THE PROPAGATION OF PREMIXED FLAMES IN CLOSED TUBES

A nonlinear evolution equation that describes the propagation of a pre mixed flame in a closed tube has been derived from the general conserv ation equations. What distinguishes it from other similar equations is a memory term whose origin is in the vorticity production at the flam e 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 alpha is above a critical value alpha(c) the planar flame is the stable equilibrium solution. Fo r alpha below alpha(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 conditi ons the results indicate that, after a short transient, a flat flame d evelops when alpha > alpha(c) and it remains flat until it reaches the end of the tube. When alpha < alpha(c), on the other hand, stable cur ved flames may develop down the tube. Depending on the initial conditi ons the flame assumes either a cellular structure, characterized by a finite number of cells convex towards the unburned gas, or a tulip sha pe characterized by a sharp indentation at the centre of the tube poin ting toward the burned gases. In particular, if the initial conditions are chosen so as to simulate the elongated finger-like flame that evo lves from an ignition source, a tulip flame evolves downstream. In acc ord with experimental observations the tulip shape forms only after th e flame has travelled a certain distance down the tube, it does not fo rm 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 fla me results from the vortical motion created in the burned gas which is a consequence of the vorticity produced at the flame front.