Application of the finite-element method to one-dimensional flame propagation problems

Abstract

A linear semidiscrete Galerkin method and an adaptive finite-element method are first used to compute the steady-state wave speed of a reaction-diffusion equation which has an exact traveling wave solution. The Galerkin method is then applied to study the propagation of a laminar flame in a closed combustor. The numerical results obtained with the Galerkin and adaptive finite-element methods are compared to those obtained with a finite-difference Crank-Nicolson scheme. The comparisons show that the finite-element methods overpredict the wave speed, whereas the Crank-Nicolson scheme underpredicts it. The Galerkin method results are closer to the exact solution than those of the Crank-Nicolson scheme for a 901 point grid. The adaptive finite element requires about 171 points to obtain a wave speed equal to that of the Galerkin method with 901 points. The application of the Galerkin method to the propagation of one-dimension al enclosed deflagrations shows that, in order to account properly for the steep temperature gradients at the flame front, at least 400 grid points are required. The largest temperature difference between the finite-difference and finite-element results is less than 2%. This difference is attributed to the oscillations present in the finite-element method, the linearization of the reaction terms and the use of linear basis.