Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue

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


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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]


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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)]


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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


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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


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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 ]


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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


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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


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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

Justin Tang's Seminar - Nonlinear dynamics and route to chaos in Fickett's detonation analogue

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