Forced liquid vibrations in prismatic tanks under vertical and horizontal loads

The method of studying forced vibrations of a liquid in rigid prismatic tanks partially filled with a liquid is offered. It is supposed that the liquid is an ideal and incompressible one, and its motion, caused by the action of external influences, is irrotational. For those assumptions there is velocity potential that satisfies the Laplace equation. The boundary value problem for this potential is formulated. On the wetted surfaces of the tank the non-penetration conditions have been chosen. On the free surface of the liquid, the kinematic and static conditions have been specified. The static condition is the equality of pressure on the free surface to atmospheric one. The liquid pressure is determined from the Cauchy-Lagrange integral. To formulate the kinematic condition an additional unspecified function describing the motion of the free surface is introduced. The kinematic condition is the equality of the velocity of the liquid described by the velocity potential, and the velocity of the free surface itself. The modes of free vibrations are used as a system of basic functions to solve the problems of forced fluid vibrations in reservoirs. Unspecified functions are presented as series of the basic functions. The coefficients of these series are generalized coordinates. Periodic excitation forces acting in the vertical and horizontal directions have been considered. The vertical excitation examination leads to appearance of additional acceleration. Therefore we obtain a system of unbounded differential equations of the Mathieu type. This allows us to investigate the phenomena of parametric resonance. The effect of parametrical resonance is considered when the vertical excitation frequency is equal to double own frequency of liquid vibrations. Dependences of change in the level of free surface via time under both separate and mutual action of horizontal and vertical forces have been obtained. The phase portraits of a dynamic system with indication of resonances are presented. The method allows us to carry out the adjustment of undesired excitation frequencies at the reservoir design stage in order to prevent the loss of stability.

Ключевые слова: призматические резервуары, идеальная несжимаемая жидкость, вертикальные и горизонтальные нагрузки, уравнения Матье, фазовые портреты 1. Formulation of the problem and its relevance Containers and reservoirs for the storage and transportation of different liquids are widely used in aerospace, chemical, oil and gas industry, power engineering, sea transport. These reservoirs and fuel tanks are usually filled with oil or other dangerously explosive, flammable or toxic substances. Fluid motion in liquid storage tanks due to intensive external loadings can be very complicated and violent. Therefore, studying the dynamic behavior of fluid in tanks is an urgent task. During past decades the significant progress in experimental technique and numerical methods based on using the computational fluid dynamics approach has been achieved. But these techniques are very expensive and timeconsuming. Therefore the linear wave models based on the potential flow assumptions are suitable as the first approximation at designing the liquid storage reservoirs. The most important problems are associated with fluid motion in reservoirs caused by external loadings, especially applied suddenly.
The methods for solving fluid oscillation problems in rigid prismatic tanks under simultaneous action of horizontal and vertical excitations are proposed in this paper.

Analysis of recent research and publications
Sloshing is a phenomenon associated with the intense movement of fluid in partially filled tanks [1]. This phenomenon can lead to negative effects caused by suddenly applied loads (earthquakes, aircraft crashes, etc.). Most studies are devoted to the analysis of free liquid vibrations [2][3][4] or excitation forces acting only in the horizontal direction [5][6][7]. Liquid vibrations in fluid-filled prismatic tanks under action of horizontal loading have been studied in [8]. The liquid motion under action of harmonic force has been considered. Kim has carried out the numerical simulation of sloshing to predict impact loads and provided comparison of various numerical techniques in [9]. The effects of sloshing have been considered for viscid liquids in [10]. The authors of [11] have used the coupled finite and boundary elements method for sloshing in 3D tanks of different configurations. Parametric instability of liquid free surface in different fluid-filled reservoirs caused by vertical excitations has been the subject of extensive research in many scientific areas since Faraday's first works [12].

The aim of the study
The aim of the study is to create the methodology for estimating the amplitude of the liquid free surface vibrations in prismatic tanks under action of various external influences.

Outline of the main research material
This paper deals with the problems of free and forced oscillations of liquids in rigid prismatic tanks. It is assumed that the external load can act either horizontally or vertically. The scheme of the tank is shown in Fig. 1. We suppose that the fluid is inviscid, incompressible, and its motion is irrotational. Under these conditions, there exists a potential of velocities φ(x,y,z,t), such that This potential satisfies the Laplace equation. The mixed boundary value problem for this equation is formulated. At the same time, non-penetration conditions are set on the lateral surfaces and the bottoms of the reservoir, and kinematic and dynamic conditions are set on the free surface. The kinematic condition is that the point on the free surface of the fluid in the reservoir at the initial time of motion remains on that surface throughout the whole movement. The dynamic condition characterizes the equilibrium of the atmospheric pressure and the fluid pressure on the free surface. The unknowns are the velocity potential  and function  that describes the level the free surface elevation. The relationship between these two functions is given by the dynamic boundary condition Firstly we obtain a relation between the velocity potential, the liquid pressure, and accelerations due to driving forces and gravity. We have where w is the acceleration of the fluid flow,  is the liquid density, and p is the fluid pressure. Therefore the acceleration of liquid particles under gravitational forces, horizontal, and vertical excitations always has the potential (an analog of the Prandtl's potential). Using equation (4.3) and assuming that the flow is irrotational, Bernoulli equation can be derived in the following form: where p 0 is the atmospheric pressure. If small oscillations of the liquid are considered then 1 2    , and we have the next expression: (4.5) Thus, for the velocity potential, we have the following boundary value problem: To solve the problem of forced oscillations we construct a system of basis functions, which are the solutions of the spectral boundary value problem under additional condition (4.7).
We present the expressions for the first 8 eigenmodes of fluid oscillations obtained in 13. This is a system of basis functions obtained as a solution to problem (4.6) -(4.7) for studying forced oscillations. In the Tab. 1 the natural frequencies of fluid oscillations in a prismatic reservoir are shown. Fig. 1 shows the numerical values of the frequencies ij and the frequency parameter  ij for a cubeshaped prismatic reservoir with geometric characteristics а=b=H=1м where  ij are found using relation (4.9).
Then, provided that H z  , we obtain  

Forced fluid oscillations in a rigid tank
Let us suppose that at the initial time, the liquid in the tank is at the state of rest. A combined periodic load is applied to the tank in horizontal and vertical directions. We compose a system of differential equations of fluid motion based on the boundary condition on the free surface Substituting equations (5.1) to equation (5.2) we obtain It should be noted that for z=H we obtain the following system of differential equations of the second order: It should be noted that ) Ф , ( l x =0, for every modes except i=2. Therefore considering the following initial data:            Fig. 6(c) corresponds to mutual action of horizontal and vertical excitations with abovementioned parameters. In this case the vibration process is unstable, both vertical and horizontal excitations are essential. The phenomenon of parametrical resonance can be observed.

Conclusion
The method for estimation of liquid vibration in prismatic reservoir under action of periodic horizontal and vertical loads is developed. The nature of the behavior of the liquid in the reservoir is established depending on the frequency of the driving forces. The effects of instability are investigated. The effect of parametrical resonance at vertical excitation frequency equals to double own frequency of liquid vibrations is considered. The most dangerous liquid vibrations occur when the frequency of horizontal excitation coincides to own frequency of liquid vibrations, and the vertical excitation frequency equals to double value of own frequency.