Combustion instability in solid rocket motors and liquid engines is a complication that continues to plague designers and engineers. Many rocket systems experience violent fluctuations in pressure, velocity, and temperature originating from the complex interactions between the combustion process and gas dynamics. During severe cases of combustion instability, fluctuation amplitudes can reach values equal to or greater than the average chamber pressure. Large amplitude oscillations lead to damaged injectors, loss of rocket performance, damaged payloads, and, in some cases, breach of case/loss of mission. Historic difficulties in modeling and predicting combustion instability haves reduced most instances of most rocket systems experiencing instability into a costly fix through testing or scrapping of the system entirely.

Figure 1. The injector faceplate of the F1 engine that powered the Saturn V rocket required over 1,300 test firings to configure the injector baffles and reduce the pressure oscillations to 10% of the chamber pressure.
During the early development of rocket propulsion technology, scientists and engineers were cued to the underlying physics at play through the measurement of vibrating test stands, observation of fluctuating exhaust plumes, and most notably, the audible tones accompanying instabilities. These observations lead the pioneers of combustion instability research to acutely focus their modeling efforts on the acoustic waves inside combustion chambers. This focus on acoustics is quite logical given that the measured frequency of oscillation often closely matches the normal acoustic modes of the combustion chamber. But this narrow focus misses contributions made by rotational and thermal waves that are a direct result of, or closely couple with, the acoustic wave. A more complete depiction of combustion instability oscillations is achieved when a global energy-based assessment is used.

Figure 2. Pressure trace of a stable (red) and unstable (blue) solid rocket motor¹.
Recent advances in energy-based modeling of combustion instabilities require an accurate determination of acoustic frequencies and mode shapes. Of particular interest are the acoustic mean flow interactions within the converging section of a rocket nozzle, where gradients of pressure, density, and velocity become large. The expulsion of unsteady energy through the nozzle of a rocket is identified as the predominant source of acoustic damping for most rocket systems. Recently, an approach to address nozzle damping with mean flow effects was implemented by French2. This new approach extends the work originated by Sigman and Zinn3 by solving the acoustic velocity potential equation (AVPE) formulated by perturbing the Euler equations4.

Figure 3
Figure 4
Determining eigenvalues of the AVPE, Figure 3 is considerably more complex than the traditionally used pressure based wave equation, Figure 4, and requires numerical approximations of the chamber flow-field and eigenvalues.

Modeling Chamber Gas Dynamics

The latest theoretical models for oscillatory disturbances in high-speed flows require a precise determination of the chamber acoustic eigenmodes. But first, a simulation of the mean flow properties of the combustion chamber must be performed. COMSOL Multiphysics provides a numerical platform for conveniently and accurately simulating both the chamber gas dynamics and internal acoustics. This finite element software package provides multiple predefined physics modules along with a generalized mathematics module.

Figure 5. Simulated liquid engine geometry with boundary conditions.
The present study employs the COMSOL Multiphysics finite element framework to model the steady flow-field parameters of a generic liquid engine using the laminar flow equations within the software’s High Mach Number Flow (HMNF) physics module. The HMNF module makes use of these fully compressible Navier-Stokes equations for an ideal gas, which are detailed in the COMSOL documentation.

In order to account for the injection of hot gas due to the burning propellant, the injector face plate is modeled with a uniform inward flow of combusted propellant gas. All other solid boundaries are modeled with the slip boundary condition, and the exit plane is modeled with the hybrid outflow condition.