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Subharmonic growth by parametric resonance

Published online by Cambridge University Press:  26 April 2006

M. R. Hajj ,
R. W. Miksad  and
E. J. Powers
M. R. Hajj
Affiliation:
College of Engineering, The University of Texas at Austin, Austin, TX 78712, USA
R. W. Miksad
Affiliation:
College of Engineering, The University of Texas at Austin, Austin, TX 78712, USA
E. J. Powers
Affiliation:
College of Engineering, The University of Texas at Austin, Austin, TX 78712, USA
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Abstract

An experimental investigation is conducted in order to quantify the nonlinear and parametric resonance mechanisms that are associated with the subharmonic growth in the transition to turbulence in plane mixing layers. Higher-order digital statistical analysis techniques are used to investigate the nonlinear and parametric mechanisms responsible for the energy transfer to the subharmonic. The results show that the dominant interaction is a parametric resonance mechanism between the fundamental and the subharmonic modes which leads to a pronounced growth of the subharmonic. Measurements also indicate that the fundamental, besides interacting with the subharmonic, is also engaged in redistributing its energy to the other Fourier components of the flow via nonlinear three-wave interactions. Local wavenumber measurements verify that frequency—wavenumber resonance matching conditions exist between the fundamental and subharmonic in the region where the subharmonic gains its energy by parametric resonance. The results are in general agreement with theoretical models by Kelly (1967), and Monkewitz (1988) on subharmonic growth.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 236 , March 1992 , pp. 385 - 413
Copyright
© 1992 Cambridge University Press

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References

Beall, J. M., Kim, Y. C. & Powers, E. J. 1982 Estimation of wavenumber and frequency spectra using fixed probe pairs. J. Appl. Phys. 53, 3933–3940.Google Scholar
Bretherton, F. P. 1964 Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20, 457–479.Google Scholar
Browand, F. K. 1966 An experimental investigation of the instability of an incompressible separated shear layer. J. Fluid Mech. 26, 281–307.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Corke, T. C. 1987 Measurements of resonant phase locking in unstable axisymmetric jets and boundary layers. In Nonlinear Wave Interactions in Fluids (ed. R. W. Miksad, T. R. Akylas & T. Herbert). The Winter Annual Meeting of ASME, AMD vol. 87, pp. 37–65.
Ho, C. M. 1982 Local and global dynamics of free shear layers. In Proc. Symp. on Numerical and Physical Aspects of Aerodynamics Flow (ed. T. Cebeci), pp. 521–533. Springer.
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
Huang, L. S. & Ho, C. M. 1990 Small-scale transition in a plane mixing layer. J. Fluid Mech. 210, 475–500.Google Scholar
Itoh, N. 1977 Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 1. An asymptotic theory valid for small amplitude disturbances. J. Fluid Mech. 82, 455–467.Google Scholar
Jones, F. L. 1983 An experimental study of nonlinear wave interactions and modulations during transition of a symmetric wake. Ph.D. dissertation, LTT-83–02, The University of Texas at Austin.
Jones, F. L., Ritz, C. P., Miksad, R. W., Powers, E. J. & Solis, S. R. 1988 Measurement of the local wavenumber and frequency spectrum in a plane wake. Exps. in Fluids 6, 365–372.Google Scholar
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657–689.Google Scholar
Kim, K. I. & Powers, E. J. 1988 A digital method of modeling quadratically nonlinear systems with a general random input. IEEE Trans. Acoust., Speech, Signal Process. 36, 1758–1769.Google Scholar
Miksad, R. W. 1972 Experiments on the non-linear stages of free shear layer transition. J. Fluid Mech. 56, 695–719.Google Scholar
Miksad, R. W. 1973 Experiments on nonlinear interactions in the transition of a free shear layer. J. Fluid Mech. 59, 1–21.Google Scholar
Miksad, R. W., Hajj, M. R. & Powers, E. J. 1989 Measurements of nonlinear transfer functions for subharmonic generation in mixing layers. AIAA 89–0980.Google Scholar
Miksad, R. W., Jones, F. L., Powers, E. J., Kim, Y. C. & Khadra, L. 1982 Experiments on the role of amplitude and phase modulations during the transition to turbulence. J. Fluid Mech. 123, 1–29.Google Scholar
Mollo-Christensen, E. 1971 Physics of turbulence flows. AIAA J. 9, 1217–1228.Google Scholar
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223–252.Google Scholar
Monkewitz, P. A. & Huerre, P. 1982 The influence of the velocity ratio on the spatial instability of mixing layers. Phys. Fluids 25, 1137–1143.Google Scholar
Nayfeh, A. H. 1987 On secondary instabilities in boundary layers. In Stability of Time Dependent and Spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini), pp. 18–49. Springer.
Nikitopoulos, D. E. & Liu, J. T. C. 1987 Nonlinear binary—mode interactions in a developing mixing layer. J. Fluid Mech. 179, 345–370.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Powers, E. J. & Miksad, R. W. 1987 Polyspectral and measurements and analysis of nonlinear wave interactions. In Nonlinear Wave Interactions in Fluids (ed. R. W. Miksad, T. R. Akylas & T. Herbert). The Winter Annual Meeting of ASME, AMD-vol. 87, pp. 37–65.
Raetz, G. S. 1959 A new theory of the cause of transition in fluid flows. Norair Rep. NOR-59–383, Hawthorne, CA.Google Scholar
Ritz, C. P. & Powers, E. J. 1986 Estimation of nonlinear transfer functions for fully developed turbulence. Physica D 20, 320–334.Google Scholar
Ritz, C. P., Powers, E. J., Miksad, R. W. & Solis, R. S. 1988 Nonlinear spectral dynamics of a transitioning flow. Phys. Fluids 31, 3577–3588.Google Scholar
Sato, H. 1959 Further investigation on the transition of two dimensional separated layers at subsonic speed. J. Phys. Soc. Japan 14, 1797–1810.Google Scholar
Stegun, G. R. & Van Atta, C. W. 1970 A technique for phase speed measurements in turbulent flow. J. Fluid Mech. 43, 689–699.Google Scholar
Stuart, J. T. 1960 On the nonlinear mechanisms of hydrodynamic stability. J. Fluid Mech. 4, 1–21.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex Pairing: the mechanism of turbulent mixing layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237–255.Google Scholar