Investigation of nonlinear sloshing effects in seismically excited tanks

Nonlinear behavior of liquid sloshing inside a partially filled rectangular tank is investigated. The nonlinearity in the numerical modeling of the liquid sloshing originates from the nonlinear terms of the governing equations of the fluid flow and the liquid free surface motion as a not known boundary condition. The numerical simulations are performed for both linear and nonlinear conditions. The computed results using linear conditions are compared with readily available exact solution. In order to verify the results of the nonlinear numerical solution, a series of the shaking table tests on rectangular tank were conducted. Having verified linear and nonlinear numerical models, they are used for computation of near wall sloshing height at a series of real scale tanks (with various dimensions) under the both harmonic and earthquake base excitation. Finally, the nonlinear effects on liquid sloshing modeling are discussed and the practical limitations of the linear solution in evaluating the response of seismically excited liquids are also addressed.     

Balancing the source terms in a SPH model for solving the shallow water equations

A shallow flow generally features complex hydrodynamics induced by complicated domain topography and geometry. A numerical scheme with well-balanced flux and source term gradients is therefore essential before a shallow flow model can be applied to simulate real-world problems. The issue of source term balancing has been exhaustively investigated in grid-based numerical approaches, e.g. discontinuous Galerkin finite element methods and finite volume Godunov-type methods. In recent years, a relatively new computational method, smooth particle hydrodynamics (SPH), has started to gain popularity in solving the shallow water equations (SWEs). However, the well-balanced problem has not been fully investigated and resolved in the context of SPH. This work aims to discuss the well-balanced problem caused by a standard SPH discretization to the SWEs with slope source terms and derive a corrected SPH algorithm that is able to preserve the solution of lake at rest. In order to enhance the shock capturing capability of the resulting SPH model, the Monotone Upwind-centered Scheme for Conservation Laws (MUSCL) is also explored and applied to enable Riemann solver based artificial viscosity. The new SPH model is validated against several idealized benchmark tests and a real-world dam-break case and promising results are obtained.     

SPH simulation of free surface flow over a sharp-crested weir

In this paper the numerical simulation of a free surface flow over a sharp-crested weir is presented. Since in this case the usual shallow water assumptions are not satisfied, we propose to solve the problem using the full weakly compressible Navier–Stokes equations with the Tait equation of state for water. The numerical method used consists of the new meshless Smooth Particle Hydrodynamics (SPH) formulation proposed by Ferrari et al. (2009) [8], that accurately tracks the free surface profile and provides monotone pressure fields. Thus, the unsteady evolution of the complex moving material interface (free surface) can been properly solved. The simulations involving about half a million of fluid particles have been run in parallel on two of the most powerful High Performance Computing (HPC) facilities in Europe. The validation of the results has been carried out analysing the pressure field and comparing the free surface profiles obtained with the SPH scheme with experimental measurements available in literature [18]. A very good quantitative agreement has been obtained.     

大型矩形渡槽-水耦合体系的动力性能分析

本文把任意拉格朗日-欧拉(ALE)方法用于分析大型矩形渡槽结构-水耦合体动力性能,并针对不同的水位、不同截面深宽比,研究渡槽结构在谐波激励、地震激励下的振动反应。研究表明,渡槽结构中水体的泼溅具有明显的非线性特征;渡槽中水位对耦合体动力特性影响极具规律性,但是随截面深宽比减小,水体振荡的非线性明显增强;渡槽结构截面深宽比选择合理,水体泼溅作用产生的倾覆力矩不会使槽体发生倾覆失稳,但是水体的大幅晃动产生的动水压力对渡槽槽身应力有显著的影响,在渡槽抗震设计中应给予足够重视。     

LARGE AMPLITUDE LIQUID SLOSHING IN SEISMICALLY EXCITED TANKS

An innovative method of analysis was developed to simulate the non-linear seismic finite-amplitude liquid sloshing in two-dimensional containers. In view of the irregular and time-varying liquid surface, the method employed a curvilinear mesh system to transform the non-linear sloshing problem from the physical domain with an irregular free-surface boundary into a computational domain in which rectangular grids can be analysed by the finite difference method. Non-linearities associated with both the unknown location of the free surface and the high-order differential terms were considered. The Crank–Nicolson time marching scheme was employed and the resulting finite difference algorithm is unconditionally stable and very lightly damped with respect to the temporal co-ordinate. In order to minimize numerical instability caused by the computational dispersion in spatially discretized surface wave, a second-order dissipation term was added to the system to filter out the spurious high-frequency waves. Sloshing effects and structural response were measured in terms of sloshing amplitude, base shear and overturning moment generated by the hydrodynamic pressure of the liquid exerted on the container walls. Simulation results of liquid sloshing induced by earthquake and harmonic base excitations were compared with those of the linear wave theory and the limitations of the latter in assessing the response of seismically excited liquids were addressed.     

渡槽中流体非线性晃动的边界元模拟

应用边界元法分析了流体非线性晃动及其对渡槽的作用效应,将所得结果与线性解析结果作了比较．数值计算表明：流体以较大幅度晃动时,线性模型结果与非线性边界元结果有较大偏差,计算有限幅流体晃动反应宜采用非线性模型．     

Nonlinear sloshing in a floating‐roofed oil storage tank under long‐period seismic ground motion

A hybrid analytical and FEM is proposed to investigate the nonlinear sloshing in a floating‐roofed oil storage tank under long‐period seismic ground motion. The tank is composed of a rigid cylindrical wall and a flat bottom, whereas the floating roof is treated as an elastic plate undergoing large deflection. The contained liquid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. The method of analysis is based on representation of the liquid motion by superposing the analytical modes that satisfy the Laplace equation and the rigid wall and bottom boundary conditions. The FEM is then applied to solve the remaining kinematic and dynamic boundary conditions at the moving liquid surface coupled with the nonlinear equation of motion of the floating roof. This requires only the discretization of the liquid surface and the floating roof into finite elements, thus leading to a computationally efficient and accurate method compared with full numerical analysis. As numerical examples to illustrate the applicability of the proposed method, two oil storage tanks with single‐deck type floating roofs damaged during the 2003 Tokachioki earthquake are studied. It is shown that the nonlinear oscillation modes with the circumferential wave numbers 0, 2 and 3 caused by the finite liquid surface elevation as well as the membrane action due to large deflection of the deck produce excessively large stresses in the pontoon, which may cause the catastrophic failure of pontoon followed by the submergence of the roof. Copyright © 2012 John Wiley & Sons, Ltd.     

Lagrangian simulations of unstable gravity-driven flow of fluids with variable density in randomly heterogeneous porous media

A new Lagrangian particle model based on smoothed particle hydrodynamics (SPH) is developed and used to simulate Darcy scale flow and transport in porous media. The method has excellent conservation properties and treats advection exactly. The Lagrangian method is used in stochastic analysis of miscible density-driven fluid flows. Results show that heterogeneity significantly increases dispersion and slows development of Rayleigh–Taylor instability. The presented numerical examples illustrate the advantages of Lagrangian methods for stochastic transport simulations.     

Investigation of pressure variations over stepped spillways using smooth particle hydrodynamics

In this paper the smoothed particle hydrodynamics, (SPH), technique is used to investigate the pressure distribution on steps located in the non-aerated flow region of a stepped spillway for different discharges typical of skimming flow conditions. The open source code 2D SPHysics has been employed after being validated against the laboratory model studies of flow over broad crested weirs and flow over stepped spillways. The numerical results, in terms of the water surface and velocity profiles at different sections, are in good agreement with the corresponding experimental results. The code is then applied to determine the pressure distribution on the vertical and horizontal step faces. Also, the aspects of the pressure pattern are described and the positions/magnitudes of the maximum and minimum pressure values are presented.     

An investigation of tuned liquid dampers equipped with damping screens under 2D excitation

This paper reports on the results of a study conducted on tanks partially filled with water, representing tuned liquid dampers (TLD), subjected to both 1D and 2D horizontal excitations. The sloshing response of the water in the tank is characterized by the free surface motion, the resulting base shear force, and evaluation of the energy dissipated by the sloshing water. A 1D non‐linear flow model capable of simulating a TLD equipped with damping screens is employed to model a 2D TLD. Application of this particular model requires the assumption that the response is decoupled and can be treated as the summation of two independent 1D TLDs. Results from the non‐linear flow model are compared with the 2D experimental shake table test results leading to a validation of the decoupled response assumption. This attractive decoupled response property allows square and rectangular tanks to be used as 2D TLDs, which can simultaneously reduce the dynamic response of a structure in two perpendicular modes of vibration. Copyright © 2005 John Wiley & Sons, Ltd.     

Shallow water SPH for flooding with dynamic particle coalescing and splitting

In this paper an adaptive algorithm for Smoothed Particle Hydrodynamics (SPH) for the Shallow Water Equations (SWEs) is presented. The area of a particle is inversely proportional to depth giving poor resolution in small depths without particle refinement. This is a particular limitation for flooding problems of interest here. Higher resolution is created by splitting the particles, while particle coalescing (or merging) improves efficiency by reducing the number of the particles when acceptable. The new particle coalescing procedure merges two particles together if their area becomes less than a predefined threshold value. Both particle splitting and coalescing procedures conserve mass and momentum and the smoothing length of new particles is calculated by minimizing the density error of the SPH summation. The new dynamic particle refinement procedure is assessed by testing the numerical scheme against analytical, experimental and benchmark test cases. The analytical cases show that with particle splitting and coalescing typical convergence rates remain faster than linear. For the practical test case, in comparison to using particle splitting alone, the particle coalescing procedure leads to a significant reduction of computational time, by a factor of 15. This makes the computational time of the same order as mesh-based methods with the advantage of not having to specify a mesh over a flood domain of unknown extent a priori.     

Weakly nonlinear approximation of periodic flow in phreatic aquifers

Published frequency-domain solutions of periodic flow in aquifers apply strictly to mathematically linear systems that arise when aquifer diffusivity is assumed constant in space and time. This assumption can be invalid in phreatic aquifers that experience spatiotemporal variation in the free surface position and consequent variation in saturated thickness. A weakly nonlinear approach to formulating and solving periodic flow problems in the frequency domain can be applied in situations where conventional linearized approximations break down. The weakly nonlinear equations provide robust approximations of the true nonlinear response and require much less computational effort and time to solve than the full nonlinear problem. Nondimensional rules of thumb are presented for choosing between linear, weakly nonlinear and nonlinear solution strategies.     

A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces

Flow on fracture surfaces has been identified by many authors as an important flow process in unsaturated fractured rock formations. Given the complexity of flow dynamics on such small scales, robust numerical methods have to be employed in order to capture the highly dynamic interfaces and flow intermittency. In this work we use a three-dimensional multiphase Smoothed Particle Hydrodynamics (SPH) model to simulate surface tension dominated flow on smooth fracture surfaces. We model droplet and film flow over a wide range of contact angles and Reynolds numbers encountered in such flows on rock surfaces. We validate our model via comparison with existing empirical and semi-analytical solutions for droplet flow. We use the SPH model to investigate the occurrence of adsorbed trailing films left behind droplets under various flow conditions and its importance for the flow dynamics when films and droplets coexist. It is shown that flow velocities are higher on prewetted surfaces covered by a thin film which is qualitatively attributed to the enhanced dynamic wetting and dewetting at the trailing and advancing contact lines. Finally, we demonstrate that the SPH model can be used to study flow on rough surfaces.     

THE USE OF VARIATIONAL METHODS AND OF ERROR DISTRIBUTION PRINCIPLES IN GROUNDWATER HYDRAULICS

Abstract

The aim of the present paper is to present some mathematical techniques for the solution of problems connected with three-dimensional steady-state groundwater flow with a free surface. The validity of Darcy's law is assumed. As no use is made of the Dupuit-Forschheimer approximation, the shape of the free surface and the velocity potential must be determined simultaneously from a non-linear boundary value problem. In order to demonstrate the use of a variational method and of error distribution principles for the solution of those problems by an example as simple as possible, we investigate the gravity flow of incompressible, homogeneous groundwater towards a circular well completely penetrating an isotropic, homogeneous, inelastic aquifer resting on a horizontal, impermeable substratum.     

Coupling of finite element and optimization methods for the management of groundwater systems

The paper describes an optimization method for the solution of groundwater management problems. The method consists of a combination of the computation of horizontal plane groundwater flow with a free surface (finite element method) and a linear optimization procedure (simplex algorithm). Considering the special structure of data which result form computing the groundwater flow with the finite element method, and modifying the simplex algorithm, the solution of management problems with complex groundwater flow is realized without any difficulties. Compared to a flow computation alone the additional effort of the optimization (computer time and scope for data storage) is only small.     

Sloshing analysis of rectangular tanks with a submerged structure by using small-amplitude water wave theory

A new sloshing analysis method for rectangular tank systems with a submerged structure are proposed by using the velocity potential and the linear water wave theory. The velocity potential functions are obtained by decomposing the surface wave into a wall-induced wave, reflected and transmitted waves, and a scattered wave. A simplified method using a response spectrum for zero damping is also proposed. The results of the simplified method are in good agreement with those of the analytical method. The sloshing response of the fluid-structure system is found to be very sensitive to the characteristics of the ground motion and the configuration of the system. Under typical earthquakes, the submerged structure shows a tendency to decrease sloshing amplitude, hydrodynamic pressure, and base shear, while it shows a tendency to increase the overturning moment. For the ground excitation dominated by low-frequency contents, the sloshing response increases significantly and the contribution of the higher sloshing modes increases. Copyright © 1999 John Wiley & Sons, Ltd.     

大型三维矩形、U形渡槽地震响应对比分析

本文应用任意拉格朗日-欧拉(ALE)有限元方法对比分析大型三维排架支撑矩形、U形渡槽结构地震响应,以获得渡槽中动水压力资料。研究表明,渡槽中水体的晃动幅度十分显著,水体的大幅晃动显著地改变两种截面形式渡槽中水压力、槽体应力值及U形渡槽动水压力和应力分布,对矩形渡槽动水压力和应力分布规律影响不大。动应力分布的改变直接影响到大型预应力U形截面渡槽结构中预应力钢筋的有效性。在渡槽的抗震设计中,应充分考虑U形截面渡槽动水压力和动应力分布变化问题,动水压力应该作为一项主要的外荷载予以考虑。     

Incompressible SPH scour model for movable bed dam break flows

In this study an incompressible smoothed particle hydrodynamics (ISPH) approach coupled with the sediment erosion model is developed to investigate the sediment bed scour and grain movement under the dam break flows. Two-phase formulations are used in the ISPH numerical algorithms to examine the free surface and bed evolution profiles, in which the entrained sediments are treated as a different fluid component as compared with the water. The sediment bed erosion model is based on the concept of pick-up flow velocity and the sediment is initiated when the local flow velocity exceeds a critical value. The proposed model is used to reproduce the sediment erosion and follow-on entrainment process under an instantaneous dam break flow and the results are compared with those from the weakly compressible moving particle semi-implicit (WCMPS) method as well as the experimental data. It has been demonstrated that the two-phase ISPH model performed well with the experimental data. The study shows that the ISPH modelling approach can accurately predict the dynamic sediment scouring process without the need to use empirical sediment transport formulas.     

Simulation of submerged granular flows by development of a two-phase incompressible explicit mesh-free method

The interaction between fluid and sediment particles is widely involved in hydraulic engineering problems. In the current study, an explicit incompressible mesh-free method in the framework of the Moving Particle Semi-implicit(MPS) method is proposed to simulate the interaction between the two phases in submerged conditions. The proposed method solves two sets of the continuity and momentum equations, respectively, for the fluid phase and the sediment phase according to the mixture theory. In th...     

Smooth Particle Hydrodynamics with nonlinear Moving-Least-Squares WENO reconstruction to model anisotropic dispersion in porous media

Most numerical schemes applied to solve the advection–diffusion equation are affected by numerical diffusion. Moreover, unphysical results, such as oscillations and negative concentrations, may emerge when an anisotropic dispersion tensor is used, which induces even more severe errors in the solution of multispecies reactive transport. To cope with this long standing problem we propose a modified version of the standard Smoothed Particle Hydrodynamics (SPH) method based on a Moving-Least-Squares-Weighted-Essentially-Non-Oscillatory (MLS-WENO) reconstruction of concentrations. This scheme formulation (called MWSPH) approximates the diffusive fluxes with a Rusanov-type Riemann solver based on high order WENO scheme. We compare the standard SPH with the MWSPH for different a few test cases, considering both homogeneous and heterogeneous flow fields and different anisotropic ratios of the dispersion tensor. We show that, MWSPH is stable and accurate and that it reduces the occurrence of negative concentrations compared to standard SPH. When negative concentrations are observed, their absolute values are several orders of magnitude smaller compared to standard SPH. In addition, MWSPH limits spurious oscillations in the numerical solution more effectively than classical SPH. Convergence analysis shows that MWSPH is computationally more demanding than SPH, but with the payoff a more accurate solution, which in addition is less sensitive to particles position. The latter property simplifies the time consuming and often user dependent procedure to define the initial dislocation of the particles.