7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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B Y : J U S T I N T A N G
S U P E R V I S O R : M A T E I R A D U L E S C U
Non-linear dynamics and route tochaos in Ficketts detonation
analogue
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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Introduction: Detonations
Reaction processes
Deflagration: Subsonic.
Detonation: Supersonic. Coupled shock and following reaction.
Density
x
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Introduction: Detonations
Detonation applications
Solid explosives
Dust powder combustion in airSafety and prevention
Astrophysical phenomenon
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Detonation Propagation
Steady Structure: ZND profile
D e ns i t y
T e m p e r a t u r e
: Induction timeit
rt : Reaction time
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Instability in Detonations
Detonation are unstable, which has lead to theformation of complex patterns 2D cellular structures
1D pulsating instability behaviour
[Austin, The role ofinstability in gaseousdetonation]
[Ng et al, Nonlinear dynamicsand chaos analysis of 1Dpulsating detonations 2005]
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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1D Pulsating Instability
It is an oscillation in detonationstrength
With increasing sensitivity ofthe reaction rate (Ea), variousmodes of oscillation arise
[Ng et al, Nonlinear dynamicsand chaos analysis of 1Dpulsating detonations 2005]
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Bifurcation behaviour
Studied numerically with the 1Dreactive Euler equations
Admit universal dynamics [Sharpe,Ng et al., Henrick et al.] Period doubling bifurcation following
Feigenbaum route to chaos
Bifurcationdiagram of 1Ddetonation[Henrick et al.(2006)]
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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Summary
1D Pulsating detonation instability
The complexity has made it difficult to identify the
governing mechanism
What is the governing mechanism
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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Approach: Ficketts model
Simplified toy-model of the 1D reactive Eulerequations
The physics are more transparent
Similar to Burgers equation for wave propagationwith added reaction
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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Pulsating Instability: Reaction Models
Ficketts model
Square wave reaction model [Fickett (1985), Hall and Ludford(1987)]
Eulers equation with reaction 2-step induction-reaction model [ Leung et al., Short and
Sharpe ]
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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Present Study
Use Ficketts model with 2-step reaction model
Determine the governing mechanism behindpulsating instability
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Governing equations: Ficketts Model
Ficketts model [1] is given for 1D inviscid reactivecompressible flow
Governing Equations:
Conservation:
Equation of State:
Reaction Rate Equation:
1. W. Fickett, Introduction to Detonation Theory ,1985
0=
+
xp
t
( )Qp += 22
1
( )
,rt=
xQ
xt
r
=
+
2
1
: Density
p: PressureQ: Heat release: Progress variabler: Reaction rate
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Characteristics Description
Characteristic can be used to describe the wave motion
Wave characteristic:
Particle paths:
==dt
dxalongrQ
dt
dp,
2
1
0, ==
dt
dxalongr
dt
d
t
x
t
x
Pressure Wave Particle Paths
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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Reaction Model
2-step Induction-Reaction model
An induction delay period begins before the exothermicity
,1
2
==
CJD
iii eKrt
( ) ,1 vrrrr Krt
==
I n d u ct io n zo n e r e a ct io n r a t e
R e a c t io n zo n e r eac t i on r a t e
: Reaction ratesensitivityparameter
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Numerical model
Exact Riemann solver with fractional step method
Programming framework: C++
Parameters: Q=5, Ki=1, Kr=2, v=0.5
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Numerical Simulation
We wanted to examine the instability behaviour
Initial conditions: 1D channel of unreacted gas
Initial steady ZND detonation profile
Shock Front
InductionZone
ReactionZone
7/30/2019 Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue
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Numerical Simulation
We varied and tracked the detonation dynamics
Modes of oscillation
Bifurcation diagram
Characteristic analysis to describe the governingmechanism
,
12
==
CJD
ii
i
eKrt
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Results
A steady solution was present
Oscillatory solution occurred for >5.7
We tracked the amplitude of the shock front
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Results: Oscillation Over Time
After 5.7 , the detonation transitioned from a stablesolution to a single mode oscillation pattern
= 4.5 = 6.8
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Results: Different Oscillation Modes
= 7.6 = 7.8
Period 21 :double period Period 22 : quadruple period
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Results: Bifurcation Behaviour
The detonation was seen to undergo period doublingbifurcations with increasing
Period of Oscillation at Bifurcations
1 2 4 3
= 5.7 6.9 7.7 8.72
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Results: Bifurcation Diagram
Period 21 (double period):
6.9 < 7.7Stable Solution:
< 5.7
Stable Period 3:
8.72
Period 22 :
7.7 < 7.9
Period 20 (single period):
5.7 < 6.9
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Instability Mechanism
x-t characteristics diagram in the frame of the CJ detonation
End ofInduction
Zone
End ofReaction
Zone
Shock Front
x
t
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Instability Mechanism
Stable SolutionAcceleration
Decelerationt
x
==dt
dxalongrQ
dt
dp,Wave Characteristic:
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Conclusion
Ficketts model did admit stable and oscillatory solutions,and followed the Feigenbaum route to chaos via perioddoubling bifurcations
We were able to get a clearer picture of the instabilitymechanisms by analysing the characteristics in thissimpler analogue, in which the dynamics of theacceleration and deceleration feedback were much moretransparent [Radulescu & Tang (2011)]
While we did qualitatively recreate the bifurcationdiagram, the mechanisms behind the bifurcations stillrequires further study
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Acknowledgements
NSERC Discovery Grant
SME4SME
http://sme4sme.ca/en