NUMERICAL MODELING OF LAMINAR COFLOW FLAMES WITH DETAILED KINETICS

In this paper a numerical algorithm for the numerical modeling of multidimensional laminar flames with detailed kinetic mechanisms is described. The proposed approach is based on the operator-splitting technique, in order to exploit the best numerical method available for the treatment of reacting, stiff processes and transport, non-stiff processes. The algorithm was implemented in the OpenFOAM® framework and was validated for a laminar coflow flame fed with ethylene. The numerical simulations (performed with a detailed kinetic mechanism with ~170 species and ~4800 reactions) showed a satisfactory agreement with the experimental data (temperature and composition along the centerline), demonstrating the feasibility of the suggested methodology. Introduction The detailed numerical simulation of multidimensional laminar flames flows with realistic chemical mechanisms represents a challenging problem and places severe demands on computational resources. The numerical description of laminar flames results in a complex mathematical model, because of the large number of chemical species involved and to the characteristics of the flames (presence of high gradient regions, flame fronts, etc). When detailed kinetic mechanisms are used, special attention has to be paid to the numerical algorithms, which must be sufficiently accurate and efficient. In particular, the numerical methods have to be able to manage the resulting set of governing very stiff equations. At the same time, the discretization has to be fine enough to treat the flame fronts and the high gradients adequately. As a consequence, the computational effort in terms of CPU time and memory requirements is considerable and in many cases prohibitive. Conventional CFD methods based on segregated algorithms have serious difficulties to treat the stiffness and the high non-linearities of the governing equations and cannot be efficiently applied in this context [1]. In order to overcome these problems, coupled methods appear to be an attractive alternative. In particular, among others, two main numerical approaches have been used for the resolution of such a stiff system of equations: (i) fully coupled algorithms [2]; (ii) segregated algorithms based on operator-splitting methods [3]. An advantage of fully coupled algorithms is that all the processes are considered simultaneously, so all physical interactions among processes are taken into account together. However, the resulting system of XXXIV Meeting of the Italian Section of the Combustion Institute 2 governing equations can be extremely large (especially when detailed kinetic mechanisms and complex, geometries are considered). When operator-splitting methods are used, the governing equations are split in subequations, usually with each having a single operator, which captures only a portion of the physics present. Splitting approaches can be conveniently applied for the numerical solution of combustion problems, by separating the stiff chemical reaction from the non-stiff transport processes. An advantage of this approach is that it avoids many costly matrix operations (typical of fully coupled algorithms) and allows the best numerical method to be used for each type of process. A disadvantage with respect to fully coupled methods is that separate algorithms can be very complex and usually differ from term to term. In the present paper the operator-splitting methodology employed by the authors in the detailed numerical simulation of laminar flames is briefly described, together with some details about its implementation in the OpenFOAM® framework [4]. Then, the results deriving from the application of this methodology to the numerical modeling with a detailed kinetic mechanism of a co-flow axis-symmetric laminar C2H4/air flame (experimentally studied in [5]) are presented. Numerical methods The OpenFOAM® framework [4] is used to solve the usual gas-phase transport equations of continuity, momentum, energy and species mass fractions (not here reported) [3]. The standard OpenFOAM® solver for compressible, unsteady, nonreacting flows (pisoFoam) was modified in order to make possible the introduction of detailed kinetic mechanisms. As reported before, an operatorsplitting approach, based on the separation of transport and reaction terms, was implemented and is briefly described in the following [6]. For a general transport/reaction system like a laminar flame described by a set of partial differential equations (PDEs), the governing PDEs can be transformed into a set of ODEs by the spatial discretization and the application of the method of lines: ( ) ( ) , d t dt = + ψ S ψ M ψ (1) where t is the time, ψ the dependent (or primary) variables (mass fractions and enthalpy), ( ) S ψ is the rate of change of ψ due to chemical reactions and ( ) ,t M ψ the rate of change of ψ due to transport processes, such as diffusion, convection, heat loss, inflow/outflow, etc. In order to solve Equations (1) numerically, the time is discretized in increments Δt, and the integration in time is then performed using the so-called Strang splitting scheme [7]. Reaction is separated from the transport process and the numerical integration is performed in 3 sub-steps: Sub-step 1. The reaction terms are integrated over a time interval Δt/2 through the solution of a ODE system: XXXIV Meeting of the Italian Section of the Combustion Institute