Nonlinear interaction of two oblique modes in a supersonic mixing layer.

Persistent Link:
http://hdl.handle.net/10150/187386
Title:
Nonlinear interaction of two oblique modes in a supersonic mixing layer.
Author:
Gartside, James Nicholas Burgess.
Issue Date:
1995
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
We 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.
Type:
text; Dissertation-Reproduction (electronic)
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Balsa, Thomas

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleNonlinear interaction of two oblique modes in a supersonic mixing layer.en_US
dc.creatorGartside, James Nicholas Burgess.en_US
dc.contributor.authorGartside, James Nicholas Burgess.en_US
dc.date.issued1995en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.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.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairBalsa, Thomasen_US
dc.contributor.committeememberFasel, Hermannen_US
dc.contributor.committeememberKerschen, Edwarden_US
dc.identifier.proquest9620442en_US
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