ON THE JOINT APPLICATION OF THE COLLOCATION BOUNDARY ELEMENT METHOD AND THE FOURIER METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTION IN FINITE CYLINDERS WITH SMOOTH DIRECTRICES

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .

INVESTIGATION OF THE INFLUENCE OF RANDOM PERTURBATIONS ON THE DYNAMICS OF THE SYSTEM IN THE SUSLOV PROBLEM

The paper considers the generalized Suslov problem with variable parameters and the influence of random perturbations on the dynamics of the system under consideration. The physical meaning of the Suslov problem is Chaplygin's sleigh, which moves along the inner side of the circle. In the case of a deterministic system, a brief review of the previously obtained results is made, the presence of chaotic dynamics in the system and such effects as the appearance of a strange attractor and noncompact (escaping) trajectories is shown. Moreover, the latter may indicate a possible acceleration in the system. The appearance of chaotic strange attractors occurs due to a cascade of bifurcations of doubling the period. We also consider the dynamics of a perturbed system which arises due to the addition of «white noise» modeled by the Wiener process to one of the equations. Changes in the dynamics of a perturbed system compared to an unperturbed one are studied: chaotization of periodic regimes, the appearance of noncompact trajectories, and the premature destruction of strange attractors. In this paper, phase portraits, maps for the period, graphs of system solutions, and a chart of dynamical regimes are constructed using the Maple software package and the software package «Computer Dynamics: Chaos» (/http://site4.ics.org.ru//chaos_pack).

APPLICATION OF THE IMPLICIT METHOD OF CHARACTERISTICS ON A REGULAR GRID FOR SOLVING THE GAS-DYNAMIC PROBLEMS

In this work, an implicit method is proposed to numerically solve a system of the onedimensional nonstationary equations of gas dynamics transformed by the method of characteristics. Internal points of the channel for a solid-propellant charge are considered at a preignition period of the solid-propellant rocket engine operation. The use of the implicit method makes it possible to calculate the values of gas-dynamic parameters at nodal points of the regular coordinate grid. Calculations of the gas-dynamic parameters both when integrating over time and along the spatial coordinate are performed with the second order of accuracy. Both subsonic and supersonic flows are studied. It is shown that, when predicting the expected pressure value during the transition from one time layer to another with the second order of accuracy, the twenty-fold efficiency of the implicit method is achieved in comparison with the explicit difference method. The trial calculation is performed.

NUMERICAL SIMULATION OF COMBUSTION OF THE COMPOSITE SOLID PROPELLANT CONTAINING BIDISPERSED BORON POWDER

A problem of combustion of the composite solid propellants containing various powders of metals and non-metals is relevant in terms of studying the effect of various compositions of powders on the linear rate of propellant combustion. One of the lines of research is to determine the effect of the addition of a boron powder on the burning rate of a composite solid propellant. This work presents the results of numerical simulation of combustion of the composite solid propellant containing bidispersed boron powder. Physical and mathematical formulation of the problem is based on the approaches of the mechanics of two-phase reactive media. To determine the linear burning rate, the Hermance model of combustion of composite solid propellants is used, based on the assumption that the burning rate is determined by mass fluxes of the components outgoing from the propellant surface. The solution is performed numerically using the breakdown of an arbitrary discontinuity algorithm. The dependences of the linear burning rate of the composite solid propellant on the dispersion of the boron particles and gas pressure above the propellant surface are obtained. It is shown that the burning rate of the composite solid propellant with bidispersed boron powder changes in contrast to that of the composite solid propellant with monodispersed powder. This fact proves that the powder dispersion should be taken into account when solving the problems of combustion of the composite solid propellants containing reactive particles.

AN ELECTROMAGNETIC CATAPULT FOR LAUNCHING HEAVY UAVs (DRONES) FROM SMALL VESSELS

The paper presents the results of mathematical modeling of the operation of a novel electromagnetic catapult design. The main elements of the latter are a single-section multi-rail accelerator with a metal armature and a pulsed energy source based on the powerful pulsed MHD generator and current-increasing transformer. The possibilities of such a scheme for accelerating bodies weighing 7 tons to speeds of about 150 km/h at a maximum permissible acceleration of 15 g are investigated. The mathematical model describes the coordinated operation of the device, starting with connecting of the pulsed MHD generator in idle mode to the primary winding of the transformer and up to the moment when the drone accelerates to a given takeoff speed. Using the proposed model, the efficiency of the electromechanical energy conversion in the developed catapult scheme is tested. The parameters of the main elements of the device, namely the length of the acceleration section of the catapult and the maximum acceleration of the drone, are determined.

COMBUSTION OF A GAS SUSPENSION OF COAL DUST IN A SWIRLING FLOW

Swirling combustion is currently one of the most important engineering problems in physics of combustion. There is a hypothesis on the increase in the combustion efficiency of reacting gas mixtures in combustion chambers with swirling flows, as well as on the increase in the efficiency of fuel combustion devices. In this paper, it is proposed to simulate a swirling flow by taking into account the angular component of the flow velocity. The aim of the study is to determine the effect of the angular component of the flow velocity on the characteristics of the flow and combustion of an air suspension of coal dust in a pipe. The problem is solved in a twodimensional axisymmetric approximation with allowance for a swirling flow. A physical and mathematical model is based on the approaches of the mechanics of multiphase reacting media. A solution method involves the arbitrary discontinuity decay algorithm. The impact of the flow swirl and the size of coal dust particles on the gas temperature distribution along the pipe is determined.

ROTATIONS AND VIBRATIONS OF FULLERENES IN THE MOLECULAR COMPLEX C20@C80

The aim of this work is to apply classical mechanics to a description of the dynamic state of C20@C80 diamond complex. Endohedral rotations of fullerenes are of great interest due to the ability of the materials created on the basis of onion complexes to accumulate energy at rotational degrees of freedom. For such systems, a concept of temperature is not specified. In this paper, a closed description of the rotation of large molecules arranged in diamond shells is obtained in the framework of the classical approach. This description is used for C20@C80 diamond complex. Two different problems of molecular dynamics, distinguished by a fixing method for an outer shell of the considered bimolecular complex, are solved. In all the cases, the fullerene rotation frequency is calculated. Since a class of possible motions for a single carbon body (molecule) consists of rotations and translational displacements, the paper presents the equations determining each of these groups of motions. Dynamic equations for rotational motions of molecules are obtained employing the moment of momentum theorem for relative motions of the system near the fullerenes’ centers of mass. These equations specify the operation of the complex as a molecular pendulum. The equations of motion of the fullerenes’ centers of mass determine vibrations in the system, i.e. the operation of the complex as a molecular oscillator.

GEOMETRIC MODELING OF A SHAPE OF PARALLELOGRAM PLATES IN A PROBLEM OF FREE VIBRATIONS USING CONFORMAL RADII

The paper considers a method of geometric modeling applied when solving basic twodimensional problems of the theory of elasticity and structural mechanics, in particular the applied problems of engineering. The subject of this study is vibrations of thin elastic parallelogram plates of constant thickness. To determine a basic frequency of vibrations, the interpolation method based on the geometric characteristic of the shape of plates (membrane, cross sections of a rod) is proposed. This characteristic represents a ratio of interior and exterior conformal radii of the plate. As is known from the theory of conformal mappings, conformal radii are those obtained by mapping of a plate onto the interior and exterior of a unit disk. The paper presents basic terms, tables, and formulas related to the considered geometric method with a comparative analysis of the curve diagrams obtained using various interpolation formulas. The original computer program is also developed. The main advantage of the proposed method of determining the basic frequency of plate vibrations is a graphic representation of results that allows one to accurately determine the required solution on the graph among the other solutions corresponding to the considered case of parallelogram plates. Although there are many known approximate approaches, which are used to solve the considered problems, only geometric modeling technique based on the conformal radii ratio gives such an opportunity.

ESTIMATION OF HETEROGENEITY OF THE ATMOSPHERIC AIR VELOCITY FIELD IN ADSORBERS OF FRONT-END PURIFICATION UNITS FOR AIR SEPARATION PLANTS

Assuming unidirectional motion of compressed atmospheric air through a vertical cylindrical adsorbent with a fixed granular layer of the front-end purification unit adsorbent, the mathematical model for estimating the heterogeneity of a hydrodynamic velocity field in the radial and axial directions in a turbulent regime is proposed. The model is based on the boundary layer approximation of the Darcy – Brinkman – Forchheimer phenomenological equation. The steady-state flow at low permeability of the granular layer is identified using the collocation method, and the approximate analytical solution is obtained which justifies the applicability of an ideal displacement mode when describing the carrier medium motion. Numerical integration of a boundary value problem of the model equation using the finite-difference method with Richardson extrapolation confirms the conclusion validity. The structure of an accelerated turbulent flow having constant flow velocity in the input section shows that for small Forchheimer coefficients, the Darcy – Brinkman equation is used to obtain the analytical ratio for calculating the length of the initial hydrodynamic section. The proposed mathematical model for estimating the heterogeneity of the velocity field in adsorbers with a stationary dispersed layer is applicable for a laminar flow regime. Testing of this approach by assessing velocity field uniformity for a mass-produced front-end purification unit of air separation plants has shown its efficiency.