Time-independent Finite Difference Research Articles

IMPROVEMENT IN CONVERGENT-TYPES OF MICROMIXERS BY THE USE OF PLATES

A time-independent finite difference method was used to solve for a fully nonlinear sloshing fluid in a tank mounted on a structure. The interaction between the sloshing fluid and the dynamic response of the structure was investigated. The dynamic response of the structure was calculated using the fourth-order Runge-Kutta method, and the numerical model was extensively validated through comprehensive data comparisons. The tank was tuned to an appropriate depth to length ratio and was used as a tuned liquid damper. Several cases, including large and small structures, were studied. In addition, a vibration control analysis of the tank-structure system in a real earthquake scenario is described in this paper. Vibration control was tested in terms of reduction in the dynamic response of the structure and reduction in the maximum energy development in the tank-structure system during excitation. The response of the tank-structure system was sensitive to the excitation frequency and water depth in the fluid tank.

Hydrodynamic forces induced by transient sloshing in a 3D rectangular tank due to oblique horizontal excitation

A time-independent finite difference method is developed to simulate fluid sloshing in a three-dimensional tank. The developed numerical scheme is verified by the rigorous benchmark tests. The experiment measurement of liquid sloshing in a 3D tank was also carried out in this study to further validate the accuracy of the present numerical results. Transient waves can change their types naturally in the time domain, especially for a tank excited by resonant frequencies. In this study, if the excitation frequencies are far from the fundamental natural frequency and there are four types of stable sloshing waves discovered due to the oblique horizontal excitation: “diagonal”, “single-directional”, “square-like”, and “irregular” waves. Besides, the swirling waves can only be generated for a partially-filled tank excited at near resonant frequency with oblique horizontal excitation. The evolution of forces induced by different sloshing waves acting on the tank walls is calculated and discussed in this work. In addition, the dynamics of sloshing force induced by swirling waves are explored in detail. The force of the single-directional waves acting on the tank bottom is time-invariant but the other types of sloshing waves show a beating phenomenon which is attributed to the momentum flux across the free surface and the vertical inertia of sloshing fluid. The effect of various oblique excitation directions of the tank on liquid sloshing is discussed as well. The horizontal hydrodynamic force of sloshing waves acting on the mid-section of the left wall of the tank is dominated by the added mass effect if the external excitation frequency is larger than 4 times the lowest natural frequency (ω1) of the tank with partially-filled fluid. On the other hand, the wave elevation of sloshing waves plays a key effect on the horizontal sloshing-induced force when the excitation frequency of the tank is less than 4ω1. A novel mechanism is presented to describe the phenomenon of alternate switch directions of the swirling waves. The relationship between the external force and the sloshing hydrodynamic force is the major factor to trigger the switch direction of the swirling waves. The influence of different base ratios of a rectangular tank on kinematic and dynamic responses of sloshing fluid is also explored in this work.

Time-Independent Finite Difference and Ghost Cell Method to Study Sloshing Liquid in 2D and 3D Tanks with Internal Structures

AbstractA finite difference scheme with ghost cell technique is used to study viscous fluid sloshing in 2D and 3D tanks with internal structures. The Navier-Stokes equations in a moving coordinate system are derived and they are mapped onto a time-independent and stretched domain. The staggered grid is used and the revised SIMPLEC iteration algorithm is performed. The developed numerical model is rigorously validated by extensive comparisons with reported analytical, numerical and experimental results. The present numerical results were also validated through an experiment setup with a tank excited by an inclined horizontal excitation or a tank mounted by a vertical baffle. The method is then applied to a number of problems including sloshing fluid in a 2D tank with a bottom-mounted baffle and in a 3D tank with a vertical plate. The phenomena of diagonal sloshing waves affected by a vertical plate are investigated in detail in this work. The effects of internal structures on the resonant frequency of a tank with liquid are discussed and the present developed numerical method can successfully analyze the sloshing phenomenon in 2D or 3D tanks with internal structures.

Analysis on Shift of Nature Modes of Liquid Sloshing in a 3D Tank Subjected to Oblique Horizontal Ground Motions with Damping Devices

The extended study of Wu et al. (2012) of sloshing fluid in tanks with internal structures from 2D to 3D is presented in the paper. The phenomenon of liquid sloshing in a 3D tank with various damping devices is solved by the time-independent finite difference method combined with the ghost (fictitious) cell approach. Two types of damping devices, a tank bottom-mounted baffle and a vertically surface-piercing plate, are considered in the study. In this work, the experimental measurement of liquid sloshing in a 3D tank with the baffle is carried out to further validate the present simulation. The comparison of the results between the experimental measurement and the present computation shows good accuracy. The effect of the vertically tank bottom-mounted baffle or the vertically surface-piercing plate on various sloshing waves for the tank under horizontal oblique excitation is discussed and investigated. The phenomena of the shift of the nature frequency of the tank with damping devices due to various oblique horizontal excitations under different sloshing waves are presented in detail. The sloshing wave type is varied due to the influence of the baffle or the plate, and the coexistence of two types of sloshing waves is found for the tank under larger excitation frequencies.

Numerical study of sloshing liquid in tanks with baffles by time-independent finite difference and fictitious cell method

The numerical analysis of liquid sloshing in tanks is a big challenge when the fully nonlinear and viscous effects are all included in the analysis. The analysis will become more complicate as the tank is attached with internal structures, such as baffles. The width of the baffle is very thin compared with the breadth length and the numerical technique used to capture the detailed flow phenomenon (vortex generation and shedding) around the baffle is very rare in the literatures. In this paper, a time-independent finite difference scheme with fictitious cell technique is used to study viscous fluid sloshing in 2D tanks with baffles. The Navier–Stokes equations in a moving coordinate system are derived and they are mapped onto a time-independent and stretched domain. The developed numerical model is rigorously validated by extensive comparisons with reported results. An experiment setup was also made to validate the present numerical sloshing results in a tank with baffles. The method is applied to a number of problems including impulsive flow past a flat plate, sloshing fluid in a 2D tank with a surface-piercing baffle, sloshing fluid in 2D tanks with bottom-mounted baffles. The effects of baffles on the resonant frequency are discussed. The present developed numerical model can successfully analyze the sloshing phenomenon in 2D tanks with internal structures and can be easily extended to 3D model.

Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional (2D) tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bangfuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.

Effects of excitation angle and coupled heave–surge–sway motion on fluid sloshing in a three-dimensional tank

Sloshing waves in moving tanks is an important engineering problem, and most studies of this phenomenon have focused on tanks that are excited by forcing motion in a limited number of directions and with fixed excitation frequencies throughout the forcing. In practice, the excitation comprises multiple degree of freedom motion that potentially couples surge, sway, heave, pitch, roll, and yaw motions. In the present study, a time-independent finite difference method is used to simulate fluid sloshing in three-dimensional tanks filled to an arbitrary depth for various excitation frequencies and multiple degree of freedom motion. The numerical scheme developed here was verified by rigorous benchmark tests. The coupled motions of surge and sway are simulated for various excitation angles, frequencies and water depths. Five kinds of sloshing waves found under coupled surge–sway motions: diagonal, single-directional, square-like, swirling, and irregular waves. The effect of excitation angle on the frequency responses of different sloshing waves is analyzed and discussed in the present study. Further, the components of horizontal force of various sloshing waves are also presented. The coupled effect of surge, sway and heave motions is also discussed, and the results show that unstable sloshing occurs when the excitation frequency of the heave motion is twice the fundamental natural frequency. Moreover, the effects of heave motion on the different types of sloshing waves are explored. It is found that heave motion causes all of the sloshing waves to change type.

A TIME-INDEPENDENT FINITE DIFFERENCE ANALYSIS OF VISCOUS FLOW ACROSS A TRANSLATING CIRCULAR CYLINDER

A time-independent finite difference scheme is developed and applied to analyze the viscous flow past a translating circular cylinder. In this scheme, the instantaneous moving cylinder face is mapped to a fixed boundary and the computational domain is, therefore, time independent. Large Reynolds number and high oscillating frequency of moving cylinder are used in the analysis. The finite difference approximation and algorithm were validated by the reported numerical simulation and flow visualization. The vorticity and streamline patterns developments were described and discussed. The surface vorticity distribution and position of separation point versus phases of oscillating stages were discussed. The flow behaviors of various amplitudes of exciting velocity and frequencies of moving cylinder are simulated and compared. The in-line forces on cylinder during cylinder oscillation were calculated and are in good agreement with those predicted by Morison’s equation. The developed scheme can be extended to analyze the viscous flow passing an arbitrary moving cylinder.

Sloshing waves and resonance modes of fluid in a 3D tank by a time-independent finite difference method

A 3D time-independent finite difference method is developed to solve for wave sloshing in a three-dimensional tank excited by coupled surge and sway motions. The 3D equations of fluid motion are derived in a moving coordinate system. The three-dimensional tank, with an arbitrary depth and a square base, is subjected to a range of excitation frequencies with motions that exhibit multiple degrees of freedom. For demonstration purposes the numerical scheme is validated by a benchmark study. Five types of sloshing waves were observed when the tank is excited by various excitation frequencies. A spectral analysis identified the resonant frequencies of each type of wave and the results show a strong correlation between resonant modes and the occurrence of the sloshing wave types. The method can be used to simulate fluid sloshing in a 3D tank with six-degrees of freedom.

Time-independent finite difference analysis of fully non-linear and viscous fluid sloshing in a rectangular tank

A novel, time-independent finite-difference method for analyzing complete two-dimensional sloshing motion (surge, heave and pitch) in a tank has been developed based on the primitive 2D Navier-Stokes equations. Both the fully non-linear free surface condition and fluid viscosity are included. The boundary of the tank is mapped onto a fixed square domain through proper mapping functions and stretched meshes are employed near boundaries in order to more accurately evaluate the large disturbance of fluid along the boundary. The sloshing displacement agrees well with previously published results. The maximum transient amplitude is much larger than that of the steady-state. Clear beating phenomenon can be found when the tank is excited by near resonance frequency. The frequency dependence and Reynolds number effects are studied. For a fixed forcing-function amplitude, the sloshing response is greatest near resonance. An analysis under coupled surge and pitch motions is also made. The coupling effect is significant and simultaneous surge, heave and pitch motions should be included in the tank sloshing analysis. A simple formula is derived to approximate the horizontal force coefficient, CF, on the tank walls. The formula implies that CF is dominated by the free surface displacement when the tank is excited by small surge frequencies. Whereas CF is attributed to added mass effects when the tank is under higher surge frequency forcing. A power spectra analysis is made to analyze the time series of sloshing displacement. For lower frequency of excitation, the system presents two peaks corresponding to the forcing frequency and fundamental frequency of the system. For higher frequency of excitation, the system shows only one major peak at the fundamental frequency. The limitations of the proposed method are also discussed.

Numerical Study of Tidal Effects on Seawater Intrusion in Confined and Unconfined Aquifers by Time-Independent Finite-Difference Method

This study presents a two-dimensional time-independent finite difference model to simulate tidal effects on the intrusion of seawater in either a confined or phreatic aquifer. The model considers a sloped beach face. The irregular beach face and varying ground water surface are mapped to a square domain and the time dependence of the moving boundary is removed by an appropriate coordinate transformation. The finite-difference expression of the governing equations is, therefore, written in a time-independent mesh system, and a time-independent finite difference scheme is established. The numerical technique is confirmed by the reported results of the traditional Henry's and modified Henry's problems. Idealized saltwater intrusion, known as Henry's problem, was extended to simulate seawater intrusion on sloped beach faces accounting for tidal effects. Tidal effects enhance the migration of seawater. The dominant feature of a confined aquifer is that seawater intrudes inland during the flood tide. The saltwater continues to intrude further inland and reaches its furthest upstream position at ebb tide. In an unconfined aquifer, the tidal fluctuation causes the variation of the intrusion profile in the vertical direction only. The fluctuation of the water table oscillates according with the fluctuating tide with a time lag, but the influence of the tide is damped out with horizontal distance from the beach. The variation of the distance through which the seawater intrudes also oscillates with the tide, with a constant time lag of 0.25T, where T is the tidal period. The shape of the aquifer also affects the intrusion of saltwater and the velocity of intrusion. The rate of seawater intrusion and the distance through which the seawater intrudes increases with the slope of the bank.

Complete 2D and Fully Nonlinear Analysis of Ideal Fluid in Tanks

A two-dimensional rectangular tank with rigid walls is used as the example for presenting the concepts of fully nonlinear and complete dimensional analysis. The container is subjected to simultaneous action of horizontal and vertical ground accelerations. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. The Euler equations as well as the nonlinear, kinematic free-surface boundary condition are used in the analysis. A time-independent finite difference method is used to evaluate the velocity fields and the surface profile of the sloshing fluid. The significance of the nonlinear effect on hydrodynamic force on tank face is studied. The comparison of numerical results evaluated by the complete two-dimensional analysis and the linear superposition (invalid when high nonlinearity is presented) of results evaluated by separated ground exciting is made. Details of the characteristics of nonlinear sloshing as well as of the beating phenomena are described. The occurrence of resonance is also studied.