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Contents Page

Contents Page .................................................................................................................................................. 1

Index Page ........................................................................................................................................................ 6

Symbols ............................................................................................................................................................ 7

Approximations ................................................................................................................................................ 8

Fundamentals ................................................................................................................................................... 9

Common Equations & Values ......................................................................................................................... 10

Nature of Fluids .............................................................................................................................................. 11 Continuum Concept ................................................................................................................................................................................................ 11 Fluid Properties ........................................................................................................................................................................................................ 11

Density ................................................................................................................................................................................................................................ 11 Gas Law .............................................................................................................................................................................................................................. 12 Viscosity .............................................................................................................................................................................................................................. 12 Surface Tension .............................................................................................................................................................................................................. 12

Types of Flow ............................................................................................................................................................................................................ 12 Laminar flow .................................................................................................................................................................................................................... 12 Turbulent Flow ............................................................................................................................................................................................................... 13 Temporal Variation ...................................................................................................................................................................................................... 13 Spatial Variation ............................................................................................................................................................................................................ 13

Governing Principles .............................................................................................................................................................................................. 13 Continuity .......................................................................................................................................................................................................................... 13 Momentum ........................................................................................................................................................................................................................ 13 Energy ................................................................................................................................................................................................................................. 13 Applications of Governing Principles .................................................................................................................................................................... 14

Example 2 – Types of Flow .................................................................................................................................................................................. 14

Hydrostatic Pressure ...................................................................................................................................... 15 Pressure ....................................................................................................................................................................................................................... 15

Absolute Pressure .......................................................................................................................................................................................................... 15 Gauge Pressure ............................................................................................................................................................................................................... 15 Atmospheric Pressure .................................................................................................................................................................................................. 15

Pressure Transmission .......................................................................................................................................................................................... 15 Pascal’s Law ..................................................................................................................................................................................................................... 15

Equation of Fluid Statics ....................................................................................................................................................................................... 15 Pressure and Temperature in the Atmosphere ........................................................................................................................................... 16 Manometers ............................................................................................................................................................................................................... 16

U-tube Manometers ...................................................................................................................................................................................................... 16 Differential Manometer .............................................................................................................................................................................................. 17

Horizontal Acceleration Effects ......................................................................................................................................................................... 17 Vertical Acceleration Effects ............................................................................................................................................................................... 17 Combined Acceleration ......................................................................................................................................................................................... 17 Example 3 - Manometer ........................................................................................................................................................................................ 18 Summary ..................................................................................................................................................................................................................... 18

Hydrostatic Forces on Surfaces ....................................................................................................................... 19 Forces on Horizontal and Surfaces ................................................................................................................................................................... 19 Line of Action – Centre of Pressure & Centre of Gravity .......................................................................................................................... 19

Complex Shapes .............................................................................................................................................................................................................. 19 Parallel Axis Theorem .................................................................................................................................................................................................. 20

Example 4 - Hydrostatics ...................................................................................................................................................................................... 20 Summary ..................................................................................................................................................................................................................... 21

Hydrostatics - Buoyancy ................................................................................................................................. 22 Forces on Floating or Submerged Bodies ...................................................................................................................................................... 22

Archimedes Principle ................................................................................................................................................................................................... 22

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Stability of Submerged Bodies ........................................................................................................................................................................... 22 Stability of Floating Bodies .................................................................................................................................................................................. 22

Stability .............................................................................................................................................................................................................................. 23 Determination of Metacentric Height .............................................................................................................................................................. 23 Example 5 – Buoyancy ........................................................................................................................................................................................... 24 Summary ..................................................................................................................................................................................................................... 24

Dynamics – Fluid Motion ................................................................................................................................ 25 Tools for Describing Fluid Motion .................................................................................................................................................................... 25

Control Surface ............................................................................................................................................................................................................... 25 Control Volume ............................................................................................................................................................................................................... 25 Continuity .......................................................................................................................................................................................................................... 25 Flow Visualization ......................................................................................................................................................................................................... 25 Streamlines ....................................................................................................................................................................................................................... 25

1, 2 & 3 Dimensional Flows ................................................................................................................................................................................. 25 Continuity – Discharge & Mean Velocity ........................................................................................................................................................ 26 Reference Frames .................................................................................................................................................................................................... 26

Eulerian .............................................................................................................................................................................................................................. 26 Lagrangian ....................................................................................................................................................................................................................... 26

Flow Acceleration .................................................................................................................................................................................................... 26 Equation of Motion .................................................................................................................................................................................................. 27

Acceleration ..................................................................................................................................................................................................................... 27 Differential form of the Continuity Equation .................................................................................................................................................... 27

Example 6 – Wave Runup ..................................................................................................................................................................................... 27 Summary ..................................................................................................................................................................................................................... 29

Dynamics – Energy Equation ........................................................................................................................... 30 Potential Energy (PE) .................................................................................................................................................................................................. 30 Kinetic Energy (KE) ...................................................................................................................................................................................................... 30 Pressure Energy .............................................................................................................................................................................................................. 30 General Energy Equation ........................................................................................................................................................................................... 30

Energy Equation {E} or Bernoulli Equation {B} .......................................................................................................................................... 30 Steady Flow Energy Equation .................................................................................................................................................................................. 30

Piezometric and Energy/Total Head Lines ................................................................................................................................................... 30 E-line V P-Line ................................................................................................................................................................................................................. 31 Piezometric Pressure Equation ............................................................................................................................................................................... 31

Pressure & Velocity ................................................................................................................................................................................................. 31 Flow Meters ................................................................................................................................................................................................................ 32

Pitot or Total Head Tubes .......................................................................................................................................................................................... 32 Venturi Meter ................................................................................................................................................................................................................... 32 Orifice Meter .................................................................................................................................................................................................................... 32

Discharge Equation for Venturi Meter or Orifice Meter .......................................................................................................................... 33 Example 7 – Flow Meters – Venturi & Orifice .............................................................................................................................................. 33 Kinetic Energy Correction .................................................................................................................................................................................... 34 Fluid Power & Work ............................................................................................................................................................................................... 35

Power between two Points ........................................................................................................................................................................................ 35 Radial Flow & the Energy Equation ................................................................................................................................................................. 35 Jet Discharge .............................................................................................................................................................................................................. 35 Example 8 – Energy & Piezometric Lines ...................................................................................................................................................... 35 Summary ..................................................................................................................................................................................................................... 36

Dynamics – Momentum Equation .................................................................................................................. 37 Momentum Equation {M} ..................................................................................................................................................................................... 37

Convective Acceleration .............................................................................................................................................................................................. 37 Temporal Acceleration ................................................................................................................................................................................................ 37

Applications ............................................................................................................................................................................................................... 37 Force on a Pipe Bend .................................................................................................................................................................................................... 37

Example 9 – Force on a Contraction ................................................................................................................................................................ 38 Jet Impact .................................................................................................................................................................................................................... 39 Jet Reaction or Thrust ............................................................................................................................................................................................ 39 Euler’s Equation ....................................................................................................................................................................................................... 39

Euler’s Equation for Steady Fluid Motion .......................................................................................................................................................... 40

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Euler & Bernoulli ............................................................................................................................................................................................................ 40 Euler’s Equation for Unsteady flow ....................................................................................................................................................................... 40

Example 10 – Head Loss at Sudden Expansion ........................................................................................................................................... 41 Summary ..................................................................................................................................................................................................................... 41

General ................................................................................................................................................................................................................................ 42

Real & Ideal Fluids .......................................................................................................................................... 43 Viscous Flow .............................................................................................................................................................................................................. 44 Laminar Flow ............................................................................................................................................................................................................. 44 Stress/Strain Relationship: Newton’s Law of Viscosity ........................................................................................................................... 44 Viscous & Inertial Forces ...................................................................................................................................................................................... 44

Turbulent Flow ............................................................................................................................................................................................................... 45 Reynolds Number ........................................................................................................................................................................................................... 45

Laminar & Turbulent Flow Regimes ................................................................................................................................................................ 46 Example 11 – Reynolds Number ....................................................................................................................................................................... 46 Parallel Flow .............................................................................................................................................................................................................. 47

General Governing Equation for Steady Parallel Laminar Flow ............................................................................................................. 48 Example 12 – Parallel Flow ................................................................................................................................................................................. 48 Laminar Flow in Pipes – Velocity Profile ....................................................................................................................................................... 49

Parabolic Profile............................................................................................................................................................................................................. 50 Laminar Flow in Pipes ........................................................................................................................................................................................... 50 Steady Flow in Pipes ............................................................................................................................................................................................... 50

Momentum Equation ................................................................................................................................................................................................... 50 Head Loss – Hagen Poiseuille Equation .............................................................................................................................................................. 51

Summary – Viscous fluids & Head Loss .......................................................................................................................................................... 51

Turbulent Flow ............................................................................................................................................... 53 Darcy-Weisbach Equation for Flow in Pipes ................................................................................................................................................ 53 Example 13 – Laminar & Turbulent Flow Head Loss in Pipes .............................................................................................................. 53 Velocity Profile in a Pipe ....................................................................................................................................................................................... 54

Viscous or Laminar Sub Layer ................................................................................................................................................................................. 55 Eddy Viscosity .................................................................................................................................................................................................................. 55

Development of Velocity Profiles in Pipes – Laminar Flow ................................................................................................................... 56 Development of Velocity Profiles in Pipes – Turbulent Flow ................................................................................................................ 56 Boundary Roughness ............................................................................................................................................................................................. 56 Friction Factors: Ks V L ....................................................................................................................................................................................... 56

Smooth Turbulent Flow .............................................................................................................................................................................................. 57 Transitional Flow .......................................................................................................................................................................................................... 57 Rough Turbulent Flow ................................................................................................................................................................................................. 57

Example 14 – Pipe Friction Factors .................................................................................................................................................................. 57 Summary ..................................................................................................................................................................................................................... 58

Pipeline & Pipe Network Design ..................................................................................................................... 59 Local Losses in Pipe Flow – Flow Separation Losses ................................................................................................................................ 59 Local Losses at an Expansion .............................................................................................................................................................................. 59

Head Loss at Sudden Expansion.............................................................................................................................................................................. 59 Local Losses – Empirical Relationships .......................................................................................................................................................... 59

Intakes................................................................................................................................................................................................................................. 60 Exits/Outfalls ................................................................................................................................................................................................................... 60 Expansion/Contraction ............................................................................................................................................................................................... 60 Bends, Valves .................................................................................................................................................................................................................... 60

Pipeline Design ......................................................................................................................................................................................................... 60 Two Cases .......................................................................................................................................................................................................................... 60

Example 15 – Pipelines.......................................................................................................................................................................................... 61 Example 16 – Pipelines.......................................................................................................................................................................................... 62 Pipe Networks ........................................................................................................................................................................................................... 62

Pipes in Series .................................................................................................................................................................................................................. 62 Pipes in Parallel .............................................................................................................................................................................................................. 62

Example 17 – Cofferdam ....................................................................................................................................................................................... 63 Hardy-Cross Technique ......................................................................................................................................................................................... 64

The Three Relationships to Satisfy ........................................................................................................................................................................ 64 Example 18 – Hardy-Cross Method .................................................................................................................................................................. 65

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Summary ..................................................................................................................................................................................................................... 66

Dynamic Fluid Loading .................................................................................................................................... 67 Design Requirements ............................................................................................................................................................................................. 67

Example Wave Conditions ......................................................................................................................................................................................... 67 Currents .............................................................................................................................................................................................................................. 67

Design Requirements - Loading Regimes ...................................................................................................................................................... 67 Ideal Flow Approach .................................................................................................................................................................................................... 68 Real Fluids & Viscous Fluids ...................................................................................................................................................................................... 68

Fluid Loading ............................................................................................................................................................................................................. 68 Pressure Drag .................................................................................................................................................................................................................. 68 Total Drag ......................................................................................................................................................................................................................... 69 Drag & Lift Force............................................................................................................................................................................................................ 69

Boundary Layer Separation ................................................................................................................................................................................. 70 Terminal Velocity ..................................................................................................................................................................................................... 70 Example 19 – Terminal Velocity ........................................................................................................................................................................ 70 Skin Friction ............................................................................................................................................................................................................... 71 Boundary Layer Drag – Surface Roughness .................................................................................................................................................. 71 Loading Regimes – Real & Viscous Flows ...................................................................................................................................................... 72 Spin & Lift – Magnus Effect .................................................................................................................................................................................. 72 Unsteady Fluid Loading – Total in-line force ............................................................................................................................................... 72

Morison equation ........................................................................................................................................................................................................... 72 Hydrodynamic mass (added mass) and an Accelerating Body ............................................................................................................. 72 Example 20 – Unsteady Loading ....................................................................................................................................................................... 73 Summary ..................................................................................................................................................................................................................... 74

Dimensional Analysis & Similarity .................................................................................................................. 75 Importance of Dimensions ................................................................................................................................................................................... 75

Fundamental Dimensions .......................................................................................................................................................................................... 75 Buckingham Method ........................................................................................................................................................................................... 75 Step-by-step Method .............................................................................................................................................................................................. 75 Common Groups ....................................................................................................................................................................................................... 76 Example 21 – Dimensional Analysis ................................................................................................................................................................ 77 Similarity ..................................................................................................................................................................................................................... 78

Geometric Similarity .................................................................................................................................................................................................... 78 Dynamic Similarity ....................................................................................................................................................................................................... 78 Perfect Similarity ........................................................................................................................................................................................................... 79

Froude Scaling ........................................................................................................................................................................................................... 79 Froude Similitude – Undistorted Models ....................................................................................................................................................... 79 Reynolds Scaling – Acceleration & Viscous Forces .................................................................................................................................... 80 Reynolds Similitude – Undistorted Models ................................................................................................................................................... 80 Scale Modelling ......................................................................................................................................................................................................... 81

Problems ............................................................................................................................................................................................................................ 81 Other Scale Relationships .......................................................................................................................................................................................... 81

Example 22 – Similitude ....................................................................................................................................................................................... 81 Summary ..................................................................................................................................................................................................................... 82

Unsteady Flow in Pipe Systems ...................................................................................................................... 84 Unsteady Flows in Closed Conduits ................................................................................................................................................................. 84 Analysis technique .................................................................................................................................................................................................. 84 Slow Variations in Discharge .............................................................................................................................................................................. 84 Time Required for Head Change ....................................................................................................................................................................... 85 More Rapid Changes in Discharge .................................................................................................................................................................... 85 Rigid Column Theory.............................................................................................................................................................................................. 85 Example 23 – Rigid Column Theory ................................................................................................................................................................. 86 Unsteady Energy Equation .................................................................................................................................................................................. 89 Effect of Acceleration ............................................................................................................................................................................................. 89 Applications of Unsteady Energy Equation .................................................................................................................................................. 89 Example 24 – Unsteady Flow with Acceleration: Flow between two Reservoirs ......................................................................... 90 Summary ..................................................................................................................................................................................................................... 91

Surge Shafts ................................................................................................................................................... 92 Analysis ........................................................................................................................................................................................................................ 92

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Frictionless Solution ............................................................................................................................................................................................... 93 Solution Including Friction .................................................................................................................................................................................. 93 Types ............................................................................................................................................................................................................................. 94 Surge Protection in Pumped Mains .................................................................................................................................................................. 94 Example 25 – Surge Shaft ..................................................................................................................................................................................... 95 Summary ..................................................................................................................................................................................................................... 96

Waterhammer – Unsteady Compressible Flow ............................................................................................... 97 Shock Waves .............................................................................................................................................................................................................. 97 Shock Wave Propagation ...................................................................................................................................................................................... 97

Steps ..................................................................................................................................................................................................................................... 98 Shock Wave Velocity ............................................................................................................................................................................................... 99 Waterhammer Pressure ........................................................................................................................................................................................ 99 Pressure Variation with Time .......................................................................................................................................................................... 100 Friction Effects ....................................................................................................................................................................................................... 100 Rate of Valve Closure ........................................................................................................................................................................................... 100 Waterhammer Theory Summary ................................................................................................................................................................... 101 Example 26 - Waterhammer ............................................................................................................................................................................ 101 Summary .................................................................................................................................................................................................................. 102

Appendix ...................................................................................................................................................... 103 Lecture Schedule ................................................................................................................................................................................................... 103 Experiment 1 – Flow Meters ............................................................................................................................................................................ 104 Experiment 2 – Sluice Gate ............................................................................................................................................................................... 105 Experiment 3 – Pipe Flow ................................................................................................................................................................................. 106 Experiment 4 – Drag on Cylinder ................................................................................................................................................................... 107 Worked Lecture Examples ...................................................................................................................... Error! Bookmark not defined. Moody Diagram ........................................................................................................................................... Error! Bookmark not defined. Drag Coefficient Charts ............................................................................................................................. Error! Bookmark not defined. AS2200-2006: Resistance Coefficients (k) of Valves and Fittings .......................................... Error! Bookmark not defined.

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Index Page accelerate total mass, 71 Acceleration, 28 added mass coefficient, 71 Adiabatic expansion, 13 Archimedes Principle, 23 atmospheric pressure variation, 17 Bernoulli Equation, 15 Boundary Layer Drag, 70 Boundary Layer Separation, 69 Buckingham Method, 74 bulk modulus of the fluid, 12 Buoyancy, 23 CD, 69 Celerity, 98 centre of buoyancy, 23 centre of gravity, 20, 23 centre of pressure, 20 Chezy equation, 52 Compressibility, 12 Continuity, 14, 26 Continuum Concept, 12 Control Surface, 26 Control Volume, 26 Convective Acceleration, 27 Currents, 66 Darcy-Weisbach, 52 Density, 12 dimensionless groups, 75 Discharge Equation for Venturi

Meter, 33 Drag, 68 Dynamic Similarity, 77 Dynamic viscosity, 13 Eddy Viscosity, 54 E-line, 32 Energy Equation, 15, 31 energy loss, 50 Euler & Bernoulli, 40 Euler’s Equation for Steady Fluid

Motion, 40 Euler’s Equation for Unsteady flow,

40 Euler’s formula, 18 Eulerian, 27 Flow Regimes, 46 Flow separation, 58 force on a vertical surface, 20 friction drag, 70 Friction Factors, 55 Froude Scaling, 78 Froude Similitude, 78 Gas Law, 17 Geometric Similarity, 77 Hagen Poiseuille Equation, 50 Hardy-Cross Technique, 63

Head Loss, 50 Head Loss at Sudden Expansion, 58 hydraulic diameter, 52 Hydraulically rough, 55 Hydraulically smooth, 55 Hydrodynamic mass, 71 I00, 24 inertia coefficient, 71 instantaneous complete closure, 99 Isothermal expansion, 13 jet discharge, 35 Jet Reaction, 39 Kinematic viscosity, 13, 44 Kinetic Energy, 14 Kinetic Energy Correction, 35 Ks V L, 55 ks values, 55 Lagrangian, 27 Laminar Flow in Pipes, 49 Laminar Sub Layer, 54 Lapse rate, 17 Lift, 68 Lift force, 68 Loading Regimes, 66 Local Losses, 58 Logarithmic Velocity profile, 54 Mach number, 75 Magnus Effect, 71 Manometers, 17 Mass flow rate, 14 metacentric height, 23 Momentum, 14 Momentum Equation, 37 Momentum flux, 14 More Rapid Changes in Discharge,

84 Morison equation, 71 Moving control volumes, 42 Orifice Meter, 33 Parabolic Profile, 49 Parallel Axis Theorem, 21 Parallel Flow, 47 Pascal’s Law, 16 penstock, 91 Perfect Similarity, 78 physical parameters, 74 Piezometric Pressure Equation, 32 Pipes in Parallel, 61 Pipes in Series, 61 Pitot or Total Head Tubes, 32 P-line, 32 Potential Energy, 14 Power, 35 Pressure, 16 Pressure Drag, 67 Pressure Energy, 14

Pressure Transmission, 16 Rate of change of momentum, 14 rate of strain of fluid, 48 Rate of Valve Closure, 99 Reference Frames, 27 Reynolds Number, 45 Reynolds Scaling, 79 Reynolds Similitude, 79 Rigid Column Theory, 84 Scale Modelling, 80 shear stress velocity, 54 shock wave, 96 Shock Wave Velocity, 98 skin coefficient, 70 Skin Friction, 70 Spatial Variation, 14 Specific Gravity, 12 Stability of submerged bodies, 23 Stagnation pressure, 68 Steady Flow Energy Equation, 31 Steady Flow in Pipes, 50 Steady Parallel Laminar Flow, 48 Step-by-step Method, 74 streakline, 26 Streamlines, 26 Streamtubes, 26 Stress/Strain Relationship, 44 Strouhal number, 71 sudden expansion, 58 Surface Tension, 13 Surge Pressure, 85 Surge shafts, 91 Temperature in the Atmosphere, 17 Temporal (local) Acceleration, 27 Temporal Variation, 14 Terminal Velocity, 69 Time Required for Head Change, 84 Total Drag, 68 Unsteady Energy Equation, 88 Unsteady Flows in Closed Conduits,

83 Unsteady Fluid Loading, 71 valve closure, 99 Variations in Discharge, 83 velocity head, 32 Velocity Profile in a Pipe, 53 Venturi Meter, 33 Viscosity, 13, 44 Viscous force, 79 Von Karman constant, 53 vortex shedding frequency, 71 Wake, 68 waterhammer theory, 96 Wave Conditions, 66 Weber number, 75 wetted area, 70 Work, 35

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Symbols

~ centre of pressure

a ~ acceleration

A ~ cross sectional area (m2)

B ~ centre of buoyancy

BG ~ height of G (centre of gravity) above B (centre

of buoyancy)

BM ~ height of M (metacentre) above B (centre of

buoyancy)

c ~ celerity

Cc ~ coefficient of contraction

Cd ~ actual discharge correction factor

CD ~ drag coefficient

Cf ~ average skin coefficient

cf ~ skin coefficient

Cm ~ inertia coefficient

CP ~ centre of pressure

Cp ~ elastic celerity

CV ~ control volume

D ~ diameter

DH ~ hydraulic diameter (used when non circular

pipes)

Ev ~ bulk (or volume) modulus of the fluid

FD ~ drag force

FL ~ lift force

G ~ centre of gravity

GM ~ metacentric height

H ~ height

hf ~ head loss due to friction

hL ~ head loss (for turbulent flow, includes friction

factor from hf)

K ~ bulk modulus of the fluid (N/m2, Pa) –

measures substance resistance to uniform

compression

k ~ Von Karman constant 0.4

kM ~ added mass coefficient

ks ~ roughness of the boundary

M ~ metacentre

p ~ pressure (N/m2, Pa)

Pw ~ wetted perimeter

Q ~ flow rate

Q ~ volumetric flow rate or discharge

R ~ gas constant for a particular gas (J Kg-1 K-1)

R ~ upthrust

S ~ specific gravity

T ~ temperature (C)

U ~ local velocity, mean flow velocity

u* ~ shear stress velocity

V ~ velocity

z ~ height

~ lapse rate

~ unit weight of fluid

H ~ surge pressure

~ turbulent eddy viscosity

~ dynamic viscosity, absolute coefficient of

viscosity (Ns/m2) or (kg/ms)

~ kinematic viscosity (nu) (m2/s)

~ density (kg/m3)

Q ~ mass flow rate (kg/s)

~ surface tension (N/m)

~ shear stress (N/m2)

~ angular velocity

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Approximations

Velocity in a river

o Non-uniform flow cross section

Width constantly changing flow velocity constantly changing

o Boundary layer

Indicated by velocity profile close to bed

Not constant; mud, sand, vegetation etc

Difficult to calculate

o Turbulent flow

o Varying bed roughness

Turbulent Pipe flow

o Steady flow to start

o Turbulent eddies created at bends

Solution

o Ignore details by time averaging

o Ignore details inside Control Volume (CV)

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Fundamentals

Main principles

o Continuity, {C}

Q UA

discharge or volume flow rate, (m3/s)

o Momentum, {M}

F ma QU = gV

Force = mass x acceleration (N)

o Energy, {E}

H p1

g

U1

2

2g z1

Sum of potential, kinetic and pressure energy = total head (m)

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Common Equations & Values

dh

dtU

1 ton = 8896N

10hPa = 1kPa

1g/cm3 = 1000kg/ m3

1L = 0.001m3 = 10-6cm3

Acceleration

a v 2

r w 2r

Atmospheric pressure = bar, atm

Centroid of a Parabola

3

8X

cos2 = 1 – sin2

dA = 2rdr

Density

o water = 1000kg/m3

o seawater = 1025kg/m3

o air = 1.2kg/m3

Differentiation of cos -sin

Differentiation of sin cos

Efficiency: Power required = power/%

Extrapolating

Y YA X XA

XB XA

YB YA

Gas Law PV = mRT

Head =

h p1 p2

g

Power = gQh = force x speed = work/unit time

Pressure

o Patmos = 101.3kPa

o Pwater = 50kPa

o Pwater vapour = 2.3kPa

o Pair = 60Pa

Rotation rate = rpm – revolutions per minute

Steady flow = no acceleration

dU

dt 0

Temperature Absolute = T+ 273

Velocity

v wr

Viscosity

o water=1x10-3kg/ms

o seawater=1.13x10-3kg/ms

o vwater=1x10-6 m2/s

o vair = 1.5x10-5 m2/s

Volume sphere =

D3

6

4

3r3

Wave Period =

2

t

Wave velocity = U cos (wt)

Weight = F = gV

Youngs Modulus

o Ewater = 2x109

o Esteel = 2.1x1011

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Nature of Fluids

Fluids = liquids & gases

Fluids can

o Flow

o Change shape

o Take up the shape of the boundaries Strong intermolecular forces at boundary

o Deform continuously and permanently under application of a shearing stress,

Fluid at rest

o no shearing forces acting

o Therefore, all forces in the fluid are perpendicular to the surfaces on which they act

Fluids in motion

o Molecules adhere to the boundaries – no slip condition

o Velocity varies away from the boundary

o Fluid element deforms

o

Continuum Concept

If there are enough molecules, the average conditions (pressure, density etc) are considered constant or change

smoothly

o Valid – liquids and most gases

Tightly packed molecules

o Not valid – rarefied gases, small number of molecules

Large spaces between molecules

Fluid Properties

Density

- Mass per unit volume, kg/m3

Typical values

o air = 1.2 kg/m3

o water = 1000 kg/m3 (15C)

Specific Gravity

o S =

fluid

water

Compressibility

o Density varies with pressure and temperature

Change in volume with a change in pressure depends on the bulk modulus of the fluid, K (N/m2)

p

K where Kwater = 2.05x109 N/m2

Force per unit area exerted on the fluid by the boundary and vice-versa is the shear stress,

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o Generally, water can be treated as incompressible

Except for very large changes in pressure (i.e. a waterhammer)

Gas Law

Air is more compressible but can be treated as incompressible at velocities much lower than the speed of sound

Ideal Gas Law

o

p RT

R = gas constant for a particular gas

Rair = 287 J Kg-1 K-1

T = temperature

Isothermal expansion – no change in temperature

Adiabatic expansion – no heat exchange out of the system

Viscosity

Measure of how easily a fluid flows

o Dynamic viscosity, , Ns/m2

Also called absolute coefficient of viscosity (kg/ms)

water = 1x10-3 kg/ms (20C)

o Kinematic viscosity, (nu) alternative, frequently used

(m2/s)

water = 1x10-6 m2/s (20C)

Newtons Law of Viscosity

o

F

A

du

dy

Newtonian fluids obey this law – most common fluids

Surface Tension

, N/m

o arises from elasticity of the surface

o reduces surface area to a minimum

o Causes capillary rise between surfaces

Manometer tubes

Errors in readings

o

Types of Flow

Laminar flow

Smooth, uniform, regular

Weight of column of fluid = surface tension force acting on wetted length

gD2

4H D cos

H 4 cos

gD so small, so cos = 1

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Turbulent Flow

Chaotic, random, dispersive

Temporal Variation

Flow variation with time

o Steady flows

Velocity and depth constant with time

o Unsteady flows

Velocity and depth vary with time

Spatial Variation

Flow variation with space

o Uniform

Flow properties constant in direction

o Non-uniform

Flow properties vary in flow direction

Governing Principles

Continuity

Continuity, {C}

Conservation of mass

o Mass flow rate = Q (kg/s)

No storage

Q1 = Q2

Incompressible fluid, or no change in density

Q1 = U1A1 = U2A2 = Q2 Continuity equation {C}

o A = cross sectional area

o U = mean flow velocity

Momentum

Momentum, {M}

o Conservation of momentum Newton’s Second Law

F = ma

o Rate of change of momentum = sum of forces

o Momentum flux = mass flow rate x velocity

Or rate at which momentum passes through a cross section

QU UA U

o Rate of change between cross sections = QU

Requires a resultant force F in the direction of motion

F QU Q U2 U1 Momentum equation {M}

Energy

Energy, total head, head, H

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o The sum of three forms of energy

Kinetic Energy, ½U2

Potential Energy, gz

Pressure Energy, p

o Conserved when no energy lost

p1 1

2U1

2 gz1 const

when divided through by g gives H in dimensions of length (m)

H p1

g

U1

2

2g z1 const

Energy Equation {E}

o also known as Bernoulli Equation {B}

Applications of Governing Principles

{C}

o eliminate unknowns

o write velocity in terms of area

{E}

o determine how velocity and pressure vary in the flow

o to find energy losses

{M}

o find forces

o determine how pressure and velocity vary if there are energy losses

Example 2 – Types of Flow

Steady flow

o Flow from a reservoir with constant head and boundary conditions (does not vary with time)

Steady & uniform flow

o Flow from a reservoir with constant head and boundary conditions into a long straight pipe (cross section not

changing, no change in flow)

Steady, non-uniform flow

o Flow from a reservoir with constant head and boundary conditions into a converging pipe or through an orifice

(conditions change along the pipe)

Unsteady, uniform flow

o Flow in an oil pipeline controlled by a variable speed pump (time varying)

Unsteady, non-uniform flow

o Wave motion

o Arterial flow

o Surge towers

o Domestic plumbing

Compressible unsteady flow or shockwave

o Power trip in hydro-electric plant (worst possible problem)

Rotational or vortex flow

o Flow into a vertical orifice

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Hydrostatic Pressure

Pressure

Fluid at rest

o Static equilibrium

o No shearing forces perpendicular or normal forces only

Shear only occurs when moving

Scalar quantity

Intensity & magnitude equal in all directions

p F

A (N/m2, Pa)

o Small forces over small areas give large pressures

Hydraulic presses, stiletto heels

o Small forces over large areas give large forces

Wind flow over roof

Absolute Pressure

Pressure in a vacuum = absolute zero

Gauge Pressure

Measured relative to local atmospheric pressure and can be positive or negative

Most common

Example

o Pressure reading is 50kPa where atmospheric pressure is 100kPa

Gauge pressure = 50kPa

Absolute pressure = 150kPa

Atmospheric Pressure

105N/m2 1 atm or 1 bar or 10m H2O

Pressure Transmission

Occurs in closed systems and can be used to amplify forces – hydraulic

controls

p1 = p2

F2 A2

A1

F1 ; where A2/A1 is the amplifier

Varying pressures (dynamic pressures) may be damped by viscous effects, but mean or static pressures are not

Pascal’s Law

o Pressure change at one point in a system is transmitted through the entire system

Equation of Fluid Statics

A fluid at rest is in equilibrium

Hydrostatic relationship for an incompressible fluid

o Vertical equilibrium

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dp

dz g

p gz const

o Horizontal equilibrium

dp

dx 0,

dp

dy 0

Implies equality of pressure at the same level in the same static fluid or in combinations of

static fluids

o PRESSURE IS ADDITIVE in the vertical for immiscible fluids (fluids that don’t mix)

Pressure and Temperature in the Atmosphere

Pressure, density & temperature vary with elevation in the atmosphere

Troposphere

o Sea level 13000m

o Temperature drops linearly with increasing elevation

Lapse rate, = drop in T with z

avg conditions = 5.87K/km

With the Gas Law, pressure and density can be calculated at any elevation

o

p RT

T T0 (z zo)

Using the hydrostatic equation gives

o

p p0

T0 z z0 T0

g

R

atmospheric pressure variation in the atmosphere

Manometers

Head & pressure measurement

o Fluid most often water, but can be any fluid

water = 1000kg/m3

= g

= unit weight of fluid

o Manometer Equation h = (S-1)h or h = (1-S)h

U-tube Manometers

A type of manometer with immiscible fluids

(will not mix), used for measuring pressures in

gases and higher pressures in liquids

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Differential Manometer

Created by connecting both ends of the manometer to the pipe

Similar in principle to the U-tube manometer

p = (2 - 1)gh = (2 - 1)h

Horizontal Acceleration Effects

A particle on the surface of a fluid under constant horizontal acceleration, a

Surface is a plane at angle to the horizontal

Vertical acceleration = 0

o No additional vertical forces, hence hydrostatic

pressure

Planes of equal pressure lie parallel to the free surface

Vertical Acceleration Effects

Pressure increases with upward acceleration, similar to going up in an elevator

Oscillatory flows (waves, surge towers) have non-hydrostatic pressure below the accelerating free surface not pure

hydrostatic

Combined Acceleration

Total acceleration = pressure gradient

o Euler’s formula

p

x ax

p

z g az

o if az is small, pressure is hydrostatic - the free surface slope gives the total horizontal acceleration

For rotating flows, the horizontal acceleration varies with radial distance from the centre of rotation – free surface

becomes parabolic – a forced vortex

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Example 3 - Manometer

In general

o

p1 p2

g z1 z2

1 2

h

S1 S2 h

Summary

Fluid at rest = no shearing forces

Hydrostatic pressure increases linearly with depth

Pressure is equal along lines of constant elevation

Pressures additive in the vertical

Manometers measure gage or relative pressures

Acceleration gives rise to additional pressure gradients

Surface slopes indicate pressure gradients