ALBERT

Description: An analytical theory is developed for the stability properties of planar fronts of premixed laminar flames freely propagating downwards in a uniform reacting mixture. The coupling between the hydrodynamics and the diffusion process is described for an arbitrary expansion of the gas across the flame. Viscous effects are included with an arbitrary Prandtl number. The flame structure is described for a large value of the reduced activation energy and for a Lewis number close to unity. The flame thickness is assumed to be small compared with the wavelength of the wrinkles of the front, this wavelength being also the characteristic lengthscale of the perturbations of the flow field outside the flame. A two-scale method is then used to solve the problem. The results show that the acceleration of gravity associated with the diffusion mechanisms inside the front can counterbalance the hydrodynamical instability when the laminar-flame velocity is low enough. The theory provides predictions concerning the instability threshold. In particular, the dimensions of the cells are predicted to be large compared with the flame thickness, and thus the basic assumption of the theory is verified. Furthermore, the quantitative predictions are in good agreement with the existing experimental data. © 1982, Cambridge University Press. All rights reserved.

Description: A new criterion for the direct initiation of cylindrical or spherical detonations by a localized energy source is presented. The analysis is based on nonlinear curvature effects on the detonation structure. These effects are first studied in a quasi-steady-state approximation valid for a characteristic timescale of evolution much larger than the reaction timescale. Analytical results for the square-wave model and numerical results for an Arrhenius law of the quasi-steady equations exhibit two branches of solutions with a C-shaped curve and a critical radius below which generalized Chapman–Jouguet (CJ) solutions cannot exist. For a sufficiently large activation energy this critical radius is much larger than the thickness of the planar CJ detonation front (typically 300 times larger at ordinary conditions) which is the only intrinsic lengthscale in the problem. Then, the initiation of gaseous detonations by an ideal point energy source is investigated in cylindrical and spherical geometries for a one-step irreversible reaction. Direct numerical simulations show that the upper branch of quasi-steady solutions acts as an attractor of the unsteady blast waves originating from the energy source. The critical source energy, which is associated with the critical point of the quasi-steady solutions, corresponds approximately to the boundary of the basin of attraction. For initiation energy smaller than the critical value, the detonation initiation fails, the strong detonation which is initially formed decays to a weak shock wave. A successful initiation of the detonation requires a larger energy source. Transient phenomena which are associated with the intrinsic instability of the quasi-steady detonations branch develop in the induction timescale and may induce additional mechanisms close to the critical condition. In conditions of stable or weakly unstable planar detonations, these unsteady phenomena are important only in the vicinity of the critical conditions. The criterion of initiation derived in this paper works to a good approximation and exhibits the huge numerical factor, 106–108, which has been experimentally observed in the critical value of the initiation energy. © 1994, Cambridge University Press. All rights reserved.

Description: 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 piston-supported 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 up by considering two different temperature sensitivities of the overall reaction rate: one governing the induction length, another one the thickness of the exothermic zone. A nonlinear integral equation for the longitudinal dynamics of overdriven detonations is obtained as an asymptotic solution of the reactive Euler equations. The analysis uses a distinguished limit based on an infinitely large temperature sensitivity of the induction kinetics and a small difference of specific heats. Comparisons with numerical calculations show a satisfactory agreement even outside the limits of validity of the asymptotic solution.

Description: This paper is concerned with the coupling mechanisms leading to the spontaneous generation of sound during flame propagation in a tube open at one end. We consider The cases of premixed gaseous combustion and of premixed spray combustion of decane droplets in air. The flame front propagates from the open to the closed end of a tube and, for a particular position, starts to amplify a longitudinal acoustic mode of the tube. We call this mode the primary acoustic instability and focus our study on the physical mechanisms responsible for sound amplification. Measured amplification rates are compared to calculated values. In the gaseous case, it is shown that the instability results from a coupling between the acoustic acceleration field and the geometry of the flame front separating the burnt gases from the denser unburnt mixture. The situation is quite different in the spray case. The primary acoustic instability is much stronger and results from a modification of the inner structure of the flame. This modification arises from the velocity lag of the droplets in the acoustic velocity field, leading to a modulation of the fuel flux at the flame.

Description: The spectrum of linear modes governing the multidimensional instabilities of gaseous detonations is revisited by combining a numerical analysis with new analytical results. In view of recent developments in nonlinear analyses for describing the cellular structure of weakly unstable detonation fronts, particular attention is paid to the neighbourhood of the instability threshold. A first objective is to check the validity domain of the analytical results and to investigate to what extent they are useful when approaching the self-sustained regime (Chapman-Jouguet conditions). A second objective is to study how the multidimensional instabilities are influenced by multiple-step chemistry. The roles of the induction period and of the stiffness of the exothermic runaway will be investigated separately.

Description: An analytical study of the asymptotic behaviour of descending spikes is carried out for the idealized limit of an inviscid, incompressible fluid without surface tension, bounded by a vacuum. A self-similar solution is obtained for the shape of the free surface at the spike tip, yielding the evolution in time of the surface curvature there. The approach to free-fall acceleration is shown to follow an inverse power law in time. Results are given for both planar (two-dimensional) and axisymmetric spikes. Potential areas of application include ablation-front dynamics in inertial-confinement fusion. © 2005 Cambridge University Press.

Description: A complete analysis of the one-dimensional vibratory instability of planar flames of premixed gases propagating in tubes is provided. The driving mechanism results from unsteady coupling between flame structure and acoustic waves through temperature fluctuations. In certain conditions, the strength of such an instability will be proved to be sufficiently strong to produce large-amplitude fluctuations as soon as the flame has travelled a distance of the order of the acoustic wavelength. Stability limits and total amplification of an initial perturbation are computed in the framework of the simple flame mode of a one-step exothermic reaction governed by an Arrhenius law with an activation energy much larger than the thermal energy. Diffusive and thermal effects within the flame are included with a Lewis number different from unity. Damping mechanisms associated with viscous and thermal dissipation at the walls, as well as with loss of acoustic energy by sound radiation from the open end of the tube, are retained. In ordinary conditions, for a reactive mixture with an effective Lewis number close to unity, the predicted instability is weak. In the framework of the simplified flame model used here, islands of strong instabilities are predicted to occur at low Mach numbers for Lewis numbers larger than unity. © 1990, Cambridge University Press. All rights reserved.