conclusion.lyx

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\begin_layout Standard
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status collapsed

\begin_layout Plain Layout

% Conclusion File
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\begin_layout Chapter
Conclusions
\begin_inset CommandInset label
LatexCommand label
name "chap:conclusions"

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\end_layout

\begin_layout Section
Summary
\end_layout

\begin_layout Standard
In this thesis, we developed and implemented a new method to study the stability
 of steady solutions of detonation models.
 Our approach is based on the linearization of the governing equations written
 in a shock-attached frame and numerical solution of the linearized problem.
 Resultant time series of detonation velocity are postprocessed using a
 novel technique of dynamic mode decomposition, which is aimed at identification
 of modes of the system that grow or decay exponentially in time with fixed
 frequencies, and these modes are identical to the normal modes in linearized
 systems.
\end_layout

\begin_layout Standard
We implement and test our algorithm using a
\begin_inset space ~
\end_inset

detonation model based on the reactive Euler equations with one irreversible
 reaction.
 Comparing our stability results with the results of linear stability analysis
 via normal-mode approach, which are widely present in the literature for
 this model, we have found excellent agreement both for growth rates and
 frequencies of the perturbations of the base flow for all cases considered.
 Computation of the neutral stability boundary for a wide range of activation
 energy 
\begin_inset Formula $E$
\end_inset

 and heat release 
\begin_inset Formula $Q$
\end_inset

 for 
\begin_inset Formula $\gamma=1.2$
\end_inset

 also have an agreement with the results of the normal-mode approach.
 Additionally, we have studied neutral stability boundaries for other values
 of 
\begin_inset Formula $\gamma$
\end_inset

 (1.1, 1.3, and 1.4) and have found that for low values of 
\begin_inset Formula $Q$
\end_inset

 the critical value of 
\begin_inset Formula $E$
\end_inset

 decreases as 
\begin_inset Formula $\gamma$
\end_inset

 increases, while for, large values of 
\begin_inset Formula $Q$
\end_inset

, the critical values of 
\begin_inset Formula $E$
\end_inset

 increase as 
\begin_inset Formula $\gamma$
\end_inset

 increases.
 Also, we have studied the dependence of the frequencies along the neutral
 stability curves on 
\begin_inset Formula $\gamma$
\end_inset

 and have found that for low values of 
\begin_inset Formula $Q$
\end_inset

 frequency is insensitive to 
\begin_inset Formula $\gamma$
\end_inset

, while, for large values of 
\begin_inset Formula $Q$
\end_inset

, frequency increases as 
\begin_inset Formula $\gamma$
\end_inset

 increases.
 One particularly interesting result was that although for 
\begin_inset Formula $\gamma\geq1.2$
\end_inset

 neutral stability curves of the fundamental mode are neutral stability
 boundaries that separate stable base flows from unstable ones, for 
\begin_inset Formula $\gamma=1.1$
\end_inset

 that was not the case: for mid-range of 
\begin_inset Formula $Q$
\end_inset

 the first overtone is more unstable than the fundamental mode, hence, the
 neutral stability boundary is determined by a union of neutral stability
 curves for these two modes.
\end_layout

\begin_layout Standard
The advantage of the present approach over the traditional normal-mode approach
 is that analytic complications of formulating boundary-value problem, namely,
 formulation of the boundary condition at the end of the reaction zone,
 are replaced by straightforward linearization with simple outflow boundary
 condition.
 However, difficulty arises in the postprocessing step as time series of
 detonation velocity inevitably contain noise due to numerical approximation,
 hence, it may become challenging to identify all normal modes correctly.
\end_layout

\begin_layout Standard
Having developed and tested this approach to linear stability analysis,
 we have considered an extension of the model based on the reactive Euler
 equations with one-step kinetics, to the model that includes two reactions:
 one is exothermic and one is endothermic.
 Analysis of the ZND solutions for this model showed that there are two
 types of admissible ZND solutions, which are the same for the region between
 the embedded sonic locus and the leading shock, and are different between
 the end of the reaction zone and the sonic locus.
 We have studied the effects of endothermicity on stability and have shown
 that, as endothermicity increases, more and more unstable modes are excited
 in the stability spectrum, rendering the base ZND flow unstable.
 We have also shown that stability spectrum is independent of the postsonic
 part of the ZND solution, which serves as a demonstration of the postulate
 that intrinsic stability of detonations is determined solely by flow between
 the sonic locus and the detonation front.
\end_layout

\begin_layout Standard
Using the particular reaction-rate expression obtained in the asymptotic
 theory of detonations, we analyze the Fickett's detonation analogue both
 for linear and nonlinear stability.
 We have shown that this model demonstrates rich nonlinear dynamics with
 multiple bifurcations and chaotic behavior.
\end_layout

\begin_layout Section
Future Research Work
\end_layout

\begin_layout Standard
The work presented in this thesis can be extended in the following directions.
\end_layout

\begin_layout Enumerate
Application of the present method of linear stability analysis to more complicat
ed models based on reactive Euler equations coupled with thermodynamic relations
 of calorically perfect gases with multistep reaction mechanisms 
\begin_inset CommandInset citation
LatexCommand citep
key "Short1996,ShortSharpe:2003:Two-step,Gorchkov2007"

\end_inset

.
\end_layout

\begin_layout Enumerate
Application of the method to more complicated models based on reactive Euler
 equations coupled with thermodynamic relations of ideal, but not calorically
 perfect, gases with multistep reaction mechanisms 
\begin_inset CommandInset citation
LatexCommand citep
key "Aslam2009,Romick2012"

\end_inset

.
\end_layout

\begin_layout Enumerate
Application of the method to detonation models for explosives with non-ideal
 equation of state
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "ShortEtAl2006"

\end_inset

.
\end_layout

\begin_layout Enumerate
Application of the method to analysis of detonation waves in different geometrie
s, such as cylindrical detonations 
\begin_inset CommandInset citation
LatexCommand citep
key "WescottEtAl2004,Kasimov2014"

\end_inset

.
\end_layout

\begin_layout Enumerate
Application of the method to detonation models with losses 
\begin_inset CommandInset citation
LatexCommand citep
key "SemenkoEtAl2016,SowEtAl2017"

\end_inset

.
\end_layout

\begin_layout Enumerate
Extension of the method to include perturbations in the direction transverse
 to the lead shock 
\begin_inset CommandInset citation
LatexCommand citep
key "ShortStewart1998,Stewart2006"

\end_inset

.
\end_layout

\begin_layout Enumerate
Thorough investigation of nonlinear dynamics of the Fickett's model with
 the reaction-rate expression
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fm:math-model:gov-eqn-lab:production-rate"

\end_inset

 by using different numerical schemes for approximation of spatial derivatives
 (including characteristic decomposition) and different time integrators
 (strong-stability-preserving Runge–Kutta schemes 
\begin_inset CommandInset citation
LatexCommand citep
key "Ketcheson2005,Gottlieb2008,Hadjimichael2013,BrestenEtAl2016"

\end_inset

 and implicit-explicit Runge–Kutta schemes 
\begin_inset CommandInset citation
LatexCommand citep
key "Pareschi2005,ZharovskyEtAl2015,BoscarinoPareschi2017"

\end_inset

) to analyze sensitivity of the nonlinear dynamics to numerical methods
 and to determine which schemes are the most robust in terms of preservation
 of the solution invariants 
\begin_inset Formula $(u>0$
\end_inset

 and 
\begin_inset Formula $0\leq\lambda\leq1)$
\end_inset

 and handling of the secondary shock waves.
\end_layout

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