### Abstract

A nonlinear evolution equation that describes the propagation of a premixed flame in a closed tube has been derived from the general conservation equations. What distinguishes it from other similar equations is a memory term whose origin is in the vorticity production at the flame front. The two important parameters in this equation are the tube's aspect ratio and the Markstein parameter. A linear stability analysis indicates that when the Markstein parameter α is above a critical value α_{c} the planar flame is the stable equilibrium solution. For α below α_{c} the planar flame is no longer stable and there is a band of growing modes. Numerical solutions of the full nonlinear equation confirm this conclusion. Starting with random initial conditions the results indicate that, after a short transient, a flat flame develops when α > α_{c} and it remains flat until it reaches the end of the tube. When α < α_{c}, on the other hand, stable curved flames may develop down the tube. Depending on the initial conditions the flame assumes either a cellular structure, characterized by a finite number of cells convex towards the unburned gas, or a tulip shape characterized by a sharp indentation at the centre of the tube pointing toward the burned gases. In particular, if the initial conditions are chosen so as to simulate the elongated finger-like flame that evolves from an ignition source, a tulip flame evolves downstream. In accord with experimental observations the tulip shape forms only after the flame has travelled a certain distance down the tube, it does not form in short tubes and its formation depends on the mixture composition. While the initial deformation of the flame front is a direct result of the hydrodynamic instability, the actual formation of the tulip flame results from the vortical motion created in the burned gas which is a consequence of the vorticity produced at the flame front.

Original language | English (US) |
---|---|

Pages (from-to) | 331-350 |

Number of pages | 20 |

Journal | Journal of Fluid Mechanics |

Volume | 336 |

DOIs | |

State | Published - Apr 10 1997 |

### Fingerprint

### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of Fluid Mechanics*,

*336*, 331-350. https://doi.org/10.1017/S0022112096004843

**The propagation of premixed flames in closed tubes.** / Matalon, Moshe; Metzener, Philippe.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 336, pp. 331-350. https://doi.org/10.1017/S0022112096004843

}

TY - JOUR

T1 - The propagation of premixed flames in closed tubes

AU - Matalon, Moshe

AU - Metzener, Philippe

PY - 1997/4/10

Y1 - 1997/4/10

N2 - A nonlinear evolution equation that describes the propagation of a premixed flame in a closed tube has been derived from the general conservation equations. What distinguishes it from other similar equations is a memory term whose origin is in the vorticity production at the flame front. The two important parameters in this equation are the tube's aspect ratio and the Markstein parameter. A linear stability analysis indicates that when the Markstein parameter α is above a critical value αc the planar flame is the stable equilibrium solution. For α below αc the planar flame is no longer stable and there is a band of growing modes. Numerical solutions of the full nonlinear equation confirm this conclusion. Starting with random initial conditions the results indicate that, after a short transient, a flat flame develops when α > αc and it remains flat until it reaches the end of the tube. When α < αc, on the other hand, stable curved flames may develop down the tube. Depending on the initial conditions the flame assumes either a cellular structure, characterized by a finite number of cells convex towards the unburned gas, or a tulip shape characterized by a sharp indentation at the centre of the tube pointing toward the burned gases. In particular, if the initial conditions are chosen so as to simulate the elongated finger-like flame that evolves from an ignition source, a tulip flame evolves downstream. In accord with experimental observations the tulip shape forms only after the flame has travelled a certain distance down the tube, it does not form in short tubes and its formation depends on the mixture composition. While the initial deformation of the flame front is a direct result of the hydrodynamic instability, the actual formation of the tulip flame results from the vortical motion created in the burned gas which is a consequence of the vorticity produced at the flame front.

AB - A nonlinear evolution equation that describes the propagation of a premixed flame in a closed tube has been derived from the general conservation equations. What distinguishes it from other similar equations is a memory term whose origin is in the vorticity production at the flame front. The two important parameters in this equation are the tube's aspect ratio and the Markstein parameter. A linear stability analysis indicates that when the Markstein parameter α is above a critical value αc the planar flame is the stable equilibrium solution. For α below αc the planar flame is no longer stable and there is a band of growing modes. Numerical solutions of the full nonlinear equation confirm this conclusion. Starting with random initial conditions the results indicate that, after a short transient, a flat flame develops when α > αc and it remains flat until it reaches the end of the tube. When α < αc, on the other hand, stable curved flames may develop down the tube. Depending on the initial conditions the flame assumes either a cellular structure, characterized by a finite number of cells convex towards the unburned gas, or a tulip shape characterized by a sharp indentation at the centre of the tube pointing toward the burned gases. In particular, if the initial conditions are chosen so as to simulate the elongated finger-like flame that evolves from an ignition source, a tulip flame evolves downstream. In accord with experimental observations the tulip shape forms only after the flame has travelled a certain distance down the tube, it does not form in short tubes and its formation depends on the mixture composition. While the initial deformation of the flame front is a direct result of the hydrodynamic instability, the actual formation of the tulip flame results from the vortical motion created in the burned gas which is a consequence of the vorticity produced at the flame front.

UR - http://www.scopus.com/inward/record.url?scp=0031126321&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031126321&partnerID=8YFLogxK

U2 - 10.1017/S0022112096004843

DO - 10.1017/S0022112096004843

M3 - Article

AN - SCOPUS:0031126321

VL - 336

SP - 331

EP - 350

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -