z3422572_Thesis

School of Mechanical and Manufacturing Engineering
Faculty of Engineering
UNSW Australia
Optimal Design and Numerical
Analysis of Axisymmetric Nozzles
By
Rahul Singh
z3422572
Thesis submitted as a requirement for the degree
of Bachelor of Engineering in Mechanical
Engineering
October 2015
Dr. J. Olsen

Rahul Singh Optimal Design and Numerical Analysis of
Axisymmetric Nozzles
i
Certificate of Originality
I, Rahul Singh declare that this submission is my own work and to the best of my
knowledge it contains no materials previously published or written by another person, or
substantial proportions of material which have been accepted for the award of any other
degree or diploma at UNSW or any other educational institution, except where due
acknowledgement is made in the thesis. Any contribution made to the research by others,
with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the
thesis.
I also declare that the intellectual content of this thesis is the product of my own work,
except to the extent that assistance from others in the project’s design and conception in
style, presentation and linguistic expression is acknowledged.
Signed ………………………………
Date ………………………………

ii
ABSRACT
Improvement in computational technology has opened up a vast area of research in
analysing design theories and their future feasibility. Method of Characteristic (MOC) has
always been an attractive choice in analysing the supersonic nozzle flow-fields.
The prime objective of the thesis is to develop a MATLAB code that uses the MOC to
produce ideal thrust optimised nozzle contours incorporating recent developments for
isentropic, inviscid, irrotational supersonic flows. For nozzles in particular and contour
design in general, the problem involves heavy user interface. For this reason, a graphical
interface has been developed and implemented in this report.
The code designs the contour and simulates the flow field for an input exit Mach number
and expansion angle. Once the boundary conditions are set, the MATLAB code then
generates the full length MOC contour. This contour can then be truncated by removing
excessive length, for use on rocket engines
The truncated contours design developed in this thesis has been validated against the
selected test cases. In order to verify and validate the MATLAB code, existing known
nozzles design cases were chosen. Results were obtained by inputting boundary conditions
into the computer program. Contour, wall pressure distribution and the thrust coefficient
comparison has been analysed in this thesis report. The results are in agreement with the
test results. In principle, MOC is truly exact only in the limit of an infinite number of
characteristic lines.

iii
Acknowledgements
My Appreciation and deepest gratitude to Dr. John Olsen, for being my supervisor and
throughout guidance in this project. I would like to extend specially thank to Mr. Kyll
Schomberg for his continuous supervision and expert advice throughout this project.
Huge thanks to thank Dom, Nikhil, Abhi, Fahim, Milen and Hilton for their help and
suggestion. It would have been a lonely semester without them. Special thanks to David
Herd and Mechanical School administration.
Finally, I would also like to thank my parents, and sister for their support and
encouragement.

iv
Nomenclature
Symbol Variable Unit
A Area 𝑚2
AR Area Ratio N/A
a Sonic Velocity 𝑚/𝑠
CD Convergent-Divergent N/A
CFD Computational Fluid Dynamics N/A
CF Thrust Coefficient Non-dimensional
γ Ratio of specific heats Non-dimensional
F Force N
Isp Specific impulse 𝑁𝑠𝑒𝑐/𝑘𝑔
λ Divergence loss Non-dimensional
LP Liquid Propulsion N/A
M Mach number Non-dimensional
MOC Method of characteristics N/A
𝑚̇ Mass flow rate 𝑘𝑔/𝑠
P Pressure Pa
PR Pressure Ratio N/A
ρ Density 𝑘𝑔/𝑚3
rt Throat radius Mm
RANS Reynolds-average Navier-Strokes N/A
R Specific Heat Ratio 𝑘𝐽/𝑘𝑔𝐾

v
S-A Spalart-Allmaras N/A
SST Shear Stress Transport N/A
T Static Temperature K
TIC Truncated Ideal Contour N/A
TOC Thrust Optimised Contour N/A
TOP Thrust Optimised Parabola N/A
μ Mach angle 𝐷𝑒𝑔𝑟𝑒𝑒𝑠
V Velocity 𝑚/𝑠
𝜈 Prandtl-Meyer angle Degrees
x, r Axial and radial Cartesian
coordinates
𝑚
u, v Axial and radial velocity
coordinates
𝑚/𝑠
Subscripts
Symbol Variable
0 Total or stagnation conditions
a Ambient conditions
c Combustion conditions
e Conditions at nozzle exit
* Conditions at nozzle throat

vi
TABLE OF CONTENTS
ABSRACT............................................................................................................................ii
Acknowledgements .............................................................................................................iii
Nomenclature....................................................................................................................... iv
TABLE OF CONTENTS ....................................................................................................vi
List of Figures...................................................................................................................... xi
List of Tables.....................................................................................................................xiii
Chapter 1 INTRODUCTION ............................................................................................... 1
1.1 Background................................................................................................................. 1
1.2 Problem Overview...................................................................................................... 2
Chapter 2 Literature Review................................................................................................. 4
2.1 Introduction ................................................................................................................ 4
2.2 One-dimensional Nozzle Flow Theory....................................................................... 5
2.2.1 Inviscid, Compressible Flow............................................................................... 5
2.2.2 Sonic velocity and Mach number........................................................................ 6
2.2.3 Stagnation............................................................................................................ 6
2.2.4 Quasi One Dimensional Flow ............................................................................. 8
2.2.5 Combustion Chamber........................................................................................ 11
2.3 Shocks....................................................................................................................... 11

vii
2.3.1 Prandtl-Meyer Expansion Waves...................................................................... 11
2.4 Nozzle Flows............................................................................................................ 12
2.4.1 Nozzle losses ..................................................................................................... 13
2.5 Nozzle Development ................................................................................................ 13
2.6 Contour Nozzle Design ............................................................................................ 15
2.6.1 Nozzle Configuration ........................................................................................ 16
2.6.2 Ideal Nozzle....................................................................................................... 16
2.6.3 TIC Nozzle ........................................................................................................ 18
2.6.4 TOC Nozzle....................................................................................................... 18
2.6.5 Parabolic Bell Nozzles (TOP) ........................................................................... 19
2.6.6 LEA TOP Nozzle .............................................................................................. 20
2.6.7 VS1 Nozzle........................................................................................................ 20
2.6.8 Summary............................................................................................................ 21
2.7 Method of characteristics ......................................................................................... 23
2.7.1 Two Dimensional Planar nozzle........................................................................ 26
2.7.2 Axisymmetric Nozzle........................................................................................ 27
2.8 Throat Modelling...................................................................................................... 29
2.9 Thrust Coefficient..................................................................................................... 30
Chapter 3 Numerical Methods............................................................................................ 33
3.1 Introduction .............................................................................................................. 33
3.2 Geometry .................................................................................................................. 33

viii
3.2.1 TIC..................................................................................................................... 33
3.2.2 LEA nozzle........................................................................................................ 34
3.2.3 VS1.................................................................................................................... 35
3.3 Numerical Model...................................................................................................... 35
3.3.1 Governing Equations......................................................................................... 36
3.3.2 Numerical Solver Methods................................................................................ 37
3.4 Meshing .................................................................................................................... 40
3.5 Boundary Conditions................................................................................................ 42
3.6 Results & Conclusion............................................................................................... 42
Chapter 4 Computer Program Development.............................................................. 44
4.1 Introduction .............................................................................................................. 44
4.2 Methodology............................................................................................................. 45
4.2.1 Axisymmetric MOC .......................................................................................... 45
4.2.2 New Method ...................................................................................................... 50
4.3 Discussion................................................................. Error! Bookmark not defined.
Chapter 5 Results and Discussion................................................................................. 53
5.1 Contour Comparison ................................................................................................ 53
5.1.1 TIC Nozzle ........................................................................................................ 53
5.1.2 LEA Nozzle....................................................................................................... 55
5.1.3 VS1 Nozzle........................................................................................................ 56
5.2 Wall Pressure Distribution ....................................................................................... 58

ix
5.2.1 TIC Nozzle ........................................................................................................ 58
5.2.2 LEA Nozzle....................................................................................................... 59
5.2.3 VS1 nozzle......................................................................................................... 60
5.3 Thrust Coefficient..................................................................................................... 62
Chapter 6 Conclusion.................................................................................................... 64
6.1 Conclusion................................................................................................................ 64
6.2 Suggestions for Future Work.................................................................................... 65
Bibliography .................................................................................................................... 66
Appendices ...................................................................................................................... 69
6.1 Appendix A .............................................................................................................. 69
6.1.1 Nozzle Full Length Contours ............................................................................ 69
6.1.2 Wall Pressure Distribution(Full Length) ........................................................... 71
6.2 Appendix B .......................................................................................................... 73
6.2.1 Code GUI......................................................................................................... 73
6.2.2 Code nozzleDesignGUI..................................................................................... 89
6.2.3 Code nozzleDesign_CurveGUI......................................................................... 94
6.2.4 Code nozzlePlotCurve_GUI............................................................................ 100
6.2.5 Code sonicline_Kliegel ................................................................................... 104
6.2.6 Code mach_Val ............................................................................................... 105
6.2.7 Code nozzleDesignAxiSharp_GUI.................................................................. 106
6.2.8 Code nozzleDesignAxiSharp_GUI.................................................................. 113

x
6.3 Appendix C............................................................................................................. 120
CFD Contours........................................................................................................... 120
6.4 Appendix D ............................................................................................................ 122
6.5 Appendix E............................................................................................................. 123
Mesh Profiles............................................................................................................ 123
6.6 Appendix F ............................................................................................................. 124
Hall Method.............................................................................................................. 124
Kliegel-Levine.......................................................................................................... 127

xi
List of Figures
Figure 2.2-1: Illustration and comparison of a supersonic nozzle and a supersonic diffuser
............................................................................................................................................ 10
Figure 2.6-1: Basic flow structure in an ideal nozzle [8] ................................................... 17
Figure 2.6-2 - Basic flow structure in a thrust optimised nozzle contour [8].................... 18
Figure 2.6-3: Mach number distribution and shock formation in conical(150
), TIC, TOC,
TOP nozzles with ε = 43.4[8]............................................................................................. 22
Figure 2.7-1 Intersecting Characteristic Lines ................................................................... 24
Figure 2.7-2: Nozzle Expansion fans ................................................................................. 25
Figure 3.2-1: Sketch of horizontal test section [19] ........................................................... 34
Figure 3.2-2: Numerical domain with outlet and inlet dimension (not to scale)[27] ......... 34
Figure 3.2-3: Numerical domain including inlet and downstream exhaust dimensions (not
to scale)[26]........................................................................................................................ 35
Figure 3.4-1: Meshing of TIC nozzle................................................................................. 41
Figure 4.2-1 An overview of method of characteristic for axisymmetric nozzle............... 48
Figure 4.2-2: Initial Characteristic mesh ............................................................................ 49
Figure 5.1-1: Truncated Nozzle Contours TIC................................................................... 54
Figure 5.1-2: Full Design Nozzle Contours TIC ................................................................ 55
Figure 5.1-3: Truncated Nozzle Contour LEA................................................................... 56

xii
Figure 5.1-4: Truncated Nozzle Contour VS1 ................................................................... 57
Figure 5.2-1 Nondimensional wall pressure, TIC .............................................................. 59
Figure 5.2-2: Nondimensional Wall Pressure, LEA........................................................... 60
Figure 5.2-3: Nondimensional Wall Pressure, VS1 ........................................................... 61
Figure 6.1-1: LEA Nozzle Contour of full length design................................................... 69
Figure 6.1-2: VS1 Nozzle Contour of full length design ................................................... 70
Figure 6.3-1: TIC Velocity Contour................................................................................. 120
Figure 6.3-2: LEA Velocity Contour................................................................................ 120
Figure 6.3-3: VSI Velocity Contour................................................................................. 121
Figure 6.4-1: Example of Planar Nozzle GUI.................................................................. 122
Figure 6.4-2: Example of Axisymmetric GUI.................................................................. 122
Figure 6.5-1: Mesh profile of LEA nozzle ....................................................................... 123
Figure 6.5-2: Mesh Profile of VS1 nozzle........................................................................ 123

xiii
List of Tables
Table 2-1: Geometrical Configuration of different nozzle................................................. 20
Table 4-1Boundary Parameters for MOC .......................................................................... 46
Table 5-1: Thrust Coefficients calculation from MOC ...................................................... 62

1
Chapter 1
INTRODUCTION
1.1 Background
The nozzle on a rocket engine directs the enormous thermal energy released by the
combustion process into a strong, directional high speed flow. The performance of the
rocket nozzle is highly dependent on the aerodynamic design and the operating conditions
with main the focus being the contour shape.
Due to high cost of transporting a payload into orbit, any increase in efficiency is highly
desired. This restricts access to the selected governments or corporations that can afford
huge expenses on satellite launch. The requirement for greater efficiency has led to several
new concepts being examined.
The propulsion system that supports the key launch vehicle functions during ascent is a
significant factor affecting cost, complexity and reliability. The performance requirement
of all launch propulsion systems is dictated by the physics of escaping Earth’s gravity.
Almost all feasible launch systems require propulsion, and their efficiency peaks when the
forwards vehicle velocity equals the exhaust velocity.
Increase in performance can be achieved by optimised nozzle contour first proposed by
Rao [12].The contour of the converging diverging (CD) nozzle is comprised of a
convergent segment, throat and a divergent segment. Each section affects the flow
differently and must be considered in the analysis of the nozzle performance.

2
One way of designing a rocket engine is by using the method of characteristics (MOC),
which allows for physical boundaries to be found directly. The MOC uses a technique of
succeeding propagation paths in order to find a solution by simultaneously solving the
partial differential equations. The propagation paths or characteristics represent geometric
form, discontinuity or physical disturbances. The method of characteristics over the years
has been extremely effective for the analysis of supersonic and inviscid flows, being both
rapid and accurate.
A viable axisymmetric method of characteristics solution for a two-dimensional,
irrotational, steady flow has been developed. For nozzles in particular and contour design
in general, the problem involves heavy user interface. For this reason, a graphical interface
has been developed and implemented in this report.
1.2 Problem Overview
The primary motivation for this research arises from the need to develop software for a
well-defined technique in nozzle analysis.
This thesis will cover implementation of MOC in both planar and axisymmetric nozzles,
analysis of the transonic region and the development of hybrid code.
Chapter 2, Literature Review, establishes the theoretical and literal background on
thermodynamics of flow and advance nozzle concepts.
Chapter 3, Numerical Methods, focuses on the computational fluid dynamics (CFD) for
the verification and validation of the developed code. The chapter discuss in brief, the
methodology, assumption and results from the analysis.
Chapter 4, Computer Program Development, explores into the methodology used to
develop the MATLAB code.

3
Chapter 5, Results and Discussion, compiles all the results to validate and verify the
working of the MATLAB code.
Chapter 6, Conclusion, summarises the major finding with recommendations for future
research improvements and possibilities.

4
Chapter 2
Literature Review
2.1 Introduction
The importance of nozzle design to modern space exploration is well defined in numerous
resources [1,3,6]. Extensive research has been conducted in making nozzles more efficient
and reliable. A nozzle’s efficiency is primarily dependent on nozzle area ratio and the
ambient operating pressure. To achieve peak efficiency, the exit pressure should be equal
to the ambient pressure.
Furthermore, numerous areas on rocket design theories and their future feasibility with
improvements in computational technology have not been analysed in detail. This
improvement in computational method opened up a vast area of research to analyse the
possibility of further development into current methodologies.
To fully explore the behaviour of the fluid within a supersonic nozzle, it is important to
fully understand the various conditions experienced by a fluid in supersonic velocity. The
purpose of this chapter is to explore the behaviour of the fluid within a supersonic nozzle,
and previous attempts to design nozzle contour and how current design techniques could
be applied for this purpose.
Although the focus of this project is specifically on ideal contour design but due to
validation and analysis of the code, other nozzle concepts will be identified and covered in
this chapter.

5
The topics cover the necessary information required to understand the Numerical methods
used to design the nozzle Contour and the thrust optimised contour.
2.2 One-dimensional Nozzle Flow Theory
To evaluate flow behaviour and nozzle performance, it is crucial to know the relationship
between thermodynamic parameters such as density, pressure, velocity and temperature.
The purpose of the present section is to review aspects of thermodynamics that are
important to compressible flows.
2.2.1 Inviscid, Compressible Flow
The air or other gases, unless entrapped is effectively incompressible in a steady flow as
long as its velocity relative to an obstacle or container walls is well below speed of sound.
But when fluid speed approaches the speed of sound, compression is unavoidable. This
section briefly discusses, concepts related compressible flow in ideal fluids specifically
through the nozzles.
In case of inviscid, incompressible flow, two basic equations namely continuity and
momentum are required. In such flows, ρ and T are assumed to be constant throughout
flow and is governed purely by mechanical laws. In contrast, ρ is a variable in
compressible flow and this introduction of a new variable requires additional governing
equation that is energy equation in this case. The five unknowns namely p, V, ρ, e and T
are evaluated by continuity, momentum, energy, state and internal energy equations.
Additional, two equations are obtained with assumption of calorically perfect gas. It is
important to note that Bernoulli’s equation is derived from the momentum and continuity
equations and thus, does not hold for compressible flow. [1,2]

6
2.2.2 Sonic velocity and Mach number
The velocity of sound or sonic velocity is a fundamental parameter in compressible flow
theory. It is equal to the propagation speed of an elastic pressure wave, sound being an
infinitesimal pressure wave. Velocity of sound in ideal gases is independent of pressure. In
definition, local speed of sound is defined as:
𝑎 = √𝛾𝑅𝑇
2.1
Mach number is another important parameter as it compares the speed of sound in fluid
and fluid flowing speed. Mach number M is a dimensionless flow parameter, defined as
𝑀 =
𝑉
𝑎
2.2
The flow with Mach number equal, greater and less than 1 is called sonic, supersonic and
subsonic respectively. Another terminology that is widely used in this report is Mach
angles and Mach wave. For supersonic flows, small disturbances are transmitted
downstream within a cone. The vertex half angle of the generated Mach cone is referred to
as Mach angles.
𝑀𝑎𝑐ℎ 𝐴𝑛𝑔𝑙𝑒 = 𝜇 = sin−1
1
𝑀
2.3
Mach wave is a pressure wave travelling with the speed of sound across the fluid at the
Mach angle. [1,3]
2.2.3 Stagnation
For compressible flow, stagnation condition at a point is properties of a flow (such as T, P,
ρ) obtained if the local flow were imagined to cease to zero velocity isentropically. In

7
practice, this would correspond to the thermodynamic state of a very low-speed flow
entering the nozzle inlet from an upstream combustion chamber. The stagnation properties
are donated by a subscript zero or referred as chamber conditions with subscript c.
When the potential energy is negligible as in high-speed flows, the stagnation enthalpy
represents the total energy of a flowing fluid stream per unit mass as:
ℎ0 = ℎ +
𝑉2
2
= ℎ(1 +
𝛾 − 1
2
𝑀2
)(𝐾𝐽/𝐾𝑔)
2.4
The stagnation enthalpy of a fluid remains constant during a steady-flow in absence of any
heat and work interactions. For a large number of practical compressible flow problems,
the temperature is moderate. The fluid is thus approximated as an ideal gas with constant
specific heat, the enthalpy can be replaces by cpT. This gives the expression of the
stagnation temperature, the temperature a fluid attains when it is brought to rest
adiabatically, shown as:
𝑐 𝑝 𝑇0 = 𝑐 𝑝 𝑇 +
𝑉2
2
or,
𝑇0
𝑇
= 1 +
𝛾 − 1
2
𝑀2
2.5
Isentropic relations [1] can be used to obtain stagnation pressure and stagnation density
expression as:
𝑃0
𝑃
= (
𝑇0
𝑇
)
𝛾/(𝛾−1)
= [1 +
𝛾 − 1
2
𝑀2
]
𝛾
𝛾−1
2.6
𝜌0
𝜌
= (
𝑇0
𝑇
)
1/(𝛾−1)
= [1 +
𝛾 − 1
2
𝑀2
]
1
𝛾−1
2.7

8
Note that in a combustion chamber the gas velocity is small and thus local combustion
pressure and temperature are essentially equal to the stagnation pressure and temperature.
2.2.4 Quasi One Dimensional Flow
In reality the flow is three dimensional and flow field variables are functions of x, y, and z.
However, when the area variation is gradual then we can neglect the changes in cross
stream directions. In such a case, axial direction component, x is large enough compared
to y and z directions and the flow field variables are assumed to vary only with x direction.
Such a flow is defined as quasi-one-dimensional flow[1,3].
The integral form of the continuity, momentum and energy equation can be evaluated for
the quasi-one-dimensional flow. The continuity for the steady state can thus be
represented as,
𝜌𝑉𝐴 = 𝑐𝑜𝑛𝑡
2.8
The differential forms of equations are essential to examine the some physical
characteristics. Differential form of the continuity equation:
𝑑(𝜌𝑉𝐴) = 0
2.9
Differential form of the momentum equation for steady, inviscid, quasi-one-dimensional
flow known as Euler’s equation can be represented as:
𝑑𝑝 = −𝜌𝑉𝑑𝑉
2.10
Differential form of energy equation is derived from the enthalpy equation, which is
represented as:

9
𝑑ℎ + 𝑉𝑑𝑉 = 0
2.11
Above three equations are differential forms of continuity, momentum and energy
equations for inviscid, adiabatic quasi one-dimensional flow. Combination of above
equations and definition of speed of sound gives area- velocity relation [1] as
𝑑𝐴
𝐴
= (𝑀2
− 1)
𝑑𝑉
𝑉
2.12
The above area-velocity relation is important to determine the conditions that define
whether a nozzle or diffuser should be converging or diverging for a given flow velocity
and the condition at the throat. The velocity 𝑉is a vector quantity and the positive value of
dV indicates an increase in velocity and a negative dV implies decrease in velocity. From
above equation and Figure 2-1 states that:
 For positive dV (increasing velocity), the subsonic flow is associated with the
decrease in area (negative dA) as seen in the converging part of the supersonic
nozzle. The quantity in parentheses is positive in supersonic flow which is
associated with increase in area (positive dA) as in diverging part of supersonic
nozzle. For sonic flow, there is no change in area which corresponds to minimum
area.
 For negative dV (decreasing Velocity), the connotation of subsonic flow is with
increasing area and supersonic flow is with decreasing as in the supersonic
diffuser. Therefore, the diverging section acts as subsonic diffuser in which the
pressure increases and velocity decreases. The sonic flow, equation gives dA =0,
which corresponds to minimum area[1].

10
Figure 2-1: Illustration and comparison of a supersonic nozzle and a supersonic diffuser
As seen in Figure 2-1, the shape of the supersonic nozzle and the supersonic diffuser is the
same, but the difference is in the inlet velocity. Nozzle achieve supersonic flow at outlet
by converging the subsonic gas to make it achieves sonic velocity at throat and then
isentropically expand it in the diverging duct. Same is true for the supersonic diffuser,
which decelerates the incoming supersonic velocity in converging duct to subsonic in the
diverging duct.
Note that the viscous terms and gravity have been neglected in the momentum equation.

11
2.2.5 Combustion Chamber
The performance of rocket engine is directly dependent on the propellants, their
combustion efficiency and expansion of gases in the process. The combustion chamber is
the section of the thrust chamber where burning of propellants takes place. The chamber is
designed to retain the mixture of propellants that ensures complete mixing and combustion
[12]. The details of thrust chamber and the processes involved are beyond the scope of this
investigation.
2.3 Shocks
Occurrence of shocks in the flow field is mainly due to the compressibility of fluid flow.
Shocks occur due to interaction between compression waves and expansion waves causing
pressure rise [5].
The oblique shock occurs inside the nozzle if the walls of the nozzle are not just right
curved. To obtain isentropic shock free flow inside the nozzle, it is essential to account for
the three- dimensionality of the actual flow [1]. These shock waves are the important
contributor in non-ideal behaviour of nozzle. The strong compression discontinuities or
shock waves exist only inside diverging section of supersonic flow. The location of
separation plane inside the diverging section or estimating pressure at that point of a
supersonic nozzle has been empirical. [3]
Figure 2-4 on page number 22, shows the formation of shocks in different contours.
2.3.1 Prandtl-Meyer Expansion Waves
In expanding flow, the flow changes in order to conserve the mass. In continuous
expanding flow, the expansion fan composed of infinite number of Mach waves called
Prandtl-Meyer expansion waves. With the assumption of no heat or work, isentropic flow

12
relations is then to be used to calculate other flow properties downstream. By taking
velocity vectors for an infinitesimal turning angle, the expression can be given as:
𝑑𝑣 = √ 𝑀2 − 1
𝑑𝑉
𝑉
2.13
The above equation 2.13 can be integrated with known condition of 𝑣 = 0 at M = 1, to get
Prandtl-Meyer function.
2.4 Nozzle Flows
In general, nozzle flow always generates forces associated to the change in the flow
momentum. In a simple rocket nozzle, thrust is produced by ejecting mass from a chamber
backwards through a special shaped nozzle. The flow in nozzle is very rapid, i.e. adiabatic
with very little frictional losses, due to nearly one-dimensional flow and favourable
pressure gradient except shocks. Thus the assumption of isentropic all along the nozzle
can be made.
The mass flow rate through a nozzle is a maximum when throat achieves sonic condition.
The maximum mass flow rate through the nozzle can be expressed as:
𝑚̇ 𝑚𝑎𝑥 = 𝐴∗
𝑃0√
𝛾
𝑅𝑇0
(
2
𝛾 + 1
)
𝛾+1
[2(𝛾−1)]
2.14
The maximum mass flow, as given in above equation 2.14 through a nozzle for a given
throat area is fixed by the stagnation pressure and temperature of the inlet flow. This
implies that the flow rate is controlled by the combustion (stagnation) temperature and
pressure conditions. An increase in T0 or decrease in P0 will decrease the mass flow rate
through nozzle and vice versa [1].

13
A relation between throat Area A*
and flow area A can be written in a non-dimensional
form in terms of area ratio or nozzle expansion ratio as:
𝐴
𝐴∗
=
1
𝑀2
[
2
𝛾 + 1
(1 +
𝛾 − 1
2
𝑀2
)]
𝛾+1
2(𝛾−1)
2.15
At any point within a steady flow if the sonic velocity is reached, then it is impossible for
a pressure disturbance to travel past the location of sonic or supersonic flow. The physical
significance of this provided that the downstream pressure is below its critical value, is
that any disturbance of the flow downstream of sonic nozzle throat has no influence on the
throat or upstream. So, any decrease in exit pressure will not impact the velocity or the
flow rate in the nozzle. This condition is often described as choking the flow. [3]
2.4.1 Nozzle losses
Friction losses occur due to viscous dissipation within the boundary layer. There is also
possibility of incomplete combustion in the combustion chamber, leading to kinetic losses.
The nozzles flow in reality is non-adiabatic and not all of the kinetic energy of flow results
in axial thrust. Due to radial outflow, actual axial speed calculated is lower with one-
dimensional exit speed. Furthermore, losses can be due to internal shock in the nozzle.
2.5 Nozzle Development
In the field of Liquid Propulsion (LP), the first serious technical mention of a LP Rocket
engine has been credited to the Russian teacher K.E. Tsiolkovsky. His investigation
described the future role of use of rocket propelled vehicles and also, mathematically
derived the laws of motion of a changing mass body. He was first to show the equation for
the Earth escape velocity and developed the equation for rocket motion, which is now

14
known as Tsiolkowsky equation. Due to lack of support and funding, he wasn’t able to
perform the experiments and generate any experimental data for his design. The first
person however to design, develop, build and launch a flying vehicle with Liquid
propelled rocket engine was an American physics professor Robert H. Goddard. [28]
The development of Method of characteristics started when Ludwig Prandtl and Theodor
Meyer first worked out a theory for centred expansion waves in 1907–1908 and laid down
the fundamentals of expansion wave theory for supersonic flow[2]. Prandtl, Jakob Ackeret
and H. Glauret then developed the basis of theoretical approach focused on linearization of
the governing equations. Major contributions in developing the theory of characteristics
were made by French mathematician Jacques Salomon Hadamard in 1903 and the Italian
Tullio Levi-Civita in 1932. Their study aided in developing theory as a means to solve
general system of first order partial differential equations[9]. It was in 1929, when Prandtl
and Adolf Busemann found exact non-linear solution to the Euler equations for two
dimensional supersonic flows. The method implemented graphically allowed location
physical boundary[5,9]. This led to design of first practical supersonic wind tunnel by
Busemann in Germany in the mid-1930s, although Prandtl had a small supersonic facility
operational in 1905 for the study of shock waves. [1]
The first application of well-defined optimization technique using the calculus of
variations to design thrust optimised nozzles was done by Guderley and Hantsch [10] in
1955 for a fixed length nozzle. This formulated direct approach of finding the exit area
and nozzle contour to produce optimum thrust for the given nozzle length and the ambient
pressure was not widely adopted until a simplified solution was presented by Rao. Rao in
1958 developed a much easier technique of formulation and solution. This approach was
widely used in 1960’s but has been known to create shocks during the ignition
transient[5,6].
G.V.R. Rao also developed a procedure for computing optimum thrust exhaust nozzle
contour for axisymmetric nozzle in an isentropic flow. This method used a combination of
Lagrangian multipliers and MOC keeping the length of the nozzle is fixed constrain [6]. A
unique contour can be generated to maximise thrust for a prescribed length was described

15
by Rao often referred to as thrust-optimised contour (TOC). Rao derive the approximation
through a list of contour points using skewed parabola which is denoted as thrust-
optimized parabola (TOP).
The supersonic axisymmetric nozzle design concept using Method of Characteristics was
first conceptualised by Haddad and Mosst[4] but this was met with limited success when
solving the pure axi-symmetric supersonic nozzle design. Another approach by Wang and
Hoffman made use of the two-step predictor corrector algorithm on pre-specified flow
field data, which has been in use since. By using the technique of choosing the
downstream characteristics points to fit the developing flow-field, Wayne and Mueller [5]
applied rotational axisymmetric method of characteristics.
A technique based on centreline pressure distribution as the governing boundary condition
was put forth by Hartfield and Ahuja[4]. This technique has shown some promising
progress by preventing characteristics to cross even while using non-viable axial
centreline pressure distribution.
2.6 Contour Nozzle Design
Conical nozzle was used almost exclusively during early application of rocket engine,
proving to be acceptable in most respects. The conical nozzle contour has a diverging
section with a constant angle increased over a nozzle area and thus can be manufactured
easily for any reasonable expansion ratio. The divergent half-angle for conical nozzle with
a 15-deg has been a standard due to good combination of performance and size. Non-axial
component of exhaust gas induces certain performance losses. The thrust efficiency factor,
λ is applied as correction factor in exit-gas momentum. The value of λ can be expressed
as:
𝜆 =
1
2
(1 + cos 𝛼)
2.16

16
Where α is half angle of the conical nozzle.
For a conical nozzle with 15-deg half angles, thrust efficiency is 98.3% as calculated from
above equation [11]. The bigger size and weight associated with small divergence angle
for a given area expansion ratio make it a poor choice of geometry. The conical nozzle
currently used for short nozzles when simple design is preferred over performance [5,8].
To improve the efficiency by turning the momentum axially, contour or bell nozzle was
developed. Ideal divergent contour produces uniform and axial flow at nozzle exit by
turning nozzle flow. The design of non-linear contour is however complicated and
incorrect design leads to occurrence of shocks within the nozzle. MOC has been a
predominant technique for designing contour nozzles. Although MOC has many
applications, however the focus within this report is specifically on application for
analysis on supersonic nozzle design.
2.6.1 Nozzle Configuration
The design of nozzle structure is subject to number of considerations, the details of these
are not part of this work. In brief, the converging section between the chamber and the
nozzle throat of any moderate curved shape or cone angle is satisfactory. This subsonic
flow section of wall contour has never been critical factor in performance.
Nozzle Contours can be classified into different types, the nozzles relevant to this will be
discussed below.
2.6.2 Ideal Nozzle
Ideal nozzle is a nozzle that produces a parallel uniform exit flow and flow through throat
expands isentropically. The nozzle exit pressure is equals to the ambient pressure and thus
produces the maximum possible thrust for a given expansion ratio ε. Momentum-flux
generated by gases exiting the rocket engine creates a reactive thrust force on the rocket
driving the rocket. The thrust force, F can be expressed as:

17
𝐹 = 𝐶 𝐹 𝑝𝑐 𝐴 𝑡 = 𝑚̇ 𝐼𝑠𝑝
2.17
Where CF is the thrust coefficient, 𝑚̇ is mass flow rate and 𝐼𝑠𝑝 is the specific impulse.
Efficiency of conversion of flow rate of propellant to thrust is often measured Specific
impulse. The ideal specific impulse can be written as:
𝐼𝑠𝑝,𝑖𝑑𝑒𝑎𝑙 =
𝐹
𝑚̇
= √
2𝛾𝑅𝑇𝑐
𝛾 − 1
[ 1 − (
𝑝 𝑒
𝑝𝑐
)
𝛾−1
𝛾
] + √ 𝑅𝑇𝑐
𝛾
[
2
𝛾 + 1
]
𝛾+1
1−𝛾
.
𝜖(𝑝 𝑒 − 𝑝 𝑎)
𝑝𝑐
2.18
An outline of an ideal nozzle contour is shown in Figure 2-2.
Figure 2-2: Basic flow structure in an ideal nozzle [8]
In diverging section ABE, the contour BE turns flow to axial direction after the initial
expansion at AB [8]. The ideal nozzle contour which turns the flow to uniform and
parallel at the exit can be designed by MOC.
Some selected nozzle designs which are significant to this study are briefly described in
this section.

18
2.6.3 TIC Nozzle
To produce uniform exhaust, ideal nozzle has to be very long in length but it is not
practical in real world application. The thrust contribution of latter part of the contour is
negligible and can be truncated [8]. A graphical technique for selecting optimum nozzle
contour within a given constrain and then an optimisation process to truncate the full
nozzle contour is described in detail by Oustlun[8]. This technique is very useful in
selecting maximum performance nozzle for a constrained size range. Truncated Ideal
nozzle contour eliminate high weight and length with a relatively small performance loss.
2.6.4 TOC Nozzle
The simplified method of designing optimised contours was presented by Rao and
therefore optimised nozzle contour is often called as Rao nozzle. This method gives
relatively higher performance for a certain length and expansion ratio.
The basic flow structure behind generating thrust optimised contour is outlined in Figure
2-3 below.
Figure 2-3 - Basic flow structure in a thrust optimised nozzle contour [8]

19
For a given throat curvature and 𝜃 𝑁, initial expansion region is constructed. The points B
and P can be found from the designed or calculated design parameters ( ε and L or ε and
ME). The diverging nozzle contour BE can then be generated by solving for the series of
left and right running characteristics. Ideal nozzle contour is obtained when point P is
equal to point K, i.e. the control surface perpendicular to centreline at exit of the nozzle.
Formation of compression waves due to more drastic turning flow thus happens in contour
nozzles as in reality point P does not equal point K [8].
2.6.5 Parabolic Bell Nozzles (TOP)
Since TOC can only be described through computing a list of contour points, Rao
proposed a skewed parabolic-geometric approximation to TOC [7]. This variant contour is
referred to as a thrust-optimized parabola (TOP) can be described as:
(
𝑟
𝑟𝑡
+ 𝑏
𝑥
𝑟𝑡
)
2
+ 𝑐
𝑥
𝑟𝑡
+ 𝑑
𝑟
𝑟𝑡
+ 𝑒 = 0
2.19
The nozzle contour is chosen by specifying the fractional nozzle length of a 15 degree
half-angle conical nozzle, and the expansion area ratio. The parabola specifies the exit
flow angle and the initial turning angle. Parabolic nozzles are presented in detail by Sutton
[3].
The proper inputs can approximate the TOC nozzle accurately without any significant
performance loss but, studies of TOP contours have shown that it resulted in a lower thrust
than an equivalent TOC [15]. Additionally, the spatial description of the TOP contour
simplifies flow control analysis which implies that nozzle area ratio can be increased and
therefore the vacuum performance through the manipulation of static wall pressure. The
problem discovered with Rao’s contours, discovered later that it produced shocks inside
the nozzle flow [16].
The resulting flow-field of TOP contour is neither ideal nor thrust optimised as contour is
a derivative of MOC design.

20
2.6.6 LEA TOP Nozzle
In brief, this nozzle is an approximation of a parabolic contour and designed as a thrust
optimised contour (TOC). The first stage engine of modern launch vehicles usually has a
thrust optimised nozzle like in Ariane 5, due to its high thrust/mass factor.
The geometrical configuration of this nozzle is given in Table 2-1.
2.6.7 VS1 Nozzle
Vulcain nozzle is used as a core stage engine on the Ariane 5 launcher. A subscale variant
of Vulcain nozzle (hereafter referred to as VS1) was designed with primary aim of
verifying the aeroelastic behaviour during parametric study of different TOP contours. The
Vulcain nozzle has shown gain in the performance given that the ideal expansion ratio is
achieved. The geometrical configuration is described in Table below.
Table 2-1: Geometrical Configuration of different nozzle
Nozzle LEA VS1
Throat diameter (m), 2rt 0.02724 0.06708
Nozzle Length (m) 15rt 0.35
Area Ratio 30.32 20
Nozzle Exit Angle 40
40
Throat wall angle 340
35.0250

21
2.6.8 Summary
The study of nozzle design therefore suggests that the current requirements for high-area-
ratio nozzles are independent of the inherent characteristics within conventional design
methods. The TIC nozzle is the only nozzle that produces shock free flow, whereas TOC,
TOP and conical an internal shock is formed in the flow-field[8]. The difference of
internal flow field in contours is essentially due to internal shock as it has a strong
influence on the shock pattern of the exhaust plume, flow separation and the side load
behaviour of the nozzle. Figure 2-4, by Ostlund [8], shows the the mach number
distribution in these nozzle types.
Shock in the nozzle contour disrupts the flow and creates large conversion losses. Non-
adaptive flow conditions i.e. overexpanded or under expanded exhaust flow results in
performance losses.

22
Figure 2-4: Mach number distribution and shock formation in conical(150
), TIC, TOC, TOP
nozzles with ε = 43.4[8]

23
2.7 Method of characteristics
Method of Characteristics is an analytical method for modelling inviscid, supersonic
flows. The method of characteristic is an exact solution unlike the finite difference
method but only valid for inviscid, nonlinear supersonic flow.
The flow field characteristic lines are defined as the direction in which flow variables
(such as P, ρ) are continuous but their derivatives are indeterminate. Along these lines, the
governing equations of flow can be transformed into ordinary differential equations called
compatibility equations. [1]
Hence, the flow-field variables can be calculated along these lines as the governing
equations are integrable with known boundary conditions and an input characteristic. This
compatibility equation varies depending on the two dimensional or axisymmetric type of
analysis [12]. The flow is assumed to be as irrotational which implies that flow-filed is
treated as with no shock waves or flow separation. The MOC resulting equations are
briefly outlined in this section, and more description can be found in several textbooks [1].
The directions of the characteristics lines are given by:
𝑑𝑦
𝑑𝑥
= tan(𝜃 ∓ 𝜇)
2.20
where θ and μ are flow angle and Mach angle respectively.
The dy/dx states that two characteristic lines run through a point with online with a slope
equal to tan(θ -μ) and the other with a slope equal to tan(θ +μ). Consider a streamline at a
given point, makes an angle θ with horizontal. The equation stated that there are two
characteristic lines one below the streamline and other above the streamline direction by
an angle μ.

24
Figure 2-5 Intersecting Characteristic Lines
In order to start calculation by method of characteristics, we have to know the flow
properties at some initial data line. From Figure 2-5, if the flow properties of point A on
RRC and B on LRC are known, then the downstream point C flow properties are given by
[12]:
𝜃 𝐶 =
1
2
[(𝐾−) 𝐴 + (𝐾+) 𝐵 + (𝐾𝑖) 𝐴 + (𝐾𝑖) 𝐵]
2.21
𝜈 𝐶 =
1
2
[(𝐾−) 𝐴 − (𝐾+) 𝐵 + (𝐾𝑖) 𝐴 − (𝐾𝑖) 𝐵]
2.22
where
𝐾− = 𝜃 + 𝜈(𝑀); 𝐾+ = 𝜃 − 𝜈(𝑀)
Fundamentally, two distinct systems for expansion fan distribution have developed over
the years; both utilize the same set of MOC equations but different methodologies.
The one system initiates the expansion fan from the throat wall of nozzle. The expansion
fan from both sides of the throat then intersects at centreline, and then hits the nozzle wall.
C
RRC
A
B
LRC
θ
θ
+µ
-µ

25
Since the nozzle is symmetrical, the system analyses top half of the nozzle, as seen in
Figure 2-6, to solve for the flow field as well as designing the contour. The contour is
designed in such a way to halt the Mach wave reflecting back to centreline. For this, angle
of the wall contour equals the flow angle calculated at the last point of a characteristic
line. This technique as described in detail by Vanco [13] is effective in case of sharped
edge throat.
Figure 2-6: Nozzle Expansion fans
The other system generates the expansion fan from points on the sonic line but
fundamentally utilises same set of equations. The discretisation of area from sonic line
originating Mach lines is quite similar to meshing process in CFD[5]. There is also third
system that has been in evaluation involving initiating expansion fans from expansion
curve as well as sonic line involving flow angle gradient.
The supersonic nozzles can be defined in two categories, planar and axisymmetric. Early
year’s nozzles were planar and thereby simplifying the calculations made as expansion of
the air was in one plane only. Back then relative low stagnation temperature and pressure

26
did not caused dimensional stability problems in the throat region. As the stagnation
temperature was raised, dimensional stability becomes more difficult to maintain in planar
nozzle. This in turn fast paced the development in axisymmetric nozzles which have
shown uniform boundary layer growth comparatively. However, disadvantages of
imperfect contour have associative adverse effect on safety and performance of the
system.
The fundamental physical aspects of the flow field is same in both two dimensional and
axisymmetric nozzles, hence the same Control Volume analysis can be used produce the
results [12]. The continuation of equations after 2.22 is different for two dimensional
planar and axisymmetric nozzles. The sections below describe each of them with
irrotational flow briefly.
2.7.1 Two Dimensional Planar nozzle
For irrotational flow in planar nozzle:
(𝐾−) 𝐴 = (𝐾+) 𝐵 = 0
2.23
The significance of the characteristic lines is that the partial differential equations defining
flow along these lines reduce to ordinary differential equations.
𝑑𝜃 = ∓√(𝑀2 − 1)
𝑑𝑉
𝑉
2.24
The above equation gives the compatibility relations only along the characteristic lines
which is identical to the Prandtl-Meyer equation. This gives us our compatibility equation,
can be described as:
𝜃 + 𝜈(𝑀) = 𝑐𝑜𝑛𝑡 = 𝐾− (Along the C- characteristic)
2.25

27
𝜃 − 𝜈(𝑀) = 𝑐𝑜𝑛𝑡 = 𝐾+ (Along the C+ characteristic)
Our compatibility relations are algebraic equations which are valid only along the
characteristic lines [1]. The solution to algebraic equations is much simpler than partial
differential equation.
As described earlier, K- along the point A and C, and K+ along point B and C are constant.
The value for unknown θ and ν for two dimensional planar nozzles becomes:
𝜃 𝐶 =
1
2
[(𝐾−) 𝐴 + (𝐾+) 𝐵]
2.26
𝜈 𝐶 =
1
2
[(𝐾−) 𝐴 − (𝐾+) 𝐵]
2.27
The flow parameters can then be evaluated from θC and νC. The Prandtl-Meyer variable ν
gives the associated value for the Mach number. This in turn can be used to calculate the
rest of the flow variables[1].
2.7.2 Axisymmetric Nozzle
The governing equation for axisymmetric throat geometries has a radial component. This
means that specific radial position is required for the throat as it relates the flow variables
at any location. The continuing equation for the MOC for axisymmetric nozzles described
as follows:
(𝐾−) 𝐴 =
1
√𝑀2 − 1 − cot θ
𝑑𝑟
𝑟
; (𝐾+) 𝐵 =
1
√𝑀2 − 1 + cot θ
𝑑𝑟
𝑟
2.28

28
The discretise equation define the location in x-r space. The compatibility equation for
axisymmetric nozzle:
(
𝑑𝑟
𝑑𝑥
)
𝑐ℎ𝑎𝑟
= 𝑡𝑎𝑛(𝛳 ± 𝛼)
2.29
𝑑(𝜃 ± 𝜈) =
1
√𝑀2 − 1 ∓ cot 𝜃
𝑑𝑟
𝑟
2.30
The above equations are for C+ and C- characteristic lines respectively.
Forward differencing and rearranging the equation:
𝑟𝑖+1 − 𝑡𝑎𝑛(𝜃𝑖 − 𝛼𝑖) . 𝑥𝑖+1 = 𝑟𝑖 − 𝑡𝑎𝑛( 𝜃𝑖 − 𝛼𝑖) . 𝑥𝑖 (For C+)
𝑟𝑖+1 − 𝑡𝑎𝑛(𝜃𝑖 + 𝛼𝑖) . 𝑥𝑖+1 = 𝑟𝑖 − 𝑡𝑎𝑛(𝜃𝑖 + 𝛼𝑖) . 𝑥𝑖 (For C-)
2.31
The variables in the above equation with subscript i+1 are unknown. These equations can
be rearranged as follows:
𝜃𝑖+1 + 𝑣𝑖+1 = (𝜃𝑖 + 𝑣𝑖) +
1
√𝑀𝑖
2−1−cotθi
𝑟 𝑖+1−𝑟 𝑖
𝑟 𝑖
(For C+)
𝜃𝑖+1 − 𝑣𝑖+1 = (𝜃𝑖 − 𝑣𝑖) −
1
√𝑀2−1+cot θi
𝑟 𝑖+1−𝑟 𝑖
𝑟 𝑖
(For C-)
2.32
The above equations are solved simultaneously to find 𝜃𝑖+1 , 𝑣𝑖+1 at the point where
characteristic lines intersect. Other flow parameters can then be evaluated from this.[14]
In axisymmetric equations, the flow direction and r-coordinate are known at the centreline,
i.e. 𝜃 = 0 𝑎𝑛𝑑 𝑟 = 0. Equation 2.32 becomes indeterminate as, 𝑐𝑜𝑡𝜃 will be infinite. The
limiting case for the dimensionless equation as described in [14] can be written as:

29
𝑑𝜃
𝑑𝑟
=
𝜃𝑖+1 − 𝜃𝑖
𝑟𝑖+1 − 𝑟𝑖
=
1
√𝑀2 − 1 + cot θ
1
𝑟
2.33
The equation relates 𝜃𝑖+1 , 𝑣𝑖+1 for the curves originating from centreline. For more
information on derivation can be found in journal by Denton & Brandon[14]. The
equation can be written as:
2𝜃𝑖+1 − 𝜈𝑖+1 = 𝜃𝑖 − 𝜈𝑖 As C+
2.34
2.8 Throat Modelling
For subsonic flow, the governing equations are elliptical in nature and their solution
depends on both upstream as well as downstream of all the boundary conditions. The
transonic expansion location is crucial for precise computation of the flow field
characteristics at the throat. The sonic velocity at throat is required for the supersonic
expansion in the diverging section. Studies have shown that an analytical method produces
generally good results in region near to the throat, especially in axisymmetric case [12].
As described earlier, MOC is only effective for the analysis of supersonic and inviscid
flows. This means that the throat region of the rocket cannot be modelled by MOC
governing equations. Therefore, a separate method must be utilised to analyse the region
of the flow within the vicinity of the nozzle throat. The designed throat region known
parameters can be then utilized as an input for the MOC algorithms.
Various expansion approaches have been developed over the years to describe the
transonic flow-field. The method for approximating the throat was first produced by Sauer
in 1944. This method was based on a power series expansion of small velocity

30
perturbations about sonic line was then extended by 1962 by considering higher order
expansions.
The approach developed by Hall [17, 18] based on the small perturbation theory for an
irrotational perfect gas. The velocity components were derived in terms of inverse powers
of the normalised throat wall radius of curvature. The equation describing Hall method is
described in Appendix F.
From analysis of Halls approach, the flow angle needs to be equal to local wall slope as
the expansion parameter (1/R) is introduced through wall boundary. It was found that Hall
method [17] could not exactly satisfy this in cylindrical coordinates. Both Sauer and Hall
have shown instability when radius of the nozzle wall at the throat is small.
An alternative approach by Kliegel-Levine has shown numerous advantages over Hall, as
it converge regardless the value of the normalized throat radius of curvature and also
produces third order axisymmetric solutions [12]. This means that this method can be
applied to nozzles with small radii of curvature and therefore used in the analysis for
modelling throat curve. The normalised throat radius of curvature has been observed to
greatly influence the shape and the position of the sonic line [17].
The equation governing this method has been explained in Appendix F
2.9 Thrust Coefficient
The thrust produced by a rocket engine is equal to integral of the axial momentum and
pressure force acting on the exit plane of the nozzle. A thrust coefficient is the coefficient
of force per unit of throat area per unit of stagnation pressure.

31
𝐶 𝐹 =
𝑚𝑣𝑒 + 𝐴 𝑒(𝑃𝑒 − 𝑃𝑎𝑚𝑏)
𝐴 𝑡 𝑃0
̇ 2.35
Where,
 𝑚̇ us the mass flow rate through the throat;
 𝑣𝑒is the mass-averaged exit velocity;
 𝐴 𝑒 and 𝐴_𝑡 are the respective exit and throat area;
 𝑃𝑒is mass average exit pressure;
 𝑃𝑎𝑚𝑏 and 𝑃0 are the ambient pressure and stagnation pressure respectively
The exit plane is the horizontal nozzle exit perpendicular to centreline at nozzle exit. As
discussed, the exit plane flow field is not claculated by the methods in this thesis. The
method suggested by Taylor [12] for both conical and optimised nozzle types considers
the forces acting on rest of the control volume. This method is true for two dimensional
and axisymmetric flows uses the wall pressure integrals in order to calculate for thrust
cofficient. The total force excerted by the flow on nozzle contour is found to be:
𝐶 𝐹,𝑊𝑎𝑙𝑙 = ∑ [ 𝑃̅
𝑛 𝑤𝑎𝑙𝑙
2
] 𝑌
2.36
And 𝑌 = {
(𝑟′𝑖
2
− 𝑟′𝑖−1
2
)/𝑃_𝑐 𝑖𝑛 𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐
(𝑟′𝑖 − 𝑟′𝑖−1)/𝑃_𝑐 𝑖𝑛 𝑝𝑙𝑎𝑛𝑎𝑟
Where, 𝑛 𝑤𝑎𝑙𝑙 is total number of points and quantities are averaged of points 𝑖 and 𝑖 − 1
and the 𝑟′ is throat normalised r value.
The total pressure in case of irrotational flow remains constant.

32
Thrust coefficient requires to be corrected for the base drag, and with the assumption that
flow does not separate within this control volume, the equation can be described as:
𝐶 𝐹,𝐵𝑎𝑠𝑒 = −
𝑃𝑏
𝑃0
.
𝐴 𝐸
𝐴 𝑡
2.37
𝑃0 is assumed to be identical to the chamber pressure 𝑃𝑐, the total thrust coefficient
produced will be:
𝐶 𝐹 = 𝐶 𝐹,𝑊𝑎𝑙𝑙 + 𝐶 𝐹,𝐵𝑎𝑠𝑒
2.38

33
Chapter 3 Numerical Methods
3.1 Introduction
Computational Fluid Dynamics (CFD) is a reliable and cost effective way to validate the
nozzles’ contour. This gives the user greater flexibility to analyse and visualise the flow
characteristics. For analysis and validation, three different nozzle geometries were chosen.
These were designed and CFD simulated by Kyll Schomberg, a UNSW PhD Candidate as
part of his study. The nozzles were selected for the proper assessment of the accuracy of
code in simulation of different nozzles.
The code accuracy is also investigated with LEA nozzle. This nozzle is an approximation
of a parabolic contour and designed as a thrust optimised contour (TOC). The first stage
engine of modern launch vehicles usually has a thrust optimised nozzle like in Ariane 5,
due to its high thrust/mass factor [20].
This section outlines of the steps taken to validate the nozzle contour developed through
MoC code.
3.2 Geometry
3.2.1 TIC
The geometry and boundary conditions on the TIC nozzle in the CFD analysis is based on
the experimental test conducted earlier. As can be seen from Figure 3-1, the flow passes a
settling chamber, and then through a cross-section construction and a bending tube section
before it accelerates in the convergent-divergent nozzle to supersonic velocity.

34
Figure 3-1: Sketch of horizontal test section [19]
Nozzle and test setup are described in more detail in Stark [19].
3.2.2 LEA nozzle
The supersonic nozzle in a rocket propulsion system is generally designed to maximise the
exhaust velocity and therefore thrust. The LEA TOP nozzle geometry was replicated using
Rao’s method. Figure 3-2, outlines the domain for the inlet and outlet dimensions. The
detail geometric description can be sourced from [27].
Figure 3-2: Numerical domain with outlet and inlet dimension (not to scale)[27]

35
3.2.3 VS1
To explore the performance of the code further, a subscale variant of the axisymmetric
Vulcain nozzles (VS1) was selected for analysis. The nozzle has been validated [26] and
shown to represent flow structure within the Vulcain nozzle. The analysis was done by
Kyll Schomber and details can be found on his journal [26]. Figure 3-3, provides detail of
the inlet and downstream exhaust dimensions.
Figure 3-3: Numerical domain including inlet and downstream exhaust dimensions (not to
scale)[26]
3.3 Numerical Model
CFD solves the governing equations of the fluid flow in discretised volumes. The
equations require that model must first split into finite volumes to solve for the set of
partial equations known as Navier-Stokes(NS) and predict the properties of the flow. The
solver then solves the equations in each volume and afterwards model condition in the
adjacent volumes. The residual reduces over iterations and thus solution converges.

36
In order to correctly apply the assumptions and boundary conditions to solve for the flow,
it is critical to understand the complexity of flow through nozzle. The nozzle has a high
pressure in the combustion chamber and the pressure gradient too is very high throughout
the flow regime. Moreover, the flow accelerates to supersonic velocity in region upstream
of the throat and thus the possibility of shocks waves exists.
3.3.1 Governing Equations
The flow field to be analysed is governed by set of equations referred to as Navier-Stokes,
which solves for flows by predicting viscous and density variations. It consists of a range
of equations including mass, momentum and continuity.
Mass Conservation
The fluid flow must satisfy the law of conservation of mass in each volume. This equation
dictates that the mass flux entering a control volume must be equal to the mass flux
leaving that control volume. The CFD software incorporates the continuity of mass
equation, can be expressed as:
𝜕𝜌
𝜕𝑡
+ 𝛻. (𝜌𝑣) = 0
3.1
The first term represent the change in fluid density over time which is equal to the
divergence of the velocity vector field. If the fluid density remains constant, then 𝛻. (𝜌𝑣)
= 0, which implies that the divergence of the velocity vector field is zero throughout the
control volume.
Conservation of Momentum
Another important key equation that must be solved in CFD is conservation of
momentum. This law states that the rate of change in momentum of an object on each

37
volume element is equal to the sum of all the forces acting on the object. In a fluid body
the forces acting on each body can be considered as sum of forces acting on the volume
from an external force known as body force and those acting on the volume due to its
surrounding referred to as surface forces. From Newton’s second law, Reynold’s
Transport Theorem and Gauss’ Divergence Theorem, the Cauchy Momentum Equation
can be obtained. The Cauchy’s equation can be written as,
𝜌 (
𝑑𝑣
𝑑𝑡
+ 𝑣. 𝛻𝑣) = 𝜌𝑓̃ + ∇𝜋̃̃
3.2
Where 𝑓̃ is the body force on the volume and 𝜋̃̃ is the stress tensor acting on the fluid. For
compressible flows, all terms must be considered including the changes in density and
viscosity.
The mass and momentum conservation equations are fundamental to the calculations of all
the solution. In current study, the fluctuation in pressure and the velocity is the main area
of focus. The energy interactions are neglected in this analysis.
3.3.2 Numerical Solver Methods
A two-dimensional, axisymmetric flow model was analysed in CFD code FLUENT15.
The fluid entering was modelled as an ideal gas with constant specific heat, viscosity and
thermal conductivity.
Spatial Discretisation
ANSYS Fluent uses a control-volume-based to convert the governing equation that can be
solved numerically. This control volume technique consists of integrating the governing
equation about each control volume. The gradients are needed for constructing values of a
scalar at the cell face as well as computing secondary diffuser terms and velocity
derivatives.

38
ANSYS Fluent allows a choice from several upwind schemes: first order upwind, second
order upwind, power law and Quick. In addition, it allows choosing the discretization
scheme of desired accuracy for the convection terms of each governing equation. With
second-order accuracy, quantities at cell faces are computed using a multidimensional
linear reconstruction approach.
For the analysis of the nozzles, a second order spatial discretisation for all convection
terms was chosen as it consists an extra truncation error term to reduce the overall residual
error term.
3.3.2.1 Turbulence Models
Turbulent flows are unpredictable and difficult to model accurately. The direct solution of
the Navier-Stroke equations is difficult or impossible in some cases to compute. This
complexity is due to the nonlinear and time dependent nature of the Navier-Stroke
equation. The primary issue with the Navier-Stroke equation is how to model the
Reynolds Stress terms. The most common method to model the Navier-Stroke equations
is Reynolds-averaging. These family of solvers known as Reynolds-averaged Navier
Strokes (RANS) models utilise Boussinesq’s approximation which states that Reynolds
stress terms are proportional to the deformation rate of element in the CFD model. [25]
In turbulence flow, vortex structure moves along the flow and hence cannot be specified
as local. This implies that upstream history of flow unlike in case of MOC is also of great
importance. For the CFD analysis in this thesis, RANS solvers have been considered
which are briefly discussed in this section.
Spalart-Almaras
The Spalart-Almaras (S-A) turbulence model is one-equation that solves for a modelled
transport equation. The S-A model only utilises on transport variable, 𝜈̃ which differs with
kinematic viscosity, µ in the near wall region. The Spalart-Allmaras model was initially

39
designed for aerospace applications with wall-bounded flows [23] however since then
shown to be useful for a variety of applications.
The S-A model is known to have difficulty with separated or recirculating flows and its
inability to rapidly accommodate changes in length scale. This might be as S-A model was
originally designed to model to solve near-wall vortices which are useful in cases when
the flow changes abruptly from wall-bounded to a free shear flow.
This model has been accepted in the industry and has a good potential to provide a
practical alternative to more sophisticated models, at least for a certain class of flows such
as one presented in this thesis report.
k-ϵ Realizable
The realizable k-ϵ model is a two equation transport model. This model uses transport
equation for the turbulent kinectic energy (k) and dissipation rate (ϵ), thus solving both of
Boussinesq’s equation. The realizable part of the model improves the standard model by
reformulating the turbulent viscosity and the mean-square velocity function in order to
solve for the dissipation rate. This modification allows computing flows with separation
and adverse pressure gradients. The key difference of this from the S-A model is on how it
calculates turbulent viscosity, in addition to S-A model inability to solve for both of
Boussinesq’s transport equation.
k-ω Shear Stress transport model
The k-ω is another two equation turbulence model, which primarily predicts the turbulent
kinectic energy k and the specific dissipation ω. The k-ω addresses the issues with the k-ϵ
model in low Reynolds number flows and flows with severe pressure gradients. The
standard k-ω model is sensitive to inlet conditions. For this reason, Shear Stress transport
(SST) is included in the analysis which allows the model to be less sensitive to free stream
turbulence properties of the Inlet, hence making it more stable. The transport equation
incorporates turbulent shear stress and cross diffusion derivative terms in additional to
standard k-ω model. The SST model also improves analysis in sub-viscous part of the

40
boundary layer, but however depends on high quality boundary layer cells. Overall, with
inclusion of the SST formulation, the near wall flow can be fully resolved unlike k-ϵ
which uses a standard wall function.
Conclusion
The CFD computations results differ with the turbulence models. The computations
conducted using SST turbulence for TIC nozzle gives under-predicted separation location.
The results for the fine mesh shows the prediction by the Spalart-Allmaras, standard kω
are in good agreement with the experimental data while k-ϵ severely under predicted [19]
the sepration point. It also shows that SA model captures both the shock location and the
pressure distribution afterwards remarkably well. The prediction by k-ϵ and SST are seen
to be rather poor, although modification by Ostlund [7].
A good agreement with the experimental data was found with standard kω and Spalart-
Allmaras models. The Spalart-Allmaras turbulence model is selected for the analyses of
the nozzles due to advantage over to calculate nozzle flows.
3.4 Meshing
The grid refinement plays crucial role in terms of computational time and the accuracy of
results. Achieving high quality mesh is one of the important aims while doing CFD. In
nozzle analysis, the mesh structure near the wall region is crucial in determining the
accurate results. Thus mesh is concentrated especially on the inlet, wall and exit where
large gradient in velocity or pressure are known to exist. Figure 3-4 shows the mesh of the
TIC nozzle as done by Schomberg as part of his study.

41
Figure 3-4: Meshing of TIC nozzle
For the nozzles analysed in this study, the convergence was accepted with consistency of
all surface monitors over 500 iterations and continuity through the variation of mass flux
in nozzle. The numerical model verification and validation was achieved by grid
independence and comparative turbulent closure model. The turbulence closure model was
selected based upon the variation in static wall pressure and through calculation of thrust
coefficient. Reynolds averaging was used due to quasi-steady nature of flow and has been
used in most publication concerning numerical simulations in complex shock pattern [22].
All parameters were discretised to second order upwind. More information on meshing
can be found on journal papers by Kyll Schomberg [26,27]. The mesh structure for LEA
and VS1 is provided in Appendix E.
Axis
Pressure Inlet
Wall

42
3.5 Boundary Conditions
For a CFD model to evaluate, required initial conditions and boundary conditions must be
established. For all the nozzles analysed, the boundary condition for the current study was
set accordingly to properly simulate the previous experiments. The inlet was assumed to
have stagnation conditions, to ensure purely pressure driven analysis. The outlet in all the
cases was assumed to be at sea level pressure of 1atm.
TIC nozzle has an inlet Stagnation Temperature of 283K and pressure of 2.5*106
N/m2
.
The specimen has a throat diameter of 20 mm with divergence length of 90mm and a
design Mach number of 5.15.
The Stagnation temperature for TIC nozzle is 288K.
The working fluid in all the cases was air as to keep the consistency with the experimental
conditions.
3.6 Results & Conclusion
The Numerical model was selected in order to replicate the physical dimensions and
operating conditions used in the experimental setup.
For LEA nozzle mesh, no significant variation in wall pressure was observed with
difference in choice of turbulence model. A good agreement was observed in static
pressure distribution with experimental data was with both SA and SST turbulence
models. The difference in thrust coefficient between SST and SA was 0.05-0.08% across
all conditions [27]. The SA model was selected to generate the results due to faster
convergence rate compared to SST.

43
The full regime flow field is characterised by internal shock which divides the core flow in
two, a high momentum flow region near the symmetry axis and a high-pressure region
near the wall. The shock emanates from a region close to the throat. The side load and the
flow separation in TOC have been experimentally and numerically studied. The further
explanation on the internal flow field is given in [20,21].
In case of VS1, significant variations in wall pressure were observed with difference in
choice of turbulence model. The comparative analysis of different models suggested that
both the SA and k-ω SST were adept at describing flow behaviour within the nozzle. The
SA turbulence model was selected on basis of small reduction in computational time [26].

44
Chapter 4
Computer Program Development
4.1 Introduction
The development of MATLAB computer program for the design of nozzles using Method
of Characteristics integrated through graphical interface is presented in this chapter. Thus,
this chapter focuses on the methodology based on literature described earlier, to develop a
computer code to answer the main research objective. The code for both planar as well as
an axisymmetric nozzle with a sharp throat and initial expansion curve has been developed
in this thesis. A new code analysing the transonic flow at the throat has discussed in this
section. Axisymmetric code developed is based on the first designed planar code. This
section mainly describes the development of axisymmetric nozzle computer program and
the new code concept. Planar code can be easily understood with the understanding of the
technique in an axisymmetric nozzle.
This effort is the continuation of works completed on planar 2D nozzle design codes [29].
The approach and code design of Olson[29], Taylor[12], Young[5] and Khare[30] have
been studied. The Taylor and Khare’s code is based on the analysis of Expansion
Deflection (ED) nozzles, therefore only the code structure was studied. Olson’s work was
based on planar 2D nozzles; hence his work was studied and the initial planar design was
built on his work with improvements in computational time by using direct equations
instead of simultaneous equation solution through inverse matrix. The work by Young
included chemical kinetics on the design of axisymmetric nozzle. His work along with
work by Denton & Brandon[14] was very important to this thesis . Furthermore, a GUI

45
was developed based on a wide range of user inputs capable of designing planar and
axisymmetric nozzle design.
4.2 Methodology
The general approach in the design of a nozzle uses Prandtl-Meyer expansion waves from
the throat region. The overall work on code developed in this thesis can be divided into
Axisymmetric MOC, New Method and GUI.
4.2.1 Axisymmetric MOC
The algorithm developed is based on the iterating the initial nozzle expansion angle until
the given exit number is reached. In this code the sonic line is assumed to be horizontally
aligned with the axis at throat. The new code described later in this section also takes into
account flow from the throat. As described earlier, two types of nozzles are associated
with axisymmetric nozzles, one with an expansion at the sharp corner downstream of
nozzle throat and other one with the expansion of curvature. Initial expansion curvature
after the throat region is similar to the initial expansion region in parabolic contour
approximation. The expansion angle increases with the increasing angle in the nozzle
expansion curvature. The equations are only required to be solved for the top half of the
nozzle due to symmetrical behaviour of the nozzle.
Input Parameters
As stated earlier, in order to start a calculation using method of characteristics, the flow
properties along with the initial data line are required. From this point, the characteristic
mesh and the associated mesh properties can be simulated downstream. The current code
is integrated through interface. The code starts with initialising the variables that would be
utilised with zero matrices. The input parameters were taken from the user through GUI
whereas constant thermodynamics parameters were hard coded in the code.

46
The program then defines the conditions of the throat and expansion arc depending on the
type of analysis. The input parameters for the test cases for the nozzle are described in
Table 4-1.
Table 4-1Boundary Parameters for MOC
Parameter TIC LEA VSI
Combustion Pressure
(Pc)(Pa)
25.5e5 4.1e6 1.75e7
Combustion
temperature (Tc)(K)
283 288 450
Desire Exit Mach
Number(Mexit)
5.15 5.245 5.5
Specific heat Ratio (γ) 1.4 1.4 1.4
The array for the intersection points (x,r), flow Mach number, flow angle (θ), Mach angle
(µ), and Prandtl-Meyer angle (ν) were first initialised.
Algorithm and interaction points
After initial expansion at throat, the same procedure is followed for both these types of
nozzles. The MOC solution thus begins with designing the curve downstream from the

47
throat. The throat point is considered as a point at the expansion region for sharp edge
throat, and hence all characteristic waves originating from this point are with increasing
expansion angles. The thermodynamic parameters at all these points are known, which are
essential to solve MOC afterwards. This process of new originating expansion waves from
the throat continues until the exit Mach number is achieved at the exit.
Starting from point (1,1), the properties along the left running characteristic (LRC) are
found using compatibility equations as described in section 2.7.2. Point (2,1) is then found
at the axisymmetric line using equation 2.34 and the flow properties along the LRC. Thus
the LRC characteristic line meets at the centreline where flow direction and r-coordinate
are known at the centreline, i.e. 𝜃 = 0 𝑎𝑛𝑑 𝑟 = 0. The LRC terminates at centreline, and
then a new characteristic line, point (1,2) originates from expansion curve with increased
Prandtl-Meyer expansion angle. Point (2,2) is then found as an intersection of LRC from
point (1,2) and newly created right running characteristic (RRC) originating from point
(2,1). LRC from point (2,2) intersect the centreline following the same process as earlier.

48
Figure 4-1 An overview of method of characteristic for axisymmetric nozzle
As we are designing for a certain Mach number, the Mach number calculated and then
compared to the exit Mach number. Moreover, the values of theta and the maximum
expansion angle are validated. The user is notified if any deviation from standard
condition happens and the possible reason. This way iteration continues until the value of
the Mach number at the centreline reaches line matches the desired exit Mach number.
Once the Mach number is reached, the contour of the nozzle is designed using RRC flow
(2,2)
(2,1)
(1,2)
(1,1)
(3,2)

49
properties along the characteristic lines from the last characteristic line. The initial
characteristic mesh or kernel for LRC characteristic is shown in Figure 4-2. The density of
the characteristic mesh can be increased with decreasing the value of the incremental
expansion angle. Note that decreasing the value of the expansion angle after certain limit
will affect the stability of the code. If said limit reaches, the user will be notified and the
program terminates itself.
Figure 4-2: Initial Characteristic mesh
The complete code for planar and axisymmetric nozzle with both sharp and expansion
throats is given in the Appendix B.
Subprograms
The subprograms were made in order to perform a particular task or function multiple
times within the program. This is to reduce the memory conjunction and the size of the
main program. Thus, the computer program is made up of its own contents as well as
independent subprograms that work together in the main program. One subprogram that is
used in this code, mach_Val, uses bisection method to calculate the Mach number
associated with the flow field expansion angle. This is very simple and robust technique,
but also is relatively slow. The program is used to obtain an approximation to a Mach
number. Every time the program is called, it tries to converge to a Mach number and

50
continues the iterative loop till the absolute difference between the Prandtl-Meyer
expansion angle calculated with Mach number and the original value of expansion angle at
the point is less than 1e-3 to 1e-5. The source code is provided in Appendix B.
4.2.2 New Method
As discussed in section 2.8, the transonic expansion location is crucial for precise
computation of the flow field inside the throat. The new proposed development will
hybridise this code with the existing axi-symmetric code. The starting position of Prandtl-
Meyer expansion is from the sonic line. As MOC is only effective for analysis in
supersonic flow, a separate code is required to analyse the region of the flow within the
vicinity of the nozzle throat. Studies have shown that an analytical method produces
generally good results in region near to the throat [12]. For sonic line, the location and
flow parameters can be evaluated by Kliegel-Levine method as described in detail in
section 2.8. A code has been created for this (Appendix B 6.2.7) and the output can be
seen as a yellow line on Figure 4-3.The designed throat region now has known parameters
that can be further utilized as an input for the new iterating MOC algorithms.
Figure 4-3: New Proposed sonic line and the originating LRC.

51
GUI development
The GUI is a graphical interface that enables a user to perform interactive tasks. This
means that user does not have to type commands or change code to accomplish the task.
This makes it accessible and easier for the user as they need not to understand the details
of how the tasks are performed.
The GUI development is particularly important as it inputs value of the user, run the
relevant program requested by the user and displays the requested profiles of the nozzle.
The GUI was developed using MATLAB Graphical User Interface Developer (GUIDE).
The component in GUIDE includes toolbars, push buttons, list boxes, toolbars and menus.
Each component in GUIDE is connected with one or more user defined routines known as
‘callback’. A user action such as pressing a push button or typing in a value triggers the
execution of each callback. GUIDE creates an associated code file containing callbacks
for the User Interface and its components. The editor in GUIDE saves two files: ‘.fig’ used
to enter inputs, and ‘.m’ file that has all the code structure. The application can be
launched from either the figure or the code.
The overall layout of the program is shown below in Figure 4-4. The features of the
program allow the user to design a nozzle as explained in the earlier sections. The program
features two dimensional plots of the contour design and the associated temperature, Mach
and pressure plots.

52
Figure 4-4: The main interface of the developed GUI at initialisation
The GUI allows selection of the analysis and nozzle type and the input the boundary
conditions. With all the inputs, the compute push buttons evaluates the solution and
generates a new figure for nozzle contour. The GUI output outlook for both analysis types
is provided in Appendix D. The working of the GUI is self-explanatory.
The complete code of GUI is attached in Appendix B.

53
Chapter 5
Results and Discussion
5.1 Contour Comparison
The primary objective of the study was to design a contour resembling ideal contour using
method of characteristics. The contours made by the computer program are referred to as
MOC nozzle for each design condition.
5.1.1 TIC Nozzle
Ideally, the MOC designed truncated contour should have a similar contour shape as the
TIC nozzle. Observatoin from Figure 5-1, gives very close agreement with MOC design
and the TIC nozzle.
The accuracy of the method of characterists increases with higher mesh density.
Increasing mesh density increases the computational time and thus a balance is required.
The MOC Design is generated with boundary conditions as described earlier and with
0.002 incremental Prandtl-Meyer expansion angle at the throat. The expansion angle
directly controls the characteristic mesh density and the accuracy of predicting the exit
Mach number. The different exansion angle converges with critical point of 0.002.

54
Figure 5-1: Truncated Nozzle Contours TIC
The shape of the parabolic contour is similar to the other two nozzles. The difference,
however is that the parabolic contour has a higher initial expansion followed by a more
drastic turning of the flow compared with the TIC nozzle. This means that Parabolic
Contour (TOP) in comparison with TIC and MOC nozzles corresponds to an initial higher
wall angle and higher Mach number downstream the throat and lower values of the wall
angle and Mach number at the exit.
Moreover, numerical data analysis of different mesh density shows that the small variation
in TIC and MOC design is primarily sourced at initial region of the design nozzle. One
factor of this behaviour can be due to the assumption of the sonic line in this analysis.
The above nozzle design is a truncated part of full MOC design, as shown in Figure 5-2 at
desired length.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Nozzle Contour TIC
MOC Design
TIC Nozzle
Parabolic Contour

55
Figure 5-2: Full Design Nozzle Contours TIC
Briefly, the analysis of the TIC nozzle contour comparision has validated and verified the
computer program contour design. Two more test cases are described below.
5.1.2 LEA Nozzle
LEA nozzle contour is an approximation of a thrust-optimised contour and thus, harder to
reproduce from the MOC equations. The contours as shown in figure Figure 5-3, plots
MOC, LEA and Parabolic Contours.
Initial Expansion region between the Parabolic Contour matches the LEA nozzle. The
LEA nozzle continues to follow the higher wall angle than the Parabolic nozzle afterwards
for some more length. This is followed by very high wall turning in LEA in effort to turn
the flow axially. The truncated MOC nozzle designed to match the performance of LEA,
has shown quite consistant wall angle. The axial turning of the flow in the MOC design
happens in the latter section, this section is plotted in Appendix A. The plot is based on the
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Nozzle Contour TIC (FULL DESIGN)
MOC Design
TIC Nozzle
Parabolic Contour

56
expansion angle value of 0.003, with any decrease in the value of the expansion angle or
higher mesh density shows minor changes in the nozzle contour numerically.
In conclusion, for the LEA nozzle more effort by the wall contour has been made to turn
the axial direction for this particular length as compared to MOC design. The difference
however in the contour of LEA nozzle and LEA MOC can be credited to the fact that LEA
is a thrust optimised contour not an ideal contour.
Figure 5-3: Truncated Nozzle Contour LEA
5.1.3 VS1 Nozzle
VS1 nozzle is a subscale model of a parabolic nozzle. Rao’s Parabolic contour has shown
consistency with the VS1 nozzle contour design for most part of the length. After three-
quarter of the total length, the VS1 contour shows a higher change in the wall angle
towards horizontal than Rao’s parabolic contour. MOC truncated contour designed with
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.00 0.05 0.10 0.15 0.20
Nozzle Contour LEA
MoC Contour
LEA Nozzle Contour
Parabolic Contour

57
the VS1 performance parameters, shows moderate changing of the wall angle. This means
that the truncated MOC contour is part of the expansion region of full length contour.
Figure 5-4: Truncated Nozzle Contour VS1
The design philosophy behind the design of the VS1 was to eliminate uncontrolled flow
separation during low altitude operations. The compressive turning of the nozzle wall is
similar to an extent with spike nozzle, improves the nozzle efficiency at low altitude. It
can also be seen from full design contour Figure 0-2, that the length of VS1 nozzle is
shorter compared to Rao’s parabolic nozzle conotur.
The reason for difference in the MOC contour and VS1 is primarily due to VS1 beign a
subscale model of a parabolic contour. Note that above contour is designed with 0.005
incremental expansion angle at the throat.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Width(m)
Length(m)
Nozzle Contour Comparision , VS1
MoC Contour
VSI Nozzle Contour
Parabolic Contour

58
5.2 Wall Pressure Distribution
To demonstrate the numerical accuracy and validity of the computer code, comparison of
wall pressure, pw normalised by ambient pressure pa versus axial location, X normalised
by throat radius, R* is shown in this section. This plot is compared to the data from the
experimental or CFD depending on nozzle type. To present the results of the wall pressure
distribution, the same three nozzles were selected. The experimental and the CFD data
was used for the TIC nozzle whereas, for other two nozzles the CFD data has already been
verified in the journal papers by Schomberg[26,27].
In nozzles, flow is introduced in the axial direction which accelerates before it passes
through the exit. The pressure along the wall decreases along the axial direction. The static
pressure rise just downstream of the nozzle can be observed in all the nozzles. This is
generally associated with the changing direction of the momentum where throat induces
strong angular motion in the flow, and this section cannot be properly predicted by MOC
solution. Note that in experimental analysis, loses resulting from the wall friction are
negligible.
5.2.1 TIC Nozzle
As can be seen in Figure 5-5, there is good agreement between the experimental data and
the CFD prediction, whereas MOC design wall pressure slightly under predicts wall
pressure distribution. A step rise in the static wall pressure can be seen with experimental
data due to flow separation. Flow separation occurs when wall pressure, pw is considerably
lower than the ambient back pressure, pa.

59
Figure 5-5 Nondimensional wall pressure, TIC
As the effect of directional momentum fades away after a short distance from throat, the
MOC wall pressure then shows close agreement with experimental and CFD data. This
validates the accuracy of the flow simulation by the MATLAB program.
5.2.2 LEA Nozzle
As we analysed in section 5.1, the contour of the LEA nozzle has shown sudden transition
just after throat as compared to TIC nozzle. This can be clearly observed from
comparative analysis of the non-dimensional wall contour of both of these. LEA wall
pressure has shown sudden decrease just downstream of throat whereas this was relatively
moderate in TIC nozzle. The normalised value of the wall pressure is very low in the LEA
nozzle; this can lead to separation and subsequent formation of recirculation zone. This
type of separation gives rise to abrupt rise of wall pressure, inducing shock.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.00 2.00 4.00 6.00 8.00 10.00
(pw/p0)
X/R*
Wall Pressure, TIC
CFD Data
Test Data
MOC

60
Figure 5-6: Nondimensional Wall Pressure, LEA
Above Figure 5-6, shows that truncated MOC contours wall pressure is in very close
approximation with the CFD data.
5.2.3 VS1 nozzle
Like the LEA nozzle, the VS1 nozzle shows sudden transition but not as extreme as LEA
nozzle. From Figure 5-7, the value of wall pressure at nozzle exit is not as low as LEA
nozzles. Thus, this nozzle is likely to have less chance of inducing shocks.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00
(pw/p0)
X/R*
Wall Pressure , LEA
CFD Data
MOC

61
Figure 5-7: Nondimensional Wall Pressure, VS1
Above Figure 5-7, shows that the truncated MOC contours’ wall pressure over predicts the
wall pressure initially and under predicts at near nozzle exit location. This discrepancy is
mainly due to difference in the contours designed by MOC and VS1.
Note that the deviation of the wall pressure around the throat is highly dependent on
upstream throat nozzle curve. In abstract, the wall pressure distribution from MOC code
shows similar results to ANSYS fluent.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.00 2.00 4.00 6.00 8.00 10.00 12.00
(pw/p0)
X/R*
Wall Pressure, VSI
CFD Data VSI
MOC

62
5.3 Thrust Coefficient
The dimensionless thrust coefficient makes it easier to compare the nozzle performance of
engines with different chamber pressure and mixture ratio. Thrust coefficient as described
in Section 2.9, can be written as:
𝐶 𝐹 = 𝐶 𝐹,𝑊𝑎𝑙𝑙 + 𝐶 𝐹,𝐵𝑎𝑠𝑒
The coefficient are calculated for the designed nozzle contours by integration of pressure.
The Pressre Ratio selection is based on previous analysis on these nozzles [25,26]. The
significance of pressure ratio is that at PR of 50, full flowing conditions could not be
achieved in LEA nozzle. However, PR of 350 represents ideal condition for the VS1
nozzle but underexpanded for theTIC nozzle (ideal 700) and overexpanded for LEA. As it
is clear from the literature, method of characteristics only solves for the full-flowing
condition of nozzle. The negative value of the 𝐶 𝐹,𝐵𝑎𝑠𝑒decreases with increasing PR.
Table 5-1: Thrust Coefficients calculation from MOC
Pressure Ratio CF ( MOC TIC ) CF (MOC LEA ) CF (MOC VS1)
50 1.08089 0.932740 1.01310
350 1.315412 1.451312 1.24643
1000 1.368178 1.507490 1.39310
From above Table 5-1, it can be observed that the increasing pressure ratio increases the
coefficient of thrust. In experimental results [25, 26], higher pressure ratio also increased
the thrust coefficient. As pointed out earlier, full-flowing condition can only achieved at a
particular pressure ratio. The actual value of CF at ideal flowing condition for VS1 nozzle

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