OX’

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∴ =

= +

i.e., > or CP is always below CG of the area as is always positive.Similarly,

3.5.3 Total Pressure on Inclined Plane Surface

Fig. 3.14 Pressure on inclined plane

dp = γ.xdA= γ.ySinθ.dAP = γ .Sinθ.= γ .Sinθ.A= γ .ATaking Moments at Oy = dp.y = γy 2 Sin θdAP.Y = γ Sinθ dAP.Y = γ.Sinθ dAYp = Ioy


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3.6 Total Pressure on Curved SurfaceConsider a curved surface ABC wholly submerged in a static mass of liquid of sp.wt.r. At any point on the curved surface, the pressure acts normal to the surface. If dA is the area of a small element of the curved surface at a depth of h below the free surface of a liquid, then dp=pdA=rhdA is the total pressure on the elementary area which is acting normal to the elementP = =

However, in the case of a curved surface, the direction of the total pressure on the elementary area varies from point to point. Therefore integration is not possible. The total pressure can be found out by resolving P on the curved surface into the horizontal and vertical component.

Fig. 3.15 Pressure on curved surface

dP = ГhdAdPH = dPSinθdPH=ГhdA SinθdPV = dP CosθdPV = Гh dA Cosθwhere θ is the inclination of the elementary area with the horizontal

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PH = = PH = =

But, dA Sinθ- vertical protection of the elementary area dAand dA Cosθ- horizontal protection of the elementary area dA

PH = = represent the total pressure on the protected area of the curved surface on a vertical plane. It will act at the CP of the plane surface.

PH = = represents the total pressure on the horizontal projection of the elementary area dA. It is equal to the weight of the liquid contained in the portion extending above the elementary area upto the free surface & Pv will act through CG of volume of liquid contained in that prism.

3.7 Pressure DiagramA pressure diagram is a graphical representation of variation of intensity of pressure over a surface. It is constructed by plotting pressure intensities at various points on the surface to some convenient scale. Pressure diagrams for horizontal, vertical, and inclined surfaces are shown in the figure given below.

Fig. 3.16 Pressure diagram


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SummaryPrinciples of fluid statics find application in engineering, related to forces acting on submerged bodies such as • gates, submarines, dams etc. and in the analysis of stability of floating bodies such as ships. If the fluid is at rest, there are no sheer stresses.Pressure is the force per unit area applied in a direction perpendicular to the surface of an object.• Pascal’s law for a fluid at rest the pressure is same in all directions. It states that the pressure at a point in a fluid • at rest is same in all the directions, i.e., when a pressure is applied at any point in a fluid at rest. The pressure is equally transmitted in all the directions and to every other point in the fluid.Pressure is used to specify the usual force per unit area at a given point, acting on a specified plane within the • fluid mass of interest. Pressure at a point, P = γh.Differential Manometer is used to measure the difference in pressure between two points in a pipe or in two • different pipes. The choice of manometric liquid depends upon the range of pressures to be measured. For large pressure differences, heavy liquid like mercury is used while for small pressure difference carbon tetrachloride is generally used. U tube differential manometer, = x

References2007 • Fluid mechanics [Online] Available at: <http://www.gaussianmath.com/fluidmech/pressurepoint/pressurepoint.html>. [Accessed on 14 April 2011.]Pressure• Measurement By Manometer [Online] Available at: < http://www.cartage.org.lb/en/themes/sciences/physics/mechanics/fluidmechanics/statics/Measurement/Measurement.htm>. [Accessed on 14 April 2011.]Fluid mechanics• – theory [Online] Available at: < https://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=fl&chap_sec=02.3&page=theory> [Accessed on 14 April 2011.]

Recommended ReadingGiles, R., Liu, C., Evett, J., 1994. • Schaum’s Outline of Fluid Mechanics and Hydraulics, 3rd ed, McGraw-Hill.Finnemore, E., Franzini, F.,2001. • Fluid Mechanics with Engineering Applications, 10th ed., McGraw-Hill Science/Engineering/Math.Zeytounian, R., 2009. • Convection in Fluids: A Rational Analysis and Asymptotic Modelling (Fluid Mechanics and Its Applications), 1st ed., Springer.

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Self Assessment______________is the force per unit area applied in a direction perpendicular to the surface of an object.1.

Pressurea. Strainb. Surface tensionc. forced.

Which of the following is a graphical representation of variation of intensity of pressure over a surface?2. Pressure and inclined planea. Pressure diagramb. Pressure on curved surfacec. Pressure on plane surfaced.

For measuring very small pressure difference or for the measurement of pressure difference with very high 3. precision, _______________are used.

micro manometersa. mechanical gaugesb. U-tube manometersc. differential manometersd.

The pressure measured above the absolute zero is_____________4. .absolute pressurea. gauge pressureb. atmospheric pressurec. pressure on plane surfaced.

_________________ states that the pressure at a point in a fluid at rest is same in all the directions.5. Kirchhoff’s Lawa. Pascal’s Lawb. Archimedes Lawc. Newton’s Lawd.


1. Pressure at a point A. Absolute pressure – Local atmospheric pressure

2. Pascal’s LawB. It consists of glass tube of specific diameter, with one end connected

to a point where pressure is to be measured and other end open to the atmosphere.

3. Gauge pressure C. force per unit area

4. Simple manometers D. the pressure at a point in a fluid at rest is same in all the directions

1-C, 2-D, 3-A, 4-Ba. 1-D, 2-A, 3-B, 4-Cb. 1-B, 2-D, 3-A, 4-Cc. 1-C, 2-A, 3-D, 4-Bd.


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The standard atmospheric pressure is the mean pressure at _____________.7. surface levela. altitudeb. water levelc. sea leveld.

Which of the following is used to measure the difference in pressure between two points in a pipe or in two 8. different pipes?

Pascala. Stokeb. Differential manometerc. U-tube manometerd.

Which of the following apparatus is used to measure pressure?9. Mechanical gaugea. Poiseb. Stokec. Manometerd.

The simplest form of manometer which is used to measure moderate pressures of liquid is ___________.10. Piezometera. Differential manometerb. U-tube manometerc. d. ∩-tube manometer

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Chapter IV

Fluid Kinematics

Aim


describe the concept of fluid motion•

introduce different types of flow patterns•

discuss continuity equation•

Objectives


highlight the concept of velocity and acceleration•

explain Langrangian and Eulerian method•

analyse rotational and irrotational flow•

Learning outcome


explain circular rotation•

elaborate flow net•

discuss the concept of velocity function and stream function•


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4.1 IntroductionKinematics is defined as the branch of science, which deals with motion of particles without considering the forces causing the motion. The motion of fluid is analysed on lines similar to one used in solid mechanics.

While studying the motion of fluid, the motion of individual fluid particle is to be considered. Because of extreme mobility of fluid particles, it is necessary to observe the motion of fluid particles at various points in space and at successive instants of time.

4.2 Description of Fluid MotionThe motion of fluid particles may be described by the following methods:

4.2.1 Langrangian MethodIn this method, the observer concentrates on the movement of a single particle. The path taken by the particle and the changes in its velocity and acceleration are studied. This method entails the following shortcomings:

Cumbersome and complex• The equations of motion are very difficult to solve and the motion is complex to understand.•

4.2.2 Eulerian MethodIn Eulerian method, the acceleration and observer concentrates on a point in the fluid system. Velocity, acceleration and other characteristics of the fluid at that particular point are studied. This method is almost exclusively used in fluid mechanics, especially because of its mathematical simplicity. In fluid mechanics, we are not concerned with the motion of each particle, but we study the general state of motion at various points in the fluid system.

4.3 VelocityIt is defined as the ratio of the distance traveled by the fluid particle in unit time interval.

Vs =

It is a vector quantity. The direction of velocity is along the tangent to the flow path. However, the distance Ds can be resolved into 3 components: Dx, Dy and Dz, along the three co-ordinate axes. Let u, v and w be velocity vectors along x, y and z direction respectively.

Fig.4.1Velocityofafluidparticle

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This shows that velocity is a function of the position of a fluid particle and time.u=f1(x,y,z,t)v=f2(x,y,z,t)w=f3(x,y,z,t)

In the vector notation = u + v + w where , and are unit vectors along x, y, z direction respectively. Depending upon whether the velocity and other flow properties are function of space and time, the fluid flows are classified as follows.

4.3.1 Steady and Unsteady FlowsSteady flow is defined as that type of flow in which the fluid characteristics like velocity, pressure, density, etc. at a point do not change with time. Thus, for steady flow, mathematically we have(∂v/∂t) at (x0,y0,z0) =0(∂p/∂t) at (x0,y0,z0)=0(∂δ/∂t) at (x0,y0,z0)=0Where (x0,y0,z0) is a fixed point in fluid field.

Unsteady flow is that type of flow, in which the velocity, pressure or density at a point changes with respect to time. Thus, mathematically, for unsteady flow,(∂v/∂t) at (x0,y0,z0) ≠0(∂p/∂t) at (x0,y0,z0)≠0(∂δ/∂t) at (x0,y0,z0)≠0

4.3.2 Uniform and Non-Uniform FlowsUniform flow is defined as that type of flow in which velocity at any given time does not change with respect to space (i.e., length of direction of the flow. Mathematically, for uniform flow(∂v/∂s) at (t0) =0∂V=Change of Velocity∂ S=Length of flow in the direction S.

Non-uniform flow is that type of flow in which the velocity at any given time changes with respect to space. Thus, mathematically, for non-uniform flow,(∂v/∂s) at (t0) ≠0

4.3.3 Laminar and Turbulent FlowsLaminar flow is defined as the type of flow in which the fluid particles move along well-defined paths or streamline and all the streamlines are straight and parallel. Thus, the particles move in laminas or layers gliding smoothly over the adjacent layer. This type of flow is also called streamline flow or viscous flow.

Turbulent flow is the type of flow in which the fluid particles move in a zigzag way. Due to movement of fluid particles, eddies formation take place which are responsible for high energy loss. For a pipe flow, the type of flow is determined by a non-dimensional number (Re = ) called Reynolds number.

where,D=Diameter of pipeV=Mean velocity of flow in pipen=Kinematic viscosity of fluid

If the Reynolds number is less than 2000, the flow is called laminar. If the Reynolds number is more than 4000, it is called turbulent flow. If the Reynolds number lies between 2000 and 4000, the flow may be laminar or turbulent.


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4.3.4 Compressible and Incompressible FlowsCompressible flow is the type of flow in which density of the fluid changes from point to point or in other words the density (r) is not constant for the fluid. Thus, mathematically,for compressible flow ρ ≠Constant.

Incompressible flow is the type of flow in which the density is constant for the fluid flow. Liquid is generally incompressible while gases are compressible. Mathematically,for incompressible flow ρ=Constant.

4.3.5 Rotational and Irrotational FlowsRotational flow is the type of flow in which the particles while flowing along streamlines also rotate about their own axis. If the fluid particles flowing along steam-lines do not rotate about their own axis, that type of flow is called irrotational flow.

4.3.6 One, Two and Three-Dimensional FlowsOne-dimensional flow is the type of flow in which the flow parameter such as velocity is a function of time and one space co-ordinate only, say x. For a steady one-dimensional flow, the velocity is a function of one-space-coordinate only. The variation of velocities in other two mutually perpendicular directions assumed negligible. Hence, mathematically,for one-dimensional flow,u = f(x), v = 0, w = 0.

where u, v and w are velocity components in x, y and z directions respectively. Two-dimensional flow is the type of flow in which the velocity is a function of time and two rectangular space co-ordinates say x and y. For a steady two dimensional flow, the velocity is a function of two space co-ordinates only. The variation of velocity in the third direction is negligible. Thus, mathematically, for two dimensional flow, u = f1(x,y), v = f2(x,y), w = 0

Three-dimensional flow is the type of flow in which the velocity is a function of time and three mutually perpendicular directions. But for a steady three dimensional flow the fluid parameters are functions of three space coordinates u = f 1(x,y) (x,y and z) only. Thus, mathematically, for three-dimensional flow, u = f1(x,y,z), v = f2(x,y,z), w = f3(x,y,z)

ExamplesSteady Uniform flow: Flow of fluid through a long pipe of constant diameter at a constant rate• Unsteady Uniform flow: Flow of fluid through a long pipe of constant diameter whose valve is being opened • or closed graduallySteady Non-uniform flow: Flow of fluid through a tapering pipe at a constant rate• Unsteady Non-uniform flow: Fluid flow through a tapering pipe at varying rate• Laminar flow: Ground water flow• Turbulent flow: High velocity flow in a conduct of large size• Compressible flow: Flow of gases in orifices.• Incompressible fluid flow: Subsonic flow in aerodynamics• Rotational flow: Fluid motion in a rotating tank• Irrotational flow: Motion of a carriage in a giant wheel• 3D flow: Fluid flow in a channel• 2D flow: Flow between two parallel plates of infinite extent• 1D flow: Flow in a pipe where average flow parameters are considered for analysis•

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4.4 Flow PatternsWhenever a fluid is in motion, its particles move along certain paths depending upon the conditions of flow. Some important flow paths are described below.

4.4.1 Stream LineA stream line is an imaginary line drawn in a flow field such that a tangent drawn at any point on this line represents the direction of the velocity vector. Thus, there can be no flow across a streamline. Considering a particle moving along a stream line for very short distance ds having its components dx, dy and dz along the three mutually perpendicular co-ordinate axes. Let the components of the velocity vector Vs along x, y and z directions be u, v and w respectively. The time taken by a fluid particle to move distance ds along the stream line with a velocity Vs is,t =

is path line which is the same as

t = = =

Hence the differential equation of the streamline may be written as,

= =

Fig. 4.2 Streamline

The characteristics of streamlines are as follows.Streamlines do not cross each other, otherwise the fluid particles will have two velocities at the point of intersection • and that is physically impossible.There cannot be any movement of fluid mass across the streamlines, i.e., the flow is only along the streamlines • and not across it.Streamline spacing varies inversely as the velocity, which means that converging of streamlines in any particular • direction shows accelerated flow in that direction.


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4.4.2 Path Line

Fig. 4.3 Path line

A path line is the locus of a fluid particle as it moves along. It is a curve traced by a single fluid particle during its motion. Thus, path line represents the trace or trajectory of a fluid particle over a period of time. Path line shows the direction of the velocity of the same fluid at successive instants of time. A path line can intersect itself at different times.

Figure shows a streamline at time t1 indicating velocity vectors for particles A and B. At times t2 and t3, the particle A is shown to occupy the successive positions. The line connecting these various positions of A represent its path line.

4.4.3 Streak Line

Fig. 4.4 Streakline

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Streak line is the instantaneous picture of the positions of all the fluid particles which have passed through a fixed point in the flow field. A line formed by the smoke particles ejected into atmosphere from a fixed nozzle constitutes the streak line. For a steady flow there is no geometric distinction between the streamlines, path lines and streak lines they are coincident if they originate at the same point. However, for an unsteady flow the path, streak and stream lines are all different.

4.4.4 Streamtube

Fig. 4.5 Streamtube

If stream lines are drawn through a closed curve, they form a boundary surface across which fluid cannot penetrate. Such a surface bounded by streamlines is a sort of tube, and is known as a streamtube.

From the definition of streamline, it is evident that no fluid can cross the boundary surface of the streamtube. This implies that mass of fluid entering the Streamtube at one end must be the same as the mass leaving it at the other end. The streamtube is generally assumed to be of a small cross-sectional area so that the velocity over it could be considered uniform. The streamtube may be of any shape regular or irregular.

4.5 Continuity Equation for Three Dimensional Flow in Cartesian CoordinateConsider an elementary element in the form of a parallel pipe with sides δx, δy and δz as shown in the fig. Let P (x, y, z) be the centroid of the element. Let u, v and w be the components of velocity vector in x, y and z directions respectively and ρu the mass density of fluid at the centroid P (x, y, z)

Fig.4.6Continuityequationforthreedimensionalflow


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Then by Principle of Conservation of Mass, we can write:Rate of inflow of mass in the element = Rate of outflow of mass in element + Rate of increased mass in the element.

Consider faces PQRS and P’Q’R’S’ perpendicular to x axis. Rate of inflow of mass in the i. element through ‘PQRS’ [ρu - ]δy.δz – (inflow)

Negative sign is used because PQRS is in negative ‘x’ direction from central section. Similarly, ii. we can write the rate of outflow of mass from the element through P’Q’R’S’

[ρu + ]δy.δz – (outflow)

Net rate of increase of mass in the element through the pair of faces PQRS and P’Q’R’S’ is given iii. by Rate of increase of mass = inflow – outflow

=- .δx δy.δz

Similarly rate of increase of mass in the element through other two pairs of faces can be written iv. as (y and z direction)

- .δx δy.δz

- .δx δy.δz

Net rate of increase of mass in the element is v.

[- - - ].δx δy.δz

Now the mass of fluid in the element = p. vi. δx δy.δz ∴Rate of increase of mass can also be written as:

(ρ.δx.δy.δz)

or, (δx.δy.δz)

vii. Equating v and vi vii.

[- - - ].δx δy.δz = (δx.δy.δz)

or, [- - - ] -

or, + + +

This is the continuity equation for three dimensional flow in Cartesian co-ordinates. This is the most general form of continuity equation applicable to steady, unsteady, uniform – non-uniform as well as compressible and incompressible flow.

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Forsteadyflow,

= 0

\ continuity equation for steady flow is

+ + = 0

Forincompressiblefluidρisconstant\ Continuity equation for steady-incompressible flow takes the form

+ + = 0 or V.V = div V = 0

Fortwodimensionalflow,

+ = 0

4.6 Continuity Equation for One Dimensional FlowIn order to derive continuity equation for one dimensional flow, consider an elementary element in the form of a tube shaped volume of fluid along a stream tube as shown in fig. The flow takes place only from the ends of the element and not through the surface of the element. Let A, V and ρ be the area of cross-section, velocity and mass density of fluid at the central section of the element. A, V and ρ are functions of space only. We can write,

Fig.4.7Continuityequationforonedimensionalflow

Rate of inflow of mass in the element through PQ i.

= ρAV - (ρAV).

Rate of outflow of mass from the element through P’Q’ ii.

= ρAV + (δAV).

Net mass of fluid remained in the region iii.

= - (ρAV)δs


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Rate of increase of mass within the region iv.

=. (ρAδs)

Equating equation (iii) and (iv) v.

(ρAδs)= - (ρAV)δs

Divide the equation by vi. δs

(ρA) + (ρAV) = 0

The above equation is continuity equation for all types of the dimensional flow.

For steady flow, vii. (ρA) = 0

(ρAV) = 0 ∴ (δAV) = 0

Steady incompressible flow viii. δ = constant A1V1=A2V2=………………AnVn=Constant

The above equation relates the cross sectional area and mean velocity of flow.ix.

The product (AV=Q) is known as discharge or rate of flow. For a constant discharge, if area of flow decreases, the velocity of flow increases.

4.7 Velocity and AccelerationLet V be the resultant velocity at any point in fluid flow, having components u, v and w along x, y and z directions respectively.u=f1 (x, y, z, t)v=f2 (x, y, z, t)w=f3 (x, y, z, t)V =

The components of acceleration of the fluid particle can be found out by partial differentiation as follows:

Now, Resultant velocity: v =

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Resultant acceleration, a =

In vector notation,

Velocity vector: = u + v + wThe velocity, in general, is a function of space (s) and time (t) i.e.,V=f(x,y,z,t)Or, V=f(s,t)

And the acceleration

a = = +

a = V +

Thus the acceleration consists of the two parts:

i. : This part is due to change in position or movement and is called convective acceleration.

∴Convective acceleration = V =

ii. : This part is with respect to time at a given location and is called local (or temporal) acceleration. ∴Local acceleration =

=

Tangential and normal accelerationA particle moving in a curved path will always have a normal acceleration an towards the center. (r being the radius of the path), though its tangential acceleration (as) may be zero as in the case of uniform circular motion.For motion along a curved path, in generala = as +an

(V )


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4.8 Rotational and Irrotational FlowsRotationalflow: Fluid motion in which the fluid particles rotate about their own axes, is known as rotational flow.

Irrotationalflow: Fluid motion in which the fluid particles do not rotate about their own axis is known as irrotational flow. A steady irrotational flow is called as potential flow.

Fluid displacement: Fluid particles can undergo the following types of displacements:pure translation• linear deformation• angular deformation• rotation•

Fig. 4.8 Fluid displacement

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These motions have been shown in diagram.

Pure translation is shown in fig. (a) In this type of motion, the fluid particle is neither rotated, nor deformed. It only moves bodily from one position to another position in a given interval, such that central lines ab and cd remain parallel.

Linear deformation is shown in fig. (b). Linear deformation takes place when liquid flows in a pipe of uniform cross-section as indicated.

In angular deformation as show in fig. (c) the two axes rotate by same amount but in opposite direction.In rotation as show in fig. (d) the two axes rotate in the same direction.

4.8.1 Components of RotationRotation of fluid particle is defined in terms of the components of rotation about x, y and z axes. Consider the rotation in xy plane about ‘z’ axis.

Rotation component about any axis may be defined as the average of the sum of angular velocities of any two infinitesimal linear elements which are perpendicular to each other and also perpendicular to axis of rotation.

Consider two line elements ‘PA’ and ‘PB’ in xy plane perpendicular to each other passing through P (x, y, z) as shown in diagram. ‘u’ and ‘v’ are velocity components in x and y directions at P.

Fig. 4.9 Rotation

Velocity of ‘A’ in y direction = v + .δx

Velocity of B in x direction = u + .δy

Since velocities of ‘P’ and ‘A’ in y direction are different, element ‘PA’ will undergo rotation. Similarly ‘PB’ also will undergo rotation. PA’ and PB’ are the positions of PA and PB after time interval dt. Now, distance travelled by ‘P’ in time dt in y direction = v.dt


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Similarly distance travelled by ‘A’ in time dt in ‘y’ direction = (

∴Distance travelled by ‘A’w.r.t. P in time dt = ( )δxδt

∴let δ angle between PA and PA’ = = )δt

∴Angular velocity of PA = ωPA = =

Similarly, angular velocity of PB = ωPB = -

(Negative sign as movement of PB is against direction of ‘u’.)

Wz = [ - ]

It may be pointed out that rotation is caused by torque. The torque causes shear stress. Shear stress is associated with viscosity. Therefore, liquids with higher viscosity have rotational flow. Fluids like water and air having low viscosity will have irrotational flow.

4.8.2 CirculationFlow of fluid, around or along a closed curve or a body situated in the flow field, is knows as circulation denoted by β. The concept of circulation is used for determining the forces acting on bodies immersed in fluids.

Fig.4.10Forceactingonbodyimmersedinfluid

Consider a closed curve C. Let it be subjected to a two dimensional fluid flow streams as shown in diagram. Let point of intersection of a stream say middle one, with the curve be p as shown. Consider an elementary surfaceδs of the curve. Let U be the velocity of the stream at P and β be the angle between U and δs as shown.Component of velocity along with δs =U cosβCirculation on the elementary point: dГ = U Cos β .dsCirculation on the entire curved surface: τ = cU cos β ds.As an illustration, let us find out circulation around rectangular closed curve ABCD shown in diagram. Let u and v be velocities of flow at A in x and y axes as shownwhich are increased to (u + ) and (v + ) at D and B respectively.

Circulation along AB: ГAB = u

Circulation along BC: ГBC = (v + )

Circulation along CD: ГCD = -(u + )

Circulation along DA: ГDA = -(v)

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Total circulation along with ABCDГ = ГAB + ГBC + ГC + ГDA

Although in the above illustration, a rectangular curve was chosen for simplicity, it holds for any shape of the curve.

The circulation per unit area is two times that rotation about an axis perpendicular to the plane of the area.

4.9VelocityPotential(Ø)It is a scalar function of space and time and its negative derivative with respect to any direction gives the velocity in that axis. It is denoted by Ø (phi). Actually, the velocity potential is a mathematical form for irrotational flow.Thus Ø = f (x, y, z, t)

- = u

And - = v

- = w

The negative sign signifies that Ø decreases with an increase in the values of x, y or z. i.e. the flow is in the direction of decreasing Ø.

For incompressible fluid, the equation of continuity for a steady flow is given by

+ + = 0

Substituting for u, v and w we get,

This equation is known as Laplace equation. Any function Ø that satisfies the above equation will correspond to some case of fluid flow. Considering the conditions for irrotational flow and substituting for u, v and w we get


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Thus wx = wy = wz = 0 which is the conditional for irrotational flow. Therefore, any function of Ø which satisfies the Laplace equation is a irrotational flow.

4.10StreamFunction(ψ)It is defined as the scaler function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angles to that direction. It is defined only for two-dimensional flow. Mathematically, for steady flow the steam function defined asΨ = f (x, y) such that

= v

and = -u

The continuity equation for two-dimensional flow is

+ = 0

Substituting u = and v = in above equation, we get,

+ = 0

i.e., + = 0

Hence, existence of ψ means possible case of fluid flows. The flow may be rotational or irrotational. The rotational component W is given by and substituting the values of u and v, we get

Wz = [ - ]

= [ + ]

Forirrotationalflow,Wz = 0Wz = 0For rotational flow, Wz= constantFrom the above discussion of velocity potential function and stream function, we arrive at the following conclusions:

Potential function (• Ø) exists only for irrotational flow. Stream function (ψ) applies to both the rotational and irrotational flows (which are steady and incompressible).In case of irrotational flow, both the stream function and velocity function satisfy Laplace equation and as such • they are interchangeable. For irrotational incompressible flow, the following relationship between Ø and ψholds good.

u = - -

v = - +

These equations, in hydrodynamics, are sometimes called Cauchy Riemann equations.

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4.11 Relation between Stream Function and Velocity PotentialØ = constant, represents a case for which the velocity potential is same at every point, hence, it represents an equipotential line. One of the properties of a stream function is that the difference of its values at two points represents the flow across any line joining the points. Thus, if two points lie on the same streamline, then, there being no flow across a streamline, the difference between the stream functions ψ1and ψ2equals zero i.e. ψ1- ψ2= 0; this means the streamline is given by ψ = constant.

Let two curves Ø = constant and ψ = constant intersect each other at any point. At the point of intersection, the slopes are:

For the curve Ø = constant: Slope = = = =

For the curve ψ= constant: Slope = = = -

Now, product of the slopes of these curves m1m2 = , = -1

It shows that these two sets of curves, viz, streamlines and equipotential lines intersect each other orthogonally at all points of intersection.

4.12 Flow NetsA grid obtained by drawing a series of streamlines and equipotential lines is known as flow net. The flow net provides a simple graphical technique for studying two dimensional irrotational flows especially in the cases where mathematical relations for stream function and velocity function are either not available or are rather difficult and cumbersome to solve.

MethodsofdrawingflownetThe following methods are used for drawing flow nets:

Graphical Method:• A graphical method consists of drawing streamlines and equipotential lines such, that they cut orthogonally and form curvilinear squares. This method consumes lot of time and requires lot of erasing to get the proper shape of a flow net.Analyticalmethod(orMathematicalanalysis):• Here, the equations corresponding to the curves Ø and ψare first obtained and the same are plotted to give the flow net pattern for the flow of fluid between the given boundary shape. This method can be applied to problems with simple and ideal boundary conditions.Electrical analogy method:• This method is a practical method of drawing a flow net for a particular set of boundaries. It is based on the fact that the flow of fluids and flow of electricity through conductor are analogues each other. These two systems are similar in the respect that electrical potential is analogues to the velocity potential, the electric current is analogous of the velocity of flow, and the homogeneous conductor is analogous to the homogeneous fluid.Hydraulic models:• Streamlines can be traced by injecting a dye in a seepage model or Heleshaw apparatus. Then, by drawing equipotential lines the flow net is completed.

UsesofflownetsThe following are the uses of flow-net analysis:

to determine quantity of seepage and uplift pressure below hydraulic structure• to determine the velocity and pressure distribution, for given boundaries of flow (provided the velocity • distributions and pressure at any reference section are known)to determine the design of the outlets for their streamlining•


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LimitationsofflownetsThe following are the limitations of flow nets:

The flow net analysis cannot be applied in the region close to the boundary where the effects of viscosity are • predominant.In case of a flow of a fluid past a solid body, while the flow net gives a fairly accurate picture of the flow pattern • for the upstream part of the solid body, it can give little information concerning the flow conditions at the rear because of separation and eddies.

Fig.4.11Limitationofflownet

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SummarySteady flow is defined as that type of flow in which the fluid characteristics like velocity, pressure, density, etc. at • a point do not change with time. Steady flow = 0. Uniform flow = 0. Incompressible fluid ρ = constantContinuity equation for 3D flow, (steady, incompressible)•

+ + = 0

1D flow, (steady, incompressible)Q = a1v1 = a2v2 = ………anvn

= constantIt is defined as the scaler function of space and time, such that its partial derivative with respect to any direction • gives the velocity component at right angles to that direction. It is defined only for two-dimensional flow.

u = - -

v = - +

ReferencesLaplace’s Eqn & Flow Nets• [Online] Available at : <http://www.ees.nmt.edu/Hydro/courses/erth441/lectures/L8_Flownets.pdf>. [Accessed on 15 April 2011.]Fluid dynamics• (Updated on 14 April 2011). [Online] Available at : <http://en.wikipedia.org/wiki/Fluid_dynamics>. [Accessed on 15 April 2011.]Fluid Mechanics• [Online] Available at : < http://www.freestudy.co.uk/fluid%20mechanics/t5203.pdf>. [Accessed on 15 April 2011.]

Recommended ReadingDurst, F., 2010. • Fluid Mechanics: An Introduction to the Theory of Fluid Flows, 1st ed., Springer.Minchin, G. M., 2009. • Uniplanar Kinematics of Solids and Fluids: With Applications to the Distribution and Flow of Electricity, Cornell University Library.Singh, V. P., 1997. • Kinematic Wave Modeling in Water Resources, Environmental Hydrology, Wiley-Interscience.


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Self AssessmentIn which method, the observer concentrates on the movement of a single particle? 1.

Langrangian methoda. Eulerian methodb. Flow nets c. Irrotational flowd.

In __________, the fluid characteristics like velocity, pressure, density, etc. at a point do not change with 2. time.

rotational flowa. steady flowb. unsteady flowc. compressible flowd.

Where does the velocity, pressure or density at a point changes with respect to time?3. Steady flowa. Unsteady flowb. Rotational flowc. Compressible flowd.

The type of flow where the velocity at any given time does not change with respect to space is called______.4. steady flowa. compressible flowb. uniform flowc. rotational flowd.

Flow between parallel plates of infinite extent is an example of___________.5. one dimensional flowa. two dimensional flowb. three dimensional flowc. compressible flowd.


1. Laminar flow A. the flow parameter such as velocity is a function of time and one space co-ordinate only

2. Compressible flow B. the particles while flowing along streamlines also rotate about their own axis.

3. Rotational flow C. density of the fluid changes from point to point or in other words the density is not constant for the fluid.

4. One-dimensional D. The fluid particles move along well-defined paths or streamline and all the streamlines are straight and parallel.

1-D, 2-C, 3-B, 4-Aa. 1-C, 2-A, 3-D, 4-Bb. 1-B, 2-A, 3-D, 4-Cc. 1-D, 2-A, 3-B, 4-Cd.

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A ___________is an imaginary line drawn in a flow field such that a tangent drawn at any point on this line 7. represents the direction of the velocity vector.

non-uniform flowa. stream line b. path linec. streak lined.

In which type of flow, the fluid particles move in a zigzag way8. ?Turbulent flow a. Laminar flowb. Transition flowc. Potential flowd.

_____________ is the instantaneous picture of the positions of all the fluid particles which have passed through 9. a fixed point in the flow field..

Path linea. Stream lineb. Streak linec. laminard.

Flow of fluid, around or along a closed curve or a body situated in the flow field is known as ___________.10. circulationa. velocityb. forcec. accelerationd.


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Chapter V

Dynamics of Fluid Flow

Aim


describe the force acting on a fluid in motion•

explain three dimensional flow of Euler’s equation of motion•

discuss streamline Euler’s equation of motion•

Objectives


highlight the concept of Bernoulli’s equation•

explain the significance of various terms in Bernoulli’s equation•

analyse the limitations of Bernoulli’s equation•

Learning outcome


describe hydraulic gradient line and total energy line•

explain energy correction factor•

elaborate the application’s of Bernoulli’s equation•

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5.1 IntroductionFluid flow dynamics include the study of motion of fluid including the study of forces causing the motion and corresponding changes in energy.

Dynamics of fluid flow is governed by Newton’s second law of motion. Newton’s second law of motion states that “the rate of change of momentum is proportional to the impressed force and takes place in the direction of that force.” It can be expressed as:

= M.aWhere = sum of all external forces in the direction of motion acting on the element of a fluid,M = mass of fluid and a = accelerationIn three co-ordinate directions, we can write

= M.ax

= M.ay

= M.az

where,Fx, Fy and Fz are components of force in x, y, z directions and ax, ay, az the components of ‘acceleration’ in x, y, z directions respectively.

5.2 Forces Acting on Fluid in MotionA fluid in motion is subjected to several forces, which result in the variation of the acceleration and energies involved in the flow phenomenon of the fluid. It is analysed by Newton’s second law of motion, which relates the acceleration with the forces. The fluid is assumed to be incompressible and non-viscous.

The various forces acting on fluid mass may be classified as:Body forces: • Force proportional to volume of body is known as Body force.For example, weight, centrifugal force, magnetic force.Surface forces:• This force is proportional to surface area, for example, pressure force, shear force, compressibility force.Line forces:• The force which is proportional to the length is known as line force. For example, surface tension. According to Newton’s second law of motion, resultant force on any fluid element must be equal to product of mass and acceleration of the element and the acceleration vector has the direction of the resultant force vector. Mathematically,

= M.a where = Resultant force, M = Mass of the fluid element and a = Acceleration in the direction of resultant force. The forces and accelerations can be resolved into three directions x, y and z and corresponding equations can be written as

= M.ax

= M.ay

= M.az

The various forces that influence the fluid motion are due to gravity, pressure, viscosity, turbulence, surface tension, and capillarity.


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Fluid motion forces

Gravityforce

PressureForce

ViscousForce

Turbulentforce

Surfacetensionforce

Compress-ibilityforce

Fig.5.1Typesofforcesthatinfluencethefluidmotion

Gravity force F• g: This is the force due to the weight of the body and Fg = M.g where M = Mass and g = Acceleration due to gravity. The gravity force per unit volume = ρ .g.Fg= mgPressure Force F• p: It is the force exerted on fluid mass if there exists a pressure gradient between two points in the direction of flow.

Fp = dAdx

Viscous force F• v: This force is due to viscosity of the flowing fluid and exists in case of all real fluids.

Fv = τA = µ A

Turbulent force F• t: This force is due to turbulence of flow which causes transfer of momentum.Surface tension force F• s: This is due to cohesive property of the fluid mass. However, this force is significant only when the depth of flow is small.Compressibility force F• e: This force is due to elastic property of the fluid and is important only either compressible fluids or in case of the flowing fluids in which elastic properties are significant. If a certain mass of fluid in motion is influenced by all the above mentioned forces, then according to Newton’s second law of motion, following equations may be written.

M.a = Fg + Fp + Fv + Ft + Fs + FeResolving forces in x, y, and z directions,M.ax = Fgx + Fpx + Fvx + Ftx + Fsx + FexM.ay = Fgy + Fpy + Fvy + Fty + Fsy + FeyM.az = Fgz + Fpz + Fvz + Ftz + Fsz + Fez

In most of the problems of fluid in motion, the surface tension forces and compressibility forces are insignificant. Hence these forces may be neglected. Then the above Equation reduces toM.ax = Fgx + Fpx + Fvx + Ftx M.ay = Fgy + Fpy + Fvy + FtyM.az = Fgz + Fpz + Fvz + Ftz

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These equations are known as Reynold’s equation of motion, which are useful for the analysis of turbulent flow. Further, for the laminar flow, turbulent forces are less significant and are neglected. The above Equation further reduces toM.ax = Fgx + Fpx + FvxM.ay = Fgy + Fpy + FvyM.az = Fgz + Fpz + Fvz

These equations are known as Navier stokes equation which is useful for study of laminar flow. Further, if the viscous forces are also of very little significance, which happens in case of ideal fluid and real fluids with low viscosity.Then,M.ax = Fgx + FpxM.ay = Fgy + FpyM.az = Fgz + Fpz

The above equation is known as Euler’s equation of motion.

5.3EulersEquationofMotionforThreeDimensionalFlow(CartesianCoordinates)Assumption:

Only two forces are acting on the element • Fluid is non viscous • Pressure force acts normal to the surface.•

Euler’s equation of motion for three dimensional flow can be derived by applying Newton’s second law of motion, considering the gravity forces and pressure forces in the three co-ordinate directions.

Consider an elementary parallelepiped with sides δx, δy and δz with it’s centroid at P(x, y, z) as shown in the figure below. Let ρ be the mass density of fluid at P and let u, v and w be the components of velocity in x, y and z directions respectively. Consider the forces acting on the parallelepiped in x direction. The forces are as shown below in the figure.

Fig. 5.2 Forces acting as the parallelepiped


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The forces are as follows:

Pressure forcesNet Pressure force in x direction = Pressure force on left face in x direction - Pressure force on right face in x direction

Fpx = δy.δz – δy.δz

or, Fpx = . δx.δy.δz

Similarly, the components of pressure force in y and z directions are given by

Fpy = .δx.δy.δz

Fpz = .δx.δy.δz

Body forcesThe body force is due to gravity or weight. The total weight of fluid in parallelepiped is (ρ.g.δx×δy δz). It’s components in x, y and z directions can be written as X(ρδx.δy δz) in x direction Y(ρδx.δy δz) in y directionAnd Z(ρδx.δy δz) in z direction

Where X, Y and Z are the components of body force (gravity force) in x, y and z directions per unit mass of fluid. Now applying Newtons Second Law of motion we get

= M.ax∴Body force in x direction + Pressure force in x direction = (Mass of fluid) x acceleration in x direction

X(ρδx.δy.δz) - .(δx.δy.δ) = (ρδx.δy δz).ax

Dividing both sides by (ρδx.δy δz).the mass of fluid in the parallelepiped. We get

X - = ax

Similarly for y and x direction,

Y - = ay

Z - = az

These equations are called as Euler’s equations of motion. Substituting values of ax, ay and az the above equations can be modified as:

These equations are applicable to steady or unsteady compressible or incompressible flow of non-viscous fluids. This Euler’s equation of motion can be integrated to give energy or Bernoulli’s equation under the following assumptions.

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There exists a force potential O such that its negative derivative with respect to any direction gives i. the component of body force in that direction.The flow is irrotationalii.

∴x = , y = , z =

wz= 0, =

wx= 0, =

wy= 0, =

= = -

Substituting the above values, we get [Ω + + - ] = 0

Integrating wrt x

Ω + + - = F1(y, z, t)

Similarly for y & t direction

Ω + + - = F2(x, z, t)

Ω + + - = F3(x, y, t)

∴F1 = F2 = F3 (LHS are same) which is possible only when it is a function of time

∴Ω + + - = F(t)

But body force is due to gravity acting along vertically downward direction

z = -g = z

-g =

Ω = gz + c∴ For steady flow

As = 0

+ + gz = constant


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5.4 Euler’s Equation of Motion along a Stream Line

Fig. 5.3 Flow in a stream tube

This is an equation of motion in which the forces due to gravity and pressure are taken into consideration. This is derived by considering the motion of a fluid element along a streamline as:Consider a stream tube in which flow is taking place in S-direction as shown in figure above Consider a cylindrical element of cross-sectional area dA and length dS. The forces acting on this element are:

Pressure force pdA in the direction of flow.• Pressure force (p + • .ds).dA, opposite to the direction of flow.Weight of fluid element ρ× gdAds. Let • θ be the angle between the direction of flow and the line of action of weight of the element,Then by Newton’s second law, Resultant force in the direction of flow = Mass of fluid element x acceleration • in the direction of flow.

∴pdA – (p + .ds) dA - ρ gdAds. Cos θ = ρ gdAds. aswhere as is the acceleration in s direction. Now as = where V is function of space and time.

∴ as = +

= V. + [ = V]

If the flow is steady, = 0

as = V.

Substituting this value of as in equation (iv)

.dsdA - ρgdAds. Cos θ = ρdAds.

Dividing by ρ dA ds,

- g Cos θ = V

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or, + g Cos θ + V = 0

but, Cos θ =

∴ + g + V = 0

or, ( + gz + ) = 0

The above Equation is known as Euler’s equation of motion along a stream line.

5.5Bernoulli’sEquationfromEuler’sEquationfor(IncompressibleFluid)Bernoulli’s equation is obtained by integrating above Euler’s equation of motion long.

+ + = constant

If flow is incompressible ρ is constant and

+ gz + = constant

or, + z + = constant

or, + z + = constant

i.e., + + z = constant

The above equation is Bernoulli’s equation in which

= = Pressure energy per unit weight of fluid or pressure head.

= Kinetic energy per unit weight of fluid or kinetic head or velocity head.

Z = Potential energy per unit weight of fluid or potential head.

5.6 Bernoulli’s Equation for Steady Compressible FlowFor isothermal process,

= constant = k

dp = k.dp

dp = dp

∴ =

According to Euler’s equation,

+ gz + = constant

+ gz + = constant


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+ gz + = constant

+ gz + = constant

For Adiabatic process,

= constant dp = k.ρk-1dρ

+ gz + = constant

+ gz + = constant

+ gz + + gz = constant

5.7SignificanceofVariousTermsinBernoulli’sTheoremMost commonly used form of Bernoulli’s Equation is,

Z + + = constant

Statement of Bernoulli’s equation:In a steady flow of an ideal fluid (incompressible and non viscous), the total head of any particle, is same along a stream line. Total head of any particle at a given instant is sum of its

Datumhead=i. = = Z Potential energy possessed by weight w Newton of fluid when raised through height z above the datum level is wz.

Pressure headii. = =

If ρ is the pressure on the fluid element of cross sectional area (dA) resulting in a pressure force p.dA which causes the fluid element to move through distance dS. Then work done or energy is p.dA.dS.

Velocity headiii. =

∴ Total head = Datum head + Pressure head + Velocity head

Total head = z + +

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AssumptionsFollowing assumptions are made in the derivation of he Bernoulli’s equation:The fluid is ideal i.e. it does not exhibit any frictional effects due to fluid viscosity.

The flow is steady i.e. at a given point; there is no variation of fluid properties with respect to time.• The flow is incompressible i.e., no variation in fluid density.• The flow is irrotational.• Flow is continuous and velocity is uniform over a section.• Only gravity and pressure forces are present. No energy in the form of heat or work is either added or subtracted • from the fluid.

5.8 Limitations of Bernoulli’s EquationThe assumption that the velocity of flow is uniform over the cross-section of pipe is not true, in practice. It has • been found that the velocity is practically zero at the boundary of the pipe walls and is maximum at the center of cross-section.In addition to gravity and pressure forces, certain other forces such as pipe friction forces act on the system.• There is bound to be certain energy loss in fluid flow.•

5.9ModificationsofBernoulli’sEquationsFor loss of head:If there is loss of energy as the fluid passes from section ‘I’ to section ‘2’, Bernoulli’s equation can be modified as follows:

+ +

(Upstream section) = +

+ + (Downstream section)

where,hL is loss of head (i.e., loss of energy per Newton of fluid). Thus, loss of energy is to be added to the energy of the downstream point (which is reached afterwards in the direction of flow).

5.10 Hydraulic Gradient Line and Total Energy Line

Fig. 5.4 Hydraulic gradient line

Hydraulic gradient line H.G.L is defined as the line joining the points representing piezometric heads, i.e., the • locus of along the flow direction, is known as hydraulic grade line.Total energy line T.E.L. is defined as the line joining the points representing the sum of piezometric• Head and kinetic head • of flowing fluid in the flow direction. Figure above indicates the hydraulic grade line and total energy line for a typical pipe line connecting two reservoirs.


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5.11 Energy Correction FactorIn the Bernoulli’s equation a term is the kinetic energy per unit weight of the fluid. This has the presumption that the velocity of flow V over a cross-section of a control fluid is uniform. This may be true for ideal fluids but in real fluids, the velocity is not uniform over a cross-section. The velocities of different particles will be different.

Consider an elementary area ‘da’ of a fluid. Suppose its velocity at an instant is u.Mass of fluid flowing/s = ρ uda.

Its kinetic energy = ρ udau2 = dau3

Total energy across the entire cross-section = = If the exact velocity profile over the cross-section is known, the true kinetic energy can be determined by the above equation.In actual practice it is convenient to use the average velocity of flow U with an energy correction factorα.

Therefore, total kinetic energy at a section of fluid = AU3

Equating the two, we have AU3 =

From which α =

With the energy correction factor α, the Bernoulli’s theorem is re-written as follows:

+ α + gZ = constant

The value of ‘α’ is always more than unity. More the velocity variation across the cross-section, greater is the value of ‘α’. For different types of flow the value of α for flow in circular pipes has been found to be as given below:For laminar flow α =2

For turbulent flow α is from 1.03 to 1.06.

5.12 Applications of Bernoulli’s EquationFollowing are the different ways of Bernoulli’s equation application:

The pitot tube

Fig. 5.5 Simple pitot tube

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The pitot tube is a simple device meant for measuring the velocity of a liquid at any point. In its simple form the pitot tube consists of a glass tube whose lower end is bent at right angles. The device is placed in a moving liquid with the lower opening directed in the upstream direction.

The liquid level in the pitot tube will depend on the velocity of the stream. The pitot tube in the form shown in the figure is meant for measuring the velocity at any point in a stream of liquid whose surface is open to the atmosphere.

Let the inlet of the pitot tube be at depth H, below the liquid surface. Consider the two points A and B. The point B is just at the inlet to the pitot tube while the point A is at the same level as that of B but at some distance from B.

Let v be the velocity at A.Pressure head at A = = HAt B, there is not velocity and the pressure head at B = = = H + h

where,h = height of the liquid level in the pitot tube above the liquid surface of the stream. The point B where the velocity has reached zero, is called the stagnation point.

Applying Bernoulli’s theorem to the points A and B.

H + = H + h

h =

v =

To take into account of losses of head in the actual cases the velocity is given by,v = Cvwhere Cv = Pitot tube coefficient.

If the pitot tube is to be used to measure the velocity of a liquid in a pipe, then we must adopt some method to know the static pressure head H. For instance, we may use a pitot tube and a vertical piezometer tube and measure the difference of the liquid levels in the two tubes. Another method is to connect the pitot tube and the piezometer tube and note the difference of liquid levels in the two tubes.

VenturimeterIt is used to measure the rate of flow of fluid through a pipe. The main principle is reducing the cross- sectional area of flow passage, a pressure difference is created and the measurements of the pressure difference enables to determine the rate of flow through the pipe, i.e., on Bernoulli’s equation component parts.

An inlet section followed by convergent cone of an angle of 20• 0

A cylindrical portion of short length, known as throat.•


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Fig. 5.6 Venturimeter

The divergent cone at venturimeter is at larger length than the convergent cone, because in convergent i. cone, according to continuity equation, fluid is accelerated while in the divergent cone fluid is retarded. The fluid is allowed to retard in a smaller length then the fluid will not remain in contact with the boundary, i.e., flow separates out and eddies are formed which causes loss of energy. To avoid this divergent cone is larger and is not used for flow measurement.

aii. 1 > a2 ∴v1 > v2 ∴P1 > P2 Therefore, pressure difference is created between the two sections which can be measured by connecting a differential manometer between two pressure taps or by erecting pressure gauge at each tap. This pressure difference enables to calculate the rate of flow.

For greater accuracy in the measurement of pressure difference, the cross sectional area of the iii. throat should be reduced considerably, so that pressure at the throat is reduced. But if the cross sectional area is reduced so much that the pressure at the throat falls below vapour pressure of the flowing fluid, the liquid may vapourise and vapour packets or air bubbles may be formed. These bubbles may contain some dissolved air which is released as pressure is reduced and it too may form air bubbles. This phenomenon know as cavitation which is reduced upto a certain limit. It is kept 1/3 to 3/4 diameter of pipe and more commonly ½ diameter of pipe. Proof:

a1, a2…………..> cross sectional area at the inlet and throat section. P1, P2: pressure at the inlet and throat section V1,V2: velocities at the inlet and there is no loss of energy between the two section 1 & 2 applying Bernoulli’s equation between 1 & 2

+ + = + + ………………. (1)

If the pipe is horizontal, then Z1=Z2

= =

is the pressure difference between the two section which is known as venturi head (h…. (2)

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∴h =

∴ = h…………………………(3)

According to continuity equation if Qth represents the rate of flow through the pipe then Q = a1v1 = a2v2

∴v1 = and v2 = ………………………… (4) Substituting

= = h

= ………………………… (5)

As the fluid is flowing there is always some loss of energy due to which through discharge is more than actual discharge \ actual discharge is obtained by multiplying Qth by a factor K known as co-efficient of discharge of venturimeter.Qa= K Qth

Qth =

= KC where c = C = constantand K = fn (Re)

fn = …………………………… (6)

If U tube manometer is used

= = h = x Sm>S

If inverted U tube manometer is used

= = h = x Sm<S

+ (Z1 – Z2) + y + x = + y + x ( )

- = h = x

Q=K C0h where K is known as coefficient of discharge of venturimeter and it depends upon d2/d1 and Re. the value of K is taken as 0.98.


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OrificeMeter

Fig.5.7Orificemeter

Another simple device used for measurement of flow in a pipe line is orifice meter or orifice plate. It is also based on Bernoulli’s equation. It works on the same principle as that of venturimeter. It consists of a flat circular plate which has a sharp edged circular orifice cut in it and is placed in the pipe such that the orifice is concentric with the pipe line. The orifice diameter d varies from 0.4 to 0.8 times the diameter of the pipe ‘D’. Generally, the diameter of the orifice is 0.5 times the diameter of the pipe (d=0.5D).

Flow rate in the pipe can be measured by observing the pressure difference between two sections, one on the upstream side of the plate and the other at vena contracta on the down steams side of the orifice plate shown in above figure.

Applying Bernoulli’s theorem to section 1-1 and section 2-2, we get

+ + = + + but Z1 = Z2

∴ = - = h

where ‘h’ is the pressure head difference between section 1-1 and 2-2 in terms of column of liquid flowing through the pipe line and can be found out with piezometers or U-tube manometer or inverted U-tube manometer as usual.

For example, h = hm for U-tube differential manometer

or, h = hm for U-tube inverted differential manometer

= 2gh …….. (a)

But by continuity equation, we can writeA.V1 = acV2where A = area of pipe

ac = area of jet at vena – contraction Cc = Coefficient of contraction a = area of orifice

But further = Cc∴ac = Cc.a∴A.V1 = Cc.a.V2

∴ V1 = Cc. .V2

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Substituting value of V1 in equation (a), we get

- . = 2gh

or, V2 =

This is the theoretical velocity at vena contracta since we have not accounted for losses. To account for losses, we make use of coefficient of velocity Cv of the orifice.

Cv =

V2 actual = Cv V2 theoretical

∴ V2 actual = Cv

∴Actual discharge A = Cc. a. V2

∴Q = a.

but Cc.Cv = Cd the coefficient of discharge for orifice

Q =

a.

but =

Q =

Q =

where is replaced by = where C is a coefficient.

The above equation can be written similar to equation for venturimeter in the form

Q = K.a where k is and is known as Orifice meter flow coefficient.


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SummaryA fluid in motion is subjected to several forces, which result in the variation of the acceleration and energies • involved in the flow phenomenon of the fluid. It is analysed by Newton’s second law of motion, which relates the acceleration with the forces. The fluid is assumed to be incompressible and non-viscous.Integration of Euler’s equation of motion, gives Bernoulli’s equation•

+ + = constant

The pitot tube is a simple device meant for measuring the velocity of a liquid at any point. In its simple form • the pitot tube consists of a glass tube whose lower end is bent at right angles. The device is placed in a moving liquid with the lower opening directed in the upstream direction. Pitot tube, v = CvIt is used to measure the rate of flow of fluid through a pipe. The main principle is reducing the cross- sectional • area of flow passage, a pressure difference is created and the measurements of the pressure difference enables to determine the rate of flow through the pipe, i.e., on Bernoulli’s equation component parts; an inlet section followed by convergent cone of an angle of 200 and a cylindrical portion of short length, known as throat. Venturimet• er is given by Qa =

ReferencesBasic equations ofmotion in fluidmechanics• [Online]. Available at: <http://www2.alterra.wur.nl/Internet/webdocs/ilri-publicaties/publicaties/Pub20/pub20-h.annex1.pdf> [Accessed on 18 April 2011.]1997, • IdealfluidsandEuler’sequation [Online]. Available at: <http://galileo.phys.virginia.edu/classes/311/notes/fluids1/fluids11/node10.html> [Accessed on 18 April 2011.]Limitations: The Limitations of Bernoulli’s Equation• [Online]. Available at: < http://www.engihub.com/?p=110> [Accessed on 18 April 2011.]

Recommended ReadingBear, J., 1988. • Dynamics of Fluids in Porous Media, Dover Publications.Saad, M.A., 1992. • Compressible Fluid Flow, 2 ed., Prentice Hall.Shapiro, A.H., 1953. • The Dynamics and Thermodynamics of Compressible Fluid Flow, 1st ed., Wiley.

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Self AssessmentWhich of the following signifies velocity head?1. a.

b.

c.

d.

What is the Bernoulli’s equation for incompressible fluid?2.

p/γ + v^2/2g + z = constanta.

b. + + z = constant

c. + + z = constant

d. + + z = constant

Which of the following is not an assumption of Bernoulli’s equation?3. The flow is steady.a. The flow is irrotational.b. The flow is incompressible.c. The flow is two-dimensional.d.

_______________is due to the viscosity of the flowing fluid.4. Gravity forcea. Pressure forceb. Viscous forcec. Turbulent forced.

Which of the following devices measure the rate of flow of fluid through a pipe?5. Gaugea. Pitot tubeb. Thermometerc. Venturimeterd.


1. Body forces A. This is the force due to the weight of the body

2. Surface forces B. The force which is proportional to the length

3. Line forces C. Force proportional to surface

4. Gravity force D. force proportional to volume of body 1-D, 2-C, 3-B, 4-Aa. 1-C, 2-A, 3-D, 4-Bb. 1-B, 2-D, 3-A, 4-Cc. 1-D, 2-A, 3-B, 4-Bd.


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____________is a simple device to measure the velocity of a liquid at any point.7. Gaugea. Orifice meterb. Venturimeterc. Pitot tubed.

Which of the following forces influence fluid motion? 8. Kinetic forcea. Gravity force b. Static forcec. Electro-magnetic forced.

What governs the dynamics of fluid flow?9. Newton’s 3rd law of motiona. Newton’s 2nd law of motionb. Kirchav’s lawc. Benoulli’s equationd.

Which of the following forces act upon fluid mass?10. Body forcea. Viscous forceb. Pressure forcec. Static forced.

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Chapter VI

Flow Through Orifices

Aim


classify orifices•

describe experimental determination of hydraulic coefficients•

discuss types of notches•

Objectives


highlight partially submerged orifice•

explain totally submerged orifice•

analyse the concept of flow over notches•

Learning outcome


discuss the concept of elbow meter•

explain rotameter•

calculate the time for emptying a tank•


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6.1 IntroductionAn orifice is a geometric opening in the side or bottom of a thin-walled tank or vessel. The opening serves as an outlet to discharge the liquid contained in the tank and is regarded an orifice only if the top edge of the opening lies below the liquid surface in the tank.

6.2ClassificationofOrificesClassification of Orifices is based on different properties. They are as follows:Sizes: The orifices are classified as small orifice or large orifice, depending upon the size of the orifice and head of liquid from the centre of orifice, H. If H > five times the depth of the orifice, then the orifice is called a small orifice. If H < 5 times the depth of the orifice, then the orifice is called as the large orifice.

Shape: Circular orifice, triangular orifice, rectangular orifice and square orifice, depending upon their cross-sectional areas are the categories based on shape.

Shape of the edge: Depending upon the shape of the edge, orifices are classified as sharp edged orifice and bell mouthed orifice.

Nature of discharge: According to nature of discharge, orifices are classified as free discharging orifices, fully submerged orifice and partially submerged orifices. All these types are shown in figure below.

Fig.6.1Typesoforifices

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Technical terms:Jet of liquid:• The continuous stream of a liquid that comes out or flows out of an orifice is known as jet of liquid.Vena contracts:• It is observed that the liquid approaching the orifice converges towards it, to form a jet whose cross-sectional area is lesser than that of the orifice.HydraulicCoefficients:• The following coefficients are known as hydraulic coefficients.Coefficientofcontraction(C• c)

Cc =

The average value of Cc = 0.64

Coefficientofvelocity(Cv)

Cv = =

In general Cv = 0.95 to 0.99 and for sharp edged orifice Cv = 0.98

Coefficientofdischarge(Cd)

Cd = =

But Actual discharge,Q = (Actual area of jet at vena contracta × actual velocity of jet)Q = (ac × V) = Cc × a (Cv )Similarly, Qth = (a. )

Cd = = Cc×Cv∴Cd = Cc×Cv

6.3ExperimentalDeterminationofHydraulicCoefficientsConsider a tank containing water at constant level maintained by a constant supply as shown in the figure below. The constant head is achieved by adjusting the inflow of water in the tank through supply pipe equal to the outflow of water through an orifice fixed on the wall of the tank.


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Fig.6.2Hydrauliccoefficientswithatankfilledwithwater

Consider a particle of water on the jet at point P. Let section c-c represent the section of vena contracta.Let H = Constant head of water above the centre line of the orifice. x = Horizontal distance of point p from vena contracta. (i.e., y axis) y = Vertical distance of P from centre line of the orifice (i.e., x – axis) t = time taken by particle to reach P from c – c

We know that the motion of the jet flows a projectile path given by,y = gx = V x tt =

Substituting in above equation for y,

y = = g =

V=

This is the equation of parabola: Hence, path of the jet is also a parabola. We know that, theoretical velocity

Vth =

Coefficient of velocity

Cv = = =

∴Cv =

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The coefficient of discharge Cd can be determined by measuring the actual quantity of discharge through orifice. Then,

Cd = =

Cc can be found by measuring the cross-sectional area at vena contracta and dividing the same by area of orifice.

Cc =

Or alternatively knowing Cd, and Cv, Cc is calculated as

∴ Cc =

6.4FlowthroughSubmergedOrificeFollowing are the types of submerged orifice:Totallysubmergedorifice

Fig.6.3Totallysubmergedorifice

Figure above shows a totally submerged orifice. Let H1 be the height of liquid on upstream side above the centre of the orifice. Let H2 be the height of liquid above the centre of the orifice on downstream side. A vena contracts is formed in this case also, and pressure there corresponds to head H2.

Applying Bernoulli’s equation at points 1 and 2.

+ + z1 = + + z2

But, = H1 + Z2 – Z1

= H2

If velocity at point 1, V1 is negligible then by substitution, velocity at vena contracta is

V2 =


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If ‘a’ is the cross-sectional area of an orifice, then theoretical discharge through the orifice

= [a ]

And actual discharge

Q = Cd a

In a totally submerged orifice, the issuing jet is inferred with the liquid present at the outlet of the orifice. This results in reducing slightly the coefficient of discharge. Hence, discharge through submerged orifice is less than that through orifice discharging freely in air.

Partiallysubmergedorifices

Fig.6.4Partiallysubmergedorifice

If the outlet side of an orifice is only partly submerged under liquid, it is known as partially submerged orifice. Upper portion of the orifice behave as an orifice discharging free and the lower portion behaves as a totally submerged orifice.

Discharge through such orifice may be determined by computing discharges separately for the two portions.

Let H1 be the height of liquid on the upstream side above the bottom edge of the orifice and H be the difference between the liquid surfaces on upstream and down stream side of orifice. Now if Q1 and Q2 are the discharge through free and submerged portions respectively, then total discharge Q = Q1 + Q2. If be is the width of an orifice, then

Q1 = b [ ]

where, represents the coefficient of discharge for free portion. Similarly Q2 for submerged portion is given by,

Q2 = (H2 - H)

where, represents coefficient of discharge for submerged portion of an orifice.∴ Total discharge is given by Q = Q1 + Q2

Q = b [ ] + (H2 - H)

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DischargethroughLargeRectangularOrificeAn orifice is considered to be large, if the available head of the liquid is less than five times the height of the orifice. We know that the velocity through an orifice varies with the available head of liquid, i.e., if the liquid is flowing through a large orifice, the velocity of the fluid particle will not be constant over the height of the orifice since there is a variation in pressure head.

Consider a large rectangular orifice fitted to one side of a tank, discharging freely into air, as shown in the figure below.

Fig.6.5Largerectangularorifice

Consider a horizontal strip of thickness dh and at a depth h from the free liquid surface, as shown in figure.Let H1 = Height of liquid above top of the orifice H2 = Height of liquid above bottom of orifice b = Width of orificeArea of strip = b x dhTheoretical velocity through the stripe = Discharge through elementary strip = dQdQ = Cd x Area of strip x Velocity= Cd×b×dh× = Cdb

By integrating this equation over the entire depth of the orifice we get

Q = ×dh = Cdb

Q = Cdb

Q = Cdb

Equation gives the discharge through large rectangular orifice. However, if the same rectangular orifice is treated as a small orifice, then there will not be considerable change in the velocity along the depth of the orifice and then the discharge may be expressed as

Q = Cd×b×dh×


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By comparing the values of Q calculated from equations, it can be shown that there is a slight difference in the values and which may however be neglected provided the head H has the minimum value. Thus, if H = d then from equation which shows that there is an error of 1% only between the values. Thus, as long as H>d, orifice may be treated as small orifice.

Q = 0.99 Cd×b×dh×

6.5 Time of Emptying a Tank

Fig.6.6Dischargetimeofwaterthroughorifice

Consider a tank of uniform sectional area A provided with an orifice of area α at the bottom. Let the head of water over the orifice fall from a value H1 to value H2 in an interval of time T seconds. Let at any instance, the head of water over the orifice be h. Let the water level fall by dh in an interval of dt seconds.∴ Quantity discharged in dt secondsAdh = Cda .dt

dt = .dh

∴Interval of time required to change the head of water from H1 to H2 is obtained by integrating the above quantity from the lower limit of h to upper limit of h.

∴T = .dh = = ( )

If the tank is to be emptied, H2 = 0, thenT =

Flow of liquid from one tank to other tankAs shown in figure above, consider liquid flowing from tank with area A1 to tank having area A2 in plan. Orifice in this case is a submerged orifice and hence the head causing the flow through the orifice is the difference of liquid levels in the tanks. It is required to find out the required to reduce the difference of liquid level from H1 to H2.

Let at any instant the difference of liquid levels in the two tanks be ‘h’ and let the liquid level in tank A1 fall by ‘dH’.

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The volume of liquid going from tank A1 to A2 is A1dH. When this volume goes to tank A2, the liquid level in tank A2 rises by , further reducing the difference of levels ‘h’.

Decrease in liquid level in tank A1 = dHIncrease in liquid level in tank A2 = .dH

Therefore, change in instantaneous head ‘h’ in time ‘dt’ is

dh = dH + .dH = .dH

Now we can writeVolume of liquid leaving the tank in time ‘dt’= Discharge through orifice in time dtor, -A1dH = Qdt

6.6 Flow over NotchesA notch may be defined as an opening provided in the side of a tank or vessel such, that the liquid surface in the tank is below the top edge of the opening. These are also provided in narrow channels to measure the discharge of liquids. As such in general, notches are used to measure the rate of flow of liquid from tank, or in a channel. An orifice can also be treated as a notch if the liquid level in the tank falls below the top edge of the orifice. The water that flows over a notch is in the form of sheet of water known as nappe or vein. The lower and upper part of sheet of water is being termed as lower and upper nappe respectively. The bottom edge of the notch over which water flows is called as the crest or sill, and its vertical distance measured from the bottom of the tank is termed as crest heights. Notches can be classified into following categories:

According to shape, they are classified as rectangular, triangular, trapezoidal, parabolic and stepped notches.• According to effect of sides on the nappe emerging from notch, Notch with end contraction and Notch without • end contraction are the categories.

6.7 Elbow Meter or Bend Meter

1

2

R

V

Fig. 6.7 Elbow meter

An elbow meter consists of a simple bend in a pipe line. Such an arrangement can be used as a discharge measuring device. Above figure shows a pipe of diameter d, having a bend in a horizontal plane of mean radius R. Consider an elemental radially situated cylinder of fluid of area da. Let the points 1 and 2 refer to the outer and inner ends of the cylinder. Let the pressures at these points be p1 and p2. Let the mean velocity of flow be v. Mass of the elemental cylinder of fluid = p.da.d


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Inward pressure force on the cylinder(p1 – p2)da

Centrifugal force on the elemental cylinder = (ρdad)For radial equilibrium, equating the above two forces,

(ρdad) = (p1 – p2)da

∴v2 =

But, ρ =

∴v2 =

6.8 RotameterThis is a direct reading discharge gauging device. This consists of a float provided within a transparent tapering vertical tube introduced in the pipe line, the flow being upwards. The float is made of a material denser than the fluid passing through the tube. The float is free to rise in the tube and its position is a function of the rate of flow. A calibration scale is etched on the transparent tube in units of the rate of flow. To provide stability to the float, it is provided with slanting slots at the upper end due to which it keeps on rotating about its vertical axis remaining in the central position.

The discharge can also be determined from the relationQ = CdA

where, Cd = Coefficient of dischargeA = Annular area between the tapering pipe and the top of the floatH = Effective head across the floatand

h = ×(S-1)

where,

S = Specific gravity of the float material

The quantity (S-1) is called the effective specific gravity of the float.

For the rotameter, Cd lies between 0.7 and 0.75. Usually the dimensions of the float are such that the top diameter of the float is equal to the bottom diameter of the tapering tube.

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SummaryClassification of Orifices is based on different properties including sizes, shape and nature of discharge.• A notch may be defined as an opening provided in the side of a tank or vessel such that the liquid surface in the • tank is below the top edge of the opening. According to shape, notches are classified as rectangular, triangular, trapezoidal, parabolic and stepped • notches.An elbow meter consists of a simple bend in a pipe line. Such an arrangement can be used as a discharge • measuring device. Rotameter is a direct reading discharge gauging device. This consists of a float provided within a transparent tapering vertical tube introduced in the pipe line, the flow being upwards.

ReferencesOrifice,NozzleandVenturiFlowRateMeters• [Online]. Available at: <http://www.engineeringtoolbox.com/orifice-nozzle-venturi-d_590.html> [Accessed on 18 April 2011.]Types of Fluid Flow Meters [Online]. Available at: <http://www.engineeringtoolbox.com/flow-meters-d_493.• html> [Accessed on 18 April 2011.]Elbow Flow Meters• [Online]. Available at: <http://www.usbr.gov/pmts/hydraulics_lab/water/elbow_meter.html> [Accessed on 18 April 2011.]

Recommended ReadingRedding, T.H., 1952. • Abibliographicalsurveyofflowthroughorificesandparallel-throatednozzles, Chapman & Hall.Rateau, A., 2010. • ExperimentalResearchesOntheFlowofSteamThroughNozzlesandOrifices:ToWhichIsAdded a Note On the Flow of Hot Water, Nabu Press.Davis, D.E and Jordan, H.H., 2010. • Theorificeasameansofmeasuringflowofwaterthroughapipe, Nabu Press.


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Self AssessmentAn ____________ is a geometric opening in the side or bottom of a thin-walled tank or vessel.1.

Elbow metera. Rotameter b. Orificec. Vena contractsd.

The continuous stream of a liquid that comes out or flows out of an orifice is known as _____________.2. jet of liquida. vena contractsb. hydraulic coefficientsc. contraction coefficientd.

If the outlet side of an orifice is ___________ submerged under liquid, it is known as partially submerged 3. orifice.

fullya. partlyb. oftenc. rarelyd.

What is the Coefficient of contraction?4.

Ca. d =

b.

c.

d.

A ____________ is an opening provided in the side of a tank or vessel such that the liquid surface in the tank 5. is below the top edge of the opening.

notcha. hydraulic coefficientsb. vena contractsc. elbow meterd.

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1. Vena contracts A. Here, the issuing jet is inferred with the liquid present at the outlet of the orifice.

2. Hydraulic Coefficients B. It is a geometric opening in the side or bottom of a thin-walled tank or vessel.

3. Totally submerged orificeC. It observes that the liquid approaching the orifice converges towards

it, to form a jet whose cross-sectional area is less than that of the orifice.

4. Orifice D. It contains coefficient of contraction, coefficient of velocity and coefficient of discharge

1-d,2-c,3-b, 4-aa. 1-c, 2-a, 3-d, 4-bb. 1-c, 2-d, 3-a, 4-bc. 1-b, 2-d, 3-a, 4-cd.

How does the lower portion of the orifice behaves?7. It behaves as Partial submerged orifice.a. It behaves as Totally submerged orifice.b. It behaves as Large rectangular Orifice.c. It behaves as Small rectangular Orifice.d.

Which of the following devices reads the discharge gauge directly?8. elbow metera. vena contractsb. orificec. rotameterd.

A/an _____________ consists of a simple bend in a pipe line.9. rotametera. vena contractsb. elbow meterc. small rectangular orificed.

If H is greater than five times the depth of the orifice, then the orifice is known as _________.10. small orificea. triangular orificeb. large orificec. circular orificed.


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Chapter VII

The Boundary Layer Theory

Aim


introduce the concept of boundary layer•

describe boundary layer thickness•

discuss the types of boundary layer thickness•

Objectives


explain laminar and turbulent boundary layers•

highlight laminar sub layer•

analyse smooth and rough boundary•

Learning outcome


explain the separation of boundary layer•

explain the zones of boundary layer•

describe methods of controlling boundary layer•

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7.1 IntroductionThe concept of boundary layer was first introduced by L. Prandtl in 1904 and since then, has been applied to several fluid flow problems. When a real fluid (viscous fluid) flows past a stationary solid boundary, a layer of fluid which comes in contact with the boundary surface adheres to it (on account of viscosity) and condition of no slip occurs (The no-slip condition implies that the velocity of fluid at a solid boundary must be same as that of boundary itself). Thus, the layer of fluid which cannot slip away from the boundary surface undergoes retardation. This retarded layer further causes retardation to the adjacent layers of the fluid, thereby developing a small region in the immediate vicinity of the boundary surface in which the velocity of the flowing fluid increases rapidly from zero at the boundary surface and approaches the velocity of main stream. The layer adjacent to the boundary is known as boundary layer.

According to boundary layer theory, the extensive fluid medium around bodies moving in fluids can be divided into following two regions.

A thin layer adjoining the boundary called the boundary layer where the viscous shear takes place.• A region outside the boundary layer where the flow behaviour is quite like that of an ideal fluid and the potential • flow theory is applicable.

Fig. 7.1 Types of boundary layer

7.2 Boundary Layer Thickness (δ)The velocity within the boundary layer increases from zero at the boundary surface to the velocity of the main stream asymptotically. Therefore, the thickness of the boundary layer is arbitrarily defined as that distance from the boundary in which the velocity reaches 99 per cent of the velocity of the free stream (u = 0.99 U). It is denoted by the symbol δ. This definition, however, gives an approximate value of the boundary layer thickness and hence, d is generally termed as nominal thickness of the boundary layer.


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Fig. 7.2 Boundary layer thickness

The boundary layer thickness for greater accuracy is defined in terms of certain mathematical expressions which are the measure of the boundary layer on the flow. The commonly adopted definitions of the boundary layer thickness are:

Displacement thickness (δ*)The displacement thickness can be defined as follows: “It is the distance, measured perpendicular to the boundary, by which the main/free stream is displaced on account of formation of boundary layer.”The displacement thickness is denoted by δ *Let fluid of density ρ flow past a stationary plate with velocity U, as shown in the fig. 7.2. Consider an elementary strip of thickness dy at a distance y from the plate. Assuming unit width, the mass flow per second through the elementary strip. = ρudyMass flow per second through the elementary strip (unit width) if the plate were not there = ρU.dyReduction of mass flow rate through the elementary strip = ρ(U - u)dy........................(i)[The difference (U-u) is called velocity of defect].Total reduction of mass flow rate due to introduction of plate

..................... (ii)

(if the fluid is incompressible)Let the plate is displaced by a distance δ * and velocity offlow for the distance δ * is equal to the main/free stream velocity (i.e., U). Then loss of the mass of the fluid/sec. flowing through the distance δ * = ρu ρ δ*..................... (iii)Equating eqns. (iii) and (ii), we get

ρu ρ δ* =

or, δ* =

Momentum thickness (θ)Momentum thickness is a measure of the boundary layer thickness. Momentum thickness is defined as the distance by which the boundary should be displaced to compensate for the reduction in momentum of the flowing fluid on account of boundary layer formation. Momentum thickness is denoted byθ.

It may also be defined as the distance measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for reduction n momentum of the flowing fluid on account of boundary layer formation.

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Mass of flow per second through the elementary strip = ρudy.Momentum/sec. of this fluid inside the boundary layer = ρudy×u = ρu2dyMomentum /sec of the same mass, before entering boundary layer = ρ uUdyLoss of momentum/sec. = ρuUdy - ρu2dy = ρu(U - u)dyTherefore, total loss of momentum/sec

= .................. (i)

Let θ = distance by which plate is displaced when the fluid is flowing with a constant velocity U.Then loss of momentum/sec. of fluid flowing through distance θ with a velocityU = ρθU2....................... (ii)Equating eqns. (i) and (ii), we have

ρθU2 =

or, θ =

Energy thickness (δe)Energy thickness is defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in K.E. of the flowing fluid on account of boundary layer formation. It is denoted by δe.

Mass of flow per second through the elementary strip = ρudy , K.E. of this fluid inside the boundary layer mu2 = ( ) u2

K.E. of the same mass of fluid before entering the boundary layer= ( ) U2

Loss of K.E. through elementary strip= ( ) U2 - ( ) u2 = (U2 – u2) dyTherefore Total loss of K.E. of fluid = (U2 – u2) dy ………………… (i)

Let δe = distance by which the plate is displaced to compensated for the reduction in K.E.Then, loss of K.E. through δe of fluid flowing with velocity U= ( ) U2 ……………….. (ii)

Equating eqns. (i) and (ii), we have

( ) U2 =

or, =

=


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7.3BoundaryLayerProfileoveraFlatPlateConsider a free stream flowing with a uniform velocity U. Assume that a thin flat plate is held amidst the stream parallel to it as shown in fig. 7.3. As the fluid approaches the leading edge of the plate, its velocity is reduced to zero at the boundary, and a very thin layer of fluid sticks to the boundary surface. The retarding force caused by skin friction of the boundary surface further causes retarding force due to shear, to act on the successive layers of the fluid as it flows downstream from the leading edge. Consequently, thickness of the boundary layer goes on increasing and it develops into a profile as shown. As a matter of fact, thickness of the boundary layer, from any point on the boundary has been arbitrarily fixed as the vertical distance from the boundary such that the velocity of flow at the end of that distance may be 99% of the mean velocity U of the free stream. The mean or uniform velocity of the free stream is known as ambient velocity. The difference of velocity between the ambient velocity or 0.99 of the ambient velocity and zero velocity at the boundary remain constant where-as thickness of the boundary layer goes on increasing toward the down stream or trailing edge of the plate, therefore, the velocity gradient in the down-stream goes on decreasing resulting in decrease in shear stress.

Fig. 7.3 Boundary layer thickness

FactorsinfluencingboundarylayerthicknessMore the distance from the leading edge x, more is the boundary layer thickness.• More the free stream velocity or the ambient velocity, lesser is the thickness.• More the kinematic viscosity, more is the thickness.• Boundary layer thickness depends upon the pressure gradient • significantly in the direction of the flow.

If the pressure decreases in the direction of the flow (i.e., is negative) as is the case in a converging flow, the resulting pressure will act against the retarding forces. Consequently, the thickness will decrease. In case of positive pressure gradients which take place when the area of flow increases, the thickness of layer increases due to more retarding force caused due to increase in pressure in the down stream. Such a situation is very important as it gives rise to back flow and separation.

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Zones of boundary layerAs indicated there are three zones of a boundary layer:

Laminar boundary layer (L.B.L)• Transitional boundary layer• Turbulent boundary layer (T.B.L.).•

The thickness of the boundary layer depends upon the density, viscosity, ambient velocity U and the distance from the leading edge of the plate. All these factors compose of the Reynolds number. Therefore, Re = = in free flow passing over a flat plate held parallel to the flow stream in it, upto certain length from the leading edge of the plate, the flow in the boundary layer is laminar irrespective to whether the ambient flow is laminar or turbulent. This zone of the boundary layer is known as laminar boundary layer as being a laminar flow, the velocity variation curve over this portion is parabolic as shown for any one point 1.

Beyond some distance from the leading edge, after the L.B.L., the flow becomes unstable and the boundary layer starts exhibiting characteristics between those of laminar and turbulent. This part as shown in the above figure is known as transitional zone. After this, the flow in the boundary layer becomes turbulent and this zone is known as turbulent boundary layer (T.B.L). The velocity boundary thickness curve for this part is logarithmic as is the usual case for a turbulent flow. The change of boundary layer from laminar to turbulent depends upon Re and other factors such as roughness of the plate, its curvature, pressure gradient, etc. However, in general, the critical Reynolds number varies from 3 x 105 to 5.5 x 105

7.4 Laminar Boundary LayerInside L.B.L., two equal and opposite forces, namely inertia forces and viscous forces act.Thickness of L.B.L

Inertia force/unit vol. = i. (mass×acceleration = ρ×volume× )

for flat plate

Viscous force/unit vol. = ii. = = =

but, = and = k

Equating the two forces iii. k

=

In this equation, k is a constant the value of which is found by Blasius, and it is 5.

= =

= 5

Coefficient of drag Civ. f = shear stress at the boundary Substituting the value of δ

= µ ≈µ

= constant.

= constant.


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v.

or, =

where, is called the local drag coefficient, denoted by Cf.

Total horizontal drag force acting on the plate Fvi. D = Where, B = width of plate; L = Length of plate

Surface area of the plate: A = BL Substituting for τ0 from eq., Average coefficient of drag CD = =

= CD× ×B×L on one side

or, = CD× ×A

Velocity distribution: Since the flow is laminar the velocity distribution in the vertical section vii. would follow parabolic law. For this, three conditions must be satisfied: • As fluid adheres to the boundary, u = 0 at y = 0 • Beyond boundary layer, the velocity is uniform U. Hence u = U at y = δ• At the edge of the boundary layer, there is no velocity gradient or =0 and = 0 at y = δ

Example: A smooth two dimensional flat plate is exposed to a wind velocity of 360 m/hr. If the laminar boundary layer exists upto a value of Re = 2×105, find the maximum distance from the leading edge upto which laminar boundary layer exists and its maximum thickness. Kinematic velocity of air is 1.49×10-5 m2/s. Solution: As given ambient velocity of wind:U = 360 km/hr = 100 m/sRe for the laminar position = 2×105

v = 1.49×10-5 m2/sSuppose x = distance from the leading edge upto which boundary layer is laminarRe =

∴2×105 =

from which,Suppose δmax= maximum thickness of boundary layer upto which the flow is laminar.

δmax = = = 0.0003m = 0.03mm

For Laminar boundary layer

δ =

=

=

Rex = Reynold’s number at a distance x & L

= from the leading edge

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7.5 Turbulent Boundary LayerAs the flow passes down the plate, the distance ‘x’ and hence Rex goes on increasing. This increases the thickness of the boundary layer and it becomes unstable. The boundary layer passes from laminar to turbulent. This transition takes place between Rex = 1.3 x 105 and 4 x 106 with the mean value of Rex = 5 x 105 which is taken as critical Reynold’s number. If plate is sufficiently long, the boundary layer on the initial part upto Rex = 5 x 105 will be laminar and thereafter it will be turbulent.

In case of turbulent boundary layer, velocity variation is logarithmic and is normally expressed in the form of power law:

=

the value of n varies from 5 to 10 depending upon Reynold’s Number.

For values of Rex from 5 x 105 to 2 x 107 ‘n’ is taken as 7 and the velocity distribution

is given by = , the power law

For Turbulent Boundary Layer and Rex between 5 x 105 to 2 x 107 , we have

=

=

=

7.6 Laminar Sub Layer

Fig. 7.4 Graphical representation of zones

If the flat plate is very smooth, even in the zone of turbulent boundary layer, there exists a very thin layer in the immediate vicinity of the boundary in which the flow is still laminar. This thin layer is known as laminar sub layer. Its thickness is represented byδ. Since the flow in the sub layer is laminar, the velocity of distribution laminar sub layer is parabolic in nature, whereas in the turbulent boundary layer, it follows one-seventh power law or logarithmic law.


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Laminar sub layer thickness δ is the value of y, distance normal to boundary where the turbulent velocity distribution and the laminar velocity distribution curve meet each other. Since the value of δ is very small, the velocity distribution in laminar sub layer is taken as straight line (linear) as against parabolic.

Nikuradse’s experiments have shown that:

δ’ = =

Where, v = kinematic viscosity of fluid = Wall shear stress Ρ = mass density of fluid

= is called as shear velocity since has dimension of velocity.

Relative magnitudes of thickness of laminar sublayer δ' and the average height of roughness projections ‘k’ can be used for defining the nature of boundary, either hydro dynamically smooth or hydro dynamically rough boundary.

7.7 Separation of Boundary Layer

Fig.7.5Velocitydistributedforflowoveracurvedsurface

Consider a flow over curved surface as shown in figure above. Pressure from A decrease till point C and further, it increases again till point E that is from point C onwards adverse pressure gradient is established. Figure shows velocity distribution at different points. As velocity increases from point A to C, boundary layer thickness decreases. From point C onwards pressure increases hence velocity decreases, fluid already retarded in the boundary layer further retards due to reduction in the velocity (increase in the pressure). As momentum of the fluid is reduced further, fluid near the boundary surface (at point D) brought to rest that is = 0 at point D. Further increase in the pressure enables the fluid near the boundary continues to flow, but flows in reverse direction (at point E). The fluid cannot follow contour of the surface and separate away from it, and is called as separation. The point at which velocity gradient = 0 is called separation point. The line joining the points of zero velocity gradient and dividing forward and reverse flow is called as separation line.

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Laminar boundary layer is more sensitive for separation than turbulent boundary layer because of two reasons: • is uniform

Velocity of fluid in laminar flow is already less which can retarded more easily due to adverse pressure • gradient.

7.8 Methods of Controlling Boundary LayerSeparation of flow results in formation of eddies and loss of energy. It is therefore necessary to avoid the separation. Following methods are used to control the separation.

Fig.7.6Separationofflowregionforenergy

Motion of solid boundaryThe formation of boundary layer is due to the difference between the velocity of the flowing fluid and that of the solid boundary. As such it is possible to eliminate the formation of boundary layer by causing the solid boundary to move with the flowing fluid. Such a motion of the boundary may be achieved by rotating a circular cylinder lying in a stream of fluid, so that as the upper side of the cylinder, where the fluid as well as the cylinder moves in the same direction, the boundary layer does not form and hence the separation is completely eliminated. However, on the lower side of the cylinder, where the fluid motion is opposite to that of the cylinder, separation would occur.

Streamlining of body shapesBy the use of suitably shaped bodies, the point of transition of the boundary layer from laminar to turbulent can be moved downstream which results in the reduction of the skin friction drag. Further more by streamlining of body shapes, the separation may be eliminated.

AccelerationofthefluidintheboundarylayerThis method consists of supplying additional energy due to the particles of fluid which are being retarded in the boundary layer. This can be achieved either by injecting fluid into the region of boundary layer from the interior of the body with the help of some suitable device or by diverting a portion of the fluid of the main stream from the region of high pressure to the retarded region of boundary layer through a slot provided in the body as in the case of slotted wing. However, the disadvantage of this method is that if the fluid is injected into a laminar boundary layer, it undergoes a transition to turbulent boundary layer which results in an increased skin friction drag.


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SuctionofthefluidfromtheboundarylayerIn this method, the slow moving fluid in the boundary layer is removed by suction through slots or through a porous surface, so that on the downstream of the point of suction, a new boundary layer starts developing which is able to withstand an adverse pressure gradient and hence separation is prevented. Moreover, the suction of the fluid from the boundary layer also delays it transition from laminar to turbulent due to which skin friction drag is reduced.

7.9 Hydro-dynamically Smooth and Rough Boundaries

Fig.7.7Developmentofboundarylayerflowinpipe

Any boundary surface of flow pressure with which fluid is in contact while flowing, can never be absolutely smooth. Surface, if observed under microscope, will contain innumerable irregularities. These are called as surface irregularities or roughness protrusions. The average height of the protrusions is denoted by k. Laminar sub layer which exhibits properties of laminar flow will always exist near the boundary. The thickness of this laminar sub layer is denoted byδ. This k and δ' will decide the behavior of the boundary of the flow passage.

Case I: δ'>KIn this case, laminar sub layer thickness δ' is having higher value than average height of protrusions (k). All the protrusions are in totally submerged condition. Laminar sub layer works as a smooth surface for the turbulent flow outside the laminar sub layer. Thus eddies which inherent characteristic of turbulent flow, generated in the flow can not penetrate in laminar flow i.e., they can’t reach surface irregularities and therefore the rough element are not exposed to flow This boundary acts now as an absolute smooth boundary and called as hydro-dynamically smooth boundary.

Case II: K>δ'As the velocity of fluid increases, the thickness of laminar sub layer goes on reducing and at one time it becomes smaller than average height protrusions. As δ' <K most of protrusions will project in the flow and wake is formed behind each roughness element and energy loss will take place. This type of boundary is called as hydro-dynamically rough boundary. In this condition (i.e., δ'<K), resistance of flow (i.e., friction factor f) is function of ration called relative roughness where L is characteristic dimensions and in case of flow through pipe, it is diameter of pipe d.

Experimental result has given the conclusion as

<0.25 = Boundary is hydro-dynamically smooth

>6 = Boundary is hydro-dynamically rough

0.25< <6 = Transition zone

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SummaryThe concept of boundary layer was first introduced by L. Prandtl in 1904 and since then, has been applied to • several fluid flow problems. The displacement thickness can be defined as follows: “It is the distance, measured perpendicular to the boundary, • by which the main/free stream is displaced on account of formation of boundary layer.”Energy thickness is defined as the distance, measured perpendicular to the boundary of the solid body, by which • the boundary should be displaced to compensate for the reduction in K.E. of the flowing fluid on account of boundary layer formation.In case of turbulent boundary layer, velocity variation is logarithmic and is normally expressed in the • form of power law: = Whenever a real fluid flows past a stationary solid boundary, boundary layer forms.• There are three zones namely, laminar, transitional and turbulent boundary layer.• Separation of boundary layer should be avoided.•

ReferencesBoundary layer• (Updated on 11 March 2011) [Online]. Available at: <http://en.wikipedia.org/wiki/Boundary_layer>. [Accessed on 19 April 2011].Schlichting, H., Gersten, K. and Gersten, K, 2000. • Boundary-layer theory, 8 ed., Springer, p- 156.1997, Grosser, W.I., • Factorsinfluencingpitotprobecenterlinedisplacementinaturbulentsupersonicboundarylayer [Online]. Available at: <http://gltrs.grc.nasa.gov/reports/1997/TM-107341.pdf>. [Accessed on 19 April 2011].

Recommended ReadingHazen, D.C., • Boundary layer control. [Online]. Available at: <http://teaching.alexeng.edu.eg/Mech/Boundary%20Layer%20Control.pdf>. [Accessed on 19 April 2011].Schlichting, H. and Gersten, K., 2000. • Boundary-Layer, 8th ed., Springer.Sobey, I.J.,2001. • Introduction to Interactive Boundary Layer Theory, Oxford University Press.


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Self AssessmentThe ________________can be defined as the distance, measured perpendicular to the boundary, by which the 1. main/free stream is displaced on account of formation of boundary layer.

boundary layera. momentum thickness b. displacement thicknessc. displacement d.

During the change of boundary layer from laminar to turbulent, the critical Reynolds number varies from 2. ______________.

3 x 10a. 4 to 5 x 104

3 x 10b. 5 to 5.5 x 105

2 x 10c. 6 to 5 x 106

5 x 10d. 6 to 8 x 106

The formation of boundary layer is due to the difference between the velocity of the flowing fluid and that of 3. the __________.

high velocitya. boundary layerb. solid boundaryc. high accelerationd.

Which of the following equation denotes momentum thickness?4. a.

b.

c.

d.

According to____________, the extensive fluid medium around bodies moving in fluids can be divided into 5. two regions.

Bernoulli's equationa. Euler's equationb. boundary layer theoryc. Newton's lawd.

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Match the following:6.

1. Motion of solid boundary A. The slow moving fluid in the boundary layer is removed by suction through slots or through a porous surface.

2. Streamlining of body shapesB. This method consists of supplying additional energy due

to the particles of fluid, which are being retarded in the boundary layer.

3. Acceleration of the fluid in the boundary layer

C. By the use of suitably shaped bodies, the point of transition of the boundary layer from laminar to turbulent can be moved downstream.

4. Suction of the fluid from the boundary layer

D. The formation of boundary layer is due to the difference between the velocity of the flowing fluid and that of the solid boundary.

1-D, 2-C, 3-B, 4-Aa. 1-C,2-A, 3-D, 4-Bb. 1-B, 2-D, 3-A, 4-Cc. 1-D, 2-C, 3-A, 4-Bd.

Who introduced the concept of boundary layer?7. Newtona. L. Prandtlb. P. Druckerc. Bernoullid.

The ____________ of the boundary can be prevented by providing a bypass in the slotted wing. 8. separationa. joiningb. startingc. endingd.

What does the T.B.L. stands for?9. Transition Boundary Layer a. Terminal Boundary Layer b. Turbulent Boundary Layer c. Thick Boundary Layer d.

____________ is defined as the distance, measured perpendicular to the boundary of the solid body, by which 10. the boundary should be displaced to compensate for the reduction in K.E. of the flowing fluid on account of boundary layer formation.

Displacement thickness a. Energy thicknessb. Momentum thicknessc. Boundary thicknessd.


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Case Study I

SuccesswithUCONTrident68AWHydraulicFluid

TargetConversion of a Billet Sawing Unit to UCON Trident 68 AW Hydraulic Fluid in Canada

ProblemA steel Mill in Canada was using three different types of hydraulic fluids in its facility; a phosphate ester fluid, a blend of phosphate ester and mineral oils and a mineral oil fluid. The company’s insurance underwriter suggested that all the mill’s hydraulic systems use fire resistant fluids or install sprinkler systems throughout in the mill. Installing sprinklers in the mill would entail using both water and dry chemical type of systems and would be very costly.

The company was also dissatisfied with the phosphate ester fluid and the blend of phosphate esters and mineral oils that they were using in their hydraulic systems, due to corrosion and sludge formation. The normal fluid maintenance required the mill to replace 50% of the fluid every 3 to 4 months, based on oil analysis results from the fluid supplier. The mill wanted a fire resistant fluid that would perform better than mineral oil and not require them to install a sprinkler system. The company required an ISO 68 hydraulic fluid for their equipment.

SolutionIn November of 2001, the mineral oil based hydraulic fluid in a Stacker Billet was replaced with UCON Trident 68 AW Hydraulic Fluid

ResultSamples were taken of the UCON Trident 68 AW Hydraulic Fluid initially at the time of conversion, after one week of operation, after one month of operation, and after three months of operation. The samples were analyzed for viscosity, pH, total acid number, metal wear, and methanol insoluables. After 3 months hours of operations, the UCON Trident fluid had no indication of metal wear or unwanted residue and showed excellent performance in this service.

UCON Trident 68 AW Hydraulic Fluid was used in this Billet Sawing Unit form November 2001 until April 2002, at which time the equipment was taken out of service and replaced with a new unit. The customer was pleased with the performance of the fluid and is interested in purchasing additional Trident fluid for use in their hydraulic systems.

Product descriptionUCON Trident AW Hydraulic Fluids are high performance hydraulic fluids designed for demanding industrial applications requiring environmental sensitivity, water solubility, fire resistance, and excellent anti-wear properties over wide temperature ranges. These polyalklene glycol (PAG) based fluids, which are available in three viscosity grades, are anhydrous (water-free) and are rated as anti-wear. UCON Trident Hydraulic Fluids do not breakdown to form sludge, and they do not hydrolyse in the presence of water. Furthermore, because of their high viscosity indices and low temperature characteristics, one UCON Trident viscosity grade fluid may replace two or three mineral oils. These hydraulic fluids are ideal for use in applications such as dockside/marine, forestry amusement, and industrial operations.

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QuestionsWhat is the problem discussed in the above case study?1. AnswerA steel Mill in Canada was using three different types of hydraulic fluids in their facility; a phosphate ester fluid, a blend of phosphate ester and mineral oils and a mineral oil fluid. The company’s insurance underwriter suggested that all the mill’s hydraulic systems use fire resistant fluids or install sprinkler systems throughout in the mill. Installing sprinklers in the mill would entail using both water and dry chemical type of systems and would be very costly.

How did the Canadian Steel Mill solve their Stacker Billet problem?2. AnswerNovember of 2001, the Steel Mill replaced the mineral oil based hydraulic fluid in a Stacker Billet with UCON Trident 68 AW Hydraulic Fluid.

What are the results after converting the Billet Sawing Unit to UCON Trident 68 AW Hydraulic Fluid in the 3. Mill?AnswerSamples were taken of the UCON Trident 68 AW Hydraulic Fluid initially at the time of conversion, after one week of operation, after one month of operation, and after three months of operation. The samples were analysed for viscosity, pH, total acid number, metal wear, and methanol insoluables. After 3 months hours of operations, the UCON Trident fluid had no indication of metal wear or unwanted residue and showed excellent performance in this service.


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Case Study II

Decreasing Design Time for Control ValvesHiter, Brazil; Control Valves

OverviewFor 40 years, Hiter has been developing and manufacturing control valves for the Brazilian and international markets. From their start in the sugar market, a commitment to quality has propelled growth into the paper and cellulose, oil, food processing, chemical and waste industries. Located in São Paulo, Brazil in a ten thousand square meter industrial park, Hiter owns the only laboratory for noise and cavitation testing in Latin America. Continuous research, modern technology and a specialized engineering team are committed to efficiently producing valves of the highest quality and reliability.

Testimonial“Using ANSYS software eased our workload. Now, each design change can easily be communicated between our solid modeler and the simulation software. We selected ANSYS Design Space and ANSYS CFX because of their state of- the-art technology. This software helps us create virtual prototypes so that we can analyze dozens of alternatives before manufacturing. If we were not using the software, it would be necessary to build dozens of prototypes (requiring up to four months per prototype) in order to develop the final design. It is also possible to customize the results in order to determine if standard criteria have been met. Using ANSYS software allows us to simulate pressure and temperature load under a wide variety of conditions and the result is reduced risk for Hiter in both design and manufacturing phases. We are very pleased with the results obtained by using ANSYS Design Space and ANSYS CFX.

ChallengeHiter requires flexible, integrated software for both fluid flow and structural analysis. This software must be capable of interacting with the solid modeler already in use in the design process. Hiter wants to virtually test many design variations of their control valves in order to increase understanding of their product and to minimize time spent by their engineering team in repetitive tasks and experimental testing.

SolutionHiter started using ANSYS Design Space and ANSYS CFX computational fluid dynamics software in 2004. Bidirectional CAD associatively allows ANSYS Design Space, ANSYS CFX and the solid modeler work together to easily exchange information. Physical tests are now required only for at the final stage of design, eliminating the trial and error process of live testing.

BenefitsUsing ANSYS software reduced the number of prototypes and physical tests and enabled the study of more design alternatives. The superior technology in ANSYS products and the knowledgeable support from ESSS (the ANSYS distributor in Brazil), is saving Hiter’s engineers significant time in the modeling and simulation process. Product development time was reduced by months.

QuestionsGive an overview of the above case study.1. What are the challenges for Hiter group mentioned in this case study?2. Briefly describe the solution to the problem discussed.3.

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Case Study III

Changing the Shape of Ships to ComeNavatek Ltd., USA, Marine Hydrodynamics

OverviewSince its founding in 1979, Navatek has been a leader in researching, developing and deploying innovative, advanced ship hull designs and associated technologies for both military and commercial use. The company, based in Honolulu, holds multiple U.S. patents relating to its underwater lifting body hull technology and is also known for its work in CFD hull optimization, drag reduction and integrated propulsion systems. Navatek’s technology prototypes have demonstrated benefits including: better sea kindliness, increased range and payload, reduce power requirements, higher seaway speeds, and reduced wake wash.

Testimonial“We are very happy with the ANSYS CFX results in all areas including forces and free surface predictions. ANSYS CFX has performed as advertised and aided us in our design with precise appendage placement based on free surface and flow analysis.”

ChallengeScale model testing is time consuming, expensive, and can be unreliable due to scaling effects. The physics of the processes involved are complex, involving transient, transitionally turbulent, multiphase flow with a free surface.

SolutionANSYS CFX offers reliable multiphase flow models which allow prediction of free surface shape, forces and effects due to cavitation. Simulation results have been validated against towing-tank experiments and have been found to show excellent agreement. CFX-Mesh allows for highly-automated, rapid creation of high-quality meshes with an inflation layer that ensures excellent near-wall resolution.

BenefitsCFD simulation allows the investigation of more design alternatives, while reducing the need for expensive towing tank tests. ANSYS CFX allows for rapid completion of what-if scenarios providing valuable insight into design variations such as appendage placement. The end result is hulls which perform better in all key areas.

QuestionsSummarise the above case study.1. What is the solution suggested?2. Briefly discuss the benefits of the study.3.


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ReferencesBasic equations ofmotion in fluidmechanics• [Online]. Available at: <http://www2.alterra.wur.nl/Internet/webdocs/ilri-publicaties/publicaties/Pub20/pub20-h.annex1.pdf> [Accessed on 18 April 2011.]Boundary layer• (Updated on 11 March 2011) [Online]. Available at: <http://en.wikipedia.org/wiki/Boundary_layer>. [Accessed on 19 April 2011].Elbow Flow Meters• [Online]. Available at: <http://www.usbr.gov/pmts/hydraulics_lab/water/elbow_meter.html> [Accessed on 18 April 2011.]Fluid dynamics• (Updated on 14 April 2011). [Online] Available at : <http://en.wikipedia.org/wiki/Fluid_dynamics>. [Accessed on 15 April 2011.]Fluid mechanics – theory• [Online] Available at:< https://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=fl&chap_sec=02.3&page=theory> [Accessed on 14 April 2011.]Fluid Mechanics• [Online] Available at :< http://www.freestudy.co.uk/fluid%20mechanics/t5203.pdf>. [Accessed on 15 April 2011.]Fluids Mechanics and Fluid Properties.• [Online] Available at: <http://www.efm.leeds.ac.uk/CIVE/CIVE1400/PDF/Notes/section1.pdf>. [Accessed on 8 April 2011.]Galileo, A. Supercomputer for everyone, 1997. • IdealfluidsandEuler’sequation [Online]. Available at: <http://galileo.phys.virginia.edu/classes/311/notes/fluids1/fluids11/node10.html> [Accessed on 18 April 2011.]Gaussian Department of Mathematics, 2007. • Fluid mechanics [Online] Available at: <http://www.gaussianmath.com/fluidmech/pressurepoint/pressurepoint.html>. [Accessed on 14 April 2011.]Grosser, W.I., 1997, • Factorsinfluencingpitotprobecenterlinedisplacementinaturbulentsupersonicboundarylayer [Online]. Available at: <http://gltrs.grc.nasa.gov/reports/1997/TM-107341.pdf>. [Accessed on 19 April 2011].Limitations: The Limitations of Bernoulli’s Equation• [Online]. Available at: < http://www.engihub.com/?p=110> [Accessed on 18 April 2011.]NTNU, 2004. • Buckingham’s pi-theorem [Online] Available at: <http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf.> [Accessed on 12 April 2011.]Orifice,NozzleandVenturiFlowRateMeters• [Online]. Available at: <http://www.engineeringtoolbox.com/orifice-nozzle-venturi-d_590.html> [Accessed on 18 April 2011.]Pressure• Measurement By Manometer [Online] Available at:< http://www.cartage.org.lb/en/themes/sciences/physics/mechanics/fluidmechanics/statics/Measurement/Measurement.htm>. [Accessed on 14 April 2011.]Prof.Wilson.J, 2006. • Laplace’s Eqn & Flow Nets [Online] Available at : <http://www.ees.nmt.edu/Hydro/courses/erth441/lectures/L8_Flownets.pdf>. [Accessed on 15 April 2011.]Properties of gases. • [Online] Available at: <http://library.thinkquest.org/19957/gaslaws/propertiesgasbody.html>. [Accessed on 8 April 2011.]Rayleigh’s method of dimensional analysis• [Online] (Updated on 15 March 2011) Available at: <http://en.wikipedia.org/wiki/Rayleigh%27s_method_of_dimensional_analysis>. [Accessed on 12 April 2011.]Reynolds number• [Online] (Updated on 11 April 2011) Available at: <http://en.wikipedia.org/wiki/Reynolds_number >. [Accessed on 12 April 2011.]Schlichting, H., Gersten, K. and Gersten, K, 2000. • Boundary-layer theory, 8 ed., Springer, p- 156.Types of Fluid Flow Meters• [Online]. Available at: <http://www.engineeringtoolbox.com/flow-meters-d_493.html> [Accessed on 18 April 2011.]Types of Fluid Motion.• [Online] Available at: <http://www.adl.gatech.edu/classes/lowspdaero/lospd2/lospd2.html>. [Accessed on 8 April 2011.]

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Recommended ReadingBear, J., 1988. • Dynamics of Fluids in Porous Media, Dover Publications.Bridgman, P. W., 2001.• Dimensional Analysis, General Books LLC.Davis, D.E and Jordan, H.H., 2010. • Theorificeasameansofmeasuringflowofwaterthroughapipe, Nabu Press.Durst, F., 2010. • Fluid Mechanics: An Introduction to the Theory of Fluid Flows, 1st ed., Springer.Finnemore, E., Franzini, F.,2001. • Fluid Mechanics with Engineering Applications, 10 ed., McGraw-Hill Science/Engineering/MathGibbings, J.C., 2011. • Dimensional Analysis, 1st ed., Springer.Giles, R., Liu, C., Evett, J., 1994. • Schaum’s Outline of Fluid Mechanics and Hydraulics, 3 ed, McGraw-Hill.Hazen, D.C., • Boundary layer control. [Online]. Available at: <http://teaching.alexeng.edu.eg/Mech/Boundary%20Layer%20Control.pdf>. [Accessed on 19 April 2011].Menzies, J. Kay and Nedderman, R. M, 1985. • Fluid mechanics and transfer processes, 2nd ed., CUP Archive. Minchin, G. M., 2009. • Uniplanar Kinematics of Solids and Fluids: With Applications to the Distribution and Flow of Electricity, Cornell University Library.Rateau, A., 2010. • ExperimentalResearchesOntheFlowofSteamThroughNozzlesandOrifices:ToWhichIsAdded a Note On the Flow of Hot Water, Nabu Press.Redding, T.H., 1952. • Abibliographicalsurveyofflowthroughorificesandparallel-throatednozzles, Chapman & Hall.Saad, M.A., 1992. • Compressible Fluid Flow, 2nd ed., Prentice Hall.Schlichting, H. and Gersten, K., 2000. • Boundary-Layer, 8th ed., Springer.Shapiro, A.H., 1953.• The Dynamics and Thermodynamics of Compressible Fluid Flow, 1st ed., Wiley.Singh, V. P., 1997. • Kinematic Wave Modeling in Water Resources, Environmental Hydrology, Wiley-Interscience.Sobey, I.J.,2001. • Introduction to Interactive Boundary Layer Theory, Oxford University Press.Som and Biswas, K, 2008. • Intro to Fluid Mechanics, 2nd ed., Tata McGraw-Hill Education.Streeter, V. Lyle, 1958. • Fluid mechanics, 2nd ed., Tata McGraw-Hill Education. Szirtes, T., 2006. • Applied Dimensional Analysis and Modeling, 2nd ed., Butterworth-Heinemann.Zeytounian, R., 2009. • Convection in Fluids: A Rational Analysis and Asymptotic Modelling (Fluid Mechanics and Its Applications), 1 ed., Springer


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Self Assessment Answers

Chapter Id 1. c 2. c 3. b 4. c 5. a 6. c 7. b 8. a 9. b10.

Chapter IId1. c2. c3. c4. a5. c6. c7. d8. a9. c10.

Chapter IIIa1. b 2. a 3. a 4. b 5. a 6. d 7. c 8. d 9. a10.

Chapter IVa1. b 2. b 3. c 4. b 5. a 6. b 7. a8. c9. a10.

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Chapter Vb 1. a 2. d 3. c 4. d 5. a 6. d 7. b 8. b 9. a 10.

Chapter VIc1. a2. b3. c4. a5. c6. b7. d8. c9. a10.

Chapter VIIc1. b2. c3. b4. c5. a6. b7. a8. c9. b10.

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