Rubric: "2.5.14. Strength and thermal conditions of aircraft (technical sciences)"



539.3 Coupled modeling of high-speed aerothermodynamics and internal heat and mass transfer in composite aerospace structures

Dimitrienko Y. I. (Bauman Moscow State Technical University), Koryakov M. N. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Zakharov A. A. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2021-3-4261


A coupled problem of high-speed aerothermodynamics and internal heat and mass transfer in heat-shielding structures of reentry spacecraft made of ablative polymer composite materials is considered. To determine the heat fluxes in the shock layer of the reentry vehicle, the chemical composition of the atmosphere is taken into account. The mathematical formulation of the conjugate problem is formulated and an algorithm for the numerical solution is proposed. An example of the numerical solution of the problem for the reentry spacecraft Stardust is presented. It is shown that taking into account chemical reactions in the gas flow around the surface of the reentry vehicle is essential for the correct determination of the gas temperature in the boundary layer. It is also shown that the developed numerical method for solving the problem makes it possible to determine the parameters of phase transformations in a heat-shielding structure depending on the heating time, in particular, it allows calculating the pore pressure field of gaseous products of thermal decomposition of a polymer composite, which, under certain conditions, can lead to material destruction.


мДимитриенко Ю.И., Коряков М.Н., Юрин Ю.В., Захаров А.А., Сборщиков С.В., Богданов И.О. Сопряженное моделирование высокоскоростной аэротермодинамики и внутреннего тепломассопереноса в композитных аэрокосмических конструкциях. Математическое моделирование и численные методы, 2021, № 3, с. 42–61.



536.2 Finite-difference approximation of mixed derivatives in mathematical physics equations

Gorskiy V. V. (Bauman Moscow State Technical University), Resh V. G. (Moscow State Textile University named after A. N. Kosygin)


doi: 10.18698/2309-3684-2021-4-5879


A qualitative numerical solution of the equations of mathematical physics is intimately connected with ensuring a high accuracy of approximation of all differential operators included in these equations. The solution of this problem for the first and second derivatives in the equations of mathematical physics, which are used to describe a wide range of scientific and technical problems has been described in numerous literary publications. At the same time, mixed derivatives are not so often present in the equations of mathematical physics and, therefore, issues related to the quality of finite-difference approximation of these derivatives are not given enough attention in literary publications. One of the main reasons for the appearance of mixed derivatives in the equations of mathematical physics is the use of an affine transformation of the coordinate system, which provides the transition to domain of a substantially simpler form. The solution of this problem is the subject of the present paper. The problem is solved by the example of approximation of mixed derivatives on rectangular domain of definition of the required function with constant steps in each direction. A detailed derivation of the finite-difference relations used for the finite-difference approximation of mixed derivatives in all typical nodes of the function domain is given, which makes it possible to develop the proposed technique on domains of different types.


Горский В.В., Реш В.Г. Конечно-разностная аппроксимация смешанных производных в уравнениях математической физики. Математическое моделирование и численные методы, 2021, № 4, с. 58–79



519.6:533.6 Numerical simulation of the flow structure near the descent vehicle and located in its wake parachute atsupersonic motion

Babakov A. V. (Institute for Computer Aided Design of the Russian Academy of Sciences)


doi: 10.18698/2309-3684-2023-3-6279


The article presents the results of a numerical study of the spatial non-stationary flow structure arising during supersonic motion in the atmosphere of the descent vehicle and parachute located in its vortex wake. The cases of the vehicle at angle of attack and various location of the parachute in relation to the vehicle are considered. For different distances between vehicle and parachute the patterns of the spatial non-stationary vortex structure of the flow occurring between them and in the near wake of the parachute are given. The significant influence of the distance between the vehicle and the parachute on the flow structure and force characteristics effect of the flow on the parachute is shown. Data on the influence of attack angle of the vehicle on the aerodynamic characteristics of parachute are presented. Numerical simulations are performed using two conservative numerical methods based on the approximation of conservation laws written in integral form for a finite volume. Calculations are based on the parallel algorithms implemented on modern supercomputer systems.


Бабаков В.А. Численное моделирование структуры потока около спускаемого аппарата и расположенного в его следе парашюта при сверхзвуковом движении. Математическое моделирование и численные методы, 2023, № 3, с. 62–79.



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.



533.6.011.5 A new dependence for the enthalpy profile in the boundary layer model

Kotenev V. P. (Bauman Moscow State Technical University), Tonkih E. G. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2022-2-7887


A new dependence for the enthalpy profile in the boundary layer model. A new analytical dependence is proposed for the enthalpy profile in the boundary layer, the use of which makes it possible to quickly and efficiently obtain the distribution of heat flux over the surface of blunted bodies


Котенев В.П., Тонких Е.Г. Новая зависимость профиля энтальпии в модели пограничного слоя. Математическое моделирование и численные методы, 2022, № 2, с. 80–89



539.3 Finite element modeling of temperature fields in thin-walled multilayer anisotropic shells

Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Koryakov M. N. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2023-1-4363


The problem of developing a model for calculating temperature fields in thin-walled multilayer curvilinear-anisotropic thin shells of arbitrary geometric shape, including composite ones, is considered. As a rule, to solve this problem, a specific coordinate notation of the equations of the theory of heat conduction is used, which creates certain difficulties for calculating complex composite shells. In this paper, it is proposed to use an invariant record of the variational formulation of problems in the theory of heat conduction, followed by the application of the finite element algorithm procedure. As a result, a matrix differential equation is derived for determining the temperature field at the nodes of a finite element mesh. A software module has been developed for the finite element solution of the problem of non-stationary thermal conductivity of shells. The module functions as part of the SMCM software package, created at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Systems, Bauman Moscow State Technical University (REC SIMPLEX). An example of solving the problem of calculating a non-stationary temperature field in a cylindrical shell with longitudinal-transverse reinforcement is given. Comparison of numerical simulation with similar calculations in the ANSYS software was carried out, which showed the high accuracy of the proposed method: the relative deviation of the results does not exceed 0,5%


Димитриенко Ю.И., Юрин Ю.В., Коряков М.Н., Маремшаова А.В. Конечно-элементное моделирование температурных полей в тонкостенных многослойных оболочечных элементах конструкций. Математическое моделирование и численные методы, 2023, No 1, с. 43–63



533.6.011.55 Numerical simulation high-speed flow around a cylindrical–conical body and a double cone

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


doi: 10.18698/2309-3684-2022-3346


The paper presents a classical validation problem of high-speed modeling. This problem is about interaction of a shock wave with a boundary layer in a laminar air flow around a cylindrical–conical body and a double cone. The main computational complexity of this problem is the detailed resolution of the near-wall region in order to further reproduce the experimental distributions of the surface characteristics of pressure and heat flux. Depending on the conditions of the undisturbed flow of the researched flow mode, the problem can have a recirculation zone, which is a vortex flow. This flow has a significant effect on the structure of the near-wall flow.


Харченко Н.А., Носенко Н.А. Численное моделирование обтекания высокоскоростным потоком цилиндрически–конического тела и двойного конуса. Математическое моделирование и численные методы, 2022, № 3, с. 33–46.



539.376 Simulation of creep in thin-walled shellsunder variable loads

Butina T. A. (Bauman Moscow State Technical University), Dubrovin V. M. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2022-1-97108


Under prolonged loading during operation, structures are subject to the phenomenon of creep, which can affect its performance. This influence depends on the load level, loading duration, operating conditions, design features, and type of material. All of these factors are taken into account in testing to obtain creep curves for a specific material and various environmental conditions corresponding to the operating conditions of the structure. The paper considers the problem of calculating the creep deformations of thin-walled cylindrical shells under the combined action of internal pressure and axial force. A model of the theory of flow with hardening under variable loading is considered. A numerical example of calculating the creep deformations of a cylindrical shell for an aluminum alloy is given


Бутина Т.А., Дубровин В.М. Моделирования ползучести тонкостенных оболочек при переменных нагружениях. Математическое моделирование и численные методы, 2022, № 1, с. 97–108.