On the dynamics and linear stability of one-dimensional steady detonation waves (original) (raw)
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Steady detonation waves for gases undergoing dissociation/recombination and bimolecular reactions
Continuum Mechanics and Thermodynamics, 2004
The one-dimensional steady propagation problem of a shock wave modelling detonation as a combustion process in a mixture of reacting gases is studied. For this purpose, suitable reactive Euler equations deduced from appropriate closures of the Boltzmann-like equations holding at a mesoscopic level are used. The considered chemical reactions driving detonation are either of dissociation/recombination or of bimolecular type. After a jump discontinuity governed by the Rankine-Hugoniot conditions, the reactions proceed until chemical equilibrium is reached. Physical consistency of solutions is discussed in terms of the relevant Hugoniot diagrams, showing different features for exothermic and endothermic processes, and determining the allowed range of detonation velocities.
Steady detonation problem for slow and fast chemical reactions
2006
Two sets of hydrodynamic equations for a mixture of four gases undergoing a bimolecular chemical reaction are discussed. The former consists in a system of balance laws for the case of a chemical relaxation time of the same order of the macroscopic processes (slow reaction). Conversely, the latter is a system of conservation laws for the case of short chemical relaxation time (fast reaction). After the analysis of the hyperbolic nature of the hydrodynamic equations, we formulate and solve the problem of the stationary propagation of a detonation wave. The differences of the shock structure in the two cases are shown by the presented numerical results.
2019
Theoretical and numerical studies of the stability of Zeldovich-von Neumann-Doring (ZND) solution describing a stationary detonation wave (DW), as well as pulsating modes of DW propagation, go back to the works of Erpenbeck [1] and Fickett [2]. A large number of subsequent fundamental studies of pulsating DW are associated with the use of a single-stage model of the kinetics of chemical reactions. On the one hand, this is due to the considerable degree of development of this model, the well-established methodology of transition to dimensionless variables using the length of the half-reaction and a large amount of accumulated material. On the other hand, it is known that this model is able to describe the main features of nonlinear dynamics of DW propagation such as one-dimensional pulsations, twodimensional detonation cells and three-dimensional spin. Among the disadvantages of the model is the impossibility of explicit separation of the induction and reaction zones, which was a bac...
Linear stability analysis of one-dimensional detonation coupled with vibrational relaxation
Physics of Fluids
The linear stability of one-dimensional detonations with one-reaction chemistry coupled with molecular vibration nonequilibrium is investigated using the normal mode approach. The chemical kinetics in the Arrhenius form depend on an averaged temperature model that consists of translational–rotational mode and vibrational mode. The Landau–Teller model is applied to specify the vibrational relaxation. A time ratio is introduced to denote the ratio between the chemical time scale and the vibrational time scale in this study, which governs the vibrational relaxation rate in this coupling kinetics. The stability spectrum of disturbance eigenmodes is obtained by varying the bifurcation parameters independently at a different time ratio. These parameters include the activation energy, the degree of overdrive, the characteristic vibrational temperature, and the heat release. The results indicate that the neutral stability limit shifts to higher activation energy on the vibrational nonequili...
Stability of detonations for an idealized condensed-phase model
Journal of Fluid Mechanics, 2008
The stability of travelling wave Chapman–Jouguet and moderately overdriven detonations of Zeldovich–von Neumann–Döring type is formulated for a general system that incorporates the idealized gas and condensed-phase (liquid or solid) detonation models. The general model consists of a two-component mixture with a one-step irreversible reaction between reactant and product. The reaction rate has both temperature and pressure sensitivities and has a variable reaction order. The idealized condensed-phase model assumes a pressure-sensitive reaction rate, a constant-γ caloric equation of state for an ideal fluid, with the isentropic derivative γ=3, and invokes the strong shock limit. A linear stability analysis of the steady, planar, ZND detonation wave for the general model is conducted using a normal-mode approach. An asymptotic analysis of the eigenmode structure at the end of the reaction zone is conducted, and spatial boundedness (closure) conditions formally derived, whose precise fo...
On the dynamics of multi-dimensional detonation
Journal of Fluid Mechanics, 1996
We present an asymptotic theory for the dynamics of detonation when the radius of curvature of the detonation shock is large compared to the one-dimensional, steady, Chapman-Jouguet (CJ) detonation reaction-zone thickness. The analysis considers additional time-dependence in the slowly varying reaction zone to that considered in previous works. The detonation is assumed to have a sonic point in the reactionzone structure behind the shock, and is referred to as an eigenvalue detonation. A new, iterative method is used to calculate the eigenvalue relation, which ultimately is expressed as an intrinsic, partial differential equation (PDE) for the motion of the shock surface. Two cases are considered for an ideal equation of state. The first corresponds to a model of a condensed-phase explosive, with modest reaction rate sensitivity, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, the first normal time derivative of the normal shock velocity, D,,, and the shock curvature, IC. The second case corresponds to a gaseous explosive mixture, with the large reaction rate sensitivity of Arrhenius kinetics, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, its first and second normal time derivatives of the normal shock velocity, b,, B,, and the shock curvature, IC, and its first normal time derivative of the curvature, k. For the second case, one obtains a one-dimensional theory of pulsations of plane CJ detonation and a theory that predicts the evolution of self-sustained cellular detonation. Versions of the theory include the limits of near-CJ detonation, and when the normal detonation velocity is significantly below its CJ value. The curvature of the detonation can also be of either sign, corresponding to both diverging and converging geometries.
Stability of viscous detonations for Majda’s model
Physica D: Nonlinear Phenomena, 2013
ABSTRACT Using analytical and numerical Evans-function techniques, we examine the spectral stability of strong-detonation-wave solutions of Majda's scalar model for a reacting gas mixture with an Arrhenius-type ignition function. We introduce an efficient energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane, and we use a numerical approximation of the Evans function to search for possible unstable eigenvalues in this region. Our results show, for the parameter values tested, that these waves are spectrally stable. Combining these numerical results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [J. Differential Equations 233 (2007), no. 2, 654-698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonation-wave solutions of the Majda model without a smallness assumption. Notably, our results indicate that, for the simplified Majda model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to "galloping" or "pulsating" solutions as is observed in the full reactive Navier-Stokes equations. This answers in the negative a question posed by Majda as to whether the scalar detonation model captures this aspect of detonation behavior.
One-dimensional nonlinear stability of pathological detonations
Journal of Fluid Mechanics, 2000
In this paper we perform high-resolution one-dimensional time-dependent numerical simulations of detonations for which the underlying steady planar waves are of the pathological type. Pathological detonations are possible when there are endothermic or dissipative effects in the system. We consider a system with two consecutive irreversible reactions A→B→C, with an Arrhenius form of the reaction rates and the second reaction endothermic. The self-sustaining steady planar detonation then travels at the minimum speed, which is faster than the Chapman–Jouguet speed, and has an internal frozen sonic point at which the thermicity vanishes. The flow downstream of this sonic point is supersonic if the detonation is unsupported or subsonic if the detonation is supported, the two cases having very different detonation wave structures. We compare and contrast the long-time nonlinear behaviour of the unsupported and supported pathological detonations. We show that the stability of the supported...
Multidimensional Detonation Solutions from Reactive Navier-Stokes Equations 1
2008
This study will describe multi-dimensional detonation wave solutions of the compressible reactive Navier-Stokes equations. As discussed in detail by Fickett and Davis [1], a steady onedimensional detonation with a spatially resolved reaction zone structure is known as ZND wave, named after Zeldovich, von Neumann, and Döring. In experiments [2] and calculations with simplified models [3], [4], [5], [6], it has been observed and predicted that these ZND waves are unstable. In the experiments, detonation in a tube with walls coated with a thin layer of soot etches detailed regular patterns on the tube walls, indicating the existence of cellular detonation wave structure. Linear analysis [3] demonstrates the fundamental instability of the one-dimensional ZND structure. This is extended by analysis of the full one-dimensional unsteady Euler equations to describe galloping detonations [4]. In two-dimensional calculations [5] it is found that complex cellular structures and transverse wave...
Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases
Journal of Fluid Mechanics, 1996
The purpose of this analytical work is twofold: first, to clarify the physical mechanisms triggering the one-dimensional instabilities of plane detonations in gases; secondly to provide a nonlinear description of the longitudinal dynamics valid even far from the bifurcation. The fluctuations of the rate of heat release result from the temperature fluctuations of the shocked gas with a time delay introduced by the propagation of entropy waves. The motion of the shock is governed by a mass conservation resulting from the gas expansion across the reaction zone whose position fluctuates relative to the inert shock. The effects of longitudinal acoustic waves are quite negligible in pistonsupported detonations at high overdrives with a small difference of specific heats. This limit leads to a useful quasi-isobaric approximation for enlightening the basic mechanism of galloping detonations. Strong nonlinear effects, free from the spurious singularities of the square-wave model, are picked ...