Abstract
We are interested in the nonlinear interaction between a pair of oblique instability waves that develop when an initially linear, spatially growing instability waves evolve downstream in a nominally two-dimensional, unbounded or semi-bounded, laminar shear flows. It is appropriate to suppose that the Reynolds number R is large enough so that the flow is nearly parallel, and we also suppose that some sort of small-amplitude harmonic excitation (i.e. an excitation of a single frequency) is imposed on the flow (see Figure 1). Then the initial motion just downstream of the excitation device will also have harmonic time dependence and, within a few wavelengths or so, will be well described by linear instability theory.
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Bibliography
Benney, D.J. & Bergeron, R.F., Jr., 1969, A new class of nonlinear waves in parallel flows. Stud. Appl. Math. 48, 181–204.
Craik, A.D.D., 1971, Non-linear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.
Goldstein, M.E. & Choi, S.W., 1989, Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97–120. Also Corrigendum, J. Fluid Mech. 216, 659-663.
Goldstein, M.E., Durbin, P.A. & Leib, S.J., 1987, Roll-up of vorticity in adverse-pressure-gradient boundary layers. J. Fluid Mech. 183, 325–342.
Goldstein, M.E. & Hultgren L.S., 1988, Nonlinear spatial evolution of an externally excited instability wave in a free shear layer. J. Fluid Mech. 197, 295–330.
Goldstein, M.E. & Leib, S.J., 1989, Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73–96.
Goldstein, M.E. & Lee, S.S., 1992, Fully coupled resonant-triad interaction in an adverse-pressure-gradient boundary layer. J. Fluid Mech. 245, 523–551.
Landau, L.D. & Lifshitz, E.M., 1987, Fluid Mechanics, 2nd edn. Pergamon.
Ryzhov, O.S., 1990, The formulation of ordered vortex structures from unstable oscillations in the boundary layers. U.S.S.R. Comput. Maths. Maths. Phys. 30, 146–154.
Smith, F.T. & Stewart, P.A., 1987, The resonant-triad nonlinear interaction in boundary-layer transition. J. Fluid Mech. 179, 227–252.
Stuart, J.T., 1960, On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.
Watson J., 1960, On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. J. Fluid Mech. 9, 371–389.
Wu, X., Lee, S.S., & Cowley S.J., 1992, On the nonlinear threedimensional instability of Stokes layers and other shear layers to pairs of oblique waves. NASA TM 105918 (ICOMP 92–20).
Zhuk, V.I. & Ryzhov, O.S., 1982, Locally non-viscous perturbations in a boundary layer with self-induced pressure. Soviet Phys. Aerodynamics 27, 177–179.
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Goldstein, M.E., Lee, S.S. (1993). Oblique Instability Waves in Nearly Parallel Shear Flows. In: Fitzmaurice, N., Gurarie, D., McCaughan, F., Woyczyński, W.A. (eds) Nonlinear Waves and Weak Turbulence. Progress in Nonlinear Differential Equations and Their Applications, vol 11. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0331-5_9
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DOI: https://doi.org/10.1007/978-1-4612-0331-5_9
Publisher Name: Birkhäuser, Boston, MA
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