Analytical and numerical solution of the pressure response to an unsteady heat release pulse in 1D

Abstract

This paper analyzes 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 an unsteady heat source is derived, and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, a Gaussian spatial distribution with either a step or a Gaussian temporal distribution, are used to model the heat source. The analytical solutions predict two 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.