Flame acceleration due to wall friction: Accuracy and intrinsic limitations of the formulations

The analytical formulations on the premixed flame acceleration induced by wall friction in two-dimensional (2D) channels [Bychkov et al., Phys. Rev. E 72 (2005) 046307] and cylindrical tubes [Akkerman et al., Combust. Flame 145 (2006) 206] are revisited. Specifically, pipes with one end closed are considered, with a flame front propagating from the closed pipe end to the open one. The original studies provide the analytical formulas for the basic flame and fluid characteristics such as the flame acceleration rate, the flame shape and its propagation speed, as well as the flame-generated flow velocity profile. In the present work, the accuracy of these approaches is verified, computationally, and the intrinsic limitations and validity domains of the formulations are identified. Specifically, the error diagrams are presented to demonstrate how the accuracy of the formulations depends on the thermal expansion in the combustion process and the Reynolds number associated with the flame propagation. It is shown that the 2D theory is accurate enough for a wide range of parameters. In contrast, the zeroth-order approximation for the cylindrical configuration appeared to be quite inaccurate and had to be revisited. It is subsequently demonstrated that the first-order approximation for the cylindrical geometry is very accurate for realistically large thermal expansions and Reynolds numbers. Consequently, unlike the zeroth-order approach, the first-order formulation can constitute a backbone for the comprehensive theory of the flame acceleration and detonation initiation in cylindrical tubes. Cumulatively, the accuracy of the formulations deteriorates with the reduction of the Reynolds number and thermal expansion.