Stability and sensitivity analysis in a simplified solid rocket motor flow

The present article aims at enhancing the computation of the global stability modes of the internal flow of solid rocket motors (SRMs) approximated by the Taylor-Culick solution. This modal approach suffers from the consequences of the non-normality of the global linearized incompressible Navier-Stokes operator, namely the lack of robustness of the eigenvalues that can lead to the computation of pseudo-modes rather than actual eigenmodes. In this respect, the effects of non-normality associated with strongly amplified eigenfunctions are highlighted on a simplified convective-diffusive stability problem with uniformly accelerated base state, the latter property being a typical characteristic of the Taylor-Culick flow. Non-convergence zones for the eigenvalues are exhibited for large Reynolds numbers and are related to the critical sensitivity to disturbances applied to one of the boundary conditions. For this reason, and according to experimental and numerical data related to the stability of simplified SRMs, a global stability analysis is performed assuming that the hydrodynamic fluctuations emerge from a geometrical defect applied at the sidewall. This comes to fix the upstream boundary condition at the abscissa of the sidewall disturbance. The resulting eigenmodes are shown to be discrete, numerically converged, well identified by a finite number of points of undefined phase of the velocity fluctuations. They marginally depend on Reynolds number variations, but are modified by changes on the boundaries location. As in the simplified problem, the inflow boundary condition is the most critical in terms of sensitivity to numerical errors, although not dramatic. Finally, the sensitivity analysis to infinitesimal base flow changes indicates that the variations applied close to the inflow boundary condition induce the largest move of the eigenvalues. In spite of the large non-normal effects induced by the large polynomial growth of the eigenfunctions, this paper shows that discrete instabilities may emerge from a wall defect, in contrast to configurations without such a geometrical perturbation whose dynamics may be rather driven by pseudo-modes.