to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

A comprehensive and critical review of closure approximations for two-equation turbulence models has been made. Particular attention has focused on the scale-determining equation in an attempt to find the optimum choice of dependent variable and closure approximations. Using a combination of singular perturbation methods and numerical computations, this paper demonstrates that: 1) conventional A:-e and A>w formulations generally are inaccurate for boundary layers in adverse pressure gradient; 2) using "wall functions'' tends to mask the shortcomings of such models; and 3) a more suitable choice of dependent variables exists that is much more accurate for adverse pressure gradient. Based on the analysis, a two-equation turbulence model is postulated that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows. With no viscous damping of the model's closure coefficients and without the aid of wall functions, the model equations can be integrated through the viscous sublayer. Surface boundary conditions are presented that permit accurate predictions for flow over rough surfaces and for flows with surface mass addition.

Reassessment of the scale-determining equation for advanced turbulence models

Laminar diffusion flamelet models in non-premixed turbulent combustion

The laminar flamelet concept covers a regime in turbulent combustion where chemistry (as compared to transport processes) is fast such that it occurs in asymptotically thin layers—called flamelets—embedded within the turbulent flow field. This situation occurs in most practical combustion systems including reciprocating engines and gas turbine combustors. The inner structure of the flamelets is one-dimensional and time dependent. This is shown by an asymptotic expansion for the Damkohler number of the rate determining reaction which is assumed to be large. Other non-dimensional chemical parameters such as the nondimensional activation energy or Zeldovich number may also be large and may be related to the Damkohler number by a distinguished asymptoiic limit. Examples of the flamelet structure are presented using onestep model kinetics or a reduced four-step quasi-global mechanism for methane flames. For non-premixed combustion a formal coordinate transformation using the mixture fraction Z as independent variable leads to a universal description. The instantaneous scalar dissipation rate χ of the conserved scalar Z is identified to represent the diffusion time scale that is compared with the chemical time scale in the definition of the Damkohler number. Flame stretch increases the scalar dissipation rate in a turbulent flow field. If it exceeds a critical value χ q the diffusion flamelet will extinguish. Considering the probability density distribution of χ , it is shown how local extinction reduces the number of burnable flamelets and thereby the mean reaction rate. Furthermore, local extinction events may interrupt the connection to burnable flamelets which are not yet reached by an ignition source and will therefore not be ignited. This phenomenon, described by percolation theory, is used to derive criteria for the stability of lifted flames. It is shown how values of ∋ q obtained from laminar experiments scale with turbulent residence times to describe lift-off of turbulent jet diffusion flames. For non-premixed combustion it is concluded that the outer mixing field—by imposing the scalar dissipation rate—dominates the flamelet behaviour because the flamelet is attached to the surface of stoichiometric mixture. The flamelet response may be two-fold: burning or non-burning quasi-stationary states. This is the reason why classical turbulence models readily can be used in the flamelet regime of non-premixed combustion. The extent to which burnable yet non-burning flamelets and unsteady transition events contribute to the overall statistics in turbulent non-premixed flames needs still to be explored further. For premixed combustion the interaction between flamelets and the outer flow is much stronger because the flame front can propagate normal to itself. The chemical time scale and the thermal diffusivity determine the flame thickness and the flame velocity. The flamelet concept is valid if the flame thickness is smaller than the smallest length scale in the turbulent flow, the Kolmogorov scale. Also, if the turbulence intensity v′ is larger than the laminar flame velocity, there is a local interaction between the flame front and the turbulent flow which corrugates the front. A new length scale L G =v F 3 /∈ , the Gibson scale, is introduced which describes the smaller size of the burnt gas pockets of the front. Here v F is the laminar flame velocity and ∈ the dissipation of turbulent kinetic energy in the oncoming flow. Eddies smaller than L G cannot corrugate the flame front due to their smaller circumferential velocity while larger eddies up to the macro length scale will only convect the front within the flow field. Flame stretch effects are the most efficient at the smallest scale L G . If stretch combined with differential diffusion of temperature and the deficient reactant, represented by a Lewis number different from unity, is imposed on the flamelet, its inner structure will respond leading to a change in flame velocity and in some cases to extinction. Transient effects of this response are much more important than for diffusion flamelets. A new mechanism of premixed flamelet extinction, based on the diffusion of radicals out of the reaction zone, is described by Rogg. Recent progress in the Bray-Moss-Libby formulation and the pdf-transport equation approach by Pope are presented. Finally, different approaches to predict the turbulent flame velocity including an argument based on the fractal dimension of the flame front are discussed.

Laminar Flamelet Concepts in Turbulent Combustion

Recent advances in droplet vaporization and combustion

The adequacy of direct one-step chemical kinetics for describing ignition and extinction in initially unmixed gases is studied through the particular case of inviscid axisymmetric stagnation-point flow. Oxidant is assumed to blow from upstream infinity at a non-gaseous reservoir of pure fuel at its boiling (or sublimating) temperature. Before reaching the reservoir the oxidant reacts with gaseous fuel flowing in the opposite direction to form product and release heat. This heat is in part conducted and diffused to the reservoir interface to transform more fuel into the gaseous state and continue the steady-state burning. Second-order Arrhenius kinetics for Lewis-number unity is examined. A critical parameter characterizing the phenomenon is shown to be the first Damkohler similarity group D1, the ratio of a time characterizing the flow to a time characterizing the chemical activity.For small D1 the reactants convect away heat without releasing the energy stored in their chemical bonds. Regular perturbation about chemically frozen flow establishes this condition as the weak burning limit. For large D1 singular perturbation describes a narrow region of intense chemical activity. For infinite D1 (indefinitely fast rate of reaction) the region is reduced to a surface of discontinuity (the thin-flame kinetics of Burke & Schumann).For intermediate D1 numerical techniques establish that a solution describing burning of moderate intensity joins the two previously mentioned asymptotic limits. It is suggested that sudden transition of the system between the various branches in this domain of intermediate D1 accounts for the phenomena of ignition and extinction of burning.

Ignition and extinction in combustion of initially unmixed reactants

Several steady state and time-dependent solutions to the compressible conservation laws describing direct one-step near-equilibrium irreversible exothermic burning of initially unmixed gaseous reactants, with Lewis-Semenov number unity, are presented. The quantitative investigation first establishes the Burke-Schumann thin-flame solution using the Shvab-Zeldovich formulation. Real flames do not have the indefinitely thin reaction zone associated with the Burke-Schumann solution. Singular perturbation analysis is used to provide a modification of the thin-flame solution which includes a more realistic reaction zone of small but finite thickness. The particular geometry emphasized is the un bounded counterflow such that there exists a spatially constant rate of strain along the flame. While the solutions
for diffusion flames under a finite tangential strain rate may be of interest in and of themselves for laminar flow, the problems are motivated by the authors' belief that they are pertinent to the study of diffusion-flame burning in transitional and turbulent shear flows.

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The Effect of Strain Rate on Diffusion Flames

Laboratory-scale experiments on wind-aided firespread across an array of very-small-diameter (1.3-4.4-mm), discrete fuel elements were carried out in a specially designed wind tunnel. The rate of firespread, ty, is inferred from the output of streamwise-disiributed, near-bed-surface thermocouples. The fuel consists of an array of identical, regularly arranged, wooden, toothpick-like Tuel elements, positioned upright in shallow holes drilled in a ceramic substrate. Adoption of this well-defined fuel bed facilitates repetition. Extensive testing suggests that νy ∼ (U/m)1/2 over a wide range of the uniform ambient wind speed U and the fuel-mass distribution m (fuel, per unit planform area of the bed, consumed with firefront passage). The effects on the firespread rate of other parameters investigated and reported include the type of wood species; fuel-element length; fuel-element diameter; fuel-bed width, including streamwise-varyjng width; enhanced moisture content of the substratum and/or fuel; sm...

Wind-Aided Firespread Across Arrays of Discrete Fuel Elements. II. Experiment

The propagation for spread rate lrofwind-aided firespreadacross an array of very-small-diarne-ter, discrete fuel elements is sought, as a step toward the objective of predicting the advance in time of a firefront through either strewn debris in a heavily-blast-damaged scenario or through brush-and-grass-type wildlands. Here, the quasisteady rate (if one exists) is sought for conditions under which there is both (1) a wind whose mean speed U is constant and whose direction is constant, and (2) a horizontal bed with a macroscopically uniform fuel distribution m. More precisely, m is the mass of fuel (per unit planform area of the bed) consumed with firefront passage; here the fuel elements are thin. so that, for conditions under which fire propagates at all, the fuel consumed is identical with the fuel loading initially present. In accord with laboratory-scale experiments in a specially dedicated firetunnel, to be reported separately, analysis suggests thai I'r varies with (U/m)1/2 over a wide (but obviousl...

Wind-aided firespread across arrays of discrete fuel elements. I, Theory

: Asymptotic expansion techniques are used in the limit of large Reynolds number, to study the structure of fully turbulent shear layers. Two cases examined are channel (or pipe) flow and two-dimensional boundary layer flow with an applied pressure gradient, upstream of any separation. Attention, for these two cases, is confined to the flow of incompressible constant-property fluids. Closure is effected through the introduction of an eddy viscosity model formulated with sufficient generality so that most existing models are special cases.

Francis E. Fendell

Papers

Ignition and extinction in combustion of initially unmixed reactants

The Effect of Strain Rate on Diffusion Flames

Wind-Aided Firespread Across Arrays of Discrete Fuel Elements. II. Experiment

Wind-aided firespread across arrays of discrete fuel elements. I, Theory

Asymptotic analysis of turbulent channel and boundary-layer flow