Lean premixed reacting flows with swirl and wall-separation zones in a contracting open circular chamber

A model problem of low-Mach-number lean premixed reacting swirling flows with wall-separation zones in a contracting circular finite-length open chamber is studied. Assuming a complete reaction with high activation energy and chemical equilibrium behind the reaction zone, a nonlinear partial differential equation is derived for the solution of the flow stream function behind the reaction zone in terms of the specific total enthalpy for a reacting flow, the specific entropy, and the circulation functions prescribed at the chamber's inlet. Four types of solutions of the resulting ordinary differential equation for the columnar flow case describe the outlet state of the flow in a long chamber. The bifurcation diagrams of steady flows as the inlet swirl level is increased at fixed chamber contraction and reaction heat release are described. The approach is applied to an inlet solid-body rotation flow 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 first appearance of wall-separation states and for the size of the separation zone as a function of the inlet swirl ratio, Mach number, chamber contraction, and heat release of the reaction. The methodology developed in this paper provides a theoretical feasibility for the development of the technology of swirl-assisted combustion where the reaction zone is supported and stabilized by a wall-separation zone.

Figure 1

Schematic diagram of a premixed reacting swirling flow with a wall-separation zone in an open contracting circular chamber.

Figure 2

The various types of solutions of the columnar problem: (a) a centerline accelerated flow state at $\sigma =-0.2,\beta \delta =1,\omega =4$; (b) a wall-separation state at $\sigma =-0.2,\beta \delta =1,\omega =4.8$; (c) a decelerated flow state at $\sigma =-0.2,\beta \delta =1.2,\omega =4$; (d) a vortex breakdown state at $\sigma =-0.2,\beta \delta =1.2,\omega =5.5$.

Figure 3

Schematic diagram of axial velocity and temperature profiles at the chamber inlet and outlet: (a) a wall-separation state at $\sigma =-0.2,\beta \delta =1,\omega =4.8$; (b) a vortex-breakdown state at $\sigma =-0.2,\beta \delta =1.2,\omega =5.5$.

Figure 4

The dividing line ${\left(\beta \delta \right)}_l$ as a function of $\sigma <0$ separating between operational conditions where reacting flows with swirl and wall-separation zones may appear and conditions where reacting flows with swirl and vortex breakdown may appear.

Figure 5

The diagrams of main parameters of reacting flow states with swirl for $\sigma =-0.2$ and $\beta \delta =1$. (a) The centerline axial velocity $w\left(0\right)$ (solid line) and related wall axial velocity $w\left(y_e\right)$ (dashed line) as a function of $\omega$. (b) The resulting location of the interface of the wall-separation zone $y_w$ as a function of $\omega$. The circles mark the swirl ratio ${\omega }_{\mathrm{WS}}$ for the first appearance of wall-separation states.

Figure 6

The critical swirl ratio ${\omega }_{\mathrm{WS}}$ for the first appearance of wall-separation states as a function of exothermicity $\beta \delta$ for contracting chambers with $\sigma =-0.2$.