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




(Robert W. MacCormack)




(Juan J. Alonso)

Approved for the University Committee on Graduate Studies.

iii

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.

vi

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.

vii

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

viii

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

ix

A.3 Contributions from the Energy Equation . . . . . . . . . . . . . . . . 84

A.4 The Viscous Adjoint Field Operator . . . . . . . . . . . . . . . . . . 87

B Verification of Transition Prediction Module 88

x

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

xi

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

xii

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



5.11 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

xiii

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

xiv

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


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]


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.


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


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.


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.


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.


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)


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)


• •(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


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


(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.


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)


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)


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


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


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.


-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


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


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,


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


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

))

. . . = . . .


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


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


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)


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.


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)


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


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)


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

.


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)


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)


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


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.


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


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


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.


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)


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


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


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


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◦


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]


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

,


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


40 42 44 46 48 50 520.5

0.55

0.6

0.65

0.7

Number of Iterations

xtran


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.



CL 0.2600 CD 0.0046 CM -0.0820 CLV 0.0000 CDV 0.0031


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


CL 0.2600 CD 0.0024 CM -0.0649 CLV 0.0000 CDV 0.0032


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


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.


CL 0.2600 CD 0.0021 CM -0.0617 CLV 0.0000 CDV 0.0032


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


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


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

++++++++++++++++++++

+++++++++++

+++++++

+++++

++++

++++

++++

+++++++

++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+++++++

+

+

+

+

++++++++++++++++++++++++++++++++++++

+++++++

+++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++



CL 0.2600 CD 0.0023 CM -0.0636 CLV 0.0000 CDV 0.0032


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

++++++++++++++++++++

+++++++++++

+++++++

+++++

++++

++++

++++

+++++++

++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++

+

+++++++

+

+

+

+

++++++++++++++++++++++++++++++++++++

++++++

++++++++

++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++



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


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.


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


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


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


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


X

Y

Z

Figure 5.15: Mesh Distribution and Divided Subdomains


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


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


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


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


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


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


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




Cp = -2.0


Cp = -2.0


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


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


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.


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




Cp = -2.0


Cp = -2.0


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


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


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


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

)

.


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


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


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)


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.


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


Figure B.3: Pressure distribution on upper surface

Figure B.4: Close look of pressure distribution near the bump on upper surface


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


Figure B.5: Convergence history of transition locations for airfoil with artificiallyintroduced bump

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