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 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).

scholarly journals МОДЕЛИРОВАНИЕ ТЕЧЕНИЯ ГАЗА В КАМЕРЕ ЖРД С УЧЕТОМ ЕЕ ОХЛАЖДЕНИЯ

2018 ◽  
Vol 26 (4) ◽  
pp. 40-45
Author(s):  
И. Д. Дубровский ◽  
В. Л. Бучарский
Keyword(s):  
Heat Exchange ◽  
Heat Balance ◽  
Gas Dynamics ◽  
Rocket Engine ◽  
Liquid Rocket Engine ◽  
System Of Differential Equations ◽  
One Dimensional ◽  
Equations Of Gas Dynamics ◽  
Liquid Rocket ◽  
The Heat Balance

The problem of one-dimensional motion of gas in thecombustion chamber of the liquid rocket engine, taking into account its heat exchange with the chamber wall and with the cooling liquid, is considered in the article. The results of calculating the cooling of the chamber obtained by numerically solving the system of differential equations of gas dynamics combined with the heat balance equation are presented. The analysis of the results allows to assert about the possibility of applying the proposed method in practice.

scholarly journals The analysis of gas dynamics equations with initial and boundary conditions and the construction of the corresponding averaged system

2013 ◽  
Vol 54 ◽  
Author(s):  
Aleksandras Krylovas ◽  
Rima Kriauzienė
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Hyperbolic System ◽  
Gas Dynamics ◽  
System Of Differential Equations ◽  
Gas Dynamics Equations ◽  
Large Domain ◽  
First Order ◽  
Initial And Boundary Conditions ◽  
Asymptotical Analysis

In this paper hyperbolic system of the first order gas dynamics PDE with initial and boundary conditions is studied. The aim of the paper is to construct the averaged system of differential equations in order to find the uniformly valid in a large domain asymptotical solution. The averaged system is a new object of asymptotical analysis.

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.

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

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.

On the Vibration of a Rotating Disk

1972 ◽  
Vol 39 (4) ◽  
pp. 1143-1144 ◽  
Author(s):  
S. Barasch ◽  
Y. Chen
Keyword(s):  
Approximate Calculation ◽  
Initial Conditions ◽  
Rotating Disk ◽  
Equation Of Motion ◽  
Outer Radius ◽  
Order Equation ◽  
System Of Differential Equations ◽  
Fourth Order Equation ◽  
First Order ◽  
Inner Radius

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

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