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Scattering of long water waves in a canal with rapidly varying cross-section in the presence of a current

Semyon Churilov

Andrei Ermakov

Germain Rousseaux

Yury Stepanyants*

  • Institute of Solar-Terrestrial Physics of the Siberian Branch of Russian Academy of Sciences, Irkutsk 664033, Russia

  • School of Agricultural, Computational and Environmental Sciences, University of Southern Queensland, QLD 4350, Australia

  • Institut Pprime, UPR 3346, CNRS - Université de Poitiers-ISAE ENSMA, 11 Boulevard Marie et Pierre Curie, Téléport 2, BP 30179, 86962 Futuroscope Cedex, France

  • Department of Applied Mathematics, Nizhny Novgorod State Technical University, Nizhny Novgorod, 603950, Russia and School of Agricultural, Computational and Environmental Sciences, University of Southern Queensland, QLD 4350, Australia

  • *yury.stepanyants@usq.edu.au
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Phys. Rev. Fluids 2, 094805 – Published 22 September, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.094805

Abstract

The analytical study of long-wave scattering in a canal with a rapidly varying cross-section is presented. It is assumed that waves propagate on a stationary current with a given flow rate. Due to the fixed flow rate, the current speed is different in the different sections of the canal, upstream and downstream. The scattering coefficients (the transmission and reflection coefficients) are calculated for all possible orientations of incident wave with respect to the background current (downstream and upstream propagation) and for all possible regimes of current (subcritical, transcritical, and supercritical). It is shown that in some cases negative energy waves can appear in the process of waves scattering. The conditions are found when the over-reflection and over-transmission phenomena occur. In particular, it is shown that a spontaneous wave generation can arise in a transcritical accelerating flow, when the background current enhances due to the canal narrowing. This resembles a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect.

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References (30)

  1. H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932).
  2. S. Massel, Hydrodynamics of the Coastal Zone (Elsevier, Amsterdam, 1989).
  3. M. W. Dingemans, Water Wave Propagation over Uneven Bottoms (World Scientific, Singapore, 1997).
  4. A. Kurkin, S. Semin, and Y. Stepanyants, Transformation of surface waves over a bottom step, Izv. Atmos. Ocean. Phys. 51, 214 (2015).
  5. R. Grimshaw, E. Pelinovsky, and T. Talipova, Fission of a weakly nonlinear interfacial solitary wave at a step, Geophys. Astrophys. Fluid Dyn. 102, 179 (2008).
  6. E. N. Churaev, S. V. Semin, and Y. A. Stepanyants, Transformation of internal waves passing over a bottom step, J. Fluid Mech. 768, R3–1 (2015).
  7. K. A. Belibassakis, Th. P. Gerostathis, and G. A. Athanassoulis, A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography, Appl. Ocean Res. 33, 384 (2011).
  8. W. L. Jones, Reflexion and stability of waves in stably stratified fluids with shear flow: A numerical study, J. Fluid Mech. 34, 609 (1968).
  9. R. Smith, The reflection of short gravity waves on a non-uniform current, Math. Proc. Camb. Phil. Soc. 78, 517 (1975).
  10. M. Stiassnie and G. Dagan, Partial reflexion of water waves by non-uniform adverse currents, J. Fluid Mech. 92, 119 (1979).
  11. K. Trulsen and C. C. Mei, Double reflection of capillary/gravity waves by a non-uniform current: A boundary-layer theory, J. Fluid Mech. 251, 239 (1993).
  12. W. G. Unruh, Experimental Black-Hole Evaporation? Phys. Rev. Lett. 46, 1351 (1981).
  13. T. Jacobson, Black hole evaporation and ultrashort distances, Phys. Rev. D 44, 1731 (1991).
  14. W. G. Unruh, Sonic analog of black holes and the effects of high frequencies on black hole evaporation, Phys. Rev. D 51, 2827 (1995).
  15. D. Faccio, F. Belgiorno, S. Cacciatori, V. Gorini, S. Liberati, and U. Moschella (eds.), Analogue Gravity Phenomenology, Lecture Notes in Physics, Vol. 870 (Springer International Publishing, Switzerland, 2013).
  16. L.-P. Euvé, F. Michel, R. Parentani, T. G. Philbin, and G. Rousseaux, Observation of Noise Correlated by the Hawking Effect in a Water Tank, Phys. Rev. Lett. 117, 121301 (2016).
  17. A. Coutant and S. Weinfurtner, The imprint of the analog Hawking effect in subcritical flows, Phys. Rev. D 94, 064026 (2016).
  18. S. Robertson, F. Michel, and R. Parentani, Scattering of gravity waves in subcritical flows over an obstacle, Phys. Rev. D 93, 124060 (2016).
  19. A. Coutant, R. Parentani, and S. Finazzi, Black hole radiation with short distance dispersion, an analytical s-matrix approach, Phys. Rev. D 85, 024021 (2012).
  20. S. Robertson, The theory of Hawking radiation in laboratory analogues, J. Phys. B: At. Mol. Opt. Phys. 45, 163001 (2012).
  21. E. R. Gazizov and D. V. Maklakov, Waveless gravity flow over an inclined step, J. Appl. Mech. Tech. Phys. 45, 379 (2004).
  22. Y. A. Stepanyants and A. L. Fabrikant, Propagation of waves in hydrodynamic shear flows, Sov. Phys. Uspekhi 32, 783 (1989).
  23. A. L. Fabrikant and Y. A. Stepanyants, Propagation of Waves in Shear Flows (World Scientific, Singapore, 1998).
  24. P. Maïssa, G. Rousseaux, and Y. Stepanyants, Negative energy waves in shear flow with a linear profile, Eur. J. Mech. B/Fluids 56, 192 (2016).
  25. M. S. Longuet-Higgins, Surface manifestations of turbulent flow, J. Fluid Mech. 308, 15 (1996).
  26. K. B. Dysthe, Lecture notes on linear wave theory, A lecture given at the Summer School, Water Waves and Ocean Currents, (2004), pp. 1–21, http://folk.uio.no/johng/info/dysthe/Nordfjordeid-versjon.pdf.
  27. A. Y. Basovich and V. I. Talanov, Transformation of short surface waves on inhomogeneous currents, Izv. Amos. Ocean. Phys. 13, 514 (1977).
  28. P. Maïssa, G. Rousseaux, and Y. Stepanyants, Wave blocking phenomenon of surface waves on a shear flow with a constant vorticity, Phys. Fluids 28, 032102 (2016).
  29. S. Churilov, A. Ermakov, and Y. Stepanyants, Wave scattering in spatially inhomogeneous currents, Phys. Rev. D 96, 064016 (2017).
  30. M. E. McIntyre, On the ‘wave momentum’ myth, J. Fluid Mech. 106, 331 (1981).

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Phys. Rev. Fluids 2, 094805– Published 22 September, 2017
  • Received 23 May 2017

©2017 American Physical Society

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