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![Page 1: 1 Linear Stability of Detonations with Reversible Chemical Reactions Shannon Browne Graduate Aeronautical Laboratories California Institute of Technology.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649ed95503460f94be768e/html5/thumbnails/1.jpg)
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Linear Stability of Detonations with Reversible Chemical Reactions
Linear Stability of Detonations with Reversible Chemical Reactions
Shannon BrowneGraduate Aeronautical Laboratories
California Institute of TechnologySupported by: NSF, ASCAdvisor: J. E. Shepherd
March 12, 2008
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What is a Detonation?What is a Detonation?
• A Strong Shock Wave Sustained by Chemical Reaction Energy
• A Three-Dimensional Phenomenon
Chapman-Jouguet Model• Purely Thermodynamic• Predicts Natural Detonation Front Velocity
ZND Model • One-Dimensional• Steady• Predicts Length Scales
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Motivation – ExperimentalMotivation – Experimental
“Weakly Unstable”
“Highly Unstable”
Austin, Pintgen, Shepherd (’97)
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MotivationMotivation
2 H2–O2–12 Ar
P1=20 kPa
θeff = 5.2
h/RT1= 24.2
C2H4–3O2–10.5 N2
P1=20 kPa
θeff = 12.1
h/RT1= 56.9
Austin, Pintgen, Shepherd (’97)
Short, Stewart (’98)α = Growth Rate
k = Wave Number
= 1.2
f = 1.2
Ea/RT1 = 50
h/RT1= 0.4
= 1.2
f = 1.2
Ea/RT1 = 50
h/RT1= 50
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In Two Dimensions
Governing EquationsGoverning Equations
Reactive Euler Equations (2+d+N Equations)
Ideal Gas Equation of State
Net Rate of Production of Species k
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Coordinate Transformation and Base Flow
Coordinate Transformation and Base Flow
Lab Frame Flat Shock Fixed Frame
Shock Velocity Perturbation
Base Flow = Steady Flow(ZND Model Detonation)
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2D Linear Perturbation Equations2D Linear Perturbation Equations
Base Flow [ZND Model] (o)
Perturbations (1)
• A & B = Convective Derivatives
• C = Source Matrix
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Single Reversible ReactionSingle Reversible Reaction
• In realistic chemical systems, reactions are reversible• Previous stability studies – all irreversible
Δs = soB – so
A
Reversibility
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Constant TCJ Family of SolutionsConstant TCJ Family of Solutions
Vary Δh/RT & Δs/R
Maintain TCJ & t1/2 = half reaction time
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Boundary Condition – x = 0Boundary Condition – x = 0
Linearly Perturbed Shock Jump Conditions
• U is a Free Parameter
• Frozen Shock – No Change in Composition
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Boundary Condition – x = ∞Radiation Condition
Boundary Condition – x = ∞Radiation Condition
• Require that all waves travel out of the domain
1. Perturbation Equation = Algebraic Eigenvalue Problem
2. Find characteristic speeds (eigenvalues)
3. Projection of solution along incoming eigenvector must be zero
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Boundary Condition – x = ∞Radiation Condition
Boundary Condition – x = ∞Radiation Condition
• Introduce near equilibrium relaxation time scale ()
• Single Irreversible Reaction – One-way coupling (Analytic Condition)
• Single Reversible Reaction – No One-way Coupling (Numerical Condition)
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Mode 1
Effect of Reversibility on Downstream State Complex Wave Speeds
Effect of Reversibility on Downstream State Complex Wave Speeds
Mode 3
Mode 2
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ImplementationImplementation
• Shooting Method– Specify ky, Guess – Integrate through domain– Root Solver (Muller’s Method) New Guess for
• Cantera Library (Goodwin)– Idealized & Detailed Mechanisms (thermodynamics + kinetics)– All derivatives computed analytically
• CVODE (Cohen & Hindmarsh) – stiff integrator
• ZGEEV (LAPACK) – complex numerical eigenvalue/eigenvector routine
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Flow ProfilesFlow Profiles
s/R = 0 s/R = -8
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Mode 2
Overdrive SeriesOverdrive Series
Mode 1
Mode 3
Conclusions
• Mode 1: Reversibility is Stabilizing
• Higher Modes: Reversibility is Destabilizing
Irreversible Reaction (Lee & Stewart ’90)