Numerical studies of flames in wide tubes: Stability limits of curved stationary flames

Abstract

Flame dynamics in wide tubes with ideally adiabatical and slip walls is studied by means of direct numerical simulations of the complete set of hydrodynamical equations including thermal conduction, fuel diffusion, viscosity, and chemical kinetics. Stability limits of curved stationary flames in wide tubes and the hydrodynamic instability of these flames (the secondary Darrieus-Landau instability) are investigated. The stability limits found in the present numerical simulations are in a very good agreement with the previous theoretical predictions. It is obtained that close to the stability limits the secondary Darrieus-Landau instability results in an extra cusp at the flame front. It is shown that the curved flames subject to the secondary Darrieus-Landau instability propagate with velocity considerably larger than the velocity of the stationary flames.

Nonlinear cellular instabilities of planar premixed flames: numerical simulations of the Reactive Navier–Stokes equations

“…Asymmetric cells of this type are seen for one case of the solution of the MichelsonSivashinsky equation (Michelson and Sivashinsky 1977), but unlike here, it was later found that this was just a transitory stage and eventually the solution evolved to a single symmetric cell across the domain (Gutman & Sivashinsky 1990). However, for a domain width similar to the one used here (in terms of λ m ) and perturbation wavelength corresponding to our λ = 24 ∼ 2λ m case, Travnikov et al (2000) also found in their Le = 1 calculations that the cell evolved from a half a cell across the domain to a single asymmetric cell. The mechanism in their purely hydrodynamic case appears to be different, in that there is no temperature overshoot, and the new cusp forms in the interior of the domain not at the symmetry line corresponding to the original crest.…”

“…The first scheme is an explicit secondorder in time and space Godunov type solver (cf. Kadowaki (1997Kadowaki ( ,1999Kadowaki ( ,2000 and Travnikov et al (2000)). In this case, the hyperbolic fluxes are evaluated using a linearized Riemann solver while the diffusive terms are approximated by central differences.…”

“…they enforced the states along y = 0 and y = W to be identical. Here we instead opt for standard symmetry (or reflective) boundary conditions (zero normal velocity) on the y-boundaries, as also used by Travnikov et al (2000). They refer to such conditions as 'adiabatic, slip walls' and thus argue that their simulations correspond to flames in tubes (or more properly, in channels).…”

“…As in Kadowaki (1997Kadowaki ( ,1999Kadowaki ( ,2000 and Travnikov et al (2000), the governing equations of the model that we solve numerically are the compressible Reactive Navier-Stokes equations for a single reaction A→B. These are, in two-dimensions,…”

“…There is also a further question that, even if Kadowaki (1997Kadowaki ( ,1999Kadowaki ( ,2000 had used much larger domains, whether these simulations were in general ran for a long enough time for nonlinear bifurcations to occur and the solution to have evolved to the truly nonlinear stationary state. Travnikov, Bychkov & Liberman (2000) performed simulations for the purely hydrodynamic case (unit Lewis number) in domain widths slightly larger than λ m , with symmetry (reflective) boundary conditions (which unlike periodic conditions, allow solutions consisting of half cells in the domain). They found that even when the calculations were initiated with half a cell across the domain (wavelength slightly greater than 2λ m ) a new cusp eventually formed and the solution evolved to a single asymmetric cell in the domain, i.e.…”