unique design discoveries for a modern mach 1.3 airliner
TRANSCRIPT
Unique Design Discoveries for a Modern Mach 1.3 Airliner Including Anomalies in the
Shock Wave Formation Along a Highly Swept Blunt Leading Edge Wing
by
Noah John Kurus
A Thesis Presented in Partial Fulfillment
of the Requirements for the Degree of
Master of Science
Approved November 2020 by the
Graduate Supervisory Committee:
Timothy Takahashi, Chair
Mary Niemczyk
David Benson
ARIZONA STATE UNIVERSITY
December 2020
i
ABSTRACT
The process of designing any real world blunt leading-edge wing is tedious and
involves hundreds, if not thousands, of design iterations to narrow down a single design.
Add in the complexities of supersonic flow and the challenge increases exponentially.
One possible, and often common, pathway for this design is to jump straight into detailed
volume grid computational fluid dynamics (CFD), in which the physics of supersonic
flow are modeled directly but at a high computational cost and thus an incredibly long
design process. Classical aerodynamics experts have published work describing a process
which can be followed which might bypass the need for detailed CFD altogether.
This work outlines how successfully a simple vortex lattice panel method CFD
code can be used in the design process for a Mach 1.3 cruise speed airline wing concept.
Specifically, the success of the wing design is measured in its ability to operate sub-
critically (i.e. free of shock waves) even in a free stream flow which is faster than the
speed of sound. By using a modified version of Simple Sweep Theory, design goals are
described almost entirely based on defined critical pressure coefficients and critical Mach
numbers. The marks of a well-designed wing are discussed in depth and how these traits
will naturally lend themselves to a well-suited supersonic wing.
Unfortunately, inconsistencies with the published work are revealed by detailed
CFD validation runs to be extensive and large in magnitude. These inconsistencies likely
ii
have roots in several concepts related to supersonic compressible flow which are
explored in detail. The conclusion is made that the theory referenced in this work by the
classical aerodynamicists is incorrect and/or incomplete. The true explanation for the
perplexing shock wave phenomenon observed certainly lies in some convolution of the
factors discussed in this thesis. Much work can still be performed in the way of creating
an empirical model for shock wave formation across a highly swept wing with blunt
leading-edge airfoils.
iii
DEDICATION
Dedicated to my loving parents who have tirelessly supported my dreams and inspired
me to pursue my ambitions of a graduate degree and graduate research.
iv
AKNOWDLEGEMENTS
A special thanks to Maurice Nayman at the Royal Military College of Canada for
assisting in this research by providing some of the CFD simulations (STAR CCM++)
contained in this thesis.
v
TABLE OF CONTENTS
Page
LIST OF TABLES ................................................................................................................ vii
LIST OF FIGURES ............................................................................................................. viii
CHAPTER
1 INTRODUCTION ........................................................................................... 1
2 PRIOR ART – BASIS FOR DESIGN ........................................................... 5
a. The Concept of the Upper and Lower Critical Mach Numbers ....... 5
b. The Lower Critical Mach Number and Critical Pressure
Coefficient .........................................................................................14
c. The Upper Critical Mach Number and Critical Pressure
Coefficient .........................................................................................17
d. Slender Body Theory Wave Drag and Its Relationship to the
Lower Critical Mach Number and Area Distributions ...................25
3 SIZING THE BASIC AIRFRAME ..............................................................33
4 COMPUTATIONAL DESIGN OF GEOMETRY...................................... 39
5 SURFACE PANEL METHOD RESULTS AND IMLPICATIONS OF
WAVE DRAG ...............................................................................................59
6 VOLUME GRID RESULTS ........................................................................ 70
7 ANAYLZING CRITICAL MACH NUMBER(S) AND PRESSURE
COEFFICIENT(S)...................................................................................... 106
vi
CHAPTER Page
8 CONCLUSIONS ....................................................................................... 119
REFERENCES .....................................................................................................................123
vii
LIST OF TABLES
Table Page
1. Wing Constraints and Final Specifications ...................................................................41
2. Defining the Section Airfoil Ordinates ..........................................................................56
3. Harris Wave Drag Results Indicating a Geometry Well Suited for Wave Drag .........68
4. VORLAX and SU2 Tabular Data ..................................................................................81
viii
LIST OF FIGURES
Figure Page
1. Rendering of Mach 1.25 Supersonic Transport1 .............................................................2
2. Proposed East Coast Routing of a Mach 1.25 Supersonic Transport1 ...........................3
3. Notional Interior of 150-seat of Supersonic Transport1..................................................3
4. Spanwise and Chordwise Flow Representation on a Swept Wing ................................6
5. Simple Sweep Theory Effective Mach Number for Different Sweep Angles ..............8
6. Critical Pressure Coefficient Based on Simple Sweep Theory for Multiple Sweep
____Angles ...............................................................................................................................9
7. Isobars Unsweeping Crossing the Line of Bi-Lateral Symmetry12..............................12
8. Flow Structure Around VORLAX Sandwich Panels....................................................13
9. Cp* vs. Mach Number for Purely 2-D Flow...................................................................15
10. Cp Trend with Increasing Mach Number19 ..................................................................16
11. Cp* Trend with Increasing Mach Number for Swept Wings7 .....................................19
12. Cp* Workup for a 57-Degree Leading Edge Swept Wing at Mach 1.25 ...................21
13. CL Workup for a 57-Degree Leading Edge Swept Wing at Mach 1.25 ....................21
14. Airbus A380 Using Negative Camber at Root of Horizontal Stabilizer and Positive
____Camber Outboard ...........................................................................................................23
15. A Visual Representation of the Mach Cone Angle .....................................................28
16. Shock Wave Formation Modeling on a Supersonic Transport ..................................29
17. Supersonic Area Rule Wave Drag Formulation..........................................................30
18. Sears-Haack Body .........................................................................................................31
ix
19. “Corona-Shocks” Forming Past the Point of Maximum Thickness26........................32
20. Initial Sizing Algorithm in ModelCenter1 ...................................................................33
21. Multi-Variate Trade Study for Airframe Parameters1 ................................................35
22. Mission Profile as Described Below for the Supersonic Transport1..........................37
23. VORLAX Panel Structure ............................................................................................43
24. Resultant VORLAX Critical Mach Number Workup ................................................46
25. Upper Surface Isobar Contours from the Low Speed VORLAX Sandwich Panel
____Solution ...........................................................................................................................46
26. Lower Surface Isobar Contours from the Low Speed VORLAX Sandwich Panel
____Solution ...........................................................................................................................47
27. Enhanced Upper Surface Isobar Indicating Line of Peak Under-Pressure from the
____Low Speed VORLAX Sandwich Panel Solution .........................................................47
28. Transverse Span Load from Low Speed VORLAX Design Solution .......................48
29. Section Lift Coefficient from Low Speed VORLAX Design Solution .....................49
30. Basic Thickness Forms .................................................................................................50
31. Basic Camber Lines ......................................................................................................51
32. Spanwise Variation in Thickness .................................................................................52
33. Spanwise Variation in Camber .....................................................................................52
34. Spanwise Variation in Incidence ..................................................................................53
35. Overall Wing and Body Geometry ..............................................................................56
36. Airfoil Definitions Along the Span of the Wing .........................................................57
Figure Page
x
37. VORLAX Flat Plate Model with the Center of Gravity Marked ...............................60
38. CL vs. α for the VORLAX Flat Plate and Sandwich Panel Model ............................61
39. Longitudinal Stability of the VORLAX Wing-Body Flat Plate Model and Sandwich
____Panel Model About its MRP (88-feet FS; Mach = 0.3) ...............................................62
40. CP Distribution at 41.8% Span Away from the Side of Body ....................................63
41. CP Distribution at 51.5% Span Away from the Side of Body ....................................63
42. CP Distribution at 61.2% Span Away from the Side of Body ....................................64
43. CP Distribution at 80.6% Span Away from the Side of Body ....................................64
44. CP Distribution at Wing Tip .........................................................................................65
45. Harris Wave Drag Input File ........................................................................................67
46. Wave Drag Results from D2500 Wavedrag Program ................................................69
47. Volume Grid Density Screenshot for the SU2 Model ................................................73
48. Surface Grid Density Screenshot for the SU2 Model .................................................73
49. Grid Convergence Study Run on Low Speed Case (M = 0.3, α = 0.2 deg) ..............74
50. Low Speed Comparison Between SU2 (Top) and VORLAX (Bottom) (M = 0.3, α =
____0.2 deg) ...........................................................................................................................75
51. CL vs. α Plot for the Low Speed Case with SU2 .........................................................76
52. CL vs. CD Plot for the Low Speed Case with SU2 ......................................................77
53. CL2 vs. CD Plot for the Low Speed Case with SU2 ....................................................77
54. CM vs. CL Plot for the Low Speed Case with SU2.....................................................78
55. Convergence History for All Low Speed SU2 Runs ..................................................80
Figure Page
xi
56. Pressure Coefficient Contour from SU2 at Mach 1.3 (α = 0.2 deg, CL = 0.178) ......83
57. Mach Number Contour from SU2 at Mach 1.3 (α = 0.2 deg, CL = 0.178)................83
58. α vs. CL Plot for the Mach 1.3 Cruise Speed Case with SU2 ....................................84
59. CL vs. CD Plot for the Mach 1.3 Cruise Speed Case with SU2 .................................85
60. CM vs. CL Plot for the Mach 1.3 Cruise Speed Case with SU2.................................85
61. Pressure Coefficient Contour from SU2 at Mach 0.95 (α = 0.2 deg, CL = 0.204) ....87
62. Mach Number Contour from SU2 at Mach 0.95 (α = 0.2 deg, CL = 0.204)..............87
63. Shock Wave Angle Inconsistency with Mach Angle at Mach 0.95 (α = 0.2 deg, CL
____= 0.204) ...........................................................................................................................88
64. Pressure Coefficient Isobar Superimposed with the Cross-Sectional Area
____Distribution and it’s Second Derivative (Mach 0.95) ..................................................89
65. α vs. CL Plot for the Mach 0.95 Case with SU2 .........................................................90
66. CL vs. CD Plot for the Mach 0.95 Case with SU2 ......................................................91
67. Pressure Coefficient Contour from SU2 at Mach 1.05 (α = 0.2 deg, CL = 0.186) ....92
68. Mach Number Contour from SU2 at Mach 1.05 (α = 0.2, CL = 0.186).....................92
69. Shock Wave Angle Inconsistency with Mach Angle at Mach 1.05 (α = 0.2 deg, CL
____= 0.186) ...........................................................................................................................93
70. CL vs. CD Plot for the Mach 1.05 Case with SU2 ......................................................93
71. Pressure Coefficient Isobar Superimposed with the Cross-Sectional Area
____Distribution and it’s Second Derivative (Mach 1.05) ..................................................94
Figure Page
xii
72. Visual Representation of Cutting Planes for Cross-Sectional Area Calculations ____
____ (Mach 1.05→ 72.2 deg) ................................................................................................95
73. Pressure Coefficient Contour from STAR CCM++ at Mach 0.95 (α = 0.2 deg, CL =
____0.230) ..............................................................................................................................96
74. Pressure Coefficient Contour from STAR CCM++ at Mach 1.05 (α = 0.2 deg, CL =
____0.209) ..............................................................................................................................97
75. Zero-Lift Drag from SU2 and Wavedrag from D2500 Superimposed ......................98
76. Aerodynamic Center and Center of Pressure for Each Mach Number and Angle of
____Attack ........................................................................................................................... 100
77. Finite Differencing in the Low Speed Stability Characteristics.............................. 101
78. Implications of Sweep Angles in Relation to Mach Lines ...................................... 102
79. Wing Geometry Superimposed on Mach Cone Angles at Mach 1.3 (Mach Lines
____Propagate from Trailing Edge Intersection) .............................................................. 103
80. Wing Geometry Superimposed on Mach Cone Angles at Mach 1.3 (Mach Lines
____Propagate from Yehudi Intersection) ......................................................................... 104
81. Pressure Coefficient Contour from SU2 at Mach 2.0 (α = 0.2 deg, CL = 0.110) ... 105
82. Critical Pressure Coefficient Workup for Mach 0.95 and a 57-Degree Leading-Edge
____Swept Wing ................................................................................................................. 108
83. Extrapolation to a Critical Pressure Coefficient of -0.48 at Mach 1.18 with 57-
____Degrees of Leading-Edge Sweep ............................................................................... 108
84. CP Distribution at 41.8% Span Away from the Side of Body Critical Pressure
____Coefficient Workup for Mach 1.05 and a 57-Degree Leading-Edge Swept Wing . 110
Figure Page
xiii
85. CP Distribution at 41.8% Span Away from the Side of Body Extrapolation to a
____Critical Pressure Coefficient of -0.61 at Mach 1.12 with 57-Degrees of Leading-
____Edge Sweep ................................................................................................................. 110
86. Difference in Observed Mach Number and Equivalent Mach Number ................. 111
87. Local Isobar Sweep Angle from the Low Speed SU2 Solution Superimposed on the
____Relevant Mach 0.95 SU2 Solution ............................................................................. 113
88. Critical Pressure Coefficient Workup for Mach 0.95 and a 25.3-Degree Pressure
____Isobar Sweep Angle .................................................................................................... 114
89. Local Isobar Sweep Angle from the Low Speed SU2 Solution Superimposed on the
____Relevant Mach 1.05 SU2 Solution ............................................................................. 116
90. CP Distribution at 41.8% Span Away from the Side of Body Critical Pressure
____Coefficient Workup for Mach 1.05 and a 49.2-Degree Pressure Isobar Sweep
____Angle ............................................................................................................................ 117
Figure Page
1
Chapter 1: Introduction
The days of supersonic flights across the globe appear to have come and
gone amidst rising fuel prices and environmental pressures making such a bold concept
no longer practical or profitable for airlines. This study focuses on the design process for
a wing for a supersonic transport category aircraft. This wing was initially designed for a
Senior Capstone Design project at Arizona State University for the Fall of 20191. The
mission drivers supporting this aircraft design were an ability to fly daytime flights across
the North Atlantic. With a design cruise speed of Mach 1.3, it is fast enough to allow a
flyer to take an early morning flight to connect into an east coast hub airport and reach
central Europe later that evening or take an early morning flight from a major East Coast
gateway city and arrive in the UK or Europe early enough to connect to an evening intra-
European flight to their final destination. Although the team did not design this aircraft to
have a low subsonic boom, it is a conceptual design that otherwise shares almost no
commonality with the Concorde3. The differences become very apparent when looking at
the full CAD model of this aircraft seen in Figure 1. The most notable design features
that make this airplane stand apart from Concorde are its lack of a pure delta wing, the
low bypass ratio turbofans mounted with normal shock inlets, and the rather large “T-
tail” setup, although the differences certainly don’t end there.
2
Figure 1. Rendering of Mach 1.25 Supersonic Transport1
The driving factors that govern the overall aerodynamic configuration, and the
detail wing design are the aircraft specific range in terms of fuel burn per seat and how
much fuel it can carry within the wing. In order to be viable, a commercial supersonic
transport aircraft must have a low enough fuel burn as to be competitive with premium
subsonic services (Figure 2). For subsonic transatlantic services, most scheduled flights
are overnight “red-eye” services with premium seating offering 180-degree reclining
seats or private suites4. For daytime supersonic transatlantic services, with a flight time
commensurate with a domestic transcontinental flight, premium seating need not be so
generous. Thus, the reference aircraft was engineered to have an interior with 34 rows of
4-abreast seats (Figure 3). Each seat is 18 inches wide and 34-inches in pitch with a 20-
inch aisle thus resembling a “domestic premium-economy” interior. This supports a 150-
seat aircraft with an approximately 170-foot long and 122-inch diameter fuselage that
features an approximately 103-foot long interior that is nearly 110-inches wide.
3
Figure 2. Proposed East Coast Routing of a Mach 1.25 Supersonic Transport1
Figure 3. Notional Interior of 150-seat of Supersonic Transport1
The following elements are discussed in this thesis:
1. Why the wing is designed to have an Upper Critical Mach Number of
1.25.
2. How and why an admittedly simple potential flow code can be used to
design a supersonic cruise wing.
3. Whether volume-grid CFD can successfully validate the potential flow
design at subcritical speeds.
4
4. If the low-speed design Cp* approach works as shown with a volume-grid
CFD analysis at supersonic flight speeds and provide insight into any
discrepancies CFD might uncover.
5
Chapter 2: Prior Art – Basis for Design
The wing design process discussed herein has its roots in an approach outlined by
Takahashi5,6, Küchemann7 and Obert8, Neumark12. The idea is to identify the critical
pressure coefficient for incipient sonic flow at the design point6,7, transform it back to a
low-speed design pressure coefficient and rigorously develop a three-dimensional loft to
develop a wing that features an elliptical transverse load distribution with isobars
matching the leading-edge sweep angle8. The Transonic Area Rule9 and its practical
implications must also be considered for the design problem.
a. The Concept of the Upper and Lower Critical Mach Numbers
While classic texts10,11 consider the “wave drag” due to the transonic and supersonic
area rule to be independent of the “drag divergence” of a 2-D wing section, the
importance of Neumark’s12 rarely cited paper comes to light.
The delay in the onset of wing-induced shock-waves, and its resulting drag
implications, was first studied in Göttingen, in Germany in the 1930’s. The pioneering
work of Busemann13 and Ludweig14 clearly demonstrated the ability for a swept-back
wing to delay shock-wave formation at high, subsonic speeds. While this empirical data
was quickly incorporated into aircraft designs, such as the Boeing B74715 it was not until
6
1954 that the fundamental problem of actually calculating critical Mach numbers and
their associated critical pressure coefficients was published in open literature.
Neumark12 aimed to solve the problem of determining the critical Mach number for
an aircraft with a fuselage and wing of arbitrary profile. Busemann13, Ludweig14 as well
as R.T. Jones16,17 expanded on this concept and considered the case an infinite straight
wing, albeit at an angle. This forms the basis for simple sweep theory where they then
resolve the flow into components parallel and perpendicular to the wing edges as seen in
Figure 4.
Figure 4. Spanwise and Chordwise Flow Representation on a Swept Wing
7
They then claim that transverse or spanwise flow components are insignificant; all
that matters is the flow component normal to the edges or the chordwise component. By
this extension, they consider the flow essentially 2-D. Simple Sweep Theory notes that
the normal component of the flow has an undisturbed velocity equal to the freestream
velocity multiplied by the cosine of leading edge sweep angle ; and the 'effective' Mach
number can be taken as the freestream Mach number multiplied by the cosine of the
leading edge sweep angle (Λ) as seen in Equation 1.
𝑀𝑎𝑐ℎ𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 = 𝑀𝑎𝑐ℎ𝑓𝑟𝑒𝑒𝑠𝑡𝑟𝑒𝑎𝑚 ∗ cos(Λ) [1]
Thus, simple sweep theory claims that if the sweep angle is 60 degrees, then the
critical Mach number is doubled since the cosine of 60 degrees is 0.5. This can also be
observed graphically in Figure 5 where the point at which the effective Mach numbers
are reaching sonic are well into the supersonic regime; this intercept is what is referred to
as the Critical Mach number in the free stream.
8
Figure 5: Simple Sweep Theory Effective Mach Number for Different Sweep Angles
However, the idea of the Critical Mach number above is assuming that the maximum
velocity at any point along the wing is keyed to the free steam velocity. For wings of
finite thickness and even more so for cambered airfoils, this is not the case. These types
of airfoils will accelerate the flow around them thus inducing local velocities that are in
excess of the freestream velocity. That being said, it is more useful to instead define a
Critical pressure coefficient that will correspond to the Critical Mach number. This
Critical pressure coefficient is what will be monitored along the wing to ensure that at no
point along the wing is the local pressure coefficient in excess of the predefined Critical
pressure coefficient based on the sweep angle in Simple Sweep theory. This critical
pressure coefficient is referred to as Cp* and Figure 6 shows how the critical pressure
changes in each free stream Mach number at different angles of sweep. Note that a
9
pressure coefficient of 0 represents a situation in which the flow cannot be sped up
around an airfoil past the free stream velocity without reaching sonic conditions. This
will be expanded upon further in this section.
Figure 6. Critical Pressure Coefficient Based on Simple Sweep Theory for Multiple
Sweep Angles
Recently, Kirkman & Takahashi20,21 demonstrated that simple sweep theory is not
formally correct. Even for idealized 2-D flow; simple sweep theory predicts significantly
inaccurate (and optimistic) performance. They noted that a more nuanced theory, found
in Küchemann7, but stemming from Neumark12 is correct. As Neumark stated “that the
problem of critical Mach numbers for swept wings is a serious scientific problem which
10
cannot be solved by an empirical 'guess' such as, for instance, the notorious” cosine law12.
The cosine law which he was referring to is the Simple Sweep theory described in the
previous paragraphs.
Neumark12 holds for proper aerodynamic design two tasks must be accomplished.
The first is to define the critical conditions of the flow, being the conditions at which, if
exceeded, trigger supersonic phenomena (i.e. shock-waves) possible, at least locally. The
second being to determine the velocity distribution over the surface of the airplane;
especially maximum incremental velocities (super-velocities) and their location, first at
low Mach numbers, i.e., in incompressible flow and later at higher Mach numbers with
the appropriate correction factors in place. Neumark’s key observation is that there are
two critical conditions associated with the flow. The first is when the local velocity of the
flow reached the local sonic condition (this is associated with the concept of the “lower
critical Mach number”), and the second, where some component of the velocity (that
“normal to the isobars”) exceeds a critical condition (this is associated with the concept
of the “upper critical Mach number”).
For the simple, infinite yawed wing of Busemann13 it is clear that 'normal to the
isobars ' means simply 'normal to the leading edge;’ because there is no plane of bi-lateral
symmetry in an infinite wing, hence Busemann’s original idea is a particular case of a
more general one. From a design perspective, it’s preferable to engineer an aerodynamic
11
shape that when pitched to the required angle-of-attack, creates wing-borne isobar
patterns that allow the airplane to fly at supersonic speed, while “pretending” to fly at
subsonic conditions.
However, since the isobars unsweep as they cross the apex of an ideal wing or
cross the fuselage on a practical wing-body configuration, the general criterion also
predicts troublesome regions on real airplanes. On an aircraft with bi-lateral symmetry,
the isobars must be un-swept along the centerline; thus the “lower critical Mach number”
must consider the full freestream flow velocity. In other words, the 'lower critical Mach
number' will always be less than 1 because there is always a section where the isobars
must be orthogonal to the free stream flow; whereas a wing with substantial sweep can
elevate the upper critical Mach number to a value considerably in excess of 1.0. This can
be observed in action in Figure 7 where the isobars clearly unsweep as they get close to
the center of the wing body.
12
Figure 7. Isobars Unsweeping Crossing the Line of Bi-Lateral Symmetry12
To solve the velocity distribution over the surface of the airplane, and determine
the basic super-velocity field, the “sandwich panel” feature in VORLAX is used to
support the design process. VORLAX18 is a generalized subsonic/supersonic vortex
lattice panel method code. While the code supports subsonic and supersonic influence
coefficients for thin, “double-impermeable” panels (conventional “vortex lattice”
paneling), it only supports subsonic solutions using the “single-impermeable” panel
elements needed to properly model a thick, cambered wing. In addition, the generalized
vortex lattice method is a potential flow model and hence lacks any ability to model
shock waves. A visual for how the sandwich panels work in provided below in Figure 8
where the velocity normal to the surface is zero on the outside of the panels but unknown
on the inside. Since they are both single-impermeable, as with any vortex lattice code, the
velocity on the inside does not affect the solution and the normal velocity on the outside
is zero.
13
Figure 8. Flow Structure Around VORLAX Sandwich Panels
Nonetheless, it is easy to build and solve thick sandwich panel models using this
code, and hence its utility will be showcased to design a subcritical or slightly
supercritical supersonic wing using it.
Since a VORLAX “thick sandwich” panel model is constrained to run at only
subsonic Mach numbers, it is possible to predict surface pressures that would otherwise
form shockwaves. Thus, the trust range of VORLAX includes almost any geometry so
long as the predicted pressures due not exceed the shockwave producing critical pressure
coefficient for a swept, finite wing.
One of the major driving factors in the design of this wing is the aforementioned
critical pressure coefficient. For a wing devoid of any sweep and having conventional
blunt airfoils, the critical pressure coefficient will occur at the point on the wing where
14
the local velocity reaches sonic first. For wings having finite thickness and no sweep, the
critical pressure coefficient will always be reached before the freestream velocity reaches
sonic. This phenomenon is the consequence of the flow being sped up around some parts
of the wing in order to create lift. The overarching goal is to prevent the wing from ever
generating its critical pressure coefficient until it reaches or exceeds Mach 1.25, so that
the design cruise point of Mach 1.3 will operate shock-free or just slightly at critical
conditions (very weak shock possible).
b. The Lower Critical Mach Number and Critical Pressure Coefficient
To begin, let us consider the critical pressure coefficient associated with purely two-
dimensional flow7.
𝐶𝑃∗ =
𝑝−𝑝0
𝑞=
2
𝛾𝑀∞[(
1+𝛾−1
2𝑀∞2
1+𝛾−1
2
)
𝛾
𝛾−1
− 1] [2]
Entering the freestream Mach number into Equation 2 this equation can be used to
determine the maximum under-pressure that can be sustained without developing locally
supersonic flow. The simple 2-D critical pressure coefficient equation lacks any sort of
sweep correction. No such correction is needed because, as Neumark notes, pressure
15
“isobar” contours must smoothly cross the line of bilateral symmetry (Figure 7)12. The
local flow field at the at the line of bilateral symmetry is independent of the sweep of the
rest of the wing.
Thus, as the freestream Mach number increases, the pressure drop arising from a
super-velocity (accelerating the local flow to an allowable shock free Mach number
above the freestream) declines. At a freestream Mach number of 1.0, for 2-D flow no
further acceleration is permissible, hence Cp* = 0 as seen in the trend shown in Figure 9.
Figure 9. Cp* vs. Mach Number for Purely 2-D Flow
16
However, to make this design problem simpler (and to make it compatible with
VORLAX thick sandwich panel geometry), it must be understood what the low-speed
(Mach ~ 0) pressure coefficient would be that would lead to locally sonic flow at some
higher freestream Mach number. This correction process was initially described in von
Kármán19; in 2-D flow, one can determine the subcritical (i.e. shock-free) pressure
coefficient at higher Mach numbers (Cp) if you know low-speed pressure-coefficient
(Cp0) and apply the Kármán-Tsien correction factor presented in Equation 2.
𝐶𝑝 =𝐶𝑝0
√1−𝑀∞2 +
𝑀∞2
1+√1−𝑀∞2(𝐶𝑝02)
[3]
Von Kármán demonstrated its utility comparing it to test data in his 1940 paper19
as seen in Figure 10.
Figure 10. Cp Trend with Increasing Mach Number19
17
Combining the ideas in Equation 2 with Equation 3, the reader can see how an
accurate measurement of low speed pressure coefficients (Cp0) can be used to determine
the critical Mach number of a geometry in Equation 4: one would replicate the process
shown in Figure 10 feeding the most negative Cp found in the low speed test into the
Kármán-Tsien equation and then observing at what corrected Mach number (M∞) would
the corrected pressure (Cp) equal the Critical Pressure Coefficient (Cp*).
𝐶𝑝0
√1−𝑀∞2 +
𝑀∞2
1+√1−𝑀∞2(𝐶𝑝02)
=2
𝛾𝑀∞[(
1+𝛾−12𝑀∞2
1+𝛾−12
)
𝛾𝛾−1
− 1] [4]
The important thing to note is that this process is only valid for purely 2-D flows.
This thesis will return to the practical implications of the Lower Critical Mach Number in
subsection (d) of this chapter.
c. The Upper Critical Mach Number and Critical Pressure Coefficient
In order to understand the process for 3-D swept wings, Kirkman & Takahashi20,21
show that the most accurate equations for incorporating sweep effects were created by
Küchemann7. As such, these equations are turned to for a more credible approach for
correcting a low-speed incompressible pressure coefficient to its equivalent at a larger
18
Mach number (see Equation 5). This equation is not such a radical departure from the
norm since an un-swept wing will have the cosine term drop out and be left with the
Kármán-Tsien correction equation.
𝐶𝑝 =𝐶𝑝0
√1−(𝑀∞ cos(Λ))2 [5]
Küchemann proposes7 and Kirkman & Takahashi21 confirm that the critical
pressure coefficient Cp* for a swept wing is governed by Equation 6.
𝐶𝑝∗ =
2
𝜆𝑀∞2 ((
2
𝜆+1)
𝛾
𝛾−1(1 +
𝜆−1
2(𝑀∞ cos(Λ))2)
𝜆
𝜆−1− 1) [6]
The reader should note that Equation 6 does not require Cp* to trend to zero when
the freestream Mach number reaches 1.0, so long as the wing incorporates sweep. In
Küchemann’s correct world view, Cp* reaches zero only when the component of the
incipient flow normal to the wing leading edge reaches Mach 1.0 as seen in Figure 127.
Thus, Küchemann provides a basis to use potential flow methods to develop accurate
shock-free supersonic lofts so long as the leading-edge flow remains subsonic7.
19
The graphical procedure found in Figure 11 can be implemented by setting
Equations 5 and 6 equal to one another. At this point since the design Mach number (M∞)
and sweep angle (Λ) are known, the incompressible pressure coefficient that will result in
the critical pressure coefficient at the design Mach number can be solved for (Cp0*).
Figure 11. Cp* Trend with Increasing Mach Number for Swept Wings7
𝐶𝑝0∗ = √1 − (𝑀∞ cos(Λ))2 (
2
𝜆𝑀∞2 ) ((
2
𝜆+1)
𝛾
𝛾−1(1 + (
𝜆−1
2) (𝑀∞ cos(Λ))2)
𝜆
𝜆−1− 1) [7]
This was transformed into code and the resultant values for the low-speed-critical
pressure at an arbitrary design Mach number are shown in Figure 12. Thus, a wing with
57-degrees of leading-edge sweep should have Cp* = -0.427 when M∞ = 1.25 and
Cp0* = -0.315 at a sufficiently low subsonic Mach number M∞ = 0.3. This wing has a
20
clearly subsonic leading edge, M1.25 * cos(57) = M0.68. Hence, if a wing is designed at
57-degrees of sweep it can be expected to develop a pressure profile at low speed where
Cp > -0.315, and can further be expected to be shock free so long as M∞ < 1.25.
Since the overall lift of a wing is the integration of the net pressures between the
upper and lower surfaces over the planform of the wing, lift follows the same sorts of
Mach number dependent characteristics. Thus:
𝐶𝐿 =𝐶𝐿0
√1−(𝑀∞ cos(Λ))2 [8]
This was also transformed into code and the resultant values for the low-speed-
critical pressure at an arbitrary design Mach number are shown in Figure 13. Thus, a
wing with 57-degrees of leading-edge sweep with a design CL ~ 0.21 at M∞ = 1.25 needs
to develop CL ~ 0.15 at low speeds.
21
Figure 12. Cp* Workup for a 57-Degree Leading Edge Swept Wing at Mach 1.25
Figure 13. CL Workup for a 57-Degree Leading Edge Swept Wing at Mach 1.25
22
A major complicating factor in the design of such a wing is the existence of three-
dimensionality in aerodynamics. There is a significant amount of aerospace engineers
who would neglect the principles of three-dimensionality for less accurate (albeit more
common) thin airfoil and simple sweep theories.
Obert has written a lot about what characteristics a “well-designed” wing or fin
ought to have in order to get the best performance out of them8. These same
characteristics can be observed in airplanes all over the world as seen in Figure 14. He
says that one of the best measures of a wing’s effectiveness can be seen in the pressure
coefficient isobars in that they need to align themselves with the leading-edge sweep of
the wing. He prescribes methods by which a wing could achieve well aligned isobars as:
o Increasing the thickness-to-chord-ratio near the wing/fuselage junction.
o Decreasing the positive camber or even apply negative camber to the root
section.
o Increasing the incidence of the root section, typically via twist.
23
Figure 14. Airbus A380 Using Negative Camber at Root of Horizontal Stabilizer and
Positive Camber Outboard
We will see in this thesis that the wing designed herein will have all of these
characteristics and achieve isobars which are properly aligned.
It is obvious that a lot of care has been taken to make sure that the modeling and
design work has a strong basis in three-dimensionality. Although to some this might seem
extraneous, the work of Jensen & Takahashi illustrates the significant departure of the
behavior of a 3-D wing section from pure 2-D models22. In fact, some of the important
takeaways from that work are:
24
1. There is no section on a real 3-D wing that behaves like its 2-D equivalent.
a. At the root of a finite wing, the defining airfoil will not exhibit the same
lift pressure properties as its 2-D equivalent would.
b. Midspan on a finite wing, the defining airfoil(s) will not exhibit the same
lift pressure properties as its 2-D equivalent would.
c. Near the tip of a finite wing, the defining airfoil will not exhibit the same
lift pressure properties as its 2-D equivalent would.
2. The effects of perturbations in the wing such as thickness, camber, and twist will
all depend where on the 3-D wing section they are applied.
3. The smaller the aspect ratio of the wing is, the more important three-
dimensionality becomes.
As this is a medium-low aspect ratio (4.596) design; the differences between the
local sectional flow and idealized 2-D could not be more marked. Takahashi & Thomas28
used a similar design process to develop the wing loft for a small, high-altitude UAV for
atmospheric science missions. They used vortex lattice panel methods, both flat thin
panels, cambered thin panel and “sandwich” thick panel elements to configure their
aerodynamic design18. As with Obert8, their design flies the fuselage at a nose-up deck
angle in cruise, the wing has reflex camber at the side-of-body and significant spanwise
taper in t/c. Their wing design was subject to four main requirements: to develop good
25
handling qualities via careful tuning of stall characteristics, to provide an elliptical
transverse load distribution to minimize induced drag, develop straight isobars to
encourage the development of “clean” flow, and a pressure distribution favorable for
attached flow at the modest Reynolds’ numbers seen in high-altitude flight. This work
demonstrates how the vortex lattice panel method code can develop a detailed
aerodynamic configuration, and included results showing positive volume grid CFD
validation of the potential flow model that should lead, with no need for re-design, to
successful flight tests.
d. Slender Body Theory Wave Drag and its Relationship to the Lower Mach Number
and Are Distributions
The concept of the lower Critical Mach Number, at first seems impossible to evade;
here it will be shown how careful and precise area-ruling methods are needed to take
advantage of sonic fuselage flow.
Recall that any shape or volume will disturb the flow, produce super-velocities and
hence lead to a design where the “Lower Critical Mach Number” is less than 1 (Figures
9, 10 and 11). However, shockwaves do not necessarily form the instant that the flow
goes supersonic.
26
NACA 113523 expands on the three primary shock waves.
1. A “Normal Shock Wave”, where quasi-2-D flow abruptly decelerates in
proximity to some sort of blockage. Typically, they form ahead of a blunt
obstruction such as a hemispherical nosecone, blunt un-swept leading-edge, or a
blunt engine inlet. The flow decelerates from the free-stream supersonic speed to
a subsonic speed; the velocity diminishes; the pressure and temperature rise.
2. An “Oblique Shock Wave”, where a three-dimensional shock wave forms
separating two regions of non-uniform flow. However, the shock transition at
each point takes place instantaneously. Oblique shock waves are triggered when
flow is forced to turn; classically, by the presence of a wedge or cone. Oblique
shock waves result in a deceleration of the flow; the downstream Mach number is
always lower than the upstream Mach number. Mathematically, an oblique shock
wave can be considered as a normal shockwave superimposed upon a field of
uniform transverse flow.
3. Expansion waves, where a shock-wave forms when supersonic flow turns around
a convex corner (which for supersonic inbound flow means to further accelerate).
The classic case for expansion waves is the “Prandtl-Meyer expansion fan,” an
infinite number of Mach waves, diverging from a sharp corner.
27
Recall that “shock waves form ahead of any body which is immersed in
supersonic flight and remain fixed relative to the body if the flight is steady.”23 Shock
waves “stand ahead of blunt shapes, but may be attached to pointed shapes.”23 The
definition of bluntness has to do with the nuanced shape of the geometry and whether the
shape supports subsonic or supersonic flow normal to its leading-edge. Thus, at a high
enough free-stream Mach number even a slender, tapered shape may trigger normal
shock waves.
For subsonic flight conditions, no shock will form ahead or attached to the leading
edge of any aerodynamic feature whether swept or un-swept. Thus, the only sorts of
shockwaves produced at these speeds are triggered by the same low-speed,
incompressible effects associated with classical “diffusion” or a slowing of the flow.
However, for a shockwave to form, the upstream flow must be supersonic, and the
downstream flow must be slower (following oblique and normal shock rules).
We can see this in action in Figure 15, the classic Mach cone angle is the angle of
an oblique shock associated with flow deceleration for 2-D body-of-revolution with
otherwise subsonic leading edges (Equation 9)24. For the F-18, at high subsonic speeds,
the oblique shock is broadly triggered by the overall flow field as it decelerates just aft of
the point of maximum cross-sectional area. Because θ ~ 70-deg, it can be inferred that the
local flow upstream of the shock is at a Mach number very close to 1.06.
28
Figure 15. A Visual Representation of the Mach Cone Angle
θ = arcsin (1
M∞) [9]
Recall from basic shock jump relations, as flow crosses a weak shock its Mach
number decreases and its density, temperature and static pressure increase. Thus, a shock
wave manifests itself as a sharp increase in pressure coefficient; Cp’s ahead of the shock
are more negative than the Cp’s behind the shock. Thus, shock waves represent a sharp
adverse pressure gradient. Figure 16 provides a visual representation for shockwave
formation at various places along an aircraft which are measured by gradients in pressure.
29
Figure 16. Shock Wave Formation Modeling on a Supersonic Transport
For supersonic flight conditions, shocks will form ahead of or possibly attached to
the leading edge of any aerodynamic feature and will also form in regions of inherent
flow deceleration. If a swept wing geometry is developed that does not exceed its upper
critical Mach number and associated upper critical pressure coefficient, in theory the
wing will be shock-free under some supersonic freestream conditions. However, the fact
that the aircraft will most assuredly exceed its lower critical Mach number and lower
critical pressure coefficient in supersonic flight must also be considered.
Slender body theory states that useful aerodynamic properties may be inferred
from turning any arbitrary shape into an equivalent body of revolution. Whitcomb
advocated the idea that shock formations and hence drag rise about relatively complex
configurations at zero lift near the speed of sound are “similar to those that would be
expected for a body of revolution with the same axial development of cross-sectional
30
areas normal to the airstream.”26 Thus, he postulated the idea of the Transonic Area Rule
that “the zero-lift drag rise of a wing-body configuration generally should be primarily
dependent on the axial development of the cross-sectional areas normal to the
airstream.”26
Classic wave drag formulations compute shock-free pressure distributions at
supersonic speeds over the equivalent body of revolution and then integrate this 1-D axial
pressure field over the projected changes in cross-sectional area, either forward or aft
facing9. In the famous Harris Wave Drag Code which is referenced in Chapter 5, the
wave drag of each equivalent body is determined by the von Karman slender-body
formula27 which gives the drag as a function of the free-stream conditions and the
equivalent-body area distribution; see Figure 17. The wave drag of the aircraft at the
given Mach number is then taken to be the integrated average of the equivalent body
wave drags.
Figure 17. Supersonic Area Rule Wave Drag Formulation
31
Harris notes that “that the supersonic area rule is not an exact theory.” He warns
that the supersonic area rule “assumes that an aircraft, which usually departs considerably
from a body of revolution, can be represented by a series of equivalent bodies of
revolution. The theory, therefore, does not account for wave reflections which may occur
due to the presence of the fuselage, wing, or tail surfaces.”27
In order to make the wave drag low, the second derivatives of cross-sectional area
with respect to length should be as small as possible. Thus, the minimum drag shape
should be long and slender with shallow second-derivatives of the cross-sectional area.
The classic “Sears-Haack” body, with minimum theoretical wave-drag has its maximum
cross-section area at its half-length (see Figure 18)13.
Figure 18. Sears-Haack Body
However, Whitcomb’s early work demonstrated secondary effects that show that
the ideal practical aircraft should not have its maximum cross-sectional area akin to the
Sears-Haack body. Schlieren images show that both a wing-body and its equivalent
body-of-revolution generate a weak “corona-shock” just aft of the point of maximum
thickness (Figure 19). These shocks shown in Figure 19 are associated with Mach jumps
32
as small as from M ~ 1.01 to M ~ 0.99. While the shocks are triggered at the aft edge of
the delta wings and in the case of the body-of-revolution, are associated with inbound
flow turning a convex corner, they are not expansion fans but are instead classical weak
shocks associated with flow deceleration.
Figure 19. “Corona-Shocks” Forming Past the Point of Maximum Thickness26
With this in mind, it will be shown that no matter how successful a swept wing
design is in not exceeding its Upper Critical Mach Number, the Lower Critical Mach
Number must be kept in mind and thus tailor the cross-sectional area of the configuration
so that the LCM associated Corona-Shock does not accidentally impinge upon the wing.
33
Chapter 3: Sizing the Basic Airframe
Kurus, et al sized the basic supersonic airframe to carry 150 passengers over
4,000-nM still air range on a flight with significant cruise segment at Mach 1.3 and
FL4001. The relatively low cruise altitude, compared to Concorde, was a byproduct of the
strong altitude and speed lapse of thrust from the reference normal shock turbofan
engines having a bypass ratio of 1.5 and fan pressure ratio of 1.8; a cycle similar to a
JT8D-200 series turbofan found on a MD-8029. The team used Phoenix Integration’s
ModelCenter (see Figure 20) to integrate a number of industry tools including EDET30
and the NPSS derived 5-column propulsion data31 to determine the optimum
configuration1.
Figure 20. Initial Sizing Algorithm in ModelCenter1
The airframe design process comprised: initial trade studies varied parameters
which define the planform to batch collect, rank and sort candidate designs whereas the
34
detail design was much less “brute force,” and used coupled tools with much more
human intervention and nuance.
Initial studies varied the reference area, sweep, taper ratio, aspect ratio (see
Figure 21). The goal of this first screening was to find the wing planforms that would
yield sufficient fuel volume for the aircraft along with drag characteristics to enable a
high enough specific range to make the aircraft economically viable. More specifically,
the wing had to have the fuel capacity to carry the desired fuel for the sizing mission plus
all reserves. Based upon a mission “fly out” beginning at MTOW with consistent OEW,
payload and fuel weights. Ensuring that the aircraft fuel burn did not exceed 0.3287-
lbm/(pax NM) over the 4,500-NM sizing mission.
35
Figure 21. Multi-Variate Trade Study for Airframe Parameters1
Figure 21 shows the scatter matrix output of a several hundred run trade study
performed using the initial trade study model. Gray points are designs that failed the
sanity checks, and colored points are set to show designs that minimize total fuel burn,
with red designs being the most economical, and blue the least. The final wing planform
is a manual refinement of a point suggested by this trade study.
The overall aircraft that came from this project has an estimated maximum takeoff
weight (MTOW) of 411,000-lbm, operating empty weight (OEW) of 207,500-lbm, wing
36
reference area (Sref) of 4473 square feet, wing aspect ratio (AR) of 4.596, and leading
edge sweep (ΛLE) of 57.1-degrees with four 56,000-lbf dry sea-level static thrust JT8D-
200 series cycle engines without afterburners mounted behind normal shock inlets. At
MTOW this represents an aircraft with average wing loading (Weight/Sref) of 92-lbf/ft2
and thrust to weight ratio (Thrust/Weight) of 0.54.
Figure 22 represents the anticipated flight profile of this aircraft as used on the
North Atlantic route. It flies 4,000-nM still air range burning 152,000-lbm fuel over the
reference mission, holding 14,000-lbm fuel reserve. For a favorable flight, for example
from IAD to LHR (with the winds) predicted flight times are around 3.5 hours. For the
long-range flight, for example from ZRH to IAD (against the winds with a substantial
subsonic leg over Europe) predicted flight times are around 6.25 hours.
37
Figure 22. Mission Profile as Described Below for the Supersonic Transport1
The sizing mission consists of the typical phases of flight starting at ZRH airport.
The airplane will perform a ground runup, followed by an initial climb to 10,000-feet at
250 KIAS, climb further to 23,000-feet at Mach 0.8, cruise overland at 25,000-feet at
Mach 0.82, climb to 40,000 feet at Mach 0.9, accelerate to Mach 1.3 and cruise at this
speed until the airplane descends to 23,000-feet and decelerate to Mach 0.82 for overland
flight, then finally land at its arrival airport at IAD. Provisions are also included for a
38
diversion by repeating climb segments and cruising subsonic for 440 NM (440 is 10% of
credit distance per 14 CFR § 121.645 + 40 NM to alternate airport BWI).
Thus, the wing design is expected for supersonic cruise at near MTOW at FL400
and Mach 1.3. Thus, the design CL in cruise is ~ 0.21. The design and validation of a
wing to these specifications will be discussed over the next few sections.
39
Chapter 4: Computational Design of Geometry
The design of this wing is not an exercise in any complicated optimization
algorithm but is rather an iterative approach where the nuance is decided upon by the
designer who needs to be minutely aware of the design goals. With each progressive
iteration tradeoffs need to be assessed on order to decide whether or not the wing is
progressing towards or away from a viable design. One such tradeoff comes while trying
to create well aligned isobars while also keeping the wing from going supercritical. In
this design it will be shown that although the wing is designed to be purely subsonic
across its entire operating range, that the wing might end up being slightly supercritical it
its fastest deign point. This is a decision similar to many others that had to be considered
and prioritized when attempting to mold the geometry.
The initial specifications for this wing are heavily coupled with the full design of
a supersonic passenger aircraft. All of the specifications presented in this section were all
derived from intensive trade studies regarding the performance required out of the
supersonic transport to meet specific mission goals. These specifications are given in
Table 1.
All of these parameters were decided upon prior to the more advanced airfoil
design on the wing but resulted in a necessary initial geometry to model in VORLAX.
Although there were a clear set of goals that the wing needed to accomplish, arriving at
40
the final configuration depended entirely on hand manipulations of the aerodynamics
tool. There are parameters which define the effectiveness of the wing’s aerodynamics.
The first is that the wing must provide a sufficient lift coefficient at cruise where the total
lift is equaling the weight of the aircraft resulting in a CL = 0.21. The wing must also not
develop strong shock waves at cruise conditions, and this is done by ensuring that the
critical Mach number is just below the cruise speed of Mach 1.3 and ensuring that local
pressure coefficients do not exceed the critical pressure coefficient. The wing must also
have a nearly elliptical transverse loading distribution; minimizing induced drag. The
wing must develop favorable stall conditions meaning that the root of the wing needs to
stall first as to maintain controllability and create a nose down pitching moment and the
aircraft must maintain control power on its wing mounted control surfaces at the onset of
a stall. Lastly and potentially most pertinent is that the pressure isobars must line up
consistently along the leading-edge of the wing as non-aligned isobars work adversely to
the effects of sweep.
41
Table 1. Wing Constraints and Final Specifications
Design Critical Mach Number 1.25
Design Cruise Alt. (ft) 40,000
MTOW (design weight) (lbm) 402,000
Wingspan (ft) 143.38
Fuselage Length (ft) 169.4
Fuselage Diameter (ft) 10.1
Wing-Fuselage Junction (ft) 40 (aft of nose)
Leading Edge Sweep (deg) 57.1
Wing Trapezoidal Planform Area
(ft2)
4473.13
Root Chord (ft) 72.74
Tip Chord (ft) 19.55
Aspect Ratio 4.596
Taper Ratio 0.465
42
The design interface is built on a basic EXCEL spreadsheet that is heavily
modified in order to iterate through multiple designs with ease. The basic wing geometry
is inputted into the spreadsheet and is subdivided into five wing subsections (see Figure
23). For each wing subsection, the user then selects which thickness and what camber
profile they want and allows for scaling of each. These thickness and camber profiles are
all stored in internal libraries on the program but are mostly simple NACA forms and
camber lines32. Once all the desired inputs are chosen, the EXCEL file then writes the
input file to VORLAX and then writes a batch file and then proceeds to run VORLAX.
From here the program then reads in the output file and the .log file containing all of the
important lift and pressure distribution data. Next, there is a data scraper written to take
the cryptic text files and format them in a way that is easily readable and accessible for
plotting. Lastly the code plots a variety of important aerodynamic data for further
analysis and design work. The only additional step taken during this analysis is that the
EXCEL sheet also creates formatted data tables for the full span pressure distributions to
be read by a MATLAB script in order to produce the isobars seen later in this thesis. As
the different iterations of the wing were being run, tweaks to airfoil shapes, camber lines,
wing twist, and scale factors were adjusted until all criteria were met.
43
Figure 23. VORLAX Panel Structure
Diving into a more detail about the design of the VORLAX model, which was the
initial basis for the entirety of the wing design, we look at the panels in Figure 23. For
each wing section, for which there are five per side, each panel is comprised of two so
called “sandwich panels.” This is significant because in order for VORLAX to take
thickness effects into account, an infinitesimally thin cambered wing panel simply cannot
be used for the desired effect. The use of sandwich panels allows the user to pass the
coordinates of the top and bottom sections of the airfoil independently. Additionally, the
user can only define the airfoil at each junction between wing sections. The airfoil profile
between any two junctions is simply a linear interpolation between the airfoils at the
endpoints. Lastly, there is a flat uncambered panel included in the model to simulate the
fuselage for more accurate lift coefficient readings but the junction between the fuselage
44
panel and the wing panel does induce some amount of numerical inaccuracy which will
be seen in the VORLAX wing isobars.
The final result of the iterative surface panel VORLAX method is described in
this section and will prove that this wing is designed to meet all operating criteria while
also aligning with the method described by Obert8.
A crucial goal of this wing was for it to remain purely subsonic at the design
conditions (Mach = 1.25) which means that it cannot develop suctions stronger than its
critical pressure coefficient of -0.315. However, the other way to look at this is to define
a critical Mach number, which is defined as the Mach number for which the wing will
reach its critical pressure coefficient and thus will start generating shock waves and
ensure that this number is greater than 1.25.
The way to solve for the critical Mach number is to set Equations 5 and 6 equal to
each other and solve for the free stream Mach number where they intersect (recall Figure
11). At the design Cp*, it’s no surprise that the critical Mach number turns out to be 1.25
which is exactly the design speed.
45
The converged iterative design did not identically satisfy this requirement;
although the design CL and the elliptical span load was still achieved with an inviscid
predicted Cp(min) value of -0.229; which correlates to an estimated Critical Mach Number
of 1.38; thus this wing is expected to be shock free at speeds less than Mach 1.38 which
is well over the design condition of 1.25 or the cruise condition of 1.3 (see Figure 24).
Looking at this same idea in the sense of pressure coefficients, we can look at the
overall pressure distributions along the top and bottom surfaces of the wing via isobar
plots extracted from the VORLAX solution. These are presented in Figures 25 for the
upper surface and 26 for the lower surface. The isobar patters on the upper surface have a
peak under-pressure of Cp ~ -0.232; and since the wing is lifting the upper surface
pressures tend lower than the lower surface pressures. There is some “noise” in the
converged solution at the wing / side-of-body junction; where the grid densities develop a
discontinuity.
46
Figure 24. Resultant VORLAX Critical Mach Number Workup
Figure 25. Upper Surface Isobar Contours from the Low Speed VORLAX Sandwich
Panel Solution
47
Figure 26. Lower Surface Isobar Contours from the Low Speed VORLAX Sandwich
Panel Solution
Figure 27: Enhanced Upper Surface Isobar Indicating Line of Peak Under-Pressure from
the Low Speed VORLAX Sandwich Panel Solution
48
We can turn to Figure 27 and examine the line of peak under pressure follows a
line that is equal to the sweep of the leading edge of the wing. This figure colors peak
under-pressure each spanwise strip as yellow. Arguably the most important aspects of
this design were ensuring that the wing makes enough lift at the design Mach number and
altitude to support steady level flight and that the loading is nearly elliptical as to avoid
large induced drag penalties. Figure 28 reveals how well this was actually accomplished
with this wing. Recall, the design targets were to achieve an integrated lift coefficient of
0.21 in cruise and 0.15 at low speed. The computed span load is plotted in blue and the
ideal is plotted as orange. The ideal ellipse is given as6:
𝐿(𝑦) = 1.226 ∗ √1 − (2𝑦
𝑏)2(𝑊
𝑆) 𝑐 ̅ [10]
Figure 28. Transverse Span Load from Low Speed VORLAX Design Solution
49
The integrated lift of the wing generally follows the elliptical shape, slightly
exceeding the target function; the ideal curve and the experimental design vary slightly -
within the 5% accuracy range. Induced drag losses from a loading of this type would
likely only be incrementally larger than a perfectly loaded wing.
Turning next to Figure 29, we can see how the section lift coefficient varies
across the span. The presence of the large Yehudi, for landing-gear and high-lift system
integration causes the sharp falloff in section lift coefficient inboard of semi-span around
the 30-foot mark. That break is also reflected in some “lumpiness” in the transverse span
loads seen earlier in Figure 28.
Figure 29. Section Lift Coefficient from Low Speed VORLAX Design Solution
50
Thus, we can see that the developed loft achieves its goals of developing the
target integrated lift coefficient, the nearly elliptical span load while not exceeding the
desired peak under-pressure, Cp*.
How was this shape developed? Recall that an iterative process was used to define
the wing in terms of 6 basis airfoils, each of which have an associated thickness (t/c),
camber (% camber) and local incidence (twist). Thus, the wing design problem is posed
in terms of 18 independent variables.
The airfoil shape is made up of scaled thickness forms (see Figure 30 for the
reference form) modified by a scaled camber line (see Figure 31).
Figure 30. Basic Thickness Forms
51
Figure 31. Basic Camber Lines
The basic thickness profiles (the NACA 64 and 65) control the location of the
maximum thickness, otherwise they are very similar. These thickness forms were then
scaled up or down as desired.
The final lofting airfoils impose a camber line upon the basic thickness form.
NACA 62, 63 and 64 cambers lines were chosen as references. These camber lines were
then scaled in order to impose more or less leading-edge droop as necessary. We can see
that all of the basis camber lines feature the majority of their camber towards the front of
the airfoil.
52
The geometry developed therefore is a prismatic loft between each of the six
defining airfoil sections. The spanwise variation of these basis functions can be best
summarized in Figures 32, 33, and 34.
Figure 32. Spanwise Variation in Thickness
Figure 33. Spanwise Variation in Camber
53
Figure 34. Spanwise Variation in Incidence
The thickness to chord ratio along the entire span is plotted in Figure 32, and they
show that the average percent thickness to chord ratio (% t/c) of the entire wing is about
5.81 % which as mentioned earlier is a byproduct of the drag estimation tool being
limited in maximum thickness capability. The other major trend is that the wing is most
thick at the side of the body which is for fuel volume purposes and for the reason that
thickness effects are nearly independent of lifting performance. Due to this independence,
the side of body thickness can be exploited in that it can be as thick as needed without
worrying about any lift generation degradation with the only limiting factor being how it
mounts on a fairly short fuselage. Moving out on the span the wing becomes less thick
and eventually levels out at a minimum value and this has less to do with aerodynamics
and is more of a structural consideration because any lifting surface can only be so thin
before it becomes infeasible from a loading point of view. This is especially true when
54
the wing tapers as much as it does on this airplane since the local chord is shrinking so
the dimensional thickness is also decreasing even with a constant percent thickness.
Obviously, none of the resultant airfoils shown in Figure 36 are symmetric which
means they have all had a camber line imposed on them. Figure 33 shows the percent
camber distribution over the whole span, but the actual camber lines chosen for the wing
vary a bit more than the thickness forms. At the root, there is a NACA62 camber-line
imposed, then at the next station a NACA220, then another NACA62, then a NACA63
for the rest. The process for choosing camber profiles involved a lot of nuance but it
came down to needing the camber very far forward closer to the side of the body and
wanting it a little further back going out towards the wing tip. The reason that the camber
is chosen this way is to ensure the elliptical loading shape and also because there is a very
strong interest in keeping the minimum pressure line along the same sweep angle as the
leading edge. The reason that the amount of camber is relatively high along the span past
nearly midspan is closely related to that same idea. Plus, the wing loses effectiveness due
to spanwise flow the further out you go on the span so increasing the amount of camber
compensates for this effect to some extent. However, what happens near the side of the
body is more interesting because there is significant negative camber at the side of the
body which eventually transitions into positive camber farther outboard on the wing. The
main reason for this is that it balances out the elliptical loading of the wing because the
ellipse naturally flattens out near side of the body meaning that the airfoil there needs
generate less lift, thus negative camber. Negative camber also provides favorable
55
spanwise flow conditions for airfoils downstream on the span. Additionally, there are
stability benefits to having the root be negatively cambered in that they create nose up
pitching moments and the rest of the positively cambered wing creates a nose down
pitching moment and this will become important in stall behavior discussions.
The wing twist distribution is decided upon last because the goal of the wing twist
is to clean up the elliptical loading distribution while also promoting favorable stall
characteristics of the wing section. The twist at the wing root is heavily “wing-tip up” and
this is necessitated from the negative camber in this region. The twist over the rest of the
span is very minimal and is only in place to make small corrections to the isobar
alignment or the elliptical lift distributions. We can see the overall loft as rendered and
ready for wing-body volume grid CFD and future wind tunnel testing in Figure 35. The
final wing design features the six resultant airfoils as shown below in Table 2, and in
graphically as Figure 36.
56
Table 2. Defining the Section Airfoil Ordinates
Figure 35. Overall Wing and Body Geometry
57
Side of Body 41.8 % Semi-span
51.5 % Semi-span 61.2 % Semi-span
80.6 % Semi-span 100 % Semi-span
58
Figure 36. Airfoil Definitions Along the Span of the Wing
59
Chapter 5: Surface Panel Method Results and Implications of Wave Drag
In this section the VORLAX solution of the “sandwich panel” model is compared
against a simplified flat plate representation (the more classic “vortex lattice” model).
The aircraft synthesis process described in Chapter 3 relied upon a flat plate model of the
entire (wing, body, horizontal tail, vertical tail) configuration for stability & control.
Data produced for the low-speed solution at “off-design” angles-of-attack will be
used a reference for further computational results (volume grid CFD) at low speeds. To
compare pitching moment, and aerodynamic center predictions, this moment-reference-
point (MRP) is kept consistent between the vortex lattice CFD, and the volume grid CFD.
The MRP for the overall configuration is found at the fuselage station (FS) of 88-feet,
which is 48-feet full scale (or 65.99% of the side-of-body chord) aft of the apex of the
wing/body junction; this is consistent with the MRP for the complete configuration1.
We can see the location of the MRP marked with an asterisk along the centerline
of Figure 37. All relevant parameters (Sref, Span, Average Chord) are the same as those
presented in Table 1.
60
Figure 37. VORLAX Flat Plate Model with the Center of Gravity Marked
Figure 38 plots CL-vs-α for the sandwich panel model (with camber, twist and
thickness) and for the flat plate model (no camber, no twist, and no thickness). The
solution was run at Mach = 0.3. The plot shows a close resemblance between the lift
curves of the flat plate model and the fully configured sandwich panel model. The main
difference being that the cambered panels increase the lift overall at 0 degrees angle of
attack. The lift curve slope predicted by 2-D thin airfoil theory is also plotted in Figure 38
and we can see that the lift curve slopes largely agree between the cambered thick model
and thin flat plate model. Both are significantly shallower for both finite wing sections.
This once again goes to show why three-dimensionality effects are so important to the
design of any finite aspect ratio wing.
61
Figure 38. CL vs. α for the VORLAX Flat Plate and Sandwich Panel Model
Figure 39 plots CL-vs-Cm for the flat plate model. The slope 𝑑𝐶𝑚
𝑑𝐶𝐿 ~ -0.10, showing
the basic wing/body ensemble has a 10% positive static margin without any horizontal
tail. When the cambered sandwich panel model is added in the resultant wing body has a
Cm0 value at a lift coefficient that is somewhere around 0.15 which isn’t too far off of the
design cruise CL. By adding a horizontal tail this can be tuned closer to the design point.
62
Figure 39. Longitudinal Stability of the VORLAX Wing-Body Flat Plate Model and
Sandwich Panel Model About its MRP (88-feet FS; Mach = 0.3)
The CP distributions found in Figures 40-44 show the chordwise pressure
distributions across the span at the low-speed design condition (CL ~ 0.153). Looking at
all of these figures together, the pressure profiles achieve maximum under-pressure near
the leading edge; this is consistent having the minimum pressure isobar lines aligned with
the leading edge; at higher speeds this pattern should cause the shock to first form near
the leading edge. The pressure profiles are a byproduct of the twist, camber, and
thickness strategies.
63
Figure 40. CP Distribution at 41.8% Span Away from the Side of Body
Figure 41. CP Distribution at 51.5% Span Away from the Side of Body
64
Figure 42. CP Distribution at 61.2% Span Away from the Side of Body
Figure 43. CP Distribution at 80.6% Span Away from the Side of Body
65
Figure 44. CP Distribution at Wing Tip
The wing twist over the span (return to Figure 34) embodies that is nearly
opposite to the camber distribution (return to Figure 33); that is no accident. At the side
of the body, there is considerable positive incidence to counteract the negative camber
that have already been imposed there. Since negatively cambered wings don’t make
nearly as much lift at any given incidence, it needs to be corrected by setting it to a higher
incidence than the rest of the span. This geometry is needed to keep the peak under-
pressure near the leading edge in the context of the overall elliptical lift distribution.
Conversely, since the rest of the wing has positive, camber, it does not require
nearly as much incidence to develop lift. Hence the wing achieves the elliptical lift
distribution through significant “wash-out.”
66
The reflex camber near the side of body should help stall characteristics. A swept
wing will exhibit favorable stall behavior if the root of the wing were to stall first. The
reason for this is simple, the root of the wing is ahead of the MRP. By stalling the only
part of the wing where the local airfoil inherently produces a nose up pitching moment
(see negative camber) it leaves only airfoils producing a nose down pitching moment (see
positive camber) which is a stable configuration. Since the VORLAX solution cannot
directly capture stall, any supporting evidence cannot be captured in Figure 38.
Although the flat panel code can be used to estimate lift and drag characteristics at
low speeds and by proxy high speeds, this code still lacks the ability to account for
compressibility effects at supersonic Mach numbers. A simple case for the model aircraft
was run through the famous Harris Wavedrag program (NASA D2500 model) in order to
gauge how well optimized the body is for drag produced by shockwave formation27.
There was no effort taken in optimizing the shape of the fuselage or wings for
wave drag properties but by following the design guidelines that this thesis details the
wave drag properties are inherently good. As briefly mentioned in Chapter 2, the
software reads in a formatted text file very similar to that of the VORLAX code, but the
key parameters are significantly different. Figure 45 shows the input file being utilized
for this basic analysis and it takes in properties such as (but not totally limited to), wing
67
planform geometry, fuselage geometry, and total cross sectional area at several
prescribed points along the length of the model.
Figure 45. Harris Wave Drag Input File
To determine the effects of wave drag this study was run over a range of Mach
numbers within (and just exceeding) the design maximum cruise speed of the airplane.
These findings are presented in Table 3 and provide validity for the argument that the
airplane model as is does not need major geometrical overhaul to keep drag due to
68
shockwaves under control. The Harris Wave Drag program provides its calculation of the
wavedrag produced at each Mach number as well as the theoretical wave drag to be
expected from an optimally shaped body with equal cross-sectional area. Note that the
validity of these results can be corroborated by the fact that actual drag is consistently
greater than the predicted minimum. Note as well that the discrepancy between the two
measurements tends toward zero as the Mach number approaches 1.0. The trend can also
be observed graphically below in Figure 46.
Table 3: Harris Wave Drag Results Indicating a Geometry Well Suited for Wave Drag
Mach Actual Predicted
Wave Drag
Predicted
Minimum Wave
Drag
Percent
Difference
1.00 0.0293 0.0293 0.004 %
1.05 0.0278 0.0272 2.236 %
1.10 0.0266 0.0257 3.275 %
1.15 0.0258 0.0249 3.471 %
1.20 0.0255 0.0245 3.652 %
1.25 0.0250 0.0239 4.291 %
1.30 0.0248 0.0232 6.460 %
1.35 0.0247 0.0224 9.282 %
1.40 0.0245 0.0216 11.882 %
69
Figure 46. Wave Drag Results from D2500 Wavedrag Program
70
Chapter 6: Volume Grid Results
This section will describe and analyze my findings with the popular SU2 CFD
package as well as a few comparisons to STAR CCM++ results that were provided by my
colleagues in academia. The CFD findings will be used to assess their validity in
comparison to what has been previously found from the low speed VORLAX results. The
model that was shown in Figure 35 has been shrunk down in this section to accommodate
for future testing in a wind tunnel but is the basis for the model being analyzed in the
CFD. Although the scale of the model is taken into account and kept consistent when
processing any CFD results, for rigors sake the CFD model dimensions are given in
Table 3. Keep in mind that the model being used in CFD is exactly a 1:144 scale of the
full-sized airplane. Another way to think about this is that 12 feet in the full-scale model
is equal to one inch in the CFD model, conveniently leaving a scaled wingspan of almost
one foot exactly.
71
Table 3. Scaled Model Dimensions
Wingspan (in) 11.95
Fuselage Length (in) 14.11 (from nose tip to tail tip)
Fuselage Diameter (in) 0.84
Leading Edge Sweep (deg) 57.1
Wing Trapezoidal Planform Area
(Reference Area) (in2)
15.5316
Root Chord (in) 3.50
Tip Chord (in) 1.63
Aspect Ratio 4.596
Taper Ratio 0.465
Wing-Body Junction (in) 3.33 (aft of tip of nose on the model)
MRP (in) 4 (aft of wing body junction)
7.33 (aft of tip of nose cone)
The same model that was presented in Figure 35 was gridded up sufficiently well
such that it could be run through SU2 (version 7.0 on Windows) at various Mach
numbers, both high and low. In modern CFD software, there are generally a good number
of solvers that can be employed to solve the flow field, all coming with their own benefits
and drawbacks. In this case, an Euler solver was chosen, which is a considerably robust
and computationally efficient method of solving the flow field. In an ideal world where I
had unlimited computing power at my disposal, more intricate and rigorous programs
such as RANS or LES would have beneficial, unfortunately this was not the case.
72
However, the Euler solver and gridding scheme that was utilized for these trials is more
than sufficient for the level of resolution and precision being sought after in the flow
field. Additionally, the Euler solver is an inviscid code which is consistent with the
VORLAX code. Additionally, this analysis required nearly 40 different runs of SU2, each
taking about 6 hours to complete, so if these had been done with a RANS or LES solver,
the time constraint would not have been realistic for the scope of this work.
Before going into detail about any of the results, we must be certain the grid density
of the model is sufficient to capture the intricacies of shock formation and wing isobar
patterns. However, the computing power that was at my disposal was not incredibly
powerful, so the number of total cells needed to be kept below about 30 million (1
million cells ~ 1 GB of RAM). Figure 47 and 48 show screenshots of the volume grid
that were used and we can observe that the grid captures the curvature of the body quite
well. Additionally, the mesh comes to a total of 28 million cells which is just small
enough to be run on my machine. Note that the model is symmetric, thus saving a bit of
computational cost. Lastly, to ensure that the grid density was adequate to capture the
flow field, I ran a small grid convergence study (4 trials in total on the Mach 0.3 design
case) to make sure that the solution was convergent over the mesh sizes used, this is seen
in Figure 49.
73
Figure 47. Volume Grid Density Screenshot for the SU2 Model
Figure 48. Surface Grid Density Screenshot for the SU2 Model
74
Figure 49. Grid Convergence Study Run on Low Speed Case (M = 0.3, α = 0.2 deg)
With the overall model setup validated, the first goal of the CFD analysis is to
validate the low speed results obtained from the sandwich model produced in the vortex
lattice potential flow code seen in Chapter 5. Specifically, we are looking to see that
pressure coefficient isobars (both magnitude and shape) are consistent between the
VORLAX and SU2 models. Additionally, we would like to show that the overall lift,
drag, and pitching moment qualities are similar between the two models.
Taking a look first at the low speed results, the first thing we notice is the similarity
of the isobars presented in Figure 50. We can clearly see that the overall isobar pattern is
very similar, and the magnitudes of the pressure coefficients generally line up on the top
75
surface. At this point we can deduce that the low speed results from the inviscid potential
flow code (VORLAX) are valid in comparison with the more complex inviscid Euler
solver in SU2. We can also confirm that the SU2 result produces a most negative wing
borne pressure coefficient of -0.27 which is compliant with the low speed critical
pressure coefficient of -0.31.
Figure 50. Low Speed Comparison Between SU2 (Top) and VORLAX (Bottom)
(M = 0.3, α = 0.2 deg)
76
From Figure 50 we can see that the pressure isobars look remarkably similar
between the two at the design configuration which is an angle of attack of 0.2 degrees,
but we would also like to understand how the results compare for a variety of off design
angles of attack. Specifically, we would like to see how the SU2 off-design cases
compare with the cases shown previously in Figures 38 and 39. By plotting the
converged solutions of these cases, we can superimpose the results from SU2 on to the
results obtained earlier from VORLAX. Figures 51, 52, 53, and 54 show the results from
SU2 superimposed on to the VORLAX results.
𝐶𝐷 = 𝐶𝐷0 +𝐶𝐿2
π(AR)e [11]
Figure 51. CL vs. α Plot for the Low Speed Case with SU2
77
Figure 52. CL vs. CD Plot for the Low Speed Case with SU2
Figure 53. CL2 vs. CD Plot for the Low Speed Case with SU2
78
Figure 54. CM vs. CL Plot for the Low Speed Case with SU2
From the figures above, we can see that the lift coefficient curve is very similar to
the sandwich panel model across the whole angle of attack range and validates that out
lift figures are reliably predicted in the VORLAX code. Next, we observe that the drag
polar is quite a bit higher at angle given lift coefficient that the VORLAX models and this
is to be expected since the fuselage is modeled as a true cylinder in SU2 while in
VORLAX it is only modeled at the long flat plane. However, even at low CL’s the
VORLAX flat plate model does do a decent job at predicting drag. Figure 53 is
interesting because it is a proxy for the induced drag and thus the Oswald efficiency
factor “e”. Equation 11 gives the classical equation for drag induced by lift and in the
case where “e” is equal to 1.0, we would be left with a slope dependent on the aspect
ratio alone. We see that the VORLAX flat plate model does in fact exhibit this
79
characteristic but the SU2 model gives a more realistic curve which is steeper than ideal,
this corresponding to an “e” which is less than 1.0. We also notice that VORLAX has a
CD0 value of zero, while the SU2 models all have some finite drag at zero lift conditions.
Lastly, the pitching moment polar is similarly in close agreement between the SU2 model
and the VORLAX model especially in the linear region of the plot corresponding to
angles of attack between -3 and +3. There is a slight break in stability at high angles of
attack, but this can be corrected for with the addition of an empennage (V-tail and H-tail).
The tabulated data from Figures 51, 52, 53, and 54 are given below in Table 4.
Lastly, as with all volume grid results, it is necessary to prove the convergence of
the low speed SU2 cases presented here and Figure 55 does exactly this by plotting the
CL vs. Iteration curve for every low speed run. We can see that in this figure every run
does have some initial oscillation but reaches steady state by the end of the simulation.
80
Figure 55. Convergence History for All Low Speed SU2 Runs
81
Table 4: VORLAX and SU2 Tabular Data
Alpha
(deg)
VORLAX
Sandwich Panel
VORLAX Simple Flat
Plate Model
SU2 Results
CL Cm CL Cm CL Cm
-5 -0.1482 0.02557 -0.3075 0.03205 -0.18062 0.02117
-4 -0.0909 0.01701 -0.24656 0.02579 -0.11653 0.01642
-3 -0.0335 0.00845 -0.18508 0.01937 -0.05166 0.01047
-2 0.0239 -0.00008 -0.12333 0.01288 0.01350 0.00389
-1 0.08138 -0.00859 -0.06169 0.00644 0.07850 -0.0027
0 0.13881 -0.01703 0 0 0.143103 -0.00897
1 0.19616 -0.02541 0.06169 -0.00644 0.20707 -0.01461
2 0.25337 -0.03369 0.12333 -0.01288 0.27006 -0.01925
3 0.3104 -0.04187 0.18508 -0.01937 0.33176 -0.0227
4 0.3672 -0.04991 0.24656 -0.02579 0.39201 -0.02499
5 0.42374 -0.05782 0.3075 -0.03205 0.45039 -0.02536
6 0.47997 -0.06557 0.3689 -0.03852 0.50535 -0.02177
7 0.53585 -0.07314 0.42969 -0.04483 0.55666 -0.01451
8 0.59134 -0.08052 0.48959 -0.05089 0.60556 -0.00476
9 0.64642 -0.0877 0.55027 -0.05727 0.65381 0.00615
10 0.70106 -0.09467 0.60929 -0.06315 0.70220 0.01751
11 -0.1482 0.02557 -0.3075 0.03205 0.75123 0.02859
12 -0.0909 0.01701 -0.24656 0.02579 0.80125 0.03898
13 -0.0335 0.00845 -0.18508 0.01937 0.85204 0.04854
14 0.0239 -0.00008 -0.12333 0.01288 0.90273 0.05726
15 0.08138 -0.00859 -0.06169 0.00644 0.95226 0.06514
82
Now that we’ve established that the low speed results are very comparable
between the two models, we want to explore the more interesting case, which is the
supersonic design case at Mach 1.3. To this point we have used previous theory from
several prior authors, and we have postulated that this aircraft should maintain a clean
(shock free) planform all the way up to Mach 1.319,20. However, the findings from SU2
reveal that this is in fact not the case. Figure 56 shows the pressure coefficient contours
for the wing at Mach 1.3 and we can observe a very strong shock wave forming at the
trailing edge of the planform. How do we know that is in fact a shock wave and not a
form of a Prandtl-Meyers expansion fan (which is not covered explicitly by included
theory)? Figure 57 shows this same contour but instead in terms of Mach number and at
the location of the pressure discontinuity, the flow suddenly slows down from about
Mach 1.7 to about Mach 1.2. The lift coefficient produced here is 0.178 which is actually
slightly lower than the predicted and required 0.21, but this pesky shock wave is certainly
responsible for some of the loss of lift at the design speed.
At this point we can compare the results of the CFD to the expected results from
the low speed extrapolation. Recall in Chapter 2 that it was asserted that in order for the
wing to be shock free at the design condition of Mach 1.3, that the pressure coefficient
cannot be any more negative than CP = -0.427. When we probe the wing for the minimum
pressure coefficient, which occurs just ahead of the trailing edge shock, its discovered to
be only -0.419, which is not exceeding the critical pressure coefficient and thus by that
logic there should be no shock formation, yet one persists.
83
Figure 56. Pressure Coefficient Contour from SU2 at Mach 1.3 (α = 0.2 deg, CL = 0.178)
Figure 57. Mach Number Contour from SU2 at Mach 1.3 (α = 0.2 deg, CL = 0.178)
84
The shockwave at the trailing edge of the wing is troubling but before we explore
that, we’d also like to get a feel of the airplane’s performance characteristics at the cruise
point. Figures 58, 59, and 60 show how the airplane performs in terms of lift coefficient,
drag coefficient, and pitching moment over a tight band of angles of attack close to the
design point. We see that once again the VORLAX flat plate model gives an accurate lift
curve slope but is offset vertically due to the lack of cambered airfoils. The pitching
moment is not very far off either and it markedly linear (this will be important in the
stability discussion at the end of this chapter).
Figure 58. α vs. CL Plot for the Mach 1.3 Cruise Speed Case with SU2
85
Figure 59. CL vs. CD Plot for the Mach 1.3 Cruise Speed Case with SU2
Figure 60. CM vs. CL Plot for the Mach 1.3 Cruise Speed Case with SU2
86
However, upon revealing that the wing is in fact NOT shock-free at the design
point, we wanted to look at results that are lower in the sonic range to see if we could get
any hints as to why, where on the wing, where in the Mach range, and how strong the
shockwave is that is forming. In order to do this, we ran the same simulation down at
Mach numbers 0.95 (just below of sonic) and 1.05 (just above of sonic) with the hopes of
capturing the early moments of shock formation and analyzing how strongly flight
performance is affected by the onset of shock waves.
Figures 61 and 62 show the relevant contours for the airplane at a free stream
Mach number of 0.95. What we notice here is that there is a very distinct shock wave
being formed at the side of body around 80% of the root chord. The shockwave is
relatively strong in that the flow goes from about Mach 1.1 to about Mach 0.8. One
interesting aspect of this shock formation is that the shock radiates outward at very close
to a 90-degree angle. Recall back to Figure 15 when it was established that the Mach
cone angle for freestream flow that is sonic (Mach = 1.0) is a 90-degree angle.
Technically, at a freestream Mach number of 0.95 there shouldn’t even be a Mach cone
but since the flow upstream of the shock is noticeably supersonic, there is in fact a Mach
cone angle associated with this region of flow. The flow in this region reaches Mach 1.18
and the associated Mach cone angle is about 58 degrees but the measured angle of the
shock wave does not line up with that as seen in Figure 63 where it’s measured to
actually be about 65.8 degrees.
87
Figure 61. Pressure Coefficient Contour from SU2 at Mach 0.95 (α = 0.2 deg, CL =
0.204)
Figure 62. Mach Number Contour from SU2 at Mach 0.95 (α = 0.2 deg, CL = 0.204)
88
Figure 63. Shock Wave Angle Inconsistency with Mach Angle at Mach 0.95 (α = 0.2
deg, CL = 0.204)
This does not seem to be a coincidence but what is more interesting is that the
shockwave does appear to be forming at the point on the fuselage where the second
derivative of the cross-sectional area is maximized in accordance with the cutting plane
for a freestream Mach number of around 1.0 (90 degrees). Figure 64 shows the
superposition of the cross-sectional area curve with one of the isobars from the Mach
0.95 case but keep in mind that the cross-sectional area plot is upside down. We can see
that the shock wave does indeed form at the point where the curve of the second
derivative is maximized. The second derivate is calculated via finite differencing schemes
as seen in Equations 12 - 14 for center points and end points (forwards, center, and
backwards respectively).
89
𝑓′′(𝑥𝑖) =𝑓(𝑥𝑖)−2𝑓(𝑓𝑖+1)+𝑓(𝑥𝑖+2)
ℎ2+ 𝑂(ℎ) [12]
𝑓′′(𝑥𝑖) =𝑓(𝑥𝑖−1)−2𝑓(𝑓𝑖)+𝑓(𝑥𝑖+1)
ℎ2+ 𝑂(ℎ2) [13]
𝑓′′(𝑥𝑖) =𝑓(𝑥𝑖−2)−2𝑓(𝑓𝑖−1)+𝑓(𝑥𝑖)
ℎ2+ 𝑂(ℎ) [14]
Figure 64. Pressure Coefficient Isobar Superimposed with the Cross-Sectional Area
Distribution and it’s Second Derivative (Mach 0.95)
90
The implications of this finding are that the formation of the shock wave may be
deeply rooted in the interaction of the fuselage geometry with the wing geometry. With
the obvious shockwave interfering with a large section of the wing, we would like to see
how the performance characteristic of the airplane are at this Mach number. Figures 65
and 66 reveal that the lift curve is actually steeper (more in line with thin airfoil theory)
than previous cases despite the large shock interference and the drag is considerably
lower than Mach 1.3 case and comparable to the Mach 0.3 case.
Figure 65. α vs. CL Plot for the Mach 0.95 Case with SU2
91
Figure 66. CL vs. CD Plot for the Mach 0.95 Case with SU2
However, as we look to Figures 67 and 68 which show the relevant contours for
the Mach 1.05 solution, we notice an inconsistency. According to Figure 15, the Mach
cone angle for this solution would be about 72.2 degrees but what we actually observe is
close to 50.2 degrees as seen in Figure 69. Additionally, the flow upstream of the shock is
probed at Mach 1.39 and this would correspond to an angle of 46.0 degrees which is
closer, but still a good bit off of what would be expected. The performance impacts here
are also a lot more significant in terms of drag as seen in Figure 70 where we see a CD0
which is much higher than the 0.95 solution for only 0.1 Mach greater in speed.
92
Figure 67. Pressure Coefficient Contour from SU2 at Mach 1.05 (α = 0.2 deg, CL =
0.186)
Figure 68. Mach Number Contour from SU2 at Mach 1.05 (α = 0.2, CL = 0.186)
93
Figure 69. Shock Wave Angle Inconsistency with Mach Angle at Mach 1.05 (α = 0.2 deg ,
CL = 0.186)
Figure 70. CL vs. CD Plot for the Mach 1.05 Case with SU2
94
However, if we look at Figure 71, which is the same idea as Figure 64, we see
once again that the shock wave in question is being formed again at the point along the
root chord where the cross sectional area’s second derivative is large. Since the Mach
cone angle in this case is 72.2 degrees, then the cross-sectional area cutting planes
according to Harris9 need to be set to this same angle. This is achieved in SolidWorks
using the “slice” tool and can be seen in Figure 72.
Figure 71. Pressure Coefficient Isobar Superimposed with the Cross-Sectional Area
Distribution and it’s Second Derivative (Mach 1.05)
95
Figure 72. Visual Representation of Cutting Planes for Cross-Sectional Area Calculations
(Mach 1.05→ 72.2 deg)
The idea that the interaction of the fuselages cross sectional area causing a blockage
or restriction in the flow field around the wing and thus triggering a shock wave is one
that is thus far supported by the findings in SU2. We can further corroborate the volume
grid CFD findings by the fact that colleagues of ours ran similar trials using an Euler
solver in STAR CCM++ to compare against and the most pertinent two are given below.
We see in Figures 73 and 74 that the STAR CCM++ results for the Mach 0.95 and Mach
96
1.05 cases are in strong agreeance with the SU2 results from this thesis, as in engineering
we like to have consistency over various independent sources, this achieves exactly that.
Figure 73. Pressure Coefficient Contour from STAR CCM++ at Mach 0.95 (α = 0.2 deg,
CL = 0.230)
Figure 74. Pressure Coefficient Contour from STAR CCM++ at Mach 1.05 (α = 0.2 deg,
CL = 0.209)
97
Now that we’ve got solutions for multiple angles of attack for multiple Mach
numbers, we can look a little more in depth at the performance of the design over its
operational envelope as a whole. First we’d like to look at how the zero-lift drag changes
with Mach number, and we can see in Figure 75 that the zero-lift drag is heavily
dependent on Mach number and increases sharply around Mach 1.0, which is consistent
with accepted classical aerodynamics. The Harris Wavedrag results are also
superimposed on this figure and overall, we can see that the drag results are largely
similar in magnitude.
Figure 75. Zero-Lift Drag from SU2 and Wavedrag from D2500 Superimposed
98
The next major concern with supersonic aircraft is the issue of longitudinal
stability, specifically with regard to the locations of the aerodynamic center (AC) and
center of pressure (CP). Recall that for typical 2-D symmetric airfoils, the aerodynamic
center will be located at the quarter chord (25% of the mean aerodynamic chord) and will
not shift due to angle of attack. For this airplane to be considered stable we would like to
see an aerodynamic center that does not change vs. angle of attack for any given Mach
number. We are also interested in the location of the center of pressure, which is where
the net lift force will act, and we would like to see this location trending towards the
aerodynamic center over the range of angles of attack. We would like to see these trend
to the sane point because the farther away the AC is from the CP the more intense
pitching moment the aircraft will experience. Equations for the two parameters are given
below in Equations 15 and 16 where MPR is moment reference point, and MAC is mean
aerodynamic chord.
𝐶𝑃 = (𝑀𝑅𝑃) − (𝐶𝑚
𝐶𝐿∗ 𝑀𝐴𝐶) [15]
𝐴𝐶 = (𝑀𝑅𝑃) − (𝑑𝐶𝑀
𝑑𝐶𝐿∗ 𝑀𝐴𝐶) [16]
Figure 76 shows the trend for each of these parameters over the full range of Mach
numbers, and since the CM vs CL plots in the range of angles of attack between 3 and -3
are linear (all resembling Figure 60), the aerodynamic centers due remain constant which
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is good for stability. The aerodynamic center does shift rearward with increasing Mach
number, but this is to be expected and legacy airplanes such as Concorde incorporated a
rear fuel tank in the tail for ballast at supersonic speeds to counter this effect. When the
CP is located aft of the AC this induces a nose down pitching moment and conversely
when the CP is located forward of the AC this induces a nose up pitching moment.
Additionally, we can see that the trend of the centers of pressure is towards the
aerodynamic center which is desirable for longitudinal stability.
Figure 76. Aerodynamic Center and Center of Pressure for Each Mach Number and
Angle of Attack
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However, you may have noticed that the pitching moment polar for the low speed
case is NOT linear for the entire band of angles of attack between -5 and 15 and using
finite differences we can get a better feel for the aerodynamic center and center of
pressure for this Mach number using finite differencing again. Even with the non-
linearity in the pitching moment polar, Figure 60 shows that the stability characteristics
are still relatively tame (even without a vertical tail) for the whole operating range. Note
that the plot starts at -3 degrees angle of attack and this is due to a sign switch in CL
causing a singularly in Equation 15 around an angle of attack of -2 degrees.
Figure 77. Finite Differencing in the Low Speed Stability Characteristics
However, there is something else that may be going on here and that it goes back to
the concept of the upper and lower critical Mach numbers. Although the leading edge of
the wing is swept to an angle that is indicative of subsonic leading-edge normal flow,
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there is existing theory that asserts that the sweep angle of the trailing edge ,might be just
as important in deciding the lower critical Mach number. In the case of this airplane, the
trailing edge has a significantly shallower sweep angle (48.7 deg) and Figure 78 (case A)
shows the potential implications of the wing geometry. It illustrates a wing whose leading
edge is swept behind the Mach line generated by the incoming flow but whose trailing
edge is not thus triggering a trailing edge shock wave.
Figure 78. Implications of Sweep Angles in Relation to Mach Lines
Figure 79 shows Figure 78 (A) recreated for this specific wing and with Mach angles
that correspond to the upper design point of Mach 1.3 (50.3-degree Mach cone angle). If
the Mach lines at this Mach number are superimposed against the wing planform, we see
that the wing does in fact replicate the situation seen in Figure 78 and thus the Mach cone
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is interfering with the platform of the wing. This wing features a sizable Yehudi and in
Figure 79 the Mach cones propagating from the point of the wing where the trailing
edges would meet if there were not a Yehudi are shown. In Figure 80 the same concept is
shown but the Mach lines instead radiate out from the end of the Yehudi and extend into
the side of body.
Figure 79. Wing Geometry Superimposed on Mach Cone Angles at Mach 1.3 (Mach
Lines Propagate from Trailing Edge Intersection)
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Figure 80. Wing Geometry Superimposed on Mach Cone Angles at Mach 1.3 (Mach
Lines Propagate from Yehudi Intersection)
The lines seen in Figures 79 and 80 show a keen resemblance to the results from
SU2 in Figures 67 and 68 but the inconsistency is that those runs were at Mach 1.05 in
which case the shock angle should have had a much different angle than what we see
above. What we would have expected to see is Figures 79 and 80 resembling Figures 56
and 57 since those were the runs at Mach 1.3 where the Mach cone angle would match,
but we do not observe much similarity between these sets of figures.
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The trend seen in the SU2 results is that the shock wave becomes less and less
prominent as Mach number increases. Following this trend, it is not a bad assumption to
assume that if the Mach number were to continue increasing that the shock might “fall
off” the wing altogether and finally leaves a shock free planform. Out of pure curiosity I
ran this wing through a Mach 2.0 simulation which is much faster than it was ever
designed for to see if a shock would still be visible. Figure 81 shows that the trend is
actually correct and the shock wave that has been evident this far has “fallen off” the
wing planform. An interesting finding that perhaps future research can expand upon.
Figure 81. Pressure Coefficient Contour from SU2 at Mach 2.0 (α = 0.2 deg, CL = 0.110)
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Chapter 7: Analyzing Critical Mach Number(s) and Pressure Coefficient(s)
To this point, this thesis has detailed the process of attempting to achieve a shock
free wing design at a supersonic Mach number of 1.3, using subsonic design methods
derived from pervious aerodynamics experts. The design presented in this thesis was
designed rigorously to the standards of a critical Mach number, and by extension, a
critical pressure coefficient such that the wing would not incur any shock waves over it’s
planform at the design cruise speed. However, Chapter 6 shows that even though the
design met the prescribed low speed criterion for a subcritical wing, shock waves still
present themselves in the supersonic flight regime and even into slightly subsonic flight
conditions as well. This finding suggests that the low speed design criterion that the wing
was designed to meet are either incomplete, or invalid altogether. In this Chapter the
contradictory results presented in Chapter 6 will be analyzed in more detail as it pertains
to critical pressure coefficients and critical Mach numbers.
First, let’s take a closer look at the Mach 0.95 solution and why this particular
result is so perplexing. This wing was designed for a cruise speed of Mach 1.3 and
according to the low speed theory, the wing should have been shock free even at this
supersonic speed, so to see shocks forming in the subsonic freestream region is quite
concerning. We have already obtained the critical pressure coefficient for the design case
of Mach 1.3 and as seen in Chapter 6 that despite not reaching the critical pressure
coefficient, a shock is forming, nonetheless.
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We can perform this same analysis to the Mach 0.95 result. See Figure 82 for the
workup of CP* for Mach 0.95, but it turns out to be almost exactly -0.75. Meaning, that
for a wing designed to operate just at critical conditions at Mach 0.95, the wing in the low
speed case should have a local pressure coefficient of -0.64 and we’d expect the Mach
0.95 solution to then exhibit a local pressure coefficient of -0.75. Going back to Figure 61
and probing the solution for the most negative pressure coefficient along the span, it turns
out to be -0.48 just forward of the visible shock wave. This finding is very interesting
because the Mach 1.3 critical pressure coefficient was only slightly off of its actual
measured pressure coefficient ahead of the shock, but in this case the pressure coefficient
is nowhere near what would be expected to cause a shock wave to form. In fact, the free
stream Mach number that would correspond to a critical pressure coefficient of -0.48 is
Mach 1.18 as seen in Figure 83. But even if this were the case, Küchemann predicts a low
speed pressure coefficient of -0.36 which if we look back to Figure 50 of the low speed
SU2 solution, the pressure coefficient never approaches that. The inconsistencies here are
numerous and quite large in magnitude as well.
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Figure 82. Critical Pressure Coefficient Workup for Mach 0.95 and a 57-Degree Leading-
Edge Swept Wing
Figure 83. Extrapolation to a Critical Pressure Coefficient of -0.48 at Mach 1.18 with 57-
Degrees of Leading-Edge Sweep
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However, the mystery deepens when the same steps are applied to the Mach 1.05
solution as in the previous paragraph. Reference Figure 84 for the critical pressure
coefficient workup for this Mach number. It turns out that the critical pressure coefficient
predicted by Küchemann is -0.61 at Mach 1.05. However, once again by probing the SU2
result from Figure 67, we find that the most negative pressure coefficient along the span
is only -0.53 suggesting again that there should not be a shock present. By extending the
same line of logic as in the previous paragraph, according to the low speed analogies
presented by Küchemann, for this flow to become critical, we’d need a low speed
pressure coefficient of at least -0.5. Looking again to Figure 50 of the low speed SU2
solution, the local pressure coefficients never even reach half of the required -0.5. The
free stream Mach number corresponding to a critical pressure coefficient of -0.53 is Mach
1.12 as seen in Figure 85. This again implies that the low speed solution would be
producing a pressure coefficient of at least -0.42 which again, isn’t even close to the
reality of things.
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Figure 84. Critical Pressure Coefficient Workup for Mach 1.05 and a 57-Degree Leading-
Edge Swept Wing
Figure 85. Extrapolation to a Critical Pressure Coefficient of -0.61 at Mach 1.12 with 57-
Degrees of Leading-Edge Sweep
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Interestingly from these results we notice that the discrepancy is decreasing as the
flow becomes more and more supersonic. In fact, by the time that the flow has reached
the cruise speed of Mach 1.3, the flow is only “acting like” a flow of Mach 1.33. This
trend is plotted below in Figure 86 and suggests that Küchemann’s theory7 may in fact
become more accurate more Mach numbers further into the supersonic region above
Mach 1.3.
Figure 86. Difference in Observed Mach Number and Equivalent Mach Number
These results above point to the concept of the Lower Critical Mach number
coming into play, because for both of the workups provided in Figures 82 and 84, they
were given a sweep angle equal to the leading-edge sweep of 57 degrees. But as
Neumark12 has previously been cited saying, it all depends on the sweep angle of the
local pressure isobars. So really, it’s not totally accurate to assume that we can use 57
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degrees in the previous workup, instead we might try to use the sweep of the local isobars
which would have the effect of decreasing the critical pressure coefficients.
This becomes a circular problem in a sense because once a shock has formed, it
will fundamentally change the wings pressure distribution and thus the wings pressure
isobar pattern. Luckily, we can assume that the pressure isobar pattern will remain similar
in shape (but different in magnitude) up until the point of a shock formation, meaning we
can use the low speed isobar pattern to determine local sweep angles.
Starting with the Mach 0.95 case, we have observed that there is a shock forming
in the presence of a most negative pressure coefficient of -0.48. Instead of solving the
difficult problem of iterating through sweep angles to find a critical isobar sweep angle,
we can instead visually obtain the isobar sweep angle from Figure 50 in the region where
the Mach 0.95 shock is forming and extrapolate from there. This is presented in Figure 87
and we see that the isobar sweep angle in the vicinity of the shock location is around 25.3
degrees. Interestingly enough, this isobar sweep angle is actually pretty similar to the
shock wave angle measured in terms of sweep from vertical, instead of the angle from
horizontal in the case of the Mach cone angle.
112
Figure 87. Local Isobar Sweep Angle from the Low Speed SU2 Solution Superimposed
on the Relevant Mach 0.95 SU2 Solution
113
With the local isobar sweep angle identified, the analysis to find critical pressure
coefficients and Mach numbers is exactly the same as above but now the sweep angle is
no longer 57 degrees, its 25.3 degrees. This workup is presented below in Figure 88 but
this figure reveals that the new low speed pressure coefficient would be -0.38 and this
corresponds to a new Mach 0.95 critical pressure coefficient of -0.74 (which is not much
different than when the sweep was set to 57 degrees). This once again poses an issue
because the low speed solution in the vicinity of the shock formation only exhibits a
pressure coefficient that is as low as -0.2 which is still significantly off of the -0.38 that
Küchemann would have everyone believe is required for shock formation. Not to
mention that the Mach 0.95 solution still isn’t producing a pressure coefficient near -0.74.
Figure 88. Critical Pressure Coefficient Workup for Mach 0.95 and a 25.3-Degree
Pressure Isobar Sweep Angle
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The next logical question to ask is: If 25.3 degrees of isobar sweep won’t trigger a
shock according to Küchemann, what angle will? As it turns out, this angle doesn’t exist
because when 25.3 degrees is substituted out for 0 degrees, indicating isobars that are
perfectly orthogonal to the oncoming flow, the Küchemann workup results in a low speed
critical pressure coefficient that is -0.24 which is still too negative compared to what the
local pressure coefficients in the SU2 solution are showing.
However, this critical pressure coefficient of -0.24 at 0-degrees of sweep is an
interesting result because the low speed SU2 solution does actually reach that number but
in a region of the span that is not affected by the Mach 0.95 shock formation. As Neumrk
as stated previously, there must be a point along the span where the isobars would have a
zero-sweep angle but normally it would be expected that only the local conditions near a
shockwave would contribute to its formation. This finding possibly indicates that a shock
wave could be triggered along the span NON-LOCALLY to the point where the pressure
coefficients are reaching the zero-sweep angle critical pressure coefficient i.e. the Lower
Critical Pressure Coefficient.
In the effort of expanding the sample size, this same analysis is performed on the
Mach 1.05 solution. Once again, the first step is to get a measure of the local isobar
pattern sweep angle out at the point where the shock wave is visually forming in the
Mach 1.05 SU2 solution. This is shown below in Figure 89 and reveals a much more
drastic sweep angle of about 49.2 degrees.
115
Figure 89. Local Isobar Sweep Angle from the Low Speed SU2 Solution Superimposed
on the Relevant Mach 1.05 SU2 Solution
116
Once again, chasing this angle through the Küchemann workup, it is found that
the critical low speed pressure coefficient turns out to be -0.44 corresponding to a nearly
identical Mach 1.05 pressure coefficient of -0.61. This workup is presented below in
Figure 90 but it is not surprising to find that the local low speed pressure coefficient in
the region of the shock is only -0.21 in the most intense areas of suction. The Mach 1.05
case similarly also does not reach its prescribed pressure coefficient of -0.61 and only
achieves around -0.53. Inconsistent results persist on both sides of this design problem.
Figure 90. Critical Pressure Coefficient Workup for Mach 1.05 and a 49.2-Degree
Pressure Isobar Sweep Angle
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For the Mach 1.05 case it is even less surprising that there is no such angle that
could produce a low speed critical pressure coefficient of -0.21 because when the angle is
set to zero all of the CP curves in plots like those found in Figure 90 have a vertical
asymptote at Mach 1.0 and thus will never intersect with the vertical line at Mach = 1.05.
This follows the rationale of this thesis because at supersonic Mach numbers, a wing or
pressure isobar that has no sweep will be devoid of any spanwise flow component and
thus it would be impossible to design a shock free supersonic wing that has zero degrees
of sweep angle. By extending the principle of the Lower Critical Mach Number, the
Mach 1.05 case must have isobars that are perfectly normal to the flow, and thus a shock
wave is inevitable no matter how much the wing is swept according to Neumark12.
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Chapter 8: Conclusion
It has been shown that low speed results from a simple, yet robust, potential flow
code (VORLAX) can be validated to a high degree of accuracy against low speed CFD
results from multiple software packages (SU2 and STAR CCM++). Sandwich panel
methods have been proven to produce accurate low speed lift coefficient vs. angle of
attack and pitching moment vs. lift coefficient plots. Drag predications are found to be
inadequate even at low speeds due to the unreliability of sandwich panel drag methods
and the lack of nuanced geometry in flat plate methods.
Pervious work by Kirkman & Takahashi20,21 has shown that Simple Sweep
Theory is in adequate for the use of supersonic design goals by extrapolating from low
speed performance solutions. Methods prescribed by Küchemann12 have been shown in
previous work to produce more reliable results but have been discovered in this work to
produce some serious inconsistencies as well. Analysis of low speed pressure coefficient
data is implicitly solved to find expected results of high transonic (Mach 0.95) and low
supersonic (Mach 1.05, Mach 1.3) pressure coefficient data. Küchemann’s theory is
employed with the goal of designing an entirely shockwave free wing planform with only
low speed data obtained from a potential flow code (VORLAX).
By finding the low speed critical pressure coefficient that would lend itself to the
high-speed critical pressure coefficient, the low speed design in this work should remain
119
subcritical all the way up to the design cruise speed of Mach 1.3 according to
Küchemann’s theory. The reality of the situation is much different, as both CFD
packages reveal significant shock formation at Mach numbers 0.95, 1.05, and even the
cruise speed of 1.3. Not only does the wing generate unexpected shock waves, the local
pressure coefficients in the presence of these shock waves are much less negative (closer
to 0) than the critical pressure coefficient values predicted by Küchemann’s theory. In
other terms, the local high-speed pressure coefficients are not reaching the critical values
identified in the low speed results but are still generating shock waves along the
planform. Interestingly, the discrepancy between Küchemann’s theory and the CFD
results decreases as the free stream Mach number increases and if the wing is run at a
Mach number of 2.0, which is well above its design point, the wing is in fact shock free.
There are a few possible explanations for the unexpected formation of these
“corona shocks” along the span that are investigated in this work. The first of which is
related to Nuemark’s7 theory of an Upper and Lower Critical Pressure Coefficient. This
theory states that the effective sweep angle that should be considered is not that of the
leading edge but is the sweep angle of the local pressure isobar patterns. Since any real
wing that is swept back with a line of bilateral symmetry must have isobars that unsweep
to perfectly orthogonal to the free stream, there will always be a point along the wing
where the effective sweep is zero degrees. A wing with zero degrees sweep cannot have a
critical pressure coefficient less than 0 at Mach 1.0, and thus will always incur locally
supercritical flow at speeds approaching and exceeding Mach 1.0.
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However, visually measuring the isobar sweep patterns and taking them as
effective sweep angles leads to more inconsistencies with Nuemark’s theory. When this
is done with the Mach 0.95 and Mach 1.05 solutions, pressure coefficients observed are
still not negative enough in magnitude to correspond with the critical pressure
coefficients calculated using the effective isobar sweep angles. There large
inconsistencies found on both ends of the solutions (both high and low speed solutions)
when local isobar sweep angles are taken into account. In fact, according to the low speed
VORLAX solution, even in the region where the isobars are perfectly orthogonal to the
flow, the pressure coefficient is still not negative enough to trigger a shock at Mach 0.95,
and yet, one persists in the SU2 solutions.
The next possible explanation investigated is the interference of the leading and
trailing edge sweep angles with the local Mach cone angle35. This investigation yields
more dead ends because the shock waves that being triggered in all of the SU2 solutions
do not match in angle with the expected Mach cone angle based on the free stream flow
velocity. Even taking the local wing borne flow speed directly ahead of the shock wave
does not any continuity with the expected Mach cone angle and the observed shock wave
angle.
The last and potentially most promising partial explanation of the shock wave
formation is based in the concept of area ruling and wing-fuselage interferences.
121
Whitcomb27 and Harris13 have asserted that for a wing-body combination where the
fuselage resembles a body of revolution, there is the potential for corona shock formation
at the point along the fuselage where the second derivative of the cross-sectional area is
maximized. This design is not a perfect body of revolution as described by Harris, but
this thesis finds some interesting truth to these claims. By taking cross sectional area cuts
at angles corresponding to the free stream Mach cone angle and using finite differencing,
an acceptably accurate representation of the second derivative can be constructed. By
superimposing these results in the SU2 results, the shock waves do indeed tend to form
very closely to the point of the cross-sectional area’s second derivative’s maximum.
Although the theory used for the wing deigns in this work does yield flight
characteristics (pitching moments and lift coefficients) that would be suitable for Mach
1.3 flight, the existence of the shock waves along the planform pose some serious design
complications. Control surface interferences and drag penalties are very prominent issues
with these unexpected corona shocks. This works has shown that the theory described
within is either incomplete or inaccurate and future designers of an aircraft of similar
purpose must take into account other factors besides currently published theory. The
concept of Lower Critical Conditions are paramount to the success of designs of this type
and are likely going to be functions of some combination of wing geometry, isobar
geometry, and wing-body area ruling, and relevant Mach cone angles.
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