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 results of the theoretical solution of the problem of braking a supersonic flow in a round pipe based on direct numerical simulation by integrating the Navier – Stokes equations without the use of additional models and empirical constants are shown. Shaded maps of density distribution depending on flow parameters are presented. The flow consists of successive rhombus-shaped shock waves distributed along the entire length of the channel. It is determined that the size of x-shaped structures depends on the flow parameters. At a lower Mach number, the rhombuses have a smaller size and, accordingly, their number increases along the length of the channel. The Reynolds number also affects the size of structures, however, it is less pronounced. With a lower Reynolds number, x-shaped structures have a smaller size. It is shown that over time the flow tends to a stationary state.

We investigate discrete-time dynamical systems generated by an infinite-dimensional nonlinear operator that maps the Banach space $l_1^{}$ to itself. It is demonstrated that this operator possesses up to seven fixed points. By leveraging the specific form of our operator, we illustrate that analyzing the operator can be simplified to a two-dimensional approach. Subsequently, we provide a detailed description of all fixed points, invariant sets for the two-dimensional operator and determine the set of limit points for its trajectories. These results are then applied to find the set of limit points for trajectories generated by the infinite-dimensional operator.

When an articulated drinking straw is slid over a rod vibrated by a motor, the straw moves up and down continuously. To clarify the mechanism for this motion, a straw was created that was narrower at the center than at the ends and the mechanism for the up-and-down motion as well as its reversal at the rod's tip was examined. The angle $\psi$ between the rod and the straw is constant because they are in contact at three points, i.e., at both ends and the center. Therefore, when the straw rotates around the rod, it will move unidirectionally according to the sign of $\psi$. This velocity is determined as a function of the straw half-length and the number of rotations of the nut to which the rod is attached. As the straw rises, the rod's tip enters the straw, and the ascent stops when $\psi$ reaches zero because the rate of ascent is proportional to $\psi$. The sign of $\psi$ then reverses to start the downward motion, and the straw returns. When it reaches a reflector disk, the sign of $\psi$ reverses again and the straw rises. Experiments were conducted to measure the velocities of three straws with different lengths, and the results showed that the theoretical velocities were greater than the experimental ones. The reason for this is assumed to be that although in theory the straw does not slip at the point of contact between its center and the rod, in reality it does slip. However, the theoretical and experimental velocities decreased in similar ways with increasing straw half-length, and the agreement between them was determined by the relative error. For half-lengths below $1.0$ cm, the average agreement was approx. $80\,\%$, and for all lengths it was approx. $73\,\%$. The agreement would have been even better if it had not been for the effects of slippage between the straw and the rod, the presence of nodes and antinodes of the rod vibration, and deformation in the central portion of straw. Considering these effects, the experimental values support the validity of the mechanistic considerations of the straw motion and the theoretically determined velocity.

In this work we focus on the chaotic diffusion in the phase space of a triaxial potential resembling an elliptical galaxy. The transport process is studied in two different action-like starting spaces in order to cope with circulating and noncirculating orbits. Estimates of the diffusion rate obtained by means of the variance approach are discussed in detail and their limitations are exposed. After revisiting the Shannon-entropy-based method from a conceptual point of view in the framework of simple arguments taken from the information theory, we apply it to measure changes in the unperturbed actions or integrals of motion of the system for different sets of small ensembles of random initial conditions. For such sets of ensembles, estimates of the Lyapunov times are also provided. The results show that, within the chaotic component of the phase space, the Lyapunov times are shorter than any physical time scale as the Hubble time, but the diffusion times are much larger than the latter. Thus, we conclude that stable chaos dominates the dynamics of realistic galactic models.