PULSED DETONATION ENGINE
NOZZLE DESIGN AND ANALYSIS
by
RAHUL KUMAR
DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctorate of Philosophy in Aerospace Engineering at
The University of Texas at Arlington
December 2019
Arlington, Texas
Supervising Committee:
Donald R. Wilson, Supervising ProfessorZhen Xue HanRatan KumarLiwei ZhangBenito Chen-Charpentier
List of Figures
1.1 Engine issues for hypersonic airbreathing propulsion systems [2] . . . . 21.2 Specific Impulse vs. Mach number . . . . . . . . . . . . . . . . . . . . 41.3 Schematic of geometry considered for this research . . . . . . . . . . . 7
2.1 One dimensional model of a detonation wave . . . . . . . . . . . . . . 132.2 Rankine-Hugoniot curve [28] . . . . . . . . . . . . . . . . . . . . . . . 142.3 Thermodynamic properties across a ZND Detonation wave[28] . . . . . 182.4 Cellular pattern of detonation . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 p− v diagram of a Humphrey Cycle [42] . . . . . . . . . . . . . . . . 223.2 Physical steps that make up the Fickett-Jacobs cycle . . . . . . . . . . . 233.3 p− v diagram showing the sequence of states and connecting paths that
make up the FJ cycle (with πc = 5) for a stoichiometric propane- airmixture at 300 K and 1 bar initial conditions. . . . . . . . . . . . . . . . 25
3.4 Thermal efficiency as a function of compression ratio (left) and combus-tion pressure ratio (right) for FJ, Humphrey, and Brayton cycles for astoichiometric propane-air mixture at 300 K and 1 bar initial condition [36] 25
3.5 p− v diagram of ZND Cycle [42] . . . . . . . . . . . . . . . . . . . . 263.6 Ideal Humphrey (1 → 2H → 3H → 1), FJ(1 → 2CJ → 3CJ → 1)
and ZND (1 → 1 → 2CJ → 3CJ → 1) cycles for a stoichiometrichydrogen/air mixture initially at STP [41]. . . . . . . . . . . . . . . . . 27
3.7 Working of a conventional PDE [39] . . . . . . . . . . . . . . . . . . . 283.8 Multi-mode Pulsed Detonation based propulsion system . . . . . . . . . 303.9 Bypass stream to control combustion properties . . . . . . . . . . . . . 31
4.1 Control volume with faces ab, bc, cd and da . . . . . . . . . . . . . . . 44
5.1 Schematic representation of initial conditions of shock tube . . . . . . . 535.2 Variation of pressure along the length of tube at t = 0.2sec . . . . . . . 545.3 Variation of density along the length of tube at t = 0.2sec . . . . . . . . 545.4 Computational domain to capture detonation phenomenon . . . . . . . 565.5 Variation of static pressure along the length of the tube for different grid
sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
ii
5.6 Variation of static temperature along the length of the tube for differentgrid sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.7 Temperature scale of detonation propagation with a grid size of 0.03mm 575.8 Detonation propagation with a grid size of 0.03mm . . . . . . . . . . . 585.9 Variation of pressure with time in H2-air reaction mechanism . . . . . . 595.10 Variation of temperature with time in H2-air reaction mechanism . . . . 605.11 Rate of change of mole fractions of species in H2-air reaction mechanism 605.12 Schematic of the computational domain for ODWE . . . . . . . . . . . 615.13 Matrix of test cases for the combination of incoming Mach number and
wedge angle [64] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.14 Variation of static pressure for different grid sizes . . . . . . . . . . . . 635.15 Pressure contour of ODWE mode . . . . . . . . . . . . . . . . . . . . . 645.16 Temperature contour of ODWE mode . . . . . . . . . . . . . . . . . . 655.17 Velocity contour of ODWE mode . . . . . . . . . . . . . . . . . . . . . 655.18 Change in Heat of reaction for ODWE mode simulation . . . . . . . . . 665.19 Change of mole fraction of different species . . . . . . . . . . . . . . . 675.20 Variation of static pressure along the length of the detonation chamber . 675.21 Variation of static temperature along the length of the detonation chamber 685.22 Schematic of the operation of NDWE mode by changing the stoichiomet-
ric ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.23 Variation of flow properties at exit of detonation chamber in a NDWE mode 71
6.1 Scramjet Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Schematic diagram of ideal, minimum length, two dimensional nozzle
designed using method of characteristics [84] . . . . . . . . . . . . . . 756.3 Schematic diagram of Single Expansion Ramp Nozzle [86] . . . . . . . 766.4 Nozzle flow exit conditions [74] . . . . . . . . . . . . . . . . . . . . . 786.5 Graphical representation of nozzle for an exit Mach number of 2.4 . . . 806.6 Nozzle contour generated using MATLAB . . . . . . . . . . . . . . . . 806.7 Comparison of values at random discrete points from MATLAB code
with Anderson [74] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.8 Mach number contour . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.9 Pressure contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.10 Velocity vector plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.11 Variation of nozzle geometry with change in number of characteristics.
Specific heat ratio = 1.4, exit Mach number = 2.4 . . . . . . . . . . . . 856.12 Variation of nozzle geometry with change in specific heat ratio. No. of
characteristics = 50, exit Mach number = 2.4 . . . . . . . . . . . . . . . 866.13 Variation of nozzle geometry with change in exit Mach number. No. of
characteristics = 50, Specific heat ratio = 1.4 . . . . . . . . . . . . . . . 87
iii
7.1 Schematic diagram of an ideal, minimum length, two dimensional exhaustnozzle designed by means of the method of characteristics [85] . . . . . 89
7.2 Geometric altitude vs flight Mach number trajectories for constant dy-namic pressure [85] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.3 Hypersonic nozzle contour using MOC . . . . . . . . . . . . . . . . . . 927.4 Tulip like structure of the expansion waves emanating from the nozzle inlet 937.5 Truncated nozzle contour . . . . . . . . . . . . . . . . . . . . . . . . . 947.6 Variation of pressure along the length of the nozzle at design point . . . 977.7 Variation of density along the length of the nozzle at design point . . . . 977.8 Variation of Mach number along the length of the nozzle at design point 987.9 Ratio of zone III static pressure to entry static pressure for ideal design
point expansion components as function of entry Mach number and exitMach number [85] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.10 Variation of pressure along the length of the nozzle at off-design condition 997.11 Variation of density along the length of the nozzle at off-design condition 1007.12 Variation of Mach number along the length of the nozzle at off-design
condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.13 Shock interactions of X-15 being fired into a wind tunnel [108] . . . . . 1027.14 Pressure contour of the formation of plumes and shocks at the design
condition and at a time instant of 0.001 sec . . . . . . . . . . . . . . . . 1037.15 Density contour of the formation of plumes and shocks at the design
condition and at a time instant of 0.001 sec . . . . . . . . . . . . . . . . 1047.16 Pressure contour of the formation of plumes and shocks at an altitude of
42 km . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.17 Density contour of the formation of plumes and shocks at an altitude of
42 km . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.18 Mach number contour of the formation of plumes and shocks at an
altitude of 42 km . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.19 Pressure contour of the formation of plume and shocks at an altitude of
34 km and at a time instant of 0.001 sec . . . . . . . . . . . . . . . . . 1077.20 Density contour of the formation of plume and shocks at an altitude of
34 km and at a time instant of 0.001 sec . . . . . . . . . . . . . . . . . 1087.21 Off-design condition: Pressure contour of the formation of plume and
shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.22 Off-design condition: Density contour of the formation of plume and
shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.23 Off-design condition: Mach number contour of the formation of plume
and shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.1 Pressure distribution at p=3 atm and T=700 K [100] . . . . . . . . . . . 114
A.1 Illustration of characteristic direction . . . . . . . . . . . . . . . . . . . 116
iv
A.2 Left and right running characteristics . . . . . . . . . . . . . . . . . . . 120
v
List of Tables
1.1 Description of numbers and symbols in Fig. 1.3a . . . . . . . . . . . . 7
4.1 Hydrogen-air reaction model . . . . . . . . . . . . . . . . . . . . . . . 43
5.1 Initial conditions for SOD Shock tube problem . . . . . . . . . . . . . . 535.2 Exit conditions at the detonation chamber . . . . . . . . . . . . . . . . 68
7.1 Flight conditions considered for the current research . . . . . . . . . . . 95
vi
ABSTRACT
Pulsed Detonation Engine
Nozzle Design and Performance Analysis
Rahul Kumar, Ph.D
The University of Texas at Arlington, 2019
Supervising Professor: Donald R. Wilson
For a hypersonic flight mission, different flight regimes have been recognized. The
successful and efficient operation of the aircraft to traverse through all the flight regimes
requires the integration of various propulsion cycles into a single flow path. Using the
phenomenon of detonation initiation and propagation, a multi-mode detonation based
propulsion concept was proposed for hypersonic flight. Of the different modes proposed,
it was recognized that the efficient operation of Normal Detonation Wave Engine (NDWE)
mode and Oblique Detonation Wave Engine (ODWE) mode played an important role as
they delivered thrust at critical parts of the trajectory. For this study, two different flight
conditions representing the ODWE mode and NDWE mode are selected along a constant
dynamic pressure trajectory of 47,880 N/m2. The ODWE mode is chosen as a design
point and the flight Mach number chosen is 15 at an altitude of 42 km. The NDWE mode
is the off-design condition and the flight Mach number representing this mode is 8.75 at
an altitude of 34 km above sea level.
An inviscid Euler simulation was carried out for the design point with an incoming
combustion chamber Mach number of 6 which leads to the oblique detonation wave
mode and the exit conditions at the expansion region are determined. The exit parameters
of the expansion region are treated as inlet conditions into the nozzle. Nozzle inlet Mach
number of 4.12 was determined from the simulation and using this Mach number, method
vii
of characteristics was used to design the nozzle contour for efficient expansion of the flow
through the nozzle. Usually, hypersonic nozzles are large and they can be truncated at
40% of original length without significant loss of thrust. The designed nozzle is truncated
at 40% of original length and CFD simulations are carried out to study the flow within
the nozzle and also the flow interactions with the external flowfield.
Using the mathematical model and geometry used for the design case, CFD simula-
tions were carried out for off-design case with an incoming combustion chamber Mach
number of 3.5. The CJ Mach number is greater than the incoming combustion chamber
Mach number, leading to a moving detonation wave. The detonation wave movement
is controlled and made to oscillate at a particular location downstream of the wedge
by varying the stoichiometric ratio of the fuel-air mixture. The exit conditions at the
expansion region are nearly constant because of this oscillation and the parameters at the
exit are used as inlet conditions into the nozzle.
Comparing the flow structures at the nozzle exhaust of the design and off-design
conditions, similar shock structures are observed however, the plume appears to be longer
and more voluminous in the design case when compared to the off-design case. Also,
the plumes traverse a longer distance downstream of the nozzle before they mix into the
external flowfield in case of the design case because of higher Mach number at the exit
of the nozzle exhaust.
This research sets the procedure to study the gas dynamic aspects of the flowfield
from the combustion chamber all the way to the nozzle exhaust for a particular inlet
combustion chamber conditions.
viii
ACKNOWLEDGEMENT
This dissertation became a reality only because of all the support and help I got from
my parents and family members. First of all, I would like to thank them for being along
side me in this phase of my life. It is also my privilege to thank my wife, Sanjana for her
encouragement while I needed the most and bearing all my hardships during this time.
It is a genuine pleasure to express my deep gratitude to my advisor Dr. Donald Wilson
for his expertise, assistance, guidance and patience throughout this journey of my PhD.
Without his help, this dissertation document would not have been possible. I would like
to thank all my committee members for sharing their insight on this subject. I would like
to extend my gratitude to Dr. Linda Wang for letting me use her lab and the computer
because of which, many simulations in this research was possible. My sincere thanks to
all the faculty and staff members of the MAE department for all the help and support.
I would like to thank all my friends in Arlington for all the fun and laughter we have
shared. Lastly, I would like to thank Texas Advanced Computing Center (TACC) at UT
Austin for all the support.
ix
Table of Contents
1 Introduction 11.1 Research Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Procedure followed to achieve research objectives . . . . . . . 61.2 Overview of the document . . . . . . . . . . . . . . . . . . . . . . . 8
2 The Detonation Phenomenon 112.1 Gas Dynamics of Detonations . . . . . . . . . . . . . . . . . . . . . . 122.2 Modes of Initiation of Detonation . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Deflagration to Detonation Transition . . . . . . . . . . . . . 162.2.2 Direct Initiation . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Shock Induced Detonation . . . . . . . . . . . . . . . . . . . 17
2.3 A Brief Review on Early Research of Detonation . . . . . . . . . . . 17
3 Pulsed Detonation Engine (PDE) 203.1 Thermodynamic Analysis of PDE . . . . . . . . . . . . . . . . . . . 20
3.1.1 Humphrey Cycle . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.2 FJ Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.3 ZND Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Conventional PDE Cycle . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Multi-Mode Pulsed Detonation based Propulsion Concept . . . . . . . 293.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Mathematical Model Development for Wedge Induced Detonation 364.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 374.1.2 Thermodynamic Properties . . . . . . . . . . . . . . . . . . . 384.1.3 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.1 Finite Volume Formulation . . . . . . . . . . . . . . . . . . . 434.2.2 Density Based Solver . . . . . . . . . . . . . . . . . . . . . . 464.2.3 Discretization Schemes . . . . . . . . . . . . . . . . . . . . . 474.2.4 Evaluation of Gradients . . . . . . . . . . . . . . . . . . . . . 494.2.5 Convective Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 50
x
5 Simulation of Wedge Induced Detonation 525.1 Shock capturing capability of FLUENT . . . . . . . . . . . . . . . . 525.2 Capturing Detonation Phenomenon in FLUENT . . . . . . . . . . . . 555.3 Chemistry Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.5 Oblique Detonation Wave Engine Mode . . . . . . . . . . . . . . . . 635.6 Normal Detonation Wave Engine mode . . . . . . . . . . . . . . . . . 69
6 Nozzle 726.1 Single Expansion Ramp Nozzle (SERN) . . . . . . . . . . . . . . . . 746.2 Design of Nozzle Contour . . . . . . . . . . . . . . . . . . . . . . . . 766.3 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.1 Verification of MOC MATLAB Code . . . . . . . . . . . . . 796.3.2 CFD Simulation of Nozzle flow . . . . . . . . . . . . . . . . 82
6.4 Parameters affecting Nozzle geometry . . . . . . . . . . . . . . . . . 84
7 Hypersonic Nozzle Design 887.1 Supersonic Inlet Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . 887.2 Hypersonic Nozzle Design using MOC . . . . . . . . . . . . . . . . . 89
7.2.1 Constant dynamic pressure trajectory . . . . . . . . . . . . . 907.2.2 Nozzle contour . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.3 CFD Simulation of Nozzle Flow . . . . . . . . . . . . . . . . . . . . 947.3.1 Design condition . . . . . . . . . . . . . . . . . . . . . . . . 957.3.2 Off-design condition . . . . . . . . . . . . . . . . . . . . . . 99
7.4 CFD simulation of nozzle exhaust . . . . . . . . . . . . . . . . . . . . 1017.4.1 Design condition . . . . . . . . . . . . . . . . . . . . . . . . 1027.4.2 Off-Design condition . . . . . . . . . . . . . . . . . . . . . . 106
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8 Future Work 113
A Method of Characteristics 115A.1 Theory of MOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.2 Determination of Characteristic Lines . . . . . . . . . . . . . . . . . 117A.3 Compatibility Equations . . . . . . . . . . . . . . . . . . . . . . . . 119
Bibliography 122
xi
Chapter 1
Introduction
October 14th 2017 marks the 70th anniversary of the historical event when the US
Air Force Major (now General) Charles E. “Chuck” Yeager piloted the Bell XS-1 aircraft,
breaking the sound barrier for the first time and reached a speed of 700 miles per hour
(Mach 1.06). In 1967, a rocket powered North American X-15 flew at Mach 6.7 which
till date, holds the world record for the highest speed ever reached by a manned, powered,
winged aircraft [1]. Missions like these, gave humanity a ray of hope in attaining a stable
long range manned hypersonic flight.
Hypersonic air-breathing aircraft provide certain potentials like long-range cruise
missiles for attack of time sensitive targets, flexible high altitude atmospheric interceptors,
responsive hypersonic aircraft for global payload delivery and reusable launch vehicles for
efficient space access [2]. However, there are also large number of technical challenges
the designers will have to address like the capture of atmospheric air and burning it in
combustors at flight speeds of the order of 6000 - 10,000 km/h [2], to generate thrust large
enough to overcome external drag, to enhance the process of mixing in the combustor
because of the short residence time of the airflow, stabilization of combustion process, an
efficient structural material of the airframe to withstand high temperatures as a result of
1
the boundary layer heating and many more. The challenges are shown schematically in
Fig. 1.1.
Figure 1.1: Engine issues for hypersonic airbreathing propulsion systems [2]
Over the last few decades, there has been significant research towards the development
of hypersonic vehicles. Few of the research programs are listed below.
• As early as 1964, US started the NASA Hypersonic Research Engine (HRE)
project which aimed at designing, developing and constructing a high performance
research ramjet/scramjet engine to fly over a speed range of Mach 4-8 using the
X-15A-2 research airplane [3].
• The National Aero-Space Plane (NASP) program (1986-1993) attempted to design
the Rockwell X-30, which was a Single-Stage-To-Orbit (SSTO) and passenger
spaceliner [4]. A detailed 1/3rd scale mockup of the X-30 was built by engineering
students at Mississippi State University’s Raspet Flight Research Lab in Starkville
[5]. However, due to technical issues and budget cuts, the program was terminated.
• An eight year NASA Hyper-X program undertook a high risk and a high pay-off
2
research program to design the X-43 research aircraft. In November 2004, the
X-43 reached a speed of Mach 9.6, setting a speed record for an air-breathing
vehicle [6].
• In the recent years, NASA, AFRL and Australia’s Defense Science and Technology
Organization (DSTO) were working with a number of partners on the HIFiRE
(Hypersonic International Flight Research Experimentation) program to advance
hypersonic flight with an aim to explore fundamental technologies needed to
achieve practical hypersonic flight [7]. The HIFiRE team was successful in their
goals such as the design, assembly and extensive pre-flight testing of the hypersonic
vehicles and design of complex avionics and flight systems. It was also the first
time a hydrocarbon-fueled Scramjet was tested and reached speed up to Mach 8
[8].
An aircraft from lift-off to hypersonic Mach number goes through a wide range
of aerothermodynamic conditions. Only a certain type of engine cycle is suitable for
a particular range of Mach numbers. The performance of different engine cycles is
determined by a parameter called specific impulse which is defined as the change in
momentum per unit of propellant consumed. Fig. 1.2 shows the specific impulse for
the various engine cycles as a function of Mach number. From around Mach 3-4, it can
be seen that ramjets become more efficient than the turbojets and turbofans but beyond
around Mach 5-6, their specific impulse decays and the scramjets deliver a higher specific
impulse. Taking the example of X-43, for the first stage from take-off, a subsonic aircraft
propulsion B-52 was used followed by The Orbital Sciences Pegasus booster rocket
which carried the X-43 to its test altitude where the boosters separated and it flew under
its own built-in engine and pre-programmed control system [9].
3
Figure 1.2: Specific Impulse vs. Mach number
To optimize the long range flights and also to fly the aircraft at a broad range of Mach
numbers, an aircraft with a combined engine cycle is required. A vehicle propulsion
system referred to as the Turbine-based Combined Cycle (TBCC) was proposed which
employed a turbine engine to produce thrust at low speeds [10] and ramjet and scramjet
combustors to accelerate above flight Mach number of 3. This system consisted of a
dual flow path: a low speed flow path for turbine engine operation and a high speed flow
path with a ramjet, scramjet or a dual-mode combustor [11] . There are several technical
challenges associated with the TBCC engines ie.,
• Achieving a stable transitioning mode between the flowpaths
• Integration of propulsion system with the airframe structure
• Achieving the required performance over the entire flight range
4
Also with the TBCC, there is an increase in the frontal projection area and the weight of
the aircraft.
Recently, detonation phenomena are being studied as a prospect for hypersonic
propulsion [12]. Pulsed Detonation Engine (PDE), an airbreathing engine which has been
proposed as an alternative to the conventional propulsion system, shows promise towards
increasing the cycle efficiency, specific thrust and reducing the specific fuel consumption
when compared to certain deflagration based systems [13].
Wilson et al [15] proposed a multi-mode pulsed detonation engine which has inte-
grated various engine cycles that potentially is suitable for efficient operation over a
broad range of Mach numbers and altitudes into a single flow path. The main applications
of this concept of engine are a trans-atmospheric flight vehicle for access to space and an
atmospheric cruise vehicle. The various modes in this proposed concept are [16]
• An ejector augmented pulse detonation rocket for take-off to moderate supersonic
Mach numbers.
• A pulsed Normal Detonation Wave Engine (NDWE) mode at combustion chamber
Mach numbers less than the Chapman-Jouguet Mach number that operate at flight
Mach numbers between 3-7 approximately.
• An Oblique Detonation Wave Engine (ODWE) mode of operation for Mach num-
bers in the air-breathing regime that operate for flight Mach numbers that result in
detonation chamber higher than the Chapman-Jouguet Mach number.
• A pure Pulsed Detonation Rocket (PDR) mode of operation at high altitude.
5
1.1 Research Proposal
Inspired by the multi-mode pulsed detonation engine concept proposed by Wilson
et al [15], the principal purpose of this research is to design an exhaust system for the
multi-mode pulsed detonation engine concept considering the Oblique Detonation Wave
Engine (ODWE) mode as design point. Since the ODWE mode is characterized by high
combustion chamber Mach number (greater than CJ Mach number), the flight Mach
number at the considered design point is also high. As a rule of thumb, the combustion
chamber Mach number is around 40-45% of the flight Mach number. In this regard, flight
Mach number of 15 is chosen at an altitude of 42 km above sea level. Fig. 1.3 shows the
schematic of the geometry considered for this research.
The analysis carried out is 2 dimensional as shown in Fig. 1.3a. For better visualiza-
tion of the geometry, a 3 dimensional view is presented in Fig. 1.3b. Referring to Fig.
1.3a, an outline of the procedure followed to conduct this research is presented in section
1.1.1.
1.1.1 Procedure followed to achieve research objectives
1. CFD simulation of the flow through the combustion chamber is carried out with
specific inflow conditions which represents the ODWE mode. The primary reason
for this simulation is to determine the inlet conditions into the nozzle.
2. The exit conditions at the expansion section marked as region 3 in Fig. 1.3a will
be used as inlet conditions into the nozzle.
3. A mathematical technique for solving partial differential equations called Method
of Characteristics is used to design the contour of the nozzle to expand the flow
from region 3 into the atmosphere.
6
(a)
(b)
Figure 1.3: Schematic of geometry considered for this research
1 Inlet into the combustion chamber
2 Wedge where the shock waves are formed
3 Expansion section
4 Nozzle inlet
5 Nozzle contour
6 Flap length
⇒ Direction of fluid flow
Table 1.1: Description of numbers and symbols in Fig. 1.3a
7
Hypersonic nozzles designed to ideal conditions tend to be long which add to the
weight of the vehicle and also significantly increase the aerodynamic drag. To
encounter this issue,the length of the nozzle can be truncated to a certain percentage
of the initial length without significant loss of thrust.
4. The characteristics emanating from the upper corner of the throat of the nozzle
deflect to the solid surface which is called the flap represented by region 6 in Fig.
1.3a. The location of the last characteristic falling on the flap is determined and the
length of the flap is the distance from the nozzle inlet to the the point where the
last characteristic reflects off from the surface.
5. Once the nozzle geometry is determined, CFD simulation is carried out to visualize
the flow through the nozzle and interaction of flow from the nozzle exit with the
atmosphere.
1.2 Overview of the document
Chapter 2 introduces the concept of detonation and discusses the hydrodynamic
equations which represent the detonation phenomenon. Detonation initiation can be
achieved by various modes and a brief description of the modes are provided. For this
research, the geometry of the combustion chamber is used to initiate the detonation which
falls into the category of the shock induced detonation mentioned in section 2.2.3. A
brief review of early research of detonation phenomenon is also discussed.
Chapter 3 introduces the application of detonation in a Pulsed Detonation Engine
(PDE). Since the application of detonation phenomenon towards propulsion was discov-
ered, various studies had been carried out to accurately ascribe the detonation propagation
phenomenon to a thermodynamic cycle. A brief description of the thermodynamic cycles
8
which have been associated with detonation are provided and the cycles are compared
with their work output and efficiencies. This discussion is important because the working
of PDE is different than the conventional jet engines and the thermodynamic analysis
shows the PDE thermodynamic cycle has higher efficiency than the conventional jet
engine (Brayton cycle) and this is an encouragement to conduct further research in
improving the PDE performance and design. Multi-mode PDE concept is discussed
which is a modification of a conventional PDE. Literature review of detonation initiation
and propagation is presented at the end of the chapter.
Chapter 4 is a discussion of mathematical formulation used to simulate the wedge
induced detonation and also the numerical techniques used to solve the mathematical
formulation.
Chapter 5 discusses the simulation results of the wedge induced detonations. The
chapter starts off with first testing the ability of the numerical solver in FLUENT to
capture shocks by simulating a SOD shock tube problem and the results are verified with
the analytical solution. Ability of FLUENT to capture detonation phenomenon too is
tested. The oblique detonation wave engine mode simulation results are presented and
discussed. The exit conditions of the detonation chamber is used to design the hypersonic
nozzle at design condition. For an off-design condition, NDWE mode is considered and
discussion of the exit conditions of NDWE is presented.
Chapter 6 discusses the importance of nozzle in the design and performance of
high speed flights. A mathematical technique called Method of characteristic (MOC) is
introduced to design the nozzle contour. Verification of the results of MATLAB code to
design a nozzle contour using MOC is also discussed. The verified MATLAB code will
then be used to design the hypersonic nozzle contour by changing the inlet conditions at
the nozzle.
Chapter 7 talks about the hypersonic nozzle contour design generation. Various
9
features of a typical hypersonic nozzle are discussed and CFD simulations are carried out
to study the flow within the nozzle at the design condition. Using the same nozzle contour,
CFD simulation is also carried out for off-design condition operating at an altitude of
34km. CFD simulation results are presented for the nozzle exhaust flow interaction with
the surrounding environment for both the design and off-design condition.
Chapter 8 talks about the future work that could be carried out to make the design of
hypersonic vehicle more realistic.
10
Chapter 2
The Detonation Phenomenon
Combustion is a phenomenon where the reactants are converted into products, releas-
ing potential energy stored in the chemical bonds of the reactant molecules and converting
them into thermal and kinetic energy. During the process of combustion, a wave is formed
across which there are large thermodynamic variations. When a combustion wave travels
at subsonic velocities with respect to the reactants, the phenomenon is known as De-
flagration. The structure of a deflagration wave consists of a precursor shock followed
by a reaction front [17]. The gradation in thermal and species concentration across the
reaction front is usually via diffusion of heat and mass. When a combustion wave travels
at supersonic speeds with respect to the reactants, then the phenomenon is referred to as
Detonation. In this phenomenon, there is a sharp increase in thermodynamic properties
across the detonation front and since it is a supersonic wave, the reactants ahead of the
wave remain undisturbed.
In detonation, the leading part of the front is a strong shock wave and as this shock
wave propagates into the undisturbed mixture ahead of it, it compresses the reactants
resulting in a steep gradient in the thermodynamic properties. The high temperature
behind the shock wave triggers chemical reactions, releasing heat and this energy released
11
in turn drives the detonation front forward. The material consumed in this process is
of the order of 103 to 108 times faster than any combustion process and this makes
the phenomenon of detonation more distinguishable than other combustion process.
An example of the usefulness of the detonation process as stated in [18], a good solid
explosive converts energy at a rate of 1010 watts per square centimeter of its detonation
front. To put it into perspective, this energy release can be compared to the total electric
generation capacity of the United States, which is about 4 X 1011 watts or a 20-m square
detonation wave operates at a power level equal to all the power received from the sun.
2.1 Gas Dynamics of Detonations
Detonation as a phenomenon, though being an unsteady three- dimensional process
with a cellular pattern, for the sake of modeling it can assumed to be a planar one
dimensional wave where the shock and reaction zone are coupled. This theory is known
as the CJ theory. A schematic representation of a one dimensional model of a detonation
wave is presented in Fig. 2.1. Considering the one dimensional conservation equations,
ρ1u1 = ρ2u2 (2.1)
P1 + ρ1u21 = P2 + ρ2u
22 (2.2)
h1 +1
2u2
1 = h2 +1
2u2
2 (2.3)
The subscripts (1) and (2) indicate the upstream and downstream conditions respectively.
In eqs. 2.1 - 2.3, P, u, ρ and h are the pressure, velocity, density and the enthalpy of the
flow respectively.
Splitting the total enthalpy to sensible enthalpy and heat of formation and assuming
12
Figure 2.1: One dimensional model of a detonation wave
constant specific heat, we get
h1 = cp1 + h01 (2.4)
h2 = cp2 + h02 (2.5)
where h0is the standard enthalpy of formation at standard state and cp is the specific heat
capacity at constant pressure.
The heat addition q is given by
q = h01 + h0
2 (2.6)
Combining eqs.2.1 and 2.2, gives the Rayleigh line which is
(p2 − p1)
(1/ρ2 − 1/ρ1)= −ρ2
1u21 (2.7)
The Rankine Hugoniot relation is given by,
γ
γ − 1(p2
ρ2
− p1
ρ1
)− 1
2(p2 − p1)(
1
ρ1
+1
ρ2
)− q = 0 (2.8)
The Rankine Hugoniot relation represents the possible solutions of pressure and density
13
for the combustion products for given reactant properties and heat addition term q. In eq.
2.8, γ is the ratio of specific heat of gas at constant pressure to specific heat at constant
volume.
Fig. 2.2 represents the Rankine - Hugoniot (R-H) curve where all the possible
solutions for any flow going through a shock or combustion wave can be found.
Figure 2.2: Rankine-Hugoniot curve [28]
As can be observed from the graph, the curve does not pass through the origin, in fact
it is fixed at a known value of (p1,1ρ1
) and this position can be considered as the origin.
Along with the fixed values, the heat addition parameter q in eq. 2.8 is also assumed to
be known. Any realizable states going from states (1) to (2) must satisfy the Rayleigh
and Rankine and Hugonoit relations. In Fig. 2.2, the intersection of tangent lines from
the origin to the curve represent stable solutions. The speed of the combustion wave can
be found using Eq. 2.7, which is
uc =1
ρ1
√p2 − p1
1ρ1− 1
ρ2
(2.9)
14
The Rayleigh lines divide the Hugonoit curve into five segments.
• Section I : Strong detonation
• Section II : Weak detonation
• Section III : Weak deflagration
• Section IV : Strong deflagration
• Section V : Physically impossible states
At a certain minimum velocity, the Rayleigh line touches the R-H curve at a point
U, which represents the upper CJ point. This is the minimum velocity solution for
detonation of a particular mixture where the burnt products travel at a speed of Mach 1
with respect to the detonation front. Above the point U, the solutions represent an over
driven detonation in which the pressure is higher than the CJ pressure. These conditions
usually occur at the onset of the detonation initiation and eventually stabilize to a CJ
detonation. Section II, which is the weak detonation and in this state, the velocities of the
burnt products are supersonic compared to the detonation wave. As a result, the burnt
products over take and weaken the detonation front, eventually leading to a deflagration
phenomenon. A weak deflagration case is when a subsonic flow upon passing through a
combustion wave accelerates to a higher subsonic velocity; but with strong deflagration,
the flow will have to accelerate to higher supersonic velocities which is impossible in
a constant area duct. Therefore, it is nearly impossible to observe strong deflagrations
experimentally [32]. In region V, it can be observed that ( 1ρ1− 1
ρ2) < 0 and p1 − p2 > 0
which makes the wave velocity imaginary. Therefore. Section V has no solutions and is
physically not realizable.
15
2.2 Modes of Initiation of Detonation
2.2.1 Deflagration to Detonation Transition
DDT is the transition of a deflagration wave in a tube to detonation. At the closed end
of the tube, a flame is introduced into the reacting mixture. The flame being subsonic,
creates disturbances ahead of the flame and these disturbances in turn help to increase
the surface area of the flame, increasing the flame velocity. As the flame propagates
along the tube, its pressure and temperature become high, increasing the energy release
rate and also resulting in formation of a strong shock wave. As the reactants reach
critical ignition conditions, one or more localized explosion pockets are formed and these
explosions creates a blast wave. The blast wave amplifies, merging the shock wave and
reaction zone into a detonation wave. Turbulence appears to the mechanism responsible
for the transition from deflagration to detonation [17]. However, there are several issues
with this process of initiation like the critical conditions necessary for the formation
and amplification of explosion centers and formation of a blast wave from a localized
explosion to propagate into a detonation wave.
2.2.2 Direct Initiation
Direct initiation is when an ignition source is placed spontaneously in a reactive
medium which leads to the formation of detonation wave. This mode of initiation was
initially used to obtain spherical detonations. Zeldovich et. al. [33] showed that there
exists a critical diameter of the ignition source below which the blast wave decays to
an acoustic wave. They also found out that there exists a critical blast energy for a
spherical detonation to be formed. They gave a criterion for the critical energy for the
blast initiation which indicates that the blast radius must be at least the induction zone
thickness of the detonation of the explosive mixture at the instant when the blast has
16
decayed to the CJ value.
2.2.3 Shock Induced Detonation
In this mode of initiation, a shock wave is used to form a detonation wave. The
reactive medium when passed through a shock wave of a particular strength, the flow
gets heated up resulting in chemical reaction behind the shock wave and the heat released
in turn drives the shock forward. The shock induced detonation in PDE is explained in
detail in chapter 3.
2.3 A Brief Review on Early Research of Detonation
Detonation was first discovered by Berthelot and Vielle [19] and Mallard and Le
Chatelier [20] while studying flame propagation and from the time of its discovery,
the phenomenon of detonation has been a fascination for both scientists and engineers.
Mallard and Le Chatelier found that the velocity of the detonation front is related to
the speed of sound of the combustion products. One of the earliest theories to predict
the detonation velocity was proposed by Chapman and Jouguet [21] which is known
as Chapman-Jouguet (CJ) theory. Their theory was based on the works of Rankine
and Hugoniot, who analyzed the conservation equations across a shock wave [22] [23].
According to the CJ theory, the entire flow field is treated as a one dimensional flow field
and the detonation front as a discontinuity across which the conservation equations of
the shock wave apply. However in 1927, Campbell and Woodhead [24] discovered that
detonations are a more complex process with the spin phenomenon and this phenomenon
is observed where the available energy and the rate of reaction are barely sufficient for
the detonation propagation in a tube. In the 1940’s, Zeldovich [25], Von Neumann [26]
and Doring [27] independently formulated a model where in the detonation wave is
17
comprised of a leading shock wave, an ignition region, followed by a reaction region.
From Fig. 2.3, the shock is assumed to move to the left into the unreacted mixture.
Figure 2.3: Thermodynamic properties across a ZND Detonation wave[28]
As the shock wave compresses the unreacted gas ahead of it, the enthalpy across
the shock wave is increased thereby sharply increasing the thermodynamic properties
like pressure, temperature and density to a particular value that depends on the shock
strength. At this state, the reactant molecules start to decompose and start forming free
radicals. It takes a certain amount of time for the chemical decomposition to occur that is
called the induction time and the region where it occurs right behind the shock wave is
called the induction region. The length of the induction zone depends on certain dynamic
parameters like the critical initiation energy [29].The properties across this region are
approximately held constant and once there are enough radicals formed, exothermic
chemical reaction takes place in the reaction zone. Because of the expansion process,
the density and pressure decrease and temperature reaches a maximum value. As shown
in Fig. 2.3, the chemical reactions cease when the properties of the reacted mixture
18
reach an equilibrium state. 1n 1961, White [30] used interferometry to discover that the
detonation front was cellular in structure with fish scale patterns. The measurements
demonstrated the presence of a secondary transverse wave pattern, almost perpendicular
to the front, with the detonation front itself being slightly curved. Further investigation
on the unsteady pattern of the detonation front was conducted by Voitsekhovskii et. al.
[31].
Fig. 2.4 shows the fish scale pattern of an unsteady detonation wave. It consists of a
leading shock, triple points, followed by a reaction zone and transverse waves trailing
downstream of the leading shock wave. The cellular pattern change with respect to time
as they propagate into the reaction zone. Behind the incident shock, there are small
pockets of unreacted gas mixture. As the shock propagates, the triple points collide,
creating explosions and the energy generated in turn drives the shock forward. Fickett and
Davis [18] have shown that the unsteady cellular structure is because of thermo-acoustic
instabilities.
Figure 2.4: Cellular pattern of detonation
19
Chapter 3
Pulsed Detonation Engine (PDE)
A PDE is a type of propulsion system that uses the phenomenon of detonation to
generate thrust. With a high thermodynamic efficiency, reduced mechanical simplicity
and high thrust to weight ratio, PDE’s have an upper hand over the conventional gas
turbine engines which operate on the constant pressure Brayton Cycle. However, several
key issues must be successfully addressed like efficient mixing of fuel and air in the
combustion chamber, prevention of autoignition, prevention of inlet unstart, integration
with inlet and a nozzle before PDE can become a reality. The following section compares
the thermodynamic cycle of a PDE cycle with that of the conventional jet engine cycles.
3.1 Thermodynamic Analysis of PDE
The combustion of fuel in detonation takes place so rapidly that the fuel-air mixture
will not have time to expand and some studies compared detonation propagation to
a constant volume Humphrey Cycle [38]. One of the first people to come up with a
nearly accurate thermodynamic cycle for detonation propagation was Zel’dovich [35]. He
concluded that the efficiency of thermodynamic detonation cycle was greater than the con-
20
stant volume Humphrey Cycle. The Humphrey Cycle under-predicted the performance
of a PDE and did not adequately capture the physics of detonation [34]. Wintenberger
et al [36] presented a physical model for the detonation cycle handling propagating
detonations in a purely thermodynamic fashion. They used the Fickett-Jacobs(FJ) cycle
to compute an upper bound to the amount of mechanical work that can be obtained from
detonating a given mass of explosive. The limitations of this cycle is that it cannot be
used to directly estimate the performance of the PDE because of the unsteadiness of
the exit flow [37]. The FJ Cycle is based on the CJ Detonation theory and hence the
physics encapsulated by the ZND model cannot be captured. Using the ZND Cycle, the
Von Neumann spike can be achieved and more accurately describe the thermodynamic
process of a ZND detonation model. Vutthivithayarak et. al [41] have compared the
performance and efficiency of Humphrey Cycle, FJ Cycle and ZND Cycle and a brief
discussion is presented in the following subsections.
3.1.1 Humphrey Cycle
Fig. 3.1 shows the p − v diagram of an ideal Humphrey Cycle. The values in the
graph are presented with respect to initial conditions and 0 represents the initial state of
the mixture. A detailed discussion is given in [42].
Ideal Humphrey Cycle consists of 4 processes:
1. Isentropic compression(0 → 1): During this process the incoming fuel-air mix-
ture is compressed isentropically, thereby increasing the total pressure and total
temperature of the gas. Entropy remains unchanged during this process.
2. Constant volume heat addition(1→ 2): This is the stage which distinguishes the
Brayton cycle from the Humphrey cycle. Heat is added at constant volume into the
combustion chamber.
21
Figure 3.1: p− v diagram of a Humphrey Cycle [42]
3. Isentropic expansion(2 → 3): In this phase, there is a reversible and adiabatic
expansion of the burnt gasses where in the stagnation temperature and pressure is
decreased as work is extracted.
4. Insentropic heat rejection(3 → 0): Heat is removed from the engine at constant
pressure and the gas in the combustion chamber exists through the nozzle generat-
ing thrust.
3.1.2 FJ Cycle
The FJ cycle is based on the piston-cylinder arrangement as shown in Fig. 3.2. The
piston and cylinder for this arrangement are considered rigid and with no mass. The
cycle of operation is as follows [36].
(a) Represents the initial state of the reactants in the piston.
(b) Reactants are isentropically compressed.
(c) The velocity up by which the piston is moved initiates a detonation and the front
moves towards to the right. The detonation products which are formed behind the
22
Figure 3.2: Physical steps that make up the Fickett-Jacobs cycle
detonation front move with a velocity up
(d) When the detonation front reaches the right end of the piston, both the pistons
together move with a velocity up.
(e) The mechanical motion of the piston-cylinder arrangement is converted to me-
chanical work by bringing the products of detonation to rest adiabatically and
isentropically.
(f) Products of detonation are isentropically expanded to ambient pressure.
(g) Heat is extracted by cooling the products to ambient temperature.
23
(h) Products are converted to reactants at constant pressure and temperature.
Wintenberger et. al [36] have discussed the thermodynamic states of the FJ cycle as
shown in Fig. 3.3 and also compared the thermal efficiencies of FJ Cycle with Humphrey
Cycle and Brayton Cycle as shown in Fig. 3.4. In Fig. 3.3,
1. Represents initial state of the reactants in the cylinder.
2. Reactants are isentropically compressed with a particular compression ratio.
3. Corresponds to the state where the entire piston-cylinder arrangement moves at a
constant velocity up (state d in Fig. 3.2).
4. The mechanical motion is converted to external work by bringing the detonation
products to rest adiabatically and reversibly.
5. Isentropic expansion to detonation products to intial pressure.
6. Heat is extracted by reversibly cooling the products at constant pressure to the
initial temperature.
Fig. 3.4 compares the thermal efficiencies of FJ Cycle with Humphrey and Brayton
Cycle as a function of compression pressure ratio πc and combustion pressure ratio
π′c. For a specific pressure ratio, it is observed that the FJ cycle has a higher thermal
efficiency followed by Humphrey cycle and the Brayton cycle. When comparing with the
combustion pressure ratio, the results seem to be reversed. The lower efficiency of the
FJ cycle is due to the lower precompression required for the FJ cycle for a given fixed
combustion pressure ratio [36].
24
Figure 3.3: p− v diagram showing the sequence of states and connecting paths that makeup the FJ cycle (with πc = 5) for a stoichiometric propane- air mixture at 300 K and 1 barinitial conditions.
Figure 3.4: Thermal efficiency as a function of compression ratio (left) and combustionpressure ratio (right) for FJ, Humphrey, and Brayton cycles for a stoichiometric propane-air mixture at 300 K and 1 bar initial condition [36]
3.1.3 ZND Cycle
The ZND Cycle accounts for the gas dynamics of the detonation propagation and the
p− v diagram is shown in Fig. 3.5:
25
Figure 3.5: p− v diagram of ZND Cycle [42]
1. (0 → 1): Compression achieved by the compressor to raise the pressure of the
reactants from P0 to P1.
2. (1 → 2): This is the combustion phase where the detonation is initiated. For
the ZND cycle, this stage is divided into 2 parts. (1 → 2a) represents the stage
where the shock front compresses the reactants and increases the pressure to Von
Neumann spike. (2a → 2) is the phase where the chemical reactions take place
and the products of detonation reach CJ condition at 2.
3. (2→ 3): Expansion of products take place and pressure P3 is equal initial pressure
P0.
4. (3→ 0): Heat is removed from the engine at constant pressure and the gas in the
combustion chamber exits through the nozzle generating thrust.
Vutthivithayarak et. al [41] have made performance comparisons between Humphrey,
FJ and ZND Cycle. They concluded that the Humphrey Cycle could not accurately
capture the physics of detonation, FJ Cycle underestimated the work output and finally,
the ZND analysis was considered most appropriate to represent a pulse detonation engine.
The comparison of all the three cycles is shown the T − s diagram in Fig. 3.6.
26
Figure 3.6: Ideal Humphrey (1→ 2H → 3H → 1), FJ(1→ 2CJ → 3CJ → 1) andZND (1 → 1 → 2CJ → 3CJ → 1) cycles for a stoichiometric hydrogen/air mixtureinitially at STP [41].
3.2 Conventional PDE Cycle
Fig. 3.7 shows the operation of a single cycle of the PDE. As in this case, the PDE
consists of a simple shock tube, closed at one end where the fuel-air mixture is introduced
and open at the other for the exhaust gases to expand into the atmosphere. A detailed
working on the PDE cycle is explained in [40] and a brief explanation of the working is
explained below.
1. The initial stage is the purging phase where the burnt products of the previous
cycle of operation are removed from the duct using a blowdown process and the
pressure in the duct is at ambient pressure
1. The initial stage is the purging phase where the burnt products of the previous
cycle of operation are removed from the duct using a blowdown process and the
pressure in the duct is at ambient pressure P0.
2. The fuel-air mixture is introduced through the left end of the tube at a particular
27
Figure 3.7: Working of a conventional PDE [39]
pressure and is let to fill the tube.
3. Once the tube is filled with the fuel-air mixture, an ignition source at the closed
end of the tube ignites the mixture, creating a blast wave thereby increasing the
temperature around the blast radius.
4. The blast wave gains momentum and after traveling a certain distance, the velocity
of the wave, temperature and pressure become apt for it to transit to a detonation
wave which becomes self-sustaining.
5. The detonation wave reaches the end of the tube and downstream of the wave,
there are expansion waves which move towards the closed end of the tube.
6. At this stage, the detonation exits the tube and the expansion wave creates a
low pressure downstream of the detonation wave which causes the products of
combustion to be expelled out of the tube.
7. This process of expelling the combustion products out of the tube, decreases the
pressure in the tube further; also, the expansion wave gets reflected form the closed
end and further pushes away the combustion product residuals out of the tube
which helps in re-filling of the tube for the next cycle.
28
3.3 Multi-Mode Pulsed Detonation based Propulsion Concept
In general, a PDE cycle operation is referred to that which is mentioned in Section 3.2.
A limitation to this case is the range of operation as the air will have to be introduced into
the tube at a low subsonic speed to reduce the total pressure loss in the fuel and oxidizer
valves. This in turn leads to the conversion of kinetic energy to thermal energy resulting
in high increase in static temperature in the flow. An upper limit of flight Mach number
around 4 is placed for this operation at which point the static temperature exceeds the
autoignition temperature of the fuel which leads to deflagration rather than detonation
combustion. Munipalli et. al [16] proposed the multi-mode detonation propulsion system
which uses a single flow path to produce thrust from take-off to hypersonic speeds. A
brief description about this concept was given in Chapter 1. This multi-mode concept
produces thrust at critical parts of the trajectory using both unsteady (Normal Detonation
Wave Engine mode) and steady (Oblique Detonation Wave Engine mode) detonation
waves and also circumvents the above limitation of introducing the fuel-air mixture at
subsonic speed into the tube. A schematic representation of multi-mode pulsed detonation
propulsion system is shown in Fig. 3.8. From take-off to moderate supersonic speeds,
an ejector augmented Pulsed Detonation Rocket (PDR) embedded in a mixing chamber
is used [43]. An ejector PDR produces more thrust than a regular PDR by using the
momentum of a secondary flow.
For the flight Mach numbers ranging from approximately 3-7, an unsteady detonation
wave is used to generate thrust. In this mode, fuel is injected at regular intervals generating
“puffs” of combustible gases which leads to the formation of detonation waves. The
detonation wave propagates upstream in the tube until the fuel concentration is zero, and
once the fuel has been used up completely, the wave recedes. Referring to Fig. 3.8(b)
which is a Normal Detonation Wave Engine mode, the incoming stoichiometric fuel-air
29
Figure 3.8: Multi-mode Pulsed Detonation based propulsion system
mixture is made to impinge on the wedge. Depending on the strength of the shock wave,
detonation gets ignited at a particular location along the wedge. In this case, the CJ
Mach number is greater than the incoming Mach number and the unsteady propagating
detonation wave travels upstream into the oncoming supersonic flow in the detonation
chamber until the detonation can no longer sustain itself. At this point, the wave is pushed
into the nozzle by the incoming air. The residence time of the detonation wave in the
tube determines the frequency at which the engine can be operated. The frequency can
be increased as the combustion chamber Mach number is increased. However, when
combustion chamber Mach number is greater than the CJ Mach number, detonation wave
can no longer propagate upstream and instead, becomes a steady detonation wave which
is setup at the wedge. Munipalli et. al. studied the variation of temperature and Mach
number in the combustion chamber sized as per the LMTAS (Lockheed Martin Tactical
Aircraft Systems) data and they found that the temperature is nominal for this mode of
operation while the Mach number seemed to have an upper limit of 6 which makes the
30
NDWE mode feasible for cruise conditions at Mach 5.5- 6. However, this mode will
have to be operated at higher altitudes because the combustion chamber Mach number
will have to be less than the CJ Mach number. Combustion chamber pressures between
0.5-1 atm were deemed feasible for the successful initiation of detonation. Another issue
addressed is the efficient mixing of fuel and air in the combustion chamber.
As the injection pressure of the fuel is often low when compared to the supersonic
stream of air, the mixing is inefficient. In order to produce a shear layer region which
generates vortices to enhance the mixing of fuel and air, the intake is split into a bypass
stream and this travels at a different velocity than the primary flow generating a vortex
field as shown in Fig. 3.9. As the fuel fraction is quiet small, the perturbations created
will entrap the fuel and the vorticity gradients help in efficient mixing of the fuel [16].
Figure 3.9: Bypass stream to control combustion properties
For a particular flight Mach number above Mach 7, the incoming Mach number in
the detonation chamber, which is denoted by M1 in Fig. 3.8(c), will be equal to the CJ
Mach number and in this case, there will be a standing detonation wave which will be
formed at a particular location upstream of the wedge. The cycle operation will be steady
in this case. When the incoming Mach number further increases, the detonation wave is
pushed towards the wedge and gets stabilized as an attached oblique detonation wave.
31
This cycle of operation, Mode 3 as shown in Fig. 3.8, will also be steady. For the wave to
be stabilized at the wedge, there are two pre-requisites. One, the fuel-air mixture is near
the stoichiometric ratio and two, the condition for shock instability of shock waves must
be satisfied by one of the reflected shocks at the design ramp angle and Mach number.
The instability condition is given by [16]
q
CpT>
(M2n1 − 1)2
2(γ + 1)M2n1
(3.1)
Where q is the heat release during chemical reaction, Cp is the specific heat at constant
pressure, γ is the ratio of specific heats of the mixture, T is the incoming flow static
temperature and Mn1 is the incoming Mach number normal to the shock. This mode of
operation is also limited to a restricted range of flight Mach numbers.
At high altitudes, in the upper layers of the atmosphere, a pure Pulsed Detonation
Rocket mode is used to generate thrust as shown in Fig. 3.8(d). To extend the operation
of a PDE to hypersonic speeds, the successful operation of mode 2 and 3 is very critical.
As mentioned earlier in this section, thrust is generated at critical parts of the trajectory in
mode 2 and also the transition to mode 3 occurs naturally when the detonation chamber
Mach number exceeds CJ Mach number for the fuel-air mixture. Due to this reason, this
research focuses only on the analysis of mode 2 and 3. However, the previous study
done so far in the multi-mode detonation concept focused mainly on the flow within the
detonation chamber and ignored the unsteady flow through the aft nozzle.
3.4 Literature Review
Although detonation as a phenomenon was discovered in the first decade of the 19th
century, there have been many stumbling blocks to harness the energy of detonation
32
for propulsion applications. In the early 1940’s, Hoffman [44] worked on developing
the concept of the Pulsed Detonation Engine and succeeded in achieving intermittent
detonations. Some of the early research works on Pulsed Detonation Engine are compiled
in [12] [45]. Supersonic combustion being an unsteady phenomenon, Roy’s proposal [46]
helped in furthering research in designing systems to stabilize combustion in supersonic
flows. Nicholls et. al [47] and Gross[48][49] have studied means to stabilize detonation
waves so that they could be used for propulsion applications. Work was also carried
out by Nicholls[50] for using intermittent or pulsed detonation waves for propulsion
applications.
Performance studies on Pulsed Detonation Engine ejectors were carried out by
Allgood et. al [51] who found that the thrust augmentation increased as the ejector length
increased and also the ejector performance was strongly dependent on the operating fill
fraction.The propagation of a detonation wave in an ejector-augmented pulse detonation
rocket fueled with hydrogen-oxygen mixture was studied for the multi-mode detonation
engine concept discussed in Section 3.3 by Yi et. al[43]. The interaction between a
primary flow from a pulse detonation rocket embedded in a mixing chamber and an
incoming secondary flow from an inlet was studied at different incoming Mach numbers,
however, no attempt was made to study the performance of the ejector-augmented
pulsed detonation rocket. An experimental study was performed to understand the
performance of ejectors for multi-cycle airbreathing PDE by Changxin et. al [52]. The
effect of ejector length and axial location of the ejector on thrust augmentation were
also investigated by them. Influence of an ejector nozzle extension on gas flow in
a PDE was investigated numerically and experimentally by Korobov et. al [53]. A
cylindrical ejector was constructed and mounted at the open end of the tube. Thrust, air
consumption, and detonation parameters were measured in single and multiple regimes
of operation and the results were verified numerically. Wilson et. al [54] studied
33
the initiation of detonation waves and wave propagation in a normal detonation wave
engine, and from their preliminary analysis showed that thrust and specific impulse are
comparable or in some cases superior to the existing RBCC engines. Li et. al [55]
investigated the Pulsed Normal Detonation Wave Engine (PNDWE) from the viewpoint
of entropy generation associated with combustion. For a given flight condition and
a given amount of heat release, they found that PNDWE is superior in theory to the
conventional subsonic combustion ramjet and scramjet. Kim et. al [56] developed a
numerical model to simulate hydrogen-air detonation wave propagation. Calculations
were performed so that a scheme with adequate temporal and spatial resolution for
modeling the physical process is selected and the calculations were compared against
CJ theory and experiment. Geometric independence of detonation wave properties was
also confirmed as a validation process of the present numerical model. Qu et. al [57]
performed a two dimensional numerical simulation to study the interaction of a gaseous
detonation wave with obliquely inclined surface. They used a weighted essentially non-
oscillatory (WENO) numerical scheme with a relatively low resolution grid. The results
showed there existed a transition region where in the initial detonation cells become
distorted and irregular before they re-obtained their regularity as the detonation wave
propagated through the converging/diverging chamber. Fan [58] numerically simulated a
detonation process occurring in a combustion chamber with variable cross-sections. Two
cases was simulated, one in which the detonation was initiated at the closed left end and
another at the open, right end. The study showed that area change gave rise to complex
wave phenomenon. The area change and wave reflections produced extreme parameters
of thrust and impulse. Wedge induced detonations were studied by Papalexandris [59] and
Walter [60]. Walter studied the interaction between the leading oblique detonation wave
stabilized by a ramp of finite length and the expansion waves generated by the sudden
deflection of the wedge surfaces. Lu et. al [61] studied the auto-ignition detonation
34
phenomenon induced by a confined wedge in a channel. The results showed that within
certain ranges of incoming flow Mach number or wedge angle, detonation could be
self-ignited in the channel and then further investigation was carried out on detonation
waves based on three different detonation initiation positions. Honghui [62] performed
a parametric study to analyze the effect of inflow pressure and Mach number on the
initiation structure and length in a wedge induced oblique detonation wave. The results
demonstrated that the transition patterns depended strongly on the incoming Mach number
while the inflow pressure had little effect on oblique shock to detonation transition. Fan
et. al [63] performed a computational study to understand the gasdynamics of wedge-
induced oblique shock and detonation wave phenomena in the flow of a combustible
mixture. The simulations were performed with wedges upto 200 semi angle and Mach
numbers ranging from 1.25-6, with other inflow parameters fixed. They developed a
matrix of test cases with different incoming Mach number and wedge angle. From the
computational domain, four flow modes were obtained namely, a propagating detonation
wave, a standing detonation wave, a propagating shock wave, and a standing shock wave
mode. They investigated the detonation modes further and found that the detonation
wave modes can be further subdivided on where the detonation is initiated [64]. Also
for a narrow range of Mach numbers for the 50 wedge, a case without combustion was
discovered. This matrix of test cases and the results provided by Fan, will be the basis of
choosing the incoming conditions and geometry for this research.
35
Chapter 4
Mathematical Model Development for
Wedge Induced Detonation
4.1 Mathematical Formulation
The following section describes the equations used to simulate flow mixture. The
laws that govern any fluid flow obey the conservation principles of mass, momentum and
energy. The assumptions made in the governing equations are
i) No body forces such as gravity and electromagnetic forces
ii) No wall shear (Inviscid flow)
iii) No heat transfer (adiabatic flow)
Fluids are composed of molecules which are discrete in nature and spread out over the
entire fluid domain. However, while mathematically modelling certain fluid phenomenon,
it can be assumed that the molecules in a medium are continuously distributed across
the entire domain. For fluids, the Knudsen number, which is the ratio of the molecular
mean free path length to a representative physical length scale, is used to asses as to what
36
extent the approximation of continuity can be made. In order to fit in large number of
molecules into the computational domain, the Knudsen number should be less than unity.
When the intermolecular forces of a gas are neglected, which is usually in the range
of temperatures between 1000K and 2500K, the internal energy, enthalpy and also the
specific heats at constant pressure and volume become a function of temperature only.
Such kind of fluids are called thermally perfect gasses.
For accurate description of the detonation phenomenon, it is important to include a
chemical kinetic model which can be used to predict as accurately as possible, the species
and mixture properties and also the chemical composition of the products of detonation.
4.1.1 Governing Equations
The governing equations for a two dimensional, unsteady, compressible flow with
chemically reacting gas mixture are described below in Cartesian coordinates.
∂U
∂t+∂F
∂x+∂G
∂x= S (4.1)
where U is a vector for conserved variables, F and G are convective fluxes and S is the
source term vector.
U =
ρs
ρu
ρv
ρE
ρYs
, F =
ρsu
ρu2 + P
ρuv
(ρE + p)u
ρuYs
, G =
ρsv
ρuv
ρv2 + P
(ρE + p)u
ρuYs
, S =
0
0
0
0
Rs
(4.2)
Subscript s=1,2,3,....,Ns where Ns is the number of species. The first Ns rows represent
species continuity, followed by the next two rows of momentum equations. The terms u
37
and v represent the x and the y velocity components respectively, ρ =∑Ns
s=1 ρs represents
the mixture density, ρs is the species density, E is the total energy per unit mass of the
mixture and Rs is the net rate of production of species due to chemical reactions and Ys
is the mass fraction species s.
Ys =ρsρ
(4.3)
Total energy E is related to total enthalpy H by
E = H − p
ρ(4.4)
The first row in Eq. (4.2) represents the continuity equation for the fluid flow, the next
two rows represent the momentum equations in the x and y direction respectively. The
third row represents the energy equation and the last row represents species continuity
equations for all chemical species in the mixture.
4.1.2 Thermodynamic Properties
Since a thermally perfect gas is assumed for each species,
e = e(T ) (4.5)
h = h(T ) (4.6)
dh = cpdT (4.7)
de = cvdT (4.8)
The enthalpy and internal energy of each species is given by
38
hs(T ) = hfs (Tref ) +
∫ T
Tref
cps(τ)dτ (4.9)
es(T ) = hfs (Tref ) +
∫ T
Tref
cvs(τ)dτ (4.10)
where hfs (Tref ) is the standard enthalpy of formation of species s at the reference tem-
perature Tref which is the change of enthalpy during the formation of 1 mole of the
substance in their standard states from its constituent elements. The heat of formation
of all the elements in their standard state is zero as there is no change involved in their
formation.
The empirical equations [87] that calculate heat capacity, enthalpy and entropy are
Heat capacity of species s,
CpsR
= a1 + a2T + a3T2 + a4T
3 + a5T4 (4.11)
Enthalpy of species s,
Hs
RT= a1 + a2
T
2+ a3
T 2
3+ a4
T 3
4+ a5
T 4
5(4.12)
Entropy of species s,
SsR
= a1lnT + a2T + a3T 2
3+ a4
T 3
4+ a5
T 4
5(4.13)
For the specific heats, enthalpy and internal energy of the mixture,
cp(T ) =Ns∑s=1
Yscps(T ) (4.14)
39
cv(T ) =Ns∑s=1
Yscvs(T ) (4.15)
h(T ) =Ns∑s=1
Yshs(T ) (4.16)
e(T ) =Ns∑s=1
Yses(T ) (4.17)
4.1.3 Chemical Kinetics
The time scales of chemical reactions involved and the fluid flow plays an important
role in modeling the physics of the flow. If the reaction time of the chemical mechanism
is large when compared to the characteristic flow time, reactions cannot occur and the
flow is assumed frozen. When the reaction time is much faster than the fluid dynamic
time, an equilibrium state is reached. However, when the time scales of both the reaction
mechanism and the fluid dynamic time scale is of the same order, a finite rate chemistry
model is taken into consideration, which is what is considered in this section.
To accurately model a detonation phenomenon where rapid chemical processes occur
at the detonation front, the conservation equations are coupled with a chemical kinetic
model. As a result of this, the mass production rate of the species in Eq. (4.2), can be
accurately determined.
The chemical reaction mechanism considered here is the stoichiometric hydrogen -
air mixture which is expressed as
2H2 + (O2 + 3.762N2)→ 2H2O + 3.762N2 (4.18)
The above equation represents the overall single step mechanism where in hydrogen is
reacted with air to produce water vapor and nitrogen. However, in actuality, the reaction
40
of products goes through a series of elementary reactions. The chemical source terms in
Eq. (4.2) is computed using Arrhenius expressions.
The Rs term is computed as
Rs = Mw,s
NR∑r=1
Rs,r (4.19)
where Mw,s is the molecular weight of species s, NR is the number of reactions and Rs,r
is the Arrhenius molar rate of production of species s in reaction r.
Rs,r = Γ(v′′
s,r − v′s,r)(kf,rNs∏j=1
[Cj,r]η′j,r)− (kb,r
Ns∏j=1
[Cj,r]v′′j,r) (4.20)
Cj,r is the molar concentration of species j in reaction r (kmol/m3), v′s,r and v′′s,r are the
stoichiometric coefficients of the reactant and product s respectively in reaction r and η′j,r,
the rate exponent of the reactant species j in reaction r.
General form of the rth reaction is:
Ns∑s=1
v′s,rMs
kf,r
kb,r
Ns∑s=1
v′′
s,rMs (4.21)
where Ms denotes species s.
Forward rate constant is computed using the Arrhenius expression
kf,r = ArTβre−
ErRT (4.22)
Ar is the pre-exponential factor, βr is temperature exponent, Er (J/kmol) is the activation
energy for the reaction and R (J/kmol-K) the universal gas constant.
41
Backward rate constant similarly is determined by
kb,r =kf,rKr
(4.23)
where Kr is the equilibrium constant of the rth reaction.
Third body reactions are taken into consideration in the chemical mechanism. Third
body reactions involve two species A and B to yield a product AB with the help of
a third body M. The third body in a reaction appears when there is a recombination
or dissociation happening in the reactions [88] [89]. In a recombination process, the
third body usually carries the excess energy released from the reaction and eventually
dissipates it as heat. In a dissociation mechanism, M provides the energy for the splitting
of the molecules.
The general form of the molar concentration of M is given by
Xs =Ns∑i=1
αs,r[Ms] (4.24)
where the αs,r is the third body efficiency.
In the present study, 11 species and 23 step hydrogen - air reaction mechanism [90]
is used for the simulation. The reaction mechanism is given below in Table (4.1)
4.2 Numerical Formulation
Numerical method used to solve differential equations presented in the previous
chapter are discussed in this section. Chosen numerical approach plays an important
role in determining accuracy and stability of the simulation because in a reacting flow
mixture, a wide range of time and length scales exist in the flow and usually they lead to
numerical stiffness [91].
42
REACTION MECHANISM Amole-cm-sec-K
b Ecal/mole
H2 +O2 OH +OH 1.70E+13 0 47780OH +H2 H2O +H 1.17E+09 1.3 3626O +OH O2 +H 4.00E+14 -0.5 0O +H2 OH +H 5.06E+04 2.67 6290H +O2 +M HO2 +MH2O enhanced by 18.6H2 enhanced by 2.86N2 enhanced by 1.26
3.61E+17 -0.72 0
OH +HO2 H2O +O2 7.50E+12 0 0H +HO2 OH +OH 1.40E+14 0 1073O +HO2 O2 +OH 1.40E+13 0 1073OH +OH O +H2O 6.09E+08 1.3 0H +H +M H2 +MH2O enhanced by 0.0H2 enhanced by 0.0
1.00E+18 -1.0 0
H +H +H2 H2 +H2 9.20E+16 -0.6 0H +H +H2O H2 +H2O 6.00E+19 -1.25 0H +OH +M H2O +MH2O enhanced by 5.0
1.60E+22 -2 0
H +O +M OH +MH2O enhanced by 5.0
6.20E+16 -0.6 0
O +O +M O2 +M 1.89E+13 0 -1788H +HO2 H2 +O2 1.25E+13 0 0HO2 +HO2 H2O2 +O2 2.00E+12 0 0H2O2+M OH+OH+M 1.30E+17 0 45500H2O2 +H HO2 +H2 1.60E+12 0 3800H2O2 +OH H2O+HO2 1.00E+13 0 1800O +N2 NO +N 1.40E+14 0 75800N +O2 NO +O 6.40E+09 1 6280OH +N NO +H 4.00E+13 0 0
Table 4.1: Hydrogen-air reaction model
4.2.1 Finite Volume Formulation
In a finite volume method, the terms in the governing equations are evaluated as
fluxes at the surface of each control volume. Also, the volume integral in the differential
equation that contain divergence terms are converted to surface integrals using the
43
divergence theorem. An important aspect and a convenience in using this approach is
the use of integral form of the equations which ensures conservation as the flux entering
a given control volume is identical to that leaving the adjacent volume and also correct
treatment of the discontinuities [92].
Considering a 2 dimensional control volume A as shown in Figure 4.1
Figure 4.1: Control volume with faces ab, bc, cd and da
The governing equations in integral form are represented as
∂
∂t
∫ΩA
UdΩ +
∫δΩA
~H.~nds =
∫ΩA
SdΩ (4.25)
where ΩA is the interior and δΩA is the boundary of control volume A, ds is an infinitesi-
mal area where ~n is normal and points outward. Also,
~H = (F,G)
The cell volume average is defined as
< U >=1
Ω
∫ΩA
UdΩ, < S >=
∫ΩA
SdΩ (4.26)
44
Substituting Eq. (4.26) in Eq. (4.25), we get
∂
∂t(< U > Ω) +
∫δΩA
(F,G).~nds =< S > Ω (4.27)
In this approach, a cell centered formulation is used, which means the flux is calcu-
lated at the center of the cell. The contribution of the flux is in all the four directions
which is ab, bc, cd and da as shown in Figure 4.1. If the center of the cell is assumed to
have indices (i, j), then,
∫δΩA
(F,G).~nds =
∫δΩAi+1/2
Fds+
∫δΩAi−1/2
Fds+
∫δΩAj+1/2
Gds+
∫δΩAj−1/2
Gds
(4.28)
The area average is defined as,
< F >i+ 12=
1
δΩAi+1
2
∫δΩAi+1/2
Fds (4.29)
< G >j+ 12=
1
δΩAj+1
2
∫δΩAj+1/2
Gds (4.30)
where δΩAi+1/2and δΩAj+1/2
represent cell face lengths.
Substituting Eqs. (4.29) and (4.30) into Eq. (4.28), the general conservation equation
in two dimensional coordinates is given by:
∂Ui,j∂t
= −
(Fi+ 1
2
δΩAi+1
2
ΩAi,j
− Fi− 12
δΩAi− 1
2
ΩAi,j
)−
(Gj+ 1
2
δΩAj+1
2
ΩAi,j
−Gj− 12
δΩAj− 1
2
ΩAi,j
)+Si,j
(4.31)
45
4.2.2 Density Based Solver
To solve the governing equations and to simulate the flow, a commercial software
called ANSYS Fluent [93] is used. There are 2 types of solvers in this software to solve
for the flow field properties.
a) Pressure based solver
b) Density based solver
In pressure based solver approach, the pressure field is extracted by solving a pres-
sure or pressure correction equation which is obtained by manipulating continuity and
momentum equations whereas in density based approach, the continuity equation is
used to obtain the density field while the pressure is determined from the equation of
state. Traditionally, a density based approach is used for high speed compressible flow
and a pressure based approach for incompressible flow. However, recently because of
improvements of numerical algorithms, both the methods have been used to solve a wide
range of flow conditions beyond what it was traditionally built for. In this simulation, a
density based approach is used to solve for the flow parameters.
The density based approach uses a control volume technique and the summary of the
solution procedure is as follows:
a) Update the fluid properties based on the current solution.
b) Solve the continuity, momentum, energy and species transport equations simulta-
neously.
c) Updating the source terms in the appropriate continuous phase equations with a
discrete phase trajectory calculation.
d) Check for convergence.
46
In order to linearize the governing equations, either an implicit or explicit approach
can be used and is applicable only to a coupled set of governing equations. In the present
simulation, an implicit approach is chosen. In this approach, the governing equations are
linearized implicitly with respect to all the dependent variables which results in a system
of linear equations with N equations for each cell in the domain, where N is the number
of the coupled equations in the set. A point implicit linear equation solver is used along
with an Algebraic MultiGrid (AMG) method to solve the resultant system of equations.
This approach solves for all the variables (p, u, v, T ) in all the cells at the same time.
4.2.3 Discretization Schemes
An analytical solution of Euler equations give a closed form solution for the flow
parameters p, u, v, ρ, etc., as a function of x, y and t which implies that solution can be
found at any of the infinite number of points in the domain. When the partial differential
equations are converted to an algebraic system of equations which can be solved at
discrete points on the domain, then equations are said to be discretized. In CFD, various
discretization schemes are used to solve for the flow variables. The equations will have
to be discretized in both space and time.
4.2.3.1 Spacial Discretization
Since the finite volume method is chosen and the domain is discretized into discrete
control volumes, the flux quantities are calculated at the cell (each discrete volume)
boundaries. The solution is known and stored only at certain points, called the cell centers
and from the cell centers the information has to extrapolated at the cell boundaries. For
the supersonic flow, information travels from upstream to downstream as in the case of
solution of hyperbolic equations. So, the calculation of the flux should be such that, only
the values from the cell center in the upstream direction must be taken into account for
47
the calculation of the flux at a given cell boundary. The numerical algorithm must also
take into account the accurate discretization of the shock wave present in the flow field.
4.2.3.2 Monotonic Upwind Scheme for Conservation Laws (MUSCL) Scheme
The MUSCL scheme is a finite volume method that can provide highly accurate
numerical solutions for a given system, even in cases where the solutions exhibit shocks,
discontinuities or large gradients. MUSCL solves the Lagrange equations, which are
Euler equations expressed in a coordinate system fixed to the fluid [94]. The third order
MUSCL approach is obtained by the blend of a central difference scheme and second
order upwind scheme which is shown below:
φf = θφf,CD + (1− θ)φf,SOU (4.32)
where θ is a value implemeted by ANSYS Fluent such as to avoid introducing a new
solution extrema [95] and φ is a scalar quantity.
φf,CD =1
2(φ0 + φ1) +
1
2(Oφ0.~r0 + Oφ1.~r1) (4.33)
where the indices 0 and 1 refer to the cells that share the face f , Oφr,0 and Oφr,1 are the
reconstructed gradients at cells 0 and 1 respectively and ~r is the vector directed from the
cell centroid toward the face centroid.
Using the second order upwind scheme, the following face value is obtained:
φf,SOU = φ+ Oφ.~r (4.34)
where φ and Oφ are the cell-centered value and its gradient in the upstream cell.
48
4.2.3.3 Temporal Discretization
When the problem to be solved is unsteady, the discretization of the governing
equations will have to be done in both space and time. Temporal discretization involves
the integration of every term in the differential equations over a particular time step.
The general form of the time dependent variable φ is given by
∂φ
∂t= F (φ) (4.35)
A second order discretization is given by
3φn+1 − 4φn + φn−1
24t= F (φ) (4.36)
An implicit time integration method is used to evaluate F (φ) at a future time level as
follows:
(φn+1 − φn)/4t = F (φn+1) (4.37)
From Eq. (4.37), φn+1 in a given cell is related to φn+1 in the neighboring cells
through F (φn+1).
φn+1 = φn +4tF (φn+1) (4.38)
4.2.4 Evaluation of Gradients
For the evaluation of gradients for structured grids, the gradient of the scalar φ can
be easily computed using the definition of the derivatives. Green Gauss theorem is used
which states that the surface integral of a scalar function is equal to the volume integral
49
of the gradient of the scalar function.
∫Ω
OφdΩ =
∫S
φndS (4.39)
where n is the surface normal pointing out from the volume. In discretized form,
OφC =1
Ω
∑f
φf ~Af (4.40)
where φf is the value of φ at the cell face centroid and φc is the value of the scalar at the
cell center.
φf is calculated using a cell based approach which is the arithmetic average of the
values of the neighboring cell centers.
φf =φc + φc1
2(4.41)
4.2.5 Convective Fluxes
Advection Upwind Splitting Method (AUSM) is used to split the flux into two separate
components so that each can be properly stenciled. The two split components are the
convective flux and pressure flux.
The AUSM scheme first computes a cell interface Mach number based on the charac-
teristic speeds from the neighboring cells. The interface Mach number is then used to
determine the upwind extrapolation for the convection part.
F = mfφ+ pi (4.42)
where mf is the mass flux through the interface, which is computed using the fourth
50
order polynomial functions of the left and right side Mach numbers. The Mach number
determined is a measure of the convective potential of the flow. Advantages of AUSM
scheme are
a) Accurate capturing of shock and discontinuities.
b) Uniform accuracy and convergence rate for all Mach numbers
c) Algorithmic simplicity
d) Free of oscillations at stationary and moving shocks.
e) Preserves positivity of scalar quantities.
51
Chapter 5
Simulation of Wedge Induced
Detonation
5.1 Shock capturing capability of FLUENT
This section tests the capability of the numerical solver in FLUENT to capture shocks.
SOD shock tube problem is used to test the accuracy of a computational code and its
ability to capture shock waves [97]. The analytical solution for this type of problem is
known and a numerical method can be verified by comparing the results to the analytical
solution of the SOD problem and get information as to how well the code captures shocks
and discontinuities.
The SOD shock tube problem consists of a rectangular tube that is closed at both
ends as shown in Fig. 5.1. It is divided into two sections by a thin diaphragm and filled
with the same gas but different thermodynamic properties in each section. The initial
conditions of the problem are given in Table 5.1.
52
Figure 5.1: Schematic representation of initial conditions of shock tube
Region 4 in Fig. 5.1, is called the driven section and region 1 the driver section. When
the diaphragm is broken, a shock wave is formed and it starts to propagate towards the
right end of the tube (lower pressure region) and in turn the expansion waves propagate
towards the left end of the tube. As the shock wave moves to the right, it increases the
pressure of the gas behind it and similarly, the expansion waves while propagating to the
left, smoothly and continuously decrease the pressure behind the expansion wave.
Region Pressure (Pa) Temperature (K) Density (Kg/m3)4 1 348.432 1.0001 0.1 278.746 0.125
Table 5.1: Initial conditions for SOD Shock tube problem
The geometry considered to test the numerical solver of FLUENT is a rectangular tube
of dimensions 100mm X 14.5mm. A grid size of 0.05mm with a time step of 10−5sec
is used. Fig. 5.2 and Fig. 5.3 show the variation of pressure and density along the x-axis.
Comparison is made with analytical result at t = 0.2 sec which is indicated by the black
line and the numerical simulation is the yellow line in the figure. It is observed that the
numerical result almost overlaps the analytical result with a slight variation along the
location of the shock wave. This is because of the numerical errors. In the present study,
a uniform and structured mesh was used. To get an accurate solution for the location of
the shock wave, an adaptive grid can be used.
53
Figure 5.2: Variation of pressure along the length of tube at t = 0.2sec
Figure 5.3: Variation of density along the length of tube at t = 0.2sec
54
However, the intention of this simulation was only to verify if FLUENT could capture
the location of the shock wave at a fairly accurate value. The simulations show that there
is a good degree of accuracy of the numerical simulation with the analytical results.
5.2 Capturing Detonation Phenomenon in FLUENT
This section focuses on the ability of the numerical solver of FLUENT to capture
important detonation characteristics. A computational grid of 100mm X 15mm is con-
sidered with both ends closed as shown in Fig. 5.4. The entire domain is filled with
stoichiometric hydrogen-air mixture at pressure and temperature of 1 atm and 700 K.
To initiate the detonation, a high enthalpy patch of pressure 40 atm and a temperature
of 4000 K is used. The values at the patched region are chosen such that the energy
release at the inlet is high enough for a detonation to be initiated. To capture the global
characteristics of the detonation phenomenon, the simulation is carried out with various
grid sizes of 0.8 mm, 0.09 mm, 0.07 mm and 0.03 mm. Fig. 5.5 and Fig. 5.6 show the
variation of pressure and temperature respectively along the length of the tube at a time
instant of 0.45ms. As the mesh gets finer, the properties move closer to the CJ value
which can be determined from NASA CEA [98]. It can be observed from Fig. 5.5 that
as the grid size gets smaller, the Von-Neummann spike gets steeper. Grid spacing of
0.03mm captures the Von-Neumann spike accurately and hence, it is used for further
simulations.
Fig. 5.8 shows the pressure contour of detonation propagation. From the figure, the
detonation front and fish scale patterns can be clearly seen. As the detonation propagates
along the length of the tube, it can be observed that towards the left end of the tube, the
fish scale patterns disintegrate. This is because they do not have enough energy to sustain
the fish scale pattern.
55
Figure 5.4: Computational domain to capture detonation phenomenon
Figure 5.5: Variation of static pressure along the length of the tube for different grid sizes
Fig. 5.7 shows the temperature scale of the detonation propagation at a time instant
of 0.45 ms. Here, the disintegration of the fish scale pattern can be clearly observed.
The temperature at this region is much lower compared to the detonation front which
implies because of significant heat release during the detonation propagation, the entropy
is higher in this region. Coming back to Fig 5.5 and 5.6, the reason for the unsteadiness
observed downstream of the CJ spike can be seen in Fig. 5.7.
56
Figure 5.6: Variation of static temperature along the length of the tube for different gridsizes
Concluding with the plots and simulation contours in this section, FLUENT is able
to capture the global characteristics of the detonation phenomenon and hence, will be
used in the simulation of Oblique Detonation wave mode.
Figure 5.7: Temperature scale of detonation propagation with a grid size of 0.03mm
57
Figure 5.8: Detonation propagation with a grid size of 0.03mm
5.3 Chemistry Solver
Modeling the chemistry plays a very important role in determining the accurate results
concerning a detonation phenomenon. The very definition of detonation wave comprises
of a shock wave coupled with a layer of reaction zone where exothermic reactions
take place. The chemical kinetics involved in describing a detonation phenomenon is
illustrated in detail in section 4.1.3.
A chemistry package called CHEMKED [99] which was designed for processing
thermodynamic and chemical kinetics data and solving problems of complex gas-phase
chemistry, was used to model the chemical equations. In order to verify the results from
CHEMKED, the rate of change of mole fraction was compared to that obtained by Yi et.
al [100].
58
Figure 5.9: Variation of pressure with time in H2-air reaction mechanism
The initial conditions to solve the chemistry using CHEMKED are a pressure of 1
atm and temperature of 1500 K, initial mole fractions corresponding to the stoichiometric
hydrogen-air reaction which is hydrogen of 0.296, oxygen of 0.148 and nitrogen of 0.556
at static condition. The hydrogen-air reaction mechanism which is tabulated in table
4.1 is used. From Fig. 5.9 and 5.10, it can be seen that around 0.1ms, the reactants
approach to an equilibrium state. The temperature and pressure become constant after
the chemical reactions take place at about 0.1 ms. The same phenomenon can also be
observed from Fig. 5.11, where the rate of mole fractions of the species involved in the
chemical reaction becomes constant after 0.1 ms, indicating the termination of chemical
reactions and reaching an equilibrium state. On verifying the chemistry package, the
CHEMKED chemistry solver will be incorporated into FLUENT for the simulation of
oblique detonation wave mode.
59
Figure 5.10: Variation of temperature with time in H2-air reaction mechanism
Figure 5.11: Rate of change of mole fractions of species in H2-air reaction mechanism
60
5.4 Geometry
To simulate the flow through the detonation chamber, the geometry considered is a two
dimensional rectangular tube followed by a wedge at a particular angle with respect to the
horizontal, along which the incoming supersonic flow is made to impinge leading to the
formation of detonation waves. Downstream of the wedge is a shock cancellation region
where the expansion waves resulting from the detonation phenomenon is propagated
into the exhaust nozzle. A schematic of the geometry used as a computational domain is
shown in Fig. 5.12.
Figure 5.12: Schematic of the computational domain for ODWE
The dimensions in Fig. 5.12 are referenced to the inlet height “h” and the direction of
propagation of the flow is indicated by the arrow.
Referring to the section 3.4, the combination of incoming Mach number and wedge
angle to generate different phenomenon of propagating and standing detonation waves
was studied by Fan et. al [64]. For ease of understanding, the matrix of test cases is
presented in Fig. 5.13. Different domains have been marked depending on the kind of
phenomenon taking place.
61
Figure 5.13: Matrix of test cases for the combination of incoming Mach number andwedge angle [64]
• Type 1 : Propagating detonation wave/shock wave mode
• Type 1’ : Propagating detonation wave/shock wave mode with wedge tip initiation
• Type 2 : No combustion mode
• Type 3 : Standing detonation wave/shock wave mode
• Type 3’: Standing detonation wave/shock wave mode with wedge tip initiation
Keeping the wedge angle fixed and varying the incoming Mach number, either the
unsteady Normal Detonation wave mode or the Oblique Detonation wave mode can be
obtained. For the oblique detonation wave mode, the incoming Mach number is fixed at
6 and wedge angle to 20 degree.
62
5.5 Oblique Detonation Wave Engine Mode
Oblique detonation mode being a steady process, the time derivative in the governing
Eq. (4.1) becomes zero. To numerically simulate this mode, a grid study is performed
with the geometry as shown in Fig. 5.12. The detonation is initiated at the wedge and the
physics of the problem has to be resolved at the wedge to get accurate results. If the grid is
too coarse at the wedge, the detonation properties along the shock do not approximate to
the CJ detonation value. For the simulations of ODWE mode, the incoming Mach number,
temperature and pressure are 6, 700K and 101325 Pa respectively with a wedge angle
of 20 degree. Simulations are carried out for various grid sizes of 0.01mm, 0.008mm,
0.002mm and 0.0002mm with a time step of 10−9sec as shown in Fig. 5.14.
Figure 5.14: Variation of static pressure for different grid sizes
Fig. 5.14 shows the variation of static pressure for different grid sizes. The wave
profiles of the grid size 0.002mm and 0.0002mm almost overlap and hence, a grid size
of 0.002mm is chosen for the simulation. Considering the geometry of Fig. 5.4 with a
63
wedge angle of 20 degree, incoming Mach number, pressure and temperature fixed at 6,
1 atm and 700 K respectively and initial mole fractions of CH2 = 0.296, CO2 = 0.148
and CN2 = 0.556, numerical simulation is carried in FLUENT. The simulation is carried
so that the exit conditions at the end of the expansion section is determined.
Fig. 5.15, 5.16 and 5.17 show the pressure, temperature and velocity contour respec-
tively of the simulation. The incoming supersonic fuel-air mixture impinges on to the
wedge which results in the formation of a shock wave at the bottom of the wedge. An
expansion fan is also observed at the tip of the wedge to compensate for the pressure
differences at that region. The shock wave which is initiated at the bottom of the wedge,
is reflected from the top surface and multiple such reflections take place along the straight
end of the downstream portion of the detonation chamber.
Fig. 5.18 shows the variation of heat of reaction along the length of the detonation
chamber. Heat of reaction represents the change in enthalpy of the chemical reaction
and it also helps in calculating the amount of energy either released or produced in a
chemical reaction.
Figure 5.15: Pressure contour of ODWE mode
64
Figure 5.16: Temperature contour of ODWE mode
Figure 5.17: Velocity contour of ODWE mode
65
Figure 5.18: Change in Heat of reaction for ODWE mode simulation
To complement the variation of heat of reaction plot, Fig. 5.19 shows the variation of
mole fraction of species. Observing Fig. 5.18, there is a spike in the change of enthalpy at
about 1.4 mm along the length of the tube which is the same location at which shock wave
is initiated by the wedge. Also from Fig. 5.19, it is evident that the change in species
concentration occur at around the location 1.4 mm along the length of the tube. This
shows that the shock wave at the wedge is indeed a detonation wave and the detonation
properties match with the CJ values which are got from NASA CEA. Fig. 5.20 and Fig.
5.21 show the variation of static pressure and temperature respectively along the length
of the detonation chamber. At around the same location where there is a spike in heat
of reaction, there is also a first pressure and temperature spike which is the result of a
shock induced detonation. The rest of the spikes represent the shock reflections along the
straight end of the detonation chamber. Since the oblique detonation mode is a steady
process, the conditions at the exit are also constant and the exit conditions are tabulated
in Table 5.2.
66
Figure 5.19: Change of mole fraction of different species
Figure 5.20: Variation of static pressure along the length of the detonation chamber
67
Pressure (Pa) 301284.41
Temperature (K) 2064.63
Density (Kg/m3) 0.34
Mach number 4.12
Mass flow rate (Kg/sec) 0.6748
Table 5.2: Exit conditions at the detonation chamber
Figure 5.21: Variation of static temperature along the length of the detonation chamber
68
5.6 Normal Detonation Wave Engine mode
In an unsteady normal detonation wave engine mode, the incoming detonation cham-
ber Mach number plays a significant role in the propagation of detonation wave. Three
possible scenarios are possible depending on the incoming detonation chamber Mach
number for a given stoichiometric condition of fuel-air mixture.
• If the incoming detonation chamber Mach number if less than the CJ Mach number,
the detonation wave tends to move upstream.
• If the incoming detonation chamber Mach number is equal to the CJ Mach num-
ber, there will a standing detonation wave formed at a particular location in the
detonation chamber.
• If the incoming detonation chamber Mach number is greater than the CJ Mach
number, detonation tends to move downstream.
The strength of the detonation wave depends on the incoming detonation chamber Mach
number, and for the propagation of the detonation wave in the chamber, the strength of
the detonation wave is varied by varying the mass fraction of the fuel while keeping other
properties like pressure, temperature and velocity constant. In the general operation in
the NDWE mode, once the detonation is initiated at a particular location in the detonation
chamber, the detonation wave propagates upstream and at this instant, the fuel injection
is turned off. The upstream traveling wave loses strength in the absence of fuel and
becomes a blast wave which eventually dies out. Ajjay [101] in his Master’s thesis,
worked on simulating a detonation propagation phenomenon where in the detonation
wave was made to oscillate about a particular location by changing the equivalence ratio
of the fuel at the detonation chamber inlet as shown schematically in Fig. 5.22.
69
Figure 5.22: Schematic of the operation of NDWE mode by changing the stoichiometricratio
Referring to Fig. 5.22, the distance between the red and the blue line in the expansion
region is the range of movement of the detonation wave by varying the stoichiometric
ratio of fuel-air mixture. For a particular inflow condition, the detonation gets initiated at
the blue line and because the CJ Mach number in this mode is greater than the incoming
combustion chamber Mach number, the detonation wave propagates upstream. When the
detonation wave reaches the red line, the stoichiometric ratio of the fuel is varied to again
push back the detonation wave until it reaches the blue line. The results are discussed
in detail in [109][101]. The geometry, governing equations and chemistry are the same
as the simulations carried for the oblique detonation wave engine mode (section 5.5),
however, the combustion chamber inlet Mach number is 3.5.
Fig. 5.23 is the variation of flow properties at the exit of the detonation chamber
with respect to time. Conditions 1 and 2 refer to the incoming flow properties into
the detonation chamber and flow properties exiting the chamber respectively. The
variations in the pressure with respect to time is result of the expansion waves which
propagate downstream into the exit of the detonation chamber. It can be observed that the
variations of parameters at the exit of the expansion section are almost constant except
70
for the pressure. Hence, time averaged parameters are considered as nozzle inlet for the
simulation of flow through the nozzle.
Figure 5.23: Variation of flow properties at exit of detonation chamber in a NDWE mode
71
Chapter 6
Nozzle
Nozzle is a tube of a varying cross sectional area which converts the thermal energy
available at the combustion chamber to kinetic energy by expansion of gases. The flow
through nozzle controls the direction of the exhaust flow and also the forces generated
by the flow is associated with a change in momentum, generating thrust. Hydrogen
fueled hypersonic airbreathing aircraft have been studied for their unique and desirable
characteristics in a variety of civil transportation and military applications [65][66][67].
Fig. 6.1 shows the design and important characteristics of a scramjet engine. The
forefront is used as an inlet compression ramp, the center portion of the body is where
the fuel is injected and combustion takes place and the complete afterbody forms the
exhaust-nozzle surface. This integrated concept is beneficial for both cruise and accel-
eration applications for a hypersonic aircraft [68][69]. Some of the advantages of an
efficient engine-airframe integration is that the forebody inlet pre-compression reduces
the required engine size and weight requirements and the afterbody nozzle area can
potentially provide very efficient exhaust-gas expansion with minimal aerodynamic drag
[70].
72
Figure 6.1: Scramjet Engine
In an ideal nozzle, the flow is parallel to the nozzle wall. The length of the nozzle is
usually dependent on the exit Mach number and for higher exit Mach numbers, the length
of the nozzle is large. For hypersonic flight, the geometric parameters of the nozzle play
a very important role in the design of the aircraft and if the length of the nozzle is large,
the drag associated with it is also large. This reduces the overall engine performance.
The forces acting on the nozzle surface generate a pitching moment which can be used
to propel the aircraft in a particular direction. Properties of an ideal two dimensional
exhaust nozzle are:
• The exhaust nozzle is two dimensional and the flow properties do not vary into
the page. The reason is that the conventional circular nozzles are heavy and they
hamper in the airframe integration of the hypersonic vehicle.
• The nozzle entry flow is assumed supersonic and the governing equations are
hyperbolic.
• The exhaust stream is assumed isentropic and calorically perfect.
73
• The design of the nozzle produces uniform, parallel flow at the desired exit Mach
number.
• The nozzle design is a minimum length nozzle.
A numerical technique called the Method of Characteristics (MOC) is usually used to
design the exhaust nozzle. The qualities of the numerical technique like the ease of
calculation of quantities of the flowfield and the nozzle geometry as discussed in the A,
makes this technique more meaningful for the design process.
6.1 Single Expansion Ramp Nozzle (SERN)
SERN is a linear expansion nozzle with a 2D configuration. Many space access
vehicles like the NASA’s X-43 make use of SERN mainly because of weight reduction
at large expansion ratios and they can be easily blended into the airframe which greatly
reduces the drag [75] [76]. It can also be compared to a single sided aerospike nozzle
because of the single expansion ramp. If the pressure ratio is high, the thrust of the SERN
is virtually unaffected by the external flow but if the pressure ratio is low, there is an
influence on the nozzle flow field by external flow [77].
There has been significant research on modeling the SERN. Murugan [78] developed
a computer code to design a scramjet nozzle using the MOC technique. The effect of
back pressure on nozzle exit at different altitudes was also studied. Ridgway et. al [79]
did a parametric study using CFD to analyze the sensitivity of the SERN’s performance
parameters with changing geometry and operating conditions. The interaction between
nozzle exhaust and external flow was studied by Thiagaraian et.al [80]. Hirschen et.
al [81] studied the performance of scramjet nozzle at different flight altitudes and run
conditions by varying the Reynold’s number, nozzle pressure ratio and the heat capacity
ratio. Schlieren photographs were also taken to visualize the flowfield and investigate the
74
influence of heat capacity ratio on the interactions between the external and the nozzle
flow. A commercial CFD software, FLUENT, was used to investigate the geometric
parameters like divergent angles, total lengths, height ratios, cowl lengths and cowl angles
on nozzle performance by Jianping et. al [82]. Huand et. al [83] used the two dimensional
coupled implicit Reynolds Averaged Navier-Stokes equation and the two equation RNG
k − ε turbulent model to numerically simulate the flowfield in a SERN. They studied
the interactions between the parametric parameters and the objective functions, like the
thrust force and lift force using the data mining technique coupled with a design of
the experiment. Their study showed that the influences on the horizontal length of the
inner nozzle, the external expansion ramp and the internal cowl expansion on thrust
force performance are substantial. Fig. 6.2 shows an ideal minimum length exhaust
Figure 6.2: Schematic diagram of ideal, minimum length, two dimensional nozzledesigned using method of characteristics [84]
nozzle designed using the Method of Characteristics. The terms “th” and “e” in Fig. 6.2
represent the throat and exit of the nozzle respectively. In terms of hypersonic expansion
system, the same result could be achieved for either the top or the bottom half of the
entry flow if the plane of symmetry was replaced by a physical surface that reflects the
75
arriving characteristics [85]. As can be observed, the characteristics do not reach beyond
a certain point in the x-direction along the plane of symmetry. The reflecting surface must
in fact extend only until the point where the characteristics arrive and also at that point,
the flow has already reached the freestream static pressure everywhere along the final
characteristic. This short reflecting surface is often referred to as a flap. Fig. 6.3 shows a
Figure 6.3: Schematic diagram of Single Expansion Ramp Nozzle [86]
schematic of the truncated version of the Fig. 6.2. The nozzle flap is of a short length and
the expansion ramp is a contour which is designed by the characteristics emanating from
the nozzle throat. The feature of this type of nozzle is that the ratio of surface area to
entry throughflow area remains about the same as for the original closed exhaust nozzle
but the ratio of the length to entry height is double that of a closed nozzle because the
length is unaffected by halving the entry height [85].
6.2 Design of Nozzle Contour
A characteristic of hypersonic vehicle design is the effective coupling and integration
of various vehicle subsystems to achieve high performance for a desired flight condition.
In this regard, generating an optimum nozzle contour is important. One of the main
reason is that the right nozzle design prevents the formation of shocks within the nozzle
76
which greatly hampers the engine performance. The nozzle contour is a prime factor in
determining the quality of flow exiting the nozzle.
The nozzle contour is designed based on the desired exit Mach number. For a
hypersonic flight, the length of the nozzle is large as shown in Fig. 6.13. To make the
nozzle length more realistic and for optimum integration of the nozzle into the airframe
model, nozzle length is truncated. As a result of this, the exit flow from the nozzle is
either under-expanded or over-expanded along the flight trajectory. If the exit pressure of
the nozzle flow is equal to the ambient pressure, then the nozzle is set to design condition
(Fig. 6.4a) . When the exit pressure of the nozzle flow is lower than the ambient pressure,
nature works in a way to match the ambient pressure by shock waves at the exit of the
nozzle. The occurrence of shock waves at the nozzle exit increases the entropy of the
flow, thereby reducing the performance measure of the nozzle by a significant amount.
This condition is called over-expansion (Fig. 6.4b). In an under-expanded case (Fig.
6.4c), the exit pressure of the nozzle is higher than the ambient pressure and hence
expansion waves occur at the nozzle exit for the nozzle exit flow to match the ambient
pressure leading to a separation zone at the nozzle exit. When a hypersonic vehicle is
designed along a trajectory, at some point of the journey the nozzle flow will be either
under-expanded or over-expanded. Based on the flight mission, if the majority of the
flying time of the vehicle is spent cruising at a particular altitude, then the nozzle can be
optimized to function at that condition. However, considering military applications, the
vehicle performance has to be optimized over a wide range of operating conditions. In
this scenario, the losses at the nozzle exit due to improper expansion of the exit flow will
have to be made minimum over the transonic region of operation of the vehicle. Hence,
for a fixed nozzle design, the following compromises are evident from a study by Migdal
et. al [96]:
77
1. A large nozzle geometry is required to design a high performance vehicle at
supersonic cruise
2. A large nozzle geometry causes severe over-expansion loses at lower than design
Mach number.
(a) Design point
(b) Over-expanded flow
(c) Under-expanded flow
Figure 6.4: Nozzle flow exit conditions [74]
Another vital contribution of the nozzle towards engine performance is the generation
of thrust. Thrust produced by nozzle is the resultant static pressure forces acting on the
nozzle wall. In order to produce desirable thrust, the pressure forces acting on the inside
of the wall must be higher than the ambient pressure. If the pressure on the external
nozzle wall is greater than the inner wall, there will be loses due to internal drag.
78
The importance of nozzle geometry in hypersonic vehicle design has been established
in this section and to design an ideal nozzle, MOC is used to design the contour to obtain
an isentropic flow within the nozzle.
6.3 Method of Characteristics
Method of characteristic is a classical and elegant numerical technique to solve
hyperbolic partial differential equations. In this numerical technique, lines called charac-
teristics are identified which are oriented along a direction such that the derivative of flow
variables are indeterminate. Characteristic lines are mach waves that carry with them
the disturbances that propagate along the flowfield. A detailed explanation of the theory
and mathematical description of MOC is given in Appendix A. One of the important
application of this method is the design of supersonic nozzle contour for isentropic flow.
6.3.1 Verification of MOC MATLAB Code
To generate the contour of the supersonic nozzle using MOC, a MATLAB was code
written and verified with a solution of two dimensional minimum-length nozzle design
for an exit Mach number of 2.4 by John D. Anderson in his book, Modern Compressible
Flow [74]. Graphical construction of the nozzle is shown in Fig. 6.5 and the contour
generated using the MATLAB code is shown in Fig. 6.6. The flow at the inlet of the
nozzle is at sonic condition. In this condition, the flow gets chocked and the expansion
wave emanating from the nozzle inlet keeps bouncing at the same location. For the
characteristics to propagate downstream into the nozzle, the first characteristic (a-1) is
chosen to be slightly inclined to the normal sonic line. In reality, the sonic line at the
throat is curved but to simplify the calculations, solution in the book by Anderson has
assumed the sonic line to be a straight line. The specific heat ratio value is taken as 1.4.
79
To verify the values of flow variables at discrete points on the characteristic net from the
MATLAB code, numbering at the intersection of characteristic points are similar to that
made by Anderson. Random intersection points are chosen to compare the values.
Figure 6.5: Graphical representation of nozzle for an exit Mach number of 2.4
Figure 6.6: Nozzle contour generated using MATLAB
Fig. 6.7a are the values from Anderson and Fig. 6.7b are the values generated from
80
MATLAB code. The first column represents the point number on the characteristic net,
second column is the flow deflection angle (theta). The values from MATLAB code are
accurate when compared to Anderson’s solution.
(a) Values from Anderson [74]
(b) Values from MATLAB code
Figure 6.7: Comparison of values at random discrete points from MATLAB code withAnderson [74]
81
6.3.2 CFD Simulation of Nozzle flow
In this section, CFD simulation is carried out using FLUENT on the nozzle generated
using the MOC technique. This is done to verify that the results from MATLAB code
matches the CFD results generated using FLUENT. The inlet boundary condition for
the nozzle is the pressure inlet condition with pressure equal to atmospheric pressure
and Mach number of 1, outlet is a pressure outlet boundary condition and the nozzle
geometry with a wall condition.
Fig. 6.8 is the variation of Mach number along the length of the nozzle. It can be
observed from the figure that the inlet Mach number is at 1 and the exit Mach number is
2.4 which is the desired exit Mach number. Fig. 6.9 shows the pressure contour, with
inlet pressure at atmospheric condition.
Figure 6.8: Mach number contour
It can observed that the flow within the nozzle is smooth without the presence of
any shock waves, representing an isentropic flow within the nozzle. Fig. 6.10 shows the
velocity vector along the length of the nozzle.
82
Figure 6.9: Pressure contour
Figure 6.10: Velocity vector plot
83
At the nozzle exit, the direction of the vectors are parallel to the x-axis which is the
orientation of the nozzle. The figures discussed in this section, show that the nozzle
simulations obey the ideal nozzle conditions and hence, the MATLAB code is also
verified by CFD simulations.
6.4 Parameters affecting Nozzle geometry
In designing the nozzle contour using MOC, three parameters that significantly affect
the nozzle geometry are specific heat ratio, number of characteristics and exit Mach
number.
Fig. 6.11 shows the variation of nozzle length and height with the change in number of
characteristics. As the number of characteristics used to design the contour increase, the
nozzle contour gets more accurate. Comparing the values of the nozzle geometry when
the number of characteristics is 10 and 40, there is a steep variation. When the number
of characteristics is 10, the angle between is each successive expansion fan emanating
from the nozzle throat is coarse and hence the flow turning along the expansion fan is not
smooth. As the number of characteristics increase, the angle between each successive
expansion fan is finer and the flow turning along these expansion fan is also smooth.
As the number of characteristics increase beyond 60, nozzle geometry does not change
significantly.
Fig. 6.12 shows the variation of nozzle geometry with change in specific heat ratio
(γ). For air at room temperature, γ is 1.4 where the rotational and translational modes
are active. When the temperature increase above 600K, vibrational energy mode gets
activated thereby increasing the number of degree of freedom of the gas molecules which
in turn decreases the value of γ. When vibrational energy mode gets activated, the
sensible energy of the gas also increases.
84
For efficient expansion of gases with high sensible heat, larger nozzle geometry is
required when compared to higher values of γ. This is the trend noticed in Fig. 6.12. Fig.
6.13 shows the variation of nozzle geometry with exit Mach number. The area ratio (ratio
of exit area of nozzle to throat area) is a parameter which is dependent on the exit Mach
number and as the exit Mach number increases, larger nozzles are required for efficient
expansion.
Figure 6.11: Variation of nozzle geometry with change in number of characteristics.Specific heat ratio = 1.4, exit Mach number = 2.4
85
Figure 6.12: Variation of nozzle geometry with change in specific heat ratio. No. ofcharacteristics = 50, exit Mach number = 2.4
86
Figure 6.13: Variation of nozzle geometry with change in exit Mach number. No. ofcharacteristics = 50, Specific heat ratio = 1.4
87
Chapter 7
Hypersonic Nozzle Design
This chapter is an integration of the results from Chapter 5 and Chapter 6. The exit
conditions of ODWE mode tabulated in Table 5.2 along with the MOC Matlab code
discussed in Chapter 6 will be used to design the nozzle contour for the ODWE mode.
The following section is a discussion of supersonic inlet Mach number nozzle.
7.1 Supersonic Inlet Nozzle
Fig. 7.1 shows a schematic of a minimum two dimensional nozzle in which H4 is the
inlet height, H10, the nozzle exit height and L is the length of the nozzle. The inlet entry
Mach number is greater than 1 and hence expansion waves emanate from the corners at
the nozzle inlet. The expansion waves generated from the nozzle inlet guides the flow to
turn in a direction parallel to the contour and at the end of the straightening section of the
nozzle, the flow is parallel and uniform at the desired exit Mach number. In the case of
hypersonic flight, only one half of the nozzle is considered and the plane of symmetry is
replaced by a solid surface which is called the flap. The expansion waves emanating from
the upper corner of the nozzle inlet strikes the surface and reflects back in the direction of
88
Figure 7.1: Schematic diagram of an ideal, minimum length, two dimensional exhaustnozzle designed by means of the method of characteristics [85]
the contour. The length of the flap is the distance from nozzle inlet to the point where the
last characteristic strikes the surface. As the inlet and exit Mach numbers increase, the
turning angle at the sharp expansion corner increases and the trailing edge of the initial
expansion fan moves away from the plane of symmetry, creating a tulip like shape at the
entrance region which is a trademark of hypersonic nozzles.
7.2 Hypersonic Nozzle Design using MOC
To design the hypersonic nozzle contour, the MOC Matlab code discussed in Section
6.3.1 and Section 6.3.2 is used. In the original code that was developed, the inlet Mach
number into the nozzle was considered to be 1 and for propagation of the flow into the
nozzle, an approximation of the initial deflection of the characteristic line was assumed.
However in case of a hypersonic nozzle, the flow into the nozzle is supersonic and the
approximation of the initial deflection of the characteristic line can be eliminated. Instead
the initial deflection of the characteristic line is the Mach wave angle of the incoming
flow.
89
7.2.1 Constant dynamic pressure trajectory
The pressure an aircraft generates as it moves through air is called dynamic pressure
(q0) which is expressed as
q0 =ρ0V0
2
2(7.1)
where V0 is the magnitude of the velocity of the atmosphere relative to the flight and ρ0
is the atmospheric density. Dynamic pressure can also be expressed in terms of Mach
number as
q0 =γ0P0M0
2
2(7.2)
where P0 , M0 and γ0 are the freestream pressure, Mach number and specific heat ratio
respectively. Two main applications of dynamic pressure in hypersonic vehicle design
are
1. Useful scaling factor in determining the pressure and forces experienced by hy-
personic vehicle, as the lift and drag of the vehicle are usually a function of the
dynamic pressure.
2. Dynamic pressure can also be used to set the vehicle’s structural limits. If q0 is too
large, structural forces and drag on the vehicle is large and if q0 is too small, then
large wing span is required for a sustained flight. Hence, it is for this reason that
hypersonic flights operate over a narrow range of dynamic pressure.
Fig. 7.2 shows the variation of standard day geometric altitude with freestream Mach
number trajectories for the expected range of values of q0. For a hypersonic flight, the
range of operation of q0 is narrow as shown by the shaded region in Fig. 7.2. In the
ODWE mode, the incoming combustion chamber Mach number was set to 6 according
to Heiser et. al [85], the incoming combustion chamber Mach number is approximately
equal to 40% free stream Mach number. Using this thumb rule, a flight Mach number
90
of 15 chosen. The dynamic pressure chosen for this research is based on the dynamic
pressure of NASA’s X-43 [102] which is 1000 psf at an altitude of 42 km above sea level.
Figure 7.2: Geometric altitude vs flight Mach number trajectories for constant dynamicpressure [85]
7.2.2 Nozzle contour
The three main parameters affecting the nozzle geometry are specific heat ratio, exit
Mach number and number of characteristics as discussed in section 6.4. Based on Fig.
6.11, the number of characteristics chosen to design the hypersonic nozzle contour is 50.
From the simulations of the combustion chamber of the ODWE mode, the specific heat
ratio at the exit of the expansion section is 1.286 which will be used to design the contour.
The ambient conditions of the nozzle exit is standard day atmospheric conditions at an
altitude of 40km and the nozzle exit Mach number is calculated using isentropic relations.
Fig. 7.3 shows the nozzle contour and the characteristic net generated using MOC
91
designed for an inlet nozzle Mach number of 4.12. Using isentropic relations, the exit
Mach number of the nozzle is calculated as 7. The inlet height is considered as 1 unit so
that the nozzle length and height could be compared to the inlet height. It can be seen
that, for an exit Mach number of 7, the nozzle length is a little more than 1000 times the
inlet height and nozzle height is 130 times the inlet height. This nozzle contour obeys
the principles of an ideal 2 dimensional nozzle and because of this, for the complete
expansion of the flow, hypersonic nozzles are generally large.
Figure 7.3: Hypersonic nozzle contour using MOC
From Fig. 6.13, comparing the nozzle geometry for exit Mach number of 4.5 to 7,
the sensitivity of the nozzle geometry increases with increase in exit Mach number from
about 5. For an exit Mach number of 5, the nozzle length is about 200 times the inlet
height and nozzle height is about 30 times the inlet height. But with increase in exit
Mach number to 7, the nozzle length is about 800 times the nozzle inlet and the nozzle
height is about 100 times inlet height. The numbers discussed with respect to the exit
Mach number and nozzle geometry show that for higher exit Mach numbers, the nozzle
gets bigger. This discussion is based on the specific heat ratio of 1.4. From Fig. 6.12, it
can be seen that the lower values of specific heat ratio need larger nozzles for the flow to
expand. Thus the nozzle geometry in Fig. 7.3 is justifiable. Fig. 7.4 shows the ”tulip”
like structure which was discussed in Section 7.1 which is a characteristic of hypersonic
92
nozzles. Also, it can be seen that the last characteristic from the nozzle inlet falls on the
x-axis at a distance 111.53 units from the origin which is 111.53 times the inlet height.
This is considered as the flap length.
Figure 7.4: Tulip like structure of the expansion waves emanating from the nozzle inlet
The geometric length scales of the nozzle mentioned in Fig. 7.3 is very large to
integrate it into an hypersonic vehicle. It adds on to the weight of the aircraft and
also increases overall the aerodynamic drag, decreasing the overall efficiency of the
hypersonic vehicle. Heiser et al [85] studied the variation of stream thrust along the
length of the nozzle. Stream thrust function is a parameter which determines the mass
flow rate specific thrust which is often used in performance evaluation. The nozzle studied
was designed for an exit Mach number of 5.5 and specific heat ratio of 1.24. According
to their study, most of the thrust generation is recovered by the early expansion process
within the nozzle. They also showed that the nozzle could be truncated at approximately
40% of the initial length without significant loss of thrust. This encourages the designers
to truncate the nozzle to a certain percentage of the initial length to maintain a balance
between the weight and overall efficiency of the nozzle. Extending their study to the
current nozzle design of the ODWE mode, the nozzle is truncated at 40% of the original
93
length and the truncated contour is as shown in Fig. 7.5. The truncated nozzle contour
will be used for further analysis in this research.
The total axial force on the internal nozzle surface is the difference between the local
stream thrust function and the entry stream thrust function, and the maximum possible
force on the internal nozzle surface is the difference between exit stream thrust function
and the entry stream function. The stream thrust fraction determines the fraction of
the available stream thrust gained by truncating the nozzle at a particular axial location.
For the current design, the stream thrust fraction is 0.82 which means that 82% of the
available thrust is recovered by truncating the nozzle at 40% of the initial length which is
a fair trade-off between the thrust and the weight of the vehicle.
Figure 7.5: Truncated nozzle contour
7.3 CFD Simulation of Nozzle Flow
For this research, two different flight conditions are considered along a constant
dynamic pressure of 47,880 N/m2, similar to that of NASA’s X-43 which is till date, the
world’s fastest aircraft reaching a Mach number of approximately 9.6. The two conditions
94
chosen are as shown in Table 7.1.
Flight Mach number Altitude (km)
15 42 Design condition
8.75 34 Off-design condition
Table 7.1: Flight conditions considered for the current research
For the design condition, the inlet Mach number into the combustion chamber is 6
which is greater than the CJ pressure, leading to an oblique detonation wave and for the
off-design condition, the inlet Mach number into the combustion chamber is 3.5 which
represents an oscillating detonation wave represented by NDWE mode. As discussed
in Section 3.3, the NDWE mode and ODWE mode play a crucial role in determining
the success in operation of the multi-mode propulsion concept as they generate thrust at
critical parts of the trajectory of the flight. In this regard, the flight conditions shown in
Table 7.1 are chosen to study the nozzle flow characteristics.
7.3.1 Design condition
With the design of the nozzle contour for the design condition obtained in the previous
section, CFD simulations will be performed in this section to analyse the flow within the
nozzle. The nozzle contour points are imported into ANSYS SpaceClaim, which is a
3D modeling application software to create the domain for the flow analysis. In order to
study the flow within the nozzle, static pressure is matched at the termination of the final
characteristic
The CFD simulation is carried out in ANSYS Fluent. For the simulation, the density
95
based approach is used. The pressure inlet boundary condition is used with the gauge
inlet pressure of 301284.41 Pa, density of 0.34 Kg/m3, velocity of 4117 m/s, a Mach
number of 4.12 and an effective gamma of 1.286 is considered. The exit condition is
modeled as pressure far-field using the atmospheric conditions at an altitude of 42km.
The nozzle contour and flap are treated as wall. The numerical method used for the
simulation is an implicit formulation with a Roe-Flux difference splitting algorithm to
calculate the fluxes. Least square cell based method is used to calculate the gradients and
a second order upwind scheme is used for the propagation of the flow within the flow
domain.
The results of the simulation are shown in Fig. 7.6, 7.7 and 7.8 which represent
the contours of variation of pressure, density and Mach number along the length of the
nozzle. Referring to the pressure contour which is Fig. 7.6, at the initial expansion
region, there is a decrease in static pressure. According to the text book, Hypersonic
Airbreathing Propulsion by Heiser et al [85], this initial expansion region is termed as
zone III. There is a study on the variation of static pressure in this zone as functions of
exit and entry Mach numbers and graph is shown in Fig. 7.9. The indices 4 represent
the nozzle inlet conditions, MY represent the exit Mach number, the points A, B and C
on the graph refers to conditions of exit Mach of 5, 5.5 and 6.5 respectively. It can be
observed that most of the inlet static pressure is removed by the initial expansion fan,
which also indicates that the vast majority of thrust is generated at the initial portion of
the nozzle.
Extending this discussion to Fig. 7.6, it can be observed that the pressure along the
initial expansion region is reduced, thereby accelerating the flow towards the nozzle exit
which can also be seen in Fig. 7.8. As the flow accelerates towards the nozzle exit, the
density of the flow decreases which can be seen in Fig. 7.7. The tulip like expansion
wave can also be observed which was discussed in Section 7.1.
96
Figure 7.6: Variation of pressure along the length of the nozzle at design point
Figure 7.7: Variation of density along the length of the nozzle at design point
97
Figure 7.8: Variation of Mach number along the length of the nozzle at design point
Figure 7.9: Ratio of zone III static pressure to entry static pressure for ideal design pointexpansion components as function of entry Mach number and exit Mach number [85]
Since the nozzle is truncated at a particular value of the original length, the incomplete
tulip like structure interacts with the freestream.
98
7.3.2 Off-design condition
For the off-design condition, the exit conditions of the combustion chamber are
calculated from the research presented by Ajjay [101][103]. The simulation is an unsteady
NDWE mode for a flight Mach number of 8.75 and at an altitude of 34 km above sea
level. Since it is an unsteady mode, the time averaged exit parameters are considered as
inlet conditions into the nozzle. For the simulation of the off-design condition, geometry
and numerical technique discussed for the design condition is used with only change in
nozzle inlet parameters. The nozzle inlet conditions for this simulation are pressure of
566,890 Pa, temperature of 2600 K and Mach number of 2.5. Fig. 7.10, 7.11 and 7.12
show the variation of pressure, density and Mach number respectively along the length
of the nozzle. The same tulip like shape is observed but in the off-design case, the tulip
like structure is formed within the nozzle. The decrease in static pressure at the initial
expansion region of the nozzle can also be observed.
Figure 7.10: Variation of pressure along the length of the nozzle at off-design condition
99
Figure 7.11: Variation of density along the length of the nozzle at off-design condition
Figure 7.12: Variation of Mach number along the length of the nozzle at off-designcondition
100
7.4 CFD simulation of nozzle exhaust
In this section, CFD results of the interaction of the nozzle exit flow with the external
freestream is presented. This study is important because the shocks occurring at the
nozzle exit tend to influence the aircraft’s aerodynamic performance. Also, depending on
the overall design of the hypersonic aircraft, the shocks emanating from the wings and
the tail may interact with the exhaust shocks and may affect the pitching moment of the
aircraft [105][106]. The exhaust plumes are dependent on the atmospheric conditions
which vary with altitude. Different shock and plume structures can be expected at
different altitude of flight operation. Typically, at lower altitude, the plumes assume
a narrow configuration as the exhaust gases do not expand much laterally against the
ambient pressure. However, at higher altitudes, the ambient pressure in low and the
plumes interacts drastically with the external freestream making the plumes voluminous
[104]. Bauer et al [107] have developed an engineering model to study the interactions
of the plumes with the surroundings with the theory of characteristics.
The performance and operation of the aircraft greatly depends on the shock interac-
tions of the aircraft structure like the wings and tail with the external freestream and also
the interaction of these resulting shocks with the shock structure at the nozzle exhaust.
As an example for this discussion, the shocks formed by X-15 being fired in a wind
tunnel is shown in Fig. 7.13. The shocks emanating from various parts of the aircraft can
be seen in Fig. 7.13. For higher Mach number, the shock waves become more steep and
the flow properties behind the shock wave increase drastically. This drastic increase has
an impact on the vehicle structure which impacts the aerodynamic forces acting on the
aircraft. Also the shock waves from the tail and the wings impact the flow structure at
the exhaust of the nozzle and further affect the lifting moment of the aircraft.
The intent of this section is to simulate the overall structure of the near field of the
101
Figure 7.13: Shock interactions of X-15 being fired into a wind tunnel [108]
nozzle exhaust. The geometry is similar to that discussed in the previous sections but
since the interactions of the exhaust flow with the external freestream is studied, the
nozzle flap length is limited to 111.3 mm as shown in Section 7.2.2. The length of the
farfield from nozzle exit is one and a half times the nozzle length for the flow structures
at the nozzle exit to be completely formed. The farfield conditions are the atmospheric
conditions at the altitude considered. The results are presented in grey scale for better
visualization of the formation of shock waves and jet stream.
7.4.1 Design condition
The CFD results of the interaction of the plumes with the surrounding atmosphere
at design condition is presented in this section. The design condition is operated at a
flight Mach number of 15 and an altitude of 42km above sea level and the standard
102
day atmospheric pressure at this altitude is 21,997.65 Pa and at this low pressure, the
simulation of the flowfield downstream of the nozzle exhaust is presented in the Fig.
7.16, 7.17 and 7.18. For better understanding of the formation of shocks and plumes,
a transient simulation is carried out from initial conditions. Fig. 7.14 and Fig. 7.15
represent the contours of pressure and density at a time instant of 0.001 sec.
Fig. 7.14 and Fig. 7.15 show the formation of the plume structure and the shock
waves developing at the edges of the nozzle. As the flow starts to develop, the shock
waves become more acute and the plume structure gets more voluminous.
Figure 7.14: Pressure contour of the formation of plumes and shocks at the designcondition and at a time instant of 0.001 sec
103
Figure 7.15: Density contour of the formation of plumes and shocks at the designcondition and at a time instant of 0.001 sec
Figure 7.16: Pressure contour of the formation of plumes and shocks at an altitude of 42km
104
Figure 7.17: Density contour of the formation of plumes and shocks at an altitude of 42km
Figure 7.18: Mach number contour of the formation of plumes and shocks at an altitudeof 42 km
105
Fig. 7.16, 7.17 and 7.18 show the pressure, density and Mach number contour of a
fully developed flow structure at a time instant of 0.0032 sec at the nozzle exhaust. The
formation of the plume ends at a location outside the nozzle. As it was discussed earlier,
the plumes get bigger as the ambient pressure decreases. The exhaust flow can be seen as
a core flow surrounded by a region of high density which is caused by the acute shock
waves emanating from the nozzle contour. In case of viscous flow, this region of high
density would be a re-circulation region where small eddies are formed. Inside the core
flow is where the plumes are formed and the core flow extends far into the atmosphere as
a stream of jet. Hence, the core as can be seen as dominated by shocks and expansion
waves. In most of the supersonic exhaust jets, the exit pressure if higher than the ambient
pressure and for this reason, flow is generally under-expanded. In perspective of gas
dynamics, the nozzle exhaust flow tends to match the ambient condition and for this
reason, an expansion wave is formed. The expansion wave moves downstream until
it interacts with the core flow. This phenomenon can also be observed in the contours
presented in this section. The plume structures and the shock waves are similar to the
CFD results obtained by Bauer et al [107] for an altitude of 42 km. The results of the
nozzle exit as shown in Fig. 7.6 can be extended into this section to understand the
formation of shocks and expansion waves better. The upper portion of the nozzle exit
has a higher pressure of around 50,000 Pa and when the flow interacts with the ambient
pressure which is at a much lower pressure, expansion wave is formed to match the
conditions.
7.4.2 Off-Design condition
The simulation results of off-design results are synchronistic to the discussion of the
interaction of plumes at the design condition except that the inlet conditions into the
nozzle are as discussed in Section 7.3.2 and also the aircraft operation is at an altitude
106
of 34 km above sea level. Since the ambient pressure is much higher than the design
condition, the structure of the plume should be much smaller. Fig. 7.19 and Fig. 7.20
show the pressure and density contour of the flow at a time instant of 0.001 sec.
From Fig. 7.19 and Fig. 7.20, the formation of the shock and expansion wave can
be seen and also at this time instant, the core structure is almost getting steady. This
is because the exhaust velocity from the nozzle is much lower when compared to the
velocity at design condition.
Fig. 7.21, 7.22 and 7.23 show the complete formation of plumes, shock and expansion
waves. The structure of the plumes, core flow and the formation of shocks and expansion
waves are similar to that discussed in the previous section except that the shocks and the
core jet coalesce and disperse into the external freestream at a much shorter distance than
at the design condition.
Figure 7.19: Pressure contour of the formation of plume and shocks at an altitude of 34km and at a time instant of 0.001 sec
107
Figure 7.20: Density contour of the formation of plume and shocks at an altitude of 34km and at a time instant of 0.001 sec
Figure 7.21: Off-design condition: Pressure contour of the formation of plume andshocks
108
Figure 7.22: Off-design condition: Density contour of the formation of plume and shocks
Figure 7.23: Off-design condition: Mach number contour of the formation of plume andshocks
109
7.5 Conclusion
The current research is a baseline analysis by setting the procedure and a template
to optimize the nozzle design. For a particular inlet combustion chamber condition,
the flowfield is simulated through the combustion chamber, where the detonation gets
initiated and the detonation products are allowed to expand in a straight channel. Using
the exit conditions of the expansion section as inlet conditions into the nozzle, flow is
simulated through the nozzle and allowed to interact with the ambient conditions. Two
combustion chamber inlet conditions are considered based on the Table 7.1.
For the design case, an incoming combustion chamber Mach number of 6 is con-
sidered which leads to a steady oblique detonation wave mode. The exit conditions of
the expansion section are considered as inlet conditions into the nozzle. The exit Mach
number of the expansion section is 4.12 and based on this Mach number and using the
method of characteristics approach, a nozzle contour is designed for efficient expansion
of the flow into the external atmosphere. CFD simulations within the nozzle contour
and also the interaction of the nozzle exhaust with the ambient condition are carried out.
For the nozzle flow, chemical reactions were not considered. The large drop in pressure
along the length of the nozzle would freeze the chemical reactions. The uniqueness of
this simulation is that, for a particular incoming combustion chamber condition, the gas
dynamics of the entire flowfield from the combustion chamber all the way to the nozzle
exhaust interacting with the ambient conditions can be visualized. The procedure and
the mathematical model used for this simulation can be used to study the downstream
flow characteristics from the inlet of the combustion chamber as a function of incoming
combustion chamber conditions. The simulation model also helps in understanding
the nozzle exhaust flow and the shock structures at the nozzle exit as a function of the
incoming combustion chamber parameters. Another important aspect of this research
110
is the approach towards nozzle design. Usually for hypersonic nozzles, a straight ramp
of a particular angle is considered for the flow expansion which leads to complex flow
patterns and flow separation at the nozzle exit. The method of characteristics approach
is a relatively lengthy procedure in designing the nozzle contour, however, the flow
emerging from the nozzle exit is uniform and also mitigates the complex shock patterns
which affect the overall performance of the aircraft.
Considering the off-design case, the incoming combustion chamber Mach number
is 3.5 which leads to the operational mode of NDWE. In this simulation, the detonation
wave was made to oscillate about a particular location downstream of the wedge by
changing the stoichiometric ratio of the incoming of the incoming fuel-air mixture
into the combustion chamber. In doing so, the exit of the expansion section will have
thermodynamic parameters which are nearly constant over a time range and this makes it
relatively easier to simulate the nozzle flow conditions using the time averaged expansion
section exit conditions as inlet conditions into the nozzle. In this case, the Mach number
of the flow entering the nozzle is 2.45.
Comparing the flow structures through the nozzle and also the flow interactions with
the ambient conditions of the design case with that of off-design, it can be noticed that
the plume formation is more voluminous and the density of the exhaust flow is lower
when compared to the off-design case. The exhaust flow of the nozzle can be divided into
two streams which is the inner core (plume) and the flow outside the region of the inner
core moving along the nozzle contour. The inner core is inviscid in nature which when
extends into the external freestream conditions, results in the formation of a jet flow as
seen in Fig. 7.13. The flow outside the inner core interacts with the external freestream
at much lower velocity compared to the inner core. This stream of flow when interacting
with the external freestream, tries to match the ambient conditions and in doing so, a
slip line is formed at the edge of the nozzle contour. Comparing the design case flow
111
structures at the nozzle exit with that of the off-design case, it can be noticed that the
plume is more voluminous and because of the high exit Mach number, the plume extends
far downstream of the nozzle exhaust than the off-design case.
112
Chapter 8
Future Work
The main intent of this research was to design the nozzle contour for a specific design
condition representing the ODWE mode and simulate the nozzle exhaust flow. To make
the study and design more realistic, following are a few suggestions.
• Designing an inlet cowl and carrying out a CFD simulation to get the inlet con-
ditions into the combustion chamber. The incoming pressure and Mach number
into the combustion chamber play an important role in the location of detonation
initiation and propagation.
• The nozzle contour length becomes large as the exit Mach number is higher which
in turn depends on the nozzle inlet Mach number. In order to optimize the nozzle
inlet Mach number, a parametric study can be done. The parameters considered
could be the wedge angle, length of the expansion region of the combustion
chamber and stoichiometric ratio of fuel-air mixture. By varying the wedge angle
and the stoichiometric ratio of fuel-air mixture, the location of the detonation
initiation can be varied and the length of the expansion region of the combustion
chamber can also be varied so that there is enough time for the detonation products
113
to expand and the Mach number entering the nozzle is reduced.
• Introducing viscosity into the simulation of the flow through the combustion
chamber can also vary the detonation initiation location and other flow properties.
Figure 8.1: Pressure distribution at p=3 atm and T=700 K [100]
Fig. 8.1 shows the initiation of detonation wave for a combustion chamber Mach
number of 4 for the inviscid flow and 2.43 for the viscous flow. As can be seen
from the figure, for the inviscid case, the detonation initiation happens by shock
interactions whereas for the viscous flow, the detonation is initiated at the wall
due to the presence of boundary layer. The mechanism of the initiation of the
detonation phenomenon, location of detonation initiation and also the propagation
velocity of the detonation wave is different for both the viscous and inviscid case.
Introducing viscosity into the simulation of nozzle flow would also affect the
generation and the strength of shock waves. The core of the nozzle exhaust would
however remain inviscid.
• The geometric dimensions of the nozzle contour designed in this research can be
used to integrate it into the aircraft structure. Once the geometric properties of the
aircraft like the wing, tail and inlet are designed, a CFD simulation could be carried
of the interactions of the ambient conditions with the entire aircraft geometry. This
way, the shock interactions generated from various parts of the geometry of the
aircraft can be studied and the performance of the aircraft can be evaluated.
114
Appendix A
Method of Characteristics
The Method of Characteristics (MOC) is a mathematical model which reduces
the partial differential equations to a family of ordinary differential equations along
characteristics lines which the solution can be integrated from some initial data [71]. The
ordinary equations can then be solved using simpler and well established methods. The
MOC has a wide range of applications and a review of MOC with applications to science
has been done by Eklund et. al [72].
MOC is based on a discretization technique along a set of characteristic lines and
the accuracy of this method depends on the number of characteristics used to solve the
problem. Recording and storage of information along these lines require large storage
and computational resources. However, since the advent of super computers, the MOC
can be used with a high degree of accuracy [73].
A.1 Theory of MOC
In a two dimensional irrotational flowfield, let V represent the velocity known at a
point in the flowfield as shown in Fig. A.1 and u, the x-component of velocity V . There
115
Figure A.1: Illustration of characteristic direction
exists a line at a particular angle to the streamline direction along which the derivatives of
the flowfield properties like velocity, pressure, temperature and density are indeterminate
and across which it may be discontinuous. Such lines are called as characteristic lines
and they make an angle µ with respect to the velocity vector V .
sinµ =u
V(A.1)
Characteristics lines are also Mach lines, which means the velocity component perpen-
dicular to the y direction is a sonic line.
sinµ =a
V=
1
M(A.2)
116
where a is the speed of sound and M is the Mach number. Therefore, µ is given by,
µ = sin−1(1
M) (A.3)
A general method to solve for the flowfield is carried out in three steps.
1. Identify the characteristic lines along which the derivatives of the variables of
flowfield are indeterminate.
2. Construct Ordinary Differential Equations (ODE) from partial differential equa-
tions that hold along the characteristic lines. Such ODE are called compatibility
equations.
3. Assuming the initial conditions are known at some point in the flowfield, solve the
compatibility equations along the characteristic lines to map the entire flowfield.
A.2 Determination of Characteristic Lines
For a two-dimensional flow, the governing non-linear equation is given by [74],
(1− Φx2
a2)Φxx + (1− Φy
2
a2)Φyy −
2ΦxΦy
a2Φxy = 0 (A.4)
where Φ is the full velocity potential. It is known that Φx = f(x, y), therefore,
dΦx =∂Φx
∂xdx+
∂Φy
∂ydy = Φxxdx+ Φyydy (A.5)
dΦy =∂Φy
∂xdx+
∂Φy
∂ydy = Φxydx+ Φyydy (A.6)
V = ui+ vj; Φx = u; Φy = v
117
Combining Eq. (A.4) - Eq. (A.6),
(1− u2
a2)Φxx + (1− v2
a2)Φyy −
2uv
a2Φxy = 0 (A.7)
(dx)Φxx + (dy)Φxy = du (A.8)
(dx)Φxy + (dy)Φyy = dv (A.9)
Solving for Φxy using Cramer’s rule,
Φxy =
∣∣∣∣∣∣∣∣∣∣1− u2
a20 1− v2
a2
dx du 0
0 dv dy
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣1− u2
a2−2uva2
1− v2
a2
dx dy 0
0 dx dy
∣∣∣∣∣∣∣∣∣∣
=N
D(A.10)
For the chosen dx and dy in the flowfield, there exists a corresponding value for the
change in velocity, du and dv. But if dx and dy are chosen such that D = 0 in Eq. (A.10),
Φxy becomes an infinite value which is physically inconsistent. To make Φxy a finite
value, N = 0. As a result of this Φxy cannot be defined in this particular direction where
the choice of dx and dy makes D = 0. Such lines along which the derivatives of flow
variables are indeterminant are called characteristic lines.
By setting D = 0 in Eq. (A.10) and simplifying using the quadratic formula,
(dy
dx)char = −uv
a2±√
(u2 + v2)/a2 − 1
(1− u2
a2)
(A.11)
where dydxchar
is the slope of the characteristic lines and Eq. (A.11) defines the character-
118
istic curves. Considering the term under the square root in Eq. (A.11),
u2 + v2
a2− 1 =
V 2
a2− 1 = M2 − 1 (A.12)
From Eq. (A.11) and Eq. (A.12), depending on the value of M, the solutions can be
classified as follows,
1. If M > 1, there are two real characteristics passing through each point in the
flowfield and Eq. (A.4) is a hyperbolic partial differentiation equation.
2. IfM = 1, there is one real characteristic passing through each point in the flowfield
and Eq. (A.4) is parabolic partial differentiation equation.
3. If M < 1, the characteristics are imaginary and Eq. (A.4) is an elliptical partial
differential equation.
A.3 Compatibility Equations
For supersonic flows with M > 1, the governing equations belong to the class of
hyperbolic partial differential equations and they are defined to have two characteristics
at each point in the flowfield, which is the left-running and right-running characteristics
as shown in Fig. A.2.
From Fig. A.2, the equation of characteristic line can also be written as
(dy
dx)char = tan(θ ± µ) (A.13)
The characteristic line associated with the angle θ + µ is called the C+ characteristic
which is the left running wave, and θ − µ is called the C− characteristic which is the
119
Figure A.2: Left and right running characteristics
right running wave. In general, the characteristic lines are generally curved as shown in
Fig. A.2.
In order to determine the compatibility equations to be solved along the characteristic
lines, D = 0 in Eq. (A.10) which yields,
(1− u2
a2)dudy + (1− v2
a2)dxdv = 0 (A.14)
dv
du=
(1− u2
a2)
(1− v2
a2)
dy
dx(A.15)
The dydx
term in Eq. (A.15), is valid along the characteristic line. Therefore, dydx
= ( dydx
)char.
Substituting Eq. (A.10) into Eq. (A.15) and after simplification, we get
dv
du=
uva2±√
u2+v2
a2− 1
1− v2
a2
(A.16)
u = V cos θ and v = V sin θ, Eq. (A.16) becomes
dθ = ±√M2 − 1
dV
V(A.17)
120
Eq. (A.17) is the compatibility equation which describes the variation of flow properties
along the characteristic lines. Eq. (A.17) when integrated, can be compared to the
Prandtl-Meyer function ν(M). Therefore, Eq. (A.17) can be replaced by,
θ + ν(M) = constant = K−(along C− characteristic) (A.18)
θ − ν(M) = constant = K+(along C+ characteristic) (A.19)
The constants K− and K+ in Eq. (A.18) and Eq. (A.19) respectively signify that they are
invariant along their respective characteristics and are known as Riemann invariants.
121
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