Exact Solutions of Gas Dynamics Equations in Series in the Lagrangian Coordinate and Their Numerical Realization

2020 ◽  
Vol 55 (6) ◽  
pp. 858-869
Author(s):  
D. V. Ukrainskii
Keyword(s):  
Exact Solutions ◽  
Gas Dynamics ◽  
Numerical Realization ◽  
Gas Dynamics Equations ◽  
In Series ◽  
Lagrangian Coordinate

scholarly journals New Exact Solutions with a Linear Velocity Field for the Gas Dynamics Equations for Two Types of State Equations

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 123
Author(s):  
Renata Nikonorova ◽  
Dilara Siraeva ◽  
Yulia Yulmukhametova
Keyword(s):  
Velocity Field ◽  
Exact Solutions ◽  
Gas Dynamics ◽  
Three Dimensional ◽  
State Equation ◽  
Linear Velocity ◽  
State Equations ◽  
Gas Dynamics Equations ◽  
Linear Velocity Field ◽  
Monatomic Gas

In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted by the system to 12 and 14 parameters, respectively. Invariant submodels of rank one are constructed from two three-dimensional subalgebras of the corresponding Lie algebras, and exact solutions with a linear velocity field with inhomogeneous deformation are obtained. On the one hand of the special state equation, the submodel describes an isochoric vortex motion of particles, isobaric along each world line and restricted by a moving plane. The motions of particles occur along parabolas and along rays in parallel planes. The spherical volume of particles turns into an ellipsoid at finite moments of time, and as time tends to infinity, the particles end up on an infinite strip of finite width. On the other hand of the state equation of a monatomic gas, the submodel describes vortex compaction to the origin and the subsequent expansion of gas particles in half-spaces. The motion of any allocated volume of gas retains a spherical shape. It is shown that for any positive moment of time, it is possible to choose the radius of a spherical volume such that the characteristic conoid beginning from its center never reaches particles outside this volume. As a result of the generalization of the solutions with a linear velocity field, exact solutions of a wider class are obtained without conditions of invariance of density and pressure with respect to the selected three-dimensional subalgebras.

scholarly journals Invariant submodel of rank 1 and two families of exact solutions of gas dynamics equations with an equation of state of the special form

2021 ◽  
Vol 2099 (1) ◽  
pp. 012017
Author(s):  
D Siraeva
Keyword(s):  
Equation Of State ◽  
Exact Solutions ◽  
Special Form ◽  
Gas Dynamics ◽  
Three Dimensional ◽  
Inhomogeneous Deformation ◽  
System Of Equations ◽  
Gas Dynamics Equations ◽  
Linear Velocity Field ◽  
Invariant Submodel

Abstract In this article, the gas dynamics equations with an equation of state of the special form are considered.The equation of state is the pressure which is equal to the sum of two functions, with one being a function of a density, and the other one being a function of an entropy. The system of equations is invariant under the action of 12-parameter transformations group. For three-dimensional subalgebra 3.32 of the 12-dimensional Lie algebra invariants are calculated, an invariant submodel of rank 1 is constructed, and two families of exact solutions are obtained. The obtained solutions specify the motion of particles in space with a linear velocity field with inhomogeneous deformation. The first family of solutions has two moments of time of particles collapse. The second family of solutions has one moment of time of particles collapse on the plane. In the simplest case of second family of solutions, a surface consisting of particle trajectories is constructed.

Systematization of relaxation gas dynamics equations. A review

2010 ◽  
Vol 45 (4) ◽  
pp. 517-536
Author(s):  
V. S. Galkin ◽  
S. A. Losev
Keyword(s):  
Gas Dynamics ◽  
Gas Dynamics Equations

Comparison of the numerical and self-similar solutions of Sedov’s problem on a point explosion in gas

2020 ◽  
Vol 15 (3-4) ◽  
pp. 212-216
Author(s):  
R.Kh. Bolotnova ◽  
V.A. Korobchinskaya
Keyword(s):  
Gas Dynamics ◽  
Numerical Solutions ◽  
Similarity Theory ◽  
Compressible Medium ◽  
Initial Moment ◽  
Point Explosion ◽  
Perfect Gas ◽  
Gas Dynamics Equations ◽  
Self Similar ◽  
Similar Solutions

Comparative analysis of solutions of Sedov’s problem of a point explosion in gas for the plane case, obtained by the analytical method and using the open software package of computational fluid dynamics OpenFOAM, is carried out. A brief analysis of methods of dimensionality and similarity theory used for the analytical self-similar solution of point explosion problem in a perfect gas (nitrogen) which determined by the density of uncompressed gas, magnitude of released energy, ratio of specific heat capacities and by the index of geometry of the explosion is given. The system of one-dimensional gas dynamics equations for a perfect gas includes the laws of conservation of mass, momentum, and energy is used. It is assumed that at the initial moment of time there is a point explosion with instantaneous release of energy. Analytical self-similar solutions for the Euler and Lagrangian coordinates, mass velocity, pressure, temperature, and density in the case of plane geometry are given. The numerical simulation of considered process in sonicFoam solver of OpenFOAM package built on the PISO algorithm was performed. For numerical modeling the system of differential equations of gas dynamics is used, including the equations of continuity, Navier-Stokes motion for a compressible medium and conservation of internal energy. Initial and boundary conditions were selected in accordance with the obtained analytical solution using the setFieldsDict, blockMeshDict, and uniformFixedValue utilities. The obtained analytical and numerical solutions have a satisfactory agreement.

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