Circulation Control Wing

7/29/2019 Circulation Control Wing

http://slidepdf.com/reader/full/circulation-control-wing 1/192

NUMERICAL SIMULATIONS OF THE AERODYNAMIC CHARACTERISTICS

OF CIRCULATION CONTROL WING SECTIONS

A Thesis

Presented to

The Academic Faculty

by

Yi Liu

In Partial Fulfillmentof the Requirements for the Degree

Doctor of Philosophy in the

School of Aerospace Engineering

Georgia Institute of Technology

April 2003

Copyright © 2003 by Yi Liu



NUMERICAL SIMULATIONS OF THE AERODYNAMIC CHARACTERISTICS

OF CIRCULATION CONTROL WING SECTIONS

Approved:

Lakshmi N. Sankar, Chairman

Krishan K. Ahuja

Robert J. Englar

D. Stefan Dancila

Richard Gaeta

Date Approved



iii

DEDICATION

To my wife, Qiang Le

And to my parents



iv

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Dr. Lakshmi N. Sankar, my

teacher and dissertation advisor, for his encouragement and support throughout the

research period. His delightful personality and detailed knowledge of this research topic

has guided me along the way. I would not be here without all that he has done.

I would also like to thank Dr. K. Ahuja, Mr. R. Englar, Dr. D. Dancila and Dr. R.

Gaeta, members of my thesis committee, for their thorough review of the thesis and for

their valuable comments. I am especially appreciative to Bob Englar for providing the

experimental data, and for helpful suggestions from his many years of experience.

I would like to acknowledge NASA Langley Research Center for sponsoring this

research under the Breakthrough Innovative Technology Program, Grant-NAG1-2146.

I thank Ms. Mary Trauner of High Performance Computing Group of the Office

of Information Technology at Georgia Tech, for her understanding and support during

my last semester of thesis work.

I would like to thank all my colleagues in the CFD lab for their warm friendship

and support during my Ph.D. studies. I would also like to thank my old friends in China,

for their constant encouragement through all these years.

Finally, I would like to thank my parents, Mr. Feng Liu and Mrs. Yuhua Peng, for

their continued support throughout my education at Georgia Tech. I would also like to

acknowledge the warm support and caring of my dear wife, Qiang Le. Without her

encouragement and enthusiasm, this work could not have been completed.



v

TABLE OF CONTENTS

DEDICATION.................................................................................................................. iii

ACKNOWLEDGEMENTS ............................................................................................ iv

TABLE OF CONTENTS ................................................................................................. v

LIST OF TABLES......................................................................................................... viii

LIST OF FIGURES......................................................................................................... ix

LIST OF NOMENCLATURE...................................................................................... xiv

SUMMARY .................................................................................................................... xix

1. INTRODUCTION......................................................................................................... 1

1.1 Motivation and Objectives........................................................................................ 1

1.2 Circulation Control Technology............................................................................... 6

1.2.1 The Circulation Control Wing Concept............................................................. 6

1.2.2 The Advanced Circulation Control Airfoil........................................................ 9

1.2.3 Applications and Benefits of the Circulation Control Wing............................ 11

1.3 Previous Research Work......................................................................................... 14

1.4 Overview of the Present Work................................................................................ 19

2. MATHEMATICAL AND NUMERICAL FORMULATION ................................ 21

2.1 The Governing Equations ....................................................................................... 22

2.1.1 Governing Equations in Cartesian Coordinates............................................... 22

2.1.2 Governing Equations in Curvilinear Coordinates............................................ 28

2.2 Numerical Procedure .............................................................................................. 34

2.2.1 Temporal Discretization................................................................................... 34



vi

2.2.2 Linearization of the Difference Equations....................................................... 36

2.2.3 Approximate Factorization Procedure ............................................................. 38

2.2.4 Spatial Discretization of the Inviscid Terms.................................................... 39

2.2.5 Spatial Discretization of the Viscous Terms.................................................... 41

2.2.6 Implementation of Low Pass Filters ................................................................ 42

2.3 Turbulence Models ................................................................................................. 45

2.3.1 Baldwin-Lomax Turbulence Model................................................................. 47

2.3.2 Spalart-Allmaras Turbulence Model................................................................ 49

2.4 Initial and Boundary Conditions............................................................................. 51

2.4.1 Initial Conditions ............................................................................................. 52

2.4.2 Outer Boundary Conditions............................................................................. 52

2.4.3 Solid Surface Conditions ................................................................................. 54

2.4.4 Boundary Conditions at the Cuts in the C Grid ............................................... 55

2.4.5 Jet Slot Exit Conditions with Given Cµ ........................................................... 56

2.4.6 Jet Slot Exit Conditions with Given Total Jet Pressure ................................... 59

3. TWO DIMENSIONAL STEADY BLOWING RESULTS ..................................... 61

3.1 Code Validations with a NACA 0012 Wing........................................................... 62

3.2 Unblown and Steady Blowing Results ................................................................... 63

3.2.1 Configuration Modeled.................................................................................... 63

3.2.2 Computational Grid ......................................................................................... 64

3.2.3 Blowing and Unblown Results Comparison.................................................... 65

3.2.4 Steady Blowing with Specified Total Pressure................................................ 69



vii

3.3 Effects of Parameters that Influence the Momentum Coefficient .......................... 70

3.3.1 Free-stream Velocity Effects with Fixed Cµ and Fixed Jet Slot Height .......... 71

3.3.2 Jet Slot Height Effects with Fixed Cµ and Fixed Free-stream Velocity .......... 72

3.4 Other Simulations for the CC Airfoil...................................................................... 73

3.4.1 Comparisons with the Conventional High-Lift System................................... 73

3.4.2 Leading Edge Blowing .................................................................................... 74

4. TWO DIMENSIONAL PULSED BLOWING RESULTS...................................... 93

4.1 Jets Pulsed Sinusoidally.......................................................................................... 94

4.2 Jets Pulsed with a Square Wave Form.................................................................... 96

4.2.1 Pulsed Jet Flow Behavior................................................................................. 96

4.2.2 Effects of Frequency at a Fixed Cµ .................................................................. 99

4.2.3 Strouhal Number Effects................................................................................ 100

4.3 Summary of Observations..................................................................................... 103

5. THREE DIMENSION CIRCULATION CONTROL WING SIMULATIONS. 117

5.1 Tangential Blowing on a Wing-flap Configuration.............................................. 118

5.2 Spanwise Blowing over a Rounded Wing-tip....................................................... 121

6. CONCLUSIONS AND RECOMMENDATIONS.................................................. 137

6.1 Conclusions........................................................................................................... 138

6.2 Recommendations................................................................................................. 141

APPENDIX A. GENERALIZED TRANSFORMATION........................................ 144

REFERENCES.............................................................................................................. 149

VITA............................................................................................................................... 159



viii

LIST OF TABLES

Table Page

4.1 The Computed Time-averaged Lift Coefficient for the Case One

(U∞ and Lref fixed, Strouhal number varying with the frequency)

105

4.2 The Computed Time-averaged Lift Coefficient for the Case Two

(Strouhal number and Lref fixed, U∞ varying with the frequency)

105

4.3 The Computed Time-averaged Lift Coefficient for the Case Three

(Strouhal number and U∞ fixed, Lref varying with the frequency)

106

5.1 The Total Lift Coefficient and Drag Coefficient for the Wing Tip

Configuration

123



ix

LIST OF FIGURES

Figure Page

1.1 Normalized Noise Levels of Aircraft by Year of Certification 2

1.2 Airframe Noise Sources 3

1.3 Boeing 737 Wing/Flap System 4

1.4 Basics of Circulation Control Aerodynamics 7

1.5 Dual Radius CCW Airfoil with LE Blowing 10

2.1 The Outer Boundary Conditions for Sample C Grid 53

2.2 The Solid Surface Boundary Conditions for Viscous Flow 55

2.3 The Wake-cut Boundary Conditions for C Grid 56

2.4 The Jet Slot Boundary Conditions 57

3.1a CP Distribution over NACA 0012 Wing Sections at 34% Span 76

3.1b CP Distribution over NACA 0012 Wing Sections at 50% Span 76

3.1c CP Distribution over NACA 0012 Wing Sections at 66% Span 77

3.1d CP Distribution over NACA 0012 Wing Sections at 85% Span 77

3.2 Lift Coefficient Distribution along Span at Angle of Attack 8 Degrees 78

3.3 The Circulation Control Wing Airfoil with 30-degree Flap 78

3.4 The Body-fitted C Grid near the CC Airfoil Surface 79

3.5 The Lift Coefficients in Different Grid Spacing Cases (Cµ = 0.15) 79

3.6 Variation of the Lift Coefficient with Momentum Coefficients at α= 0 80



x

Figure Page

3.7 The Variation of the Lift Coefficient with Angle of Attack 80

3.8 The Streamlines over the CC Airfoil at Two Instantaneous Time Step 81

3.9 Time History of the Lift Coefficient for the Unblown Case

(U∞ = 94.3 ft/sec)

82

3.10 Time History of the Lift Coefficient for the Unblown Case

(U∞ = 220 ft/sec)

82

3.11 The FFT of the Lift Coefficient Variation with Time (U∞ = 220 ft/sec) 83

3.12a Streamlines over the TE of the CC Airfoil (Unblown Case) 84

3.12b Streamlines over the TE of the CC Airfoil (Blowing Case) 84

3.13 The Cµ Variation with the Total Jet Pressure for Steady Blowing Case 85

3.14 The Lift Coefficient Variation with Cµ for Steady Blowing Case 85

3.15 Lift Coefficient vs. Free-stream Velocity

(Cµ = 0.1657, h = 0.015 inch and V∞, exp = 94.3 ft/sec)

86

3.16 Drag Coefficient vs. Free-stream Velocity


86

3.17 Mass Flow Rate vs. Free-stream Velocity


87

3.18 Lift Coefficient vs. Jet Slot Height (V∞ = 94.3 ft/sec) 87

3.19 Drag Coefficient vs. Jet Slot Height (V∞ = 94.3 ft/sec) 88



xi

Figure Page

3.20 The Efficiency vs. Jet Slot Height (V∞ = 94.3 ft/sec) 88

3.21 The Mass Flow Rate vs. Jet Slot Height (V∞ = 94.3 ft/sec) 89

3.22 The Shape of the Multi-element Airfoil and the Body-fitted Grid 89

3.23 The Drag Polar for the Multi-element Airfoil and the CC Airfoil 90

3.24 The Efficiency (Cl /Cd+ Cµ) for the Multi-element Airfoil and the CC

Airfoil

90

3.25a The Grid for the Leading Edge Blowing Configuration 91

3.25b The Grid Close to the Leading Edge Jet Slot 91

3.25c The Grid Close to the Trailing Edge Jet Slot 91

3.26 Lift Coefficient vs. The Angle of Attack 92

3.27 Drag Coefficient vs. The Angle of Attack 92

4.1 The Time History of the Momentum Coefficient

(Sinusoidal Wave, Frequency = 400 Hz, Cµ,0 = 0.04)

107

4.2 The Time History of the Lift Coefficient


107

4.3 The Time History of the Mass Flow Rate


108

4.4 Time-averaged Lift Coefficients vs. Frequency 108

4.5 Time-averaged Mass Flow Rate vs. Frequency 109

4.6 The Time History of the Momentum Coefficient

(Square Wave Form, Frequency = 40 Hz, Cµ,0 = 0.04)

109



xii

Figure Page

4.7 The Time History of the Lift Coefficient


110

4.8 The Time History of the Mass Flow Rate


110

4.9 The Incremental Lift Coefficient vs. Time-averaged Momentum

Coefficient

111

4.10 The Incremental Lift Coefficient vs. Time-averaged Mass Flow Rate 111

4.11 Time-averaged Mass Flow Rate vs. Time-averaged Momentum

Coefficient

112

4.12 The Efficiency vs. Time-averaged Momentum Coefficient 112

4.13 The Efficiency vs. Time-averaged Mass Flow Rate 113

4.14 Time-averaged Lift Coefficient vs. Pulsed Jet Frequency

(Ave. Cµ,0 = 0.04)

113

4.15 The Efficiency vs. Pulsed Jet Frequency (Ave. Cµ,0 = 0.04) 114

4.16 Time History of the Lift Coefficient for a 40Hz Pulsed Jet 114

4.17 Time History of the Lift Coefficient for a 200Hz Pulsed Jet 115

4.18 Time-averaged Lift Coefficient vs. Frequency 115

4.19 Time-averaged Lift Coefficient vs. the Frequency & Strouhal Number 116

5.1 The Wing-flap Tangential Blowing Configuration 119

5.2 The Grid of the 3-D Wing-flap Configuration with a 300 Partial Flap 125



xiii

Figure Page

5.3 The Lift Coefficient Distribution along Span for the Wing-flap

Configuration

126

5.4 The Vorticity Contours for Noblowing Case 127

5.5 The Vorticity Contours for Constant Blowing Case 128

5.6 The Vorticity Contours for Gradual Blowing Case 129

5.7 The Wing Tip Configuration 122

5.8 The H-Grid for the Wing Tip Configuration

(Side View at Spanwise Station)

130

5.9 The O-Grid around the Rounded Wing Tip (Front View) 130

5.10 The Surface Grid for the Rounded Wing Tip 131

5.11 The Detailed Grid Close to the Jet Slot 131

5.12 The Vorticity Contours around the Wing Tip (x/C = 0.81) 132



5.15 The Velocity Vectors around the Wing Tip (x/C = 0.81) 135

5.16 The Lift Coefficient Distribution along Span for Wing Tip

Configuration

136

5.17 The Drag Coefficient Distribution along Span for Wing Tip

Configuration

136



xiv

LIST OF NOMENCLATURE

a Speed of sound

a jet Speed of sound of the jet

A jet Area of the jet slot

A, B, C Flux Jacobian matrices

C p Specific heat at constant pressure

C v Specific heat at constant volume

C l, C L Lift coefficient

C d , C D Drag coefficient

C µ Jet momentum coefficient

C µ ,0 Time-averaged momentum coefficient for pulsed jets

E t Total energy per unit volume

E, F, G Inviscid flux matrices

f Frequency

F + Non-dimension frequency

F kleb Klebanoff intermittency correction

J Jacobian of transformation

k Thermal conductivity

K c Clauser’s constant

Lref Reference length

lm Mixing length



xv

m Mass flow rate

n Normal vector of cell surface

M∞ Free-stream Mach number

M jet Jet Mach number

O(.) Order of variable

P Pressure

P jet Pressure at the jet slot exit

P0 Total pressure

P0, jet Total pressure at the jet slot exit, Duct pressure

Pr Prandtl number

q State variable vector

q x, q y, q z Heat transfer by conduction

R, S, T Viscous flux matrices

Re Reynolds number

S Wing area

Str Strouhal number

t Time in the physical domain

T Temperature

T jet Temperature at the jet slot exit

T 0, jet Total temperature at the jet slot exit, Duct temperature

x, y, z Cartesian coordinates

u, v, w Cartesian velocities



xvi

U , V , W Contravariant velocities

U jet Jet velocity from CFD calculation

V a Jet velocity obtained in experiments

∆ Forward difference operator

∇ Backward difference operator

α Angle of attack

δ Central difference operator

δ ij Kronecker Delta function

ε Turbulent dissipation rate

γ Specific heat ratio

λ Second coefficient of viscosity

λ ξ, λ η, λ ζ Eigenvalues

µ Coefficient of viscosity

ν Kinematic viscosity

ρ Density

ρ jet Jet density

τ Non-dimensional time

τ ij Viscous stress

ω Vorticity

ξ , η, ζ Computational domain coordinates



xvii

Subscripts

i, j, k Indices in three coordinate directions

t Derivative with respect to physical time

w Variable on the wall surface

τ Derivative with respect to time in (ξ, η, ζ) coordinates

ξ , η, ζ Derivatives with respect to generalized coordinates

x, y, z Derivatives with respect to Cartesian coordinates

∞ Free-stream value

ref Reference value of non-dimension

jet Variable at the jet slot

Superscripts

n, n+1 Time level

* Non-dimensional variable

^ Variable in the computational domain

- Mean value of the flow variables

‘ Fluctuation quantity after average

Acronyms and Abbreviations

2-D Two dimensional

3-D Three dimensional

ADI Alternating Direction Implicit



xviii

AF Approximate Factorization

BVI Blade Vortex Interaction

CC Circulation Control

CCW Circulation Control Wing

CFD Computational Fluid Dynamics

DNS Direct Numerical Simulation

DTNSRDC David Taylor Naval Ship Research and Development Center

FAA Federal Aviation Administration

FFT Fast Fourier Transformation

GTRI Georgia Tech Research Institute

HBPR High Bypass-ratio

HSCT High Speed Civil Transport

LE Leading Edge

LES Large Eddy Simulation

NACA National Advisory Committee for Aeronautics

NASA National Aeronautics and Space Administration

RANS Reynolds-Averaged Navier-Stokes

RHS Right Hand Side

STOL Short Take-off and Landing

TE Trailing edge



xix

SUMMARY

Circulation Control technology is a very effective way of achieving very high lift

coefficients needed by aircraft during take-off and landing. This technology can also

directly control the flow field over the wing. Compared to a conventional high-lift

system, a Circulation Control Wing can generate the same high lift during take-off/

landing with fewer or no moving parts and much less complexity.

In this work, an unsteady three-dimensional Navier-Stokes analysis procedure has

been developed and applied to CCW configurations. This method uses a semi-implicit

ADI scheme that is second or fourth order accurate in space, and first order in time. The

solver can be used in both a 2-D and a 3-D mode, and can thus model airfoils as well as

finite wings. The jet slot location, slot height, and the flap angle can all be varied easily

and individually in the grid generator and the flow solver. Steady jets, pulsed jets, the

leading edge and trailing edge blowing can all be studied with this solver.

The effects of 2-D steady jets and 2-D pulsed jets on the aerodynamic

performance of CCW airfoils have been investigated. It is found that a steady jet can

generate very high lift at zero angle of attack without stall, and that a small amount of

blowing can eliminate the vortex shedding, a potential noise source. A thin jet is also

found to be more beneficial than a thick jet from an aerodynamic design perspective,

although the power requirements of generating thin jets can be high. It is also found that

the pulsed jet can achieve the same high lift as the steady jet but at less mass flow rates,



xx

especially at a high frequency, and that the Strouhal number has a more dominant effect

on the pulsed jet performance than just the frequency.

Three-dimensional simulations have also been done for two cases. The first is a

streamwise tangential blowing on a wing-flap configuration. It is demonstrated that a

gradually varied CC blowing can totally eliminate the flap-edge vortex, thus reducing the

flap-edge noise. The second case involves spanwise tangential blowing over a rectangular

wing with a rounded wing tip. It is found that CC blowing can not totally cancel or

eliminate the tip vortex. However, it can control and modify the location of the tip vortex,

and increase the vertical clearance between the wing and the tip vortex, thus reducing the

blade vortex interaction and the BVI noise.



1

CHAPTER I

INTRODUCTION

1.1 Motivation and Objectives

During the past three decades, there has been a significant increase in air travel,

and thus a rapid growth in commercial aviation. At the same time, environmental

regulations and restrictions on aircraft operations have become a critical issue that

threatens to affect and limit the growth of commercial aviation. For instance, the Federal

Aviation Administration (FAA) and similar agencies in other countries have issued

stringent regulations on the legal use and operation of airports that satisfy community

concerns [1, 2]. In particular, the noise pollution from aircraft, especially around the

airport, has become a major problem that needs to be solved. Thus, reducing aircraft

noise has become a priority for airlines, aircraft manufacturers, and NASA researchers. In

response to this challenge, NASA has proposed a plan to double aviation system capacity

while reducing perceived noise by a factor of two (10dB) by 2011, and to triple system

capacity while reducing perceived noise by a factor of four (20dB) by 2025 [3].

In general, the aircraft noise may be divided into two major categories based on

noise sources. The first is jet engine noise, which is primarily produced from fan and

exhaust, although other components such as compressors, turbines, and combustors also



2

contribute to this. The second is the airframe noise, which is generated by components

such as fuselage, wing, under-carriages, slat and flap edges, etc. In the case of jet engines,

due to improvements to the technology from the early turbojet engines to current

generation high bypass-ratio (HBPR) turbofan engines, today’s new jet transport

airplanes are about 20dB quieter than those introduced in the 1960s [4]. Figure 1.1

indicates the noise levels of aircraft as a function of the year they were first introduced

into service. It is clearly seen that the noise has been reduced significantly by the use of

better jet engines over the past forty years.

777-200

A330-300

MD-90-30

MD-11

A320-200

747-400

A300-600R

767-300ER

A310-300757-200

MD-87MD-82

B-747-300

A300B4-620

A310-222

MD-80

B-747-200

B-747-SPDC-10-40

B-747-200

A300

B-747-200

B-747-100

B-737-200

B-737-200

B-727-200

DC9-10

B-727-100

B-727-100

-20.0

-10.0

0.0

10.0

1960 1970 1980 1990 2000 2010 2020

Year of Certification

Average NoiseLevel Relative

to Stage 3(EPNdB)

HistoryJT3D, JT8D, JT9D,CF6,CFM56 CurrentJT8D-200,PW2000,PW4000,V2500,CF6,GE90 Future Goals

Stage 2

Stage 3

Stage 4

NegotiatedOut of Service*

Impact of current Noise

Reduction program goal of 5 dB

A v e r a g e i n S e r v i c e

Impact of achieving NASA goal

(additional 5 dB)

777-200

A330-300

MD-90-30

MD-11

A320-200

747-400

A300-600R

767-300ER

A310-300757-200

MD-87MD-82

B-747-300

A300B4-620

A310-222

MD-80

B-747-200

B-747-SPDC-10-40

B-747-200

A300

B-747-200

B-747-100

B-737-200

B-737-200

B-727-200

DC9-10

B-727-100

B-727-100

-20.0

-10.0

0.0

10.0

1960 1970 1980 1990 2000 2010 2020

Year of Certification

Average NoiseLevel Relative

to Stage 3(EPNdB)

HistoryJT3D, JT8D, JT9D,CF6,CFM56 CurrentJT8D-200,PW2000,PW4000,V2500,CF6,GE90 Future Goals

Stage 2

Stage 3

Stage 4

NegotiatedOut of Service*

Impact of current Noise

Reduction program goal of 5 dB

A v e r a g e i n S e r v i c e

Impact of achieving NASA goal

(additional 5 dB)

Figure 1.1: Normalized Noise Levels of Aircraft by Year of Certification [4]

Airframe-generated noise can be the dominant component of the total noise

radiated from commercial aircraft, particularly for large aircraft and especially during the

landing approach when the engines are at a relatively low power setting and the high-lift



3

systems are fully deployed. Figure 1.2 from Chapter 7 of Reference [5] illustrates some

of the airframe noise sources, which include the fuselage, main wing, landing gear, and

high-lift devices, etc. Since the mid-80s, many researchers have pointed out that the

airframe noise predominantly emanates from high-lift devices and the landing gear of

subsonic aircraft [6, 7]. Depending on the type of aircraft, the dominant source varies

between flap, slat, and landing gear. Recent studies by Davy and Remy [8] on a scaled

model of Airbus aircraft also indicate that high-lift devices and landing gear are major

sources of airframe noise when the aircraft is configured for landing. The studies

mentioned here are just a few examples of the work that has been done in this area, and

point to high-lift system as a dominant noise source. For a more detailed review of

airframe noise studies, the reader is referred to Reference [9].

Figure 1.2: Airframe Noise Sources [5]



4

Current generation large commercial aircraft are dependent on components that

can generate high lift at low speeds during take-off or landing in order to use existing

runways. As shown in Figure 1.3 from Englar’s paper [10], conventional high-lift

systems include flaps, slats, with the associated flap-edges and gaps. In addition to their

contribution to noise, these high-lift systems also add to the weight of the aircraft, and are

costly to build and maintain.

Figure 1.3: Boeing 737 Wing/Flap System [10]

An alternative to the conventional high-lift systems is the Circulation Control

Wing (CCW) technology. This technology and its aerodynamic benefits have been



5

extensively investigated over many years by Englar et al at Georgia Tech through

experimental studies [11, 12]. A limited number of numerical analyses [11, 13, 14] have

also been done. Work has also been done on the acoustic characteristics of Circulation

Control Wings [15]. These studies indicate that very high CL values (as high as 8.5 at

α=0°) may be achieved with Circulation Control (CC) wings. Because many mechanical

components associated with the high-lift system are no longer needed, the wings can be

lighter and less expensive to build. Some of the major airframe noise sources, such as

flap-edges, flap-gaps, and trailing/leading edge flow-separation can all be eliminated with

the use of CCW systems.

To further understand the aeroacoustic characteristics and benefits of the

Circulation Control Wing, Munro, Ahuja and Englar [16, 17, 18, 19] have recently

conducted several acoustic experiments comparing the noise levels of a conventional

high-lift system with that of an advanced CC wing at the same lift setting. The present

Computational Fluid Dynamics (CFD) study [20] is intended to be a complement to this

work, and to numerically investigate the aerodynamic characteristics and benefits

associated with the CC airfoil. CFD studies such as the one presented here can also help

the design of CCW configurations.



6

1.2 Circulation Control Technology

1.2.1 The Circulation Control Wing Concept

Conventional airfoils, such as the NACA series airfoils, all have a sharp trailing

edge. The Kutta condition [21] will be readily satisfied for this kind of the airfoil, and

determines the circulation over the airfoil at a given free-stream condition and angle of

attack. This sharp trailing edge design is very efficient for fixing circulation and lift, and

is widely used both in nature and on man-made lifting surfaces. However, there are two

limitations associated with it. First, the lift generated by a sharp trailing edge airfoil is

only a function of angle of attack, camber, and free-stream conditions, and it can not be

otherwise controlled. Secondly, the maximum lift achieved is limited, because the

adverse pressure gradient on the upper surface eventually causes boundary layer

separation and static stall with the increase in angle of attack. Thus, in order to obtain the

high lift coefficient required during take-off or landing, high-lift devices must be used on

a commercial aircraft. However, a high-lift system always contains many moving parts,

and results in a significant weight penalty, and noise.

The Circulation Control (CC) airfoil overcomes these drawbacks in another way.

It takes advantage of the Coanda effect by blowing a small, high-velocity jet over a

highly curved surface, such as a rounded trailing edge. Since the airfoil trailing edge is

not sharp, the Kutta condition is not fixed and the trailing edge stagnation point is free to

move along the surface. In addition, the upper surface blowing near the trailing edge

energizes the boundary layer, increasing its resistance to separation. With blowing, the



7

trailing edge stagnation point location moves toward the lower airfoil surface, thus

changing the circulation for the entire wing and increasing lift. Since the jet flow mass

rate is readily controlled, this results in direct control of the separation point location, and

thus the circulation and lift, as suggested by the name of this concept. Figure 1.4 shows a

typical traditional CC airfoil with a rounded trailing edge.

Figure 1.4: Basics of Circulation Control Aerodynamics [10]

The Coanda effect is named after the Romanian inventor Henri Coanda [22] who

had discovered and used it in the 1930s. As shown in Figure 1.4, this effect is due to a

balance within the jet sheet between the pressure gradient normal to the surface and the

centrifugal force caused by the streamline curvature. The curved trailing edge region is

thus known as the Coanda Surface. In general, the Coanda effect will move the stagnation

point aft, and delay the separation. Eventually, the momentum within the jet and the

boundary layer will decrease, and the adverse pressure gradient along the surface will

increase. It is this adverse pressure gradient which eventually causes the jet to separate

and leave the surface. The location of the separation point will depend upon several



8

parameters, including the jet momentum coefficient Cµ, the turbulent characteristics of

the jet and the Reynolds number [23]. The jet momentum coefficient Cµ is defined as,

qS

Vm

C

jet

=µ , and will be discussed in the next chapter.

Also as shown in Figure 1.4, at low momentum coefficients, especially when Cµ

is less than 0.01 [23], the tangential blowing will add some energy to the slow moving

flow near the surface. This will delay or eliminate the separation as in conventional

boundary layer control. If the momentum coefficients are high, the lift of the wings will

be increased significantly. The lift augmentation, which is defined as ∆CL / ∆Cµ, and as a

measure of the effectiveness of the blowing in generating lift, can exceed 80. This latter

effect of generating lift via blowing in the manner described above is referred to as

Circulation Control. The Circulation Control concept is superior to boundary layer

control. While boundary layer control aims to eliminate or postpone separation, CC aims

to increase the maximum CL value. This reduces the take-off and landing velocity by a

factor of 2 or so, thereby reducing the runway distance. This is achieved without the

penalty of noise associated with high-lift systems.

The physics of CC airfoils is highly complex and nonlinear. Wood [24], however,

suggests that there are two characteristics of a CC airfoil that determine its performance.

The first is the velocity difference between the jet and the external flow. The second

characteristic is the outer boundary layer momentum deficit. Specifically, Wood also

suggests that the ratio of the jet momentum to outer boundary layer momentum deficit

determines the lift increment due to blowing. Based on this theory and some experimental



9

evidence, Wood predicts that, for low Cµ values, CL varies linearly with Cµ, while for

higher Cµ values, the µ≈ CCL with low free-stream velocities. Eventually, however,

the lift will cease to increase with the momentum coefficient. This phenomenon is known

as jet stall, and is defined as the condition at which ∆CL / ∆Cµ = 0.

1.2.2 The Advanced Circulation Control Airfoil

The earlier designs of the CC airfoils used rounded trailing edges with large

radius to maximize the lift benefit. However, these designs also produced very high drag

[25]. In particular, the high drag associated with the blunt, large radius trailing edge can

be prohibitive under cruise conditions when Circulation Control is no longer necessary.

One way to reduce the drag is to reduce the trailing edge radius. This, however,

causes a loss of lift compared to a large radius configuration. It was also found that the

small radius CC airfoil with larger slot height could cause jet detachment and sudden lift

loss at higher momentum coefficients

[25]. Thus a compromise was needed. The

advanced CC airfoil, i.e., a circulation hinged flap [11, 25, 26], was developed to replace

the original rounded trailing edge CC airfoil.

The advanced CC airfoil developed by Englar is shown in Figure 1.5. The upper

surface of the CCW flap is a large-radius arc surface, but the low surface of the flap is

flat. The flap could be deflected from 0 degrees to 90 degrees. When an aircraft takes-off

or lands, the flap is deflected. Then this large radius on the upper surface produces a large

jet turning angle, leading to a high lift. When the aircraft is in cruise, the flap is retracted

and a conventional sharp trailing edge shape results, greatly reducing the drag. This kind



10

of flap does have some moving elements, which increase the weight and complexity over

an earlier CCW design shown in Figure 1.4. But overall, the hinged flap design still

maintains most of the Circulation Control high lift advantages, while greatly reducing the

drag in cruising condition associated with the rounded trailing edge CCW designs.

Figure 1.5: Dual Radius CCW Airfoil with LE Blowing [10]

The CCW flap is similar to a blown flap. However, it is important to note that

compared to a flat blowing surface in the case of a blown flap, the upper surface is highly

curved for the CCW flap. The curvature is either a curve built from a single radius, or

from multiple radii. A dual-radius configuration is shown in Figure 1.5. The size of the

CCW flap is also much smaller than the blown flap. The governing difference between

CCW flap and the blown flap is that, for CCW flap, there will be a continuously curved

surface downstream of the tangentially blowing jet, and the force modification and high

lift are mainly produced by changes to the jet blowing parameters. On the other hand, for

a blown flap, the surface downstream of the blowing jet is flat, and lift is produced with

the change of the angle of the sharp flap trailing edge or the jet angle relative to the

chord-line.



11

1.2.3 Applications and Benefits of the Circulation Control Wing

Circulation Control technology has many potential applications for both fixed and

rotary wing aircraft, as well as ground vehicles. All of these applications take advantage

of the high lift benefits and the ability of directly controlling the flow field associated

with the CC technology.

For fixed wing vehicles, the high lift generated by CC wings makes them ideal

candidates for short take-off and landing (STOL) and high lift aircraft. To find ways of

improving the aircraft operation from carriers, the Navy sponsored a full-scale flight test

program on an A-6/CCW STOL demonstrator in 1979 [27, 28]. The airfoil used was a

rounded trailing edge CC airfoil. Using only available bleed air from the engines, it could

achieve CL values that were 120% higher than a conventional Fowler flap, or a 140%

increase in the usable lift coefficient at take-off/approach angles of attack. The

researchers were also aware of the drag penalty, and improvements with use of smaller

cylinder trailing edges and hinged flaps have been recommended [25, 29].

For commercial aircraft, compared to a conventional high-lift system, the

advanced CCW flap system can give the same high lift in take-off/landing and small drag

in cruise, but greatly reduce the complexity and weight of the wing. The manufacturing

cost will also be significantly reduced. An experimental and computational study by

Englar et al [11, 12] was conducted to evaluate the effectiveness of applying this concept

to an Advanced Subsonic Transport. As shown in Figure 1.3, a typical wing such as that

found on a Boeing 737 has 15 moving parts. A CCW system, on the other hand, will have

a maximum of 3 components per wing even with leading edge blowing. Using only fan



12

bleed air, the CCW flap will give at least triple the usable lift at taking off and will reduce

the ground roll compared to the conventional high-lift system. Recently, experimental

evaluations were also conducted on the use of blown high-lift devices and control

surfaces on the High Speed Civil Transport (HSCT) aircraft [30]. These studies found

that the advanced pneumatic high-lift devices produced large lift increases as well as

significant drag reductions, and confirmed the effectiveness of combined pneumatic high-

lift devices and control surfaces on these HSCT aircraft.

The ability of controlling the lift directly without angle of attack change gives the

CC airfoil potential of being used on rotary wing aircraft as well. This concept allows the

use of higher harmonic control of helicopters, where cyclic lift variations are usually at

frequencies higher than one per revolution. Suppression of these high frequency

components can result in considerable reduction of rotor vibration, fatigue and noise. In

1979, a CC rotor flight demonstrator based on a Kaman H-2 helicopter was tested [31,

32]. Instead of using a conventional mechanical cyclic and collective blade pitch control

system, a pneumatic aerodynamic and control system was applied. It was found that the

CC rotor had the potential of eliminating the mechanical blade lift and control devices in

hover and forward flight, and also had the ability of achieving higher harmonic control. It

also suggested that the elimination of the angle of attack control could also result in

reducing the hub complexity, number of mechanical parts, size, and drag. However, due

to a control system phasing problem, the flight test envelope was limited.

Another application of the CCW technology is the X-wing stopped rotor aircraft.

In this design, a four-blade CC rotor would be used during vertical take-off and landing,



13

and the rotor assembly would be locked into a stationary position during forward flight,

and function as a fixed wing [33]. Since the wing area was relatively small, the

Circulation Control technology had been used to achieve high lift coefficient during both

the rotary wing and fixed wing modes. Because the rotor blades in such a design need to

be functional both in rotary as well as in fixed forward flight mode, they must be fore-

and-aft symmetrical. CC airfoils, with their rounded leading and trailing edges, are

ideally suited to this application. During the rotary wing mode, as mentioned above, the

cyclic variation of the lift coefficient may be controlled by variation of the jet momentum

coefficient, rather than pitching motion. This concept was tested full-scale in the NASA

Ames wind tunnel and successfully completed the transition from hover to forward flight.

Besides these applications in flying vehicles, a number of non-flying applications

have also been investigated, where the Circulation Control technology was used to

modify or control the flow field around moving objects. One investigation by Englar [34]

is to improve the performance, economics and safety of heavy vehicles (i.e. large

tractor/trailer trucks). There are many other potential applications for the Circulation

Control or Pneumatic Aerodynamic technology besides these mentioned above. The

reader is strongly referred to the Reference [10], which summarizes many of this effort

from beginning to the year 2000.



14

1.3 Previous Research Work

Circulation Control research based on the Coanda effect originates back in the

1930s [22]. Because of the great benefits of the Circulation Control technology, many

experimental and numerical studies have since then been done to investigate the

characteristics and performance of CC airfoils.

The early research work was done in England. Cheeseman et al [35] applied

blowing to helicopter rotors. Kind [36] gave a simplified calculation method for

Circulation Control by tangential blowing around a bluff trailing edge. After 1970, this

concept was pursued in the United States by Navy researchers. The David Taylor Naval

Ship Research and Development Center (DTNSRDC) became a major center for

Circulation Control research. Experiments by Williams and Howe [37], Englar [38, 39],

Abramson [40], Abramson and Rogers [41], and others examined the effect of a wide

range of parameters on Circulation Control airfoils, including geometric factors such as

the thickness, camber, angle of attack, and free-stream conditions such as Mach number.

For a summary of this research work for the years 1969 through 1983, the reader is

referred to Reference [42]. This work by Englar et al also provides a summary of CC-

related research conducted by other agencies outside the Navy.

In addition to these basic aerodynamic experiments, recently, many studies have

been focused on CCW applications for the rotary and fixed wing aircraft. Some of the

studies were mentioned in section 1.2.3, and Reference [10] gives a detailed description

of many such studies made until the year 2000.



15

Compared to so many aerodynamic studies, however, the acoustic studies for the

CCW are very limited. Salikuddin, Brown and Ahuja [15] experimentally examined the

changes in noise produced by an upper surface with and without blowing. Carpenter et al

[43] have experimentally investigated the noise emitted from supersonic jet flows over

axisymmetric Coanda surfaces. Howe [44] recently analytically studied the noise

generated by a hydrofoil with a Coanda wall jet Circulation Control device. Munro and

Ahuja [16, 17, 18, 19] compared the noise characteristics of a CC wing and a

conventional flap wing at the same lift setting, and studied the fluid dynamics and

aeroacoustics of a high aspect-ratio jet. It was found that a CC wing had a significant

acoustic advantage over a conventional wing for the same lift performance.

There have only been a limited number of computational studies of CCW

configurations. In the earliest studies, panel methods combined with boundary layer

analysis and wall jet models were used. Some good results were obtained by using a

potential flow solver developed by Dvorak et al [45, 46], but the solutions did not appear

to have the accuracy needed for CC airfoil designs.

Determining the performance of CC airfoils using analytical or numerical

methods has proven to be extremely difficult due to the viscous flow region that needs to

be modeled. The flow over a CC airfoil is greatly complicated by the rounded trailing

edge or Coanda surface, and the introduction of the jet blowing. There are strong

interactions between the jet region and the overall flow due to circulation coupling. An

accurate analysis of the flow field requires a procedure that accounts for this highly-

coupled nature of the viscous and inviscid flow regions. This could not be done by the



16

simple potential methods until the 1980s. Due to this highly-coupled, nonlinear viscous

behavior of the flow field, the Navier-Stokes equations present the best prospects of

modeling this problem. However, even the Navier-Stokes methods are challenged due to

the lack of accurate turbulence models for highly curved flows with strong adverse

pressure gradients.

Many numerical studies were conducted during the 1980s, which examined the

possibility of using Navier-Stokes equations to predict the characteristics of CC airfoils.

Berman [47] of DTNSRDC computed the flow over the aft 50% chord of a CC airfoil

using a MacCormack explicit solver with the Baldwin-Lomax turbulence model [48]. The

results showed trends consistent with the experiments. However, the magnitudes of the

computed pressure coefficients were not as large as those found experimentally. Pulliam

et al. [49] also employed an implicit formulation of the Navier-Stokes equations with the

Baldwin-Lomax turbulence model to compute the flow over CC airfoils. Their results

faithfully reproduced the experimental results of Abramson and Rogers [41] for the

higher blowing rates, although they also concluded that better turbulence models were

needed. Viegas et al [50] computed the flow field over the trailing edge of the CC airfoil

used in the experiment of Spaid and Keener [51], and a good agreement with the

measured pressure distribution was also obtained. Shrewsbury [52, 53, 54] of Lockheed

Martin used an implicit formulation of the compressible Reynolds-average Navier-Stokes

equations (RANS) with a modified form of the Baldwin-Lomax turbulence model. The

turbulence model included a correction by Bradshaw [55] to account for the curvature of

the Coanda surface. This method performed well and provided lift and pressure



17

distribution results in close agreement with the experimental data. Shrewsbury [53] also

concluded that better turbulence models were needed to more accurately calculate the

flow field characteristics around CC airfoils. These studies have demonstrated that the

Navier-Stokes equations can indeed provide good estimates of the lift, pressure

distribution, and pitch moments of CC airfoils at various flight conditions provided the

turbulence model is able to give a reasonable good estimate of the jet separation point

from the Coanda surface.

More recently, the solutions of Navier-Stokes equations have also been used to

predict the static and dynamic performance of CC airfoils. Shrewsbury [13, 14, 56]

conducted a study of an oscillating CC airfoil to determine the dynamic stall

characteristics. Williams and Franke [57] also developed a computational procedure

based on Navier-Stokes equations to predict the aerodynamic performance of a CC airfoil

for a range of jet blowing rates. The results were shown to be dependent on an empirical

curvature constant in the Baldwin-Lomax turbulence model to account for the curved

flow over the blunt trailing edge, and the development of accurate turbulence models for

CC airfoils was also recommended. Linton [58] computed the post jet-stall behavior of a

CC airfoil using a fully implicit Navier-Stokes code and the Baldwin-Lomax and κ -ε

turbulence models. Numerical solutions for the stalled and unstalled flow over a CC

airfoil were obtained, and it was found that the post-stall behavior of a CC airfoil was a

highly regular periodic oscillation. Liu et al [59] investigated the unsteady flow around a

CC airfoil with a Navier-Stokes method. The calculations included the flow around a CC

airfoil with a pulsating jet, the flow around an oscillating CC airfoil, and the flow around



18

an oscillating airfoil with pulsed jet. Wang and Sun [60] also studied the Circulation

Control with multi-slot blowing. It was found that at small and medium Cµ, the multi-slot

CC blowing could increase the lift of the airfoil and reduce the amount of the energy

expenditure, so that it could improve the aerodynamic performance of CC airfoils at

higher Mach number by avoiding the “compressibility stall”. All of above studies were

based on the traditional rounded trailing edge CC airfoils. For the advanced small hinge-

flap CC airfoil, Smith et al [61] calculated the pressure coefficient distribution over a

dual-radius CC airfoil with aft CCW flap at 900 and Krueger flap at 600. The agreement

with the experimental data was quite good.

Around 2000, thanks to great improvements in computer speed, more complicated

and accurate methods began to be used to numerically investigate the Circulation Control

or separation control phenomena. Slomski et al [62] investigated the influence of

turbulence models on the performance of CC airfoils. Instead of using the traditional

Baldwin-Lomax model, three advanced turbulence models were used: the standard κ -

ε model, the modified κ -ε model, and a full Reynolds stress model. It was found that for

small Cµ, the κ -ε and modified κ -ε turbulence models could predict the lift generated by

the CC airfoil reasonably well. However, at higher Cµ, only the Reynolds stress

turbulence model could capture the physics of the Circulation Control problem, allowing

a reasonable prediction of the lift. Large-eddy Simulation (LES) [63] and Direct

Numerical Simulation (DNS) [64] methods have also been reported in last two years,

primarily for the numerical investigation of boundary layer separation control.



19

1.4 Overview of the Present Work

The main objectives of the current study are to numerically investigate the

aerodynamic characteristics and benefits of Circulation Control Airfoils/Wings. The

present study is aimed at understanding the physical phenomena associated with the CC

concept, and at extending these 2-D studies to 3-D applications, and to pulsed jet

configurations. Specifically this work is aimed at answering the following questions,

which have not been fully numerically investigated to the knowledge of the author:

SCan pulsed jets be used to replace steady blowing to generate the same high lift

with relative lower mass flow rate? If so, what is the optimum value for jet

blowing coefficient Cµ, pulsed jet frequency, wave shape and duty cycle? What

are the benefits and drawbacks of pulsed jets relating to steady jets?

SIn many instances, it may be desirable to retrofit an existing wing with Circulation

Control. What are the aerodynamic benefits and drawbacks? For example, can the

vortices generated at flap edges be reduced in strength or altogether eliminated

using Circulation Control and why? Can the tip vortex of a wing be weaken or

eliminated by jet blowing over the rounded wing tip?

Of course these issues can, and should also be, studied in good quality wind

tunnels. CFD provides a powerful way of taking a first look at the problems before an

experiment is designed. Thus it is hoped that one of the benefits of this work will be a

comprehensive matrix of calculations that will assist the experimental aerodynamics

researchers in designing the experiments.



20

The CC airfoil configuration used in present study is the advanced hinge-flap CC

wing tested in References [9, 11, 12] and [16, 17, 18, 19]. This configuration was chosen

for the following reasons. This design is proven to be aerodynamically highly beneficial

during both take-off/landing and cruise conditions, and also less noisy than a

conventional high-lift system. An extensive set of experimental data for the two-

dimensional steady blowing is available for comparison and validation.

The rest of this thesis is organized as follows. In Chapter II, the mathematical and

numerical formulation of the governing equations is presented. It also includes the

mathematical representation of the turbulence models. The initial and boundary

conditions, which include the jet exit slot boundary condition, are addressed at the end of

Chapter II. The numerical results for the two-dimensional steady blowing are presented

in Chapter III. It includes a validation study for a NACA0012 wing, and comparisons

with the experimental measurements for steady jets. The effects of several parameters on

the static performance of the CC airfoil are also included. Simulations of the use of

pulsed jets on CCW configurations are given in Chapter IV. In particular, the wave form

and frequency effects on pulsed jet performance are discussed. Some preliminary results

for three-dimensional Circulation Control wing simulations are presented in Chapter V.

They include effects of tangential blowing on a wing-flap configuration to eliminate the

flap-edge vortex, and a spanwise blowing over a rounded wing tip to control the tip

vortex. Finally, the conclusions and recommendations for the further improvement of the

CC technology studies are given in Chapter VI.



21

CHAPTER II

MATHEMATICAL AND NUMERICAL FORMULATION

In order to analyze the flow field around the Circulation Control Airfoils/Wings,

solution of two-dimensional or three-dimensional Navier-Stokes equations is required.

Because of the complexity of wing/airfoil configurations and the strong viscous effects, it

is impossible to obtain an analytical solution of the Navier-Stokes equations for practical

configurations. Thus numerical techniques have to be used to solve those equations. In

this chapter, the governing equations and the numerical procedures employed in the

present study are documented. The formulation given below has been applied to many

fixed wing and rotorcraft studies by Sankar and his co-workers [65, 66, 67, 68].

In section 2.1, the governing equations for the three-dimensional unsteady

compressible flow are presented in Cartesian coordinates and Curvilinear coordinates

separately. The numerical discretization procedure and the alternating directing implicit

(ADI) scheme used to solve the governing equations are given in section 2.2. The

turbulence models used in the present study are discussed in section 2.3. Finally the

initial conditions, boundary conditions, and the special jet slot boundary condition

applied to the solver are described in section 2.4.



22

2.1 The Governing Equations

Navier-Stokes equations are a set of partial differential equations for the

conservation of mass, momentum, and energy. These may be derived by applying the

principle of classical mechanics and thermodynamics. These equations are based on

Newton’s hypothesis, that the normal and shear stresses are linear functions of the rates

of deformation, and that the thermodynamic pressure is equal to the negative of one-third

the sum of the normal stresses.

2.1.1 Governing Equations in Cartesian Coordinates

The divergence form of three-dimensional compressible Navier-Stokes equations

in Cartesian coordinates without external body forces or outside heat addition can be

written as [69]:

zyxzyxt ∂∂+

∂∂+

∂∂=

∂∂+

∂∂+

∂∂+

∂∂ TSRGFEq (2.1)

Here q is the flow vector or the unknown flow variables, which include the density and

velocities. E, F and G are the inviscid flux vectors and R, S and T are the viscous flux

vectors in the x, y and z directions, respectively.



23

The flow vector and the inviscid flux terms are:

ρ

ρ

ρ

ρ

=

tE

w

v

u

q

+

ρ

ρ

+ρ

ρ

=

u)pE(

uw

uv

pu

u

t

2

E

+

ρ

+ρ

ρ

ρ

=

v)pE(

vw

pv

uv

v

t

2F

+

+ρ

ρ

ρ

ρ

=

w)pE(

pw

uw

uw

w

t

2

G

(2.2)

Here, E t is the total energy, and it can be expressed as:

+++ρ= )wvu(

2

1TCE 222

vt (2.3)

In above equations, the density ρ , the velocity components (u, v, w) in the ( x, y, z)

directions and total energy E t are the unknown flow parameters. The pressure is related to

the total energy and velocities by the following equations:

RTp ρ= (2.4)

and:

++ρ−−γ = )wvu(

2

1E)1(p 222

t (2.5)



24

In Equation (2.5), γ is the specific heat ratio. Since the working fluid is air, a value of 1.4

is used.

The viscous terms in equation (2.1) are:

τ

τ

τ

=

5x

xz

xy

xx

E

0

R ,

τ

τ

τ

=

5y

yz

yy

yx

E

0

S ,

τ

τ

τ

=

5z

zz

zy

zx

E

0

T (2.6)

As stated earlier, the Newtonian fluid assumption has been made to link the stress

tensor with the pressure and velocity components [70]. Then the following relations can

be obtained:

3,2,1 j,ixu

xu

xu

quE

ij

k

k

i

j

j

iij

iij j5i

=δ∂∂λ+

∂∂+∂∂µ=τ

−τ=

(2.7)

where δ ij is the Kronecker delta function; Subscripts “1, 2, 3” represent the tensors in the

x, y, and z directions. The effect of fluid compressibility is expressed by the dilatation

term in conjunction with the second coefficient of viscosity λ . In the current study, the

fluid is assumed to be in a state of local thermodynamic equilibrium [71], i.e., Stoke’s

hypothesis [72] is used to relate the first and second viscosity coefficients in the above

equation (2.7). Thus,



25

µ−=λ3

2 (2.8)

The stress terms and heat transfer terms in equation (2.6) can now be written as follows:

∂

∂−

∂∂

−∂∂

µ=τ

∂∂

+∂∂

µ=τ

∂∂−

∂∂−

∂∂µ=τ

∂∂

+∂∂

µ=τ

∂∂

+∂∂

µ=τ

∂∂

−∂∂

−∂∂

µ=τ

y

v

x

u

z

w2

3

2

y

w

z

v

z

w

x

u

y

v2

3

2

x

w

z

u

x

v

y

u

z

w

y

v

x

u2

3

2

zz

yz

yy

xz

xy

xx

(2.9)

xxzxyxx5x qwvuE +τ+τ+τ=

yyzyyxy5y qwvuE +τ+τ+τ=

zzzyzxz5z qwvuE +τ+τ+τ=

(2.10)

Under local equilibrium conditions, Fourier’s law [73] is used to relate the heat transfer

rates q x , q y and q z with the temperature gradient:



26

z

Tk q

y

Tk q

x

Tk q

z

y

x

∂∂−=

∂∂

−=

∂∂

−=

(2.11)

The thermal conductivity, k , can be related to the molecular viscosity using the

kinetic theory of gases [74]:

Pr

C

Pr

Ck VP µγ

=µ

= (2.12)

where C p is the specific heat at constant pressure. For a calorically perfect gas, it is a

constant and defined as1

RCP −γ

γ = . Here R is the gas constant and γ is the specific heat

ratio, which is equal to 1.4 for air. Furthermore, Pr is the Prandtl number; and Pr = 0.72

for air.

The local speed of sound is given by:

( ) ( )

++−

ρ−γ γ =γ = 222t wvu

2

1E1RTa (2.13)

In numerical simulations, it is convenient if all quantities in the Navier-Stokes

equations are non-dimensionalized by some reference values. The advantage in doing

this is that the number of parameters in the flow reduces to a few, such as Mach number,



27

Reynolds number, and Prandtl number. Also, by non-dimensionalizing the equations, the

flow variables will be of the order of O(1).

The following reference values have been used in the present studies:

etemperaturFreestream,TT

itycosvisFreestream,

densityFreestream,

soundof speedFreestream,aV

airfoiltheof ChordL

ref

ref

ref

ref

ref

∞

∞

∞

∞

=

µ=µ

ρ=ρ

=

=

(2.14)

The non-dimensional flow variables are expressed as follows:

2

ref ref

t*

t

ref

*

2

ref ref

*

ref

*

ref

*

ref

*

ref

*

ref

*

ref ref

*

ref

*

ref

*

ref

*

V

EE

T

TT

V

pp

V

ww

V

vv

V

uu

V / L

tt

L

zz

L

yy

L

xx

ρ==

ρ=

ρρ

=ρ

µµ

=µ===

====

(2.15)

where the non-dimensional variables are denoted by an asterisk.

Substituting the non-dimensional variables in equation (2.15) into the equation

(2.1), an equation very similar to (2.1) is obtained, but there are two non-dimensional

coefficients that appear in front of the inviscid and viscous terms. These coefficients are

the Mach number and Reynolds number, which are defined as follows:

∞

∞∞

∞

∞∞ µ

ρ=

γ = ref

L

LVRe

RT

VM

ref (2.16)



28

For a detailed description and expression of the non-dimensional equations, the reader is

referred to Reference [69]. In the following discussions, all variables (ρ , u, v, p etc) are

non-dimensional. The asterisk has been dropped for convenience.

2.1.2 Governing Equations in Curvilinear Coordinates

To obtain solutions for the flow past arbitrary geometries and handle arbitrary

motions, a body-fitted coordinate system is desired so that the boundary surfaces in the

physical plane can be easily mapped onto planes or lines in the computational domain.

The compressible Navier-Stokes equations can be written in terms of a generalized non-

orthogonal curvilinear coordinate system ( ξ ,η , ζ ) using the generalized transformation

described in Appendix A:

t

)t,z,y,x(

)t,z,y,x(

)t,z,y,x(

=τ

ζ=ζ

η=η

ξ=ξ

(2.17)

Applying the transformation to the equation (2.1), the following non-dimensional

governing equations in the curvilinear coordinate system can be obtained.

)ˆˆˆ

(Re

Mˆˆˆˆ ςηξ

∞

ςηξτ ++=+++ TSRGFEq (2.18)

Here,



29

ρρρρ

=

e

w

v

u1

ˆJ

q (2.19)

J is the Jacobian of transformation, and it is given by:

)zxzx(y)zxzx(y)zxzx(y

1

ηξξηςξςςξηςηηςξ −+−+−=J (2.20)

The quantities GFE ˆ,ˆ,ˆ and TSR ˆ,ˆ,ˆ are related to their counterparts E, F, G, and R, S, T

as follows:

( )tzyx

1ˆ ξ+ξ+ξ+ξ= qGFEJ

E

( )tzyx1ˆ η+η+η+η= qGFEJ

F

( )tzyx

1ˆ ζ+ζ+ζ+ζ= qGFEJ

G

( )zyx

1ˆ ξ+ξ+ξ= TSRJ

R

( )zyx

1ˆ η+η+η= TSR

J

S

( )zyx

1ˆ ζ+ζ+ζ= TSRJ

T

(2.21)



30

In numerical simulations, the contravariant velocities U , V , and W are used as the

velocity components in the generalized coordinates, which are related to the original

velocities (u,v and w) as:

zyxt

zyx

wvu

)zw()yv()xu(U

ξ+ξ+ξ+ξ=

ξ−+ξ−+ξ−= τττ

zyxt

zyx

wvu

)zw()yv()xu(V

η+η+η+η=

η−+η−+η−= τττ

zyxt

zyx

wvu

)zw()yv()xu(W

ζ+ζ+ζ+ζ=

ζ−+ζ−+ζ−= τττ

(2.22)

where:

zyxt

zyxt

zyxt

zyx

zyx

zyx

ς−ς−ς−=ς

η−η−η−=η

ξ−ξ−ξ−=ξ

τττ

τττ

τττ

(2.23)

The contravariant velocity components U, V and W are in directions normal to the

constant ζηξ and, surfaces, respectively. The quantity ( xτ , yτ and zτ ) is the velocity of

any points on the “grid” in an initial frame. In the present work, the body is not in motion

and these velocities are zero.

The inviscid GFE ˆ,ˆ,ˆ and viscous TSR ˆ,ˆ,ˆ flux vectors in the transformed

coordinate system are:



31

( )

ξ−+

ξ+ρξ+ρξ+ρ

ρ

=

pUpe

pwU

pvU

puU

U

1

t

z

y

x

JE (2.24a)

( )

η−+µη+ρη+ρη+ρ

ρ

=

pVpe

pwV

pvV

puV

V

1ˆ

t

z

y

x

JF (2.24b)

( )

ζ−+ζ+ρζ+ρζ+ρ

ρ

=

pWpe

pwW

pvW

puW

W

1ˆ

t

z

y

x

JG (2.24c)

ξ+ξ+ξτξ+τξ+τξτξ+τξ+τξτξ+τξ+τξ

=

5zz5yy5xx

zzzyzyxzx

yzzyyyxyx

xzzxyyxxx

EEE

0

1ˆJ

R (2.24d)



32

η+η+ητη+τη+τητη+τη+τη

τη+τη+τη

=

5zz5yy5xx

zzzyzyxzx

yzzyyyxyx

xzzxyyxxx

EEE

0

1ˆJ

S (2.24e)

ζ+ζ+ζτζ+τζ+τζτζ+τζ+τζ

τζ+τζ+τζ

=

5zz5yy5xx

zzzyzyxzx

yzzyyyxyx

xzzxyyxxx

EEE

0

1ˆJ

T (2.24f)

The viscous flux terms with the shear stresses in the transformed coordinates are:

( )[ ]

( )[ ]

( )[ ]

)wwwvvv(

)wwwuuu(

)vvvuuu(

)vvv()uuu(www23

2

)www()uuu(vvv23

2

)www()vvv(uuu232

yyyzzzyz

xxxzzzxz

xxxyyyxy

yyyxxxzzzzz

zzzxxxyyyyy

zzzyyyxxxxx

ζ+η+ξ+ζ+η+ξµ=τ



ζ+η+ξ−ζ+η+ξ−ζ+η+ξµ=τ



ζηξζηξ

ζηξζηξ

ζηξζηξ

ζηξζηξζηξ

ζηξζηξζηξ

ζηξζηξζηξ

(2.25)

where the Stokes hypothesis for bulk viscosity has been used, as discussed earlier.



33

The auxiliary functions are:

( )

∂ζ∂

ζ+∂η∂

η+∂ξ∂

ξ−γ

µ+τ+τ+τ=

2

x

2

x

2

xxzxyxx5x

aaa

1PrwvuE

( )

∂ζ∂

ζ+∂η∂

η+∂ξ∂

ξ−γ

µ+τ+τ+τ=

2

y

2

y

2

yyzyyxy5y

aaa

1PrwvuE

( )

∂ζ∂

ζ+∂η∂

η+∂ξ

∂ξ

−γ µ

+τ+τ+τ=2

z

2

z

2

zzzyzxz5z

aaa

1PrwvuE

(2.26)

The quantities ξ x, ξ y, ξ z etc. are called the metrics of transformation, Pr is Prandtl

number, a is the speed of sound, and a2

= γ RT . γ is the specific heat ratio and a value of

1.4 has been given for air. Those values can be earlier computed on a body fitted grid

[69], as discussed later.

The time derivativeτ∂

∂in the transformed plane is related to the time derivative in

the physical planet∂∂ as follows:

ς∂∂

ς+η∂∂

η+ξ∂

∂ξ+

τ∂∂

=∂∂

ςηξttt

,,z,y,xt (2.27)

with ξ t , ηt and ζ t appropriately defined as shown in equation (2.23). If the body is not

moving, or the grid is not moving,τ∂

∂=

t∂∂

.



34

2.2 Numerical Procedure

The governing equations (2.1) can be solved analytically only for some very

simple cases. For most aerodynamic applications, the equations have to be solved

numerically. There are two major approaches to solve these equations numerically in

Computational Fluid Dynamics (CFD). One is the Finite Difference method, in which the

governing equations in the continuous domain are transformed into a computational

domain with uniform grid spacing and then discretized. The second is the Finite Volume

method, in which the conservation principles are applied to a fixed region in physical

space known as a control volume. The governing equations are thus represented in

integral forms for finite volume method, which are discretized directly in the physical

domain.

In the present work, a semi-implicit finite difference scheme based on the

Alternating Direction Implicit (ADI) [75, 76, 77] method was used. A brief description of

this finite difference scheme will be given in this section. For a detailed review of the

development of the finite difference scheme and the ADI method, the reader is referred to

the Reference [78].

2.2.1 Temporal Discretization

The unsteady compressible Navier-Stokes equations are a mixed set of

hyperbolic-parabolic equations. Since the governing equations are parabolic in time, they

may be solved with advancing in time using a stable, dissipative scheme. Because of the



35

time step restriction from stability considerations for the explicit method, an implicit

first-order accurate scheme is used:

)(Oˆ

ˆˆ

1n

n1n

τ∆+τ∂

∂

τ∆+=

++ q

qq (2.28)

Here, the superscript ‘n+1’ represents the new, or unknown time level, and the

superscript ‘n’ represents the previous, or known time level. A second order scheme

could also have been used, but the first-order scheme gives better stability properties. If

the time step is small enough to maintain the stability for Navier-Stokes equations, a

satisfactory temporal accuracy is still achieved.

The Navier-Stokes equation (2.18) then can be written in a semi-discrete form as:

n1n1n

)ˆˆˆ(Re

M)ˆˆˆ(

ˆTSRGFE

qζηξ

∞+

ζηξ

+

δ+δ+δ+δ+δ+δ−=τ∂

∂

(2.29)

where, for example, Eξδ is a numerical approximation to the derivative ξ∂∂E , and

standard fourth-order or second-order central differencing is used to calculate these

derivatives. The viscous terms are evaluated explicitly by using the old time level values,

and added to the right-hand side of the equation. The details of the spatial discretization

procedure will be discussed later.

Substitute equation (2.29) into equation (2.28), the following equation is obtained:

)ˆˆˆ(Re

M)ˆˆˆ(ˆˆ

nnn1n1n1nn1n

TSRGFEqq ζηξ∞+

ζ+

η+

ξ+

δ+δ+δτ∆+δ+δ+δτ∆−= (2.30)



36

Note that the inviscid terms are at the new time level ‘n+1’ (or treated implicitly),

while the viscous terms are explicitly computed using known information at the old time

level ‘n’. For this reason, this approach is a semi-implicit scheme.

2.2.2 Linearization of the Difference Equations

Because the flux vectors E , F , and G are nonlinear, the algebraic equations

shown in (2.30) for the unknown vector, 1n+q , are nonlinear. However, this non-linearity

may be removed, while maintain the temporal accuracy, by using a linearization

procedure. In the present study, following the method proposed by Beam and Warming

[79], those nonlinear flux vectors are linearized about the non-linear solution at an earlier

time level ‘n’ as follows:

)(O)ˆˆ]([ˆˆ

)(O)ˆˆ]([ˆˆ

)(O)ˆˆ]([ˆˆ

2n1nnn1n

2n1nnn1n

2n1nnn1n

τ∆+−+=

τ∆+−+=

τ∆+−+=

++

++

++

qqCGG

qqBFF

qqAEE

(2.31)

where [A], [B] and [C] are the Jacobian matrices:

q

GC

q

FB

q

EA

ˆ

ˆ][and

ˆ

ˆ][

ˆ

ˆ][

∂∂

=∂∂

=∂∂

= (2.32)

For the Euler equations, these 5*5 matrices can be evaluated analytically and are

given by Pulliam [80]. The viscous terms are modeled explicitly, and no linearization is

needed.



37

The detailed forms of those matrices are shown below:

σθ+ξθσ−ξθσ−ξθσ−ξ−φθσξ−γ ξ−Θσξ−ξσξ−ξθ−φξσξσξ−ξ−γ ξ−Θσξ−ξθ−φξ

σξσξ−ξσξ−ξ−γ ξ−Θθ−φξξξξξ

=

tzyx

2

zzzyzx

2

z

yyzyyx2

y

xxzxyx

2

x

zyxt

wEvEuE)E(

w)2(vwuww

wvv)2(uvv

wuvuu)2(u

0

][A (2.33)

where:

2

t

zyx

2222

eE

1

wvu

2 / )wvu)(1(

φ−ργ

=

θ+ξ=Θ

−γ =σ

ξ+ξ+ξ=θ

++−γ =φ

(2.34)

The matrices [B] and [C] may be similarly conducted if η and ζ are used in the above

equations, respectively, instead of ξ .

After substituting the linearized flux vectors equations (2.31) into (2.30), the

following systems of linear equations for 1n+q can be achieved:

nn1nnnn )ˆˆ)]((+[ RHSqqCBAI =−δ+δ+δτ∆ +ζηξ (2.35)

The RHS in above equation is the steady state portion of the governing equations,

and is known as the “residual”. For Navier-Stokes calculations, the residual is given by:

)ˆˆˆ(Re

M)ˆˆˆ( nnnnnnn TSRGFERHS ζηξ

∞ζηξ δ+δ+δτ∆+δ+δ+δτ∆−= (2.36)



38

In steady flow problems, the residual should go to zero asymptotically after a sufficiently

long period of time, starting from an arbitrary initial value for the flow variables.

By defining n1n1nqqq −=∆ ++ in equation (2.35), the so called “delta form”

equation is obtained:

n1nnnn ˆ)](+[ RHSqCBAI =∆δ+δ+δτ∆ +ζηξ (2.37)

There are several advantages using the delta form of the equations. First, the delta

form is convenient and makes the equations analytically simpler. Secondly, the delta

form algorithm is easier to code and modify, and also provides a steady-state solution that

is independent of the time step. The boundary conditions are also more easily applied in

the delta form. However, the delta form equations are slightly less stability than the non-

delta form [78]. But the instabilities of both forms are very weak, and can be easily

controlled by taking somewhat smaller time steps.

2.2.3 Approximate Factorization Procedure

The matrix form on the left hand side of equation (2.35) sparsely links a cell (or

node) to its six neighbors, yet the dimension of the matrix is large. If classical finite

difference methods are used to discretize this matrix, a seven-diagonal equation will be

obtained. A direct inversion of this system is so costly that it would negate the advantages

of an implicit scheme. To simplify the inversion of this system, without reducing the

accuracy of the method, an approximate factorization scheme by Beam and Warming

[79] was used in the present studies:

)(o)]()][()][([)](+[ 2τ∆+δτ∆+δτ∆+δτ∆+≅δ+δ+δτ∆ ζηξζηξ CIBIAICBAI (2.38)



39

This factorization method has not reduced the temporal accuracy of the method,

but it has changed a large, sparse matrix into the product of three easily inverted

tridiagonal matrices. Thus, the computational efficiency is greatly increased.

After the approximate aactorization (AF) procedure, the system equations for

1nq +∆ are solved by three successive block-tridiagonal inversions:

21nn

12n

n1n

ˆˆ)]([

ˆˆ)]([

ˆ)]([

qqCI

qqBI

RHSqAI

∆=∆δτ∆+

∆=∆δτ∆+

=∆δτ∆+

+ζ

η

ξ

(2.39)

2.2.4 Spatial Discretization of the Inviscid Terms

The right hand side of equation (2.37), nRHS , contains inviscid derivative terms

such as Eξδ , where E includes the flux of mass, momentum and energy, and viscous

terms such as Rξδ . To numerically model those derivatives, a spatial discretization is

required.

For those inviscid terms, a standard second or fourth order central differencing is

used.

Second Order:ξ∆

−=δ −+

ξ2

ˆˆˆ k , j,1ik , j,1i EEE (2.40a)

and

Fourth Order:ξ∆

+−+−=δ −−++

ξ12

ˆˆ8ˆ8ˆˆ k , j,2ik , j,1ik , j,1ik , j,2i EEEEE (2.40b)



40

It is possible to obtain fourth-order spatial accuracy without increasing the coding

work too much, while preserving the block-tridiagonal nature of the system.

In order to further save the computation time, in the present study, just two

directions, streamwise and normal directions, are treated implicitly, while the spanwise

term is treated semi-implicitly. That is, the values obtained from the “n” and “n+1” time

levels in η direction are used in the right hand side of the equation (2.37), instead of the

left hand side. Thus, equation (2.37) yields:

1n,n1nnn ˆ)]][+[++

ζξ =∆τδ∆+τδ∆ RHSqCIAI (2.41)

and the right hand side becomes:

)ˆˆˆ(Re

M)ˆˆˆ( 1n,n1n,n1n,nn1n,nn1n,n +

ζ+

η+

ξ∞

ζ+

ηξ+ δ+δ+δτ∆+δ+δ+δτ∆−= TSRGFERHS

(2.42)

where the term, 1n,nˆ +ηδ F , means using the latest value available (in the new time level or

the old time level) to evaluate the flux term. This type of semi-implicit difference method

was first used by Rizk and Chausee [81] with the Beam and Warming algorithm. This

algorithm is unconditionally stable from the stability analysis. However, due to the non-

linearity in η direction, this scheme is best suitable for the geometries and grids in which

the spacing is much larger in one direction (η direction here) than others. For fixed wings

and rotor blades, in spanwise direction, the spacing of the grid is generally much larger

than it in the streamwise and normal directions. Thus the semi-implicit scheme works

very well for these applications.



41

Treatment of the spanwise (or η-derivative) in this scheme leads to the following

equations that require just two tridiagonal matrix inversion:

11nn

1n,n1n

ˆˆ)]([

ˆ)]([

qqCI

RHSqAI

∆=∆δτ∆+

=∆δτ∆++

ζ

+ξ (2.43)

Notice that in equation (2.43), the inversion directions have been uncoupled,

hence the name “Alternating Direction”. In general, the two inversions are called as the

ξ-sweep/i-sweep and ζ-sweep/k-sweep, respectively. These inversions are done at each

fixed span station. The calculations are done at one η-plane at a time, sweeping from the

root to beyond the tip in the span direction. The marching direction is reversed after each

iteration to avoid any dependence the solution may have on the sweep direction.

2.2.5 Spatial Discretization of the Viscous Terms

As mentioned before, the viscous terms of the Navier-Stokes equations are

evaluated explicitly and added to the right hand side of the equation (2.37). This approach

also allows very easy modeling of the Euler as well as Navier-Stokes equations, and

saves a lot of computer time. It has been found by Tannehill et at [82] that, for high

Reynolds numbers, an artificial explicit treatment of the viscous terms is stable provided

a suitable low pass filter is used which filters out the high spatial frequency noise in the

solution at each time step, before moving onto the next step.



42

Unlike the inviscid derivative terms, the derivatives of the viscous terms are

differenced about the half points. A typical term such as Rξδ is written as

( ) ξ∆−−+

2 / 1i2 / 1iˆˆ RR . Because the R itself contains the derivatives of the velocities, such

asx

u

∂∂

, the use of half points differencing limits the second-order differences to three

points in each coordinate direction, while second-order spatial accuracy is maintained.

For example,x

u

∂∂

can be expressed in the transformed coordinates as:

xxxuuu

xu ζ

ζ∂∂+η

η∂∂+ξ

ξ∂∂=

∂∂ (2.44)

At half points (i+1/2, j, k), the standard central differences are used:

ξ∆−

=ξ∂

∂ +

+

i1i

k , j,2 / 1i

uuu (2.45)

2.2.6 Implementation of Low Pass Filters

The use of central difference equations in the above numerical procedure can lead

to an odd-even decoupling which manifests itself as high frequency saw-tooth like waves.

These waves are non-physical, and must be filtered out with the use a “low-pass filter”

before they grow and contaminate the solution. In some literatures, these low pass filters,

which dissipate the energy contained in high frequency waves, are also called artificial

dissipation terms.



43

Artificial dissipation was first used by Von Neumann and Richtmyer [83] in 1950.

A second-order low pass filter was used for the unsteady 1-D flow equations to capture

the shock. By second order, it is meant that these filter terms are proportional to O(∆2),

where ∆ is the grid spacing. Since then, artificial dissipation has been successfully used

by many researchers for the numerical solution to virtually all types of flow problems,

such as, Lax and Wendoff [84], Lapdius [85], Lindmuth and Killeen [86], and McDonald

and Briley [87]. For a detailed material review of related studies, the reader is also

referred to Reference [78].

Based on Beam and Warming [79] and Steger [88] ‘s work, a set of fourth-order

low-pass filter terms has been added explicitly to the right hand side of the governing

equations. The magnitude of these terms is of order O(∆4), and will drop to zero as the

grid is refined and as the grid space goes to zero. Also, second-order low-pass filter terms

have been added implicitly to the left hand side.

After adding the implicit and explicit low-pass filter terms, the above block-

tridiagonal equation (2.43) becomes:

11n

,II

n

EE

1n,n1

,II

n

ˆˆ]D)C([

Dˆ]D)([

qqI

RHSqAI

∆=∆τε∆+δτ∆+

τε∆−=∆τε∆+δτ∆+

+ζζ

+ξξ

(2.46)

where DE is the fourth-order explicit filter term, and ξ,ID and ζ,ID are implicit filter

terms in the ξ and ζ directions, respectively. The coefficients Iε and Eε are user-input to

control the amount of filtering. Excessive filtering can filter out physical meaningful

information such as vorticity, tip vortex etc. Thus, these coefficients must be kept small.



44

In present study, a non-linear filter with eigenvalue scaling was used. Thus the

fourth-order explicit low pass filter term is defined as:

ςηξ ++= ,E,E,EE DDDD (2.47)

where

])[(])[(D n

j,i

)4(

j,i

1

j,i j,i,

n

j,i

)2(

j,i

1

j,i j,i,

1

j,1i j,1i,,E qJqJJ ξξ−

ξξξξ−

ξ−++ξξξ ∆∇ελ∆∇+∆ελ+λ∇−= (2.48)

In above equation, ξ∇ represents a forward difference in ξ direction, and ξ∆ denotes a

backward difference in ξ direction. The λξ is the largest eigenvalue of the flux matrix A,

which is defined as follows:

2 / 12

z

2

y

2

x )(aU ξ+ξ+ξ+=λξ (2.49)

and the coefficients )2(

j,iε and )4(

j,iε are defined as follows:

j,1i j,i j,1i

j,1i j,i j,1i

j,i

)2( j,i4

)4( j,i

j,1i j,i j,1i2

)2(

j,i

pp2p

pp2pand

)k ,0max(

),,max(k

−+

−+

+−

++

+−=σ

ε−=ε

σσσ=ε

(2.50)

The typical values of the constant k 2 and k 4 are 0.25 and 0.01. In the η and ζ directions,

the explicit filter terms DE,η and DE,ζ are defined in a similar manner, with the scaling

factors λη and λζ, respectively.



45

The second-order, implicit filter terms in equation (2.46) are defined as:

)(D j,i j,i,

1

j,i,I JJ ξξξ−

ξ ∆∇λ−=

)(D j,i j,i,1 j,i,I JJ ςςς−ς ∆∇λ−=

(2.51)

For a detailed description of the non-linear low pass filter terms, the reader is

referred to the Reference [56].

2.3 Turbulence Models

In many practical CCW applications, the Reynolds number based on the airfoil

chord is usually very high, and the flow region is turbulent. Although the Navier-Stokes

equations can be used to solve turbulent flows from first principles, extremely small grid

sizes are required to accurately simulate the instantaneous flow quantities and capture the

smallest scale eddies. Thus solving the turbulent flow behavior by a direct numerical

simulation (DNS) requires very large computer resources [89]. To reduce the

computational time, the RANS (Reynolds Average Navier-Stokes System of equations)

are employed in this work. These equations are derived by decomposing the flow

variables in the conservation equations into time-mean and fluctuating components, and

then time averaging the entire Navier-Stokes equations.

The time-averaged Navier-Stokes equations lead to the Reynolds stresses ( vu ′′ ,

wu ′′ , wv ′′ , 2u′ , 2v′ , 2w′ ), which can not be solved directly from the equations,

and must be modeled.



46

To model these Reynolds Stresses, based on the theory that the stresses are

proportional to the strain, a simple algebraic model is used. For example, the Reynolds

Stress vu ′′ can be expressed as:

∂∂

+∂∂

ρµ

=′′−x

v

y

uvu T

(2.52)

where vu ′′ is the time-averaged values of the product v,u ′′ , and v,u ′′ are the

instantaneous velocity fluctuations about the mean velocities of u and v, respectively. The

term µ T / ρ is the turbulent viscosity coefficient, and is also called as the eddy viscosity.

In the present RANS equations based modeling of the turbulent flow, (µ + µ T ) is

used in the dimensional form of Navier-Stokes equations instead of µ . Also in the energy

equation, µ /Pr is replaced by (µ /Pr + µ T /Pr T ), where the turbulent Prandtl number, Pr T , is

given as 0.91, and Pr is 0.72 for air. Since µ T is an unknown parameter and depends on

the turbulent flow field, a turbulence model is needed to evaluate the value of the eddy

viscosityµ T .

There are many turbulence models used in CFD to simulate the turbulent flow

[90]. However, most of them are just good under some specific flow situations. A proper

choice of turbulence models is important and can have a large effect on the accuracy of

the simulations. The turbulence models used in this work are the Baldwin-Lomax Model

(zero-equation model) [48], and the Spalart-Allmaras Model (one-equation model) [91].



47

2.3.1 Baldwin-Lomax Turbulence Model

The Baldwin-Lomax turbulence model is a two-layer algebraic model. It does not

require solving any transport equation. Thus it is called a zero-equation model.

In Baldwin-Lomax model, the eddy viscosity is treated differently in the “inner”

and “outer” layers. In the inner layer close to the wall, the eddy viscosity µT is given by:

( ) ωρ=µ=µ 2

minnerTT l for d<dc (2.53)

where d is the distance from the surface of the body, and d c is the value of d for which

( ) ( )outerTinnerT µ=µ . The quantity, lm, is the Prandtl mixing length, which is the product of

the distance from the wall and Van Driest damping factor. The expression for lm is:

µ

τρ−−κ =

w

wwm

26

zexp1zl (2.54)

Here, k is the Von Karman constant, set to 0.4. The variable z represents the physical

distance from the nearest wall. The subscript ‘w’ refers to conditions at the wall, and τ w is

the shear stress at the wall.

In equation (2.53), the magnitude of the local mean vorticity ω is defined as:

222

y

u

x

v

x

w

z

u

z

v

y

w

∂∂

−∂∂

+

∂∂

−∂∂

+

∂∂

−∂∂

=ω (2.55)

In the outer layer, the eddy viscosity is computed with the following equation:



48

( ) klebwakecpcouterlayerT FFCK ρ=µ (2.56)

where:

= max

2

dif max

maxmaxwake F

Uz25.0

,FzminF

µ

τρ−−ω= )

26

zexp(1z)z(F

w

ww

min

222

max

222

dif wvuwvuU ++−++=

6

max

kleb

z

z3.05.51

1F

+

=

(2.57)

In equation (2.57), the constant K c = 0.0168 is the Clauser’s constant, and C cp =

1.6 is an empirical constant. The Klebanoff intermittency correction, F kleb, and the

function F wake are based on a formulation given by Cebeci [92]. The quantity zmax is the

distance from the wall where F(z) reaches the maximum value of F max.

In this turbulence model, the distribution of vorticity has been used to determine

length scales. So, the necessity of finding the boundary-layer thickness used in models

such as the Cebeci-Smith model is removed. It is seen that many empirical constants are

used in this model. These constants were obtained by the original developers based on

simple benchmark calibrations. Thus there is a limitation of applying this model to real

configurations. This is common to all turbulence models. In the present work, the

constants from the original work by Baldwin and Lomax were used without

modifications.



49

2.3.2 Spalart-Allmaras Turbulence Model

In the Spalart-Allmaras Turbulence Model, a partial differential equation is solved

that models the production, dissipation, diffusion, and transport of an eddy-viscosity like

quantity at each time step. Thus this is a one-equation model.

The turbulent eddy viscosity µ t is equal to ρν t , and ν t is given by:

ν ν

=χ+χχ

−= ν= ν~

,c

1f ,f ~3

1v

3

3

1v1vt (2.58)

where ν is the molecular viscosity. The working variable, ν~ , is governed by the

transport equation.

2

1t

2

2t2

1bw1w

2

2b2t1b

Uf d

~f

cf c

])~(c~)~.(([1~S

~)f 1(c

Dt

~D

∆+

ν

κ −

+ ν∇+ ν∇ ν+ ν∇σ

+ ν−= ν

(2.59)

Here,

,f d

~SS

~2v22κ

ν+= (2.60)

Also,

1v

2vf 1

1f χ+χ

−= (2.61)

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



50

The function f w in the destruction term is given by the following expression:

61

6

3w

6

6

3ww

cg

c1gf

+

+= (2.62)

where

)rr(crg 6

2w −+=

22dS~

~r

κ ν

≡

(2.63)

For large values of r , f w asymptotically reaches a constant value; therefore, large

values of r can be truncated to 10 or so.

In simple zero equation models, the transition region is abruptly modeled as a

single line or plane. Upstream of this line, the flow is laminar, and the eddy viscosity is

only computed downstream. To better represent the transition from the laminar flow to

turbulent conditions, the Spalart-Allmaras model has a set of terms to provide control

over the laminar regions of the shear layers. The first of these terms is the f t2 function,

which goes to unity upstream of the transition point.

)cexp(cf 2

14132t λ−= (2.64)

A trip function f t1 is obtained from the following equation:

[ ]

+

∆ω

−= 2

t

2

t

2

2

2

t2tt1t1t dgd

Ucexpgcf (2.65)

where )z

U,1.0min(g

t

t ∆ω∆= and t

ω is the wall vorticity at the trip point and d t is the



51

distance from the field point to the trip point, a user specified transition location, and

U ∆ is the difference between the velocity at the field point and that at the trip point. Use

of the trip function allows the eddy viscosity to vary gradually in the transition region.

However, the user still needs to specify the transition location, or compute it using a

criterion, such as Michal’s [93] or Eppler’s [94] transition model.

The wall boundary condition is ν~ = 0. In the free-stream and outer boundary ν~ =

10 ν , and this value is also used as the initial conditions.

The constants used in this model were given by Spalart et al, based on many

successful numerical tests [91]. These constant values used in our work are:

cb1 = 0.1335, cb2 = 0.622, σ =2 /3, κ = 0.41,σ

++

κ =

)c1(cc 2b

2

1b1w

cw2 = 0.3, cw3 = 2, cv1 = 7.1, ct1 = 1, ct2 = 2, ct3 = 1.1 , ct4 = 2

These are the constants in the original work of Spalart and Allmaras, and no attempt was

made to change these constants for the present application.

2.4 Initial and Boundary Conditions

Because the governing equations (2.1) are parabolic with respect to time and

elliptic in space, initial and boundary conditions are required to solve these equations. In

general, the initial flow conditions are set to free-stream values inside the flow field,

which is enough to get the final convergence solution with the time marching scheme.

The boundary conditions must be carefully specified to obtain meaningful solutions, and



52

their implementation is usually based on physics. For instance, “non-slip” conditions are

appropriate for the viscous surface, and “slip” conditions may be used in an “inviscid”

simulation. For the CCW simulations, jet slot exit boundary condition must be specified

to simulate the jet flow effects.

2.4.1 Initial Conditions

Because the numerical scheme for equation (2.1) uses a time-marching technique,

the solution of the equations at new time “n+1” level depends upon the values at the old

time “n” level, and the boundary conditions. Thus a meaningful initial condition must be

specified before the calculation starts.

In the present study, at the start of the calculation, the airfoil or wing is

impulsively started from rest. The flow properties everywhere in the system are assumed

to be uniform. Thus, the free-stream properties are specified as the initial conditions

everywhere.

2.4.2 Outer Boundary Conditions

The outer boundary is usually placed far from the airfoil surface, at least six

chords away. One common boundary condition is to assume that the outer boundary is a

permeable surface, where instability waves emitted from the body are free to pass and are

not reflected back. In this study, a non-reflecting boundary condition is used at the outer

boundary as shown in Figure 2.1.



53

Figure 2.1: The Outer Boundary Conditions for Sample C Grid

The boundary conditions must allow the correct number of characteristic waves to

leave, since each characteristic wave that leaves the computational domain corresponds to

one piece of the physical information such as isentropic/acoustic waves, entropy and

vorticity, etc, leaving the domain. The number of the waves (acoustic, entropy, vortical)

depends on the flow conditions on the outer boundary. To satisfy the non-reflecting

boundary conditions, one quantity should be extrapolated from the information inside for

every wave that leaves the domain.

For example, at the subsonic-inflow boundary (upstream), one characteristic

should be allowed to leave. Thus density is extrapolated from the interior while four other

quantities (ρu, ρv, ρw, and Et), are fixed to the free-stream values. However, at the

subsonic-outflow boundary (downstream), four characteristics should leave, so the four

quantities (ρ,ρu, ρv, and ρw) are extrapolated from the interior, while the pressure, p, is

fixed to free-stream value. Many researchers have also used Riemann equations to

specify these boundary conditions.

nu1

a2+

−γ leaves

nu1

a2

−−γ enters

No vorticity or entropyenters from upstream since

the boundary faces uniform

flow.

Vorticity, Entropy, Acoustic

properties (Riemann invariant

nu1

a2+

−γ ) are allowed to

leave the domain

Downstream pressure field caninfluence upstream components.Riemann invariant

nu1

a2 −−γ

is allowed to

enter in.



54

2.4.3 Solid Surface Conditions

On the airfoil surface, the boundary conditions must be properly specified for

accurate solutions. In inviscid flows, in the absence of transpiration, the flow must be

tangent to the airfoil surface:

0b =• nV

(2.66)

where n is the unit vector normal to surface and bV

is the velocity vector.

In viscous flows, the “no-slip” conditions is applied, which is state that all

components of the velocity with respective to the airfoil surface are zero at the surface:

0b =V

(2.67)

The density at the airfoil surface is extrapolated from the interior using the

following expression:

3 / )4(or0n3i2i1i ρ−ρ=ρ≈∂

ρ∂ (2.68)

where, “i1” represents the point on the surface, “i2” is the point next to the surface, and

“i3” is the point next to the point “i2” in normal direction.

Away from jet slots, the pressure at the surface is also determined from the

specification that the pressure gradient at the surface be zero. That is:

0n

P=

∂∂

(2.69)



55

The numerical expression of this is:

3 / )PP4(P 3i2i1i −= (2.70)

Thus, for a viscous flow without jet blowing, the boundary conditions shown in

Figure 2.2 are applied.

Figure 2.2: The Solid Surface Boundary Conditions for Viscous Flow

2.4.4 Boundary Conditions at the Cuts in the C Grid

When C-type grids are used, there will be a branch cut across the wake region to

maintain a simply connected region. Since physically the flow variables are continuous

across this cut, the properties on this cut are specified as averages of the variables one

point above and one point below the cut line. The grid should contain sufficient

resolution in this region to avoid the errors introduced by this condition.

u = v = w = 0; No slip

0n

P =∂∂

(Simplification of normal

momentum equation)

0n

T

=∂

∂

Adiabatic wall



56

Figure 2.3: The Wake-cut Boundary Conditions for C Grid

As shown in the Figure 2.3, at the wake cut, the points B and C are at the same

physic location that happens to fall on both sides of the cut. Thus CB qq = . Continuity of

properties is ensured by setting )qq(2

1qq DACB +== .

2.4.5 Jet Slot Exit Conditions with Given C

In most Circulation Control Wing studies, the driving parameter is the momentum

coefficient, C µ , defined as follows.

SV2

1

UmC

2

jet

∞∞

µ

ρ=

(2.71)

Here, the jet mass flow rate is given by:

jet jet jet AUm ρ= (2.72)

where A jet is the area of the jet slot, and S is the area of the whole wing section. In 2-D

simulations, A jet is the height of the jet slot and S is the chord of the CC airfoil.



57

In the present study, the following boundary conditions are specified at the slot

exit: the total temperature of the jet T 0, which is approximately equal to the total

temperature of free-stream, the momentum coefficient C µ as a function of time, and the

flow angle at the exit. In this simulation, the jet velocity direction is normal to the jet slot

exit and tangential to the surface. Since the jets are nearly always under-expanded, the jet

slot exit location will be assumed as the minimum area of the nozzle, i.e., the throat. The

physics of the jet slot boundary conditions are shown in Figure 2.4.

Figure 2.4: The Jet Slot Boundary Conditions

For subsonic jets, one characteristic can propagate upwind into the slot. Thus the

pressure at the jet exit is extrapolated from the outside values using the same constraints

as equation (2.70). Then the static pressure at the jet slot exit can be obtained as:

3 / )PP4(PP 3i2i1i jet −== (2.73)

Flow angularity depends on slot geometries

P0 = Total pressure depends on upstream conditions

T0 = Total temperature also depends on

upstream conditions

These are

specified.

In subsonic jets, P must be continuous.

BCA P

2

PP=

+

In supersonic jets, P

should be specified.



58

From equation (2.71), the momentum coefficient can also be expressed as:

SV2

1

AUC

2

jet

2

jet jet

∞∞

µ

ρ

ρ= (2.74)

From the ideal gas law and the equation of state, the following relations can be

obtained:

)TT(R1

2U jet jet,0

2

jet −−γ γ

= and jet

jet

jetRT

P=ρ (2.75)

Substituting equation (2.75) into (2.74), another expression for C µ with just one unknown

parameter can be obtained:

jet

jet jet,0 jet

2

jet

T

)TT(P

SV2

1

A

1

2C

−

ρ−γ γ

=∞∞

µ (2.76)

The only unknown variable is T jet , which can be easily solved from equation (2.76).

After the T jet is calculated, the other jet flow variables, such as U jet and ρ jet , can be

obtained from equation (2.75). These parameters are also non-dimensionalized by

corresponding reference values before used in the solver as the boundary conditions.

For supersonic jets, no information can be propagated upstream into the slot, thus

the extrapolation of jet exit pressure from the outside points is not correct. Because the jet

slot is assumed to be the throat of the nozzle, the local Mach number at the jet slot should

be unity. And the jet velocity at the exit should be equal to the local speed of sound.



59

From the isentropic relations for the total temperature and jet exit temperature:

2

jet

jet

jet,0M

2

11

T

T −γ += (2.77)

where M jet = 1, and the T jet can be easily solved as the T 0,jet is known.

After the T jet is obtained, other jet flow quantities could be determined for the

supersonic flow from the equations (2.74) and (2.75).

2.4.6 Jet Slot Exit Conditions with Given Total Jet Pressure

In experiments, C µ could not be directly measured from the wind tunnel. Instead,

it is the ratio of the jet total pressure to free-stream static pressure,∞P

P jet,0, and the jet total

temperature to free-stream temperature,∞T

T jet,0, that are specified as the blowing

conditions. Then the momentum coefficient is calculated from the measured data.

Again, the momentum coefficient C µ is defined as:

qC

VmC a

=µ (2.78)

where m is the mass flow rate of the jets defined as equation (2.72), q is the free stream

dynamic pressure, which is equal to2V

2

1∞∞ρ , and V a is the jet slot velocity obtained

assuming that the flow was expanded to the free-stream pressure.



60

Then the local Mach number at the jet exit slot is determined from the isentropic

relationship:

− −γ ==γ −γ

1PP

12MM

1

jet

jet,01i jet (2.79)

If M jet is less than 1.0, which indicates a subsonic jet, then the local density can be

obtained from P jet by the following relationship:

∞

∞∞

∞ ρ−γ +

−γ +

=ρ

=ρ

P

)M2

11(

)M2

11(

T

TPP

2

jet

2

,0

jet,0

1i

1i

jet

jet (2.80)

and the local speed of sound:

1i

1i

1i jet

Paa

ργ == (2.81)

with the local values of the Mach number, speed of sound and pressure, using the

equation of state with the geometric considerations, the other flow properties u, v, w, and

total energy can be easily obtained.

If M jet is great than 1.0, which indicates a supersonic jet, then the Mach number is

constrained to 1.0 as mentioned above, and the local pressure is calculated from the

following expression instead of equation (2.73).

1

jet,01i jet

2

1PPP

−γ γ

−

+γ ==

(2.82)

The local density, speed of sound and velocities (u, v, w) etc can subsequently be

determined using the same method as above.



61

CHAPTER III

TWO DIMENSIONAL STEADY BLOWING RESULTS

In the following studies, an unsteady three-dimensional compressible Navier-

Stokes solver based on the numerical scheme and boundary conditions described in

Chapter II is being used. The solver, called GT-CCW3D, can model the flow field over

isolated wing-alone configurations with or without Circulation Control jets. Both 3-D

finite wings and 2-D airfoils may be simulated with the same solver, and both leading

edge blowing and trailing edge blowing can be simulated.

In this chapter, the results of two-dimensional unblown and steady blowing cases

are presented. First, this code is validated with a rectangular wing with NACA0012

airfoil sections, and the results are compared with the experimental measurement. Next,

the flow field over the CC airfoil with steady blowing is simulated and compared with the

unblown case. After validation of the analysis through a comparison of the lift

coefficients at different momentum coefficients with the experimental data, results are

presented on the effects of control parameters such as the momentum coefficient, the total

pressure, the free-stream velocity, and jet slot heights, etc, on the performance of the CC

airfoil. Finally, a series of studies, comparing the CC airfoil to the conventional high-lift

system, and the leading edge blowing, are presented.



62

3.1 Code Validations with a NACA 0012 Wing

Prior its use to model CCW configurations, the Navier-Stokes solver is validated

by modeling the viscous subsonic flow over a small aspect-ratio wing made of NACA

0012 airfoil sections. The wing aspect ratio is 5 and the angle of attack is 8 degrees. The

free-stream Mach number is 0.12, and the Reynolds number based on wing chord is 1.5

million. Measured surface pressure data and the lift coefficient distribution along span for

this wing at these conditions have been documented by Bragg and Spring [95].

Figure 3.1 shows the computed and measured surface pressure distributions at

four spanwise stations (34%, 50%, 66% and 85% SPAN) on a 121*21*41 coarse grid.

There were 121 points in the wrap-around C-direction, 21 points along the span, and 41

points in the direction normal to the wing. Good agreement with measurements is

observed at 34%, 50% and 66% span locations. However, at the 85% span location, the

calculated lift is over-predicted because the grid spacing is sparse.

A fine grid, which has the dimension of 151*51*51, has been used for the grid

study case. As shown in Figure 3.2, the lift coefficient at each spanwise station is much

closer to the measured data for the fine grid case (151*51*51) than the coarse grid

(121*21*41) case. The turbulence model effects have also been studied with this case. It

is found that the Spalart-Allmaras model gives somewhat better predictions of lift

distribution along the span, particularly at the tip region, compared to the Baldwin-

Lomax model. However, the computed load around the wing tip region is still over-

predicted compared to the experimental data, because the very strong tip vortex could not



63

be accurately captured by the Reynolds average Navier-Stokes codes with the Baldwin-

Lomax turbulence model. To exactly capture the tip vortex, a better turbulence model and

an even fine mesh in the tip region are likely required.

3.2 Unblown and Steady Blowing Results

The 3-D solver validated above can also be used to study flow over 2-D airfoils.

In the 2-D mode, the spanwise derivative is set to zero, and the flow properties at only

one span location need to be calculated. The solver in 2-D mode was used to study the

effects of steady jets on the performance of CC airfoils. The effects of pulsed jets will be

discussed in the next chapter.

3.2.1 Configuration Modeled

As mentioned earlier, a supercritical airfoil with a simple hinged dual-radius

CCW flap shown in Figure 3.3 designed by Englar et al [11] was used in all the

simulations shown here. The jet slot is located at 88.75% chord length at the upper

surface of the airfoil, and the jet slot height is about 0.2% of the chord length. The CCW

flap is just aft of the jet slot, and is fixed at 30 degrees. From existing experimental data

[11], it is known that this CCW hinged flap design at this (lower) flap angle of 30° can

maintain most of the advantages of increased circulation attributable to the Coanda effect

at a lower drag, compared to conventional CC airfoils with a rounded trailing edge and/or

CCW flap airfoils with larger flap angles. According to recent aeroacoustic studies [9],



64

the tone noise emitted from the 30-degree flap is also much less than that from the 90-

degree flap CCW airfoil.

3.2.2 Computational Grid

A hyperbolic three-dimensional C-H grid generator was used in all calculations.

The three-dimensional grid is constructed from a series of two-dimensional C-grids with

an H-type topology in the spanwise direction. The grid is clustered in the vicinity of the

jet slot and the trailing edge to accurately capture the jet behavior over the airfoil surface.

For 2-D studies, the grid at a single span station was used in the solver.

The near field grid is shown in Figure 3.4. The slot location, slot height, and flap

angle can all be varied easily and individually in the grid generator and the flow solver.

The construction of a high-quality grid about CC airfoils is made difficult by the presence

of the jet channel that originates in an interior plenum. Shrewsbury [52, 53, 54] and

Williams et al [57] solved this problem by treating the jet slot as a grid-aligned boundary.

Pulliam et al [49] used an innovative spiral grid topology as well as a multi-block grid.

Berman [47] used a non-rectangular computational domain. In the present study, the

method similar to Shrewsbury was used, with the jet slot boundary condition described in

Chapter II. The grid close to the jet slot is clustered to accurately simulate the jet flow

behavior. In the present study, it was found that at least 7 points should be used across the

jet slot.

The grid close to the trailing edge of the CC airfoil should also be adequately

clustered, since the attached jet flow will turn at the corner of the trailing edge if the Cµ is



65

high, and the turning angle will affect the total circulation and the lift over the CC airfoil.

As shown in Figure 3.5, a grid-sensitivity study has been done to investigate the effect of

grid spacing near the trailing edge on the lift coefficient. It is found that a spacing of

0.001 chord length between the trailing edge point and the first point in the wake is

needed to correctly capture the high lift over the CC airfoil at a large momentum

coefficient, which is equal to 0.15 in this case. In the following studies, the trailing edge

spacing is always at 0.001 chord length, and 51 points have been placed in the wake

region. Thus, the total dimension of the grid is 221*51, with 221 points in the streamwise

direction and 51 points in the normal direction.

3.2.3 Blowing and Unblown Results Comparison

In the steady blowing studies, the flow conditions are the same with the

experimental studies of Englar et al [11]. The free-stream velocity was approximately

94.3 ft/sec (28.74 m/sec) at a dynamic pressure of 10 psf and an ambient pressure of 14.2

psia (0.979 bar). The free-stream density was about 0.00225 slugs/ft3 (1.1596 kg/m3), and

the chord of the CC airfoil is about 8 inch (0.2032 m). These conditions are translated

into a free-stream Mach number 0.0836 and a Reynolds Number of 395,000 in current

numerical simulations.

Figure 3.6 shows the variation of lift coefficient with respect to Cµ at a fixed angle

of attack (α=0 degree) for the CC airfoil with a 30-degree flap. Excellent agreement with

measured data from experiment by Englar [11] is evident. It is seen that very high lift can

be achieved by Circulation Control technology with a relatively low Cµ. A lift coefficient



66

as high as 4.0 can be obtained at a Cµ value of 0.33, and the lift augmentation ∆Cl / ∆Cµ is

greater than 10 for this 30-degree flap configuration.

Figure 3.7 shows the computed Cl variation with the angle of attack, for a number

of Cµ values, along with measured data. It is found that the lift coefficient increases

linearly with angle of attack until stall, just as it does for conventional sharp trailing edge

airfoils. This is expected due to the increase in circulation with the angle of attack.

However, due to the presence of the jet blowing, and the change of the rearward

stagnation point location, the relation of lift coefficient with angle of attack for CC

airfoils is quite complex.

Also as shown in Figure 3.7, the increase of lift with angle of attack breaks down

at high enough angles. This is due to static stall, and is much like that experienced with a

conventional airfoil, but occurs at very high C l,max values, thanks to the beneficial effects

of Circulation Control. The calculations also correctly reproduce the decrease in the stall

angle observed in the experiments at high momentum coefficients. Unlike conventional

airfoils, this is a leading edge stall. The computed stall angle is lower than the

experimental measurement, possibly due to the relatively simple turbulence model used,

which may not be accurate enough to capture the separation behavior of the flow at high

angles of attack. Figure 3.8 shows the streamlines around the CC airfoil at an angle of

attack of 6 degrees, and Cµ = 0.1657. In this case, a leading edge separation bubble

forms, and then spreads over the entire upper surface, resulting in a loss of lift. However,

the flow is still attached at the trailing edge because of the strong Coanda effect. The

leading edge stall at high Cµ may be explained as follows. As Cµ increases, the



67

circulation around the airfoil increases, leading to large pressure suction levels over the

upper surface. As angle of attack (α) increases, this large suction level generates a steep

adverse pressure gradient near the leading edge, leading to local separation bubbles, and

ultimately stall. Of course, with CC airfoils, it is seldom necessary to operate at high

angles of attack since high lift is easily achieved at low α values and modest amounts of

blowing.

These simulations also give some insight into the physics of the flow. For

example, consider a typical case at α = 0°. Without any blowing, trailing edge separation

and vortex shedding occur over the CCW flap, and the lift coefficient varies from 0.768

to 0.854 as shown in Figure 3.9. The measured data have an average of 0.878. When CC

blowing is applied with a moderate Cµ of 0.1657, the 2-D lift coefficient increases to a

value of 3.07. This is in excellent agreement with the measured value of 3.097. These

values can be attained in conventional wings only with the use of complex flaps and at a

higher flap angle or a higher angle of attack, which would considerably increase the

mechanical complexity and weight of the wing. For comparison, a 30-degree Fowler flap

on this same airfoil experimentally yielded only a lift coefficient of 1.8 at α = 0° [11].

In Figure 3.9, it is seen that the variation is periodic with a dimensional frequency

around 400Hz at a free-stream velocity of 94.3 ft/sec. This is due to the vortex shedding

over the trailing edge flap. In the acoustic experimental work of Munro [9, 16], it was

also found that there was a strong tone noise at a specified frequency for the unblown

case. To get more understanding into this vortex shedding frequency, a simulation at free-

stream velocity of 220 ft/sec, which is the same as in the acoustic experimental study [9],



68

has also been done. As shown in Figure 3.10, Cl also varies periodically with time, and

the extracted frequency is about 1080 Hz (Strouhal number = f*Chord/U∞ = 3.27). The

experimental studies [9] indicated that vortex shedding was present at a higher frequency

of 1600 Hz (Strouhal number = 4.84). To explain this difference, a Fast Fourier

Transform (FFT) has been done to transfer the computed periodic variation of the lift

coefficient with time into the frequency domain. As shown in Figure 3.11, in the

frequency domain, there is a dominant peak frequency at about 1080 Hz, which matches

the extracted frequency from the calculations. But there are secondary peak frequencies

due to the complex non-linearity of the flow, one of which occurs at 1600 Hz, which is

the same with the acoustic measurement. However, the flow characteristics for the vortex

shedding and separation are likely too complicate to be properly simulated here simply

with a Reynolds-averaged Navier-Stokes equation with a simple turbulence model. For

more accurate results, more advanced methods and improved boundary conditions may

be necessary. One of the methods is to use a higher order interpolation across the wake

cut boundary. It is suggested by Dancila and Vasilescu [96] that this method can

smoothly capture the vortex passage across the wake cut boundary, and give a better

prediction of the vortex shedding.

Figure 3.12 shows the streamlines around the trailing edge of the CC airfoil for

the blowing and unblown cases at a typical instance in time. It is clearly seen that the

trailing-edge vortex shedding, a potential source of noise, can be eliminated by even

blowing a small amount of jets.



69

3.2.4 Steady Blowing with Specified Total Pressure

In above CFD simulations, the moment coefficients Cµ, was directly specified as

the boundary condition using the method described in Chapter II. However, in

experiments, Cµ can not be directly specified, and it is instead calculated from the

measured jet total pressure in the pneumatic chamber. In this section, some calculations

have been done with the specified total jet pressure as the boundary condition, and the

results have been compared with the ones obtained from previous simulations, where the

momentum coefficient is specified as the boundary condition. In this case, the control

parameter is P0,jet /P∞, instead of Cµ, and the Cµ is subsequently calculated asqCVmC a=µ ,

where m is the mass flow rate and V a is the slot velocity obtained from the assumption

that the flow were expand to the free-stream pressure. Thus, from equations (2.79) to

(2.82), a relation between Cµ and P0,jet /P∞ can be obtained as follows:

− =

γ −γ

∞µ 0.1P

P

GC

1

jet,0

(3.1)

where, G is a constant coefficient that is dependent on the area of the jet slot, the free-

stream dynamic pressure and the area of the wing.

The Cµ variation with P0,jet is shown in Figure 3.13, and the lift coefficient

variation with Cµ is shown in Figure 3.14. From those figures, it is seen that the C µ is a

unique function of the total jet pressure. The predicted Cl values by changing the total jet

pressure are similar in behavior to the results computed by changing the specified

momentum coefficient. Both these results are very close to the experimental



70

measurements at the same Cµ. Thus it is reasonable to use Cµ as the driving parameter in

the numerical studies, instead of varying the jet total pressure as in experiments.

3.3 Effects of Parameters that Influence the Momentum Coefficient

As mentioned in Chapter II, the driving parameter of the Circulation Control is

the momentum coefficient, Cµ, which is defined as follows.

SV2

1

Um

C2

jet

∞∞

µ

ρ=

=SV

2

1

AU

2

jet

2

jet jet

∞∞ρ

ρ (3.2)

Thus, besides the jet velocity, the momentum coefficient is also a function of the area of

jet slot and the free-stream velocity.

In general, it is assumed in experiments and numerical simulations that the lift of

CC airfoils achieved is the same for a given Cµ. But some questions have arisen during

the studies: 1) What happens to the lift and drag if one doubles or halves the blowing slot

height or free-stream velocity with the same Cµ? 2) Would a thin wall jet be more

beneficial than a thicker, slower jet at the same Cµ?

To answer these questions, some simulations have been done to investigate the

effects of those parameters that influence the momentum coefficient, on the performance

of CC airfoils at a fixed Cµ. In following sections, the effects of two most important

parameters, the free-stream velocity and the jet slot area (or jet slot height for a 2-D

airfoil), have been investigated.



71

3.3.1 Free-stream Velocity Effects with Fixed C and Fixed Jet Slot Height

At first, a simulation was done to study the effects of the free-stream velocities on

the lift and drag coefficients for the 2-D steady blowing. In this case, the jet momentum

coefficient, Cµ, is fixed at 0.1657, and the jet slot height is also fixed at 0.015 inch, which

is about 0.2% of the chord. However, the free-stream velocities are varying from 0.5 to

1.8 times of the experimental free-stream velocity, which is equal to 94.3 ft/sec, thus the

jet velocity will vary with the free-stream velocity to keep a constant Cµ.

As shown in Figures 3.15 and 3.16, for a given momentum coefficient, the lift

coefficient and drag coefficient do not vary significantly with the change of the free-

stream velocity except at the very low free-stream velocities. The reason for the

production of low lift and high drag at low free-stream velocities is that the jet velocity is

too low to generate a sufficiently strong Coanda effects that eliminates separation and the

vortex shedding. It can be concluded that the performance of CC airfoils is independent

of the free-stream velocity under the fixed Cµ and fixed jet slot height conditions, and that

Cµ is an appropriate driving parameter for CC blowing if the slot-height is fixed. From

Figure 3.17, the total mass flow rate increases linearly with the increase in the free-stream

velocity. This is because the Cµ is non-dimensionalized by the free-stream dynamic

pressure, which includes the free-stream velocity. Thus the jet velocity and the mass flow

rate have to be increased with the free-stream velocity to keep a constant Cµ when the jet

slot height is also fixed.



72

3.3.2 Jet Slot Height Effects with Fixed C and Fixed Free-stream Velocity

According to the acoustic measurements [9], the jet slot height has a strong effect

on the noise produced by the CC airfoil, and a larger jet slot will reduce the noise at the

same momentum coefficient compared to a smaller one. To investigate the effect of jet

slot heights on the aerodynamic characteristics of CC airfoils, simulations at several slot

heights (from 0.006 inch to 0.018 inch) have been done, at a fixed low Cµ (Cµ =0.04) and

a fixed high Cµ (Cµ =0.1657) value, and at a constant free-stream velocity of 94.3 ft/sec.

From Figure 3.18, it is found that a higher lift coefficient can be achieved with a

smaller slot height even for the same momentum coefficient, and that the lift coefficient

is decreased by 20% as the slot height is increased from 0.006 inch to 0.018 inch. A

similar behavior is seen for the drag coefficient as shown in Figure 3.19. Thus the

efficiency of the airfoil, which is defined as Cl /(Cd+Cµ), and is corrected by adding Cµ to

the drag considering the momentum induced by the jet flow, does not vary much with the

change of the jet slot height. As shown in Figure 3.20, the efficiency decreases by about

7.6% for Cµ =0.1657 case, and increases by about 5.3% for Cµ =0.04 case when the slot

height is changed. However, as shown in Figure 3.21, the mass flow rate, which measures

the total amount of the jet needed, is increased by at least more than 60% when the slot

height is increased from 0.006 inch to 0.018 inch, due to the larger jet slot area.

Since it is always preferable to obtain higher lift with as low a mass flow rate as

possible, a thin jet is more beneficial than a thick jet. However, a higher pressure is

required to generate a jet issuing through a smaller slot than through a larger slot at the

same momentum coefficient. The power needed by a compressor to produce the required



73

high pressures can thus increase, neglecting any beneficial effects of Circulation Control

for very thin jets. In general, within the range of power consumption, a smaller jet slot

height is preferred from an aerodynamic design perspective. However, as mentioned

above, a larger jet slot height is preferred from an acoustic design perspective. Thus, a

compromise should be made for the jet slot height between the aerodynamic and acoustic

considerations of CC airfoils.

3.4 Other Simulations for the CC Airfoil

3.4.1 Comparisons with the Conventional High-Lift System

Some preliminary calculations were also done for a high-lift system configuration

with the same supercritical airfoil and a 30-degree Fowler flap to determine how this

configuration performs relative to a CC airfoil configuration. The solver used to simulate

the high-lift systems is also developed at Georgia Tech, and has been validated by

Bangalore and Sankar [97, 98] for a number of applications. The multi-element airfoil

configuration and the grid close to the surface are shown in Figure 3.22.

Figure 3.23 shows the airfoil drag polar, i.e., lift variation with drag, for the multi-

element airfoil and the CC airfoil with blowing. For both these cases, the flap angle is

fixed at 30 degrees. For the multi-element airfoil, the lift and drag are varied with the

change of the angle of attack. But for the CC airfoil, the angle of attack is fixed at 0

degree, and the lift and drag variations are achieved by changing the jet momentum

coefficient. It is seen that the CC airfoil configuration has a consistently lower drag at a



74

given lift compared to the multi-element airfoil, and that it can achieve very high lift

without stall. Figure 3.24 shows the efficiency, Cl /Cd+Cµ for these two configurations,

and it is seen that the CC airfoil is much more efficient than a multi-element airfoil at the

same lift coefficient.

3.4.2 Leading Edge Blowing

As mentioned in section 3.2.3, and as shown in Figure 3.7, the stall angles of CC

airfoils are quickly decreased with the increase in the jet momentum coefficient. The

same behavior happens for the conventional two-element high lift airfoil studied earlier.

To avoid leading edge stall, a third element, the slat, is usually added to the two-element

airfoil, giving a three-element high-lift configuration. Both experiments [99] and CFD

simulations [100] show that a slat can control the flow field around the leading edge of

the airfoil, and greatly increase the stall angle. However, the slat will also add more

moving parts and weight to the wing.

Leading edge blowing is an effective way of alleviating stall and achieving the

desired performance at higher angle of attack. To understand the effects of the leading

edge blowing, a dual-slot CC airfoil was designed, and simulations have been done for

both the leading edge (LE) blowing and trailing edge (TE) blowing cases. Figure 3.25

shows the grid for the leading edge blowing configuration. The jet slot height at the LE is

half that of the slot at the TE.

Figure 3.26 shows the lift coefficient variation with the angle of attack for three

different combinations of LE and TE blowing. For the first case, there is only a TE



75

blowing with Cµ= 0.08, and it is seen that the stall angle is very small, at approximately 5

degrees. But if a small amount of LE blowing is used (Cµ= 0.04), while keeping the TE

blowing at Cµ= 0.08 as before, the stall angle will be greatly increased (from 5 degrees to

12 degrees). If more LE blowing is used, e.g. a LE blowing of Cµ= 0.08 and a TE

blowing of Cµ= 0.04, the stall angle will be increased to more than 20 degrees, but the

total lift is decreased at the same angle of attack compared to the previous case even

when the total momentum coefficients (Cµ,LE + Cµ,TE) of the both cases are the same,

which is equal to 0.12 here. Figure 3.27 shows the drag coefficient variation with the

angle of attack. It shows the same behavior as the lift coefficient. An increase in TE

blowing produces higher drag.

In conclusion, the leading edge blowing is seen to increase the stall angle,

replacing the slat, while the trailing edge blowing could produce higher lift. Leading edge

blowing can also reduce the large nose down pitch moment due to the high lift and the

large level of suction peak in the vicinity of the slot. In general, operating at high angles

of attack is not necessary for CC airfoils since high lift can be readily achieved with low

angles of attack and a moderate amount of blowing. But in simulations where the CC

airfoil must operate at high angles of attack, a combination of leading edge and trailing

edge blowing could be used to achieve the best performance.



76

34% SPAN

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Chord

Cp

Exp CFD

Figure 3.1a: Cp Distribution over NACA 0012 Wing Sections at 34% Span

50% SPAN

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Chord

C p

Exp CFD

Figure 3.1b: Cp Distribution over NACA 0012 Wing Sections at 50% Span



77

66% SPAN

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Chord

C p

Exp CFD

Figure 3.1c: Cp Distribution over NACA 0012 Wing Sections at 66% Span

85% SPAN

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

CHORD

C p

Exp CFD

Figure 3.1d: Cp Distribution over NACA 0012 Wing Sections at 85% Span



78

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Span, Y/C

C l

Exp 8 DEG

CFD, BL Model, Coarse Grid

CFD, SA Model, Coarse Grid

CFD, BL Model, Fine Grid

Figure 3.2: Lift Coefficient Distribution along Span at Angle of Attack 8 Degrees

(Rectangular Wing with NACA 0012 Airfoil Sections)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Jet Slot Location

Figure 3.3: The Circulation Control Wing Airfoil with 30-degree Flap



79

Figure 3.4: The Body-fitted C Grid near the CC Airfoil Surface

0

0.5

1

1.5

2

2.5

3

0 4000 8000 12000 16000

Iterations

C l

Dx=0.001, Cl_ave=2.96

Dx=0.002, Cl=2.88

Dx=0.005, Cl=2.53

DX

Figure 3.5: The Lift Coefficients in Different Grid Spacing Cases (Cµ = 0.15)

(Dx: The distance between the trailing edge point (A) and the first point in the wake (B))



80

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

C

C l

Cl, Measured

Cl, Computed

Figure 3.6: Variation of the Lift Coefficient with Momentum Coefficients at α=0°

0

1

2

3

4

-4 -2 0 2 4 6 8 10 12 14 16Angle of Attack

L i f t C o e f f i c i e n t , C l

EXP, Cmu = 0.0

EXP, Cmu = 0.074

EXP, Cmu = 0.15

CFD

C =0.1657

C =0.074

C =0.0

Figure 3.7: The Variation of the Lift Coefficient with Angle of Attack



81

Figure 3.8: The Streamlines over the CC airfoil at Two Instantaneous Time Step

(Cµ = 0.1657, Angle of Attack = 60)



82

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0 1 2 3 4 5 6 7 8 9 10

Time (msec)

C l

t = 1.578693 msec t = 4.128484 msec t = 6.678274 msec

Figure 3.9: Time History of the Lift Coefficient for the Unblown Case

(U∞=94.3 ft/sec)

0.845

0.85

0.855

0.86

0.865

0.87

0.875

0.88

0.885

0 1 2 3 4 5 6 7 8 9 10

Time (msec)

C l

Figure 3.10: Time History of the Lift Coefficient for the Unblown Case

(U∞=220 ft/sec)



83

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500

Frequency (Hz)

F F T

Figure 3.11: The FFT of the Lift Coefficient Variation with Time

(U∞=220 ft/sec)

Dominant Vortex

Munro[9]’s measurement = 1600Hz



84

Figure 3.12a: Streamlines over the TE of the CC Airfoil

(Unblown Case, 30-degree Flap)

Figure 3.12b: Streamlines over TE of the CC Airfoil(Blowing Case, Cµ=0.04, 30-degree Flap)



85

0

0.2

0.4

0.6

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Pjet-total / Pinf

C

Figure 3.13: The Cµ Variation with the Total Jet Pressure for Steady Blowing Case

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

C

C l

Cl, Measured

Cl, Computed by Specified Cmu

Cl, Computed by Specified Jet TotalPressure

Figure 3.14: The Lift Coefficient Variation with Cµ for Steady Blowing Case



86

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2(Vinf in CFD) / (Vinf in Exp.)

C l

Figure 3.15: Lift Coefficient vs. Free-stream Velocity


0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(Vinf in CFD) / (Vinf in Exp.)

C d

Figure 3.16: Drag Coefficient vs. Free-stream Velocity




87

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(Vinf in CFD) / (Vinf in Exp.)

M a s s F l o w

R a t e

Figure 3.17: Mass Flow Rate vs. Free-stream Velocity


0

1

2

3

4

0.006 0.009 0.012 0.015 0.018

Jet Slot Height (inch)

L i f t C o e f f i c i e n t

Cmu = 0.04

Cmu = 0.1657

Figure 3.18: Lift Coefficient vs. Jet Slot Height

(V∞= 94.3 ft/sec)



88

0

0.05

0.1

0.15

0.2

0.25

0.006 0.009 0.012 0.015 0.018


D r a g C o e f f i c i e n t

Cmu = 0.04

Cmu = 0.1657

Figure 3.19: Drag Coefficient vs. Jet Slot Height

(V∞= 94.3 ft/sec)

0

5

10

15

20

0.006 0.009 0.012 0.015 0.018


E f f i c i e n c y C l / ( C d + C m u )

Cmu = 0.04

Cmu = 0.1657

Figure 3.20: The Efficiency vs. Jet Slot Height

(V∞= 94.3 ft/sec)



89

0

0.0005

0.001

0.0015

0.002

0.0025

0.006 0.009 0.012 0.015 0.018


M a s s F l o w

R a t e ( s l u g s / s e c )

Cmu = 0.04

Cmu = 0.1657

Figure 3.21: The Mass Flow Rate vs. Jet Slot Height

(V∞= 94.3 ft/sec)

-0.5

0

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Figure 3.22: The Shape of the Multi-element Airfoil and the Body-fitted Grid

(30-degree Fowler flap)



90

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Drag Coefficient, Cd


Multi-element Airfoil with 30 degrees fowler flap

CCW Airfoil with 30 degrees flap, Cd not corrected

CCW Airfoil with 30 degrees flap, Cd corrected withCd + Cmu

Figure 3.23: The Drag Polar for the Multi-Element Airfoil and the CC Airfoil

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4

Lift Coefficient, Cl

E f f i c i e n c y ,

L / D R a t i o

Multi-element Airfoil with 30 degrees fowler flap

CCW Airfoil with 30 degrees flap, Cd corrected withCd + Cmu

Figure 3.24: The Efficiency (Cl /Cd+Cµ) for the Multi-Element Airfoil and the CC Airfoil



91

Figure 3.25 (a) The Grid for the Leading Edge Blowing Configuration

(b) (c)

Figure 3.25 (b): The Grid Close to the Leading Edge Jet Slot

Figure 3.25 (c): The Grid Close to the Trailing Edge Jet Slot



92

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12 14 16 18 20 22 24

Angle of Attack (degrees)

L i f t C o e f f i c i e n t , C l LE Blowing, C = 0.08

TE Blowing, C = 0.04

LE Blowing, C = 0.04




Figure 3.26: Lift Coefficient vs. The Angle of Attack

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16 18 20 22 24

Angle of attack

D r a g C o e f f i c i e n t , C

d






TE Blowing, Cµ = 0.08

Figure 3.27: Drag Coefficient vs. The Angle of Attack



93

CHAPTER IV

TWO DIMENSIONAL PULSED BLOWING RESULTS

During the past five years, there has been increased interest in the use of pulsed

jets, and "massless" synthetic jets for flow control and performance enhancement.

Wygnansky et al [101, 102] studied the effects of the periodic excitation on the control of

separation and static stall. Lorber et al [103], and Wake et al [104] have studied the use of

directed synthetic jets for dynamic stall alleviation of the rotorcraft blade. Hassan [105]

has studied the use of synthetic jets placed on the upper and lower surfaces of an airfoil

surface as a way of achieving desired changes in lift and drag, offsetting vibratory

airloads that otherwise would occur during blade-vortex interactions. Pulsed jets and

synthetic jets have also been used to affect mixing enhancement, thrust vectoring, and

bluff body flow separation control. In 1972, Olyer and Palmer [106] experimentally

studied the pulsed blowing of blown flap configurations. More recently, some numerical

simulations employing a pulsed jet have also been reported for separation control of high-

lift systems [107], and traditional rounded trailing edge CC airfoils with multi-port

blowing [108]. Most of the studies above were focused on the use of low momentum

coefficient or zero-mass blowing to control the boundary layer separation or static and

dynamic stall. Only a few studies [106] considered the use of pulsed jets for lift

augmentation, at smaller mass flow rates compared to steady jets.



94

The present computational studies were aimed at answering the following

questions: Can pulsed jets be used to achieve desired increases in the lift coefficient at

lower mass flow rates relative to a steady jet? What are the effects of the pulsed jet

frequency on the lift enhancement at a given time-averaged Cµ? What is the optimum

wave shape for the pulsed jet, i.e. how should it vary with time?

In the calculations below, the angle of attack was set at zero, and the dual-radius

CC airfoil flap angle was fixed at 30 degrees. The shape of the CC airfoil, free-stream

Mach number, slot height, chordwise location of the slot, and Reynolds number were all,

likewise, held fixed as in the steady jet studies mentioned in Chapter III. In the present

studies, the following variation of the momentum coefficient with time was assumed:

( ) )]t(F1[CtC 0, += µµ (4.1)

where, Cµ ,0 is the time-averaged momentum coefficient, which is also the value of the

steady jet used for comparison. F(t) is a function of time, which varies from –1 to 1, and

determines the temporal variation of the pulsed jet.

4.1 Jets Pulsed Sinusoidally

Prior to the use of square wave form pulsed jets, a set of preliminary calculations

were done using a sinusoidal function form pulsed jet, i.e, F(t) is equal to sin(2πft) in

equation (4.1). It was found that this sinusoidal form was not an effective wave shape to

use compared to the square wave form.



95

Figure 4.1 shows a typical sinusoidal variation of the momentum coefficient with

time. The frequency is defined as the number of cycles per second. For the 400 Hz pulsed

jet, the time period between the peaks is 0.0025 seconds. Since the time-averaged Cµ,0 is

0.04, Cµ sinusoidally changes between a maximum value of 0.08, and a minimum of

zero. Figure 4.2 shows the lift coefficient variation with time for this sinusoidal form

pulsed jet. It is seen that the lift coefficient variation also follows a periodic variation like

sinusoidal form, no appreciable improvement in Cl compared to the steady jet. The mass

flow rate variation with the time is shown in Figure 4.3. It is seen that the change of mass

flow rates is also periodic, and that the average mass flow rate is less than the steady

flow, but very close to it.

Figure 4.4 shows the time-averaged lift coefficient of the sinusoidal pulsed jet as a

function of the frequency, with a comparison to the square wave pulsed jet and the steady

jet. It is seen that the differences between the sinusoidal wave and square wave pulsed jet

are small as far as the average values of Cl are concerned. Also a higher Cl can be

achieved at higher frequency in both cases. However, as shown in Figure 4.5, the mass

flow rate required for the sinusoidal pulsed jet is much higher than that for the square

wave jet. For this case with an average Cµ,0 of 0.04, the sinusoidal pulsed jet requires

92% of the steady jet mass flow, while the square wave pulsed jet requires just 73% of

the steady jet mass flow to achieve nearly the same lift as the sinusoidal pulsed jet. Since

the main advantage of the pulsed jets is to produce the comparable lift at lower mass flow

rates, the square wave pulsed jet is seen to be more efficient for the practical applications.



96

4.2 Jets Pulsed with a Square Wave Form

Improved results were obtained when the function F(t) in equation (4.1) was

chosen to be a square wave with a 50% duty cycle. Under this setting, F(t) equals to +1

for half the cycle, and F(t) equals to -1 for the other half of a cycle, as shown in Figure

4.6. It shows a square wave pulsed jets with frequency at 40 Hz, and the average

momentum coefficient is 0.04. The frequency f indicates the number of cycles that the jet

was turned on and off per second. Note that the instantaneous Cµ is zero during one half

of the cycle, and equals 2 Cµ,0 during the other half of the cycle. Thus the time-averaged

value is Cµ,0, which is also the value of the steady jet used for comparison.

Figure 4.7 shows the lift coefficient variation with time for this square wave

pulsed jet. It is seen that the Cl variation of the square wave pulsed jet is neither

sinusoidal nor like a square wave because of a time delay that exists in reducing or

increasing the circulation when the jet is turned off or on. At this low frequency of 40 Hz,

over large portion of the time the beneficial effects of Circulation Control are lost, and

the airfoil behaves like a conventional trailing edge stalled airfoil. However, the mass

flow rate variation, as shown in Figure 4.8, is still like a square wave, and the average

mass flow rate is lower than the steady jet.

4.2.1 Pulsed Jet Flow Behavior

Figures 4.9 and 4.10 show the variation of the time-averaged incremental lift

coefficient ∆Cl over and above the base-line unblown configuration at three frequencies,



97

40 Hz, 120 Hz and 400 Hz. Figure 4.9 shows the variation with the average momentum

coefficient Cµ,0, and Figure 4.10 shows the variation with the average mass flow rate.

Figure 4.11 shows the relation between the average mass flow rate and the average

momentum coefficient. It is seen that the average mass flow rate is the same for pulsed

jets at different frequencies with a given Cµ,0. Mass flow rate is a linear function of the jet

velocity. Since the average momentum coefficient is independent of the frequency, the

average jet velocity and the mass flow rate do not depend on the frequency as well.

Figures 4.12 and 4.13 show the behavior of the time-averaged lift-to-drag ratio

Cl /(Cd+Cµ,0) with Cµ,0 and mass flow rate, respectively. As done previously, the drag

coefficient has been corrected by adding Cµ,0 to account for the momentum imparted by

the jet into the wake. For comparison, the corresponding values of the steady jet with the

same Cµ,0 are also shown in these figures.

For a given value of Cµ,0, a steady jet gives a higher value of ∆Cl compared to a

pulsed jet as shown in Figure 4.9. This is to be expected because the pulsed jet is

operational only half the time during each cycle as where the steady jet is continuously

on. The benefits of the pulsed jet are more evident in Figure 4.10. At a given mass flow

rate, it is seen that the time-averaged values of lift are higher for the pulsed jet compared

to the steady jet, especially at higher frequencies. This superior performance of the pulsed

jet can be explained as follows. The momentum coefficient is proportional to the square

of the jet velocity, where as the mass flow rate is proportional to jet velocity V jet. As a

consequence, doubling the instantaneous momentum coefficient to twice its average

value increases the instantaneous mass flow rate only by a factor of square root of 2,



98

compared to a steady jet. Thus, the mean mass flow rate of the square wave form pulsed

jet is just about 70% of the mean mass flow rate of a steady jet at the same average Cµ,0

value. The Coanda effect, on the other hand, is dependent on the jet velocity squared, and

greatly benefits from these brief increases in the momentum coefficient. This leads to

higher lift for the same mass flow rate, compared to a steady jet as seen in Figure 4.10.

Since the mass flow rate is not a function of frequency as shown in Figure 4.11, a much

higher lift can be achieved at higher frequencies for the same mass flow rate.

At first glance, Figure 4.9 and Figure 4.10 will appear to show opposite trends.

Figure 4.10 appears to favor high frequencies – i.e. ∆Cl increases as frequency increases,

and pulsed jet produces a higher ∆Cl than a steady jet. This appears to be consistent with

experiments [106]. However, Figure 4.9 appears to show the opposite trend – steady jet

appears to be always more efficient than a pulsed jet, and produces a large ∆Cl.

To resolve this “apparent” inconsistency between Figure 4.9 and 4.10, four points

A, B, C, D are shown in Figure 4.9. These points are at the same mass flow rate of

0.00088 slug/sec. It is seen that point A is above point B. That is, a steady jet is indeed

able to produce a higher ∆Cl than a low frequency 40 Hz jet. This is because the flow

separates over a period of time before a new cycle of blowing begins, destroying the lift

generation. However, points C and D (120 and 400 Hz jets) are higher than point A. In

these cases, bound circulation over the airfoil has not been fully shed into the wake

before a new cycle begins. The time-averaged lift at the same specified averaged mass

flow rate is thus higher compared to a steady jet. This is consistent with Figure 4.10.



99

The lift-to-drag ratio for the steady jet is, however, still better compared to the

pulsed jet case as seen in Figures 4.12 and 4.13, partly because the Cd values have been

augmented by the momentum coefficient Cµ,0.

4.2.2 Effects of Frequency at a Fixed C

As mentioned above, the frequency has a strong effect on the performance of the

CC airfoil. To further investigate this, pulsed jet simulations have also been done at a

fixed time-averaged value of Cµ,0 equal to 0.04, while parametrically changing the

frequency f . Figures 4.14 and 4.15 show the variation of the average lift coefficient and

the efficiency with the frequency, respectively. It is seen that higher frequencies are, in

general, preferred over lower frequencies. For example, as shown in Figure 4.14, when

the frequency is equal to 400 Hz, the square form pulsed jet only requires 73% of the

average steady jet mass flow rate while it can achieve 95% of the lift achieved with a

steady blowing.

Examination of the flow field over an entire cycle indicates that it takes some time

after the jet has been turned off before all the beneficial circulation attributable to the

Coanda effect is completely lost. If a new blowing cycle could begin before this occurs,

the circulation will almost instantaneously reestablish itself as shown in Figures 4.16 and

4.17. At high enough frequencies, as a consequence, the pulsed jet will have all the

benefits of the steady jet at considerably lower mass flow rates.

For the 40 Hz jet, as shown in Figure 4.16, it is found that it takes about 0.00335

seconds (for a 8 inch chords airfoil at a free-stream velocity of 94.3 ft/sec) before the



100

Coanda benefit is lost completely. After that, the flow behaves like a conventional airfoil

and vortex shedding occurs during the rest time of the duty cycle until the jet is turned on

again. However, it just takes 0.00137 seconds to regain the Coanda effect after the jet is

turned on. This behavior is also observed as the frequency increased to 200 Hz, as shown

in Figure 4.17. During the first half of the duty cycle, when the jet is turned off, it is seen

that the lift coefficient is always decreasing but has not reach a minimum as in the 40Hz

case. It takes about 0.00248 seconds for the Coanda effect to be lost, which is just equal

to the jets-off time of the duty cycle. However, during the second half of the duty cycle,

when the jet is turned on, it is seen that it just takes 0.00113 seconds for lift coefficient to

reach the 98% of the maximum value, and the airfoil operates at this value for the

remainder of the duty cycle, for about 0.00137 seconds. The average lift coefficient will

much higher for a 200 Hz pulsed jet than that for a 40 Hz pulsed jet. As stated earlier,

these two cases have the same time-averaged mass flow rate. Thus, the 200 Hz pulsed jet

performs better than the 40Hz pulsed jet.

4.2.3 Strouhal Number Effects

For aerodynamic and acoustic studies, the frequency is usually expressed as non-

dimensional quantity called the Strouhal number. A simulation has been done to calculate

the average lift generated by the pulsed jet at fixed Strouhal numbers to answer the

following question: which of them, the frequency or the Strouhal number, has the a more

dominant effect on the pulsed jet performance?



101

The Strouhal number is defined as following:

∞

=U

Lf Str ref

(4.2)

In equation (4.2), f is the frequency of the pulsed jets. Lref is the reference value of the

length, which is the chord length of the airfoil, and U ∝ is the free-stream velocity. In

some applications, the vertical length of the flap has been chosen as Lref . Here the chord

length of the airfoil is used to simplify the analysis. In the present study, for the baseline

case, the Lref is 8 inches, and the U ∝ is equal to 94.3 ft/sec. For a 200 Hz pulsed jet, the

Strouhal number is equal to 1.41.

It should be noticed that another dimensionless frequency, F +, has also been used

in many pulsed jet and synthetic jet studies [101, 102, 108], which is defined as follows:

∞

+ =U

Lf F f

(4.3)

Here, L f is the length of the flap chord. In this case, the L f is 1 inch, and the F + is about

0.17625. Since F + is linearly related to the Strouhal number, the present discussion will

just focus on the Strouhal number.

From equation (4.2), besides the frequency, there are other two parameters that

could affect the Strouhal number, which are the free-stream velocity and Lref (Chord of

the CC airfoil). Thus, three cases have been studied. In the first case, as shown in Table

4.1, the free-stream velocity and the Chord of the CC airfoil are fixed, and the Strouhal

number is varied with the change of frequency. In the second case, as shown in Table

4.2, the Strouhal number is fixed at 1.41 and the chord of the CC airfoil is also fixed. The



102

frequency is varied along with the free-stream velocity to achieve the same Strouhal

number. In the third case, as shown in Table 4.3, the Strouhal number is fixed at 1.41 and

the free-stream velocity is also fixed, while the frequency is varied along with the chord

of the CC airfoil. The Mach number and Reynolds number are also functions of the free-

stream velocity and the airfoil chord, and were changed appropriately. The time-averaged

momentum coefficient, Cµ,0, is fixed at 0.04 in these studies. Figure 4.18 shows the lift

coefficient variation with the frequency for these three cases.

From tables 4.2 and 4.3, it is seen that the computed time-averaged lift coefficient

varies less than 2% when the Strouhal number is fixed. Table 4.2 indicates that the same

Cl can be obtained at a much lower frequency with a smaller free-stream velocity as long

as the Strouhal number is fixed. Table 4.3 shows that for a larger configuration, the same

Cl can be obtained at a lower frequency provided the Strouhal number is fixed. Table 4.1,

on the other hand, shows that varying the frequency and Strouhal number while holding

the other variables fixed can lead to a 12% variation in C l. Thus, it can be concluded the

Strouhal number has a more dominant effect on the average lift coefficient of the pulsed

jet than just the frequency.

Figure 4.19 shows that the lift coefficient is general increased with the Strouhal

number as it does with the frequency, when the momentum coefficient, the free-stream

velocity, and the chord of the airfoil are fixed. When the Strouhal number is about 2.8,

the square form pulsed jet can achieve 95% of the lift achieved with a steady blowing

while using only 73% of the average steady jet mass flow rate.



103

4.3 Summary of Observations

In Chapter III and IV, a number of studies have been presented on the effects of

the steady and pulsed jets on the behavior of CC airfoils. Before moving onto 3-D

configurations, it is worthwhile to summarize some observations from 2-D simulations.

S Very high lift could be achieved by CC blowing with a relative low momentum

coefficient, and the trailing edge vortex shedding, a potential noise source, can be

eliminated by the CC blowing.

SThe stall angle of the CC airfoil is decreased with the increase in the momentum

coefficient, and it is a leading edge stall.

SUnder steady blowing conditions, the momentum coefficient has a unique relation

with the jet total pressure. The variation of Cl with Cµ is the same as the variation

of Cl with P0,jet /P∞. Thus, it is reasonable to just vary Cµ as the driving parameter

for CCW computational studies. In experiments, it is of course more convenient

to vary P0,jet as the driving parameter.

SAt a fixed momentum coefficient, the performance of the CC airfoil does not vary

with changes to the free-stream velocity and free-stream Reynolds number. As a

result, one can study CCW performance in low speed tunnels with small models.

SBetter performance is achieved for a CC airfoil with a smaller jet slot height than

the one with a larger jet slot height. In practice, thin jets may require high plenum

pressure, which translates into higher power requirements of compressors that

will supply the high pressure air.



104

SCompared to a conventional multi-element airfoil, the CC airfoil can achieve a

higher L/D at the same lift coefficient, and it can generate much higher lift

coefficient prior to stall.

SLeading edge blowing can increase the stall angle, and allow the CC airfoil to

operate at high angles of attack.

SThe sinusoidal pulsed jet is not very effective compared to a square wave form

pulsed jet due to higher mass flow rates required with sinusoidal jets.

SThe square wave form pulsed jet can generate the same lift of the steady jet at a

much lower mass flow rate, and the performance of the pulsed jet improves with

the increase in frequency.

SThe Strouhal number has a more dominant effect on the performance of the

pulsed jet than just the frequency. For a larger configuration or at a small free-

stream velocity, the same lift coefficient can be obtained at a lower frequency

provided the Strouhal number is fixed. This means low frequency actuators that

are more readily available may be used on full-scale aircraft.



105

Table 4.1: The Computed Time-averaged Lift Coefficient for the Case one

(U∞ and Lref fixed, the Strouhal number varying with the frequency)

Baseline Half Frequency Double Frequency

Frequency (Hz) 200 100 400

Free-Stream Velocity

U∞ (ft/sec) 94.3 94.3 94.3

Chord of the Airfoil

Lref (inch) 8 8 8

Strouhal Number 1.41 0.705 2.82

Computed Average Lift

Coefficient (Cl) 1.6804 1.5790 1.8026

Computed Average

Mass Flow Rate (slugs/sec) 0.0006194 0.0006200 0.0006210

Table 4.2: The Computed Time-averaged Lift Coefficient for the Case Two

(Strouhal number and Lref fixed, the U∞ varying with the frequency)

Baseline Half Velocity Double Velocity



U∞ (ft/sec) 94.3 47.15 118.6


Lref (inch) 8 8 8



Coefficient (Cl) 1.6804 1.6601 1.7112

Computed Average




106

Table 4.3: The Computed Time-averaged Lift Coefficient for the Case Three

(Strouhal number and U∞ fixed, the Lref varying with the frequency)

Baseline Double Chord Half Chord



U∞ (ft/sec) 94.3 94.3 94.3


Lref (inch) 8 16 4



Coefficient (Cl) 1.6804 1.7016 1.6743

Computed Average




107

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.21 0.212 0.214 0.216 0.218 0.22Real Time (sec)

M o m e n t u m C

o e f f i c i e n t , C

Sin Form Wave Pulsed Jets

Steady Jets

DT = 0.0025

Figure 4.1: The Time History of the Momentum Coefficient


0

0.5

1

1.5

2

2.5

0.21 0.212 0.214 0.216 0.218 0.22Real Time



Steady Jets

Figure 4.2: The Time History of the Lift Coefficient




108

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.21 0.212 0.214 0.216 0.218 0.22

Real Time

M a s s F l o w

R a t e


Steady Jets

Figure 4.3: The Time History of the Mass Flow Rate


0

0.4

0.8

1.2

1.6

2

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Frequency (Hz)


Steady Jet, Cmu=0.04

Sinusoidal Pulsed Jet, Ave. Cmu=0.04

Square Wave Pulsed Jet, Ave. Cmu=0.04

Figure 4.4: Time-averaged Lift Coefficient vs. Frequency



109

0

0.0002

0.0004

0.0006

0.0008

0.001

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Frequency (Hz)

M a s s F l o w

R a t e ( s l u g / s e

c )


Sinusoidal Form Pulsed Jet, Ave. Cmu=0.04

Square Wave Form Pulsed Jet, Ave. Cmu=0.04

Figure 4.5: Time-averaged Mass Flow Rate vs. Frequency

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.44 0.45 0.46 0.47 0.48 0.49Real Time

M o m e n t u m C

o e f f i c i e n t , C

Square Wave Pulsed Jet

Steady JetDT = 0.025 sec

Figure 4.6: The Time History of the Momentum Coefficient




110

0

0.5

1

1.5

2

2.5

0.44 0.45 0.46 0.47 0.48 0.49

Real Time



Steady Jet

Figure 4.7: The Time History of the Lift Coefficient


0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.44 0.445 0.45 0.455 0.46 0.465 0.47 0.475 0.48 0.485 0.49

Real Time

M a s s F l o w

R a t e


Steady Jet

Figure 4.8: The Time History of the Mass Flow Rate




111

0

0.5

1

1.5

2

2.5

3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Time-Averaged Momentum Coefficient, C 0

C l

Steady Jet

Pulsed Jet , f = 40Hz

Pulsed Jet, f = 120 Hz


A

B

C

D

Figure 4.9: The Incremental Lift Coefficient vs. Time-averaged Momentum Coefficient

0

0.5

1

1.5

2

2.5

3

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

Time Averaged Mass Flow Rate (slug/sec)

C l

Steady Jet




A

B

C

D

Figure 4.10: The Incremental Lift Coefficient vs. Time-averaged Mass Flow Rate



112

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14


T i m e A v e r a g e d M a s s F l o w

R a t e

( s l u g / s e c ) Steady Jet




Figure 4.11: Time-averaged Mass Flow Rate vs. Time-averaged Momentum Coefficient

0

5

10

15

20

25

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14



C l / ( C d + C

)

Steady Jet




Figure 4.12: The Efficiency vs. Time-averaged Momentum Coefficient



113

0

5

10

15

20

25

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

Time Averaged Mass Flow Rate (slug/sec)


C l / ( C d + C )

Steady Jet




Figure 4.13: The Efficiency vs. Time-averaged Mass Flow Rate

0

0.4

0.8

1.2

1.6

2

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Frequency (Hz)



Pulsed Jet, Ave. Cmu=0.04

Figure 4.14: Time-averaged Lift Coefficient vs. Pulsed Jet Frequency (Ave. Cµ,0 = 0.04)



114

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Frequency (Hz)


L / D



Figure 4.15: The Efficiency vs. Pulsed Jet Frequency (Ave. Cµ,0 = 0.04)

0

0.5

1

1.5

2

2.5

0.455 0.46 0.465 0.47 0.475 0.48 0.485 0.49 0.495

Real Time (sec)


DT-cycle = 0.02501 sec

DT-down = 0.00335 sec

DT-up = 0.00137 sec

Figure 4.16: Time History of the Lift Coefficient for a 40 Hz Pulsed Jet



115

0

0.5

1

1.5

2

2.5

0.21 0.212 0.214 0.216 0.218 0.22

Real Time (sec)


DT-cycle = 0.00501 sec

DT-down = 0.00248 secDT-up

= 0.00113

Figure 4.17: Time History of the Lift Coefficient for a 200 Hz Pulsed Jet

1.2

1.4

1.6

1.8

2

50 100 150 200 250 300 350 400 450

Frequency

L i f t C o e f f i c i e n t , C

l

Case 1

Case 2

Case 3

Figure 4.18: Time-averaged Lift Coefficient vs. Frequency

(Case 1. Strouhal number was not fixed; U∞ and Lref were fixed)

(Case 2. Strouhal number and Lref were fixed; U∞ was not fixed)

(Case 3. Strouhal number and U∞ were fixed; Lref was not fixed)



116

0

0.4

0.8

1.2

1.6

2

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Frequency (Hz)




0

Stouhal Number ( f * Chord / Vinf)

2.8281.414

Figure 4.19: Time-averaged Lift Coefficient vs. Frequency & Strouhal Number



117

CHAPTER V

THREE DIMENSION CIRCULATION CONTROL WING

SIMULATIONS

The previous two chapters dealt with the use of Circulation Control for enhancing

the lift characteristics of 2-D airfoil configurations. It was demonstrated that both steady

and unsteady (pulsed) jets are effective in achieving high values of lift without resort to

the use of complex multi-element configurations.

Circulation Control has a number of other uses. It may be used to modify the

spanwise lift distribution of wing sections, effectively altering the span loading of lift

forces. Since the trailing vortex structures are directly affected by, and related to the

bound circulation, one can modify the strength (or spatial distribution) of trailing vortex

structures, including the strong vortex that forms at the wing tips.

This chapter addresses the uses and benefits of 3-D Circulation Control. Two

cases have been studied. The first is a streamwise tangential blowing on a wing-flap

configuration. The second is a spanwise tangential blowing over a rectangular wing with

a rounded wing tip. Some interesting results have been obtained for both cases,

demonstrating that there are many potential practical applications for the Circulation

Control technology, beyond high lift applications.



118

5.1 Tangential Blowing on a Wing-flap Configuration

As mentioned in Chapter I, the flap edge vortex is always a strong source of the

airframe noise, especially when high lift devices are fully deployed during take-off or

landing. According to Prandtl’s classic lifting-line theory [21], a trailing vortex will be

generated whenever there is a change in the bound circulation over the wing. For a wing-

flap configuration, the lift and hence the bound circulation is much higher over the flap

than on the main wing. Thus the circulation will not be continuous at the interface

between the wing and the flap, and a very strong vortex will be generated here. These

vortices have been seen in many experiments and flight tests. For example, the

experimental [109] and computational [110] studies indicated that a very strong vortex

was generated at the flap-edge due to the sudden increase in the lift. This vortex, due to

its interaction with the flap gap, will generate a strong noise, commonly labeled as “flap-

edge noise”.

A number of approaches have been proposed to eliminate this noise source.

Vortex fences and serrated flap edges have been proposed and tested. These devices add

to the weight and cost of manufacturing of the wing. Because these are passive devices,

they can be at best optimized for a single operating condition (e.g. a specified flap angle,

flow angle of attack, and free-stream velocity), and can not be expected to work for all

conditions.



119

C

This region is modeled as

shown in next figure

2-D BC

Symmetry BC

Small blowing to suppress vortex

shedding

15 C 5 C 5 C

C

This region is modeled as

shown in next figure

2-D BC

Symmetry BC

Small blowing to suppress vortex

shedding

15 C 5 C 5 C

Figure 5.1: The Wing-flap Tangential Blowing Configuration

The purpose of present research is to determine if the Circulation Control

technology may be used to modify the lift distribution along the span, thereby weakening

or eliminating the flap-edge vortex. Figure 5.1 shows a sketch of this concept – a wing-

flap configuration with tangential blowing over the main wing. Only the left half of this

wing-flap configuration has been simulated, and the flow has been assumed symmetric.

In this region, the wing section within the first five chord-length from the central

boundary has a 30-degree flap, and there is a weak jet blowing (Cµ ≅ 0.01) over the flap

to suppress the vortex shedding. The other part of the wing has no flap, but a scheduled

CC blowing is put in this section of the wing to generate high lift that is comparable with

the lift generated by the 30-degree flap.

Figure 5.2 shows details of the grid around this configuration. There are two

regions that are very important in these simulations. Region A is the interface



120

between the blowing section of the main wing and the unblown section of the main wing,

and region B is the interface between the blowing section of the main wing and the wing-

flap section.

Three cases have been studied. In the first case, there is no blowing on the main

wing, so it is just a regular wing-flap configuration. In the second case, there is a constant

blowing, which means the Cµ is constant along the span, over some sections of the main

wing (from 15C to 20C). Finally, a gradual blowing case has been studied, where the C µ

is gradually increased along the span over some sections of the main wing (from 10C to

20C). Figure 5.3 shows the lift coefficient distribution along the span of this wing-flap

configuration for these three cases. When there is no blowing, a steep jump in lift

coefficient is found at the interface between the main wing and the flap. It is expected

because the sectional lift generated in the vicinity of the 30-degree flap is much higher

than the main wing. In the second case, when a constant blowing is put over a section of

the main wing, the lift at these stations will be greatly increased due to the Coanda effect.

Thus in Region B, the difference of lift between the blowing section of the main wing

and the flap will be reduced, but a jump in the lift is still found at the interface between

the blowing section of the main wing and the unblown section. In the third case involving

the gradually blowing, it is seen that the lift is smoothly increased along the span, from

0.25 to 1.4 over the flap without a sudden change. This is due to the gradual increase in

the blowing momentum coefficient, Cµ.

According to the lift distribution and the Prandtl’s lift ling theory, case 1 and 2

should generate strong vortices in Region B and A, respectively, while in case 3, there is



121

a weakening or a total elimination of the flap-edge vortex. This has been observed in the

vorticity contours shown in Figures 5.4, 5.5 and 5.6. It should be noted that the

disturbance along the boundary of the flap-edge is due to the grid discontinuity along the

interface between the main wing and the flap, which is not a vortex.

In summary, the preliminary conclusions for the 3-D tangential streamwise

blowing over the wing-flap configuration are: 1) the flap-edge vortex is generated by the

suddenly increase in the lift along the flap-edge interface; 2) CC blowing with a constant

momentum coefficient can not eliminate the flap-edge vortex, but can weaken and move

the location of this vortex from the flap-edge towards the main wing; 3) a gradually

varying CC blowing can totally eliminate the vortex. It should be noted that this is just a

preliminary simulation, and that the model used here is very simple. To fully understand

the effect of the CC blowing on the flap-edge vortex, more detailed simulations are

recommended.

5.2 Spanwise Blowing over a Rounded Wing-tip

The vortex over the wing tip region is also a strong noise source. In rotor wing

applications, this vortex can interact with other blades, giving rise to blade vortex

interaction (BVI) noise. Tip vortex is generated by the pressure differences between the

upper and lower surface of the lift wing. Since in general, the pressure at the lower

surface is much higher than that at the upper surface, the vorticity of the fluid particles

within the boundary layer at the lower surface will flow around the wing tip, roll-up, and



122

form a tip vortex. The tip vortex formation may be drastically altered by generating a

flow in a direction opposite to that of the boundary layer. To investigate the feasibility of

this concept, a wing-tip configuration has been studied on the effects of tangential

spanwise blowing on the flow field around the wing-tip region.

Figure 5.7 below shows a sketch of this concept for a rounded wing tip. The wing

is a simple rectangular wing with NACA 0012 airfoil sections, but the wing tip is round.

The angle of attack was 8 degrees, giving rise to sufficient lift and a strong tip vortex.

The jet slot is located above the rounded wing tip edge, and the jet is coming in the

spanwise direction. Figures 5.8 through 5.11 show the configuration and the body fitted

grid in the vicinity of the rounded wing tip and the jet slot.

Figure 5.7: The Wing Tip Configuration

Three cases have been studied. In the first case, there is no blowing, simulating a

rectangular wing with a rounded wing tip. In the second case, there is a small amount of

blowing with Cµ = 0.04. In the third case, there is a stronger blowing with Cµ = 0.18.

Figures 5.12, 5.13 and 5.14 show the vorticity contours around the wing tip region at

three different streamwise locations, which are x/c = 0.81, 1.0 and 1.50, respectively.

From those figures, it is seen that there is a strong tip vortex if there is no blowing, which



123

is expected. If there is a small amount of blowing over the wing tip in the opposite

direction, the tip vortex will be pushed away from the wing tip, but the vortex could not

be eliminated. Even when the blowing is increased, the tip vortex is just pushed down

and far away from the wing. Another weaker vortex with an opposite rotation direction

has been generated. Figure 5.15 shows the velocity flow field around the wing tip region

at x/c = 0.81. It shows the same qualitative behavior as the vorticity contour.

Figures 5.16 and 5.17 show the lift and drag coefficients distribution along span

for this wing tip configuration. It is seen that the tangential blowing over the wing tip can

also increase the lift around whole wing, especially when there is a strong CC blowing.

The calculated overall lift coefficient and drag coefficient for the whole wing are tabled

as follows:

Table 5.1: The Total Lift Coefficient and Drag Coefficient for the Wing Tip

Configuration

Total LiftCoefficient

CL

Total DragCoefficient

CD

Computed Drag from theInviscid Relation

CD,C = (CL)2 /(πÆe)

Noblowing Case 0.4850 0.02997 0.02997

Less Blowing,

Cm = 0.04

0.5215 0.03078 0.03465

More Blowing,

Cm = 0.18

0.6064 0.04342 0.04685

where Æ is the aspect-ratio of the wing, which is equal to 4 for this configuration, and the

e is the efficiency of the lift distribution, which is set at 0.6246 from the noblowing case



124

calculation. It is seen that the total drag of the wing has been reduced by about 10% by

CC blowing, when the drag has been corrected for the increase in C L.

The preliminary conclusions for the 3-D spanwise blowing over a rounded wing

tip configuration is that the jet blowing around the rounded wing tip can modify and

change the location of the tip vortex. It can not totally cancel or eliminate the tip vortex,

but can change or increase the vertical clearance between the wing and the vortex. Since

the blade vortex interaction of rotors is strongly influenced by the clearance between the

following blades and the tip vortex, this approach does have the potential of reducing

BVI noise. It can also slightly reduce the drag of the whole wing tip configuration by

pushing the tip vortex away from the wing, and increasing the aspect-ratio.



125

Figure 5.2: The Grid of the 3-D Wing-flap Configuration with a 300 Partial Flap

Region A

Region B

0 20 251510



126

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25

Span, Y/C


Noblowing on Main Wing

Constant Blowing on Main Wing

Gradual Blowing on Main Wing

Figure 5.3: The Lift Coefficient Distribution along Span for the Wing-flap Configuration



127

Case 1: Noblowing Case

Figure 5.4: The Vorticity Contours for Noblowing Case

Flap Main Wing

Unblown SectionBlowing Section



128

Case 2: Constant Blowing Case

Figure 5.5: The Vorticity Contours for Constant Blowing Case

Flap Main Wing




129

Case 3: Gradual Blowing Case

Figure 5.6: The Vorticity Contours for Gradual Blowing Case

Flap Main Wing




130

Figure 5.8: The H-Grid for the Wing Tip Configuration (Side View at Spanwise Station)

Figure 5.9: The O-Grid around the Rounded Wing Tip (Front View)



131

Figure 5.10: The Surface Grid for the Rounded Wing Tip

Figure 5.11: The Detailed Grid Close to the Jet Slot



132

Figure 5.12: The Vorticity Contours around the Wing Tip (x/C = 0.81)

No-Blowing

Case

Less Blowing

Case (C = 0.04)

More Blowing

Case (C = 0.18)



133


No-Blowing

Case

Less Blowing

Case (C = 0.04)

More Blowing

Case (C = 0.18)



134


No-Blowing

Case

Less Blowing

Case (C = 0.04)

More Blowing

Case (C = 0.18)



135

Figure 5.15: The Velocity Vectors around the Wing Tip (x/C = 0.81)

No-Blowing

Case

Less Blowing

Case (C = 0.04)More Blowing

Case (C = 0.18)



136

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2

Span, Y/Ytip

C l

Noblowing

Blowing, Cmu = 0.04

Blowing, Cmu = 0.18

Figure 5.16: The Lift Coefficient Distribution along Span for Wing Tip Configuration

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.2 0.4 0.6 0.8 1 1.2

Span, Y/Ytip

C d

Noblowing

Blowing, Cmu = 0.04

Blowing, Cmu = 0.18

Figure 5.17: The Drag Coefficient Distribution along Span for Wing Tip Configuration



137

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

In the present study, a three-dimensional unsteady Reynolds-average Navier-

Stokes analysis capable of modeling the Circulation Control Wings/Airfoils has been

developed. The method uses a semi-implicit finite difference scheme to solve the

governing equations on a body-fitted grid. A zero-equation Baldwin-Lomax and a one-

equation Spalart-Allmaras turbulence model have been implemented in the solver to

account for the turbulence effects. Physically appropriate boundary conditions are used to

model the jet exhausts from the slot located over the CCW flap. The solver can be used

both in a 2-D mode to compute the CC airfoil performance and in a 3-D mode for

studying CC wings.

The configuration selected is an advanced hinge-flap CC airfoil developed by

Englar, which has been extensively tested by Georgia Tech Research Institute (GTRI).

Prior to this work, the numerical studies for this kind of CC airfoils have been very

limited.

Prior to its use, a code-validation study has been done for a rectangular wing with

NACA 0012 airfoil sections. Subsequently, two-dimensional steady blowing simulations

have been done and compared with experimental data. The influence of some parameters

such as the slot-height and free-steam velocity on the performance of the CC airfoil has



138

also been studied. The CCW configuration has also been compared with the baseline

unblown configuration and a conventional high-lift system. The effects the 2-D pulsed jet

on the CC airfoil performance have also been investigated. The effects of the wave shape

and the frequency of the pulsed jet have been specifically studied.

Finally, some simulations have been done for two three-dimensional

configurations with the use of the Circulation Control technology. The first involves a

streamwise tangential blowing on a wing-flap configuration to eliminate the flap-edge

vortex. The second study deals with the use of a spanwise blowing on a rounded wing-tip

configuration to control the tip vortex.

In this chapter, the conclusions of this research are presented in Section 6.1. The

recommendations for the future work are given in Section 6.2.

6.1 Conclusions

The investigation in the present study leads to the following major conclusions:

1. Navier-Stokes simulations are necessary for the CC wings/airfoils studies due to

the complexity of the flow field and the strong viscous effects. The results

indicate that this approach is an efficient and accurate way of modeling CC

airfoils with steady and pulsed jets.

2. The Circulation Control Technology is a useful way of achieving very high lift at

even zero angle of attack. It can also eliminate the vortex shedding in the trailing

edge region, which is a potential noise source. The lift coefficient of the CC



139

airfoil is also increased with the angle of the attack as the conventional sharp

trailing edge airfoil. However, the stall angle of the CC airfoil is decreased

quickly with the increase in the blowing momentum coefficient. This stall

phenomenon occurs in the leading edge region, and may be suppressed by leading

edge blowing. In practice, because high CL values are achievable at low angles of

attack, it may seldom be necessary to operate CC wings at high angles of attack.

However, because there is always a large nose down pitch moment for the CC

airfoil, leading edge blowing is generally used to reduce this pitch moment for the

large amount of blowing case even at zero angle of attack.

3. The jet momentum coefficient varies uniquely with the total pressure of the jet

plenum. The behavior of computed lift coefficient is similar whether the

momentum coefficient or total jet pressure is varied. In experimental studies, it

may be more convenient to vary the jet total pressure, thereby changing the

momentum coefficient.

4. When the momentum coefficient is fixed, the computed lift coefficient does not

vary with the free-stream velocity. However, at a fixed Cµ, Cl is influenced by the

jet slot height. A thin jet from a smaller slot is preferred since it requires much

less mass flow, and has the same efficiency in generating the required Cl values as

a thick jet. From a practical perspective, much higher plenum pressure may be

needed to generate thin jets for a given Cµ. This may increase the power

requirements of compressors that provide the high pressure air.



140

5. The CC airfoil with trailing edge blowing can generate higher lift and avoid static

stall compared to a conventional Fowler flap airfoil. It also achieves higher

efficiencies (Cl /(Cd+Cµ)) without the moving parts associated with the high-lift

system.

6. Sinusoidal pulsed jet was found not to be very effective compared to the square

wave pulsed jet due to higher mass flow rates required. A square wave shape

pulsed jet configuration gives larger increments in lift over the baseline unblown

configuration, when compared to the steady jet with the same time-averaged mass

flow rate. Pulsed jet performance is improved at higher frequencies due to the fact

that the airfoil has not fully shed the bound circulation into the wake before a new

pulse cycle begins.

7. The non-dimensional frequency, Strouhal number, has a more dominant effect on

the performance of the pulsed jet than just the frequency. Thus, the same

performance of a pulsed jet could be obtained at lower frequencies for a larger

configuration or at smaller free-stream velocities provided the Strouhal number is

kept the same. Furthermore, at a Strouhal number of 2 or above, the ∆Cl due to

pulsed jets is nearly 90% of ∆Cl achieved with the steady jet, while the mass flow

rate required is only 70% of the steady jet. Of course, an optimal Strouhal number

may be dependent on other physical parameters such as slot height, flap angle and

flap chord, etc. Nevertheless, it is clear that Strouhal number, and not the

frequency, is the dominant parameter.



141

8. From the preliminary studies about the three-dimensional CC wing

configurations, it is found that a gradual streamwise tangential CC blowing near

the flap-edge can weaken or totally eliminate the flap-edge vortex, a strong noise

source. Spanwise tangential blowing over a rounded wing tip can push the tip

vortex down away from the wing tip. Thus an effective control of the tip vortex

position is feasible with Circulation Control.

6.2 Recommendations

While a number of computational issues have been addressed in this work,

additional work remains to be done before CFD based analysis such as the present work

can be confidently used to design CCW system. Research in following areas is

recommended:

1. Turbulence models are very important for the CC wing study, especially in the

area where strong separation and vortex shedding are present. The Baldwin-

Lomax and the Spalart-Allmaras turbulence models did a satisfactory job of

modeling the flow when the flow is attached and when there was no separation,

especially for the advanced CC airfoil with a sharp trailing edge flap. However, to

accurately simulate the strong tip vortex, the vortex shedding of the unblown

configuration, and the traditional rounded trailing edge CC airfoil, a systematic

study of improved turbulence models is necessary. Furthermore, many of the

existing models were developed or calibrated using steady flow data. Further



142

calibrations or adjustments of the constants in these models may be necessary for

modeling the pulsed jet and unsteady flow.

2. The pulsed jet is a very effective way of obtaining the same high lift as a steady

jet while requiring lower mass flow rates. However, the desired high frequencies

are hard to achieve in experiments, especially when the test configuration is small.

Methods of improving the pulsed jet performance at low frequencies will be very

useful. One possibility is to vary the total jet pressure periodically instead of the

momentum coefficient. A second possibility is to change the slot height

dynamically while keeping a constant jet total pressure to generate a square wave

pulsed jet. However, the computational grid needs to be dynamically modified

with the change of the slot height. The current solver could not deal with this grid,

but this method is highly recommended for the future research of the pulsed jet.

3. There are many potential applications of the Circulation Control technology for

practical three-dimensional configurations beyond what has been studied in this

work. Some applications include: (a) drag reduction of bluff bodies and vehicles

such as trucks and automobiles, (b) Modification to the leading edge and exhaust

flow around engine nacelles, (c) suppression of vortex shedding from automobile

antennae and mirrors. The potential of this concept is limitless and should be

further explored, just keeping in mind the effect and cost involved in having a

readily available air source with some pressure.



143

4. This work has addressed only the aerodynamic benefits of Circulation Control

wings. Elimination of high lift devices with this simple yet powerful approach can

reduce high lift system noise. However, it must be remembered that the high

speed steady or pulsed jet itself can be a source of noise. Thus a combined

aerodynamic/aeroacoustic analysis from a system wide perspective is necessary.

A companion experimental work by Munro [9], also funded by NASA (our

sponsor), looks at the issue of CCW noise in a careful manner. The numerical

studies of the aeroacoustic characteristics of CCW airfoils are recommended.

In conclusion, a first principle-based approach for modeling the Circulation

Control wing/airfoils has been developed and validated. It is hoped that this work will

serve as a useful step for the further investigations in this exciting area.



144

APPENDIX A

GENERALIZED TRANSFORMATION

The generalized transformation is used to transform the governing equation from

the physical domain (x, y, z, t) to the computational domain (ξ, η, ζ, τ). In general, the

coordinates (ξ, η, ζ) in computational domain are assumed to be uniform spacing, and

they are the functions of (x, y, z) as follows:

t

)t,z,y,x(

)t,z,y,x(

)t,z,y,x(

=τ

ζ=ζ

η=η

ξ=ξ

(A.1)

For the time derivatives, the ξt, ηt and ζt are given in terms of the grid velocity xτ,

yτ and zτ as:

z

z

y

y

x

x

zz

yy

xx

zz

yy

xx

t

t

t

∂

ς∂−

∂

ς∂−

∂

ς∂−=ς

∂η∂

−∂η∂

−∂η∂

−=η

∂ξ∂

−∂ξ∂

−∂

ξ∂−=ξ

τττ

τττ

τττ

(A.2)

and the xτ, yτ and zτ are the grid velocity, and equal to zero if there is no body or grid

moving.



145

For the spatial derivatives, using the chain rule of the partial differentiation, the

partial derivatives for coordinates (x, y, z) become:

ς∂∂

∂ς∂

+η∂∂

∂η∂

+ξ∂

∂∂ξ∂

=∂∂

ς∂∂

∂ς∂

+η∂∂

∂η∂

+ξ∂

∂∂ξ∂

=∂∂

ς∂∂∂ς∂+η∂∂∂η∂+ξ∂∂∂ξ∂=∂∂

zzzz

yyyy

xxxx

(A.3)

In above equation, for simplicity, the following expressions will be used for the

derivatives:

ς∂∂

=η∂

∂=

ξ∂∂

=∂ς∂

=ς∂

ς∂=ς

∂ς∂

=ς

ς∂∂

=η∂

∂=

ξ∂∂

=∂η∂

=η∂η∂

=η∂η∂

=η

ς∂∂

=η∂

∂=

ξ∂∂

=∂ξ∂

=ξ∂ξ∂

=ξ∂ξ∂

=ξ

ςηξ

ςηξ

ςηξ

zz,

zz,

zzand

z,

y,

x

yy,

yy,

yyand

z,

y,

x

xx,

xx,

xxand

z,

y,

x

zyx

zyx

zyx

(A.4)

Then, the differential expressions by the partial difference can be written as:

dzdydxd

dzdydxd

dzdydxd

zyx

zyx

zyx

ς+ς+ς=ς

η+η+η=η

ξ+ξ+ξ=ξ

(A.5)



146

Equation (A.5) could be written in a matrix form as:

ςςς

ηηηξξξ

=

ς

ηξ

dz

dy

dx

d

d

d

zyx

zyx

zyx

(A.6)

Similarly, the (dx, dy, dz) can be expressed by (dξ, dη, dζ) as the following

matrix form:

ςηξ

=

ςηξ

ςηξ

ςηξ

d

d

d

zzz

yyy

xxx

dz

dy

dx

(A.7)

Combining equation (A.6) and (A.7), the following equation can be obtained:

−−−−−−−−−

=

=

ςςςηηηξξξ

ηηξηξξηξηηξ

ξξςξςςξςξξς

ςςηςηηςηςςη

−

ςηξ

ςηξ

ςηξ

xyxzxzxzyzy

xyxzxzxzyzy

xyxzxzxzyzy

J

zzz

yyy

xxx1

zyx

zyx

zyx

(A.8)

Thus, the derivatives of (ξ, η, ζ) can be expressed as:

)yxyx(J

)zxzx(J)zyzy(J

)yxyx(J

)zxzx(J)zyzy(J

)yxyx(J

)zxzx(J)zyzy(J

z

yx

z

yx

z

yx

ξηηξ

ηξξηξηηξ

ςξξς

ξςςξςξξς

ηςςη

ςηηςηςςη

−=ς

−=ς−=ς

−=η

−=η−=η

−=ξ

−=ξ−=ξ

(A.9)



147

where J is the Jacobian Matrix of the transformation, which is given as:

)zyzy(x)zyzy(x)zyzy(x[ / 1

zzz

yyy

xxx

/ 1J

zyx

zyx

zyx

ξηηξςξςςξηηςςηξ

ςηξ

ςηξ

ςηξ

−+−−−=

=

ςςς

ηηηξξξ

= (A.10)

ξξξ z,y,x etc could be obtained by the differencing method. For instance, using

the second-order central difference, ξξξ z,y,x etc are expressed as follows:

ς∆

−=

ς∂∂

=

η∆

−=

η∂

∂=

ξ∆−

=ξ∂

∂=

ς∆−

=ς∂

∂=

η∆−=

η∂∂=

ξ∆−

=ξ∂

∂=

ς∆−

=ς∂

∂=

η∆−

=η∂

∂=

ξ∆−

=ξ∂

∂=

−+ς

−+η

−+ξ

−+ς

−+η

−+ξ

−+ς

−+η

−+ξ

2

zzzz

2

zzzz

2

zzzz

2

yyyy

2yyyy

2

yyyy

2

xxxx

2

xxxx

2

xxxx

1k , j,i1k , j,i

k , j,i

k , j,i,

k ,1 j,ik ,1 j,i

k , j,i

k , j,i,

k , j,1ik , j,1i

k , j,i

k , j,i,

1k , j,i1k , j,i

k , j,i

k , j,i,

k ,1 j,ik ,1 j,i

k , j,i

k , j,i,

k , j,1ik , j,1i

k , j,i

k , j,i,

1k , j,i1k , j,i

k , j,i

k , j,i,

k ,1 j,ik ,1 j,i

k , j,i

k , j,i,

k , j,1ik , j,1i

k , j,i

k , j,i,

(A.11)



148

As for the fourth-order central difference, the following expressions could be

obtained for ςηξ x,x,x :

ς∆

+−+−=

ς∂∂

=

η∆

+−+−=

η∂∂

=

ξ∆+−+−

=ξ∂∂

=

−−++ς

−−++η

−−++

ξ

12

xx8x8xxx

12

xx8x8xxx

12

xx8x8xx

x

2k , j,i1k , j,i1k , j,i2k , j,i

k , j,i

k , j,i,

k ,2 j,ik ,1 j,ik ,1 j,ik ,2 j,i

k , j,i

k , j,i,

k , j,2ik , j,1ik , j,1ik , j,2i

k , j,i

k , j,i,

(A.12)

The similar equations could be obtained for ςηξ y,y,y and ςηξ z,z,z with y and

z, respectively.

Note that in most practical applications, the grid in the computational domain is

assumed to be uniform, thus the grid spacing ∆ξ , ∆η, and ∆ζ are equal to one.



149

REFERENCES

1. “Noise Standards: Aircraft Type and Airworthiness Certification,” Federal

Aviation Regulations Part 36, Federal Aviation Administration, November 1969.

2. “Policies and Procedures For Considering Environmental Impacts,” Federal

Aviation Administration Order 1050.1D, December 1986.

3. Goldin, D. S., et al., National Aeronautics and Space Administration Strategic

Plan, NPD 1000.1B, pp. 42-43, September 2000.

4. Huff, D., “Recent Progress in Engine Noise Reduction for Commercial Aircraft

Applications.” Seminar, School of Aerospace Engineering, Georgia Institute of

Technology, April 8, 2003.

5. Smith, M. J. T., Aircraft Noise, Cambridge University Press, 1989.

6. Crighton, D. G., “Aircraft Noise in Aeronautics of Flight Vehicles: Theory andPractice,” Vol. 1: Noise Sources, NASA PR 1258, pp. 391-447, 1991.

7. Sen, R., “A Study of Unsteady Fields Near Leading-edge Slats,” AIAA Paper 97-1696, 1997.

8. Davy, R. and Remy, H., “Airframe Noise Characteristics on a 1/11 Scale AirbusModel,” AIAA Paper 98-2335, June 1998.

9. Munro, S., Jet Noise of High Aspect-Ratio Rectangular Nozzles with Application

to Pneumatic High-Lift Devices, Ph.D Thesis, Georgia Institute of Technology,2002.

10. Englar, R. J., “Circulation Control Pneumatic Aerodynamics: Blown Force andMoment Augmentation and Modification; Past, Present & Future”, AIAA Paper

2000-2541, June 2000.

11. Englar, Robert J., Smith, Marilyn J., Kelley, Sean M. and Rover, Richard C. III.,

“Application of Circulation Control to Advanced Subsonic Transport Aircraft,

Part I: Airfoil Development,” Journal of Aircraft , Vol.31 No.5, pp. 1160-1168,Sep. 1994.



150

12. Englar, Robert J., Smith, Marilyn J., Kelley, Sean M. and Rover, Richard C. III.,

“Application of Circulation Control to Advanced Subsonic Transport Aircraft,

Part II: Transport Application,” Journal of Aircraft , Vol.31, No.5, pp. 1169-1177,Sep. 1994.

13. Shrewsbury, G. D. and Sankar, L. N., “Dynamic Stall of an OscillatingCirculation Control Airfoil,” International Symposium on Nonsteady Fluid

Dynamics, oronto, Canada, June 4-7, 1990, Proceedings. New York, American

Society of Mechanical Engineers, pp. 15-22, 1990.

14. Shrewsbury, G. D. and Sankar, L. N., “Dynamic Stall of Circulation Control

Airfoils,” AIAA Paper 90-0573, January 1990.

15. Salikuddin, M., Brown, W. H. and Ahuja, K. K., “Noise From a Circulation

Control Wing with Upper Surface Blowing,” Journal of Aircraft , Vol.24, pp55-

64, Jan. 1987.

16. Munro, S., Ahuja, K., and Englar, R., “Noise Reduction Through Circulation

Control Technology,” AIAA Paper 2001-0666, Jan. 2001.

17. Munro, S. and Ahuja, K. K., “Aeroacoustics of a High Aspect-Ratio Jet,” AIAA

paper 2003-3323, 2003, presented at the 9th

AIAA/CEAS AeroacousticsConference and Exhibit, Hilton Head, South Carolina, 12-14 May 2003.

18. Munro, S. and Ahuja, K. K., “Fluid Dynamics of a High Aspect-Ratio Jet,”

AIAA paper 2003-3129, 2003, presented at the 9th

AIAA/CEAS AeroacousticsConference and Exhibit, Hilton Head, South Carolina, 12-14 May 2003.

19. Munro, S. and Ahuja, K. K., “Development of a Prediction Scheme for Noise of High-Aspect Ratio Jets,” AIAA paper 2003-3255, 2003, presented at the 9 th

AIAA/CEAS Aeroacoustics Conference and Exhibit, Hilton Head, South

Carolina, 12-14 May 2003.

20. Liu, Y., Sankar, L. N., Englar, R. J. and Ahuja, K. K., “Numerical Simulations of

the Steady and Unsteady Aerodynamic Characteristics of a Circulation ControlWing Airfoil,” AIAA paper 2001-0704, January 2001.

21. Anderson, J. D. Jr., Fundamentals of Aerodynamics, Second Edition, McGraw-

Hill, Inc. 1991.

22. Metral, A. R., “On the Phenomenon of Fluid Veins and their Application, the

Coanda Effect”, AF Translation, F-TS-786-RE, 1939.



151

23. Levinsky, E. S. and Yeh, T. T., “Analytical and Experimental Investigation of

Circulation Control by Means of a Turbulent Coanda Jet,” NASA CR-2114,

September, 1972.

24. Wood, N. J., “Circulation Control Airfoils: Past, Present and Future,” AIAA

paper 85-0204.

25. Englar, R. J. and Huson, G. G., “Development of Advanced Circulation Control

Wing High Lift Airfoils,” AIAA paper 83-1847, presented at AIAA AppliedAerodynamics Conference, July 1983.

26. Nichols, J. H. Jr, Englar, R. J., Harris, M. J. and Huson, G. G., “ExperimentalDevelopment of an Advanced Circulation Control Wing System for Navy STOL

Aircraft,” AIAA paper 81-0151, presented at AIAA Aerospace Science Meeting,

January 1981.

27. Englar, R. J., Hemmerly, R. A., Moore, H., Seredinsky, V., Valckenaere, W. G.and Jackson, J. A., “Design of the Circulation Control Wing STOL Demonstrator

Aircraft,” AIAA paper 79-1842, August, 1979.

28. Pugliese, A. J. and Englar, R. J., “Flight Testing the Circulation Control Wing,”

AIAA paper 79-1791, August 1979.

29. Englar, R. J., “Low-speed Aerodynamic Characteristics of a Small Fixed-

Trailing-Edge Circulation Control Wing Configuration Fitted to a Supercritical

Airfoil,” David Taylor Naval Ship Research and Development Center,DTNSRDC Report ASED-81/08, March 1981.

30. Englar, R. J., Niebur, J. S. and Gregory, S. D., “Pneumatic lift and control surfacetechnology for high speed civil transports,” Journal of Aircraft , vol. 36, no.2,

pp.332-339, Mar.-Apr., 1999.

31. Wilkerson, J. B., Reader, K. R. and Linck, D. W., “The Application of

Circulation Control Aerodynamics to a Helicopter Rotor Model,” American

Helicopter Society Paper AHS-704, May 1973.

32. Wilkerson, J. B., Barnes, D. R. and Bill, R. A., “The Circulation Control Rotor

Flight Demonstrator Test Program,” American Helicopter Society Paper AHS-

7951, May 1979.

33. Williams, R. M., Leitner, R. T. and Rogers, E. O., “X-Wing: A New Concept in

Rotary VTOL,” Presented at AHS Symposium on Rotor Technology, August1976.



152

34. Englar R. J., “Development of Pneumatic Aerodynamic Devices to Improve the

Performance, Economics, and Safety of Heavy Vehicles,” SAE Paper 2000-01-

2208, June 2000.

35. Cheeseman, I. C. and Seed, A. R., “The Application of Circulation Control by

Blowing to Helicopter Rotors,” Journal of the Royal Aeronautical Society, Vol.71, July 1966.

36. Kind, R. J., A Proposed Method of Circulation Control, Ph.D Thesis, CambridgeUniversity, England, 1967.

37. Williams, R. M. and Howe, H. J., “Two-Dimensional Subsonic Wind TunnelTests on a 20% thick, 5% Cambered Circulation Control Airfoil,” NSRDC TN

AL-176, Aug. 1970.

38. Englar, R. J., “Two-Dimensional Transonic Wind Tunnel Tests of Three 15-

Percent Thick Circulation Control Airfoils,” NSRDC Rept. ASED-182, Dec.1970.

39. Englar, R. J., “Circulation Control for High Lift and Drag Generation on STOL

Aircraft,” Journal of Aircraft , Vol. 12, No. 5, pp.457, May 1975.

40. Abramson, J., “Two Dimensional Subsonic Wind Tunnel Evaluation of a 20

percent Thick Circulation Control Airfoil,” DTNSRDC Report ASED-311, 1975.

41. Abramson, J. and Rogers, E. O., “High-Speed Characteristics of CirculationControl Airfoils,” AIAA paper 83-0265, 1983.

42. Englar, R. J. and Applegate, C. A., “Circulation Control – A Bibliography of DTNSRDC Research and Selected Outside References (January 1969 through

December 1983),” DTNSRDC Report 84-052, 1984.

43. Carpenter, P. W., Bridson, D. W. and Green, P. N., “Features of Discrete Tones

Generated by Jet Flow over Coanda Surfaces,” AIAA paper 86-1685, 1986.

44. Howe, M. S., “Noise Generated by a Coanda Wall Jet Circulation Control

Device,” Journal of Sound and Vibration, Vol. 249, No.4, pp. 679-700, Jan. 2002.

45. Dvorak, F. A. and Kind, R. J., “Analysis Method for Viscous Flow OverCirculation-Controlled Airfoils,” AIAA paper 78-157, January 1978.

46. Dvorak, F. A. and Choi, D. H., “Analysis of Circulation Controlled Airfoils inTransonic Flow,” Journal of Aircraft , Vol. 20, pp.331-337, April 1983.



153

47. Berman, H. A., “A Navier-Stokes Investigation of a Circulation Control Airfoil,”

Presented at AIAA 23rd

Aerospace Sciences Meeting, Reno, NV, Jan. 1985,

AIAA paper 85-0300.

48. Baldwin, B. S., and Lomax, H., “Thin Layer Approximation and Algebraic

Model for Separated Turbulent Flows,” AIAA Paper 78-257, Jan. 1978.

49. Pulliam, T. H., Jespersen, D. C. and Barth, T. J., “Navier-Stokes Computations

for Circulation Control Airfoils,” AIAA paper 85-1587.

50. Viegas, J. R., Rubesin, M. W. and MacCormack R. W., “ Navier-Stokes

Calculations and Turbulence Modeling in the Trailing Edge Region of ACirculation Control Airfoil,” Proceedings of the Circulation Control Workshop,

pp.1-22, 1986.

51. Spaid, F. W. and Keener, E. R., “Boundary-layer and Wake Measurements on a

Swept, Circulation-control Wing,” Proceedings of the Circulation ControlWorkshop, pp. 239-266, 1986.

52. Shrewsbury, G. D., “Analysis of Circulation Control Airfoils using an Implicit

Navier-Stokes Solver,” AIAA paper 85-0171, January 1985.

53. Shrewsbury, G. D., “Numerical Evaluation of Circulation Control Airfoil

Performance Using Navier-Stokes Methods,” AIAA paper 86-0286, January

1986.

54. Shrewsbury, G. D., “Numerical Study of a Research Circulation Control Airfoil

Using Navier-Stokes Methods,” Journal of Aircraft , Vol. 26, No. 1, pp.29-34,

1989.

55. Bradshaw, P., “Effects of Streamline Curvature on Turbulent Flow (applied to

boundary layer conditions on wing sections and turbomachine blades),” AGARD-AG-169, 1973.

56. Shrewsbury, G. D., Dynamic Stall of Circulation Control Airfoils, Ph.D Thesis,Georgia Institute of Technology, 1990.

57. Williams, S. L. and Franke, M. E., “Navier-Stokes Methods to Predict

Circulation Control Airfoil Performance,” Journal of Aircraft , Vol. 29, No.2,pp.243-249, March-April 1992.

58. Linton, S. W., “Computation of the Poststall Behavior of a Circulation ControlledAirfoil,” Journal of Aircraft , Vol. 31, No. 6, pp. 1273-1280, Nov. – Dec. 1994.



154

59. Liu, J., Sun, M., and L. Wu, “Unsteady Flow about a Circulation Control Airfoil”,Science in China (Series E), Vol. 39, No. 1, Feb. 1996.

60. Wang, C., and Sun, M., “Aerodynamic Properties of Circulation Control Airfoil

with Multi-slot Blowing”, Acta Aerodynamica Sinica, vol. 17, no. 4, pp.378-383,

1999.

61. Englar, R. J., Smith, M. J., Kelley, S. M. and Rober, R. C. III, “Development of

Circulation Control Technology for Application to Advanced Subsonic TransportAircraft,” AIAA paper 93-0644, presented at 31st Aerospace Sciences Meeting &

Exhibit, Reno, NV, January 1993.

62. Slomski, J. F., Gorski, J. J., Miller, R. W. and Marino, T. A., “Numerical

Simulation of Circulation Control Airfoils as Affected by Different Turbulence

Models,” AiAA paper 2002-0851, 2002.

63. Yang, Z. Y. and Voke, P. R., “Large-eddy Simulation of Boundary-layerSeparation and Transition at a Change of Surface Curvature,” Journal of Fluid

Mechanics, Vol. 439, pp. 305-333, 2001.

64. Li, J. and Liu, C., “Direct Numerical Simulation for Flow Separation Control

with Pulsed Jets,” 41st

Aerospace Sciences Meeting and Exhibit, Reno NV, 2003,AIAA paper 2003-0611.

65. Kwon, J. and Sankar, L.N., “Numerical Study of the Effects of Icing on Finite

Wing Aerodynamics,” AIAA Paper 90-0757.

66. Bangalore, A., Phaengsook, N. and Sankar, L. N., “Application of a Third Order

Upwind Scheme to Viscous Flow over Clean and Iced Wings,” AIAA Paper 94-0485.

67. Sankar, L. N. and Wake, B., “Solutions of the Navier-Stokes Equations for theFlow about a Rotor Blade,” In National Specialists' Meeting on Aerodynamics

and Aeroacoustics, Arlington, TX, Feb. 25-27, 1987.

68. Sankar, L. N., Ruo, S. Y., Wake, B. E. and Wu, J., “An Efficient Rrocedure for

the Numerical Solution of Three-dimensional Viscous Flows,” Computational

Fluid Dynamics Conference, 8th, Honolulu, HI, June 9-11, 1987, AIAA Paper 87-

1159.

69. Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., Computational Fluid

Mechanics and Heat Transfer , Second edition, Taylor & Francis, Washington,1997.



155

70. Schlichting, H., Boundary-Layer Theory, 7th ed., translated by J. Kestin,

McGraw-Hill, New York, 1979.

71. Vincenti, W. G., and Kruger, C. H., Introduction to Physical Gas Dynamics, John

Wiley, New York, 1965.

72. Stokes, G. G., “On the Theories of Internal Friction of Fluids in Motion,” Trans.

Cambridge Phil. Soc., Vol. 8, pp. 287-305, 1845.

73. Fourier, J., The Analytical Theory of Heat , 1822, trans. By A. Freeman, 1878;

reprinted by Dover, New York, 1955.

74. Chapman, S., and Cowling, T. G., The Mathematical Theory of Non-uniform

Gases, Cambridge University Press, Cambridge, 1939, 1952.

75. Douglas, J., “On the Numerical Integration of ut = uxx + uyy by Implicit

Methods,” Journal of Society of Industrial and Applied Mathematics, Vol. 3,No.1, March 1955.

76. Peaceman, D. W. and Rachford, H. H., “The Numerical Solution of Parabolic and

Elliptic Differential Equations,” Journal of Society of Industrial and Applied

Mathematics, Vol. 3., No.1, March 1955.

77. Briley, W. and McDonald, H., “Solution of Multi-Component Navier-Stokes

Equations by Generalized Implicit Method,” Journal of Computational Physics,

Vol.24, p. 372, 1977.

78. Wake, B. E., Solution Procedure for the Navier-Stokes Equations Applied to

Rotors, Ph.D Thesis, Georgia Institute of Technology, Atlanta, GA 1987.

79. Beam R. and Warming R. F., “An Implicit Finite Difference Algorithm for

Hyperbolic Systems in Conservation Law form”, Journal of Computational

Physics, Vol.22, Sep. 1976, pp.87-110.

80. Pulliam, T. H., and Chaussee, D. S., “A Diagonal Form of An ImplicitApproximation Factorization Algorithm,” Journal of Computational Physics, Vol.

39, pp. 347-363, 1981.

81. Rizk, Y. M. and Chausee, D. S., “Three-Dimensional Viscous-FlowComputations Using a Directionally Hybrid Implicit-Explicit Procedure,” AIAA

Paper 83-1910, 1983.



156

82. Tannehill, J. C., Holst, W. L. and Rakish, J. V. “Numerical Computation of Two-

Dimensional Viscous Blunt Body Flows with an Impinging Shock,” AIAA

Journal, Vol. 14, No.2, pp.204, 1976.

83. Von Neumann, J. and Richtmyer, R. D. “A Method for the Numerical Calculation

of Hydrodynamic Shocks,” Journal of Applied Physics, Vol. 21, pp. 232, 1950.

84. Lax, P. D. and Wendoff, B., “ Systems of Conservation Laws,” Communications

in Pure and Applied Mathematics, Vol. 13, pp. 217, 1960.

85. Lapidus, L., “A Detached Shock Calculation by Second-Order Finite-

Differences,” Journal of Computational Physics, Vol. 2, pp. 154, 1967.

86. Lindmuth, I. And Killeen, J., “Alternating Direction Implicit Techniques for

Two-Dimensional Magnetohydrodynamics Calculations,” Journal of

Computational Physics, Vol. 13, pp. 372, 1977.

87. McDonald, H. and Briley, W., “Three-Dimensional Supersonic Flow of a

Viscous or Inviscid Gas,” Journal of Computational Physics, Vol. 19, pp. 150-157, 1975.

88. Steger, J. L., “Implicit Finite-Difference Simulation of Flow about Arbitrary Two-Dimensional Geometry,” AIAA Journal, Vol. 16, No. 7, 1978.

89. Kim, J., Moin, P., and Moser, R., “Turbulence Statistics in Fully Developed

Channel Flow at Low Reynolds Number,” Journal of Fluid Mechanics, Vol. 177,1987, pp. 133-166.

90. Pope, S. B., Turbulent Flows, Cambridge University Press, 2000.

91. Spalart, P. R., and Allmaras, S. R., “A One-Equation Turbulence Model for

Aerodynamic Flows,” AIAA Paper 92-0439, Jan. 1992.

92. Cebeci, T., “Calculation of Compressible Turbulent Boundary Layers with Heat

and Mass Transfer,” AIAA Paper 70-741, AIAA 3rd

Fluid and Plasma DynamicsConference, Los Angeles, CA, June 1970.

93. Michel, R., and Arnal, D., “Investigation of the Conditions for Tripping

Transition with Roughness Elements and their Influence on Boundary LayerDevelopment,” ESA Boundary Layer Control by Transition Fixing (ESA-TT-

909), pp.103-113, 1984.

94. Eppler, R., and Fasel, H., “Laminar-turbulent Transition,” Proceedings of the

Symposium, Stuttgart, West Germany, September 16-22, 1979.



157

95. Bragg, M. B., and Spring, S. A., “An Experimental Study of the Flow Field about

an Airfoil with Glaze Ice,” Presented at the AIAA 25th

Aerospace Science

Meeting, Reno, Nevada, AIAA paper 87-0100, Jan 12-15, 1987.

96. Dancila, D. S. and Vasilescu, R., “Modeling of Piezoelectrically Modulated and

Vectored Blowing For a Wing Section,” AIAA paper 2003-0219, presented at41st AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, Jan. 6-9, 2003.

97. Bangalore, A. K., Sankar, L. N., and Tseng, W., “A Multi-zone Navier-StokesAnalysis of Dynamic Lift Enhancement Concepts,” AIAA, Aerospace Sciences

Meeting & Exhibit, 32nd, Reno, NV, Jan. 10-13, 1994, AIAA Paper 94-0164.

98. Bangalore, A. K., and Sankar, L. N., “Numerical Analysis of Aerodynamic

Performance of Rotors with Leading Edge Slats,” AIAA Applied Aerodynamics

Conference, 13th, San Diego, CA, June 19-22, 1995, AIAA Paper 95-1888.

99. Valarezo, W. O., Dominik, C. J., McGhee, R. J., and Goodman, W. L., “HighReynolds Number Configuration Development of a High-Lift Airfoil,” AGARD

Conference Proceedings 515, AGARD-CP-515, No.10, 1993.

100. Cock, K. de, “High-Lift System Analysis Method using Unstructured Meshes,”

AGARD Conference Proceedings 515, AGARD-CP-515, No.12, 1993.

101. Seifert, A., Darabi, A. and Wygnanski, I., “Delay of Airfoil Stall by Periodic

Excitation,” AIAA Journal of Aircraft , Vol. 33, No. 4, 1996.

102. Wygnanski, I, “Some New Observations Affecting the Control of Separation by

Periodic Excitation,” AIAA 2000-2314, FLUIDS 2000 Conference and Exhibit,

Denver, CO, 19-22 June, 2000.

103. Lorber, P.F., McCormick, D., Anderson, T., Wake, B.E., MacMartin, D., Pollack,

M., Corke, T., and Breuer, K., “Rotorcraft Retreating Blade-Stall Control,” AIAA2000-2475, FLUIDS 2000 Conference and Exhibit, Denver, CO, 19-22 June,

2000.

104. Wake, B. and Lurie, E. A., "Computational Evaluation of Directed Synthetic Jets

for Dynamic Stall Control,” AHS International Annual Forum, 57th, Washington,

DC, May 9-11, 2001, Proceedings (A02-12351 01-05), Alexandria, VA, AHS

International, 2001.

105. Hassan, A., and Janakiram, R. D., “Effects of Zero-Mass Synthetic Jets on the

Aerodynamics of the NACA 0012 Airfoil,” Journal of the American Helicopter

Society, Vol. 43, No. 4, October, 1998.



158

106. Oyler, T.E., and W.E. Palmer, “Exploratory Investigation of Pulse Blowing for

Boundary Layer Control,” North American Rockwell Report NR72H-12, Jan.

1972.

107. Schatz, M., and Thiele, F., “Numerical Study of High-Lift Flow with Separation

Control by Periodic Excitation,” AIAA Paper 2001-0296, Jan. 2001.

108. Sun, M., and Hamdani, H., “Separation Control by Alternating Tangential

Blowing/Suction at Multiple Slots,” AIAA Paper 2001-0297, Jan. 2001.

109. Radeztsky, R. H., Singer, B. A. and Khorrami, M. R., “Detailed Measurements of

a Flap Side-edge Flow Field,” AIAA paper 98-0700, Jan. 1998.

110. Khorrami, M. R., Singer, B. A. and Radeztsky, R. H., “Reynolds Averaged

Navier-Stokes Computations of a Flap Sied-edge Flow Field,” AIAA paper 98-

0768, Jan. 1998.



159

VITA

Yi Liu was born in Hunan province, China on June 1, 1973. He graduated from

Beijing University of Aeronautics & Astronautics, China, with a Bachelor of Science

(B.Sc) degree in Aerospace Engineering in July 1994. He worked as an aerospace

engineer in the Nanhua Jet Engine Research Institute from 1994 to 1995. Then he

continued his graduate study in the Department of Jet Propulsion of Beijing University of

Aeronautics & Astronautics, China, and earned a Master of Science (M.Sc) degree in

April 1998. In September 1998, he joined the Ph.D program in the School of Aerospace

Engineering at the Georgia Institute of Technology, Atlanta, Georgia. He is a student

member of the American Helicopter Society and the American Institute of Aeronautics

and Astronautics.



160

1 “Noise Standards: Aircraft Type and Airworthiness Certification,” Federal Aviation

Regulations Part 36, Federal Aviation Administration, November 1969.

2 “Policies and Procedures For Considering Environmental Impacts,” Federal Aviation

Administration Order 1050.1D, December 1986.

3 Goldin, D. S., et al., National Aeronautics and Space Administration Strategic Plan,

NPD 1000.1B, pp. 42-43, September 2000.

4 Huff, D., “Recent Progress in Engine Noise Reduction for Commercial Aircraft

Applications.” Seminar, School of Aerospace Engineering, Georgia Tech, April 8,

2003.

5 Smith, M. J. T., “Aircraft Noise,” Cambridge University Press, 1989.

6 Crighton, D. G., “Aircraft Noise in Aeronautics of Flight Vehicles: Theory and

Practice,” Vol. 1: Noise Sources, NASA PR 1258, pp. 391-447, 1991.

7 Sen, R., “A Study of Unsteady Fields Near Leading-edge Slats,” AIAA 97-1696, 1997.

8 Davy, R. and Remy, H., “Airframe Noise Characteristics on a 1/11 Scale Airbus

Model,” AIAA Paper 98-2335, June 1998.

9 Munro, S., “Jet Noise of High Aspect-Ratio Rectangular Nozzles with Application to

Pneumatic High-Lift Devices,” Ph.D Thesis, Georgia Institute of Technology, 2002.

10 Englar, R. J., “Circulation Control Pneumatic Aerodynamics: Blown Force and

Moment Augmentation and Modification; Past, Present & Future”, AIAA Paper 2000-

2541, June 2000.



161

11 Englar, Robert J., Smith, Marilyn J., Kelley, Sean M. and Rover, Richard C. III.,

“Application of Circulation Control to Advanced Subsonic Transport Aircraft, Part I:

Airfoil Development,” Journal of Aircraft, Vol.31 No.5, pp. 1160-1168, Sep. 1994.

12 Englar, Robert J., Smith, Marilyn J., Kelley, Sean M. and Rover, Richard C. III.,

“Application of Circulation Control to Advanced Subsonic Transport Aircraft, Part II:

Transport Application,” Journal of Aircraft, Vol.31, No.5, pp. 1169-1177, Sep. 1994.

13 Shrewsbury, G. D. and Sankar, L. N., “Dynamic stall of an oscillating circulation

control airfoil,” International Symposium on Nonsteady Fluid Dynamics, Toronto,

Canada, June 4-7, 1990, Proceedings. New York, American Society of Mechanical

Engineers, pp. 15-22, 1990.

14 Shrewsbury, G. D. and Sankar, L. N., “Dynamic Stall of Circulation Control Airfoils,”

AIAA Paper 90-0573, January 1990.

15 Salikuddin, M., Brown, W. H. and Ahuja, K. K., “Noise From a Circulation Control

Wing with Upper Surface Blowing,” Journal of Aircraft, Vol.24, pp55-64, Jan. 1987.

16 Munro, S., Ahuja, K., and Englar, R., “Noise Reduction Through Circulation Control

Technology,” AIAA Paper 2001-0666, Jan. 2001.

17 Munro, S. and Ahuja, K. K., “Aeroacoustics of a High Aspect-Ratio Jet,” AIAA paper

2003-3323, 2003, presented at the 9th AIAA/CEAS Aeroacoustics Conference and

Exhibit, Hilton Head, South Carolina, 12-14 May 2003.

18 Munro, S. and Ahuja, K. K., “Fluid Dynamics of a High Aspect-Ratio Jet,” AIAA

paper 2003-3129, 2003, presented at the 9th AIAA/CEAS Aeroacoustics Conference

and Exhibit, Hilton Head, South Carolina, 12-14 May 2003.



162

19 Munro, S. and Ahuja, K. K., “Development of a Prediction Scheme for Noise of High-

Aspect Ratio Jets,” AIAA paper 2003-3255, 2003, presented at the 9 th AIAA/CEAS

Aeroacoustics Conference and Exhibit, Hilton Head, South Carolina, 12-14 May 2003.

20 Liu, Y., Sankar, L. N., Englar, R. J. and Ahuja, K. K., “Numerical Simulations of the

Steady and Unsteady Aerodynamic Characteristics of a Circulation Control Wing

Airfoil,” AIAA paper 2001-0704, January 2001.

21 Anderson, J. D. Jr., “Fundamentals of Aerodynamics”, Second Edition, McGraw-Hill,

Inc. 1991.

22 Metral, A. R., “On the Phenomenon of Fluid Veins and their Application, the Coanda

Effect”, AF Translation, F-TS-786-RE, 1939.

23 Levinsky, E. S. and Yeh, T. T., “Analytical and Experimental Investigation of

Circulation Control by Means of a Turbulent Coanda Jet,” NASA CR-2114,

September, 1972.

24 Wood, N. J., “Circulation Control Airfoils: Past, Present and Future,” AIAA paper 85-

0204.

25 Englar, R. J. and Huson, G. G., “Development of Advanced Circulation Control Wing

High Lift Airfoils,” AIAA paper 83-1847, presented at AIAA Applied Aerodynamics

Conference, July, 1983.

26 Nichols, J. H. Jr, Englar, R. J., Harris, M. J. and Huson, G. G., “Experimental

Development of an Advanced Circulation Control Wing System for Navy STOL

Aircraft,” AIAA paper 81-0151, presented at AIAA Aerospace Science Meeting,

January, 1981.



163

27 Englar, R. J., Hemmerly, R. A., Moore, H., Seredinsky, V., Valckenaere, W. G. and

Jackson, J. A., “Design of the Circulation Control Wing STOL Demonstrator Aircraft,”

AIAA paper 79-1842, August, 1979.

28 Pugliese, A. J. and Englar, R. J., “Flight Testing the Circulation Control Wing,” AIAA

paper 79-1791, August, 1979.

29 Englar, R. J., “Low-speed Aerodynamic Characteristics of a Small Fixed-Trailing-

Edge Circulation Control Wing Configuration Fitted to a Supercritical Airfoil,” David

W. Taylor Naval Ship Research and Development Center, DTNSRDC Report ASED-

81/08, March, 1981.

30 Englar, R. J., Niebur, J. S. and Gregory, S. D., “Pneumatic lift and control surface

technology for high speed civil transports,” Journal of Aircraft, vol. 36, no.2, pp.332-

339, Mar.-Apr., 1999.

31 Wilkerson, J. B., Reader, K. R. and Linck, D. W., “The Application of Circulation

Control Aerodynamics to a Helicopter Rotor Model,” American Helicopter Society

Paper AHS-704, May, 1973.

32 Wilkerson, J. B., Barnes, D. R. and Bill, R. A., “The Circulation Control Rotor Flight

Demonstrator Test Program,” American Helicopter Society Paper AHS-7951, May,

1979.

33 Williams, R. N., Leitner, R. T. and Rogers, E. O., “X-Wing: A New Concept in

Rotary VTOL,” Presented at AHS Symposium on Rotor Technology, August, 1976.



164

34 Englar R. J., “Development of Pneumatic Aerodynamic Devices to Improve the

Performance, Economics, and Safety of Heavy Vehicles,” SAE Paper 2000-01-2208,

June, 2000.

35 Cheeseman, I. C. and Seed, A. R., “The Application of Circulation Control by

Blowing to Helicopter Rotors,” Journal of the Royal Aeronautical Society, Vol. 71,

July, 1966.

36 Kind, R. J., “A Proposed Method of Circulation Control,” Ph.D Thesis, Cambridge

University, England.

37 Williams, R. M. and Howe, H. J., “Two-Dimensional Subsonic Wind Tunnel Tests on

a 20% thick, 5% Cambered Circulation Control Airfoil,” NSRDC TN AL-176, Aug.

1970.

38 Englar, R. J., “Two-Dimensional Transonic Wind Tunnel Tests of Three 15-Percent

Thick Circulation Control Airfoils,” NSRDC Rept. ASED-182, Dec. 1970.

39 Englar, R. J., “Circulation Control for High Lift and Drag Generation on STOL

Aircraft,” Journal of Aircraft, Vol. 12, No. 5, pp.457, May 1975.

40 Abramson, J., “Two Dimensional Subsonic Wind Tunnel Evaluation of a 20 percent

Thick Circulation Control Airfoil,” DTNSRDC Report ASED-311, 1975.

41 Abramson, J. and Rogers, E. O., “High-Speed Characteristics of Circulation Control

Airfoils,” AIAA paper 83-0265, 1983.

42 Englar, R. J. and Applegate, C. A., “Circulation Control – A Bibliography of

DTNSRDC Research and Selected Outside References (January 1969 through

December 1983),” DTNSRDC Report 84-052, 1984.



165

43 Carpenter, P. W., Bridson, D. W. and Green, P. N., “Features of Discrete Tones

Generated by Jet Flow over Coanda Surfaces,” AIAA paper 86-1685, 1986.

44 Howe, M. S., “Noise Generated by a Coanda Wall Jet Circulation Control Device,”

Journal of Sound and Vibration, Vol. 249, No.4, pp. 679-700, Jan. 2002.

45 Dvorak, F. A. and Kind, R. J., “Analysis Method for Viscous Flow Over Circulation-

Controlled Airfoils,” Journal of Aircraft, Vol. 16, No. 1, January 1979.

46 Dvorak, F. A. and Choi, D. H., “Analysis of Circulation Controlled Airfoils in

Transonic Flow,” Journal of Aircraft, Vol. 20, April 1983.

47 Berman, H. A., “A Navier-Stokes Investigation of a Circulation Control Airfoil,”

AIAA paper 85-0300.

48 Baldwin, B. S., and Lomax, H., “Thin Layer Approximation and Algebraic Model for

Separated Turbulent Flows,” AIAA Paper 78-257, Jan. 1978.

49 Pulliam, T. H., Jespersen, D. C. and Barth, T. J., “Navier-Stokes Computations for

Circulation Control Airfoils,” AIAA paper 85-1587.

50 Viegas, J. R., Rubesin, M. W. and MacCormack R. W., “ Navier-Stokes Calculations

and Turbulence Modeling in the Trailing Edge Region of A Circulation Control

Airfoil,” NASA CP-2432.

51 Spaid, F. W. and Keener, E. R., “Boundary-layer and wake measurements on a swept,

circulation-control wing,” AIAA, Aerospace Sciences Meeting, 25th, Reno, NV, Jan.

12-15, 1987, AIAA Paper 87-0156.

52 Shrewsbury, G. D., “Analysis of Circulation Control Airfoils using an Implicit Navier-

Stokes Solver,” AIAA paper 85-0171, January 1985.



166

53 Shrewsbury, G. D., “Numerical Evaluation of Circulation Control Airfoil Performance

Using Navier-Stokes Methods,” AIAA paper 86-0286, January 1986.

54 Shrewsbury, G. D., “Numerical Study of a Research Circulation Control Airfoil Using

Navier-Stokes Methods,” Journal of Aircraft, Vol. 26, No. 1, pp.29-34, 1989.

55 Bradshaw, P., “Effects of Streamline Curvature on Turbulent Flow (applied to

boundary layer conditions on wing sections and turbomachine blades),” AGARD-AG-

169, 1973.

56 Shrewsbury, G. D., “Dynamic Stall of Circulation Control Airfoils,” Ph.D Thesis,

Georgia Institute of Technology, 1990.

57 Williams, S. L. and Franker, M. E., “Navier-Stokes Methods to Predict Circulation

Control Airfoil Performance,” Journal of Aircraft, Vol. 29, No.2, pp.243-249, March-

April 1992.

58 Linton, S. W., “Computation of the Poststall Behavior of a Circulation Controlled

Airfoil,” Journal of Aircraft, Vol. 31, No. 6, pp. 1273-1280, Nov. – Dec. 1994.

59 Liu, J., Sun, M., and L. Wu, “Unsteady Flow about a Circulation Control Airfoil”,

Science in China (Series E), Vol. 39, No. 1, Feb. 1996.

60 Wang, C., and Sun, M., “Aerodynamic Properties of Circulation Control Airfoil with

Multi-slot Blowing”, Acta Aerodynamica Sinica, vol. 17, no. 4, pp.378-383, 1999.

61 Englar, R. J., Smith, M. J., Kelley, S. M. and Rober, R. C. III, “Development of

Circulation Control Technology for Application to Advanced Subsonic Transport

Aircraft,” AIAA paper 93-0644, presented at 31st Aerospace Sciences Meeting &

Exhibit, Reno, NV, January 1993.



167

62 Slomski, J. F., Gorski, J. J., Miller, R. W. and Marino, T. A., “Numerical Simulation

of Circulation Control Airfoils as Affected by Different Turbulence Models,” AiAA

paper 2002-0851, 2002.

63 Yang, Z. Y. and Voke, P. R., “Large-eddy simulation of boundary-layer separation

and transition at a change of surface curvature,” Journal of Fluid Mechanics, Vol. 439,

pp. 305-333, 2001.

64 Li, J. and Liu, C., “Direct Numerical Simulation for Flow Separation Control with

Pulsed Jets,” 41

st

Aerospace Sciences Meeting and Exhibit, Reno NV, 2003, AIAA

paper 2003-0611.

65 Kwon, J. and Sankar, L.N., “Numerical Study of the Effects of Icing on Finite Wing

Aerodynamics,” AIAA Paper 90-0757.

66 Bangalore, A., Phaengsook, N. and Sankar, L. N., “Application of a Third Order

Upwind Scheme to Viscous Flow over Clean and Iced Wings,” AIAA Paper 94-0485.

67 Sankar, L. N. and Wake, B., “Solutions of the Navier-Stokes equations for the flow

about a rotor blade,” In National Specialists' Meeting on Aerodynamics and

Aeroacoustics, Arlington, TX, Feb. 25-27, 1987.

68 Sankar, L. N., Ruo, S. Y., Wake, B. E. and Wu, J., “An efficient procedure for the

numerical solution of three-dimensional viscous flows,” Computational Fluid

Dynamics Conference, 8th, Honolulu, HI, June 9-11, 1987, AIAA Paper 87-1159.

69 Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., “Computational Fluid

Mechanics and Heat Transfer,” Second edition, Taylor & Francis, Washington, 1997.



168

70 Schlichting, H., “Boundary-Layer Theory,” 7th

ed., translated by J. Kestin, McGraw-

Hill, New York, 1979.

71 Vincenti, W. G., and Kruger, C. H., “Introduction to Physical Gas Dynamics,” John

Wiley, New York, 1965.

72 Stokes, G. G., “On the Theories of Internal Friction of Fluids in Motion,” Trans.

Cambridge Phil. Soc., Vol. 8, pp. 287-305, 1845.

73 Fourier, J., “The Analytical Theory of Heat”, 1822, trans. By A. Freeman, 1878;

reprinted by Dover, New York, 1955.

74 Chapman, S., and Cowling, T. G., “The Mathematical Theory of Non-uniform Gases,”

Cambridge University Press, Cambridge, 1939, 1952.

75 Douglas, J., “On the Numerical Integration of ut = uxx + uyy by Implicit Methods,”

Journal of Society of Industrial and Applied Mathematics, Vol. 3, No.1, March 1955.

76 Peaceman, D. W. and Rachford, H. H., “The Numerical Solution of Parabolic and

Elliptic Differential Equations,” Journal of Society of Industrial and Applied

Mathematics, Vol. 3., No.1, March 1955.

77 Briley, W. and McDonald, H., “Solution of Multi-Component Navier-Stokes

Equations by Generalized Implicit Method,” Journal of Computational Physics, Vol.24,

p. 372, 1977

78 Wake, B. E., “Solution Procedure for the Navier-Stokes Equations Applied to Rotors,”

Ph.D Thesis, Georgia Institute of Technology, Atlanta, GA 1987.



169

79 Beam R. and Warming R. F., “An Implicit Finite Difference Algorithm for Hyperbolic

Systems in Conservation Law form”, Journal of Computational Physics, Vol.22, Sep.

1976, pp.87-110.

80 Pulliam, T. H., and Chaussee, D. S., “A Diagonal Form of An Implicit Approximation

Factorization Algorithm,” Journal of Computational Physics, Vol. 39, 1981, pp. 347-

363.

81 Rizk, Y. M. and Chausee, D. S., “Three-Dimensional Viscous-Flow Computations

Using a Directionally Hybrid Implicit-Explicit Procedure,” AIAA Paper 83-1910,

1983.

82 Tannehill, J. C., Holst, W. L. and Rakish, J. V. “Numerical Computation of Two-

Dimensional Viscous Blunt Body Flows with an Impinging Shock,” AIAA Journal,

Vol. 14, No.2, p.204, 1976.

83 Von Neumann, J. and Richtmyer, R. D. “A Method for the Numerical Calculation of

Hydrodynamic Shocks,” Journal of Applied Physics, Vol. 21, p. 232, 1950.

84 Lax, P. D. and Wendoff, B., “ Systems of Conservation Laws,” Communications in

Pure and Applied Mathematics, Vol. 13, p. 217, 1960.

85 Lapidus, L., “A Detached Shock Calculation by Second-Order Finite-Differences,”

Journal of Computational Physics, Vol. 2, p. 154, 1967.

86 Lindmuth, I. And Killeen, J., “Alternating Direction Implicit Techniques for Two-

Dimensional Magnetohydrodynamics Calculations,” Journal of Computational Physics,

Vol. 13, p. 372, 1977.



170

87 McDonald, H. and Briley, W., “Three-Dimensional Supersonic Flow of a Viscous or

Inviscid Gas,” Journal of Computational Physics, Vol. 19, p. 150, 1975.

88 Steger, J. L., “Implicit Finite-Difference Simulation of Flow about Arbitrary Two-

Dimensional Geometry,” AIAA Journal, Vol. 16, No. 7, 1978.

89 Kim, J., Moin, P., and Moser, R., “Turbulence Statistics in Fully Developed Channel

Flow at Low Reynolds Number,” Journal of Fluid Mechanics, Vol. 177, 1987, pp. 133-

166.

90 Pope, S. B., “Turbulent Flows,” Cambridge University Press, 2000.

91 Spalart, P. R., and Allmaras, S. R., “A One-Equation Turbulence Model for

Aerodynamic Flows,” AIAA Paper 92-0439, Jan. 1992.

92 Cebeci, T., “Calculation of Compressible Turbulent Boundary Layers with Heat and

Mass Transfer,” AIAA Paper 70-741, AIAA 3rd Fluid and Plasma Dynamics

Conference, Los Angeles, CA, June 1970.

93 Michel, R., and Arnal, D., “Investigation of the conditions for tripping transition with

roughness elements and their influence on boundary layer development,” ESA

Boundary Layer Control by Transition Fixing (ESA-TT-909), pp.103-113, 1984.

94 Eppler, R., and Fasel, H., “Laminar-turbulent transition,” Proceedings of the

Symposium, Stuttgart, West Germany, September 16-22, 1979.

95 Bragg, M. B., and Spring, S. A., “An Experimental Study of the Flow Field about an

Airfoil with Glaze Ice,” Presented at the AIAA 25th Aerospace Science Meeting, Reno,

Nevada, AIAA paper 87-0100, Jan 12-15, 1987.



171

96 Dancila, D. S. and Vasilescu, R., “Modeling of Piezoelectrically Modulated and

Vectored Blowing For a Wing Section,” AIAA paper 2003-0219, presented at 41st

AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, Jan. 6-9, 2003.

97 Bangalore, A. K., Sankar, L. N., and Tseng, W., “A multi-zone Navier-Stokes analysis

of dynamic lift enhancement concepts,” AIAA, Aerospace Sciences Meeting &

Exhibit, 32nd, Reno, NV, Jan. 10-13, 1994, AIAA Paper 94-0164.

98 Bangalore, A. K., and Sankar, L. N., “Numerical analysis of aerodynamic

performance of rotors with leading edge slats,” AIAA Applied Aerodynamics

Conference, 13th, San Diego, CA, June 19-22, 1995, AIAA Paper 95-1888.

99 Valarezo, W. O., Dominik, C. J., McGhee, R. J., and Goodman, W. L., “High

Reynolds Number Configuration Development of a High-Lift Airfoil,” AGARD

Conference Proceedings 515, AGARD-CP-515, No.10, 1993.

100 Cock, K. de, “High-Lift System Analysis Method using Unstructured Meshes,”

AGARD Conference Proceedings 515, AGARD-CP-515, No.12, 1993.

101 Seifert, A., Darabi, A. and Wygnanski, I., “Delay of Airfoil Stall by Periodic

Excitation,” AIAA Journal of Aircraft, Vol. 33, No. 4, 1996.

102 Wygnanski, I, “Some New Observations Affecting the Control of Separation by

Periodic Excitation,” AIAA 2000-2314, FLUIDS 2000 Conference and Exhibit,

Denver, CO, 19-22 June, 2000.

103 Lorber, P.F., McCormick, D., Anderson, T., Wake, B.E., MacMartin, D., Pollack,

M., Corke, T., and Breuer, K., “Rotorcraft Retreating Blade-Stall Control,” AIAA

2000-2475, FLUIDS 2000 Conference and Exhibit, Denver, CO, 19-22 June, 2000.



104 Wake, B. and Lurie, E. A., "Computational Evaluation of Directed Synthetic Jets for

Dynamic Stall Control," to appear in the American Helicopter Society Annual Forum,

2001.

105 Hassan, A., and Janakiram, R. D., “Effects of Zero-Mass Synthetic Jets on the

Aerodynamics of the NACA 0012 Airfoil,” Journal of the American helicopter Society,

Vol. 43, No. 4, October, 1998.

106 Oyler, T.E., and W.E. Palmer, “Exploratory Investigation of Pulse Blowing for

Boundary Layer Control,” North American Rockwell Report NR72H-12, Jan. 1972.

Circulation Control Wing

Documents

Transcript of Circulation Control Wing