Prandtl's essentials of fluid mechanics herbert oertel

Applied Mathematical Sciences
Volume 158
Editors
S.S. Antman J.E. Marsden L. Sirovich
Advisors
J.K. Hale P. Holmes J. Keener
J. Keller B.J. Matkowsky A. Mielke
C.S. Peskin K.R.S. Sreenivasan
This page intentionally left blank
Herbert Oertel
Editor
Prandtl’s Essentials
of Fluid Mechanics
Second Edition
With Contributions by M. Bo
¨
hle, D. Etling, U. Mu
¨
ller,
K.R.S. Sreenivasan, U. Riedel, and J. Warnatz
Translated by Katherine Mayes
With 530 Illustrations
Herbert Oertel
Institut fu
¨
r Stro
¨
mungslehre
Universita
¨

t Karlsruhe
Kaiserstr. 12
Karlsruhe D-76131
Germany

Ludwig Prandtl, em. Prof. Dr. Dr Ing. e.h.mult., Universita
¨
tGo
¨
ttingen, Dir. MPI fu
¨
r
Stro
¨
mungsforschung, † 1953.
Editors:
S.S. Antman J.E. Marsden L. Sirovich
Department of Mathematics Control and Dynamical Division of Applied
and Systems, 107-81 Mathematics
Institute for Physical Science California Institute of Brown University
and Technology Technology Providence, RI 02912
University of Maryland Pasadena, CA 91125 USA
College Park, MD 20742-4015 USA
USA

Mathematics Subject Classification (2000): 76A02, 76-99
Library of Congress Cataloging-in-Publication Data
Oertel, Herbert.
Prandtl’s essentials of fluid mechanics / Herbert Oertel
p. cm.

Includes bibliographical references and index.
ISBN 0-387-40437-6 (alk. paper)
1. Fluid mechanics. I. Title.
TA357.O33 2003
620.1′06—dc22 2003059136
ISBN 0-387-40437-6 Printed on acid-free paper.
Origin al ly published in the German language by Vieweg Verlag/GWV Fachv er la ge GmbH, D-651 89 Wies-
baden, Germany , a s “Herbe rt Oertel (Hsrg. ): Fu
¨
hrer durch die Stro
¨
mungsle hre . 10. Auflage (10th Edition)”
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Preface
Ludwig Prandtl, with his fundamental contributions to hydrodynamics, aero-
dynamics, and gas dynamics, greatly influenced the development of fluid me-
chanics as a whole, and it was his pioneering research in the first half of the

last century that founded modern fluid mechanics. His book F¨uhrer durch
die Str¨omungslehre, which appeared in 1942, originated from previous publi-
cations in 1913, Lehre von der Fl¨ussigkeit und Gasbewegung, and 1931, Abriß
der Str¨omungslehre. The title F¨uhrer durch die Str¨omungslehre,orEssentials
of Fluid Mechanics, is an indication of Prandtl’s intentions to guide the reader
on a carefully thought-out path through the different areas of fluid mechan-
ics. On his way, the author advances intuitively to the core of the physical
problem, without extensive mathematical derivations. The description of the
fundamental physical phenomena and concepts of fluid mechanics that are
needed to derive the simplified models has priority over a formal treatment
of the methods. This is in keeping with the spirit of Prandtl’s research work.
The first edition of Prandtl’s F¨uhrer durch die Str¨omungslehre was the
only book on fluid mechanics of its time and, even today, counts as one of
the most important books in this area. After Prandtl’s death, his students
Klaus Oswatitsch and Karl Wieghardt undertook to continue his work, and to
add new findings in fluid mechanics in the same clear manner of presentation.
When the ninth edition went out of print and a new edition was desired
by the publishers, we were glad to take on the task. The first four chapters of
this book keep to the path marked out by Prandtl in the first edition, in 1942.
The original historical text has been linguistically revised, and leads, after the
Introduction, to chapters on Properties of Liquids and Gases, Kinematics of
Flow, and Dynamics of Fluid Flow. These chapters are taught to science and
engineering students in introductory courses on fluid mechanics even today.
We have retained much of Prandtl’s original material in these chapters, but
added a section on the Topology of a Flow in Chapter 3 and on Flows of Non-
Newtonian Media in Chapter 4. Chapters 5 and 6, on Fundamental Equations
of Fluid Mechanics and Aerodynamics, enlarges the material in the original,
and forms the basis for the treatment of different branches of fluid mechanics
that appear in subsequent chapters.
The major difference from previous editions lies in the treatment of addi-

tional topics of fluid mechanics. The field of fluid mechanics is continuously
VI Preface
growing, and has now become so extensive that a selection had to be made.
I am greatly indebted to my colleagues K.R. Sreenivasan, U. M¨uller, J. War-
natz, U. Riedel, D. Etling, and M. B¨ohle, who revised individual chapters in
their own research areas, keeping Prandtl’s purpose in mind and presenting
the latest developments of the last sixty years in Chapters 7 to 14. Some of
these chapters can be found in some form in Prandtl’s book, but have un-
dergone substantial revisions; others are entirely new. The original chapters
on Wing Aerodynamics, Heat Transfer, Stratified Flows, Turbulent Flows,
Multiphase Flows, Flows in the Atmosphere and the Ocean, and Thermal
Turbomachinery have been revised, while the chapters on Fluid Mechanical
Instabilities, Flows with Chemical Reactions, and Biofluid Mechanics of Blood
Circulation are new. References to the literature in the individual chapters
have intentionally been kept to those few necessary for comprehension and
completion. The extensive historical citations may be found by referring to
previous editions.
Essentials of Fluid Mechanics is targeted to science and engineering stu-
dents who, having had some basic exposure to fluid mechanics, wish to attain
an overview of the different branches of fluid mechanics. The presentation
postpones the use of vectors and eschews the use integral theorems in order
to preserve the accessibility to this audience. For more general and compact
mathematical derivations we refer to the references. In order to give students
the possibility of checking their learning of the subject matter, Chapters 2
to 6 are supplemented with problems. The book will also give the expert in
research or industry valuable stimulation in the treatment and solution of
fluid-mechanical problems.
We hope that we have been able, with the treatment of the different
branches of fluid mechanics, to carry on the work of Ludwig Prandtl as he
would have wished. Chapters 1–6, 8, 9, and 13 were written by H. Oertel

Jr., Chapter 7 by K.R. Sreenivasan, Chapter 10 by U. M¨uller, Chapter 11 by
J. Warnatz and U. Riedel, Chapter 12 by D. Etling, and Chapter 14 by M.
B¨ohle. Thanks are due to those colleagues whose numerous suggestions have
been included in the text.
I thank Katherine Mayes for the translation and typesetting of the En-
glish manuscript and U. Dohrmann for the completion of the text files. The
extremely fruitful collaboration with Springer-Verlag also merits particular
praise.
Karlsruhe, June 2003 Herbert Oertel
Contents
Preface V
1. Introduction 1
2. Properties of Liquids and Gases 17
2.1 Properties of Liquids 17
2.2 State of Stress 18
2.3 Liquid Pressure 21
2.4 Properties of Gases 26
2.5 Gas Pressure 29
2.6 Interaction Between Gas Pressure and Liquid Pressure 32
2.7 Equilibrium in Other Force Fields 35
2.8 Surface Stress (Capillarity) 39
2.9 Problems 42
3. Kinematics of Fluid Flow 47
3.1 Methods of Representation 47
3.2 Acceleration of a Flow 51
3.3 Topology of a Flow 52
3.4 Problems 59
4. Dynamics of Fluid Flow 63
4.1 Dynamics of Inviscid Liquids 63
4.1.1 Continuity and the Bernoulli Equation 63

4.1.2 Consequences of the Bernoulli Equation 67
4.1.3 Pressure Measurement 75
4.1.4 Interfaces and Formation of Vortices 77
4.1.5 Potential Flow 80
4.1.6 Wing Lift and the Magnus Effect 93
4.1.7 Balance of Momentum for Steady Flows 95
4.1.8 Waves on a Free Liquid Surface 103
4.1.9 Problems 113
4.2 Dynamics of Viscous Liquids 118
4.2.1 Viscosity (Inner Friction), the Navier–Stokes Equation 118
VIII Contents
4.2.2 Mechanical Similarity, Reynolds Number 122
4.2.3 Laminar Boundary Layers 123
4.2.4 Onset of Turbulence 126
4.2.5 Fully Developed Turbulence 136
4.2.6 Flow Separation and Vortex Formation 144
4.2.7 Secondary Flows 151
4.2.8 Flows with Prevailing Viscosity 153
4.2.9 Flows Through Pipes and Channels 160
4.2.10 Drag of Bodies in Liquids 165
4.2.11 Flows in Non-Newtonian Media 175
4.2.12 Problems 180
4.3 Dynamics of Gases 186
4.3.1 Pressure Propagation, Velocity of Sound 186
4.3.2 Steady Compressible Flows 190
4.3.3 Conservation of Energy 195
4.3.4 Theory of Normal Shock Waves 196
4.3.5 Flows past Corners, Free Jets 200
4.3.6 Flows with Small Perturbations 203
4.3.7 Flows past Airfoils 207

4.3.8 Problems 213
5. Fundamental Equations of Fluid Mechanics 217
5.1 Continuity Equation 217
5.2 Navier–Stokes Equations 218
5.2.1 Laminar Flows 218
5.2.2 Reynolds Equations for Turbulent Flows 225
5.3 Energy Equation 230
5.3.1 Laminar Flows 230
5.3.2 Turbulent Flows 234
5.4 Fundamental Equations as Conservation Laws 236
5.4.1 Hierarchy of Fundamental Equations 236
5.4.2 Navier–Stokes Equations 237
5.4.3 Derived Model Equations 240
5.4.4 Reynolds Equations for Turbulent Flows 247
5.4.5 Multiphase Flows 248
5.4.6 Reactive Flows 251
5.5 Differential Equations of Perturbations 253
5.6 Problems 258
6. Aerodynamics 265
6.1 Fundamentals of Aerodynamics 265
6.1.1 Bird Flight and Technical Imitations 266
6.1.2 Airfoils and Wings 268
6.1.3 Airfoil and Wing Theory 276
6.1.4 Aerodynamic Facilities 290
Contents IX
6.2 Transonic Aerodynamics 292
6.2.1 Swept Wings 294
6.2.2 Shock–Boundary-Layer Interaction 297
6.2.3 Flow Separation 304
6.3 Supersonic Aerodynamics 306

6.3.1 Delta Wings 307
6.4 Problems 314
7. Turbulent Flows 319
7.1 Fundamentals of Turbulent Flows 319
7.2 Onset of Turbulence 320
7.2.1 Linear Stability 321
7.2.2 Nonlinear Stability 323
7.2.3 Nonnormal Stability 324
7.3 Developed Turbulence 326
7.3.1 The Notion of a Mixing Length 326
7.3.2 Turbulent Mixing 328
7.3.3 Energy Relations in Turbulent Flows 329
7.4 Classes of Turbulent Flows 331
7.4.1 Free Turbulence 331
7.4.2 Flow Along a Boundary 334
7.4.3 Rotating and Stratified Flows, Flows with Curvature
Effects 337
7.4.4 Turbulence in Tunnels 340
7.4.5 Two-Dimensional Turbulence 344
7.5 New Developments in Turbulence 348
7.5.1 Lagrangian Investigations of Turbulence 353
7.5.2 Field-Theoretic Methods 354
7.5.3 Outlook 354
8. Fluid-Mechanical Instabilities 357
8.1 Fundamentals of Fluid-Mechanical Instabilities 357
8.1.1 Examples of Fluid-Mechanical Instabilities 357
8.1.2 Definition of Stability 363
8.1.3 Local Perturbations 366
8.2 Stratification Instabilities 367
8.2.1 Rayleigh–B´enard Convection 367

8.2.2 Marangoni Convection 379
8.2.3 Diffusion Convection 382
8.3 Hydrodynamic Instabilities 388
8.3.1 Taylor Instability 388
8.3.2 G¨ortler Instability 393
8.4 Shear-Flow Instabilities 395
8.4.1 Boundary-Layer Flows 396
8.4.2 Tollmien–Schlichting and Cross-Flow Instabilities 403
X Contents
8.4.3 Kelvin–Helmholtz Instability 419
8.4.4 Wake Flows 422
9. Convective Heat and Mass Transfer 427
9.1 Fundamentals of Heat and Mass Transfer 427
9.1.1 Free and Forced Convection 427
9.1.2 Heat Conduction and Convection 429
9.1.3 Diffusion and Convection 430
9.2 Free Convection 431
9.2.1 Convection at a Vertical Plate 431
9.2.2 Convection at a Horizontal Cylinder 437
9.3 Forced Convection 438
9.3.1 Pipe Flows 438
9.3.2 Boundary-Layer Flows 442
9.3.3 Bodies in Flows 449
9.4 Heat and Mass Exchange 449
9.4.1 Mass Exchange at the Flat Plate 450
10. Multiphase Flows 453
10.1 Fundamentals of Multiphase Flows 453
10.1.1 Definitions 454
10.1.2 Flow Patterns 457
10.1.3 Flow Pattern Maps 457

10.2 Flow Models 460
10.2.1 The One-Dimensional Two-Fluid Model 461
10.2.2 Mixing Models 464
10.2.3 The Drift–Flow Model 466
10.2.4 Bubbles and Drops 468
10.2.5 Spray Flows 471
10.3 Pressure Loss and Volume Fraction in Hydraulic Components 474
10.3.1 Friction Loss in Horizontal Straight Pipes 475
10.3.2 Acceleration Losses 479
10.4 Propagation Velocity of Density Waves and Critical Mass
Fluxes 483
10.4.1 Density Waves 483
10.4.2 Critical Mass Fluxes 486
10.4.3 Cavitation 493
10.5 Instabilities in Two-Phase Flows 497
11. Reactive Flows 503
11.1 Fundamentals of Reactive Flows 503
11.1.1 Rate Laws and Reaction Orders 503
11.1.2 Relation Between Forward and Reverse Reactions 504
11.1.3 Elementary Reactions and Reaction Molecularity 505
11.1.4 Temperature Dependence of Rate Coefficients 508
Contents XI
11.1.5 Pressure Dependence of Rate Coefficients 510
11.1.6 Characteristics of Reaction Mechanisms 512
11.2 Laminar Reactive Flows 517
11.2.1 Structure of Premixed Flames 517
11.2.2 Flame Velocity of Premixed Flames 520
11.2.3 Sensitivity Analysis 521
11.2.4 Nonpremixed Counterflow Flames 523
11.2.5 Nonpremixed Jet Flames 525

11.2.6 Nonpremixed Flames with Fast Chemistry 526
11.2.7 Exhaust Gas Cleaning with Plasma Sources 527
11.2.8 Flows in Etching Reactors 530
11.2.9 Heterogeneous Catalysis 531
11.3 Turbulent Reactive Flows 532
11.3.1 Overview and Concepts 532
11.3.2 Direct Numerical Simulation 533
11.3.3 Turbulence Models 535
11.3.4 Mean Reaction Rates 536
11.3.5 Eddy-Break-Up Models 542
11.3.6 Large-Eddy Simulation (LES) 542
11.3.7 Turbulent Nonpremixed Flames 543
11.3.8 Turbulent Premixed Flames 554
11.4 Hypersonic Flows 560
11.4.1 Physical-Chemical Phenomena in Reentry Flight 560
11.4.2 Chemical Nonequilibrium 561
11.4.3 Thermal Nonequilibrium 564
11.4.4 Surface Reactions on Reentry Vehicles 567
12. Flows in the Atmosphere and in the Ocean 571
12.1 Fundamentals of Flows in the Atmosphere and in the Ocean . 571
12.1.1 Introduction 571
12.1.2 Fundamental Equations in Rotating Systems 571
12.1.3 Geostrophic Flow 574
12.1.4 Vorticity 576
12.1.5 Ekman Layer 579
12.1.6 Prandtl Layer 582
12.2 Flows in the Atmosphere 584
12.2.1 Thermal Wind Systems 584
12.2.2 Thermal Convection 588
12.2.3 Gravity Waves 590

12.2.4 Vortices 592
12.2.5 Global Atmospheric Circulation 598
12.3 Flows in the Ocean 600
12.3.1 Wind-Driven Flows 601
12.3.2 Water Waves 603
12.4 Application to Atmospheric and Oceanic Flows 606
XII Contents
12.4.1 Weather Forecast 606
12.4.2 Greenhouse Effect and Climate Prediction 608
12.4.3 Ozone Hole 612
13. Biofluid Mechanics of Blood Circulation 615
13.1 Fundamentals of Biofluid Mechanics 615
13.1.1 Respiratory System 618
13.1.2 Blood Circulation 620
13.1.3 Rheology of the Blood 625
13.2 Flow in the Heart 626
13.2.1 Physiology and Anatomy of the Heart 627
13.2.2 Structure of the Heart 630
13.2.3 Excitation Physiology of the Heart 634
13.2.4 Flow in the Heart 637
13.2.5 Cardiac Valves 642
13.3 Flow in Blood Vessels 645
13.3.1 Unsteady Pipe Flow 649
13.3.2 Unsteady Arterial Flow 650
13.3.3 Arterial Branches 653
14. Thermal Turbomachinery 655
14.1 Fundamentals of Thermal Turbomachinery 655
14.2 Axial Compressor 659
14.2.1 Flow Coefficient, Pressure Coefficient, and Degree of
Reaction 659

14.2.2 Method of Design 663
14.2.3 Subsonic Compressor 666
14.2.4 Transonic Compressor 668
14.3 Centrifugal Compressor 672
14.3.1 Flow Physics of the Centrifugal Compressor 672
14.3.2 Flow Coefficient, Pressure Coefficient, and Efficiency . . 676
14.3.3 Slip Factor 678
14.4 Combustion Chamber 679
14.4.1 Flow with Heat Transfer 679
14.4.2 Geometry of the Combustion Chamber 681
14.5 Turbine 682
14.5.1 Basics 682
14.5.2 Efficiency, Flow Coefficient, Work Coefficient, and
Degree of Reaction 683
14.5.3 Impulse and Reaction Stage 684
Selected Bibliography 687
Index 715
1. Introduction
The development of modern fluid mechanics is closely connected to the name
of its founder, Ludwig Prandtl. In 1904 it was his famous article on fluid
motion with very small friction that introduced boundary-layer theory. His
article on airfoil theory, published the following decade, formed the basis
for the calculation of friction drag, heat transfer, and flow separation. He
introduced fundamental ideas on the modeling of turbulent flows with the
Prandtl mixing length for turbulent momentum exchange. His work on gas
dynamics, such as the Prandtl–Glauert correction for compressible flows, the
theory of shock waves and expansion waves, as well as the first photographs
of supersonic flows in nozzles, reshaped this research area. He applied the
methods of fluid mechanics to meteorology, and was also pioneering in his
contributions to problems of elasticity, plasticity, and rheology.

Prandtl was particularly successful in bringing together theory and exper-
iment, with the experiments serving to verify his theoretical ideas. It was this
that gave Prandtl’s experiments their importance and precision. His famous
experiment with the tripwire, through which he discovered the turbulent
boundary layer and the effect of turbulence on flow separation, is one ex-
ample. The tripwire was not merely inspiration, but rather was the result of
consideration of discrepancies in Eiffel’s drag measurements on spheres. Two
experiments with different tripwire positions were enough to establish the
generation of turbulence and its effect on the flow separation. For his experi-
ments Prandtl developed wind tunnels and measuring apparatus, such as the
G¨ottingen wind tunnel and the Prandtl stagnation tube. His scientific results
often seem to be intuitive, with the mathematical derivation present only to
provide service to the physical understanding, although it then does indeed
deliver the decisive result and the simplified physical model. According to a
comment by Werner Heisenberg, Prandtl was able to “see” the solutions of
differential equations without calculating them.
Selected individual examples aim to introduce the reader to the path to
understanding of fluid mechanics prepared by Prandtl and to the contents and
modeling in each chapter. As an example of the dynamics of flows, the differ-
ent regimes in the flow past a vehicle, an incompressible flow (hydrodynamics,
Chapter 4), and in the flow past a wing, a compressible flow (aerodynamics,
Chapter 6) are described.
2 1. Introduction
In the flow past a vehicle, we differentiate between the free flow past the
surface and the flow between the vehicle moving with velocity u
∞
and the
street at rest. At the stagnation point, where the pressure is at its maximum,
the flow divides, and is accelerated along the hood and past the spoiler along
the base of the vehicle. This leads to a pressure drop and to a negative

downward pressure to the street, as shown in Figure 1.1. The flow again
slows down at the windshield, and is decelerated downstream along the roof
and the trunk. This leads to a pressure increase with a positive lift, while the
negative downward pressure on the street along the lower side of the vehicle
remains.
The viscous flow (Section 4.2) on the upper and lower sides of the vehicle
is restricted to the boundary-layer flow, which passes over to the viscous wake
at the back edge of the vehicle. The flow in the wind tunnel experiment is
made visible with smoke, and this shows that downstream from the back of
the automobile, a backflow region forms. This is seen in the figure as the
black region. Outside the boundary layer and the wake, the flow is essentially
inviscid (Section 4.1).
In order to be able to understand the different flow regimes, and therefore
to establish a basis for the aerodynamic design of a motor vehicle, Prandtl
worked out the carefully prepared path (Chapters 2 to 4) from the properties
of liquids and gases, to kinematics, and to the dynamics of inviscid and viscous
flows. By following this path, too, the reader will successively gain physical
understanding of this first flow example.
The second flow example considers the compressible flow past a wing with
a shock wave (Sections 4.3 and 6.2). The free flow toward the wing has the
velocity of a civil aircraft u
∞
, a large subsonic velocity. Figure 1.2 shows
-
+
-
-
-
-
+

+
Grenzschicht
reibungsfreie Umströmung
Nachlauf
u
inviscid flow
8
boundary layer
wake
u
wake flow visualization
Fig. 1.1. Flow past a vehicle
1. Introduction 3
the flow regimes on a cross-section of the wing and the negative pressure
distribution, with the flow again made visible with small particles. From the
stagnation point, the stagnation line bifurcates to follow the suction side (up-
per side) and the pressure side (lower side) of the wing. On the upper side,
the flow is accelerated up to supersonic velocities, an effect that is connected
with a large pressure drop. Further downstream, the flow is again decelerated
to the subsonic regime via a compression shock wave. This shock wave inter-
acts with the boundary layer and causes it to thicken, leading to increased
drag.
On the lower side the flow is also accelerated from the stagnation point.
However, the acceleration in the nose region is not as great as that on the
suction side, and so no supersonic velocities occur along the pressure side.
From about the middle of the wing onwards, the flow is again decelerated.
The pressures above and below then approach one another, leading to the
wake region downstream of the trailing edge.
A thin boundary layer is formed on the suction and pressure sides of the
wing. The suction and pressure side boundary layers meet at the trailing edge

and form the wake flow downstream. As in the example of the flow past a
motor vehicle, both the flow in the boundary layers and the flow in the wake
are viscous. Outside these regions the flow is essentially inviscid.
The pressure distribution in Figure 1.2 results in a lift, which, for the
wing of the civil aircraft, has to be adapted to the number of passengers to
be transported. In designing the wing, the design engineer has to keep the
drag of the wing as small as possible to save fuel. This is done by shaping
the wing appropriately.
Nachlauf
u
Grenzschicht
reibungsfreie Umströmung
Stoss
p
-c
-1
x
-
-
+
+
x
8
inviscid flow
boundary layer
wake
p
u
− c
− 1

flow visualization
shock
Fig. 1.2. Flow past a wing
4 1. Introduction
Different equations for computing each flow result from the different prop-
erties of each flow regime. To good approximation, the boundary-layer equa-
tions hold in the boundary-layer regime. In contrast, computing the wake
flow and the flow close to the trailing edge is more difficult. In these regimes,
the Navier–Stokes equations have to be solved. The inviscid flow in the re-
gion in front of the shock can be treated using the potential equation, a
comparatively simple task. The inviscid flow behind the shock outside the
boundary layer has to be computed with the Euler equations, since the flow
there is rotational. In the shock-boundary-layer interaction region, again the
Navier–Stokes equations have to be solved.
In contrast to Prandtl’s day, numerical software is now available for solv-
ing the different partial differential equations. Because of this, in Chapter
5 we present the fundamental equations of laminar and turbulent flows as a
basis for the following chapters dealing with the different branches of fluid
mechanics. Following the same procedure as Prandtl, the mathematical so-
lution algorithms and methods are to be found by referral to the texts and
literature cited.
As will be shown in Chapters 6 to 14, in spite of numerically computed
flow fields, it is necessary to consider the physical modeling in the different
regimes. There are still no closed theories of turbulent flows, of multiphase
flows, or of the coupling of flows with chemical reactions out of thermal
or chemical equilibrium. For this reason, Prandtl’s method of intuitive con-
nection of theory and experiment to physical modeling is still very much
up-to-date.
The fascinating complexity of turbulence has attracted the attention of
scientists for centuries (Chapter 7). For example, the swirling motion of fluids

that occurs irregularly in space and time is called turbulence. However, this
randomness, apparent from a casual observation, is not without some order.
Turbulent flows are a paradigm for spatially extended nonlinear dissipative
systems in which many length scales are excited simultaneously and coupled
strongly. The phenomenon has been studied extensively in engineering and
in diverse fields such as astrophysics, oceanography, and meteorology.
Figure 1.3 shows a turbulent jet of water emerging from a circular orifice
into a tank of still water. The fluid from the orifice is made visible by mixing
small amounts of a fluorescing dye and illuminating it with a thin light sheet.
The picture illustrates swirling structures of various sizes amidst an avalanche
of complexity. The boundary between the turbulent flow and the ambient is
usually rather sharp and convoluted on many scales. The object of study is
often an ensemble average of many such realizations. Such averages obliterate
most of the interesting aspects seen here, and produce a smooth object that
grows linearly with distance downstream. Even in such smooth objects, the
averages vary along the length and width of the flow, these variations being
a measure of the spatial inhomogeneity of turbulence. The inhomogeneity is
typically stronger along the smaller dimension of the flow. The fluid velocity
1. Introduction 5
measured at any point in the flow is an irregular function of time. The degree
of order is not as apparent in time traces as in spatial cuts, and a range of
intermediate scales behaves like fractional Brownian motion.
In contrast, Figure 1.4 shows homogeneous and isotropic turbulence pro-
duced by sweeping a grid of bars at a uniform speed through a tank of still
water. Unlike the jet turbulence of Figure 1.3, turbulence here does not have
a preferred direction or orientation. On average, it does not possess signifi-
cant spatial inhomogeneities or anisotropies. The strength of the structures,
such as they are, is weak in comparison with such structures in Figure 1.3.
Homogeneous and isotropic turbulence offers considerable theoretical simpli-
fications, and is the object of many studies.

In many fluid-mechanical problems, the onset of turbulent flows is due to
instabilities (Chapter 8). An example of this is thermal cellular convection in
a horizontal fluid layer heated from below and under the effect of gravity. The
base below the fluid has a higher temperature than the free surface. Above
a critical temperature difference between the free surface and the base, the
fluid is suddenly set into motion and, as in Figure 1.5, forms hexagonal cell
structures in the center of which fluid rises and on whose edges the fluid
sinks. The phenomenon is known as thermal cellular convection. If the fluid
is covered by a plate, periodically spaced rolling structures are formed without
surface tension instead of hexagonal cells. The reason for the instabilities is
Fig. 1.3. Turbulent jet of water Fig. 1.4. Homogeneous and isotropic
turbulent flow
6 1. Introduction
the same in both cases. Cold, denser fluid is layered above warmer fluid,
and this tends to flow toward lower layers. The smallest perturbation to this
layering leads to the onset of the equalizing motion, as long as a critical
temperature difference is exceeded.
The transition to turbulent convection flow takes place with increasing
temperature difference via several time-dependent intermediate states. The
size of the hexagonal structures or the long convection rolls changes, but the
original cellular structure of the instability can still be seen in the turbulent
convection flow.
Convection flows with heat and mass transport are treated in Chapter 9.
These occur frequently in nature and technology, and it is in this manner
that heat exchange in the atmosphere determines the weather. The example
of a tropical cyclone is shown in Figure 1.10. The extensive heat adjustment
between the equator and the North Pole leads to convection flows in the
oceans, such as the Gulf Stream (Figure 1.11). Convection flows in the cen-
ter of the Earth are also the cause of continental drift and are responsible
for the Earth’s magnetic field. Flows in energy technology and environmen-

tal technology are connected with heat and mass transport, and with phase
transitions, as in steam generators and condensers. Convection flows are used
in cooling towers to transport the waste heat to power stations. Other ex-
amples of convection flows are the propagation of waste air and gas in the
atmosphere and of cooling and waste water in lakes, rivers, and oceans, heat-
free surfaces
rigid boundaries
hexagons
rolls
Fig. 1.5. Thermal cellular convection
1. Introduction 7
ing engineering and air-conditioning technology in buildings, circulation of
fluids in solar collectors and heat accumulators.
Figure 1.6 shows experimental results on thermal convection flows. In con-
trast to forced convection flows, these are free convection flows, where the flow
is due to only lift forces. These may be due to temperature or concentration
gradients in the gravitational field. A heated horizontal circular cylinder ini-
tially generates a rising laminar convection flow in the surrounding medium,
which is at rest, until the transition to turbulent convection flow is caused by
thermal instabilities. Similar thermal convection flows occur at vertical and
horizontal heated plates.
The multiphase flow (Chapter 10) is the flow form that appears most
frequently in nature and technology. Here the word phase is meant in the
thermodynamic sense and implies either the solid, liquid, or gaseous state,
any of which can occur simultaneously in a one-component or multicompo-
nent system of substances. Impressive examples of multiphase flows in nature
are storm clouds containing raindrops and hailstones, and snow dust in an
avalanche or a cloud of volcano ash.
In power station engineering and chemical process engineering, multiphase
flows are an important means of transporting heat and material. Two-phase,

or binary, flows determine the processes in the steam generators, condensers,
and cooling towers of steam power stations. The cooling-water rain falling
down out of a wet cooling tower is shown in Figure 1.7. The water drops
lose their heat by evaporation to the warmed rising air. Multiphase, multi-
component flows are used in the extraction, transportation, and processing
of oil and natural gas. These flow forms are also very much involved in distil-
lation and rectification processes in the chemical industry. They also appear
as cavitation effects on underwater wing surfaces in fast flows. The example
in Figure 1.8 shows a cavitating underwater foil. Phenomena of this kind are
heated cylinder vertical plate horizontal plate
Fig. 1.6. Thermal convection flows
8 1. Introduction
Fig. 1.7. Wet cooling tower
highly undesirable in flow machinery since they can lead to serious material
damage.
Turbulent reactive flows are very important for a great number of appli-
cations in energy, chemical, and combustion technology. The optimization
of these processes places great demands on the accuracy of the numerical
simulation of turbulent flows. Because of the complexity of the interaction
between turbulent flow, molecular diffusion, and chemical reaction kinetics,
improved models to describe these processes are highly necessary.
Turbulent flames are characterized by a wide spectrum of time and length
scales. The typical length scales of the turbulence extend from the dimen-
sions of the combustion chamber right down to the smallest vortex in which
turbulent kinetic energy is dissipated. The chemical reactions that cause the
combustion have a wide spectrum of time scales. Depending on the overlap-
ping of the turbulent time scales with the chemical time scales, there are
regimes with a strong or weak interaction between chemistry and turbulence.
Because of this, a joint description of turbulent diffusion flames generally
always requires an understanding of turbulent mixing and combustion.

A complete description of turbulent flames therefore has to resolve all
scales from the smallest to the largest, which is why a numerical simulation
of technical combustion systems is not possible on today’s computers and
Fig. 1.8. Cavitation at an underwater
foil
1. Introduction 9
why averaging techniques in the form of turbulence models have to be used.
However, if such turbulence models are to describe such aspects of technical
application as mixing, combustion, and formation of emissions realistically,
it is necessary to be able to better determine the parameters of such models
from detailed investigations.
One promising approach is the use of direct numerical simulation, the
generation of artificial laminar and turbulent flames with the computer. For
a small spatial area, the conservation equations for reactive flows are solved,
taking all turbulent fluctuations into account, and thus describing a small
but realistic section of a flame. This can then be used to describe real flames.
The formation of closed regions of fresh gas that penetrate into the ex-
haust are an interesting phenomenon of turbulent premixed flames. The time
resolution of this transient process can be investigated by means of direct
numerical simulation and is important in determining the region of validity
of current models and the development of new models to describe turbulent
combustion. Figure 1.9 shows the concentration of OH and CO radicals, as
well as the vortex strength in a turbulent methane premixed flame.
Many different flows in nature (Chapter 12) can be seen on Earth and in
space. The flow processes in the atmosphere stretch from small winds to the
tropospherical jet stream of strong winds surrounding the globe. One par-
ticularly impressive atmospheric phenomenon is the tropical cyclone, known
in the Caribbean and the United States under the name hurricane. Hurri-
canes form in the summer months above the warm waters off the African
coast close to the equator and move with a southeasterly flow first toward

the Caribbean and then northeastwards along the east coast of the United
States. Wind speeds of up to 300 km/h can occur in these tropical wind
storms, with much resulting damage on land. An example of a cyclone is
shown in Figure 1.10. This figure shows the path and a satellite image of
Hurricane Georges which passed over the Caribbean islands and the south-
east coast of the United States in July 1998, and continued its path as a
low-pressure region across the Atlantic as far as Europe.
OH concentration CO concentration vorticity
Fig. 1.9. Turbulent premixed methane flame
10 1. Introduction
Fig. 1.10. Path of Hurricane Georges 1998
The flow processes in the ocean extend from small phenomena such as
water waves to large sea currents. An example of the latter is the Gulf Stream,
which as a warm surface current can be tracked practically from the African
coast, past the Caribbean to western and northern Europe. Thanks to its
relatively high water temperature, it ensures a mild climate along the British
and Norwegian coasts. In order to compensate the warm surface current
directed towards the pole, a cold deep current forms, and this flows from the
north Atlantic along the east coast of North and South America, toward the
south. Both of these large flow systems are shown in Figure 1.11.
In contrast to the previous examples of flows, biofluid mechanics in Chap-
ter 13 deals with flows that are characterized by flexible biological surfaces.
One distinguishes between flows past living beings in the air or in water, such
as a bird in flight or a fish swimming, and internal flows, such as the closed
gulf stream
ice field
Fig. 1.11. Large ocean currents in the Atlantic
1. Introduction 11
blood circulation of living beings. An example is the periodically pulsating
flow in the human heart.

The heart consists of two separate pump chambers, the left and right
ventricles. The right ventricle is filled with blood low in oxygen from the
circulation around the body, and on contraction it is emptied into the lung
circulatory system. The reoxygenated blood in the lung is passed into the
circulation around the body by the left ventricle. A simple representation
of the flow throughout one cardiac cycle is shown in Figure 1.12. The atria
and ventricles of the heart are separated by the atrioventricular valves, which
regulate the flow into the ventricles. They prevent backward flow of the blood
during contraction of the ventricles. During relaxation of the ventricles, the
pulmonary valves prevent backward flow of the blood out of the lung arteries,
while the aortal valves prevent backward flow out of the aorta into the left
ventricle.
During the cardiac cycles, the ventricles undergo a periodic contraction
and relaxation, ensuring the pulsing blood flow in the circulatory system
around the body. This pump cycle is associated with changes in pressure in
the ventricles and arteries. The pressure differences control the opening and
closing of the cardiac valves. In a healthy heart, the pulsing flow is laminar
and does not separate. Defects in the pumping behavior of the heart and
ventrical relaxation
mitral valve open
ventrical contraction
outward flow
aortic valve open
inward flow
Fig. 1.12. Flow in the heart during one cardiac cycle
Fig. 1.13. Velocity measurements in the heart by means of echocardiography,
University Clinic, Freiburg, 2001
12 1. Introduction
heart failure lead to turbulent flow regimes and backflow in the ventricles,
increasing flow losses in the heart.

Knowledge of the unsteady three-dimensional flow field is necessary for
medical diagnosis. Measurement of the velocity field takes place in clinical
practice by means of ultrasonic echocardiography. Figure 1.13 shows in four
separate pictures the three-dimensional reconstruction of the left ventricle
close to the aortal and central valves during one cardiac cycle. The section
of the three-dimensional contour of the left ventricle is shown surrounded in
black (right). The left atrium and the aorta (left), as well as the upper section
of the right ventricle (left), can be seen. Isolines of the measured velocity field
are shown. Dark gray indicates negative inward flow velocities, and light gray,
positive outward flow velocities. The magnitude of the velocity is denoted by
thin isotachic lines.
The first image shows the inward flow process in the left ventricle. The
mitral valve is open and the aortal valve closed. Large inward flow velocities
directed downward and with a maximal velocity of about 0.5m/s can be seen.
When the ventricle contracts, the aortal and mitral valves are closed. The left
ventricle is completely filled with blood, and the flow velocities measured are
very small and are not necessarily due to the blood flow. The velocities shown
might also be due to the relative movement of the heart to the ultrasonic
probe of the echocardiography. As the blood flows out of the ventricle, the
mitral valve is closed and the aortal valve open. Since the flow is directed
transversly to the ultrasonic Doppler beam, velocities directed downward are
evaluated as the blood flows into the aorta. As the ventricle relaxes, both
cardiac valves are closed. The flow into the left atrium can be seen.
The velocity fields measured give the doctor important information for a
medical diagnosis. However, they are at present insufficient for a quantitative
analysis of heart diseases with respect to higher flow losses in the heart. Sup-
plementing ultrasonic echocardiography, flow simulation presents a method
to determine the unsteady three-dimensional flow field quantitatively. The
simulation results will be described in Section 13.2.4.
The flow phenomena already discussed in relation to the flow past wings

and vehicles can also occur in flows through turbomachines. In order to clar-
ify this, let us consider the flow processes through a fan jet engine which
generates the thrust for civil aircraft.
Figure 1.14 shows a section of a modern fanjet engine. The front blades
form the so-called fan, which mainly generates the thrust for the entire jet
engine. The fan is driven by a gas turbine found inside the jet engine (also
called the core engine). A very small part of the thrust is generated by the
exhaust jet momentum leaving the gas turbine. The flow through the gas
turbine will be discussed in detail in Chapter 14, Thermal Turbomachinery.
The fanjet engine is a flow machine in which almost all phenomena of fluid
mechanics occur that have to be taken into account in the development of such
machinery. The blades of the fan are in a large subsonic Mach number flow of