| Unstable waves may occur as a result of acoustic and/or vortical (hydrodynamic) oscillations. If these two different types of waves are coupled together, their physical interactions lead to extremely complicated phenomena. Theoretically, there exists an infinite number of frequencies for both acoustic and vortical oscillations. Realistically, however, only a limited number of combined frequencies are excited. Our objective is, then, to determine the combined nature of acoustic and vortical frequencies at which instabilities may arise. This subject is important in rocket motor chambers when the vortical field is coupled with acoustic pressure oscillations. In the past, the acoustic combustion instability was studied independently of the vortical instability induced by vortex motions. This paper is intended to combine the two different sources of energy everywhere within the spatial domain and to determine the effect of one upon the other. This can be achieved by calculating the mean flow velocities and vorticities and their fluctuating parts of velocities and vortices, as well as the fluctuating pressure. To elucidate this coupling mechanism, however, a very simple model is first introduced. The Orr-Sommerfeld equation is utilized to determine the wave numbers and the unsteady stream functions from which vortically coupled acoustic instability growth constants are calculated. This process demonstrates that there are two different frequencies, acoustic and vortical, various combinations of which contribute to either damping or amplification. Finally, the limitation of the Orr-Sommerfeld equation is removed by numerical solution of the perturbed vorticity transport equation using finite elements. It is found that stability boundaries for coupled acoustic and vortical oscillations are somewhat similar to the classical hydrodynamic stability boundaries, but they occur in the form of multiple islands. |
