Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-22T16:18:00.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation

Published online by Cambridge University Press:  10 April 2009

MOHAMMAD-REZA ALAM ,
YUMING LIU  and
DICK K. P. YUE
MOHAMMAD-REZA ALAM
Affiliation:
Department of Mechanical Engineering, Center for Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
YUMING LIU
Affiliation:
Department of Mechanical Engineering, Center for Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
DICK K. P. YUE*
Affiliation:
Department of Mechanical Engineering, Center for Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a direct numerical method to study the general problem of nonlinear interactions of surface/interfacial waves with variable bottom topography in a two-layer density stratified fluid. We extend a powerful high-order spectral (HOS) method for nonlinear gravity wave dynamics in a homogeneous fluid to the case of a two-layer fluid over non-uniform bottom. The method is capable of capturing the nonlinear interactions among large number of surface/interfacial wave mode and bottom ripple components up to an arbitrary high order. The method preserves exponential convergence with respect to the number of modes of the original HOS and the (approximately) linear effort with respect to mode number and interaction order. The method is validated through systematic convergence tests and comparison to a semi-analytic solution we obtain for an exact nonlinear Stokes waves on a two-layer fluid (in uniform depth). We apply the numerical method to the three classes of generalized Bragg resonances studied in Alam, Liu & Yue (J. Fluid Mech., vol. 624, 2009, p. 225), and compare the perturbation predictions obtained there with the direct simulation results. An important finding is possibly the important effect of even higher-order nonlinear interactions not accounted for in the leading-order perturbation analyses. To illustrate the efficacy of the numerical method to the general problem, we consider a somewhat more complicated case involving two incident waves and three bottom ripple components with wavenumbers that lead to the possibility of multiple Bragg resonances. It is shown that the ensuing multiple (near) resonant interactions result in the generation of multiple new transmitted/reflected waves that fill a broad wavenumber band eventually leading to the loss of order and chaotic motion.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 624 , 10 April 2009 , pp. 225 - 253
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alam, M.-R. 2008 Interaction of waves in a two-layer density stratified fluid. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009 Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis. J. Fluid Mech. 624, 225–253.CrossRefGoogle Scholar
Ardhuin, F. & Magne, R. 2007 Scattering of surface gravity waves by bottom topography with a current. J. Fluid Mech. 576, 235–264.CrossRefGoogle Scholar
Argyris, J., Faust, G. & Haase, M. 1994 An Exploration of Chaos. North-Holland Publishing Co.Google Scholar
Athanassoulis, G. A. & Belibassakis, K. A. 1999 A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. J. Fluid Mech. 389, 275–301.CrossRefGoogle Scholar
Ball, F. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19, 465–478.CrossRefGoogle Scholar
Boegman, L., Imberger, J., Ivey, G. N. & Antenucci, J. P. 2003 High-frequency internal waves in large stratified lakes. Limnol. Oceanogr. 48 (2), 895–919.CrossRefGoogle Scholar
Cummins, P. F., Vagle, S., Armi, L. & Farmer, D. M. 2003 Stratified flow over topography: upstream influence and generation of nonlinear internal waves. R. Soc. Lond. Proc. Ser. A Math. Phys. Engng Sci. 459 (2034), 1467–1487.CrossRefGoogle Scholar
Dommermuth, D. G. & Yue, D. K. P. 1987 A higher-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.CrossRefGoogle Scholar
Dommermuth, D. G. & Yue, D. K. P. 1988 The nonlinear three-dimensional waves generated by a moving surface disturbance. In Proceedings of 17th Symposium on Naval Hydrodynamics, Hague, Netherlands, pp. 918, 923. National Academy Press.Google Scholar
Goda, Y. & Suzuki, Y. 1976 Estimation of incident and reflected waves in random wave experiments. In Proceedings of 15th Coastal Engng Conference, Honolulu, Hawaii. Pp. 828–845.Google Scholar
Hill, D. F. & Foda, M. A. 1996 Subharmonic resonance of short internal standing waves by progressive surface waves. J. Fluid Mech. 321, 217–233.CrossRefGoogle Scholar
Hill, D. F. & Foda, M. A. 1998 Subharmonic resonance of oblique interfacial waves by a progressive surface wave. R. Soc. Lond. Proc. Ser. A Math. Phys. Engng Sci. 454 (1972), 1129–1144.CrossRefGoogle Scholar
Jamali, M. 1998 Surface wave interaction with oblique internal waves. PhD thesis, The University of British Columbia, Vancouver, Canada.Google Scholar
Joyce, T. M. 1974 Nonlinear interactions among standing surface and internal gravity waves. J. Fluid Mech. 63 (04), 801–825.CrossRefGoogle Scholar
Liu, Y. & Yue, D. K. P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–326.CrossRefGoogle Scholar
Miropol'sky, Y. Z. 2001 Dynamics of Internal Gravity Waves in the Ocean, Atmospheric and Oceanographic Sciences Library, vol. 24. Kluwer Academic Publishers.Google Scholar
Temam, R. 1997 Applied Mathematical Sciences, vol. 68: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer-Verlag.Google Scholar
Wen, F. 1995 Resonant generation of internal waves on the soft sea bed by a surface water wave. Phys. Fluids 7 (8), 1915–1922.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
Zhu, Q., Liu, Y. & Yue, D. K. P. 2008 Resonant interactions between Kelvin ship waves and ambient waves. J. Fluid Mech. 597, 171–197.CrossRefGoogle Scholar