A priori analysis of turbulent flamelet combustion in a premixed spherical flame kernel R.J.M. Bastiaans * , J.A. van Oijen and L.P.H. de Goey Section Combustion Technology, Department of Mechanical Engineering Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands Abstract Three-dimensional direct numerical simulations are performed of turbulent combustion of initially spherical flame kernels. The chemistry is described by a progress variable which is attached to a flamelet library. In order to progress towards turbulent closure of such a system the validity of the flamelet approach must be studied. Then the turbulent closure of the equation for the progress variable should be investigated. The influence of flame stretch and curvature on the local mass burning rate is studied and compared to an analytical model. Then first investigations of the relative importance of the unresolved turbulent flux compared to the source term of the progress variable are presented. 1 Motivation and objective The present research is concerned with the direct nu- merical simulation (DNS) and analysis of turbulent prop- agation of premixed flame kernels. The simulations are direct in the sense that the smallest scales of motion are fully resolved, while the chemical kinetics are solved in advance and parameterized in a table by the method of the flamelet generated manifolds (FGM) [12]. The state of the reactions are assumed to be directly linked to a single progress variable. The conservation equation for this progress variable is solved using DNS, with the un- closed terms coming from the table. This allows the use of detailed chemical kinetics without having to solve the individual species conservation equations. In the present study only a single progress variable is used, which shows good results, however in [13] it is shown that adding a sec- ond progress variable even increases the accuracy with an order of magnitude. Question is whether the FGM method is able to handle flame stretch and curvature effects. If this is the case it can be stated that the method is able to deal with turbulent combustion in the flamelet regime. In that case it is also possible to use the flamelets in integrations over the flame surface density, which is used to close filtered equations in turbulence modelling. These points are the topic of the present paper. Stretch Flame stretch is an important parameter that is recog- nized to have a determining effect on the burning velocity in premixed flames. In the past this effect has not been taken into account in the flamelet approach for turbulent combustion in a satisfying manner. The laminar burning velocity, which is largely affected by stretch, is an impor- tant parameter for modelling turbulent combustion. Flame stretch is also responsible for the creation of flame surface * Corresponding author: r.j.m.bastiaans@tue.nl Associated Web site: http://www.combustion.tue.nl Proceedings of the European Combustion Meeting 2005 area, affecting the consumption rate as well. In the turbu- lent case, stretch rates vary significantly in space and time. An expression for the stretch rate is derived directly from its mass-based definition in [8], K = 1 M dM dt , (1) where M is the amount of mass in an arbitrary control volume moving with the flame velocity M = Z V (t) ρdV, (2) resulting in a relation for the stretch rate to be ρK = ∂ ∂x i (ρs L n i ). (3) Here n i is the normal to the flame. On the basis of this definition, a model for the influence of stretch and curvature on the mass burning rate has been developed, which in principle also holds for strong stretch in turbu- lent flames. In a numerical study [9], it was shown that this model, with a slight reformulation, shows good agree- ment with calculations for spherically expanding laminar flames. This formulation, for the ratio of the actual mass burning rate at the inner layer, m in , relative to the unper- turbed mass burning rate at the inner layer, m 0 in (for unity Lewis numbers), reads m in m 0 in =1 -Ka in , (4) with the Karlovitz integral being a function of flame stretch (1), flame surface area, σ, and a progress variable, Y , Ka in := 1 σ in m 0 in   s b Z s u σρKY ds - s b Z s in σρKds   . (5) The integrals have to be taken over paths normal to the flame and s u , s b and s in are the positions at the unburned