Biglobal Stability of Compressible Flowfields. Part 2: Application to Solid Rocket Motors

Abstract

This work investigates the unsteady wave behavior in a simulated, planar rocket motor using a compressible biglobal stability approach. This approach has the ability to capture both hydrodynamic and vorticoacoustic fields en meme temps, thus providing a complete solution that inherently accounts for the much sought after hydrodynamic-acoustic wave coupling. The present model is compared and validated against existing analytical models, especially those that are employed in the field of combustion instability. In this sequel, we begin by validating the compressible formulation by showcasing its ability to reproduce, in the absence of a mean flowfield, the precise acoustic frequencies and mode shapes predicted by pure acoustic theory. We also show that the rotational layers corresponding to the hydrodynamic fluctuations are consistent with those obtained analytically, when a vortico-acoustic eigenmode is considered, i.e., one that emerges in the vicinity of a pure acoustic tone. For example, the penetration depth of the vorticoacoustic fluctuations is found to be fairly constant when the penetration number is kept fixed, thus exhibiting similar behavior to that associated with the analytically derived wave formulation. The vortico-acoustic waves are also shown to be most appreciable in the vicinity of the walls, in full conformance to the behavior displayed by parietal vortex shedding. As for the strictly hydrodynamic modes, which appear at much lower frequencies, these are manifested mostly in the core region, where the mean flow velocity is most pronounced. For this reason, patterns associated with mean flow breakdown are first realized in the midsection plane, where the velocity is highest. Having vetted the overall oscillatory behavior, the work proceeds by evaluating the influence of propellant grain properties and aspect ratio on the stability of the compressible flowfield, with the main emphasis being placed on the geometric configuration of a slab rocket motor. At the outset, the modal analysis that we perform is shown to predict with a high degree of precision both longitudinal and transverse modes, thus giving rise to a more comprehensive stability framework. Several parametric test cases are systematically used to demonstrate that hydrodynamic modes tend to dominate for higher injection velocities and longer chamber lengths. As a windfall, the work provides a clear physical explanation for the observed jump in amplified frequencies near acoustic modes, as seen in a variety of experiments. Whenever possible, our results are compared and validated against analytical approximations of the vorticoacoustic wave motion, thus confirming that our approach is capable of resolving rotational and viscous wall effects. As for the strictly hydrodynamic component, improved agreement is achieved between our predictions and those reported in the VECLA experiments at ONERA. By retaining the influence of the mean flow on unsteady chamber acoustics, our technique straightforwardly displays a slight frequency shift from the Helmholtz type resonance modes. This behavior corresponds to a yet unexplained phenomenon that is repeatedly captured in live experiments. As one would expect, our findings indicate that the frequency shift widens with successive increases in the mean flow velocity. Moreover, the modal analysis is seen to extend over both the longitudinal and transverse modes, thus providing the full spectrum of system modes associated with either acoustics or hydrodynamics. In this manner, the present study enables us to gain deeper physical insight into hydrodynamic-acoustic interactions, which can lead to vortex synchronization and frequency shifting as observed in the amplified frequencies of static test firings. Finally, the study shows that inclusion of compressibility effects into a hydrodynamic stability framework leads to an accurate determination of both vortico-acoustic and hydrodynamic modes; this in turn suggests that all fluctuations are of a hydrodynamic origin, be it compressible or incompressible.