Modelling and solutions to the linear stability of a detonation wave in the kinetic frame

Abstract

The analysis of linear stability of a steady detonation wave is formulated for the first time at the kinetic level in the frame of the Boltzmann equation extended to reacting gases. Within this context and for a reversible reaction, the stability problem is carried out, in agreement with most classical papers on gas detonation, through a normal mode approach for the one-dimensional disturbances of the steady wave solution, and an acoustic radiation condition at the final equilibrium as closure condition. The proposed modelling leads to an initial value problem, constituted by the linearized reactive Euler equations in the perturbed shock frame with related Rankine-Hugoniot conditions, which can be solved by means of a proper numerical technique. An application is provided for an elementary bimolecular reaction.

On the dynamics and linear stability of one-dimensional steady detonation waves

“…Typically, when studying the propagation of steady detonation waves starting from a kinetic frame, an Eulerian regime with fast chemical reaction is assumed. This has been considered in [4] and [10], for example, where the assumed chemical regime is defined by the Maxwellian distributions of mechanical equilibrium and the elastic and reactive time scales are of the same order. The only non-equilibrium effect described in this regime is the relaxation of the mixture towards a chemical and thermal equilibrium state.…”

“…Closure condition. The required closure condition, which gives the dispersion relation for the normal modes (30), is the acoustics radiation condition adopted in many previous works on detonation stability as, for example, in [10,18,34,37,41,42,47]. Such a condition states that the inherent instability of the detonation wave solution results exclusively from the interplay between the leading shock and the reaction zone and cannot be affected by further disturbances travelling towards the shock from a great distance from the reaction zone.…”

“…Kinetic approaches to chemically reactive fluids and direct simulations of the Monte Carlo method have been widely used in the literature, with the aim of studying fluid dynamical applications in which the chemical reaction plays a crucial role [1][2][3][4][5][6][7][8][9][10][11][12]. Starting from a kinetic approach, a consistent macroscopic picture can be obtained in terms of the corrected balance equations that include the non-equilibrium effects due to the chemical reaction.…”

“…Concerning the detonation stability analysis in the context of the kinetic theory for chemically reacting gases, the hydrodynamic linear stability of steady detonation waves has been investigated for the first time in [10]. The emphasis of [10] is on the mathematical formulation and solutions to the stability problem in the kinetic frame, for a quaternary gas mixture with a reversible bimolecular chemical reaction. Some numerical results and visualizations are shown regarding the time evolution of the eigenfunctions for both instability and stability pictures as well as at conditions of neutral stability.…”

“…However, the considered kinetic modelling does not include the effects of the reaction heat, nor those of the activation energy, so that the stability picture remains incomplete. Starting from the kinetic formulation proposed in [10], the steady detonation solution characterized in [19] is revisited here, with the aim of investigating its linear stability. The main objective is to develop a detailed hydrodynamic stability analysis, and investigate the influence of the reaction heat and activation energy on the stability behaviour.…”