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Analysis and Modelling of

K. Hanjalic S. KenjerešM.J. Tummers H.J.J. Jonker

Physical Transport Phenomena

Analysis and Modelling ofPhysical Transport Phenomena

The cover pictures represent the instantaneous temperature fields in the prox-imity of hot and cold walls (inside thermal boundary layers) in Rayleigh-Benardconvection of air at Ra = 109, large-eddy simulation. S. Kenjeres, 2006.

Analysis and Modelling of

Physical Transport Phenomena

K. Hanjalic, S. Kenjeres, M.J. Tummers, H.J.J. Jonker

Department of Multi-Scale PhysicsFaculty of Applied SciencesDelft University of Technology

VSSD

c� VSSDFirst edition 2007, corrected 2009

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Printed versionISBN-13 978-90-6562-165-8Electronic versionISBN-13 978-90-6562-166-5

NUR 924

Key words: physical transport phenomena

Contents

Preface xi

I Fundamental Equations 1

1 Fundamental Equations of Transport Phenomena - Field description 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Conservation laws for a control volume in differential form . . . . . . . . 3

1.2.1 Source terms and constitutive relations . . . . . . . . . . . . . . 81.2.2 Common form of the differential conservation law . . . . . . . . 10

1.3 Classification of equations . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . 131.5 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . 14

II Analytical Methods 17

2 Analytical Methods 192.1 Partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Eigenfunctions and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 Spherical symmetry . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Cylindrical symmetry: Bessel functions . . . . . . . . . . . . . . . . . . 282.5 Laplace-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Error- and Gamma-function . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6.1 Error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6.2 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 Combination of variables and dimension analysis . . . . . . . . . . . . . 382.8 Integral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.9.1 Heat storage in the ground . . . . . . . . . . . . . . . . . . . . . 442.9.2 Steel cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.9.3 Chemical reaction on a sphere . . . . . . . . . . . . . . . . . . . 45

v

vi Analysis and Modelling of Physical Transport Phenomena

2.10 Answers to exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.10.1 Heat storage in the ground . . . . . . . . . . . . . . . . . . . . . 452.10.2 Steel cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.10.3 Chemical reaction on a sphere . . . . . . . . . . . . . . . . . . . 50

3 Transport in Stagnant Media 533.1 Stationary problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 553.2.2 Solutions for sources . . . . . . . . . . . . . . . . . . . . . . . . 563.2.3 Summation of sources and Green’s functions . . . . . . . . . . . 593.2.4 Line sources and point sources . . . . . . . . . . . . . . . . . . . 613.2.5 Penetration theory and Duhamel’s theorem . . . . . . . . . . . . 623.2.6 Contact temperature . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Moving front problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.1 Non-stationary heat conduction with phase changes . . . . . . . . 653.3.2 Moving fronts in mass transport problems . . . . . . . . . . . . . 673.3.3 Integral method for solidification problems with convective heat

transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4 Diffusion equation with source terms . . . . . . . . . . . . . . . . . . . . 713.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5.1 Colour photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.5.2 Apollo heat shield . . . . . . . . . . . . . . . . . . . . . . . . . 733.5.3 Decomposing apple . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.4 Wall temperature for type-3 boundary condition . . . . . . . . . . 75

3.6 Answers to exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.6.1 Colour photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.6.2 Apollo heat shield . . . . . . . . . . . . . . . . . . . . . . . . . 773.6.3 Decomposing apple . . . . . . . . . . . . . . . . . . . . . . . . . 783.6.4 Wall temperature for type-3 boundary condition . . . . . . . . . . 80

4 Momentum Transport 834.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Incompressible flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.1 Potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.2 Creeping flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.1 Boundary-layer equations . . . . . . . . . . . . . . . . . . . . . 904.3.2 Solving the boundary-layer equations . . . . . . . . . . . . . . . 93

5 Transport in Flowing Media 975.1 Stationary transport in flows with a uniform velocity profile . . . . . . . . 97

5.1.1 Transport in a plug flow along a flat plate . . . . . . . . . . . . . 985.1.2 Diffusion of heat in a plug flow through a pipe . . . . . . . . . . 100

Contents vii

5.2 Leveque-problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3 Diffusion of heat in a laminar pipe flow . . . . . . . . . . . . . . . . . . 1035.4 Natural convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5.1 Electrochemical detector . . . . . . . . . . . . . . . . . . . . . . 1065.5.2 Infinitely long vertical tank . . . . . . . . . . . . . . . . . . . . . 106

5.6 Answers to exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.6.1 Electrochemical detector . . . . . . . . . . . . . . . . . . . . . . 1085.6.2 Infinitely long vertical tank . . . . . . . . . . . . . . . . . . . . . 112

III Numerical Methods 115

6 Numerical Heat and Fluid Flow 1176.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.2.1 Steady one-dimensional (1D) diffusion . . . . . . . . . . . . . . 1206.2.2 An iterative procedure for nonlinear equations . . . . . . . . . . . 1226.2.3 Boundary conditions for heat conduction . . . . . . . . . . . . . 1226.2.4 Basic principles of the discretisation procedure . . . . . . . . . . 1236.2.5 Unsteady one-dimensional (1D) diffusion . . . . . . . . . . . . . 1246.2.6 Unsteady two-dimensional (2D) diffusion . . . . . . . . . . . . . 1266.2.7 Unsteady three-dimensional (3D) diffusion . . . . . . . . . . . . 1276.2.8 Iterative solving of system of linear discretised equations . . . . . 1286.2.9 Measures for facilitating convergence . . . . . . . . . . . . . . . 129

6.3 Convection and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.3.1 Steady one-dimensional (1D) convection and diffusion . . . . . . 1306.3.2 The central differencing scheme (CDS) . . . . . . . . . . . . . . 1306.3.3 The upwind differencing scheme (UDS) . . . . . . . . . . . . . . 1316.3.4 The hybrid (flux-blending) scheme . . . . . . . . . . . . . . . . . 1326.3.5 The exact solution . . . . . . . . . . . . . . . . . . . . . . . . . 1326.3.6 The exponential scheme . . . . . . . . . . . . . . . . . . . . . . 1336.3.7 The generalised formulation of differencing schemes . . . . . . . 1346.3.8 Numerical (”false”) diffusion . . . . . . . . . . . . . . . . . . . . 1366.3.9 UDS versus CDS for one-dimensional convection-diffusion . . . 1386.3.10 Higher order differencing schemes . . . . . . . . . . . . . . . . 1396.3.11 Total variation diminishing schemes (TVD) . . . . . . . . . . . . 1416.3.12 Unsteady two-dimensional (2D) convection and diffusion . . . . . 1456.3.13 Unsteady three-dimensional (3D) convection and diffusion . . . . 147

6.4 Calculation of the velocity field . . . . . . . . . . . . . . . . . . . . . . . 1486.4.1 The discretised form of the momentum equation . . . . . . . . . 1506.4.2 The pressure-correction equation . . . . . . . . . . . . . . . . . . 1516.4.3 The SIMPLE algorithm . . . . . . . . . . . . . . . . . . . . . . 152

viii Analysis and Modelling of Physical Transport Phenomena

6.4.4 Pressure-velocity coupling for collocated grid arrangement . . . . 1536.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.6 Chart diagram of a numerical code for CFD . . . . . . . . . . . . . . . . 155

IV Turbulence and Transport Phenomena 159

7 Turbulence: Some Features and Rationale for Modelling 1617.1 The phenomenon of turbulence . . . . . . . . . . . . . . . . . . . . . . . 1617.2 Solution to turbulence: needs for modelling and simulations . . . . . . . 1627.3 Statistical description of turbulence: Reynolds decomposition . . . . . . . 166

7.3.1 Reynolds averaging . . . . . . . . . . . . . . . . . . . . . . . . . 1687.3.2 Reynolds-averaged conservation equations . . . . . . . . . . . . 170

7.4 Some features of turbulence . . . . . . . . . . . . . . . . . . . . . . . . 1727.5 Characterisation and scales of turbulence . . . . . . . . . . . . . . . . . . 179

7.5.1 Measure of turbulence: intensity . . . . . . . . . . . . . . . . . . 1797.5.2 Turbulence scales . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.5.3 Turbulence Reynolds numbers . . . . . . . . . . . . . . . . . . . 1817.5.4 Another look at the prospects of Direct Simulation . . . . . . . . 1837.5.5 Two-point correlations . . . . . . . . . . . . . . . . . . . . . . . 1847.5.6 Turbulence Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . 186

8 Generic Flows and Similarity Analysis 1918.1 Generic turbulent flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.1.1 Homogeneous turbulent flows . . . . . . . . . . . . . . . . . . . 1938.1.2 Thin shear flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1948.1.3 More complex generic flows . . . . . . . . . . . . . . . . . . . . 195

8.2 Turbulent wall boundary layer . . . . . . . . . . . . . . . . . . . . . . . 1978.2.1 Similarity analysis and velocity distribution . . . . . . . . . . . . 1978.2.2 Temperature distribution . . . . . . . . . . . . . . . . . . . . . . 201

9 Turbulence Models for RANS 2059.1 The closure problem and modelling principles . . . . . . . . . . . . . . . 205

9.1.1 Scope and limitations of RANS (SPCM) . . . . . . . . . . . . . 2069.1.2 Desirable features of Turbulence Models . . . . . . . . . . . . . 2079.1.3 Some modelling principles and rules . . . . . . . . . . . . . . . . 207

9.2 Eddy-viscosity/diffusivity models (EVM, EDM) . . . . . . . . . . . . . . 2089.2.1 Turbulent viscosity/diffusivity; classification of models . . . . . . 210

9.3 Algebraic models, mixing length . . . . . . . . . . . . . . . . . . . . . . 2119.4 Differential eddy viscosity/diffusivity models . . . . . . . . . . . . . . . 212

9.4.1 One- and two-equation differential EVM/EDM . . . . . . . . . . 2139.4.2 The k-equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2149.4.3 The ε-equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

9.5 The k� ε model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Contents ix

9.5.1 Modelling the k-equation . . . . . . . . . . . . . . . . . . . . . . 2209.5.2 The modelled ε equation for high-Re-number flows . . . . . . . . 2229.5.3 Determining the coefficients . . . . . . . . . . . . . . . . . . . . 2239.5.4 Summary of the high-Re-number k� ε model in integral form . . 2259.5.5 Boundary conditions: Wall Functions . . . . . . . . . . . . . . . 2269.5.6 Low-Re-number (near-wall) k� ε models . . . . . . . . . . . . . 228

9.6 Other two-equation eddy-viscosity models . . . . . . . . . . . . . . . . . 2319.7 Limitations of two-equation linear EVMs . . . . . . . . . . . . . . . . . 2339.8 Non-linear eddy-viscosity models (NLEVM) . . . . . . . . . . . . . . . 2349.9 Second-moment closure models . . . . . . . . . . . . . . . . . . . . . . 236

Literature 239

Index 241

Preface

These lecture notes contain the course material Advanced Physical Transport Phenomena,offered in the Master’s programme in Applied Physics at Delft University of Technology.The notes follow in part the concept and content of the book Fysische Transportverschi-jnselen II (in Dutch) by Hoogendoorn and Van der Meer (Delft University Press, 1991).However, a significant amount of new material on turbulent flows, convective processesand numerical methods has now been included. The course aims at providing graduatestudents with an overview of analytical, numerical and modelling methods for solvingproblems of heat and fluid flow, following a unified and comparative approach.

The course is divided into four parts. The first part gives the conservation laws formass, momentum and energy in general differential forms, accompanied with the relevantconstitutive relations, physical and mathematical classification of equations, and theirboundary conditions. This concise introduction is just a generalisation of the macroscopicconservation laws considered in basic courses on Physical Transport Phenomena at thebachelor’s level.

Part II covers a number of classical analytical methods for solving some generic prob-lems in heat, mass and momentum transfer. In addition to providing insight into thebasic physics of transport phenomena, this part is meant to encourage students to masterthe analytical tools and to use analytical approaches for gaining a physical intuition bysolving elementary problems in idealized situations. It also illustrates the limitations andconstraints of analytical methods in solving complex problems in transport phenomena.

Part III introduces numerical methods for computer-aided solutions of complex prob-lems that are not tractable by analytical approaches. It is, in fact, an introduction tocomputational fluid dynamics (CFD) and computational heat and mass transfer (CHMT),which have recently emerged as major tools for solving heat and fluid flow problems inengineering and environmental applications. With that in mind, the focus is on the finite-volume discretization of the conservation laws, which is the main approach in industrialCFD and CHMT. In addition to introducing basic concepts of equation discretization andtheir numerical solution, this chapter aims at illustrating the potential but also the limita-tions and inherent snares of computational methods. Rather than providing a full coverageof various schemes and solution methods, the chapter aims at developing a critical attitudeamong students and an ability to recognize potential errors and numerical contaminations.

Part IV deals with turbulent convection, considered to be the most widespread modeof transport in real life problems, but also the most challenging both for analytical and

xi

xii Analysis and Modelling of Physical Transport Phenomena

computational treatment. Basic notions on turbulence relevant to its modelling are firstintroduced, followed by statistical averaging of the conservation equations and their in-terpretation. Major features of turbulence are briefly outlined, followed by definition ofthe characteristic scales. A series of generic turbulent flows is then introduced, with fo-cus on flows partially or fully bounded by solid walls. This should provide students withbasic physical insight based solely on similarity and scaling arguments. Limitations ofnumerical simulation are also discussed, together with the need for mathematical mod-elling of turbulence and the associated turbulent transport of momentum, heat and mass.The last section in this part covers the basics of turbulence modelling, its scope and limi-tations. The practice and the rationale of turbulence modelling are illustrated by detailedderivation of the k� ε model and its closure. This is accompanied by physical inter-pretations, which provide an insight into modelling arguments, levels of approximationand model limitations. The section is closed with a brief overview of other two-equationeddy-viscosity/diffusivity models and a brief introduction into non-linear eddy-viscositymodels and second-moment closures.The lecture notes contain a number of worked-out examples, especially in Part II (analyt-ical methods).

Authors, December 2008, Delft, The Netherlands

Part I

Fundamental Equations

Chapter 1

Fundamental Equations of TransportPhenomena - Field description

1.1 Introduction

Physical Transport Phenomena is the common name for processes involving transfer ofmass, heat and momentum. While each of these phenomena has evolved into a separatescientific (and engineering) discipline on its own, taught as separate courses and coveredin numerous textbooks and monographs, they also have much in common. The funda-mental principles and physical laws governing these phenomena and their mathematicaldescription can often be treated in a unified and comparative manner. Such an approachhas been adopted in this course, and is especially suited for students in general appliedsciences.

The principal aim of a course in Transport Phenomena is to understand the underlyingphysics and to master methods that can be used to predict the effects of heat, mass andmomentum transport in various situations. The prediction tasks essentially focus on eval-uating heat, mass and momentum fluxes and their integrals - total heat and mass transferand forces on target surfaces, and are usually related to solid walls bounding the systemunder consideration. But in order to solve the problems, we need to consider field vari-ables such as temperature, species concentration, fluid velocity. These are governed by theconservation laws and complementary constitutive relations, from which the fundamentalequations are derived. The basic conservation laws in integral form (for macroscopic sys-tems) are considered in the undergraduate courses in Thermodynamics, Fluid Mechanicsand Transport Phenomena and here we give a brief overview of the general formulationand then move on to differential forms that describe the motion of a continuum matter andassociated transport of heat and mass.

1.2 Conservation laws for a control volume in differential form

We first introduce the notion of a conserved (or conservable) variable, as a quantity whoseidentity in the original or transformed form can be followed and described by the basicphysical conservation laws:

� conservation of mass

� conservation of momentum (linear, angular) which is the main focus of Solid andFluid Mechanics;

� conservation of energy - mechanical (kinetic, potential), thermal or total, which isin essence the First law of Thermodynamics

3

4 Analysis and Modelling of Physical Transport Phenomena

Other conservation laws can also be formulated, e.g. conservation of entropy (SecondLaw of Thermodynamics). It is noted here that the notions ’conserved’ or ’conservable’quantity, is, strictly speaking, true only if the source term in the conservation equations iszero. In that case the ’conserved variable’ remains indeed conserved in a closed system,just transported by fluid motion and molecular effects from one place to another withinthe system.

It is recalled that all conservation laws have been postulated for a certain mass (defin-able quantity of matter under study). Such a mass system is often referred to as a closedsystem (control mass, mass system), defined by a finite number of state properties. Con-servation laws for a control mass can be formulated in a general (’generic’) mathematicalform defining a finite change or a time rate of the variable Φs associated with the masssystem:

�ΔΦ�s � ϒs or� in time

�ΔΦΔt

�s�

ϒs

Δt; with lim

Δt�0

�ΔΦΔt

�s�

�dΦdt

�s� ϒs (1.1)

where Φs is a conservable quantity in the system (an extensive property, which depends onthe size of the system, i.e on the amount of matter considered), such as mass m, momen-tum �M, energy E, and ϒ is the source/sink of Φ. A dot over the symbols denotes the timerate. Everything outside and beyond this system represents its environment. The controlmass can be finite, in which case we talk about the integral form of the conservation law,or infinitesimally small (infinitesimal continuum element) when the conservation laws arepostulated in a differential mathematical form.

Note that an extensive property can be expressed as a product of mass and the corre-sponding intensive property, Φ � mφ , or, if φ varies within the system (as it is usually thecase):

Φ ��

sφdm �

�sφρdV (1.2)

where φ is an intensive property of a system (independent of its size or its extent).

Table 1.1 summarises the variables for the three basic conservation laws,

Φ φ ϒ ϒmass m 1 0 (or r) 0 (or r)momentum �M � m�v �v Σ�FΔt Σ�Fenergy E � me e Q�W Q�W

Table 1.1 Basic conservation laws.

where

� r denotes the reaction source or sink (r is the reaction rate in time) for mass m incase we consider mass conservation of a reacting substance (note that r � 0 for aninert substance or for the total mass),

1 Fundamental Equations of Transport Phenomena - Field description 5

� �v denotes the fluid velocity vector and Σ�F is the sum of all forces acting on the masssystem

� Q and W denote the amount of energy exchanged with the environment in form ofheat and work respectively, whereas Q and W denote the time rates of Q and W .

It is also recalled that the above formulation of the conservation laws is applicable only toa closed mechanical or thermodynamical system within whose boundaries, fixed or vari-able in time, there exists always the same mass. Studies of such systems focus essentiallyon the interaction of the mass (and associated momentum, energy, entropy) in the systemwith its environment.

Most problems of interest in transport phenomena are associated with open systems,identifiable with engineering or environmental systems, equipment, device, machine, ora part of them, through which a fluid flows. In such cases it is convenient to apply theconservation laws to a control volume bounded by the control surface that coincides withthe boundary surface of the system (equipment, device, or part of it), which we want tostudy. A control volume can be fixed in the adopted coordinate frame, can move withits own velocity, expand or contract (with a local velocity of the control surface). Bydefinition, an open system should have some parts of the control surface open, throughwhich the matter (fluid) flows in and out of the control volume and caries fluid properties.Just as the control mass, the control volume can be of a finite size (macroscopic or integralsystems) or be infinitesimally small, Figure 1.1. In fact, ”a point” in space at which aproperty is defined (e.g. fluid velocity, temperature, pressure, density) is an infinitesimallysmall control volume. In a continuous matter (continuum) the size of the infinitesimalcontrol volume must be sufficiently small so that it can be regarded as ”a point” both in themathematical sense (associated with the variable in differential conservation equations)and in a practical physical realm (corresponding to the size of an instrument sensor bywhich that property is measured). It must, however, be also sufficiently large to containsufficient number of molecules to allow the unique definitions of fluid properties, e.gdensity, pressure, at a point.

The control mass approach, suited for describing and analysing movement of discretemass systems (e.g. material particles, or bodies) is also called the Lagrangean descrip-tion, whereas the control volume approach, suited for describing flow of a continuum,especially in differential form (for an infinitesimal control volume) is called the Euleriandescription.

The application of the generic conservation laws (1.1) to an open system requires itstransformation to account for the inflow and outflow of the considered quantity Φ intoand out of the control volume by inflowing and outflowing fluid through the open parts ofthe system control surface. Such a transformation, or direct derivation of the conservationlaws for a finite-size (macroscopic) control volume can be found in basic textbooks inThermodynamics, Fluid Mechanics and Transport Phenomena, and here it will suffice tolist the generic form of the conservation law for a control volume with identifiable inletsdenoted by ”i” and exits denoted by ”e” for an extensive thermodynamic and dynamic


Figure 1.1 Above: macroscopic control mass (closed system) and control volume (opensystem); below: differential control volume

property Φ:

dΦCV

dt� ∑

iΦi � ∑

eΦe � ϒ (1.3)

Since we are interested here in the differential form of the conservation laws, we recallfirst that

Φ ��

mφdm �

�V

ρφdV Φ ��

Aφdm �

�A

ρφdV ��

Aρφ ��v �d�A�

ϒ ��

mγdm �

�V

ργdV ϒ ��

mγdm �

�V

ργdV

where dm � ρ ��v �d�A� and dV � ��v �d�A�

are the mass and volume flow rates respectively through the elementary surface d�A.


A control volume can always be identified with a mass system at a particular timeinstant, hence the definition of an extensive property Φ as an integral of its intensiveproperty φ over a mass or volume can be regarded as identical at that time instant, buttheir changes in time, i.e. the time rates are different,�

dΦdt

�mass system

��

dΦdt

�control volume

The extensive properties in each term of the generic formulation (1.3) can now bereplaced by volume/surface integrals in terms of intensive properties. So, the time rate ofchange term becomes

dΦCV

dt�

ddt

�CV

ρφdV ��

CV

∂ �ρφdV �

∂ t(1.4)

Note that the ordinary time derivative d�dt is replaced by the partial derivative ∂�∂ t aftercommutation of the sequence with the integral because it is now applied to field propertiesρ and φ , which depend also on space variables.

The inflow and outflow are expressed as

∑i

Φi ��

Ai

ρφ ��v �d�A� and ∑e

Φe ��

Ae

ρφ ��v �d�A� (1.5)

or,

∑i

Φi � ∑e

Φe � ��

ACV

ρφ ��v �d�A� ��

CV∇ � �ρφ�v�dV (1.6)

where the integral over the control surface was converted into an integral over the controlvolume using the Green-Gauss theorem. Note that the sign of the surface integral isdetermined by the scalar product of the velocity vector and the outward-pointing unitnormal vector of the surface element, yielding ��v �d�A�i � 0 and ��v �d�A�e � 0.

Equation (1.4) now becomes:

�CV

∂ �ρφ dV �

∂ t��

�CV

∇ � �ρφ�v�dV ��

CVρ γ dV (1.7)

For an infinitesimally small control volume (a point in space) dV � dxdydz we can omitthe

�CV sign and divide the equation by dV , yielding the generic differential form of the

conservation laws1:

∂ �ρφ �∂ t

�� ∇ � �ρφ�v� � ρ γ (1.8)

For convenience we will present all equations both in vector and in index notation (thelatter applicable only to Cartesian coordinate systems), i.e. for velocity we shall use

1The generic differential form of the conservation laws can be derived directly from the Reynolds TransportTheorem for conservation of a continuum field variable φ (any continuous summable function in space and time,see advanced textbooks).


�v�vx�vy�vz� and vi�v1�v2�v3�2:

∂ �ρφ �∂ t

�� ∇ � �ρφ�v� � ρ γ

∂ �ρφ �∂ t

�� ∂∂x j

�ρφv j � � ρ γ (1.9)

It is noted that equation (1.9) is given in the so-called strong conservative form of thedifferential conservation law (recognizable by the density being lumped together with thevariable φ under all differentiation operators) implying the conservation of φ per unitvolume. Other forms are also in use, which could be more suitable for solving problemswith constant or mildly varying fluid density. For example, applying equation (1.9) tomass conservation, i.e. φ � Φ�m � 1 and γ � 0, leads to the continuity equation indifferential form:

∂ρ∂ t

�� ∇ � �ρ�v �

∂ρ∂ t

�� ∂∂x j

�ρv j � (1.10)

Using the continuity equation, the differential conservation law can be transformed into asimpler, more convenient form, the so-called weak conservative form

ρDφDt

� ρ∂φ∂ t

� ρ ��v �∇�φ � ρ γ

ρDφDt

� ρ∂φ∂ t

� ρ v j∂φ∂x j

� ρ γ (1.11)

whereDDt

�∂∂ t

� v j∂

∂x jis the material (substantial) derivative , denoting the total rate

of change of the variable φ along a streamline (felt by an observer traveling with the fluidelement by its velocity).

1.2.1 Source terms and constitutive relations

The ”source” γ can be associated with the system mass, the so-called mass source(s) γm

and with the system surface, the so-called surface (diffusion) source(s) γA, so that the totalsource can be written as a sum of the two source types,

ϒ ��

Vγm ρ dV �

�A

γA dA ��

CV�ρ γm � ∇ � γA �dV (1.12)

2Note also other common notations for the velocity vector and its components: �v�u�v�w� or Ui�U�V�W� areused often in the analysis of simpler flows especially when one or two velocity components are zero (two- andone-dimensional flows), and Ui�U1�U2�U3� is more common in the treatise of turbulent flows.


For example:

� γm stands for an internal source per unit mass of heat due to chemical reaction,combustion, electric or magnetic heating in the energy equation, whereas in themomentum equation it denotes the gravitational, Coriolis, centrifugal or electro-magnetic force ;

� γA represents the diffusion flux of heat through the control surface in the energyequation, and a force on control surface due to pressure and viscous stresses in themomentum equation.

The surface source is in essence the molecular flux through the surface and can often beexpressed in terms of the gradient of the property considered , i.e.

γA � �Γ∇φ

where Γ is the molecular transport coefficient. The above equation connects the contin-uum property φ with their molecular (material) constitution (represented by the transportcoefficient Γ) and is called the constitutive relation or equation. The constitutive equa-tions associated with the three conservation laws here considered are:

� Conduction flux (Fourier’s law):

γA � ��q � λ ∇T or� � qi � λ∂T∂xi

(1.13)

where λ [W m�1 K�1] is the thermal conductivity.

� Mass molecular diffusion flux (Fick’s law):

γA � ��m�� ∇c or � m��i � �

∂c∂xi

(1.14)

where �m�� [kg m�2 s�1] is the mass flux of a species in kg (or moles) per unit areaand in unit time, C [kg m�3] (or [mol m�3]) is the species concentration and� [m2

s�1] is the mass diffusivity.

� Momentum flux, i.e. the stress (Newton’s law of viscosity)3:

γA � T � �pI�μ �∇ ��v��∇ ��v�T � 23

∇ ��vI� or�

τi j � �pδi j � μ

�∂vi

∂x j�

∂v j

∂xi� 2

3

∂vk

∂xkδi j

�

� � pδi j �2μ �Si j�13

Skk δi j� (1.15)

3Note that the capital boldface letters denote Cartesian second-order tensors.


where μ [kg m�1 s�1] is the dynamic molecular viscosity, Si j �12

�∂vi

∂x j�

∂v j

∂xi

�is the rate of strain (rate of deformation of a fluid element), and I denotes the unitsecond-order tensor with its Cartesian components δi j (known also as Kroneckerdelta).

1.2.2 Common form of the differential conservation law

Now we can write the common differential form of the conservation law:

ρDφDt

� ρ∂φ∂ t

� ρ ��v �∇�φ � ρ γm � ∇ � γA

ρDφDt

� ρ∂φ∂ t

� ρ v j∂φ∂x j

� ρ γm �∂ γA

∂x j(1.16)

Inserting the appropriate variables and the corresponding source terms from Table 1.1,defining the unit mass sources γm and using Eqns (1.13), (1.14) and (1.15) for the surfacesources γA, yield the common conservation equations:

� Momentum (Navier-Stokes) equations, φ ��v or vi:

ρD�vDt

� ρ∂�v∂ t

� ρ ��v �∇��v � ρ�g � ∇ �T

� ρ�g�∇pI � ∇ ��

μ�∇ ��v��∇ ��v�T �� 2

3∇ ��v I

��

ρDvi

Dt� ρ

∂vi

∂ t� ρ v j

∂vi

∂x j� ρgi �

∂τi j

∂x j

� ρgi�∂ p∂xi

�∂

∂x j

�μ

�∂vi

∂x j�

∂v j

∂xi� 2

3

∂vk

∂xkδi j

��(1.17)

� Energy equation, written in terms of enthalpy, φ � h � cp T :

ρDhDt

� ρ�

∂h∂ t

� ��v �∇�h

�� ρ qg � ∇ � � λ

cp∇h�

ρDhDt

� ρ

�∂h∂ t

� v j∂h∂x j

�� ρ qg �

∂∂x j

�λcp

∂h∂x j

�(1.18)

which for a constant specific heat cp [J kg�1 s�1] can also be written in terms oftemperature φ � T , by dividing Eq. (1.18) by ρcp:

DTDt

�

�∂T∂ t

� ��v �∇�T

��

qg

cp� ∇ � �α∇T �


DTDt

�

�∂T∂ t

� v j∂T∂x j

��

qg

cp�

∂∂x j

�α

∂T∂x j

�(1.19)

where α � λ��ρcp� [m2s�1] is the thermal diffusivity, and qg [W kg�1] is theinternal heat source.

� Concentration equation (for each species in a multi-component system) φ �C:

ρDCDt

� ρ�

∂C∂ t

� ��v �∇�C

�� ρ r � ∇ � ��∇C�

ρDCDt

� ρ

�∂C∂ t

� v j∂C∂x j

�� ρ r �

∂∂x j

��

∂C∂x j

�(1.20)

1.3 Classification of equations

Conservation equations in differential form can often be simplified for specific problemsby omitting various terms that are negligible in that specific situation. In many casesthis makes it possible to solve the equations either by analytical tools or by simpler com-putational methods. Moreover, simplification of equations makes it possible to identifyclearly the major physical phenomenon governing the problem under investigation. Insome cases the simplification leads to well established forms of equations known underseparate names and used in other areas of science. We discuss classification of generalpartial differential equations (PDE) from two points of view, using physical criteria basedon the physical meaning of terms in the equations, and mathematical criteria that primarilydefine the method of solution.

Physical criteria

A precise physical meaning can be assigned to each of the terms in the general differentialconservation equations:

ρDφDt � ��

� ρ∂φ∂ t � ��

� ρ ��v �∇�φ � ��

� ρ γm��

� ∇ � �Γ∇φ � � ��

(1.21)

where:

� = material (substantial) derivative (the total change of φ along the streamline)

� = local time rate of change (felt by the observer at a fixed position in an inertialcoordinate frame)

� = convection (rate of change felt by the observer moving with the fluid particle atlocal velocity�v)

� = diffusion (flux of φ through the surface of the elementary control volume)

� = source of φ


In the absence of various terms, the equation reduces to special forms. Assumingfor clarity that in all cases the material properties remain constant, so that Γ=const, thefollowing well-known equations are obtained:

� For stationary transport (∂�∂ t � 0) without flow (�v � 0, solids, stagnant liquids)and without internal source, � � �� 0� � � 0� and equation (1.21) reducesto the Laplace (potential) equation

∇2φ � 0 (1.22)

� For non-stationary (transient) transport without flow and without source:�� 0� �� 0� we obtain the Diffusion (Conduction) equation

ρ∂φ∂ t

� Γ∇2φ (1.23)

� For stationary transport with flow, but without internal source, � � 0, � � 0,equation (1.21) is known as the Convection-diffusion equation

ρ ��v �∇�φ � Γ∇2φ (1.24)

When the time-rate term � is added, the equation is known as the Unsteady convection-diffusion equation.

Mathematical criteria

In the mathematical sense, second-order partial differential equations can be classifiedinto three types according to the equation discriminant. Consider, as an illustration, ageneral form of such an equation for two-dimensional flows

Auxx �Buxy �Cuyy �Dux �Euy �Fu � 0 (1.25)

where, for brevity, subscripts are used to denote differentiation with respect to particularcoordinates, i.e. ux � ∂u�∂x, uxy � ∂ 2u�∂x∂y

Depending on the system discriminant, equations are classified as:

1. Elliptic, if

B2�4AC � 0

Example: The Laplace equation

uxx �uyy � 0

2. Parabolic, if:

B2�4AC � 0

Example: Diffusion equation (with y � t)

Auxx�uy � 0


3. Hyperbolic, if:

B2�4AC � 0

Example: wave propagation equation (e.g. momentum equation for supersonicflow)

It is noted that this classification has major implications on the selection of the appropriatenumerical or analytical methods for solving the equations. For example, in problems gov-erned by parabolic equations (e.g. fluid flow and scalar convection in wall boundary layersand jets) there is no influence of flow properties upstream from the point of considerationand one can simply march along the flow and compute all variables in one sweep, start-ing from the given inflow conditions and side boundary conditions. In contrast, problemsgoverned by elliptic equations are influenced by conditions on all boundaries surroundingthe solution domain and the numerical solutions are much more demanding, requiringusually an iterative solution process. Hyperbolic problems resemble the parabolic ones,but the propagation of a variable occurs along specific lines called the characteristics.For that reason, different numerical solvers have been developed for different families ofproblems governed by each type of equations, though general solution methods are alsoavailable.

1.4 Boundary and initial conditions

Transport processes are described by partial differential equations of the second order inspace, and first order in time: Hence,

� two boundary conditions are needed for each dependent variable in terms of spacecoordinates, and

� for non-stationary problems we need to know the initial conditions.

Two types of boundary conditions can be used (depending on the problem)

� Dirichlet boundary conditions: the value of φ is defined on the boundary, i.e. φb �

φb�x�y�z� t�

� Neumann boundary conditions: the flux of φ related to its gradient in the directionnormal to the boundary is given, i.e.

γb ��Γ�

∂φ∂n

�b� f �x�y�z� t�φb� (1.26)

For the momentum equation the no-slip boundary condition at a solid wall impliesthat the fluid velocity equals the wall velocity (if any), i.e. �v�w � 0 for a stationary wall,and �v�w � vw for a wall moving with the velocity vw. For the energy equation, the heatflux at the wall can be defined in two ways (depending on whether the heat exchange atthe boundary occurs by convection or by radiation - or both)


� Convective flux

qb ��λ�

∂T∂n

�b� f �x�y�z� t�Tb� � h�x�y�z� t��Tb�T0� (1.27)

� Radiative flux

qb ��λ�

∂T∂n

�b� f �x�y�z� t�Tb� � G�x�y�z� t�σ�T 4

b �T 40 � (1.28)

where subscript b denotes the boundary of the solution domain (a solid wall or a freesurface) and T0 is the reference fluid temperature, h [W m�2 K�1] is the heat transfercoefficient and σ [W m�2 K�4] stands for the Stefan-Boltzmann constant.

1.5 Coordinate transformations

The differential conservation equations for mass (1.10), momentum (1.17), energy (1.18and 1.19), and species concentration (1.20) do not depend on the coordinate system aslong as the vector notation is used. However, the conservation equations with the indexnotation (that are given below the vector notation) are valid only in a Cartesian coordinatesystem. When solving problems involving heat, mass, and/or momentum transfer, onehas to select a reference coordinate system. Tables 1.2 and 1.3 give the various terms inconservation equations for Cartesian, cylindrical and spherical coordinate systems. Theconservation equations on a general curvilinear coordinate system can be derived by usingmethods described by e.g. Bird et al. (2002), Chung (1996).


Car

tesi

anco

ordi

nate

scy

lind

rica

lcoo

rdin

ates

sphe

rica

lcoo

rdin

ates

Te x

T xe y

T ye z

T ze r

T re

1 rT

e zT z

e rT r

e1 r

Te

1rs

inT

vv x x

v y yv z z

1 rrrv

r1 r

vv z z

1 r2rr2

v r1

rsin

vsi

n1

rsin

v

2 T2T x2

2T y2

2T z2

1 rr

rT r

1 r22T 2

2T z2

1 r2r

r2T r

1r2

sin

sin

T1

r2si

n2

2T 2

oror

2T r2

1 rT r

1 r22T 2

2T z2

2 rT r

2T r2

1 r22T 2

cos

r2si

nT

1r2

sin2

2T 2

Tx

T xy

T yz

T z1 r

rr

T r1 r

rT

zT z

1 r2r

r2T r

1rs

insi

n rT

1rs

inrs

inT

vT

v xT x

v yT y

v zT z

v rT r

v rT

v zT z

v rT r

v rT

vrs

inT

D Dt

tv x

xv y

yv z

zt

v rr

v rv z

zt

v rr

v rv

rsin

Tabl

e1.

2C

oord

inat

etr

ansf

orm

atio

ns.

X

Z

YP

X

Z

Y

Pr

X

Z

Yy

Pz

xr


Car

tesi

anco

ordi

nate

s

x-co

mp.

ρ

� ∂vx

∂t

�

v x∂v

x∂x

�

v y∂v

x∂y

�

v z∂v

x∂z

� ��

∂p ∂x

�

μ

� ∂2v x

∂x2

�

∂2v x

∂y2

�

∂2v x

∂z2

� �

ρgx

y-co

mp.

ρ

� ∂vy

∂t

�

v x∂v

y∂x

�

v y∂v

y∂y

�

v z∂v

y∂z

� ��

∂p ∂y

�

μ

� ∂2v y

∂x2

�

∂2v y

∂y2

�

∂2v y

∂z2

� �

ρgy

z-co

mp.

ρ

� ∂vz

∂t

�

v x∂v

z∂x

�

v y∂v

z∂y

�

v z∂v

z∂z

� ��

∂p ∂z

�

μ

� ∂2v z

∂x2

�

∂2v z

∂y2

�

∂2v z

∂z2

� �

ρgz

Cyl

indr

ical

coor

dina

tes

r-co

mp.

ρ

� ∂vr

∂t

�

v r∂v

r∂r

�

v θ r∂v

r∂θ

�

v2 θ r

�

v z∂v

r∂z

� ��

∂p ∂r

�

μ

� ∂ ∂r

� 1 r∂ ∂r

�rv r

��

1 r2∂2

v r∂θ

2

�

2 r2∂v

θ∂θ

�

∂2v r

∂z2

� �ρg

r

θ-co

mp.

ρ

� ∂vθ

∂t

�

v r∂v

θ∂r

�

v θ r∂v

θ∂θ

�

v rv θr

�

v z∂v

θ∂z

� ��

1 r∂

p∂θ

�

μ

� ∂ ∂r

� 1 r∂ ∂r

�rv θ

��

1 r2∂2

v θ∂θ

2

�

2 r2∂v

r∂θ

�∂2

v θ∂z

2

� �

ρgθ

z-co

mp.

ρ

� ∂vz

∂t

�

v r∂v

z∂r

�

v θ r∂v

z∂θ

�

v z∂v

z∂z

� ��

∂p ∂z

�

μ

� 1 r∂ ∂r

� r∂vz

∂r

� �

1 r2∂2

v z∂θ

2

�

∂2v z

∂z2

� �

ρgz

Sphe

rica

lcoo

rdin

ates

r-co

mp.

ρ

� ∂vr

∂t

�

v r∂v

r∂r

�

v θ r∂v

r∂θ

�

v ϕrs

inθ

∂vr

∂ϕ

�

v2 θ

�

v2 ϕr

� ��

∂p ∂r

�

μ

� ∇2v r

�

2 r2v r

�2 r2

∂vθ

∂θ

�

2 r3v θ

cotθ

�

2r2

sin

θ∂ v

ϕ∂ϕ

� �

ρgr

θ-co

mp.

ρ

� ∂vθ

∂t

�

v r∂v

θ∂r

�

v θ r∂v

θ∂θ

�

v ϕrs

inθ

∂vθ

∂ϕ

�

v rv θr

�

v2 ϕco

tθr

� ��

1 r∂

p∂θ

�μ

� ∇2v θ

�

2 r2∂v

r∂θ

�

v θr2

sin2

θ

�

2co

sθr2

sin2

θ∂ v

ϕ ϕ

� �

ρgθ

ϕ-c

omp.

ρ

� ∂vϕ

∂t

�

v r∂v

ϕ∂r

�

v θ r∂v

ϕ∂θ

�

v ϕrs

inθ

∂vϕ

∂ϕ

�

v ϕv r r

�

v θv ϕ r

cotθ

� ��

1rs

inθ

∂p

∂ϕ

�

μ

� ∇2v ϕ

�

v ϕr2

sin2

θ

�

2r2

sin

θ∂ v

r∂ϕ

�

2co

sθr2

sin2

θ∂v

θ∂ϕ

� �

ρgϕ

Tabl

e1.

3C

oord

inat

etr

ansf

orm

atio

nsfo

rN

avie

r-S

toke

seq

uatio

ns.