MIXED LIQUID FLOWS IN CYLINDRICAL RESERVOIR WITH VERTICAL CROSSROADS

Abstract

The methods of studying free and forced vibrations of a liquid in rigid cylindrical tanks without partitions and at presence of vertical partitions with partial filling by a liquid are offered. It is supposed that the fluid is ideal and incompressible one, and its motion, caused by the action of external influences, is non-vortex. In these conditions, there exists a velocity potential that satisfies the Laplace equation. The boundary value problem for this potential is formulated. On the wetted surfaces of the shell as boundary conditions for the solution of the boundary value problem, the conditions of non-penetration are chosen. On the free surface of the liquid, the kinematic and static conditions are specified. The static condition consists in the equality of pressure on the liquid free surface with to atmospheric one. The liquid pressure is determined from the linearized Cauchy-Lagrange integral. To formulate the kinematic condition, an additional unknown function is introduced, which describes the motion of the free surface. The kinematic condition is the equality of the velocity of the liquid, which is described by the velocity potential, and the velocity of the free surface itself. The method to determine eigenfrequencies and modes for the considered shells is described. These modes are used as a system of basic functions in solving problems of forced fluctuations of fluid in reservoirs. Unknown functions are depicted in the form of series for the received basic functions. The coefficients of these serias are generalized coordinates. Periodic excitation forces acting in the vertical and horizontal directions are considered. If vertical excitation is studied, this leads to appearance of additional acceleration. Here we obtain a system of unbounded differential equations, each of which is the equation of Mathieu. This allows us to investigate the phenomena of parametric resonance. It is shown that the installation of vertical partitions moves the spectrum of resonant frequencies towards high frequency oscillations. The questions of convergence of the method have been clarified. Dependences of change in the level of free surface in time under the condition of horizontal force of excitation were obtained. The phase portraits of a dynamic system with indication of resonances are presented. The method allows as to carry out the adjustment of undesired excitation frequencies at the design stage in order to prevent loss of stability.