The effect of structure on the stability of detonations I. Role of the induction zone

Buckmaster, J.; Ludford, G. S. S.

doi:10.1016/s0082-0784(88)80400-4

Cited by 44 publications

(65 citation statements)

References 4 publications

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“…Indeed, treatments of the exact problem [36,40] show that for increasing activation energy, a tremendous number of unstable modes are available to most planar detonations, all other parameters staying fixed. In 1986 Buckmaster and Ludford [57] carried out an asymptotic analysis that considered long transverse wave length disturbances and slow temporal variation, both O1=E in the limit of large activation energy, for the basic problem. They found an unstable nonoscillatory root and concluded that in this limit detonation is always unstable.…”

Section: Large Activation Energy Asymptoticsmentioning

confidence: 99%

State of Detonation Stability Theory and Its Application to Propulsion

Stewart

Kasimov

2006

Journal of Propulsion and Power

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We present an overview of the current state of detonation stability theory and discuss its implications for propulsion. The emphasis of the review is on the exact or asymptotic treatments of detonations, including various asymptotic limits that appear in the literature. The role that instability plays in practical detonation-based propulsion is of primary importance and is largely unexplored, hence we point to possible areas of research both theoretical and numerical, that might help improve our understanding of detonation behavior in propulsion devices. We outline the basic formulation of detonation stability theory that starts from linearized Euler equations, describe the algorithm of solution, and present an example that illustrates typical results.shock velocity vector E = activation energy e = internal energy per unit mass including chemical energy f = degree of overdrive H = enthalpy flux across the shock k = transverse wave number k ! = reaction rate constant M = local Mach number relative to the shock M = mass flux across the shock n = shock-attached frame axial coordinate n = unit normal to the shock P = momentum flux across the shock p = pressure Q = heat release per unit mass q = state vector t = unit tangent vector to the shock U = x component of particle velocity in steady shock frame U 1 = x component of particle velocity in unsteady shock frame u = particle velocity vector in laboratory frame u 1 = x component of particle velocity in laboratory frame u 2 = y component of particle velocity v = specific volume x = laboratory frame axial coordinate y = transverse coordinate = perturbation growth rate = ratio of specific heats = reaction progress variable = density = thermicity = shock displacement from the steady position along x axis ! = reaction rate University. He has published extensively in the areas of combustion theory, asymptotic analysis, theory for multidimensional detonation, detonation stability, continuum mechanics of multiphase flows, phase transformations, modeling of energetic materials, solid rocket motor combustion, advanced level set methods, and computational modeling of reactive flow. He is a past associate editor for SIAM Journal of Applied Mathematics, and was a founding (and still serves as a) board member for Combustion Theory and Modelling. In 1987 he was a founder of the SIAM Conferences on Numerical Combustion that are still held worldwide. He was the concept originator and principal investigator for the University of Illinois Center for Advanced Rockets (CSAR).

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Section: Large Activation Energy Asymptoticsmentioning

confidence: 99%

State of Detonation Stability Theory and Its Application to Propulsion

Stewart

Kasimov

2006

Journal of Propulsion and Power

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“…In one form or another, such condition has been used by many researchers investigating stability of detonation waves (e.g. Erpenbeck 1962, Buckmaster & Ludford 1986, Lee & Stewart 1990). The radiation condition has a different form in cylindrical coordinates than in Cartesian coordinates, but it expresses the same physical condition.…”

Section: The Radiation Conditionmentioning

confidence: 99%

Spinning instability of gaseous detonations

Kasimov

Stewart

2002

J. Fluid Mech.

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We investigate hydrodynamic instability of a steady planar detonation wave propagating in a circular tube to three-dimensional linear perturbations, using the normal mode approach. Spinning instability is identified and its relevance to the well-known spin detonation is discussed. The neutral stability curves in the plane of heat release and activation energy exhibit bifurcations from a low-frequency to high-frequency spinning modes as the heat release is increased at fixed activation energy. With a simple Arrhenius model for the heat release rate remarkable qualitative agreement is found with experiments that vary the mixture dilution, initial pressure, and tube diameter. The analysis contributes to an explanation of the spin detonation which has essentially been absent since the discovery of the phenomenon over seventy years ago.

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“…This problem is proposed by Buckmaster [5], Buckmaster and Ludford [7], and Buckmaster, Dold, and Schmidt-Lainé [6] in the study of detonation waves. In these works c and l are real positive constants and f is given by the formula f (s) = ln e s − 1…”

Section: Introduction Mathematical Formulation Of the Detonation Promentioning

confidence: 99%

“…Writing the Rankine-Hugoniot relations, developing all the variables in the high energy asymptotics, and supposing the wall perfectly reflecting, Buckmaster [5], Buckmaster and Ludford [7], and Buckmaster, Dold, and Schmidt-Lainé [6] derived the following evolution equation for the variable u = 1 + h/K where K is a positive constant:…”

Section: Introduction Mathematical Formulation Of the Detonation Promentioning

confidence: 99%

Quenching for a One-Dimensional Fully Nonlinear Parabolic Equation in Detonation Theory

Vázquez¹,

Gerbi²,

Galaktionov³

2001

SIAM J. Appl. Math.

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Abstract. We study and describe the quenching phenomenon for the fully nonlinear parabolic equation ut + 1 2 (ux) 2 = f (c u uxx) + ln u, x ∈ (0, l), t > 0, which for f (s) = ln[(e s − 1)/s] represents the evolution of the perturbations of the Zel'dovich-von Neuman-Doering square wave occurring during a detonation in a duct. In the general case, the function f : R → R is smooth and satisfies the parabolicity condition f ′ (s) > 0 in R, c and l are positive constants, and we impose Neumann boundary conditions ux(0, t) = ux(l, t) = 0 for t > 0 and take initial data u(x, 0) > 0 with inverse bell-shaped form.The phenomenon of quenching is characterized by the existence of a finite time T at which the solution u ceases to exist as a classical solution because minxu(x, t) → 0 as t → T ; then the equation degenerates and forms a singularity at the level u = 0, due to the presence of the logarithmic zero-order term.We first exhibit conditions on f and u(x, 0) which imply the presence of this type of singularity. Next we derive estimates on u , uxx in order to study the behavior of the profile in the neighborhood of the time T . We then find the asymptotic scaling factors, which are universal, and the asymptotic profile which is given in the rescaled coordinates by a parabola with a free constant to adjust. For this purpose we use the theory of stability of ω-limit sets of infinite-dimensional dynamical systems under asymptotically small perturbations. In this problem the perturbation is singular but exponentially vanishing as t → T . Finally, we prove that the present model does not admit any extension beyond the singularity, i.e., for t > T .

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The effect of structure on the stability of detonations I. Role of the induction zone

Cited by 44 publications

References 4 publications

State of Detonation Stability Theory and Its Application to Propulsion

State of Detonation Stability Theory and Its Application to Propulsion

Spinning instability of gaseous detonations

Quenching for a One-Dimensional Fully Nonlinear Parabolic Equation in Detonation Theory

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