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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)