Modeling and Dynamics of Nonlinear Acoustic Waves in a Combustion Chamber

Abstract

Future combustors designed for better efficiency and lower pollutant emission are expected to operate closer to their stability boundary, thereby increasing the risk of encountering combustion instability. Onset of combustion instability leads to limit cycle oscillations in the acoustical fluctuations that can often reach amplitudes large enough to cause severe damage. Active control strategies are, therefore, being considered to prevent combustion instabilities, but their development requires nonlinear models that can faithfully capture the combustor system dynamics. A framework for the approximate analysis of the nonlinear acoustics in a combustion chamber exists, which includes all relevant linear contributions and also second order gasdynamic nonlinearities. Nonlinear combustion effects in the form of pressure and velocity coupling models have also been incorporated into the analysis with the aim of capturing the phenomenon of triggered instability, where the acoustical fluctuations are linearly stable to small perturbations, but show a limit cycle behavior for larger perturbations. However, several questions such as those relating to 1) modal truncation of the equations for the acoustic dynamics, 2) absence of triggered limit cycles in the formulation with only second order gasdynamic nonlinearities, and 3) the form of the velocity coupling function, including the need for a threshold character, have not been satisfactorily resolved. In this paper, we address some of these questions on modeling and dynamics of acoustic waves in combustion chambers, using the approximate analysis, that have remained unanswered over the years.

Verification and Validation of Rocket Stability Integral Transformations

“…For instance this study does not attempt to convert the terms corresponding to particle damping, distributed combustion or velocity coupling. [35][36][37][38][39][40][41][42][43] These remain to be addressed or resolved as needed.…”

Volume-to-Surface Transformations of Rocket Stability Integrals

“…8 These are covered quite thoroughly by Culick 8 and others. [9][10][11] As for the velocity coupling correction that has often been cited in the literature, [12][13][14][15][16][17][18][19][20] it is accounted for systematically in the current (rotational) formulation. This will be explained next.…”

“…5 In the previous, irrotational formulation, velocity coupling had to be introduced a posteriori (having been invented in the form of an empirical relation) to compensate for the inability of the onedimensional acoustic model to permit a relationship between axial and radial velocity fluctuations at the propellant surface. [12][13][14][15][16][17][18][19][20] In the present analysis, velocity coupling is not only built into the model a priori, it is obtained from the fundamental requirement to satisfy mass conservation, thus obviating the need for guesswork, experimentation, curve fitting, or trial.…”

“…By using the divergence theorem, the first term of the volumetric integral is converted into surface form viz. (17) as shown in Eqs. (5)-(7), the normal projections of û…”

Thermoacoustic Instability in a Rijke Tube with a Distributed Heat Source

“…Hysteresis phenomenon was reported in the stability boundary and limit cycles were predicted as observed in experiments. Ananthkrishnan et al [20] captured the dynamics of the nonlinear acoustic waves with reduced-order models via truncating the modal expansions and determined the number of modes required for accurate predictions. Balasubramanian and Sujith [21] studied the role of nonnormality and nonlinearity in thermoacoustic system in a Rijke tube and found out that the nonnormality could result in transient growth of oscillations which can trigger nonlinearities in the system.…”