76th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 19–21, 2023;
Washington, DC
Session X05: Thermoacoustics
8:00 AM–10:23 AM,
Tuesday, November 21, 2023
Room: 102A
Chair: Sarma Rani, University of Alabama in Huntsville
Abstract: X05.00002 : Triggered Instabilities in a Rocket Motor. Part I. On the Role of Acoustic and Combustion Nonlinearities
8:13 AM–8:26 AM
Abstract
Presenter:
Swarnalatha K. V.
(University of Alabama in Huntsville)
Author:
Swarnalatha K. V.
(University of Alabama in Huntsville)
We investigate the role of acoustic and combustion nonlinearities in the occurrence of triggered longitudinal instabilities in a cylindrical combustion chamber with axially varying mean properties. By applying the Galerkin method of weighted averaging, the conservation equations are reduced to nonlinear ordinary differential equations governing the time-dependent amplitudes of the natural modes of the chamber. The amplitude equations include linear and nonlinear self- and cross-coupling among the modes, as well as the quadratic and cubic acoustic nonlinearities. The combustion mass and energy source terms are modeled using a pressure exponent--timelag ($ extsf{n}$-$ar{ au}$) model, which gives rise to the combustion nonlinearities. The effects of acoustic nonlinearities on triggering are first studied by dropping the combustion terms. It is seen that a one-mode system has two linearly stable fixed points, while a two-mode system has three stable fixed points. We demonstrate quantitatively that the acoustic nonlinearities can result in a triggered bifurcation between any two stable fixed points and in either direction. The direction of bifurcation is conclusively established by considering the dynamical equations governing the perturbations imposed on the reference fixed point, and then varying the magnitude of perturbations to illustrate triggering to all other stable fixed points. We compute the basin of attraction for a visual representation of the initial conditions (or perturbations) leading to a triggered bifurcation, or to unstable oscillations growing without limit. The inclusion of combustion nonlinearities has a marginal effect on the number and coordinates of the stable fixed points, at least for small values of the control parameter determining the magnitude of combustion nonlinearities. However, even for small values of this parameter, it is seen that the combustion terms render the fixed points extremely sensitive to the magnitude of perturbations, which is visually captured in the basin of attraction. Based on these results, a novel nonlinear dynamics perspective is proposed for instability, which is more nuanced than the widely accepted mechanism of the constructive interference of acoustic and heat-release fluctuations.