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Nonlinear roll-up of externally excited free shear layers

Published online by Cambridge University Press:  21 April 2006

M. E. Goldstein  and
S. J. Leib
M. E. Goldstein
Affiliation:
National Aeronautics and Space Administration, Lewis Research Center, Cleveland, OH 44135, USA
S. J. Leib
Affiliation:
Case Western Reserve University, Cleveland, OH 44106, USA and NASA Resident Research Associate
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Abstract

We consider the effects of strong critical-layer nonlinearity on the spatially growing instabilities of a shear layer between two parallel streams. A composite expansion technique is used to obtain a single formula that accounts for both shear-layer spreading and nonlinear critical-layer effects. Nonlinearity causes the instability to saturate well upstream of the linear neutral stability point. It also produces vorticity roll-up that cannot be predicted by linear theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 191 , June 1988 , pp. 481 - 515
Copyright
© 1988 Cambridge University Press

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References

Benney, D. J. & Bergeron, R. F. 1969 A new class of non-linear waves in parallel flows. Stud. Appl. Maths 48, 181–204.Google Scholar
Benney, D. J. & Maslowe, S. A. 1975 The evolution in space and time of nonlinear waves in parallel shear flows. Stud. Appl. Maths 54, 181–205.Google Scholar
Browand, F. K. & Ho, C. M. 1983 The mixing layer: an example of quasi two-dimensional turbulence. J. Méc. Théor. Appl. 2, 99–120.Google Scholar
Corcos, G. M. & Sherman, F. S. 1976 Vorticity concentration and the dynamics of unstable free-shear layers. J. Fluid Mech. 73, 241–264.Google Scholar
Cohen, J. 1985 Instabilities and resonances in turbulent free shear flows. Ph.D. thesis, University of Arizona.
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413.Google Scholar
Davis, R. E. 1969 On high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36, 337–346.Google Scholar
Gaster, M., Kit, E. & Wygnanski, I. 1985 Large-scale structures in a forced turbulent mixing layer. J. Fluid Mech. 150, 23–39.Google Scholar
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.Google Scholar
Haberman, R. 1972 Critical layers in parallel shear flows. Stud. Appl. Maths 51, 139–161.Google Scholar
Haynes, P. H. 1985 Nonlinear instability of a Rossby-wave critical layer. J. Fluid Mech. 161, 493–511.Google Scholar
Hickernell, F. J. 1984 Time-dependent critical layers in shear flows on the beta-plane. J. Fluid Mech. 142, 431–449.Google Scholar
Ho, C.-M. & Huang, L.-S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Huerre, P. 1977 Nonlinear instability of free shear layers. In Laminar-Turbulent Transition, AGARD CP, pp. 224–229.
Huerre, P. 1980 The nonlinear stability of a free shear layer in the viscous critical layer regime. Phil. Trans. R. Soc. Lond. A 293, 643–672.Google Scholar
Huerre, P. 1987 On the Landau constant in mixing layers. Proc. R. Soc. Lond. A 409, 369–381.Google Scholar
Huerre, P. & Scott, J. F. 1980 Effects of critical layer structure on the nonlinear evolution of waves in free shear layers. Proc. R. Soc. Lond. A 371, 509–524.Google Scholar
James, M. L., Smith, G. M. & Wolford, J. C. 1977 Applied Numerical Methods for Digital Computation. Harper and Row.
Killworth, P. D. & McIntyre, M. E. 1985 Do Rossby-wave critical layers absorb, reflect, or over-reflect? J. Fluid Mech. 161, 449–492.Google Scholar
Maslowe, S. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405–432.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech. 19, 543–556.Google Scholar
Miura, A. & Sato, T. 1978 Theory of vortex nutation and amplitude oscillation in an inviscid shear instability. J. Fluid Mech. 86, 33–47.Google Scholar
Monkewitz, P. A. & Huerre, P. 1982 The influence of the velocity ratio on the spatial instability of mixing layers. Phys. Fluids 25, 1137–43.Google Scholar
Oster, D. & Wygnanski, I. 1982 The forced mixing layer between parallel streams. J. Fluid Mech. 123, 91–130.Google Scholar
Plaschko, P. & Hussain, A. K. M. F. 1984 A spectral theory for weakly nonlinear instabilities of slowly divergent shear flows. Phys. Fluids 27, 1603–1606.Google Scholar
Robinson, J. L. 1974 The inviscid nonlinear instability of parallel shear flows. J. Fluid Mech. 63, 723–752.Google Scholar
Ko, D. Ru-Sue, Kobata, T. & Lees, L. 1970 Finite disturbance effect on the stability of a laminar incompressible wake behind a flat plate. J. Fluid Mech. 40, 315–341.Google Scholar
Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Geophys. Astrophys. Fluid Dyn. 9, 185–200.Google Scholar
Stewartson, K. 1981 Marginally stable inviscid flows with critical layer. IMA J. Appl. Maths 27, 133–175.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. J. Fluid Mech. 9, 353–370.Google Scholar
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Warn, T. & Warn, H. 1978 The evolution of a nonlinear critical level. Stud. Appl. Maths 59, 37–71.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. J. Fluid Mech. 9, 371–389.Google Scholar
Wygnanski, I. J. & Petersen, R. A. 1987 Coherent motion in excited free shear flows. AIAA J. 25, 201–212.Google Scholar