All issues

The paper has methodical character; it is devoted to three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Special attention was payed to the matter conservation law and the offensive effect of excessive hexagonal symmetry.

It has been shown that in contrary to finite-difference approach, in spite of terminological equivalence of CA local transition function to the pattern of computing double layer explicit method, CA approach contains the replacement of matrix technique by iterative ones (for instance, sweep method for three diagonal matrixes). This suggests that discretization of boundary conditions for CA-cells needs more rigid conditions.

The correct local transition function (LTF) of the boundary cells, which is valid at least for the boundaries of the rectangular and circular shapes have been firstly proposed and empirically given for the hexagonal grid and the conservative boundary conditions. The idea of LTF separation into «internal», «boundary» and «postfix» have been proposed. By the example of this problem the value of the Courant-Levy constant was re-evaluated as the CA convergence speed ratio to the solution, which is given at a fixed time, and to the rate of the solution change over time.

The work is dedicated to investigation of possibility to increase the efficiency of solving external aerodynamic problems. Methodical questions of using locally-adaptive grids and wall functions for numerical simulation of turbulent flows past flying vehicles are studied. Reynolds-averaged Navier–Stokes equations are integrated. The equations are closed by standard $k–\varepsilon$ turbulence model. Subsonic turbulent flow of perfect compressible viscous gas past airfoil RAE 2822 is considered. Calculations are performed in CFD software FlowVision. The efficiency of using the technology of smoothing diffusion fluxes and the Bradshaw formula for turbulent viscosity is analyzed. These techniques are regarded as means of increasing the accuracy of solving aerodynamic problems on locally-adaptive grids. The obtained results show that using the technology of smoothing diffusion fluxes essentially decreases the discrepancy between computed and experimental values of the drag coefficient. In addition, the distribution of the skin friction coefficient over the curvilinear surface of the airfoil becomes more regular. These results indicate that the given technology is an effective way to increase the accuracy of calculations on locally-adaptive grids. The Bradshaw formula for the dynamic coefficient of turbulent viscosity is traditionally used in the SST $k–\omega$ turbulence model. The possibility to implement it in the standard $k–\varepsilon$ turbulence model is investigated in the present article. The calculations show that this formula provides good agreement of integral aerodynamic characteristics and the distribution of the pressure coefficient over the airfoil surface with experimental data. Besides that, it essentially augments the accuracy of simulation of the flow in the boundary layer and in the wake. On the other hand, using the Bradshaw formula in the simulation of the air flow past airfoil RAE 2822 leads to under-prediction of the skin friction coefficient. For this reason, the conclusion is made that practical use of the Bradshaw formula requires its preliminary validation and calibration on reliable experimental data available for the considered flows. The results of the work as a whole show that using the technologies discussed in numerical solution of external aerodynamic problems on locally-adaptive grids together with wall functions provides the computational accuracy acceptable for quick assessment of the aerodynamic characteristics of a flying vehicle. So, one can deduce that the FlowVision software is an effective tool for preliminary design studies, for conceptual design, and for aerodynamic shape optimization.

When a supersonic air flow interacts with a transverse secondary jet injected into this flow through an orifice on a flat wall, a special flow structure is formed. This flow takes place during fuel injection into combustion chambers of supersonic aircraft engines; therefore, in recent years, various approaches to intensifying gas mixing in this type of flow have been proposed and studied in several countries. The approach proposed in this work implies using spark discharges for pulsed heating of the gas and generating the instabilities in the shear layer at the boundary of the secondary jet. Using simulation in the software package FlowVision 3.13, the characteristics of this flow were obtained in the absence and presence of pulsed-periodic local heat release on the wall on the windward side of the injector opening. A comparison was made of local characteristics at different periodicities of pulsed heating (corresponding to the values of the Strouhal number 0.25 and 0.31). It is shown that pulsed heating can stimulate the formation of perturbations in the shear layer at the jet boundary. For the case of the absence of heating and for two modes of pulsed heating, the values of an integral criterion for mixing efficiency were calculated. It is shown that pulsed heating can lead both to a decrease in the average mixing efficiency and to its increase (up to 9% in the considered heating mode). The calculation method used (unsteady Reynolds-averaged Navier – Stokes equations with a modified $k-\varepsilon$ turbulence model) was validated by considering a typical case of the secondary transverse jet interaction with a supersonic flow, which was studied by several independent research groups and well documented in the literature. The grid convergence was shown for the simulation of this typical case in FlowVision. A quantitative comparison was made of the results obtained from FlowVision calculations with experimental data and calculations in other programs. The results of this study can be useful for specialists dealing with the problems of gas mixing and combustion in a supersonic flow, as well as the development of engines for supersonic aviation.

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.

Based on the modeling method using a mesh system, an algorithm is implemented for solving a unsteady problem with moving bodies The algorithm takes into account the movement and rotation of bodies according to a given law of motion. The algorithm is applied to analysis the flow around an infinite composed of cylinders with an elliptical cross-section, which either move across the flow or rotate with a change in the angle of attack. To simulate the flow of bodies with a sharp edge, characteristic of the profiles of gas turbine machines, an algorithm for constructing a mesh of type C with the inclusion of a certain area behind the profile is implemented. The program for modeling the flow near the profile is implemented within the framework of models of Euler equations, Navier – Stokes equations in the approximation of a thin layer with laminar viscosity and turbulent viscosity in the framework of an algebraic viscosity model. The program has also been adapted to solve the problems of internal gas dynamics of turbomachines. For this purpose, the method of setting the boundary conditions at the entrance and exit from the calculated area from the velocity to the pressure drop, as well as at the lateral boundaries from the free flow to the periodicity, was changed. This made it possible to simulate the flow of gas in the inter-blade channels of compressors and turbines of gas turbine engines. To refine the algorithm, a series of calculations of the aerodynamic parameters of several turbine cascades in various subsonic and supersonic modes and their comparison with the experiment were carried out. Calculations of turbine grating parameters were carried out within the framework of the inviscid and viscous gas model. The calculation and experiment were compared by the distribution of gas parameters near the profile, as well as by the energy losses of the flow in the cascade. Calculations have shown the applicability and correctness of the program to solve this class of problems. To test the program on the problems of external subsonic aerodynamics, calculations of the aerodynamic characteristics of an isolated airfoil in an undisturbed flow were performed. The results obtained allow us to assert the applicability of the hybrid grid method to various classes of problems of applied gas dynamics.

The paper reviews the existing approaches to calculating the destruction of solids. The main attention is paid to algorithms using a unified approach to the calculation of deformation both for nondestructive and for the destroyed states of the material. The thermodynamic derivation of the unified rheological relationships taking into account the elastic, viscous and plastic properties of materials and describing the loss of the deformation resistance ability with the accumulation of microdamages is presented. It is shown that the mathematical model under consideration provides a continuous dependence of the solution on input parameters (parameters of the material medium, initial and boundary conditions, discretization parameters) with softening of the material.

Explicit and implicit non-matrix algorithms for calculating the evolution of deformation and fracture development are presented. Non-explicit schemes are implemented using iterations of the conjugate gradient method, with the calculation of each iteration exactly coinciding with the calculation of the time step for two-layer explicit schemes. So, the solution algorithms are very simple.

The results of solving typical problems of destruction of solid deformable bodies for slow (quasistatic) and fast (dynamic) deformation processes are presented. Based on the experience of calculations, recommendations are given for modeling the processes of destruction and ensuring the reliability of numerical solutions.

One of the destroying natural processes is the initiation of the regional seismic activity. It leads to a large number of human deaths. Much effort has been made to develop precise and robust methods for the estimation of the seismic stability of buildings. One of the most common approaches is the natural frequency method. The obvious drawback of this approach is a low precision due to the model oversimplification. The other method is a detailed simulation of dynamic processes using the finite-element method. Unfortunately, the quality of simulations is not enough due to the difficulty of setting the correct free boundary condition. That is why the development of new numerical methods for seismic stability problems is a high priority nowadays.

The present work is devoted to the study of spatial dynamic processes occurring in geological medium during an earthquake. We describe a method for simulating seismic wave propagation from the hypocenter to the day surface. To describe physical processes, we use a system of partial differential equations for a linearly elastic body of the second order, which is solved numerically by a grid-characteristic method on parallelepiped meshes. The widely used geological hypocenter model, called the “double-couple” model, was incorporated into this numerical algorithm. In this case, any heterogeneities, such as geological layers with curvilinear boundaries, gas and fluid-filled cracks, fault planes, etc., may be explicitly taken into account.

In this paper, seismic waves emitted during the earthquake initiation process are numerically simulated. Two different models are used: the homogeneous half-space and the multilayered geological massif with the day surface. All of their parameters are set based on previously published scientific articles. The adequate coincidence of the simulation results is obtained. And discrepancies may be explained by differences in numerical methods used. The numerical approach described can be extended to more complex physical models of geological media.

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.

The paper presents the physical and mathematical formulation of the problems of internal ballistics of an artillery shot for a charge consisting of a set of powder tubes and their stress-strain state. Combustion and movement of a bundle of powder tubes along the barrel channel is modeled by an equivalent tubular charge of all-round combustion. It is assumed that the equivalent tube moves along the axis of the bore. The speed of movement of an equivalent tubular charge and its current position are determined from Newton’s second law. When calculating the flow parameters, two-dimensional axisymmetric equations of gas dynamics were used, for the solution of which an axisymmetric orthogonalized difference grid is constructed, which adapts to the flow conditions. The control volume method is used to numerically solve the system of gas-dynamic equations. The gas parameters at the boundaries of the control volumes are determined using a self-similar solution to the Godunov’s problem of the decay of an arbitrary discontinuity. The stress-strain state is modeled for a separate burning powder tube located in the field of gas-dynamic parameters. The calculation of the gas-dynamic parameters of the shot is carried out without taking into account the deformed state of the powder elements. The behavior of powder elements during firing is considered under these conditions. The finite element method with the division of the calculation area into triangular elements is used to solve the problem of elasticity. In the process of powder tube burnout, the computational grid on each time layer of the dynamic problem is completely updated due to a change in the boundaries of the powder element due to combustion. The paper shows the time dependences of the parameters of the internal ballistics process and the stress-strain state of powder elements, as well as the distribution of the main parameters of the flow of combustion products at different points in time. It has been established that the tubular powder elements during the shot experience significant deformations, which must be taken into account when solving the basic problem of internal ballistics. The data obtained give an idea of the level of equivalent stresses acting at various points of the powder element. The results obtained indicate the relevance of the conjugate formulation of the problem of gas dynamics and the stress-strain state for charges consisting of tubular powders, since this allows a new approach to the design of tubular charges and opens up the possibility of determining the parameters on which the physics of the combustion process of gunpowder significantly depends, therefore, and the dynamics of the shot process.

The given work is devoted aerodynamic properties of system of the bodies which are flowed round by a supersonic stream. The question on reduction of mutual influence with increase in the size characterising scattering of elements of system is considered. The method of construction of a grid is applied to current modeling from a set of grids. One of grids, regular with rectangular cells, is responsible for an interference between bodies
and serves for the description of an external nonviscous current. Other grids are  connected with surfaces of streamline bodies and allow to describe viscous layers about streamline bodies. These grids are imposed on the first, without combination of any knots. Boundary conditions are realized through interpolation of functions on borders from one grid on another.