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COMBUSTION INSTABILITY | ROCKET PROPULSION

FIGURE 2. Simulated liquid engine geometry with boundary conditions.

                                                                                                            Mach 1 surface  FIGURE 5. Comparison of the rst
                                                                                                                            tangential eigenmode calculated using the
FIGURE 3. Velocity streamlines plotted over chamber pressure. The Mach 1 surface is                                         classic homogeneous wave equation (left),
plotted in magenta.                                                                                                         and the AVPE (right) of a half period (T)
                                                                                                                            of oscillation.
                                                    mode, with the imaginary
                                                                                                                            conditions, can now be more accurately
                                                    part de ning the frequency                                              modeled before testing.

                                                    of oscillation. The complex                                             ð CONTINUED WORK

                                                    eigenvectors represent the                                              A more complete depiction of
                                                                                                                            combustion instability includes
                                                    spatial amplitude and phasing                                           rotational oscillations and thermal
                                                                                                                            oscillations in conjunction with chamber
                                                    of the acoustic wave.                                                   acoustics. Rotational oscillations
                                                                                                                            occur as a direct result of the acoustic
                                                    Comparing the acoustic                                                  oscillation, where thermal waves can
                                                                                                                            also be present in the absence of
                                                    mode shapes derived using                                               acoustic uctuation. Continued work
                                                                                                                            using COMSOL Multiphysics will focus
                                                    the classic homogeneous                                                 on solving the viscous rotational wave
                                                                                                                            that accompanies all
FIGURE 4. Acoustic analysis geometry with boundary  wave equation (Helmholtz
conditions.                                         equation) to those derived                                              acoustic oscillations. v
                                                    using the AVPE demonstrates
                                                                                                                            This article was written by Sean R.
                                                    the bene ts of higher-                                                  Fischbach, Marshall Space Flight Center/
                                                                                                                            Jacobs ESSSA Group, MSFC, Huntsville, AL.
                                                    delity models that correctly
                                                                                                                            REFERENCES
This condition prevents disturbances          represent the underlying physics (see
                                                                                                                            1. F. S. Bloomshield, Lessons Learned in Solid
downstream of the sonic plane from            Figure 5). Inclusion of mean ow terms                                         Rocket Combustion Instability, 43rd AIAA
                                                                                                                            Joint Propulsion Conference, AIAA-2007-
propagating back upstream. The                in the AVPE accurately models the phase                                       5803, Cincinnati, OH, July 2007.
                                                                                                                            2. J. C. French, Nozzle Acoustic Dynamics
diverging section of the nozzle is            shift caused by the steady gas ow.                                            and Stability Modeling, Vol. 27, Journal of
                                                                                                                            Propulsion and Power, 2011.
acoustically silent and does not affect       Phasing is extremely important since                                          3. R. K. Sigman and B. T. Zinn, A Finite
                                                                                                                            Element Approach for Predicting Nozzle
the chamber acoustics. The simulation         combustion instability models make use                                        Admittances, Vol. 88, Journal of Sound and
                                                                                                                            Vibration, 1983, pp. 117-131.
geometry is truncated at the nozzle           of temporal and spatial integration of                                        4. L. M. B. C. Campos, On 36 Forms of the
                                                                                                                            Acoustic Wave Equation in Potential Flows
sonic line, where a zero ux boundary          the acoustic eigenvectors.                                                    and Inhomogeneous Media, Vol. 60, Applied
                                                                                                                            Mechanics Reviews, 2007, pp. 149-171.
condition is self-satisfying (see Figure 4).  Utilizing COMSOL Multiphysics to

The remaining boundaries are modeled simulate the rocket gas dynamics and

with a zero ux boundary condition,            acoustic eigenmodes provides a more

assuming zero acoustic absorption on          accurate mode shape over previous

all surfaces.                                 techniques. The higher- delity acoustic

The eigenvalue analysis produces              representation is easily incorporated

complex eigenmodes and eigenvalues            into combustion instability models to

representing each acoustic mode and           give rocket designers and engineers

its complex conjugate. The real part          greater predictive capabilities. The

of the complex eigenvalue represents          inclusion of damping devices, such

the temporal damping of the acoustic          as baf es, or changes in operating

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