Development of the wave motion induced by near-bottom periodic disturbances in a two-layer shear current

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The behavior of wave motion arising in an ideal incompressible homogeneous fluid under switching-on periodic bottom disturbances is studied in the linear approximation for the two-dimensional non-stationary problem. In the undisturbed state, the velocities of two-layer fluid flow are linear functions of the vertical coordinate in each of the layers with different gradients and coincide on the boundary of the layers. The upper boundary of fluid can be either free or bounded by the rigid cover. The dispersion dependences and the group velocities of the appearing wave modes are determined. The vertical displacements of the free surface and the interface between the layers are calculated. A comparison with the solution for a single-layer fluid is carried out.

Full Text

Restricted Access

About the authors

I. V. Sturova

Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences

Author for correspondence.
Email: sturova@hydro.nsc.ru
Russian Federation, Novosibirsk

References

  1. Черкесов Л.В. Неустановившиеся волны. Киев: Наук.думка, 1970. 196 с.
  2. Черкесов Л.В. Гидродинамика поверхностных и внутренних волн. Киев: Наук.думка, 1976. 364 с.
  3. Черкесов Л.В. Гидродинамика волн. Киев: Наук.думка, 1980. 259 с.
  4. Ellingsen S.Å., Li Y. Approximate dispersion relations for waves on arbitrary shear flows // J. Geophysical Research: Oceans. 2017. V. 122. Р. 9889–9905.
  5. Zhang X. Short surface waves on surface shear // J. Fluid Mech. 2005. V. 541. P. 345–370.
  6. Longuet-Higgins M.S. Instabilities of a horizontal shear flow with a free surface// J. Fluid Mech. 1998. V. 364. P. 147–162.
  7. Стурова И.В. Действие пульсирующего источника в жидкости при наличии сдвигового слоя // Изв. РАН. МЖГ. 2023. № 4. С. 14–26.
  8. Суворов А.М. Генерация внутренних волн в потоке двухслойной жидкости со сдвигом скорости // Цунами и внутренние волны. Севастополь: МГИ АН УССР, 1976. С. 170–178.
  9. Кейз К.М. Гидродинамическая устойчивость как задача с начальными данными // В сб.: Гидродинамическая неустойчивость. М.: Мир, 1964. С. 37–46.
  10. Wahlén E. On rotational water waves with surface tension // Phil. Trans. R. Soc. A. 2007. V. 365. № 3. P. 2215–2225.
  11. Tyvand P.A., Torheim T. Surface waves from bottom vibrations in uniform open-channel flow // Eur. J. Mech. B./Fluids. 2012. V. 36. P. 39–47.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1. Flow diagram in an undisturbed state: a – two–layer liquid; b - single-layer liquid.

Download (69KB)
Indexing metadata
3. Fig. 2. Areas of unstable values of wave numbers depending on the velocity of the shear flow V0: 1, 2 – cases 1 and 2.

Download (58KB)
Indexing metadata
4. Fig. 3. Dispersion dependences (a, c) and group velocities (b, d): curves 1-3 correspond to the mode number for case 1 (a, b) and case 2 (c, d). Curves 4 show the dispersion dependence and group velocity of a single wave mode in the considered cases for a two-layer liquid under pressure a solid lid.

Download (255KB)
Indexing metadata
5. Fig. 4. Elevations of the free surface (a, b) and the interface of the layers (c–e) at t = 12 c for case 1 (a, c, e) and case 2 (b, d, e) in the presence of a free surface (a–d) and a solid cover (e, e). Vertical arrows indicate the positions of the wavefronts for different modes.

Download (417KB)
Indexing metadata
6. Fig. 5. Dispersion dependences (a) and group velocities (b): 1 – n = 1 and n = 3 for a two–layer liquid in case 1; 2 and 3 - modes n = 1 and n = 2 for a single-layer liquid in cases 3 and 4, respectively.

Download (157KB)
Indexing metadata
7. Fig. 6. Vertical displacements of the free surface (a, c, e, w) and on the horizon y = - H1 (b, d, e, h) at t = 12 c for a single-layer liquid, cases 3-6, respectively. Vertical arrows indicate the positions of the wavefronts for various models.

Download (473KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences