Load motion on an ice cover in the presence of a liquid layer with velocity shear

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Abstract

The behavior of an ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a pressure domain that moves rectilinearly at a constant velocity in the presence of a current with velocity shift in the upper layer is studied. It is assumed that the ice deflection is steady in the coordinate system moving with the load. The Fourier transform method is used within the framework of the linear wave theory. The critical velocities, the deflection of ice cover, and the wave forces are studied depending on the current velocity gradient, the shear layer thickness, the direction of motion, and the compression ratio.

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About the authors

L. А. Tkacheva

Институт гидродинамики им. М.А. Лаврентьева СО РАН

Author for correspondence.
Email: tkacheva@hydro.nsc.ru
Russian Federation, Новосибирск

References

  1. Ткачева Л.А. Движение нагрузки по ледяному покрову при наличии течения со сдвигом скорости // Изв. РАН. МЖГ. 2023. № 2. С. 113–122.
  2. Thompson P.D. The propagation of small surface disturbances through rotational flow // Ann. NY Acad. Sci. 1949. V. 51. P. 463–474.
  3. Abdullah A.J. Wave motion at the surface of a current which has an exponential distribution of vorticity // Ann. NY Acad. Sci. 1949. V. 51. P. 425–441.
  4. Fenton J.D. Some results for surface gravity waves on shear flows // J. Inst. Maths. Applics. 1953. V. 1. P. 1–20.
  5. Peregrine D.H. Interaction of water waves and currents // Adv. Appl. Mech. 1976. V. 16. P. 9–117.
  6. Kirby J.T., Chen T.M. Surface waves on vertically sheared flows: Approximate dispersion relation // J. Geophys. Res. 1989. V. 94. P. 1013–1027. doi: 10.1029/jc094ic01p01013.
  7. Skop R.A. Approximate dispersion relation for wave-current interactions // J. Waterw., Port, Coastal, Ocean Eng. 1987. V. 113. P. 187–195.
  8. Swan С., James R. A simple analytical model for surface water waves on a depth-varying current // Appl. Ocean Res. 2001. V. 22. P. 331–347.
  9. Stewart R.H., Joy J.W. HF radio measurements of surface currents // Deep Sea Res. 1974. V. 21. P. 1039–1949.
  10. Shrira V.I. Surface waves on shear currents: Solution of the boundary-value problem // J. Fluid Mech. 1993. V. 252. P. 565–584.
  11. Thompson P.D. The propagation of small surface disturbances through rotational flow // Ann. NY Acad. Sci. 1949. V. 51. P. 463–474.
  12. Герценштейн С.Я., Ромашева Н.Б., Чернявский М.В. О возникновении и развитии ветрового волнения // Изв. РАН. МЖГ. 1988. № 3. С. 163–169.
  13. Longuet-Higgins M.S. Instabilities of a horizontal shear flow with a free surface // J. Fluid Mech. 1998. V. 364. P. 147–162.
  14. Smeltzer B.K., Ellingsen S.A. Surface waves on arbitrary vertically-sheared currents // Phys. Fluids. 2017. V. 29. P. 047102.
  15. Zhang X. Short surface waves on surface shear // J. Fluid Mech. 2005. V. 541. P. 345–370.
  16. Стурова И.В. Действие пульсирующего источника в жидкости при наличии сдвигового слоя // Изв. РАН. МЖГ. 2023. № 4. С. 14–26.
  17. Brown M.K. A quadratically convergent Newton-like method upon Gaussian elimination // SIAM Numer. Anal. 1969. V. 6. № 4. P. 560–569.
  18. Лайтхилл Дж. Волны в жидкостях. М.: Мир, 1981.

Supplementary files

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2. Fig. 1. An example of eigenvectors of the dispersion relation for the system of equations (2.3), (2.4).

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3. Fig. 2. Wave number curves for an ice cover of 50 m thickness, 1–3 –

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4. Fig. 3 Dependence of critical velocities on the thickness of the shear layer at (a, b) and (c, d), y = 0 (a, c) and y = p (b, d): curves 1–5 correspond to the values

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5. Fig. 4. Ice cover deflection at –Q = 0 (a, b) and –Q = 1 (c, d), y = 0 (a, c) and y = p (b, d): 1 – S = 0; 2, 3 – S = 0.3, 0.5 for two layers H1 = 5 m, H2 = 45 m; 4, 5 – S = 0.3, 0.5 for one layer H1 = H = 50 m.

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6. Fig. 5. Dimensionless wave force coefficients A1, A2 depending on the load movement speed at –Q = 0, y = 0 (a), y = p/2 (b) and y = p (c): 1 – S = 0; 2, 3 – S = 0.3, 0.5 for two layers: H1 = 5 m, H2 = 45 m; 4, 5 – S = 0.3, 0.5 for one layer: H = 50 m.

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7. Fig. 6. Dimensionless wave force coefficients A1, A2 depending on the load movement speed at = 1, y = 0 (a), ψ = p/2 (b) and ψ = π (c): 1 – S = 0; 2, 3 – S = 0.3, 0.5 for two layers: H1 = 5 m, H2 = 45 m; 4, 5 – S = 0.3, 0.5 for one layer: H = 50 m.

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8. Fig. 7. Contribution of additional roots to wave forces depending on the speed of load movement at H1 = 5 m, ψ = π/2,  = 0 (a),  = 1 (b): 1–2 – S = 0.3, 0.5.

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