Abstract
The study on the global instability of a Stokes layer, which is a typical unsteady flow, is usually a paradigm for understanding the instability and transition of unsteady flows. Previous studies suggest that the neutral curve of the global instability obtained by the Floquet theory is only mapped out in a limited range of wave numbers (0.2 ≤ α ≤ 0.5). In this paper, the global instability is investigated with numerical simulations for all wave numbers. It is revealed that the peak of the disturbances displays irregularity rather than the periodic evolution while the wave number is beyond the above range. A “neutral point” is redefined, and a neutral curve of the global instability is presented for the whole wave numbers with this new definition. This work provides a deeper understanding of the global instability of unsteady flows.
Similar content being viewed by others
References
Collins, J. I. Inception of turbulence at the bed under periodic gravity waves. J. Geophys. Res., 68, 6007–6014 (1963)
Davis, S. H. and von Kerczek, C. A reformulation of energy stability theory. Arch. Rat. Mech. Anal., 52, 112–117 (1973)
Cowley, S. J. High frequency Rayleigh instability of Stokes layers. Stability of Time-Dependent and Spatially Varying Flows (eds. Dwoyer, D. L. and Hussaini, M. Y.), Springer, New York, 261–275 (1987)
Von Kerczek, C. and Davis, S. H. Linear stability theory of oscillatory Stokes layers. J. Fluid Mech., 62, 753–773 (1974)
Hall, P. The linear instability of flat Stokes layers. J. Fluid Mech., A359, 151–166 (1978)
Akhavan, R., Kamm, R. D., and Shapiro, A. H. An investigation of transition to turbulence in bounded oscillatory Stokes flows II: numerical simulations. J. Fluid Mech., 225, 423–444 (1991)
Blennerhassett, P. J. and Bassom, A. P. The linear stability of flat Stokes layers. J. Fluid Mech., 464, 393–410 (2002)
Hall, P. On the instability of Stokes layers at high Reynolds numbers. J. Fluid Mech., 482, 1–15 (2003)
Blennerhassett, P. J. and Bassom, A. P. The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech., 556, 1–25 (2006)
Luo, J. and Wu, X. On the linear stability of a finite Stokes layer: instantaneous versus Floquet modes. Phys. Fluids, 22, 1–13 (2010)
Thomas, C., Bassom, A. P., Blennerhassett, P. J., and Davies, C. Direct numerical simulations of small disturbances in the classical Stokes layer. J. Eng. Math., 68, 327–338 (2010)
Thomas, C., Bassom, A. P., and Blennerhassett, P. J. The linear stability of oscillating pipe flow. Phys. Fluids, 24, 1–10 (2012)
Thomas, C., Davies, C., Bassom, A. P., and Blennerhassett, P. J. Evolution of disturbance wavepackets in an oscillatory Stokes layer. J. Fluid Mech., 752, 543–571 (2014)
Thomas, C., Bassom, A. P., Blennerhassett, P. J., and Davies, C. The linear stability of a Stokes layer subjected to high-frequency perturbations. J. Fluid Mech., 764, 193–218 (2015)
Hino, M., Sawamoto, M., and Takasu, S. Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech., 75, 193–207 (1976)
Hino, M., Kasguwayanagi, M., Nakayama, A., and Hara, T. Experiments on the turbulence statistics and the structure of a reciprocating oscillatory flow. J. Fluid Mech., 131, 363–400 (1983)
Jensen, B. L., Sumer, B. M., and Fredsoe, J. Turbulent oscillatory boundary layers at high Reynolds numbers. J. Fluid Mech., 206, 265–297 (1989)
Akhavan, R., Kamm, R. D., and Shapiro, A. H. An investigation of transition to turbulence in bounded oscillatory Stokes flows I: experiments. J. Fluid Mech., 225, 395–422 (1991)
Malik, M. R. Numerical methods for hypersonic boundary layer stability. J. Comput. Phys., 86, 376–413 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (No. 11202147) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120032120007)
Rights and permissions
About this article
Cite this article
Kong, W., Luo, J. Global instability of Stokes layer for whole wave numbers. Appl. Math. Mech.-Engl. Ed. 37, 999–1012 (2016). https://doi.org/10.1007/s10483-016-2113-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-016-2113-8