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## Study of the effects of unsteady heat release in combustion instability

Rocket combustors and other high-performance chemical propulsion systems are prone to combustion instability. Recent simulations of rocket combustors using detailed chemical kinetics show that the constant pressure assumption used in classical treatments may be suspect due to high rates of heat release. This study is a exploration on the effects of these extraordinary rates of heat addition on the local pressure field, and interactions between the heat release and an acoustic field.

The full problem is decomposed into simpler unit problems focused on the particular interactions of physical phenomena involved in combustion instability. The overall strategy consists of analyzing fundamental problems with simplified scenarios and then build up the complexity by adding more phenomena to the analysis. Seven unit problems are proposed in this study.

The first unit problem consists of the pressure response to an unsteady heat release source in an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, one a Gaussian spatial distribution with a step temporal profile, and the other a Gaussian spatial distribution with a Gaussian temporal distribution, are used to model the heat source. The analytical solutions predict two different regimes in the pressure response depending on the Helmholtz number, which is defined as the ratio of the acoustic time over the duration of the heat release pulse. A critical Helmholtz number is found to dictate the pressure response regime. For compact cases, in the subcritical regime, the amplitude of the pressure pulse remains constant in space. For noncompact cases, above the critical Helmholtz number, the pressure pulse reaches a maximum at the center of the heat source, and then decays in space converging to a lower far field amplitude. At the limits of very small and very large Helmholtz numbers, the heat release response tends to be a constant pressure process and a constant volume process, respectively. The parameters of the study are chosen to be representative of the extreme conditions in a rocket combustor. The analytical models for both heat source profiles closely match the simulations with a slight overprediction. The differences observed in the analytical solutions are due to neglecting mean flow property variations and the absence of loss mechanisms. The numerical simulations also reveal the presence of nonlinear effects such as weak shocks that cannot be captured by the linear acoustic wave equation.

The second unit problem extends the analysis of the pressure response of an unsteady heat release source to an unconfined three-dimensional domain. An analytical model based on the spherical acoustic wave equation with an unsteady heat source is derived and then compared against results from highly-resolved three-dimensional numerical simulations. Two different heat release profiles, a three-dimensional Gaussian spherical distribution with either a step or a Gaussian temporal distribution, are used to model the heat source. Two different regimes in the pressure response depending on the Helmholtz number are found. This analysis also reveals that whereas for the one-dimensional case the pressure amplitude is constant over the distance, for the three-dimensional case it decays with the radial distance from the heat source. In addition, although for moderate heat release values the analytical solution is able to capture the dynamics of the fluid response, for large heat release values the nonlinear effects deviate the highly-resolved numerical solution from the analytical model.

The third unit problem studies the pressure response of a fluctuating unsteady heat release source to an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, a flat spatial distribution with sinusoidal temporal profile and a Gaussian spatial distribution and sinusoidal temporal profile, are used to model the heat source. For both cases, the acoustically compact and noncompact regimes depending on the Helmholtz number are analyzed. While in the compact regime the amplitude of the pressure is constant over the distance, in the noncompact regime the amplitude of the pressure fluctuation is larger within the heat source area of application, and once outside the heat source decays to a far field pressure value. In addition, the analytical model does not capture the nonlinear effects present in the highly-resolved numerical simulations for large rates of heat release such as the ones present in rocket combustors.

Finally, the last four unit problems focus on the interaction between unsteady heat release and the longitudinal acoustic modes of a combustor. The goal is to assess and quantify how pressure fluctuations due to unsteady heat release amplify a longitudinal acoustic mode. To investigate the nonlinear effects and the limitations based on the acoustic wave equation, the analytical models are compared against highly-resolved numerical simulations. The fourth unit problem consists of the pressure response to a moving rigid surface that generates a forced sinusoidal velocity fluctuation in a one-dimensional open-ended cavity. The fifth unit problem combines an analytical solution from the velocity harmonic fluctuation with an unsteady heat pulse with Gaussian spatial and temporal distribution developed in the first unit problem. The choice of an open-ended cavity simplifies the analysis and serves as a stepping stone to the sixth unit problem, which also includes the pressure reflections provoked by the acoustic boundaries of the duct. This sixth unit problem describes the establishment of a 1L acoustic longitudinal mode inside a closed duct using the harmonic velocity fluctuations from the fourth unit problem. A wall on the left end of the duct is only moved for one cycle at the 1L mode frequency to establish a 1L mode in the initially quiescent fluid. The last unit problem combines the analytical solution of the 1L mode acoustic field developed in the sixth unit problem with an unsteady heat pulse with Gaussian spatial and temporal distribution, and also accounts for pressure reflections. The derivation of the present analytical models includes the identification of relevant length and time scales that are condensed into the Helmholtz number, the phase shift between the longitudinal fluctuating pressure field and the heat source, and ratio of the fluctuating periods. The analytical solution is able to capture with an acceptable degree of accuracy the pressure trace of the numerical solution during the fist few cycles of the 1L mode, but it quickly deviates very significantly from the numerical solution due to wave steepening and the formation of weak shocks. Therefore, models based on the acoustic wave equation can provide a good understanding of the combustion instability behavior, but not accurately predict the evolution of the pressure fluctuations as the nonlinear effects play a major role in the combustion dynamics of liquid rocket engines.

The full problem is decomposed into simpler unit problems focused on the particular interactions of physical phenomena involved in combustion instability. The overall strategy consists of analyzing fundamental problems with simplified scenarios and then build up the complexity by adding more phenomena to the analysis. Seven unit problems are proposed in this study.

The first unit problem consists of the pressure response to an unsteady heat release source in an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, one a Gaussian spatial distribution with a step temporal profile, and the other a Gaussian spatial distribution with a Gaussian temporal distribution, are used to model the heat source. The analytical solutions predict two different regimes in the pressure response depending on the Helmholtz number, which is defined as the ratio of the acoustic time over the duration of the heat release pulse. A critical Helmholtz number is found to dictate the pressure response regime. For compact cases, in the subcritical regime, the amplitude of the pressure pulse remains constant in space. For noncompact cases, above the critical Helmholtz number, the pressure pulse reaches a maximum at the center of the heat source, and then decays in space converging to a lower far field amplitude. At the limits of very small and very large Helmholtz numbers, the heat release response tends to be a constant pressure process and a constant volume process, respectively. The parameters of the study are chosen to be representative of the extreme conditions in a rocket combustor. The analytical models for both heat source profiles closely match the simulations with a slight overprediction. The differences observed in the analytical solutions are due to neglecting mean flow property variations and the absence of loss mechanisms. The numerical simulations also reveal the presence of nonlinear effects such as weak shocks that cannot be captured by the linear acoustic wave equation.

The second unit problem extends the analysis of the pressure response of an unsteady heat release source to an unconfined three-dimensional domain. An analytical model based on the spherical acoustic wave equation with an unsteady heat source is derived and then compared against results from highly-resolved three-dimensional numerical simulations. Two different heat release profiles, a three-dimensional Gaussian spherical distribution with either a step or a Gaussian temporal distribution, are used to model the heat source. Two different regimes in the pressure response depending on the Helmholtz number are found. This analysis also reveals that whereas for the one-dimensional case the pressure amplitude is constant over the distance, for the three-dimensional case it decays with the radial distance from the heat source. In addition, although for moderate heat release values the analytical solution is able to capture the dynamics of the fluid response, for large heat release values the nonlinear effects deviate the highly-resolved numerical solution from the analytical model.

The third unit problem studies the pressure response of a fluctuating unsteady heat release source to an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, a flat spatial distribution with sinusoidal temporal profile and a Gaussian spatial distribution and sinusoidal temporal profile, are used to model the heat source. For both cases, the acoustically compact and noncompact regimes depending on the Helmholtz number are analyzed. While in the compact regime the amplitude of the pressure is constant over the distance, in the noncompact regime the amplitude of the pressure fluctuation is larger within the heat source area of application, and once outside the heat source decays to a far field pressure value. In addition, the analytical model does not capture the nonlinear effects present in the highly-resolved numerical simulations for large rates of heat release such as the ones present in rocket combustors.

Finally, the last four unit problems focus on the interaction between unsteady heat release and the longitudinal acoustic modes of a combustor. The goal is to assess and quantify how pressure fluctuations due to unsteady heat release amplify a longitudinal acoustic mode. To investigate the nonlinear effects and the limitations based on the acoustic wave equation, the analytical models are compared against highly-resolved numerical simulations. The fourth unit problem consists of the pressure response to a moving rigid surface that generates a forced sinusoidal velocity fluctuation in a one-dimensional open-ended cavity. The fifth unit problem combines an analytical solution from the velocity harmonic fluctuation with an unsteady heat pulse with Gaussian spatial and temporal distribution developed in the first unit problem. The choice of an open-ended cavity simplifies the analysis and serves as a stepping stone to the sixth unit problem, which also includes the pressure reflections provoked by the acoustic boundaries of the duct. This sixth unit problem describes the establishment of a 1L acoustic longitudinal mode inside a closed duct using the harmonic velocity fluctuations from the fourth unit problem. A wall on the left end of the duct is only moved for one cycle at the 1L mode frequency to establish a 1L mode in the initially quiescent fluid. The last unit problem combines the analytical solution of the 1L mode acoustic field developed in the sixth unit problem with an unsteady heat pulse with Gaussian spatial and temporal distribution, and also accounts for pressure reflections. The derivation of the present analytical models includes the identification of relevant length and time scales that are condensed into the Helmholtz number, the phase shift between the longitudinal fluctuating pressure field and the heat source, and ratio of the fluctuating periods. The analytical solution is able to capture with an acceptable degree of accuracy the pressure trace of the numerical solution during the fist few cycles of the 1L mode, but it quickly deviates very significantly from the numerical solution due to wave steepening and the formation of weak shocks. Therefore, models based on the acoustic wave equation can provide a good understanding of the combustion instability behavior, but not accurately predict the evolution of the pressure fluctuations as the nonlinear effects play a major role in the combustion dynamics of liquid rocket engines.