Flame behavior during propagation in small isothermal tubes characterized by different degrees of the end opening

In the present work, we study numerically freely propagating flame in the stoichiometric propane-air mixture. The isothermal small tubes with one end fully open and the second one characterized by different degrees of opening are examined. The degree of opening of the tubes was equal to: 0% (completely closed), 25%, 50%, 75% and 100% (fully opened) of the tube cross-sectional area. Several mechanisms, such as thermal expansion of the burned gas which can leave the tube freely (fully opened left end of the tube), frictional forces and heat losses at the tube walls, movement of the unburned mixture generated by the propagating flame, occur simultaneously during flame propagation. Only the partial opening of the right end of the tube limits the flow of the unburned gas outside this end. This results in an additional pressure gradient and a rapid increase in velocity near the right end. The heat losses to walls cause that behind the flame pressure drops up to negative values. During propagation flames change their shapes and elongates. As a result of these mechanisms, we observe that flame speed change is almost linear for greater parts of tubes. This increase depends on the right end opening and tube diameter. Our examinations show that flame speed loses its linear relationship only for the bigger tube and right end opening in the range 75-100%. It suggests that heat losses to the wall restraint a sudden flame acceleration.