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     Historic dif culties in modeling         by Sigman and Zinn3 by solving the          COMSOL nite element framework to
 and predicting combustion instability        acoustic velocity potential equation        model the steady ow- eld parameters
 have reduced most instances of rocket        (AVPE) formulated by perturbing the         of a generic liquid engine using the
 systems experiencing instability to a        Euler equations4.                           High Mach Number Laminar Flow
 costly x through testing (see Figure 1),                                                 physics interface, which makes use of
 or to scrapping of the system entirely.         Determining eigenvalues of the           the fully compressible Navier-Stokes
                                              AVPE, where ψ is the complex acoustic       equations for an ideal gas together
“A more complete                              potential, λ the complex eigenvalues,       with conservation of energy and
                                              c the speed of sound, and M the             mass equations.
 depiction of                                 Mach vector,
 combustion instability                                                                      In order to account for the injection
 oscillations is                              is considerably more complex than the       of hot gas due to the burning
 achieved when a                              traditionally used pressure-based           propellant, the injector face plate is
 global energy-based                          wave equation,                              modeled with a uniform inward ow of
 assessment is used.”                                                                     combusted propellant gas (see Figure 2).
                                              and requires numerical approximations       All other solid boundaries are modeled
    During the early development of           of the chamber ow eld and                   with the slip boundary condition, and
 rocket propulsion technology scientists      eigenvalues.                                the exit plane is modeled with the
 and engineers were cued to the                                                           hybrid out ow condition, which means
 underlying physics at play through the       ð MODELING CHAMBER                          that both subsonic and supersonic ows
 measurement of vibrating test stands,        GAS DYNAMICS                                are supported.
 observation of uctuating exhaust
 plumes, and, most notably, the audible       The latest theoretical models for              Results from the mean ow analysis
 tones accompanying instabilities. These      oscillatory disturbances in high-speed      are reviewed to ensure a valid and
 observations lead the pioneers of                                                        converged solution. Mean ow
 combustion instability research to focus       ows require a precise determination       parameters such as pressure, density,
 their modeling efforts on the acoustic       of the chamber acoustic eigenmodes.         velocity, and speed of sound are needed
 waves inside combustion chambers.            But rst, a simulation of the mean ow        to model the AVPE. The values of the
                                              properties of the combustion chamber        mean ow in the converging section of
    This focus on acoustics is quite logical  must be performed.                          the nozzle, near the sonic choke plane,
 given that the measured frequency                                                        are of considerable interest. The sonic
 of oscillation often closely matches            COMSOL Multiphysics® software            plane, where the Mach number is equal
 the normal acoustic modes of the             provides a numerical platform for           to 1, creates an acoustic barrier in the
 combustion chamber. But this narrow          conveniently and accurately simulating
 focus misses contributions made by                                                         ow. In order to create an accurate
 rotational and thermal waves that are a      FIGURE 1. Pressure trace of a stable (red)  geometry for the acoustic analysis, the
 direct result of, or closely coupled with,   and unstable (blue) solid rocket motor1.    sonic plane (pictured in magenta in
 the acoustic wave. A more complete                                                       Figure 3) is extracted from the mean
 depiction of combustion instability          both the chamber gas dynamics and
 oscillations is achieved when a global       internal acoustics. This nite element         ow analysis.
 energy-based assessment is used.             software package provides many
                                              prede ned physics along with a              ð MODELING CHAMBER
    Recent advances in energy-based           generalized mathematics interface.          ACOUSTICS
 modeling of combustion instabilities
 require an accurate determination               The present study employs the            The Coef cient Form PDE (Partial
 of acoustic frequencies and mode                                                         Differential Equation) mathematics
 shapes. Of particular interest are the                                                   interface of COMSOL Multiphysics
 acoustic mean ow interactions within                                                     is used to determine the complex
 the converging section of a rocket                                                       eigenvalues of the AVPE. Mean ow
 nozzle, where gradients of pressure,                                                     terms in the AVPE are supplied by the
 density, and velocity become large. The                                                  solution from the mean ow analysis.
 expulsion of unsteady energy through                                                     Gas dynamics within the combustion
 the nozzle of a rocket is identi ed as                                                   chamber play a key role in de ning the
 the predominate source of acoustic                                                       boundary conditions for the acoustic
 damping for most rocket systems.                                                         analysis. Within the converging and
                                                                                          diverging section of the rocket nozzle,
    Recently, an approach to address                                                      gradients of chamber pressure, velocity,
 nozzle damping with mean ow effects                                                      and density grow theoretically in nite
 was implemented by French2. This new                                                     at the sonic plane where the Mach
 approach extends the work originated                                                     number is equal to 1. Downstream of
                                                                                          the sonic plane, acoustic disturbances
                                                                                          are convected with the mean ow at
                                                                                          speeds greater than the speed of sound.

Originally published in the December 2015 edition of NASA Tech Briefs magazine.           COMSOL.COM                                9
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