Three methods for constructing an approximate Riemann solver for the Soave – Redlich – Kwong real gas model are presented: linearization of nonlinear equations, cubic interpolation, and local approximation of the equation of state by a two-term equation of state. These methods are tested by considering the problem of the decay of a discontinuity in a pipe in an axisymmetric setting for the low-molecular and high-molecular substances, including a region of nonclassical gas behavior. It is demonstrated that the linearization method is reasonable only for the testing problems. The method of approximation by cubic splines is acceptable for complex three-dimensional nonstationary calculations. However, it is found that the bicubic interpolation method does not work well for flows with large pressure drops. The local approximation method is the most economical and universal for practical calculations. It has been used for numerical modeling of real gas flows through a safety valve. The results of calculations for hydrogen and water vapor in a wide range of pressure variation are presented. The method of local approximation of the equation of state allows one to describe all features of gas flows for complex problems.

This paper presents a survey of original methods for solving problems of current interest concerning numerical simulation of the dynamics of operation of a direct-acting relief valve, as formulated and tested by Professor V.A.Tenenev, Doctor of Physics and Mathematics. New methods (not based on experimental data) are proposed to solve the problem of selecting an initial clearance and initial conditions for the dynamic characteristics of disk motion in a spring-loaded relief valve. A method due to V. A.Tenenev for constructing a computational dynamical grid for a three-dimensional analysis of the complete cycle of valve operation (“open-closed”) is presented. Approaches and methods for reducing the dimensionality of the problem of operation of the relief valve are discussed. Methods of taking into account the influence of the gas-dynamic feedback on the working processes in relief valves are developed and presented. Methods, numerical schemes and algorithms for taking into account the real properties of substances in simulating the operation of the valve are presented.

The formation of a supersonic gas target for lasers that operate in the extreme ultraviolet wavelengths is considered. The gas target is generated in the interaction zone of two opposite supersonic gas jets. The emission properties of inert gas targets were investigated experimentally. The distributions of the emission radiation intensity for argon, krypton and carbon dioxide were obtained and the shapes of the emission zone were detected.
The experimental conditions were reproduced in numerical experiments. The mathematical model of viscous compressible gas was used to model the gas dynamics of supersonic gas jets. The problem was solved in a two-dimensional axisymmetric setting for argon. The obtained distributions of the main gasdynamic quantities made it possible to detail the flow features and estimate the size of the emission zone, as well as the density level corresponding to this zone. It was demonstrated that the results of calculations qualitatively agree with the experimental data. In addition, it was found that the density level of the emission region with the required extreme ultraviolet intensity factor can be obtained by monitoring the total pressure.

This paper considers krypton flow in a micronozzle with a cylindrical tube. A standardized conical nozzle elongated with cylindrical portion performs gas discharge into a vacuum chamber at a pressure of $10^{−2}$ Pa. Under such conditions, a low temperature area is formed in the central part of the jet with gas condensation. The particles are entrained by the gas flow. The portion with a constant section behind the nozzle should focus the supersonic flow part and the condensed particle flow and also decrease particle dispersion behind the nozzle throat.
The paper expresses a mathematical model of homogeneous gas motion with respect to formation processes and the growth of condensation nuclei. Since the condensed particles are small, the research is carried out with a single velocity motion model. The results obtained have shown that the application of the cylindrical tube leads to nonlinear flow effects. The flow responds to: the geometrical exposure related to flow transition from the conical diverging nozzle into the cylindrical tube, heat exposure and mass outflow due to particle formation and growth, and considerable friction force exposure due to the small sizes of the channel. The sum total ofthese factors leads to an insignificant deceleration of the supersonic flow part and highly impacts condensation.

The paper presents a modification of the digital method by S. K. Godunov for calculating real gas flows under conditions close to a critical state. The method is generalized to the case of the Van der Waals equation of state using the local approximation algorithm. Test calculations of flows in a shock tube have shown the validity of this approach for the mathematical description of gas-dynamic processes in real gases with shock waves and contact discontinuity both in areas with classical and nonclassical behavior patterns. The modified digital scheme by Godunov with local approximation of the Van der Waals equation by a two-term equation of state was used for simulating a spatial flow of real gas based on Navier – Stokes equations in the area of a complex shape, which is characteristic of the internal space of a safety valve. We have demonstrated that, under near-critical conditions, areas of nonclassical gas behavior may appear, which affects the nature of flows. We have studied nonlinear processes in a safety valve arising from the movement of the shut-off element, which are also determined by the device design features and the gas flow conditions.