Valentin Tenenev

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 gives a spatial mathematical formulation of the problem of internal ballistics based on the Navier – Stokes equations, taking into account the swirl of the flow due to the rotation of the projectile. The k-e model of turbulent viscosity is used. The control volume method is used for the numerical solution of systems of equations. The gas parameters at the boundaries of the control volumes are determined by the method of S. K. Godunov using a self-similar solution to the problem of the decay of an arbitrary discontinuity. The MUSCL scheme is used to increase the order of approximation of the difference method. For equations written in a cylindrical coordinate system, an orthogonal difference grid is constructed using the complex boundary element method. A comparative analysis of the results obtained with different approaches to modeling the process of an artillery shot is given. Quantitative data are presented on the influence of factors not previously taken into account on the characteristics of the process.

This paper addresses the problem of balancing an inverted pendulum on an omnidirectional platform in a three-dimensional setting. Equations of motion of the platform – pendulum system in quasi-velocities are constructed. To solve the problem of balancing the pendulum by controlling the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is investigated under different initial conditions taking into account a necessary stop of the platform or the need for continuation of the motion at the end point of the trajectory. It is shown that the solution of the problem in a two-dimensional setting is a particular case of three-dimensional balancing.

This paper is concerned with assessing the correctness of applying various mathematical models for the calculation of the hydroshock phenomena in technical devices for modes close to critical parameters of the fluid. We study the applicability limits of the equation of state for an incompressible fluid (the assumption of constancy of the medium density) to the simulation of processes of the safety valve operation for high values of pressures in the valve. We present a scheme for adapting the numerical method of S. K. Godunov for calculation of flows of incompressible fluids. A generalization of the method for the Mie – Grüneisen equation of state is made using an algorithm of local approximation. A detailed validation and verification of the developed numerical method is provided, and relevant schemes and algorithms are given. Modeling of the hydroshock phenomenon under the valve actuation within the incompressible fluid model is carried out by the openFoam software. The comparison of the results for the weakly compressible and incompressible fluid models allows an estimation of the applicability ranges for the proposed schemes and algorithms. It is shown that the problem of the hydroshock phenomenon is correctly solved using the model of an incompressible fluid for the modes characterized by pressure ratios of no more than 1000 at the boundary of media discontinuity. For all pressure ratios exceeding 1000, it is necessary to apply the proposed weakly compressible fluid approach along with the Mie – Grüneisen equation of state.

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.

This paper addresses the problem of plane-parallel motion of the Zhukovskii foil in a viscous fluid. Various motion regimes of the foil are simulated on the basis of a joint numerical solution of the equations of body motion and the Navier – Stokes equations. According to the results of simulation of longitudinal, transverse and rotational motions, the average drag coefficients and added masses are calculated. The values of added masses agree with the results published previously and obtained within the framework of the model of an ideal fluid. It is shown that between the value of circulation determined from numerical experiments, and that determined according to the model of and ideal fluid, there is a correlation with the coefficient $\mathcal{R} = 0.722$. Approximations for the lift force and the moment of the lift force are constructed depending on the translational and angular velocity of motion of the foil. The equations of motion of the Zhukovskii foil in a viscous fluid are written taking into account the found approximations and the drag coefficients. The calculation results based on the proposed mathematical model are in qualitative agreement with the results of joint numerical solution of the equations of body motion and the Navier – Stokes equations.

This paper is concerned with the process of the free fall of a three-bladed screw in a fluid. The investigation is performed within the framework of theories of an ideal fluid and a viscous fluid. For the case of an ideal fluid the stability of uniformly accelerated rotations (the Steklov solutions) is studied. A phenomenological model of viscous forces and torques is derived for investigation of the motion in a viscous fluid. A chart of Lyapunov exponents and bifucation diagrams are computed. It is shown that, depending on the system parameters, quasiperiodic and chaotic regimes of motion are possible. Transition to chaos occurs through cascade of period-doubling bifurcations.

An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier–Stokes equations and equations of motion. A non-stationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid, which is caused by the motion of internal material points, in a gravitational field is explored. The possibility of motion of a body in an arbitrary given direction is shown.

We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are represented in the form of the Kirchhoff equations. In the case of piecewise continuous controls, the integrals of motion are indicated. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. An optimal control problem for several types of the inputs is then solved using genetic algorithms.

In the paper we consider the motion of a rigid body in a boundless volume of liquid. The body is set in motion by redistribution of internal masses. The mathematical model employs the equations of motion for the rigid body coupled with the hydrodynamic Navier–Stokes equations. The problem is mostly dealt with numerically. Simulations have revealed that the body’s trajectory is strongly governed by viscous effects.