Numerical simulation of flame instability in lean H 2 / air mixtures

This paper reports on studies of the instability of lean hydrogen/air laminar premixed flames with the equivalence ratio of 0.6. Two intrinsic instability mechanisms are involved in the flames, hydrodynamic instability and thermal-diffusive instability. Numerical simulations employing detailed chemical kinetics and transport properties are carried out to simulate the initial linear growth of instability as well as the nonlinear evolution of flame instability under two different pressures, 5 atm and 25 atm. A small perturbation of sine wave with a prescribed wavelength and small amplitude is first placed on the initial planar laminar flame front. Then, the evolution of the sine wave shaped flame front is simulated. Chaotic flame fronts are captured whose nonlinear evolution is shown to be very sensitive to initial perturbations. In the initial linear growth stage, the amplitude of the initial sine wave flame front grows exponentially. Later on, the flame front develops into wrinkling surface with different wavelengths and amplitudes. Preference of cellular flame cells with even or odd modes has been observed which is keen on initial perturbation with small wave numbers. With smaller even number of waves in the initial perturbation of the flame front, e.g. six waves, the even wave modes are amplified and odd wave modes are suppressed; whereas, with bigger number of initial waves, e.g. eleven waves, the even and odd wave modes can compete fairly with each other. After a long time, the long wavelength mode dominates. In high-pressure flames, the flames are thinner, both of hydrodynamic and thermal-diffusive instability are enhanced, and the flame fronts are more chaotic. Introduction Intrinsic flame instability is a phenomenon frequently seen in laminar flames [1-6], e.g. in flames with non-unity Lewis numbers. It has implication in engineering applications, e.g. safety in nuclear power plants [1]. There are several mechanisms of intrinsic flame instability: the hydrodynamic instability (also known as Landau-Darrieus (LD) instability) induced by thermal expansion from exothermal reactions [2, 3], thermal-diffusive (TD) instability owing to differential diffusion between thermal diffusion and mass diffusion [4, 5], and RayleighTaylor instability (with a larger scale) due to buoyancy force. Without considering turbulence and body force, hydrodynamic instability and thermal-diffusive instability result in multidimensional unstable cellular shaped flames frequently. These have been observed in the experiments and have been reviewed by several researchers [7-11]. Formation of cellular flame from an initial planar flame has been studied theoretically. Usually a dispersion relation is found from linear stability analysis, which states the influence of parameters contained in models on stability at the linear stage of flame evolution. Considering both instabilities, hydrodynamic and thermal-diffusive, the analyses based on asymptotic linear stability theories assuming high activation energy have been performed [1216]. Recently, finite activation energy linear stability analyses have been carried out [17-19], which provide conditions closer to reality. Beyond the onset of instability, linear stability theories fail to simulate the flame front evolution. Weakly nonlinear flame front evolution theories with asymptotic limits of high activation energy have been developed by Sivashinsky et al., e.g. Kuramoto–Sivashinsky (KS) equation containing purely thermal-diffsive nature, Michelson–Sivashinsky (MS) equation accommodating purely hydrodynamic instability, and for a mixed problem, a third equation which combined by the KS and MS equations [20]. These equations are solved using numerical methods to study the nonlinear evolution of flame [21-23]. MS equation has been extended to higher order models [24, 25]. Numerical simulations of the nonlinear evolution of flame have been carried out by solving full Navier-Stokes (NS) equations for either compressible flows [26-29] with oneor two-step reaction chemistry models, or incompressible flows [30-32] with detailed chemistry model, in order to capture cellular instability more accurately, and to examine and develop linear and nonlinear theoretical models. The approach is used in this work. Complex flame dynamics is invoked by hydrodynamic instability, damped or enhanced by thermal-diffusive instability whose key parameter is the effective Lewis number, or equivalently, effective Markstein number [11]. For premixed mixture of subunity Lewis number, e.g. H2/air, detailed numerical simulations performed with oneor two-step reaction chemistry models [26-29] have shown very intense nonlinear behaviors of flames including cell-merging and splitting, symmetry breaking bifurcation, local quenching and cell lateral movement. Only a few numerical simulations have been performed with detailed chemistry mechanisms and transport properties with subunity Lewis number [32-34], where Soret effect [33] and turbulence effect [34] are studied. Altantzis et al. [32] used low Mach number code with spectral element method and Li’s detailed H2 chemistry [35] model to simulate 2D planar lean premixed H2/air flames. They reported the results of linear instability, the influence of width of computational domain on the nonlinear flame front evolution, and the behavior of flame front in a wide domain. The results show that TD instability is increased with larger domain width. In this work, we investigate the nonlinear evolution of 2D planar lean H2/air premixed flame in a wide region which can correctly predict multicell interaction, validated by Yuan et al. [26]. This is motivated by the fact that systematic and general interpretation has not established yet in the nonlinear evolutions of flame fronts with coupling LD instability and TD instability. In addition, we consider the long time evolution of flame front at the condition of high pressure. The content of this paper is as follows: Firstly, the computational setup is presented. The simulation case is similar to the one of large computational domain [32]. Results are analyzed and compared with the previous results of [32]; then high pressure case is simulated and pressure dependence is investigated; finally, dependence of flame instability on the initial perturbation is studied. Numerical methods Laminar lean hydrogen/air premixed flame propagation in a two-dimensional domain is considered. The flow speed is low and essentially it can be assumed as low Mach number flow. The physical pressure can be decomposed into two parts, one thermodynamic pressure that is uniform throughout the cylinder and one hydrodynamic pressure that appears in the momentum equation. The mixture considered is in gas phase, without body force, and with negligible thermal radiation. The governing equations are made up of the transport equations of species mass fraction and energy, the Navier-Stokes equations, and the continuity equation,