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

Bibliography

[1] J. Urzay, “Supersonic combustion in air-breathing propulsion systems for hyper-

sonic flight,” Annual Review of Fluid Mechanics, vol. 50, pp. 593–627, 2018.

[2] D. M. V. Wie, S. M. D’Alessio, and M. E. White, “Hypersonic airbreathing

propulsion,” John Hopkins APL Technical Digest, vol. 26, no. 4, pp. 430–437,

2005.

[3] E. H. Andrews and E. A. Mackley, “NASA’s Hypersonic Engine Project: A

review,” Hampton, 1994.

[4] R. P. Hallion, J. Becker, J. Vitalli, and J. Young, “The hypersonic revolution.

volume 2. from scramjet to the national aero-space plane,” Defense Technical

Information Center, OH, 1995.

[5] D. Andersen, X-30 national aero-space place mockup rolls out, https://www.

nasa.gov/home/hqnews/1992/92-086.txt, Accessed: 10-18-2019.

[6] Y. Gibbs, NASA armstrong fact sheet: Hyper-x program 28 feb 2014, https:

//www.nasa.gov/centers/armstrong/news/FactSheets/FS-

040-DFRC.html, Accessed: 10-18-2019.

[7] K. Bowcutt, A. Paull, D. Dolvin, and M. Smart, “Hifire: An international collab-

oration to advance the science and technology of hypersonic flight,” Brisbane,

QLD, Australia , 2011.

122

[8] K. Barnstorff, Hifire scramjet research flight will advance hypersonic technology,

https://www.nasa.gov/topics/aeronautics/features/

hifire.html, Accessed: 10-18-2019.

[9] Y. Gibbs, Past projects: X-43a hypersonic flight program, https://www.

nasa.gov/centers/dryden/history/pastprojects/HyperX/

index.html, Accessed: 10-18-2019.

[10] H. William, Hypersonic Airbreathing Propulsion. AIAA Education Series, 1994.

[11] T. J. Stueber, D. K. Le, and D. R. Vrnak, “Hypersonic vehicle propulsion system

control model development roadmap and activities,” NASA Glenn Research

Center, Cleveland, OH, United States, Tech. Rep., Jan. 2009.

[12] K. Kailasanath, “Review of propulsion applications of detonation waves,” AIAA

Journal, vol. 38, no. 9, pp. 1698–1708, 2000.

[13] F. K. Lu, “Prospects for detonations in propulsion,” Gyeongju, Korea: 9th Inter-

national Symposium on Experimental and Computational Aerothermodynamics

of Internal Flows (ISAIF9), Sep. 2009.

[14] W. Huang, M. Pourkashanian, and D. B. Ingham, “Flow-field analysis of a

typical hydrogen-fueled dual-mode scramjet combustor,” Journal of Aerospace

Engineering, pp. 336-346, 2012.

[15] D. R. Wilson and F. K. Lu, Multi-mode pulsed detonation propulsion system,

US6857261B2, Feb. 2005.

[16] R. Munipalli, V. Shankar, D. R. Wilson, H. Kim, F. K. Lu, and P. E. Hagseth,

“Pulsed detonation based multimode engine concept,” AIAA, 2001-1786.

[17] J. H. S. Lee, The Detonation Phenomenon. Cambridge University Press, 2008.

123

[18] W. Fickett and W. Davis, Detonation: Theory and Experiment, ser. Dover books

on physics. Dover Publications, 2000, ISBN: 9780486414560.

[19] M. Berthelot and P. Vielle, “On the velocity of propagation of explosive processes

in gases,” C. R. Hebd. Seances Acad. Sci., vol. 93, no. 2, pp. 18–21, 1881.

[20] E. Mallard and H. L. Chatelier, “Sur la vitesse de propagation de l’inflammation

dans les melanges gazeux explosifs,” Comptes Rendus Academie des Sciences,

vol. 93, pp. 145–148, 1881.

[21] D. L. Chapman and B. (Oxon.), “On the rate of explosion in gases,” The London,

Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 47,

no. 284, pp. 90–104, 1899.

[22] W. J. M. Rankine, “On the thermodynamic theory of waves of finite longitudinal

disturbance,” Philosophical Transactions of the Royal Society of London, vol. 160,

pp. 277–288, 1870, ISSN: 02610523.

[23] H. Hugoniot, “Propagations of movements in bodies and specially in ideal gases,”

Journel de l’Acole polytechnique, vol. 57, pp. 1–97, 1887.

[24] C. Campbell and D. W. Woodhead, “The ignition of gases by an explosion

wave. Part I. Carbon monoxide and hydrogen mixtures,” J. Chem. Soc., vol. 129,

pp. 3010–3021, 0 1926.

[25] Y. B. Zeldovich, Theory of combustion and Detonation of Gases. Moscow:

Izdatel’stvo Akademii Nauk SSSR, 1944.

[26] J. V. Neuman, “Theory of detonation waves,” Office of Scientific Research and

Development, Washington D. C., Tech. Rep. 238, Apr. 1942.

[27] M. Berthelot and P. Vielle, “On the detonation process in gases,” Annalen Der

Physik, 5e Folge, vol. 43, no. 2, pp. 421–436, 1943.

124

[28] K. Kuo, Principles of combustion. John Wiley, 2005, ISBN: 9780471046899.

[29] J. H. Lee, “Dynamic structure of gaseous detonation,” in Dynamic Structure of

Detonation in Gaseous and Dispersed Media, A. A. Borissov, Ed. Dordrecht:

Springer Netherlands, 1991, pp. 1–25, ISBN: 978-94-011-3548-1.

[30] D. R. White, “Turbulent structure of gaseous detonation,” The Physics of Fluids,

vol. 4, no. 4, pp. 465–480, 1961.

[31] B. V. Voitsekhovskii, V. V. Mitrofanov, and M. E. Topchiyan, “Structure of the

detonation front in gases (survey),” Combustion, Explosion and Shock Waves,

vol. 5, no. 3, pp. 267–273, Jul. 1969, ISSN: 1573-8345.

[32] S. Turns, An introduction to combustion: concepts and applications, ser. McGraw-

Hill series in mechanical engineering. McGraw-Hill, 1996, ISBN: 9780079118127.

[33] Y. B. Zeldovich, S. Kogarko, and N. Simonov, “An experimental investigation of

spherical detonation of gases,” Sov. Phys. Tech. Phys., vol. 1, pp. 1689–1713, Jan.

1956.

[34] Y. Wu, F. Ma, and V. Yang, “System performance and thermodynamic cycle

analysis of airbreathing pulse detonation engines,” Journal of Propulsion and

Power, vol. 19, no. 4, pp. 556–567, 2003.

[35] B. Zeldovich, “On the theory of the propagation of detonations in gaseous

systems,” Journal of Experimental and Theoretical Physics, vol. 10, pp. 542–568,

1940.

[36] E. Wintenberger and J. Shepherd, “On the theory of the propagation of detona-

tions in gaseous systems,” Journal of Propulsion and Power, vol. 22, pp. 694–698,

2006.

125

[37] E. Wintenberger, J. M. Austin, M. Cooper, S. Jackson, and J. E. Shepherd, “Reply

to comment on ”analytical model for the impulse of single-cycle pulse detonation

tube” by w.h. heiser and d.t. pratt,” Journal of Propulsion and Power, vol. 20,

no. 1, pp. 189–191, 2004.

[38] J. A. C. Kentfield, “Thermodynamics of airbreathing pulse-detonation engines,”

Journal of Propulsion and Power, vol. 18, no. 6, pp. 1170–1175, 2002.

[39] Pulsed Detonation Engines, http://arc.uta.edu/research/pde.

htm, Accessed: 2018-10-30.

[40] T. Bussing and G. Pappas, “An introduction to pulse detonation engines,” in 32nd

Aerospace Sciences Meeting and Exhibit.

[41] R. Vutthivithayarak, E. M. Braun, and F. K. Lu, “On thermodynamic cycles for

detonation engines,” in 28th International Symposium on Shock Waves, K. Kontis,

Ed., Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, pp. 287–292, ISBN:

978-3-642-25685-1.

[42] R. Bellini, “Ideal cycle analysis of a regenerative pulse detonation engine for

power production,” PhD thesis, The University of Texas at Arlington, 2010.

[43] T.-H. Yi, D. Wilson, and F. Lu, “Detonation wave propagation in an ejector-

augmented pulse detonation rocket,” in 44th AIAA Aerospace Sciences Meeting

and Exhibit.

[44] N. Hoffmann, “Reaction propulsion by intermittent detonative combustion,” Ger-

man Ministry of Supply, AI152365 Volkenrode Translation, 1940.

[45] K. M. Pandey and P. Debnath, “Review on recent advances in pulse detonation

engines,” Journal of Combustion, vol. 2016, p. 16, 2016.

126

[46] M. Roy, “Propulsion par statoreacteur a detonation (detonation ramjet propul-

sion),” Comptes Rendus Hebdo-madaires des Seances de lAcademie des Sciences,

vol. 222, no. 2, pp. 31–32, 1946.

[47] J. Nicholls and E. Dabora, “Recent results on standing detonation waves,” Sym-

posium (International) on Combustion, vol. 8, no. 1, pp. 644–655, 1961, Eighth

Symposium (International) on Combustion, ISSN: 0082-0784.

[48] R. A. Gross, “Recent advances in gaseous detonation,” ARS Journal, vol. 29,

no. 3, pp. 173–179, 1959. DOI: 10.2514/8.4713.

[49] R. A. Gross and W. Chinitz, “A study of supersonic combustion,” Journal of the

Aerospace Sciences, vol. 27, no. 7, pp. 517–524, 1960. DOI: 10.2514/8.8620.

[50] “Intermittent detonation as a thrust-producing mechanism,” Journal of Jet Propul-

sion, vol. 27, no. 5, pp. 534–541, 1957. DOI: 10.2514/8.12851.

[51] D. Allgood, E. Gutmark, J. Hoke, R. Bradley, and F. Schauer, “Performance

studies of pulse detonation engine ejectors,” Journal of Propulsion and Power,

vol. 24, no. 6, pp. 1317–1323, 2008.

[52] P. Changxin, F. Wei, Z. Qun, Y. Cheng, C. Wenjuan, and Y. Chuanjun, “Experi-

mental study of an air-breathing pulse detonation engine ejector,” Experimental

Thermal and Fluid Science, vol. 35, no. 6, pp. 971–977, 2011, ISSN: 0894-1777.

[53] A. E. Korobov and S. V. Golovastov, “Investigation of the effect of the ejector

on the performance of the pulse detonation engine nozzle extension,” vol. 653,

p. 012 065, Nov. 2015.

[54] D. Wilson, F. Lu, H. Kim, and R. Munipalli, “Analysis of a pulsed normal

detonation wave engine concept,” in 10th AIAA/NAL-NASDA-ISAS International

Space Planes and Hypersonic Systems and Technologies Conference.

127

[55] J.-l. Li, W. Fan, H. Qiu, C.-j. Yan, and Y.-Q. Wang, “Preliminary study of a pulse

normal detonation wave engine,” Aerospace Science and Technology, vol. 14,

no. 3, pp. 161–167, 2010, ISSN: 1270-9638.

[56] H. Kim, Y. F. K. Lu, D. A. Anderson, and D. R. Wilson, “Numerical simulation

of detonation process in a tube,” CFD J, pp. 227–241,

[57] Q. Qu, B. C. Khoo, H.-S. Dou, and H. M. Tsai, “The evolution of a detonation

wave in a variable cross-sectional chamber,” Shock Waves, vol. 18, no. 3, p. 213,

Jul. 2008, ISSN: 1432-2153.

[58] H. Y. Fan and F. K. Lu, “Numerical simulation of detonation processes in a

variable cross-section chamber,” Proceedings of the Institution of Mechanical

Engineers, Part G: Journal of Aerospace Engineering, vol. 222, no. 5, pp. 673–

686, 2008.

[59] M. V. Papalexandris, “A numerical study of wedge-induced detonations,” Com-

bustion and Flame, vol. 120, no. 4, pp. 526–538, 2000, ISSN: 0010-2180.

[60] M. T. Walter and L. F. Figueira Da Silva, “Numerical study of detonation sta-

bilization by finite length wedges,” AIAA Journal, vol. 44, no. 2, pp. 353–361,

2006.

[61] F. K. Lu, H. Fan, and D. R. Wilson, “Detonation waves induced by a confined

wedge,” Aerospace Science and Technology, vol. 10, no. 8, pp. 679–685, 2006,

ISSN: 1270-9638.

[62] H. Teng, H. D. Ng, and Z. Jiang, “Initiation characteristics of wedge-induced

oblique detonation waves in a stoichiometric hydrogen-air mixture,” Proceedings

of the Combustion Institute, vol. 36, no. 2, pp. 2735–2742, 2017, ISSN: 1540-

7489.

128

[63] H. Y. Fan and F. K. Lu, “Numerical modelling of oblique shock and detonation

waves induced in a wedged channel,” Proceedings of the Institution of Mechanical

Engineers, Part G: Journal of Aerospace Engineering, vol. 222, no. 5, pp. 687–

703, 2008.

[64] H. Fan and F. Lu, “Numerical study of reactive flow past a wedge in a channel,”

in 43rd AIAA Aerospace Sciences Meeting and Exhibit.

[65] A. NAGEL and J. BECKER, “Key technology for airbreathing hypersonic air-

craft,” in 9th Annual Meeting and Technical Display.

[66] J. V. Becker, “New approaches to hypersonic aircraft,” The organization, Rome,

Italy: Seventh Congress of the International Council of the Aeronautical Sciences,

Jul. 1970.

[67] J. V. Becker and F. S. Kirkman, “Hypersonic transports,” Vehicle Technology for

Civil Aviation - The Seventies and Beyond, NASA SP-292, 1971, pp. 429–445.

[68] P. J. Johnston, J. M. Cubbage, and J. P. Weidner, “Studies of engine-airframe

integration on hypersonic aircraft,” Journal of Aircraft, vol. 8, no. 7, pp. 495–501,

1971.

[69] C. J. M. and K. F.S, “Investigation of engine-exhaust-airframe interference on

a cruise vehicle at mach 6,” NASA Langley Research Center; Hampton, VA,

United States, Tech. Rep. 2, Jan. 1971.

[70] W. J. Small, J. P. Weidner, and P. Johnston, “Scramjet nozzle design and analysis

as applied to a highly integrated hypersonic research airplane,” NASA Langley

Research Center, Tech. Rep., Nov. 1976.

[71] M. Abbott, An Introduction to the Method of Characteristics. American Elsevier,

1966.

129

[72] M. Eklund, M. Alamaniotis, H. Hernandez, and T. Jevremovic, “Method of

characteristics a“ a review with applications to science and nuclear engineering

computation,” Progress in Nuclear Energy, vol. 85, pp. 548–567, 2015, ISSN:

0149-1970.

[73] C. Tai, Y. Zhao, and K. Liew, “Parallel-multigrid computation of unsteady incom-

pressible viscous flows using a matrix-free implicit method and high-resolution

characteristics-based scheme,” Computer Methods in Applied Mechanics and

Engineering, vol. 194, no. 36, pp. 3949–3983, 2005, ISSN: 0045-7825.

[74] J. Anderson, Modern Compressible Flow: With Historical Perspective, ser. Aero-

nautical and Aerospace Engineering Series. McGraw-Hill Education, 2003, ISBN:

9780072424430.

[75] B. D.L. and L. J.A., “Integration of advanced exhaust nozzles,” AGARD-CP-301,

Tech. Rep., Sep. 1981.

[76] Integration of Turbo-Expander- and Turbo-Ramjet-Engines in Hypersonic Ve-

hicles, vol. Volume 2: Aircraft Engine; Marine; Microturbines and Small Tur-

bomachinery, Turbo Expo: Power for Land, Sea, and Air, V002T02A007, Jun.

1992.

[77] P. Perrier, M. Rapuc, P. Rostand, R. Hallard, D. Regard, A. Dufour, and O.

Penanhoat, “Nozzle and afterbody design for hypersonic airbreathing vehicles,”

in Space Plane and Hypersonic Systems and Technology Conference.

[78] T. Murugan, “Scramjet nozzle design using method of characteristics,” PhD

thesis, MasteraTMs thesis, Indian Institute of Technology, Kanpur, India, 2003.

130

[79] A. Ridgway, A. Sam, and A. Pesyridis, “Modelling a hypersonic single expansion

ramp nozzle of a hypersonic aircraft through parametric studies,” Energies,

vol. 11, no. 12, p. 3449, 2018.

[80] V. Thiagarajan, S. Panneerselvam, and E. Rathakrishnan, “Numerical flow vi-

sualization of a single expansion ramp nozzle with hypersonic external flow,”

Journal of visualization, vol. 9, no. 1, pp. 91–99, 2006.

[81] C. Hirschen and A. Gulhan, “Influence of heat capacity ratio on pressure and

nozzle flow of a scramjets,” Journal of Propulsion and Power, vol. 25, no. 2,

pp. 303–311, 2009.

[82] L. Jianping, S. Wenyan, X. Ying, and L. Feiteng, “Influences of geometric param-

eters upon nozzle performances in scramjets,” Chinese Journal of Aeronautics,

vol. 21, no. 6, pp. 506–511, 2008.

[83] W. Huang, Z.-g. Wang, D. B. Ingham, L. Ma, and M. Pourkashanian, “Design

exploration for a single expansion ramp nozzle (sern) using data mining,” Acta

Astronautica, vol. 83, pp. 10–17, 2013.

[84] R. Vos and S. Farokhi, Introduction to transonic aerodynamics. Springer, 2015,

vol. 110.

[85] W. H. Heiser and D. T. Pratt, Hypersonic airbreathing propulsion. Aiaa, 1994.

[86] Y. Yu, J. Xu, J. Mo, and M. Wang, “Numerical investigation of separation pattern

and separation pattern transition in overexpanded single expansion ramp nozzle,”

The Aeronautical Journal, vol. 118, no. 1202, pp. 399–424, 2014.

[87] B. J. McBride, S. Gordon, and M. A. Reno, “Coefficients for calculating thermo-

dynamic and transport properties of individual species,” 1993.

131

[88] S. Turns, An Introduction to Combustion: Concepts and Applications, ser. McGraw-

Hill series in mechanical engineering. McGraw-Hill, 2012, ISBN: 9780071086875.

[89] R. Kee, M. Coltrin, and P. Glarborg, Chemically Reacting Flow: Theory and

Practice. Wiley, 2005, ISBN: 9780471461302.

[90] R. J. Kee, F. M. Rupley, E. Meeks, and J. A. Miller, Chemkin-iii: A fortran

chemical kinetics package for the analysis of gas phase chemical and plasma

kinetics,” sandia national laboratories report, 1996.

[91] T. R. Bussing and E. M. Murman, “Finite-volume method for the calculation of

compressible chemically reacting flows,” AIAA journal, vol. 26, no. 9, pp. 1070–

1078, 1988.

[92] D. Anderson, J. C. Tannehill, and R. H. Pletcher, Computational fluid mechanics

and heat transfer. CRC Press, 2016.

[93] ANSYS R© CFD, Release 18.1.

[94] C. Laney, Computational Gasdynamics, ser. Computational Gasdynamics. Cam-

bridge University Press, 1998, ISBN: 9780521625586.

[95] Ansys Fluent 12.0 Theory guide, chapter 18.3.1. 2009.

[96] D. Migdal and J. J. Horgan, “Thrust nozzles for supersonic transport aircraft,”

1964.

[97] G. A. Sod, “A survey of several finite difference methods for systems of nonlinear

hyperbolic conservation laws,” Journal of Computational Physics, vol. 27, no. 1,

pp. 1–31, 1978, ISSN: 0021-9991.

[98] B. J. McBride, M. J. Zehe, and S. Gordon, “Nasa glenn coefficients for calculating

thermodynamic properties of individual species,” 2002.

132

[99] M. Jelezniak and I. Jelezniak, “Chemked: Chemical kinetics of gas phase reac-

tions,” 2007.

[100] T.-H. Yi, D. Anderson, D. Wilson, and F. Lu, “Numerical study of two-dimensional

viscous, chemically reacting flow,” in 17th AIAA Computational Fluid Dynamics

Conference, 2005, p. 4868.

[101] A. Omprakas, “Numerical simulation of unsteady normal detonation combustion,”

Master’s thesis, University of Texas at Arlington, 2018.

[102] M. C. Davis and J. T. White, “X-43a flight-test-determined aerodynamic force

and moment characteristics at mach 7.0,” Journal of spacecraft and rockets,

vol. 45, no. 3, pp. 472–484, 2008.

[103] R. Kumar, A. Omprakas, and D. Wilson, “Numerical simulation of normal

detonation wave engine,” in AIAA Scitech 2019 Forum.

[104] H. Yoshihara, “Gasdynamics of rocket exhaust plumes,” in The Middle Ultravio-

let: Its Science and Technology, 1966, p. 269.

[105] R. Castner, A. Elmiligui, and S. Cliff, “Exhaust nozzle plume and shock wave in-

teraction,” in 51st AIAA Aerospace Sciences Meeting including the New Horizons

Forum and Aerospace Exposition, 2013, p. 12.

[106] R. Castner, K. Zaman, A. Fagan, and C. Heath, “Wedge shock and nozzle exhaust

plume interaction in a supersonic jet flow,” Journal of Aircraft, vol. 54, no. 1,

pp. 125–134, 2017.

[107] C. Bauer, A. Koch, F. Minutolo, and P. Grenard, “Engineering model for rocket

exhaust plumes verified by cfd results,” in 29th International Symposium on

Space Technology and Science, 2013.

133

[108] W. Stillwell, X-15 Research Results: With a Selected Bibliography, ser. NASA

SP. Scientific, Technical Information Division, National Aeronautics, and Space

Administration, 1965. [Online]. Available: https://books.google.com/

books?id=fpxPAQAAIAAJ.

[109] R. Kumar, A. Omprakas, and D. Wilson, “Numerical simulation of normal

detonation wave engine,” in AIAA Scitech 2019 Forum.

134