Equations Of Gas Dynamics Research Articles

Lie group approach for constructing all reciprocal transformations: The two‐dimensional stationary gas dynamics equations

Recently, an infinitesimal approach for finding reciprocal transformations has been proposed. The method uses the group analysis approach and consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). Similar to the classical group analysis, this approach can be also applied for finding all reciprocal transformations (not only composing a group) of the equations under study. The present paper provides this algorithm. As an illustration, the method is applied to the two‐dimensional stationary gas dynamics equations. Equivalence group, group of reciprocal transformations, and completeness of all discrete reciprocal transformations are presented in the paper. The results are stated in form of a theorem.

14-moment maximum-entropy modeling of collisionless ions for Hall thruster discharges

Ions in Hall effect thrusters are often characterized by a low collisionality. In the presence of acceleration fields and azimuthal electric field waves, this results in strong deviations from thermodynamic equilibrium, introducing kinetic effects. This work investigates the application of the 14-moment maximum-entropy model to this problem. This method consists in a set of 14 partial differential equations (PDEs) for the density, momentum, pressure tensor components, heat flux vector, and fourth-order moment associated with the particle velocity distribution function. The model is applied to the study of collisionless ion dynamics in a Hall thruster-like configuration, and its accuracy is assessed against different models, including the Vlasov kinetic equation. Three test cases are considered: a purely axial acceleration problem, the problem of ion-wave trapping, and finally the evolution of ions in the axial-azimuthal plane. Most of this work considers ions only, and the coupling with electrons is removed by prescribing reasonable values of the electric field. This allows us to obtain a direct comparison among different ion models. However, the possibility to run self-consistent plasma simulations is also briefly discussed, considering quasi-neutral or multi-fluid models. The maximum-entropy system appears to be a robust and accurate option for the considered test cases. The accuracy is improved over the simpler pressureless gas model (cold ions) and the Euler equations for gas dynamics, while the computational cost shows to remain much lower than direct kinetic simulations.

Mathematical model to study the keyhole formation in pulsed Nd:YAG laser welding of SS 316L material and its experimental verification

A mathematical model to study keyhole formation and its propagation in the material is developed for laser welding performed in an open atmosphere. The present model overcomes the limitations of existing models in assuming sonic vapor jet velocity to calculate vaporization-induced recoil pressure responsible for keyhole formation. In the present model, the exact value of vapor jet velocity is calculated using gas dynamics equations. The minimum threshold value of absorbed laser beam intensity required to perform keyhole welding irrespective of laser pulse duration for laser beam radius of 0.6 mm has been found to be 0.8 × 105 W/cm2 and is in good agreement with the experimental value. In between conduction mode welding and keyhole mode welding, a transition mode exists where a keyhole mechanism develops itself and melt displacement is not considerable in this zone. Weld penetration occurs mainly through heat diffusion in this transition mode. The predicted values for keyhole penetration velocity are also in good agreement with the experimental values. At a longer pulse duration, the model over-predicts the keyhole penetration velocity as compared to the experimental value due to nonconsideration of vapor plasma absorption of the laser beam.

A Semi-Analytical Method to Investigate Fractional-Order Gas Dynamics Equations by Shehu Transform

This work aims at a new semi-analytical method called the variational iteration transformation method for solving nonlinear homogeneous and nonhomogeneous fractional-order gas dynamics equations. The Shehu transformation and the iterative technique are applied to solve the suggested problems. The proposed method has an advantage over existing approaches because it does not require additional materials or computations. Four problems are used to test the authenticity of the proposed method. Using the suggested method, the solution proves to be more accurate. The proposed method can be implemented to solve many nonlinear fractional order problems because it has a straightforward implementation.

Homotopi pertürbasyon Elzaki dönüşümü yöntemi ile doğrusal olmayan zaman-kesirli kısmi diferansiyel denklemler için yeni yaklaşık analitik çözümler

Some nonlinear time-fractional partial differential equations are solved by homotopy perturbation Elzaki transform method. The fractional derivatives are defined in the Caputo sense. The applications are examined by homotopy perturbation Elzaki transform method. Besides, the graphs of the solutions are plotted in the MAPLE software. Also, absolute error comparison of homotopy perturbation Elzaki transform method and homotopy perturbation Sumudu transform method solutions with the exact solution of nonlinear time-fractional partial differential equations is presented. In addition, this absolute error comparison is indicated in the tables. The novelty of this article is the first analysis of both the gas dynamics equation of Caputo fractional order and the Klein-Gordon equation of Caputo fractional order via this method. Thus, homotopy perturbation Elzaki transform method is quick and effective in obtaining the analytical solutions of time-fractional partial differential equations.

Natural Gas Short-Term Operation Problem With Dynamics: A Rank Minimization Approach

Natural gas-fired generation units can hedge against the volatility in the uncertain renewable generation, which may occur during very short periods. It is crucial to utilize models capable of correctly capturing the natural gas network dynamics induced by the volatile demand of gas-fired units. The Weymouth equation is commonly implemented in literature to avoid dealing with the mathematical complications of solving the original governing differential equations of the natural gas dynamics. However, it is shown in this paper that this approach is not reliable <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in the short-term operation problem</i> . Here, the merit of the non-convex transient model is compared with the simplified Weymouth equation, and the drawbacks of employing the Weymouth equation are illustrated. The results demonstrate how changes in the natural gas demand are met by adjustment in the pressure within pipelines rather than the output of natural gas suppliers. This work presents a convex relaxation scheme for the original non-linear and non-convex natural gas flow equations with dynamics, utilizing a rank minimization approach to ensure the tightness. The proposed method renders a computationally efficient framework that can accurately solve the non-convex non-linear gas operation problem and accurately capture its dynamics. Also, the results suggest that the proposed model improves the solution optimality and solution time compared to the original non-linear non-convex model. Finally, the scalability of the proposed approach is verified in the case study.

A Reliable Way to Deal with Fractional-Order Equations That Describe the Unsteady Flow of a Polytropic Gas

In this paper, fractional-order system gas dynamics equations are solved analytically using an appealing novel method known as the Laplace residual power series technique, which is based on the coupling of the residual power series approach with the Laplace transform operator to develop analytical and approximate solutions in quick convergent series types by utilizing the idea of the limit with less effort and time than the residual power series method. The given model is tested and simulated to confirm the proposed technique’s simplicity, performance, and viability. The results show that the above-mentioned technique is simple, reliable, and appropriate for investigating nonlinear engineering and physical problems.

Well-balanced positivity preserving adaptive moving mesh central-upwind schemes for the Saint-Venant system

We extend the adaptive moving mesh (AMM) central-upwind schemes recently proposed in Kurganov et al. [Commun. Appl. Math. Comput. 3 (2021) 445–479] in the context of one- (1-D) and two-dimensional (2-D) Euler equations of gas dynamics and granular hydrodynamics, to the 1-D and 2-D Saint-Venant system of shallow water equations. When the bottom topography is nonflat, these equations form hyperbolic systems of balance laws, for which a good numerical method should be capable of preserving a delicate balance between the flux and source terms as well as preserving the nonnegativity of water depth even in the presence of dry or almost dry regions. Therefore, in order to extend the AMM central-upwind schemes to the Saint-Venant systems, we develop special positivity preserving reconstruction and evolution steps of the AMM algorithms as well as special corrections of the solution projection step in (almost) dry areas. At the same time, we enforce the moving mesh to be structured even in the case of complicated 2-D computational domains. We test the designed method on a number of 1-D and 2-D examples that demonstrate robustness and high resolution of the proposed numerical approach.

Thermal and aerodynamic parameters radiation chambers of the hydrocarbon pyrolysis furnace

THE PURPOSE. To carry out mathematical modeling of interrelated physicochemical processes in radiation chambers of tubular furnaces of pyrolysis of hydrocarbons. Perform numerical calculations of heat and mass transfer in the furnace furnace chamber with a multi-tiered arrangement of wall burners on the side lined walls.METHODS. Using our own package of applied programs based on the numerical solution of a system of differential equations of energy conservation and equations of radiation gas dynamics, the fields of velocities and temperatures of natural gas combustion products in the radiation chamber of a hydrocarbon pyrolysis tubular furnace are calculated. Interrelated processes of natural gas combustion, radiant-convective heat transfer, and turbulent flow of flue gases take place in the radiation chamber. These processes are modeled by two-dimensional equations of the combustion model, energy transfer by radiation, and equations of motion. In the side walls of the radiation chamber on both sides in eight horizontal tiers there are wall burners in the amount of 128 pieces. Combustion products emerging from the nozzles of these burners create complex velocity and temperature fields in the chamber volume. Due to the radiation of flue gases, the smallest soot particles and hot lined walls of the radiation chamber, heat flows are formed to the tubular screen, where hydrocarbons are pyrolyzed to produce ethylene.RESULTS. As a result of calculations, the velocity fields, the temperature in the volume of the radiation chamber and the concentration of the main components of the combustion products were obtained. Local values of surface densities of radiant heat fluxes to reaction pipes for methane and propane pyrolysis furnaces are calculated. Comparisons of some of the results obtained with the readings of the devices of existing technological installations are carried out.CONCLUSION. Calculations show that the developed software package makes it possible to obtain realistic values of local velocities and temperatures in the radiation chambers of tube furnaces, surface densities of convective and radiant heat fluxes to boundary surfaces, and other thermal and gas-dynamic parameters in the furnace volume. With a multi-row arrangement of a large number of wall burners of low thermal power, a complex field of temperature and velocities of combustion products in the furnace volume is formed on the side walls of the radiation chamber of the furnace. At the same time, this arrangement of the burners ensures a fairly uniform distribution of heat flows over the surface of the tubular screen of the hydrocarbon pyrolysis furnace.

CAE studies of river ice cover aerodynamics

The paper provides an overview of autumn-winter movements and ice breaking on river shipping routes. The range of tasks of ensuring the safety of ice navigation is highlighted, which requires the predict of wind loads on hummocky ice fields of small floe fractions. As a complex argument for the characterization of such a field, the coefficient of its windage is proposed. The statistical function of the distribution of the height of the hummocks is revealed.&#x0D; To speed up the process of three-dimensional description of the geometry of the ice surface with a given law of the distribution of hummocks, the joint use of CAD and CAE systems is proposed. A series of CAE «purges» of model ice fields was carried out. The dependence of aerodynamic loads on the direction of atmospheric flow has been confirmed by the analysis of the results of CAE-modeling of wind impact on small floe ices. The conclusion is made about the limitations of the application of the classical equation of gas dynamics in relation to the studied ice.

On Thermodynamically Compatible Finite Volume Schemes for Continuum Mechanics

In this paper we present a new family of semidiscrete and fully discrete finite volume schemes for overdetermined, hyperbolic, and thermodynamically compatible PDE systems. In the following, we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics as well as the more complex Godunov--Peshkov--Romenski (GPR) model of continuum mechanics, which, with the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first-order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the entropy inequality as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper [Dokl. Akad. Nauk SSSR, 139(1961), pp. 521--523] exactly at the discrete level. All other terms in the governing equations of the more general GPR model, including nonconservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing PDEs.

A Piecewise-Linear Reconstruction to Reduce the Dissipation of the HLL Method in Solving the Gas Dynamics Equations

A Piecewise-Linear Reconstruction to Reduce the Dissipation of the HLL Method in Solving the Gas Dynamics Equations

Turbulent Cosmic Ray–Mediated Shocks in the Hot Ionized Interstellar Medium

The structure of shocks and turbulence are strongly modified during the acceleration of cosmic rays (CRs) at a shock wave. The pressure and the collisionless viscous stress decelerate the incoming thermal gas and thus modify the shock structure. A CR streaming instability ahead of the shock generates the turbulence on which CRs scatter. The turbulent magnetic field in turn determines the CR diffusion coefficient and further affects the CR energy spectrum and pressure distribution. The dissipation of turbulence contributes to heating the thermal gas. Within a multicomponent fluid framework, CRs and thermal gas are treated as fluids and are closely coupled to the turbulence. The system equations comprise the gas dynamic equations, the CR pressure evolution equation, and the turbulence transport equations, and we adopt typical parameters for the hot ionized interstellar medium. It is shown that the shock has no discontinuity but possesses a narrow but smooth transition. The self-generated turbulent magnetic field is much stronger than both the large-scale magnetic field and the preexisting turbulent magnetic field. The resulting CR diffusion coefficient is substantially suppressed and is more than three orders smaller near the shock than it is far upstream. The results are qualitatively consistent with certain observations.

Group analysis of the two‐dimensional two‐temperature gas dynamics equations

The two‐dimensional two‐temperature nonstationary gas dynamics equations with the relaxation Landau–Teller equation are considered in the paper. The group analysis method is applied to the study these equations. The admitted Lie groups are found for two modifications of relaxation time. Optimal systems of one‐ and two‐dimensional subalgebras are constructed for the group corresponding to the case of varying relaxation time which is closer to the case of ideal gas dynamics. Solutions with two and one independent invariant variables are derived using the optimal systems of admitted Lie algebras. The corresponding reduced systems were numerically calculated and compared with numerical solutions of analogous problems for classical gas dynamics equations. The differential conservation laws which present the fundamental properties of two‐temperature gas dynamics equations are also obtained.

Conservation laws of the two-dimensional relativistic gas dynamics equations

Two-dimensional relativistic gas dynamics equations are considered in the paper. Using a suitable Lagrangian found by solving the Helmholtz problem, the equations under study are reduced to the Euler–Lagrange equations. The admitted Lie algebra of these equations is derived. Applying Noether’s theorem with the found Lagrangian and the admitted generators, conservation laws in Lagrangian variables are obtained. Counterparts of the found conservation laws in Eulerian coordinates are also presented.

High Order Semi-implicit WENO Schemes for All-Mach Full Euler System of Gas Dynamics

In this paper, we propose high order semi-implicit schemes for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. The schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes. Besides, they can well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes.

Adaptation of conformable residual series algorithm for solving temporal fractional gas dynamics models

In this paper, we introduced, discussed, and investigated analytical-approximate solutions for nonlinear time fractional gas dynamics equations in terms of conformable differential operator. The proposed algorithm relies upon the conformable power series method and residual error of the generalized Taylor series in terms of the conformable sense. This technique provides analytical solutions in the form of rapid and accurate convergent series in terms of the multiple fractional power series with easily computable components. In this direction, error estimation and convergence analysis for solutions of fractional gas dynamics equations are provided as well. Eventually, several physical examples are tested to justify the theoretical portion and give a clear explanation of dynamic systems for the proposed model for different orders of fractional case The obtained numeric-analytic results indicate that the current algorithm is simple, effective, and profitably dealing with the complexity of many nonlinear fractional dispersion problems.

In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. An extension to second-order of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic is presented. This extension is based on the combination of the first-order in-cell reconstruction with the standard MUSCL-Hancock reconstruction. The first-order strategy allowed in particular to capture exactly the isolated shocks and this new second-order extension keep this property. Moreover, the well-balanced property of the method is also studied. Several numerical tests are proposed to validate the methods for the Coupled-Burgers system, Gas dynamics equations in Lagrangian coordinates and the modified shallow water system.

On Solutions of Fractional-Order Gas Dynamics Equation by Effective Techniques

In this work, the novel iterative transformation technique and homotopy perturbation transformation technique are used to calculate the fractional-order gas dynamics equation. In this technique, the novel iteration method and homotopy perturbation method are combined with the Elzaki transformation. The current methods are implemented with four examples to show the efficacy and validation of the techniques. The approximate solutions obtained by the given techniques show that the methods are accurate and easy to apply to other linear and nonlinear problems.

The modeling of two-dimensional vortex flows in a cylindrical channel using parallel calculations on a supercomputer

Objectives . The study aimed to examine vortex structures formed during the interaction of incident and reflected shock waves in a cylindrical channel. The shock wave was described by the Hugoniot relations, which make it possible to determine the parameters of the gas behind the shock front by a given Mach number and the values of the gasdynamic parameters ahead of the pressure jump. The propagation of a strong shock wave (Mach number was 20) in argon was simulated. Methods . The methods of mathematical modeling were used. A parallel algorithm for solving two-dimensional equations of gas dynamics in cylindrical coordinates (r, z, t) was developed and a new version of the NUTCY_ps program created. The calculations were performed on an MVS-100K supercomputer. Results. Two methods of parallelization when solving a system of equations were considered. Using a specific task as an example, a comparison of the effectiveness of these methods was conducted. A parallel algorithm was developed and a program was upgraded for solving two-dimensional equations of gas dynamics in cylindrical coordinates (r, z are spatial coordinates, t is time). Numerical calculations were performed to simulate: 1) the shock wave incidence to and reflection from a metal screen; 2) the propagation of the shock wave through a hole in the screen; 3) the propagation of the shock wave through a cylindrical channel and its reflection from the bottom of the channel and interaction with the incident wave. The results obtained by the parallel supercomputer with different numbers of processors are presented. It is shown that using 16 processors, it is possible to reduce the computation time for getting a solution for the test problem by approximately 12 times. Conclusions . It is shown that the interaction of incident shock wave and the one reflected at an angle leads to the formation of regions with low and high gas densities, as well as vortex flows. The vortex interaction area (turbulence zone) gets a complex shape. The article discusses the possibility of carrying out full-scale experiments in shock tubes or using a laser shock tube. Such studies would make it possible to compare experimental data with the results of numerical simulation and, on their basis, to develop more advanced models of turbulent motions.