Incompressible Fluid Dynamics

8/11/2019 Incompressible Fluid Dynamics

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Lecture Notes inIncompressible Fluid Dynamics:

Phenomenology,Concepts

and Analytical Tools.

Jacques LewalleSyracuse University


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Figure 1: Internal waves in a cloud, indicating warmer air above

Figure 2: Internal waves in a cloud layer at sunrise

c Jacques Lewalle 2006


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Contents

0.1 What is different about these notes? . . . . . . . . . . . . . . 90.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

0.2.1 Part I: Introduction . . . . . . . . . . . . . . . . . . . . 130.2.2 Part II: Basic concepts and equations . . . . . . . . . . 140.2.3 Part III: Approximations . . . . . . . . . . . . . . . . . 14

1 Motivation 171.1 Internal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1.1 A simple problem . . . . . . . . . . . . . . . . . . . . . 171.1.2 Entrance ow . . . . . . . . . . . . . . . . . . . . . . . 191.1.3 Transition to turbulence . . . . . . . . . . . . . . . . . 211.1.4 Pipe exit and secondary ow . . . . . . . . . . . . . . . 241.1.5 Advanced problems . . . . . . . . . . . . . . . . . . . . 251.1.6 Food for thought . . . . . . . . . . . . . . . . . . . . . 25

1.2 External ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.1 Control volume analysis . . . . . . . . . . . . . . . . . 281.2.2 Potential ow model . . . . . . . . . . . . . . . . . . . 291.2.3 Phenomenology . . . . . . . . . . . . . . . . . . . . . . 301.2.4 Food for thought . . . . . . . . . . . . . . . . . . . . . 33

1.3 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Kinematics 352.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.1 Intrinsic notations . . . . . . . . . . . . . . . . . . . . 362.1.2 Component notations . . . . . . . . . . . . . . . . . . . 372.1.3 Index notations . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Eulerian vs. Lagrangian descriptions . . . . . . . . . . . . . . 422.2.1 Lagrangian description of motion . . . . . . . . . . . . 432.2.2 Eulerian description . . . . . . . . . . . . . . . . . . . 48

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4 CONTENTS

2.2.3 Pathlines, streaklines and streamlines . . . . . . . . . . 49

2.2.4 Caution! . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3 Differential concepts and their geometry . . . . . . . . . . . . 562.3.1 Differentials and vectors . . . . . . . . . . . . . . . . . 56

2.4 Mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.4.1 Vector potential and stream function . . . . . . . . . . 60

2.5 Flow about a point: Helmholtz . . . . . . . . . . . . . . . . . 612.5.1 Rate of strain . . . . . . . . . . . . . . . . . . . . . . . 612.5.2 Local rotation and vorticity . . . . . . . . . . . . . . . 64

2.6 Kinematic decompositions of a velocity eld . . . . . . . . . . 662.7 Vorticity, , 2 , etc. . . . . . . . . . . . . . . . . . . . . . 67

2.7.1 Biot-Savart relation . . . . . . . . . . . . . . . . . . . . 672.7.2 Vorticity, streamfunctions, and more . . . . . . . . . . 692.7.3 Vorticity and boundary conditions . . . . . . . . . . . . 712.7.4 Discrete vortices vs. continuous vorticity . . . . . . . . 712.7.5 Vorticity and circulation . . . . . . . . . . . . . . . . . 722.7.6 The Helmholtz theorems . . . . . . . . . . . . . . . . . 722.7.7 Helicity, Lamb vector . . . . . . . . . . . . . . . . . . . 742.7.8 Famous vortices . . . . . . . . . . . . . . . . . . . . . . 74

2.8 Advanced topics and ideas for further reading . . . . . . . . . 80

3 Dynamics 853.1 Newtonian dynamics of continua . . . . . . . . . . . . . . . . . 853.2 Stress at a point, Newtonian uids . . . . . . . . . . . . . . . 88

3.2.1 The Navier-Stokes equations . . . . . . . . . . . . . . . 903.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . 923.2.3 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 923.2.4 Vorticity in NS . . . . . . . . . . . . . . . . . . . . . . 94

3.3 Vorticity equation . . . . . . . . . . . . . . . . . . . . . . . . . 963.4 Energy and dissipation . . . . . . . . . . . . . . . . . . . . . . 96

3.4.1 Bernoulli with losses . . . . . . . . . . . . . . . . . . . 983.5 Enstrophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.6 Structure of the equations . . . . . . . . . . . . . . . . . . . . 993.7 Incompressible ow approximation . . . . . . . . . . . . . . . 100

3.7.1 Small Mach number ows . . . . . . . . . . . . . . . . 1023.7.2 Boussinesq approximation . . . . . . . . . . . . . . . . 102

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.9 Advanced topics and ideas for further reading . . . . . . . . . 103


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CONTENTS 5

4 Dimensionless expressions 105

4.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 1054.2 Non-dimensionalization of equations . . . . . . . . . . . . . . . 1104.3 Dimensionless Equations and Scaling Analysis . . . . . . . . . 1124.4 Rational approximations . . . . . . . . . . . . . . . . . . . . . 114

4.4.1 Large Re . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.5 Advanced topics and ideas for further reading . . . . . . . . . 117

5 Inviscid Flows and Irrotational Flows 1215.1 Inviscid ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2 Bernoullis equation . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2.1 The strong form . . . . . . . . . . . . . . . . . . . . . . 1225.2.2 The weaker form . . . . . . . . . . . . . . . . . . . . . 1235.2.3 Bernoulli and energy . . . . . . . . . . . . . . . . . . . 1235.2.4 Simplied Croccos equation . . . . . . . . . . . . . . . 1265.2.5 Applications of Bernoullis equation . . . . . . . . . . . 1285.2.6 Lagranges theorem . . . . . . . . . . . . . . . . . . . . 1285.2.7 Creation of vorticity . . . . . . . . . . . . . . . . . . . 1285.2.8 Kelvins theorem . . . . . . . . . . . . . . . . . . . . . 1295.2.9 Helmholtzs vortex theorems . . . . . . . . . . . . . . . 132

5.3 Irrotational ow . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.3.1 2D potential lines and streamlines . . . . . . . . . . . . 1365.3.2 Superposition of 2D potentials . . . . . . . . . . . . . . 1375.3.3 F=ma!? . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.3.4 DAlemberts paradox . . . . . . . . . . . . . . . . . . 143

5.4 Viscous potential ows . . . . . . . . . . . . . . . . . . . . . . 1445.5 Advanced topics and ideas for further reading . . . . . . . . . 144

6 Stokes Flows 1476.1 Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.1.1 Kinematic reversibility . . . . . . . . . . . . . . . . . . 1506.2 Stokes two problems . . . . . . . . . . . . . . . . . . . . . . . 151

6.3 Stokes ow around a sphere . . . . . . . . . . . . . . . . . . . 1536.3.1 Nonlocal effects . . . . . . . . . . . . . . . . . . . . . . 1576.3.2 Application: Slender bodies . . . . . . . . . . . . . . . 158

6.4 Cylinder: Stokes paradox . . . . . . . . . . . . . . . . . . . . 1586.5 Reynolds lubrication theory . . . . . . . . . . . . . . . . . . . 1596.6 Lagrangian turbulence, chaotic mixing . . . . . . . . . . . . . 161


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7 Interlude 1657.1 The approximations . . . . . . . . . . . . . . . . . . . . . . . . 1657.2 Non-local effects . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.3 The role of vorticity . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Narrow Flows 1678.1 Flat plate boundary layers . . . . . . . . . . . . . . . . . . . . 167

8.1.1 Control volume analysis . . . . . . . . . . . . . . . . . 1708.1.2 Scaling: mass balance . . . . . . . . . . . . . . . . . . 1718.1.3 Scaling: streamwise momentum . . . . . . . . . . . . . 1728.1.4 Scaling: transverse momentum . . . . . . . . . . . . . . 1728.1.5 Scaling: pressure . . . . . . . . . . . . . . . . . . . . . 1738.1.6 The ZPG BL equations . . . . . . . . . . . . . . . . . . 1748.1.7 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.2 Integral method: von K arm an s approach . . . . . . . . . . . 1798.3 Vorticity and circulation in ZPGBL . . . . . . . . . . . . . . . 1818.4 Descriptive: BL transition . . . . . . . . . . . . . . . . . . . . 1828.5 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.5.1 Streamwise development . . . . . . . . . . . . . . . . . 1878.5.2 Similarity proles . . . . . . . . . . . . . . . . . . . . . 1888.5.3 Entrainment . . . . . . . . . . . . . . . . . . . . . . . . 189


9 Flow Separation and Secondary Flow 1919.1 Curved channel . . . . . . . . . . . . . . . . . . . . . . . . . . 1919.2 Vorticity reversal in 2D separation . . . . . . . . . . . . . . . . 1959.3 Introduction of vorticity . . . . . . . . . . . . . . . . . . . . . 1999.4 Advanced topics and ideas for further reading . . . . . . . . . 201

10 Rotating Flows 203

10.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 20310.1.1 Coriolis force . . . . . . . . . . . . . . . . . . . . . . . 20510 .2 Scal ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20510.3 Geostrophic approximation . . . . . . . . . . . . . . . . . . . . 207

10.3.1 Taylor columns . . . . . . . . . . . . . . . . . . . . . . 20910.4 Rossby waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 210


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CONTENTS 7


11 Linearization 21511.1 Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

11.1.1 Tsunami speed . . . . . . . . . . . . . . . . . . . . . . 21811.1.2 Internal waves . . . . . . . . . . . . . . . . . . . . . . . 22211.1.3 More advanced topics . . . . . . . . . . . . . . . . . . . 223

11.2 Inviscid linear stability: Kelvin-Helmholtz . . . . . . . . . . . 22311.2.1 Setting up the problem . . . . . . . . . . . . . . . . . . 22311.2.2 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 22511.2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 22611.2.4 Stability as equilibrium . . . . . . . . . . . . . . . . . . 228

12 Additional Reading 23312.1 About this course . . . . . . . . . . . . . . . . . . . . . . . . . 23312.2 Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235


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Introduction

0.1 What is different about these notes?

These are lecture notes: not a textbook, even less a reference book. Supple-mentary material, in the form of specic reading from existing texts and owvisualization (movies and still photographs), allows these notes to remainconcise.

This material has been taught as an entry-level course for graduate stu-dents in Mechanical Engineering and related programs for a few years. WhileFluid Dynamics is a well-established discipline, its focus has shifted over theyears, and the range of applications has diversied. The biggest change re-sults from the widespread availability of desktop CFD packages in engineeringpractice, which makes it possible to obtain solutions to complex problems.The dependence on applied mathematics (as well as inclination for it andlevel of prociency) has decreased accordingly among typical students, butit could be argued that physical insight is more important than ever. Actualsolutions, numerical or analytical or experimental, have built-in approxima-tions and blind spots: with increasingly complex problems, it is important torecognize these limitations, their possible effects on the solutions and theirinterpretation. Analysis and insight are more difficult to combine in complexows.

Thus, I attempted to strike a new balance between the rich phenomenol-ogy of incompressible ows, as presented e.g. by Tritton, and the wealth of

analytical methods found in references such as Batchelor, Panton and Currie.There is also merit in a very selective approach such as Acheson. So, I havedevised my own mix of these excellent texts, and bits and pieces of them willbe recognized in this material. The classroom presentation will rely heavilyon movies (NSF series: GI Taylor, A Shapiro, F Abernathy, E Taylor, etc.)and ow-vizualization still pictures (Van Dykes Album of Fluid Motion, the

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Figure 3: Overview of this approach


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0.1. WHAT IS DIFFERENT ABOUT THESE NOTES? 11

Gallery of Fluid Motion in Physics of Fluids, and the many pictures on eu-

ids.com), for the double purpose of using the wonder of ow phenomena asmotivation, and to start the process of analysis and interpretation.The main innovation in these notes is the graphic representation of re-

lations between ideas. For years, I have observed that students need helpplacing ideas in context, need bridges between mathematics and reality. Asystematic approach is provided by mind-mapping , a graphical techniquerelated to free-association in psychology and to left vs. right brain thinking,used rst in the context of business schools (N. Margulies, R. Carter) butwith versatile pedagogical merit (I learned about it from Mrs. Kate Regan,our sons fth grade teacher). In this incarnation of the method, I ask the

students to map the web of concepts related to a given topic, so as to improveawareness of connections, missing links, and analogies.Take the example (Fig. 4) of what might have been learned about vis-

cosity at the undergraduate level: at the mention of the word viscosity, anentire context should come to mind. A list of keywords (here: denition,no-slip, friction and wall stress, Reynolds number, never Bernoulli, etc.), isa skeleton for any written text. But conventional presentations (text or listof keywords) are by necessity sequential. A 2-dimensional layout, empha-sizing relations between keywords, is apparently read by a different part of the brain: adding graphically expressive features and personal emphasis isan important part of the exercise. Drawing a mind-map, rst collectivelyin class and then individually as assignments, helps students create a con-text. The question of Are we overlooking anything? is as important aswhat is in the picture, but the context should be limited to items directlyrelated to the central topic, leaving further branchings (e.g. details aboutBernoulli, wall stress, etc.) for other diagrams. The end-result should be amental landscape, readable at a glance, showing connections and mismatches,eventually supporting the analytical developments and the interpretation of results. The frustration of where to start? in traditional problems may giveway to excitement at the realization that the handle is usually easier to ndon the mind map: it connects the various pieces of the problem statement.

The version printed here is somewhat limited in expressive value: addingcolors, texture, etc., is more easily done by hand, although software withclip-art libraries is now available. These maps of ideas can be more or lessdetailed: one can, for example, trace vorticity throughout this text and geta very different picture in Chapter 3, chapter 4, chapter 7 or 8: context iseverything.


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Figure 4: Two versions of a mind-map based on the same keywords


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0.2. OVERVIEW 13

While ideas and techniques need to be presented in detail, some ques-

tions are left unanswered in these notes so as not to spoil the fun of real-timethinking in the classroom. In the selection of material, fundamentals andvariety were more important than completeness or depth: especially withpointers to specic topics, going to the library for additional information isa normal step at any level of learning. The goal is to lead the students toa fundamental understanding of reasonably complex ows, including the na-ture of approximations, possible shortcomings and opportunities for furtherdevelopment. Formal manipulations are but one aspect of the learning: thesequence of ideas, the relations with other elds (elasticity, thermodynam-ics, etc.), and the ability to approach the wealth of applications, are more

important.

0.2 Overview

The help of a publisher would make two obvious improvements manageable:the professional drafting of gures, and the inclusion of copyrighted pho-tographs and movie clips in a multimedia package. The expansion of thesenotes to include other topics would rely on contribution from co-authors: thisis my selection.

The course is in three parts. The students should review independentlysome undegraduate topics, such as the Reynolds transport theorem, Bernoullisequation (without and with losses) and dimensional similarity. These meth-ods are assumed to be known, regardless of the students varied backgrounds.

0.2.1 Part I: Introduction

Ch. 1 is a short version of Trittons idea, to illustrate the limitations of the undergraduate tools of analysis for both internal and external ows.

These topics involve a rich phenomenology for which new analytical toolsare required. This is the vehicle for two main ideas. First, the need tounderstand the phenomena in question and their side-effects is motivationfor the introduction of more advanced concepts in Part II. Second, the mapsof ideas are introduced in a known context, and they summarize the manyloose ends.


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0.2.2 Part II: Basic concepts and equations

Chs. 2 and 3 are rather conventional in scope, with very few attempts atoriginality (the relation between Biot-Savart and the inverse Laplacian beingan exception). Kinematics (Ch. 2) formalize the tools that do not involve anyreference to forces (Newtons second law): the geometry of motion. At onelevel, this is descriptive, including ow visualization and plotting of solutionsfor various purposes of understanding and communication. At another level,it is deeply analytical, with more partial derivatives than some students feelcomfortable with. The teachers challenge is to link graphical and analyticalaspects into one unied context. Chapter 3 is devoted to dynamics, i.e. theform taken by Newtons second law in continua (Cauchys equation), thenmore precisely in inviscid (Eulers equations) and Newtonian incompressibleows (Navier-Stokes equations). Formal similarities for the balance of mo-mentum, mass and energy are exploited, and the governing equations arederived for related quantities such as pressure, energy and vorticity.

0.2.3 Part III: Approximations

Very few exact solutions are known for the Navier-Stokes (NS) equations.For the Couette-Poiseuille class of fully-developed parallel steady ows, con-vective terms cancel out and the simple balance of pressure and viscous forcesis featured in undergraduate texts. Unsteady problems of simple geometry(Stokes problems: impulsively started and oscillating plates in a viscousuid) are also well known. But there are remarkably few exact solutions of the NS equations including nonlinearities, and they are all time-independent.Finding closed-form exact solutions is not what this eld is about (althoughif you do, you will be famous).

Of course, the naive student will think in terms of just solving the NSequations numerically (or experimentally). But even computers and softwarerelate poorly to brute force: it is necessary to make approximations, and tounderstand their implications (strengths and shortcomings). Experimentally,

not all variables can be controlled or measured, and again approximationsneed to be interpreted correctly. The third part of the course is a surveyof approximation methods to be developed more fully in specialized courses.More importantly, equations such as NS can yield a lot of information evenif we cannot solve them: it is all about extracting information and puttingit in context.


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0.2. OVERVIEW 15

One overriding technique is presented rst (Ch. 4): suitable scaling of the

equations brings out dimensionless coefficients to the various terms, some of which may be relatively small and might be neglected (one needs to be carefulwith this approach: very large Reynolds number is associated with turbu-lence, while innite Reynolds number may yield unique solutions). Then,in turn, we explore the features of inviscid and of irrotational ow (Ch. 5),the small Re limit (Stokes ow in Ch.6), narrow ows (boundary layer fam-ily) in Ch. 8, rotating ows (small Rossby number) in Ch. 10, and linearwaves and stability (Ch. 11); and examine the phenomena responsible forow separation and secondary ows (Ch. 9). The balance between depthof coverage and breadth of ideas is constrained by the 1-semester format:

this particular selection is but one choice among other possibilities. Manycolleagues and students have contributed encouragements and constructivecriticism. All suggestions were thankfully considered, even if they were notadopted. There would be more questionable statements about vector nota-tions and continuum mechanics without Prof. AJ Levys input, and moretypographical errors without A Nicolais, Y Wangs and Prof. Carlos Perezspatient proofreading. I will happily correct any errors introduced since, as Ibecome aware of them. And the other shortcomings are all mine, I am afraid.

JL


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Chapter 1

Motivation

This chapter, inspired by Trittons book, presents the need for more advancedtools than are typically covered at the undergraduate level in a typical En-gineering curriculum in the USA. For internal ows, the canonical case isthe ow in a circular pipe. The student should be familiar with the laminar(Poiseuille) parabolic velocity prole, with the Moody diagram for pressureloss in a pipe, and with the modied Bernoulli equation including frictionand minor losses. So we understand laminar friction and pressure drop, buthow about entrance losses? Secondary ows at the entrance and exit? Flowsin curved pipes? Pulsatile ow?

At the undergraduate level, external ows are studied in relation withdrag in uniform ows (usually involving empirical dimensionless curves of drag coefficients, and a description of ow separation and the drag crisis),at plate boundary layers, and possibly some simple potential ows such asthe ow around a cylinder. In this chapter, we revisit these topics to motivatethe introduction of the more advanced concepts required to understand e.g.ow separation. We will end up with many questions marks, many unresolveddifficulties to be addressed later in the course.

1.1 Internal Flow1.1.1 A simple problem

There are a number of practical design issues related to the pipe ow issuingfrom a head tank (Fig. 1.1). There are also some added features (e.g. about

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18 CHAPTER 1. MOTIVATION

Figure 1.1: A simple pipe ow set-up

entrance pressure drop) that can be explained on the basis of undergraduatetools. And an engineer should not overlook these approaches: simple rst!But there are also a number of facts that show the need for more advancedconcepts, more sophisticated tools (and more advanced mathematics).

On the practical side, it would be important to maintain constant owrate. This can be achieved in several ways:

use a constant-displacement pump, insulated from the pipe by a baffledplenum; or

use an overow pipe, so the supply rate into the head tank needs nomonitoring; etc.You might want to discuss alternative designs. By running this experimentover a wide range of Reynolds numbers, and making sure the pipe is longenough so entrance and exit losses can be neglected (how long is that?), onecan collect the data represented on the Moody diagram (Fig. 1.2), here shownfor a smooth pipe. The analysis was carried out in the previous chapter.


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1.1. INTERNAL FLOW 19

102

103

104

105

106

107

108

103

102

101

100

Reynolds number

F r i c

t i o n

f a c

t o r

Figure 1.2: Sketch of Moody diagram for pipe friction

Refer to the previous chapter for the basic analysis, using Bernoullisequation and control volume in the pipe. For automated calculations, onecan make use e.g. of Colebrooks formula for the friction factor f

1 f = 2.log10 (

e

3.7 +

2.51ReD f ), (1.1)

where e is the pipes relative roughness (Fig. 1.2). We now turn to some of the details.

1.1.2 Entrance ow

At one level, an improved analysis in the entrance region of the ow uses el-ementary tools, even if the execution (e.g. to calculate the entrance length)can be rather complicated (can be found in uids or heat transfer undergrad-uate texts: it can be done). At another level, as we pay closer attention, theshortcomings of the analysis are more obvious.


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Figure 1.3: Entrance region of pipe

Entrance pressure drop

Assume a sharp leading edge to the pipe; a boundary layer (Fig. 1.3) devel-ops on the curved wall (which appears at as long as /x remains small: thisrequires some thought, along the lines of the earth is at). The main differ-ence with the ZPG boundary layer is that the freestream, near the centerof the pipe, cannot be deected outward, since there is no outward. Referto Ch. for the context: the same web of ideas is present here. The defectin mass ux in the wall region (as the ow changes from top-hat entrance

prole to no-slip boundary layer prole) must be made up in the core region,even before it experiences direct effects of viscosity.

In other words, the core velocity must increase - the ow is accelerated.Since F = ma , there must be a force. By default, pressure is the only avail-able source, therefore pressure must drop in the entrance region to overcomethe growth of the boundary layers! This can explain entrance losses, up to apoint, in laminar ow. Putting this in equations, a variety of methods apply:integral (von K arm an method), and series expansions for the solution, aretextbook material. Once the idea is clear, the execution is a matter of skilland perseverance.

Vena contracta and recirculation

The problem is actually made more complicated by the fact that, possibly inrelation with the nite thickness of the pipe wall which should be consideredas a bluff body, the ow separates from the wall at the leading edge (Fig. 1.4).


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Figure 1.4: Vena contracta and recirculation

The streamlines are no longer nearly parallel, and an annular recirculationregion appears near the wall. Within this region, the ow is even goingupstream near the wall!

In fact, once we examine the entrance region (or many other simpleproblems) in more detail, some of the features can be understood based onundergraduate material (mass ux, acceleration, pressure drop), and othersclearly cannot (streamlines, separation and reattachment, secondary ow,pressure recovery, etc.) (Fig. 1.5).

These questions are important because similar phenomena occur in avariety of applications. Flow separation is very important in aerodynamics(reduces the effective cross-section in turbine passages, and reduces lift). Sec-ondary ows matter in environmental ows (pollutants can remain trappedin such regions), and in wake formation (see below). The representation of streamlines needs analytical support, or even a clear denition, so we cantrust that the software is capable of catching such ow features. And soon: undergraduate uid mechanics cannot go beyond the descriptive levelfor these phenomena, we need something better.

1.1.3 Transition to turbulence

This last remark is also true for the transition to turbulence (Fig. 1.7).From Stewarts movie on turbulence, the initial clip about pipe ow is rel-evant here (Fig. 1.6). Undergraduates know about the jump on Moodysdiagram, about the magic number Re 2000 (or is it 2300??). A descriptionof the phenomenon includes, possibly, dye injection to visualize the ow (arethey streamlines?), the appearance of wavy ripples, the abrupt but inter-


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Figure 1.5: Pipe entrance: some ideas


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R. Stewarts movie (excerpt): Turbulence We only cover the pipe ow experiment: transition

The set-up: viscosity as control parameter

Visualization of pressure drop

Velocity proles

Transition, pressure drop vs. viscosity

Secondary ow with laminar discharge

Figure 1.6: Transition in pipe ows, laminar discharge secondary ow: ex-cerpt from Stewarts movie on turbulence.

Figure 1.7: Transition in pipe ow

mittent transition sending turbulent puffs down the pipe, the possibility of relaminarization, the intense mixing by the turbulence, the increase in drag,etc. Critical Re, what does it depend on?

(engineering: design away from critical point to have predictable behav-

ior)Beyond the empirical observations, we do not yet have the tools to under-

stand any of the qualitative or quantitative features of transition in a pipe.Therefore we cannot begin to understand which features would occur in othertypes of ows, or how software might include these important phenomena,or whether it should...


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Figure 1.8: Secondary laminar ow at pipe exit

1.1.4 Pipe exit and secondary ow

A simpler but equally puzzling situation occurs at the pipe exit. Again, wecan inject dye near the centerline or the wall to get an idea of what happens how do we do this numerically?

The turbulent case is, in a way, simpler: with momentum thoroughlymixed across the pipe, and the dye with it, relatively simple jet comes outof the pipe: still turbulent inside, with surface tension effects (no coveredin this course, what a pity!) affecting interface geometry (jet break-up intodrops? surely this could matter for some applications.)

But let us focus on the laminar ow discharge (Fig. 1.8). The jet isnow made of three distinct parts: the main jet following a roughly parabolictrajectory as in the turbulent case, and a puzzling stream falling at a muchsteeper angle, the two connected by a thin lm of water (or whatever theliquid is) (surface tension again! Look at the pictures in Van Dykes book orin the Gallery of Fluid Motion in Physics of Fluids).

Dye injection shows that the minor stream collects the low-speed liquid,not only from the bottom of the pipe but also from the top. An end-view of the ow shows that the dye lament wraps around the central jet, unable tomaintain near-horizontal motion. We would expect this from free particlescoming out at different speeds, but how does a continuous jet do it? Itsparticles cannot fall through the high-speed core, they have to go around it!


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Figure 1.9: Laminar liquid jet discharge: some ideas

and why do the uid particles sort themselves into two main speed categories?(Fig. 1.9.)

1.1.5 Advanced problemsCurved/helical pipes and secondary ows, pulsatile ows, turbulent ows,non-circular cross-section, variable viscosity,...

1.1.6 Food for thoughtOther undergraduate tools are clearly insufficient to understand the com-plexity of phenomena.

For example, consider minor losses in pipe systems: one level of engi-neering solution is to look up the appropriate loss coefficients K m in a table.But how trustworthy are these numbers in compressible ow? in two-phaseows? in ows where variations in viscosity (with temperature) are large?What are the relevant ideas that would raise a red ag or give us condencein the simple tools?


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Figure 1.10: Minor losses: questions and phenomena


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1.2. EXTERNAL FLOW 27

Figure 1.11: Mixing in a control volume: some ideas

The same might be said of such a common (and powerful) tool as controlvolume analysis. Satisfying the basic conservation laws (mass, momentum,energy, entropy) overall for a macroscopic system of interest sweeps underthe rug a number of details about some mechanisms that might affect design.Suppose the pipe is a chemical reactor: surely mixing would be an importantconcern, as might energy dissipation and pressure drop. Is the friction coef-cient (Moody chart) valid regardless of temperature or other conditions of reaction? Is it safe to extrapolate data (with the implication that no qual-itative changes would occur)? If mixing is desirable, is it sensible to add agrid? or some other device? Would better reaction yield be obtained withoutturbulence at all?

1.2 External owFlows around obstacles (such as aircraft wings, tennis balls or underwater

robot arms) offer a variety of problems that can be organized in order of dif-culty and generality. Canonical ows combine key common elements of theow physics with a simple-enough conguration so that the understanding of the simple case does shed light on the more complicated reality. Thus, uni-form ows around circular cylinders and spheres are a starting point towardmore complicated (realistic) situations. They can be studied analytically


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of the problem (are we assuming that the entire solution, including the wake,

is symmetric?), extending upstream and downstream of the cylinder over alength L, and considerably wider than the wake half-width (say, w + d at thedownstream end). There is no ow in the z -direction (why is that?).

Mass balance (per unit density) is summarized in the equation (Reynoldstransport theorem again! a basic tool)

0 = (U dA)= U 0 (w + d) + L (v.dx) + U 0 d + ([U 0 u]dy) + 0

=

L(v.dx)

w

0u(x) dy (1.3)

which, again, shows a ow away from the plane of symmetry as a result of the defect in mass ux in the wake.

For the momentum balance, we have to acknowledge that the cylinder isapplying a drag force F (negative x direction) on the uid; pressure is uniformat the control surface, located far enough from the cylinder so streamwiseviscous forces are negligible. Then

F

= U 0 [U 0 (w + d)] + L U 0 (v.dx) + w + d

0(U 0 u(x, y))

2 dy + 0

= U 20 w + U

0 u(x) w0 f dy + w

0(U 0 u)

2 dy

= w

0u2 dy U 0

w

0u dy (1.4)

ConsequentlyF

12 U

20

= 2 w

0

uU 0

(1 uU 0

)dy (1.5)

which shows, as for the boundary layer, that the net force is related to thedefect in momentum ux.

1.2.2 Potential ow modelAn exact solution (Fig. 1.13) can be obtained if we assume that the ow ispotential, i.e.

u = (1.6)


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3 2 1 0 1 2 33

2

1

0

1

2

3

x

y

Potential flow around a cylinder

Figure 1.13: Irrotational ow past a cylinder.

for some velocity potential . The solution is found in undergraduate texts,See more details in Ch. 5. This elementary solution accounts for frontstagnation pressure, reasonable streamlines, acceleration toward the mid-section of the cylinder, etc. Its failure to account for drag (DAlembertsparadox, see Ch. 5) was resolved by Prandtls discovery of the boundarylayer. But the model shows no prospect of being extended to account forsome of the phenomenology: Karm an vortex street, unsteady wake, dragcrisis...

1.2.3 PhenomenologyA dramatic feature of bluff-body wakes at low Reynolds numbers is the sep-

aration of the boundary layer from the obstacle surface, and (in the caseof the cylinder) formation of the K arm an vortex street (Fig. 1.14): a stag-gered array of counter-rotating vortices illustrated e.g. on the cover of VanDykes book. As the Reynolds number increases, the shear layers becomeincreasingly unstable, and soon the wake becomes turbulent. Even so, inthe intermediate wake, ow structures reminiscent of the vortex street are


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Figure 1.14: Karm an vortex street behind a cylinder

Figure 1.15: Wake structure

observed (Fig. 1.15).

Moving from the descriptive to the quantitative, the drag curve also pointsto the need for deeper understanding. The variations in the drag coefficient(Fig. 1.16) at low and moderate Reynolds numbers can be tied to some

of the wake behavior described above. But the sudden dip in the drag isassociated to a different phenomenon: the boundary layer itself becomesturbulent before it separates from the surface (Fig. 1.17), thus delayingseparation. Such behavior is important in design (from dimples in golf ballsto vortex generators on aircraft wings), but cannot really be understoodbased solely on undergraduate analysis tools.


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Figure 1.16: Drag coefficient for a cylinder in a uniform ow.

Figure 1.17: Effect of BL transition on wake formation


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1.3. PERSPECTIVE 33

1.2.4 Food for thoughtMost students would have heard about the transition Reynolds numbers:2000 for pipe ow, 500,000 for at plate boundary layers: we need to under-stand the meaning of such numbers, and why they are so different.

Problems1. Derive the Poiseuille solution for channel ow; express the wall shear

stress and friction coefficient.

2. Read paper or reference textbook on helical pipes: map the ideas rel-

ative to new phenomena/concepts/questions relative to this chapter.(do not write or solve equations).

3. Ditto with pulsatile ow

4. Select one of the following problems, nd relevant documentation at thelibrary, and identify to what extent the phenomena can or cannot bedescribed by undergraduate methods: pulsatile ow in a pipe, curvedtrajectories of balls (baseball, ping-pong, tennis or soccer), wake of anairfoil with lift, wind gusts in downtown canyons.

1.3 PerspectiveNeed to understand vorticity, separation, secondary ows, transition, ...


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Figure 1.18: Wake formation: some ideas


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Chapter 2

Kinematics

Kinematics is a study of the geometry of motion independently of appliedforces. It consists of descriptive tools and various constraints that limit inimportant ways how the uid can respond to the application of forces.

We will learn about vectors, with velocity as the foremost example; aboutvarious algebraic, differential and integral operations on vectors; about suchvectors as attributes of a particle (Lagrangian description) or of a point inspace (Eulerian description); about rate-of-strain and vorticity, maybe themost important new concept in the whole course; about mass balance andits consequences; and about ow decompositions, vortices as ow structures,and much more.

Particularly relevant movies are: Klines Flow visualization, as back-ground for all others; Lumleys Eulerian and Lagrangian descriptions, andShapiros Vorticity.

2.1 VectorsClassical mechanics are rmly grounded in 3-dimensional space, in whichvectors are ubiquitous objects. Sure enough, scalar quantities such as energy

and pressure are important, as seen in Bernoullis equation; but mechanicsis dominated by the use of vectors for position, momentum, rotation, etc.

Vectors are mathematical objects endowed with a direction and a mag-nitude they may also have a point or line of application. In this course,we will use three equivalent notations for vectors, each with advantages de-pending on the problem at hand. It is important that the student learn to

35


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2.1. VECTORS 37

Without implication about a particular system of axes, this notation will

be used by default, but is sometimes awkward when tensors of orders largerthan vectors are being used, e.g.

u or (a b), (2.9)for which alternative notations are clearer.

2.1.2 Component notationsComponent notations are most popular at the undergraduate level, but suf-fer from runaway bulk! We will use them in this course only when differentdirections contain different physics that should be reected in the notations,for example in boundary layers where the streamwise (along the wall), trans-verse (normal to the wall) and spanwise (along the wall, normal to the othertwo) directions are very distinct.

The use of non-Cartesian coordinates (polar, cylindrical, spherical, etc.)is less popular for beginners, but is actually implied by the above reason-ing: boundary conditions may well dictate which coordinate system to usefor easier handling! We will use non-Cartesian coordinates as needed, andencourage the students to overcome any reluctance in this regard as soonas possible. Pointers will be included at appropriate places throughout thechapters.

Coordinate axes may be Cartesian ( ex , ey ,ez), equivalent to ( i, j ,k); orcylindrical-polar ( er ,e, ez); among many options. The component of a vectoralong such an axis is the projection of the vector on the axis, e.g.

ux = u ex . (2.10)Then, we can write

u = uxex + uyey + uzez . (2.11)

It is sufficient to write the vector as the ordered set of its components, forexample (ux ,uy ,uz) for which the alternative notation ( u,v,w) is also common.Multiplication by the unit vectors is implied, and should be restored whenin doubt. This notation can then be used to evaluate vector operations. Thedot product reduces to

u v = uxvx + uyvy + uzvz . (2.12)


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38 CHAPTER 2. KINEMATICS

Figure 2.1: Vector notations


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2.1. VECTORS 39

For the cross product u = r , its x-component isux = yr z zr y , (2.13)

and similar expressions are obtained by even permutation of the subscripts.The vector-operator is best expressed in Cartesian coordinates. Then,

its components are ( x , y, z) and we can write (in a convenient mix of notations)

u = xux + yuy + zuz . (2.14)Similarly, the x-component of = u is

x = yuz

zuy, (2.15)

and the Laplacian is the familiar

2 = 2xx + 2yy +

2zz . (2.16)

There is no general expression for in non-Cartesian coordinates; the con-text of gradient, divergence, curl or Laplacian is necessary for specic termsto make sense, and the corresponding expressions can be found in manyreference books.

2.1.3 Index notationsA more concise version of component notations is possible when the Carte-sian axes are assumed, but without any particular orientation. Then, thesubscripts x, y and z are just an ordered triad, and 1, 2 and 3 turns out tofacilitate the sums. With unit vectors of the form ei for i = 1, 2, 3, we have

u = uiei (2.17)and the triad of components can be written simply as ( u i). Then for a dotproduct

u v = i uivi = uivi (2.18)where the summation over repeated indices is implied (Einsteins convention).Some textbooks and websites use index notations while keeping explicitlythe unit vectors ei : this defeats the purpose of compact notations. Anycomponent with a free index is understood to multiply the corresponding


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unit vector, with summation implied: ui stands for all components and unit

vectors together. The use of ei will be penalized in this course.Concise notations can save work, but require neat handling. Studentsshould be careful to adhere to the following rules:

any repeated index in a product implies summation; therefore, that in-dex pair should be used only once in an expression, but can be renamedat will (dummy index) to avoid conicts with other pairs. Thus

uivi = ukvk ; (2.19)

no index may appear 3 or more times in a product; in rare instanceswhen this occurs, the use of more detailed notations is required, andsummation is usually restored explicitly to avoid confusion

an index appearing once denotes the component of the vector (freeindex); it must be the same for all terms in a sum, since they all arefactors of the same implied ei ; thus

a i = bi + ck (2.20)

is meaningless.

They can also be generalized very usefully to higher-order objects: whilea vector requires a single index, a matrix requires 2, etc. The economy of notation is most obvious in such context, as we will see in later chapters.Two symbols are particularly important: the Kronecker symbol

ij (2.21)

has value 1 if i = j and 0 otherwise. It is a representation of the identitymatrix, since

ij u j = ui . (2.22)

Note also that, in 3 dimensions, ii = 3. (2.23)

More complicated, the permutation symbol

ijk (2.24)


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2.1. VECTORS 41

has value 1 if (i,j,k) is an even permutation of (1,2,3), -1 if an odd permu-

tation, and zero otherwise (i.e. if any two indices are equal). is used torepresent the cross product:

ui = ( r )i = ijk j r k . (2.25)This should be veried component by component (yes, do it!) One notewor-thy relation, used for handling double cross products, is

ijk ilm = jl km jm kl . (2.26)

Note that the order of the indices of can be rearranged based on

ijk = kij = jik (2.27)and similar alternatives

With these tools, we can write the differential operators as

( f )i = if (2.28)

for the gradient; the divergence takes the form

u = iu i . (2.29)The Laplacian is

2 = 2ii . (2.30)

Finally, for the curli = ijk j uk . (2.31)

The order of indices of is important: rst index for the free index, second

for the derivative, third for the vector of which the curl is taken (and youcan rotate the indices afterward, of course).Using index notations requires practice (beyond assigned problems), but

is well worth the effort when mastered. In this course, all three notationswill be used depending on the context: the idea is to save work and facilitateformal manipulations.


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J.L. Lumleys movie (excerpt): Eulerianvs. Lagrangian

No dynamics! This is kinematics Velocity of particle, velocity at a point Convective derivative Pathlines, streaklines, timelines, streamlines

Unsteady ow: wave eld

Figure 2.2: Eulerian and Lagrangian descriptions, from Lumleys movie.

Tensor notations

In the case of non-Cartesian coordinates, index notations can be generalizedas tensor notations 1. Added complications include co- and contravariance,and the use of the local metric tensor (responsible for the strange factors

in the expression of curl, div and grad). In uid dynamics, this can beencountered in some treatments of computational grid generation, in thestudy of non-Newtonian uids, and a few other applications, but will not beneeded in this course.

2.2 Eulerian vs. Lagrangian descriptionsViewing of Lumleys movie, as summarized on Fig. 2.2.

Newtons mechanics, with its discovery of the point particles as a usefulconcept, traces the motion of material points. These points are endowedwith mass, but otherwise have no properties, and the purpose is to study themotion of these simple objects. This simple program was expanded later toinclude rigid bodies (motion of and around a point, moments of inertia) and

1 Even more elegant notation is made possible by using exterior products and differentialforms; I am not aware that these have been used in mainstream uid dynamics.


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2.2. EULERIAN VS. LAGRANGIAN DESCRIPTIONS 43

continua: this idea will be expanded in the next Chapter.

Consider a traffic ow, and you are seated in one particular car, which isassimilated to a point particle. As the car moves and stops, it experiencesforces that change its speed and direction. The point of application of theforce changes with the location of the car, and the associated accelerationsare those of your car. This is the Lagrangian description of the traffic ow,very close in spirit to Newtons initial formulation. It deals with the motionof material points .

An alternative, called Eulerian description, looks at the traffic ow fromthe viewpoint of the bystander: motion at a geometrical point regardlessof which particle is moving by. If you stand at a traffic light, the vehicles

will experience periodic accelerations and decelerations, as the light turnsfrom red to green and back again. This periodicity, a dominant featureof what happens at this intersection, is not experienced in the Lagrangiandescription. Thus, the two descriptions will give rather different views of the motion of traffic. Steady or periodic ow are Eulerian properties.The only steady ow in the Lagrangian description is uniform, i.e. at restis a suitable frame of reference, and of no practical interest. Mathematicaldescription and display of results are equivalent but show a different aspect of the same reality. The choice of formulation may be dictated by convenienceor habit, but the results must be interpreted accordingly and converted if necessary: experimental ow visualization is naturally Lagrangian, whereasanalytical and/or numerical solutions yield Eulerian properties rst. In thecase of traffic ow, it would be necessary to know the state of (Lagrangian)motion of all vehicles at various times in order to reconstruct the Eulerianmotion at a given point; and conversely. The purpose of this section is todevelop the required tools and to provide illustrations of the main steps;only the simplest cases can be treated in closed form, but the procedureis identical for the numerical processing of more complex experimental ornumerical data.

2.2.1 Lagrangian description of motionFor the purposes of presentation, we will focus on the description of a ve-locity eld; appropriate changes can be made for other properties such asacceleration, pressure, temperature, etc. In the Lagrangian description,

u(t, x 0, t0) = ddt

x(t, x 0, t0) (2.32)


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Figure 2.3: Lagrangian, Eulerian, experimental, numerical


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describes the velocity of the particle that was at location x0 at time t = t0

and is now at x(t). Notations such as x0 reinforce this idea, but can becumbersome; as we omit some details, the student should retain the implieddependences in their thinking. We can also write

x

x0dx = x x0 =

t

0u(, x0, t0)d (2.33)

From elementary mechanics, we know that, as time t varies, this is the trajectory of the particle, in parametric form as time changes. Elimination of time between the components of the trajectory gives one (2D) or two (3D)equations for the trajectory components in terms of each other. Its calcula-tion requires the knowledge of the history of velocity for this particle. In theexample of the traffic ow, knowing history of your velocity enables you toevaluate your current location relative to the point of origin. This is true foreach vector component. Mapping ocean currents can be done by tracking thetrajectory of a buoy. In uid mechanics, a pathline is equivalent terminologyfor trajectory, and its reference to a particle makes it a Lagrangian concept.Experimentally, you might mark one particle to be very bright, and take avery long exposure that will record its trajectory: if it is a uid particle, youhave generated a pathline (but if it is a rey, you only have a trajectory of the y, which is not a picture of the motion of the air).

A different concept is generated by the gradual release of particles (e.g.

smoke or dye) from one point (Fig. 2.5). The resulting line is not a pathline,but a streakline (and they coincide if the trajectories are independent of thetime of release t0). Its equation should make reference to the release from agiven point x0 at successive instants in the past

x = x0 + t

t 0u(, x 0, t0)d (2.34)

Here, the parameter along the streakline is the release time t0, with thematerial points closer to x0 having been released most recently. We notethat the equations for pathlines and streaklines are sections through thesame family of solutions, with different parameters being held xed. The

streakline is commonly achieved by the release of dye, bubbles, smoke orother particles as markers of the uid particles (question: are they goodmarkers?)

To illustrate all this, use the ow around a sphere, in the frame of referenceof the sphere and for a sphere in free fall, from Trittons book. See alsoexample below, and in Curries book.


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Figure 2.4: Particle pathline

Figure 2.5: Streaklines


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S. Klines movie: Flow vizualization

Hydrogen bubbles: water electrolysis

Bubbles: good tracers?

Lagrangian or Eulerian? Types of lines

Path lines Streaklines

Time lines

Streamlines?

Steady / unsteady

Other ow markers: dyes, particles Also note:

Break of symmetry in cylinder wake, in diffuser

Instability in free shear layers (lifting airfoil wake, etc)

Figure 2.6: S. Klines movie on ow visualization: what do we really see?


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Figure 2.7: Eulerian velocity vectors from particle traces

2.2.2 Eulerian description

It is not uncommon for experimentalists to observe many pieces of trajectoriesat once: seeding many bright particles in a ow, and taking an exposure of short but nite time , the collection of vectors (for many initial locationsx0 is

x x0 = t+ dt

tu(t, x 0, )d u(t, x 0, t )dt (2.35)

for each x0 gives a nite difference approximation of the velocity at eachpoint, leading to the Eulerian description of the eld (Fig. 2.7). Note thattime t denotes the running time as well as the release time.

In the Eulerian description, the emphasis shifts away from the individualparticles (still necessary to assign a velocity!) in favor of the distribution of values (e.g. or velocity) in space, i.e. the velocity eld. This is the approach

taken in PIV and related methods.Given an experimental sample such as on the gure, the eye (actually:

our brains, processing visual information and superimposing patterns) in-terpolates lines that are at every point tangent to the velocity vectors: thestreamlines. In the plane, the equation of streamlines, given a velocity eldin Cartesian coordinates, is easily constructed: at every point, the slope of


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Figure 2.8: Direction of a streamline

the line must match the direction of the velocity vector

dydx |streamline =

vu

(2.36)

In 3D, a line requires 2 equations: it can be seen that the streamline isdetermined by

dxu

= dy

v =

dz w

(2.37)

With the velocity vector tangent at each point to pathlines and to stream-lines, the distinction between the two types of lines is emphasized. For thestreamline, the direction of the line results from a snapshot: simultaneousvelocity vectors at each point. For the pathline, we have the history of mo-tion of a particle. Once again, for steady ow, they coincide; in unsteadyow, they can be unrecognizable from each other (Tritton). When comparing

experimental and numerical results, this can complicate the interpretation.

2.2.3 Pathlines, streaklines and streamlinesThe conversion from Lagrangian to Eulerian (and conversely) is a usefulexercise. An example is given in Currie (p38-42). Here, we will treat a few


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examples of increasing complexity. Students should carefully mark what is

general and what is specic to the problem at hand, and not use relations outof context; this is also a cue about brushing up on differential equations andelementary calculus. The pivotal item is velocity: the Eulerian velocity isalso the velocity of a particle that happens to be at the point of observation.

Example 1: Steady ow

Consider the velocity eld (Eulerian: why?) given by the expressions

u = xyv = y. (2.38)

which is independent of time. The equations of the (Eulerian: why?) stream-lines is easily obtained

dxu

= dxxy

= dy

y (2.39)

and are readily integrated:

y = y0 + ln( xx0

) (2.40)

Switching to the Lagrangian concepts requires an important change: theparticle trajectories are generated by updating their Eulerian locations to

match the location of each material point. Henceu =

dxdt

= x(t)y(t)

v = dydt

= y(t). (2.41)

Because the y-equation can be integrated separately, this problem is relativelysimple. We get

y = y0et t 0 (2.42)

which is substituted into the x-equation, and gives

ln ( xx0 ) = y0(et t 0 1). (2.43)

There are two parameters ( t and t0) for the location of a particle associ-ated with x0, y0. If we eliminate t, we obtain the trajectory

y = y0 + ln( xx0

) (2.44)


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identical to the streamlines through the same point. If instead we eliminate

t0, thus obtaining the equation of the streakline through the same point atthe current time t, the result is again the same, a feature of steady ows.The distinction between streamlines, pathlines and streaklines can be

subtle: the correct interpretation of the relations depends on identifying therelevant parameters.

Example 2: Unsteady ow

This distinction is highlighted in unsteady ows, where the three types of lines are usually different. We start from the Eulerian eld

u = x sin (t)v = y (2.45)

Within the Eulerian framework, we obtain the streamlines

ln ( xx0

) = sin (t) ln( yy0

), (2.46)

which vary in time.Switching to the Lagrangian framework, we have for the particle motion

dxdt

= x(t) sin (t)

dydt = y(t) (2.47)

The two variables are separated, allowing for simple solutions:

ln ( xx0

) = cos(t0) cos(t)ln (

yy0

) = t t0 (2.48)Two families of lines are included here: if we eliminate t, we have the path-lines x(y |x0, y0, t 0):

ln ( xx0

) = cos(t0) cos(t0 + ln( yy0

)); (2.49)

whereas if we eliminate t0, we get the streaklines

ln ( xx0

) = cos(t) + cos(t ln ( yy0

)); (2.50)

The 3 families of lines (streamlines, pathlines and streaklines) are illus-trated on Fig. 2.9


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0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Figure 2.9: Streamlines (solid lines) at times t = 0 (black), / 4 (red), / 3(magenta) and / 2 (blue); Pathlines (dashed) through the point (1 , 1) attimes t0 = 0, / 4, / 3 and / 2; Streaklines (dotted) through the point (1 , 1)at times t = 0, / 4, / 3 and / 2.


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Figure 2.10: The oscillating hose

Example 3: Unsteady ow

In this example, we go from Lagrangian to Eulerian. In an approximationof an oscillating garden hose (Fig. 2.10), the water motion is limited tothe horizontal x y plane, ignoring the effect of gravity. At time t0, thewater leaves the nozzle (origin of coordinates) a constant speed U , but at anoscillating angle = 0 sin t0. If the angular amplitude 0 of oscillations issmall, we have (at rst order in 0)

u0(t0) = U (2.51)

andv0(t0) = U 0sin (t0) (2.52)

as the initial conditions for the trajectory of a drop leaving the nozzle at timet0.

In the absence of applied forces, the trajectory of a drop is easily calcu-lated:

x(t, t 0) = 0 + u0(t0)( t t0) (2.53)and

y(t, t 0) = 0 + v0(t0)( t t0). (2.54)


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This is the parametric representation of pathlines for this particular problem,

with t as the parameter and t0 as an additional label (note the different roleplayed by t and t0 below). We can eliminate the parameter t easily:

t t0 = xu0

= yv0

(2.55)

ory =

v0u0

x = x 0sin (t0) (2.56)

The pathlines are straight lines issuing from the origin, with a slope depend-ing on t0.

Streaklines are more difficult to obtain: the collection of drops that left

the nozzle at different times in the past. This requires the elimination of t0from the individual trajectories. We have

x = ( t t0)U t0 = t xU

(2.57)

which is substituted in the equation for y

y = x 0(t xU

)sin ((t xU

)) . (2.58)

This is the equation of streaklines at a given time t.Note which substitution changes the formulae from a particle trajectory

into a streakline! Obtaining the streakline in the form y(x; t) could be quitedifficult without the initial linearization. Yet, taking a snapshot of the waterdrops would achieve the same result. This illustrates how seemingly simpleexperimental results can be analytically difficult to reproduce. The concep-tual key is in the role of the various parameters.

Finally, the streamlines are obtained easily: at any instant, the velocityvector is radial from the origin, so that the streamlines are indistinguishablefrom the pathlines:

dxu

= dy

v (2.59)

gives againy = v

ux (2.60)

The superposition of pathlines and streamlines arises because the Eulerianvelocity eld is independent of time. Similarly, if the Lagrangian velocity istime-independent, the pathlines and streaklines coincide. All three lines areidentical in steady ow, as seen above.


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Figure 2.11: Pathlines, streaklines and streamlines: the parameters


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Stagnation point...

x

y OS

3 2 1 0 1 2 33

2

1

0

1

2

3...is not Galileaninvariant

x

y O

S

3 2 1 0 1 2 33

2

1

0

1

2

3

Figure 2.12: Stagnation point (rst frame) is modied by the superpositionof a uniform speed (second frame). Streamlines are blue.

2.2.4 Caution!The streamlines, pathlines and streaklines are very dependent on the frameof observation. From a moving reference frame even in uniform motion(Galilean invariance: see Ch. 10 for rotating systems with ctitious forces),the appearance of the lines can be dramatically affected.

For example, consider the simple potential ow (See Ch. 5) with a stag-nation point at the origin. From a reference point moving at uniform velocityat 45 degrees to the axes, the addition of the corresponding velocity changesthe streamlines as shown on Fig 2.12. The problem of describing ow topol-ogy in a way that is independent of the point of observation goes beyond thescope of this course.

2.3 Differential concepts and their geometry

2.3.1 Differentials and vectorsThe gradient of a function f requires no explanation at this level. It is avector (direction, magnitude), represented by f which has various repre-sentations depending on the coordinate system (Cartesian, polar, etc.) Intwo dimensions, think of a pressure eld p(x, y). The gradient will have adirection normal to the isolines p = C This interpretation translates easily


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2.3. DIFFERENTIAL CONCEPTS AND THEIR GEOMETRY 57

Figure 2.13: Gradient is normal to isolines of a function

to 3D. Turn your attention to one component, e.g. in the y-direction, of thisgradient: the projection of the gradient on the unit vector ey , it measures therate of increase of p in the y direction. This is the basis for the denition of the directional derivative. Given a unit vector e in some direction denotedby , the directional derivative of p in the -direction is naturally

p = e p, (2.61)or, regardless of the eld to which it is applied

= e . (2.62)Learn to recognize the combinations vector , whenever it occurs.Now consider a function f (r, t ). The vector position has coordinatesr = ( x(t), y(t), z (t)) that move with the uid. If we look at the evolution of f , its time derivative is obtained through the chain rule

dt f = t f + xfd t x + yfd t y + zfd tz = t f + u x f + v yf + w zf

= t f + u f (2.63)This material derivative, i.e. time derivative following a uid particle, issometimes denoted as D t for emphasis. The alert student has spotted the di-


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Figure 2.14: Denition sketches for Gauss and Stokes theorems

rectional derivative u in the direction of streamlines, also called convectivederivative.We already saw Gausss divergence theorem:

F dV =

F dA (2.64)

or, in index notations

iF idV = F idAi (2.65)Another important theorem is due to Stokes

F dr = ( F ) dA (2.66)In the case where the vector eld is velocity, Stokes theorem introduces twoimportant concepts. Vorticity is dened as

=

u. (2.67)

Vorticity will be one of dominant new concepts throughout this course. Re-lated to vorticity by Stokes theorem is the concept of circulation

= u dr (2.68)


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2.4. MASS BALANCE 59

2.4 Mass balanceWe start from the integral representation

t dV + (U dA) = 0 , (2.69)assume that the control volume is xed in space, and make use of Gaussdivergence theorem to obtain

t dV + (U ) dV = ( t + (U )) dV = 0 (2.70)Since this holds for any xed control volume, the integrand must vanish atevery point: therefore

t + (U ) = 0 (2.71)

Note how much simpler this derivation is, compared to the use of materialderivatives following a uid particle. Then

dt = 0 (2.72)

because the density changes as the particles volume does: the divergence of

velocity accounts for this.Since we limit ourselves to the study of incompressible ows, where den-sity can be treated as constant, the continuity equation reduces to

U = 0 or iui = 0 (2.73)This relation is a kinematic constraint on changes in the velocity compo-

nents. For example, as a unidirectional ow approaches stagnation point andits velocity decreases, it must develop a component of velocity normal to theupstream motion for mass to be conserved. It is benecial to compare theresults of control volume analysis and continuity equation in this regard.

One of the great simplications in incompressible ow, relative to com-pressible ows, is related to the use of the simple (linear) continuity equation just derived. It is interesting to note (proof in Ch. 3) that, in the case of natural convection, the continuity equation can be applicable even thoughthe variations in density drive the motion. This non-trivial result is knownas the Boussinesq approximation.


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2.4.1 Vector potential and stream functionIt is known from vector calculus that any divergence-free vector can be ex-pressed as the curl of another vector (and conversely). Therefore, wheneverthe continuity equation applies, there is a vector A (called the vector poten-tial) such that

U = A (2.74)The vector potential will be used extensively later. Note that arbitrary con-stants, or even the gradient of any smooth eld, can be added to each com-ponent of A without affecting the velocity eld. This exibility is used toensure that A = 0 without loss of generality (gage invariance).A particular case of considerable importance relates to the 2D ows (inthe x-y plane; axisymmetric versions can be found below). Then, inspectionof the components of A shows that only the z-component of A is non-zero.In this case, we denote

Az = (x, y) (2.75)and we have

u = y (2.76)and

v = x. (2.77)Substitution into the continuity equation shows that mass balance is satised.

Lines of constant are of particular interest. It so happens that thedirectional derivative of along streamlines is zero:

U = u x + v y = uv + vu = 0. (2.78)This means that lines of constant are the streamlines. For incompressible

ow, nding the streamfunction and its contour lines is the fastest way to plotstreamlines. Alternatively, the above relation can be read as the projectionof on the velocity vector, and they are orthogonal. In other words again,it means that there is no mass ux across lines of constant , which canbe used as impermeable boundaries as long as no viscous effects need to bedescribed (slip allowed along the boundary). Furthermore, we see that

d = xdx + ydy = vdx + udy = U dA (2.79)so that the difference in streamfunction value is proportional to the massux through the streamtube. Or again, we can say that the velocity ina streamtube is inversely proportional to its width, and velocity can onlyvanish if the streamtube expands into a reservoir of innite size.


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2.5. FLOW ABOUT A POINT: HELMHOLTZ 61

Figure 2.15: Streamfunction and 2D velocity

2.5 Flow about a point: HelmholtzConsider a velocity eld (the following applies to any vector eld) ui(xk)(the use of a free index for xk should convey that any velocity componentcan depend on all coordinates.) In the vicinity of this point (Fig. 2.16), sayat xk + xk , a Taylor series will show the variations in eld values. Keepingonly rst order terms, we have

ui(xk + xk) = ui(xk) + x j j u i |xk + ...= ui |xk +

12

x j ( j u i + iu j ) + 12

x j ( j ui iu j ) + ...= ui |xk + x j sij |xk + x j r ji |xk + ... (2.80)

which can be interpreted as a sum of three contributions: a translation withthe point of reference, a straining motion, and a rotation: this is shown inthe next few subsections.

2.5.1 Rate of strainWith the understanding that all properties are evaluated at xk , we haveintroduced the symmetric part of the velocity derivative

s ij = 12

( j ui + iu j ) (2.81)


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Figure 2.16: Flow about a point

which is called the rate of strain. (This is similar to the strain in solid me-chanics, except this applies to velocities instead of displacements.) Becauseof incompressibility, its trace vanishes

s ii = 0 (2.82)

The eigenvalues and eigenvectors of a matrix are scalars s and unit vectorsei , respectively, such that

s ij e j (k) = s(k)ei(k) (2.83)

The parenthesis indicates that k is a label distinct from the indices andnot subject to implied summation. Interpreting the denition, applying thematrix to the vector does not change its direction, but can change its mag-nitude by a factor s. A 3x3 matrix has 3 such combinations of directionsand stretching, some of which can be complex-valued. But since sij is real

and symmetric, we know from linear algebra that its eigenvalues and eigen-vectors are all real. Combining this with the vanishing trace, which is thesum of the eigenvalues, we see that at least one eigenvalue of rate of strainis positive, corresponding to an extensional axis, and at least one value isnegative, corresponding to a compression axis, while the middle eigenvalueadjusts the sum of all eigenvalues to be zero.


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L

U

e(1)

e(2)

Figure 2.17: Couette ow and principal axes

The eigenvectors are orthogonal to each other and can be adopted as localcoordinates: they are called principal axes. In such a coordinate system, thematrix is diagonal, i.e.

s ij = ij s (i) (2.84)

Example: Couette ow

In the Couette ow shown on the gure, the velocity eld is given by

u(x, y) = U L

y and v(x, y) = 0 (2.85)

You can verify that this 2D ow is incompressible. The successive matricesis easily constructed from their denitions:

( j u i) = U L

0 10 0 (2.86)

Then

(s ij ) = U 2L 0 11 0 (2.87)

It is then a simple matter to obtain the two eigenvalues

s = U 2L

(2.88)


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corresponding to the eigenvectors

e = 1 2 1

1 (2.89)

Rate of strain takes a square (cube in 3D) with diagonals along the princi-pal axes and makes it into a diamond with its long diagonal on the extensionalaxis. For innitesimal stretching, this conserves area (incompressible).

2.5.2 Local rotation and vorticityThe antisymmetric part of the velocity derivative is

r ij = 12( iu j j u i), (2.90)

in which only 3 components are independent. rij could therefore be arrangedas a vector. In fact, it can be seen that

r ij = 12 ijk

k (2.91)

wherei = ijk j uk or = U (2.92)

is the vorticity vector. The corresponding contribution to the motion arounda point (Helmholtz above) is then written as

ui(xk + xk) = ui |xk + x j sij |xk + 12 ijk j |xk xk (2.93)In the + 12 dr term, we recognize the expression for rigid-body rotationaround the (local) origin, with angular velocity 12 . Pay attention to thefactors 1/2 in the rate-of-strain and rotation rate; pay attention to signs,they matter.

Example: Couette ow

Pursuing the example of the Couette ow, the vorticity vector is seen to have

only one component normal to the plane of the ow, with magnitude = yu =

U L

(2.94)

uniform throughout the ow eld. Note that vorticity and the implied rota-tion do not imply curved streamlines or trajectories! It is about the relativemotion of nearby points.


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A.H. Shapiros movie: Vorticity Kinematics

Denition, vorticity meter

Potential vortex

Vorticity and circulation: Stokes theorem

Start-up vortex

Dynamics

Bernoullis equation: conditions for B = p + 12 V 2 + g z to

be constant

Croccos equation: how B changes

Kelvins theorem: inviscid, Lagrangian, vortex stretching

Potential ow, singular vortices Model ( = 0)

Motion of vortex pair, induced velocity

Helmholtzs theorems (material lines, vortex tubes, vortex linesend only at boundary) How to introduce vorticity

Singularities

p and interface unsteadiness

rotational forces

pressure gradient along boundary (viscous effect) See Ch. 9

Figure 2.18: Kinematics and dynamics of vorticity and circulation, A.Shapiros movie.


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Figure 2.19: Rotation in Couette ow

2.6 Kinematic decompositions of a velocityeld

Also due to Helmholtz, the kinematic decomposition of velocity is frequentlyused:

u = u + u (2.95)

The rotational part u is obtained from the vorticity distribution through theBiot-Savart relation (below). Then, boundary conditions must be adjustedby the uniquely dened potential ow u (see Ch. 5). Helmholtz provedthat such a decomposition is always possible for incompressible ow. Hodgefurther proved that the decomposition is unique, and that the potential androtational components are orthogonal in a mathematical sense (see Chorin& Marsden, or Majda and Bertozzi).

For an alternative decomposition of velocity, due to Clebsch, see e.g.Panton Section 14.4 p.355.


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2.7. VORTICITY, , 2, ETC. 67

2.7 Vorticity,

,

2, etc.

2.7.1 Biot-Savart relationThe Biot-Savart relation rst came up in relation with the magnetic eldinduced by a current: it turns out the equations are identical to ours, withvorticity instead of current and velocity instead of magnetic induction. Themost common form for 3-D elds is (e.g. Panton Section 14.2 p.351)

u = 14 r

r 3 dV . (2.96)

Here,r = | x x | (2.97)

is the distance between the vorticity element at x and the point x where itinduces velocity. (Note that f (r ) = f (r ) for any function of r = | r |2).The velocity eld can be rewritten in equivalent forms, by integrationby parts, provided the boundary terms vanish fast enough at innity (seeBatchelor, section 2.9 for the effect of boundaries). First we get

u = 14 1r dV

=

1

4

r dV

= A (2.98)where we recognize the vector potential for which 2A = . Also, inte-grating by parts

u = 14 1r dV

= 14 1r dV

=

1

4

r

dV . (2.99)

Here = 2u = (2.100)

2 In these notes, the notation r will be used for the radial distance in po-lar/spherical/cylindrical coordinates, or for the distance between two points: the studentshould pay attention to the context.


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denes exion. So, the Biot-Savart formula reconstructs velocity from

its Laplacian (for incompressible elds), or the velocity potential from itsLaplacian: it gives one explicit form of the inverse Laplacian, since

u = 2. (2.101)

This allows us to view the Biot-Savart integration kernels

G(x, x ) = 14r

(3D) and ln r

2 (2D) (2.102)

as Greens functions for the Poisson equation. The Poisson equation is of

the form 2f = s, (2.103)

where s is a source term. Then, Greens function gives

f = G(x, x )s dx (2.104)the solution f in terms of the superposition of effects of localized source termsdistributed throughout the eld. Greens function is solution of the equation

2G(x, x ) = (x ), (2.105)

where (x) is Diracs distribution.For example, a 2D potential vortex (see Ch. 5 for details) is given by

(x, y) = (x, y) (2.106)

which is innite at the origin, zero everywhere else, and integrates to 1. Youcan think of the 2D version of as the limit of a normalized Gaussian bellshape of unit integral as its scale becomes smaller (and its magnitude larger):

(x, y) = limr 0

1

4re

x 2 + y 2

4 r 2 (2.107)

Then the streamfunction is the z component of the vector potential ( 2 =

), so that we reconstruct by using Greens function:(x, y) =

12 ln r (x x , y y ) dx dy = 12 ln x2 + y2 (2.108)


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2.7. VORTICITY, , 2, ETC. 69(avoid confusion with 2 expressions of r: distance between x and x or polar

distance...)Greens function (and the Biot-Savart relation in particular) introducesremote effect of sources in the solution to a problem.While the use of Greensfunction to solve problems goes beyond the scope of this course, it is impor-tant to realize the non-local properties it introduces in the ows. Vorticityinduces velocity at large distances because of the non-local dependence of solutions of the Laplace and Poisson equations; other examples include thepressure (Ch. 3), and effects of diffusion (Ch. 8).

2.7.2 Vorticity, streamfunctions, and moreAs seen above, it is a mathematical fact that

u = 0 u = A (2.109)It can also be shown that there is no loss of generality in selecting the vectorpotential to be incompressible (gage invariance). 3 Then, the vector identity

( ) = 2() (2.110)(for incompressible vector elds only!) (check the general relation in yourfavorite reference book). In particular:

2A = (2.111)In conjunction with the Biot-Savart equation, this is the motif in a pattern

of relations between

vectors: the vector potential A, velocity U , vorticity , and exion = 2U differential operators: the curl , the Laplacian 2, and their inverses

The pattern is summarized on the gure. It is good to remember that theinverse curl and inverse Laplacian are non-local, i.e. they induce eld prop-erties far away from the sources.

3 Suppose that A is a vector potential, with A = b = 0. Adding a gradient does notchange the resulting velocity eld, so we can consider any A = A + B as equivalent. Wethen adjust B so that A

= b + 2 B = 0, i.e. B = 2 b, a familiar operation.


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Figure 2.20: Kinematic relations


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2.7. VORTICITY, , 2, ETC. 712.7.3 Vorticity and boundary conditionsConsider a conguration with solid boundaries, and a certain distributionof vorticity (respectively, exion). The Biot-Savart relation gives an inducedvelocity eld, which will not, in general, satisfy the boundary conditions. Letus call U the error on the velocity at the boundary. If we add to the velocityeld a potential component u (resp.: a component of zero Laplacian) whichhas value U at the boundary, we have a well dened problem of solvingthe Laplace equation with Neumann (resp., Dirichlet) boundary conditions,and a unique solution with zero vorticity (resp., exion). Thus, the additionof a potential component is related to the presence of boundaries and theenforcement of boundary conditions.

Part of Ch. 6 (Potential ows) could actually be moved here: assumingirrotational ow (zero vorticity) is a kinematic condition. That correspondingmaterial is conventionally coupled with inviscid ow, and this Chapter isbulky enough as it is.

2.7.4 Discrete vortices vs. continuous vorticityThis section can be omitted without loss of continuity in the text. It has abearing on vortex-based numerical methods, as well as on various modelingapproaches in turbulence (see Saffmans book for vorticity, Chorins book for

applications to turbulence). The point is made here in 2D, the same appliesin 3D.Consider a collection of point vortices

i = i (x i) (2.112)

The overall streamfunction is obtained by summation

= ii2 (x i) ln r dA = i

i2

ln r i = i (2.113)which obeys the Laplace equation except at the singular points. So theinduced velocity eld is irrotational.

The same is not true, in general, for continuous distributions of vorticity(x) over a nite region. Indeed, for

= (x )

2 ln r dA , (2.114)


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the vorticity is

= 2 = (x

)2 2 ln r dA , (2.115)which is, in general, non-zero even in the region where Gamma vanishes. So,for continuous distributions of vorticity, the induced velocity is not irrota-tional.

2.7.5 Vorticity and circulationVorticity and circulation are not equivalent concepts, but are obviously re-lated through Stokes theorem:

= dA (2.116)where the integral is over any simple surface supported by a closed contour.We can also write

Incompressible Fluid Dynamics

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