Volume 220, Nº 3 (2017)

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.

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.

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.

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.

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.