Nonlinear Interactions Between Oblique Instability Waves on Nearly Parallel Shear Flows
Asymptotic methods are used to describe the nonlinear self interaction between pairs of oblique instability modes that eventually develops when initially linear, spatially growing instability waves evolve downstream in nominally two-dimensional, unbounded or semi bounded, laminar shear flows. The first nonlinear reaction takes place locally within a so-called “critical layer” with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves together with an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy a pair of integral differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.
KeywordsVortex Convection Explosive Vorticity Lution
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- 10.R. R. Mankbadi, X. Wu, and S. S. Lee, “A critical-layer analysis of the resonant triad in boundary-layer transition.” To be published in J. Fluid Mech (1993).Google Scholar
- 11.W. H. Reid, “The stability of parallel flows,” in Basic Developments in Fluid Dynamics, M. Holt, ed., Academic Press, 249 (1965).Google Scholar
- 13.M. E. Goldstein, “Nonlinear interactions between oblique instability waves on nearly parallel shear flows.” To be published in Physics of Fluids A, 1994.Google Scholar
- 17.X. Wu, “On critical-layer and diffusion-layer nonlinearity in the three-dimensional stage of boundary-layer transition.” To be published in Proc. Roy. Soc. A. (1993).Google Scholar
- 18.D. Wundrow, L. S. Hultgren, and M. E. Goldstein, “Interaction of oblique instability waves with a nonlinear plane wave.” Submitted to J. Fluid Mech. (1993).Google Scholar
- 21.C-L Chang and M. R. Malik, “Oblique mode breakdown in a supersonic boundary layer using nonlinear PSE,” in Instability, Transition, and Turbulence, M. Y. Hussaini, A. Kumar, and C. L. Street, eds., Springer-Verlag (1992).Google Scholar
- 22.A. Thumm, W. Wolz, and H. Fasel, “Numerical simulation of spatially growing three- dimensional disturbance waves in compressible boundary layers,” Proceedings of the Third IUTAM Symposium on Laminar-Turbulent Transition, Toulouse, France, September 11–15, 1989.Google Scholar
- 23.T. Herbert, “Boundary-layer transition-analysis and prediction revisited,” AIAA Paper 91–0737 (1991).Google Scholar
- 25.G. S. Raetz, “A new theory of the cause of transition in fluid flows,” Northrop Corp, NOR-59–383 BLC-121 (1959).Google Scholar
- 32.M. E. Goldstein and S. S. Lee, “Oblique instability waves in nearly parallel shear flows,” in Nonlinear Waves and Weak Turbulence with Applications in Oceanography and Condensed Matter Physics, N. Fitzmaurice, D. Gurarie, F. McCaughan and W. A. Woyczynski, eds., Birkh user-Boston, 159 (1993).Google Scholar