NLF WING DESIGN BY ADJOINT METHOD AND AUTOMATIC ...
Transcript of NLF WING DESIGN BY ADJOINT METHOD AND AUTOMATIC ...
NLF WING DESIGN BY ADJOINT METHOD
AND AUTOMATIC TRANSITION PREDICTION
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND
ASTRONAUTICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Jen-Der Lee
April 2009
c© Copyright by Jen-Der Lee 2009
All Rights Reserved
ii
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(Antony Jameson) Principal Adviser
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(Robert W. MacCormack)
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(Juan J. Alonso)
Approved for the University Committee on Graduate Studies.
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To My Parents,
Ming-Dau Lee and Lien-Hua Hong
iv
v
Abstract
This dissertation describes the application of optimization technique based on control
theory for natural-laminar flow airfoil and wing design in viscous compressible flow
modeled by the Reynolds averaged Navier-Stokes equations. A transition prediction
module which consists of a boundary layer method and two eN -database methods
for Tollmien-Schlichting and crossflow instabilities are coupled with flow solver to
predict and prescribe transition locations automatically. Results of the optimization
will demonstrate that an airfoil can be designed to have the desired favorable pressure
distribution for laminar flow and the new airfoil can be redesigned for higher Mach
number for performance benefits while still maintaining reasonable amount of laminar
flow. For 3D wing, the redesigned wing configuration will demonstrate an overall
improvement not only at a single design point, but also at off-design conditions.
The results prove the feasibility and necessary of incorporating laminar-turbulent
transition prediction with flow solver in natural laminar-flow wing design.
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Acknowledgements
I would like to express gratitude to my adviser, Professor Antony Jameson, who share
his professional and personal experience with me throughout my study at Stanford.
He has always been very patient and encouraged me to pursue what I really interested
in. I thank him for believing my potential and providing me the opportunity to work
with him.
The study of Ph.D. is a long journey and I realized I could not have finished it
without the supports I received from my family, friends, colleagues, and aero/astro
staffs. A special thanks to my best friend, Ja-Wei Chen, who has been providing me
supports since elementary school. I would also like to thank Jing-Jing Yang’s family
for inviting me to their house during special Chinese holidays and Ralph Levine for
providing me a place to stay when I needed it most. I would like to express my
appreciation to my colleagues, Ki Hwan Lee, Nawee Butsuntorn, Kui Ou, Rui Hu,
Aaron Katz,and Chunlei Laing, and friends, Yen-Yen Lee, Kuo-Jen Teng and Yin-Hsi
Kuo, for their supports. I would like to pay my very special thanks to my girl friend,
Pei-Chi Huang, for her unbounded love.
I would also like to thank Professor Juan Alonso and Professor Robert MacCor-
mack for participating in my dissertation committee.
Above all, I would like to thank the most important two person in my life, my
father and mother, for years of dedication and confidence in me. I would like to
dedicate this dissertation to them.
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Contents
Abstract vi
Acknowledgements vii
1 Introduction 1
1.1 History of Laminar Flow Control . . . . . . . . . . . . . . . . . . . . 2
1.2 Airfoil Design Methodology . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Current Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Governing Equations and Discretization 9
2.1 Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Numerical Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Discretization of the Convective Flux . . . . . . . . . . . . . . 12
2.2.2 Discretization of the Viscous Flux . . . . . . . . . . . . . . . . 14
2.3 Artificial Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Upwinding and CUSP Schemes . . . . . . . . . . . . . . . . . 17
2.3.2 Implementation of Limiters . . . . . . . . . . . . . . . . . . . 21
2.4 Time Integration and Convergence Acceleration . . . . . . . . . . . . 22
2.4.1 Time stepping scheme . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Multigrid method . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.3 Local time stepping and Residual smoothing . . . . . . . . . . 27
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3 Design via Control Theory 28
3.1 Formulation of Adjoint Method . . . . . . . . . . . . . . . . . . . . . 28
3.2 Design using Euler equations . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Numerical Discretization of the Adjoint Equations . . . . . . . 35
3.2.2 Adjoint Boundary Conditions . . . . . . . . . . . . . . . . . . 36
3.3 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Steepest Descent . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 Gradient Smoothing . . . . . . . . . . . . . . . . . . . . . . . 39
4 Transition Prediction 41
4.1 Transition Analysis Overview . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 The eN -database Method . . . . . . . . . . . . . . . . . . . . . 42
4.2 Transition Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 Streamwise Amplification Factor Calculation . . . . . . . . . . 44
4.2.2 Crossflow Amplification Factor Calculation . . . . . . . . . . . 45
4.3 Transition Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Transition Prescription on Surface . . . . . . . . . . . . . . . . 46
4.3.2 Transition Prescription in Flow Domain . . . . . . . . . . . . 47
4.4 Coupling of Transition Prediction Module with RANS Solver . . . . . 48
5 NLF Airfoil and Wing Design Results 50
5.1 Verification of Boundary-Layer Parameters and Transition Locations . 50
5.2 Natural-Laminar-Flow Airfoil Design . . . . . . . . . . . . . . . . . . 52
5.3 Natural-Laminar-Flow Wing Calculation . . . . . . . . . . . . . . . . 62
5.4 Natural-Laminar-Flow Wing Design . . . . . . . . . . . . . . . . . . . 69
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Conclusion 78
A Derivation of Viscout Adjoint Terms 80
A.1 Transformation to Primitive Variables . . . . . . . . . . . . . . . . . . 81
A.2 Contributions from the Momentum Equations . . . . . . . . . . . . . 82
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A.3 Contributions from the Energy Equation . . . . . . . . . . . . . . . . 84
A.4 The Viscous Adjoint Field Operator . . . . . . . . . . . . . . . . . . 87
B Verification of Transition Prediction Module 88
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List of Tables
5.1 Comparison of Predicted Transition Locations with Experimental Results 52
5.2 Case 1: Comparison of Aerodynamic Coefficients , M = 0.69, CL = 0.26 62
5.3 Case 2: Comparison of Aerodynamic Coefficients, M = 0.70, CL = 0.38 62
5.4 Case 3: Comparison of Aerodynamic Coefficients, M = 0.70, CL = 0.50 64
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List of Figures
1.1 Variation of drag coefficient with Reynolds number for a smooth flat
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Anticipated fuel saving as a function of range [31] . . . . . . . . . . . 3
1.3 The X-21 Maximum Laminar Flow Areas, M∞ = 0.75, Alt.=40,000 ft. 5
2.1 Coordinate transformation from physical to computational domain . . 11
2.2 Discretization of inviscid flux . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Discretization of viscous flux . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Multigrid W-cycle. E, evaluate the change in the flow for one step; C,
collect the solution; T, transfer the data without updating the solution. 26
4.1 Schematic Diagram of Turbulent Subdomains Surrounded in Laminar
Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Coupling Structure of Flow Solver and Transition Prediction Module 49
5.1 Displacement Thickness, δ⋆, on Upper Surface for NLF(1)-0416 Airfoil,
M∞ = 0.3, Re∞ = 4 · 106, α = 2.03◦ . . . . . . . . . . . . . . . . . . . 51
5.2 Momentum Thickness, θ, on Upper Surface for NLF(1)-0416 Airfoil,
M∞ = 0.3, Re∞ = 4 · 106, α = 2.03◦ . . . . . . . . . . . . . . . . . . . 51
5.3 Convergence History of Transition Locations, xtran,upper = 0.348, xtran,lower =
0.587, for NLF(1)-0416 Airfoil, M∞ = 0.3, Re∞ = 4 · 106, α = 2.03◦ . . 53
5.4 Pressure Distribution for Designed NLF Airfoil, M∞ = 0.69, Re∞ =
11.7 · 106, Cltarget= 0.26 . . . . . . . . . . . . . . . . . . . . . . . . . 54
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5.5 Convergence History of Transition Locations, M∞ = 0.69, Re = 11.7 ·106, xtran,upper = 0.51, xtran,lower = 0.546 . . . . . . . . . . . . . . . . 55
5.6 Number of design iterations: 0 . . . . . . . . . . . . . . . . . . . . . . 56
5.7 Number of design iterations: 30, M∞ = 0.72, Re = 12 ·106, Cltarget= 0.26 56
5.8 Off-design Condition at M∞ = 0.69, Cltarget= 0.26 . . . . . . . . . . . 57
5.9 Off-design Condition at M∞ = 0.70, Cltarget= 0.26 . . . . . . . . . . . 58
5.10 Off-design Condition at M∞ = 0.71, Cltarget= 0.26 . . . . . . . . . . . 58
5.11 Comparison of optimized airfoil profiles between automatic-transition
prediction and full-turbulence model. M∞ = 0.72, Cltarget= 0.26 . . . 59
5.12 Comparison of optimized airfoil profiles at upper-rear portion between
automatic-transition prediction and full-turbulence model. . . . . . . 60
5.13 Pressure distribution for automatic-transition prediction case, M∞ =
0.72, Cltarget= 0.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.14 Pressure distribution for full-turbulence case, M∞ = 0.72, Cltarget=
0.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.15 Mesh Distribution and Divided Subdomains . . . . . . . . . . . . . . 63
5.16 Pressure Distribution on Upper Surface, M = 0.69, CL = 0.26 . . . . 65
5.17 Pressure Distribution on Lower Surface, M = 0.69, CL = 0.26 . . . . . 65
5.18 Initial and Final Transition Locations on Upper Surface,M = 0.69, CL =
0.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.19 Initial and Final Transition Locations on Lower Surface, M = 0.69, CL =
0.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.20 Shear Stress, τ , Distribution on Upper Surface, M = 0.69, CL = 0.26 . 67
5.21 Shear Stress, τ , Distribution on Lower Surface, M = 0.69, CL = 0.26 . 67
5.22 CD v.s. Mach number . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.23 Range parameter v.s. Mach number . . . . . . . . . . . . . . . . . . . 69
5.24 Full turbulence design for NLF 3D wing. Dashed lines and solid
lines represent pressure distribution of the baseline NLF wing and re-
designed configuration respectively . . . . . . . . . . . . . . . . . . . 70
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5.25 Automatic transition prediction design for NLF 3D wing. Dashed lines
and solid lines represent pressure distribution of the baseline NLF wing
and redesigned configuration respectively . . . . . . . . . . . . . . . . 71
5.26 Convergence history of the NLF wing cost function . . . . . . . . . . 72
5.27 Comparison of drag coefficient as a function of Mach number between
the baseline and redesigned NLF wing . . . . . . . . . . . . . . . . . 74
5.28 Comparison of range parameter as a function of Mach number between
the baseline and redesigned NLF wing . . . . . . . . . . . . . . . . . 74
5.29 Redesign of 3D wing with new cost function. Dashed lines and solid
lines represent pressure distribution of the baseline NLF wing and re-
designed configuration respectively . . . . . . . . . . . . . . . . . . . 75
5.30 Comparison of final transition lines on upper surface . . . . . . . . . 76
5.31 Comparison of final transition lines on lower surface . . . . . . . . . . 76
B.1 Bump location on upper surface . . . . . . . . . . . . . . . . . . . . . 89
B.2 Close look of bump location and transition location on upper surface 89
B.3 Pressure distribution on upper surface . . . . . . . . . . . . . . . . . 90
B.4 Close look of pressure distribution near the bump on upper surface . 90
B.5 Convergence history of transition locations for airfoil with artificially
introduced bump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
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Chapter 1
Introduction
The operating cost of airlines is directly proportional to the amount of fuel consumed
during the operation and airlines could spend on the order of millions of dollars more
for the increase of fuel price even by one dollar. For a transport aircraft at cruise
condition, a significant amount of drag comes from the skin friction and this force
has to be overcome by the thrust provided by engines by means of burning fuel in
order to maintain level flight. Hence, any reduction of skin friction will directly result
in a reduction in operating cost for airlines and also more affordable ticket price for
travelers. From the environmental point of view, reduced drag means a reduction
of total fuel consumption for a given flight distance and this reduces the amount of
engine emissions and air pollution.
Based on the physical origins of the drag components, the drag can be divided
into the following components [34]:
• Skin Friction Drag
The drag force on the body resulting from the viscous shear stress acting over
the wetted surface area. The skin friction drag counts about 50% of drag for
airplane in cruise.
• Pressure Drag
Due to the viscous effect, a boundary layer is developed on the surface of aircraft.
The existence of boundary layer causes imbalances of pressure on forward and
1
CHAPTER 1. INTRODUCTION 2
aft surface of aerodynamic body and creates the pressure drag.
• Vortex Drag
Vortex drag is produced by the generation of trailing vortex wake donwstream
of a lifting system with finite span.
• Wave Drag
For airplanes flying at transonic and supersonic speeds, the presence of shock
waves produces the wave drag. It is the result of shock losses and the influence
of shock wave on boundary layer.
1.1 History of Laminar Flow Control
The magnitude of skin friction in cruise is highly dictated by the types of boundary
layer, laminar or turbulent flow. The laminar boundary layer is intrinsically unstable
and difficult to maintain under most of flight conditions. Whether the flow is laminar
or turbulent directly depends on the Reynolds number and premature transition could
happen due to many reasons, e.g. surface roughness, free stream disturbances, and
large wing sweep which causes crossflow to develop and disturbs the laminar boundary
layer. The advantage of laminar boundary layer over turbulent one is its low skin
friction coefficient. Figure 1.1 shows the variation of drag coefficient with Reynolds
number for a flat plate and the advantage on drag reduction is evident if one can
maintain the the boundary layer to be laminar. Based on studies conducted by Boeing
Company and others [31] on the amount of fuel saving for subsonic transport aircraft,
figure 1.2 shows the percentage fuel saving as a function of ranges in nautical miles.
The extent of fuel saving for laminar flow wing varies significantly and depends on
the types of laminar-flow control employed, e.g. passive or active control, the extend
of aerodynamic surface to be laminarized, and the range of operation. It is clear that
a considerable amount of fuel saving can be achieved for long range operations.
The early researches on laminar flow dated back to 1930’s when researchers around
the world tried varieties of approaches attempting to delay the boundary layer transi-
tion from laminar to turbulent. In general, three techniques are available and they are
CHAPTER 1. INTRODUCTION 3
105
106
107
108
109
10−3
10−2
Reynolds number, ReL
CD
Drag coefficient v.s. Reynolds number for a smooth plate
LaminarTurbulent
Figure 1.1: Variation of drag coefficient with Reynolds number for a smooth flat plate
Figure 1.2: Anticipated fuel saving as a function of range [31]
CHAPTER 1. INTRODUCTION 4
classified as passive, active, and hybrid laminar flow control methods. In the passive
approach, the boundary layer is stabilized by modifying the shape of wing to create
favorable pressure gradient over large portion of aerodynamic surface and this is also
known as natural laminar flow (NLF). Based on this concept, researchers in NACA
designed the NACA 6-series NLF airfoils and the North American P-51 Mustang was
the first aircraft intensionally designed to take the advantage of laminar flow. How-
ever, the wing behaved like traditional wings in real flight conditions due to the fact
that the wing was not manufactured with sufficiently smooth and wave-free surface
which is crucial for natural laminar flow wing. In 1939, a series of fight tests on
NLF concept was conducted on a B-18 fitted with 17 × 10-foot NACA 35-215 airfoil
wooden glove and reached transitional Reynolds number about 11.3 million at 42.5%
chord [58], a record in NLF which was not to be surpassed over 40 years until the
NASA F-111 flight test [53, 40, 2]. After the World War II, flight tests of the King
Cobra and Hurricane [47] were conducted to invest the practicality of NLF technol-
ogy, but concerns on the abilities to manufacture and maintain a sufficiently smooth
wing surface defer the real application of NLF airfoils on aviation [45].
In contrast to passive laminar flow control, active laminar flow control, also known
as LFC, stabilizes the boundary layer by the usage of surface slots or small perfora-
tions to remove small amount of boundary layer by suction. This type of laminar flow
control is necessary in order to extend the laminar flow to a larger distance in the
adverse pressure gradient region and plays a crucial role in controlling the crossflow
instabilities induced by the wing sweep. Flight tests employing different LFC tech-
niques were conducted in British and U.S. [2, 32, 3], and extensive laminar flow was
achieved [33] at the end of the X-21 program. Two WB-66D airplanes were modified
by the Northrop Corporation under sponsorship of the Air Force with slotted suc-
tion wing to conduct laminar flow control research and figure 1.3 shows the extent of
laminar flow achieved in flight test at M∞ = 0.75 and Alt. = 40,000 ft.
Hybrid laminar flow control, HLFC, combines both NLF in regions where favorable
pressure gradient exists and LFC in regions where crossflow effect and adverse pressure
gradient dominate. Although the net effect of HLFC is not as effective as LFC, the
gains are still huge as seen from figure 1.2 and is easier to implement on airplanes.
CHAPTER 1. INTRODUCTION 5
Figure 1.3: The X-21 Maximum Laminar Flow Areas, M∞ = 0.75, Alt.=40,000 ft.
1.2 Airfoil Design Methodology
This first approach to airfoil design is also known as direct method. Based on the mis-
sion requirements, one starts with an already available airfoil geometry, e.g. NACA
airfoil, designed for similar mission, determines the characteristics of this airfoil, and
fixes unsatisfactory characteristics by adjusting the camber, leading edge radius, and
thickness distribution. While this approach is straightforward, it often requires the
designers having considerable amount of experiences.
The second approach is known as inverse design method. In 1945, Lighthill [36]
developed the exact inverse method for two-dimensional incompressible potential flow
using conformal mapping. It is a single-point inverse method and the shape of airfoil is
calculated by the prescribed velocity distribution, which has to satisfy three integral
constraints, around the circle in the transformed plane. Eppler [7, 9, 8] further
extended the concept and resulted in a method capable of multi-point design. Now
designers can divide the airfoil into segments and design each segment independently
for each condition. The advantage of this approach is that an airfoil can be designed
in few seconds on a modern laptop computer once the desired velocity distribution
CHAPTER 1. INTRODUCTION 6
is defined and prescribed, but it only applies to low Mach number and inviscid flow
because of the incompressible and potential flow assumption.
The last approach is by gradient-based numerical optimization, where a proper
cost function is defined and the shape of airfoil is repeatedly modified until the cost
function reduces to a certain level. This approach is quite general and can be applied
to varieties of flows governed by different types of flow governing equations. The
drawback of numerical optimization is the necessary of large numbers of flow evalua-
tion in order to obtain the sensitivity of the cost function with respect to each design
variable, and this is a formidable task in airfoil and wing design using Navier-Stokes
equations.
1.3 Current Approach
The current approach chosen in this study is a gradient-based numerical optimization
technique. In optimum shape design problems, the true design space is a free surface
which has infinite number of design variables and will require N+1 flow evaluations for
N design variables in order to calculate the required gradients necessary for gradient-
based optimization technique. Here we treat the wing as a device which controls the
flow to produce lift with minimum drag and apply the theory of optimal control of
systems governed by partial differential equations. By using the optimum control
theory, we can find the Frechet derivative of the cost function with respect to the
shape by solving an adjoint equation problem. The total cost, which is independent
of number of design parameters, is one flow plus one adjoint evaluation and this
makes this technique very attractive for the optimum shape design. After the Frechet
derivative has been found, we can make an improvement by making a modification in
a descent direction and the process repeats. Since this method was first proposed by
Jameson [17], it has been proved to be very effective in wing shape optimization [18,
25].
In the flow calculation and shape optimization of laminar-flow airfoils by RANS
equations, it is necessary to prescribe the locations where the flow transitions from
laminar to turbulent and apply a turbulence model in the turbulent flow region.
CHAPTER 1. INTRODUCTION 7
The transition locations are critical in order to obtain accurate result, e.g. the drag
coefficient, and those information are usually provided by the assumed transition
locations based on the engineering judgement or experimental data if it is available.
However, at the initial design stage, this information is usually not available and a
direct numerical simulation at such high Reynolds number is not practical. Hence it
is necessary to acquire the information of transition locations based on the solutions
of a RANS solver and transition prediction method which is much less expensive than
direct numerical simulation.
In this dissertation, the eN -database method, a method based on linear stability
theory and experimental data, and has been proven [35] to provide reasonably accu-
rate transition locations, is chosen for the streamwise transition prediction. For a 3D
swept wing, the pressure varies not only in the streamwise direction, but also in the
spanwise direction. This variation of pressure in the spanwise direction consequently
results in the development of secondary flow, or crossflow, in the boundary layer. The
velocity profile of crossflow causes instability to develop in the boundary layer and
provokes the transition of boundary layer from laminar to turbulent. This kind of
instability is known as crossflow instability and much more difficult to predict than
Tollmien-Schlichting instability. However, as streamwise instability, there exists some
criteria that can be used at initial design stage and a similar amplification factor for
crossflow, NCF , can be calculated for crossflow transition prediction.
1.4 Outline
Chapter 2 describes the flow governing equations, numerical discretization, time-
stepping scheme, and convergence acceleration used in this dissertation. Chapter 3
introduces the concept of adjoint method and presents a detailed derivation of ad-
joint equation and its corresponding adjoint boundary condition for 2D airfoil inverse
design. Chapter 4 describes the transition prediction methodologies for streamwise
and crossflow instabilities and the coupling of transition prediction module with flow
solver. Chapter 5 shows the results of the verification of transition locations using the
current method for 2D airfoil and 3D wing design using viscous compressible flow.
CHAPTER 1. INTRODUCTION 8
The necessity of prescribing laminar-turbulent transition locations in flow simulation
in order to obtain more accurate aerodynamic coefficients will also be shown. Finally,
chapter 6 concludes this dissertation.
Chapter 2
Governing Equations and
Discretization
2.1 Flow Equation
For the NLF wing design to be representative of real cases, it is essential to have a
suitable mathematical model that is able to describe the complex flow field around a
3D wing geometry. In this dissertation, the Navier-Stokes equations, which describes
the conservation of mass, momentum, and energy , has been used as the mathematical
model for the flow equation.
It proves convenient to use x1, x2, x3 and u1, u2, u3 to represent Cartesian coordi-
nates and its corresponding velocity components and to adopt the convention that a
repeated index “i” implies summation over i = 1 to 3. Then the three-dimensional
Navier-Stokes equations can be written as
∂w
∂t+∂fi
∂xi
=∂fvi
∂xi
in D, (2.1)
9
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 10
where the state vector w, inviscid flux vector f , and viscous flux vector fv are de-
scribed respectively by
w =
ρ
ρu1
ρu2
ρu3
ρE
, fi =
ρui
ρuiu1 + pδi1
ρuiu2 + pδi2
ρuiu3 + pδi3
ρuiH
, fvi =
0
τijδj1
τijδj2
τijδj3
ujτij + κ ∂T∂xi
. (2.2)
In these definitions, ρ is the density, E is the total energy per unit mass, and δij is
the Kronecket delta function. The pressure is determined by the equation of state
p = (γ − 1)ρ
{
E − 1
2(uiui)
}
, (2.3)
where γ is the ratio of the specific heats and the stagnation enthalpy is given by
H = E +p
ρ.
The viscous stress may be written as
τij = µ
(
∂ui
∂xj
+∂uj
∂xi
)
+ λδij∂uk
∂xk
, (2.4)
where µ and λ are the first and second coefficients of viscosity. The coefficient of
thermal conductivity and the temperature are computed as
κ =cpµ
Pr, T =
p
Rρ, (2.5)
where Pr is the Prandtl number, cp is the specific heat at constant pressure, and R
is the gas constant.
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 11
2.2 Numerical Discretization
The flow equations are discretized by a semi-discrete cell-centered finite volume
scheme. The finite volume scheme has the advantage of preserving the global con-
servation of mass, momentum, and energy at discrete level and can be applied to
arbitrary complex geometries. This section describes the numerical discretization
implemented in the flow solver for the 2D case.
For applications using a discretization on a body conforming structured mesh,
it is useful to transform the flow equations from physical coordinates (x1, x2) to
computational coordinates (ξ1, ξ2) as shown in figure 2.1.
x1
x2
ξ1
ξ2
x1 = x1(ξ1, ξ2)x2 = x2(ξ1, ξ2)
Figure 2.1: Coordinate transformation from physical to computational domain
Define the metrics as
Kij =
[
∂xi
∂ξj
]
, J = det(K), K−1ij =
[
∂ξi∂xj
]
,
and
Sij = JK−1ij
In a finite volume discretization, the elements of S are just the face areas of the
computational cells projected in the x and y directions and also note
∂Sij
∂ξi= 0 (2.6)
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 12
which represents the fact that the sum of the face areas over a closed volume is zero,
as can be readily verified by a direct examination of metric terms.
Multiplying equation 2.1 by J , applying the chain rule, and using 2.6, the Navier-
Stokes equations can now be written in computational space as
∂(Jw)
∂t+∂Fi
∂ξi=∂Fvi
∂ξiin Dξ, (2.7)
where the inviscid and viscous flux contributions in computational domain are defined
by Fi = Sijfj and Fvi = Sijfvj . Define the residual at the center of cell (i, j) as
R(w)ij =∂F1
∂ξ1+∂F2
∂ξ2− ∂Fv1
∂ξ1− ∂Fv2
∂ξ2(2.8)
and equation 2.7 in each computational cell can be written as
∂(Jw)ij
∂t+R(w)i,j = 0 (2.9)
Each partial derivative in equation 2.8 represents the net flux across each cell in each
computational direction and can be computed by a central second order discretization.
2.2.1 Discretization of the Convective Flux
The convective flux term, for example in ξ1 direction, in equation 2.8 is discretized as
∂F1
∂ξ1=Fi+ 1
2,j − Fi− 1
2,j
∆ξ1
where Fi+ 1
2,j and Fi− 1
2,j are the convective flux evaluated at cell interfaces. For the
finite-volume scheme with flow variables saved at cell centers, the flow variables can
be regarded as cell-averaged values and the convective flux at cell interface as shown
in figure 2.2 can be calculated by
Fi+ 1
2,j =
1
2(Fi,j + Fi+1,j) . (2.10)
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 13
• •(i, j) (i+ 1
2, j) (i+ 1, j)
Figure 2.2: Discretization of inviscid flux
and
Fi,j = (S11f1 + S12f2)
=
S11
ρu
ρu2 + p
ρuv
ρuH
+ S12
ρv
ρuv
ρv2 + p
ρvH
i,j
=
ρ (S11u+ S12v)
ρu (S11u+ S12v) + pS11
ρv (S11u+ S12v) + pS12
ρH (S11u+ S12v)
i,j
(2.11)
Define flux velocity as
Q− = (S11u+ S12v)i,j
Q+ = (S11u+ S12v)i+1,j
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 14
Then the flux vector can be calculated as
Fi,j =
ρQ−
ρuQ− + pS11
ρvQ− + pS12
ρHQ−
i,j
, and Fi+1,j =
ρQ+
ρuQ+ + pS11
ρvQ+ + pS12
ρHQ+
i+1,j
(2.12)
2.2.2 Discretization of the Viscous Flux
The numerical evaluation of the viscous fluxes at the cell interface is done by first
evaluating the viscous fluxes, fvi+ 1
2,j± 1
2
and gvi+1
2,j± 1
2
, at the end points (vertex) of the
edge. Second, the viscous flux Fvi+ 1
2,j
at cell interface as illustrated by the blue arrow
in figure 2.3 is computed by the following formula
Fvi+ 1
2,j
= S11i+ 1
2,jfv
i+ 12
,j+ S12
i+ 12
,jgv
i+ 12
,j, (2.13)
where fvi+ 1
2,j
and gvi+ 1
2,j
represent the fluxes at the mid-point of the cell face. These
fluxes are computed by averaging the fluxes at cell vertex (i + 12, j ± 1
2) and can be
written as
fvi+ 1
2,j
=1
2
(
fvi+ 1
2,j+ 1
2
+ fvi+ 1
2,j− 1
2
)
gvi+ 1
2,j
=1
2
(
gvi+ 1
2,j+ 1
2
+ gvi+ 1
2,j− 1
2
)
.
From equation 2.2, the viscous fluxes at the cell vertex can be written explicitly as
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 15
(i, j) (i + 1, j)
(i + 1, j + 1)(i, j + 1)
(i, j − 1) (i + 1, j − 1)
(i + 12 , j + 1
2)
(i + 12 , j − 1
2)
(i + 12 , j)
Figure 2.3: Discretization of viscous flux
fvi+ 1
2,j+ 1
2
=
0
τxx
τxy
uτxx + vτxy + k ∂T∂x
i+ 1
2,j+ 1
2
gvi+1
2,j+ 1
2
=
0
τyx
τyy
uτyx + vτyy + k ∂T∂y
i+ 1
2,j+ 1
2
,
and hence this requires us to evaluate the stress tensor and the heat flux components
of the viscous fluxes at the end points of the edge. The velocity components, u and v,
the coefficient of thermal conductivity k are calculated by averaging the cell-centered
values of four cells containing the common vertex(
i+ 12, j + 1
2
)
as shown in figure 2.3.
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 16
For example,
ui+ 1
2,j+ 1
2
=1
4(ui,j + ui+1,j + ui,j+1 + ui+1,j+1) . (2.14)
From equation 2.4, the evaluation of stress tensor requires an estimate of the partial
derivative of velocity. The following illustrates the steps for calculating normal stress
tensor, τxx, and other components of stress tensor can be calculated in a similar
fashion. Following equation 2.4, the normal stress at cell vertex(
i+ 12, j + 1
2
)
can be
written explicitly as
τxx = 2µ
[
∂u
∂x
]
+ λ
{[
∂u
∂x
]
+
[
∂v
∂y
]}
, (2.15)
where every term in equation 2.15 is evaluated at cell vertex. The first and second
coefficient of viscosity are calculated by using equation 2.14. The first coefficient of
viscosity is a combination of the laminar and turbulent viscosity coefficients and is
defined as
µtotal = (µlam + µturb) . (2.16)
The laminar coefficient of viscosity is calculated by the Sutherland equation
µlam = C1T
3
2
T + C2, (2.17)
where C1 and C2 are constants for a given gas. For air at moderate temperatures,
C1 = 1.458 × 10−6kg/(
ms√
◦K)
and C2 = 110.4K. The coefficient of turbulent
viscosity is calculated by the Baldwin-Lomax turbulence model [1]. Then the velocity
gradients at the vertex are calculated by applying the Gauss divergence theorem
to the auxiliary control volume formed by the four cells sharing the same vertex(
i+ 12, j + 1
2
)
as illustrated in figure 2.3. For a vector function−→F =
−→F (x, y), the
Gauss divergence theorem states
∫
V ol
(▽ · −→F ) dV =
∮ −→F · n dS (2.18)
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 17
Now let−→F = (u(x, y), 0) and apply Gauss theorem to the funtion of interest, we have
[
∂u
∂x
]
i+ 1
2,j+ 1
2
=1
V
4∑
k=1
uk Sxk,
where V is the volume of the auxiliary control volume and k represents the value
evaluated at the middle point of each edge that forms the auxiliary control volume.
For example,
u1 =ui+1,j+1 + ui+1,j
2
Sx1= yi+1,j+1 − yi+1,j.
Each element of the stress tensor can be calculated in the similar fashion and the
viscous flux Fvi+ 1
2,j
at cell interface can then be evaluated by using equation 2.13.
2.3 Artificial Dissipation
The spacial discretization presented in previous sections is equivalent to second or-
der central difference scheme on a Cartesian mesh. It is well known that the central
difference scheme permits odd-even decoupling of the solution and generates oscilla-
tions around shock waves. In order to eliminate oscillations around discontinuities,
it is necessary to include artificial dissipation [42]. Over those years, Jameson et. al.
have developed numerous shock capturing algorithms, e.g. JST, SLIP, and USLIP
schemes [30, 19, 21], and different forms of flux-splitting schemes were implemented
and tested with SLIP and USLIP schemes [55, 56]. This section describes the up-
winding and shock capturing schemes used in this study.
2.3.1 Upwinding and CUSP Schemes
Consider the general one dimensional conservation law for a system of equations
written as∂w
∂t+
∂
∂xf(w) = 0. (2.19)
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 18
Here the state and flux vectors are
w =
ρ
ρu
ρE
, f =
ρu
ρu2 + p
ρuH
,
and those variables are defined in section 2.1. In a steady flow H is constant and this
remains true for the discrete scheme only if the numerical diffusion is constructed so
that it is compatible with this condition.
Approximate equation (2.19) over the interval (0, L) on a mesh with an interval
∆x by the semi-discrete scheme
∆xdwj
dt+ hj+ 1
2
− hj− 1
2
= 0, (2.20)
where wj represents the volume-averaged discrete solution in cell j and hj+ 1
2
is the
numerical flux evaluated at cell interface between cells j and j + 1. Suppose the
numerical flux is approximated as
hj+ 1
2
=1
2(fj + fj+1) − dj+ 1
2
, (2.21)
where dj+ 1
2
is the diffusive flux added to eliminate oscillations around discontinuities
in the discrete solution and fj represents the flux vector evaluated at the state wj. A
general form of diffusive flux can be written as
dj+ 1
2
=1
2αj+ 1
2
Bj+ 1
2
(wj+1 − wj) , (2.22)
where the matrix Bj+ 1
2
controls the properties of the scheme, and the scaling factor
αj+ 1
2
is introduced for convenience. The first scheme is the scalar diffusion and
Bj+ 1
2
= I.
While the formulation for the scalar diffusion is straightforward, it has been proven
[21] that this scheme cannot support a perfect discrete shock with a single interior
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 19
point. The characteristic upwind scheme is produced by setting
Bj+ 1
2
= |Aj+ 1
2
| = T |Λ|T−1,
where Aj+ 1
2
is an estimate of the Jacobian matrix ∂f
∂wwith the property that
Aj+ 1
2
(wj+1 − wj) = fj+1 − fj,
and T is the similarity transformation matrix which composes the eigenvectors of
Aj+ 1
2
in its columns. The notation |Aj+ 1
2
| is used to represent the absolute value of
Aj+ 1
2
which is defined to be the matrix obtained by replacing the eigenvalues by their
absolute values.
The Convective Upwind and Split Pressure (CUSP) Scheme is obtained by defining
the diffusive flux as
dj+ 1
2
=1
2α∗c (wj+1 − wj) +
1
2β (fj+1 − fj) , (2.23)
where the factor c is included so that α∗ is dimensionless. The flux vector f can be
decomposed as
f = uw + fp, (2.24)
where
fp =
0
p
up
. (2.25)
Then
fj+1 − fj = u (wj+1 − wj) + w (uj+1 − uj) + fpj+1− fpj
, (2.26)
where u and w are the arithmetic averages
u =1
2(uj+1 + uj) , w =
1
2(wj+1 + wj) .
Scheme of this class are fully upwind in supersonic flow if one takes α∗ = 0 and
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 20
β = sign(M) when the absolute value of the Mach number M exceeds 1. To support a
stationary shock with a single interior point, α∗ and β can not be chosen independently
and has to satisfy
α∗c = (1 + β) (c− u) , 0 < u < c,
which leads to a one-parameter family of schemes once α∗ is chosen. If the convec-
tive terms are separated by splitting the flux according to equations (2.24), (2.25),
and (2.26), then the total effective coefficient of convective diffusion is
αc = α∗c+ βu.
The choice αc = u leads to low diffusion near a stagnation point, and also leads to
a smooth continuation of convective diffusion across the sonic line since α∗ = 0 and
β = 1 when |M | > 1. The scheme must also be formulated so that the cases of u > 0
and u < 0 are treated symmetrically. This leads to the diffusion coefficients
α = |M | (2.27)
β =
+max(
0, u+λ−
u−λ−
)
if 0 ≤M ≤ 1
−max(
0, u+λ+
u−λ+
)
if − 1 ≤M ≤ 0
sign (M) if |M | ≥ 1,
(2.28)
where M = uc
and λ± = u ± c. Near a stagnation point α may be modified to
α = 12
(
α0 + |M |2
α0
)
if |M | is smaller than a threshold α0. The expression for β in
subsonic flow can also be expressed as
β =
{
max (0, 2M − 1) if 0 ≤M ≤ 1
min (0, 2M + 1) if − 1 ≤M ≤ 0.
The coefficients α(M) and β(M) are displayed in figure 2.4 for the case when α0 = 0.
The cutoff of β when |M | < 12, together with α approaching zero as |M | approaches
zero, is also appropriate for the capture of contact discontinuities.
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 21
-1 1
1
α(M)
M -1 1
1
-1
M
β(M)
Figure 2.4: Diffusion Coefficients
2.3.2 Implementation of Limiters
By limiting the action of anti-diffusive terms, high resolution schemes which guarantee
the preservation of the positivity can be constructed in the case of a scalar conserva-
tion law. Typically, these schemes compare the slope of the solution at nearby mesh
intervals. The fluxes appearing in the CUSP scheme have different slopes approaching
from either side of the sonic line, and use of limiters which depends on comparisons
of the slopes of these fluxes can lead to a loss of smoothness in the solution at the
entrance to supersonic zones.
An alternative formulation is to form the diffusive flux from left and right states at
the cell interface. These are interpolated or extrapolated from nearby data, subject
to limiters to preserve monotonicity. Define
R (u, v) = 1 −∣
∣
∣
∣
u− v
|u| + |v|
∣
∣
∣
∣
q
, (2.29)
where q is a positive power. Then R(u, v) = 0 when u and v have opposite sign. Now
define the limited average as
L (u, v) =1
2R (u, v) (u+ v) . (2.30)
Let w(k) represents the kth element of the solution vector w. Now define left and
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 22
right states of dependent variable as
w(k)L = w
(k)j +
1
2L(
∆w(k)
j+ 3
2
,∆w(k)
j− 1
2
)
w(k)R = w
(k)j+1 −
1
2L(
∆w(k)
j+ 3
2
,∆w(k)
j− 1
2
)
,
where
∆wj+ 1
2
= wj+1 − wj. (2.31)
To implement the CUSP scheme the pressure pL and pR for the left and right states
are calculated from wL and wR. Then the diffusive flux is calculated by replacing wL
for wj and wR for wj+1 to give
dj+ 1
2
=1
2α∗c (wR − wL) +
1
2β (f(wR) − f(wL)) .
2.4 Time Integration and Convergence Accelera-
tion
2.4.1 Time stepping scheme
Consider the semi-discrete system
dw
dt+R (w) = 0, (2.32)
where w is the vector of flow variables and R (w) is the residuals resulting from spacial
discretization of the flow equations. In the case of steady state calculation, the order
of accuracy is immaterial and hence the scheme can be designed to maximize the
stability region.
Let us consider a semi-discretization of the linear model problem
∂v
∂t+ a
∂v
∂x= 0
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 23
with central differences and third order artificial diffusion ∼ ∆t3 ∂3u∂x3
∆tduj
dt=λ
2(uj+1 − uj−1) + λ (uj+2 − 4uj+1 + 6uj − 4uj−1 + uj−2)
where λ is the CFL number
λ =a∆t
∆x.
With the substitution of a Fourier mode u(x, t) = u(t)eipx, the resulting Fourier
symbol has an imaginary part proportional to the wave speed , and a negative real
part proportional to the diffusion. Thus the permissible CFL number depends on
the stability interval along the imaginary axis, as well as the negative real axis. To
achieve large stability intervals along both axes it pays to treat the convective and
dissipative terms in a distinct fashion [29]. Accordingly the residual is split as
R (w) = Q (w) +D (w) ,
where Q (w) is the convective part and D (w) is the dissipative part. Denote the time
level n∆t by a superscript n. Then the multistage time stepping scheme is formulated
as
w(0) = wn
w(1) = w0 − α1∆t(
Q(0) +D(0))
w(2) = w0 − α2∆t(
Q(1) +D(1))
. . .
w(k) = w0 − αk∆t(
Q(k−1) +D(k−1))
. . .
w(n+1) = wm,
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 24
where the superscript k denotes the k-th stage, αm = 1, and
Q(0) = Q(
w0)
, D(0) = β1D(
w0)
. . .
Q(k) = Q(
w(k))
D(k) = βk+1D(
w(k))
+ (1 − βk+1)D(k−1).
The coefficients αk are chosen to maximize the stability interval along the imaginary
axis, and the coefficients βk are chosen to increase the stability interval along the
negative real axis.
These schemes do not fall within the standard framework of Runge-Kutta schemes,
and they have much larger stability regions. Two schemes which have been found to
be particularly effective are tabulated below. The first is a four-stage scheme and its
coefficients areα1 = 1
3β1 = 1.00
α2 = 415
β2 = 0.50
α3 = 59
β3 = 0.00
α4 = 1 β4 = 0.00
The second is a five-stage scheme with three evaluation of dissipation and its coeffi-
cients areα1 = 1
4β1 = 1.00
α2 = 16
β2 = 0.00
α3 = 38
β3 = 0.56
α4 = 12
β4 = 0.00
α5 = 1 β5 = 0.44
2.4.2 Multigrid method
The concept of accelerating solution to steady-state by introducing multiple grids was
first proposed by Fedorenko [10]; however, this theory only holds for elliptic equations.
In 1982, Ni [43] applied the multigrid to Euler equations and various multigrid time-
stepping schemes have been proposed and implemented [12, 15, 16, 27, 38] to Euler
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 25
and Navier-Stokes equations since then. The basic idea of a multigrid time stepping
scheme is to transfer some of the task of tracking the evolution of the system to
a sequence of successively coarser meshes. In the case of an explicit time stepping
scheme, this permits the use of successively larger time steps without violating the
stability bound. Suppose that successively coarser grids are formed by agglomerating
fine grids cells in group of four (eight in three dimensions) and subscript k denotes
the k-th grid level. First the solution vector on grid k must be initialized as
w(0)k = Tk,k−1wk−1,
where wk−1 is the current solution on grid k − 1, and Tk,k−1 is a transfer operator
defined by
Tk,k−1wk−1 =
∑
Vk−1wk−1
Vk
,
where the sum is over the constituent cells on grid k − 1, and V is the cell area or
volume. Next it is necessary to transfer a residual forcing function such that the
solution on grid k is driven by the residuals calculated on grid k − 1. This can be
accomplished by setting
Pk = Qk,k−1Rk−1 (wk−1) − Rk(w(0)k ),
where Qk,k−1 is another transfer operator defined by
Qk,k−1Rk−1 =∑
Rk−1.
Then Rk(wk) is replaced by Rk(wk) + Pk in the time stepping scheme. Thus, the
multi-stage scheme is reformulated as
w(1)k = w
(0)k − α1∆tk
(
Rk
(
w(0)k + Pk
))
. . . = . . .
w(q+1)k = w
(0)k − αq+1∆tk
(
Rk
(
w(q)k + Pk
))
. . . = . . .
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 26
Figure 2.5: Multigrid W-cycle. E, evaluate the change in the flow for one step; C,collect the solution; T, transfer the data without updating the solution.
The result w(m)k then provides the initial data for grid k+1. Finally the accumulated
correction on grid k has to be transferred back to grid k − 1. Let w+k be the final
value of wk resulting from both the correction calculated in the time step on grid k
and the correction transferred from grid k + 1. Then one sets
w+k−1 = wk−1 + Ik−1,k
(
w+k − w0
k
)
,
where wk−1 is the solution on grid k − 1 after the time step on grid k − 1 and before
the transfer from grid k, and Ik−1,k is an interpolation operator. A multigrid W-cycle
illustrated in figure 2.5 proves to be a particularly effective strategy for managing the
work split between the meshes. In a three-dimensional case the number of cells is
reduced by a factor of eight on each coarser grid. On examination of the figure, it
CHAPTER 2. GOVERNING EQUATIONS AND DISCRETIZATION 27
can be seen that the work measured in units corresponding to a step on the fine grid
is of the order of
1 + 2/8 + 4/64 + . . . < 4/3,
and consequently the very large effective time step of the complete cycle costs only
slightly more than a single time step in the final grid.
2.4.3 Local time stepping and Residual smoothing
If the final steady state flow field is the only desired result, one can, instead of using
the minimum time step for each computational cell, advance the flow solution at each
cell’s stability limit. This often leads to faster convergence for the solution of the
steady state Euler and Navier-Stokes equations and has been used in this study.
The rate of convergence of multigrid scheme can be further improved by implicit
residual smoothing. The idea is to increase the time step limit by replacing the
residual at one cell with a weighted average of the residuals at the neighboring cells.
The average is calculated implicitly by the following formula
(1 − ǫiδxx)(1 − ǫjδyy)(1 − ǫkδzz)Ri,j,k = Ri,j,k, (2.33)
whereǫi, ǫj, and ǫk control the amount of smoothing, and Ri,j,k is the updated residual
obtained by solving equation 2.33 in each coordinate direction. A detailed discus-
sion of the overall benefit of this acceleration technique is provided by Jameson and
Baker [24].
Chapter 3
Design via Control Theory
The use of control theory for shape optimization of systems governed by elliptic
equations was first proposed by Pironneau [46]. In 1988, Jameson [17] extended this
idea to optimal aerodynamic design for transonic flow and with his associates has
successfully applied it to optimal aerodynamics design problems governed by Euler
and Navier-Stokes equations [18, 25, 22, 23, 26, 28] since then. The advantage of
this approach is that the required sensitivity information for large number of design
variables can be obtained by one flow equation plus one adjoint equation and the idea
is presented in the following.
3.1 Formulation of Adjoint Method
Suppose the aerodynamic performance index can be expressed by a cost function
I =
∫
B
M(w, S) dBξ +
∫
D
P(w, S) dDξ,
containing both boundary and field contributions where dBξ and dDξ are the surface
and volume elements in the computational domain. In general, both M and P are
functions of the flow variables w and the metrics S defining the computational domain.
The design problem is now treated as a control problem where the control is done
by varying the shape of the boundary to minimize the cost function subject to the
28
CHAPTER 3. DESIGN VIA CONTROL THEORY 29
constrained defined by equation 2.7. A variation in shape δS will result in a variation
of flow solution δw and in turn produce a variation in the cost function
δI =
∫
B
δM(w, S) dBξ +
∫
D
δP(w, S) dDξ, (3.1)
and δM and δP can be split as
δM =
[
∂M∂w
]
I
δw + δMII ,
δP =
[
∂P∂w
]
I
δw + δPII ,
(3.2)
where we use subscripts I and II to distinguish between the contributions associated
with the variation of the flow solution δw and those associated with the metric vari-
ations δS. Thus[
∂M∂w
]
Iand
[
∂P∂w
]
Irepresent the variations of M and P with metrics
fixed and δMII and δPII represent the contribution from the metric variations.
The variation of the flow solution δw can be obtained by taking the variation of
the constraint equation. Taking a variation of equation 2.7 at steady state,
δR =∂
∂ξiδ (Fi − Fvi) (3.3)
Here, both δFi and δFvi can also be split into contributions from δw and δS as
δFi =
[
∂Fi
∂w
]
I
δw + δFiII
δFvi =
[
∂Fvi
∂w
]
I
δw + δFviII .
(3.4)
The inviscid contributions are easily evaluated as
[
∂Fi
∂w
]
I
= Sij
∂fi
∂w, δFiII = δSijfj .
The variation of viscous contributions are complicated by the additional level of
derivatives in the stress and heat flux terms and detailed derivation are given by
CHAPTER 3. DESIGN VIA CONTROL THEORY 30
Appendix A.Multiplying equation 3.3 by a co-state vector ψ, which will play the role as La-
grange multiplier, and integrating over the domain gives
∫
D
ψT ∂
∂ξiδ (Fi − Fvi) dDξ = 0. (3.5)
Assuming ψ is differentiable and integrating equation 3.5 by parts to give
∫
B
niψT δ (Fi − Fvi) dBξ −
∫
D
∂ψT
∂ξiδ (Fi − Fvi) dDξ = 0. (3.6)
By using the relationship 3.4 and regrouping terms containing δw and δS, equation 3.6
becomes
∫
B
niψT
([
∂Fi
∂w
]
I
−[
∂Fvi
∂w
]
I
)
δw dBξ −∫
D
∂ψT
∂ξi
([
∂Fi
∂w
]
I
−[
∂Fvi
∂w
]
I
)
δw dDξ
+
∫
D
ψT δRII dDξ = 0,
(3.7)
where
∫
D
ψT δRII dDξ =
∫
B
niψT (δFiII − δFviII) dBξ −
∫
D
∂ψT
∂ξi(δFiII − δFviII) dDξ.
Since the left hand side of equation 3.7 equals zero, it may be subtracted from the
variation of cost function (3.1) to give
δI =
∫
B
{[
∂M
∂w
]
I
− niψT
([
∂Fi
∂w
]
I
−[
∂Fvi
∂w
]
I
)}
δw dBξ
+
∫
D
{[
∂P
∂w
]
I
+∂ψT
∂ξi
([
∂Fi
∂w
]
I
−[
∂Fvi
∂w
]
I
)}
δw dDξ
+
∫
B
{
δMII − niψT (δFiII − δFviII)
}
dBξ
+
∫
D
{
δPII +∂ψT
∂ξi(δFiII − δFviII)
}
dDξ.
(3.8)
CHAPTER 3. DESIGN VIA CONTROL THEORY 31
Since ψ is an arbitrary differentiable function, it may be chosen such that equation 3.8
no longer depends on δw which requires the reevaluation of flow solution for each per-
turbation of design variable and the gradient of cost function can then be calculated
directly from the evaluation of the variations of metric terms.
The elimination of δw from field integral in equation 3.8 produces the adjoint
equation governing ψ
∂ψT
∂ξi
([
∂Fi
∂w
]
I
−[
∂Fvi
∂w
]
I
)
+
[
∂P
∂w
]
I
= 0 in Dξ. (3.9)
The corresponding adjoint boundary condition is obtained by eliminating δw from
boundary integral in equation 3.8 to produce
niψT
([
∂Fi
∂w
]
I
−[
∂Fvi
∂w
]
I
)
=
[
∂M
∂w
]
I
on Bξ. (3.10)
The remaining terms from equation 3.8 then yield a simplified expression for the cost
function which defines the gradient
δI =
∫
B
{
δMII − niψT (δFiII − δFviII)
}
dBξ
+
∫
D
{
δPII +∂ψT
∂ξi(δFiII − δFviII)
}
dDξ.
(3.11)
The detailed formulation for the gradient depends on the way in which the bound-
ary shape is parameterized as a function of the design variables, and the way in which
the mesh is deformed as the boundary is modified. Using a relationship between the
mesh deformation and the surface modification, the field integral is reduced to a sur-
face integral by integrating along the coordinate lines emanating from the surface and
δI is finally reduced to
δI =
∫
B
GδFdBξ (3.12)
where F represents the design variables and G is the gradient, which is a function
defined over the boundary surface.
CHAPTER 3. DESIGN VIA CONTROL THEORY 32
3.2 Design using Euler equations
This section describes the derivation of adjoint equation and its corresponding bound-
ary condition for two-dimensional flow modeled by Euler equations. For flow modeled
by Navier-Stokes equations, the derivation is further complicated by extra level of
derivatives in the stress and heat flux terms and is explained in detail in Appendix A.
For an airfoil designed to meet a desired pressure distribution, a natural choice of
cost function is
I =1
2
∫∫
B
(p− pd)2 dS
where pd is the desired surface pressure. For simplicity, the airfoil is transformed from
a physical domain to a computational domain and, in this computational domain, the
cost function is transformed to
I =1
2
∫∫
Bw
(p− pd)2|S2| dξ (3.13)
where
|S2| =√
S2jS2j .
A variation of surface shape δ|S2| results in variation of pressure δp through the
equation of state (2.3) and hence a variation in the cost function
δI =
∫∫
Bw
(p− pd)δp|S2| dξ +1
2
∫∫
Bw
(p− pd)2δ|S2| dξ. (3.14)
At steady state, the Euler equation can be written in computational domain as
∂Fi
∂ξi= 0 in Dξ. (3.15)
Taking a variation of equation 3.15, multiplying it by a co-state vector, and integrating
over the domain result in∫
D
ψT ∂δFi
∂ξidDξ = 0. (3.16)
CHAPTER 3. DESIGN VIA CONTROL THEORY 33
Assuming ψ is differentiable, the above equation may be integrated by parts to give
∫
B
niψT δFi dξ −
∫
D
∂ψT
∂ξiδFi dDξ = 0, (3.17)
where
δFi = δSijfj + Sij
∂fj
∂wδw.
The boundary integral consists of contributions from airfoil surface and far fields. If
the variations of the mesh is such that δSij is negligible and ψ is chosen such that
niψTSij
∂fj
∂wδw = 0,
the only contribution from boundary integral is from airfoil surface. Because the
airfoil is restricted to the η = 0 surface, the only non-zero vector is n2 = 1 and
equation 3.17 becomes
∫
Bw
ψT δF2 dξ −∫
D
∂ψT
∂ξi
(
δSijfj + Sij
∂fj
∂wδw
)
dDξ = 0. (3.18)
Because there is no flow through the boundary at η = 0, so
U2 = 0 and δU2 = 0
when the boundary shape is modified. Consequently δF2 is reduced to
δF2 =
0
S21
S22
0
δp+
0
δS21
δS22
0
p. (3.19)
Because the right hand side of equation 3.18 equals to zero, it can be subtracted from
CHAPTER 3. DESIGN VIA CONTROL THEORY 34
equation 3.14 to give
δI =
∫
Bw
{(p− pd) |S2| − (ψ2S21 + ψ3S22)} δp dξ
+
∫
D
∂ψT
∂ξi
(
Sij
∂fj
∂w
)
δw dDξ +
∫
D
∂ψT
∂ξi(δSijfj) dDξ
+1
2
∫
Bw
(p− pd)2δ|S2| dξ −
∫
Bw
(ψ2δS21 + ψ3δS22) p dξ.
(3.20)
Because ψ is an arbitrary differentiable function, the dependence of δI on δw and δp
can be eliminated by choosing ψ to satisfy the adjoint equation
[
Sij
∂fj
∂w
]T∂ψ
∂ξi= 0 in Dξ, (3.21)
and its corresponding adjoint boundary condition
p− pd = ψ2S21
|S2|+ ψ3
S22
|S2|on Bw. (3.22)
Defining the components of surface normal vector as
nj =S2j
|S2|,
the adjoint boundary condition can be expressed as
ψj+1nj = p− pd. (3.23)
This amounts to a transpiration boundary condition on the co-state variables corre-
sponding to the momentum components. Note that it imposes no restriction on the
tangential component of ψ at the boundary.
Now the variation of the cost function reduced to
δI =1
2
∫
Bw
(p− pd)2δ|S2| dξ −
∫
Bw
(ψ2δS21 + ψ3δS22) p dξ
+
∫
D
∂ψT
∂ξi(δSijfj) dDξ,
(3.24)
CHAPTER 3. DESIGN VIA CONTROL THEORY 35
which is independent of the variations of the flow variables δp and δw.
3.2.1 Numerical Discretization of the Adjoint Equations
The adjoint differential equations for the Euler formulation have been given by equa-
tion 3.21. To find the solution of the adjoint equations, introduce a time-like derivative
term, which will vanish at the steady state solution of equation 3.21. Thus the adjoint
equations 3.21 can be written as
∂ψ
∂t− CT
i
∂ψ
∂ξi= 0 in Dξ, (3.25)
where
Ci = Sij
∂fj
∂w.
The convective adjoint flux is discretized using a second order central discretization.
Expand equation 3.25 for a two-dimensional problem
∂ψ
∂t− CT
1
∂ψ
∂ξ1− CT
2
∂ψ
∂ξ2= 0, (3.26)
and then the adjoint equation can be discretized as
V∂ψij
∂t=
1
2
[
CT1i,j
(ψi+1,j − ψi−1,j) + CT2i,j
(ψi,j+1 − ψi,j−1)]
+ di+ 1
2,j − di− 1
2,j + di,j+ 1
2
− di,j− 1
2
,
where V is the cell area and di± 1
2,j± 1
2
are the artificial dissipation terms. The Jacobian
fluxes can be expanded as
CT1i,j
= S11AT1i,j
+ S12AT2i,j
CT2i,j
= S21AT1i,j
+ S22AT2i,j,
where
AT1i,j
=
[
∂f
∂w
]T
, AT2i,j
=
[
∂g
∂w
]T
.
CHAPTER 3. DESIGN VIA CONTROL THEORY 36
3.2.2 Adjoint Boundary Conditions
For the inverse design problem with cost function
I =1
2
∫∫
B
(p− pd)2 dS,
where pd is the desired pressure, the adjoint boundary condition given by equa-
tion (3.23) is
ψj+1nj = p− pd,
where
nj =S2j
|S2|is the surface normal vector components. In order to make use of the summation
convention, it is convenient to set φj = ψj+1 for j = 1, 2 and θ = ψ4. Now the adjoint
boundary condition can be restated as
φjnj = p− pd, j = 1, 2, (3.27)
and it states the normal component of φ is equal to the difference between current
and desired target pressure. Equation (3.27) does not constrain the tangential compo-
nents, φ1 and φ4, of φ vector, and assign a zero value does not violate equation (3.27).
However, this results in poor convergence for the adjoint equation. Different treat-
ments of boundary conditions for the tangential component of the φ can be used, and
the one currently used in this research is based on the studies conducted by [49] For a
cell centered finite volume scheme, equation (3.27) is approximated at boundary cells
by(
φ2j + φ1
j
2
)
nj = p− pd j = 1, 2, (3.28)
where the superscript 2 and 1 represent the cells above and below the wall bound-
ary, respectively. Additional conditions representing the equivalence of tangential
components of φ above and below the wall are given by
φ2j − (φ2
ini)nj = φ1j − (φ1
ini)nj (3.29)
CHAPTER 3. DESIGN VIA CONTROL THEORY 37
From equation (3.28) and (3.29), the value of φj below the wall boundary can be
expressed as
φ1j = φ2
j + 2[
(p− pd) − φ2ini
]
nj . (3.30)
For the first and last costate variables, the discrete boundary condition are given
by
ψ11 = ψ2
1
θ1 = θ2.
For an inverse design problem, a set of satisfactory boundary condition at the wall
may be formulated as
ψ11 = ψ2
1
ψ12 = ψ2
2 + 2[
(p− pd) − n1ψ22 − n2ψ
23
]
n1
ψ13 = ψ2
3 + 2[
(p− pd) − n1ψ22 − n2ψ
23
]
n2
ψ14 = ψ2
4.
(3.31)
3.3 Optimization Algorithms
General gradient-based optimization procedures typically involve the calculation of
the gradient and line searches along the direction of steepest descent. This section de-
scribes the steepest descent, continuous descent, and gradient smoothing optimization
algorithms.
3.3.1 Steepest Descent
Let us define the objective function as I(x) and x represents the current design point,
where x ∈ Rn and f : R
n → R is a smooth function. We want to choose a direction
d and a step size α such that
I (x + αd) ≤ I (x) . (3.32)
CHAPTER 3. DESIGN VIA CONTROL THEORY 38
Then by Taylor’s theorem
I (x + αd) = I (x) + αdT▽I (x) +O
(
α2)
. (3.33)
For small enough α, O (α2) can be neglected and the variation of I can be represented
as
δI = I (x + αd) − I (x)
∼= αdT▽I (x) .
(3.34)
For a reduction in I, we need to choose d to be a descent direction such that
αdT▽I (x) < 0. (3.35)
In the steepest descent method, the search direction, d, is chosen to be the negative
of the gradient at each iteration
d = −▽I (x) , (3.36)
and equation 3.34 becomes
δI ∼= αdT▽I (x)
= −α ‖ ▽I(x) ‖2< 0.(3.37)
For a line search method, the step size α is chosen such that the maximum reduction
of the cost function I is obtained and new design obtained by
xk+1 = x
k − α▽I(xk). (3.38)
The determination of optimum step size, α, requires extra evaluations of the cost
function, which is very expensive in the case for the design using Euler or Navier-
Stokes equations. To avoid line searches, an alternative approach is to use a continu-
ous descent process. The basic idea is to treat the search process in equation 3.38 as
CHAPTER 3. DESIGN VIA CONTROL THEORY 39
a time dependent process in pseudo time. Rearrange equation 3.38 as
xk+1 − x
k
α= −▽I. (3.39)
In the limit as α → 0, this can be represented as
dx
dt= −▽I,
where α corresponds to a forward Euler discretization. The continuous descent pro-
cess has been analyzed by [25], and Jameson and Vassberg [14] provide a stability limit
for the Brachistochrone problem, where the time step is dominated by the parabolic
term in the continuous gradient formula.
3.3.2 Gradient Smoothing
The gradient ▽I obtained from section 3.2 is generally of a lower smoothness class
that the shape x. Hence it is necessary to restore the smoothness. Instead of taking
the step
δx = −α▽I(x),
a smoothed gradient ▽I is used. The smoothed gradient can be calculated from
▽I − ∂
∂xǫ∂
∂x▽I = ▽I,
where ǫ is the smoothing parameter and ▽I = 0 at end points. The the variation of
cost function becomes
δI = +
∫∫
▽Iδx dx
= −α∫∫
(
▽I − ∂
∂xǫ∂
∂x▽I
)
▽I dx
= −α∫∫
▽I2dx+ α
∫∫ (
∂
∂xǫ∂
∂x▽I
)
▽I dx.
CHAPTER 3. DESIGN VIA CONTROL THEORY 40
Integrating the second term by parts and applying ▽I = 0 at end points,
δI = −α∫∫
▽I2dx− α
∫∫
ǫ
(
∂▽I
∂x
)2
dx
= −α∫∫
(
▽I2+ ǫ
(
∂▽I
∂x
)2)
dx
< 0.
For a positive ǫ, the variation of the cost function is less than zero and this guarantees
an improvement unless ▽I and hence ▽I are zero. The gradient smoothing procedure
ensures that the new shape remains smooth which is critical in aerodynamic shape
optimization problems.
Chapter 4
Transition Prediction
The laminar-turbulent transition in the boundary layer is a very complex problem
and still an active area of research. It normally started with the development of small
disturbances in the laminar boundary layer and those disturbances grow as they
propagate downstream. Finally, the entire flow transits from laminar to turbulent.
The transition process is affected not only by the Reynolds number, but also by other
parameters such as pressure distribution, surface roughness, and level of disturbances
in the free stream.
There are many different types of laminar instabilities, e.g. Tollmien-Schlichting
waves, attachment line instability, Gortler vortices, and crossflow vortices. Schlicht-
ing [50] and White [59] provide detailed discussion of these instabilities. This chapter
describes two transition prediction methodologies that are suitable for engineering
design applications for Tollmien-Schlichting and crossflow instabilities.
4.1 Transition Analysis Overview
The most straightforward method for the transition analysis can be done by a direct
numerical simulation (DNS). DNS involves the numerical simulation of the three-
dimensional, time-dependent, turbulent flows governed by the Navier-Stokes equa-
tions. Even with today’s computational power, DNS is limited to simple geometry
41
CHAPTER 4. TRANSITION PREDICTION 42
with relatively low Reynolds number. For a complex geometry at high Reynolds num-
ber, a brute force DNS is infeasible and a different approach is needed for laminar-
turbulent transition analysis.
A step down from direct numerical simulation (DNS) is the solution of stability
equations. These equations are set of parabolic-type partial differential equations
obtained by subtracting the steady mean flow terms from Navier-Stokes equations
and describe the unsteady disturbances. They are also known as parabolized stabil-
ity equations (PSE) and have the advantage that efficient space-marching numerical
schemes can be devised. Although the PSE has the advantage of efficiently numerical
computation of laminar instabilities, the main drawback is that the solution of PSE
requires the initial conditions describing the birth of laminar instabilities which are
usually not available. On the other hand, the linear stability theory (LST) can be
used for transition prediction without extensive understanding of initial conditions.
Linear stability theory states that the initial disturbances grow or decay linearly in
steady laminar flow and the flow will remain laminar if the initial disturbances decay.
In the derivation of linear stability equations, each flow variable is decomposed into a
mean-flow term plus a fluctuation term and substitute into flow equations. Because
the fluctuations are assumed to be small, there products can be neglected. With the
additional assumption of parallel flow, a set of partial differential equations describing
the grow or decay of the disturbances can be derived. A detailed derivation can be
found in [4]. The difficulty of using the LST in transition prediction is that the user
has to monitor results, discard non-physical solutions, and modify inputs, and this
interactive procedure makes LST not suitable at initial design stage.
4.1.1 The eN-database Method
Both PSE and LST do not predict the transition locations, but they predict the
growth of instabilities in the laminar boundary layer. In industrial design applica-
tions, the most widely used method for streamwise transition prediction is the eN -
database method. This is a method based on linear stability theory and experimental
data. In the 1950s, van Ingen [57] and Smith and Gamberoni [51], independently
CHAPTER 4. TRANSITION PREDICTION 43
used the results from the linear stability theory and compared them with experi-
mental data of viscous boundary layers. They found that transition from laminar to
turbulent frequently happens when the amplification of disturbance calculated from
linear stability theory reaches about 8100. This corresponds to eN where N equals
to 9 and this is the well known criterion for Tollmien-Schlichting instabilities . The
present authors choose the eN -database method for streamwise transition prediction
because it has been proven [35] to provide reasonably accurate transition locations
on airfoils. For 3D swept wing, the variation of pressure in the spanwise direction
causes crossflow to develop in the boundary layer and this results in the crossflow
instability. This kind of instability is much more difficult to predict than streamwise
instability; however, there exists some criteria that can be used at initial design stage
and a similar N factor for crossflow, NCF , can be calculated for crossflow transition
prediction.
4.2 Transition Prediction
The first step in transition prediction using eN -database method is to calculate vis-
cous laminar boundary-layer parameters. In [48, 41], RANS solvers were used to
provide high accuracy boundary-layer parameters, e.g. displacement thickness, δ⋆,
and momentum thickness, θ, which are necessary for eN -database method,
δ⋆ =
∫ δe
0
(1 − U(y)
Ue
)dy
θ =
∫ δe
0
U(y)
Ue
(1 − U(y)
Ue
)dy, (4.1)
where δe is the edge of boundary layer. For this method to be successful, the edge
of boundary layer needs to be located. This can be achieved by first calculating
boundary-layer edge velocity, Ue, with pressure distribution and isentropic relation-
ship and, once the edge velocity is defined, the edge of boundary layer, δe, is located
at the location where U(y) intersects with Ue in the direction normal to the surface.
After the edge of boundary layer has been located, the boundary-layer parameters
CHAPTER 4. TRANSITION PREDICTION 44
can be calculated by equation 4.1. The use of a RANS solver to provide viscous
data is straightforward; however, it is necessary to have large number of mesh points
imbedded inside boundary layer and expensive grid adaptation may also be needed.
To reduce the computational cost of resolving the boundary layer, a compressible
laminar boundary-layer method for swept, tapered wings [13] was chosen by the au-
thors to produce highly accurate integral boundary-layer parameters for eN -database
method.
4.2.1 Streamwise Amplification Factor Calculation
With the availability of high quality boundary-layer parameters provided by the
boundary-layer code, the next step toward transition prediction is to calculate ampli-
fication factor for Tollmien-Schlichting waves, NTS, base on boundary-layer param-
eters. This can be accomplished by using parametric fits to the amplification rates
of TS waves and this has been done by Drela and Gleyzes et al [6, 11]. The current
authors use the parametric fitting results from [54] who introduces the ratio of wall
temperature to the external temperature, Tw/Te, as new parameter to account for
the stabilizing effect of compressible boundary layer. The TS amplification factor can
then be calculated by
NTS =
∫ Reθ
Reθ0
dnts
dReθ
dReθ, (4.2)
whereReθ
= momentum thickness Reynolds number,
Reθ0= critical Reynolds number = f
(
Hk,Tw
Te
)
,
dnTS
dReθ
= g(
Hk,Tw
Te
)
, and
Hk = kinematic shape factor =∫
(1− UUe
) dy∫
UUe
(1− UUe
) dy
At each station, the above parameters are calculated and the critical point is
reached when Reθ> Reθ0
. After the critical point is reached, Equation 4.2 is used to
integrate the amplification rate to give the amplification factor at the current station
and transition is predicted when NTS reaches about 9.
CHAPTER 4. TRANSITION PREDICTION 45
4.2.2 Crossflow Amplification Factor Calculation
For crossflow instability calculations, one of the most widely used methods is based
on the work of Owen and Randall [44] who suggest that crossflow Reynolds number
Rcf =ρe|wmax|δcf
µe
(4.3)
is the crucial parameter for cross flow instability. In the above definition, wmax is the
maximum velocity in the crossflow velocity profile and δcf , the crossflow thickness,
is the height where the crossflow velocity is about 1/10th of wmax. Malik et al. [37]
state that the transition occurs when the critical Reynolds number
Rcrit = 200
(
1 +γ − 1
2M2
e
)
(4.4)
is reached. Instead of simply using equation 4.4 as crossflow instability criterion, the
parametric fitting results from [54] are used in this work. The amplification rate, α,
of crossflow instability can be expressed as
α = α
(
Rcf ,wmax
Ue
, Hcf ,Tw
Te
)
. (4.5)
Those parameters are calculated at each station and the amplification rate, α, is
integrated
NCF =
∫ x
x0
α dx (4.6)
starting from x0 to give the croosflow amplification factor at current station, where
x0 is the location at which crossflow Reynolds number exceeds its critical value
Rcf0 = 46Tw
Te
. (4.7)
CHAPTER 4. TRANSITION PREDICTION 46
4.3 Transition Prescription
To simulate flow around a wing which comprises both laminar and turbulent flows,
it is necessary to divide the flow domain into laminar and turbulent subdomains and
apply turbulence model to turbulent flow subdomain. The current turbulence model
used in the RANS solver is the Baldwin-Lomax model [1] with total viscosity defined
as
τij = (µlam + µturb)
{
∂ui
∂xj
+∂uj
∂xi
− 2
3[∂uk
∂xk
]δij
}
(4.8)
where µlam is the coefficient of laminar viscosity and µturb is the coefficient of eddy vis-
cosity. The laminar-turbulent prescription is done by setting µturb = lt switch(x) µturb,
where lt switch(x) is the laminar-turbulent switch and its value depends on the lo-
cation of x according to
lt switch(x) =
{
= 0 if x ∈ laminar
= 1 if x ∈ turbulent(4.9)
4.3.1 Transition Prescription on Surface
The first step in transition prescription is to split the airfoil surface into laminar and
turbulent patches and this is achieved from the results of transition prediction mod-
ule. The transition prediction module uses the pressure coefficients provided by the
RANS solver as inputs, splits the airfoil into upper and lower surfaces from stagnation
point, analyzes each surface separately, and the results are the transition locations
on upper, xtran upper, and lower surface, xtran lower. Given the transition locations on
upper and lower surfaces of airfoil, the lt switch on the surface is set according to:
Upper surface:
xstag 6 x < xtran upper ⇒ lt switch = 0
x ≥ xtran upper ⇒ lt switch = 1
CHAPTER 4. TRANSITION PREDICTION 47
Lower surface:
xstag 6 x < xtran lower ⇒ lt switch = 0
x ≥ xtran lower ⇒ lt switch = 1
4.3.2 Transition Prescription in Flow Domain
With the laminar-turbulent patches defined on the surface of airfoil, the next step
is to define laminar-turbulent regions in the flow field. This is done by projecting
the turbulent patches into the flow field in the direction normal to airfoil surface and
the extent of turbulent zones is defined at the edge, which is a distance dedge normal
to the surface and can be controlled in the input file, of viscous layer. The result
is turbulent subdomains surrounded by laminar zones and is shown schematically in
Figure 4.1.
X
Y
0 0.2 0.4 0.6 0.8 1
-0.2
0
0.2
turb.
turb.
xtran_upper
xtran_lower
stagnationpoint
Figure 4.1: Schematic Diagram of Turbulent Subdomains Surrounded in LaminarZones
CHAPTER 4. TRANSITION PREDICTION 48
4.4 Coupling of Transition Prediction Module with
RANS Solver
The flow and adjoint solver chosen in this research are based on these developed
by Jameson [20, 26] and the flow solver solves the steady state RANS equations
on structured meshes with multistep time stepping scheme. Rapid convergence to
a steady state is achieved via variable local time stepping, residual averaging, and
multi-grid scheme.
The RANS solver is coupled with transition prediction module which consists of
a laminar boundary-layer code and two transition prediction methods for Tollmien-
Schlichting and crossflow instabilities. The complete coupling of transition prediction
module with RANS solver is summarized as following and shown schematically in
Figure 4.2.
1. The RANS solver starts its flow iterations with prescribed transition locations
setting far down stream on upper and lower surfaces of airfoil, e.g. 80% from
the leading edge.
2. With this fixed transition locations, the RANS solver iterates until the density
residual drops below certain level and the iteration on RANS solver is then
suspended.dρ
dt≤ dρ
dt limit
3. The transition prediction module is called. The surface pressure distribution
from RANS solver at current iteration is used as input for laminar boundary-
layer code to calculate all of the boundary-layer parameters which are necessary
for two eN -database methods.
4. With the calculated highly accurate boundary-layer parameters, Equations 4.2
and 4.6 are used to calculate amplification factors for T-S and C-F instabilities
and transition locations on both upper and lower surfaces can be determined.
The calculated transition locations are then fed into RANS solver and transi-
tion prescriptions on airfoil surfaces and in flow domains are performed. This
CHAPTER 4. TRANSITION PREDICTION 49
completes one iteration of transition prediction module.
5. The control of the program now returns back to the RANS solver and the flow
solver iterates again. With each successive flow iteration, the transition predic-
tion module is called and the determination of transition locations becomes an
iterative procedure. This is continued until the convergence criteria
|xtran(k) − xtran(k − 1)| ≤ δ
is reached, where k is the current iteration and δ is a small value, and this
condition is checked for Ncheck repeated times to prevent premature termination
of transition prediction.
repeated untilConvergence
Flow Solver
Adjoint Solver
Gradient Calculation
Shape & GridModification
QICTP boundary layer code
Transition Prediction Method
Transition Prediction ModuleCP
Xtran
Design Cycle
Figure 4.2: Coupling Structure of Flow Solver and Transition Prediction Module
Chapter 5
NLF Airfoil and Wing Design
Results
In this chapter, we first present results of verification of boundary-layer code and
transition locations tested on a benchmark case using the methodology described in
chapter 4 and then a natural-laminar-flow airfoil and wing design using Reynolds
averaged Navier-Stokes equations will be demonstrated. The results demonstrate
that it is necessary to prescribe the laminar-turbulent transition locations in order
to obtain more realistic results, e.g. the drag coefficient and lift-to-drag ratio, in
natural-laminar-flow wing design.
5.1 Verification of Boundary-Layer Parameters and
Transition Locations
The accuracy of the boundary-layer parameters calculated by the QICTP [13] code is
compared with the SWPTPR [54] and DLR Tau codes [41], where the NLF(1)-0416
airfoil at specific flight condition was used as a test case. Figures 5.1 and 5.2 show
the comparisons of calculated incompressible displacement thickness and momentum
thickness and, they are both in good agreement.
50
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 51
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5x 10
−3
x/c
δ*/c
SWPTPRDLR TauQICTP
Figure 5.1: Displacement Thickness, δ⋆, on Upper Surface for NLF(1)-0416 Airfoil,
M∞ = 0.3, Re∞ = 4 · 106, α = 2.03◦
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8x 10
−3
x/c
θ/c
SWPTPRDLR TauQICTP
Figure 5.2: Momentum Thickness, θ, on Upper Surface for NLF(1)-0416 Airfoil,
M∞ = 0.3, Re∞ = 4 · 106, α = 2.03◦
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 52
Table 5.1: Comparison of Predicted Transition Locations with Experimental Results
xtran upper xtran lower
Current Method 0.348 0.587Experiment 0.35 0.6
The calculations of laminar boundary layer commonly terminate on the approach
to flow separation and this can be clearly seen on both figures. This early termi-
nation of the boundary layer calculation, in general, does not pose a problem for
transition prediction because the calculated transition locations are located upstream
of the termination locations. In the case where the boundary-layer calculation does
terminate before reaching the limiting N factor, the transition location is set at the
location where boundary-layer calculation terminates, and this transition is classified
as transition due to laminar separation.
The transition locations predicted with current method are compared with the
experimental results from Somers [52] and the results are in good agreements as
can be seen from Table 5.1. In this case, the initial transition locations are set
at 70% from the leading edge on both upper and lower surfaces of airfoil, and the
transition prediction module is turned on after the density residual drops below a
certain level. Figure 5.3 shows the convergence history of transition locations and it
is clear that transition locations converge to their final values in about ten iterations
after the transition prediction module is turned on. Natural-laminar-flow over a wing
is very sensitive to small unevenness or surface contamination, and the verification of
transition location due to surface waviness is given in Appendix B.
5.2 Natural-Laminar-Flow Airfoil Design
The design targets of this natural-laminar-flow airfoil are based on the specifications
of the Honda lightweight business jet [39] at its cruise condition. The initial shape of
the airfoil is designed by using the adjoint method with Navier-Stokes equations [26]
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 53
42 44 46 48 50 52 54 56
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Number of Iterations
Xtran
upper surfacelower surface
Figure 5.3: Convergence History of Transition Locations, xtran,upper =0.348, xtran,lower = 0.587, for NLF(1)-0416 Airfoil, M∞ = 0.3, Re∞ = 4 ·106, α = 2.03◦
and
I =1
2
∮
B
(p− pd)2 dS
is used as the cost function. This corresponds to an inverse design problem and the
shape of airfoil is modified to match the desired target pressure, pd. The pressure
coefficient of this designed airfoil at cruise condition is shown in figure 5.4 and does
demonstrate a reasonable amount of laminar flow on both surfaces. The convergence
history of transition locations for the designed airfoil is shown in figure 5.5 with
the final transition locations located at 0.510 and 0.546 on upper and lower surface,
respectively.
With certain assumptions, a good estimate of range performance is provided by
the Breguet range equation
R =V L
D
1
SFClog
W1
W2
,
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 54
NLF AIRFOIL MACH 0.690 ALPHA -1.214 RE 0.117E+08
CL 0.2600 CD 0.0023 CM -0.0765 CLV 0.0000 CDV 0.0034
GRID 512X64 NDES 0 RES0.527E-03 GMAX 0.000E+00
0.1E+
010.8
E+00
0.4E+
00-.2
E-15
-.4E+
00-.8
E+00
-.1E+
01-.2
E+01
-.2E+
01
Cp
+++++++++++++++++++++++
+++++++++
+++++++
+++++
++++
++++
++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++
+
+++++++
++
+
+
++++++++++++++++++++++++++++++++++++
++++++++
++++++++
++++++++
++++++++
+++++++++
++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.4: Pressure Distribution for Designed NLF Airfoil, M∞ = 0.69, Re∞ =11.7 · 106, Cltarget
= 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 55
40 42 44 46 48 50 520.5
0.55
0.6
0.65
0.7
Number of Iterations
xtran
upper surfacelower surface
Figure 5.5: Convergence History of Transition Locations, M∞ = 0.69, Re = 11.7 ·106, xtran,upper = 0.51, xtran,lower = 0.546
where V is the speed, L/D is the lift to drag ratio, SFC is the specific fuel con-
sumption of the engines, W1 is take-off weight, and W2 is the landing weight. From
aerodynamic point of view, this suggests that designer should try to increase the
speed until the onset of drag rise in order to maximize range. The authors believe
that the new designed airfoil can be further optimized for a higher Mach number to
improve the range parameter, M∞L/D, and still maintain a reasonable amount of
laminar flow at the same time. The design Mach number is increased from 0.69 to
0.72 and the adjoint optimization technique is used to minimize drag and keep the
same amount of lift. In this case, the adjoint method is mainly used to minimize
the wave drag resulting from the existence of shock wave due to higher flying Mach
number. Figure 5.6 and 5.7 shows the pressure distributions at new design Mach be-
fore and after optimizations, respectively. As expected, there is a strong shock wave
on the top of airfoil surface due to the increase of Mach number and this results in
significant increase of wave drag. After 30 design cycles, the shock wave is completely
eliminated and this greatly reduce the inviscid drag from 46 counts to 24 counts. The
new designed airfoil has M∞L/D = 33.4, which is much better than the one flying at
M∞ = 0.69 with M∞L/D = 31.4.
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 56
NLF AIRFOIL MACH 0.720 ALPHA -1.302 RE 0.120E+08
CL 0.2600 CD 0.0046 CM -0.0820 CLV 0.0000 CDV 0.0031
GRID 512X64 NDES 0 RES0.167E-02 GMAX 0.000E+00
0.1E
+01
0.8E
+00
0.4E
+00
-.2E
-15
-.4E
+00
-.8E
+00
-.1E
+01
-.2E
+01
-.2E
+01
Cp
+++++++++++++++++++++
+++++++++
++++++
+++++
++++
++++
+++++++
++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+
+
+
+
+++++
++
+
+
+++++++++++++++++++++++++++++++++++++
+++++++
++++++
++++++++
++++++++
++++++
++++++
++++++++++++++
+
+
+
+
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.6: Number of design iterations: 0
NLF AIRFOIL MACH 0.720 ALPHA -0.980 RE 0.120E+08
CL 0.2600 CD 0.0024 CM -0.0649 CLV 0.0000 CDV 0.0032
GRID 512X64 NDES 0 RES0.492E-03 GMAX 0.000E+00
0.1E
+01
0.8E
+00
0.4E
+00
-.2E
-15
-.4E
+00
-.8E
+00
-.1E
+01
-.2E
+01
-.2E
+01
Cp
+++++++++++++++++++
+++++++++
+++++++
+++++
++++
++++
++++
++++
++++++
++++
++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+
+
+
++++++
++
+
+
++++++++++++++++++++++++++++++++++++
++++++
+++++++
+++++++++
++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.7: Number of design iterations: 30, M∞ = 0.72, Re = 12 · 106, Cltarget= 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 57
Natural-laminar-flow airfoils may have undesirable characteristics, such as forma-
tion of shock waves, when flying at off-design conditions. The new designed airfoil is
then tested at three off-design flight conditions to make sure that the new design does
not exhibit undesirable characteristics. Figures 5.8-5.10 show the pressure distribu-
tions at those off-design conditions and they do demonstrate that the new design is
satisfactory at both design and off-design conditions.
NLF AIRFOIL MACH 0.690 ALPHA -0.863 RE 0.120E+08
CL 0.2600 CD 0.0021 CM -0.0617 CLV 0.0000 CDV 0.0032
GRID 512X64 NDES 0 RES0.531E-03 GMAX 0.000E+00
0.1E
+01
0.8E
+00
0.4E
+00
-.2E-
15-.4
E+00
-.8E+
00-.1
E+01
-.2E+
01-.2
E+01
Cp
++++++++++++++++++++
+++++++++++
+++++++
+++++
++++
++++
++++
+++++++
++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+++++++
+
+
+
+
+++++++++++++++++++++++++++++++++++++++
++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.8: Off-design Condition at M∞ = 0.69, Cltarget= 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 58
NFL AIRFOIL MACH 0.700 ALPHA -0.895 RE 0.120E+08
CL 0.2600 CD 0.0022 CM -0.0625 CLV 0.0000 CDV 0.0032
GRID 512X64 NDES 0 RES0.513E-03 GMAX 0.000E+00
0.1E
+01
0.8E
+00
0.4E
+00
-.2E-
15-.4
E+00
-.8E+
00-.1
E+01
-.2E+
01-.2
E+01
Cp
++++++++++++++++++++
+++++++++++
+++++++
+++++
++++
++++
++++
+++++++
++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+++++++
+
+
+
+
++++++++++++++++++++++++++++++++++++
+++++++
+++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.9: Off-design Condition at M∞ = 0.70, Cltarget= 0.26
NLF AIRFOIL MACH 0.710 ALPHA -0.935 RE 0.120E+08
CL 0.2600 CD 0.0023 CM -0.0636 CLV 0.0000 CDV 0.0032
GRID 512X64 NDES 0 RES0.497E-03 GMAX 0.000E+00
0.1E
+01
0.8E
+00
0.4E
+00
-.2E-
15-.4
E+00
-.8E+
00-.1
E+01
-.2E+
01-.2
E+01
Cp
++++++++++++++++++++
+++++++++++
+++++++
+++++
++++
++++
++++
+++++++
++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+++++++
+
+++++++
+
+
+
+
++++++++++++++++++++++++++++++++++++
++++++
++++++++
++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.10: Off-design Condition at M∞ = 0.71, Cltarget= 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 59
To demonstrate the effects of including transition prediction model, figure 5.11
and 5.12 show the comparison of optimized airfoil profiles calculated from automatic-
transition prediction and full-turbulence model for M∞ = 0.72 and Cltarget= 0.26.
The cost function used in this case is the drag coefficient which is directly related to
the surface pressure distribution. Because of the differences in the pressure distribu-
tion between automatic-transition prediction and full-turbulence model as shown in
figure 5.13 and 5.14, the computed gradients are also different and this results in the
differences in the optimized airfoil profiles, especially in the upper-rear portion of the
airfoil as can be clearly seen in figure 5.12.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
x/c
y/c
auto−transition
full−turbulence
Figure 5.11: Comparison of optimized airfoil profiles between automatic-transitionprediction and full-turbulence model. M∞ = 0.72, Cltarget
= 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 60
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
x/c
y/c
auto−transition
full−turbulence
Figure 5.12: Comparison of optimized airfoil profiles at upper-rear portion betweenautomatic-transition prediction and full-turbulence model.
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 61
HONDA AIRFOIL MACH 0.720 ALPHA -0.815 RE 0.120E+08
CL 0.2600 CD 0.0024 CM -0.0583 CLV 0.0000 CDV 0.0032
GRID 512X64 NDES 0 RES0.493E-03 GMAX 0.000E+00
0.1E
+01
0.8E
+00
0.4E
+00
-.2E-
15-.4
E+00
-.8E+
00-.1
E+01
-.2E+
01-.2
E+01
Cp
+++++++++++++++++++
++++++++++
+++++++
++++++
++++
++++
++++
+++++
+++++
++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+++++++
+
+++++++
+
+
+
+
++++++++++++++++++++++++++++++++++++
+++++++
+++++++++
++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.13: Pressure distribution for automatic-transition prediction case, M∞ =0.72, Cltarget
= 0.26
HONDA AIRFOIL MACH 0.720 ALPHA -0.541 RE 0.120E+08
CL 0.2600 CD 0.0045 CM -0.0496 CLV 0.0000 CDV 0.0055
GRID 512X64 NDES 0 RES0.400E-03 GMAX 0.000E+00
0.1E
+01
0.8E
+00
0.4E
+00
-.2E-
15-.4
E+00
-.8E+
00-.1
E+01
-.2E+
01-.2
E+01
Cp
+++++++++++++++++++++
++++++++++
+++++++
++++++
++++
++++
++++
++++++
++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+++++++
+
++++++
++
+
+
+
+++++++++++++++++++++++++++++++++++++
+++++++
+++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Figure 5.14: Pressure distribution for full-turbulence case, M∞ = 0.72, Cltarget= 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 62
5.3 Natural-Laminar-Flow Wing Calculation
The wing used in the 3D computation is a semi-span, swept, tapered wing with taper
ratio λ = 0.278. The leading and trailing edge of the wing are swept at ΛLE = 16.69◦
and ΛTE = 1.67◦, respectively, and cross sections are made up of airfoils designed at
M∞ = 0.69 from section 5.2.
The mesh used in this computation is a C-type structured mesh with total number
of 786432 cells in the flow domain. The wing is defined by 128 cells looping around
the airfoil from the bottom of trailing edge to the top of trailing edge and has 33
airfoil sections along the span direction. To speed up the computation, the domain
is divided into subdomains and a 3D RANS solver paralleled by MPI is used to solve
the flow field to steady state. Figure 5.15 shows the distribution of mesh lines and
divided subdomains used in this computation.
Three different target lift coefficients and their corresponding flight Mach numbers
were studied. The target lift coefficients were achieved by constantly adjusting the
angle of attack during flow iterations. Tables 5.2-5.4 summarize the comparison of the
aerodynamic coefficients for the results obtained from automatic transition prediction
and 100% full turbulence for three cases studied here.
Table 5.2: Case 1: Comparison of Aerodynamic Coefficients , M = 0.69, CL = 0.26
CL CDpressCDfric
CDtotL/Dpress L/D
Auto 0.258 58.0 39.5 97.5 44.54 26.49100% 0.259 71.6 61.3 133 36.14 19.47
Table 5.3: Case 2: Comparison of Aerodynamic Coefficients, M = 0.70, CL = 0.38
CL CDpressCDfric
CDtotL/Dpress L/D
Auto 0.379 89.1 40.3 129.4 42.59 29.31100% 0.379 103.3 60.7 164.0 36.71 23.12
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 63
X
Y
Z
Figure 5.15: Mesh Distribution and Divided Subdomains
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 64
Table 5.4: Case 3: Comparison of Aerodynamic Coefficients, M = 0.70, CL = 0.50
CL CDpressCDfric
CDtotL/Dpress L/D
Auto 0.50 130.2 40.4 170.7 38.33 29.25100% 0.50 146.1 60.2 206.3 34.16 24.19
Figures 5.16-5.17 show the contour plots of computed pressure coefficient on upper
and lower surface, respectively, for M∞ = 0.69 and CL = 0.26 and it can be seen that
the variations of pressure are mainly in the streamwise direction, but not much in
chordwise direction.
In these calculations, the initial transition locations are set at 80% from the wing
leading edge. For streamwise instability, NTS = 9, which is well-known and accepted,
was chosen as limiting N factor. Depending on the levels of surface roughness, the
N factor for crossflow instability varies in a wide range. Based on the results from
Crouch and Ng [5] and assumed surface roughness level, NCF = 8 was chosen in this
study. During the flow iteration, the density residual is monitored and transition
prediction module is turned on after the residual drops below certain level. Each
wing section is analyzed individually, the new transition locations are calculated, and
transition prescription is applied according to section 4.3.
Figures 5.18-5.19 show the initial and final transition locations on upper and lower
surface, respectively, for M∞ = 0.69 and CL = 0.26. Except at few inboard sections,
the majority of transitions are due to Tollmien-Schlichting instability. In Figures 5.20-
5.21, the contours of wall shear stress are shown and it can be clearly seen that there
is a rise of shear stress downstream of transition lines.
Figure 5.22 shows the variations of drag coefficient as Mach number increases for
both 100% turbulence and automatic transition prediction cases. Although the airfoils
used for the wing section were designed at M∞ = 0.69, the drag increases slowly until
M∞ = 0.72. Beyond this Mach number, there is a relatively larger drag increment
due to the formation of shock waves on the upper surfaces. One of the most important
performance requirements for an executive jet is the the cruise efficiency, which can
be measured by the range parameter M · (L/D). The range parameter as a function
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 65
X-0.500.511.52
Z
0
1
2
CP10.90.80.70.60.50.40.30.20.10
-0.1-0.2-0.3-0.4-0.5-0.6-0.7
Figure 5.16: Pressure Distribution on Upper Surface, M = 0.69, CL = 0.26
X-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Z
0
1
2
CP0.980.860.740.620.50.380.260.140.02
-0.1-0.22-0.34-0.46-0.58-0.7
Figure 5.17: Pressure Distribution on Lower Surface, M = 0.69, CL = 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 66
X-1 -0.5 0 0.5 1 1.5 2
Z
0
1
2
initial
final
Upper surface
Figure 5.18: Initial and Final Transition Locations on Upper Surface, M = 0.69, CL =0.26
X-1 0 1 2
Z
0
0.5
1
1.5
2
2.5
Lower surface
final
initial
Figure 5.19: Initial and Final Transition Locations on Lower Surface, M = 0.69, CL =0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 67
X-1-0.500.511.52
Z
0
1
2
TAUW
0.00190.00180.00170.00160.00150.00140.00130.00120.00110.0010.00090.00080.00070.00060.00050.00040.00030.00020.0001
Figure 5.20: Shear Stress, τ , Distribution on Upper Surface, M = 0.69, CL = 0.26
X-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Z
0
1
2
TAUW
0.00190.00180.00170.00160.00150.00140.00130.00120.00110.0010.00090.00080.00070.00060.00050.00040.00030.00020.0001
Figure 5.21: Shear Stress, τ , Distribution on Lower Surface, M = 0.69, CL = 0.26
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 68
of Mach number for current wing is shown in Figure 5.23 and it does demonstrate
a satisfactory characteristic around the designed Mach number. In fact, the range
parameter keeps increasing until M∞ = 0.72 before the formation of shock waves. It is
also evident from figures 5.22 and 5.23 that one does need to prescribe the transition
locations in order to obtain more realistic results in laminar-flow calculations.
0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.880
90
100
110
120
130
140
150
160
170
180
190
M
CD
(co
unts
)
Full turbulence
Tran Prediction
Figure 5.22: CD v.s. Mach number
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 69
0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.810
11
12
13
14
15
16
17
18
19
M
ML/
D
Full turbulence
Tran Prediction
Figure 5.23: Range parameter v.s. Mach number
5.4 Natural-Laminar-Flow Wing Design
As for the 2D airfoil design case, the 3D wing can be designed for higher cruise
Mach number to further improve the range performance. From figure 5.22, it is
clear that there is a sudden increase in drag at M∞ = 0.74 due to the formation of
relatively strong shock waves on the upper surface of the wing. The design target
Mach number is then increased from M∞ = 0.69 to M∞ = 0.74 and the adjoint
optimization technique is used to eliminate shock waves at new design Mach number.
Figure 5.24 and 5.25 display the pressure distributions of the baseline NLF wing
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 70
IAI-NLF5 Mach: 0.740 Alpha:-0.015 CL: 0.258 CD: 0.01352 CM:-0.1669 Design: 20 Residual: 0.1833E-02 Grid: 257X 65X 49
Cl: 0.233 Cd: 0.02408 Cm:-0.1077 Root Section: 6.2% Semi-Span
Cp = -2.0
Cl: 0.281 Cd: 0.00292 Cm:-0.1257 Mid Section: 49.2% Semi-Span
Cp = -2.0
Cl: 0.210 Cd:-0.00372 Cm:-0.1050 Tip Section: 92.3% Semi-Span
Cp = -2.0
Figure 5.24: Full turbulence design for NLF 3D wing. Dashed lines and solid linesrepresent pressure distribution of the baseline NLF wing and redesigned configurationrespectively
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 71
IAI-NLF5 Mach: 0.740 Alpha:-0.230 CL: 0.257 CD: 0.01016 CM:-0.1715 Design: 20 Residual: 0.9618E-03 Grid: 257X 65X 49
Cl: 0.229 Cd: 0.02295 Cm:-0.1124 Root Section: 6.2% Semi-Span
Cp = -2.0
Cl: 0.282 Cd: 0.00135 Cm:-0.1338 Mid Section: 49.2% Semi-Span
Cp = -2.0
Cl: 0.208 Cd:-0.00549 Cm:-0.1104 Tip Section: 92.3% Semi-Span
Cp = -2.0
Figure 5.25: Automatic transition prediction design for NLF 3D wing. Dashed linesand solid lines represent pressure distribution of the baseline NLF wing and redesignedconfiguration respectively
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 72
0 5 10 15 20 25101
102
103
104
105
106
107
108
109
110
111
Design iteration
CD
(co
unts
)
Convergence history
Figure 5.26: Convergence history of the NLF wing cost function
and redesigned configuration after 20 design cycles for full turbulence and automatic
transition prediction cases respectively. In both cases, the shock waves are completely
eliminated and directly result in a reduction in drag.
The convergence history of drag minimization with automatic transition prediction
is shown on figure 5.26. The initial oscillations of drag coefficient is due to the
formation of two relatively weak shock waves on the top of the wing and they are
completely removed after 10 design iterations.
By eliminating shock waves at M∞ = 0.74, the new designed wing does demon-
strate an improvement in terms of drag coefficient. For wing design, one seeks not
only an improvement at a single design point, but also requires the new design to per-
form not worse than the original design at off design conditions. Figure 5.27 shows
the comparison of drag coefficient between original and new designed wing and the
new wing clearly demonstrates an improvement over a wide range of cruising Mach
number. The comparison of range parameter as a function of Mach number is shown
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 73
on figure 5.28 and an overall improvement is also evident.
It can be seen from figure 5.24 and 5.25 that the optimized wings have shifted
their suction peaks forward during the drag minimization process. This is because
only the drag coefficient, CD, is used as the cost function, and this forward movement
of suction peaks might result in early laminar-turbulent transition. To mitigate the
adverse effects of forward movement of suction peaks, a new cost function which
is used to minimize drag and, at the same time, try to maintain suction peaks at
40 ∼ 50% chord is used. Figure 5.29 shows the pressure distribution with the new
cost function and it is evident that the suction peaks are moved backward as compared
with figure 5.25. Figure 5.30 and 5.31 show the comparisons of final transition lines
calculated from two different cost functions for upper and lower surface. It can be seen
that the new cost function results in a further delay of laminar-turbulent transition
due the the effect of backward movement of suction peaks, and this delayed boundary-
layer transition also results in further reduction in drag.
5.5 Discussion
It can be seen from the results of this chapter that the predicted drag coefficient
and lift-to-drag ratio are very different between full-turbulence and laminar-turbulent
transition model. The difference in drag comes from the contributions of both pressure
and skin friction drag. The higher skin friction drag in full turbulence case is due to the
fact that the complete wing surface is submerged in high velocity gradient turbulent
flow and high shear stress is applied to the complete wetted area of wing surface; in
contrast, only part of the wing is subjected to high shear stress in laminar-turbulent
case and this directly results in lower skin friction drag. The effect of turbulent
boundary layer is not only on the skin friction drag, but also on the pressure drag as
well. The existence of boundary layer creates pressure imbalance in the drag direction
and greater imbalance of pressure is created if the flow is full turbulence than if the
flow comprises both laminar and turbulent regions. This is the reason that there are
also differences in pressure drag in table 5.2-5.4.
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 74
0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.890
100
110
120
130
140
150
M
CD
(co
unts
)
Orig. Wing
New Design
Figure 5.27: Comparison of drag coefficient as a function of Mach number betweenthe baseline and redesigned NLF wing
0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.813
14
15
16
17
18
19
M
ML/
D
Orig. Wing
New Design
Figure 5.28: Comparison of range parameter as a function of Mach number betweenthe baseline and redesigned NLF wing
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 75
IAI-NLF5 Mach: 0.740 Alpha:-0.281 CL: 0.260 CD: 0.00998 CM:-0.1763 Design: 20 Residual: 0.2216E-02 Grid: 257X 65X 49
Cl: 0.235 Cd: 0.02290 Cm:-0.1201 Root Section: 6.2% Semi-Span
Cp = -2.0
Cl: 0.284 Cd: 0.00134 Cm:-0.1397 Mid Section: 49.2% Semi-Span
Cp = -2.0
Cl: 0.205 Cd:-0.00525 Cm:-0.1112 Tip Section: 92.3% Semi-Span
Cp = -2.0
Figure 5.29: Redesign of 3D wing with new cost function. Dashed lines and solid linesrepresent pressure distribution of the baseline NLF wing and redesigned configurationrespectively
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 76
X-1 -0.5 0 0.5 1 1.5 2
Z
0
0.5
1
1.5
2
2.5
Upper surface
Final transition lineOrig. design
Final transition lineNew design
Figure 5.30: Comparison of final transition lines on upper surface
X-1 0 1 2
Z
0
1
2
Lower surface
Final transition lineNew design
Final transition lineOrig. design
Figure 5.31: Comparison of final transition lines on lower surface
CHAPTER 5. NLF AIRFOIL AND WING DESIGN RESULTS 77
For both 2D and 3D cases, the redesigned airfoil and wing configurations demon-
strate satisfactory improvements not only at a single design point, but also at off-
design conditions. The results show that it is feasible and necessary to incorpo-
rate adjoint optimization technique with laminar-turbulent transition prediction in
natural-laminar-flow wing design.
Chapter 6
Conclusion
This dissertation focuses on the application of optimization technique based on control
theory for natural-laminar-flow airfoil and wing design in viscous compressible flow
modeled by the Reynolds averaged Navier-Stokes equations. A transition prediction
module which consists of a boundary layer method and two eN -database methods for
Tollmien-Schlichting and crossflow instabilities were coupled with the flow solver to
predict and prescribe transition locations automatically.
The results of this study demonstrate that the coupling of a 2D RANS flow solver
with a transition prediction module provides reasonable accurate transition locations.
By using the adjoint method to provide the gradient information which is necessary
for gradient-based optimization technique, an airfoil can be designed to have a desired
favorable pressure distribution for laminar flow and the new airfoil can be redesigned
for higher Mach number for performance benefits while still maintains reasonable
amount of laminar flow.
For 3D wing configurations, the difference in aerodynamic coefficients are evident,
and this indicates the necessary of incorporating transition prediction mechanism with
the flow solver in order to obtain more realistic results. The airfoil sections for the
3D wing have thickness-to-chord ratio about 15% which gives the airplane sufficient
fuel volume for the required range. The redesigned configuration not only has a
reduction in drag and improvement on range parameter at the design Mach number,
but also has an overall improvement over wide range of off-design Mach numbers.
78
CHAPTER 6. CONCLUSION 79
Although the predicted transition locations are not as accurate as airfoils due to
complicated nonlinear interactions between streamwise and crossflow instabilities,
the results are still reasonable and can be used for the estimation of aerodynamic
performance coefficients at the initial design stage for industrial applications. It is
important to notice that different choices of limiting N factors and parametric fitting
formula will results in different wing section profiles and aerodynamic performance
coefficients. This dissertation presents a methodology for natural-laminar-flow wing
design which can be further improved by replacing the parametric fitting formula in
the transition prediction module with better transition prediction model in the future.
Appendix A
Derivation of Viscout Adjoint
Terms
In computational coordinates, the viscous terms in the Navier-Stokes equations have
the form∂Fvi
∂ξi=
∂
∂ξi(Sijfvj) . (A.1)
Taking the variation δw resulting from a shape modification of the boundary, intro-
ducing a co-state vector ψ, and integrating equation A.1 by parts following the steps
outlined by equation 3.3 to 3.8 produces
∫
Bw
ψT (δS2jfvj + S2jδfvj) dBξ −∫
D
∂ψT
∂ξi(δSijfvj + Sijδfvj) dDξ, (A.2)
where the shape modification is restricted to the coordinates ξ2 = 0 so that n1 = n3 =
0 and n2 = 1. Furthermore, it is assumed that the boundary contributions at the
far field may either be neglected or else eliminated by a proper choice of boundary
conditions.
The viscous adjoint terms will be derived under the assumption that the viscosity
and heat conduction coefficients µ and k are essentially independent of the flow, and
their variations may be neglected. This simplification has been successfully used for
many aerodynamic problems of interest. In the case of some turbulent flows, there
is the possibility that the flow variations could result in significant changes in the
80
APPENDIX A. DERIVATION OF VISCOUT ADJOINT TERMS 81
turbulent viscosity, and it may then be necessary to account for its variation in the
calculation.
A.1 Transformation to Primitive Variables
The derivation of the viscous adjoint terms is simplified by transforming to the prim-
itive variables
wT = (ρ, u1, u2, u3, p)T , (A.3)
because the viscous tresses depend on the velocity derivatives ∂U−i∂xj
, while the heat
flux can be expressed as
κ∂
∂xi
(
p
ρ
)
.
where κ = kR
= γµ
Pr(γ−1). The relationship between the conservative and primitive
variables is defined by the expressions
δw = Mδw, δw = M−1δw
which make use of the transformation matrices M = ∂w∂w
and M−1 = ∂w∂w
. These
matrices are provided in transposed form for future convenience
MT =
1 u1 u2 u3uiui
2
0 ρ 0 0 ρu1
0 0 ρ 0 ρu2
0 0 0 ρ ρu3
0 0 0 0 1γ−1
M−1T=
1 −u1
ρ−u2
ρ−u3
ρ
(γ−1)uiui
2
0 1ρ
0 0 − (γ − 1)u1
0 0 1ρ
0 − (γ − 1)u2
0 0 0 1ρ
− (γ − 1)u3
0 0 0 0 γ − 1
APPENDIX A. DERIVATION OF VISCOUT ADJOINT TERMS 82
The conservative and primitive adjoint operators L and L corresponding to the vari-
ations δw and δw are then related by
∫
D
δwTLψ dDξ =
∫
D
δwT Lψ dDξ, (A.4)
with
L = MTL,
so that after determining the primitive adjoint operator by direct evaluation of the
viscous portion of equation 3.9, the conservative operator may be obtained by the
transformation L = M−1TL. Since the continuity equation contains no viscous terms,
it makes no contribution to the viscous adjoint system. Therefore, the derivation pro-
ceeds by first examining the adjoint operators arising from the momentum equations.
A.2 Contributions from the Momentum Equations
In order to make use of the summation convention, it is convenient to set ψj+1 = φj
for j = 1, 2, 3. Then the contribution from the momentum equation is
∫
B
φk (δS2jσkj + S2jδσkj) dBξ −∫
D
∂φk
∂ξi(δSijσkj + Sijδσkj) dDξ (A.5)
The velocity derivative in the viscous stresses can be expressed as
∂ui
∂xj
=∂ui
∂ξl
∂ξl∂xj
=Slj
J
∂ui
∂ξl
with corresponding variations
δ∂ui
∂xj
=
[
Slj
J
]
I
∂
∂ξlδui +
[
∂ui
∂ξl
]
II
δ
(
Slj
J
)
.
APPENDIX A. DERIVATION OF VISCOUT ADJOINT TERMS 83
The variations in the stresses are then
δσkj =
{
µ
[
Slj
J
∂
∂ξlδuk +
Slk
J
∂
∂ξlδuj
]
+ λ
[
δjkSlm
J
∂
∂ξiδum
]}
I
+
{
µ
[
δ
(
Slj
J
)
∂uk
∂ξl+ δ
(
Slk
J
)
∂uj
∂ξl
]
+ λ
[
δjkδ
(
Slm
J
)
∂um
∂ξl
]}
II
.
As before, only those terms with subscript I, which contain variations of the flow
variables, need be considered further in deriving the adjoint operator. The field
contributions that contain δui in equation (A.5) appear as
−∫
D
∂φk
∂ξiSij
{
µ
(
Slj
J
∂
∂ξlδuk +
Slk
J
∂
∂ξlδuj
)
+ λδjkSlm
J
∂
∂ξlδum
}
dDξ.
This may be integrated by parts to yield
∫
D
δuk
∂
∂ξl
(
SljSij
µ
J
∂φk
∂ξi
)
dDξ
+
∫
D
δuj
∂
∂ξl
(
SlkSij
µ
J
∂φk
∂ξi
)
dDξ
+
∫
D
δum
∂
∂ξl
(
SlmSij
λδjkJ
∂φk
∂ξi
)
dDξ,
where the boundary integral has been eliminated by noting that δui = 0 on the solid
boundary. By exchanging indices, the field integrals may be combined to produce
∫
D
δuk
∂
∂ξlSlj
{
µ
(
Sij
J
∂φk
∂ξi+Sik
J
∂φj
∂ξi
)
+ λδjkSim
J
∂φm
∂ξi
}
dDξ,
which is further simplified by transforming the inner derivatives back to Cartesian
coordinates∫
D
δuk
∂
∂ξlSlj
{
µ
(
∂φk
∂xj
+∂φj
∂xk
)
+ λδjk∂φm
∂xm
}
dDξ. (A.6)
The boundary contributions that contain δui in equation (A.5) may be simplified
using the fact that∂
∂ξlδui = 0 if l = 1, 3
APPENDIX A. DERIVATION OF VISCOUT ADJOINT TERMS 84
on the boundary B so that they become
∫
B
φkS2j
{
µ
(
S2j
J
∂
∂ξ2δuk +
S2k
J
∂
∂ξ2δuj
)
+ λδjkS2m
J
∂
∂ξ2δum
}
dBξ. (A.7)
Together, (A.6) and (A.7) comprise the field and boundary contributions of the mo-
mentum equations to the viscous adjoint operator in primitive variables.
A.3 Contributions from the Energy Equation
In order to derive the contribution of the energy equation to the viscous adjoint terms
it is convenient to set
ψ5 = θ, Qj = uiσij + κ∂
∂xj
(
p
ρ
)
,
where the temperature has been written in terms of pressure and density using equa-
tion (2.5). The contribution from the energy equation can then be written as
∫
B
θ (δS2jQj + S2jδQj) dBξ −∫
D
∂θ
∂ξi(δSijQj + SijδQj) dDξ. (A.8)
The field contributions that contain δui, δp, and δρ in equation (A.8) appears as
−∫
D
∂θ
∂ξiSijδQj dDξ = −
∫
D
∂θ
∂ξiSij{δukσkj + ukδσkj
+ κSlj
J
∂
∂ξl
(
δp
ρ− p
ρ
δρ
ρ
)
} dDξ.
(A.9)
The term involving δσkj may be integrated by parts to produce
∫
D
δuk
∂
∂ξlSlj
{
µ
(
uk
∂θ
∂xj
+ uj
∂θ
∂xk
)
+ λδjkum
∂θ
∂xm
}
dDξ, (A.10)
where the conditions ui = δui = 0 are used to eliminate the boundary integral on B.
Notice that the other term in (A.9) that involves δuk need not be integrated by parts
APPENDIX A. DERIVATION OF VISCOUT ADJOINT TERMS 85
and is merely carried on as
−∫
D
δukσkjSij
∂θ
∂ξidDξ. (A.11)
The terms in expression (A.9) that involve δp and δρ may also be integrated by
parts to produce both a field and a boundary integral. The field integral becomes
∫
D
(
δp
ρ− p
ρ
δρ
ρ
)
∂
∂ξl
(
SljSij
κ
J
∂θ
∂ξi
)
dDξ
which may be simplified by transforming the inner derivative to Cartesian coordinates
∫
D
(
δp
ρ− p
ρ
δρ
ρ
)
∂
∂ξl
(
Sljκ∂θ
∂xj
)
dDξ. (A.12)
The boundary integral becomes
∫
B
κ
(
δp
ρ− p
ρ
δρ
ρ
)
S2jSij
J
∂θ
∂ξidBξ. (A.13)
This can be simplified by transforming the inner derivative to Cartesian coordinates
∫
B
κ
(
δp
ρ− p
ρ
δρ
ρ
)
S2j
∂θ
∂xj
dBξ, (A.14)
and identifying the normal derivative at the wall
∂
∂n= S2j
∂
∂xj
, (A.15)
and the variation in temperature
δT =1
R
(
δp
ρ− p
ρ
δρ
ρ
)
, (A.16)
to produce the boundary contribution
∫
B
kδT∂θ
∂ndBξ. (A.17)
APPENDIX A. DERIVATION OF VISCOUT ADJOINT TERMS 86
This term vanishes if T is constant on the wall but persists if the wall is adiabatic.
There is also a boundary contribution left over from the first integration by parts
(A.8) which has the form∫
B
θδ (S2jQj) dBξ, (A.18)
where
Qj = k∂T
∂xj
,
since ui = 0. Notice that for future convenience in discussing the adjoint boundary
conditions resulting from the energy equation, both the δw and δS terms correspond-
ing to subscript classes I and II are considered simultaneously. If the wall is adiabatic
∂T
∂n= 0,
so that using (A.15),
δ (S2jQj) = 0,
and both the δw and δS boundary contributions vanish.
On the other hand, if T is constant ∂T∂ξl
for l = 1, 3, so that
Qj = k∂T
∂xj
= k
(
Slj
J
∂T
∂ξl
)
= k
(
S2j
J
∂T
∂ξ2
)
.
Thus, the boundary integral (A.18) becomes
∫
B
kθ
{
S2j2
J
∂
∂ξ2δT + δ
(
S2j2
J
)
∂T
∂ξ2
}
dBξ. (A.19)
Therefore, for constant T , the first term cooresponding to variations in the flow field
contributes to the adjoint boundary operator and the second set of terms coorespond-
ing to metric variations contribute to the cost function gradient.
All together, the contributions from the energy equation to the viscous adjoint
operator are the three field terms ( A.10), ( A.11), and ( A.12), and either of two
boundary contributions ( A.17) or ( A.19), depending on whether the wall is adiabatic
or has constant temperature.
APPENDIX A. DERIVATION OF VISCOUT ADJOINT TERMS 87
A.4 The Viscous Adjoint Field Operator
Collecting together the contributions from the momentum and energy equations, the
viscous adjoint operator in primitive variables can be expressed as
(
Lψ)
1= − p
ρ2
∂∂ξl
(
Sljκ∂θ∂xj
)
(
Lψ)
i+1= ∂
∂ξl
{
Slj
[
µ(
∂φi
∂xj+
∂φj
∂xi
)
+ λδij∂φk
∂xk
]}
+ ∂∂ξl
{
Slj
[
µ(
ui∂θ∂xj
+ uj∂θ∂xi
)
+ λδijuk∂θ∂xk
]}
for i = 1, 2, 3
−σijSlj∂θ∂ξl
(
Lψ)
5= 1
ρ∂
∂ξl
(
Sljκ∂θ∂xj
)
.
The conservative viscous adjoint operator may now be obtained by the transformation
L = M−1TL.
Appendix B
Verification of Transition
Prediction Module
Natural-laminar-flow over a wing is very sensitive to small unevenness or surface
contamination, and premature laminar-turbulent transition could occur due to the
imperfection of manufacture. To test the capability of current transition prediction
module to detect the surface unevenness, a small artificial bump is introduced on
the airfoil upper surface as shown in figure B.1 and B.2. It can be seen from both
figures that the size of the bump is relatively small, and this is used to simulate
the imperfection of manufacture or surface contamination. Figure B.3 and B.4 show
the pressure distribution, and it can be clearly seen from fiture B.4 that there is a
local pressure disturbance due to the existence of the bump. Figure B.5 shows the
convergence history of the transition location, and the transition prediction module
indeed detects the existence of the bump and set the transition location in the rear
part of the bump. In the case of without a bump, the transition would happen
further downstream than the current location. This study simulates the imperfection
of airfoil surface and demonstrates the capability of transition prediction module to
detect the transition location.
88
APPENDIX B. VERIFICATION OF TRANSITION PREDICTION MODULE 89
Figure B.1: Bump location on upper surface
Figure B.2: Close look of bump location and transition location on upper surface
APPENDIX B. VERIFICATION OF TRANSITION PREDICTION MODULE 90
Figure B.3: Pressure distribution on upper surface
Figure B.4: Close look of pressure distribution near the bump on upper surface
APPENDIX B. VERIFICATION OF TRANSITION PREDICTION MODULE 91
40 42 44 46 48 50 52 540.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Number of flow Iteration
Tra
nsiti
on lo
catio
ns
upper surfacelower surface
Figure B.5: Convergence history of transition locations for airfoil with artificiallyintroduced bump
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