Vortex damping of sloshing in tanks with baffles

The problem of damping the sloshing in tanks with sharp-edged baffles (thin inserts which partially span a longitudinal or transverse cross-section) is considered. Separation of the boundary layer and the formation of vortices occur at these sharp edges. It is assumed that the domains where there is significant vortex motion of the fluid are localized in small neighbourhoods of the sharp edges of the baffles. The non-linear vortex damping is determined from the distribution of the velocity intensity factors at these sharp edges in the same way as the linear damping, caused by the dissipation of energy in a boundary layer close to a wall, is determined from the fluid velocity distribution on the walls of a cavity. Both of the above-mentioned distributions are calculated by solving the same boundary-value problem on the oscillations of an ideal fluid. The second of the distributions characterizes the singular properties of the solutions of this problem on particular lines. A method based on the variation of the area of the baffles, which simplifies the calculation of the velocity intensity factors is described. The distinctive features arising when the method of finite elements is used are considered. The results of numerical calculations of the damping of sloshing in a cylindrical tank with a ring baffle are compared with experimental data.     

拓扑优化技术在抑制流体晃荡中的数值模拟研究

流体晃荡问题广泛存在于船舶与海洋工程领域,任何部分载液的储罐运载装备在运动过程中均存在晃荡问题.当外界激励频率接近液舱内流体自由液面的固有频率时,很容易产生剧烈的晃荡,产生极大的冲击力,进而引起结构损害.因此,研究有效的减晃方案,以抑制流体晃荡带来的冲击具有重要意义.该文研究了基于自主研制的数值程序模拟长方体液舱内的流体晃荡问题.该数值程序采用有限差分法求解均质不可压缩的三维非定常Navier-Stokes方程,利用VOF/PLIC方法对自由液面进行捕捉,并结合基于最优控制理论的拓扑优化程序对液舱内隔板进行优化设计.数值计算了液舱内固定形状的双隔板以及拓扑优化的双隔板的晃荡问题,分析了增设双隔板后流场的运动学和动力学特性.结果表明,拓扑优化后的双隔板抑制流体晃荡的效果更好,为船舶与海洋工程领域和航空航天领域中的晃荡问题提供了一种新的研究思路.     

An analytical solution for free liquid sloshing in a finite-length horizontal cylindrical container filled to an arbitrary depth

A general series-type theoretical formulation based on the linearized potential theory, the method of separation of variables, and the translational addition theorem for cylindrical Bessel functions is developed to study three-dimensional natural sloshing in a partially filled horizontally-mounted circular cylindrical tank of finite span. Assuming time-harmonic variations, the potential solutions associated with the Symmetric/Antisymmetric (S/A) modes of free liquid surface oscillations are first analytically expanded as series of bounded spatial functions with unknown modal coefficients. The impenetrability conditions of the rigid end-plates along with the free surface dynamic/kinematic boundary condition are then imposed. The zero-normal-velocity requirement of the lateral tank boundary is subsequently applied by innovative use of Graf's translational addition theorem for modified cylindrical Bessel functions. After truncation, four independent sets of homogeneous algebraic equations are obtained that are then numerically worked out for the natural sloshing eigen-frequencies and free surface oscillation mode shapes. Extensive numerical data include the first thirty six longitudinal/transverse Antisymmetric/Symmetric (AA, SA, AS, SS) dimensionless sloshing frequencies, for a wide range of liquid fill depths and container span to radius ratios. Also, the influence of fill depth on the free surface oscillation mode shapes is addressed through selected 2D images. Comprehensive numerical simulations illustrate the strong effects of container length and liquid fill depth on the calculated sloshing frequencies. It is revealed that the frequency branches with the same transverse mode number form a cluster that progressively merge together amid the tank fill-depth limits as the tank span ratio increases. On the other hand, when the tank length substantially decreases, the number of “frequency cross-overs” between various frequency clusters at certain liquid fill depths considerably increases. Moreover, primary advantages of proposed methodology in comparison to other approximate/numerical methods are explicitly pointed out, convergence of solution is tested, and accuracy/reliability of the results is demonstrated by comparisons with available data.     

低重环境下液体非线性晃动的稳态响应

在俯仰激励作用下,圆柱贮箱中液体晃动存在平面运动、旋转运动和平面运动中的旋转运动等,而这些运动的稳定、不稳定区间的分界线与贮箱的半径、充液深度、重力强度、表面张力系数和晃动阻尼等基本系统参数有关．据此,首先建立了液体非线性晃动的微分方程组,并借助变分原理建立了液体压力体积分形式的Lagrange函数;然后将速度势函数在自由液面处作波高函数的级数展开,通过变分从而导出自由液面运动学和动力学边界条件非线性方程组;最后用多尺度法求解非线性方程组,就重力强度对圆柱形贮箱中液体非线性晃动的全局稳态响应的影响进行了详细的理论分析,并发现系统软硬特性的变化、跳跃和滞后等非线性现象．     

Dynamic responses of shear flows over a deformable porous surface layer in a cylindrical tube

This paper investigates dynamic responses of a viscous fluid flow introduced under a time dependent pressure gradient in a rigid cylindrical tube that is lined with a deformable porous surface layer. With the Darcy’s law and a linear elasticity assumption, we have solved the coupling effect of the fluid movement and the deformation of the porous medium in the Laplace transform space. Governing equations are deduced for the solid displacement and the fluid velocity in the porous layer. Analytical solutions in the transformed domain are derived and the time dependent variables are inverted numerically using Durbin’s algorithm. Interaction between the solid and the fluid phases in the porous layer and its effects on fluid flow in tube are investigated under steady and unsteady flow conditions when the solid phase is either rigid or deformable. Examples are presented for flows driven by a Heaviside or a sinusoid pressure gradient. Significant effects of the porous surface layer on the flow in the tube are observed. The analytical solutions can be used to test more complicated numerical schemes.     

Numerical modeling of baffle location effects on the flow pattern of primary sedimentation tanks

Computational fluid dynamics (CFD) is used extensively by engineers to model and analyze complex issues related to hydraulic design, planning studies for future generating stations, civil maintenance and supply efficiency. In order to find the optimal position of a baffle in a rectangular primary sedimentation tank, computational investigations are performed. Also laboratory experiments are conducted to verify the numerical results and the measured velocity fields which were by Acoustic Doppler Velocimeter (ADV) are used. The GMRES algorithm as a pressure solver was used in the computational modeling. The results of computational investigations performed in the present study indicate that the favorable flow field (uniform in the settling zone) would be enhanced for the case that the baffle position provide small circulation regions volume and dissipate the kinetic energy in the tank. Also results show that the GMRES algorithm can obtain the good agreement between the results of numerical models and experimental tests.     

Optimal rigid body motions,part 1: Approximate formulation

Optimal rigid body angular motions are investigated in the absence of direct control over one of the angular velocity components, via an approximate dynamic model. An analysis of first-order necessary conditions for optimality with the proposed model reveals that, over a large range of boundary conditions, there are, in general, several distinct extremal solutions. A classification in terms of subfamilies of extremal solutions is presented. Second-order necessary conditions are investigated to establish local optimality for the candidate minimizers.This work was supported in part by DARPA Contract No. ACMP-F49620-87-C-0116 and by Air Force Grant AFOSR-89-0001.     

Sloshing characteristics in half-full horizontal elliptical tanks with vertical baffles

The effects of surface-piercing or bottom-mounted vertical baffles on two-dimensional liquid sloshing characteristics in a half-full non-deformable horizontal cylindrical container of elliptical cross section is investigated. The problem solution is achieved by employing the linear potential theory in conjunction with the successive conformal transformation technique, leading to matrix eigenvalue problems on simple (rectangular and semi-infinite strip) regions. Plots of the resonant eigen-frequencies as a function of the baffle extension at selected container aspect ratios are presented and discussed for the three lowest antisymmetric and symmetric transverse oscillation modes. Also, the effects of vertical baffles on the hydrodynamic pressure mode shapes and sloshing flows are examined through appropriate 2D images. The surface-piercing vertical baffle is demonstrated to be an effective tool in reducing the antisymmetric sloshing frequencies, especially for lower aspect ratio tanks and higher modes. On the other hand, the bottom-mounted baffle is shown to have a great influence on the higher antisymmetric slosh modes only when its tip approaches the liquid free surface. Limiting cases are considered and good agreements with available analytic and numerical solutions as well as experimental data are obtained.     

Effects of Finite Water Depth on Natural Frequencies of Suspended Water Tanks

Linear sloshing problems (inviscid irrotational flows) in suspended tanks are revisited, with an intention to address some issues in the previous study based on shallow water wave theory. Time‐periodic solutions are considered, which describe the synchronized oscillation of the water and tank, reached after the initial transient dies out. The solutions are developed for arbitrary water depths, and separate explicitly the propagating and evanescent wave components of the fluid motion, illuminating clearly the physics and converging rapidly. At the limit of infinite string length, these solutions describe the sloshing motions in tanks that are free to oscillate horizontally on a frictionless plane. Various effects on the lowest sloshing mode are discussed, emphasizing the physical interpretations and examining the limitations of the shallow water approximations. Comparisons with existing laboratory experiments are made, showing agreements with the analysis.     

Attitude stabilization of a rigid body in conditions of decreasing dissipation

The paper presents the problem of triaxial stabilization of the angular position of a rigid body. The possibility of implementing a control system in which dissipative torque tends to zero over time and the restoring torque is the only remaining control torque is considered. The case of vanishing damping considered in this study is known as the most complicated one in the problem of stability analysis of mechanical systems with a nonstationary parameter at the vector of dissipative forces. The lemma of the estimate from below for the norm of the restoring torque in the neighborhood of the stabilized motion of a rigid body and two theorems on asymptotic stability of the stabilized motion of a body are proven. It is shown that the sufficient conditions of asymptotic stability found in the theorems are close to the necessary ones. The results of numerical simulation illustrating the conclusions obtained in this study are presented.     

On the fall of a heavy rigid body in an ideal fluid

We consider a problem about the motion of a heavy rigid body in an unbounded volume of an ideal irrotational incompressible fluid. This problem generalizes a classical Kirchhoff problem describing the inertial motion of a rigid body in a fluid. We study different special statements of the problem: the plane motion and the motion of an axially symmetric body. In the general case of motion of a rigid body, we study the stability of partial solutions and point out limiting behaviors of the motion when the time increases infinitely. Using numerical computations on the plane of initial conditions, we construct domains corresponding to different types of the asymptotic behavior. We establish the fractal nature of the boundary separating these domains.     

Waves in an inhomogeneous fluid in the presence of a dock : PMM vol. 39, n 6, 1975, pp. 1140–1142

We investigate the propagation of waves generated by oscillations of a section of the bottom of a tank through a two-layer fluid, in the presence of a dock. Wave motions in an inhomogeneous fluid generated by displacement of a section of the bottom of a tank were studied in [1] where the upper surface of the fluid was assumed either to be completely free, or completely covered with ice. In the present paper we use the method given in [2] to investigate a similar problem under the assumption that the fluid surface is partly covered with an immovable rigid plate. The expressions obtained for the velocity potential are used to determine the form of the free surface and of the interface. We show that when the fluid is inhomogeneous, the wave amplitude on the free surface increases, while the presence of a plate reduces the amplitude of the surface waves, as well as of the internal waves in the region between the plate and the oscillating section of the bottom.     

Seismic-generated unsteady motions in shallow basins and channels. Part II: Numerical modelling

In this paper, unsteady motions generated by seismic-type excitation are simulated by a 2D depth-averaged mathematical model based on the classic shallow water approximation. A suitable time-dependent forcing term is added in the governing equations, and these are solved by a MUSCL-type shock-capturing finite volume scheme with a splitting treatment of the source term. The HLL approximate Riemann solver is used to estimate the numerical fluxes. The accuracy of the numerical scheme is assessed by comparison with novel exact solutions of test cases concerning sinusoidally-generated sloshing in a prismatic tank, a rectangular open channel, and a parabolic basin. A sensitivity analysis is performed on the influence of the relevant dimensionless parameters. Moreover, numerical results are validated against experimental data available in literature concerning shallow water sloshing in a swaying tank. Finally, real‐scale applications to a reservoir created by a dam and an urban water-supply storage tank are presented. The results show that the model provides accurate solutions of the shallow water equations with a seismic-type source term and can be effectively adopted to predict the main flow features of the unsteady motion induced by horizontal seismic acceleration when the long wave assumption is valid.     

On the motion of rigid bodies in a viscous fluid

We consider the problem of motion of several rigid bodies in a viscous fluid. Both compressible and incompressible fluids are studied. In both cases, the existence of globally defined weak solutions is established regardless possible collisions of two or more rigid objects.     

Existence of Weak Solutions Up to Collision for Viscous Fluid‐Solid Systems with Slip

We study in this paper the movement of a rigid solid inside an incompressible Navier‐Stokes flow within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no‐slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents obtaining global H¹ bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions inappropriate. © 2014 Wiley Periodicals, Inc.     

Long Nonlinear Waves in Resonance with Topography

The evolution of periodic long surface waves over a periodic bottom topography resonant with the waves is studied. Coupled Korteweg–de Vries equations are derived and describe the evolution in terms of interaction between right- and left-traveling waves. The coupling arises from the cumulative effect of wave scattering. We discuss the various conserved quantities of the system and compute solutions for the initial value problem and for the time-periodic problem of fluid "sloshing" in a tank. Some three-dimensional extensions are discussed.     

Fluid-structure interaction of an elastic pipe immersed in a coaxial rigid pipe: a model for the Tullio Phenomenon

An axisymmetric, elastic pipe is filled with an incompressible fluid and is immersed in a second, coaxial rigid pipe which contains the same fluid. A pressure pulse in the outer fluid annulus deforms the elastic pipe which invokes a fluid motion in the fluid core. It is the aim of this study to investigate streaming phenomena in the core which may originate from such a fluid-structure interaction. This work presents a numerical solver for such a configuration. It was developed in the OpenFOAM software environment and is based on the Arbitrary Lagrangian Eulerian (ALE) approach for moving meshes. The solver features a monolithic integration of the one-dimensional, coupled system between the elastic structure and the outer fluid annulus into a dynamic boundary condition for the moving surface of the fluid core. Results indicate that our configuration may serve as a mechanical model of the Tullio Phenomenon (sound-induced vertigo). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modeling of the movement of a rigid particle in viscous fluid

Modeling the movement of a rigid particle in viscous fluid is a problem physicists and smathematicians have tried to solve since the beginning of this century. A general model for an ellipsoidal particle was first published by Jeffery in the twenties. We exploit the fact that Jeffery was concerned with formulae which can be used to compute numerically the velocity field in the neighborhood of the particle during his derivation of equations of motion of the particle. This is our principal contribution to the subject. After a thorough check of Jeffery's formulae, we coded software for modeling the flow around a rigid particle based on these equations. Examples of its applications are given in conclusion. A practical example is concerned with the simulation of sigmoidal inclusion trails in porphyroblast.     

Viscoelastic body colliding against a rigid wall with and without dry friction effects

The impact of a body against a rigid wall is studied. The body consists of a rigid block and a viscoelastic rod described by the distributed-order fractional model and in particular its solid-like and fluid-like special cases. Translatory motion of a body is studied in two cases: without and with the influence of dry friction. When present, dry friction is modeled by the Coulomb friction law. The problem is treated analytically by the use of the Laplace transform method and solutions are obtained in a convolution form.     

On conditions of existence of permanent rotations of connected rigid bodies system about vertical vector

In contrast to the classical problem of motion of a heavy rigid body about a fixed point where the permanent rotations are well known and completely investigated [7, 3] as the most simple and good visually demonstrated type of motions, in multibody mechanics under an increasing of quantity of the system bodies, mechanical parameters and the order of differential motion equations the study of such motions is more complicated problem. The problem on permanent rotations of two connected rigid bodies under influence of gravity force was investigated in [2, 4]. In this paper a system consisting of arbitrary constant quantity, n, of heavy rigid bodies which are sequentially jointed in a chain is considered. The conditions of existence of motions when each body permanently rotates about the vertical vector are determined. These conditions are analyzed in a general case when the bodies angular velocities are different.