Gary J. Sharpe

Transverse waves in numerical simulations of cellular detonations

In this paper the structure of strong transverse waves in two-dimensional numerical simulations of cellular detonations is investigated. Resolution studies are performed and it is shown that much higher resolutions than those generally used are required to ensure that the flow and burning structures are well resolved. Resolutions of less than about 20 numerical points in the characteristic reaction length of the underlying steady detonation give very poor predictions of the shock configurations and burning, with the solution quickly worsening as the resolution drops. It is very difficult and dangerous to attempt to identify the physical structure, evolution and effect on the burning of the transverse waves using such under-resolved calculations. The process of transverse wave and triple point collision and reflection is then examined in a very high-resolution simulation. During the reflection, the slip line and interior triple point associated with the double Mach configuration of strong transverse waves become detached from the front and recede from it, producing a pocket of unburnt gas. The interaction of a forward facing jet of exploding gas with the emerging Mach stem produces a new double Mach configuration. The formation of this new Mach configuration is very similar to that of double Mach reflection of an inert shock wave reflecting from a wedge.

Numerical simulations of pulsating detonations: I. Nonlinear stability of steady detonations

Very-long-time numerical simulations of an idealized pulsating detonation with one irreversible reaction having an Arrhenius form are performed using a hierarchical adaptive second-order Godunov-type scheme. The initial data are given by the steady solution and the truncation error produces the perturbation to trigger the instability. The detonation is allowed to run for thousands of half-reaction times of the underlying steady wave to ensure that the final amplitudes and periods of the nonlinear oscillations are achieved. Thorough resolution studies are performed for various representative regimes of the instability. It is shown that to obtain quantitatively good solutions over 50 numerical grid points in the half-reaction length of the steady detonation are required, while to obtain a converged solution over 100 points are required, even near the stability boundary. This is much higher resolution than has generally been used in previous papers in either one or two dimensions. Resolutions of less than approximately 20 points per half-reaction length give very poor predictions of the periods and amplitudes near the stability boundary or entirely spurious solutions for more unstable detonations. The evolution of the converged solutions as the activation energy increases, and the detonation becomes more unstable, is also investigated.

Pulsating instability of detonations with a two-step chain-branching reaction model: theory and numerics

The nonlinear dynamics of Chapman–Jouguet pulsating detonations are studied both numerically and asymptotically for a two-step reaction model having separate induction and main heat release layers. For a sufficiently long main heat release layer, relative to the length of the induction zone, stable one-dimensional detonations are shown to be possible. As the extent of the main reaction layer is decreased, the detonation becomes unstable, illustrating a range of dynamical states including limit-cycle oscillations, period-doubled and four-period solutions. Keeping all other parameters fixed, it is also shown that detonations may be stabilized by increasing the reaction order in the main heat release layer. A comparison of these numerical results with a recently derived nonlinear evolution equation, obtained in the asymptotic limit of a long main reaction zone, is also conducted. In particular, the numerical solutions support the finding from the analytical analysis that a bifurcation boundary between stable and unstable detonations may be found when the ratio of the length of the main heat release layer to that of the induction zone layer is O(1/ε), where ε (≪1) is the inverse activation energy in the induction zone.

Linear stability of planar premixed flames: Reactive Navier–Stokes equations with finite activation energy and arbitrary Lewis number

A numerical shooting method for performing linear stability analyses of travelling waves is described and applied to the problem of freely propagating planar premixed flames. Previous linear stability analyses of premixed flames either employ high activation temperature asymptotics or have been performed numerically with finite activation temperature, but either for unit Lewis numbers (which ignores thermal-diffusive effects) or in the limit of small heat release (which ignores hydrodynamic effects). In this paper the full reactive Navier–Stokes equations are used with arbitrary values of the parameters (activation temperature, Lewis number, heat of reaction, Prandtl number), for which both thermal-diffusive and hydrodynamic effects on the instability, and their interactions, are taken into account. Comparisons are made with previous asymptotic and numerical results. For Lewis numbers very close to or above unity, for which hydrodynamic effects caused by thermal expansion are the dominant destablizing mechanism, it is shown that slowly varying flame analyses give qualitatively good but quantitatively poor predictions, and also that the stability is insensitive to the activation temperature. However, for Lewis numbers sufficiently below unity for which thermal-diffusive effects play a major role, the stability of the flame becomes very sensitive to the activation temperature. Indeed, unphysically high activation temperatures are required for the high activation temperature analysis to give quantitatively good predictions at such low Lewis numbers. It is also shown that state-insensitive viscosity has a small destabilizing effect on the cellular instability at low Lewis numbers.

Detonation ignition from a temperature gradient for a two-step chain-branching kinetics model

The evolution from a linear temperature gradient to a detonation is investigated for combustible materials whose chemistry is governed by chain-branching kinetics, using a combination of high-activation-temperature asymptotics and numerical simulations. A two-step chemical model is used, which captures the main properties of detonations in chain-branching fuels. The first step is a thermally neutral induction time, representing chain initiation and branching, which has a temperature-sensitive Arrhenius form of the reaction rate. At the end of the induction time is a transition point where the fuel is instantaneously converted into chain-radicals. The second step is the main exothermic reaction, representing chain termination, assumed to be temperature insensitive. Emphasis is on comparing and contrasting the results with previous studies that used simple one-step kinetics. It is shown that the largest temperature gradient for which a detonation can be successfully ignited depends on the heat release rate of the main reaction. The slower the heat release compared to the initial induction time, the shallower the gradient has to be for successful ignition. For example, when the rate of heat release is moderate or slow on the initial induction time scale, it was found that the path of the transition point marking the end of the induction stage should move supersonically, in which case its speed is determined only by the initial temperature gradient. For steeper gradients such that the transition point propagates subsonically from the outset, the rate of heat release must be very high for a detonation to be ignited. Detonation ignition for the two-step case apparently does not involve the formation of secondary shocks, unlike some cases when one-step kinetics is used.

The structure of planar and curved detonation waves with reversible reactions

The structure of both steady planar and slowly varying weakly curved detonations with one reversible reaction are investigated. For reactive systems with reversible reactions there are two distinguished sound speeds in the equilibrium fluid; the frozen sound speed and the equilibrium sound speed. According to the Chapman–Jouguet condition, self-sustaining, steady and planar detonation waves in such systems are equilibrium sonic at the end of the reaction zone. In this paper, it is shown that for any small, but nonzero, curvature of the front, the solution passes through a frozen sonic point where the thermicity simultaneously vanishes, the so-called generalized Chapman–Jouguet condition. Hence, the structure for the steady, planar wave, which is frozen subsonic throughout and is equilibrium sonic at the end of the reaction zone, is a singular limit of the structure of curved detonation waves as the curvature tends to zero. Since in any real detonation there will always be some curvature of the front, howe...

Shock-induced ignition for a two-step chain-branching kinetics model

The ignition of reactive materials by a shock wave, when the chemistry is governed by chain-branching kinetics, is investigated using a combination of high activation temperature asymptotics and numerical simulations. A two-step chemical model is used. The first step is a thermally neutral induction time, representing chain initiation and chain branching, which has a temperature-sensitive Arrhenius form of the reaction rate. At the end of the induction time is a transition point where the fuel is instantaneously converted into chain radicals. The second step is a temperature-insensitive exothermic reaction, representing chain recombination. It is found that the initiation process is qualitatively different from that for a temperature-sensitive one-step reaction considered previously. Three different cases, when the rate of heat release is slow, comparable and fast compared to the initial induction time, are considered. In each case ignition first occurs at the piston and the transition point (which marks the end of the induction zone and the start of the main heat release zone) initially propagates away from the piston at subsonic speeds, so that pressure and temperature disturbances from the exothermic region overtake the transition path and accelerate it. For rapid rates of heat release, a secondary shock is very promptly formed near the piston, which is subsequently amplified into a strong detonation propagating through the induction zone behind the leading shock. However, unlike for one-step kinetics the formation of the secondary shock does not involve quasisteady weak detonations. For moderate rates of heat release a secondary shock still eventually forms at the front of the disturbed region of the induction zone behind the leading shock, but a detonation is not formed until after the collision of the shocks. When the rate of heat release is slow, the transition point is continuously accelerated, but its speed remains subsonic until disturbances due to the heat release overtake the shock. No secondary shock forms for this case, completely unlike the case for one-step kinetics.

Numerical simulations of pulsating detonations: II. Piston initiated detonations

Extremely long time, high-resolution one-dimensional numerical simulations are performed in order to investigate the evolution of pulsating detonations initiated and driven by a constant velocity piston, or equivalently by shock reflection from a stationary wall. The results are compared and contrasted to previous simulations where the calculations are initiated by placing a steady detonation on the numerical grid. The motion of the piston eventually produces a highly overdriven detonation propagating into the quiescent fuel. The detonation subsequently decays in a quasi-steady manner towards the steady state corresponding to the given piston speed. For cases where the steady state is one-dimensionally unstable, the shock pressure begins to oscillate with a growing amplitude once the detonation speed drops below a stability boundary. However, the overdrive is still being degraded by a rarefaction which overtakes the front, but on a time-scale which is very long compared with both the reaction time and the period of oscillation. As the overdrive decreases, the detonation becomes more unstable as it propagates and the nature (e.g. period and amplitude) of the oscillations change with time. If the steady detonation is very unstable then the oscillations evolve in time from limit cycle to period doubled oscillations and finally to irregular oscillations. The ultimate nature of the oscillations asymptotically approaches that of the saturated nonlinear behaviour as found from calculations initiated by the steady state. However, the nonlinear stability of the steady detonation investigated in previous calculations represents only the very late time (O(105) characteristic reaction times) behaviour of the piston problem.