Combustion waves for gases (
            <i>Le</i>
            = 1) and solids (
            <i>Le</i>
            →∞)

Weber, R. O.; Mercer, G. N.; Sidhu, Harvinder; Gray, B.F.

doi:10.1098/rspa.1997.0062

Cited by 102 publications

(62 citation statements)

References 9 publications

(13 reference statements)

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“…Recently, a similar system of two equations has been studied by Weber, Mercer, Sidhu and Gray [22]. Their model also balances thermal diffusion with exothermic heat production, and includes the effects of fuel consumption and diffusion.…”

Section: Introductionmentioning

confidence: 99%

A combustion wave of permanent form in a compressible gas

Forbes

Derrick

2001

ANZIAM J.

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A model is presented for describing the propagation of a one-dimensional wave of permanent form in a compressible gas in a pipe. Energy is lost to the system through the walls of the pipe, but the combustion wave produces heat through an exothermic chemical reaction. The full set of equations for the model is reduced to a phase-plane system, and it is shown that, for small amplitude waves, a weakly non-linear analysis leads to a temperature profile that is a classical solitary wave. A novel shooting method is developed for the full non-linear problem, and this confirms and extends the solitary-wave solution, up to a value of the temperature amplitude at which the wave begins to develop a shock. The jump conditions across the shock are presented, and numerical integration is used to continue the solution for temperature amplitudes at which a shock is present. An example is given of the extreme situation in which the shock is so strong that all the fuel behind the shock is exhausted.

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Section: Introductionmentioning

confidence: 99%

A combustion wave of permanent form in a compressible gas

Forbes

Derrick

2001

ANZIAM J.

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“…For convenience we assume u a = 0 . This assumption does not affect the qualitative properties of the travelling waves [7] and circumvents the cold boundary problem [16].…”

Section: Mathematical Modelmentioning

confidence: 99%

“…A real world example of such a configuration would be a long, imperfectly insulated cylinder containing a fuel-oxygen mixture, with an appropriate a priori averaging over the transverse spatial dimension of C777 the flame front. Assuming that the rate of the exothermic combustion is adequately described by the Arrhenius law, the equations describing conservation of energy and mass lead to the following (non-dimensionalised) reaction-diffusion pde system [16]:…”

Section: Mathematical Modelmentioning

confidence: 99%

Analysing instability of combustion waves using the Evans function

Sharples¹,

Sidhu²,

Gubernov³

2011

ANZIAMJ

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We consider travelling wave solutions of a reaction-diffusion system corresponding to a single step, homogeneous, premixed combustion scheme with Newtonian heat loss and general Lewis number. Particular attention is paid to unstable combustion wave regimes, especially those associated with oscillatory behaviour. The instability analysis is conducted with the use of Evans function techniques, which we use to derive eigenvalues of the linear stability problem via the argument principle and Nyquist plots. These techniques permit the study of transitions to different modes of unstable behaviour in great detail. Threshold values of the parameters corresponding to Hopf and Bogdanov-Takens bifurcation are established and it is shown that for certain parameter values the system exhibits a period doubling route to chaotic behaviour.

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“…Actually, temperature is spread by molecular processes and turbulent flows, so it has a random character and is modelled by a diffusion process. Furthermore, the fire is an energy source, and a reaction-diffusion equation follows from conservation of energy and fuel on the basis of the combustion wave approach (Weber et al, 1997). Two-equation models concerning the average temperature field T (x, t) and the fuel mass fraction Y (x, t), Y ∈ [0, 1], have been developed and analysed in the literature (see e.g.…”

Section: Reaction-diffusion Equation Modellingmentioning

confidence: 99%

“…One of these modelling approaches is based on evolution equations of the reaction-diffusion type (e.g. Weber et al, 1997;Asensio and Ferragut, 2002;Mandel et al, 2008;Babak et al, 2009), and the other is based on the front tracking technique named the level-set method (Sethian and Smereka, 2003): see for example Mallet et al (2009), Rehm andMcDermott (2009), andMandel et al (2011).…”

Section: Introductionmentioning

confidence: 99%

Modelling wildland fire propagation by tracking random fronts

Pagnini

Mentrelli

2014

Nat. Hazards Earth Syst. Sci.

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Abstract. Wildland fire propagation is studied in the literature by two alternative approaches, namely the reactiondiffusion equation and the level-set method. These two approaches are considered alternatives to each other because the solution of the reaction-diffusion equation is generally a continuous smooth function that has an exponential decay, and it is not zero in an infinite domain, while the levelset method, which is a front tracking technique, generates a sharp function that is not zero inside a compact domain. However, these two approaches can indeed be considered complementary and reconciled. Turbulent hot-air transport and fire spotting are phenomena with a random nature and they are extremely important in wildland fire propagation. Consequently, the fire front gets a random character, too; hence, a tracking method for random fronts is needed. In particular, the level-set contour is randomised here according to the probability density function of the interface particle displacement. Actually, when the level-set method is developed for tracking a front interface with a random motion, the resulting averaged process emerges to be governed by an evolution equation of the reaction-diffusion type. In this reconciled approach, the rate of spread of the fire keeps the same key and characterising role that is typical of the level-set approach. The resulting model emerges to be suitable for simulating effects due to turbulent convection, such as fire flank and backing fire, the faster fire spread being because of the actions by hot-air pre-heating and by ember landing, and also due to the fire overcoming a fire-break zone, which is a case not resolved by models based on the level-set method. Moreover, from the proposed formulation, a correction follows for the formula of the rate of spread which is due to the mean jump length of firebrands in the downwind direction for the leeward sector of the fireline contour. The presented study constitutes a proof of concept, and it needs to be subjected to a future validation.

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Combustion waves for gases ( Le = 1) and solids ( Le →∞)

Cited by 102 publications

References 9 publications

A combustion wave of permanent form in a compressible gas

A combustion wave of permanent form in a compressible gas

Analysing instability of combustion waves using the Evans function

Modelling wildland fire propagation by tracking random fronts

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