Vortex breakdown in premixed reacting flows with swirl in a finite-length circular open pipe

Abstract

A global analysis of steady states of low Mach number inviscid premixed reacting swirling flows in a straight circular finite-length open pipe is developed. We focus on modelling the basic interaction between the swirl and heat release of the reaction. For analytic simplicity, a one-step first-order Arrhenious reaction kinetics is considered in the limit of high activation energy and infinite Peclet number. Assuming a complete reaction with chemical equilibrium upstream and downstream of the reaction zone, a nonlinear partial differential equation is derived for the solution of the flow stream function downstream of the reaction zone in terms of the specific total enthalpy, specific entropy and circulation functions prescribed at the inlet. Several types of solutions of the nonlinear ordinary differential equation for the columnar flow case describe the outlet states of the flow in a long pipe. These solutions are used to form the bifurcation diagram of steady reacting flows with swirl as the inlet swirl level is increased at a fixed heat release from the reaction. The approach is applied to two profiles of inlet flows, the solid-body rotation and the Lamb-Oseen vortex, both with constant profiles of the axial velocity, temperature and mixture reactant mass fraction. The computed results provide theoretical predictions of the critical inlet swirl levels for the appearance of vortex breakdown states and for the size of the breakdown zone as a function of the inlet flow swirl level, Mach number and heat release of the reaction. For the inlet solid-body rotation, flow is decelerated to breakdown as the inlet swirl is increased above the critical swirl level, and there is a delay in the appearance of breakdown with the increase of the heat release of the reaction. For the inlet Lamb-Oseen vortex at low values of heat release, the critical swirl for breakdown is decreased with the increase of heat release while, at high values of heat release, the appearance of breakdown is delayed to higher incoming flow swirl levels with the increase of heat release. The analysis sheds light on the global dynamics of low Mach number reacting flows with swirl and vortex breakdown and on the interaction between vortex breakdown and heat release that affects the shape of the reaction zone in the domain.