Computational fluid dynamics for engineers, springer (2005), 3540244514

T.Cebeci J.RShao
F. Kafyeke E. Laurendeau
Computational Fluid Dynamics for Engineers


HORIZONS
PUBLISHING
Long Beach, California
Heidelberg, Germany


Tuncer Cebeci Jian P. Shao
Fassi Kafyeke Eric Laurendeau

Computational
Fluid Dynamics
for Engineers
From Panel to Navier-Stokes Methods
with Computer Programs

With 152 Figures, 19 Tables, 84 Problems
and a CD-ROM

Springer
HORIZONS
PUBLISHING


Tuncer

Fassi

Cebeci

The Boeing Company
Long Beach, CA 90807-5309, USA
and
810 Rancho Drive
Long Beach, CA 90815, USA


Eric
Jian P. Shao
The Boeing Company
Huntington Beach, CA 92647, USA


Kafyeke

Advanced Aerodynamics Department
Bombardier Aerospace
400 Cote Vertu Road West
Dorval, Quebec, Canada H4S 1Y9
fassi.kafyeke @ aero.bombardier.com
Laurendeau

Advanced Aerodynamics Department
Bombardier Aerospace
400 Cote Vertu Road West
Dorval, Quebec, Canada H4S 1Y9

ISBN 0-9766545-0-4 Horizons Publishing Inc., Long Beach
ISBN 3-540-24451 -4 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2005923905
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Horizons Publishing Inc., 810 Rancho Drive, Long Beach,
CA 90815, USA) except for brief excerpts in connection with reviews or scholarly analysis. Use
in connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
© Horizons Publishing Inc., Long Beach, California 2005
Printed in Germany
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Please note: All rights pertaining to the Computer Programs are owned exclusively by the authors
and Horizons Publishing Inc. The publisher and the authors accept no legal responsibility for any
damage caused by improper use of the programs. Although the programs have been tested with
extreme care, errors cannot be excluded.
Typeset in MS Word by the authors. Edited and reformatted by Kurt Mattes, Heidelberg, Germany,
using LMEX.

Printing and binding: Strauss GmbH, Morlenbach, Germany
Cover design: Erich Kirchner, Heidelberg, Germany
Printed on acid-free paper

54 3 2 1 0

Preface

History reminds us of ancient examples of fluid dynamics applications such as
the Roman baths and aqueducts that fulfilled the requirements of the engineers
who built them; of ships of various types with adequate hull designs, and of wind
energy systems, built long before the subject of fluid mechanics was formalized
by Reynolds, Newton, Euler, Navier, Stokes, Prandtl and others. The twentieth
century has witnessed many more examples of applications of fluid dynamics
for the use of humanity, all designed without the use of electronic computers.
They include prime movers such as internal-combustion engines, gas and steam
turbines, flight vehicles, and environmental systems for pollution control and
ventilation.
Computational Fluid Dynamics (CFD) deals with the numerical analysis of
these phenomena. Despite impressive progress in recent years, CFD remains
an imperfect tool in the comparatively mature discipline of fluid dynamics,
partly because electronic digital computers have been in widespread use for less
than thirty years. The Navier-Stokes equations, which govern the motion of
a Newtonian viscous fluid were formulated well over a century ago. The most
straightforward method of attacking any fluid dynamics problem is to solve these
equations for the appropriate boundary conditions. Analytical solutions are few
and trivial and, even with today's supercomputers, numerically exact solution
of the complete equations for the three-dimensional, time-dependent motion of
turbulent flow is prohibitively expensive except for basic research studies in simple configurations at low Reynolds numbers. Therefore, the "straightforward"
approach is still impracticable for engineering purposes.
Considering the successes of the pre-computer age, one might ask whether it
is necessary to gain a greater understanding of fluid dynamics and develop new
computational techniques, with their associated effort and cost. Textbooks on
fluid dynamics reveal two approaches to understanding fluid dynamics processes.
The first is to devise useful correlations through a progression from demonstrative experiments to detailed experimental investigations that yield additional

VI

Preface

understanding and subsequent improvement of the processes in question. The
second is to solve simplified versions of fluid dynamics equations for conservation
of mass, momentum and energy for comparatively simple boundary conditions.
There is great advantage in combining both approaches when addressing complex fluid dynamics problems, but interaction between these two approaches has
been limited until recently by the narrow range of useful solutions that could
be obtained by analytic methods or simple numerical computations. It is evident, therefore, that any method for increasing the accuracy of computational
methods by solving more complete forms of the conservation equations than has
been possible up to now is to be welcomed. The numerical approaches of CFD
have, in most cases, proven much more powerful than the closed-form analytical
solutions of the past. As an example, the flow through the blade passage of a
gas turbine is three-dimensional, and, even if we ignore the problem of modeling the behavior of turbulence, the corresponding equations can only be solved
by numerical methods; even the inviscid flow in an axisymmetnc engine intake
cannot be calculated by purely analytic methods. Thus, without computational
fluid dynamics, we cannot calculate detailed flow characteristics essential to
improving understanding and supporting the design process.
It should be recognized that both experimental and computational fluid
dynamics require resources. The cost of experiments in some cases can be prohibitive as, for example, with extensive flight tests of airplanes, full-scale tests
of a gas turbine, or destructive testing of expensive components. In such cases,
it may be possible to reduce the number of experimental tests by using CFD,
since only a relatively small number of experiments are required to check the
accuracy of the numerical results. Of course, the cost of obtaining accurate
numerical solutions of differential equations may also be large for a complex
flow, but still are usually much less than the cost of the additional experiments
that would otherwise be required. In reality, the most cost-effective approach
to solving a fluid dynamics problem is likely to be a combination of measurements and calculations. Both are subject to uncertainties, but the combination

of these two approaches can result in a more cost-effective and more reliable
design than by using only one approach or the other, and thus may be necessary to meet today's more stringent requirements for improved performance and
reduced environmental impact, along with technical innovation and economy.
This book is an introduction to computational fluid dynamics with emphasis
on the solution of conservation equations for incompressible and compressible
flows with two independent variables. From the range of formulations in CFD,
such as finite-difference, finite volume, finite element, spectral methods and
direct numerical simulation, it concentrates on the first two, which are widely
used to solve engineering problems. The restriction to two-dimensional flow and
the omission of finite element, spectral methods and direct numerical simulation
are imposed to facilitate understanding and to allow the essential material to be


Preface

VII

presented in a book of modest size. The discussions, however, are general in this
introductory book and apply to a variety of flows, including three-dimensional
flows.
The format of the book assures that essential topics are covered in a logical
sequence. The Introduction of Chapter 1 presents some examples to demonstrate the use of computational fluid dynamics for solving engineering problems
of relevance. Chapter 2 presents the conservation equations; it is comparatively
brief since detailed derivations are available elsewhere. The third chapter introduces important properties of turbulent flows, and exact and modeled forms of
the turbulence equations with explanations to justify the assumptions of the
models.
Chapters 4 and 5 provide an introduction to the numerical methods for solving the model equations for conservation equations which are useful for modeling
the behavior of the more complete and complicated parabolic, hyperbolic and
elliptic partial-differential equations considered in subsequent chapters. Chapter
4 discusses the numerical methods for the model parabolic and elliptic equations and Chapter 5 the model hyperbolic equations and include many computer

programs.
The calculation of solutions for inviscid and boundary-layer equations is addressed in Chapters 6 and 7. Chapter 6 discusses finite-difference and panel
methods for solving the Laplace equation and include computer programs for
single and multi-element airfoils. Chapter 7 discusses the solution of laminar
and turbulent boundary-layer equations for a prescribed external velocity distribution and specified transition location and includes a computer program
based on Keller's finite-difference method.
The prediction of the onset of transition from laminar to turbulent flow has
traditionally been achieved by correlations which are known to have limited
ranges of applicability. The use of the e n -method, based on the solutions of the
stability equations, has been proposed as a more general approach. Chapter 8
describes the solution of the stability equations and provides a computer program for solving the Orr-Sommerfeld equation and computing transition with
the e n -method. It also presents applications of the stability/transition program,
together with the computer programs of Chapters 6 and 7, to demonstrate how
problems of direct relevance to engineering can be addressed by this approach.
Chapter 9 presents grid generation methods and is followed by Chapters
10 to 12 which describe methods for solving Euler (Chapter 10), incompressible Navier-Stokes (Chapter 11) and compressible Navier-Stokes equations.
Again computer programs are included in each chapter and summarized in
Appendix B.
A one semester course for advanced undergraduate and first-year graduate
students would include a brief reading of Chapter 1 followed by Chapters 2, 4, 5
and 10 which include an extensive number of example problems and associated


Preface

VIII

computer programs arranged to provide the student a better understanding of
the computational tools discussed. Parts of the material in Chapters 3, 6, 7 to
9 and 11 and 12 can be covered in a second semester course, with parts of the

material in those chapters serving as useful information/reference.
A list of related and current books and solution manuals, including the one
for the present book, published by Horizons and Springer-Verlag, is available
on the Horizons Web site,
/>The authors would like to express their appreciation to several people who
have given thoughts and time to the development of this book. The first and
second authors in particular want to thank Herb Keller of the California Institute of Technology, Jim Whitelaw of Imperial College, and Hsun Chen of the
California State University, Long Beach. They also want to thank K. C. Chang
for proof reading the manuscript and making many useful suggestions. The third
and fourth authors like to thank Bombardier Aerospace for supplying some of
the applications cited in the text. Thanks are also due to Kurt Mattes for his
excellent typing and Karl Koch for the production of the book.
Finally we would like to thank our wives, Sylvia Cebeci, Jennifer Shaw,
Nathalie David and Solange Lusinde, and our children for their understanding
and the hours they relinquished to us. Their continuous support and encouragement are greatly appreciated.
Long Beach, April 2005

Tuncer Cebeci
Jian P. Shao
Fassi Kafyeke
Eric Laurendeau


Contents

1.

2.

Introduction

1.1
Skin-Friction Drag Reduction
1.1.1
Laminar Flow Control
1.1.2
Calculations for NLF and HLFC Wings
1.2
Prediction of the Maximum Lift Coefficient
of Multielement Wings
1.3
Aircraft Design and Power Plant Integration
1.4
Prediction of Aircraft Performance Degradation Due to Icing . .
1.4.1
Prediction of Ice Shapes
1.4.2
Prediction of Aerodynamic Performance
Characteristics
1.5
Aerodynamics of Ground-Based Vehicles
1.5.1
Applications of CFD to Automobiles
References

1
2
3
6

28

34
36
39

Conservation Equations
2.1
Introduction
2.2
Navier-Stokes Equations
2.2.1
Navier-Stokes Equations: Differential Form
2.2.2
Navier-Stokes Equations: Integral Form
2.2.3
Navier-Stokes Equations: Vector-Variable Form
2.2.4
Navier-Stokes Equations: Transformed Form
2.3
Reynolds-Averaged Navier-Stokes Equations
2.4
Reduced Forms of the Navier-Stokes Equations
2.4.1
Inviscid Flow
2.4.2
Stokes Flow
2.4.3
Boundary Layers
2.5
Stability Equations
2.6

Classification of Conservation Equations

41
41
42
42
48
50
51
55
57
60
62
62
64
67

10
19
23
26


X

Contents

2.7
Boundary Conditions
References

Problems

70
72
73

3.

Turbulence M o d e l s
3.1
Introduction
3.2
Zero-Equation Models
3.2.1
Cebeci-Smith Model
3.2.2
Baldwin-Lomax Model
3.3
One-Equation Models
3.4
Two-Equation Models
3.5
Initial Conditions
References

81
81
83
83
85

87
88
90
93

4.

N u m e r i c a l M e t h o d s for M o d e l Parabolic
and Elliptic Equations
4.1
Introduction
4.2
Model Equations
4.3
Discretization of Derivatives with Finite Differences
4.4
Finite-Difference Methods for Parabolic Equations
4.4.1
Explicit Methods
4.4.2
Implicit Methods: Crank-Nicolson
4.4.3
An Implicit Method: Keller's Box Method
4.5
Finite-Difference Methods for Elliptic Equations
4.5.1
Direct Methods
4.5.2
Iterative Methods
4.5.3

Multigrid Method
References
Problems

95
95
96
98
100
100
105
109
113
115
121
127
132
132

N u m e r i c a l M e t h o d s for M o d e l H y p e r b o l i c Equations
5.1
Introduction
5.2
Explicit Methods: Two-Step Lax-Wendroff Method
5.3
Explicit Methods: MacCormack Method
5.4
Implicit Methods
5.5
Upwind Methods

5.6
Finite-Volume Methods
5.7
Convergence and Stability
5.8
Numerical Dissipation and Dispersion: Artificial Viscosity
References
Problems

141
141
146
148
149
152
157
165
170
173
174

5.


Inviscid Flow Equations for Incompressible F l o w s
6.1
Introduction
6.2
Laplace Equation and Its Fundamental Solutions
6.3

Finite-Difference Method
6.4
Hess-Smith Panel Method
6.5
A Panel Program for Airfoils
6.5.1
MAIN Program
6.5.2
Subroutine COEF
6.5.3
Subroutine GAUSS
6.5.4
Subroutine VPDIS
6.5.5
Subroutine CLCM
6.6
Applications of the Panel Method
6.6.1
Flowfield and Section Characteristics
of a NACA 0012 Airfoil
6.6.2
Flow Over a Circular Cylinder
6.6.3
Multielement Airfoils
Appendix 6A Finite Difference Program for a Circular C y l i n d e r . . . .
Appendix 6B Panel Program for an Airfoil
6B.1 MAIN Program
6B.2 Subroutine COEF
6B.3 Subroutine VPDIS
Appendix 6C Panel Program for Multielement Airfoils

6C.1 MAIN Program
6C.2 Subroutine COEF
6C.3 Subroutine VPDIS
6C.4 Subroutine CLCM
References
Problems

179
179
179
182
189
194
195
196
196
196
196
197
197
198
201
202
203
203
203
203
203
203
204

204
204
204
204

Boundary-Layer Equations
7.1
Introduction
7.2
Standard, Inverse and Interaction Problems
7.3
Numerical Method for the Standard Problem
7.3.1
Numerical Formulation
7.3.2
Newton's Method
7.4
Computer Program BLP
7.4.1
MAIN
7.4.2
Subroutine INPUT
7.4.3
Subroutine IVPL
7.4.4
Subroutine GROWTH
7.4.5
Subroutine COEF3
7.4.6
Subroutine SOLV3

211
211
212
216
218
220
222
222
222
225
225
226
226


Contents

7.4.7
Subroutine O U T P U T
7.4.8
Subroutine EDDY
7.5
Applications of BLP
7.5.1
Similar Laminar Flows
7.5.2
Nonsimilar Flows
References
Problems

229
229
229
230
231
237
237

Stability and Transition
8.1
Introduction
8.2
Solution of the Orr-Sommerfeld Equation
8.2.1
Numerical Formulation
8.2.2
Eigenvalue Procedure
8.3
e n -Method
8.4
Computer Program STP
8.4.1
MAIN
8.4.2
Subroutine VELPRO
8.4.3
Subroutine CSAVE
8.4.4
Subroutine NEWTON

8.4.5
Subroutine NEWTONI
8.5
Applications of STP
8.5.1
Stability Diagrams for Blasius Flow
8.5.2
Transition Prediction for Flat Plate Flow
8.5.3
Transition Prediction for Airfoil Flow
References
Problems

243
243
246
247
249
253
256
257
257
258
258
258
259
259
259
261
261

262

Grid
9.1
9.2
9.3
9.4

263
263
264
267
268

Generation
Introduction
Basic Concepts in Grid Generation and Mapping
Stretched Grids
Algebraic Methods
9.4.1
Algebraic Grid Generation Using TYansfmite
Interpolation
9.5
Differential Equation Methods
9.6
Conformal Mapping Methods
9.6.1
Parabolic Mapping Function
9.6.2
Wind Tunnel Mapping Function

9.7
Unstructured Grids
9.7.1
Delaunay Triangulation
9.7.2
Advancing Front Method
References

271
277
282
283
285
288
289
292
293


Contents

XIII

10. Inviscid Compressible Flow
10.1 Introduction
10.2 Shock Jump Relations
10.3 Shock Capturing
10.4 The Transonic Small Disturbance (TSD) Equation
10.5 Model Problem for the Transonic Small Disturbance Equation:
Flow Over a Non-Lifting Airfoil

10.5.1 Discretized Equation
10.5.2 Solution Procedure and Sample Calculations
10.6 Solution of Full-Potential Equation
10.7 Boundary Conditions for the Euler Equations
10.8 Stability Analysis of the Euler Equations
10.9 MacCormack Method for Compressible Euler Equations
10.10 Model Problem for the MacCormack Method:
Unsteady Shock Tube
10.10.1 Initial Conditions
10.10.2 Boundary Conditions
10.10.3 Solution Procedure and Sample Calculations
10.11 Model Problem for the MacCormack Method:
Quasi 1-D Nozzle
10.11.1 Initial Conditions
10.11.2 Boundary Conditions
10.11.3 Solution Procedure and Sample Calculations
10.12 Beam-Warming Method for Compressible Euler Equations . . . .
10.13 Model Problem for the Implicit Method: Unsteady Shock Tube
10.13.1 Solution Procedure and Sample Calculations
10.14 Model Problem for the Implicit Method: Quasi-ID Nozzle
10.14.1 Solution Procedure and Sample Calculations
References
Problems

295
295
296
299
301

315
316
317
318
320
321
321
322
325
326
326

11. Incompressible N a v i e r - S t o k e s E q u a t i o n s
11.1 Introduction
11.2 Analysis of the Incompressible Navier-Stokes Equations
11.3 Boundary Conditions
11.4 Artificial Compressibility Method: INS2D
11.4.1 Discretization of the Artificial Time Derivatives
11.4.2 Discretization of the Convective Fluxes
11.4.3 Discretization of the Viscous Fluxes
11.4.4 System of Discretized Equation
11.5 Model Problem: Sudden Expansion Laminar Duct Flow
11.5.1 Discretization of the Boundary Conditions

327
327
328
329
331
331

332
334
335
336
337

302
303
304
308
309
311
312
313
314
314
314


XIV

Contents

11.5.2 Initial Conditions
11.5.3 Solution Procedure and Sample Calculations
11.6 Model Problem: Laminar and Turbulent Flat Plate Flow
11.7 Applications of INS2D
References
Problems

338
339
342
344
350
351

12. Compressible Navier—Stokes E q u a t i o n s
12.1 Introduction
12.2 Compressible Navier-Stokes Equations
12.2.1 Practical Difficulties
12.2.2 Boundary Conditions
12.3 MacCormack Method
12.4 Beam-Warming Method
12.5 Finite Volume Method
12.6 Model Problem: Sudden Expansion Laminar Duct Flow
12.6.1 Initial Conditions
12.6.2 Boundary Conditions
12.6.3 Solution Procedure and Sample Calculations
Appendix 12A Jacobian Matrices of Convection
and Diffusion Terms E, F , Ev and Fv
Appendix 12B Treatment of the Region Close to the Boundaries
for Eq. (12.5.4)
References
Problems

353
353
354
354

355
356
357
361
365
365
365
367

370
374
375

Appendix A Computer Programs
on t h e A c c o m p a n y i n g C D - R O M

377

Appendix B Computer Programs
Available from t h e First A u t h o r

381

Subject I n d e x

391

367

^

Introduction

In this chapter we present five examples to demonstrate the application of
CFD techniques to solve real engineering problems. These examples are taken
from the literature and encompass flows which make use of solutions of inviscid, boundary-layer and Navier-Stokes equations. For some of these flows, the
reduced forms of the conservation equations, such as inviscid and boundarylayer equations are more appropriate, and for others more general equations are
needed. In this way, the scope of this book is clarified further with additional
terminology and fluid-dynamics information.
The first example, discussed in Section 1.1, addresses the application of CFD
to reduce the drag of a wing by adjustment of pressure gradient by shaping and
by suction through slotted or perforated surfaces. The drag of an aircraft can
be reduced in a number of ways to provide increased range, increased speed,
decreased size and cost, and decreased fuel usage. The adjustment of pressure
gradient by shaping and using laminar boundary-layer control with suction are
two powerful and effective ways to reduce drag. This is demonstrated with a
calculation method for natural laminar flow (NLF) and hybrid laminar flow
control (HLFC) wings.
The second example, discussed in Section 1.2, addresses the calculation of the
maximum lift coefficient of a wing which corresponds to the stall speed, which
is the minimum speed at which level flight can be maintained. A calculation
method is described and used to predict the maximum lift coefficient of a highlift system; this coefficient plays a crucial role in the takeoff and landing of an
aircraft.
Aircraft design was traditionally based on theoretical aerodynamics and wind
tunnel testing, with flight-testing used for final validation. CFD emerged in the
late 1960's. Its role in aircraft design increased steadily as speed and memory
of computers increased. Today CFD is a principal aerodynamic technology for
aircraft configuration development, along with wind tunnel testing and flighttesting. State-of-the-art capabilities in each of these technologies are needed to
achieve superior performance with reduced risk and low cost.

2

1. Introduction

The third example, discussed in Section 1.3, deals with aircraft design and
power plant integration.
The fourth example, discussed in Section 1.4, corresponds to a calculation
method for predicting the performance degradation of an aircraft due to icing.
A NACA icing research aircraft is chosen to compare the calculated results
with measurements. The calculations are first performed by computing the ice
shapes that form on the leading edges of the lifting surfaces of the aircraft and
are followed by flowfield calculations to predict the loss in lift and increase in
drag due to ice.
The fifth example, discussed in Section 1.5 is the application of CFD to
ground-based vehicles, in particular to automobile aerodynamics development.
The use of CFD in this area has been continuously increasing because the aerodynamic characteristics have a significant influence on the driving stability and
fuel consumption on a highway. Since the aerodynamic characteristics of automobiles are closely coupled with their styling, it is impossible to improve them
much once styling is fixed. Therefore, it is necessary to consider aerodynamics
in the early design stage.
CFD also finds applications in internal flows and has been used to solve
real engineering problems such as subsonic, transonic and supersonic inlets,
compressors and turbines, as well as combustion chambers and rocket engines.
These applications are, however, beyond the scope of this book and the reader
is referred to the extensive literature available on these problems.

1.1 Skin-Friction Drag Reduction
There are several techniques for reducing the skin-friction drag of bodies. While
the emphasis in this section is on aircraft components, the arguments apply

equally to the reduction of skin-friction drag on all forms of transportation,
including underwater vehicles. The importance of the subject has been discussed
in a number of articles; a book edited by Bushnell and Heiner [1] summarizes
the research in this area and the reader is referred to this book for an in-depth
review of viscous drag reduction and for discussions of the possible savings which
can occur from the reduction of the drag. As an example of the argument in
support of the importance of the calculation methods used for reducing skinfriction drag, it is useful to point out that a three-percent reduction in the
skin-friction drag of a typical long-range commercial transport, which burns
around ten million gallons of fuel per year, at 50 cents per gallon, would yield
yearly savings of around $ 150,000.
There have been many suggestions for reducing the skin-friction drag on
aircraft components including extension of regions of laminar flow, relaminarization of turbulent flow and modification to the turbulence characteristics of the
near-wall flow. In general, these attempts to control the flow depend on changes


1.1 Skin-Friction Drag Reduction

3

to the wall boundary conditions including variations of longitudinal and transverse surface curvatures, the nature of the surface and heat and mass transfer
through the surface. A partial exception is the use of thin airfoils (LEBUs) in
the outer region of the boundary layer to break up the large eddy structure of
turbulent flow [1].
In this section the discussion is limited to laminar flow control (LFC) and
the reader is referred to [1] for a discussion of other techniques for reducing
the skin-friction drag. In subsection 1.1.1, a brief description of laminar flow
control first by "Adjustment of Pressure Gradient by Shaping," then by "Suction
Through Slotted or Perforated Surfaces" is given. This subsection is followed by
a description and application of a calculation method to natural laminar flow
(NLF) and hybrid laminar flow control (HLFC) wings (Subsection 1.1.2).

1.1.1 Laminar Flow Control
A d j u s t m e n t of P r e s s u r e Gradient by Shaping
Laminar flow on a two-dimensional or axisymmetric body can be achieved by
designing the geometry so that there are extensive regions of favorable pressure
gradients. This technique is frequently referred to as natural laminar flow (NLF)
control and may be implemented on a wing or a body of revolution by bringing
the point of maximum thickness as far aft as possible. Typical airfoil sections
designed for this purpose are shown on Fig. 1.1 and the location of the onset
of transition, where laminar flow becomes turbulent flow, can be estimated by

L R N ( l ) - 1010
LOW ALTITUDE

N L F ( l ) - 1015
HIGH ALTITUDE

NLF(1)-0414F
GENERAL AVIATION

HSNLF(1)-0213F
BUSINESS JET

NLF(2)-0415
COMMUTER

SCLFC(1)-0513
TRANSPORT

Fig. 1.1. Typical NLF airfoils for a wide range of applications. SCLFC denotes supercritical LFC airfoil.

1. Introduction

4

using the e n -method discussed in Chapter 8. The success of this technique and
of the calculation method also depends on factors besides the pressure gradient
including surface roughness, surface waviness, freestream turbulence, and the
concentration of a second phase such as rain or solid particles in water, all of
which can play a role in triggering transition [2]. The influence of these factors
can usually be avoided by careful design, for example by keeping the surface
waviness and roughness below the allowable limits.
A number of modern low-speed aircraft make use of extended regions of natural laminar flow on their wings [1] but transonic cruise, and the swept wings
required for this configuration, introduce further complications. In particular,
flow from the fuselage boundary layer can introduce instabilities which result
in turbulent flow along the attachment line of the wing [2], or a favorable pressure gradient on the upper surface can result in a shock wave which interacts
with the boundary-layer to cause turbulent flow. The first problem depends on
the Reynolds number, sweep angle and curvature of the leading edge and it
is possible to shape the leading edge of the wing so that the attachment-line
flow is laminar. In this case it is likely that, depending on the sweep angle, the
flow may become turbulent away from the attachment line due to the crossflow
instability discussed in [2]. In subsection 1.1.2 calculations are presented for a
typical NLF wing in incompressible flow to demonstrate the role of sweep angle
and crossflow on transition.
Extending the region of natural laminar flow on fuselages in order to reduce
the fuselage drag is also important, as indicated by the examples of Fig. 1.2,
relevant to transport aircraft [1]. It should be pointed out that the total skinfriction drag of a modern wide-body transport aircraft is about 40% of the
total airplane drag, with approximately 3% from nacelles and pylons, 15% from
fuselage, 15% from wing, and 8% from empennage. Thus, nacelles and pylons
account for about 8% of the total skin-friction drag, while the fuselage, wing and

empennage account for 38%, 35% and 20%, respectively. For smaller airplanes,
such as the MD-80 and 737, the portion of the total skin-friction drag is usually
higher than for wide bodies.

Table 1.1. Drag coefficients for an
axisymmetric body with a fineness
ratio 6.14 at a = 0, RL = 40.86 x
106 [1].
XtT

Cd x 102

0.322
0.15
0.10
0.05

2.60
3.43
3.62
3.74


1.1 Skin-Friction Drag Reduction

5

All-turbulent surfaces

Laminar lifting surfaces

Nacelles and misc.
Fuselage
Empennage
Wing

5.2%
48.7%
14.3%
31.8%

7.6%
70.2%
6.9%
15.3%

Nacelle and others
Fuselage
Empennage
Wing
Total profile CD

0.0010
0.0092
0.0027
0.0060
0.0189

0.0010
0.0092

0.0009
0.0020
0.0131

Fig. 1.2. Profile drag buildup for all-turbulent transport jet and airplane with laminar
lifting surfaces [1].

Table 1.1 shows the reduction in drag coefficient which can be achieved on
an axisymmetric body by control of the location of the onset of transition: as
an example, a delay of transition by 27% of the body length reduces the drag
coefficient by some 30%. As in the case of wings, the onset of transition on
fuselages and bodies of revolution can be estimated by an extension of the enmethod discussed in Chapter 8 from two-dimensional flows to three-dimensional
flows discussed in [2,3].
S u c t i o n T h r o u g h S l o t t e d or Perforated Surfaces
The attainment of laminar flow by adjustment of pressure gradient by shaping
becomes increasingly more difficult as the Reynolds number increases because
the boundary layer becomes relatively thinner and, as a result, more sensitive to
roughness and small disturbances. Thus, there are practical limits to maintaining natural laminar flow at high Reynolds numbers because the effort spent to
maintain extremely smooth surfaces may be negated by the increased sensitivity
to external factors over which one has little control.
The next technique to maintain laminar flow is the use of active laminar flow
control by suction which thins the boundary-layer, generates a fuller velocity
profile and leads to increased boundary-layer stability. The use of suction at the
leading edge of a wing, through slots or perforated material, can overcome the
tendency for the cross-flow velocity to create a turbulent boundary-layer flow
beginning at the attachment line [1], see also [2]. The technique is referred to
as hybrid laminar-flow control (HLFC) since it combines suction mass transfer
with the arrangement of the airfoil (see Fig. 1.3) so as to impose a favorable
longitudinal pressure gradient. This type of LFC is applicable to a wide range of
small to moderate sized aircraft. The perforated plate makes use of holes of the

order of 0.004 inches in diameter with a pitch-to-diameter ratio of around ten


1. Introduction

6

- SUCTION

Fig. 1.3. A typical airfoil section for hybrid laminar flow control (HLFC).

and cleaning of the holes can be accomplished by reversing the mass flow while
the aircraft is stationary. Extensive wind-tunnel tests have been reported by
Pfenninger [1] who made use of vertical slot widths graded from 0.008 to 0.003
inches depending on the thickness of the boundary-layer and a pitch which
varied from 3 to 0.6 inches depending on the static pressure. Difficulties were
experienced with the effective roughness created by the edges of the slots, but
the system was made to operate satisfactorily so that the effects of the crossflow velocity were removed in that the flow around the leading edge remained
laminar. Again, stability (Chapter 8) and boundary-layer (Chapter 7) theories
can be used in the design of the HLFC wing, as discussed in the following
subsection.
1.1.2 Calculations for N L F and H L F C W i n g s
A calculation method (Chapter 4 of [1]) based on the solutions of the panel,
boundary-layer and stability equations for three dimensional flows can be used
to demonstrate the effects of sweep, angle of attack, and suction on transition.
A wing with a cross section of the NACA 6-series laminar flow airfoil family
developed in the late thirties is chosen for this purpose. Its particular designation
is NACA 65-412 where the first digit designates the airfoil series and the second
indicates the extent of the favorable pressure gradient in tenths of chord on both
upper and lower surfaces at design condition; the third digit gives the design

lift coefficient and the last two digits denote the thickness in percent of the
chord. The camber line used to generate this airfoil has the NACA designation
a = 1.0 which means that the additional loading due to camber is uniform along
the chord. It also happens that the use of this particular camber line results in
an airfoil which has its design lift coefficient at zero angle of attack and all
calculations presented here were performed at this angle of attack. The results


1.1 Skin-Friction Drag Reduction

7

1.2
1.0

WU«

0.8
0.6
0.4
0.2

0.0
0.0

0.2 0.4

0.6
x/c

0.8

1.0

F i g . 1.4. Variation of inviscid velocity distribution with sweep angle for
the NACA 65-412 wing.

correspond to a Reynolds number of 10 7 , based on the total freest ream velocity
Voc and chord c, and for several sweep angles ranging from 0° to 50°. The
inviscid velocity distribution was computed from the Hess panel method [4, 5]
which is an extension of the two-dimensional panel method of Section 6.4 to
three-dimensional flows and the boundary-layer calculations were performed
by a boundary-layer method for three-dimensional flows which is an extension
of the two-dimensional boundary-layer method of Chapter 7 [2,4]. Transition
calculations are performed by using the e n -method for three-dimensional flows
which is an extension of the e n -method for two-dimensional flows discussed in
Chapter 8 [2,3].
Figure 1.4 shows the inviscid velocity distribution Ue/u^ for the upper surface of the wing for A = 20°, 30° and 40° and, as can be seen, the flow has a favorable pressure gradient up to around 50-percent chord, followed by an adverse
pressure gradient. We expect that the cross-flow instability will be rather weak
at lower sweep angles, so that transition will take place in the region where the
flow deceleration takes place. With increasing sweep angle, however, crossflow
instability [2] will begin to dominate and cause transition to occur in the region
of acceleration. The results of Fig. 1.5 for A = 20° confirm this expectation and
indicate that amplification factors computed with different frequencies reach
values of n as high as 6.75 at x/c = 0.44 but not a value of n = 8 as required
to indicate transition (Chapter 8). Additional calculations show that transition
occurs at x/c = 0.65 and is not caused by crossflow instability. The results for
A = 40°, shown in Fig. 1.6, however, indicate that crossflow instability makes
its presence felt at this sweep angle, causing transition to occur at x/c = 0.08
corresponding to a radian disturbance frequency of 0.03740. The location of the

critical frequency is at x/c = 0.0046, very close to the attachment line of the
wing.
Calculations performed for A = 30°, 35° and 50° indicate results similar to
those for A = 40° in that the transition location moves closer to the leading edge


1. Introduction

n

0.2

0.3 0.4
x/c
Fig. 1.5. Amplification factors for several
frequencies for A — 20°. The numbers 1 to
7 show different frequencies used for each
amplification calculation (Chapter 8).

0.3
x/c

F i g . 1.6. Amplification factors for several
frequencies for A = 40°.

with increasing sweep angle, occurring at x/c = 0.22 for A = 30°, at x/c = 0.12
for A = 35° and at x/c = 0.05 at A = 50°.
Figures 1.7 to 1.9 show the calculated amplification factors for the same
wing with suction, which is a powerful means of maintaining laminar flow over
the whole wing. In practice, however, this is difficult to achieve because of the

need for ailerons, flaps and openings for inspection and maintenance. Clearly a
suction system adds to the complexity, weight and cost of a design. Increasing
suction rates requires larger ducting system and more power so that at some
point all available space in the wing may be used up and the higher suction drag
will produce diminishing returns. Increased suction also makes the boundarylayer thinner, which in turn reduces the critical height of roughness that will
cause transition. If suction is applied through discrete holes or slots and is not

10
/ NO SUCTION
9
n 8 !"
7
^^~ S 4
6 5 4 - — SI
3 -/
2 7
1
1
1
1
1
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6

10 -NO SUCTION
9
n 8
7
/ / s i
/ ^ S ?

6
5 -/
4 -/
3
r~ S 2 /
2
1
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6

Fig. 1.7. Effect of suction on amplification
rates for A = 30°.

F i g . 1.8. Effect of suction on amplification
rates for A = 40°.

^_i___

1

x/c

i

i

i

i

i

i

x/c


1.1 Skin-Friction Drag Reduction

9

Table 1.2. Suction rates vw = v ^ / V ^ used in the
stability calculations. SI, S2 and S3 are applied to
the whole wing while S4 to S8 are applied to the
first 5% chord of the wing. S9 is applied to the
first 10% of the wing.
vw x 104
51
52
53
54

-3
-5
-7
-3

vw x 104
S5
S6

S7
S8
S9

-5
-7
-10
-12
-12

distributed over the area, increased suction velocities may cause the suction
holes or slots to become critical themselves and act as sources for disturbances.
It is important that the suction system be carefully designed by calculating
minimum suction rates to maintain laminar flow. In addition, the suction rate
distribution must be optimum. A calculation method, such as the one described
in Chapter 4 of [1] and [3], is capable of determining the minimum and optimum
suction rates for the ducting system. Table 1.2 lists the suction distributions
used in the calculations presented here. For simplicity, two types of suction
distributions are considered: the first with uniform suction on the whole wing
and the second with uniform suction over the front portion of the wing only,
e.g. 5% chord from the leading edge.
Figure 1.7 shows the amplification factors for three frequencies: one without
suction, and the other two for two types of suction, SI and S4 for A = 30°. As
can be seen, a small suction level of vw = —0.0003 either over the whole wing,
SI, or over the front 5% chord of the wing, S4, is sufficient to maintain laminar
flow until separation or transition occurs at x/c — 0.58 for S4 and at x/c = 0.78
for SI. The calculations for SI produce a low value of n — 3 at x/c — 0.34 and
indicate that the suction rate is excessive at this sweep angle.
Figure 1.8 shows the results for A = 40° for which case a suction level of
vw — —0.0003 for SI yields a maximum value of n = 6 at x/c = 0.20 and a

suction level corresponding to S2 yields a maximum value of n = 3 at x/c = 0.12.
Both cases eliminate transition which occurs at x/c = 0.08 without suction, but
the latter also eliminates the occurrence of separation while the former delays
the separation until x/c = 0.78. To avoid excessive suction, two additional cases
corresponding to S5 and S7 were considered and it was observed that transition
takes place at x/c = 0.22 for S5, and the maximum value of n is equal to 6.7 at
x/c = 0.52 for S7 which shows that the crossflow instabilities can be eliminated
in the front portion of the wing. It is interesting to note that the small bump near
x/c = 0.05 along the curve for S7 shown in Fig. 1.8 is caused by the switch-off
of suction at x/c = 0.05.


1. Introduction

10

cq
0 rN u su U 1 KJ1N
9
8 -,
7
//S2
/
6 _1
5
4
/ S3 /
3
2
1

/
| ^S
0
0.1
0.2 0.3 0.4 0.5 0.6
00

x/c

F i g . 1.9. Effect of suction on amplification rates
for A = 50°.

As expected, it is more difficult to avoid the crossflow instabilities for A =
50° because of the high sweep, and Fig. 1.9 shows that only suction levels
corresponding to S2 and S3 can eliminate transition. However, if suction is
switched off at 5% chord from the leading edge, transition occurs even if a high
suction level of vw = —0.0012 is applied. In order to laminarize the flow, it is
necessary to extend the range of suction at a suction level of vw = —0.0012 for
the first 10% chord of the wing, case S9, leading to transition at x/c — 0.48
which is 8% upstream of the separation location. Further extensions of the
suction area will eliminate transition before separation occurs. From the results
corresponding to S8 and S9, it can be seen that the growth of the disturbances
can be prevented only in the range over which suction is applied for A = 50°.
Once the suction is switched off, the disturbances grow with almost constant
speed and cause transition to occur downstream, indicating the difficulty of
laminarizing the flow on a highly swept-back wing.

1.2 Prediction of t h e M a x i m u m Lift Coefficient
of Multielement Wings
In aircraft design it is very important to determine the maximum lift coefficient

as accurately as possible, since this lift coefficient corresponds to the stall speed,
which is the minimum speed at which controllable flight can be maintained. Any
further increase in angle of incidence will increase flow separation on the wing
upper surface, and the increased flow separation results in a loss in lift and a
large increase in drag.
The high-lift system of an aircraft plays a crucial role in the takeoff and
landing of an aircraft. Without high-lift devices, the maximum lift coefficient,
(Czjmax? attainable by a high-aspect-ratio wing is about five times the incidence
(in radians) at incidences up to stall. Typical values of (C^Jmax are commonly
in the range of 1.0 to 1.5. The addition of high-lift devices such as flaps and