Browsing by Author "Clarke, J. F."
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Item Open Access CFD modelling of solid propellant ignition(Cranfield University, 1996-07) Lowe, C.; Clarke, J. F.; Toro, E. F.Solid propellant is the highly energetic fuel burnt in the combustion chamber of ballistic weapons. It is manufactured, for this purpose, in either granular or stick form. Internal ballistics describes the behavior within the combustion chamber throughout the ballistic cycle upto projectile exit from the muzzle of the gun barrel. Over the last twenty years this has been achieved by modelling the process using two-phase flow equations. The solid granules or sticks constitute the first phase, which can be assumed to be incompressible over typical pressure ranges within the chamber. The gas-phase is composed of both the original ambient gas contained around the propellant and additional gas produced by the propellant gasifying on heating. Equations can be derived that describe the conservation of mass, momentum and energy in terms of average flow variables. The equations are a highly non-linear system of partial-differential- equations. High-speed flow features are observed in internal ballistics and ordinary fini te- difference methods are unsuitable numerical methods due to inaccurate prediction of discontinuous flow features. Modern shock-capturing methods are employed, which solve the system of equations in conservation form, with the ability to capture shocks and contact discontinuities. However, although the numerical solutions compare well with experiment over the bulk of the combustion chamber, the ignition models used in internal ballistics are unreliable. These are based on either gas or solid-surface temperature achieving some empirically measured 'ignition temperature' after which the propellant burns according to an empirical pressure dependent burning law. Observations indicate that this is not an adequate representation of ignition. Time differences between first solid gasification and ignition imply two distinct processes occurring. ]Further, ignition occurring in gas-only regions indicates that ignition is controlled by a gas-phase reaction. This thesis develops simple ideas to describe possible mechanisms for these physical observations. The aim is to provide an improved model of the ignition of solid propellant. A two stage reaction process is described involving endothermic gasification of the solid, to produce a source of reactant gas, followed by a very exothermic gas-phase ignition reaction. Firstly the gas-phase ignition is considered. A very simple reaction is suggested which is assumed to control the combustion of reactant gas, produced by solid gasification. Ignition is, by definition, the initiation of this exothermic reaction. Chemical kinetics are included in the gas-phase flow equations to explore the evolution of the reactant gas that is subject to changes in temperature and pressure. By assuming spatial uniformity, analytical solutions of the problem are deduced. The physical interpretation of the solution is discussed, in particular, the relationship between temperature, reactant concentration and ignition is explored. Numerical methods are required to solve the one-dimensional flow equations. Development of suitable CFD methods provides a method of solution. Finite-volume schemes, based on the original work by Godunov, are used to solve the conservation form of the equations. A simple test problem is considered whereby reactant gas is injected into a cylindrical combustion chamber. By examining the resulting flow histories, valuable information is gathered about the complicated coupling of chemistry and flow. Chemistry is included into a system of two-phase flow equations. By using standard averaging methods along with an equation for gas-phase species, equations are derived that describe the rate of change of average flo%v variables for both gas and particle phases. Numerical schemes are developed and some of the difficulties involved in two-phase flow systems, that are not an issue in single-phase flow, are presented. An internal ballistics application is considered as a test case and the solution discussed. The other important reaction involved in the combustion cycle, solid gasification, is explored. The model is based on detailed description of interphase mass and energy transfer at the solid-gas interface. This involves the solution of the heat conduction equation with a moving boundary that divides the solid and gas regions. Similar numerical schemes are constructed to solve the equations. Finally, this model is coupled with the equations of gas-phase reaction. This describes the complete cycle whereby increases in gas temperature cause the solid to increase in temperature and gasify. Subsequent gas-phase combustion of the reactant gases produces heat-transfer between the solid and gas and continues to accelerate gasification. Eventually this results in selfsustained combustion of the solid propellant.Item Open Access Compressible Flow Produced by Distributed Sources of Mass: An Exact Solution(Aerodynamics, Cranfield Institute of Technology, 1987-07) Clarke, J. F.The paper considers the case of a one-dimensional isentropic unsteady compressible flow that is driven entirely by a distribution of sources in the left-hand half space of an unbounded domain. The right-hand half-space contains no sources, so that source-strength drops discontinuously to zero as one crosses from left to right-hand space. Exact solutions are obtained for those parts of the flow that remain isentropic.Item Open Access An elementary study of gas injection and sublimation into a simple shear layer(College of Aeronautics, 1960-02) Clarke, J. F.Item Open Access The flow of chemically reacting gas mixtures(College of Aeronautics, 1958-11) Clarke, J. F.Suitable forms of the equations for the flow of an inviscid, non-heat- conducting gas in which chemical reactions are occurring are derived, The effects of mass diffusion and non-equilibrium amongst the internal modes of the molecules are neglected. Special attention is given to the speeds of sound in such a gas mixture and a general expression for the ratio of frozen to equilibrium sound speeds is deduced. An example is given for the ideal dissociating gas. The significance of the velocity defined by the ratio of the convective derivatives of pressure and density is explained. It is the velocity which exists at the throat of a convergent-divergent duct under maximum mass flow conditions, and it is shown that this velocity depends on the nozzle geometry as well as on the 'reservoir' conditions. As an illustration the phenomena of sound absorption and dispersion are discussed for the ideal dissociating gas. The results can be concisely expressed in terms of the frozen and equilibrium sound speeds, the frequency of the (harmonic) sound vibration and a characteristic time for the rate of progress of the reaction.Item Open Access Heat conduction through a gas with one inert internal mode(College of Aeronautics, 1960-05) Clarke, J. F.The rate of energy transfer between parallel flat plates is evaluated when the (stagnant) gas between them is polyatomic with one inert internal mode. Deviations of the thermal conductivity from the complete equilibrium (Eucken) value are expressed in terms of the inert mode relaxation time and the effectiveness of the walls in exciting or de-exciting this mode. The results are obtained via a linear theory consistent with small temperature differences between the plates. It is found that the Eucken-value of conductivity could be exceeded if the relaxation times are non-zero and the plates very effective in exciting the inert mode. 1Nhen relaxation times are very short the effect of the walls on the energy transfer rate is small, but the malls make their presence felt by distorting the temperature profiles in "boundary layers" adjacent to the walls which are of order VDT in thickness (D = diffusion coefficient, r = relaxation time). This result is analogous to Hirschfelderts (1956) for the case of chemical reactions. For experimental measurement of conductivity in a hot wire cell type of apparatus it is shown that extrapolation of measured reciprocal conductivities to zero reciprocal pressure should lead to the full Eucken value. It is also shown that the slope of reciprocal apparent (measured) conductivity versus reciprocal pressure curves is a function of relaxation time as well as of the accommodation coefficients. It is quite possible that the relaxation effect here is comparable with the temperature jump effects, even for rotation in diatomic molecules.Item Open Access Natural co-ordinates and high speed flows, a numerical method for reactive gases(1992) Dawes, A. S.; Clarke, J. F.A new high resolution, space marching, numerical method, based on operator splitting, for high speed, steady, planar two dimensional, Inviscid, chemically reacting (real gas) flow fields as described. A simple model for chemical non-equilibrium was used based on Lighthill's ideal dissociating gas model for diatomic Oxygen. The method is formulated within the natural co-ordinate space consisting of streamlines and their normals. Boundaries are treated exactly and the variation of the chemistry shown to be along the streamline co-ordinate. Operator splitting is used to split the full problem up into two consisting of the wave and chemical contributions. We use the Riemann problem based Random Choice Method (RCM) for solving the homogeneous Euler equations with frozen initial data for the wave contribution. The definition of the Riemann problem for the natural co-ordinate space is described. The RCM does not suffer from numerical smearing or spurious oscillations like other methods and this will be a positive advantage for chemical non-equilibrium due to the sudden rise in temperature across a shock wave. The flow field changes due to the chemical non-equilibrium are included by solving numerically a system of stiff ordinary differential equations. This was done using the stiff solver LSODE. The algorithm is first validated for inert flow fields. This is done by making comparisons between theoretical and experimental results for several classic high speed configurations. The method is then validated for chemical non-equilibrium by making comparisons with results from other numerical methods. Finally, the qualitative effect of these real gas effects on the aerodynamic characteristics of s simple re-entry profile (modelling the Space Shuttle) was investigated.Item Open Access Numerical computation of two-dimensional unsteady detonation waves in high energy solids(1990-01-01T00:00:00Z) Clarke, J. F.; Karni, S.; Quirk, J. J.Item Open Access On a problem involving heat conduction through a polyatomic gas(College of Aeronautics, 1961-05) Clarke, J. F.A heat conduction problem is set up which, in essence, simulates the conditions arising when a plane shock wave reflects from a co-planar solid boundary. The gas is assumed to be polyatomic, with one 'significantly relaxing' internal energy mode. The quantity of primary interest is the temperature of the solid at the interface, since this can be observed experimentally without much difficulty. Solutions are obtained for this quantity which cover a range of practically plausible relaxation times and 'wall effect' parameters. It is essential to include proper temperature jump boundary conditions for both active and relaxing (or inert) energy modes. Thus it is necessary to know accommodation coefficients for these modes of energy storage. The temperature jump effects are found to dominate the (interface) solid's temperature/ time history, with relaxation effects playing a very secondary role. The theoretical results are compared with some experimental observations and encouraging agreement is found. As a result of this agreement it proves possible to estimate the accommodation coefficient for the active modes (in this case for the combination platinum/air), the pressure being about 15 atmospheres. The pressure sensitivity of accommodation effects is commented on.Item Open Access Reaction-resisted shock fronts(College of Aeronautics, 1961-05) Clarke, J. F.It is shown that shock waves whose structure is determined solely by the effects of chemical reactions (reaction-resisted shock fronts) are possible and completely analogous to relaxation - resisted waves. A single dissociation reaction is considered and numerical results indicate that such waves could be observed experimentally. Bulk viscosities equivalent to reaction effects are possibly 10[to the power of 2] or more times shear viscosity values. (Examples are based on Lighthill's ideal dissociating gas).Item Open Access Relaxation effects on the flow over slender bodies(College of Aeronautics, 1961-03) Clarke, J. F.The effects of heat capacity lag on the flow over slender bodies are examined via an extension of Ward's (1949) generalised 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. 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 in 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 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.Item Open Access Studies in hypersonic viscous interactions(1972-12) Murthy, A. V.; Clarke, J. F.The literature on the leading edge viscous interaction phenomena has "been reviewed. Theoretical studies of the leading edge bluntness and the boundary - layer displace¬ment effects have been made by using a combination of tangent wedge + centrifugal force pressure law relation and also using the 2nd order tangent wedge formula suggested by Lees . These analyses, when applied to concave surfaces show a mild oscillatory behaviour compared to the highly oscillatory results of Cheng's theory. Further, an integral approach due to Chernyi has been applied to concave surface flows and the damping of the oscillations present in Cheng’s theory is demonstrated. Experiments have been conducted at a Mach number Of 1 2 .2 in a gun tunnel, on flatplates and on a concave surface with a sharp leading edge. The results are compared with the predictions of various interaction theories.Item Open Access Theoretical analysis of Gas Dynamic disturbances in an explosive atmosphere(Cranfield University, 1984-02) Ashdown, P. D.; Clarke, J. F.Various problems, which examine the propagation of gas dynamic disturbances, through an explosive atmosphere, are considered. The first set studies a model relaxing gas, and asymptotic methods are employed. A high frequency expansion is used to investigate piston oscillations in an infinite half space. The first two terms in the velocity perturbation are found in the acoustic case. The amplitude and frequency change on a wavelet are given; the wave number alters from wavelet to wavelet. For an enclosed volume the multi-time method is employed. When a standing wave exists in the vessel the frequency changes: when the vessel oscillates the wave number changes. The situation when forced oscillations at a natural frequency of the container is discussed. Also finite amplitude oscillations in a vessel are considered by using the multi-time method. An integral equation for the amplitude growth is found. A numerical solution of outward wave propagation, in spherical and cylindrical coordinates until shock formation, is given. The second group of problems considers a multi-component gas which can be analysed numerically. The effect of the homogeneous explosion in amplifying or damping a weak ii discontinuity is simulated. Thus proposals for reaction schemes can be analysed. It is found there is a relation- ship between amplification/damping and strong/weak ignition, in a mixture of hydrogen and oxygen in a shock tube. The reactions liberating a significant amount of energy in the chemical reaction, are the reactions causing greatest amplification.Item Open Access Weak compression waves in relaxing gases(Cranfield University, 1972) Scott, E. M.; Clarke, J. F.; Busing, J. R.Studies have been made of the structure of weak compression (shock) waves in relaxing gases. These studies have been primarily concerned with the development of separate theoretical and experimental techniques. These techniques are not inter-related in general except in the overall context of Vibrational relaxation. The theoretical studies have been concerned with the influence on the structure of weak normal shock waves of translational non-equilibrium. bimodal relaxation and second order unimodal relaxation. The affect of translational nonequilibrium on the relaxation process has been studied by forming an asymptotic expansion in the ratio of the viscous length to the relaxation length. The perturbation scheme was singular and required the application at the method of matched asymptotic expansions. Bimodal relaxation has been studied by forming an asymptotic expansion in the ratio of the energy of the secondary vibrational mode to the total Vibrational energy. The addition of a second order term to the rate equation describing the behaviour of a single vibrational mode has also been studied by forming an asymptotic expansion. In this case. the perturbation parameter was the ratio of the two relaxation times concerned. The experimental studies have been concerned with the production and study or weak normal shock waves in the Cranfield Institute or Technology 2" shock tube. A time resolved quantitative schlieren system has been used for the study of the weak normal shock waves. This particular system had been developed previously for this purpose. and further developments and refinements have been made to it. Experimental studies have been made with the schlieren system of the structure or strong incident shock waves in carbon dioxide. The vibrational relaxation time of carbon dioxide determined in this way for translational temperatures from 300 o K to 1200 o K has been found to be in reasonable agreement with measurements made elsewhere. A technique has been developed for the production of weak incident shock waves in t.i.1e shock tube, which involved the positioning of a perforated plate in the channel of the shock tube. The vibrational relaxation time of carbon dioxide determined in this way for translational o temperatures of approximately 300 K has been found to be in good agreement with measurements made elsewhere. Good agreement has also been obtained between the experimentally measured density gradient profiles and theoretical profiles. The curvature of the shock waves obscured the transition from a fully dispersed to a partly dispersed shook wave. Further improvements and refinements have been made to a technique which had been developed previously for the propagation of weak normal shock waves through the reflected shock region of the shock tube flow. This technique was used to study the behaviour of fully dispersed shock waves at high temperatures. The relaxation time of carbon dioxide determined in this way for temperatures from 300 o K to 600o K has been found to be in good agreement with measurements made elsewhere.