Rubric: "1.1.9. Mechanics of liquid, gas and plasma (physical and mathematical sciences)"



534.131.2 Modeling of dynamic stability of thin-walled structures partially filled with liquid under hydrostatic action

Park S. (Moscow Aviation Institute (National Research University)), Grigor’ev V. G. (Moscow Aviation Institute (National Research University))


doi: 10.18698/2309-3684-2022-3-317


In this paper, we consider the problem of stability of a thin-walled shell structure with two hemispherical bottoms of the same thickness, partially filled with liquid, which is immersed in an external liquid medium and is under hydrostatic pressure. The dynamic characteristics of such a structure containing a limited volume of liquid under internal pressure and hydrostatic pressure are obtained. The developed program for calculating the dynamic characteristics of axisymmetric shell structures containing liquid is based on the finite element method. The finite elements have an annular shape when rotated around the axis of symmetry. The program is implemented in Excel spreadsheet using Visual Basic for Applications (VBA). It allows to calculate the natural frequencies of thin-walled elastic structures interacting with an arbitrary number of liquids, considering the influence of the static stress-strain state caused by hydrostatic and internal pressure and other external forces that do not violate the axial symmetry of the problem. At a fixed value of the internal pressure, the calculation of the lowest natural frequencies of vibrations with different numbers of waves along the circumference is performed. By successive refinement, the critical thickness of the shell is determined, at which at least one of the natural frequencies reaches zero. The internal pressure p varies from 0 to 1 atm. in increments of 0,1 atm. and the calculations are repeated to obtain each critical value. At each pressure value, curves are plotted on the graph "number of waves — natural frequency". On the coordinate plane "pressure — shell thickness" the boundary of the instability region is constructed.


Пак Сонги, Григорьев В.Г. Моделирование динамической устойчивости тонкостенных конструкций, частично заполненных жидкостью, при гидростатическом воздействии. Математическое моделирование и численные методы, 2022, № 3, с. 3–17.



519.6:532.5 The Mathematical modeling of a spots chain with different intervals in a stratified

Gushchin V. A. (Institute for Computer Aided Design of the Russian Academy of Sciences), Smirnova I. A. (Institute for Computer Aided Design of the Russian Academy of Sciences)


doi: 10.18698/2309-3684-2023-3-8091


A comparison is made of the spots chain dynamics of mixed liquid in a stratified environment, with different initial intervals between spots. For mathematical modeling, the SMIF method (splitting method for studying incompressible fluid flows) is used. The finitedifference scheme of the method has the second order of approximation in terms of spatial variables, minimal scheme dissipation and dispersion, and has the property of monotonicity. The results of comparing the dynamics of the stream function depending on the initial interval between spots are presented for the same values of the Reynolds and Froude numbers.


Гущин В.А., Смирнова И.А. Математическое моделирование цепочки пятен с различными интервалами в стратифицированной жидкости. Математическое моделирование и численные методы, 2023, № 3, с. 80–91.



532.5+519.6 Mathematical modeling of laminar and turbulent filtration processes of piquid incompressible medium in porous mesh materials

Gordonov A. O. (АО ГНЦ "Центр Келдыша"), Laptev I. V. (АО ГНЦ "Центр Келдыша"), Sidorenko N. Y. (АО ГНЦ "Центр Келдыша"/Moscow Institute of Physics and Technology), Ivanov M. Y., Malahov A. S., Resh G. F.


doi: 10.18698/2309-3684-2023-2-6789


The problems of mathematical modeling of three–dimensional laminar and turbulent motion of a viscous incompressible fluid in multilayer permeable structures – porous mesh materials are considered. Each layer of the material is a woven metal mesh with square cells of micron sizes. Porous mesh materials are widely used in space, chemical, oil and gas, nuclear and other industries, for example, as hydraulic filters. Such materials have a complex internal structure and a variety of possible geometric configurations. Therefore, in the general case, the nature of the functional dependence of the hydraulic resistance that a material sample exerts on the flow of fluid flowing in its pore channels from the Reynolds number is not known. To determine this dependence on the existing material, as well as to create a material with a predetermined hydraulic resistance, computational fluid dynamics tools were used. The domestic engineering analysis system "Logos" and the author's program code developed in Keldysh Research Center were used. The physical parameters of liquid mass transfer in a porous filter material and its hydraulic resistance are determined by the methods of control volumes on an unstructured computational grid for integrating the Navier-Stokes equations and Lattice Boltzmann Method. It is established that the theoretical methods used allow us to estimate from above the functional dependence of the hydraulic resistance of a porous mesh material on the Reynolds number in the range of values from 0.01 to 500. To verify the mathematical model an experimental setup was made with the help of which a cycle of hydraulic spills of sample of porous mesh material was performed. The numerical solutions obtained are consistent with the available analytical dependencies obtained in the works of domestic and foreign scientists and the results of experimental studies.


Городнов А.О., Лаптев И.В., Сидоренко Н.Ю., Иванов М.Ю., Малахов А.С., Реш Г.Ф. Математическое моделирование процессов ламинарной и турбулентной фильтрации жидкой несжимаемой среды в пористых сетчатых материалах. Математическое моделирование и численные методы, 2023, № 2, с. 67–89.



551.5; 519.6 Organization of numerical experiments on a joint global model of the atmosphere and ocean

Parkhomenko V. P. (ФИЦ ИУ РАН/Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2022-4-3147


A three-dimensional hydrodynamic model of the global climate is presented, including a block of the atmosphere general circulation, a block of the ocean in geostrophic approximation with a frictional term in the horizontal momentum equations with a real distribution of depths and continents, a block of the sea ice evolution. Calculations of possible climate change up to 2100 are given on the basis of several CO2 growth scenarios. A significant decrease in the meridional flow of water in the Atlantic has been established during the implementation of a strong CO2 growth scenario. Numerical experiments have been carried out to identify potential hysteresis associated with attenuation, up to blocking (under certain conditions) Atlantic meridional thermohaline circulation.


Пархоменко В.П. Организация численных экспериментов на совместной глобальной модели атмосферы и океана. Математическое моделирование и численные методы, 2022, № 4, с. 31–4



532.59 Numerical study of the effect of an internal wave on the surface of a two-layer flow flowing around a point source

Nosov V. N. (Vernadsky Institute of Geochemistry and Analytical Chemistry, RAS), Savin A. S. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2023-3-92104


The flow around a point source localized in the lower layer of a two-layer liquid with a free surface is considered. Expressions are obtained for the perturbation of the free surfaceof the liquid associated with the manifestation of an internal wave. The source operating in the liquid is represented as a superposition of point pulse sources. This approach made it possible to find the perturbation of the flow surface as a superposition of perturbations caused by point pulse sources. The approximation used is quite justified in cases of modeling real sources of disturbances located at considerable depths, since such sources cause small disturbances of the sea surface. It is established that the internal waves appearing on the flow surface form a wedge-shaped structure. The angle of the wedge solution of the internal waves coming to the surface decreases with increasing flow velocity. The dependence of the angle of the wave wedge solution on the Froude number determined by the flow velocity and thickness of the upper liquid layer is found. The considered problem is of theoretical and practical interest, since more complex models of real disturbances of the surface of the marine environment during the flow of various inhomogeneities can be constructed as superpositions of model elementary disturbances from point sources.


Носов В.Н., Савин А.С. Численное исследование воздействия внутренней волны на поверхность двухслойного потока, обтекающего точечный источник. Математическое моделирование и численные методы, 2023, № 3, с. 92–104.



621.74.043 Mathematical modeling of the process of mixing of liquid metal in the mold of a continuous steel casting plant

Odinokov V. I. (Komsomolsk-na-Amure State University), Evstigneev A. I. (Komsomolsk-na-Amure State University), Dmitriyev E. A. (Komsomolsk-na-Amure State University), Karpenko V. A. (Komsomolsk-na-Amure State University)


doi: 10.18698/2309-3684-2023-3-1841


Experimental studies of the flow of liquid metal in the UNRS crystallizer are a long, complex and time-consuming process. Therefore, mathematical modeling by numerical methods is increasingly used for this purpose. A new technology for pouring liquid metal into a crystallizer is proposed. The original patented design of the device, consisting of a direct-flow and rotating gluhodon glasses, is given. The main results of studies of the melt flow in the crystallizer volume are presented. The objects of research were the hydrodynamic and thermal fluxes of liquid metal of a new process of casting steel into a rectangular section mold of the UNRS, and the result was a spatial mathematical model describing the fluxes and temperatures of liquid metal in the mold. To model the processes occurring during the flow of metal in the crystallizer, a specially created software package was used. The theoretical calculations are based on the fundamental equations of hydrodynamics, the equations of mathematical physics (the equation of thermal conductivity taking into account mass transfer) and a proven numerical method. The studied area was divided into elements of finite dimensions, for each element the resulting system of equations was written in a difference form. The result of the solution is the fields of velocities and temperatures of the metal current in the volume of the mold. According to the developed numerical schemes and algorithms, a calculation program has been compiled. An example of calculating the casting of steel into a mold of rectangular cross-section, flow diagrams of liquid metal along various sections of the mold is given. Vector flows of liquid metal in different sections of the mold at different numbers of revolutions of the jacket with vertical ribs are clearly presented. Areas of different turbulence have been identified. The mixing mode at n = 30 rpm is considered optimal. At n = 50 rpm, liquid metal is ejected into the slag bath.


Одиноков В.И., Евстигнеев А.И., Дмитриев Э.А., Карпенко В.А. Математическое моделирование процесса перемешивания жидкого металла в кристаллизаторе установке непрерывной разливки стали. Математическое моделирование и численные методы, 2023, № 3, с. 18–41.