scholarly journals CABARET scheme on moving grids for two-dimensional equations of gas dynamics and dynamic elasticity

2021 ◽  
pp. 306-321
Author(s):  
Н.А. Афанасьев ◽  
П.А. Майоров
Keyword(s):  
Gas Dynamics ◽  
Ideal Gas ◽  
Two Dimensional ◽  
Arbitrary Lagrangian Eulerian ◽  
Cabaret Scheme ◽  
One Dimensional ◽  
Equations Of Gas Dynamics ◽  
Dynamic Elasticity ◽  
First Case ◽  
Lagrangian Variables

Схема КАБАРЕ, являющаяся представителем семейства балансно-характеристических методов, широко используется при решении многих задач для систем дифференциальных уравнений гиперболического типа в эйлеровых переменных. Возрастающая актуальность задач взаимодействия деформируемых тел с потоками жидкости и газа требует адаптации этого метода на лагранжевы и смешанные эйлерово-лагранжевы переменные. Ранее схема КАБАРЕ была построена для одномерных уравнений газовой динамики в массовых лагранжевых переменных, а также для трехмерных уравнений динамической упругости. В первом случае построенную схему не удалось обобщить на многомерные задачи, а во втором — использовался необратимый по времени алгоритм передвижения сетки. В данной работе представлено обобщение метода КАБАРЕ на двумерные уравнения газовой динамики и динамической упругости в смешанных эйлерово-лагранжевых и лагранжевых переменных. Построенный метод является явным, легко масштабируемым и обладает свойством временн´ой обратимости. Метод тестируется на различных одномерных и двумерных задачах для обеих систем уравнений (соударение упругих тел, поперечные колебания упругой балки, движение свободной границы идеального газа). The conservative-characteristic CABARET scheme is widely used in solving many problems for systems of differential equations of hyperbolic type in Euler variables. The increasing urgency of the problems of interaction of deformable bodies with liquid and gas flows requires the adaptation of this method to Lagrangian and arbitrary Lagrangian-Eulerian variables. Earlier, the CABARET scheme was constructed for one-dimensional equations of gas dynamics in mass Lagrangian variables, as well as for three-dimensional equations of dynamic elasticity. In the first case, the constructed scheme could not be generalized to multidimensional problems, and in the second, a time-irreversible grid movement algorithm was used. This paper presents a generalization of the CABARET method to two-dimensional equations of gas dynamics and dynamic elasticity in arbitrary Lagrangian-Eulerian and Lagrangian variables. The constructed method is explicit, easily scalable, and has the property of temporal reversibility. The method is tested on various one-dimensional and two-dimensional problems for both systems of equations (collision of elastic bodies, transverse vibrations of an elastic beam, motion of the free boundary of an ideal gas).

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

Two-dimensional quasi-stationary flows in gas dynamics

1968 ◽  
Vol 64 (4) ◽  
pp. 1099-1108 ◽  
Author(s):  
A. G. Mackie
Keyword(s):  
Differential Equations ◽  
Unsteady Flow ◽  
Exact Solutions ◽  
Gas Dynamics ◽  
Two Dimensional ◽  
Independent Variables ◽  
Stationary Flows ◽  
First Case ◽  
Special Cases ◽  
Flow Round

In this paper we are concerned with the two-dimensional, unsteady flow of an inviscid, polytropic gas whose adiabatic index γ lies between 1 and 3. We recall that comparatively early in the study of gas dynamics we encounter two exact solutions of gas dynamic problems. One, in one-dimensional unsteady flow, is the expansion of a semi-infinite column of gas which is initially at rest behind a piston which, at time t = 0, begins to move with constant speed away from the gas. The second, in two-dimensional, steady, supersonic flow, is the Prandtl–Meyer flow round a sharp convex corner. Both of those flows may be regarded as special cases of more general exact solutions which are obtained by the method of characteristics (see, for example, Courant and Friedrichs(1)). On the other hand, each may be obtained directly from the appropriate equations by making use of the fact that, in so far as neither problem contains any characteristic length parameter in its formulation, the principle of dynamic similarity can be used to reduce the system of partial differential equations to one of ordinary differential equations. In the first case the independent variables x and t occur only in the combination x/t and in the second the independent variables x and y occur only in the combination x/y. Interesting and instructive as the derivation of these solutions from such principles may be, it is an unfortunate fact that they are the only non-trivial solutions of the respective equations. This is not altogether surprising as the equations are ordinary with (in this case) a limited number of non-trivially distinct solutions.

Barochronous shear gas motion

2019 ◽  
Vol 14 (4) ◽  
pp. 274-278
Author(s):  
Yu.V. Yulmukhametova
Keyword(s):  
Gas Dynamics ◽  
Ideal Gas ◽  
Linear Functions ◽  
System Of Differential Equations ◽  
Spatial Variables ◽  
3 Dimensional ◽  
First Order ◽  
Equations Of Gas Dynamics ◽  
Shear Motion ◽  
Cartesian Coordinate

The equations of ideal gas dynamics admit an 11-dimensional Lie algebra of first-order differentiation operators. All subalgebras of this algebra are listed. Khabirov S.V. for all 48 types of 4-dimensional subalgebras, the bases of point invariants are calculated and three 4-dimensional subalgebras are considered that produce regular partially invariant solutions in Cartesian, cylindrical and spherical coordinates, respectively. In this paper, we pose the problem of finding the solution of 3-dimensional equations of gas dynamics in a Cartesian coordinate system with an arbitrary equation of state, built on invariants of a 4-dimensional subalgebra. The basic operators of the considered subalgebra are combinations of translations and Galilean transfers. The invariants of this subalgebra define a representation of the solution for unknown hydrodynamic functions. Speed components are linear functions in terms of spatial variables. Moreover, density and pressure depend only on time. After substituting the solution representation, we studied the compatibility of the resulting system of differential equations. The system is collaborative and has an exact solution. Such a solution describes the isentropic barochronous shear motion of a gas. The equations of the world lines of motion of gas particles are found. The moments of particle collapse are established. There were two of them. The equations of collapse surfaces are found and written. For the flat case, several statements about the nature of the motion of gas particles are proved.

scholarly journals Asymptotical approximation of one dimensional gas dynamics problem

2013 ◽  
Vol 54 ◽  
Author(s):  
Aleksandras Krylovas ◽  
Olga Lavcel-Budko
Keyword(s):  
Asymptotic Approximation ◽  
Gas Dynamics ◽  
Resonant Interaction ◽  
Ideal Gas ◽  
Time Interval ◽  
One Dimensional ◽  
Long Time

We analyze nonlinear one dimensional gas dynamics system. The constructed asymptotic approximation which describes periodic acoustics waves resonant interaction is uniformly valid in the long time interval. The results allow to determine the resonance conditions for the emergence and summarizes the previous analyzed polytropic ideal-gas case.

Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

2012 ◽  
Vol 11 (4) ◽  
pp. 1144-1168 ◽  
Author(s):  
Juan Cheng ◽  
Chi-Wang Shu
Keyword(s):  
Spherical Symmetry ◽  
Gas Dynamics ◽  
Control Volume ◽  
Two Dimensional ◽  
Cylindrical Coordinates ◽  
One Dimensional ◽  
First Order ◽  
Conservation Of Momentum ◽  
Numerical Fluxes ◽  
Geometric Conservation Law

AbstractIn, Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work to the first order control volume scheme of Maire in to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

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