Reactive linearized equations of perturbed compressible variables for low-Mach number variable-density flows

A hydrodynamic/acoustic splitting approach is proposed to study noise emitted from reactive variable-density flows. A simulation using the variable-density low-Mach number equations provides a solution of the hydrodynamic motions of the flow (base-flow), and a set of equations for perturbed variables is additionally solved to capture the acoustic motions. A rigorous derivation of these equations for the assumed base-flow is given, which compared to its non-reacting counterpart includes additional terms related to variable-density flows. Two different test cases are presented. First, the Kirchhoff vortex is simulated to highlight instability issues related to constant-density flows. Second, an academic test case for variable-density flows is proposed in the form of a reacting dipole, which is used to underline the stability of the proposed perturbed equations in such a scenario. Various intermediate forms of the derived perturbation equations are juxtaposed and analyzed with respect to their stability for these two test cases, and assumptions made in their derivation are numerically justified.