The chain of embedded invariant submodels for conic motions

2019 ◽  
Vol 14 (4) ◽  
pp. 268-273
Author(s):  
T.F. Mukminov
Keyword(s):  
Gas Dynamics ◽  
Continuous Medium ◽  
Ideal Gas ◽  
Optimal System ◽  
Higher Dimension ◽  
Cylindrical Coordinates ◽  
Galilean Group ◽  
Independent Variables ◽  
Time Translation ◽  
Uniform Extension

The equations of continuum mechanics are invariant in relation to the Galilean group generalized by extention. Its 11-dimensional Lie algebra has many subalgebras, which form the optimal system of dissimilar subalgebras. Subalgebras from the optimal system form the graph of embedded subalgebras. There are many chains of subalgebras in the graph. We consider the chain of embedded subalgebras containing operators of space and time translation, the rotation and uniform extension of all independent variables for the models of the continuous medium mechanics. We choose concordant invariants for each subalgebra from the chain. The chain of invariant submodels is constructed in a cylindrical coordinates based on chosen invariants. It is proved that solutions of a submodel constructed on a subalgebra of higher dimension will be part of solutions of submodels constructed on subalgebra of smaller dimensions for the considered chain. Thus, the chain of embedded invariant submodels is constructed by the example of equations of ideal gas dynamics.

scholarly journals Variational principle for electromagnetic hydrodynamics of plasma

2021 ◽  
pp. 1-48
Author(s):  
Mikhail Borisovich Gavrikov
Keyword(s):  
Variational Principle ◽  
Gas Dynamics ◽  
Continuous Medium ◽  
Variational Principles ◽  
Necessary Conditions ◽  
Ideal Gas ◽  
Embedding Theorem ◽  
Action Functional ◽  
Central Result ◽  
Over Time

Within the framework of the Lagrangian approach, a mathematically correct procedure is given for obtaining the necessary conditions of locextr from the variational principle for a number of continuous media that fill a certain geometric region at each moment of time and does not leave it over time. The proposed method is based on the geometric properties of the set of diffeomorphisms of the indicated domain, which is the configuration space of a continuous medium, and on a certain embedding theorem. The efficiency of the method is demonstrated by examples of obtaining the necessary locextr conditions from variational principles for ideal gas dynamics and classical MHD plasma. The central result of the work is the construction of the variational principle, in particular, the Lagrangian and the action functional for the theory of electromagnetic hydrodynamics of plasma.

scholarly journals Propagation of Blast waves in a Non-Ideal Magnetogasdynamics

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 458 ◽  
Author(s):  
Astha Chauhan ◽  
Rajan Arora ◽  
Mohd Siddiqui
Keyword(s):  
Magnetic Field ◽  
Gas Dynamics ◽  
Approximate Analytical Solution ◽  
Ideal Gas ◽  
Specific Conductance ◽  
Blast Waves ◽  
Highly Nonlinear ◽  
Vast Literature ◽  
Flow Variables ◽  
The Magnetic Field

Blast waves are generated when an area grows abruptly with a supersonic speed, as in explosions. This problem is quite interesting, as a large amount of energy is released in the process. In contrast to the situation of imploding shocks in ideal gas, where a vast literature is available on the effect of magnetic fields, very little is known about blast waves propagating in a magnetic field. As this problem is highly nonlinear, there are very few techniques that may provide even an approximate analytical solution. We have considered a problem on planar and radially symmetric blast waves to find an approximate solution analytically using Sakurai’s technique. A magnetic field has been taken in the transverse direction. Gas particles are supposed to be propagating orthogonally to the magnetic field in a non-deal medium. We have further assumed that specific conductance of the medium is infinite. Using Sakurai’s approach, we have constructed the solution in a power series of ( C / U ) 2 , where C is the velocity of sound in an ideal gas and U is the velocity of shock front. A comparison of obtained results in the absence of a magnetic field within the published work of Sakurai has been made to generate the confidence in our results. Our results match well with the results reported by Sakurai for gas dynamics. The flow variables are computed behind the leading shock and are shown graphically. It is very interesting that the solution of the problem is obtained in closed form.

scholarly journals Collapsing Cavities and Converging Shocks in Non-Ideal Materials

2019 ◽  
Vol 72 (4) ◽  
pp. 501-520 ◽  
Author(s):  
Zachary M Boyd ◽  
Emma M Schmidt ◽  
Scott D Ramsey ◽  
Roy S Baty
Keyword(s):  
Gas Dynamics ◽  
Nonlinear Eigenvalue Problem ◽  
Ideal Gas ◽  
Test Problems ◽  
Infinite Dimensional ◽  
Cavity Collapse ◽  
Scaling Solution ◽  
Metal Deformation ◽  
Special Case ◽  
The Ideal

Summary As modern hydrodynamic codes increase in sophistication, the availability of realistic test problems becomes increasingly important. In gas dynamics, one common unrealistic aspect of most test problems is the ideal gas assumption, which is unsuited to many real applications, especially those involving high pressure and speed metal deformation. Our work considers the collapsing cavity and converging shock test problems, showing to what extent the ideal gas assumption can be removed from their specification. It is found that while most materials simply do not admit simple (that is scaling) solutions in this context, there are infinite-dimensional families of materials which do admit such solutions. We characterize such materials, derive the appropriate ordinary differential equations and analyze the associated nonlinear eigenvalue problem. It is shown that there is an inherent tension between boundedness of the solution, boundedness of its derivatives and the entropy condition. The special case of a constant-speed cavity collapse is considered and found to be heuristically possible, contrary to common intuition. Finally, we give an example of a concrete non-ideal collapsing cavity scaling solution based on a recently proposed pseudo-Mie–Gruneisen equation of state.

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.

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).

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.

Investigation of the influence of boundary conditions in the numerical solution of a vortex tube model

2019 ◽  
Vol 14 (2) ◽  
pp. 89-100
Author(s):  
M.R. Minibaev ◽  
C.I. Mikhaylenko
Keyword(s):  
Boundary Conditions ◽  
Turbulent Flows ◽  
Gas Dynamics ◽  
Vortex Tube ◽  
Computational Simulation ◽  
Ideal Gas ◽  
Tube Model ◽  
Gas Dynamics Problems ◽  
Dynamics Problems ◽  
Physical Quantities

The applicability of various boundary conditions in the computational simulation of a Ranque–Hilsch vortex tube is investigated. A review of existing works on the effect of geometry and various thermodynamic parameters on the efficiency of the pipe is made. The substantiation of the possibility of introducing additional computational domains when moving the boundaries to study the influence of boundary conditions when modeling gas dynamics problems is given. To simulate the dynamics of a gas in a vortex tube, a mathematical model is written that includes the Navier–Stokes system of equations describing a compressible viscous fluid, which is closed by the equation of state of an ideal gas. Existing methods for calculating turbulent flows are considered. The applicability of various semi-empirical models of turbulence for modeling a vortex tube is described. The possibility of using the selected k−ε model and its description is argued. The boundary conditions characteristic of the vortex tube model are described, and the boundary conditions most combined in the simulation of gas dynamics problems are also shown. Presents a grid that takes into account the area formed by the removal of boundaries. The solution is based on the sonicFoam algorithm in the OpenFOAM package. Utilities of the postprocessor are used when preparing the model for calculations on a high-performance cluster and utilities for averaging the obtained physical quantities. The simulation results for different combinations of boundary conditions and models with remote boundaries are given. Comparison of the results obtained. It is shown that the geometrical dimensions have a strong influence on the operation of the pipe; the correct choice of boundary conditions makes it possible to obtain the values of physical quantities that are closest to the known experimental ones. Moving the boundaries away from direct exits provides an opportunity to more accurately estimate the effects that arise near the real boundaries of the vortex tube, especially affecting the magnitude of the Ranque–Hilsch effect.

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