Vol 220, No 3 (2017)
- Year: 2017
- Articles: 10
- URL: https://journals.rcsi.science/1072-3374/issue/view/14799
Article
Ivan Oleksandrovych Lukovs’kyi (On his 80th Birthday)
Multimodal Method in Sloshing
Abstract
The multimodal method reduces the sloshing problem with free surface to a (modal) system of nonlinear ordinary differential equations. The method was originally proposed for nonimpulsive hydrodynamic loads. However, recently it has been successfully extended to the case of sloshing-induced slamming. In the 1950–1960’s, this method was used in the computational fluid dynamics (CFD) but later was replaced by the algorithms developed in the 1990-2000’s. At present, the method plays a dual role: first, as a unique analytic tool for the investigation of nonlinear sloshing regimes, their stability, and chaos as well as for simulations when traditional CFD fails (e.g., in the case of containers with perforated screen) and, second, as a source of construction of the modal systems, which are analogs of the Korteweg–de-Vries, Boussinesq, and other equations but for bounded volumes of liquid. We present a survey of the state-ofthe-art of the problem, describe the existing modal systems, and formulate open problems.
Solutions of the Laplace Equation Satisfying the Condition of Impermeability on a Spherical Segment
Abstract
We construct a set of harmonic functions satisfying the condition of impermeability on a spherical segment. These functions can be used as a functional basis for the construction of approximate solutions of the boundary-value problems of liquid sloshing. The indicated harmonic functions are obtained as a result of the Kelvin inversion of the auxiliary functions satisfying the corresponding boundary condition on an interval of the horizontal line. The constructed system of functions is applied to the determination of natural sloshing frequencies in a spherical tank.
Self-Induced Oscillations of a Jet Flowing Over the Wedge. The Mechanism of Appearance of the Feedback
Abstract
As a result of the direct numerical analysis of the nonstationary system of Navier–Stokes equations, we solve the problem of submerged jet formed by a narrow channel (nozzle) and flowing over a sharp rigid wedge. The time dependences of the vortex and pressure fields in the course of the transient processes and the process of stationary self-induced oscillations of the jet are established. It is shown that the role of hydrodynamic feedback channels in the transient process is played by the vortex formations generated at the time when the jet meets the wedge and moving against the jet flow. In the process of steady self-induced oscillations, the hydrodynamic feedback channel is formed due to the pressure drop on the wedge faces and periodic changes in its sign. This leads to the formation of a periodic flow in the medium from one face of the wedge to the other face and, hence, to the periodic transverse deflections of the jet. The comparison of the theoretically estimated frequency of self-induced oscillations of the jet with the experimentally obtained value reveals their coincidence with graphical accuracy.
Superexponentially Convergent Algorithm for an Abstract Eigenvalue Problem with Applications to Ordinary Differential Equations
Abstract
A new algorithm for the solution of eigenvalue problems for linear operators of the form A = A + B (with a special application to high-order ordinary differential equations) is proposed and justified. The algorithm is based on the approximation of A by an operator \( \overline{A}=A+\overline{B} \) such that the eigenvalue problem for Ā is supposed to be simpler than for A: The algorithm for this eigenvalue problem is based on the homotopy idea and, for a given eigenpair number, recursively computes a sequence of approximate eigenpairs that converges to the exact eigenpair with a superexponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The case of multiple eigenvalues of the operator Ā is emphasized. Examples of eigenvalue problems for the high-order ordinary differential operators are presented to support the theory.
Two-Point Problem for Systems Satisfying the Controllability Condition with Lie Brackets of the Second Order
Abstract
We study a two-point control problem for systems linear in control. The class of problems under consideration satisfies a controllability condition with Lie brackets up to the second order, inclusively. To solve the problem, we use trigonometric polynomials whose coefficients are computed by expanding the solutions into the Volterra series. The proposed method allows one to reduce the two-point control problem to a system of algebraic equations. It is shown that this algebraic system has (locally) at least one real solution. The proposed method for the construction of control functions is illustrated by several examples.
Stabilization by a Measurable Output and Estimation of the Level of Attenuation for Perturbations in Control Systems
Abstract
We establish new criteria for the output stabilization in linear control systems with the help of static and dynamic regulators. It is shown that the stabilization algorithms derived from these criteria can be applied to a certain class of nonlinear control systems. We propose some algorithms for the construction of the regularities of control guaranteeing the required estimates of the weighted level of attenuation of input signals. The obtained results are illustrated by an example of a system stabilizing a one-link robot-manipulator.
Propagation of Nonlinear Surface Gravity Waves on the Basis of a Model Degenerated in the Parameter of Dispersion
Abstract
We obtain equations generalizing the previously known results on the propagation of nonlinear waves in water of variable depth. To this end, we use the method of power series, which enables us to decrease the dimensionality of the problem and asymptotically construct some weakly dispersive but strongly nonlinear models close to the hyperbolic models of propagation of waves on water. The model has a broader range of application as compared with the available experimental and numerical results.
Nonaxisymmetric Vibrations of a Shell of Revolution Partially Filled with Liquid
Abstract
We propose an algorithm for finding the frequencies and modes of natural vibrations of the shells of revolution partially filled with liquid. The problem of perturbed motion of a liquid is solved under the assumption that its free surface remains flat and perpendicular to the axis of the shell. The solution is based on the use of the method of decomposition of the domain of integration of the equations of the theory of shells in combination with the variational method and the approximate construction of the inverse operator for the hydrodynamic part of the problem. We construct a generalized functional with respect to displacements of the shell for which the conditions of conjugation of the solutions in subdomains are included in the natural boundary conditions. The obtained numerical results are compared with the available exact solutions of the problem under consideration with regard for the wave motions of liquid in a shell in the form of circular cylinder.
Differential and Variational Formalism for an Acoustically Levitating Drop
Abstract
We consider the most general problem of waves on the interface of two ideal fluids regarded as an ullage gas and a liquid, respectively. Separating the fast and slow time scales, we develop the differential and variational formalism for an acoustically levitating drop and determine its time-averaged shape (vibroequilibrium state of the drop). The vibroequilibrium states of the drop may differ from the spherical shape. Stable vibroequilibria are associated with the local minima of the quasipotential energy whose analytic form is also established.