NUMERICAL SIMULATIONS OF THE AERODYNAMIC CHARACTERISTICS OF
CIRCULATION CONTROL WING SECTIONS
A Thesis
Presented to
The Academic Faculty
by
Yi Liu
In Partial Fulfillment
of 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
DEDICATION
To my wife, Qiang Le
And to my parents
iii
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.
iv
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 ............................................................................... 5
1.2.1 The Circulation Control Wing Concept ............................................................. 5
1.2.2 The Advanced Circulation Control Airfoil ........................................................ 8
1.2.3 Applications and Benefits of the Circulation Control Wing ............................ 10
1.3 Previous Research Work ......................................................................................... 13
1.4 Overview of the Present Work ................................................................................ 18
2. MATHEMATICAL AND NUMERICAL FORMULATION ................................ 21
2.1 The Governing Equations ....................................................................................... 21
2.1.1 Governing Equations in Cartesian Coordinates ............................................... 22
2.1.2 Governing Equations in Curvilinear Coordinates ............................................ 28
2.2 Numerical Procedure .............................................................................................. 33
2.2.1 Temporal Discretization................................................................................... 34
v
2.2.2 Linearization of the Difference Equations ....................................................... 35
2.2.3 Approximate Factorization Procedure ............................................................. 38
2.2.4 Spatial Discretization of the Inviscid Terms .................................................... 38
2.2.5 Spatial Discretization of the Viscous Terms .................................................... 40
2.2.6 Implementation of Low Pass Filters ................................................................ 41
2.3 Turbulence Models ................................................................................................. 44
2.3.1 Baldwin-Lomax Turbulence Model ................................................................. 46
2.3.2 Spalart-Allmaras Turbulence Model................................................................ 48
2.4 Initial and Boundary Conditions ............................................................................. 50
2.4.1 Initial Conditions ............................................................................................. 51
2.4.2 Outer Boundary Conditions ............................................................................. 51
2.4.3 Solid Surface Conditions ................................................................................. 52
2.4.4 Boundary Conditions at the Cuts in the C Grid ............................................... 54
2.4.5 Jet Slot Exit Conditions with Given C ........................................................... 55
2.4.6 Jet Slot Exit Conditions with Given Total Jet Pressure ................................... 58
3. TWO DIMENSIONAL STEADY BLOWING RESULTS ..................................... 60
3.1 Code Validations with a NACA 0012 Wing........................................................... 60
3.2 Unblown and Steady Blowing Results ................................................................... 62
3.2.1 Configuration Modeled .................................................................................... 62
3.2.2 Computational Grid ......................................................................................... 62
3.2.3 Blowing and Unblown Results Comparison .................................................... 64
3.2.4 Steady Blowing with Specified Total Pressure ................................................ 67
vi
3.3 Effects of Parameters that Influence the Momentum Coefficient .......................... 68
3.3.1 Free-stream Velocity Effects with Fixed C and Fixed Jet Slot Height .......... 69
3.3.2 Jet Slot Height Effects with Fixed C and Fixed Free-stream Velocity .......... 70
3.4 Other Simulations for the CC Airfoil...................................................................... 71
3.4.1 Comparisons with the Conventional High-Lift System ................................... 71
3.4.2 Leading Edge Blowing .................................................................................... 72
4. TWO DIMENSIONAL PULSED BLOWING RESULTS...................................... 91
4.1 Jets Pulsed Sinusoidally .......................................................................................... 92
4.2 Jets Pulsed with a Square Wave Form .................................................................... 93
4.2.1 Pulsed Jet Flow Behavior................................................................................. 94
4.2.2 Effects of Frequency at a Fixed C .................................................................. 96
4.2.3 Strouhal Number Effects.................................................................................. 98
4.3 Summary of Observations..................................................................................... 100
5. THREE DIMENSION CIRCULATION CONTROL WING SIMULATIONS . 115
5.1 Tangential Blowing on a Wing-flap Configuration .............................................. 115
5.2 Spanwise Blowing over a Rounded Wing-tip ....................................................... 119
6. CONCLUSIONS AND RECOMMENDATIONS .................................................. 135
6.1 Conclusions ........................................................................................................... 136
6.2 Recommendations ................................................................................................. 138
APPENDIX A. GENERALIZED TRANSFORMATION........................................ 141
REFERENCES .............................................................................................................. 146
VITA............................................................................................................................... 155
vii
LIST OF TABLES
Table
4.1
Page
The Computed Time-averaged Lift Coefficient for the Case One
105
(U and Lref fixed, Strouhal number varying with the frequency)
4.2
The Computed Time-averaged Lift Coefficient for the Case Two
105
(Strouhal number and Lref fixed, U varying with the frequency)
4.3
The Computed Time-averaged Lift Coefficient for the Case Three
106
(Strouhal number and U fixed, Lref varying with the frequency)
5.1
The Total Lift Coefficient and Drag Coefficient for the Wing Tip
Configuration
viii
123
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 (C = 0.15)
79
3.6
Variation of the Lift Coefficient with Momentum Coefficients at = 0
80
ix
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
82
(U = 94.3 ft/sec)
3.10
Time History of the Lift Coefficient for the Unblown Case
82
(U = 220 ft/sec)
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 C Variation with the Total Jet Pressure for Steady Blowing Case
85
3.14
The Lift Coefficient Variation with C for Steady Blowing Case
85
3.15
Lift Coefficient vs. Free-stream Velocity
86
(C = 0.1657, h = 0.015 inch and V, exp = 94.3 ft/sec)
3.16
Drag Coefficient vs. Free-stream Velocity
86
(C = 0.1657, h = 0.015 inch and V, exp = 94.3 ft/sec)
3.17
Mass Flow Rate vs. Free-stream Velocity
87
(C = 0.1657, h = 0.015 inch and V, exp = 94.3 ft/sec)
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
x
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+ C) for the Multi-element Airfoil and the CC
90
Airfoil
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
107
(Sinusoidal Wave, Frequency = 400 Hz, C,0 = 0.04)
4.2
The Time History of the Lift Coefficient
107
(Sinusoidal Wave, Frequency = 400 Hz, C,0 = 0.04)
4.3
The Time History of the Mass Flow Rate
108
(Sinusoidal Wave, Frequency = 400 Hz, C,0 = 0.04)
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
109
(Square Wave Form, Frequency = 40 Hz, C,0 = 0.04)
xi
Figure
4.7
Page
The Time History of the Lift Coefficient
110
(Square Wave Form, Frequency = 40 Hz, C,0 = 0.04)
4.8
The Time History of the Mass Flow Rate
110
(Square Wave Form, Frequency = 40 Hz, C,0 = 0.04)
4.9
The Incremental Lift Coefficient vs. Time-averaged Momentum
111
Coefficient
4.10
The Incremental Lift Coefficient vs. Time-averaged Mass Flow Rate
111
4.11
Time-averaged Mass Flow Rate vs. Time-averaged Momentum
112
Coefficient
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
113
(Ave. C,0 = 0.04)
4.15
The Efficiency vs. Pulsed Jet Frequency (Ave. C,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
xii
Figure
5.3
Page
The Lift Coefficient Distribution along Span for the Wing-flap
126
Configuration
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
130
(Side View at Spanwise Station)
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
5.13
The Vorticity Contours around the Wing Tip (x/C = 1.0)
133
5.14
The Vorticity Contours around the Wing Tip (x/C = 1.50)
134
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
136
Configuration
5.17
The Drag Coefficient Distribution along Span for Wing Tip
Configuration
xiii
136
LIST OF NOMENCLATURE
a
Speed of sound
ajet
Speed of sound of the jet
Ajet
Area of the jet slot
A, B, C
Flux Jacobian matrices
Cp
Specific heat at constant pressure
Cv
Specific heat at constant volume
Cl, CL
Lift coefficient
Cd , CD
Drag coefficient
C
Jet momentum coefficient
C,0
Time-averaged momentum coefficient for pulsed jets
Et
Total energy per unit volume
E, F, G
Inviscid flux matrices
f
Frequency
F+
Non-dimension frequency
Fkleb
Klebanoff intermittency correction
J
Jacobian of transformation
k
Thermal conductivity
Kc
Clauser’s constant
Lref
Reference length
lm
Mixing length
xiv

m
Mass flow rate

n
Normal vector of cell surface
M
Free-stream Mach number
Mjet
Jet Mach number
O(.)
Order of variable
P
Pressure
Pjet
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
qx, qy, qz
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
Tjet
Temperature at the jet slot exit
T0, jet
Total temperature at the jet slot exit, Duct temperature
x, y, z
Cartesian coordinates
u, v, w
Cartesian velocities
xv
U, V, W
Contravariant velocities
Ujet
Jet velocity from CFD calculation
Va
Jet velocity obtained in experiments

Forward difference operator

Backward difference operator

Angle of attack

Central difference operator
ij
Kronecker Delta function

Turbulent dissipation rate

Specific heat ratio

Second coefficient of viscosity
, , 
Eigenvalues

Coefficient of viscosity

Kinematic viscosity

Density
jet
Jet density

Non-dimensional time
ij
Viscous stress

Vorticity
, , 
Computational domain coordinates
xvi
Subscripts
i, j, k
Indices in three coordinate directions
t
Derivative with respect to physical time
w
Variable on the wall surface

Derivative with respect to time in (, , ) coordinates
, , 
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
xvii
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
xviii
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, especially at a high frequency, and that the Strouhal number has a more
xix
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.
xx
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 contribute to this. The
second is the airframe noise, which is generated by components such as fuselage, wing, under-
1
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.
History
Current
Future Goals
JT3D, JT8D, JT9D,CF6,CFM56 JT8D-200,PW2000,PW4000,V2500,CF6,GE90
10.0
Stage 2
B-737-200
B-737-200
Negotiated
Out of Service*
DC9-10
0.0
B-727-200A
v
in erag
Se e
B-747-100
rvi
ce
B-747-200
B-727-100
B-727-100
DC-10-40
Average Noise
Level Relative
to Stage 3
(EPNdB)
B-747-200
Stage 3
B-747-200
B-747-SP
MD-80
A300
A310-222
A300B4-620
B-747-300
MD-82
A300-600R
MD-87
MD-11 A330-300
757-200
A310-300
A320-200
767-300ER
777-200
747-400
Stage 4
MD-90-30
-10.0
Impact of current Noise
Reduction program goal of 5 dB
Impact of achieving NASA goal
(additional 5 dB)
-20.0
1960
1970
1980
1990
2000
2010
2020
Year of Certification
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 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
2
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]
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,
3
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 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 =0°) may be achieved with Circulation
4
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.
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
5
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 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.
6
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 parameters, including the jet momentum coefficient C , the
turbulent characteristics of the jet and the Reynolds number [23]. The jet momentum coefficient
C is defined as, C 
 Vjet
m
qS
, and will be discussed in the next chapter.
Also as shown in Figure 1.4, at low momentum coefficients, especially when C is less
than 0.01 [23], the tangential blowing will add some energy to the slow moving flow near the
7
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/C, 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 evidence, Wood predicts that, for low C 
values, CL varies linearly with C, while for higher C values, the C L  C  with low freestream 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/C = 0.
1.2.2 The Advanced Circulation Control Airfoil
The earlier designs of the CC airfoils used rounded trailing edges with large radius to
8
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. 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 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.
9
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.
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
10
aircraft operation from carriers, the Navy sponsored a full-scale flight test program on an A6/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 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
11
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, 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
12
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.
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
13
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.
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.
14
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 simple potential methods until the 1980s. Due to this highlycoupled, 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
15
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 Reynoldsaverage 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
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
16
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 BaldwinLomax and - 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 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 C, 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 -model, the modified - model, and a
full Reynolds stress model. It was found that for small C, the - and modified- turbulence
models could predict the lift generated by the CC airfoil reasonably well. However, at higher C ,
17
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.
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 2D 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:

Can 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
C, pulsed jet frequency, wave shape and duty cycle? What are the benefits and
drawbacks of pulsed jets relating to steady jets?

In 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?
18
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.
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
19
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.
20
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.
2.1 The Governing Equations
21
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]:
q E F G R S T






t x y z x y z
(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.
The flow vector and the inviscid flux terms are:
22
 
 
u 
 
q  v 
 
w 
 
E t 
(2.2)
u



u 2  p 




E  uv



uw



(E t  p)u 
v



uv



 2

F  v  p 


vw



(E t  p) v 
w



uw





G  uw



w 2  p 


( E t  p) w 
Here, Et is the total energy, and it can be expressed as:
1


E t   C v T  ( u 2  v 2  w 2 ) 
2


(2.3)
In above equations, the density , the velocity components (u, v, w) in the (x, y, z)
directions and total energy Et are the unknown flow parameters. The pressure is related to the
total energy and velocities by the following equations:
p  RT
(2.4)
1


p  (   1) E t  (u 2  v 2  w 2 )
2


(2.5)
and:
In Equation (2.5),  is the specific heat ratio. Since the working fluid is air, a value of 1.4 is used.
23
The viscous terms in equation (2.1) are:
0 
 
 xx 
 
 
R   xy  ,
 
 xz 
 
E x 5 
0 
 
 
 yx 
 
S   yy  ,
 
 yz 
 
E 
 y5 
0 
 
 zx 
 
 
T   zy 
 
 zz 
 
E z 5 
(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:
E i 5  u j ij  q i
 u
u j 
   u k  ij
ij   i 
 x

x k
 j x i 
i, j  1, 2, 3
(2.7)
where 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 . 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,
2
 
3
(2.8)
The stress terms and heat transfer terms in equation (2.6) can now be written as follows:
24
 xx 
2  u v w 

 2


3  x y z 
 u v 
 xy     
 y x 
 u w 
 xz    
 z x 
2  v u w 

 yy   2 

3  y x z 
(2.9)
 v w 
 yz    

 z y 
 zz 
2  w u v 
 2

 
3  z x y 
E x 5  u xx  v xy  w xz  q x
E y5  u xy  v yy  w yz  q y
(2.10)
E z 5  u xz  v yz  w zz  q z
Under local equilibrium conditions, Fourier’s law [73] is used to relate the heat transfer rates qx,
qy and qz with the temperature gradient:
T
x
T
q y  k
y
T
q z  k
z
q x  k
25
(2.11)
The thermal conductivity, k, can be related to the molecular viscosity using the kinetic
theory of gases [74]:
k
C P C V

Pr
Pr
(2.12)
where Cp is the specific heat at constant pressure. For a calorically perfect gas, it is a constant
and defined as CP 
R
. Here R is the gas constant and  is the specific heat ratio, which is
 1
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:
E

1
a  RT     1 t  u 2  v 2  w 2 
 2



(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, 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:
26
L ref  Chord of the airfoil
Vref  a  , Freestream speed of sound
 ref    , Freestream density
(2.14)
 ref    , Freestream vis cos ity
Tref  T , Freestream temperatur e
The non-dimensional flow variables are expressed as follows:
x* 
x
L ref
y* 
y
L ref
z* 
z
L ref
t* 
u* 
u
Vref
v* 
v
Vref
w* 
w
Vref
* 

 ref
* 

 ref
p* 
p
2
 ref Vref
E *t 
Et
2
 ref Vref
T* 
T
Tref
L ref
t
/ Vref
(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:
M 
V
RT
Re Lref 
 V L ref

(2.16)
For a detailed description and expression of the non-dimensional equations, the reader is referred
to Reference [69]. In the following discussions, all variables (, u, v, p etc) are non-dimensional.
The asterisk has been dropped for convenience.
27
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 NavierStokes equations can be written in terms of a generalized non-orthogonal curvilinear coordinate
system (  ,  ,  ) using the generalized transformation described in Appendix A:
   ( x , y, z , t )
  ( x , y, z, t )
   ( x , y, z, t )
(2.17)
t
Applying the transformation to the equation (2.1), the following non-dimensional
governing equations in the curvilinear coordinate system can be obtained.
ˆ  M  (R
ˆ  Fˆ  G
ˆ  Sˆ  T
ˆ )
qˆ   E






Re
(2.18)

 u 
1  
qˆ   v 
J 
w
 
 e 
(2.19)
Here,
28
J is the Jacobian of transformation, and it is given by:
J
1
y (x  z   x z  )  y (x  z   x  z  )  y (x z   x  z  )
(2.20)
ˆ and R
ˆ , Sˆ , T
ˆ are related to their counterparts E, F, G, and R, S, T as
The quantities Eˆ , Fˆ , G
follows:
ˆ  1 E  F  G  q 
E
x
y
z
t
J
1
Fˆ  E x  F y  Gz  q t 
J
ˆ  1 E  F  G  q 
G
x
y
z
t
J
(2.21)
ˆ  1 R  S  T 
R
x
y
z
J
1
Sˆ  Rx  S y  Tz 
J
ˆ  1 R  S  T 
T
x
y
z
J
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:
29
U  ( u  x  ) x  ( v  y  ) y  ( w  z  ) z
  t  u x  v y  w z
V  (u  x  )x  ( v  y  )y  ( w  z  )z
 t  ux  vy  wz
(2.22)
W  ( u  x  )  x  ( v  y  ) y  ( w  z  )  z
  t  u x  v y  w z
where:
 t  x   x  y  y  z  z
 t   x   x  y   y  z  z
(2.23)
 t  x   x  y  y  z  z
The contravariant velocity components U, V and W are in directions normal to the constant
,  and  surfaces, respectively. The quantity (x, y and z) 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.
ˆ and viscous R
ˆ , Sˆ , T
ˆ flux vectors in the transformed coordinate
The inviscid Eˆ , Fˆ , G
system are:
U


 uU   p 
x

 1 

vU


p
E 

y
J
wU   z p 


e  p U   t p 
30
(2.24a)
V


 uV   p 
x


ˆF  1  vV   y p 
J
wV  z p 


e  p V   t p 
(2.24b)
W


 uW   p 
x

1 
ˆ
G   vW   y p 
J
wW   z p 


e  p W   t p 
(2.24c)
0


       
x xx
y xy
z xz 
1 

ˆ
R    x  xy   y  yy   z  yz 
J
    y  yz   z  zz 
 x xz

 x E x 5   y E y 5   z E z 5 
(2.24d)
0


       
y xy
z xz 
 x xx
ˆS  1  x  xy  y  yy  z  yz 
J
   y  yz  z  zz 
 x xz

x E x 5  y E y 5  z E z 5 
(2.24e)
31
0


       
y xy
z xz 
 x xx
ˆT  1   x  xy   y  yy   z  yz 
J
    y  yz   z  zz 
 x xz

 x E x 5   y E y 5   z E z 5 
(2.24f)
The viscous flux terms with the shear stresses in the transformed coordinates are:






2
 2u   x  u   x  u   x   ( v   y  v   y  v   y )  ( w   z  w   z  w   z )
3
2
 yy   2v   y  v   y  v   y   (u   x  u   x  u   x )  ( w   z  w   z  w   z )
3
2
 zz   2w   z  w   z  w   z   (u   x  u   x  u   x )  ( v   y  v   y  v   y )
3
 xy  (u   y  u   y  u   y  v   x  v   x  v   x )
 xx 
(2.25)
 xz  (u   z  u   z  u   z  w   x  w   x  w   x )
 yz  ( v   z  v   z  v   z  w   y  w   y  w   y )
where the Stokes hypothesis for bulk viscosity has been used, as discussed earlier.
The auxiliary functions are:
E x 5  u xx  v xy  w xz 
 a 2

a 2
a 2 
  x

 x
 x
Pr  1 


 
E y 5  u xy  v yy  w yz 
 a 2

a 2
a 2 
  y

 y
 y
Pr  1  

 
E z 5  u xz  v yz  w zz 
 a 2

a 2
a 2 
  z

 z
 z
Pr  1 


 
32
(2.26)
The quantities x, y, z etc. are called the metrics of transformation, Pr is Prandtl
number, a is the speed of sound, and a2 = RT.  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 derivative
physical plane

in the transformed plane is related to the time derivative in the


as follows:
t

t

x , y ,z




 t
 t
 t
 ,,



(2.27)
with t, t and t appropriately defined as shown in equation (2.23). If the body is not moving, or
the grid is not moving,


=
.
  t
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
33
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 hyperbolicparabolic 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 time step restriction
from stability considerations for the explicit method, an implicit first-order accurate scheme is
used:
qˆ
n 1
qˆ
 qˆ  

n 1
 O()
n
(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.
34
The Navier-Stokes equation (2.18) then can be written in a semi-discrete form as:
qˆ

n 1
ˆ)
ˆ   Fˆ   G
  (  E


n 1

n
M
ˆ   Sˆ   T
ˆ
(  R

 )
Re
(2.29)
where, for example,   Ê is a numerical approximation to the derivative Ê  , 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 righthand 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:
ˆ n 1 )   M  ( R
ˆ n 1   Fˆ n 1   G
ˆ n   Sˆ n   T
ˆ n)
qˆ n 1  qˆ n  (  E





Re
(2.30)
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 Ê , F̂ , and Ĝ are nonlinear, the algebraic equations shown in
(2.30) for the unknown vector, q n1 , 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:
35
Eˆ n 1  Eˆ n  [ A n ](qˆ n 1  qˆ n )  O( 2 )
Fˆ n 1  Fˆ n  [B n ](qˆ n 1  qˆ n )  O( 2 )
(2.31)
ˆ n 1  G
ˆ n  [C n ](qˆ n 1  qˆ n )  O( 2 )
G
where [A], [B] and [C] are the Jacobian matrices:
[ A] 
ˆ
E
qˆ
[B ] 
Fˆ
qˆ
and
[C] 
ˆ
G
qˆ
(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.
The detailed forms of those matrices are shown below:
t
x
y
z
0 

   2  u    (   2)u
 y u   x v
 z u   x w
 x  
x
 x
2
 x v   y u
   y (   2) v
 z v   y w
 y 
[A]    y   v
 2

 x w  z u
 y w  z v
   z (   2) w
z 
 z   w
 ( 2  E )
 x E  u
 y E  v
 z E  w
 t  

(2.33)
where:
 2  (   1)( u 2  v 2  w 2 ) / 2
  x u  y v  z w
   1
(2.34)
  t  
E
e
 2

36
The matrices [B] and [C] may be similarly conducted if  and  are used in the above equations,
respectively, instead of .
After substituting the linearized flux vectors equations (2.31) into (2.30), the following
systems of linear equations for q n1 can be achieved:
[I + ( An  Bn   Cn )](qˆ n1  qˆ n )  RHS n
(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:
ˆ n )   M  ( R
ˆ n   Fˆ n   G
ˆ n   Sˆ n   T
ˆ n)
RHS n  (  E





Re
(2.36)
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 q n 1  q n 1  q n in equation (2.35), the so called “delta form” equation is
obtained:
[I + ( An  Bn   Cn )]qˆ n1  RHS n
(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.
37
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:
[I + ( A  B   C)]  [I  ( A)][I  (B)][I  ( C)]  o(2 )
(2.38)
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 q n 1 are
solved by three successive block-tridiagonal inversions:
[I  (  A n )]qˆ 1  RHS n
[I  (  B n )]qˆ 2  qˆ 1
(2.39)
[I  (  C n )]qˆ n 1  qˆ 2
2.2.4 Spatial Discretization of the Inviscid Terms
The right hand side of equation (2.37), RHS n , contains inviscid derivative terms such as
  Ê , where Ê includes the flux of mass, momentum and energy, and viscous terms such as
38
  R̂ . 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:
ˆ 
E
ˆ
ˆ
E
i 1, j,k  E i 1, j,k
(2.40a)
2
and
Fourth Order:
ˆ 
E
ˆ
ˆ
ˆ
ˆ
E
i  2 , j, k  8E i 1, j, k  8E i 1, j, k  E i  2 , j, k
12
(2.40b)
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 semiimplicitly. That is, the values obtained from the “n” and “n+1” time levels in  direction are used
in the right hand side of the equation (2.37), instead of the left hand side. Thus, equation (2.37)
yields:
[I +  A n ][I   Cn )]qˆ n1  RHS n ,n1
(2.41)
and the right hand side becomes:
ˆ n )  
ˆ n   Fˆ n ,n 1   G
RHS n ,n 1  (  E


M
ˆ n ,n 1   Sˆ n ,n 1   T
ˆ n ,n 1 )
(  R


Re
(2.42)
where the term, Fˆ n ,n 1 , 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
39
unconditionally stable from the stability analysis. However, due to the non-linearity in 
direction, this scheme is best suitable for the geometries and grids in which the spacing is much
larger in one direction ( 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.
Treatment of the spanwise (or -derivative) in this scheme leads to the following
equations that require just two tridiagonal matrix inversion:
[I  (  A n )]qˆ 1  RHS n ,n 1
[I  (  C )]qˆ
n
n 1
 qˆ
(2.43)
1
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 -sweep/i-sweep
and -sweep/k-sweep, respectively. These inversions are done at each fixed span station. The
calculations are done at one -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
40
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.
Unlike the inviscid derivative terms, the derivatives of the viscous terms are differenced


ˆ
ˆ
about the half points. A typical term such as   R̂ is written as R
i1/ 2  R i1/ 2  . Because the
R̂ itself contains the derivatives of the velocities, such as
u
, the use of half points differencing
x
limits the second-order differences to three points in each coordinate direction, while secondorder spatial accuracy is maintained. For example,
u
can be expressed in the transformed
x
coordinates as:
u u
u
u

x 
x 
x
x 


(2.44)
At half points (i+1/2, j, k), the standard central differences are used:
u
u  ui
 i1
 i1/ 2, j,k

2.2.6 Implementation of Low Pass Filters
41
(2.45)
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.
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:
[I  (  A n )   I D I, ]qˆ 1  RHS n ,n 1   E D E
[I  (  C n )   I D I, ]qˆ n 1  qˆ 1
42
(2.46)
where DE is the fourth-order explicit filter term, and D I, and D I, are implicit filter terms in the
 and  directions, respectively. The coefficients  I and  E 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.
In present study, a non-linear filter with eigenvalue scaling was used. Thus the fourthorder explicit low pass filter term is defined as:
D E  D E ,  D E ,   D E , 
(2.47)
DE,    [( ,i1, jJ i11, j   ,i, jJ i,1j )i(,2j) qin, j ]    [( ,i, jJ i,1j )i(,4j) qin, j ]
(2.48)
where
In above equation,   represents a forward difference in  direction, and   denotes a backward
difference in  direction. The  is the largest eigenvalue of the flux matrix A, which is defined
as follows:
   U  a(2x  2y  2z )1/ 2
(2.49)
and the coefficients i(,2j) and i(,4j) are defined as follows:
 i(,2j)  k 2 max(  i1, j ,  i , j ,  i1, j )
 i(,4j)  max( 0, k 4   i(,2j) )
and  i , j 
(2.50)
p i1, j  2p i , j  p i1, j
p i1, j  2p i , j  p i1, j
The typical values of the constant k2 and k4 are 0.25 and 0.01. In the  and  directions, the
43
explicit filter terms DE, and DE, are defined in a similar manner, with the scaling factors  and
, respectively.
The second-order, implicit filter terms in equation (2.46) are defined as:
DI,  J i,1j ( ,i, j  J i, j )
(2.51)
1
i, j
DI,  J ( ,i, j   J i, j )
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.
44
The time-averaged Navier-Stokes equations lead to the Reynolds stresses ( u v , u w  ,
v w  , u  2 , v 2 , w2 ), which can not be solved directly from the equations, and must be
modeled.
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 u v can be
expressed as:
 u v 
T

 u v 
 y  x 


(2.52)
where u v  is the time-averaged values of the product u , v , and u , v are the instantaneous
velocity fluctuations about the mean velocities of u and v, respectively. The term T/ is the
turbulent viscosity coefficient, and is also called as the eddy viscosity.
In the present RANS equations based modeling of the turbulent flow, ( + T) is used in
the dimensional form of Navier-Stokes equations instead of . Also in the energy equation, /Pr
is replaced by (/Pr + T/PrT), where the turbulent Prandtl number, PrT, is given as 0.91, and Pr
is 0.72 for air. Since T is an unknown parameter and depends on the turbulent flow field, a
turbulence model is needed to evaluate the value of the eddy viscosity 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)
45
[48], and the Spalart-Allmaras Model (one-equation model) [91].
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 T is given by:
 T   T inner  l 2m 
for d<dc
(2.53)
where d is the distance from the surface of the body, and dc is the value of d for which
 T inner   T outer . 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:

  z w  w
l m  z 1  exp 
 26 w




(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 w is the shear stress
at the wall.
In equation (2.53), the magnitude of the local mean vorticity  is defined as:
46
2
 w v   u w   v u 
  
    
    
 y z   z x   x y 
2
2
(2.55)
In the outer layer, the eddy viscosity is computed with the following equation:
T outerlayer  KcCcpFwakeFkleb
(2.56)
where:
2

0.25z max U dif
Fwake  min  z max Fmax ,
Fmax





 z w  w 
F(z)  z  1  exp(
)
26 w 

U dif 
u
2
 v2  w 2
Fkleb 
  u
2
max
 v2  w 2

(2.57)
min
1
 0.3z 

1  5.5
 z max 
6
In equation (2.57), the constant Kc = 0.0168 is the Clauser’s constant, and Ccp = 1.6 is an
empirical constant. The Klebanoff intermittency correction, Fkleb, and the function Fwake 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 Fmax.
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
47
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.
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 t is equal to t, and t is given by:
t  ~
f v1 , f v1  1 
~
3

,


3
3
  c v1

(2.58)
where  is the molecular viscosity. The working variable, ~
 , is governed by the transport
equation.
D~

~
1
 c b1 (1  f t 2 )S ~
  [.((  ~
 )~
  c b 2 (~
)2 ] 
Dt

2
c b1   ~


2
c w1f w   2 f t 2   d   f t1U

 
(2.59)
Here,
~
S  S
~

f ,
2 2 v2
 d
Also,
48
(2.60)
f v2  1 

1  f v1
(2.61)
where S is the magnitude of the vorticity, and d is the distance to the closest wall.
The function fw in the destruction term is given by the following expression:
 1  c6 
f w  g  6 w63 
 g  cw3 
1
6
(2.62)
where
g  r  c w 2 (r 6  r )
~

r~ 2 2
S d
(2.63)
For large values of r, fw 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 ft2 function, which goes to unity upstream of the
transition point.
f t 2  c13 exp( c142 )
A trip function ft1 is obtained from the following equation:
49
(2.64)

2
f t1  c t1g t exp   c t 2 t 2 d 2  g 2t d 2t
U


where g t  min( 0.1,

(2.65)

U
) and  t is the wall vorticity at the trip point and dt is the distance
t z
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 ~
 = 0. In the free-stream and outer boundary ~
 =  10 ,
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,
cw2 = 0.3,
cw3 = 2,
 =2 /3,
cv1 = 7.1,
 = 0.41,
ct1 = 1,
c w1 
ct2 = 2,
c b1 (1  c b 2 )

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
50
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 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.
51
Vorticity, Entropy, Acoustic
properties (Riemann invariant
2a
 u n) are allowed to
 1
leave the domain
2a
 u n leaves
 1
2a
 u n enters
 1
Downstream pressure field can
influence upstream components.
Riemann invariant
No vorticity or entropy
enters from upstream since
the boundary faces uniform
flow.
2a
 u n is allowed to
 1
enter in.
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 (u,
v, w, and Et), are fixed to the free-stream values. However, at the subsonic-outflow boundary
(downstream), four characteristics should leave, so the four quantities (u, v, and w) 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.
2.4.3 Solid Surface Conditions
52
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:

Vb  n  0
(2.66)

where n is the unit vector normal to surface and Vb 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:

Vb  0
(2.67)
The density at the airfoil surface is extrapolated from the interior using the following
expression:

 0 or i1  (4i 2  i 3 ) / 3
n
(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:
P
0
n
(2.69)
The numerical expression of this is:
53
Pi1  (4Pi 2  Pi 3 ) / 3
(2.70)
Thus, for a viscous flow without jet blowing, the boundary conditions shown in Figure
2.2 are applied.
u = v = w = 0; No slip
P
 0 (Simplification of normal
n
momentum equation)
T
0
n
Adiabatic wall
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.
54
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 q B  q C . Continuity of properties is
1
ensured by setting q B  q C  (q A  q D ) .
2
2.4.5 Jet Slot Exit Conditions with Given C
In most Circulation Control Wing studies, the driving parameter is the momentum
coefficient, C, defined as follows.
C 
 U jet
m
1
  V2S
2
(2.71)
Here, the jet mass flow rate is given by:
   jet U jet A jet
m
(2.72)
where Ajet is the area of the jet slot, and S is the area of the whole wing section. In 2-D
simulations, Ajet is the height of the jet slot and S is the chord of the CC airfoil.
55
In the present study, the following boundary conditions are specified at the slot exit: the
total temperature of the jet T0, which is approximately equal to the total temperature of freestream, the momentum coefficient C 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.
P0 = Total pressure depends on upstream conditions
T0 = Total temperature also depends on
upstream conditions
Flow angularity depends on slot geometries
These are
specified.
In subsonic jets, P must be continuous.
PA  PC
 PB
2
In supersonic jets, P
should be specified.
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:
Pjet  Pi1  (4Pi 2  Pi 3 ) / 3
56
(2.73)
From equation (2.71), the momentum coefficient can also be expressed as:
C 
 jet U 2jet A jet
1
  V2S
2
(2.74)
From the ideal gas law and the equation of state, the following relations can be obtained:
U 2jet 
2
R (T0, jet  Tjet )
 1
and
 jet 
Pjet
RT jet
(2.75)
Substituting equation (2.75) into (2.74), another expression for C with just one unknown
parameter can be obtained:


2  A jet  Pjet (T0, jet  Tjet )


C 
  1  1  V 2S 
Tjet

  
2

(2.76)
The only unknown variable is Tjet, which can be easily solved from equation (2.76).
After the Tjet is calculated, the other jet flow variables, such as Ujet and 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.
From the isentropic relations for the total temperature and jet exit temperature:
57
T0, jet
Tjet
1 
 1 2
M jet
2
(2.77)
where Mjet = 1, and the Tjet can be easily solved as the T0,jet is known.
After the Tjet 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 could not be directly measured from the wind tunnel. Instead, it is the
ratio of the jet total pressure to free-stream static pressure,
free-stream temperature,
T0, jet
T
P0, jet
P
, and the jet total temperature to
, that are specified as the blowing conditions. Then the momentum
coefficient is calculated from the measured data.
Again, the momentum coefficient C is defined as:
C 
 Va
m
qC
(2.78)
 is the mass flow rate of the jets defined as equation (2.72), q is the free stream dynamic
where m
pressure, which is equal to
1
  V2 , and Va is the jet slot velocity obtained assuming that the
2
flow was expanded to the free-stream pressure.
Then the local Mach number at the jet exit slot is determined from the isentropic
58
relationship:
M jet  M i1 
 1




2  P0, jet  

 1



  1  Pjet 


(2.79)
If Mjet is less than 1.0, which indicates a subsonic jet, then the local density can be obtained from
Pjet by the following relationship:
Pjet
 jet
 1 2
(1 
M )
Pi1 T0, jet
P
2


i1 T0, (1    1 M 2 )  
jet
2
(2.80)
and the local speed of sound:
a jet  a i1  
Pi1
 i1
(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 Mjet 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
Pjet  Pi1  P0, jet 
 2 


 1
(2.82)
The local density, speed of sound and velocities (u, v, w) etc can subsequently be
determined using the same method as above.
59
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.
3.1 Code Validations with a NACA 0012 Wing
60
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 SpalartAllmaras 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 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
61
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 2D 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], 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
62
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 C  is 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
63
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 C at a fixed angle of
attack (=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 C. A lift coefficient as high as 4.0 can be
obtained at a C value of 0.33, and the lift augmentation Cl/C is greater than 10 for this 30degree flap configuration.
Figure 3.7 shows the computed Cl variation with the angle of attack, for a number of C
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
64
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 Cl,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 C = 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 C may be explained as follows. As C increases, the circulation
around the airfoil increases, leading to large pressure suction levels over the upper surface. As
angle of attack ( 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  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  = 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
65
moderate C 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  = 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], 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
66
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.
3.2.4 Steady Blowing with Specified Total Pressure
In above CFD simulations, the moment coefficients C, was directly specified as the
boundary condition using the method described in Chapter II. However, in experiments, C 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 C, and the C is subsequently
calculated as C 
 Va
m
 is the mass flow rate and Va is the slot velocity obtained from
, where m
qC
the assumption that the flow were expand to the free-stream pressure. Thus, from equations
(2.79) to (2.82), a relation between C and P0,jet/P can be obtained as follows:
67
 1



P


0
,
jet

  1.0
C  G 
 P 



(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 C variation with P0,jet is shown in Figure 3.13, and the lift coefficient variation with
C is shown in Figure 3.14. From those figures, it is seen that the C 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 measurements at the same C . Thus it is reasonable to
use C 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, C, which is defined as follows.
 jet U 2jet A jet
=
C 
1
1
  V2S
  V2S
2
2
 U jet
m
(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
68
airfoils achieved is the same for a given C. 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 freestream velocity with the same C? 2) Would a thin wall jet be more beneficial than a thicker,
slower jet at the same C?
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 C. 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.
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, C ,
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 freestream velocity to keep a constant C.
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 C and
fixed jet slot height conditions, and that C is an appropriate driving parameter for CC blowing if
69
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 C is non-dimensionalized by the freestream 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 C when the
jet slot height is also fixed.
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 C (C =0.04) and a fixed high C (C
=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+C), and is corrected by adding C 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 C  =0.1657 case, and increases by about 5.3% for C
=0.04 case when the slot height is changed. However, as shown in Figure 3.21, the mass flow
70
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 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-
71
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 given lift compared to the multi-element airfoil,
and that it can achieve very high lift without stall. Figure 3.24 shows the efficiency, C l/Cd+C for
these two configurations, and it is seen that the CC airfoil is much more efficient than a multielement 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.
72
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 blowing with C=
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 (C= 0.04), while keeping the TE blowing at C= 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 C= 0.08 and a TE blowing of C= 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 (CLE + C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.
73
-3.5
-3
34% SPAN
-2.5
-2
Exp
-1.5
Cp
CFD
-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
Figure 3.1a: Cp Distribution over NACA 0012 Wing Sections at 34% Span
-3
-2.5
50% SPAN
-2
-1.5
Exp
CFD
Cp
-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
Figure 3.1b: Cp Distribution over NACA 0012 Wing Sections at 50% Span
74
-3
-2.5
66% SPAN
-2
-1.5
-1
Cp
Exp
CFD
-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
Figure 3.1c: Cp Distribution over NACA 0012 Wing Sections at 66% Span
-2.5
-2
85% SPAN
Exp
-1.5
CFD
Cp
-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
Figure 3.1d: Cp Distribution over NACA 0012 Wing Sections at 85% Span
75
1
0.8
Cl
0.6
0.4
Exp 8 DEG
CFD, BL Model, Coarse Grid
0.2
CFD, SA Model, Coarse Grid
CFD, BL Model, Fine Grid
0
0
0.2
0.4
0.6
0.8
1
Span, Y/C
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
Jet Slot Location
0.2
0.1
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.2
-0.3
-0.4
-0.5
Figure 3.3: The Circulation Control Wing Airfoil with 30-degree Flap
76
Figure 3.4: The Body-fitted C Grid near the CC Airfoil Surface
3
2.5
2
Cl
DX
1.5
1
Dx=0.001, Cl_ave=2.96
Dx=0.002, Cl=2.88
0.5
Dx=0.005, Cl=2.53
0
0
4000
8000
12000
16000
Iterations
Figure 3.5: The Lift Coefficients in Different Grid Spacing Cases (C = 0.15)
(Dx: The distance between the trailing edge point (A) and the first point in the wake (B))
77
5
4
Cl
3
2
Cl, Measured
1
Cl, Computed
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
C
Figure 3.6: Variation of the Lift Coefficient with Momentum Coefficients at =0°
4
EXP, Cmu = 0.0
EXP, Cmu = 0.074
C=0.1657
EXP, Cmu = 0.15
3
CFD
Lift Coefficient, Cl
C=0.074
2
C=0.0
1
0
-4
-2
0
2
4
6
Angle of Attack
8
10
12
14
Figure 3.7: The Variation of the Lift Coefficient with Angle of Attack
78
16
Figure 3.8: The Streamlines over the CC airfoil at Two Instantaneous Time Step
(C = 0.1657, Angle of Attack = 60)
79
0.9
0.88
t = 1.578693 msec
t = 4.128484 msec
t = 6.678274 msec
0.86
0.84
Cl
0.82
0.8
0.78
0.76
0.74
0.72
0.7
0
1
2
3
4
5
6
7
8
9
10
Time (msec)
Figure 3.9: Time History of the Lift Coefficient for the Unblown Case
(U=94.3 ft/sec)
0.885
0.88
0.875
Cl
0.87
0.865
0.86
0.855
0.85
0.845
0
1
2
3
4
5
6
7
8
9
10
Time (msec)
Figure 3.10: Time History of the Lift Coefficient for the Unblown Case
(U=220 ft/sec)
80
30
Dominant Vortex
25
FFT
20
15
Munro[9]’s measurement = 1600Hz
10
5
0
0
500
1000
1500
2000
Frequency (Hz)
Figure 3.11: The FFT of the Lift Coefficient Variation with Time
(U=220 ft/sec)
81
2500
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, C=0.04, 30-degree Flap)
82
0.6
C
0.4
0.2
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Pjet-total / Pinf
Figure 3.13: The C Variation with the Total Jet Pressure for Steady Blowing Case
5
4
Cl
3
2
Cl, Measured
Cl, Computed by Specified Cmu
1
Cl, Computed by Specified Jet Total
Pressure
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
C
Figure 3.14: The Lift Coefficient Variation with Cfor Steady Blowing Case
83
4
Cl
3.5
3
2.5
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.)
Figure 3.15: Lift Coefficient vs. Free-stream Velocity
(C = 0.1657, h = 0.015 inch and V, exp = 94.3 ft/sec)
0.2
Cd
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(Vinf in CFD) / (Vinf in Exp.)
Figure 3.16: Drag Coefficient vs. Free-stream Velocity
(C = 0.1657, h = 0.015 inch and V, exp = 94.3 ft/sec)
84
2
0.004
0.0035
0.0025
0.002
0.0015
0.001
0.0005
0
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.)
Figure 3.17: Mass Flow Rate vs. Free-stream Velocity
(C = 0.1657, h = 0.015 inch and V, exp = 94.3 ft/sec)
4
Cmu = 0.04
Cmu = 0.1657
3
Lift Coefficient
Mass Flow Rate
0.003
2
1
0
0.006
0.009
0.012
0.015
Jet Slot Height (inch)
Figure 3.18: Lift Coefficient vs. Jet Slot Height
(V= 94.3 ft/sec)
85
0.018
0.25
Cmu = 0.04
Cmu = 0.1657
Drag Coefficient
0.2
0.15
0.1
0.05
0
0.006
0.009
0.012
0.015
0.018
Jet Slot Height (inch)
Figure 3.19: Drag Coefficient vs. Jet Slot Height
(V= 94.3 ft/sec)
20
Efficiency Cl/(Cd+Cmu)
15
10
Cmu = 0.04
Cmu = 0.1657
5
0
0.006
0.009
0.012
0.015
Jet Slot Height (inch)
Figure 3.20: The Efficiency vs. Jet Slot Height
(V= 94.3 ft/sec)
86
0.018
0.0025
Cmu = 0.04
Cmu = 0.1657
Mass Flow Rate (slugs/sec)
0.002
0.0015
0.001
0.0005
0
0.006
0.009
0.012
0.015
0.018
Jet Slot Height (inch)
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)
87
3.5
3
Lift Coefficient, Cl
2.5
2
1.5
Multi-element Airfoil with 30 degrees fowler flap
1
CCW Airfoil with 30 degrees flap, Cd not corrected
CCW Airfoil with 30 degrees flap, Cd corrected with
Cd + Cmu
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Drag Coefficient, Cd
Figure 3.23: The Drag Polar for the Multi-Element Airfoil and the CC Airfoil
25
Efficiency, L/D Ratio
20
15
10
Multi-element Airfoil with 30 degrees fowler flap
5
CCW Airfoil with 30 degrees flap, Cd corrected with
Cd + Cmu
0
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
Figure 3.24: The Efficiency (Cl/Cd+C) for the Multi-Element Airfoil and the CC Airfoil
88
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
89
4
3.5
LE Blowing, C = 0.04
TE Blowing, C = 0.08
Lift Coefficient, Cl
3
LE Blowing, C = 0.08
TE Blowing, C = 0.04
2.5
LE Blowing, C = 0.00
TE Blowing, C = 0.08
2
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
18
20
22
24
22
24
Angle of Attack (degrees)
Figure 3.26: Lift Coefficient vs. The Angle of Attack
0.25
LE Blowing, C = 0.00
TE Blowing, C= 0.08
LE Blowing, C = 0.04
TE Blowing, C = 0.08
Drag Coefficient, Cd
0.2
0.15
LE Blowing, C = 0.08
TE Blowing, C = 0.04
0.1
0.05
0
0
2
4
6
8
10
12
14
16
18
20
Angle of attack
Figure 3.27: Drag Coefficient vs. The Angle of Attack
90
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 bladevortex 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.
The present computational studies were aimed at answering the following questions: Can
91
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 C? 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:
C t   C,0 [1  F( t )]
(4.1)
where, C,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(ft) 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.
Figure 4.1 shows a typical sinusoidal variation of the momentum coefficient with time.
92
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 C0 is 0.04, C 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 C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.
4.2 Jets Pulsed with a Square Wave Form
93
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 C is zero during one half of the cycle, and equals 2 C during the other
half of the cycle. Thus the time-averaged value is C, 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, 40 Hz, 120 Hz and
400 Hz. Figure 4.9 shows the variation with the average momentum coefficient C,, 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
94
mass flow rate is the same for pulsed jets at different frequencies with a given C . 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+C) with C and mass flow rate, respectively. As done previously, the drag
coefficient has been corrected by adding C to account for the momentum imparted by the jet
into the wake. For comparison, the corresponding values of the steady jet with the same C  are
also shown in these figures.
For a given value of C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 Vjet. 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, 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
C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
95
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.
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 C0.
4.2.2 Effects of Frequency at a Fixed C
96
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 timeaveraged value of C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 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
97
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 nondimensional 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?
The Strouhal number is defined as following:
Str 
f L ref
U
(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
98
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:
F 
f Lf
U
(4.3)
Here, Lf is the length of the flap chord. In this case, the Lf 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 frequency is varied along with the freestream 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, C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
99
Strouhal number is fixed. Table 4.3 shows that for a larger configuration, the same C l 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 Cl. 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.
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.

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.

The stall angle of the CC airfoil is decreased with the increase in the momentum
coefficient, and it is a leading edge stall.

Under steady blowing conditions, the momentum coefficient has a unique relation with
100
the jet total pressure. The variation of Cl with C is the same as the variation of Cl with
P0,jet/P. Thus, it is reasonable to just vary C as the driving parameter for CCW
computational studies. In experiments, it is of course more convenient to vary P 0,jet as the
driving parameter.

At 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.

Better 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.

Compared 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.

Leading edge blowing can increase the stall angle, and allow the CC airfoil to operate at
high angles of attack.

The 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.

The 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.

The Strouhal number has a more dominant effect on the performance of the pulsed jet
101
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.
102
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
200
100
400
94.3
94.3
94.3
Lref (inch)
8
8
8
Strouhal Number
1.41
0.705
2.82
1.6804
1.5790
1.8026
0.0006194
0.0006200
0.0006210
Frequency (Hz)
Free-Stream Velocity
U (ft/sec)
Chord of the Airfoil
Computed Average Lift
Coefficient (Cl)
Computed Average
Mass Flow Rate (slugs/sec)
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
200
100
400
94.3
47.15
118.6
Lref (inch)
8
8
8
Strouhal Number
1.41
1.41
1.41
1.6804
1.6601
1.7112
0.0006194
0.0003070
0.001288
Frequency (Hz)
Free-Stream Velocity
U (ft/sec)
Chord of the Airfoil
Computed Average Lift
Coefficient (Cl)
Computed Average
Mass Flow Rate (slugs/sec)
103
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
200
100
400
94.3
94.3
94.3
Lref (inch)
8
16
4
Strouhal Number
1.41
1.41
1.41
1.6804
1.7016
1.6743
0.0006194
0.001240
0.0003100
Frequency (Hz)
Free-Stream Velocity
U (ft/sec)
Chord of the Airfoil
Computed Average Lift
Coefficient (Cl)
Computed Average
Mass Flow Rate (slugs/sec)
104
0.09
DT = 0.0025
sec
0.08
Sin Form Wave Pulsed Jets
Steady Jets
Momentum Coefficient,C
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0.21
0.212
0.214
0.216
0.218
0.22
Real Time (sec)
Figure 4.1: The Time History of the Momentum Coefficient
(Sinusoidal Wave, Frequency = 400 Hz, C,0 = 0.04)
2.5
Lift Coefficient, Cl
2
1.5
1
Sin Form Wave Pulsed Jets
Steady Jets
0.5
0
0.21
0.212
0.214
0.216
0.218
Real Time
Figure 4.2: The Time History of the Lift Coefficient
(Sinusoidal Wave, Frequency = 400 Hz, C,0 = 0.04)
105
0.22
0.0014
Sin Form Wave Pulsed Jets
Steady Jets
0.0012
Mass Flow Rate
0.001
0.0008
0.0006
0.0004
0.0002
0
0.21
0.212
0.214
0.216
0.218
Real Time
Figure 4.3: The Time History of the Mass Flow Rate
(Sinusoidal Wave, Frequency = 400 Hz, C,0 = 0.04)
2
Lift Coefficient, Cl
1.6
1.2
0.8
Steady Jet, Cmu=0.04
Sinusoidal Pulsed Jet, Ave. Cmu=0.04
0.4
Square Wave Pulsed Jet, Ave. Cmu=0.04
0
0
20
40
60
80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400
Frequency (Hz)
Figure 4.4: Time-averaged Lift Coefficient vs. Frequency
106
0.22
0.001
0.0006
0.0004
Steady Jet, Cmu=0.04
Sinusoidal Form Pulsed Jet, Ave. Cmu=0.04
0.0002
Square Wave Form Pulsed Jet, Ave. Cmu=0.04
0
0
20
40
60
80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400
Frequency (Hz)
Figure 4.5: Time-averaged Mass Flow Rate vs. Frequency
Square Wave Pulsed Jet
0.09
Steady Jet
DT = 0.025 sec
0.08
0.07
Momentum Coefficient,C 
Mass Flow Rate (slug/sec)
0.0008
0.06
0.05
0.04
0.03
0.02
0.01
0
0.44
0.45
0.46
0.47
0.48
Real Time
Figure 4.6: The Time History of the Momentum Coefficient
(Square Wave Form, Frequency = 40 Hz, C,0 = 0.04)
107
0.49
2.5
Lift Coefficient, Cl
2
1.5
1
Square Wave Pulsed Jet
0.5
Steady Jet
0
0.44
0.45
0.46
0.47
0.48
0.49
Real Time
Figure 4.7: The Time History of the Lift Coefficient
(Square Wave Form, Frequency = 40 Hz, C,0 = 0.04)
Square Wave Pulsed Jet
0.0014
Steady Jet
0.0012
Mass Flow Rate
0.001
0.0008
0.0006
0.0004
0.0002
0
0.44
0.445
0.45
0.455
0.46
0.465
0.47
0.475
0.48
0.485
Real Time
Figure 4.8: The Time History of the Mass Flow Rate
(Square Wave Form, Frequency = 40 Hz, C,0 = 0.04)
108
0.49
3
Steady Jet
2.5
Pulsed Jet , f = 40Hz
Pulsed Jet, f = 120 Hz
Cl
2
Pulsed Jet, f = 400 Hz
1.5
D
C
A
1
B
0.5
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time-Averaged Momentum Coefficient, C0
Figure 4.9: The Incremental Lift Coefficient vs. Time-averaged Momentum Coefficient
3
Steady Jet
2.5
Pulsed Jet , f = 40Hz
Pulsed Jet, f = 120 Hz
Cl
2
Pulsed Jet, f = 400 Hz
1.5
D
C
A
1
B
0.5
0
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
Time Averaged Mass Flow Rate (slug/sec)
Figure 4.10: The Incremental Lift Coefficient vs. Time-averaged Mass Flow Rate
109
0.002
Time Averaged Mass Flow Rate (slug/sec)
0.0018
Steady Jet
0.0016
Pulsed Jet , f = 40Hz
0.0014
Pulsed Jet, f = 120 Hz
0.0012
Pulsed Jet, f = 400 Hz
0.001
0.0008
0.0006
0.0004
0.0002
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time-Averaged Momentum Coefficient, C0
Figure 4.11: Time-averaged Mass Flow Rate vs. Time-averaged Momentum Coefficient
25
Efficiency, C l/(Cd+C)
20
15
10
Steady Jet
Pulsed Jet , f = 40Hz
5
Pulsed Jet, f = 120 Hz
Pulsed Jet, f = 400 Hz
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Time-Averaged Momentum Coefficient, C0
Figure 4.12: The Efficiency vs. Time-averaged Momentum Coefficient
110
0.14
25
Efficiency, C l/(Cd+C)
20
15
10
Steady Jet
Pulsed Jet , f = 40Hz
5
Pulsed Jet, f = 120 Hz
Pulsed Jet, f = 400 Hz
0
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
Time Averaged Mass Flow Rate (slug/sec)
Figure 4.13: The Efficiency vs. Time-averaged Mass Flow Rate
2
Lift Coefficient, Cl
1.6
1.2
0.8
Steady Jet, Cmu=0.04
0.4
Pulsed Jet, Ave. Cmu=0.04
0
0
20
40
60
80
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400
Frequency (Hz)
Figure 4.14: Time-averaged Lift Coefficient vs. Pulsed Jet Frequency (Ave. C0 = 0.04)
111
20
18
16
12
10
8
Steady Jet, Cmu=0.04
6
Pulsed Jet, Ave. Cmu=0.04
4
2
0
0
20
40
60
80
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400
Frequency (Hz)
Figure 4.15: The Efficiency vs. Pulsed Jet Frequency (Ave. C0 = 0.04)
2.5
DT-cycle = 0.02501 sec
2
Lift Coefficient, Cl
Efficiency, L/D
14
1.5
1
0.5
DT-up = 0.00137 sec
DT-down = 0.00335 sec
0
0.455
0.46
0.465
0.47
0.475
0.48
0.485
0.49
0.495
Real Time (sec)
Figure 4.16: Time History of the Lift Coefficient for a 40 Hz Pulsed Jet
112
2.5
DT-cycle = 0.00501 sec
1.5
1
DT-down = 0.00248 sec
DT-up
= 0.00113
0.5
0
0.21
0.212
0.214
0.216
0.218
0.22
Real Time (sec)
Figure 4.17: Time History of the Lift Coefficient for a 200 Hz Pulsed Jet
2
1.8
Lift Coefficient, Cl
Lift Coefficient, Cl
2
1.6
Case 1
Case 2
Case 3
1.4
1.2
50
100
150
200
250
300
350
400
Frequency
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)
113
450
2
Lift Coefficient, Cl
1.6
1.2
Pulsed Jet, Ave. Cmu=0.04
0.8
Steady Jet, Cmu=0.04
0.4
0
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
Frequency (Hz)
0
1.414
2.828
Stouhal Number ( f * Chord / Vinf)
Figure 4.19: Time-averaged Lift Coefficient vs. Frequency & Strouhal Number
114
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.
5.1 Tangential Blowing on a Wing-flap Configuration
115
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.
116
Symmetry BC
15 C
C
5C
5C

Small blowing to suppress vortex
shedding
This region is modeled as
shown in next figure
2-D BC
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 (C  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 between the blowing section of
the main wing and the unblown section of the main wing, and region B is the interface
117
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 C 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 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 30degree 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, C.
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 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
118
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 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
119
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 C = 0.04. In the third case, there is a stronger blowing with C = 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 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.
120
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 Lift
Total Drag
Computed Drag from the
Coefficient
Coefficient
Inviscid Relation
CL
CD
CD,C = (CL)2/(Æe)
Noblowing Case
0.4850
0.02997
0.02997
Less Blowing,
0.5215
0.03078
0.03465
0.6064
0.04342
0.04685
Cm = 0.04
More Blowing,
Cm = 0.18
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 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 CL.
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
121
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.
122
0
10
15
20
25
Region B
Region A
Figure 5.2: The Grid of the 3-D Wing-flap Configuration with a 300 Partial Flap
123
1.6
Noblowing on Main Wing
1.4
Constant Blowing on Main Wing
1.2
Lift Coefficient, Cl
Gradual Blowing on Main Wing
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
Span, Y/C
Figure 5.3: The Lift Coefficient Distribution along Span for the Wing-flap Configuration
124
Case 1: Noblowing Case
Main Wing
Flap
Blowing Section
Unblown Section
Figure 5.4: The Vorticity Contours for Noblowing Case
125
Case 2: Constant Blowing Case
Main Wing
Flap
Blowing Section
Unblown Section
Figure 5.5: The Vorticity Contours for Constant Blowing Case
126
Case 3: Gradual Blowing Case
Main Wing
Flap
Blowing Section
Unblown Section
Figure 5.6: The Vorticity Contours for Gradual Blowing Case
127
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)
128
Figure 5.10: The Surface Grid for the Rounded Wing Tip
Figure 5.11: The Detailed Grid Close to the Jet Slot
129
No-Blowing
Case
More Blowing
Case (C= 0.18)
Less Blowing
Case (C= 0.04)
Figure 5.12: The Vorticity Contours around the Wing Tip (x/C = 0.81)
130
No-Blowing
Case
More Blowing
Case (C= 0.18)
Less Blowing
Case (C= 0.04)
Figure 5.13: The Vorticity Contours around the Wing Tip (x/C = 1.0)
131
No-Blowing
Case
More Blowing
Case (C= 0.18)
Less Blowing
Case (C= 0.04)
Figure 5.14: The Vorticity Contours around the Wing Tip (x/C = 1.50)
132
No-Blowing
Case
Less Blowing
Case (C= 0.04)
More Blowing
Case (C= 0.18)
Figure 5.15: The Velocity Vectors around the Wing Tip (x/C = 0.81)
133
1.4
Noblowing
1.2
Blowing, Cmu = 0.04
Blowing, Cmu = 0.18
1
Cl
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Span, Y/Ytip
Figure 5.16: The Lift Coefficient Distribution along Span for Wing Tip Configuration
0.2
0.18
Noblowing
0.16
Blowing, Cmu = 0.04
0.14
Blowing, Cmu = 0.18
Cd
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
0.8
1
1.2
Span, Y/Ytip
Figure 5.17: The Drag Coefficient Distribution along Span for Wing Tip Configuration
134
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 bodyfitted 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 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
135
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 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
136
may be suppressed by leading edge blowing. In practice, because high C L 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 C, 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
C. This may increase the power requirements of compressors that provide the high
pressure air.
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+C)) 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
137
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.
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
138
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
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
139
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.
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.
140
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 (, , ). In general, the coordinates
(, , ) in computational domain are assumed to be uniform spacing, and they are the functions
of (x, y, z) as follows:
   ( x , y, z , t )
  ( x , y, z, t )
   ( x , y, z, t )
(A.1)
t
For the time derivatives, the t, t and tare given in terms of the grid velocity x, y and
z as:
 t  x 



 y
 z
x
y
z
t   x 



 y
 z
x
y
z
 t  x 



 y
 z
x
y
z
(A.2)
and the x, y and z are the grid velocity, and equal to zero if there is no body or grid moving.
For the spatial derivatives, using the chain rule of the partial differentiation, the partial
141
derivatives for coordinates (x, y, z) become:
      



x x  x  x 
      



y y  y  y 
(A.3)
      



z z  z  z 
In above equation, for simplicity, the following expressions will be used for the
derivatives:
x 



, y 
, z 
x
y
z
and x  
x
x
x
, x 
, x 



x 



, y 
, z 
x
y
z
and y  
y
y
y
, y 
, y 



x 



, y 
, z 
x
y
z
and z  
z
z
z
, z 
, z 



(A.4)
Then, the differential expressions by the partial difference can be written as:
d   x dx   y dy   z dz
d   x dx   y dy  z dz
d   x dx   y dy   z dz
Equation (A.5) could be written in a matrix form as:
142
(A.5)
 d    x
d  
   x
 d    x
y
y
y
 z  dx 

z  dy 
 z   dz 
(A.6)
Similarly, the (dx, dy, dz) can be expressed by (d, d, d) as the following matrix form:
dx   x 
dy    y
   
 dz   z 
x
y
z
x    d 

y   d
z    d 
(A.7)
Combining equation (A.6) and (A.7), the following equation can be obtained:
 x

 x
x

y
y
y
z 

z  
 z 
x 

y
z

x
y
z
1
x 

y   J
z  
 y z   y z 

 y z  yz
 y z   yz 

x  z   x z 
x z  x z
x z   x  z 
x  y  x  y 

x  y   x  y  (A.8)
x  y   x  y  
Thus, the derivatives of (, , ) can be expressed as:
 x  J (yz  y z  )
 y  J (x  z   x z  )
z  J (x  y  x  y )
x  J ( y  z   y  z  )
y  J (x  z   x  z  )
z  J ( x  y   x  y  )
 x  J ( y z   yz  )
 y  J (x z   x  z  )
 z  J (x  y  x  y )
where J is the Jacobian Matrix of the transformation, which is given as:
143
(A.9)
x
y
z
x
x
x
J  x
y
z  1 / y 
y
y
x
y
z
z
z
z
(A.10)
1 / [x  ( y z   y  z  )  x  ( y  z   y  z  )  x  ( y  z   y z  )
x  , y  , z  etc could be obtained by the differencing method. For instance, using the
second-order central difference, x  , y  , z  etc are expressed as follows:
x ,i , j,k 
x i1, j,k  x i1, j,k
x

 i , j,k
2
x ,i , j,k 
x i , j1,k  x i , j1,k
x

 i , j,k
2
x  ,i , j,k 
x i , j,k 1  x i , j,k 1
x

 i , j,k
2
y ,i , j,k 
y i1, j,k  y i1, j,k
y

 i , j,k
2
y ,i , j,k 
y i , j1,k  y i , j1,k
y

 i , j,k
2
y  ,i , j,k 
y i , j,k 1  y i , j,k 1
y

 i , j,k
2
z ,i , j,k 
z i1, j,k  z i1, j,k
z

 i , j,k
2
z ,i , j,k 
z  ,i , j,k 
(A.11)
z i , j1,k  z i , j1,k
z

 i , j,k
2
z i , j,k 1  z i , j,k 1
z

 i , j,k
2
As for the fourth-order central difference, the following expressions could be obtained for
144
x , x , x :
x ,i , j,k 
 x i 2, j,k  8x i1, j,k  8x i1, j,k  x i2, j,k
x

 i , j,k
12
x ,i , j,k 
 x i , j 2,k  8x i , j1,k  8x i , j1,k  x i , j2,k
x

 i , j,k
12
x  ,i , j,k 
 x i , j,k  2  8x i , j,k 1  8x i , j,k 1  x i , j,k 2
x

 i , j,k
12
(A.12)
The similar equations could be obtained for y  , y  , y  and 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  , and  are equal to one.
145
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154
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.
155
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64 Li, J. and Liu, C., “Direct Numerical Simulation for Flow Separation Control with Pulsed
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167