AP08856926 – Determination of the aerodynamic characteristics of aircraft using a non-stationary nonlinear one-dimensional system of moment equations

The main objective of the project: Study existence-uniqueness of solution initial and boundary value problem for nonlinear 1dimensional nonstationary system of moment equations under macroscopicboundary conditions arising from approximation of microscopic Maxwell boundary condition for particledistribution function,solving initialandboundary value problem for nonlinear 1dimensional nonstationary system of moment equations under macroscopicconditions on a moving boundary bynumerical method

The relevance of the project: Boltzmann’s moment system equations are intermediate between Boltzmann’s (the kinetic theory) and hydrodynamic levels of description of condition of the rarefied gas, and form the earlier non-studied class of nonlinear equations in partial derivatives. Existence of such class of equations was noticed by Grad in 1949. He received moment system through decomposition of particles distribution function by Hermitte polynomials near the local Maxwell’s distributions. But Grad’s moment system hasn't been used in practice and not studied because of the complexity of the differential part. Questions of approximation of homogeneous boundary condition for particles distribution function in case of full nonlinear Boltzmann equation were studied in works, where was proved correctness of initial-boundary value problems for non-stationary nonlinear three-dimensional system of Boltzmann’s moment equations in an arbitrary approximation. In, it is assumed that the gas moves in a limited region with a fixed boundary, which corresponds to the solution of the Boltzmann equation with a parameter depending on a constant boundary temperature. The problem of approximating the Maxwell microscopic boundary condition on a fixed boundary in the case of a non-stationary one-dimensional nonlinear Boltzmann equation was solved in. The moment equations, taking into account the speed of movement and the surface temperature of the aircraft, are a nonlinear hyperbolic system of partial differential equations, and the differential part depends on such unknown parameters as the speed and surface temperature of the aircraft. In addition, the moments of the collision integral are quadratic forms containing the products of the moments of the particle distribution function. According to the microscopic boundary condition of Maxwell, part of the molecules is reflected from the boundary specularly, and part is diffuse with the Maxwell distribution, and the boundary is mobile. The issues of approximating the Maxwell microscopic boundary condition on a moving boundary in the case of a non-stationary one-dimensional nonlinear Boltzmann equation, which depends on the speed of the aircraft, have not yet been solved. The correctness of initial and boundary value problems for a non-stationary nonlinear one-dimensional system of moment equations, which depends on the speed of motion and the surface temperature of aircraft, under macroscopic boundary conditions on a moving boundary is studied for the first time.

Scientific adviser:  Doctor of Physics and Mathematics, Professor, Auzhan Sakabekov

Results obtained: Within the research project, an initial-boundary value problem for the one-dimensional non-stationary nonlinear Boltzmann equation was formulated, taking into account the velocity of the aircraft and Maxwell-type temperature-dependent boundary conditions. Nonlinear systems of moment equations incorporating velocity and surface temperature parameters, as well as their derivatives, were derived. The microscopic Maxwell boundary conditions for a moving boundary were approximated depending on the order of approximation. The existence and uniqueness of the solution for the third-order approximation were proven. Numerical methods, algorithms, and software were developed to solve both direct and inverse problems, including the determination of aerodynamic characteristics of the aircraft. Flight parameters and atmospheric properties such as density and temperature were calculated. The obtained results contribute to the advancement of rarefied gas dynamics and can be applied in aerospace engineering and remote sensing applications.

List of publications with links to them

1. Imansakipova B., Aitkazinova Sh., Sakabekov A., Shakiyeva G., Imansakipova M., Taukebayev O. Improving the accuracy of predicting the hazard of the earth’s surface failure formation during underground mining of mineral deposits // Mining of Mineral Deposits. – 2021. – Vol. 15, No. 4. – P. 15–24. – URL: https://www.scopus.com/sourceid/21100914192; http://mining.in.ua/2021vol15_4_3.html
2. Сакабеков А., Аужани Е. Применение уравнения Больцмана для определения аэродинамических характеристик летательных аппаратов // Вестник КазНИТУ. – 2021. – Т. 143, № 1. – С. 57–64. – URL: https://vestnik.satbayev.university/index.php/journal/article/view/511
3. Sakabekov A., Auzhani Y., Madalieva S. Application of moment equations for calculating the aerodynamic characteristics of aircraft in the transition regime // Proceedings of the 3rd International Applied Mathematics, Modeling and Simulation Conference (AMMS 2021). – Paris, France, 2021.
4. Sakabekov A., Auzhani Y., Dauletov A. Investigation of the aerodynamic characteristics of aircraft in a rarefied gas flow by the moment method // Pre-RGD 32 Online Workshop on Recent Hot Topics in Rarefied Gas Dynamics. – Seoul, South Korea, 2021. – URL: http://www.rgd32.org/pdfviewer/web/viewer.asp?session=S-10&file=/DATA/workshop/S-10.pdf
5. Sakabekov A., Auzhani Y., Yergazina R. About macroscopic boundary conditions for three-dimensional nonlinear nonstationary Boltzmann's moment system of equations // Proceedings of the 10th International Conference on Pure and Applied Mathematics (ICPAM 2021). – Athens, Greece, 2021.
6. Сакабеков А., Даулетов А. Смешанная задача для нестационарной нелинейной одномерной системы моментных уравнений в третьем приближении при макроскопических граничных условиях Максвелла-Аужана // Труды международной научной конференции «Современные проблемы физико-математических наук и междисциплинарные исследования». – Атырау, 2021. – 5 с. – URL: https://pps.kaznu.kz/ru/Main/FileShow2/190426/83/3/1129/2021//
7. Mekebai N., Yergazina R., Sakabekov A. Boltzmann’s six-moment one-dimensional nonlinear system of equations with the Maxwell-Auzhan boundary conditions // Материалы международной научно-практической конференции «Modern problems of natural sciences and interdisciplinary research». – Атырау, 2021. – P. 547–550. – URL: https://pps.kaznu.kz/ru/Main/FileShow2/186897/72/3/713/2021//
8. Sakabekov A., Madalieva S., Yergazina R. Investigation of aerodynamic characteristics of aircrafts in a rarefied gas flow using the moment method // International Journal of Mathematics and Mathematical Sciences. – 2022. – Vol. 2022. – Article ID 6943602. – URL: https://www.scopus.com/sourceid/130103#tabs=1; https://www.hindawi.com/journals/ijmms/2022/6943602/; https://doi.org/10.1155/2022/6943602
9. Sakabekov A., Auzhani Y. The solvability of mixed value problem for first and second approximations of one-dimensional nonlinear system of moment equations with macroscopic boundary conditions // Journal of Nonlinear Mathematical Physics. – 2022. – Vol. 29. – P. 124–148. – URL: https://www.scopus.com/sourceid/28549#tabs=1; https://link.springer.com/article/10.1007/s44198-021-00024-7
10. Мадалиева С.Н., Ергазина Р.А., Акимжанова Ш.А. Применение моментных уравнений для расчета аэродинамических характеристик летательных аппаратов в переходном режиме // Сатпаевские чтения. – Алматы, 2022. – URL: https://official.satbayev.university/ru/materialy-satpaevskikh-chteniy
11. Sakabekov A., Yergazina R., Auzhani Y. Application of the moment system of equations in the second approximation to determine the speed and surface temperature of the aircraft // Proceedings of the International Symposium “Rarefied Gas Dynamics-32”. – Seoul, 2022. – URL: http://rgd32.org/program.asp
12. Sakabekov A., Madalieva S., Yergazina R. Determining the aerodynamic characteristics of an aircraft in a rarefied gas flow // Dynamical Systems, Modeling, and Mathematical Sciences International Conference. – Dubai, UAE, 2022.