Theoretical Analysis of Fully Conservative Difference Schemes with Adaptive Viscosity

For the equations of gas dynamics in Eulerian variables, a family of two-layer in time completely conservative difference schemes with space-profiled time weights is constructed. Considerable attention is paid to the methods of constructing regularized flows of mass, momentum, and internal energy that do not violate the properties of complete conservatism of difference schemes of this class, to the analysis of their amplitudes and the possibility of their use on non-uniform grids. Effective preservation of the balance of internal energy in this type of divergent difference schemes is ensured by the absence of constantly operating sources of difference origin that produce "computational"entropy (including those based on singular features of the solution). The developed schemes can be easily generalized in order to calculate high-temperature flows in media that are nonequilibrium in temperature (for example, in a plasma with a difference in the temperatures of the electronic and ionic components), when, with the set of variables necessary for describing the flow, it is not enough to equalize the total energy balance.

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

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).

INFLUENCE OF WALL SCATTERING EFFECT ON ELECTRONS GAS DYNAMICS PARAMETERS IN ELECTRIC PROPULSION THRUSTERS WITH CLOSED ELECTRON DRIFT

The analysis is represented of some works devoted to the mathematical modeling of processes in plasma-ion thrusters and Hall effect thrusters. It is shown that the common in these works is the use of approximate forms of the equations of gas dynamics, which are applicable to the description of relatively dense gases, but not to analyze the processes in the rarefied plasma of electric propulsion thrusters. As a result, the above mathematical models do not represent the processes that are significantly responsible for the values of the thruster operating parameters.Authors try to partially correct this drawback by insertion into the initial approximate forms of the equations written for a point in the plasma volume, the parameters that actually represent the boundary effects and should be written not in the equations of gas dynamics themselves, but in the boundary conditions for these equations.The most complete forms of the necessary equations are given in this paper. It is shown that it is necessary to take into account electrons thermal conductivity as well as at least one (radial-azimuth) component of viscosity tensor to describe the "wall scattering" effect.It is concluded that the most productive approach in mathematical modeling is to write the most complete forms of equations with their subsequent simplification – removing the terms responsible for the processes recognized on the basis of primary numerical estimates as such, which can be neglected.

Barochronous shear gas motion

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.