Представлен метод динамической локальной адаптации градуированных декартовых деревьев для численного решения задач газовой динамики. Локальный вейвлетный анализ газодинамического поля на базе неравномерных B-сплайнов применяется независимо к каждой ячейке расчетной сетки и позволяет выделить негладкие или существенно нелинейные участки решения (или наоборот, достаточно гладкие и линейные) и модифицировать сетку для расчета следущего шага по времени так, чтобы у разномасштабных особенностей течения было адекватное сеточное разрешение. В комбинации с другими методами вычислительной газовой динамики, такими как метод свободный границы, представленный метод позволяет эффективно решать нестационарные задачи с обтеканием движущихся тел. На ряде таких задач продемонстрирована работа предложенного варианта вейвлетной адаптации. Библ. 37. Фиг. 5.

A method for dynamic local adaptation of graded Cartesian trees for the numerical solution of fluid dynamics problems is presented. Local wavelet analysis of a gas-dynamic field based on nonuniform B-splines is applied independently to each cell of the computational grid and makes it possible to identify nonsmooth or significantly nonlinear sections of the solution (or, vice versa, sufficiently smooth and linear ones) and modify the grid to calculate the next time step so that nonuniformly scaled flow features had adequate grid resolution. In combination with other computational fluid dynamics methods, such as the free boundary method, the presented technique allows one to effectively solve nonstationary problems involving flow around moving bodies. The operation of the proposed version of wavelet adaptation is demonstrated using a number of such problems.

The unsteady flow structure evolution during an interaction of supersonic under expanded jet, a blunt body and periodic energy input has been investigated using the free boundary method on multilevel Cartesian grids with local adaptation based on the wavelet analysis.The grid is restructured according to the various occurring discontinuities.Periodic energy input first leads to an increase in body drag (due to energy source shockwave reaching the body) then to decrease to a level lower than if no energy input was present.A large amount of discontinuities and their interaction with each other can be observed.Due to the flow around the body being non-uniform (presence of under expanded jet and low-pressure chamber), shock waves occurring from energy input become curved.Shock waves from previous energy inputs interact with those from the next ones and contact discontinuities, which results in structures similar to the Richtmyer-Meshkov instability.The flow dynamics are illustrated with a series of images and animations which show the distribution of density and pressure, stream lines and mesh structure.

The problem of a supersonic flow around a system of bodies freely moving in a gas flow is considered. The mathematical model consists of Euler’s equations for a region filled with gas, supplemented by Newton’s equations for describing the motion of rigid bodies under the influence of pressure. The computational algorithm uses locally adaptive Cartesian grids, in which the adaptation is based on wavelet analysis. The interaction of gas and solids is modeled using the free boundary method. The capabilities of the software-implemented code are demonstrated on the problem of the rise of a dust grain under the action of a shock wave and on the simulation of the motion of a system of bodies in a two-dimensional supersonic flow of an inviscid gas. Quantitative and qualitative results are obtained on the velocity of a dust grain and on the evolution of the initial configurations of bodies, which refine the known results.

The features of a flow generated during the injection of a supersonic small body (pellet) from the channel of a spherically blunted cylinder into a supersonic flow are studied. The cylinder’s diameter is 35 times larger than the diameter of the injected body. The numerical simulation is carried out using multilevel Cartesian grids with the local adaptation based on the wavelet analysis. The body’s motion is simulated by the free boundary method. The dynamics of the moving body’s interaction with the bow shock from the cylinder, the formation of the reverse flow region between the bodies, its evolution and disappearance, and the subsequent establishment of a stationary flow are studied. The decrease of the main body’s drag to 20% of the initial value is demonstrated.

The visual and physical features of the flow forming around spherically blunted cylinder during pellet (millimeter-sized solid body) injection towards the flow with supersonic speed are considered. The structure of this time dependent flow is very complicated and new ideas are used in the study. Numerical simulation of moving bodies is made by the free boundary method (version of immersed boundary method) on multilevel Cartesian grids with local adaptation based on the wavelet analysis. Dynamics of a moving body interaction with the bow shock, formation of the reverse flow region between bodies, its deformation and disappearance, and subsequent establishment of a stationary flow are studied. Reduction of main body drag to the level of 20% of the original is obtained. In this process, several specific stages can be found. When the front part of the pellet is in the subsonic flow behind the front bow shock wave it has little effect on the outside flow. Then pellet interacts with the bow shock wave and deforms it. Recirculation zone forms between bodies. It grows to a certain size, after which, because of the pellet wake intensity lessening due to the pellet's increasing distance from the main body, it begins to decrease and is eventually blown away from the front part of the main body. Stationary flow close to the initial one (i.e., before pellet injection) is established around the main body. Flow dynamics are illustrated by a series of images and animations which show the distribution of density and pressure, stream lines and mesh structure.

The prominent researcher Boris Nikolaevich Chetverushkin, an authority in applied mathematics, member of the Russian Academy of Sciences (RAS) and its Presidium, and director of the Keldysh Institute of Applied Mathematics of the RAS, observed his 70th birthday on 26 January 2014. Chetverushkin is the author of fundamental and internationally acknowledged results in applied mathematics, parallel computing, and mathematical modelling. He is chairman of the Russian National Committee on Industrial and Applied Mathematics (which represents the Russian Federation in SIAM) and a member of the European Community on Computational Methods in the Applied Sciences. He has been honoured as an invited speaker at many world and European congresses and conferences on computational methods, computational fluid dynamics, computational mechanics, computational aeroacoustics, and other areas. He was born in Moscow, in a family of physicians. In 1960 he enrolled in the Moscow Institute of Physics and Technology (MIPT), and graduated in 1966. V. Ya. Goldin, one of the pioneers of computational mathematics in our country and an active participant of the Atomic Project, supervised his diploma work (1963–1966) and then guided his studies in the framework of the postgraduate programme of MIPT. Their research at that time was mainly focused on topical problems in radiative gas dynamics, including some aspects important for the defense industry. In 1968 Chetverushkin was hired at the Institute of Applied Mathematics of the Academy of Sciences of the USSR and in 1971 defended his Ph.D. thesis, “Methods for the numerical solution and investigation of some problems in radiative gas dynamics”. This was the time when his research style took shape, characterized by a deep and comprehensive understanding of the processes and phenomena under investigation.