The reactive Rayleigh-Bénard problem with throughflow

A. Bayliss*, T. K. Ma, B. J. Matkowsky, C. W. Wahle

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We consider the Rayleigh-Bénard problem for a combustible gaseous mixture in a two-dimensional channel formed by upper and lower horizontal plates, with periodic boundary conditions employed in the horizontal direction. Fresh unburned fluid is injected through the upper plate and burned products removed through the lower plate. The lower plate is heated to the burning temperature of the mixture so that, for appropriate injection rates, a flame forms within the channel. Using the Boussinesq model we describe the results of numerical computations to illustrate fluid-flame interactions. The model we consider is motivated by recent experiments on flames in the thin annular region between two closely spaced, finite, coaxial cylinders, formed by upper and lower horizontal plates. Here, the angular direction in the annulus corresponds to the horizontal direction in the channel. Our model results if the curvature of the cylinders is ignored, as is reasonable for relatively large cylindrical radii. Instabilities can ensue even if thermal expansion is ignored, as in the diffusional thermal model where density variations are not accounted for. Moreover, fluid instabilities, e.g., convection cells, can ensue even for nonreactive mixtures. We consider parameter regimes in which the instabilities and resulting patterns that we observe are due to fluid-flame interactions. In addition to stationary flat and cellular flames, we compute a variety of nonstationary, cellular flames, with associated convection rolls, including (i) flames with periodic and aperiodic episodes of cell creation (splitting) and annihilation (merging) at selected locations, (ii) heteroclinic cycles in which long periods of nearly quiescent cellular behavior are interrupted by episodes of cell splitting and merging, (iii) traveling and modulated traveling waves of cellular flames, (iv) traveling blinking modes in which a traveling array of cells is periodically interrupted by merging/splitting events which alternate between the two halves of the spatial domain when viewed in a moving coordinate system, (v) breathing cellular flames in which cells whose mean location is fixed in space oscillate periodically, and (vi) chaotic flames. We also describe perforated flames which exhibit incomplete consumption of the deficient component locally in space and time.

Original languageEnglish (US)
Pages (from-to)1103-1142
Number of pages40
JournalSIAM Journal on Applied Mathematics
Issue number4
StatePublished - 2000


  • Boussinesq
  • Buoyancy
  • Combustion
  • Rayleigh-Bénard convection

ASJC Scopus subject areas

  • Applied Mathematics


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