Hydrodynamically unstable premixed flames in gravitational fields: Effect of gas expansion

The hydrodynamic instability is a long wave instability that results in premixed flames from gas expansion caused by the heat released during combustion. It was discovered independently by Darrieus and Landau treating the flame as a structureless surface of discontinuity, and bears their names suitably. Investigations that followed their pioneer work revealed that diffusion effects within the flame zone may, for mixtures deficient in their less mobile reactants, stabilize short wavelength disturbances. Thus, unless confined to a narrow domain, a planar flame remains unconditionally unstable. Landau has also examined the role of gravity and concluded that the long wavelength disturbances are stabilized by gravitational forces when the flame is propagating downwards. In the present paper, we study the composite effects of thermal expansion, differential diffusion, and gravity on the flame dynamics. We use a hydrodynamic theory that properly resolves the flame structure and extracts an explicit expression for the flame speed with a coefficient that lumps all the physicochemical properties of the combustible mixture. This nonlinear model enables studying both, the initial flame development by means of a linear stability analysis and the nonlinear development using a hybrid Navier–Stokes/flame-tracking methodology. The linear analysis yields explicit conditions for the growth rate of disturbances of arbitrary wavelength, and provides the critical conditions for the instability onset. The consequence of the instability has been determined numerically by examining the long time flame evolution, focusing on the flame morphology and the overall propagation speed. We examine both, downward and upward propagation and exploit the following three parameters: the unburned-to-burned density ratio or thermal expansion parameter, the Markstein number that depends on the composition of the mixture and the diffusive properties of the reactants, and the ratio of the gravitational to the inertial forces, or the gravitational parameter. Our results are summarized by means of bifurcation diagrams that elucidate the role of these parameters on the flame propagation. Novelty and significance This paper consists of a comprehensive stability analysis, linear and nonlinear of planar flames propagating upwards and downwards. The linear analysis covers a wide range of the three relevant parameters: the unburned-to- burned density ratio or thermal expansion parameter, the Markstein number that depends on the pressure level, the composition of the mixture, and the diffusive properties of the reactants, and the ratio of the gravitational to the inertial forces, or reciprocal of the Froude number. The nonlinear analysis is the first that considers realistic values of the gas expansion and describes the evolving flame morphology and the flame propagation speed for a wide range of Markstein numbers under different Froude numbers.