19 th Australasian Fluid Mechanics Conference Melbourne, Australia 8-11 December 2014 Lattice Boltzmann Modeling of Autoignition in Non-uniformly Heated Mixtures V. Novozhilov 1 1 Centre for Environmental Safety and Risk Engineering Victoria University, Victoria 8001, Australia Abstract Lattice Boltzmann (LB) computational model is used to simulate autoignition (thermal explosion) development in mixtures with non-uniform initial temperature conditions. The paper demonstrates LB modeling application to practically important reacting flow problem. Complications to the classical thermal explosion problem arise in the presence of dynamical heat exchange (natural and/or forced convection), combined with non-uniform initial conditions in the reacting mixture. The present study reports critical conditions for thermal explosion in such circumstances. Introduction The problem of autoignition (also referred to as thermal explosion or thermal runaway) has been studied for decades in various formulations, e.g. [1-4]. Nevertheless, complications of this problem arising in the presence of dynamical heat exchange, either self-exerted by natural convection, or introduced by forced convection has been poorly investigated. Further complications arising from non-uniform initial conditions in reacting mixture have never been studied. The question of how the initial non- uniformities in the mixture temperature field affect autoignition development is of practical importance. The present paper addresses this issue in the case of natural convection conditions. As is always the case in the theory of thermal explosion, critical conditions for autoignition are of primary interest. In the view of this, essentially an induction period of the thermal explosion is being modeled. The model predicts natural convective flows developing at this stage, associated chemical reaction and energy dissipation rates, as well as the onset of the thermal explosion (for super-critical conditions). The conditions leading to autoignition are formulated in terms of the critical Frank-Kamenetskii parameter. Effects of convection on the critical conditions are described using ratios of the respective critical Frank-Kamenetskii parameters to the ones corresponding to no-convection conditions. Effects of non- uniform initial temperature distributions are described in terms of wavelength and amplitude of the temperature perturbation. The parameters that are varied in simulations are flow Rayleigh numbers, as well as geometrical parameters describing initial non-uniformities in the mixture temperature field. The Lattice Boltzmann (LB) method [5,6] is used as numerical technique in the present study. This approach becomes increasingly attractive as a fast and efficient method of solution of partial differential equations. However, application of LB method to combustion problems remains very limited, e.g. [7- 10]. The present study demonstrates the potential of LB method in application to such problems. In-house CFD code LBMComb [10] is used. Mathematical Model Flow is considered in the Boussinesq approximation. The rationale for this assumption is that effectively only induction period of thermal explosion is considered. Separation of explosion and no-explosion regimes becomes evident before temperature variations become large enough to fail the Boussinesq approximation. The set of governing equations is therefore as follows:   0 u t       (1)   1 Re g u u u p u Ar e t             (2)   1 1 exp 1 u Pe Ar t Ar                         (3) This non-dimensional form of equations is obtained by choosing the particular length scale L , velocity scale Lg , time scale 1 Lg  , density scale 0  and pressure scale 0 Lg  . p  is pressure deviation from the background level, Re and Pe are Reynolds and Peclet numbers, respectively, g e is a unit gravity vector. Excess temperature is defined as   0 0 T T ArT  where 0 RT Ar E  is Arrhenius number. The expression for the chemical source follows standard reaction rate dependence on temperature. Reactants consumption is neglected as only the induction stage of thermal explosion is considered. The Frank-Kamenetskii parameter   1 1 1 0 0 exp p QB LT Ar Ar c g         is defined in a slightly different manner compared to conventional. Here Q is the heat of reaction and B is the pre-exponential factor. The Lattice Boltzmann equation is used here with the Bhatnagar- Gross-Krook (BGK) collision model. Accordingly, solutions of the continuum, momentum and energy equations (1-3) are approximated by the families of distribution functions , iu f and , i f  evolving by