Analysis of Low Speed Unsteady Airfoil Flows

T.Cebeci M.Platzer H.Chen K.-C. Chang J.P.Shao Analysis of Low-Speed Unsteady Airfoil Flows HORIZONS PUBLISHING Long ...

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T.Cebeci M.Platzer H.Chen K.-C. Chang J.P.Shao Analysis of Low-Speed Unsteady Airfoil Flows

HORIZONS PUBLISHING Long Beach, California Heidelberg, Germany

Tuncer Cebeci Max Platzer Hsun Chen Kuo-Cheng Chang Jian P. Shao

Analysis of Low-Speed Unsteady Airfoil Flows With 131 Figures, 3 Tables, and a CD-ROM

HORIZONS PUBLISHING

Springer

Tuncer

Hsun

Cebeci

The Boeing Company Long Beach, CA 90807-5309, USA and 810 Rancho Drive Long Beach, CA 90815, USA [email protected] Max

Platzer

Naval Postgraduate School Monterey, CA 93943, USA and 3070 Hermitage Road Pebble Beach, CA 93953, USA [email protected]

ISBN 0-9668461 -8-4 ISBN 3-540-22932-9

Chen

Department of Mechanical and Aerospace Engineering California State University, Long Beach 1250 Bellflower Blvd. Long Beach, CA 90840-8304, USA hhchen @ csulb.edu Kuo-Cheng Chang The Boeing Company Huntington Beach, CA 92647, USA kuo-cheng.chang @ boeing.com Man P. Shao The Boeing Company Huntington Beach, CA 92647, USA [email protected]

Horizons Publishing Inc., Long Beach Springer Berlin Heidelberg New York

Library of Congress Control Number: 2004116174 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 IATgX. Printing and binding: Strauss GmbH, Morlenbach, Germany Cover design: Erich Kirchner, Heidelberg, Germany Printed on acid-free paper

5 4 3 210

Preface

The standard textbooks on aerodynamics usually omit any discussion of unsteady aerodynamics or, at most, consider it only in a single chapter, based on two justifications. The first is that unsteady aerodynamics should be regarded as a specialized subject required "only" in connection with understanding and analyzing aeroelastic phenomena such as flutter and gust response, and therefore should be dealt with in related specialist books. The second reason appears to be reluctance to discuss aerodynamics with the inclusion of the time-dependent terms in the conservation equations and the boundary conditions for fear that added complications may discourage the reader. We take the opposite view in this book and argue that a full understanding of the physics of lift generation is possible only by considering the unsteady aerodynamics of the starting vortex generation process. Furthermore, certain "steady" flows are inherently unsteady in the presence of flow separation, as for example the unsteady flow caused by the Karman vortex shedding downstream of a cylinder and "static" airfoil stall which is an inherently unsteady flow phenomenon. Therefore, it stands to reason that a unified treatment of aerodynamics that yields steady-state aerodynamics as a special case offers advantages. This reasoning is strengthened by the developments in computational fluid dynamics over the past forty years, which showed that accurate steady-state solutions can be obtained efficiently by solving the unsteady flow equations. We have, however, chosen to concentrate on unsteady low-speed flows over airfoils in order to present a reasonably comprehensive coverage while limiting the size of the book. This implies that the content is restricted to the discussion of two-dimensional incompressible flows and, as a consequence, the book is structured as described in the following paragraphs. The introductory first chapter describes the physics of unsteady flows by explaining the unsteady flow mechanisms underlying the generation of lift on two-dimensional airfoils and finite-span wings and the generation of thrust on flapping airfoils and wings. This is followed by a demonstration that airfoils

VI

Preface

capable of pitch and plunge oscillations can extract energy from the air stream instead of generating thrust, provided the amplitudes of oscillation and the phasing between the pitch and plunge oscillations reach certain critical values. This phenomenon can lead to the destruction of an aircraft wing within seconds due to explosive flutter. The possibility of airfoil flutter due to pitch oscillations only is then considered and, in this case, the unsteady aerodynamic effects caused by the vortex shedding from the airfoil are shown to be an essential component in the explanation of this phenomenon. An understanding and incorporation of unsteady flow concepts is also required for the determination of the loads caused by wind gusts. The chapter ends by drawing attention to the dynamic airfoil stall and stall flutter phenomena, as caused by flow separation effects. It is well recognized that the "rational" analysis of separated flows, i.e., an analysis other than empirical or semi-empirical, needs to be based on the solution of the viscous flow equations stemming from the Navier-Stokes equations. In addition, most flows of practical importance are partly or fully turbulent and therefore require the use of Reynolds averaging in order to evolve a practically useful computational tool. For these reasons, the second chapter begins with a presentation of the Navier-Stokes equations and their Reynolds-averaged form. Furthermore, since many flows can be analyzed efficiently by the use of reduced forms of the Navier-Stokes equations, the thin-layer Navier-Stokes, boundary layer and inviscid flow equations are also included in this chapter. Since inviscid, boundary layer, and Navier-Stokes methods are now widely used, separate chapters are devoted to describe the three methods for the computation of steady and unsteady airfoil flows. The computation of inviscid airfoil flows benefited enormously, both conceptually and computationally, by the introduction of the so-called panel method, pioneered at the Douglas Aircraft Company in the 1960s. Thus, a panel method for the calculation of the flow over an airfoil executing a general time-dependent motion is described in chapter three. It is known that the viscous flow effects can be included with the pressure distribution obtained from an inviscid flow solution as input into the boundary layer equations. This concept can be further refined by interaction between the inviscid and boundary layer computations, thus making it possible to analyze mildly separated flows as described in chapter five. The fourth, sixth and seventh chapters describe applications of the inviscid, boundary layer and viscous-inviscid interaction codes, respectively, to provide the reader with an appreciation for the usefulness and range of validity of each method by comparing the computations with available experimental results. The eighth and ninth chapters consider the analysis of strongly viscous and separated flows by means of the Reynolds averaged Navier-Stokes equations by describing first the various solution methods for both incompressible and compressible flows and then presenting applications and comparisons in the ninth chapter. In this chapter we concentrate on the analysis of dynamic airfoil stall and show the importance

Preface

VII

of transitional flow effects. The final tenth chapter documents the details of the computer programs given on the accompanying CD-ROM. We hope that this structure will allow particular parts of the book to be read and used independently from others. Thus, readers interested only in unsteady inviscid panel methods may want to use only the third and fourth chapters, while those wishing to use boundary layer and viscous-inviscid interaction codes will find the fifth, sixth, and seventh chapter beneficial. Readers who want to inform themselves merely about the applicability and range of validity of the inviscid, boundary layer and Navier-Stokes methods may want to limit themselves to chapters four, six and nine. This book is an outgrowth of a collaboration between the authors over many years. As is evident from the cited references, most examples are taken from papers produced by them. The first author and his colleagues would like to express their appreciation to several people who have given thought and time to the development of methods discussed in this book. In particular, they want to thank the late Keith Stewartson of Unversity College, London, James Whitelaw of Imperial College, London, and A. A. Khattab of the California State University at Long Beach. The second author is especially indebted to Drs. Kevin D. Jones, John A. Ekaterinaris, Ismail Tuncer, Joseph C. S. Lai, Wolfgang Sanz, and M. S. Chandrasekhara for their contributions. He also gratefully acknowledges the support provided by the Naval Postgraduate School, the National Research Council, the Office of Naval Research, the Naval Research Laboratory, the Naval Air Systems Command, and the Air Force Institute of Technology. Finally, our most sincere thanks go to our spouses for their steadfast support and encouragement, which made it possible for us to complete this project. November, 2004

Tuncer Cebeci Max Platzer Hsun Chen Kuo-Cheng Chang Jian P. Shao

Contents

1.

P h y s i c s of U n s t e a d y Flows 1.1 Introduction 1.2 Lift Generation 1.3 Thrust Generation by Airfoil Oscillation 1.4 Power Extraction 1.5 Single-Degree-of-Freedom Airfoil Flutter 1.6 Airfoil Gust Response 1.7 Dynamic Airfoil Stall 1.8 Stall Flutter 1.9 Summary References

1 1 5 8 10 13 15 16 18 19 19

2.

The 2.1 2.2 2.3 2.4

Differential Equations of Fluid Flow Introduction Navier-Stokes Equations Reynolds-Averaged Navier-Stokes Equations Reduced Forms of the Navier-Stokes Equations 2.4.1 Parabolized and Thin-Layer Navier-Stokes Equations . . . 2.4.2 Inviscid Flow Equations 2.4.3 Boundary-Layer Equations References

21 21 22 24 25 25 26 27 29

3.

Panel M e t h o d s 3.1 Introduction 3.2 HSPM 3.3 Extension of HSPM to Unsteady Airfoil Flows 3.3.1 Influence Coefficients 3.3.2 Solution Procedure 3.3.3 Velocity Potential and Pressure Distribution

31 31 31 38 40 42 45

X

Contents

3.4 Extension of HSPM to Unsteady Flow over Finite-Span Wings . References

47 47

4.

Applications of Panel M e t h o d 4.1 Introduction 4.2 Analysis of Lift Generation 4.3 Analysis of Thrust Generation or Power Extraction 4.4 Analysis of Oscillating Airfoils 4.5 Analysis of Torsional Airfoil Flutter 4.6 Analysis of Airfoil Gust Response References

49 49 49 53 54 56 57 58

5.

Boundary-Layer M e t h o d s 5.1 Introduction 5.2 Standard, Inverse and Interaction Problems 5.2.1 Standard Problem 5.2.2 Inverse Problem 5.2.3 Interaction Problem 5.3 Solution of the Standard Problem for Two-Dimensional Steady Flows 5.3.1 Numerical Formulation 5.3.2 Newton's Method 5.3.3 Block-Elimination Method 5.4 Solution of the Inverse Problem for Two-Dimensional Steady Flows 5.5 Solution of the Standard and Inverse Problems for Two-Dimensional Unsteady Flows 5.5.1 Initial Conditions 5.5.2 Transformed Equations 5.5.3 Numerical Method: Flows without Reversal 5.5.4 Numerical Method: Flows with Reversal References

59 59 61 61 62 63

6.

Applications of Boundary-Layer Flows W i t h o u t Separation 6.1 Introduction 6.2 Unsteady Boundary-Layers with in External Velocity 6.2.1 Laminar Flows 6.2.2 Turbulent Flows 6.3 Boundary-Layer Motion Started 6.3.1 Impulsively Started Flat

64 66 69 71 72 76 77 78 79 83 91

Methods: 93 93 Fluctuations

Impulsively from Rest Plate

93 93 96 97 97

Contents

XI

6.3.2

Impulsively Started Circular Cylinder: Unsteady Separation

References 7.

99 103

Applications of Boundary-Layer M e t h o d s : Flows w i t h Separation 7.1 Introduction 7.2 Separation and Reattachment Near the Leading Edge of a Thin Oscillating Airfoil 7.2.1 Model Problem 7.2.2 Initial Conditions 7.2.3 The Question of Singularity on an Oscillating Airfoil. . . . 7.2.4 Interaction as an Answer to the Question of Singularity . 7.3 Steady Airfoil Flows 7.3.1 Airfoils at Low Reynolds Numbers 7.3.2 Airfoils at High Reynolds Numbers 7.4 Unsteady Airfoil Flows 7.4.1 Results of Unsteady Flows 7.4.2 Initiation of Dynamic Stall on a Pitching Airfoil 7.4.3 Summary References

106 106 108 112 118 123 125 130 133 139 148 149 151

8.

Navier-Stokes Methods 8.1 Introduction 8.2 Navier-Stokes Equations 8.2.1 Vector-Variable Form 8.2.2 Transformed Form 8.3 Turbulence Models 8.3.1 Algebraic Models 8.3.2 One-Equation Models 8.3.3 Two-Equation Models 8.4 Numerical Methods: Incompressible Flows 8.4.1 Vorticity-Streamfunction Formulation 8.4.2 Velocity-Vorticity Formulation 8.4.3 Pseudo-Compressibility Formulation 8.5 Numerical Methods: Compressible Flows References

153 153 154 156 158 161 162 166 167 172 173 173 175 181 188

9.

Applications of N a v i e r - S t o k e s M e t h o d s 9.1 Introduction 9.2 Laminar Flow Calculations for Incompressible Flows 9.3 Laminar Flow Calculations for Compressible Flows

191 191 192 192

105 105

XII

Contents

9.4

Laminar and Turbulent Flow Calculations for Incompressible and Compressible Flows 9.5 Effect of Transition 9.6 Flapping-Wing Flight 9.7 Three-Dimensional Dynamic Stall Calculations References

196 200 204 206 208

10. C o m p a n i o n C o m p u t e r P r o g r a m s 10.1 Introduction 10.2 Hess-Smith Panel Method (HSPM) for Steady Flows 10.3 Interactive Boundary-Layer Program 10.3.1 Input 10.3.2 Output 10.3.3 Test Cases References

211 211 211 212 213 214 215 223

Subject I n d e x

225

Physics of Unsteady Flows

1.1 Introduction Standard textbooks on aircraft aerodynamics either omit any discussion of unsteady aerodynamic effects or, at most, devote a single chapter to it. A more detailed discussion of unsteady aerodynamics is usually found in textbooks on aeroelasticity, as for example in the books by Dowell et al. [1] and Bisplinghoff et al. [2]. This is because a complete understanding and analysis of aircraft flutter and dynamic response phenomena cannot be attained without the proper unsteady aerodynamic analysis methods. This state of affairs is somewhat unfortunate because it generates the impression that unsteady aerodynamics is a highly specialized discipline which is needed only for the prediction of aeroelastic phenomena. The purpose of this book is to show that the study of unsteady aerodynamics yields many benefits beyond acquiring an ability to analyze flutter and dynamic response phenomena. Foremost among these benefits is the insight gained into the physics of lift generation by considering the flow changes due to incidence changes of an aircraft wing. Some introductory textbooks merely invoke the steady Bernoulli equation and the location of the rear stagnation point at the airfoil trailing edge to explain the generation of a pressure difference between upper and lower surface. Many standard texts, for example Anderson [3], Bertin [4] and Kuethe & Chow [5] discuss the generation of a starting vortex in connection with the Kelvin-Helmholtz theorem but do not present a detailed explanation of the generation of the starting vortex and the resulting flow and pressure changes on the airfoil. As a result, most aeronautical engineering students and practitioners have a rather incomplete understanding of the physics of lift generation. In fact, a very recent text by Anderson & Eberhardt [6] attempts to convince pilots and flight enthusiasts that lift is caused by the Coanda effect,

1. Physics of Unsteady Flows

2

i.e., the air's tendency to stick to the airfoil and thus to bend the air around the wing. The reason for the omission of a more detailed explanation of the physics of lift generation is, of course, the need for a discussion of the unsteady viscous flow phenomena underlying the formation of the starting vortex. This requires knowledge of both unsteady and viscous concepts - topics which are usually deferred to specialized advanced courses. The fundamental equations describing unsteady viscous flow processes are the Navier-Stokes equations - a system of nonlinear partial differential equations which started to become amenable to solution only recently with the availability of powerful high-speed computers. However, these recent developments of numerical solutions for unsteady inviscid and viscous flows now make it possible to visualize the essential flow processes in great detail in "numerical wind tunnels" and thus to retrace the steps taken by the pioneers of modern aerodynamics, starting with Ludwig Prandtl's flow visualization studies at his water channel in 1903 and his introduction of the boundary layer concept in 1904. Our approach therefore is to demonstrate the insights which can be achieved by studying unsteady low-speed airfoil flows. The restriction to this class of flow problems is quite intentional in order to limit the size of the book. Threedimensional unsteady low-speed flows and high subsonic, transonic and supersonic unsteady flows therefore are deferred to future treatises. It is instructive to recall the major historical developments of unsteady lowspeed airfoil aerodynamics. Prandtl again must be regarded as the pioneer of this field [7]. In a lecture at a scientific congress in Innsbruck, Austria, in 1922 he proposed to attack the problem of incompressible flow past an oscillating airfoil by neglecting the influence of viscosity and thus to take the Laplace equation as the governing equation. He pointed out that, according to Kelvin's theorem, every change in lift must be accompanied by the detachment of a vortex from the airfoil's trailing edge. He then proceeded to sketch a small perturbation approach to solve the problem of incompressible flow past an airfoil which executes small amplitude sinusoidal oscillations. In the same year, Prandtl's doctoral student W. Birnbaum presented a solution to this problem using a series approximation of the resulting integral equation for the unknown vortex distribution. In his dissertation, Birnbaum [8] showed that the parameter k k = ujc/uoo

(1.1.1)

where uu is the circular frequency of oscillation, c the airfoil chord and UQO the flow speed, has a special significance. He called it the reduced frequency. It is a measure of the unsteadiness and is an important similarity parameter. This can be understood from Fig. 1.1. The oscillating airfoil sheds a vortical wake which has a certain wavelength. Hence the reduced frequency compares this wavelength with the airfoil chord because during one oscillation a vortex

1.1 Introduction

3

(a)

V

> * y

rv (b)

*J

^

Fig. 1.1. Computed vortical wake due to sinusoidal plunge oscillation (a) k — 0.5, (b) k = 1.0.

shed from the trailing edge travels the distance UOQ/UJ. Therefore the higher the reduced frequency the smaller the wave length, as illustrated in Fig. 1.1. Note that the computed vortical wavelength shed from an airfoil plunging with a reduced frequency k = 1.0 is half that in Fig. 1.1a where the reduced frequency is k = 0.5. At about the same time, H. Wagner in Berlin studied in his doctoral dissertation the problem of an airfoil which is suddenly set in motion with constant velocity in an incompressible inviscid flow [9]. He solved this problem quite elegantly using Betz' method of conformal mapping. The "Wagner function", Fig. 1.2, shows that the lift of a flat plate right after start of the motion is half the steady value, asymptotically reaching this value. In the following years, Kiissner in Gottingen and Theodorsen at NACA Langley succeeded to develop solutions for incompressible inviscid flow past thin oscillating airfoils valid for arbitrary reduced frequencies based on Prandtl's original small perturbation proposal [10, 11]. This oscillatory thin airfoil theory is still of considerable value today. The next big advance in the analysis of inviscid unsteady flows came with the availability of sufficient computing power in the 1960s to solve large systems of linear equations in a reasonable amount of time. Hess and Smith [12] at the Douglas Aircraft Company pioneered the use of so-called panel methods to model inviscid steady incompressible airfoil flows by distributing a finite number of sources and vortices on the airfoil surface, thereby making it possible to account for airfoil geometry effects. Satisfying the flow tangency condition on each panel together with the K u t t a condition of zero pressure difference at the trailing edge yields a system of equations for the unknown source and vortex

1. Physics of Unsteady Flows

4

C-L steady

1.0 r 0.8 h

^

0.4 f0.2 k

0

2

4

6

8

10

12

14

16

18

20

r, distance traveled, in semichords

Fig. 1.2. Wagner's function for an incompressible fluid.

strengths. In the accompanying CD-ROM we present a computer program for calculating airfoil flows with the Hess-Smith panel method (HSPM). The approach of Hess and Smith was extended to the analysis of unsteady inviscid incompressible airfoil flows by Giesing [13], also at the Douglas Aircraft Company. In subsequent years, a number of investigators have built upon this work and have developed versatile computer programs for this class of airfoil flow problems. In Chapter 3 we describe an unsteady panel method for airfoils developed by Teng and Platzer at the Naval Postgraduate School in 1987 [14]. In the 1960s it was also realized that the computation of viscous effects could be greatly improved by the development of finite difference solutions for incompressible boundary layer flows. One of the present authors, Cebeci [15], was among the pioneers of such solutions, first for the direct computation of airfoil boundary layers in response to a given pressure distribution and, later on, by developing viscous-inviscid interaction methods which enable the prediction of separation bubbles, mildly separated flows and the complete lift and drag characteristics of airfoils in incompressible steady flows. In Chapter 5 we discuss this interactive boundary-layer (IBL) method which combines the panel method of Chapter 3 with an inverse boundary-layer method. The accompanying CDROM contains the computer program for this IBL method. Finally, in the 1980s solutions of the Navier-Stokes equations became possible which removed the limitations of viscous-inviscid interaction methods. The rapidly developing computing power over the past twenty years and numerical methods (Chapter 8) made it possible to migrate Navier-Stokes computations from supercomputers to generally available desk-top workstations and apply them to rather complex airfoil flows including dynamic stall (Chapter 9). As a result, the aerodynamicist interested in the analysis of low-speed unsteady airfoil flows can now accomplish this task with three basic methods, depending on the complexity of the flow problem, namely with panel methods,

1.2 Lift Generation

5

Fig. 1.3. Visualization of starting vortex due to sudden angle of attack change [18].

viscous-inviscid interaction methods, and Navier-Stokes methods. Therefore it is instructive to discuss the physical aspects of the most important unsteady airfoil and wing problems before proceeding to the presentation of the computational methods.

1.2 Lift Generation Kutta [16] and Joukowski [17] were the first ones to recognize and derive the fundamental relationship L = QUoor (1.2.1) between the lift L generated by an airfoil in low-speed flow and the circulation r using steady-state potential flow analysis. Unfortunately, the physics of lift generation remains relatively obscure as long as one limits oneself to steady flow considerations only. A full understanding of the underlying physics can be achieved by studying the flow changes which occur in response to a sudden change in the airfoil's incidence angle. Consider a symmetric airfoil, say a NACA 0012 profile, at zero angle of attack. As shown by Prandtl in his water channel flow visualization studies, Fig. 1.3, it is easy to visualize the formation and separation of a counterclockwise vortex at the trailing edge if the airfoil is suddenly moved to a positive angle of attack. This phenomenon can be explained by the flow processes occurring in the upper and lower surface boundary layers of the airfoil. At zero angle of attack the upper boundary layer contains clockwise vorticity, the lower layer contains counterclockwise vorticity. At the trailing edge the two boundary layers merge to form a wake with the typical velocity defect distribution indicative of a net drag. Sudden rotation of the airfoil to a positive angle of attack initiates the formation of additional clockwise and counterclockwise vorticity in the upper and lower

6

1. Physics of Unsteady Flows

Fig. 1.4. Inviscid flow without circulation.

boundary layers, respectively, such that the counterclockwise vorticity of the lower layer predominates for a while and accumulates into a distinct trailing edge vortex until a new equilibrium is established. It is important to note the crucial role of the pointed trailing edge. Vortex formation at the trailing edge and hence lift generation is greatly diminished by rounding the airfoil trailing edge. This brief description shows that the explanation of lift generation is inherently tied to an understanding of vortex generation in a boundary layer and the merging of the two layers at a sharp trailing edge in response to a sudden change in approach flow angle. Obviously, these are unsteady viscous flow processes which are difficult to analyze without resort to the full Navier-Stokes equations. However, it is clear that viscosity prevents the flow to "go around" the sharp trailing edge as is possible for an inviscid flow, shown in Fig. 1.4 where the rear stagnation point is on the upper airfoil surface thus creating a flow without circulation. Instead, the sharp trailing edge is instrumental in generating a so-called starting vortex and a vorticity distribution in the two airfoil boundary layers with a net positive total vorticity (i.e., circulation). It is remarkable that this process can be modelled quite satisfactorily with potential flow tools if the assumption is made that the rear stagnation point coincides with the sharp trailing edge. Invoking the Kelvin-Helmholtz vortex conservation laws for inviscid incompressible flows then makes it possible to develop a reasonably accurate unsteady airfoil theory, as first pioneered by Prandtl [7], Birnbaum [8] and Wagner [9]. Naturally, such a theory can be expected to describe the real viscous flow only if the boundary layers on both airfoil surfaces are quite thin, i.e., if the Reynolds numbers are quite high. Airfoil flow analysis at low Reynolds numbers, say below one million based on airfoil chord, or at high angles of attack near stall at any Reynolds number requires viscous flow analysis tools. Having emphasized the fundamental importance of the Kutta-Joukowski law, Eq. (1.2.1) and the inherent connection between lift and vortex generation, the question arises about the connection between lift and Newton's Second Law. Unsurprisingly, Newton was the first one to propose that the normal force

1.2 Lift Generation

7

F i g . 1.5. Schematic for Newtonian impact theory.

experienced by a flat plate at positive angle of attack (Fig. 1.5) is due to the deflection of the flow impinging on the lower surface, leading to the formula N = gu^S sin2 a

(1.2.2)

where S is the plate area and a is the angle of attack. It is now known that this formula predicts the plate normal force quite well at hypersonic flight speeds. Newton had no knowledge of vortices and of the laws governing their behavior which were established by Helmholtz and Lord Kelvin almost two hundred years later. Looking at the s t e a d y flow field generated by a lifting airfoil in low-speed flow, long after the starting vortex has been carried downstream, it is indeed difficult to detect a relationship between lift and vorticity and, even more so, between lift and momentum change. The answers to these questions were provided by Prandtl who recognized the need to account for lift changes along the span of a finite-span wing. Every such change is accompanied by the generation of a trailing vortex line in the streamwise direction. The trailing vortex sheet captures a certain amount of flow and gives it a downward momentum. The induced downward velocity is related to the wing lift by Prandtl's famous formula w/Uoo

= CL/TTAR

(1.2.3)

L = 2TVS2QU00W

(1-2.4)

which can also be rewritten as

with s denoting the wing semi-span. It shows that the wing gives an air mass flowing through a circle with a radius equal to the semi-span s with velocity UQQ the downward velocity 2w (this being the well known additional result given by Prandtl that the induced velocity at infinite downstream distance is twice the induced velocity at the wing). Hence Prandtl's lifting line theory is in complete

8

1. Physics of Unsteady Flows

agreement with Newton's Second Law that the lift exerted on the wing is the reaction to a momentum change experienced by the air. Prandtl's theory also shows that lift generation is inherently linked to vortex generation. Lift can only be maintained if the airplane continues to generate trailing vortices thus requiring a power plant to overcome the vortex drag needed to generate lift. Prandtl's formula also demonstrates the asymptotic nature of the two-dimensional lift theory. As the aspect ratio is allowed to become infinite, the induced velocity becomes zero giving the impression that no momentum change is required to generate lift. However, this is deceiving because a twodimensional airfoil generates a finite lift by giving an infinite amount of air a zero downward momentum.

1.3 Thrust Generation by Airfoil Oscillation Over the millenia birds and various insects have developed an ability to use their wings as fully integrated lift and propulsion devices. This ability greatly impressed early flight pioneers, such as Lilienthal in the 1890's, but no viable theory existed. A first attempt to explain the generation of thrust by means of wing flapping was first made by Knoller [19] and Betz [20] in 1909 and 1912, respectively. They proposed to use quasi-steady arguments. Consider an airfoil flying with the velocity UOQ. If the airfoil starts to descend with the velocity w it acquires an angle of attack a = w/uoo (1.3.1) for small values of w compared to u^. Consequently, the airfoil generates a lift force which has a small thrust component in the direction of i ^ . Sinusoidal up and down motion (plunge or heave motion) then generates a sinusoidally varying small thrust force, as shown in Fig. 1.6, provided the viscous drag force

Fig. 1.6. Elementary explanation of thrust generation due to a sinusoidally plunging airfoil.

1.3 Thrust Generation by Airfoil Oscillation

9

Fig. 1.7. Vortex street indicative of thrust production (a) Panel code computation and schematic (b) Flow visualization [21].

is sufficiently small. However, as already pointed out, every angle of attack change produces a starting vortex which is being shed from the trailing edge. Sinusoidal plunge motion therefore produces a vortex street consisting of alternating clockwise and counterclockwise vortices. This phenomenon was first analyzed by Birnbaum [8] who derived an analytical expression for the resulting thrust. Using the panel method of Chapter 3 without viscous effects the vortex street is found to have the characteristics of a "reverse" Karman vortex street, such that the upper row has counterclockwise and the lower row clockwise vortices, as shown in Fig. 1.7. Close inspection shows that such a vortex street induces a time-averaged jet-like flow. This is to be expected because the airfoil

10

1. Physics of U n s t e a d y Flows

experiment panel method

1.5

1.0 U

2.C

/U„

F i g . 1.8. C o m p a r i s o n of t i m e averaged velocity profiles [21].

thrust must show up as flow momentum excess downstream of the airfoil. The flapping airfoil therefore acts like a "jet engine". Measurements of the jet-like flow and the inviscid unsteady panel method predictions [21] are in remarkable agreement, as shown in Fig. 1.8.

1.4 Power Extraction Birds usually flap their wings in a more complicated manner, using at minimum a combined pitch and plunge motion. One might therefore conclude that pitch and plunge are a requirement for efficient propulsion. Insight into this question can be obtained from panel method calculations for combined pitch and plunge motions where the pitch and plunge amplitudes of oscillation and the phase angle between these two motions are varied systematically. However, even before resorting to such calculations, the essential physics of the problem can already be deduced from Fig. 1.9, where Figs. 1.9a and 1.9b show the pure plunge and pitch motions. Figs. 1.9d and 1.9e, on the other hand, show two fundamentally different cases of combined pitch and plunge motion. The phase angle between pitch and plunge is 90 degrees in both cases. Note that the airfoil is at zero pitch angle at its maximum up or down position. However, in Fig. 1.9e the pitch amplitude is large enough to generate a lift which is in the same direction as the airfoil motion throughout the complete oscillation cycle. Hence positive work is done by the airflow on the airfoil. This case illustrates the classical two-degreeof-freedom bending-torsion flutter of a conventional high-aspect ratio wing. As

1.4 Power Extraction

T^-

b).

11

- * *

Fig. 1.9. Various plunge/pitch motions.

1.0

180

270

360

phase a n g l e , (> | Fig. 1.10. Thrust, power, propulsive efficiency as a function of phase angle between airfoil pitch and plunge oscillations [22].

12

1. Physics of Unsteady Flows

2.0

T5 0)

1.0

•P

rti U

-H

T5

a

-H

CO

ri

0.0 GarrickPanel Ctx2 0 Cpx2 0 Tl ii

0

90

I

i

i

I

180 270 phase angle, 0

i

i

I

360

Fig. 1.11. Thrust, power, propulsive efficiency as a function of phase angle between airfoil pitch and plunge oscillations [22].

discussed in detail by Theodorsen [11], such a wing can be excited into very dangerous "explosive" flutter for many combinations of bending and torsion frequencies, elastic axis and center of gravity locations of the wing section. Complete details can be found, for example, in [2]. The unsteady panel method calculations shown in Fig. 1.10 are for the case shown in Fig. 1.9d. Plotted are the propulsive efficiency 77 and the thrust ct and power cp coefficients based on Garrick's calculations [23], using Theodorsen's flat-plate theory, and the unsteady panel method computations. It is readily seen that the maximum propulsive efficiency occurs at a phase angle of 90 degrees. In contrast, Fig. 1.11 depicts the calculations for the case shown in Fig. 1.9e. The thrust and power coefficients are seen to be negative at phase angles near 90 degrees and the "propulsive efficiency" assumes values greater than one, indicating the extraction of power. The question therefore occurs whether power extraction can even occur if the airfoil has only one degree of freedom, i.e., either the plunge or pitch degree of freedom shown in Figs. 1.9a and 1.9b. The answer to this question is given in the next section.

1.5 Single-Degree-of-Freedom Airfoil Flutter

13

1.5 Single-Degree-of-Freedom Airfoil F l u t t e r It is easy to see that an airfoil which executes a sinusoidal plunge motion generates a sinusoidally varying lift force which always opposes the airfoil motion. Therefore, the plunge oscillation of an elastically supported airfoil will always damp out. However, this conclusion does not hold if we consider the airfoil shown in Fig. 1.12 which is elastically supported by a torsion spring and thus can execute a pure pitch oscillation about the airfoil leading edge. Again, it is important to recall that the airfoil is shedding starting vortices as it is excited into a pitch oscillation. The effect of these vortices on the pressure distribution is illustrated in Fig. 1.13. The upper and lower surface pressure distributions are shown as the symmetric NACA 0012 airfoil rotates counterclockwise through the mean position. If the airfoil were held steady at zero angle of attack, then the two pressure distribution would coincide. The difference in pressure distributions therefore reflects the lag effect induced by the pitch oscillation and the

Fig. 1.12. Rigid airfoil mounted on torsion spring in two-dimensional flow. -0.5

-o lower -o upper -Slower -• upper

0.5

surface,k=0.04 surface, k=0.04 surface, k=0.4 surface, k=0.4

1 ^ 0.2

0.4

0.6

0.8

x/c Fig. 1.13. Pressure distributions on a pitching airfoil at two different frequencies.

1. Physics of Unsteady Flows

14

vortices shed from the trailing edge. Note the important differences in induced pressures at the two reduced frequencies k — 0.04 and k = 0.4. For the lower reduced frequency oscillation the lower surface is at a higher pressure than the upper surface and therefore the pressure difference distribution between lower and upper surface produces a moment in the direction of the motion, hence generating a moment which reinforces the motion during each pitching cycle. The opposite effect is seen at the higher reduced frequency. Here the upper surface is at a higher pressure than the lower surface (except near the leading edge) and thus a moment is generated which opposes the pitching oscillations. The same effect occurs, of course, when the airfoil rotates clockwise through the mean position. The upper and lower surface pressure distributions will be reversed, generating again a reinforcing moment at the low frequency and an opposing moment at the high frequency. Hence, at low frequencies the pitch oscillation generates a negative aerodynamic damping moment which, although quite small, is sufficient to induce a dynamically unstable (flutter) motion. It is important to understand that this instability is caused by the time lag effect produced by the vortex shedding. Obviously, if the airfoil were held at a steady zero angle of attack, no pitching moment would be generated. If the airfoil oscillates relatively slowly, the wavelength of the shed vortex wake is quite large and therefore only vortices of the same sign close to the trailing edge have a sufficiently strong effect to induce the pressures shown in Fig. 1.13 for the lower frequency case. An increase of the frequency of oscillation changes the wavelength so that both clockwise and counterclockwise vortices come into play. They induce the pressures shown in Fig. 1.13 for the higher frequency case

k = 0.1 (clockwise) k = 0.154 k = 0.2 (counterclockwise)

0.2 0.1

—

^Sx. \ V v < %V?

Cm 0.0 -0.1 -0.2 -ft 3

-10.0

:

l _

i

1

-5.0

0.0

•

!

5.0

i

1

10.

a Fig. 1.14. Effect of reduced frequency on the computed pitching moment hysteresis [24].

1.6 Airfoil Gust Response

15

which add up to a positive pitch damping moment. Accurate prediction of the vortex shedding therefore is essential to predict potentially disastrous flutter phenomena. It is illustrative to display the pitching moment as a function of angle of attack over the complete cycle. Figure 1.14 shows the two cases. It is seen that for the lower frequency case, the pitching moment loop is clockwise, for the higher frequency case it is counterclockwise. Since the work per cycle of pitching motion is given by the cyclic integral of the product of the pitching moment and differential pitch angle the enclosed area represents the work done by the air on the airfoil or vice versa. A counterclockwise enclosure of the area represents negative work, which is equivalent to positive damping and hence a stable motion. Conversely, a clockwise enclosure represents positive work, negative damping, and hence flutter.

1.6 Airfoil Gust Response Consider the case of an airfoil entering the step gust shown in Fig. 1.15. For simplicity, the airfoil is assumed to have only the plunge degree of freedom. Adopting quasi-steady reasoning it is tempting to assume that the airfoil instantaneously acquires the full lift force corresponding to the new angle of attack wo/uoo where WQ is the gust velocity. The normal acceleration and stress experienced by the airfoil using this type of analysis, however, is highly conservative, thus leading to a wing design which is heavier than needed to withstand expected wind gusts during normal flight operations. As first shown by Wagner [9], the lift response to a sudden change in angle of attack occurs as shown in Fig. 1.2. The physical reason for this gradual lift build-up is the shedding of the starting vortex from the trailing edge which prevents the attainment of the steady-state lift as long as the starting vortex is still relatively close to the airfoil. Kuessner [25] analyzed the case of a flat-plate airfoil which penetrates the step gust shown in Fig. 1.15 and found the lift build-up shown in Fig. 1.16. The Wagner and Kuessner functions shown in Fig. 1.16 are the two fundamental lift response functions needed to perform a more realistic gust response analysis. Both functions were derived for flat-plate airfoils. Calculations with the unsteady panel method of Chapter 3 make it possible to account for airfoil geometry effects.

1

I

I

t

/

[

w0

Sharp-edged gust

Fig. 1.15. Airfoil penetrating a sharp-edged gust.

16

1. Physics of Unsteady Flows

Fig. 1.16. Wagner's function 0(s) for indicial lift and Kussner's function ip(s) for lift due to a sharpedged gust, s denotes distance traveled in semichord lengths.

1.7 Dynamic Airfoil Stall A full understanding of s t a t i c airfoil stall can only be achieved by studying the changes in the boundary layer on the airfoil's suction surface. The onset of flow separation usually occurs either near the leading or trailing edge. Leading edge stall typically is preceded by the formation of a separation bubble which starts to burst as soon as a critical incidence angle is exceeded. Trailing edge stall progresses from the trailing edge toward the leading edge. The details of the onset of s t a t i c stall are strongly dependent on Reynolds number and airfoil geometry. Good progress has been achieved in predicting the onset of s t a t i c stall and the initiation of dynamic stall by the use of IBL theory as discussed in Chapter 7. A complete calculation of airfoil dynamic stall can be achieved with reasonable accuracy as discussed in Chapter 9. A radically different stall behavior is observed if the airfoil is subjected to rapid changes in incidence angle. Kramer [26] was the first one to note that the airfoil generates substantially larger lift for a short period of time than can be obtained quasi-statically. The reasons for this lift overshoot phenomenon can be understood by examining the sequence of events depicted in Fig. 1.17. The rapid rotation of the airfoil to a higher incidence angle, at first, is not accompanied by an immediate pressure change on the upper surface corresponding to the new incidence angle because of the lag time between airfoil motion and pressure response. As a consequence, for a short while the airfoil "sees" a lower angle of attack than the geometric angle of attack and therefore boundary layer separation is delayed. This stage is followed by the formation of a vortex near the leading edge which grows and moves downstream over the upper airfoil surface. During this phase of the dynamic stall process additional lift is produced which may amount to between 50% to 100% of the maximum static lift value. Due to this movement of the dynamic stall vortex the center of pressure moves

17

1.7 Dynamic Airfoil Stall

(a) STATIC STALL ANGLE EXCEEDEO (b) FIRST APPEARANCE OF FLOW REVERSAL ON SURFACE

(c) LARGE EDOIES APPEAR IN BOUNDARY LAYER

(h)

/I

,

M 7/ *'\\7 (b) v \ y

z

(J UJ

a a O

(a)

LL

-J

<

\,

(1)

oc o

^

^

X

(dl

(e) VORTEX fORMS NEAR LEADING EDGE /

TOi01

/

z

C

1

L.

i

(d) FLOW REVERSAL SPREADS OVER MUCH OF AIRFOIL CHORD

(f) LIFT SLOPE INCREASES

>

ig) MOMENT STALL OCCURS

(h) LIFT STALL BEGINS (i) MAXIMUM NEGATIVE MOMENT (i) FULL STALL

(k) BOUNDARY LAYER REATTACHES FRONT TO REAR 10 15 20 INCIDENCE, a, deg

25

(I) RETURN TO UNSTALLED VALUES

F i g . 1.17. Dynamic stall events on a NACA-0012 airfoil at low free-stream Mach number [27].

downstream causing a significant increase in nose-down pitching moment. This is the start of the moment stall which is followed by a rapid loss of lift (lift stall) as soon as the dynamic stall vortex passes the trailing edge. The final stage then is complete flow separation, followed by flow reattachment as soon as the angle of attack is reduced below the static stall angle. As reviewed by Carr [27] and

1. Physics of Unsteady Flows

18

Ekaterinaris and Platzer [24], the precise details of this dynamic stall sequence strongly depend on the airfoil geometry and the Reynolds and Mach number to which the airfoil is exposed. This is especially true for the flow reattachment process which still defies successful prediction. However, as discussed in Chapter 9, considerable progress has been achieved in recent years using Navier-Stokes methods.

1.8 Stall Flutter In Section 1.5 we explained the mechanism leading to single-degree-of-freedom torsional flutter of an airfoil in attached incompressible flow. The flow mechanism inducing this type of flutter was found in the vortex shedding from the airfoil's trailing edge. However, the above described dynamic stall phenomenon may also lead to single degree of freedom flutter and may occur on helicopter and turbomachinery blades. Consider Fig. 1.18, taken from reference [28], which depicts the effect of frequency and mean incidence angle on the pitching moment

a-6 M-0.2

*M-°°

aM

-15°

0.10

™EXPERIMENT 0

k

f-4 0.224

' \ : THEORY -0.01

-0.20 0.10

5 « |

£^

f-8 k - 0.45

0 -o.io i.

-0.10

f-16 k = 0.90

^

l ^*
-0.10 -

8

-

4

0

4

8

8

12

16

20

24

a, (degrees) Fig. 1.18. Effect of frequency and mean incidence angle on moment hysteresis loops [28].

1.9 Summary

19

CM hysteresis loop of an airfoil which is pitching about zero mean incidence ( % = 0) on the left and about a mean incidence angle ( % = 15) on the right. In both cases, the amplitude of oscillation, a, is 6°. The moment hysteresis loops on the left are similar to those shown in Fig. 1.14, as one would expect in attached flow. All the loops are for sufficiently high reduced frequency so that they are counterclockwise and hence stabilizing. On the other hand, at a mean incidence of 15 degrees, the two lowest frequency loops show a figure eight type behavior. As already explained, system stability is related to work per cycle of motion. For a reduced frequency of 0.45 the area enclosed by the clockwise loop clearly exceeds the counterclockwise loop area, indicating an energy balance where work is done by the air on the airfoil. The loops shown in Fig. 1.18 were obtained by forcing the airfoil into the pitch motion. However, if the airfoil had been free to oscillate, the pitch amplitude would have increased to a so-called limit cycle where the separated flow aerodynamics had changed to the point of achieving a true energy balance.

1.9 Summary This brief overview of some of the most important unsteady airfoil flow problems shows the need for prediction methods which range from purely inviscid flow methods suitable for the prediction of attached high Reynolds number flows to viscous-inviscid interaction methods suitable for mildly separated flows and fully viscous (Navier-Stokes) methods suitable for separated flow problems and low Reynolds number flows. The book therefore is organized to treat these three classes of flow problems and their prediction methods.

References [1] Dowell, E. H. (editor), Curtiss, H.C., Scanlan, R. H. and Sisto, F.: "A Modern Course in Aeroelasticity," Sijthoff & Noordhoff, 1978. [2] Bisplinghoff, R. L., Ashley, H. and Halfman, R. L.: "Aeroelasticity," Addison-Wesley, 1955. [3] Anderson, J.D.: "Fundamentals of Aerodynamics," 3rd edn., McGraw-Hill, 2001. [4] Bertin, J. J.: "Aerodynamics for Engineers," 4th edn., Prentice Hall, 2001. [5] Kuethe, A.M. and Chow, C.Y.: "Foundations of Aerodynamics," 3rd edn., Wiley, 1976. [6] Anderson, D . F . and Eberhardt, S.: "Understanding Flight," McGraw-Hill, 2001. [7] Prandtl, L.: "Uber die Entstehung von Wirbeln in der idealen Fliissigkeit, mit Anwendung auf die Tragfliigeltheorie und andere Aufgaben," Hydro- und Aerodynamik, Berlin, Julius Springer Verlag, pp. 18-33, 1924. [8] Birnbaum, W.: "Das ebene Problem des schlagenden Fliigels," Zeitschrift fur angewandte Mathematik und Mechanik (ZAMM), Vol. 4, pp. 277-292, 1924.

20

1. Physics of Unsteady Flows

[9] Wagner, H.: "Uber die Entstehung des dynamischen Auftriebs an Tragfliigeln," Zeitschrift fur angewandte Mathematik und Mechanik (ZAMM), Vol. 5, pp. 17-35, 1925. Kiissner, H.G.: "Zusammenfassender Bericht iiber den instationaren Auftrieb von Fliigeln," Luftfahrt-Forschung, Vol. 13, pp. 410-424, 1936. Theodorsen, T.: "General Theory of Aerodynamic Instability and the Mechanism of Flutter," NACA Report 496, 1935. Hess, J. L. and Smith, A. M. O.: "Calculation of Potential Flow about Arbitrary Bodies," Progress in Aerospace Sciences, Vol. 8, Pergamon Press, 1966. Giesing, J. P.: "Nonlinear Two-dimensional Unsteady Potential Flow with Lift," J. of Aircraft, pp. 135-143, March-April 1968. Teng, N.H.: "The Development of a Computer Code (U2DIIF) for the Numerical Solution of Unsteady, Inviscid and Incompressible Flow over an Airfoil," M.S. Thesis, Naval Postgraduate School, Monterey, CA, 1987. Cebeci, T.: An Engineering Approach to the Calculation of Aerodynamic Flows. Horizons Pub., Long Beach, Calif, and Springer, Heidelberg, 1999. Kutta, M.W.: "Auftriebskrafte in stromenden Fliissigkeiten," Illustrierte Aeronautische Mitteilungen, Vol. 6, pp. 133-135, 1902. Joukowski, N.: "On the Adjunct Vortices," Obshchestvo liubitelei estestvoznaniia, antropolgii I etnografii, Moskva, Izvestiia, 112, Transactions of the Physical Section, 13, pp. 12-25, 1907. Prandtl, L. and Tietjens, O.G.: "Applied Hydro- and Aeromechanics," McGraw-Hill Book Company, New York and London, 1934. Knoller, R.: "Die Gesetze des Luftwiderstandes," Flug- und Motortechnik (Wien), Vol. 3, No. 21, pp. 1-7, 1909. Betz, A.: "Ein Beitrag zur Erklarung des Segelfluges," Zeitschrift fur Flugtechnik und Motorluftschiffahrt, Vol. 3, pp. 269-272, 1912. Jones, K.D., Dohring, C M . and Platzer, M.F.: "Experimental and Computational Investigation of the Knoller-Betz Effect," AIAA Journal, Vol. 36, No. 7, pp. 1240-1246, May 1998. Jones, K. D., Lund, T. C. and Platzer, M. F.: "Experimental and Computational Investigation of Flapping Wing Propulsion for Micro Air Vehicles," Chapter 16 in "Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications," Vol. 195, Progress in Astronautics and Aeronautics, American Institute of Aeronautics and Astronautics, 2001. Garrick, I.E.: "Propulsion of a Flapping and Oscillating Airfoil," NACA Report 567, 1936. Ekaterinaris, J. A. and Platzer, M.F.: "Computational Prediction of Airfoil Dynamic Stall," Progress in Aerospace Sciences, Vol. 33, pp. 759-846, 1997. Kiissner, H. G.: "Untersuchung der Bewegung einer Platte beim Eintritt in eine Strahlgrenze," Luftfahrtforschung, Vol. 13, p. 435, 1936. Kramer, M.: "Die Zunahme des Maximalauftriebes von Tragfliigeln bei plotzlicher AnstellwinkelvergroBerung," Zeitschrift fur Flugtechnik und Motorluftschiffahrt, Vol. 23, pp. 185-189, 1932. Carr, L. W.: "Progress in Analysis and Prediction of Dynamic Stall," J. Aircraft, Vol. 25, pp. 6-17, 1988. Carta, F.O.: "Aeroelasticity and Unsteady Aerodynamics," Chapter 22, The Aerothermodynamics of Aircraft Gas Turbine Engines, AFAPL-TR-78-52, 1978.

The Differential Equations of Fluid Flow

2.1 Introduction The differential equations of fluid flow are based on the principles of conservation of mass, momentum and energy and are known as the Navier-Stokes equations. For incompressible flows and for flows in which the temperature differences between the surface and freestream are small, the fluid properties such as density Q and dynamic viscosity \i in the conservation equations are not affected by temperature. This assumption allows us to ignore the conservation equation for energy and concentrate only on the conservation equations for mass and momentum. In this chapter we first discuss the conservation equations for mass and momentum (Section 2.2), and we shall refer to them as the Navier-Stokes equations. Since most flows are turbulent with fluctuations of velocity over a range of frequencies, the solution of the Navier-Stokes equations of Section 2.2 presents a formidable challenge and is unlikely to be achieved for the boundary conditions of real engineering flows in the foreseeable future. For this reason, it is common practice to average the equations. The resulting equations, usually called Reynolds-Averaged Navier-Stokes (RANS) equations, include correlation of fluctuation terms, as discussed briefly in Section 2.3 and these require turbulence models discussed briefly in Chapter 5 and in detail in [1,2]. Depending on the flow conditions, it is appropriate and sometimes necessary to use the reduced forms of the Navier-Stokes equations as discussed in Section 2.4. These simplified equations, especially for inviscid flows (subsection 2.4.2) and for boundary-layer flows (subsection 2.4.3), reduce the complexity of solving the Navier-Stokes equations, provide substantial savings of computer time and in some situations permit accurate analytical and numerical solutions to the conservation equations.

2. The Differential Equations of Fluid Flow

22

2.2 Navier-Stokes Equations The Navier-Stokes equations may be obtained by using infinitesimal or finite control volume approaches, and the governing equations can be expressed in differential or integral forms. The derivation of these equations can be found in various textbooks [3-5]. Here we briefly discuss them in differential form obtained by using an infinitesimal control volume moving along a streamline with a velocity vector V(u,v,w) equal to the flow velocity at each point. For a three-dimensional incompressible flow, they are given by the following continuity and three momentum equations: Continuity equation du dx

dv dy

dw _ dz

x-component of the momentum equation Du

dp

+t fdaxx

[-lt

xy +t daL + daxz+

^

<2 2 2)

^fl ^

'-

^/-component of the momentum equation Dv

dp

t

(dcFyX

^ = -^H^

dayy

±+

+

dayz

l>f ^ »

+

rt

(2 2 3)

-'

z-component of the momentum equation Dw

dp

+t

(dozx

+t

dazy

°-m=-t Vw ^ where D/Dt

i +

daz

l£)

+

°I'

(2 2 4

-'>

represents the substantial derivative given by

Equations (2.2.2) to (2.2.4) make use of Newton's second law of motion with their left-hand sides representing mass acceleration per unit volume and their right-hand sides representing the sum of net forces per unit volume acting on the fluid which consists of surface and body forces. Typical body forces are gravity forces or electrical forces. Surface forces arise because of molecular stresses in the fluid (such as pressure, p, which is present in a fluid at rest and acts normal to a surface) and viscous stresses which act normal to a surface or tangentially (shear stress). The first term on the right-hand side of Eqs. (2.2.2)-(2.2.4) denotes the net pressure force per unit volume and the minus sign arises because, by definition, a positive pressure acts inward. The second, third and fourth terms denote the viscous forces per unit volume, and they arise as a result of the different components of normal and shear stresses. The first subscript to the symbol a represents the direction of the stress and the second

2.2 Navier-Stokes Equations

23

the direction of the surface normal. By convention, an outward normal stress acting on the fluid in the control volume is positive, and the shear stresses are taken as positive on the faces furthest from the origin of the coordinates. Thus axy acts in the positive x direction on the visible (upper) face perpendicular to the y axis; a corresponding shear stress acts in the negative x direction on the invisible lower face perpendicular to the y axis. Sometimes it is more convenient to write the viscous terms in the momentum equations in tensor notation as

with i, j — 1, 2,3 for three-dimensional flows; for example, i = 1, j = 1, 2, 3 for Eq. (2.2.2). For a constant density "Newtonian" viscous fluid, the normal viscous stresses aij (i — j) and shear stresses <JIJ (i ^ j) are obtained from the viscous stress tensor given by

+

(227)

^=H^ ^)

--

Sometimes Eq. (2.2.7) is written as

where Sij defined by s

= %J

1 fdui_ 2 \dxj

duj dxi

is called the rate of strain tensor. According to Eq. (2.2.7), the normal viscous stress axx and the shear stresses axy and oxz in Eq. (2.2.2) are given by _ du "** = 2 / ^ ,

ffxj,

=

M

(du dv\ ^ _ + _ j ,

(du dw\ ^ - + —j

,n n . (2.2.8)

with similar expressions for the viscous stress tensor terms in Eqs. (2.2.3) and (2.2.4). In terms of Eq. (2.2.7), the Navier-Stokes equations can be simplified considerably so that, for example, the x-momentum equation, (2.2.2) for a Newtonian fluid becomes ^ = - ^ + ^ u + fx (2.2.9) Dt

Q OX

with similar expressions for the y- and z-components obtained from Eqs. (2.2.3) and (2.2.4). The resulting equations can be written in vector form as ^ = --Vp + vV2V Dt g with V 2 denoting the Laplacian operator

„2

d2

d2

dx

2

+f

d2

dy

1

dz2

(2.2.10)

2. The Differential Equations of Fluid Flow

24

2.3 Reynolds-Averaged Navier-Stokes Equations The Navier-Stokes equations of the previous section also apply to turbulent flows if the values of fluid properties and dependent variables are replaced by their instantaneous values. A direct approach to solving the equations for turbulent flows is to solve them for specific boundary conditions and initial values that include time-dependent quantities. Mean values are needed in most practical cases, so an ensemble of solutions of time-dependent equations is required. Even for the most restricted cases, this approach, referred to as direct numerical simulation (DNS) and discussed in [6], becomes a difficult and extremely expensive computing problem because the unsteady eddy motions of turbulence appear over a wide range of scales. The usual procedure is to average the equations rather than their solutions, as discussed in detail in [3, 5, 6] and briefly below. In order to obtain the conservation equations for turbulent flows, we replace the instantaneous quantities in the equations by the sum of their mean and fluctuating parts. For example, the instantaneous values of the u, v and w velocities and pressure p are expressed by the sum of their mean iZ, v, iD, p and fluctuating parts v!', v' and w1', p' u = u + u',

v = v + ?/,

w = w + wf,

p =p+p

(2.3.1)

where, for example, u is the ensemble average of u defined by u =

1 N lim — V^ ui

with Ui denoting the sample and TV the number of samples. With the help of the continuity equation, (2.2.1), one can now write the lefthand sides of the momentum equations in conservation form and introduce the above relations into the continuity and momentum equations. After averaging and making use of the substantial derivatives given by Eq. (2.2.5), the Reynolds averaged Navier-Stokes (RANS) equations for three-dimensional incompressible flow can be written in the following form [4]: du dx

dv dy

dw dz

•TZ = "I! + "V2,B + "• - "I
2.4 Reduced Forms of the Navier-Stokes Equations

25

It is common to drop the overbars on u, v, w andp; this results in a continuity equation identical to that given by Eq. (2.2.1), and the left-hand sides of the momentum equations, Eqs. (2.3.3) to (2.3.5), become identical to the equations for laminar flow. The right-hand sides of the momentum equations also resemble the right-hand sides of Eqs. (2.2.2) to (2.2.4) with the addition of the Reynolds normal and shear stress terms; in our previous notation the Reynolds stresses in Eq. (2.3.3) represent the turbulent contributions to ^ &XX-) ®xy a n d (Jxzi

respectively. The mean viscous contributions are still given by Eq. (2.2.7) and are based on the mean-velocity components. Equations (2.2.2) to (2.2.4) thus apply to both laminar and turbulent flows, provided that the so-called "stress tensor", cr^-, including the viscous contributions, is written as

or a

ij = aij + aij (2.3.7) where now G\- denotes the Reynolds stresses so that for three-dimensional flows Glxx = -QU'2,

axy

= alyx

= -QU'V',

Gxz = G\x = -QU'W',

Glyy = -Qv'2,

Glyz =

Gzy

= —QV'W', GZZ = —QW'2, and G\- is the viscous stress tensor as given by Eq. (2.2.7) for a Newtonian fluid. The Reynolds stress terms in the momentum equations introduce additional unknowns into the conservation equations. To proceed further, additional equations for these unknown quantities, or assumptions regarding the relationship between the unknown quantities to the time-mean flow variables, are needed. This is referred to as the "closure" problem in turbulent flows: we shall discuss turbulence modeling briefly in Sections 5.1 and 8.3. For a detailed discussion, the readers is referred to [1,2].

2.4 R e d u c e d F o r m s of t h e N a v i e r - S t o k e s E q u a t i o n s The conservation equations can be reduced to simpler forms by examining the relative magnitudes of the terms in the equations. In the application of this procedure, known as "order-of-magnitude" analysis, to two-dimensional steady flows discussed in this section, it is common to introduce two length scales L and 6 (which are, respectively, parallel and normal to the wall) to assume a typical velocity to be of order ue, and to estimate the relative magnitudes of inertia, pressure, and viscous and body force terms in the Navier-Stokes equations. 2.4.1 Parabolized a n d Thin-Layer N a v i e r - S t o k e s Equations Using order of magnitude arguments, the conservation equations of the previous section can be simplified by neglecting some of the viscous terms. For example, in some three-dimensional flows the viscous terms dGxx/dx1 dGyx/dx and

2. The Differential Equations of Fluid Flow

26

dozxjdx in Eqs. (2.2.2) to (2.2.4) are omitted, and the resulting form of the Navier-Stokes equations, referred to as "parabolized Navier-Stokes equations," are solved together with the continuity equation. In other flows the NavierStokes equations are simplified further by retaining only the viscous terms with derivatives in the coordinate direction normal to the body surface y or, for free shear flows, the direction normal to the thin layer. This is referred to as the thin-layer Navier-Stokes approximation and leads to the following equations for three-dimensional unsteady flows with the continuity equation remaining unaltered: Du dp d2u d —r-7 „ /„ , , x

*m=-irx+^-%uv+gfx Dv

dp

d2v

+

v

d —^

Dw

dp

+fi

d2w

.

+gfy

em=-i ^-% e

(2A1)

d —,—r

1/v/+efz

i*=-£ w-%

A

^

(2A2) „

,„ , ^

(2A3)

Blottner [7] provides a good review of the significance of these equations which, along with additional assumptions, are used in the parabolized Navier-Stokes solution procedure. Note that these are not the boundary-layer equations (subsection 2.4.3): we do not neglect dp/dy, for instance. 2.4.2 Inviscid Flow Equations One simplification of the Navier-Stokes equations assumes all cr-stresses to be negligible, which corresponds to inviscid flow. With the neglect of viscous forces, Eq. (2.2.10) becomes DV 1 __Vp + / (2.4.4) Dt g which is known as the Euler equation. For a steady flow with no body forces, the Euler equation reduces to (V-VW

= -—

(2.4.5) Q

If we take a dot product of the above equation with a differential length of a streamline ds, assume g constant (incompressible flow), and integrate the Euler equations along a streamline, we get P+

7>QV2

= constant

(2.4.6)

where V2 = u2 + v2 + w2. This equation is the well known Bernoulli equation. Additional simplifications arise if the flow is assumed irrotational, which is defined by zero vorticity u = V x V = 0 (2.4.7)

2.4 Reduced Forms of the Navier-Stokes Equations

27

This condition implies the existence of a scalar function , called the velocity potential, defined by V = V0 (2.4.8) In this case the continuity equation, Eq. (2.2.1), can be combined with Eq. (2.4.8) to give Laplace's equation V20 = 0

(2.4.9)

which provides a good approximation to some real incompressible flows at high Reynolds numbers where the viscous effects are negligible, as is sometimes the case when there is no flow separation on the body, as will be discussed for twodimensional unsteady flows in Chapters 3 and 4. For some problems, discussed in Chapter 5, viscous effects can be introduced into the solution of Eq. (2.4.9). This approach, sometimes referred to as the interactive boundary-layer theory, can be used to solve many engineering problems efficiently, and accurately as discussed in detail in [8] and in Chapter 7 for two-dimensional unsteady flows. 2.4.3 Boundary-Layer Equations Another simplification of the Navier-Stokes equations occurs when the ratio of boundary-layer thickness 6 to a reference length L, 6/L, is sufficiently small because terms that are smaller than the main terms by a factor of 6/L can be neglected. For example, for two-dimensional steady flows, it is assumed that u ~ ixe,

p ~ gue, vP ^^/2

x ~ L,

y ~ 6

^^v

(2.4.10a) (2.4.10b)

These assumptions lead to v ~ ^y-

(2.4.10c)

and the pressure variation across the boundary-layer to be of 0(6) so that in comparison with the streamwise variation, dp/dy is small and can be neglected. This is referred to as the boundary-layer approximations. Thus for two-dimensional steady flows, the Navier-Stokes equations reduce to

du dx

du dy

1 dp g dx

Jr=0

d2u dy2

d (—j—j^ dy

(2.4.13)

dy for laminar and turbulent flows with the Reynolds shear stress term —gufv' equal to zero for laminar flows.

2. The Differential Equations of Fluid Flow

28

A comparison of Eqs. (2.4.12) and (2.4.13) with the x- and y-momentum Navier-Stokes equations for two-dimensional flows shows that with the boundarylayer approximations we have reduced some of the viscous terms in the xmomentum equation and neglected the variation of p with y. Note that, unlike the Navier-Stokes equations, p is no longer an unknown but has been absorbed into the boundary conditions by equating dp/dx to the value in the freestream where Bernoulli's equation applies. dp

due

.

A H jX

,

A

Equation (2.4.12) now becomes du u

du +v

d2u

due u

+ v

irx lfy= ^

d .—1—i. {uv)

W-dy

.

(2A15)

For two dimensional unsteady laminar and turbulent flows, Eq. (2.4.11) remains the same but the momentum equation, Eq. (2.4.15), becomes du du du due due d2u d u u + ^Z + ^ = "ST + e~ZZ + v^z e 2 - ^ ( « V ) dt dx dy dt dx dy dy"

(2.4.16)

The boundary-layer equations are parabolic partial differential equations and are much easier to solve and less costly than the Reynolds averaged NavierStokes equations which are elliptic. These equations are also associated with the inviscid flow equations discussed in the previous subsection. The approximations used to obtain these, however, are not valid in the some region of the flow. The boundary-layer equations apply close to the surface of a body and in wakes which form behind the body. The inviscid flow equations apply outside the boundary-layer. For external flows, there are three boundary conditions for the velocity field that must be specified, two at the wall and the other at the boundary-layer edge y — 6. The conditions at the wall involve the specification of normal (v) and tangential (u) components of velocity and at the edge the specification of the external velocity which is a function of x for steady flows and x,t for unsteady flows, that is, 2/= 0, y = S,

u = 0,

v = vw(x)

u = ue(x, i)

(2.4.17a) (2.4.17b)

Here 6 is sufficiently large so that dimensionless du/dy at the boundary-layer edge is small, say around 1 0 - 4 . The transpiration velocity, vw(x), may be either suction or injection. On a nonporous surface it is equal to zero. The boundary-layer equations also require initial conditions in the (x, y)plane for steady flows. For time-dependent flows, they also require initial conditions in the (£, 7/)-plane in addition to those in the (x, y)-plane as discussed in subsection 5.5.1.

References

29

References [1] Wilcox, D.C., Turbulence Modeling for CFD, DCW Industries, Inc. 5354 Palm Av., La Canada, Ca. 1998. [2] Cebeci, T., Analysis of Turbulent Flows, Elsevier, London, 2004. [3] Anderson, J., Fundamentals of Aerodynamics, McGraw-Hill, 1991. [4] Cebeci, T. and Cousteix, J., Modeling and Computation of Boundary-Layer Flows, Horizons, Long Beach, CA, and Springer, Heidelberg, 1998. [5] Cebeci, T., Convective Heat Transfer, Horizons, Long Beach, CA, 2002. [6] Rai, M. M. and Moin, P., Direct Simulations of Turbulent Flow Using Finite-Difference Schemes, J. Comp. Phys., 96, p. 15, 1991. [7] Blottner, F.G., Significance of the Thin-Layer Navier-Stokes Approximation. In: Numerical and Physical Aspects of Aerodynamic Flows III, p. 184 (ed. T. Cebeci), Springer-Verlag, NY, 1986. [8] Cebeci, T., An Engineering Approach to the Calculation of Aerodynamic Flows, Horizons, Long Beach, CA and Springer, Heidelberg, 1999.

""•'"Bfc.

» Panel Methods

3.1 I n t r o d u c t i o n Panel methods are ideal for calculating the flowfield over an airfoil executing unsteady time-dependent motion in an inviscid incompressible medium. The unsteady motion on the airfoil causes continuous vortex shedding at the trailing wake, which distinguishes unsteady flows from steady flows. Therefore, before we discuss panel methods for unsteady flows, it is useful to discuss panel methods for steady flows. In Section 3.2 we discuss the Hess and Smith panel method (HSPM) for steady flows [1], and in Section 3.3 we discuss its extension to airfoils subject to ramp-type or harmonic oscillation motions. In the accompanying CD-ROM, we present a computer program for HSPM. This unsteady flow method was first developed by Teng [2] and was later modified by Jang [3] and Jang and Cebeci [4] for viscous effects so that, as in the calculation method for steady flows, the modified unsteady panel method can be used interactively with the boundary-layer method of Chapter 5 as described in Chapter 7.

3.2 H S P M We consider an airfoil at rest in an onset flow of velocity V ^ . We assume that the airfoil is at an angle of attack, a (the angle between its chord line and the onset velocity), and that the upper and lower surfaces are given by functions Yu(x) and Y/(#), respectively. These functions can be defined analytically, or (as often is the case) by a set of (x, y) values of the airfoil coordinates. We denote the distance of any field point (x,y) measured from an arbitrary point, b, on the airfoil surface by r, as shown in Fig. 3.1. Let ft also denote the unit vector

32

3. Panel Methods

source and vorticity distributions

yr Y uOO

control points

y - Y«(x)

boundary

Fig. 3.1. Panel representation of airfoil surface and notation for an airfoil at incidence a.

normal to the airfoil surface and directed from the body into the fluid and ?, a unit vector tangential to the surface, and assume that its inclination to the x-axis is given by 9. It follows from Fig. 3.1 that with i and j denoting unit vectors in the x- and ^/-directions, respectively, n = — sin 9 i + cos 9 j

(3.2.1)

t = cos 9 i -h sin 9 j If the airfoil contour is divided into a large number of small segments, ds, then we can write dx = cos 9 ds (3.2.2) dy = sin 9 ds We next assume that the airfoil geometry is represented by a finite number (TV) of short straight-line elements called panels, denned by (N -f l)(xj,yj) pairs called boundary points. It is customary to input the (x, y) coordinates starting at the lower surface trailing edge, proceeding clockwise around the airfoil, and ending back at the upper surface trailing edge. If we denote the boundary points by (xi, yi), (x 2 ,2/2), • • •, (x/v, 2/jv), (xN+i,yN+i) (3.2.3) then the pairs (xi, y\) and (x^v+i, VN+I) a r e identical and represent the trailing edge. It is customary to refer to the element between (xj,yj) and ( X J + I , T / J + I ) as the j - t h panel, and to the midpoints of the panels as the control points. Note from Fig. 3.1 that as one traverses from the z-th boundary point to the (i + 1)th boundary point, the airfoil body is on the right-hand side. This numbering sequence is consistent with the common definition of the unit normal vector rti and unit tangential vector t{ for all panel surfaces, i.e., Hi is directed from the body into the fluid and U from the z-th boundary point to the (i + l)-th boundary point with its inclination to the x-axis given by 9{.

3.2 HSPM

33

In the HS panel method, the velocity V at any point (x, y) is represented by V = U+v

(3.2.4)

where U is the velocity of the uniform flow at infinity U = Voo(cosai + sin a j )

(3.2.5)

and v is the disturbance field due to the body which is represented by two elementary flows corresponding to source and vortex flows. A source or vortex on the j - t h panel causes an induced source velocity vs at (x, y) or an induced vortex velocity vv at (x, y), respectively, which are obtained by taking gradients of potential source
(3.2.7)

both centered at the origin, so that, with integrals applied to the airfoil surface, V[X,%y)

= / vsqj(s)dsj

+ / vvTj(s)dsj

(3.2.8)

Here qjdsj is the source strength for the element dsj on the j - t h panel. Similarly, Tjdsj is the vorticity strength for the element dsj on the same panel. Each of the N panels are represented by similar sources and vortices distributed on the airfoil surface. The induced velocities in Eq. (3.2.8) satisfy the irrotationality condition and the boundary condition at infinity d<j)

dip

u = — = — = l/oocosa ox oy

(3.2.9a)

v

= IT = - ? = ^oosina (3.2.9b) oy ox For uniqueness of the solutions, it is also necessary to specify the magnitude of the circulation around the body. To satisfy the boundary conditions on the body, which correspond to the requirement that the surface of the body is a streamline of the flow, that is, ib = constant

or

7— = 0 (3.2.10) on at the surface on which n is the direction of the normal, the sum of the sourceinduced and vorticity-induced velocities and freestream velocity is set to zero in the direction normal to the surface of each of the Af panels. It is customary to choose the control points to numerically satisfy the requirement that the resultant flow is tangent to the surface. If the tangential and normal components of the total velocity at the control point of the z-th panel are denoted by (Vl)i

34

3. Panel Methods

and (Vn)i, respectively, the flow tangency conditions are then satisfied at panel control points by requiring that the resultant velocity at each control point has only (K*)j, and (Vn)i = 0 i = l, 2 , . . . , TV (3.2.11) Thus, to solve the Laplace equation with this approach, at the z-th panel control point we compute the normal {Vn)i and tangential (Vt)i^ (i = 1, 2 , . . . , iV) velocity components induced by the source and vorticity distributions on all panels, j (j = 1, 2 , . . . , iV), including the z-th panel itself, and separately sum all the induced velocities for the normal and tangential components together with the freestream velocity components. The resulting expressions, which satisfy the irrotationality condition, must also satisfy the boundary conditions discussed above. Before discussing this aspect of the problem, it is convenient to write Eq. (3.2.4) expressed in terms of its velocity components (Vn)i and (Vt)i by TV

TV A

(Oi = E

(V^

l%

+ E

Z J + Voo sm(a

3=1

3=1

N

N

= £

MjQj + E

(3.2.12a)

B T

B r

hJ

(3.2.12b)

+ ^oo cos(a - Oi)

where A?-, Bf-, A1--, B\- are known as influence coefficients defined as the veIJ '

Lj

lJ

IJ

locities induced at a control point {xrni^yrni)\ more specifically, Af- and A\denote the normal and tangential velocity components, respectively, induced at the z-th panel control point by a unit strength source distribution on the j - t h panel, and Bf- and B\- are those induced by unit strength vorticity distribution on the j - t h panel. The influence coefficients are related to the airfoil geometry and the panel arrangement; they are given by the following expressions: A?- — I

/)Au

sin(#i — 6j In ^ ± i + cos(0i T i,3

2TT

1

i ^ 3

(3.2.13)

i = j r

sm(0i — 0j)/3ij — cos(9i — 9j) Ini,J + l h3

(3.2.14)

J

i = j -At

TV = AV-.

(3.2.15)

Here ^ i + l ) 2 + (Vn

r

i,j+l 'i,3

(xri

1/2

Vj+l) _ -,1/2

{ymi - yj)

3.2 HSPM

35

x

rrii = 7>(Xi +

X

^+l)'

Vrrii = ^ ( ^ + 2/*+l)

1

ft = t a n ' ( W t i J l W ^

fl

= tan-l

( V£H^*A

\Xi+i-XiJ

\Xj+i-XjJ

Ptj = tan" 1 (y^~_y^\ \Xrrii

(3.2.16)

- tan" 1

x

j+l/

(**-_"') \Xrrii

Xj J

Regardless of the nature of qj(s) and Tj(s), Eq. (3.2.12) satisfies the irrotationality condition and the infinity boundary condition, Eq. (3.2.9). To satisfy the requirements given by Eq. (3.2.11) and the condition related to the circulation, it is necessary to adjust these functions. In the approach adopted by Hess and Smith [1], the source strength qj(s) is assumed to be constant over the j-th panel and is adjusted to give zero normal velocity over the airfoil, and the vorticity strength TJ is taken to be constant on all panels (TJ = r) and its single value is adjusted to satisfy the condition associated with the specification of circulation. Since the specification of the circulation renders the solution to be unique, a rational way to determine the solution is required. The best approach is to adjust the circulation to give the correct force on the body as determined by experiment. However, this requires advance knowledge of that force, and one of the principal aims of a flow calculation method is to calculate the force and not to take it as given. Thus, another criterion for determining circulation is needed. For smooth bodies such as ellipses, the problem of rationally determining the circulation has yet to be solved. Such bodies have circulation associated with them and resulting lift forces, but there is no rule for calculating these forces. If, on the other hand, we deal with an airfoil having a sharp trailing edge, we can apply the Kutta condition [5,6]. It turns out that for every value of circulation except one, the inviscid velocity is infinite at the trailing edge. The Kutta condition states that the particular value of circulation that gives a finite velocity at the trailing edge is the proper one to choose. This condition does not include bodies with nonsharp trailing edges and bodies on which the viscous effects have been simulated by, for example, surface blowing, as discussed [4]. Thus, the classical Kutta condition is of strictly limited validity. It is customary to apply a "Kutta condition" to bodies outside its narrow definition, but this is an approximation; nevertheless the calculations are often in close accord with experiment. In the panel method, the K u t t a condition is indirectly applied by deducing another property of the flow at the trailing edge that is a direct consequence of the finiteness of velocity; this property is used as "the Kutta condition." Properties that have been used in lieu of "the K u t t a condition" in panel methods include the following: (a) A streamline of the flow leaves the trailing edge along the bisector of the trailing-edge angle.

3. Panel Methods

36

(b) Upper and lower surface total velocities approach a common limit at the trailing edge. The limiting value is zero if the trailing-edge angle is nonzero. (c) Source and/or vorticity strengths at the trailing edge must satisfy conditions to allow finite velocity. Of the above, property (b) is more widely used. At first it may be thought that this property requires setting both the upper and lower surface velocities equal to zero. This gives two conditions, which cannot be satisfied by adjusting a single parameter. The most reasonable choice is to make these two total velocities in the downstream direction at the 1st and N-th panel control points equal so that the flow leaves the trailing edge smoothly. Since the normal velocity on the surface is zero according to Eq. (3.2.11), the magnitude of the two tangential velocities at the trailing edge must be equal to each other, that is, (3.2.17)

-(V'h

(V% =

Introducing the flow tangency condition, Eq. (3.2.11), into Eq. (3.2.12a) and noting that TJ = r, we get N

TV

A

E

T

B

ii + V™ s i n ( a " °i) = °> * = 1, 2 , . . . , iV

iJ
3=1

(3.2.18)

3=1

In terms of the unknowns, qj (j = 1 , 2 , . . . , TV) and r, the Kutta condition of Eq. (3.2.17) and Eq. (3.2.18) for a system of algebraic equations which can be written in the following form, Ax = b (3.2.19) Here A is a square matrix of order (TV + 1), that is an

aij

ai2

aiTv

a2j

«21

ai,JV+l «2,7V+1

&2N

A =

an

Ui,N+l

d%2

(3.2.20)

a

iN

^TVl

&7V2

a/v+1,1

a/v+i,2

&NN a

N+lj

0>N,N+1 a

N+l,N+l

and x — ( # i , . . . , qi,..., tfyv, r)T and b — ( 6 i , . . . , 6^,..., b/v, 6 T V + I ) T with denoting the transpose. The elements of the coefficient matrix A follow from Eq. (3.2.18)

TV

<MT+i = E

B

5'

i

= 1 »2,...,iV

(3.2.21b)

3.2 HSPM

37

Afj are given by Eq. (3.2.13) and B% by Eq. (3.2.15). The relation in Eq. (3.2.20) follows from the definition of x where r is essentially #JV+ITo find ajv+i,j ( J = 1 , . . . , AT) and ajv+^jv+i m the coefficient matrix A, we use the K u t t a condition and apply Eq. (3.2.17) to Eq. (3.2.12b) and, with r as a constant, we write the resulting expression as N

N

J2 AjQj + r Y^ B\j + Voo cos(a - 0i) 3=1

3=1 " N

I^

N

A^jqj

|j'=l

or as

B

+TJ2

Nj

+ ^oo cos(a -

9N)

3=1

N

N

J=l

3=1

(3.2.22b)

= -V'oo cos(a - 0i) - 1/oo cos(a - 6N)

so that,

(3.2.23a) l

aN+hj

= A[j + A Nj,

j = 1, 2 , . . . , TV

N

aN+l,N+l = z2(Blj

+

(3.2.23b)

B

Nj)

3=1

where now A\- and AlN- are computed from Eq. (3.2.14) and B\- and BlN- from Eq. (3.2.15). The components of b again follow from Eqs. (3.2.18) and (3.2.21). From Eq. (3.2.18), h = - K o sin(a - 0i), i = 1 , . . . , TV (3.2.24a) and from Eq. (3.2.22), bN+i = -Voo cos(a - 9\) -Voo cos(o; - 9N)

(3.2.24b)

With all the elements of a^ determined from Eqs. (3.2.21) and (3.2.23) and the elements of b from Eq. (3.2.24), the solution of Eq. (3.2.19) can be obtained with the Gaussian elimination method [7]. The elements of x are given by

Xi —

1 ,(<-!)

(t-i)

N+l

b)

(i-l), ij "

N + !,...,!

(3.2.25)

where (fc-l) (fc-l) (fc) _ (fc-l) _ a „„ ik %' _ a i j (k-l)akj

a fcfc

fc= 1 , . . . , T V j = fc + l , . . . , 7 V + l i = fc + 1 , . . . , 7 V + 1 (o) a

ij

=a*3

(3.2.26a)

3. Panel Methods

38

F i g . 3.2. Panel-method representation of unsteady potential flow at time step tk-

(k-1)

/c = l , . . . , T V

& (/c) = ^-l)_^_^-l) 5

i = k + 1

a

,

m

.

m

,

N + 1

(3<2

.26b)

bf] = bi

kk

3.3 E x t e n s i o n of H S P M t o U n s t e a d y Airfoil F l o w s In the extension of HSPM to unsteady flows, the airfoil surface is again represented by singularity distributions of source strength qj (j = 1, 2 , . . . , TV) and vorticity r leading to (TV + 1) unknowns as in steady flow. For unsteady flows, however, these unknowns are time dependent. For this reason, we introduce a subscript k as the time-step index so that the inviscid flow solutions can be obtained at successive time-steps tk (k = 1, 2 , . . . ) , starting from to = 0. At each time-step, the airfoil is again represented by surface singularity distributions of source strength (qj)k (j = 1, 2 , . . . , TV) and vorticity rk. Since the total circulation in the flowfield must be preserved according to the Helmholtz theorem, any changes in the circulation r on the airfoil surface must be balanced by an equal and opposite change in the vorticity in the wake which causes vortex shedding at the trailing edge of the airfoil. In the unsteady panel method, the vortex shedding is represented by an additional wake element, often referred to as "the shed vorticity panel" attached to the trailing edge with uniform vorticity distribution (rw)kl see Fig. 3.2. If the length of the shed vorticity panel is denoted by Ak and its inclination angle to the x-axis by 0k and the overall circulation on the airfoil surface at time-step tk by rk(= rk£) with £ denoting the perimeter of the airfoil, then the Helmholtz theorem can be expressed as rk + Ak(rw)k

=

rk_i

or Ak(rw)k = rk~i

~ rk = Z(rk_i - Tfe)

(3.3.1)

Here rk_i and rk_\ represent the overall circulation on the airfoil surface and the vorticity strength, respectively, already determined at time-step tk-\. From time-step tk to £&+!, we assume that the shed vorticity panel is detached from the trailing edge and convects downstream as a concentrated free

3.3 Extension of HSPM to Unsteady Airfoil Flows

39

vortex, with circulation Ak(rw)k. The convection velocity of the free vortex is given by the local flow velocity at the center of the vortex. At the same time, a new shed vorticity panel is formed and the vortex shedding process is repeated. As a result of this continuous vortex shedding, a string of concentrated core vortices is formed in the wake behind the airfoil as shown in Fig. 3.2. The presence of the shed vorticity panel and the wake core vortices influences the upstream flow. As a result, the wake core vortices are convected under the influence of the freestream and the singularity distributions on the airfoil surface including the shed vorticity panel. This coupling process increases the complexity of the unsteady flow model and the solution procedure recommended is an iterative one. In addition to the surface singularities (qj)k (j — 1> 2 , . . . , N) and r^, there are three additional unknowns (rw)k, Ak and 0k due to the vortex shedding process. The Helmholtz theorem, Eq. (3.3.1) provides one additional relation in conjunction with the flow tangency conditions and the K u t t a condition. Two additional relations are required and can be obtained from the assumptions recommended by Basu and Hancock [8]: 1. The shed vorticity panel is oriented along the direction of the local flow velocity at the panel midpoint. 2. The length of the shed vorticity panel is equal to the product of the local flow velocity at the panel midpoint and the size of the time-step. With these assumptions, the two additional relations can be written as t^n0k

=

^ k \Vw)k

(3.3.2)

and Ak = [tk - tk-i)[(uw)l + (vw)l}1/2 (3.3.3) are where (uw)k and (vw)k the flow velocity components at the panel midpoint in the x- and ^/-directions, respectively. The flow tangency conditions remain the same as those for steady flows given by Eq. (3.2.11) [(Vnh]k = 0 i = l , 2 , . . . , J V (3.3.4) However, the K u t t a condition must now include the rate of change of velocity potential. According to the unsteady Bernoulli's equation, the Kutta condition can be expressed by [1,2]

[iy^iM - [{v*) N\k

d{(j>N - 0 0 dt

fdr\

=2UA —z

<33 5)

'-

Using a backward finite-difference approximation for the rate of change of total circulation on the airfoil surface, Eq. (3.3.5) can be written as [{VWl

- [{V)N]l

= 2^"f*-1 = 2 ^ ^ Z ^ i

(3.3.6)

3. Panel Methods

40

3.3.1 Influence Coefficients

The influence coefficients, Afp A\^ B1^, B\- for the source and vorticity distributions given by Eqs. (3.2.13)-(3.2.16) for steady flows remain the same for unsteady flows. These coefficients are time-independent since they are functions of geometrical parameters which are fixed for a rigid airfoil. Additional influence coefficients involving the shed vorticity panel and the wake core vortices are introduced for unsteady flows. These coefficients, however, are time-dependent and must be computed at each time-step; they are defined as follows: {BfnJrl)k

= normal velocity component induced at the z-th panel control point by unit strength vorticity distribution on the shed vorticity panel at time-step £&.

(Bjn+1)k

= tangential velocity component induced at the i-th panel control point by unit strength vorticity distribution on the shed vorticity panel at time-step t^.

( ^ H _ I j)k

= x-velocity component induced at the shed vorticity panel midpoint by unit strength source distribution on the j-th panel at time-step t^.

(^n+i j)k

~ 2/ _ve l° c ity component induced at the shed vorticity panel midpoint by unit strength source distribution on the j - t h panel at time-step t^.

(B^+1j)k

= x-velocity component induced at the shed vorticity panel midpoint by unit strength vorticity distribution on the j-th panel at time-step £&.

(^n+i j)k

~ 2/-velocity component induced at the shed vorticity panel midpoint by unit strength vorticity distribution on the j-th panel at time-step t^.

(A°^ -)k

= x-velocity component induced at the center of the h-th core vortex by unit strength source distribution on the j-th panel at time-step £&.

(Avh -)k

= y-velocity component induced at the center of the h-th core vortex by unit strength source distribution on the j-th panel at time-step £&.

(B% -)k

= x-velocity component induced at the center of the h-th core vortex by unit strength vorticity distribution on the j-th panel at time-step £&.

(J3^ -)k

= y-velocity component induced at the center of the h-th core vortex by unit strength vorticity distribution on the j-th panel at time-step tk.

3.3 Extension of HSPM to Unsteady Airfoil Flows

41

(Bhn+i)k

=

3> v e l°tity component induced at the center of the h-th core vortex by unit strength vorticity distribution on the shed vorticity panel at time-step tk.

(B^ n+i)fc

=

2/ _ve l° c ity component induced at the center of the h-th core vortex by unit strength vorticity distribution on the shed vorticity panel at time-step tk.

(^Tm)fc

=

norma

(^im)k

=

tangential velocity component induced at the i-th panel control point by unit strength ra-th core vortex at time-step tk.

(Cn+i m)k

— x-velocity component induced at the shed vorticity panel midpoint by unit strength ra-th core vortex at time-step tk.

l velocity component induced at the i-th panel control point by unit strength ra-th core vortex at time-step tk.

(^n+i m)k ~ y~Ye^oc^y component induced at the shed vorticity panel midpoint by unit strength ra-th core vortex at time-step tk. (CfLrn)k

= x-velocity component induced at the center of the h-th core vortex by unit strength ra-th core vortex at time-step tk.

(C^ m)k

= y-velocity component induced at the center of the h-th core vortex by unit strength ra-th core vortex at time-step tk.

The calculation of the new A and B influence coefficients is similar to the calculation of the old A and B influence coefficients. For example, (B™n+1)k and (Bjn+1)k are computed exactly the same way as B7^- and B\- are computed from Eqs. (3.2.13) and (3.2.15) with subscript n-\-l replacing j . Similarly, (A°^+1 -)k and (A^ -)k are calculated from Eq. (3.2.14) with 8{ set to zero and subscript i appropriately replaced. The relations used to calculate the C coefficients are different from those used for A and B coefficients since they are velocities induced by core vortices. It follows from Fig. 3.3 that the normal and tangential velocity components induced at the i-th panel control point by unit strength ra-th core vortex at time-step tk are (rn {C m)k

x _ ^ ~

COs[g,- (flj,m)fc] 2n(rhm)k

(3 3 7)

' -

and (rt

x _

{Ci m)k

'

where

-

sin [0j - (0i, m ) fc ]

M^m)k

(3 3 8)

' '

42

3. Panel Methods

A

( r UTl)k

(xm i? y mi )

Fig. 3.3. Evaluation of the influence coefficients at ( x m i , y m i ) due to core vortex at {xm^ym)k- Note that Xmi^ymi are the coordinates of the z-th control point as in steady flows (Chapter 2).

- h ^m = ^-coordinate of m-th core vortex at time-step t^.

(3.3.9)

Vm = y-coordinate of m-th core vortex at time-step t^. aOj = +t a n - l (Vi+l '

{Qi,m)k = tan"

~Vi

1 / Vrrii — V\ b

rrii

Similarly, (C^+lm)k anc

and {Cfirn)k are com

are computed by Eq. (3.3.7) while

u

(^n+im)fc ^ (^hm)k P t e d by Eq. (3.3.8) if ^ is set equal to zero and the subscript i appropriately replaced. 3.3.2 Solution P r o c e d u r e In terms of the influence coefficients, the flow tangency conditions given by Eq. (3.3.4) can be written as TV

E i=

N A

iMh

+ Tk Yl Brj + ( S t r e a m j=1

Vi + £(C£m)*(rm_i-rm) = o ra=l

(3-3-10) ; = I,2,...,JV

3.3 Extension of HSPM to Unsteady Airfoil Flows

43

Here Vstream denotes the unsteady stream velocity seen by an observer situated at a coordinate system fixed on the airfoil. For an airfoil executing an unsteady motion, including linear translation and angular rotation about a pivot point, ^stream = ^ o o + [Ul + Vj] + Q{yi -

(3.3.11)

Xj)

where —(Ui + Vj) is the translation velocity of the airfoil and Q is the rotational velocity. The above equation sums all the induced velocities due to an individual singularity with the stream velocity and sets it to zero to satisfy the flow tangency conditions. Substituting (rw)k from Eq. (3.3.1), collecting like terms and rearranging, Eq. (3.3.10) becomes N

N

J2^?j(Qj)k = rk -A-( ?,n+l)k ^k B

~ 2^

B

(^stream '

ij

n

i)k

3=1

3=1

(3.3.12)

fc-1

~ ( ^-J

T

A:-l(^z>+l)/c ~ ^2 (C^m)k(Fm-l

~ Frr

771=1

1,2,...,N Eq. (3.3.12) results in a system of algebraic equations which can be written in the form Ax = rk-b + c (3.3.13) where A is an TV x N matrix, x = [(qi)k, te)fc, • •., (tf/v)fc]T, & = (J>iM,--,bN)T T and c = (ci, C 2 , . . . , CTV) . The elements of A, the components of b and c are given by n • — An (3.3.14a) N

~7~(^i,ra+l)fc ~ ^k

(3.3.14b) 3= 1 k-

stream • ^

-

(-7-

\^k/

T

k-l(B?n+ i)fc

i

- m y= ,1

V^i,m)lA^m- - 1

-rm)

(3.3.14c) We note that the components of b and c are known only if the shed vorticity panel at time-step tk is established, i.e., if Ak and Ok are known. Therefore, the iterative solution procedure can be formulated as follows: 1. Find the locations of the wake core vortices downstream according to the local flow velocities at their centers with respect to an inertial coordinate system. 2. Compute the coordinates of these core vortices relative to the coordinate system fixed on the airfoil executing an unsteady motion.

3. Panel Methods

44

3. Start the iteration cycle for the current time-step by assuming values of Ak and 0k. Except for the first time-step, values at previous time steps can be used as initial guesses. 4. Compute all the influence coefficients needed in Eq. (3.3.12). 5. Solve Eq. (3.3.13) by the Gaussian elimination method (see Section 3.2) to obtain (qj)k m terms of rk\ i.e., find x — rkA~xb + A~1c. 6. Solve for rk using the Kutta condition of Eq. (3.3.6). Note that Eq. (3.3.6) is a nonlinear algebraic equation with rk as the only unknown. 7. Once rk is solved, (qj)k can be determined. The velocity components {uw)k and (vw)k at the midpoint of the shed vorticity panel can also be calculated. 8. The values of Ak and Ok are updated by Eqs. (3.3.2) and (3.3.3). 9. Repeat the iteration cycle from steps 3 to 8 until the values of Ak and 0k converge. 10. Compute the tangential velocities and the velocity potential at all panel control points to determine the pressure distribution which can be integrated to give force and moments. 11. Compute the local flow velocities at the centers of all the core vortices that are convected downstream. Proceed to the next time-step. The calculation of tangential velocities at the trailing edge panels, the velocity components (uw)k and (vw)k at the midpoint of the shed vorticity panel and the local flow velocities at centers of all core vortices required in this iteration process can be obtained by first computing the tangential velocities [(V*)i]fc (i = 1, 2 , . . . , N) at the panel control points by: N

N

3=1

3= 1

fc-i + (rw)k

• (B\n+l)k

+ ^

(3.3.15) (Ctm)k(rm-1

~

rm)

i = 1,2,... ,7V The components of the flow velocities at the centers of all core vortices are then calculated with respect to the inertial coordinate system by: TV

{uh)k = Yl(AhM
+ rk J2(Bh,3h

+ (^oo • i)k + (rw)k(Bxh^1)k

3= 1 k-1

+

(Ch,m)k(Fm-l

Y^ 7n=l,

ra//i

~

r

m)

(3.3.16)

3.3 Extension of HSPM to Unsteady Airfoil Flows

45

TV

(vh)k =

E^Vkiijh N

T

+ * E K , A +
(3-3-17)

fc-1

+

E

(Ch,Jk +

(rm-i-rm)

m=l,m^h

The above equations can also be used to obtain the velocity components {uw)k and (vw)k by replacing h with n -f 1 in Eqs. (3.3.16) and (3.3.17). 3.3.3 Velocity Potential and P r e s s u r e D i s t r i b u t i o n In a steady flow problem, it is not necessary to calculate the velocity potential (p since the pressure distribution can be obtained by evaluating the induced velocity from the concept of influence coefficients. However, this is not the case for an unsteady flow and it is necessary to use the unsteady Bernoulli equation written with respect to the airfoil-fixed coordinates system as [2]. _

n

V-Voo

_ / V k r e a m A 2 _ f V_\2

Cp

-(l/2)QV£-{

VoG

)

\VJ

_ JL ^

VI dt

/o o 10^ {3 SAS)

'

where Ktream is defined according to Eqs. (3.3.11) and V is the total velocity with respect to the airfoil-fixed coordinate system computed from V2 = (V1)2 + (Vn)2

(3.3.19)

Eq. (3.3.18) shows that in order to compute the pressure distribution on the airfoil surface, the rate of change of velocity potential must be evaluated. Using a backward finite-difference approximation for d(j)/dt, the pressure coefficients at the i-th panel control point at time-step t^ can be written as u n

xi

[(Cp)i]k

[(^stream)z]fc

=

—^l

[{V )i]k

vT

2

~ vg

(
*fc-*fc-i

,Q

Q onx

(3 3 20)

--

where [ ( V * ) ^ is obtained from Eq. (3.3.15). It is necessary to calculate the velocity potential at all the panel control points at each time-step. Instead of solving the Laplace equation, the velocity potential can be evaluated by integrating the velocity field in two stages, first from upstream at infinity to the airfoil leading edge, then along the airfoil surface from the leading edge to each panel control point. Care must be taken here to include only the velocity contribution due to singularities. The integration of the velocity field can be made by choosing an arbitrary straight line extending upstream to infinity from the leading edge of the airfoil along a direction parallel to Foo and infinity can be set at ten chord lengths

46

3. P a n e l M e t h o d s

upstream of the leading edge since the velocities at far-field points induced by the singularities are small enough to be neglected. This line is divided into z panels with element lengths near the leading edge comparable to the panel sizes near the leading edge along the airfoil. However, the panel size is progressively increased to take advantage of the inversely decaying induced velocities at larger distances. The tangential components of the induced velocities at the midpoints of these panels are computed using the influence coefficients analogous to those used on the airfoil panels. Using subscript / to denote these panel midpoints, influence coefficients ( ^ / j ) / ^ (B}j)k^ (^/,n+i)fc a n d (^/,m)fc a r e c o m P u t e d by the same expressions used to calculate the ^4's, B ' S and C's coefficients with cos0j replaced by — cos a, sin Oi replaced by — sin a and subscript i replaced by / . With these influence coefficients, the tangential velocities are obtained from

k-i +

(3.3.21) (Ctf,m)k(Fm-l

E

~

r

m)

vn—\

f =

l,2,...,z

The velocity potential at the airfoil leading edge (i.e.) is the sum of the products of the induced velocity at each panel and the panel length. (4>i,.h = E [ ( ^ ) / U ( * / + i " xf)2 + (1//+1 " Vf?]1'2

(3-3.22)

Similarly, for the line integral over the airfoil surface, the tangential component of the induced velocity at the z-th panel control point is computed from N

N

i=\_x + E

i=i (Ctm)k(rm-1

(3.3.23) ~

r

m)

m=l

Performing the line integration by summation, the velocity potential at the i-th boundary point on the airfoil is given by: (le.h + E

[(^%]fc[(xi+1-xJ)2 + (yi+1-yi)2]1/2

j=k.e. {(pnodeijk

—

(0Le.)*- E

[(^),]A,[(^+l - ^ )

2

+ (2/i+l - % ) 2 ] 1 / 2

j=i

1 < i < ii.e. (3.3.24)

3.4 Extension of HSPM to Unsteady Flow over Finite-Span Wings

47

Finally, the velocity potential at the i-th panel control point is OMfc = 2^node^k

+ (^nodez+l)/c]

(3.3.25)

In summary, the tangential velocity [ ( V ^ ) ^ , the stream velocity [(Vstream)i]fc at the i-th panel control point, and the velocity potential ((pi)k are evaluated from Eqs. (3.3.15), (3.3.11) and (3.3.25), respectively. The pressure coefficient Cp is then calculated by Eq. (3.3.20). The aerodynamic coefficients C^C^ and Cm are calculated by integrating the pressure distributions as in the steady flow problem.

3.4 Extension of H S P M t o U n s t e a d y Flow over Finite-Span Wings The extension of HSPM to unsteady flow over finite-span wings and more general three-dimensional configurations follows the same methodology described in Section 3.3 for two-dimensional airfoil flows. It is described in considerable detail by Katz and Plotkin [9]. The wake shedding is again accomplished by satisfying the Kutta condition. At each time step a new wake panel row is created. Once a wake panel is shed, its strength remains unchanged and the wake vortex moves with the local velocity. The induced velocity is then computed at each wake panel. Repetition of this procedure then facilitates the computation of the wake development and rollup. Further details can be found in the manuals for two commercial codes, i.e., CM ARC [10] and US AERO [11]. The former code is based on the PMARC code (Panel Method Ames Research Center) which was developed at the NASA Ames Research Center [12]. The latter code is an extension of the first commercially available panel code for steady three-dimensional flow calculations VSAERO to unsteady three-dimensional flow calculations.

References [1] Hess, J. L. and Smith, A. M. O., "Calculation of potential flow about arbitrary bodies," Progress in Aerospace Sciences, Vol. 8, Pergamon Press, N.Y., 1966. [2] Teng, N.H., "The development of a computer code (U2DIIF) for the numerical solution of unsteady, inviscid and incompressible flow over an airfoil," M.S. Thesis, Naval Postgraduate School, Monterey, CA 1987. [3] Jang, H.M., "A viscous-inviscid interactive method for unsteady flows," Ph.D. thesis, University of Michigan, 1990. [4] Jang, H. M. and Cebeci, T., "An interactive boundary-layer method for unsteady flows," Rept. AE-91-5, Aerospace Engineering Dept., California State University, Long Beach, May 1991. [5] Anderson, J., Aerodynamics, McGraw-Hill, NY, 1988.

48

3. Panel Methods

[6] Moran, J., An Introduction to Theoretical and Computational Aerodynamics, John Wiley, NY, 1984. [7] Isaacson, E. and Keller, H. B., Analysis of Numerical Methods, John Wiley, NY, 1966. [8] Basu, B. C. and Hancock, G. J., "The unsteady motion of a two-dimensional airfoil in incompressible inviscid flow," J. Fluid Mech., Vol. 87, pp. 157-168, 1987. [9] Katz, J. and Plotkin, A., "Low Speed Aerodynamics. From Wing Theory to Panel Methods, " McGraw-Hill, 1991. [10] Garrison, P. and Pinella, D., "CMARC User's Guide," AeroLogic, Inc., 1996. [11] "USAERO, A Time-Stepping Analysis Method for the Flow about Multiple Bodies in General Motion, User's Manual Version 4.2," Analytical Methods, Inc., 2002. [12] Ashby, D. L., Dudley, M. R., Iguchi, S. K., Browne, L., Katz, J., "Potential Flow Theory and Operation Guide for the Panel Code PMARC-12," NASA TM 102851, 1992.

Applications of Panel M e t h o d

4.1 Introduction In this chapter the panel method described in Chapter 3 is applied to the basic unsteady aerodynamic problems described in the first chapter, namely the problems of lift and thrust generation, airfoil flutter and gust response. However, before discussing these applications, it is instructive to recall the need to differentiate between streamlines, path lines and streak lines in unsteady flows in contrast to steady flows where all three lines collapse into a single line. A streamline is a curve whose tangent at any point is in the direction of the velocity vector at that point. If the flow is unsteady, the streamline pattern is different at different times because of the change in velocity vectors with time. The path line is the trace of the fluid particle as it moves downstream with the local flow velocity whereas the streak line is obtained by connecting the location of those particles which moved through a given point in space at consecutive times. Smoke particles released at a given point at different times for the purpose of flow visualization therefore represent streak lines. Figure 4.1 shows the panel method computed streak lines, streamlines and path lines due to a ramp- type change in angle of attack of a symmetric airfoil. Note that the formation of the starting vortex is most clearly visible using streak lines.

4.2 Analysis of Lift Generation As briefly discussed in Chapter 1, the lift generation on a low-speed airfoil is inherently linked to the formation of a starting vortex which forms at the airfoil trailing edge. As shown in Section 3.3, in potential flow this process can be simulated by prescribing the condition of zero pressure difference between the upper and lower surface panels at the trailing edge and by satisfying Kelvin's circulation theorem. Since the circulation around an airfoil in zero motion is certainly zero, the total circulation after the start of the airfoil motion around

50

4. Applications of Panel Method

Streaklines

Streamlines

'

•

.

*

$

:

•

•

Pathlines Fig. 4 . 1 . Example of streaklines, streamlines, pathlines.

a fluid line which contains both the airfoil and the starting vortex must remain zero. Hence the circulation around the airfoil must be of equal but opposite sign than the circulation around the starting vortex. Once the starting vortex has been swept sufficiently far downstream, the familiar steady- state flow field around a lifting airfoil is obtained. However, it is again instructive to visualize the motion of individual fluid particles. Consider the two neighboring fluid particles shown in Fig. 4.2. One is flowing just above the stagnation streamline, the other one just below this line. Note that the two fluid particles arrive at different times at the trailing edge. The upper one flows at a larger speed than the lower one and arrives at the trailing edge before the lower one. Applying Bernoulli's equation immediately leads to the recognition that the pressures on the upper surface are lower than on the lower surface (on an airfoil with positive total lift). Therefore, often the Bernoulli equation alone is used for the explanation of lift generation. However, such an explanation omits the inherently unsteady flow process leading to the generation of the starting vortex which in turn leads to the generation of circulation and hence lift on the airfoil. This can easily be seen if the flow over an infinitely thin cambered airfoil is considered. Here the neighboring fluid particles flowing very close to the upper and lower surfaces

4.2 Analysis of Lift Generation

51

Fig. 4.2. Visualization of fluid particles flowing over an airfoil at steady angle of attack.

move along essentially identical streamlines and the application of the Bernoulli equation would lead to zero pressure difference between the upper and lower surfaces. Yet, the airfoil generates lift due to the generation of a starting vortex at an earlier time! Consider now the problem of lift response due to a sudden change in angle of attack. Using the unsteady panel method described in the previous chapter the responses shown in Fig. 4.3 are obtained. It is seen that the results obtained with the panel method for the NACA 0001 airfoil closely reproduces Wagner's function for the flat plate shown in Fig. 1.2. On the other hand, the response for the NACA 0012 airfoil is significantly different at the start of the motion, demonstrating the advantage of the panel method to account for airfoil geometry effects. The panel method also provides the flexibility to prescribe more general angle of attack changes, such as ramp functions. The lift responses due to purely sinusoidal oscillations of thin airfoils were first obtained by Theodorsen [1] and Kuessner [2] in the 1930s. Again, the advantage of the unsteady panel method is the possibility to study the dependence of such responses on airfoil shape and thickness and amplitude of oscillation.

Wagner f u n c t i o n NACA 0012 NACA 0 0 0 1

0.2

5

10

15

20

x, distance traveled, semichords Fig. 4 . 3 . Comparison of Wagner's function with panel method predictions for the NACA 0001 and NACA 0012 airfoils.

4. Applications of Panel Method

52

1.0 r 1 \i

r

0.9

1 \

o.i

f^ l V.

—

r thee>ry ,_, ixnea UPOT, NACA 0015 UPOT,NACA 0012 UPOT, NACA 0009 UPOT, NACA 0003

0.7

0.6 0.5

•

0.4 0.0

i

1.0

i

i

i

2.0

3.0

i

i

4.0

k

F i g . 4.4. Effect of airfoil thickness on propulsive efficiency [3]. 1.0 linear theory UPOT, h=0.1 UPOT, h=0.2 UPOT, h=0.4

0.9 0.8 0.7

I 0.6 0.5 0.4 0.3 0.0

1.0

2.0 k

3.0

"> I 4.0

Fig. 4.5. Effect of plunge amplitude on propulsive efficiency [3].

For example, in Figs. 4.4 and 4.5 the propulsive efficiencies of a sinusoidally plunging airfoil computed with the unsteady panel code, sometimes referred to as unsteady potential (UPOT) code, are compared with linear theory. It is seen that the effect of thickness and amplitude becomes increasingly important with increasing frequency.

4.3 Analysis of Thrust Generation or Power Extraction

53

4.3 Analysis of Thrust Generation or Power Extraction The unsteady panel method of Chapter 3 can readily be applied to obtain the force component in the flow direction, i.e., the thrust force which is produced by a sinusoidal plunge or pitch oscillation of the airfoil in dependence of the reduced frequency of oscillation and the amplitude of oscillation. Consider an airfoil in sinusoidal plunge motion. The panel method predicts a reverse Karman vortex street of the type previously shown in Fig. 1.7. The generation of this type of vortex arrangement with counterclockwise vortices above the mean line and clockwise vortices below the mean line can be understood by recalling that the maximum induced angle of attack occurs as the airfoil moves through the mid-point. Hence the airfoil experiences a reduction in the negative angle of attack during the second part of the upstroke and an increasingly positive angle of attack during the first part of the downstroke, thus shedding counterclockwise starting vortices which accumulate to the counterclockwise vortices shown in Fig. 1.7. During the second part of the downstroke the positive angle of attack decreases and during the first part of the upstroke an increasingly negative angle of attack is generated, hence the shedding of clockwise vorticity which accumulates in a clockwise vortex located below the mean line. It is also readily seen that this vortex arrangement induces an increased velocity between the upper and lower vortices, hence a time-averaged jet-like flow, as shown in Fig. 1.8, and therefore a net thrust. A systematic computational analysis [3] shows that harmonically plunging airfoils in potential flow generate thrust over the whole frequency range such that the thrust increases nonlinearly with frequency as shown in Fig. 4.6, whereas a harmonically pitching airfoil generates thrust only above a certain critical frequency. The predictions of U P O T agree well for small amplitudes of oscillations and thin airfoils with the predictions of / 2.0

Gh=0.1, -> Jh=0.2, v - x h=0.4, h=0.1, 11=0.2, 11=0.4,

/

Garrick Garrick Garrick UPOT UPOT UPOT

/

/ / /

U

1.0

/

0 . 0 HXT^'^—HjT" 0.0 1.0

/

/

/

/ // //

/ //

/

/• /

/

/

-j t—i—I-^-J—i—1__ 2.0 3.0 k

L_I

Fig. 4.6. Effect of plunge amplitude on thrust [3].

i_

4.0

54

4. Applications of Panel Method

Fig. 4.7. Vortex rings generated by sinusoidally plunging finite-span wing.

Garrick [4] who used Theodorsen's linearized theory. It is also possible to study combined pitch and plunge oscillations with a phase angle between the pitch and plunge motions. As already explained in Section 1.4, this type of combined motion produces either thrust or extracts power from the air stream, i.e., causes flutter, depending on the amplitudes of oscillation, as shown in Figs. 1.9, 1.10 and 1.11. Although beyond the scope of this book, it is instructive to study the phenomenon of thrust generation for a finite-span wing in pure harmonic plunge oscillation. Using the three-dimensional panel method PMARC, briefly described in the previous chapter, one observes the formation of the vortex rings shown in Fig. 4.7. The starting vortices of alternating sign obtained in the twodimensional calculations now are connected by trailing vortices, as one would expect.

4.4 Analysis of Oscillating Airfoils The panel method makes it possible to study the precise time-lag effects caused by airfoil oscillation [5]. For example, consider the computed pressure distributions near the leading edge of a NACA 0012 airfoil, shown in Fig. 4.8. In this figure the origin corresponds to the leading edge, positive distance represents the upper surface, and negative distance the lower surface. The airfoil is either held steady at zero angle of attack or is pitching harmonically with an amplitude of 10 degrees at a reduced frequency of k = 0.1. The pressure distributions are plotted at that instant of time when the airfoil pitches through zero

4.4 Analysis of Oscillating Airfoils

55

1.0 steady xp=-l.0 xp=-.5 xp=0.0

0.5

u 0.0

-0.5

-0.2

-0.1 0.0 0.1 arc-length from LE

0.2

Fig. 4.8. Leading edge pressure distributions [5].

-0.2

-0.1

0.0

0.1

0.2

a r c - l e n g t h from LE Fig. 4.9. Leading edge pressure distributions [5].

angle of attack, from positive to negative angle of attack. The dependence on pitch axis location is shown by plotting the pressures for five different pitch axis locations (trailing edge xp = 1.0, mid-chord xp = 0.5, leading edge xp = 0.0, one-half chord xp — —0.5 and one chord xp = —1.0 upstream of the leading edge. In the same figure, the pressure distribution induced by a harmonically plunging NACA 0012 airfoil is also shown. The non-dimensional plunge amplitude was chosen as 1.76 so that the pitching and plunging airfoils experience the same effective angle of attack. The pressure distribution on the plunging airfoil is plotted when it is in the maximum downward position. One observes that the pressure is lower on the upper surface and higher on the lower surface than on the steady airfoil, thus demonstrating a significant pressure lag effect.

4. Applications of Panel Method

56

This effect increases with increasing downstream location of the pitch axis. One can also see that the pressure on the plunging airfoil is almost identical to that of the pitching airfoil for a mid-chord pitch axis, but there is a 90 degree phase lag between the pitch and plunge motion. The pressure lag effect is also noticeable in Fig. 4.9 where the airfoil is at the maximum positive angle of attack during the pitch oscillation and the plunging airfoil moves through the mean position. The suction peak is much reduced in comparison to the steady case. As discussed in more detail by Jones and Platzer [5], the well known delay in stall onset due to airfoil oscillation can be directly related to this pressure lag effect.

4.5 Analysis of Torsional Airfoil Flutter Consider the airfoil shown in Fig. 1.12. The airfoil is mounted on a torsion spring so that it can oscillate in the single-degree-of-freedom torsion mode. As is well known, the equation of motion describing this problem is the mass-springdamper equation Id-^+Ka

= M

(4.5.1)

where / is the airfoil's moment of inertia about the pitch axis, K is the spring constant and M is the aerodynamic moment acting on the airfoil. The possibility of nutter can be analyzed in either the frequency or the time domain. Choosing the frequency domain method it is assumed that the airfoil oscillates harmonically in time at constant pitch amplitude. Hence, using the unsteady panel method, the aerodynamic moment can be computed as a function of reduced frequency, yielding a component which is in phase with the angle of attack variation (the real part of the moment) and a component which is out of phase (the imaginary part). The in-phase component adds stiffness to the system, whereas the out-of-phase component is the only "damping" controlling the oscillation. Both the linearized theory of Theodorsen and the panel method predictions show that the damping changes sign at a reduced frequency of about 0.08 (based on half-chord) and becomes negative at frequencies below this value. For low frequency oscillations, i.e., for weak torsion springs, there exists therefore the possibility of negative aerodynamic damping which may induce nutter. It is important to recognize that the magnitude of the damping moment is quite small, yet the mere fact that it is negative is sufficient to drive the pitching oscillation to ever higher amplitudes. Also, it is important to note that a quasi-steady aerodynamic analysis would fail to predict this type of flutter. For the complete solution of the equation of motion we refer to the classical texts, e.g., Bisplinghoff et al. [6], pp. 527-532. The second way to analyze this problem is to use the time-domain method [7]. Here the equation of motion is solved in a time-stepping manner. The aero-

4.6 Analysis of Airfoil Gust Response

57

Fig. 4.10. Torsional oscillation as a function of torsion spring constant [7].

dynamic moment is recomputed for every time step using the unsteady panel method. The airfoil response to an initial deflection then can be obtained as a function of time in dependence of the parameters governing this problem (spring constant, moment of inertia, flow speed and density). The change in response from a damped oscillation through the flutter condition (the oscillation just maintaining itself at constant amplitude) to a rapidly divergent oscillation is shown Fig. 4.10. Note the change in reduced torsional frequency from ka = 0.15 to ka = 0.04 to obtain these three responses. Also note the self-excited nature of this motion. Putting the airfoil at an initial angle of attack is entirely sufficient to initiate the flutter oscillation because the airfoil keeps generating an aerodynamic moment which reinforces the pitch motion and thus enables the airfoil to extract energy from the flow. The cause for this destabilizing aerodynamic moment is the pressure lag effect which produces the pressure distributions shown in Fig. 1.13. As already explained in Section 1.5 a plot of the aerodynamic moment as a function of angle of attack during one cycle then yields a clockwise hysteresis loop which indicates positive work being performed by the flow on the airfoil.

4.6 Analysis of Airfoil Gust Response The lift build-up in response to a prescribed angle of attack change can also be obtained quite rapidly using the panel method of Chapter 3. Figure 4.11 shows the computed lift build-up on the NACA 0012 airfoil due to a ramp motion of different pitch rates. Again, the lag effect is exhibited by observing that the lift build-up occurs more slowly as the pitch rate increases.

4. Applications of Panel Method

58

2.0

"-:- rzft —* 1.5 r h

CI 1.0

i

r''

\ 1 k=0.0 terO.01 k=0.1 k«0.2 k=0.4

0.5

0.0 L B L . 0.0

•,

i

i

i

60.0

120.0

180.0

time-step

Fig. 4.11. Effect of reduced frequency on the unsteady lift at ramp motion.

References [1] Theodorsen, T.: General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA Report 496, 1935. [2] Kuessner, H.G.: Zusammenfassender Bericht iiber den instationaren Auftrieb von Fliigeln. Luftfahrt-Forschung 13, 410-424, 1936. [3] Jones, K. D. and Platzer, M.F.: Numerical Computation of Flapping-Wing Propulsion and Power Extraction. AIAA 97-0826, 35th Aerospace Sciences Meeting, Reno, Nevada, January 1997. [4] Garrick, I. E.: Propulsion of Flapping and Oscillating Airfoil. NACA Report 567, 1936. [5] Jones, K.D. and Platzer, M.F.: A fast Method for the Prediction of Dynamic Stall Onset on Turbomachinery Blades. ASME Paper 97-GT-101, presented at the International Gas Turbine Congress, Orlando, Florida, June 2-5, 1997. [6] Bisplinghoff, R. L., Ashley, H., and Halfman, R. L.: Aeroelasticity. Addison-Wesley, 1955. [7] Jones, K. D. and Platzer, M. F.: Time-Domain Analysis of Low-Speed Airfoil Flutter. AIAA Journal 34(5), 1027-1033, May 1996.

Boundary-Layer Methods

5.1 Introduction This chapter is concerned with the solution of the boundary-layer equations of subsection 2.4.3 for boundary conditions that include a priori specification of the external velocity distribution either from experimental data or from inviscidflow theory (called the standard problem), a priori specification of an alternative boundary condition which may be a displacement thickness distribution (called the inverse problem), or the determination of the freestream boundary condition by iteration between solutions of inviscid and boundary-layer equations (called an interaction problem). Section 5.2 describes the standard, inverse and interaction problems. It is followed by two sections that describe the numerical procedures used to solve two-dimensional steady boundary-layer equations in standard (Section 5.3) and inverse (Section 5.4) modes. Section 5.5 describes the numerical procedures used to solve the two-dimensional boundary-layer equations in standard and inverse modes. In the solution of the boundary-layer equations for turbulent flow, we make use of the eddy-viscosity (em) concept -

QU'V'

= Em

(5.1.1)

so that the momentum equation given by Eq. (2.4.16) can be written as du dt

du dx

du _ due dy dt

e

due dx

d / du\ dy \ dy J

where

&=l + £ +,

4, = ^

(5.1.3)

5. Boundary-Layer Methods

60

In this way the solution procedure of the momentum equation and continuity equation, Eq. (2.4.11), which is renumbered for convenience du dx

dv dy

(5.1.4)

is the same for laminar and turbulent flows with an algebraic eddy-viscosity formulation. The solution producedure is general and can be used to solve the turbulent boundary-layer equations with turbulence models other than those based on an algebraic eddy viscosity formulation as discussed in [1]. In this book we use the Cebeci-Smith (CS) algebraic eddy viscosity formulation described in detail in [1]. This model treats a turbulent boundary layer as a composite layer consisting of inner and outer regions with separate expressions in each region. In the inner region, the eddy-viscosity formula is defined by Lz

\£m)i

du 7tr dy

0 < y < yc

(5.1.5)

where L = K,y[l -

(5.1.6a)

exp(-y/A)} 1/2

(5.1.6b)

A = 26* @ / max

In the outer region, the eddy-viscosity formula is defined by r6

(£m)o = a / (ue - u) dy 7tr7 Jo 0.0168

Vc
(5.1.7) (5.1.8)

a =

The parameter F is related to the ratio of the product of the turbulence energy by normal stresses to that by shear stress evaluated at the location where the shear stress is maximum. As discussed in [1], it is given by F =

l_f3fdu/dx\

\du/dy)

(_e^v)n

Here the parameter (3 is a function of Rt = Tw/(—Qufv')max, by Rt < 1.0 l + 2Rt{2-Rt)

(5.1.9) which is represented

(5.1.10) l + Rt Rt > 1 0 Rt For TW < 0, Rt is set equal to zero. The intermittency expression 7 is based on Fiedler and Head's correlation and is given by

P={

5.2 Standard, Inverse and Interaction Problems

7 =

2

61

(5.1.11)

y/2a

Here Y and a are general intermittency parameters with Y denoting the value of y where 7 = 0.5 and a denoting the standard deviation. The condition used to define yc in Eqs. (5.1.5) and (5.1.7) is the continuity of the eddy viscosity, so that £m is defined by (em)i from the wall outward (inner region) until its value is equal to that given for the outer region by (£ m ) G . The expression 7t r models the transition region and is given by 7tr = 1 - exp

-G(x

- xtT) /

\

Jxtr

—

(5.1.12)

u

e J

Here xtr denotes the onset of transition and G is defined by G

=§2^lU

(5-1-13)

where C is 60 for attached flows and RXtr = (uex/u)tT is the transition Reynolds number. In the low Reynolds number range from Rc = 2 x 10 5 to 6 x 10 5 , the parameter C is given by C2 = 213(logi? Xtr - 4.7323)

(5.1.14)

5.2 Standard, Inverse and Interaction Problems 5.2.1 Standard P r o b l e m Solutions to Eqs. (5.1.2) and (5.1.4) are usually obtained for prescribed boundary conditions given by Eqs. (2.4.17), which for the case of a nonporous wall 2/ = 0, y = 5,

u = v = 0 u = ue(x,t)

(5.2.1a) (5.2.1b)

This procedure, sometimes referred to as the standard problem can be used to solve the boundary-layer equations, either for steady or unsteady flows, provided that there is no flow separation. In two-dimensional steady flows, boundary-layer separation corresponds to vanishing wall shear. When this occurs at some allocation, the solutions break down and convergence cannot be obtained. This is referred to as the singular behavior of the boundary-layer equations at separation. For laminar flows, the behavior of the wall shear rw close to the separation point xs has been shown to be of the form, £ ) - ( . . - * > ' "

(5.2*)

5. Boundary-Layer Methods

62

by Goldstein [2], who considered a uniformly retarted flow past a semi-infinite plate and showed that, with the relation given above, there is no real solution downstream of separation; the normal velocity component v becomes infinite at xs. Prandtl [3] pointed out that the pressure distribution around the separation point cannot be taken arbitrarily and must satisfy conditions associated with the existence of reverse flow downstream of separation. For unsteady flows, the boundary-layer equations are also singular at flow separation, which does not correspond to vanishing wall shear. Its identificiation and definition is discussed in Chapter 7. 5.2.2 Inverse P r o b l e m The boundary-layer equations are not singular at separation when the external velocity pressure is computed as part of the solution. Catherall and Mangier [4] were the first to show that for a two-dimensional steady laminar flow, the modification of the external velocity distribution near the region of flow separation leads to solutions free of numerical difficulties. Prescribing the displacement thickness as a boundary condition, that is y = <5,

6*(x) = given

(5.2.3)

in addition to those given by Eq. (5.2.1), they were able to integrate the boundary-layer equations through the separation location and into a region of reverse flow without any evidence of singularity at the separation point. This procedure for solving the boundary-layer equations for a prescribed displacement thickness distribution, with the external velocity or pressure computed as part of the solution is known as the inverse problem. This observation of Catherall and Mangier has led to other studies by various investigators of inverse solutions of the boundary-layer equations: these are obtained by prescribing distributions of displacement thickness or wall shear. Furthermore, it has been demonstrated in [5] that for flows with separation bubbles, these solutions are in good agreement with the solutions of the Navier-Stokes equations. It is now well established that the boundary-layer equations, either for twoor three-dimensional, steady and unsteady flows, are not singular at separation when the external velocity or velocities are computed as part of the solution [6]. We shall demonstrate this for two-dimensional unsteady flows in Chapter 7. A problem associated with the use of these inverse techniques for external flows is the lack of a priori knowledge of the required displacement thickness or wall shear. The appropriate value must be obtained as part of the overall problem interaction between the boundary layer and the inviscid flow. For two-dimensional steady flows, two procedures have been developed to couple the solutions of the inviscid and viscous flow equations. In the first procedure, developed by LeBalleur [7] and Carter and Wornom [8], the solution

5.2 Standard, Inverse and Interaction Problems

63

of the boundary-layer equations is obtained by the standard method, and a displacement-thickness, 8*°(x), distribution is determined. If this initial calculation encounters separation, 6*°(x) is extrapolated t o the trailing edge, and one complete cycle of the viscous and inviscid calculation is performed. This will, in general, lead to two different external velocity distributions, uev(x) derived from the inverse boundary-layer solution, and ue[(x) derived from t h e updated approximation to the inviscid velocity past the airfoil with the added displacement thickness. A relaxation formula is introduced to define an updated displacement-thickness distribution,

«*(x) = «*°(x)

(5.2.4) eiO)

where u is a relaxation parameter, and the procedure is repeated with this updated mass flux. In the second approach, developed by Veldman [9], the external velocity ue(x) and the displacement thickness 6*(x) are treated as unknown quantities, and the equations are solved in the inverse mode simultaneously in successive sweeps over the airfoil surface. For each sweep, the external boundary condition for the boundary-layer equations in dimensionless form, with ue(x) normalized with i^oo, is written as ue(x) = ue(x) + 6ue(x) (5.2.5) where u®(x) denotes the inviscid velocity and 6ue the perturbation due t o the displacement thickness, which is calculated from the Hilbert integral xb 1

6uL{ e

I/A ( t l e 0 _*U 7r J da

(5.2.6)

x—a

The above expression is based on the thin airfoil approximation with the term d/d<j(ue6*) denoting the blowing velocity used to simulate the boundary layer in the interaction region (x a , x^). This approch is more general and will be used for two-dimensional unsteady flows as we shall discuss in this and the following chapters. 5.2.3 Interaction P r o b l e m Predicting the flowfield by solutions based on inviscid-flow theory is usually adequate as long as the viscous effects are negligible. A boundary layer that forms on the surface causes the irrotational flow outside it t o be on a surface displaced into the fluid by a distance equal t o the displacement thickness <$*, which represents the deficiency of mass within the boundary layer. Thus, a new boundary for the inviscid flow, taking the boundary-layer effects into consideration, can be formed by adding 6* to the body surface. The new surface is called

64

5. Boundary-Layer Methods

the displacement surface and, if its deviation from the original surface is not negligible, the inviscid flow solutions can be improved by incorporating viscous effects into the inviscid flow equations [10]. A convenient and popular approach described in detail in [10] for aerodynamic flows, is based on the concept that the displacement surface can also be formed by distributing a blowing or suction velocity on the body surface. The strength of the blowing or suction velocity v^ is determined from the boundarylayer solutions according to u~8* (5.2.7) dx where x is the surface distance of the body, and the variation of v^ on the body surface implies that the boundary of the potential flow becomes the displacement (<$*) surface and thereby takes account of the viscous effects in the potential-flow solution. This approach, which can be used for both steady and unsteady flows, is used in this chapter and the following chapter to adhere the interaction problem for an airfoil in steady and unsteady two-dimensional flows. For a given airfoil geometry and freest ream flow conditions, the inviscid velocity distribution is first obtained with the panel method of Chapter 4. The boundary-layer equations are then solved in the inverse mode so that the blowing velocity distribution can be computed from Eq. (5.2.7), as shown in Fig. 5.1. With the off-body Kutta condition satisfied at a distance equal to <$*, and with the nonzero blowing velocity condition at the surface, the panel method of Chapter 4 is used to obtain an improved inviscid flow solution. With one cycle corresponding to the solution of inviscid and viscous flow equations, this procedure is repeated for several cycles until convergence is obtained, which is usually based on the lift and total drag coefficients of the airfoil, as we shall discuss in Chapter 6. Vb

Inviscid Method

\ /

/ \ Viscous Effect

Boundary Layer Method

j

\ / Fig. 5 . 1 . Interactive boundary-layer scheme.

5.3 Solution of the Standard Problem for Two-Dimensional Steady Flows Consider the boundary-layer equations and their boundary conditions given in subsection 2.4.3 for two-dimensional steady flows. With the eddy viscosity

5.3 Solution of the Standard Problem for Two-Dimensional Steady Flows

65

concept, Eq. (5.1.1), the momentum equation, (2.4.15) can be written as du du due d f du\ b ^~ + v^~ = ue^~ + v^~ ^~ (5.3.1) ox ay ox oy \ oy J where b is given by Eq. (5.1.3). Since the boundary-layer equations are parabolic, the solution procedure of Eqs. (5.1.4) and (5.3.1) and their boundary conditions, that is, y = 0, u = 0, v = 0 (5.3.2a) u

y = 6,

u = ue(x)

(5.3.2b)

employs a marching procedure. It begins with initial conditions at x = XQ and proceeds in the positive x-direction with dependent variables determined in sequence normal to the flow at each x-station subject to the boundary conditions. In general the boundary-layer thickness 8(x) increases with increasing downstream distance for both laminar and turbulent flows, and small steps must be taken in the streamwise direction to maintain computational accuracy. Transformed coordinates employing similarity variables are advantageous in the solution of boundary-layer equations since in terms of transformed variables, the boundary-layer thickness r]e remains nearly constant for laminar flows and grows considerably less in turbulent flows than 8{x) when physical variables are used. This also allows larger steps to be taken in the streamwise direction. Furthermore, and as shown below, they can also be used to generate initial conditions needed for the solution of the boundary-layer equations. As discussed in [10], there are several transformations that can be used for this purpose, and even through most of them are developed for laminar flows, they can also be used for turbulent flows. The Falkner-Skan transformation is a convenient choice and it is defined by rj=\

—V,

*l> = y/uevxf{x,ri)

(5.3.3)

Using this transformation and with i = j ,

^e = — ,

(5.3.4)

Eqs. (5.3.1) and (5.3.2) can be written as

(bf'Y + ^ff"

+ m[l - (f'f] - * ( / ' ^ - / " | 0

(5-3.5)

77 = 0,

/ ' = 0,

(5.3.6a)

r) = rk,

/' = 1

(5.3.6b)

Here a prime denotes differentiation with respect to 77 and r]e corresponds to a transformed boundary-layer thickness, ^/ue/ux6^ or yJueRi,/^8/L\ RL is a

5. Boundary-Layer Methods

66

Reynolds number based on reference velocity UQQ and length L; and m is a dimensionless pressure-gradient parameter defined by

ue at, The velocity components u and v are related to the dimensionless stream function /(£,77) by u = uef,

v = -y/u^(—^

— y/^+

— +f'—)

(5.3.8)

For laminar flows with external velocity of the form ue = Cxm', with C and m constants, and with boundary conditions on / and / ' independent of x, the left-hand side of Eq. (5.3.5) reduces to the well known Falkner-Skan equation r+rn+lffll

+ m[i_{f)2]

= 0

( 5 3 9 )

Its solutions, which are limited to —0.0904 < m < oo can be used to generate the initial conditions needed for the solution of the boundary-layer equations, as we shall see later. There are several numerical methods that can be used to solve the boundarylayer equations. Finite-difference methods offer the greatest flexibility, and those of Crank-Nicolson and Keller discussed in [10,11] have been widely used. The latter method provides significant advantages over the former method and will be used to solve the boundary-layer equations here. One of the requirements of Keller's box method is that the governing equations are written as a first-order system. First, derivatives of / with respect to r\ are introduced as new functions, and the resulting first-order equations are approximated on an arbitrary rectangular net, Fig. 5.2, with "centered-difference" derivatives, which are averaged at the midpoints of the net rectangle to obtain finite-difference equations with a truncation error of second order. The resulting difference equations are implicit and nonlinear, are linearized by Newton's method described in subsection 5.3.2 and are solved by the block-elimination method discussed in subsection 5.3.3. 5.3.1 N u m e r i c a l Formulation To express Eqs. (5.3.5) and (5.3.6) as a first-order system, define f = u

(5.3.10a)

v! = v

(5.3.10b)

and the momentum equation and its boundary conditions become

67

5.3 Solution of t h e S t a n d a r d P r o b l e m for T w o - D i m e n s i o n a l S t e a d y Flows

1

L

- x —i

*v> 1,.,'

~p.

>— •

\

en-1

^'

^

—u *

- <j>-

--M

1M

6

e

3 n-1

e n-l/2

' f2 ^n

F i g . 5 . 2 . N e t rectangle for difference a p p r o x i m a t i o n s .

(M' + ^ f v 77 = 0,

+ m ( l - u*) = { ( u | f - , g ) 14 = 0,

f = fw(t)]

V = Ve,

(5.3.10c)

U=l

(5.3.11)

Next denote the net points of the net rectangle shown in Fig. 5.2 by = r"

1

£° = 0,

r

7/o = 0,

rij=rij-i+hj,

+ *n,

n=l,2,...,7V j = l,2, . . . , J

(5.3.12)

and write the difference equations that are to approximate Eqs. (5.3.10) by considering one mesh rectangle as shown in Fig. 5.2. We write the finite-difference approximation of the ordinary differential equations (5.3.10a) and (5.3.10b) for the midpoint (£ n , 77j_i/2) of the segment P1P2 using centered-difference derivations. J

1

3

""+"7-1 - ; j n

(5.3.13a)

hj

«?

-1

(5.3.13b)

= vj - 1 / 2

h*

Similarly, the partial differential equation (5.3.10c) is approximated by centering about the midpoint ( £ n - 1 / / 2 , Vj-1/2) of the rectangle PIPZP^PA, and this is done n_1 2 in two steps. In the first step it is centered about the point (£ ' ,?7) without specifying 77. If its left-hand side is denoted by L, then the finite-difference approximation to Eq. (5.3.10c) is

2

(Ln + Ln - l \

tn-1/2

,71-1

.,"-1/2

- ^ - 1 / 2

r-/

n—1

^n

(5.3.14) With

68

5. Boundary-Layer Methods £n-l/2 an = ^— ,

Rn~L

mn

= -Ln~l

Ln~l

_|_ ^ +an,

ai =

+ an[(fv)n-L

a2 = mn + an

- (u2)n

= [(bv)' + ^ ^ f v

] - mn

+ m ( l - u2)]n~l

(5.3.15a)

(5.3.15b)

(5.3.15c)

Eq. (5.3.10c) can then be written as [(bv)'}71 + ai(fv)n

- a2{u2)n

+ an{vn~lfn

- / n " V ) = Rn~l

Next, center Eq. (5.3.16) about the point ( £ n _ 1 V = Vj-i/2

and

g

(5.3.16)

,7?7_i/2); that is, choose

et

V X ( W - & i-i^-i) + «i(/«)"-i/2 - Mu2)]_1/2

where R

U/2 = ~LT-\i2 + ^ 1 ( ^ ) ^ / 2 -

LnJ~-l l / 2

fa2)^

- ™n

(5-3-18a) in-1

fy ^ M J -

fy-ifj-i)

+ —PT"(f v )j-i/2 2

+ M 1 - (^ 2 )j-i/2]

(5.3.18b) Equations (5.3.13) and (5.3.17) are imposed for j = 1, 2 , . . . , J at given 77 and the transformed boundary-layer thickness, r]ei is sufficiently large so that u —> 1 asymptotically. The latter requirement is usually satisfied when |^(rye)| is less than about 1 0 - 3 . The boundary conditions [Eq. (5.3.11)] yield, at £ = £ n , / o n = 0,

uS = 0,

unj = l

(5.3.19)

The next step is to solve the algebraic equations represented by Eqs. (5.3.13) and (5.3.17) with the boundary conditions of Eqs. (5.3.19). Since these equations are nonlinear, the linearization procedure using Newton's method described in the following subsection, is introduced before they are solved by the block elimination discussed in subsection 5.3.3.

5.3 Solution of the Standard Problem for Two-Dimensional Steady Flows

69

5.3.2 N e w t o n ' s M e t h o d Assume / j 1 - 1 , n™'1 and v™~1 to be known for 0 < j < J; then Eqs. (5.3.13), (5.3.17) and (5.3.19) form a system of 3 J + 3 equations for the solution of 3 J + 3 unknowns (/j 2 , u7*, vj), j = 0 , 1 , . . . , J . We use Newton's method to solve this nonlinear system. Thus, the iterates [/• , u^ , v^ ], v — 0 , 1 , 2 , . . . , are introduced with initial values {y — 0) equal to those at the previous ^-station, £ n _ 1 . For higher iterates we set /j

= fj

+ °fj

'

U

^}

j

+ ^}

V

'

j'

^J

+

^j (5.3.20) We insert the right-hand sides of these expressions in place of / j 1 , iif and ^ in Eqs. (5.3.13) and (5.3.17) and drop the terms that are quadratic in
~ y (^J +

6UJ_I

SU

J-I)

= (ri)j

(5.3.21a)

—^~{^ v j + ^ j - i ) = {^s)j-i "2

+ ( s 2 ) j ^ j _ i + {S3)j6fj +

(5.3.21b)

(s4)j8fj-i

+ (s$)j6uj + (se)jSuj-i

= (r2)j

(5.3.21c)

where

(ri)j = /jr\ - / j " 5 + Mj-1/2 (r 3 ),_i = ^

(5.3.22a)

- uf + h3vf}l/2

(5.3.22b)

- «2(- 2 )t ) i/ 2 + « n ( ^ r i / 2 / ] - ) 1 / 2 - / r i / 2 ^ 1 / 2 ) ]

(5-3-22C)

As will be shown later, in writing the system given by Eqs. (5.3.21), a certain order has been chosen to ensure that the Ao-matrix in Eq. (5.3.26a) is not singular. The coefficients of the linearized momentum equation are («i)i - hfbf

+ f

/j"> - f / ? _ " %

(5.3.23a)

v™

(«2)i - - V ^ - l

+ f/)-! " T^'l/2

(*3)i = f ^ + YV?-W

( 5 - 3 - 23b ) (5 3 23c)

'-

5. Boundary-Layer Methods

70

Mj = YVA

(5.3.23d)

+ T Vi/2

(5.3.23e) (5.3.23f) The boundary conditions, Eq. (5.3.19), become (5.3.24) The linear system given by Eqs. (5.3.21) and (5.3.24) has a block tridiagonal structure and can be expressed in matrix-vector form and be written as (5.3.25)

>6 = r

The coefficients matrix A has the structure given by Eq. (5.3.27) where A/, Bj and Cj represent 3 x 3 matrices given by

A0 =

1 0 0 0 1 0 0 - 1 - ^

1

^i =

-%

0

( * 3 ) j (S5)j

0

1 < j < J

(Sl)j

-1

-1.

- ^ (5.3.26a)

1

- ^

(S3)j

(s$)j

0

1

0 (si)j

B,

0

-1 - ^ (s4)j (s6)j 0 0

0 (s2)j 0

1<3<J(5.3.26b)

Cj =

0 0 0 0 0 0 0 1 -^±i

0<j < J-l.

(5.3.26c)

The compound vectors 8 and r and coefficient matrix A are given by A0 C 0 Bi Ai Cx

£o 61 6 =

r =

B3

Aj Bj-\

6j

rj

(5.3.27)

C, Aj_i Bj

Cj_x Aj

where the three-dimensional vectors, 6j and r j , for each value of j , are

63 = SUj 8vi

(nh (r2)j

fo)i

0 < j < J.

(5.3.28)

5.3 Solution of the Standard Problem for Two-Dimensional Steady Flows

71

Note that the first two rows of Ao and Co and the last row of Bj and Aj correspond to the boundary conditions. To solve the boundary-layer equations for different boundary conditions, only the matrix rows mentioned above need altering. As as result of these boundary conditions, we note from Eq. (5.3.24) that (ri)o = (72)0 = 0 and {r^)j = 0 so that 0 0 0-3)0

^0

(ri)j rj

=

(5.3.29)

iri)j 0

5.3.3 Block-Elimination M e t h o d The solution of Eq. (5.3.25) can be obtained efficiently and effectively by using the block-elimination method described in [1]. According to this method, the solution of Eq. (5.3.25) is obtained by forward and backward sweeps summarized below. Forward Sweep: A0 = A0 r

jAj-i

= Bj,

3=

A3 = Aj - '

r C

3 3-^

1,2,-- -,J

J = 1,2,.. -,J

(5.3.30)

wo = ro Wj = 13 -

r

j^J~

l<j<J

(5.3.31)

Backward Sweep: AJSJ

AjSj

(5.3.32)

= WJ

= WJ •

d6i2j+i>

3 = J-l,J-2,...,0.

The amount of algebra required to solve the above recursion formulas given by Eqs. (5.3.30) to (5.3.32) depends on the order of the matrices Aj, Bj, Cj. When the order is small, the matrices i~j, Aj and the vector Wj can be obtained by relatively simple expressions, as discussed in [1] when the matrices are 2 x 2 , 3 x 3 and 4 x 4 . However, this efficient procedure becomes increasingly tedious as the order of the matrices increases and requires the use of an algorithm that reduces the algebra internally. A general algorithm, called the "matrix solver", is discussed in [1, 2] can be used for this purpose.

72

5. Boundary-Layer Methods

5.4 Solution of t h e Inverse P r o b l e m for Two-Dimensional Steady Flows The solution procedure discussed for the standard problem can be extended by using the procedure described in subsection 5.2.2. Of the two procedures, the procedure of Veldman has substantial advantages over that based on the relaxation formula, Eq. (5.2.4), and will be used here. In both cases, however, modifications are needed to the solution procedure of the standard problem. When the viscous effects on the edge boundary condition are accounted for by Veldman's procedure, the boundary conditions for the inverse problem can be written as u = 0, v =0 (5.4.1a) y = 0, y = 61

u = uei

ue = u® + Sue(x)

(5.4.1b)

with 6ue(x) given by Eq. (5.2.6). As in the case of the standard problem, it is convenient and prudent to use transformed variables, but since ue{x) is also an unknown in the inverse problem, slightly changes are made in the Falkner-Skan transformation given by Eqs. (5.3.3), replacing ue(x) by UQQ and redefining new variables Y and F by Y =^uOQ/uxy,

*P = ^I^cF(ZJY)

(5.4.2)

so that Eqs. (5.1.4) and (5.3.1), with a prime now denoting differentiation with respect to Y, can be written in a form analogous to that given by Eq. (5.3.5),

The boundary conditions, except for Eq. (5.3.6b), can be written as y = 0,

F ' = 0,

Y = ye,

F = 0

(5.4.4a)

F'e = ue(0,

(5.4.4b)

The boundary condition corresponding to Eq. (5.4.4b) is obtained by applying a discretization approximation to the Hilbert integral z-l

TV

c D

MC) = u°(?) + CaDi + J2 a J + E

c D

a i

(5A5)

where the subscript i denotes the ^-station where the inverse calculations are performed, Cij is a matrix of interaction coefficients obtained by the procedure described in [1] and D is given by D = ue8*. In terms of transformed variables, the parameter D becomes

D = -^LUoo

= J4~ (yeue - Fe) V RL

(5.4.6)

5.4 Solution of the Inverse Problem for Two-Dimensional Steady Flows

73

and the relation between the external velocity ue and displacement thickness 6* provided by the Hilbert integral can then be written in dimensionless form as Y = Ye,

Ft(?)

- A[F e F e '(f) - F e ( 0 ] =

9i

(5.4.7)

where

* = Cijyfsi/RL i-l

(5.4.8a) TV

c

3i = «e(f) + E ^ A + E

c D

( 5A8b )

a i

In an inverse method which makes use of the Hilbert integral formulation, the boundary-layer calculations must be repeated on the body. Each boundarylayer calculation, starting at £ = £° and ending at £ = £N, is called a sweep. In sweeping through the boundary layer, the right-hand side of Eq. (5.4.5) uses the values of <5* from the previous sweep when j > i and the values from the current sweep when j < i. Thus, at each ^-station the right-hand side of Eq. (5.4.5) provides a prescribed value for the linear combination of ue(^1) and S*(£l). After convergence of the Newton iterations at each station, the summations of Eq. (5.4.5) are updated for the next ^-station. Note that the Hilbert integral coefficients Cij have been computed and stored at the start of the boundarylayer calculations. For the inverse problem, as described in Section 5.2, ue(£) is computed as part of the solution procedure. This is done by treating it as an unknown, denoting ue(£) by w(£), remembering that the external velocity is a function of £ only, and noting that w = 0 (5.4.9) As in the case of the standard problem, new variables U(£,Y), V(£,Y) are introduced and Eq. (5.4.3) and its boundary conditions, Eq. (5.4.4) and (5.4.7), are expressed as a first-order system,

(bVy + \FV = t y = o, Y

= ye,

u = w,

F' = U

(5.4.10a)

7' = V

(5.4.10b)

'

(U

dU

dF\

dw

F = U=0 XF + (1 - XYe)w =

(5.4.10c) (5.4.11a)

9i

(5.4.11b)

Finite-difference approximations to Eqs. (5.4.9) and (5.4.10) are written in a similar fashion to those expressed in the original Falkner-Skan variables, yielding expressions again similar in form to those of Eqs. (5.3.13) and (5.3.17).

74

5. Boundary-Layer Methods h~l(w]

- w^_{) = 0

h~\F?

- Ff_x)

hj\Uf

hj\b?V?

- qVjLi)

(5.4.12a)

= U?_1/2

(5.4.12b)

n

- U?_{) = V3 _1/2

(5.4.12c)

+ an[(w2)]_1/2

+ Q + a " ) (FV)]_1/2

-

(U2)]_1/2]

where now ^-1,2 L

T-\l2

= -Ll-l/2

= [hjHbjVj

+ <*n[{FV)£l/2

- bj-iVj-!)

- {U*)»ZI,2\

+ \{FV)3_l/2

(5.4.13a)

- an{w2)3_l/2]n-1

(5.4.13b)

Comparing Eq. (5.4.10c) with Eq. (5.3.10c) and Eq. (5.4.12d) with Eq. (5.3.17), one can see that if m is set to zero in Eq. (5.3.10c) or (5.3.17) so that ai

= ^-+an,

a2 = an

(5.4.14)

and include the underlined term in Eqs. (5.4.12d) and (5.4.13b) and the unknowns F , [/, V are replaced by / , u, v the resulting equations and the previous ones can both be used to solve the standard and inverse problems subject to the boundary conditions of Eqs. (5.3.19) and (5.4.11). The linearized form of Eqs. (5.4.12) can also be expressed in a way that can be used in both standard and inverse problems so that Eq. (5.4.12a) becomes 8WJ

—

6WJ-I

= Wj-i — wj = (r/Cjj-i

(5.4.15)

The linearized forms of Eq. (5.4.12b,c) are similar to those given Eqs. (5.3.21a,b); the linearized term of Eq. (5.4.12d) can be expressed in the form of Eq. (5.3.21c) provided that two terms are added to its left-hand side, (S7)JSWJ

+ (s$)j6wj_i

(5.4.16a)

with (57)j and (sg)j defined by (s7)j = anWj,

(sg)j = anWj-i

(5.4.16b)

The coefficients (s\)j to (SQ)J defined by Eqs. (5.3.23) remain unchanged except for the modifications indicated by Eq. (5.4.14) and (r2)j now given by

(r2)j = R]:l/2

- [hjHbjVj - bjVj-i) + {l2+

an){FV)3_1/2

+ « n [ ( A - i / 2 " ( f V 1 / 2 ] + ""(Vp'/iFj-i/*

~

F^/2V3_l/2)] (5.4.17)

5.4 Solution of the Inverse Problem for Two-Dimensional Steady Flows

75

The linearized form of the boundary conditions given by Eq. (5.4.11) become 6F0 = 6U0 = 0;

8Uj - Swj = wj-

Uj,

nSFj

+ l2bWj

= 73 (5.4.18)

nfs=9i-

{liFj

+ 7 2 wj)

(5.4.19)

where 71 = A,

12 = 1- A l j ,

The linear system given by Eqs. (5.3.21) with modifications given by Eq. (5.4.16a) (with / j , Uj and Vj expressed in capital letters) and Eqs. (5.4.15) and the linearized boundary conditions of Eqs. (5.4.18) and (5.4.19) are written in the same matrix-vector form as Eq. (5.3.25) in the following ordered form. 6F0

6u0

SV0

6w0

8F3

8Uj

6Vj

8Wj

b.c b.c

:

1

0

0

0

0

0

0

0

8F0

(^1)0

:

0

1

0

0

0

0

0

0

8U0

(7-2)0

Eq (5.3.21b)

:

0

- 1

-h1 2

0

0

1

-h1 2

0

6V0

03)0

Eq (5.4.15)

:

0

0

0

- 1

0

0

0

1

8w0

(7-4)0

1

-h3 2

-H3

Eq (5.2.21a)

- 1

Eq (5.3.21c) and (5.3.17a)

0

0

(s&)j

(s2)j

(s&)j

0

0

0

0

0

- 1

0

0

0

0

0

0

Mi

2

(35) j

Mj

-1

b.c. b.c.

\ («4)./

(SG).J

:

0

0

0

0

8Fj

SUj

8wj

6Vj

0

0

0

0

0

8Fj

(ri)j

(s7)j

0

0

0

0

8U3

(r2)j

0

0

1

0

8V3

0

- 1

0

0

0

1

8w3

(r±)j

0

0

1

0

0

8Fj

(ri)j

(s7).J

6Uj

(r2)j

0

(si) j -hj + 1 2

( s 2).7

0 0

(«8).7 '• (S3)J

0 0

2

(s$)j

(si)j

0 1

0

; 7i : 0

72 0 - 1

=

(rs)j

8Vj

(i"3)j

8wj

(r4)j

(5.4.20) with 6j and 7%- now defined by 6Fj *i

fo)i fo)i

8V3

(5.4.21)

6Wn

and Aj, Bj, Cj matrices becoming 4 x 4 matrices denned by 1 0 0 0

A0

0 1 -1 0

0 0 hi 2

0

hi

11. 1 0 2 (*3)i (*5)j ( « l ) i -1 - ^ 0 0 0 0

0 0 0 - ]

0 (S7)j

0 -1

l<j<J-l (5.4.22a)

Aj

=

1

-fy

0

(S3)j

(Sbjj

(Sl)j

71 0

0

0 1 0

0 (S7)j

72 - 1

h

- 1 - ^ 0 0 (s )j (se)j («2)i {ss)j 4 B; = 0 0 0 0 0 0 0 0

1<3<J (5.4.22b)

5. Boundary-Layer Methods

76

0 0 0 0

Cj

0 0 0 0 1 - ^ 0 0

0 0 0 1

(5.4.22c)

Note again that the first two rows of AQ and Co and the last two rows of Bj and Aj correspond to the boundary conditions given Eq. (5.4.18). As a result ( n ) o = (r 2 )o = 0 (rs)j = 73,

(r4)j

= WJ-UJ

(5.4.23a) (5.4.23b)

The remaining elements of the rj vector follow from Eqs. (5.3.22), (5.4.15) and (5.4.17) so that, for 1 < j < J , ( r i ) j , (7*2)j, (rs)j-i are given by Eqs. (5.3.22a), (5.4.17) and (5.3.22b), respectively. For the same j-values, (r<±)j-i is given by the right-hand side of Eq. (5.4.15). The parameters 71, 72 und 73 determine whether the system given by the linearized form of Eqs. (5.4.12) and their boundary conditions is to be solved in standard or inverse form. For an inverse problem, they are represented by the expressions given in Eq. (5.4.19) and for a standard problem by 71 = 0, 72 = 1.0 and 73 = 0. The solution of Eq. (5.3.25), with 6j and rj defined by Eq. (5.4.21) and A/, Bj and Cj matrices given by Eqs. (5.4.22), can again be obtained by the blockelimination of subsection 5.3.3. The resulting algorithm, called SOLV4, is given in the accompanying diskette.

5.5 Solution of the Standard and Inverse Problems for Two-Dimensional Unsteady Flows A major problem in the calculation of boundary-layers is to include regions of flow reversal. As we discussed in Section 5.2, in a two-dimensional steady flow with a prescribed external velocity distribution, ue(x), the problem is associated with separation; in general the solutions come to an end if and when the skin friction vanishes due to the appearance of a singularity centered at the wall. In a three-dimensional steady flow [3] under similar conditions, the problem is associated with cross-flow reversals and with separation. Even though the flow remains attached in the generally accepted terminology and the solutions of the governing equations are not singular at the vanishing of zero cross-flow skin friction, numerical instabilities result from integration opposed to the flow direction and require special numerical procedures be devised to obtain stable and accurate solutions. When the streamwise skin-friction vanishes at the wall, the solutions, as in two-dimensional steady flows, come to an end due to the existence of a singularity. In general this situation occurs after regions of negative crossflow velocity in the flow; an accurate calculation of the flow under these

5.5 Solution of the Standard and Inverse Problems

77

conditions is crucial in defining the flow separation line in three-dimensional flows. As in two-dimensional steady flows, the calculation of separated flows (negative streamwise skin-friction) requires the solution of the governing equations in the inverse mode. The computational problems associated with unsteady boundary-layers are similar to those for three-dimensional steady boundary-layers. In two-dimensional unsteady flows with no separation, the problem is roughly analogous to a three-dimensional reverse cross-flow problem if we associate time with the direction of a mainstream with a unit velocity component. The unsteady boundarylayer is then the cross-flow velocity. As in three-dimensional steady flows [10], the computation of unsteady flows under these conditions requires the development of special procedures to avoid numerical instabilities which result from flow reversal. The flow separation, if there is one, again occurs after some regions of flow reversal develop within the boundary-layer, and unlike steady two-dimensional flows, the breakdown of the solution does not always coincide with the vanishing of the skin-friction nor it is centered at the wall; to avoid the singularity and be able to obtain solutions of separated flows, it is again necessary to solve the unsteady boundary-layer equations in the inverse mode. In this section we extend the numerical method of Sections 5.3 and 5.4 for steady flows to standard and inverse problems in unsteady flows. Whether we solve the standard or the inverse problem for unsteady flows, the solution procedure requires initial conditions in the (t, y) and (x, y) planes as discussed in subsection 5.5.1. It is again convenient and efficient to use transformed variables in both standard and inverse problems as discussed in subsection 5.5.2. Subsections 5.5.3 and 5.5.4 describe the numerical procedures used to solve Eqs. (5.1.2) and (5.2.1) in the contexts of flows without reversal (subsection 5.5.3) and flows with reversal (subsection 5.5.4). 5.5.1 Initial Conditions The determination of initial conditions in the (x, y) and (£, y) planes is important. Sometimes it can be arbitrary and sometimes approximate. For example, if initial conditions in the (x, y) plane are arbitrary, the values of du/dt at t = 0 are nonzero; this implies an inviscid acceleration and, as a consequence, a slip velocity develops at the wall and is smoothed by an inner boundary-layer initially of thickness (vt)1'2 in which viscous forces are important. Thus a double structure develops in the boundary-layer and may be treated by the numerical method described in [4]. However, if interest is centered on the solution at large times, this feature may be reduced in importance by requiring that the initial velocity distribution satisfies the steady-state equation with the instantaneous external velocity. In addition, it is necessary to smooth out the external velocity ue(x, t) so that due/dt = 0 at t = 0 and then standard numerical methods may

5. Boundary-Layer Methods

78

be used and are stable. The use of a smoothing function makes for some loss of accuracy at small values of £, but the error soon decays to zero once the required value of ue is specified. In this chapter we assume that steady-state conditions prevail at t = 0 and generate the initial conditions in the (x, y) plane by the procedure described in Section 5.3. The calculation of initial conditions in the (£, y) plane at some x = xo when the conditions at a previous time-line are known, can introduce different problems. In principle, solutions can be determined at the next time-line by an explicit method but, if stability problems are avoided by the use of an implicit method, there arises the problem of generating a starting profile on the desired time-line. This requires the use of a special numerical procedure, such as the one discussed in subsection 7.2.2. An alternative is a quasi-steady approach in the immediate vicinity of the stagnation point and provided there are no flow reversals, this is simpler. For unsteady flows with low frequency, it is expected that the assumption of quasi-steady flow in the region near the stagnation point will not affect the overall solution. 5.5.2 Transformed Equations As in two-dimensional steady flows, it is again desirable and convenient to express the unsteady boundary-layer equations in tranformed variables. Standard P r o b l e m When the initial conditions are generated in the (£, y) plane for t > 0 with the procedure to be described in subsection 7.2.2, it is more convenient to use a modified form of the Falkner-Skan transformation given by Eq. (5.3.3). With the definition of a dimensionless distance Y and stream function / ( £ , y ) by

V = (UouL)1/2/^, Y)

Y = J^-y,

(5.5.1)

and a prime denoting differentiation with respect to Y, Eqs. (5.1.2) to (5.1.4) can be written as

(6/} +

^

+w

+f

Y = 0;

/ = /' = 0

(5.5.3a)

f' = w

(5.5.3b)

^ =^

Y = Ye;

^~f

oc

(5 5 2)

--

with the definition of (5.5.4) Eq. (5.5.2) can also be written as

5.5 Solution of the Standard and Inverse Problems

79

(5.5.5)

Inverse P r o b l e m To obtain the solution of the unsteady boundary layer equations for the inverse problem, we use the transformation given by Eq. (5.4.2) with Uoo replaced by UQ and express them in the form

The wall and edge boundary conditions given by Eqs. (5.4.4) remain the same except that F, U, V are represented by / , u and v. Equation (5.4.7) becomes Y = Ye,

fi(C) - \[Yefi(?)

- fe(e)]

= 9r

(5.5.7)

where gi is given by Eq. (5.4.8b) and A by Eq. (5.4.8a). 5.5.3 N u m e r i c a l M e t h o d : Flows w i t h o u t Reversal To discuss the numerical method for unsteady flows, let us assume that quasisteady approach is being used. Again we write Eq. (5.5.6) as a first-order system by using the new variables iz(£, 77), v(£, 77) defined by Eqs. (5.3.10a) and (5.3.10b)

du

(bv)' +^fv = i u

v

df

rr dt

dw

dw

(5.5.

To write the difference equations for the inverse problem represented by Eqs. (5.4.9) and (5.3.10a,b), we consider the net cube shown in Fig. 5.3 and denote the net points

(n-l,i,j)

uG)

OUj)

A

(n,i-lj)

(n-l,i-lj) . x(i) (n-l,i,j-l)J/_

(n,i,j-l)

_

/ / / (n-l,i-l,j-l)

(n,i-l,j-l)

^T

§(n)

F i g . 5.3. Net cube for the difference equations for two-dimensional unsteady flows.

5. Boundary-Layer Methods

80

£° = o,

e = c _ 1 + kn,

n = l , 2 , . ..,N

vo = o,

nj = nj-i + hj,

J = 1,2,.. . , J

r° = 0,

Ti

* = 1,2,.. • , /

=Ti~±

+T.

(5.5.9)

where the definitions of kni hj are identical to those in Eq. (5.3.12) and rz- = AT{. The difference approximations to Eqs. (5.4.9) and (5.3.10a,b) are obtained by averaging about the midpoint (^ n ,T z ,^j_i/2)) n,i

wJ

W

n,i

J

,

(5.5.10a)

=o

hj n.i

hj

,

n.i

n.i

n.i

_

n.i

V

i

hj

(5.5.10b)

Vi/2

2

.

n.i

+ V l 2

_

n,i i-V2

(5.5.10c)

The difference approximations to Eq. (5.5.8) are obtained by centering all quantities at the center of the cube ( £ n _ 1 , r 2 - 1 / 2 , ^ - 1 / 2 ) by taking the values of each, say ^ j , at the four corners of the box, that is, n-1/2,7-1/2

V J- / n.i

.

n,2 — l .

= 4(9/ + 5 / 1

n—l,i — l .

+ 9,

n—l,i\

+ 9;

9,-1/2 = 2 ^ J + 9 J - I ) = 4(9"'* l/2

+

)

(5.5.11)

Qj-l/2)

71— 1,Z— 1 Here by g 234 we mean ^q2M „v. ^.v.^. g?n.i -"'"—1 + q'j "'* " + q'j 3i "'*, the sum of the values of ^- at three of the four corners of the face of the box. In terms of this notation, the finite-difference approximation can be written as:

71— 1,:

jr _

tn-1/2

+ 2(^-1/2 (u2)n

/•71—1 M

3-1/2

j

i~l/2

2rCn

_ Wj-1/2

J

•0-1/2

j-l/2 ^71

(5.5.12) l/2Mi-l/2

_

tn-1/2

"j-1/2

(™ 2 )?-l/2 - ( ^ ) j n - l / 2 , * } - l / 2 " ^ - 1 / 2 AKj)

+

Ti

5.5 Solution of t h e S t a n d a r d a n d Inverse P r o b l e m s

81

where -n Qj-i/2

n,i—1/2 1 / n.i , n.i—1 . n.i , n.i—1\ J- / n.i . n,i—l \ = Qj-i/2 = i^i + 9j + «i-i + «i-i ) = 2(gi-V2 + ^-1/2)

-i n—1/2,z J- / n,i . n—l,i , n.i . n—l,i\ ^ / n,z 9 , - 1 / 2 = Qj-l/2 = ~A^3 + «i + « i - l + « i - l ) = 2 (
W

-i j-l/2

n—l/2,i 1 / n,z . n — \,i \ j-l/2 = 2(^-1/2 + ^-1/2)

W

=

(^2\n

n—l,i \ S-l/2)

1 / n,z . 234 \ =i(Wi-l/2+%-l/2)'

3-l/2=Wj_1/2 W

, +

_ /

2\n,i-l/2 _

1 r/ ? | ,2\rM

, /

2\n,z-l 1

(5.5.13) Making use of the relations defined by Eqs. (5.5.11) and (5.5.13), after some algebra and rearranging. Eq. (5.5.12) can be written in a form similar to Eq. (5.4.12d) for two-dimensional steady flows, u — IIYL

h3

\n,i

/L

\n4

1 1 1 / r

[{bv)y -{bv)jll] 1

fM

\n,i

n(

1 ^

2\n,i

n.i

±{fv)j'_1/2-an{uz)Jl/2-aluj>_l/2

+ r n4

-cnii

1

234

pn.i

i

/ r cr 1 o\

+ a2^:i/2 + -yfy-i^/j-i^ + ^_1/2/,:1/2] + an(wz).'_

1/2

+ aiiy'

/2

(5-5.13)

= /?i

tn-1/2

a i = 2^

(5.5.14a)

«2 = Y ( ^ - V 2 - 2 ^ I % )

(5.5.14b)

tn-1/2

a3 = ^ r A = hf[(bv)f±

(5.5.14c) - [bv)f_\] +

\(fv)?_\/2

- a»[(«2)£j^ - 2 ^ ) ^ - "1(1*^ - 2 ^ ) + «i(%nri/2 -

2

*i-i/2)

+ a"[(«,2)^ -

v*\/2(f^-2f^/2)} 2 ( ^ ) ^ 1 (5.5.14d)

Equations (5.5.10) and (5.5.13) are again imposed for j = 1, 2 , . . . , J at given n, i. They are accompanied with boundary conditions for the inverse problem by Eqs. (5.4.18). The nonlinear system is again linearized by Newton's method and written in a form similar to the linear system for steady flows. The linearized equations for Eqs. (5.5.10) are given by Eqs. (5.4.15) and (5.3.21a,b). For Eq. (5.5.13), it is given in the same form as Eq. (5.3.21) with, again, two terms are added to the left-hand side,

5. Boundary-Layer Methods

82

(si)j8vj

+ (s2)jSvj-i

+ (se)j6uj-i

+ (ss)jSfj +

+ ( s 4 ) j « / j - i + {sb)jSuj +

+

(S7)J6WJ

(SS)JSWJ-I

This time, however, the coefficients for (r2)j and (si)j to (ss)j they are given by

+

a

V >

f<">

+ ,234

(5.5.15)

= (r2)j

f<"> 1

are

different;

(5 5 16)

' '

(^•-V^n/j-Uf^/^

(5.5.17b)

( « 0 ; - \vf + °^[vf}l/2

(5.5.17c)

Mj

= \vf},

( ^

= -oTuf

+ £ [ » < % + i^1/2] -^

(s6)j = -anuf\ (s7); = «

n

^

+ vf_\/2]

}

-^ + ^

(s 8 ), = a - u - j i + ^

(5.5.17d) (5.5.17e) (5.5.17f) (5.5.17g) (5.5.17h)

The boundary conditions for the inverse problem are identical to those in Eqs. (5.4.18) to (5.4.19) provided that A is defined by Eq. (5.4.19). The solution of the linear system given by Eqs. (5.5.10) and (5.5.15) subject to the boundary conditions in Eqs. (5.4.18) is identical to the procedure described in Section 5.4 for the inverse problem, provided that initial conditions in the (x, y) plane at r = 0 and those in the (£, y) plane at £ = £o are specified. In the quasi-steady approach, the solutions in the (x,y) plane at r = 0 are obtained by solving the steady flow equations, Eq. (5.4.3), which is a reduced form of the time-dependent equation (5.5.1) at r = 0. The initial conditions in the (£, y) plane (see subsection 7.7.2) are obtained from the solution of Eq. (5.5.2) for a given external velocity distribution for a specified range of £ values and for all desired time intervals at r > 0.

5.5 Solution of the Standard and Inverse Problems

83

5.5.4 N u m e r i c a l M e t h o d : Flows w i t h Reversal When there is flow reversal across the shear layer, it is necessary to modify the standard box of the previous section in order to avoid the numerical instabilities resulting from integration opposed to the flow direction. A convenient procedure is to include the zig-zag formulation of Krause et al. [11]. In common with the often-used Crank-Nicolson method, this scheme is easy to employ, particularly since the orientation of the numerical mesh is chosen a priori. This advantage has a corresponding and potentially dangerous drawback in the presence of large reverse flows, since the mesh ratio must be related to the velocity according to the famous Courant, Friedricks, Lewey (CFL) condition [2,12] if stability is to be achieved. For a fixed grid chosen a priori, this condition may be violated as the flow velocities are determined in ever increasing computational domains. Thus a natural boundary limiting the domain in which stable computations can be made is also determined a priori. One way to avoid the above limitation is to allow the grid to be determined along with the flow calculations and we shall describe a numerical scheme which does this. The grid spacings and orientation are adjusted depending upon the magnitude and direction of the local velocity so that the CFL condition is satisfied. The scheme is thus in some sense intelligent in that it maximizes the domain in which the computations can be carried out. For completeness we first describe the zig-zag scheme and then proceed with the intelligent scheme which will be referred to as the characteristic box scheme. Zig-Zag B o x To solve Eq. (5.5.12) with the zig-zag box scheme, we start with the first-order system given by Eqs. (5.3.10a,b) and (5.4.9). The main difference between the standard box of the previous subsection and the zig-zag box scheme of this subsection lies on how the difference equations are written for Eq. (5.5.8); the remaining three equations, Eqs. (5.3.10a,b), (5.4.9), and boundary conditions remain unchanged. To write the difference equations for Eq. (5.5.8) centered at P (see Fig. 5.4), we use quantities centered at P , Q and i?, where

Tl(j) A

,x(i) -+ £(n) En

Fig. 5.4. Finite-difference molecule En+i for the zig-zag box.

5. Boundary-Layer Methods

84

P = (CTi-V^Vj-ip), R

=

(Zn-l/2,T\Vj_1/2),

Q=

(5.5.18)

ic+i/2y-ifrij_i/2)

and, for convenience, use the following model equation for Eq. (5.5.8),

(by)' + -fv = t

du_dl +£ ["at V d£

(5.5.19)

and write Eq. (5.5.19) in the form

(bvy(p) + yV(p) du du df, (R) - 6v(Q)^-(Q) Z(P) 0u(Q) — (Q) + 4m(R) — OH du

9f,

-

4>v{R)it(R)

d(,

(5.5.20) where •n+l

e

•ra-1

e

•C

£n+l _ tn—1'

^

(5.5.21)

£ n + l _ en—I

The difference equations for Eq. (5.5.20) are /,

\Tl.i—1/2

{bv)j

/ L

\7l,2—1/2

- (fo)j-i hi ra,i

+

i -^ / i- \n,i—1/2 0(/V)i-l/2

_

n—l,i

n+1,2

fl,n-l/2,t"j-l/2~"j-l/2

^-1/2

jT

n-l/2,jJj-l/2-Jj-l/2

0

J-l/2 71,2

r

^j-1/2

—1

, n+l/2,i-l V l / 2 _ + ^-1/2 jfe" ;

n,z—1

Vl/2

n+l wi+l,i—1 rn,i—l ~ Jj-1 /2 .n+lfaii-lJj-1/2 1/2 ^n+1

_

tin n.i—1

(5.5.22)

"j-1/2

A similar procedure can obviously be applied to Eq. (5.5.8). The resulting algebraic system is again nonlinear and can be linearized with Newton's method. The linear system is identical to the linear system discussed in subsection 5.5.3, except for the coefficients (r2)j and (s\)j to (SS)J; they are given by ( r 2 ) , = - f t - [h-l[{bv){p 2\M

"l («)}_!/:

- (bv)
+ "ll«j_l/2^-l/2

„2\M

a

J-l/2

+

V

hUvf;ll/2 (")

3uj-l/2

j-l/2Jj-l/2i

(");

+ a i K ) y i / 2 + a2^_ 1/2

(5.5.23)

5.5 Solution of the Standard and Inverse Problems

85

Wi-vT + X'-f (so)

. _ _^-U(")

2 JJ-i/2

(5.5.24a)

Ol .(„) ' 2 " / i- 1 /2

(5.5.24b)

, I f (") _ ^ 2

71—1,1 l

(5.5.24c)

(«3)j = 4«i" ) + Y [ ^ - l / 2 + V l / 2 J 4"J-I

-a\u Oe)j

3

-a\u.

a\w

(5.5.24e)

2

(")

(s 7 )j = aiuA(") + (ss)j

(5.5.24d)

' 2 l i-V2 a .(") 2

(5.5.24f) CH2

(5.5.24g)

2 CC2

(") 'i-i

(5.5.24h)

T

where n,i— 1 ft = hf[(bv)y-1 - (bv)]^1} + l(fv)^ J-l/2

+ 20i

^

«ra+l

n + l / 2 , i — 1 / rn+l,i— 1 _ rn,i—IN _ ^-1/2 Wj-1/2 -/J-1/2J

/ 2\n— l,i

,

n,i—l

,

71+1/2,2—1/ «7l+l

w,3-1/2 -, ; 0

j-l/2

^j-l/2

71,2—1 \

^•-1/2^

2\n—\,i

+ OLl(U )j_lj2+OL2Uj_1/2-OLl{w

-20-

71+1/2,2-1/ 71+1,1-1 l

) J -_ 1 ^ 2

71+1,2—1

(w,3-1/2 -, ; 0

72,2— 1 \

WALI/2) 1/2'J

-

71,2—1

a2w ' j - l / 2

(5.5.25a)

£72

(5.5.25b)

ai ^72

«2

(5.5.25c)

> ^ ^2 £72

«3

is

rn—l,i

(5.5.25d)

Characteristic B o x The characteristic box scheme, developed by Cebeci and Stewartson [3], is based on the solution of the governing equations along local streamlines. This method uses Keller's method and allows the grid to be determined along with the flow calculations. It is described here for two-dimensional unsteady flows with flow reversal and separation. To solve Eq. (5.5.6) subject to its wall boundary conditions u0 = 0,

#o = 0

5. Boundary-Layer Methods

86

and edge boundary conditions given by Eq. (5.5.7) with the characteristic box scheme, we define 6' by 9

'=2U

+^

(5.5.26a)

and write Eq. (5.5.6) as (bv)> + ve

+

t(-+w-)=t(-+

u-

j

(5.5.26b)

Noting the definition of local streamlines, which in our notation dr _ d£

(5.5.27)

T ~V

and denoting the streamline distance by ip and the angle that it makes with the r-axis by (f) (see Fig. 5.5), we write Eq. (5.5.26b) as (by)'+ £P + v0 = £\—

(5.5.28)

where

( 3

A = y/l + u2

(5.5.29a)

0 = tan-1 u

(5.5.29b)

dw = ^

+

1 dw2 2~Ol

. „^ x (5 5 29c) - -

The solution of the transformed boundary layer equations is obtained by defining two regions depending on the sign of the streamwise velocity u. In the region 1, u is negative, and the solution of the boundary layer equations is obtained from Eqs. (5.3.10b), (5.5.26a) and (5.5.28) subject to wall boundary conditions. The difference approximations to Eqs. (5.3.10b) are given by Eq. (5.5.10c) and (5.5.26a) by Qn-\/2,i-l/2

Qn-l/2,i-\/2

hj

J

V

= 8(^-l/2 + V l / 2 + Uj-l/2 +

hj

+ uj-l/2

)

(5-5-30)

/ n,i n—l,z , n,i—1 n—l,z—1\ ^n-l/2 ^"j-1/2 ~ Uj-l/2 + "j-1/2 ~ "j-1/2 > Zfvn

The difference approximation to Eq. (5.5.28) are obtained along the streamline distance Aipj at point B (see Fig. 5.5)

87

5.5 Solution of t h e S t a n d a r d a n d Inverse P r o b l e m s

A

h-k.^-

Xi+l

x

x

*X

Known

°

Unknown

3f

^r

Ti-i

-A&

Ti-2

£n-l

x(n)

^"+1

£n

F i g . 5 . 5 . N o t a t i o n for t h e characteristic scheme.

/ L \n,i

/ L \n,i

\m,i—l \m,z—i

/i

m, i—1 \m,i-

/i

+

+ ^Bi^f- 1 / 2

+ (^m + ^mK-m n,z £ # /

-f

An,Z

.

_

(5.5.31)

m,i— 1

,i-K V l / 2 ^ V l / 2

A 771,Z-

(Vl/2 + Vl/2^

^_1/2

where ^j-1/2

(5.5.32) COS(/)J_1/2

The relation between 0?_1,2 and those values of 6 centered at (n—1/2, i —1/2) and ( n - 3 / 2 , i - 1/2) are nn-l/2,i-l/2 _ J-l/2 cn-1/2 _

B

9 0-1/2

^n-3/2,i-l/2 J-l/2 , . £ _ ^n-3/2x cn-3/2

fln-3/2,i-l/2 J-l/2

+

(5.5.33a)

or B ^ - 1 / 2 - = -2 ( ^ + ^ _ 1 ) / 1 + / 2

(5.5.33b)

where £# _ £n-3/2 1

tn-1/2 _ £n-3/2'

h = {i-h)onjzll^i-1'2

(5.5.34)

The pressure gradient term {3 is given by

/3f_lrt = ( ^

+

1

(dwr j-i/2

(5.5.35a)

5. B o u n d a r y - L a y e r M e t h o d s

n,i—l ;

n,i J-l/2

.* 2A

n—l,z

w

A/2

| A

j-ih

n—l,i—l

w

3-\h (5.5.35b)

n-2,i-l T

<9w2\

5

"arj.... J-l/2

5l

' ^

2\n,i—1 >i-l/2 ^ R (W ) j - l / 2 + ( ^ + #2 7)

2\n4 i / ^-1/2

2

+ B3

•

W

) j - l / 2 + (W

)j-l/2

)j-l/2

(5.5.35c) ^ _

(

1

r

- i

)

(

^ _

-

r

2

)

2

(e-e-^ce-e- ) (gB n_1

_r)(eB_gn-2) — £ n )(£ n _ 1 — £ n ~ 2 )

2

(£

3

(£ n _ 2 — £ n )(£ n _ 2 — £ n _ 1 ) otB

Bl =

en—2

(e-e^xe-e- 2 ) 2 gB

^

(5.5.36a)

- gn _

f

-2

( Cn — l _ cn\Un-2 trt,$tn. — cl _ (Cn-2

(5.5.36b)

tiCn-1\

As before the system given by Eqs. (5.5.10c), (5.5.30), (5.5.31) and the boundary conditions are linearized with Eq. (5.5.10c) given by Eq. (5.3.21b), the linearized form of Eq. (5.5.30) is h

— + h{6uj + 6UJ-I) = (r 3 )j-i

(5.5.37)

where

6l= (r3)j-i =

-(^+W -

Vj-l

l_nn-l/2,i-l/2 2

2VV n,i

_ tn-l/2

\ _

kn _

n—l,i

u" j - l / 2 ~ Uj-l/2

(5.5.38)

, n,i—1 _ n—l,i—l-i + "j-l/2 ~ "j-1/2

After linearization, Eq. (5.5.31) is expressed in the same form as Eq. (5.5.15) with different coefficients for (r2)j and (s$j to (ss)j where

5.5 Solution of the Standard and Inverse Problems

89

<•>>> = £ + 2«f-v»

(5.5.39a)

(5.5.39b) /

\

J- / n.i

, m,i—l\

(5.5.39c)

(S3)j = 2 ^ - 1 / 2 + V l / 2 )

(5.5.39d)

(S4)j = (S3)j ' \

{S5)i

A

B

n

^

"2« ^ X71'^

(«0; = - ^

B

A

_1_ \

^ ~ 1 \

n

/

5^

772,2 — 1 '

j - l / 2 1 _ KB^I ( UJ~l/2 ~ " j - 1 / 2 2 Aj I ^j-1/2 ^j-1/2

j-l/2

+ A

r n

n,i _ m,t—1 "j-1/2 ~ "j-1/2

_L_ X 7 7 " 1 ^ - 1

j - l / 2 + \ J --l / 2 J _ l ^ B " j - l 2 A,-_i A*l>j-l/2

^ - 1 / 2

(5.5.39g)

(«8)i = ^

(

^1 , S i

/?

iX2)i =

+ m,i—l\fiB

+ ( ^ : i / 2 + Vj21/2)0j-l/2 £^ \n,z

? IVl/2

(5.5.39h)

\77i,z—1

\m,i—1

/i7

(5.5.40)

+ 2^i-l/2

n,z I I I jT_ \ ra,z—1 ^ - 1 / 2 "

^-1/2

/i

- (Hj-i

(Hj

71.

m,z—1 j-l/2

U

M3-l/2

The wall boundary conditions are 69o = 0 6UQ

(5.5.41)

=0

In the region 2, u is positive and the solution of the boundary layer equations is obtained by applying the regular box scheme t o Eqs. ( ) and (5.5.4a,b) subject t o the edge boundary conditions given by ( ). After linearization, they are expressed in a form similar to those in region 1. For example, Eq. (5.5.26b) can be written as /, ^/

n

^

w

^dw2

.du

£du2

5. Boundary-Layer Methods

90

(bv)j -

(bv)j-

r

- i

1

/ 2

. i", 0 ^-1/2^-1/2

n-l/2,i-l

n 1 1 2 - ^- /' - /( ^ ) j _2\ 1;2

/ ( ^ )2\n,i-l/2 j:1/2

Z

(5.5.42)

/CTT,

»-l/2,i _ , n-l/2,i-l ^n-l/2Mi-V2 "j-1/2

^_

1 / 2

( ^ - 1 / 2 _

^2^-1^-1/2

+

7"?;

where

n-l/2,i 3-1/2

| tn-l/2

/C72

z

( ) = [( ) n " M + ( ) n " M + ( ) n ' 1 - 1 + ( ) n " 1 - i - 1 ]/4

After linearization, it again can be expressed in the same form as Eq. (5.5.15) with {s\)j to (ss)j and (rz)j given by (5.5.43a)

(-1); = bf. + \0i-H2

(5.5.43b) (s3)j

=

(5.5.43c)

2VJ_1/2

(S4)j = {S3)j tn-1/2 £ra-l/2 Ui (*5)j = -

"*

£71-1/2 (^)j = £71-1/2 (s7)j =

1 £71-1/2

s

( $)j

=

(5.5.43d) (5.5.43e)

£71-1/2 7 %-l

(5.5.43f)

£71-1/2 ; Wj

(5.5.43g)

£71-1/2

+

(5.5.43h)

-Wj-l ^71

(bv)j - ( M ; - i

(r 2 )i = - 4

+

Vj-i/20j-l/2

n-1/2 n-l/2,i-l ! £ » - l / 2 «;..J - ,{„ l/2 - ^ - - x / 2 -i /o / 2 \ n — 1 / 2 , i / 9\n_1^—1/2 ^"-1/2 ( w 2 ) ^ ^ ' - ( « ^;

J-l/2

+

fCn

n-l/2,i _ n-l/2,t-l _ £n-l/2"j-l/2 "j-1/2

r

- l / 2 («2)^/V2 " ( t * 2 ) ^ " 1 / ^

(5.5.44)

References

91

The wall boundary conditions are given by Eq. (5.5.41) and the edge conditions are given by Eq. (5.4.7) with D and A now defined by D = ue6*^RL

(5.5.45a)

* = Cuy/l

(5.5.45b)

The solution of the linear system given by Eqs. (5.3.21b), (5.5.37), (5.5.15) with coefficients (s\)j to (s$)j and {r2)j given by Eqs. (5.5.39) and (5.5.40) for region 1 and by Eqs. (5.5.43) and (5.5.44) for region 2, subject to the wall edge boundary conditions given by Eqs. (5.5.41) and (5.4.7) is obtained with the block elimination method described in subsection 5.3.3.

References [I] Cebeci, T., Analysis of Turbulent Flows, Elsevier, London, 2004. [2] Cebeci, T., Shao, J. P., Kafyeke, F. and Laurendeau, E., Computational Fluid Dynamics for Engineers, Horizons Pub., Long Beach, Calif., and Springer, Heidelberg, 2005. [3] Cebeci, T. and Cousteix, J., Modeling and Computation of Boundary-Layer Flows, Horizons Pub., Long Beach, Calif., and Springer, Heidelberg, 1998. [4] Catherall, D. and Mangier, K.W., "The interpretation of two-dimensional laminar boundary layer equations past the point of vanishing skin friction," J. Fluid Mech., Vol. 26, p. 163, 1966. [5] Briley, W. R., "A numerical study of laminar separation bubbles using the NavierStokes equations," J. Fluid Mech., Vol. 47, pp. 713-736, 1971. [6] McDonald, H. and Briley, W. R., "A survey of recent work on interacted boundarylayer theory for flows with separation," in: Numerical and Physical Aspects of Aerodynamic Flows, II, ed. T. Cebeci, Springer, Heidelberg, 1983. [7] LeBalleur, J. C , "New possibilities of viscous-inviscid numerical techniques for solving viscous flow equations with massive separation," in: Numerical and Physical Aspects of Aerodynamic Flows, IV, ed. T. Cebeci, Springer, Heidelberg, 1990. [8] Carter, J. and Wornom, S. F., "Solutions for incompressible separated boundary layers including viscous-inviscid interaction," in: Aerodynamic Analysis Requiring Advanced Computers, NASA SP-347, p. 125, 1975. [9] Veldman, A. E. P., "A numerical method for the calculation of laminar incompressible boundary layers with strong inviscid interaction," NLRTR 79023L, 1979. [10] Cebeci, T., An Engineering Approach to the Calculation of Aerodynamic Flows, Horizons Pub., Long Beach, Calif., and Springer, Heidelberg, 1999. [II] Krause, E., Hirschel, E. H. and Bothmann, Th., Die numerische Integration der Bewegungsgleichungen dreidimensionaler laminarer kompressibler Grenzschichten, Fachtagung Aerodynamik, Verling, 1968, DGLP-Fachzeitschrift, Band 3. [12] Isaacson, E. and Keller, H. B., Analysis of Numerical Method, John Wiley, New York, 1966.

Applications of Boundary-Layer Methods: Flows W i t h o u t Separation

6.1 I n t r o d u c t i o n In this chapter we discuss the applications of the boundary-layer method to laminar and turbulent flows without flow separation; flows with separation are addressed in the following chapter. In Section 6.2 we discuss laminar and turbulent flows on a flat plate with fluctuations in external flow and in Section 6.3 the development of unsteady laminar boundary-layers when the body is given, impulsively, a free-stream velocity. We consider two cases, the first corresponding to impulsive motion of a flat plate and the second a circular cylinder. The latter has been used extensively as a model problem to study the nature of solutions in the presence of flow reversal, flow singularity and separation.

6.2 U n s t e a d y B o u n d a r y - L a y e r s w i t h F l u c t u a t i o n s in E x t e r n a l V e l o c i t y There are several unsteady boundary-layer problems in which it is necessary to account for the fluctuations in the external flow. These fluctuations may change both in direction and in magnitude. A simpler case is one in which the external flow fluctuates only in magnitude and not in direction. In this section we consider the solution of such problems for laminar (subsection 6.2.1) and turbulent (subsection 6.2.2) flows. 6.2.1 Laminar Flows The problem of laminar flows has been studied previously by Lighthill [1] for a flat-plate flow. According to his analysis for an external flow in the form

6. Applications of Boundary-Layer Methods: Flows Without Separation

94

ue(t) = UQ(1 + pcosout)

(6.2.1)

the reduced skin-friction coefficient (cf/2)y/R^ is given by two separate formulas depending on whether the local Strouhal number u (= UJX/UQ) is much smaller or much greater than one, (Rx = UQX/V). cf

j

f 0.332 + B(0.498 cos wt - 0.849a; sin ut) w « l = < ffi 2 2) X 2 [0.332 + Bu1'2 cos(o;£ + TT/4) £ > 1 In many problems it is often desirable to find the phase angle, 0, between the external flow and, say the reduced skin-friction coefficient, which in terms of the transformed variables defined by Eq. (5.3.3) is f!^. To determine this numerically for a fixed x = xo, let us consider the general case in which the external flow is given by — A//?

ue(x,t)

= izo 0*0(1 + Bcosujt)

(6.2.3)

and let f!£(x, t) — g(x, t). We use the following procedure to determine the phase angle between ue(xo,t) and g(xo,t). We first compute ^o(^o) and g(xo) from 1 o(xo) = y,

u

rto+P

ue(x0,t)dt Jto I rto+P 9(xo) = -5 / 9(xo,t)dt

(6.2.4a)

F

(6.2.4b)

Here P denotes the period defined by P = 2it/uj and to must be taken sufficiently large so that all transient effects have decayed. Usually to = P is adequate. From Eq. (6.2.3) we can write [A = uo(xo)B] ue(xo, t) — uo(xo) = Acosut

(6.2.5)

Similarly, we can write g(x0,t)

-g(x0)

= Ccos[ujt + (p(xo)}

^ . {0.2.0)

= C[cosu;tcos(/)(xo) — smujtsm(j)(xo)] with (J>(XQ) denoting the phase angle between ue and g at x = XQ. If we take the product of Eqs. (6.2.5) and (6.2.6) and integrate the resulting expressions, we find cos0(xo) to be given by ., v j{°+P{[ue(xo,t) cos 0(x o ) = - ^

- u0(x0)}[g(x0,t) j ^ M

-

g(x0)}}dt (6 2 7)

- '

Here A2 = - / 71" Jtn

[« e (xo, t) -

UO(XQ)}2

dt

(6.2.8a)

6.2 Unsteady Boundary-Layers with Fluctuations in External Velocity

C2

to+P

* J to

[g(x0lt) -

g(x0)]2dt

95

(6.2.8b)

Using this procedure we find that the phase angle [between the external velocity and reduced skin friction Eqs. (6.2.2) and (6.2.3) respectively] according to Lighthill's analysis is 1 (l + 2.9064a>)1/2

" ^ *

0.707

UJ > 1

(6.2.9a)

COS0 =

(6.2.9b)

Figure 6.1 shows a comparison of results obtained by the numerical solution of the boundary-layer equations described in Chapter 5 and those obtained from Eq. (6.2.9). The numerical calculations were made by Cebeci [1] for an external flow given by Eq. (6.2.1) with B = 0.150 and UQ = 19.5. A total of 41 t-stations and 26 x-stations with At = 0.20 and Ax = 0.1 were taken. Initially, r]e was taken as 10 with Ar\ = 0.25. These numerical calculations are in good agreement with those computed by Ackerberg and Phillips [2]. They show that if B < 0.150, Lighthill's low-frequency approximation is accurate for UJ < 0.20i£o/^ and his high-frequency approximation is accurate for UJ > 2.6uo/x. Oscillating laminar flow in a pipe, which is of great importance in the study of viscous attenuation of sound waves, is discussed by Rosenhead [3]. An analytic solution of the Navier-Stokes equation can be obtained in the form of a Bessel

CO

Fig. 6 . 1 . Variation of phase angle
6. Applications of Boundary-Layer Methods: Flows Without Separation

96

function. For high frequencies of oscillation the viscous layer is confined close to the pipe wall, corresponding to the reduction of boundary-layer thickness with increasing frequency in oscillatory external flows.

6.2.2 Turbulent F l o w s The experimental data on unsteady turbulent boundary-layers are very limited. Karlsson's flow [4] consists of an oscillating freest ream in a zero pressured gradient turbulent flow. In order to compare the numerical calculations with this data, we take the external flow to be the same as that of Eq. (6.2.3) and use the same eddy-viscosity formulation given by Eqs. (5.1.5) and (5.1.7).

1.2r

0.8

0.4

2.0 3.0 y (INCHES)

4.0

y (INCHES) 2.0 3.0

4.0

(a) Fig. 6.2. (a) Comparison of calculated and experimental velocity profiles at Rs* = 3.6 x 10 3 . Symbols denote u/uQ for values of B = 0.292, 0.202, and 0.147 from Karlsson [4]. The solid line denotes the numerical solutions of [5] and (b) Experimental data from Karlsson are for UJ/2IT = 4.0 cycles s _ 1 : UoJ/UQ = 26.4% (circles), 13.65% (triangles) and 6.2% (squares). The solid lines denote the numerical solutions of [5] for u)J/UQ = 26.4%.

Figure 6.2 shows the results for Karlsson's data. These calculations were started as laminar at x = 0 for the external flow given by Eq. (6.2.3) with uo = 19.5 ft s - 1 . The zAx-spacing for turbulent flows was 1ft. The turbulentflow calculations were started at x — 0.1ft and at x = 12 ft, the experimental data [corresponding to R$* (= uoS* jv) of 3.6 x 103] were matched as shown in Fig. 6.2a. Figure 6.2b shows a comparison between calculated and experimental values of the in-phase and out-of-phase components that is, u^ COS( u^J and w1' s'm(/)/uoo , respectively according to Karlsson's notation.

6.3 Boundary-Layer Motion Started Impulsively from Rest

97

6.3 Boundary-Layer Motion Started Impulsively from Rest In problems involving boundary-layer motion from rest we are interested in the development of the unsteady boundary-layer when the body is given, impulsively, a free-stream velocity. In general, the external velocity can be expressed as (0 t<0

ue(x,t) = {

(6.3.1)

The governing boundary-layer equations are given be Eqs. (2.4.11) and (2.4.16). They are subject to the conditions given by Eq. (2.4.17). At the beginning of the motion the boundary-layer is of zero thickness. For this reason we can set the convective terms in Eq. (2.4.15) equal to zero to get du

= V

ai

d2u

W

(6 3 2)

-"

The solution of Eq. (6.3.2) subject to Eq. (2.4.17) is u = ueerf(()

(6.3.3)

Here the variable £ and the error function are given by

C = -^=

erf (C) = 4= / C e~? d(

(6.3.4)

This problem was first considered by Lord Rayleigh [5] for a plate that moved impulsively from rest; it was extended to the case of a semi-infinite plate by Howarth [6]. To find the development of the flow for later times (t —> oo and steady-state conditions), we need to solve the system given by Eqs. (2.4.11), (2.4.15) and (6.3.1). To illustrate the calculation of such flows we shall now consider two simple cases, the impulsive motion of a flat plate (subsection 6.3.1) and that of a circular cylinder (subsection 6.3.2). 6.3.1 Impulsively Started Flat P l a t e The development of a boundary-layer over an impulsively started semi-infinite flat plate has two features. Initially, the flow is identical to that given by Eq. (6.3.3) and the flow tends ultimately to that given by Blasius by solving the laminar boundary-layer equations for a flat plate

/'" + //" = o The interest in the calculation of this flow and flows of this type is to describe the evolution of the flow from the initial to the steady state.

6. Applications of Boundary-Layer Methods: Flows Without Separation

98

The problem was first studied by Stewartson [7]. He found that the structure of this boundary-layer at a fixed distance x from the leading edge is different, according to r ^ 1, where r = uet/x or, more generally, t/ f£ dx'/ue. If r oo. This means that the disturbance caused by the presence of the leading edge travels down the boundary-layer with the maximum local velocity at any station of x, ue. Hence the effect of the leading edge is felt when r > x/ue or, in general, r > J dx/ue. Note that there is a fundamental difference between sharp-edged plates and bodies with a front stagnation point near which ue ~ x, so that the integral becomes infinite [8]. The first numerical solutions to the problem were obtained by Hall [9]. He solved the systems Eqs. (2.4.11) and (2.4.16) subject to the sharp-edged plate problem. At x — xo and t > ta he assumed initial-velocity profiles u = ua{t,y) and used the Rayleigh solution [Eq. (6.3.3)] at t = ta and x > x$. Then making use of a similarity condition that the solutions must satisfy at some x-station, he used an iteration procedure to improve (or update) his initial-velocity profiles, u = ua(t,y). He observed that his iteration procedure converged rapidly. — 1/2

Figure 6.3 shows the variation of wall shear parameter f!^ [= Rx x/ue (du/dy)w] with dimensionless time t* (= u^t/u). Note the rapidity of the transition from the Rayleigh to the Blasius state. Mathematically, an infinite time is required for the attainment of the steady state. Numerically, for all practical purposes the steady state is reached by t* — 4, as shown in Fig. 6.3. The figure 0.61

0.5

/; 0.41

0.3'

4

Fig. 6.3. Variation of wall-shear parameter f^ with dimensionless time for an impulsively started flat-plate flow. Calculations were made by Hall [9].

6.3 Boundary-Layer Motion Started Impulsively from Rest

99

also indicates that there is no appreciable departure from the Rayleigh state until t* = 2. These solutions were later confirmed by Dennis [10], who solved the governing equations in the similarity form. 6.3.2 Impulsively Started Circular Cylinder: U n s t e a d y Separation As was discussed in Section 5.2, for a prescribed external velocity distribution ue(x), the solutions of the two-dimensional boundary-layer equations for steady flows become singular when the wall shear vanishes. The physical interpretation of this result is that ue(x) is incorrectly chosen and that in order to determine the properties of the fluid motion at high Reynolds-number flow, the boundarylayer must not be considered separately from the external inviscid flow: instead, it is necessary to use an interactive theory such as the one described in Chapter 5 that allows for the mutual interaction between the boundary-layer and the inviscid flow. The generalization of this result to unsteady laminar flows has attracted many workers to conduct extensive studies on the flow over a circular cylinder started impulsively from rest. Calculations with laminar boundary-layer equations were made to see whether the breakdown of the solution will always coincide with the vanishing of wall shear or away from the wall. Here we shall refer to the occurrence of the singularity as separation and discuss the progress made in our understanding of the concept of unsteady separation in laminar flows together with presenting the solutions for flow over a circular cylinder impulsively started from rest. We start the discussion with the incompressible boundary-layer equations given by Eqs. (2.4.11) and (2.4.16) for laminar flows and assume the following boundary conditions 2/ = 0,

u = v = 0,

y —> oo,

u —> ue{x, t)

(6.3.5)

It was suggested by Moore [11], Rott [12], and Sears [13] that for laminar flows, the solution of the system given by Eqs. (2.4.11), (2.4.16) and (6.3.5) will indicate flow separation if and when there is a point xs{t),ys(i) in the flowfield such that u = 0,

-^ = 0 (6.3.6) dy i.e. u = du/dy = 0 at the separation point when viewed by an observer traveling with the speed of separation. This, the Moore, Rott and Sears (MRS) criterion, is consistent with the Goldstein singularity (subsection 5.2.1) for a steady flow past a fixed wall, but permits ys to be nonzero. Its plausibility in more general flows was strengthened by the investigation of Telionis and Werle [14] and Williams and Johnson [15] in which the unsteady problems were reduced to steady problems with moving walls.

100

6. Applications of Boundary-Layer Methods: Flows Without Separation

A further generalization is to ask whether in any flow started from rest at t = 0 there exists a t* such that for t < t* the solution is smooth throughout the entire domain of integration in the x, y plane but for t > t* it can only be found in a portion of the domain bounded in some way by a singularity in the solution. If so, one might ask whether the boundary is the line y = ys, and whether the MRS criterion holds there. Were this the case, there would be profound implications for the theory of high Reynolds-number flows. The classical view is that they evolve smoothly as t increases, but the boundarylayer in reversed-flow regions grows exponentially with time, leading eventually to significant changes in the inviscid flow. This view would need changing, and for t > t* the boundary-layer would abruptly make a noticeable impact on the external flow in a way that is by no means understood, but might well be by initiating a jet of fluid into it, as has been suggested by Shen [16]. The questions we have raised are also important from a practical point of view in the problem of dynamic stall. Here Carr, McAlister and McCroskey [17] have observed large eddies to break away from the boundary of slowly oscillating airfoils, but only after flow reversal has occurred in the boundary-layer. These may be associated with the occurrence of a singularity in the solution of the unsteady equations. These considerations have led to renewed interest in the laminar boundarylayer on a circular cylinder moving with a uniform speed after an impulsive start for which ue = sinx in Eq. (6.3.5). For this problem, the steady-state solution has a singularity at x = xs = 1.82 = 104°, and for xs < x < IT it does not exist. The previous views of the unsteady solution were that they rapidly approached the steady state if x < xSj but for x > xs, the boundary-layer increased in thickness exponentially with time, the flow there being largely an inviscid eddy but with a thin subboundary layer below it, moving fluid from the rear stagnation point at x = 7r to x = S. Careful computations for t < 2 were carried out by a number of authors and lent strong support to this description. An important contribution to our understanding of unsteady separation in two-dimensional laminar flows was made by van Dommelen and Shen [18]. They used a procedure new to boundary-layer theory, namely the Lagrangian method. Their calculations agreed well with others for t < 1 but at higher values of £, they began to differ. For example, they observed the development of a hump in the displacement thickness <5*(x, i) in the neighborhood of x — .z, i.e., a little way into the reversal-flow region, consistent with others. They were, however, able to extend their calculations into higher values of t and show that a very sharp singularity developed at t = 3.04 and x = 1.939. They examined the analytic structure of the singularity and concluded that the MRS criterion is satisfied but it is not of the Goldstein type. Calculations using the characteristic box scheme of subsection 5.5.4 have also been performed for this flow [19]. In these calculations, the finite-difference mesh was allowed to vary across the shear layer so as to maintain the angle and

6.3 Boundary-Layer Motion Started Impulsively from Rest

101

so that with time [see Eq. (5.5.29b)]

(j) < tan - 1 £-

(6.3.7)

is satisfied. The resulting values of kn for the predetermined values of steps in the redirection are shown in Table 6.1 and, as can be seen, they become extremely small at r — 3.0. These calculations made use of increments in £, r\ and r of 101, 161 and 435 respectively, and could have been extended beyond r = 3.1 but at considerable expense, as witnessed by the small and decreasing values of kn. Table 6.1. The distribution of step sizes in r and £.

0

-+ 1

1 1.5 2.3 2.73 3.024

-> 1.5 -+2.3 -> 2.73 ^3.024 -* 3.1

rz

€

k>n

0

0.05 0.02 0.01 0.005 0.002 0.001

^0.54 ^0.57 ^0.58 -+0.60 -> 0.612 ^0.64 -+0.67 ^0.72 -> 1.0

0.54 0.57 0.58 0.60 0.612 0.64 0.67 0.72

0.02 0.01 0.0025 0.0020 0.0015 0.0020 0.0025 0.01 0.02

Figures 6.4 to 6.6 display the variations of dimensionless displacement thickness, Zi*, local skin-friction coefficient, c% and dimensionless displacement velocity, vw, where these are defined with R^ = UQL/V by A*

A

y/RE = -—-

L

f°° (, /

Jo

u\ 1

V

*

)dy,

UeJ

2TW r— = —2 VRL, J QUQ

cf

_

vw

vw

= — U0

d = -TZ(UA

)

d£

(6.3.8) It is of particular note that the displacement thickness is close to monotonic with the small maximum and minimum for r = 3.1 at which the calculations were terminated. The previous results of Cebeci [20], shown in dashed lines, reveal the maxima which stemmed from the use of the zig-zag box scheme which did not meet the requirements imposed by the CFL condition. The distributions of local skin-friction coefficients of Fig. 6.5 show trends which are similar to those of the previous results but with differences in magnitude consistent with those of Fig. 6.4. It should be noted that the results of Figs. 6.4 and 6.5 are identical with those previously obtained up to the value of 9 at which the displacement thickness gradient reaches its maximum and for values of r less than around 2.75. The differences for large values of 9 and r are

6. Applications of Boundary-Layer Methods: Flows Without Separation

102

CHARACTERISTIC BOX ZIC-ZAGBOX

100

120

140

160

180

0(DEG)

Fig. 6.4. Variation of displacement thickness for the impulsively started circular cylinder.

CHARACTERISTIC BOX ZIG-ZAG BOX

Fig. 6.5. Variation of local skin-friction coefficient £*f for the impulsively started circular cylinder.

associated with the numerical procedure and, in particular, with its ability to satisfy the CFL condition as discussed previously. The dimensionless displacement velocity, vw, is shown in Fig. 6.6 together with the locus of points corresponding its maxima which increases with time and decreasing angle. At r = 3.0, the calculated value of 9 is 111.5 and corresponds very closely to that determined by van Dommelen and Shen who terminated their calculations at this time. As the peak in the displacement velocity moves upstream with increasing time, the location at which the skin-friction coeffi-

References

103

MAX@ 0=110.9°

60

50 h

40 h

4.5

30 H

20

10

105

no

115

120

6>(DEG) Fig. 6.6. Variation of displacement velocity for the impulsively started circular cylinder. The inset shows the variation of the location of maximum displacement velocity 6lu with circles indicating the computed values, the dashed lines indicate the linear extrapolation of 0ni and the solid line a conjectured variation of 0m to steady state, [20].

cient becomes zero also moves upstream but at a slower rate and towards its steady-state value of 105°. Figure 6.6 also shows that it is desirable to perform calculations at higher values of 9 so as to confirm the conjecture t h a t the only singularity is associated with the steady-state solution. To make a conclusive judgement, calculations should be performed up to r = 4.1 but, as Table 6.1 suggests, the required time steps are likely to be very small. As a result, the computer time likely to be required to reach r = 4.1 will be considerable.

References [1] Cebeci, T., "Calculation of unsteady two-dimensional laminar and turbulent boundary layers with fluctuations in external velocity," Proc. Roy. Soc. Ser. A, 1977. [2] Ackerberg, R. C. and Phillips, J.H., "The Unsteady Laminar Boundary Layers on a Semi-Infinite Flat Plate Due to Small Fluctuations in the Magnitude of the Freestream Velocity," J. Fluid Mech., 5 1 , p. 137, 1972. [3] Rosenhead, L., Laminar Boundary Layers, Clarendon Press, Oxford, 1963. [4] Karlsson, S.K.F., "An Unsteady Turbulent Boundary Layer," J. Fluid Mech., 5, p. 622, 1959.

104

6. Applications of Boundary-Layer Methods: Flows Without Separation

[5] Lamb, H., Hydrodynamics, University Press, Cambridge, 1932. [6] Howarth, L., "Rayleigh's problem for a semi-infinite plate," Proc. Cambridge Phil. Soc. Vol., 46, p. 127, 1950. [7] Stewartson, K., "On the impulsive motion of a flat plate in a viscous fluid," Quart. J. Mech. Appl. Math., 4, p. 192, 1951. [8] Stewartson, K., "On the impulsive motion of a flat plate in a viscous fluid, pt. 2," Quart J. Mech. Appl. Math., vol. 26, p. 143, 1973. [9] Hall, M.G., "A numerical method for calculating unsteady two-dimensional laminar boundary layers," Ing. Arch., vol. 38, p. 97, 1969. [10] Dennis, S.C.R., "The motion of a viscous fluid past an impulsively started semiinfinite flat plate," J. Inst. Math. Applicat., vol. 10, pp. 105-117, 1972. [11] Moore, F.K., "On the Separation of the Unsteady Laminar Boundary Layer," In: Boundary Layer Research, (ed. H. Gortler), pp. 296-311, 1957. [12] Rott, N., "Unsteady Viscous Flow in the Vicinity of a Stagnation Point," Quarterly J. of Appl. Math., 13, pp. 444-451, 1956. [13] Sears, W. R., "Some Recent Developments in Airfoil Theory," J. Aero. Sci., 23, pp. 490-499, 1956. [14] Telionis, D. P. and Werle, M. J., "Boundary Layer Separation from Downstream Moving Boundaries," ASME J. Appl. Mech., 40, pp. 369-374, 1973. [15] Williams, J. C. Ill and Johnson, W. D., "Semi-similar Solutions to Unsteady BoundaryLayer Flows Including Separation," AIAA J., 12, pp. 1388-1393, 1974. [16] Shen, S. F., "Unsteady Separation According to the Boundary-Layer Equations," Adv. Appl Mech., 18, pp. 177-200, 1978. [17] Carr, L.W., McAlister, K.W. and McCroskey, W. J., "Analysis of the Development of dynamic Stall Based on Oscillating Airfoil Experiments," NASA TN D-8382, 1977. [18] Van Dommelen, L. L. and Shen, S. F., "The Spontaneous Generation of the Singularity in a Separating Laminar Boundary Layer," J. Comp. Phys. 38, pp. 125-140, 1981. [19] Cebeci, T., "Unsteady Boundary-Layers with an Intelligent Numerical Scheme," J. Fluid Mech., 163, pp. 129-140, 1986. [20] Cebeci, T., "The laminar boundary layer on a circular cylinder started impulsively from rest," J. Comp. Phys., 31, pp. 173-172, 1979.

Applications of Boundary-Layer Methods: Flows with Separation

7.1 Introduction The calculation method of the previous chapter is limited to flows without separation. As discussed in Section 5.2, the boundary-layer equations for steady flows become singular at the vanishing of wall shear, and the solutions break down. To avoid the singularity and be able to obtain solutions of flows with separation, it is necessary to solve the equations in the inverse mode. The same is true for unsteady flows when the boundary-layer solutions experience a singularity as discussed in Section 5.5 and again require an inverse boundary-layer method to continue the calculations past the singularity. Unfortunately our knowledge of unsteady flows, particularly those for boundary-layer equations, is less sound than of steady ones. As we dicussed in Section 5.2, in the classical boundary-layer theory, the point of zero wall shear stress coincides with the point of separation, and the equations have singular behavior, but this is not the case in the unsteady theory. As described in [1,2], for example, flow reversals can occur in unsteady flows without flow separation. It is useful to distinguish one from the other so that flow separation can properly be identified in an unsteady flow when the boundary-layer calculations are performed for a prescribed pressure distribution. It is also useful and necessary to explore the accuracy of the numerical method used to solve the time-dependent boundary-layer equations, especially in regions of flow reversal and separation. In the following section, we consider a model problem for a thin oscillating airfoil and discuss the evolution of the unsteady boundary layers in the vicinity of its leading edge and the ability of the calculation method to deal with the movement of the stagnation point and with regions of reverse and separated flow. In Section 7.3, we discuss the application of the boundary-layer method to several steady airfoil flows and in Section 7.4 we extend the application to

106

7. Applications of Boundary-Layer Methods: Flows wth Separation

oscillating airfoils and airfoils subject to ramp-type of motion and compare the calculated results with experimental data.

7.2 Separation and Reattachment Near the Leading Edge of a Thin Oscillating Airfoil The lift and drag characteristics of airfoils at moderate Reynolds numbers can be affected by separation bubbles which occur close to the leading edge as discussed in [3] and, at high angles of attack, can increase in size to cause stall. The added complexity of unsteady motion such as that associated with the rotor blades of helicopters implies that the flow characteristics are influenced by amplitude and frequency and that, in particular, the stall characteristics can be considerably modified. The investigations reported in [4-10] examined these effects over limited ranges of the parameters, and that of [6,10] provides detailed information on the mechanism of dynamic stall of an oscillating airfoil. It appears that stall is associated with flow reversals in the unsteady boundarylayer and that these may translate downstream or upstream depending upon various parameters including the radius of the leading edge of the airfoil. At some stage in the cycle, stall occurs and is preceded by a vortex which forms close to the surface and is probably associated with a breakdown of the unsteady boundary-layer. The above physical problems involve laminar, transitional and turbulent flow and their representation requires a calculation procedure that can provide accurate solutions to conservation equations in all regions of flow as well as appropriate transition and turbulence models. Here in this section we are concerned with the ability and accuracy of the calculation method of Chapter 5 to represent the regions of reverse flow and its use in examining the nature of solutions for parameters close to those associated with stall. The emphasis is on regions of flow close to the leading edge of a model problem corresponding to a thin oscillating airfoil with calculations performed (a) for a prescribed pressure gradient and (b) with interaction between solutions of the viscous and inviscid flow equations. With the configuration chosen, an analytical solution for the potential flow equations is available and the inviscid flow method of Chapter 3 is not needed. 7.2.1 M o d e l P r o b l e m Local regions of separated flow are common on thin airfoils at angle of attack and can be important particularly because of their association with the phenomenon of stall. For thin airfoils, the laminar boundary layer grows from the stagnation point and is subjected to an adverse pressure gradient near the leading edge, which causes separation and subsequent transition to turbulent flow.

7.2 Separation and Reattachment Near the Leading Edge

107

.V < i

^^~^ r

"Tl^-^^

$

a^

_v

"""•"-—-^

____--^

^—_______

^~>*^'\<x

-+ -

a

^

Fig. 7 . 1 . Flow configuration for the model problem.

Reattachment occurs and a turbulent boundary-layer extends to the trailing edge. As the angle of attack is increased, the separation region moves towards the leading edge, and eventually the bubble of the separated flow bursts. Reattachment does not occur and stall sets in. To study the leading edge separation problem on an airfoil, we consider an ellipse whose center is located at the origin and at an angle of incidence a in a uniform stream of velocity UQQ^ and with 9 denoting the polar angle as shown in Fig. 7.1. The ellipse is represented by x = — a cos 0,

y = at\ sin 0,

— IT < 6 < n

(7.2.1)

and inviscid flow theory provides the external velocity distribution U°e = U^l

+ h)

sin(# + a) + sin a

[*? + (!-*?) sin2 0]V2

(7.2.2)

for a thickness ratio, t\ (= b/a). In the region close to the leading edge, Eq. (7.2.1) becomes x

= -a{\

- \92[l + O(tf)]},

y = ati0[l + 0{t\)\

Neglecting terms where the relative error is 0(t\), mated by the parabola x = - a ( l - \e2),

(7.2.3)

the ellipse can be approxi-

y = ati0

(7.2.4)

and, with (7.2.5)

e= h the relations given by Eq. (7.2.3) can be written as x = -a{\

- ^2*i),

V = at\i

(7.2.6)

Expanding Eq. (7.2.2) and discarding second-order terms leads to u" «oo(l+*l) where £o =

2a/t\.

£ + £o (l+^2)1/2

(7.2.7)

108

7. Applications of Boundary-Layer Methods: Flows wth Separation

The surface distance s or the dimensionless distance £ ( = ^) required for the boundary-layer equations comes from Eq. (7.2.5) as ft C = tj

(l+£2)1/2d£

(7.2.8) Jo The above model for the leading edge of a thin airfoil can be extended for an oscillating airfoil by writing Eq. (7.2.7) in the form

where A and £o denote parameters that need to be specified and CJ* is a dimensional frequency. The parameter £o can also be regarded as a reduced angle of attack. The term fo(l + Asmu*t) can be interpreted as an effective angle of attack, aefi(t). Eq. (7.2.8) becomes

" g f o * ) = (1 + ^2)1/2

With this definition,

(7-2-10)

7.2.2 Initial Conditions In order to solve the unsteady boundary-layer equations (5.1.2) and (5.1.4) subject to the boundary conditions given by Eqs. (5.2.1), initial conditions are needed in the (x,y) and (t,y) planes as discussed in subsection 5.5.1. If steadyflow conditions prevail at t = 0, the initial conditions in the (x, y) plane can be obtained easily for both surfaces of the airfoil by solving the steady boundarylayer equations for laminar flow. There is no problem with the initial conditions for these equations since the calculations start at the stagnation point where they admit similarity solutions. The generation of the upstream boundary conditions for Eqs. (5.1.2) and (5.1.4) requires special numerical procedures as discussed in subsection 5.5.1 and 5.5.5. In order to explain the difficulties, it is instructive to see what happens to the stagnation point as a function of time. For this purpose let us consider Eq. (7.2.9). Since u® — 0 at the stagnation point, its location, £ s , based on the external streamlines is given by & = - f o ( l + 4sina;*t)

(7.2.11)

Figure 7.2 shows the variation of the stagnation point with time for one cycle according to Eq. (7.2.10), with A = 1, CJ* = 7r/4. We see that when t = 2, the stagnation point £ s is at — 2£o, and when t = 6, it is at 0, etc. If £ s were fixed, we could assume that u = 0 at £ = £ s for all time and all y, but this is not the case. It is also possible to assume that the stagnation point is coincident with

7.2 Separation and Reattachment Near the Leading Edge

109

Fig. 7.2. Variation of stagnation point with time for one cycle according to Eq. (7.2.10) with A = 1, LU* =

$3^0

zero ^-velocity for a prescribed time, but we should note that the stagnation point defined by Eq. (7.2.10) is based on the vanishing of the external velocity. For a time-dependent flow, this does not imply that the u-velocity must be zero across the layer at a given ^-location and specified time; indeed flow reversals can occur owing to the movement of the locus of zero u-velocity across the layer and can cause numerical instabilities which require the use of special numerical schemes described below. As in subsection 5.5.4, we again linearize the difference approximations and, with those for Eq. (5.3.13b) given by Eq. (5.3.21b), with (rs)j-i called ( n ) j , the linearized form for Eq. (5.5.26b) is written in the form + {s2)j6vj-i

{SI)J6VJ

+ (ss)jSuj

+ (s4)j6uj-i

= (r2)j_i

(7.2.12)

where 1

(si)j -(s2)j

=

K +2 -'j-1

) 777,2—1

J-V2

,

l/jm,i-l 777, i—\

(s 3 )j =

A

i-i/2

IM 'j-1/2

2 A

f-l/2

u j-l/2~Uj-l/2 Aip' i - i / 2

(7.2.13)

777, i — 1

\

(*4)j

B

A

j-l/2

IM

(to) (^)j-i

J-1/2

(H--i

=

ui - i / 2

1 Uj

"f-i/ 2 (6u

V777,Z—1

+

"i-l/2

Z\t/>

'i-i/2

/i

\m,i — 1

(fa)j-l 777, i — 1

77,Z

,

/

77,2

,

777,i — 1\/)777,Z— 1

+ (Vl/2+Vl/2)V-1/2

U

A• i - i / 2 "

J-l/2

^'-1/2

Zty' j - 1 / 2

•2/3*

110

7. Applications of Boundary-Layer Methods: Flows wth Separation

The solution of the linear system given by Eqs. (5.3.21b) and (7.2.12) subject to the boundary conditions 6u0 = 0,

6uj = 0

(7.2.14)

is obtained with the block elimination method of subsection 5.3.3. We note that the matrices resulting from the application of the Newton's method are 2 x 2 matrices. Once an approximate solution at point 1 is obtained, the system given by Eqs. (5.5.26a) and (5.5.28) is solved to obtain Uj and Vj at point 2 and 0j at i —1/2, n —1/2. This procedure is also repeated at point 3 so that a new estimate for 6j at B can be obtained from the computed values of 9j at (n — 3/2, i — 1/2), and (n—1/2, i —1/2). The 2 x 2 system is then solved again to obtain an improved solution at point 1. The procedure is repeated until the solutions converge and experience has shown that two to three iterations are sufficient. One method for obtaining the solutions on the next time-line in the (x,y) plane at a specified ^-location is the characteristic method discussed in subsection 5.5.4. To describe the solution procedure, let us consider Fig. 7.3 and denote the edge stagnation line computed according to Eq. (7.2.10). The first velocity profile on either side of the edge stagnation point can be calculated iteratively by first assuming that 0 in Eq. (5.5.28) at point B is known and is equal to its value at 0 m > n - 1 . This assumption decouples Eq. (5.5.4) from Eq. (5.5.5) and reduces the problem to one with two unknowns. Equation (5.5.5) is again written as a first-order system u' = v (bv)'+

V0

= \ ^ i -/SB dip

where t(i)

Fig. 7.3. Notation and finite-difference molecnle for the characteristic box 2.

(7.2.15a) (7.2.15b)

7.2 Separation and Reattachment Near the Leading Edge

111

B

*•-''£)"•;(£'

(7.2.16)

The difference approximations to Eq. (7.2.11a) are given by Eq. (5.3.13b) and to Eq. (7.2.11b) by / L \nil

/ L x^M

(Hj

cu \m,i—L p2"1

-(Hjii

(Hj_

n,i tt

2

=A g.1/0 ^ " "i-1/2"

n m.i—1 Uj 1/2

-

^Ai-l/2

- (M,--i

-0.22 -0.20 -0.18

0

0.02

+ (^-1/2 + ^ - 1 / 2 ) ^ - 1 / 2

-2/3fl

E

-0.12 -0.10 -0.08

-0.02

(i \m,i-1'1 - (by)™:

(7.2.17)

R

y

-0.12 -0.10 -0.08 -0.06 -0.04

-0.16 -0 14 -0.12 -0.10 -0.08

Fig. 7.4. Velocity profiles in the immediate neighborhood of the stagnation line at different times for UJ = 7r/4 and A = 1. The dashed lines indicate the locus of zero u-velocity across the layer.

Figure 7.4 shows the computed velocity profiles in the immediate neighborhood of the stagnation region at different times when the external stagnation point varies with time according to Fig. 7.2. We note that for steady state the velocity profiles on either side of the stagnation point, defined by the vanishing of external velocity, behave "well" as expected. However, for unsteady flow, the stagnation point varies with time and flow reversals begin to develop in the velocity profiles. The changes in the behavior of the velocity profiles on either side of the vanishing ue become more drastic as t increases.

7. Applications of Boundary-Layer Methods: Flows wth Separation

112

7.2.3 T h e Q u e s t i o n of Singularity o n an Oscillating Airfoil Before we examine the solutions of the boundary-layer equations for the flow on an oscillating airfoil which can give rise to extensive regions of flow reversal and separation, it is useful to examine the variation of external velocity distribution of the model problem for steady flow and discuss the studies conducted for this problem at several angles of attack [11]. Figure 7.5 shows a typical variation of the external velocity distribution, u®(£,t) with £ for a reduced angle of attack of £o = 1-3. As call be seen, the external velocity increases linearly from the stagnation point (^ = ^ s ) to a maximum and then decelerates before it reaches a uniform value. In the region of deceleration, the flow may separate depending on the reduced angle of incidence £o? and the uniform external velocity downstream of the leading edge allows the flow to reattach, provided that £o is not too large. The studies conducted in [9] showed that the solutions were well behaved and unseparated provided £o was less than 1.155. At higher angles separation occurred with an associated singularity which was overcome by the use of the inverse boundary-layer procedure described in Chapter 5, and results were obtained for small regions of separated flow. There is, however, a limiting size of separation bubble beyond which solutions could not be obtained and this may be related to the physical phenomenon of non-existence of large laminar separation bubbles as discussed in [10]. To study the question of singularity on an oscillating airfoil, calculations were made for three values of reduced frequency UJ in the standard mode with upstream boundary conditions in the (t,y) plane and those in the (x,y) plane at t = 0 determined as described in subsection 7.2.2. With £o = 1, A = — ^, dimensionless time r defined by ^°°v x t and the values of a; = 0.001, 0.01 and 0.10, the maximum value of effective angle of attack, ae& is sufficient to 1.8

r

S

Fig. 7.5. Variation of u° with f.

7.2 Separation and Reattachment Near the Leading Edge

113

provoke separation with a strong singularity. For example, the maximum value of aeff is 1.5 at UJT = 270° and the flow conditions closely resemble a steady separated flow at the smaller frequencies UJ = 0.001 and 0.01. Since the value of aeff corresponding to steady flow separation is 1.155, we would expect the calculations to break down before UJT = 270° owing to the singularity. For the higher frequency case (UJ = 0.10), we expect the solutions to break down later than (UJT = 270°) with flow reversals occurring in the range 270° < UJT < 360°. The calculations were arranged parallel to those previously performed for a circular cylinder [2] and reported in [12]. Thus both the zig-zag and the characteristic-box schemes were used first with time and distance steps that were chosen arbitrarily and subsequently with values in agreement with the stability criterion. The results of Fig. 7.6 for UJ = 0.10 and A — — ^ were obtained with the zig-zag box scheme described in subsection 5.5.4 for a A£ (= ^ f ) spacing specified such that Z\£n = 0.01 up to £ = 1.7, Z\£n = 0.005 for 1.7 < £ < 4 and A£n = 0.01 for 4 < £ < 8; the time steps T{ were 10 degrees for 0 < UJT < 260°, 5 degrees for 260° < UJT < 295°, and 1.25 degrees for 295° < UJT < 360° [13]. The calculations broke down at UJT = 310°, indicating flow separation at this location. Figure 7.6a shows that the variation of the displacement thickness, with R defined by R = 2au00/u^

<$* =

6* 'Z2(l + *i) 1/2 — at i

is generally smooth except in the neighborhood of £ = 2.12 and for UJT = 308.75°. The first sign of irregularity is the steepening of the slope of 6* when UJT = 300°, and a local maximum of 6* occurs at £ = 2.12 when UJT = 308.75°. When the same results are plotted for a displacement velocity, (d/d£)(ue6*) (Fig. 7.6b), we observe that the steepening of the displacement velocity near £ = 2.12 is dramatic as the peak moves from £ = 2.125 to 2.08 with UJT changing from 300° to 308.75°. It should be noted that the maximum value of displacement velocity moves towards the separation point with increasing UJT and the same behavior occurs for the circular cylinder as discussed in [12]. As shown in Fig. 7.6c, the wall-shear parameter f!^ shows no signs of irregularity for UJT < 308.75° but a deep minimuin in f!£ occurs near £ = 2.15, i.e. near the peak of 6*. The calculations that led to the results in Fig. 7.6 were repeated with the characteristic-box scheme using the same coarse variations of T{ and A^n and the results were identical with those obtained with the zig-zag scheme up to UJT = 280°. At UJT = 282.5°, the solutions of the zigzag scheme were smooth and free of wiggles but those of the characteristic-box scheme exhibited oscillations in f!^ which led to their breakdown. The solutions with the zig-zag scheme, however, continued without numerical difficulties

114

7. Applications of Boundary-Layer Methods: Flows wth Separation

Fig. 7.6. Variation of (a) displacement thickness <5*, (b) displacement velocity d/d£(ue6*), and (c) wall-shear parameter fw with £ for the oscillating airfoil; A = | , UJ = 0.1.

until o;r = 310°, where oscillations appeared and led to the breakdown of the solutions at the next time step. The characteristic box was used subsequently with values of A^n fixed as before and with values of T{ determined so that (3 = - ^ (see Fig. 5.5), is always less than unity, that is tan0 < 1

(7.2.18)

^n

We shall refer to this requirement as the stability criterion of the characteristic box. The distribution of time steps determined in accord with this stability requirement is shown in Table 7.1. This procedure avoided the breakdown of the solutions and, as can be seen from Fig. 7.7, the maximum value of (3 increases considerably with UJT SO that the solutions required correspondingly smaller values of the time step. It is interesting to note that the wall-shear distributions of

7.2 Separation and Reattachment Near the Leading Edge

115

u.or

0.5 \

0A\

0.3 \

0.2 \

0.1

0

260

280

300

320 <0T

340

360

Fig. 7.7. Effect of the coarse and fine meshes on the variation of the stability parameter (3 with LOT.

Table 7.1. The distribution of step sizes in LOT for UJ — 0.1 in accordance with the requirements of the stability parameter /3. LOT

0-240° 240-255° 255-261° 261-265° 265-284° 284-305° 305-320° 320-360°

n

10° 5° 3° 2° 1° 0.5° 0.25° 0.5°

Fig. 7.8 are uninfluenced by the mesh at UJT = 280° and 310° but, for UJT > 310°, the coarse mesh leads to large values of (3 and breakdown of the solutions. Figure 7.9a shows the distributions of displacement thickness for values of UJT from 260° up to 360° and completes the cycle. The results up to 300° were identical with those of Fig. 7.6a with rapid increase of the displacement thickness corresponding to the increasing extent of flow reversal, as shown by the wallshear distributions of Fig. 7.9b. It can also be seen from this figure that the maximum displacement thickness and minimum wall shear move upstream with increasing UJT for values of UJT up to 324.5°. Figures 7.10 and 7.11 show the distributions of wall shear and displacement thickness for two smaller frequencies, UJ = 0.01 and 0.001. As expected, the critical value of the reduced angle that corresponds to separation is smaller than that for the higher frequency and closer to that of the steady state, £ = 1.16. For UJ = 0.01, the breakdown of the solutions occurs at UJT = 226°, which cor-

7. Applications of Boundary-Layer Methods: Flows wth Separation

116

0.7 0.05,

'

0.6

-FINE MESH -COARSE MESH

0.5

0.4

0.3

0.2h

L

OJT = 280°

Fig. 7.8. Effect of the coarse and fine meshes on the variation of the (a) stability parameter j3 and (b) wall shear f'J, with £.

260 270 285 300.25 331.5 360

o (a)

Fig. 7.9. Results obtained with the characteristic-box scheme for u — 0.1. Variation of (a) displacement thickness 8* and (b) wall shear f^ with £.

7.2 Separation and Reattachment Near the Leading Edge

117

0.15

o.io L

10

0.05

T(DEG)

-0.05 h

-0.10

2

-0.15

(a)

(b)

3 5

Fig. 7.10. Variation of (a) wall shear f", and (b) displacement thickness 6* with £ for a; = 0.01.

0.15

0.10 WT_[DE6J

0.05

202 202.5 203 203.5

—» —s —N —v

4

0

»

1

i

2

5

v/ /

V^

' 4

i

5

•0.05

(a)

(b)

Fig. 7.11. Variation of (a) wall shear f'^ and (b) displacement thickness 6* with £ for u; = 0.001.

7. Applications of Boundary-Layer Methods: Flows wth Separation

118

responds to an effective reduced angle of aeff = 1.360; for UJ = 0.001, the corresponding values are UJT — 204° and aeff = 1.203. We also note from Figs. 7.10a,b that the flow is a "little" unsteady even at these frequencies, and the solutions do not break down with the appearance of flow reversal, which increases in extent as UJ changes from 0.001 and 0.01. 7.2.4 Interaction as an A n s w e r t o t h e Q u e s t i o n of Singularity The interaction procedure discussed in subsection 5.2.3 has been applied to the model problem discussed in the previous subsection with the standard method. Before we present the results for this case, it is useful to point out that again we solve Eq. (5.5.2) subject to the boundary conditions given by Eq. (5.5.3a) and (5.5.7). The dimensionless distance Y and stream function / ( ^ , F, r ) defined in Eq. (5.5.1) and £ defined in Eq. (5.3.4) are now defined by V =

V2

2*?

*, a

*=-^, ati

R =

2

- ^ v

i>=[(l+t1)au00vti}1/2f(li,T1,T)

(7.2.19a) (7.2.19b)

with f and w in Eq. (5.5.2) now denoting /' =

7T

T>

w

=

TT

T

( 7 ' 2 - 2 °)

In contrast to the standard problem which makes the implicit assumption of infinite Reynolds number, the interaction requires a finite Reynolds number R and a thickness ratio t\ in this case. For this reason, the calculations are performed for a specified value e defined by 1/2

1 i?(l + *i)

(7.2.21)

In all cases shown, the calculations made use of time steps determined by the characteristic box scheme, in agreement with the stability requirement. The present calculations were performed in the following way. For all values of time, with UJT ranging from 0° to 360°, the standard method and the leadingedge region procedure described in subsection 7.2.2 were used to generate initial conditions at a short distance from the leading edge, 0.5. With these initial conditions and for each value of UJT, the inverse method was used to calculate the unsteady flow from £ = 0.5 to 10, for the specified value of e. In general three sweeps were required where flow reversal was encountered and a single sweep sufficed where it did not. It is to be expected that the value of e will influence the number of sweeps and, since it is linked to physical parameters, will affect the singularity and the size of the bubble.

7.2 Separation and Reattachment Near the Leading Edge

119

0.15

OT

(PEG)

10

(b)

3

4

Fig. 7.12. Effect of interaction on the variation of (a) wall shear f^ and (b) displacement thickness S* for uo = 0.001 and £ = 3 x 10~ 3 .

0.10

10

(b)

Fig. 7.13. Effect of interaction on the variation of (a) wall shear / ^ and (b) displacement thickness 8* for LJ = 0.01 and £ = 3 x 10~ 3 .

120

7. Applications of Boundary-Layer Methods: Flows wth Separation

Figures 7.12 and 7.13 show the results for UJ = 0.001 and 0.01 with e = 3 x 10~ 3 . They are nearly the same as those obtained by the standard method and shown in Figs. 7.10 and 7.11 prior to flow reversal where the influence of the Reynolds number is small and increase after flow reversal. In the case of UJ = 0.001, for example, the standard method predicts flow reversal around aeff = 1.19 (see Fig. 7.11), and with interaction (Fig. 7.12) this effective angle is between 1.219 and 1.254. The maximum negative value of the wall-shear parameter f/£ obtained with the standard method is around —0.03 at aeQ = 1.199 and may be compared with the maximum value of f!^ of 0.14 at aeff = 1.286 obtained with interaction. As expected, the interaction allows the calculations to be performed at higher angles of attack than those achieved with the standard method. For UJ = 0.001, the maximum aeff for which calculations can be performed with the standard method is 1.199 with breakdown occurring at aeff = 1.209; the corresponding values with interaction are 1.286 and 1.287. Comparison of wallshear results with both procedures and UJ = 0.001 indicates that the extent of the recirculation region Z\£ is around 0.5 for the standard case, and around 2.5 for the interactive case. The solutions do not have a singularity in the former case but do contain flow reversals, and this suggests that time-dependent flows can be calculated without using an inverse procedure. As the angle of attack exceeds aeff = 1.199 for UJ = 0.001, a singularity develops and requires an inverse procedure as in two-dimensional steady flows. This procedure allows the calculation of larger regions of reverse flow where the flow is now separated. We see a similar picture with the greater unsteadiness corresponding to UJ = 0.01, for which the standard method allows calculations up to an effective angle of attack of 1.354 (Fig. 7.10a), a value considerably higher than 1.199 obtained at UJ = 0.001. The first flow reversal occurs shortly after aeff = 1.294 and breakdown occurs at aeff = 1.360 with maximum negative wall shear values of —0.14 at aeff = 1.354 and —0.035 at aeff = 1.315. The extent of the maximum reverse-flow region is now 1.5, considerably larger than UJ = 0.001, and indicates that the more unsteady nature of the flow produces a bigger region of reverse flow free from singularities. For this value of UJ, the interactive scheme increases the value of aeff for which solutions can be obtained to 1.424 with breakdown occurring shortly after this value, at 1.428 (see Fig. 7.13). The first flow reversal occurs after aeff = 1.315 with maximum negative wall shear equal to —0.19 at aeff = 1.424, and the extent of the recirculation region has now increased by about 30%. Comparison of maximum wall shear values f^ at a similar value of aeff indicates that those computed with the interactive scheme are lower than those with the standard scheme so that, for example, the interactive scheme gives (/^)max — —0.04 at aeQ = 1.36 compared with —0.14 at aeff = 1.354 with the standard method (Fig. 7.10a). Figures 7.13 and 7.14 show that the size of the reverse-flow region increases with Reynolds number but the effective angle of attack for which solutions can

7.2 Separation and Reattachment Near the Leading Edge

o

121

2

3

5

(b)

Fig. 7.14. Effect of interaction on the variation of (a) wall shear f^ and (b) displacement thickness <5* with £ for UJ = 0.01 and e — 1 0 - 3 .

r

f

1

1

1

\ V w 3\(\/

uV/w \YY L

i 4

i 5

"T

(PEG)

260

vV-280 OV— 309 V—339 \—360

0

-0.3

(a)

2

3

(b)

Fig. 7.15. Effect of interaction on the variation of (a) wall shear / ^ and (b) displacement thickness <5* with £ for CJ = 0.1 and e = 3 x 1 0 - 3 .

be obtained is only slightly reduced, changing from 1.428 for e = 3 x 1 0 - 3 to around 1.415 for e = 1 0 - 3 . It is interesting to note that the interactive solutions do not have any flow reversal at aeQ = 1.315 with e = 3 x 1 0 - 3 . Figures 7.15 and 7.16 show the results for cu = 0.1 with values of e of 3 x 1 0 - 3 and 10~ 3 and they are again similar to those obtained by the standard method, as shown in Fig. 7.9, prior to flow reversal where the influence of Reynolds number is small. After flow reversal, the differences between the results obtained

122

7. Applications of Boundary-Layer Methods: Flows wth Separation

Fig. 7.16. Effect of interaction on the variation of (a) wall shear f^ and (b) displacement thickness <5* with £ for UJ = 0.1 and e — 1 0 - 3 .

with the standard and interactive methods increase as the Reynolds number decreases. It is clear that the solutions are free from the numerical "wiggles" encountered when the stability criterion was not met. Comparison of results obtained at the two Reynolds numbers for UJ = 0.1 indicates that the interaction does not reduce the maximum negative value of the wall-shear parameter as it did with lower frequencies. For example, f^ at UJT = 360° is around —0.19 with the standard scheme, and around —0.30 at e = 3 x 1 0 - 3 , and around —0.35 at e = 1 0 - 3 with the interactive method. The maximum value of negative wall shear calculated with interaction is considerably greater than its corresponding value obtained with the standard method at the end of one complete cycle. Furthermore, the behavior of the wall shear is not monotonic without interaction so that, for example, / ^ reaches a maximum value equal to —0.25 around UJT = 331° and then decreases to —0.195 at UJT = 360°. With interaction this is not the case, with the continuously increasing with UJT. The results of Figs. 7.10, 7.15 and 7.16 are for an unsteady flow and they are free from singularities. For this reason, even though the results in the reverseflow region are different, owing to the Reynolds number effect, the extent of the reverse-flow region is essentially the same and is consistent with the results obtained at lower frequencies in the absence of flow separation even though the extent of the reverse-flow region is reduced at the lower Reynolds numbers. Since the calculations began at UJT = 0° with solutions obtained by solving steady-state equations, it was necessary to confirm the extent of their influence. As a consequence, calculations were performed for a second cycle and gave

7.3 Steady Airfoil Flows

123

Fig. 7.17. Effect of interaction on the variation of (a) wall shear f'^ and (b) displacement thickness 6* for a steady flow at e — 3 x 1 0 - 3 .

results at UJT = 720° that were identical with those at UJT = 360°, confirming that the flow is cyclic. Examination of the results showed that the influence of the initial conditions die out rapidly and have no influence on the solutions presented here. The results obtained with UJ = 0.001 can usefully be compared with the steady-state results of [11] shown in Fig. 7.17. We might expect that the small unsteadiness associated with this frequency will lead to results very similar to those of steady state. Inspection of Figs. 7.11 and 7.17 show that, although this is correct in general terms, the answers are quantitatively different. As can be seen, the maximum effective angle at which solutions can be obtained is greater in the unsteady case by some 7%. There are differences in the two calculation procedures but it is unlikely that they are responsible for this difference. On the other hand, it is possible that the difference in the negative wall-shear values may have been influenced by the use of the FLARE approximation in the steadystate solutions. Nevertheless, the unsteady nature of the flow with UJ = 0.001 is clear, in spite of this very low reduced frequency.

7.3 Steady Airfoil Flows The boundary layer method of Chapter 5 is applicable to both steady and unsteady flows with separation. For t = 0, the initial conditions are generated for steady flows and boundary calculations are performed for both laminar and turbulent flows with the onset of transition location specified. In this section we describe its application to steady airfoil flows at low and high Reynolds numbers and postpone its application to unsteady airflow flows to Section 7.4. The panel method of Chapter 3 provides the external velocity distribution for a specified airfoil geometry and identifies the airfoil stagnation point. The inverse boundary layer of Chapter 5 then provides solutions on the upper and

124

7. Applications of Boundary-Layer Methods: Flows wth Separation

lower surfaces from the stagnation point. If the onset of the transition location is not specified, the laminar flow solutions can be used to calculate it by either empirical expressions or the e n -method based on the linear stability theory. In the former case, a useful correlation is based on a combination of Michel's method and Smith's e correlation curve. It is given in [14] as a connection between RQ (= ue6/v) and Rx (= uex/v) at transition (see Fig. 7.18). R0tr = 1.174

1+

22,000 \ Rx

m,

46

(7.3.1)

R„

-- Smith --• Michel

-- Eq. (7.3.1)

10* XLV

Fig. 7.18. Empirical transition location curves for twodimensional incompressible flows. The symbols denote various experimental data.

Thus, the boundary layer development on the airfoil is calculated for a laminar flow starting at the stagnation point so that both RQ and Rx are calculated. They are usually beneath the curve given by Eq. (7.3.1), and if transition occurs, then the calculated RQ and Rx values will be over this curve. The onset of transition is then determined by interpolating the last values of RQ and Rx and the first high values of RQ and Rx which intersects the transition curve. Sometimes, and especially at higher angles of attack, laminar separation takes place before transition location can be computed and in this case, the separation location may be assumed to correspond to that of the onset of transition. When the Reynolds number is low and the angles of attack are low to moderate, it is necessary to use the e n -method. Of course, this method can also be used when the Reynolds number is high. In either case, the computed laminar velocity profiles are analyzed with the linear stability method and growth rates are determined for different physical frequencies as described in [3,14]. Transition is assumed to take place when the integrated amplification factor reaches to an n-value between 8 and 10. At higher angles of attack, as in the case of

7.3 Steady Airfoil Flows

125

high Reynolds number flows, the prediction of the onset of transition with the e n -method must be replaced with the assumption that the onset of transition corresponds to the separation location. Once the onset of transition location is known, the viscous flow solutions are obtained for both laminar and turbulent flows on the airfoil and in the wake, with calculations taking place separately on the upper and lower surfaces of the airfoil and wake. For a given external velocity distribution, these calculations are repeated. Each boundary-layer calculation, starting at the stagnation point and ending at some specified ^-location in the wake is called a sweep. In sweeping through the boundary-layer, the right-hand side of Eq. (5.4.5) uses the values of <5* from the previous sweep when j > i and the values from the current sweep when j < i. Thus, at each ^-station the right-hand side of Eq. (5.4.5) provides a prescribed value for the linear combination of ue{^1) and <$*(£*). After convergence of the Newton iterations at each station, the summations of Eq. (5.4.5) are updated for the next ^-station. Note that the Hilbert integral coefficients Cij have been computed and stored at the start of the boundarylayer calculations. At the completion of the boundary-layer sweeps on the airfoil and in the wake, boundary-layer solutions are available on the airfoil and in the wake. The blowing velocity on the airfoil v^ is computed from Eq. (5.2.7) and a jump in the normal velocity component Avi in the wake is computed from Avi

=

jL{UeS*u)

+

jL{UeSf)

(7.3.2)

and they are used to obtain a new distribution of external velocity u®(x) from the inviscid method. As before, the onset of transition location is determined from the laminar flow solutions and the boundary-layer calculations are performed on the upper and lower surfaces of the airfoil and in the wake by making several specified sweeps. This sequence of calculations is repeated for the whole flowfield until convergence is achieved. 7.3.1 Airfoils at Low R e y n o l d s N u m b e r s The low Reynolds number airfoils have both civil and military applications including remotely piloted vehicles, propeller and wind turbine aerodynamics, aircraft with high-aspect ratio wings, and ultralight human-powered vehicles, as evidenced in a review article by Lissaman [15], two proceedings volumes edited by Mueller [16,17], and one by the Royal Aeronautical Society [18]. The behavior of these airfoils differs from those at high Reynolds numbers, discussed in subsection 7.3.2, in that rather large separation bubbles can occur some way downstream of the leadiug edge with transition taking place within the bubble prior to reattachment. The length of the bubble increases with a decreasing

126

7. Applications of Boundary-Layer Methods: Flows wth Separation

Reynolds number and strongly influences the aerodynamic characteristics of the airfoils. The results presented here were obtained with the computer program described and given in [19]. Similar calculations, however, can also be performed with the computer program given in the accompanying CD-ROM provided the onset of transition is calculated with the e n -method which is necessary for airfoils at low and moderate angles of attack. At higher angles, it is sufficient to assume the transition location to correspond to the location of flow separation. R e s u l t s for Eppler Airfoil Figures 7.20 and 7.21 and Table 7.2 show the results for the Eppler airfoil (Fig. 7.19) which is a low-drag airfoil. The experimental data for this airfoil was obtained by McGhee et al. [20] in the Langley Low-Turbulence Pressure Tunnel (LTPT). The tests were conducted over a Mach number range from 0.03 to 0.13 and a chord Reynolds number range from 60 x 10 3 to 460 x 10 3 . Lift and pitching-moment data were obtained from airfoil surface pressure measurements and drag data from wake surveys. Oil flow visualization was used to determine laminar-separation and turbulent-reattachment locations. In [20], the calculations for this airfoil were performed for chord Reynolds numbers of 10 5 , 2 x 10 5 , 3 x 10 5 , and 4.6 x 10 5 and for a range of angles of attack up to stall. Figures 7.20 and 7.21 show a comparison between calculated and measured results for a chord Reynolds number of 2 x 10 5 . Figure 7.20 presents a comparison between measured and calculated distributions of pressure and local skin-friction coefficients for angles of attack of 0°, 4° and 8°. As can be seen, at lower angles of attack, the separation bubble is long and located away from the leading edge. It becomes smaller with increasing angle of attack and moves towards the leading edge. Figure 7.21 shows a similar comparison for the lift and drag coefficients up to and beyond the stall angle. In general, the calculated results agree remarkably well with the measured ones. Further details of the results shown in Figs. 7.20 and 7.21 are presented in Table 7.2. The calculated values of the chordwise location of laminar separation (LS), turbulent reattachment (TR), and the onset of transition are given for several angles of attack. The experimental results of this table are subject to some uncertainty because of difficulties associated with the surface visualization technique. With this proviso, comparison between measured and calculated values must be considered outstanding. It should be noted that when there is a separation bubble, the transition location obtained from the e n -method occurs within the bubble in all cases, and, in accord with experimental observation, leads to reattachment some distance downstream.

7.3 Steady Airfoil Flows

127

Fig. 7.19. Eppler 387 airfoil.

-1.0,

Cr

0.0

0.2

0.4 0.6 x/c

0.8

-0.01 0.0

1.0

0.2

0.4

0.6

0.8

1.0

x/c

(a)

0.0

0.2

0.4

0.6

0.8

L0

-0.005 0.0

;

0.2

x/c

;

0.4

0.6

L

J

0.8

1.0

x/c

(b) 0.015 0.010 Cf 0.005 0.000

0.0

0.2

0.4

0.6 x/c

(c)

0.8

1.0

-0.005 '

0.0

0.2

0.4

0.6

0.8

1.0

x/c

Fig. 7.20. Comparison of calculated (solid lines) and measured (symbols) pressurecoefficient and local skin-friction-coefficient distributions for the Eppler airfoil at (a) a = 0°, (b) a = 4°, and (c) a = 8° for Rc = 2 x 10 5 .

7. Applications of Boundary-Layer Methods: Flows wth Separation

128

Q]D

15 ;

Q08

10

°°

/

Q03 -

/o

Q01 -

1° /°

ca

Q5 QC2

y Q0 -20

°~o^^ i

i ..__!_

20

i.

60

1Q0

L._..J

QOO -50

i

MO

_]_

Q0

a (a)

1

50 a

L_

B0

:

_._j

150

(b)

Fig. 7.21. Comparison of calculated {solid lines) and measured (symbols) (a) lift, and (b) drag coefficients for the Eppler airfoil at Rc — 2 x 10 5 . Table 7.2. Experimental and calculated chordwise laminar separation (LS), and turbulent reattachment (TR), and transition locations on the upper surface of the Eppler airfoil for Rc = 2x 10 5 . Calculated

Experiment

a

LS

(x/c)tr

TR

LS

TR

-2 0 2 4 5 6

0.56 0.51 0.46 0.415 0.40 0.39

0.748 0.688 0.624 0.564 0.526 0.467

0.835 0.785 0.716 0.65 0.60 0.52

0.53 0.48 0.43 0.40 0.38 0.37

0.80 0.74 0.67 0.62 0.59 0.55

R e s u l t s for N A C A 65-213 Airfoil In [21], Hoheisel et al. conducted studies for the NACA 65-213 airfoil at zero angle of attack for a chord Reynolds number of 2.4 x 10 6 . For this flow, separation occurred from x/c of around 0.6 to 0.8. The experiments were carried out in the wind tunnel of the French-German Institute at St. Louis which had a freestream turbulence intensity of 0.2%. The airfoil and velocity profiles and distributions of external velocity and skin-friction coefficient are shown in Figs. 7.22 and 7.23. In Fig. 7.23, we see a slightly lower calculated variation in external velocity between the x/c-stations corresponding to laminar separation and turbulent reattachment which were reported to occur at 0.609 and 0.774c, respectively. The velocity profiles imply separation at a similar location with

7.3 Steady Airfoil Flows

129

SEPARATION BUBBLE

E 0.774

0.030

0.020

A x/c = 0.367 o

B 0.483 o

F 0.810

G 0.841

H 0.946

0.730

C 0.609

y/c

0.010

0.000

0

0.5 1.0

Fig. 7.22. Comparison of calculated (solid lines) and measured (symbols) velocity profiles for the NACA 65-213 airfoil.

Hi

7.5

A

5.0 c f -10"

-J

0.2

0.4

0.6 x/c

8

2.5

1.0

Fig. 7.23. Comparison of calculated (solid lines) and measured (symbols) external velocity and skin-friction distributions for the NACA 65-213 airfoil.

reattachment slightly downstream of the measured value. The location of onset of transition was calculated at x/c of 0.721, which is consistent with an experimental measurement of maximum fluctuations in the near-wall region at the reattachment location of around 0.774.

7. Applications of Boundary-Layer Methods: Flows wth Separation

130

7.3.2 Airfoils at H i g h R e y n o l d s N u m b e r s The computer program described in Chapter 10 can be used to calculate steady airfoil flows at high Reynolds numbers for a specified or calculated transition location. Here we present results for two NACA airfoils for ranges of angle of attack including stall with the onset of transition calculated from Eq. (7.3.1) at low to modest angles of attack. At higher angles of attack, it is assumed to correspond to the separation location. In performing these calculations for either airfoils, either at high or low Reynolds numbers, it is usually sufficient to neglect the wake effect and perform the calculations on the airfoil only, provided that there is no or little flow separation on the airfoil. With flow separation, the relative importance of including the wake effect in the calculations depends on the flow separation as shown in Fig. 7.24. Figure 7.24a shows the computed separation location on a NACA 0012 airfoil at a chord Reynolds number, i? c , of 6 x 10 6 . When the wake effect is included, separation is encountered for angles of attack greater than 10°, and attempts to obtain results without consideration of the wake effect lead to erroneously large regions of recirculation that increases with angle of attack, as discussed in [3]. Figure 7.24b shows that the difference in displacement thickness at the trailing edge is negligible for a = 10° but more than 30% for a = 16°. Since the viscous effects are introduced into the panel method through the blowing velocity and off-body Kutta condition, both of which involve (5*, the accuracy of the inviscid flow depends on the accuracy of determining <5* everywhere in the flowfield, especially at the airfoil trailing edge and in the wake. Results for the NACA 0012 airfoil, with calculations including the wake effects, are shown in Figs. 7.25 to 7.27 with Figs. 7.25 and 7.26 corresponding to a chord Reynolds number of 3 x 10 6 and Fig. 7.27 to Reynolds numbers of 1.0

0.06

- - -^ V

0.9

^\. N

\\

0.8

0.05

\^ \\

\

0.04

\

\ \

05

'8

-

\\

0.02

\\ \

L

1

10

1

1

1

12

1

14

.

1

16

.

0.01 1

0.00

18

a

(a)

0.03

C

0.7 0.6

6

*

0.0

0.2

0.4

0.6

0.8

1.0

x/c

(b)

Fig. 7.24. Wake effect on (a) flow separation and (b) displacement thickness - NACA 0012 — with wake; , without wake.

7.3 Steady Airfoil Flows

131

0.040

0.030 h

0 0 2 0

Cd

0.010

0.0

5.0

10.0

15.0

0.000

20.0

(a)

(b)

Fig. 7.25. Comparison between calculated (solid lines) and experimental values (symbols) of: (a) ci vs a, and (b) cd vs cj. NACA 0012 airfoil at Rc = 3 x 10 6 .

6x 10 6 and 9x 10 6 . Figure 7.25 shows the variation of the lift and drag coefficients of the NACA 0012 airfoil for a chord Reynolds number of 3 x 10 6 . As can be seen from Fig. 7.25a, viscous effects have a considerable effect on (c/) m a x of the airfoil, which occurs at a stall angle of around 16°, and the calculated results agree well with measurements [22]. Figure 7.25b shows the variation of the drag coefficient with lift coefficient. As can be seen, the measurements of drag coefficients do not extend beyond an angle of attack of 12 degrees and at smaller angles agree well with the calculations. The nature of the lift-drag curve is interesting at higher angles of attack with the expected increase in drag coefficient and reduction in lift coefficient for post-stall angles.

Cf

0.0

0.2

0.4

0.6

1.0

x/c

(a)

(b)

Fig. 7.26. Variation of (a) local skin-friction coefficient and (b) dimensionless displacement thickness distribution. NACA 0012 airfoil at Rc = 3 x 10 6 .

7. Applications of Boundary-Layer Methods: Flows wth Separation

132

Figure 7.26 shows the variation of the local skin-friction coefficient Cf and dimensionless displacement thickness 6* /c distribution, for the same airfoil at the same Reynolds number. As can be seen from Fig. 7.26a, flow separation occurs around a = 10° and its extent increases with increasing angle of attack. At an angle of attack a = 18°, the flow separation on the airfoil is 50% of the chord length. The variation of dimensionless displacement thickness along the airfoil and wake of the airfoil shown in Fig. 7.26b indicates that, as expected, displacement thickness increases along the airfoil, becoming maximum at the trailing edge, and decreases in the wake. For a = 10°, 6*/c at the trailing edge is around 2% of the chord, becoming 4% at a = 14° and 7% at a = 16°. With increase in angle of attack, the trailing-edge displacement thickness increases significantly, becoming 9% of the chord at a = 17° and 14% at a = 18°. However, what is quite interesting, aside from this rather sharp increase in displacement thickness, is the behavior of the maximum value of the displacement thickness. While for angles of attack up to and including stall angle, a = 16°, its maximum value is at the trailing edge, at higher angles of attack corresponding to post-stall, its maximum is after the trailing edge. Figure 7.27, together with Fig. 7.25, shows the effect of the Reynolds number on the lift coefficient. In accord with the measurements, the calculation method satisfactorily accounts for the effects of Reynolds number. The results show that the maximum lift coefficient, ( q ) m a x , increases with an increase in Reynolds number in agreement with measurements. Figures 7.28 and 7.29 correspond to the NACA 23012 airfoil for which measurements have been reported in [22] with Fig. 7.28 corresponding to Reynolds numbers of 3 x 10 6 . In common with the results of Fig. 7.25, the results of Fig. 7.28 show a fairly gentle approach to stall, which in this case occurs around

2.0

ty io

(a)

(b)

Fig. 7.27. Effect of Reynolds number on the lift coefficient. NACA 0012 airfoil at (a) Rc = 6 x 10 6 , and (b) Rc = 9 x 10 6 .

7.4 Unsteady Airfoil Flows

133

0.040

2.0

0.030

C

t

1.0

Cd

0.020 1-

0.010

0.0 0.0

; 5.0

...

i 10.0

,

1-. 15.0

.

i 20.0

(a)

o.OOO ' 0.0

'

i0.5

. . .i - . - . i 1.0 1.5

.

-i 2.0

(b)

Fig. 7.28. Comparison between calculated (solid lines) and experimental (symbols) values of (a) a vs a, and (b) cd vs Q . NACA 23012 airfoil.

C/

1.0

C/

l.o

0.0

(a)

5.0

10.0

15.0

20.0

(b)

F i g . 7.29. Effect of Reynolds number on the lift coefficient. NACA 23012 airfoil at (a) Rc = 6x 10 6 , and (b) Rc = 8.8 x 10 6 .

17 degrees. The calculated drag coefficients are in good agreement with the measured values but to a lesser degree than those obtained for the NACA 0012 airfoil. Figures 7.28a and 7.29 show the Reynolds number effect on the lift coefficient. As in the case of the NACA 0012 airfoil, again this effect is well reflected in the calculations.

7.4 U n s t e a d y Airfoil Flows We now apply the computer program of Chapter 10 to oscillating airfoils and airfoils subject to a ramp-type motion with the onset of transition location either calculated or specified. The calculations again involve sweeps both on the

134

7. Applications of Boundary-Layer Methods: Flows wth Separation

airfoil and its wake. After each sweep, the displacement thickness is updated so that the upstream and downstream viscous effects can be included in the next sweep. In addition to improving the boundary-layer calculations in this way, the viscous effects on the pressure distribution are also accounted for via the blowing velocity in the panel method. Moreover, for the flow over a lifting body, the Kutta condition and wake vorticity effects are related to the pressure distribution on the body surface. The inviscid airfoil calculations require that the airfoil trailing edge is closed. If this is not the case, in subroutine close-geom we modify the trailing edge of the airfoil slightly as shown in Fig. 7.30. The normal distance between the original and modified airfoil denoted by AY for upper (AYU) and lower {AY{) surfaces corresponds to the displacement thickness <5* which later is used to calculate an equivalent blowing velocity in subroutine close-geom. The inviscid-viscous iteration procedure is set up in the following manner. The panel method first provides the pressure or edge velocity distribution u®(x,t) for the boundary-layer method, and then the boundary-layer method computes the blowing-velocity distribution, which represents the boundary-layer effects on the outer flow for the panel method. With these viscous effects, the panel method then provides the boundary-layer method with a pressure distribution which involves the boundary-layer effects from the previous iteration. This procedure is repeated until the solutions converge. The procedure for generating the upstream initial conditions, however, is different than the one described in subsection 7.2.2. While that procedure is an accurate one, studies indicated that it was necessary to use too many time steps to maintain the computational accuracy. To speed up the calculations, an approximate procedure was developed. According to this procedure a stagnation point (ue = 0) is determined at each time step and a new surface distance starting at the stagnation point up to the trailing edge was computed. The distance for each time step was normalized so that dimensionless surface distance became zero at the stagnation point and unity at the trailing edge. A new grid was then generated for all angles of attack for which a new stagnation point was determined. A typical evolution of the solutions during the invisicd-viscous iteration procedure is shown in Figs. 7.31 and 7.32. The calculations were performed for the Sikorsky SSC-A09 airfoil section at a chord Reynolds number of 2 x 10 6 and an angle of attack of 15° which is beyond that of maximum lift [23]. In order to

Fig. 7.30. Modified airfoil to close the trailing edge.

7.4 Unsteady Airfoil Flows

135

no. of iteration 1/1^20 30 40 lower surface

1.2

no. of iteration

A ''/»

,1 ,10 ,30 ,30 ,40

y/,40 1.1 1

1.0

upper surface

V

» wake

lower surface .

*v

>40 X

•^*t^»

20

• ^ " ^ ^

•••***

0.0

(rt

•'V\ \ MO M

1

'

OL7

aa

1

^ ^ ^

ao x to

I

i.i

1.2

Fig. 7.31. The variation of the (a) displacement thickness, (b) local skin-friction coefficient, and (c) boundary-layer edge velocity (only the trailing edge regions are shown) distributions with iteration number, — steady, a = 15°, Rc — 2 x 10 6 .

improve the rate of convergence, the calculations began with the displacement thickness and blowing velocity distributions obtained from a converged solution at an angle of attack of 14° so that the solutions indicated by iteration 1 in Fig. 7.31 include viscous effects. Figure 7.31a shows the upper surface, lower surface and wake displacement thickness distributions. The displacement thickness on the lower surface and for the forward 0.4 of chord on the upper surface, do not change with the iterations, whereas the trailing-edge displacement thickness on the upper surface increases by 50 percent during the computation. Figure 7.31b illustrates the change in local skin-friction coefficients with iteration number on the two surfaces; the only important changes are in the vicinity of separation. The variation of the velocity distribution, uei with iteration is shown in Fig. 7.31c, and it should be noted that the discontinuities in the ue and <5* distributions at the trailing edge are gradually eliminated as the iteration number increases.

136

7. Applications of Boundary-Layer Methods: Flows w t h Separation

~20 30 Number of Iterations

40

F i g . 7 . 3 2 . T h e variation of liftcoefficient with iteration n u m b e r , — s t e a d y flow, a = 15°, Rc = 2 x 10 6 .

The above calculations were reported in [23]. In general, twenty inviscidviscous iterations were required for convergence of the solutions involving stall, and fewer iterations were sufficient where stall did not occur. The variation of lift coefficient with iteration is shown in Fig. 7.32, and the large change in the first iteration is due to the initial guess. As discussed in the previous section, the interactive boundary-layer calculations can be performed for an airfoil with and without consideration of its wake. Before we present the results for an airfoil operating under unsteady flow conditions, as we did in steady flows, we consider the wake effect on the solutions. To this end, we examine the Sikorsky SSC-A09 airfoil undergoing a harmonic oscillation a = 5° + 7°sincjt at a very low reduced frequency, k (= UC/UQO) — 1 x 10~ 5 . The lift coefficients of Fig. 7.33a show that for a chord number, i? c , of 2 x 10 6 , the roles of wall boundary layer and viscous wake are opposite to that the former reduces the lift coefficient while the latter increases it, and the magnitudes of the effect increase at high angles of attack. The drag coefficient Cd (form drag which is about onehundreth of q ) is increased by both the wall boundary-layer and the viscous wake, as shown in Fig. 7.33b. Figure 7.33c shows the displacement thicknesses on the upper and lower surfaces at the trailing edge for one cycle of motion, and Fig. 7.33d shows the displacement thickness distribution on the upper surface at various angles of attack. Both figures show that the viscous wake reduces the displacement thickness which is built up by the viscous effects in the wall boundary layer. The differences between the displacement-thickness distributions also affect the locus of the stagnation point in time, Fig. 7.34, since the effective surfaces of the airfoil formed by adding the displacement-thickness distribution to the airfoil surface, are different.

137

7.4 Unsteady Airfoil Flows

0.012

Panel method.

r

0.011 0.010 0.009

>

Interactive method *•

0.008

/

/ 0

0.007

with viscous wake^

/

0.006

without viscous weJce.

0.005

>

0.004 0.003 0.002 0.001 0.000

_^~--^ *-«... —•-...•—•o—- ..—-o

*"

L

-0.001 -0.002 x.

-0.003 -2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

-2.0

12.0

-1.0

0.0

1.0

2.0

3.0

4.0

.....

5.0

.

.

.

_l

1

6.0

7.0

8.0

9.0

10.0

1

. _

11.0

12.0

0.040

/ / / /

0.035

i

upper surface 0.030

Y^without viscous wake ^^-with viscous wake

•

If ' // // " A

/

J

^

0.025

\\ \\ \\ \\ \\

with viscous wake,

lower surface

5.0

10.0

20.0

J.JH uliiiliiJ L

25.0

/

/•'•

//Ys, »•

0.010

0.005

"*CC^,

——t^^r^

O.(HN) 0.0

/»•»«•

0.015

s*

\

\T

/

1

without viscous wake

0.020

30.0

35.0

40.0

45.0

50.0

55.0

60.0

65.0

0.1

0.0

0.1

0.2

0.3

\

.

,

.

,

,

,

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 7.33. The effect of viscous wake on (a) lift coefficient and (b) drag coefficient distributions, (c) displacement-thickness distribution at trailing edge and (d) displacementthickness distribution on the upper surface.

Fig. 7.34. The effect of viscous wake on the locus of stagnation point.

The second case illustrates the role of the wake at angles of attack beyond stall for ramp-type motion with the onset of transition imposed near the stag-

1

138

7. Applications of Boundary-Layer Methods: Flows wth Separation

with viscous wake

Fig. 7.35. The effect of viscous wake on lift-coefficient distribution. Ramp-type motion with constant pitch rate, A = 0.00001, a = 0° to 16°.

nation point. The lift coefficient curves with and without viscous wake are displayed in Fig. 7.35. The study of the viscous wake effects in the slow motion cases, as described above, provides a quantitative standard for evaluating the viscous wake effects in more severe unsteady flows. When the airfoil performs slow movements, the vorticities shed to the wake are weak and the wake, due to the viscosity, behaves like a distribution of sinks along the dividing line. Increasing the severity of the unsteady motion increases the strength of the vorticities shed to the wake. The wake of a real flow over an airfoil can, therefore, be described as including vorticities and sinks which mix and interact with each other. To model the unsteady wake by a viscous-inviscid interactive method, some assumptions are necessary. In the study reported in [23], it was assumed that the two elements of the unsteady wake model, vorticities and sinks, are distinct and do not influence each other. This means that the vorticities shed via the potential flow go downstream according to their own local velocities without the influence of sinks, and the sinks computed from the viscous wake distribute on the instantaneous wake dividing line, the location of which is evaluated without the influence of vorticities. This model satisfies the steady-flow condition as the unsteady motion approaches zero and should capture most of the characteristics of the wake as the unsteady motion of the airfoil increases. Some instantaneous wake dividing lines for the Sikorsky SSC-A09 airfoil executing a harmonic motion, a = 5° + 10°sina>£

7.4 Unsteady Airfoil Flows

139

unsteady

Fig. 7.36. Comparison of instantaneous wake dividing lines for the SSC-A09 airfoil.

at k = 0.001, 0.01, 0.1 and 0.5 are shown in Fig. 7.36 and the wake locations for frequencies below 0.1 are very close to the steady-wake locations. Those with k = 0.5 are quite different from the others in the far wake region, but not in the near wake region where the wake-viscosity effects are most important. 7.4.1 R e s u l t s of U n s t e a d y Flows The accuracy of an interactive method for simulating the unsteady flow depends on several factors which include the inviscid-flow method, the modification of the inviscid-flow method to accept viscous effects, the boundary-layer scheme as well

140

7. Applications of Boundary-Layer Methods: Flows wth Separation

as the numerical scheme used to solve the equations, and the coupling procedure between the inviscid- and viscous-flow calculations. The results discussed in the previous section and in [3] show that these factors are properly represented in the steady-flow calculations so that the steady-flow model can now be used as a base for extension to unsteady-flow calculations. The overall features of a subsonic flow over an airfoil executing an unsteady motion are primarily characterized by the extent of flow reversal (or separation). For testing the performance of the method, calculations are performed in the order of increasing levels of flow complexity, starting first with flows without, and then with reversal and in subsection 7.4.2 are followed by flows with substantial regions of flow separation. Flows W i t h o u t Reversal R e g i o n We again consider the same Sikorsky airfoil in which the change in angle of attack takes place according to a(t) = 5 0 + 5°sincj* at three reduced frequencies, k — 1 x 1 0 - 4 , 0.1 and 0.5 for a chord Reynolds number of 2 x 10 6 . The interactive flow calculations are performed with the unsteady panel method of Chapter 3 and the inverse boundary-layer method described in Chapter 5. At first the solution of the outer inviscid flow which pilots the computation of inner viscous flow, was investigated. Figure 7.37 displays the lift, drag and pitch moment coefficients for the first cycle of airfoil motion at k — 0.5. The results show that the inviscid-flow solutions generally are not sensitive to the size of time step except that there are irregular solutions near the beginning of the motion. The irregularity is due to the discontinuity caused by the sudden change of the flowfield from steady to unsteady regimes. Figure 7.38 shows that, in the highest frequency case, the initial unfavorable effects die out in one cycle. In the following computations, 25 time steps were used in one cycle of motion and the viscous-inviscid interactive calculations started at the beginning of the third cycle. The results with the interactive method shown in Fig. 7.39 were first obtained with the inverse boundary-layer calculations performed for steady flow calculations. Even though at each time the inviscid flow was calculated with the unsteady panel method, the viscous effects were calculated with the boundarylayer equations in which the time-dependent terms, du/dt and due/dt were neglected. This quasi-steady model is valid when du -

and

due ~ ^ ^ u

e

due —

Repeating the calculations in which the viscous effects are computed with the unsteady boundary-layer equations and comparing the results with those ob-

141

7.4 U n s t e a d y Airfoil Flows

0.03

r
0.02

^ V

X a/

0.01

1

0.00

/ f

-0.01

j&

-0.02 1

1 0 1

2 3 4 5 6 7

8 9

10 11

- 1 0 1

2 3 4

5 6 7 a

(b)

8 9

10 11

0.04

0.02 time steps/cycle

0.00

a

-0.02

25

—.__.. 49 —

-0.04:

73

-0.06 -0.08

- 1 0 1

2

3 4 5

6 7 8 9

10 11

(C) F i g . 7 . 3 7 . T h e effect of t i m e - s t e p on t h e potential-flow a e r o d y n a m i c coefficients of (a) lift, ( b ) d r a g , a n d (c) pitch m o m e n t .

tained with the quasi-steady model showed that the solutions of both interactive methods were identical (Fig. 7.39). The lift coefficient curve for the lowest reduced frequency, k — 0.0001, shows that the solutions are essentially steady, Fig. 7.39. In this and the following figures, the dashed curves represent the periodic inviscid results (the third cycle) and the solid curves represent the corresponding results of viscous-inviscid interaction. Figures 7.40 and 7.41 display the lift, drag and pitch moment coefficient curves for the other two frequency cases. We can see from the results that the hysteresis effects increase with the increase of the reduced frequency. Since the boundary-layer calculations started from steady state and suddenly joined the inviscid flow calculation, the viscous-inviscid results show a little discontinuity at the beginning and the end of the cycle. One very interesting aspect of the lift coefficient curve is that at k = 0.1 the lift coefficient of the upward stroke is lower than that of the downward stroke, but it is just opposite at k = 0.5. This phenomenon can be explained by comparing the pressure coefficient distributions in the upward and downward strokes at a certain angle of attack (a — 7.5° purely inviscid flow, for example), as shown in Fig. 7.42 for different frequencies.

142

7. Applications of B o u n d a r y - L a y e r M e t h o d s : Flows w t h S e p a r a t i o n

0.2

-2.0

0.0

2.0

4.0

6.0

10.0

8.0

12.0

a

F i g . 7 . 3 8 . T h e d e v e l o p m e n t of t h e potential-flow lift coefficient from s t e a d y t o periodic s t a t e .

0.06 0.04

Q

1.6

Panel method Interactive method

0.02

1.4 0.00 Panel method , Interactive method v

1.2

-0.02

1.0 0.8 0.6

0.04

10

12

Panel method Interactive method '

0.00

0.2

v

-0.02

^

r

-0.04

-0.2 4 (a)

6

a

0.02

0.4 f-

0.0

-0.04 (b) -

6

a

8

10

12

•0.06 -2

(c)

0

6

10

12

a

F i g . 7 . 3 9 . Variations of (a) lift, (b) d r a g a n d (c) pitching m o m e n t coefficients w i t h angle of a t t a c k for t h e SSC-A09 airfoil.

143

7.4 U n s t e a d y Airfoil Flows

0.06 0.04

1.6

Panel method Interactive methodN

0.02

1.4 Panel method

1.2

0.00

•£

-0.02 (-

1.0

Interactive method >

0.8

(b)

-0.041 -2

4

0

a

6

10

12

10

12

0.04 r

0.6

0.02

Panel method Interactive method

0.4 0.00

0.2

1

m

X—

-0.02 \

0.0 -0.04

-0.2 4

6

10

a

(a)

12 -0.06 4

(c)

a

6

8

F i g . 7 . 4 0 . Variation of (a) lift, (b) d r a g , a n d (c) pitch m o m e n t coefficients w i t h angle of a t t a c k . - U n s t e a d y , k — 0.1. 0.06 0.04

1.6

0.02

1.4

panel method Interactive method^

0.00

1.2

panel method

1.0

Interactive method

I

/J 7

'j

/f

0.8

•0.02

y''/'

-0.04 -2.0

(b)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

10.0

12.0

a

0.04

Interactive method

0.6 0.4 0.2

P'

0.02

0.00
s

panel method ^ N

/

Y

0.0 -0.2 2.0

(a)

- J -0.04 0.0

2.0

4.0

6.0

8.0

10.0

12.0

a

-0.06

(c)

-2.0

0.0

2.0

4.0

6.0

8.0

a

F i g . 7 . 4 1 . Variation of (a) lift, (b) d r a g , a n d (c) pitch m o m e n t coefficients w i t h angle of a t t a c k . - U n s t e a d y , k — 0.5.

144

7. Applications of Boundary-Layer Methods: Flows wth Separation

At k = 1 0 - 4 , no hysteresis effects appear and the pressure-coefficient distributions of upward and downward strokes are identical to the steady one. Increasing k to 0.1, the hysteresis effects appear mainly on the fore part of the airfoil. The pressure peak at a = 7.5° | is lower than the corresponding steady one, whereas that at a = 7.5° [ is higher, so that the lift coefficient in the downward stroke is higher than that in the upward stroke. Increasing k generates more unsteady motion of the airfoil aft (since it is farther from the pivot point) so that the effects on the pressure distribution there are more enhanced, as shown in Fig. 7.42c, for k = 0.5. From the pressure distribution, we can see that in the downward stroke the negative contribution to the lift coefficient in the aft part cannot be balanced by the positive contribution from the fore part so that the Q in the downward stroke is lower. The drag-coefficient curves at k = 0.1 and 0.5, Figs. 7.40b and 7.41b, show that the upward stroke motion has higher drag than the downward stroke, and the unsteady motion could induce thrust (or negative drag) in the downward stroke. At k = 0.1 the viscosity increases the drag coefficient, Fig. 7.40b, but at higher frequency, k = 0.5, it sometimes reduces the drag coefficient. The ratio between the drag and its corresponding lift coefficients is less than 1 percent at k = 0.1 and this ratio tends to increase as k increases. When an airfoil performs an unsteady motion, there is energy transfer between the airfoil and its surrounding fluid, and this energy transfer is indicated by the pitching moment. The pitching moment at k = 0.1 within one cycle of motion is shown in Fig. 7.40c. The negative pitch moment in the upward stroke indicates that the airfoil motion is against the aerodynamic moment induced by the surrounding fluid, whereas in the downward stroke the aerodynamic moment is favorable to the airfoil motion. Therefore, the energy is transferred to and absorbed from the fluid in the upward and the downward strokes, respectively. The area enclosed by the pitch-moment curve indicates the net energy transferred to the fluid in each cycle of harmonic oscillation. The viscosity enhances the hysteresis effects, Fig. 7.40c, so that more energy is required to execute a cycle of motion. In the very slow pitching case, k = 10~ 4 , both the inviscid and the viscous-inviscid interactive curves show almost no hysteresis effects, Fig. 7.39c, so that the energy transferred to the fluid in one cycle is negligible. At k — 0.5, the pitching moment in most of the downward stroke becomes positive, which indicates that the airfoil motion is against the aerodynamic moment not only in the upward stroke but also in most of the downward stroke. In this case, the viscosity also increases the hysteresis effects and the energy transferred to the surrounding fluid in one cycle of motion. The boundary-layer development can be represented by the displacement thickness distributions; some features of them are displayed in Figs. 7.43 and 7.44. The displacement thickness distributions on the upper and lower sides of the airfoil and wake at angles of attack a = 5°, 7.5° and 10° in the upward

7.4 Unsteady Airfoil Flows

145

(c)

Fig. 7.42. The inviscid-flow pressure-coefficient distributions for the SSC-A09 airfoil at a = 7.5° of the harmonic oscillation, with k = (a) 1 0 - 4 (b) 0.1, and (c) 0.5.

and downward strokes are shown in Fig. 7.43 for k = 0.5. At a certain angle of attack, the upper-side displacement thickness in the downward stroke is higher than that in the upward stroke. The influence of the reduced frequency on the displacement thickness distribution can be clearly represented by the trailingedge value, as shown in Fig. 7.44. The maximum displacement thickness shifts to the downward stroke as the reduced frequency increases, as in steady flows discussed in the previous section. Flows W i t h Reversal R e g i o n In the upward stroke of the harmonic oscillation, the pitch rate decreases when the airfoil approaches the maximum angle. At high angles of attack, decreasing the pitch rate speeds up the rate of expansion of the trailing-edge reversal or separation region. To suppress this region and delay the occurrence of stall, a high pitch rate must be maintained at high angles of attack. To study the behavior of an airfoil under these conditions we consider the same airfoil executing a ramp-type motion from 5° to 16° with a constant pitch rate of A(ac/uOQ) = 0.02. In Fig. 7.45, the lift-coefficient curves computed by using the unsteady and quasi-steady approaches for A = 0.02 are compared with the solution computed

146

7. Applications of B o u n d a r y - L a y e r M e t h o d s : Flows w t h S e p a r a t i o n

o.oi 8 -i

0.014

5* 0.010 H

0.006 H

0.002

F i g . 7 . 4 3 . C o m p a r i s o n of t h e displacement-thickness d i s t r i b u t i o n s of t h e SSC-A09 airfoil in t h e u p w a r d a n d d o w n w a r d strokes.

0.024,

0.020

0.016

K*

0.012

0.008 K

0.004

F i g . 7 . 4 4 . T h e effect of red u c e d frequency on t h e trailingedge d i s p l a c e m e n t thickness of t h e SSC-A09 airfoil.

by the quasi-steady approach for A = 10~ 5 . The quasi-steady solution at A = 0.02 has a higher maximum lift at higher angle of attack than that at A = 1 0 - 5 . Due to the retarding effects from the lower angle of attack and smaller streamwise pressure gradient, the unsteady-flow solution shows no stall within

7.4 Unsteady Airfoil Flows

147

the angles of attack computed. The different flowfield structures due to different pitch rates also reflect on the locus of the leading-edge stagnation point, as shown in Fig. 7.46. The stagnation point is frozen when there is stall but keeps moving toward the lower surface when the stall is suppressed. Figures 7.47a and 7.47b display the displacement thickness distributions for two angles of attack. At an angle of attack before static stall, a = 13°, the difference between the three distributions in Fig. 7.47a are much less than those shown in Fig. 7.47b for the solutions after static stall at a = 15°. The variation of displacement thickness with the reduced frequency can be represented by the trailing edge values, Fig. 7.48. From Figs. 7.47c and 7.48, we can conclude that increasing the pitch rate reduces the displacement thickness on the upper side of the flowfield and hence the boundary-layer effects on the outer inviscid flow. The local skin-friction coefficient distributions on the upper and lower surfaces of the airfoil computed with the quasi-steady approach at a = 13° are shown in Fig. 7.49. The Cf distribution on the lower surface are influenced very little with pitch rate, but those on the upper surface are influenced with the reversal region suppressed by the increasing pitch rate. Figure 7.50 shows the results in which the boundary-layer calculations are performed with unsteady mode for A = 0.02. With increase in angle of attack, the friction coefficient on the upper surface decreases, whereas on the lower surface it increases, and a reversal flow region starts growing from the trailing edge at the angle of attack of 12.7°. The wiggle on the cj distribution near the trailing edge is enhanced as

2.0 |

1.8

- Unsteady, A * 0.02 v Quasi-steady, A = 0.02

1.6

c,

Quasi-steady ^s. A - 0.0OOOU £

•**••

1.4

Jr \

1.2

1.0

0.8

na i 5.0

I

10.0

15.0

I.

I

20.0

Fig. 7.45. The effects of pitch rate and numerical scheme on the lift coefficient distribution. Ramp-type motion with constant pitch rate Rc = 2 x 10 6 .

7. Applications of Boundary-Layer Methods: Flows wth Separation

148

1.005

r

Fig. 7.46. The effects of pitch rate and numerical scheme on the locus of leading-edge stagnation point. Ramp-type motion with constant pitch rate Rc = 2 x 10 .

the reversal region increases. The extension of the calculation method requires a procedure which is able to avoid the wiggles in the solutions as discussed in the next subsection. 7.4.2 Initiation of D y n a m i c Stall on a P i t c h i n g Airfoil The computer program of Chapter 10 has also been applied to study the initiation of dynamic stall on the Sikorsky airfoil subject to a ramp-time motion with a pitch rate of 0.02. The experimental data due to Lorber and Carta [28] indicates that vortex initiates around 18 to 19 degrees of angle of attack. The calculations confirm this and indicate how the trailing-edge separation causes the initiation of the vortex. To elaborate further on this point, let us consider the distribution of the local skin-friction values on the upper surface of this airfoil at several angles of attack, Fig. 7.51. It is clear from this figure that for a < 12°, there is no flow separation on the airfoil which has a steady stall angle of around 14°. The unsteadiness causes the stall angle to increase to around 30°, according to experiments. The flow behavior on the airfoil begins to change quickly, however, once the trailing-edge separation takes place for a > 12°. At a = 17.09°, there is no leading-edge flow separation but only a trailing-edge separation which occurs around 12%. At the next angle of attack, a = 17.59°, leading-edge separation takes place close to 12% chord with a bubble reattaching around 33% and is followed by a trailing-edge separation at 70%. The explosive nature of the leading-edge separation bubble, which is all turbulent, becomes more obvious at the next a = 17.85°, where leading-edge separation takes place around 10% but the reattachment of the bubble moves to 50% chord, a bubble of

7.4 Unsteady Airfoil Flows

149

o.io 0.09 0.08 0.07 0.06 0.05 0.04

, Quasi-steady, A « 0.00001

0.03

/Quasi-steady, A » 0.02

0.02 0.01

(a)

0.00 0.5

0.0

2.5

3.0

3.5

4.0

3.5

4.0

Quasi-steady, A « 0.00001 Quasi-steady, A » 0.02

Fig. 7.47. The effects of pitch rate and the boundary-layer approach on the displacement thickness distributions of the SSC-A09 airfoil subject to ramp-type motion with constant pitch rate (a) a = 13° and (b) a — 15°.

40% in extent, followed by trailing-edge separation moving to 60% chord. Very shortly thereafter, less than one degree increase in angle of attack, the leadingedge separation bubble disappears with complete flow separation taking place at around 5% chord. 7.4.3 S u m m a r y The following principal conclusions may be drawn from the studies reported in this section. (a) In the harmonic oscillation cases, hysteresis effects are evident as the reduced frequency increases. The hysteresis phenomena appear in the solutions of both viscous and inviscid flows and are qualitatively represented by the calculation method.

7. Applications of Boundary-Layer Methods: Flows wth Separation

150

0.10

I I

Quasi-steady, A = 0.00001

0.08 h

0.06

Quasi-steady, A = 0.02

[

/

/ /

\

//

\/ I

0.04 r Unsteady, A = 0.02

y S

/

>^

0.02 h

0.00

I

56

'

7°

9°

11°

13°

15°

Fig. 7.48. The effects of pitch rate and the boundary-layer approach on the trailing edge displacement thickness distribution of the SSC-A09 airfoil subject to a ramp-type motion with constant pitch rate.

0.009 V 0.008 [ 0.007 0.006

C,

0.005

lower surface

0.004 Quasi-steady, A = 0.00001

0.003 0.002

upper surface urface

/

[uasi-steady, A - 0.02 QUi Unsteady, A = 0.02

0.001 h 0.000

Fig. 7.49. The effects of pitch rate and the boundary-layer approach on the local skinfriction coefficient distribution of the SSC-A09 airfoil subject to a ramp-type with constant pitch rate.

(b) In the case of ramp-type motion with a high constant pitch rate, the solutions correctly show that the thick trailing-edge separation region in steady flow is suppressed to a thin reversal region and, at the same time, stall is delayed. (c) Due to the neglect of unsteady terms, du/dt and duejdt, the quasi-steady approach does not properly simulate unsteady flows at high pitch rates. (d) The prediction of the onset of dynamic stall with the calculation method agrees with the data of Lorber and Carta for the Sikorsky airfoil and shows

References

151

Fig. 7.50. The friction coefficient distribution of the SSC-A09 airfoil subject to a ramptype motion with constant pitch rate, A = 0.02.

Fig. 7.51. Initiation of leading-edge vortex with trailing-edge separation on the Sikorsky airfoil subject to a ramp-type motion with a pitch rate of 0.02, Rc = 2 x 10 6 .

that for this airfoil the initiation of dynamic stall is caused by the trailingedge separation.

References [1] Sears, W. R. and Telionis, D. P., "Boundary-layer separation in unsteady flow," SI AM J. Appl Math. 18, 215-235, 1975. [2] Cebeci, T., "The laminar boundary layer on a circular cylinder started impulsively from rest," J. Comp. Phys. 3 1 , 153-172, 1979.

152

7. Applications of Boundary-Layer Methods: Flows wth Separation

[3] Cebeci, T., An Engineering Approach to the Calculation of Aerodynamic Flows, Horizons PubL, Long Beach, Calif, and Springer, Heidelberg, Germany, 1999. [4] Patterson, M. T. and Lorber, P. F., "Computational and experimental studies of compressible dynamic stall," Fourth Symposium on Numerical and Physical Aspects of Aerodynamic Flows, Long Beach, CA, 16-19 Jan., 1989. [5] Carr, L. W., "Progress in analysis and prediction of dynamic stall," J. of Aircraft 25, 6-17, 1988. [6] McCroskey, W. J., McAlister, K.W., Carr, L. W. and Pucci, S.L., "An experimental study of dynamic stall on advanced airfoil sections," NASA TM 84245, 1982. [7] Carr, L.W. and Chandrasekhara, M.S., "Design and development of a compressible dynamic stall facility," AIAA Paper 89-0647, 1989. [8] Ham, N. D., "Aerodynamic loading on a two-dimensional airfoil during dynamic stall," AIAA J. 6, 1927-1934, 1968. [9] Mehta, U. B., "Dynamic stall of an oscillating airfoil," AGARD Conf. P r o c , No. 227, Paper No. 23, 1977. [10] Walker, J. M., Helin, H. E. and Strickland, J. H., "An experimental investigation of an airfoil undergoing large-amplitude pitching motions," AIAA J. 12, 1141-1142, 1985. [11] Cebeci, T., Stewartson, K. and Williams, P. C , "Separation and reattachment near the leading edge of a thin airfoil at incidence," AGARD CP 291, Paper 20, 1980. [12] Cebeci, T., "Unsteady boundary-layers with an intelligent numerical scheme," J. Fluid Mech., Vol. 163, pp. 129-140, 1986. [13] Cebeci, T., Khattab, A. A. and Schimke, S.M., "Separation and reattachment near the leading edge of a thin oscillating airfoil," J. Fluid Mech. 188, 253-274, 1986. [14] Cebeci, T. and Cousteix, J., Modeling and Computation of Boundary-Layer Flows, Horizons PubL, Long Beach, Calif, and Springer, Heidelberg, Germany, 1998. [15] Lissaman, P. B.S., "Low-Reynolds-Number Airfoils," Annual Review of Fluid Mech. 15, 223-239, 1983. [16] Mueller, T. J., eds., Proceedings of the Conference on Low Reynolds Number Airfoil Performance, UNDAS-CP-778123, June 1985. [17] Mueller, T. J., eds., Proceedings of the Conference on Low Reynolds Number Aerodynamics. University of Notre Dame, June 1989. [18] Proceedings of the International Conference on Aerodynamics at Low Reynolds Numbers. The Royal Aeronautical Society, October 1986. [19] Cebeci, T., Stability and Transition: Theory and Application, Horizons PubL, Long Beach, Calif, and Springer, Heidelberg, 2004. [20] McGhee, R. J., Jones, G.S. and Jouty, R., "Performance Characteristics from Wind Tunnel Tests of a Low Reynolds-Number Airfoil," AIAA paper 88-0607, 1988. [21] Hoheisel, H., Hoeger, M., Meyer, P. and Koerber, C , "A Comparison of Laser-Doppler Anemometry and Probe Measurements Within the Boundary-layer of an Airfoil at Subsonic Flow," in Laser Anemometry in Fluid Mechanics - II, Selected Papers from the Second Intl. Symp. on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 1984. LADOAN, pp. 143-157. [22] Abbott, J.H. and von Doenhoff, A.E., Theory of Wing Sections, Dover, 1959. [23] Jang, H.M., "A viscous-inviscid interactive method for unsteady flows," Ph.D. Dissertation, Univ. of Michigan, 1990. [24] Lorber, P. F. and Carta, F.O., "Unsteady stall penetration experiments at high Reynolds number," United Technologies Research Center Rept. R87-956939-3, 1987.

Navier—Stokes Methods

8.1 I n t r o d u c t i o n Numerical methods for the solution of boundary layer equations were discussed in Chapter 5 and here the discussion is extended to the Navier-Stokes equations for incompressible and compressible flows. Forms of the equation appropriate for numerical methods are presented in Section 8.2 and turbulence models including those based on algebraic and one and two transport equations are introduced in Section 8.3. Brief discussions of the numerical methods for incompressible and compressible flows are provided in Sections 8.4 and 8.5 respectively and the reader is referred to [1,2] for further information. Most numerical methods use body fitted C-type grids clustered at the body surface in the normal direction. For airfoil flows with deep dynamic stall, as discussed in Chapter 9, depending on the Reynolds number the leading and trailing edges are usually refined and the spacing of the first grid point at the surface in the normal direction varies from 0.00005 to 0.000005 chord lengths. Also, the grid boundaries in most computations are typically located 15 to 20 chords away from the airfoil in all directions. Further details about the grids used for numerical solution of equations appropriate to laminar and turbulent flows with dynamic stall flow fields are available in the references given in Chapter 9. The unsteady flow solutions are usually obtained on a moving grid and the orientation of the moving frame with respect to the fixed frame changed at each instant of time. Thus, all metrics are recomputed after the grid has moved to the new location at each time step. The boundary conditions for unsteady flow calculations are often updated explicitly so that, for subsonic inflow-outflow, the flow variables at the boundaries are evaluated using one-dimensional Riemann-invariant extrapolation. There is one incoming and three outgoing characteristics at the inflow boundary and three variables, the density, normal velocity, and pressure, are specified with

8. Navier-Stokes Methods

154

the fourth variable, the axial velocity, extrapolated from the interior. There are one incoming and three outgoing characteristics at the outflow boundary and only one quantity, the pressure, is specified while the others are extrapolated from the interior. First-order extrapolation is used for the density. A non-slip condition is applied for the velocities on the body surface, and the contravariant velocity components in the body-fixed coordinate system are set equal to zero. It should be noted that the surface velocity is non-zero because of the body motion through the unsteady metrics. Averaging of the flow variables at the wake cut is used for C-type grids. For additional details, the reader is referred to [3].

8.2 Navier-Stokes Equations For compressible flows, the Navier-Stokes equations are similar to those given by Eqs. (2.2.1) to (2.2.4) for incompressible flows. Since the fluid properties now also vary with temperature, the continuity and momentum equations are coupled to the energy equation, and the solution of the energy equation provides the temperature distribution in the flowfield. These equations are discussed in some detail in several references, see for example [1,2], and are summarized below for an unsteady compressible three-dimensional flow. The continuity equation is § f + V-(eY) = 0

(8.2.1a)

For a Cartesian coordinate system, it becomes

The momentum equations are identical to those given by Eqs. (2.2.2) to (2.2.4) provided that, with 6{j denoting the Kronecker delta function (6{j = 1, if i = j and 6ij = 0 if i ^ j ) , the viscous stress tensor aij is written as (Tij =

fl

/dui \ dxj

9UJ\

dxi J

2 3

lJ

duk dxk

(i,j,k

= 1,2,3)

(8.2.2)

The energy equation can be written either in terms of total energy per unit ume, Et, (

/2\

\ 2"

(8- 2 '3)

as

dEt + V • EtV = qh - V • q + V • (77y- • V) + gf • V dt or in terms of internal energy per unit mass, e, as

(8.2.4a)

8.2 Navier-Stokes Equations

155

Dc Q-jX+P&-V)

du' + (Jrj - ^

= qh-V-q

(8.2.4b)

In the above equations, U{j represents the stress tensor given by Ilij = -p6ij + (Tij and Gij (dui/dxj) $ = fj,

represents the dissipation function

dx J 'du Kdz

(8.2.5a)

\dy J

\dz

^w\2_
dv_ dy

J

\dx

dy J

\dy

dw\2 dz J

dz J (8.2.5b)

Finally, we need an equation of state for the fluid to relate p, g and e. The commonest example is the perfect gas law p = (7 -

1)CVQT

where e

=

cvi i

7 — Cp/Cy

The terms on the left-hand side of the energy equation given by Eq. (8.2.4a) represent the rate of increase of total energy in the control volume (per unit volume) and the rate of total energy lost by convection through the control volume (per unit volume), respectively. The first term on the right-hand side of the equation represents the heat produced per unit volume by external agencies, the second term represents the rate of heat lost by conduction through the control volume (per unit volume), and the third and fourth terms represent the work done on the control volume by the surface forces and body forces, respectively (per unit volume). In the numerical solution of the conservation equations, it is often preferable to express them in "divergence form" or "conservation form" to avoid numerical difficulties that may arise in some flows, such as flows containing shock waves, when the nondivergence form or the nonconservation form of the equations is used. In conservation form the coefficients of the derivative terms can be constant or variable; if variable, their derivatives do not appear in the equation. For example, the continuity equation (8.2.1a) is in the conservation form. However, if it is written as dg dg du dg dv dg dw ^ 7 + U1T + P^T + V7T + ^7T + W7T + ^IT = °ot ox ox oy oy oz oz it is in nonconservative form. The momentum equations in conservation form are dgu d , 2 ~wr + w-(&u2 +P~

\ 9 , d N Vxx) + -W-{QUV - crxy) + —(guw

- axz) = gfx

(8.2.6)

(8.2.7)

8. N a v i e r - S t o k e s M e t h o d s

156

dgv d . ~dT + ~dx^guv~ayx^

d . o d~^gv +p~ayy'

N

-^T - + -~-(Quw -azx)

x

9 ,

+

~Q~Z^VW

. x ~ Gvz) = efy

+ -^-{gvw - azy) + ^-(0™ +p-<7zz)

/ n n n, (8-2-8)

= QJz (8.2.9)

where the components of the viscous stress tensor follow from Eq. (8.2.2),

Gxx =

2

20

/ <9?i

<9i>

/ cfe (Q2 — ~ V dy f dw

<9^ ^ — - — j dx dz du dv

r{ di~dy-

2 Gyy = -H 3 2

°xy = M |

dz J =

^ dx ( dv

=

Ti) (8.2.10)

— ®yx

CTxz = M I fly*

dw\

dw \

/X (

CTzx

= cr22/

Similarly, the energy equation can be written in conservation form as dEt dt

8Q - QifxU + fyV + fzU)) dt d_ + -^z{Etu + pu- uaxx - vaxy - waxz + qx) dx

(8.2.11)

9 ,„ + -^-{EtV

+ PV - UOXy - VOyy ~ W(JyZ + qy)

+ -x-(Etw

+pw - uaxz - vayz - wazz + qz) = 0

8.2.1 Vector-Variable Form As we shall see in Sections 8.4 and 8.5, before the application of the numerical methods to the conservation equations, it is convenient to combine the continuity, momentum and energy equations into a compact vector-variable form. With I denoting a length scale, the speed of sound, a, denoting the velocity scale, the parameters £>, u, v, p, o~xx, axy, o~yy, Et, /i, t and x, y, with oo referring to freestream quantities, can be expressed in nondimensional form as Q

„ i

Qoo

_ —

u

^

Gxx

2 ' QoO^QO

„ •>

&oo

z.

_

x

v

—

v

v^ =

p i

y

2

&oo

U

xy

2 ' QoO^QO

'

£?oo&oo

~

_ yy

—

Q~yy 2 ' QoO^QO

/o

9 1 9

\

\O.A.X£)

8.2 Navier-Stokes Equations

Et 2 Qooaoo

Et

157

tac

t

V

:

I

~ x x = -V ,

'

„ y y = 7

Then the compressible Navier-Stokes equations in a Cartesian coordinate system, given by Eqs. (8.2.1b), (8.2.7), (8.2.8) and (8.2.4a) can be written in dimensionless form, without body forces or external heat addition for two-dimensional flows, as

dQ

d^

a^ _ J_ /d^

dt

dx

dy

dFz

Re \ dx

dy

(8.2.13)

For simplicity, the ~ will be dropped in dimensionless quantities, and the Reynolds number Re is defined by c

Re =

(8.2.14)

Moo

and the Q, E, F, Ev, Fv vectors by gu gu2 + p guv (Et+p)u

Q

Q

gu QV

E

Et

,

gv guv F = gv2 + p {Et+p)v

(8.2.15a)

0 ®XX

Ev

Fv =

®xy

Hx

u

xy

(8.2.15b)

a

yy

A/

with 7 denoting the ratio of specific heats and a the speed of sound, which for ideal gases is given by a2 = jp/'g. The viscous stresses are 2

(du dv\ 2— dx dy J dv du\ } G vv = oM 2 3 " \"dy ~~ &c) du dvs a.xy V dy dx ?xx = -V

3.2.16)

and we also write Px = uaxx + vaxy (3 — Uaxy

+

M

d , 2

(a')

Pr(7 — 1) dx d M + V Gyy +

Pr( 7 -1) V

(8.2.17)

8. Navier-Stokes Methods

158 8.2.2 Transformed Form

The Navier-Stokes equations discussed above and expressed for a Cartesian coordinate system are valid for any coordinate system. In many problems it is more convenient to write the equations in general curvilinear coordinates by using a coordinate transformation from the rectangular Cartesian form. To illustrate the procedure, consider a two-dimensional unsteady flow and introduce the generic transformation r = t (8.2.18) r) =

7](x,y,t)

For an actual application, the transformation in Eq. (8.2.18) must be given in some analytical or numerical form. Often the transformation is chosen so that the grid spacing in the curvilinear space is uniform and of unit length, that is Ar\ = 1, A£ = 1. This produces a computational space £,77 with a rectangular domain and with a regular uniform mesh so that the differencing schemes used in the numerical formulation are simpler. The original Cartesian space is usually referred to as the physical domain. Using the chain rule of differential calculus, we can write d_ _ d_ Ft " ~frr

+

d_d£ d_dr^_d_ ~di~dt + lh)~di " ~frr

d dx

<9<9£ <9£ dx

d drj _ d drj dx d^

d dy

d d£ <9£ dy

d dr] _ d drj dy <9£

+

d_ dp

d drj y

d dr]

+

d_ 'drp1 (8.2.19)

y

or in compact form as d

dt d dx d dy

3 1 1 Ct vt 1 dr = 0 £r % 0

Zy Vy\

d

(8.2.20)

d \ 9rj

In a similar manner, the second derivatives that appear in the momentum and energy equations can be expressed in transformed variables. They are, however, somewhat more involved than those in Eq. (8.2.19). For example, with the chain rule, it can be shown that

8.2 Navier-Stokes Equations

d_ 2

dx

2

dy

O2 dxdy

159

d_ xx +

dC

^

d2

u_2

Vxx

+

dr)

x+

de^

o d2 dr]dZ

2

drj

d2 d_ d\2 + aJlyy + Q£2^V + Q^V drj

d d2 d - <9£ fiAxy + drj *Vxy + fitf^V d?

+

2

,

d2 f]y^y dr)d£,

d2 (Vxty + ixVy) dr)d£, drj' (8.2.21) In terms of the transformation denned by Eq. (8.2.18), the vector form of the transformed Navier-Stokes equations, Eq. (8.2.13), can be written as

aQ dQ dQ d£ + t +Vt +ix Or * di dr] d£ Re\^x

8£

+T1x

dV

J* ;VxVy

+

ae dr]

OF

+Vx

+l;y

+

+T]y

Z dk +Vy

dt

dF

y

'dr,

drj

(8.2.22)

The coefficients of the derivatives in Eq. (8.2.19) with respect to £, 77, namely £t) ^a;; Cy; Vt, Vx a n d rjy are metric terms which can be obtained from the transformation given by Eq. (8.2.18). If the relations between the independent variables in physical space and transformed space are given analytically, then the metrics can be obtained in closed form. In general, however, we usually are provided with just the (x, y) coordinates of grid points and numerically generate the metrics using finite-difference quotients. Reversing the role of the independent variables in the chain rule formulas, Eq. (8.2.20) becomes d_ dr d_

d dx

d_ dt d dy

d_ dx d_ drj

d dy d__ dx

d__

(8.2.23)

which can be written in matrix form d dr d d drj

d 1 xT yT | dt = 0 x5 y J d dx d \ dy

(8.2.24)

Solving Eq. (8.2.24) for the curvilinear derivatives in terms of the Cartesian derivatives yields

8. Navier-Stokes Methods

160

d dt d dx d dy

1

J

1

{xvyT - xTyv)

(xTy£ -

d_ dr d_

yTx^)

0

yv

-V(,

0

— Xr)

X£

dt

(8.2.25)

d_ drj

where J - 1 is the inverse Jacobian determinant defined by J~l

dx dy ^ d£ dx dy = xgyr, - xvy^

= d(x,y) d&V)

(8.2.26)

drj drj

Evaluating Eq. (8.2.25) for the metric terms by comparing to the matrix of Eq. (8.2.20), we find that & = (xvyT - xryv)/J Vr]

ix = J - 1 '

\

ve- 1

Vx

J

l

rjt = (xTy^ - yTx^)/J X y = £y

'

J -1 '

Vy y

Z

(8.2.27)

J-1'

We note from Eq. (8.2.27) that if we are given the inverse transformation t =T x = x(^,r/,r)

(8.2.28)

so that £, rj and r are now the independent variables, then the metric coefficients in Eqs. (8.2.22) can be obtained from the relations given by Eq. (8.2.27). At this point, we notice that Eqs. (8.2.22) are in a weak conservation form. That is, even though none of the flow variables (or more appropriately, functions of the flow variables) occur as coefficients in the differential equations, the metrics do. However, as discussed by Pulliam [3], the expressions d_ dr

^>+l(§K(?) d_

k

d_ / % \ drj \ J J '

k

d_(V\ dr)\J J

J

and

d_

J are defined as invariants of the transformation and are analytically equal to zero. Equations (8.2.22) can then be expressed in the strong conservation form and written as

8.3 Turbulence Models

161

dQ dr

dE_ d£

&F _ }_(dE^ dr) Re I d£

dFy drj

(8.2.29)

where gU QUU + £xp QVU + £yP U(Et+p)-£tP

Q

Q = J~l

QU QV

E = J~l

,

Et

,

F = J~1

QV QUV + T]xp QVV + rjyp V(Et+p)-ritP (8.2.30a)

with the contravariant velocities U and V defined by U = £t+€xU + ZyV,

V = T)t+ T]XU + TJyV

(8.2.30b)

The viscous flux terms are Ev = J

(fixEv + £,yFv),

Fv = J

(rjxEv + VyEv)

(8.2.30c)

with Ev and Fv given by Eq. (8.2.15b). The stress terms, such as aXXl ayy, etc. are also transformed in terms of the £ and rj derivatives where

a

yy = 3 [ ~ 2 ( ^ ^ + VxUrj) + 4(£ y v € + VyVri)]

(8.2.31)

°xy = v(€yv>£ + r)yUv + £XV£ + rjxvv) Px = uaxx + vaxy

+

(3y = uaxy + w y y +

\

9 / 2x

^ / 2x

P r ( 7 - 1) P r ( 7 - 1)

(8.2.32) '3^

'dr?

8.3 Turbulence Models There are several approaches for modeling Reynolds stresses in boundary-layer and Navier-Stokes equations as discussed in, for example, [4,5]. The most common approach is to define an eddy viscosity, emi in the same form as the laminar viscosity. Thus, for a two-dimensional incompressible flow. QUrv' =

Q£m

du dy

(8.3.1)

Another approach is to use the mixing length, /, concept and express the Reynolds shear stress by gufv' — gl

duV dy)

(8.3.2)

162

8. Navier-Stokes Methods

The specification of £ m or I may be made in terms of algebraic equations or in terms of a combination of algebraic and differential equations and this has given rise to terminology involving the number of differential equations. Thus, the turbulence models may be described in terms of zero, one and two differential equations. For a two-dimensional boundary-layer flow, the zero-equation approach usually treats a turbulent boundary layer as a composite layer with separate expressions for em or I in each region. The Cebeci-Smith (CS) model discussed in Section 5.1 is a typical example of this approach. In the one-differential equation approach the eddy viscosity is written, with c^ denoting a constant, as em - c^kl'2l (8.3.3) with k obtained from a differential equation which represents the transport of turbulence energy and I from an algebraic formula. In the two-differential equation approach, the eddy viscosity is written as

with k and e obtained from differential equations which represent the transport of turbulence energy and its rate of dissipation. Here we present a brief description of turbulence models used in NavierStokes methods for dynamic-stall calculations. 8.3.1 Algebraic M o d e l s Of the several algebraic models, the Cebeci-Smith (CS) model discussed in Section 5.1 is the most widely used turbulence model for boundary-layer flows. Extensive studies, mostly employing boundary-layer equations, showed that while many wall boundary-layer flows can satisfactorily be calculated with the original version of this model, improvements were needed for flows which contain regions of strong pressure gradient and flow separation, for example, flows either approaching stall or post-stall. The main weakness in this model is the parameter a used in the outer eddy viscosity formula, Eq. (5.1.7), taken as 0.0168. Experiments indicate that in strong pressure gradient flows, the extent of the law of the wall region becomes smaller; to predict flows under such conditions, it is necessary to have a smaller value of a in the outer eddy viscosity formula. The question is how to relate a to the flow properties so that the influence of strong pressure gradient is included in the variation of a. Cebeci and Chang [6] investigated this question and related a to the expression given by Eq. (5.1.8). In addition, they improved the intermittency expression 7 used in Eq. (5.1.7) replacing the Klebanoff's formula for 7 with the one given by Fiedler and Head, Eq. (5.1.11).

8.3 Turbulence Models

163

Johnson—King M o d e l Another improvement to the original Cebeci-Smith model was made by Johnson and King [7] by adopting a nonequilibrium eddy-viscosity formulation em in which the CS model serves as an equilibrium eddy viscosity (em)eq distribution. An ordinary differential equation (ODE), derived from the turbulence kinetic energy equation, is used to describe the streamwise development of the maximum Reynolds shear stress, —(gufvf)m, or (—u'vf)m for short, in conjunction with an assumed eddy-viscosity distribution which has y ( - w V ) m as its velocity scale. In the outer part of the boundary layer, the eddy viscosity is treated as a free parameter that is adjusted to satisfy the ODE for the maximum Reynolds shear stress. More specifically, the nonequilibrium eddy-viscosity distribution is defined again by separate expressions in the inner and outer regions of the boundary layer. In the inner region, (£ m )i is given by (e m )i = (e m i )i(l " 72) + 72(£m-x)J-K where (£ mi )i is given either by (hiy)2du/dy (£mi)j-K

(8-3.5)

or uTy. The expression (emi)j-K

= D2hiyum

is

(8.3.6)

where um = max(w T , y(—ufv')m)

(8.3.7a)

and D is a damping factor similar to that defined by Eq. (5.1.6a) D = 1 - exp f - ^ ( - ^ )

m

-^\

(8.3.7b)

with the value of A+ equal to 17 rather than 26, as in Eq. (5.1.6a). The parameter 72 in Eq. (8.3.5) is given by 72 = tanh (jj\

(8.3.7c)

where, with ym corresponding t o the y-location of maximum turbulent shear stress, (—u'v f ) m , L'c =

U

^

Lm

(8.3.8)

with 2 2 M

f0.4 9m

^ ° '

1 0.09.5

ym > 0.2256 .

(8 .3. 9 )

In the outer region, ( e m ) 0 is given by ( e m ) 0 = <7(0.0168«e$*7)

(8.3.10)

8. Navier-Stokes Methods

164

where a is a parameter to be determined. The term multiplying a on the righthand side of Eq. (8.3.10) is the same as the expression given by Eq. (5.1.7) without 7t r and with a = 0.0168. The nonequilibrium eddy viscosity across the whole boundary-layer is computed from '(e m )i" £ m = (£ m )otanh (8.3.11) (£m)o J The maximum Reynolds shear stress (—u'v')m is computed from the turbulence kinetic energy equation using assumptions similar to those used by Bradshaw et al. [8]. After the modeling of the diffusion, production and dissipation terms and the use of ( - « V ) >m r ai = 0.25 /Cn

f f

the transport equation for (—u v )m locity at y m , is written as

with um now denoting the streamwise ve-

-£(-=V)m = ° l ( - " ^ ) m [ ( - ^ V ) ^ - GV)JL/2] - ^Dm ^m^m

(XX

(8.3.12)

^m

where, with c^if = 0.5, the turbulent diffusion term along the path of maximum {—u'v') is given by

m

"

ai6

[0.7 -

(y/6)n

(-ttV)r (-u'vf) m,eq

1/2^

(8.3.13)

To use this closure model, the continuity and momentum equations are first solved with an equilibrium eddy viscosity (em)eq distribution such as in the CS model, and the maximum Reynolds shear stress distribution is determined based on (£ m )eq, which we denote by ( - u V ) m ) e q . Next the location of the maximum Reynolds shear stress is determined so that ym and um can be calculated. The transport equation for ( - w V ) m is then solved to calculate the nonequilibrium eddy-viscosity distribution em given by Eq. (8.3.11) for an assumed value of a so that the solutions of the continuity and momentum equations can be obtained. The new maximum shear stress term is then compared with the one obtained from the solution of Eq. (8.3.12). If the new computed value does not agree with the one from Eq. (8.3.12), a new value of a is used to compute the outer eddy viscosity and eddy-viscosity distributions across the whole boundary-layer so that a new ( - w V ) m can be computed from the solution of the continuity and momentum equations. This iterative procedure of determining a is repeated until (—ufvf)m is computed from the continuity and momentum equations agrees with that computed from the transport equation, Eq. (8.3.12).

8.3 Turbulence Models

165

Baldwin-Lomax Model Due to its simplicity and its good success in external boundary-layer flows, the CS model with modifications has also been used extensively in the solution of the Reynolds-averaged Navier-Stokes equations for turbulent flows. These modifications are described below. Baldwin and Lomax [9] adopt the CS model, leave the inner eddy viscosity formula given by Eq. (5.1.5) essentially unaltered, but in the outer eddy viscosity, Eq. (5.1.7), use alternative expressions for the length scale <5* of the form (e m )o = a c i 7 7 / m a x F m a x (8.3.14a) or (e m )o = a c i T ^ d i f f T ^ 1

(8.3.14b)

-^max

with c\ = 1.6 and C2 = 0.25. The quantities F m a x and ?/ max are determined from the function F = y ( g ) [1 - e~y/A] (8.3.15) with F m a x corresponding to the maximum value of F that occurs in a velocity profile and ym8iX denoting the ^/-location of F m ax- ^diff is the difference between maximum and minimum velocity in the profile ^diff = ^ m a x - ^ m i n

(8.3.16)

where umm is taken to be zero except in wakes. In Navier-Stokes calculations, Baldwin and Lomax replace the absolute value of the velocity gradient du/dy in Eqs. (5.1.5) and (8.3.15) by the absolute value of the vorticity |u;|, \du dv\ (8.3.17a) \dy dx\ and the intermittency factory 7 in Eq. (5.1.7) is written as -1

1 = 1 + 5.5

(8.3.17b) I/max

with C3 = 0.3. The studies conducted by Stock and Haase [10] clearly demonstrate that the modified algebraic eddy viscosity formulation of Baldwin and Lomax is not a true representation of the CS model since their incorporation of the length scale in the outer eddy viscosity formula is not appropriate for flows with strong pressure gradients. Stock and Haase proposed a length scale based on the properties of the mean velocity profile calculated by a Navier-Stokes method. They recommend computing the boundary-layer thickness 6 from «=1.936ymax

(8.3.18)

8. Navier-Stokes Methods

166

where yma^ is the distance from the wall for which y\du/dy\ or F in Eq. (8.3.15) has its maximum. With 6 known, ue in the outer eddy viscosity formula, Eq. (5.1.7) is the u at y = <5, and 7 is computed from 7 = 1 + 5.5

(De

(8.3.19)

based on KlebanofFs measurements on a flat plate flow and not from Eq. (8.3.17b). The displacement thickness 6* for attached flows is computed from its definition, 6* = [

( l ~ — ) dy

(8.3.20a)

and, for separated flows from 6* = [ (l-—) dy Jyu=o V ueJ

(8.3.20b)

either integrating the velocity profile from y = 0, or y — yu=o to <5, or using the Coles velocity profile. The results obtained with this modification to the length scale in the outer CS eddy viscosity formula improve the predictions of the CS model in Navier-Stokes methods as discussed in Stock and Haase [10]. A proposal which led to Eq. (8.3.18) was also made by Johnson [11]. He recommended that the boundary-layer thickness 6 is calculated from 6 = 1.2y1/2

(8.3.21)

where 2/1/2 = y

at

-^-=0.5.

(8.3.22)

The predictions of the original and modified CS models were also investigated by Cebeci and Chang [6] by using the Navier-Stokes method of Swanson and Turkel [12] as well as by the interactive boundary-layer method of Cebeci (a — 0.0168) for steady flows [13]. The models considered include the original CS model, BL model, modifications to the BL model and the JK model. The reader is referred to [5,13] for a comparison of these models. 8.3.2 O n e - E q u a t i o n M o d e l s Unlike the CS model which uses algebraic expressions for eddy-viscosity, the oneequation model of Spalart and Allmaras [14] uses a transport equation for eddy viscosity. This model is local (i.e., the equation at one point does not depend on the solution at other points), and therefore compatible with grids of any structure and Navier-Stokes solvers in two or three-dimensions. The wall and freestream boundary conditions are trivial. The model yields relatively smooth

8.3 Turbulence Models

167

laminar-turbulent transition at the specified transition location. Its defining equations are as follows. £m = hfVl (8.3.23) Di>t

r.

c

„ , ~„

/

ch, „ \ fvt\2

„

I

d

di>t~

= h [i - ft2]Sh - (cwjw - -^ft2) ( 4 J + - dx _| {y + h) dx _ k k

Dt

+

cb2 dvt dvt a dxk dxk

(8.3.24)

Here cbl = 0.1355, Cbi

C

W\

+ (i +

q>2 = 0.622,

b2)

x3 X3 + c

3

a =

3.3.25a)

c

a

v\ —

cUl = 7.1,

cw2 = 0.3, X

M

~ i

vt

/«

i + xU,'

g = r + cW2(r% o = o -\

ft2 =ct3e

cW3 = 2,

-r),

ct x

*

,

l + c!103
=

2772^2'

K = 0.41

ct3 = 1.1,

y

^hj**\V

Q4 = 2

(8.3.25b)

1/6

(8.3.25c) (8.3.25d) (8.3.25e) (8.3.25f)

where d is the distance to the closest wall and S is the magnitude of the vorticity, O

J/

U

' dUj _ _ - I (
_

2 \dxj

du &U± j\ dxi )'

The wall boundary condition is i>t = 0. In the freest ream and as initial condition 0 is best, and values below ^ are acceptable [14]. 8.3.3 T w o - E q u a t i o n M o d e l s Over the years a number of two-equation models have been proposed. A description of most of these models is given in detail in [4,5]. Here we consider three of the more popular, accurate and widely used models for dynamic-stall calculations. They include the k-e model of Jones and Launder [15], the k-uo model of Wilcox [4] and the SST model of Menter which blends the k-e model in the outer region and k-uj model in the near wall region [16]. k-e M o d e l The k-e model is a popular and widely used two-equation eddy viscosity model. With £m given by Eq. (8.3.4), the kinetic energy k and rate of dissipation e are obtained from the turbulence kinetic energy equation,

8. Navier-Stokes Methods

168 Dk ~Dt

d dxk

v+

dk dxk

°k

+ ^r

/ dui \dxj

duj dxi

duj dxj

(8.3.26)

and from the dissipation equation De ~Dt

d dxk

de dxk

v+

+ c£

dm k

dx.

duj \ du, J dx

^€2

(8.3.27)

a£ = 1.3

(8.3.28)

+ dxj

The parameters cM, c £ l , c £2 , ak and a£ are given by 0.09,

1.44,

^ei

c£2 = 1.92,

ak = 1.0,

These equations apply only to free shear flows. For wall boundary-layer flows, they require modifications to account for the presence of the wall as discussed in [4, 5]. To account for the presence of the wall, it is necessary to include lowReynolds-number effects into the k-e model. There are several approaches that can be used for this purpose. For a review of these models, see [4, 5,13]. An extension of the k-e model, used in time-dependent Navier-Stokes equations, was developed by Yakhot et al. [17]. With techniques from renormalization group theory they proposed the so-called RNG k-e model. In this model, k and e are still given by Eqs. (8.3.26) and (8.3.27). The only different occurs in the definitions of the parameters given by Eq. (8.3.28). In the RNG k-e model, they are given by 3 C/ ,A (1 - A/A 0 ) c e i = 1.42, c £9 = 1.68 + 1 + 0.012A3 k (8.3.29) A0 = 4.38, A c^ = 0.085 V2 SijSjit ak = 0.72,

a£ = 0.72,

s

ij

1 /duj 2 dxn

~

dun dx

krw M o d e l Like the k-e model, k-uo model is also very popular and widely used. Over the years, this model has gone over many changes and improvements as described in [4, 5]. The most recent model is due to Wilcox [4] and is given by the following defining equations. With em defined by k em = (8.3.30) the turbulence kinetic energy and specific dissipation rate equations are Dk Dt DUJ

Dt

d dxk

d dxk

[('

v H

dk dxk

v H duu

dxk\

+

+

„

R

dm l k dxk

uu dui a-Rik^— k oxk

^ 7 ^ - ^ k u (3u2

J.3.31) (8.3.32)

8.3 Turbulence Models

169

where Rik is given by *Hj — ^m

dxj

(8.3.33)

dxi

and 13 25'

a

A)

P = Mp, 9 125'

P* = Pof(S, °k = 2,

iJijlJjfcDfci

1 + 70Xa 1 + 80Xu

Xu

0Wa

(8.3.34a) (8.3.34b)

Xfc<0 9

^5-ioo'

7

^-

1 + 680*2 "2 > I l + 400Xf

v

1 dk da; — u)3 dxj dxj

> n

Xfc > U

(8.3.34c)

The tensors f2{j and S^ appearing in Eq. (8.3.34b) are the mean rotation and mean-strain-rate tensors, respectively, defined by

na = - (—

du

3

i

1 2

o

(duk \dXl

duj

(8.3.35) 2 \dXj The parameter Xu is zero for two-dimensional flows. This model takes the length scale in the eddy viscosity as 13

+ dxj.

(8.3.36a)

/ = UJ

and calculates dissipation e from

P*uk

(8.3.36b)

Wilcox's model equations have the advantage over the k-e model that they can be integrated through the viscous sublayer, without using damping functions. At the wall the turbulent kinetic energy k is equal to zero. The specific dissipation rate can be specified in two different ways. One possibility is to force uo to fullfill the solution of Eq. (8.3.32) as the wall is approached [4]: CJ

6v —^ as y -> 0

(8.3.37)

The other [4] is to specify a value for uo at the wall which is larger than uow > lOOi?^ where fiw is the vorticity at the wall. Menter [16] applied the condition of Eq. (8.3.37) for the first five grid points away from the wall (these points were always below y+ = 5). He repeated some of his computations with uow — 1000i7^ and obtained essentially the same results. He points out that the second condition is much easier to implement and

8. Navier-Stokes Methods

170

does not involve the normal distance from the wall. This is especially attractive for computations on unstructured grids. SST Model The SST model of Menter [16] combines several desirable elements of existing two-equation models. The two major features of this model are a zonal weighting of model coefficients and a limitation on the growth of the eddy viscosity in rapidly strained flows. The zonal modeling uses Wilcox's k-uo model near solid walls and Launder and Sharma's k-e model near boundary layer edges and in free shear layers. This switching is achieved with a blending function of the model coefficients. The shear stress transport (SST) modeling also modifies the eddy viscosity by forcing the turbulent shear stress to be bounded by a constant times the turbulent kinetic energy inside boundary layers. This modification, which is similar to the basic idea behind the Johnson-King model, improves the prediction of flows with strong adverse pressure gradients and separation. In order to blend the k-uj model and the k-e model, the latter is transformed into a k-uu formulation. The differences between this formulation and the original k-uj model are that an additional cross-diffusion term appears in the cj-equation and that the modeling constants are different. Some of the parameters appearing in k-uj model are multiplied by a function F\ and some of the parameters in the transformed k-e model by a function (1 — F\) and the corresponding equations of each model are added together. The function F\ is designed to be a value of one in the near wall region (activating the k-uj model) and zero far from the wall. The blending takes place in the wake region of the boundary layer. The SST model also modifies the turbulent eddy viscosity function to improve the prediction of separated flows. Two-equation models generally underpredict the retardation and separation of the boundary layer due to adverse pressure gradients. This is a serious deficiency, leading to an underestimation of the effects of viscous-inviscid interaction which generally results in too optimistic performance estimates for aerodynamic bodies. The reason for this deficiency is that two-equation models do not account for the important effects of transport of the turbulent stresses. The Johnson-King model has demonstrated that significantly improved results can be obtained with algebraic models by modeling the transport of the shear stress as being proportional to that of the turbulent kinetic energy. A similar effect is achieved in the SST model by a modification in the formulation of the eddy viscosity using a blending function F
aik

max(aiu;, QF2)

(8.3.38)

8.3 Turbulence Models

171

where a\ = 0.31. In turbulent boundary layers, the maximum value of the eddy viscosity is limited by forcing the turbulent shear stress to be bounded by the turbulent kinetic energy times a i , see subsection 8.3.1. This effect is achieved with an auxiliary function F 2 and the absolute value of the vorticity J?. The function F 2 is defined as a function of wall distance y as (8.3.39a)

F 2 = tanh(arg 2 ) where Vk arg 2

max

500i/'

(8.3.39b)

y2uu

omuy

The two transport equations of the model for compressible flows are defined below with a blending function F\ for the model coefficients of the original UJ and e model equations. d dxk

Dgk

DQUJ

dk

d (fJL dxk [

+

+ 2(1

F\)Q°UJ2

where

duo

, 7 sm

Q£m
dxk

1 dk

du{ dxk

Ri,

+ Rik

UJ

duk ^ dxj,

duj dxk

•P*QUk d

^i dxk

p

•PQUJ2

duj

dxk

(8.3.40)

(8.3.41)

dxk

2du T,^—^ik 3 dx

~ 7>QkSik

(8.3.42)

The last term in Eq. (8.3.41) represents the cross-diffusion (CD) term that appears in the transformed adequation from the original ^-equation. The function F\ is designed to blend the model coefficients of the original k-uj model in boundary layer zones with the transformed k-e model in free shear layer and freest ream zones. This function takes the value of one on no-slip surfaces and near one over a larger portion of the boundary layer, and goes to zero at the boundary layer edge. This auxiliary blending function F\ is defined as F\ = tanh(argf) arg x = mm max

/

(8.3.43)

y/k 500i/' 0 . 0 9 ^ ' y2u )

^gcr^k

'CD^

2

(8.3.44)

where C D ^ is the positive portion of the cross-diffusion term of Eq. (8.3.41 1 dk CDk(JJ

=

max

2QG{JJ2 UJ2 — UJ

dxk

duj

,10 dxk

-20

(8.3.45)

The constants of the SST model are /3* = 0.09,

K

= 0.41

5.3.46)

8. Navier-Stokes Methods

172

The model coefficients /?, 7, ak and au denoted with the symbol cf) are defined by blending the coefficients of the original k-u model, denoted as c/>i, with those of the transformed k-e model, denoted as 02. 0 = ^i0i + ( l - W 2

(8.3.47)

where (f)= j ^ , (7^/3, 7} with the coefficients of the original models defined as inner model coefficients akl = 0.85,

aUl = 0.5,

71 =

Pi

K2

Pi = 0.075 = 0.553

(8.3.48)

outer model coefficients ak2 = 1.0,

Ou<2 — 0.856,

_ p2

72 =

~ P*

O^K2

\/W

P2 = 0.0828

= 0.440

(8.3.49)

The boundary conditions of the SST model equations are the same as those for the k-uj model. For the numerical implementation of the SST model equations to NavierStokes equations, the reader is referred to [16].

8.4 Numerical Methods: Incompressible Flows The primary problem with time-accurate solutions of the incompressible NavierStokes equations is the difficulty of coupling changes in the velocity field with changes in the pressure field while satisfying the continuity equation. The incompressible flow equations include pressure in a non-time-dependent form because the continuity equation has a non-evolutionary character. The non-timedependent form of pressure is the source of difficulties of numerical schemes which must treat continuity with special techniques. The alternative vorticitystreamfunction (subsection 8.4.1) and velocity-vorticity (subsection 8.4.2) formulations of the incompressible flow equations do not have the same problems as the primitive variable formulation, but their application is straightforward only for two-dimensional flows. As an alternative to the above two methods the artificial compressibility or pseudo-compressibility method (subsection 8.4.3) can be used. This method was initially introduced by Chorin [18] for the solution of steady-state incompressible flows, and it was also extended [19] to time-accurate incompressible flow

8.4 Numerical Methods: Incompressible Flows

173

solutions. This method can be used for the solution of unsteady flows when a pseudo-time derivative of pressure is added to the continuity equation. This term directly couples the pressure with velocity and allows to advance the equations in physical time by iterating until a divergent-free velocity field is obtained at the new physical time level. This method has been used successfully for the solution of high Reynolds number, two- and three-dimensional steady and unsteady flows [20-22] and it is an excellent method for the solution of incompressible Navier-Stokes equations. 8.4.1 Vorticity-Streamfunction Formulation The vorticity-streamfunction formulation is limited to two-dimensional flows and requires the definition of a streamfunction ^ as u = d&/dy,

v = -dV/dx

(8.4.1)

Then the vorticity UJZ is obtained by uz = - V 2 ^

(8.4.2)

This equation and the vorticity transport equation constitute the vorticitystreamfunction formulation which has been utilized by Ghia et al. [23] for the solution of incompressible two-dimensional dynamic stall flows. In these investigations the following streamfunction along with the appropriate boundary conditions for a moving reference frame was implemented # = y + ^o + ^ D

(8-4.3)

In this equation, y is the vertical coordinate, $o a n integration constant representing instantaneous displacement of the zero streamline at infinity due to the inviscid lift generation and ^rj a disturbance streamfunction due to deviation from uniform flow. In the vorticity-streamfunction formulation the pressure does not appear explicitly and can be computed from the following Poisson pressure equation obtained by taking the divergence of the momentum equation V-

9v ^ _, . 1 _ x — + v • V v + £2(t) x v + — ( V x a?) = - V • (Vp) ut Re

(8.4.4)

8.4.2 Velocity-Vorticity Formulation The velocity-vorticity formulation consists of the vorticity transport equation and an integral equation for velocity. Since the vorticity field is closely related to the viscous stress and vorticity is absent in the potential flow zone, the vorticity transport equation needs only to be solved in the viscous flow zone. In addition, attached boundary layer and detached recirculating flow regions in the viscous

174

8. Navier-Stokes Methods

flow zone may be treated individually. The velocity vector in the viscous flow zone can be evaluated explicitly from the integral equation for velocity. The velocity-vorticity formulation thus confines the computations into the viscous flow zone only. The confinement of the computations and the zonal approach greatly reduce the computational demands and lead to an efficient zonal solution procedure. The vorticity transport equation in a rotating reference frame attached to the solid body is ^

= -(Vk-VR)w + V|(i/e«)

(8.4.5)

where the subscript CR' indicates that the velocity vector and the differentiations are defined in the rotating reference frame. The vorticity transport equation describes the kinetic transport of vorticity. All the vorticity in the fluid domain originates from the fluid boundary in contact with a solid, and spreads into the fluid domain by diffusion and is then transported away from the solid boundary by both convection and diffusion processes. These processes produce a region of non-zero vorticity around the solid body. However, in high Reynolds number flows, a large region of the flow domain is vorticity free and Eq. (8.4.5) needs to be solved only in the region containing vorticity. Furthermore, in high Reynolds number flows, the vortical region can conveniently be separated into attached and detached flow regions. In the attached flow region, the flow direction is tangential to the solid surface and the diffusion of vorticity along the flow direction is negligible compared to its convection. In this region, the boundary-layer simplification which neglects the streamwise diffusion of vorticity is justified. The relationship between the vorticity and velocity fields at any given instant of time constitutes the kinematic aspect of the problem. Wu [24] has shown that Eq. (8.4.5) can be recast into an integral representation for the velocity vector in terms of the vorticity field and the velocity boundary conditions, V(r) = - i - / " ° * ( 2n JR \-r2\

r

W J -

/

(Vo-°o)(-r)-(Voxno)x(-,) I - rl (8.4.6) where r is the position vector, the subscript '0' denotes the variables and the integration in the ro space. R is the flow domain, B is the boundaries of the flow domain and n is the unit normal vector on B directed away from the domain R. In particular, the flow domain for the airfoil problem is doubly connected and B consists of Bs, the solid boundary and B^, the far-field boundary. It should be noted that the far-field boundary conditions are applied analytically at the limit as TQ goes to infinity. As mentioned earlier, Eq. (8.4.5) needs to be solved only in the region containing vorticity. Thus, the velocity values during the numerical solution are needed only in this region. Since the velocity vector at any point in the flowfield can be evaluated explicitly by Eq. (8.4.6), computations are conveniently 2TT JB

8.4 Numerical Methods: Incompressible Flows

175

continued only to the vortical flow zone. The vorticity free potential fow zone is excluded from the computations. Equation (8.4.5) is parabolic in time and elliptic in space. At any instance of time, a boundary value problem is solved and the vorticity values on the inner and outer boundaries need to be prescribed. The outer boundary is taken to be outside the region containing vorticity. However, if the outer boundary cuts through the vortical wake, then the gradient of vorticity in the direction normal to the boundary is assumed to be zero, which allows vortices to pass through the downstream boundary at the local velocity. The inner boundary is the solid surface, BSl and the vorticity is generated continuously at this surface. In [25], it has been shown that the solid boundary vorticity values can be determined through the kinematic relationship between the instantaneous velocity and vorticity fields. It should be noted that the kinematic relationship, Eq. (8.4.6), is valid in both the solid region S and the fluid region R. Therefore, the vorticity field throughout S and R must be such that the velocity in S as computed by Eq. (8.4.6) should agree with the prescribed solid body motion. Equation (8.4.6) applied on Bs and the principle of conservation of vorticity give rise to the relations suitable for the evaluation of the surface vorticity distribution

l//y(-/'U = -V(r., *)-;!/ 27r7 5 +

2

\-YS\

+

.

Uo X (

2TT 7 R - B +

+ J - / (Vo-"o)(-r*)r(VoX"Q)x(-r*) 2TT JB

| - r

5

" ,r s ) dR0

|-rs|2 dBo

( g 4 J }

r

where B + is a boundary adjacent to B s where the surface vorticity distribution is evaluated. uos is the surface vorticity and rs is a point on B s . With prescribed V ( r s , t) and having evaluated the interior vorticity distribution in R — B + , the surface vorticity distribution, ous, can be evaluated through Eq. (8.4.7). Note that Eq. (8.4.7) is a vector equation and either component can be used to solve for UJS.

8.4.3 P s e u d o - C o m p r e s s i b i l i t y Formulation The peseudo-compressibility formulation, first introduced by Chorin [18], has the advantages that it couples the pressure and velocity fields directly at the same time level and produces a hyperbolic system of equations. The resulting artificial waves provide a mechanism for propagating information throughout the domain and drive the divergence of velocity towards zero. An appropriate method with which to apply finite differencing to a hyperbolic system uses the direction of signal propagation to bias the differencing stencil. Hence, upwind differencing schemes have recently been developed for the compressible Euler and Navier-Stokes equations [1]. The convective terms are differenced by an

8. Navier-Stokes Methods

176

upwind method that is biased by the signs of the eigenvalues of the local flux Jacobian. This is done by casting the governing equations in their characteristic form and then forming the differencing stencil such that it accounts for the direction of wave propagation. This method has been successfully applied to both two- and three-dimensional steady and unsteady flows, and here we describe it for two-dimensional incompressible unsteady flows. For simplicity, we consider the equations expressed in Cartesian coordinates for unsteady flows rather than in transformed form (subsection 2.1.4) for steady and unsteady flows. For incompressible unsteady flows, Eq. (8.2.15a) can be written as

Ev) + where

u u =

U

E--

V

Ev =

^-(F-Fv)=0

dy

2

+p~ uv

zx

F =

Fv =

J*

(8.4.8)

uv _v2 -\- p

(8.4.9a)

TXy

(8.4.9b)

T

. yy,

This viscous normal and shear stresses are given by Eq. (2.1.8) 2/i

du dx

T

yx

>xy

— fl>

du dy

dv dx

'yy

= 2/i

dv dy

(8.4.10)

The time derivatives in the momentum equations are differenced by using a second-order three point implicit formula 1.5fZn+1 -2un + At

71+1

0.5un-1

zn+l

_

&-*•>+&"

Fv)

$.4.11) To solve Eq. (8.4.11) at time level n + 1, we use the pseudo-compressibility method in which we add a time derivative of pressure to the continuity equation and write it as dp du dv n-hl, ra+1 .4.12) -/? dx dy\ and iteratively solve Eqs. (8.4.11) and (8.4.12) so that ^ n + 1 ^ m + 1 approaches un+l. Here the superscript m denotes the pseudo-time iteration count. We represent the pseudo-time derivative with a backward-difference formula w n+l,raH-l

_

yjn+1,m

du

= -p dx

'AT

dv 71+1,777+1 dy\

(8.4.13)

and write Eq. (8.4.11) as 1_mn+l,m+l

_

At

15un+l,m

-71+1,777+1

=

1.5un+1'm

- 2un + Q.bu71'1

At

5.4.14)

8.4 Numerical Methods: Incompressible Flows

177

The above two equations can be written as J t r ( £ > n + 1 , m + 1 ~ £> n + 1 ' m ) - _fl"+i,m+i _ ^ ( i < 5 £ P + i , ™ - 2Dn +

0.5D71-1) (8.4.15)

where

r

(3v ' F = uv _v2 + p_ '

' 0u '

p~ D = u _v _

2

u

E =

+p uv

(8.4.16a)

0 Ev

T

Fv

xx

(8.4.16b)

•xy yy

T

T

xy

R=l(E-Ev)

+

ly(F-Fv)

(8.4.16c)

r 1 L5 ~At

Ur

(8.4.16d) L5

By using Taylor series expansion and noting that R is a function of D, we can write / r ) J/ t? \\n + 1 > m f 7-)n+l,m+l njn+l,m+l n + i , m + i _ pn+l,m nn+l,m _,,_ / ^ —" | / jjn-\-1 ,m-\-i _ n n + l , m \

/o ^ i y \

Substituting Eq. (8.4.17) into (8.4.15), we get /tr(Dn+l,m+l

_

Dn+l,m)

+ (^

(Z? n+l,ra+l _ 7-^n+l,ra\

)

,n—1\

zir Factoring out (^p+i.m+i - £p+i,™) n+l,m

^n+i.m^

=

we

have

4 D n + 1 ' m = - i T l + 1 ' m - - ^ ( 1 . 5 D n + 1 , m - 2 I > n + 0.5L> n - 1 ) (8.4.18a)

or BADn+hm

=

_Rn+l,m

_ ^.^Dn+l,m

_

2E)n

+ 0.5jC>n-l)

( 8 .4.18b)

where B =

SdR\n+1>m'

(8.4.18c)

8. Navier-Stokes Methods

178

Before we describe the upwind method discussed in [....], the Jacobian matrices of the convective flux E and F are defined by

dE dD

"0 1

(3 2u

Lo

v

0" 0 u_

B =

'0 0 [l

OF dD

0/3 v u 0 2v

(8.4.19)

and the eigenvalues and eigenvectors of A and B are written as (i) ,(1)

u 0 0

0 u + c\ 0

0 0 u - ci J

,( 2 )

"i; 0 .0

0 v + c2 0

0 0 v-c2

»i

A1 = X^AXi

= ,(2)

A2 = X^BX2

=

A(2) 1

Xx

x2

1

2^1

0 ci/3 0 ii(ci+iO+/? 2(3 v(ci +1/) 0 -2/3 0

S.4.20)

-ci/3 u{c\-u)+p v(c\ — u)

c2/3

-c 2 /3

1^(C2 + v)

u(c2

-

v)

v(c2 +v) + (3 v(c2 - v) + (3 J

where

ci = \]u2 + (3

(8.4.21a)

C2 = yv2

(8.4.21b)

and

The inverse matrices X^ —v

x;

C\ — U

and X ^

+ P

are given by

-uv u2 + 01 0 p 0 p

—u

x? =

C2 -

/3

-V2

V

0 0

vu

P

13 (8.4.22) As described in [2], A and B can be decomposed as the sum of two parts, one part corresponding to positive eigenvalues and the other to negative eigenvalues. — C\ — U

A+ + A~ where

_ -C2 -

B = B+ + B'

V

(8.4.23)

8.4 Numerical Methods: Incompressible Flows

179

r

AW±|AW| 2 A<1)±|A<1)|

A±=Xi

X -1 2

AJ 2) ±|AJ 2) | B±=X2

(8.4.24) Xo

2 A< 2 >±|A< a >|

Application of the upwind method discussed in [2] to the derivates of the convective flux in the x- and ^/-direction yields, BE _

E

i+l/2,j ~ Ei-1/2J

dx

OF _

dy

Ax

Here Ei+i/2j

is

a

E

ij+l/2 ~ Ay

E

iJ-l/2

(8.4.25)

numerical flux at j and i + 1/2 is the discrete spatial index ls a

for the x-direction. Similarly i^j+i/2

numerical flux at i and j ' + 1/2. They

are given by - $1+1/2,3}

(8.4.26a)

Fij+1/2 = J [ £ ( A , J + i ) + £ ( A j - i ) - ij+l/2]

(8.4.26b)

E

i+l/2j

= 2 I ^ ( ^ + l j ) + E(Di-l,j)

When h+l/2,j = °

a n d

ij + l/2 = °

the convective flux terms in Eq. (8.4.25) are represented by a second-order accurate central difference scheme. A first-order accurate upwind difference scheme is given by (8.4.27a) h+l/2,j = AEf+l/2j ~ AEi+l/2j PiJ+1/2 =

AF

i\j+l/2

~

(8.4.27b)

AF

i~j + l/2

where AE^ and AF^ are the flux differences across the positive or negative traveling waves. As discussed in [2], they are given by AE

tl/2J

^

+

l / 2

= ^±(A+1/2J)^A+1/2)J

(8.4.28a)

=

(8.4.28b)

i?±

( A , i + l / 2 ) ^ A , J + l/2

Here A+i/2,j = ^ ( A + i , j + A,j)

ADi+1/2j

= A+i j

Aj+1/2 = ^ ( A j + i + A,j)

^ A j + 1 / 2 = A,j+i ~

^hj

(8.4.29a)

D

(8.4.29b)

1,3

8. Navier-Stokes Methods

180

After approximating the derivatives of the convective flux in the x- and ydirections with the upwind method, we approximate the viscous fluxes, ^L and Q&- with second-order accurate central differences, dEv _ (Ev)i+i/2,j

~

(Ev)i-i/2j

dx

Ax

0FV _ (Fv)ij+l/2

~

dy

.4.30a)

(Fv)ij-l/2

.4.30b)

Ay

With Eqs. (8.4.25) and (8.4.30), the residual vector ij n + 1 > m -term in Eq. (8.4.15) can be written as Rn+l,m

i,J

E

_

~ Ei-l/2J

i+l/2j

~

F

ij+l/2

F

~

Ax

iJ-l/2

Ay y

E

E

_ ( v)i+l/2j

F

- ( v)i-l/2,j

_ ( v)ij+l/2

(Fv)i,j-l/2

-

Ax

(8.4.31)

Ay

The calculation of the exact Jacobian matrix of residual vector can be very expensive, particularly when higher order upwind methods are used. Therefore, it is more economical to approximate ( § § ) n + 1 ' m from the residual i ? n + 1 ' m resulting from the first order upwind method. Applying the first order upwind to residual Rn+1^ results nn+l,m ^

Si+lJ - gi-lj _ ^ £ l / 2 J ~

ij

AE

^1/2J

~

2Ax F i

r

1

AF+.,

^J~l

- / 9 - AF~

M + l/2

/0

_ ( v)i+l/2,j

+ ^-1/2,,

- AF+

M + l/2

2Ay E

tl/2J

2Ax

^ - F

M+ l

AE

, / 9 + AF7.

2,j-l/2

1

._

M-l/2

2Ax E

- ( v)i-1/2J

F

i v)iJ

+ l/2 -

Ax

(Fv)iJ-l/2

Ay

(8.4.32) The exact Jacobian matrix of the residual given by Eq. (8.4.31) forms a banded matrix B and can be written in the form dR dR dR dR dR 8R\ iJ ij ij ij hJ n n n n , U , . . . , U , dD 'nr^k ) o ? ^ , . . . , U, 3D J dDij-i' ' ' ' i~i,j ' 9Dij ' dDi+ij ' ' " ' ' ' dDij+i (8.4.33) As discussed in [....], the discrete form of Eq. (8.4.18) can be written as n

B[V, 0 , . . . , 0, X, F, Z, 0 , . . . , 0, W] A D n + 1 ' m r =

_jRn+l,m _ ^ ( 1 # 5 j D n + l , m _

2L,n +

Q^TI-IN

(8.4.34)

J AV where V, X, Y, Z, and W denote vectors of 3 by 3 blocks which lie on the diagonals of the banded matrix, with the Y vector on the main diagonal. These vectors can be approximated as

8.5 Numerical Methods: Compressible Flows

<9i?n+1'm

^dRiJ

x

Y<=

1

1

~l(

dRn+Un

181

A

A+

,

+A~

A

)+ — I

—

x

dDiA

'i,3

" 2^+1/2j+A-l/2j B

5

A+1/2J

A

B

B

+ i | j + l / 2 + £ - l / 2 ~ iJ+l/2

\+l,j dDi+li

l+l 3 ~ 2[{Ai+hJ '

AA

i-l/2,j

-

(8.4.35)

i,j-l/2)

A

t+l/2j+Ai+i/2j> + r+l/2,j) i+V2,j

R/™dx RJn

QRn+l,m

The solution of Eq. (8.4.34) is obtained with the ADI method discussed in [2]. Thus, we first write Eq. (8.4.34) for a given time (n + l , r a + 1). Along the x-direction, Eq. (8.4.34) is written as B[X,Y,Z]AD L

J

= - i ? - ^ ( 1 . 5 D n + 1 ' m - 2 L > n + 0.5Dn-1) ; ^V

(8.4.36)

-ADj-iV-ADij+iW and is solved with the block elimination method discussed in subsection 5.5.3. Similarly, Eq. (8.4.34) is written along the y-direction, B[V, Y, W)AD L

J

= -R - ^ ( 1 . 5 D n + 1 ' m - 2Dn + AV - XADi-x, ~ ZADi+i,

Q.bD71'1) )

(8.4.37)

and again solved with the block elimination method. Both equations (8.4.36) and (8.4.37) are solved iteratively until convergence. Then the pseudo-time step is incremented and the above procedure is repeated for the next pseudo-time step until convergence. After convergence at the pseudo-time step, the computations move to next time step. For additional details, the reader is referred to [2].

8.5 Numerical M e t h o d s : Compressible Flows There are several Navier-Stokes methods for compressible flows. They employ implicit, explicit finite-difference schemes and finite-volume methods. Perhaps so far the most widely used one for time-dependent flows is the method based on the Beam-Warming algorithm [26] in which the spatial derivatives are represented by central differences or upwind methods. In the latter case, there are

8. Navier-Stokes Methods

182

several choices and the most popular ones include the Steger-Warming fluxvector splitting [27], the implicit forms of approximate Riemann solvers of Roe [28] and Osher [29]. Here we describe t h e numerical method which employs central differences for the spatial derivatives and refer the reader to [1,2] for upwind methods. For simplicity we again consider Cartesian coordinates. Before we discuss the numerical method for two-dimensional unsteady compressible flows, we first describe it for the time-dependent Euler equations onedimensional in space, dQ dE _ $.5.1) dt dx where Q=

Q QU

qi

~

Et E =

(8.5.2a)

Q2 Q3

QU

ei

QU2 + p

62

(Et+p)u\

^3

(8.5.2b)

A general form employing the Beam-Warming algorithm to Eq. (8.5.1) is

{i+OQ

(l+20Q n +^Q n " 1 = -At

71+1

,dEn+l dx

. n BEn + (1—6> + 0) ~ dx

n 1

tdE

-

dx (8.5.3) For second-order accuracy in time, the parameters (£, #, 0) are related by

i-e +

I

(8.5.4a)

and if, in addition, S=

(8.5.4b)

20--

the method is third-order accurate. Several well-known methods are special cases of the general two-step method given by Eq. (8.5.3): they are summarized in Table 8.1. For further details, see Hirsch [1]. Table 8.1. Partial list of one- and two-step methods according to Eq. (8.5.3).

0 1 2

1 1 2

0

0 0 0

0 0 0

1 2 1

1 2

9

0

Scheme

Accuracy in Time

Euler explicit scheme One-step implicit trapezoidal scheme Euler implicit scheme Two-step implicit trapezoidal scheme Explicit leapfrog scheme

O(At) 0(At2) O(At) 0(At2) 0(At2)

8.5 Numerical Methods: Compressible Flows

183

A particular family of schemes, extensively applied, are those with Equation (8.5.3) then becomes ,dEn+1 dx

(i + 0Qn+1 - (i + 20<2n + ZQn~l = -At

n . n,dE + ( i - 0 ) dx

0.

(8.5.5)

which can also be written as AQn = -

At (1 + 0

d . ._ , — (AE*)-,. dx

dEn +

At „

£

(1 + 0 dx

(1 + 0

AQ n - l

(8.5.6)

where /AQ" = Q"+! - Qn n

n+l

AE

= E

(8.5.7a)

n

-

E

(8.5.7b)

For £ = 0, we obtain the two-level, one-step scheme, namely the generalized trapezoidal method discussed in [1,2]. r)

AQn = -At6^-{AEn) ox

r)En

-

~

At^$ox

(8.5.8)

which reduces to the Crank-Nicolson method for 9 — 1/2. An essential aspect of the implicit methods is connected to the linearization process of the flux derivative dEn+l/dx, which is almost always carried out by using the linearization scheme first introduced by Briley and McDonald [30]: the fluxes at time level (n + 1) are obtained from En+l

= En + At(^j = En + AtiA-^-\

+

= En + An{Qn+1

- Qn) +

= En + AnAQn or

0(At2)

+

AEn

0(At2) 0(At2)

+

0{At2)

(8.5.9a)

+

0(At2)

(8.5.9b)

= AnAQn

where An = (8.5.10) Substituting Eq. (8.5.9b) into Eq. (8.5.6) yields At

/ + (1 +

0

£A» dx

AQn

where / is the identity matrix,

=

e (i + 0

AQ

n-l

At

8En

(1+0

dx

(8.5.11)

184

8. Navier-Stokes Methods 0 0N 1 0 0 1

1 I 0 ,0

/=

3.5.12)

This three-level Beam and Warming scheme, often called the A-iorra (deltaform), contains Q n " \ Qn and Qn+1. For £ = 0 and $ = 1/2, it reduces to the one-step trapezoidal (Crank-Nicolson) scheme. At

r

d

„r

AQn =

-At

8En dx

(8.5.13)

In the extension of this procedure to a two-dimensional flow, Eq. (8.2.13), we write Eq. (8.5.6) as AQn

= -

OAt

l(AEn

+

At At \d ^n l + £ [dx V ~

6At

ly{Ar) ££\ Re/

i

\d

d

pn

d dy

+

T^—,AQn~l

1+e

(8.5.14)

The first term on the right-hand-side of Eq. (8.5.14) AEn is given by Eq. (8.5.9b) and, similarly, the second term can be written as AFn

= BAQn

+

0(At2

(8.5.15)

where Bn =

(8.5.16)

9Q,

Since E™ and F™ are functions of Q, Q and Q , they can be written as ~y

Ev = Vi(Q,Q) + V2(Q,QJ

(8.5.17a)

~y

Fu^W1(Q,Qx)

+

W2(Q,Qy)

(8.5.17b)

where

Vi

V2 =

flUy

lJ'vuy ~

2

^uvy

3.5.18a)

8.5 Numerical Methods: Compressible Flows

185

0

0 \±vx

2

Wi

6

fiuvx

4

, w2 = 2

-

M««y + ^ ™ , +

^vux

(7

_

i ^ P r ^ (8.5.18b)

Then

d dx

A

^=i

%-AF$ = %-AW? dy dy As with Eq. (8.5.9a), we can write yn+l

=

yn

d

+ At

(8.5.19a)

ox +

yn

(8.5.19b)

%-AW$ dy

+

(^2)

0

and, with the chain rule and noting that V™ is a function of Q and Qx,

(8.5.20)

d

= V? + At (P9Q ,

n

9x

where dVi ~dQ

pn

dVi

Rn

(8.5.21)

Analogous to Eq. (8.5.9b), we can write Ayn

=

pnAgn

+

Rn Agn

+

0 (^2)

or AVI1 = (Pn - Rl)AQn

+

{RAQ)l

(8.5.22)

Similarly, noting that AW% is a function of Q and Qy, AW? = (Mn - N^)AQn

+ (NAQ)%

0W2

dW2

(8.5.23)

where Mn = Since AV? the terms

and AW?

Nn =

are functions of Q, Qx ( = dQ/dx)

(8.5.24) and Q

(=

in Eq. (8.5.19) represent mixed derivatives and are approximated by

dQ/dy),

8. Navier-Stokes Methods

186 8_ AVi dx

d_ AV?-1 dx

+

%-AW? dy

= %-AW?~x dy

+

0(Atz

(8.5.25a)

0(At2)

(8.5.25b)

Equations (8.5.9a), (8.5.15), (8.5.22), (8.5.23), and (8.5.25) may now be substituted into Eq. (8.5.14) to yield 9 At

M-Ny

d_ dx

Re

d I' AY?'1'

OAt

dx \ At

+1+

^ dx

-En

+

)

dy

_J_d_N

Re

\AQU

Redy

(AW™~V

d_ +

Re

Re dx ^y \

Re

Re

d_ + dy

.pn

+1+

Re

AQ

n-\

^ (8.5.26)

where I is the unity matrix given by 10 0 0 0 10 0 0 0 10 0 0 0 1

(8.5.27)

and the Jacobian matrices A, P, R, Rx, B, M, N, Ny are given in Appendix 8A. We note that in Eq. (8.5.26) P-Rx Re

d_ A dx

1 d R)AQn, Redx

J_d_ N\AQn

M-Ny Re

d_ B dy

Re dy

(8.5.28a)

are equivalent to P-Rx Re

d_ dx

J__d_

d_

R)AQr

Redx'

B

dy L

M-N, Re

y

J_d_ N) Re dy

AQr'

(8.5.28b) The left-hand-side of Eq. (8.5.26) can be factored and expressed in the form OAt [I} +

dx V

J

6 At

[/] +

T

At

+

i + £

1

± (lA] _ W - fe !(>«>

[M] - [Ny] Re

d_ -E dx 6At 1 d_ {AY- n-U l + <£Re~ dx

V2\n

+

d

n

Re dx2 \R]

Re

Vi + Re

_J_d^_

n

_J_d? Re dy 2 M "

d_ dy

+ 5JW-1)

AQr'

Re (AQn-L)

(8.5.29)

8.5 Numerical Methods: Compressible Flows

187

T h i s e q u a t i o n is now w r i t t e n in a slightly different form by i n t r o d u c i n g a new variable AQn~1/2

AQ n-l/2

= <[!} +

OAt 1+

£

d (m

[M] - [Ny]

B]

aj V

r>„ R e »„,2 dy2 I J

R^~

\AQn (8.5.30)

E q u a t i o n (8.5.29) b e c o m e s

[I] +

OAt

Hi

d flA] Ox \[Al

dx

OAt 1

d2

d (Avr1) + [dx

AQ n-l/2

n

Re dx2 [R}

*£*M'-**

At

+ 1 + £ Re

1

[P]-[Rx] Re

(8.5.31) Re

w

§i(Awr1)

,ra-l\

T h e choice of r e p r e s e n t i n g t h e s p a t i a l derivatives in E q s . (8.5.30) a n d (8.5.31) w i t h c e n t r a l differences h a s t h e d r a w b a c k t h a t t h i s scheme does n o t h a v e t h e ability t o d i s t i n g u i s h u p s t r e a m from d o w n s t r e a m influences. As a result, t h e solutions p r o d u c e oscillations in t h e vicinity of s h a r p flow g r a d i e n t s a n d discontinuities. T h e s e n u m e r i c a l oscillations c a n b e d a m p e d o u t by a d d i n g artificial dissipation or by using u p w i n d m e t h o d s discussed, for e x a m p l e , in [1,2]. T h e a d d i t i o n of artificial dissipation t e r m s t o E q s . (8.5.30) a n d (8.5.31) m a y b e in t e r m s of explicit fourth-order a n d implicit second-order t e r m s . In t h e former case, ee6x, ee6$ (8.5.32) is included on t h e r i g h t - h a n d - s i d e of E q . (8.5.31). Here <54 d e n o t e s a c e n t r a l difference o p e r a t o r defined by Sxu = ui+2j 6„,u

-

4v,i+ij

•6uij

Uij+2 ~ 4:Ui 4^z,j+i + 6uij

-Aui-ij

+Ui-2,j

- 4v>ij-i

+ Uij-2

(8.5.33a) (8.5.33b)

O n m e s h p o i n t s adjacent t o t h e b o u n d a r i e s , E q s . (8.5.33) are r e p l a c e d by a second-order four-point differences, SxU2j

= u±j — 4i£3j + 5^2,j — 2u\ j

8yUi,2 = U%A ~ 4^i,3 + 5 ^ , 2 - 2 ^ , i

(8.5.34a) (8.5.34b)

T h e implicit second-order d i s s i p a t i o n t e r m s are £'2

£iOx,

(8.5.35)

£i8y

w h e r e <52 d e n o t e s c e n t r a l difference o p e r a t o r s defined by b2xu = Ui+ij

- 2uij

+

Ui-ij

(8.5.36a)

8. Navier-Stokes Methods

188

6yU = Wij+i - 2uij +

(8.5.36b)

Uij-i

The above explicit and implicit dissipation terms are usually selected according to (8.5.37a)

0<ee<

8(1+0

and (8.5.37b)

2£ R

to ensure stability and optimal convergence rate [2]. With the addition of the above explicit and implicit dissipation terms, Eqs. (8.5.30) and (8.5.31) become 6 At

m +T + Z

d_ dx

1 d2 [RT-eJlil] Re dx2

[P] - \Rx Re

[A]

At dx \ 0At 1 + £Re

+l

Re

/

dy

l(Avr>}+fy(Awr)

-F +

W1 + Re

+

__ i

AQn~l>2

W2^n

{AQn-l)-ee{6t

+ 8Ay)Qn (8.5.38)

6At [I\ +

d_ rol [B] dy

n

[M]-[Ny]\

2

1 d

i t e " J -Re~^

[7V]

_

"

^

AQr

AQn-1'2 (8.5.39) As discussed in [2], after the spatial derivatives are represented by secondorder central differences and the artificial dissipation terms are added, the resulting equations can be expressed in a block-tridiagonal form that is solved with an ADI scheme. For further details, the reader is referred to [2].

References [1] Hirsch, C : Numerical Computation of Internal and External Flows. John Wiley & Sons, 1990. [2] Cebeci, T., Shao, J. P., Kafyeke, F. and Laurendeau, E.: Computational Fluid Dynamics for Engineers. Horizons Publ., Long Beach, Calif, and Springer, Heidelberg, 2005. [3] Pulliam, T. H.: Efficient solution methods for the Navier-Stokes equations. Lecture Notes for the Von Karman Institute for Fluid Dynamics Lecture Series: Numerical Techniques for Viscous Flow Computation in Turbomachinery Bladings, Brussels, Belgium, Jan. 20-24, 1986. [4] Wilcox, D. C : Turbulence Modeling for CFD. DCW Industries, Inc. 5354 Palm Drive, La Canada, Calif., 1998. [5] Cebeci, T.: Analysis of Turbulent Flows. Elsevier, London, 2004.

References

189

[6] Cebeci, T. and Chang, K.C.: An Improved Cebeci-Smith Turbulence Model for Boundary-Layer and Navier-Stokes Methods. 20th Congress of the International Council of the Aeronautical Sciences, paper No. ICAS-96-1.7.3, Sorrento, Italy, 1996. [7] Johnson, D. A. and King, L. S.: Mathematically Simple Turbulence Closure Model for Attached and Separated Turbulent Boundary Layers. AIAA J., 23, No. 11, 1684-1692, 1985. [8] Bradshaw, P., Ferriss, D. H. and Atwell, N.P.: Calculation of Boundary-Layer Development Using the Turbulent Energy Equation. J. Fluid Mech., 23, 593, 1967. [9] Baldwin, B. S. and Lomax, H.: Thin Layer Approximation of Algebraic Model for Separated Turbulent Flows. AIAA paper No. 78-257, 1978. [10] Stock, H. W. and Haase, W.: Determination of Length Scales in Algebraic Turbulence Models for Navier-Stokes Methods. AIAA J., 27, No. 1, 5-14, 1989. [11] Johnson, D.A.: Nonequilibrium Algebraic Turbulence Modeling Considerations for Transonic Airfoils and Wings. AIAA paper No. 92-0026, 1992. [12] Swanson, R. C. and Turkel, E.: A Multistage Time-Stepping Scheme for the NavierStokes Equations. AIAA paper No. 85-0035, 1985. [13] Cebeci, T.: Turbulence Models and Their Application. Horizons Publ., Long Beach, Calif, and Springer, Heidelberg, 2004. [14] Spalart, P. R. and Allmaras, S.R.: A One-Equation Turbulence Model for Aerodynamics Flows. AIAA Paper 92-0439, 1992. [15] Jones, W. P. and Launder, B.E.: The Prediction of Laminarization with a TwoEquation Model of Turbulence. International Journal of Heat and Mass Transfer, 15, 301-314, 1972. [16] Menter, F.R.: Two-Equation Eddy Viscosity Turbulence Models for Engineering Applications. AIAA J., 32, 1299-1310, 1994. [17] Yakhot, V. and Orszag, S. A.: Renormalization Group Analysis of Turbulence, 1. Basic Theory, J. Scientific Computing, 1, 3-51, 1986. [18] Chorin, A. J.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12, 1967. [19] Chorin, A. J.: Numerical solution of the Navier-Stokes equations. Math. Comput. 22, 745-762, 1968. [20] Roger, S. E. and Kwak, D.: An upwind differencing scheme for the incompressible Navier-Stokes equations. AIAA J. 8, 43-64, 1991. [21] Roger, S. E. and Kwak, D.: An upwind differencing scheme for the time accurate incompressible Navier-Stokes equations. AIAA J. 28(2), 253-262, 1990. [22] Roger, S. E. and Kwak, D. and Kiris, C : Steady and unsteady solutions of the incompressible Navier-Stokes equations. AIAA J. 4(4), 603-610, 1991. [23] Ghia, K.N., Yang, J, Oswald, G. A. and Ghia, U.: Study of dynamic stall mechanism using simulation of two-dimensional Navier-Stokes equations. AIAA Paper, 91-0546, 1991. [24] Wu, J.C.: Theory of aerodynamic force and moment in viscous flows. AIAA Paper, 80-0011, 1980. [25] Wu, J.C.: Fundamental solutions and numerical methods for flow problems. Int. J. Numer. Methods Fluids 4, 185-201, 1984. [26] Beam, R. M. and Warming, R. F.: An implicit factored scheme for the compressible Navier-Stokes equations. AIAA J. 16, 393-401, 1978. [27] Steger, J. L. and Warming, R. F.: Flux vector splitting of the inviscid gas dynamic equations with applications to finite-difference methods. J. Comput. Phys. 40, 263293, 1981. [28] Roc, P. L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357-372, 1981.

190

8. Navier-Stokes Methods

[29] Osher, S. and Solomon, F.: Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comput. 38(158), 339-374, 1982. [30] Briley, W. R. and McDonald, H.: Solution of the three-dimensional Navier-Stokes equations by an implicit technique. Proc. Fourth International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics, Vol. 35, Springer, Berlin 1975.

I Applications of Navier—Stokes Methods

9.1 I n t r o d u c t i o n Navier-Stokes (NS) methods are more general than those based on interactiveboundary-layer (IBL) theory and can be used to solve some airfoil flows that IBL methods cannot. For example, as discussed in Chapter 7, the IBL method can predict the initiation of dynamic stall of an airfoil but cannot predict the details of the downstream separated flow and, therefore overall lift and drag. Also, the prediction of airfoil dynamic stall for oscillatory, ramp and more complex motions at various freestream speeds and Reynolds numbers requires the use of NS methods. In this chapter we discuss applications of NS methods to incompressible and compressible laminar and turbulent flows, and consider the role transition plays in the calculations. Dynamic stall of airfoils and wings in laminar flows is mainly of theoretical interest because fully laminar flow cannot be sustained at chord Reynolds numbers, RCl above 10 4 . Early solutions of the NS equations were, however, obtained for laminar flows (Sections 9.2 and 9.3), partly because of the lack of measurements for most low Reynolds number conditions and partly because of the belief that the laminar flow calculations could capture flow-field features similar to the available flow visualizations at higher Reynolds numbers. There have been several experimental and numerical investigations of dynamic stall of airfoils in turbulent flows because dynamic stall of airfoils, helicopters and turbomachinery blades and aircraft wings for turbulent high-speed flow is of primary interest in practical applications. Thus in Section 9.4 we discuss several examples for incompressible and compressible laminar and turbulent flows. Helicopter rotors and propeller blades as well as most turbomachinery blades operate at Reynolds numbers between 10 5 and several millions and, as a result, the flows are mainly turbulent. However, there is always a small lami-

192

9. Applications of Navier-Stokes Methods

nar/transitional flow region near the leading edge unless the flow is tripped. In Section 9.5, therefore, we discuss the effect of including transition from laminar to turbulent flow in calculations of dynamic stall of airfoils. Sections 9.2 to 9.5 address the prediction of dynamic stall flow fields of airfoils, which is the area that has received the most attention. Unsteady threedimensional flows have been considered only recently, even though they are important in industrial applications and a brief discussion of these studies is given in Section 9.7. The chapter ends with a summary of problem areas to be addressed in dynamic stall computations. For a more detailed recent review of the challenges posed by the computational prediction of airfoil dynamic stall, see [1].

9.2 Laminar Flow Calculations for Incompressible Flows Laminar incompressible flow solutions over a modified NACA-0012 airfoil were obtained by Mehta [2] using the vorticity-stream function formulation briefly discussed in subsection 8.4.1. The computations were conducted at Rc = 5 x 10 3 and 10 x 10 3 for oscillatory motion given by a = 10°[l + coso;t] for reduced frequencies k = 0.5 and 1.0. The development of the unsteady flowfield and the loads was investigated and good qualitative agreement with water tunnel flow visualization was obtained. Sample results of Mehta's computations are shown in Fig. 9.1. Incompressible laminar dynamic stall flowfields were also investigated by Ghia et al. [3] using the vorticity-streamfunction formulation. In [4], solutions were obtained for pitch up motion of a NACA-0015 airfoil at Rc = 10 x 10 3 and 4.5 x 10 3 which compared favorably with smoke flow visualizations as shown in Fig. 9.2.

9.3 Laminar Flow Calculations for Compressible Flows One of the first studies of dynamic stall for a compressible laminar flow with a Navier-Stokes method was conducted by Sankar and Tassa [5]. Their method used the numerical method of Beam and Warming similar to the one discussed in Section 8.5. Calculations for Reynolds numbers between 5 x 10 3 and 10 4 were performed and the effects of compressibility on dynamic stall were investigated for Moo = 0.2 and 0.4. Figure 9.3 shows the lift and pitching moment hysteresis loops for a reduced frequency k = 1.0 and M ^ = 0.4. They show similar trends with the high Reynolds number turbulent flow measurements of McCroskey [6].

9.3 Laminar Flow Calculations for Compressible Flows

193

Fig. 9 . 1 . Computed instantaneous streamlines, velocity vectors, vorticity countours, and surface pressure distribution: Incompressible flow calculation, Rc = 5 x 10 3 , k = 1.0, a = 10° + 10°cosu;£.

Dynamic stall in laminar flow was further investigated by Visbal [7] and Visbal and Shang [8]. Figure 9.4 demonstrates the effect of pitching axis location on the development of the dynamic stall flowfield for Rc = 4.5 x 10 4 at M^ = 0.2. These results are in good agreement with flow visualization experiments. Investigations of dynamic stall in laminar flow were also conducted in [9,10] to identify the onset of leading-edge separation. Solutions were obtained for a pitching airfoil. The effect of compressibility was investigated and complex recirculatory flow patterns at the airfoil leading edge were obtained. Figure 9.5 shows the instantaneous streamlines for Rc = 10 4 , of a NACA 0012 airfoil at a = 18° rapidly pitching with reduced frequency k = 0.2, for M ^ = 0 . 2 and 0.5. The results show that an increase in Mach number from 0.2 to 0.5 delays the formation of the recirculatory region. Also for low pitch rates trailing-edge dynamic stall was obtained, while for higher pitch up rates, the leading-edge dynamic stall started at the leading edge. Similar results were also obtained in

9. Applications of Navier-Stokes Methods

194

(a) 8 = 24°

(b) TIME - 1.701 ,

(c) 0 = 27°

(e) « = 36°

(d) * »oui«>tC

TIME = 2.201 ;

8-18

246°

9 = 23.976°

riME

2 701

8

^9 706°

<^«*« (g) 8 = 40° f

«»«..

TIME - 3.301 ; 8 ~ 16 581

Fig. 9.2. Computed vorticity contours for a NACA 0015 airfoil at ramp motion: Rc 4.5 x 10 3 .

9.4 Laminar and Turbulent Flow Calculations

0.00

195

4.00

Fig. 9 . 3 . Computed lift and pitching moment hysteresis loops for the dynamic stall of a NACA 0012 airfoil: M ^ = 0.4, Rc = 5,000, k = 1.0.

111 ^

w^^L

\

^y&fiy>&

W^ a = 40° t

•

i

a)nj = 0.2, ^ 0 / c = 0.25

b)flJ=0.2,A-0/c = 0.75 Fig. 9.4. Effect of pitching axis location on the evolution of the vorticity field.

[9], with a zonal grid having very high grid resolution near the leading edge and a more accurate finite difference scheme.

196

9. Applications of Navier-Stokes Methods

0.5, Rc = 104, Ac = 0.1.

9.4 Laminar and Turbulent Flow Calculations for Incompressible and Compressible Flows Dynamic-stall flow field calculations for laminar and turbulent incompressible flows were conducted by Wu and his collaborators using the vorticity-velocity formulation briefly described in subsection 8.4.2 [11-13]. Fully turbulent solutions for the lift, drag and pitching moment hysteresis loops at Rc = 1 x 10 6 for a sinusoidal oscillation on a NACA 0012 airfoil with a = 15° + 10° sin ut for reduced frequencies k = 0.2, 0.3 and 0.5 are shown in Fig. 9.6 using the Baldwin-Lomax turbulence model discussed in subsection 8.3.1. Also shown is the comparison with the experiment. Compressible, unsteady, turbulent flow computations for the NACA 0012 airfoil were performed by Sankar and Tassa [5] already more than twenty years ago. Dynamic stall of a SSC-A09 Sikorsky airfoil executing rapidly pitching and oscillatory motions was investigated by Ekaterinaris [14] for a turbulent flow of Moo = 0.2 and Rc = 2 x 10 6 , again using the Baldwin-Lomax turbulence model. An implicit finite-difference scheme was used for the numerical solution of the compressible Navier-Stokes equations. The computed and measured lift and pitching moment hysteresis loops for oscillatory motion a = 10° + 10° sin out are shown in Fig. 9.7. In the same study the effects of compressibility and pitch rate were also investigated, showing that an increase in pitch rate delays the stall, Fig. 9.8, whereas an increase in Mach number promotes it, Fig. 9.9. The same cases were also investigated by Patterson and Lorber [15] who reported similar results.

9.4 Laminar and Turbulent Flow Calculations

M l

j

< ^

< /

ift

oH . O t

jt

*»

o*

y^

-^

0.0

J

5.0

j

*r",*r

_^i

0.0

^•^

_

* s

VN.

\

/ X/f

/'/""""N

^s^

£

J/j

.*""»".

0.4

j~ 0

Drag Coefficient

S/<*r /

*

<J>

\

''r

M~

c

t.

•*"'

^\

1

o

197

tO.O

,

,

,

15.0

20.0

25.0

1

^

^

C4 1

J

30.0

0.0

a degrees

,

5.0

,

10.0

,

,

15.0

,

20.0

25.0

1

30.0

a degrees

c a> od OM

Full Viscous Flow Solution Experimental Data

Oo — I

c a>*>

£6 o «

0.0

S.O

10.0

15-0

20.0

25.0

30.0

a degrees Fig. 9.6. Comparison of the computed load hysteresis loops with the measurements: a 10°(1 + sinwt), k = 0.4, Rc = 3 x 10 6 . 2.5

. 1 -j

T

COMPUTED _ 2.0 o uu

O

EXPERIMENT, UNSTEADY

-A-- EXPERIMENT, STEADY -.1

I- 2

O u. H 1.0

s -.3 O 10

15

adcg

COMPUTED EXPERIMENT

-.5 10

15

20

ocdeg

Fig. 9.7. Comparison of the computed unsteady lift and pitching moment hysteresis loops with the measurements: a = 10° + 10° sino;*, k = 0.2, Rc = 2 x 10 6 .

Fung and Carr [16] drew attention to the fact that at high incidences the flow becomes locally supersonic and the local supersonic flow region may be terminated by a shock close to the leading edge even though the freestream Mach number is only 0.2. Currier and Fung [17] also pointed out that the dependency

9. Applications of Navier-Stokes Methods

198

COMPUTED, k = 0.0f

3.0 o 2.5

EXPERIMENT, k - 0 . 0 f COMPUTED, k « 0.02

o

•

ui 2.0

EXPERIMENT, k = 0.02

— « - - STEADY EXPERIMENT

o cc SI* 0

5

10

15

20

25

30

.5

a deg Fig. 9.8. Effect of reduced frequency on the unsteady lift: Rc — 2 x 10 6 , M ^ = 0.2. COMPUTED, M = 0.2

3.U

EXPERIMENT, M = 0.2 2.5 <

COMPUTED, M * 0.4 E X P E R I M E N T S = 0.4

°2.0 UJ

O

O 15-

4

LL

t 1.0

^

•J

.5

\

S^

^ ^ ^

o

°

o

£^ 10

15

20

30

a deg Fig. 9.9. Effect of Mach number on the unsteady lift.

on reduced frequency can be vastly different for airfoils with different leadingedge shapes. The ability of algebraic turbulence models in these early numerical investigations of dynamic stall in turbulent high Reynolds number flows to properly simulate the massively separated flow at high angles of incidence was, of course, regarded as doubtful. Therefore, turbulence models based on transport equations discussed in subsections 8.3.2 and 8.3.3 were examined. Wu and Sankar [11] used the k-e model (subsection 8.3.3) to compute deep stall of a NACA 0012 airfoil, but computed load hysteresis loops did not show significant improvement, as seen in Fig. 9.10. Similar studies for a rapidly pitching airfoil were performed by Rizetta and Visbal [18]. An even more challenging problem is posed by the phenomenon of light dynamic stall. It occurs when the airfoil oscillates around a mean angle of attack close to static stall with a small oscillation amplitude. Depending on the airfoil shape, Reynolds and Mach numbers, mean angle, oscillation amplitude and reduced frequency the pitching moment hysteresis loop may indicate either positive or negative airfoil damping. As pointed out by Carta and Lorber [19] and explained in Section 1.8, negative damping may induce stall flutter. Accurate

9.4 Laminar and Turbulent Flow Calculations

199

BL MODEL K - £ MODEL

Fig. 9.10. Effect of turbulence modeling on the computed load hysteresis loops: NACA 0012 airfoil, M^ = 0.283, a = 15° + 10° sin a;*, k = 0.3, Rc = 3.45 x 10 6 .

computation of the pitching moment hysteresis loops is, therefore, critical to avoid stall flutter on helicopter and turbomachinery blades. Computations of light dynamic stall of a NACA 0012 airfoil with the Johnson-King model described in subsection 8.3.1, were carried out by Dindar and Kaynak [20] and Clarkson et al. [21]. The Reynolds number of the experiment was Rc = 3 x 10 6 , the reduced frequency k = 0.2 and the oscillatory motion a = 10° + 5° sin a;* In both solutions similar grid densities were used and the oscillation amplitude was increased to 5.5° in order to promote separation because for the amplitude of the experiment (a = 5°), very little separation at the trailing edge region was observed. The computed lift and pitching moment hysteresis loops, shown in Figs. 9.11 and 9.12, indicate not only the strong dependence on the turbulence model, but also an inability to predict the experimental moment hysteresis loop. Note that in addition to the Baldwin-Lomax and Johnson-King models Clarkson et al. [21] also tested the RNG model [22]. Therefore, to improve the prediction of dynamic stall, Srinivasan et al. [23] performed a more systematic study of the effect of turbulence modeling on dynamic stall computations, using algebraic and one-equation turbulence models for deep and light dynamic stall of the NACA 0015 airfoil tested by Piziali [24] at M = 0.3 and Reynolds number R = 2 million. Note that the flow was tripped at the leading edge to ensure a fully turbulent flow. The comparison of the computed hysteresis loops [23] with the measurements [24], Figs. 9.13 and 9.14, shows that the one-equation models are more appropriate for the computation of deep stall. However, the light dynamic stall clearly is much more challenging, as is also apparent from Fig. 9.15, where significant differences in the predicted flow reversal points are plotted depending on the different turbulence models.

9. Applications of Navier-Stokes Methods

200

C L - & Hysteresis

C - Qt Hysteresis

2.001

© Experiment - - BL model — JK Model

0 Experiment - - BL model — JK Model

• .-lz*

-02'

Fig. 9.11. Effect of turbulence modeling on the computed load hysteresis loops: M^ = 0.3, a = 10° + 5.0° sinut, k = 0.2, NACA 0012 airfoil Rc = 3 x 10 6 . r~ •

Measured

[_

J K Model

i

U N G Model

% -*r»»t^- , *ri"-—-^T "» ^^y.

^

y-|J j" » ~ ~ ^

«P* •

t

• 1

•

• • 1

1

J_

J

1

L_

1

10 11 a deg

1

1

12

13

• • •

1

14

15

16

Fig. 9.12. Effect of turbulence modeling on the computed load hysteresis loops: M^ = 0.3, a = 10° + 5.0° sine;*, k - 0.2, NACA 0012 airfoil Rc = 3 x 106 for 161 x 64 C-grid.

9.5 Effect of Transition Experimental investigations [25] of a NACA 0012 airfoil at a Reynolds number of 5.4 x 10 5 showed that at the leading edge the flow is transitional and a separation bubble develops as the angle of attack increases, as was discussed in subsection 7.3.1. Ekaterinaris et al [26] argued that, consistent with the IBL results and the results shown in Figs. 9.11 and 9.12, a fully turbulent computation cannot predict the flow physics at the leading edge, but that instead the transition from laminar to turbulent flow must be incorporated. In their computations the location of the transition onset was obtained with Michel's method, given by Eq. (7.3.1), and the computation of the transitional flow region was obtained by an effective eddy viscosity. Scaling of the computed turbulent eddy viscosity in the transitional flow region was obtained by multiplying the computed eddy

9.5 Effect of Transition

201

RNG"} j^g

> Turb. model

SA J Experiments-Piziali

Fig. 9.13. Effect of turbulence modeling on the computed load hysteresis loops for light stall Moo = 0.3, a = 11° + 4.2° sinut, k = 0.2, NACA 0013 airfoil, Rc = 2 x 10 6 , 361 x 71 grid.

BL ) RNG J K > Turb. model BB

SA J o

Experiments-Piziali

14

16 a (deg)

18

20

Fig. 9.14. Effect of turbulence modeling on the computed load hysteresis loops for deep 6 stall Moo = 0.3, a = 15° + 4.2° sin a;*, k = 0.2, NACA 0015 airfoil, Rc - 2 x 10 , 361 x 71 grid.

202

9. Applications of Navier-Stokes Methods

Leading Edge

Trailing Edge

Fig. 9.15. Reversed flow point obtained from computations with different turbulence models: M ^ = 0.3, a = 15° +4.2° sin cut, k = 0.2, NACA 0015 airfoil, Rc = 2x 10 6 , 361 x71 grid.

viscosity with the Chen-Thyson intermittency function given by Eqs. (5.1.12) and (5.1.13). This approach yielded separation bubbles which were in fairly good agreement with the experimentally observed ones. The transition onset location and the transition length were found to be critically important parameters. More recently, Sanz and Platzer [27] improved this approach by adopting the Solomon-Gostelow-Walker transition length model [28]. Ekaterinaris and Platzer [29] showed that it may be sufficient to specify the onset of transition at points downstream from the maximum adverse pressure gradient locations on both the suction and pressure surfaces and to compute the effective eddy viscosity in the transition region, using the Baldwin-Barth turbulence model [30], by imposing zero production at the transition onset so that the eddy viscosity increases progressively to the fully turbulent flow value. An application of this approach to the case of light dynamic stall of the Sikorsky SC-1095 airfoil measured by Carta and Lorber [19] is shown in Fig. 9.16. It is seen that the pitching moment hysteresis loop is a clockwise loop, indicating that energy is transferred from the airstream to the airfoil, thereby inducing stall flutter, as explained in Section 1.8. The reduced frequency is 0.188, the Reynolds number 6.5 x 10 5 and the Mach number 0.18. It is seen that the transitional flow calculation is in substantial agreement with the experiment, predicting instability, whereas a fully turbulent calculation does not even predict the correct behavior. The explanation for this discrepancy can be found by inspecting the pressure

9.5 Effect of Transition

o — •

203

Experiment, Carta & Lorber Fully Turbulent Transitional

0.05-1 t*>00 QQ9,<^ fXPqK
o

<*>*

-0.05 n

*,t

v-o*° °°\

-0.10 -0.15

(c) 10

-0.20

11

12

13

Angle of Attack, deg.

Fig. 9.16 , Effect of transition on the computed loads for SC-1095 Airfoil, Moo = 0.3, k = 0.188, . R e - 0 . 6 x 10 6 .

o —

Experiment McCroskey Computed, Transitional Computed, Fully Turbulent

o —

Experiment, McCroskey Computed, Transitional Computed, Fully Turbulent

a =13.0° down ,'<.'.-V/rnnUj » l l .

-2 10I

a =11.0° down

o.o

0.2

0.4

0.6

x/c Fig. 9.17. Effect of leading-edge transition on the load hysteresis effects: Moo = 0.3, a 12° + 2.0° sinu>£, k - 0.2, NACA 0012 airfoil, Rc = 4 x 10 6 .

204

9. Applications of Navier-Stokes Methods

a

deg

a deg

Fig. 9.18. Effect of leading-edge transition on the computed surface pressure coefficient distribution: M ^ = 0.3, a = 10° + 5.0° sin out, k = 0.2, NACA 0012 airfoil, Rc = 4 x 10 6 .

distributions shown in Fig. 9.17 where light dynamic stall of a NACA 0012 airfoil is computed at a Reynolds number of 4 x 10 6 . It is seen that the differences between the fully turbulent and the transitional flow calculations are minor during the upstroke. However, as the airfoil reaches the maximum incidence of 14 degrees and as it starts the downstroke, the transitional flow calculations predict a much lower suction peak than the fully turbulent ones. Hence it is the difference in flow behavior near the leading edge which is responsible for the differences in pitching moment hysteresis loops. It is remarkable that this effect occurs not only at Reynolds numbers below one million (as one would expect) but also at high Reynolds numbers. The hysteresis loops of the NACA 0012 airfoil are shown in Fig. 9.18. Although transitional calculations yield the qualitatively correct behavior (in contrast to fully turbulent calculations), good quantitative agreement is still elusive because the flow reattachment process is not captured sufficiently well, as is evident from the pressure distributions shown in Fig. 9.17.

9.6 Flapping-Wing Flight As shown by Jones et al. [31,32] flapping airfoils start to shed dynamic stall vortices from the leading edges as soon as the product of flapping amplitude and frequency exceeds a critical value. Hence the analysis of the dynamic stall phe-

9.6 Flapping-Wing Flight

205

/ia=0.0125

Fig. 9.19. Unsteady particle traces vs flow-visualization data behind a NACA 0012 airfoil oscillated in plunge at k = 7.85.

nomenon is acquiring further importance in addition to the "classical" dynamic stall problems encountered on helicopter and turbomachinery blades which may cause the destruction of such blades. In contrast, there is growing evidence that the dynamic stall vortices generated by flapping wings may enhance the flight performance of birds, insects and micro air vehicles. As discussed in Section 1.3, flapping airfoils generate a "reverse" Karman vortex street, as shown in Fig. 1.7. This vortex street can be modelled quite well with inviscid panel codes. Further insight can be gained by computing the traces

9. Applications of Navier-Stokes Methods

206

h = 0.07 h = - 0.37

h = - 0.59

h = 0.59

Fig. 9.20. Unsteady particle traces over a NACA 0012 airfoil oscillated in plunge at k = 0.8 and hQ = 0.60.

of particles released anywhere in the flowfield. Such a particle tracing procedure was developed by Tuncer [33] which was applied by Tuncer and Platzer [34] to assist in visualizing the vortex wake behind a sinusoidally plunging airfoil. As the plunge amplitude is increased the vortex shedding becomes more distinct and the upper row of counterclockwise and lower clockwise vortices (indicative of thrust production) agree well with the flow visualization experiment of Lai and Platzer [35], as shown in Fig. 9.19. An increase in plunge amplitude beyond the value shown in Fig. 9.19 eventually leads to the shedding of vortices from the airfoil leading edge. Such a case is shown in Fig. 9.20. It is obvious that the phasing of the dynamic stall vortices in response to pitch and plunge amplitude and frequency and the interaction between the dynamic stall vortices shed from two airfoils arranged in tandem (as for example on the dragonfly) is critical to achieve optimal thrust and lift performance. Therefore, several research groups are currently conducting such computations using Navier-Stokes methods.

9.7 Three-Dimensional Dynamic Stall Calculations The discussion of three-dimensional flow problems is largely beyond the scope of this book. Therefore we merely mention the three-dimensional dynamic stall

9.7 Three-Dimensional Dynamic Stall Calculations

207

1.4 1.2 C

O —

|

Experiment, Pizialt Computation

1.0 0.8 0.6

(

10

12

14

16

a deg 0.12

0.06 i

0.08

0.04 \

Cd

'm 0.02 \

0.CK

o.oo

0.00 6

8

10

12

14

6

16

8

10

12

14

16

a deg

a deg

Fig. 9 . 2 1 . Comparison of the computed load hysteresis at 50% span location with the experiments: M^ = 0.3, a = 15° + 4.2° sinu;*, k = 0.2, NACA 0015 airfoil, Rc = 2 x 10 6 .

1.2 Experiment, Pixiaii

1.0 C,

Computation

0.8

0.6 0.4 6

8

10

12

14

16

a deg.

0.12

0.06 1

0.04 A

0.08

Cd

f

m 0.02

0.04

0.00

0.00 8

10

12

a deg.

14

16

8

10

12

14

16

a deg.

Fig. 9.22. Comparison of the computed load hysteresis at 80% span location with the experiments: M ^ = 0.3, a = 15° + 4.2° sinu;*, k = 0.2, NACA 0015 airfoil, Rc = 2 x 10 6 .

208

9. Applications of Navier-Stokes Methods

O

Experiment, Piziali Computation

10

12

a deg. Fig. 9.23. Comparison of the computed load hysteresis at 89% span location with the experiments: M^ = 0.3, a = 15° + 4.2° sin a;*, k = 0.2, NACA 0015 airfoil, Rc = 2 x 10 6 .

experiments of Piziali [24] for rectangular wings and Lorber et al. [36] and Lorber [37] for swept wings. Three-dimensional dynamic stall computations are still quite time consuming. Nevertheless, as shown by Ekaterinaris [38], reasonably good agreement could be obtained with Piziali's measurements at 50% span, Fig. 9.21 which, however, started to deteriorate closer to the wing tip at 80% span, Fig. 9.22, and even more so at 89% span, Fig. 9.23. Again, these results show that Navier-Stokes methods have the potential to provide important insight into the nature of the dynamic stall phenomenon.

References [1] Ekaterinas, J. A. and Platzer, M. F., Computational prediction of airfoil dynamic stall. Prog. Aerospace Set. 3 3 , 759-846, 1997. [2] Mehta, U. B., Dynamics stall of an oscillating airfoil, AGARD CP 227, Paper No. 23, 1-32, 1977. [3] Ghia, K. N., Yang, J., Oswald, G. A. and Ghia, U., Study of dynamic stall mechanism using simulation of two-dimensional Navier-Stokes equations, AIAA Paper, 91-0546, 1991. [4] Ghia, K.N., Yang, Y., Oswald, G. A. and Ghia, U., Study of the role of unsteady separation in the formation of dynamic stall vortex, AIAA Paper 92-0196, 1992. of two-dimensional Navier-Stokes equations, AIAA Paper, 91-0546, 1991. [5] Sankar, L. N. and Tassa, W., Compressibility effects of dynamic stall of a NACA 0012 airfoil, AIAA J. 19(5), 557-568, 1981.

References

209

[6] McCroskey, W. J., The phenomenon of dynamic stall. NASA TM-81264, 1981. [7] Visbal, M.R., Effect of compressibility on dynamic stall of a pitching airfoil, AIAA Paper 88-0132, 1988 [8] Visbal, M. R. and Shang, J.S., Investigation of the flow structure around a rapidly pitching airfoil, AIAA J. 27(8), 1044-1055, 1989. [9] Choudhuri, P. G. and Knight, D. D., Effects of compressibility, pitch rate, and Reynolds number of unsteady incipient leading-edge boundary layer separation over a pitching airfoil, J. Fluid Mech. 308, 195-217, 1996. Reisenthal, P. H., Further results on the Reynolds number scaling of incipient leadingedge stall, AIAA Paper 95-0780, 1995. Wu, J. C. and Sankar, N.L., Evaluation of three turbulence model for the prediction of steady and unsteady airloads, AIAA paper 89-0609, 1989. Wu, J.C., Fundamental solutions and numerical methods for flow problems. Int. J. Numer. Methods Fluids 4, 185-201, 1984. Tuncer, T. H., Wu, J. C. and Wang, C. M., Theoretical and numerical studies of oscillating airfoils. AIAA J. 28(9), 1615-1624, 1990. Ekaterinaris, J. A., Compressible studies on dynamic stall, AIAA Paper 89-0024, 1989. Patterson, M. T. and Lorber, P. F., Computational and experimantal studies of compressible dynamic stall, J. Fluids Struct. 4, 259-285, 1990. Fung, K.Y. and Carr, L.W., Effects of compressibility on dynamic stall, AIAA J. 26(2), 306-308, 1991. Currier, J. and Fung, K. Y., An analysis of the onset of dynamic stall, AIAA Paper 91-0003, 1991. Rizetta, D. P. and Visbal, M. R., Comparative numerical study of two turbulence models for airfoil static and dynamic stall, AIAA J. 3 1 , 784-786, 1993. Carta, F. O. and Lorber, P. F., Experimental study of the aerodynamics of incipient torsional stall flutter, J. Propulsion Power 3(2), 164-170, 1987. Dindar, M. and Kaynak, U., Effect of turbulence modeling on dynamic stall of a NACA 0012 airfoil, AIAA Paper 92-0027, 1992. Clarkson, J.D., Ekaterinaris, J. A. and Platzer, M.F., Computational investigation of airfoil stall flutter, In Atassi, H. M. (ed.) Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines and Propellers, pp. 415-432, Springer, Berlin, 1993. Yakhot, V. and Orszag, S.A., Renormalization group analysis of turbulence. J. Scientific Computing 1, 1-36, 1986. Srinivasan, G. R., Ekaterinaris, J. A. and McCroskey, W. J., Dynamic stall of an oscillating wing, Part I: evaluation of turbulence models, Comput. Fluids 24(7), 833-861, 1995. Piziali, R. A., An experimental investigation of 2D and 3D oscillating wing aerodynamics for a range of angle of attack including stall, NASA Technical Memorandum 4632, 1993. Chandrasekhara, M. S. and van Dyken, R. D., LDV measurements in dynamically separated flows, SPIE Proc. Laser Anemometry Adv. Appl. 2052, 305-312, 1993. Ekaterinaris, J. A., Chandrasekhara, M. S. and Platzer, M. F., Analysis of low Reynolds number airfoils, J. Aircraft 32(3), 625-630, 1995. Sanz, W. and Platzer, M.F., On the Navier-Stokes calculation of separation bubbles with a new transition model, ASME J. of Turbomachinery 120, 36-42, 1998. Solomon, W. J., Walker, G.J. and Gostelow, J. P., Transition length production for flows with rapidly changing pressure gradients, ASME J. of Turbomachinery 118, 744-751, 1996.

210

9. Applications of Navier-Stokes Methods

[29] Ekaterinas, J. A. and Platzer, M.F., Progress in the analysis of blade stall flutter, Proc. Symp. Unsteady Aerodynamics and Aeroelasticity of Turbomachines, edited by Y. Tanida and M. Namba, pp. 287-302, Elsevier 1995. [30] Baldwin. B. S. and Barth, T. J., A one-equation turbulence transport model for high Reynolds number wall-bounded flows. NASA TM 102847, 1990. [31] Jones, K.D., Dohring, C M . and Platzer, M.F., Experimental and computational investigation of the Knoller-Betz effect, AIAA J. 36, 1240-1246, 1998. [32] Jones, K.D., Lund, T. C. and Platzer, M.F., Experimental and computational investigation of flapping wing propulsion for micro air vehicles, chapter 16 in "Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications," Vol. 195, Progress in Astronautics and Aeronautics, American Institute of Aeronautics and Astronautics, 2001. [33] Tuncer, I. H., A particle tracing method for 2-D unsteady flows on curvilinear grids, ASME Paper FEDSM97-3071, June 1997. [34] Tuncer, I. H. and Platzer, M. F., Computational study of flapping airfoil aerodynamics, J. of Aircraft 37, 514-520, 2000. [35] Lai, J . C . S . and Platzer, M.F., Jet characteristics of a plunging airfoil, AIAA J. 37, 1529-1537, 1999. [36] Lorber, P. F., Carta, F. O. and Covino, A.F., An oscillating three-dimensional wing experiment: compressibility, sweep, rate, waveform and geometry effects on unsteady separation and dynamic stall, UTRC Report R92-958325-6, November 1992 and AIAA Paper 91-1795, June 1991. [37] Lorber, P. F., Compressibility effects on the dynamic stall of a three-dimensional wing, AIAA Paper 92-0191, 1992. [38] Ekaterinaris, J. A., Numerical investigation of dynamic stall of an oscillating wing, AIAA J. 33(10), 1803-1808, 1995.

ill'

Mi

m Companion M Computer Programs

10.1 I n t r o d u c t i o n In this chapter we describe two computer programs. The computer program of Section 10.2 is for steady airfoil flows based on the Hess-Smith panel method (HSPM). The computer program in Section 10.3 is based on the interactive boundary layer method discussed in Chapters 4 and 5 and is applicable to both steady and unsteady laminar and turbulent flows, including separation. Several sample calculations for steady and unsteady airfoils are also presented. Both computer programs are given on the accompanying CD-ROM.

10.2 Hess-Smith Panel Method (HSPM) for Steady Flows The computer program for steady inviscid flows has a MAIN program and four subroutines, COEF, GAUSS, VPDIS and CLCM. MAIN contains the following input information: NODTOT X(I), Y(I) ALPHA FMACH

Number of panels along the airfoil surface Airfoil coordinates, x/c, y/c, normalized with respect to its chord c Angles of attack, a Free-stream Mach number, M ^

MAIN also contains the logic of the calculations. The panel slopes are calculated from Eq. (3.2.2). Subroutine COEF is called to compute the elements aij of the coefficient matrix A from Eq. (3.2.21) and (3.2.23) and the elements of 6. Note that N + 1 corresponds to KUTTA and N to NODTOT. Once x is determined by subroutine GAUSS so that source strengths qi (i = 1, 2 , . . . , N) and vorticity r on the airfoil surface are known, the tangential velocities are

10. Companion Computer Programs

212

obtained with the help of Eq. (3.2.12b). This subroutine also determines the distribution of the dimensionless pressure coefficient Cp (= CP) defined by Cp = ,f / 0 ,;~ 2 (1/2)^

(10.2.1a)

which in terms of velocities can be written as

CP = l - ( 0

(10.2.1b)

It is common to use panel methods for low Mach number flows by introducing compressibility corrections which depend upon the linearized form of the compressibility velocity potential equation and are based on the assumption of small perturbations and thin airfoils [1]. A simple correction formula for this purpose is the Karman-Tsien formula which uses the tangent gas approximation to simplify the compressible potential-flow equations. According to this formula the effect of Mach number on the pressure coefficient is estimated from C Cp

= (3 + [Ml/(l

•

+ (3)}(Cpi/2)

(1

°-2-2)

and the corresponding velocities are computed from V

Foe

(1 - A)(F/F 0 0 ) l

i - Kv/Vo,

Here Cpi denotes the incompressible pressure coefficient, M^ Mach number and

(10.2.3) the freestream

Subroutine CLCM is used to compute the airfoil characteristics corresponding to lift (CL) and pitching moment (CM) coefficients.

10.3 I n t e r a c t i v e B o u n d a r y - L a y e r P r o g r a m The interactive boundary-layer program contains both the unsteady panel method of Chapter 3 modified for viscous effects and the boundary-layer method of Chapter 5. It can be used for computing low speed unsteady airfoil flows subject to ramp type or harmonic oscillation motions with the interaction scheme described in subsection 5.2.3. The program contains a MAIN program and several subroutines. A brief description of some of the subroutines is given below. Additional information is provided on the accompanying CD-ROM.

10.3 Interactive Boundary-Layer Program

MAIN

213

Contains the logic of the calculations which begin by calling Subroutine INPUT to read in the airfoil geometry (see subsection 10.3.1) to set up the panels for inviscid and viscous flow calculations so that inviscid/viscous iterations can be performed.

Subroutines: CLOSE-GEOM If the airfoil has an open trailing edge, this subroutine is used to close the trailing edge. BLOWNG Calculates blowing velocity, v^, on the airfoil, Eq. (5.2.7), and along the wake. VELWK Calculates the total velocity and pressure coefficient at each control point along the upper and lower wakes separately. The normal and tangential components of the total velocities are computed from Eqs. (3.3.15). TEWAK Determines the strength of the vortex shedding and the velocity at the trailing edge as described in subsection 3.3.2. 10.3.1 Input Input t o t h e Inviscid P r o g r a m • • •

Airfoil name Number of panels along the airfoil. Airfoil geometry coordinates normalized with respect to its chord, c, x / c ,

•

y/cAirfoil motion: The program deals with three types of airfoil motion.

1. Ramp-type of motion with constant pitch rate A (=

{

a0

^ ^ )

t <0

do + \OLf — ao)— 0
0 < t < tf

Here UJ is the reduced frequency, 77-^, t dimensionless time, t^u^o/c, tf = — ^ ^ with t c y c i e denoting the total number of cycles. Parameters UJ* and t* denote dimensional frequency and time, respectively. Parameters ptch(A), delhx, delhy and phase defined below are set equal to zero.

214

10. Companion Computer Programs

3. Translational harmonic oscillation The instantaneous translations in the x- and y-directions are defined by ux = Ax oj cos(ojt + (/))

uy =

Ayujcos(ujt)

Here UJ is the reduced frequency and 0 is the phase shift defined by *

180

with Ax, Ay and
•

•

Chord Reynolds number Rc x 10" 6 ( = ^ ) Inviscid flow Rc = 0 Viscous flow Rc = input Boundary-layer calculations can be performed using the quasi-steady flow assumption or for unsteady flow. The parameter UNSTEADY is 0 for quasisteady flow and 1 for unsteady flow. The parameter zgzg is used to choose the numerical method for the boundarylayer calculations, zgzg is 1 for zig-zag box and 0 for backward box scheme.

Additional input information is given in the accompanying CD-ROM. 10.3.2 O u t p u t The output of the program includes the four files described below. • • •

•

Unit 6 - Standard output file. Unit 77 - Summary file: Includes lift, q, and total drag, c^, and pitching moment c m , coefficients and stagnation point. Unit 50 - Summary file: Output for inviscid flow results which include pressure coefficient C p , dimensionless edge velocity, u^ju^, along the displacement surface, blowing velocity distribution v\>{x) on the body and along the wake. This output is for every station for each a. Unit 12 - Summary file: Output for boundary-layer results for every a. It includes boundary-layer x-grid, boundary-layer edge velocity Ue/uoc, skinfriction coefficient cy, wake center-line velocity i^/i^oo, dimensionless displacement <5*/c, and momentum thickness 6/c distributions and transition locations.

10.3 Interactive Boundary-Layer Program

215

10.3.3 Test Cases The accompanying CD-ROM contains six test cases described below each with complete input and output. Test Case 1 To demonstrate the use of HSPM, we consider a NACA 0012 airfoil that is symmetrical with a maximum thickness of 0.12c. Table 10.1 given in the CD-ROM defines the airfoil coordinates for 184 points in tabular form. This corresponds to N O D T O T = 183. Note that the x/c and y/c values are read in starting on the lower airfoil to the upper surface TE. The calculations are performed for angles of attack of a — 0°, 8° and 16°. In identifying the upper and lower surfaces of the airfoil, it is necessary to determine the x/c-locations where ue ( = ue/uoo) = 0. This location, called the stagnation point, is easy to determine since the ue values are positive for the upper surface and negative for the lower surface. In general it is sufficient to take the stagnation point to be the x/c-location where the change of sign to ue occurs. For higher accuracy, if desired, the stagnation point can be determined by interpolation between the negative and positive values of Ug as a function of the surface distance along the airfoil. Figures 10.1a and 10.1b (see also the accompanying CD-ROM) show the variation of the pressure coefficient Cp and external velocity ue on the lower and upper surface of the airfoil as a function of x/c at three angles of attack starting from 0°. As expected, the results show that the pressure and external velocity distribution on both surfaces are identical to each other at a — 0°. With increasing incidence angle, the pressure peak moves upstream on the upper surface and downstream on the lower surface. In the former case, with the pressure peak increasing in magnitude with increasing a, the extent of the flow 4.0

-14.0

\ 1

-10.0

„

\1

(X — K) n — 9. (X — O

\

a = 16

|

-6.0

fi

i

a = 0

"x . \

a =8 a = 16

2.0

\ V

V

0.0 \ \ -2.0

l-\^^ 1

r\ f\

0.0 (a)

"~~ 1

0.2

-"" ~" ~~ - — .

1

..-x_.

0.4

1

0.6 x/c

.

1

0.8

1

1

1.0

-2.0 0.0 (b)

i

0.2

.

i

.

0.4

i

0.6

.

i

.

0.8

i

1.0

x/c

Fig. 10.1. Distribution of (a) pressure coefficients and (b) dimensionless external velocity distributions on the NACA 0012 airfoil at a = 0°, 8° and 16°.

216

10. Companion Computer Programs

deceleration increases on the upper surface, and increases the region of flow separation on the airfoil. On the lower surface, on the other hand, the region of accelerated flow increases with incidence angle which leads to regions of more laminar flow than turbulent flow. Test Case 2 The unsteady interactive boundary-layer (UIBL) program can be used for steady and unsteady airfoil flows. This test case demonstrates its application to steady flows at high Reynolds numbers (see subsection 7.3.1) with the onset of transition location computed with Michel's method, Eq. (7.31), except where the boundary layer separates upstream of this location, in which case transition is assumed to correspond to the separation point. It can also be used for low Reynolds number flows provided the onset of transition is computed with the e n -method since the transition location is usually inside the region of separated flow (subsection 7.3.2)

o.o*0

• 5

(a)

• 10 ex ( d e g r e e s )

' 15

* 20

o.o 0 (b)

5

10 a

15

20

(degrees)

Fig. 10.2. Comparison between calculated (solid lines) and experimental values (symbols) of the lift coefficient. NACA 0012 airfoil, (a) Rc = 3 x 10 6 , (b) Rc = 6 x 10 6 .

Figure 10.2 shows the variation of the lift coefficient with angle of attack for chord Reynolds numbers Rc of 3 x 10 6 and 6 x 10 6 . The calculated results agree well with experimental data as well as with those computed (see Fig. 7.25a) with the airfoil code for steady flows described in [1]. Test Case 3 As discussed in Section 4.6, the unsteady panel method uses steady-state conditions to initiate the time-dependent inviscid flow solutions. This procedure is quite acceptable at low frequencies but not at higher frequencies. To illustrate this, the inviscid panel method is used to calculate the variation of the lift coefficient with angle of attack for a NACA 0012 airfoil undergoing a rotational

10.3 Interactive Boundary-Layer Program

217

harmonic motion, a = 5° + ll°sinu;£ Figure 10.3 shows the results for three cycles for uo = 0.001. Figure 10.3a shows that the solutions that originate at a — 5° (t = 0) have a small spike which continues up to 11° and is absent at higher angles so that the solutions are independent of the initial conditions. Figure 10.3b shows that the valve of the lift coefficient at the beginning of the first cycle is different from that at the end of the third cycle and, although the difference is small it increases with increasing frequency. This is illustrated in Fig. 10.4 for UJ — 0.01 where the spike in the lift coefficient is very different than its value at the end of the third cycle. Figures 10.5, 10.6 and 10.7 show the results for u = 0.1, 0.25, 0.5 respectively, and that in order to obtain truly unsteady flow solutions, it is necessary to run at least two cycles. This is important in performing inviscid-viscous interactions since the spike due to steady-state effects can cause the boundary-layer solutions to break down.

- 6 - 4 - 2

0

2

(a)

4

6

8

10

12

14

16

a (degrees)

0.0

0.5

1.0

(b)

1.5

2.0

2.5

3.0

cycle

Fig. 10.3. Variation of lift coefficient a with (a) angle of attack a and (b) cycle, u — 0.001.

- 6 - 4 - 2 (a)

0

2

4

6

a (degrees)

8

10

12

14

16

0.0 (b)

0.5

1.0

1.5

2.0

2.5

3.0

cycle

Fig. 10.4. Variation of lift coefficient ci with (a) angle of attack a and (b) cycle, uo = 0.01.

10. Companion Computer Programs

218

(a)

a (degrees)

(b)

cycle

Fig. 10.5. Variation of lift coefficient Q with (a) angle of attack a and (b) cycle, UJ = 0.1.

Fig. 10.6. Variation of lift coefficient Q with (a) angle of attack a and (b) cycle, u = 0.25.

Fig. 10.7. Variation of lift coefficient c\ with (a) angle of attack a and (b) cycle, UJ = 0.5.

10.3 Interactive Boundary-Layer Program

219

Test Case 4 This case considers the NACA 0012 airfoil but with the boundary-layer calculations employing quasi-steady and unsteady flow assumptions for reduced frequencies of 0.0001 (Fig. 10.8), 0.01 (Fig. 10.9), 0.1 (Fig. 10.10), 0.25 (Fig. 10.11) and 0.5 (Fig. 10.12), for a chord Reynolds number Rc of 3 x 10 6 with a\ = 5, Aa = 11 and ATS = 30. In all cases, the results indicate the strong effect of the viscosity and the effect of quasi-steady and unsteady flow assumptions on the solutions.

- 6 - 4 - 2

0

2

(a)

4 a

6

8

10

12

14

16

(degrees)

Fig. 10.8. Variation of lift coefficient a with (a) angle of attack a and (b) cycle, uo — 0.0001.

- 6 - 4 - 2 (a)

0

2

4 a

6 (degrees)

8

10

12

14

16

2.0 0>)

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

cycle

Fig. 10.9. Variation of lift coefficient a with (a) angle of attack a and (b) cycle, u> = 0.01.

10. Companion Computer Programs

220

(a)

a (degrees)

cycle

Fig. 10.10. Variation of lift coefficient c\ with (a) angle of attack a and (b) cycle, UJ = 0.1.

Fig. 10.11. Variation of lift coefficient a with (a) angle of attack a and (b) cycle, u = 0.25.

- 6 - 4 - 2

0

2

4

6

a (degrees)

8

10

12

14

16

2.0

(b)

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

cycle

Fig. 10.12. Variation of lift coefficient Q with (a) angle of attack a and (b) cycle, CJ = 0.5.

10.3 Interactive Boundary-Layer Program

221

Test Case 5 This test case considers the NACA 0012 airfoil subjected to a ramp-type harmonic motion. The boundary-layer calculations involve quasi-steady and unsteady flow assumptions and were performed for constant pitch rates A of 0.0001 (Fig. 10.13a), 0.01 (Fig. 10.13b), 0.05 (Fig. 10.13c), and 0.1 (Fig. 10.13d) at a chord Reynolds number Rc of 3 x 10 6 with ax = 5, Aa = 11 and ATS = 30. The results indicate that viscous effects have a strong effect at the lowest pitch rate with significant differences from those computed with the panel method. There is practically no difference between the lift coefficients computed with quasisteady or full unsteady flow assumptions at low to modest angles of attack, but there are differences at higher angles of attack for which the quasi-steady flow assumption shows a stronger viscous effect. As the pitch rate increases the differences between the inviscid and viscous lift coefficients reduce, with quasi-steady flow assumption again indicating a stronger viscous effect.

2

4

6

(a)

8

10

12

14

2

4

6

8

10

a (degrees)

2

16

a (degrees)

4

6

^'

12

14

2

16

(d)

8

10

12

14

16

12

14

16

a (degrees)

4

6

8

10

a (degrees)

Fig. 10.13. Variation of lift coefficient ci for ramp-type harmonic motion for constant pitch rates of A (a) 0.0001, (b) 0.01, (c) 0.05 and (d) 0.1.

10. Companion Computer Programs

222

Test Case 6 Here, the NACA 0012 airfoil is assumed to undergo translational harmonic motion with a = 6°, Rc = 3 x 10 6 , Ax = 0.1, Ay = 0.3, 0 = 10 for reduced frequencies u = 0.025 and 0.25. Figures 10.14 and 10.15 show Q , UX and i ^ as a function of cycle for the two reduced frequencies and that the quasi and unsteady flow assumptions agree very well with each other and the viscous lift coefficients differ from those given by the inviscid flow theory. Note that the inviscid flow solutions were obtained for two cycles in order to reduce the steady-state effects and the viscous flow solutions were started at the end of the first cycle.

0.60 * 1.0

(a)

•

'

'

'

'

'

'

'

•

'

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

cycle

-0.008-' 1.0

(b)

'

•

'

•

'

'

'

•

'

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

cycle

Fig. 10.14. Variation of (a) lift coefficient c/ and (b) velocity translation (ux,uy) translational harmonic motion with u = 0.025.

(a)

c

ycle

(b)

' 2.0

for

cycle

Fig. 10.15. Variation of (a) lift coefficient c\ and (b) velocity translation (ux,uy) translational harmonic motion with UJ = 0.25.

for

References

223

References [1] Cebeci, T.: Stability and Transition: Theory and Application. Horizons PubL, Long Beach, Calif, and Springer, Heidelberg, 2004.

Subject Index

Airfoil chord 2 Algebraic turbulence model 201, Angle of attack 7, 8, 13

164, 200,

Baldwin-Lomax (BL) turbulence model 167, 199, 201 Bernoulli equation 26, 28, 39, 50, 51 Block elimination method 71 Blowing velocity 64 Boundary layer 2, 4, 5, 6, 16, 28, 31, 59, 77, 94, 96, 98, 107, 108, 127, 136 Boundary layer equations 28, 59, 61, 62, 63, 65, 66, 86, 93, 95, 96, 107, 110, 155 Cebeci-Smith (CS) turbulence model 59, 164 CFL condition 82, 83, 103 Circular frequency 2 Circulation 5, 6, 35, 38, 39, 49, 50 Closure problem 25 Compressible flow 183, 184, 193, 194, 198 Conservation equations 21, 24, 157, 158 Continuity equation 22, 24, 25, 26, 27, 156, 158, 166, 178 Direct numerical simulation (DNS) 24 Displacement thickness 59, 62, 63, 102, 103, 115, 116, 117, 119, 121, 123, 124, 125, 132, 133, 134, 136, 137, 138, 139, 146, 147, 148, 149, 151, 152, 168 Dissipation function 157 Dynamic stall 16, 17, 18, 150, 153, 155, 194, 195, 197, 200, 203, 204, 206, 207, 208 Eddy viscosity 59, 60, 61, 96, 164, 165, 166, 167, 168, 171, 201 Energy equation 156, 158 Ensemble average 24 Euler equation 26, 177, 184

Flow reversal 82, 101, 107, 108, 111, 114, 115, 120, 122, 147, 204 Flow separation 16, 62, 93, 99, 100, 107, 108, 109, 114, 115, 127, 128, 129, 132, 140, 150 Flow tangency condition 34, 36, 42 Fluid particle 50, 51 Flutter 10, 12, 13, 14, 18, 54, 56 Gust response

15, 57

Hilbert integral 63, 72, 73, 127 Hysteresis 18, 19, 146, 197, 198, 199, 200, 201, 202, 203, 205, 209, 210 Incompressible flow 26, 31, 100, 101, 174, 193, 195, 198 Influence coefficient 34, 36, 40, 42, 44, 45, 46 Interactive boundary layer theory 4, 16, 27, 63, 142, 143, 144, 145 Intermittency 60, 61, 167, 202 Inviscid flow equations 26, 28, 31, 64 Irrotational flow 26, 33, 35, 63 Johnson-King (JK) turbulence model 165, 172, 201 Jones-Launder (k-s) turbulence model 169, 200 Keller's box method 66, 85 Kelvin-Helmholtz theorem 1, 6, 38, 49 Kuessner's function 15, 16 Kutta condition 3, 35, 37, 39, 64 Kutta-Joukowski law 5, 6 Laminar flow 25, 27, 28, 60, 65, 93, 95, 96, 100, 101, 108, 110, 193, 198 Laplace's equation 2, 27, 34 Laplacian operator 23 Lift generation 5, 6, 49 Lifting-line theory 7

226

Menter (SST) turbulence model 169, 172 Mixing length 163 Momentum equation 22, 24, 25, 59, 60, 68, 156, 158, 166, 178 Navier-Stokes equations 2, 4, 5, 6, 19, 21, 22, 23, 24, 25, 26, 27, 28, 155, 156, 159, 160, 161, 174, 177, 193 Newtonian fluid 23, 25 Newton's method 66, 68, 81, 84, 112 Newton's second law of motion 6, 8, 22 Normal stress 22, 23 Panel method 4, 10, 31, 32, 33, 34, 38, 46, 47, 49, 51, 53, 56, 64, 136 Parabolized Navier-Stokes equations 25, 26 Path line 49, 50 Perfect gas law 157 Pitch motion 10, 11, 13, 14, 19, 54, 55, 56 Pitch damping 14, 56 Plunge motion 8, 9, 10, 13, 15, 51, 54, 55, 56 Power extraction 9, 53 Reduced frequency 2, 54, 138, 143 Reynolds number 6, 16, 19, 27, 61, 99, 101, 120, 123, 124, 127, 128, 130, 132, 136 Reynolds stress 25 Reynolds-averaged Navier-Stokes equations (RANS) 21, 24, 28 RNG turbulence model 170, 201 Separation bubble 16, 62, 108, 109, 114, 127, 201, 202 Shear stress 22, 23, 61, 62, 99, 116, 117, 118, 121, 122, 123, 124, 125, 165, 166, 178 Skin friction 76, 77, 94, 102, 103, 128, 129, 130, 131, 133, 134, 137, 149, 152, 153

Subject Index

Source flow 33, 34, 35, 36, 38, 40, 41, 44 Spalart-Allmaras (SA) turbulence model 168 Stall 16, 108, 132, 134, 150 Stall flutter 18 Starting vortex 1, 2, 5, 7, 9, 13, 15, 49, 50, 54 Step gust 15 Stream function 175, 194 Streamline 26, 49, 50, 51, 195, 198 Strain tensor 23 Streak line 49, 50 Stress tensor 25, 157, 158, 163 Strouhal number 94 Substantial derivative 22, 24 Suction velocity 64 Thin-layer Navier-Stokes equations 25 Thrust generation 8, 53, 54 Transition 59, 61, 108, 125, 126, 127, 128, 130, 131, 132, 169, 194, 201, 204, 205, 206 Transpiration velocity 28 Turbulence model 108 Turbulent flow 24, 25, 27, 28, 59, 60, 65, 96, 109, 125, 128 Velocity potential 27, 45, 46 Viscous force or stress 22, 159, 178 Viscous-inviscid interaction method 4, 5, 136, 142, 143, 144, 145 Vortex 2, 6, 7, 8, 9, 13, 16, 31, 33, 34, 35, 36, 38, 39, 40, 41, 53, 54, 150, 206, 207 Vortex street 9, 53, 206, 207 Vortical wake 2, 3, 14, 206, 207 Vorticity 5, 6, 26, 140, 167, 171, 175, 176, 177, 194, 195, 196, 197 Wagner's function 3, 15, 16, 51 Wave length 2, 3, 14 Wilcox (k-u) turbulence model 169, 170, 171

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