ABSTRACT
Title of thesis: APPLICATION OF UNCERTAINTY QUANTIFI-
CATION OF TURBULENCE INTENSITY ON
AIRFOIL AERODYNAMICS
Atif Salahudeen, Masters of Science, 2017
Thesis directed by: Dr. James Baeder
Department of Aerospace Engineering
Traditional CFD results have a number of freestream inputs. In the physical
world, these input conditions often have some uncertainty associated with them.
However, this uncertainty is often omitted from the CFD results. The effects of
uncertainty in CFD can be determined through application of Uncertainty
Quantification (UQ). The primary objective of the present work is to determine
the effect of uncertainty in freestream turbulence intensity (FSTI) on the
coefficients of lift, drag, and moment for four different airfoils: S809, NACA
0012, SC1095, and RC(4)-10. In this work, the Monte Carlo method is used to
calculate the sensitivities of the aerodynamic coefficients to Gaussian
distributions of uncertainty in FSTI over a range of angles of attack (AOA) at
various Reynolds numbers and Mach numbers. However, the Monte Carlo
method would require hundreds of thousands of CFD calculations in order to
converge to the correct results. A surrogate surface is therefore generated using a
parametric study using the in-house flow solver OVERTURNS. Rather than run
a separate CFD run for each Monte Carlo run, all of the results can be attained
virtually instantaneously via the surrogate surface.
The UQ analysis shows how varying these parameters affects the sensitivies of
the aerodynamic coefficients to uncertainty in FSTI. In most cases, the response
is nearly Gaussian and the mean response is not too different from the discrete
FSTI response without uncertainty. However, the output standard deviation for
drag and pitching moment can become large when the transition location
changes rapidly with changing FSTI.
APPLICATION OF UNCERTAINTY QUANTIFICATION
OF TURBULENCE INTENSITY ON AIRFOIL
AERODYNAMICS
by
Atif Salahudeen
Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Masters of Science
2017
Advisory Committee:
Dr. James Baeder, Chair/Advisor
Dr. Inderjit Chopra
Dr. Anubhav Datta
© Copyright by
Atif Salahudeen
2017
Acknowledgments
First and foremost I would like to express my gratitude to God Almighty for
showering me with His blessings and enabling me to achieve all that I have
achieved.
This thesis is the result of an amazing supporting cast whose help was invaluable.
I would like to thank my advisor, Dr. James Baeder, for his wisdom, guidance,
and patience. I would have never been able to overcome the many challenges in
my research without his help. I would also like to thank my committee members,
Dr. Inderjit Chopra and Dr. Anubhav Datta, for the time they spent to serve on
my committee as well as the invaluable feedback they provided.
Next, I would like to thank Dr. Nishan Jain, Bumseok Lee, Dr. Camli Badrya,
and Dylan Jude for all of their help with this research, their constructive
criticism, and for their patience with my endless questions. I would also like to
thank all of my friends, especially my good friend Luthfe Siddique, for making
both my undergraduate and graduate experiences at the University of Maryland
truly enjoyable.
Finally, I am forever indebted to my parents, grandmothers, and sister. Without
their unwavering love and support, I would not be where I am today.
ii
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Boundary Layer Transition . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Laminar, Turbulent, and Transitional Flow . . . . . . . . . . 4
1.2.2 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Physical Mechanisms of Transition . . . . . . . . . . . . . . 8
1.2.5 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.6 Types of Transition . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.7 Transition Modeling . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . 17
iii
CONTENTS CONTENTS
1.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.3 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . 19
1.3.4 Stochastic Collocation . . . . . . . . . . . . . . . . . . . . . 20
1.3.5 Surrogate Surfaces . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Scope and Organization of Thesis . . . . . . . . . . . . . . . . . . . 25
2 Numerical Implementation 27
2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Non-Dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Reynolds-Averaged Navier-Stokes Equations . . . . . . . . . . . . . 33
2.4 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Numerical Algorithms in OVERTURNS . . . . . . . . . . . . . . . 36
2.6.1 Inviscid Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . 37
iv
CONTENTS CONTENTS
2.6.2 Viscous Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . 39
2.6.4 Transition Modeling . . . . . . . . . . . . . . . . . . . . . . 41
2.6.5 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Uncertainty Quantification Methodology 48
3.1 Parametric Sweeps . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Surrogate Surface Generation . . . . . . . . . . . . . . . . . . . . . 52
3.3 UQ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Surrogate Surface Validation . . . . . . . . . . . . . . . . . . 56
4 Parametric Sweep Results 59
4.1 Computational Meshes . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 S809 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 NACA 0012 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 SC1095 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 RC(4)-10 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
v
CONTENTS CONTENTS
5 Uncertainty Quantification Results 115
5.1 S809 Airfoil Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 NACA 0012 Airfoil Results: Reynolds Number Sweep . . . . . . . . 124
5.3 SC1095 Airfoil Results: Mach Number Sweep . . . . . . . . . . . . 132
5.4 RC(4)-10 Airfoil Results: Mach Number Sweep . . . . . . . . . . . 138
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6 Conclusions 147
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2 Key Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.1 General Observations . . . . . . . . . . . . . . . . . . . . . . 149
6.2.2 S809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.2.3 NACA 0012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2.4 SC1095 and RC(4)-10 . . . . . . . . . . . . . . . . . . . . . 152
6.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.4 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 154
vi
CONTENTS CONTENTS
Appendices 157
A Complete Parametric Sweep Results 158
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
vii
List of Tables
2.1 Piecewise Linear Correlations between Reθt and FSTI . . . . . . . . 43
3.1 Freestream Conditions for Parametric Sweeps . . . . . . . . . . . . 49
3.2 FSTI values for the quadrature for mean FSTI = 1%, σ = 0.3333 . 54
3.3 Number of FSTI Input Values Required for Consistency . . . . . . . 55
3.4 Comparison of Expected QOIs and Standard Deviations for for S809
Airfoil at Mach = 0.2, Re = 2× 106, AOA = 8 degrees . . . . . . . 58
4.1 Freestream Conditions for Parametric Sweeps . . . . . . . . . . . . 60
4.2 NACA 0012 - Mach = 0.2, Re = 1×106, Mach = 0.2 - Peak-to-Peak
Averaged Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 RC(4)-10 - Peak-to-Peak Averaged Cases . . . . . . . . . . . . . . . 102
viii
List of Figures
1.1 Skin friction over flat plate, Tu = 2.0% (reproduced from [6]) . . . . 7
1.2 Boundary layer transition (reproduced from Comsol website) . . . . 7
1.3 S809 Airfoil at Mach = 0.2, Re = 2× 106, AOA = 1 degree, FSTI
= 0.05% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Natural transition due to Tollmien-Schlichting waves (reproduced
from White 1991) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Flow structure of laminar-separation bubble (reproduced from Hor-
ton 1968) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Transformation from physical domain to computational domain (re-
produced from Blazek 2006) . . . . . . . . . . . . . . . . . . . . . . 35
2.2 C-topology mesh around NACA 0012 . . . . . . . . . . . . . . . . . 36
2.3 Computational Cell(reproduced from [36]) . . . . . . . . . . . . . . 37
2.4 Transition Model Validation for S809 Airfoil . . . . . . . . . . . . . 45
ix
LIST OF FIGURES LIST OF FIGURES
3.1 Initial Turbulence Sweep for S809 Airfoil at Mach = 0.2, Re = 2×106 50
3.2 Final Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106 51
3.2 Final Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2×106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 UQ Analysis for Drag - S809 Airfoil, Mach = 0.2, Re = 2 × 106,
Mean FSTI = 1.0%, Std Dev = 0.3333% . . . . . . . . . . . . . . . 54
3.4 Comparison of Surrogate Results and CFD Results for S809 Airfoil
at Mach = 0.2, Re = 2× 106, AOA = 8 degrees . . . . . . . . . . . 57
4.1 Computational Meshes for Airfoil Parameter Sweeps . . . . . . . . . 61
4.2 Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106 . . 62
4.2 Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2×106 (cont.) 63
4.3 S809 Airfoil at Mach = 0.2, Re = 2.0 × 106 - Aerodynamic Coeffi-
cients vs. AOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 S809 Airfoil at Mach = 0.2, Re = 2.0 × 106 - Skin Friction and
Pressure Plots - Separation Bubble . . . . . . . . . . . . . . . . . . 65
4.5 Separation Bubble for S809 Airfoil for AOA = 0°, FSTI = 1.0%,
Mach = 0.2, Re = 2.0× 106 . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
0.5× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
x
LIST OF FIGURES LIST OF FIGURES
4.6 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
0.5× 106 (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Pressure and Friction Plots for NACA 0012 Airfoil, Re = 0.5× 106,
Mach = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.8 Zoomed in Skin Friction Plots for NACA 0012 Airfoil, Re = 0.5×106,
Mach = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.9 NACA 0012 - Coefficient of Lift, Mach = 0.2 . . . . . . . . . . . . . 73
4.10 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
1× 106, Mach = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.11 Convergence Oscillations - NACA 0012 - Mach = 0.2, Re = 1×106,
AOA = 5°, FSTI = 0.1% . . . . . . . . . . . . . . . . . . . . . . . . 76
4.12 Skin Friction Plots for NACA 0012 Airfoil, Re = 1.0× 106, Mach =
0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.13 Zoomed in Skin Friction Plots for NACA 0012 Airfoil, Re = 1.0×106,
Mach = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.14 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
2× 106, Mach = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.15 Skin Friction Plots for NACA 0012 Airfoil, Re = 2.0× 106, Mach =
0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.16 NACA 0012 Airfoil at Mach = 0.2, Re = 2× 106, Mach = 0.2 . . . 81
4.17 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 4×106 83
xi
LIST OF FIGURES LIST OF FIGURES
4.17 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
4× 106 (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.18 Skin Friction Plot for NACA 0012 Airfoil, Re = 4.0× 106, Mach =
0.2, AOA = 12 degrees . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.19 Parametric Sweep for SC1095 Airfoil at Mach = 0.2, Re = 2× 106 . 86
4.20 SC1095 Airfoil at Mach = 0.2, Re = 2× 106 . . . . . . . . . . . . . 87
4.21 Skin Friction Plots for SC1095 - Mach = 0.2 . . . . . . . . . . . . . 88
4.22 Parametric Sweep for Drag Coefficient - SC1095 Airfoil at Mach =
0.4, Re = 2× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.23 Mach Contours for SC1095 Airfoil at Mach = 0.4, Re = 2× 106 . . 90
4.24 Parametric Sweep for Drag Coefficient - SC1095 Airfoil at Mach =
0.6, Re = 2× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.25 Mach Contours for SC1095 Airfoil at Mach = 0.6, Re = 2 × 106,
AOA = 6 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.26 Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.6, AOA
= 6 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.27 Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2× 106 . 94
4.27 Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2× 106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.28 Mach Contours for SC1095 Airfoil at Mach = 0.7, Re = 2 × 106,
AOA = 6 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
xii
LIST OF FIGURES LIST OF FIGURES
4.29 Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.7, AOA
= 6 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.30 Parametric Sweep for SC1095 Airfoil at Mach = 0.8, Re = 2× 106 . 97
4.31 Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.8, AOA
= 2 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.32 Mach Contours for SC1095 Airfoil at Mach = 0.8, Re = 2 × 106,
AOA = 6 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.33 Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.8, AOA
= 6 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.34 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.2, Re = 2× 106 101
4.35 RC(4)-10 Airfoil at Mach = 0.2, Re = 2× 106 . . . . . . . . . . . . 102
4.36 Skin Friction Plots for RC(4)-10 - Mach = 0.2, Re = 2× 106 . . . . 103
4.37 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.4, Re = 2× 106 105
4.38 Skin Friction Plots for RC(4)-10 - Mach = 0.4 . . . . . . . . . . . . 106
4.39 Comparison of Plots for SC1095 Airfoil and RC(4)-10, Mach = 0.4,
AOA = 2 degrees, FSTI = 0.5% . . . . . . . . . . . . . . . . . . . . 107
4.40 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.6, Re = 2× 106 108
4.41 Skin Friction Plots for RC(4)-10 - Mach = 0.6 . . . . . . . . . . . . 109
xiii
LIST OF FIGURES LIST OF FIGURES
4.42 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.7, Re = 2× 106 110
4.43 Skin Friction Plot for RC(4)-10 - Mach = 0.7 - AOA = 6 degrees . . 111
4.44 Mach Plot for RC(4)-10 - Mach 0.7 - AOA = 6° - FSTI = 0.5% . . 111
4.45 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.8, Re = 2× 106 113
4.46 Plots for RC(4)-10 Airfoil, Mach = 0.8, Re = 2× 106 . . . . . . . . 114
5.1 Distributions of Inputs and Outputs for S809 Airfoil at AOA = 2°,
Mach = 0.2, Re = 2× 106, Input σ = 0.3333% . . . . . . . . . . . . 117
5.2 Monte Carlo Results for S809 Airfoil for Mean Input FSTI 1.0%
and Standard Deviation 0.3333% . . . . . . . . . . . . . . . . . . . 119
5.3 UQ Results for S809 Airfoil at Mach = 0.2, Re = 2 × 106, Input
σ = 0.3333% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 UQ Results for S809 Airfoil at Mach = 0.2, Re = 2 × 106, Mean
FSTI = 1.0% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5 Drag Polar for S809 Airfoil at Mach = 0.2, Re = 2× 106 . . . . . . 123
5.6 Standard Deviations for NACA 0012 Airfoil at Mach = 0.2, Mean
FSTI = 1%, Input σ = 0.3333% . . . . . . . . . . . . . . . . . . . . 125
5.7 Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re =
500,000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.8 Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re =
500,000 - AOA 8° versus 12° . . . . . . . . . . . . . . . . . . . . . . 127
xiv
LIST OF FIGURES LIST OF FIGURES
5.9 Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re =
1× 106 - AOA 3° - 5° . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.10 Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re =
2× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.11 Pressure and Skin Friction Plots for NACA 0012 Airfoil at Mach =
0.2, Re = 2× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.12 Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re =
4× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.13 UQ Results for SC1095 Airfoil at Re = 2× 106, Mean FSTI = 1%,
Input σ = 0.3333% . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.14 Mach Contours for SC1095 Airfoil at 6° AOA, Re = 2× 106 . . . . 136
5.15 Skin Friction Plots for SC1095 Airfoil, Mach = 0.7 - 0.8, Re = 2× 106137
5.16 UQ Results for RC(4)-10 Airfoil at Re = 2×106, Mean FSTI = 1%,
Input σ = 0.3333% . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.17 Skin Friction Plots for RC(4)-10 Airfoil, Mach = 0.2 - 0.4, Re =
2× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.18 Skin Friction Plots for RC(4)-10 Airfoil, Mach = 0.6, Re = 2× 106 144
5.19 Skin Friction Plots for RC(4)-10 Airfoil, Mach = 0.7 - 0.8, Re =
2× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.1 Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106 . . 159
xv
LIST OF FIGURES LIST OF FIGURES
A.1 Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2×106 (cont.)160
A.2 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
0.5× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
A.2 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
0.5× 106 (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.3 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 1×106162
A.3 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
1× 106 (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.4 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 2×106163
A.4 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
2× 106 (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.5 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 4×106165
A.5 Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re =
4× 106 (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.6 Parametric Sweep for SC1095 Airfoil at Mach = 0.2, Re = 2× 106 . 166
A.6 Parametric Sweep for SC1095 Airfoil at Mach = 0.2, Re = 2× 106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.7 Parametric Sweep for SC1095 Airfoil at Mach = 0.4, Re = 2× 106 . 168
A.7 Parametric Sweep for SC1095 Airfoil at Mach = 0.4, Re = 2× 106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
xvi
LIST OF FIGURES LIST OF FIGURES
A.8 Parametric Sweep for SC1095 Airfoil at Mach = 0.6, Re = 2× 106 . 169
A.8 Parametric Sweep for SC1095 Airfoil at Mach = 0.6, Re = 2× 106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.9 Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2× 106 . 171
A.9 Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2× 106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
A.10 Parametric Sweep for SC1095 Airfoil at Mach = 0.8, Re = 2× 106 . 172
A.10 Parametric Sweep for SC1095 Airfoil at Mach = 0.8, Re = 2× 106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
A.11 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.2, Re = 2× 106 174
A.11 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.2, Re = 2×106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
A.12 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.4, Re = 2× 106 175
A.12 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.4, Re = 2×106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
A.13 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.6, Re = 2× 106 177
A.13 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.6, Re = 2×106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A.14 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.7, Re = 2× 106 178
xvii
LIST OF FIGURES LIST OF FIGURES
A.14 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.7, Re = 2×106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.15 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.8, Re = 2× 106 180
A.15 Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.8, Re = 2×106
(cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
xviii
Nomenclature
√
a Speed of sound = γRT
c Chord length
D
Cd Sectional drag coefficient = 1ρ
2 ∞U
2
∞c
τw
Cf Sectional friction coefficient = 1ρ∞U22 ∞
L
Cl Sectional lift coefficient = 1ρ U2 c
2 ∞ ∞
p− p∞
Cp Pressure coefficient = 1ρ
2 ∞U
2
∞
d Nearest wall distance
D Drag force
e Internal energy
δ∗
H Shape factor =
θ
1 [ ]
k Turbulent kinetic energy = (u′)2 + (v′)2
2
L Reference length / Lift force
M Mach number
p Static pressure
Pr Prandtl number
r Distance along rotor blade non-dimensionalized by rotor radius
ρ∞U∞L
Re Reynolds number =
µ∞
ρ∞U∞θ
Reθ Reynolds number based on momentum thickness =
µ∞
Reθt Reθ at transition onset location
Reθt Transport variable for Reθt
LIST OF FIGURES LIST OF FIGURES
ρd2Ω
Rev Vorticity Reynolds number =
µ
t Time √
2k/3
Tu∞ Freestream turbulence intensity =
U∞
u, v Flow velocity components along Cartesian coordinate directions
u′, v′ Turbulent fluctuation velocities along Cartesian coordinate directions
u′u′i j Time-average of Reynolds stress tensor components
U∞ Total freestream velocity magnitude
Ue Boundary layer edge velocity magnitude
x, y Cartesian coordinate directions
Symbols
δ Boundary layer thickness
δ∗ Displacement thickness
γ Ratio of specific heats / Intermittency
ρθ2 dU
λ Pressure gradient parameter =
µ ds
µ Molecular viscosity / Mean
µt Turbulent/Eddy viscosity
ω Specific dissipation rate
Ω Vorticity magnitude
ρ Density
σ Standard deviation
τw Wall shear stress
θ Momentum thickness
Subscripts
e Boundary layer edge
i Inviscid quantities
v Viscous quantities
∞ Freestream flow variables
Abbreviations
xx
LIST OF FIGURES LIST OF FIGURES
AOA Angle of Attack
CFD Computational Fluid Dynamics
DES Detached Eddy Simulation
FSTI Freestream Turbulence Intensity
MUSCL Monotone Upstream-centered Schemes for Conservation Laws
NREL National Renewable Energy Laboratory
OVERTURNS Overset Transonic Unsteady Rotor Navier-Stokes
PDF Probability Density Function
QOI Quantity of Interest
RANS Reynolds-Averaged Navier Stokes
UQ Uncertainty Quantification
V& V Verification and Validation
xxi
Chapter 1
Introduction
1.1 Motivation
The field of computational fluid dynamics (CFD) has witnessed advances in 
several different areas, including advanced mesh generation techniques, increased 
solution accuracy, and reduced computational costs. However, while CFD is able 
to provide quantitative data through numerical simulation, the effect of uncertain-
ties in the input quantities is often omitted. Therefore, reliability tests often resort 
to physical tests rather than CFD. Yet, it is possible to provide this information 
through numerical simulation via uncertainty quantification (UQ). UQ is the sci-
ence of determining how uncertainties in inputs propagate through a system and 
affect the final results. It draws heavily from statistics and can be applied to CFD 
to provide additional information to augment solution results, such as confidence
1
CHAPTER 1. INTRODUCTION
intervals and statistical variance. This information can aid in sensitivity analysis,
design, and reliability analysis.
UQ can be defined within the framework of Verification and Validation (V&V).
Verification is the process of determining whether the equations modeling a system
are being solved correctly. This field essentially deals with the mathematics of the
model. When applied to CFD, verification traditionally deals with mesh refinement
and/or use of high order methods. On the other hand, validation compares how
well the model compares against the real world. The UQ analysis presented in this
work falls under the validation regime, as it attempts to take physical uncertainties
into account for CFD simulations.
The need for UQ in CFD stems from uncertainties present in the real world
that are essentially ignored by traditional CFD analysis. While these uncertainties
may be small in magnitude, they may lead to varying results. The use of UQ allows
for improved decision-making, more accurate validation, and more robust designs.
In terms of decision-making, UQ can determine how confident one can be that the
results obtained from a CFD simulation are correct, given certain uncertainties in
the input. UQ can also be used in validation to bridge the gap between experiments
and simulations by accounting for the physical uncertainties which are absent from
the simulations. Finally, UQ can be utilized to determine the trade offs between
an optimal design with high sensitivity to certain parameters and a sub-optimal
design that is not as sensitive to those parameters. All of these benefits can be
primarily achieved by determining the true mean and quantifying the variance
present for quantities of interest.
Application of Uncertainty Quantification to Turbulence Intensity 2
CHAPTER 1. INTRODUCTION
One particular area where UQ can be utilized is boundary layer transition
modeling in CFD. The transition from laminar to turbulent flow is a very signifi-
cant phenomenon, especially for rotorcraft. This transition results in an increase
in viscous drag over various rotorcraft components, as well as an increase in heat
transfer, and possible flow separation. It is therefore desirable to design airfoils
and aircraft which promote laminar flow and delay the onset of transition.
Transition is highly dependent on a number of factors, including the freestream
turbulence intensity (FSTI), stream-wise pressure gradient, surface curvature and
roughness, and compressibility. It is therefore important to determine how uncer-
tainties in these quantities propagate and how sensitive aerodynamic quantities
such as lift, drag, and moment are to these uncertainties. Determining this sen-
sitivity will allow for better decision-making, validation, and design. The present
work will focus on the effect of uncertainty in FSTI specifically.
Drag is subject to strong non-linear effects; this makes UQ extremely impor-
tant when applied to quantities affecting drag, like FSTI. As mentioned previously,
UQ can be used to determine the sensitivity of lift, drag, and moment to input
quantities such as FSTI. In addition, UQ can be used to calculate the expected
aerodynamic coefficients, given specified uncertainty in the input. Since the aero-
dynamic quantities are not necessarily linear as FSTI varies, it cannot be assumed
that the expected quantities are equivalent to the values obtained using the ex-
pected FSTI. UQ can therefore be used to determine how valid such an assump-
tion would be as well as calculate more accurate values of lift, drag, and moment,
given the input uncertainty. This is achieved in the present work by applying the
Application of Uncertainty Quantification to Turbulence Intensity 3
CHAPTER 1. INTRODUCTION
UQ techniques (to be described later) to determine the true means and expected
standard deviations for the coefficients of lift, drag, and moment, given a specific
uncertainty distribution in FSTI.
1.2 Boundary Layer Transition
When air flows over a surface, a boundary layer forms over the surface. The
boundary layer is a thin region in contact with the surface and is formed due
to presence of friction and viscosity. The friction/viscosity present between the
surface and the air slows the air down and prevents it from moving at freestream
conditions. The boundary layer is defined as the region extending up to the point
where the flow velocity is equal to 99% of the freestream velocity. It may initially
be in a laminar state. However, as the air moves over the surface, the boundary
layer may transition from laminar flow to turbulent flow. For a description of
laminar and turbulent flow, see section 1.2.1 below. There are a number of factors
that may contribute to the transition from laminar to turbulent flow, including
freestream turbulence intensity, surface roughness, surface curvature, vibrations,
and adverse pressure gradients. In order to minimize drag, it is desirable to delay
the onset of boundary layer transition.
1.2.1 Laminar, Turbulent, and Transitional Flow
Laminar flow is a term used in fluid dynamics to describe a flow in which the
fluid moves in parallel layers. There is no lateral mixing, cross-currents, eddies,
Application of Uncertainty Quantification to Turbulence Intensity 4
CHAPTER 1. INTRODUCTION
or swirls present in this flow as the layers flow without any interaction with each
other. The flow can be characterized as “smooth.” Typically, laminar flow occurs
at low flow velocities.
On the other hand, turbulent flow is characterized as “rough”, with irregu-
lar/chaotic flow paths. The turbulence is due to the inertial forces from the veloc-
ity of the flow dominating the viscous forces of the flow. Turbulent flows contain
unsteady vortices of varying sizes, resulting in the rotationality of the flow. The
flow creates eddies, which are made up of smaller eddies, which in turn are made up
of even smaller eddies, and so on. Thus, in turbulent flow, there is mixing between
the various layers in the boundary layer. These eddies dissipate kinetic energy as
heat, resulting in an increase in skin friction drag when compared to laminar flow.
This increase in drag is the primary reason why laminar flow, turbulent flow, and
the transition between them, are important to the aerospace field.
1.2.2 Reynolds Number
The Reynolds number is a non-dimensional parameter that is used as a general
measure of turbulence. The Reynolds number is defined as the ratio of the inertial
forces of a fluid to the viscous forces of the fluid and is defined by Eq. 1.1:
ρUL
Re = (1.1)
µ
In this equation, ρ is the fluid density, U is the fluid velocity, and L is the char-
acteristic length, and µ is the fluid’s dynamic viscosity. For flows with a low
Application of Uncertainty Quantification to Turbulence Intensity 5
CHAPTER 1. INTRODUCTION
Reynolds number, the viscous forces are dominant and the Reynolds number is
laminar. Turbulent flows are dominated by inertial forces and therefore have high
Reynolds numbers.
1.2.3 Transition
Transitional flow describes the flow as it changes from laminar to turbulent
flow or vice versa. As the flow moves across a surface, it may begin to undergo
transition from laminar flow to turbulent flow. Usually transition is defined as the
region during which the skin friction coefficient (Cf ) increases to its fully turbulent
value. Fig 1.1 illustrates how transition affects the skin friction over a flat plate.
The momentum thickness Reynolds number (Reθ) also increases during transition.
The increase in skin friction leads to an increase in drag and a transfer of heat to
the surface. In addition, the flow may eventually reverse and the boundary layer
may separate from the surface [1]. During transition, the velocity of the flow no
longer varies uniformly across the boundary layer and oscillations begin to appear
in the flow. The flow is no longer steady with respect to time. Small eddies begin
to form in the flow. As the eddies grow, the flow transitions into turbulent flow.
This is illustrated in Fig. 1.2. Fig. 1.3 shows a comparison between the eddy
viscosity plots for a fully turbulent case and a case with transition. As can be seen
from the figure, the eddy viscosity decreases when the transition model is applied
when compared to the fully turbulent solution.
Application of Uncertainty Quantification to Turbulence Intensity 6
CHAPTER 1. INTRODUCTION
Figure 1.1: Skin friction over flat plate, Tu = 2.0% (reproduced from [6])
Figure 1.2: Boundary layer transition (reproduced from Comsol website)
Application of Uncertainty Quantification to Turbulence Intensity 7
CHAPTER 1. INTRODUCTION
(a) Eddy viscosity field plot - fully turbulent (b) Eddy viscosity field plot - transition
Figure 1.3: S809 Airfoil at Mach = 0.2, Re = 2 × 106, AOA = 1 degree, FSTI =
0.05%
1.2.4 Physical Mechanisms of Transition
One way to approach the transition of a laminar boundary layer is to treat
the transition as a “non-linear response of a very complicated oscillator - the lam-
inar boundary layer - to a random forcing function” with “infinitesimal amplitude
compared with the appropriate laminar-flow quantities” [2]. This forcing func-
tion’s characteristics (i.e. amplitude, frequency, and phase) are determined by
receptivity, a concept introduced by Morkovin [3]. Receptivity refers to the way
in which a disturbance enters the boundary layer. The disturbances induce a re-
sponse. The response amplitude grows until it reaches the “breakdown phase”.
During the breakdown phase, spots of intermittent turbulence develop in the flow
and begin to grow. As these spots grow, they begin to merge with one another
until the entire flow becomes turbulent [4].
Application of Uncertainty Quantification to Turbulence Intensity 8
CHAPTER 1. INTRODUCTION
1.2.5 Intermittency
One way to quantify the transition from laminar to turbulent flow is to use
the intermittency factor, γ [5]. As the flow transitions, intermittent turbulent
spots appear and grow. γ is defined as the fraction of time any point spends in
turbulent flow (i.e. one of these intermittent turbulent spots). When γ is equal to
one, this means that airflow at this location is now fully turbulent, whereas when
γ is equal to zero, the flow is laminar.
1.2.6 Types of Transition
There are a number of different types of transition, including natural tran-
sition, separation-induced transition, bypass transition, and crossflow transition.
Each type of transition will be reviewed briefly, with the exception of crossflow
transition, since the scope of the present work is limited to two-dimensional air-
foils.
Natural Transition
Natural transition is the most common form of transition for aircraft. For
this type of transition, transient growth is negligible. The initial disturbances are
small and breakdown is reached through the growth of the disturbances via linear
processes. These processes include Tollmien-Schlichting, Gortler, and crossflow in-
stabilities [6]. Tollmien-Schlichting waves are free disturbances that are the normal
Application of Uncertainty Quantification to Turbulence Intensity 9
CHAPTER 1. INTRODUCTION
modes of the boundary layer. Gortler and cross-flow instabilities appear in three-
dimensional flows. Gortler instabilities are induced by concave surface curvature
while crossflow instabilities manifest due to sweep and pressure gradients. Linear
stability theory and the Orr-Sommerfeld equations [7, 8] can be used to predict
this kind of transition accurately. A depiction of natural transition induced by
Tollmien-Schlichting waves is shown in Fig. 1.4.
Figure 1.4: Natural transition due to Tollmien-Schlichting waves (reproduced from
White 1991)
Separation-Induced Transition
Another type of transition is separation-induced transition. This process
occurs in the presence of a strong adverse pressure gradient. The pressure gradient
causes the laminar boundary layer to separate over a “laminar-separation bubble”
(Fig. 1.5). The boundary layer then undergoes rapid transition before reattaching
as a turbulent boundary layer. For low Reynolds numbers, the separation bubble
Application of Uncertainty Quantification to Turbulence Intensity 10
CHAPTER 1. INTRODUCTION
can extend up to 50% of the airfoil chord, resulting in significant amounts of viscous
drag. Stall is induced when the adverse pressure gradient is increased to the point
where the bubble “bursts,” causing the flow to remain unattached.
Figure 1.5: Flow structure of laminar-separation bubble (reproduced from Horton
1968)
Bypass Transition
When the initial disturbance amplitude is relatively large, the transition may
bypass the stage of linear growth and directly reach the breakdown phase [3]. This
is known as bypass transition. The large initial disturbances associated with this
type of transition are typically due to a high level of freestream turbulence intensity
(FSTI) or surface roughness. In bypass transition, the intermittent turbulent spots
form quickly. This type of transition has been observed experimentally and is
most common in turbomachinery, where high FSTI levels are present in the wake
of upstream blade rows. However, the phenomenon cannot be described by linear
stability theory due to the non-linear growth of the disturbances. Bypass transition
is still not well understood and is a topic of ongoing research.
Application of Uncertainty Quantification to Turbulence Intensity 11
CHAPTER 1. INTRODUCTION
1.2.7 Transition Modeling
In CFD, transition models predict a “transition zone”, in which the flow
is in between the laminar and turbulent states. The transition models are then
incorporated into the flow solvers in order to simulate transition within the solver.
There are a number of different transition models in use today, ranging from
models based on hydrodynamic stability theory [9, 12] to correlation-based mod-
els [6, 16]. The first category involves using linear stability analysis to determine
where transition occurs. The complete unsteady Navier-Stokes equations are lin-
earized locally. The Orr-Sommerfeld stability equations can then be obtained,
which are used to check if the disturbance amplitude grows or decays. The change
in disturbance amplitudes determines when transition occurs. On the other hand,
correlation models use the relationship between transition location and certain
quantities, such as intermittency and transition momentum thickness Reynolds
number (Reθt) to predict the transition location [6].
There are a number of transition models currently in use in CFD. These models
include the eN model [12], the K-V model [10], the Langtry-Menter Model [16],
and the Medida model [6] which is derived from the Langtry-Menter model. A
brief description of each of these models is presented here.
The eN Transition Model
The eN transition model is a linear stability based model that utilizes the
Orr-Sommerfeld equations to calculate the growth of spatial disturbances. The
Application of Uncertainty Quantification to Turbulence Intensity 12
CHAPTER 1. INTRODUCTION
model compares the initial unstable amplitude to the amplitude corresponding
to the most unstable frequency. Transition occurs when the latter exceeds the
former by more than eN . Typically, N ranges between 8 and 10 [10]. This
method is derived from an eigenvalue analysis of the Orr-Sommerfeld equations,
with the assumption that viscous parallel flow is present. However, this assumption
is not necessarily valid. This is addressed by the linear Parabolized Stability
Equations (PSE) method. A non-linear PSE method has also been developed
to take into account bypass transition, roughness-induced transition, and other
non-linear phenomena [11].
In order to implement this model, the laminar velocity and temperature profiles
have to be calculated at different locations in the flow. The local linear stability
equations (or the PSE) are then solved in order to obtain the local amplification
rates. N is then calculated by integrating the local amplification growth rate
along the streamline. Transition takes place once N exceeds a certain value [12].
Some versions of the model use a database to evaluate the instability data. The
inputs to the database are determined via two equations: 1) the von Karman
integral relation and 2) the kinetic energy thickness equation. These equations
allow for two N -factors to be calculated and used to estimate the growth rate in
the database [13].
The K-V Transition Model
Kapsalis et al published a paper in 2016 proposing a new transition model.
The model, dubbed the “K-V transition model”, is a linear stability theory model
Application of Uncertainty Quantification to Turbulence Intensity 13
CHAPTER 1. INTRODUCTION
which utilizes the velocity shape factor Λ to determine the instability point. Like
the eN model, the various parameters of the laminar boundary layer are calculated
along the surface. After this, the critical Reynolds number, Reδ1,cr, can be calcu-
lated at each point based on the known distribution of Λ. Reδ1 is the Reynolds
number based on displacement and is defined by Eq. 1.2:
Ubδ1
Reδ1 = (1.2)
ν
where Ub is the velocity at the edge of the boundary layer, δ1 is the displacement,
and ν is the air kinematic viscosity. The critical Reynolds number is the value
which correlates to the instability/inflection point as Λ varies. Once Reδ1,cr is
computed, it can be used to calculate Reθt along the airfoil. Each point is then
assumed to be the transition point, and the value DReθt is calculated. This value
represents the difference in the Reynolds number based on momentum thickness
at the transition point and the instability point. The transition point is then
determined to be the point when DReθt exceeds the critical value Reθt,cr. In
the work of Kapsalis, this method is shown to be more accurate than the two
equation implementation of the eN model. In addition, the K-V model only uses
one parameter (Λ) and therefore is able to converge faster than the eN model [10].
The Langtry and Menter Transition Model
In 2009, Langtry and Menter published a paper establishing a new correlation-
based transition model. It has become one of the most widely used transition im-
plementations in industry [6]. The model has also been adapted to create a hybrid
Application of Uncertainty Quantification to Turbulence Intensity 14
CHAPTER 1. INTRODUCTION
method, where this model is used to calculate the amplification factor N from the
eN model [14].
This model utilizes the two equation Shear Stress Transport (SST) k − ω tur-
bulence model in order to calculate the turbulence. The key concept behind this
model is the use of the vorticity Reynolds number Rev , which was derived by
Van Driest and Blumer [15]. Langtry and Menter were able to establish an empir-
ical correlation between the local boundary layer quantities and Rev . They were
therefore able to avoid the requirement of integrating the boundary-layer velocity
profile. Instead, they calculate Reθ with the simple equation:
max(Rev)
Reθ = (1.3)
2.193
Rev can be easily calculated at each mesh point, allowing this model to be efficient.
Meanwhile, Reθt,cr can be calculated with experimental correlations and solving
a transport equation. Another transport equation is then used to solve for the
intermittency, using Rev [16].
The model then uses a transport equation to solve for intermittency. The local
intermittency values are used in conjunction with the k − ω turbulence model.
The transition onset location is determined using equation 1.3. Using correlations
based on the local transition momentum thickness Reynolds number, Langtry and
Menter were able to determine the Flength function, which sets the length of the
transition zone, and Fonset, which enables the transition from laminar to turbulent
flow. Meanwhile, the intermittency production term is calculated using Flength and
Application of Uncertainty Quantification to Turbulence Intensity 15
CHAPTER 1. INTRODUCTION
Reθt,cr. A second transport equation is used to solve for the transport variable for
the momentum thickness Reynolds number, Reθt. A transport equation is used
rather than a correlation based on experiment in order to account for the large
variance of turbulence intensity in some applications. However, the local Reθt value
is calculated using experimental correlations. These correlations take turbulence
intensity into account and require an iterative process in order to compute the
final value of Reθt [16].
The Medida Transition Model
The Medida model [6] is a derivation of the Langtry-Menter model. The
model has been modified to be compatible with the one equation Spalart-Allmaras
(S-A) turbulence model rather than the k − ω two-equation turbulence model.
Another modification is the adjustment of the correlations between Reθt and the
turbulence intensity. The model implements a linear piece-wise form of the cor-
relations which better account for data from the T3-series zero-pressure gradient
flat plate experiments and laminar flow simulations. The intermittency produc-
tion and destruction terms have been adjusted in order to improve intermittency
recovery in the turbulent boundary layer. The model assumes constant freestream
turbulence intensity in the flow field and omits separation-induced transition. In
addition, the destruction term is not scaled by intermittency when using the term
in the S-A turbulence model [6].
Application of Uncertainty Quantification to Turbulence Intensity 16
CHAPTER 1. INTRODUCTION
1.3 Uncertainty Quantification
1.3.1 Definitions
In order to provide a comprehensive understanding of uncertainty quantifi-
cation (UQ), it is necessary to review the definitions of certain terms and concepts:
Uncertainty: Uncertainty is defined as a potential deficiency that is due to a lack
of knowledge. Uncertainty can be divided into two categories: aleatory uncertainty
and epistemic uncertainty [18].
Aleatory uncertainty: Aleatory uncertainty is uncertainty associated with the
inherent randomness/variation in a population. This type of uncertainty is also
known as irreducible uncertainty, since this uncertainty cannot be eliminated from
the model [19]. An example of aleatory uncertainty in CFD is the variation of
an input parameter, such as free stream Mach number. This uncertainty usually
has a probability distribution associated with it, such as a normal or uniform
distribution.
Epistemic uncertainty: Epistemic uncertainty is uncertainty stemming from a
model’s failings due to a lack of knowledge. Research into the area where knowledge
is lacking can result in an improved model, which can in turn reduce or even
eliminate epistemic uncertainty [19]. One current source of epistemic uncertainty
in CFD is turbulence model assumptions [18].
Application of Uncertainty Quantification to Turbulence Intensity 17
CHAPTER 1. INTRODUCTION
Sensitivity analysis: Sensitivity analysis (SA) is the study of how much an out-
put varies due to a change in a certain input. SA can be categorized as either local
or global. Local SA measures how much the output changes due to infinitesimal
changes in the input. On the other hand, global SA measures how the output
changes when the input is varied over all feasible values. [20]
Non-intrusive and intrusive techniques: Non-intrusive techniques are UQ
techniques which do not require the modification of the deterministic function,
model, and/or codes that are used. These techniques are relatively simple to
implement since they simply require the selection of certain inputs into the model
without any “tinkering under the hood” of the CFD solvers. On the other hand,
intrusive methods modify the actual mathematical equations being solved by the
CFD codes.
1.3.2 Overview
UQ falls under the framework of Verification and Validation [17]. Verifica-
tion addresses errors that occur when solving equations and deals primarily with
the numeric solvers and algorithms using numerical analysis. On the other hand,
validation deals with how accurate the numerical simulations are when compared
to the physical world and experimental data. When conducting experiments, un-
certainty bars are provided along with the data to quantify the repeatability and
measurement errors associated with the experiment. UQ can be used to calculate
computational uncertainty bars that are analogous to the uncertainty bars from ex-
periments as well as quantify the numerical errors attributed to the computational
Application of Uncertainty Quantification to Turbulence Intensity 18
CHAPTER 1. INTRODUCTION
solver [18].
When applying UQ to CFD, it is useful to consider each CFD simulation as
a function y = g(~x), where ~x is a vector of inputs to the CFD simulation, such
as angle of attack, freestream Mach number, Reynolds number, and FSTI and y
represents the quantities of interest (QOI), such as the coefficients of lift, drag, and
moment. Uncertainties in ~x result in ~x becoming a stochastic variable. The un-
certainty propagates through the system, resulting in y also becoming a stochastic
variable. The actual UQ techniques depend on the type of uncertainty. Aleatory
and epistemic uncertainties are each handled differently in UQ. The present work
focuses on aleatory uncertainties attributed to variation in FSTI. For aleatory
uncertainty, UQ can be used to determine the distribution of the output and/or
statistics related to this distributions, such as the expected mean and standard
deviation [20]. The following sections will review the different UQ techniques
typically used to obtain these quantities.
1.3.3 Monte Carlo Method
The Monte Carlo method [21] is one of the oldest UQ techniques. It is a
simple non-intrusive sampling method that utilizes the law of large numbers to
converge to the correct solution. The law of large numbers states that as the
frequency of a probabilistic event increases, the mean of the results will converge
to the expected/theoretical mean. The Monte Carlo method involves randomly
sampling the input parameters; the inputs can have a uniform distribution, a
normal distribution, or any other distribution desired.
Application of Uncertainty Quantification to Turbulence Intensity 19
CHAPTER 1. INTRODUCTION
The randomly generated samples are then used as the inputs for the system
and the desired output quantities are computed. This is done repeatedly until the
average of the outputs converges. This converged mean output is the expected
value. The simplicity of this method allows for easy implementation and universal
application. However, the method is slow to converge and may require an excessive
amount of samples to converge to the correct solution [18]. For CFD simulations
that take some time to run, this may make the Monte Carlo method infeasible
from a computational cost standpoint without the use of a surrogate surface (see
section 1.3.5).
1.3.4 Stochastic Collocation
Stochastic collocation is the application of quadrature to integrate integrands
composed of random variables. This can be used to determine the expectation of
a random variable. The goal of stochastic collocation is to calculate the expected
mean output values E[y] and the variance of the output values V ar[y] (using the
notation introduced in section 1.3.2). One way to determine these quantities is to
use the probability density function (PDF) fy. If the PDF has been computed,
E[y] and V ar[y] can be calculated as follows:
∫ ∞
E[y] = zfy(z)dz (1.4)
∫ −∞∞
V ar[y] = (z − E[y])2fy(z)dz = E[y2]− (E[y])2 (1.5)
−∞
These integrals can be computed numerically through quadrature techniques.
Application of Uncertainty Quantification to Turbulence Intensity 20
CHAPTER 1. INTRODUCTION
Consider the case where there is a single uncertain parameter ξ. Stochastic collo-
cation is achieved by expressing the integrals in Eq. 1.4 and 1.5 as weighted sums
of the integrands evaluated at a number of locations on the ξ-axis. These loca-
tions are referred to as abscissas. The abscissas and weights are typically chosen
by selecting a basis function which is easy to integrate, usually a polynomial. The
accuracy of the stochastic collocation method increases as the number of abscissas
increases [18].
Gaussian quadratures are often used as they are highly accurate. For Gaussian
quadrature, equation 1.4 is rewritten in terms of a weighted sum:
∫ b ∑n
y(ξ)f(ξ)dξ = wky(ξk) +Rn(y) (1.6)
a k=1
where ξk are the zeros of the chosen orthogonal polynomial for the quadrature, wk
are the weights, which are calculated using the weighting function f . n is the total
number of abscissas used for the quadrature and Rn is the remainder term, which
also denotes the order of accuracy of the method.
Like the Monte Carlo method, stochastic collocation is non-intrusive and rel-
atively simple to implement. Once the abscissas have been determined, the CFD
simulation is simply run using the abscissas as the inputs. Thus, stochastic collo-
cation usually requires far fewer CFD runs to achieve convergence.
Application of Uncertainty Quantification to Turbulence Intensity 21
CHAPTER 1. INTRODUCTION
Gauss-Hermite Quadrature
The Gauss-Hermite quadrature is used to approximate integrals of the form
[22]: ∫ ∞ ∑n
2
e−ζ φ(ζ)dζ ≈ wkφ(ζk) (1.7)
−∞ k=1
where wi and ζi are defined by the Hermite polynomial and M is the number
of nodes used for the quadrature. The Hermite polynomial can be expressed
recursively as follows [23]:
∑n/2 1
Hn(x) = n! (−1)m (2x)n−2m (1.8)
m!2m(n− 2m)!
m=0
ζi are defined by the zeros of 1.8 and wi is defined by [25]:
− √2n 1n! π
wi = (1.9)
n2[H − (ζ )]2n 1 i
If the uncertain parameter X is characterized with a normal distribution with a
mean µ and standard deviation σ, the Gauss-Hermite quadrature can be used
to compute E[y] and V ar[y] by applying a simple transformation ζ = (x −
√ √ √
µ)/( 2σ), φ(ζ) = g(µ+ 2σζ)/ π (g(x) is defined as the CFD simulation). Using
this transformation and the abscissas and weights defined by the Hermite polyno-
mial, E[y] and V ar[y] can be written as follows:
∫ ∞ ∑n √
E[y] = √ 1 e−(x−µ)2/2σ2 wkg(x)dx ≈ √ g(µ+ 2σζk) (1.10)
−∞ 2πσ2 πk=1
Application of Uncertainty Quantification to Turbulence Intensity 22
CHAPTER 1. INTRODUCTION
∑n
≈ √wkV ar[y] (g(ζk)− E[y])2 (1.11)
π
k=1
1.3.5 Surrogate Surfaces
One of the main drawbacks of both the Monte Carlo method and stochastic
collocation is the computational time required to compute the mean and variance
of the QOIs. Stochastic collocation methods can require 5-10 CFD runs for a single
case, while Monte Carlo sampling may require tens of thousands of runs to achieve
convergence. This can make these methods cost-prohibitive unless combined with
a surrogate surface. A surrogate surface approximates the relationship between
the output QOIs of the CFD simulation and the uncertain input parameters [20].
Each QOI requires its own surrogate surface. Although several CFD simulations
are necessary to create the surrogate surfaces, this method provides a way to
perform Monte Carlo sampling and stochastic collocation inexpensively.
There are several methods that can be used to generate the surrogate surface.
The simplest method is to run a parametric sweep through the uncertain variable
and extrapolate between the points. More complex methods include using radial
basis functions [26], support vector regression [27], polynomial chaos [28, 29], and
Kriging [30–33].
Application of Uncertainty Quantification to Turbulence Intensity 23
CHAPTER 1. INTRODUCTION
1.4 Thesis Contributions
The purpose of this thesis is to examine the effects of introducing uncertainty
into the input FSTI and investigate the sensitivity of the aerodynamic coefficients
of lift, drag, and moment to this uncertainty. This was achieved by applying UQ
techniques such as stochastic collocation and surrogate surfaces to determine the
expected mean values of QOIs and the expected standard deviations for these
QOIs. A comparative analysis was used in conjunction with the calculated stan-
dard deviations to determine the sensitivities of the QOIs to uncertainty in FSTI.
All CFD simulations in this study used the in-house flow solver OVERTURNS
[34] with a γ−Reθ−SA transition model. Four airfoils were selected to be used in
this study: 1) NACA 0012, 2) S809, 3) SC1095, and 4) RC(4)-10. These airfoils
were selected due to their widespread usage in the field. For each airfoil, a para-
metric sweep of FSTI was conducted in order to generate the surrogate surface.
Stochastic collocation was performed for a test case using both the surrogate sur-
face and actual CFD runs in order to validate the surrogate surface. Once this was
complete, stochastic collocation was applied to the surrogate surface to compute
the expected means and standard deviations of the QOIs over a range of angles of
attack. For the SC1095 and RC(4)-10, both of which often operate in transonic
flows, the freestream Mach number was varied between 0.2 and 0.8 as well.
The contributions of the present work is to lay out the foundation and process
for generating “uncertainty reference tables” for various airfoils. This data can
Application of Uncertainty Quantification to Turbulence Intensity 24
CHAPTER 1. INTRODUCTION
be used to provide more accurate data when there is known uncertainty in the
freestream conditions. These tables will also highlight operating conditions under
which there is greater sensitivity to this uncertainty. Such information would be
relevant for determining confidence in a CFD simulation, providing more accurate
validation, and creating more robust designs.
1.5 Scope and Organization of Thesis
This thesis is concerned with the application of UQ techniques for airfoils
undergoing transition with an uncertain FSTI value. This thesis details the mo-
tivation for this research, the UQ techniques used, the general approach, and the
results of applying the UQ techniques to four airfoils. The rest of the thesis is
organized as follows:
ˆ Chapter 2 describes the computational methodology used for this study.
The chapter reviews how the flow solver OVERTURNS solves the Reynolds
Averaged Navier Stokes (RANS) equations. In addition, the discretizations
methods, mesh descriptions, and boundary conditions used by the solver are
described.
ˆ Chapter 3 reviews the UQ methodology used by the present work. This
includes descriptions of the parametric studies, the generation of the sur-
rogate surfaces, and the details of the quadratures used for the stochastic
collocation.
Application of Uncertainty Quantification to Turbulence Intensity 25
CHAPTER 1. INTRODUCTION
ˆ Chapter 4 presents the results of the parametric studies and the generation
of the surrogate surfaces for the NACA0012, S809, SC1095, and RC(4)-10.
The FSTI is varied from 0.05% to 5% while the angle of attack is varied from
-4 degrees to 12 degrees.
ˆ Chapter 5 presents the results of the UQ analysis. These results include
the expected mean values of the QOIs and the standard deviations for these
QOIs. A comparison of these standard deviations over the range of angles of
attack for specific mean FSTI values will be used to demonstrate the changes
in sensitivity for the QOIs.
ˆ Chapter 6 summarizes the key results presented in this study and provides
recommendations for future work.
Application of Uncertainty Quantification to Turbulence Intensity 26
Chapter 2
Numerical Implementation
In this chapter, an overview of the CFD analysis used for this work is presented. 
The Overset Transonic Unsteady Rotor Navier-Stokes (OVERTURNS) flow solver 
[34] was used to run all simulations. This chapter will cover the partial differential 
equations (PDEs) which govern the flow and how OVERTURNS solves these PDEs 
to obtain the flow field solution. This includes the turbulence and transitioning 
modeling used by OVERTURNS to obtain the solution. The mesh generation 
process used for the present work is also discussed. Finally, sample results which 
validate the approach are presented.
27
CHAPTER 2. NUMERICAL IMPLEMENTATION
2.1 Governing Equations
In order to determine flow characteristics, the governing equations, the Navier-
Stokes equations, must be solved. The Navier-Stokes equations are coupled partial
differential equations relating the velocity, density, pressure, and temperature. The
equations are derived using conservation of mass, momentum, and energy and are
dependent on both time t and the spatial coordinates x and y.
The equations can be written in “strong conservation form” in terms of the
viscous and inviscid fluxes through the control volumes as follows:
∂Q ∂Fi ∂Gi ∂Fv ∂Gv
+ + = + (2.1)
∂t ∂x ∂y ∂x ∂y
where Q is the conserved variables vector, Fi and Gi are the inviscid flux vectors,
and Fv and Gv are the viscous flux vectors. Q is defined by Eq. 2.2:
 ρ 

 ρu

Q =  ρv 
 (2.2)
E 
where ρ is the fluid density, u and v are the components of the fluid velocity in
the x and y directions. E is the total energy per unit volume and is calculated as
follows: [ ]
1
E = ρ e+ (u2 + v2) (2.3)
2
Application of Uncertainty Quantification to Turbulence Intensity 28
CHAPTER 2. NUMERICAL IMPLEMENTATION
where e is the energy per unit mass.
The inviscid fluxes are defined by Eqs. 2.4 and 2.5, and the viscous fluxes are
defined by Eqs. 2.6 and 2.7

 ρu 

 
 ρuu+ p

Fi =   (2.4) ρuv (E + p)u 
 
 ρv

 ρvu

Gi =   (2.5) ρvv + p (E + p)v 
 
 0 

τ xx
Fv = 
(2.6)
 τ 

yx 
uτxx + vτ

yx − qx
   0 
 τ

xy
Gv =  (2.7) τyy uτxy + vτ yy − qy
Application of Uncertainty Quantification to Turbulence Intensity 29
CHAPTER 2. NUMERICAL IMPLEMENTATION
In the above equations, qx and qy are the heat conduction terms. They are
defined as:
∂T
qj = k (j = x, y) (2.8)
∂xj
where k is the coefficient of thermal conductivity and T is the temperature of the
fluid. τij is the viscous stress tensor and is defined by Stokes’ hypothesis [45]:
[( ) ]
∂ui ∂uj − 2 ∂ukτij = µ + δij , δij = 1 if i=j; δij = 0 if i 6= j (2.9)
∂xj ∂xi 3 ∂xk
µ is the coefficient of molecular viscosity and is given by Eq. 2.10:
3
T 2
µ = C1 (2.10)
T + C2
√
For air at standard temperature and pressure (STP), C1 = 1.4×10−6 kg/(ms K)
and C2 = 110.4 K. In order to solve this system, one more equation is needed:
p = ρRT (2.11)
Eq. 2.11 is the equation of state for ideal gases. R is the gas constant. The fluid
in the present work is air at STP; it can therefore be assumed to be a calorically
perfect gas, where the specific heats remain constant:
R γR
cv = ; cp = (2.12)
γ − 1 γ − 1
where γ is the ratio of specific heats. For air at STP, γ = 1.4. In addition, the
Application of Uncertainty Quantification to Turbulence Intensity 30
CHAPTER 2. NUMERICAL IMPLEMENTATION
following equations are applicable to calorically perfect gases:
e = cvT (2.13)
p = (γ − 1)ρe (2.14)
e = cvT (2.15)
Using the above relations, Eq. 2.3 can be re-written as follows:
p 1
E = + ρ(u2 + v2) (2.16)
γ − 1 2
2.2 Non-Dimensionalization
OVERTURNS solves the governing equations in their non-dimensionalized
form. Non-dimensionalization results in all the flow variables to be of the same
magnitude, thereby reducing numerical errors due to mathematical operations be-
tween numbers of very different values. In addition, non-dimensionalization allows
for the parameter like Reynolds number and Mach number to be varied indepen-
dently. Non-dimensionalized reference variables are denoted by the * superscript
and are defined below:
∗ x ∗ y tx = , y = , t∗ = (2.17)
L L L/a∞
u∗
u v µ
= , v∗ = , µ∗ = (2.18)
a∞ a∞ a∞
Application of Uncertainty Quantification to Turbulence Intensity 31
CHAPTER 2. NUMERICAL IMPLEMENTATION
∗ ρ ∗ p Tρ = , p = , T ∗ = (2.19)
ρ∞ p∞a2∞ T∞
In the above equations, L represents the chord length of the airfoil and a∞ is
the freestream speed of sound. The governing equations in terms of the non-
dimensionalized variables are obtained by substituting Eqs. 2.17 - 2.19 into Eq.
2.1. The dimensional governing equations are the same as the dimensionless ones,
with the exception of the viscous stress tensor and thermal conduction terms. The
modified terms are defined here:
[ ]
µM∞ ∂ui ∂uj
τij = + −
2 ∂uk
δij (2.20)
Re∞ ∂xj ∂xi 3 ∂xk
µM∞ ∂T
qj = − (2.21)
Re∞Pr(γ − 1) ∂xj
In the above equations, all the variables are non-dimensional, so the * superscript
has been dropped for the sake of convenience. The above equations introduces
three new dimensionless quantities:
ρ∞V∞L
Reynolds number: Re∞ = (2.22)
µ∞
V∞
Mach number: M∞ = (2.23)
a∞
µcp
Prandtl number: Pr = (2.24)
√ k
where V∞ is the freestream velocity magnitude ( u2∞ + v
2
∞). For air at STP, the
Prandtl number, Pr = 0.72.
Application of Uncertainty Quantification to Turbulence Intensity 32
CHAPTER 2. NUMERICAL IMPLEMENTATION
2.3 Reynolds-Averaged Navier-Stokes Equations
In order to model boundary layer transition, it is necessary to modify the
governing equations into a form which is compatible with turbulence models. This
can be achieved using the Reynolds-Averaged Navier Stokes (RANS) approach.
Using this approach, the dependent variables are broken down into two compo-
nents: the mean component and the fluctuating component, as shown in Eq. 2.25:
u = u+ u′ v = v + v′ ρ = ρ+ ρ′ p = p+ p′ T = T + T (2.25)
The governing equations in terms of these components are then time-averaged.
Eq. 2.26 is used to time average each quantity:
∫
1 t0+∆t
f = fdt (2.26)
∆t t0
where f is the time-averaged quantity and ∆t is the time period. This time period
must be small sufficiently small such that the mean flow variation is small, yet
large when compared to the period of turbulent fluctuations. By definition, the
mean of the fluctuating component (f ′) is equal to zero:
∫
1 t0+∆t
f ′ = f ′dt = 0 (2.27)
∆t t0
The RANS equations are obtained by substituting Eq. 2.25 into the unsteady
Navier-Stokes equations (Eq. 2.1). The RANS governing equations are identical
Application of Uncertainty Quantification to Turbulence Intensity 33
CHAPTER 2. NUMERICAL IMPLEMENTATION
to the Navier-Stokes equation with the addition of fluctuating terms. These terms
can be grouped into a stress tensor, known as the Reynolds Stress Tensor (τij):
(τij)turb = −ρu′u′i j (2.28)
In order to solve the RANS equations, the Reynolds Stress tensor must be put in
terms of the mean quantities. There are various turbulence models which are used
to do so. The present work utilizes the Spalart-Allmaras turbulence model [35],
which is discussed in detail in Sec. 2.6.3.
2.4 Coordinate Transformation
Eq. 2.1 is the Cartesian form of the Navier-Stokes equations. However,
in order to apply these equations to the computational meshes used by OVER-
TURNS, it becomes necessary to apply a curvilinear coordinate transformation.
This transformation maps the Cartersian coordinates x and y which may not be
uniformly spaced, onto the computational domain (ξ and η) for an equally-spaced
computational mesh (see Fig. 2.1). By applying chain rule differentiation to Eq.
2.1, one obtains the following:
∂Q̃ ∂F̃ ∂G̃
+ + = 0 (2.29)
∂t ∂ξ ∂η
Application of Uncertainty Quantification to Turbulence Intensity 34
CHAPTER 2. NUMERICAL IMPLEMENTATION
where,
1
Q̃ = Q
J
1
F̃ = [ξtQ+ ξx(Fi − Fv) + ξy(Gi −Gv)]
J (2.30)
1
G̃ = [ηtQ+ ηx(Fi − Fv) + ηy(Gi −Gv)]
J
1
H̃ = [ζtQ+ ζx(Fi − Fv) + ζy(Gi −Gv)]
J
and J is the coordinate transformation Jacobian (i.e. the determinant of the 3×3
∂(ξ, η)
matrix .
∂(x, y)
Figure 2.1: Transformation from physical domain to computational domain (re-
produced from Blazek 2006)
2.5 Grid Generation
The present work studies airflow over two-dimensional airfoils. It was there-
fore necessary to generate body-fitted meshes for these airfoils. The meshes used
in this work were C-topology structured meshes, as pictured in Fig. 2.2. These
meshes were created using hyperbolic grid generation. The mesh is made up of
three types of boundaries: solid wall, farfield, and wake-cut.
Application of Uncertainty Quantification to Turbulence Intensity 35
CHAPTER 2. NUMERICAL IMPLEMENTATION
(a) C-mesh (b) Close-up of C-mesh
Figure 2.2: C-topology mesh around NACA 0012
2.6 Numerical Algorithms in OVERTURNS
This section describes the numerical algorithms used to discretize the govern-
ing equations, model turbulence and transition, and handle boundary conditions.
OVERTURNS solves the curvilinear form of the RANS equations using a cell-
averaged finite volume approach with a cell-vertex scheme. The flow variables are
stored at each grid point, and the control volumes are computational cells which
are centered around the grid points, as shown in Fig. 2.3. In order to determine
the time rate of change in the conserved quantities at each timestep, the inviscid
and viscous fluxes are calculated at the interfaces of the cell and integrated over
the cell faces. Eq. 2.29 can therefore be converted into a semi-discrete form:
∂Q̃ F̃ 1 − F̃ 1 G̃− j+ j− − k+
1 − G̃k− 1
= 2 2 2 2 (2.31)
∂t ∆ξ ∆η
Application of Uncertainty Quantification to Turbulence Intensity 36
CHAPTER 2. NUMERICAL IMPLEMENTATION
Figure 2.3: Computational Cell(reproduced from [36])
where j and k are the indices corresponding to the ξ and η directions respectively.
The computational cell boundaries are therefore defined by (j± 1 , k± 1). Note that
2 2
this is the two-dimensional form of the equation. The next two sections describe
the discretization techniques used to compute the inviscid and viscous fluxes.
2.6.1 Inviscid Fluxes
Evaluating the inviscid fluxes (Eqs. 2.4 and 2.5) in OVERTURNS is a two-
step process. First, the primitive variables must be reconstructed at the cell faces.
Once this has been achieved, the fluxes can be evaluated at the cell faces using the
reconstructed primitive variables. All CFD solutions in this work were obtained
using a Monotone Upstream-Centered Scheme for Conservation Laws (MUSCL)
[37] to reconstruct the primitive variables for both the right and left sides of each
face (qL , qR1 − 1 ). MUSCL calculates the primitive variables at each face using thei+ i
2 2
Application of Uncertainty Quantification to Turbulence Intensity 37
CHAPTER 2. NUMERICAL IMPLEMENTATION
following equations:
[ ]
L 1 1q 1 = q + φi (qi+1 − qi) + (qi − qi−1) (2.32)i+
2 3 6
[ ]
qR− 1 = q −
1 1
φ (q
i i i+1
− qi) + (qi − qi−1) (2.33)
2 3 6
where φ us Koren’s differentiable limiter [38]:
3∆qi∇qi + 
φi = (2.34)
2(∆qi −∇q )2i + 3∆qi∇qi + 
In the above equation,  represents a small value used to avoid division by zero. ∆
and∇ are forward and backward operators: ∆qi = (qi+1−qi) and∇qi = (qi−qi−1).
Once the primitive variables are reconstructed, they are used to evaluate the
inviscid fluxes using Roe’s flux difference splitting scheme [39] with an entropy fix:
F (qL) + F (qRL R ) q
R − qL
F (q , q ) = − |Ã(qL, qR)| (2.35)
2 2
In the above equation, FL and FR are the left and right state fluxes, and à is the
Roe-averaged Jacobian matrix. Numerical dissipation is taken into account by the
second term on the right hand side of the above equation. In order to prevent
entropy violations, Harten’s entropy correction to the flux Jacobian eigenvalues is
applied [40]: |λ|, if |λ| > δ|λ| =  (2.36)λ2 + δ2
[ , if |λ| ≤ δ2δ ]
where δ = max 0, (λi+ 1 − λi), (λi+1 − λi+ 1 ) . λ denotes Roe-averaged eigenvalues.
2 2
Application of Uncertainty Quantification to Turbulence Intensity 38
CHAPTER 2. NUMERICAL IMPLEMENTATION
2.6.2 Viscous Fluxes
In order to evaluate the viscous fluxes (Eqs. 2.6 and 2.7), in the curvilinear
form of the governing equations, the following must be evaluated:
( )
∂ ∂β
α (2.37)
∂ξ ∂η
OVERTURNS evaluates the above expression with a second order central differ-
encing scheme:
([ ] [ ])
1 βj+ 1 ,k+1 − βj+ 1 ,k βj− 1 ,k − βj− 1 ,k−1
α 2 2 2 2j+ 1∆ξ ,k
− αj− 1 ,k (2.38)
2 ∆η 2 ∆η
δj,k + δj+1,k
where δj+ 1 ,k = and δ = (α, β).
2 2
2.6.3 Turbulence Modeling
OVERTURNS is able to incorporate turbulence into the CFD solutions by
solving the RANS governing equations using turbulence models (see Sec. 2.3).
The present work uses the one-equation Spalart-Allmaras turbulence model [35].
The model is based on the Boussinesq eddy viscosity hypothesis [41, 42], which
proposes the following relationship between the Reynolds stress tensor and the
mean strain rate:
[( ) ]
′ ′ 2 ∂ui ∂uj 2 ∂uk(τ ij)turb = −ρuiuj = ρkδij − µt + − δij (2.39)3 ∂xj ∂xi 3 ∂xk
Application of Uncertainty Quantification to Turbulence Intensity 39
CHAPTER 2. NUMERICAL IMPLEMENTATION
where µt is the turbulent/eddy viscosity and k is the turbulent kinetic energy:
1 [ ]
k = (u′)2 + (v′)2 + (w′)2 (2.40)
2
Using Eq. 2.39, the total viscous stress tensor to be used in the RANS equations
can be calculated:
[( ) ]
2 − ∂ui ∂uj 2 ∂uk(τ ij)total = ρkδij (µ+ µt) + − δij (2.41)
3 ∂xj ∂xi 3 ∂xk
The Spalart-Allmaras model uses a transport equation for the eddy viscosity,
evaluating the diffusion, production, and destruction terms for eddy viscosity:
Dν̃ [ ]
= Pν −
1
Dν + ∇((ν + ν̃) + c 2b2(∇ν̃) (2.42)
Dt σ
ν̃
Pν = cb1Ω̃ν̃ and Dν = c f
2
w1 w[ ] (2.43)
d
Ω is the vorticity magnitude. Ω̃ is given by:
( )
ν̃ χ
Ω̃ = max Ω + fv2, 0.3Ω , fv2 = 1− (2.44)
κ2d2 1 + χfv1
The wall damping function, fw, is given by:
[ ] 1
1 + c6 6
f w3 6
ν̃
w = g , g = r + cw2(r − r), r = (2.45)
g6 + c6 Ω̃κ2 2w3 d
The constants in the above equations are defined as follows: cb1 = 0.1355, σ =
2/3, cb2 = 0.622, κ = 0.41, cw1 = c
2
b1/κ +(1+cb2)/σ, cw2 = 0.3, cw3 = 2.0, cv1 = 7.1.
Application of Uncertainty Quantification to Turbulence Intensity 40
CHAPTER 2. NUMERICAL IMPLEMENTATION
The eddy viscosity transport variable, ν̃ is set equal to zero at smooth wall bound-
aries, equal to 0.1 at the inflow boundaries, and extrapolated from the interior at
the outflow boundaries.
For the present work, convection terms were discretized using a first order
upwind scheme, while diffusion terms were discretized using a second order central
scheme. The algebraic system of equations was solved using the Diagonalized
Alternating Direction Implicit (DADI) approximate factorization method [43].
2.6.4 Transition Modeling
The present work utilizes the γ − Reθ−SA transition model, as developed
by Medida [6]. The transition model solves two transport equations in order to
generate an intermittency and transition momentum thickness flow field around the
airfoil. The model is coupled with the Spalart-Allmaras one equation turbulence
model (see Sec. 2.6.3) by multiplying the eddy viscosity production term from the
SA model with the intermittency factor, γ. Therefore, the effects of turbulence
only appear as the intermittency factor approaches one.
The transition model defines the intermittency factor transport equation as
follows: [ ]
D(ργ) − ∂ ∂γ= Pγ Dγ + (µ+ µt) (2.46)
Dt ∂xj ∂xj
The source term is given by:
[ ]
Ω 1.0
Pγ = ρ Fonset Gonset max , (2.47)
Flength Flength, min
Application of Uncertainty Quantification to Turbulence Intensity 41
CHAPTER 2. NUMERICAL IMPLEMENTATION
If γ > 1.0, Pγ = (1.0− γ)Pγ (2.48)
Flength = 40.0, Flength, min = 2.5 (2.49)
The destruction term is given by:
Dγ = ρ Ω γ(1.0−Gonset) (2.50)
Fonset = max(Fonset2 − Fonset3, 0) (2.51)
Rev
Fonset1 = (2.52)
2.193Reθc
Fonset2 = min(max(F
4
onset1, Fonset1), 4.0) (2.53)
Fonset3 = max(2− (0.25RT )3, 0) (2.54)
ρd2S µt
Rev = , RT = (2.55)
µ µ
The critical Reynolds number can be calculated using the following numerical
correlation:
Reθc = 0.62Reθt (2.56)
The transport equation for the transition momentum thickness Reynolds num-
ber, Reθt is given by: [ ]
D(ρReθt) ∂ ∂Reθt
= Pθt + 2.0(µ+ µt) (2.57)
Dt ∂xj ∂xj
Application of Uncertainty Quantification to Turbulence Intensity 42
CHAPTER 2. NUMERICAL IMPLEMENTATION
FSTI% Reθt,new
0.01 1800.0
0.03 1135.0
0.51 894.0
1.33 392.0
2.00 252.0
5.25 165.0
6.5 100.0
Table 2.1: Piecewise Linear Correlations between Reθt and FSTI
The production term is given by:
ρ
Pθt = 0.03 (Reθt −Reθt)(1.0− Fθt) (2.58)
t ( )
d 4
Fθt = min e
−( δ ) , 1.0 (2.59)
Reθtµ 50Ωd
θBL = ; δBL = 7.5θBL; δ = δBL (2.60)
ρU U
Piecewise linear correlations, as defined by Table 2.1, are multiplied by the
F(λθ) equations to calculate Reθt values outside of the boundary layer. This takes
the FSTI and streamwise pressure gradient effects. The F(λθ) equations are given
below:
 1.51− [−12.986λ − 123.66λ2 − 405.689λ3]e−[Tuθ 1.5 ]θ θ , λθ ≤ 0
F (λθ) =  (2.61)1 + 0.275[1− e[−35.0λθ]]e−[Tu0.5 ], λθ > 0
where the pressure gradient parameter, λθ, is defined by:
ρθ2 dU
λθ = (2.62)
µ ds
Application of Uncertainty Quantification to Turbulence Intensity 43
CHAPTER 2. NUMERICAL IMPLEMENTATION
dU
is the streamwise acceleration:
ds
dU u dU v dU
= + (2.63)
ds U dx U dy
[ ]
dU 1 du dv
= 2u + 2v (2.64)
dx 2U [ dx dx]
dU 1 du dv
= 2u + 2v (2.65)
dy 2U dy dy
√
U = u2 + v2 (2.66)
In order to evaluate F(λθ), an iterative process is used with the initial value of
λθ = 0.0. The freestream Reθt value is set equal to Reθt inf . Reθt inf can be computed
using the above equations with λθ equal to zero.
The intermittency transport equation is solved for the intermittency factor γ,
which can then be multiplied by production term of the eddy viscosity in the SA
model. Eq. 2.42 therefore is rewritten to include the intermittency term:
Dν̃ 1 [ ]
= γPν −Dν + ∇((ν + ν̃) + c 2b2(∇ν̃) (2.67)
Dt σ
Transition Model Validation
In order to verify the transition model’s accuracy in OVERTURNS, a test
case was run using the S809 airfoil. The case was run with an FSTI of 0.05%,
an angle of attack of 1 degree, and a Reynolds number of 2 million. The case
was compared against both experimental results as well as the results obtained by
Application of Uncertainty Quantification to Turbulence Intensity 44
CHAPTER 2. NUMERICAL IMPLEMENTATION
(a) Coefficient of Pressure (b) Coefficient of Skin Friction
Figure 2.4: Transition Model Validation for S809 Airfoil
Medida [6]. Fig. 2.4 shows the coefficient of pressure and skin friction. As can be
seen from the figure, there is good agreement between the results of the present
work, the results from Medida’s work, and experimental results.
2.6.5 Time Integration
Although the present work only deals with steady cases, for the sake of
completeness, a short overview of time integration is provided here. OVERTURNS
utilizes BDF2 [44], a second order implicit backwards-in-time method as its time-
marching scheme to integrate the semi-discrete form of the governing equations
(Eq. 2.31). Backwards-in-time implicit methods converge well and are usually
more stable, making these methods a logical choice when dealing with boundary
layer flows. The BDF2 scheme is implemented using Eq. 2.68:
∂Q̃n+1 F̃
n+1
1 − F̃ n+1 G̃n+1 n+1j+ j− 1 k+ 1 − G̃k− 1
= − 2 2 − 2 2 + S̃n+1 (2.68)
∂t ∆ξ ∆η j,k
Application of Uncertainty Quantification to Turbulence Intensity 45
CHAPTER 2. NUMERICAL IMPLEMENTATION
where the LHS can be discretized using Eq. 2.69 :
∂Q̃n+1 3Q̃n+1 − 4Q̃n + Q̃n−1
= (2.69)
∂t 2∆t
Eq. 2.68 is a non-linear equation; a Taylor series expansion about Q̃n can be
applied to the fluxes to linearize the equation:
F̃ n+1 = F̃ n + Ã∆Q̃+O(∆t2) (2.70)
G̃n+1 = G̃n + B̃∆Q̃+O(∆t2) (2.71)
where the solution update ∆Q̃ = Q̃n+1− Q̃n, and A and B are the flux Jacobians,
∂F̃ ∂G̃
defined as and respectively. The source term S can also be in linearized
∂Q̃ ∂Q̃
using the same method. The linearization does not decrease the time-accuracy of
the scheme since the linearized terms are second-order accurate. The linearizations
are then applied to Eq. 2.68, resulting in the following equation:
[I + ∆t(δ Ãnξ + δηB̃
n)]∆Q̃ = −∆t[δ nξF̃ + δηG̃n] (2.72)
In the above equation, the left-hand side controls the convergence and stability of
the equation, while the right-hand side represents the physics of the problem.
Eq. 2.72 can be written as a banded matrix of algebraic equations. OVER-
TURNS solves this system using an approximate factorization method. In order
to invert the system, the Lower-Upper Symmetric Gauss-Seidel (LUSGS) method
Application of Uncertainty Quantification to Turbulence Intensity 46
CHAPTER 2. NUMERICAL IMPLEMENTATION
is utilized. Using this method, Eq. 2.72 can be rewritten as:
[L+D + U ]∆Q̃n ≈ [D + L]D−1[D + U ]∆Q̃n = −∆t[RHS]n (2.73)
In the above equation, L, U , and D represent the lower, upper, and main diagonal
of the left-hand side of the equations, respectively and are defined as follows:
L = ∆t(−Ã+j−1,k − B̃
+
j,k−1) (2.74)
U = ∆t(Ã− −j+1,k + B̃j,k+1) (2.75)
D = I + ∆t(Ã+ − Ã− + B̃+ −j,k j,k j,k − B̃j,k) (2.76)
A two step process is used to obtain the solution update ∆Q̃:
[D + L]∆Q = −∆t[RHS]n (2.77)
[D + U ]∆Q̃ = DQ (2.78)
Application of Uncertainty Quantification to Turbulence Intensity 47
Chapter 3
Uncertainty Quantification
Methodology
This chapter details the uncertainty quantification methodology used in the
present work. The following steps were taken for each airfoil studied:
1. Run Parametric Sweep: Parametric studies were run varying FSTI and
the angle of attack. For some airfoils, Reynolds number or Mach number
was also varied. Details of the specific parameters being varied are provided
in Table 3.1.
2. Generate Surrogate Surface: Once the parametric studies were run, a
piecewise cubic Hermite interpolating polynomial (PCHIP) was used to in-
terpolate between the points in the parametric sweeps and generate the sur-
rogate surface.
48
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
Airfoil Mach Number Reynolds Number (×106)
S809 0.2 2.0
NACA 0012 0.2 0.5, 1.0, 2.0, 4.0
SC1095 0.2, 0.4, 0.6, 0.7, 0.8 2.0
RC(4)-10 0.2, 0.4, 0.6, 0.7, 0.8 2.0
Table 3.1: Freestream Conditions for Parametric Sweeps
3. Perform UQ Analysis - Monte Carlo Simulation: The Monte Carlo
method was used in conjunction with the surrogate surface in order to com-
pute the expected means and standard deviations of the aerodynamic coef-
ficients, given an input FSTI and standard deviation. These computations
were repeated over a range of angles of attack. For some of the airfoils, the
Mach or Reynolds number was also varied. The standard deviations of the
aerodynamic coefficients indicate the sensitivity of each parameter to un-
certainty in the FSTI. By computing the standard deviations for different
freestream conditions, it is possible to see how the freestream parameters
affect this sensitivity.
3.1 Parametric Sweeps
Based off of preliminary sweeps, the FSTI range of 0.05% to 5% was identified
as the area of interest for this study. An initial increment of 0.5% was used for
the first sweep. Three angles of attack were initially chosen: 0 degrees, 4 degrees,
and 8 degrees. Each case was then run in OVERTURNS until convergence. The
results of this initial sweep are shown in Fig. 3.1 for the S809 airfoil. This initial
sweep showed that the QOIs experienced rapid change between 0.5% and 2%
Application of Uncertainty Quantification to Turbulence Intensity 49
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
(a) Lift coefficient (b) Drag coefficient
(c) Moment coefficient
Figure 3.1: Initial Turbulence Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106
FSTI. Additional points were therefore added to the sweep in this region until the
parametric sweep was well resolved. In addition, the angle of attack range was
expanded, from -4 degrees to 12 degrees, using 1 degree increments. The final
parametric sweep is shown in Fig. 3.2 for the S809 airfoil.
Application of Uncertainty Quantification to Turbulence Intensity 50
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
(a) Lift coefficient
(b) Drag coefficient
Figure 3.2: Final Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106
Application of Uncertainty Quantification to Turbulence Intensity 51
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
(c) Moment coefficient
Figure 3.2: Final Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106
(cont.)
3.2 Surrogate Surface Generation
Once the parametric turbulence sweep was complete, a surrogate surface
was generated using the “PCHIP” function in MATLAB [46]. PCHIP stands for
Piecewise Cubic Hermite Interpolating Polynomial. This function uses a Her-
mite polynomial P (x) to interpolate points from a given data set, resulting in a
monotonic cubic interpolation. The polynomial generated has a continuous first
derivative dP . The PCHIP function selects the slopes at each data point such
dx
that the interpolating polynomial preserves the shape of the data and respects
monotonicity. Therefore the extrema of the data are also the extrema of P (x).
In this work, the PCHIP function was invoked using the range of 0% to 5.0%
Application of Uncertainty Quantification to Turbulence Intensity 52
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
FSTI, with intervals of 0.001% as the input interpolation points for the PCHIP
function. The CFD results from the parametric sweep were used as the data to be
interpolated. The solid lines in Fig. 3.2 represent the generated surrogate surface.
It can be seen that PCHIP smoothly fills in the data between the CFD outputs.
3.3 UQ Analysis
Once the surrogate surface has been generated, the next step is to compute
the expected means and standard deviations. This can be achieved through either
a Monte Carlo approach or stochastic collocation. The Monte Carlo method was
deemed better suited for the current work since the use of the surrogate surface
allowed for a large number of runs to be conducted without a large computational
penalty. While stochastic collocation would have been more efficient, it was not
possible to use more than 5 nodes for quadrature with the desired mean input
FSTI (1%) and standard deviation (0.3333%). Increasing the number of nodes
with these inputs would have required CFD simulations to be run with negative
FSTI values, which was not possible (see Table 3.2). Therefore, a Monte Carlo
approach was taken. As shown by Fig. 3.3, using the Monte Carlo approach results
in the means and standard deviations to converge to the correct values. Since the
stochastic collocation method is limited to 5 nodes, it is not as accurate. Note
that the present work assumes that the uncertainty in FSTI results in a normal
distribution of freestream FSTI values.
Application of Uncertainty Quantification to Turbulence Intensity 53
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
Node 5 Nodes - FSTI (percent) 6 Nodes - FSTI (percent)
1 4.777 ∗ 10−2 -0.1080
2 0.548 0.3703
3 1.0 0.7945
4 1.452 1.2055
5 1.952 1.6297
6 - 2.1080
Table 3.2: FSTI values for the quadrature for mean FSTI = 1%, σ = 0.3333
Figure 3.3: UQ Analysis for Drag - S809 Airfoil, Mach = 0.2, Re = 2× 106, Mean
FSTI = 1.0%, Std Dev = 0.3333%
Application of Uncertainty Quantification to Turbulence Intensity 54
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
Airfoil No. of Inputs
S809 800,000
NACA 0012 350,000
SC1095 550,000
RC(4)-10 650,000
Table 3.3: Number of FSTI Input Values Required for Consistency
3.3.1 Monte Carlo Method
This section details how the Monte Carlo method was applied to determine
the expected mean and standard deviations for the aerodynamic coefficients. First
a vector of normally distributed FSTI input values was generated using the MAT-
LAB normal random number generation function ’normrnd’. The number of input
values was experimentally determined such that the results were consistent. This
number differed per airfoil; the number of input values are listed in Table 3.3. Once
this vector of inputs was generated, the aerodynamic coefficients were calculated
at these input values using the surrogate surface. The results were then averaged
and the standard deviation was calculated.
3.4 Validation
In order to be confident of the results of the process detailed above, it was
necessary to validate the method. This process included validating the surrogate
surface and verifying that the results were reasonable when compared to discrete
FSTI.
Application of Uncertainty Quantification to Turbulence Intensity 55
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
3.4.1 Surrogate Surface Validation
In order to avoid the high costs of thousands of CFD simulations, surrogate
surfaces were used in place of actual CFD results. The surrogate surfaces were
generated using actual CFD results from parametric sweeps, as detailed in Sec. 3.1
- 3.2. In order to verify that the surrogate surfaces could be relied upon to provide
accurate results, both stochastic collocation and Monte Carlo simulations were
performed using both the surrogate surface. This was compared against stochastic
collocation using actual CFD simulations for a test case for the S809 airfoil. Note
that the Monte Carlo method was not used with actual CFD results due to its
excessively high computational cost. A five node Gauss-Hermite quadrature was
performed using a mean FSTI value of 1% and a standard deviation of 0.3333.
The 5 node quadrature therefore required 5 CFD simulations at the FSTI values
specified in Table 3.2. The results of the 5 CFD simulations are compared to the
results generated from the surrogate surface in Fig. 3.4. The figure shows that the
surrogate surface agrees well with the actual CFD results.
Once the aerodynamic coefficients were obtained from the surrogate surface
results, the expected mean and standard deviations could be calculated for the
test case (AOA = 8 degrees, mean FSTI = 1%, σ = 0.3333). The results of
this quadrature are presented in Table 3.4 using the actual CFD results and the
surrogate surface. The table shows that the final output of stochastic collocation
using the surrogate surface is comparable to the results of stochastic collocation
using actual CFD results. A Monte Carlo simulation using 10,000 runs was also
performed using the surrogate surface. The results of this Monte Carlo simulation
Application of Uncertainty Quantification to Turbulence Intensity 56
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
(a) Lift coefficient (b) Drag coefficient
(c) Moment coefficient
Figure 3.4: Comparison of Surrogate Results and CFD Results for S809 Airfoil at
Mach = 0.2, Re = 2× 106, AOA = 8 degrees
Application of Uncertainty Quantification to Turbulence Intensity 57
CHAPTER 3. UNCERTAINTY QUANTIFICATION METHODOLOGY
Stochastic Stochastic Monte Carlo -
Collocation - Collocation - Surrogate - 10,000
CFD - 5 Nodes Surrogate - 5 Nodes Runs
Coefficient Expected Standard Expected Standard Expected Standard
Value Deviation Value Deviation Value Deviation
Lift 1.077963 0.003297 1.077950 0.003234 1.077877 0.003214
Drag 0.0140970 3.082e-04 0.0140959 3.035e-04 0.0141036 3.005e-04
Moment -0.0633449 4.180e-04 -0.0633469 4.138e-04 -0.0633367 4.073e-04
Table 3.4: Comparison of Expected QOIs and Standard Deviations for for S809
Airfoil at Mach = 0.2, Re = 2× 106, AOA = 8 degrees
are also included in Table 3.4. The table shows that the results are all in general
agreement. Thus it was deemed that Monte Carlo with a surrogate surface could
be used for the uncertainty quantification of FSTI on airfoil aerodynamics.
Application of Uncertainty Quantification to Turbulence Intensity 58
Chapter 4
Parametric Sweep Results
While the main purpose of the parametric sweeps was to generate the surrogate
surfaces for the UQ analysis, many cases of interest were identified and investi-
gated. This chapter presents selected results which featured behavior worth noting
and exploring. Note that the complete parametric sweep results can be found in
Appendix A. Four airfoils were used: 1) S809, 2) NACA 0012, 3) SC1095, and
4) RC(4)-10. FSTI was varied from 0.05% to 5%. However, this section focuses
on the results up to 3% FSTI. The angle of attack was varied from -4° to 12° for
the S809, SC1095, and RC(4)-10 airfoil, with the exception of certain cases where
the convergence of the CFD cases became problematic due to the high angles of
attack, as discussed later. For the NACA 0012, the angle of attack was varied
from 0° to 12°. The negative angles of attack were omitted since the NACA 0012
is a symmetric airfoil. Mach number was also varied for the SC1095 and RC(4)-10
airfoils. In order to explore the effect of varying Reynolds number, four Reynolds
59
59
CHAPTER 4. PARAMETRIC SWEEP RESULTS
Airfoil Mach Number Reynolds Number (×106)
S809 0.2 2.0
NACA 0012 0.2 0.5, 1.0, 2.0, 4.0
SC1095 0.2, 0.4, 0.6, 0.7, 0.8 2.0
RC(4)-10 0.2, 0.4, 0.6, 0.7, 0.8 2.0
Table 4.1: Freestream Conditions for Parametric Sweeps
numbers were used for the NACA 0012. Table 3.1 provides the freestream Mach
numbers and Reynolds numbers for each parametric sweep.
4.1 Computational Meshes
All the meshes used for this study are two-dimensional with C-topology
grids. 491 points are used in the wrap-around direction on the airfoil surface,
while 131 points are used in the wall-normal direction. These dimensions were
chosen to ensure sufficient discretization for result accuracy while minimizing the
computational time required to run CFD simulations. The outer boundary was
placed 40 chord lengths away from the airfoil to ensure that the outer domain did
not interfere with the CFD solution. The meshes for each airfoil are presented in
Fig. 4.1.
4.2 S809 Airfoil
The S809 airfoil is a thick airfoil (maximum thickness of 21% of chord) used
specifically in horizontal axis wind turbines. The NREL Phase VI model wind
Application of Uncertainty Quantification to Turbulence Intensity 60
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) S809 (b) NACA 0012
(c) SC1095 (d) RC(4)-10
Figure 4.1: Computational Meshes for Airfoil Parameter Sweeps
Application of Uncertainty Quantification to Turbulence Intensity 61
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Lift coefficient
(b) Drag coefficient
Figure 4.2: Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106
Application of Uncertainty Quantification to Turbulence Intensity 62
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(c) Moment coefficient
Figure 4.2: Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2×106 (cont.)
turbine primarily uses this airfoil for its turbine blades [47]. The airfoil is designed
to promote laminar flow over the airfoil for over 50% of the airfoil at low angles
of attack. This design makes the S809 airfoil popular for transition studies. The
parameter sweep for the S809 airfoil was run at a freestream Mach number of 0.2
and a Reynolds number of 2 million.
Fig. 4.2 presents the parametric sweep for the S809 airfoil. Observe that lift
remains relatively constant as FSTI is varied, although there is a slight decrease
in lift as FSTI increases for the high AOA. Drag increases as FSTI increases for
all AOA, with the maximum amount of change occurring near 0° AOA. The pitch
moment also increases as FSTI increases for all AOA.
Fig. 4.3a depicts the coefficient of drag versus the angle of attack for various
Application of Uncertainty Quantification to Turbulence Intensity 63
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Coefficient of Drag (b) Coefficient of Moment
Figure 4.3: S809 Airfoil at Mach = 0.2, Re = 2.0×106 - Aerodynamic Coefficients
vs. AOA
FSTI values up to 2%. The graph allows one to clearly visualize the “drag bucket.”
This drag bucket phenomenon is caused by the transition from laminar to turbulent
flow. When the flow is laminar, the drag does not change substantially, even
as angle of attack increases, creating the bottom of the bucket. When the flow
transitions to turbulent flow, the drag increases sharply and continues to increase
as angle of attack increases, creating the sides of the bucket. One can note that
for low FSTI values (0.05% - 1.0%), the laminar portion of the graph is much
larger. At higher FSTI values (1.5% and 2%), the laminar region is much shorter
and steeper. One can also note that at low angles of attack, there is a significant
difference between the values of drag between FSTI values of 1% and 2%. However,
as AOA increases, the curves for all the FSTI values approach the same drag
coefficient value. A similar trend can be seen for the coefficient of moment, as
illustrated in Fig. 4.3b, where the 0.05%, 0.5%, and 1.0% FSTI curves fall on top
of each other at the plotted moderate AOA.
Application of Uncertainty Quantification to Turbulence Intensity 64
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Skin Friction - AOA = 0° (b) Pressure - AOA = 0°
(c) Skin Friction - AOA = 8° (d) Pressure - AOA = 8°
Figure 4.4: S809 Airfoil at Mach = 0.2, Re = 2.0×106 - Skin Friction and Pressure
Plots - Separation Bubble
Application of Uncertainty Quantification to Turbulence Intensity 65
CHAPTER 4. PARAMETRIC SWEEP RESULTS
Figure 4.5: Separation Bubble for S809 Airfoil for AOA = 0°, FSTI = 1.0%, Mach
= 0.2, Re = 2.0× 106
Fig. 4.4 shows the coefficients of pressure and skin friction along the airfoil for
an angle of attack of 0° and 8° AOA. Note that the curve at the top portion of the
graph represents the upper surface of the airfoil, while the curve at the bottom
of the graph represents the lower surface. Also note the sign convention for skin
friction plots: airflow moving from left-to-right over the airfoil results in positive
skin friction on the upper surface and negative skin friction on the lower surface.
The transition from laminar to turbulent flow can be determined from the skin
friction coefficient plots. The steep increase in the magnitude of the skin friction
coefficient indicates the transition location. As can be seen from the plots, as the
FSTI value increases, the transition location moves toward the leading edge of the
airfoil.
In addition, the skin friction plots can be used to determine the type of transi-
tion. Observe the skin friction coefficient plot for AOA of 0°. For the lower values
of FSTI (0.05% and 0.5%), it can be seen that the friction becomes negative on the
Application of Uncertainty Quantification to Turbulence Intensity 66
CHAPTER 4. PARAMETRIC SWEEP RESULTS
upper surface right before transition. The opposite occurs on the bottom surface.
This indicates that the air flow has reversed direction in this region. This indi-
cates a separation bubble has formed, and the air is circulating in this bubble, as
pictured in Fig. 4.5. The laminar separation bubble results in separation-induced
transition (see sec. 1.2.6). Once the flow transitions, the turbulent boundary layer
then reattaches to the airfoil.
The separation bubbles can also be seen on the pressure plots, albeit more
subtly. There is a small pressure drop at the bubble, since suction pressure de-
creases due to the adverse pressure gradient. Since the location of the bubble varies
slightly for each FSTI case, the pressure plot varies slightly at the location of the
separation bubble. The separation bubbles have been circled on the figure. One
can notice that the transition location does not move as much between 0.05% and
1.0%, when compared to the change in transition location between 1.5% and 2.0%.
On the other hand, at 0° AOA for 1.5% and 2.0% FSTI, there is no negative skin
friction for the upper surface and no positive skin friction for the lower surface.
Rather, the skin friction just increases steeply. This indicates natural transition
has occurred. The flow remains attached in this case as well.
At 8° AOA, all five cases undergo separation-induced transition, with a very
small separation bubble near the leading edge at the top of the airfoil. At the
bottom surface, another separation bubble is present near mid-chord, except for
the 2.0% FSTI case, which undergoes natural transition here. Again, there is only
a small change in transition location as FSTI varies. The lack of change in the
transition location at 8° AOA explains why, in Fig. 4.3a, there is only a small
Application of Uncertainty Quantification to Turbulence Intensity 67
CHAPTER 4. PARAMETRIC SWEEP RESULTS
difference in drag at the high angles of attack, but greater difference at lower
angles of attack. At low angles of attack, the type of transition changes from
separation-induced to natural transition. At high angles of attack, separation-
induced transition occurs predominantly.
It can be noted that there is a significant difference between the 0° and 8° AOA
plots. The transition location moves from mid-chord to the leading edge of the
airfoil and a much larger pressure gradient is present for the 8° AOA case on the
upper surface. On the lower surface, the transition location only moves slightly
aft from 0° to 8° AOA because of the separation induced transition near midchord
at lower FSTI.
4.3 NACA 0012 Airfoil
The NACA 0012 airfoil is a symmetrical airfoil with a maximum thickness of
12% of the chord. This airfoil was used for helicopter rotors for a number of years.
The large amount of experimental data available for the NACA 0012 makes it an
ideal ”yardstick” to compare other airfoils against. In this study, the NACA 0012
airfoil was used for the Reynolds number sweep using a constant Mach number of
0.2. The results of the parametric studies for the four Reynolds numbers selected
are presented in Figs. 4.6 - 4.17.
Application of Uncertainty Quantification to Turbulence Intensity 68
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Lift coefficient
(b) Drag coefficient
Figure 4.6: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 0.5×106
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(c) Moment coefficient
Figure 4.6: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 0.5×106
(cont.)
Re = 500,000
Fig. 4.6 shows the results of the parametric sweep for the NACA 0012 airfoil
at a Reynolds number of 500,000. Most of the variations are relatively smooth.
Similar to the S809, the lift curves remain relatively constant as FSTI varies. At
low AOA, drag also does not change much at low values of FSTI up to 1.0%.
From 1.0% to 2.0%, drag increases noticeably with FSTI. At high AOA, the drag
increases as FSTI increases. Moment decreases as FSTI increases, with the excep-
tion of AOA 3° to 6°. At these AOA, moment first increases to a peak between
FSTI values of 1.25% and 1.75% before decreasing again as FSTI increases.
Fig. 4.7 presents the pressure and friction plots for 5°, 6°, and 12° AOA. As
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Pressure - AOA = 5° (b) Skin Friction - AOA = 5°
(c) Pressure - AOA = 6° (d) Skin Friction - AOA = 6°
(e) Pressure - AOA = 12° (f) Skin Friction - AOA = 12°
Figure 4.7: Pressure and Friction Plots for NACA 0012 Airfoil, Re = 0.5 × 106,
Mach = 0.2
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(a) AOA = 5° (b) AOA = 6°
(c) AOA = 12°
Figure 4.8: Zoomed in Skin Friction Plots for NACA 0012 Airfoil, Re = 0.5× 106,
Mach = 0.2
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(a) FSTI = 0.05% (b) FSTI = 3.0%
Figure 4.9: NACA 0012 - Coefficient of Lift, Mach = 0.2
can be seen from the skin friction graphs, the flow remains laminar on the bottom
surface, with the exception of the 2% FSTI case for 5° and 6° AOA. Fig. 4.8 zooms
in on the separation bubble region. At this Reynolds number, at the lower FSTI
values, two separation bubbles form. After the first bubble, the flow reattaches
without becoming fully turbulent, before forming a second separation bubble. The
flow finally transitions fully after this second bubble. At 5° AOA, only the 0.05%
and 0.5% FSTI cases exhibit this double separation bubble, while at 6° AOA, the
phenomenon occurs for all the FSTI values besides 2.0%. At 12° AOA, all five
FSTI values have the double separation bubble. This double separation bubble
does not occur at Reynolds numbers of 2 million and higher, as will be shown in
later sections.
Since the two separation bubbles are near the leading edge of the airfoil on the
upper surface, they effectively add positive camber to the airfoil. This results in
an increase in suction pressure at the leading edge, which in turn increases the lift.
This is why the suction peak in the pressure graphs is slightly higher for the cases
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
which have multiple separation bubbles. This also accounts for the increase in lift
for the Re = 500,000 case at the moderate angles of attack, as shown in Fig. 4.9.
As FSTI varies, the bubble locations move, which is why there is a steep increase
in pitching moment at the moderate angles of attack for low FSTI values in Fig.
4.6c. The steep increase is limited to the region with the separation bubbles. At
the higher FSTI values, the flow undergoes natural transition and the moment
begins to decrease as FSTI increases.
Re = 1 × 106
The results of the sweep for a Reynolds number of 1 million are presented in
Fig. 4.10. For this sweep, certain cases did not converge smoothly to a single
value, but rather resulted in repeating oscillations, as can be seen in Fig. 4.11. In
order to obtain the final aerodynamic coefficients, the results were averaged from
peak to peak for the last three to five oscillations. Table 4.2 lists the cases where
this method was applied. The different methods used to obtain the quantities of
interest resulted in a noticeable jump in values in the moment coefficient graph in
Fig. 4.10.
Fig. 4.12 shows the skin friction plots for 5°, 6°, and 12° AOA cases. Similar
trends to the Reynold number of 500,000 cases can be seen. Fig 4.13 shows the
portion of the skin friction plots where the separation bubbles are present. The
double separation bubble phenomenon is present for an FSTI value of 0.05% at 5°
AOA, for FSTI values of 0.05% - 1.0% at 6° AOA, and for all of the FSTI values
at 12° AOA except for FSTI of 1.5%. One noticeable difference between Reynolds
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Drag coefficient
(b) Moment coefficient
Figure 4.10: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 1×106,
Mach = 0.2
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Density Residual (b) Coefficient of Lift
Figure 4.11: Convergence Oscillations - NACA 0012 - Mach = 0.2, Re = 1× 106,
AOA = 5°, FSTI = 0.1%
numbers 500,000 and 1 million is that the bottom surface is no longer laminar for
the Reynolds number of 1 million and instead undergoes natural transition for all
of the AOA. It can also be noted that as angle of attack increases, the transition
location moves closer to the leading edge on the upper surface and closer to the
trailing edge on the bottom surface. One can also see that, as FSTI varies for the
5° AOA case, there is a greater change in transition location, but for 6° and 12°
AOA, there is only a small amount of change in transition location.
Re = 2 × 106
Fig. 4.14 shows the sweep results for a Reynolds number of 2 million. When
compared to the lower Reynolds number, one can observe that there is more
change in drag as FSTI increases, especially at the higher angles of attack. At
low Reynolds number, the drag is relatively constant at these higher angles. This
can be explained by examining the skin friction plots in Fig. 4.15. As can be seen
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
Angle of Attack (degrees) FSTI (percent)
4 0.1 - 0.4
5 0.05 - 0.4
Table 4.2: NACA 0012 - Mach = 0.2, Re = 1 × 106, Mach = 0.2 - Peak-to-Peak
Averaged Cases
(a) AOA = 5° (b) AOA = 6°
(c) AOA = 12°
Figure 4.12: Skin Friction Plots for NACA 0012 Airfoil, Re = 1.0× 106, Mach =
0.2
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) AOA = 5° (b) AOA = 6°
(c) AOA = 12°
Figure 4.13: Zoomed in Skin Friction Plots for NACA 0012 Airfoil, Re = 1.0×106,
Mach = 0.2
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Drag coefficient
(b) Moment coefficient
Figure 4.14: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 2×106,
Mach = 0.2
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) AOA = 6° (b) AOA = 8°
(c) AOA = 10° (d) AOA = 12°
Figure 4.15: Skin Friction Plots for NACA 0012 Airfoil, Re = 2.0× 106, Mach =
0.2
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) L/D vs Angle of Attack (b) Cd vs Cl
Figure 4.16: NACA 0012 Airfoil at Mach = 0.2, Re = 2× 106, Mach = 0.2
from these plots, transition on the upper surface occurs at the leading edge at
these angles. Since transition occurs so close to the leading edge, the flow is able
to stay attached after transition. The turbulent flow therefore has a greater effect
on the skin friction, and therefore the drag. On the other hand, at low Reynolds
numbers, some separation occurs, allowing the drag to remain constant.
It can also be noted that the sharp increases in moment are no longer present
for a Reynolds number of 2 million. Instead, as angle of attack increases, there is
a sudden decrease in moment which becomes steeper and occurs at higher FSTI
values as angle of attack increases. When examining the skin friction plots, it be-
comes apparent that this sudden drop in moment is correlated to transition on the
bottom surface. At the lower FSTI values, the flow has a small separation bubble
near the trailing edge of the bottom surface; for the higher FSTI values, natural
transition occurs near mid-chord. The separation bubble results in a virtual in-
crease in reverse camber, which in turn affects the moment coefficient. Therefore,
there is sharp drop in moment when comparing the low FSTI cases which have
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
this trailing edge separation bubble and the higher FSTI values which have natural
transition.
Compare the pitching moment for the S809 airfoil (Fig. 4.2) and the pitch-
ing moment for the NACA 0012 at a Reynolds number of 2 million (Fig. 4.14).
Note that even though the freestream Mach and Reynolds numbers are identical,
the trends are significantly different. The NACA 0012 airfoil pitching moment
decreases as FSTI increases, including the region of sharp decreasing pitching mo-
ment. On the other hand, the pitching moment increases as FSTI increases for
the S809 airfoil.
Fig. 4.16 shows the L/D curve and drag polar for various values of FSTI. At
the lower FSTI values, there is a larger peak in the L/D curve but a faster drop
off. The higher FSTI values stay flatter. All of the FSTI curves approach a similar
value at the high angles of attack.
Re = 4 × 106
The results for the Reynolds number of 4 million sweep are shown in Fig.
4.17. This case is very similar to the case for a Reynolds number of 2 million case.
There is a large change in drag at the higher angles of attack due to the large
region of attached flow. However, there is a noticeable difference in the moment
coefficient plot. The sharp decrease in moment has shifted left (i.e. at lower FSTI
values). Fig. 4.18 shows the skin friction for an angle of attack of 12 degrees.
When comparing this case against lower Reynolds numbers, one can see that the
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Drag coefficient
Figure 4.17: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 4×106
transition location on the bottom surface has moved forward. Also, the 1% FSTI
case also naturally transitions at this Reynolds number.
4.4 SC1095 Airfoil
The SC1095 airfoil was designed by Sikorsky Aircraft for use on the Black
Hawk UH-60A helicopter. It is a cambered airfoil witha maximum thickness of
9.5%. The airfoil can be considered to be representative of modern airfoil design
[48]. In this study, the Mach number was varied for this airfoil from 0.2 to 0.8.
The results of the parametric study are presented in Figs. 4.19 - 4.30. Note that
for the sweeps with a Mach number of 0.6 or greater, the results have been omitted
for high angles of attack (i.e. over 6 degrees). These cases did not result in stable
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(b) Moment coefficient
Figure 4.17: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 4×106
(cont.)
Figure 4.18: Skin Friction Plot for NACA 0012 Airfoil, Re = 4.0 × 106, Mach =
0.2, AOA = 12 degrees
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
steady solutions. This is due to unsteady transonic effects due to shock induced
separation, which are beyond the scope of this work.
Mach 0.2
Fig. 4.19 shows the results for a Mach number of 0.2 and a Reynolds num-
ber of 2 million. The plots show trends similar to the NACA 0012 at the same
conditions. Comparing the results of both the NACA 0012 and SC1095 against
the S809 airfoil, one can see that minimum drag for the NACA 0012 and SC1095
airfoils is lower than the S809 airfoil. This makes sense, since the NACA 0012 and
SC1095 airfoils are used for helicopter rotors and therefore drag minimization was
a significant design consideration. On the other hand, the S809 airfoil’s primary
design objective was promoting laminar flow.
Fig. 4.20 shows the L/D plot and drag polar for various FSTI values. From the
L/D plot, it can be seen that the low FSTI curves (0.05% and 0.5%) are close to
each other, indicating there is not much variance. This is attributed to the smaller
drag bucket (shown in the drag polar graph) when compared to NACA 0012. This
smaller drag bucket is due to the camber of the SC1095 airfoil.
Fig. 4.21 shows the skin friction plots for the cases at Mach 0.2. Once again,
the graphs show similar trends to the NACA 0012 results (see Fig. 4.15). However,
for the SC1095 airfoil, at the high angles of attack and lower values of FSTI, the
separation bubble does not occur right at the leading edge, but rather slightly
later.
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Drag coefficient
(b) Moment coefficient
Figure 4.19: Parametric Sweep for SC1095 Airfoil at Mach = 0.2, Re = 2× 106
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) L/D vs Angle of Attack (b) Cd vs Cl
Figure 4.20: SC1095 Airfoil at Mach = 0.2, Re = 2× 106
Mach 0.4
Fig. 4.22 shows the drag coefficient for the Mach 0.4 sweep. Note the
large increase in drag between AOA of 10° and 12°. Fig. 4.23 shows the Mach
contours at these angles of attacks for FSTI values of 0.5% and 2.0%. From these
plots, it can be seen that there is not much difference between the 0.5% and 2.0%
cases. However, when comparing the 10° and 12° AOA cases, the Mach number is
observed to be higher for the 12 degree case near the leading edge of the airfoil.
The boundary layer is also observed to be much thicker for the 12° AOA case.
This thicker boundary layer negatively affects the effective camber at the trailing
edge. This results in a small drop off in lift and a large jump in drag.
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) AOA = 4° (b) AOA = 6°
(c) AOA = 8° (d) AOA = 10°
(e) AOA = 12°
Figure 4.21: Skin Friction Plots for SC1095 - Mach = 0.2
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
Figure 4.22: Parametric Sweep for Drag Coefficient - SC1095 Airfoil at Mach =
0.4, Re = 2× 106
Mach 0.6
Fig. 4.24 shows the coefficient of drag for the Mach 0.6 sweep. The graph
resembles the graphs for the Mach 0.4 cases. However, the jumps in drag have
increased between each angle of attack. The magnitude of the jump between 5°
and 6° AOA for Mach 0.6 is of the same magnitude as the jump between 10° and
12° AOA for Mach 0.4. It can also be noted that for each curve, drag remains
fairly constant before a noticeable jump as FSTI increases. After this jump, drag
remains constant. Fig. 4.25 compares the Mach contours for FSTI values of 0.5%
and 2.0% for the 6° AOA case. The spike in Mach along the airfoil indicates that
a shock has occurred there. However, similar to the Mach 0.4 case, there is not
much change between FSTI values of 0.5% and 2.0%.
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) AOA = 10°, FSTI = 0.5% (b) AOA = 12°, FSTI = 0.5%
(c) AOA = 10°, FSTI = 2.0% (d) AOA = 12°, FSTI = 2.0%
Figure 4.23: Mach Contours for SC1095 Airfoil at Mach = 0.4, Re = 2× 106
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
Figure 4.24: Parametric Sweep for Drag Coefficient - SC1095 Airfoil at Mach =
0.6, Re = 2× 106
Fig. 4.26 shows the pressure and skin friction plots. Observe that the upper
surface is virtually unaffected as FSTI increases. However, the transition location
moves forward as the FSTI increases. For this case, there is a large jump in the
transition location between 1% and 1.5%. This corresponds to the increase in drag
that was noted earlier in Fig 4.24.
Mach 0.7
Fig. 4.27 presents the results of the Mach 0.7 sweep. Note that the lift
actually decreases between 4° and 6° AOA. There is also a very large increase in
drag and a large decrease in moment between 4° and 6° AOA. Fig. 4.28 shows the
Mach contours for the 6° case for FSTI = 0.5% and 2.0%. Unlike the previous
Mach numbers, there is a noticeable, albeit still small, difference between the two
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) FSTI = 0.5% (b) FSTI = 2.0%
Figure 4.25: Mach Contours for SC1095 Airfoil at Mach = 0.6, Re = 2×106, AOA
= 6 degrees
(a) Coefficient of Pressure CP (b) Skin Friction Cf
Figure 4.26: Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.6, AOA =
6 degrees
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
FSTI values. The shock is stronger for the 2.0% FSTI case. The pressure and skin
friction plots are shown in Fig. 4.29. Observe that the low FSTI values result in
a small reduction in the suction peak on the upper surface.
Mach 0.8
The lift and moment curves for the Mach 0.8 sweep are presented in Fig. 4.30.
There are large jumps in lift between -2°, 0°, and 2° AOA. On the other hand,
the lift for 2°, 4°, and 6° AOA are all bunched together. The 2° AOA case is
of particular interest, especially between 1.0% and 1.25%. At this point, the lift
increases before decreasing again, unlike any of the other angles of attack. The
moment also decreases before increasing in this range. The pressure and skin
friction plots are provided for this case in Fig. 4.31. The plots reveal why such
rapid change in the aerodynamic coefficients occurs between FSTI values of 1.0%
and 1.25%. The plots show that the shock has moved downstream. For the 1.0%
case, it is evident that the flow becomes turbulent after the shock. The drop in
skin friction on the upper surface of the 1.0% FSTI case indicates the shock has
occurred, and then the increase in skin friction marks the transition to turbulent
flow. The FSTI value of 1.25% leads to flow transition at the shock. The 1.25%
FSTI case undergoes transition before the shock. This results in very different skin
friction profiles for each case (and lift and moment curves), even though FSTI is
only varying by a small amount.
Figs. 4.32 and 4.33 show the Mach contours, pressure, and skin friction plots
for the 6° AOA cases. The Mach contours show that at an FSTI of 0.5%, there is a
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Lift coefficient
(b) Drag coefficient
Figure 4.27: Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2× 106
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(c) Moment coefficient
Figure 4.27: Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2 × 106
(cont.)
(a) FSTI = 0.5% (b) FSTI = 2.0%
Figure 4.28: Mach Contours for SC1095 Airfoil at Mach = 0.7, Re = 2×106, AOA
= 6 degrees
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Coefficient of Pressure CP (b) Skin Friction Cf
Figure 4.29: Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.7, AOA =
6 degrees
weaker shock that occurs closer to the leading edge, which results in a longer sep-
aration bubble. The skin friction plot shows that the 2.0% case transitions before
the shock, the 1.5% case transitions at the shock, and the low FSTI cases undergo
separation-induced transition, as evidenced by the separation bubble. Once again,
these differences are responsible for the change in moment in Fig. 4.30.
4.5 RC(4)-10 Airfoil
The RC(4)-10 airfoil was designed by NASA for the inboard section of he-
licopter rotors [49]. The airfoil was designed to optimize the maximum lift coef-
ficient in order to increase the lift load capacity of the retreating rotor blade. In
the present work, the Mach number was varied for this airfoil from 0.2 to 0.8. The
results of the parametric studies are presented in Figs. 4.34 - 4.45. Note that,
like the SC1095 airfoil, the results have been omitted for high angles of attack (i.e.
Application of Uncertainty Quantification to Turbulence Intensity 96
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Lift coefficient
(b) Moment coefficient
Figure 4.30: Parametric Sweep for SC1095 Airfoil at Mach = 0.8, Re = 2× 106
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Coefficient of Pressure CP (b) Skin Friction Cf
Figure 4.31: Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.8, AOA =
2 degrees
(a) FSTI = 0.5% (b) FSTI = 2.0%
Figure 4.32: Mach Contours for SC1095 Airfoil at Mach = 0.8, Re = 2×106, AOA
= 6 degrees
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Coefficient of Pressure CP (b) Coefficient of Skin Friction Cf
Figure 4.33: Pressure and Friction Plots for SC1095 Airfoil, Mach = 0.8, AOA =
6 degrees
over 7 degrees) at high Mach numbers. Also note that, similar to the NACA 0012,
certain cases did not converge to a single value, but rather resulted in repeating
oscillations (see Fig. 4.11). The values were averaged peak-to-peak for the last
three to five oscillations. The cases which were averaged are listed in Table 4.3.
Mach 0.2
Fig. 4.34 shows the results for Mach 0.2 sweep for the RC(4)-10. The drag
coefficient plot looks fairly similar to the SC1095 sweep at the same conditions
(Fig. 4.19), with the exception of the 2° AOA case, which has a sharp increase
in drag around 0.5% FSTI. The 2° and 4° AOA cases also exhibit an increase in
moment before decreasing, which was not present in the SC1095 case.
Fig. 4.35 shows the L/D and drag polar plots for this sweep at various FSTI
values. Comparing against the SC1095 results (Fig. 4.20), one notes that the
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
RC(4)-10 is able to achieve higher L/D values as a more modern airfoil. However,
besides this, the trends are fairly similar between the two airfoils, although the
drag bucket is much sharper for the RC(4)-10 at low FSTI values.
Fig. 4.36 presents the skin friction plots for the Mach 0.2 sweep. One thing to
note is the large change in transition location for the 2° AOA case. As the angle
of attack increases, the change in the transition location becomes smaller.
Mach 0.4
The results for the Mach 0.4 sweep are presented in Fig. 4.37. Note in Fig.
4.37, that there is a noticeable jump in drag between 0.6% and 0.75% FSTI for
the 2° AOA case. However, it is possible that this is due to the fact that the
values obtained for the FSTI values from 0.3% to 0.6% at this angle of attack
were obtained using peak-to-peak averaging. However, one thing to note when
examining the moment graph is that even the lower FSTI values for the 2° AOA
case (which were not averaged) are greater than the 4° AOA case. Also, the 4°
and 6° AOA cases show the trend of an increasing and then decreasing moment
coefficient, which was seen previously.
Fig. 4.38 presents the skin friction plots for the Mach 0.4 sweep. Once again, it
can be seen that as angle of attack increases, the change in transition location de-
creases. One interesting case is the 0.5% FSTI case at 2° AOA. This case behaves
very differently to the other cases, and starts undergoing transition before relami-
narizing on the bottom surface. On the other hand, the flow at the FSTI value of
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Drag coefficient
(b) Moment coefficient
Figure 4.34: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.2, Re = 2× 106
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
Mach Number Angle of Attack (degrees) FSTI (percent)
0.2 1 0.5
1 0.1-0.4
0.4
2 0.3-0.6
0.6 2 0.05
Table 4.3: RC(4)-10 - Peak-to-Peak Averaged Cases
(a) L/D vs Angle of Attack (b) Cd vs Cl
Figure 4.35: RC(4)-10 Airfoil at Mach = 0.2, Re = 2× 106
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) AOA = 2° (b) AOA = 4°
(c) AOA = 8° (d) AOA = 12°
Figure 4.36: Skin Friction Plots for RC(4)-10 - Mach = 0.2, Re = 2× 106
Application of Uncertainty Quantification to Turbulence Intensity 103
CHAPTER 4. PARAMETRIC SWEEP RESULTS
0.05% remains laminar for the entire bottom surface while the cases with higher
values of FSTI undergo transition as usual. This relaminarization phenomenon
was not observed for the same conditions for the SC1095 airfoil.
In order to investigate further, the pressure, skin friction, and intermittency
were compared, as shown in Fig. 4.39. A plot of the airfoils’ geometries is also
provided in the figure. The intermittency plot confirms that the bottom surface
of the RC(4)-10 relaminarizes in the midst of transition, whereas the SC1095
undergoes full transition. A comparison of the pressure plots reveals that the
RC(4)-10 does not have as favorable of a pressure gradient. When comparing
the two airfoil profiles, it can be seen that the RC(4)-10 is slightly flatter on the
bottom surface when compared to the SC1095. In addition, Fig. 4.1 shows that the
RC(4)-10 has extra camber at the leading edge compared to the SC10095. These
geometrical features lead to the relaminarization at these freestream conditions.
Mach 0.6
The results for the Mach 0.6 are presented in Fig. 4.40. The results appear
to be fairly similar to the previous results. Fluctuations are again present for the
moment coefficient at 4° AOA. However, there is no cross-over between 2° and 4°
AOA. Fig. 4.41 shows the skin friction plots at 2° and 4° AOA. Both the 0.05%
and 0.5% FSTI cases at 2° AOA relaminarize after beginning transition. At 4°,
the large change in moment can be explained by the small separation bubble near
the bottom of the trailing edge for the 0.5% and 0.05% cases, which is not present
at the higher FSTI cases.
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(a) Drag coefficient
(b) Moment coefficient
Figure 4.37: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.4, Re = 2× 106
Application of Uncertainty Quantification to Turbulence Intensity 105
CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) AOA = 2° (b) AOA = 4°
(c) AOA = 6°
Figure 4.38: Skin Friction Plots for RC(4)-10 - Mach = 0.4
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(a) Airfoil Geometry (b) Intermittency
(c) Coefficient of Pressure CP (d) Coefficient of Skin Friction Cf
Figure 4.39: Comparison of Plots for SC1095 Airfoil and RC(4)-10, Mach = 0.4,
AOA = 2 degrees, FSTI = 0.5%
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Drag coefficient
(b) Moment coefficient
Figure 4.40: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.6, Re = 2× 106
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(a) AOA = 2° (b) AOA = 4°
Figure 4.41: Skin Friction Plots for RC(4)-10 - Mach = 0.6
Mach 0.7
Fig. 4.42 shows the results of the Mach 0.7 sweep. It is fairly similar to
the Mach 0.6 sweep. However, there is one prominent difference. At 6° AOA, the
moment is significantly lower, dropping below the 0° AOA case at some points.
Fig. 4.43 shows the skin friction plot for the 6° AOA case. One notices that after
the shock, the friction remains negative, indicating that the flow never reattaches
after the shock. The Mach contour of the 0.5% case, as shown in Fig. 4.44 confirms
this.
Mach 0.8
Fig. 4.45 presents the results of the Mach 0.8 sweep. One thing to note is
that the -4° AOA case has a very large moment coefficient compared to the other
angles of attack. Also, the 6° AOA case has similar behavior to the Mach 0.7 case.
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
(a) Drag coefficient
(b) Moment coefficient
Figure 4.42: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.7, Re = 2× 106
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Figure 4.43: Skin Friction Plot for RC(4)-10 - Mach = 0.7 - AOA = 6 degrees
Figure 4.44: Mach Plot for RC(4)-10 - Mach 0.7 - AOA = 6° - FSTI = 0.5%
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CHAPTER 4. PARAMETRIC SWEEP RESULTS
The pressure, skin friction, and Mach contours are shown for both angles of attack
in Fig. 4.46.
The skin friction plot for the -4° AOA case shows that there is separation
induced transition on the upper surface for the lower FSTI values. This results
in a small increase in suction pressure at the separation bubble, whereas the high
FSTI values undergo natural transition and therefore do not have this increase
in pressure. The -4° AOA case also shows a strong shock on the bottom surface
followed by a region with a thin layer of separation.
For the 6° AOA case, the skin friction plot shows that the flow remains laminar
on the bottom surface, even after the shock. This behavior differs greatly from
the SC1095 at the same conditions (see Fig. 4.33). Once again, this is due to the
difference in the geometries of the airfoils, which were noted earlier.
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(a) Drag coefficient
(b) Moment coefficient
Figure 4.45: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.8, Re = 2× 106
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(a) CP - AOA = -4° (b) CP - AOA = 6°
(c) Cf - AOA = -4° (d) Cf - AOA = 6°
(e) Mach Plot - AOA = -4° - FSTI = 0.5% (f) Mach Plot - AOA = 6° - FSTI = 0.5%
Figure 4.46: Plots for RC(4)-10 Airfoil, Mach = 0.8, Re = 2× 106
Application of Uncertainty Quantification to Turbulence Intensity 114
Chapter 5
Uncertainty Quantification
Results
This chapter presents the results of the UQ analysis. In order to obtain these 
results, a Monte Carlo approach was used, as described in section 3.3.1. Results 
are shown for a mean FSTI of 1% with a standard deviation of 0.3333%, unless 
stated otherwise. A standard deviation of 0.3333 with a mean of 1.0% means that 
99.73% of the input values are expected to be within 0% and 2.0%. The results 
consist of the expected means and standard deviations for each of the four airfoils 
for various Mach and Reynolds numbers, as specified in Table 3.1. The standard 
deviations can be used to determine the sensitivity of the coefficients of lift, drag, 
and momentum as various parameters are varied. These parameters include FSTI, 
angle of attack, Mach number, and Reynolds number. An increase in standard 
deviation indicates that the quantity of interest is more sensitive to uncertainty
115
CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
in FSTI. Trends in standard deviation can simply be explained by examining how
the aerodynamic coefficients change as FSTI is varied at each AOA. However, this
section explains these trends further by correlating the trends in sensitivity of the
quantities of interest to the physics of the airflow.
5.1 S809 Airfoil Results
In order to demonstrate the importance of UQ analysis, Monte Carlo simu-
lations were run for the S809 airfoil at an AOA of 2° with an input FSTI standard
deviation of 0.3333%. Two mean input values were used for FSTI: 1.5% and 4.0%.
Fig. 5.1 shows the distributions of the input FSTI values as well as the output
moment coefficient. As illustrated by the figure, the mean input of 4.0% results in
a normal distribution for the moment coefficient. However, when the mean input
is changed to 1.5%, the output is not normally distributed. Instead, there is a
second peak in the distribution near a pitching moment coefficient value of -0.059.
This shows that even with a normally distributed input, the outputs could vary
considerably. This also demonstrates why a large number of runs are required
for the Monte Carlo simulation to reach statistical convergence. By applying UQ
analysis using a Monte Carlo simulation, the true mean and standard deviation
can be determined.
Monte Carlo simulations were run for the S809 airfoil for AOA from -4° to 12°,
using different mean input FSTI values and different input standard deviations.
Fig. 5.2 shows the results for a mean FSTI input of 1.0% and an input standard
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(a) Mean Input FSTI = 4.0%
(b) Mean Input FSTI = 1.5%
Figure 5.1: Distributions of Inputs and Outputs for S809 Airfoil at AOA = 2°,
Mach = 0.2, Re = 2× 106, Input σ = 0.3333%
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
deviation of 0.3333%. The error bars on the figure represents the output standard
deviation. The values obtained for the discrete 1.0% FSTI value (i.e. without any
input uncertainty) are shown for comparison. As shown in the figure, the discrete
values are close in value to the values obtained with input uncertainty in FSTI,
especially for lift and moment. However, there is a noticeable difference between
the discrete values and the values obtained via Monte Carlo for the coefficient of
drag, especially at low angles of attack. Note that the standard deviations for
lift are also very small. On the other hand, the standard deviations for drag and
moment are much larger with respect to the magnitude of the drag and moment
coefficient, respectively. Variation in standard deviation values can be observed
clearly as AOA increases.
Plots in Fig. 5.3 shows the results of varying the mean of the input FSTI value
between 1.0% to 2.0%. This sweep was performed to narrow down the region of
interest. In order to better visualize the trends present, the difference between
the expected mean of the aerodynamic coefficients with uncertainty in the input
FSTI and the discrete value obtained if no uncertainty was present is plotted
against AOA in one plot, and the standard deviations are shown in a separate
plot. Note that standard deviation is essentially a measure of sensitivity to the
input uncertainty in FSTI. Proceeding forward, sensitivity is used interchangeably
with the output standard deviation. The plots of the aerodynamic coefficients
show that the trends vary as the mean input is increased. As was shown in the
parameter sweeps for the S809 airfoil (see Fig. 4.2), the aerodynamic coefficients
do not vary linearly with FSTI. Therefore changing the mean input FSTI value
results in variation in the difference between the discrete values and mean Monte
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Lift coefficient (b) Drag coefficient
(c) Moment coefficient
Figure 5.2: Monte Carlo Results for S809 Airfoil for Mean Input FSTI 1.0% and
Standard Deviation 0.3333%
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Lift coefficient
(b) Drag coefficient
(c) Moment coefficient
Figure 5.3: UQ Results for S809 Airfoil at Mach = 0.2, Re = 2 × 106, Input
σ = 0.3333%
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Lift coefficient
(b) Drag coefficient
(c) Moment coefficient
Figure 5.4: UQ Results for S809 Airfoil at Mach = 0.2, Re = 2× 106, Mean FSTI
= 1.0%
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
Carlo outputs.
As can be seen from the standard deviation plots, the sensitivity of the quan-
tities of interest are lower for a mean FSTI of 2.0%. The trends from the S809
parametric sweep support these results (see Fig. 4.2). As FSTI increases above
2.0%, the trends become relatively linear and do not vary as much. Therefore, the
sensitivity of the quantities of interest to uncertainty in FSTI decreases for input
FSTI values of 2.0% or greater. Fig. 5.3 shows that at low angles of attack (up
to 3°), there is greater sensitivity for a mean FSTI of 1.5%, while at moderate to
high angles of attack, sensitivity is greater for the mean input 1.0% case.
Fig. 5.4 shows the Monte Carlo results for different input standard deviations.
As expected, decreasing the input standard deviation decreases the sensitivity
to the aerodynamic coefficients and the difference between the discrete and mean
coefficients. An input standard deviation of zero means that there is no uncertainty
present. As the input standard deviations approach zero, the expected means
approach the discrete values and the standard deviations approach zero. However,
it is interesting to note that at high angles of attack (9° or greater), the curves
approach a single value. This indicates that at angles of attack of 9° or greater,
an increase in the amount of uncertainty in FSTI does not impact the final results
as much as it would at angles of attack below 9°.
When comparing the difference in coefficients to the standard deviation plots,
note that the trends and peaks do not necessarily correlate. The two types of
graphs provide different information. While the difference between the discrete
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
Figure 5.5: Drag Polar for S809 Airfoil at Mach = 0.2, Re = 2× 106
and mean values may be important for practical application for a specific case,
it does not provide any information about the sensitivity of each coefficient to
the input uncertainty, which is the focus of the present work. The plots of the
difference between the discrete and mean coefficients is therefore omitted when
discussing the other three airfoils.
The standard deviations graphs show that all three coefficients have peaks in
sensitivity at an angle of attack of 4° and after this point, the standard deviations
decrease monotonically. This peak can be attributed to the “drag bucket” for
the S809 airfoil (see the drag polar in Fig. 5.5) [50]. At angles of attack below
4°, the flow remains laminar over a significant portion of the leading edge of the
airfoil. Since the pressure drag is low at these low angles of attack, the overall drag
remains low, creating the laminar drag bucket. For these low angles of attack, small
changes in FSTI result in small changes in the drag. At angles of attack above 6°,
the pressure drag dominates the drag term and therefore changes in FSTI have
relatively small impacts on the quantities of interest. However, for moderate angles
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
of attack (i.e. near 4°-6°), the viscous drag, which is affected by FSTI, is the main
contributor to the total drag. Small changes in FSTI therefore result in greater
changes in the aerodynamic coefficients for these moderate angles of attack.
5.2 NACA 0012 Airfoil Results: Reynolds Num-
ber Sweep
Fig. 5.6 presents the standard deviations of the aerodynamic coefficients for
the NACA 0012 airfoil. The Reynolds number has been varied from 500,000 to 4
million. There are a number of conclusions regarding the sensitivity to FSTI that
can be reached by examining the trends in the standard deviation graphs. Observe
that as the Reynolds number increases, the sensitivity curves change shape. This
indicates that there is not a simple relationship between Reynolds number and
sensitivity to uncertainty in FSTI. Detailed analysis for each Reynolds number is
presented in the following sections.
Re = 500,000
Referring back to Fig. 5.6, a trend can be observed in the standard deviations
of lift and moment coefficients for a Reynolds number of 500,000. There is a region
of sharp increase in sensitivity between 2° and 5° AOA followed by a region of
sharp decrease. This is attributed to the double separation bubble phenomenon
mentioned in sec. 4.3. Fig. 5.7 shows skin friction plots for FSTI values of 0.3%,
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Lift coefficient (b) Drag coefficient
(c) Moment coefficient
Figure 5.6: Standard Deviations for NACA 0012 Airfoil at Mach = 0.2, Mean
FSTI = 1%, Input σ = 0.3333%
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) FSTI = 0.3%
(b) FSTI = 0.4%
(c) FSTI = 0.875% (d) FSTI = 1.0%
Figure 5.7: Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re = 500,000
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) AOA = 8° (b) AOA = 12°
Figure 5.8: Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re = 500,000
- AOA 8° versus 12°
0.4%, 0.875%, and 1.0%. Zoomed in portions of the 0.3% and 0.4% are provided
to see the separation bubbles more clearly. The 0.3% FSTI case has the double
separation bubble on the upper surface from 3° to 5° AOA. However, if the FSTI
is slightly increased to 0.4%, then the double separation bubble appears for 4° and
5° AOA. At 3° AOA, there is only a single separation bubble. Likewise, 0.875%
and 1.0% FSTI can be compared. At 5° AOA, there is a double separation bubble
for 0.875% FSTI case but not for 1.0% FSTI. As mentioned in sec. 4.3, the double
separation bubble increases the effective camber of the airfoil, thereby increasing
lift and moment. Since these small changes in FSTI at AOA between 2° and
5° can result in the appearance/disappearance of the double separation bubble,
the coefficients of lift and moment are more sensitive to uncertainty in FSTI in
this AOA range. As FSTI approaches 1.0% (which is the mean input FSTI), the
separation bubble appears/disappears near 5° AOA, which is why there is a peak
at sensitivity at 5° AOA. The sensitivity decreases at 6° AOA because the double
separation bubble is present for all values of FSTI up to 1.5% (not shown in figure).
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
Fig. 5.8 presents the skin friction plots at 8° and 12° AOA, focusing on the
double separation bubble region. This figure explains why the sensitivity of the
lift coefficient increases from 8° to 12° AOA. For FSTI values of 0.75% to 1.5% at
8° AOA, the values of the skin friction coefficient at the region of the separation
bubble are close. However, at 12° AOA, there is a significant difference between
FSTI values of 1.25% and 1.0%. This also explains the peak in sensitivity for the
drag coefficient as well.
Re = 1 × 106
When examining the sensitivity curves for a Reynolds number of 1 million
in Fig. 5.6, the most distinctive feature is the sharp peak in sensitivity at 4° AOA
for lift and moment. Fig. 5.9 shows the skin friction plots from 3° to 5° AOA. At
3° AOA, the 1.0% and 1.25% FSTI cases undergo natural transition on the lower
surface. This differs from the 0.75% FSTI case, which has a small separation
bubble on the lower surface near the trailing edge. Compare this against the plot
for 5° AOA. Here, there is a separation bubble near the trailing edge of the lower
surface for both the 0.75% and 1.0% FSTI cases. However, at 4°, the 1.0% FSTI
case does not behave like either of the other two FSTI cases at the lower surface.
This unique behavior is responsible for the spike in sensitivity for lift and moment
at 4° AOA. A small separation bubble at the lower surface near the trailing edge
increases the negative effective camber of the airfoil, resulting in significant changes
in lift and moment.
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) AOA = 3° (b) AOA = 4°
(c) AOA = 5°
Figure 5.9: Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re = 1×106
- AOA 3° - 5°
Re = 2 × 106
In Fig. 5.6, the standard deviation of lift and moment at a Reynolds number of
2 million increase smoothly to a peak at 8° before decreasing gradually. Fig. 5.10
shows the skin friction plots for 4°, 8°, and 12° AOA respectively. When comparing
the transition location on the lower surface, note that the transition location has
the largest change for the 8° AOA case, which is why the sensitivity peaks at this
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) AOA = 4° (b) AOA = 8°
(c) AOA = 12°
Figure 5.10: Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re =
2× 106
point for lift and moment.
When examining the sensitivity of the drag coefficient in Fig. 5.6, one notes
that the sensitivity generally decreases as AOA increases. However, there is a sharp
peak in sensitivity at 11° AOA. Upon examining the pressure and skin friction plots
from 10° to 12° AOA (see Fig. 5.11), the cause of this peak is not readily apparent.
Further work is needed to explain the reason for this peak.
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(a) AOA = 10°
(b) AOA = 11°
(c) AOA = 12°
Figure 5.11: Pressure and Skin Friction Plots for NACA 0012 Airfoil at Mach =
0.2, Re = 2× 106
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
Re = 4 × 106
Going back to Fig. 5.6, one can observe that the standard deviation for lift
at a Reynolds number of 4 million is fairly smooth, with a peak near 10° AOA.
However, the sensitivity remains almost constant above 10°. Also note that the
sensitivity of lift and moment at a Reynolds number of 4 million is lower than
the sensitivity at a Reynolds number of 2 million. It is also observed that the
sensitivity peaks at 6° AOA for both drag and moment. Fig. 5.12 shows the skin
friction plots at 5°, 6°, 7°, 10°, and 12° AOA. However, the friction plots do not
provide a clear explanation for the trends in aerodynamic coefficients; more work
is required to determine the actual cause.
5.3 SC1095 Airfoil Results: Mach Number Sweep
Fig. 5.13 presents the standard deviations of the aerodynamic coefficients. The
standard deviations correlate to the sensitivity of each coefficient to the uncertainty
in FSTI. It can be observed from the figure that standard deviation curves for Mach
0.2, 0.4, and 0.6 follow similar trends for all three aerodynamic coefficients, with
the exception of AOA between 3° and 6°. The similarities at these Mach numbers
is due to the incompressibility of air at subsonic speeds. At Mach 0.7 and 0.8,
transonic flow begins to appear, and compressibility effects change the trends in
sensitivity.
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) AOA = 5° (b) AOA = 6°
(c) AOA = 7° (d) AOA = 10°
(e) AOA = 12°
Figure 5.12: Skin Friction Plots for NACA 0012 Airfoil at Mach = 0.2, Re =
4× 106
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Lift coefficient (b) Drag coefficient
(c) Moment coefficient
Figure 5.13: UQ Results for SC1095 Airfoil at Re = 2 × 106, Mean FSTI = 1%,
Input σ = 0.3333%
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
Mach 0.2, 0.4, and 0.6
For Mach 0.2 and 0.4, the flow remains subsonic and the aerodynamic co-
efficients follow similar trends. At Mach 0.6, the flow is subsonic at AOAs below
3°. However, there are some differences between Mach 0.2, 0.4, and 0.6 between
3° and 6°. In this region, lift and moment are significantly more sensitive to FSTI
uncertainty for Mach 0.2 compared to Mach 0.4 and 0.6. Also, the sensitivity of
drag decreases over AOA 3° to 6° for Mach 0.2 and 0.4, but increases for Mach
0.6. Referring to Fig. 5.14, which depicts the Mach contours for AOA 6° at Mach
0.4 and 0.6, one can see that at Mach 0.4, there is no observable differences in the
Mach contour plots between FSTI 0.5% and 2.0%. On the other hand, for Mach
0.6, the shock is observed to be stronger at FSTI 2.0% compared to 0.5%. The
increase in drag sensitivity for Mach 0.6 between 3° and 6° can be attributed to
this increase in shock strength, which would increase the drag over the airfoil.
Mach 0.7 and 0.8
Mach 0.7 and 0.8 differ in behavior compared to Mach 0.2 to 0.6. At almost
all AOAs, the sensitivity of lift and drag are greater for Mach 0.7 and 0.8 than for
Mach 0.2 to 0.6. For Mach 0.7, the sensitivity for lift and drag increases sharply
as AOA increases between 0° and 4° AOA before decreasing sharply. For Mach
0.8, the sharp increase in sensitivity occurs at 2° AOA.
Fig. 5.15 shows the skin friction plots for Mach 0.7 at 2°, 4°, and 6°; and for
Mach 0.8 at 0°, 2°, and 6° AOA. For Mach 0.7 at an AOA of 2°, all the cases
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Mach = 0.4, FSTI = 0.5% (b) Mach = 0.4, FSTI = 2.0%
(c) Mach = 0.6, FSTI = 0.5% (d) Mach = 0.6, FSTI = 2.0%
Figure 5.14: Mach Contours for SC1095 Airfoil at 6° AOA, Re = 2× 106
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Mach 0.7 - AOA = 2° (b) Mach 0.8 - AOA = 0°
(c) Mach 0.7 - AOA = 4° (d) Mach 0.8 - AOA = 2°
(e) Mach 0.7 - AOA = 6° (f) Mach 0.8 - AOA = 6°
Figure 5.15: Skin Friction Plots for SC1095 Airfoil, Mach = 0.7 - 0.8, Re = 2×106
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
undergo transition after the shock. For Mach 0.8 at an AOA of 0°, all of the cases
undergo transition after the shock except the 1.5% FSTI case, which transitions
completely before the shock. For Mach 0.7 at an AOA of 4° and Mach 0.8 at
an AOA of 2°, there is one FSTI case which is mid-way through transition at
the shock. For Mach 0.7, this is the 1.5% FSTI case; for Mach 0.8, this is the
1.12% case. Note that for both cases where transition is occurring at the shock,
the transition location differs from all of the other FSTI cases. Since these FSTI
values are relatively close to 1.0%, this results in an increase in sensitivity for lift
and drag for a mean input of 1.0%. As the AOA increases, the sensitivity decreases
once again, as all of the FSTI cases exhibit the same behavior, as shown in the
Mach 0.7 and 0.8 plots at an AOA of 6°.
5.4 RC(4)-10 Airfoil Results: Mach Number Sweep
Fig. 5.16 presents the standard deviations of the aerodynamic coefficients for
the RC(4)-10. The Mach number was varied from 0.2 to 0.8. Both the Mach 0.2
and 0.4 standard deviation curves follow similar trends. Similar to the SC1095
cases, at these Mach numbers, the flow remains subsonic. However, the sensitivity
results for Mach 0.6 to Mach 0.8 become more erratic, once compressibility effects
come into effect. Compare the sensitivity curves for the RC(4)-10 against the
SC1095 sensitivity plots in Fig. 5.13. While the trends are similar for Mach 0.2
and 0.4, the sensitivities are very different for Mach 0.7 and 0.8.
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Lift coefficient (b) Drag coefficient
(c) Moment coefficient
Figure 5.16: UQ Results for RC(4)-10 Airfoil at Re = 2× 106, Mean FSTI = 1%,
Input σ = 0.3333%
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
Mach 0.2 and 0.4
The sensitivities for Mach 0.2 and 0.4 are very close to each other for all
three coefficients, with a few exceptions:
ˆ Lift and moment coefficients between 5° and 8° AOA
ˆ Drag coefficient at 1° AOA
Fig. 5.17 presents skin friction plots for 1°, 6°, and 7° AOA for both Mach 0.2 and
0.4. Comparing the 1° AOA plots, one can see that at Mach 0.2 for an FSTI value
of 0.5%, the flow relaminarizes on the lower surface but does not relaminarize at
the same conditions for Mach 0.4. Since laminar flow results in significantly less
drag than turbulent flow, the drag coefficient is much more sensitive to uncertainty
in FSTI at Mach 0.2 than at 0.4. However, when examining the skin friction plots
for AOA 6° and 7°, the graphs do not differ significantly between Mach 0.2 and
0.4. Additional investigation is required to determine why there is a difference in
the sensitivity of lift and moment between Mach 0.2 and 0.4 for AOA 5° to 8°.
Mach 0.6 to 0.8
When examining Fig. 5.16, one can observe that the Mach 0.6 to 0.8 sensi-
tivity curves are not as smooth as the Mach 0.2 and 0.4 curves; rather, they are
characterized by jumps and fluctuations in sensitivity as AOA varies. Seven points
of interest have been identified to explore further:
1. Mach 0.6: Peak in sensitivity of the moment coefficient at AOA of 2°.
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Mach 0.2 - AOA = 1° (b) Mach 0.4 - AOA = 1°
(c) Mach 0.2 - AOA = 6° (d) Mach 0.4 - AOA = 6°
(e) Mach 0.2 - AOA = 7° (f) Mach 0.4 - AOA = 7°
Figure 5.17: Skin Friction Plots for RC(4)-10 Airfoil, Mach = 0.2 - 0.4, Re =
2× 106
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CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
2. Mach 0.6: Minimum sensitivity of the moment coefficient at AOA of 3°.
3. Mach 0.6: Peak in sensitivity of the moment coefficient at AOA of 4°.
4. Mach 0.7: Peak in sensitivity of the moment coefficient at AOA of 4°.
5. Mach 0.7: Peaks in sensitivity of the lift and drag coefficient at AOA of 5°.
6. Mach 0.8: Peak in sensitivity of the moment coefficient at AOA of 1°.
7. Mach 0.8: Peak in sensitivity of the drag coefficient at AOA of 5°.
Fig. 5.18 provides the skin friction plots for the three Mach 0.6 cases mentioned
above, while Fig. 5.19 presents the skin friction plots for the Mach 0.7 and 0.8
cases. For the Mach 0.6 cases, at an AOA of 2° and an FSTI of 0.5%, the flow
relaminarizes. This results in a peak in sensitivity for the moment coefficient.
At AOA 3° and 4°, this relaminarization is not present. However, at AOA 4°, a
separation bubble forms on the lower surface near the trailing edge for the FSTI
0.5% and 0.75% cases. At AOA 3°, the separation bubble only forms on the 0.5%.
Since the mean input is 1.0%, the change in the transition mode between 0.75%
and 1.0% FSTI results in a peak in the moment coefficient sensitivity.
Fig. 5.19 shows the skin friction plots for the four Mach 0.7 and 0.8 cases of
interest. For Mach 0.7 at an AOA of 4°, both the upper and lower surfaces show
different methods of transition. On the upper surface, at FSTI values of 0.5% to
1.0%, the flow transitions after the shock. At 1.25% FSTI, the flow transitions at
the shock, while at 1.5% FSTI, the flow transitions before the shock. On the lower
surface, at 0.5% FSTI, there is a small separation bubble at the trailing edge. For
the FSTI values greater than 0.5%, the flow undergoes natural transition. The
small changes in FSTI result in completely different methods of transition on both
the upper and lower surface, resulting in a peak in sensitivity for the moment
Application of Uncertainty Quantification to Turbulence Intensity 142
CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
coefficient for Mach 0.7.
The Mach 0.7 case at an AOA of 5° also features different methods of transition
as FSTI varies. On the upper surface, the flow transitions at the shock at an FSTI
of 1.5% and transitions after the shock for all the other FSTI values. However,
unlike the 4° AOA case, there is no separation bubble on the lower surface. On the
lower surface, the 0.5% and 0.75% cases remain laminar, while the 1.0%, 1.25%,
and 1.5% cases undergo natural transition. The moment sensitivity decreases due
to the absence of the separation bubble on the lower surface, but the changes in
the transition location on the lower surface result in a peak in sensitivity for both
lift and drag.
For the Mach 0.8 case, at an AOA of 1°, the flow on the upper surface transitions
before the shock for FSTI values of 1.25% and 1.5%. For FSTI values of 0.5%,
0.75%, and 1.0%, the flow transitions after the shock. The change in transition
method results in a peak in moment sensitivity. For the Mach 0.8 case at an AOA
of 5°, the relaminarization of the 0.5% FSTI case is responsible for the peak in
drag sensitivity.
5.5 Summary
This chapter presented the results of the Monte Carlo simulation for each
of the S809, NACA 0012, SC1095, and RC(4)-10 airfoils. The importance of UQ
analysis was demonstrated by showing how a normally distributed FSTI input
Application of Uncertainty Quantification to Turbulence Intensity 143
CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) AOA = 2° (b) AOA = 3°
(c) AOA = 4°
Figure 5.18: Skin Friction Plots for RC(4)-10 Airfoil, Mach = 0.6, Re = 2× 106
Application of Uncertainty Quantification to Turbulence Intensity 144
CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
(a) Mach 0.7 - AOA = 4° (b) Mach 0.7 - AOA = 5°
(c) Mach 0.8 - AOA = 1° (d) Mach 0.8 - AOA = 5°
Figure 5.19: Skin Friction Plots for RC(4)-10 Airfoil, Mach = 0.7 - 0.8, Re =
2× 106
Application of Uncertainty Quantification to Turbulence Intensity 145
CHAPTER 5. UNCERTAINTY QUANTIFICATION RESULTS
could result in a non-Gaussian output, using a case from the S809 airfoil as an
example. The mean input FSTI and standard deviations were varied for the S809
airfoil to show how this would affect the difference between discrete and mean
outputs and the output standard deviations. For the NACA 0012, the Reynolds
number was varied. The standard deviations of the aerodynamic coefficients fea-
tured spikes in sensitivity at certain AOA. The reasoning behind these peaks was
explained for some of these cases, while the cause for some of the peaks require
additional research. The results for the SC1095 and RC(4)-10 were also presented
for Mach 0.2 to 0.8. Both airfoils showed similar sensitivity trends at Mach 0.2
and 0.4, when the flow is transonic. However, they exhibited different behavior at
Mach 0.6 to 0.8. The physics behind some of the more interesting features of the
graphs were explained.
Application of Uncertainty Quantification to Turbulence Intensity 146
Chapter 6
Conclusions
6.1 Summary
This thesis studied the effects of uncertainty on the aerodynamic coefficients 
of four airfoils. The study focused on the sensitivities of the aerodynamic coef-
ficients to uncertainty in FSTI and examined how varying the mean FSTI, the 
input standard deviation, the Reynolds number, and the Mach number affected 
the sensitivity through UQ analysis.
The first chapter highlighted the importance of UQ and the motivation behind 
this work, before discussing boundary layer transition and providing an overview of 
UQ. The chapter went over the differences between laminar, turbulent, and tran-
sitional flow and described the physical mechanisms of transition and the different
147
CHAPTER 6. CONCLUSIONS
types of transition. The chapter then provided a brief overview of a few transition
models that are currently used by CFD solvers. The chapter then described the
field of UQ as well as different UQ techniques that were used for the present work.
The second chapter provided an overview of the CFD analysis used in this
work. This included a description of the governing equations that are solved by
the OVERTURNS flow solver and how these equations are non-dimensionalized.
The RANS equations were then derived from the non-dimensionalized governing
equations. A brief description of grid generation is provided. The chapter then
reviewed how OVERTURNS uses numerical algorithms to solve the governing
equations, including the incorporation of the turbulence and transition models.
The third chapter went through the UQ methodology used in this work. The
chapter described how parametric sweeps were used to generate a surrogate sur-
face to use in place of CFD calculations. The chapter then explained why the
Monte Carlo method was chosen over stochastic collocation. Finally, the chapter
described how the Monte Carlo method was implemented and provided validation
results for the surrogate surface and the overall UQ analysis.
The fourth chapter presented the results of the parametric sweeps for the S809,
NACA 0012, SC1095, and RC(4)-10. Results of interest were identified and the un-
derlying physics were examined in order to understand the results. This uncovered
cases with multiple laminar separation bubbles and cases with relaminarization.
At higher Mach numbers, some cases with shock induced separation (with transi-
tion before or after the shock) were observed.
Application of Uncertainty Quantification to Turbulence Intensity 148
CHAPTER 6. CONCLUSIONS
Finally, the fifth chapter went over the results of the UQ analysis. The Monte
Carlo method was used to determine the expected means and standard deviations
of the aerodynamic coefficients for each airfoil. The standard deviations provided
a measure of the sensitivity of each aerodynamic coefficient to the uncertainty in
FSTI. Cases were selected to be explored further to attempt to understand why
the coefficients were more or less sensitive for a particular Mach/Reynolds number
and AOA.
6.2 Key Observations
This section lists the key observations gleaned from the present work. Both
general observations and specific results for each airfoil have been highlighted.
6.2.1 General Observations
ˆ Although the input distribution of FSTI values was normally distibuted, the
resulting output distributions were not always Gaussian.
ˆ Mean values of lift and pitching moment were close to the values obtained
with discrete FSTI inputs (i.e. without uncertainty). However, the mean
drag values differed from the drag values obtained with discrete FSTI inputs
when inside the drag bucket AOA range.
ˆ Even when the mean outputs were close to the values obtained with discrete
Application of Uncertainty Quantification to Turbulence Intensity 149
CHAPTER 6. CONCLUSIONS
FSTI, the standard deviations of the aerodynamic coefficients were large
enough to be statistically relevant.
6.2.2 S809
ˆ The UQ analysis performed on the S809 airfoil demonstrated that the sensi-
tivity trends of the aerodynamic coefficients is dependent on the mean FSTI
value. Although this was expected, the S809 cases provided confirmation
as well as showed how much the sensitivity could change. For example, the
standard deviation for the coefficient of lift was reduced by over a factor of
4 when the input mean FSTI value was changed from 1.0% to 2.0% for an
AOA of 4°.
ˆ The S809 case was also used to show how reducing the input standard devia-
tion affects the sensitivity of the aerodynamic coefficients. While the trends
remained the same, the standard deviations of the aerodynamic coefficients
were reduced, which was expected. However, as the angle of attack increases,
the difference in standard deviations decrease, indicating that varying FSTI
at these high angles of attack does not impact the results as much.
ˆ All three aerodynamic coefficients have peaks in standard deviation at an
AOA of 4°. This is due to the drag bucket phenomenon. Near 4° AOA, the
viscous drag increases rapidly in response to small increases in FSTI.
Application of Uncertainty Quantification to Turbulence Intensity 150
CHAPTER 6. CONCLUSIONS
6.2.3 NACA 0012
ˆ At Reynolds numbers of 500,000 and 1 million, a double separation bubble
forms at certain AOA and FSTI values. This double separation bubble is
responsible for fluctuations in sensitivity for the aerodynamic coefficients.
For a given AOA, the double separation bubble will appear for a certain
threshold of FSTI values. If the FSTI is increased/decreased slightly at the
edges of this threshold, the double separation bubble will form or disappear,
resulting in a relatively large change in drag. When the double separation
bubble occurs near the leading edge, it also increases the effective camber,
leading to changes in lift and moment as well. Therefore as threshold val-
ues approach the mean input FSTI value, the sensitivity of the coefficients
increase.
ˆ At Reynolds numbers of 2 million and 4 million, as AOA increases, the
transition location approaches the leading edge of the airfoil. At high AOA,
the flow transitions close to or at the leading edge and is therefore able to
remain attached to the airfoil after transitions. This leads to increases in
drag as FSTI increases. At Reynolds numbers of 500,000 and 1 million, at
high angles of attack, the drag remains relatively constant as FSTI increases,
since the flow separates after transition.
ˆ Change in transition location affects the sensitivity of the aerodynamic coef-
ficients. The more that the transition locations for a set increase/decrease in
FSTI, the more that the sensitivity of the aerodynamic coefficients change.
Application of Uncertainty Quantification to Turbulence Intensity 151
CHAPTER 6. CONCLUSIONS
ˆ At Reynolds numbers of 2 million and 4 million, sharp peaks occur at certain
AOA. The causes for these peaks were not able to be determined; further
work is required to provide an explanation.
6.2.4 SC1095 and RC(4)-10
ˆ The RC(4)-10 exhibits an interesting phenomenon at AOA of 2° for Mach
0.4 and Mach 0.6. At an FSTI value of 0.5% (and 0.05% at Mach 0.6), the
flow on the lower surface transitions, but then relaminarizes. This does not
occur for the SC1095. This relaminarization is attributed to the increased
camber of the leading edge and the flatter lower surface of the RC(4)-10.
ˆ For Mach numbers 0.2 and 0.4, the sensitivity trends are very similar. This is
due to the fact that at these Mach numbers and AOA, the flow is subsonic and
incompressible. However, for Mach 0.6 to 0.8, the flow becomes transonic
and is no longer incompressible. This results in large fluctuations in the
sensitivity of the aerodynamic coefficients to uncertainty in FSTI. At these
Mach numbers, the trends differ between the SC1095 and the RC(4)-10.
ˆ Spikes in sensitivity occur when small changes in FSTI near the mean input
FSTI of 1.0% result in the transition location moving before, after, or at
the shock location. Each scenario results in a different skin friction profile,
which affects all three aerodynamic coefficients.
Application of Uncertainty Quantification to Turbulence Intensity 152
CHAPTER 6. CONCLUSIONS
6.3 Contributions
Currently, UQ is not widely implemented in CFD results. Most CFD papers
do not address the role of uncertainty in the published results. However, incor-
porating UQ analysis into CFD results can provide additional information to be
used for design and reliability analysis. In addition, it will make CFD results more
accurate by taking into account uncertainties that exist in the physical world in
CFD calculations.
The present work takes the first step into incorporating UQ analysis for CFD
results for uncertainty in FSTI specifically and lays the foundation for establishing
a UQ framework for flow transition studies in general. Specifically, the sensitivity
analysis provided in this work can be used in conjunction with CFD simulations
in order to obtain more accurate results. The sensitivity analysis can also be
used in rotor design, allowing designers to select airfoils which are less sensitive
to uncertainty in FSTI at the expected operating conditions of the rotor. The
present work also provides the UQ framework so that a similar approach can be
taken for different airfoils as well as different input quantities besides FSTI.
Finally, this work required extensive parametric studies, varying FSTI, AOA,
Reynolds number, and Mach number. The result was a treasure trove of data,
from which several key points were highlighted. There has been little to no work
previously that has examined small changes in FSTI values to this extent. While
the present work has highlighted a number of key observations, there is always
Application of Uncertainty Quantification to Turbulence Intensity 153
CHAPTER 6. CONCLUSIONS
more that can be gleaned from all of the data.
6.4 Recommendations for Future Work
The work presented in this thesis represents only a small subsection of the
work possible for applying UQ to CFD results. There are numerous ways to build
upon this research and expand further. Some of the possible ways to do so are
listed below:
ˆ The sensitivity analysis unearthed a number of features that remain unex-
plained. Further work is required to determine why at certain conditions
there are peaks in the sensitivity of the aerodynamic coefficients to uncer-
tainty in FSTI.
ˆ The present work assumed that the uncertainty in FSTI was Gaussian. How-
ever, the methodology presented here can easily be applied to any other
probability distribution. Future work could include experimental studies to
make the probability distribution of FSTI more representative of the physical
world. This would improve the accuracy of the UQ analysis.
ˆ While examining the results of the parametric studies, a number of key
findings were highlighted. However, there were many additional findings that
were left unaddressed. A thorough review of the parametric study results
would yield additional findings.
Application of Uncertainty Quantification to Turbulence Intensity 154
CHAPTER 6. CONCLUSIONS
ˆ The present work was limited to isolated two-dimensional airfoils. However,
the UQ methodology can be applied to FSTI for three dimensional objects,
such as rotors as a whole and/or fuselages.
ˆ The present work was concerned with the effect of uncertainty in FSTI on
the aerodynamic coefficients. A possible subject of future work would be
to utilize the distributions of sectional aerodynamics due to uncertainty in
FSTI in a comprehensive rotor analysis. The effect of uncertainty can be
propagated forward to see the impacts on performance, vibrations, and/or
acoustics.
ˆ While the present work was solely concerned with uncertainty in FSTI, the
UQ methodology established here can be applied to other input variables.
One example would be surface roughness, which is another factor which im-
pacts transition onset. There are transition models, including the model used
by OVERTURNS, which incorporate surface roughness in the model. The
UQ methodology can therefore be applied in a similar fashion to uncertainty
in surface roughness.
ˆ The current work can be utilized as a small component in a much larger
project to formalize the use of UQ in CFD. Such a project could take the form
of a tool that could automate the UQ analysis and provide sensitivities of the
aerodynamic coefficients for requested flow conditions. However, this would
require that parametric studies are already conducted for the requested flow
conditions and airfoil in use. Another possible form would be to use the
current methodology to generate “uncertainty reference tables” for various
airfoils.
Application of Uncertainty Quantification to Turbulence Intensity 155
CHAPTER 6. CONCLUSIONS
Application of Uncertainty Quantification to Turbulence Intensity 156
Appendices
157
Appendix A
Complete Parametric Sweep 
Results
A.1 Introduction
In this work, parametric sweeps are used to generate the surrogate surfaces 
used for UQ analysis (see Sec. 1.3). The complete results of the parametric sweeps 
are provided for reference here for all four airfoils used in this study. The freestream 
conditions for the parametric sweeps are provided in Table 3.1.
158
(a) Lift coefficient
(b) Drag coefficient
Figure A.1: Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2× 106
159
(c) Moment coefficient
Figure A.1: Parametric Sweep for S809 Airfoil at Mach = 0.2, Re = 2×106 (cont.)
(a) Lift coefficient
Figure A.2: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 0.5×106
160
(b) Drag coefficient
(c) Moment coefficient
Figure A.2: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 0.5×106
(cont.)
161
(a) Lift coefficient
(b) Drag coefficient
Figure A.3: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 1×106
162
(c) Moment coefficient
Figure A.3: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 1×106
(cont.)
(a) Lift coefficient
Figure A.4: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 2×106
163
(b) Drag coefficient
(c) Moment coefficient
Figure A.4: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 2×106
(cont.)
164
(a) Lift coefficient
(b) Drag coefficient
Figure A.5: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 4×106
165
(c) Moment coefficient
Figure A.5: Parametric Sweep for NACA 0012 Airfoil at Mach = 0.2, Re = 4×106
(cont.)
(a) Lift coefficient
Figure A.6: Parametric Sweep for SC1095 Airfoil at Mach = 0.2, Re = 2× 106
166
(b) Drag coefficient
(c) Moment coefficient
Figure A.6: Parametric Sweep for SC1095 Airfoil at Mach = 0.2, Re = 2 × 106
(cont.)
167
(a) Lift coefficient
(b) Drag coefficient
Figure A.7: Parametric Sweep for SC1095 Airfoil at Mach = 0.4, Re = 2× 106
168
(c) Moment coefficient
Figure A.7: Parametric Sweep for SC1095 Airfoil at Mach = 0.4, Re = 2 × 106
(cont.)
(a) Lift coefficient
Figure A.8: Parametric Sweep for SC1095 Airfoil at Mach = 0.6, Re = 2× 106
169
(b) Drag coefficient
(c) Moment coefficient
Figure A.8: Parametric Sweep for SC1095 Airfoil at Mach = 0.6, Re = 2 × 106
(cont.)
170
(a) Lift coefficient
(b) Drag coefficient
Figure A.9: Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2× 106
171
(c) Moment coefficient
Figure A.9: Parametric Sweep for SC1095 Airfoil at Mach = 0.7, Re = 2 × 106
(cont.)
(a) Lift coefficient
Figure A.10: Parametric Sweep for SC1095 Airfoil at Mach = 0.8, Re = 2× 106
172
(b) Drag coefficient
(c) Moment coefficient
Figure A.10: Parametric Sweep for SC1095 Airfoil at Mach = 0.8, Re = 2 × 106
(cont.)
173
(a) Lift coefficient
(b) Drag coefficient
Figure A.11: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.2, Re = 2× 106
174
(c) Moment coefficient
Figure A.11: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.2, Re = 2× 106
(cont.)
(a) Lift coefficient
Figure A.12: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.4, Re = 2× 106
175
(b) Drag coefficient
(c) Moment coefficient
Figure A.12: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.4, Re = 2× 106
(cont.)
176
(a) Lift coefficient
(b) Drag coefficient
Figure A.13: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.6, Re = 2× 106
177
(c) Moment coefficient
Figure A.13: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.6, Re = 2× 106
(cont.)
(a) Lift coefficient
Figure A.14: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.7, Re = 2× 106
178
(b) Drag coefficient
(c) Moment coefficient
Figure A.14: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.7, Re = 2× 106
(cont.)
179
(a) Lift coefficient
(b) Drag coefficient
Figure A.15: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.8, Re = 2× 106
180
(c) Moment coefficient
Figure A.15: Parametric Sweep for RC(4)-10 Airfoil at Mach = 0.8, Re = 2× 106
(cont.)
181
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