Inertia effect on a structure of pressure-driven flames in porous media

The phenomenon of a subsonic pressure-driven flame in an inert porous medium filled with a flammable gaseous mixture is considered in the present work. The paper focuses on the analysis of the impact of the inertia of a fluid on the fine structure of the flame front and its velocity. In the frame of reference attached to an advancing combustion wave and after a suitable non-dimensionalization the corresponding mathematical model includes three highly nonlinear ordinary differential equations (ODEs). The system is converted into a singularly perturbed system of ODEs by a suitable choice of new phase coordinates and then treated analytically along the lines of a suggested asymptotic machinery (modified version of the method of invariant (integral) manifolds - MIM). According to the MIM, an arbitrary solution of the initial system of ODEs under consideration is represented as a trajectory in the phase space. It is shown that two principally different parts of the trajectory exist: fast motion from the initial point to a slow curve and another fast motion from the matching point lying on the slow curve to the singular (final) point of the system. The first stage is associated with a pre-heat subzone of the flame, whereas the second is interpreted as a reaction subzone. It is demonstrated that the matching point of these two parts of the trajectory plays a crucial role in the description of the flame characteristics. The proposed analytical approach allows one to obtain an analytical expression for the effect of inertia on the flame velocity. The theoretical predictions show reasonably good agreement with data resulting from direct numerical simulations.