Smouldering front shapes for steady propagation in a stratified medium

The steady propagation of a smouldering reaction front parallelto the faces of a solid reactive slab has been considered withthe density of the reactive material changing with distancefrom one surface. Such slow steady propagation is known to existin peat bogs where smoulder can take place over many monthsPage et al.(2002, Nature, 420, 61–65). As smoulder progressesit leaves behind a porous matrix through which oxidizer is ableto diffuse to the reaction front. At the front, oxidizer andfuel combine stoichiometrically and the concentration of oxidizeris reduced to zero. In the analysis to be presented the Pecletnumber, based on an assumed constant smoulder speed, is small.The equations and boundary conditions for the oxidizer in theporous region are solved to first order by a complex variablemethod and hodograph transformation. The solution allows theshape of the smouldering front and the oxidizer distributionbehind the front to be determined. Example curves for particularfuel distributions are given. An analysis shows how furtherterms in the expansion of oxidizer concentration in powers ofthe Peclet number may be obtained.     

Diffusion-controlled smoulder propagation in a semi-infinite solid: The asymptotic temperature distribution

The authors consider the steady propagation of a two-dimensionaldiffusioncontrolled smouldering reaction front parallel to theplane boundary of a semi-infinite nonporous reactant. The reactionfront is assumed to be a sheet of line heat sources of variablestrength. The distribution of oxidizer concentration and temperaturein the porous burnt char and of the temperature in the reactantis determined in the form of an asymptotic expansion involvinga similarity variable. The temperature on the reaction frontin the asymptotic region is found to be constant to high order.The dependence of this temperature on the Lewis numbers associatedwith the reactant and the char is found to be in general agreementwith observations.     

Transient diffusion-controlled smoulder propagation: a similarity solution approach

One-, two- and three-dimensional time-dependent smoulder propagationthrough solid reactants with plane bounding surfaces is considered.Propagation is maintained by the diffusion of oxidizer fromthe boundaries to the smouldering reaction zone. The resultingburnt solid is assumed to be porous and the unreacted solidis taken to be sufficiently dense for no oxidizer to be present.The activation energy of the reaction is taken to be so largethat the exothermic reaction term has a delta-function behaviour.This enables the reaction zone to be approximated by a narrowreaction front and results in the equations of heat and masstransfer being decoupled away from the front. The assumption,based on experimental observations, that the reaction frontspropagate with a speed proportional to t^–1/2, where tis the time, permits the introduction of similarity variables.The resulting intermediate asymptotic equations, lying betweenthose for very small and very large times, are solved and theequation of the smouldering fronts determined for each geometryis considered.     

Diffusion-controlled smoulder propagation in a semi-infinite solid: A numerical solution

The propagation of a thin two-dimensional smouldering reactionfront (SRF) parallel to a plane surface in a semi-infinite solidfuel is considered. Combustion is controlled by oxidizer diffusionthrough the porous burnt material behind the SRF. The modelgives rise to a free boundary problem as the position of theSRF is to be determined as part of the solution. The Burke-Schumanncondition of the oxidizer concentration vanishing on the SRFis adopted. Different forms of the plane surface boundary conditionare considered. The governing linear p.d.e., which holds inan unknown domain, is converted into a nonlinear p.d.e. overa fixed domain by Alt's method, in which the free boundary conditionsare satisfied by a jump condition. The position of the SRF andthe oxidizer concentration distribution behind it are determined.It is shown that the shape of the SRF is asymptotically parabolic,consistent with analytic and experimental results. Contoursof equal oxygen concentration are presented.     

Wave front analysis of weak nonlinear waves in a two-phase flow of gas and dust particles

The non-linear behavior of waves including the characteristic front, the expansion wave front and the shock front, in a mixture of gas and dust particles has been studied. Such waves are conceived of as produced by a piston moving with a small velocity as compared with the speed of sound. The trajectories of these waves and the particle paths in the physical plane are determined. The effect of solid particles and the adiabatic heat exponent on the wave propagation is also investigated.     

Fractographic study of macrocrack propagation in oriented Kapron

The fracture surface of an oriented Kapron monofilament has been studied at electron-microscope and optical magnifications with the object of establishing the details of the micromechanism of crack propagation associated with the fracture of polymeric materials. Microparabolic figures can be observed in the specular zone of the fracture surface. This relief may be assumed to originate in the interaction of the main crack and the submicroscopic cracks present in loaded polymers. In the region of high main-crack velocities it is possible to observe a self-oscillatory motion of the tip of the main crack leading to the formation on the fracture surface of a system of bands parallel to the main crack front.Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 4, pp. 645–648, July–August, 1971.     

Aplicación del método level set para modelar el proceso de combustión premezclada en un motor Otto de dos tiempos

In this paper, the numerical method level set has been used to model the combustion process in an Otto two-stroke engine. The level set has been implemented in a CFD (Computational Fluid Dynamics) software based in finite volumes. The pressure and temperature fields have been obtained, such as the propagation of the flame front. In order to validate this model, the numerically obtained results have been compared with experimental data, verifying a satisfactory concordance between both of them. Besides, the level set method has been compared with other numerical procedure, showing the difference between both results.     

Remarks on the wave front of polarization: Polarized solutions for the evolution three-dimensional Lame system

Some remarks complementary to the work of N. Dencker in [1] concerning the wave front of polarization are given. The theorem due to N. Dencker about the propagation of a wave front of polarization is applied to solutions for the evolution of the three-dimensional Lame system. The propagation of polarized solutions is established.     

The segment projection method for interface tracking

There has recently been important progress in the development of front tracking and level set methods for the numerical simulation of moving interfaces. The segment projection method is a new technique for computational geometry. It can be seen as a compromise between front tracking and level set methods. It is based on the regular mathematical representation of a manifold as an atlas of charts. Each chart or segment is evolved independently by a partial differential equation that is discretized on an Eulerian grid. The connectivity of the segments is handled by an appropriate data structure and by numerical interpolation. The method is presented and its properties are analyzed. Applications to multiphase flow, epitaxial growth, and high‐frequency wave propagation are given. © 2002 Wiley Periodicals, Inc.     

Applications of Distributional Derivatives to Wave Propagation

First the concepts of the surface distributions are explained.Thereafter the first and higher-order distributional derivativesare derived for a function of several variables. These conceptsare then used to study the propagation of wave fronts in continuummechanics. Explicit formulas for the jump relations for thefirst and second-order partial derivatives are obtained acrossthe wave front. A systematic method is then developed such thatthe algebraic work for the study of waves becomes very simple.A few illustrations in wave propagation are presented. At theend it is pointed out how this method can be effectively appliedin the derivation of the jump relations for single and double-layerpotentials in the theory of harmonic functions.     

Comparisons between thermodynamic and one-dimensional combustion models of spark-ignition engines

A one-dimensional combustion model, employing a constant eddy diffusivity and a one-step chemical reaction, has been developed and applied to study the flame propagation in a spark-ignition engine. Calculations have been made at 1600 and 4200 rev min⁻¹ under fuel rich conditions and compared with available engine pressure data. One- and two-zone thermodynamic models have also been developed and applied to study the combustion process in the engine. The thermodynamic models have been compared with the one-dimensional model results and comparisons include the average mixture temperature, the temperatures of the burned and unburned gases and the flame surface area. These comparisons indicate that the one-dimensional model predictions are very sensitive to the eddy diffusivity and reaction rate data. The two-zone thermodynamic model predicts, first, a monotonically increasing flame surface area with time and, then, a monotonically decreasing surface area, whereas the one-dimensional model always predicts a monotonically increasing flame surface area. The average mixture temperature predicted by the one-zone thermodynamic model is higher than those of the two-zone and one-dimensional models during the compression stroke, while that of the one-dimensional model is higher than the temperatures predicted by the one- and two-zone models during the expansion stroke. The one-dmensional model predicts an accelerating flame even when the front approaches the cold cylinder wall. This yields a faster fuel consumption rate than those predicted by the one- and two-zone thermodynamic models which predict smoother burned fuel mass profiles.     

Properties of simple model problems for reacting gas flows

The compressible Navier–Stokes equations for reacting gases are extremely complex. Simpler models have been considered, and for these completely non-physical propagation speeds have been observed. These model problems are stiff, meaning that several different scales are present in the solution. Numerical solution of non-reacting flows almost always involves addition of extra dissipation. It will be shown that this action will render a totally wrong propagation speed for a simple model equation of reacting flows. This problem will be accentuated by increasing stiffness of the problem. Existence and uniqueness of a solution to this model equation is proved. The dependence of the propagation speed on the viscosity and a term governing the stiffness (comparable to the reaction rate for a more complete model) is investigated. A remedy for the wrong propagation speed for this simple model equation is proposed such that the speed is correct although the front is smeared out.     

Ray equations of a weak shock in a hyperbolic system of conservation laws in multi-dimensions

In this paper we give a complete proof of a theorem, which states that ‘for a weak shock, the shock ray velocity is equal to the mean of the ray velocities of nonlinear wavefronts just ahead and just behind the shock, provided we take the wavefronts ahead and behind to be instantaneously coincident with the shock front. Similarly, the rate of turning of the shock front is also equal to the mean of the rates of turning of such wavefronts just ahead and just behind the shock’. A particular case of this theorem for shock propagation in gasdynamics has been used extensively in applications. Since it is useful also in other physical systems, we present here the theorem in its most general form.     

Wave front tracing and asymptotic stability of planar travelling waves for a two-dimensional shallow river model

The propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two-dimensional shallow river model. In this paper, we consider the time-asymptotic stability of weak planar travelling waves for a two-dimensional shallow river model with Darcy's law. We derive an effective parabolic equation to analyze the wave front motion. By employing weighted energy estimates, we show that weak planar travelling waves are time-asymptotically stable under sufficiently small perturbations.     

Simulation of a turbulent premixed open V-shaped flame using contour advection with surgery

Despite its capability of high spatial resolution, simulation of turbulent flows with traditional Lagrangian (front tracking) scheme is often discouraged by numerical instability caused by clustering of marker nodes and topological changes of fronts. Contour advection surgery (CAS), being a robust front tracking scheme, can limit the growth of front complexity during simulation without jeopardizing accuracy or efficiency. This allows it to open up an advantage over traditional front-tracking schemes. It has already been demonstrated that CAS, with incorporation of the reaction sheet model, can accurately simulate the propagation and advection of a turbulent premixed V-shaped flame. In this study, it is further tested with 10 values of vortex circulation. A range of upstream turbulence levels of 1.8–19.8% was obtained. Results indicate that upstream turbulence increase the average flame length, flame zone area and the overall burning rate. Flame surface density Σ was also estimated. Maximum values of Σ obtained lie in the range 0.1–1.4 mm⁻¹. Skewness towards the burnt region was observed in all profiles of Σ. Similar to results from laboratory experiments, it was found that Σ values decreases with upstream turbulence. From this study, the ability of CAS to cope with intense turbulence is demonstrated and a better quantitative understanding on the scheme has also been acquired.     

Relaxation schemes for curvature‐dependent front propagation

In this paper we study analytically and numerically a novel relaxation approximation for front evolution according to a curvature‐dependent local law. In the Chapman‐Enskog expansion, this relaxation approximation leads to the level‐set equation for transport‐dominated front propagation, which includes the mean curvature as the next‐order term. This approach yields a new and possibly attractive way of calculating numerically the propagation of curvature‐dependent fronts. Since the relaxation system is a symmetrizable, semilinear, and linearly convective hyperbolic system without singularities, the relaxation scheme captures the curvature‐dependent front propagation without discretizing directly the complicated yet singular mean curvature term. © 1999 John Wiley & Sons, Inc.     

Front Propagation in Stirred Media

The problem of asymptotic features of front propagation in stirred media is addressed for laminar and turbulent velocity fields. In particular we consider the problem in two dimensional steady and unsteady cellular flows in the limit of very fast reaction and sharp front, i.e., in the geometrical optics limit. In the steady case we provide an analytical approximation for the front speed, v _f, as a function of the stirring intensity, U, in good agreement with the numerical results. In the unsteady (time-periodic) case, albeit the Lagrangian dynamics is chaotic, chaos in the front dynamics is relevant only for a transient. Asymptotically the front evolves periodically and chaos manifests only in the spatially wrinkled structure of the front. In addition we study front propagation of reactive fields in systems whose diffusive behavior is anomalous. The features of the front propagation depend, not only on the scaling exponent ν, which characterizes the diffusion properties, \({( \langle (x(t) - x(0))^2 \rangle \sim t^{2\nu} )}\) , but also on the detailed shape of the probability distribution of the diffusive process.     

Numerical inversion of Laplace transform by the method of asymptotic extension of the interval in dynamic viscoelasticity problems

Conclusions 1. A numerical investigation into the process of wave propagation in infinite and finite viscoelastic rods has been carried out by means of the method of asymptotic extension of the interval.2. It is established that use, in the calculations, of kernels of relaxation with weak singularity does not give rise to a stress jump at the wave front.3. The effect of parameters of the Rzhanitsyn kernel on the "erosion" of the wave front has been investigated. It is discovered that the instant of occurrence of a stress that is appreciably different from zero, at points far away from the beginning of the rod, is determined by the long-term modulus of elasticity of the material of the rod.4. The solution of the problem concerned with the propagation of a load impulse of finite duration shows a decrease of the maximum value of the stress in the rod, when the duration of the applied impulse decreases, and an intense change in the shape of the impulse while it propagates along the rod, as a consequence of wave dispersion.P. Stuchka Latvian State University, Riga. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 864–870, September–October, 1976.