Numerical analysis of a two-dimensional nonsteady detonations

Taki, Shiro; Fujiwara, Toshi

doi:10.2514/6.1976-404

Cited by 9 publications

(11 citation statements)

References 2 publications

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“…The earliest simulations of one-dimensional pulsating detonation were performed by Fickett and Wood in 1966 [1]. In 1978, Taki and Fujiwara [2] performed the first two-dimensional simulations of cellular detonation. As computational capacity has grown, simulations have been made that incorporate higher grid resolution and more detailed models.…”

Section: Introductionmentioning

confidence: 99%

Mode selection in weakly unstable two-dimensional detonations

Taylor

Kasimov

Stewart

2009

Combustion Theory and Modelling

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A formulation of the reactive Euler equations in the shock-attached frame is used to study the two-dimensional instability of weakly unstable detonation through direct numerical simulation. The results are shown to agree with the predictions of linear stability analysis. Comparisons are made with linear perturbation growth rates and oscillation frequencies as a function of transverse disturbance wavelength. The perturbation eigenfunctions predicted by linear stability analysis are directly validated through numerical simulation. Three regimes of unstable behavior -linear, weakly nonlinear, and fully nonlinear -are explored and characterized in terms of the power spectrum of the normal shock velocity for a Chapman-Jouguet detonation with weak heat release. IntroductionNumerical simulations of detonation have been conducted by many researchers over the years. The earliest simulations of one-dimensional pulsating detonation were performed by Fickett and Wood in 1966 [1]. In 1978, Taki and Fujiwara [2] performed the first two-dimensional simulations of cellular detonation. As computational capacity has grown, simulations have been made that incorporate higher grid resolution and more detailed models. Despite the significant advances made, however, many fundamental problems remain to be investigated further. In particular, the physical nature of the detonation instability, nonlinear growth of the instability, physical mechanisms of cell formation and the nature of cell dynamics demand further elucidation theoretically as well as by means of accurate numerical simulation. Our present paper is concerned primarily with the latter.Recently, methods of simulating shocks and detonations in a reference frame attached to the shock surface have been developed [3,4]. Rather than attempting to capture the evolving shock front numerically, as in conventional simulations, these shock-attached frame methods map the shock surface to a fixed location in the computational domain and solve for the shock state as part of the numerical solution. These methods allow a sharp representation of the lead shock, avoid the numerical smearing inherent in shock capturing methods, and can be much more accurate than conventional methods of simulation. These methods are ideal for the study of detonation stability since they are posed in the same frame

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Section: Introductionmentioning

confidence: 99%

Mode selection in weakly unstable two-dimensional detonations

Taylor

Kasimov

Stewart

2009

Combustion Theory and Modelling

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“…Complex time-dependent and multi-dimensional structures tend to develop [14,26]. Numerical simulations of the equations of reactive gas dynamics are able to reproduce at a qualitative level the complex structures observed in experiments (see, e.g., [39,1,30]). However, obtaining physical insights into the basic mechanisms of the instability requires simplified modeling and remains challenging.…”

Section: Introductionmentioning

confidence: 99%

Study of a Model Equation in Detonation Theory

Faria¹,

Kasimov²,

Rosales³

2014

SIAM J. Appl. Math.

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Here we analyze properties of an equation that we previously proposed to model the dynamics of unstable detonation waves [A. R. Kasimov, L. M. Faria, and R. R. Rosales, Model for shock wave chaos, Phys. Rev. Lett., 110 (2013), 104104]. The equation is utIt describes a detonation shock at x = 0 with the reaction zone in x < 0. We investigate the nature of the steady-state solutions of this nonlocal hyperbolic balance law, the linear stability of these solutions, and the nonlinear dynamics. We establish the existence of instability followed by a cascade of period-doubling bifurcations leading to chaos. Introduction.A detonation is a shock wave that propagates in a reactive medium where exothermic chemical reactions are ignited as a result of the heating by the shock compression. The energy released in these reactions, in turn, feeds back to the shock in the form of compression waves and thus sustains the shock motion. The dynamics of such shock-reaction coupling is highly nonlinear due to the sensitivity of the chemical reactions to temperature, making the problem significantly more challenging than shock dynamics in nonreactive media. A steady planar detonation wave is rarely observed in experiments. Complex time-dependent and multidimensional structures tend to develop [14,26]. Numerical simulations of the equations of reactive gas dynamics are able to reproduce at a qualitative level the complex structures observed in experiments (see, e.g., [40,1,30]). However, obtaining physical insights into the basic mechanisms of the instability requires simplified modeling and remains challenging.In one dimension, the instabilities of the reactive shock wave manifest themselves in the form of a "galloping detonation" [15,14], wherein the shock speed oscillates around its steady value. It has been shown through extensive numerical experiments that as the activation energy, E, a parameter in the equations measuring the temperature sensitivity of the chemical reactions, is varied, the shock speed transitions from a constant to an oscillatory function. Further increase of E leads to a period-doubling bifurcation cascade, which ultimately results in the shock moving at a chaotic speed [29,20]. The mechanism for such instabilities is still not completely understood.In this paper, we show that the model introduced in [23], which consists of a single nonlocal partial differential equation (PDE), is capable of reproducing the complexity observed in one-dimensional simulations of reactive Euler equations. The model possesses traveling wave solutions precisely analogous to the ZND theory (named after Zel'dovich [44], von Neumann [41], and Döring [6], who independently developed the

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“…Usually (see e.g. [17] ; [24] ), the relation (13) is used to model equation an accumulation of active radicals in a particle during a chemical reaction. At the initial time, ω = 0, and the induction period stops at ω = 1.…”

Section: Modelmentioning

confidence: 99%

Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov׳s chemical source term

Ibragimov

Galiakberova

2016

International Journal of Non-Linear Mechanics

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Cite this article as: R.N. Ibragimov, N.H. Ibragimov and L.R. Galiakberova, Conservation laws and invariant solutions of the nonlinear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov's chemical source term, International Journal of Non-Linear Mechanics, http://dx. AbstractThe nonlinear governing gas dynamics equations that are used as a descriptor of a rotating detonation engine are investigated from the group theoretical standpoint. The equations incorporate approximation of the Korobeinkov's chemical reaction model that are used to describe the two-dimensional detonation field on a surface of a two-dimensional cylindrical chamber without thickness. The transformations that leave the equations invariant are found. On the basis of these transformations, the conservation equations were constructed and the invariant solutions were obtained for specific form of the equation of state, for which the equations are nonlinearly self-adjoint. The invariant solutions are given in terms of the functions that satisfy nonlinear ordinary differential equations. The above reduction simplifies the analysis of the original nonlinear system of partial differential equations on a surface of rotating cylinder.

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Numerical analysis of a two-dimensional nonsteady detonations

Cited by 9 publications

References 2 publications

Mode selection in weakly unstable two-dimensional detonations

Mode selection in weakly unstable two-dimensional detonations

Study of a Model Equation in Detonation Theory

Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov׳s chemical source term

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