applied hydrodynamics

Applied Hydrodynamics

2009 by Taylor & Francis Group, LLC

Dedication/Ddicace

To Ya Hui.Pour Bernard, Nicole et Andr.


Applied Hydrodynamics:An Introduction to Idealand Real Fluid Flows

Hubert ChansonProfessor of Civil Engineering, School of Engineering,The University of Queensland, Brisbane QLD 4072, Australia


CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

2009 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business

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Table of Contents

List of Symbols VII

Acknowledgements XI

About the Author XIII

1 Introduction 1

2 Fundamental Equations 7

Part I Irrotational Flow Motion of Ideal Fluid

I-1 Introduction to Ideal Fluid Flows 17

I-2 Ideal Fluid Flows and Irrotational Flow Motion 29

I-3 Two-Dimensional Flows (1) Basic Equations and Flow Analogies 55

I-4 Two-Dimensional Flows (2) Basic Flow Patterns 81

I-5 Complex Potential, Velocity Potential and Joukowski Transformation 137

I-6 Joukowski Transformation, Theorem of Kutta-Joukowski andLift Force on Airfoil 167

I-7 Theorem of Schwarz-Christoffel, Free Streamlines and Applications 185

Part II Real Fluid Flows: Theory and Applications

II-1 Introduction 229

II-2 Turbulence: An Introduction 241

II-3 Boundary Layer Theory Application to Laminar BoundaryLayer Flows 265

II-4 Turbulent Boundary Layers 295


VI Table of Contents

Appendix A Glossary 331

Appendix B Constants and Fluid Properties 357

Appendix C Unit Conversions 363

Appendix D Mathematics 367

Appendix E The Software 2D Flow+ 381

Appendix F Whirlpools in the World 383

Appendix G Examples of Civil Engineering Structures in theAtmospheric Boundary Layer 389

Appendix H Boundary Shear Stress Measurements with Pitot Tubes 397

Assignment A Application to the Design of the Alcyone 2 399

Assignment B Applications to Civil Design on the Gold Coast 405

Assignment C Wind Flow Past a Series of Circular Buildings 413

Assignment D Prototype Freighter Testing 417

References 421

Citation Index 439

Subject Index 441

Suggestion/Correction Form 447


List of Symbols

A cross-section area (m2);C celerity (m/s);CD drag coefficient;

CD = Drag12

V2O chordfor a two-dimensional object;

CL lift coefficient;

CL = Lift12

V2O chordfor a two-dimensional object;

Cc contraction coefficient;Cd discharge coefficient;Cv energy loss coefficient;DH hydraulic diameter (m):

DH = 4 cross sectional areawetted perimeter =4 A

Pw

d flow depth (m);e internal energy per unit mass (J/kg);Fp pressure force (N);Fv volume force (N);f Darcy coefficient (or head loss coefficient, friction factor);fvisc viscous force (N);g gravity constant (m/s2): g= 9.80m/s2 (in Brisbane);H 1. total head (m) defined as:

H = P g + z +

V2

2 g2. piezometric head (m) defined as:

H = P g + z

h height (m);K 1. vortex strength (m2/s) or circulation;

2. hydraulic conductivity (m/s);k permeability (m2);k constant of proportionality;ks equivalent sand roughness height (m);L length (m);P absolute pressure (Pa);


VIII List of Symbols

Pd dynamic pressure (Pa);Ps static pressure (Pa);Q discharge (m3/s);q discharge per meter width (m2/s);R 1. circle radius (m);

2. cylinder radius (m);R1 radius (m);Ro gas constant: Ro= 8.3143 J/Kmole;r polar radial coordinate (m);T thermodynamic (or absolute) temperature (K);U volume force potential;V velocity (m/s);v specific volume (m3/kg):

v = 1

W complex potential: W=+ i;w complex velocity: w=Vx + iVy;x Cartesian coordinate (m);y Cartesian coordinate (m);z 1. altitude (m);

2. complex number (Chapters 5 & 6); velocity potential (m2/s); circulation (m2/s); specific heat ratio:

= CpCv

1. strength of doublet (m3/s);2. dynamic viscosity (N.s/m2 or Pa.s);

kinematic viscosity (m2/s):

=

= 3.141592653589793238462643; polar coordinate (radian); density (kg/m3); surface tension (N/m); shear stress (Pa);o average shear stress (Pa); 1. speed of rotation (rad/s);

2. hydrodynamic frequency (Hz) of vortex shedding; stream function vector; for a two-dimensional flow in the {x,y} plane:

= (0, 0, );

two-dimensional flow stream function (m2/s);


List of Symbols IX

Subscriptn normal component;o reference conditions: e.g., free-stream flow conditions;r radial component;s streamwise component;x x-component;y u-component;z z-component; ortho-radial component;

Notes1. Water at atmospheric pressure and 20.2 Celsius has a kinematic viscosity of

exactly 106 m2/s.2. Water in contact with air has a surface tension of about 0.073N/m.

Dimensionless numbersCa Cauchy number (Henderson 1966):

Ca = V2

ECOwhere Eco is the fluid compressibility;

CD drag coefficient for a structural shape:

CD = o12

V2= shear stress

dynamic pressure

where O is the shear stress (Pa);Note: other notations include Cd and Cf;

Fr Reech-Froude number:

Fr = Vg dcharac

Note: some authors use the notation:

Fr = V2

g dcharac = V2 A

g A dcharac =inertial force

weight

M Sarrau-Mach number:

M = VC

Nu Nusselt number:

Nu = ht dcharac

= heat transfer by convectionheat transfer by conduction

where ht is the heat transfer coefficient (W/m2/K) and is the thermal conductivity


X List of Symbols

Re Reynolds number:

Re = V dcharac

= inertial forcesviscous forces

Re shear Reynolds number:

Re= V ks

St Strouhal number:

St = dcharacVo

We Weber number:

We = V2

dcharac= inertial forces

surface tension forces

Note: some authors use the notation:

We = V

dcharac

CommentsThe variable dcharac characterises the geometric characteristic length (e.g. pipediameter, flow depth, sphere diameter, . . .).


Acknowledgements

The author wants to thank especially Professor Colin J. Apelt, University ofQueensland, for his help, support and assistance all along the academic careerof the writer. He thanks also Dr Sergio Montes, University of Tasmania for hispositive feedback and advice on the course material.

He expresses his gratitude to the following people who provided photographs andillustrations of interest:

Mr Jacques-Henri Bordes (France);Mr and Mrs J. Chanson (Paris, France);Ms Y.H. Chou (Brisbane, Australia);Mr Francis Fruchard (Lyon, France)Prof. C. Letchford, University of Tasmania;Dassault Aviation;Dryden Aircraft Photo Collection (NASA);Equipe Cousteau, France;NASA Earth Observatory;Northrop Grumman Corporation;Officine Maccaferri, Italy;Rafale International;Sequana-Normandie;VF communication, La Grande Arche;Vought Retiree Club;Washington State Department of Transport;

Last, but not least, the author thanks all the people (including colleagues, formerstudents, students, professionals) who gave him information, feedback and com-ments on his lecture material. In particular, he acknowledges: Mr and Mrs J. Chanson(Paris, France); Ms Y.H. Chou (Brisbane, Australia); Mr G. Illidge (University ofQueensland); Mrs N. Lemiere (France); Professor N. Rajaratnam (University of Alberta,Canada); Mr R. Stonard (University of Queensland).


About the Author

Hubert Chanson, Professor inHydraulic Engineering andApplied Fluid Mechanics

Hubert Chanson received a degree of IngnieurHydraulicien from the Ecole Nationale SuprieuredHydraulique et de Mcanique de Grenoble(France) in 1983 and a degree of Ingnieur GnieAtomique from the Institut National des Scienceset Techniques Nuclaires in 1984. He worked forthe industry in France as a R&D engineer at theAtomic Energy Commission from 1984 to 1986,and as a computer professional in fluid mechan-ics for Thomson-CSF between 1989 and 1990.From 1986 to 1988, he studied at the Universityof Canterbury (New Zealand) as part of a Ph.D.project.

Hubert Chanson is a Professor in Hydraulic Engi-neering and Applied Fluid Mechanics at the Uni-

versity of Queensland since 1990. His research interests include design of hydraulicstructures, experimental investigations of two-phase flows, coastal hydrodynam-ics, water quality modelling, environmental management and natural resources.He authored single-handedly seven books: Hydraulic Design of Stepped Cascades,Channels, Weirs and Spillways (Pergamon, 1995), Air Bubble Entrainment inFree-Surface Turbulent Shear Flows (Academic Press, 1997), The Hydraulics ofOpen Channel Flow: An Introduction (Butterworth-Heinemann, 1999 & 2004),The Hydraulics of Stepped Chutes and Spillways (Balkema, 2001), EnvironmentalHydraulics of Open Channel Flows (Elsevier, 2004) and Applied Hydrodynamics:an Introduction to Ideal and Real Fluid Flows (CRC Press, 2009). He co-authoredthe book Fluid Mechanics for Ecologists (IPC Press, 2002,2006) and he edited twofurther books (Balkema 2004, IEaust 2004). His textbook The Hydraulics of OpenChannel Flow: An Introduction has already been translated into Chinese (HydrologyBureau of Yellow River Conservancy Committee) and Spanish (McGraw Hill Interamer-icana), and it was re-edited in 2004. His publication record includes more than 450international refereed papers and his work was cited over 2,000 times since 1990.Hubert Chanson has been active also as consultant for both governmental agenciesand private organisations.

The International Association for Hydraulic engineering and Research (IAHR) pre-sented Hubert Chanson the 13th Arthur Ippen award for outstanding achievements inhydraulic engineering. This award is regarded as the highest achievement in hydraulicresearch. TheAmerican Society of Civil Engineers, Environmental andWater ResourcesInstitute (ASCE-EWRI) presented him with the 2004 award for the best practice paperin the Journal of Irrigation and Drainage Engineering (Energy Dissipation and AirEntrainment in Stepped Storm Waterway by Chanson and Toombes 2002). In 1999he was awarded a Doctor of Engineering from the University of Queensland foroutstanding research achievements in gas-liquid bubbly flows.


XIV About the Author

He has been awarded eight fellowships from the Australian Academy of Science. In1995 he was a Visiting Associate Professor at National Cheng Kung University (TaiwanR.O.C.) and he was Visiting Research Fellow at Toyohashi University of Technology(Japan) in 1999, 2001 and 2008. In 2004 and 2008, he was a visiting Research Fellowat Laboratoire Central des Ponts et Chausses (France). In 2008, he was an invitedProfessor at the Universit de Bordeaux, Laboratoire des Transferts, Ecoulements,Fluides, et Energetique.

Hubert Chanson was invited to deliver keynote lectures at the 1998 ASME FluidsEngineering Symposium on Flow Aeration (Washington DC), at the Workshop onFlow Characteristics around Hydraulic Structures (Nihon University, Japan 1998), atthe first International Conference of the International Federation for Environmen-tal Management System IFEMS01 (Tsurugi, Japan 2001), at the 6th InternationalConference on Civil Engineering (Isfahan, Iran 2003), at the 2003 IAHR BiennialCongress (Thessaloniki, Greece), at the International Conference on Hydraulic Designof Dams and River Structures HDRS04 (Tehran, Iran 2004), at the 9th InternationalSymposium on River Sedimentation ISRS04 (Yichang, China 2004), at the Interna-tional Junior Researcher and Engineer Workshop on Hydraulic Structures IJREW06(Montemor-o-Novo, Portugal 2006), at the 2nd International Conference on Estu-aries & Coasts ICEC-2006 (Guangzhou, China 2006), at the 16th Australasian FluidMechanics Conference 16AFMC (Gold Coast, Australia 2007), at the 2008 ASCE-EWRIWorld Environmental andWater Resources Congress (Hawaii, USA 2008), at the2nd International Junior Researcher and Engineer Workshop on Hydraulic StructuresIJREW06 (Pisa, Italy 2008), and at the 11th Congrs Francophone des TechniquesLaser CTFL 2008 (Poitiers, France 2008). He gave invited lectures at the InternationalWorkshop on Hydraulics of Stepped Spillways (ETH-Zrich, 2000), at the 2001 IAHRBiennial Congress (Beijing, China), at the International Workshop on State-of-the-Art in Hydraulic Engineering (Bari, Italy 2004), and at the Australian Partnership forSustainable Repositories Open Access Forum (Brisbane, Australia 2008). He lecturedseveral short courses in Australia and overseas (e.g. Taiwan, Japan, Italy). HubertChanson chairs the Organisation of the 34th IAHR Biennial Congress in Brisbane,Australia in June 2011.

His Internet home page is {http://www.uq.edu.au/e2hchans}. He also developed agallery of photographs website {http://www.uq.edu.au/e2hchans/photo.html} thatreceived more than 2,000 hits per month since inception. His open access publicationwebpage is the most downloaded publication record at the University of Queenslandopen access repository: {http://espace.library.uq.edu.au/list.php?browse=author&author_id=193}.

{http://www.uq.edu.au/ Gallery of photographs in watere2hchans/photo.html} engineering and environmental fluid mechanics

{http://www.uq.edu.au/ Internet technical resources in watere2hchans/url_menu.html} engineering and environmental fluid mechanics

{http://www.uq.edu.au/ Reprints of research papers in watere2hchans/reprints.html} engineering and environmental fluid mechanics

{http://espace.library.uq.edu.au/ Open access publications at UQeSpacelist.php?browse=author&author_id=193}


CHAPTER 1

Introduction

Fluid dynamics is the engineering science dealing with forces and energies generatedby fluids in motion. The study of hydrodynamics involves the application of the funda-mental principles of mechanics and thermodynamics to understand the dynamics offluid flowmotion. Fluid dynamics and hydrodynamics play a vital role in everyday lives.Practical examples include the flow motion in the kitchen sink, the exhaust fan abovethe stove, and the air conditioning system in our home. When we drive a car, the airflow around the vehicle body induces some drag which increases with the square ofthe car speed and contributes to fuel consumption. Engineering applications encom-pass fluid transport in pipes and canals, energy generation, environmental processesand transportation (cars, ships, aircrafts). Other applications includes coastal struc-tures, wind flow around buildings, fluid circulations in lakes, oceans and atmosphere(Fig. 1), even fluid motion in the human body.

The challenges are gigantic. The range of the relevant time scales in fluid motionis huge from less than 1 millisecond for the viscous dissipation time scale in a smallstream to 24 h 50min for a tidal cycle andmore than 50 years for the deep sea currentscontrolling the balance between oxygen and carbon dioxide in Earths atmosphere.Fluid mechanics and hydraulic engineering may have direct impacts onto the worldpolitics, as illustrated by past armed conflicts around water systems. In the Bible,a wind-setup effect allowed Moses and the Hebrews to cross shallow water lakesand marshes during their exodus. Droughts were artificially introduced: e.g., duringthe siege of the ancient city of Khara Khoto (Black City) in AD 1372, the Chinese armydiverted the Ezen river (also called Hei He river Black River by the Chinese) supplyingwater to the city. Artificial flooding created by dam and dyke destruction played a rolein History : e.g., the war between the cities of Lagash and Umma (Assyria) aroundBC 2,500 was fought for the control of irrigation systems and dykes. The 21st centuryis facing political instabilities centred around water systems, and the scope of therelevant problems is broad and complex: e.g., water quality, pollution, flooding,drought. An example is the disaster of the Aral Sea with the formation of the NorthAral Sea and South Aral Sea since 1987, and some recovery of the North Aral Seafollowing the completion of the Kok-Aral Dam in 2005 (Fig. 2). Natural disasters mayalso impact on armed conflicts. Japan was saved from an invading Mongol-Koreanfleet destroyed during a typhoon (divine wind) in 1281. The 2004 Indian Oceantsunami disaster had some direct impact on the armed conflicts in Aceh and Sri Lanka(Le Billon and Waizenegger 2007).


2 Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows

Figure 1Geophysical flow past a

circular disturbance :Karman street of cloudvortices in the wake of

two islands parts ofAlaskas Aleutian

archipel (Image providedby USGS EROS Data

Center Satellite SystemsBranch as part of theEarth as Art II image

series, Courtesy of NASAEarth Observatory) Thevortices were created by

the prevailing easterlywinds encountering theAleutian Islands Wind

flow from top right tobottom left (See Plate 1)

Civil, environmental and mechanical engineers require basic expertise in hydrody-namics, turbulence, multiphase flows and water chemistry. The education of thesefluid dynamic engineers is a challenge for present and future generations. Althoughsome introduction course is offered at undergraduate levels, most hydrodynamicsubjects are offered at postgraduate levels only, and they rarely develop the com-plex interactions between air and water. Too many professionals and governmentadministrators do not fully appreciate the complexity of fluid flow motion, nor theneeds for further higher education of quality. Todays engineering problems requireengineers with hydrodynamic expertise for a brand range of situations ranging fromdesign and evaluation to maintenance and decision-making. These challenges implya sound understanding of the physical processes and a solid grasp of the physicallaws governing fluid flow motion.

Notes

1. Located in the Gobi desert, Khara Khoto was ruled by the Mongol king KharaBator (Webster 2002).


Introduction 3

Figure 2Aral Sea on 15 April2005 (Courtesy of NASAEarth Observatory, NASAimage created usingdata provided courtesyof the MODIS RapidResponse Team at NASAGSFC) Once Earths 4thlargest lake, the Aral Seahad been steadilyshrinking since the1960s as a result ofirrigation projectsdiverting the main riverssustaining the sea; thelake shrunk to a quarterof its original size In2007, the North Aral Seastarted some recoveryfollowing theconstruction of a dam in2005 between the Southand North Aral Seas (seeArrow); the damseparates the North AralSea from its saltier andmore polluted southernhalf (See Plate 2)

2. Man-made flooding of an army or a city, by building an upstream dam and destroy-ing it, was carried out by the Assyrians (Babylon, Iraq BC 689), the Spartans(Mantinea, Greece BC 38584), the Chinese (Huai river, AD 51415). It may beadded the aborted attempt to blow up Ordunte dam, during the Spanish civil war,by the troops of General Franco, and the anticipation of German dam destruc-tion at the German-Swiss border to stop the crossing of the Rhine river by theAllied Forces in 1945 (R 1946). A related case was the air raids of the dam bustercampaign conducted by the British in 1943.

3. In the Greek mythology, Hercules diverted two rivers to wash the stables of KingAugeas.

STRUCTURE OF THE BOOK

This text deals with the topic of applied hydrodynamics. The lecture material isregrouped into two complementary sections: ideal fluid flow and real fluid flow.The former deals with two- and possibly three-dimensional fluid motions that arenot subjected to boundary friction effects, while the latter considers the flow regionsaffected by boundary friction and turbulent shear.



Section I develops the basic theory of fluid mechanics of ideal fluid with irrotationalflow motion. Under an appropriate set of conditions, the continuity and motionequations may be solved analytically. This technique is well-suited to two-dimensionalflows in regions where the effects of boundary friction are negligible: e.g., outside ofboundary layers. The outcomes include the entire flow properties (velocity magnitudeand direction, pressure) at any point. Although no ideal fluid actually exists, manyreal fluids have small viscosity and the effects of compressibility may be negligible.For fluids of low viscosity the viscosity effects are appreciable only in a narrow regionsurroundings the fluid boundaries. For incompressible flow where the boundary layerremains thin, non-viscous fluid results may be applied to real fluid to a satisfactorydegree of approximation. Applications include the motion of a solid through an idealfluid which is applicable with slight modification to the motion of an aircraft throughthe air, of a submarine through the oceans, flow through the passages of a pumpor compressor, or over the crest of a dam, and some geophysical flows. While thecomplex notation is introduced in the chapters I-5 to I-7, it is not central to the lecturematerial and could be omitted if the reader is not familiar with complex variables.

In Nature, three types of shear flows are encountered commonly : (1) jets and wakes,(2) developing boundary layers, and (3) fully-developed open channel flows. Section IIpresents the basic boundary layer flows and shear flows. The fundamentals of bound-ary layers are reviewed. The results are applied to both laminar and turbulent boundarylayers. Basic shear flow and jet applications are developed. The text material aims toemphasise the inter-relation between ideal and real-fluid flows. For example, the cal-culations of an ideal flow around a circular cylinder are presented in Chapter I-4 andcompared with real-fluid flow results (Fig. 1). Similarly, the ideal fluid flow equationsprovide the boundary conditions for the developing boundary layer flows (Chap. II-3and II-4). The calculations of lift force on air foil, developed for ideal-fluid flows (Chap.I-6), give good results for real-fluid flow past a wing at small to moderate angles ofincidence (Fig. 3).

Figure 3Hand glider (delta wing)

above la Ville Pichard,Plneuf-Val-Andr on

11 Apr. 2004


Introduction 5

The lecture material is supported by a series of appendices (A to H), while somemajor homework assignments are developed before the bibliographic references.The appendices include some glossary of scientific terms and technical terminology,basic fluid properties, unit conversion tables, mathematical aids, an introduction to anideal-fluid flow software, and some presentation of relevant real-fluid flows. Furtherlinks to two specialised hydrodynamic softwares are included in the relevant booksections.

Computational fluid dynamics (CFD) is largely ignored in the book. It is a subject initself and its inclusion would yield a too large material for an intermediate textbook.In many universities, computational fluid dynamics is taught as an advanced post-graduate subject for students with solid expertise and experience in fluid mechanicsand hydraulics. In line with the approach of Liggett (1994), this book aims to providea background for studying and applying CFD.

The lecture material is designed as an intermediate course in fluid dynamics for seniorundergraduate and postgraduate students in Civil, Environmental, Hydraulic andMechanical Engineering. Basic references on the topics of Section I include Streeter(1948) and Vallentine (1969). The first four chapters of the latter reference providessome very pedagogical lecture material for simple flow patterns and flow net analy-sis. References on the topics of real fluid flows include Schlichting (1979) and Liggett(1994). Relevant illustrations of flow motion comprise Van Dyke (1982), JSME (1988)and Homsy (2000, 2004)

Warnings

Sign conventions differ between various textbooks. In the present manuscript, thesign convention may differ sometimes from the above references.


CHAPTER 2

Fundamental Equations

Summary

After a short paragraph on fluid properties, the fundamental equations ofreal fluid flows are detailed. The particular case of ideal fluid is presented inthe next chapter (Chap. I-1).

1 INTRODUCTION

A problem may be analysed for a system of constant mass, or volume for anincompressible flow motion. The description of the flow motion is called a systemapproach and the basic equations are the integral forms of the continuity, momen-tum and energy principles. The technique yields global results without entering intothe details of the flow field at the small scale. For example, the application of theintegral form of the conservation of mass and conservation of momentum to theflow underneath a vertical sluice gives an expression of the pressure force applied tothe gate (Fig. 1A), without the need to know the pressure distribution on the gateupstream face.

Another technique, called field approach, gives a description of the flow field (pres-sure, velocity) at each point in the coordinate system. It is based upon the differentialforms of the basic principles: conservation of mass, of momentum and of energy. Forthe example of the flow beneath a vertical sluice gate, the field approach yields theentire pressure and velocity field at any point. The method is well-suited to study theflow field in the vicinity of the gate opening where the fluid is rapidly accelerated andthe pressure distribution is not hydrostatic (Fig. 1B). In this text, we will use primarilya field approach to gain a complete solution of the two- or three-dimensional flowproperties.



(A) Integral approach

1 2

d1

FG

d2

(B) Streamline and equipotential lines

Figure 1Flow beneath a vertical

sluice gate

2 FLUID PROPERTIES

The density of a fluid is defined as its mass per unit volume.

All real fluids resist any force tending to cause one layer to move over another butthis resistance is offered only while movement is taking place. The resistance to themovement of one layer of fluid over an adjoining one is referred to the viscosity ofthe fluid. Newtons law of viscosity postulates that, for the straight parallel motion ofa given fluid, the tangential stress between two adjacent layers is proportional to thevelocity gradient in a direction perpendicular to the layers:

= Vy

(1)

where is the dynamic viscosity of the fluid (Fig. 2).


Fundamental Equations 9

y

Control volume

V

Figure 2Sketch of atwo-dimensional flowpast a solid boundary

Notes

1. Isaac Newton (16421727) was an English mathematician (see Glossary).

2. The kinematic viscosity is the ratio of viscosity to mass density:

=

3. A Newtonian fluid is one in which the shear stress, in one-directional flow, is pro-portional to the rate of deformation asmeasured by the velocity gradient across theflow (i.e. Equation (1)). The common fluids such as air, water and light petroleumoils, are Newtonian fluids. Non-Newtonian fluids will not be considered any further.

4. Basic fluid properties including density and viscosity of air and water, includingfreshwater and seawater, are reported in Appendix B. Tables for unit conversionare presented in Appendix C.

3 REAL FLUID FLOWS

3.1 Presentation

All fluid flow situations are subjected to the following relationships: the first andsecond laws of thermodynamics, the law of conservation of mass, Newtons law ofmotion and the boundary conditions. Other relations (e.g. state equation, Newtonslaw of viscosity) may apply.

3.2 The continuity equation

The law of conservation of mass states that themass within a system remains constantwith time (disregarding relativity effects):

DM

Dt= D

Dt

x

y

z

dx dy dz = 0 (2)

where M is the total mass, t is the time, and x, y and z are the Cartesian co-ordinates.For an infinitesimal small control volume the continuity equation is:

t+ div( V ) = 0 (3a)



whereV is the velocity vector and div is the divergent vector operator. In Cartesian

coordinates, it yields:

t+

i=x,y,z

( Vi)xi

= 0 (3b)

where Vx, Vy and Vz are the velocity components in the x-, y- and z-directions respec-tively. For an incompressible flow (i.e. = constant) the continuity equation becomes:

divV = 0 (4a)

and in Cartesian coordinates:

i=x,y,z

Vixi

= 0 (4b)

Notes

1. The word Cartesian is named after the Frenchman Descartes. It is spelled with acapital C. Ren Descartes (15961650) was a French mathematician, scientist, andphilosopher who is recognised as the father of modern philosophy.

2. In Cartesian coordinates, the velocity components are Vx, Vy, Vz:

V = (Vx, Vy, Vz)

3. The vector notation is used herein to lighten the mathematical writings. The readerwill find some relevant mathematical aids in Appendix D.

4. For a two-dimensional and incompressible flow the continuity equations is:

Vxx

+ Vyy

= 0

5. Considering an incompressible fluid flowing in a pipe the continuity equation maybe integrated between two cross sections of areas A1 and A2. Denoting V1 andV2 the mean velocity across the sections, we obtain:

Q = V1 A1 = V2 A2

The above relationship is the integral form of the continuity equation.

3.3 The motion equation

Equation of motionNewtons second law of motion is expressed for a system as:

D

Dt(M V ) =

F (5)



where

F refers to the resultant of all external forces acting on the system (including

body forces such as gravity) andV is the velocity of the centre of mass of the system.

The forces acting on the control volume are (a) the surface forces (i.e. shear forces) and(b) the volume force (i.e. gravity). For an infinitesimal small volume, the momentumequation is applied to the i-component of the vector equation:

D( Vi)Dt

=( Vi)

t+

j=x,y,zVj ( Vi)

xj

= FVi +

j=x,y,z

ij

xj(6)

where Fv is the resultant of the volume forces, is the stress tensor (see Notes below)and i,j= x,y,z. If the volume forces FV are derived from a potential U (U=g z forthe gravity force), they can be rewritten as:

FV = grad U (7)

where grad is the gradient vector operator. In Cartesian coordinates, it yield:

Fvx = U

x

Fvy = U

y

Fvz = U

z

For a Newtonian fluid the shear forces are (1) the pressure forces and (2) the resul-tant of the viscous forces on the control volume. Hence, for a Newtonian fluid, themomentum equation becomes:

D( V )Dt

= Fv grad P + f visc (8a)

wherefvisc is the resultant of the viscous forces on the control volume and P is the

pressure. In Cartesian coordinates, it yields:

D( Vi )Dt

= FVi P

xi+ f visci (8b)

Assuming a constant viscosity over the control volume and using the expressions ofshear and normal stresses in terms of the viscosity and velocity gradients, the equationof motion becomes

D( Vi)Dt

= Fvi P

xi 2

3

j=x,y,z

2Vjxi xj

+

j=x,y,z

(2Vi

xj xj+

2Vjxj xi

)

(8c)



Notes

1. The gravity force equals:

FV = grad(g z)U = g x

where the z-axis is positive upward (i.e. U= g z) and g is the gravity acceleration(Appendix B).

2. The i-component of the vector of viscous forces is:

fi = div i =

j=x,y,z

ij

xj

3. For a Newtonian fluid the stress tensor is (Streeter 1948, p. 22):

ij = P ij + ij

ij = 2 3 e ij + 2 eij

where P is the static pressure, ij is the shear stress tensor, ij is the identity

matrix element: ii = 1 and ij = 0 (for i different of j), eij = 12 (

Vixj

+ Vjxi

)and

e= divV =j=x,y,z

Vixi

.

Note that for an incompressible flow the continuity equation gives: e= divV = 0

4. The equations of motion were first rigorously developed by Leonhard Euler andare usually referred as Eulers equations of motion.

5. Leonhard Euler (17071783) was a Swiss mathematician and a close friend ofDaniel Bernoulli (Swiss mathematician and hydrodynamist, 17001782).

3.3.2 Navier-Stokes equationFor an incompressible flow (i.e. = constant) the derivation of the equations ofmotionyields to the Navier-Stokes equation:

DV

Dt= Fv grad p + V (9a)

In Cartesian coordinates, it becomes:

Vi

t+ Vj

j=x,y,z

Vixj

= Fvi

P

xi+

j=x,y,z

2Vixj xj

(9b)



Dividing by the density, the Navier-Stokes equation becomes:

DViDt

= Fvi 1

P

xi+

Vi (9c)

where =/ is the kinematic viscosity.

For a two-dimensional flow and gravity forces, the Navier-Stokes equation is:

(Vxt

+ Vx Vxx

+ Vy Vxy

)

= x

(P + g z) + (

2Vxx x

+ 2Vx

y y

)(10a)

(Vyt

+ Vx Vyx

+ Vy Vyy

)

= y

(P + g z) + (

2Vyx x

+ 2Vy

y y

)(10b)

where z is taken as a coordinate which is positive vertically upward. Then (dz/dx) isthe cosine of the angle between the x-axis and the z-axis, and similarly (dz/dy) for they-axis and z-axis.

Notes

1. The viscous force term is a Laplacian:

Vi = 2Vix2

+ 2Viy2

+ 2Viz2

= Vi = div grad Vi

2. The equations were first derived by Navier in 1822 and Poisson in 1829 by anentirely different method. They were derived in a manner similar as above bySaint-Venant in 1843 and Stokes in 1845.

3. Henri Navier (17851835) was a French engineer who primarily designed bridgebut also extended Eulers equations of motion. Simon Denis Poisson (17811840)was a French mathematician and scientist. He developed the theory of elasticity,a theory of electricity and a theory of magnetism. The Frenchman Adhmar JeanClaude Barr De Saint-Venant (17971886) developed the equations of motionof a fluid particle in terms of the shear and normal forces exerted on it. GeorgeGabriel Stokes (18191903), British mathematician and physicist, is known for hisresearch in hydrodynamics and a study of elasticity (see Glossary).


PART I

Irrotational Flow Motion of Ideal Fluid

Spiral around anindustrial chimney toreduce wind loadvibrations


CHAPTER I-1

Introduction to Ideal Fluid Flows

Summary

After a short paragraph on the ideal fluid properties, the basic equations aresimplified for the case of an ideal fluid flow.

1 PRESENTATION

Although no ideal fluid exists, many real fluid flows have small viscosity and the effectsof compressibility are negligible. When the boundary layer regions remain thin, thenon-viscous fluid results may be applied to real fluid flow motion to a satisfactorydegree of approximation. Applications include the motion of a solid through an idealfluid which is applicable with slight modification to the motion of an aircraft throughthe air, of a submarine through the oceans, flow through the passages of a pump orcompressor, or over the crest of a dam, and some geophysical flows.

2 DEFINITION OF AN IDEAL FLUID

An ideal fluid is defined as a non-viscous and incompressible fluid. That is, the fluidhas zero viscosity and a constant density:

= constant (1.1)

= 0 (1.2)

An ideal fluid flow must satisfy: (1) the continuity equation, (2) the equations ofmotion at every point at every instant and (3) neither penetration of fluid into norgaps between fluid and boundary at any solid boundary.

Remember

An ideal fluid has zero viscosity (= 0) and a constant density (= constant). Since ithas a nil viscosity, an ideal fluid cannot sustain shear stress.



3 IDEAL FLUID FLOWS

3.1 Presentation

For fluids of low viscosity the effects of viscosity are appreciable only in a narrowregion surroundings the fluid boundaries, and ideal fluid flow results may be applied.Converging or accelerating flow situations generally have thin boundary layers butdecelerating flows may have flow separation and development of large wake that isdifficult to predict with non-viscous fluid equations.

Figure 1.1 illustrates a number of engineering flow applications where both ideal-fluid and real-fluid flow regions are observed. Figure 1.1A presents the supercriticalflow downstream of a sluice gate. A boundary layer develops from about the venacontracta downstream of the gate. Figure 1.1B shows a turbulent flow past a smoothflat plate. At the plate, the velocity is zero and the flow region affected by thepresence of the plate is called the boundary layer, while ideal fluid flow calculationsmay applied to the flow outside of the boundary layer. Figure 1.1C presents the flowregion around an immersed body (e.g. a torpedo), while Figure 1.1D illustrates theeffect of incidence angle on flow separation behind an aerofoil. Figures 1.1E and 1.1Fshow the flow around high-speed catamaran and the wake region behind it.

In each example, the assumption of ideal-fluid flow is reasonable outside of theboundary layers and outside of the wake regions.

(A) Flow downstream of a sluice gate

x

Sluicegate

Developing boundary layer

Ideal fluid flow (potential flow)

(B) Uniform flow past a flat plate

Vo

Free-stream velocity

V

y

Developing boundary layerFlat plate

Outer edge ofboundary layer

Ideal fluid flow(potential flow)

Figure 1.1Examples of ideal fluid

flow situations inengineering applications


Introduction to Ideal Fluid Flows 19

(C) Flow regions around an immersed body

Vo

Stagnationpoint

Free-streamvelocity Separation region

(wake)

Boundary layers

Ideal fluid flow(potential flow) Separation

points

(D) Separation caused by increased angle of incidence of an aerofoil

Vo

Vo

Vo

Stagnation point

Free-stream velocity

Stagnationpoint

Separation region(wake)

Separationpoint

Stagnationpoint

Separation region(wake)Separation point

Figure 1.1(Continued)

Notes

1. A boundary layer is the flow region next to a solid boundary where the flow field isaffected by the presence of the boundary and where friction plays an essential part(Chap. II-3 and II-4). A boundary layer flow is characterised by a range of velocitiesacross the boundary layer region from zero at the boundary to the free-streamvelocity at the outer edge of the boundary layer.

2. A wake is the separation region downstream of the streamline that separates froma boundary. It is also called a wake region.



(E) Flow past a catamaran ferry on the Brisbane river (Australia) on 5May 2002 Note the white waters

behind highlighting the ship wake

(F) Wake behind a high-speed catamaran sailing at 22 knots on 24 Dec. 2001 Looking from the

catamaran stern




3. In a boundary layer, a deceleration of fluid particles leading to a reversed flowwithin the boundary layer is called a separation. The decelerated fluid particles areforced outwards and the boundary layer is separated from the wall. At the pointof separation, the velocity gradient normal to the wall is zero:(

Vxy

)y=0

= 0

In a boundary layer, the separation point is the intersection of the solid boundarywith the streamline dividing the separation zone and the deflected outer flow. Theseparation point is a stagnation point.

4. A stagnation point is defined as the point where the velocity is zero. When astreamline intersects itself, the intersection is a stagnation point. For irrotationalflow a streamline intersects itself at right-angle at a stagnation point.

Application

For an ideal fluid flow (i.e. neglecting separation), the lift force on a planewing may be analytically estimated as:

Lift = 12

V2 2 L W sin (1.3)

where L is the wing chord, W is the wing span and is the angle ofincidence. This relationship is derived from the Kutta-Joukowski law for two-dimensional cylinders and airfoils without camber (see Chapter I-6). Twopractical applications are detailed in Table 1.1.

At take off, the lift on a commercial jumbo jet is generated by the largewing surface area. In a military fighter, the lift is caused by the large angleof incidence and high take-off speed. Figure 1.2 illustrates two aircrafts witha relatively high incidence angle to increase their lift force at take-off andlanding.

Note that these ideal fluid flow calculations neglects the effect of separation.Figure 1.1D illustrates the effect of the angle of incidence on flow separation.

Notes

1. The chord or chord length is the straight line distance joining the leading andtrailing edges of an airfoil.

2. The span is the maximum lateral distance (from tip to tip) of an airplane.

3. The angle of incidence or angle of attack is the angle between the approachingflow velocity vector and the chordline.



Aircraft Wing Angle of Take-offmass Chord L span W incidence speed V Lift force

Aircraft kg m m deg. m/s N

Jet fighter 16 E+3 5 9 20 98 5.6 E+5Boeing 747 300 E+3 9 60 10 83 2.4 E+6

Table 1.1Lift force for two types

of aircraft

(A) Rafale fighter plane at takeoff from Charles de Gaulle aircraft carrier (Courtesy of Dassault Aviation

and Vronique Almansa)

(B) Airbus A300-600ST Beluga at landing at Nantes airport on 10 August 2008

Figure 1.2Aircrafts at take off and

landing

3.2 Fundamental equations for ideal fluid flows

The fundamental equations developed in Chapter 2 may be simplied for an ideal fluidflow. The continuity and Navier-Stokes equations are presented below.

3.2.1 Continuity equationFor an incompressible flow, the continuity equation is:

divV = 0 (1.4a)




i=x,y,z

Vixi

= 0 (1.4b)

Notes

1. This expression is valid for viscous and non-viscous fluids, and steady and unsteadyflows.

2. For a two-dimensional flow the continuity equation in Cartesian coordinatesbecomes:

Vxx

+ Vyy

= 0

3.2.2 Navier-Stokes equationFor an ideal fluid flow the Navier-Stokes equation becomes:

DV

Dt= Fv grad P (1.5a)


DViDt

= Fvi P

xi(1.5b)

For a two-dimensional flow and gravity forces, the Navier-Stokes equation written inCartesian coordinates is:

(Vxt

+ Vx Vxx

+ Vy Vxy

)=

x(P + g z) (1.6a)

(Vyt

+ Vx Vyx

+ Vy Vyy

)=

y(P + g z) (1.6b)

where z is taken as a coordinate which is positive vertically upward.

In polar system of coordinates, the Navier-Stokes equation is:

(Vrt

+ Vr Vrr

+ Vr

Vr

V2

r

)=

r(P + g z) (1.7a)

(Vt

+ Vr Vr

+ Vr

V

+ Vr Vr

)= 1

r

(P + g z) (1.7b)

where Vr and V are the radial and ortho-radial velocity components (Fig. 1.3).



y

r

M

Vy

x

Vx

Vr

VV

Figure 1.3Definition sketch of

radial and ortho-radialvelocity components

EXERCISES (CHAPTER 1)

1.1 Physical properties of fluids

Give the following fluid and physical properties (at 20 Celsius and standard pressure)with a 4-digit accuracy.

Value Units

Air density:Water density:Air dynamic viscosity:Water dynamic viscosity:Gravity constant (in Brisbane):Surface tension (air and water):

SolutionSee Appendix B.

1.2 Basic equations (1)

What is the definition of an ideal fluid?

What is the dynamic viscosity of an ideal fluid?

Can an ideal fluid flow be supersonic? Explain.


From what fundamental equation does the Navier-Stokes equation derive: a- conti-nuity, b- momentum equation, c- energy equation?

Does the Navier-Stokes equation apply for any type of flow? If not for what type offlow does it apply for?

In a fluid mechanics textbook, find the Navier-Stokes equation for a two-dimensionalflow in polar coordinates.



SolutionFor an incompressible flow the continuity equation is:

divV = Vx

x+ Vx

y= 0

The Navier-Stokes equation for a two-dimensional flow is:

(Vxt

+ Vx Vxx

+ Vy Vxy

)=

x(P + g z) +

(2Vxx x

+ 2Vx

y y

)

(Vyt

+ Vx Vyx

+ Vy Vyy

)=

y(P + g z) +

(2Vyx x

+ 2Vy

y y

)

The relationships between Cartesian coordinates and polar coordinates are:

x = r cos y = r sin r2 = x2 + y2 = tan1

(yx

)

The velocity components in polar coordinates are deduced from the Cartesiancoordinates by a rotation of angle :

Vx = Vr cos V sin Vy = Vr sin + V cos Vr = Vx cos + Vy sin V = Vy cos Vx sin

The continuity and Navier-Stokes equations are transformed using the followingformula:

dr

dx= cos dr

dy= sin

d

dx= sin

r

d

dy= cos

r

and

dx = dr cos r d sin dy = dr sin + r d cos

dr = dx cos + dy sin d = 1r

(dy cos dx sin )

In polar coordinates the continuity equations is:

1

r (r Vr)

r+ 1

r V

= 0

and the Navier-Stokes equation is:

(Vrt

+ Vr Vrr

+ Vr

Vr

V2

r

)

= r

(P + g z) + (Vr Vrr2

2

r2 V

)



(Vt

+ Vr Vr

+ Vr

V

+ Vr Vr

)

=1r

(P + g z) + (V + 2r2

Vr

Vr

)

where:

Vr = 1r

r

(r Vr

r

)+ 1

r2

2Vr2

+ 2Vrz2

V = 1r

r

(r V

r

)+ 1

r2

2V2

+ 2Vz2


For a two-dimensional flow, write the Continuity equation and the Momentumequation for a real fluid in the Cartesian system of coordinates.

What is the name of this equation?

Vxx

+ Vyy

= 0

Does the above equation apply to any flow situation? If not for what type of flowfield does the equation apply for?

Write the above equation in a polar system of coordinates.

For a two-dimensional ideal fluid flow, write the continuity equation and the Navier-Stokes equation (in the two components) in Cartesian coordinates and in polarcoordinates.

SolutionFor an incompressible flow the continuity equation is:

divV = Vx

x+ Vy

y= 0

The Navier-Stokes equation for a two-dimensional flow of ideal fluid is:

(Vxt

+ Vx Vxx

+ Vy Vxy

)=

x(P + g z)

(Vyt

+ Vx Vyx

+ Vy Vyy

)=

y(P + g z)

In polar coordinates the continuity equations is:

1

r (r Vr)

r+ 1

r V

= 0



and the Navier-Stokes equation is:

(Vrt

+ Vr Vrr

+ Vr

Vr

V2

r

)=

r(P + g z)

(Vt

+ Vr Vr

+ Vr

V

+ Vr Vr

)= 1

r

(P + g z)

1.5 Mathematics

If f is a scalar function, and F a vector, verify in a Cartesian system of coordinates that:

curl(

grad f ) = 0

div(curl

F ) = 0

1.6 Flow situations

Sketch the streamlines of the following two-dimensional flow situations:

A. A laminar flow past a circular cylinder,

B. A turbulent flow past a circular cylinder,

C. A laminar flow past a flat plate normal to the flow direction,

D. A turbulent flow past a flat plate normal to the flow direction.

In each case, show the possible extent of the wake (if any). Indicate clearly in whichregions the ideal fluid flow assumptions are valid, and in which areas they are not.


CHAPTER I-2

Ideal Fluid Flows and IrrotationalFlow Motion

Summary

The concept of ideal fluid flow is defined and the main equations are pre-sented. Then the concepts of irrotational flows, stream function, and velocitypotential are developed. At last energy considerations are discussed.

1 PRESENTATION

1.1 Ideal fluid

An ideal fluid is one that is frictionless and incompressible. It has zero viscosity andit cannot sustain a shear stress at any point.

An understanding of two-dimensional and three-dimensional flow of ideal fluidprovides the engineer with a much broader approach to real-fluid flow situations.Although no ideal fluid actually exists, many real fluids have small viscosity and theeffects of compressibility may be small. Applications include the motion of a solidthrough an ideal fluid are applicable with slight modification to the motion of an air-craft through the air, of a submarine through the oceans, flow through the passagesof a pump or compressor, or over the crest of a dam.

Prandtls hypothesis states that, for fluids of low viscosity, the effects of viscosity areappreciable only in a narrow region surrounding the fluid boundaries. For incom-pressible flow situations in which the boundary layer remains thin, the ideal fluid flowresults may be applied to flow of a real fluid to a satisfactory degree of approximationoutside the boundary layer.

An additional assumption of irrotational flow will be introduced and developed.



Note

Ludwig Prandtl (18751953) was a German physicist and aerodynamist. He intro-duced the concept of boundary layer in On Fluid Motion with Very Small Friction(1904):

Prandtl, L. (1904). Uber Flussigkeitsbewegung bei sehr kleiner Reibung. Verh. IIIInt. Math. Kongr., Heidelberg, Germany.

1.2 Fundamental equations

An ideal fluid must satisfy the following basic principles and associated boundaryconditions.

(a) The continuity equation:

divV = 0 (2.1a)

whereV is the velocity vector. In Cartesian coordinates, it yields:

Vxx

+ Vyy

+ Vzz

= 0 (2.1b)

where x, y and z are the Cartesian coordinates.

(b) The equations of motion at every point at every instant:

DV

Dt= Fv grad P (2.2)

where is the fluid density, Fv is the resultant of the volume forces and P is thepressure. In Cartesian coordinates, it yields:

DVxDt

= Fvx P

x(2.3a)

DVyDt

= Fvy P

y(2.3b)

DVzDt

= Fvz P

z(2.3c)

(c) The solid boundary condition: that is, neither penetration of fluid into nor gapbetween fluid and boundary.

The unknowns in an ideal fluid flow situation with given boundaries are thevelocity vector and the pressure at every point at every instant.


Ideal Fluid Flows and Irrotational Flow Motion 31

2 IRROTATIONAL FLOWS

2.1 Introduction

The assumption that the flow is irrotational provides a means to integrate the motionequations if the volume forces are derivable from a potential (i.e. gravity force). Thiswill be developed in the Velocity potential paragraph. The concept of rotation andvorticity are introduced before the irrotational flow condition is defined.

2.2 Rotation and vorticity

The rotation component of a fluid particle about an axis (e.g. the z axis) is defined asthe average angular velocity of any two infinitesimal linear elements in the particlethat are perpendicular to each other and to the axis (e.g. the z axis).

The rotation vector is related to the velocity vector as:

= 12

curl V

= 12

(

Vzy

Vyz

)i + 1

2(

Vxz

Vzx

)j + 1

2(

Vyx

Vxy

)k (2.4)

where curl is the curl vector operator (App. D).

The vorticity vector is defined as twice the rotation vector:

Vort = curl V =

(Vzy

Vyz

)i +

(Vxz

Vzx

)j +

(Vyx

Vxy

)k (2.5)

Note

For a two-dimensional flow in the {x, y} plane, the rotation vector and the vorticityvector are perpendicular to the {x, y} plane. That is, their components in the x- andy-directions are zero.

2.3 Irrotational flow

An irrotational flow is defined as a flow motion in which the rotation and hencevorticity are zero everywhere:

curl

V = 0 (2.6)

In Cartesian coordinates. the condition of irrotational flow becomes:(Vzy

Vyz

)= 0 (2.7a)

(Vxz

Vzx

)= 0 (2.7b)

(Vyx

Vxy

)= 0 (2.7c)



The individual particles of a frictionless incompressible fluid initially at rest cannot becaused to rotate. Considering a small free body of fluid in the shape of a sphere(Fig. 2.1), the surfaces forces must act normal to its surface since the fluid is friction-less: i.e., there is no shear stress. Hence they act through the centre of the sphere.The volume force also acts at the centre of mass. Simply no torque can be exerted onthe sphere and the spherical volume of fluid remains without rotation if it was initiallyat rest.

Gravity force(volume force)

Pressure force(surface force)

Centre ofgravity

Figure 2.1Forces acting on a

spherical control volumeof an ideal fluid

Conversely, once an ideal fluid has rotation, there is no way of altering it as no torquecan be exerted on an elementary sphere of the fluid. Rotation, or lack of rotation ofthe fluid particles, is a property of the fluid itself and not of its position in space.

Notes

1. For a two-dimensional flow, the condition of irrotationality is:

Vyx

= Vxy

2. There may be isolated points or lines in an irrotational flow where the abovecondition (Eq. (2.6)) is not satisfied. Such points or lines are known as singularities.They are points or lines where the velocity is zero or theoretically infinite.

3. The French mathematician Joseph-Louis Lagrange showed that Eulers equationsof motion (Chapter 2, paragraph 3.3.1) could be solved analytically for irrotationalflow motion of ideal fluid. He argued on the permanence of irrotational fluidmotion by introducing the velocity potential (paragraph 4, in this chapter).

4. Leonhard Euler (17071783) was a Swiss mathematician.

3 STREAM FUNCTION STREAMLINES

3.1 Stream function

For an incompressible flow the continuity equation is:

divV = 0 (2.1)



This relationship is equivalent to:

V = curl (2.8)

because: div(curl

F )= 0 (see Appendix D).

For a two-dimensional flow, the vector becomes:

= (0, 0, ) (2.9)

where (x, y, t) is called the stream function. The use of the continuity equationimplies that:

D = Vx dy + Vy dx

is an absolute differential which satisfies:

Vx = y

(2.10a)

Vy = x

(2.10b)

for a two-dimensional flow.

Since the stream function satisfies the continuity equation, it does exist.

Notes

1. The stream function was introduced by the French mathematician Joseph-LouisLagrange (Lagrange 1781, pp. 718720, Chanson 2007).

2. Joseph-Louis Lagrange (17361813) was a French mathematician and astronomer.During the 1789 Revolution, he worked on the committee to reform the metricsystem. He was Professor of mathematics at the cole Polytechnique and coleNormale from the start in 17941795.

3. The stream function is defined for steady and unsteady incompressible flowsbecause it does satisfy the continuity principle.

4. Stream functions may also be defined for three-dimensional incompressible flows.The stream function is then a vector:

= (x, y, z)

5. The stream function has the dimensions L2T1 (i.e. m2/s).



6. For a two-dimensional flow, the relationship between velocity and stream functionin polar coordinates is:

Vr = 1r

V = r

3.2 Streamlines

A streamline is the line drawn so that the velocity vector is always tangential toit (i.e. no flow across a streamline). For a two-dimensional flow the streamlineequation is:

dy

dx= tan = Vy

Vx(2.11a)

which may be rewritten as:

dx

Vx= dy

Vy(2.11b)

If the values of Vx and Vy are substituted a function of in the streamline equation(i.e. Vx dy+Vy dx=0), it follows that, along a streamline, the total differentialof the stream function is:

D = x

dx + y

dy = 0 (2.12)

Simply, along a streamline, the stream function is constant.

For a two-dimensional flow, the volumetric flow rate q (m2/s) between twostreamlines is:

q = = V n (2.13)

where V is the velocity magnitude between the streamlines and n is the distancebetween two adjacent streamlines (Fig. 2.2).

c

c

c

c 2*

c

n

s

Streamlines

Equi-potentials

Flow direction

Figure 2.2Sketch of streamlines

and equipotentials



Some important characteristics of streamlines are (Vallentine 1969, p. 14):

1. There can be no flow across a streamline.

2. Streamlines converging in the direction of the flow indicate a fluid acceleration.

3. Streamlines do not cross.

4. In steady flow the pattern of streamlines does not change with time.

5. Solid stationary boundaries are streamlines provided that separation of the flowfrom the boundary does not occur.

Notes

1. The concept of streamline was first introduced by the Frenchman J.C. de Borda(Fig. 2.3).

2. Jean-Charles de Borda (17331799) was a French mathematician and militaryengineer. He achieved the rank of Capitaine de Vaisseau and participated tothe American War of Independence with the French Navy. He investigated theflow through orifices and developed the Borda mouthpiece. During the FrenchRevolution, he worked with Joseph-Louis Lagrange and Pierre-Simon Laplace(17491827) on the metric system.

3. The definitions of stream function and streamline do not rely on the irrotationalflow assumption.

Figure 2.3Jean-Charles de Borda(17331799)



4. The concept of stream function is valid for any incompressible flow because sincethe stream function satisfies the continuity equation, it does exist.

Note on the stagnation pointA stagnation point is defined as a point where the velocity is zero. The stagna-tion point is a flow singularity. For a two-dimensional flow the stream function at astagnation point is such as:

Vx = y

= 0 (2.14)

Vy = x

= 0

It can be shown that, at a stagnation point, the streamline (= 0) crosses itself: i.e.,a double point. For irrotational flow, the two branches of the streamline cut at rightangles.

3.3 Stream function of irrotational flow

For two-dimensional flows, the irrotational flow condition (Eq. (2.7)) applies:

Vyx

= Vxy

Substituting the stream function , it gives:

= 2

x2+

2

y2= 0 (2.15)

where is the Laplacian differential operator (App. D). Only one function satisfiesboth the Laplace equation (i.e. = 0) and the boundary conditions for a particularflow pattern.

Applications

1. The stream function of an uniform flow parallel to the x-axis is:

= V y

2. The stream function of a radial flow to a point outlet at the origin (flowrate q) is:

= q2

where is the angular coordinate.



3. The stream function of a flow past a cylinder (radius R) at the origin in afluid of infinite extent, and whose undisturbed velocity is Vo, is:

= Vo y (

1 R2

x2 + y2)

4. Equation (2.15) is a Laplace equation in terms of the stream function .

5. The solution to the Laplace equation is uniquely determined if (a) thevalue of the function is specified on all boundaries (i.e. Dirichlet boundaryconditions) or (b) the normal derivative of the function is specified on allboundaries (Neumann boundary conditions).

4 VELOCITY POTENTIAL

4.1 Definition

The velocity potential is defined as a scalar function of space and time (x,y,z,t)such that its negative derivative with respect to any direction is the fluid velocity inthat direction:

Vx = x

(2.16a1)

Vy = y

(2.16a2)

Vz = z

(2.16a3)

In vector notation, it becomes:

V = grad (2.16b)

Notes

1. The velocity potential was first introduced by the French mathematician Joseph-Louis Lagrange in 1781 in his paper Mmoire sur la thorie du mouvement desfluides (Lagrange 1781, Liggett 1994) (Fig. 2.4).

2. The velocity potential function only exists for irrotational flows (see nextparagraph).

3. Velocity potential functions are defined for two-dimensional and three-dimensional, steady and unsteady flows. The velocity potential is always a scalar.

4. The velocity potential function has the dimension L2T1 (i.e. m2/s) that is thesame as for the two-dimensional flow stream function .



Figure 2.4Joseph-Louis Lagrange

(17361813)

5. In polar coordinates, the relationship between the velocity vector and the velocitypotential is:

Vr = r

V = 1r

Vz = z

6. Although many applications of the velocity potential only exists for irrotationalflows of ideal fluids, J.L. Lagrange. demonstrated that the velocity potential existsfor irrotational flow motion of both ideal- and real-fluids (Chanson 2007).

4.2 Velocity potential and irrotational flow

The existence of a velocity potential implies irrotational flow motion. Indeed, if thevelocity potential exists, the vorticity may be rewritten as:

Vort = curl V = curl grad = 0 (2.17)

since:curl

grad= 0 (see Appendix D). The assumption of irrotational flow and the

assumption that a velocity potential exists are one and the same thing. In conclusion,the velocity potential function exists if and only if flow is irrotational. Corollary a flowis irrotational if and only if a velocity potential function exists.



For an incompressible flow, the continuity equation may then be rewritten in termsof the velocity potential:

div(grad ) = = 0 Continuity equation (2.18)

The continuity equation (Eq. (2.18)) is a Laplace equation in terms of the velocitypotential which must be satisfied at every point throughout the fluid.

In summary, considering the irrotational flow motion of an ideal fluid, it was shownthat:

(a) the stream function exists since it satisfies the continuity equation;

(b) the velocity potential exists;

(c) the condition of irrotational flow written in term of the stream function yields toa Laplace equation: = 0; and

(d) the continuity equation can be rewritten in term of the velocity potential asanother Laplace equation: = 0.

A velocity potential can be found for each stream function. If the stream func-tion satisfies the Laplace equation (Eq. (2.15)), the velocity potential also satisfies it(Eq. (2.18)) but with different boundary conditions. Hence the velocity potential maybe considered as the stream function for another flow case. The velocity potential and the stream function are called conjugate functions.

Properties of the Laplace equation

A interesting property of any linear homogenous differential equation, including theLaplace equation, is the principle of superposition. If any two functions are solu-tions, their sum, or any linear combination, is also a solution. Hence, if 1 and 2satisfy the Laplace equation subject to the respective boundary conditions B1(1)and B2(2), any linear combination of these solutions: (a1 + b2) satisfies theLaplace equation and the boundary conditions a B1(1)+ b B2(2). This is theprinciple of superposition.

Another property of the Laplace equation is the unicity of its solution for awell-definedset of boundary conditions.

Notes

1. An important difference between velocity potential and stream function lies in thefact that the velocity potential exists for irrotational flows only, while the streamfunction is not restricted to irrotational flows but to incompressible fluids.

2. Pierre-Simon Laplace (17491827) was a French mathematician, astronomer andphysicist. He is best known for his investigations into the stability of the solarsystem.



4.3 Application to the equations of motion: the Bernoulli equation

The equations of motion are:

Vit

+

j=x,y,zVj Vi

xj=

xi

(P

+ U

)(2.2)

where U is the volume force potential:Fv =grad U.

Substituting the irrotational flow conditions (Eq. (2.6)):

Vkxj

= Vjxk

for j = k

and the velocity potential (Eq. (2.16)):

Vi = xi

the motion equations may be expressed as (Streeter 1948, p. 24):

xi

t+ V

2j

2+

j=x,y,z

P

+ U = 0

(2.19)

Notes

1. The gravity force is a volume force:

Fv = grad(g z)

where the z axis is positive upward: i.e., U= g z.

2. If the only volume force is the gravity force, the integrated form of the Bernoulliequation is:

ddt

+ V2

2+ P

+ g z = F(t)

Introducing the gravity force, the integration with respect to xi yields:

t

+ V2

2+ P

+ g z = Fi(xj, xk, t) (2.20)

where i,j,k= x,y,z and V is defined as the magnitude of the velocity defined as:V2 =V2x +V2y +V2z . The integration of the three motion equations are identical andthe left-hand sides of the equations are the same:

Fx(y, z, t) = Fy(x, z, t) = Fz(x, y, t)



The final integrated form of the three equations of motion is the Bernoulli equationfor unsteady flow:

ddt

+ V2

2+ P

+ g z = F(t) (2.21)

containing an arbitrary function of time F(t) that is independent of space (x,y,z).

Notes

1. Daniel Bernoulli (17001782) was a Swiss mathematician and hydrodynamist, whodeveloped the equation in his Hydrodynamica textbook (1st draft in 1733) (Carvill1981, p. 3).

2. In steady flow the Bernoulli equation reduces to:

V2

2+ P

+ g z = Constant

Application

Usually, it is convenient to write the pressure P as:

P = Ps + Pd (2.22)where Ps is the static (ambient) pressure and Pd the dynamic pressure (i.e.pressure due to changes in velocity). Assuming that the gravity is the onlyvolume force acting, Equation (2.21) becomes:

t

+ V2

2+ Pd

+ Ps

+ g z = F(t)

If the pressure distribution is hydrostatic:(Ps

+ g z)

= constant

then the Bernoulli equation becomes:

t

+ V2

2+ Pd

= F(t) (2.22)

For steady flows with hydrostatic pressure distribution, the Bernoulli equationbecomes:

V2

2+ Pd

= constant (2.23)

If the dynamic pressure and velocity are known at one point (Pdo , Vo), thevariation in pressure may be determined if the change in velocity magnitudeis known:

Pd = Pdo + V2o2

(

1 (

V

Vo

)2)(2.23b)



4.4 Discussion

In the irrotational flowmotion of ideal fluid, the problem resolves itself into a geomet-rical problem: to find the velocity potential , or the stream function , that satisfiesthe continuity principle and the boundary conditions. As the continuity equation andboundary equations are both kinematical (i.e. contain no density term), the magni-tude and directions of the velocity at all points are independent of the particular fluid.Basically the solution of the velocity field is independent of the fluid properties.

Once the velocity is known, the pressure may be determined from the Bernoulliequation (Eq. (2.23)).

Remarks

In ideal fluid flows with irrotational motion, the basic equations may be solved analyt-ically (Chapters I-3 and I-4). They yield the velocity field. In turn the pressure field maybe calculated using the Bernoulli equation. The technique is particularly useful to pre-dict hydrodynamic forces on structures in converging or accelerating flow conditions.In real fluid flows, it is usual to combined ideal-fluid and real-fluid flow calculationsto assess fluid-structure interactions (see Chapters II-3 and II-4). An example is windloading on buildings and man-made structures. Figure 2.5 illustrates severe damageduring wind storm episodes.

(A) Damage in La Rochelle marina after a strong wind storm on 2628 December 1999 (Courtesy ofJ.H. Bordes) During the storm event, wind speeds exceeded 100 knots (51m/s), combined with hightidal range and storm surge effects

Figure 2.5Wind storm damage

(See Plate 3, 4, 5)



(B) Fallen tree n the Brisbane valley (Australia) after a wind storm in March 1998 (Courtesy of ProfC. Letchford) Wind gusts in excess of 200 km/h were experienced in the valley, causing extensivedamage

(C) Damaged transmission tower (275 kV double circuit electricity transmission tower) in March 1998(Courtesy of Prof C. Letchford) Repair crews are visible in background




5 ENERGY CONSIDERATIONS

5.1 Energy equation for non-viscous fluids

Considering a control volume [CV], the total kinetic energy Ek in the fluid is:

Ek =

CV

V2

2 dVolume = 1

2

V2 dx dy dz (2.24)

The total potential energy Ep of the fluid region is:

Ep =

CV

U dVolume = 12

U dx dy dz (2.25)

where U is the potential of the volume forces (i.e. potential energy per unitmass).

For incompressible fluids the energy equation becomes (Streeter 1948, p. 31;Vallentine 1969, p. 46):

D

Dt(Ek + Ep) =

CS

P Vn dArea (2.26)

where Vn is the velocity of the boundary normal to itself in the direction of the fluidand [CS] is the control surface. Hence for an incompressible frictionless fluid, the totalincrease in kinetic plus potential energy equals the work done by the pressures on itssurface.

Note

This result is valid for both rotational and irrotational flows.

5.2 Kinetic energy in irrotational flows

When the flow motion is irrotational (i.e. V=), the use of the Greens theorempermits the derivation of an expression of the kinetic energy term:

Ek = 2

() () dx dy dz = 2

ddn

dS (2.27)

where is the velocity potential,. The velocity of the fluid along the inward normal,n, to an element of surface dS is: Vn =d/dn.

The surface integral (i.e. right term of the above equation) represents the work doneby an impulsive pressure () in starting the motion from rest.



Notes

1. George Green (17931841) was an English mathematician.

2. Greens theorem states that: A dS =

div

A dx dy dz

where A is a vector whose components are any finite, single-valued, differentiablefunctions of space in a connected region completely bounded by one or moreclosed surfaces S of which dS is an element and the direction normal to the surfaceelement dS is directed into the region (Vallentine 1969, p. 250). Greens theoremcan be regarded as a transformation of a surface integral into a volume integral.

5.3 Discussion

The following theorems are proved as a consequence of the above kinetic energyconsiderations and they are limited to irrotational flows of ideal fluid (Streeter 1948,p. 37; Vallentine 1969, p. 49):

(a) Irrotational motion is impossible if all of the boundaries are fixed.

(b) Irrotational motion of a fluid will cease when the boundaries come at rest.

(c) The pattern of irrotational flowwhich satisfies the Laplace equation and prescribedboundary conditions is unique and is determined by themotion of the boundaries.

(d) Irrotational motion of a fluid at rest at infinity is impossible if the interiorboundaries are at rest.

(e) Irrotational motion of a fluid at rest at infinity is unique and determined by themotion of the interior solid boundaries.

6 CONCLUSION

For an ideal fluid (i.e. incompressible and frictionless) and for an irrotational flowmotion (i.e. a velocity potential exists), the solution of the flow problem is a solutionof the Laplace equation (Eq. (2.6) or (2.16)) in term of or that also satisfies theboundary conditions.

The velocity field may be obtained from the definition of the velocity potential (orstream function) and the pressure distribution throughout the fluid is deduced fromthe Bernoulli equation.

EXERCISES (CHAPTER 2)

2.1 Quiz

What is the definition of the velocity potential?

Is the velocity potential a scalar or a vector?



Units of the velocity potential?

What is definition of the stream function? Is it a scalar or a vector? Units of thestream function?

For an ideal fluid with irrotational flow motion:

Write the condition of irrotationality as a function of the velocity potential.

Does the velocity potential exist for 1- an irrotational flow and 2- for a real fluid?

Write the continuity equation as a function of the velocity potential.

Further, answer the following questions:

What is a stagnation point?

For a two-dimensional flow, write the stream function conditions.

How are the streamlines at the stagnation point?

Solution

V =grad

A scalar.

The stream function and the velocity potential are in m2/s.

The stream function is a vector. It does exist for incompressible flow because itsatisfies the continuity equation.

The existence of a velocity potential implies irrotational flow. Indeed the vorticitybecomes:

Vort = curl V = curl grad = 0

as:curl

grad = 0 (see Appendix D). exist for both ideal and real-fluids with

irrotational flow motion,

The assumption of irrotational flow and the assumption that a velocity potentialexists are one and the same thing (Streeter 1948, p. 23).

Conclusion: Velocity potential functions exist if and only if flow is irrotational. Corollaryflow is irrotational if and only if velocity potential functions exist.

div (grad )== 0



A stagnation point is defined as a point where the velocity is zero.

For a two-dimensional flow the stream function at a stagnation point is such as:

Vx = y

= 0 Vy = x

= 0

It can be shown that at a stagnation point the streamline = 0 crosses itself (i.e.a double point).

2.2 Basic applications

1. Considering the following velocity field:

Vx = y z tVy = z x tVz = x y t

Is the flow a possible flow of an incompressible fluid?

Is the motion irrotational? If yes: what is the velocity potential?

2. Draw the streamline pattern of the following stream functions:

= 50 x (1) = 20 y (2) = 40 x 30 y (3) = 10 x2 (from x = 0 to x = 5) (4)

Solution (1)The equations satisfy the equation of continuity for:

Vxx

= Vyy

= Vzz

= 0

so that:

Vxx

+ Vyy

+ Vzz

= 0

The components of the vorticity are:

(Vzy

Vyz

)= (x t x t) = 0

(Vxz

Vzx

)= (y t y t) = 0



(Vyx

Vxy

)= (z t z t) = 0

hence vorticity is zero and the field could represent irrotational flow. The velocitypotential would then be the solution of:

x= Vx = y z t = x y z t + f1(y,z,t)

y= Vy = x z t = x y z t + f2(x,z,t)

z= Vz = x y t = x y z t + f3(x,y,t)

and hence:

= x y z t + f (t)

is a possible velocity potential.

Solution (2)1. Vertical uniform flow: Vo =+50m/s

2. Horizontal uniform flow: Vo =+20m/s

3. Uniform flow: Vo = 50m/s and = 127 degrees

4. Non uniform vertical flow

2.3 Two-dimensional flow

Considering a two-dimensional flow, find the velocity potential and the streamfunction for a flow having components:

Vx = 2 x y(x2 + y2)2

Vy = x2 y2

(x2 + y2)2

SolutionIn polar coordinate the velocity components are:

Vx = 2 cos sin r2 Vy =cos2 sin2

r2

Using:

Vr = Vx cos + Vy sin V = Vy cos Vx sin



we deduce:

Vr = sin r2 V =cos

r2

In polar coordinates the velocity potential and stream function are:

Vr = r

V = 1r

Vr = 1r

V =

r

Hence:

= sin r

+ constant

2.4 Applications

(a) Using the software 2DFlowPlus (App. E), investigate the flow field of a vortex(at origin, strength 2) superposed to a sink (at origin, strength 1). Visualise thestreamlines, the contour of equal velocity ad the contour of constant pressure.

Repeat the same process for a vortex (at origin, strength 2) superposed to a sink (atx=5, y= 0, strength 1). How would you describe the flow region surroundingthe vortex.

(b) Investigate the superposition of a source (at origin, strength 1) and an uniformvelocity field (horizontal direction, V= 1). How many stagnation point do youobserve? What is the pressure at the stagnation point? What is the half-Rankinebody thickness at x=+1? (Youmay do the calculations directly or use 2DFlowPlusto solve the flow field.)

(c) Using 2DFlowPlus, investigate the flow past a circular building (for an ideal fluidwith irrotational flow motion). How many stagnation points is there? Comparethe resulting flow pattern with real-fluid flow pattern behind a circular bluff body(search Reference text in the library).

(d) Investigate the seepage flow to a sink (well) located close to a lake. What flowpattern would you use?

Note: the software 2DFlowPlus is described in Appendix E. It may be down-loaded from {http://www.dynaflow-inc.com/Products/Software/DFlow/2dflow.htm}.(See also {http://www.uq.edu.au/e2hchans/reprints/book15.htm}).

2.5 Laplace equation

What is the Laplacian of a function? Write the Laplacian of the scalar function in Cartesian and polar coordinates.

Rewrite the definition of the Laplacian of a scalar function as a function of vectoroperators (e.g. grad, div, curl).



Solution

(x, y, z) = (x, y, z) = div grad (x, y, z) = 2

x2+

2

y2+

2

z2

Laplacian of scalar

F (x, y, z) = F (x, y, z) = i Fx + j Fy + k Fz Laplacian of vector

(r, , z) = 1r

r

(r

r

)+ 1

r2

2

2+

2

z2Polar coordinates

It yields:

f = div grad f

F = grad divF curl(curl F )

where f is a scalar.

Note the following operations:

(f + g) = f + g(

F + G ) = F + G(f g) = g f + f g + 2 grad f grad g

where f and g are scalars.


For a two-dimensional ideal fluid flow, write:

a. the continuity equation,

b. the streamline equation,

c. the velocity potential and stream function,

d. the condition of irrotationality and

e. the Laplace equation

in polar coordinates.

SolutionContinuity equation

Vxx

+ Vyy

= 0 1r

(r Vr)r

+ 1r

V

= 0



Momentum equation

Vxt

+ Vx Vxx

+ Vy Vxy

= x

(P

+ g z

)

Vyt

+ Vx Vyx

+ Vy Vyy

= y

(P

+ g z

)

Vrt

+ Vr Vrr

+ Vr

Vr

V2

r=

r

(P

+ g z

)

Vt

+ Vr Vr

+ Vr

V

+ Vr Vr

= 1r

(P

+ g z

)

Streamline equation

Vx dy Vy dx = Vr r d V dr = 0

Velocity potential and stream function

Vx = x

= y

Vr = r

= 1r

Vy = y

= +x

V = 1r

= +

r

Q = Q =

Condition of irrotationality

Vyx

Vxy

= 0 Vr

1r

Vr

= 0

Laplace equation

2

x2+

2

y2= 0

2

r2+ 1

r2

2

2= 0

2

x2+

2

y2= 0

2

r2+ 1

r2

2

2= 0


Considering an two-dimensional irrotational flow of ideal fluid:

write the Navier-Stokes equation (assuming gravity forces),

substitute the irrotational flow condition and the velocity potential,



integrate each equation with respect to x and y,

what is the final integrated form of the three equations of motion?

This equation is called the Bernoulli equation for unsteady flow.

For a steady flow write the Bernoulli equation. When the velocity is known, howdo you determine the pressure?

Solution

1. Vxt

+ Vx Vxx

+ Vy Vxy

= x

(P

+ g z

)

Vyt

+ Vx Vyx

+ Vy Vyy

= y

(P

+ g z

)

2. Substituting the irrotational flow conditions:

Vxy

= Vyx

and the velocity potential:

Vx = x

Vy = y

the equations may be expressed as (Streeter 1948, p. 24):

x

(

t+ V

2x

2+ V

2y

2+ P

+ g z

)= 0

y

(

t+ V

2x

2+ V

2y

2+ P

+ g z

)= 0

3. Integrating with respect to x and y:

t

+ V2

2+ P

+ g z = Fx(y, t)

t

+ V2

2+ P

+ g z = Fy(x, t)

where V is defined as the magnitude of the velocity: V2 =V2x +V2x . The integrationof the threemotion equations are identical and the left-hand sides of the equationsare the same:

Fx(y, t) = Fy(x, t)


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