Theoretical analysis of the mass balance equation through a flame at zero and non-zero Mach numbers

This paper discusses a classical paradox in thermoacoustics when jump conditions are derived for acoustic waves propagating through a thin flat flame. It shows why volume conservation must be used for perturbations at zero Mach number (continuity of v′=u′A) while mass conservation is used at non-zero Mach numbers (continuity of m′=ρ¯0u′A+u¯0ρ′A). First, from the three-dimensional mass balance equation, a quasi one-dimensional mass balance equation is obtained for surface-averaged quantities. Then it is demonstrated that the acoustic and entropy disturbances are coupled and need to be solved together at the flame front because singularities in the entropy profile affect mass conservation. At non-zero Mach number, the entropy generated in the thin flame is convected by the mean flow: no singularity occurs and leads to the classical mass conservation at the interface. However, at zero Mach number, the flow is frozen and entropy spots are not convected downstream: they produce a singularity at the flame front due to the mean density gradient, which acts as an additional source term in the mass conservation equation. The proper integration of this source term at zero Mach number leads, not to the mass, but to the volume flow rate conservation of perturbations. A balance equation for the volume flow rate has been also derived. This equation couples the volume flow rate and the mean and fluctuating pressure. This latter equation degenerates naturally toward the volume flow rate conservation at the flame interface at zero Mach number because of the pressure continuity. This theoretical analysis has been compared to LEE (Linearized Euler Equation) simulations of stable flames and a good agreement is found for the entropy fluctuations shape and the conserved quantities.