Mark Short

Cellular detonation stability. Part 1. A normal-mode linear analysis

A detailed investigation of the hydrodynamic stability to transverse linear disturbances of a steady, one-dimensional detonation in an ideal gas undergoing an irreversible, unimolecular reaction with an Arrhenius rate constant is conducted via a normal-mode analysis. The method of solution is an iterative shooting technique which integrates between the detonation shock and the reaction equilibrium point. Variations in the disturbance growth rates and frequencies with transverse wavenumber, together with two-dimensional neutral stability curves and boundaries for all unstable low- and higher frequency modes, are obtained for varying detonation bifurcation parameters. These include the detonation overdrive, chemical heat release and reaction activation energy. Spatial perturbation eigenfunction behaviour and phase and group velocities are also obtained for selected sets of unstable modes. Results are presented for both Chapman–Jouguet and overdriven detonation velocities. Comparisons between the earlier pointwise determination of stability and interpolated neutral stability boundaries obtained by Erpenbeck are made. Possible physical mechanisms which govern the wavenumber selection underlying the initial onset of either regular or irregular cell patterns are also discussed.

Experimental observations of methane–oxygen diffusion flame structure in a sub-millimetre microburner

We examine the structure of confined, laminar methane–oxygen diffusion flames in an alumina microburner with a sub-millimetre dimension. To minimize termination of gas-phase combustion via surface radical quenching, the reactor walls are chemically treated and annealed. We show, through chemiluminescent images, that gas-phase methane–oxygen diffusion flames exist in the microburner without the need for catalytic reaction. However, their structure differs from the continuous laminar diffusion flame profiles that we would expect in a similar burner configuration on a macroscopic scale. Instead, we observe a sequence of isolated reaction zones structures (flame cells) that form along the length of the microburner combustion channel aligned in the direction of the gas flow. This form of cellular diffusion flame instability appears to be unique to wall-confined combustion in microscale devices. The number of flame cells observed depends on the inlet gas velocities and initial mixture strengths.

The generation of axial vorticity in solid-propellant rocket-motor flows

We examine small deviations from axial symmetry in a solid-propellant rocket motor, and describe a ‘bath-tub-vortex’ effect, in which substantial axial vorticity is generated in a neighbourhood of the chamber centreline. The unperturbed flow field is essentially inviscid at modest Reynolds numbers, even at the chamber walls, as has long been known, but the inviscid perturbed flow is singular at the centreline, and viscous terms are required to regularize it. We examine perturbations sufficiently small that a linear analysis is valid everywhere (e Re small, where e is a measure of the perturbation amplitude and Re is a Reynolds number), and larger perturbations in which a nonlinear patch is created near the centreline of radius O (√e). Our results provide an explanation of swirl experimentally observed by others, and a cautionary note for those concerned with numerical simulations of these flows, whether laminar or turbulent.

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.

Cellular instabilities, sublimit structures and edge-flames in premixed counterflows

We examine twin premixed flames in a plane counterflow and uncover, in the parameter space, a hitherto unknown domain of cellular instability. This leads us to hypothesize that for small Lewis numbers a two-dimensional (2D) steady solution branch bifurcates from the one-dimensional (1D) solution branch at a neutral stability point located near the strain-induced quenching point. Solutions on this 2D branch are constructed indirectly by solving an initial-value problem in the edge-flame context defined by the multiple-valued bistable 1D solution. Three kinds of solution are found: a periodic array of flame-strings, a single isolated flame-string and a pair of interacting flame-strings. These structures can exist for values of strain greater than the 1D quenching value, corresponding to sublimit solutions.

The multi-dimensional stability of weak-heat-release detonations

The stability of an overdriven planar detonation wave is examined for a one-step Arrhenius reaction model with an order-one post-shock temperature-scaled activation energy 0 in the limit of a small post-shock temperature-scaled heat release β. The ratio of specific heats, y, is taken such that (γ - 1) = O(1 Under these assumptions, which cover a wide range of realistic physical situations, the steady detonation structure can be evaluated explicitly, with the reactant mass fraction described by an exponentially decaying function. The analytical representation of the steady, structure allows a normal-mode description of the stability behaviour to be obtained via a two-term asymptotic expansion in β. The resulting dispersion relation predicts that for a finite overdrive f, the detonation is always stable to two-dimensional disturbances

The chemical-gas dynamic mechanisms of pulsating detonation wave instability

The chemical–gas dynamic mechanisms behind the instability and failure of a one–dimensional pulsating detonation wave driven by a three–step chain–branching reaction are revealed by direct numerical simulation. Two types of pulsating instability observed experimentally are explained. The first involves regular oscillations of the detonation front, where the instability is driven by low–frequency finite–amplitude compression and expansion waves in the chain–branching induction zone between the main reaction layer and the detonation shock. For irregular oscillations of the front, the instability mechanism first involves a decoupling between the shock and main reaction layer. Subsequently, the main reaction layer accelerates, drives a compression wave ahead of it, and undergoes a transition to detonation. This internal detonation wave overtakes the lead detonation shock, generating a new high–pressure detonation, which rapidly decays. A smaller–amplitude pressure oscillation occurs during the decay with a mechanism reminiscent of that observed for the previous regular oscillation, before the detonation and main reaction layer once again decouple and the instability cycle is repeated. For failure scenarios, the shock temperature is observed to drop to the cross–over temperature for the chain–branching reaction, causing the main reaction layer to decouple and retreat indefinitely from the detonation shock.

A nonlinear evolution equation for pulsating Chapman-Jouguet detonations with chain-branching kinetics

A nonlinear evolution equation for pulsating Chapman-Jouguet detonations with chain-branching kinetics is derived. We consider a model reaction system having two components: a thermally neutral chain-branching induction zone governed by an Arrhenius reaction, terminating at a location where conversion of fuel into chain radical occurs; and a longer exothermic main reaction layer or chain-recombination zone having a temperature-independent reaction rate. The evolution equation is derived under the assumptions of a large activation energy in the induction zone and a slow evolution time based on the particle transit time through the induction zone, and is autonomous and second-order in time in the shock velocity perturbation. It describes both stable and unstable solutions, the latter leading to stable periodic limit cycles, as the ratio of the length of the chain-recombination zone to chain-induction zone, the exothermicity of reaction, and the specific heats ratio are varied

Stability of detonations for an idealized condensed-phase model

The stability of travelling wave Chapman-Jouguet and moderately overdriven detonations of Zeldovich-von Neumann-Dtype is formulated for a general system that incorporates the idealized gas and condensed-phase (liquid or solid) detonation models. The general model consists of a two-component mixture with a one-step irreversible reaction between reactant and product. The reaction rate has both temperature and pressure sensitivities and has a variable reaction order. The idealized condensed-phase model assumes a pressure-sensitive reaction rate, a constant-γ caloric equation of state for an ideal fluid, with the isentropic derivative γ =3 , and invokes the strong shock limit. A linear stability analysis of the steady, planar, ZND detonation wave for the general model is conducted using a normal- mode approach. An asymptotic analysis of the eigenmode structure at the end of the reaction zone is conducted, and spatial boundedness (closure) conditions formally derived, whose precise form depends on the magnitude of the detonation overdrive and reaction order. A scaling analysis of the transonic flow region for Chapman- Jouguet detonations is also studied to illustrate the validity of the linearization for Chapman-Jouguet detonations. Neutral stability boundaries are calculated for the idealized condensed-phase model for one- and two-dimensional perturbations. Comparisons of the growth rates and frequencies predicted by the normal-mode analysis for an unstable detonation are made with a numerical solution of the reactive Euler equations. The numerical calculations are conducted using a new, high-order algorithm that employs a shock-fitting strategy, an approach that has significant advantages over standard shock-capturing methods for calculating unstable detonations. For the idealized condensed-phase model, nonlinear numerical solutions are also obtained to study the long-time behaviour of one- and two-dimensional unstable Chapman-Jouguet ZND waves.

Edge-flame structure and oscillations for unit Lewis numbers in a non-premixed counterflow

We examine the structure and oscillatory instability of low Peclet number, non-premixed edge-flames in a fixed rectangular channel, closed at one end, with constant side-wall mass injection, one surface supplying fuel, the other oxidizer. Both reactant components have unit Lewis numbers. This study is motivated by issues regarding the nature of combustion that may occur in a propellant crack formed at the interface between the fuel-binder and oxidizer in a heterogeneous propellant. Flux conditions are imposed on the fuel and oxidizer at the injection walls, while the temperature on the boundary walls is held constant. For situations in which steady burning occurs, the flame has two components: an edge that faces towards the closed end of the channel and a trailing one-dimensional diffusion flame. A large Damköhler number study of the trailing, planar, strained diffusion flame structure is conducted and a new solvability condition uncovered in this limit, whereby the flame may not exist if the supply mixture strength is sufficiently far from stoichiometric. Numerical calculations also reveal that axial oscillations of the edge-flame in the channel may occur, but for a finite range of mixture strengths sufficiently far from stoichiometric values. The importance of mixture strength, heat losses to the relevant injection surface and the channel end-wall, and the role of the injection surface reactant flux conditions in inducing the oscillations are emphasized. Finally we explore the effect on the combustion structure of varying Peclet number and of different injection velocity magnitudes on the side-wall surfaces.