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 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.