1. 1 Linear Stability of Detonations with Reversible Chemical Reactions Shannon Browne Graduate Aeronautical Laboratories California Institute of Technology Supported by: NSF, ASC Advisor: J. E. Shepherd March 12, 2008

2. 2 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

3. 3 Motivation – Experimental “Weakly Unstable” “Highly Unstable” Austin, Pintgen, Shepherd (’97)

4. 4 Motivation 2 H 2 –O 2 –12 Ar P 1 =20 kPa θ eff = 5.2  h/RT 1 = 24.2 C 2 H 4 –3O 2 –10.5 N 2 P 1 =20 kPa θ eff = 12.1  h/RT 1 = 56.9 Austin, Pintgen, Shepherd (’97) Short, Stewart (’98) α = Growth Rate k = Wave Number  = 1.2 f = 1.2 E a /RT 1 = 50  h/RT 1 = 0.4  = 1.2 f = 1.2 E a /RT 1 = 50  h/RT 1 = 50

5. 5 In Two Dimensions Governing Equations Reactive Euler Equations (2+d+N Equations) Ideal Gas Equation of State Net Rate of Production of Species k

6. 6 Coordinate Transformation and Base Flow Lab Frame Flat Shock Fixed Frame Shock Velocity Perturbation Base Flow = Steady Flow (ZND Model Detonation)

7. 7 2D Linear Perturbation Equations Base Flow [ZND Model] ( o ) Perturbations ( 1 ) A & B = Convective Derivatives C = Source Matrix

8. 8 Single Reversible Reaction In realistic chemical systems, reactions are reversible Previous stability studies – all irreversible Δs = s o B – s o A Reversibility

9. 9 Constant T CJ Family of Solutions Vary Δh/RT & Δs/R Maintain T CJ & t 1/2 = half reaction time

10. 10 Boundary Condition – x = 0 Linearly Perturbed Shock Jump Conditions U is a Free Parameter Frozen Shock – No Change in Composition

11. 11 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

12. 12 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)

13. 13 Mode 1 Effect of Reversibility on Downstream State Complex Wave Speeds Mode 3 Mode 2

14. 14 Implementation Shooting Method –Specify k y, 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

15. 15 Flow Profiles  s/R = 0  s/R = -8

16. 16 Mode 2 Overdrive Series Mode 1 Mode 3 Conclusions Mode 1: Reversibility is Stabilizing Higher Modes: Reversibility is Destabilizing  Irreversible Reaction (Lee & Stewart ’90)