Showing papers in "Combustion Theory and Modelling in 1998"
TL;DR: In this article, the mean structure of turbulent bluff-body jets and flames is compared with predictions made using standard turbulence models, and three mixing layers are identified in the recirculation zone; an outer vortex close to the air coflow and an inner vortex between the outer vortex and the jet.
Abstract: The mean structure of turbulent bluff-body jets and flames is presented. Measurements of the flow and mixing fields are compared with predictions made using standard turbulence models. It is found that two vortices exist in the recirculation zone; an outer vortex close to the air coflow and an inner vortex between the outer vortex and the jet. The inner vortex is found to shift downstream with increasing jet momentum flux relative to the coflow momentum flux and gradually loses its circulation pattern. The momentum flux ratio of the jet to the coflow in isothermal flows is found to be the only scaling parameter for the flow field structure. Three mixing layers are identified in the recirculation zone. Numerical simulations using the standard k-ϵ and Reynolds stress turbulence models underpredict the length of the recirculation zone. A simple modification to the C1 constant in the dissipation transport equation fixes this deficiency and gives better predictions of the flow and mixing fields. The mixed-is-b...
215 citations
TL;DR: In this article, the influence of thermal diffusion on the structure of hydrogen-air and methane-air flames is investigated numerically using complex chemistry and detailed transport models, and the results show that thermal diffusion is important for an accurate prediction of flame structure.
Abstract: The influence of thermal diffusion on the structure of hydrogen-air and methane-air flames is investigated numerically using complex chemistry and detailed transport models. All the transport coefficients in the mixture, including thermal diffusion coefficients, are evaluated using new algorithms which provide, at moderate computational costs, accurate approximations derived rigorously from the kinetic theory of gases. Our numerical results show that thermal diffusion is important for an accurate prediction of flame structure. §E-mail address: ern@cermics.enpc.fr ∥E-mail address: giovangi@cmapx.polytechnique.fr
171 citations
TL;DR: In this article, the authors studied flame propagation in a strained mixing layer formed between a fuel stream and an oxidizer stream, which can have different initial temperatures, and found that the Lewis numbers affect the propagation of the triple flame in a way similar to that obtained in the studies of stretched premixed flames.
Abstract: We have studied flame propagation in a strained mixing layer formed between a fuel stream and an oxidizer stream, which can have different initial temperatures. Allowing
the Lewis numbers to deviate from unity, the problem is first formulated within the framework of a thermo-diffusive model and a single irreversible reaction. A compact formulation is then derived in the limit of large activation energy, and solved analytically for high values of the Damkohler number. Simple expressions describing the flame shape and its propagation velocity are obtained. In particular, it is found that the Lewis numbers affect the propagation of the triple flame in a way similar to that obtained in the studies of stretched premixed flames. For example, the flame curvature determined by the transverse enthalpy gradients in the frozen mixing layer leads to flame-front velocities which grow with decreasing values of the Lewis numbers.
The analytical results are complemented by a numerical study which focuses on preferentialdiffusion effects on triple flames. The results cover, for different values of the fuel Lewis number, a wide range of values of the Damkohler number leading to propagation speeds which vary from positive values down to large negative values.
107 citations
TL;DR: The role of wall quenching of radicals in ignition, extinction and autothermal behavior of premixed H2-air flames impinging on a flat surface was studied using numerical bifurcation techniques, with detailed gas-phase chemistry and surface radical recombination reactions.
Abstract: The role of wall quenching of radicals in ignition, extinction and autothermal behaviour of premixed H2–air flames impinging on a flat surface was studied using numerical bifurcation techniques, with detailed gas-phase chemistry and surface radical recombination reactions. Quenching out of radicals was found to retard the system at ignition due solely to the kinetics of the surface reactions. While kinetically extinction is also retarded, the thermal feedback from the wall recombination of radicals can render the flame more stable and lead to a higher wall heat flux as a function of wall temperature compared to an inert surface under some conditions. It is also shown that the combined kinetic and thermal effects of wall radical quenching can expand the autothermal regime. Implications for estimating flammability limits near reactive surfaces of tubes are finally discussed. M This article features multimedia enhancements available from the abstract page in the online journal; see http://www.iop.org.
93 citations
TL;DR: Local rectangular refinement (LRR) solution-adaptive gridding method produces robust unstructured rectangular grids, utilizes novel multiple-scale finite-difference discretizations, and incorporates a damped modified Newton's method for simultaneously solving systems of governing elliptic PDEs as discussed by the authors.
Abstract: Within realistic combustion devices, physical quantities may change by an order of magnitude over an extremely thin flamefront, while remaining nearly unchanged throughout large areas nearby. Such behaviour dictates the use of adaptive numerical methods. The recently developed local rectangular refinement (LRR) solution-adaptive gridding method produces robust unstructured rectangular grids, utilizes novel multiple-scale finite-difference discretizations, and incorporates a damped modified Newton's method for simultaneously solving systems of governing elliptic PDEs. Here, the LRR method is applied to two axisymmetric laminar flames: a premixed Bunsen flame with one-step chemistry and a diffusion flame employing various complex chemical mechanisms. The Bunsen flame's position is highly dependent upon grid spacing, especially on coarse grids; it stabilizes only with adequate refinement. The diffusion flame results show excellent agreement with experimental data for flame structure, temperature and major sp...
84 citations
TL;DR: In this article, a method called rate-controlled constrained equilibrium (RCE) was developed based on the assumption that complex chemical systems evolve through a sequence of quasi-equilibrium states deterministically.
Abstract: A method called rate-controlled constrained equilibrium has been developed. It is based on the assumption that complex chemical systems evolve through a sequence of quasi-equilibrium states determi...
65 citations
TL;DR: In this article, the intrinsic low-dimensional manifolds of premixed CH4-air flames are analyzed and it is shown that in typical flame applications only a small domain of the state space is accessed.
Abstract: The intrinsic low-dimensional manifolds of strained and unstrained premixed CH4-air flames are analysed. We show that in typical flame applications only a small domain of the state space is accessed. This is found by simple one-dimensional flame computations using different strain rates. Secondly, in the different flame regions a different number of time scales is rate limiting. This means that in different regions there exist different numbers of relaxed modes corresponding to equilibration processes (species in quasi-steady states, reactions in partial equilibrium). It is shown that even in critical zones of the flame, such as the pre-heating zone, a considerable number of time scales can be decoupled from the equation system, guaranteeing an accurate description of the flame front by automatically simplified kinetics.
41 citations
TL;DR: In this article, a simplified two-step model for the combustion of energetic solids has been used to investigate the effect of gas-phase activation energy on flame structure and burning rate and the role of gas versus condensed-phase kinetics in determining burning rate.
Abstract: A simplified two-step kinetics model for the combustion of energetic solids has been used to investigate the effect of gas-phase activation energy on flame structure and burning rate and the role of gas- versus condensed-phase kinetics in determining burning rate. The following assumptions are made: a single-step, unimolecular, high activation energy decomposition process which is overall relatively energetically neutral, is followed by a highly exothermic single-step, bimolecular, gas-phase reaction with arbitrary activation energy, E˜ g. The results show that at extremely low ( 1012 Pa) pressures the burning rate is controlled by the condensed-phase reaction kinetics for any E˜g. At intermediate pressures (105-1010 Pa) gas reaction kinetics contribute strongly to the burning rate. In this pressure range the value of E˜g plays an important function in determining the role of gas- and condensed-phase reactions: for high E˜g a gas-phase kinetically controlled regime exists; for low E˜g bo...
36 citations
TL;DR: In this paper, a reduced model for premixed gas filtration combustion where the nonlinear effects are discarded everywhere but in the reaction rate term and where the only accounted for effect of the porous medium is its resistance to the gas flow is explored.
Abstract: A reduced model for premixed gas filtration combustion where the nonlinear effects are discarded everywhere but in the reaction rate term and where the only accounted for effect of the porous medium is its resistance to the gas flow is explored. While ruling out formation of shock waves the model appears rich enough to cover detonation-like phenomenon with barodiffusion acting as a driving agency. It is shown that depending on the initial conditions this creeping detonation mode is evoked either immediately or emerges after some time delay as a product of an abrupt transition from the low-velocity deflagration. The transition is triggered by a localized thermal explosion in the extended (friction-induced) preheat zone gradually formed ahead of the advancing deflagration.
34 citations
TL;DR: In this paper, the authors derived an expression for the acoustic pressure radiated in the far field of a compact region of fluid where low Mach number non-isomolar combustion takes place.
Abstract: This paper shows that the change in the number of moles of species during combustion can make a strong contribution to the acoustic power radiated by turbulent flames and cannot be systematically neglected. Starting from standard conservation equations, we derive an expression for the acoustic pressure radiated in the far field of a compact region of fluid where low Mach number non-isomolar combustion takes place. In this formulation, the contributions from ‘molar’ and thermal expansion appear explicitly. We also give a formulation in which the sound emission arising from purely non-stationary and from purely convective effects appear independently. As an application of the theory, we derive the acoustic power emitted by a premixed flame in the flamelet regime. Numerical evaluations show that the contribution of molar expansion to the acoustic power is between 2 and 5.6 dB (260% increase) for some common hydrocarbon-oxygen flames.
32 citations
TL;DR: In this article, an original physical model of self-ignition in a combustible gas mixture containing liquid fuel droplets is developed, where the droplets are small enough for the gas-droplet mixture to be considered as a fine mist such that individual droplet burning is subsumed into a well-stirred, spatially invariant burning approximation.
Abstract: An original physical model of self-ignition in a combustible gas mixture containing liquid fuel droplets is developed. The droplets are small enough for the gas-droplet mixture to be considered as a fine mist such that individual droplet burning is subsumed into a well-stirred, spatially invariant burning approximation. A classical Semenov-type analysis is used to describe the exothermic reaction, and the endothermic terms involve the use of quasi-steady mass transfer/heat balance and the Clausius-Clapeyron evaporative law. The resulting analysis predicts the ignition delay which is a function of the system parameters. Results are given for typical dynamical regimes. The case of different initial temperatures for droplets and gas is highly relevant to gas turbine lean blow-out and re-ignition.
TL;DR: In this article, the authors present results of a numerical study of a free-interface problem modelling self-propagating high-temperature synthesis (solid combustion) in a one-dimensional infinite medium.
Abstract: This paper presents results of a numerical study of a free-interface problem modelling self-propagating high-temperature synthesis (solid combustion) in a one-dimensional infinite medium. Evolution of the free interface exhibits a remarkable range of dynamical scenarios such as finite and infinite sequences of period doubling; the latter leading to chaotic oscillations, reverse sequences and infinite period bifurcation that may replace the supercritical Hopf bifurcation for some interface kinetics. Solutions were verified by using different numerical methods, including reduction to an integral equation for which convergence to the solutions has been demonstrated rigorously. Therefore, the ability of the free-interface model to generate the dynamical scenarios observed previously in models with a distributed reaction rate should be regarded as firmly established.
TL;DR: In this article, a detailed analysis of piecewise constant three-state weak solutions of the Fickett-Majda detonation model equations is carried out and it is shown that the threshold temperatures needed to effectively suppress the undesired numerical artefacts are considerably lower than any temperature actually found in the reaction zone of resolved detonation.
Abstract: This paper contributes to the topic of unphysical one-cell-per-time-step travelling combustion wave solutions in numerical computations of detonation waves in the presence of stiff chemical source terms. These false weak detonation solutions appear when a gas-dynamics–chemistry operator-splitting technique is used in conjunction with modern shock-capturing schemes for compressible flow simulations. A detailed analysis of piecewise constant three-state weak solutions of the Fickett–Majda detonation model equations is carried out. These structures are idealized analogues of the fake numerical solutions observed in computations. The analysis suggests that the problem can be cured by introducing a suitable ignition temperature below which the chemistry is frozen. It is found that the threshold temperatures needed to effectively suppress the undesired numerical artefacts are considerably lower than any temperature actually found in the reaction zone of a resolved detonation. This is in contrast to earlier suggestions along the same lines in the literature and it allows us to propose the introduction of such a low and otherwise irrelevant ignition temperature threshold as a routine measure for overcoming the problem of artificial weak detonations. The criterion for choosing the ignition temperature is then extended to the reactive Euler equations and extensive computational tests for both the model and the full equations demonstrate the effectiveness of our strategy. We consider the behaviour of a first-order Godunov-type scheme as well as its second-order extension in space and time using van Leer's MUSCL approach and Strang splitting.
TL;DR: In this paper, a theoretical two-phase flow analysis is presented to describe the quasi-steady propagation, across a pressure jump, of a multi-phase deflagration in confined porous energetic materials, which is particularly applicable to those types of porous energetic solids, such as degraded nitramine propellants, that can experience significant gas flow in the solid preheat region.
Abstract: A theoretical two-phase-flow analysis is developed to describe the quasi-steady propagation, across a pressure jump, of a multi-phase deflagration in confined porous energetic materials. The difference, or overpressure, between the upstream (unburned) and downstream (burned) gas pressure leads to a more complex structure than that which is obtained for an unconfined deflagration in which the pressure across the multi-phase flame region is approximately constant. In particular, the structure of such a wave is shown by asymptotic methods to consist of a thin boundary layer characterized by gas permeation into the unburned solid, followed by a liquid/gas flame region, common to both types of problems, in which the melted material is preheated further and ultimately converted to gaseous products. The effect of gas flow relative to the condensed material is shown to be significant, both in the porous unburned solid as well as in the exothermic liquid/gas melt layer, and is, in turn, strongly affected by the overpressure. Indeed, all quantities of interest, including the burned temperature, gas velocity and the propagation speed, depend on this pressure difference, leading to a significant enhancement of the burning rate with increasing overpressure. In the limit that the overpressure becomes small, the pressure gradient is insufficient to drive gas produced in the reaction zone in the upstream direction, and all gas flow relative to the condensed material is directed in the downstream direction, as in the case of an unconfined deflagration. The present analysis is particularly applicable to those types of porous energetic solids, such as degraded nitramine propellants, that can experience significant gas flow in the solid preheat region and which are characterized by the presence of exothermic reactions in a bubbling melt layer at their surfaces. 7 refs., 6 figs.
TL;DR: In this article, it was shown that the discrepancy between the PDE and the ODEs is due to the instability of exact pole solutions for the partial differential equation (PDE) describing the dynamics of a hydrodynamically unstable planar flame front.
Abstract: It is well known that the partial differential equation (PDE) describing the dynamics of a hydrodynamically unstable planar flame front has exact pole solutions for which the PDE reduces to a set of ordinary differential equations (ODEs). The paradox, however, lies in the fact that the set of ODEs does not permit the appearance of new poles in the complex plane, or the formation of cusps in the physical space, as observed in experiments. The validity of the PDE itself has thus been questioned. We show here that the discrepancy between the PDE and the ODEs is due to the instability of exact pole solutions for the PDE. In previous work, we have reported that most exact pole solutions are indeed unstable for the PDE but, for each interval of relatively small length L, there remains one solution (up to translation symmetry) which is neutrally stable. The latter is a one-peak, coalescent solution for which the poles (whose number is maximal) are steady. The front undergoes bifurcations as the length of the dom...
TL;DR: In this article, the response of a premixed flame in stagnation-point flow with an imposed oscillating strain rate has been examined and a detailed solution is constructed in the burned and unburned gas regions and includes the flame response to the imposed fluctuations as well as the resulting displacement of the incident flow.
Abstract: The response of a premixed flame in stagnation-point flow with an imposed oscillating strain rate has been examined. This configuration is of fundamental interest and has potential application to turbulent combustion modelling. Of interest are flames which stand well clear of the front stagnation point of a bluff body. Under these conditions the flame can be treated as a surface of density discontinuity. A detailed solution is constructed in the burned and unburned gas regions and includes the flame response to the imposed fluctuations as well as the resulting displacement of the incident flow. Our analysis accounts for the full coupling between the flame and the underlying flow field and, unlike most previous studies, is not restricted to small-amplitude oscillations.
TL;DR: In this article, the flow induced in a layer of liquid fuel at sub-flash temperature by the thermocapillary forces associated with the spreading of a flame that heats and vaporizes the liquid is analyzed numericall...
Abstract: The flow induced in a layer of liquid fuel at sub-flash temperature by the thermocapillary forces associated with the spreading of a flame that heats and vaporizes the liquid is analysed numericall...
TL;DR: In this article, the authors investigated salient features caused by the differential diffusion effect in nitrogen-diluted hydrogen diffusion flames and derived a simplified transport equation for the z H parameter to explain qualitatively the behaviours exhibited in the numerical solutions.
Abstract: Laminar flame calculations have been made for a Tsuji counterflow geometry to investigate salient features caused by the differential diffusion effect in nitrogen-diluted hydrogen diffusion flames. A strong dependence of the differential diffusion parameter z H on fuel dilution is found, where z H is the difference of the mixture fractions based on H and O elements. The strain rate, however, appears to have a relatively minor impact on z H. A simplified transport equation for the z H parameter has been derived to explain qualitatively the behaviours exhibited in the numerical solutions. Two source terms of z H are identified in the transport equation; one is due to mixing among species of different diffusion coefficients and the other one is associated with chemical reactions of H2. More importantly, the second source term is found to be dominant in reacting flows, and it increases with inert gas dilution. This feature causes the differential diffusion parameter to increase with the amount of fuel dilutio...
TL;DR: In this paper, a simple model where this hypothesis can be shown to be qualitatively false is presented. But the model does not consider that small scales participate in the renormalization of the turbulent flame speed, but do not renormalize the turbulence at larger scales.
Abstract: A common hypothesis in the study of turbulent premixed flames has been to consider that small scales participate in the renormalization of the turbulent flame speed, but do not renormalize the turbulence at larger scales. We study here a simple model where this hypothesis can be shown to be qualitatively false.
TL;DR: In this article, it was shown that the adiabatic high Lewis number premixed gas flame spreading through a large-scale zero-mean time-independent periodic shear flow constitutes a bistable system with a hysteretic transition between stable propagation modes.
Abstract: It is shown that the adiabatic high Lewis number premixed gas flame spreading through a large-scale zero-mean time-independent periodic shear flow constitutes a bistable system with a hysteretic transition between stable propagation modes. A mildly non-adiabatic flame may be quenched provided the flow-field intensity exceeds a certain critical value. The study is motivated by the experimentally known phenomenon of flame extinction by turbulence.
TL;DR: In this paper, two gaseous streams approaching each other from opposite directions in a counterflow are considered, and two opposed streams each carry a distribution of liquid droplets, and the sprays vaporize and the vaporized fuel and oxidizer gases diffuse and convect toward a chemical reaction region near the stagnation plane, where the reactants burn.
Abstract: In this spray model we consider two gaseous streams approaching each other from opposite directions in a counterflow. The two opposed streams each carry a distribution of liquid droplets. The sprays vaporize, and the vaporized fuel and oxidizer gases diffuse and convect toward a chemical reaction region near the stagnation plane, at which the reactants burn. A set of steady-state ordinary differential equations is derived to describe the temperature of the gas flow and the mass fractions of each reactant. We solve the differential equations in three consequent cases, each more complicated than the previous one: (i) fast vaporization and fast chemistry; (ii) finite-rate vaporization and fast chemistry; and (iii) finite-rate vaporization and finite-rate chemistry. Comparisons are made of our model results to previous fuel-spray-only and purely gaseous counterflow diffusion flame models. The parametric dependences of vaporization-zone movement, flame movement, temperature rise and degree of reactant leakage ...
TL;DR: The chain-branching process leading to ignition in the high-temperature laminar wake that forms at the trailing edge of a thin splitter plate separating a stream of hydrogen from the stream of oxygen is investigated in this paper.
Abstract: The chain-branching process leading to ignition in the high-temperature laminar wake that forms at the trailing edge of a thin splitter plate separating a stream of hydrogen from a stream of oxygen is investigated with a reduced chemistry description that employs H as the only chain-branching radical not in steady state. The analysis presented covers ignition events occurring in the Rott–Hakkinen and Goldstein regions, where self-similar solutions for the different flow variables are available. It is found that the initiation reactions, which create the first radicals, are only important in a relatively small initial region, becoming negligible downstream as the radical mole fractions increase to values larger than the ratio of the characteristic branching time to the characteristic initiation time, a very small quantity at temperatures of practical interest. As a result, most of the ignition history is controlled by the autocatalytic branching reactions, giving rise to a radical pool that increases expon...
TL;DR: In this paper, the dynamical behavior of an isolated combustible gas bubble surrounded by unlimited inviscid liquid is analyzed in the case of large activation energy and using spatially uniform assumptions.
Abstract: The dynamical behaviour of an isolated combustible gas bubble surrounded by unlimited inviscid liquid is analysed in the case of large activation energy and using spatially uniform assumptions. The pressure effect is crucial in this problem because of the limited gas volume. The mathematical model used is a system of three nonlinear ordinary differential equations including the energy equation, the concentration equation and the Rayleigh equation. The thermal behaviour is classified into slow and explosive regimes, and the thermal explosion criterion is obtained analytically, along the lines of the classical Semenov theory. The system is shown to reveal temperature and volumetric oscillations, the amplitude and frequency of which depend strongly on the intensity of the thermal process. In particular, the amplitude of slow and explosive regimes differs by at least an order of magnitude.
TL;DR: In this paper, a P1 approximation to the radiation transport equation was developed to be consistent with the exact transport equation for scenarios based upon the large-eddy simulation (LES) approach for fires developed at NIST.
Abstract: Three-dimensional simulations of fires cannot be performed on present-day computers without devising simplifications to the governing equations. One such method is the large-eddy simulation (LES) approach for fires developed at NIST. This method results in computationally efficient fire simulations in which the buoyancy-generated motion of hot gases and smoke is driven by Lagrangian particles that carry the heat released by combustion. Complex geometries are represented by blocking cells interior to a rectangular domain. A P1 approximation to the radiation transport equation was developed to be consistent with the exact transport equation for scenarios based upon this model. An isolated fire plume above a semi- infinite solid with a constant absorption coefficient in each half-space was studied as an example. A direct elliptic solver required only a fraction of the total LES computational cost. Radiative fluxes and intensities from the numerical and exact solutions to the P1 approximation were in excellent agreement.
TL;DR: In this article, the fast-time instability of Linan's premixed-flame regime is revisited in order to resolve the unrealistic result, previously obtained by Peters, that the inner reaction zone becomes unstable under all subadiabatic conditions.
Abstract: The fast-time instability of Linan's premixed-flame regime is revisited in order to resolve the unrealistic result, previously obtained by Peters, that the inner reaction zone becomes unstable under all subadiabatic conditions. The problem is posed as that of finding the stable range of the heat-loss parameter, defined as the ratio of the downstream heat loss to the total chemical energy influx, near the adiabatic condition. Central to the analysis is rescaling near the adiabatic condition by employing a distinguished limit that the heat-loss parameter is of the order of the inverse of the Zel'dovich number, which enables us to take into account the stabilizing effect of the outer diffusive layers on the inner reaction zone. For a general diffusion flamelet model, the critical value of the heat-loss parameter at the neutral-stability condition is obtained to form a bound for the stable subadiabatic range of the heat–loss parameter.
TL;DR: In this article, a model for the heating and ignition of a combustible solid by microwave energy is formulated and analyzed in the limit of small inverse activation energy and small Biot number B.
Abstract: A model for the heating and ignition of a combustible solid by microwave energy is formulated and analysed in the limit of small inverse activation energy and small Biot number B. The high activation energy limit implies that the heating process is effectively inert until the temperature within the material reaches a critical ignition value, while the small Biot number limit implies that during this stage spatial variations in temperature throughout the material are always small. Analysis of the inert stage includes determination of the dynamics of inert hot-spots. As the ignition temperature is approached chemical energy is released rapidly in the form of heat, and the evolution then enters an ignition stage which develops on a fast time-scale. A reduced system is derived governing small-amplitude departures of the temperature from the inert value during the ignition stage under the significant scaling relation between the expansion parameters, which is shown to be ~ B. This reduced system recovers both ...
TL;DR: In this article, the effect of fluid leakage into insulation material is considered, where the combustible fluid spread over the inner fibres of the material is both slowly oxidizing and undergoing evaporation/desorption.
Abstract: The effect of fluid leakage into insulation material is considered, where the combustible fluid spread over the inner fibres of the material is both slowly oxidizing and undergoing evaporation/desorption. It is found, in particular, that the constant supply of fuel can lead to unexpected oscillatory behaviour in regions of parameter space which would otherwise have been considered safe. In particular, the case of an extremely slow rate of supply of fluid can be the most deceptive in terms of the possibility of ignition. *An early version of this paper was presented at the 16th ICDERS, Cracow, Poland, 4–8 August 1997.
TL;DR: In this article, a self-similar focusing structure for a Cauchy initial-value problem with φ(x, t=0)=φi(x) in which φi possesses a smooth global maximum is presented.
Abstract: Finite time blow-up in the semilinear reactive-diffusive parabolic equation φ1=µφxx+eφ is examined in the limit of weak diffusion μ<<1, for a Cauchy initial-value problem with φ(x, t=0)=φi(x) in which φi(x) possesses a smooth global maximum. An asymptotic description of the evolution of φ is obtained from the initial time through blow-up using singular perturbation techniques. Near blow-up, an exact self-similar focusing structure for φ, identical to that previously associated with non-diffusive thermal runaway, is shown to be appropriate. However, in an exponentially small layer close to the blow-up time, the focusing structure must be modified to ensure a uniformly valid solution. This modification uncovers the asymptotically self-similar focusing structure previously recognized for blow-up in equations of the form φ1=φxx+eφ. In contrast to previous studies, however, the structure arises here as a natural consequence of removing the non-uniformity in the expansions which occurs exponentially close to bl...