Equations Of Gas Dynamics Research Articles

Construction of Exact Solutions of the System of One-Dimensional Gas Dynamics Equations without Gradient Catastrophe

The system of equations that describes one-dimensional polytropic gas flows is considered. The invariants up to the second order of characteristics of the considered system of equations are classified. The method of reducing the Cauchy problems to systems of ordinary differential equations is proposed. Examples of the solutions without gradient catastrophe are constructed using invariants of characteristics supplementary to the Riemann invariants.

AN ANALYTICAL STUDY OF SPACE-TIME FRACTIONAL ORDER GAS DYNAMIC EQUATIONS

In this article, the Sumudu transform with iterative method is implemented to obtain approximate analytical solutions in series form to non-linear homogeneous and non-homogeneous space-time fractional gas dynamic equations. The fractional derivatives presented here are in the Caputo sense. Furthermore, the findings of this study are graphically represented and the solution graphs demonstrate a strong connection between the approximate and exact solutions.

Моделирование взаимодействия лазерного излучения с криогенной водородной пленкой на octree-сетках блочного типа

The paper presents a technique for simulating the laser-matter interaction. The technique is based on a single-temperature plasma model and includes schemes for solving three-dimensional gas dynamics equations, a nonlinear heat equation, and a method for calculating the thermal contribution of laser radiation by tracing rays through a computational grid. The schemes are implemented using the technology of local adaptation of computational grids of the octree type. Results of three-dimensional simulation of the effect of a high-power nanosecond laser pulse on a layer of cryogenic hydrogen are presented.

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 волн давления при дефлаграционном горении облаков топливно- воздушных смесей

The paper examines the assumptions underlying a whole group of methods for estimating the parameters of shock waves (pressure waves) during the combustion of spherical fuel-air clouds - the so-called parametric formulas. The formulas allow to determine the parameters of pressure waves from the given energy release and combustion rate. To do this, a complete system of gas-dynamic equations is solved using high-precision numerical simulation. In this case, the combustion front is explicitly distinguished in the form of an infinitely thin discontinuity, on which the laws of conservation of mass, momentum, and energy are satisfied. The results of such a numerical calculation are compared with calculations using the corresponding parametric formulas. Clouds of a stoichiometric mixture hydrogen-air are considered. It is shown that the parametric formulas describe the expansion of clouds of hot products with a good degree of accuracy. Here, there is complete agreement with the direct numerical simulation. It is also shown that the parameters of pressure waves are satisfactorily described both in the calculation by parametric formulas and in direct numerical simulation. In this case, within the cloud, there is a greater increase in pressure than the parametric formulas predict. This is explained by the fact that the assumptions used in the formulas regarding the behavior of the flame front as an impenetrable piston do not fully reflect the physical processes: the combustion front is a semipermeable piston, and some of the mass of the initial mixture still penetrates through it. The third factor considered, which determines the achieved pressures, is the rarefaction wave that occurs in the air after the cloud burns out. This wave propagates in combination with the leading pressure front formed during combustion initiation. The extent of this complex remains practically constant, and its attenuation during spherical propagation is adequately described by the parametric formulas.

Group classification of one-dimensional equations of relativistic gas dynamics

Analysis of the one-dimensional relativistic gas dynamics equations is presented in the paper. The equations contain the internal energy as an arbitrary element. Depending on the form of this function the equations have different symmetries. Complete group classification of the one-dimensional relativistic gas dynamics equations with respect to an admitted Lie group is obtained in the present paper. The classification divides the equations into several classes. Analysis of invariant solutions is also provided.

THE FUNCTIONALITY ESTIMATOR METHOD OF THE PGS GAS PIPELINES COMPENSATORS

The offered operative method of an estimation of a functioning state of jacks of gas highways of pneumatic hydraulic systems (PHS), with the help, so-called, a power intensity indicator. The algorithm in the form of a number of the equations of gas dynamics for definition of current parametres of a gas stream is made. On concrete examples of designs of jacks, the sequence and results of the calculations are shown, allowing to reveal their most intense working sites, and necessity of their corresponding completion. Besides, for the purpose of economy and convenience of carrying out of technological operations of purges of knots of automatics or highways important parities for replacement of expenses, for example, helium on gases more simple and accessible to work (air, nitrogen) are received. The received recommendations and expressions are successfully confirmed in practice during working out and manufacturing of subsystems PHS of some LV «Zenith», «Cyclone», etc.

ENO-ET: a reconstruction scheme based on extended ENO stencil and truncated highest-order term

High-order numerical methods for hyperbolic balance laws must circumvent Godunov’s theorem, for which non-linear spatial reconstruction is a successful procedure; this allows the design of non-linear semi-discrete and fully discrete high-order finite volume methods. The Essentially-Non-Oscillatory (ENO) method pioneered by Harten and collaborators in 1987 was a fundamental breakthrough in providing the bases for constructing schemes of both high-order of accuracy in smooth regions and essentially non-oscillatory at discontinuities. Several variations of ENO have since appeared in the literature, notably the Weighted Essentially-Non-Oscillatory (WENO) method of Jiang and Shu. In the present article, after revisiting the main elements of reconstruction methods, including their analysis, we propose a new, conservative ENO-type reconstruction procedure, called ENO-ET, that preserves the main advantages of the classical ENO and successfully overcomes its well-known weaknesses. The newly proposed reconstruction scheme is systematically assessed and compared with the best existing reconstruction schemes. We do so in two ways: (i) the methods are purely regarded as local, non-linear interpolation objects and (ii) the reconstruction methods are implemented in the context of high-order fully-discrete ADER numerical schemes for solving hyperbolic balance laws. The resulting schemes are of arbitrary order of accuracy in both space and time. In this paper the methods are implemented up to seventh order of accuracy and tested on carefully chosen numerical examples. Tests solved include the linear advection equation, the Euler equations of gas dynamics and the blood flow equations. Particular features of interest here are: (a) attaining theoretically expected convergence rates for smooth solutions arising from demanding initial conditions, (b) compute essentially non-oscillatory profiles for discontinuous solutions and (c) attain successfully steady-steady state solutions by time marching. As compared to existing methods, the newly proposed ENO-ET scheme performs very well for all these problems.

Symmetries and Casimirs of radial compressible fluid flow and gas dynamics inn > 1 dimensions

Symmetries and Casimirs are studied for the Hamiltonian equations of radial compressible fluid flow inn>1dimensions. An explicit determination of all Lie point symmetries is carried out, from which a complete classification of all maximal Lie symmetry algebras is obtained. The classification includes all Lie point symmetries that exist only for special equations of state. For a general equation of state, the hierarchy of advected conserved integrals found in the recent work is proved to consist of Hamiltonian Casimirs. A second hierarchy that holds only for an entropic equation of state is explicitly shown to comprise non-Casimirs, which yield a corresponding hierarchy of generalized symmetries through the Hamiltonian structure of the equations of radial fluid flow. The first-order symmetries are shown to generate a non-abelian Lie algebra. Two new kinematic conserved integrals found in the recent work are likewise shown to yield additional first-order generalized symmetries holding for a barotropic equation of state and an entropic equation of state. These symmetries produce an explicit transformation group acting on solutions of the fluid equations. Since these equations are well known to be equivalent to the equations of gas dynamics, all of the results obtained forn-dimensional radial fluid flow carry over to radial gas dynamics.

Two-dimensional axisymmetric radiation hydrodynamics model of moderate-intensity nanosecond laser-produced plasmas

A two-dimensional axisymmetric radiation hydrodynamics model has been proposed to simulate nanosecond laser ablation of a solid target in ambient argon, air and helium at different pressures. The heat conduction equation used to simulate the conduction of laser deposition energy in the target and gas dynamic equations to describe the interaction between laser and vapor plasma and the evolution of plasma are coupled through the Knudsen layer relations at the target-vapor interface. A collisional-radiative model including 12 atomic processes is used to calculate the population of atomic energy levels and fractional ion abundance. The internal energy and pressure of the plasma are expressed by the equations of state based on a real gas approximation, which divides the internal energy into the ionization energy, thermal energy, and excitation energy of atoms and ions. The distributions of the temperature, pressure, density and velocity of the target and plasma are calculated by using this model, and the results are analyzed. Experimental results of multiple diagnostic tools including fast photography, shadowgraphy images, spatio-temporally resolved optical emission spectroscopy and laser interferometry, are used to benchmark the simulation results, and satisfactory consistencies are obtained. The model provides a numerical tool to interpret experimental data of a moderate-intensity nanosecond laser ablated solid target when the temperature of the target surface does not reach the critical value.

Dynamics of Interaction between Two Soliton Clouds

On the basis of relationship between the kinetic equation for two soliton clouds in the theory of the Korteweg-de Vries equation and equations of the Chaplygin gas dynamics it is shown that the existence of waves propagating without a change in their form is a fundamental property of the nonlinear dynamics of soliton gases. The solutions of several typical problems in the soliton gas dynamics are considered and characteristic features of such dynamics, which make it possible to estimate the effects of interaction of soliton gases, are indicated.

Mathematical and computer simulation of particle redistribution and inertial swarming in dispersed systems

<span>This work is devoted to modeling the motion of a dispersed phase in a gas flow in two cases: i) the dynamic characteristics of the gas flow (velocity components, stream functions and vorticity, ii) are found by numerically solving the Navier-Stokes equations with the subsequent construction of the droplet trajectory; study of the coalescence of droplets in a given model gas flow with analysis of the swarming of dispersed particles. In the first approach, the size and location of vortex and stagnant zones in a channel with a streamlined plate and with a turning flow are determined by numerically solving the equations of gas dynamics in Helmholtz variables. To study the coalescence of drops that occurs when the trajectories of two or more drops come into contact, a second approach has been developed-a computer model of inertial swarming in dispersed systems using the Delphi10 programming environment based on stochastic lattice algorithms and computable lattice enlargement. This approach is needed to pre-define the media flow rate profile. Also in this article, were analyzed the grouping and redistribution of particles for the manifestation of the "swarm of drops" effect.</span>

On L2-Dissipativity of a Linearized Scheme on Staggered Meshes with a Regularization for 1D Barotropic Gas Dynamics Equations

On L2-Dissipativity of a Linearized Scheme on Staggered Meshes with a Regularization for 1D Barotropic Gas Dynamics Equations

Mathematical and information models of decision support systems for explosion protection

This paper is dedicated to the issue of mathematical and information modeling of the combustion-to-explosion transition that makes it possible to create an adequate mathematical and information support for decision support systems (DSS) for automatedcontrol of explosive objects.A simple mathematical model for the transition of combustion to explosion is constructed. This model is based on solving mathematical problems of the hydrodynamic stability of flames and detonation waves. These problems are reduced to solving eigenvalue problems for linearized differential equations of gas dynamics. Mathematical model is universal enough. It provides opportunities for making simple analytical estimates for the explosive induction distance and the time of the shock wave formation. The possibilities of the transition of slow combustion to both a deflagration explosion and a detonation wave are considered. Theoretical estimates of the explosive induction distance and the time of the combustion-to-explosion transition are obtained. These estimates are expressed by algebraic a formula, the use of which save computer resources and does not require significant computer time. The application of fuzzy logic makes it possible to use the proposed mathematical model of the combustion-to-explosion transition for real potentially explosive objects in industry and transport.Mathematical models of potentially explosive objects are based on combination of the fuzzy logic and classical mathematical methods. These models give possibilities for creating corresponding information models. Thus mathematical and information support of DSS for automated control systems of explosive objects is developed. The main advantage of these DSS is that it makes it possible for decision makers to do without experts. In particular, developed mathematical and information models create the base forsoftware of DSS for explosion safety of grain elevators. Appropriate software is developed and some calculations are performed. These calculations are useful not only from the point of view of testing the proposed method of mathematical modeling of a grain elevator as a potentially explosive object or testing the software itself, but also from the point of view of the grain elevator designing.

Improvements of the fifth-order WENO-JS-type scheme with normalized smoothing factor for gas dynamic Euler equations

In this paper, we developed a novel high-resolution fifth-order WENO-JS-type scheme that can achieve the optimal accuracy order at or near the arbitrary order critical points. To enhance the accuracy of the weight function of WENO-JS, we normalized the local smoothness factor to obtain a series of normalized variables. By introducing a mapping function that can increase the accuracy of the normalization variables, we developed a new normalized WENO-JS-type scheme called WENO-NSI. The WENO-NSI scheme can improve the shock capture ability by selecting the apposite parameter λ in the mapping function. Numerical experiments of one- and two-dimensional gas dynamic Euler equations for the present WENO-NSI scheme are compared with those of the WENO-JS/Z/M/D/IM and TENO-Z schemes. The results show that the present scheme can achieve theoretical accuracy at or near the arbitrary order critical points, and it has a better shock capturing ability than other schemes.

Local characteristic decomposition based central-upwind scheme

We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontinuities, arising even for smooth initial conditions. In order to reduce the diffusion in the original central-upwind schemes, we use a local characteristic decomposition procedure to develop a new class of central-upwind schemes. We apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics to illustrate the performance on a variety of examples. The obtained numerical results clearly demonstrate that the proposed new schemes outperform the original central-upwind schemes.

Physics-based computational modeling and time-resolved imaging of plasma plume generated by nanosecond laser interaction with a bed of micro metallic powder

Although continuous lasers see frequent applications in selective laser melting or sintering, lasers with short pulse durations (such as those with nanosecond scale durations) possess potential advantages that may include good spatial resolutions and small heat-affected zones. Previous work on selective laser sintering with nanosecond lasers has been reported, particularly for laser micro sintering. At a sufficiently high intensity, a nanosecond (ns) laser pulse can generate a plasma plume from its irradiated metallic powder bed surface. The plasma plume evolution, such as the pressure it generates on the surface of the powder bed, may significantly influence the sintering process in the powder bed. However, physics-based modeling work for plasma generated by nanosecond-pulsed laser interaction with a metallic micro powder bed has been seldomly seen in literatures according to the knowledge of the authors’. Such modeling work has been reported in this paper for a ~4-ns laser pulse interaction with a cobalt micro powder bed, integrated with time-resolved plasma imaging using an intensified CCD (ICCD) camera with nanosecond scale gate widths for the model validation. For the conditions investigated, the model-predicted plasma plume evolutions agree reasonably well with the ICCD imaging results for the given period of comparison. This has reasonably supported the hypothesis posed in this paper (which has been rarely tested according to the knowledge of the authors’) that a short ns laser pulse-induced plasma from a metallic micro powder bed below the critical temperature can be reasonably well described by solving gas dynamic equations in the gaseous phase together with solving the heat transfer equation in the powder bed condensed phase, where the coupling is via the Knudsen layer (KL) relations for vaporization at the interface between the two phases. The model calculations show that in comparison with the bulk cobalt situation, the ns laser pulse can induce more significant surface vaporization from the cobalt powder bed, leading to a plasma plume with typically higher peak temperatures and densities in the simulated period. The plume can generate a short total pressure pulse with a ~491-MPa peak magnitude on the surface of the powder bed.

Theoretical study of stability of nodal completely conservative difference schemes with viscous filling for gas dynamics equations in Euler variables

For the equations of gas dynamics in Eulerian variables, a family of twolayer time-fully conservative difference schemes (FCDS) with space-profiled time weights is investigated. Nodal schemes and a class of divergent adaptive viscosities for FCDS with spacetime profiled weights connected with variable masses of moving nodal particles of the medium are developed. Considerable attention is paid to the methods of constructing regularized flows of mass, momentum and internal energy that preserve the properties of fully conservative difference schemes of this class, to the analysis of their stability and to the possibility of their use on uneven grids. The effective preservation of the internal energy balance in this class of divergent difference schemes is ensured by the absence of constantly operating sources of difference origin that produce “computational” entropy (including entropy production on the singular features of the solution). Developed schemes may be used in modelling of hightemperature flows in temperature-disequilibrium media, for example, if it is necessary to take into account the electron-ion relaxation of temperature in a short-living plasma under conditions of intense energy input.

Application of invariants of characteristics to construction of solutions without gradient catastrophe

The one-dimensional system of equations of isentropic gas dynamics is considered. First-order invariants of characteristics of this system are classified. Second-order invariants of characteristics are classified for polytropic processes. The infinite sequence of Darboux integrable systems is described. The approach to construction of smooth solutions without gradient catastrophe is proposed. Examples of solutions without gradient catastrophe are presented.

Cavitation bubble collapse and rebound in water: Influence of phase transitions

The influence of the mass transfer across the bubble surface due to evaporation and condensation on the cavitation (vapor) bubble collapse and rebound in water under room conditions has been studied. The bubble is spherical with an initial radius of 1.92mm. The dynamics of the vapor in the bubble and the surrounding liquid is governed by the gas dynamics equations. The effects of the liquid viscosity and heat conductivity of both fluids are taken into account. The liquid and vapor states are mainly described by the known wide-range equations by Nigmatulin and Bolotnova. The mass transfer is governed by the accommodation coefficient αac in the Hertz–Knudsen–Langmuir formula. It has been found that at αac in the range 0.001−0.075 the vapor in the bubble is homobaric at collapse. As αac increases in the range αac>0.075, an isentropic compression wave convergent to the bubble center is formed in the bubble. At focusing of this wave at the bubble center, a divergent (reflected from the center) isentropic compression wave occurs. With rising αac the intensities of these convergent and divergent waves increase. The divergent wave partially penetrates to the liquid in the form of an isentropic divergent wave. With growing αac the penetrated wave becomes steeper. At αac≈0.25 the divergent isentropic wave in the bubble transforms into a shock wave during its propagation to the bubble surface. This shock wave partially penetrates to the liquid in the form of a divergent shock wave. At small αac the outgoing pulse in liquid is shockless. Starting with αac≈0.03, it becomes discontinuous. In the range 0.03≤αac≤0.12 the shock pulse results from only large pressure gradients in the liquid in the vicinity of the bubble at the beginning of rebound. In the range 0.12<αac<0.25 the discontinuous outgoing pulse is formed from the divergent isentropic compression wave arisen in liquid as a result of partial escape of the isentropic wave out of the bubble. At αac>0.25 the shock pulse is formed directly on the bubbles surface as a result of the partial penetration of the divergent shock wave from the bubble to liquid. The obtained numerical results are in good agreement with available numerical and experimental data.