Qualitative modeling of the dynamics of detonations with losses

We consider a simplified model for the dynamics of one-dimensional detonations with generic losses. It consists of a single partial differential equation that reproduces, at a qualitative level, the essential properties of unsteady detonation waves, including pulsating and chaotic solutions. In particular, we investigate the effects of shock curvature and friction losses on detonation dynamics. To calculate steady-state solutions, a novel approach to solving the detonation eigenvalue problem is introduced that avoids the well-known numerical difficulties associated with the presence of a sonic point. By using unsteady numerical simulations of the simplified model, we also explore the nonlinear stability of steady-state or quasi-steady solutions.

[1]  B. M. Fulk.  MATH , 1992 .

[2]  M R Flynn,et al.  Self-sustained nonlinear waves in traffic flow. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Rupert Klein,et al.  The Relation Between Curvature, Rate State-Dependence, and Detonation Velocity , 1993, SIAM J. Appl. Math..

[4]  R. Rosales,et al.  Model for shock wave chaos. , 2012, Physical review letters.

[5]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[6]  Y. Zeldovich,et al.  On the theory of the propagation of detonation in gaseous systems , 1950 .

[7]  Wildon Fickett,et al.  Detonation in miniature , 1979 .

[8]  A. Kasimov,et al.  Set-valued solutions for non-ideal detonation , 2013, 1312.2180.

[9]  D. Scott Stewart,et al.  The Dynamics of Detonation in Explosive Systems , 2007 .

[10]  W. C. Davis,et al.  Detonation: Theory and Experiment , 1979 .

[11]  Rodolfo Ruben Rosales,et al.  Study of a Model Equation in Detonation Theory , 2014, SIAM J. Appl. Math..

[12]  Nikolaos Nikiforakis,et al.  Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations , 2005 .

[13]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[14]  I︠a︡. B. Zelʹdovich,et al.  Theory of detonation , 1960 .

[15]  M. Radulescu,et al.  Nonlinear dynamics of self-sustained supersonic reaction waves: Fickett's detonation analogue. , 2011, Physical review letters.

[16]  John H. S. Lee.  The Detonation Phenomenon , 2008 .

[17]  Tariq D. Aslam,et al.  Simulations of pulsating one-dimensional detonations with true fifth order accuracy , 2006, J. Comput. Phys..

[18]  J. Davenport.  Editor , 1960 .

[19]  Gary J. Sharpe,et al.  Linear and nonlinear dynamics of cylindrically and spherically expanding detonation waves , 2005, Journal of Fluid Mechanics.

[20]  A. Higgins.  Steady One-Dimensional Detonations , 2012 .

[21]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[22]  Asymptotic theory of evolution and failure of self-sustained detonations , 2005, Journal of Fluid Mechanics.

[23]  Andrew J. Majda,et al.  Weakly Nonlinear Detonation Waves , 1983 .

[24]  G. Sivashinsky,et al.  Combustion waves in hydraulically resisted systems , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[25]  D. Scott Stewart,et al.  Theory of Detonation Shock Dynamics , 2012 .

[26]  D. C. Bull,et al.  Initiation of spherical detonation in hydrocarbon/air mixtures , 1978 .

[27]  M. Radulescu,et al.  Dynamics of shock induced ignition in Fickett’s model: Influence of χ , 2013 .