Generalized Galerkin Approach for the Study of Nonlinear Thermoacoustic Instabilities

Abstract

A Galerkin expansion approach to the study of nonlinear thermoacoustic instabilities is presented within a finite element framework. The foundation of the approach lies with weighted residual and spectral methods that are connected to the present framework in the context of the viscous Burgers equation. Moving to a thermoacoustic instability analysis, an eigenvalue problem for a convected wave equation leads to a set of basis functions defined on a finite element grid. Before the set of basis functions is used in the solution of the perturbed nonlinear Navier-Stokes equations, orthonormality is ensured by applying the modified Gram-Schmidt process. Results are verified with a well established finite volume tool, and the nonlinear stability of a slab rocket motor with two different outflow boundary conditions is investigated. In addition, the first components of a comprehensive linear stability analysis are also developed, namely, the solution of the convected wave equation, and the determination of the vortical disturbances that are driven by the acoustic oscillations and the need to satisfy the viscous no-slip condition. A streamline upwind/Petrov-Galerkin (SUPG) finite element method is employed for solving the set of equations describing the vortical disturbances. Results are shown to be in agreement with established analytical approximations. Additionally, two possible mechanisms for the inception of parietal vortex shedding (PVS) type instabilities are suggested in the context of acoustically driven vortical waves.