and Computational Methods

#### 519.8 A Diffusion Model of Cluster Evolution in a Heat-Resistant Nickel Alloy Metal Melt

**Tyagunov A. G. (Ural Federal University), Zeyde K. M. (Universidad Politècnica Salesiana/University of Genoa), Milder O. B. (Ural Federal University), Tarasov D. A. (Ural Federal University)**

doi: 10.18698/2309-3684-2023-2-332

In this work, a mathematical model of the thermo-temporal evolution of a cluster in the melt of a heat-resistant nickel alloy ZhS6U is constructed. An initial-boundary value problem with a moving boundary is formulated, for the solution of which numerical modeling is used by the particle trajectory method, and a number of classical physical theories are used to describe evolutionary processes. To check the accuracy of the model, a physical experiment is involved in constructing polytherms and isotherms of the electrical resistance of the alloy under consideration. It has been confirmed that the Brownian diffusion model and Drude's theory of conductivity are applicable to describe both the temporal and temperature evolution of a cluster. The approach to modeling based on "hard balls" also justified itself. According to the simulation results, in the time range from 1690 to 1752 K, the number of particles in the cluster varies from 5000 to 2000, the average dynamic viscosity of the cluster varies from 3 to 2 * 1010 Pa * s, however, it is assumed that the central part is much denser than periphery. The cluster radius varies from 24 to 18 A, and the radius of the free zone around the cluster varies from 56 to 43 A. The directions of further development of the model are determined.

Тягунов А.Г., Зейде К.М., Мильдер О.Б., Тарасов Д.А. Диффузионная модель эволюции кластера в металлическом расплаве жаропрочного никелевого сплава. Математическое моделирование и численные методы, 2023, № 2, с. 3–32.

#### 539.3 Asymptotic theory of thin multilayer micropolar elastic plates

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Boyko S. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-2-3366

The problem of development of a theory for calculating the stress-strain state of thin multilayer elastic plates in the moment (micropolar) theory, is considered. The solution of this problem is built using an asymptotic analysis of the general equations for a 3-dimensional quasi-static problem of the moment theory of elasticity. The asymptotic analysis is carried out with respect to a small parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local problems of the moment theory of elasticity are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged system of equations for multilayer plates is presented. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 9 components of the stress tensor and the moment stress tensor (in general) in the plate. As a special case, the problem of calculating the stress-strain state of a centrally symmetrical hingedly fixed plate when bending under the action of a uniformly distributed pressure. A complete analytical solution of this problem for all non-zero components of the stress tensor and the moment stress tensor is obtained. A numerical analysis of the solution of the problem for a single layer plate for the stress tensor is carried out, basing on the obtained expressions. A comparative analysis of the obtained results with similar calculations for the classical theory of elasticity is carried out, with revealing of similarities and differences for all components of the stress tensor.

Димитриенко Ю.И., Бойко С.В. Асимптотическая теория многослойных тонких микрополярных упругих пластин. Математическое моделирование и численные методы, 2023, № 2, с. 33–66.

#### 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.

#### 533.6.011.5 Heat transfer modeling on the surface of a sphere in a gas flow

**Kotenev V. P. (Bauman Moscow State Technical University), Sysenko V. A.**

doi: 10.18698/2309-3684-2023-2-9099

The simple analytical formula for calculation of laminar specific heat flow (divided by corresponding value at the critical point) brought to sphere surface streamlined by supersonic gas flow are received in this work. The analysis of the results shows that the use of the presented formula gives the values of the specific heat flow with greater accuracy than the known approximate formulas. The comparing of the relative heat flow with the accurate computational results of solving the Navier-Stokes equations also confirm the effectiveness of the presented method. It is proposed to formulate a special rule of local spheres for a quick evaluation of the specific heat flow on the surfaces of other blunted bodies with different generators in the future.

Котенев В.П., Сысенко В.А. Новая зависимость профиля энтальпии в модели пограничного слоя. Математическое моделирование и численные методы, 2023, № 2, с. 90–99

#### 533.6.011.35 Determination of distributed aerodynamic characteristics of an axisymmetric body of the SOCBT configuration under turbulent flow by a transonic flow

**Kharchenko N. A. (Central Aerohydrodynamic Institute (TsAGI)/MEPhI/Moscow Aviation Institute (National Research University)), Nikonov A. M. (Bauman Moscow State Technical University/Central Aerohydrodynamic Institute (TsAGI))**

doi: 10.18698/2309-3684-2023-2-100128

The article presents the validation problem of transonic simulation of turbulent airflow of an axisymmetric body of the SOCBT configuration. The main computational complexity of the problem under consideration is the detailed resolution of the flow in the wall region to describe the turbulent boundary layer and further reproduce the experimentally obtained distributions of the pressure coefficient on the surface of the SOCBT configuration body.

Харченко Н.А., Никонов А.М. Определение распределенных аэродинамических характеристик осесимметричного тела конфигурации SOCBT при турбулентном обтекании трансзвуковым потоком. Математическое моделирование и численные методы, 2023, № 2, с. 100–128.

#### 338.001.36 Mathematical model for the formation of supply chains of raw materials from the commodity exchange under risk based on the profit trajectory for previous periods redundancy

**Rogulin R. S. (Vladivostok State University)**

doi: 10.18698/2309-3684-2023-2-129154

The formation of the supply chain of raw materials is closely related to the production problems of woodworking enterprises. Construction of supply chains for raw materials and the optimal calculation of daily production have been hot topics since the beginning of the second industrial revolution. This article discusses the enterprise of the Primorsky Territory of the woodworking industry, which does not have plots for rent. The purpose of the work is to solve the problem of building a supply chain of raw materials, taking into account the daily loading of production facilities and finding the optimal solution. The source of raw materials is the commodity exchange, where lots appear daily randomly in different mining regions. In the scientific literature, there are many ways to calculate the best profit value, taking into account many restrictions, but they do not take into account many features that are important for woodworking enterprises. Based on a review of the scientific literature, this article presents a mathematical model that acts as a decisionmaking mechanism on each individual day, and it differs in that it can take into account the coefficient of useful volume of raw materials that will reach the warehouse and travel time. The model was tested on the data of the Russian Commodity and Raw Materials Exchange and a company in Primorsky Krai. The result of testing the model is the calculated optimal profit trajectory for each set of data on the volume of raw materials, the time of lots in transit, as well as many important indicators for any production: profit volume, production volume of goods. The analysis of the received solutions showed that there are difficulties in planning supply chains and production volumes. The regions are analyzed as sources of raw materials, from which regions and when it is worth buying raw materials. The shortcomings and positive aspects of the mathematical model are given

Рогулин Р.С. Математическая модель формирования цепочек поставок сырья с товарно-сырьевой биржи в условиях риска с опорой на траекторию прибыли за предыдущие периоды. Математическое моделирование и численные методы, 2023,№ 2, с. 129–154.

#### 519.8 Simulation of the confrontation between the two sides, taking into account redundancy

**Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-2-155163

Based on the method of dynamics of averages, a model of two parties confrontation has been developed taking into consideration the bringing up of reserves by one of the parties. It is established that timely supply of reserves can significantly affect the course of the process and its final result. It is also shown that the use of the reserve at the beginning of the action significantly increases the capabilities of a group.

Чуев В.Ю., Дубограй И.В. Моделировании противоборства двух сторон c учетом резервирования. Математическое моделирование и численные методы,2023, № 2, с. 155–163