1. Aerodynamic Shape Optimization via
Control Theory of Helicopter
Rotor Blades using a Non-Linear
Frequency Domain Approach
Charles A. Tatossian
Supervised by Siva K. Nadarajah
Masters of Engineering
Mechanical Engineering Department
McGill University
Montreal,Quebec
2008-02-18
A thesis submitted to McGill University in partial fulfilment of the requirements
for the degree of Masters of Engineering
c Charles A. Tatossian, Siva K. Nadarajah - Feb. 2008
3. ACKNOWLEDGEMENTS
I am profoundly indebted to my mentor, Professor Siva K. Nadarajah, who gener-
ously contributed to my own understanding of fluid mechanics since 2004. Profes-
sor Nadarajah’s contribution, whether it was through course notes, discussions, or
back of the envelope calculations, is entirely reflected in this thesis. His extensive
knowledge, enthusiasm, and expertise has given me the confidence to pursue the
unimaginable, and I truly thank him for his time and friendship.
This research has benefited from the generous support of the Natural Sciences
and Engineering Research Council of Canada (CGS-M) and by McGill’s Recruit-
ment Excellence Fellowship. McGill has offered me a great experience so far and I
am grateful to be able to share my passion for engineering with such a remarkable
university. Special thanks to Professor Pascal Hubert, Professor John Lee, Profes-
sor Evgeny Timofeev, Professor Peter Bartello, and Professor Dan Mateescu.
I am extremely grateful to my colleagues at the McGill Computational Fluid
Dynamics Laboratory. A very special thanks to Jean-Sebastien Cagnone, Arash
Mousavi, Patrice Castonguay, Olivier Soucy, and Robert Ritlop for their endless
support and friendship. I’d also like to thank the lab administrators, Yves Simard
and Patrice Hamelin, and lab director, Professor Fred Habashi, for provided me
with a pleasant and stimulating work environment.
Finally, I would like to express my gratitude towards my father, Armand Tatossian
R.C.A.; to my sister, Anais Tatossian; and to my girlfriend, Angie Chiazzese, for
their unconditional love. Their support have continually encouraged me to develop
my potential to its fullest. I would also like to thank my aunt, Mary Tatossian, and
iii
4. her boyfriend, Ivan Sarkissian, for taking care of my father throughout his illness.
Last but not least, I appreciate the time spent with Paul Hebert and Malcolm
Cairns, who both proved to be a great source of inspiration.
iv
5. ABSTRACT
This study presents a discrete adjoint-based aerodynamic optimization algorithm
for helicopter rotor blades in hover and forward flight using a Non-Linear Frequency
Domain approach. The goal is to introduce a Mach number variation into the
Non-Linear Frequency Domain (NLFD) method and implement a novel approach
to present a time-varying cost function through a multi-objective adjoint boundary
condition. The research presents the complete formulation of the time dependent
optimal design problem. The approach is firstly demonstrated for the redesign of
a NACA 0007 and a NACA 23012 helicopter rotor blade section in forward flight.
A three-dimensional inviscid Aerodynamic Shape Optimization (ASO) algorithm
is then employed to validate and redesign the Caradonna and Tung experimental
blade. The results in determining the optimum aerodynamic configurations require
an objective function which minimizes the inviscid torque coefficient and maintains
the desired thrust level at transonic conditions.
v
6. ABR´EG´E
Cette ´etude pr´esente un algorithme adjoint discret d’optimisation a´erodynamique
pour les h´elicopt`eres en vol stationnaire et en mouvement utilisant une approche
bas´ee sur un domaine `a fr´equences non-lin´eaires (NLFD). L’objectif est d’introduire
une variation de nombre de Mach dans le NFLD et d’implanter une nouvelle ap-
proche pour pr´esenter la fonction de coˆut variable `a travers une condition fronti`ere
adjointe multi-objective. La recherche pr´esente la formule compl`ete du design opti-
mal d´ependant du temps. Dans un premier temps, l’approche est d´emontr´ee pour
le redesign d’une g´eom´etrie NACA 0007 et NACA 23012. Dans un deuxi`eme temps,
un algorithme ASO `a trois dimensions non visqueux est employ´e pour valider et
redesigner les pales exp´erimentales Caradonna et Tung. Les r´esultats servant `a
d´eterminer les configurations a´erodynamiques optimales n´ecessitent une fonction
objective qui minimalise le coefficient couple non visqueux et maintient le niveau
de la pouss´ee axiale `a des conditions transsoniques.
vi
9. LIST OF TABLES
Table page
4–1 Weights Variation for Subsections (4.3.1), (4.3.2), and (4.3.3) . . . . 53
4–2 Synopsis of Results for Redesign of NACA 0007 . . . . . . . . . . . . 57
4–3 Weights Variation for Subsection (4.4.1) . . . . . . . . . . . . . . . . 60
4–4 Weights Variation for Subsection (4.4.2) . . . . . . . . . . . . . . . . 61
ix
10. LIST OF FIGURES
Figure page
1–1 Vortex Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1–2 Combination of Rotational & Translational Flow in Forward Flight 5
1–3 Stall and Compressibility Limits to allow Higher Forward Flight . . 6
2–1 Simplified diagram of the characteristics lines . . . . . . . . . . . . . 21
2–2 Simplified dataflow diagram of the time advancement scheme il-
lustrating the pseudo spectral approach used in calculating the
non-linear spatial operator R. . . . . . . . . . . . . . . . . . . . . 31
2–3 Simplified dataflow diagram of a 4 level W-multigrid cycle. . . . . . 34
2–4 Example of Domain Decomposition for a Two-Dimensional C-grid . 35
4–1 Comparison of the Lift, Drag, and Moment Hysteresis for a NACA
0007 at a Pitch of 2.89◦
± 2.41◦
, Mach of 0.6 ± 0.1, and Reduced
Frequency of 0.081 . . . . . . 64
4–2 Comparison of the Lift, Drag, and Moment for the NLFD and Time
Accurate Approaches, for a NACA 0007 at a Pitch of 2.89◦
±2.41◦
,
Mach of 0.6 ± 0.1, and Reduced Frequency of 0.081 . . . . . . . . 64
4–3 Comparison of Pressure Distribution between the NLFD and Time
Accurate Approach for a NACA 0007 at a Pitch of 2.89◦
± 2.41◦
,
Mach of 0.6 ± 0.1, and Reduced Frequency of 0.081 . . . . . . . . 65
4–4 Comparison of Pressure Distribution between the NLFD, Time
Accurate, and Steady-State Solution at Low Reduced Frequencies,
for a NACA 0007 at a Pitch of 2.89◦
± 2.41◦
, Mach of 0.6 ± 0.1 . 65
4–5 Comparison of the Lift, Drag, and Moment Hysteresis for a NACA
23012 at a Pitch of 5.50◦
± 4.00◦
, Mach of 0.441 ± 0.17, and
Reduced Frequency of 0.151 . . . . . . . . . . . . . . . . . . . . . 66
4–6 Real, Imaginary, and Mean Pressure Distributions for a NACA
23012 at a Pitch of 5.50◦
± 4.00◦
, Mach of 0.441 ± 0.17, and
Reduced Frequency of 0.151 . . . . . . . . . . . . . . . . . . . . . 67
x
11. 4–7 Maintaining cl while Minimizing cd via Fixed Weights for a NACA
0007 at a Pitch of 3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and Reduced
Frequency of 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4–8 Comparison of Methods for Maintaining cl while Minimizing cd for
a NACA 0007 at a Pitch of 3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and
Reduced Frequency of 0.005 . . . . . . . . . . . . . . . . . . . . . 68
4–9 Maintaining cl and cm while Minimizing cd via Fixed Weights for a
NACA 0007 at a Pitch of 3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and
Reduced Frequency of 0.005 . . . . . . . . . . . . . . . . . . . . . 69
4–10 Maintaining cl while Minimizing cd via Time-Varying Weights for a
NACA 0007 at a Pitch of 3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and
Reduced Frequency of 0.005 . . . . . . . . . . . . . . . . . . . . . 70
4–11 Synopsis of the Multi-Objective Functions; Comparison of the Lift,
Drag, and Moment Hysteresis for a NACA 0007 at a Pitch of
3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and Reduced Frequency of 0.005 71
4–12 Synopsis of the Multi-Objective Functions; Comparison of the Lift,
Drag, and Moment Convergence History for a NACA 0007 at a
Pitch of 3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and Reduced Frequency
of 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4–13 Gradient Comparison - Advancing Side of the Blade for a NACA
0007 at a Pitch of 3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and Reduced
Frequency of 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4–14 Gradient Comparison - Retreating Side of the Blade for a NACA
0007 at a Pitch of 3.00◦
± 3.00◦
, Mach of 0.6 ± 0.3, and Reduced
Frequency of 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4–15 Maintaining Lift and Minimizing Drag for a NACA 23012 at a Pitch
of 5.50◦
± 4.00◦
, Mach of 0.481 ± 0.25, and Reduced Frequency of
0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4–16 Comparison of Pressure Distributions and Airfoil Geometry for a
NACA 23012 at a Pitch of 5.50◦
± 4.00◦
, Mach of 0.481 ± 0.25,
and Reduced Frequency of 0.005 . . . . . . . . . . . . . . . . . . 74
4–17 Maximizing Lift and Minimizing Drag for a NACA 23012 at a Pitch
of 5.50◦
± 4.00◦
, Mach of 0.481 ± 0.25, and Reduced Frequency of
0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4–18 Comparison of Pressure Distributions and Airfoil Geometry for a
NACA 23012 at a Pitch of 5.50◦
± 4.00◦
, Mach of 0.481 ± 0.25,
and Reduced Frequency of 0.005 . . . . . . . . . . . . . . . . . . 76
xi
12. 5–1 Periodic Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5–2 Simplified dataflow diagram of the periodic boundary interchange . 81
5–3 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5–4 Structured 257 × 65 × 49 C-H grid . . . . . . . . . . . . . . . . . . . 91
5–5 Example of 3, 5, and 7 Time Steps . . . . . . . . . . . . . . . . . . 91
5–6 Convergence history of NLFD flow solver (collective pitch of 0◦
and
tip Mach of 0.520) . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5–7 Comparison of surface pressure distribution at z/R = 0.80 (collective
pitch of 0◦
and tip Mach of 0.520) . . . . . . . . . . . . . . . . . . 92
5–8 Comparison of surface pressure distribution at z/R = 0.89 (collective
pitch of 0◦
and tip Mach of 0.520) . . . . . . . . . . . . . . . . . . 92
5–9 Comparison of surface pressure distribution at z/R = 0.96 (collective
pitch of 0◦
and tip Mach of 0.520) . . . . . . . . . . . . . . . . . . 92
5–10 Convergence history of NLFD flow solver (collective pitch of 8◦
and
tip Mach of 0.439) . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5–11 Comparison of surface pressure distribution at z/R = 0.80 (collective
pitch of 8◦
and tip Mach of 0.439) . . . . . . . . . . . . . . . . . . 93
5–12 Comparison of surface pressure distribution at z/R = 0.89 (collective
pitch of 8◦
and tip Mach of 0.439) . . . . . . . . . . . . . . . . . . 93
5–13 Comparison of surface pressure distribution at z/R = 0.96 (collective
pitch of 8◦
and tip Mach of 0.439) . . . . . . . . . . . . . . . . . . 93
5–14 Convergence history of the thrust coefficient (collective pitch of 8◦
and tip Mach of 0.439) . . . . . . . . . . . . . . . . . . . . . . . . 94
5–15 Comparison of spanwise sectional lift distribution (collective pitch
of 8◦
and tip Mach of 0.439) . . . . . . . . . . . . . . . . . . . . . 94
5–16 Convergence history of NLFD flow solver (collective pitch of 8◦
and
tip Mach of 0.877) . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5–17 Comparison of surface pressure distribution at z/R = 0.80 (collective
pitch of 8◦
and tip Mach of 0.877) . . . . . . . . . . . . . . . . . . 95
5–18 Comparison of surface pressure distribution at z/R = 0.89 (collective
pitch of 8◦
and tip Mach of 0.877) . . . . . . . . . . . . . . . . . . 95
5–19 Comparison of surface pressure distribution at z/R = 0.96 (collective
pitch of 8◦
and tip Mach of 0.877) . . . . . . . . . . . . . . . . . . 95
xii
13. 5–20 Convergence history of NLFD adjoint solver (collective pitch of 8◦
and tip Mach of 0.877) . . . . . . . . . . . . . . . . . . . . . . . . 96
5–21 Comparison of gradients for various number of time steps at z/R =
0.80 (collective pitch of 8◦
and tip Mach of 0.877) . . . . . . . . . 96
5–22 Comparison of gradients for various number of time steps at z/R =
0.89 (collective pitch of 8◦
and tip Mach of 0.877) . . . . . . . . . 96
5–23 Comparison of gradients for various number of time steps at z/R =
0.96 (collective pitch of 8◦
and tip Mach of 0.877) . . . . . . . . . 96
5–24 Convergence of Thrust and Torque at a Collective Pitch of 8◦
and
Tip Mach of 0.877 . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5–25 Convergence of Thrust and Torque at a Collective Pitch of 8◦
and
Tip Mach of 0.860 . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5–26 Comparison of Initial and Final Pressure Distributions at Various
Span Stations at a Collective Pitch of 8◦
and Tip Mach of 0.860 . 98
5–27 Comparison of Initial and Final Pressure Distributions at Various
Span Stations at a Collective Pitch of 8◦
and Tip Mach of 0.877 . 99
5–28 Comparison of Initial and Final Airfoil Shapes at a Collective Pitch
of 8◦
and Tip Mach of 0.860 . . . . . . . . . . . . . . . . . . . . . 100
5–29 Comparison of Initial and Final Airfoil Shapes at a Collective Pitch
of 8◦
and Tip Mach of 0.877 . . . . . . . . . . . . . . . . . . . . . 100
xiii
14. CHAPTER 1
Introduction
Rotorcraft aerodynamicists continually challenge themselves to improve the aero-
dynamic performance of today’s helicopter blade designs. Modern designs in ap-
plied aerodynamics can be resolved via two fundamental approaches: theoretical
fluid dynamics or computational fluid dynamics. Although both methodologies
have been vigorously scrutinized to study the interaction between the air and solid
bodies, computational fluid dynamics has always been characterized as a supple-
mental aid in the design process of rotorcrafts rather than serving as a direct tool.
Despite the recent efforts in ameliorating existing steady flow aerodynamic shape
optimization algorithms, there remains a considerable need to develop innovative
and cost-effective optimal control techniques for the design of aerodynamic surfaces
subjected to unsteady loads.
In the 1980s, the rotorcraft community lacked the level of sophistication to build
CFD applications which promptly fitted within industry standards, mainly due to
the unsteady nature of the problem [1]. A lot of progress has been made since
then; however, the need to build cost-effective CFD solutions still remains, espe-
cially due to the complex, interdependent, and computationally demanding aspects
of the research. So far, researchers have applied aerodynamic shape optimization
algorithms to numerous steady-state problems, ranging from the design of two-
dimensional airfoils to full aircraft configurations to decrease drag, increase range,
promote lift, etc. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. However, unlike fixed wing
aircrafts, helicopter rotors operate in an unsteady flow and are constantly sub-
jected to unsteady loads. The goal of this chapter is to review the critical points
1
15. of rotorcraft design as well as bring the reader up to date with current literature
on the topic. The chapter is divided into two parts. Section (1.1) and (1.2) will
investigate the current two- and three-dimensional aerodynamic issues related to
rotorcraft applications, while section (1.3) will examine the design capabilities and
desired level of performance.
1.1 Helicopters in Hover Flight
In hover flight conditions, the change in freestream
Figure 1–1: Vortex Formation
Mach number and pitching angle is negligible;
and therefore, the local flow throughout the blade’s
cycle is characterized as axisymmetric and quasi-
steady. Reliable predictions of helicopter rotors
in hover remain challenging, primarily due to
the heavy dependency on the resolution of the
blade-vortex interaction near the tip region of
the blade. This interaction strongly influences
the incoming flow and consequently alters the effective angle of attack. The pre-
diction of the rotor wake structure requires the treatment of three-dimensional,
non-linear compressible blade flow, at a fairly high resolution.
Numerical simulations of three-dimensional flow over helicopter rotors have been
attempted and perfected by a number of researchers. Substantial efforts were made
by Caradonna and Isom in the mid 1970s [13, 14], where a three-dimensional un-
steady form of the transonic small-perturbation potential (TSP) equations were
used to model a helicopter rotor. Despite the fact that, at the time, there were
significant uncertainties about the unsteady nature of the transonic simulations
(lack of experimental data, negligence of the close proximity of the rotor to its
2
16. own wake, etc.), Caradonna managed to bring confidence to the rotorcraft commu-
nity by explicitly illustrating the potential of CFD methods for drag reduction. In
the late 1970s, Caradonna and Phillippe [15] validated their TSP computational
model using a stiff 2-blade rotor system equipped with micro-transducers. The
comparison against the experimental data was in good agreement; and thus, the
potential for predicting rotor aerodynamics via CFD methods was becoming abun-
dantly clear [16].
Despite the fact that TSP algorithms continued to flourish in the mid 80s, it be-
came apparent that the methodology was fairly limited in terms of accuracy (the
majority of potential-flow algorithms required external wake models) [16]. The
TSP methods were gradually replaced with three-dimensional Euler solvers, as
presented by Roberts and Murman [17] in 1985 and by Agarwal and Desse [18] in
1987. Roberts and Murman were able of efficiently capture the wake-induced inflow
with a fair amount of precision via a finite volume scheme. The conclusion drawn
from their study suggested using an approach which eliminated both the truncation
and artificial diffusion of the rolled up wake. However, the majority of compressible
flowsolvers, regardless of the numerical methodology, introduces a certain amount
of numerical dissipation, which requires to be explicitly added to avoid numeri-
cal instability. Agarwal and Desse pushed the development further and presented
an Euler solver capable of calculating transonic flows on rotor blades in hover or
forward flight. The governing equations were solved via Jameson’s finite-volume
explicit Runge-Kutta time-stepping scheme, and the rotor effects were modeled via
a correction term applied to the effective angle of attack along the blades.
3
17. The first RANS calculations for hovering helicopter rotors were performed by Wake
and Sankar [19] in 1987. They proceeded via a time-accurate hybrid implemen-
tation of the Alternating-Direction Implicit (ADI) scheme, which was originally
employed by Sanker et al. [20] a year earlier. Their semi-implicit scheme produced
results in an efficient manner, capable of capturing the influence of the rotor wake
at large time step intervals. The RANS approach provided the developers with an
inherent mechanism to model complex rotor wake systems, since the tip vortex is
a viscous-generated phenomena. However, the non-linearity nature of the problem
required exceedingly large computer resources; and, from that point onward, the
rotorcraft community agreed to focus their attention on reducing the computa-
tional loads while simultaneously perfecting the large-scale convection of the wake.
A lot of work has been done in the field of helicopter simulation since then, namely
from Srinivasan et al. [21], Ahmad and Strawn [22], and Pomin and Wagner [23].
For further information, Lieshman et al. [24] presented an extensive list of articles
related to computational methods for helicopter aerodynamics.
Several grid generation tools emerged in the mid 1990s and early 2000s to better
suit the rotor and wake topologies. Overset grids provided code developers, such as
Moulton et al. [25], with the ability to generate high quality configurations near the
body-boundary layer. These overset grids also delivered better results when solv-
ing the hover wake convection problem. In terms of computational loads, Wissink
et al. [26] introduced a parallel version of the transonic unsteady rotor Navier-
Stokes code (TURNS) designed for distributed-memory architectures. Alonso et
al. [27] followed with their parallel version of ROTOR87, which incorporated a full
time-accurate compressible multiblock Euler solver with aeroelastic effects. Oscil-
lating airfoils and dynamic stall computations were also of interest, especially for
4
18. analyzing the behavior of various turbulence models for detached flows. Substan-
tial efforts were performed by Ko et al. [28], who identified which turbulence model
produced the most accurate and economical results for dynamic stall computations.
1.2 Helicopters in Forward Flight
As the helicopter moves forward, the rotor blade comes across a combination of
rotational and translational flow, and the symmetry formed during hover vanishes.
Local supersonic zones that terminate at a shock wave develop on the advancing
side, while at the retreating phase, the blade’s velocity relative to the air decreases
and the blade approaches the stall angle. This causes significant flow separation to
occur on the upper surface of the blade which in turn produces a loss in lift. All
these issues have to be carefully taken into consideration to fully comprehend the
performance of helicopter rotor blades and to truly appreciate the simplicity of the
final design.
!"#$%&'%()*+$",),-.,/',%&,0)1/$%02%'&)3+24 5,6/,$6'%()*+$",),-.,/',%&,0)5,#,/0,")3+24
Figure 1–2: Combination of Rotational & Translational Flow in Forward Flight
From an industrial standpoint, the development of two-dimensional rotorcraft al-
gorithms remains a reliable and inexpensive strategy; and therefore, there is a
constant need to develop rapid tools to design helicopter blade sections. For the
5
19. past thirty years, the rotorcraft community presented an extensive list of papers,
reports, and articles related to two-dimensional computational methods for heli-
copter aerodynamics [16], and researchers have developed ingenious ways to identify
and solve two-dimensional rotor issues. At the 38th Cierva Memorial Lecture, in
1997, Wilby [29] illustrated that the substantial change in pitching moment during
stall enforced a greater constraint on the design space of the retreating phase of
the rotor. The substantial change in pitching moment caused large fluctuations
on the blade pitch control mechanism, which in turn dramatically shortened the
fatigue life of the rotor. One solution to this problem involved the introduction of
the reflex camber towards the rear portion of the upper surface. The reflex camber
counteracted the pitching moment and consequently improved the performance of
the blade profile.
Figure 1–3: Stall and Compressibility Limits to allow Higher Forward Flight
1.3 Aerodynamic Shape Optimization for Helicopter Rotor Blades
In terms of designing helicopter rotor blades, there are two desirable requirements
that can be identified. The first criterion involves the ability to accurately and
rapidly predict the flow past a rotor blade at different flight regimes. The sec-
ond requirement involves a rapid design environment of the configuration at hand,
6
20. preferably via an automatic design method which incorporates shape optimization.
Although substantial progress has been made toward the second objective with the
introduction of fast optimization techniques such as the adjoint method, various
challenges still remain regarding the capability to predict the flow in a cost-effective
manner. The goal of this subsection is to highlight the assets of the adjoint solver
for optimization purposes; and to demonstrate the benefits of using the NLFD
method for optimizing periodic problems.
In the last few years, automatic gradient-based optimization techniques have had
a significant impact on the design process of rotorcraft blades. An essential step
in these automatic optimization procedures involves the accurate calculation of
the gradient. The gradient provides information about the sensitivity of the cost
function with respect to the variations of the design parameters. Finite differ-
ence methods are commonly used to calculate the sensitivity derivatives of the
aerodynamic cost function; however, the computational cost associated with these
methods remain problematic, especially for large numbers of design variables. The
accuracy of the gradient is sensitive to the step size, and requires N + 1 flow cal-
culations for N derivatives. An alternative solution to the finite difference method
involves calculating the gradient by solving an adjoint equation via control theory.
The computational cost associated with the control theory technique is significantly
lower, despite the fact that there is the additional operating cost of solving the ad-
joint equation. Once the adjoint equation has been solved, the cost of obtaining the
sensitivity derivatives of the objective function with respect to each design variable
is considered negligible. In fact, the total cost of calculating these gradients is
independent of the number of design variables. The total cost associated with the
control theory technique consists of one flow solution and one adjoint solution.
7
21. Presently, a very limited number of research work for aerodynamic shape optimiza-
tion of helicopter blades exists. The transition of adjoint-based algorithms orig-
inally developed for fixed-wing towards three-dimensional unsteady viscous flows
has been slow primarily due to the demanding computational cost. Nevertheless,
Nadarajah and Jameson [30, 31] have pursued the development of optimum shape
design for two- and three-dimensional unsteady flows using the time accurate ad-
joint based design approach. Nadarajah derived and applied the time accurate ad-
joint equations (both the continuous and discrete) to the redesign of an oscillating
airfoil in an inviscid transonic flow. The approach utilized a dual time stepping [32]
technique that implements a fully implicit second order backward difference for-
mula to discretize the time derivative. Typical runs required 15 periods with 24
discrete time steps per period and 15 multigrid cycles at each time step. In 1999,
Yee et al. [33] suggested a new optimization method that was able to handle the
dynamic response of airfoils undergoing unsteady motion. Yee introduced a novel
objective function which constrained the design space to a specific point, via RSM
methods. A weighting function was also introduced to determine the specific az-
imuthal location at which the performance peaked. The conclusion drawn from the
study indicated that the overall performance of the optimized rotor airfoils were
superior to that of the baseline. His efforts generated blade profiles which avoided
stall incidence on the retreating side of the blade, while simultaneously achieved a
low wave drag coefficient on the advancing phase. In 2006, Lee et al. [34], presented
the redesign of the Caradonna-Tung and UH-60 rotors via a continuous adjoint ap-
proach. Very encouraging results were obtained at a substantial computational
expense. The redesigned shape achieved a 11% reduction in time-averaged torque
while maintaining the time-averaged thrust within 4% of the original value. Lee also
managed to implement a solution-adaptive mesh refinement method to efficiently
capture the blade’s tip vortex while simultaneously reduce the computational cost.
8
22. In 2007, Morris et al. [35] provided a generic optimization tool which allowed high-
fidelity aerodynamic optimization of two- and three-dimensional blades. Heavily
constrained two-dimensional cross-sections showed significant improvements over
existing designs, with reduced blade twist and control loads. Despite these efforts,
the optimization of unsteady flows continues to present severe challenges in terms
of computational cost. Although the adjoint approach presents a computational
viable technique for ASO of helicopter blades, the operating cost remains large due
to the demanding resources of the flow solver.
There has been much effort focused on the development of efficient and practical
alternatives to the study of unsteady problems. One potential approach is to use pe-
riodic methods. Linearized frequency domain and deterministic stress methods[36]
are examples of periodic methods that are widely used in industry. Unfortunately,
the inability of these methods to accurately model the solution becomes evident
for systems that contain strong nonlinearities. The Harmonic Balance technique,
a pseudo-spectral approach initially proposed by Hall[37] and later modified by
McMullen[38, 39], has been validated against both the Euler and Navier-Stokes
equations for a number of unsteady periodic problems and has been shown to ac-
count for strong nonlinearities. The cost associated with spectral methods like
McMullen’s Non-Linear Frequency Domain method (NLFD) is proportional to the
cost of the steady state solution multiplied by the number of desired temporal
modes. For inviscid flow, McMullen[40] has shown that to accurately model an
oscillating airfoil pitching about its quarter chord, a temporal resolution of only 1
mode above the fundamental frequency (or equivalently 3 time samples per period)
is needed using the NLFD method versus the 45 time samples needed with a back-
ward difference formulation of the time derivative[32]. These results demonstrate
the potential of the method to provide significant reduction in computational cost
9
23. for the analysis and design of more realistic problems such as helicopter rotors,
turbomachinery, and other unsteady devices operating in the transonic regime.
Nadarajah and Jameson [41] extended their two-dimensional optimum shape de-
sign for unsteady flows from a time accurate scheme to the NLFD approach and
in the process developed the NLFD adjoint equations. The method was further
extended for three-dimensional inviscid and viscous flows and a complete compar-
ison between the two techniques [42, 43] was presented. The conclusion drawn
from their study showed that the computational cost associated with the NLFD
approach was significantly lower than with the time accurate scheme.
The modeling of unsteady aerodynamic design sensitivities using either the har-
monic balance technique or the non-linear frequency domain approach have also
been investigated by Duta et al. [44] and Thomas et al. [45]. Duta et al. [44] have
presented a harmonic adjoint approach for unsteady turbomachinery design. The
aim of the work was to reduce blade vibrations due to flow unsteadiness. The
research produced adjoint methods that were based on a linearized analysis of
periodic unsteady flows. Thomas et al. [45] presented a viscous discrete adjoint ap-
proach for computing unsteady aerodynamic design sensitivities. The adjoint code
was generated from the harmonic balance flow solver with the use of an automatic
differentiation software compiler.
In terms of analyzing helicopter rotor blades in the frequency domain, Choi et
al. [46] presented a time-spectral approach to simulate a four-bladed UH-60A rotor
in 2007. Choi noticed that nine time steps was sufficient to simulate simple flight
test cases; however, the flowsolver required up to fifteen time steps to efficiently cap-
ture complex configurations, such as low speed transition and dynamic stall. She
also established that the rotor analysis with all four blades was more accurate than
10
24. the single blade analysis with wake modeling. Overall, the conclusion drawn from
Choi’s research suggested that the time-spectral method provides great potential
for adjoint-based optimization algorithms for periodic problems when compared
to time-accurate schemes. Later that year, Kumar and Murthy [47] developed a
framework for two- and three- dimensional helicopter blades via a frequency do-
main approach. They presented several simulations for the Caradonna-Tung rotor
in both hovering and forward flight conditions; claiming one to two orders of mag-
nitude faster than the time domain solutions. Kumar and Murthy pursued their
research in 2008 by implementing a multiblade coordinate system (MCS) [48] into
their existing frequency domain framework. With the MCS in place, fewer har-
monics around the azimuth were required to accurately calculate the flow around
the helicopter blade. Finally, Butsuntorn and Jameson [49] recently proposed a
Time Spectral method to simulate time-periodic unsteady three-dimensional ro-
torcraft flow for both Euler and Reynolds averaged Navier-Stokes equations. They
also formulated two new methods for vorticity confinement: one which used the
local velocity magnitude to scale the confinement parameter and another which
used a helicity to determine the strength of the confinement term. The conclusion
drawn from the research showed that the simulation of rotorcraft flows can cohere
to engineering accuracy without the need of massive computing resources.
1.4 Objective of the Research
The objective of this study is to develop a cost-effective design framework for heli-
copter rotor blades. This will be done by extending our two- and three-dimensional
NLFD adjoint-based design framework for helicopter rotor blades in hover and for-
ward flight, and by investigating its impact on the overall change in aerodynamic
performance. The motivation of the research has been fueled both by the success
of our current capability for automatic shape optimization for unsteady flows and
11
25. the future potential of the NLFD method. This research produces unbiased in-
formation via a non-linear frequency domain flowsolver which can then be used
to solve complex aerodynamic optimization problems. Our long-term goal would
be to extend these new configurations to a wider group outside the research envi-
ronment such that they would ultimately fit within industry standards. The work
presented in this thesis represents my contribution towards the project. The work
can be split into two categories:
Simulation and Design of Helicopter Blade Sections
• Validate the Non-Linear Frequency Domain method.
• Develop a variation in Mach into the existing NLFD framework.
• Develop and Implement of two innovative multi-objective cost functions to
increase the performance of helicopter blade sections.
Simulation and Design of Helicopter Blades
• Implement hover boundary conditions and periodic boundary conditions for
the three-dimensional NLFD framework.
• Generate hexa meshes via Pointwise.
• Implement a multi-objective cost function to reduce the torque coefficient
and to maintain the thrust coefficient.
• Demonstrate the approach for the redesign of hovering helicopter blades.
The study is divided in six chapters. In the first three chapters, the reader is made
aware of some fundamental concepts of computational fluid dynamics, including the
conservative form of the field equations, the Baldwin-Lomax and Spalart-Allmaras
turbulence models, the spatial and temporal discretizations, multigrid and implicit
smoothing, etc. The discrete unsteady adjoint equations are introduced in chapter
3 to draw attention, early on, to some of the interesting effects of aerodynamic
shape optimization. Both the time accurate and pseudo-spectral approaches are
12
26. investigated. The multi-objective functions used in this research will be carefully
selected to address the current rotor issues; thus, drag minimization will be empha-
sized on the advancing side of the blade, while lift conservation will be accentuated
on the retreating side.
Chapter 4 introduces a Mach number variation into the current two-dimensioanl
NLFD framework; and makes use of the time-varying cost function derived in chap-
ter 3. The approach for the redesign of a NACA 0007 and a MBB BO-105 helicopter
rotor blade sections is demonstrated at the end of chapter 4. Chapter 5 presents
the results obtained for three-dimensional geometries. To simulate and redesign
the flow over hovering three-dimensional helicopter rotors, a number of additional
developments have been added to the present framework, such as hover boundary
conditions, periodic boundary conditions and new adjoint boundary conditions.
The flow solver is initially validated for the Caradonna-Tung experimental rotor
blade at a tip Mach number of 0.439 and 0.877 and collective pitch of 8 degrees
and the blade is subsequently redesigned using the NLFD adjoint-based algorithm.
13
27. CHAPTER 2
The Euler and Navier-Stokes Equations
This chapter develops a treatment of the topics discussed in the introduction,
starting with the fundamental concepts of fluid mechanics and thermodynamics.
The emphasis is on demonstrating how these fundamentals concepts govern all
compressible fluid behavior. Section (2.1) formulates the conservation laws for a
Newtonian fluid; while section (2.2) explores a more rigorous, formal treatment of
these laws by presenting the limiting forms of the Navier-Stokes equations and the
various turbulence models used in this study. Finally, section (2.3) illustrates the
underlying numerical discretization, which includes the spatial discretization of the
convective, dissipative, and viscous fluxes; as well as the temporal discretization.
2.1 Conservation Laws for a Newtonian Fluid
The following subsections serve as an introduction for the subsequent parts. Sub-
section (2.1.1) introduces the Reynolds’ Transport theorem while subsections (2.1.2)
to (2.1.4) derive the integral and conservative form of the field equations.
2.1.1 Reynolds’ Transport Theorem
The laws of Newtonian mechanics and thermodynamics are generally given by
identifying a particular group of particles and investigating its properties through
time, via a Lagrangian approach. However, in fluid mechanics, it is more conve-
nient to treat the medium as a continuum and evaluate the time evolution of the
dynamic and thermodynamic state of a fluid at a fixed point in space, through a
Eulerian description. Our first task consists of converting the time derivatives of a
14
28. Lagrangian description into Eulerian form, using the Reynolds’ Transport theorem,
dX
dt
=
∂X
∂t
+
CS
xρu · ds. (2.1)
2.1.2 Conservation of Mass
From the definition of (2.1), the time rate of change associated with a control mass
system equals the time rate of change of the mass for a control volume, plus the
net rate of mass flow through the control surface. Since no mass can be created
nor destroyed, the time rate of change of M equals 0, as shown below,
dM
dt
=
∂
∂t CV
ρdv +
CS
ρu · ds = 0. (2.2)
Using Gauss’ Divergence theorem, equation (2.2) can be written in differential form,
CV
∂
∂t
ρ + · (ρu) dv = 0,
∂
∂t
ρ + · (ρu) = 0. (2.3)
It is more convenient to rewrite equation (2.3) using the material derivative,
Dρ
Dt
+ ρ · (u) = 0. (2.4)
2.1.3 Conservation of Momentum
Newton’s second law of motion for a system states that the sum of forces acting on
a control volume equals the rate of change of momentum inside the control volume
plus the rate at which momentum is being convected out of the control volume.
∂
∂t CV
ρudv +
CS
ρ(uu) · ds =
CV
ρg + τ dv. (2.5)
15
29. Using Gauss’ Divergence theorem, equation (2.5) can be written as,
CV
∂(ρu)
∂t
+ · (ρuu) dv =
CV
ρg + τ dv,
ρ
∂u
∂t
+ (u · )u = ρg + τ. (2.6)
Once again, it is more convenient to write equation (2.6) in material derivative
form,
ρ
Du
Dt
= ρg + · τ. (2.7)
2.1.4 Conservation of Energy
For the conservation of energy, the time rate of total energy change of the mass
inside the control volume equals the rate of work done by the body force, plus the
rate of work done by the surface forces, plus the heat transferred to the system due
to conduction,
∂
∂t CV
ρ dv +
CV
· (ρ u)dv =
CV
ρg · u +
CV
· (τu) +
CV
· (κ T), (2.8)
where = e + u2
2
+ gz. The coefficient of thermal conductivity, κ, and the temper-
ature, T, are computed as
κ =
cpµ
Pr
, T =
p
Rρ
, (2.9)
where Pr is the Prandtl number, cp is the specific heat at constant pressure, and
R is the gas constant. Using the definition of the convective derivative, equation
(2.8) can be conveniently rewritten as:
ρ
D
Dt
= ρg · u + · (τu) + · (κ T). (2.10)
16
30. 2.2 Conservative Form of the Field Equations
The next step consists of deriving the tensorial form of the Navier-Stokes equa-
tions for a viscous, heat conducting, compressible fluid. Although this procedure
will not be shown in this work (only the final result is displayed), it remains an
important aspect of the research since it helps us understand the foundations of the
subject as well as its corresponding assumptions. We should appreciate the simpli-
fication techniques, such as dimensional analysis and scaling laws, which provide
us with additional information about the relative importance of each term in the
Navier-Stokes equations. Subsections (2.2.1) and (2.2.2) examine the simplified
Navier-Stokes equations and its corresponding boundary conditions; while subsec-
tions (2.2.4) and (2.2.5) demonstrate the various turbulence models used in this
work.
2.2.1 The Navier-Stokes Equations
Consider the tensorial form of the continuity, momentum, and energy equations for
an unsteady, viscous, heat conducting, compressible fluid,
Dρ
Dt
+ ρ
∂ui
∂xi
= 0, (2.11)
ρ
Dui
Dt
=
−∂p
∂xi
+
∂τvis
ij
∂xj
, (2.12)
ρ
De
Dt
+ ρ
D1
ρ
Dt
=
∂
∂xi
κ
∂T
∂xi
+ τvis
ij
∂ui
∂xj
, (2.13)
where τvis
ij represents the viscous stress tensor of a Newtonian fluid,
τvis
ij = µ
∂ui
∂xj
+
∂uj
∂xi
−
2
3
µ
∂uk
∂xk
δij.
Here, µ represents the sum of the laminar coefficient of viscosity and the eddy
viscosity and 2
3
µ corresponds to Stokes’ approximation of the second coefficient of
17
31. viscosity. The Kronecker delta function is designated by the variable δij.
Since the flow variables are stored in memory via arrays, it is more appropriate to
express the Navier-Stokes equations in matrix form,
∂w
∂t
+
∂fi
∂xi
=
∂fvi
∂xi
in D, (2.14)
where the state vector w, inviscid flux vector f and viscous flux vector fv are
described respectively by
w =
ρ
ρu1
ρu2
ρu3
ρE
, fi =
ρ(ui − bi)
ρu1(ui − bi) + pδi1
ρu2(ui − bi) + pδi2
ρu3(ui − bi) + pδi3
ρE(ui − bi) + pui
, and fvi =
0
τvis
ij δj1
τvis
ij δj2
τvis
ij δj3
ujτvis
ij + κ ∂T
∂xi
.
In these definitions, ρ represents the density, ui denotes the Cartesian velocity
components of the fluid, and E characterizes the total energy. The investigation
of unsteady flows requires us to subtract the Cartesian velocity components of the
boundary, bi, from the fluid velocity. The pressure is determined by the equation
of state
p = (γ − 1) ρ E −
1
2
(uiui) ,
and the stagnation enthalpy is given by
H = E +
p
ρ
,
where γ is the ratio of the specific heats.
18
32. 2.2.2 Boundary Conditions
For inviscid flows, the zero-slip condition across the surface is not respected. The
lack of friction forces drives the need for the velocity vector to become tangent to
the surface. This implies that the dot product between the velocity vector, v, and
the unit normal vector at the surface, n, must equate zero.
v · n = 0 at the surface. (2.15)
For viscous flows, the relative velocity between the fluid and its corresponding
surface is assumed to be zero. Notice the emphasis on the term relative; since, in
this work, the unsteadiness of our flow causes our velocity vector to include a mesh
component, which must be carefully addressed. The zero-slip condition forces the
Cartesian velocity modulus to become
u = 0; v = 0; w = 0; at the surface. (2.16)
At the far-field boundary, we can simplify the complexity of the problem and
assume an isentropic, inviscid, non-heat conducting, and irrotational fluid. This
suggests using a characteristic based boundary condition via Riemann invariants,
where we extrapolate the outward waves based on interior information and compute
the incoming waves using the freestream conditions. The Riemann invariants can
be expressed as
R∞ = v∞ · n −
2c∞
γ − 1
; Re = ve · n −
2ce
γ − 1
.
The normal velocity and its corresponding speed of sound can be written as
va =
1
2
(R∞ + Re); ca =
γ − 1
4
(Re − R∞).
19
33. A more detailed derivation of the characteristic based boundary condition is pre-
sented in the next subsection.
2.2.3 Riemann Invariants
This subsection derives the Riemann invariants from the inviscid, adiabatic govern-
ing equations. Consider the following inviscid and adiabatic governing equations:
∂ρ
∂t
+ · (ρV ) = 0, (2.17)
ρ
∂V
∂t
+ V · V = − p, (2.18)
De
Dt
+ ρ
Dp
Dt
= 0. (2.19)
Using the isentropic property p
ργ = constant and the internal energy e = 1
(γ−1)
p
ρ
,
we can rewrite equation (2.19) as
De
Dt
=
1
γ − 1
1
ρ
Dp
Dt
−
p
ρ2
Dρ
Dt
,
0 =
1
ρ
1
(γ − 1)
Dp
Dt
−
p
ρ2
1
(γ − 1)
− 1
Dρ
Dt
,
0 =
1
ρ
Dp
Dt
−
γp
ρ2
Dρ
Dt
. (2.20)
Equation (2.20) can be manipulated further, via c2
= γp
ρ
,
1
c2
Dp
Dt
+ ρ
∂u
∂x
= 0. (2.21)
Equations (2.21) and (2.18) yield a pair of non-linear partial differential equations,
which must be solved in terms of space, x, and time t.
∂p
∂t
+ u
∂p
∂x
+ ρc2 ∂u
∂x
= 0;
∂u
∂t
+ u
∂u
∂x
+
1
ρ
∂p
∂x
= 0. (2.22)
20
34. space
time
C+
C- Boundary
Ub, Cb
Ue, Ce U∞, C∞
∂+ @ e = ∂+ @ b ∂- @ ∞ = ∂- @ b
Figure 2–1: Simplified diagram of the characteristics lines
The first task consists of replacing this pair of partial differential equations with a
total differential equation. To accomplish this, consider the undetermined multi-
pliers α and β:
α
∂p
∂t
+ αu
∂p
∂x
+ αρc2 ∂u
∂x
+ β
∂u
∂t
+ βu
∂u
∂x
+
β
ρ
∂p
∂x
= 0. (2.23)
Divide by α, and factorize the derivative,
∂
∂t
+ u +
β
αρ
∂
∂x
p +
β
α
∂
∂t
+ u +
β
α
ρc2 ∂
∂x
u = 0. (2.24)
Thus, β
α
must equate,
u +
β
αρ
= u +
β
α
ρc2
,
β
α
= ±ρc.
Substituting the coefficients back, we obtain
∂
∂t
+ (u ± c)
∂
∂x
p ± ρc
∂
∂t
+ (u ± c)
∂
∂x
u = 0. (2.25)
This implies that, along the curves where ∂x
∂t
= u ± c, the information carries the
21
35. following property:
dp ± ρcdu = 0,
dp
ρc
± du = 0,
2
(γ − 1)
dc ± du = 0,
which, when integrated, yields the left and right Riemann invariants,
2
(γ − 1)
c ± u = constant. (2.26)
Again, it is worth reminding ourselves that the velocity and speed of sound at the
boundary can be determined using the relations found in (2.26), via:
R∞ = v∞ · n −
2c∞
γ − 1
; Re = ve · n −
2ce
γ − 1
;
va =
1
2
(R∞ + Re); ca =
γ − 1
4
(Re − R∞).
2.2.4 Baldwin-Lomax Turbulence Model
The Baldwin-Lomax turbulence model [50] is an algebraic model that uses a two-
layer eddy diffusivity formulation,
µt =
µtinner
if yn ≤ ycrossover,
µtouter if yn > ycrossover,
(2.27)
where yn is the shortest distance to the wall, and ycrossover is the smallest value of
yn at which µtinner
= µtouter . The eddy viscosity of the inner layer is defined as
µtinner
= ρl2
Ω. (2.28)
where l, the length scale of the turbulence in the inner region, and Ω, the magnitude
of the vorticity, are written as
l = kyn 1 − e
−y+
A+
; Ω =
δui
δxj
−
δuj
δxi
.
22
36. The distance from the wall, y+
, is represented by
y+
=
√
ρwallτwall
µwall
yn.
The eddy viscosity of the outer layer is defined by
µtouter = KCcpρFwakeFkleb, (2.29)
with functions Fwake and Fkleb
Fwake = min(ymaxFmax, Cwkymax
U2
max
Fmax
); Fkleb = 1 + 5.5
Cklebyn
ymax
6 −1
.
The function Fmax is determined by maximizing
ynΩ 1 − e
−y+
A+
.
Finally, the remaining coefficients are specified as
Ccp = 1.6; Cwk = 1.0; Ckleb = 0.3; K = 0.0168; k = 0.4; A+
= 26.0.
2.2.5 Spalart-Allmaras Turbulence Model
The Spalart-Allmaras turbulence model [51] is a one-equation model for the deter-
mination of the kinematic turbulent viscosity. The kinematic turbulent viscosity,
υ, can be expressed using the eddy viscosity, ˜υ:
υt = ˜υfv1,
where fv1 is expressed as
fv1 =
χ3
χ3 + cv1
; cv1 = 7.1; χ :=
˜υ
υ
.
The eddy viscosity, ˜υ, takes the form of a scalar partial differential equation given in
Cartesian coordinates. The expression may be split into components of production,
23
37. diffusion, and destruction,
D˜υ
Dt
= cb1 1 − ft2
˜S˜υ
production
+
1
σ
· ((υ + ˜υ) ˜υ) + cb2( ˜υ)2
diffusion
− cw1fw −
cb1
κ2
ft2
˜υ
d
2
destruction
+ft1∆U2
,
where the auxiliary relations are defined via,
ft1 = ct1gte
„
−ct2
ω2
t
∆U2
[d2+gtd2
t ]
«
; ft2 = ct3e(−ct4χ2)
; fv2 = 1 −
χ
1 + χfv1
;
fw = g
1 + c6
w3
g6 + c6
w3
1
6
; g = r + cw2(r6
− r) gt = min(0.1, ∆U/ωt∆x);
r ≡
˜υ
˜Sκ2d2
; ˜S ≡ S +
˜υ
κ2d2
fv2; S = 2ΩijΩij Ωij =
1
2
δui
δxj
−
δuj
δxi
;
with corresponding closure coefficients,
σ = 2/3; κ = 0.41; cb1 = 0.1355; cb2 = 0.6220;
cw1 = cb1 κ2
+ (1 + cb2)/σ; cw2 = 0.3; cw3 = 2.0; cv1 = 7.1;
ct1 = 1.0; ct2 = 2.0; ct3 = 1.1; ct4 = 2.0.
The variable d corresponds to the normal wall distance, κ represents the von Kar-
man constant, Ωij stands for the rotation tensor, and S is the vorticity of the fluid.
When the flow is fully turbulent, the trip term can be discarded and the equation
above reduces to
D˜υ
Dt
= cb1
˜S˜υ
production
+
1
σ
· ((υ + ˜υ) ˜υ) + cb2( ˜υ)2
diffusion
− {cw1fw}
˜υ
d
2
.
destruction
Using the definition of the material derivative, the advection contribution can be
paired up with the diffusion terms. Note that the production and destruction terms
solely depend on the cell of interest, while the grad from the advection and diffusion
expression forms a stencil which depend on a preceding and succeeding cell,
δ˜υ
δt
= M(˜υ)˜υ
advection/diffusion
+ P(˜υ)˜υ
production
− D(˜υ)˜υ,
destruction
(2.30)
24
38. where,
M(˜υ)˜υ = −(u · )˜υ +
1
σ
· ((υ + ˜υ) ˜υ) + cb2( ˜υ)2
,
P(˜υ)˜υ = cb1
˜S˜υ,
D(˜υ)˜υ = {cw1fw}
˜υ
d
2
.
The transport equations of the turbulence model are calculated apart from the
Navier-Stokes equations, via an Alternating Direction Implicit (ADI) scheme based
on an approximate factorization [51]. Equation (2.30) is evaluated only on the fine
mesh using the latest information from the flow governing equations. At a solid
wall boundary, the boundary condition sets ˜v = 0. A short numerical discretization
is presented in Appendix A.
2.3 Numerical Discretization
For this research, it is convenient to adopt a structured body-fitted coordinate
system, since it allows us to accurately resolve the flow near the physical location
of the boundary. The structured body-fitted coordinate system also enables us
to simplify the evaluation of gradients, fluxes, and boundary conditions. Each
cell is distinctively identified by its corresponding Cartesian coordinate address
xi,j,k, yi,j,k, zi,j,k. The physical space (x1, x2, x3) is transformed into a linear address
space, also known as the computational domain (ξ1, ξ2, ξ3). The shape of the grid
cells consists of quadrilaterals for two-dimensional flows and hexahedrons for three-
dimensional flows. The transformation to the computational coordinate system is
defined by the following metrics,
Kij =
∂xi
∂ξj
, J = det (K) , K−1
ij =
∂ξi
∂xj
.
25
39. The Navier-Stokes equations can then be written in computational space as
∂ (Jw)
∂t
+
∂ (Fi − Fvi
)
∂ξi
= 0 in D,
where the inviscid and viscous flux contributions are now defined with respect to the
computational cell faces by Fi = Sijfj and Fvi
= Sijfvj
. The quantity Sij = JK−1
ij
represents the projection of the ξi cell face along the xj axis. A finite volume
scheme is derived by applying Jameson’s Dual Time Stepping equation directly
to a collection of control volumes. This gives rise to a set of ordinary differential
equations of the form
d(V w)
dt
+ R(w) = 0 in D, (2.31)
where V is the cell volume, and R(w) is the residual. The residual is evaluated
by summing up the convective, dissipative, and viscous fluxes across the cell faces.
The simulations contained in this research are restricted to rigid mesh translation
and therefore the cell volumes remain constant in time.
2.3.1 Spatial Discretization
The convective flux is represented in discrete form for each computational cell using
a central second-order discretization. The values of the flow variables are stored
as conservative variables at the cell centers and can be regarded as cell averages.
Accordingly, the convective flux at the cell face is computed by taking the average
of the flux contributions from each cell across the cell face. The spectral radius
of the flux Jacobian matrix are rescaled in the each direction to better handle the
high aspect ratio cells in the vicinity of the blade tip. Further details can be found
in Jameson’s ’Analysis and design of numerical schemes for gas dynamics’ [52].
26
40. To eliminate odd-even decoupling of the solution and overshoots before and after
shock waves, the convective flux is added to a diffusion flux. Two artificial dissi-
pation schemes were used in this research. The first artificial dissipation scheme
consists of a blended first and third order flux, originally introduced by Jameson et
al. [53]. This artificial dissipation scheme is known as the Jameson-Schmidt-Turkel
(JST) scheme, and can be summarized in the form,
d = (2)
∆x3 Λ
p
∂2
p
∂x2
∂w
∂x
− (4)
∆x3
Λ
∂3
w
∂x3
. (2.32)
The left term in equation (2.32) is a first order scalar diffusion term, where the
modulus is scaled by the normalized second difference of the pressure and serves to
damp oscillations around shock waves. (4)
is the coefficient for the third derivative
of the artificial dissipation flux. The coefficient is scaled such that it is zero at
regions of large gradients, such as shock waves and eliminates odd-even decoupling
elsewhere.
The second artificial dissipation scheme is derived for the three-dimensional hover
calculations, presented in chapter 5. The scheme is formed using the Convec-
tive Upwind Split Pressure (CUSP) scheme, which was originally introduced by
Jameson [54] and later modified by Tatsumi et al. [55]. The CUSP scheme is also
composed of an average of fluxes, but presents less dissipation than the JST scheme.
Less dissipation consequently implies a higher resolution of the captured vortex.
As for the viscous discretization, the viscous stress tensor is computed at the ver-
tex via a discrete form of the Gauss theorem to the auxiliary control volume. The
viscous flux is then averaged at both ends of the edge. Refer to Nadarajah [31] for
further information of this procedure.
27
41. 2.3.2 Temporal Discretization: Dual Time Stepping
The derivation of the Dual Time Stepping scheme starts with equation (2.31) and
incorporates an unsteady term via a pseudo time step,
∂ (Jw)
∂τ
+
∂ (Jw)
∂t
+
∂ (Fi − Fvi
)
∂ξi
= 0 in D. (2.33)
The next step consists of performing a chain rule on equation (2.33),
∂ (Jw)
∂τ
+
∂ (Jw)
∂ξi
∂ξi
∂t
+
∂ (Fi − Fvi
)
∂ξi
= 0 in D. (2.34)
Transforming equation (2.34) in computational space yields
∂ (Jw)
∂τ
+
∂ (Jw)
∂ξi
∂ξi
∂t
+
∂
∂ξi
J
∂ξi
∂xj
(fj − fvj) = 0 in D,
∂ (Jw)
∂τ
+
∂
∂ξi
Jw
∂ξi
∂xj
∂xj
∂t
+ J
∂ξi
∂xj
(fj − fvj) = 0.
Let ∂ξi
∂t
be defined as the mesh velocity in the computational domain; a further
arrangement of the flux terms produces the unsteady Navier-Stokes equations,
∂ (Jw)
∂τ
+
∂(Fi − Fvi
)
∂ξi
= 0. (2.35)
Note that the mesh velocity term is incorporated within the flux contributions.
Equation (2.35) can be approximated via
∂
∂τ
[w
(n+1)
i,j V
(n+1)
i,j ] + R(w
(n+1)
i,j ) = 0, (2.36)
which in part can be estimated using a second order expansion,
3
2∆τ
[w
(n+1)
i,j V
(n+1)
i,j ] −
2
∆τ
[w
(n)
i,j V
(n)
i,j ] +
1
2∆τ
[w
(n−1)
i,j V
(n−1)
i,j ] + R(w
(n+1)
i,j ) = 0. (2.37)
We solve for each time step via an explicit multistage modified Runge-Kutta
scheme. It is important to note that a third order formula could have been used;
28
42. however, the idea was set aside due to the compromised stability of the scheme.
W(0)
= W(n)
,
W(1)
= W(0)
− α1∆tR(W(0)
),
W(k)
= W(0)
− αk∆tR(W(k−1)
),
W(n+1)
= W(M)
.
The residual R can be split into convective and dissipative fluxes,
R(k)
= C(W(k)
) + βkD(W(k)
) + (1 − βk)D(k−1)
.
It is important to notice that the artificial dissipation is present only three times
throughout the five stages (β2 and β4 are zero). The coefficients for the scheme are
described as
α1 =
1
4
; α2 =
1
6
; α3 =
3
8
; α4 =
1
2
; α5 = 1,
β1 = 1; β2 = 0; β3 = 0.56; β4 = 0; β5 = 0.44.
2.3.3 Temporal Discretization: NLFD Approach
The derivation of the NLFD method starts with equation (2.31) and assumes that
the vector of flow variables w and local residual R can be represented by separate
Fourier series:
w =
N
2
−1
k=−N
2
ˆwkeikt
; R =
N
2
−1
k=−N
2
ˆRkeikt
, (2.38)
where i =
√
−1. The Fourier representations are then substituted into the semi-
discrete form of the governing equations as described in equation (2.31) to yield,
V
d
dt
N
2
−1
k=−N
2
ˆwkeikt
+
N
2
−1
k=−N
2
ˆRkeikt
= 0 in D. (2.39)
29
43. As a result of the time derivative,
ˆR∗
k = ikV ˆwk + ˆRk, (2.40)
forms the new unsteady residual in the frequency domain for each wavenumber and
must be solved iteratively. The solver attempts to find a solution, w, that drives
this system of equations to zero for all wavenumbers. The nonlinearity of the un-
steady residual and, hence, bulk of our computing effort stems from the spatial
operator, R.
There are two approaches to calculating the spatial operator expressed in the fre-
quency domain. The first uses a complex series of convolution sums to calculate
ˆRk directly from ˆwk. This approach is discarded due to its massive complex-
ity (considering artificial dissipation schemes and turbulence modeling) and cost
that scales quadratically with the number of modes N. This paper implements a
pseudo-spectral approach that relies on the computational efficiency of the Fast
Fourier Transform (FFT). To calculate R in the frequency domain, several trans-
formations between the physical and frequency domains are performed by FFT.
The computational cost of this transform scales like N log(N), a significant savings
over a similar method that uses convolution sums and scales like N2
. A diagram
detailing the transformations used by the pseudo spectral approach at each stage
of the modified multistage Runge-Kutta scheme is provided in figure (2–2). The
pseudo-spectral approach begins by initializing the state vector, w(t), at all time
instances. At the first iteration, w(t) will take on the value of the initial condition
and for subsequent iterations, the values are based on the previous iterations. At
each of these time instances the steady-state operator R(w(t)) can be computed
by summing the convective and artificial dissipation fluxes as mentioned in the
previous subsection. A FFT is then used to transform the state vector and spatial
30
44. R (w(t))
w(t) ˆwk
ˆRk
ikV ˆwk
+ ˆR∗
k ˆwk
Update
FFT
FFT
w(t)
Inverse
FFT
Update Wall and Far-Field
Boundary Conditions
Repeat Runge-Kutta Stage
Figure 2–2: Simplified dataflow diagram of the time advancement scheme illustrat-
ing the pseudo spectral approach used in calculating the non-linear spatial operator
R.
operator to the frequency domain where ˆwk and ˆRk are known for all wavenumbers.
The unsteady residual ˆR∗
k can then be calculated by adding ˆRk to the spectral
representation of the temporal derivative ikV ˆwk. The iteration is advanced and the
ˆwk is updated. Using an inverse FFT, ˆwk is then transformed back to the physical
space resulting in a state vector w(t) sampled at evenly distributed intervals over
the time period. The wall and far-field boundary conditions are imposed within
the time domain. Consistent with the time accurate approach, we can numerically
integrate our residual in fictitious time t∗
resulting in the following equation:
V
d ˆwk
dt∗
+ ˆR∗
k = 0.
An unsteady residual exists for each wavenumber used in the solution and the
pseudo-time derivative acts as a gradient to drive the absolute value of all of these
components to zero simultaneously. A modified five stage Runge-Kutta time inte-
gration scheme is applied to march the solution to a steady state solution. Local
time stepping, residual averaging, and multigrid are employed to accelerate the con-
vergence. In synopsis, the NLFD approach can be implemented into any existing
time-accurate flow solver with the following two primary modifications. First, the
state vector must be modified to allow an additional dimension to hold the value
at various time steps. This is unlike the time accurate approach, where the second
31
45. order backward difference approach only required to store the state vector at three
time levels. For the NLFD approach if three or greater time steps are employed,
the memory cost increases. Second, ’calls’ to FFT routines must be implemented
within the time-stepping scheme to perform both the FFT and inverse FFT.
An important limitation of the current implementation is the assumption that the
cell volume, V , is constant in time. In the work presented in this paper, a rigid
body translation and rotation is employed, and, therefore, this assumption is valid.
An extension of the method for unsteady motions where small grid deformations
are present, the volume, V (t), will be a function of time and must be included in
equation (2.39) within the time derivative. However, for cases where large grid
deformations are observed that might include grid adaptation which potentially
could increase or decrease the number of cells to improve the quality of the grid for
the flow solver, an interpolation of the state vector, w(t), to a common fixed point
in space might be needed to transform the state vector to the frequency domain.
Nevertheless, such an approach would reduce the order of the method and cast the
approach as a reduced order model.
The three-dimensional cases attempted in this work (chapter 5) are limited to the
simulation of rotor blades in a hover condition, where the collective pitch is fixed
at all phases of the rotation. Therefore, an alternate cost effective approach to
compute the flow would be to solve for the absolute velocities in the rotational
frame of reference without physically rotating the computational grid such as that
proposed by Holmes and Tong [56]. This approach requires the addition of a source
term to account for the rotor angular velocity and is computational very attractive.
However, it is our intention to further develop the technique and demonstrate the
framework for rotor blades in forward flight, where the addition of a freestream
32
46. Mach number as well as a variable collective pitch would require a full unsteady
flow solver, as presented in the two-dimensional test cases (chapter 4).
2.3.4 Implicit Residual Smoothing
The maximum CFL number of the Runge-Kutta time-stepping scheme can be influ-
enced through residual smoothing. The smoothing encourages the explicit scheme
to adopt an implicit characteristic, which in part allows higher values of the CFL
number. The technique also assists the damping of the high-frequency error com-
ponents of the residual, which is particularly important for the multigrid method.
The residual smoothing technique was introduced by Jameson and Baker [57] in
1983. For the one-dimensional problem, the standard formulation for the implicit
residual averaging reads as
− ˆRi−1 + (1 + 2 ) ˆRi − ˆRi+1 = Ri, (2.41)
where must satisfy the following inequality
≥
CFL
CFL*
2
− 1 . (2.42)
The objective of equation (2.41) is to solve for ˆR. Here, the CFL variable represents
the new Courant number, and CFL* consists of the Courant number without the
influence of the implicit residual averaging technique.
2.3.5 Multigrid Method
Brandt’s work [58] in 1972 brought multigrid to the forefront of convergence ac-
celeration tools. The multigrid method was further perfected by Jameson and
Martinelli in the mid 1980s for hyperbolic problems [59, 60]. The advantage of
performing multigrid cycles resides in damping low-frequency numerical errors.
33
47. CALCULATE FLOW
COLLECT RESIDUAL
COARSEN MESH
Figure 2–3: Simplified dataflow diagram of a 4 level W-multigrid cycle.
On a given fine mesh, high-frequency errors are damped at a faster rate than those
of low-frequency; and therefore, the speed of convergence would dramatically be
enhanced if one can encourage the suppression of low-frequency errors. As the
fine grid results are transferred to a coarser grid, the low-frequencies errors are
converted into the high-frequency spectrum. These high frequencies are then at-
tenuated during the coarse of the iterative process.
In this research, the multigrid cycle starts with the application of a 5-stage Runge-
Kutta time stepping scheme on the fine mesh. The flow variables are transfered to
a coarser mesh through the use of cell volumes as weights,
w
(0)
2h =
Vi,jw
(m)
h
Vi,j
. (2.43)
The residuals are also converted via the simple summation,
R2h(wm+1
h ) = Rh(wm+1
h ). (2.44)
The next step consists of transferring a residual forcing function to let the coarse
grid solution be driven by the transfered residuals from the fine grid, as shown
below,
P2h = FCOLL · R2h(w
(m+1)
h ) − R2h(w
(0)
2h ). (2.45)
34
48. The FCOLL variable is primarily used for validation purposes; setting it to zero will
rid the multigrid cycle. The forcing function is ultimately added to the coarse grid
residual via the modified Runge-Kutta time stepping scheme,
w(k)
= w(0)
− αk∆t(R2h(w(k−1)
) + P2h). (2.46)
Once the time-stepping scheme has been completed on the coarse grid, the correc-
tion is interpolated back to the fine mesh, via
wnew
h = w
(0)
h + Ih
2h(w
(m)
2h − w
(0)
2h ), (2.47)
where Ih
2h represents the prolongation operator. When moving up to a finer mesh,
the corrections are interpolated without performing a flow evaluation via straight
injection.
2.3.6 Parallelization
A domain decomposition scheme was used to efficiently decrease the algorithm’s
computational time. The exchange between processors was performed using a
Message Passing Interface (MPI) library. The two-dimensional code typically ran
on 4 processors, while the three-dimensional algorithm ran on either 12 or 24
Figure 2–4: Example of Domain Decomposition for a Two-Dimensional C-grid
35
49. processors, depending on the grid size. Load balancing was carefully taken into
account by dividing each block with approximately the same number of cells. The
partitions were also performed to maximize the levels of the multigrid scheme: each
block was capable of performing 4-level W-multigrid cycles.
36
50. CHAPTER 3
The Discrete Unsteady Adjoint Approach
Section (3.1) introduces the optimization algorithm for a time-dependent design
problem. Although the NLFD adjoint derivation does not respect some of the the-
ory presented in (3.1), it provides a good overview of the optimization methodology
used in this research. Sections (3.2) and (3.3) present a more formal treatment of
the time accurate and NLFD adjoint equations; while section (3.4) explores and
develops the multi-objective cost functions presented in this work.
3.1 Formulation of the Time-Dependent Optimal Design Problem
The aerodynamic properties that define the cost function are dependent of the flow
field variables, w, and the physical location of the boundary, f. The cost function
shown below is formulated for a time-dependent optimal design problem. It is
expressed as the time averaged of a function L plus a function M,
I =
1
T
tf
0
L(w, f)dt + M(w(tf )),
where the function L depends on the flow solution w, and the shape function, f;
while the function M depends on the time dependent flow solution, w(tf ). Assume
that the following equation defines the time-dependent flow solution
V
∂w
∂t
+ R(w, f) = 0,
where V is the cell volume and R represents a residue containing the convective,
dissipative and viscous fluxes. A change in f results in a change in
δI =
1
T
tf
0
∂LT
∂w
δw +
∂LT
∂f
δf dt +
∂MT
∂w
δw(tf ),
37
51. in the cost function. The variation in the flow solution is
V
∂
∂t
δw +
∂R
∂w
δw +
∂R
∂f
δf = 0.
The ∂MT
∂w
δw(tf ) term shown in the differential of I requires the recalculation of
the flow field. We can bypass this problem by incorporating a Lagrange multiplier
to the time-dependent flow equation. With the Lagrange multiplier in place, the
governing equations will be introduced as a constraint such that that the final
expression of the gradient will not require the reevaluation of the flow field. It
is important to note the NLFD scheme does not include the ∂MT
∂w
δw(tf ) term,
and therefore, control theory is not necessarily required; however, for all intents
and purposes, we will proceed with the derivation to give us a feel of how the
optimization algorithm works. Denote the Lagrange multiplier as ψ and integrate
it over time,
1
T
tf
0
ψT
V
∂
∂t
δw +
∂R
∂w
δw +
∂R
∂f
δf = 0.
Subtract the equation above from the variation of the cost function to arrive at the
following equation
δI =
1
T
tf
0
∂LT
∂w
δw +
∂LT
∂f
δf dt +
∂MT
∂w
δw(tf )
−
1
T
tf
0
ψT
V
∂
∂t
δw +
∂R
∂w
δw +
∂R
∂f
δf dt.
The next step consists of integrating the
tf
0
ψT
V ∂
∂t
δwdt term by parts,
δI =
1
T
tf
0
∂LT
∂w
+ V
∂ψT
∂t
− ψT ∂R
∂w
δwdt +
∂MT
∂w
−
V
T
ψT
(tf ) δw(tf )
+
1
T
tf
0
∂LT
∂f
− ψT ∂R
∂f
δfdt.
Choose ψ to satisfy the adjoint equation,
V
∂ψ
∂t
=
∂R
∂w
T
ψ −
∂L
∂w
,
38
52. with the terminal boundary condition
ψ(tf ) =
∂M
∂w
.
Then, the expression for δI can be written as
δI = GT
δf,
where
GT
=
1
T
tf
0
∂LT
∂f
− ψT ∂R
∂f
dt.
The sensitivity derivatives are determined by the solution of the adjoint equation
in reverse time from the terminal boundary condition and the time-dependent so-
lution of the flow equation. These sensitivity derivatives are then used to get a
direction of improvement and steps are taken until convergence is achieved. The
computational costs of unsteady optimization problems are directly proportional
to the desired number of time steps. The unsteady flow calculation can be obtained
either by the use of implicit time-stepping schemes or a NLFD approach.
3.2 Formulation of the Time Accurate Discrete Adjoint Equations
The goal of the following three sections is to derive an expression for the variation
in the cost function, and understand its influence on the discrete unsteady time
accurate and NLFD adjoint equations, as well as its influence on the gradient
calculation.
δI = δIc
Section 3.4
−
1
T
tf
t=0 Ω
ψT
ijδR∗
ij(w).
Sections 3.2 & 3.3
Recall the time-dependent flow solution, presented in chapter 2,
∂
∂τ
[w
(n+1)
i,j V
(n+1)
i,j ] + R(w
(n+1)
i,j ) = 0,
39
53. and consider the modified residual δR∗n+1
ij (w), mentioned in equation (2.37),
R∗n+1
ij (w) =
3
2∆the
wn+1
ij V n+1
ij −
2
∆t
wn
ijV n
ij +
1
2∆t
wn−1
ij V n−1
ij + Rn+1
ij (w).
The adjoint equations are solved using a either a pseudo-spectral or time accurate
approach, similar to the one applied to the flow equations. Here, the unsteady
residual at each time step is differentiated to produce the discrete adjoint fluxes.
Taking the variation of R∗n+1
ij (w) yields
δR∗n+1
ij (w) =
3
2∆t
δwn+1
ij V n+1
ij −
2
∆t
δwn
ijV n
ij +
1
2∆t
δwn−1
ij V n−1
ij + δRn+1
ij (w),
where,
δR(w)n+1
i,j = δhn+1
i+1
2
,j
− δhn+1
i−1
2
,j
+ δhn+1
i,j+1
2
− δhn+1
i,j−1
2
,
and δhn+1
i+1
2
,j
= δFn+1
i+1
2
,j
−δDn+1
i+1
2
,j
−δFn+1
vi+ 1
2 ,j
. The next step consists of pre-multiplying
δR∗n+1
ij (w) by the Lagrange multiplier, ψ, and integrating over the domain, Ω, to
give
1
T
tf
t=0 Ω
ψT
ijδR∗
ij(w) = ...+ψTn+1
ij δR∗n+1
ij (w)+ψTn+2
ij δR∗n+2
ij (w)+ψTn+3
ij δR∗n+3
ij (w)+...
Substitution of δR∗n+1
ij in the summation produces the discrete time accurate ad-
joint equation,
1
T
tf
t=0 Ω
ψT
ijδR∗
ij(w) = ... + ψTn+1
ij
3
2∆t
V n+1
δwn+1
i,j −
2
∆t
V n
δwn
i,j +
1
2∆t
V n−1
ij δwn−1
ij + δRn+1
i,j
+ψTn+2
ij
3
2∆t
V n+2
δwn+2
i,j −
2
∆t
V n+1
δwn+1
i,j +
1
2∆t
V n
ij δwn
ij + δRn+2
i,j
+ψTn+3
ij
3
2∆t
V n+3
δwn+3
i,j −
2
∆t
V n+2
δwn+2
i,j +
1
2∆t
V n+1
ij δwn+1
ij + δRn+3
i,j
+...
40
54. which can be rearranged in terms of δwn+1
ij
1
T
tf
t=0 Ω
ψT
ijδR∗
ij(w) = ... +
3
2∆t
ψTn+1
ij −
2
∆t
ψTn+2
ij +
1
2∆t
ψTn+3
ij δwn+1
ij V n+1
ij
+ψTn+1
ij δwRn+1
ij (w) + ψTn+1
ij δf Rn+1
ij (w) + ... (3.1)
Note that the first two terms will be introduced in the source of the adjoint equa-
tion, while the remaining expression will influence the calculation of the gradient.
Recall that time dependent discrete Navier Stokes equations are introduced into
∂I as a constraint such that
δI = δIc −
1
T
tf
t=0 Ω
ψT
ijδR∗
ij(w)
Substituting equation (3.1) into the equation above yields
δI = δIc ... −
3
2∆t
ψTn+1
ij −
2
∆t
ψTn+2
ij +
1
2∆t
ψTn+3
ij δwn+1
ij V n+1
ij
−ψTn+1
ij δwRn+1
ij (w) − ψTn+1
ij δf Rn+1
ij (w) − ...
The next step consists of expanding equation (3.2) and extract the δwn+1
i,2 terms to
produce the boundary source term,
δI = δIc ... −
3
2∆t
ψTn+1
ij −
2
∆t
ψTn+2
ij +
1
2∆t
ψTn+3
ij δwn+1
ij V n+1
ij −ψTn+1
ij δwRn+1
ij (w)
−
3
2∆t
ψTn+1
i2 −
2
∆t
ψTn+2
i2 +
1
2∆t
ψTn+3
i2 δwn+1
i2 V n+1
i2 −ψTn+1
i2 δwRn+1
i2 (w)
−ψTn+1
ij δf Rn+1
ij (w) − ...
The expansion of δwRn+1
i2 is required in order to construct the boundary source term
for the time accurate discrete adjoint equation. Although the expansion of δwRn+1
i2
will not be shown in this work, the reader is encouraged to refer to Nadarajah et
41
55. al. [41] for further information regarding the rest of the derivation of δwRn+1
i2 .
ψTn+1
i,2 δwRn+1
i,2 = −
1
2
ATn+1
i− 1
2
,2
(ψn+1
i,2 − ψn+1
i−1,2)
+ATn+1
i+1
2
,2
(ψn+1
i+1,2 − ψn+1
i,2 ) + BTn+1
i, 5
2
(ψn+1
i,3 − ψn+1
i,2 )
−∆yξψn+1
2i,2
+ ∆xξψn+1
3i,2
δwn+1
i,2 .
The time accurate discrete adjoint equation can then be written as
∂ψn+1
i,2
∂τ
−
3
2∆t
ψTn+1
ij −
2
∆t
ψTn+2
ij +
1
2∆t
ψTn+3
ij V n+1
−
1
2
ATn+1
i−1
2
,2
(ψn+1
i,2 − ψn+1
i−1,2) + ATn+1
i+1
2
,2
(ψn+1
i+1,2 − ψn+1
i,2 )
+BTn+1
i, 5
2
(ψn+1
i,3 − ψn+1
i,2 ) − Φ = 0,
where Φ is the source term for our cost function,
Φ = ∆yξψn+1
2i,2
− ∆xξψn+1
3i,2
+ contribution of cost function δIc
Section 3.4
3.3 Formulation of the NLFD Discrete Adjoint Equations
The NLFD discrete adjoint equation can be developed using two separate ap-
proaches. In the first approach, we first take a variation of the unsteady residual
ˆR∗
k represented in equation (2.40) with respect to the state vector ˆwk and shape
function f, to produce
δ ˆR∗
k = ikV δ ˆwk + δ ˆRk.
The next step, would be to expand δ ˆRk as a function of ˆwk. As mentioned ear-
lier, this approach would require a series of convolution sums to express δ ˆRk as a
function of δ ˆwk. This method was not implemented due to its computational cost
and added complexity. Instead, the adjoint equations were solved using a pseudo-
spectral approach similar to the one applied to the flow equations. In the latter
approach, the NLFD adjoint equations are developed from the semi-discrete from
42
56. of the adjoint equation, which can be as expressed as
V
∂ψ
∂t
+ R(ψ) = 0,
where R(ψ) is the sum of all the spatial operators, both convective and dissipative,
used in the discretized adjoint equations. Refer to Nadarajah [31] for a detailed
derivation of these spatial operators and boundary conditions. Next, we assume
that the adjoint variable and spatial operator can be expressed as a Fourier series:
ψ =
N
2
−1
k=−N
2
ˆψkeikt
; R(ψ) =
N
2
−1
k=−N
2
R(ψ)keikt
. (3.2)
The derivation of the NLFD adjoint then follows that of the NLFD flow equations.
The NLFD adjoint equations are expressed as
V
∂ ˆψk
∂τ
+ R(ψ)
∗
k = 0.
where R(ψ)
∗
k = ikV ˆψk +R(ψ)k. The pseudo-spectral approach illustrated in figure
(2–2) is employed in the NLFD adjoint code to form the unsteady residual. This
term in conjunction with a pseudo time derivative provides an iterative solution
process consistent with that documented for the flow equations.
3.4 Formulation of the Cost Function
This section is divided into three parts. Subsections (3.4.1) and (3.4.2) primarily
deal with the formulation of two cost functions presented in chapter 4. The first
set of results demonstrated in chapter 4 require an objective function which mini-
mizes the time-averaged drag and maintains the desired lift and moment levels at
transonic conditions. The second set of results presented in chapter 4 explores a
cost function which maximizes the time-averaged lift while simultaneously reduces
the time-averaged drag. Subsection (3.4.3) derives the cost function of chapter 5,
43
57. which primarily deals with conservation of thrust and reduction of torque.
3.4.1 Preserving cl and cm while Minimizing cd
The NLFD adjoint boundary conditions consists of a time-varying multi-objective
cost function. In order to fully appreciate the performance of the helicopter rotor
blades, the goal of the objective function will be to minimize the time-averaged
drag coefficient on the advancing side of the rotor blade, and to maintain the lift
coefficient on the retreating side. Moreover, in order to reduce any torsional load
generated by the incidence variation, the objective function will also include a
moment contribution. Since symmetric blade geometries do not carry any signifi-
cant torsional loads when compared to cambered airfoils, the goal of the objective
function will be to maintain the time-averaged coefficient of moment to its initial
value. The time-varying weights have been carefully selected to address current
rotor issues; thus, drag minimization will be emphasized on the advancing side of
the rotor via a cosinusoidal function, while lift conservation will be accentuated on
the retreating side via a sinusoidal function. Refer to subsection (4.3.3) for more
information regarding these time-varying penalty functions.
Ic =
tf
t=0
1(t)(clt − cl)2
+ 2(t)cd + 3(t)(cmt − cm)2
∆t, (3.3)
where the lift, drag, and moment coefficient can be expressed as
cl = cn cos α − ca sin α, (3.4)
cd = cn sin α + ca cos α, (3.5)
cm =
i
cni
(xi − xac) + cai
(yi − yac) . (3.6)
44
58. Substituting equations (3.4),(3.5),(3.6), back in (3.3) yields
Ic =
tf
t=0
1(t)(clt − {cn cos α − ca sin α})2
∆t
+
tf
t=0
2(t) {cn sin α + ca cos α} ∆t
+
tf
t=0
3(t)(cmt −
i
{cni
(xi − xac) + cai
(yi − yac)})2
∆t. (3.7)
Equation (3.7) can be recast in terms of pressure, p, and geometry, x, y,
Ic = 1(t) clt −
1
1
2
γp∞M2
∞c
tf
t=0
UTE
i=LTE
(pi − p∞)(−
∆xi
∆si
cos α −
∆yi
∆si
sin α ∆si∆t
2
+ 2(t)
1
1
2
γp∞M2
∞c
tf
t=0
UTE
i=LTE
(pi − p∞)(
∆yi
∆si
cos α −
∆xi
∆si
sin α) ∆si∆t
+ 3(t) cmt −
1
1
2
γp∞M2
∞c
tf
t=0
UTE
i=LTE
(pi − p∞)(−
∆xi
∆si
(xi − xac) +
∆yi
∆si
(yi − yac)) ∆si∆t
2
.
A variation in the cost function results to a variation in the pressure, ∆p, and in
the geometry, ∆x, ∆y. Thus, taking the derivative of the expression above yields
two categories of terms. The first set of terms is dependent on the flow, and will
ultimately influence the calculation of the Lagrange multiplier,
δIcflow
=
2 1(t)(clt − cl) −Γ
tf
t=0
UTE
i=LTE
(
δp
δw
)
δwi2 + δwi1
2
−
∆xi
∆si
cos α −
∆yi
∆si
sin α ∆si∆t
+ 2(t) Γ
tf
t=0
UTE
i=LTE
(
δp
δw
)
δwi2 + δwi1
2
(
∆yi
∆si
cos α −
∆xi
∆si
sin α ∆si∆t
+2 3(t)(cmt − cm) −Γ
tf
t=0
UTE
i=LTE
(
δp
δw
)
δwi2 + δwi1
2
(−
∆xi
∆si
+
∆yi
∆si
ϑ) ∆si∆t ,
+ negligible terms...
45
59. The second category is dependent on the shape function, and will ultimately con-
tribute to the calculation of the gradient,
δIcshape
=
2 1(t)(clt − cl) −Γ
tf
t=0
UTE
i=LTE
pi2 + pi1
2
− p∞ (−δ∆xi cos α − δ∆yi sin α) ∆t
+ 2(t) Γ
tf
t=0
UTE
i=LTE
pi2 + pi1
2
− p∞ (δ∆yi cos α − δ∆xi sin α ∆t
+2 3(t)(cmt − cm) −Γ
tf
t=0
UTE
i=LTE
pi2 + pi1
2
− p∞ (−δ∆xi + δ∆yiϑ) ∆t
+ negligible terms...
where,
Γ =
−1
1
2
γp∞M2
∞c
; = (xi − xac); ϑ = (yi − yac).
3.4.2 Maximizing cl while Minimizing cd
The design stage can be further developed in chapter 4 by imposing a multi-
objective cost function which maximizes the time-averaged lift and minimizes the
time-averaged drag. Note that in this case, the penalty functions are not dependent
of time; instead, they are determined by the geometry of the blade at every design
cycle.
Ic =
tf
t=0
{− 1cl + 2cd}∆t,
=
tf
t=0
UTE
i=LTE
− 1
1
c
− cp
∆yi
∆si
sin α +
1
c
− cp
∆xi
∆si
cos α ∆si∆t
+
tf
t=0
UTE
i=LTE
2
1
c
cp
∆yi
∆si
cos α +
1
c
− cp
∆xi
∆si
sin α ∆si∆t,
where,
1 =
γ
cl + βcd
; 2 =
cl + βcd
γ
; γ = cl +
cl
cd
cd; β ∈ {R+
}.
46
60. The β weight controls the strength of the drag contribution: a high β emphasizes
drag minimization; while a low β promotes maximum lift. The weight also acts
as a buffer which prevents our design algorithm from oscillating erratically. We
could ultimately rid the oscillations by replacing our steepest decent optimization
method with a conjugate gradient with line search approach.
3.4.3 Preserving ct while Minimizing cd
For chapter 5, the objective function consists of a weighted sum of the difference
of the current and initial thrust coefficients and the torque coefficient as written
below,
Ic =
tf
t=0
{ 1 (ct − ctinitial
)2
+ 2cq}∆t.
where 1 and 2 are the weights for the objective function. The magnitude of the
weights were based on trial and error and the final values will be presented and
discussed in chapter 5. The initial thrust coefficient, ctinitial
, is set at the end of the
initial flow solver cycle as the goal of the objective is to maintain the current level
of thrust while reducing the torque coefficient.
3.5 Design Procedure
The UFSYN103P developed by Nadarajah and Jameson employs a non-linear fre-
quency domain method in the solution of the unsteady Navier-Stokes equations.
The NLFD adjoint based design procedures require the following steps:
1. Periodic Flow Calculation. A set of multigrid cycles is used to drive
the unsteady residual to a negligible value for all the modes used in the
representation of the solution.
2. Adjoint Calculation. The adjoint equation is solved by integrating in re-
verse time. With minor modifications, the NLFD numerical scheme employed
47
61. to solve the flow equations is used to solve the adjoint equations in reverse
time.
3. Gradient Evaluation. An integral over the last period of the adjoint solu-
tion is used to form the gradient. This gradient is then smoothed using an
implicit smoothing technique.
4. Wing Shape Optimization. The blade shape is then modified in the direc-
tion of improvement using a smoothed steepest descent method as described
in the previous section.
5. Grid Modification. The internal grid is modified based on perturbations on
the surface of the blade. The method modifies the grid points along each grid
index line projecting from the surface. The arc length between the surface
point and the far-field point along the grid line is first computed, then the
grid point at each location along the grid line is attenuated proportional to
the ratio of its arc length distance from the surface point and the total arc
length between the surface and the far-field.
6. Repeat the Design Process. The entire design process is repeated until
the objective function converges.
48
62. CHAPTER 4
Aerodynamic Shape Optimization for Helicopter Blade Sections
This chapter is divided into four sections. Sections (4.1) and (4.2) document the
numerical validation for a NACA 0007 and a MBB BO-105 helicopter rotor blade
section, while sections (4.3) and (4.4) examine the approach for their respective re-
design. Given the derivation provided in the previous sections, the adjoint bound-
ary condition can easily be modified to admit other figures of merit. Note that
the shape of the blade section is constrained such that the maximum thickness to
chord ratio remains constant between the initial and final designs.
4.1 Validation of a Pitching NACA 0007 with Variable Mach Number
The following subsection contains a code validation for a NACA 0007 airfoil un-
dergoing both a pitching angle variation and sinusoidal variation of the freestream
Mach number. The study establishes the required number of time steps needed
to resolve the flow field by juxtaposing the lift, drag and moment hysteresis loops.
A comparison between the NLFD and the time accurate approaches is presented
to demonstrate and emphasize the advantage in computational cost of the NLFD
technique. The test case itself is a proof of concept for the NLFD method, since it
demonstrates a cost effective technique to design rotor blade sections.
The two-dimensional airfoil undergoes a change in angle of attack and Mach number
as a function of time, via
α(t) = αo + αm sin(ωt),
M(t) = Mo − Mm sin(ωt).
49
63. For the cases presented in this subsection, the mean angle of attack, αo, is 2.89◦
.
The deflection angle, αm, is set to ±2.41◦
. The reduced frequency, Ωr = ωc
2V∞
, is
set to 0.081, with a mean Mach number, Mo, of 0.6, and a variation of ±0.1. The
airfoil is pitched about the 25% of the chord.
Figure (4–1) demonstrates the number of time steps required for the NLFD flow
solver to reach a converged hysteresis loop. For viscous pitching airfoils, McMullen
illustrated that one temporal mode was sufficient to provide a convergent solution
to accurately plot the lift hysteresis. [39] However, further investigation clarified
that the presence of a second harmonic was necessary to accurately plot the drag
variation, due to the presence of a double peak in the sinusoidal shape. [41] In
this study, the combined variation in both the pitching angle and Mach number
produces a sinusoidal solution that requires more than two modes to capture. Fig-
ure (4–1) illustrates that at least five time steps, or two harmonics, is required to
attain a general trend and nine time steps, or four harmonics, is needed to attain
a converged solution.
Previous work [61, 62] revealed that the time accurate solutions required 5 periods
until the transient errors were sufficiently diminished. The decay of these transients
errors represents the dominant computational cost for the time accurate scheme,
when compared to the NLFD algorithm. We should appreciate the fact that the
NLFD method does not produce any transient errors; and thus, its solution after
1 period is equivalent to the 5 periods time accurate outcome, in terms of global
convergence. This dramatically reduces the number of multigrid cycles needed to
obtain a converged solution. This is demonstrated in figure (4–2), where the lift,
drag, and moment hysteresis loops of the 9 time steps NLFD test case is compared
with the 12, 24, and 36 time steps time accurate solutions. The hysteresis loop
50
64. for all four cases compare very well; the NLFD lift, drag, and moment variations
closely follow the time accurate solution with 24 time steps. Figure (4–3) further
demonstrates the validation presented in figure (4–2) with a comparison of the
pressure distributions at three separate phases between the NLFD and time accu-
rate solutions. The pressure distributions are in good agreement with one another,
with an almost point to point match. We observe a small discrepancy in the region
of the shock wave; however, the location and strength of the shock are in agree-
ment. Figure (4–4-a) investigates the behavior of the blade’s cross-sections when
subjected to very low rotational speeds. This exercise validates the implementa-
tion of the oscillating pitch with the variable Mach number. As shown from the
figure, the low reduced frequency pressure distributions compare very well to those
of the steady-state, with an almost point to point match. This statement is further
demonstrated in figures (4–4-b) and (4–4-c), where we repeat the same exercise for
different phases of the rotor.
4.2 Validation of a Pitching NACA 23012 with Variable Mach Number
In this subsection, we solve for the flow around a pitching NACA 23012 undergoing
a sinusoidal variation in Mach number. The change in angle of attack and Mach
number can be defined as
α(t) = αo + αm sin(ωt),
M(t) = Mo − Mm sin(ωt).
where the mean angle of attack, αo, is 5.5◦
. The deflection angle, αm, is set to
±4.0◦
about the quarter chord. The reduced frequency, ωc
2V∞
, is set to 0.151, with
a mean Mach number, Mo, of 0.441, and a variation, Mm, of ±0.17.
Figures (4–5-a), (4–5-b), and (4–5-c) demonstrate the number of time steps re-
quired for the NLFD flowsolver to reach a converged hysteresis loop. Similarly to
51
65. the previous test case, at least seven time steps, or three harmonics, are required
to attain the general trend for lift and drag; and nine time steps, or four harmon-
ics, are needed to attain a converged solution for moment. The eleven time steps
curve confirms that nine time steps are enough to capture all three figures of merit.
Figure (4–5-d) illustrates the convergence history for the NLFD scheme. For this
particular test case, the 0th, 1st, and 2nd modes all converge at the same rate,
with the exception of the NLFD 3 time steps, which proceeds at a faster pace. The
0th mode approaches 1×10−4
, while the 1st mode moves toward 5×10−3
, and the
2nd mode closes in on 1 × 10−2
.
Figures (4–6-a) establish the number of time steps required for the NLFD flow-
solver to reach a converged pressure distribution. In most cases, five time steps,
or two harmonics, is necessary to obtain a general trend; and seven time steps, or
three harmonics, is required to achieve a converged distribution. Figures (4–6-c)
and (4–6-d) further investigate the pressure distributions by decomposing them in
their respective real and imaginary first harmonic parts. Notice the dramatic dif-
ference between the three and five time steps curves. Moreover, the real component
requires seven time steps to complement the overall trend, while nine time steps
confirm the convergence. This further demonstrates the need for nine time steps
to obtain accurate results. The imaginary component does not play a significant
role in setting the time step benchmark. Finally, figure (4–6-d) illustrates the con-
verged NLFD 9 time steps pressure distributions for various phases. Notice how
the leading edge pressure peak oscillates as the blade rotates.
4.3 Redesign of a NACA 0007 Airfoil
The following subsections contain a redesign of a NACA 0007 airfoil undergoing
both a pitching angle variation and sinusoidal variation of the freestream Mach
52
66. number. The redesign of the NACA 0007 airfoil requires an objective function
which minimizes the time-averaged drag coefficient and maintains the desired lift
and moment levels at transonic conditions.
I =
1
T
N
n=1
{ n
1 (t)(cn
lt
− cn
l )2
+ n
2 (t)cn
d + n
3 (t)(cn
mt
− cn
m)2
}∆t
Subsection 1 2 3
§4.1.2 3.0 5.5 8.5 0.333 0
§4.1.3 8.5 0.333 0.0005 0.0040
§4.1.4 8.5 ± f(t) 0.333 ± f(t) 0
Table 4–1: Weights Variation for Subsections (4.3.1), (4.3.2), and (4.3.3)
4.3.1 Preserving cl while Minimizing cd
For the design test cases, the mean angle of attack, αo, is 3.0◦
, with a deflection
angle, αm, of ±3.0◦
. The reduced frequency, Ωr = ωc
2V∞
, is set to 0.005; while the
mean Mach number, Mo, is set to 0.6, with a variation of ±0.3. The objective of
the redesign is to achieve a low time-averaged drag coefficient while simultaneously
maintain the lift coefficient at every phase of the period. We will first demonstrate
the effect of imposing weights of increasing magnitude on the lift contribution of
the multi-objective cost function. We will then compare these results with another
lift conservation method. Note that for this specific test case, the weights from the
cost function do not vary with time, and the moment conservation is not respected.
Figure (4–7) illustrates the convergence of the time-averaged lift and drag as we
impose weights of increasing magnitude for the lift coefficient. In all cases, the
drag minimization contribution is present with a weight of 1
3
, while the moment
contribution is switched off with a weight of 0. These coefficients are maintained
through the inclusion of these terms within the objective function as shown in
the section (3.4), titled ’Formulation of the Cost Function’. As shown from the
53
