A numerical study of the diffusive-thermal instability of opposed nonpremixed tubular flames

The diffusive-thermal (D-T) instability of opposed nonpremixed tubular flames near extinction is investigated using two-dimensional (2-D) direct numerical simulations together with the linear stability analysis. Two different initial conditions (IC), i.e. the perturbed IC and the C-shaped IC are adopted to elucidate the effects of small and large amplitude disturbances on the formation of flame cells, similar to conditions found in linear stability analysis and experiments, respectively. The characteristics of the D-T instability of tubular flames are identified by a critical Damköhler number, Da_C, at which the D-T instability first occurs and the corresponding number of flame cells for three different tubular flames with different flame radii. It is found that Da_C predicted through linear stability analysis shows good agreement with that obtained from the 2-D simulations performed with two different ICs. The flame cell number, N_cell, from the 2-D simulations with the perturbed IC is also found to be equal to an integer close to the maximum wavenumber, k_max, obtained from the linear stability analysis. However, N_cell from the 2-D simulations with the C-shaped IC is smaller than k_max and N_cell found from the simulations with the perturbed IC. This is primarily because the strong reaction at the edges of the horseshoe-shaped cellular flame developed from the C-shaped IC is more likely to produce larger flame cells and reduce N_cell. It is also found that for cases with the C-shaped IC, once the cellular instability occurs, the number of flame cells remains constant until global extinction occurs by incomplete reaction manifested by small Da. It is also verified through the displacement speed, S_d, analysis that the two edges of the horseshoe-shaped cellular flame are stationary and therefore do not merge due to the diffusion-reaction balance at the edges. Moreover, large negative S_d is observed at the local extinction points while small positive or negative S_d features in the movement of flame cells as they adjust their location and size towards steady state.