Simulation of sloshing motions in fixed and vertically excited containers using a 2-D inviscid σ-transformed finite difference solver (original) (raw)
Related papers
Nonlinear Sloshing in Fixed and Vertically Excited Containers
5th International Symposium on Fluid Structure International, Aeroeslasticity, and Flow Induced Vibration and Noise, 2002
ABSTRACT Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.
Nonlinear Free-Surface and Viscous-Internal Sloshing in 2-D Numerical Wave Tanks
Volume 3: Materials Technology; Ocean Engineering; Polar and Arctic Sciences and Technology; Workshops, 2003
This paper examines free-surface and internal-pycnocline sloshing motions in 2-D numerical wave tanks subjected to horizontal base excitation. In all of the cases studied, the rectangular tank of liquid has a width-to-depth ratio of 2. The first set of results are based on an inviscid, fully nonlinear finite difference free-surface model. The model equations are mapped from the physical domain onto a rectangular domain. Case studies at and off resonance are presented illustrating when linear theory is inadequate. The next set of results are concerned with analyzing internal waves induced by sloshing a density-stratified liquid. Nonlinear, viscous flow equations are solved. The influence of the side-wall boundary layers on sloshing motions as well as the onset of internal breaking of the primary sloshing mode are discussed. The frequencies that characterize the motion of internal waves are also reported.
Transient Response of Sloshing Fluid in a Three Dimensional Tank
Journal of marine science and technology, 2012
Sloshing waves in moving tanks have been studied numerically, theoretically and experimentally in the past several decades. Most reported studies have been for tanks excited by forcing motion in a limited number of directions and with fixed excitation frequencies throughout the forcing. In the present study, a time-independent finite difference method is used to simulate fluid sloshing in the three-dimensional tanks with arbitrary depths and the tanks are subject to a range of excitation frequencies with motions that exhibit multiple degrees of freedom. The developed numerical scheme is verified by rigorous benchmark tests, and the advantage and efficiency of the method is also discussed. The wave motions that arise for a variety of water depths and a range of excitation frequencies are presented and discussed. The coupled motions of surge and sway are simulated with various excitation angles and frequencies. The 'diagonal', 'single-directional', 'square-like', 'swirling' and 'chaotic' waves are successfully obtained in this study and the transient response of sloshing waves in the tank is discussed in detail.
Classification of three-dimensional nonlinear sloshing in a square-base tank with finite depth
Journal of Fluids and Structures, 2005
The paper classifies steady state three-dimensional resonant waves in a square-base tank by using the asymptotic modal system proposed by the authors in 2003. The effective frequency domains of stable steady state motions are analysed versus mean fluid depths and forcing amplitude. The results are validated by experiments both qualitatively and quantitatively. r (O.M. Faltinsen). related to this study are reviewed by and . Three different approaches to theoretical sloshing modelling are distinguished. One of them focuses on low-order asymptotic mathematical theories and appropriate Hamiltonian formalism for the system of ordinary differential equations governing the dominating standing waves. Another approach is based on computational fluid dynamics (CFD) [see surveys by , Gerrits (2001), Celebi and Akyildiz ]. The third approach deals with multimodal/pseudospectral methods. Such methods are able to provide in different versions both analytical and numerical studies and 'build a bridge' between the first and second ones. All three approaches have their advantages and disadvantages from mathematical, physical and engineering points of view outlined in details by the mentioned surveys. Links, common features and differences should be demonstrated. This is a difficult problem with regard to the lower-order mathematical theories (first approach) and modal/pseudospectral methods [see some details given by ; Hill ]. Both approaches reduce the original free boundary problem to systems of ordinary differential equations with finite nonlinear kernel and often focus on nonlinear steady state waves. The difference is that the modal methods account for the full set of activated modes and their arbitrary initial perturbation, while the first approach studies the behaviour of the leading modes. Generally speaking, the multidimensional modal approach is more general, because under some additional asymptotic assumptions a corresponding low-order Hamiltonian system can be derived from the modal systems. The opposite is not true. AlthoughHill presented a version of single-dominant theory of two-dimensional sloshing, where the behaviour of some higher modes can be restored, his scheme is invalid for arbitrary initial conditions (for the higher modes) and requires the single harmonic forcing in a very small vicinity of the primary resonance.
Sloshing effects in periodically and seismically excited tanks
Proceedings of the 5th Intl. World Congress on Computational Mechanics
A fully nonlinear finite difference model has been developed based on inviscid flow equations. Numerical experiments of sloshing wave motion are undertaken in a 2-D tank which is moved both horizontally and vertically. Results of liquid sloshing induced by harmonic and earthquake base excitations are presented for small to steep non-breaking waves for steepness up to 0.3. Good agreement for small horizontal forcing amplitude is achieved between the numerical model and first order small perturbation theory. For large horizontal forcing, nonlinear effects are captured by the numerical model. The effect of the simultaneous vertical and horizontal excitation in comparison with the pure horizontal motion is examined. It is shown that vertical excitation causes the instability of the combined motion for a certain set of frequencies and amplitudes of the vertical motion. It is also found that in addition to the resonant frequency of the pure horizontal excitation, two additional resonance frequencies exist due to the combined motion of the tank. The dependence of the nonlinear behaviour of the solution on the wave steepness is discussed. It is found that for the present problem nonlinear effects become important when the steepness reaches about 0.1.
Journal of Computational Physics, 2005
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 nonlinear 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, C F , on the tank walls. The formula implies that C F is dominated by the free surface displacement when the tank is excited by small surge frequencies. Whereas C F 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.
The semi-Lagrangian procedure is widely used for updating the fully-nonlinear free surface in the time domain. However, this procedure is only available to cases when the body surface is vertical near the waterline. Present study introduces an improved semi-Lagrangian procedure which removes this ‘vertical-wall’ limitation. Coupling with the boundary element method, the improved semi-Lagrangian procedure is applied to the simulation of fully-nonlinear sloshing waves in non-wall-sided tanks. From the result comparison with the open source CFD software OpenFOAM, it is confirmed that this numerical scheme could guarantee a sufficient accuracy. Further series studies on 2D and 3D fully-nonlinear sloshing waves in wedged tanks are performed. Featured phenomena are observed which are distinct from those in wall-sided tanks. Application of an improved semi-Lagrangian procedure to fully-nonlinear simulation of sloshing in non-wall-sided tanks. Available from: https://www.researchgate.net/publication/274193712\_Application\_of\_an\_improved\_semi-Lagrangian\_procedure\_to\_fully-nonlinear\_simulation\_of\_sloshing\_in\_non-wall-sided\_tanks [accessed May 14, 2015].
Tank Sloshing Interaction With Elastic Support Structure
23rd International Conference on Offshore Mechanics and Arctic Engineering, Volume 3, 2004
ABSTRACT A fully nonlinear 2-D σ-transformed finite difference solver has been developed based on inviscid flow equations in rectangular tanks. The fluid equations are coupled to an elastic support structure. Sloshing motion are simulated during structural vibration cycles at and outside resonance. The wave tank acts as a Tuned Liquid Damper (TLD). The TLD response is highly nonlinear due to the liquid sloshing. The solver is valid at any water depth except for small depth when shallow water waves and viscous effects would become important. Results of liquid sloshing induced by horizontal base excitations are presented for small to steep non-breaking waves. The effectiveness of the TLD is discussed through predictions of coupling frequencies of the tank-structural system for different tank sizes and mass ratios between fluid and structure. Good agreement is achieved between numerical model and first-order theory. It was found that the system response is extremely sensitive to small changes in forcing frequency. Furthermore, the solver removes the need for free-surface smoothing for the cases considered herein. The numerical model provides a quick and accurate way of determining system eigenfrequencies which can be hard to identify and interpret in physical experiments. Therefore the numerical solver could serve as a valuable guidance to physical experiments. The present studies can easily be expanded to include multiple wave tanks to investigate tank interaction effects, and thus cover suppression of a wider range of frequencies.
Sloshing motions in excited tanks
Journal of Computational Physics, 2004
A fully non-linear finite difference model has been developed based on inviscid flow equations. Numerical experiments of sloshing wave motion are undertaken in a 2-D tank which is moved both horizontally and vertically. Results of liquid sloshing induced by harmonic base excitations are presented for small to steep non-breaking waves. The simulations are limited to a single water depth above the critical depth corresponding to a tank aspect ratio of h s =b ¼ 0:5. The numerical model is valid for any water depth except for small depth when viscous effects would become important. Solutions are limited to steep non-overturning waves. Good agreement for small horizontal forcing amplitude is achieved between the numerical model and second order small perturbation theory. For large horizontal forcing, nonlinear effects are captured by the third-order single modal solution and the fully non-linear numerical model. The agreement is in general good, both amplitude and phase. As expected, the third-order compared to the second-order solution is more accurate. This is especially true for resonance, high forcing frequency and mode interaction cases. However, it was found that multimodal approximate forms should be used for the cases in which detuning effects occur due to mode interaction. We present some test cases where detuning effects are evident both for single dominant modes and mode interaction cases. Furthermore, for very steep waves, just before the waves overturn, and for large forcing frequency, a discrepancy in amplitude and phase occurs between the approximate forms and the numerical model. The effects of the simultaneous vertical and horizontal excitations in comparison with the pure horizontal motion and pure vertical motion is examined. It is shown that vertical excitation causes the instability associated with parametric resonance of the combined motion for a certain set of frequencies and amplitudes of the vertical motion while the horizontal motion is related to classical resonance. It is also found that, in addition to the resonant frequency of the pure horizontal excitation, an infinite number of additional resonance frequencies exist due to the combined motion of the tank. The dependence of the non-linear behaviour of the solution on the wave steepness is discussed. It is found that for the present problem, non-linear effects become important when the steepness reaches about 0.1, in agreement with the physical experiments of Abramson [Rep. SP 106, NASA, 1966].