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As part of the paper, a physical and theoretical analysis of the impact processes of various factors of a highaltitude and high-energy nuclear explosion on the asteroid in extra-atmospheric conditions of open space is done. It is shown that, in accordance with the energy and permeability of the plasma of explosion products, X-ray and gamma-neutron radiation, a layered structure with a different energy density depending on angular coordinates is formed on the surface of the asteroid. The temporal patterns of the energy transformation for each layer is clarified and the roles of various photo- and collision processes are determined. The effect of a high-speed plasma flow is erosive in nature, and the plasma pulse is transmitted to the asteroid. The paper presents that in a thin layer of x-ray absorption, the asteroid substance is heated to high temperatures and as a result of its expansion, a recoil impulse is formed, which is not decisive due to the small mass of the expanding high-temperature plasma. Calculations shows that the main impulse received by an asteroid is associated with the entrainment of a heated layer of a substance formed by a neutron flux (7.5 E 10¹⁴ g E cm/s). It is shown that an asteroid with a radius of ~100 m acquires a velocity of . 100 cm/s. The calculations were performed taking into account the explosion energy spent on the destruction of the amorphous structure of the asteroid material (~1 eV/atom = 3.8 E 10¹⁰ erg/g) and ionization in the region of the high-temperature layer. Based on a similar analysis, an approximation is obtained for estimating the average size of fragments in the event of the possible destruction of the asteroid by shock waves generated inside it under the influence of pressure impulses. A physical experiment was conducted in laboratory conditions, simulating the fragmentation of a stone asteroid and confirming the validity of the obtained dependence on the selected values of certain parameters. As a result of numerical studies of the effects of the explosion, carried out at different distances from the surface of the asteroid, it is shown that taking into account the real geometry of the spallation layer gives the optimal height for the formation of the maximum asteroid momentum by a factor of 1.5 greater than similar estimates according to the simplified model. A two-stage concept of the impact of nuclear explosions on an asteroid using radar guidance tools is proposed. The paper analyzes the possible impact of the emerging ionization interference on the radar tracking of the movement of large fragments of the asteroid in the space-time evolution of all elements of the studied dynamic system.

The work is devoted to the development of a computational algorithm of the Cartesian grid method for studying the interaction of a shock wave with moving bodies with a piecewise linear boundary. The interest in such problems is connected with direct numerical simulation of two-phase media flows. The effect of the particle shape can be important in the problem of dust layer dispersion behind a passing shock wave. Experimental data on the coefficient of aerodynamic drag of non-spherical particles are practically absent.

Mathematical model is based on the two-dimensional Euler equations, which are solved in a region with varying boundaries. The defining system of equations is integrated using an explicit scheme and the Cartesian grid method. The computational algorithm at the time integration step includes: determining the step value, calculating the dynamics of the body movement (determining the force and moment acting on the body; determining the linear and angular velocities of the body; calculating the new coordinates of the body), calculating the gas parameters. At each time step, all cells are divided into two classes – external (inside the body or intersected by its boundaries) and internal (completely filled with gas). The solution of the Euler equations is constructed only in the internal ones. The main difficulty is the calculation of the numerical flux through the edges common to the internal and external cells intersected by the moving boundaries of the bodies. To calculate this flux, we use a two-wave approximation for solving the Riemann problem and the Steger-Warming scheme. A detailed description of the numerical algorithm is presented.

The efficiency of the algorithm is demonstrated on the problem of lifting a cylinder with a base in the form of a circle, ellipse and rectangle behind a passing shock wave. A circular cylinder test was considered in many papers devoted to the immersed boundary methods development. A qualitative and quantitative analysis of the trajectory of the cylinder center mass is carried out on the basis of comparison with the results of simulations presented in eight other works. For a cylinder with a base in the form of an ellipse and a rectangle, a satisfactory agreement was obtained on the dynamics of its movement and rotation in comparison with the available few literary sources. Grid convergence of the results is investigated for the rectangle. It is shown that the relative error of mass conservation law fulfillment decreases with a linear rate.

In this paper, the structure of a shock wave in a binary gas mixture is studied on the basis of direct solution of the Boltzmann kinetic equation. The conservative projection method is used to evaluate the collision integral in the kinetic equation. The applied evaluation formulas and numerical methods are described in detail. The model of hard spheres is used as an interaction potential of molecules. Numerical simulation is performed using the developed simulation environment software, which makes it possible to study both steady and non-steady flows of gas mixtures in various flow regimes and for an arbitrary geometry of the problem. Modeling is performed on a cluster architecture. Due to the use of code parallelization technologies, a significant acceleration of computations is achieved. With a fixed accuracy controlled by the simulation parameters, the distributions of macroscopic characteristics of the mixture components through the shock wave front were obtained. Computations were conducted for various ratios of molecular masses and Mach numbers. The total accuracy of at least 1% for the local values of molecular density and temperature and 3% for the shock front width was achieved. The obtained results were compared with existing computation data. The results presented in this paper are of theoretical significance, and can serve as a test computation, since they are obtained using the exact Boltzmann equation.

The study of the development of π-periodic perturbations of a converging spherical shock wave leading to cumulation limitation is performed. The study is based on 3D hydrodynamic calculations with the Carnahan – Starling equation of state for hard sphere fluid. The method of solving the Euler equations on moving (compressing) grids allows one to trace the evolution of the converging shock wave front with high accuracy in a wide range of its radius. The compression rate of the computational grid is adapted to the motion of the shock wave front, while the motion of the boundaries of the computational domain satisfy the condition of its supersonic velocity relative to the medium. This leads to the fact that the solution is determined only by the initial data at the grid compression stage. The second order TVD scheme is used to reconstruct the vector of conservative variables at the boundaries of the computational cells in combination with the Rusanov scheme for calculating the numerical vector of flows. The choice is due to a strong tendency for the manifestation of carbuncle-type numerical instability in the calculations, which is known for other classes of flows. In the three-dimensional case of the observed force, the carbuncle effect was obtained for the first time, which is explained by the specific nature of the flow: the concavity of the shock wave front in the direction of motion, the unlimited (in the symmetric case) growth of the Mach number, and the stationarity of the front on the computational grid. The applied numerical method made it possible to study the detailed flow pattern on the scale of cumulation termination, which is impossible within the framework of the Whitham method of geometric shock wave dynamics, which was previously used to calculate converging shock waves. The study showed that the limitation of cumulation is associated with the transition from the Mach interaction of converging shock wave segments to a regular one due to the progressive increase in the ratio of the azimuthal velocity at the shock wave front to the radial velocity with a decrease in its radius. It was found that this ratio is represented as a product of a limited oscillating function of the radius and a power function of the radius with an exponent depending on the initial packing density in the hard sphere model. It is shown that increasing the packing density parameter in the hard sphere model leads to a significant increase in the pressures achieved in a shock wave with broken symmetry. For the first time in the calculation, it is shown that at the scale of cumulation termination, the flow is accompanied by the formation of high-energy vortices, which involve the substance that has undergone the greatest shock-wave compression. Influencing heat and mass transfer in the region of greatest compression, this circumstance is important for current practical applications of converging shock waves for the purpose of initiating reactions (detonation, phase transitions, controlled thermonuclear fusion).

We study the computational properties of a parametric class of finite-volume schemes with customizable dissipative properties with splitting by physical processes into Lagrangian, Eulerian, and the final stages (the hybrid large-particle method). The method has a second-order approximation in space and time on smooth solutions. The regularization of a numerical solution at the Lagrangian stage is performed by nonlinear correction of artificial viscosity. Regardless of the grid resolution, the artificial viscosity value tends to zero outside the zone of discontinuities and extremes in the solution. At Eulerian and final stages, primitive variables (density, velocity, and total energy) are first reconstructed by an additive combination of upwind and central approximations weighted by a flux limiter. Then numerical divergent fluxes are formed from them. In this case, discrete analogs of conservation laws are performed.

The analysis of dissipative properties of the method using known viscosity and flow limiters, as well as their linear combination, is performed. The resolution of the scheme and the quality of numerical solutions are demonstrated by examples of two-dimensional benchmarks: a gas flow around the step with Mach numbers 3, 10 and 20, the double Mach reflection of a strong shock wave, and the implosion problem. The influence of the scheme viscosity of the method on the numerical reproduction of a gases interface instability is studied. It is found that a decrease of the dissipation level in the implosion problem leads to the symmetric solution destruction and formation of a chaotic instability on the contact surface.

Numerical solutions are compared with the results of other authors obtained using higher-order approximation schemes: CABARET, HLLC (Harten Lax van Leer Contact), CFLFh (CFLF hybrid scheme), JT (centered scheme with limiter by Jiang and Tadmor), PPM (Piecewise Parabolic Method), WENO5 (weighted essentially non-oscillatory scheme), RKGD (Runge –Kutta Discontinuous Galerkin), hybrid weighted nonlinear schemes CCSSR-HW4 and CCSSR-HW6. The advantages of the hybrid large-particle method include extended possibilities for solving hyperbolic and mixed types of problems, a good ratio of dissipative and dispersive properties, a combination of algorithmic simplicity and high resolution in problems with complex shock-wave structure, both instability and vortex formation at interfaces.

For a non-homogeneous model transport equation with source terms, the stability analysis of a linear hybrid scheme (a combination of upwind and central approximations) is performed. Stability conditions are obtained that depend on the hybridity parameter, the source intensity factor (the product of intensity per time step), and the weight coefficient of the linear combination of source power on the lower- and upper-time layer. In a nonlinear case for the non-equilibrium by velocities and temperatures equations of gas suspension motion, the linear stability analysis was confirmed by calculation. It is established that the maximum permissible Courant number of the hybrid large-particle method of the second order of accuracy in space and time with an implicit account of friction and heat exchange between gas and particles does not depend on the intensity factor of interface interactions, the grid spacing and the relaxation times of phases (K-stability). In the traditional case of an explicit method for calculating the source terms, when a dimensionless intensity factor greater than 10, there is a catastrophic (by several orders of magnitude) decrease in the maximum permissible Courant number, in which the calculated time step becomes unacceptably small.

On the basic ratios of Riemann’s problem in the equilibrium heterogeneous medium, we obtained an asymptotically exact self-similar solution of the problem of interaction of a shock wave with a layer of gas-suspension to which converge the numerical solution of two-velocity two-temperature dynamics of gassuspension when reducing the size of dispersed particles.

The dynamics of the shock wave in gas and its interaction with a limited gas suspension layer for different sizes of dispersed particles: 0.1, 2, and 20 ìm were studied. The problem is characterized by two discontinuities decay: reflected and refracted shock waves at the left boundary of the layer, reflected rarefaction wave, and a past shock wave at the right contact edge. The influence of relaxation processes (dimensionless phase relaxation times) to the flow of a gas suspension is discussed. For small particles, the times of equalization of the velocities and temperatures of the phases are small, and the relaxation zones are sub-grid. The numerical solution at characteristic points converges with relative accuracy $O \, (10^{-4})$ to self-similar solutions.

In a number of fundamental and practical problems, it is necessary to describe the dynamics of the motion of complexshaped particles in a high-speed gas flow. An example is the movement of coal particles behind the front of a strong shock wave during an explosion in a coal mine. The paper is devoted to numerical simulation of the dynamics of translational and rotational motion of a square-shaped body, as an example of a particle of a more complex shape than a round one, in a supersonic flow behind a passing shock wave. The formulation of the problem approximately corresponds to the experiments of Professor V. M. Boiko and Professor S. V. Poplavski (ITAM SB RAS).

Mathematical model is based on the two-dimensional Euler equations, which are solved in a region with varying boundaries. The defining system of equations is integrated using an explicit scheme and the Cartesian grid method which was developed and verified earlier. The computational algorithm at the time integration step includes: determining the step value, calculating the dynamics of the body movement (determining the force and moment acting on the body; determining the linear and angular velocities of the body; calculating the new coordinates of the body), calculating the gas parameters. To calculate numerical fluxes through the edges of the cell intersected by the boundaries of the body, we use a two-wave approximation for solving the Riemann problem and the Steger – Warming scheme.

The movement of a square with a side of 6 mm was initiated by the passage of a shock wave with a Mach number of 3,0 propagating in a flat channel 800 mm long and 60 mm wide. The channel was filled with air at low pressure. Different initial orientation of the square relative to the channel axis was considered. It is found that the initial position of the square with its side across the flow is less stable during its movement than the initial position with a diagonal across the flow. In this case, the calculated results qualitatively correspond to experimental observations. For the intermediate initial positions of a square, a typical mode of its motion is described, consisting of oscillations close to harmonic, turning into rotation with a constant average angular velocity. During the movement of the square, there is an average monotonous decrease in the distance between the center of mass and the center of pressure to zero.

The paper is devoted to the study of wave and relaxation effects during the pulsed outflow of a gas mixture with a high content of solid particles from a cylindrical channel during its initial partial filling. The problem is formulated in a two-speed two-temperature formulation and was solved numerically by the hybrid large-particle method of the second order of approximation. The numerical algorithm is implemented in the form of parallel computing using basic Free Pascal language tools. The applicability and accuracy of the method for wave flows of concentrated gas-particles mixtures is confirmed by comparison with test asymptotically accurate solutions. The calculation error on a grid of low detail in the characteristic flow zones of a two-phase medium was 10^-6 . . . 10^-5.

Based on the wave diagram, the analysis of the physical pattern of the outflow of a gas suspension partially filling a cylindrical channel is performed. It is established that, depending on the degree of initial filling of the channel, various outflow modes are formed. The first mode is implemented with a small degree of loading of the high-pressure chamber, at which the left boundary of the gas-particles mixture crosses the outlet section before the arrival of the rarefaction wave reflected from the bottom of the channel. At the same time, the maximum value of the mass flow rate of the mixture is achieved. Other modes are formed in cases of a larger initial filling of the channel, when the rarefaction waves reflected from the bottom of the channel interact with the gas suspension layer and reduce the intensity of its outflow.

The influence of relaxation properties with changing particle size on the dynamics of a limited layer of a gas-dispersed medium is studied. Comparison of the outflow of a limited gas suspension layer with different particle sizes shows that for small particles (the Stokes number is less than 0.001), an anomalous phenomenon of the simultaneous existence of shock wave structures in the supersonic and subsonic flow of gas and suspension is observed. With an increase in the size of dispersed inclusions, the compaction jumps in the region of the two-phase mixture are smoothed out, and for particles (the Stokes number is greater than 0.1), they practically disappear. At the same time, the shock-wave configuration of the supersonic gas flow at the outlet of the channel is preserved, and the positions and boundaries of the energy-carrying volumes of the gas suspension are close when the particle sizes change.

The review is based on some of the authors ’early works of particular scientific, methodological and practical interest and the greatest attention is paid to recent works, where quite detailed numerical studies of not only single, but also double and multiple explosions in a wide range of heights and environmental conditions have been performed . Since the shock wave of a powerful explosion is one of the main damaging factors in the lower atmosphere, the review focuses on both the physical analysis of their propagation and their interaction. Using the three-dimensional algorithms developed by the authors, the effects of interference and diffraction of several shock waves, which are interesting from a physical point of view, in the absence and presence of an underlying surface of various structures are considered. Quantitative characteristics are determined in the region of their maximum values, which is of known practical interest. For explosions in a dense atmosphere, some new analytical solutions based on the small perturbation method have been found that are convenient for approximate calculations. For a number of conditions, the possibility of using the self-similar properties of equations of the first and second kind to solve problems on the development of an explosion has been shown.

Based on numerical analysis, a fundamental change in the structure of the development of the perturbed region with a change in the height of the explosion in the range of 100–120 km is shown. At altitudes of more than 120 km, the geomagnetic field begins to influence the development of the explosion; therefore, even for a single explosion, the picture of the plasma flow after a few seconds becomes substantially three-dimensional. For the calculation of explosions at altitudes of 120–1000 km under the guidance of academician A. Kholodov. A special three-dimensional numerical algorithm based on the MHD approximation was developed. Numerous calculations were performed and for the first time a quite detailed picture of the three-dimensional flow of the explosion plasma was obtained with the formation of an upward jet in 5–10 s directed in the meridional plane approximately along the geomagnetic field. After some modification, this algorithm was used to calculate double explosions in the ionosphere, spaced a certain distance. The interaction between them was carried out both by plasma flows and through a geomagnetic field. Some results are given in this review and are described in detail in the original articles.

The article is devoted to the numerical study of shock-wave flows in inhomogeneous media–gas mixtures. In this work, a two-speed two-temperature model is used, in which the dispersed component of the mixture has its own speed and temperature. To describe the change in the concentration of the dispersed component, the equation of conservation of “average density” is solved. This study took into account interphase thermal interaction and interphase pulse exchange. The mathematical model allows the carrier component of the mixture to be described as a viscous, compressible and heat-conducting medium. The system of equations was solved using the explicit Mac-Cormack second-order finite-difference method. To obtain a monotone numerical solution, a nonlinear correction scheme was applied to the grid function. In the problem of shock-wave flow, the Dirichlet boundary conditions were specified for the velocity components, and the Neumann boundary conditions were specified for the other unknown functions. In numerical calculations, in order to reveal the dependence of the dynamics of the entire mixture on the properties of the solid component, various parameters of the dispersed phase were considered — the volume content as well as the linear size of the dispersed inclusions. The goal of the research was to determine how the properties of solid inclusions affect the parameters of the dynamics of the carrier medium — gas. The motion of an inhomogeneous medium in a shock duct divided into two parts was studied, the gas pressure in one of the channel compartments is more important than in the other. The article simulated the movement of a direct shock wave from a high-pressure chamber to a low–pressure chamber filled with a dusty medium and the subsequent reflection of a shock wave from a solid surface. An analysis of numerical calculations showed that a decrease in the linear particle size of the gas suspension and an increase in the physical density of the material from which the particles are composed leads to the formation of a more intense reflected shock wave with a higher temperature and gas density, as well as a lower speed of movement of the reflected disturbance reflected wave.