J. Fluid Mech. (2000), vol. 415, pp. 227{260. Printed in the United Kingdom
c
© 2000 Cambridge University Press
227
Theoretical and numerical study of a
symmetrical triple flame using the parabolic
flame path approximation
By S A N D I P G H O S A Ly AND L U C V E R V I S C H
LMFN, INSA-Rouen, UMR { CNRS 6614/CORIA,
Saint-Etienne-du-Rouvray, 76801 Cedex, France
(Received 5 December 1997 and in revised form 8 February 2000)
In non-premixed turbulent combustion the reactive zone is localized at the stoichio-
metric surfaces of the mixture and may be locally approximated by a diusion flame.
Experiments and numerical simulations reveal a characteristic structure at the edge
of such a two-dimensional diusion flame. This ‘triple flame’ or ‘edge flame’ consists
of a curved flame front followed by a trailing edge that constitutes the body of
the diusion flame. Triple flames are also observed at the edge of a lifted laminar
diusion flame near the exit of burners. The speed of propagation of the triple flame
determines such important properties as the rate of increase of the flame surface in
non-premixed combustion and the lift-o distance in lifted flames at burners. This
paper presents an approximate theory of triple flames based on an approximation
of the flame shape by a parabolic prole, for large activation energy and low but
nite heat release. The parabolic flame path approximation is a heuristic approxima-
tion motivated by physical considerations and is independent of the large activation
energy and low heat release assumptions which are incorporated through asymptotic
expansions. Therefore, what is presented here is not a truly asymptotic theory of triple
flames, but an asymptotic solution of a model problem in which the flame shape is
assumed parabolic. Only the symmetrical flame is considered and Lewis numbers are
taken to be unity. The principal results are analytical formulas for the speed and
curvature of triple flames as a function of the upstream mixture fraction gradient in
the limit of innitesimal heat release as well as small but nite heat release. For given
chemistry, the solution provides a complete description of the triple flame in terms of
the upstream mixture fraction gradient. The theory is validated by comparison with
numerical simulation of the primitive equations.
1. Introduction
In many engineering applications of turbulent combustion, fuel and oxidizer are
not perfectly mixed before entering the combustion chamber. Large-scale unsteady
movements together with micro-mixing mechanisms subsequently bring fuel and
oxidizer into contact where they react within a thin reaction zone (Bray 1996) that
may be locally approximated by a diusion flame. A typical example is given in
gure 1 which shows an instantaneous reaction rate prole from a two-dimensional
y Present address: Northwestern University, Department of Mechanical Engineering, 2145 Sheri-
dan Road, Evanston, IL 60208, USA.

228 S. Ghosal and L. Vervisch
Triple-layer
system
Diffusion flame
Edge flame
Figure 1. Isolines of reaction rate in a two-dimensional simulation of combustion in pockets of
fuel and hot air in freely decaying turbulence (Vervisch & Poinsot 1998).
simulation of the propagation of combustion in pockets of fuel and hot air in freely
decaying turbulence (Vervisch & Poinsot 1998).
A generic picture of such a laminar diusion flame was given by Li

n

an & Crespo
(1976) for a counterflowing fuel and oxidizer stream with a single-step chemical
reaction. The flame is localized at the stoichiometric surface where fuel and oxidizer
are mixed in stoichiometric proportions, and their analysis provides a full description
of the flame in terms of the local mixture fraction gradient normal to the flame
front. In this quasi-steady diusion flame, the gradients of chemical species and
temperature are such that the amount of heat diusing away from the reaction zone
is exactly balanced by the heat produced by combustion. Should the local gradient
of temperature become too large, the rate of chemical reactions is not able to keep
up with the heat losses and quenching occurs.
Applying this description to the above picture of non-premixed turbulent combus-
tion, one expects that the reaction zone would be conned to the highly convoluted
stoichiometric surface; however, the reaction rate would not be uniform over the
surface. Instead, there would be zones where excessive thermal gradients cause local
extinction of the flame. This is shown schematically in gure 2. Such enhanced ther-
mal gradients are expected to occur in a turbulent fluid where velocity fluctuations
would cause the flame to stretch. When these thermal gradients are subsequently
reduced below the quenching limit through turbulent fluctuations, the diusion flame
may propagate along the stoichiometric surface, re-igniting the quenched zones. The
characteristic flame structure that is observed at the edges of the stoichiometric sur-