EFFECT OF EXTERNAL PULSE ON SOLID PROPELLANT ROCKET INTERNAL BALLISTICS

Niraj Solanki

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Aerospace Studies

University of Toronto

O Copyright by Niraj Solanki, 2000

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Abstract An introductory evaluation of the effects of an extemally induced pulse on the intemal

ballistics of solid propellant rocket motors (SRMs) is performed in this study. The main reason

for the study was to demonstrate numerically that the axial wave motion generated inside the

rocket combustion charnber resulting f'rorn an extemal pulse has enough strength to cause a

motor to go into nonlinear combustion instability. In doing so, this study modelled N o practical

scenarios that would induce a pulse inside the SRM. The first scenario modelled was one in

which a missile in its mission flight encounters a blast wave resulting from a nuclear explosion at

a distance. The pulse generated within the SRM as a resuit of missile/blast wave interaction is

evaluated. Another scenario modelled was a case in which the SRM is being tested on the staiic

thrust test-stand when suddenly the load-ceil fails causing a sudden acceleration of the motor,

resulting in the motor impacting an end-wall from which it may bounce back and forth until it

reaches a new equilibriurn position. The effect of a pulse induced through this scenario is

evaluated. Further, a pulsing unit is designed for pulsing an SRM artificially to test the effect of

extemally induced pulses on the intemal ballistics of SRMs.

The results of the project demonstrate that the tint scenario creates weak pressure waves

inside the SRM, however, they are strong enough in some situations to be above the minimum

criterion required to cause combustion instability. The result of the second scenario also

produces weak pressure waves inside the SRM, however they are stronger in cornparison to the

firsi scenario. It indicates that the extemally induced pulse resulting fiom this scenario will more

likely cause nonlinear combustion instability in inherently susceptible SRM designs.

Acknowledgments

I would like to thank Dr. David Greatrix (my thesis coordinator) for allowing me to work with him on such an interesting and challenging project. His constant encouragement, motivation and financial aid in the fom of a stipend is acknowledged here with thanks. In addition to this, his help in debugging the code and aid in understanding of the various aspects of theory associated with solid propellant rocket motors is greatly appreciated. Also, 1 would like to thank him for his patience and devotion towards helping me complete my project. 1 think he is the best supervisor a graduate student can have and for this reason 1 don? think I can explain in words how much his help and kindness meant to me.

1 would also like to thank Prof. Gonlieb (my thesis CO-supervisor) for his constant encouragement and motivation to work hard. In addition to this, his financial assistance in paying for my expenses to present a paper resulting fiom this thesis project is acknowledged here with thanks. Furthennore, I would like to thank him for teaching me FORTRAN programrning and an excellent graduate course in unsteady gasdynamics which were the basis for this project and hence very crucial to my understanding and analyses of this thesis project.

I would also like to thank Mr. Jerry Karpynczyk for providing me encouragement and motivation and not to forget, he was the person who recommended me to join Dr. Greatrix's research team. Thank you Jerry for your advice.

1 would also like to thank my family for their constant support and encouragement to punue my graduate studies. Without their support this would not have been possible.

Last but not least 1 would like to thank God, without whose mercy I would not have met these great people and have hished my thesis in such a short time. So thank you God.

iii

Table of Contents

Abstract Acknow ledgements List of Tables List of Figures Nomenclature

Chapter 1 Introduction 1.1 Introduction 1.2 Factors Encouraging Nonlinear Combustion Instability 1.3 Methods Used for Stability Ratings 1.4 Present Study on Extemal Pulse Effects

Chapter 2 Scenario 1 - Missile-Blast Wave Interaction 2.1 Introduction 2.2 Introduction to Blast Waves 2.3 Missile Characteristics 2.4 Modelling of Missile-Blast Wave Interaction 2.5 Modelling Intemal Ballistics of a Solid Rocket Motor

2.5.1 Model ling of Core Flow Inside the Chamber 2.5.2 Determination of Axial Acceleration With Inclusion of Structural

Defonnation 2.5.3 Modelling of Radial Vibration 2.5.4 Model for Propellant Buming Rate 2.5.5 Solution Procedure Using Random Choice Method

Chapter 3 Example Results for Scenario 1 3.1 introduction 3.2 Results and Discussion 3.3 Remarks Concerning Pulse Strength

Chapter 4 Scenario 2 - Test Stand Failure 4.1 introduction 4.2 Modelling of SRM/Wall Interaction 4.3 Results and Discussions 4.4 Rernarks Conceming Pulse Strength

rn . . 11

S. .

111

vi vii

Chapter 5 Design of Pulse Generators 5.1 introduction 5.2 Current Pulsing Units 5.3 Design Requirements 5.4 Proposed Design

Chapter 6 Conclusion and Future Work 6.1 Conclusion 6.2 Future Work and Recornrnendations

REFERENCES

List of Tables Table 2.1 Motor characteristics

List of Figures Figure 1.1 Figure 1.2 Figure 2.1 Figure 2.2 Figure 2.3a Figure 2.3b Figure 2.4

Figure 2.5 Figure 2.6

Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.1 1 Figure 2.12 Figure 2.13 Figure 2.14 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.1 1 Figure 3.12 Figure 3.13 Figure 3.14

Figure 3.15

Figure 3.16

Nonlinear vs. normal operation of an SRM Different grain shapes and their corresponding t h s t profiles Blast wave profile Blast wave profile as a fùnction of distance Peak overpressures on the ground for a 1 -kiloton burst Positive phase duration of overpressure on the ground for 1 -kiloton burst Rate of decay of pressure with time for various values of the peak overpressure Generic missile used in this study Missile h g coefficient as a function of Mach number for various angles of attack Pre-blast wave interaction shock geometry Blast wave interaction pattern with supersonic vehicle Interaction time calculation geometry General missile and blast wave flow geometry Cylindrical-grain SRM Missile mode1 for deformation analysis Displays first two steps of RCM Displays last two steps of RCM Blast wave profile at sea level- 10 1.3 kPa peak overpressure Blast wave profile at sea level - 202.6 kPa peak overpressure Velocity profile behind 10 1.3 kPa peak overpressure blast at sea level Velocity profile behind 202.6 kPa peak overpressure blast at sea level Head-end pressure profile with missile at sea-level, 0 = 0°, M = 2.10, 202.6 kPa overpressure Head-end pressure profile with missile at sea-level, 0 = 0°, M = 3.08 1, 202.6 kPa overpressure Head-end pressure profile with missile at sea-level, 8 = 0°, M = 3.57, 202.6 kPa overpressure Head-end pressure profile with missile at sea-level, 0 = oO, M = 5.5 1 1, 202.6 kPa overpressure Head-end acceleration profile, M = 3.08 1,8 = 0°, sea-level, 202.6 kPa overpressure Head-end acceleration profile, M= 5.5 1 l , û = O*, sea-level, 202.6 kPa overpressure Head-end pressure profile with missile at sea-level, M = 3.08 l ,û = 22.5" Head-end pressure profile with missile at sea-level, M =3.08 1,8 = 45' Head-end pressure profile with missile at sea-level, M =3.08 1,0 = 90' Variation of missile angle of attack with tirne for 8 = 90°, sea level, M = 3.081 Head-end acceleration profile when 0 = 90°, sea-level, 202.6 kPa overpresswe Missile acceleration ~rofile when 0 = 90°, sea-level, 202.6 kPa overpressure

vii

Figure 3.17 Figure 3.18 Figure 3.19 Figure 3.20 Figure 3.2 1 Figure 3.22 Figure 3.23 Figure 3.24 Figure 3.25 Figure 3.26 Figure 3.27 Figure 3.28

Figure 3.29 Figure 4.1 a Figure 4. l b Figure 4.1 c Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.1 1 Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 Figure 4.1 8 Figure 4.19 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9

Head-end pressure profile , O = 0°, sea level, 202.6 kPa overpressure Headsnd pressure profile with missile of 1059 d s , 0 = O0 Head-end pressure profile with missile of 1062 d s , 0 = 0' Head-end pressure profile with missile of 1059 m/s, 0 = 22.5' Head-end pressure profile with missile of 1062 mfs, 0 = 22.5" Head-end pressure profile with missile at sea level, M= 3.07,û = 0' Head-end chamber pressure profile, M = 5.5 1 1,0 = O", sea-level Head-end pressure profile, M= 5.5 1 1, sea level, & = 0.1 Head-end acceleration profile, M = 5.5 1 1,8 = 0°, sea-level, 5, = 0. I Head-end pressure profile, M= 5.5 1 1, sea level, b = 0.1 Head-end pressure profile. M= 5.5 1 1, sea level, k = 1 x log Nlm Noule-end charnber pressure profile, sea-level, 8 = 0°, 202.6 kPa overpressure Missile acceleration profile, sea-level, 0 = 0°, 202.6 kPa overpressure Prior to load-ce11 failure Upon load-ce11 failure - model of the wall and motor in f?ee flight Shows model of wall and SRM interaction d = 5 cm, kW = 5.01 x 10'~/m, Gw = 0.0 d = 5 cm, kW = 2.0 x 108~/rn, 6, = 0.0 d = 5 cm, k,=3.0 x 108~/rn, ~ , = 0 . 0 d = 5 cm, kW = 8.01 x 108~/rn, 6, = 0.0 Acceleration profile with d = 5 cm, kW = 8.01 x 108 Nlm, c, = 0.0 Acceleration profile with d = 5 cm, kW = 3.0 x logN/m, <,,, = 6.0 Acceleration profile with d = 5 cm, k W = 5.01 x 10'NIm. Cw = 0.0 Head-end acceleration profile with d = 5 cm, kW = 8.0 1 x 108 N/m, GW = 0.0 Head-end acceleration profile with d = 5 cm, kW = 3.0 x 10' Nlm, &,,, = 0.0 Head-end acceleration profile with d = 5 cm, kW = 5.01 x 10' N/m, Gw = 0.0 d = 10 cm, kW = 3.0 x 1o8N/rn, cw = 0.0 d = 15 cm, kW= 3.0 x 1o8N/rn, Cw =0.0 Acceleration profile with d = 10 cm, kW = 3.0 x 108 N/m, Gw = 0.0 Acceleration profile with d = 15 cm, kW = 3.0 x 10' N/m, Cw = 0.0 d = 5 cm, kW = 3.0 x 108~/m, 5, = 0.5 d = 5 cm, kW = 3.0 x loaN/m, &, = 1.0 d = 5 cm, kW = 3.0 x 108 N/m, c, = 4.0 d = 5 cm, kW = 3.0 x 1o8N/rn, cw = 0.0 Explosive bomb Simplifieâ schematic of pyrotechnic pulser unit Pressure history for pyrotechnic pulsers Simplified schematic of low-brisance unit Low-brisance pressure-tirne profile Simplified schematic of piston pulser unit Pressmtime history of piston pulser Setup for intemal pulse simulation Setup for extenial pulse simulation

viii

Nomenclature

Arabic missile cross-sectional area or local cote cross-sectional area inside the chamber chamber head end cross-sectional area c hamber wall cross-sectional area chamber nozzle end cross-sectional a m grain port ana nozzle throat area core flow sound speed axial or longitudinal nodal acceleration maximum drop in head-end acceleration upon blast wave interaction r :mal acceleration of propellant/casing assembly sound speed behind the blast wave as a bction of time free Stream sound speed at normal atmospheric conditions sound speed right behind the blast wave shock pressure-dependent buming rate coefficient thrust coefic ient missile drag coefficient specific heat of solid particles specific heat at constant pressure of combustion products at the flame specific heat of solid propellant damping constant missile drag or drag force acting on the particle due to viscous interaction between jwticulate phase and gas inside the chamber missile drag just d e r complete submergence inside the blast wave missile drag just bcfore interaction with blast wave distance between SRM and the end wall or the local grain port diarneter combustion chamber inner diameter initial port diarneter nozzle boa t diarneter elastic modulus of SRM casing total energy per unit volume of gas in the combustion chamber total energy per unit volume of particdate phase in the combustion chamber head-end force on the SRM nozzle-end force on the SRM Darcy-Weisbach fiction factor naturd axial frequency of the core flow cavity natural axial structural frequency of SRM casing naturai frequency of the SRMIwall system accelerative mas flux component SRM casing waii thickness or effective convective heat transfer coefficient for erosive buming convective heat transfer coefficient for zero-transpiration

rb rm ro

rp SRM

orientation correction factor thermal conductivity of combustion products at the flame payload spring constant wall spring constant effective length of the chamber missile length propellant grain length missile flight Mach number relative missile Mach number due to blast wave flow superimposed on the missile blast wave shock Mach number flow Mach nunber right behind the blast wave shock average mass of a particle mass of the payload mass of SRM time level or pressure-dependent buming rate exponent gas Prandtl number local static chamber pressure blast wave peak overpressure blast wave overpressure as a function of time mean combustion chamber pressure free stream static pressure or core flow pressure ahead of compression wave static pressure right behind the blast wave shock or right behind the core flow compression wave heat transfer rate f'kom core flow to the particles in the combustion chamber specific gas constant Reynolds number based on core diameter inner propeilant radius outer propellant radius radial distance fiom chamber wall mid-surface to the motor centreline overall propellant buming rate wall mid-surface distance from centreline of the motor under no charnber pressure base propellant buming rate pressure dependent buming rate solid propellant rocket motor thnist or average temperature inside the combustion chamber or period of core flow cavity waves propellant flame temperature initial propellant temperature particle temperature propellant surface temperature fiee stream temperature temperature right behind the blast wave shock time blast wave arriva1 time missile-blast wave interaction time blast wave positive phase duration

flow velocity of the gas inside the combustion chamber missile velocity particle core flow velocity relative flow velocity over the missile flow velocity behind the blast wave as a function of time total velocity of propellant inflow above the flame (p, rd p) flow velocity right behind the blast wave shock distance measured fiom motor's head-end

Greek a missile angle of attack ai initial missile angle of attack prior to blast wave amival a, fraction of particle m a s in solid propellant a, angle between missile velocity and relative flow velocity E propellant surface roughness height during combustion P heat flux coefficient AH, net surface heat of reaction Ap initial pulse amplitude in the core flow gas Ar incremental displacement of the propellant surface with respect to casing displacement Arai, incremental displacernent of casing

time increment increment of core flow gas velocity due to extemal pulse distance between nodes reference energy film thickness specific heat ratio, 1.4 for atmospheric air and 1.2 for gas inside the combustion chamber local wall dilation terrn SRM structural displacement head-end displacement of the SRM structure payload mass displacement absolute viscosity of combustion products at flame and in core flow natural angular frequency of propellantlcasing assembly vibrating radially natural angular structural frequency of SRM structure

natural angular frequency of the payload spring-damper system

natural angular frequency of the SRM/wall system

acceleration orientation angle (degrees) displacement angle (degrees) density ofgas inside the combustion cbamber SRM casing material density particle core flow density propellant density flow density behind the blast wave as a function of time fm Stream density flow density right b e h d the blast wave shock

buming rate temperature sensitivity, as a pressure-dependent constant angle between blast wave and missile body axis Poisson's ratio random position (O<c~l) in random choice method SRM casing axial damping ratio payload damping ratio SRM casing radial damping ratio wall damping ratio

xii

Chapter 1 Introduction

1.1 Introduction Solid-propellant rocket moton (SRMs) are occasionally subjected to oscillatory

acceleration environments during their fiee flight or during static test stand firings12. These

structural oscillations would disturb the steady-state combustion process in the charnber, in some

cases initiating axial pressure waves that propagate back and forth along the chamber length.

The resulting unsteady flow field interacts with the combustion process and causes an increase in

the propellant burning rate3. Conditions may be such that these initial pressure waves grow

rather than decay. The waves may become strong enough to be identified as evidence of

sustained axial combustion instability. A proper coupling is required between the oscillatory

wave motion and the combustion process to maintain the sustained wave motion. As a result of

this sustained wave motion, the t h s t and base charnber pressure may also rise above nominal,

and in conjunction with strong motor vibration, this may in turn lead to a substantial degradation

in engine and vehicle performance. The rise in the base chamber pressure could go beyond the

structural design limit of the motor and result in chamber structural failure leading to the loss of

the SRM.

In flight, this sort of combustion instability c m be triggered via random unintentional

events fiom within the combustor, such as an expulsion of an igniter, an insulation hgment or

srna11 piece of propellant grain through the nozzle section of the motor? It could also be

triggered via sudden burning of the accumulated seactants in the combustor. In static test stand

firings, symptoms associated with axial combustion instability could be initiated via unexpected

synchronized vibration of the test stand or as a result of b s t load ce11 failure causing sudden

motor acceleration. It is worth noting that a SRM may be stable in the absence of the tnggering

disturbance but once pulsed could go violently unstable.

The development of sustained compression waves inside the combustion chamber of an

S M generally leads to degraded vehicle As a result of this, the designers and

users alike of SRMs would like to know what factors encourage the development of these high

amplitude pressure waves. Efforts have been made and research is still in progress tociay to

model the intemal ballistics of rocket motor in order to explain the phenornena that leads to

combustion instability. At present, SRMs are pulsed artificially to determine their susceptibility

to unstable cornbu~tion~~ 6. The motor's instability region is mapped out experimentally and then

evaluated against standard criteria. However, the triggering pulse shape, strength and its

duration encountered in practice are not fully known, hence, making it difficult to detemine with

any accuracy the motor's instability region6. If researchers could accurately model the wave

development inside the rocket charnber, it would mitigate the need for expensive expenments to

detemine a motor's stability. It would enable designers of SRMs to come up with better, more

stable designs without carrying out time-consuming experiments. The final designs could then

be tested experimentally for stability.

There are two types of combustion instability. The first type is called linear while the

other is nonlinear combustion instabilitys. In lincar combustion instability, a small amplitude

pulse triggers the SRM into combustion instability. For example, the pulse üiggered by the

ignition process dunng siart-up may be small but strong enough for a particular design of motor

to cause it to go into linear combustion instability. In this type of instability, a small amplitude

pulse triggers the development of sustained weak pressure waves inside the chamber. In

nonlinear combustion instability a large amplitude pulse triggers the development of sustained

high amplitude shock-fionted waves. Nonlinear instability may result from either ejection of

material through the noule, or via the delivery of a high amplitude pulse to the motor from the

surrounding environrnents Say due to an explosion at a distance. Generally, both types of

instability lead to sirnilar wave development inside the chamber. A motor that is linearly stable

is harder to be pulsed into nonlinear combustion instability.

At present, there is more interest in nonlinear combustion instability. The presence of

nonlinear combustion instability in an SRM c m be identified if there exist pressure oscillations

leading to a limiting amplitude, bas the presence of bigh amplitude shock-fionted pressure

waves, and has a mean pressure shift (dc rise) due to the waves h i d e the chamber. Figure 1.1

compares the head-end pressure-tirne profile for stable versus unstable operation of an SRM'.

The diagram displays the limiting amplitude (piim) and dc pressure nse bdC) typical of nonlinear

combustion instability.

In order for the motor to remain stable, the gasdynamic waves in the chamber must

decay. To achieve this goal some SRMs have stability devices such as baffles and resonance

rods7. These devices damp the pressure waves and return the motor to its steady-state operating

conditions. Inert or reactive particles added to the intemal flow is another common means for

stabilising motor operationa.

, dc Pressure shift

Pressure

Figure 1.1 Nonlinear vs normal operation of an S R M

1.2 Factors Encouraging Nonlinear Combustion Instabüity There are many facton involved in aiding the development of compression waves leading

to nonlinear combustion instability in SRMs. It cannot be said that a particular factor alone is a

prime cause of this sort of instability; rather it is a complex combination of these facton which

lead to instability. The following will illustrate the key factors known to aid development of

nonlinear combustion instability.

1) Chamber Pressure: It has been proven by various experiments that chamber pressure does

affect wave development. The higher the mean chamber pressure, the easier it is to trigger

rocket motor into nonlinear combustion instabilig 'O. It is possible for a motor to remain

stable to a pulse at low chamber pressure but to become unstable by the same pulse at higher

pressure.

Triggering pulse time lag after ignition: The time at which the motor is pulsed also affects

the motor's ability to return to its normal operating conditions. The greater the time elapsed

between ignition and pulsing, the easier it is to pulse some motors into nonlinear combustion

instability? This is largely due to the fact that at later times the charnber pressure is higher

for progressive-buniing motor grains, corresponding to factor (1).

PropeUant burning rate: The €aster the propellant burning rate, the harder it is to pulse the

motor into nonlinear instability4.

Propellant grain sbape and interna1 area transition: Propellant grain shape may aiso

affect the compression wave development. Different grain geometries may produce different

wave reflection characteristics that can have a profound effect on sustained axial wave

strength. Figure 1.2 (obtained fiom Ref. I l ) shows different grain dcsigns and their

corresponding thnist profiles. Also, area transitions within the propellant grain may

significantly affect the stability of a rocket motor as demonstrated by Greatrix and

Propel lan t

IF, @ lh Tirne

A Section A-A C Time [el A @ 1 h-7

Section A-A Time

A Section A-A Time

.-

Figure 1.2 Different grain shapes and their cocresponding thnist profiles

- - Section A-A

Leagth of combustion chamber: The length of combustion charnber also affects wave

developrnent. Typically, a longer charnber length is more susceptible to axial combustion 13.15 instability, althougb on occasion the reverse has been observed .

Amplitude and shape of triggering pulse: Whether the motor will go into combustion

instability or not is also influenced by the amplitude of the triggering pulse. The larger the

amplitude of the ûiggenng pulse the greater the probability of getting a sustained wave inside

the charnber8* '. Also, different triggering pulse shapes exist and their impact on rocket motor

stability may depend on their shape and duration, along with the motor design.

Type of propeliant: There are 2 basic types of propellant generally used for SRMs:

composite and double-base". The double-base propellants are composed of two primary

compounds, nitmglycenne and nitrocellulose. Each of these is sel f-sufficientl y combustible.

The composite propellants include a granular oxidizer such as ammonium perchlorate or

ammonium nitrate, which is mixed into an organic fuel binder such as asphalt, synthetic

rubber, or various plastics. The individual propellant composition affects the amount of

energy and mass released in the flow thereby infiuencing wave development. It is possible to

trigger a rnotor into combustion instability by changing its propellant while applying the

same ûiggenng pulse.

Percentage of stability additives in propellant: As the percentage of stability additives

such as zirconium carbide (2s) increases in the propellant, the more resistant the motor

becomes to nonlinear combustion9* ". However, it does not mean that a motor with stability

additives in the propellant will not go into nonlinear combustion instability. Experiments by

Blomshield et aL8 have proven that a motor with stability additives can be pulsed into

nonlinear instability.

Nozzle throat to grain-port area ratio: As the nozzle throat to grain port area decreases, the

susceptibility to nonlinear combustion instability increases3* 12. This corresponds to the

noule convergence more closely acting as an end-wall with respect to axial wave reflection.

10) Motor oscüiatioa frequency: If the motor structural oscillation frequency is close to the

natural frequency of the motor interna1 cavity, thcn there is more chance of the motor going

into nonlinear combustion instability?

11)Motor Miss: Lightweight motors are more susceptible to combustion instabiiity in

cornparison to heavy motors4.

1.3 Methods Used for Stability Rating The primary purpose of stability rating tests are to assure that a rocket engine will

perform ail its mission requirements without sustaining an uncontrollable instability. To

accomplish this, the stability rating tests are intended to simulate al1 disturbing forces that rnay

be encountered naturally by the engine. However, the types of disturbances to which an engine

might be subjected are never completely known. This makes the achievement of this goal very

difficult. However, there are two broad categones of stability rating techniques that

experimentally detemine an engine's stability6. They are as follows:

This method relies on the sponianeous occurrence of unstable combustion dunng normal

operation of the rocket motor. One spontaneous instability rating technique nomally used

during early development stages of an engine is the observation of the percentûge of tests in

which instability is triggered naturally during normal operation of the rocket motor. If the

naturally occurring disturbances are quite frequent such that the motor will become unstable on

numerous occasions, this clearly indicates that the design needs improvement. This method is

not very attractive for rating if the system is moderately to very stable, since, it would require a

large number of engine tests6. For this reason artificial pulsing meîhods are preferred which are

discussed later. Another rating technique for spontaneous instabilities employs sysiematic

variation of normal operating conditions until a region of spontaneous instability is found and its

boundaries are mapped out. The proximity of normal operating conditions to stability boundary

is presumed to be a direct indicator of the combustor's resistance to the occurrence of

spontaneous instability. The stability rating then detemines how the boundaries of the unstable

region shift with changes in operating conditions. Changes in operating conditions include

variation of factors mentioned in section 1.2.

The advantage of using rating techniques that rely on spontaneous occurrences of

instability is that the ratings are associated with the naturally occurring disturbances. Also in this

type of rating method the combustion gas flow patterns are not disnapted and contaminated by

foreign bodies or substances as typical of artificial pulsing methods6. Further, the stability

boundaries that are mapped out are quite reproducible. The disadvantage of this type of rating

techniques is that it requires carrying out a large nurnber of tests to obtain a single rating. Also,

this method fails to indicate the magnitude of the triggering pulse that leads to instability.

In this method, finite amplitude disturbances are introduced into the rocket motor

artificially to initiate instability. Regions of instability are mapped out by varying the pulse

strength and initiation time of the disturbing pulse. There are four different pulsing devices

generally used to obtain stability ratings in SRMs. These are explosive bombs", pyrotechnie

pulsers, low-brisance pulsers, and piston puisers'6. How these artificial pulsing units operate are

explained in Chapter 5 .

Explosive bombs produce sphencal pressure pulses that are not directionai, however, the

other three methods produce a pressure pulse that is strongest along the longitudinal a i s of the

motor and hence are used to test a motor's longitudinal mode of instability. Artificial pulsing

methods are most commonly used in industry to test a motor's stability to nonlinear combustion.

The advantage of this method is that the time and amplitude of the disturbing pulse can be

controlled6. In addition to this, the motor can be pulsed many times during the firing to obtain

stability ratings, thus reducing the cost. The disadvantage of the artificial pulsing methods is that

it does not completely simulate the naturally occurring disturbances inside the rocket motor.

Furthemore, the artificial pulsing methods induce foreign highflow temperature gases and

particles into the combustion chamber, thus distorting the core flow

1.4 Present Study on External Pulse Effects An important aspect of the present study is that solid propellant rocket motors rnay be

triggered into nonlinear combustion instability during their normal operation by an unexpected

extemal pulse, rather than by an internal distutbance initiation such as those mentioned in

Section 1.1. Most of the present studies to mode1 and experimentally test SRMs for nonlinear

combustion instability have been devoted to pulse generation fiom within the motor combustion

chamber. The effect of an externally induced pulse on the intemal ballistics of SRMs have been

overlooked in al1 these studies. In order to make stable SRMs, it is important that they be tested

for al1 the possible triggering pulses which may be encountered during their normal operation.

nierefore, it is the intent of the present study to investigate the effect of external pulse delivery

on the internal ballistics of SRMs. in doing so, two practical scenarios are numerically modelled

in some detail. The first scenario investigates the effect of a pulse delivered by a blast wave

Erom an external atmospheric explosion to an SRM within an intercepter missile in its ôpe flight.

This blast wave could be generated at a distance as a result of the exploding of combat fightea,

buildings, or incoming missiles that may be encountered by a missile during its mission flight in

a combat situation. The second scenario involves modelling the extemal pulse delivered to the

motor by the sudden failure of a thnist load-ce11 in a static test-stand fiMg. Due to the

component failure, the SRM suddenly accelerates and travels a specified distance before it hits

an end-wall and then oscillates back and forth until its new equilibrium position is attained. The

resulting intemal axial wave motion and its effects on the internal ballistics of the SRM are

evaluated for both scenarios noted above. The pulse strength generated inside the SRM as a

result of these scenarios is compared to the typical minimum pulse strength required to cause

combustion instability. Typically, pulse strength of more than 0.1% of base chamber pressure is 17,18 suficient to cause combustion instability in inherently susceptibly SRMs .

In addition to this, the study also looks at potential experimental methods that could be

used to test a rocket motor's stability artificially by simulating naturally occming extemal and

intemal pulses. It should be noted that the current pulsing units are inadequate for testing rocket

rnotor stability due to an extemal pulse, since their main focus is to simulate the nanirally

occuning inteml pulses. Further, the current intemal pulsing devices, as those mentioned in

section 1.3, contaminate the core flow by introducing foreign higMow temperature particles

along with disrupting the combustion flow pattern. It is therefore important to design a pulsing

unit that will eliminate the addition of foreign substances inside the combustion chamber and

more closely simulate the reality in order to obtain a better understanding of combustion

instability phenomena.

It should be mentioned here that this is an introductory snidy of evaluating the potential

effects of extemal pulse application on the intemal ballistics of SRM. Hence, this study is

primarily one-dimensional and models only the axial wave generated inside the SRM resulting

fiom the extemally induced pulse fkom the scenarios mentioned above. In addition to this, the

study does not include the effects of radial and transverse waves generated inside the combustion

chamber due to extemal pulse application.

In this study, an existing FORTRAN code written by Dr. D. R &eatrix4 for modelling

the internal ballistics of an SRM was modified to simulate the above mentioned scenarios. Since

the code was too long, it has not ken included in this thesis report.

Chapter 2 Scenario 1 - Missile-Blast Wave Interaction

2.1 Introduction This chapter is devoted towards the modelling of an extemal pulse that is delivered to the

solid propellant rocket motor (SRM) within an interceptor missile dunng its flight mission via a

blast wave created by an extemal explosion in the atmosphere. The characteristics of the blast

wave are such that it results in the application of a rapid transient decelerative load on the missile

(and motor). The intemal ballistic response of the SRM and rocket motor structural deformation

are included in the analysis. This chapter begins with a brief introduction to blast wave

formation and the resulting wave profile. The method for modelling blast wave profiles for the

purpose of this study is outlined. The pertinent missile characteristics used in this study are also

presented. A generic interceptor missile similar in characteristics to the Patriot surface-to-air

missile was defined for this study. This chapter then proceeds to descnbe how the blast wave

and missile interaction was modelled, and in turn how the non-steady one-dimensional

hydrodynamic conservation equations were solved using the random choice method (RCM).

2.2 Introduction to Blast Waves Blast waves are generated by explosions of nuclear weapons or by conventional high

explosives such as TNT. For this study we will focus on the blast waves generated by detonation

of nuclear weapons since this type of explosion is more likely to be encountered for the scenario

being investigated. Blast waves created by conventional explosives are similar to nuclear blast

waves except for the fact that the latter releases more energy and therefore produces stronger

blast waves. Approximately 50% of the energy released h m a nuclear detonation goes into the

formation of a blast wave. This arnount decreases with increasing altitude, however, there is not

a substantial change in energy below 30480 m (10,000 fi)".

When nuclear explosions occur, iremendous amounts of energy are liberated rapidly

within a limited space. This sudden liberation of energy causes a considerable pressure and

temperature rise at the point of detonation. The temperature nse is near tens of million degrees

centigrade in the immediate vicinity of the explosion. This in turn causes vaporisation of al1

matenals into hot compressed gases. These high-temperature compressed gases expand rapidly

by moving out radially at high velocities thus initiating a pressure wave called a shock wave. A

shock wave in air is generally referred to as a blast wave. The characteristic profile of the blast

wave is such that there is a sudden increase in static pressure at the shock tiont and a graduai

decay behind it as shown in Figure 2.1. Also associated with the blast wave is a sudden increase

in flow velocity (leading to an increase in dynarnic pressure), density, temperature, and sound

speed behind the shock front. Wave profiles for the above mentioned flow properties are similar

in shape to the static overpressure profile of the blast wave.

The maximum pressure of the blast wave (located at the shock front) is called the peak

overpressure. Overpressure refers to the jump in pressure above the normal atrnospheric

pressure. Similarly the drop in pressure below the atrnospherk pressure is referred to as

underpressure. The time duration for which the overpressure exists is called the positive phase

or compression phase. The positive phase causes the most damage. The time duration for the

underpressure is called the negative or expansion phase. During this phase the flow reverses and

moves towards the explosion. The positive phase can last from a couple of milliseconds up to

approximately 10 seconds.

.

"

-5 O 0.5 1 1 .S

Time (s) Figure 2.1 Blast wave profile

hock Front

Positive Phase

In order to mode1 a blast wave it is important to understand the relationship between peak

overpressure, positive phase duration, the weapon yield, atrnospheric conditions (changes with

altitude) and the distance fkom detonation. The blast wave peak overpressure decreases with

increasing distance from an explosion of a given yield, while the reverse is tme for positive

phase duration as shown in Figure 2.2. Also, for a specified yield of an explosion and distance,

the blast wave peak overpressure generally decreases with increasing altitude of the burst point

while the positive phase duration increases. Further, as the weapon yield increases, the positive

phase duration and peak overpressure at a given distance also increase. As can be seen there is

an inverse relationship between the peak overpressure and positive phase duration. As one

increases the other decreases. In addition to this, it is clearly show that for a given peak

overpressure, the corresponding positive phase duration is going to be the same for different

yield weapons (only the distance at which this overpressure is felt is different) as long as the

energy formation mechanism does not change sub~tantial l~~~. This last point is very important in

modelling the blast wave profile since in this stuciy, peak overpressure was chosen arbitrarily

regardless of the weapon yield, altitude and distance fiom the explosion centre and its

corresponding positive phase duration was determined as a function of the overpressure.

explosion

Figure 2.2 Blast wave profile as a function of distance

In this study, the positive phase duration correspondhg to a given peak overpressure was

determined using the curves shown in Figure 2.3. These were obtained fiom Ref. 19 in which

they combined theory and achial experiments to produce these cwes . Furthemore, the rate of

decay corresponding to a given peak overpressure was determined using Figure 2.4, obtained

also fiom Ref. 19. Equations were derived such that the decay of the blast wave closely rnatched

that of Figure 2.4 for a given peak overpressure, and the corresponding positive phase as

obtained fiom Figure 2.3.

-

a) Peak overpressures on the ground for a 1-kiloton burst

b) Positive p Figure 2.3

Distance fiom ground zero (fi)

Distance fiom ground z m (fi)

tse duration of overpressure on the ground for 1-kiloton burst

A

Figure 2.4 Rate of decay of pressure with time for various values of the peak overpressure

0.4 0.6 Normalized Time, t/tc

For example, the blast wave profile for a peak overpressure of 30 psi and the corresponding

positive phase duration of 0.167 seconds (obtained from Figure 2.3 at sea-level i.e. height of

burst = O fi) was modelled using Equation 2.1 where pb is the peak static overpressure, t is the

iime from the point of blast wave arriva1 and L is the positive phase duration. This equation was

was obtained from Ref.19 which States that this equation gives a good correlation to the actual

blast profiles provided the peak overpressures are less than 10 psi. For peak overpressures of

more than 10 psi, this study added some correction factors to this equation to match the rate of

decay to that of Figure 2.4.

Knowing the initial pressure discontinuity due to the blast, the blast wave shock front Mach

nurnber (M,) was determined using:

where pz is static pressure right behind the shock front aad pl is the nomal atmospheric pressure

(free stream pressure) ahead of the blast wave. in addition to this, the temperature, density, and

sound speed ratio across the shock was computed using Equations 2.3 to 2S2*. Subscripts 1 and

2 represent flow properties ahead and right behind the shock wave respectively:

Furthemore, the flow Mach number (M2) and flow velocity (u2) behind the shock was computed

using Equation 2.6 and 2.7:

u2 = M,a, . (2-7)

The variation of the abovementioned flow properties with time behind the blast wave was

assumed to be isentropic and was computed using isentropic flow relationsZo (Equations 2.8 and

2.9). The accuracy of this assumption improves at lower blast wave strengths.

The flow velocity behind the shock as a fwiction of time was obtained using the method of

characteristics for non-steady isentropic flows as shown Ui Equation 2.10:

The wave profile for the flow properties behind the shock was similar in shape with the blast

wave pressure profile.

2.3 Missile C haracteristics For this study a generic interceptor missile was defmed. tt is 5.2 metres long and 4 1 cm

in body diameter with four fuis. Figure 2.5 shows the generic missile. This missile contains 345

kg of payload weight (mPl). This included the weight of electronics, warhead, etc. I t is propelled

by a solid propellant rocket motor with characteristics shown in Table 2.1. A typical composite

propellant, nonaluminized ammonium perchlorate/hydroxyl-terminated polybutadiene

(APMTPB), was used.

I Payload SRM -

Figure 2.5 Generic missile used in this study

The missile's drag coefficient (Cd) for different Mach numbers (M) and angles of attack (a) was

obtained using the Digital DATCOM~' aerodynamic design software. Curves were modelled

using the points obtained by this software and their corresponding equations were derived.

Figure 2.6 shows the modelled c w e s for drag coefficient as a function of Mach number for

different angles of attack for the generic missile.

Chamber wall thickness (h) 1 2mm I

Chamber wall material

Elastic Modulus for chamber wall matenal (EJ

Steel

200 x 10' ~ / r n '

1 Chamber wall Poisson's ratio (v,) 1 0.3

Chamber wall axial damping ratio (k) Chamber wall radial damping ratio (I;)

1 Grain shape Cylindrical I

4.0

O. 1

Propellant

Propellant grain length (Lp)

APMTPB

2.65 rn

Initial port diarneter (di)

Imer casing diarneter (df)

1 Propellant specific heat (Cs) 1 1500 Jlkg-K 1

40 cm

41 cm

Nozzle throat diarneter (4) Propellant/nozzle contraction length ratio

Pressure-dependent buming rate (r,)

Propellant density (pJ

I l cm

9: 1

0.09 [p(kPa)lo-' cm/s ,

1750 kg/m3

Propellant flame temperature (Tf)

1 Initial propellant temperature (Ti) 1 294 K 1

3000 K

Propellant surface temperature (T,)

1 Propellant surface roughness (E) 1 4 0 0 p I

1000 K

1 Gas specific heat (C,) 1 2000 Jlkg-K 1 Gas Prandtl number (Pr)

Specific gas constant (R)

1 Gas specific heat ratio (y) 1 1.2 I

0.8

320 Jkg-K

Gas thermal conductivity (k)

Gas absolute viscosity (p)

1 Particle rnass fraction (a,) 1 0% l

0.2 W/m-K

8.075~ 1 o - ~ kglm-s

1 Thnist Coefficient (CF) 1 1.3

Table 2.1 Motor characteristics

I

Figure 2.6. Missile drag coefficient as a function of Mach number for various angles of attack

a = O" O 1 I I 1 L

O 1 2 3 4 5 6 7 8 Missile Mach Number

2.4 Modelling of Missile-Blast Wave Interaction When a vehicle interacts with a blast wave there is a sudden increase in vehicle drag

causing the vehicle to decelerate and then gradually accelerate under motor thmst as the vehicle

passes through the blast wave. The effect of this transient acceleration pulse transmitted to the

vehicle is of interest here. In this study, the determination of missile acceleration,deceleration

due to the interaction with the blast wave is divided into three distinct phases. The first phase

deals with pre-blast wave drag calculations. The second phase models vehicle drag for the

duration in which the blast wave peak overpressure moves from missile nose tip to the nonle

exit tip. During this period a complex interaction of missile body shock and blast wave takes

place. in the final phase the missile is completely submerged within the blast wave.

In the pre-blast wave phase (shown in Figure 2.7, obtained fiom Ref. 22), the missile

drag @) was calculated using Equation (2.1 1) where pl is fkee Stream pressure as shown in

Figure 2.7 and A is the missile transverse cross-sectional area:

D = o . ~ c , I > , M ~ A . (2. I 1)

in this study, it was assumed that the missile was flying through quiescent air before interacting

with a blast wave. The cimg coefficient was obtained fiom Figure 2.6.

Figure 2.7

In the second phase a complex interaction takes place between the missile body shock

and the blast wave. This interaction is quite dificult to mode1 exactly since there is a complex

pattem of wave reflection from the missile body surface, which depends on the angle of vehicle

entry into the blast wave. The reflected wave fiom the missile surface could be either regular or

Mach reflection which again depends on the angle of vehicle entry into the blast wave2'. The

body shock angle also changes with time since the flow properties ahead of the missile are

changing. In addition to this, things get more compiicated when the reflected shock from the

missile body interacts with the new body shock. This may cause another shock to be transmitted

and reflected fkom the missile surface. This entire interaction takes place in a very small amount

of time - in the order of milliseconds. Figure 2.8 shows a typical wave interaction pattem

resulting fiom a supersonic vehicle interaction with a blast waveu.

Figure 2.8 Blast wave interaction pattern with supersonic vehicle

In order to simulate this cornplex interaction, this study linearly increased drag on the missile

From its initial h g (when blast wave is at the nose tip, pre-blast wave conditions) to the final

drag (blast wave located at the novle exit tip) when it is completely submerged inside the blast

wave. This linear approximation for drag increase is reasonable since the interaction duration is

very small. To approximate the interaction time, the missile Mach number was assumed to be

constant during for that period. Figure 2.9 shows a general case used to determine the interaction

period. In this diagram the missile is initially at some angle of incidence 0 to the blast wave

when it strikes, changing the vehicle's angle of attack with the oncoming induced tlow behind

the blast wave front.

I

Blast wave location at start of interaction

Blast wave +location at end

of interaction

Figure 2.9 interaction time calculation geometry

Interaction time (cm) was derived fiom Figure 2.9 using Equation 2.12, where M, is the blast

shock Mach nurnber and ai is the fkee Stream sound speed before interaction.

- L, cos B + (2h +d ,) sin B tint -

(M, + w u ,

Drag was linearly increased for this interaction p e n d using Equation 2.13 in which Di is vehicle

drag at the beginning of interaction and Df is the fmal drag at the end of interaction:

where Df = 0.7Cd ( p , + p , ) ~ I A and Di is the sarne as given in Equation 2.1 1.

In the final phase, drag was calculated using Equation 2.14:

D = 0 . 7 ~ ~ ( p , + p, ( t ) ) ~ f A . (2.14)

In this equation overpressure was added to atmosphenc pressure since the free stream conditions

were varying with time due to the blast wave. Furthemore, the relative missile Mach numbet

(MJ was used, since the induced flow associated with the blast wave was superimposed on the

flow passing over the missile.

The missile angle of attack in the final phase was derived as a function of time fiom

Figure 2.10. In this diagram, the variables ai, a, and 0 represent the initial missile angle of

attack, the new missile angle of attack resulting from additional flow from the blast wave, and

the angle between the blast wave induced flow and the missile body axis respectively. From this

diagram it is obvious that the angle of attack (a) can be defined as a linear combination of ai and

a, as given by Equation 2.15.

a-a, +ai . (2.15)

Since the initial missile angle of attack ai and the blast wave angle 0 are known, the relative flow

over the vehicle (u,) can be determined using the Cosine Law as given by Equation 2.16.

Variables LI, and u(t) are missile velocity and blast wave fiow velocity respectively. Both of

them are varying with time.

The angle defimd as a, must be computed in order to find the new missile angle of attack, hence,

it can be defined as:

u; +u; - ( ~ ( t ) ) ' a, = cos-'

2urum 1

Figure 2.10 General missile and blast wave flow geometry

2.5 Modelling Interna1 Ballistics of a Solid Rocket Motor In order to model the intemal ballistics of the rocket motor, a model is required that

incorporates both the nonsteady gasdynamic flow within the motor's core (including solid-fluid

interaction) and the propellant bumhg process at the core periphery. This section is divided into

parts in order to clarify how the intemal ballistics of the SRM was modelled.

2.5.1 Modelling of Core Flow Inside the Chamber For a cylindncal grain SRM, the two-phase (gas and particles) subsonic core flow along

the length of the propellant grain and nozzle convergence, and the supersonic flow along the

noule divergence, can be modelled adequately via the non-steady one-dimensional

hydrodynamic conservation equationsu. The conservation equations and solution procedures

c m be simplified if the reference fiame is h e d to the accelerating rocket motor structure.

Figure 2.1 1 shows a standard configuration setup for the solution procedure of the intemal

ballistics analysis, with the axial direction x comrnencing fiom the head end of the motor, i.e., x

= O at the head end of the motor.

Figure 2.1 1 Cy lindrical-grain SRM

With this setup, the appropriate equations for conservation of mas, momentum and energy of

the compressible gas phase of the core flow can be expressed by Equations 2.18 to 2.20, and for

the piuticulate phase by Equations 2.2 1 to 2.2323.

& + a(ue+ up) 4 5 - v' 1 dA =-(1-a,)p,(C,T, +L)--- (ue+up)-(-+KI@-pco, 4 5 --(u,D+Q) PP

at & d 2 A & d

In the above equations rb is the local propellant buming rate at the distance x fiorn the

head-end, a, is longitudinal acceleration of the gas resulting fiom the oscillation of the motor in

the axial direction, d is the local hydraulic diameter of the core flow, A is the local core cross-

sectional area, a, is the mass fraction of inert particulates within the propellant, and p, is the

solid propellant density. Variables p, u, p, and e are the local gas density, velocity, pressure, and

energy per unit volume (e = p/(y-1) +1/2 p2) respectively. Variables C,, Tf, and v, are the gas

specific heat, flame temperature, and flame front velocity. Variables p,, u,, and e, are the

particle core flow density, velocity, and energy per unit volume (e, = p,C,T,+IIî pput with TP

and Cm being the particle temperature and specific heat) respectively. The viscous interaction

between the gas and particulate phases is represented by h g force D, and heat transfer from the

core flow to the particles is defined by Q. The variable K represents the radial dilation of the

fiow at a given section due to casing/propellant wall rnovement above and beyond that due to

propellant regression under burning. It is expressed as K = 11A ' d N d t.

The importance of the ternis on the right hand side of the non-steady equations of motion

are descnbed here to clarify where the terrns are coming fiom. Each term is designated by a

letter. In Equation 2.18 term (a) represents the mass addition terni for the gaseous combustion

products evolving h m the local propellant surface, where La, represents the mass fiaction of

solid propellant composed of energetic material. Term (b) represents a mass flux dilatation

decrease due to the expanding cross-sectional area as the propellant sunace regresses and due to

propellantlcasing radial vibration. Tenn (c) represents a mass flux decrease or increase

depending on whether the port is diverging or converging axially, and it is important when the

propellant surface regresses nonuniformerly with distance. in Equation 2.19, tetm (d) represents

the effect of an axially converging or diverging port area on the gas momentun. Term (e)

represents the dilatation reduction of mornentum flux due to propellant surface regression and

radial casing/propellant vibration. T e m (f) and (g) represent the reduction in gas momentum

due to axial acceleration and due to drag of particles in the flow respectively. In Equation 2.20,

term (h) represents the enthalpy addition from the burning propellant surface to the core gas.

Term (i) represents the effect of axially converging or diverging port area and term (j) represents

the combination effect of dilating area due to propellant burning and radial vibration on the

enthalpy flux. Term (k) represents the Ioss of energy due to axial acceleration and terni (1)

represeots the loss of energy due to particle drag and heat transfer fiom gas to particles.

Beginning with the particle phase equatiom of motion, term (a) in Equation 2.21 gives

the particle mass addition fiom the pyrolyzhg propellant sutface and term (b') represents the

dilatation loss on the pamcle mass flux due to propellant surface regression and raâial propellant

Icasing wall movement. The effect of axial area change on the particle mas flw is given by

tetm (c'). In Equation 2.22, terni (d) indicates the effect of axial area change on particle

momentum. Tem (e') represents the radial dilation loss on momentun flw . Tenn (0 represents

the loss of particle momentum as a result of axial acceleration and the term (g) represents the

momentum imparted by the gas to particles. In Equation 2.23, tenn (h') the inflow of particle

enthalpy fiom the burning propellant surface. Tenn (i') represents the effect of axially diverging

or converging port area on particle enthalpy and tenn (j') represents the loss of enthalpy flux due

to dilating area as a result of propellant burn back and radial vibration. Term (k') represents the

loss of particle energy due to axial acceleration and the final tem (I') represents the energy

imparted by the gas to the particle through both drag and heat tnuisfer.

In most SRM design studies in one dimension, the particle phase equations are not

included at al1 for calculating chamber flow behaviour. This is partly due to the fact that the

range and magnitudes of particle diameters may not be well known. In addition to this, the

loading percentages of particulates in solid propellants are in many cases low and cm be

justifiably neglected. As a result, the presence of two-phase flow is commonly taken into

account only when calculations for two-âimensional flow through a nozzle are perfomed, since

the presence of particies can significantly affect the predicted thrust. This study perfomed

calculations based on a propellant with no particdate loading, however, the above mentioned

equations of motion for the particulate phase, and terms including variables of the particle phase

in the gas phase equations, have been included here for completeness.

in order to perform the intemal ballistics analysis of an SRM, two boundary conditions

are required. The first is that the flow velocity at the head-end (x = O) of the motor is zero. The

second is that the flow at the nozzle throat must be choked that is, the flow Mach nurnber u l a

is Wty.

2.5.2 Determination of Axial Acceleration With Inclusion of

Structural Deformation

Axial acceleration of the core gas (a3 due to motor oscillations appeeis in the momenhun

and energy equations as a body force contribution within a fixed Eulerian teference and is a

function of time and position of the SRM structure. The axial acceleration effectively included

the rigid vehicle acceleration supenmposed with the acceleration of each node on the structure

due to stnichiral deformation. In order to determine the overall acceleration, the missile was

treated as a prismatic beam as show in Figure 2.12, and beam theory was used to determine

nodal displacement as a function of time and the corresponding nodal acceleration.

F

- Figure

Cap mass Nozzle mass

7

Payioad II

2.12 Missile

Equivalent chamber x-section as beam

for deformation analysis

4

Equation 2.24 is the structural wave equation24 that was used to detemine the nodal

displacement at a given axial location :

where the natural angular fiequency of the chamber is detennined via,

Variables h, E,, and h in the above two equations are charnber wall thickness, elastic modulus of

chamber material, and nodal displacement on the structure. Equation 2.26 is the boundary

condition applied at the head-end of the SRM.

where the head-end force is detemined using the following:

In Equation in 2.27, D is the drag force of the vehicle, hi is the head-end node displacement, and

c is the damping constant as defined below:

= 2mp,%$Cp, 9

where is the payload damping ratio equivalent to 4.0 and a, is the natural angular fiequency

of the payload mass spring-damper system defined in Equation 2.29 in which kPi is the spring

constant equal to 1 x 10' N/m: -

The boundary condition applied at the node-end of the SRM is given by:

where the force at the nozzle-end F, is defined as follows:

in which the first term T is the thrust force given byl':

T =&&A, ?

and the second tenn represents integration of pressure force at the nonle exit. In the pressure

integration tenn, variable A, is the grain port area at the n o d e entrance and A, is the nozzle

throat cross-sectional area.

A second order finite difference solution technique was applied to the structural wave

equation (Equation 2.24) and a first order finite differeace solution was applied to the boundary

conditions of Equations 2.26 and 2.30. In hm, the axial acceleration ( a, ) at eac h node of the

structure was determined by applying a second order finite difference solution to Equation 2.33

where n+l, n and n-l represent next time level, current time level and previous tirne level

respectively :

In this study, it was assumed that a fiee slip condition existed between the gas core and

structure's periphery, bence the nodal acceleration of the structure is the same as core gas axial

acceleration (note that this core acceleration is only due to the motor/structural oscillation). This

axial acceleration was then superimposed ont0 the bulkaveraged acceieration of the core fiow

which was accelerating with respect to the rnotor.

In order to advance in time, it was found that the wave equation as given by Equation

2.24, had to have time increment At less then the stability criterion At as defined belod5:

At S [(1 -vJ

In order to make certain that the stability criterion was met, 1/10" of the At that was obtained

fiom the above equation was used in this study to advance in time.

2.5.3 Modelling of Radial Vibration The defonnation dynamics of the casing/propellant assembly was essentially modelled as

an independent ring elernent at each node dong the motor length. In this study, the propllant

was treated as incompressible, thus only the mass was accounted for in deriving the following

ordinary differential equation, which modelled the radial movement of the casing's thin wallL6:

Even though this is a simplified model, it shows good comlation to the experimental results at

later times into the firings2'. in the above equation, r is the radial distance of the wall's mid-

surface nom the motor centreline, r, is the wall mid-surface radius under no chamber pressure

and & is the d i a l darnping ratio. This equation is solved using a backward difference scheme.

The fundamentai natural frequency a>, of the casing/propellant assembly was obtained via:

where Ri is the radius of the propellant core penphery and pe is the chamber wall matenal

density. The normal acceleration (a,) was determined via finite differencing technique as:

When the casing expands/conîrac ts the displacement of the propellant surface may no t

be the same as that of the casing and depends on the propellant web thickness. A simple

volurnetnc displacement of the propellant has been assumed in this study. The incremental

displacement of the propellant surface with respect to the casing's displacement c m be

determined using:

2.5.4 Mode1 for Propellant Burning Rate Expressions for the overall propellant buming rate are intrinsically coupled to the

gasdynamic flow equations, and these equations must be defined before the core flow can be

computed. In this study the propellant pyrolysis rate was a function of local chamber pressure,

core flow velocity and normal acceleration directed into the propellant due to radial casing

vibration.

The pressure-dependent buming rate (r,) was determined using the St. Robert relation as 11.28, given by Equation 2.39 .

r, = Cp" , (2.39)

where p is the local static pressure and variable C is given by:

c = c. exp(?(~, -T,)) ,

in which Ti and a, are the initial propellant temperature and pressure dependent propellant

buming rate temperature sensitivity respectively. Tio and Co are reference values; for example,

Ti. is the room temperature. The constants Co and n are usually determined experimentally from

separate buming rate motor or strand bumer tests. The exponent n is invariable under most

conditions for a given propellant fomulation.

The buniing rate due io core flow velocity was determined via the quasi-steady (rapid

kinetic rate) buming-rate model developed by Greaûix and ~ottl ieb~' . This model is essentially

a convective heat transfer model, which stipulates that the velocity induced increase in the

pymlysis rate of the propellant is due primarily as a result of increased heat transfer. The

velocity augrnented burning rate is also known as erosive burning. In their model the overall

surface regression (rb) is defined as:

where ro is the base buming rate which includes pressure-dependent and normal-acceleration

related buming rate component while the remaining term is the contribution of velocity-induced

burning rate. In this equation, Ts is the propellant burning surface temperature and Tr is the

flarne temperature. The propellant specific heat and surface heat of reaction are denoted as Cs

and AH, respectively. The effective heat transfer coefficient h is detennined as follows:

h = ~ , ' p C p

exp(psrpCp 1 h') - 1 '

for which the zero-transpiration heat transfer coefficient h' is given as a hinction of gas thermal

conductivity k, Reynolds number Red, Prandtl nurnber Pr, fiction factor f and propellant surface

roughness E, as follows:

where

The normal acceleration (a,) related burning rate due to radial casing vibration is

detennined via the phenomenological model developed by ~reatr ix '~ (which is based on the

concept that the compression of the combustion zone is the dominant mechanism for burning rate

increase due to normal acceleration) which gives overall burning rate (includes base buming rate

r,, under zero acceleration plus the effect of normal acceleration) as follows:

where is the heat flux coefficient defmed as:

and 6. is the reference energy film thickness detennined via:

The variable Ga is the accelerative mass flux through which the compressive effect is stipulated

and is detennined using:

in which R is the specific gas constant and Od is the augmentation orientation angle given as:

which is needed to compensate for the effect of longitudinal acceleration on normal acceleration.

in this equation 4 is the acceleration orientation angle defined as tan"([a, th]) and K is an overall

correction factor dependent upon experimental information. Ln this study K was equal to 8.

Equations 2.41 and 2.45 are solved iteratively to obtain an overall burning rate.

This study employs the convective heat transfer feedback model of erosive b d n g as

given by Equation 2.41 because it contains most of the factors that are known to affect erosive

combustion phenornenon (e.g., propellant suiface roughness e, port diameter d, fiction factor f),

and also experiments have show support for this rn~del'~. The convective heat transfer model

for erosive burning w d in this study is nominally restricted to SRMs with moderate to large

length-to-diameter ratios because the core flow in ihis model is assumed to be a fblly developed

turbulent flow, under mass addition From the bounding propellant surface. However, one may

note that the SRM used in this study does not have a piuticularly large length-to-diameter ratio

(UDz6.6) hence, it may seem tenuous (or consemative, given higher shear flow and heat

transfer in developing flow) to use this erosive buming model in this study. It should be noted

here that the core-flow velocities observed in this study were not sigaificant1y hi@, hence, the

increase in propellant b&g rate due to erosive buming would not be large relative to other

factors.

Solution Procedure Using Random Choice Method The equations of motion (Equations 2.18 to 2.20) for continuity, axial momentum, and

energy for a single-phase flow, structural deformation equations (axial and radial), propellant

buming rate expression (Equation 2.41 and Equation 2.45), along with equation of state for the

gas @ = pRT) and the core flow boundary conditions mentioned above were solved in

conjunction using the random-choice method (RCM). The RCM is an explicit finite volume

method of integrating hyperbolic sets of partial differential equations, which solves Riemann

problems for the core gas between two adjacent nodes, and then random samples these solutions

in order to make an assignment to a node at the next time levet'. The time step is controlled by

the Courant Fnedrichs-Lewy condition, that is Af = 1 / 2(1ul+ a),, , since the RCM method is

conditionally stable. The higher order RCM has been utilised in this study. In this scheme, the

inhomogeneous equations of motion are solved via a generalized Riemann solution approach (as

opposed to solving the homogeneous equations of motion (left side of Eqs. 2.18 to 2.20) in the

fint-order RCM). However, only the source terms due to grain and noule area transitions, and

propellant buming, are included in the general Riemann solution. The remainder of the source

tems due to axial acceleration and radial dilatation at a given core section are incorporated into

the Riemann solution via the operator splitting technique. The main purpose for using the higher

order RCM method as opposed to the first order method originally developed by ~ l i m m - ' ~ was to

reduce the low level background noise contributed mainly by the inhomogeneous source terms

due to grain and novle area transition and propellant gas influx. By inclusion of this source

terms into the Riemann problem as done by Ben-Artzi and ~aicovid) , the background noise

was largely removed.

There are four main steps that are to be followed in this upgraded RCM rnethods4. The

first step (1) is essentially the sarne as the standard RCM method in that you select a random

position & between lefl and right nodes at the start of a given time step (-4) as show in

Figure 2.13. Then gradients in the flow properties are established between the intermediate

randomly chosen position and the left and right nodes. In Ben-Artzi and Falcovitzss' work3' they

suggested that the linear gradient assumption would be adequate in the standard Godunov

scheme. However, due the nonlinear nature of the flow, and the need for specific definition for

the flow properties at the randomly chosen position (especially when the flow is in the transonic

regime near the nozzle throat), this study applied the quasi-steady flow equations to obtain the

gradient in the flow properties between the randomly chosen position and the left and right

nodes. In a transient flow condition it is expected that a discontinuity in the flow properties on

the either side of the intermediate position will exist. This jump condition on the either side of

the intemediate position (as shown in Figure 2.13) will create a wave motion essentially

behaving like a shock tube problem. At times, the quasi-steady calculations will reveal choked

flow conditions before reaching the intermediate position. In these cases, it is suggested that an

isentropic compression or rarefaction wave either lehard or nghtward moving depending on

the application be placed beîween the node and the intermediate position so that the flow just

becomes sonic at the random position.

The second step (11) is to obtain the planar Riemann solution of the resulting wave

profile due the discontinuity resulting fiom the flow properties as mentioned above. The flow

properties are detemined at the mid node position (i + 1/2) at the end of time step (n + 112) and

are stored for M e r calculations. As show in the example of Figure 2.13 the flow properties

between the right-moving contact surface and the rightward-moving shock are selected. It

should be noted that the node shifts by M 2 when moving up in time by the increment At (as

shown in Figure 2.13). This time increment is refered as the half tirne step.

In the third step (III), the higher order method described by Ben-Artzi and ~a l cov i t z~~ is

used to implement the inhomogeneous source term effect on the planar solution. The main

concept in this is that the temporal derivative that is used to correct the global planar wave

solution tends to become linear when the time increment becomes small, hence, leading to higher

accuracy. In order to estimate the temporal denvatives, a weak-wave Eulerian scheme was

applied as shown in Figure 2.14, where a lehard and nght moving characteristic wave is shown

as opposed to the leftward-moving rarefraction wave and right moving shock wave. Based on

the derivation of Ben-Artzi and ~alcovitz~', with the inclusion of factors such as mass idlux

kom the propellant and the changing cross-sectional area, a matching of flow condition fiom leA

to nght through the weak waves and contact surface allowed for the determination of tirne-based

denvatives arising from the random position I;. However, the mid nodepoint will in general not

coincide with 5, therefore the influence of the temporal derivatives that correct for wave-based

inhomogeneous effects will in general occur over a shorter pend of tirne increment Aba from

the full time increment At. In summarizhg the above process, one can find for example the

wave corrected flow property of gas density via:

where pi is the selected planar solution of the Riemann problem at i + %, and the temporal

derivative is evaluated the start of the tirne step (spatially, at 5, with the possibility of selecting

left or right of the contact surface for estimation).

In the final step (IV), aAer obtaining the wave-corrected values for p', and u', these

properties are modified for the background quasi-steady flow changes fiom the random position

(i + 6) to the rnid node (i +1/2) in a similar manner as in step (1). Care must be taken when

keeping track of flow properties at specific positions of interest, so as not to cause any

artificially-induced noise. Also, as in step (1), if the local quasi-steady flow chokes before

reaching the rnid node position then an appropriate isentropic wave is placed such that the flow

just becomes sonic at the mid-node.

. X iaX (itQAx ( W 2 ) L l x (i+ 1 )Llx

Figure 2.13 Displays first two steps of RCM

Figure 2.14 Displays last two steps of RCM

Chapter 3 Example Results for Scenario 1

3.1 Introduction This chapter is devoted to the analysis of the results obtained for the fiee flight case

scenario in which a blast wave is encountered by a missile. Various parameters have been varied

to demonstrate the influence of this sort of extemal pulse on the intemal ballistics of solid

propellant rocket motors (SRMs). The influence of missile flight Mach number, blast wave

angle, blast wave static peak overpressure, altitude, effect of chamber wall and payload mass

damping ratio, payload spring constant and the effect of radial structural vibration has been

analysed in this study. In addition to this, the effect on a non-regressing propellant grain (fiozen

grain) has also been studied to simulate a cold-flow experimental case.

In this study, two different strengths of blast waves were used. One blast wave had a

static peak overpressure of 101.3 kPa while the other had an overpressure of 202.6 kPa. In al1 of

the runs, a blast wave peak overpressure of 202.6 kPa was used except for one nin, which was

done to compare the effect of peak overpressure on the intemal ballistics of an SRM. Figures 3.1

and 3.2 depict 101.3 kPa and 202.6 kPa peak overpressw blast wave profiles that were used. It

must be mentioned here that the profiles shown in these figures are for sea level atmosphenc

pressure. For higher altitudes, the profiles were the sarne except for the fact that the curves were

shifted so that the base pressure matched the local atmospheric pressure at that altitude. The

peak overpressure and the positive phase duration were kept constant at different altitudes. Note

that the blast wave with 101.3 Wa peak overpressure has a longer positive phase duration

compared to the 202.6 kPa peak overpressure case. Figure 3.3 and 3.4 show sample profiles of

the flow velocity behind different strengîh blast waves. They have been computed here for sea

level conditions. It is worth noting tbat stronger blast waves have a higher flow velocity behind

them.

Furthemore, the variable M in the figures associated with this chapter, is the missile

flight Mach number just before the blast wave is encountered. in al1 of the simulations, the blast

wave was encountered by the missile at a time equal to 0.5 seconds. It should also be noted that

the standard sign convention used in this study has the left-hand direction taken as negative and

the right-hand direction taken as positive. In this study, the missile was travelling towards the lefi

when a blast wave travelling towards it (cases considered ranged fiom head-on at zero degrees to

coming from the side at ninety degrees) was encountered.

Figurë3. 1 Blast wave profile at sea level - 10 1.3 kPa peak overpressure

1 Figure 3.2 Blast wave profile at sea level - 202.6 kPa peak overpressure

Flow Valocity Bahincl the 101.3 kPr Park O w m s u m Blrat Wme ir.. Time

I Figure 3.3 Velocity profile behind 101.3 kPa peak overpressure blast at sea level

Flow Voiocity 0ehind 202.6 kP8 Paik Owipr#rum 01ast Warm a. Time 300 - 1 r I i r 1 T I

I \ i - - - - ' - ' -- - - - ' - - - - - 2 - . - - - - - . - . - . ' . - - - - l

', 1 i \ 1

i j

Figure 3.4 Velocity profile behind 202.6 kPa peak overpressure blast wave at sea level

3.2 Results and Discussion

The missile's flight Mach number appears to influence the wave development inside the

motor chamber. Figures 3.5 to 3.8 depict head-end chamber pressure for different missile Mach

numbers when a blast wave of 202.6 kPa peak overpressure was encountered head-on at sea

level. The results demonstrate that at greater missile flight Mach numbers, the interna1 pressure

waves generated are stronger. In the case when the blast wave arrives at a missile Mach number

of 2.10 (Figure 3 . 9 , there is barely any pressure wave generated inside the combustion chamber.

in contrast to this when the sarne blast wave strikes the missile at a flight Mach number of 5.5 1 1

(Figure 3.8), the initial pressure wave generated by this interaction has an amplitude of

approximately 13 kPa, whicb eventually decays as the wave moves back and forth along the

chamber length. It is also evident from the Figures 3.5 to 3.8 that as these generated pressure

waves become stronger with increasing vehicle Mach number, they cause an increase in the base

chamber pressure. The increase seen in these results are quite srna11 but evident. The reason for

the different strength pressure waves inside the chamber with different flight Mach numbers is

that the vehicle expenences different decelerative forces as can be seen by cornparhg Figures

3.9 and 3.10. Figure 3.9 shows that at a missile flight Mach number of 3.081, the head-end

acceleration drops by approximately 13.5 g upon blast wave interaction, while at a higher flight

Mach number of 5.5 1 1 the head-end acceleration drops by approximately 22.5 g. This stronger

drop in acceleration causes stronger wave development inside the charnber. The higher

decelerative force at higher Mach numbers cm be attributed to the fact that the vehicle

experiences a larger drag force at higher speeds. This can be illustrated if one observes the

Equation 2.1 1 along with Figure 2.6 carefully. Equation 2.1 1 shows that the drag force is a

function of drag coefficient, missile Mach number and atmospheric pressure. Hence, as the

flight Mach nurnber increases, the drag force on the missile increases. However, one may note

fiom Figure 2.6 that the drag coefficient decreases with increasing missile Mach number. Since

the decrease in drag coefficient is small, the effect of it on drag force is not felt substantially as

the effect of the square of missile Mach number. Further, when the blast wave is encountered

the atmospheric pressure increases. This increase in pressure combined with increase in Mach

number of the flow over the vehicle, increases the vehicle drag, creating a stronger decelerative

force on the missile,

Head-End C h m k r Pmrurm m. ilme, M = 2.10 15.56 r t

I

Figure 3.5 Head-end pressure profile with missile at sea-level, 8 = O*, 202.6 kPa overpressure

Hmd-End Chambar Pmciium w. Tirne. M = 3.081 15.56 r 1 r 1

1

1

15.54 . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . t ..;..! I +A*- :

,.bW !

C ut,--- 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.52 'C i . . . . . . . . . . .

O

a i ' -.v "7" , 1

E id- -Y 7 15.5r-. . . A . . - . . . . . . . . . . . . . . . . . . . . . . . ;-.+. -- . . . . . . . . . . . . . . . . . . . . . . . 4

I I dn

a -+-' 4-

? ' .,.*- , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y l s . 4 a t . . . . . . . . . . . . . . . . . . . . . -/< -t

f ! ,bb' .,-' 1 "+*-* '

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 . 4 . . -

I -v 1 2.- 1

a-.

i V- *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 . 4 $s'- . . . . : -

I

15.42 1

0.49 0.5 0.51 0.52 0.53 O. 54 0.55 nmm (s)

J

Figure 3.6 Head-end pressure profile with missile at sea-level, 8 = 0°, 202.6 kPa overpressure

Head-End Chamkr Pm81ure W. Time. M = 3.57 15.56,

I

1 1

Figure 3.7 Head-end pressure profile with missile at sea-level, 0 = 0°, 202.6 kPa overpressure

Hed-End Chamber P m s u n u. lime . M = 5-59 1 15.56 1 T r 1 r I 1

1 1 I

+'i 15.54 . . . . . . . . . . . . . . . . . . . . - ' 4

4

*' ... - .- I

II . 15-52 . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . - . . . . . . . . . . . . I.ctr,.-: . . . . . . . . . . . . . 4

a -l. %- z = 1

4.L 5 t s . s ~ . . . . . . . . . i . . . . . . . -:- . . . . - . . . . . - * . .*: . . . . . . - . . . . . . - 1 . . . . - . . . j

$ 1 ,. -5, a

-1 D -La ,'

1 !

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.aC. - - . - - - - * - - - - - - . i 1 , - - ? -t

h

, -\* ; . - r. - 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.481- 1

"d 1 r w I ..+'- ' t' O

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.44 t? -,

1 15.42

1 1

0.49 0.5 0.51 0.52 O. 53 0.54 0.55 Tlme (s)

Figure 3.8 Head-end pressure profile with missile at sea-level, 8 = 0°, 202.6 kPa overpressure

Head-End Accdmtlon m. Tlme -28 1 r 1 b t

3.9 Head-end acceleration profile, M = 3.08 1,8 = O*, sea-level, 202.6 kPa overpressure

r - - - . . 4

4 2 l 1 0.1 0.2 0.3 0.4 O. 5 0.6 0.7 0.8 0.9 1

Timm (m)

Figure 3.10 Head-end acceleration profile, M= 5.5 1 1,8 = 0°, sea-level, 202.6 kPa overpressure

The angle at which the blast wave strikes the missile also influences the pressure wave

strength that is generated inside the combustion charnber. Four different blast wave angles, O",

22S0, 45', and 90' were studied here to illustrate this. Figure 3.6 and Figures 3.1 1 to 3.13 show

the head-end charnber pressure profiles at sea level when the blast wave strikes the missile at a

flight Mach number of 3.081. It can be seen fiom these graphs that as the blast wave angle

increases ftom 0' (head-on case) to 90' (blast wave coming from directly undemeath the

missile), the pressure wave generated inside the chamber becomes stronger. The initial pressure

wave amplitude for the blast wave angle of 0' is barely visible on the graph (Figure 3.6)

compared to the initial pressure wave amplitude of approximately 20 kPa generated by a 90"

blast wave angle. The increase in the pressure wave strength inside the combustion chamber

resulting nom different blast wave angles is due to the sudden increase in the vehicle angle of

attack. The increase in angle of attack causes the missile drag to increase which in tum causes a

higher decelerative force (compare Figures 3.9 and 3.15) which leads to stronger wave

generation inside the combustion charnber. It is obvious that higher blast wave angles cause the

missile to reach higher angles of attack, therefore, it is not surprising to see stronger waves

generated inside the chamber with higher blast wave angles. Figure 3.14 shows the missile angle

of attack as a function of time for a blast wave angle of 90'. In this figure the missile is initially

at zero angle of attack when the 90' blast wave strikes it at 0.5 seconds. The missile angle of

attack irnmediately reaches approximately 16' after which it decays to its original position. The

head-end acceleration and missile acceleration are s h o w in Figures 3.15 and 3.16 for this case.

It is worth noting that when the blast wave stnkes, the missile acceleration drops by only 45.5 g

compared to its head-end acceleration &op of about 102 g. This difference is because the

transient structural deformation resulting from the increased decelerative force is taken into

account for head-end acceleration, whereas the other result is for net rigid-body vehicle

acceleration.

In order to study the influence of the blast wave on the intemal ballistics of the SRM, two

different blast waves were applied head-on to the missile at a flight Mach number of 5.5 1 1 at sea

level. By comparing Figures 3.8 and 3.17, it is evident that higher peak overpressure blast waves

create stronger pressure waves inside the charnber. This is because the stronger blast waves

apply a greater decelerative force on the missile, transmitting a stronger pulse.

D = 0 . 7 ~ ~ ( p , + p, ( t ) ) ~ f A (3.1)

From Equation 3.1 (insert From Chapter 2), it c m be deduced that the higher overpressure blast

wave is in fact supposed to produce a greater h g force (hence greater decelerative force) on the

missile cornpared to a lower overpressure blast wave. Since stronger blast waves are

accompanied with higher flow velocities behind them and have higher overpressures (pb), they

result in greater relative flow Mach number over the missile body a d attain greater atmospheric

pressures, hence fiom Equation 3.1 it cm be seen that they will produce larger drag force

transmitting a stronger pressure pulse to the core flow of the combustion charnber.

1 Figure 3.1 1 Head-end pressure profile with missile at sea-level, M =3.08 l ,û = 22.5'

Heod-End Charnôor Pmwm u. ilme with Blirt W i u Angle a 45

1 L

Figure 3.12 Head-end pressure profile with missile at sea-level, M t3.08 l ,û = 45'

H..d-€nd Charnôor Pmrum u. Tirna with Blr i t W8w Angle = 80

9- 15.*t,**cr--. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - - P l

i l I

15.42 ' 0.48 O. 5 0.51 0.52 0.53 0.54 O. 55 0.56

Time (s)

Figure 3.13 Head-end pressure profile with missile at sea-level, M =3.08 l,û = 90"

Mlsrik Angk of At tuk m. trme 16 y r

1 I T 1

I I 1

" t . . . - . . . . . . . . . . . . . . . . . . . . . . . . . , _ _ _ - * . . . . . . . . . . . . . . . . . . . . . . . . f I l

. . . . . . . . 1 : 1

12 i.. I......'......'......,............................. I

1

I l

Figure 3.14 Variation of missile angle of attack with time for 9 = 90°, sea level, M = 3.08 1

L 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 O. 8 0.9 1 Tirne (s)

I L

Figure 3.15 Head-end acceleration profile when 0 = 90°, sea-level, 202.6 kPa overpressure

neid-End Cham ber Pressure va Time. M m S 51 1 15 5 6 r 1 T l I I 1

1 1 1

f

Figure 3.17 Head-end pressure profile , O = 0°, sea level, 101.3 kPa overpressure

Effect of Altitude:

In order to study the effect of altitude on the intemal ballistics of the SRM, the missile

velocity was set to 850.725 m/s at the start of the run and afler 0.5 seconds the blast wave was

applied. The missile velocity for al1 the different altitudes was approximate l y 1055 t 7 rn/s when

the blast wave of 202.6 kPa peak overpressure arriveci. Three different missile flight altitudes

were used, namely sea level, 10,000 m (32,800 fi), and 16,000 m (52,493 fi). The results in

Figure 3.6 and Figures 3.18 and 3.19 seem to indicate that as the altitude increases, the pressure

pulse generated inside the combustion chamber becomes stronger. In Figure 3.6 the peak initial

pressure pulse amplitude is about 4 kPa at sea level while the initial pressure amplitude is about

5 kPa for 10,000 m altitude and approximately 7 kPa for 16,000 m. Since, the initial peak

pressure amplitudes are very close in value for the different altitudes, it may not seem very

evident to the reader that the altitude does affect the strength of the wave generated inside the

chamber. Howwer, if one studies Equation 3.1 caref'utly then it will become evident that the

altitude does have an impact on the pulse generated inside the combustion chamber. From

Equation 3.1, it can be seen that the drag force increases with increasing flight Mach number and

atmospheric pressure. At higher altitudes the missile has a higher flight Mach number since the

sound speed decreases with increasing altitude (remember that the missile has roughly the sarne

velocity at different altitudes). In addition to this, the blast wave shock Mach number (and the

flow Mach nurnber behind the blast wave) is greater at higher altitudes since the pressure ratio

across the blast wave shock (p2/p1) is higher due to lower atmospheric pressure at higher

altitudes (blast wave peak overpressure taken to be constant for different altitudes). Hence,

during the blast wave interaction with the missile, the relative flow Mach number over the

missile body (Md is greater at higher altitudes thereby resulting in a greater drag force jfrom

Equation 3.1) and a greater decelerative force, hence a greater pulse transmitted to the flow

inside the chamber. It should also be noted that the drop in atmospheric pressure at higher

altitudes is more than compensated for by the increase in and square of the relative flow Mach

number over the missile at higher altitudes in Equation 3.1.

To show by means of another example that altitude is a factor, another set of results are

displayed here in which a blast wave of 202.6 kPa overpressure is encountered at a blast angle

(8) of 22.5" by the missile at the above mentioned altitudes. As before, the initial flight velocity

of the missile was set to 850.725 mis and afier 0.5 seconds the blast wave was applied. Figures

3.1 1, 3.20 and 3.2 1 show the results obtained for this case. They clearly indicate that the initial

pulse strength increases with increasing altitude. It should be noted here that some readers may

feel that demonstrating the effect of altitude on the intemal ballistics of SRM by use of blast

wave angle (O) greater than zero may not be justified because this causes the missile to be at a

different angle of attack (since a different flight Mach nurnber and blast wave shock Mach

nurnber resuit at different altitudes). hence the cornparison is not equal. However, this has been

done here to illustrate that the altitude is a factor that causes different angles of attack for the

missile and thereby increases the drag force. As the altitude increases, the missile is subjected to

higher angles of attack upon blast wave interaction since the blast wave shock Mach number

(MJ and missile flight Mach number are higher at higher altitudes and thus their vector addition

results in a greater angle of attack of the vehicle. Higher angles of attack combined with higher

relative flow over the missile results in a greater drag force (fiom Equation 3.1) aml ultimately a

greater pressure pulse inside the combustion chamber.

1 1

Figure 3.18 Head-end pressure profile with missile velocity of 1059 m/s, 0 = 0"

-&End Charnôor PmSum u. Time for Altitudm = 16.m rn 15.56 u 1 7

I Figure 3.19 Head-end pressure profile with missile velocity of 1062 d s , 0 = O0

15.42 0.49 0.5 0.5t 0.52 0.53 O. 54 0.55

Tirna (s)

l Figure 3.20 Head-end pressure profile with missile velocity of 1059 m/s, 0 = 22.5'

Head-End Chrmbr Pm8aum w. Tims br Altitude = 16.000 m

I I Figure 3.21 Head-end pressure profile with missile velocity 1062 mls, 8 = 22.5'

For the fiozen grain evaluation, a run was done with 202.6 kPa peak overpressure blast

wave striking the missile at a flight Mach number of 3.07 at sea level. Figure 3.22 shows the

result obtained. The pressure-time profile obtained seems sirnilar in profile to what one would

get in a cold flow experiment. From the graph it can be seen that the charnber pressure does not

retwn to its base pressure of about 14.743 MPa after the disturbing pulse is applied. It levels off

at a slightly higher pressure. This may be due in part to an early blast wave arriva1 before full

equilibriurn had been achieved in the computation. Generally, the pressure should retum to the

base charnber pressure after the blast wave decays completely, if the motor has not achieved

nonlinear combustion instability. The peak initial pressure amplitude appears to be

approximately 4 kPa for both the frozen gain and the regressing grain (Figure 3.6). This is what

one would expect since the fiozen grain does not affect the initial peak pressure amplitude. The

initial pressure pulse amplitude is only affected by the decelerative force encountered by the

missile and the base pressure in the chamber. It is worth noting that the Frozen grain case had a

slightly lower chamber pressure compared to the regressing p i n case when the blast wave was

applied, however the resulting peak amplitude came out to be approximately the same since the

chamber pressures were only different by 5%. Hence, the initial peak pressure amplitude of the

fiozen grain case should be 5% less than the regressing grain case but this difference is so small

that it is not visible in the graphs.

To study the effect of radial vibration of the structure, a scenario with a 202.6 kPa peak

overpressure blast wave striking the missile head-on at a flight Mach number of 5.51 1 was

simulated at sea level with no tadial structural vibration. Comparing the result for this case as

shown in Figure 3.23 to the case with radial vibration in Figure 3.8, one would notice no

difference in the results. This is because at higher base buming rates (which are used in this

study) the effect of radial vibration on the combustion process is reduced significantly, and

strong pressure waves are not developed as a result inside the combustion chamber. If a lower

base buming rate had ken used, then the SRM would have potentially entered combustion

instability for some cases. The main intent of the cumnt study was to show primarily the

gasdynamic influence of an extemal pulse on the motor, without over-estimation of the initial

pressure wave generated due to other instability-related rnechanisms.

Head-End Chamber Pisrsum Profile Ath Frozen Grain 14.749 r 2 I 1 1 r I 1 1

! 1

i 1 1

14.747 ?. : , : : . . . . . . . . . ' . . . . . ' .

l . . . . . . . . . . . . . . . a . . . . . . . . - . . . . ...:....:....

I

1 Figure 3.22 Head-end pressure profile with missile at sea level, M= 3.07,8 = 0'

HmabEnd ChimWr Plsrnuts w. Time Wilhout Wi8l Vlbntlon 1 1

I

. .-f . . . . . . . . . . . . . . , ,' #.%k~' ! l

'a%%*' 1

C

Figure 3.23 Head-end chamber pressure profile, M = 5.5 1 1,0 = 0°, sea-level

To study the effect of chamber wall damping ratio, a run was perfomed with 202.6 kPa

peak overpressure blast wave hitting the missile at a flight Mach number of 5.5 11 at sea level.

The chamber wall damping ratio was set to 0.1. Note that in al1 the simulations done in this

study, the default charnber wall damping ratio was 4.0. The head-end chamber pressure profile

obtained for this case is shown in Figure 3.24. If one compares Figure 3.24 with Figure 3.8

(same case but with T, = 4.0) then it is evident that a lower chamber wall damping ratio creates a

stronger pressure wave inside the combustion chamber. This is expected since the SRM

structure will deflect and vibrate more in cornparison to the highly damped case, thereby

transmitîing a stronger pulse. This can be demonstrated by comparing the head-end acceleration

profiles for high (5, = 4.0) and low (& = 1.0) damping cases. Figures 3.10 and 3.25 show the

head-end acceleration profile for high and low damping ratio cases respectively. It is evident

from these graphs that the head-end of highly damped chamber casing experiences only about

22.5 g decelerative force upon interaction with blast wave while the under-damped casing

experiences about 120 g decelerative force which is considerably higher. Thus the pulse

generated for the under-damped chamber casing is expected to be higher and this is seen in the

results. It is also interesting to note that a secondary wave is visible in Figure 3.24, which is not

seen in the highly damped case. This second wave's frequency is about 795 Hz. If one

calculates the natural axial fiequency (Le ) of the chamber wall using

where o, is found to be 4995.619lrads using Equation 2.25, then it is found that fmc is equal to

795 Hz. This indicates that the second wave generated with lower damping ratio is actually due

to the axial structural vibration. It is quite obvious that this second wave could not be seen with

higher chamber wall damping because the higher damping ratio effectively eliminates secondary

structural vibration after shock amival.

H.&nd Chimkr Presaum m. Time 15.56 1 1 1 8 1

:J 15.51 . . . . . . . . .

"- - . +'-*j ,.r -

I L , ,-- I

1 '+ 1 15-52 - - - - - - - - - : - . - - . - - . - -:- - - - - - - - - -: - - - - - - - - . : - - .;--:- . - . - - - - 4

, 4 1 8 "-0

1 - 2 l

.i L- I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 -,, .A-. 4 8 \-,

, ' P ? ., .. ,-v

l

i ', ' ' ,

I . .<

. . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . 15.48C- . . . . - . . . . . . - 1 . , & . , 7 - . *

I I : i \ - - - y

! , + ,, +

I - . 1

* , 4

, s . a C . . . . .\- . . . . . . . . . . . . . . . . . . . .:. . . . . . . . . . . . . . . . . . . . - .A'

c* i

l .-. "-- 1 *' ' y*-v

1 ls.up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A i l

t

t Figure 3.24 Head-end chamber pressure profile, M = 5.5 1 1,8 = 0°, sea-level, = 0.1

Mid-End Accdantlon u. Tlm. mr 1 I

I Figure 3.25 Head-end acceleration profile, M = 5.5 1 l , û = 0°, sea-level, 6 = O. 1

ct o

To study the effect of payload damping ratio on the intemal ballistics of SRM, a run was

done with 202.6 kPa peak overpressure blast wave hitting the missile at a flight Mach number of

5.5 11 at sea level. A payload damping ratio of 0.1 was used in this simulation. The default

value for C;,, in this study was 4.0. Figure 3.26 shows the head-end pressure profile obtained for

this case. If one compares Figure 3.26 to Figure 3.8, a very slight difference in pressure increase

is detectable in the case with lower darnping ratio. In fact the initial pressure wave amplitude in

the chamber with a lower damping ratio is about 2 kPa higher in amplitude in cornparison to the

highly damped case. The increase in amplitude is expected since the push of the payload mass

on the head-end of the motor will be higher with lower payload damping in cornparison to a

higher damping ratio. It can be concluded here that the effect of wall damping ratio on the pulse

strength increment is small. Changing the damping ratio from 0.1 to 4.0 only changes the pulse

strength by about 2 kPa.

1 L

Figure 3.26 Head-end chamber pressure profile, M = 5.5 1 l , û = 0°, sea-level, Spi = 0.1

In order to study the effect of payload spring constant on the intemal ballistics of SRM, a

nui was done with 202.6 kPa peak overpressure blast wave hitting the missile at a flight Mach

nurnber of 5.51 1 at sea level. The payload spring constant of 1 x 108 N/m was used in this

simulation. It should be mentioned here that the spring constant of 1 x 10' Nlm was used in al1

the simulations in this study. Figure 3.27 shows the head-end pressure profile for the result

obtained in this case. If one compares Figure 3.27 to Figure 3.8 (kPi = l x 16 N/m in this figure),

it is evident that stiffer springs reduce the wave strength generated inside the combustion

chamber. The initial pressure wave amplitude is reduced by about 3 kPa in magnitude for a

payload spring constant of I x 1o8 N/m in cornparison to a sofler payload spring constant of 1 x

10' Nlm. The reduction in pressure wave amplitude is expected since a stiffer spring would

reduce the push of the payload mass on the head-end of the motor.

to the Pre

With the use of the method of characteristics, an approximation of the initial pressure

wave amplitude generated inside the combustion chamber can be calculated. To demonstrate

this, a sarnple calculation is done here and compared with the result obtained in this study. The

sarnple calculation is done here for the case when a blast wave strikes the missile at a flight

Mach nurnber of 5.5 11 at sea level. The initial pressure amplitude for this case can be seen in

Figure 3.8.

The sound speed (a) in the charnber 1s calculated using

where R is the specific gas constant (320 Skg. K), y is the specific heat ratio equivalent to 1.2 in

this study and T is the average temperature inside the combustion chamber of about 3000 K.

Substituting these in Equation 3.3, sound speed inside the chamber is obtained:

a = 1073.3 1 m/s.

The natural frequency of the axial wave (f.) generated inside the chamber can be calculated

using Equation 3.4:

where L, is the effective combustion charnber length of 2.65 m. Note that one complete

wavelength is when the wave moves from the head-end of the charnber to the noule-end and

retum to the head-end again. This requires the wave to travel twice the chamber length as

shown in Equation 3.4. It should also be noted that the wave inside the chamber is assumed to

be moving at the sound speed of the gas in the combustion chamber since the waves are very

weak.

1073.3 1 fn = (2)(2.65)

= 202.51 Hz.

The p e n d (T) of the wave inside the chamber cm be calculated using Equation 3.5:

The change in the velocity Au of the gas inside the chamber due to the extemal pulse can be

approximated using Equation 3 6:

where a,- is the maximum change in acceleration of the missile due to the blast wave

interaction. In this equation only half the period is used since the rarefaction wave generated at

the node-end due to the blast wave takes only half a period to arrive at the head-end which

causes the pressure at the head-end to &op. As can be seen in Figure 3.28, the pressure at the

nozzle-end drops when the blast wave strikes, generating a rarefaction wave fiom this end. This

is because upon the &val of the blast wave, the missile experiences a sudden drop in its

acceleration which causes the gas inside the chamber to move forward hence, an increase in

pressure is observed at the head-end and a decrease in pressure is observed at the nonle-end.

This results in a compression wave moving fiom the head-end of the chamber towards the

noule-end and a rarefaction wave moving fiom the nonle end to the head-end. When the

rarefaction wave arrives at the head-end, it causes the pressure to drop and when the original

compression wave retums after reflecting from the nozzie end it causes the pressure at the head-

end to increase again. The pressure oscillations seen in the pressure profile cwes of the

c hamber pressure are due to these waves travening the c hamber.

From Figure 3.29 the change in acceleration can be seen to be approximately 27.5 g.

Hence,

It should be noted here that the rigid body acceleration &op is used as opposed to the head-end

acceleration &op in Equation 3.5 for a,- . Ideally head-end acceleration value is supposed to

be used but since the duration of the peak acceleration &op is veiy minute, the effect on wave

developrnent is essentially equivalent to the rigid body acceleration &op.

By use of the method of characteristics and assuming the pressure waves are weak (which

they are in this study), the pressure ratio across the compression wave (pz/pi) that is generated at

the head-end can be calculated using:

The pressure pulse amplitude can be calculated using the foiiowing:

where pl is the chamber pressure when the blast wave hits and is obtained fiom Figure 3.8.

Ap = 1 S.46(l .O00745 - 1) = 0.0 1 15 MPa

This compares well with the peak amplitude observed in Figure 3.6 of about 0.012 MPa.

Noulm-End Chrmkr Pmrure W. Tirne. M = 5.511 i5.w T ,

1

1 1 Figure 3.28 Nozzle-end chamber pressure profile, sea-level, 0 = O*, 202.6 kPa overpressure

l .Y * ' Y-f

15.44 - . . . . . . . . , l u . 7 . + . . . . . . . ' . . . . . . . . . . ' . . . . . . . . . ' . . I

15.42

. . . . . . . . . . . . . . . . . . ," : y,

1 ri- /-.

t 1 1 I

0.48 O. 5 0.51 0.52 0.53 0.54 0.55 nmo (m)

. . . . . . . . . . . . . i . . . . . . . . . . . .~ . . . . . . . . . . . . ' . . . r . -7 '

l

. .....................,.............,........... . 1

. . . . . : . . . . . . . . . . . .:1. . . . . . : . . . . . . . . . . . ..:. . . . . . , - - - - - ' i 1

1 \

. . . . . . . . . ; . . ; - . F - . - . - . . . . . . ; - . . - . - *. . . . . . . . . - - - - 1 1 l

I ' . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . i Y : . ', aj -28 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . -:- . . . . '. . . . . . . . . . . . . . . . . . . . .

7

-- - --

Figure 3.29 Missile acceleration profile, sea-level, 0 = 0°, 202.6 kPa overpressure

3.3 Remarks Concerning Pulse Strength From studying the effects of various parameten on the intemal ballistics of SRM, it is

evident fiom the results that the wave strength generated inside the SRM for this scenario is

weak (pressure ratio across the compression wave generated inside the chamber (pz/pi) very

close to 1). Even though the initial peak pressure amplitude generated inside the chamber is

small in amplitude it does however exceed the minimum cnterion required to cause combustion 17, 18 instability in the rocket motor. According to Hessler , the standard criterion used in industry

suggests that a pulse strength of more than O. 1% of the base chamber pressure is a minimum for

an SRM to enter combustion instability if it is inherently susceptible to instability. Hence taking

a closer look at the results obtained in this study, it is evident that high angle missilehlast wave

interactions and head-on cases with low SRM casing axial damping exceed this minimum cut-off

required to trigger combustion instability. In addition to this, it is evident that the high missile

flight Mach number cases with head-on biast wave interactions at sea level with high axially

damped SRM casings are very close to these limit requirements. If the missile was at hi&

altitude with these flight Mach ambers and with lower axial casing damping then it is possible

that these would have also be above the minimum criterion. Hence, these results indicate that

blast wave interaction with missiles in their mission flight could have devastathg effects on their

performance due to the possibility of the SRM suffering combustion instability. It should also be

mentioned here that the results in this report indicate that the SRM does not suffer combustion

instability due to the various strengths of pulse application. The main reason for this was that a

higher base buming rate was used in this study, which causes the waves to decay since the effect

of radial vibration of the casing on the combustion process is significantly reduced. If a lower

base buming rate had been used, then the motor would have potentiaiiy entered sonibustion

inshbility. n i e objective of the present study was to demonstrate pnrnarily the gasdynamic

influence of the extemal pulse on the motor, and not compound the pressure wave result with

instability-related mechanisms that might enhance the initial wave magnitudes.

Chapter 4 Scenario 2 - Test Stand Failure

4.1 Introduction This chapter is devoted towards modelling of an extemal pulse that is delivered to the

solid propellant rocket motor ( S M ) when a failure of the thrust load-ce11 occurs during its static

test-stand firing. The motor is suddenly accelerated as a result of this test-stand component

failure and eventually collides with an end-wall, from which it bounces back and forth until it

attains its new equilibrium position. The effect of this son of pulse on the intemal ballistics of

SRM is analysed in this chapter. In this study, the structural deformation of both the motor and

the end wall is incorporated. This chapter begins by illustrating how this scenario was modeled

and how the resulting non-steady one-dimensional hydrodynamic conservation equations were

solved using the random choice method (RCM) described in Chapter 2. The results obtained

from this scenario are in tuni analysed and discussed.

4.2 Modelling of SRMWall Interaction The simulation for this scenario is essentially the same as the case in Chapter 2 except for

the fact that the motor is not in flight, hence the boundary condition at the head-end of the motor

is changed. The equations for the non-steady conservation of mass, momentum, and energy for

the compressible gas phase (Equations 2.18 to 2.20) as described for the free flight case are used.

In addition to this, the capid kinetic buming rate mode1 (see section 2.5.4), structural wave

equation (Equation 2.24), and radial vibration mode1 equation (Eqwtion 2.35) are used in

modelling this scenario. The method of the solution procedure using the RCM was identical to

that described in Chapter 2. Figure 4.1 depicts a simplified schematic of the SWwal l

interaction. There are thtee head-end boundary conditions for this case which depend on:

1) Prior to load-ce11 failure as in Figure 4.l(a).

Rocket Motor

a) Prior to load-ce11 failure

End-wall modeled as a spring-damper system Rocket Motor

Motor in flight or 1

b) Upon load-ce11 failure - model of the wall and motor in fiee flight

bouncing off

I l

Wall modeled as a spring-damper system

-F Damper

Cap Mass /

Nozzle Mass I

I

C) Shows model of wall and SRM interaction

Figure 4.1 Diagram of the SRM/wall interaction model

2) After load ce11 failure but either before hitting the wall or after bouncing off the wall as shown

in Figure 4.l(b)

3) When the SRM attaches to the wall as shown in Figure 4.l(c).

The motor was treated here as a beam and its head-end boundary condition is given by

Equation 4.1 which is the sarne as Equation 2.26, the head-end boundary condition for the ûee-

flight case. However, the force acting on the head-end (Fhe) is different in this scenario:

Prior to load-ce11 failure, Fhe is taken to be equal to the -Fm. F, is defined in Equation

2.3 1 which is essentially the surnmation of the thnist force and the pressure force acting on the

nozzle-end. From Figure 4.1(a) it can be seen that the force acting on the head-end of the motor

must be equal and opposite to the force acting on the nozzle-end so that the motor does not

move, hence this was implemented in the boundary condition.

In the second case in which the SRM is free, Fhe is defined as:

F, =-PA, (4.2)

Equation 4.2 only accounts for the force at the head end due to intemal pressure since no other

force is acting on it. It should be noted that the drag force has been neglected here since the

distance to the end wall and the peak velocity attained by the rnotor are really small hence it is

justifiable to neglect drag force on the motor.

In the third case, where the SRM was attached to the wall (the wall was modelled as a

spring-damper system as shown in Figure 4.1 (c)), Fhe was defined as:

where d is the distance fiom the SRM's initial position to the wall andtu,, is the natural angular

fiequency of the SRM mass spring-damper system defined as:

In this study, it was assumed the SRM attachesldetaches

uncompressed position.

(4.4)

tohom the wall spring at its

4.3 Results and Discussion In this sîudy, the effect of wall spring constant (kW), the wall damping ratio ((;,), and the

distance (d) the SRM to bas to travel before encountenng the end wall has been evaluated. In

addition to this, the effect of a frozen grain on the intemal ballistics of SRM has also been

studied here for academic interest. In al1 of the simulations, the thrust load-ce11 fails at 0.5

seconds. The standard sign convention used in this study has the lefi-hand direction taken as

negative and the right-hand direction as positive. The SRM moves towards the left upon load-

ce11 failure prior to encountenng the end-wall.

To study the influence of the wall spring constant, mns were done with four different

values of spnng constant without any wall damping, and with the distance to the wall (d) set at 5

cm. Figures 4.2 to 4.5 depict the head-end pressure profiles for these cases. The results indicate

that the wall spnng constant does influence the pressure pulse generated inside the combustion

chamber of the SRM. It c m be noted fiom these profiles that as the wall spring constant

increases, the pressure waves. criss-cross the chamber become closer. This is because with the

stiffer wall spring the SRM bounces off sooner compared to the soAer spring since the spring

does not cornpress much hence the SRM travels lesser distance before bouncing off. Therefore,

with the stiffer wall spring the SRM hits and bounces off the wall and hits the wall again in

shorter period of time compared to softer spring thereby generating more waves inside the

chamber per unit time. It is also worth noting that with higher values of wall spring constant, the

pressures waves generated inside the charnber settie out sooner. This can attributed to the fact

that when the SRM collides with a wall with a stiffer spring, it does not bounce off the wall as

many times as it does with a sofier spring since it loses a substantial amount of energy in the first

few impacts with the wall. if one compares Figures 4.6 and 4.7, then it can be seen that the drop

in second and third SRM acceleration peaks resulting from the impact with the wall is higher for

the stiffer spring, indicating an increased loss in energy for stiffer springs in cornparison to sofier

springs. Also, it can be seen that the peaks in these curves are getting closer hence indicating

that with each bounce the SRM is travelling less distance before impacting the wall again.

However, the spacing between acceleration peaks is less for a stiffer spring (Figure 4.6)

compared to that with a sofier spring (Figure 4.7), hence, indicating that sofier springs push the

motor further in cornparison to stiffer springs.

The results also indicate that as the wall spring constant increases, the strength of the

waves generated inside the chamber increases but after reaching a certain value of the wall

spring constant (in this study kW = 3.0 x 10' N/m) a M e r increment in the wall spring constant

decreases the strength of the waves. This phenomenon can be explained by studying the change

in the natural frequency of the SRM-walVspring system with the change in wall spring constant

(kW):

Equation 4.5 shows that the natural frequency of the SM-walllspring system is a function of the

wall spring. Hence, by varying the value of kW, it is possible to have it coincide with the natural

frequency of the core flow inside the combustion charnber. Thus, the stronger pulse resulting in

the chamber for kW equivalent to 3.0 x 10' N/m implies that this value must be very close to the

natural fiequency of the core flow in the charnber. In Chapter 3 it was shown that the natural

frequency of the core flow cavity of the SRM is about 202.5 1 Hz. This corresponds to a kW value

of 2.0 x10' Nlm. The head-end pressure profile for this case is shown in Figure 4.3. If one

compares this with Figure 4.4, then it is evident that stronger waves are forming inside the

chamber when kW is 3.0 x 10' Nlm, which corresponds to a natural frequency ( f n ) of 247.76

Hz. Hence, the system seerns to achieve resonance at a higher frequency than that calculated in

Chapter 3. This could be atüibuted to a nwnber things. One thing that could cause the

difference in fiequency to that calculated in Chapter 3 is that the SRM is not completely

oscillating at the core flow axial fiequency since it occasionally detaches fiom the wall. Hence,

the SRM is only oscillating at the core flow hquency based on Equation 4.5 when it is attached

to the wall. Another factor that may be causing a higher pressure amplitude in the chamber at

higher than the natural core flow frequency of the SRM, is the timing between impact and the

compression wave pend inside the chamber. It is possible to expect that at kW equal to 3.0 x 10*

Nlm, the timing of the impact to the wall and the compression wave arriva1 to the head-end of

the chamber coincided thus creating a stmnger wave inside the charnber.

Figures 4.6 to 4.8 show the S M acceleration profile for different spring constants. It

can be seen that upon the fmt impact with the end-wall, the highest decelerative force

encountered by the SRM is with a stiffer spring. It experiences a force of 2980 g for a stiffer

sprhg (Figure 4.6) in contrast to about 1300 g for a softer spring (Figure 4.8). Figures 4.9 to

4.1 1 show the head-end acceleration profiles for three different spring constants. In these

figures, the initial spike in the head-end acceleration (peaking to about -1500 g) at 0.5 seconds is

due to the sudden loss of the load-ce11 force at the head-end, hence, the intemal chamber pressure

causes the motor to stretch. It can be seen from Figure 4.9 that the head-end experiences a

substantial amount of g force (approximately 4300 g) at the head-end upon the initial impact,

with a wall modelled using a very stiff spring.

Figure 4.2 d = 5 cm, kW = 5.0 1 x IO', 5, = 0.0

b 4

Figure 4.3 d = 5 cm, kW = 2.0 x 10' N/m, c, = 0.0

Figure 4.4 d = 5 cm, kW = 3.0 x 10% Nlm, 6, = 0.0

Hed-End Chrmbm Prssaurs u. Tims

14.7 . . . . . . . . . - - - - - . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 - - . . , . - - -<

14.5 . . . . . . . . . . . . . . . . . . . . . . . . . - . . . - . . . . . . . - . - - - - - . . . . . . . . . . . . - t

14.4 I

0.45 0.5 0.55 O. 6 0.65 0.7 O. 75 lime (r )

I

Figure 4.5 d = 5 cm, kW = 8.01 x 108 N/m, 5, = 0.0 J

SRM Accelarallon w. lima

1 -

l -

J00. I 1 1

0.45 0.5 0.55 0.6 0.65 0.7 0.75 Timm (8)

Figure 4.6 Acceleration profile with d = 5 cm, kW = 8.01 x 10' Nlm, 5, = 0.0

SRM Accaim8llon w+ Tïme

-/ 1

I Figure 4.8 Acceleration profile with d = 5 cm, kW = 5.01 x 10' N/m, c, = 0.0

. . . - - . . . . . . . . . . . - . i

-1500 1

0.45 0.5 0.56 0.6 0.85 0.7 0.75 Tirna (m)

I Figure 4.9 Head-end acceleration profile with d = 5 cm, kW = 8.01 x 108 Nlm, 6, = 0.0

1

l Figure 4.10 Head-end acceleration profile with d = 5 cm. kW = 3 .O x 10' Nim, 5. = 0.0

Head-End Accdmtton w. Tms 4000 1 1 1 1 I

! ..;. .:. .:.. : . . ..... . . . . . . . . ....... . . ...... , . . . . . . . . . . . . . . . . i

' 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

4

F, Y

5 2WOt. . . . . . . ..; . . . . . . . . . . . . . . . . . . . . . . . . . - . - . . . ' - - * . - - . . - . - . . - - . . ' - -? E i 1 I g ,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' t 4 I

. . . . . . . . . . . . . ? 1 -

1 U r , 1 . . , ? ' * . . ', - , . , a ,k... . . . . . . . . . . . . . . . . . . . . . . ..I ...,.: .,,. . . . . . . . . , . ...... . . , . : . - * . . - . ., 1 f I , . l I

I 1 1 . , ,

O C - ' . . ! 4 ' ) - " ,- , . - - . ' . - * - + - 8 . ' . " . . ' , , . . , - L - , . . - . - , - - ,

I pu*-'\ 1- \-.- "& $ *-,: ' . * p , , , L . , 1- h - - t . , i- l , L \ - " 1

~ i & r e 4.1 1 ~ead-end acceleration profile with d = 5 cm, kW = 5.0 1 x 10' Nh, &, = 0.0

To study the effect of distance to the end-wall, three distances of 5 cm, 10 cm, and 15 cm

were simulated with a wall spring constant of 3 .O x 108 N/m. Figures 4.4,4.12 and 4.13 depict

the head-end pressure profiles obtained Eom these simulations. It can be seen fiom these graphs

that the initial peak pressure amplitude increases with the increasing wall distance. This is

expected since the SRM gains greater momenhun with increasing wall distance hence

experiences a greater decelerative force as can be seen by comparing Figures 4.7,4.14 and 4.15.

The pressure profiles also depict that the waves tend to decay more afier the first impact with the

end-wall, with increasing distance. This is because in the case with greater distance to the end-

wall, it takes more time for the SRM to hit the wall again after each bounce. This increase in

time between impact msults in an increase in settling time for the waves inside the chamber, for

the greater distance to the end-wall case in cornparison to shorter distances. In addition to this, it

is worth noting that the pressure waves become strmger in the case with IO-cm wall distance

compared to 5-cm and l51m case. This may be due to synchronization between the frequency

at which the SRM impacts the wall and the intemal core flow frequency.

Figure 4.12 d = 10 cm, kW = 3.0 x 10' N/m, c,,, = 0.0

I Figure 4.13 d = 15 cm, kW = 3.0 x 10' ~ l m , <, = 0.0

1

I Figure 4.14 Acceleration profile with d = 10 cm, kW = 3.0 Nlm x 108, &, = 0.0

SRM Accekritlon w. Tirna 4000, 1 1 1

I Figure 4.15 Accelerabion profile with d = 15 cm, kW = 3.0 x 108 N/m, Cr = 0.0

1

To study the effect of wall damping, four runs were done with the damping ratio equal to

0, 0.5, 1 and 4. These represented undamped, under-darnped, critically-damped and over-

darnped systems. Figure 4.2, and Figures 4.16 to 4.18 display the head-end pressure-time profile

obtained for these simulations. They clearly indicate that the initial spike in pressure upon first

impact with the wall tends to decrease with increasing wall damping ratio. It is also worth noting

that with a higher damping ratio, the pressure waves resulting from impacting the wall tend to

decrease with time except for the over-darnped case as shown in Figure 4.18. In this graph it can

be seen that very sharp transient spikes are superimposed on the pressure waves, which tend to

increase the pressure wave strength in cornparison to the under-darnped and cntically-damped

cases. nie initial developrnent of this secondary wave is evident in the system that is under-

darnped. However, the strength of this secondary wave is very small in this case. It seems that

the strength of this secondary wave increases with increasing damping ratio. The wave also

seems to appear earlier in cases with higher damping ratio. It is not clear as to what is causing

this secondary wave pattern.

To study the effict of a fiozen grain (non-regressing propellant), a run was done with the

motor placed at 5 cm from the end-wall and the wall modelled as having a spring constant of 3.0

x 10' N/m and no damping. The result obtained for this scenario is s h o w in Figure 4.19. If one

compares this with a regressing grain case as s h o w in Figure 4.4, then it is evident that the

strength of pressure waves generated in the fiozen grain case is slightly lower than the regressing

grain case. This is because the chamber pressure for the fiozen propellant-grain simulation was

slightly lower. If they were ai the same pressure than the pressure wave amplitudes would have

been identical. This is because the pulse strength is not a function of grain regression rate. I! is

also evident in Figure 4.19 that afier the disturbance fades away, the motor returns to its base

operating pressure.

Hoad-End Chamber Prai rurr vs. Timr 1 6 . 4 , T r 1

t Figure4.16d= 5 cm, kW = 3.0 x 1 0 * ~ / m , &=OS

1

Haad-End Chamber Pmrure W. Tme 16.4 r I

L Figure4.17d=5 cm, kW =3.Ox I O ~ N I ~ , rw= 1.0

I

Figure4.18 d = 5 cm, kW = 3.0 x 10%/m, 5, =4.0

Figure 4.19 d = 5 cm. kW= 3.0 x 10'N/m, Sv= 0.0

4.4 Remarks Concerning of Pulse Strength From the results of this scenario, it is evident that the waves generated inside the

combustion chamber can be of substantial amplitude. The initial peak pressure amplitudes seen

in this scenario are larger than the missilefblast wave interaction scenario. It is also clear fiom

the results that this scenario is capable of ttiggering some SRMs into combustion instability 17, 18 based on Hessler s information on the minimum criterion for pulse strength requirement

(more than 0.1% of base chamber pressure to trigger combustion instability in inherently

susceptible SRMs). A pulse strength weaker than this criterion is assumed to be ineffective

within the inherent flow noise due to combustion and turbulence.

Furthemore, it should be noted here that the SRM has been stable for various strengths

of pulses used in this study, some of which exceed the criterion noted above. The reason for the

motor not entering instability is because a higher base burning rate has been used in this study.

A higher base bming rate makes the SRM more stable by reducing the motor's ability to sustain

pressure waves through the combustion process.

Chapter 5 Design of Pulse Generator

5.1 Introduction In the design phase, solid propellant rocket motors (SRMs) are tested for combustion

instability. In order to investigate if a particular design of motor is stable, various pulses of

different strengths and duration are applied. The pulse shape and strength are typically modelled

to simulate the naturally occumng disturbances within the motor. In this respect, most of the

current motor designs are evaluated based on pulses that are generally triggered fiom within the

SRM, and have overlooked the possibility of a rocket motor sustaining combustion instability

due to extemal pulses. In order to come up with a stable design, it is important that when testing

motor stability experimentally, that all possible pulses that could be encountered during its

operation be applied. The limitations of the current testing mechanisms to satisfy this objective

has inspired an interest in this study to come up with some introductory ideas as to how this

could be achieved. This chapter is devoted towards studying the present designs for pulse

generation for combustion instability testing, and proposing a design that may be more suitable

for general stability evaluation.

5.2 Current Pulsing Units There are currentiy four types of pulsing units that are generally used for testing

combustion instability in tactical-size motors. They are explosive bombs, pyrotechnic pulsers,

low-brisance pulsers, and piston pulsers. Each of these pulsing units is described here briefly.

Explosive Bombs

These are essentially small pyrotechnic charges, wbich are placed at strategic locations

inside the rocket motor. M e n the combustor is operating, these charges are detonated at

predetetrnined tirnes". The detonatioos of these charges produce a pressure! pulse inside the

chamber. The impact of this pulse on the stability of the motor is analysed. To vary the strength

of the pressure pulse, the charge quantity or the explosive material itself is varied. Figure 5.1

shows the explosive bomb.

charge Bomb casing Lead wire

Figure 5.1 Explosive bombs

The disadvantage of this method is that it may disrupt the normal chamber flow prier to

detonation. It is also dificult to install the explosive inside the chamber since the lead wires for

igniting the charge have to be given access through the engine wall. There also exist possibility

of ejecting live bombs fiom the chamber hence it can be dangerous. In addition to this, the

detonation produces non-uniform beating inside the chamber in the vicinity of the explosion and

also has the potential of causing stnictunil damage to the structure.

Pyrotechnie Pulser

A simplified schematic of a pyrotechnic pulser test setup is shown in Figure 5 . f In this

type of pulser unit a pyrotechnic charge is ignited. This causes an increase in pressure in the

driver, which in tum causes the rupture of the burst diaphragm. As the diaphragm bursts, the

combustion proàucts plus a hction of the remaining unburned pyrotechnic charge expand into

the pulser charme1 and are vented into the combustion chamber. The diaphragms are designed

such that they can withstand a certain arnount of pressure before they burst. Hence, by selecting

clifletent strength diaphragms the pressure pulse amplitude can be varied. A typical pressure

history obtained by this type of pulser unit is shown in Figure 5.3 16. The main feature of the

pulse produced by the pymtechnic pulsers is that there is a very fast increase in pressure

followed by an exponential decay.

Rocket Motor charge igmrea Pressure pulse in bere

I \ /-

Driver Channel

Figure 5.2 Sirnplified schematic of pyrotechnic pulser unit

Diap hragm

Figure 5.3 Pressure history for pyroiechnic pulsers

A rupture pressure

2 W rn P! &

P cc

The disadvantage of this method is that it introduces fonign particles inside the operating

combustion chamber along with distorthg the core flow pattern due to extemal flow passing

thiough the chamber. in addition to this, th- exists a temperature gradient between the gases

coming in the chamber nom the pulser unit and the gases in the hot combustor. Therefore, in

essence the pyrotechnic pulser unit in fact alters the tme environment within the combustor;

hence, the stability rating obtained by this method may not be very accurate since the ûue

environment was not modelled.

Low-Btisance Pulser

In order to simulate nahiral triggering phenornena such as the ejection of an inert material

through the noale, a pulser with a longer rise and decay time is required. Nonnally, this type of

pulse could be produced by ejecta pulsers, which eject piugs into the motor. However, it is not

possible to use ejecta pulsers with real motod6, hence low-brisance pulsers are used. A

simplified schematic of this pulser is shown in Figure 5.4. In this pulser unit, a pyrotechnic

charge is ignited, which causes the pressure to Rse leading to the rupture of the diaphragrn. The

pressure generated by the expanding combustion products of the pyrotechnic charge acts on the

piston base. The piston is driven back and the vent begins to open. As the piston traverses the

vent opening, the combustion products flow into the vent and to the motor. Figure 5.5 shows the

pressure history for low-brisance pulser. By changing the parameters that affect the piston

velocity, the pulse rise and decay tirne can be varied.

P yrotechnic charge ignited in here

/ "a, I Explosion ~ e a d s t o p product s t Rocket Motor

?igure 5.4 Simpüfied schematic of low-brisance unit

Figure 5.5 Low-bnsance pressure-time profile

The disadvantage of this method is basically the same as pyrotechnic pulsen. It too

introduces foreign particles, and produces transient flow pattern distortion.

Piston Pulsers

In order to allow for more variation in the range of pulse characteristics for test and

research on motors, piston puisers are used. The simplified schematic of a piston pulser is

shown in Figure 5.6. The mechanism for the illustrated piston pulser unit is similar to the above

mentioned pulser units. In this pulser unit, a pyrotechnic charge is ignited, which causes the

pressure in the igniter charnber to rise and ultimately rupture the diaphragrn. AAer the rupture of

the burst diaphragrn, the combustion gases and the unburned fraction of the charge enter the

breech volume. As the pressure increases the piston is accelerated into the bore volume. The

gases in the bore are compressed by the piston and are vented to the motor. An ideal pressure-

time history is show in Figure 5.7. The nse and decay times for the pulse can be vaiied by

changing the charge and breech volume.

Burst Piston Bore volume Rocket Motor

ignited here I Breech volume

'. Teflon stopper

Figure 5.6 Simplified schematic of piston pulser unit

Figure 5.7 Pressure-time history of piston pulser

Again, the disadvantage of this method is sirnilar to the above mentioned pulser units.

Like hem, this too introduces extemai flow in the combustor. However, in this method, the

combustion products from the pyrotechnie charge explosion are not vented into the combustion

chamber.

5.3 Design Requirements In order to corne up with suitable design requirements for a pulsing uiit, the natural

operating condition of the motor should be kept in mind. Normally, rocket motors are propelling

vehicles when either a pulse fiom within or fiom some extemal source such as an explosion is

delivered to them. A design that can simulate this is going to give us a good idea as to whether

the motor will become unstable or not.

In cunent pulsing units for testing a motor's instability resulting fiom a pulse from within

the chamber, a flow of gas is vented to the combustor. It is obvious that in reality, no extemal

gases get inside the operating combustor. So, as part of the design requirement for combustion

instability testing, it should be made certain that a pulse is delivered to the motor without

introducing foreign particles and lower (or higher) temperature gases into the combustion

chamber.

In addition, current pulser units are not adequate to test motor stability due to a pulse

delivered by extemal sources. In order to detennine if a particular design of motor is stable, it

should be tested for ail the possible pulses it would encounter during its operation. It is fairly

obvious that extemal pulse applications fiom blast waves may be strong enough to render a

motor unstable, hence, overlooking this pulse application on a given design motor would be

wrong. In order to simulate an extemal pulse application, one must make sure that the motor is

not fixed (as in many intemal pulse simulations) but fiee to oscillate upon extemal pulse

application. In reality, when an extemal pulse is encountered by the motor, the entire motor

oscillates and not just the head end, therefore it is important that the motor is free to oscillate in

order to simulate the real environment.

nie final design requirement is that the pulser unit should be capable of delivering both

types of pulses (one that simulates extemal and other that simulates internai pulse applications)

with minor component changes.

5.4 Proposed design The shplified schematic for the proposed design for combustion instability testing is

shown in Figures 5.8 and 5.9. Figure 5.8 is the setup that is recommended to test for a natural

triggering pulse nom within the motor such as an expulsion of an igniter f'ragment or propellant

grain section while Figure 5.9 is for an external pulse application. Both the designs are

essentially the same except the head-end of the rocket motor is modified for the intemal pulse

simulation. It should be noted that the rnotor is fiee to oscillate in both designs cornpared to the

cunent internal pulsing methods where the motor is fixed.

In this pulser unit, the tank is pressurized until the burst diaphragm ruptures. This causes

a pressure pulse inside the channel, which in turn acts on the piston and pushes it forward. The

piston strikes the head end of the motor transmitting the desired pulse and is retracted back by

the spring. The valve in the channel opens at the predetermined time so that the air çan be let

out. In the case of internal pulse simulation, the piston inside the channel strikes another piston

mechanism attached to the head-end of the motor. The striking force causes the head-end

piston's other end to move inside the combustion chamber thus hansrnithg an intemal pulse.

The head-end piston is retracted by the springs attached to it. The motot is then left to oscillate

freely under variable pressure forces and motor thnist. For the extemal pulse simulation, the

piston inside the channel pushes on the head-end of the motor and then retracts back allowing the

motor to oscillate back and forth. This in turn generates wave motion inside the chamber that is

closer to reality. There are darnpen attached on the piston end face so that when the motor

retums to its original position due to the thnist, there is not too much extemally induced

oscillation.

The proposed design suggests using pressurized air as oppose to using pyrotechnie

charges because this is safer a way to produce pulses (since it does not involve explosions). It

should also be noted that the design proposed here is the author's introductory idea and critical

CFD and vibrational analysis would be done before final prototype implernentation.

Pressurized Valve to let air in here air out

Piston moves inside the motor generatin a pulse f

Burst diaphragm limiter Motor fiee to oscillate

Figure 5.8 Setup for intemal pulse simulation

Pressurized Valve to Piston movement Motor fiee

Burst diaphragm externally induced oscillation of motor

=igue 5.9 Setup for extemal pulse simulation

Chapter 6

Conclusion

6.1 Conclusion This study was primarily concerned with the effects of an extemally induced pulse on the

intemal ballistics of solid propellant rocket motoa (SRMs). The main reason for the study was

to demonstrate numerically that the wave motion generated inside the rocket combustion

chamber resulting from an external pulse may be of sufficient strength to cause a motor to go

into nonlinear combustion instability. In doing so, this study modelled two practical scenarios

that would most likely be encountered by a tactical size rocket motor. The first scenario dealt

with modelling a missile passing through a blast wave generated at a distance due to a nuclear

explosion. How this scenario was modelled was explained in detail in Chapter 2. Effects of

various pertinent parameters on the pulse strength generated inside the rocket motor were studied

and analysed in Chapter 3. Another scenario studied was the case in which the SRM is on the

static thnist test stand when suddenly the load-ceIl fails causing the motor to accelerate forward

and hit an end-wall. The SRM would then bounce back and forth until it attains its new

equilibrium position. The impact of this sort of externally induced pulse was studied in Chapter

4. in Chapter 5, pulsing unit designs were proposed for testing mcket motors experimentally for

combustion instability due to extemally and intemally induced pulses, since the present pulsing

methods have some inherent disadvantages.

From the results obtained for the missilehlast wave interaction scenario, it can be

concluded that the pressure pulse generated inside the combustion chamber is weak. However,

based on the information obtained fiom ~essler"* 18, the standard criterion used in the indusüy

suggests that a pulse sangth of more than 0.1% of the base chamber pressure may be sufficient

to cause a motor to go into combustion instability. Mse strengths weaker than this cnterion are

assumed to be ineffective within the inherent flow noise due to combustion and turbulence.

Hence, taking another look at the results obtained fiom this scenario clearly indicates that when a

missile encounters a blast wave at higher blast wave angles, then the pulse generated inside the

charnber clearly exceeds the minimum criterion. The minimum initial pressure pulse amplitude

required to be generated inside the SRM to cause combustion instability based on the above

mentioned cntenon is about 15.455 kPa in this study. Therefore, even though the pulse

hansmitted inside the charnber seems a little weak, it is still potentially strong enough to cause

combustion instability inside the rocket motor if the motor is susceptible to nonlinear combustion

instabilty. It should also be noted that the effect of different blast wave angles on the intemal

ballistics of the SRM was studied at sea ievel, yet we know from the results that the pulse

strength increases at higher altitudes, thus there exists the possibility of even stronger pressure

pulses being generated inside the combustion chamber. In addition to this, it should also be

mentioned that the motor casing was highly darnped (intemal axial mode) which reduced the

pulse strength inside the charnber. Hence, it can be concluded that if a missile in its mission

flight at high altitude encounters a blast wave at a high blast wave angle and if its casing material

is such that it is not highly damped axially, then it is possible that the motor propelling the

missile will enter nonlinear combustion instability (if inherently susceptible to this instability).

Therefore, the results of this study imply that an externally induced pulse to an SRM by means of

a blast wave is strong enough to cause some motor designs to possibly enter combustion

instability . The results fiom the static thnist test-stand failure scenario indicate that a strong pressure

pulse may be transrnitted to the combustion chamber. The smallest initial pressure pulse

amplitude generated in this scenario was about 400 kPa. This is substantially higher than the

minimum criterion required to cause combustion instability in an SRM. in fact, expenmental

results obtained by Blomshield et al? indicate that a 10 psi (69 kPa) pressure pulse cm cause a

tactical sue cylindrical grain motor to enter into nonlinear combustion instability. Thus, there is

more likelihood tbat an extemally induced pulse resulting fiom an SRM colliding with an end-

wall will lead to nonlinear combustion instability inside a rocket motor susceptible to this

instability.

It should be noted here that the pulsing of the motor for various cases in both the

scenarios in the present shidy never caused the motor to enter combustion instability. This was

because in this study the base burning-nite was chosen to be high which reduces the propellant's

sensitivity to normal acceleration. This is a primary mechanism for driving axial instability in

the present model. Hence, if a lower base buniing rate had been used, then it would have been

possible to trigger the present SRM into nonlinear combustion instability with the pulses

encountered in this study. There also exist other instability dnving mechanisms (combustion and

gasdynamic related) in practice, which are susceptible to pulses of lower strength. Some

examples of these mechanisrns are pressure-coupled combustion response (in which a higher

chamber pressure typically requires lower pulse strengths and visa versa), velocity-coupied

combustion response (erosive burning related), and axial lightweight motor oscillation enhanced

wave development4.

The present study also concludes that in order to design a stable motor, it is important to

test an SRM's stability to extemally and intemally induced pulses rather than just testing its

stability based on an intemally induced pulse. The results of this study indicate that overlooking

an extemal pulsing evaluation may not be a good idea. In light of this, the current study

proposed some introductory ideas as to how S W s could be tested for combustion instability due

to extemally induced pulses, since current pulsing units are not typically designed for this. A

pulsing unit design was proposed that eliminated the addition of extemal flow inside the

combustion charnber, thus preventing the introduction of lowerhigher temperature foreign

particles and gases that might distort the tme stability behaviour of the motor.

6.2 Future Work and Recommendations Since the cunent work was primarily dealing with axial waves generated inside the

combustion charnber due to an extemal pulse, it is suggested thai the effects of tangential and

radial waves developing inside the SRM be included in the analysis of external pulse

applications, to gain a fùller understanding of their impact. Furthemore, there exists a need for

more analysis to be done on the proposed pulser unit design. It is suggested that the new

prototype pulser unit ôe implemented and used to test the results obtained in this study

experimentally, thereby validating the results reported bere. in addition to this, it is

recomrnended tbat a sidesn external pulsing unit be designed and irnplemented to test for an

SRM's stability to transverse wave generation. It should also be noted here that in the

rnissilehlast wave interaction scenario, the pssibility o f radial and tangential waves being

generated inside the rocket motor during missilehlast wave interaction was not modelled in this

study; hence, by testing this experimentally by a new side-on pulsing unit, this will give more

insight on the pulse strength developed inside the SRM. It might be worth noting that at the

present time, there exist no pulser units that test for an SRM's stability to transverse waves

generated inside its combustion chamber by external side-on pulsing methods.

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