NUMERICAL INVESTIGATION AND FLOW INTERACTIONS OF A MIXEDCOMPRESSION HYPERSONIC INLET
A Thesis by
Pierre-Andre Bes
Bachelor of Science, University of Miami, 2006
Submitted to the Department of Aerospace Engineering
and the faculty of the Graduate School of
Wichita State University
in partial fulfillment of
the requirements for the degree of
Master of Science
December 2014
© Copyright 2014 by Pierre-Andre Bes
All Rights Reserved
NUMERICAL INVESTIGATION AND FLOW INTERACTIONS ON A MIXEDCOMPRESSION HYPERSONIC INLET
The following faculty members have examined the final copy of this thesis for form and
content, and recommend that it be accepted in partial fulfillment of the requirement for
the degree of Master of Science with a major in Aerospace Engineering.
___________________________________
Klaus Hoffmann, Committee Chair
___________________________________
Hamid Lankarani, Committee Member
___________________________________
Roy Myose, Committee Member
Accepted for the College of Engineering
____________________________________
Royce Bowden, Dean
Accepted for the Graduate School
____________________________________
Abu Masud, Interim Dean
iii
DEDICATION
To my family and the people dearest to my heart whose endless support is the only reason
this thesis ever came to completion.
iv
The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge.
v
ACKNOWLEDGEMENTS
I would like to express my most sincere appreciation to my thesis advisor, faculty
member, and great friend, Dr. Klaus Hoffmann for his help and guidance throughout my
graduate studies. I would also like to thank my loved ones and closest friends who never
stopped supporting me during the most difficult and stressful times of my studies.
vi
ABSTRACT
A numerical investigation of a fixed geometry mixed-compression scramjet inlet
is presented in this paper to illustrate the compression process and flow interactions of a
hypersonic inlet prior to supersonic combustion. Through the use of an AUSM
(Advection Upstream Splitting Method) differencing scheme, applied for both inviscid
and turbulent scenarios, this analysis explores the complex phenomena associated with
hypersonic flows such as shock-shock and shock-boundary layer interactions. Particular
attention is placed in the vicinity of the inlet throat area, where such interactions may
cause flow separation and give rise to further complications such as inlet unstart. The
inlet geometry is a modified version of the hydrogen-fueled axisymmetric scramjet used
on the Hypersonic Flying Laboratory (HFL) named “Kholod”, designed and tested by
NASA and the Central Institute of Aviation Motors (CIAM) on February 12, 1998. The
inlet is computationally solved using the commercial software FLUENT 6.3 and the
computational grids were generated using the grid generator GAMBIT 2.4.6. Grid nodes
were clustered near the critical flow path areas in order to accurately capture any viscous
interactions within the flow field. Solutions presented in this paper assumed free stream
properties equivalent to an altitude of 10,000 meters, at 0° angle of attack, with Mach
numbers of 5, 7, and 9, all solved through both inviscid and turbulent models.
vii
PREFACE
My study and research on hypersonics and hypersonic applications have
undoubtedly been the most interesting and enriching of my entire studies in Aerospace
Engineering. This captivating branch of aerodynamics has unlocked a whole new set of
challenges and ideas that even today’s brightest engineers and scientists have yet to fully
conquer. My passion for flight science and space, my interest in learning, and most
importantly the people dearest to my heart have all kept me motivated and determined to
complete this project with the hope that I will one day have the opportunity to work in
such a fascinating and promising field.
viii
TABLE OF CONTENTS
Chapter
1.
INTRODUCTION………………….……………………………………...…..….1
1.1
1.2
1.3
1.4
1.5
1.6
2.
Page
Historical Overview of High Speed Flight……………………………..…1
Hypersonic Aerodynamics Technical Background………...…….……...15
1.2.1 What is Hypersonics?..…………………………………………..15
1.2.1.1 Shock Layers…………………...………………………...16
1.2.1.2 Entropy Layers………...…………………....…………....17
1.2.1.3 Viscous Interactions……………………………….……..18
1.2.1.4 High Temperature Effects……………………...…….…..19
1.2.1.5 Low Density Effects………………………….………….21
1.2.2 Governing Equations for Hypersonic Aerodynamics…………....22
1.2.2.1 Hypersonic Oblique Shock Equations…………………...22
1.2.2.2 Hypersonic Expansion Wave Equations…………………27
1.2.2.3 Navier-Stokes Equations………………...……………….28
1.2.2.4 Boundary-Layer Equations………………………………31
Scramjet Engines………………………………………………………...32
1.3.1 Inlet………………………………………………………………35
1.3.2 Isolator………………………………………………………...…36
1.3.3 Combustion Chamber……………………………………………37
1.3.4 Exhaust Nozzle………………………………………………..…39
Brayton Cycle………………………………………………...............….40
Relevant Research…………………………………………………...…...43
Objectives for Research Paper…………………………………………...45
HYPERSONIC INLET CHARACTERISTICS….…………………….………..46
2.1
2.2
Basic Functions………………….……………………………………….46
Compression Process and High Temperature Effects……………….…...47
2.2.1 Scramjet Forebody Leading-Edge…….………………………….….48
2.2.2 External Compression Ramps…………………………….……….…50
2.2.3 Scramjet Cowl Leading-Edge…………………………………..……52
2.3
Inlet Performance..…………………...…………………………..………53
2.3.1 Air Capture Ratios and Design Point……… ………………………..53
2.3.2 Compression Efficiency and Contraction Ratios…………………….56
2.4
Flow Interactions and Inlet Starting………………………...……………59
2.4.1 Shock-Shock Interactions……………………………………………59
2.4.2 Shock-Layer Interactions…………………………………………….62
2.4.3 Inlet Starting and Kantrowitz Limit……………………………….....64
3.
NUMERICAL METHODOLOGY & SETUP………………………………......68
ix
TABLE OF CONTENTS (continued)
Chapter
Page
3.1
Main Aspect of CFD……………………………………………...….…..68
3.1.1 Introduction…………………………………………………………..68
3.1.2 CFD Software and Finite-Volume Method…………………………..69
3.1.3 FLUENT Code……………………………………………………….71
3.1.4 CFD Grid…………………………………...………………………..73
3.1.5 Turbulence Models………………………………………………..…76
3.1.5.1 Spalart-Allmaras Model……………………………..…….....…..77
3.1.5.2 k-ω SST Model…………………………..…………………....…78
3.1.5.3 k-ε Model………………………………..…………………….....80
3.2
Inlet Grid Generation……………………..……………………………...81
3.2.1 Viscous Mesh Characteristics...………..……………………...…..…82
3.2.2 Inviscid Mesh Characteristics……………………....……………..…90
3.3
FLUENT Initial and Boundary Conditions………………………………92
3.4
FLUENT Setup & Solver Properties……………………………...…..…95
4.
SIMULATION RESULTS………………….……………………………..…...102
4.1
Inviscid Results…………………………………...……………….……103
4.1.1 Inviscid Results – Flight Condition 1………………………..……..104
4.1.2 Inviscid Results – Flight Condition 2…………….....................…...106
4.1.3 Inviscid Results – Flight Condition 3……………........................…108
4.1.4 Inviscid Results – Discussion of Numerical Results...……………..110
4.1.5 Inviscid Results – Analysis of Numerical Results…..……………...113
4.1.6 Inviscid Results – Comparison to Analytical Results……………....120
4.1.7 Inviscid Results – Efficiency and Operability…………………..….124
4.2
Turbulent Results……………………………….………………………127
4.2.1 Turbulent Results – Flight Condition 1………………………….…128
4.2.1.1 Spalart-Allmaras Model………………………………………...128
4.2.1.2 K-Epsilon Model……………………………………………..…130
4.2.1.3 K-Omega SST Model………………………………………..…132
4.2.2 Turbulent Results – Flight Condition 2………………………….....134
4.2.2.1 Spalart-Allmaras Model……………………………………...…134
4.2.2.2 K-Epsilon Model……………………………………………..…136
4.2.2.3 K-Omega SST Model…………………………………………..138
4.2.3 Turbulent Results – Flight Condition 3………………………….…140
4.2.3.1 Spalart-Allmaras Model………………………………………...140
4.2.3.2 K-Epsilon Model………………………………………………..142
4.2.3.3 K-Omega SST Model…………………………………………..144
4.2.4 Turbulent Results – Discussion of Numerical Results……………..146
4.2.5 Turbulent Results – Analysis of Numerical Results……………..…149
x
TABLE OF CONTENTS (continued)
Chapter
4.2.6
4.2.7
5.
Page
Turbulent Results – Comparisons to Inviscid & Published
Results………………………………………………………………155
Turbulent Results – Efficiency………………..……………………156
CONCLUSION………………………………………………………………....159
BIBLIOGRAPHY………………………………………………………………………161
xi
LIST OF TABLES
Table
Page
1.
High Temperature Effects on Air………………..………………………………20
2.
Scramjet Flight Conditions……………………………………………..…….….93
3.
Computational Time and Iterations………………………………….................102
4.
Summary of FLUENT Numerical Results (Inviscid Flow)………………..…...117
5.
Summary of Analytical Results (External Ramp)…………………………...…121
6.
Summary of Analytical Results (Cowl)...………………………..……………..122
7.
Compression Efficiency for Inviscid Flight Conditions .…………...…….……124
8.
Inlet Internal Contraction Ratio …………………………………....……..……125
9.
Summary of FLUENT Numerical Results (Turbulent Flow)..………………....151
10.
Result Comparisons to Published Experimental Results.……………………....155
11.
Compression Efficiency for Turbulent Flight Conditions……………………...158
xii
LIST OF FIGURES
Figure
Page
Figure 1: Bell X-1 Rocket…………………………………………………………….….1
Figure 2: XLR-11 Rocket Engine………………………………………………………..2
Figure 3: Lockheed F-104 Starfighter……………………………………………………3
Figure 4: General Electric J79 Turbojet Engine…………………………………………4
Figure 5: Lockheed “Blackbird” SR-71………………………………………………….5
Figure 6: Pratt & Whitney J58-P4 engines………………………………………………6
Figure 7: Dr. Goddard & World’s First Liquid-Fueled Rocket………………………….8
Figure 8: V-2 Rocket Used During World War II……………………………………….9
Figure 9: V-2/WAC Corporal or “Bumper” Launch on 24 February 1949……………...10
Figure 10: Vostok 1 spaceship…………………………………………………………...11
Figure 11: X-15 Hypersonic Aircraft…………………………………………………….12
Figure 12: The Space Shuttle…………………………………………………………….13
Figure 13: CFD results of Space Shuttle heating during re-entry……………………......14
Figure 14: Thin shock layer from hypersonic flow…………………………………..….16
Figure 15: Entropy layer on from a blunt leading edge………………………………….17
Figure 16: Temperature distribution profile in hypersonic boundary layer……………...18
Figure 17: Effect of viscous interactions on boundary layer…………………………….19
Figure 18: Shock layer temperature vs. freestream velocity (at 52,000m)………...…….20
Figure 19: Oblique shock-wave geometry……………………………………………….23
Figure 20: θ-β-M Diagram………………………………………………………..….….25
Figure 21: Attached and detached oblique shocks……………………………………….26
xiii
LIST OF FIGURES (continued)
Figure
Page
Figure 22: Strong and weak shock solutions…………………………………………….27
Figure 23: Expansion Wave……………………………………………………………...28
Figure 24: Normal and shear stresses on a fluid element…………………………….….30
Figure 25: Thin boundary-layer vs. length of the body………………………………….31
Figure 26: NASA X-43a (left) and Boeing X-51 “Waverider” (right)……………….….33
Figure 27: Typical hypersonic vehicle mission profile………………………….……….34
Figure 28: Typical scramjet engine…………………………………………………...….34
Figure 29: Shock-train in isolator………………………………………………………..36
Figure 30: Supersonic combustion in a scramjet engine………………………………...38
Figure 31: Typical nozzle on a hypersonic vehicle……………………………………...39
Figure 32: Scramjet engine reference stations…………………………………………...40
Figure 33: Brayton cycle for scramjet engine……………………………………………41
Figure 34: Hypersonic mixed-compression inlet (2-D representation) …………………46
Figure 35: Hypersonic inlet shocks & compression process…………………………….47
Figure 36: Hypersonic Vehicle Forebody Leading-edge “Bow” Shock…………………48
Figure 37: Aerodynamic heating for slender & blunt vehicles…………………………..49
Figure 38: Forebody leading-edge high temperature effects…………………………….50
Figure 39: External ramp compression system ………………………..………………...51
Figure 40: External ramp compression system (Isentropic) …………………………….52
Figure 41: Cowl Leading Edge Shock and Internal Compression Process…………..….52
Figure 42: Undersped Inlet………………………………………………………...…….54
xiv
LIST OF FIGURES (continued)
Figure
Page
Figure 43: Inlet at Design Point………………………………………………………….55
Figure 44: Oversped Inlet…………………………………………………………….….55
Figure 45: Undersped vs Oversped Inlets………………………………………………..56
Figure 46: Inlet Contraction Ratios Areas……………………………………………….58
Figure 47: Potential Shock-Shock Interactions on Inlet………………………………....59
Figure 48: Interaction between oblique shocks…………………………………….…....60
Figure 49: Type-IV Shock-Shock Interaction………………………………………...….61
Figure 50: Regions of Potential Shock-Layer Interactions……………………………....62
Figure 51: Shock-Wave Boundary-Layer Interaction………………………………...….63
Figure 52: Unstarted Inlet Flow Characteristics……………………………………...….65
Figure 53: Possible Inlet Starting Solutions……………………………………………..67
Figure 54: Continuous vs. Discrete Domain………………………………………….….68
Figure 55: Control Volume Element………………………………………………….….70
Figure 56: 2D Mapping from Physical to Computational Space……………………..….73
Figure 57: 2D Structured Grid……….…………………………………………………..74
Figure 58: Typical Unstructured Triangular Mesh………………………………...…….75
Figure 59: 3-D Representation of Hypersonic Inlet………………………………..…….81
Figure 60: 2-D Representation of Hypersonic Inlet (Dimensions in mm) …………...….82
Figure 61: Inlet Viscous Mesh………………………………………………………..….83
Figure 62: Inlet Hybrid Mesh………………………………………………………..…..85
Figure 63: Boundary Layer Mesh Specifications…………………………………….....86
xv
LIST OF FIGURES (continued)
Figure
Page
Figure 64: Boundary Layer Mesh at Cowl Leading Edge…………………………..….87
Figure 65: Grid Clustering……………………………………………………………….88
Figure 66: Quad Element Aspect Ratio…………………………………………..…..….89
Figure 67: Tri Element Skewness………………………………………………...…..….89
Figure 68: Inlet Inviscid Mesh………………………………………………………..….91
Figure 69: Inviscid Mesh at the Walls………………………………………………..….92
Figure 70: Inlet Boundary Conditions………………………………………………..….94
Figure 71: Flow solver parameters……………………………………………….……..96
Figure 72: Spalart-Allmaras turbulence model parameters…………………………….97
Figure 73: K-Epsilon turbulence model parameters………………………………...….98
Figure 74: K-Omega SST turbulence model parameters……………………………….99
Figure 75: Solution control parameters………………………………………………....100
Figure 76: Material Properties………………………………………………………….101
Figure 77: Mach number contours for inlet – Flight Condition 1 (Inviscid) ……….….104
Figure 78: Mach number contours for throat region – Flight Condition 1 (Inviscid)
…………………………………………………………………………………….…….104
Figure 79: Static pressure contours for inlet – Flight Condition 1 (Inviscid) ………….105
Figure 80: Static pressure contours for throat region – Flight Condition 1 (Inviscid)
………………………………………………………………………………………..…105
Figure 81: Mach number contours for inlet – Flight Condition 2 (Inviscid) ….……….106
Figure 82: Mach number contours for throat region – Flight Condition 2 (Inviscid)
…………………………………………………………………………………………..106
xvi
LIST OF FIGURES (continued)
Figure
Page
Figure 83: Static pressure contours for inlet – Flight Condition 2 (Inviscid) ………….107
Figure 84: Static pressure contours for throat region – Flight Condition 2 (Inviscid)
………………………………………………………………………………….……….107
Figure 85: Mach number contours for inlet – Flight Condition 3 (Inviscid) …………..108
Figure 86: Mach number contours for throat region – Flight Condition 3 (Inviscid)
…………………………………………………………………………………….…….108
Figure 87: Static pressure contours for inlet – Flight Condition 3 (Inviscid) ………….109
Figure 88: Static pressure contours for throat region – Flight Condition 3 (Inviscid)
……………………………………………………………………………………….….109
Figure 89: Mach number contours for the Mach 5 flight condition (External
Compression) …………………………………………………………………………..110
Figure 90: Mach number contours for the Mach 5 flight condition (Internal compression)
……………………………………………………………………………………….….111
Figure 91: Wall Mach Number Plots for Inviscid Flight Conditions……………….….114
Figure 92: Wall Static Pressure Plots for Inviscid Flight Conditions……………….….115
Figure 93: Shock interactions and reflections within cowl region (Flight Condition 1)
…………………………………………………………………………………….…….118
Figure 94: Shock interactions and reflections within cowl region (Flight Condition 3)
…………..…………………………………………………………………………...….119
Figure 95: Nomenclature for analytical calculations (External Ramp) ………………..121
Figure 96: Nomenclature for analytical calculations (Cowl) …………………………..122
Figure 97: Numerical vs. Analytical Results…………………………………………...123
Figure 98: Inlet contraction ratios for starting conditions………………………..…….126
Figure 99: Mach number contours for inlet – Flight Condition 1 (Spalart-Allmaras)….128
xvii
LIST OF FIGURES (continued)
Figure
Page
Figure 100: Mach number contours for throat region – Flight Condition 1 (SpalartAllmaras)……………………………………………………………………………..…128
Figure 101: Static pressure contours for inlet – Flight Condition 1 (SpalartAllmaras)………………………………………………………………………………..129
Figure 102: Static pressure contours for throat region – Flight Condition 1 (SpalartAllmaras)………………………………………………………………………………..129
Figure 103: Mach number contours for inlet – Flight Condition 1 (K-Epsilon)……….130
Figure 104: Mach number contours for throat region – Flight Condition 1 (KEpsilon)………………………………………………………………………………....130
Figure 105: Static pressure contours for inlet – Flight Condition 1 (K-Epsilon)………131
Figure 106: Static pressure contours for throat region – Flight Condition 1 (KEpsilon)………………………………………………………………………………....131
Figure 107: Mach number contours for inlet – Flight Condition 1 (K-Omega SST)…..132
Figure 108: Mach number contours for throat region – Flight Condition 1 (K-Omega
SST)…………………………………………………………………………………….132
Figure 109: Static pressure contours for inlet – Flight Condition 1 (K-Omega SST)….133
Figure 110: Static pressure contours for throat region – Flight Condition 1 (K-Omega
SST)…………………………………………………………………………………….133
Figure 111: Mach number contours for inlet – Flight Condition 2 (Spalart-Allmaras)...134
Figure 112: Mach number contours for throat region – Flight Condition 2 (SpalartAllmaras)………………………………………………………………………………..134
Figure 113: Static pressure contours for inlet – Flight Condition 2 (SpalartAllmaras)………………………………………………………………………………..135
Figure 114: Static pressure contours for throat region – Flight Condition 2 (SpalartAllmaras)………………………………………………………………………………..135
Figure 115: Mach number contours for inlet – Flight Condition 2 (K-Epsilon)……….136
xviii
LIST OF FIGURES (continued)
Figure
Page
Figure 116: Mach number contours for throat region – Flight Condition 2 (KEpsilon)…………………………………………………………………………………136
Figure 117: Static pressure contours for inlet – Flight Condition 2 (K-Epsilon)………137
Figure 118: Static pressure contours for throat region – Flight Condition 2 (KEpsilon)…………………………………………………………………………………137
Figure 119: Mach number contours for inlet – Flight Condition 2 (K-Omega SST)…..138
Figure 120: Mach number contours for throat region – Flight Condition 2 (K-Omega
SST)…………………………………………………………………………………….138
Figure 121: Static pressure contours for inlet – Flight Condition 2 (K-Omega SST)….139
Figure 122: Static pressure contours for throat region – Flight Condition 2 (K-Omega
SST)…………………………………………………………………………………….139
Figure 123: Mach number contours for inlet – Flight Condition 3 (Spalart-Allmaras)...140
Figure 124: Mach number contours for throat region – Flight Condition 3 (SpalartAllmaras)………………………………………………………………………………..140
Figure 125: Static pressure contours for inlet – Flight Condition 3 (SpalartAllmaras)………………………………………………………………………………..141
Figure 126: Static pressure contours for throat region – Flight Condition 3 (SpalartAllmaras)………………………………………………………………………………..141
Figure 127: Mach number contours for inlet – Flight Condition 3 (K-Epsilon)……….142
Figure 128: Mach number contours for throat region – Flight Condition 3 (KEpsilon)………………………………………………………………………………....142
Figure 129: Static pressure contours for inlet – Flight Condition 3 (K-Epsilon)……....143
Figure 130: Static pressure contours for throat region – Flight Condition 3 (KEpsilon)………………………………………………………………………………....143
Figure 131: Mach number contours for inlet – Flight Condition 3 (K-Omega SST)…..144
xix
LIST OF FIGURES (continued)
Figure
Page
Figure 132: Mach number contours for throat region – Flight Condition 3 (K-Omega
SST)…………………………………………………………………………………….144
Figure 133: Static pressure contours for inlet – Flight Condition 3 (K-Omega SST)….145
Figure 134: Static pressure contours for throat region – Flight Condition 3 (K-Omega
SST)………………………………………………………………………………….....145
Figure 135: Mach contour results for the Mach 5 flight condition (External
Compression)…………………………………………………………………………...146
Figure 136: Mach number contours for the Mach 5 flight condition (Internal
compression)…………………………………………………………………………....147
Figure 137: Inlet lower surface Mach number………………………………………….150
Figure 138: Turbulent Parameter Y+ Value (Flight Condition 1)……………………...153
Figure 139: Turbulent Parameter Y+ Value (Flight Condition 2)……………………...153
Figure 140: Turbulent Parameter Y+ Value (Flight Condition 3)……………………...154
xx
NOMENCLATURE
CFL
Courant Number
CFD
Computational Fluid Dynamics
FC
Flight Condition
Ft
feet
Km/h
Kilometers per hour
Mph
Miles per hour
M
Mach number
M∞
Mach number at freestream conditions
M th
Mach number at throat
ρ
Density
ρ∞
Density at freestream conditions
Pa
Pascal
P
Static pressure
P∞
Static pressure at freestream conditions
Pt
Total pressure
PDE
Partial differential equations
Re
Reynold’s number
T
Static temperature
T∞
Static temperature at freestream conditions
Tt
Total Temperature
cp
Specific heat at constant pressure
cv
Specific heat at constant volume
xxi
NOMENCLATURE (continued)
γ
Ratio of c p over c v
h
Enthalpy

V
Velocity
V
Speed
u
x-component of velocity vector
v
y-component of velocity vector
w
z-component of velocity vector
υ
Volume
e
Entropy
δ
Boundary-Layer thickness
π
Total pressure ratio
ψ
Cycle static temperature ratio
ηc
Adiabatic Compression Efficiency
xxii
1
1.1
INTRODUCTION
Historical Overview of High Speed Flight
The desire to fly higher and faster has been the driver behind the majority of
breakthroughs in the history of flight. With the Wright’s Brothers’ first flight in 1903 and
Dr. H. Goddard’s first liquid-fueled rocket launch in 1926, it wasn’t long before speeds
approaching and even exceeding the speed of sound (i.e. “supersonic”) were reached (as
a reference, the speed of sound under normal conditions is around 1236 km/h or 768
mph, also referred to as “Mach 1”). In 1947, Air Force Captain Chuck Yeager performed
the very first manned supersonic flight onboard the “Bell X-1” Experimental Rocket
Plane (See Figure 1). The rocket airplane was air-launched at an altitude of 7000 m from
the bomb bay of a Boeing B-29 Superfortress Bomber, then using its rocket engine to
climb further and reach its mission altitude at 13000 m. Flight records showed that the
Bell X-1 flew at a top speed of 1127 km/h or 700 mph (Mach 1.06), becoming the first
airplane to ever fly faster than the speed of sound.
Figure 1: Bell X-1 Rocket
1
As for its propulsion system, the Bell X-1 was equipped with a four-chamber
XLR-11 rocket engine built by Reaction Motors, Inc (See Figure 2), allowing it to
increase its thrust incrementally by simply igniting one of its additional combustion
chambers. It used a mix of ethyl alcohol mixed with water for fuel, and liquid oxygen as
its oxidizer delivering 6700 Newtons of thrust per engine.
Figure 2: XLR-11 Rocket Engine
As the battle for air supremacy and military power went on, many more
technological advancements in aircraft, weapons, and propulsion design were made the
following years and the development as well production of supersonic vehicles,
2
particularly military fighter jets were growing exponentially. Examples include the
Lockheed F-104 Starfighter supersonic interceptor aircraft (See Figure 3), developed for
the United States Air Force in 1958 and retired in 2004, and capable of reaching Mach
2.2.
Figure 3: Lockheed F-104 Starfighter
On May 18, 1958, an F-104 set a world speed record of 1,404.19 mph (Mach 1.83), and
on Dec. 14, 1959 a world altitude record of 103,395 feet. The Starfighter was the first
aircraft to hold simultaneous official world records for speed, altitude and time-to-climb.
The F-104 was equipped with two General Electric J79 turbojet engines (See Figure 4),
which were side-mounted to the fuselage of the aircraft with fixed inlet cones and
geometry optimized for supersonic speeds.
3
Figure 4: General Electric J79 Turbojet Engine
In 1964, one of the most secretive yet exceptional and revolutionary breakthrough
in the history of flight for supersonic aircrafts was the development of the Lockheed SR71 or “Blackbird” (See Figure 5). This advanced strategic reconnaissance aircraft served
with the United States Air Force from 1964 to 1998 and was designed for Mach 3+ flight
with a flight crew of two in tandem cockpits, with the pilot in the forward cockpit and the
Reconnaissance Systems Officer (RSO) monitoring the surveillance systems and
equipment from the behind cockpit. Some of the SR-71’s reconnaissance equipment
included signals intelligence sensors, a side-looking radar and a photo camera. In addition
to its titanium alloy airframe required to withstand the high temperatures associated with
supersonic flight, stealth and threat avoidance were the key factors in the design of the
SR-71. Radar stealth studies and technologies at the time showed that a shape with
flattened, tapering sides would reflect most radar energy away from the radar beams'
place of origin. In addition, radar-absorbing materials were incorporated into sawtooth-
4
shaped sections of the aircraft's skin and cesium-based substances were mixed with the
fuel to reduce the visibility of the exhaust plumes to radars.
Figure 5: Lockheed “Blackbird” SR-71
In addition to being a milestone for this era, the SR-71’s formidable propulsion
system unit was the backbone of this aircraft. The Blackbird was equipped with two Pratt
& Whitney J58-P4 engines (See Figure 6) capable of producing 32500 pounds of thrust
(145000 Newtons), which allowed cruising speeds around 2457.6 mph (Mach 3.2). The
complexity and innovative technology behind the design of this air-breathing engine is
what differentiated the SR-71 from other supersonic aircrafts. The PW J58-P4 engine is a
hybrid engine, combining both turbojet and ramjet technology. At subsonic and low
supersonic speeds, compression of the air and energy from fuel combustion is obtained
from the turbojet engine. However, at higher supersonic speeds, the engine relies on the
ramjet technology by compressing the air though the shock cones and burning fuel in the
5
afterburner without using the compressor blades from the turbojet engine (being critical
elements in this paper, definitions and theory behind ramjets, scramjets, and inlet air
compression at high speeds are explained in details in later sections.).
Figure 6: Pratt & Whitney J58-P4 engines
The SR-71 was the world's fastest and highest-flying operational manned aircraft
throughout its career, breaking the world record for its class in 1976 with an altitude
record of 85,069 feet (25,929 m) and a speed record of 2193.2 mph (Mach 2.86).
Many more recent examples can be cited, displaying even further innovation and
ingenuity in terms of stealth, speed, and other critical factors in aerial combat. It should
be understood that with higher speeds comes not only more complex design concerns, but
greater safety concerns as well, since the human body can only handle certain speeds
before requiring special equipment to withstand the high “g-forces” associated with those
very high speeds. A simple example would be to examine the suits worn by astronauts
6
operating a space shuttle during the launch phase, and re-entry. These suits are
specifically designed to handle the enormous forces applied to their body during those
particular moments. As impairing as these suits can be, it is obvious that it would give
any fighter pilot a major disadvantage in combat even if the aircraft could reach such
high speeds, in addition to a great loss in maneuverability. That is why, more and more
“unmanned” vehicles and drones are now being developed and tested to reach very high
speeds within and outside the Earth’s atmosphere without the concern of putting a pilot’s
life at risk.
In summary, tremendous efforts were made since the Wright’s Brothers first flight
in 1903, particularly in the military to develop manned flying vehicles equipped with airbreathing engines capable of breaking the speed of sound and sustain supersonic flight
around Mach 2 or 3 granting air superiority. One may ask, what are the highest speeds
ever reached by flying vehicles when the human factor was never of concern and no
pilots were onboard, and how early were those speeds attained?
Dr. Robert H. Goddard (1882-1945) who is now considered one of the founding
fathers of modern rocketry, built and successfully launched the world’s first liquid-fueled
rocket in 1926 (See Figure 7). While the highest altitude reached was 2.6 km and fastest
speed recorded was 885 km/h (Mach 0.7), his revolutionary work gave rise to the
possibility of space travel and the concept of multi-stage rockets and missiles. His 1919
monograph titled “A Method of Reaching Extreme Altitudes”, along with his concepts of
three-axis controls, gyroscopes, and steerable thrust applied to rockets provided groundbreaking ideas for future rocket development.
7
Figure 7: Dr. Goddard & World’s First Liquid-Fueled Rocket
Following Dr. Goddard’s work, great improvements were made to the design of
rockets and missiles. Developed and used by Germany during World War II against ally
forces, the V-2 rocket was a single stage liquid propellant rocket used as a combatballistic missiles during the war (See Figure 8). Using a 75% ethanol/water mixture for
fuel and liquid oxygen for oxidizer, it was capable of reaching a maximum altitude of 206
km if launched vertically and a top speed of 5760 km/h (Mach 4.7). After launch, the V-2
propelled itself up for approximately 1 minute, then following engine shutdown, a
programmed motor altered the pitch angle allowing the rocket to continue its trajectory
based on a ballistic free-fall path.
8
Figure 8: V-2 Rocket Used During World War II
Even post-war, the V-2 remained subject to more experiments, improvements,
and testing. As part of program called “Bumper” developed to achieve high speeds and
high altitudes, the V-2 rocket was modified by the Unites States to carry an additional
slender, smaller size rocket at its tip called the WAC Corporal in an effort to illustrate the
use and efficiency of multistage rockets. On February 24, 1949 after its liftoff from the
test firing range in New Mexico (See Figure 9), the V-2 rocket (first stage) reached an
altitude of 160 km at a speed of 5600 km/h (Mach 4.6), at which point the WAC Corporal
rocket (2nd stage) was ignited and the first stage V-2 rocket jettisoned. The WAC
Corporal rocket then accelerated to a maximum speed of 8240 km/h (Mach 6.7) to reach
an altitude of 390.4 km. Once the WAC reached its maximum altitude and its engine shut
down, the rocket began nosing down and re-entered the Earth’s atmosphere at speeds
exceeding 8000 km/h (Mach 6.5). This day marked the very first hypersonic flight
achieved by a human-made object, where speeds greater than Mach 5 were reached.
Naturally, under the excessive temperatures experienced during re-entry the majority of
the WAC Corporal melted and only a few parts of the rockets were recovered later on.
9
Figure 9: V-2/WAC Corporal or “Bumper” Launch on 24 February 1949
It has been established and accepted as a conventional rule of thumb that when a
flow reaches or exceeds Mach 5, its flight regime is no longer characterized as supersonic
but is referred to as “hypersonic”. As a more scientifically correct definition, hypersonic
vehicle would be best defined as that regime where certain physical flow phenomena
become progressively more important as the Mach number is increased to higher values
(Anderson, 2006). Those specific flow phenomena are discussed and examined in later
sections.
The very first human spaceflight in history was carried out by Russian Major Yuri
Gagarin, as part of the Vostok 1 mission aboard the Vostok 3KA spacecraft on April 12,
1961. This flight marked the very first manned orbital flight in history, and the first
experience of hypersonic flight by a human being. The mission consisted of a single orbit
10
around the Earth, which took 108 minutes from launch to landing. During re-entry, the
Vostok 1 reached speeds around 19025 mph (Mach 25) and Major Gagarin endured as
many as 10 g while remaining conscious. Even though the capsule landed 280 km west of
the planned landing site, the successful landing was a revolutionary moment in the
history of space and high speed flight for the human kind.
Figure 10: Vostok 1 spaceship
Figure 10 shows the Vostok 1 spacecraft where the re-entry “capsule” module of
the vehicle separated about 8000 km above ground before entering the Earth’s
atmosphere, and afterwards deploying its parachute to land safely. The heatshield used on
the capsule module to withstand the extremely high temperatures during re-entry was a
resin (such as phenolic) covered with a striated material like asbestos.
11
As the next revolutionary high speed aircraft and part of the X-plane series of
experimental flying vehicles, the North American X-15 rocket-powered aircraft (See
Figure 11) was developed in 1959 and operated by the United States Air Force and
NASA. The X-15 was designed to be a hypersonic drop-launched manned vehicle that
would remain at high altitudes and reach hypersonic speeds. It had a long and cylindrical
fuselage, multiple thick wedge-fin stabilizers, rear fairings with a thick wedge tail to
maintain stability at hypersonic speeds. Even as of today, the X-15 holds the official
record for the fastest speed ever reached by a manned aircraft, reaching the edge of outer
space and managing to return safely. Its fastest speed recorded was 7274 km/h (Mach
5.9).
Figure 11: X-15 Hypersonic Aircraft
12
As a conclusion to this historical overview of high speed flight, a few words
should be said about the most breathtaking and revolutionary flying vehicle ever created
in the history of mankind, the space shuttle (See Figure 12).
Figure 12: The Space Shuttle
As part of a program for a system of reusable spacecrafts, the Space Shuttle was
crewed orbital spacecraft designed and manufactured by the United States that started in
1969 and led to operational space flights in 1982. It was composed of three main
components, the first component was the reusable Orbiter Vehicle which was itself
equipped with three main engines fueled by liquid hydrogen and liquid oxygen, and
which also contained the crew and payload. The second component was an expandable
dark orange external tank containing the liquid hydrogen fuel and liquid oxygen oxidizer.
The third component was a pair of reusable solid rocket boosters used for the first two
minutes of the Space Shuttle flight, providing as much as 83 % of the liftoff thrust. Due
to its extremely high complexity and for the purpose of this paper, it should mainly be
13
remembered that Space Shuttle experienced its greatest aerodynamics heating and speeds
during re-entry at an altitude of about 120 km, reaching speeds of about 30,000 km/h
(Mach 25), with surface temperatures of about 1650 ⁰C. Computer simulations and CFD
softwares have been largely used to predict and analyze aerothermal effects of re-entry
and other critical phases of the space shuttle mission. A sample CFD post processing
image is shown in Figure 13, which illustrates aerodynamic heating of the space shuttle
orbiter upon re-entry.
Figure 13: CFD results of Space Shuttle heating during re-entry
In conclusion, for the last century through technological advances, ingenuity,
time, and resources, the human kind has managed to go from barely being airborne for a
few seconds to making hypersonic space flights on a regular basis. While several
challenges remain unsolved and not fully understood, the next century promises to be just
as fascinating where higher speeds will be reached, and greater distances will be attained
within and outside Earth’s atmosphere.
14
1.2
Hypersonic Aerodynamics Technical Background
This following sections assume the reader is already familiar with rudimentary
aerodynamics and thermodynamics principles.
Reaching and sustaining flight at supersonic speeds means additional complexity and
constraints to the design of the vehicle because of the physical phenomena taking place at
those speeds, namely shockwaves and aerodynamic heating on the surface of the vehicle.
Proper design and choice of materials for the vehicle are critical to the safety of the pilot,
integrity of the structure, and overall success of the mission. However, when the Mach
number is further increased, the flow speed no longer falls under the supersonic regime as
additional physical flow phenomena now begin to occur and must be taken into account
in the analysis and the overall design of the vehicle.
1.2.1
What is Hypersonics?
A basic definition of hypersonic flows can be expressed as flows reaching or
exceeding speeds of Mach 5. Even though this has been somewhat accepted as a
“reasonable” rule of thumb, it remains relatively vague in today’s scientific standards. A
more precise and scientifically correct definition of Hypersonics would be the flight
regime where physical flow phenomena such as Thin Shock Layers, Entropy Layers,
Viscous Interactions, High Temperature Flows, and Low-Density effects become
progressively more important as the Mach number is increased to higher values. For
supersonic flows, the phenomena just listed are not as critical and can be analytically
disregarded. These particular physical flow phenomena that constitute hypersonic flows
are described in the following subsections:
15
1.2.1.1 Shock Layers
Let’s consider a wedge with a sharp leading edge under hypersonic flow, as
illustrated in Figure 14.
Figure 14: Thin shock layer from hypersonic flow
From oblique shock theory, we know that among the flow properties that change
from going across an oblique shockwave, the density of the flow increases across the
shock. As the Mach number increases, not only does the density across the shockwave
increases but the oblique shock wave gets very close to the surface of the body, causing
thinner space between the shock and the body. The flowfield between the shockwave and
the body surface is defined as the “shock layer”, and can be extremely thin for hypersonic
flows where the Mach number is very large. At high enough altitudes where the density is
low (and therefore the Reynold’s Number is also low and viscosity will be high), thin
shock layers can cause severe interactions between the inviscid flowfield behind the
oblique shock wave and the highly viscous boundary layer on the body surface.
16
1.2.1.2 Entropy Layers
If we now look at a similar wedge under hypersonic flow but this time with a
blunt leading edge instead of sharp, (See Figure 15) because of the flow being at a very
high Mach number, there will be an oblique shock very close to the surface of the wedge.
Oblique shock theory tells us that the shock created will be detached from the surface of
the body due to the blunt leading edge, with a shock detachment distance d as indicated.
Around the nose region, the shock is almost normal and therefore considered a strong
shock which will have a larger entropy increase than that of the oblique portion further
away from the centerline of the wedge. Due to the strong entropy gradients generated
around the nose area, an entropy layer is formed and propagates downstream from the
leading edge along the surface of the body. The boundary layer develops inside the
entropy layer and interactions of the two can sometimes result in complications when
performing boundary layer analysis.
Figure 15: Entropy layer on from a blunt leading edge
17
1.2.1.3 Viscous Interactions
As a consequence of thin shock layers, major interactions between the external
inviscid flow behind the shock and the boundary layer take place, called viscous
interactions. Due to the viscous effects of the boundary layer, when a high-speed flow
reaches its edge, some of the flow kinetic energy is converted into internal energy in the
form of heating, raising the temperature of the boundary layer, which in turn increases its
viscosity coefficient and therefore its thickness as well. A typical temperature profile in a
hypersonic boundary layer demonstrating this phenomenon is shown in Figure 16.
Figure 16: Temperature distribution profile in hypersonic boundary layer
It can be proved analytically that the thickness of a compressible boundary layer
is proportional to M ∞ 2 at hypersonic speeds ( δ ∝ M ∞ 2/√[Re x ] ) which further illustrates
the thickening of the boundary layer at high Mach numbers. A thick boundary layer
induced by a hypersonic flowfield deflects the inviscid flow within the shock layers,
which makes the body appear thicker than it really is and creates a strong curved shock
downstream of the leading edge. This principle is illustrated in Figure 17.
18
Figure 17: Effect of viscous interactions on boundary layer
As a result, the surface pressure P in the vicinity of the leading edge is
significantly greater than the freestream pressure P ∞ . The surface pressure P and
freestream pressure P ∞ only begin to get closer in value (P/ P ∞ ≈ 1.0) much further
downstream of the leading edge. Those large pressure gradients around the leading edge
have dramatic effects on temperature and the overall aerodynamic heating at the body
surface.
The majority of problems in hypersonic aerodynamic research and development
arise from the aerodynamic heating and large temperature rises at the body surface due to
those viscous interactions.
1.2.1.4 High Temperature Effects
The viscous interactions just discussed are responsible for the significant
aerodynamic heating and high temperatures within the shock layers near the surface of
the body, particularly in the vicinity any leading edge. As the upstream Mach number
increases, we know from oblique shock theory that the temperature behind the shock
wave also increases. For hypersonic speeds, the extreme temperatures experienced can
get so high that it can disturb the vibrational energy at the molecular level and cause the
molecules to dissociate, and even ionize.
19
TABLE 1: HIGH TEMPERATURE EFFECTS ON AIR
Temperature [K]
< 800
800
2000
4000
9000
Chemical Reaction
No chemical reactions of the gas
Ratio of specific heats assumed constant (γ = c p /c v = 1.4)
Air (or Gas) molecules begin to vibrate. Ratio of specific heat no longer
constant and becomes function of temperature, γ = f(T), and then both
temperature and pressure if the temperature is further increased, γ = f(T,p)
Oxygen molecules (O 2 ) dissociate
Nitrogen molecules (N 2 ) dissociate
Nitric oxide (NO) forms
Oxygen and nitrogen atoms ionize and produce free electrons that absorb
radio-frequency radiation, causing communications blackouts during
atmospheric entries. Radiative heating occurs and becomes a critical
contribution of the total aerodynamic heating
As summarized in Table 1, when the temperature increases to extremely high
values, the air or gas begin to experience “real-gas” effects and the initial assumption
made of the ratio of specific heats being constant is no longer valid. Instead, the specific
heat ratio becomes a function of temperature at first (~800K) and then pressure if the
temperature is increased further.
Figure 18: Shock layer temperature vs. freestream velocity (at 52,000m)
20
As illustrated in Figure 18, assuming a constant specific heat ratio when dealing
with hypersonic flows can yield unrealistically high temperatures within the shock layers.
As such, it is essential to include “real-gas” or high temperature effects when analyzing
hypersonic flows in order to obtain accurate results of temperatures and other flow
properties. These high temperature effects not only dominate the boundary layer but the
shock layer as well, and the vehicle is said to be travelling through a chemically reacting
flow. After molecule dissociation has occurred (2000K for oxygen and 4000K for
nitrogen) and if the temperature goes up further and reaches around 9000K, ionization
then takes place. The gas becomes a partially ionized plasma, consisting of nitrogen
atoms N and oxygen atoms O, ions N+ and O+, and electrons e-. It is also important to
note that high temperature chemically reacting flows also impact aerodynamic forces (lift
and drag) and moments experienced by the vehicle.
It is now clear that the study of hypersonics and high temperature effects go hand
in hand and it is absolutely vital to include chemically reacting effects to the analysis of
hypersonic flows to obtain accurate shock layer temperatures. Aerodynamic heating will
always take priority during the design and production phase of any hypersonic capable
vehicle, even at the expense of a less streamlined, less aerodynamically efficient design.
1.2.1.5 Low-Density Effects
When very high altitudes are reached and the air density diminishes, the distance
between air molecules increases and each molecule begins to affect the aerodynamic
properties of a body. As a result, it is no longer a valid assumption to consider air as a
continuum and equations and principles of aerodynamics must then be re-evaluated using
kinetic theory. This section of aerodynamics is referred to as low-density flows. Since
21
low density effects only become of importance at altitudes of 300,000 ft or higher (about
92 km), it does not apply to hypersonic vehicles travelling below that.
1.2.2
Governing Equations for Hypersonic Aerodynamics
The purpose of this section is to list some of the fundamental mathematical
relationships that define the behavior of hypersonic flows, and as such, any particular
topic not included within the following subsections is either assumed to remain the same
as supersonic flows or simply not relevant to the purpose of this paper.
Because the solutions provided in this report are of numerical nature, it is possible
to use the basic hypersonic shock wave equations given below to obtain analytical results
for inviscid cases as a mean of verification against the CFD results. The Navier-Stokes
equations describe the general equations of motion for a fluid flow, which are embedded
in the numerical code and solved for the inlet problem in this report. Post processing
results are provided displaying the solution of the Navier-Stokes equations for each case
run. Finally, in addition to inviscid cases, viscous effects are also considered in this
problem giving rise to shock-layer interactions. As a result, basic hypersonic boundary
layer equations are also provided.
1.2.2.1 Hypersonic Oblique Shock Equations
When a flow at supersonic (or hypersonic) speed comes across a compression
corner or wedge, we know from oblique shock theory that a shock wave is created, which
is a very thin region where large changes in flow properties such as pressure, density,
temperature, etc. take place. An oblique shock wave and corresponding geometry is
illustrated in Figure 19.
22
Figure 19: Oblique shock-wave geometry
Basic shock wave relations can be derived from the integral forms of the
continuity, momentum, and energy equations (Integral form of the Navier-Stokes
equations) when applied to a control volume parallel to the velocity and tangent to the
oblique shock wave. The integral form of the continuity, momentum, and energy
equations are given in equation (1.1) through equation (1.3), respectively.
Continuity Equation :
 
∂
ρ
υ
ρ
d
V
+
∫CS ⋅ ndS = 0
∂t C∫υ
Momentum Equation :

  

∂
ρ
V
d
υ
+
ρ
V
V ⋅ n dS = ∫ − Pn dS
∫
∫
∂t Cυ
CS
CS
Energy Equation :
(
)
(1.1)
(1.2)
 
1  
1    
∂
ρ
(
)
υ
ρ
(
)
e
V
V
d
e
V
V
V
n
dS
P
V
+
⋅
+
+
⋅
⋅
=
−
⋅ n dS
∫
∫
2
2
∂t C∫υ
CS
CS
23
(1.3)
Assuming a uniform steady, inviscid, adiabatic flow and not considering body
forces in the equations yields the following oblique shock wave relations for hypersonic
(and supersonic) flows, based on Figure 19 nomenclature:
Continuity Equation : ρ1u1 = ρ 2 u 2
(1.4)
Momentum Equation (Tengantial ) : w1 = w2
(1.5)
Momentum Equation ( Normal ) : P1 + ρ1u12 = P2 + ρ 2 u 22
u12
u 22
Energy Equation : h1 +
= h2 +
2
2
Mach Number : M n1 = M 1 sin β
(1.6)
(1.7)
(1.8)
Mach Number : M n 2 = M 2 sin( β − θ )
(1.9)
Because equations (1.4), (1.6), and (1.7) contain normal components of velocity
only, it can be concluded that the same governing equations are applicable to normal
shock waves. For a calorically perfect gas, equations (1.4), and (1.6) through (1.9) can be
manipulated algebraically to obtain the following important results for hypersonic
shocks,
M n22
(γ − 1) 2 2
(γ − 1) 2
M n1 1 +
M 1 sin β
2
2
=
=
(γ − 1)
(γ − 1)
γM n21 −
γM 12 sin 2 β −
2
2
1+
ρ2
(γ + 1) M 12 sin 2 β
=
ρ1 (γ − 1) M 12 sin 2 β + 2
P2
2γ
= 1+
( M 12 sin 2 β − 1)
P1
γ +1
(1.10)
(1.11)
(1.12)
  (γ − 1) M 12 sin 2 β + 2 
T2 P2 ρ1 
2γ
= ⋅
= 1 +
( M 12 sin 2 β − 1) ⋅ 
.
2
2
T1 P1 ρ 2  γ + 1
  (γ + 1) M 1 sin β 
24
(1.13)
Since it is common to express pressure distributions in terms of non-dimensional
terms, we can also develop an equivalent expression for the pressure coefficient behind
an oblique shock wave,
Cp =
P2 − P1
1
4
=
(sin 2 β − 2 ) .
q1
γ +1
M1
(1.14)
Finally, we know from oblique shock theory that the relationship between the Mach
Number M 1 , the shock angle β, and the deflection angle θ can be expressed by a θ-β-M
relation, which is an important equation for the analysis of oblique shock waves.
 M 2 sin 2 β − 1 
tan θ = 2 cot β  2 1

 M 1 (γ + cot 2 β ) + 2 
(1.15)
The results of equation (1.15) are plotted in Figure 20.
Figure 20: θ-β-M Diagram
25
As it can be observed on Figure 20, there is a maximum deflection angle θ max for
any upstream Mach number. When θ > θ max , no solution exists for a straight oblique
shock and the shock is no longer attached to the body. Instead there will be a detached
curved shock wave, also called “Bow Shock”, as shown in Figure 21.
Figure 21: Attached and detached oblique shocks
When θ < θ max (shock is attached), two solutions are possible for a given M 1 , a
“weak” shock solution (smaller β) and a “strong” shock solution (larger β), as shown in
Figure 22. Nevertheless, the weak shock solution usually occurs and it is safe to make
this assumption in a problem unless proof of the contrary is available.
26
Figure 22: Strong and weak shock solutions
Figure 20 also reveals that θ max increases as M 1 is increased, which means that
there can be an attached oblique shock solution for higher deflection angles at high Mach
numbers, up to the maximum possible deflection angle of 45.5 degrees. A general
statement can be made for attached oblique shock waves; given a particular fixed
upstream Mach number, if the corner deflection angle θ increases, the shock angle β will
also increase, and per Figure 20 the attached shock will become stronger until θ max is
reached causing the shock to be detached from the surface from that point on.
1.2.2.2 Hypersonic Expansion Wave Equations
As it is known from basic compressible fluid flow, an expansion wave will be
generated when a supersonic (or hypersonic) flow passes through a convex corner, where
the flow turns away from itself. A typical expansion wave is presented in Figure 23.
27
Figure 23: Expansion Wave
The following compressible flow equations can be used in conjunction with Appendix C
of Ref. [20] to calculate the flow properties across such expansion shock.
υ (M ) =
γ +1
γ −1 2
(
tan −1
M − 1) − tan −1 M 2 − 1
γ −1
γ +1
θ = ν (M 2 ) −ν (M 1 )
(1.16)
(1.17)
Note that equation (1.16) is called the Prandtl-Meyer function, along with equation
(1.17), which apply to calorically perfect gases. Moreover, because an expansion wave is
isentropic, the stagnation properties across the shock are constants, such that T 01 =T 02 and
P 01 =P 02 .
1.2.2.3 Navier-Stokes Equations
The general governing equations of fluid dynamics, called the Navier-Stokes
equations represent the mathematical product of the motion of a fluid element. The
system of equations is made up of the continuity equation (conservation of mass), the
28
momentum equations (Newton’s second law of motion), and the energy equation (first
law of thermodynamics). The Navier-Stokes equations are listed in equations (1.18)
through (1.22) and are written for a compressible, unsteady, viscous, three-dimensional
flow in Cartesian coordinates for non-reacting gases (γ = 1.4).
Continuity Equation :
( )

∂ρ 
+ ∇ ⋅ ρV = 0
∂t
(1.18)
X − Momentum Equation : ρ
Du
∂P ∂τ xx ∂τ yx ∂τ zx
=−
+
+
+
Dt
∂x
∂x
∂y
∂z
(1.19)
Y − Momentum Equation : ρ
Dv
∂P ∂τ xy ∂τ yy ∂τ zy
=−
+
+
+
Dt
∂y
∂x
∂y
∂z
(1.20)
Z − Momentum Equation : ρ
∂P ∂τ xz ∂τ yz ∂τ zz
Dw
=−
+
+
+
∂z
∂x
∂y
Dt
∂z
(1.21)
Energy Equation : ρ
⋅
D(e + V 2 / 2)
∂  ∂T  ∂  ∂T  ∂  ∂T 
 + k
= ρ q+  k

 + k
Dt
∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

 ∂ (uτ xx ) ∂ (uτ yx ) ∂ (uτ zx ) ∂ (vτ xy ) ∂ (vτ yy ) ∂ (vτ zy ) ∂ ( wτ xz )
(1.22)
− ∇ ⋅ pV +
+
+
+
+
+
+
∂x
∂x
∂y
∂z
∂x
∂y
∂z
∂ ( wτ yz ) ∂ ( wτ zz )
+
+
∂y
∂z
The nine components of stress due to viscous effects, namely the normal and
shear viscous stresses τ xx , τ yy , τ zz , τ xy (same as τ yx ), τ xz (same as τ zx ), τ yz (same as τ zy )
are illustrated in Figure 24 and are part of the momentum and energy equations.
29
Figure 24: Normal and shear stresses on a fluid element
The normal and shear stresses can be expressed in terms of viscosity coefficient μ
and are listed in equation (1.23) though (1.28).
τ xx = − µ ∇ ⋅ V + 2µ
2
3
(
 
)
∂u
∂x
(1.23)
τ yy = − µ ∇ ⋅ V + 2µ
2
3
(
 
)
∂v
∂y
(1.24)
2
3
(
 
)
∂w
∂z
(1.25)
τ zz = − µ ∇ ⋅ V + 2µ
 ∂v
∂u 
 ∂w
∂v 
τ xy = τ yx = µ  + 
 ∂x ∂y 
(1.26)
τ yz = τ zy = µ 
+ 
 ∂y ∂z 
(1.27)
 ∂u ∂w 
+

 ∂z ∂x 
(1.28)
τ zx = τ xz = µ 
30
Due to the complexity of this system of partial differential equations, there is no exact
analytic solution to the full Navier-Stokes equations, however, assumptions can be made
to obtain approximate viscous flow results and numerical solutions can also be obtained
through the use of CFD as it is the case in this paper.
1.2.2.4 Boundary-Layer Equations
The Navier-Stokes equations listed in Section 1.2.2.2 are in the most general form
for the motion of a fluid flow elements, but are not solvable as is and require a great deal
of computer resources to obtain the closest to exact possible answers. However, a simpler
system of equations can be obtained and solved for viscous flow solutions. From
boundary-layer theory, an assumption typically made is that the thickness of the
boundary-layer is considered very small compared to the length of the body, as expressed
per Equation (1.29) and illustrated in Figure 25.
δ << c
(1.29)
Figure 25: Thin boundary-layer vs. length of the body
A second assumption made from boundary-layer theory is that the Reynolds
number is considered large, as expressed in Equation (1.30).
31
1
= O(δ 2 )
Re ∞
(1.30)
After some mathematical manipulations, the boundary-layer theory assumptions based on
Equations (1.29) and (1.30) allow the simplification of the general Navier-Stokes
equations listed in Equations (1.18) through (1.22). This new set of simplified equations,
called the “boundary-layer equations” logically apply to a boundary-layer, and are listed
in Equations (1.31) though (1.34).
Continuity Equation :
∂ (ρu ) ∂ (ρv )
+
=0
∂x
∂y
X − Momentum Equation : ρu
Y − Momentum Equation :
(1.31)
dP
∂u
∂u
∂  ∂u 
+ ρv
= − e +  µ 
∂x
∂y
dx ∂y  ∂y 
(1.32)
∂P
= 0 (1.33)
∂y
2
dp
 ∂u 
∂h
∂h ∂  ∂T 
 + u e + µ   (1.34)
Energy Equation : ρu
+ ρv
=  k
∂x
∂y ∂y  ∂y 
dx
 ∂y 
1.3
Scramjet Engines
“Scramjets” or supersonic combustion ramjets are high speed air-breathing engines
where heat addition due to combustion of air and fuel takes place at supersonic speeds,
suppressing the need for a choking mechanism as it is typically the case for a
conventional ramjet engine. The scramjet propulsion unit is usually installed on the lower
surface of hypersonic-type vehicles to produce a well-integrated engine-vehicle design.
Primary applications of scramjets have been mostly military with research and
32
development in missiles, and unmanned drones or aircrafts, such as the NASA X-43 or
Boeing X-51A “Waverider”, shown in Figure 26.
Figure 26: NASA X-43a (left) and Boeing X-51 “Waverider” (right)
Scramjet engines cannot operate from takeoff, produce no thrust at zero airspeed,
and have poor to no performance at subsonic speed because the dynamic pressure isn’t
high enough to increase the cycle pressure to its most efficient operational value. For this
reason, scramjet-equipped hypersonic vehicles require several types of engine operations
to reach the required speed to be in scramjet mode and begin producing thrust, which
usually occurs somewhere between Mach 5 and 6. One common method is to combine
the scramjet to a detachable rocket engine, which is then air-launched and will accelerate
the hypersonic vehicle to a sufficiently high Mach number to allow for scramjet engine
start. Such mission profile is illustrated in Figure 27 (presented here for the X-43A
hypersonic vehicle).
33
Figure 27: Typical hypersonic vehicle mission profile
Engineers have been researching and developing other methods to bring the hypersonic
vehicle to scramjet speeds, such as incorporating a gas-turbine engine directly inside the
body, along with various geometry changes to the vehicles, all of which can greatly
increase the efficiency and operational range capability of the scramjet, but at the cost of
a much more complex end product.
A typical scramjet engine is shown in Figure 28.
Figure 28: Typical scramjet engine
The scramjet propulsion unit is composed of four main components, an air intake or
“inlet”, an isolator, a combustion chamber, and a nozzle. Each main component and their
general process of operation is described in the following subsections and illustrated in
34
Figure 28. Because this paper primarily deals with the inlet portion of the scramjet
engine, Section 1.3.1 is only meant as a simple introduction to inlets, whereas Section 2
goes in much more detail.
1.3.1
Inlet
When accelerated to operational scramjet speed around Mach 5 or so, the free-
stream airflow with very high dynamic pressure and high kinetic energy begins to enter
the scramjet inlet (Shown in Figure 28). With its specifically designed converging
geometry, the inlet compresses and decelerates the air to a lower supersonic speed
through a series of oblique shock waves before reaching the isolator.
As a result of this external-internal or “mixed” compression, no mechanical compressor
is necessary to compress the air unlike other conventional jet engines. It is important to
note that because of this lack of moving parts or rotating machinery in the engine, the
conceptual design of the scramjet is much simpler than most other air-breathing engines.
In addition, the maximum cycle temperature once limited due to the presence of
compressors and turbines now increases, which results in a better efficiency.
Viscous interactions discussed in Section 1.2.1.3 typically occur at the inlet where thick
high-speed boundary layers have developed on the forebody of the vehicle due to high
aerodynamic heating and low density effects. This compression process is critical to the
overall health of the scramjet and is discussed further in Section 2.
35
1.3.2
Isolator
Immediately following the inlet, the air enters a short duct that ends right before
the combustion chamber called the isolator (See Figure 28), an essential component of a
scramjet engine.
During combustion and the generated heat-release taking place in the combustor,
large pressure rises and boundary-layer separations on the surface of the combustion
chamber typically occur. These phenomena can create an adverse pressure gradient effect
that further compresses the flow with a risk of propagating upstream of the combustion
chamber. When a boundary-layer is present and interacts with the adverse pressure
gradient, separation occurs and a “shock-train” is generated in the isolator as a result of
the adjustment to the pressure rise in the combustion chamber.
Figure 29: Shock-train in isolator
As shown in Figure 29, a shock-train is a complex flow structure occurring in the
isolator characterized by a series of oblique and normal shock waves through which the
flow Mach number drops and the pressure increases. The separation or mixing region
near the wall essentially balances the pressure gradient across the isolator’s length
through the shear stress. This separation starts when the first oblique shock creates a
sufficiently high pressure rise to separate the boundary-layer.
36
When designing isolators, particularly when considering its length, it is crucial to
prevent the initial shock wave from propagating upstream into the inlet, which would
severely disrupts the flow and at worst cause an inlet unstart (Inlet starting condition is
discussed in Section 2.4.3).
1.3.3
Combustion Chamber
Due to the high complexity associated with the supersonic combustion process
and the fact this isn’t the primary area of interest in this paper, only the general principles
are presented here to provide the reader with a basic understanding of what goes on in the
combustion chamber.
When designing a scramjet combustion chamber, three critical criteria must be met to
obtain an efficient combustion process:
1. Fuel and air mixing
2. Ignition and flame stability
3. Proper operation for different flight conditions
Figure 30 illustrates a simplified supersonic combustion process in a scramjet engine.
37
Figure 30: Supersonic combustion in a scramjet engine
As explained in previous sections and shown in Figure 30, for a typical scramjet engine,
incoming hypersonic air endures a compression phase by going through the inlet
followed by the isolator, which both decelerate the flow to a lower supersonic speed
before it reaches the combustion chamber. A fuel injection strut is carefully placed in the
combustion chamber then injects hydrogen fuel (in gaseous state) into the supersonic
flow, which allows for fuel-air mixing. Once fuel-air mixing has been achieved (at the
molecular level), burning (or combustion) takes place in a region slightly downstream of
the fuel injector. The combination of fuel injection and combustion typically creates
disturbances associated with the pressure rise that are prone to generate complex shock
patterns, or “shock-train”, as discussed in Section 1.3.2. This is when the presence of the
isolator becomes critical, so that the combustion-induced shock waves do not propagate
back upstream to disrupt the flow at the inlet. Following combustion, the burned gases
will expand through a diverging nozzle at the back of the scramjet engine, which will
produce thrust.
38
1.3.4
Exhaust Nozzle
Once the combustion process has ended, the high potential energy of the flow is
accelerated through an expansion process and converted into kinetic energy. Because the
expansion required for hypersonic flight is substantial with a high pressure ratio, the
nozzle occupies a large portion of the vehicle and usually incorporates its lower surface
as part of the external nozzle. A typical nozzle can be seen on Figure 28 and Figure 31.
Figure 31: Typical nozzle on a hypersonic vehicle
Due to extensive length and geometry of the nozzle, the pressure distribution along the
afterbody generates significant pitch moments and lift during the expansion process.
Several factors affect the nozzle efficiency, such as the geometry itself and the properties
of the flow right after combustion occurred. In addition to significant losses in thrust due
to friction along the nozzle walls, any combustion deficiencies or incompleteness could
lead to potential freezing of any dissociated flow. Finally, the possibility of various
propulsion modes and cycles for the vehicle makes an optimum design of such nozzles
even more challenging.
39
1.4
Brayton Cycle
Scramjet engines fall into the category of Brayton cycles. The Brayton cycle is a
thermodynamic cycle consisting of two isobaric processes (constant pressure processes)
and two adiabatic processes (no heat loss). Although typically adapted for gas-turbine
engines, it is also applicable to scramjet engines. The two adiabatic processes can be
changed to isentropic processes (adiabatic and reversible) for an idealized scramjet
engine.
0
1
2
3
4
10
Figure 32: Scramjet engine reference stations
Figure 32 represents a simple schematic of a typical hypersonic vehicle and its
corresponding engine station number, each associated with a specific thermodynamic
process that can ultimately be traced back to the Brayton cycle. It becomes useful to plot
the changes experienced by the gas during each thermodynamic process, which helps
better visualize the thermodynamics of the engine. Figure 33 represents a plot of the
temperature versus the entropy of the gas, or “T-s diagram”.
40
P3
T
4
COMBUSTION
EXPANSION
P0
3
3’
10
10’
COMPRESSION
IDEAL
ACTUAL
0
s
Figure 33: Brayton cycle for scramjet engine
Station 0 corresponds to the incoming air with freestream properties. Stations 1 through 3
denote the compression process associated with the air entering the inlet (Station 1) and
then the isolator (Station 2). As the incoming hypersonic air is slowed down to
supersonic speeds through a series of oblique shockwaves (as explained in Section 1.3.1),
the high kinetic energy of the air is converted to potential energy and the static pressure
increases. For an idealized scramjet engine, this process is assumed to be isentropic
(Station 0 through 3’), which greatly simplifies the analysis of the scramjet engine and
respective calculations. In reality, the compression process is not isentropic and there is
in fact an increase in entropy of the flow as indicated by the solid line in Figure 33 due to
the expected losses from the inlet and isolator. Viscous losses, boundary-layer
interactions, and heat lost to the walls all make up some of the main inefficiencies
41
encountered in the inlet and isolator, causing the compression process to be nonisentropic. Stations 3 through 4 (Station 3’ through 4 for an idealized scramjet) represent
the entrance to the combustion chamber where heat is released through fuel combustion.
How much heat is released in the combustion chamber depends on the type of fuel used
(typically hydrogen or JP-4), the fuel-air ratio, and the overall efficiency of the fuel-air
mixing process. Whether analyzing an idealized or actual scramjet cycle, the pressure can
be assumed to remain constant during the combustion process (isobaric process) where
losses such as friction, Rayleigh losses, and heat transferred to the walls can be neglected
without introducing significant errors to the cycle analysis. Stations 4 through 10 (Station
4 through 10’ for an idealized scramjet) represent the expansion process between the
internal and external nozzle, where the hot exhaust gas with high potential is converted
back to kinetic energy through a diverging section. As illustrated in Figure 33, this
process can be idealized and considered isentropic, but in actuality, some losses are
encountered in the nozzle during the expansion process. Such irreversibilities are due to
friction, viscous dissipations, and heat transferred to the walls of the nozzle. There is also
a risk of additional energy loss if chemical equilibrium is not achieved (following
combustion) before expansion begins, which could cause some of the dissociated gas
particles to freeze. Finally, station 10 through 0 (station 10’ through 0 for an idealized
scramjet) completes the thermodynamic cycle through an isobaric process where heat is
rejected from the system, which represents the difference in thermodynamic conditions
between the nozzle exit (station 10 or 10’) and the freestream (station 0).
.
42
1.5
Relevant Research
As explained in the previous sections, the flow exiting the inlet gets fed
downstream to the remaining components of the scramjet, which denotes the importance
of the inlet for the overall engine functionality. As a result, much of today’s research on
hypersonics focuses on inlet design and ways to optimize the flow inside exiting the inlet.
When looking at past numerical as well as experimental research for 2D hypersonic
inlets, multiple validation studies and experiments on fixed or variable geometry inlets,
Busemann inlets, axisymmetric inlets, etc., have been conducted. Scientists around the
world have been trying to better understand and resolve the “unstart problem”
encountered during internal compression when the throat area is too small compared to
the capture area, causing severe flow spillage and exit of the shock system at the throat.
Saha and Chakraborty (Ref. [1]) presented a CFD validation study for a mixedcompression hypersonic inlet in effort to investigate the inlet starting characteristics. 3D
Reynolds averaged unsteady Navier Stokes equations with SST turbulence model are
solved for this particular inlet. Some of the parameters selected for the CFD software
include density based solver with 2nd order Roe-Flux difference splitting scheme for
space discretization and 2nd order implicit Euler scheme for time discretization. The mesh
is composed of hexahedral elements with clustering near the intake entry and throat area,
yielding a y+ value of less than 20 in the foremost region reaching up to 40 in the
downstream region. For different free stream Mach number ranging from 3 to 8 for both
adiabatic and isothermal conditions with zero angle of attack, reasonable agreement is
obtained with the experimental data with no separation bubble occurrence for the
isothermal cases. It was observed that wall boundary conditions for temperature,
43
adiabatic, or isothermal, had noticeable effects on determining starting the starting Mach
number.
S. Das and J.K. Prasad (Ref. [2]) conducted a numerical study for a 2D mixedcompression supersonic air intake with different cowl lip deflection angles with and
without back pressure. Also, computations were run with a bleed region. Their objective
was to investigate the behavior of the flow for a two-dimensional intake configuration
with variable cowl lip angle and capture its effect for possibility of improving
performance. They ran their numerical simulations using the commercial software
FLUENT through an explicit coupled solver with upwind discretization scheme for flow
and a 4-stage multigrid implementation for a faster convergence. A standard k-ω
turbulence model based on the Wilcox k-ω model was adopted for this simulation, which
is designed to be applied throughout the boundary layer and is applicable to wallbounded flows as well as free shear flows. Good comparison with experimental data were
obtained, and the results for free exit flow showed an improvement in performance for
increased cowl deflections. For pressurized exit flow, a cowl deflection angle of 2⁰ gave
the best increase in performance.
R. Sivakumar and V. Babu (Ref. [3]) ran a series of numerical simulations for a 3D non reacting flow in the engine inlet section of a concept hypersonic air-breathing
vehicle. Again, commercial CFD software FLUENT was used to conduct the numerical
tests, in an effort to examine the viability of numerical simulations compared to
experimental data. Turbulence models such as Spalart-Allmaras and K-Epsilon were used
to accomplish 3-D as well as 2-D runs, respectively. The computational meshes consisted
of mixed unstructured grids with quadrilateral and triangular elements and various
44
degrees of refinement ranging from 79729 cells to 119098 cells, and y+ values from 60 to
less than 30, respectively. With 119098 cells and a y+ of less than 30, the boundary layer
could be resolved and adequate solutions of the flow field were obtained. With multiples
angles of attack varying from 4 to 12 degrees, results showed good comparisons with
experimental data and predicted locations and strengths of shockwaves very well.
1.6
Objectives for Research Paper
This research paper focuses on the numerical analysis of hypersonic flows on an
axisymmetric mixed-compression scramjet inlet to examine the location and strength of
shock-layer interactions. This in turn can allow for further analysis to be completed and
address potential unstart issues, inlet performance and compression efficiency. Published
experimental and other relevant numerical data for similar studies are utilized for
comparison and validation of the results obtained. Comparisons between inviscid,
turbulent, and published results are provided to validate the numerical results obtained.
45
2
2.1
HYPERSONIC INLET CHARACTERISTICS
Basic Functions
For any air-breathing engine, an inlet’s primary purpose is to capture and compress
incoming air as required by the remaining components in the engine, while decelerating
the flow to the necessary engine entrance air speed with a minimum total pressure loss.
As introduced in Section 1.3.1, the design of the inlet for a scramjet engine is vital for
proper operation and good efficiency of any air-breathing hypersonic vehicle. Among the
various types of inlet currently existing, this paper primarily focuses on mixedcompression hypersonic inlets.
Figure 34: Hypersonic mixed-compression inlet (2-D representation)
Figure 34 illustrates a conventional mixed-compression hypersonic inlet, which
consists of a forebody exterior compression ramp and a throat area where further internal
compression can take place. Recall from Section 1.3 that the scramjet engine does not
46
have any moving or rotating machinery such as compressors, turbines, etc., therefore the
air entering the engine must be compressed entirely by the inlet.
2.2
Compression Process and High-Temperature Effects
Due to the simplistic design of the scramjet engine and because combustion must
occur at supersonic speeds, the scramjet engine must first be accelerated to the
appropriate speed by means of a separate propulsion system before it can produce any
thrust. Given this fact, air compression can be achieved by purposely designing the inlet
to produce a series of shock waves that will compress the air as it enters the inlet and
travels downstream (Recall from Section 1.2.2.1 that pressure of the air increases across
oblique shockwaves).
Figure 35: Hypersonic inlet shocks & compression process
As illustrated in Figure 35, the compression process can be sub-divided into three
consecutive stages since the air get compressed in three main areas of the inlet, the
47
forebody leading-edge, the external ramps, and the cowl leading-edge. Each one is
discussed separately in the following sub-sections.
2.2.1
Scramjet Forebody Leading-Edge
The first part of the compression process begins at the vehicle forebody leading-
edge where a strong initial vehicle “bow” shockwave is generated, causing the pressure to
greatly increase across the shock. This phenomenon is once again sketched in Figure 36.
Figure 36: Hypersonic Vehicle Forebody Leading-edge “Bow” Shock
As mentioned in Section 1.2.1.4, high-temperature effects are also significant to
the vehicle body with high upstream Mach numbers and the presence of shock waves. In
order to reduce aerodynamic heating at the surface of the body, a blunt nose is chosen for
the forebody leading-edge as opposed to a sharper, more slender geometry. From Section
1.2.2.1 and Figure 21, a wedge-type geometry such as the inlet forebody leading-edge is
associated with a strong detached shock, which occurs when the surface deflection angle
θ exceeds the critical angle θ max (given in Figure 20). The shockwave generated results in
a large pressure increase across the shock (desired for airflow compression), but more
48
importantly, because the curved shock is detached from the body, a greater portion of the
total energy (kinetic energy from the high velocity and potential energy from the high
altitude) gets dumped into the airflow in the shock layer instead of the body surface.
Consequently, less energy is transferred to the surface of the vehicle itself in the form of
heating. This contrast in nose shapes and aerodynamic heating is presented in Figure 37.
Figure 37: Aerodynamic heating for slender & blunt vehicles
It is clearly essential to choose a blunt-shaped leading edge to help reduce
aerodynamic heating and prevent material meltdowns when dealing with hypersonic
flows. The massive amount of kinetic energy it carries gets converted into internal energy
of the gas across the bow shock, causing very high temperatures in the shock-layer region
near the body. Downstream of the nose region where the bow shock layer temperature
has cooled down, a boundary-layer also exists with high temperatures at the surface of
the body due to viscous dissipations (see Section 1.2.1.3). In extreme cases when the
Mach number on the outer edge of the boundary layer is still very high, the extreme
49
viscous dissipations can cause the boundary layer to become chemically reacting, with
possible flow dissociation (See Section 1.2.1.4). These aerodynamic heating phenomena
are summarized in Figure 38.
Figure 38: Forebody leading-edge high temperature effects
2.2.2
External Compression Ramps
Referring back to Figure 35, after the airflow crosses the vehicle leading-edge
shock and begins the compression process, a second round of compression can be
obtained by modifying the geometry of the inlet forebody surface into a series converging
smaller ramps. This is shown in Figure 39.
50
Figure 39: External ramp compression system
Total pressure is an important indicator of the efficiency of a fluid flow and the
capacity of the gas to perform useful work, which means that a loss in total pressure is
always classified as an inefficiency. It is in general more efficient to compress and slow
down the airflow using a system of weaker oblique shocks such as the configuration
shown in Figure 39 as opposed to one stronger (normal) shockwave. As the upstream
Mach number increases, a stronger shock would result in a greater total pressure loss and
higher entropy increase, both undesirable in this situation. While it is therefore much
more beneficial to employ a system of multiple oblique shock waves to slow down the
flow and provide adequate compression, it should be remembered that more compression
ramps result in a longer inlet, and therefore a heavier vehicle.
Whereas Figure 39 represents a more realistic configuration, it is possible (although
very difficult in practice) to create an idealized isentropic external compression system,
where the compression ramp is designed as a single contoured surface generating a series
of infinitesimal oblique shocks. An isentropic compression surface would still yield by
definition the desired static pressure increase, but with no entropy increase and no loss in
total pressure to the flow. An example of this scenario is illustrated in Figure 40.
51
Figure 40: External ramp compression system (Isentropic)
2.2.3
Scramjet Cowl Leading-Edge
The cowl represents the last portion of the inlet and the final stage of compression
in combination with the isolator further downstream. Despite having been slowed down
by the first two compression processes, the incoming flow is still travelling at very high
supersonic speeds and a blunt-shaped leading-edge is necessary to reduce aerodynamic
heating around the cowl tip just like the scramjet forebody leading-edge discussed in
Section 2.2.1.
Figure 41: Cowl Leading Edge Shock and Internal Compression Process
52
As illustrated in Figure 41 for this particular scenario, one of the external
compression waves impinges at the tip of cowl-leading edge. The shock generated by the
cowl leading-edge impinges on the other side at the boundary layer and then gets
reflected into a weaker shock. A series of reflected shocks typically follows within the
internal duct allowing for further compression.
Further discussed in Section 2.4, impingement points between shock waves and
boundary layers are expected during any hypersonic vehicle design and require particular
attention. Such physical phenomena are a fundamental aspect of hypersonic flows, called
shock-shock and shock-layer interactions.
2.3
2.3.1
Inlet Performance
Air Capture Ratios and Design Point
The main stages involved in the compression process for a mixed-compression
hypersonic inlet have been described in Section 2.2, and all involve the generation of
shock waves through geometry selections and variations. For every inlet designed there is
always an ideal speed attainable, a “design” condition or “design point”. This design
condition corresponds to a particular upstream Mach number for which the inlet is
designed to operate at its best efficiency with minimum total pressure loss. Upstream
Mach numbers below, at, or above the design point each yield different results, with
different physical phenomena.
Figure 42 illustrates a situation where the upstream Mach number is below the
design point and the captured mass flow rate (A c ) is less than the maximum possible
mass flow rate reached at ideal conditions. The external compression shocks generated
53
will propagate away from the cowl leading-edge, causing “spillage”, meaning that a
portion of the captured flow (A c -A o ) will spill around the cowl and not go inside the inlet
duct. The spilled flow causes an additional source of drag to the vehicle. The condition
just described and shown in Figure 42 is referred to as an “undersped” inlet, where the air
capture ratio, defined as
Air Capture Ratio =
Ao
Ac
(1.35),
is less than 1.
Figure 42: Undersped Inlet
As the upstream Mach number increases, the design point is reached when the
external compression shock waves begin to impinge on the cowl leading edge, called
“shock-on-lip” condition, and no flow spillage is present. Mathematically speaking, this
means that the air capture ratio A o /A c is at unity, where the entire captured mass flow rate
is now ingested in the inlet duct, yielding optimal conditions. Naturally, modifying the
inlet geometry will also change the design Mach number since the generated shocks may
no longer focus on the cowl leading-edge. The “shock-on-lip” design condition is
represented in Figure 43.
54
Ac = Ao
Figure 43: Inlet at Design Point
Finally, as the vehicle’s Mach number increases beyond the design point, the
external compression shocks generated on the vehicle leading edge and external ramps
are ingested in the inlet duct. This situation will lead to undesirable viscous shock
interactions inside the duct, causing a severe degradation of the flow and decrease in
vehicle performance. Also, the area capture ratio does not change once the Mach number
goes beyond design conditions. This situation, known as an “oversped” inlet is illustrated
in Figure 44.
Figure 44: Oversped Inlet
55
In summary, the change in air capture ratio will result in different physical
phenomena taking place at the inlet duct, called undersped or oversped inlet, depending
whether the upstream Mach number falls below, at, or above the scramjet vehicle design
point. A graphical representation of this concept is presented in Figure 45.
Figure 45: Undersped vs Oversped Inlets
Some more complex variable geometry hypersonic inlets are equipped with a moveable
cowl leading edge, allowing for better prevention of shock system and adaptation to a
greater range of upstream Mach numbers.
2.3.2
Compression Efficiency and Contraction Ratios
Several parameters have been defined to calculate the efficiency of the compression
system of an inlet. The first critical measure is the total pressure ratio or total pressure
recovery ratio, that is, the ratio of the total pressure at the end of the isolator over the
56
freestream total pressure. Using the engine stage numbers from Figure 32 and Figure 33,
the total pressure ratio is mathematically defined in equation (1.36).
γ −1


1+
M 32 
Pt 3 P3 
2
Total Pr essure Ratio = π c =
=


Pt 0 P0  γ − 1 2 
1+
M0
2


γ
γ −1
(1.36)
Losses in total pressure ratio are classified as inefficiencies of the gas to do useful work,
which result in loss of the vehicle axial momentum and decrease in performance. Even
though the total pressure ratio has been accepted as a universal measure of performance
for subsonic and supersonic engine compression systems, its use for hypersonic flows is
only occasional because of the complex chemical effects associated with stagnation flow
properties. This causes the stagnation pressure to be a much more complicated function
of the flow conditions.
A second important parameter used to evaluate inlet efficiency is called the
adiabatic compression efficiency η c , which greatly affects the engine overall efficiency.
This adiabatic compression starts from the freestream static temperature T 0 to the
temperature at the combustion chamber entrance T 3 , and can be seen from Figure 33 (as
denoted from station 0 to station 3). Losses due to shockwaves and wall friction are
considered irreversibilities causing entropy increases between the freestream flow up to
the combustion chamber. The adiabatic compression efficiency is mathematically defined
as
57
 1
ψ − 
πc
Adiabatic Compression Efficiency = η c =
ψ −1
γ
 γ −1


≤ 1.0
(1.37),
where π c is the total pressure ratio as defined in equation (1.36), and ψ is the cycle static
temperature ratio defined as
 γ −1 2 
1+
M0 
T3 
2
ψ=
=

T0  γ − 1 2 
M3
1+


2
(1.38).
Also important for inlet performance calculations are contraction ratios, which are ratios
between the captured area (A c ), the effective captured area (A o ), the throat area at cowl
entrance (A 2 ), and the minimum inlet throat area before the combustor (A 3 or A th ).
Contraction ratios give a mathematical sense of how much and how quickly the inlet is
converging. The various areas used for calculating contraction ratios are shown in Figure
46 and their respective mathematical relationships are listed in equations (1.39) through
(1.41).
Figure 46: Inlet Contraction Ratios Areas
58
Geometric Contraction Ratio =
Ac
Ath
(1.39)
Effective Contraction Ratio =
Ao
Ath
(1.40)
Internal Contraction Ratio =
A2
Ath
(1.41)
As explained in Section 2.4, the internal contraction ratio happens to be a key element
used to determine inlet starting conditions via the Kantrowitz limit.
2.4
2.4.1
Flow Interactions and Inlet Starting
Shock-Shock Interactions
Among the many physical phenomena that must be considered when designing a
hypersonic vehicle and particularly the inlet, shockwaves impinging on other shockwaves
or “shock-shock” interactions is one of them. These shock-shock interactions commonly
occur with supersonic and hypersonic inlets and require special attention. Typical shockshock interactions encountered on a mixed-compression hypersonic inlet are shown in
Figure 47 and described thereafter.
Figure 47: Potential Shock-Shock Interactions on Inlet
59
From Figure 47, one type of possible shock-shock interactions encountered on a
hypersonic inlet would be two oblique shockwaves generated on the external
compression ramp intersecting each other. Such scenario is likely to happen when
multiple external ramps are used with increasingly steeper turning angles, as illustrated in
more details on Figure 48.
Figure 48: Interaction between oblique shocks
When the two oblique shocks intersect each other, a single refracted shock is generated
from the intersection point. Streamlines going through the two oblique shocks will
experience a different increase in entropy than that going through the single refracted
shock. As a result of this entropy difference between the two regions, a slip line must be
present downstream of the intersection point. A weak reflected shock is also generated
adjusting the flow going through the two oblique shocks to satisfy the physical conditions
that must hold across a slip line (static pressures and velocity directions are the same).
As seen on Figure 47, the second type of shock-shock interactions likely to be
encountered on hypersonic inlets is an external compression shock or forebody leadingedge shock impinging on the cowl leading edge. Recall from section 2.3 that the design
60
condition (or design point) for conventional hypersonic inlets is reached when the
vehicle’s forebody leading edge shock wave impinges on the cowl leading-edge. This
leads to an efficient containment of the flow entering the inlet duct, preventing spillage
and further complications. However, as presented in Section 2.2.3, there is also a
detached curved shock wave right upstream of the cowl leading-edge due to its blunt
shape in an effort to reduce aerodynamic heating at the cowl tip. As a result, the incoming
forebody leading-edge shockwave impinges on the cowl leading-edge shockwave as
opposed to its surface, creating a shock-shock interaction at this location. The most
common shock-shock interaction of this kind is called a “Type IV” shock-shock
interaction, as shown in Figure 49 (See Ref. [5] for the other possible types of shockshock interactions).
Figure 49: Type-IV Shock-Shock Interaction
61
A “type-IV” shock-shock interaction creates a supersonic jet that impinges on the surface
of the cowl leading-edge. This in turn causes a large increase in pressure and
aerodynamic heating on the cowl leading-edge tip, which if not seriously taken into
account can result in severe damage to the vehicle at this particular location
compromising the vehicle and the mission.
2.4.2
Shock-Layer Interactions
As briefly discussed in Section 1.2.1.3, hypersonic flows generate thicker, high
temperature boundary layers along the walls of the vehicle. When a shockwave impinges
on one of these high speed boundary layers, the interaction leads to a complex flow
phenomenon called “shock-layer” interaction. The increase in aerodynamic heating at the
point of impingement can be extremely severe, making shock-layer interactions a key
aspect of hypersonic flows. Such interactions can take place along the inlet external
compression ramp where oblique shocks are generated or at the throat where the cowlleading edge shock can impinge on the boundary-layer growing inside the duct, as
illustrated in Figure 50.
Figure 50: Regions of Potential Shock-Layer Interactions
62
When a shockwave impinges on a hypersonic boundary layer, it causes an abrupt increase
in pressure on the boundary layer flow near the surface. If the shock is strong enough,
this large pressure increase may cause a severe adverse pressure gradient on the boundary
layer, causing it to separate from the surface. Although the boundary layer typically reattaches itself to the surface further downstream, there is now a region of reversed flow
on the surface, called a “separation bubble”. This is illustrated in Figure 51.
Figure 51: Shock-Wave Boundary-Layer Interaction
During a typical shock-layer interaction as shown above, the incident shock impinges on
the boundary layer and reaches its subsonic portion. The high-pressure increase
associated with the shock gets fed upstream causing an adverse pressure gradient,
resulting in a separation of the boundary layer before the impingement point. This in turn
creates an induced separation shock right where the boundary layer begins to separate,
intersecting the incident shock (creating a slip line, not shown in Figure 51). The
separated boundary layer eventually reattaches to the body surface further downstream,
63
generating a reattachment shock. Because the boundary layer is much thinner at the
reattachment point and the pressure much higher, the inlet surface at this location
undergoes a peak in aerodynamic heating, capable of doing severe damage to the vehicle
if ignored.
2.4.3
Inlet Starting and Kantrowitz Limit
When considering the hypersonic inlet as a whole and adding all possible physical
phenomena that may occur (discussed in previous sections), the question remains,
namely, will the inlet operate (efficiently) or fail? The hypersonic inlet’s capability to
capture the required amount of air and operate efficiently depends primarily on the
vehicle’s Mach number, its geometry (contraction ratio, compression ratio, air capture
area, etc…), and how efficiently the engine can circulate the airflow through all the main
components up to the exhaust nozzle (in order to limit potential back pressure increases).
Satisfying these conditions gives rise to the “Inlet Starting Problem”, where “started” and
“unstarted” flow conditions can make the difference between a successful scramjet
engine design and a complete flight failure.
For proper scramjet operation, an inlet must be in a started mode, meaning that
the airflow entering the inlet remains supersonic with steady flow conditions throughout
the internal duct, and the physical flow phenomena taking place in the internal portion of
the inlet do not affect the airflow capture negatively.
Conversely, an unstart, or unstarted inlet, is the term used to describe a chaotic
and violent flow breakdown situation where undesirable physical flow phenomena
occurring within the inlet duct have prevented the inlet capture area from accommodating
the incoming flow, which results in excessive spillage. Typically, throat over-contraction
64
and back pressure increases from the other scramjet engine components are the main
causes of inlet unstarts. During an unstart, the shock system within the inlet duct can
become disgorged by the inlet and interfere with the thick incoming hypersonic boundary
layer, causing separation with regions of reversed flow around the throat area, preventing
once again sufficient air capture. In essence, inlet unstarts lead to a supersonic choking
phenomenon at the throat with mismatching upstream and downstream mass flow rates.
Figure 52 is a simplified pictorial representation of an unstarted inlet.
Figure 52: Unstarted Inlet Flow Characteristics
Extensive research has shown that inlet starting depends on parameters such as
internal contraction ratio (as defined in Section 2.3.2), local Mach number, and total
pressure losses. For any given upstream Mach number and specific inlet geometry, there
exists a range of contraction ratios that will allow (or prevent) inlet starting. To determine
this range of allowable contraction ratios, the Kantrowitz limit is used (Ref. [6]), which
gives a preliminary estimate of the internal contraction ratios that will permit inlet selfstart. The Kantrowitz limit applies to perfect gases by assuming the presence of a normal
shock in the isolator section that would get pushed back upstream toward the throat under
back-pressure conditions, allowing the inlet to remain self-started as long as the normal
65
shock remains in the duct, with its limit being at the throat. This relationship essentially
calculates the isentropic, one-dimensional, internal area ratio that produces sonic flow at
the throat. The Kantrowitz limit is given per equation (1.42), where A 2 and A th are
described in Figure 46.
 A2

 Athroat


 Kantrowitz
 γ +1 2 
γ
1 
1+ 
M 2 
2

γ
−
1
γ
−
1




(γ + 1)M 2
γ +1
1
2



=

 
 
2
2
γ +1
M 2  (γ − 1)M 2 + 2   2γM 2 − (γ − 1)  



2


γ +1
2 (γ −1)
(1.42)
This equation assumes that the inlet contraction begins at station 2 and ends at the throat
station, as shown in Figure 46. The Mach number M 2 should be changed accordingly if
the inlet contraction starts at a different point.
In the case where the flow is or assumed to be isentropic throughout the
compression process, Equation (1.42) can be simplified into Equation (1.43).
 A0

 Athroat
γ +1

1  2  γ − 1 2  2(γ −1)

=
 γ + 1 1 + 2 M 0  
M


0 
 Isentropic
(1.43)
Both relationships are represented graphically in Figure 53 in terms of the inverse
of the contraction ratios, illustrating the families of possible inlet starting solutions
governed by the Kantrowitz and isentropic contraction limits. In practical scramjet flight
applications however, it is uncommon to use inlet contraction ratios higher than the
Kantrowitz limit. In practical applications it is generally preferred to use a contraction
ratio falling between the Kantrowitz and the isentropic limit, where both started and
unstarted solutions are possible. For flights at high Mach numbers, the contraction ratio
66
can decrease past the Kantrowitz limit because the shock system is composed of oblique
shock waves, therefore producing smaller total pressure losses.
Figure 53: Possible Inlet Starting Solutions
67
3
3.1
3.1.1
NUMERICAL METHODOLOGY & SETUP
Main Aspects of CFD
Introduction
The hypersonic inlet presented in this paper is analyzed numerically through the
use of Computational Fluid Dynamics methods. As it is presented in Section 1.2.2.3, the
governing equations of fluid dynamics called the Navier-Stokes equations (Equations
1.18 through 1.22) are used to analyze the motion of a fluid element. In their complete
form, the Navier-Stokes equations form a system of non-linear partial differential
equations that are not solvable analytically. It is however possible to obtain approximate
solutions through computer methods, giving rise to Computational Fluid Dynamics, or
CFD. The idea behind CFD is to take the continuous domain where the flow variables (P,
T, v, ρ, e) are defined at every point of the given problem and replace it with a discretized
domain using a grid, where the flow variables are defined only at the grid points. This
difference is illustrated in Figure 54.
Figure 54: Continuous vs. Discrete Domain
68
In a discrete domain, the values of the flow variables outside of the grid points can be
calculated by interpolating the values at the grid points. By using a discrete domain, the
non-solvable system of coupled partial differential equations obtained in a continuous
domain turns into a system of algebraic equations. Such systems are solved using matrix
algebra involving strenuous and repetitive calculations, which is why the use of a
computer code can facilitate this process tremendously. The idea of the discretized
domain can then be translated into much larger scales and applications, where not only
grid nodes can be used as discrete points but also 2D and 3D grid elements, called “cells”
are considered in the discrete domain.
3.1.2
CFD Software and Finite-Volume Method
For our mixed-compression hypersonic inlet, the commercial CFD software
FLUENT is used (Ref. [10], and also presented in Section 3.1.3) for three of its main
turbulence models as well as its inviscid model. The FLUENT code solves the
conservation equations using a finite volume method, where the fluid domain is
discretized into large number of control volume elements through grid generation, called
cells. A typical 2D model would contain quadrilateral and triangular elements, where a
3D model will contain elements such as hexahedral, tetrahedral, or prisms. In a finitevolume approach, the integral forms of the Navier-Stokes equations are applied to the
control volume of each cell resulting in an average value of the fluid properties at a crosssection. A physical representation of a typical infinitesimal fluid element (dV) contained
in a control volume moving along a streamline is presented in Figure 55, with velocity
vector
, fluid density ρ, unit vector n̂ normal to an infinitesimal element (dS) of the
control surface.
69
Figure 55: Control Volume Element
With reference to Figure 55, the integral forms of the governing equations of fluid motion
can be obtained using the general “transport” equation derived for any extensive property
Ф (with ϕ being the extensive property per unit mass), listed here as Equation (1.44).
(1.44)
Physically speaking, the transport equation listed above represents the transfer of mass,
momentum, and energy by molecules travelling from one location to another within the
flow through diffusion and convection. Mathematically, the transport equation describes
the relationship existing between the rate of change of the variable of interest (extensive
property) and the time rate of change of the extensive property inside the control volume
with the addition of the net flux of the extensive property across the control surface. As a
result, the conservation laws can be applied to Equation (1.44), resulting in the integral
forms of the Navier-Stokes equations, listed in equations (1.45) through (1.47).
70
Conservation of Mass (Scalar Equation):
(1.45)
Conservation of Linear Momentum (Vector Equation):
(1.46)
Conservation of Energy (Scalar Equation, E = total energy):
(1.47)
3.1.3
FLUENT Code
The FLUENT CFD software (Ref. [10]) used to solve the hypersonic inlet in this
paper uses the Navier-Stokes equations in integral forms, but because of their similar
form based on the transport equation described in Section 3.1.2, the entire system of
integral forms of the equations of conservation of mass, momentum, and energy can be
conveniently rearranged and suited for CFD use in FLUENT, summarized by Equation
(1.48).
(1.48)
Equation (1.48) represents the entire system of the governing equations described
previously in terms of vector fluxes (matrix form) where W is the solution vector, F is the
convective flux, G is the viscous flux, and H is the source term, given by
71
72
3.1.4
CFD Grids
A collection of multiple discrete points is called a grid or a mesh. CFD solvers
require a discrete representation of the geometry of the problem, and the creation of a
grid (or mesh) represents all the elements on the model where the flow is solved. Grid
generation is essential in CFD, so essential that no CFD analysis should be fully trusted
until the grid has been optimized. Also, the numerical error decreases as the number of
grid points (and therefore elements) is increased around the object. Two types of grids are
possible, structured and unstructured. A structured grid typically applies to a simple
geometry and is represented by regular connection of quadrilateral elements in a 2D
domain or hexahedral elements for a 3D model. The computer essentially transforms the
curvilinear mesh in the physical domain into a uniform Cartesian mesh, which allows for
a faster CFD processing because all the grid points can be easily identified with reference
to the appropriate grid lines. This transition from a physical domain to a computational
domain for a structured grid is graphically represented in Figure 56.
Figure 56: 2D Mapping from Physical to Computational Space
73
As such, a typical two-dimensional structured grid will generally be better and more
efficient to use for non-complex geometry. An example of a two-dimensional structured
grid generated for an airfoil can be seen in Figure 57.
Figure 57: 2D Structured Grid
The second type of grid used is called an unstructured grid and is characterized by
irregular connectivity between grid points. Such grids apply to more complex geometries
that may contain for examples sharp angles and other similar topologies where a
structured grid would simply not work. Unstructured grids comprise of triangular
elements for two-dimensional geometries and tetrahedral elements in three dimensions.
This allow for a far greater level of flexibility for mesh adaptation when dealing with
complicated models since there is no structure of coordinate lines given by the grid and
mesh can be refined wherever it is needed. The meshing and mapping process is faster
when generating an unstructured mesh and its use has been widely accepted in CFD.
However, when compared to structured grids, unstructured grids have larger storage
74
requirements since the neighborhood connectivity must be explicitly stored and are more
difficult to program if coding your own grid generator. An example of an unstructured
mesh of a flying vehicle is presented in Figure 58.
Figure 58: Typical Unstructured Triangular Mesh
For both structured and unstructured grids, the more discrete points around the
object, the denser the grid, and the more accurate the final solution will be at the cost of
more computer resource and time. The various types of grids that can be used along with
the shapes of the elements will vary depending on the object, the physics of problem, and
the resources available. Figure 58 also illustrates the principle of grid clustering, where
additional finer grid elements are generated closer to the area of interest in order to
capture a more accurate fluid flow solution, whereas coarser elements are being used
further away from the area of interest, preventing unnecessary calculations and use of
computer resources. For the hypersonic inlet used in this paper, the grid generator
GAMBIT is used to produce the mesh. Section 3.2 will go into the details of the mesh
used to analyze the inlet.
75
3.1.5
Turbulence Models
Turbulent flows are characterized by large chaotic and abrupt fluctuations of
pressure and velocity in space and time, in contrast to laminar flows where velocity fields
vary smoothly and the kinetic energy dies out due to fluid viscosity. Each type of flow
has its pros and cons depending on the situation but it is ultimately a phenomenon that
can we cannot decide on and control. However, Mother Nature works in such a way that
a system left to act on its own will generally move towards a state of disorder, which
means work has to be exerted on the system to reinstate or maintain order in the system.
In the case of fluid flow, a laminar flow at the leading edge of a vehicle will generally
transition to turbulent flow downstream due to the influence of friction and other present
instabilities in the laminar flow. Simulating numerically such phenomena and solving
directly for the Navier-Stokes equations for any practical problem is computationally
extremely demanding. To circumvent this problem, the Reynolds-averaged Navier-Stokes
(RANS) equations were developed, which provide time-averaged solutions to the general
Navier-Stokes equations primarily used to describe turbulent flow. This modification to
the general equations of motion for fluid flow yields new variables to be solved for,
which are implemented in the various CFD turbulence models.
FLUENT provides several turbulence models that can be used to simulate
turbulent flow around the hypersonic inlet. The Spalart-Allmaras, k-ω SST, and k-ε
models are the three turbulence models used in this paper to analyze the inlet and provide
numerical solutions. Each turbulence model is briefly described in the following subsections.
76
3.1.5.1 Spalart-Allmaras Model
The Spalart-Allmaras turbulence model (Ref. [12]) used in the CFD software FLUENT to
analyze the hypersonic inlet is a one equation model that solves the transport equation for
the variable
, which is then used to find the turbulent kinematic viscosity, also known
as the eddy viscosity. This is accomplished using the Boussinesq approach (Ref. [13]),
which relates the Reynolds stresses in the RANS equations to the average velocity
gradient of the flow. Incorporating the eddy-viscosity variable to the general transport
equation given in equation (1.44) yields the following Spalart-Allmaras transport
equation
(1.49)
where
is used to calculate the eddy viscosity
given by
(1.50)
with
(1.51)
(1.52)
Description of the remaining constants and functions used in equations (1.49) can be
found in Ref. [14], page 48. From equation (1.49),
corresponds to the diffusion term,
is the production term (of turbulent
77
viscosity),
is the destruction term (of turbulent viscosity) that occurs
in the near-wall region due to wall blocking and viscous damping, and
is a trip
term that can generally be ignored when dealing with fully turbulent problems.
3.1.5.2 k-ω SST Model
Two-equation turbulence models were developed to better represent the complex
physical phenomena involved with turbulent flow fields, such as separation and
unsteadiness. The shear stress transport (SST) k-ω turbulence model (Ref. [15] and Ref.
[16]) is a two-equation eddy-viscosity model that includes one equation for the turbulent
kinetic energy k, and a second equation for the specific turbulent dissipation rate ω. The
model combines the accurate formulation of the standard k-ω model in the near-wall
region but also switches to a k-ε behavior in the free stream region, making it a versatile
turbulence model applicable to various types of flows such as adverse pressure gradient
flows and transonic shock waves. Per Ref. [10], the SST k-ω model contains the
following adjustments from the standard version:
•
The standard k-ω model and the transformed k-ε model are both multiplied by a
blending function and both models are added together. The blending function is
designed to be one in the near-wall region, which activates the standard k-ω
model, and zero away from the surface, which activates the transformed kε model.
•
The SST model incorporates a damped cross-diffusion derivative term in
the ω equation.
78
•
The definition of the turbulent viscosity is modified to account for the transport of
the turbulent shear stress.
•
The modeling constants are different.
As such, the SST k-ω two-equation model is given by
(1.53)
and
(1.54)
where P k (production of turbulence) and the function
are defined as
(1.55)
(1.56)
The constant F 1 from equation (1.54) is equal to 1 when associated with the k-ω model
and 0 when associated with the k-ε model.
The turbulent viscosity for the SST k-ω model is given by
(1.57)
where
79
(1.58)
(1.59)
3.1.5.3 k-ε Model
Just like the SST k-ω model, the k-ε turbulence model (Ref. [17]) is a twoequation model that gives a representation of turbulence properties by means of
two transport equations, which account for convection and diffusion of turbulent energy.
The two partial differential equations are derived for the turbulent kinetic energy k which
determines the energy in the turbulence, and the turbulent dissipation ε which determines
the scale in the turbulence. Both are defined as
(1.60)
and
(1.61)
An important assumption of the k-ε turbulence model is the turbulent viscosity is
isotropic, where the ratio between the Reynolds stress and the mean rate of deformation
is the same in every direction. The standard k-ε two-equation turbulence model is
expressed by the turbulent kinetic equation as well as the dissipation rate equation as
follow:
(1.62)
80
(1.63)
The K-Epsilon model has been shown to be useful for free-shear layer flows with
relatively small pressure gradients. Similarly, for wall-bounded and internal flows, the
model gives good results only in cases where mean pressure gradients are small. A drop
in accuracy has been noticed experimentally for flows containing large adverse pressure
gradients.
3.2
Inlet Grid Generation
As it was introduced in Section 3.1.4, grid generation is a fundamental aspect of
CFD analysis and the mesh generated for the computational analysis of our hypersonic
inlet is no exception.
Figure 59: 3-D Representation of Hypersonic Inlet
Modeled using the commercial software CATIA V5, Figure 59 represents a threedimensional representation of the axisymmetric hypersonic inlet analyzed in this paper. A
81
cut is then taken along the axis of symmetry, giving a two-dimensional representation of
the inlet as shown in Figure 60 with corresponding dimensions.
Figure 60: 2-D Representation of Hypersonic Inlet (Dimensions in mm)
The commercial grid generation software GAMBIT is used to generate the mesh required
to analyze the inlet. Since the two-dimensional inlet contains fairly irregular geometry
including sharp corners, an unstructured grid is used to represent the majority of the
computational domain.
3.2.1
Viscous Mesh Characteristics
When viscous effects are considered, a structured grid is added close to the walls of
the inlet with much finer and clustered elements in an effort to represent properly the
effects of the boundary layer and capture accurately any physical phenomena taking place
near the walls. As a result, the final mesh used for this inlet is a hybrid mesh, combining
both structured and unstructured grid in the same computational domain. The viscous
mesh generated is presented in Figure 61.
82
Figure 61: Inlet Viscous Mesh
83
The mesh shown in Figure 61 is the result of an effort to find an optimal mesh for
analyzing the hypersonic inlet. It is composed of 378554 elements, most of which are
triangular elements occupying the majority of the computational domain. In order to
capture the proper effects of the boundary layer along the surface of the inlet, a
refinement of the mesh near the wall is needed. As a result, a separate mesh is created
connecting the unstructured mesh just discussed to the walls of the inlet. When running
fully turbulent models, the generation of a boundary layer mesh becomes important in
CFD because of the ability of the CFD model to properly capture the turbulent boundary
layer. In Boundary layer theory, the law of the wall explains that a turbulent boundary
layer can be divided into inner and outer regions. Very close to the wall, the flow is
nearly laminar and the molecular viscosity is a main driver of momentum transfer, this
region is defined as the viscous sublayer. Still within the boundary layer but further away
from the wall where turbulent viscosity drives the velocity profile, there is an outer or
fully turbulent layer. Finally, in between the viscous sublayer and fully turbulent layer,
there is a blending region where molecular and turbulent viscosities affect the flow
equally, called the buffer layer. This division of the turbulent boundary layer is typically
represented using the non-dimensional parameter y+, which represents the spatial
coordinate direction normal to the wall. Y+ is given by
y+ = y
u*
ν
(1.64)
where y is the normal distance from the wall to the center point of the nearest element,
u * is the friction velocity (a function of the wall shear stress), and ν is the kinematic
viscosity. The various regions within the turbulent boundary layers can be identified
based on the value of y+ and the following approximate categories:
84
Viscous Sublayer: y+ < 8
(1.65)
Buffer Zone: 8 < y+ < 50
(1.66)
Fully Turbulent Zone : y+ > 50
(1.67)
For the hypersonic inlet analyzed in this paper, the boundary layer mesh is
generated using a structured grid (quad elements) then connected to the unstructured
mesh, therefore creating a hybrid mesh as shown in Figure 62. Adding a separate
structured grid to represent the boundary layer allows the user to specify the exact
spacing of the mesh elements adjacent to the walls.
Figure 62: Inlet Hybrid Mesh
The parameters used to generate the boundary layer mesh are defined as follows:
85
Figure 63: Boundary Layer Mesh Specifications
The initial row height, “a”, is the distance between the edge to which the boundary layer
is attached and the first full row of mesh nodes. The growth factor is defined by the ratio
of “b” over “a” where b is the distance between the first and second full rows. The total
number of rows is identified by “n”, and finally the total depth of the boundary layer is
calculated and displayed a “D”. For the boundary layer mesh used and shown in Figure
62, a = 0.001 mm, b/a = 1.4, n = 14, and D = 0.275 mm. In addition, due to the sharp
corner at the cowl leading edge, an additional option in GAMBIT was selected, called
“Wedge corner shape”, which allows to control the shape of the boundary layer mesh in
the region surrounding a corner that connects two edges to which boundary layers are
attached. Selecting this option yielded the following result, shown in Figure 64.
86
Figure 64: Boundary Layer Mesh at Cowl Leading Edge
Finally, a sizing function was added to the mesh in order to cluster the grid points
much closer to the wall and throat region, which are the main regions of interest where
viscous interactions and shock interactions are likely to happen. The mesh generation
software GAMBIT has grid clustering capabilities through sizing functions, which allows
the user to control the size of mesh intervals for edges or faces, similar to the boundary
layer function used earlier. In the mesh created for this inlet, my initial element size starts
at 0.02 mm and coarsens to a maximum element size of 15 mm, at a growth rate of 1.07
(7%). This seems to provide an adequate balance between the clustering in the areas of
interest and the total amount of elements generated. The final clustering generated can be
clearly observed on Figure 61, and is presented again for convenience on Figure 65.
87
Figure 65: Grid Clustering
As mentioned before, the quality of the mesh plays an essential role in the stability and
accuracy of the numerical computation. Two major quality criteria are used to evaluate
the mesh, the aspect ratio and skewness of the grid elements. The aspect ratio is a
measure of the elongation of the element; it is the ratio of the longest to the shortest side
in a cell. In an ideal mesh, the aspect ratio of an element will be equal to 1.0. Having high
aspect ratio elements could result in errors in the numerical computation and yield
incorrect results. The difference between an ideal and poor aspect ratio is illustrated in
Figure 66.
88
Figure 66: Quad Element Aspect Ratio
Just like aspect ratio, the skewness of a grid element is a primary factor of the mesh
quality and determines how close to ideal (equilateral triangle) a tri element is. A value of
0 indicates an ideal equilateral tri element with no skewness, whereas a value of 1
represents a degenerate cell with the worst possible skewness. As a rule of thumb, the
skewness of a tri element should not exceed 0.85. Figure 67 is a representation of the
skewness of a mesh triangular element.
Figure 67: Tri Element Skewness
When analyzing the mesh generated for the hypersonic inlet for quality through
GAMBIT and also through the FLUENT grid checker, no warnings were reported. The
highest aspect ratio quad element reported is 313 and the worst skewness is 0.50. While
89
the aspect ratio of 313 may seem high, the number obtained can easily be justified. The
high aspect ratio elements happen to be within the boundary layer mesh adjacent to the
wall near the inlet forebody leading edge (as opposed to far-field), which is not high
gradient location since most of the shock-layer interactions will take place around the
inlet throat. Moreover, high aspect ratio elements are frequent when seeking very low y+
values and can be ignored if in non-critical locations as explained in Ref. [18].
3.2.2
Inviscid Mesh Characteristics
For comparison purposes, inviscid numerical results are included for each flight
condition listed in Table 2. Since viscous effects are no longer considered for these runs,
the mesh does not need to account for the presence of a boundary layer and can therefore
be slightly less dense in proximity of the walls. The inviscid mesh generated is shown in
Figure 68. The inviscid mesh generated contains 354628 triangular elements with
clustering near the walls to properly capture the presence of shockwaves or any flow
separation. Similar to the sizing function generated for the viscous mesh, the initial
element size starts at 0.02 mm and coarsens to a maximum element size of 15 mm, at a
growth rate of 1.07 (7%).
90
Figure 68: Inlet Inviscid Mesh
91
As mentioned earlier, because there is no boundary layer present, there is no need
to create a boundary layer mesh with much finer elements at the walls. As such, only the
unstructured mesh with triangular elements is contained within the computational domain
for the inviscid case as opposed to the hybrid mesh for the viscous cases. For comparative
purposes with the viscous mesh, a zoomed-in picture of the walls for the inviscid grid is
presented in Figure 69.
Figure 69: Inviscid Mesh at the Walls
A quality check of the inviscid grid reveals that the highest skewness detected is 0.58 and
aspect ratio 1.66, which are both relatively healthy values for triangular elements.
3.3
FLUENT Initial and Boundary Conditions
As explained in Section 3.1, using a CFD solver allows the user to calculate an
approximate solution to the Navier-Stokes equations, which are once again a set of nonlinear partial differential equations. In order to obtain a set of unique solutions to those
92
PDEs, additional conditions must be provided to determine the arbitrary functions that
result from the integration of those PDEs. Those conditions are called initial, and
boundary conditions.
Initial conditions are mathematical representations of the initial state of the
problem, where the dependent variable is given at some initial state. For the hypersonic
inlet presented in this paper, the initial conditions are represented by the flight conditions
in Table 2.
TABLE 2: SCRAMJET FLIGHT CONDITIONS
HYPERSONIC INLET FLIGHT CONDITIONS
Flight
Condition
No.
Mach
No.
Flight
Altitude
(m)
Static
Pressure
(Pa)
Static
Temperature
(K)
Total
Pressure
(Pa)
Total
Temperature
(K)
1
5.0
10000.0
26436.0
223.15
13987176.0
1339.0
2
7.0
10000.0
26436.0
223.15
109441948.0
2410.0
3
9.0
10000.0
26436.0
223.15
557892131.0
3838.0
Any solution of a set of PDEs requires boundary conditions, which impose a
requirement (to the flow and thermal variables) on the boundary of the physical domain
that must be satisfied by the dependent variable or its derivative. For the hypersonic inlet
presented in this paper, boundary conditions represent the flow and thermal variables on
the boundaries of the inlet computational domain. The boundary conditions selected on
FLUENT for the inlet are described below per Ref. [10] and represented on Figure 70.
93
Figure 70: Inlet Boundary Conditions
Pressure inlet boundary conditions are suitable for both incompressible and
compressible flows and are used to represent fluid pressure at the flow inlet and other
scalar properties such as total pressure and temperature (which defines the initial Mach
number), static pressure, flow direction, and turbulence parameters. For the inlet
presented in this paper, the pressure inlet parameters used reflect the flight conditions
from Table 2 for the inviscid and turbulent cases. It is important to note that initial
condition for inlet (initial static pressure, velocity, turbulent viscosity) is set to be that at
the pressure inlet boundary condition, which means FLUENT is taking the incoming
airflow properties as the initial guess for the solution flow field.
Wall boundary conditions are used to bound fluid and solid regions. By default, a
no-slip boundary condition at the walls is enforced for viscous flows, which can be
circumvented by applying a tangential velocity component in terms of the translational or
rational motion of the wall boundary. For heat transfer calculations, thermal boundary
conditions can be specified at the wall (No heat transfer included in our wall boundary
94
conditions). Other inputs such as wall motion, shear conditions, wall roughness, etc, can
be included if desired but are not applicable for the inlet presented.
Symmetric boundary conditions are used when the physical geometry of interest
and the expected pattern of the flow/thermal solution have mirror symmetry. No inputs
need to be specified here.
Pressure far-field boundary conditions are used to represent free-stream
conditions of the flow, where static conditions, Mach number, and flow direction are
specified. To effectively simulate free-stream flow properties, it is important to place the
far-field boundary condition far enough from the zone of interest.
Finally, the pressure outlet boundary condition is used to represent the behavior of
the flow at the exit of the inlet. While a static pressure is needed for subsonic flow,
supersonic flows will extrapolate the static pressure from the flow within the inlet to that
location. In addition, there’s an option to specify backflow conditions if reverse flow is
expected from the isolator or combustion chamber (no reversed flow assumed for our
inlet).
3.4
FLUENT Setup & Solver Properties
The following FLUENT parameters were selected for running the inlet and can be
justified using Ref. [10]:
95
Flow Solver:
Figure 71: Flow solver parameters
96
Viscous Model – Sparlart-Allmaras
Figure 72: Spalart-Allmaras turbulence model parameters
Note that the default values for the model constants were unchanged.
97
Viscous Model – K-Epsilon
Figure 73: K-Epsilon turbulence model parameters
Note that the default values for the model constants were unchanged.
98
Viscous Model – K-Omega SST
Figure 74: K-Omega SST turbulence model parameters
Note that the default values for the model constants were unchanged.
99
Solution Controls for Turbulence Models
Figure 75: Solution control parameters
All simulations were executed using first-order accurate solutions, using a CFL number
of 1.0. The implicit solver was run in conjunction with an advection upstream splitting
method (AUSM) differencing scheme. Under-relaxation factors were unchanged.
100
Material Properties
Figure 76: Material Properties
Ideal-gas option is selected for the density calculation of compressible flows, which
calculates the density based on the ideal-gas law. Because chemical effects are not
considered for our inlet, the specific heat ratio γ is assumed constant at 1.4, therefore Cp
is also assumed constant. The kinetic theory option is selected for the gas thermal
conductivity, which in conjunction with a constant specific heat Cp and a constant
molecular weight results in a constant Prandtl number for the gas. Finally, to calculate the
effects of temperature on viscosity, the Sutherland’s Law (Ref. [19]) is used.
101
4
SIMULATION RESULTS
The following sections present the FLUENT numerical results obtained for the
flight conditions listed in Table 2. To provide the reader with the realization that
numerical simulations of this type are computationally demanding and require time to
converge, a summary of the computational runs with the number of iterations and total
time to convergence is presented in Table 3.
TABLE 3: COMPUTATIONAL TIME AND ITERATIONS
FLUENT ITERATIONS & COMPUTATION TIME TO CONVERGENCE
Flight
Mach
Condition
No.
No.
Inviscid
Spalart-Allmaras K-Epsilon Standard
K-Omega SST
1
5
11880 iterations
19h 29 min
18603 iterations
40h 34min
25918 iterations
32h 34min
28688 iterations
49h 23min
2
7
13872 iterations
17h 36 min
15626 iterations
43h 34min
15889 iterations
46h 38min
16119 iterations
56h 07min
3
9
13111 iterations
18h 14 min
15898 iterations
46h 44min
14970 iterations
47h 30 min
15118 iterations
56h 42min
Inviscid analysis results are presented first in this chapter as they provide an accurate
representation of fluidflow behavior without considering viscous effects. In addition, the
hypersonic shockwave equations listed in Section 1.2.2.1 can be used as a mean of
validation against the results obtained from FLUENT.
102
4.1
Inviscid Results
Mach number and static pressure contour profiles superimposed on the physical
domain for each inviscid flight condition in Table 2 are presented here in sections 4.1.1
through 4.1.3, demonstrating the expected variations within the flow field due to the
shock systems. As the flow enters the throat region, interesting shock systems and
interactions develop as expected from theory (Section 2.2.3 and 2.4.1). To better
visualize the shocks developed within the cowl duct, magnified screenshots of the throat
region are also provided.
103
4.1.1
Inviscid Results –Flight Condition 1
5.00
4.85
4.70
4.55
4.40
4.25
4.10
3.95
3.80
3.65
3.50
3.35
3.20
3.05
2.90
2.75
2.60
2.45
2.30
2.15
2.00
MACH NUMBER
INVISCID - MACH 5
Figure 77: Mach number contours for inlet – Flight Condition 1 (Inviscid)
MACH NUMBER
INVISCID - MACH 5
5.00
4.85
4.70
4.55
4.40
4.25
4.10
3.95
3.80
3.65
3.50
3.35
3.20
3.05
2.90
2.75
2.60
2.45
2.30
2.15
2.00
Figure 78: Mach number contours for throat region – Flight Condition 1 (Inviscid)
104
STATIC PRESSURE (Pa)
INVISCID - MACH 5
Figure 79: Static pressure contours for inlet – Flight Condition 1 (Inviscid)
STATIC PRESSURE (Pa)
INVISCID - MACH 5
766000
729000
692000
655000
618000
581000
544000
507000
470000
433000
396000
359000
322000
285000
248000
211000
174000
137000
100000
63000
26000
Figure 80: Static pressure contours for throat region – Flight Condition 1 (Inviscid)
105
766000
729000
692000
655000
618000
581000
544000
507000
470000
433000
396000
359000
322000
285000
248000
211000
174000
137000
100000
63000
26000
4.1.2
Inviscid Results –Flight Condition 2
7.00
6.77
6.54
6.31
6.08
5.85
5.63
5.40
5.17
4.94
4.71
4.48
4.25
4.02
3.79
3.56
3.33
3.10
2.87
2.64
2.42
MACH NUMBER
INVISCID - MACH 7
Figure 81: Mach number contours for inlet – Flight Condition 2 (Inviscid)
MACH NUMBER
INVISCID - MACH 7
7.00
6.77
6.54
6.31
6.08
5.85
5.63
5.40
5.17
4.94
4.71
4.48
4.25
4.02
3.79
3.56
3.33
3.10
2.87
2.64
2.42
Figure 82: Mach number contours for throat region – Flight Condition 2 (Inviscid)
106
STATIC PRESSURE (Pa)
INVISCID - MACH 7
Figure 83: Static pressure contours for inlet – Flight Condition 2 (Inviscid)
STATIC PRESSURE (Pa)
INVISCID - MACH 7
2160000
2060000
1950000
1840000
1730000
1630000
1520000
1410000
1310000
1200000
1090000
987000
880000
774000
667000
560000
453000
346000
239000
133000
25800
Figure 84: Static pressure contours for throat region – Flight Condition 2 (Inviscid)
107
2160000
2060000
1950000
1840000
1730000
1630000
1520000
1410000
1310000
1200000
1090000
987000
880000
774000
667000
560000
453000
346000
239000
133000
25800
4.1.3
Inviscid Results –Flight Condition 3
MACH NUMBER
INVISCID - MACH 9
Figure 85: Mach number contours for inlet – Flight Condition 3 (Inviscid)
MACH NUMBER
INVISCID - MACH 9
9.00
8.67
8.34
8.00
7.67
7.34
7.01
6.68
6.34
6.01
5.68
5.35
5.02
4.69
4.35
4.02
3.69
3.36
3.03
2.70
2.36
Figure 86: Mach number contours for throat region – Flight Condition 3 (Inviscid)
108
9.00
8.67
8.34
8.00
7.67
7.34
7.01
6.68
6.34
6.01
5.68
5.35
5.02
4.69
4.35
4.02
3.69
3.36
3.03
2.70
2.36
STATIC PRESSURE (Pa)
INVISCID - MACH 9
Figure 87: Static pressure contours for inlet – Flight Condition 3 (Inviscid)
STATIC PRESSURE (Pa)
INVISCID - MACH 9
4660000
4430000
4200000
3970000
3740000
3500000
3270000
3040000
2810000
2580000
2340000
2110000
1880000
1650000
1420000
1190000
954000
722000
490000
258000
26200
Figure 88: Static pressure contours for throat region – Flight Condition 3 (Inviscid)
109
4660000
4430000
4200000
3970000
3740000
3500000
3270000
3040000
2810000
2580000
2340000
2110000
1880000
1650000
1420000
1190000
954000
722000
490000
258000
26200
4.1.4
Inviscid Results – Discussion of Numerical Results
For all three flight conditions presented, initial observations of the Mach contours
and static pressure contours reveal that all initial conditions for flow properties specified
in FLUENT and listed in Table 2 match the results obtained. As theory predicts, the
hypersonic fluidflow travels downstream crossing multiple external compression waves,
causing increases in static pressure reductions in Mach number.
A description of the shock system on the external portion of the inlet is provided in
Figure 89 for flight condition 1, which clearly indicates the sudden Mach number
fluctuations due to the expected presence of shockwaves. Note that the flight condition 1
Mach number contours are used as an example here but similar shock systems exist for
flight conditions 2 and 3, which can be identified through similar observations.
Cowl / throat
shock system
Ramp compression
shocks
Leading-edge
shock
Figure 89: Mach number contours for the Mach 5 flight condition (External Compression)
The shocks along the external ramps are very much visible and agree with our theoretical
expectations and the boundary conditions selected to reflect the physics of the problem
are adequate. When looking at flight conditions 1 & 2 (Figure 77 through Figure 84) with
110
initial Mach numbers of 5 and 7 respectively, we can see that the leading edge shock does
not impinge on the cowl leading edge tip causing spillage around the cowl and additional
drag for the scramjet vehicle as described in Section 2.3.1 (referred to as an “undersped”
inlet). As predicted by theory and explained in Section 1.2.2.1, it can be seen that the
oblique shocks produced at the leading edges and compression corners become weaker
and have smaller shock angles at higher Mach numbers. Consequently, those weak
shocks can be seen to be much closer to the walls as the Mach number increases. This is
also confirmed by looking at the static pressure contours for the same flight conditions
(Figure 79, Figure 83, and Figure 87) which shows that the pressure gradient across the
oblique shocks diminishes with a higher Mach number. As the flow enters the cowl duct,
both compression and expansion shockwaves are generated due to the cowl leading edge
and overall geometry of the throat region. As it was done for the external portion of the
inlet, a description of the internal compression shock system at the cowl is presented in
Figure 90 for flight condition 1.
Compression shocks
Leading-edge
shock
Expansion
shocks
Shock-expansion
interactionMACH Reflected
NUMBER
INVISCID - shocks
MACH 5
5.00
4.85
4.70
4.55
4.40
4.25
4.10
3.95
3.80
3.65
3.50
3.35
3.20
3.05
2.90
2.75
2.60
2.45
2.30
2.15
2.00
Figure 90: Mach number contours for the Mach 5 flight condition (Internal compression)
111
From the theory discussed in Section 2.2.3 and the results obtained from FLUENT, both
appear to agree quite well when looking at the shock system and fluidflow behavior. The
cowl static pressure contours from Figure 80, Figure 84, and Figure 88 also show the
areas of high pressure gradients following an oblique shock or a reflected shock
generated within the cowl duct.
Similar fluid flow behavior and shock patterns are observed for flight condition 3
results with yet one important additional characteristic to be noted. What makes flight
condition 3 results different from flight conditions 1 & 2 is the “shock-on-lip” condition
taking place (described in Section 2.3.1) where the scramjet forebody leading edge shock
impinges on the cowl leading edge. This phenomenon is clearly observed in Figure 85
and Figure 86. This means that design conditions for this particular inlet geometry
(Figure 60) have been reached at flight condition 3, or around Mach 9. Reaching this
condition means that the totality of the captured mass flow rate is contained within the
inlet duct and the inlet air capture ratio A o /A c from Equation (1.35) is equal to or
extremely close to 1.0. This is a desired condition since the drag generated from spillage
in flight conditions 1 & 2 will no longer be present. Even though we are not considering
material choices and respective thermal properties for the inlet in this numerical analysis,
it is important to remember that the impingement of the scramjet leading-edge with the
cowl leading edge is likely to produce a strong shock-shock interaction as the cowl tip,
which as explained in section 2.4.1 will result in high pressure gradients and high
aerodynamics heating at that location. It is therefore critical to protect this area of the
inlet by selecting the proper thermally resistant material or by adding heat shields.
112
4.1.5
Inviscid Results – Analysis of Numerical Results
First off, the graphical results from FLUENT presented in Figure 77 through Figure 88
present the values of the Mach number and static pressure at any location along the inlet
based on the legends provided. Since we are dealing with inviscid flow, the Mach number
and static pressure across each shock at the walls will not be affected by any boundary
layer. Therefore, by using FLUENT’s plotting tool, we can obtain exact results of the
Mach number and static pressure at the walls of interest, namely and the external
compression ramps and internal throat walls. This will supplement all inviscid numerical
results already obtained and allow for accurate comparisons against analytical results.
Moreover, in combination with the graphical results shown above, the plots will reveal
the exact x-location of any originating or impinging shocks. Each shock will have a
different size jump or step, depending on its strength. Plots of the Mach number and
static pressure along the external compression walls are shown in Figure 91 and Figure
92.
113
Figure 91: Wall Mach Number Plots for Inviscid Flight Conditions
114
Figure 92: Wall Static Pressure Plots for Inviscid Flight Conditions
115
Both Mach number and static pressure plots presented above present an excellent
addition to the inlet results from Figure 77 to Figure 88. These plots allow us to obtain
accurate locations as well as strengths of the shocks indicated by the change (increase or
decrease) in Mach number and static pressure. At this point, looking at the plots from
Figure 91 and Figure 92 and those from Figure 77 to Figure 88 side by side is
recommended to better associate each fluctuation in Mach number or pressure on the x-y
plots with its respective shock. As it was mentioned in the theory section, using a system
of weaker oblique shocks to compress the flow is recommended for better efficiency as
opposed to using a one strong (normal) shock, which would increase the total pressure
loss as well as temperature across the shock at a much greater rate.
Using the FLUENT post-processing tools and Figure 60 to obtain exact xlocations of the compression and expansion corners along with Figure 91 and Figure 92
as references, the following table summarizes the inviscid numerical results for the Mach
number and static pressure for the fluidflow during the mixed-compression process. Note
that due to the multiple shock interactions and reflections occurring within the inlet duct,
the Mach number and static pressure can vary greatly between the upper half (closer to
the cowl lower surface) of the internal inlet and its lower half (closer to the lower ramp)
at the same x-location. Both values are therefore provided for clarity (named “Upper” and
“Lower”), which can be matched using the appropriate plot from Figure 77 through
Figure 88. The last column indicates the final values after the desired compression of the
flow that will continue through the isolator for further compression and then enter the
combustion chamber.
116
TABLE 4: SUMMARY OF FLUENT NUMERICAL RESULTS (INVISCID FLOW)
External Inlet
3.98
5.00
Internal Inlet (Cowl Duct)
3.61
3.29
FC 2
Static
Pressure (Pa)
Mach No.
26436.0
9.00
5.19
121900.3
6.17
133352.0
4.64
207361.2
5.45
199990.0
X = 335.0 mm - Oblique Shock
7.00
81523.0
X = 267.0 mm - Oblique Shock
Mach No.
26436.0
X = 0.0 mm - Leading Edge Shock
Static
Pressure (Pa)
4.18
378282.8
4.85
FC 3
Static
Pressure (Pa)
26436.0
142126.4
327630.9
605887.7
117
X = 422.0 mm - Expansion - Shock Interaction
FC 1
Upper
= 2.72
Upper
= 2.96
Lower
= 3.66
Lower
= 4.07
Upper
= 436923.0
Upper
= 274031.6
Lower
= 118544.0
Upper
= 3.45
Lower
= 4.68
Upper
= 933778.3
Lower
= 185996.0
Upper
= 3.39
Lower
= 66714.5
Upper
= 3.35
Lower
= 5.28
Upper
= 933778.3
Lower
= 57804.7
Upper
= 3.59
2.83
355477.5
Upper
= 3.58
Lower
= 5.92
Upper
= 570569.7
Lower
= 1553369.3
Upper
= 3.65
Lower
= 6.51
Lower
= 5.45
Lower
= 5.78
Upper
= 1765291.0
Upper
= 2553685.4
Upper
= 2229052.4
Lower
= 281254.7
Lower
= 234878.6
Lower
= 95750.2
3.26
X = 465.0 mm - FC 2 = Oblique Shock
X = 471.0 mm - FC 3 = Oblique Shock
Mach No.
Free
Stream
Conditions
X = 457.0 mm - FC 1 = Shock Interactions / Reflections
X = 457.0 mm - FC 2 & FC 3 = Expansion Shock
Flow
Parameter
X = 442.0 mm - Expansion - Shock Interaction
Flight
Condition
(TABLE
2)
1040605.8
3.59
1997171.7
Table 4 has been generated to better understand the effects of the various shocks on the
flowfield based on the numerical results generated by FLUENT. Such information
become useful when comparing numerical results to analytical results or experimental
results for validation purposes. Interesting observations can be made from Table 4 at the
cowl region when comparing all three flight conditions.
For flight condition 1, the cowl leading-edge shock interacts with both expansion fans
located at x = 422.0 mm and x = 442.0 mm and impinges on ramp surface where the third
expansion fan is generated at x = 457.0 mm. There is clearly a shock-expansion
interaction between the cowl leading-edge shock and the expansion wave generated at
that location, this is indicated below in Figure 93. Immediately downstream of the cowl
leading-edge impingement point, a reflected shock is generated despite the presence of
the expansion corner, causing a pressure and temperature increase at that location. As a
result of the combination of these physical phenomena and the constantly changing Mach
number and static pressure at that particular location, Table 4 reflects the (approximate)
values of the Mach number and pressure downstream of these interactions. These
fluctuations can still be observed from Figure 91 and Figure 92.
Reflected
shock
STATIC PRESSURE (Pa)
Shock-Expansion interactions
INVISCID - MACH 5
766000
729000
692000
655000
618000
581000
544000
507000
470000
433000
396000
359000
322000
285000
248000
211000
174000
137000
100000
63000
26000
Figure 93: Shock interactions and reflections within cowl region (Flight Condition 1)
118
For flight conditions 2 & 3, the same cowl-leading edge shock is generated but this time,
due to the increased upstream pressure and the fact that both external compression shocks
impinge inside the cowl duct, the throat shock system is much more ingested within the
cowl duct. As a result, the cowl leading-edge shock impinges the lower surface further
downstream than for flight condition 1, followed by a similar reflected shock. Note that
the pressure drop associated with the third expansion fan at x = 457.0 mm is now clearly
visible now that there is no longer a shock interaction at that location. This is illustrated
in Figure 94 for flight condition 3, but is also applicable for flight condition 2 in Figure
84.
Reflected
shock
Cowl-leading edge shockSTATIC PRESSURE (Pa)
INVISCID - MACH 9
4660000
4430000
4200000
3970000
3740000
3500000
3270000
3040000
2810000
2580000
2340000
2110000
1880000
1650000
1420000
1190000
954000
722000
490000
258000
26200
Figure 94: Shock interactions and reflections within cowl region (Flight Condition 3)
119
4.1.6
Inviscid Results – Comparison to Analytical Results
For comparison and validity purposes, analytical results are generated and compared to
the numerical results obtained in FLUENT and summarized in Table 4. Analytical
calculations are achieved using the hypersonic shockwave equations from Section 1.2.2.1
and Section 1.2.2.2 along with the isentropic flow properties (Appendix A of Ref. [20]),
normal shock properties (Appendix B of Ref. [20]), and Prandtl-Meyer Function
properties (Appendix C of Ref. [20]). The nomenclature used in the analytical
calculations is shown in Figure 95 and Figure 96, and the results for the analytical results
are presented in Table 5 and Table 6.
120
M4,P4
M3,P3
β2
M2,P2
β3
θ3
θ2
M1,P1
β1
θ1
Figure 95: Nomenclature for analytical calculations (External Ramp)
TABLE 5: SUMMARY OF ANALYTICAL RESULTS (EXTERNAL RAMP)
External Ramp
Inviscid
Flight
Condition
(Per
TABLE 2)
1
M1
P1
(Pa)
β1
θ1
P2 / P1
P2
(Pa)
M2
β2
θ2
P3 / P2
P3
(Pa)
M3
β3
θ3
P4 / P3
P4
(Pa)
M4
5.00
26436.0
19.38
10.0
3.04368
80462.7
4.00
18.02
5.00
1.61992
130343.2
3.64
19.51
5.00
1.55777
203044.7
3.32
2
7.00
26436.0
16.38
10.0
4.38065
115806.9
5.23
14.56
5.00
1.85117
214378.2
4.68
15.87
5.00
1.74419
373916.3
4.22
3
9.00
26436.0
14.90
10.0
6.08235
160793.0
6.23
12.81
5.00
2.06003
331238.4
5.50
14.02
5.00
1.90562
631214.5
4.90
121
M6,P6
M5,P5
θ5
M4,P4
θ4
Figure 96: Nomenclature for analytical calculations (Cowl)
TABLE 6: SUMMARY OF ANALYTICAL RESULTS (COWL)
Cowl
Inviscid
Flight
Condition
(Per TABLE
2)
1
M4
P4 /
P 04
P4
(Pa)
θ4
M5
P5 /
P 05
P5
(Pa)
θ5
M6
P6 /
P 06
P6
(Pa)
3.32
0.0168
203044.7
6.16
3.71
0.0098
118442.7
6.09
4.16
0.0053
64055.8
2
4.22
0.0049
373916.3
6.16
4.77
0.0025
190773.6
6.09
5.44
0.0011
83940.4
3
4.90
0.0021
631214.5
6.16
5.61
0.0009
270520.5
6.09
6.05
0.0006
180347.0
122
Table 5 and Table 6 reveal that there is a strong agreement between the numerical results
obtained and the calculated analytical results. This can be further observed by looking at
the numerical vs. analytical plot from Figure 97. Again, these comparisons apply to the
inviscid flight conditions 1, 2, and 3 from Table 2.
Figure 97: Numerical vs. Analytical Results
The results above appear to agree, although it was wise to compare the numerical and
analytical results up until the shock-expansion interactions and shock reflections become
too dominant within the cowl duct, which would prevent any kind of accurate results
from the hypersonic shockwave equations. Even though turbulent numerical results are
provided in later sections, being able to validate numerical results using known theory is
important. It will also be interesting to compare inviscid results with fully turbulent
results and see how the boundary layer affects the behavior of the flow.
123
4.1.7
Inviscid Results – Efficiency and Operability
It was discussed in Section 2.3.2 that the total pressure ratio and adiabatic
compression efficiency were important factors in measuring the efficiency of an inlet and
its compression process for subsonic and supersonic flow conditions. Even though we
discussed their less frequent reliability for hypersonic flows due the undesirable
stagnation chemical effects adding great complexity to its total pressure dependencies,
both are still calculated here for reference and observation. Using equation (1.36) through
equation (1.38), the following efficiency parameters are calculated for all three inviscid
flight conditions using the numerical results from Table 4:
TABLE 7: COMPRESSION EFFICIENCY FOR INVISCID FLIGHT CONDITIONS
P∞
(Pa)
P th
(Pa)
FC1 26436.0 355477.5
FC2 26436.0 1040605.8
FC3 26436.0 1997171.7
M∞
5
7
9
M th
ɣ
2.83 1.4
3.26 1.4
3.59 1.4
Total
Pressure
Ratio
at Throat,
πc
0.7220
0.5133
0.3101
Cycle Static
Adiabatic
Temperature Compression
Ratio, ψ
Efficiency, η c
2.306
3.455
4.808
0.9253
0.9145
0.8956
Table 7 reveals some very interesting results. First, it can be seen that the total pressure
ratio drop significantly as the free stream Mach number increases to its design point
around Mach 9. Such increasing stagnation pressure losses will eventually reduce the
vehicle’s axial momentum and its overall system performance. It is clear that the total
pressure ratio continuously drops as the free stream Mach number is increased and the
shock system within the inlet become much stronger. It is reasonable to assume that the
total pressure losses will be even more dramatic once the free stream Mach number
124
exceeds the design point and oversped conditions are attained since the ingested
compression shocks will generate additional strong shock interactions within the inlet
duct. Looking at the adiabatic compression efficiency demonstrate why these
compression efficiency parameters are not always considered for hypersonic flows.
Indeed, the adiabatic compression efficiencies obtained in Table 7 between flight
conditions 1 and 2 show that a 1.1% drop in compression efficiency (from 0.9253 to
0.9145) results in a 20.9 % increase in total pressure loss (from 0.7220 to 0.5133).
Similarly between flight conditions 2 and 3, a 1.9% drop in adiabatic compression
efficiency (from 0.9145 to 0.8956) yields an additional 20.3% increase in total pressure
loss (from 0.5133 to 0.3101). Such a dramatic non-linear behavior of these relationships
is the reason why these compression efficiency parameters are not as critical for
hypersonic compression systems as they are for subsonic and supersonic flow conditions.
Lastly, because inlet starting is such a critical aspect of inlet, the internal
contraction ratio is calculated in Table 8 for our inlet and compared with the allowable
solutions based on the Kantrowitz limit from Figure 53 for all three flight conditions.
TABLE 8: INLET INTERNAL CONTRACTION RATIO
Cowl
Entrance
Area, A2
Throat
Area, Ath
Internal
Contraction
Ratio, A2 / Ath
Inverse of
Contraction
Ratio, Ath / A2
17.8
11.0
1.62
0.62
Using Figure 53 and the Kantrowitz limit from equation (1.42), the following plot in
Figure 98 shows where the contraction ratio for our inlet geometry falls compared to the
boundaries of the permissible contraction ratios.
125
Figure 98: Inlet contraction ratios for starting conditions
Based on Figure 98, it can be seen that all three flight conditions fall within the
Kantrowitz limit and the maximum contraction ratio limit (Isentropic line). While this is
reasonable, our inlet has a fixed geometry and overspeeding may be required in order to
ensure proper inlet starting. One way would be to have the hypersonic vehicle attached to
another propulsion system (a rocket for example) that will accelerate the scramjet above
the Kantrowitz limit to start the inlet, which could then decelerate back to its design
Mach number. Even when the inlet has started, the physical phenomena discussed earlier
such as shockwaves interactions with the expected thick hypersonic boundary layers can
be disruptive enough to cause areas of flow separations, resulting in an inlet unstart.
126
4.2
Turbulent Results
In addition to inviscid results as presented in Section 4.1, fully turbulent viscous
results have also been generated using the flight conditions on Table 2 and turbulent
models from Section 3.1.5. Assuming fully turbulent flow and including viscous effects
provides a more realistic analysis of the inlet than inviscid results alone. It is important to
remember that the no-slip condition applies with the presence of a boundary-layer and the
velocity is expected to be zero at walls. Lastly, the results presented below reflect the
three different flight conditions listed in Table 2 for three different FLUENT turbulent
models using first-order accuracy. Despite the differences in each turbulent model as
explained in Section 3.1.5, similarities in results are expected.
127
4.2.1
Turbulent Results –Flight Condition 1
4.2.1.1 Spalart-Allmaras Model
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
MACH NUMBER
SPALART-ALLMARAS - MACH 5
Figure 99: Mach number contours for inlet – Flight Condition 1 (Spalart-Allmaras)
MACH NUMBER
SPALART-ALLMARAS - MACH 5
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
Figure 100: Mach number contours for throat region – Flight Condition 1 (Spalart-Allmaras)
128
740000
704000
667000
631000
595000
559000
522000
486000
450000
413000
377000
341000
304000
268000
232000
196000
159000
123000
86600
50300
14000
STATIC PRESSURE (Pa)
SPALART-ALLMARAS - MACH 5
Figure 101: Static pressure contours for inlet – Flight Condition 1 (Spalart-Allmaras)
STATIC PRESSURE (Pa)
SPALART-ALLMARAS - MACH 5
740000
704000
667000
631000
595000
559000
522000
486000
450000
413000
377000
341000
304000
268000
232000
196000
159000
123000
86600
50300
14000
Figure 102: Static pressure contours for throat region – Flight Condition 1 (Spalart-Allmaras)
129
4.2.1.2 K-Epsilon Model
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
MACH NUMBER
K-EPSILON - MACH 5
Figure 103: Mach number contours for inlet – Flight Condition 1 (K-Epsilon)
MACH NUMBER
K-EPSILON - MACH 5
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
Figure 104: Mach number contours for throat region – Flight Condition 1 (K-Epsilon)
130
742000
705000
669000
633000
596000
560000
523000
487000
451000
414000
378000
342000
305000
269000
232000
196000
160000
123000
87000
50600
14200
STATIC PRESSURE (Pa)
K-EPSILON - MACH 5
Figure 105: Static pressure contours for inlet – Flight Condition 1 (K-Epsilon)
STATIC PRESSURE (Pa)
K-EPSILON - MACH 5
742000
705000
669000
633000
596000
560000
523000
487000
451000
414000
378000
342000
305000
269000
232000
196000
160000
123000
87000
50600
14200
Figure 106: Static pressure contours for throat region – Flight Condition 1 (K-Epsilon)
131
4.2.1.3 K-Omega SST Model
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
MACH NUMBER
K-OMEGA SST - MACH 5
Figure 107: Mach number contours for inlet – Flight Condition 1 (K-Omega SST)
MACH NUMBER
K-OMEGA SST - MACH 5
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
Figure 108: Mach number contours for throat region – Flight Condition 1 (K-Omega SST)
132
730000
694000
659000
624000
589000
554000
519000
483000
448000
413000
378000
343000
308000
272000
237000
202000
167000
132000
96600
61400
26300
STATIC PRESSURE (Pa)
K-OMEGA SST - MACH 5
Figure 109: Static pressure contours for inlet – Flight Condition 1 (K-Omega SST)
STATIC PRESSURE (Pa)
K-OMEGA SST - MACH 5
730000
694000
659000
624000
589000
554000
519000
483000
448000
413000
378000
343000
308000
272000
237000
202000
167000
132000
96600
61400
26300
Figure 110: Static pressure contours for throat region – Flight Condition 1 (K-Omega SST)
133
4.2.2
Turbulent Results –Flight Condition 2
4.2.2.1 Spalart-Allmaras Model
MACH NUMBER
SPALART-ALLMARAS - MACH 7
7.00
6.65
6.30
5.95
5.60
5.25
4.90
4.55
4.20
3.85
3.50
3.15
2.80
2.45
2.10
1.75
1.40
1.05
0.70
0.35
0.00
Figure 111: Mach number contours for inlet – Flight Condition 2 (Spalart-Allmaras)
MACH NUMBER
SPALART-ALLMARAS - MACH 7
7.00
6.65
6.30
5.95
5.60
5.25
4.90
4.55
4.20
3.85
3.50
3.15
2.80
2.45
2.10
1.75
1.40
1.05
0.70
0.35
0.00
Figure 112: Mach number contours for throat region – Flight Condition 2 (Spalart-Allmaras)
134
2090000
1980000
1880000
1780000
1670000
1570000
1470000
1360000
1260000
1160000
1060000
952000
849000
746000
643000
541000
438000
335000
232000
129000
26200
STATIC PRESSURE (Pa)
SPALART-ALLMARAS - MACH 7
Figure 113: Static pressure contours for inlet – Flight Condition 2 (Spalart-Allmaras)
STATIC PRESSURE (Pa)
SPALART-ALLMARAS - MACH 7
2090000
1980000
1880000
1780000
1670000
1570000
1470000
1360000
1260000
1160000
1060000
952000
849000
746000
643000
541000
438000
335000
232000
129000
26200
Figure 114: Static pressure contours for throat region – Flight Condition 2 (Spalart-Allmaras)
135
4.2.2.2 K-Epsilon Model
MACH NUMBER
K-EPSILON - MACH 7
Figure 115: Mach number contours for inlet – Flight Condition 2 (K-Epsilon)
MACH NUMBER
K-EPSILON - MACH 7
7.00
6.65
6.30
5.95
5.60
5.25
4.90
4.55
4.20
3.85
3.50
3.15
2.80
2.45
2.10
1.75
1.40
1.05
0.70
0.35
0.00
Figure 116: Mach number contours for throat region – Flight Condition 2 (K-Epsilon)
136
7.00
6.65
6.30
5.95
5.60
5.25
4.90
4.55
4.20
3.85
3.50
3.15
2.80
2.45
2.10
1.75
1.40
1.05
0.70
0.35
0.00
2090000
1980000
1880000
1780000
1670000
1570000
1470000
1360000
1260000
1160000
1060000
952000
849000
746000
643000
541000
438000
335000
232000
129000
26200
STATIC PRESSURE (Pa)
K-EPSILON - MACH 7
Figure 117: Static pressure contours for inlet – Flight Condition 2 (K-Epsilon)
STATIC PRESSURE (Pa)
K-EPSILON - MACH 7
2090000
1980000
1880000
1780000
1670000
1570000
1470000
1360000
1260000
1160000
1060000
952000
849000
746000
643000
541000
438000
335000
232000
129000
26200
Figure 118: Static pressure contours for throat region – Flight Condition 2 (K-Epsilon)
137
4.2.2.3 K-Omega SST Model
MACH NUMBER
K-OMEGA SST - MACH 7
Figure 119: Mach number contours for inlet – Flight Condition 2 (K-Omega SST)
MACH NUMBER
K-OMEGA SST - MACH 7
7.00
6.65
6.30
5.95
5.60
5.25
4.90
4.55
4.20
3.85
3.50
3.15
2.80
2.45
2.10
1.75
1.40
1.05
0.70
0.35
0.00
Figure 120: Mach number contours for throat region – Flight Condition 2 (K-Omega SST)
138
7.00
6.65
6.30
5.95
5.60
5.25
4.90
4.55
4.20
3.85
3.50
3.15
2.80
2.45
2.10
1.75
1.40
1.05
0.70
0.35
0.00
2090000
1980000
1880000
1780000
1670000
1570000
1470000
1360000
1260000
1160000
1060000
952000
849000
746000
643000
541000
438000
335000
232000
129000
26200
STATIC PRESSURE (Pa)
K-OMEGA SST - MACH 7
Figure 121: Static pressure contours for inlet – Flight Condition 2 (K-Omega SST)
STATIC PRESSURE (Pa)
K-OMEGA SST - MACH 7
2090000
1980000
1880000
1780000
1670000
1570000
1470000
1360000
1260000
1160000
1060000
952000
849000
746000
643000
541000
438000
335000
232000
129000
26200
Figure 122: Static pressure contours for throat region – Flight Condition 2 (K-Omega SST)
139
4.2.3
Turbulent Results –Flight Condition 3
4.2.3.1 Spalart-Allmaras Model
MACH NUMBER
SPALART-ALLMARAS - MACH 9
9.00
8.55
8.10
7.65
7.20
6.75
6.30
5.85
5.40
4.95
4.50
4.05
3.60
3.15
2.70
2.25
1.80
1.35
0.90
0.45
0.00
Figure 123: Mach number contours for inlet – Flight Condition 3 (Spalart-Allmaras)
MACH NUMBER
SPALART-ALLMARAS - MACH 9
9.00
8.55
8.10
7.65
7.20
6.75
6.30
5.85
5.40
4.95
4.50
4.05
3.60
3.15
2.70
2.25
1.80
1.35
0.90
0.45
0.00
Figure 124: Mach number contours for throat region – Flight Condition 3 (Spalart-Allmaras)
140
4510000
4280000
4060000
3830000
3610000
3390000
3160000
2940000
2710000
2490000
2270000
2040000
1820000
1590000
1370000
1150000
922000
698000
474000
250000
26200
STATIC PRESSURE (Pa)
SPALART-ALLMARAS - MACH 9
Figure 125: Static pressure contours for inlet – Flight Condition 3 (Spalart-Allmaras)
STATIC PRESSURE (Pa)
SPALART-ALLMARAS - MACH 9
4510000
4280000
4060000
3830000
3610000
3390000
3160000
2940000
2710000
2490000
2270000
2040000
1820000
1590000
1370000
1150000
922000
698000
474000
250000
26200
Figure 126: Static pressure contours for throat region – Flight Condition 3 (Spalart-Allmaras)
141
4.2.3.2 K-Epsilon Model
9.00
8.55
8.10
7.65
7.20
6.75
6.30
5.85
5.40
4.95
4.50
4.05
3.60
3.15
2.70
2.25
1.80
1.35
0.90
0.45
0.00
MACH NUMBER
K-EPSILON - MACH 9
Figure 127: Mach number contours for inlet – Flight Condition 3 (K-Epsilon)
MACH NUMBER
K-EPSILON - MACH 9
9.00
8.55
8.10
7.65
7.20
6.75
6.30
5.85
5.40
4.95
4.50
4.05
3.60
3.15
2.70
2.25
1.80
1.35
0.90
0.45
0.00
Figure 128: Mach number contours for throat region – Flight Condition 3 (K-Epsilon)
142
4500000
4280000
4060000
3830000
3610000
3380000
3160000
2940000
2710000
2490000
2260000
2040000
1820000
1590000
1370000
1150000
922000
698000
474000
250000
26200
STATIC PRESSURE (Pa)
K-EPSILON - MACH 9
Figure 129: Static pressure contours for inlet – Flight Condition 3 (K-Epsilon)
STATIC PRESSURE (Pa)
K-EPSILON - MACH 9
4500000
4280000
4060000
3830000
3610000
3380000
3160000
2940000
2710000
2490000
2260000
2040000
1820000
1590000
1370000
1150000
922000
698000
474000
250000
26200
Figure 130: Static pressure contours for throat region – Flight Condition 3 (K-Epsilon)
143
4.2.3.3 K-Omega SST Model
9.00
8.55
8.10
7.65
7.20
6.75
6.30
5.85
5.40
4.95
4.50
4.05
3.60
3.15
2.70
2.25
1.80
1.35
0.90
0.45
0.00
MACH NUMBER
K-OMEGA SST - MACH 9
Figure 131: Mach number contours for inlet – Flight Condition 3 (K-Omega SST)
MACH NUMBER
K-OMEGA SST - MACH 9
9.00
8.55
8.10
7.65
7.20
6.75
6.30
5.85
5.40
4.95
4.50
4.05
3.60
3.15
2.70
2.25
1.80
1.35
0.90
0.45
0.00
Figure 132: Mach number contours for throat region – Flight Condition 3 (K-Omega SST)
144
4500000
4270000
4050000
3830000
3600000
3380000
3160000
2930000
2710000
2480000
2260000
2040000
1810000
1590000
1370000
1140000
920000
697000
473000
250000
26200
STATIC PRESSURE (Pa)
K-OMEGA SST - MACH 9
Figure 133: Static pressure contours for inlet – Flight Condition 3 (K-Omega SST)
STATIC PRESSURE (Pa)
K-OMEGA SST - MACH 9
4500000
4270000
4050000
3830000
3600000
3380000
3160000
2930000
2710000
2480000
2260000
2040000
1810000
1590000
1370000
1140000
920000
697000
473000
250000
26200
Figure 134: Static pressure contours for throat region – Flight Condition 3 (K-Omega SST)
145
4.2.4
Turbulent Results – Discussion of Numerical Results
A first look at the FLUENT results from Sections 4.2.1 through 4.2.3 reveal very
similar shock patterns compared to the inviscid results presented in Sections 4.1.1
through 4.1.3. It can therefore always be beneficial to look at invisid results first, which
are easier to calculate (and require less computational time) to obtain a good guess of
where the shocks will occur and predict where shock interactions are likely to take place
once viscous effects are considered. Using the flight condition 1 K-Epsilon model as an
example (taken from Figure 103), a description of the shock systems encountered on the
external inlet for turbulent cases is presented in Figure 135.
Cowl / throat
shock system
Boundary-Layer
Ramp compression
shocks
Leading-edge
shock
MACH NUMBER
K-EPSILON - MACH 5
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
Figure 135: Mach contour results for the Mach 5 flight condition (External Compression)
As it was the case for the inviscid results, the vehicle’s leading edge shock is clearly
visible, followed by the two external ramp compression shocks, and then finally a series
of shock systems at the throat. Using the legends provided in each picture from Section
4.2.1 through 4.2.3, the sudden changes in Mach number and static pressure can be
quantified as the flow travels downstream across every shockwave. Here again, note that
146
even though this example is taken for flight condition 1, using the K-Epsilon turbulent
model, similar shockwaves and phenomena can be identified for other turbulent models,
and different flight conditions. As expected for viscous results, a boundary-layer can be
seen along the walls (external ramp and cowl) where the no-slip condition applies and the
velocity decreases to zero. As the flow reaches the throat region, a combination of
expansion waves and additional compression shocks are generated, which interact with
each other along with the boundary layer, causing shock-shock and shock-layer
interactions (discussed in Section 2.4). As it was done for the external portion of the inlet,
a description of the internal compression shock system within the cowl duct is presented
here in Figure 136 for flight condition 1, K-Epsilon turbulent case.
Cowl
Leading-edge
shock
Boundary-Layer
Shock-Expansion
Interaction
Expansion
Shock
Shock-Layer
Interaction
MACH NUMBER
K-EPSILON - MACH 5
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
Figure 136: Mach number contours for the Mach 5 flight condition (Internal compression)
Similar observations to the inviscid cases can be made regarding the originating and
impingement points of the shocks along the external and internal inlet walls. The scramjet
leading-edge shock does not come in contact with the cowl-leading edge for flight
conditions 1 and 2 (as shown in Figure 99 through Figure 122), which causes spillage
147
around the cowl and therefore additional drag (described as an “undersped” inlet from
Section 2.3.1). This is not the case for flight condition 3 (from Figure 123 through Figure
134), where a shock-on-lip state is reached and none of the captured mass flow rate gets
spilled around the cowl. The same conclusion as the inviscid case can be drawn here for
turbulent runs, where flight condition 3 from Table 2 in combination with the selected
inlet geometry from Figure 60 appear to result in design conditions with a design Mach
number of around 9. As expected, there are however great differences in flow behavior
within the cowl duct due to the presence of the boundary-layer. It can be observed from
Figure 136 that in addition to the same shock-shock and shock-expansion interactions
occurring for inviscid flow, the impingement of the cowl-leading edge shock on the lower
surface creates a shock-layer interaction. When the impinging shock is strong enough as
it appears to be the case here, a separation bubble is generated (theory presented in
section 2.4.2) where an abrupt pressure increase on the boundary layer due to the
impinging shock causes the boundary layer to temporarily separate from the surface
therefore generating an area of reversed flow. This is confirmed on Figure 136 by
observing the Mach contour variations, which show the subsonic region right at the
impingement point of the cowl shock at the lower surface. Fortunately the separation
bubble does not seem to cause an unstart as it only occupies a relatively small portion of
the throat area and no additional spillage is present. When comparing all three turbulent
models used to produce the results from Sections 4.2.1 through 4.2.3, a couple of small
differences appear between each model, such as the thickness of the boundary-layer, the
strength and effect of shockwaves on other shocks or on the boundary-layer. Ultimately,
the shock systems are similar for all three models, but the separation bubble for example
148
seems to be captured better on K-Epsilon model than the on the other two models. While
one turbulent model may over-estimate the interactions present, the other may underestimate those. Since these runs are based off first-degree accuracy, the fine details that
differentiate each model are more difficult to spot.
4.2.5
Turbulent Results – Analysis of Numerical Results
The FLUENT numerical results presented in Sections 4.2.1 through 4.2.3 indicate
Mach number and static pressure fluctuations along the entire portion of the inlet.
Similarly to inviscid results shown in Sections 4.1.1 through 4.1.3, the associated legends
can be used to provide a visual estimation of the Mach number or static pressure at any
particular location. In addition, FLUENT’s plotting tools can also be used to generate x-y
plots of those same parameters along the inlet walls as it was done in Figure 91 and
Figure 92. However, along the lower surface of the inlet, the no-slip condition applies for
the viscous cases and the velocity of the flow is expected to be zero in the presence at the
boundary layer. As a verification, a plot of the Mach number for flight condition 1 (Table
2) at the lower surface of the inlet is presented here for all three viscous models against
the inviscid case to illustrate the effect of the no-slip condition.
149
Figure 137: Inlet lower surface Mach number
As predicted, all viscous cases flow fields are subsonic and more precisely close to zero
velocity at the surface because of the presence of the boundary-layer, where the no-slip
condition applies. Note that the inviscid case flight conditions were all plotted in Figure
91, clearly indicating the location of oblique shocks, expansion waves, and other shockshock interactions along the lower surface of the inlet. Based on Figure 137, the use of
such x-y plots for viscous cases is simply useless as the shocks generated by the
compression and expansion corners will form on the boundary-layer and not the walls. To
obtain exact values of the Mach number and static pressure for the fluiflow within the
capture tube, FLUENT’s post-processing tool can be used to extract such data as it was
done in Table 4. The summarized turbulent numerical results for all three flight
conditions listed in Table 2 are presented in Table 9.
150
TABLE 9: SUMMARY OF FLUENT NUMERICAL RESULTS (TURBULENT FLOW)
Flow
Parameter
Freestream Conditions
Mach No.
5.00
3.92
3.63
3.28
Upper = 2.68
Lower = 3.23
2.78
2.63
Static Pressure (Pa)
26436.0
83236.6
133899.9
206275.9
Upper = 459592.1
Lower = 133899.6
379978.5
481305.0
Mach No.
5.00
3.93
3.58
3.28
Upper = 2.73
Lower = 3.23
2.73
2.53
Static Pressure (Pa)
26436.0
83342.4
134260.6
207000.9
Upper = 468866.1
Lower = 134260.6
367029.7
497962.2
Mach No.
5.00
3.97
3.59
3.30
Upper = 2.75
Lower = 3.38
2.83
2.66
Static Pressure (Pa)
26436.0
79021.5
128248.7
205610.7
Upper = 451741.7
Lower = 121216.2
381417.1
451741.7
Mach No.
7.00
5.18
4.63
4.20
Static Pressure (Pa)
26436.0
Mach No.
7.00
Static Pressure (Pa)
26436.0
Mach No.
7.00
Static Pressure (Pa)
26436.0
Mach No.
Internal Inlet (Cowl Duct)
FC 1
K-E
K-Ω SST
118978.0
5.19
4.63
222029.3
4.69
365726.8
4.20
366301.1
4.18
X = 422.0 mm - Shock-Expansion Interaction
K-E
5.19
221698.6
X = 335.0 mm - Oblique Shock
FC 2
118821.3
X = 267.0 mm - Oblique Shock
SA
X = 0.0 mm - Leading Edge Shock
K-Ω SST
Upper = 3.40
Lower = 4.45
Upper = 962414.9
Lower = 201123.1
Upper = 3.40
Lower = 4.44
Upper = 943388.4
Lower = 201419.0
Upper = 3.39
Lower = 4.36
Upper = 982427.6
Lower = 201020.2
Upper = 1127018.6
Lower = 98245.8
Upper = 3.59
Lower = 4.51
Upper = 1087660.2
Lower = 98367.6
Upper = 3.16
Lower = 4.49
Upper = 1146934.4
Lower = 98203.4
3.77
3.28
633207.7
1024141.3
3.83
654844.7
3.67
118766.8
221583.6
365527.0
9.00
6.12
5.37
4.79
4.76
Upper = 3.55
Lower = 5.21
4.38
Static Pressure (Pa)
26436.0
182924.8
317293.5
630820.6
317293.5
Upper = 2377614.4
Lower = 182924.8
1347454.0
Mach No.
9.00
6.19
5.46
4.81
4.79
Upper = 3.43
Lower = 5.22
4.40
Static Pressure (Pa)
26436.0
182897.1
317216.6
630628.6
317216.6
Upper = 2108142.4
Lower = 182897.1
1346999.0
Mach No.
9.00
6.18
5.43
4.84
4.81
Upper = 3.50
Lower = 5.18
4.42
Static Pressure (Pa)
26436.0
182652.6
316762.5
629685.5
316762.5
Upper = 2239003.9
Lower = 182652.6
1344938.1
SA
FC 3
Upper = 3.28
Lower = 4.57
K-E
K-Ω SST
151
694540.7
X = 465.0 mm - FC 2 = Shock-Layer Interaction / Shock Reflection
X = 471.0 mm - FC 3 = Shock-Layer Interaction / Shock Reflection
SA
X = 457.0 mm = Multiple Shock Interactions
External Inlet
X = 442.0 mm = Multiple Shock Interactions
Flight
Turbulent
Condition
Model
(See Table 2)
3.28
1025829.5
3.23
1023554.3
3.55
2064087.3
3.59
2108142.5
3.58
2060190.8
Similar to the inviscid flow summarized numerical results from Table 4, due to the
multiple shock interactions and reflections occurring within the inlet duct, the Mach
number and static pressure can vary greatly between the upper half (closer to the cowl
lower surface) of the internal inlet and its lower half (closer to the lower ramp) at the
same x-location. Both values are therefore provided for clarity (named “Upper” and
“Lower”), which can be matched using the appropriate plot from Figure 99 through
Figure 134. The last column indicates the final values after the desired compression of
the flow that will continue through the isolator for further compression and then enter the
combustion chamber. The FLUENT numerical results summarized in Table 9 indicate
that all three turbulent models yield comparable values for the Mach number and static
pressure. Although one would expect the simple one-equation Spalart-Allmaras model to
underestimate the flow around regions of high vorticity affected by severe shock
interactions, it appears to match quite well with the K-Epislon and K-Omega SST
models. One possible reason for seeing those similarities in the results could be the used
of first-order accuracy solutions, which could potentially hide some of the finer shocklayer and shock-shock interactions along the walls. In order to provide additional validity
to those numerical results, and in particular to insure that the boundary-layer is properly
captured, y+ values for all three turbulent models have been generated. As it was
discussed in Section 3.2.1, an additional boundary-layer mesh was added to the inlet
mesh with finer elements at the walls in order to properly capture any potential shocklayer interactions. The y+ value plots for all three flight conditions and all three turbulent
models are presented in Figure 138 through Figure 140.
152
Figure 138: Turbulent Parameter Y+ Value (Flight Condition 1)
Figure 139: Turbulent Parameter Y+ Value (Flight Condition 2)
153
Figure 140: Turbulent Parameter Y+ Value (Flight Condition 3)
Based on Section 3.2.1 and Figure 138 through Figure 140, it can be observed that the y+
values for all cases stay within the viscous sub-layer region at y+ < 8. This viscous
sublayer is very thin and corresponds to the region where the turbulent shear stress is less
than 10% of the wall shear stress, with dominating viscous effects. The areas of jumps
correspond to the places where oblique shockwaves occur, which result in an
instantaneous increase in air density, driving the y+ value up.
154
4.2.6
Turbulent Results – Comparisons to Inviscid & Published Results
For reference and additional validity, inviscid, turbulent, and some published
experimental results are compared against each other for various flight conditions. Since
most hypersonic inlet applications are military programs or still being heavily researched
on today, limited experimental results have been published and fully released to the
public. Table 10 presents those results, where the freestream conditions are compared,
along with the flow conditions at the inlet exit.
PUBLISHED
TURBULENT
INVISCID
TABLE 10: RESULT COMPARISONS TO PUBLISHED EXPERIMENTAL RESULTS
Result Source
M∞
P ∞ (Pa)
M th
P th (Pa)
FC1 - TABLE 4
5
26436
2.83
355477.5
FC2 - TABLE 4
7
26436
3.26
1040605.8
FC3 - TABLE 4
9
26436
3.59
1997171.7
FC1 - TABLE 9
5
26436
2.61
477003.0
FC2 - TABLE 9
7
26436
3.26
1024508.3
FC3 - TABLE 9
9
26436
3.57
2077473.5
Ref. [21]
6.5
3968
2.67
< 500000.0
Ref. [22]
6.5
830
2.99
Not published
Ref. [23]
4.0
Not provided
2.40
Not published
Ref. [24]
7.0
2511
2.20
Not published
Ref. [25]
7.0
170
2.50
Not published
155
It appears from Table 10 that the inlet exit Mach numbers are relatively comparable,
which brings validity to our numerical results, although most experimental testing was
done at higher altitudes (given the lower initial static pressures). In addition, other
experimental research was conducted to address issues of inlet unstarts and possible
mitigation methods.
4.2.7
Turbulent Results – Efficiency
As it was mentioned in Section 2.3.2, efficiency parameters such as total pressure
ratio and compression efficiency are indicators of much useful work can be done by the
gas. Each is important in evaluating the efficiency of a hypersonic inlet and its
compression process. As it was done for inviscid conditions in Section 4.1.7, similar
calculations are carried out for turbulent numerical results and presented in Table 11. As
it can be seen from the results, each turbulent model presents comparable total pressure
ratios and compression efficiencies for the same flight condition. When comparing
turbulent efficiency results from Table 11 to inviscid efficiency results from Table 7, the
fully turbulent cases appear to have slightly lower efficiencies than their inviscid
counterparts. This is expected since viscous effects are considered and the shock-layer
interactions will cause additional losses within the internal inlet. In addition, Table 11
also confirms that as the Mach number increases, the shock systems and viscous
interactions become stronger, resulting in higher losses and a dramatic reduction in total
pressure ratio. As mentioned earlier, whether viscous effects are considered or not, Table
7 and Table 11 both indicate that the adiabatic compression efficiency drops very little
compared to a drastic reduction in total pressure ratio. These non-linear patterns are due
to the stagnation effects on hypersonic flow which involve chemical effects (excitement
156
of molecules), making the total pressure a much more complicated parameter to analyze.
These efficiency calculations can therefore still be used for hypersonic flowfields, but are
better suited for subsonic and supersonic conditions.
157
TABLE 11: COMPRESSION EFFICIENCY FOR TURBULENT FLIGHT CONDITIONS
FC1
FC2
FC3
Turbulent Model
P∞
(Pa)
P th
(Pa)
M∞
M th
ɣ
Total Pressure
Ratio
at Throat, π c
Cycle Static
Temperature Ratio,
ψ
Adiabatic
Compression
Efficiency, η c
Spalart-Allmaras
26436.0
481305.0
5.00
2.63
1.4
0.7192
2.517
0.9349
K-Epsilon
26436.0
497962.2
5.00
2.53
1.4
0.6373
2.631
0.9158
K-Omega SST
26436.0
451741.7
5.00
2.66
1.4
0.7071
2.484
0.9299
Spalart-Allmaras
26436.0
1024141.3
7.00
3.28
1.4
0.5201
3.427
0.9154
K-Epsilon
26436.0
1025829.5
7.00
3.28
1.4
0.5209
3.427
0.9156
K-Omega SST
26436.0
1023554.3
7.00
3.23
1.4
0.4832
3.499
0.9076
Spalart-Allmaras
26436.0
2064087.3
9.00
3.55
1.4
0.3029
4.886
0.8953
K-Epsilon
26436.0
2108142.5
9.00
3.59
1.4
0.3273
4.808
0.9013
K-Omega SST
26436.0
2060190.8
9.00
3.58
1.4
0.3154
4.827
0.8979
158
5
CONCLUSION
The numerical investigation presented for the chosen mixed-compression hypersonic
inlet yielded several important results and observations:
-
The inlet geometry defined in Figure 60 was derived from a CIAM/NASA
axisymmetric inlet model. Based on the numerical results provided, the inlet
reached its design Mach number around Mach 9 since this is where the shock-on
lip condition is seen in all flight condition 3 cases, inviscid and turbulent.
-
Conducting inviscid runs provided a simple way of obtaining a first set of
important information about the behavior of flowfield along the inlet. Such
information includes: location and strength of every compression or expansion
wave generated, location of shock-shock interactions, information about where
shock-layer interactions are likely to occur (once viscous effects are considered),
information about on and off-design conditions, and a first set of compression
efficiency results.
-
The turbulent cases presented appeared to provide adequate results based on
comparisons between each turbulent model, inviscid models, and published
results. As predicted by theory and inviscid results, a shock-layer interaction takes
place at the impingement point of the cowl-leading edge shock on the boundarylayer, which appears to generate a separation bubble on the wall within the inlet
duct. The K-Epsilon turbulence model appears to provide the most clarity in the
results given that all cases were run with first-order accuracy. As the freestream
159
Mach number is increased, the cowl leading edge gets ingested further within the
throat and so does the separation bubble as a result.
-
Although it is not as obvious from the numerical results obtained because those
reflect first-order accuracy, published experimental and numerical results
demonstrate that the choice of turbulent models can have a large impact on the
analysis and predictions of shock-layer interactions, which in turn will affect the
inlet starting characteristics.
-
The efficiency results generated seem to agree with theory where the total
pressure ratios worsens as the Mach number is increased, due to the additional
losses associated with stronger flow interactions. Moreover, the calculation of the
adiabatic compression efficiency parameter demonstrate that there is a non-linear
relationship with the corresponding total pressure ratio, which is why such
efficiency parameters are not as critical for hypersonic regimes as they are for
subsonic and supersonic flight conditions. This agrees with Heiser and Pratt’s
discussion from Section 5.4 in Ref. [8].
-
The numerical results obtained and summarized in Table 4 and Table 9
demonstrate strong agreements with each other, and against published results.
-
Ideas of further work for this inlet would include running all flight conditions
with higher-order turbulent models to better capture shock-layer interactions and
yield more accurate results. Also, it would be interesting to obtain experimental
results for this particular geometry to determine the accuracy of the numerical
results and how each model compares against one another.
160
BIBLIOGRAPHY
161
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