and Computational Methods

doi: 10.18698/2309-3684-2020-4-6172

In connection with the implementation of programs for the development of vast Arctic spaces, adopted in several countries, the attention of many researchers is attracted by the properties of the ice sheets of the seas and land bodies of water. At the same time, the following trend can be noted. If earlier theoretical works related to mathematical modeling of the ice sheet dynamics were mainly devoted to the propagation of free waves, then in recent years the work aimed at studying the processes of wave generation on the ice sheet under the influence of various sources of disturbances has clearly prevailed. To date, analytical solutions have been obtained for a number of problems concerning the generation of waves on the ice sheet by model sources of disturbances that are identical to some point hydrodynamic features, for example, point sources or dipoles. In this case, the ice was considered as a thin elastic plate floating on the surface of the water. Even in such an idealized setting, it was possible to reveal far from obvious properties of the ice cover. Modeling of sources of fluid perturbations by point hydrodynamic features was previously used in classical hydrodynamics to calculate perturbations occurring on the surface of a fluid. This approach has also shown its effectiveness in the problems of ice cover perturbations. A significant advantage of the method of modeling the sources of fluid disturbances using various systems of point hydrodynamic features can be attributed to the absence of the need to set boundary conditions in the area of localization of the sources of disturbances. Continuously distributed sources of disturbances can be approximated with varying accuracy in the form of a superposition of point hydrodynamic features, which makes it possible to model many processes occurring in the aquatic environment, for example, the flow around the bottom irregularities, the release of matter, the displacement of the bottom sections, etc. Thus, model sources of perturbations with point localization are of interest both from the point of view of modeling more complex sources, and from the point of view of obtaining the simplest estimates of practical significance. In this paper, we con-sider the spatial problem of perturbation of the ice cover by a point source localized in the thickness of an infinitely deep liquid, and having an intensity that varies according to the harmonic law. A numerical study of the amplitude-frequency characteristics of the ice cover of different thickness under the influence of such a source is carried out. The main attention is paid to the disturbances of the ice cover that occur directly above the source. The frequencies of the source intensity fluctuations to which the ice cover responds to the greatest extent are determined. The dependences of such frequencies on the thickness of the ice cover are obtained.

[1] Ilyichev A.T. Uedinyonnye volny v modelyah gidromekhaniki [Solitary waves in models of hydromechanics]. Moscow, Fizmatlit Publ., 2003, 256 p.

[2] Lamb G. Gidrodinamika [Hydrodynamics]. Moscow, Leningrad, Gostekhizdat Publ., 1947, 928 p.

[3] Kochin N.E., Kibel I.A., Roze N.V. Teoreticheskaya gidromekhanika. T.1. [Theoretical Hydromechanics. Vol. 1.] Leningrad, Moscow, Gostekhizdat Publ., 1948, 535 p.

[4] Kochin N.E. O volnovom soprotivlenii i pod"yomnoj sile pogruzhennyh v zhidkost' tel [On the wave resistance and lifting force of bodies immersed in a liquid]. Trudy konferencii po teorii volnovogo soprotivleniya [Proceedings of the Conference on the theory of wave resistance], 1937, pp. 65–134.

[5] Keldysh M.V., Lavrentiev M.A. O dvizhenii kryla pod poverhnost'yu tyazheloj zhidkosti [On the movement of a wing under the surface of a heavy liquid]. Trudy konferencii po teorii volnovogo soprotivleniya [Proceedings of the Conference on the theory of wave resistance], 1937, pp. 31–64.

[6] Keldysh M.V. Zamechaniya o nekotorykh dvizheniyakh tyazheloy zhidkosti [Notes on some movements of a heavy liquid]. Izbrannye trudy. Mekhanika [Remarks on some motions of a heavy fluid. Selected works. Mechanics]. Moscow, Nauka Publ., 1985, pp. 100–103.

[7] Sretenskiy L.N. Teoriya volnovykh dvizheniy zhidkosti [Theory of wave motions of a fluid]. Moscow, Nauka Publ., 1977, 815 p.

[8] Milne-Thompson L.M. Teoreticheskaya gidrodinamika [Theoretical hydrodynamics]. Moscow, Mir Publ., 1964, 660 p.

[9] Chowdhury R.G., Mandal B.N. Motion due to fundamental singularities in finite depth water with an elastic solid cover. Fluid Dynamics Research, 2006, vol. 38, iss. 4, pp. 224–240.

[10] Lu D.Q., Dai S.Q. Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice–cover. Archive of Applied Mechanics, 2006, vol. 76, iss. 1–2, pp. 49–63.

[11] Lu D.Q., Dai S.Q. Flexural– and capillary–gravity waves due to fundamental singularities in an inviscid fluid of finite depth. International Journal of Engineering Science, 2008, vol. 46, iss. 11, pp. 1183–1193.

[12] Stepanyants Y.A., Sturova I.V. Waves on a compressed floating ice plate caused by motion of a dipole in water. Journal of Fluid Mechanics, 2020, vol. 97, art no. A7.

[13] Li Z.F, Wu G.X., Shi Y.Y. Interaction of uniform current with a circular cylinder submerged below an ice sheet. Applied Ocean Research, 2019, vol. 86, pp. 310–319.

[14] Das D., Sahu M. Wave radiation by a horizontal circular cylinder submerged in deep water with ice–cover. Journal of Ocean Engineering and Science, 2019, vol. 4, iss. 1, pp. 49 – 54.

[15] Li Z.F., Wu G.X., Ji C.Y. Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack. Journal of Fluid Mechanics, 2018, vol. 845, pp. 682–712.

[16] Collins C.O., Rogers W.E., Lund B. An investigation into the dispersion of ocean surface waves in sea ice. Ocean Dynamics, 2017, vol. 67, iss. 2, pp. 263–280.

[17] Sturova I.V. Unsteady three–dimensional sources in deep water with an elastic cover and their applications. Journal of Fluid Mechanics, 2013, vol. 730, p. 392–418.

[18] Savin A.S., Gorlova N.E., Strunin P.A. Numerical simulation of the point pulse source impact in a liquid on the ice cover. Маthematical Modeling and Coтputational Methods, 2017, no. 1, pp. 78–90.

[19] Savin A.A., Savin A.S. Waves generated on an ice cover by a source pulsating in fluid. Fluid Dynamics, 2013, vol. 48, iss. 3, pp. 303–309.

[20] Dimitrienko Y.I., Gubareva E.A., Yurin Y.V. Asymptotic theory of thermocreep for multilayer thin plates. Маthematical Modeling and Computational Methods, 2014, no. 4, pp. 18–36.

[21] Dimitrienko Y.I., Yurin Y.V. Finite element simulation of the rock stress-strain state under creep. Маthematical Modeling and Coтputational Methods, 2015, no. 3, pp. 101–118.

[22] Savin A.S., Sidnyaev N.I., Tedeluri M.M. Study of the underwater explosion impact on the ice cover. Engineering Journal: Science and Innovation: Electronic Science and Engineering Publication, 2021, no. 2 (110). DOI: 10.18698/2308-6033-2021-2-2052

Савин А.С., Сидняев Н.И., Теделури М. М. Численное исследование амплитудно–частотной характеристики ледяного покрова, возмущаемого погруженным пульсирующим источником. Математическое моделирование и численные методы, 2020, № 4, с. 61–72.

Количество скачиваний: 327