Nonlinear interaction of two oblique modes in a supersonic mixing layer.
Committee ChairBalsa, Thomas
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PublisherThe University of Arizona.
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AbstractWe study the spatial instability of a pair of symmetric oblique modes in the long-wave limit using multiple time scales and matched asymptotic expansions. The base flow is a supersonic mixing layer that satisfies Miles' condition, namely that the Mach number is chosen so that the corresponding vortex sheet is marginally stable. We introduce the small disturbance far upstream where nonlinear terms are small and study the growth of the disturbance as it convects downstream. Even though we are using a long wavelength, the growth rate is small enough so that the disturbance is essentially periodic from one wavelength to the next. We use linear analysis to solve the problem initially, and then determine a distinguished scaling between the small characteristic frequency and the small characteristic amplitude in order to do the nonlinear analysis. The nonlinear analysis results in a nonlinear integro-differential equation for the amplitude of the disturbance that must be solved numerically. When the angle of obliqueness is than π /4, the growth of the disturbance is much larger than the growth predicted by linear theory, and the growth becomes unbounded at a singularity located at a finite distance downstream. The result is similar to the findings of other analyses done near the neutral point by other authors. When the angle of obliqueness is larger than π /4, the amplitude of the disturbance experiences a series of modulations in the downstream direction; however, the the overall growth of the amplitude is again much larger than the growth predicted by linear theory, but is not as large as the growth experienced for angles less than π/4. The modulations in the amplitude are caused by the nonlinear term in the integro-differential equation; when the angle of obliqueness is less than π /4, the nonlinear term enhances growth, but when the the angle is larger than π/ 4, the nonlinear term alternately enhances and dampens growth as the disturbance convects downstream. So, in either case, the nonlinear interaction between two oblique modes causes the disturbance to grow much more quickly than the growth predicted by linear theory alone.
Degree ProgramApplied Mathematics