Combustion and Flame 173 (2016) 235–244 Contents lists available at ScienceDirect Combustion and Flame journal homepage: www.elsevier.com/locate/combustﬂame Formalism for spatially averaged consumption speed considering spherically expanding ﬂame conﬁguration Alexandre Lefebvre, Hakim Larabi, Vincent Moureau, Ghislain Lartigue, Emilien Varea, Vincent Modica, Bruno Renou ∗ Normandie Univ., INSA Rouen, UNIROUEN, CNRS, CORIA, Rouen 76000, France a r t i c l e i n f o Article history: Received 8 April 2016 Revised 31 May 2016 Accepted 29 August 2016 Available online 13 September 2016 Keywords: Spherically expanding ﬂame Consumption speed Markstein length a b s t r a c t The determination of laminar burning velocity is a complex task. Even though its deﬁnition is well es- tablished, achieving a good and reliable value of this quantity seems to be too dependent on the exper- imental procedures. In this study, we report a rigorous derivation of the relation between the spatially averaged consumption speed and the absolute ﬂame speed for spherically expanding ﬂame conﬁgura- tions. The general expression of the consumption speed in conﬁned geometries makes it possible to re- trieve classical deﬁnitions developed in the literature over the years. It has been highlighted that the an- alytical development for the consumption speed is free from restrictive assumptions or approximations (stretch or thermodynamical gas states) that are generally made in classical approaches. The develop- ment brings up to identify two equivalent radii, one from a surface and one from a volume integration, respectively. The analytical developments are tested using a 3D DNS including full transport and complex chemistry. CH 4 /air ﬂame at three equivalence ratios (lean, stoichiometric and rich) and a stoichiomet- ric iso-octane/air ﬂame are tested. Results show that any species, reactants or products, can be used to evaluate the analytical expression of the consumption speed. © 2016 The Combustion Institute. Published by Elsevier Inc. All rights reserved. 1. Introduction Recent advances in numerical simulations have demonstrated the ability of CFD codes to simulate and accurately predict com- plex combustion processes from fuel injection to pollutant emis- sions. It is worth noting that pollutant emission regulation is a key parameter in designing combustion systems such as aircraft engines, helicopter turbines or industrial burners. With the emer- gence of new fuels, the corresponding kinetic schemes need to be developed and validated for large ranges of operating conditions, expressed in terms of equivalence ratio, pressure and temperature. Kinetic schemes are generally validated based on ignition delays time, major and/or minor species proﬁles and unstretched laminar burning velocity S 0 l data. Laminar burning velocity S 0 l is a fundamental ﬂame property which depends on the fuel/air mixture and its initial thermody- namic conditions: pressure, temperature and equivalence ratio. It represents the rate at which the fresh gases are consumed through the ﬂame front considering a 1D unstretched propagating planar premixed ﬂame. By integrating the transport equation of fuel mass ∗ Corresponding author. Fax: +33 2 32 95 97 80. E-mail address: renou@coria.fr (B. Renou). fraction over the ﬂame domain, one can easily derive the expres- sion of the laminar burning velocity for 1D planar ﬂames [1] as: S 0 l = 1 ρ u ( Y th f ,b − Y f ,u )  +∞ −∞ ˙ ω f dx, (1) where ρ u is the fresh gas density, ˙ ω f is the fuel reaction rate, and Y f,u and Y th f ,b are the fuel mass fractions in the fresh and burned gases, respectively. 1 The laminar burning velocity S 0 l is a consump- tion speed and corresponds to the fuel mass rate which enters the ﬂame front. It is worth noting that the expression of the laminar burning velocity is valid for any species k. For the 1D geometrical conﬁguration, S 0 l also corresponds to the ﬂame displacement speed relative to the fresh gases, S 0 d,u . The latter represents the motion of the ﬂame and is deﬁned as the speed of an iso-surface relative to the ﬂow of reactants [1–3] as: S 0 d,u = S f − u g,u , (2) where u g,u refers to as the fresh gas side velocity. S f is the abso- lute ﬂame speed or propagation speed. Since the ﬂow accelerates 1 The upper script th denotes that the burned gases are taken at equilibrium con- ditions. In the following, the density ρ th b and the mass fraction of a specie k in burned gases Y th k,b will be obtained from equilibrium code. http://dx.doi.org/10.1016/j.combustﬂame.2016.08.024 0010-2180/© 2016 The Combustion Institute. Published by Elsevier Inc. All rights reserved.