A small-disturbance model of transonic combustion
Abstract
A new small-disturbance model for a steady, lean, premixed combustion at transonic speeds in a channel of slightly varying area is presented. Attention is confined to dilute premixtures so that exothermicity is weak. The study uses a distinguished limit type of analysis where the nonlinear interactions between (i) the near-sonic speed of the flow, (ii) the small changes in geometry from a straight channel, and (iii) the small heat release due to the one-step first-order Arrhenius chemical reaction, are explored. The asymptotic analysis results in the similarity parameters that govern the reacting flow field. Also, the flow can be described by a nonhomogeneous (extended) transonic small-disturbance (TSD) equation which is coupled with an ordinary differential equation for the calculation of the reactant mass fraction in the combustible gas. An iterative numerical scheme which combines the Murman and Cole method for the solution of the TSD equation with Simpson's integration rule for the estimation of the reactant mass fraction is developed. It is demonstrated that steady-state solutions of the compressible reacting flow problem with detonations behind shock waves can be found. The model is used to study transonic combustion at various inlet Mach numbers, amounts of incoming reactant mass, reaction rates, and channel geometries.