Fluid DynamicsThis book presents a focused, readable account of the principal physicaland mathematical ideas at the heart of fluid dynamics. Graduate studentsin engineering, applied mathematics, and physics who are taking their firstgraduate course in fluids will find this book invaluable in providing thebackground in physics and mathematics necessary to pursue advancedstudy.

The book includes a detailed derivation of the Navier-Stokes and energyequations, followed by many examples of their use in studying thedynamics of fluid flows. Modern tensor analysis is used to simplify themathematical derivations, thus allowing a clearer view of the physics.Peter S. Bernard also covers the motivation behind many fundamentalconcepts such as Bernoulli’s equation and the stream function.

Many exercises are designed with a view toward using MATLAB or itsequivalent to simplify and extend the analysis of fluid motion, includingdeveloping flow simulations based on techniques described in the book.

Peter S. Bernard has 35 years’ experience teaching graduate-level fluidmechanics at the University of Maryland. He is a Fellow of the AmericanPhysical Society and Associate Fellow of the American Institute ofAeronautics and Astronautics. In addition to his many research articlesdevoted to the physics and computation of turbulent flow, he is thecoauthor of the highly regarded volume Turbulent Flow: Analysis,Measurement, and Prediction.

Fluid Dynamics

Peter S. BernardUniversity of Maryland

32 Avenue of the Americas, New York, NY 10013-2473, USA

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© Peter S. Bernard 2015

This publication is in copyright. Subject to statutory exception and to theprovisions of relevant collective licensing agreements, no reproduction ofany part may take place without the written permission of CambridgeUniversity Press.

First published 2015

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication DataBernard, Peter S., author.Fluid dynamics / Peter S. Bernard, University of Maryland. pages cmIncludes bibliographical references and index.ISBN 978-1-107-07157-5 (hardback)1. Fluid dynamics. I. Title.QC151.B387 2015532′.05–dc23 2014044742

ISBN 978-1-107-07157-5 Hardback

Additional resources for this publication at www.cambridge.org/bernard

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party Internet Web sites referred toin this publication and does not guarantee that any content on such Websites is, or will remain, accurate or appropriate.

To my wife, Susan Bradshaw Sullivan

Contents

Preface

1 Introduction1.1 What Is a Fluid?1.2 Molecular Structure and the Continuum Hypothesis1.3 Dilatation and Vorticity1.4 The Big Picture1.5 Vector and Tensor Analysis

1.5.1 Vectors1.5.2 Tensors1.5.3 Skew Tensors1.5.4 Gradient Tensor1.5.5 Basis Vectors and Change of Coordinates

2 Eulerian and Lagrangian Viewpoints, Paths, and Streamlines2.1 Eulerian versus Lagrangian Viewpoints2.2 Fluid Particle Paths2.3 Curves2.4 Streamlines

3 Stream Function3.1 Two-Dimensional Planar Flow3.2 Axisymmetric Flow

4 Helmholtz Decomposition4.1 Three-Dimensional Flow4.2 Bounded Domains4.3 Two-Dimensional Flow

5 Sources, Sinks, and Vortices5.1 Sources and Sinks in Two Dimensions5.2 Point Vortices

5.3 Accommodating Boundaries in Two Dimensions5.4 Sources and Sinks in Three Dimensions

6 Doublets and Their Applications6.1 Three-Dimensional Source/Sink Doublet6.2 Doublets in Two Dimensions

7 Complex Potential7.1 Connection to Complex Analysis7.2 Flows Derived from a Power Law7.3 Forces in 2D Potential Flows7.4 Inviscid Flow Past a Cylinder

8 Accelerating Reference Frames8.1 Orientation8.2 Position Vector8.3 Velocity8.4 Acceleration and Fictitious Forces

9 Fluids at Rest9.1 Forces in a Fluid at Rest

9.1.1 Micromanometer9.1.2 Force on a Dam

9.2 Buoyancy9.3 Accelerating Fluids at Rest

9.3.1 Accelerating Fish Tank9.3.2 Rotating Bucket

10 Incompressibility and Mass Conservation10.1 Some Useful Mathematics10.2 Incompressibility10.3 Mass Conservation

11 Stress Tensor: Existence and Symmetry11.1 Existence of the Stress Tensor11.2 Symmetry of the Stress Tensor

12 Stress Tensor in Newtonian Fluids

12.1 Relative Fluid Motion at a Point12.2 The Stress Tensor

13 Navier-Stokes Equation13.1 Rate of Change of Momentum13.2 Surface Forces13.3 The Navier-Stokes Equation

14 Thermodynamic Considerations14.1 Overview14.2 First Law of Thermodynamics14.3 Perfect Gases

15 Energy Equation

16 Complete Equations of Motion16.1 Differential Equations of Fluid Flow16.2 Bernoulli Equation

16.2.1 Bernoulli Equation for Steady Flow16.2.2 Bernoulli Equation for Nonsteady Flow16.2.3 Crocco’s Relation

16.3 Control Volume Equations16.3.1 Mass Conservation16.3.2 Momentum Conservation16.3.3 Conservation of Angular Momentum16.3.4 Conservation of Energy

17 Applications of Bernoulli’s Equation and Control Volumes17.1 Fluid Impinging on a Plate17.2 Draining a Tank17.3 Water Sprinkler

18 Vorticity18.1 Vorticity Equation18.2 Vortex Stretching and Reorientation18.3 Kelvin’s Circulation Theorem18.4 2D Vortex Methods18.5 Simulation of a Wing Wake

19 Applications to Viscous Flow19.1 The Reynolds Number19.2 Unidirectional Flow19.3 Flow in a Narrow Gap19.4 Stokes Flow Past a Sphere

19.4.1 Problem Formulation19.4.2 An Equation for the Stream Function19.4.3 Solution for Stokes Flow19.4.4 Forces on the Sphere19.4.5 Self-Consistency of the Solution

19.5 Motion of a Sphere at Higher Reynolds Numbers

20 Laminar Boundary Layers20.1 Boundary Layer Scaling20.2 Blasius Boundary Layer20.3 Falkner-Skan Boundary Layers

21 Some Applications to Convective Heat and Mass Transfer21.1 A Thermal Boundary Layer21.2 Monte Carlo Schemes for Modeling Convective Diffusion

21.2.1 Probabilistic Interpretation of Diffusion21.2.2 Monte Carlo Model of Diffusion21.2.3 Monte Carlo Simulation Including Convection21.2.4 Monte Carlo Solution to a Thermal Boundary Layer

Appendix A: Equations in Curvilinear CoordinatesA.1 Polar CoordinatesA.2 Cylindrical CoordinatesA.3 Spherical Coordinates

Appendix B: TensorsB.1 Divergence of a TensorB.2 Vector CrossB.3 Principal Directions

BibliographyIndex

Preface

This book is inspired by a graduate-level course in fluid dynamics that Ihave taught at the University of Maryland for many years. The typicalstudent taking this course, which is the starting point for graduate studiesin fluid mechanics, has had one undergraduate course on fluids and alimited exposure to vector and tensor analysis. Consequently, the goal ofthis book is to provide a background in the physics and mathematics offluid mechanics necessary for the pursuit of advanced studies and researchat the graduate level. It is my experience that an effective route to theseobjectives is via a synthesis of the best features of two very excellentbooks, namely, An Introduction to Fluid Dynamics by George Batchelor,which presents the physics of fluid mechanics with exceptional clarity, andAn Introduction to Continuum Mechanics by M. E. Gurtin (and nowexpanded and revised as The Mechanics and Thermodynamics of Continuaby Gurtin, Fried, and Anand), which demonstrates the advantages of directtensor notation in simplifying the expression of physical laws. Thus, to alarge extent, this book combines the physics of Batchelor with themathematics of Gurtin. The hope is that, in this way, an environment iscreated that helps make the subject of fluid dynamics clear, focused, andreadily understandable. As a practical matter, this book should serve as aneffective stepping-stone for new graduate students to enhance theiraccessibility to the books by Batchelor and Gurtin as well as those bymany others.

Stylistically, this book follows an arc through the material that buildssteadily toward the derivation and then application of the Navier-Stokesequations. The sequence of topics is also chosen so as to provide somesignificant exposure to examples of fluid flow and problem solving, beforea relatively long and unavoidable set of chapters that deal in detail withthe derivation of the flow equations. Most of what is in this book iscovered in a one-semester course at Maryland, and no attempt is made to

provide the depth of topics covered by Batchelor or Gurtin nor thecomprehensive treatment of the subject matter typically found in otheradvanced textbooks. After studying this book, it is hoped that students willbe well prepared to venture in any number of directions into morespecialized and advanced topics in fluid dynamics.

Among the topics in the book, some represent a review of subjectsnormally encountered in undergraduate fluids courses (e.g., Chapter 9, onfluids at rest). This is intended to keep the book self-contained, to aid inthe review of this material and as a needed introduction to these topics forthe occasional applied math or other nonengineering student who hasnever previously studied fluid mechanics.

The problems at the end of the chapters attempt to reflect the graduatelevel of the book by pursuing directions that are often somewhatchallenging rather than repeating the formulaic engineering problems thatare traditional to the undergraduate curriculum. For many of the problems,students are strongly encouraged to take full advantage of high-ordercomputer languages such as MATLAB to help derive relations viasymbolic manipulation, to solve algebraic and differential equations, andto calculate and plot numerical results. For example, in the case ofMATLAB, facility with using commands such as diff , int, solve, dsolve,subs, ode45, and bvp4c greatly reduces the labor necessary to solve manyproblems in this book. In some cases, without the power of the symbolicsolvers, the difficulty in obtaining solutions can be quite formidable ifattempted with pencil and paper.

Some of the material in the book is specifically designed to be alaunching point for writing computer code (e.g., with MATLAB) thatsolves interesting flow problems and displays results in the form ofanimations. Such material includes Sections 18.4 and 18.5, on the discretevortex method; Section 19.5, on the motion of a sphere and other bodies;and Section 21.2, on the use of the Monte Carlo method for simulatingscalar transport in fluid flows. In each of these cases, the numericalsimulations can be carried out with a modest investment in programmingyet bring to life intriguing aspects of fluid flow.

The author would like to express his great appreciation to ProfessorBruce Berger for his many contributions toward improving the quality andclarity of the exposition in this book. I also appreciate the insights of CarlBiagetti of the Space Telescope Science Institute and graduate student EricLeonard in reading some of the chapters.

1Introduction

1.1 What Is a Fluid?

What distinguishes fluids from solids? Because we all have an intuitiveunderstanding of the difference, the question is really about preciselyidentifying the formal differences between them. In this regard, it is usefulto consider the different ways that fluids and solids react to applied forces.For example, in the case of a solid, the material offers resistance if wepress down on it. If the applied force is not so large as to shatter the solid,then it is clear that the solid is quite capable of resisting the force so as toreach a state of equilibrium. The solid arrives at a state where it ceases tomove or deform.

Consider now specifically the case of a gas, as shown in Fig. 1.1. In thefigure, a piston is pushing down on the gas, and although initially the gasmight compress because of the applied force, it is also able to eventuallyreach a point for any given applied force where it does not compressfurther. In other words, the gas is capable of resisting the downward forcein the same way as the solid. We may conclude that resistance to a normalforce is not a good candidate for framing the distinguishing properties offluids and solids.

Figure 1.1. A gas can successfully resist an applied normal force toachieve equilibrium.

The situation for fluids and solids is different if we consider an appliedshear force, as in the experiment indicated in Fig. 1.2. Following theapplication of a shear force to the top surface of the solid, as shown in Fig.1.2(a), an equilibrium is reached in which the body has deformed a fixedamount. Alternatively, if the container holds a fluid, as in Fig. 1.2(b), and ashear force is applied to the top lid, the fluid cannot prevent the lid fromsliding to the side. This is true no matter how small the applied force maybe. This is not to say that the fluid does not offer resistance – it does – butthe resistance it offers cannot be enough to create a stationary equilibrium.It is this very different behavior between solids and fluids thatdistinguishes one from the other in a formal sense. Unlike solids, fluids areunable to prevent the motion caused by a shearing force, no matter howsmall the shearing force might be.

Figure 1.2. Response of solids and fluids to a shearing force.

1.2 Molecular Structure and the Continuum Hypothesis

Fluids and solids behave differently because of the differences in theirmolecular structures. Whereas solids tend to have a fixed arrangement ofmolecules, in a liquid or gas, molecules are in continuous relative motion.In the case of gases, the molecules generally move at very high velocities.For example, in hydrogen at room temperature, the mean speed of themolecules is approximately m/s, or faster than 4000 MPH, withmuch of that motion associated with the internal energy of the gas. Themolecules tend to be separated from each other by a distance on the orderof the mean free path. As they move, they encounter other molecules in theform of collisions, in fact, having a vast number of collisions per second, inthe neighborhood of under typical atmospheric conditions. Thecollisions are efficient at rapidly spreading information among themolecules about changes to their external conditions. For example, if heat

is added at one place in the fluid so that the local molecules at that positionmove faster, then the energy contained in this motion spreads rapidly tomore distant molecules via collisions.

During the compression of a gas, such as occurs in Fig. 1.1, themolecules nearest to the piston react first to the new circumstances by alocal increase in pressure and density, a state that the rest of the gasbecomes aware of and reacts to through the passage of information bycollisions. The news of the change in properties generally travels at thespeed of sound, which is on the order of 340 m/s in air.

Liquids also can compress under an applied force, although in this case,the molecules are much closer together than in a gas and are better able toresist compression. In fact, the molecules in a liquid are separated fromeach other by the distance at which the repulsive strong force originating inquantum effects between molecules crosses to a weak attractiveelectrochemical force. Changing the average distance between molecules inthese circumstances requires a large applied force. For a liquid, the mutualjostling of molecules in close proximity owing to the molecular forcesprovides the means by which there is a rapid flow of information aboutchanged circumstances from one place to another in the fluid.

The subject of fluid mechanics as it is often applied to the study ofengineering flows is not concerned directly with the molecular structureunderlying liquids and gases. Instead, attention is focused on fieldquantities, such as the velocity field that is defined at all positions

covering a region of flow at times contained within a specifiedinterval. In three dimensions, has scalar components in a rectangularCartesian coordinate system that will be denoted here either as

or , depending oncircumstances. Similarly, the position vector has components

, which will also be frequently referred to as .Among the many other fields of interest in fluid mechanics are the density

, the pressure with dimension of force/area, and the internalenergy/mass . The connection between a field quantity, such as

, and the relevant property of the molecules of which the fluid is

composed is for the most part conceptional. In other words, it may beassumed that field quantities represent a theoretical average of therespective property held by the molecules within a small sensing volumearound the point at given time . The average is expected to beindependent of the size of this volume for a range of scales that are small incomparison to the scale at which variations in the flow property might benoticeable in the macroscopic world, say, a fraction of a millimeter. Thiscondition is well met under most circumstances. The precise size of thesensing volume is immaterial, however, because it is not used in derivingthe fluid equations, nor does the determination of fluid flow depend onknowing it.

Figure 1.3 shows the result of a model computation of using MATLABfor different sizes of the sensing volume. In the figure, 3 million points arerandomly scattered within a unit square, and the average number of pointscontained within a box of decreasing size is counted and then used toestimate the local density. It is seen that fluctuates rapidly as the boxsize is varied through small values, indicating that this is a range where thesensing volume is too small. The density estimate is seen to be constant atlarger sensing volumes, and this is the value of the density that would bereferred to in a field quantity such as . For even larger sensingvolumes, it can be imagined that the average density will begin to changeonce again as it becomes affected by measurable (i.e., human scale)gradients in the local fluid properties.

Figure 1.3. Dependence of the density at a point on the sensing area for3 million points randomly distributed in a unit square.

The fact that it is not necessary to have a definite relationship betweenfield quantities and molecular averages is the substance of the continuumhypothesis. In effect, it is assumed that the behavior of a real fluid withproperties such as and can be modeled by the dynamics of animaginary fluid that is for the most part continuous and lacks anymolecular structure. This means that, if at time , one were to measure

and in a real fluid using an appropriate sensing volume and thenadvance these quantities further in time using laws derived for thecontinuum fluid, then at subsequent times, and will once again beequal to their respective averages computed from the local molecules. The

way in which molecular effects do enter into the equations for thecontinuum variables, as seen later, is via material properties such as theviscosity, , and specific heat, . The magnitude of such variables as wellas their functional dependence on quantities such as the temperature varyfrom one material to the next because of differences in molecular structureand need to be independently assigned values to make quantitativepredictions of fluid fields in a continuum model.

1.3 Dilatation and Vorticity

The velocity field is fundamental to the description of fluid flowand is often the primary means by which the flow field is understood.Alternatively, two fundamental field quantities formed out of derivatives of

are seen to play many important roles in characterizing fluid motion.The first is the divergence of the velocity field, , which is referred toas the dilatation and is often denoted in this book as . In rectangularCartesian coordinates,

(1.1)

where here and henceforth summation from 1 to 3 is implied in repeatedindices such as in the expression appearing in Eq. (1.1). InSection 10.2, it is shown that zero dilatation is precisely the condition thatis necessary and sufficient for the flow to be incompressible, that is, for thevolume of any particular element of fluid to remain the same as it movesthrough the flow.

The second fundamental quantity that can be computed from is thevorticity field, , defined as the curl of the velocity field

(1.2)

or, in index notation,

(1.3)

where the alternating tensor

(1.4)

Cyclic order means that , andanticyclic order means that . Acalculation shows that

In two-dimensional flow, in which case and the velocity field is, according to Eqs. (1.5), the only nonzero component of

vorticity is , that is, , or more simply, . Inthis case, it is seen that the vorticity vector is oriented perpendicular to theplane of motion.

A subsequent discussion in Section 12.1 shows that the vorticity may beunderstood physically as being twice the instantaneous angular velocity ofa local fluid element under the hypothetical circumstance that it is

instantaneously frozen and allowed to move subsequently as a solid body.Thus, vorticity is generally associated with rotational motion in a fluid,though it is not necessarily true that the fluid is rotating when the vorticityis nonzero. The degree to which and are fundamental properties offluid motion is made evident in Chapter 4, where it is seen that, takentogether, they can be used to determine the velocity field with which theyare associated.

1.4 The Big Picture

The motion of fluids is determined by a few main physical principleswhose mathematical and physical elucidation forms the central part of thisbook. At their most fundamental level, the physical ideas that need to bedeveloped concern how the mass, momentum, and internal energy of amaterial element of fluid varies in time. A material element, as illustratedin Fig. 1.4, is an identified volume of fluid that retains its identity as itmoves through the surrounding fluid even while distorting in various ways.In essence, a material fluid element is the equivalent for fluids of thedeformable bodies whose dynamics are studied in the discipline of solidmechanics. The very different constitutive properties of fluids and solids,including the way in which they respond to strain, means that theirresulting equations of motion are different, even if the governing physicallaws are the same.

Figure 1.4. Motion of a material fluid element in time with .

A material fluid element can be described mathematically by the volumein space, , that it occupies at time as it travels downstreamfrom a given initial location coinciding with volume, . A particularmaterial element is distinguished from all other elements by the volume itoccupies at the initial instant . From the field quantities

, and , one can specify the mass of the materialelement at time as

(1.6)

the momentum as

(1.7)

and the internal energy as

(1.8)

The notation for an element of volume in the integrationemphasizes that is the integration variable. Sometimes, as appropriate inother contexts, the simpler notation is used to indicate a volumeelement. To a large extent, the dynamics of fluid flow is determined by thefundamental laws governing the quantities in Eqs. (1.6)–(1.8).

The first physical law governing the motion of fluids is that of massconservation, expressed as

(1.9)

specifying that the mass of a material element does not change in time.Nuclear decay is not being considered here. The second essential physicalrequirement is that the momentum of the material element should satisfyNewton’s second law of motion, which posits that, in an inertial frame ofreference,

(1.10)

where the right-hand side denotes the sum of forces that act on the materialvolume as it moves. Among the forces are body forces, such as gravity, thateffect the entire material element and surface forces acting on the boundaryof the material element that arise from its interaction with the surroundingfluid or solid boundaries.

The third essential relation is the first law of thermodynamics, whichgoverns changes in the internal energy of the fluid element as it moves.

This may be expressed as

(1.11)

where the terms on the right-hand side, respectively, denote the rate atwhich heat (i.e., internal energy) is added or lost to the material elementand the rate at which work is done on the element that directly affectsinternal energy.

A considerable part of this book is devoted to converting the essentialideas contained in Eqs. (1.9)–(1.11) into the form of differential equationsthat may be used practically in a variety of ways to solve problemsinvolving fluid flow. To accomplish this goal requires developing a numberof supporting physical ideas together with the mathematics that allows thedevelopment to proceed. In support of this main objective, it is helpful firstto spend a number of chapters introducing ancillary ideas related to suchaspects of fluid motion as its kinematics and statics. After the fundamentallaws are derived, our interest shifts to examining in detail the solutions to anumber of representative flows.

1.5 Vector and Tensor Analysis

A goal of this book is to present many of the equations of fluid dynamicsand their derivations in a relatively unencumbered mathematical form. Thiseffort is helped greatly by often relying on direct notation for vectors andtensors, that is, notation lacking the use of indices.1 We alreadyencountered the difference between direct and index notation in Eq. (1.1),where the first expression for the divergence of the velocity in directnotation is equated to the same quantity in index notation. In fact, the latteris the form the divergence takes specifically for rectangular Cartesiancoordinates. In other coordinate systems, such as cylindrical or spherical,the expression would look different, as may be seen in Appendix A, where anumber of vector operators as well as the equations of motion are writtenout in alternative coordinate systems. This point makes clear the value of

direct notation, as the vector expression is not specific to any onecoordinate system.

A number of other mathematical quantities besides areencountered in this book, for which both direct and index notation makesense, depending on the context. In fact, it will be seen that sometimesproofs of identities can be accomplished more simply in index notationthan in direct notation, and we do not hesitate to take advantage of thiswhen it is appropriate.

This introductory section gives a brief description of some importantmathematical entities that play a major role in much of the subsequentdevelopment. Additional concepts are introduced as needed in the course ofthe book, including in Appendix B, where we describe some specializedconcepts that are best left out of the main line of development.

1.5.1 Vectors

We have already encountered the velocity field that, for any fixedtime, is an example of a vector field, that is, a vector function of vectorposition in three dimensions. At any point , three orthogonal directionscan be specified and a right-handed set of orthogonal unit basis vectors

can be chosen. Such vectors are referred to as beingorthonormal, because besides being mutually perpendicular to each other,they are unit vectors. In some contexts, the basis vectors will be indicatedas . Orthonormality means that , where

(1.12)

is the Kronecker delta. Right-handedness means that with in cyclic order. For a particular choice of basis vectors, any vector

field has components defined via

(1.13)

and one may write

(1.14)

The basis vectors may either be independent of position, as inrectangular Cartesian coordinates, or position dependent, as in cylindricalor spherical coordinate systems. For convenience, vectors are sometimesreferred to via their components, as in .

The dot or inner product of vectors and is given in terms of theircomponents by the scalar

(1.15)

which represents a projection of either vector in the direction of the other.The magnitude or length of a vector is defined using its components as

(1.16)

The cross-product of two vectors is the vector with components

(1.17)

which may be shown to have orientation normal to the plane formed by and and magnitude equal to the area of the parallelogram formed by thevectors. To evaluate expressions such as for three vectors

, and (see Problem 1.6), the identity

(1.18)

is of considerable help.

1.5.2 Tensors

The concept of a vector field is a special case of a more general entityknown as the tensor that, in a number of guises, plays a major role inexpressing the physics of fluid flow. For example, the stress tensor ,which is an example of a second-order tensor, appears in a natural waywhen accounting for the surface force fields appearing in Newton’s law ofmotion. Though we have not referred to them as tensors, in fact, scalarsmay be regarded as zeroth-order tensors and vectors as first-order tensors.Although tensors of third, fourth, and higher order exist, here, forsimplicity, second-order tensors are referred to as tensors because most ofour use of tensors will be of this variety. In this case, our formal definitionof a tensor is that it is a linear map from vectors to vectors. In other words,if is a tensor and is a vector, then the quantity is a vector.Linearity in the mapping means that for any two vectors and andconstant , it follows that

(1.19)

This relation means that the vector on the left-hand side resulting from operating on the vector is equal to the linear combination ofvectors on the right-hand side.

Tensors have components, as do vectors. In this case, a tensor hascomponents defined via the relation

(1.20)

where it is seen that the term in parentheses, namely, , is a vector, soits dot product with makes sense, in fact, yielding the scalar componentof the tensor on the left-hand side. Whereas a vector has threecomponents, a second-order tensor has nine. It is often convenient torepresent vectors as matrices and tensors as matrices. Likematrices, tensors have a transpose that is defined according to

.

A useful operation on tensors, which linearly maps them into the realnumbers, is the trace, denoted and defined by

(1.21)

that is, the sum of the diagonal components. Analogous to vectors, tensors and have an inner product defined as

(1.22)

where is a tensor formed from the product of the tensors and (see Problem 1.8). The identity tensor, denoted as , has the property that

for all vectors . It is readily shown that the matrix ofcomponents of is the Kronecker delta and, moreover, that .

Tensors may be formed out of vectors via the tensor product. In this, forany two vectors , the quantity is a tensor known as the tensorproduct of and . It is defined by the property that for any vector , itreturns a vector aligned in the direction by the formula

(1.23)

Using this definition, it is easy to verify that the components of the tensor are given by

(1.24)

In fact, it is not hard to show that any tensor with components canbe written as

(1.25)

with repeated indices summed.

1.5.3 Skew Tensors

A tensor T that satisfies

(1.26)

is said to be symmetric. Alternatively, a tensor satisfying

(1.27)

is said to be skew or antisymmetric. In this case, its components satisfy for all values of . Consequently,

(1.28)

and

so that, in fact, there are only three independent elements in a skew tensor.

Making the definitions

and substituting for the components of T using Eqs. (1.28) and (1.30), it isseen that

(1.31)

For an arbitrary vector , a calculation gives

(1.32)

From Eq. (1.17), it may be seen that the right-hand side of Eq. (1.32) isidentical to , so it has been established that for any skew tensor ,

(1.33)

where is a uniquely defined vector referred to as the axial vectorassociated with .

It is worth noting that the components of as given in Eqs. (1.30) canbe derived formally from Eq. (1.33). Thus, according to Eqs. (1.20) and(1.33),

(1.34)

Switching to index notation and using the definition of the cross-product inEq. (1.17), Eq. (1.34) becomes

(1.35)

To solve for , multiply both sides of Eq. (1.35) by , apply Eq.(1.18), and, after changing indices, obtain

(1.36)

as the axial vector. This is fully consistent with Eqs. (1.30).

One last comment is that the existence of the axial vector associated with is equivalent to stating that

(1.37)

where the right-hand side is a tensor referred to as the vector cross. Thevector cross is defined according to the property that forall vectors (see Eq. A.13). For skew, Eq. (1.37) follows byestablishing that for any vector (see Problem 1.9).

1.5.4 Gradient Tensor

Another tensor that is commonly employed in fluid mechanics is thegradient of a vector field. Thus, for a vector field its gradient is thetensor field denoted as . To get at the definition of consider thebehavior of within a short distance from , namely with now held fixed. is defined as the linear mapping on vectors(i.e., a tensor) that provides the best local approximation to the vectorfunction . Formally, makes the right-hand side of

(1.38)

“closest” to the left-hand side. To find the components of set where , divide Eq. (1.38) by and rearrange terms to get

(1.39)

Dotting this equation by , taking the limit as and noting thedefinition of partial derivative results in

(1.40)

Thus, it is seen that is the tensor whose components are the velocityderivatives .

Applying the trace to the tensor yields, according to Eq. (1.40),

(1.41)

which is the same as the divergence of the vector. In fact, this shows thatthe formal definition of the divergence of a vector is as the trace of thegradient tensor:

(1.42)

1.5.5 Basis Vectors and Change of Coordinates

Thus far a particular set of basis vectors, , has been assumed to havebeen selected from which vector and tensor components are determinedfrom Eqs. (1.13) and (1.20), respectively. If a different set of basis vectors,say, , are selected, then the components of vectors and tensors are given,respectively, by the relations

(1.43)

(1.44)

How the components associated with different sets of basis vectors arerelated to each other is an important question that we now consider.

Two sets of orthonormal basis vectors and , can bebrought into coincidence by a rotation that brings one set of vectors to theother. Such rotations can be actualized by the use of a rotation tensor thathas the effect of rotating vectors without a change to their magnitude. Forexample, this is illustrated in Fig. 1.5 for the two-dimensional case, where

is the rotation tensor that effects the mapping

(1.45)

by rotating the vectors through an angle .

Figure 1.5. Two-dimensional depiction of basis vectors and coordinateaxes. The vectors are rotated through an angle by the rotation tensor

so as to coincide with .

Some insight into the nature of the rotation tensor can be had byconsidering the relationship between and in some detail. Viewingthese vectors from a geometrical perspective, as shown in Fig. 1.6, theprimed basis vectors can be expressed in terms of the unprimed vectorsaccording to the relations

where the angle represents the difference in orientation of the vectors.

Figure 1.6. Resolution of and into the and directions.

Equations (1.45) and (1.46) can be used together to solve for thecomponents of . Thus, from the definition in Eq. (1.20), a calculationgives the matrix

(1.47)

that maps vectors by rotating them through degrees in the anticlockwisedirection. For example, the vector gets mapped by Eq. (1.47) to thevector , as expected.

The inverse matrix to Q, namely, , is a rotation in the reversedirection through the angle . In view of the identity

(1.48)

it is evident that

(1.49)

so that is an orthogonal tensor and Eq. (1.49) makes clear that

(1.50)

In three dimensions, may be thought of as a sequence of three separaterotations around the three coordinate axes. Moreover, Eqs. (1.49) and(1.50) hold in this case as well.

Using , the relationships between vector and tensor components canbe determined. Thus, depending on which basis vectors are used, a givenvector can be written as

(1.51)

or

(1.52)

Taking a dot product of Eq. (1.52) with and noting Eqs. (1.13), (1.45),and (1.20) yields the mapping between components

(1.53)

A similar sequence of steps can be pursued for tensors, as indicated inProblem 1.10, that results in the mapping between tensor components

(1.54)

Much of this discussion of the coordinate mappings will prove to be usefulin a somewhat different guise in Chapter 8, when differences in vectors andtensors seen by different observers are considered. Appendix B includes afurther discussion of vector and tensor invariants that follows from theresults in this chapter.

Problems

1.1 Use MATLAB or equivalent to make a version of Fig. 1.3 based on ananalysis of particles distributed in 3D. Determine the range of sensingvolume that is consistent with the continuum hypothesis.

1.2 From the number density of gas molecules in air, estimate a lengthscale at which the average fluid properties will no longer depend on thesize of the sensing volume.

1.3 Compute the dilatation and vorticity corresponding to the velocity field where and are constants.

1.4 Compute the dilatation and vorticity corresponding to the velocity field .

1.5 Demonstrate that right-handedness of the unit vectors is consistent withthe identity

(1.55)

1.6 Evaluate the components of for vectors with thehelp of Eq. (1.18).

1.7 Prove Eq. (1.25). Note: to prove that tensors and are equal, showthat for all vectors .

1.8 If and are tensors, their product, , is the tensor defined by , where is a vector. Why does this make sense? Find

the components of .

1.9 Prove Eq. (1.37) by using the definition of given in Eq. (1.36) andthe fact that is skew.

1.10 Derive the component transform law Eq. (1.54) for tensors. This canbe done by starting with Eq. (1.25) written in primed components and usingEq. (1.45) in the same way that it was used in driving Eq. (1.53).

1 An exhaustive and lucid treatment of vector and tensor analysis throughdirect notation may be found in Gurtin (1981) and Gurtin, Fried &Anand (2010).

2Eulerian and Lagrangian Viewpoints, Paths, and

Streamlines

2.1 Eulerian versus Lagrangian Viewpoints

In the previous chapter, a number of field quantities were introduced,including the velocity field . This means of describing the flowmathematically, in which quantities at a given time are specifiedthroughout the flow field, is called the Eulerian perspective. In contrast tothis point of view is the Lagrangian perspective based on considering themotion of fluid particles within the fluid. A fluid “particle” is, in essence, avanishingly small material element that is identified by its position, say, ,at a given time, . The Lagrangian point of view then considers themotion of all fluid particles, each of which has a path in time denoted by

that satisfies the initial condition .

In the Lagrangian framework, the velocity of the fluid is known from thevelocity of all the particles within it any given time. In other words, to getthe velocity at a given point at a given time, one seeks out the velocity ofthe fluid particle residing at that point at the particular time. This velocityis

(2.1)

An important and very helpful relation unites the Eulerian and Lagrangiandescriptions in the equation

(2.2)

to the effect that the velocity of a fluid particle given on the left-hand sideis equal to, on the right-hand side, the Eulerian velocity field evaluated at the location of the fluid particle . Equation (2.2) willbe seen to have many important uses in deriving the equations of fluidmechanics.

2.2 Fluid Particle Paths

One immediate application of Eq. (2.2) is as the defining equation for fluidparticle paths within a given velocity field . Thus, from Eq. (2.2), aparticle path is determined as the solution tothe three coupled ordinary differential equations

(2.3)

with initial condition at a specified reference time . In twodimensions, the paths are computed from just two coupled equations.

As an example of computing paths, consider the two-dimensionalvelocity field shown in Fig. 2.1 as a quiverplot. The coupled equations giving the particle paths , asdetermined from (2.3), are

with initial condition . The solution is readily calculated tobe

which is also plotted in Fig. 2.1 for the case when . Clearly, the path in the figure is consistent

with the indicated velocity field.

Figure 2.1. A particle path computed in the velocity field .

In the example just given, the velocity field was steady, that is, it did notchange in time. Equation (2.3) applies equally well in the nonsteady caseas, for example, for the velocity field , where timenow makes an explicit appearance. In this case the equations determiningthe particle path are

Choosing the initial condition at to be , acalculation yields

which is illustrated in Fig. 2.2. For velocity fields for which explicitformulas are available yet the integration implied by Eq. (2.3) cannot becarried out, the corresponding particle paths can be determined using anumerical solver for ordinary differential equations such as ode45 inMATLAB. A more common scenario is when the velocity data are given ona spatial mesh of coordinates, in which case numerical solvers can still beused, but the velocity data must be presented to them using an interpolationscheme. Several problems at the end of the chapter illustrate these ideas.

Figure 2.2. The path corresponding to Eqs. (2.7) contains a loop-the-loop.

2.3 Curves

The fluid particle paths discussed in the previous section are examples ofcurves in two- or three-dimensional space in which time acts as aparameter. As varies, so does the position along the paths. Moregenerally, other quantities can serve as parameters for describing curves,as, for example, the coordinate itself for a path in three dimensionsgiven by . Similarly, the coordinate could be chosen asparameter so that curves are given by . Another oftenuseful choice of parameter is , the arclength along the curve, that is, thetrue physical distance from a starting point. Parameters such as or aremore general than ones like or because the latter are not well suited

for representing curves that double back on themselves, as in the loop-the-loop shown in Fig. 2.2. This curve was computed using as a parameter,but if one attempted to use a parameter such as , the function would be multivalued in part of the domain. Computing such a functionis inconvenient because it requires considerable care in identifying theindividual branches of the path.

For a curve , where denotes an arbitrary parameter, the tangentvector at any point along its length is given by

(2.8)

This relation is derived by noting that in the limit as , the vector shown in Fig. 2.3 will become increasingly aligned

with the local direction of the curve. Dividing this by and taking thelimit as yields Eq. (2.8). As an example, consider a curve in twodimensions given by , where is designated asbeing the parameter and is constant. A tangent vector anywhere alongthe curve is , as shown in Fig. 2.4(a) for .As a second example, letting be the parameter in the curve

yields the tangent vector at any point ,as illustrated in Fig. 2.4(b).

Figure 2.3. Construction of the tangent vector to a curve .

Figure 2.4. Tangent vectors to curves.

In general, tangent vectors computed from Eq. (2.8) are not unit vectors,though they are easily converted to them through normalization, as in

(2.9)

where

(2.10)

is the magnitude of the vector . In the special case in which thecurve parameter is arclength, then

(2.11)

To prove this, consider the estimation of arclength illustrated in Fig. 2.5,where distance along the curve is approximated by summing over straight line segments connecting positions along the curve. In this case,

(2.12)

where . A Taylor series expansion gives

(2.13)

in which case

(2.14)

and in the limit as , the identity results:

(2.15)

Differentiating this equation with respect to yields

(2.16)

Finally, choosing so that , and therefore , Eq.(2.16) yields (2.11). Thus, if is chosen as the parameter, the tangentvectors are naturally unit vectors.

Figure 2.5. Summing segments to compute arclength.

A useful consequence of Eqs. (2.9) and (2.11) is in showing that pathintegrals along a curve defined using arclength such as

(2.17)

where denotes a vector field evaluated on the path,retain the same form under a change in parameter. Thus, for an arbitraryparameter related to through the mapping , the curve is given by

(2.18)

and consequently,

(2.19)

Defining and changing the variable of integration inEq. (2.17) to and using Eq. (2.19) gives

(2.20)

This means that the path integral in Eq. (2.17) that is defined usingarclength has the same form in all parameterizations even if isnot a unit vector.

One final observation about curves is that in many instances, a given 2Dfunction, say, , can be used to define curves implicitly via itsisocontours. In other words, the equation

(2.21)

where is a constant, can be used to solve for as a function of , orvice versa. Depending on which option is pursued, curves of the form

for which

(2.22)

or for which

(2.23)

are defined.

An derivative of Eq. (2.22) gives

(2.24)

which means that is perpendicular to the tangent vector to theimplicitly defined curve . More generally, for any curve

that renders constant so that

(2.25)

differentiation with respect to shows that

(2.26)

and clearly is perpendicular to the lines defined by constant values.

The implicit definition of curves is employed to great advantage in thenext chapter. As an example of what is entailed in this discussion, considerthe curves defined by the function . Choosing asparameter as in Eq. (2.22) gives the family of curves fordifferent values of . A calculation with Eq. (2.9) shows that they haveunit tangent vectors . The are normal to thedirection given by .

2.4 Streamlines

A streamline is a line that is everywhere tangent to the local velocity vectorin a moving fluid at a fixed instant of time. Streamlines generally exist forall velocity fields, and in the case of steady flow, it should be clear thatstreamlines and pathlines are identical. If is assumed to denote astreamline at a given time , then it follows from its definition that thetangent to must be parallel to the local velocity vector, that is,

(2.27)

where the dependence of and on is implied even though it is notindicated explicitly. This relation can be turned into a set of equations withwhich to solve for in a number of different ways. One approach is toadopt Eq. (2.3) so that with the understanding that the velocity fieldis to be kept stationary at its value at a fixed time of interest. In this case,the streamlines are the paths passing through the frozen velocity field.

Another possibility is to choose as the parameter, in which case thestreamlines are determined from the coupled equations

(2.28)

because both sides are unit vectors. More generally, one can equate the left-and right-hand sides of Eq. (2.27) as unit vectors, yielding the coupleddifferential equations

(2.29)

from which the equations

(2.30)

are readily derived. By choosing the parameter , for example, so that and thus , Eq. (2.30) leads to the coupled equations

for streamlines in the form

For two-dimensional flow, Eqs. (2.31) reduce to the single equation for thestreamlines , given by

(2.32)

Relationships equivalent to those in Eqs. (2.31) and (2.32) can be derived if or is taken as parameter.

As an example of computing streamlines, consider the two-dimensionalvelocity field1 .According to Eq. (2.32), the streamlines are computed from theequation

(2.33)

with the result that

(2.34)

where is a constant. This means that the streamlines are circles aroundthe origin at radii specified by . By using Eq. (2.32) in this example, thedetermination of the streamlines is much simplified compared to the effortrequired with parameters such as or in which two equations must besolved (see Problem 2.7).

Problems

2.1 Find explicit formulas for particle paths corresponding to the velocityfield where

are constants. This may be done symbolically, for example, usingthe dsolve command in MATLAB. Relate and to and (seeProblem 1.3). Plot and discuss what the pathlines look like for the caseswhen and when .

2.2 Find explicit formulas for particle paths corresponding to the velocityfield where

are constants. This may be done symbolically, for example, usingthe dsolve command in MATLAB. Compute and for this flow, andthen plot the pathlines for the case when .

2.3 Compute and plot pathlines for the velocity field using a numerical ordinary

differential equation solver, such as ode45 in MATLAB.

2.4 Use an interpolation routine such as interp2 in MATLAB together withode45 to compute particle paths for a velocity field that is only knownthrough discrete data on a 2D mesh of points. For example, create a 2Dmesh using meshgrid, evaluate a velocity field such as that in Problem 2.3on the mesh, and then compute the particle paths. Compare the pathscomputed using interpolation to those relying on the analytic formulas.Study the effect of mesh spacing on accuracy.

2.5 A sphere is defined by setting the 3D function equal to a constant, say, , in which case

is the radius. Create a line of latitude lying on the surface of the sphere anduse it to calculate a unit tangent vector to the surface. Create a secondcurve on the sphere that is a line of longitude, and where it intersects thefirst line, evaluate its tangent vector. Use the tangent vectors at the point ofintersection of the curves to generate a vector perpendicular to the surface.

2.6 Constant values of a function such as implicitly definesurfaces in 3D. Show that is a vector perpendicular to isosurfaces of

. Hint: consider Problem 2.5.

2.7 Solve for the streamlines corresponding to by using the fact that

they are also pathlines. Thus, use time as the parameter. Hint: the pathequations can be solved by first integrating their ratio, with the resultproviding a means of simplifying the coupled system to a solvable form.

1 It is to be understood here and henceforth that where appropriate, velocityformulas are assumed to be multiplied by a unit constant that brings themto the proper dimensions of length/time.

3Stream Function

Visualization of the collection of streamlines in a flow at a given time isvery useful for gaining a sense of how the fluid is moving. While generallythe computation of streamlines requires integration of the kinds ofdifferential equations discussed in Section 2.4, for some categories of flowsthat vary in only two coordinate directions, it turns out that their entirefamily of streamlines can be computed relatively easily as the isocontoursof a special function known as the stream function. Because of this propertyand others that will become apparent later, the stream function, when itexists, offers considerable benefit toward the analysis and understanding offluid flows. This chapter considers the derivation of the stream function intwo common flow circumstances.

3.1 Two-Dimensional Planar Flow

In two-dimensional planar flow, a stream function, , is constructedby using the idea that its isocontours need to be streamlines, in other words,lie everywhere parallel to the velocity field so there is no flow across them.To implement this idea, consider the area flux of fluid crossing an arbitrarycurve, , with denoting arclength, as shown in Fig. 3.1. Theflux of fluid area across the curve can be computed by summing up thecontribution from each small portion of the curve. As illustrated in thefigure, a unit normal vector is associated with a short line segment oflength along the curve. In time an area of fluid equal to

crosses the line segment. Note that if one imagines the fluidhaving a unit depth into the plane, then the area flux can be alternativelyviewed as being a volume flux across the surface. Dividing by establishes that the flux of area/length-sec is given by . Summing theflux contributions from each piece of a curve yields the total flux ofarea/sec across the line as the path integral

(3.1)

Clearly, this integral is zero along streamlines because in thatcase.

Figure 3.1. Flux across a line segment.

The discussion thus far suggests that a reasonable direction to pursue interms of finding a function, say, , that would be constant onstreamlines is via the definition

(3.2)

where the integral is carried out over a specified path for which is arclength, and

and . If the path is astreamline, then will remain constant on the line, as is hoped for.Moreover, it must be the case that the integral in Eq. (3.2) is the same forall paths having the end point , because if not, then defined by Eq. (3.2) would not be a well-defined, single-valued function.We now consider the circumstances necessary for to be such afunction.

Consider two paths linking the points and , as shown in Fig.3.2(a). If the integration over the paths are equal, then

(3.3)

Because the integrals in this relation represent area flux across the paths,equality in this case would seem to require that the fluid be incompressible,because otherwise the area flux would change from one path to the next. Toshow this formally, reverse the direction over which path 2 is traversed soits integral in Eq. (3.3) changes sign. Bringing this integral to the left-handside of the equation then results in a contour integral over path 1 followedby path 2 in reverse direction (see Fig. 3.2(b)), which is equivalent to thestatement that the contour integral over the closed loop formed from thetwo paths is zero, that is,

(3.4)

It is thus seen that path independence in the integral in Eq. (3.2) isequivalent to a zero contour integral in Eq. (3.4).

Figure 3.2. Integration paths.

Applying the divergence theorem (Hildebrand 1976) to Eq. (3.4) yields

(3.5)

where the integration on the right-hand side is over the area enclosed bythe contour made up of paths 1 and 2. Note that the divergence of thevelocity field (i.e., the dilatation as discussed in Section 1.3) has appearedin the area integral. The contour integral on the left-hand side will be zerofor all closed contours only if the integral on the right-hand side is zero forany region in the flow domain. The latter condition can occur only if is zero everywhere, because otherwise, by continuity, an area could befound where and is of the same sign so that its integral wouldnot be zero. It is thus seen that the integral in Eq. (3.2) is path independentso that is a single-valued function only when and theflow is incompressible.

For a particular path used in evaluating Eq. (3.2), is a unit tangent vector. The unit vector is

perpendicular to the plane of motion and when combined with producesan outward-oriented normal vector to the path in the form ,assuming the path is followed counterclockwise. A calculation gives

, as illustrated in Figure 3.3. It follows bydifferentiating Eq. (3.2) with respect to and substituting for that

(3.6)

Gathering together like terms on the left-hand side gives

(3.7)

Because the path used in establishing Eq. (3.7) is arbitrary, the only waythat this relation can be satisfied for all paths is if the coefficients of

and are separately equal to zero. This means that

showing that the velocity field can be determined from knowledge of ,and vice versa. Note as well that by substituting Eqs. (3.8) into

(3.9)

it is verified that these relations are fully consistent with the assumedincompressibility of the flow.

Figure 3.3. Tangent and outward normal unit vectors to a curve.

We may show using Eq. (3.8) that all the lines produced by constant are indeed streamlines. This depends on the fact that according to Eqs.(3.8), . Moreover, as shown in Section 2.3, is alsoperpendicular to the curve defined implicitly via . Because

is normal to both the velocity vector and the lines of constant , thelatter must be parallel to the former, that is, they are streamlines.

In summary, it has been established that Eq. (3.2) satisfies therequirements of a stream function for 2D incompressible flow. As apractical matter, Eqs. (3.8) offer the best means of either figuring out thevelocity components from a given stream function or re-creating thatbelongs to a given velocity field. A number of examples will be seen insubsequent chapters where the stream function can be determined a priorito then be used in obtaining the velocity field.

The constancy of on streamlines together with the interpretation of theintegral in Eq. (3.2) as the area flux means that the difference in the valueof the stream function between adjacent streamlines must be equal to thetotal area/sec of fluid traveling between them. This is clear if the integral isevaluated on a path traveling between the streamlines. The illustration inFig. 3.4 shows that the streamlines can be useful for intuiting the speed ofrelative motion within the flow domain. Where streamlines approach eachother, the fluid must travel faster to preserve the constant area flux.

Figure 3.4. The area flux is constant between streamlines.

It is expected that is constant on solid boundaries since the flownearby must be parallel to the surface. Across openings within a surfacewhere there is inflow or outflow, as in Fig. 3.5, the stream function varies

across the opening from one constant value, in this case on the lowerwall, to on the upper wall, with the difference between them,

(3.10)

equal to the area flux through the opening. Within the opening itself,

(3.11)

However, the unit vector , which is perpendicular to ,points in the direction across the opening so that

. By introducing the directional derivative

(3.12)

which gives the rate of change of a function with respect to acoordinate in the direction it follows from Eq. (3.11) that

(3.13)

across the opening. Assuming that is given, Eq. (3.13) can beintegrated across the opening to give a boundary condition on for use innumerical computations.

Figure 3.5. Flow across an inlet into a domain. Here .

A simple example illustrating how the stream function can be determinedfor a given incompressible velocity field is that of uniform constant flow

in the direction. In this case, integration of Eq. (3.8a) yields

(3.14)

which also satisfies the requirement that . Lines of constant arestraight lines parallel to the axis. As a second example, consider the two-dimensional velocity field

which a direct calculation shows to be incompressible so that a streamfunction exists. Substituting Eq. (3.15a) into Eq. (3.8a) and integratingyields , where remains to bedetermined. Substituting this expression into Eq. (3.8b) and comparing toEq. (3.15b) reveals that is constant and thus of no consequence. Thefinal result is that

(3.16)

as the stream function associated with Eqs. (3.15). The lines of constant are circles around the origin.

By inspection, it may be noticed that Eq. (3.8) is equivalent to thestatement that

(3.17)

In fact, this relation is formally deduced in the next chapter. Our interesthere in Eq. (3.17) is that it provides a simple means of establishing theconnection between the stream function and the velocity components innonrectangular Cartesian coordinates. For example, in polar coordinates forwhich

Eq. (A.10) applied to Eq. (3.17) shows that

An example of the use of Eqs. (3.19) is provided by Eq. (3.16) for which . In this case, and , which may be

shown (see Problem 3.1) to be consistent with Eqs. (3.15).

Later, in Chapter 10, it is shown that conservation of mass in steady,compressible, two-dimensional planar flow is governed by the relation

(3.20)

whose similarity with Eq. (3.9) suggests that a stream function may existfor this case as well. In fact, as considered in Problem 3.3, it may be shownthat there is a stream function for this flow. Thus, as far as planar flow isconcerned, both unsteady, incompressible flow and steady, compressibleflows may be described via stream functions.

3.2 Axisymmetric Flow

A three-dimensional velocity field is referred to as being axisymmetric ifan axis exists such that the motion appears to be the same when viewed onany plane that intersects the symmetry axis. In this case the flow isessentially two-dimensional because it does not depend on the azimuthalcoordinate in a cylindrical coordinate system , with corresponding to the axis of symmetry, nor does it have a velocitycomponent in this direction. Cylindrical coordinates are related torectangular Cartesian coordinates via the mapping

as shown in Fig. 3.6. Points are located in three dimensions via theirdistance along the symmetry axis given by and their polar coordinates,

in a plane parallel to the x-y plane at the location of the point. Asin the planar two-dimensional case, such axisymmetric flows have a streamfunction if they are incompressible. The derivation of this stream function,which is known as the Stokes stream function, is now considered.

Figure 3.6. A point is resolved into cylindrical coordinates asgiven in Eqs. (3.21).

In axisymmetric flow, the streamlines represent surfaces of revolutionaround the symmetry axis through which fluid does not cross. Proceeding

analogously to what was done in defining the stream function for 2D planarflow in Eq. (3.2), the definition is made that

(3.22)

where the integration is over the surface of revolution formed by , asshown in Fig. 3.7. Here, is the starting point ofthe curve and is the end point. In contrast to the area fluxin Eq. (3.2), the integration in Eq. (3.22) is the flux of volume through thesurface of revolution created by the curve appearing in the integration.Clearly, is constant on surfaces created by streamlines because

will be zero in this case. Moreover, is well defined if Eq.(3.22) is independent of the path running between the two end points.

Figure 3.7. Flux through a surface of revolution.

Omitting details of the argument, which is similar to that in the previoussection, it may be shown that path independence in the present context isequivalent to showing that the volume flux is zero through a toroidal region

created by revolving a closed contour around the axis, as shown in Fig.3.8. According to the divergence theorem in three dimensions the total fluxthrough the surface can be converted to an integral over the containedvolume, yielding

(3.23)

so that in order for to be a well-defined function.

Figure 3.8. Toroidal region formed from two paths .

Now limiting the discussion to incompressible axisymmetric flow,differentiating Eq. (3.22) with respect to and following a similarargument as led to Eqs. (3.8) yields

In this, a factor of has been absorbed into the definition of . Thishas to be taken into account when interpreting the volume flux betweenstreamlines.

Equations (3.24) are useful in determining from a given velocity field,or vice versa. As a simple example, consider uniform flow

, where is a constant. In this case, Eq. (3.24b) impliesthat . Differentiation with respect to andsubstituting into Eq. (3.24a) shows that is constant, in which case,

(3.25)

is the stream function for uniform flow expressed in cylindricalcoordinates. Several other examples of the stream function foraxisymmetric flows are considered in later chapters.

If a flow is axisymmetric around an axis, then it can also be representedas two-dimensional in a spherical coordinate system , where corresponding to in the cylindrical system is the azimuthalcoordinate around the axis. In fact, a stream function in the form can be derived as an alternative to for incompressibleaxisymmetric flows. As shown in Fig. 3.9 where cylindrical and sphericalcoordinates are compared, the axis in the cylindrical coordinate system,which is oriented upward in this illustration, corresponds to the and

directions in the spherical system. With as distance to the origin, , as shown in the figure. Similarly, is the projection of

outward from the axis so that .

Figure 3.9. Relationship between cylindrical and spherical coordinates.

The mapping between spherical and cylindrical coordinates may besummarized from this discussion as

By projecting into the and coordinate directions as shown in thefigure, the relationship between spherical and rectangular Cartesiancoordinates is derived in the form

In fact, these relations also follow directly by substituting Eqs. (3.26) into(3.21).

To connect to the velocity field, Eq. (3.23) must be rewritten inspherical coordinates (see Problem 3.7) and the steps that led to Eqs. (3.24)repeated. The result is

For uniform flow in the direction, the velocity vector can be resolvedinto the and directions according to .In this case it follows from Eq. (3.28a) that

(3.29)

a relation that can also be obtained from Eq. (3.25) by using Eq. (3.26a).Other examples of the use of spherical coordinates in axisymmetric floware considered in following chapters.

Problems

3.1 Show that map to Eqs. (3.15) via Eq. (1.53).

3.2 Consider a velocity field in polar coordinates given by for . Show that, apart from a

singularity on the line , the flow is incompressible, and find thestream function. Plot some streamlines and describe what the flow is.Explain how the flow is changing with so that the area flux betweenstreamlines is held constant.

3.3 Motivated by Eq. (3.20), define a stream function analogous toEq. (3.2) that is based on computing mass flux across pathlines instead ofarea flux. From this, prove that is well defined, and derive velocityformulas similar to Eqs. (3.8).

3.4 Find the stream function corresponding to the component ofvelocity given by

(3.30)

where , and are constants. Constrain by the condition that . Then determine and the vorticity . Make a

contour plot of and a quiver plot of in the region ,revealing that this flow has the appearance of a narrow jet. Plot , and

across the jet at several locations, and describe the flow field. Tosimplify the calculation, use a symbolic program such as that containedwithin MATLAB.

3.5 Compute the velocity components corresponding to the streamfunction

(3.31)

and evaluate them at and when . Make a contour plot of and a quiver plot of the velocity components in the quadrant

. Argue that this is the flow due to a vertical wall slidingacross a horizontal surface at velocity as viewed from the perspective ofthe moving wall.

3.6 Show that for axisymmetric flow in cylindrical coordinates, , where .

3.7 Derive Eqs. (3.28). Hint: on the semi-infinite plane (bounded onone side by the axis), define a curve in spherical coordinatesfor which acts as the parameter. In a rectangular Cartesiancoordinate system in this plane, the curve may be expressed as

(3.32)

as may be seen from Eqs. (3.27) by setting . Define via

(3.33)

where is arclength along the curve and the integral is the flux throughthe surface of revolution with local radius . Compute thetangent to the curve in Eq. (3.32) and from this compute a unit normalvector . Show that , where and

. Compute , and change the integrationvariable to with the help of Eq. (2.16). Finally, differentiate Eq. (3.33)with respect to .

3.8 Show that for axisymmetric flow in spherical coordinates, , where .

4Helmholtz Decomposition

It was mentioned in Section 1.3 that the dilatation and vorticity – obtainedfrom the divergence and curl of the velocity field, respectively – provideinformation about two essential aspects of fluid motion. The dilatationindicates the degree to which the flow is locally compressible, whereas thevorticity expresses the possibility of rotational motion. In fact, knowledgeof these two properties is sufficient to determine the velocity field to whichthey correspond. In other words, given and over a flowdomain at a given time, one can reconstruct the unique velocity field thathas these given dilatation and vorticity values. The relation that makes thispossible, known as the Helmholtz decomposition (Aris 1990), is the focusof this chapter. Subsequent chapters demonstrate a number of usefulapplications of this result.

First to be considered is the Helmholtz decomposition for flow in aninfinite three-dimensional region that is herein denoted as . Followingthis, the modification necessary to include a boundary will be considered,and then some attention will be paid to the special case of the Helmholtzdecomposition in two-dimensional flow.

4.1 Three-Dimensional Flow

The starting point for decomposing the velocity field is to define a vectorfield via the vector Poisson equation

(4.1)

for a given velocity field . For convenience, time dependence is notindicated explicitly in or , because the decomposition is meant tohold at any given time. The vector Laplacian on the left-hand side of Eq.(4.1) is formally defined via the relation

(4.2)

in which a divergence is being taken of the second-order gradient tensor that was defined previously in Eq. (1.40). Section B.1 provides a

discussion of the meaning of the divergence of a tensor. It suffices for thepresent purposes to mention that in the case of rectangular Cartesiancoordinates, the divergence of a tensor having components is thevector with components . Because has components

, it follows that

(4.3)

in which the right-hand side is just the Laplacian of the th scalarcomponent of , so that, in fact,

(4.4)

The unique solution (Riley, Hobson & Bence 2006) to Eq. (4.1) is

(4.5)

where is the variable of integration and is anelement of volume. From , define a scalar field

(4.6)

that will be referred to henceforth as the scalar potential, as well as a vectorpotential

(4.7)

In view of the identity

(4.8)

that holds for any vector field , it is clear that

(4.9)

so is divergence free.

Substituting Eqs. (4.6) and (4.7) into the vector identity

(4.10)

(see Problem 4.1) and noting Eq. (4.1), it has been deduced that

(4.11)

In this case, uniqueness of implies uniqueness of and . Taking adivergence of Eq. (4.11) and using the identity Eq. (4.8) applied to yields

(4.12)

with the dilatation defined previously in Eq. (1.1). Similarly, the identity

(4.13)

for any scalar field means that a curl of Eq. (4.11) gives

(4.14)

The left-hand side is the vorticity defined in Eq. (1.2), whereas applyingEq. (4.10) to and using Eq. (4.9) shows that

(4.15)

One implication of this discussion that is evident from Eqs. (4.11) and(4.15) is that flows lacking vorticity, referred to as irrotational flows, canbe expressed entirely as the gradient of a potential. Conversely, flows withvorticity cannot be expressed this way.

The solutions to Eqs. (4.12) and (4.15) are, respectively,

(4.16)

and

(4.17)

In fact, Eqs. (4.16) and (4.17) can also be derived by substituting Eq. (4.5)into the definitions of and given in Eqs. (4.6) and (4.7), respectively.

The last step in the derivation of the Helmholtz decomposition is tosubstitute Eqs. (4.16) and (4.17) into Eq. (4.11), yielding the infinite spacerepresentation of the velocity:

(4.18)

That this velocity field has the desired and is apparent from the factthat it satisfies Eq. (4.11) together with Eqs. (4.12) and (4.15). Using Eq.(4.18), a means has been found to determine from given and .

4.2 Bounded Domains

More often than not, boundaries are present in flows, in which case somespecific modifications to Eq. (4.18) are necessary if the idea of theHelmholtz decomposition is to remain valid in such cases. In particular, thepresence of a fixed solid boundary means that the velocity normal to theboundary must be zero in view of the nonpenetration condition. Moregenerally, it can be assumed that the normal velocity at the boundary is alsoprescribed at inlet and outlet flow regions. To be applicable to boundedflows, the Helmholtz decomposition must be modified to accommodate thespecified wall-normal velocity conditions at these points.

The full range of possible bounding surfaces to a fluid domain includesgeometries whose analysis is somewhat technical and well beyond theinterests of this book. In the interest of simplicity, only simply connectedflow domains are considered here. Any closed contour within a simplyconnected domain may be deformed to a point while remaining completelywithin the flow domain. Though this assumption simplifies the analysis, itshould not be construed as implying that the Helmholtz decompositioncannot be generalized to include more complex geometrical situations.

With this understanding, consider a fixed domain on which and are known functions. On the bounding surface to , say, , the normalvelocity

(4.19)

is given. If and are extended to all of by being set to zero outsideof , then the integrals in Eqs. (4.16) and (4.17) remain well defined, andit is clear that the velocity field given by Eq. (4.18) retains the correctdivergence and curl within the domain . However, there is no reason whythis velocity field should satisfy the desired boundary conditions, so tocorrect for this, an additional term may be added to Eq. (4.18) or,equivalently, to Eq. (4.11) to accommodate the boundary condition. In thiscase, we now have

(4.20)

Taking a divergence of this relation and applying Eq. (4.8) as well as thefact that the desired dilatation is provided for by the first term on the right-hand side yields

(4.21)

Similarly, taking a curl of Eq. (4.20), applying Eq. (4.13), and the fact thatthe correct vorticity is already given by the second term on the right-handside yields

(4.22)

In addition to the conditions in Eqs. (4.21) and (4.22), must beconstrained in such a way as to enforce the boundary condition on . Thismeans, after substituting Eq. (4.20) into (4.19), that

(4.23)

at all points on the bounding surface.

For a vector field satisfying Eq. (4.22), a scalar function, say, ,must exist such that

(4.24)

In fact, this result is already implicit in Eqs. (4.11) and (4.18) because if , then , so that . It is instructive to prove

this fundamental result directly. Thus, for a given point , select a path , with arclength, , and , and

define

(4.25)

where the integral is along the path and is the corresponding unittangent vector. As in our previous deliberations with regards to the streamfunction in Chapter 3, path independence of the integral in Eq. (4.25) isguaranteed if

(4.26)

for any contour forming a circuit. In fact, Stokes theorem(Hildebrand 1976) says that

(4.27)

where is a surface in three dimensions with boundary given by the pathin the contour integral. In view of Eq. (4.22), it is clear that Eq. (4.26) issatisfied and thus defined in Eq. (4.25) is independent of and isthen a single-valued function of . Taking a derivative of Eq. (4.25) withrespect to and rearranging the terms to one side of the equation yields

(4.28)

Because the path is arbitrary, each of the terms in parentheses in Eq. (4.28)must be separately equal to zero, so Eq. (4.24) has been proven.

The required value of is determined by the need to satisfy Eq. (4.21)and the boundary condition Eq. (4.23). Substituting Eq. (4.24) into (4.21)yields

(4.29)

and is seen to be a potential function that satisfies the Laplaceequation. Introducing the directional derivative defined in Eq. (3.12) so that

(4.30)

it follows from Eq. (4.23) that satisfies the Neumann boundarycondition

(4.31)

The decomposition of the velocity field for three-dimensional regionsincluding boundaries has now been shown to take the form

(4.32)

with and computed from Eqs. (4.16) and (4.17), respectively, and determined as the solution to Eqs. (4.29) and (4.31). In some cases it isconvenient to combine the two scalar potentials appearing in Eq. (4.32) intoa single potential , which, for convenience, may be referred to as

. In this case,

(4.33)

and is determined as the solution to

(4.34)

with boundary condition

(4.35)

4.3 Two-Dimensional Flow

The Helmholtz decomposition can also be developed for two-dimensionalflow in the plane where . In this case, ,

, and the equivalent of Eq. (4.11) is

and , respectively, are determined as the solution to the Poissonequations

(4.37)

and

(4.38)

on the infinite plane. Assuming that the functions on the right-hand side ofthese relations are bounded in an appropriate way at infinity, it may beshown that Eqs. (4.37) and (4.38), respectively, have the solutions

(4.39)

and

(4.40)

Equations (4.39) and (4.40) can also be obtained by integrating Eqs. (4.16)and (4.17), respectively, for and taking the limit as .This entails neglecting terms that become infinite with but ultimatelymake no contribution to because they are constant. The analogousrelations to Eq. (4.18) for velocity in the present case are

Similar to the three-dimensional case, Eqs. (4.36) must be modified inthe presence of boundaries by the addition of a potential function sothat

satisfies the two-dimensional form of the Laplace equation (4.29) withthe boundary condition

(4.43)

which comes from reducing Eq. (4.31) to the two-dimensional case. Theunit normal vector can be recast as a unit tangent to theboundary in the guise of . In this case,

(4.44)

where is arclength along the boundary. Consequently, the boundarycondition for is

(4.45)

If the flow is incompressible, then the terms containing , which dependon not being zero, drop out of Eq. (4.42) so that

and the boundary condition Eq. (4.45) becomes

(4.47)

It was previously shown in Section 3.1 that for incompressible 2D flow, astream function exists satisfying Eqs. (3.8). The similarity of theseequations with the terms depending on in Eq. (4.46) is evident.Moreover, it was shown in Eq. (3.13) that the stream functionautomatically satisfies the normal flow boundary conditions. This meansthat if , then Eq. (4.47) reduces to

(4.48)

so that is superfluous in this case, because the solution to Eq. (4.29)with boundary condition Eq. (4.48) is a constant. To summarize: for 2Dincompressible flow, may be identified with the stream function and

may be dispensed with. Moreover, the equivalency of and intwo-dimensional incompressible flow means that, in this case, Eq. (4.11)gives

(4.49)

a result that was anticipated in Eq. (3.17).

A final observation is that 2D flows lacking both dilatation and vorticity referred to as potential flows may be described via a potential

function as in

(4.50)

However, because such flows are incompressible, they also can berepresented using a stream function with velocities determined via Eqs.(3.8). It is evident from these relations that

(4.51)

which means, according to Section 2.3, that lines of constant and lines ofconstant must always intersect at right angles. As an example, foruniform flow where is given by Eq. (3.14), it is easy to see that

(4.52)

so that streamlines and potential lines form a grid of and coordinatelines that clearly intersect at right angles. More examples of this can beseen subsequently.

Problems

4.1 Verify Eq. (4.10) by expanding out the right-hand side using indexnotation.

4.2 Given a stream function in axisymmetric flow where and are constants, show that the

flow is irrotational, and find the corresponding potential . Describe theflow field.

5Sources, Sinks, and Vortices

The Helmholtz decomposition provides a means for determining thevelocity field from a knowledge of the dilatation and vorticity fields. Themathematical form of the decomposition whether in 3D, Eq. (4.18), or 2D,Eq. (4.41), motivates the definitions of several kinds of idealized velocityfields that can be of great use in representing flows of inviscid fluids. Thischapter and the next consider some of the principal ideas that come fromthis line of development.

5.1 Sources and Sinks in Two Dimensions

In this section and the next, we consider how the terms depending ondilatation and vorticity in Eqs. (4.41) can be adapted to create physicallyinteresting flow fields. First consider Eq. (4.39), representing the potentialgenerated by the dilatation field. If the nonzero dilatation field in a flowhappens to be restricted to a small circle of radius , with area surrounding a point , then Eq. (4.39) provides a means of obtaining thecomplete effect of this localized dilatation on the flow field. In particular,if is small enough and a point is far enough away from , then theintegration in Eq. (4.39) can be simplified by extracting the log functionevaluated at . Defining

(5.1)

then is given approximately by

(5.2)

which is singular at , where the approximation used in theconstruction of the equation fails.

Equation (5.2) becomes more legitimate as a numerical approximation as, but under normal circumstances will go to zero together with

, so the resulting potential is of no interest. Alternatively, if it is assumedthat increases as in such a way that remains nonzero andconverges to a constant , then in this limit, a potential field iscreated of the form given by Eq. (5.2). This idealized flow that has singular(and thus unphysical) behavior at approximates the potentialassociated with a physical dilatation field concentrated within a smallregion with total strength . In view of the way that Eq. (5.2) has beenderived from the Helmholtz decomposition, it is clear that outside of thesingular point the flow is incompressible and irrotational. Depending on thesign of , Eq. (5.2) is called a 2D source or sink .

According to Eq. (4.11), the velocity field corresponding to the source orsink is

(5.3)

which, depending on the sign of , is radially outward or inward from thepoint . From the divergence theorem, it follows that

(5.4)

so the source/sink strength can be interpreted as the flux of area/seceither leaving ( ) or entering ( ) the point . The linearity ofthe Helmholtz decomposition means that an arbitrary number of sourcesand sinks with different values of and can be linearly combined tocreate additional flow fields.

In the case of a lone source or sink, it is convenient to express thepotential and velocity in polar coordinates , centered at .1 In polarcoordinates, Eq. (5.2) becomes

(5.5)

whose isocontours are circles around the origin, as shown in Fig. 5.1. Inpolar coordinates,

(5.6)

and so that

Substituting Eq. (5.5) into Eqs. (5.7) yields the velocity components

Using Eqs. (5.8), a calculation gives

(5.9)

which is consistent with Eq. (5.4).

Figure 5.1. Streamlines (solid) and potential lines (dashed) correspondingto a 2D source/sink. As shown in Section 5.2, the lines have the oppositemeaning for a point vortex.

The dilatation field associated with Eq. (5.2) can be shown by directcalculation to be zero everywhere, except at , where it is

indeterminate. For example, in polar coordinates,

(5.10)

and after substituting Eq. (5.8), it is seen to be zero for . It followsthat the source/sink flow should have a stream function, everywhere, exceptat its origin.

To find the stream function, it is convenient to do the calculation in polarcoordinates. In this case, equating Eq. (5.8a) to (3.19a), integrating withrespect to , and noting that yields

(5.11)

or, in rectangular Cartesian coordinates, for a source/sink at point ,

(5.12)

is the stream function counterpart to Eq. (5.2). As expected, the lines ofconstant given by Eq. (5.11) progress radially outward from the origin,as seen in Fig. 5.1. Moreover, consistent with the discussion in Section 4.3,the lines of constant and intersect at right angles.

5.2 Point Vortices

In the same way that the potential in Eq. (4.39) was used to construct pointsource/sink flows, the potential in Eq. (4.40) can be made the basis of whatis known as a point vortex. For simplicity, assume that the fluid isincompressible and inviscid so that can be identified with the stream

function, . This assumption does not exclude the possibility that pointsources and sinks are also present because they have zero dilatationeverywhere, except at discrete points.

Point vortices can be derived by a vortical counterpart to the argumentused in deriving source/sinks. In this case it is assumed that vorticity isconcentrated in a small circle of radius and area around a point.Where source/sinks have a strength , which represents an integral overthe local dilatation field, the equivalent for the vorticity is the circulation,

, defined in the 2D case as

(5.13)

whose similarity with the definition of is apparent. Although a morecomplete discussion of circulation is deferred to Chapter 18, it may benoted here that can be defined for any surface in 3D via

(5.14)

with the local normal to the surface. This relation simplifies toEq. (5.13) if the area lies in a coordinate plane and the flow is 2D. If a 3Dsurface is bounded by a closed curve, as shown in Fig. 5.2, then Stokestheorem implies that

(5.15)

where the last term is a contour integration along the boundary curve and is a unit tangent vector to this curve. A consequence of Eq. (5.15) is that is the same for all surfaces that share a common bounding curve.

Figure 5.2. A three-dimensional surface and bounding curve asconsidered in Stokes theorem.

It should now be evident that the same kind of argument made inSection 5.1 concerning two-dimensional source/sinks can be made for theintegral term in Eq. (4.40). Thus, shrink the small circular region inEq. (5.13) to a point while maintaining the same circulation so that thevorticity within the circle approaches infinity. The end result is a flow fieldwith stream function

(5.16)

which is referred to as a point vortex. Similar to the source/sinksconsidered previously, a point vortex is a potential flow that has vanishingvorticity and dilatation everywhere, except at the place where it is centered.Equation (5.16) has an identical form as Eq. (5.2), so it is no surprise thatthe streamlines corresponding to a point vortex are circular lines around theorigin, the same as the potential lines for a source/sink. Positive circulationcorresponds to counterclockwise motion consistent with the right-handscrew rule. The velocity components corresponding to Eq. (5.16) can beobtained conveniently using Eq. (4.49). In the present case, they are foundto be

(5.17)

For a point vortex located at the origin, switching to polar coordinatesgives

(5.18)

and, consequently, from Eqs. (3.19a) and (3.19b),

A calculation confirms that this velocity field is self-consistent in the sensethat

(5.20)

Finally, from Eqs. (5.19) and Eq. (4.50) written in polar coordinates, thevelocity potential for a point vortex is determined to be

(5.21)

The similarity between source/sink and point vortices in two dimensions issummarized in Table 5.1, where it is clear that apart from a sign change, thepotential of one is equivalent to the stream function of the other, and viceversa. This is reflected in the complementary roles of the circles and radial

lines in Fig. 5.1. Note as well that the area flux between vortex streamlinesin Fig. 5.1 has been set to a constant, so that the unequal spacing of circularstreamlines reflects the increase in velocity of a point vortex as the originis approached.

Table 5.1 Potential and stream function for a 2D source/sink and pointvortex

Source/sink Vortex

5.3 Accommodating Boundaries in Two Dimensions

Our consideration of source/sinks and point vortices in the last two sectionswas done without taking into account the presence of solid boundaries. Forexample, the flow that is contained in Fig. 5.1 for either a point source/sinkor vortex is incompatible with the presence of a straight wall because thenonpenetration boundary condition will be violated. A recipe for fixing thisis provided by the discussion in Section 4.3 showing how the velocity fieldarising from given dilatation and vorticity fields can be modified to takeinto account a proscribed normal velocity at boundaries. In essence, onetakes the solution appropriate to an unbounded region and adds to it apotential flow with vanishing dilatation and vorticity in the region ofinterest that forces the boundary condition Eq. (4.43) to be met.

In general, the necessary potential flow needed to satisfy thenonpenetration boundary condition can be determined as the solution to theLaplace equation with Neumann boundary condition. In some cases,however, when the boundary is of simple shape, it is possible to directlyproscribe the desired potential field that will satisfy the boundaryconditions. Some examples of this situation are now considered.

For a single straight bounding surface, as shown in Fig. 5.3,nonpenetration in the case of a solitary source at can bearranged by placing an image source at the location . Thefigure shows how the contributions of the two sources to the normalvelocity on the surface cancel, leading to a streamline pattern thatconforms to the surface. Mathematically, the stream function and its imagetake the form

(5.22)

with the image source having the role of in Eqs. (4.46). In particular,by being located outside the flow domain, which is in this case, theimage source does not contribute to the dilatation and vorticity of the flowfield. Evaluation of Eq. (5.22) at gives so that this line isindeed a streamline.

Figure 5.3. Streamlines corresponding to a source and its image reflectedthrough a straight solid wall. Velocity contributions from the source and itsimage cancel at the boundary and elsewhere sum to point in the direction ofthe local fluid motion.

Flows containing point vortices can also be adapted to the presence ofsimple boundaries by the use of images. In this case, the strength of theimage vortex must be opposite in sign to that of the original vortex toachieve nonpenetration. Thus, for a single vortex located at in relationship to a wall at , the appropriate stream function is

(5.23)

The streamlines associated with this function are shown in Fig. 5.4, whereit is seen that the two vortices combine to prevent flow through theboundary at . Several problems at the end of the chapter consider theimage systems for source/sinks and vortices corresponding to geometriesother than the flat plate.

Figure 5.4. Streamlines corresponding to a vortex and its negative imagereflected across a straight boundary.

In some instances, boundary conditions can be satisfied via thesuperposition of sources and sinks along the boundary, as in the case of aplanar jet flow entering the half-plane through an opening at

, as illustrated in Fig. 5.5. The velocity is taken to beuniform entering the flow region at speed , and the effects of viscosityare ignored. This is a potential flow that is determined uniquely by theassigned normal velocity along the boundary.

Figure 5.5. Potential planar jet entering a half region derived bysuperimposing sources across the inlet.

The desired inlet velocity can be approximated by placing equallyspaced sources across the opening, with each one having strength .In this way, over each small interval at the opening, there is an areaflux entering the flow domain with effective average velocity .An equal flux of fluid travels to the left and is not of interest. Exactsatisfaction of the boundary condition can be met in the limit as .

The potential due to the N sources, assuming they are equally spacedacross the opening, is

(5.24)

where . After taking the limit as , Eq. (5.24) becomes the integral

(5.25)

which may be integrated in closed form, giving

Taking appropriate derivatives of Eq. (5.26) gives the velocity field for thejet as

(5.27)

and

(5.28)

A stream function can be computed for this flow (see Problem 5.12) andwith it the streamlines shown in Fig. 5.5 are determined.

The velocity formulas (5.27) and (5.28) show that the velocitycomponents are undefined at the edge of the opening (i.e.,

). This is a property of inviscid flow around sharp edges,as seen in Chapter 7. It is also the case that the vertical velocity is nonzero

across the opening. In fact, the process of determining a potential flow thatsatisfies the given normal velocity at the inlet is done without imposing acondition on the tangential velocity at the inlet. The result for the presentcase is a slight tangential velocity across the opening. If a potential flow inthis situation did satisfy the condition across the opening, itwould consist of a uniform flow containing straight horizontal streamlinesemanating from the opening and zero velocity elsewhere in the domain. Inthis unphysical flow, the velocity would be singular at the lines spanning the flow domain.

5.4 Sources and Sinks in Three Dimensions

The procedure for developing 2D source/sinks is readily generalized tothree dimensions, with Eq. (4.16) now playing the role formerly played byEq. (4.39). In this case, the integral of the dilatation field within a smallsphere of radius , volume , and surface surrounding a point iskept constant as , yielding the 3D potential function

(5.29)

where is the strength of the source/sink. Similar to thetwo-dimensional case, is equal to the volume flux of fluid leaving orentering the sphere surrounding . As before, this interpretation followsfrom the calculation

(5.30)

where the last expression is the total flux through the boundary. Thoughsingular and hence unphysical near the source/sink location at , Eq. (5.29)nonetheless well approximates the flow from a source or sink at more

distant points. A computation gives the velocity field associated withEq. (5.29) as

(5.31)

which is radially outward from the point .

In terms of local spherical coordinates , centered at , Eq. (5.29)may be written as

(5.32)

Substituting this into Eq. (A.14), which gives the form of in sphericalcoordinates, and noting that , reveals that

, and the radial velocity is

(5.33)

This result is consistent with Eq. (5.30) because in this case, sothat

(5.34)

Apart from the singularity at the center of the source/sink, the flow isincompressible. Moreover, it is axisymmetric around any axis drawnthrough its center. Consequently, there is a stream function corresponding

to Eq. (5.32). Equating Eq. (5.33) to the stream function formula for theradial velocity in spherical coordinates (3.28a) yields

(5.35)

from which an integration gives

(5.36)

as the stream function for a 3D source/sink in spherical coordinates.

It is of interest also to represent the source/sink flow in cylindricalcoordinates. Starting with the identity

(5.37)

derived from Eqs. (3.26a) and (3.26b) as illustrated in Fig. 3.9, it followsfrom Eq. (5.32) that

(5.38)

Using the expression for in cylindrical coordinates in Eq. (A.7), acalculation shows that the velocity components for source/sink flow are

where, as expected, the swirling velocity . Finally, to compute thestream function in cylindrical coordinates, set the velocities in Eqs. (5.39)to their counterparts in Eqs. (3.24), integrate either one of these relations,and substitute the result in the other, yielding

(5.40)

The streamlines corresponding to Eq. (5.40) are radially outward and thepotential lines associated with Eq. (5.38) are circular, as shown in Fig. 5.6.The numerical increments of and are uniformly spaced. In the case ofthe stream function, this means that there is an equal volume flux of fluidleaving between the surfaces of revolution generated by the streamlines.For this to happen, the streamlines must take on the unequal distributionshown in the figure. It may be noted as well that the spacing of thepotential lines varies more rapidly for the 3D source/sink than they do inthe 2D case shown in Fig. 5.1. This is a consequence of the versus

behavior of .

Figure 5.6. Streamlines (solid) and potential lines (dashed) in cylindricalcoordinates corresponding to a 3D source/sink.

Problems

5.1 A point source of strength is located at in the half-plane above the boundary and is free to move via the action of itsimage source. Compute the path of the source as a function of time.

5.2 Develop the image system needed to represent a 2D source/sink and apoint vortex near a corner, that is, in a domain for which the and coordinate axes are solid boundaries.

5.3 A point vortex is initially located at the point near a 90 cornerand is free to move under the velocity field it generates. Compute the path

of the vortex. To simplify the analysis, assume that the pathcan be expressed parametrically in terms of via .Differentiate the identity and use the path equations to geta solvable equation for . What is the asymptotic distance of thevortex above the boundary as ? Obtain an implicit solutionfor as a function of by integrating the equation using a codesuch as MATLAB.

5.4 A point source is initially located at the position near a 90corner and is free to move under the velocity field it generates consistentwith the nonpenetration condition. Develop the equations governing itsmotion and combine them to show that

(5.41)

and then use this to reduce the and equations to an easilyintegrable form. Solve for and and comment on the sourceposition for large time.

5.5 Find the image system necessary to maintain the nonpenetrationboundary condition for a point vortex contained between two parallelplates.

5.6 Show that for a point source positioned between parallel plates apart,the wall-normal velocity at its location produced by its image system(similar to that in Problem 5.5) is given by .

5.7 Determine the image system required to enforce the nonpenetrationboundary condition for a source/sink in a rectangular cavity.

5.8 Consider the 2D flow formed by the sum of a source of strength located at the origin and a uniform flow of speed in the direction.Find the stagnation point (i.e., a point where the velocity vector is zero) andcompute the shape of the streamline that runs through it. Show that theflow looks like that impinging on an infinitely long plate (in the direction) of thickness with rounded front end. Plot somestreamlines using MATLAB or equivalent. (Using , coordinates mightbe easiest.)

5.9 Consider a point vortex of strength situated at above aflat, solid wall at . Use an image vortex to satisfy the nonpenetrationboundary condition. Why are the vortices moving? Use Eq. (5.19b) to showhow fast they are moving and in what direction. Add a uniform flow to thevelocity field produced by the vortex and its image that stops the motion ofthe vortices. For the uniform flow plus two vortex system, find thelocations of the stagnation points on the axis. Find the stream functionfor this flow and show that the streamline has an oval shape. UseMATLAB or equivalent to compute and draw some streamlines.

5.10 Consider a source of strength located at a point outside ofa cylinder of radius . Show that an image system that enforces thenonpenetration condition on the surface of the cylinder is given by the sumof a source of strength located at the point and a sink ofstrength located at the origin. Compute the velocity components,and from these compute the radial velocity component and show that it iszero for . Use a code such as MATLAB to make contour plots ofstreamlines for the flow around the cylinder.

5.11 Show that if a point vortex is located at a point outside of acylinder of radius , then by adding an image vortex of opposite strengthat the point the surface of the cylinder is a streamline. Make acontour plot of the streamlines using MATLAB or equivalent. A vortex canbe added to the flow at the center of the cylinder without changing the non-penetration boundary condition. What strength of vortex added at the center

causes the velocity on the surface of the cylinder to be zero at ? Plotthe streamlines for this case.

5.12 Compute the stream function corresponding to Eq. (5.26) for theuniform planar jet. This may be done relatively simply using symbolicintegration as in MATLAB. Verify the velocity formulas in Eqs. (5.27) and(5.28). Make a quiver plot of the velocity field corresponding to Fig. 5.5.

5.13 Compute the stream function and velocity components correspondingto a planar jet with parabolic velocity acrossthe opening to the half-plane . Plot the streamlines andmake a quiver plot of the velocity and compare this to the equivalentresults for a uniform entrance flow as in Problem 5.12. What can be saidabout the velocity distribution across the opening, and particularly at thecorners?

5.14 Consider a point vortex initially located at ofcirculation together with a second point vortex of strength initiallylocated at . Show that for having the same sign, thetwo vortices orbit around a fixed point located on the line connecting them.Show that the angular velocity of rotation andfind the location of the center of rotation. If the circulations are of oppositesign, the vortices rotate around a point on the extension of the line betweenthem. Find this point and comment on what happens in the special case thatthe vortices are of equal and opposite strength. Write a code (e.g., usingMATLAB) that computes and plots the paths of two arbitrary point vorticesincluding one or more tracer particles to help visualize the nearby flowfield.

5.15 What is the stream function corresponding to a 3D source at point and sink at ? Note that this flow is axisymmetric about the

axis. Use MATLAB or equivalent to make a contour plot of thestreamlines.

5.16 The stream function for a line source can be determined by integratingthe stream function for a point source Eq. (5.40) along a segment of the axis. With the help of a code such as MATLAB, construct the 3D streamfunction in cylindrical coordinates consisting of a uniform flow of velocity

, a line source of strength per unit length located between , and a sink of strength per unit length located

between . For the particular case when , , and , evaluate the velocity component on the axis and find the

stagnation point. Evaluate at the stagnation point and show that it iszero. Compute the stream function and velocity components on a grid ofvalues in the plane and make a velocity quiver plot superimposed ona plot of streamlines revealing the flow around a body with the shape of the

streamline.

5.17 In a rubber duck race, many rubber ducks are simultaneously put intoa river at a starting line, and the first duck to cross a finish line downstreamis the winner. Write a code to model a rubber duck race using MATLAB orequivalent. First, create a “turbulent river” by randomly placing vortices into a region between parallel walls at and

. The velocity of the modeled river consists of a constant velocity inthe streamwise direction plus contributions from the vortices includingappropriate image vortices to prevent ducks from crossing the sideboundaries. To eliminate the singularity in the formulas in Eq. (5.17), add asmall constant to the denominator. Release ducks at and determinethe first one to cross at . Plot the results of the race.

1 Depending on the context, it should be straightforward to distinguish theseparate meanings of as coordinate angle and as the dilatation.

6Doublets and Their Applications

6.1 Three-Dimensional Source/Sink Doublet

Source/sink doublets, also known as dipoles, are singular flows that can beconstructed as the limiting case of a special superposition of thesource/sink flows considered in the previous chapter. Thus, consider a flowconsisting of a source of strength placed at a point and asink of the same strength at , as shown in Fig. 6.1. Such a flow isaxisymmetric about the axis connecting the two points. As long as ,the source and sink produce a flow such as that in the figure. However, ifthe source and sink are brought together so that , then their inducedvelocity fields cancel and there is no flow. Alternatively, if it is imaginedthat as the source and sink are brought together, is increased inmagnitude so that as , then a nontrivial flowresults: the source doublet.

Figure 6.1. Streamlines of a source and sink in 3D prior to carrying outthe limiting process that creates a source doublet.

The potential of a source doublet is determined by noting Eq. (5.29) andcarrying out the limiting process

(6.1)

Substituting Taylor series expansions about the location in Eq. (6.1)and taking the limit yields

(6.2)

The corresponding velocity field is computed as the gradient of Eq. (6.2),in which case

(6.3)

The singular behavior of the doublet at is evident in both Eqs. (6.2)and (6.3).

The doublet strength, , indicates the axis of symmetry of the doublet.The end result of the limiting process for the source and sink in Fig. 6.1 isto create a flow field, shown in Fig. 6.2, where the velocity vectors,normalized to unit magnitude everywhere, are plotted. It is seen that theflow exits the doublet aligned according the direction of , sweeps around,and enters the doublet location in the same direction at which it leaves.

Figure 6.2. Quiver plot of the normalized velocity vectors of a 3Dsource/sink doublet.

In the case of a source doublet at the origin of a spherical polarcoordinate system, the corresponding formulas for and can beobtained using the coordinate mapping in Eq. (3.27). Thus, for a doublet inwhich and ( ) pointing toward the direction sothat , Eq. (6.2) yields

(6.4)

From Eq. (A.14), the corresponding velocity components in sphericalcoordinates are found to be

Because the doublet flow is axisymmetric around the axis defined by thechoice of , a stream function can be found for either cylindrical orspherical coordinates based on this orientation. For the case just consideredin Eqs. (6.4) and (6.5), setting (6.5a) equal to Eq. (3.28a), integrating, andsubstituting the result into Eq. (3.28b) and noting Eq. (6.5b) yields

(6.6)

as the stream function of a doublet discharging into the direction.

For completeness, it may be noted that after a transformation fromspherical to cylindrical coordinates, Eqs. (6.4) and (6.6) yield the potentialand stream function of the doublet, respectively, in cylindrical coordinatesas

(6.7)

and

(6.8)

Moreover, Eqs. (3.24) together with (6.8) yield the velocity components

Figure 6.3 contains a plot of the streamlines of the doublet that may becontrasted with those of the source/sink system in Fig. 6.1. In this example,with , the flow leaves and enters the doublet in the direction.

Figure 6.3. Streamlines of a 3D source doublet with strength .

The value of doublets comes in their use in constructing more complexflows. For example, it may be noted that for the example in Fig. 6.3, theflow along the axis is in the negative direction. Moreover, byevaluating Eq. (6.9b) with , it is clear that the velocity is symmetricabout , has a negative singularity at , and decreases inmagnitude to zero as . In view of these properties, if a uniformflow in the positive direction is added to the doublet flow, there will be

two symmetrically placed points on the axis where the flow is notmoving, that is, there will be two stagnation points. To investigate thenature of the flow field having this property, it is helpful to determine thecorresponding stream function by combining Eq. (6.6) with the streamfunction of uniform flow Eq. (3.29), yielding

(6.10)

from which it follows that

(6.11)

in which the length scale

(6.12)

has been defined.

Equation (6.11) shows that the zero streamline contains the sphere ofradius and the lines and that make up the axis. Becausethe flow is axisymmetric about the axis, it is evident that the stagnationpoints lie on the surface of the sphere, forward and aft. This is also evidentfrom calculating the velocity components, which, according to Eqs. (6.11)and (3.28), are given by

where everywhere on the surface of the sphere defined by .Moreover, Eq. (6.13b) shows that at and .

The conclusion may be reached that the flow associated with Eq. (6.11) isthat of a uniform inviscid flow past a sphere. Far from the sphere, thevelocity on the axis is , which progressively falls to zero at thestagnation point at on the surface. The velocity recovers from thestagnation point at , returning toward as . The forwardand aft symmetry of the flow is an unphysical artifact of the failure to takeinto account the action of viscosity.

The maximum velocity on the surface of the sphere occurs at the sidewhere and has magnitude . Streamlinescorresponding to Eq. (6.11) are plotted in Fig. 6.4, where it is to beunderstood that this pattern applies on any plane containing the central axisof the doublet. It may be noted as well that any of the streamlines in Fig.6.4 can be taken to be a physical boundary. In that case, Eq. (6.11) mightrepresent inviscid flow over a smooth bump or cavity. Moreover, the flowof interest can be either on one or both sides of the boundary surface that isselected. For example, in the case of the sphere, there is flow in its interiorgiven by Eq. (6.11), though this is not usually of significant physicalinterest.

Figure 6.4. Streamlines corresponding to Eq. (6.11) in inviscid sphereflow.

To an observer moving at the speed of the incoming wind in the direction, the sphere appears to be moving through a stationary fluid in

the opposite direction with velocity . If the path of the center of thesphere is with , then its velocity potentialat any time may be written using Eqs. (6.2) and (6.12) as

(6.14)

because with the change of perspective to a moving observer, thecontribution of the uniform flow is negated. By generalizing the flow sothat is time dependent, Eq. (6.14) corresponds to the potential of anaccelerating sphere in inviscid flow.

6.2 Doublets in Two Dimensions

In the same way that the potential for a source and sink can be combined toform a doublet in 3D, the same steps can be applied in two dimensionsusing the 2D source/sink potential given by Eq. (5.2). The result is

(6.15)

For the particular case (see Problem 6.3 for the general case)and using polar coordinates centered at , the doublet has the potential

(6.16)

and velocity components

Equations (3.19) together with Eqs. (6.17) allow for the determination ofthe stream function, which is found to be

(6.18)

The streamlines and potential lines of a source doublet have the samepattern, except rotated by 90 , as illustrated in Fig. 6.5.

Figure 6.5. Streamlines (solid) and potential lines (dashed) for a 2Dsource doublet.

Point vortices of equal and opposite circulation can be combined by thesame procedure used in making a 2D source doublet to produce what isknown as a vortex doublet. Thus, plus and minus vortices are placed atlocations and , and the limits are applied to the stream function determined from Eq. (5.16), yielding

(6.19)

In view of the relationship between the potential and stream function forsources, sinks and vortices summarized in Table 5.1, and the connectionbetween potential and streamlines in Fig. 6.5, it is clear that a vortexdoublet of strength is identical to a source doublet of strength satisfying the conditions and with rotated 90counterclockwise with respect to . Thus, to create a vortex doublet withthe same flow as a source doublet for which requires that

.

Just as a 3D doublet combined with uniform flow yields the potentialflow past a sphere, so too a 2D source or vortex doublet together withuniform flow results in the flow past a circle or cylinder. In particular,expressing Eq. (3.14) in polar coordinates and adding it to Eq. (6.18) yields

(6.20)

where . At , the stream function , so thecircle is a streamline. is also zero on the lines . A calculationwith Eq. (3.19) gives the velocity components for the cylinder flow as

Clearly on the cylinder surface. On the line , the flowapproaches the cylinder with velocity

(6.22)

reaching a stagnation point on the cylinder surface. The flow recovers backto the free stream velocity along the line. In this inviscid solution,the tangential velocity on the surface is nonzero with .The maximum velocity is achieved at the sides where . In thenext chapter, the cylinder flow will be revisited in the context of studyingits pressure field and the forces acting on it.

Problems

6.1 Show the details of how Eq. (6.1) leads to the doublet potential in Eq.(6.2).

6.2 Derive the stream function and velocity field corresponding to thepotential flow past a sphere in cylindrical coordinates.

6.3 Derive the 2D potential in polar coordinates for a doublet oriented inthe direction and show that this reduces to Eq. (6.16) when .Compute the corresponding stream function for this case.

6.4 Derive the stream function for a vortex doublet given in Eq. (6.19).

6.5 The first term in the stream function

(6.23)

may be recognized as being the inviscid flow at speed past a cylinder ofradius . Verify that the second term, depending on , is also constant onthe boundary . Compute and on the boundary and show that

the wall-normal velocity is zero while the tangential velocity slips past thesurface. Show that the vorticity is constant and equal to everywhere.Convert to coordinates and compute . Explain the nature of thevelocity field far upstream of the cylinder that contributes to the vorticity.Make a contour plot of and a quiver plot of .

7Complex Potential

The study of two-dimensional potential flows of the form

(7.1)

that can be determined as the solution to the Laplace equation

(7.2)

with appropriate Neumann boundary conditions is greatly enhanced by thefact that such flows fall within the realm of complex analysis, that is, thetheory of functions of a complex variable. Several far-reaching theorems ofcomplex analysis provide practical benefit in characterizing potentialflows. The point of this chapter is to make clear this connection and toprovide a selected introduction to some of its consequences.

7.1 Connection to Complex Analysis

The relationship between 2D potential flows and complex analysis arisesbecause of the formulas relating and contained in

that are a consequence of Eqs. (3.8) and (7.1). In fact, real-valued functions and that satisfy the equations

which are known as the Cauchy-Riemann equations, can be combined tocreate a function of a complex variable in the form

(7.5)

where for this chapter (Ahlfors 1979). In view of Eqs. (7.3), and satisfy the Cauchy-Riemann equations so there exists a function

, referred to as the complex potential, such that

(7.6)

This means that all potential flows have a complex potential .Conversely, any given function can be used to create a streamfunction and scalar potential belonging to a potential flow field. Basicresults concerning the properties of functions of a complex variable areapplicable to 2D potential flows by virtue of the connection arising throughthe complex potential.

Complex functions defined as in Eq. (7.6) are analytic, that is, they havethe remarkable property of being differentiable to all orders. Moreover,because , and , and derivatives of Eq. (7.6),respectively, give

In view of the relation

(7.8)

which follows from Eq. (7.1) and either of Eqs. (7.7), is referred toas the complex velocity field.

It is interesting to see what complex potentials correspond to thepotential flows that have been considered in previous chapters. Oneparticularly simple example is the case of uniform flow , forwhich, according to Eqs. (3.14) and (4.52),

(7.9)

and Eq. (7.8) is clearly satisfied. As a second example, consider thepotential and stream function of a point source given in Table 5.1. In thiscase,

(7.10)

However,

(7.11)

because

and

(7.13)

Furthermore, , and consequently, the complex potentialof a 2D source/sink is given by

(7.14)

In the same vein, the potential and stream function for a point vortex inTable 5.1 can be added together, yielding

(7.15)

as the complex potential of a point vortex. As another example, Eqs. (6.16)and (6.18) for the potential and stream function of a source doublet forwhich may be combined, giving

(7.16)

showing that the source doublet has complex potential

(7.17)

As shown previously in Eq. (6.20), the flow past a cylinder of radius can be formed from adding a doublet to a uniform flow.

In the present context, using Eq. (7.9) and (7.17), this yields the complexpotential

(7.18)

The result in Eq. (7.18) is, in fact, a special case of a more general resultknown as the circle theorem (Milne-Thomson 1968), which concerns themodification of complex potentials that are defined over the entire plane soas to accommodate the insertion of a fixed circular boundary. The circletheorem holds that for any potential not having singularities within adistance of the origin, the potential given by

(7.19)

will have the boundary of the circle of radius as the streamline .Here the overbar denotes a complex conjugate so that, for example,

. in Eq. (7.19) may be interpreted as the complexpotential resulting from adding a circle of radius to the flow belongingto . Specifically, for uniform flow where ,

(7.20)

which clearly results in Eq. (7.18).

7.2 Flows Derived from a Power Law

Just as given scalar potentials and stream functions can be combined tocreate complex potentials, so too the real and imaginary part of a givencomplex potential can be interpreted as being the potential and streamfunction, respectively, of a flow field. In some cases the velocity field is ofobvious interest. As an example of the latter, consider the power law

(7.21)

for any exponent and real scalar . This is a generalization of thepotential in Eq. (7.9) for uniform flow. Replacing in Eq. (7.21) using Eq.(7.11) yields

(7.22)

and

(7.23)

The complex velocity in this case is

(7.24)

so that the velocity components are

To make sense of the flows corresponding to Eq. (7.21), consider thestreamlines formed by constant values of the stream function. In particular,for , the streamlines are determined by the condition that

, which means that they are the lines , …,and so forth. These are rays emanating outward from the origin, asillustrated in Fig. 7.1. By virtue of the periodicity of the sine functionappearing in , the flow patterns that occur in two adjacent regions sharinga common streamline are mirror images of each other. In somecases the physical interpretation of the flow is more meaningful if the twoadjacent sectors are considered together as one flow domain. In such casesit is useful to define the angle

(7.26)

which is also illustrated in Fig. 7.1.

Figure 7.1. Definition of the angles and formed by streamlines associated with that appear as straight linesemanating from the origin.

The velocity components in polar coordinates as computed from Eq.(7.23) using Eqs. (3.19) are

The first of these shows that alternates in sign from one radialstreamline to the next as varies in increments of . Moreover, issymmetric and is antisymmetric across the radial streamline separatingtwo adjacent regions, so their velocity fields are mirror images. With thisunderstanding, it is straightforward to gain a sense of the flow given by thestreamlines plotted in Fig. 7.2 for three particular cases, namely, when

, and , corresponding, respectively, to or . Depending on the sign of , Fig. 7.2(a) shows floweither symmetrically entering or leaving a corner along the axis; Fig.7.2(b) shows either flow impacting normal to a flat wall and turningsideways or else two columns of fluid meeting along a flat surface andbeing turned away from the wall; and finally, Fig. 7.2(c) shows either aflow impinging on a wedge-shaped object or else two separate streamsmerging in the wake of the wedge. In all cases the option exists of changingthe interpretation of these flows to focus on just half of the flows seen inthese images. In this case, Fig. 7.2(a) shows flow into a cavity along onewall and out on the other; Fig. 7.2(b) shows flow into a right-angledobstacle or corner; and Fig. 7.2(c) shows flow onto or off of a ramp.

Figure 7.2. Streamlines corresponding to : (a) ; (b) ; (c) .

For , Eq. (7.25) shows that , which is uniform flowover a flat plate. For , becomes a reflex angle, and there is flowover an edge, as is illustrated in Fig. 7.3 for or . For

, it does not make sense to consider two adjacent regions betweenzero streamlines.

Figure 7.3. Streamlines corresponding to .

The variation of the velocity along the wall for the various cases of Eq.(7.21) that have been considered can be obtained from Eq. (7.27a). For theflows in Fig. 7.2, the radial velocity varies according to ,respectively. Each of these is zero at . More generally, it is evidentthat the velocity at the point is zero for , infinite for ,and equal to for . Thus, for all cases of flow into a corner, the

velocity is zero at the corner itself, while for flow around a corner, as inFig. 7.3, the speed of the potential flow is infinite at this point. Whereverthe velocity is not bounded, the potential flow cannot be physical. In the

case, it is generally expected that the flow would separate fromthe surface downstream of the sharp edge, a phenomenon that cannot bemodeled without taking viscosity into account. The velocity singularity inthe potential solution reflects the sharp, unphysical turn of the flow aroundthe edge point.

The flows for each have a stagnation point on the wall, though itis only in the case of the flow impinging on a flat wall in Fig. 7.2(b), when

, that the stagnation point is not in a corner formed from twointersecting straight boundaries. When the stagnation point lies on thesmooth wall, it may be noticed that the streamline that intersects the wall atthis point does so at a right angle. In fact, this is a general result for anypotential flow. Streamlines that intersect a smooth wall at a stagnationpoint must do so at a right angle.

To prove this last statement, consider a streamline thatcoincides with a solid inviscid boundary formed by the surface andthat has a stagnation point at . Note that a stagnation point on anarbitrary smooth boundary can be brought to this form by using a localrectangular Cartesian coordinate system centered at the stagnation point. Itmay be shown (the details are left for Problem 7.3) that the leading termsof a local Taylor series expansion of in this case are

(7.28)

Consequently, close to the stagnation point, so that higher order terms inEq. (7.28) can be neglected, the streamlines corresponding to aregiven by and , which clearly intersect at a right angle at theboundary.

7.3 Forces in 2D Potential Flows

The flows that are conveniently created with the use of the complexpotential are irrotational and inviscid so they do not give rise to viscousforces acting either internally or upon bounding surfaces. However pressureforces do exist in potential flows, and it is of some interest to see how theycontribute to the forces on immersed bodies. Determination of the pressurefield generally requires obtaining a solution to the dynamics of fluid flowexpressed in the form of a system of differential equations that will not beconsidered until Chapter 13. However, for the potential flows of interesthere, it turns out that the pressure can be determined entirely fromknowledge of the velocity via an important result known as Bernoulli’sequation. In this, the quantity

(7.29)

where is the gravitational acceleration and is the coordinate in thevertical direction, is constant throughout potential flow fields. Thus, eventhough the velocity of a potential flow is known through kinematicalconsiderations alone, nonetheless, Bernoulli’s equation provides dynamicalinformation about the pressure field. We take advantage of this convenientresult here and defer derivation of Bernoulli’s equation until Section 16.2.

The knowledge of the pressure that comes from Bernoulli’s equation canbe used to determine the total pressure force acting on an arbitrary two-dimensional body in inviscid flow. Thus, consider a body defined by aclosed curve with general parametrization , as illustratedin Fig. 7.4. In this discussion, denotes the vertical direction, and thegravitational term in Eq. (7.29) is omitted. The force of the fluid outsideand adjacent to a small section of arclength making up the boundary isgiven by , where is an outward pointing normal. In this, a unitthickness out of the plane is assumed for the boundary but not indicatedexplicitly. Integrating over the surface gives the total pressure force on thebody as

(7.30)

The integration variable in Eq. (7.30) can be conveniently changed to byimitating the mapping used in deriving Eq. (2.20). First note that a unittangent vector to the boundary is given by

(7.31)

and thus a unit normal vector pointing outward (i.e., ) is

(7.32)

Assume maps to along the boundary. Substitute Eq. (7.32) into(7.30), change variables to , and use the identity (2.16) to give

(7.33)

Figure 7.4. A general 2D body bounded by a closed curve ).

By convention, the force in the direction to which the incoming flowpoints, the direction in the present case, is the drag, , whereas theforce in the vertical direction is the lift, . Thus

(7.34)

so that, according to Eq. (7.33),

(7.35)

and

(7.36)

Our goal now lies in converting Eq. (7.33) into the form of a contourintegral in the complex plane that will have the benefit of allowing for arelatively easy evaluation of lift and drag in many circumstances.

To develop the desired relation, first note that the and componentsof the boundary defining the body in Fig. 7.4 can be used to create a path,

, in the complex plane according to

(7.37)

The complex conjugate of satisfies

(7.38)

and thus

(7.39)

By definition, the contour integral with respect to of a complex function is given by

(7.40)

Defining pressure as a function of via the relation

(7.41)

substituting for in Eq. (7.40), and using the definitions Eqs.(7.35) and (7.36) gives

(7.42)

For the flow of interest here, the velocity is uniform and equal to farfrom the body. In regions of uniform velocity, the pressure is constanteverywhere and may be taken to be . In this case, pressure at thesurface of the body is given from Bernoulli’s equation as

(7.43)

where gravitational effects are omitted. The magnitude of the complexvelocity defined in Eq. (7.8) is

(7.44)

from which it follows that

(7.45)

Consequently, the Bernoulli equation implies that the pressure is given by

(7.46)

Because the integral of a constant over a closed contour is zero, the termscontaining and in Eq. (7.46) do not make contributions to Eq.(7.42) after substituting for . Consequently,

(7.47)

It proves useful in what follows to convert the integral in Eq. (7.47) toone with respect to . To accomplish this, first expand out Eq. (7.47)using the definition of to get

(7.48)

It was noted previously in Eq. (7.31) that

(7.49)

is a vector in the complex plane that is tangent to the boundary of the bodyin Fig. 7.4. At the same time, the surface of the body is coincident with astreamline, and the velocity field itself is tangent to the body.Consequently, the complex conjugate of Eq. (7.8) is a vector in the complexplane that is tangent to the velocity field, and

(7.50)

Because

(7.51)

is real, Eq. (7.50) with Eq. (7.51) implies that

(7.52)

is real. In this case, the fact that

(7.53)

for any two complex numbers means that

(7.54)

because real numbers are equal to their own complex conjugates.Substituting Eq. (7.54) into (7.48) and simplifying yields

(7.55)

a result known as the Blasius theorem. It shows that the drag and lift on abody in 2D potential flow can be computed by evaluating the indicatedcomplex contour integral.

The special advantage of Eq. (7.55) lies in the fact that it is in the form ofa contour integral over a complex function. Such integrals can be evaluatedrelatively easily in many cases using a fundamental result of complexanalysis known as the residue theorem (Ahlfors 1979) to the effect that

(7.56)

where the complex numbers are the “residues” of the function and the summation is limited to that collection of residues contained withinthe area swept out by the contour. Residues, if they exist, are associatedwith singular points of that have the specific behavior

(7.57)

Assuming that are the collection of points satisfying Eq. (7.57)within a contour of interest, can be expressed in the form

(7.58)

where is a function that is free of residues. Equation (7.56) comesabout because it may be shown that the contour integral of is zero,while integration of each of the terms in the sum in Eq. (7.58) gives thevalue . Because Eq. (7.55) ties the drag and lift to a contour integral

of , it is only necessary to find the residues of this expression tofind the forces on a body. These ideas are illustrated in the next section.

7.4 Inviscid Flow Past a Cylinder

The complex potential for frictionless, incompressible flow past a cylinderwas given in Eq. (7.18). From this the pressure on the cylinder surface canbe computed from Eq. (7.46) and the total forces from Eq. (7.55). Thus acomputation yields the complex velocity

(7.59)

from which it is found from Eqs. (7.11) and (7.45) that

(7.60)

Substituting Eq. (7.60) into Eq. (7.43) gives the pressure coefficient as

(7.61)

A contour plot of Eq. (7.61) is given in Fig. 7.5, which shows how thepresence of the cylinder perturbs the otherwise constant pressure field.Lines of are seen to come in from the far field and intersect withthe boundary at four points where . In the sector in front of thecylinder, the pressure rises to a maximum. On either side where the flow isaccelerated around the cylinder, the pressure decreases below the ambient.The pressure in the back shows a recovery to the ambient that is symmetricto that in the front.

Figure 7.5. Normalized pressure contours for the flow past a cylinder ofunit diameter. Solid lines, ; dotted lines, .

The pressure on the surface of the cylinder is of particular interest.Evaluating Eq. (7.61) at gives

(7.62)

which is plotted in Fig. 7.6. It is seen that is a maximum at the frontand rear stagnation points where and reaches a minimum at thesides where . Also included in the figure is a typical pressureresult for a high-speed viscous flow taken from physical experiments(Roshko 1961). Though the inviscid solution is for the most part physicallycorrect over the front half of the cylinder, the fact that this pressure field issymmetric fore and aft means that it is entirely unphysical in the rear of thecylinder, where the experiment shows a large region of approximatelyconstant pressure. The latter is a result of the way in which vorticitygenerated at the surface by viscosity separates into the wake of the cylinder.

Figure 7.6. Pressure coefficient in cylinder flow. Solid line, inviscid flow;circles, measured pressure coefficient in high-speed cylinder flow (

).

Without asymmetry in the streamwise direction, the inviscid pressurefield displayed in Figs. 7.5 and 7.6 cannot produce a net pressure force inthe streamwise direction. Because pressure is the only force acting on thesurface of the cylinder in the inviscid analysis, it means that a zero dragforce is predicted. In contrast, the experimentally determined pressuredistribution is considerably higher in the front than the back so that therewill be a nonzero drag force. Besides the drag due to the pressuredistribution, drag will also arise in the viscous shear stress at the surface.

It is instructive to apply the force law Eq. (7.55) to the example ofcylinder flow. Thus, from Eq. (7.59),

(7.63)

from which it is seen that there are no residuals to be concerned with andcertainly none within the interior of the cylinder. According to Eq. (7.55),this means that there is no drag or lift.

The cylinder flow can be generalized slightly by adding a point vortexwith circulation to the center of the cylinder. This creates a swirlingflow in the clockwise direction around the cylinder, whose circularstreamline at the cylinder surface combines with that from the cylinderflow to maintain the boundary of the cylinder as a streamline, even if theflow is changed elsewhere. Formally, the potential Eq. (7.15) for the vortexis added to that of the cylinder to yield the complex potential

(7.64)

A calculation then gives

(7.65)

By inspection, it is seen that Eq. (7.65) has as a residue at ,which is within the contour formed by the cylinder surface. A calculationusing Eq. (7.55) then reveals that

(7.66)

and after simplifying the right-hand side, it is seen that, as can beanticipated by symmetry, even with the vortex present, the drag remainszero, but now there is a nonzero lift given by

(7.67)

This result is an example of the Kutta-Zhukhovsky lift theorem, whichstates, in essence, that the lift force to be expected from a body isproportional to the net circulation of the flow around it.

The prediction of zero drag (known as d’Alembert’s paradox) and the liftformula given in Eq. (7.67) are general results for the potential flow pastarbitrary two-dimensional bodies for which singular flows, if they arepresent, such as due to sources/sinks, point vortices, and doublets, arecentered at points within the body. In this case, because of the residuetheorem, the contour integral in Eq. (7.55) is unchanged if it is carried outon a path far from the body. Moreover, it is possible to derive a generalexpression for valid at large (Batchelor 1967; Kundu &Cohen 2004) in which the only term that has a nonzero residue in theBlasius formula is that having the form of a point vortex in which thecirculation corresponds to the net circulation around the body. Using theseresults Eq. (7.67) can be shown to be independent of the exact shape of thebody.

Examination of the streamline patterns associated with Eq. (7.64) fordifferent values of provides an explanation for the appearance of liftwhen . As shown in Fig. 7.7, three different flow patterns occurdepending on the strength of rotation. For the fore and aftstagnation points shift symmetrically downward toward the lower half ofthe cylinder. At , there is just a single stagnation point at thebottom of the cylinder, and for , the sole stagnation point hasmoved off and below the cylinder remaining on the symmetry axis.Because the maximum pressure has to appear at stagnation pointsaccording to the Bernoulli equation, there is an asymmetry to the pressureforce that provides lift.

Figure 7.7. Streamlines in cylinder flow with circulation showing theeffect of on the stagnation points: (a) ; (b) ; (c)

.

Equation (7.67) is of particular consequence in airfoil theory, whoseconnection to what has been derived in this chapter depends on extendingthe potential flow modeling to airfoil shapes by a technique in complexanalysis known as conformal mapping. In essence, simple geometric shapessuch as the circle are mapped to more complex shapes such as an airfoil,and the velocity field of one gets mapped into the velocity field of the

other. In this case, Eq. (7.67) becomes a predictive tool for determining liftwhen an independent means of determining is available. For example, inthe case of the classic Joukowski airfoil (Batchelor 1967), the requirementthat the flow separates smoothly from the rear of the airfoil (the Kuttacondition) can be used to find the value of . Even though this kind ofanalysis is inviscid and thus fundamentally limited, potential flow theory,particularly for streamlined bodies, can be useful for determining boundaryconditions just off the surface of the body as part of a viscous computationof the near-wall flow. This kind of calculation is illustrated in Section 20.3.

Problems

7.1 Show that Eq. (7.17) generalizes to if thedoublet is oriented in the direction.

7.2 Plot the variation of and on a circular arc of radius spanningthe flow between straight boundaries in Fig. 7.2(a). Compute the flux offluid into and out of the corner.

7.3 Derive Eq. (7.28) for two-dimensional irrotational flow at a stagnationpoint on a smooth solid surface. Take into account the properties ofpotential flow, the geometrical constraint, and the fact that the point inquestion is a stagnation point.

7.4 Compute the stream function and velocity potential corresponding tothe complex potential in Eq. (7.64) and show that the stream function isconstant for .

7.5 In Problem 6.5 the stream function contained the term . Show that the flow corresponding to this term is

incompressible and so it may be taken to be a stream function in its ownright. Show that this flow is irrotational, and then compute its potential .Using and , determine the complex potential, and note that it is

(i.e., it has the form of a quadrupole).

7.6 Compute the pressure coefficient for the cylinder flow containing apoint vortex at the origin. Make a contour plot corresponding to the threedifferent cases shown in Fig. 7.7. Indicate the point of maximum pressurein each plot.

7.7 Consider a vortex of strength in two dimensions located at in the plane above a solid inviscid boundary at . In

addition to the point vortex, there is a uniform flow of speed . Calculatethe pressure on the solid surface. Compute the locations of the pressuremaximum and minimum on the surface.

7.8 Consider the upper half of a solid cylinder of radius and density lying on a flat surface in inviscid flow obeying Eq. (7.18) with velocity andpressure far upstream given by and , respectively. Because thepressure is a minimum on the top surface of the cylinder, if is largeenough, a lift force can be created that causes the cylinder to lift off thesurface. Find the speed at which this occurs, assuming that is the densityof the fluid.

7.9 Use the circle theorem to verify the result in Problem 5.10 concerningthe inviscid flow due to a source outside a cylinder of radius .

7.10 Consider a doublet located at the point on the positive axisof the complex plane and oriented such that it has potential

, . Assuming that , use the circletheorem to derive the potential for this doublet in the presence of a solidcircle of radius centered at the origin. Use the Blasius theorem (Eq. 7.55)to show that there is a nonzero drag force acting to move the cylindertoward the doublet.

7.11 Find the stream function corresponding to the complex potential . Use MATLAB or equivalent to compute the

velocity components. Make a contour plot of and a quiver plot of thevelocity field in the upper half-plane . What is this flow field?

7.12 Consider the complex potential . To get thestream function, solve for as a function of , and from this develop anequation for . Plot lines of constant , and show that these may beinterpreted as flow entering or leaving the half-plane through a slitbetween . Compute the complex velocity, and from this show that

across the opening. Determine the normal velocity across theopening, plot it, and show that the flow is leaving the domain through theslot. Note that, unlike Fig. 5.5 and Problem 5.13, the velocity across theopening in this problem is not specified a priori.

7.13 Let a complex potential be defined implicitly by the relation(Helmholtz 1868)

(7.68)

Use this to derive equations for and as functions of and . Usethese to show that is the streamline corresponding to andthat the streamlines corresponding to are the lines

, respectively. Eliminate from the equations for and and get an equation for . For fixed values of , this gives

streamlines in the form for . Show that the streamlinesare reflections across of the corresponding streamlines. Fora number of values of such that , evaluate and plot.Show that the streamlines indicate flow in or possibly out of a channel ofwidth projected into the plane. By differentiating Eq. (7.68), derive theformula

(7.69)

for the component of velocity. Show that for and for so that Eq. (7.68) represents flow entering the plane

through a channel.

7.14 Consider the complex potential defined via . Show that thisrepresents flow with elliptical streamlines around the line .Derive explicit formulas for the velocity components , andmake a quiver plot in the first quadrant. What can you say about thevelocity at ?

8Accelerating Reference Frames

Newton’s law governing the motion of the center of mass of an extendedrigid body of mass undergoing an acceleration and acted upon by asum of forces has the form

(8.1)

for inertial, that is, nonaccelerating, observers. By an “observer” is meantsomeone with a particular frame of reference from which the dynamics ofan object are measured and the laws of motion are expressed. In severalsubsequent contexts, our interest will lie in the study of flows that are bestconsidered from the point of view of an accelerating or noninertialobserver for which the laws of motion based on Eq. (8.1) will not applyunless an appropriate modification is made. This chapter considers theformal steps that need to be taken to develop an appropriately generalizedexpression of Newton’s law.

To account for acceleration, it is necessary to consider the ways in whichdifferent observers express and compare the values of vectors and tensorsas seen from their own frames of reference. With this knowledge, the lawsof motion can be expressed in a way that is consistent with the experienceof all observers in the sense that they arrive at the same physical laws withdifferences in expression attributable to their varying frames of reference.

An essential aspect of being an observer, besides having a specificlocation in space, is having a sense of orientation that goes with thereference frame. In fact, all the flows considered thus far, whethermentioned explicitly or not and regardless of which particular coordinate

system was used in their analysis, had a set of directions withcorresponding coordinates that established a sense of orientationto the way in which the flow is viewed. In general, both the orientation andthe origin of the reference frame may differ from one observer to the next.An observer can be accelerating because the origin of his reference frameis accelerating or because his orientation is changing in time. For example,a stationary observer rotating at constant angular velocity around an axisis accelerating because of her changing orientation. Whether theacceleration of an observer is in the rectilinear movement of his origin, orin the rotation of his defining basis vectors, such observers are noninertial,and for them, Eq. (8.1) must be modified in some special ways if it is toreflect the observer’s view of the dynamics.

The route toward obtaining the laws of motion for noninertial observersis by finding a relationship between the acceleration of objects as seen byinertial and noninertial observers. When incorporated into Newton’s law,the resulting relation then applies to the dynamics of an object as viewedby a general noninertial observer. The acceleration law, whose derivationis the focus of this chapter, will be derived as a consequence of firstconsidering how the position and then the velocity of a moving object canbe expressed consistently among the various frames of reference ofarbitrary observers.

8.1 Orientation

Consider two observers with the first one being an inertial observer andthe second one free to be either inertial or noninertial. For this discussion,the directions of the unit basis vectors associated with and perceived byobserver 1 and fixed in their reference frame are taken to be ,and the basis vectors associated with and perceived by observer 2 andfixed in their reference frame are taken to be . It should beemphasized that although the sets of basis vectors associated with eachobserver generally do not point in the same directions, nonetheless each ofthe observers has the same relationship toward her own set of basis vectors

as the other. Of course, each of the observers will see the other observer’sbasis vectors pointing differently than his own.

Because the observers base their descriptions of vectors and tensors onthe orientation of their basis vectors as defined within their referencesframes, they measure the components of any given vector or tensordifferently from each other. For example, as illustrated in Fig. 8.1 for twodimensions, a vector is shown together with the coordinate axes andbasis vectors for two different observers. How appears to the twoobservers is given in Fig. 8.2, with denoting the vector as measured byobserver 1 and the vector as measured by observer 2. It is clear that theobservers form very different impressions as to the orientation of , withobserver 1 measuring it above the axis and observer 2 measuring itbelow the axis. Our interest now lies in seeing how these differingrepresentations of the vector can be reconciled.

Figure 8.1. Observer orientation and a vector .

Figure 8.2. Interpretations of the vector : (a) observer 1; (b) observer2.

A formalism for relating to can be deduced by reconsidering in anew light some of the results in Section 1.5 concerning the effect of basisvectors on the components of vectors measured by a single observer. Thusobserver 1 can measure the basis vectors associated with the frame ofreference of observer 2 and then use these as a second set of basis vectorswithin their own frame of reference. As was shown previously inEq. (1.45), a rotation tensor can be found that maps between the twosets of basis vectors, specifically,

(8.2)

Visually, is a tensor that rotates until it is coincident with (seeFig. 1.5).

In the case of an arbitrary vector as considered above, if themeasurements of this vector by the two observers are plotted togetherwithin the reference frame of observer 1, the result appears as illustrated

in Fig. 8.3. In this, a rotation by the same tensor appearing in Eq. (8.2)maps to , as in

(8.3)

The consistency of this result with Eq. (8.2) becomes apparent if it isrealized that in both relations, operates on the vector as seen byobserver 2, yielding the vector seen by observer 1. In the case of Eq. (8.2),the vector being observed is the basis vector of observer 2: observer 1 seesthis vector as , while observer 2 sees it as . The mapping by compensates for the different orientations of the observers that makestheir measurements differ from each other.

Figure 8.3. Vector as seen by the two observers, as depicted in thereference frame of observer 1.

In the reference frame of observer 1, can be expressed viacomponents pertaining to both sets of basis vectors. In the first instance, itis

(8.4)

so that are the components as measured using the basis vector . Inthe second instance, as was done previously in Eq. (1.52), can bedefined so that

(8.5)

where the are the components using the basis vectors . MultiplyingEq. (8.5) by and applying Eqs. (8.3) and (8.2) gives

(8.6)

as the vector measured by observer 2 and written in terms of the referenceframe of observer 1.

It is straightforward to show that the previous results for the mappingbetween components given in Eq. (1.53) apply equally well in the presentsituation. Consequently, it may be shown that

(8.7)

or, conversely,

(8.8)

These relations connect the components of the measured vector as seen bythe two observers. Extension of these mappings for the case of tensorcomponents can be derived in the same form as in Eq. (1.54) so that

(8.9)

where , defined according to

(8.10)

is the tensor measured by observer 2 and expressed in the reference frameof observer 1. Switching to direct notation, Eq. (8.9) can be written as

(8.11)

which is a mapping between the tensors measured by the two observers.

8.2 Position Vector

The first step toward understanding how acceleration is seen differently bydifferent observers is to consider how two observers report the position ofa moving object such as a small material element of fluid. Figure 8.4shows an object moving along a path that is seen by two differentobservers. The first observer, the inertial observer, measures the objectposition at a time as the vector extending from his origin to theobject. Similarly, observer 2 measures the object as being at the position

extending between her origin and the object. The vector is theposition of the origin of observer 2 as measured by observer 1.

Figure 8.4. The connection between position vectors of a moving objectas reported by observer 1, , and observer 2, ; is the positionof the origin of observer 2 as seen by observer 1.

The vectors and are measured by observer 1 in accordance with thefixed basis vectors of his reference frame. Alternatively, as measuredby observer 2 is in terms of the possibly time-dependent basis vectors ofher reference frame. maps to its measured form in the referenceframe of observer 1, assuming that the origins of the observers coincide.When this does not occur, as shown in Fig. 8.4, it is necessary to take intoaccount the differing origins of the observers so that

(8.12)

In this relation, are in terms of the reference frame and basisvectors of observer 1. If , Eq. (8.12) reduces to (8.3). Note, aswell, that by allowing to be time dependent, provision is made forthe most general case wherein the orientation of the second observerchanges in time.

In the following, it will be seen to be useful to make the definition

(8.13)

so that Eq. (8.12) can be written alternatively as

(8.14)

The difference between this equation and Eq. (8.12) is that all of its termsare measured from the same orientation, namely, that of observer 1. Inparticular, unlike , the orientation underlying is fixed in time aslong as observer 1 is an inertial observer. This will be seen to be animportant consideration when arriving at the form of the equations ofmotion for noninertial observers.

8.3 Velocity

A relation linking velocities seen by the observers can be obtained bytaking a time derivative of Eq. (8.12), yielding

(8.15)

where the over-dots signify time differentiation and, for simplicity, thetime dependence of the variables is not indicated explicitly. The term onthe left-hand side of Eq. (8.15) is the velocity of the moving object as seenby observer 1 and will henceforth be denoted as . Similarly, ,appearing in the last term in Eq. (8.15), is the velocity measured in thereference frame of observer 2. By being multiplied by , it is broughtback to the basis vectors of observer 1 and hence is consistent with theother terms. The vector in Eq. (8.15) is the velocity of the origin ofobserver 2 as seen by observer 1.

The second term on the right-hand side of (8.15) can be recast usingEq. (1.50) as

(8.16)

Moreover, taking a time derivative of Eq. (1.50) yields

(8.17)

and in view of the identity for any two tensors ,Eq. (8.17) gives

(8.18)

so that it is seen that is skew. As shown in Section 1.5, this meansthat there exists an axial vector such that

(8.19)

for all vectors . Incorporating this into Eq. (8.16), it is found that

(8.20)

so the velocity transformation law is deduced to be of the form

(8.21)

Note that, once again, the rotation tensor appears in front of in thesecond term on the right-hand side so as to allow the entire relation to beconsistently in terms of the basis vectors of observer 1. This thenmotivates the definition

(8.22)

and an alternative form of Eq. (8.21) is

(8.23)

All quantities in Eq. (8.23), including and , are vectors as seen withthe orientation of the first observer.

According to Eq. (8.21) or (8.23), the velocity of a moving object seenby the two observers differs because of the relative movement of theirorigins, as contained in the first term on the right-hand side, as well as thefact that the second observer rotates at the instantaneous rate , as seenby the first observer. As an example of what this relation means, considerthe case of a dot affixed to a vinyl record revolving on a turntable in the

plane with constant angular velocity

(8.24)

in radians/second around the vertical axis, as illustrated in Fig. 8.5.Observer 1 is fixed at the center of the turntable and can watch the dot spinaround. A second observer has her origin also at the center of the disk butturns with the turntable. Since the origins of the two observers areidentical and never change, it is the case that .

Figure 8.5. Rotating and fixed observers on a turntable.

To the observer attached to the record, the dot is located at a fixedlocation and is stationary so that . From the point of view of thenonrotating observer, the dot is at the continually changing position

whose rotational movement is accounted for by a steadychange in . Because the turntable spins at a constant rate , can begiven explicitly as

(8.25)

which is an appropriate generalization of Eq. (1.47) for rotation in threedimensions around the axis. Here due to the constancy of .

It is expected that given in Eq. (8.24) should be the rotation vectorassociated with . This is indeed the case, and it is instructive to verifythis result explicitly. Thus, by direct calculation, it may be shown that

(8.26)

in which case,

(8.27)

for any vector . In fact, with given by Eq. (8.24), a calculation revealsthat

(8.28)

which is identical to Eq. (8.27). Clearly it has been proven that definedin Eq. (8.24) is the rotation vector associated with .

Putting these developments together, the standard result is obtained tothe effect that at any instant of time

(8.29)

where , the position of the dot as measured by observer 1 pointsradially outward and revolves in time about the axis. The cross productin Eq. (8.29) between a vector normal to the plane of motion and a radialvector means that the velocity is in a direction tangent to the circular path

at whatever position the dot occupies. The magnitude of the velocity of therotating point is , where .

8.4 Acceleration and Fictitious Forces

Turning attention now to the derivation of the acceleration law, a timederivative of (8.21) yields

(8.30)

As before, is the acceleration measured in the reference frame ofobserver 1, is the acceleration measured in the reference frame ofobserver 2, and is the acceleration of the origin of observer 2 asmeasured by observer 1. Some of the terms on the right-hand side ofEq. (8.30) can be simplified, as done previously. Thus, substituting as before, the term

(8.31)

and similarly,

(8.32)

The result is

(8.33)

Similar to and , it is helpful to define

(8.34)

as the acceleration measured using the orientation of observer 1 located atthe origin of observer 2. Using Eq. (8.34) and the previous definitions,Eq. (8.33) becomes

(8.35)

Before considering the meaning of the terms in this relation, it is useful toplace it into the context in which it will later be used in the force law fornoninertial observers.

The importance of Eq. (8.35) is in connecting the acceleration of aninertial observer for which Newton’s law Eq. (8.1) applies to theacceleration for a noninertial observer. Thus substituting for inEq. (8.1) using (8.35) Newton’s law is generalized to noninertial observersin the form

(8.36)

in which all vectors are in terms of the fixed, inertial orientation ofobserver 1. By substituting for , and using Eqs. (8.13), (8.22),and (8.34), respectively, Eq. (8.36) expresses Newton’s law explicitly interms of the primed vectors that are measured with respect to the frame ofreference of observer 2. However, such an equation is made unwieldy bythe presence of , which accounts for the changing orientation betweenthe observers.

It is evident from Eqs. (8.1) and (8.36) that although the inertial andnoninertial observer both experience the same effect on accelerationcaused by the sum of forces, the noninertial observer must also contendwith an array of additional forcelike effects that originate in the many

terms in the acceleration formula. Although these may be felt by thenoninertial observer as additional forces, they are in truth fictitious forcesbecause they arise only as a consequence of the acceleration associatedwith the noninertial reference frame of the observer.

Among the fictitious forces, the term containing accounts forchanges in the rotation vector over time and is of narrow interest. Morecommonly encountered in physical applications are the centrifugal forceterm

(8.37)

and the Coriolis force term

(8.38)

which naturally arise in rotating coordinate systems, such as the diskconsidered in the previous section. To illustrate the physical meaning andorigin of these fictitious forces, consider a small object of mass fixedat a point on a disk rotating according to . To the rotatingobserver, the point is measured to be at the fixed location . For the fixedobserver, it is at the rotating point . Letting where

, a calculation gives so that

(8.39)

points radially outward with magnitude at whatever position theobject occupies. Thus the rotating observer perceives an outwardcentrifugal force at all times that must be balanced with an equal andopposite force toward the axis to maintain the stationary position of theobject. Of course the fixed observer does not sense a centrifugal force.Instead, from his perspective, the force exerted toward the axis, known as

the centripetal force, is what is required so that the object travels in acircular orbit instead of flying off in a straight line.

The Coriolis force arises when an object on the disk changes positionwithin the rotating coordinate system. For example, suppose that theobject on the turntable were to move radially outward at velocity , asseen by the rotating observer. Then is the outward pointingvelocity at any instant of time as seen by the fixed inertial observer. TheCoriolis force in this case is

(8.40)

From the point of view of the rotating observer, this is a force that pushesan object sideways in the direction whenever it attempts to moveoutward. Thus, to be able to move radially outward on a turntable, as seenby the rotating observer, a compensating force in the direction of rotationmust be supplied. This phenomenon is a natural consequence of the factthat the velocity of the turntable relative to the fixed ground increases withradial distance, and an accelerative force is required to maintain theposition of the object on a radial line as it moves outward.

Centrifugal and Coriolis forces are an important part of the analysis ofgeophysical flows, such as the motion of the atmosphere on the rotatingearth. While such flows will not be considered here, we later develop, inChapter 13, the form of the governing fluid equations suitable tononinertial observers by incorporating the developments of this chapter.The present results are also used in the next chapter to study flows that arestationary only from the viewpoint of accelerating observers and inSection 17.3 to study the operation of a simple lawn sprinkler.

9Fluids at Rest

While fluids are most often encountered in a state of motion, there aretimes and circumstances when from the perspective of a particular observerthe fluid is at rest. In this case it can often be relatively easy to determineattributes of the fluid, such as the pressure distribution, that are needed insuch practical applications as determining the force of water on a dam orsubmerged body. Several standard problems where fluids are at rest areillustrated in this chapter, including, in the last two sections, problemswhere the fluid is stationary only from the perspective of an acceleratingobserver. In these cases, the results of the previous chapter play animportant role in the analysis.

9.1 Forces in a Fluid at Rest

The essential importance of Newton’s laws of motion to computing fluidflow was noted in Eq. (1.10). When the fluid is not moving, there is nochange to the momentum of fluid elements, so the left-hand side ofNewton’s law is zero and the relevant physics consists of a zero sum of theforces acting on fluid elements.

Consider a material fluid element such as was discussed in Section 1.4and is now illustrated in Fig. 9.1. The forces acting on this fluid elementconsist of body forces, such as gravity, that act everywhere over the volumeoccupied by the element as well as surface forces that act at the boundingarea of the volume. For a stationary fluid, the only surface force acting onmaterial elements is the pressure, and this must act normally, because if itdid not, then there would be an implied shearing force that would lead tofluid motion as discussed in Section 1.1. How fluid motion gives rise tocontributions to the surface forces other than pressure will be considered

later in Chapters 11 and 12. In the present case, it is only the pressure forcethat balances with the body force so as to result in a nonmoving fluid inequilibrium.

Figure 9.1. Forces on a static material fluid element of mass .

Letting denote the gravitational acceleration (taken in this text toalways be with constant), the total gravitational force onthe material element is

(9.1)

where is the density of the fluid and is the volume occupied by thematerial fluid element. Because is constant, Eq. (9.1) gives

(9.2)

where is the mass of the fluid element.

At any point on the surface of the material element, a force is exertedon it by the fluid outside the element. This force has its origin in themolecular structure of the fluid, whether it be a momentum flux producedby molecules in a gas or strong localized forces between molecules in aliquid. Within the framework of the continuum model, regardless of itsexact molecular cause, the surface force/area can be denoted by a vector

in which allowance is made for the fact that the local surfaceforce depends on the orientation of the surface by including the unit normalvector in the arguments of . Figure 9.2 illustrates the generalrelationship between and . In Chapter 11 it is shown that the apparentcomplexity of , in which it is nominally a vector function of sevenvariables, is greatly simplified because a tensor field exists, called thestress tensor, , having the property that

(9.3)

Thus the dependence of the surface force on the directionality of thesurface is readily accommodated, leaving only the determination of as aprerequisite to being able to compute surface forces.

Figure 9.2. Local force on the surface of a material fluid element.

While a complete discussion of , including its derivation, is the subjectof Chapters 11 and 12, it suffices for the present to mention that the stresstensor has the form

(9.4)

where is the pressure and is the identity tensor. Tensor , referred toas the deviatoric part of the stress tensor, is nonzero only in moving fluidscontaining relative motion. The tensor accounts for deviations awayfrom the isotropy of the term. For the present restriction tostationary fluids, it is the case that

(9.5)

and the surface force in Eq. (9.3) reduces entirely to a pressure force actingnormal to the surface according to

(9.6)

For a moving fluid, additional effects appear in that are likely to meanthat is not aligned with . The minus sign in Eq. (9.6) is consistent withthe expectation that the pressure on a surface is toward the surface,opposite to the direction of . Moreover, the pressure itself is independentof . If it were not, then a small fluid element at that point would besubject to a net force that would cause its movement.

The total surface force acting on a material volume can be computedusing Eq. (9.6) as

(9.7)

The force balance on a stationary material fluid element is thus

(9.8)

Further progress in analyzing this balance can be had by converting thesecond term into a volume integral through the identity

(9.9)

which is a variant of the divergence theorem known as the gradienttheorem (Hildebrand 1976). In this case, Eq. (9.8) becomes

(9.10)

Because the material element is arbitrary, the only way that Eq. (9.10) canbe true in all cases is for the integrand to be zero. Thus we have establishedthat the force balance for a stationary fluid as seen by an inertialobserver – is

(9.11)

The argument pursued here, in which terms are collected as a volumeintegral over an arbitrary material element, thus implying that theintegrand must vanish, is one that will be employed often in derivingsubsequent dynamical laws of fluid dynamics.

In view of the definition of , the pressure can only be expected to varyin the vertical direction according to the relation

(9.12)

Depending on the functional dependence of with altitude, this equationcan be integrated to give the pressure. For example, the simple case ofconstant density yields

(9.13)

where is a known reference pressure. In Eq. (9.13), thequantity is known as the specific gravity and is often denoted by thesymbol . Because is positive upward, it is clear that the pressureincreases linearly with depth of the fluid or, alternatively, decreases withaltitude. Choices of density for which there is a height dependence yieldother variations of pressure with vertical distance. Among the latterpossibility is having the density related to pressure through an equation ofstate. Some examples of this are given as problems at the end of thechapter.

For the common case of a container of liquid such as water surroundedby a gas such as air, the pressure across the top surface of the liquid isequal to that of the adjoining air. The pressure of the air is generallyreferred to as being atmospheric pressure. Continuity of the pressure at theinterface between the liquid and air assuming that effects such assurface tension can be ignored is necessary at all times, becauseotherwise the pressure jump would cause an immediate violent accelerationof local fluid elements. It may also be noted that the pressure change withinthe air is usually much less than that within the liquid because of thecomparatively lower density of air. Thus the assumption of a constantatmospheric air pressure above the liquid is a reasonably convenientboundary condition for determining the pressure within the liquid.

9.1.1 Micromanometer

Knowledge of how the pressure varies in a fluid at rest can be used toexplain the working of devices such as the micromanometer shown in Fig.9.3, whose purpose is to measure the pressure of the air in the bulb onthe left-hand side. The micromanometer uses two equally sized reservoirsof cross-sectional area containing equal amounts of a liquid withspecific gravity . Between the reservoirs and lying in a pipe thatconnects them of cross-sectional area is a heavier fluid withspecific gravity . The right-hand side of the system is opento the atmosphere with pressure at the free surface.

Figure 9.3. A micromanometer. The fluid in the reservoirs has specificgravity , which is just slightly smaller than the corresponding value in the lower connecting pipe.

When differs from atmospheric pressure, the levels of the two fluidsin the reservoirs are no longer equal, as seen in Fig. 9.3 for the case when

. The reservoir geometry and the choice of liquids is designed totranslate small pressure differentials into large, easily measureddifferences in the height of the lower fluid column between the two sides,as is now considered.

Analysis of the manometer proceeds by connecting the pressure on oneside of the device to the other. Taking as constant in the region abovethe left reservoir, the pressure increases proportional to over a depth,say, , until the interface between the two fluids is encountered. Thepressure at this location is identical to that shifted horizontally to points

under the right-side reservoir. From here, ascending vertically, the pressuredecreases through the column of length of the fluid until theinterface between fluids is encountered on this side. Continuing to ascend,the pressure decreases through a column of the fluid that, compared tothe column on the left-hand side, is decreased by and increased by until atmospheric pressure is encountered at the top of the reservoir on theright-hand side.

Equating the two separate computations of the pressure at the commonpoint at the bottom of the fluid on the left-hand side results in theexpression

(9.14)

It may be noticed that the terms depending on the unknown length cancel from both sides so it is not necessary to know its value. Thequantities and are related by mass conservation of the two fluids.Thus whatever differential in the amount of fluid there is in the tworeservoirs must equal the volume of the fluid differential between the twosides of the pipe containing the heavy fluid . This means that

. The final result is then

(9.15)

For the case where and so the two fluid havealmost the same density, Eq. (9.15) shows that small pressure differentialscan be associated with large changes in , making the instrument verysensitive.

9.1.2 Force on a Dam

Having information about the pressure allows for the computation of theforces on submerged surfaces. One such common situation, as illustrated in

Fig. 9.4, is determining the forces on a dam. Here it is assumed that thedam is 2D and that its boundary is given parametrically by the curve

, . Formally, the force on the dam is given by

(9.16)

where the integral is in terms of arclength and is a normal vector tothe boundary curve. The integration variable in Eq. (9.16) can beconveniently changed to , as was done in deriving Eq. (7.33), giving

(9.17)

Figure 9.4. Forces on a dam.

Because is a function of only, the first term on theright-hand side of Eq. (9.17), representing the force in the horizontaldirection, can be simplified by a change of integration variable from to

. In this case it is found that

(9.18)

whose value is clearly independent of the curve . This result isexpected because it is a fact of experience that asymmetric bodies do notexperience a net horizontal force when submerged in a fluid. In contrast, itis evident that the vertical force on the surface given by the second term inEq. (9.17) does depend on the details of the particular surface in question.For any fluid for which the pressure can be determined, it isstraightforward to evaluate Eq. (9.18).

9.2 Buoyancy

One important consequence of the change in pressure with vertical distancein a fluid is that any body that is fully or partially submerged within it feelsa greater pressure on its bottom than its top. This pressure differencecreates a net upward force on the body that is referred to as the buoyancyforce. In many circumstances, objects immersed in fluid are in anequilibrium state in which they are at rest with respect to the fluid. Forexample, an object of mass placed into a container of water, dependingon its weight relative to the magnitude of the buoyancy force, may floateither fully or partially submerged without movement, or it may lie at reston the bottom of the container. The latter case, of course, occurs when thebuoyancy force is insufficient to counterbalance the weight of the object. Inanother example, a helium balloon released into the atmosphere will travelupward until it reaches an equilibrium position where its weight is balancedby that of the local buoyancy force.

It may seem at first sight that the buoyancy force should be difficult todetermine for objects of a general shape, put it turns out that it is often easyto compute. In fact, using Eqs. (9.9) and (9.11), the buoyancy force for abody occupying a volume with outward normal on its surface, , is

(9.19)

where is the mass of the displaced fluid. Thus the buoyancyforce points opposite to gravity and is equal to the weight of the displacedfluid, a result that is known as Archimedes’ principle. As expected, Eq.(9.19) shows that there are no lateral forces on the submerged body. Thedisplaced fluid may be all of one fluid, as in the case of an objectsubmerged in water or a balloon in the atmosphere, or made up of liquidand the gas above it for an object floating on the surface of the liquid (seeFig. 9.5(a)). Usually the weight of the displaced gas is negligible comparedto that of the displaced liquid and may be neglected when computing thebuoyancy force.

Figure 9.5. A fully or partially submerged object of mass canachieve static equilibrium by (a) floating, when , where is themass of the displaced fluid; (b) sinking until a solid surface is reached,when ; and (c) rising until the object is restrained by a cable orother solid object, .

When an object immersed in fluid is floating either fully or partiallysubmerged, buoyancy and gravity forces are in balance, in which case

(9.20)

or simply . It is clear from this that a helium balloon floats at analtitude where the density of air is approximately equal to that of thehelium within the balloon.

In the case , equilibrium is reached when the object eithersinks or rises to a point where its motion is constrained by contact with asolid boundary. This could be a rock at the bottom of a pool, as in Fig.9.5(b), or a helium balloon tethered by a string, as in Fig. 9.5(c).

9.3 Accelerating Fluids at Rest

In some cases a fluid appears to be stationary from the point of view of anaccelerating observer. To be able to analyze such problems, it is necessaryto adapt Eq. (8.36) to a fluid at rest. To accomplish this, note that thedifferential balance of forces previously derived in Eq. (9.11) lacked anacceleration term due to the assumed absence of fluid motion. For the moregeneral case of a noninertial observer, the missing acceleration term needsto be replaced using Eq. (8.35) under the assumption that ,because the noninertial observer sees the flow as stationary. The result is

(9.21)

which modifies Eq. (9.11) by the inclusion of the rectilinear accelerationand centrifugal force terms. Neither of these effects are incompatible withthe possibility that the fluid lacks relative motion.

9.3.1 Accelerating Fish Tank

An example of a stationary fluid flow problem where the rectilinearacceleration term plays a role is that of an open fish tank of length carried in the back of a pickup truck, as shown in Fig. 9.6(a). Initially,before the truck begins to move, the fluid is at rest with uniform depth that is a distance below the height of the tank walls. Under a constantacceleration in the direction, , the water in the tank reaches a static

configuration, as seen by an observer traveling with the truck. Note that, inthis example, inertial observers cannot help but consider the truck to bemoving, so for them the fish tank contains moving fluid. The goal here is tofigure out the equilibrium water depth in the tank and the condition on that guarantees that water does not spill out of the back of the tank.

Figure 9.6. Static fluid undergoing rectilinear acceleration.

The acceleration, , in Eq. (9.21) is that of the origin of the acceleratingcoordinate system that is fixed to the truck. In this case, Eq. (9.21)specialized to the problem at hand yields

which shows that the pressure gradient is constant everywhere.Consequently, the lines of constant pressure are straight lines perpendicularto the direction given by the gradient. By direct integration of Eq. (9.22a)and its substitution into (9.22b), it is found that

(9.23)

where is a constant. An appropriate boundary condition with which todetermine is the expectation that the pressure is atmospheric, say, , atthe top surface of the fluid. According to the coordinate system shown inFig. 9.6, the depth of the water at remains at during accelerationbecause of symmetry and the fact that the surface is a straight line. Thus

, in which case Eq. (9.23) becomes

(9.24)

The lines of constant pressure are tilted straight lines, as shown in Fig.9.6(b). Clearly the effect of the acceleration is to reorient the initiallyhorizontal pressure contours into a direction normal to

.

The free surface is the line of constant pressure for which .In this case, Eq. (9.24) gives the surface as

(9.25)

The highest point of fluid is at the rear of the tank, where . Fluidspills out when or

(9.26)

9.3.2 Rotating Bucket

A second example where an accelerating observer may see a static fluidconcerns the case of a rotating bucket of radius filled with liquid, asshown in Fig. 9.7. In this instance, from an initial depth, , when the liquid

is at rest, a constant angular velocity around the axis is applied to thebucket until a new static configuration is reached in which the fluidrevolves as a solid body. The goal is to find the form of the free surface inthe bucket for the new equilibrium condition. Once again, to a fixedobserver on the ground, the fluid is in continuous motion, whereas to anobserver rotating with the bucket, the fluid is stationary.

Figure 9.7. Static fluid in a rotating bucket.

To specialize Eq. (9.21) to the present case, assume that , andset the position vector in the centrifugal term to be . Acalculation then gives

(9.27)

Consequently, Eq. (9.21) in cylindrical coordinates simplifies to

Integrating Eq. (9.28a) and substituting into Eq. (9.28b) gives the pressuredistribution as

(9.29)

where is a constant that can be determined from the fact that thepressure is atmospheric at the free surface.

To compute , assume for the moment that at equilibrium, the depth ofthe fluid at is . A calculation using Eq. (9.29) yields

, in which case

(9.30)

The location of the free surface is determined by setting , yielding

(9.31)

The length can now be determined by equating the volume of fluid inthe bucket at rest, namely, , with that in the equilibrium state in Fig.9.7(b). The latter may be computed by integrating the height of the freesurface on radial rings progressing outward from the center of the bucket tothe walls. Thus

(9.32)

and after substituting Eq. (9.31) and integrating, it is found that

(9.33)

Consequently, the pressure distribution is

(9.34)

and the height of free surface is

(9.35)

Equation (9.34) makes clear that the surfaces of constant pressure areparabolas of revolution around the axis, as shown in Fig. 9.7(b). Finally,

according to Eq. (9.35), fluid will spill over the edge of the tank when so that, to prevent spillage, it must be the case that

(9.36)

Problems

9.1 Find the pressure as a function of altitude for a gas satisfying theperfect gas law under isothermal conditions. Here is the gasconstant and is the temperature.

9.2 Find the pressure as a function of altitude for a perfect gas whosetemperature varies according to the formula , where isvertical distance and and are constants.

9.3 Say a fluid has a variable density with the vertical coordinate and and constants. A cube of volume andconstant density floats immersed in the fluid. Find the locationwhere it floats.

9.4 Say a fluid has the pressure field specified in Problem 9.2 and thatwithin it a cube with sides of length and constant density is floating.Derive an equation whose solution gives the location where the cube floats.Solve the equation for the case when .

9.5 Assume that the density of a liquid in a square container of crosssection dimension varies in the vertical, , direction according to thefunction , where is a constant. Assume thatthe bottom of the container is at and that the top of the containerat is open to the atmosphere. What is the pressure as a function of ? What is the magnitude of the total horizontal force acting on any sidewallof the container?

9.6 Consider the manometer shown in Fig. 9.8 containing liquid of density . The left-hand side is open to the atmosphere and has cross-sectional

area , while the closed right-hand side has cross-sectional area with . Initially the pressure is on the surface on the left-hand side

and the differential in height of fluid in the two sides is , while thecolumn of air on the closed end of the right-hand side has length .Assuming the entrapped air on the right-hand side obeys the isothermalperfect gas law and a pressure is applied to the open end on theleft, find the new length of the air column, , as a function of the scaledpressures and and the parameters

and . Make plots of the variation of as afunction of for different values of , and discuss.

Figure 9.8. Entrapped gas on the right side is compressed as pressure isapplied on the left side.

9.7 Suppose that a fully submerged balloon is tethered to the bottom centerof the fish tank discussed in Section 9.3.1. Explain what the effect of thepressure distribution will be on the position of the balloon during theacceleration of the fish tank.

9.8 Reconsider the problem of a fish tank lying in a pickup truck for thecase of a liquid whose density obeys the relation ,

where is the minimum distance of the point to the free surface ofthe liquid and is a constant. What is the maximum pressure and wheredoes it occur? Hint: show that the free surface is a straight line, and mapthe governing equations into a coordinate system for which the free surfaceis a coordinate line.

9.9 Consider a sealed circular tank of cross-sectional area , with verticalaxis of length , containing grams of a gas that obeys the perfect gaslaw. If the tank is accelerated uniformly upward along its vertical axis atrate , find the density distribution within the tank assuming isothermalconditions.

9.10 A circular tank on a spaceship of cross-sectional area oriented withits axis in the direction, as shown in Fig. 9.9, contains grams of aheavy gas satisfying the perfect gas law with constanttemperature . The tank is free of the effects of gravity and contains asealed lid that rests on the top surface of the gas and is free to move in the

direction. Initially, the gas is at uniform conditions with pressure andoccupies volume , as indicated in Fig. 9.9(a). The container is thenplaced on a rocket accelerating in the direction at a constant rate .Assuming that the gas achieves a new static equilibrium condition whileaccelerating and that the pressure at the upper surface of the gas remains at

, find the height of the gas, , in the container and the maximumpressure in the tank.

Figure 9.9. Compression of a gas during acceleration.

9.11 Consider a sealed cylindrical tank filled with water that is spinning asa solid body around the vertical axis. Assume that a Ping-Pong ball istethered to the bottom center of the tank so it is fully submerged and is freeto move where it can. Where does the Ping-Pong ball end up, and why?

9.12 Consider a cylinder filled with fluid whose axis is pointing in thehorizontal direction. The cylinder is spinning at a constant rate around its axis and the fluid inside rotates as a solid body. Find the pressureas a function of the and coordinates to within an arbitrary constantand show that the lines of constant pressure are concentric circles. What isthe location of the center of the circles?

9.13 Consider a rectangular fish tank of length , sides of height ,and unit width. At rest, the tank constains water to a depth . Derive Eq.(9.26) by equating the net horizontal pressure force on the water in the tankto its mass times acceleration.

10Incompressibility and Mass Conservation

We turn attention now to the derivation of the Navier-Stokes andcompanion equations that govern the dynamics of fluid flow. In essence,what must be done is to convert the basic physical laws as expressed inEqs. (1.9)–(1.11), into differential equations that are useful in solving forthe details of any flow field. A number of mathematical and physicalissues need to be resolved to accomplish this goal. In particular, the timerate of change of the flow properties indicated on the left-hand sides of theequations must be further elaborated and the forces and thermodynamicprocesses on the right-hand sides of the momentum and energy laws mustbe given expression. As a first step, this chapter considers themathematical machinery necessary to develop the left-hand sides of thegoverning laws and then uses these results to study the changes in volumeof fluid elements and to fully analyze the mass conservation law aspreviously expressed in Eq. (1.9). On the way to this result, wedemonstrate the relationship between dilatation and compressibility.

10.1 Some Useful Mathematics

A fundamental part of the laws discussed in Section 1.4 is the notion ofthe rate of change in time of the mass, momentum, and energy of amaterial fluid element. The natural movement and distortion of fluidelements as time goes on is reflected in the time dependence of . As aconsequence, time differentiation of the flow properties integrated over amaterial element, such as the mass

(10.1)

needs to take into account the time dependence of the domain ofintegration.

A means of accommodating time-dependent integration limits lies in thefact that all the information about is contained in the Lagrangian mappingof fluid particle paths represented by . Thus, if at time , the material fluid

element is defined as the fluid contained within the region , then the entirecollection of points within will move according to to a new region inspace at time that is filled out by the volume . The important point is thatas sweeps out the region , sweeps out . In effect, for each fixed is acoordinate mapping from to . By transforming the integration variable inEq. (10.1) to , it becomes possible to remove the need to take the time-derivative of a time dependent limit of 3D integration.

The general rule for the transformation of the variables of integration of a3D integral of an arbitrary function is given by the change of variablesformula (Gurtin 1981) to the effect that, for a mapping that maps a set toa set ,

(10.2)

where is the Jacobian of the mapping defined as the determinant of thematrix . Written out in detail,

(10.3)

Within the context in which we need to apply Eq. (10.2), represents themapping for a fixed . Consequently, is also time dependent, and Eq.(10.2) adopted to the present circumstances has the form

(10.4)

where provision for the possible time dependence of has been made. Hereand henceforth, for simplicity, the dependence of is not indicatedexplicitly. The term on the right-hand side in Eq. (10.4) is such that, unlikethe term on the left-hand side, a time derivative can pass through the fixedintegration limit . The advantage gained by this, however, is at theexpense of needing to now compute the time derivative if Eq. (10.4) is tobe useful for our purposes. In fact, a succinct formula for is readilyobtainable via direct calculation, as is now demonstrated for the two-dimensional case. An analogous 3D calculation can be done that followsthe same steps and results in the identical formula.

Thus consider the 2D Jacobian matrix

(10.5)

A time derivative gives

(10.6)

Now consider the identity

(10.7)

which is just one of the relations forming the fundamental connectionbetween the Lagrangian and Eulerian viewpoints that was previouslydiscussed in Eq. (2.2). Taking a derivative of Eq. (10.7) with respect to yields

(10.8)

and similar relations may be derived for the other second-order derivativeterms in Eq. (10.6). Substituting these four relations into Eq. (10.6) andcanceling a number of terms yields the result

(10.9)

It may be noticed that the first expression on the right-hand side is just thedivergence of the velocity field, , while the second is just itself.Consequently, it has been shown that

(10.10)

a result that may also be shown to hold in three dimensions. Equation(10.10), together with (10.4), provides a means for accommodatingdifferentiation of moving boundaries, as is demonstrated in oursubsequent derivation of the fundamental equations of fluid flow.

10.2 Incompressibility

A good place to start implementing the results of the previous section is incomputing the rate of change of the volume of a material fluid element.

The volume satisfies the identity

(10.11)

and a time derivative of this relation will reveal the circumstances inwhich the volume of a material element may change in time. Proceedingas suggested by the discussion in the previous section, Eq. (10.4) impliesthat

(10.12)

so that the region of integration is now fixed in time. Computing the timederivative of Eq. (10.12) yields

(10.13)

Applying Eq. (10.4) in the reverse direction to the last expression in(10.13) results in

(10.14)

showing that there is a direct connection between the rate at which thevolume of a fluid element changes in time and the divergence of thevelocity field.

If a fluid is incompressible, it means that the volume of any materialelement cannot change in time, so . In this case, Eq. (10.14) implies that

(10.15)

and because this holds for arbitrary , it must be the case that

(10.16)

everywhere. If not, then due to the continuity of , a material element couldbe selected surrounding a point where the divergence is all of one sign andnot zero. In this case Eq. (10.15) would be violated. Thus it has beenshown that the defining property of incompressible flow is the conditionof zero dilatation.

To get at the physical meaning of the dilatation, consider the case when isvery small. In this circumstance the right-hand side of Eq. (10.14) can beapproximated as , and it follows from Eq. (10.14) that

(10.17)

with equality in the limit as . Equation (10.17) shows that the dilatation ata point can be interpreted physically as the fractional rate of change of thelocal fluid volume surrounding that point.

Common practical situations where Eq. (10.16) is not satisfied includehigh-speed flight when airplanes approach or surpass the sound speed,regions adjacent to revolving rotor blades on helicopters, turbines, internalcombustion engines, and numerous other technologies. It also can be saidthat, in essence, a nonzero dilatation field lies at the heart of such singularflows as source/sinks and doublets considered previously. In this book, theapplications of the fluid equations given in subsequent chapters will belimited to incompressible flows for which Eq. (10.16) is satisfied.

10.3 Mass Conservation

The methodology developed in Section 10.1 and applied to deriving Eq.(10.14) in the previous section is now applied to deriving the differentiallaw for mass conservation corresponding to Eq. (1.9). The first step isapplying Eq. (10.4) to the mass of a material fluid element giving

(10.18)

where now the time dependence of in the integrand must beacknowledged. Using Eq. (10.10), the last expression can be readilydifferentiated in time, yielding

(10.19)

where it is to be understood that , and are evaluated at . Once again, fromEq. (2.2), the time derivative of Lagrangian position in Eq. (10.19) can bereplaced via

(10.20)

In this case all the terms in Eq. (10.19) consist of Eulerian quantitiesevaluated spatially at and all have as a common factor. This means thatthe change of coordinates can be applied in the reverse direction, yielding

(10.21)

By the same reasoning as was done in regard to Eqs. (10.15) and (10.16),it may be concluded that the integrand in Eq. (10.21) must be identicallyzero. Thus the differential form of the mass conservation equation is

(10.22)

a relation also known as the continuity equation.

Some alternative forms of (10.22) are worth mentioning. In particular, thefirst two terms on the right-hand side are often given the notation

(10.23)

which is referred to as the substantial or total derivative. This expressioncaptures, in this instance, the rate of change of the density “along fluidparticle paths.” In other words, because is the density of the fluid particlemoving along path , the derivative of this expression, with the help of Eq.(10.20), is (10.23). Thus Eq. (10.23) is the time rate of change of thedensity of a fluid particle as it moves. The first term on the right-handside of Eq. (10.23) captures the change in time of the overall density field,while the second term records changes in density created by themovement of the fluid particle through a nonuniform density field. UsingEq. (10.23), it follows that the equation of mass conservation can bewritten alternatively as

(10.24)

or

(10.25)

The dilatation thus also can be understood as being minus the fractionalrate of change of the density along fluid particle paths.

If the fluid is incompressible, then Eq. (10.25) implies that

(10.26)

meaning that the density does not change along particle paths. In fact, thereverse holds true: if the density does not change on particle paths, thenthe fluid is incompressible. Note that constancy of the density on particlepaths does not mean that the fluid density must be constant everywhere.There can be flows that are incompressible and yet the density field isnonuniform.

Another useful way to write Eq. (10.22) is to combine the last two termson the right-hand side, yielding

(10.27)

The expression is the flux of mass/area-sec, so that in this form, after anappropriate integration, the continuity equation can be manipulated intothe form of a control volume equation, as considered in Section 16.3.

Problems

10.1 Show that in incompressible flow at a solid boundary with orientation,

(10.28)

that is, the derivative of in the direction (see Eq. 3.12) is zero. Assumethe flow is viscous so that the nonslip boundary condition applies on solidwalls. Hint: note the identity for any scalar function and constant vector .

10.2 With the help of Problem 10.1 and the nonslip boundary condition atsolid surfaces, show that in incompressible flow, at a solid boundary withorientation . Hint: use index notation.

11Stress Tensor: Existence and Symmetry

Our development of Newton’s second law of motion thus far may besummarized in the relation

(11.1)

equating the rate of change of momentum of a material element to the sumof body and surface forces. Some inkling of how the surface force can beaccounted for via the stress tensor was discussed in our previousconsideration of fluids at rest in Chapter 9. In this chapter and the next, theform taken by the stress tensor is worked out for moving fluids. Then wewill be in a position to complete the derivation of the momentum equationin Chapter 13.

11.1 Existence of the Stress Tensor

When surface forces were last encountered in Section 9.1, it was pointedout that , representing the force/area on a surface oriented in the

direction at a point at time , can be greatly simplified in the case ofa moving fluid by the introduction of the stress tensor. As was writtenpreviously,

(11.2)

containing the tensor acting on the vector to yield the force/area onthe left-hand side of the equation. Because can be determined at any

point in the flow without regard to the orientation of the surface of interest,Eq. (11.2) is a source of great simplification in the calculation of surfaceforces.

The reason why a stress tensor satisfying Eq. (11.2) exists has to do withthe balance of forces on a vanishingly small material volume of fluid underthe geometric fact that volume shrinks to zero faster than surface area. Forexample, in the case of a sphere, as its radius , the ratio of volume toarea also. The end result is to create abalance of surface forces that implies the existence of .

A formal proof of the existence of the stress tensor is done convenientlyby considering the balance of forces on a small tetrahedral element, asshown in Fig. 11.1, and then letting its dimension shrink to zero. The topsurface of the tetrahedron is oriented in the direction, has surface area

, and intersects the axes at , respectively. Denoting theprojection of the top surface into the th direction as , it can be shownthat

(11.3)

Indeed, define the vectors and thatframe the tilted surface as shown in Fig. 11.1(b), and let denote theangle between and . The cross product is in the direction of

and its magnitude satisfies

(11.4)

which is the area of the parallelogram formed by the vectors. This area isequal to . Consequently, the unit vector is given by

(11.5)

where the numerator has been evaluated directly from the definitions of thevectors. Equation (11.3) now follows from (11.5) by multiplying throughby and recognizing that and so forth. Equation (11.3) alsoshows, after taking a dot product of both sides with , that the unit vector

has components

(11.6)

Figure 11.1. Geometry for proving the existence of .

A force balance on the tetrahedron in Fig. 11.1(a) that takes into accountbody and surface forces yields

(11.7)

with representing the mass of the tetrahedral fluid element and its volume. The size of the tetrahedron is assumed to be sufficiently

small so that spatial variations in need not be considered. Consequently,just the appropriate normal vector and not spatial position is indicated in

the arguments of . Dividing Eq. (11.7) by and taking the limit as gives

(11.8)

where the area ratios have been simplified using Eq. (11.6). The left-handside of Eq. (11.8) must be zero, as alluded to previously. Moreover, notethat

(11.9)

for any because the force of the fluid exerted on one side of a surfacemust be equal and opposite to that acting on the other side. If these forceswere not in balance, then infinitesimal material fluid elements in theneighborhood of the surface would be accelerated at great speed.

Now using Eq. (11.9), it follows that

(11.10)

Equation (11.10) can be rewritten in the form

(11.11)

where, for emphasis, the summation is indicated explicitly. Motivated bythis relation and introducing the tensor product (see Eq. 1.23), the stresstensor is defined as

(11.12)

since the right-hand side of this relation acting on is just Eq. (11.11).Moreover, as must be the case, the definition in Eq. (11.12) is independentof the choice of orthonormal basis vectors (see Problem 11.1). Clearly Eq.(11.2) is satisfied by Eq. (11.12) and the existence of the stress tensor isassured.

If the vector in Eq. (11.2) is chosen to be , then

(11.13)

is the force/area on a surface oriented in the th direction. Taking a dotproduct of Eq. (11.13) with and recalling Eq. (1.20), it follows that

(11.14)

Thus represents the force/area in the th direction acting on a surfacewhose outward normal is in the th direction. The force that is referred toin Eq. (11.14) is that of the fluid to which the vector points on the fluidfrom which it points. An illustration of the relationship between and thesurface forces on a small cube is shown in Fig. 11.2. Though there appearsto be nine independent stress components, in fact, there are only six,because the next section shows that is a symmetric tensor, in which casethe shear stresses occur in equal pairs.

Figure 11.2. Surface forces on a cube.

11.2 Symmetry of the Stress Tensor

The symmetry of the stress tensor is a consequence of the local balance offorce moments acting on a vanishingly small material element. To showthis, it is necessary first to derive an equation governing the moments on anarbitrary material fluid element, a relation that has intrinsic interest of itsown. From this equation, the symmetry of can be deduced as the resultof a limiting process in which the size of the material element is shrunk tozero.

Consider an arbitrary material fluid element and break it up into avery large number of tiny prismatic subelements that account for thewhole of , as shown in Fig. 11.3. The subelements have any number offlat sides, and the collection of element faces that form the exterior surfacecollectively approximates the outer surface. The finer the collection of

subelements, the better the volume is resolved by the subelements and thebetter the exterior surface is resolved by subelement faces.

Figure 11.3. Division of a material element into prismatic subelements.

Let the th subvolume have volume , density , and acceleration and let denote the number of its facets. Moreover, let the th facet ofthe th subvolume have outward normal vector and area . Thenthe approximate momentum balance for this small subvolume is

(11.15)

where is assumed to be evaluated at whatever face is indicated and is notsummed.

Now select a point within the material element to act as an origin and let be the moment arm from this point to the center of the th subvolume or its

corresponding facets. Take a cross product of with Eq. (11.15) and sumover all the subvolumes, yielding

(11.16)

Every interior face contained on the right-hand side in the double sumappears twice: once from each of the two adjacent subvolumes. Because thevectors for this pair are equal and opposite, the contribution of the twofaces cancel. All that survives on the right-hand side are the contributionsfrom the faces that are on the exterior of the fluid element.

Taking the limit of Eq. (11.16) as and gives

(11.17)

as an equation governing the moments of the forces acting on and within afluid element. Our goal now is to consider the implication of Eq. (11.17) inthe limit as the material element shrinks to zero around the origin. Beforethis can be done, it is necessary to convert the area integral on the right-hand side to a volume integral so its magnitude can be compared directlywith that of the term on the left-hand side.

With this end in mind, take a dot product of Eq. (11.17) with an arbitraryconstant vector yielding

(11.18)

Moving inside the integral on the right-hand side and using the identity

(11.19)

(see Problem 11.2) brings the term into a form to which the divergencetheorem can be applied. The result is

(11.20)

Shifting to index notation in the integrand on the right-hand side andcarrying out the differentiation yields

(11.21)

where the identity

(11.22)

has been used. Introducing the tensor divergence and the vector cross (seeAppendix B), Eq. (11.21) can be rewritten in direct notation as

(11.23)

where the tensor inner product Eq. (1.22) appears in the last term.

Substituting (11.23) into Eq. (11.20) and rearranging terms, it has thusbeen shown that

(11.24)

The expression on the left-hand side of the equation represents the balanceof forces and acceleration in Newton’s second law of motion as shown inthe next chapter and is thus identically zero. This means that the integrandon the right-hand side must be zero. Another way to arrive at the sameconclusion is to note that the terms on the left-hand side of Eq. (11.24)contain a factor of in the integrand that is not shared by the expression onthe right-hand side. This means that as the volume shrinks around theorigin, the right-hand side reduces to zero at the rate versus for the left-hand side. For very small , the right-hand side of Eq. (11.24) cannot bebalanced by the left-hand side and hence must itself be zero. In either case,

(11.25)

and because the vector is arbitrary, it must be true that

(11.26)

This relation proves the symmetry of the stress tensor, because, forexample, if , it implies that

(11.27)

and similarly for the other components.

Having established the existence and symmetry of the stress tensor, itnow remains to discuss its form. In essence, a constitutive law is soughtthat connects with that part of the physics of a moving fluid that gives riseto surface forces. This is accomplished in the next chapter.

Problems

11.1 Let represent a different set of orthonormal basis vectors than . Showthat

so that is uniquely defined by Eq. (11.12). Hint: use Eq. (11.11).

11.2 Using index notation, prove Eq. (11.19).

12Stress Tensor in Newtonian Fluids

12.1 Relative Fluid Motion at a Point

Though the stress tensor at any point in a moving fluid depends generallyon the local behavior of the fluid velocity field, it is only certain aspectsof the local fluid motion that contribute to the stress. Thus it is importantto first gain some awareness of the types of local fluid motions that arepossible in the neighborhood of any point in a flow field. To accomplishthis, the velocity field may be represented around a point in the form ofthe Taylor’s series expansion

(12.1)

for small values of . Here the velocity gradient tensor was definedpreviously in Eq. (1.40).

It is helpful to make the decomposition

(12.2)

where is the rate of strain tensor

(12.3)

and is the rotation tensor

(12.4)

It may be noticed that is a symmetric tensor and is an antisymmetrictensor. With these definitions, Eq. (12.1) becomes

(12.5)

where higher-order terms are omitted because just the local linearbehavior of the velocity field is considered here. The first term on the

right-hand side of Eq. (12.5) does not depend on and clearly accounts foruniform translation in the vicinity of at the magnitude and direction ofthe local velocity at . The physical meaning of the remaining terms is nowconsidered.

Defining and switching to index notation, the identity

(12.6)

can be verified by carrying out the derivatives in the last term using theidentity

(12.7)

and noting that is evaluated at and is not a function of . Defining

(12.8)

as a scalar potential, Eq. (12.6) may be expressed simply as

(12.9)

As noted in Problem 2.6, is in a direction perpendicular to the surfaces ofconstant . For a general set of coordinate directions, will have nominallynine nonzero components. However, because is symmetric, it can beshown (see Section B.3) that a set of orthogonal basis vectors, say, , existsfor which only the diagonal components of (i.e., ) are nonzero. Thedirections associated with this choice of basis vectors are referred to asthe principal directions of the tensor .

For the purpose of exploring the motions implied by Eq. (12.9), it isconvenient to work in principal coordinates, so in this case, Eq. (12.8)simplifies to

(12.10)

and a calculation with Eq. (12.9) gives

The motion implied by Eqs. (12.11) is illustrated in Fig. 12.1 on a 2Dcross section. It may be noticed that the velocity field in the vicinity ofeach axis has a direction parallel to that axis. For example, in thisillustration, along the line where , can be either in the plus or minus direction, while . Similar behavior is true for the other directions. As aconsequence, a material element that begins as a sphere will tend todeform into an ellipsoid whose orientation, as determined by its major andminor axes, lines up with the directions of the principal axes. For theparticular velocity field shown in Fig. 12.1(a), an initially spherical regiondeforms into an ellipsoid, as seen in Fig. 12.1(b). This kind of motion isreferred to as a pure straining motion.

Figure 12.1. Deformation of a local fluid element experiencing a purestaining motion.

The pure straining motion associated with can be further separated intotwo fundamental motions via the decomposition

(12.12)

in which the first term, depending on , is isotropic (i.e., has no directionalpreference) and the second term contains all anisotropic effects. Because has components , Eq. (12.12) may be rewritten in index notation as

(12.13)

The significance of the decomposition in these relations derives from thefact that the trace of is accounted for fully by the first term on the right-hand side, leaving the second term traceless. This result follows simply bynoting the identity . In fact, it may be deduced from Eqs. (1.40) and (12.3)that

(12.14)

so that the decomposition in Eq. (12.12) is essentially betweencompressible and incompressible motions.

Substituting Eq. (12.13) into (12.8) results in the decomposition

(12.15)

where

(12.16)

and

(12.17)

Moreover, defining

(12.18)

and

(12.19)

where the gradient is with respect to the coordinates, it follows that

(12.20)

The 3D surfaces that are implied by constant values of are spheres, so thevelocity field is radially inward or outward. By a direct calculation thattakes advantage of Eqs. (12.7) and (12.14), it follows that

(12.21)

In other words, accounts for a uniform expansion or compression at thepoint with the same dilatation rate as experienced by the velocity field .

As for , a computation shows it to have zero dilatation and so it is anincompressible flow. Moreover, has the same principal directions as sothat when is expressed using principal axes, Eq. (12.17) becomes

(12.22)

In view of Eq. (12.19), a calculation gives

which is a pure straining motion without change of volume. If are notidentical, then two of the directions must have velocity of opposite sign.In this case, must be such as to cause an initially spherical region todeform into an ellipsoid very much the same way as depicted in Fig. 12.1.

An additional contribution to the local fluid motion is associated with thelast term in Eq. (12.5), which is now considered. For convenience, thisflow component is referred to as

(12.24)

Because the tensor is skew, the discussion in Section 1.5.3 suggests thatthere exists an axial vector, say, , such that

(12.25)

for any vector . Moreover, according to Eq. (1.36),

(12.26)

which, after substituting from Eq. (12.4) and using the definition ofvorticity in Eq. (1.3), gives

(12.27)

(see Problem 12.2). In other words, the axial vector is one-half thevorticity. From Eqs. (12.24), (12.25), and (12.27), it follows that

(12.28)

showing that has the character of solid body rotation with angularvelocity .

In the absence of vorticity, Eq. (12.28) makes clear that there cannot be arotational component to the local velocity field. Moreover, this relationalso provides insight into the physical meaning of the vorticity. In fact, the

vorticity is seen to be twice the angular velocity that a material fluidelement would experience if it were to suddenly freeze and by so doingacquire the capacity to rotate as a solid body. It is important to note thatthe presence of vorticity does not imply the necessity of rotationalmovement. In fact, the complete velocity field also reflects thecontribution from , so that even if , the overall flow field may have norotation. A simple example of this occurs in unidirectional flow where thevelocity field is of the form and the vorticity is nonzero. All thestreamlines are straight lines, and there is no rotation (see Problem 12.3).

Collecting together the results in Eqs. (12.18), (12.19), and (12.28), it isfound that the local velocity at a point can be decomposed into the form

(12.29)

The terms on the right-hand side consist of, in turn, a uniform translation,an isotropic expansion/compression, a pure straining motion withoutchange of volume, and solid body rotation. The next section considers theextent to which these different motions do or do not contribute to thepresence of surface forces in a moving fluid.

12.2 The Stress Tensor

In the same way that the rate-of-strain tensor was decomposed into anisotropic and anisotropic part, the analysis of the stress tensor is aided bythe same kind of division. In particular, as suggested in Eq. (9.4), thedeviatoric part of the stress tensor may be introduced so that

(12.30)

where

(12.31)

is traceless and records how departs or deviates from the isotropic formgiven by the first term on the right-hand side of Eq. (12.30).

On a surface with orientation ,

(12.32)

where the first term on the right-hand side contributes a force/area alignednormal to the surface in the same way as was encountered previously inthe context of the pressure in a fluid at rest in Eq. (9.5). In fact, thepressure may be defined by the relation

(12.33)

and it is of interest to see the motivation behind this definition.

Consider the average normal surface force, say, , acting on a small sphereof radius around a point in the fluid. Mathematically, this is given by

(12.34)

where is the area of the sphere of radius , is an element of solid angle,and is an outward unit normal vector to the sphere; is the force/area onthe surface of the sphere and is the normal component of the surfaceforce. For small , Eq. (12.34) is well approximated by

(12.35)

where is evaluated at the center of the sphere.

In spherical coordinates,

(12.36)

as may be seen from Fig. 12.2, so that . With these expressions, theintegral in Eq. (12.35) can be evaluated for all combinations of indices,with the result that

(12.37)

Substituting this into Eq. (12.35) yields

(12.38)

establishing through Eq. (12.33) that the pressure in a moving fluid isdefined as minus the average normal force acting at a point. This is apurely mechanical definition that skirts consideration of thethermodynamic concept of pressure as a force produced by a molecularsystem in equilibrium. In fact, it is possible that the two concepts ofpressure may yield different values from one another. This point will bereturned to when the energy equation is considered in Chapter 15.

Figure 12.2. Decomposition of a unit normal vector into rectangularCartesian coordinates (dashed).

Now consider the contribution of the deviatoric stress to the surface force,namely, . Because is symmetric, so too is . Consequently, principaldirections can be found that are common to both and for which may beexpressed as a diagonal tensor. Assuming that in the principal coordinatesthe diagonal components of consist of , it follows that

The significance of Eq. (12.39) is that only normal forces act on surfacesaligned with the principal directions. For example, choosing , theforce/area on a surface with outward normal is

(12.40)

which is in the direction. A similar result holds for the and directions.This proves, as illustrated in Fig. 12.3 for the equivalent 2D case, that anonzero deviatoric stress will cause an initially spherical material elementto deform into an ellipsoid oriented along the principal directions. In fact,because the sum of the diagonal elements in Eq. (12.39) is zero, the forcesacting on the sphere in the principal directions must include at least oneforce pushing inward and one outward so that the material element mustdeform into an ellipsoid.

Figure 12.3. Deformation of a local fluid element caused by surfaceforces.

The similarity between Figs. 12.1 and 12.3 is evident and motivates theidea that, in some sense, and should share the same principal axes. Thisdoes turn out to be the case for a large class of fluids, known as Newtonianfluids, that include common fluids such as air and water. Newtonian fluidslack an intrinsic directional preference among the molecules of whichthey are composed, such as may be the case if long-chain polymers arepresent, which can add a directional bias to the way that responds to thelocal velocity field. In this volume, our consideration of is strictly limitedto Newtonian fluids, and for this case, the tensor relationship between andthe underlying fluid motion must be such as to treat all directions equally.Application of this principle is an integral part of deriving the form of thestress tensor.

The starting point for deriving the constitutive law is to posit the existenceof a linear tensorial relationship between and the velocity gradient tensor.This kind of connection is motivated by the physics of nondense gaseswhere the cause of the internal surface forces lies in molecular momentumtransport. Models of the process by which a net momentum flux occurs inthis case depend on the diffusion and mixing of molecules carryingmomentum down a gradient. The distance over which mixing takes placein a gas, which is proportional to the mean free path, is so exceedinglysmall that linear variation in the velocity field is all that can be expectedto be relevant to momentum transport. Moreover, the continuumhypothesis suggests that whatever constitutive law should be true fornondense gases should also be true for all Newtonian fluids, includingliquids. To formally encompass this viewpoint in the most general setting,it is thus assumed that

(12.41)

where is a fourth-order tensor that remains to be determined by applyinga number of constraints to this relation, which are dictated by thecharacter of a Newtonian fluid.

The first step in determining is to insist that it has a form that does notallow directional biases to be transmitted to the stresses from the velocityderivative field. In other words, it should be isotropic in the sense that depends on in the same way for all orientations. Tensors with such anisotropic property are limited to being appropriate combinations of theidentity tensor with scalar coefficients depending at most on thecorresponding tensor invariants. For the fourth-order tensor of interesthere, a standard result (Aris 1990) is that must be of the form

(12.42)

where , and are scalar functions of the invariants of .

Because is symmetric, it follows from (12.41) that

(12.43)

For this relation to hold for arbitrary velocity gradient fields, it must bethat is constrained so that

(12.44)

Substituting Eq. (12.42) into (12.44) gives

(12.45)

which requires that

(12.46)

Substituting the decomposition of in Eq. (12.2) into Eq. (12.41) gives

(12.47)

Considering the second term on the right-hand side in the light of Eqs.(12.42) and (12.46) yields

(12.48)

The first two terms on the right-hand side cancel because , while the lastterm is zero because . Consequently, the part of the local fluid motion thatis a solid body rotation makes no contribution to the deviatoric stress in aNewtonian fluid.

Now consider the contribution to arising from the first term on the right-hand side of Eq. (12.47). Substituting for and taking advantage of thesymmetry of gives

(12.49)

Previously it was observed that , in which case Eq. (12.49) implies that

(12.50)

so that

(12.51)

Thus our final result for the deviatoric part of the stress tensor is

(12.52)

or, more succinctly,

(12.53)

Equation (12.53) is fully consistent with the expectation that and sharethe same principal directions.

Finally, substituting Eqs. (12.33) and (12.53) into Eq. (12.30) shows thatthe stress tensor for a Newtonian fluid obeys the constitutive law

(12.54)

Here is the dynamic viscosity that is a material property of fluids in thesense that it varies from fluid to fluid depending on their individualmolecular makeups. For example, in the case of nondense gases, an

analysis of the physics of momentum transport (Chapman &Cowling 1991) establishes that

(12.55)

where is the mean free path, is the mean speed of the molecules, and isan empirically determined constant. For most fluids, due to the absence ofan adequate molecular theory, must be determined empirically.

Problems

12.1 Prove that and , defined in Eq. (12.18) and (12.19), respectively,satisfy and .

12.2 Prove that Eq. (12.27) follows from Eq. (12.26).

12.3 Decompose the velocity field according to Eq. (12.29) and visualizethe flow in each part via a quiver plot. Find the principal directions at anypoint.

12.4 Prove that in incompressible flow, . Then, taking advantage ofProblem 10.2, show that at a solid boundary with orientation ,

(12.56)

12.5 Say that a velocity field is given by , where . Calculate thecorresponding components of the rate of strain tensor and the stresstensor.

12.6 For the velocity field , calculate the components of the stress tensor.Is the flow field compressible, or not?

13Navier-Stokes Equation

The last two chapters have produced an expression for the stress tensor thatcan be used to determine the surface forces in a moving fluid. Thus all theparts are now in place to be able to complete the development of Newton’slaw of motion from the point where it was last encountered in Eq. (11.1).The remaining tasks consist of treating the left-hand side of the equationand placing the surface force expression into a final form so that adifferential equation can be derived.

13.1 Rate of Change of Momentum

In Chapter 10 it was shown how to take time derivatives of time-dependentintegration limits by use of the change of variables formula Eq. (10.2) andthe equation for the Jacobian derivative (10.10). The same steps can berepeated here in the case of the rate of change of the total momentum of amaterial fluid element. To simplify the derivation somewhat, it isconvenient to initially proceed using index notation and then return todirect notation in the final formulas.

Define

(13.1)

as the th component of the momentum of a material fluid element. By thechange of variable formula Eq. (10.2),

(13.2)

Carrying out the differentiation and applying Eq. (10.10) yields

(13.3)

where all quantities formed of and are evaluated at andthe notation

(13.4)

has been adopted to represent the total derivative of consistent with itsdefinition in Eq. (10.23). As in the discussion in Chapter 10, maybe interpreted as the rate of change of velocity along fluid particle paths.

It may be noticed that the first and third terms in the integrand on theright-hand side of Eq. (13.3) sum to zero because together they form thecontinuity equation Eq. (10.24) multiplied by the common factor .Consequently, Eq. (13.3) becomes

(13.5)

and switching the integral back to gives

(13.6)

a result that is often referred to as the Reynolds transport theorem.

In direct notation, Eq. (13.4) becomes

(13.7)

where the second term on the right-hand side reflects the fact that is the th component of the tensor . Clearly is a

vector consistent with the other terms in the equation. ConvertingEq. (13.6) to direct notation gives the rate of change of the momentum of amaterial fluid element as

(13.8)

13.2 Surface Forces

The complete surface force acting on a material element is obtained byintegrating the local force over the surface area, yielding

(13.9)

To be able to complete the development of the momentum equation, thisrelation must be converted to the form of a volume integral. One approachtoward accomplishing this is to introduce a constant vector and then notethe identity

(13.10)

which is easily proven using index notation. In this case,

(13.11)

with the last equality coming from the divergence theorem. Once againresorting to index notation to simplify the calculation, and using thesymmetry of , it follows that

(13.12)

The quantity , which in direct notation is given by , is thedivergence of the tensor , as was encountered previously and is discussedin Section B.1.

In terms of , Eq. (13.12) becomes

(13.13)

and using this in Eq. (13.11) gives

(13.14)

from which it follows that

(13.15)

because the vector is arbitrary. Equation (13.15) is, in essence, ageneralization of the divergence theorem for the case of tensors that allowsfor the effect of the surface forces acting on a material fluid element to beaccounted for as a volume integral.

13.3 The Navier-Stokes Equation

Gathering together from Eqs. (13.8) and (13.15) the expressions that havebeen derived for the terms in Eq. (11.1), it follows that

(13.16)

Rearranging this into a single integral gives

(13.17)

and because the material element is arbitrary, the integrand in Eq. (13.17)must be zero. This gives the differential form of the momentum equation as

(13.18)

and substituting for using Eq. (12.54) gives

(13.19)

Using index notation, it is a simple matter to show that the pressure termsimplifies according to

(13.20)

The last term in Eq. (13.19) originates in the deviatoric part of anddepends on . Although there are circumstances where is not constant,for example, in a nonisothermal flow for which viscosity is sensitive totemperature, the flows of main interest in this exposition will not have avariable viscosity. Consequently, the equations developed here will for themost part be in a form based on the assumption that is constant. Greatergenerality is easily accomplished by including viscosity gradients in thelast term in Eq. (13.19), and an indication of what this implies is illustratedafter first discussing the constant viscosity case.

Assuming constant , consider the terms and appearing in Eq. (13.19). The second of these simplifies in the same waythat the pressure term does and is equal to . The term in indexnotation gives

(13.21)

where the last term in this equation is clearly . The expression ,or in direct notation, in Eq. (13.21) is the Laplacian of a vectorfunction, as was encountered previously in Eqs. (4.1) and (4.2).

Substituting the various results into Eq. (13.21) gives the Navier-Stokesequation in the form

(13.22)

with the only limitation being that of constant viscosity. If the fluid motionis further limited to being incompressible so that , then the last termin Eq. (13.22) vanishes and the Navier-Stokes equation takes the form

(13.23)

In index notation, assuming rectangular Cartesian coordinates, Eq. (13.23)is

(13.24)

Appendix A provides expressions for the Navier-Stokes equation in polar,cylindrical, and spherical coordinates. In these and other orthogonalcoordinate systems, the appropriate fluid equations are derived by applyingthe classical formulas for curvilinear coordinates (Hildebrand 1976; Rileyet al. 2006) to the expressions for vectors and tensors in Eq. (13.23).

Another direction of increased generality is to adapt the Navier-Stokesequation to noninertial observers. This is readily accomplished using theresults of Chapter 8 by taking advantage of the fact that may beinterpreted as the acceleration of a fluid particle. Consequently, Eq. (8.35)can be written as

(13.25)

which relates the acceleration of a fluid particle as seen by an inertialobserver on the left-hand side to that of an arbitrary observer given by thelast term on the right-hand side. As noted in Chapter 8, the starredcomponents refer to the vectors seen from the origin of the acceleratingobserver and written in terms of the fixed orientation of an inertialobserver.

Substituting Eq. (13.25) for the acceleration into Eq. (13.23) gives theNavier-Stokes equation appropriate to a noninertial observer inincompressible flow as

In essence, Eq. (13.26) is an adaptation of Eq. (8.36) for the case of amoving fluid. The fictitious force terms on the right-hand side, such as thecentrifugal and Coriolis forces in the last two terms, respectively, can playimportant roles in the analysis of flows as seen by a rotating observer.

Finally, it is interesting to note how the Navier-Stokes equation ismodified by the fact of a variable viscosity. Thus, for a general flow in thiscase, Eq. (13.19) yields

(13.27)

Under the assumption that viscosity varies only through temperaturechanges, as in , then the gradient of viscosity in Eq. (13.27) canbe replaced by

(13.28)

In circumstances such as this, where determination of the velocity fielddepends on knowing , it is generally necessary to couple the solution ofthe Navier-Stokes equation to that of the energy equation. Other commonexamples where the behavior of needs to be taken into account includeflows with convective heat transfer, compressibility, and combustion. Thenext two chapters show how the laws of thermodynamics may be used todevelop an energy relation whose solution when coupled to the Navier-Stokes equation enables the treatment of the many applications wheretemperature and density changes are an important part of the flow physics.

14Thermodynamic Considerations

14.1 Overview

Thermodynamics is largely a study of the equilibrium states of fluidshaving spatially uniform properties such as temperature, pressure, anddensity. The notion of equilibrium refers ultimately to the idea that themolecules composing the fluid are in a statistical equilibrium insofar asthe distribution of molecular velocities and other characteristics isconcerned. At first sight, it may seem that the laws of thermodynamicsshould be incompatible with the analysis of flowing fluids where thepressure varies everywhere and there very well might be spatial gradientsof temperature and density. Nonetheless, by careful consideration of howthe laws of thermodynamics pertain to material fluid elements, therelevancy and application of thermodynamics to flowing fluids can bewell justified.

Changes to a thermodynamic system over time that maintain theequilibrium are reversible processes, whereas those that disrupt theequilibrium, say, by the addition of currents evident in a moving fluid, areirreversible. For a thermodynamic system, such as a fluid in a closedcontainer, to remain in equilibrium despite outside influences, such ascompression caused by the movement of a piston or the purposefuladdition of heat, the mechanism by which the molecules adjust to theimposed changes must act faster than the changes themselves. In gases,with each molecule experiencing approximately molecular collisions persecond, information about changing circumstances can spread extremelyquickly throughout the gas. Apart from extreme conditions, as in a shockwave, the gas has the capacity to maintain equilibrium during the changeof its gross properties such as pressure. Similarly, in a liquid, the strongintermolecular forces bring about rapid passage of information throughoutthe fluid that maintain equilibrium.

Although it is not expected that thermodynamic equilibrium is maintainedwithin a moving fluid if it is taken as a whole, it is not unreasonable to

expect that equilibrium can be maintained for individual material fluidelements if they are small enough. Specifically, they need to be small soas to have homogeneous thermodynamic properties yet large enough tocontain a very large number of interacting molecules. Such athermodynamic entity has the ability to maintain equilibrium even as itresponds to local changes in pressure, temperature, and density. In fact,such material fluid elements can be expected to transit through a series ofequilibrium states even as they encounter ever changing fluid propertieson their path through the flow field.

An energy equation for moving fluids that expresses the first law ofthermodynamics is derived by considering the physics of small materialfluid elements that can be legitimately regarded as equilibriumthermodynamic systems. An essential aspect of this is to accommodatethe effect of the surrounding fluid as it does work on the materialelements and supplies heat through conduction. The result is an energyequation that applies throughout the nonequilibrium fluid field that occursin applications. Before taking up the derivation of the energy equation inthe next chapter, it is first necessary to review some basic resultsconcerning the thermodynamics of equilibrium systems. The form of thefirst law of thermodynamics developed here will then find application inthe next chapter, when the energy equation for a moving fluid isconsidered.

14.2 First Law of Thermodynamics

Consider the fluid in a container whose properties are homogeneous and inequilibrium. Such a system can be described by any two thermodynamicproperties such as temperature, , pressure, , or density, , where is thespecific volume. Having picked any two properties, these may beincorporated into equations of state that determine the remainingproperties. For example, from knowledge of and for a perfect gas, theequation of state

(14.1)

where is the gas constant, allows for the determination of pressure. Notethat for the purposes of this section, is to be regarded as thethermodynamic pressure, which is often, but not necessarily, identical tothe mechanical pressure introduced previously in Eq. (12.33).

The internal energy/mass of the system, , accounts for energy in thevarious vibrational and rotational states of molecules as well as the energyof translational motion over and above that contained in their mean drift.The latter is the macroscopic velocity field that contributes to the kineticenergy/mass in the form

(14.2)

As mentioned in Section 1.2, the extraordinarily high velocity ofmolecules in a gas means that the aggregate internal energy of themolecules will be much greater than their kinetic energy.

The first law of thermodynamics states that

(14.3)

to the effect that the rate of change of the internal energy/mass is equal tothe sum of the rate of gain of energy via conduction to the system, , andthe work done on the system that changes internal energy, . It should beemphasized that for the purposes of the discussion in this chapter, it istaken for granted that the thermodynamic system in question is devoid ofspatial gradients. Because of this, there is no practical difference betweenconsidering changes of energy/mass, as in Eq. (14.3), or considering thechange in energy of a particular material element within the system.

Over a given time period, the left-hand side of Eq. (14.3) will register achange in the state of the system. What change has occurred depends onthe particular history of how the terms on the right-hand side havebehaved over this time interval. The end result is a particular path betweenthe beginning and end states of the internal energy. For example, if thechange to a system is adiabatic and thus lacks heat conduction, the pathfollowed is one for which . Other commonly encountered processes entailconstant pressure or constant specific volume.

The work referred to in Eq. (14.3) is reversible work of compression thatis applied in such a way as to avoid inhomogeneities in the system. Figure14.1 illustrates the process behind this. A gas of mass and initial depth contained in a chamber with uniform properties is compressed slowly bylowering a piston from the top. The cumulative work performed on thesystem over time is the pressure force on the top surface times how far thesurface has moved. Assuming the area of the piston is and that the surfacehas moved through a distance in time , then the total work done is andthe work/mass-sec is

(14.4)

where is the density of the gas. In the limit of small changes, as , thisgives

(14.5)

because .

Figure 14.1. Reversible work of compression.

The total mass before and after compression is unchanged so that

(14.6)

Expanding the terms on the right-hand side in Taylor series around thetime and taking the limit as gives the identity

(14.7)

Furthermore,

(14.8)

and after substituting Eqs. (14.7) and (14.8) into Eq. (14.5), it is found that

(14.9)

Using Eq. (14.9), the first law is now in the form

(14.10)

The heat transfer term, , can be developed further via the second law ofthermodynamics, which states, in effect, that a property of the fluid exists the entropy denoted per unit mass as such that for a reversible addition ofheat,

(14.11)

Clearly is constant for reversible, adiabatic processes, which are thusreferred to as being isentropic. In contrast, it may be shown that alwaysincreases for irreversible processes. Using Eq. (14.11), our statement ofthe first law becomes

(14.12)

Although each of the terms on the right-hand side applies only toreversible processes, their combination in the first law applies under allcircumstances. In essence, this follows because Eq. (14.12) links togethervariables that depend only on the state of the system and not the processby which the beginning and end states are accessed. Thus, even though theindividual expressions making up the right-hand side of Eq. (14.12) willnot apply to irreversible processes, their sum still adds up to . Furtherprogress in adapting Eq. (14.12) for later use as an equation to solve forfluid flow requires expressing the entropy change in terms of measurablequantities.

Toward this end, consider the entropy to be a function of and so that . Itfollows that

(14.13)

where, consistent with standard thermodynamics notation, the fact that Sis taken to be a function of and as against a different pair of variables isreinforced by the use of a subscript to denote the variable that is beingheld fixed in the partial differentiations. The usefulness of Eq. (14.13)

depends on further elucidating the derivative expressions on the right-hand side. These are now considered.

It is helpful at this point to introduce the specific heat, , that is defined forany particular reversible process associated with , as

(14.14)

Thus is the ratio of the rate at which heat/mass is added to the systemover the rate of change of the system temperature. For a small timeinterval , the amount of heat/mass added is and the temperature change is . In this case, the specific heat is approximately given by

(14.15)

which has the interpretation that the specific heat is numerically equal tothe amount of heat/mass added to effect a 1 rise in temperature.

The specific heat depends on the particular process by which the heat isadded to the system. Two principal specific heats, denoted as and , arerespectively defined for processes that involve holding either the pressureor specific volume constant while heat is added. A process for which isconstant satisfies , and in this case, Eq. (14.13) gives

(14.16)

Moreover, it also follows from Eqs. (14.11) and (14.14) that, for aconstant pressure process,

(14.17)

Consequently, equating the last two equations gives

(14.18)

Now turning attention to the expression in Eq. (14.13), define theHelmholtz free energy,

(14.19)

Using Eqs. (14.11) and (14.12), it follows that

(14.20)

Choosing and as the independent thermodynamic variables so that and ,it follows that

(14.21)

and

(14.22)

Substituting these into Eq. (14.20) yields

(14.23)

Equating terms in Eq. (14.23) containing common factors of and ,respectively, yields

(14.24)

and

(14.25)

A derivative of Eq. (14.25) must equal a derivative of Eq. (14.24)because this will bring the left-hand sides of the equations to the commonsecond derivative term . Carrying out this operation and simplifying theresult gives

(14.26)

which is one of the four well-known Maxwell’s thermodynamic relations(Batchelor 1967).

The coefficient of thermal expansion, , is defined via

(14.27)

and after substituting this into Eq. (14.26), it follows that

(14.28)

Now substituting Eqs. (14.18) and Eq. (14.28) into (14.13) gives

(14.29)

which, when substituted into Eq. (14.12) and replacing by , gives our finalresult for the first law as

(14.30)

The advantage of this expression is that it expresses the rate of change ofinternal energy of a system in terms of measurable state variables andtheir rates of change as well as measurable fluid properties and .

In the next chapter, Eq. (14.30) is incorporated into the development ofthe energy equation for general fluid flows. The relevance of Eq. (14.30)to nonequilibrium fluid systems containing currents is in predicting thebehavior of small material fluid elements within the system, as mentionedpreviously. Thus the time derivatives appearing in Eq. (14.30) can bereinterpreted as being substantial derivatives that express the rate ofchange of fluid properties during the motion of material fluid elements. Inthis case we are justified in adapting Eq. (14.30) to our purposes bywriting it in the form

(14.31)

Equation (14.31) can be simplified slightly by using Eq. (10.24) to replacethe density derivative. Moreover, in anticipation of our subsequent use ofthis relation within the energy equation, it is helpful to adopt the notation for the pressure so as to emphasize that this is the equilibriumthermodynamic pressure and not the mechanical pressure appearingwithin the stress tensor. With these changes, a final statement of the firstlaw of thermodynamics is

(14.32)

This relation will be instrumental in completing our construction of adifferential form of the energy equation in the next chapter.

14.3 Perfect Gases

When applying Eq. (14.30), or ultimately Eq. (14.32), to liquids and gases,some simplifications can be made to accommodate their specialthermodynamic properties. In the case of perfect gases, in particular, anumber of simplifications can be derived that will be drawn on in laterdiscussion. These are now considered.

In the case of perfect gases for which Eq. (14.1) holds, it may be shown(see Problem 14.3) that

(14.33)

Substituting this into Eq. (14.30) and taking advantage of Eq. (14.1) toreplace yields

(14.34)

If is taken to be a function of and , then

(14.35)

and Eq. (14.34) implies that

(14.36)

This result is consistent with the expectation (Batchelor 1967) that in aperfect gas, the internal energy is a function of temperature only, so that

(14.37)

In this case, Eq. (14.36) can be written as

(14.38)

For a constant process so that , Eq. (14.10) shows that , and appealing toEq. (14.14), it follows that

(14.39)

which in view of Eq. (14.37) means that

(14.40)

Equations (14.38) and (14.40) establish Carnot’s relation for perfect gasesto the effect that

(14.41)

The implication of these results for perfect gases is that the energyequation (14.30) for this case simplifies to the form

(14.42)

Assuming further that the specific heats and can be taken to be constantover a range of temperatures, as is approximately true for air at moderatetemperatures, then local integration of Eq. (14.42) gives

(14.43)

where is a constant. Furthermore, a calculation with Eq. (14.29) usingEqs. (14.1) and (14.41) shows that the entropy varies according to

(14.44)

where and is a constant. An implication of Eq. (14.44) is that for anisentropic process, which would include frictionless and nonconductingflow, constancy of means that the term is a constant, in which case

(14.45)

where are reference pressures and densities, respectively. A system orflow field for which the density depends solely on the pressure or,equivalently, the pressure is solely a function of density is referred to as

being barotropic. In this case, as occurs in Eq. (14.45), it may be writtenthat

(14.46)

Clearly the isentropic change of a perfect gas with constant specific heatsis a barotropic process.

Problems

14.1 Substituting Eq. (14.10) into (14.14) gives

(14.47)

Use this to derive

(14.48)

Hint: use to replace and in Eq. (14.47) and set . Then simplify the resultby taking appropriate derivatives of the identities

(14.49)

and

(14.50)

where and denote thermodynamic functions of to distinguish them fromfunctions of . Use Eq. (14.48) to verify Eq. (14.38).

14.2 By imitating the approach in Problem 14.1, show that

(14.51)

follows from Eq. (14.47) when .

14.3 Verify that Eqs. (14.33), (14.38), (14.40), and (14.44) hold for perfectgases.

15Energy Equation

We now consider the first law of thermodynamics as it pertains to theinternal energy of an arbitrary material fluid element

(15.1)

According to Eq. (1.11), the first law is

(15.2)

where the terms on the right-hand side are now to be understood,respectively, as the heat transfer and work done on the element as it movesthrough a flowing fluid. In other words, these processes are connected tothe conditions of the external environment of the material fluid element.The end result is an energy equation relevant to the entire fluid and allmaterial elements within it. The work term in Eq. (15.2) is specificallythat which affects internal energy of the fluid element, for it will be seenshortly that not all the work done on fluid elements causes changes ininternal energy. Some of it may, in fact, change the kinetic energy of theelement without affecting internal energy.

The left-hand side of Eq. (15.2) can be simplified using the sameprocedure that has been illustrated in the previous chapters. The result isthat

(15.3)

The heat conduction term on the right-hand side is readily accounted forusing Fourier’s law to the effect that the flux of energy/area-sec byconduction through the fluid is given by

(15.4)

where is the thermal conductivity in units of watts/(m K). Because isperpendicular to surfaces of constant , heat flows down the steepest decent

of the temperature field. The total flux of energy into the material element(hence the need for a minus sign) is then given by

(15.5)

which, after applying the divergence theorem and substituting Eq. (15.4),yields

(15.6)

The work done on a material element can be accomplished by both bodyforces and surface forces with the result being changes in the internaland/or kinetic energy. For the purposes of expressing the first law ofthermodynamics, as mentioned previously, it is necessary to limit thework term in Eq. (15.3) to include just the work that affects internalenergy. To accomplish this, it is clearly required to be able to decide whichkind of energy is affected by which particular part of the work term. Itwill soon become apparent, however, that this separation of effects is notdifficult to figure out once the work done by the surface forces isexamined in some detail.

To compute the total work done by surface forces on a material element,consider a small piece of the surface that has area and orientation , asshown in Fig. 15.1. The local surface force on this subelement is , andtaking a dot product of this with a unit vector in the direction to which thesurface is moving, namely, , gives the component of surface force actingin the direction of motion as

(15.7)

Figure 15.1. The work by surface forces on a fluid element of area andorientation depends on the component of in the direction of .

In time , the surface moves the distance

(15.8)

so consequently, the total work done on the subelement is

(15.9)

Dividing by to get the work per second and summing the contributionsfrom all pieces of the surface results in

(15.10)

as the total work of surface forces on the material element.

With the goal of turning Eq. (15.10) into a volume integral, consider theidentity

(15.11)

which is easily verified using index notation. Substituting Eq. (15.11) intoEq. (15.10) yields

(15.12)

and finally, using the divergence theorem, it is obtained that

(15.13)

where the integrand in this expression may be interpreted as the work/vol-sec performed by surface forces.

To interpret Eq. (15.13) further, it is necessary to carry out thedifferentiation within the integrand. Proceeding now by switching to indexnotation and using the symmetry of , it follows that

(15.14)

Returning to direct notation, Eq. (15.14) becomes

(15.15)

where the first term on the right-hand side contains the tensor divergencedefined in Section B.1 and the second incorporates the tensor innerproduct defined in Eq. (1.22). According to Eq. (15.15), the work done bysurface forces naturally divides into two terms. It remains to be seen ifthis demarcation is between work affecting internal and kinetic energy,and if so, which term affects which kind of energy.

A relatively easy way to answer the question of how to interpret the termson the right-hand side of Eq. (15.15) is to consider the equation governingthe dynamics of the kinetic energy/mass, . This relation is derived bytaking a dot product of the momentum equation (13.18) with . In fact,

(15.16)

and so the kinetic energy equation is

(15.17)

The right-hand side contains two work terms that affect kinetic energy.The first is work done by the gravitational force, which, like all bodyforces, affects only kinetic energy and not internal energy. The last termon the right-hand side is identical to the integrand of the first term on theright-hand side of Eq. (15.15), implying that this part of the work done bysurface forces must affect kinetic energy. The structure of this term isconsistent with this interpretation, because depends on having gradientsof the stress tensor that may be associated with local force imbalances thataccelerate fluid elements. As fluid particles accelerate, their kineticenergy changes.

It should be evident now that the second term on the right-hand side ofEq. (15.15) is work that affects internal energy and not kinetic energy. Anadditional motivation for this conclusion lies in the mathematical form ofthe integrand that involves the product of the stress tensor with gradientsof the velocity field. Nonzero velocity gradients imply the existence of avariation in the velocity from one side of a fluid element to the other, andif so, there is an implied deformation of the fluid element. In other words,the fluid element can be imagined to “squish” and otherwise undergo achange in shape that requires work by the surface force to accomplish.

There is no net acceleration of the fluid element implied by this process,and all the work done in this case leads to changes in the internal energyof the molecules.

The considerations up to this point serve to establish that the division ofthe work in Eq. (15.15) is between work of surface forces affecting kineticand internal energies, respectively. Putting this result into Eq. (15.3)together with Eq. (15.6) yields the first law of thermodynamics in theform

(15.18)

and gathering terms together, this becomes

(15.19)

As before, because the material element is arbitrary, the integrand ofEq. (15.19) must be zero, thus yielding

(15.20)

as the differential form of the energy equation.

Further interpretation of the surface work term in Eq. (15.20) can be hadby substituting for using Eq. (9.4), in which case

(15.21)

Working in index notation and taking advantage of the symmetry of andthe definition of in Eq. (12.3), it can be proved by direct computation that

(15.22)

Moreover, substituting for from Eq. (12.53) and using the fact that , acalculation gives

(15.23)

It is clear from Eq. (1.22) that for any tensor so that .

Defining

(15.24)

as deformation work/mass-sec and substituting into Eq. (15.21), it followsthat

(15.25)

This expression gives some insight into the nature of the work by surfaceforces as they affect the internal energy. In particular, represents theirreversible dissipation of mechanical energy/mass-sec into heat. Thisdeformation work acts to heat up the fluid at the expense of the kineticenergy of motion. In exceptional circumstances, might be quitesignificant, as in the heat generated by the rapid shearing of oil in thenarrow confines of a journal bearing. The first term on the right-hand sideof Eq. (15.25) accounts for the reversible work of isotropicexpansion/compression in a compressible flow.

Substituting Eq. (15.25) into (15.20) gives the energy equation as

(15.26)

The final step in developing this equation is to replace the explicitdependence on on the left-hand side with the thermodynamic relationEq. (14.32) that depends on measurable variables such as , and . With thissubstitution, Eq. (15.26) becomes

(15.27)

where the third term on the right-hand side reflects the fact that we havedistinguished between thermodynamic and mechanical notions ofpressure. This term will cancel if the two interpretations of pressure leadto the same result. Although this is usually the case, there are someextreme circumstances, for example, when large pressure changes travelthrough a fluid in a shock wave, when the thermodynamic pressure mayvary from the mechanical pressure. This can be due to a time lag in the

return of the fluid to molecular statistical equilibrium after the shockpasses.

The difference in pressure definitions can be accommodated empirically,consistent with the modeling of the stress tensor in Eq. (12.41), bypositing that it depends tensorially on the velocity derivatives. Becausethe pressure difference is a scalar, it is appropriate to assume that

(15.28)

where is an isotropic tensor. For second-order tensors, this means that

(15.29)

where the scalar is known by various names such as the expansionviscosity, the expansion damping coefficient, the bulk viscosity, or thesecond coefficient of viscosity. Substituting Eq. (15.29) into (15.28) givesthe pressure difference as

(15.30)

With this result, Eq. (15.27) becomes

(15.31)

where it is seen that accompanying the deformation work term that canonly enhance internal energy is a strictly positive term (the expansiondamping) that reflects the use of Eq. (15.30).

Often in the case of liquids, and sometimes for small temperaturevariations in gasses, it can be assumed that the thermal conductivity is aconstant, so that may be taken outside the gradient operator inEq. (15.31). Moreover, as mentioned, it is often justified to set and toignore the deformation work. With these assumptions, the energy equationbecomes

(15.32)

Specifically for liquids, the last term in Eq. (15.32) can be neglected dueto the small value of the coefficient of thermal expansion and the generalincompressibility of the fluid. In this case, the classic convective diffusionequation

(15.33)

results, for which

(15.34)

is the thermal diffusivity. In the absence of a velocity field, Eq. (15.33)also accounts for the temperature field in a solid.

For a gas satisfying the perfect gas law with constant, it is appropriate tobase the conversion of Eq. (15.26) into a temperature equation by the useof Eq. (14.42) rather than Eq. (14.30). In this case, the energy equationtakes the form

(15.35)

If the flow is also incompressible, then the last term drops out, and theenergy equation once again has the form of a simple convective diffusionequation, as in Eq. (15.33), though in this case, .

Having considered the kinetic and internal energy equations separately, itshould be remarked that they can be combined into an equation governingthe total energy in the fluid. Thus adding Eqs. (15.17) and (15.20) resultsin the relation

(15.36)

where the terms on the right-hand side include the work of gravity, heatconduction, and, in the last term, the total work of the surface forces. Thebalance of total energy in Eq. (15.36) is often referred to as a statement ofthe first law of thermodynamics. However, the fact that the kinetic energyequation has independent status as a consequence of the Navier-Stokesequation suggests that Eq. (15.36) is best viewed as a secondary result

deriving from combining the kinetic energy balance and the first law ofthermodynamics in the form of the internal energy equation in (15.26).

Problems

15.1 Derive Eqs. (15.22) and (15.23).

15.2 For incompressible flow, use index notation to prove the identity .

15.3 For steady, incompressible, unidirectional flow (i.e., flow for whichthe velocity field varies only in a direction normal to the motion) andneglecting gravity show that for a region with boundary that

(15.37)

where is the deformation work term defined in Eq. (15.24). Show thatwhen applied to the circular flow between two concentric cylinders, as in ajournal bearing, Eq. (15.37) implies that the total rate at which the fluid isheated by deformation work is equivalent to the sum of the torques timesangular velocity applied at the moving boundaries.

16Complete Equations of Motion

We have now completed the major goal outlined in Section 1.4 ofdeveloping a system of partial differential equations with which to predictfluid flow based on the physical laws of mass, momentum, and energyconservation. The solutions to most fluid flow problems, in principle, areobtainable from these equations though the fact that they are complicated,coupled, and nonlinear means that it is rare that they can be solvedwithout the use of numerical techniques. In fact, a vast industry ofcomputational fluid dynamics software has arisen to provide numericalsolutions on computers to the governing differential equations of fluidflow. Some of the flows where an exact or approximate analytic treatmentof the governing differential equations is possible are illustrated in laterchapters.

It is also the case that the fundamental equations provide a launchingpoint for a variety of simplified analyses of fluid flows that can be verypowerful tools for getting at the essence of the physics of fluid motion andproviding useful predictive capability. In this chapter, after first gatheringtogether the complete system of equations that has been derived, theessentials of two very helpful techniques for fluid flow analysis aredeveloped from the exact equations. One encompasses the Bernoulliequation, and the other is an approach based on the use of controlvolumes. A subsequent chapter illustrates the use of these methods insolving some particular flows of interest.

16.1 Differential Equations of Fluid Flow

For convenience, the complete system of equations with which to solvefor the dynamics of fluid flow are now gathered together. In the generalcase of compressible flow, we have the continuity equation (10.24)

(16.1)

the momentum equation (13.22)

(16.2)

and the energy equation (15.31)

(16.3)

These represent five coupled equations, and when supplemented by anadditional equation of state, such as the perfect gas law in Eq. (14.1),provide six equations with which to solve for the six unknowns consistingof the density, three velocity components, the pressure, and temperature.

For the common occurrence of incompressible flow, the complete systemof equations typically simplifies to four coupled equations for the pressureand three velocity components. In this case, the governing equations aremass conservation in the form

(16.4)

and the momentum equation

(16.5)

Assuming that such effects as a temperature-dependent viscosity are notpresent, Eqs. (16.4) and (16.5) are decoupled from consideration of theenergy equation.

During the development of the equations of motion in previous chapters, itwas shown how the Navier-Stokes equation can be generalized to includesystem acceleration in Eq. (13.26) and variable viscosity in Eq. (13.27). Inaddition, several special forms of the energy equation were considered inEqs. (15.32) and (15.35). Many other forms of the equations are possible,for example, based on alternatives to Eq. (12.54) that are appropriate tonon-Newtonian fluids such as polymers. Generalizations to the energyequation allow for treatment of combustion and chemical reactions. Formany of these situations, additional physics needs to be brought to bear,such as equations for the conservation of individual species,generalizations of Fourier law of heat conduction, and source terms. All

these many directions are appropriate to more advanced or specializedstudies for which the analysis of fluid flow is one essential aspect.

16.2 Bernoulli Equation

A family of relations, known as the Bernoulli equations, express theconservation of what amounts to a sum of energy-like terms either alongstreamlines or within either a part or whole of the flow domain. Thisknowledge can be brought to bear in finding approximate solutions to avariety of flow problems with considerably less difficulty than would beentailed in trying to obtain the solution to the complete governingdifferential equations. The following discussion considers a Bernoulliequation for steady flow that reduces in different cases to simplerexpressions as well as an equation specifically for nonsteady irrotationalflow. Crocco’s relation is also derived, which provides circumstanceswhen Bernoulli’s equation is particularly easy to apply in steady flow.

16.2.1 Bernoulli Equation for Steady Flow

A relation that may be interpreted as giving the rate of change of the totalenergy along fluid particle paths was given in Eq. (15.36). For steadyflows, the particle paths are unchanging and are identical to thestreamlines. In this context, Eq. (15.36) can therefore be viewed as givinginformation about the changes to the energy along streamlines. In fact,this line of reasoning can be considerably extended by adding potentialenergy and other effects to the kinetic and internal energies to therebyobtain a more comprehensive idea of what is conserved along streamlinesunder different flow circumstances. The relations that are derived, knownas variants of the Bernoulli equation, can sometimes provide a very usefulunderstanding of the flow field without having to solve the flow equationsin detail.

To derive the Bernoulli equation, consider the energy equation (15.36) inthe form

(16.6)

in which a substitution has been made for using Eq. (9.4). It is often thecase that gravity can be determined from the gradient of a potential, inwhich case

(16.7)

For example, in the vicinity of the earth’s surface, , assuming that is thevertical coordinate. If it is further assumed that does not change in time,that is, , as is certainly true for most applications, then it can be said that

(16.8)

Substituting Eq. (16.8) into Eq. (16.6) and moving it to the left-hand sidegives

(16.9)

wherein the equation now is a statement concerning the variation ofinternal, kinetic, and potential energies along particle paths.

The pressure work term on the right-hand side of Eq. (16.9) can berewritten using the identity

(16.10)

in which case, Eq. (16.9) becomes

(16.11)

This equation describes how the quantity

(16.12)

representing the sum of energies plus the flow work term , changes alongfluid particle paths. Note that an alternative to Eq. (16.12) is to introducethe enthalpy/mass, , in which case, .

For steady flow, the pressure time derivative term on the right-hand sideof Eq. (16.11) vanishes. The remaining two terms on the right-hand side

account for physical phenomena originating at the molecular level, whichis reflected in the appearance of the viscosity and thermal conductivity. Incases where such diffusive and dissipative actions are insignificant, thenEq. (16.11) becomes for steady flow

(16.13)

Because by definition is a unit tangent vector at any point along astreamline, after dividing Eq. (16.13) by , it may be inferred that thedirectional derivative of along streamlines vanishes, in other words,

(16.14)

where is distance (i.e., arclength). Integrating Eq. (16.14) along astreamline from an arbitrary point where shows that remains at this valueeverywhere along the streamline. Thus one statement of the Bernoulliequation is that if the flow is steady, nonconducting, and inviscid, then

(16.15)

along streamlines. Depending on the intended application, the usefulnessof this relation may depend on there being a point on the streamline whereeach of the quantities making up is known so that can be determined.

In the special case of perfect gases with constant specific heats,Eqs. (14.41) and (14.43) imply that apart from an immaterial constant, theenthalpy is

(16.16)

and when used in conjunction with Eq. (16.15), it shows that the Bernoulliequation takes the simple and practical form

(16.17)

Here, and in other simplifications of Eq. (16.12) to follow, is to beinterpreted as being the sum of the terms in the relevant form of theBernoulli equation at a specified location. An alternative form ofEq. (16.17) derived with the help of Eqs. (14.1) and (14.41) is

(16.18)

In incompressible flow wherein , Eq. (15.26) shows that is constant onstreamlines if the flow is also steady, frictionless, and nonconducting. Thismeans that for steady, incompressible, nonconducting and inviscid flow,the internal energy can be removed from the Bernoulli constant so that, infact,

(16.19)

is unvarying on streamlines. Many common applications of the Bernoulliequation fit the requirements of Eq. (16.19), as is seen in a number ofsubsequent examples. A particularly important use of Eq. (16.19) is indetermining the pressure field in flows for which the velocity fieldhappens to be known from other considerations. This scenario wasillustrated previously in Chapter 7 for flows for which the velocitypotential was available analytically or could be computed.

16.2.2 Bernoulli Equation for Nonsteady Flow

An alternative form of the Bernoulli equation can be derived fornonsteady, irrotational flow that is either barotropic or has constantdensity. Consider the momentum equation (16.2) simplified to inviscidflow and known as the Euler equation:

(16.20)

Because the flow is assumed to be irrotational, , and as shown in Section4.2, the velocity is given by a potential according to . Moreover, thevector identity

(16.21)

can be readily derived using index notation, as suggested in Problem 16.1.With the help of this relation simplified by the fact that , and the use ofEq. (16.7) for , Eq. (16.20) becomes

(16.22)

A Bernoulli-type equation can be derived from this expression forcircumstances when the right-hand side can be written as an exactdifferential. This clearly occurs when the density is constant, and as willnow be shown, is also true when the flow is barotropic.

For barotropic flow, it follows from Eq. (14.46) that

(16.23)

Now define a function via the relation

(16.24)

in which the integration is along a path between and , is a unit tangentvector to the path, and is arclength. In fact, the barotropic condition asexpressed in Eq. (16.23) can be used to prove that is independent of thepath used in its definition and hence is a well-defined function (seeProblem 16.2). Taking a derivative of Eq. (16.24) with respect to yields

(16.25)

so that, in fact,

(16.26)

Substituting this into Eq. (16.22) gives

(16.27)

where if the density is constant. Note that while may be time dependent,this is immaterial, because such variations can be absorbed into in such away that the velocity, which is computed from by spatial differentiation,is unaffected (Batchelor 1967).

The knowledge that the sum on the left-hand side of Eq. (16.27) isconstant throughout the flow field as time changes is useful in a numberof contexts. One example is in the analysis of gravity waves on the freesurface of a liquid (Gill 1982). Another significant application concernsthe force on accelerating bodies in an inviscid flow for which is known as

a function of time. In such cases, Eq. (16.27) can be solved for thepressure, and the time-dependent force acting on the body can bedetermined. It is interesting to note that despite the assumption of inviscidflow, acceleration of the body leads to the presence of a net drag force. Anexample of using Eq. (16.27) in this way may be found in Problem 17.8.

16.2.3 Crocco’s Relation

The worth of the Bernoulli equation in connecting the values of betweenpoints on a streamline in steady flow depends on having a location in theflow where the Bernoulli constant can be determined for each streamline.Sometimes the problem of finding the Bernoulli constants for streamlinesis made easy because circumstances are such that many streamlines sharethe same Bernoulli constant. This section considers some situations wherethis property of the Bernoulli constant might occur.

Previously, in Eq. (14.12), the first law of thermodynamics was derived inthe form

(16.28)

Equation (16.28) holds for the reversible changes of a system as itdevelops in time. Now imagine a steady flow in which there arenonconstant spatial distributions of the thermodynamic variables. For anyfluid path through the flow, the state variables must change relative toeach other according to the restriction of Eq. (16.28). In a steady flow, thisis equivalent to the requirement that

(16.29)

because it guarantees that Eq. (16.28) holds along any fluid particle path.

From the definition of in Eq. (16.12), it follows that

(16.30)

Substituting for in this using Eq. (16.29) and expanding out the pressureterm it is found that

(16.31)

In steady, frictionless flow, the Euler equation (16.20) simplifies to

(16.32)

in which case, using Eqs. (16.21) and (16.32) in (16.31), the equation

(16.33)

known as Crocco’s relation, is derived. An important use of Eq. (16.33)lies in providing information about circumstances when , and thus islocally constant. All streamlines that intersect such a region will share thesame Bernoulli constant, thus enhancing the ease with which the Bernoulliequation can be used and interpreted elsewhere in the flow.

Crocco’s relation implies a number of different scenarios where . Forexample, in reversible adiabatic flow, the entropy is constant so that .Then, if the flow is also irrotational (i.e., ), is constant everywhere. Theseare conditions that are relevant to many practical flows. Less likely aresituations where, for example, the last term in Eq. (16.33) is zero because and are everywhere parallel.

16.3 Control Volume Equations

The solutions of the differential equations in Section 16.1 where they canbe obtained by numerical or analytical means provide a complete set offield quantities for describing fluid flow. Alternatively, there are oftencircumstances where it is not necessary to know all the details of themotion; rather, one is interested in primarily gross properties of the flowsuch as the total force acting on an object or the average heat transferfrom a surface. If knowledge of the local flow is not required, then insome cases integrated forms of the equations of motion such as thecontrol volume (CV) method discussed here can be employed to obtainthe desired information without solving the differential equations. Whentheir use can be justified, such integral methods can be relatively easy toapply, maintain useful accuracy, and save the expense and complexity ofnumerically solving the differential equations.

For the CV approach, a fixed volume in space is selected, usually with acertain goal in mind. Among the considerations involved in choosing a CVare the ease with which it can accommodate known features of the flow,such as boundary conditions, as well as arranging for the extraction ofdesired information such as the force on a body. Unlike a material fluidelement, the CV is usually assumed to remain fixed for all time, whereasthe particular fluid that is within it at any one moment changes. For thecontrol volume illustrated in Fig. 16.1, fluid freely flows into and out ofthe CV through the control surface (CS) that makes up its boundary.

Figure 16.1. A control volume, CV, with control surface, CS. Fluid freelyflows into and out of the CV.

The CV equations are obtained by integrating the equations of motionover the control volume. The remainder of this section considers thederivation of the CV equations for mass, momentum, and energy as wellas for angular momentum. The importance and usefulness of theserelations will be illustrated in the context of several inviscid flows in thenext chapter and again in later chapters when viscous flows areconsidered.

16.3.1 Mass Conservation

The first step in deriving a control volume form of mass conservation is tointegrate the continuity equation, written in the flux form in Eq. (10.27),over the CV. The result is

(16.34)

The term on the right-hand side can be transformed into a surface integralvia the divergence theorem. Moreover, because the control volume isfixed for all time, the time derivative in the integral on the left-hand sideof the equation can be moved outside the integration. The result is thestandard control volume form of the mass conservation equation

(16.35)

to the effect that the rate of change of total mass inside the control volumeon the left-hand side is equal to the net gain or loss of mass due to itsconvection into or out of the CV on the right-hand side. Note that theminus sign in front of the flux term reflects the fact that points outwardfrom the CV so that fluid traveling in this direction is exiting.

A potential advantage of the CV idea lies in the fact that the flux term onthe right-hand side of Eq. (16.35) only requires knowledge of the flowfield at the boundaries of the CV, that is, on the CS. In fact, it may be fareasier to supply this boundary data than it is to have detailed informationabout the flow in the interior of the CV. It is also evident that ifEq. (16.35) is used as a predictive tool, then it is only reasonable to expectit to provide general information such as the total flux of mass throughindividual surfaces and not the local flux at any given location on theboundary.

16.3.2 Momentum Conservation

A control volume form of the momentum equation (13.18) is derived by asimilar technique as done for mass conservation. However, beforeintegrating the momentum equation, it is helpful to place it into a moreconvenient form. Thus, considering just the left-hand side of Eq. (13.18),add to that the continuity equation Eq. (10.22) multiplied by , which istantamount to adding zero. This results in the expression

(16.36)

where the terms have been gathered into two groups. The terms in the firstgroup combine to form

(16.37)

while those in the second group may be shown to be equal to

(16.38)

as considered in Problem 16.4. The expression in Eq. (16.38) is thedivergence of the tensor that is formed out of the tensor product of withitself.

Using the results in Eqs. (16.37) and (16.38) in Eq. (13.18) gives

(16.39)

Now integrating Eq. (16.39) over the control volume, it is found that

(16.40)

As before, the time derivative in the term on the left-hand side can bemoved outside the integral. The second term on each side of the equationis in divergence form and can be transformed into an integral over thecontrol surface by the same analysis as led to Eq. (13.15). The result is

(16.41)

According to the definition of the tensor product in Eq. (1.23),

(16.42)

so that the first term on the right-hand side becomes

(16.43)

which is the net flux of momentum entering the control volume.

The vector in the last term in Eq. (16.41) is the force/area acting on thesurface of the control volume wherein is the outward unit normal vector.Substituting for using Eq. (9.4) and using (16.43), the control volumemomentum equation becomes

(16.44)

which states that the rate of change in time of the momentum in thecontrol volume, on the left-hand side, is respectively equal to the net gainor loss of momentum by its convective flux across the control surface, the

total gravitational force acting on the fluid in the control volume, the netpressure force acting on the control surface, and the net viscous force onthe control surface.

A common situation where Eq. (16.44) may be used is where part of theboundary is a solid surface. In this case, because is pointing out of thecontrol surface into the solid, the force in Eq. (16.44) along this part of thecontrol surface is that of the solid acting on the control volume. This isconsistent with the pressure term having the form and thus pointingtoward the CV. It is instructive to consider the viscous contribution at asolid surface in more detail.

For simplicity, consider incompressible flow parallel to a flat, solidsurface forming the plane at that is the lower boundary of a controlvolume, as shown in Fig. 16.2. The local normal pointing out of thecontrol volume at the surface is . With a local velocity field given by , acalculation reveals that the only nonzero component of the deviatoricstress tensor at the wall is

(16.45)

where

(16.46)

is the shear stress at the surface. According to Eq. (1.25), Eqs. (16.45) and(16.46) imply that

(16.47)

at the surface, in which case

(16.48)

In other words, the viscous force/area at the control surface is a retardingshear force that acts to slow the motion of the fluid flowing past thesurface. The negative of Eq. (16.48) is a force in the positive direction onthe solid surface caused by viscous shearing of the fluid above it.

Figure 16.2. Viscous force/area acting on the control surface of a controlvolume.

The fact that the control volume analysis includes the integral of surfaceforces is one of the main reasons why the method can be very useful. Thusthe entire force on a surface can be found in some cases without knowingthe detailed flow behavior anywhere along the boundary.

16.3.3 Conservation of Angular Momentum

Although a control volume analysis of the momentum balance can giveinformation about the total forces acting on surfaces, it is sometimesuseful to have information about the moments of the forces on a surface aswell. For example, it could be helpful to imagine that the total force actingon a surface does so collectively from a single point. In this case, thelocation of the point can be determined so as to conserve the angularmomentum produced by the true distributed force on the surface.Information about the moments acting on a surface can be obtained via acontrol volume form of the angular momentum equation that is nowconsidered.

A differential equation for angular momentum can be derived by takingthe moment of the momentum equation (16.39) with a position vector extending from a chosen origin. The result is

(16.49)

To get a CV form of this equation, each of the terms must be prepared forintegration. The first term on the left-hand side can be consolidated to theform

(16.50)

because is not time dependent.

The next term on the left-hand side of Eq. (16.49) can be written as

(16.51)

where the tensorial symmetry of has been used. Verification of thisidentity is readily done by converting to index notation. For the last termon the right-hand side of Eq. (16.49), using index notation and thesymmetry of , it may be shown that

(16.52)

The left-hand side of this equation is the cross product of two vectors. Theright-hand side requires some explanation. The expression is to beinterpreted as the product of the tensor (i.e., the vector cross defined inSection B.2) with . The right-hand side of Eq. (16.52) is thus thedivergence of the tensor , which is clearly a vector the same as the left-hand side of the equation.

Collecting the results in Eqs. (16.50), (16.51), and (16.52) together, theangular momentum equation now has the form

(16.53)

Integration of Eq. (16.53) over a control volume and converting thedivergence forms in the second term on each side into integrals over thecontrol surface using Eq. (13.15) yields

(16.54)

Substituting Eq. (9.4) for and moving the time derivative outside of theintegrand in the first term on the left-hand side gives the control volumeform of the angular momentum equation as

This states that the rate of change in time of the total angular momentumin the control volume is equal to the net gain or loss of angularmomentum by its convection into or out of the control volume, the netangular momentum attributable to the gravitational force, and, in the lasttwo terms, the net moments exerted on the control surface by pressure andviscous forces. Careful consideration in selecting the CV in applications

can help in simplifying the evaluation of the terms in the equation. Twoexamples of the application of Eq. (16.55) can be found in Chapter 17.

16.3.4 Conservation of Energy

The development of a control volume form of the energy equation can bepursued in a number of different directions, depending on which particularvariant of the differential form of the energy equation one starts with. Aparticularly useful result is based on the equation for total energy given inEq. (16.11). As was done in deriving CV equations for mass andmomentum, it is useful to convert the convective terms on the left-handside into a divergence form by the trick of adding times the massconservation equation Eq. (10.22) to the left-hand side. Collecting termsin the same way as was done in previous calculations, it is found that

(16.56)

where it may be noticed that the first term on the left-hand side lacks thepressure because the corresponding term has canceled out with the sameexpression on the right-hand side of Eq. (16.11).

Integrating Eq. (16.56) over a control volume and converting the terms indivergence form to surface integrals gives

This states that the rate of change of the sum of the internal, kinetic, andpotential energies in the control volume on the left-hand side is equal to,on the right-hand side, respectively, the net gain or loss of energy viaconvection (including the flow work by the pressure), the work done byviscous forces on the control surface, and the net gain of heat byconduction through the boundary. Unlike the Bernoulli equation, where theassumption of steady flow has to be made to justify dropping the pressuretime derivative term, the control volume energy equation applies tononsteady flows because no assumption regarding the pressure timederivative has been made. In the same vein, the assumption of inviscidand nonconducting flow is not made for the CV energy equation, so it isapplicable, in principle, to complex engineering flows such as piping

systems and turbines, where the differential equations would be especiallydifficult to solve numerically.

Problems

16.1 Prove the identity in Eq. (16.21) by expanding out in index notationand using Eq. (1.18).

16.2 Use the same kind of argument as was given in Section 4.2 to showthat for a barotropic flow, Eq. (16.24) defines a path-independent function.

16.3 Show that for steady, frictionless, and nonconducting flow,Eq. (15.26) implies that , where is arclength along a streamline. Integratethis result and show that an alternative form of Eq. (16.15) is

(16.58)

where evaluation of the integral in this expression depends on the detailsof how and are related. When is constant, Eq. (16.19) follows. Inbarotropic flow for which Eq. (14.45) applies, show that Eq. (16.18)follows.

16.4 Prove the identity

16.5 Show how Eq. (16.29) follows from (16.28) by expressingLagrangian quantities in terms of their corresponding Eulerian fieldsevaluated on particle paths.

17Applications of Bernoulli’s Equation andControl Volumes

Three classical problems that are amenable to solution without solving thefull equations of motion are considered here. These help to illustratesituations wherein the approximate techniques based on Bernoulli’sequation and the use of control volumes can be used to great advantage.Many other interesting cases can be found in the literature. A number ofother applications may be found in the problems at the end of the chapter.

17.1 Fluid Impinging on a Plate

A careful blend of the control volume approach with the Bernoulliequation can be used effectively to explain a considerable amount of thephysics entailed in the impingement of a uniform, planar, liquid jetstriking a tilted surface, as shown in Fig. 17.1. The surface is held on ahinge located opposite the midpoint of the incoming jet and is free torotate. Which way will the plate move, and what will be its finalequilibrium position? Such questions can be answered by obtaininginformation about the total forces and moments acting on the plate, anobjective that can be met using the control volume approach withoutsolving for the detailed flow field.

Figure 17.1. Two-dimensional jet impinging on a plate.

At a sufficient distance upstream of the plate, the impinging jet isassumed to have speed uniformly across its width. On encountering theplate, the jet is expected to divide with part going upward and partdownward. In what proportion it divides is one of the results that is to befound. Each portion of the split jet at some distance downstream from thepivot point may be assumed to reach a state where the motion is parallelto the plate. To enable the use of the Bernoulli equation, it is furtherassumed that the flow in the jet is frictionless, an assumption that can beexpected to be reasonably well satisfied away from the immediate surfaceof the plate where viscous action is inevitable.

The pressure outside the liquid jet can be assumed to be atmospheric, say, , everywhere. In particular, the two outer streamlines of the jet that defineits shape must have pressure . The Bernoulli equation applied to thesestreamlines suggests that the incoming speed of the jet, , is maintainedeverywhere along them. Thus the outermost velocity will be at pointsdownstream of the hinge, where the separate streams are parallel to thesurface.

When the flow settles in to being parallel to the plate, the inviscid 2Dmomentum equation (16.5) for the wall normal velocity componentreduces to

(17.1)

where and are coordinates parallel and normal to the plate, respectively.Equation (17.1) means that across the outer regions of the jet, and in thiscase, the Bernoulli equation implies that the exiting jets are uniform atvelocity . In other words, regardless of what happens in the region near thehinge, the velocity and pressure return to their incoming values at thosepoints where the flow is parallel to the surface.

The analysis thus far suggests that there is enough information about thevelocity and pressure fields at the inlet and outlet regions of the jet flow tobe able to take advantage of a control volume analysis. The first step inapplying this method is to decide on the control volume. In the presentcase, the CS needs to run along the plate surface so that the total force andmoments on the plate can be determined. A reasonable choice of CV is

that given by dashed lines in Fig. 17.2. In this, the control surface runsalong the plate and intersects the incoming and exiting jets at right angles,an approach that will be seen to make evaluation of the flux termsrelatively easy.

Figure 17.2. Control volume for the impinging jet. It is convenient to alignthe coordinates as shown with unit vectors in the directions, respectively.

Having selected a CV now consider the statement of mass conservation inEq. (16.35) for this case. Because the flow is steady, this equation reducesto

(17.2)

where the three control surfaces are numbered as in Fig. 17.2. For this 2Dproblem, a unit depth is assigned to the flow field so that the area integralover the control surfaces is reduced to 1D integrals along its length. Toevaluate Eq. (17.2), it is evident that for , is normal to the surface and thelocal velocity is . Similarly, at , there is and , while at , there is and .Assuming that the width of the incoming jet is and that the width of theoutgoing jets are and , respectively, then it is apparent from Eq. (17.2)that

(17.3)

Of course this result depends on the assumption that the flow is paralleland constant in the outflow jets.

The use of a coordinate system aligned with the plate means that thecontrol volume momentum equation will yield balances in directionsparallel and normal to the plate. In the present case, the control volumemomentum equation (16.44) simplifies to

(17.4)

The expression in the last term on the right-hand side is the pressuredifferential with respect to the ambient and is referred to as the gagepressure. The integral in this term is over the plate only, because thecontribution of the constant pressure on all other surfaces cancels out.Note, as well, that the viscous force term on the surface, which would actin a direction parallel to the plate, has been omitted in keeping with thefrictionless model that is being employed.

The constant flow conditions existing at the inlet and outlet locations onthe control surface means that Eq. (17.4) can be simplified further to

from which scalar relations for the momentum balance in two directionscan be derived. With defined relative to the plate orientation, it can beshown that , , and . Substituting these vectors into Eq. (17.5) producesmomentum balances in the and directions, respectively, in the form

(17.6)

and

(17.7)

Equations (17.3) and (17.6) form a coupled system that can be solved for and , yielding

This result answers the question of how the jet is apportioned between theparts traveling up and down after striking the plate. For , the division isequal because the plate is vertical to the oncoming stream. Consistent withour intuition, an increasing share of the fluid travels upward as the platevaries from the vertical to a horizontal position when .

The right-hand side of Eq. (17.7) is the total force that the plate exerts onthe control volume. This is in the direction and depends on the speed andcross-sectional area of the incoming jet. This force has to be supplied tothe CV to change the direction and hence the momentum of the incomingfluid. The force of the jet on the plate is equal and opposite to the force on

the CV and thus points in the direction. This force is a maximum equal to at because all the fluid is diverted by 90 in this case. The force is zerowhen the plate is parallel to the flow.

Consideration of the moments on the plate via the control volume form ofthe angular momentum equation can answer the question as to what theplate will do if it is free to swivel. Before proceeding with this formalanalysis, however, it is of interest to note that the subsequent motion ofthe plate can be deduced through a simple heuristic argument. Thus,because the fluid runs outward along the plate in both directions, theremust be a stagnation point somewhere along the plate that marks thedivide between fluid moving up and down. By the Bernoulli equation, thepressure is a maximum at this point, so the pressure distribution along theplate, and hence the force on the plate, must be generally larger on theside of the hinge that contains the stagnation point. This imbalance inpressure force should cause the plate to swivel accordingly.

A consequence of the greater flow upward is that the central streamline inthe oncoming jet must go upward, as seen in Fig. 17.3. This means that thestreamline that arrives at the stagnation point must originate below thelevel of the pivot point. Moreover, it was shown in the discussionsurrounding Eq. (7.28) that in irrotational flow, streamlines that intersect astagnation point lying on a solid wall must do so at a right angle. Thecombination of having to start below the hinge point and arrive at thesurface at a right angle means that the stagnation streamline has tointersect the wall at a point below the hinge, as depicted in Fig. 17.3. Inthis case, it is expected that the effect of the jet striking the plate,assuming it is free to rotate, is to move it toward a vertical orientationwith .

Figure 17.3. The streamline arriving at the stagnation point isperpendicular to the surface. The pressure has a maximum at thestagnation point, causing a moment that turns the plate toward thevertical.

To verify the last conclusion, consider the force moments induced on theplate by the jet as determined from a control volume analysis based on Eq.(16.55). In the present circumstances, this takes the form

where the coordinate system is assumed to be the same as previously.Moreover, the origin of the moment arm is taken to coincide with theorigin of the coordinates at the hinge. This is an attractive choice because,by symmetry, the first term on the right-hand side of Eq. (17.9) will notcontribute to the moment balance in this case.

To evaluate the remaining terms on the right-hand side of Eq. (17.9), set and carry out the indicated operations. For example, for the outlet , acalculation gives

Of course, this and the remaining terms represent moments perpendicularto the plane of motion. The third term on the right-hand side of Eq. (17.9)is evaluated in the same way as the second term. For the last term on theright-hand side, the calculation is made that

(17.11)

Using these results, Eq. (17.9) reduces to a single scalar equation

(17.12)

for the total moment of the pressure force acting on the control volumeabout the pivot point. Because , this is negative (i.e., in the clockwisedirection). The negative of this is the moment of the fluid on the plate,which is our main interest, and after substituting for and using Eqs.(17.8), it is found that

(17.13)

which is positive for and thus clearly acting to turn the plate into thevertical direction. Owing to symmetry when , there is no moment.

Some additional insight into the response of the plate to the jet can be hadby computing the effective distance below the hinge, say, , where the totalpressure force on the plate should act to induce the same moment as thedistributed force. By definition, satisfies

(17.14)

and after substituting Eqs. (17.7) and (17.13) yields

(17.15)

If is not small, then is large, and the plate would have to be very long forthe flow considered here to be a relevant model.

The tendency for flat plates to align themselves normal to an oncomingstream is a fact of common experience. For example, if one drops a rigidplate into a swimming pool, instead of slicing quickly through the water tothe bottom, the plate tends to find an orientation parallel to the bottom ofthe pool and fall relatively slowly. In fact, from the perspective of theplate, the oncoming flow is not unlike that considered in this section.

17.2 Draining a Tank

Another interesting problem that illustrates both the use of the Bernoulliequation and the control volume approach concerns the draining of a tank.Thus consider the tank shown in Fig. 17.4 of radius filled with a liquid toheight above a small circular nozzle of radius out of which the fluid isdraining. Assume that so that the tank drains slowly. Different exitnozzles are possible, and the interest here is in computing how the exitingflow rate depends on the type and size of the tap used to drain the tank.

Figure 17.4. Fluid draining from a tank.

Figures 17.5 and 17.6 present the details of two different nozzle types thatwill be considered. These include a smooth exit nozzle in Fig. 17.5 thathas been created by punching out the tank wall from the inside out. The

opposite situation is shown in Fig. 17.6, where the opening has been madeby punching in from the outside and is referred to as a Borda mouthpiece.

Figure 17.5. Detail of the exit flow near the smooth tap.

Figure 17.6. Detail of the exit flow near the Borda mouthpiece.

The assumption that the tank drains slowly means that, for all intents andpurposes, it may be supposed that the flow field at any instant of time iswell approximated as being steady state. The instantaneous streamlinesmust go from the top surface to the exit plane in a quasi-steady state as thetank drains, and it is reasonable to assume that the principle driving forcesare gravity and pressure with a minimal role for viscosity. With theseassumptions, the use of the Bernoulli equation can be justified to linkconditions along streamlines in the tank.

At the top of the tank, the velocity of the fluid may be assumed to beapproximately zero, while the pressure is taken to be atmospheric, . Inview of the elevation of the top surface of the tank over the reference levelat the tap, the Bernoulli constant on the top surface is , a value shared byall streamlines that originate there. Along the streamlines, as they firstdescend within the tank, the reduction in altitude is matched by anincrease in pressure. As the vicinity of the exit is approached, the velocityaccelerates and the pressure begins to decrease back toward the ambientpressure outside the exit plane. To the extent that the exiting jet can beassumed to depart in parallel lines, as shown in the figures, the sameargument as used in the previous section implies that the pressure does notvary across the exiting jet. In other words, the pressure is for allstreamlines at the exit. The velocity of the fluid at the exit, say, , is thusthe same for all streamlines and, using the Bernoulli equation

(17.16)

is seen to have the value

(17.17)

It is interesting to note that though the Bernoulli analysis can provide theexit velocity, which is essential for computing the rate at which the tankdrains, the exact variation of pressure and velocity along the streamlinesis unattainable by this kind of methodology. That information for aninviscid model requires solving the Laplace equation for the scalarpotential , taking into account the boundary conditions. From , thevelocity can be determined, and then the pressure using the Bernoulliequation.

The rate at which fluid departs the tank depends on the radius of theexiting jet where it is a parallel stream and has speed given by Eq. (17.17).In the case of the smooth tap in Fig. 17.5, the flow exits parallel to and ofradius equal to that of the mouthpiece itself. In this case, the volumetricflow rate at the exit is

(17.18)

For the Borda mouthpiece, the exit geometry does not allow for thedeparting jet to remain attached to the boundary. Instead it contracts into aparallel stream of a reduced radius , where can be found using a CVanalysis that takes advantage of the special geometrical features of theBorda mouthpiece. Thus define a control volume that includes thecomplete interior of the tank and the boundary of the exiting jet up to apoint where it is a parallel stream of radius . Because the fluid exiting theBorda mouthpiece enters the tap from a location offset from the wall ofthe tank, the fluid next to the interior wall of the tank is left relativelyundisturbed as the tank drains. This means that the pressure along theinterior tank wall is for the most part only a function of depth.Consequently, when evaluating the pressure force on the control surface,there is complete cancellation of the pressure force between each smallarea of the boundary and a small area diametrically opposite to it on theother side of the tank, with the sole exception of the nozzle. At the nozzleexit, the pressure is , and this contributes to the pressure integral pointinginto the tank. Across from this point on the opposite tank wall, it is fair tosay that the fluid is virtually stagnant, so its pressure is approximately

and the contribution to the pressure integral from this region is thus oriented in a direction pointing toward the opening.

Now we are in a position to apply the CV momentum equation (16.44) forthe component in the direction of the exiting jet. Substituting the pressurecontributions into this relation and noting that the momentum flux term is yields

(17.19)

from which it follows that

(17.20)

so that, in this case,

(17.21)

It will thus take twice as long to drain the tank with the Borda mouthpiecethan with the smooth, outward projecting nozzle.

17.3 Water Sprinkler

An application that brings to the fore the use of the control volume formof the angular momentum equation is that involving the operation of asimple rotary lawn sprinkler. As shown in Fig. 17.7, the jetting of waterout of the nozzles at the ends of the arms of length results in a spinning ofthe shaft, which is assumed to be at a constant angular velocity perpendicular to the plane of motion. It proves to be advantageous toanalyze the sprinkler flow from the reference frame of an observerrotating with the sprinkler. In this case, the flow is steady and the observersees the fluid exit the nozzle at the angle , as shown in the figure. Inparticular, the shaft of the sprinkler is aligned in the direction. We assumethat there is a flow rate of of water leaving each end of the shaft and thatthe exit plane of the nozzle has area . To the observer fixed on the rotatingshaft, the symmetry of the exiting flow suggests that there is nothing to begained by applying the control volume momentum equation since it wouldonly show that the axis of the sprinkler does not move. On the other hand,

the control volume angular momentum equation should be useful for tyingthe rotation of the sprinkler to the motion of fluid through the device.

Figure 17.7. A simple lawn sprinkler.

The angular momentum equation (16.55) is limited to inertial observers soit must be appropriately generalized to accommodate the noninertialobserver attached to the sprinkler. In fact, the steps leading from theNavier-Stokes equation to Eq. (16.55) can be readily applied to Eq.(13.26) instead, thereby producing the desired generalized form of the CVangular momentum equation. In particular, taking a cross product of themoment arm with the fictitious force terms in Eq. (13.26) and integratingover the control volume introduces additional terms into Eq. (16.55)representing the moments of the fictitious forces. These are body forcesand remain as integrals over the control volume. For the limited case of anobserver rotating at constant angular velocity, only the centrifugal andCoriolis forces need be included from among the fictitious forces. In thiscase, the generalized control volume angular momentum equation is

where the last two terms are derived from the centrifugal and Coriolisterms, respectively.

Depending on how the control volume is selected, different kinds ofinformation about the sprinkler can be obtained. As our interest concernsthe overall operation of the sprinkler, it is fitting that a control volume bechosen that encompasses the entire sprinkler arm, as against, for example,using a control volume that just contains the fluid. An appropriate controlvolume is shown in Fig. 17.7, which cuts through the shaft of the sprinklerat the axis and through the exit stream of the fluid. Because the sprinklerarm is stationary to the observer, the nonsteady term on the left-hand sideof Eq. (17.22) can be omitted. Also, for simplicity, the effects of gravityon the right-hand side can be ignored.

Considering the remaining terms, the first term on the right-hand siderepresents the flux of angular momentum through the exit plane. The

moment arm to this location is essentially , while and . Consequently, theflux term is evaluated to be

(17.23)

with a second equal contribution coming from the second nozzle. Notethat only the part of the exiting velocity that is normal to the axis of theshaft makes a contribution to the moment.

The third term on the right-hand side of Eq. (17.22) fails to make a netcontribution to the balance of angular momentum because it can beassumed that the pressure is uniform and atmospheric at all points on thecontrol surface, including the exit jets from the nozzle that may influencethe component of Eq. (17.22). The next term, depending on the deviatoricstress, can be viewed as the force moment or torque exerted on the CSwhere it cuts through the sprinkler shaft. In the absence of details of theconstruction of and force distribution in the shaft, it is adequate to refer tothis contribution as , with the minus sign suggesting that, for the mostpart, this moment contribution acts as a resistance to the motion of thesprinkler arm.

The centrifugal term in Eq. (17.22) fails to make a contribution in view ofthe calculation

(17.24)

where the moment arm here is with . Lastly is the Coriolis term, whichgives

(17.25)

for each arm of the sprinkler. This is negative, and its presence reflects theneed to accelerate the fluid as it moves outward, as discussed in Section8.4.

Assembling the terms, it is found that the balance of moments is

(17.26)

In this, the jetting of momentum outward from the nozzles adds angularmomentum to the CV, which is necessary to accelerate the fluid as itmoves outward in the sprinkler arms and to overcome the resistance toturning of the shaft. According to Eq. (17.26), the sprinkler rotates at theangular velocity

(17.27)

a rate that is enhanced by the jetting of the fluid out of the sprinkler andreduced by the moment needed to overcome the resistance in the shaft. If can be neglected, for example, if the shaft were frictionless, then thesprinkler would rotate with angular velocity

(17.28)

If the sprinkler is being turned through an external agent, then and islarger than the value in Eq. (17.28). For , the sprinkler cannot moveunassisted because the water is ejected radially.

The exit velocity of the fluid as seen by an observer fixed to the groundcan be determined with the help of Eq. (8.23). Using polar coordinates inthe reference frame of the fixed observer, the exit velocity of the fluidseen by the rotating observer is . Consequently,

(17.29)

In the absence of resistance in the sprinkler, Eq. (17.28) implies that the term in Eq. (17.29) is zero so that the fluid is seen to eject out radially.The presence of resistance in the shaft means that is smaller than thevalue in Eq. (17.28) so that there is a velocity component tangential to theexit plane and the fluid angles backward while traveling outward. Finally,if , the fluid would be flung forward as well as outward.

Problems

17.1 Consider a siphon used for draining a liquid of density from a tank,as shown in Fig. 17.8. Assuming the validity of Bernoulli’s equation forthe flow within the siphon, and noting that the pressure everywhere within

the siphon must be above the vapor pressure of the liquid to preventcavitation, determine the maximum length, , that the siphon can extendbelow the surface of the liquid.

Figure 17.8. A siphon; is limited by the condition that the pressure in thesiphon must remain above the vapor pressure of the liquid.

17.2 Determine the pressure coefficient for the inviscid flow past asphere, and show that it is equal to on the surface. Make a contour plot of , and indicate where the pressure maxima are.

17.3 Consider a point source in three dimensions located at above a solidboundary at . In addition to the point source, there is a uniform flow ofspeed . Calculate and make a contour plot of the pressure on the solidsurface, indicating where the maximum and minimum pressure arelocated. Calculate the and velocity components on the surface and add aquiver plot of these to the pressure plot.

17.4 A round jet of liquid of diameter , density , and speed impingesnormally on a large, round, flat disk. If the disk is of radius , what is thethickness of the layer exiting in the radial direction? Find the forcerequired to hold the disk stationary.

17.5 Consider the draining of a large tank of cross-sectional area througha small, smooth orifice of radius , as was considered in Section 17.2.Generalize the analysis by including the velocity of the free surface in theuse of the Bernoulli equation. If the initial depth of fluid is a distance above the level of the orifice, compute how long it takes to drain the tankto a depth of . What is the effect on the prediction of the time if thevelocity of the free surface is not taken into account in the Bernoulliequation?

17.6 Consider the potential flow past a hemisphere of radius lying on aflat plate with a uniform stream of velocity and density blowing over it.Show that the density of the material in the hemisphere, , must be greaterthan for the hemisphere not to fly off the plate.

17.7 Consider a circular, vertical pipe of cross-sectional area extendingover the interval with positive upward. For most of the length of the pipe, is constant and equal to . In the vicinity of , the pipe constricts to an area over a distance that is much smaller than . Assume that the pipe, whichwas initially filled with fluid, is open to the atmosphere on both ends sothat it is draining and the level of fluid decreases in time. Let representthe volume flux of fluid leaving the pipe at time so that the velocity,assuming it is uniform over any cross section of the pipe, is given by . Usethe nonsteady Bernoulli equation to derive a differential equation for inthe form , where and the over-dot denotes time differentiation. Note that can be used to find . Solve for numerically (e.g., using ode45 inMATLAB), and discuss the behavior of the solution in time.

17.8 The potential of a sphere of radius moving in the positive directionwith speed is given by the negative of Eq. (6.14). Assuming that thesphere is accelerating, so that is increasing, and the fluid has constantdensity, compute the instantaneous pressure with the help of Eq. (16.27),and then integrate over the sphere to get the drag force. Note that in thiscalculation, a time derivative of Eq. (6.14) yields two terms: onedepending on and one on . It can be shown (Batchelor 1967) that theformer does not contribute to drag consistent with d’Alembert’s paradoxfor steady flow and may be ignored. The term containing contributes todrag as if it were an additional inertial term with a multiplicative factor(known as the added mass) equal to . This force arises from the need toaccelerate the nearby fluid as the body passes.

18Vorticity

The significance of the vorticity field has been stressed at a number ofpoints in the discussion thus far. For example, the Helmholtzdecomposition showed that the vorticity can be used with the dilatation tore-create the velocity field. Furthermore, vorticity does often concentrateinto small regions, as in a tornado, and so by following the vorticitydynamics, one has an economical and insightful means of capturing theessence of the flow field. Another significant example is in the case ofturbulent flows, where in many ways the dynamics of the vorticity fieldoffers the most direct means of understanding such key processes in theflow as momentum transport and the passage of energy between scales.

An equation that governs the physics of the vorticity field in a movingfluid can be obtained by taking a curl of the Navier-Stokes equation(16.2). Under many circumstances, as in the case of incompressible flow,the pressure is removed by the action of taking the curl, so that theresulting vorticity equation offers a route toward determining the flowwithout directly taking the pressure into account. This may represent anadvantage in the analysis of some flows by providing a morestraightforward approach toward understanding the physics of the flowthan can occur with the coupled velocity and pressure. In particular, thepressure continuously adapts to the evolution of the velocity field so thatit may be more difficult to establish the origin and cause of flowbehaviors in this case than through examining the vorticity field and itsevolution. In flows for which vortical structures dominate the physics, asin turbulent flow, there very well may be a definite advantage to analyzingthe flow through numerical schemes that model the vorticity equation. Inthis case, the terms in the vorticity equation represent the physicalprocesses that are most relevant for understanding the dynamics of theflow field.

18.1 Vorticity Equation

In preparation for constructing a differential equation for the vorticity, itis helpful to rewrite Eq. (16.2) by replacing using (16.21), replacing using Eq. (4.10), and adopting Eq. (16.7) for the gravity term. The result is

(18.1)

where is the kinematic viscosity. The vorticity equation follows by takinga curl of Eq. (18.1), which for the present analysis will be assumed, in theinterest of simplicity, to have constant . Because the curl operatorcommutes with a time derivative, the first term on the left-hand sidebecomes the time rate of change of vorticity. Moreover, in view of theidentity equation (4.13), the curl of the terms in Eq. (18.1) containing thegradients , or must vanish.

In the case of the third term on the left-hand side of Eq. (18.1), by a directcalculation with the help of index notation, it may be shown that

(18.2)

Moreover, a curl of the last term on the right-hand side of Eq. (18.1)results in an expression that was encountered previously in the identityEq. (4.10). In the present context, because , it follows that

(18.3)

Collecting the various results together gives the vorticity equation incompressible flow as

(18.4)

The left-hand side of the equation is the material derivative of vorticity, ,that accounts for the changes in vorticity along fluid particle paths. Ofparticular interest is the first term on the right-hand side of the equationaccounting for vortex stretching and reorientation. The next sectiondescribes some of the unique physics associated with this term, which hasno direct counterpart in the dynamics of the velocity field.

The second term on the right-hand side is the viscous diffusion term. Ingeneral, vorticity creation at solid surfaces by friction and its subsequentdiffusion into the flow is the principal means by which vorticity appearsin flow fields. After vorticity is already present, convection and vortexstretching may strongly affect the vorticity distribution, but it must firstbe generated, and this is most commonly accomplished by the viscousterm.

To the extent that the flow is locally compressible, the third term on theright-hand side of Eq. (18.4) accounts for an increase or decrease inpreexisting vorticity depending on whether the flow is experiencing alocal compression or expansion, respectively. Finally, the last term on theright-hand side of Eq. (18.4) is identically zero if the density is constant orif the flow is barotropic so that Eq. (16.26) holds. When this term is notzero, as it may be in nonbarotropic flow, it represents an alternative meansfor vorticity production, known as baroclinic vorticity creation, and canoccur, for example, in the unequal heating of the atmosphere far from theinfluence of solid surfaces.

For flows with constant density, Eq. (18.4) simplifies to

(18.5)

In this case, without the action of viscosity at solid surfaces, there will beno vorticity in the flow. This explains why inviscid flow is generallyirrotational unless some other mechanism is presented for vorticitycreation. Equation (18.5) is of great interest for its applicability toincompressible turbulent flow, where it forms the basis for developingnumerical models that make the vorticity field the central focus.

In two-dimensional flow where and there is just a lone scalar componentof vorticity which is oriented in the direction normal to the plane ofmotion – Eq. (18.5) reduces to

(18.6)

The vortex stretching term vanishes in this case, implying that flows forwhich vortex stretching is an important phenomenon cannot be well

modeled as two-dimensional. The most significant example of this isturbulent flow, which is intrinsically three-dimensional and for whichvortex stretching is an essential part of the physics.

18.2 Vortex Stretching and Reorientation

The vortex stretching term appearing in Eq. (18.4) (or 18.5) is anessential and important component of the physics of the vorticity fieldthat deserves some elaboration. Toward this end, consider two materialpoints and in close proximity that are moving about in the velocity field .Define

(18.7)

as a vector pointing between the material points, and let denote the smalldistance between the points; is a unit vector pointing in the direction of ,so that

(18.8)

Because travels with the local fluid velocity, it satisfies

(18.9)

in which case, after integration from an initial time , when the particle isat to a general time , one has

(18.10)

For a small time interval , (18.10) suggests that is well approximated by

(18.11)

By the same argument, a good approximation for is

(18.12)

and subtracting Eq. (18.11) from this gives

(18.13)

which specifies how the vector pointing between the two points changesover a short time interval. Equation (18.7) implies that , and because byassumption is small, it is legitimate to write the Taylor series expansion:

(18.14)

Substituting this into (18.13) gives

(18.15)

Replacing in Eq. (18.15) using (18.8), dividing by , adding to both sidesof the equation, and rearranging terms gives

(18.16)

Taking the limit of this relation as and rearranging terms gives

(18.17)

where

(18.18)

is the fractional rate of change of the length of an infinitesimal linesegment oriented in the direction. Parameter is clearly independent of theexact value of as long as is sufficiently small. We now use Eq. (18.17) toexplain some of the physics of vortex stretching.

Our interest in Eq. (18.17) lies in what it says about the time evolution ofline segments oriented in the direction of the local vorticity vector. In thiscase, substituting into Eq. (18.17) and multiplying through by gives

(18.19)

The stretching term in Eq. (18.4) is seen to be decomposed here into thesum of two physical processes involving the evolution of a local materialline element. The first term on the right-hand side of Eq. (18.19) is in the

direction of the local vorticity vector and is proportional to the fractionalrate at which a local fluid line element is stretching (or contracting). Theeffect on the local vorticity in this case is to make it either stronger orweaker, depending on the sign of .

To interpret the second term on the right-hand side of Eq. (18.19), notethat differentiation with respect to time of the identity

(18.20)

yields

(18.21)

This means that is perpendicular to the direction of . Its role in Eq.(18.19) is in accounting for the shearing of vorticity out of the directioninto orthogonal directions. In this way, the vorticity vector changes itsorientation as the flow evolves. The total effect of is to simultaneouslystretch and reorient the vorticity field.

To see the implications of (18.19) in a more concrete setting, imagine thatthe vorticity vector at a point lies in the direction so that and . At thepoint in question, apart from the viscous terms, the vorticity equation foreach component takes the form

According to Eq. (18.19), the vortex stretching terms in each of theseequations can be expressed as

The first of these relations shows that the stretching of material fluidelements has the consequence of enhancing or diminishing the preexisting component. In this, because is evidently equivalent to , it can beinterpreted as the fractional rate of change in the length of a small linesegment. Equations (18.23b) and (18.23c) refer to the creation of and vorticity, respectively, out of the preexisting vorticity by shearingassociated with the velocity gradients and . As the initial vorticityreorients, and components appear as its projection into the and

directions, respectively. Figure 18.1 illustrates the process for theproduction of from . In this, is the component of projected into the direction, as is seen in Eq. (18.23b).

Figure 18.1. Shearing of in a velocity field to create vorticity.

In the general case, arbitrarily oriented vorticity filaments aresimultaneously stretched or compressed and reoriented by the shearingmotions. The propensity for vorticity to stretch and reorient is the maindriving force behind the appearance and maintenance of turbulence inflowing fluids. In essence, this physical process is the means by whichenergy is transferred to small scales, where the action of viscous forces insmoothing the flow and dissipating energy become important. Adiscussion of these and other aspects of turbulent flow may be found in anumber of books (e.g., Bernard & Wallace 2002; Pope 2000).

18.3 Kelvin’s Circulation Theorem

There are many occasions where vorticity of dynamical significance isdistant from the influence of viscous effects adjacent to solid boundaries.For example, vorticity generated at wing surfaces of an airplane andtraveling into the wake may be well organized to the point of creatingsignificant eddies that have an influence on trailing aircraft. Onceorganized, vortices may persist in a recognizable form for considerableperiods of time and move over long distances until they dissipate ordistort away from their original shape. The capacity for vortical objects toremain distinguishable over significant periods of time, particularly whenthey are removed from the direct effects of viscosity in the vicinity of aboundary, can be explained to some extent as being a consequence ofKelvin’s circulation theorem, which is now considered.

Kelvin’s circulation theorem concerns the time history of the circulationassociated with a closed loop in a 3D fluid that moves with the localvelocity field. It may be remembered from Eq. (5.15) that the circulationof a particular closed circuit at a particular time is given by

(18.24)

where is arclength and is the unit tangent vector. Without loss ofgenerality, as shown in Eq. (2.20), the circulation can be expressed using ageneral parameterization in the form

(18.25)

with representing the closed circuit. Kelvin’s circulation theorem assertsthat remains constant in time for any closed circuit convecting withininviscid flow that either has constant density or is barotropic.

To prove Kelvin’s theorem, let represent the time history of the fluidcircuit incorporated in Eq. (18.25). At each fixed , as varies, sweeps outthe closed circuit, while for fixed , represents a path line varying with .The circulation of the curve at any time is given by Eq. (18.25) as

(18.26)

In this, the velocity is evaluated at the location occupied by the closedloop at any given time.

Taking a time derivative of Eq. (18.26) yields

(18.27)

Because

(18.28)

the first expression in parentheses in the integrand on the right-hand sideof Eq. (18.27) is the substantial derivative of the velocity. The Navier-Stokes equation, simplified to the Euler equation (16.20) under theassumption of frictionless flow, can be used to replace the substantialderivative in Eq. (18.27), yielding

(18.29)

The identity

(18.30)

derived from Eq. (18.28) applies to the last term in the integrand on theright-hand side of Eq. (18.29), and with this it follows that

(18.31)

By being an exact differential, integration of this term around a closedcircuit is zero. Similarly, the middle term in Eq. (18.29) can be written inthe form of an exact differential via

(18.32)

so this term makes no contribution to .

Now consider the term in Eq. (18.29) containing the pressure. If isconstant, then it can be taken inside the gradient leading to the exactdifferential

(18.33)

Alternatively, in barotropic flow, Eq. (16.26) holds so that

(18.34)

In either case, the pressure term will not contribute to the right-hand sideof Eq. (18.29) so that

(18.35)

and Kelvin’s circulation theorem is proved.

The significance of Kelvin’s theorem becomes apparent when consideringa structural entity such as the vortex tube depicted in Fig. 18.2. Vortextubes are constructed by first defining a closed circuit and then computingthe collection of vortex lines that pass through the circuit. A vortex line isa line that is everywhere tangent to the vorticity field. In somecircumstances, as when there is strong rotation around a central axis, thevortex tube formed by a closed circuit around the axis of rotation will be

tornado-like, that is, it will have vorticity aligned along its axis with fluidswirling around it.

Figure 18.2. A vortex tube formed by the collection of vortex linesintersecting a closed loop.

As a consequence of the way in which vortex tubes are defined, it may beshown that the circulation is the same for any closed contour lying withinthe tube surface and surrounding the central axis of the vortex tube. Toprove this, note that the area between two such circuits can be chosen tobe the surface of the vortex tube. Moreover, letting and represent areashaving the two circuits as boundaries, then the area formed by and thesides of the tube between the circuits also constitutes an area having thefirst circuit as boundary. Consequently,

(18.36)

However, the vorticity at any point in the surface of the vortex tube isoriented in a direction tangent to the surface. Thus, the normal to thesurface of a vortex tube at any point is normal to the local vorticity field,and consequently, in this case. This means that the first integral on theright-hand side of Eq. (18.36) is zero, and so the circulation is constantalong the length of a vortex tube.

If a vortex tube has constant circulation along its length and the flow isinviscid, then Kelvin’s circulation theorem supplies the additionalinformation that the circulation of the tube does not change in time as itmoves through the fluid. Thus, in the absence of viscous diffusion, vortextubes will persist in the flow without loss of their strength. Besidesdramatic vortical events such as tornadoes, the persistence of tube-likevortical structures in approximately inviscid flow conditions is a hallmarkof many other flows, including high-speed jets, shear layers, and vortexrings. In turbulent flows, tube-like vortical objects of a range of scales arepresent that drive many of the essential physical processes. In this case,however, inviscid interactions between vortical objects is generally apowerful effect that disrupts the coherency of the vortex tubes so they are

unlikely to remain as an identifiable object for very long. Conversely,mechanisms exist in turbulent flow to continuously replenish the supplyof tube-like vortical objects.

By considering an infinitesimal vortex tube, it may be shown that vortexlines convect with the corresponding material fluid lines that they occupy.Moreover, because the vortex lines cannot abruptly end in the interior ofthe flow, vortex tubes must nominally either form closed loops, as in avortex ring, or end at boundaries. However, in turbulent and viscous flowregions, mechanisms are in place to disperse vortex lines, therebyeffectively disrupting the coherency of vortex tubes within the flowinterior.

18.4 2D Vortex Methods

When solving for flows that are intrinsically vortical in nature, there canbe advantages to using a numerical method that directly treats thevorticity equation. This section presents one such scheme, known as thevortex method, that models two-dimensional flows as a grid-freecollection of convecting vortical elements. It is applied in the next sectionto the computation of a model aircraft wake flow that also serves toillustrate some important aspects of analyzing fluid motion from theperspective of the vorticity field. Though attention is confined here totwo-dimensional, inviscid flow, generalization of the methodology toinclude viscosity (Chorin 1973) and three dimensions (Bernard 2013) isalso possible.

To begin, consider a two-dimensional, incompressible flow field specifiedby its vorticity distribution. Imagine that the flow domain is subdividedinto small regions and the vorticity is gathered up locally in eachsubregion to create a collection of point vortices at locations withcirculations . According to the Helmholtz decomposition discussed inChapter 4, the velocity field can be recovered from the vorticity field bysumming over the contributions made by each point vortex according toEqs. (5.17). Consequently, the velocity field at a point produced by thecollection of vortices is

These relations reflect the kinematics of the flow field. To capture thedynamics, it is necessary to determine the time evolution of the vorticesas they move under their mutually induced velocity field. An appeal toKelvin’s theorem provides justification for assuming that the circulationof each of the point vortices that collectively make up the flow fieldremains unchanged in time.

The motion of the vortices is determined as the solution of the coupled setof ordinary differential equations

for , which are derived by evaluating Eqs. (18.37) at the locations of thevortices. In this, each vortex does not contribute to its own motion. Theexclusion of self-induction in this model is necessitated by the neglect ofthe details of the flow surrounding each of the vortices. In other words, theact of lumping the vorticity into a point is done at the expense of losinginformation about the behavior of the local vorticity field around eachvortex location, including how the local vorticity field may contribute toits own motion. To improve accuracy, either more sophisticated analyticalor numerical models of the induced velocity can be used or the number ofpoint vortices can be increased so that each one draws from a smaller area.

One drawback of the model contained in Eqs. (18.38) is that whenever twovortices get close to each other, their mutually induced velocity becomeslarge and unphysical. This is purely a consequence of the modelinginherent in replacing a continuous vorticity field with equivalent pointvortices. One way to circumvent the singularity is to base the localvelocity calculation on a model, known as the Rankine vortex, in whichthe local vorticity that has been gathered into a point vortex is insteadassumed to be spread uniformly over a small circle of radius centered atthe point. In this case, the appropriate vorticity magnitude of each vortexis determined by the relation

(18.39)

which ensures that the circulation of the disk is the same as that of thepoint vortex. In terms of a local polar coordinate system around any of thevortices, the vorticity is related to the velocity according to

(18.40)

because . Integrating this in the radial direction gives

(18.41)

and Fig. 18.3 shows how this smoothing removes what would otherwise bea singularity in the velocity field. By applying Eq. (18.41) locally aroundeach vortex in the velocity computation, the implementation of Eqs.(18.38) becomes a well-behaved and useful means of accommodating thephysics of 2D inviscid flows containing vorticity. The next sectionpresents an application of these ideas.

Figure 18.3. Smoothing of the point vortex velocity field. Dashed linesshow the velocity singularity corresponding to a point vortex.

18.5 Simulation of a Wing Wake

As an interesting application of the vortex method, consider the flow atspeed in the direction past a finite wing of span centered between and in the direction, as shown in Fig. 18.4. By design, fluid travels faster overthe top surface than the bottom surface of the wing, meaning a higherpressure underneath according to the Bernoulli equation, and thus yieldinga net lift force. A contour drawn around the wing at any fixed spanwise location has a net positive circulation that comes about because thepositive spanwise vorticity generated by the faster moving fluid travelingover the top of the wing is stronger than the negative spanwise vorticitygenerated on the bottom of the wing. The vortex lines indicated in Fig.18.4 have an orientation consistent with this physical picture. Anotherimportant aspect of the figure is its depiction of how the spanwisevorticity on the wing turns into the streamwise direction in anantisymmetric distribution around the mid-plane as the fluid moves intothe wake. It is this streamwise vorticity that is to be gathered into pointvortices for the purpose of simulating the wake flow.

Figure 18.4. View from above of vortex lines representing the net vorticity(sum of bottom and top surfaces) shed from a wing in forward flight.Spanwise vortex lines over the wing reorient to form an antisymmetricdistribution of streamwise () vorticity in the wake.

If one looks upstream toward the wing from a point downstream, as shownin Fig. 18.5, the wake vorticity in Fig. 18.4 is aligned in the flow directionand may be collected into a distribution of vortices. On the top surface ofthe wing where the flow is accelerating, the streamlines tend to convergetoward the middle, whereas below the wing, where the flow is slowed,they diverge outward. This basic flow pattern leads to a downwashthrough the middle of the wing and an upflow at the sides and particularlyaround the outer edges of the wing. In effect, a swirling motion is createdthat is naturally represented in the immediate neighborhood of the wingby the streamwise vortices depicted in Fig. 18.5. The vortices havepositive circulation for and negative circulation for and originate in thereorientation of the vortex lines, as shown in Fig. 18.4. The vortices areconcentrated toward the wing tips where the most intense swirling flowoccurs in the form of tip vortices that bracket the wake region on eitherside.

Figure 18.5. End on view of the initial vortices in the simulation.Converging flow over the top and diverging flow over the bottom of thewing produce swirling tip vortices in the wake.

The circulation of a circuit around the wing at any spanwise location is amaximum at the central plane and decreases symmetrically to zero at thewing tips. This is consistent with the idea that the complete set ofspanwise vortex lines, as seen in Fig. 18.4, are present only at the center ofthe wing. The further one proceeds toward the wing tips, the more vortexlines and their circulation have turned downstream, thus lowering thecirculation on a circuit around the wing. One commonly considereddistribution of circulation of this type, and that will be considered here, isthat of an elliptically loaded wing, in which case,

(18.42)

Here is taken to be the spanwise coordinate and the vertical coordinate inthis purely two-dimensional model. The maximum circulation at thecenter of the wing is , while is clearly zero at the wing tips . In keepingwith our discussion surrounding Kelvin’s theorem, the decrease incirculation along the wing tip is exactly counterbalanced by a rise ofstreamwise circulation in the wake. Thus, as the vorticity sheds into thewake, it changes into the streamwise direction in such a way as to accountfor the loss of circulation at the wing. The result, a short distance into thewake, is a model of the streamwise vorticity in the form of the distributionof streamwise vortices shown in Fig. 18.5. By computing the dynamics ofthese streamwise vortices via the vortex method, a model is created forthe evolution of the wake behind a wing.

The initial streamwise position of the point vortices is assumed to be closeenough to the wing so that the wake has not begun to deform consistentwith the flow pattern in Fig. 18.5. The vortices are imagined to be orientedin the streamwise direction, and their time history is to be understood ascapturing what happens to the vortical wake in a fixed streamwise positionas the aircraft moves off upstream. For the model calculation, vortices areplaced at positions along the line . It proves best in terms of numericalaccuracy to assign equal circulation to the vortices on each half of thewing. Because the circulation drops from to zero between and , each ofthe vortices on this side of the wing is assigned circulation . Between and , the vortices are given the negative circulation . The sense of rotation forthe vortices is consistent with the central downdraft indicated in Fig. 18.5.

Because the circulation in Eq. (18.42) varies elliptically, and each vortexis to have the same circulation, the locations of the vortices are notuniform across the span. For the half wing , the positions of the vorticesare taken to be . Assuming that the th vortex takes its circulation from theregion where and , it follows that

(18.43)

Using Eq. (18.42), the positions can be determined recursively from theformula

(18.44)

In other words, starting with , Eq. (18.44) can be solved for , and thenfrom this value, can be computed, and so forth up to .

The initial location of the th vortex, , can be determined as the effectivecentroid of the rate at which circulation is reoriented to the directionwithin the interval . In other words,

(18.45)

where the minus sign on the right-hand side takes into account the factthat the positive circulation assigned to the streamwise vortex has comefrom a loss of circulation from the wing. The left-hand side of Eq. (18.45)is the weighted average of the circulation density, whereas the right-handside is the total circulation times the distance from necessary to equatethe two sides. The integral in Eq. (18.45) is readily obtained in closedform so that Eq. (18.45) can be solved to give the set of initial vortexlocation for one side of the vortex sheet. The remaining vortices areplaced at with negative circulation.

With the specified initial vortex positions and strengths, the algorithmdescribed in the previous section can be implemented. Figure 18.6illustrates the computed solution at time for a case in which , , . Thestream function is plotted together with the locations of the vortices at thistime. The roll-up of the vortical wake into the kind of tip vortices seenbehind aircraft is evident. If one equates distance behind the aircraft, inthis case, to time according to the relation

(18.46)

then the collection of results at different times can be joined together tocreate a representation of a rolled-up vortex sheet extending backwardfrom the wing. Problems 18.7 and 18.8 suggest some other directions thatcan be taken with this calculation of the wing wake.

Figure 18.6. Vortices and stream function in a simulated airplane wake.

Problems

18.1 Compute the stream function corresponding to a Rankine vortex.

18.2 Consider a 3D Rankine vortex in the form of a cylinder of radius oriented in the vertical direction. In other words, for constant in theregion for all . Using the Navier-Stokes equations for steady inviscid flowin cylindrical coordinates, compute the pressure field up to an arbitraryconstant . For any fixed value of , a surface can be defined. If it isimagined that this surface is the dividing plane between water below andair above, then describes the free surface of water containing a Rankinevortex. Show that the surface of the liquid is depressed in the region of thevortex and has the shape given by

(18.47)

Make a plot of .

18.3 Consider a spanwise flow with corresponding streamwise vorticity that is subjected to the imposed irrotational flow , constant, associatedwith a stagnation line at . Use Eq. (18.5) to derive an equation for , notingthe roles played by and in this relation. Solve for , and explain what thesolution represents.

18.4 Consider swirling flow around the axis with velocity field uponwhich is superimposed an irrotational flow , constant, that stretchesmaterial fluid elements aligned along the axis. Show that this velocityfield is incompressible. Specialize the equations for to this case. Showthat is the only nonzero component of vorticity and convert the equationto one for . Integrate this relation, and find the distribution of vorticity inthe flow field and explain what it signifies.

18.5 For the same flow conditions as in Problem 18.4, derive an equationfor directly from Eq. (18.5) with the help of Eqs. (18.2), (18.3), and(A.10). Integrate the equation with respect to and show that it leads to the equation for this flow.

18.6 Consider the two-dimensional velocity field in the region . Show thatit is incompressible, and calculate the stream function. Compute thevorticity field , and show that it satisfies Eq. (18.6) so it is an exactsolution to the Navier-Stokes equation. Determine the pressure from the momentum equations. Plot streamlines and explain what this flow is.

18.7 Write a computer code (e.g., using MATLAB) that implements thevortex method solution of the wake flow discussed in Section 18.5.Enhance the solution by adding in a ground plane using image vortices toenforce the nonpenetration boundary condition as well as a side wind.Study the effect of initial release height over the ground and the amplitudeof the side wind on the history of the vortices.

18.8 Add a calculation and animation in time of the pressure on the solidboundary in the vortex wake calculation in Problem 18.7. Because farfrom the vortices, a convenient velocity to use in scaling the pressure isthe maximum initial velocity on the boundary.

19Applications to Viscous Flow

Although the differential equations of fluid flow summarized in Section16.1 contain all the physics that is needed to uniquely determine fluidmotion, these equations do not as a general rule lend themselves to solutionby any means other than numerical approximation. The problem mostlystems from the nonlinear convective term wherein the rate of change of themomentum of fluid particles depends on the momentum of the fluidparticles themselves. As momentum changes, so too does the rate ofchange of momentum. Analytical solutions containing this physics are ararity, so most examples of exact closed form solutions to fluid flow are insituations where the nonlinear term happens to be identically zero or nearzero or can be simplified in some way.

A large part of the physics of the momentum equation is tied up in therelative importance of the viscous forces versus the inertia of the fluidparticles. In fact, the pressure force can be generally expected to align withwhichever of the remaining forces are dominant. Consequently, thequestion of whether the nonlinear terms in the Navier-Stokes equation canbe ignored boils down to a discussion of whether these terms are dominantor subordinate to the viscous terms. A dimensionless parameter known asthe Reynolds number expresses this ratio of forces and is thus an arbiter inmaking decisions as to whether the viscous and convective terms need tobe kept or omitted in different situations.

The next section provides a discussion of the Reynolds number that will beuseful in setting up a context for making approximations that lead toanalytical solutions of viscous flow problems. Following this, severalexamples of flows are considered wherein the convective terms areidentically zero or else close to zero because of the geometry of the flow orbecause the Reynolds number permits this approximation. The last sectionwill briefly consider some of the issues encountered in solving for themotion of bodies when analytical solutions are not available.

19.1 The Reynolds Number

For any specific flow problem, it is generally the case that one can identifyrelevant scales of motion such as a length and velocity scale, and ,respectively, that typify the kind of motion that is present. These can formthe basis of an examination of the magnitudes of the forces within the fluidthat affect the flow field. In particular, to study the relative magnitudes ofthe convection and viscous forces, one can nondimensionalize the ratio ofthese quantities and by so doing reveal information about which of theterms is dominant.

Proceeding formally, define the nondimensional variables , and and usethem to affect the nondimensionalization

(19.1)

Here the Reynolds number

(19.2)

appears naturally in the last term. In fact, if and have been chosencorrectly, then the starred terms multiplying in Eq. (19.1) will be of , sothat gives an indication of the magnitude of the ratio of the convection todiffusion terms. For small , viscous terms dominate whereas for large ,convection is dominant.

It is helpful to place the previous result in the context of the entiremomentum equation. Thus, in addition to the scalings given thus far, alsodefine , which is justified by being dimensionally correct. In fact, thisscaling which fits in with the kind of pressure variation seen in theBernoulli equation is not entirely general. Other pressure scalings mightbe justified in other contexts, as seen below, though this scaling suffices forthe present purposes. With the proposed scalings, a calculation gives thescaled momentum equation in the form

(19.3)

Consistent with Eq. (19.1), the factor of appears in front of the viscousterm. Equation (19.3) suggests that for high Reynolds number flow, theremay be circumstances where the viscous term can be omitted from theNavier-Stokes equation because it will be much smaller than the other

terms. In fact, high Reynolds number flows are usually turbulent becausethe smoothing effect of the viscous term is overwhelmed by the inertia offluid particles leading to flow instability.

No matter how high the Reynolds number is, it is a fact of experience thatthere will always be regions sufficiently close to boundaries where viscousterms are as important as the convective terms. This aspect of the physicsis not picked up in Eq. (19.3) only because the scaling of length was notsufficiently precise to differentiate between directions tangential andnormal to a wall. This defect is remedied in Chapter 20, where flowadjacent to solid surfaces and referred to as boundary layer flow isconsidered.

For flow at very low Reynolds numbers, known as creep flow, viscosity isthe dominant force in determining the motion, and the last term inEq. (19.3) is much larger than the convective term. The resulting equationsare linear and often solvable. Note that it cannot be assumed fromEq. (19.3) that the pressure term is unimportant in creep flow. Rather, adifferent pressure scaling is warranted and, if used, will show that thepressure term is comparable to the viscous term. This will become evidentin Section 19.4, when low Reynolds flow past a sphere is considered.

19.2 Unidirectional Flow

Among the kinds of flows where exact solutions to the Navier-Stokesequation can be found are flows that are unidirectional. This means that theflow is in one direction and varies only in perpendicular directions. In thiscase, inspection of the nonlinear term in the Navier-Stokes equation showsit to be identically zero. A classic example of unidirectional flow, known aschannel flow, occurs between two parallel plates when the Reynoldsnumber is not so large (generally less than approximately 2000) as to meanthat three-dimensional instabilities are present that can cause the flow tobecome turbulent. Taking the flow domain to encompass the region , asshown in Fig. 19.1, the velocity field is exclusively in the direction andvaries only in the direction so that . If the flow is caused by a pressuregradient down the length of the channel, the resulting motion is referred toas Poiseulle flow. Alternatively, if there is no pressure gradient but thewalls of the channel move in their own plane, then the flow is known as

Couette flow. A flow combining both scenarios is possible, and also gravitymay be responsible for the motion if the channel is tilted. All of these casesare easily solved.

Figure 19.1. Poiseulle flow in a channel has a parabolic velocity profile. Abalance of forces in a control volume analysis of the shaded region showsthat the difference in pressure force is balanced by the viscous resistance atthe solid boundaries.

For Poiseulle flow, the Navier-Stokes equations in the and directions,respectively, after dropping the terms that are identically zero, are

(19.4)

(19.5)

The second of these shows that the pressure is constant across the channel.Moreover, an derivative of Eq. (19.5) makes clear that the streamwisepressure gradient itself, which appears in Eq. (19.4), does not vary acrossthe channel. Integrating Eq. (19.4) with respect to and applying nonslipboundary conditions to the effect that gives the velocity field as

(19.6)

The pressure gradient is negative to drive the flow in the positive direction. Equation (19.6) shows that the velocity has a parabolic profileand has a maximum at the center of the channel equal to

(19.7)

Defining the mean velocity as

(19.8)

it is found by substituting Eq. (19.6) into (19.8) that

(19.9)

so that

(19.10)

In coordinates scaled by and , it follows from Eq. (19.6) that the velocityfield is

(19.11)

A useful perspective with which to view the physics of Poiseulle flow is byconsidering the balance of forces on a control volume, as shown inFig. 19.1, which encompasses a finite section of the channel of streamwiseextent . Specializing Eq. (16.44) to this case establishes that there arepressure forces acting on each end of the CV and viscous forces at thewalls. The latter may be computed as minus the shear stress that the fluidexerts on the wall boundary times the length of the wall. In this case, on thelower wall

(19.12)

and minus this on the upper wall. The CV equation then gives, after somerearrangement,

(19.13)

in which the net positive pressure force on the left-hand side is necessary toovercome the frictional resistance given on the right-hand side. ClearlyEq. (19.13) is consistent with Eq. (19.12) and the constancy of the pressuregradient.

Additional insight into channel flow may be obtained via the CV energyequation (see Problem 19.12). Closely related to channel flow is that in acircular pipe considered in Problem 19.3. Several other unidirectionalflows are the subject of problems at the end of the chapter.

19.3 Flow in a Narrow Gap

An interesting application of the unidirectional flow idea(Reynolds 1886; Batchelor 1967) is in the approximate solution to the

motion of a heavy, slightly tilted block of length moving on a thin layer ofviscous fluid at speed , as illustrated in Fig. 19.2. This is a model of theflow encountered in such familiar circumstances as a person sliding on askimboard along the surf line at a beach. It is a fact of common experiencethat as long as the rider has sufficient speed, he will glide relatively easilyover the sand. This is the result of a substantial pressure force between theskimboard and ground whose origin and magnitude can be explainedthrough a simplified analysis of the flow in the gap depicted in Fig. 19.2.

Figure 19.2. Block sliding along a plate in a reference frame attached to theblock. The boundary moves to the right at speed .

Even though the gap region is not technically a channel flow because thebounding walls are tilted toward each other, nonetheless, if it is assumedthat the tilt of the block is very slight, then the flow locally at any point inthe gap may have the character of a channel flow. In such local regions,there can be assumed to be an approximately constant pressure gradient,though this is not meant to imply that the pressure gradient is the sameeverywhere in the gap. In fact, unlike a pressure-driven channel flow, theflow in the gap is driven by the movement of the block, and the pressuredistribution is not known a priori but must be solved for instead. Forexample, the pressure at either end of the block is atmospheric and thus isentirely unlike a channel flow where the pressure decreases monotonicallyin the flow direction.

To the extent that the block is tilted, there is a definite velocity normal tothe surface. A central aspect of the modeling that is proposed here is thatthis slight velocity, whose existence technically negates the unidirectionalassumption, is nonetheless so small as to be inconsequential to theessential physics of the problem. In the next chapter, when boundary layersare discussed, the wall normal velocity will be given a more nuanced andaccurate treatment. Here we are content to accept the fact that theunidirectional model remains a reasonable approximation in any localregion of the gap.

In a coordinate system fixed to the moving block, the top wall of the localchannel-like region is fixed while the lower wall moves at speed . Thismeans that the flow in the gap can be considered to be locally acombination of Poiseulle and Couette flows. The gap thickness is assumedto vary linearly between a maximum thickness of where the fluid entersthe gap to a minimum of where it exits, as shown in Fig. 19.2. This meansthat as a function of , the gap is

(19.14)

where

(19.15)

and is assumed to be very small. A calculation using the unidirectionalmomentum equation gives the result

(19.16)

where the dependence of is caused by the variation of the pressuregradient in the gap, as well as the fact that is a function of .

At this point in the analysis, the pressure gradient remains unknown.However, an additional condition on the fluid motion that allows for thedetermination of is that the area flux of fluid through the gap, say, , isconstant at any location. This follows trivially from the control volumemass conservation equation. A calculation gives

(19.17)

and from this it is found that

(19.18)

Taking the pressure at to be , and integrating Eq. (19.18) from to , givesthe pressure distribution as

(19.19)

Because the pressure is also equal to at , this condition providesa means of predicting the area flux under the block. The result is

(19.20)

Now substituting Eq. (19.20) back into (19.19) gives the pressuredistribution as

(19.21)

A plot of the pressure scaled by its maximum is shown in Fig. 19.3 for thecase when . Consistent with our expectations, the maximumpressure occurs toward the end of the gap before the exit. A calculationshows this to be at , which is in theexample. The cumulative effect of this pressure is felt in the total pressureforce on the underside of the block, which may be computed to be

(19.22)

The total force upward depends on the inverse of squared and canbecome very large when the tilt is small compared to the length . Forexample, in the case of a 1 m square block being moved at speed m/s, in an oil with viscosity kg/m-s, m, ratio

, so that the tilt angle is , then the total pressure force issufficient to hold up a block weighing 426 kg. Alternatively, assuming theliquid is saltwater with viscosity kg/m-s and a person istraveling at speed m/s on a 1 m 0.5 m skimboard, then the weightthat can be supported is 50 kg.

Figure 19.3. Pressure distribution underneath the sliding block.

19.4 Stokes Flow Past a Sphere

An especially interesting example of three-dimensional viscous flow is thatpast a sphere in uniform motion. In this discussion, the sphere is assumedto be fixed with the surrounding fluid flowing past it. Depending on howfast the oncoming flow is, a number of different regimes are possible withdramatic changes in the physics occurring from one regime to the next. Atthe lowest range of speeds and Reynolds numbers, that is, creep flow, theinertial terms in the momentum equation contribute so little to thedynamics that they can generally be ignored, thus rendering the equationslinear and solvable. Unlike unidirectional flows where the nonlinear termsare identically zero, here they are only approximately so. In this section, ananalytical solution is derived for the low Reynolds number flow past a

sphere, known as the Stokes flow. The next section considers some aspectsof the sphere flow at higher Reynolds numbers.

19.4.1 Problem Formulation

The flow past a sphere at low , though ostensibly three-dimensional,can be approached as a two-dimensional problem by taking advantage of itsaxisymmetry. Indeed, this property was used in Section 6.1 to obtain thestream function for the inviscid flow past a sphere. In the present case, asimilar strategy will be pursued, though this time, the dynamical equationsfor the motion will be solved in place of merely developing a kinematicalsolution for the flow.

While one may wish to attack the solution of the sphere flow in primitivevariables, that is, in terms of the velocity and pressure, an easier route isthrough a vorticity formulation. In fact, while there are two nonzerovelocity components, (i.e., in the and directions), there is only onenonzero vorticity component, namely, , which points in the azimuthaldirection. This means that one less differential equation needs to be solved,thus greatly simplifying the problem. On top of this, the equation forvorticity is decoupled from the pressure, leading to additionalsimplifications of the analysis.

In the creep flow model, the absence of the inertial terms in the momentumequation means that the equivalent vorticity equation is lacking both theconvection and vortex stretching terms, because these both originate in theconvective term in the Navier-Stokes equation. Moreover, because onlysteady flow is being considered, the governing equation (18.5) reduces to

(19.23)

which is just the vector form of Laplace’s equation. In view of Eq. (18.3),this can be written equivalently as

(19.24)

19.4.2 An Equation for the Stream Function

The solution to Eq. (19.24) for the sphere must be such as to satisfy therelevant velocity boundary conditions to the effect that the velocity on thesphere surface is zero and the far-field velocity in the streamwise directionis . Because the flow is axisymmetric, a stream function exists thatserves to enable a connection between the vorticity and conditions on thevelocity.

The vorticity can be related to the stream function through the vectorpotential in three dimensions, as given in Eq. (4.15). Because isdivergence free, according to Eq. (4.10),

(19.25)

It was noted in Problem 3.8 that for axisymmetric flow,

(19.26)

and substituting this into the left-hand side of (19.25) and computing thedouble curl using the formula (A.17), it follows that

(19.27)

where the operator is defined according to

(19.28)

In view of Eqs. (19.25) and (19.26), Eq. (19.27) leads to the result

(19.29)

because only the term is nonzero in each expression.

Equation (19.29) is particularly useful because it provides an easy way ofexpressing the content of Eq. (19.24), which governs the sphere flow. In

particular, substituting into Eq. (19.24) and using (19.29) gives

(19.30)

However, Eq. (19.27) holds for any value of . In particular, it remainstrue if “ ” in Eq. (19.27) is replaced with “ .” Doing so, Eq. (19.30)via Eq. (19.27) simplifies to

(19.31)

as an expression of the vorticity equation governing sphere flow.

19.4.3 Solution for Stokes Flow

Although the solution of a fourth-order partial differential equation likeEq. (19.31) is challenging, in the present case, consideration of the far-fieldboundary condition provides a means for substantially simplifying theanalysis. Thus the velocity field as should asymptote to that of auniform flow at speed . The stream function for that case was given inEq. (3.29), and so the boundary condition is

(19.32)

This relation suggests that the dependence of on may be limited tothe factor , in which case the assumption is made that

(19.33)

where now only the function has to be determined from the boundaryconditions and differential equation. A more formal way to viewEq. (19.33) is as a choice necessitated by the fact that solutions to thevector Laplace equation (19.23) are constrained to take the form of anexpansion in solid spherical harmonic functions (seeBatchelor 1967; Carrier & Pearson 1988), and only one such function, theone chosen in Eq. (19.33), has a dependence compatible withEq. (19.32).

Comparing Eq. (19.32) to (19.33), it is apparent that must satisfy thecondition

(19.34)

Additional conditions come from setting the velocities at thesurface of the sphere. In terms of , this means that at the radius of thesphere, say, ,

It is readily shown by substituting Eq. (19.33) into (19.35a) and (19.35b)that must also satisfy

(19.36)

and

(19.37)

respectively.

Now substituting Eq. (19.33) into Eq. (19.31) gives

(19.38)

suggesting that the assumptions concerning the dependence of thesolution and its separability from the dependence is correct. Satisfactionof Eq. (19.38) implies that is determined as a solution of theequidimensional equation

(19.39)

Such equations have homogeneous solutions of the form , and acalculation reveals that the most general homogeneous solution ofEq. (19.39) is

(19.40)

where are constants to be determined from the boundaryconditions. Applying Eq. (19.34) to Eq. (19.40) implies that and

. A calculation with Eqs. (19.36) and (19.37) determines theremaining constants as and so that

(19.41)

Substituting Eq. (19.41) into (19.33) gives the stream function for lowReynolds number flow past a sphere as

(19.42)

A contour plot of Eq. (19.42) for the same contour levels as in the case ofinviscid flow in Fig. 6.4, is given in Fig. 19.4. This shows some qualitativedifferences, particularly in the fact that the streamlines are spreadsomewhat further from the sphere and more obviously anticipate thepresence of the sphere than the inviscid case. This and the forward and aftsymmetry are consistent with the physical picture of a source of vorticity atthe boundary from which vorticity travels outward by viscous diffusion. Ifconvective effects were kept in the vorticity equation, then the vorticityfield would not be symmetric.

Figure 19.4. Streamlines of Stokes flow past a sphere.

Before computing the exact expression for the vorticity in Stokes flow,consider the equation formed by substituting Eq. (19.33) into Eq. (19.29),namely,

(19.43)

According to Eq. (19.41), has terms depending on the powers of given by , and . If each of these functions is substituted intoEq. (19.43) in the place of , it is found that only the term with

makes a contribution to the vorticity. This suggests that it isappropriate to rearrange the terms in Eq. (19.42) into the form

(19.44)

where the first term on the right-hand side contributes toward anirrotational flow and the second term is entirely responsible for thevorticity field in Stokes flow. Of course, it takes both terms together tosatisfy the boundary condition that at the sphere surface. FromEq. (19.43), the last term in Eq. (19.44) leads to the vorticity in the form

(19.45)

19.4.4 Forces on the Sphere

With the goal of computing the forces on the sphere, the pressure can becomputed by adapting Eq. (18.1) to the present case of incompressible,steady flow with negligible convective terms and gravity. This gives

(19.46)

from which it is found, after expressing both sides of Eq. (19.46) inspherical coordinates and using Eq. (19.45), that

Integrating either one of these relations and substituting into the otherdetermines the pressure to be

(19.48)

where the constant is the pressure far from the body. It may be noticedthat the pressure field is not symmetric forward and aft so that, unlike theinviscid case, there will be some drag produced by the pressuredistribution. Equation (19.48) can be rearranged to the form

(19.49)

where . Apropos of a previous remark in Section 19.1concerning the pressure scaling, this equation shows that the pressure inStokes flow scales like rather than the appropriate to highReynolds number flow.

The drag force on the sphere is the total force component in the streamwisedirection acting on the surface of the sphere. Taking as the streamwisedirection with oriented toward positive , the drag force is

(19.50)

where is an element of surface area, is the unit vector normalto the surface, and Eq. (9.4) has been used so that is the deviatoric stresstensor.

The constant, , in the expression for the pressure does not make acontribution to the integral because it cancels forward and aft and can beignored. The calculation of in the present case can be much simplifiedby taking advantage of an identity considered previously in Eq. (12.56) ofProblem 12.4. According to this result, for incompressible flow on thesurface of the sphere,

(19.51)

Substituting Eq. (19.45) and noting that , it follows that

(19.52)

on the surface of the sphere.

Evaluating and as is implied by Fig. 1.6,and substituting Eqs. (19.48) and (19.51) into the integrand in (19.50),gives

(19.53)

so that, in fact, the local force/area contributing to the drag force bypressure and viscous terms is independent of position on the sphere.

It is then a simple matter to evaluate

(19.54)

by factoring the constant integrand out of the integral. Noting that the areaof the sphere is , it is found that

(19.55)

as Stokes law for the drag on a sphere at low Reynolds number. In terms ofthe drag coefficient

(19.56)

where normalization is by the projected area , the Stokes drag law on asphere is the well-known result

(19.57)

19.4.5 Self-Consistency of the Solution

The velocity field corresponding to Stokes flow may be computed fromEqs. (3.28) after substituting Eq. (19.42), giving

These relations are a consequence of the assumption that the convectiveterm in the momentum equation can be neglected in favor of the viscousterm. It is natural to wonder if the result contained in Eqs. (19.58) isconsistent with this model. In fact, consistency can be tested bysubstituting Eqs. (19.58) into the ratio of the convection and viscous termsand verifying that the magnitude of the result is small. Such a calculationshows that the convection term is small compared to the diffusion termonly in the vicinity of the sphere. At increasing distances from the sphere,vorticity diffusion decreases and a point is reached, several diametersaway, where convection and diffusion are comparable.

A means of setting up the low Reynolds number sphere calculation so thatit is self-consistent everywhere was developed by Oseen(Batchelor 1967; Oseen 1910) based on a slight generalization of the

Stokes model that includes a more careful consideration of the convectionterm. In this the decomposition

(19.59)

is made where the first term on the right-hand side, which vanishes farfrom the sphere, is omitted, while the second term, which is kept, suppliesthat part of the convection whose absence leads to the inconsistency in theStokes solution. This approximation results in the Oseen equation

(19.60)

which is linear and thus amenable to some analysis. The solution toEq. (19.60) is identical to the Stokes solution in the neighborhood of thesphere, yet is also fully consistent with the assumptions behind Eq. (19.60)in the sense that the nonlinear part of the convection term that has beenomitted is everywhere small compared to the terms that are included.

Some insight into the nature of the solution to Eq. (19.60) can be had byconsidering the equivalent vorticity form of the equation that is derived bytaking its curl. In this case, it is found that Eq. (19.23) generalizes to

(19.61)

which is the convective diffusion equation. Wherein the Stokes solutionhad the character of radial diffusion of vorticity from a source on thesphere surface, the presence of the convective term in Eq. (19.61) meansthat as the vorticity field diffuses from the sphere, it is convecteddownstream in the direction of . The presence of the vortical wakemeans that the solution to this more general problem lacks the fore/aftvorticity symmetry of the simpler Stokes flow. The greater physicalaccuracy of the Oseen model translates into marginally more accurateprediction of the drag coefficient. In particular, the formula

(19.62)

may be derived, which gives a small improvement in drag prediction forReynolds numbers up to the order of unity.

19.5 Motion of a Sphere at Higher Reynolds Numbers

For Reynolds numbers larger than those considered in the previous section,the inertial term in the Navier-Stokes equation becomes large enough sothat analytical treatment of the flow past a sphere is no longer possible. Toget information about the complete flow in this case requires the use ofcomputation. In the best of circumstances, numerical simulations can beperformed that accurately solve the Navier-Stokes equation and providedetailed information about the flow and forces affecting the sphere. Theeffectiveness of computational methods depends to a large extent onwhether the resolution of the numerical mesh is sufficient to capture thesmallest dynamically important scales of motion. Beginning as low as aReynolds number of approximately 800, turbulence effects appear withinthe flow that include dissipative phenomena at increasingly small scales.As a result, the cost of a fully resolved, accurate computation becomesprogressively more expensive as the influence of turbulence increases untilsimulation of the flow is not practical.

Physical experiments have no essential limitation on Reynolds number, sothey can be applied to the study of sphere flow at a wide range ofconditions, including high Reynolds numbers. The information they canprovide, however, is often limited to average flow properties, such as thetotal force acting on the sphere, rather than detailed informationconcerning the velocity field and distribution of forces over the spheresurface. Through physical experiment, the drag coefficient for a smoothsphere has been determined over an extensive Reynolds number range, asshown in Fig. 19.5, where Eq. (19.57) is also included for comparison. in this figure, which is taken from an empirical fit to the experimental data(Morrison 2013a,b), falls to a value near 0.4 by and remainsrelatively constant until approximately , where it begins a

precipitous drop, followed by a subsequent increase beginning after .

Figure 19.5. Drag coefficient on a sphere. Dashed line is the Stokes flowsolution.

The drop in drag coefficient seen in Fig. 19.5 translates into an absolutedrop in the total drag on the sphere, as may be seen in Fig. 19.6 for the caseof a smooth sphere of radius in. (i.e., the size of a golf ball) travelingin air. Unlike Stokes flow, where viscous drag is larger than the pressuredrag, the later dominates at high Reynolds numbers, and so to understandthe cause of the change in drag in Fig. 19.6, it is helpful to contrast thebehavior of the average pressure field on the surface of the sphere forReynolds numbers before and after the loss in drag. This is indicated inFig. 19.7, showing the scaled pressure distribution on the surface of the

sphere at Reynolds number in the so-called subcriticalregime just before the drag falls, and , which is within therelatively low drag, postcritical zone. For comparison, the inviscid pressurefield as computed in Problem 17.2 is also plotted. To varying degrees, bothsets of pressure data match the inviscid result on the front of the spherebefore deviating at different points to approximately constant values. Theflow at the lower Reynolds number departs the inviscid wall pressuresignificantly upstream of the point where it happens for the high Reynoldscase.

Figure 19.6. The drag (in newtons) on a golf ball–sized smooth spheretraveling in air.

Figure 19.7. Pressure coefficient on the boundary of a sphere with denoting the front. , sub-critical flow at ; , post-criticalflow at ; —, potential flow solution. (Data fromAchenbach, 1972).

For the sub- and postcritical Reynolds numbers considered in Fig. 19.7, thepressures at the back of the sphere beyond are approximately thesame. Because these pressure magnitudes are much less than thecomparable values on the front of the sphere, they lead to a significantcontribution to the total drag in both cases. The difference between the dragat the two Reynolds numbers largely reflects the dramatically differentpressure distributions on the sides of the sphere. In particular, the pressureremains higher for the subcritical case than the supercritical flow, while thelatter stages a qualitatively different recovery after the midpoint at .The greater net pressure differential between symmetrically placed points

before and after the midpoint of the sphere for the subcritical flow leads toa larger force in this case.

The flow phenomena responsible for the pressure trends seen in Fig. 19.7have to do with the locations where the boundary layer “separates” fromthe surface. Flow separation happens when the relatively slow moving fluidnext to a boundary has insufficient inertia to overcome a rising pressuregradient. The acceleration of the flow as it moves around the spherenaturally creates a rising pressure beyond the midpoint. When the boundarylayer on the sphere is laminar, as it is for the subcritical flow, there is atendency for the flow to separate just ahead of the midpoint, consistentwith the trend in the pressure distribution seen in Fig. 19.7. When theReynolds number is higher, in the postcritical range, the boundary layertransitions to turbulence upstream of the midpoint, thus providing amechanism to bring high-momentum fluid closer to the surface, better forresisting the adverse pressure gradient. Consequently, flow separation isdelayed until a point on the back of the sphere, as is implied by thepressure trend in the figure at .

For a smooth sphere, the transition is natural in the sense that it occursthrough the unstable growth of very small perturbations that are alwayspresent in flows. If larger perturbations are forced on the boundary layer,as, for example, by the dimples of a golf ball, then the flow in the near-wallboundary layer is encouraged to transition to turbulence at a lowerReynolds number. With this change in transition, the drop in inFig. 19.5 is shifted to lower Reynolds numbers because the separationmechanism is changed to the supercritical mode in this case. For a smoothsphere of radius 1.68 in. traveling at the realistic speed of a golf ball, say,100 MPH, a computation gives , a value that is just on thesubcritical side of the drag relation. After placing dimples on the sphere soit is like a golf ball, this same Reynolds number will lie in the postcriticalrange of the drag curve, leading to an absolute drag reduction.Consequently, golf balls travel much farther than equivalent smooth balls.

Knowledge of the drag coefficient of a sphere, or more generally the dragand lift coefficients of any particular object, can be exploited to predict theflight of a body in a given wind. For example, consider a sphere moving

along the path in the plane with velocity field(i.e., wind) . In this coordinate system, and are in thehorizontal and vertical directions, respectively. For simplicity, assume thatthe sphere is sufficiently heavy so that the buoyancy force can be omitted,in addition to forces deriving from the acceleration of the body. In thiscase, the movement of the sphere depends on gravity and the drag forceonly.

The drag on the sphere is in the opposite direction of the relative velocity,defined as

(19.63)

This is the velocity of the sphere as measured by an observer moving withthe local wind. If the sphere travels with the local wind velocity, itexperiences no drag, whereas if the sphere is slower than the wind, the dragforce pushes the sphere in the direction of its motion.

For the general movement of a body, the drag depends on via the relation

(19.64)

where is an empirically determined drag coefficient, is the projected areaof the body in the relative flow direction, and

(19.65)

Taking into account the directionality of the motion of the body means thatthe vector drag force is given by

(19.66)

where is a unit vector in the direction of the relative velocity. This leads tothe result that

(19.67)

A simple model for the dynamics of a sphere of mass and radius in a windthen is

(19.68)

where can be computed dynamically from the Reynolds number using thedata in Fig. 19.5.

There are many ways that the modeling in Eq. (19.68) can be improved andgeneralized. Besides including acceleration and buoyancy forces, it may bethe case that the force distribution leads to a net moment about an axis inwhich case the angular momentum of the body needs to be taken intoaccount. For example, if the sphere were spinning, as is often the case forgolf balls, then it would develop a lift force acting to the side as it travelsdue to the Magnus effect (Batchelor 1967), which has to be accounted forin the governing equations. Many bodies develop lift forces through theirshape, and this may be accommodated in the dynamics via formulas thatare close relatives to Eq. (19.67). For example, for planar motion of a bodyin which the lift force projects into the positive direction, the vector liftforce to be added to the right-hand side of Eq. (19.68) takes the form

(19.69)

points perpendicular to the direction of motion. The factor ensures avertical force regardless of the direction of travel. When the body moves inthe opposite direction of the wind, is enhanced and lift is increased. InEq. (19.69), is a lift coefficient that is generally found by empirical means.Some applications incorporating Eq. (19.68) and (19.69) that can be readilysolved via MATLAB or equivalent are suggested in the exercises.

Problems

19.1 Consider the steady unidirectional flow of two fluids having the samedensity but different viscosities in a channel of thickness . The lower fluidoccupies and has viscosity , whereas the upper fluid has viscosity . Findthe velocity distribution across the channel and the ratio of volume fluxes.Assume a constant pressure gradient in the channel. The fluids satisfy thenonslip condition at their respective boundaries and, in addition, their

velocities and shear stresses are equal at the interface between them. Whymust this be so? What is the maximum velocity in the channel, and wheredoes it occur?

19.2 Consider fully developed laminar flow of water of thickness travelingdown an inclined plane at angle with respect to the horizontal. Find thevelocity distribution, the volume flow rate per unit width of fluid, , and thestress on the wall. How does the thickness of the layer depend on and theother parameters? It may be assumed that the shear stress of the water iszero at the air/water interface. Why is this a good assumption? Describe thepressure distribution in the water.

19.3 Compute the axial velocity corresponding to laminar flow in a pipe ofradius driven by a constant pressure gradient . Show that the maximumvelocity is twice the average velocity. Compute the stream functionsatisfying the condition that it is zero on the pipe wall. Verify thecontention surrounding Eqs. (3.24) that at gives the flow rate through thepipe. Compute the vorticity distribution and then the shear stress at thewall with the help of Eq. (12.56). Finally, verify that the pressure drop overa section of pipe is equal to the total shearing force at the boundary.

19.4 Compute the laminar flow in the axial direction through the annularspace formed between two concentric tubes with radii . Find the radiuswhere the maximum velocity is reached, the volume rate of flow, and thedistribution of shear stress. Assume a constant pressure gradient along thetube.

19.5 Calculate the velocity field in the annular gap between two concentriccylinders of radii in the case when the flow is created by the steadyrotation of the inner cylinder at angular velocity . Compute the deformationwork/mass-sec given in Eq. (15.24) and use this to evaluate the totalenergy/sec produced in a journal bearing of radius cm, gap cm, axiallength of cm, revolving at RPM and containing oil with Pa-sec. Finally,compute the pressure field in the gap up to an arbitrary constant.

19.6 Compute and plot as a function of for for Stokes flow past a sphere,the inviscid flow past a sphere, and the difference between the two. How

many radii outward from the sphere is it until the Stokes solution is within5 percent of the inviscid result?

19.7 Make a contour plot of the vorticity surrounding the sphere in Stokesflow and consider how it decays with distance from the sphere.

19.8 Show that the total viscous drag on a sphere in Stokes flow is twicethe pressure drag. Explain this result by considering a plot of the separatelocal contributions to drag as a function of (taking into account thedependence of area elements on ).

19.9 What is the steady velocity with which a small spherical particle ofradius and made of a material with density will fall in a quiescent fluid,assuming that the Stokes drag applies? Include the effect of buoyancy, andexplain why the contribution of the pressure to the buoyancy force can becomputed separately from that included in the drag force. Find a conditionon the parameters in the problem that justifies the use of the Stokes dragformula so that the solution is self-consistent. For the case of a sphere ofice falling through air, find the size range at which the Stokes drag law isself-consistent.

19.10 Compute the transient motion of a small sphere dropped from rest ina quiescent fluid of density and viscosity until it reaches the constantterminal velocity that was computed as part of Problem 19.9. Forsimplicity, neglect acceleration forces such as the added mass. The spherehas radius and is made of a material with density . How long does it takeuntil an ice sphere with cm dropping within air is within 1 percent of itsterminal velocity?

19.11 Use the control volume momentum equation applied to the regiondirectly underneath the tilted block in Fig. 19.2, to show that the slightcomponent of the pressure force in the direction of motion balances withthe sum of the frictional resistance from both walls.

19.12 Use Eq. (15.26) as the basis for developing a control volumeequation for just the internal energy in a channel flow. Interpret what itsays about the physics of channel flow. In a similar vein, apply Eq. (15.37)

to a section of the channel flow and provide a physical interpretation of theresult.

19.13 Consider the laminar flow driven by a pressure gradient betweentwo flat porous walls separated by the distance . Assume that in addition tothe steady velocity field in the direction, there is also a constant velocity arising from fluid traveling out of one porous wall and into the other one.Despite the porosity of the walls, it may be assumed that satisfies thenonslip condition at both bounding planes. Solve for and show that itconverges to Eq. (19.6) in the limit as . Make a plot of for several valuesof for fixed pressure gradient and viscosity. Explain the trend in thevelocity.

19.14 A film of fluid flows downward due to gravity on the outside of acylinder of radius . Assuming laminar flow, how thick is the layer of fluidif the volume flux down the cylinder is ? Hint: assume the outer surface ofthe fluid is at and then find the value of .

19.15 Revisiting Problem 3.5, concerned with the flow past a moving wall,compute the vorticity and show that its Laplacian is 0. Evaluate the ratio ofthe convection and diffusion terms in the and equations, as was done forStokes flow past a sphere, and then make a contour plot of the ratio.Interpret this as to the circumstances, such as flow region, for which it islegitimate to assume that the convection terms are negligible incomparison to the viscous term.

19.16 Consider an incompressible fluid in a narrow gap of thickness between two circular disks of radius , with one on top of the other. Assumethat the top disk is moving toward the stationary lower disk at speed sothat the fluid between the disks flows out radially. In a cylindricalcoordinate system centered at the disks, the flow domain consists of theregion with . By a simple scaling argument, reminiscent of that used in thestudy of boundary layers and described in the next chapter, it may beshown that the governing flow Eqs. (A.11) and (A.13) simplify to

(19.70)

(19.71)

where it is also assumed that the Reynolds number is sufficiently low sothat the nonlinear advection terms can be neglected. Taking note ofEq. (19.71), solve Eq. (19.70) for by applying no slip boundary conditionsat . Using the expression for , integrate the continuity equation determinedfrom Eq. (A.9) across the gap to obtain an equation that can be integratedto give with the boundary condition . From this, show that the totalpressure force acting against the moving plate is .

19.17 Use Eq. (19.68) as the basis for computing (e.g., using MATLAB) themotion of a smooth sphere with diameter m and mass kg that match theproperties of a golf ball. In this, solve the nondimensional equation derivedusing as the length scale and as the time scale so that the gravity term inthe vertical direction is . Use an analytical formula for (e.g., seeMorrison 2013a). How many meters does the ball travel if it leaves the teeat 100 MPH at a 45 angle with no wind present? How is this distanceaffected by a constant horizontal wind of 10 MPH in the positive andnegative directions? Make a comparative plot of the path of the sphere inthe three cases.

19.18 Modify the code developed in Problem 19.17 to mimic the effect ofroughness on a true golf ball by artificially shifting the curve in Fig. 19.5by adding a constant to the instantaneous Reynolds number. See the effecton distance if the Reynolds number is increased by 100,000 and by200,000, make a comparison plot, and comment on the result. Make a plotof during the flight of the sphere in each case so as to observe the effectivedrag coefficients.

19.19 Modify the code developed in Problem 19.17 to include the lift termgiven in Eq. (19.69). Compute the trajectory of the sphere for 0.05 and 0.2,discuss the results, and make a comparison plot of the paths.

19.20 Compute the path of a bungee jumper of mass 60 kg who dives off abridge from a point = (0, 100 m) above the ground with initial velocity m/s, . Assume the local wind in the region is that of a downdraft consistentwith Fig. 7.2(b) and given by , with m. The density of the air kg/m and theviscosity is kg/m-s. The bungee jumper is tied to the jump point via amassless elastic cable of rest length m that supplies a force to the jumperaccording to Hooke’s law: , where is the spring constant and (i.e., there is

no force for a slack cable). Assume the drag coefficient on the person is and her projected area is 1 m. Plot and make an animation of the path of thejumper and investigate the influence of the parameters and . In particular,consider values m/s and N/m. Base your code on the nondimensional formof Eq. 19.68, including the force from the bungee cord. Scale the equationusing the length scale and time scale so that the non-dimensionalgravitational force is 1.

19.21 Compute the motion, , of a kite released from the origin at into aconstant horizontal wind . Assume the kite is tethered to the ground at theorigin by a massless elastic cable of rest length m that supplies a force tothe kite according to Hooke’s law: , where N/m is the spring constant and (i.e., there is no force for a slack cable). Solve a nondimensional form ofEq. (19.68) generalized to include the lift force in Eq. (19.69) as well as thecable force. Assume the mass of the kite kg, the projected area m, and, forsimplicity, and independent of the orientation of the kite with respect tothe wind. The air may be assumed to have density kg/m and viscosity kg/m-s. Study the effect of wind speed on the equilibrium height of thekite. Make an animation of the simulation.

19.22 Compute and study the motion of a sphere of diameter , projectedarea , and mass moving in the velocity field , that corresponds to the flowin Fig. 7.2c. Here, is a constant length, and is a constant velocity. Assumethat so that the flow is toward the inclined surface. Use Eq. (19.68) withoutthe effects of gravity as the basis for computing the motion of the sphere(e.g., using MATLAB). Solve the nondimensional equations scaled by and , which may be seen to depend on the parameter which represents, inessence, the ratio of the mass of the fluid displaced by the sphere to its ownmass. Also appearing is in the Reynolds number for the relative fluidvelocity on which depends. For simplicity assume that the parameter isunity. Start the sphere at a point , at half the speed of the local velocityfield and determine if and where the sphere impinges on the inclinedsurface for different values of and . Plot the paths superimposed on aquiver plot of the underlying velocity field. Use an analytical formula for as a function of Reynolds number (e.g., see Morrison 2013a). Provide aphysical explanation for the observed dependencies.

20Laminar Boundary Layers

It was shown in Chapter 7 that inviscid flow theory predicts the absence ofdrag on the flow past solid bodies. This unphysical result merely focusesattention on the fact that when fluid flows around an object, there arefrictional effects originating in the interactions of the molecules of thefluid with the solid surface that cannot be ignored. The presence ofviscosity creates a resistance to motion that is the origin of the drag force.A functionally useful boundary condition that reflects this phenomenon isthe nonslip condition to the effect that the fluid velocity tangential to asolid boundary must move with the corresponding velocity of the surface.

The Navier-Stokes equation accommodates the frictional force byinclusion of momentum diffusion at a rate determined by the viscosity inproduct with the velocity derivatives. If an object is moving swiftlythrough fluid and the Reynolds number is large, the convective terms inthe Navier-Stokes equation are expected to be dominant over the viscousterms. For the viscous effect not to be irrelevant, the velocity gradientsthat underlie viscous momentum diffusion must be very large, and thiscan only happen in a relatively thin region near the body, where thevelocity rapidly changes from that of the object to that of the free stream.Such regions are referred to as boundary layers, and their study is ofconsiderable importance because many flows are at high speed.

The simplest example of a boundary layer is that forming on a flat platesuch as that shown in Fig. 20.1. The oncoming fluid is assumed to be at auniform velocity , and a zero velocity condition develops on the platesurface, leading to the reduction of momentum in a thin layer near theboundary. In essence, momentum diffusing to the surface is lost, givingthe impression of a spreading boundary layer. This boundary layer isequivalent to that formed by a plate moving at speed into a quiescentfluid and is known as a zero-pressure-gradient boundary layer becausepressure variation along the plate is not present to drive the flow. In moregeneral situations, the flow outside the immediate wall vicinity may varyalong the wall and be accompanied by changes in pressure. Depending on

the Reynolds number, boundary layers may also occur in these moregeneral circumstances.

Figure 20.1. Blasius boundary layer forming on a flat plate.

In the simple scenario of a zero-pressure gradient boundary layer, there isno length scale that is forced by the geometry of the problem. Forexample, a giant and an elf will see no qualitative difference in theirperception of the geometry of the flow. Oftentimes in situations like this,it is suspected that a similarity solution exists, that is, a solution that is thesame for all observers so long as they interpret the result using their ownparticular notions as to distance, velocity, and time. The zero-pressuregradient boundary layer depicted in Fig. 20.1 does turn out to have asimilarity solution known as the Blasius boundary layer. The next sectiontreats this well-known result. Following this, a generalization of theBlasius boundary layer known as the Falkner-Skan boundary layer isderived for a class of flows in which the outer fluid velocity varies as apower law, as discussed in Chapter 7. To demonstrate the versatility of theboundary layer idea, some results for a self-similar thermal boundarylayer are discussed in the next chapter.

20.1 Boundary Layer Scaling

Prior to developing a similarity solution to the boundary layer flow, it isnecessary to consider the relative magnitude of the various terms in theNavier-Stokes equation with a view toward simplifying its statementconsistent with the boundary layer assumption. This scaling analysis isakin to what was pursued in Section 19.1, with the difference being a morecareful consideration of the variation of flow properties in differentdirections. In essence, boundary layers are characterized by the conditionthat the flow is rapid and the region affected by the solid surface is thin.As will be seen, these two defining concepts produce a reduced form ofthe Navier-Stokes equation that turns out to have similarity solutions forboundary layer flows.

Our interest here lies in the steady, two-dimensional, incompressibleNavier-Stokes equations in the and directions, given respectively fromEq. (16.5) as

(20.1)

and

(20.2)

The relative importance of the various terms in these equations can beassessed by nondimensionalizing the dependent and independentvariables. A length scale in the streamwise direction is defined as thedistance from the leading edge of the plate to a point downstream, say, , sothat is the nondimensional coordinate. Because it will turn out to benecessary to distinguish variations normal to the wall from those along theboundary a separate scale, is defined as appropriate to the wall-normaldirection so that . Clearly velocities in the streamwise direction should bescaled via so that . In more general examples of boundary layers, mayvery well vary with position along the body.

In the case of the direction normal to a surface, it is not a priori evidentwhat an appropriate velocity scale should be. Thus, for convenience, it isset to with the understanding that it remains to be determined. Finally, thetime is scaled according to and, because it is generally the case that thepressure adjusts to whatever motion is present, it is assumed that . Notethat the pressure scaling can be justified on both dimensional grounds andthe fact that this scaling is consistent with the pressure variations impliedby the Bernoulli equation for high Reynolds number flow.

Because two-dimensional incompressible flow is being considered, thevelocity field is constrained by the the mass conservation equation in theform

(20.3)

Applying the previously defined scalings to this equation and thengathering the coefficients in the second term, it follows that

(20.4)

When the scalings are chosen appropriately, as is assumed here, thestarred quantities in a relation such as Eq. (20.4) can be expected to havemagnitude on the order of 1, that is, . Because there are just two terms inEq. (20.4), they must both be of the same order of magnitude. The onlyway this can be true in the present case is if the coefficient of the secondterm is . In other words, it must be that satisfies

(20.5)

and for convenience, this relation is taken to be equality so that the scaledcontinuity equation is

(20.6)

Equation (20.5) settles the question as to what an appropriate value of should be.

Now consider the scaling of the momentum equations. After substitutingfor the dimensional quantities and dividing Eq. (20.1) by and (20.2) by , itis found that

(20.7)

and

(20.8)

respectively, where the Reynolds number

(20.9)

appears in both equations.

The definition of the boundary layer suggests that must be large becauseonly in this way can the lateral diffusive spread of momentum be confinedto a thin region. Considering the different terms in Eq. (20.7), it may benoted that the convection and pressure terms are of . Among the two

diffusion terms, the diffusion in the streamwise direction is proportionalto and so must be quite small. The last term also depends on the inverseof the Reynolds number, though it is multiplied by the square of the ratio .In our concept of the boundary layer, viscous diffusion normal to theboundary is an important part of the physics. To make sure that this isreflected in the equations of motion, the coefficient of this term must be .Consequently, we are led to the conclusion that

(20.10)

an equation that shows how “thick” the boundary layer is at thedownstream point . This relation shows that the boundary layer grows insize with the square root of . As a result of this analysis, it may beconcluded that the appropriate form of the momentum equation forboundary layers is

(20.11)

For the balance of momentum normal to the wall, Eq. (20.10) implies thatthe pressure gradient term is , with the remaining terms either or smaller.This implies that the essential boundary layer physics to be taken from the momentum equation is that

(20.12)

because none of the other terms are able to balance this. Thus, at any position along the wall, the pressure can be assumed to stay constantacross the boundary layer. Taking an derivative of Eq. (20.12) andreversing the order of differentiation shows that is also constant acrossthe boundary layer.

20.2 Blasius Boundary Layer

Owing to the absence of a pressure gradient, the governing momentumequation for the Blasius boundary layer is

(20.13)

The concept of a similarity solution is based on the hypothesis that thestreamwise velocity field, even though it depends on both and , should beof the form

(20.14)

where is a dimensionless similarity variable, is an appropriate lengthscale that varies with downstream distance, and is a function that willvary from 0 at the wall to 1 in the free stream. The idea here is that thevelocity at every cross section through the boundary layer looks the samewith just the scale changing. For Eq. (20.14) to be true, no geometricalfeatures can be present that would have an influence on the form of .

To obtain a similarity solution of Eq. (20.13), it is necessary to reach adecision concerning as well as obtain an expression for the normalvelocity component that also appears in the governing equation. The firstof these points is easily solved by taking advantage of the fundamentalresult contained in Eq. (20.10). Thus, if the distance in that relation ischanged to , then a measure of the thickness of the boundary layer at thedistance from the leading edge is

(20.15)

and is the natural choice for the normalizing scale . In this case adimensionless similarity variable is defined via

(20.16)

and Eq. (20.14) can be written succinctly as

(20.17)

An appropriate similarity form for can be found through the artifice ofintroducing the stream function from which both and can be computed.Thus, according to the relation , Eq. (20.17) may be integrated withrespect to , yielding

(20.18)

and after transforming the variable of integration from to , it follows that

(20.19)

where the boundary is taken to be the streamline.

For simplicity in what follows, define

(20.20)

in which case Eq. (20.19) gives

(20.21)

According to Eq. (20.20),

(20.22)

so that Eq. (20.17) is now

(20.23)

Finally, a calculation using Eq. (20.21) and the relation yields

(20.24)

where

(20.25)

Substituting Eqs. (20.23) and (20.24) into (20.13), carrying out theindicated differentiations, and simplifying the result yields the Blasiussimilarity boundary layer equation

(20.26)

The fact that this relation is entirely in terms of functions of means thatthe hypothesis concerning the existence of a similarity solution turned outto be true. In other words, if an equation with a definite dependence had

resulted, it would have meant that either the choice of similarity variablewas wrong or that a similarity solution did not exist.

Accompanying Eq. (20.26) must be three boundary conditions because itis a third-order equation. The condition that the surface is the zerostreamline means that

(20.27)

whereas the fact that is zero on the surface means that

(20.28)

A third boundary condition is that the velocity must relax to for largeenough , implying that

(20.29)

Solutions to Eq. (20.26) with boundary conditions (20.27)–(20.29) arereadily obtainable numerically (see Problem 20.1). Figure 20.2 contains aplot of the predicted velocity field that agrees well with data fromphysical experiments and numerical computations of the boundary layer.The physical extent of the region along a flat plate for which the Blasiussolution is valid extends typically as high as , after which the capacity ofthe flow to dampen natural disturbances is reduced to the point that theflow becomes three-dimensional and transitions toward turbulence. Theexact upper streamwise limit of the Blasius solution depends on thesmoothness of the incoming stream and the geometry of the plate,including the nature of the leading edge and the degree of roughness.Considerable latitude in transition occurs depending on these factors,which act to encourage or discourage perturbations leading to thetransition to turbulence.

Figure 20.2. Velocity and vorticity in the Blasius boundary layer: solidline, ; dashed line, .

Because approaches unity for large , it means that in the range of wherethis occurs, or

(20.30)

where it may be determined from the numerical solution that . In theregion where Eq. (20.30) is valid, the asymptotic wall-normal velocity isfound from Eq. (20.24) to be

(20.31)

Consequently, as the fluid encounters the flat plate, the loss of momentumis accompanied by an outward diversion of the fluid that diminishes withdownstream distance according to .

Among the interesting aspects of the Blasius solution is the information itgives about the vorticity in the boundary layer and the drag force on theflat plate. The vorticity in this two-dimensional flow is , and a calculationusing Eqs. (20.23) and (20.24) yields

(20.32)

The term in parentheses containing , which originates in the streamwiseshearing associated with , grows downstream to be the dominantcontributor to vorticity. A plot of the vorticity when is shown in Fig. 20.2.Because drops to zero by the outer edge of the boundary layer, thevorticity is confined to the immediate vicinity of the wall.

At the wall surface, the numerical solution shows that

(20.33)

in which case Eq. (20.32) gives the surface vorticity as

(20.34)

This relation suggests that the vorticity goes to infinity at the leading edgeof the plate, an unphysical result that is a consequence of ignoring thedetails of the flow as it encounters the plate tip. For example, the Blasius

result requires the assumption of a vanishing pressure gradient that issurely violated at the leading edge besides whatever errors might accruefrom ignoring the effects of streamwise diffusion.

Another interesting aspect of the Blasius boundary layer concerns thediffusion of vorticity from the surface whose rate depends on . In fact, atthe wall surface,

(20.35)

a result that follows from the definition of vorticity with the help of thecontinuity equation. Evaluating Eq. (20.13) on the boundary surfacereveals that the right-hand side of Eq. (20.35) is zero, so that it may beinferred that there is zero viscous flux of vorticity from the solid boundaryinto the Blasius boundary layer. Because there is vorticity in the boundarylayer nonetheless, the question arises as to where the vorticity comesfrom. In fact, a computation shows that is singular at , so that under theidealized constraints of similarity, all the vorticity enters the Blasiusboundary layer from the leading edge and then diffuses outwardthroughout the boundary layer as it convects downstream. In a morerealistic model that includes the pressure gradient term near the leadingedge, it may be expected that there will be a nonzero vorticity flux at thewall in the initial upstream region of the boundary layer.

The formula (20.34) for the vorticity on the wall surface can bemanipulated to get the local wall shear as

(20.36)

From this, the drag on one side of a flat plate of length can be computedfrom

(20.37)

In terms of the drag coefficient

(20.38)

it is found that

(20.39)

The downstream growth in the thickness of the boundary layer is of someinterest. Several precise definitions of the boundary layer thickness arepossible, such as , which is the location where the velocity is within 1percent of the free stream value. Because the numerical solution showsthat when , Eq. (20.16) implies that

(20.40)

In contrast to , which depends on an arbitrary choice of a percentage of thefree stream velocity, the boundary layer thickness can be based on lengthscales that characterize differences between flow near a wall with andwithout viscosity. Thus the actual mass flux past the solid wall in theBlasius boundary layer is . Because , the mass flux is less than what wouldoccur in the same geometry if the flow were inviscid. The displacementthickness, , is defined as the distance the plate needs to be offset (in thepositive direction) so that the inviscid flux matches the true flux. In otherwords,

(20.41)

Adding to both sides of the equation, moving the integral on the left-handside to the right side, and dividing by gives

(20.42)

as the displacement thickness. Because the velocity of the boundary layerrises to , the integrand in this relation goes to zero for large enough , andthus is well defined.

In addition to , a momentum thickness can be defined in a similar way. Inthis case, the true momentum flux along the wall is . This is smaller thanthe momentum flux that would occur if it were assumed that the true massflux were present, but that the momentum it carried were traveling atspeed . The momentum thickness is the height a fluid layer traveling at

speed would have to be to carry this difference in momentum. This meansthat

(20.43)

or

(20.44)

For the same reason as in the case of , the momentum thickness is welldefined.

To evaluate for the Blasius boundary layer, first change the variable ofintegration in Eq. (20.42) to , in which case,

(20.45)

After substituting Eq. (20.23) into the integrand on the right-hand side, theintegral becomes

(20.46)

which is equal to , according to Eq. (20.30). Consequently,

(20.47)

The momentum thickness can be computed by changing the integrationvariable to , substituting Eq. (20.23), replacing with the help of Eq.(20.26), and evaluating the integral using Eqs. (20.30) and (20.33). Theresult is

(20.48)

When formed into Reynolds numbers, such as as in , and are useful forproviding a means of comparing boundary layers appearing in differentwind tunnels and other disparate contexts.

The integral character of and suggests that they may show up in a CVanalysis of the boundary layer, as is the case for in the momentum

integral equation for laminar boundary layers developed by von Karman(von Kármán 1921). In this, for a zero-pressure gradient boundary layer,Eq. (20.13) is integrated normal to the wall to a point in the free stream,say, , where the streamwise velocity is everywhere equal to . The result is

(20.49)

Replacing the integrand in the second term on the left-hand side via theidentity

(20.50)

in which Eq. (20.3) has been used, and carrying out the integration, Eq.(20.49) becomes

(20.51)

An integration of Eq. (20.3) across the boundary layer gives

(20.52)

Substituting this into Eq. (20.51), commuting the derivative with theintegral, and rearranging yields

(20.53)

where the constancy of has been used. Because for , the integration canbe taken out to infinity, and then appealing to Eq. (20.44), it is derived that

(20.54)

This expression relates the growth in to the wall shear stress, orequivalently, the decrease in momentum flux parallel to the wall thataccompanies the frictional resistance of the solid boundary (see Problem20.6). A generalization of Eq. (20.54) to accelerating boundary layers,such as are discussed in the next section, is considered in Problem 20.4.

20.3 Falkner-Skan Boundary Layers

Section 7.2 considered a family of inviscid flows with complex potential given in Eq. (7.21) that were seen to contain streamlines that could bematched up with the boundaries of simple geometrical shapes such as asharp corner and a wedge. If these flows are considered to be occurring ata high Reynolds number in a viscous fluid, then boundary layers similar tothe Blasius flow will be present. In fact, the Blasius solution is theboundary layer appropriate to the special case of Eq. (7.21) for which ,corresponding to uniform flow past a straight wall.

For the family of inviscid flows associated with Eq. (7.21), the velocitytangential to the boundary can be computed from Eq. (7.25a) by choosingthe coordinate to be along the surface. This requires setting andassociating with . In this case the velocity accelerates or decelerates nextto the boundary according to the exponent in the streamwise velocity

(20.55)

where is a constant, and for simplicity the definition has been made. Theboundary layer forming in response to Eq. (20.55) as external velocity willreflect the streamwise change in the outer flow field. This includes boththe local velocity itself, which changes along the wall, and the local valueof the pressure gradient that appears in the boundary layer form of theNavier-Stokes equation (20.11)

The assumption that the boundary layers are thin offers an effective meansof generalizing the Blasius result to include this wider class of flows. Inthe first place, the velocity along the wall in the potential solution can betaken to be a good approximation of the velocity just outside the boundarylayer forming on the solid surfaces. Second, outside the boundary layer,the flow is effectively inviscid, and so the Bernoulli equation applies,which is tantamount to supplying information about the pressure justoutside the viscous wall layer. Because the pressure is not expected to varyacross the boundary layer at any fixed streamwise location, it is seen thata means exists of accounting for the pressure term in the governingboundary layer equation.

To implement this strategy, assume that at points upstream of thebeginning of the boundary layer, the velocity and pressure are constant

and equal to and , respectively. Then, along the outer edge of theboundary layer, the Bernoulli equation holds to the effect that

(20.56)

Taking a streamwise derivative yields

(20.57)

where Eq. (20.55) has been used. The dependence of the pressure gradientin Eq. (20.57) is then . Moreover, when , the pressure gradient is negativeand thus favors acceleration of the fluid, whereas it is positive anddecelerates the fluid when .

With the use of Eq. (20.57), the boundary layer equation (20.11) becomes

(20.58)

and the goal now is to seek a similarity solution to this equation. In fact,the argument justifying the selection of the similarity variable defined inEq. (20.16) still holds in this case with the understanding that the symbol appearing in this and other relations has an dependence as given in Eq.(20.55). A nonconstant does not impact the determination of and in Eqs.(20.21) and (20.23), respectively, so these hold unchanged. The samecannot be said for because the dependence of needs to be taken intoaccount in computing .

After a lengthy computation in which each expression in Eq. (20.58) isconverted to similarity form and simplified, it is found that

(20.59)

a relation known as the Falkner-Skan equation. When , this simplifies tothe Blasius equation (20.26). As in the case of the Blasius solution, thefact that all quantities in Eq. (20.59) are in terms of suggests that asimilarity solution should exist.

Numerical solutions to Eq. (20.59) are readily obtained, for example,using MATLAB. Several cases of the computed velocity are shown in

Fig. 20.3, corresponding to different values of . Among these, the casewith shows the effect of strong acceleration driven by a favorablepressure gradient in thinning the boundary layer and increasing thevorticity near the wall surface. In contrast, the relatively smalldecelerations associated with and lead to a significant slowing andthickening of the boundary layer. All Falkner-Skan solutions with have aninflection point in the velocity distribution at a location where . This isreadily explained by the fact that when , Eq. (20.58) evaluated at showsthat at the wall surface. Conversely, as the free stream is approached forany value of , , because the convergence of means the velocity slope isdiminishing in this location. Consequently, for , changes sign at somepoint in the boundary layer, and this is an inflection point. For , theinflection point is at the wall surface, consistent with the observation inthe previous section that for the Blasius boundary layer, the vorticityderivative is zero at this location.

Figure 20.3. Velocity in the Falkner-Skan boundary layer: dotted line, ;solid line, (Blasius solution); dashed line, ; dash-dotted line, .

The particular solution with shown in the figure has the property that atthe wall surface. Any further reduction in leads to reverse flow along thewall and a rise in the external flow to greater than unity, an effect that canbe dismissed as unphysical. Perhaps the most interesting lesson from theFalkner-Skan solution is just how sensitive the boundary layer is toadverse pressure gradients. The fluid next to the wall has low momentumand, when faced with even a relatively small opposing pressure force, isquite liable to separate from the surface, thereby undermining theconditions necessary to have a boundary layer.

The displacement thickness can be evaluated in the present case from theexpression in Eq. (20.42). Here is defined in terms of , so the actual dependence of is given by

(20.60)

For , corresponding to the impingement of fluid onto a flat wall, as wasconsidered in Fig. 7.2(b), the displacement thickness is a constant. In thisspecial case, the thinning of the boundary layer caused by the linearlyincreasing speed of the outer fluid velocity balances the tendency ofviscous diffusion to widen the boundary layer so there is no change in itsthickness. In general, the boundary layer thins with streamwise distancefor and thickens for , as in the Blasius boundary layer when . Similarbehavior is expected for .

Another interesting result concerns the wall shear stress . A calculationgives

(20.61)

from which it may be ascertained that the dependence of the shear stressobeys

(20.62)

This shows that for the particular case of an accelerating boundary layerwhen , depicted in Fig. 7.2(c), the shear stress is constant along the wall.For this flow, there is an exact balance between the tendency for the shearstress to rise due to the acceleration of the outer flow and the capacity ofviscous diffusion to spread momentum away from the surface.

Problems

20.1 Solve Eq. (20.26) numerically (e.g., using the MATLAB commandbvp4c) for the Blasius velocity field and plot.

20.2 Give the details of the derivation of the governing similarity equation(20.59) for the Falkner-Skan boundary layer.

20.3 Find numerical solutions to the Falkner-Skan equation for differentvalues of . For each of these, evaluate the constant in the expression for and .

20.4 Beginning with Eqs. (20.11) and (20.57), show that the generalizationof the momentum balance in Eq. (20.54) for a general external velocityfield is given by

(20.63)

20.5 The integral momentum balance in Eq. (20.54) can be used to findapproximate solutions to the boundary layer. In this, a polynomial ortrigonometric function representing is assumed for the region , where isthe boundary layer thickness. Substituting this into the integralmomentum equation and evaluating terms yields an equation for whosesolution can be used to predict , , and . Implement this method for thecrude choice

(20.64)

and for the somewhat better guess of a cubic polynomial in satisfying theboundary conditions at and at . Compare the results in both cases to theexact solution via a velocity plot.

20.6 Apply the CV mass conservation and momentum equations to acontrol volume covering a segment of the Blasius boundary layer betweena point and and extending outward from the wall to a point where .Physically interpret the balance of terms in each equation. Show that thestreamwise CV momentum equation converges to Eq. (20.54) as .

21Some Applications to Convective Heat andMass Transfer

Energy and mass transfer within flowing fluids is ubiquitous and ofessential importance in a wide range of technologies. This chapterprovides an introduction to some of the ideas involved in analyzing theeffects of convection on the development of the energy or mass field. Italso develops an opportunity to further explore the properties of the manyflows that have been discussed previously by seeing their influence on thedissemination of contaminant fields.

The dynamics of the velocity field can be either coupled or decoupledfrom the transport of heat or mass if they are present. For example, innonisothermal flows with temperature-dependent viscosity,compressibility, combustion, and other effects, it is likely that the velocityfield can only be determined simultaneously with the thermal field.Conversely, this may not be the case for a plume of sufficiently dilutecontaminant species or of thermal energy released into a flow field. In thischapter, our focus is confined to the latter situation, in which the heat ormass added to the flow acts passively, so determination of the velocityfield can be done without regard to the energy or mass concentration field.

The first scenario of interest concerns a generalization of the boundarylayer analysis of the previous chapter to include the presence ofsimultaneous fluid and thermal boundary layers. Following this is apresentation of a numerical scheme known as the Monte Carlo method,which, with a small expenditure of programming effort, can be adapted tothe solution of a wide range of convection diffusion problems that cannotbe solved analytically. The effectiveness of the Monte Carlo method isdemonstrated for the same thermal boundary layer considered in the nextsection. Some additional applications of the Monte Carlo scheme to flowsconsidered in this book are suggested in the problem section at the end ofthe chapter.

21.1 A Thermal Boundary Layer

Many practical applications involving convective heat transfer containboundary layers where fluid and thermal fields develop simultaneouslyalong solid boundaries. Some of the central aspects of such phenomenacan be illustrated in the context of a similarity solution for the thermalfield developing within the Blasius boundary layer. Specifically, considerthe thermal field generated over a flat plate that is heated to a constanttemperature higher than the ambient . The heat produced at the surfacediffuses and convects into the flow, creating a thermal boundary layer.

The lateral spread by diffusion of momentum and heat depends on thekinematic viscosity and thermal diffusivity, respectively. In fact, the ratioof these quantities, known as the Prandtl number,

(21.1)

plays an important role in determining the relative size of the thermal andmomentum boundary layers, as illustrated in Fig. 21.1. Here , suggestingthat thermal diffusion is greater than momentum diffusion, so the thermalboundary layer is wider than the underlying Blasius boundary layer.

Figure 21.1. The thermal boundary layer () grows faster than the Blasiusboundary layer () when .

For the thermal field of interest here it is assumed that the Blasiussolution is fixed in time and is the velocity field appearing in thegoverning convective heat equation (15.33). Simplifying the latter via theboundary layer assumption and assuming steady flow gives

(21.2)

The boundary condition at the wall is , while at the outer edge of thethermal boundary layer, . There is also an upstream condition that .

To set the problem up for a similarity solution, it is necessary to scale thetemperature field similar to what was done for the velocity in Eq. (20.23).Because the difference in temperatures between the plate and the ambientis fixed in the streamwise direction, the choice of scaled variable

(21.3)

leads to the boundary conditions

(21.4)

(21.5)

and

(21.6)

which are identical to the conditions on for the Blasius solution.Therefore, taking advantage of the same reasoning that applied to theBlasius boundary layer, it is natural to take Eq. (20.16) as the similarityvariable and to define

(21.7)

Substituting Eq. (21.7) into (21.2) and using the Blasius formulas for thevelocity components gives

(21.8)

Solutions for depend on , which is a given function determined a priori aspart of the analysis of the Blasius boundary layer. The presence of thePrandtl number is consistent with the idea that it crucially decides therelative characteristics of the temperature boundary layer in comparisonto that of the momentum boundary layer. If , so that momentum andenergy diffusion are at the same rate, then Eq. (21.8) becomes identical toEq. (20.26) as long as

(21.9)

Moreover, because shares the same boundary conditions as , it is clearthat for , Eq. (21.9) satisfies all the requirements needed to be a solutionto Eq. (21.8). Consequently, the scaled temperature profiles in the thermalboundary layer are identical to the scaled velocity in the Blasius boundarylayer for the case.

For all other Prandtl numbers, Eq. (21.8) is readily integrated twice toyield the exact solution

(21.10)

Equation (21.10) can be evaluated numerically, and some of its predictionsare shown in Fig. 21.2. Compared to the unit Prandtl number case, thethermal boundary layer thins for high and thickens for low , consistentwith the relative magnitudes of and . A thermal boundary layer thickness can be defined analogously to the Blasius boundary layer thickness. Thus,for example, the outer edge of the thermal boundary layer can be definedas the value of at which the scaled temperature is within 1 percent of itsfree stream value. In this case, a calculation for the curves in Fig. 21.2shows that

(21.11)

where for , respectively.

Figure 21.2. Computed profiles; dashed line, ; solid line, ; dotted line, .

Of particular interest in heat transfer problems is predicting the flux ofheat into or out of a bounding surface. Energy enters or leaves the flow viamolecular diffusion. To enable empirical treatment of the heat transferrate, it is common practice to introduce a heat transfer coefficient, , that isdefined in the present context via the relation

(21.12)

The idea here is that the left-hand side is the exact energy flux from thesurface, whereas the right-hand side introduces the parameter in productwith the overall temperature difference that drives the energy flux into thefluid. For complicated flows where the left-hand side may be difficult todetermine, empirical measurements of may suffice to supply generalinformation about the energy flux.

For the self-similar thermal boundary layer considered here, the left-handside of Eq. (21.12) is known so that can be determined. It is commonpractice to quote values of in the context of the nondimensional Nusseltnumber defined by

(21.13)

where is an appropriate length scale. In the present context, it is useful totake as the length scale so that applying Eq. (21.3) to (21.12) gives

(21.14)

and with the help of Eq. (21.10), it follows that

(21.15)

This may be simplified somewhat by restricting the Prandtl number rangeto , in which case it may be shown (Kays, Crawford & Weigand 2005)that, to good approximation,

(21.16)

When , this result is consistent with Eq. (20.36) deriving from theequivalence between and contained in Eq. (21.9). According to Eq.(21.15) or (21.16), decreases with , consistent with the expectation thatthe temperature gradient at the wall surface is reduced as the fluid in theboundary layer heats up with downstream distance.

21.2 Monte Carlo Schemes for Modeling ConvectiveDiffusion

Among the many numerical schemes that can be used to simulate flowswith passive heat or mass transfer, the Monte Carlo method eschews theuse of a numerical mesh in favor of incorporating the random convectivemotion of particles of energy or mass to approximate the governingconvective diffusion equation. The latter, in the case of the energy field, is

(21.17)

which was previously encountered as Eq. (15.33). In this, because thepresence of heat is of no consequence for the velocity field, is assumed tobe known for all time. A companion equation to Eq. (21.17) for masstransfer governs the dynamics of the mass concentration (in units ofmass/vol) of a dilute contaminant species contained in the fluid. In thiscase, the thermal diffusivity is replaced by the mass diffusivity . Theanalogue to the Prandtl number for mass transfer is the Schmidt number .

21.2.1 Probabilistic Interpretation of Diffusion

Whether of energy or mass, the contaminant field in the proposed MonteCarlo method is represented by discrete elements of energy or mass thatconvect and diffuse in the flow in such a way that the governingconvective diffusion equation is satisfied in an average sense. To motivatethis approach, it is convenient to first consider the case of mass diffusionin a quiescent fluid in 3D that is governed by the diffusion equation

(21.18)

Imagine that at the point , at time , a unit mass of contaminant is placedinto the flow. In this case, the initial condition can be taken to be the Diracdelta function

(21.19)

which is a generalized function (Zauderer 2006) defined by the propertythat for any smooth function ,

(21.20)

In effect, is zero everywhere, except at , where it is “infinite” inmagnitude. According to Eq. (21.20),

(21.21)

so Eq. (21.19) is consistent with the idea that a unit of mass is depositedinto the flow at a single point.

The solution to Eq. (21.18) with initial condition (21.19) may be obtainedreadily by a Laplace transform (Carrier & Pearson 1988), yielding

(21.22)

that satisfies for all time

(21.23)

Thus the total mass remains constant in time as it diffuses. It may benoticed that has the same form as the probability density function of aGaussian distribution (Riley et al. 2006) with zero mean and variance,given by

(21.24)

In this case, Eq. (21.23) is a statement that the total probability is unityand stays so independently of . This connection between the timedevelopment of the concentration field of unit mass placed into the flow,and its reinterpretation as the probability density function of a Gaussianprocess that evolves in time with a variance given by Eq. (21.24), suggeststhat diffusion of a contaminant can be modeled in some sense as aGaussian process. Implementation of this idea results in the Monte Carloscheme.

21.2.2 Monte Carlo Model of Diffusion

To develop a numerical approach to modeling diffusion, break up theinitial unit mass into pieces, with a large number, so that each piece hasmass . Let each of the particles take a random hop from a Gaussian

distribution with mean zero and variance given by Eq. (21.24). Cover theflow region by a fine cubic mesh with points at the center of the th box.Let represent the number of particles that have arrived in thecorresponding box and let be the volume of the box. The total mass ineach box is , and so the mass concentration in each box is

(21.25)

Because the sum of all the is , it is clear that Eq. (21.25) satisfies thediscrete analogue of (21.23). As and , it may be shown that theapproximation in Eq. (21.25) converges to the exact solution Eq. (21.22).

Instead of breaking up the initial unit mass into pieces, an equivalentformulation is to allow the mass to take a random hop as a whole, butrepeat the experiment times, resulting in a distribution of locations wherethe unit mass has traveled. These data can then be used to evaluate adiscrete approximation to via Eq. (21.25) in the same way as before,except that is now interpreted as being the number of realizations forwhich the mass has arrived in a given box. The end result is the same, withconvergence to the Gaussian distribution assured as the number of trials isincreased to infinity.

A common scenario in studying diffusion is for the contaminant to beinitially distributed over a region with concentration that is not confinedto a single point. In this more general case, a Fourier transform (Rileyet al. 2006) yields the exact solution to Eq. (21.18) as the convolutionintegral

(21.26)

It is easily checked that Eq. (21.19) and (21.22) represent a special case ofEq. (21.26). According to Eq. (21.26), the concentration at a point at time is the result of a sum of contributions from contaminant that has diffusedto from all points of its original domain. A viable means ofapproximating the physical process captured in Eq. (21.26) is to subdividethe flow domain into small cubes of size , collect the mass of contaminantin each cube – for example, in the th cube, it may be approximated as –and then apply either variant of the Monte Carlo scheme described

previously to advance in time the mass in each individual cube. Forexample, the mass of each cube can be moved as a whole via a randomjump from a Gaussian distribution with variance given by Eq. (21.24). Anestimate of is then given by

(21.27)

where is a volume surrounding the point and is the total mass of theparticles that are within at time . Depending on the desired accuracy, thecalculation can be repeated multiple times and averaged. The morerealizations and the finer the initial division of the domain, the closer theresult can be expected to approximate the exact solution.

To illustrate the algorithm developed thus far, consider satisfying Eq.(21.18) with initial state

(21.28)

in the form of a step distribution. Substituting Eq. (21.28) into Eq. (21.26)and integrating yields the exact solution

(21.29)

where the error function is defined as

(21.30)

The numerical scheme is implemented by subdividing the region into equal elements of length so that each has mass (because ). For a fixed ,the segments take a random hop from a Gaussian distribution withvariance given by Eq. (21.24). Taking and , the predicted concentrationfield at is compared to the exact solution in Fig. 21.3. In this, the region is subdivided into cells of length 0.5 and the number of elements arrivingin each cell is counted and multiplied by to get the contained mass, andthen Eq. (21.27) is used to compute the local concentration. Clearly theMonte Carlo approach yields the correct result.

Figure 21.3. Monte Carlo scheme applied to one-dimensional diffusion ina quiescent fluid; solid line, exact solution; circles, simulation.

21.2.3 Monte Carlo Simulation Including Convection

Our interest now lies in extending the Monte Carlo scheme to includeconvection so that Eq. (21.17) can be modeled. This generalization isreadily accomplished by implementing two essential changes. In the firstplace, the modeling of diffusion via random hops of the contaminantparticles is supplemented by simultaneous convective movementsdetermined by the local velocity field. Second, the movement of theparticles is done in a series of small time steps so as to accommodate thechanging velocity field experienced by the particles as they convect. Sucha process can be shown (Hammersley & Handscomb 1964) to provide asolution to the convective diffusion equation in the limit as the length ofthe time step diminishes to zero.

To be specific about the scheme, the positions of the particles that are inthe flow at any time are advanced according to the algorithm

(21.31)

where the , are the locations of the particles at time and is a vectorconsisting of independent random hops in the and directions from aGaussian distribution with variance

(21.32)

The particles might be in the flow because they were created at the initialtime to represent a given initial state or because they enter the flowduring the calculation so as to satisfy boundary conditions.

Though the discussion thus far has centered on the case of mass diffusion,it also readily applies to the energy field by assuming that the convectingand diffusing particles contain energy as against mass. Assuming that thefluid is initially at a reference temperature, say, , then it is convenient tolet represent the energy carried by the particles over and above that

associated with the temperature . In this case, a measure of the localinternal energy/volume carried by the particles in a small volume is givenby

(21.33)

where the sum is over the particles contained in . The contribution of theenergy to the local temperature can be determined by recognizing thatthis energy is equal to the product of the mass in , namely, , times theenergy/mass given by , where, generally, for liquids and for gasses. Theend result is that

(21.34)

Through this relation, the energy distribution computed by the MonteCarlo scheme produces an estimate of the temperature field satisfying theconvective diffusion equation in the form of Eq. (21.17).

21.2.4 Monte Carlo Solution to a Thermal Boundary Layer

To apply the Monte Carlo method to the solution of the thermal boundarylayer considered in Section 21.1, a more convenient scaled temperaturethan in Eq. (21.3) is

(21.35)

In terms of , the boundary conditions consist of 1 on the flat plate and 0far from the plate, so the problem is one of capturing the transfer ofenergy from the plate into the surrounding fluid. Although the previousanalytical approach solved the steady state problem, the numericalcalculation naturally simulates the transient field developing as aconsequence of impulsively heating the plate to 1 at from an initial statewith everywhere. After some time, an equilibrium temperature field isachieved, which may be averaged over an extended time interval toproduce a good approximation to the thermal boundary layer given in Eq.(21.10).

The physical region of interest in the simulation is taken to extend fromthe leading edge of the plate until a distance downstream. The numericalcalculation in scaled coordinates includes the region , and the plate isdivided into equal pieces of length . The outer flow velocity is used toscale velocities, and time is scaled according to . Particles in the flowmove via Eq. (21.31), written in dimensionless variables, and the varianceof the random hop in this case is given by

(21.36)

Energy enters the flow by diffusion at each of the points at every timestep. The flux of internal energy into the flow at is given by

(21.37)

with the temperature gradient evaluated at the wall. The total internalenergy released as a new particle at the th location at every time step is

(21.38)

Note that for this two-dimensional problem, the right-hand side of Eq.(21.38) is assumed to be multiplied by a unit length in the direction sothat has units of internal energy.

To estimate the temperature derivative in Eq. (21.37) at the surface of theplate, a simple approach that works well is to use a finite differenceapproximation

(21.39)

where is a small distance from the surface at . In this case, substitutingEq. (21.39) into (21.37) and the result into (21.38) gives

(21.40)

as the energy that must be entering the flow at the th location at each timestep. At , the temperature is everywhere, and so in this case, Eq. (21.40)becomes

(21.41)

which is the initial amount of energy transferred to each particle at theimpulsive start of the computation.

It proves convenient to use as the basis for scaling the numericalproblem. In this case, individual particles have the nondimensional energy

(21.42)

Moreover, using Eqs. (21.40) and (21.41) with Eq. (21.42) gives

(21.43)

as the scaled energy put into the flow at each location at each time step.The temperature in Eq. (21.43) is computed from a scaled form of Eq.(21.34) in which is centered around the point . The result is

(21.44)

under the assumption that . At the start of the calculation, vanisheseverywhere, except on the plate, so that Eq. (21.43) correctly gives . Forstability of the numerical algorithm, should be bounded by 1, and thismeans that should be large enough so that the sum in Eq. (21.44) is over areasonably large sample size at any given time.

The Monte Carlo algorithm may be summarized as consisting of threesteps that march the solution forward in time. In the first place, Eq.(21.43) is used to determine the energy of the new particles to be enteredinto the flow. Second, the particles are moved according to the scaled formof Eq. (21.31), and third, Eq. (21.43) is evaluated with the help of Eq.(21.44). Particles that move below the plate are reflected back to positive so that the preexisting particles do not contribute to the energy flux at thewall. In this way, the value of the gradient in Eq. (21.37) that is desired ismaintained. In fact, a Monte Carlo simulation for an insulated boundarywhere can be modeled using particle reflection as the boundary condition.

A converged calculation of the thermal boundary layer using and timestep is reached by and contains approximately 18,500 particles. A view ata fixed time showing just one-tenth of the particles is given in Fig. 21.4.Long time averaging of the solution yields temperature data that can becompared to the similarity solution. For example, Fig. 21.5 compares thetemperature at different distances above the wall as a function of . In this,the Monte Carlo solution has been averaged over the time interval from ,allowing for relatively smooth statistics. As seen in the figure, the MonteCarlo result agrees closely with the similarity solution.

Figure 21.4. Particles of internal energy in a thermal boundary layer atequilibrium.

Figure 21.5. Comparison of for a thermal boundary layer with : circles,Monte Carlo computation; solid line, similarity solution.

The Monte Carlo method can be used to estimate the Nusselt number byemploying Eq. (21.37) with the definition of the heat transfer coefficientin Eq. (21.12). The result after applying the appropriate scalings is

(21.45)

Figure 21.6 compares the computed Nusselt number to that predicted inEq. (21.16). Apart from the relatively slow convergence of the MonteCarlo solution, it is clear that the method is successful in capturing thetrend. This gives confidence that the Monte Carlo scheme can be appliedto many problems where analytical solutions are not available and anumerical approach is necessary.

Figure 21.6. Nusselt number for a thermal boundary layer on a constanttemperature flat plate: circles, Monte Carlo simulation; solid line,similarity solution .

Problems

21.1 Compute the self-similar thermal boundary layer developing for aconstant temperature plate for low Prandtl number flow when it can beassumed that the thermal boundary layer is entirely immersed in aconstant velocity . Show that, in this case, .

21.2 Compute the high Prandtl number self-similar thermal boundarylayer for flow past a constant temperature plate when it can be assumedthat the thermal boundary layer is entirely immersed in the linear part ofthe velocity field adjacent to the wall (note Eq. 20.33 in this regard). Showthat, in this case, . Hint: use MATLAB or equivalent to get a numericalvalue of the integral .

21.3 Implement the Monte Carlo scheme for the case of a unit mass ofcontaminant placed into the inviscid flow past a cylinder. Set the initiallocation of the mass upstream of the cylinder and assume a zero fluxcondition at the surface of the cylinder. Make an animation of the result.

21.4 Generalize the Monte Carlo scheme described in Section 21.2.4 forthe case of a thermal boundary layer forming on a constant-temperatureplate of length for which the boundary layer flow is the Falkner-Skansolution with . In this, scale lengths with , velocities with , so that , , , anddefine . Show that , and show that the velocities of the particles can beobtained from and with . Use the code to verify the formula (Eckert 1942) for the case when and .

21.5 Consider a laminar zero-pressure gradient boundary layer over a flatplate of length with constant external velocity from which heat isdiffusing at every point into the ambient fluid of temperature at theconstant heat flux rate, . Simulate the temperature field using a MonteCarlo scheme and compare the predicted Nusselt number against theanalytically derived result (Kays et al. 2005). Approach the problem as inthe constant-temperature case in Section 21.2.4, in which lengths arescaled by , velocities by , and take and . Scale energy based on so that thenondimensional energy of each particle put into the flow is unity. Derivethe formula

(21.46)

for the Nusselt number using the defining relation for the heat transfercoefficient together with Eq. (21.34) and with Eqs. (21.37) and (21.39) toaccount for . Show that a scaled relation for the surface temperature isgiven by

(21.47)

Adapting Eq. (21.34) to the present case and using Eq. (21.47), show thatthe temperature field can be computed from

(21.48)

Plot the computed wall temperature from Eq. (21.47) and make a contourplot of temperature from Eq. (21.48).

Appendix AEquations in Curvilinear Coordinates

The need for the fluid flow equations expressed in polar, cylindrical, andspherical coordinates is encountered in a number of contexts in the courseof this book. Here, for convenience, a number of the most useful of suchrelations are presented.

A.1 Polar Coordinates

In a two-dimensional polar coordinate system r, θ with corresponding unitvectors , , the gradient of a scalar is

(A.1)

and its Laplacian is

(A.2)

The divergence of a vector is

(A.3)

and the curl of v is

(A.4)

where k is perpendicular to the r − θ plane.

The incompressible Navier-Stokes equation for velocity components ur,uθ, apart from a gravitational force, is

(A.5)

(A.6)

A.2 Cylindrical Coordinates

In a three-dimensional cylindrical coordinate system R, α, Z withcorresponding unit vectors , , , the gradient of a scalar is

(A.7)

and its Laplacian is

(A.8)

The divergence of a vector is

(A.9)

and the curl of v is

(A.10)

The incompressible Navier-Stokes equation for velocity components uR,uα, uZ, apart from a gravitational force, is

(A.11)

(A.12)

(A.13)

A.3 Spherical Coordinates

In a three-dimensional spherical coordinate system r, θ, withcorresponding unit vectors r, , , the gradient of a scalar is

(A.14)

and its Laplacian is

(A.15)

The divergence of a vector is

(A.16)

and the curl of v is

(A.17)

The incompressible Navier-Stokes equation for velocity components ur,uθ, u, apart from a gravitational force, is

(A.18)

(A.19)

(A.20)

Appendix BTensors

Some important aspects of vector and tensor analysis that were touched onwithin previous chapters of the book but deserve more elaboration aregiven here. Complete accounts of these and many other aspects of tensoranalysis are worthy of study and can be found in Gurtin (1981) and Gurtinet al. (2010).

B.1 Divergence of a Tensor

The concept of the divergence of a tensor arose in several instances inChapters 4, 11, 13, 15, and 16, where it was mentioned that for any tensor , is a vector with components

(B.1)

in a rectangular Cartesian system. In some sense, this represents a logicalprogression from the case of the divergence of a vector field, , which is ascalar as given in Eq. (1.41). In fact, this connection is evident in theformal definition of , which is now considered.

For a given constant vector and tensor field ,

(B.2)

is clearly a vector field, because for each value of , it is a vector. Taking adivergence of gives a scalar field

(B.3)

where the dependence of and is not indicated explicitly. For each fixed ,the operation in Eq. (B.3) returns a scalar. Moreover, it is evident that

(B.4)

is a linear map on vectors.

Any linear map that takes vectors to scalars must be in the form of aninner-product with a vector. In other words, it must be the case that avector exists, say, , such that

(B.5)

To prove this, note that

(B.6)

and define the vector

(B.7)

From the definition of the dot product, it is then seen that Eq. (B.5) holds.This argument shows that a unique vector exists such that

(B.8)

The vector is defined as the divergence of , so that satisfies

(B.9)

for all vectors .

To find the components of , set in Eq. (B.9) so that

(B.10)

By definition, the right-hand side of this equation is

(B.11)

and using Eq. (1.20), it follows that

(B.12)

B.2 Vector Cross

In the course of establishing the symmetry of the stress tensor in Section11.2, notably in Eq. (11.23), and in deriving the control volume angularmomentum equation in Section 16.3.3, specifically Eq. (16.52), the vectorcross appeared, whose meaning is now more fully explained.

The vector cross is a tensor whose definition is associated with thelinearity of the cross product of two vectors. Thus, for vectors and , if isheld fixed in , then this represents a linear map on vectors . In this case,the linear mapping is a tensor and is referred to as the vector cross anddenoted . To get the components of , it is straightforward to apply thedefinition Eq. (1.20) along with the definition of the cross product in Eq.(1.17) to conclude that

(B.13)

Having defined the vector cross as a tensor, then quantities, such as , thatappear to consist of a vector in cross product with a tensor are in fact theproduct of tensors and hence tensors in their own right. The divergence of , as appears in Eq. (16.52), can now be regarded according to thedefinition of the tensor divergence in Eq. (B.9). In this case, itscomponents according to Eq. (B.12) are

(B.14)

which a calculation shows to be consistent with Eq. (16.52).

B.3 Principal Directions

An essential aspect of the analysis in Chapter 12 concerning how the localfluid motion and the rate of strain tensor fit in with the stress tensorinvolved viewing these quantities in principal directions. Here weestablish some of the basic results that underlie the existence of principaldirections and show how these pertain to the discussion of thetransformation properties of vectors and tensors that was given in Sections1.5 and 8.1.

First it is helpful to summarize some of the main results consideredpreviously in regard to mappings between components corresponding to

different sets of basis vectors. Thus, for a given reference frame and set oforthonormal basis vectors , a vector and tensor may be expressed as

(B.15)

and

(B.16)

in which and are the respective components. Alternatively, if the basisvectors are employed, then the components will change so that, in thiscase,

(B.17)

and

(B.18)

The components for the two different sets of basis vectors can be relatedthrough the equations

(B.19)

and

(B.20)

where is a tensor that rotates vectors without a change in magnitude.

Using the basis vectors , the vector and tensor may be defined by

(B.21)

and

(B.22)

As shown in Chapter 8, these may be interpreted as how and appear,respectively, to an observer with a reference frame having the orientation .With these definitions, the mappings

(B.23)

and

(B.24)

can be defined consistent with Eqs. (B.19) and (B.20).

In Eq. (1.16), the magnitude of a vector was defined as

(B.25)

The dot product of two vectors is given formally by the relation

(B.26)

where the right-hand side is interpreted as matrix multiplication of the matrix times the matrix . A calculation using Eq. (B.23) then gives

(B.27)

so that the magnitude of a vector is an invariant in the sense that it isunaffected either by a change in basis vectors within a fixed referenceframe or between observers with different orientations; is the only suchinvariant of vectors.

A similar consideration for the case of tensors shows that they have threeindependent invariants that are traditionally characterized as the principalinvariants

(B.28)

where refers to the determinant of the components of (Riley et al. 2006).By “invariant” is meant that the values of , and remain the same if thetensor components of are substituted for those of . Proof that the first two

of Eqs. (B.28) are invariant follows from Eq. (B.24) and the definition ofthe trace. The third relation is proved using Eq. (B.24) and the identity

(B.29)

for any two matrices .

For any tensor and scalar , it may be proven (Gurtin 1981) that

(B.30)

By definition, the eigenvalues of are determined from the condition

(B.31)

and in view of Eq. (B.30), it is seen that they are the roots of the cubicequation

(B.32)

The invariance of the coefficients in Eq. (B.32) makes clear that theeigenvalues of are independent of orientation of the basis vectors.

Corresponding to the eigenvalues of are eigenvectors such that

(B.33)

If is real and symmetric, it can be shown (Riley et al. 2006) that there arethree real roots to Eq. (B.32), and corresponding to these eigenvalues, theeigenvectors are mutually orthogonal and can be chosen to beorthonormal. Moreover, if the basis vectors are taken to be theeigenvectors of , then will be a tensor with diagonal components, that is, ,and zero for all other entries.

Because the stress tensor is real and symmetric, the previous discussionapplies to this case. Consequently, an orthonormal set of eigenvectors canbe found that forms a set of basis vectors for which

(B.34)

with real. Equation (B.34) together with (1.20) means that the componentmatrix of in terms of the basis vectors is a diagonal matrix with maindiagonal components

(B.35)

and the principal directions of consist of the basis vectors .

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Index

Archimedes’ principle, 97 axial vector, 11, 119

barotropic flow, 138, 151 basis vector, 84 Bernoulli equation, 148

incompressible flow, 150 nonsteady flow, 151 perfect gas, 150

Blasius theorem, 75 boundary layer, 211

Blasius solution, 214 Falkner-Skan, 221 momentum integral equation,220 Monte Carlo simulation, 233 scaling, 212 thermal, 225, 233 thickness, 219 vorticity, 217

buoyancy, 96

Carnot’s relation, 138 Cauchy-Riemann equations, 65 centrifugal force, 90, 169 centripetal force, 90 change of variables formula, 105 circulation, 47, 180 coefficient of thermal expansion, 136 complex potential, 65, 71 complex velocity, 66 continuity equation, 108 continuum hypothesis, 3

control volume angular momentum equation, 157,169 definition, 153 energy equation, 158 mass conservation equation, 153 momentum equation, 154

convective diffusion equation, 145 coordinates

cylindrical, 31, 33 spherical, 33

Coriolis force, 90, 169 Crocco’s relation, 152 cross product, 8 cylinder flow, 75

complex potential, 67 inviscid, 63 with swirl, 77

cylindrical coordinates, 31, 33

d’Alembert’s paradox, 78, 172 dam, 95 derivative

substantial, 108 total, 108, 127

diffusion equation, 229 diffusivity

mass, 229 thermal, 145, 226

dilatation, 4 physical meaning, 107

dipole, 57 Dirac delta function, 229 direct notation, 7 displacement thickness, 219 dot product, 8 doublet

2D source, 61

3D source, 57 complex potential, 67 vortex, 62

drag coefficient in boundary layer, 218 coefficient in sphere flow, 202 coefficient in Stokes flow, 200 definition, 72 Oseen solution for sphere, 201 pressure, 202 viscous, 202

energy control volume energy equation,158 equation, 140 equation for gases, 145 equation for internal energy, 144 equation for kinetic energy, 142 equation for liquids, 145 flux, 140

enthalpy, 149 entropy, 134 Euler equation, 151 Eulerian viewpoint, 15

first law of thermodynamics, 7 fish tank, 98 flow

barotropic, 138, 151 doublet, 57 in a narrow gap, 191 past a cylinder, 63 point vortex, 44 Poiseulle, 190 potential, 42 power law, 67 Rankine vortex, 182

separation, 204 source/sink, 44 sphere at high Reynolds number,201 steady, 17 Stokes flow past a sphere, 194 unidirectional, 190 uniform, 29

fluid definition, 1

force centrifugal, 90, 169 centripetal, 90 Coriolis, 90, 169 drag, 72 lift, 72 sphere, 199 surface, 127

Fourier’s law, 140

gas, 2 energy equation, 145 perfect, 132, 137

Gaussian process, 230 golf ball, 205

Helmholtz decomposition, 36

incompressibility, 106 index notation, 7 inner product, 8, 9 isotropy, 93

Jacobian, 105 jet

impinging on a plate, 160 planar, 49

Kelvin’s circulation theorem, 178

Kronecker delta, 8, 9 Kutta-Zhukhovsky lift theorem, 78

Lagrangian viewpoint, 15 Laplacian, 36 lift

definition, 72 Kutta-Zhukhovsky theorem, 78

liquid, 3

Magnus effect, 206 manometer, 94 mass

concentration, 229 conservation, 6 control volume equation, 153 diffusivity, 229 flux, 108

mass conservation, 107 material fluid element, 6 mean free path, 2, 123, 125 momentum

control volume angular momentum equation, 157, 169 control volume equation, 154 rate of change, 126 thickness, 219

Monte Carlo scheme, 229

Navier-Stokes equation, 128 incompressible, 129 noninertial observer, 129 variable viscosity, 130

Newton’s second law, 7 Newtonian fluid, 122, 123

constitutive law, 124 Nusselt number, 228, 235

observer, 82

definition, 82 inertial, 82 noninertial, 82, 98, 129, 168

orientation, 83

paths, 15 parameter, 18

point vortex, 46, 67 Poisson equation, 36, 41 position vector, 85 potential

complex, 65, 71 flow, 42 power law, 67 scalar, 37 vector, 37

Prandtl number, 226 pressure

coefficient, 75 cylinder, 76 definition, 120 mechanical, 121 thermodynamic, 121

principal directions, 117, 122 process

adiabatic, 134 isentropic, 134, 138

pure straining motion, 118

reference frame, 82 residue theorem, 75 Reynolds number, 188 Reynolds transport theorem,127 right-handedness, 8 rotating bucket, 99 rotation

solid body, 119

tensor, 12, 84, 116

scalar potential, 37 Schmidt number, 229 second law of thermodynamics, 134 sensing volume, 3 shear force, 1 simply connected, 38 solid body rotation, 119 source/sink

2D, 44, 66 3D, 51 images, 49

specific heat, 134 sphere

high Reynolds number flow, 201 inviscid flow, 61 Stokes flow, 194

spherical coordinates, 33 sprinkler, 168 stagnation point, 71 steady flow, 17 stream function, 24

axisymmetric, 30 planar, 24

streamlines, 21 stress tensor, 92, 120

deviatoric, 93, 120 existence, 110 symmetry, 112

substantial derivative, 108

tangent vector, 18 tank

Borda mouthpiece, 167 draining through a nozzle, 165

tensor

antisymmetric, 10 components, 9 definition, 9 divergence, 127 gradient, 11 gradient tensor, 36 identity, 9 inner product, 9 principal directions, 117 product, 9 rate of strain, 116 rotation, 12, 116 skew, 10 stress, 92, 120 symmetric, 10 trace, 9 vector cross, 11

thermal diffusivity, 145, 226 thermodynamics, 131

first law, 132, 140 second law, 134

total derivative, 108, 127 trace, 9 transition, 204 turbulence, 202

uniform flow, 29, 32, 42, 66

vector, 8 potential, 37 axial, 11, 119 basis, 8, 12 components, 8 cross, 11 cross product, 8 dot product, 8 inner product, 8

magnitude, 8 orthogonal, 8 orthonormal, 8 tangent, 18

velocity complex, 66 decomposition, 116, 120 moving observer, 86

viscosity bulk, 144 dynamic, 124

volume rate of change, 106

vortex doublet, 62 image, 49 line, 180 method, 180 point, 46, 67 Rankine, 182 simulation of wing wake,183 stretching, 175 tube, 179

vorticity, 5, 119, 173 2D equation, 175 baroclinic creation, 174 boundary layer, 217 equation, 174

work deformation, 144 reversible, 133 reversible work compression/expansion, 144 surface forces, 141