System Of Gas Dynamics Equations Research Articles

MATHEMATICAL MODELS, NUMERICAL METHODS AND RESULTS OF STUDY OF TRANSSONIC FLOW AROUND PROFILES OF BLADE SYSTEMS

The aerodynamic characteristics of blade profiles in a transonic flow of an ideal gas are considered, which are obtained on the basis of the numerical solution of the system of gas dynamics equations (laws of conservation) in integral form. The solution is found by stabilizing over time, which avoids complications in the numerical method for a stationary system of variable-type equations with unknown lines of change from elliptic to hyperbolic and vice versa. The peculiarities of the position of the shock wave (jump) depending on the algorithm for calculating the parameter values on the faces of the cells of the difference grid in the finite-volume algorithm are pointed out. The calculation of numerical flows on the faces of the volume is based on the solution of the problem of the decay of an arbitrary discontinuity of parameters on them in a one-dimensional direction along the normal to the face. The choice of the working difference scheme is based on the indicators of accuracy (order of approximation) and complexity (explicit and implicit) of transition to the next time layer. The development of difference schemes of a higher order of approximation allows to obtain more detailed information about gas-dynamic flows on real grids, which was unattainable when using schemes of the first order of accuracy with high scheme (approximation) viscosity and inconsistency of the propagation of disturbances on a heterogeneous background of parameter values (dispersion calculation effects). The results of calculations using S.K. Godunov's scheme of a higher order of approximation, aerodynamic characteristics of the rigid profile and their comparison with experimental data are given. The characteristics of a profile that performs incoherent angular and vertical oscillations according to a given law are studied.

ПОЛУАНАЛИТИЧЕСКИЙ МЕТОД РЕШЕНИЯ УРАВНЕНИЙ ГАЗОВОЙ ДИНАМИКИ В ПЕРЕМЕННЫХ ЭЙЛЕРА

This paper presents a semi-analytical method for solving a system of equations of gas dynamics in Eulerian coordinates. Since only spatial derivatives are replaced by finite differences, the system of gas dynamic equations is reduced to a system of ordinary differential equations on a spatial grid. An approximate analytical solution of this system of differential equations for a small time-interval is used to describe the dynamics of a gas in the entire required time interval. Verification was carried out on one-dimensional test problems on the decay of an arbitrary discontinuity and the propagation of stationary shock waves of various intensities. To compare one-dimensional problems, the solution of test problems is given by the simple-to-implement basic particle-in-cell method. It is shown that the semianalytical method has high accuracy of calculations, and is also the most universal method for calculating applied problems.

Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $\tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(\tau^{\alpha/2})$ in the norm of $C(0,T;L^2(\mathbb{R}^n))$, for an initial function $\mathbf w_0$ in the Sobolev space $H^\alpha(\mathbb{R}^n)$, $\alpha=1,2$, or the Nikolskii space $H_2^{\alpha}(\mathbb{R}^n)$, $0<\alpha<2$, $\alpha\neq 1$, and under appropriate assumptions on the free term of the first-order system. For ${\alpha=1/2}$ a wide class of discontinuous functions $\mathbf w_0$ is covered. Estimates for derivatives of any order with respect to $x$ for solutions and of order $O(\tau^{\alpha/2})$ for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented. Bibliography: 34 titles.

Моделирование границ восходящих закрученных потоков

The work is devoted to approximate methods for solving the system of gas dynamics equations describing the gas motion in ascending swirling flows. The approximate solution of the system of partial differential equations describing the motion of the contact surface - the outer boundary of the ascending swirling flow and gas parameters on it is constructed in the paper. It is assumed that gravitational and Coriolis forces act on the ascending swirling flow, and a stationary gas flow runs over its outer boundary. In the form of the system of partial differential equations, the law of motion of the outer boundary of the ascending swirling flow is written. The approximate solution for the law of motion of the outer boundary and gas parameters on it is constructed in the form of series segments.

Математическое моделирование течений самогравитирующегося газа с помощью нестационарных автомодельных переменных

The paper considers one-dimensional isentropic flows of an ideal gas gravitating by Newton. As a mathematical model, the system of equations of gas dynamics is used, taking into account the effect of gravity according to Newton. In the system of gas dynamics equations, a self-similar feature is introduced into the variable x, and for the resulting system, the Cauchy problem with data on the sound characteristic is solved. The solution of the initial boundary value problem is constructed in the form of a convergent power series. Some of the coefficients of the series are found when solving algebraic equations, the rest of the coefficients of the series are found when integrating ordinary differential equations. Then, in the form of the system of ordinary differential equations, the law of motion of the gas-vacuum boundary is written out. The obtained analytical solutions can be used to develop new approximations of boundary and initial conditions for numerical simulation of a gravitational rarefaction wave for a long period of time.

On mathematical modeling of a hypersonic flow past a thin wing with variable shape

The present work is devoted to a further study of the spatial flowing of a thin wing of variable shape by a hypersonic flow of non-viscous gas. The head shock wave is considered to be attached to the leading edge of the wing. The use of the thin shock layer method for solving the system of gas dynamics equations makes it possible to build a mathematical model of the flow in question. It should also be noted that the analysis of boundary conditions makes it possible to determine the structure of the expansion of the quantities sought in a series and to construct approximate analytical solutions. In this case, in determining the first approximation corrections, two equations are integrated independently of the other equations. The application of the Euler — Ampere transform allows to construct a solution depending on two arbitrary functions and an unknown of the shock front. To determine these functions, previously it was obtained an integro-differential system of equations. This paper proposes one of the variants of the semi-inverse method for constructing the solution of this system, in which the form of one of arbitrary function is given. This approach allows you to additionally set the equation for the leading edge of the wing, and in the case when the head wave is attached along the entire leading edge and the inclination of the wing surface on it. The variant of the semi-inverse method presented in the work for the non-stationary spatial problem of the flow has made it possible to obtain a particular solution, which is a model for various flow regimes around the wing. Formulas are obtained to determine: the shape of the front of the shock wave, the shape of the surface of the streamlined body, the distance between the shock wave and the surface of the body, and the flow parameters on the wing surface.

Calculation of the expansion dynamics of evaporated tungsten under the action of a laser pulse

The paper is devoted to the numerical implementation of a model of the dynamics of the tungsten vapors flow evaporating from the sample surface. To calculate the speed and mass flow rate of the substance evaporating from the sample surface, a system of gas dynamics equations is numerically solved. The boundary conditions for the gas velocity and density on the heated surface have a great influence on the solution of the problem. Boundary conditions for temperature are obtained as a result of solving the two-phase Stefan problem in a cross-section of the sample. The aim of the study is to model the erosion of the sample surface and penetration of heat flow into the material.

PARTICULAR SOLUTIONS OF A LINEARIZED SYSTEM OF GAS DYNAMICS EQUATIONS IN THE ABSENCE OF GRAVITY TAKING INTO ACCOUNT THE ACTIONS OF THE CORIOLIS FORCE

For the system of gas dynamics equations, taking into account the action of the Coriolis force, linearization is carried out in case when the effect of gravity is not taken into account. Particular solutions in the form of running waves are found.

Solution of the system of gas-dynamic equations for the processes of interaction of vibrators with the air

The modern practice of using vibratory machines involving small seeds of low weight faces such an undesirable phenomenon as the effect exerted on the kinematics of vibrational movement of particles of fractions of the seed mixture by the aerodynamic forces and momenta. The periodic movement of air relative to the working planes of a vibratory machine arises due to fluctuations in the packets of these planes, which form flat aerodynamic channels. Consequently, the issues of studying the processes of interaction between the working bodies of vibratory machines and the air environment, aimed to justify their structural improvements, appear relevant. Existing mathematical models, which assess the parameters of air movement relative to the working planes of vibratory machines, produce only a generalized pattern and are flat. This paper proposes a statement, as well as an estimated finite difference scheme, of solving a three-dimensional boundary value problem on calculating the field of velocities and pressures in the region of air, located between two parallel synchronously oscillating planes. The problem employs a system of differential equations to describe the flow of the perfect gas. The finite difference scheme has been solved by a sweep method.Using the sweep method to solve these kinds of problems makes it possible to ensure the convergence and stability of estimation schemes, regardless of the step and other parameters of the grid applied. A variant of the calculation has been given, which demonstrated the feasibility of the proposed method for the assigned boundary conditions and parameters of the vibrational mode of machine operation. It has been established that in the working space enclosed between two oscillating planes there are both vertical (transverse) and horizontal (longitudinal) components of air velocity, which change over time

Analytical Solutions for Nonlinear Systems of Conformable Space-Time Fractional Partial Differential Equations via Generalized Fractional Differential Transform

The present paper introduces approximate analytical solutions for nonlinear systems of conformable space-time fractional partial differential equations (CSTFPDEs) by using a generalized conformable fractional partial differential transform (GCFPDT). The GCFPDT is a modified version of the conformable fractional partial differential transform introduced in our recent work and it can be used as an efficient alternative transform to find analytical solutions for nonlinear systems of CSTFPDEs available in the literature. The convergence and error estimation of the proposed GCFPDT are also considered. Moreover, approximate analytical solutions to nonlinear system of gas dynamic equations, nonlinear system of KdV equations, and nonlinear system of approximate long water wave equations in the sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the effectiveness and efficiency of the proposed GCFPDT.

Impulse source of high energy neutrons emitted by fusion reactions after compression of d--t gas by cumulative detonation waves

We develop the physical model and the system of equations for the impulse neutron source (INS) of high-energy neutrons ($\sim$14 MeV) emitted by fusion reactions during compression of D-T gas by cumulative detonation waves. The system of INS equations includes a system of gas dynamic equations that takes into account the energy transfer by radiation, equations for the radiation flux, the equation of the shock adiabat (the Hugoniot adiabat) for a compressed gas, and the equation for the neutron yield. We perform the INS dynamics simulation for the spherical and cylindrical geometries, and calculate maximum temperatures of D-T plasma, its density and neutron yield in the pulse. The obtained temperature estimates and simulation results show that the thermonuclear fusion temperatures are reached within this approach, and the fusion reactions proceed. Their yield determines the yield of neutrons.

MATHEMATICAL MODELING AND CALCULATION OF GAS DYNAMIC CHARACTERISTICS OF FREE THERMAL AIR VORTEX

The article analyzes experimental and analytical studies of ascending swirling air flows. In experimental works such flows are considered from the point of view of the direction of twist, the thermal regimes of heating the underlying surface, the estimation of integral parameters, the method of influence on them, and various methods of visualization. In analytical papers, by constructing solutions of the system of gas dynamics equations, the emergence of a twist of the corresponding direction is proven when there is a gas flow into a vertical cylinder of nonzero radius. In addition, in the numerical modeling of thermal ascending swirling flows, a feature was observed in the behavior of a moving gas at the initial moments of flow formation when the underlying surface was heated locally. This feature consists in the appearance on the boundary of the heating region of counter propagating gas flows with opposite directions of twist. The paper presents the results of numerical simulation of three-dimensional unsteady flows of a compressible viscous heat-conducting gas in thermal swirled vortices with local heating of the underlying surface, taking into account the action of gravity and Coriolis forces.

The compression of gas followed by expansion

We consider the system of gas dynamics equations with the state equation of the monatomic gas. The system admits a group of transformations with a 14-dimensional Lie algebra. A projective operator is specific to this algebra. We consider the invariant submodel on two-dimensional subalgebra containing the projective operator. For the vorticity-free motions, the submodel is reduced to an overdetermined system of three equations. A particular solution is found for it, physical interpretation is given, and trajectories of gas particles are depicted. The solution gives the gas compression followed by expansion.

A Unified CFD Based Approach to a Variety of Condensation Processes in a Viscous Turbulent Wet Steam Flow

A family of quasi linear mathematical models is presented and calculations made for viscous turbulent wet steam flow with a variety of condensation phenomena. These models can be applied to the analysis of equilibrium condensation, homogeneous (spontaneous) condensation, heterogeneous condensation on extraneous particles, and condensation of charged dispersed phase moving in an electrostatic field. The unified model is represented by coupled systems of gas dynamic equations for viscous turbulent two-phase flow, kinetic and electro-kinetic equations tracing out combined processes of size and charge growth, and electromagnetic field equations described an electric field with an account of self-induced in-part by a moving electrical cluster. The numerical procedure is time marching, monotone, implicit, of second order accuracy by space and time coordinates, and exhibits high resolution shock capturing ability. Viscous flow field calculations made with this procedure reveal significant influence on condensation by the shear boundary layers and wakes. Distributions of cooling rate, droplet radius and parameters of the bulk flow are predicted. Verification of the codes against known experimental data is presented.

Numerical simulation of nonequilibrium flows by using the state-to-state approach in commercial software

Nonequilibrium flows of a two-component oxygen mixture O2/O behind a shock wave are studied with due allowance for the state-to-state vibrational and chemical kinetics. The system of gas-dynamic equations is supplemented with kinetic equations including contributions of VT (TV)-exchange and dissociation processes. A method of the numerical solution of this system with the use of the ANSYS Fluent commercial software package is proposed, which is used in a combination with the authors’ code that takes into account nonequilibrium kinetics. The computed results are compared with parameters obtained by solving the problem in the shock-fitting formulation. The vibrational temperature is compared with experimental data. The numerical tool proposed in the present paper is applied to study the flow around a cylinder.

Technology of Gasification of Liquefied Natural Gas

A flow diagram of gasification of a cryogenic liquid, which is based on the utilization of the liquid′s internal energy to obtain a vapor phase, has been presented. The limiting steam fractions of the two-phase flow in a gasifier have been evaluated as applied to the problems of gasification of methane. Consideration has been given to the conditions of phase separation in the field of mass forces. A numerical scheme of solution of a system of gasdynamic equations for the two-phase flow in a cylindrical coordinate system in a three-dimensional formulation has been implemented. The results of numerical modeling of the conditions of precipitation of particles with a diameter of 2 to 10 μm from a swirling dispersed flow have been presented; the role of the particle size in the dynamics of the process of phase separation has been established.

Invariant submodels of rank 3 and rank 2 monatomic gas with the projective operator

The system of gas dynamics equations with the state equation of the monatomic gas admits a group of transformations with a 14-dimensional Lie algebra. A projective operator is specific to this algebra. We consider all one-dimensional subalgebras containing the projective operator. Invariants are calculated and invariant submodel of rank 3 is constructed for each of subalgebras. All submodels are stationary type. They are reduced to the canonical form. Area hyperbolicity of obtained system were specified. Integral entropy is obtained along the flow lines. An ordinary differential equation to the invariant functions is obtained along the flow lines (analogue of a Bernoulli integral for stationary motions). We consider all two-dimensional subalgebras containing projective operator. Invariant submodel of rank 2 stationary type is constructed for each of subalgebras. Submodels are reduced to the canonical form.

КОНЕЧНО-РАЗНОСТНАЯ СХЕМА ПОЛУРАСЩЕПЛЕНИЯ МАТРИЧНЫХ КОЭФФИЦИЕНТОВ И ЕЕ ПРИМЕНЕНИЕ ДЛЯ ОПИСАНИЯ РАСПРОСТРАНЕНИЯ ЦУНАМИ

The well-known finite-difference scheme of Moretti of splitting the matrix coefficients of the system of gas dynamics equations involves writing equations in a special form – pressure and internal energy are excluded from the equation using the equations of state for an ideal gas. In this paper the author proposes a modification of Moretti scheme as a finite-difference scheme of semi-splitting of matrix coefficients which do not intend to constitute a system of equations in a special form. The semi-splitting scheme allows solving equations of hyperbolic equations of state of any type, for example, even those in tabular form. For one-dimensional equations of propagation of circular wave on the water surface, that are the equations of hyperbolic type, the results of the calculations of the problem of the propagation of a surface wave in the ocean and the output of wave on the shore of the ocean area are given according to one-dimensional computational code of the shallow water theory. Verification of semi-splitting finite-difference scheme is performed by comparing the calculation results for the problem of the propagation of a single surface wave in the ocean and the problem of propagation of a wave train on the ocean surface with the results of calculations of the same problems cited in the work by C. Mader. To calculate the wave setup on the shore the approach is used in the computational code of the shallow water theory, which is described in the work by An. G. Marchuk, A.A. Anisimov. By comparing the calculation results with analytical solutions, the suitability of the computational code proposed in this work for the problem solution of the sea wave setup on land is demonstrated. Further development work is seen in the development of a two-dimensional program for calculating the surface wave propagation.

One method for solving systems of nonlinear partial differential equations

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

Thermal Characteristics of Air in the Problem of Hypersonic Motion of Bodies in the Earth’s Atmosphere

Introduction. The air fl ow around bodies moving in the atmosphere with hypersonic velocities is accompanied by a sharp increase in the air temperature due to the high energy of the fl ow and its dissipation in strong shock waves (SW) characteristic of such fl ows [1]. As the velocity of motion increases, there is an increase in the air temperature behind the shock wave, which leads to the initiation of a number of high-temperature processes such as vibrational excitation of molecules and their dissociation and the formation of molecular and atomic ions. These physicochemical processes have their own relaxation times. When these times are much longer than the characteristic time of the change in the fl ow parameter (in the stationary fl ow the above situation is equivalent to a relaxation length larger than the linear size), the thermodynamical properties of air can be described in the ideal gas approximation with a constant adiabatic index. In the opposite case of a short relaxation time, the thermal parameters of the medium can be assumed to be in equilibrium. To describe the intermediate region where the times are comparable, it is necessary to use physical and chemical kinetics equations [2]. It is known that the above high-temperature processes can not only infl uence locally the fl ow parameters, but also determine its character in general. Nevertheless, considering the hypersonic fl ow without account for the nonequilibrium processes is justifi ed [3], since it permits forming an approximate hydrodynamic fl ow pattern. The present paper gives the results of calculations of the equilibrium properties of a high-temperature air and describes a model of the processes of physicochemical kinetics, which, along with the gas dynamics equations, forms the basis for simulating the hypersonic motion of bodies in the Earth’s atmosphere. Equilibrium Characteristics of Air. In the ideal gas approximation, the system of gas-dynamic equations is closed by the equation of state