J. F. Clarke

Numerical computation of two-dimensional unsteady detonation waves in high energy solids

We are concerned with theoretical modelling of unsteady, two-dimensional detonation waves in high energy solids. A mathematical model and a numerical method to solve the associated hyperbolic system of equations are presented. The model consists of the Euler equations augmented bby extra conservation laws and source terms to accoxint for chemical reaction and tracking of materials. Both the thermodynamics and the chemistry are treated in a simple way. Using a detonation analogue due to Fickett, we test several numerical methods and assess their performance in modelling the essential features of detonation waves. The numerical method selected for the full model is an extension of the conservative, shock capturing technique of Roe, together with an adaptive mesh refinement procedure that allows the resolution of fine features such as reaction zones. Results for some typical tests problems are presented. Starting in 1946 as the College of Aeronautics, the Cranfield Institute of Technology was granted university status in 1969. In 1993 it changed its name to Cranfield University.

The structure of a steady high-speed deflagration with a finite origin

A theoretical study is made of the structure of a steady planar deflagration downstream of a specific origin location from which a compressible reactive gas flow emanates. The chemistry is modelled by a high-activation-energy Arrhenius reactionrate law without the introduction of an ignition temperature. Chemically derived heat addition is significant relative to the initial thermal energy of the flow. Perturbation methods, based on the limit of high activation energy, are used to construct solutions for sub- and supersonic values of the Mach number [ ] at the origin. With the exception of a thin layer adjacent to the origin in which very small changes occur, the structure of the deflagration is determined by a fundamental balance of convection, reaction and compressibility effects. Transport processes have an insignificant effect on the energetics of the flow. The upstream portion of the deflagration is dominated by an ignition event reminiscent of the induction period of an adiabatic thermal explosion. Subsequently in the neighbourhood of a well-defined ignition delay (or explosion) location a very rapid reaction takes place with order-unity changes in all the dependent variables. Compressibility effects are shown to be the source of basic limitations on the maximum temperature rise permitted in a flow with a particular value of [ ]. Chapman–Jouguet deflagrations are found to appear when the chemical heat addition is maximized for a given [ ]. Subsonic combustion is shown to exist for fairly general initial conditions at the origin. In contrast, a purely supersonic reaction is found to be possible only for specifically defined values of the initial strain rate and temperature gradient which would be difficult to control in the experimental environment.

Chemical amplification at the wave head of a finite amplitude gasdynamic disturbance

Consider a background state which consists of a spatially uniform chemically reactive mixture in a general state of disequilibrium. The analytical method of characteristics is used to show that a plane finite amplitude disturbance propagates through this system at the frozen sound speed and, if the degree of disequilibrium is sufficient, is amplified by the chemical reaction. Some comments are made about the time to shock-wave formation and its relation to the homogeneous explosion ignition time, and also about expansion waves, which are found to have a tendency towards fixed-strength ‘quenching waves’, their strength being proportional to the extent of the ambient disequilibrium.

Relaxation effects on the flow over slender bodies

The effects of heat-capacity lag on the flow over slender bodies are examined by means of an extension of Wards (1949) generalized treatment of the slender-body problem. The results are valid for smooth bodies of arbitrary cross-sectional shape and attitude in the complete Mach number range up to, but not including, hypersonic conditions. Transonic flow can be treated owing to the presence of a dissipative mechanism in the basic differential equation, but the results in this Mach number range are probably of limited practical value. The results show that cross-wind forces are unaffected to a first approximation, but that drag forces comparable with laminar skin-friction values can arise as a result of the relaxation of the internal degrees of freedom. The magnitude and sign of these effects depend strongly on body shape and free-stream Mach number. Results are given for the surface pressure coefficient, and the variations of translational and internal mode temperature on and near the body are also found. The influence of these latter effects on heat transfer to the body is discussed.

Combustion theory: a report on Euromech 203

The 203rd Euromech Colloquium, on developments in the theoretical modelling of homogeneous and heterogeneous combustion was held at Cranfield Institute of Technology from 2 to 4 December 1985. The emphasis of the meeting was on the ability of analytic, numerical and approximate methods to predict, or interpret, events which occur in the laboratory or in the field. There were forty-five participants in the Colloquium from six different countries.

Numerical studies of a detonation analogue

Abstract As an aid to deciding a computational strategy for problems involving strong detonation waves, we have applied a variety of numerical techniques to a mathematical problem devised by Fickett, which exhibits many of the essential computational difficulties and possesses analytical solution corresponding to overdriven and underdriven reacting flows. Classical shock capturing methods such as those of MacCormack and Godunov, do not produce acceptable solutions. Better solutions can be provided by Flux Difference Methods (Roes method) and by Random Choice Methods (still in one dimension), especially by using a new variant that yields second order accuracy. Adaptive gridding techniques for these methods are currently being investigated with encouraging preliminary results.

Some gas flows which obey Charles’ Law

The effects of substantial temperature and hence density changes on a low-speed ‘incompressible’ flow can be modelled by adopting Charles’ Law as one of the equations of state. It is found that planar radial inflows or outflows constitute a group of solutions which are self-similar for arbitrary temperature variations. When the temperature is written as a separable function of radius and polar angle, ordinary differential equations result. Permissible solutions include some with discontinuities in the temperature gradient across a radial line (streamline); this is a rough model of a diffusion flame and it is used to illustrate some features of a variable-density flow in a channel with radial walls in the presence of such a ‘flame’. Exact analytical solutions are given for the situation in which temperature increases linearly with radius; no boundary layers appear for either outflow or inflow. Approximate analytical solutions are presented for the case of a relatively rapid inflow with temperature independent of the radius; a velocity boundary layer exists at the walls and in the neigh bourhood of the ‘flame’, although the latter is of small velocity amplitude.