CC BY
4This work was supported by the Russian Foundation for Basic Research (grant 19-01-00595).
References
1. Agoshkov V. I., Zalesny V. B., Sheloput T. O. Variational Data Assimilation in Problems of Modeling Hydrophysical
Fields in Open Water Areas // Izvestiya, Atmospheric and Oceanic Physics. 2020. V. 56, N. 3. P. 253-267.
Construction of a model of the Earth's ion-magnetosphere, which determines the dynamics
of the propagation of a radar pulse
O. V. Shestakova
Moscow Aviation Institute (National Research University)
Email: Olay.sova@yandex.ru
DOI 10.24412/cl-35065-2021-1-01-49
The expansion of the scale of the tasks solved by the supporting space systems requires the improvement
of methods and algorithms used for processing trajectory measurements in order to increase their reliability
and accuracy of determining the motion of the aircraft.
Various factors influence the accuracy of the estimation of motion parameters.
To solve the set tasks, it is necessary to build a mathematical model of the Earth's ion-magnetosphere,
close to the real one. In this case, it is necessary to take into account the various properties and characteristics
of the Earth's ion-magnetosphere.
Effect of the shape of moving external load on ice deflections and strains distribution in a frozen channel
T. A. Sibiryakova, N. A. Osipov, K. A. Shishmarev
Altai State University, Barnaul
Email: shishmarev.k@mail.ru
DOI 10.24412/cl-35065-2021-1-01-50
Deflections of an ice cover in a frozen channel caused by a moving load are considered. The channel has a
rectangular cross-section. The fluid in the channel is inviscid, incompressible and covered with ice. The ice cov-
er is modeled as a thin elastic plate with constant thickness. The flow caused by the ice deflections is potential.
The problem is solved within the linear theory of hydroelasticity [1]. The load is modeled by a pressure distri-
bution and moves along the channel at a constant speed. The load is of arbitrary shape and vary in calcula-
tions. The problem using the Fourier transform along the channel is reduced to the problem with respect to
the deflections profile across the channel, which is solved by the method of normal modes [2]. The solution is
obtained as the sum of the integrals of the inverse Fourier transform. It is shown that parallel or sequential
motion of loads can increase stresses in the ice cover both on the channel walls and between loads. The nu-
merical and analytical study of the considered problem is presented. The problem of the motion of an external
load along unbounded ice was studied in [1], along the central line of the channel in [3].
The work is supported by the State Assignment of the Russian Ministry of Science and Higher Education entitled
`Modern methods of hydrodynamics for environmental management, industrial systems and polar mechanics' (Govt. con-
tract code: FZMW-2020-0008, 24 January 2020).
References
1. Squire V. Moving loads on ice plates. Kluwer Academic Publishers, 1996.
2. Korobkin A.A., Khabakhpasheva T.I.. Plane problem of asymmetrical wave impact on an elastic plate // J. Of
Applied Mechanics And Technical Physics. 1998. V.39(5). pp. 782-791.
3. Shishmarev K. A., Khabakhpasheva T. I., Korobkin A. The response of ice cover to aload moving along a frozen
channel // Applied Ocean Research. 2016. V.59. pp. 313-326.
Solution of advection-diffusion-reaction problems on a sphere
Yu. N. Skiba
Universidad Nacional Autonoma de Mexico
Email: skiba@unam.mx
DOI 10.24412/cl-35065-2021-1-01-51
The new algorithm proposed in [1] is applied for solving linear advection-diffusion-reaction problems and
nonlinear diffusion problems on a sphere. Discretization of differential problems in space is performed by the
finite volume method using the Gauss theorem for each grid cell. For time discretization, the method of sym-
metrized two-cycle componentwise splitting and the Crank-Nicholson scheme are used. The obtained numeri-
cal method has the second order of approximation in space and time. It is implicit and unconditionally stable;
in addition, operator splitting provides a direct (non-iterative) and fast implementation of all implicit schemes.
The theoretical results are confirmed numerically by simulating various linear processes on the sphere (diffu-
sion in a spherical sector, diffusion flux through the pole, advection flux through the pole, dispersion of a pol-
lutant emitted from multiple point sources) and nonlinear diffusion processes (spiral waves, nonlinear tem-
perature waves, HS, LS and S blow-up combustion modes and solutions of the Gray-Scott model). Numerical
experiments [2] show a high accuracy and efficiency of the method that correctly describes the processes of
advection-diffusion on a sphere (including processes near the poles and through the poles) and the mass bal-
ance of matter in forced and dissipative discrete systems. Moreover, in the absence of external forcing and
dissipation, the method conserves both the total mass and the L2-norm of the solution.
The author thanks the National System of Researchers (SNI, CONACYT, Mexico) for grant 14539.
References
1. Skiba Yu. N. A non-iterative implicit algorithm for the solution of advection�diffusion equation on a sphere // Int. J.
Numer. Methods Fluids. 2015. V. 78, N. 5. P. 257-282.
2. Skiba Yu. N., Cruz-Rodriguez R.C., Filatov D.M. Solution of linear and nonlinear advection-diffusion problems on a
sphere // Numer Methods Partial Differential Eq. 2020. V. 36, N. 6. P. 1922-1937.
Mathematical modeling of the flow in a wake behind an impact of an elastic body on a thin liquid layer
K. A. Shishmarev1, K. E. Naydenova1, T. I. Khabakhpasheva2, A. A. Korobkin3
1Altai State University, Barnaul
2Lavrentyev Institute of Hydrodynamics SB RAS
3University of East Anglia, Norwich, UK
Email: shishmarev.k@mail.ru
DOI 10.24412/cl-35065-2021-1-01-52
The study is motivated by the results of experiments on droplet deposition in an annular gas-liquid flow
and mass transfer between gas and liquid film [1]. A two-dimensional problem of the impact of an elastic body
on a thin layer of liquid is considered. The body is modeled by a thin plate. At the beginning of impact the liq-
uid layer is at rest. At the initial moment of time, the elastic plate touches the liquid at a single point. After the
impact, the plate moves in the positive x-direction and penetrates into the liquid layer (Oxy is the Cartesian
coordinate system). The coupled problem of deflections of an elastic plate and the fluid motion under the
plate is studied in [2] in the leading approximation. After determining the deflections of the plate and the be-
