Nonlinear Dynamics of Self-Sustained Supersonic Reaction Waves: Fickett’s Detonation Analogue

Radulescu, Matei I.; Tang, J.

doi:10.1103/physrevlett.107.164503

Cited by 24 publications

(32 citation statements)

References 13 publications

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“…The one-dimensional, inviscid scalar Burgers equation with a source term was proposed by Fickett [15] and Majda [36] as an analog to the reactive Euler equations in order to explore detonation dynamics. In recent years, study of this scalar analog system has generated a number of interesting results, including the existence of pulsating, chaotic solutions [48,27,61,14]. The study of Mi & Higgins [38] was a precursor to the present study using the inviscid Burgers equation with periodically spaced δ-function sources that were triggered by the passage of the leading shock front.…”

Section: Discussionmentioning

confidence: 95%

Effect of spatial discretization of energy on detonation wave propagation

Timofeev

Higgins

2017

J. Fluid Mech.

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Detonation propagation in the limit of highly spatially discretized energy sources is investigated. The model of this problem begins with a medium consisting of a calorically perfect gas with a prescribed energy release per unit mass. The energy release is collected into sheet-like sources that are now embedded in an inert gas that fills the spaces between them. The release of energy in the first sheet results in a planar blast wave that propagates to the next source, which is triggered after a prescribed delay, generating a new blast, and so forth. The resulting wave dynamics as the front passes through hundreds of such sources is computationally simulated by numerically solving the governing one-dimensional Euler equations in the lab-fixed reference frame. Two different solvers are used: one with a fixed uniform grid and the other using an unstructured, adaptively refined grid enabling the limit of highly concentrated, spatially discrete sources to be examined. The two different solvers generate consistent results, agreeing within the accuracy of the measured wave speeds. The average wave speed for each simulation is measured once the wave propagation has reached a quasi-periodic solution. The effect of source delay time, source energy density, specific heat ratio, and the spatial discreteness of the sources on the wave speed is studied. Sources fixed in the lab reference frame versus sources that convect with the flow are compared as well. The average wave speed is compared to the ideal Chapman-Jouguet (CJ) speed of the equivalent homogenized media. Velocities in excess of the CJ speed are found as the sources are made increasingly discrete, with the deviation above CJ being as great as 15%. The deviation above the CJ value increases with decreasing values of specific heat ratio γ. The total energy release, delay time, and whether the sources remain lab-fixed or are convected with the flow do not have a significant influence on the deviation of the average wave speed away from CJ. A simple, ad hoc analytic model is proposed to treat the case of zero delay time (i.e., source energy released at the shock front) that exhibits qualitative agreement with the computational solutions and may explain why the deviation from CJ increases with decreasing γ. When the sources are sufficiently spread out so as to make the energy release of the media nearly continuous, the classic CJ solution is obtained for the average wave speed. Such continuous waves can also be shown to have a time-averaged structure consistent with the classical ZND structure of a detonation. In the limit of highly discrete sources, temporal averaging of the wave structure shows that the effective sonic surface does not correspond to an equilibrium state. The average state of the flow leaving the wave in this case does eventually reach the equilibrium Hugoniot, but only after the effective sonic surface has been crossed. Thus, the super-CJ waves observed in the limit of highly discretized sources can be understood as weak detonations due to the...

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Section: Discussionmentioning

confidence: 95%

Effect of spatial discretization of energy on detonation wave propagation

Timofeev

Higgins

2017

J. Fluid Mech.

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“…where t is time, and thermicity is defined as Figure 1 illustrates the structure of the self-supported detonation wave [17]. The detonation structure consists of pressure waves originating from the back (left), travelling along characteristics dx/dt = ρ and amplifying according to the characteristic equation (4).…”

Section: Model: Reactive Burger's Equationmentioning

confidence: 99%

“…The model neglects the rear facing pressure waves of gas dynamics, hence significantly simplifying the mathematical complexity of the description. The model retains the important physics of reactive compressible flows and its complex dynamics, namely that pressure waves receive amplification, modulated by the local rate of energy release, and can form shocks [17]. Fickett has already demonstrated how the model can reproduce the complex steady structure of eigenvalue detonations in the presence of one exothermic and one endothermic reaction [16].…”

Section: Introductionmentioning

confidence: 99%

Multiplicity of detonation regimes in systems with a multi-peaked thermicity

Lau-Chapdelaine

Zhang

Radulescu

2019

Combustion Theory and Modelling

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The study investigates detonations with multiple quasi-steady velocities that have been observed in the past in systems with multi-peaked thermicity, using Fickett's detonation analogue. A steady state analysis of the travelling wave predicts multiple states, however, all but the one with the highest velocity develop a singularity after the sonic point. Simulations show singularities are associated with a shock wave which overtakes all sonic points, establishing a detonation travelling the highest of the predicted velocities.Under a certain parameter range, the steady-state detonation can have multiple sonic points and solutions. Embedded shocks can exist behind sonic points, where they link the weak and strong solutions. Sonic points whose characteristics do not diverge are found to be unstable, and to be the source of the embedded shocks. Numerical simulations show that these shocks are only quasi stable. This is believed to be due to the reaction rates having been chosen to be independent of hydrodynamics which permits shocks anywhere behind a sonic point. * slauc076@uottawa.ca † matei@uottawa.ca arXiv:1503.03588v1 [physics.flu-dyn]

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“…shocks and shock waves) have counterparts in the analog systems as well. Recently it was also shown that the Fickett model applied to a spatially uniform reactive media governed by a two-step chemical kinetics [6] has the ability to reveal some key nonlinear dynamics also observed in the Euler system [7], including the period-doubling bifurcation leading to chaos. Using a similar approach, detonation instability originated from the inclusion of a loss term such as friction or curvature can also be recovered [8].…”

Section: Introductionmentioning

confidence: 98%

Investigation of detonation velocity in heterogeneous explosive system using the reactive Burgers’ analog

Labbio¹,

Kiyanda²,

et al. 2016

AIP Conference Proceedings

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In this study, the applicability of the Chapman-Jouguet (CJ) criterion is tested numerically for heterogeneous explosive media using a simple detonation analog. The analog system consists of a reactive Burgers' equation coupled with an Arrhenius type reaction wave, and the heterogeneity of the explosive media is mimicked using a discrete energy source approach. The governing equation is solved using a second order, finite-volume approach and the average propagation velocity of the discrete detonation is determined by tracking the leading shock front. Consistent with previous studies, the averaged velocity of the leading shock front from the unsteady numerical simulations is also found to be in good agreement with the velocity of a CJ detonation in a uniform medium wherein the energy source is spatially homogenized. These simulations have thus implications for whether the CJ criterion is valid to predict the detonation velocity in heterogeneous explosive media.

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Nonlinear Dynamics of Self-Sustained Supersonic Reaction Waves: Fickett’s Detonation Analogue

Cited by 24 publications

References 13 publications

Effect of spatial discretization of energy on detonation wave propagation

Effect of spatial discretization of energy on detonation wave propagation

Multiplicity of detonation regimes in systems with a multi-peaked thermicity

Investigation of detonation velocity in heterogeneous explosive system using the reactive Burgers’ analog

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