Low Steeping Waves Simulation in a Vertical Excited Container Using σ Transformation

2009 ◽  
Author(s):  
M. Eswaran ◽  
U. K. Saha
Keyword(s):  
Free Surface ◽  
Structural Integrity ◽  
Breaking Waves ◽  
Rectangular Domain ◽  
Practical Significance ◽  
Computational Domain ◽  
Road Transportation ◽  
Central Difference ◽  
Flow Equations ◽  
Vertical Excitation

A fluid partially occupying a moving tank undergoes wave motions (sloshing). These motions generate severe hydrodynamic loads that can be dangerous for structural integrity and stability of rockets, satellites, LNG ships, trucks and even stationary petroleum containers. Free surface motions of the liquid in partially filled tanks under gravity are of practical significance particularly in marine and road transportation applications. For this reason, liquid sloshing has always been a research subject attracting great concern during the last several decades. In this paper, a fully non-linear finite difference model has been developed based on the inviscid flow equations, and a simple mapping function was used to remove the time-dependence of the free surface in the fluid domain. The time-varying fluid surface can be mapped onto a rectangular domain by the σ-transformation. This method is a simple way to simulate non-breaking waves quickly and accurately especially that has a low steepness. The fluid motion is solved in a unit square mesh in the transformed flow domain (i.e., computational domain). The fourth order central difference scheme and the Gauss–Seidel point successive over-relaxation iterative procedure are used to capture the free surface wave profiles and the free surface elevation plots of the fluid domain. Difference between the peaks and troughs of waves are discussed for the case of vertical excitation of first three natural frequency of the tank. Phase-plane diagrams are drawn to show the non-linearity of the motion of time dependent free surface. The results agree well with the previously published results.

Waves Simulation in an Excited Cylindrical Tank Using σ-Transformation

2010 ◽  
Author(s):  
M. Eswaran ◽  
Ujjwal K. Saha
Keyword(s):  
Free Surface ◽  
Excitation Frequency ◽  
Time History ◽  
Breaking Waves ◽  
Liquid Sloshing ◽  
Practical Significance ◽  
Road Transportation ◽  
Viscous Effects ◽  
Liquid Free Surface ◽  
Small Depth

Free surface motions of the liquid in partially filled tanks under gravity are of practical significance particularly in marine and road transportation applications. For this reason, liquid sloshing has always been a research subject attracting great concern during the last several decades. Numerical experiments of sloshing wave motion are undertaken in a 2-D tank which is moved horizontally. Results of liquid sloshing induced by sinusoidal base excitations are presented for small to steep non-breaking waves. 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. In this paper, the semi-circular domain with time-varying fluid surface was mapped onto a rectangular domain by the σ-transformation. Based on the inviscid flow equations, a fully non-linear finite difference model has been developed. The simulations are limited to a half-filled container. The liquid free surface elevation and wave phase-plane diagram have been plotted for different tank excitation frequency. It has been observed that while increasing the tank frequency, the liquid wave height in the tank changes according to the system natural frequency. Finally, the proposed computational scheme has been applied to a real engineering problem to capture the irregular behavior of liquid free surface inside the tank. For this, acceleration-time history of EW and NS components of the EL-Centro earthquake, California has been studied and analyzed.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

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.

scholarly journals A numerically exact non-reflecting boundary condition applied to the acoustic Helmholtz equation

Geophysics ◽  
2021 ◽  
pp. 1-42
Author(s):  
Wim A. Mulder
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Helmholtz Equation ◽  
Matrix Equation ◽  
Direct Method ◽  
Rectangular Domain ◽  
Translation Invariance ◽  
Computational Domain ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix

When modeling wave propagation, truncation of the computational domain to a manageable size requires non-reflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to infinity. Constant extrapolation in the direction perpendicular to the boundary provides the material properties in the exterior. Domain decomposition can split the enlarged domain into the original one and its exterior. Because the boundary-value problem for the latter is translation-invariant, the boundary Green functions obey a quadratic matrix equation. Selection of the solvent that corresponds to the outgoing waves provides the input for the remaining problem in the interior. The result is a numerically exact non-reflecting boundary condition on one side of the domain. When two non-reflecting sides have a common corner, translation invariance is lost. Treating each side independently in combination with a classic absorbing condition in the other direction restores translation invariance and enables application of the method at the expense of numerical exactness. Solving the quadratic matrix equation with Newton's method turns out to be more costly than solving the Helmholtz equation and may select unwanted incoming waves. A proposed direct method has a much lower cost and selects the correct branch. A test on a 2-D acoustic marine seismic problem with a free surface at the top, a classic second-order Higdon condition at the bottom and numerically exact boundaries at the two lateral sides demonstrates the capability of the method. Numerically exact boundaries on each side, each computed independently with a free-surface or Higdon condition, provide even better results.

Nonlinear Free-Surface and Viscous-Internal Sloshing in 2-D Numerical Wave Tanks

2003 ◽  
Author(s):  
Daniel T. Valentine ◽  
Jannette B. Frandsen
Keyword(s):  
Free Surface ◽  
Internal Waves ◽  
Side Wall ◽  
Rectangular Domain ◽  
Surface Model ◽  
Flow Equations ◽  
Numerical Wave ◽  
Model Equations ◽  
Nonlinear Free Surface ◽  
Free Surface Model

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.

Nonlinear Free-Surface and Viscous-Internal Sloshing

2003 ◽  
Vol 127 (2) ◽  
pp. 141-149 ◽  
Author(s):  
Daniel T. Valentine ◽  
Jannette B. Frandsen
Keyword(s):  
Free Surface ◽  
Rectangular Domain ◽  
Shear Instability ◽  
Surface Model ◽  
Primary Wave ◽  
Flow Equations ◽  
Model Equations ◽  
State Of Rest ◽  
Nonlinear Free Surface ◽  
Free Surface Model

This paper examines free-surface and internal-pycnocline sloshing motions in two-dimensional numerical wave tanks subjected to horizontal 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. Two types of breaking are discussed. One is associated with a shear instability which causes overturning on the lee side of a wave that moves towards the center of the container; this wave is generated as the dominant sloshing mode recedes from the sidewall towards the end of the first sloshing cycle. The other is associated with the growth of a convective instability that initiates the formation of a lip of heavier fluid above lighter fluid behind the crest of the primary wave as it moves up the sidewall. The lip grows into a bore-like structure as it plunges downward. It falls downward behind the primary wave as the primary wave moves up the sidewall and ahead of the primary wave as this wave recedes from the sidewall. This breaking event occurs near the end of the first cycle of sloshing, which is initiated from a state of rest by sinusoidal forcing.

The Introduction of Micro Cells to Treat Pressure in Free Surface Fluid Flow Problems

1995 ◽  
Vol 117 (4) ◽  
pp. 683-690 ◽  
Author(s):  
Peter E. Raad ◽  
Shea Chen ◽  
David B. Johnson
Keyword(s):  
Fluid Flow ◽  
Free Surface ◽  
Pressure Field ◽  
Flow Simulation ◽  
New Method ◽  
Computational Domain ◽  
Critical Feature ◽  
Flow Problems ◽  
Macro Cell ◽  
Marker And Cell Method

A new method of calculating the pressure field in the simulation of two-dimensional, unsteady, incompressible, free surface fluid flow by use of a marker and cell method is presented. A critical feature of the new method is the introduction of a finer mesh of cells in addition to the regular mesh of finite volume cells. The smaller (micro) cells are used only near the free surface, while the regular (macro) cells are used throughout the computational domain. The movement of the free surface is accomplished by the use of massless surface markers, while the discrete representation of the free surface for the purpose of the application of pressure boundary conditions is accomplished by the use of micro cells. In order to exploit the advantages offered by micro cells, a new general equation governing the pressure field is derived. Micro cells also enable the identification and treatment of multiple points on the free surface in a single surface macro cell as well as of points on the free surface that are located in a macro cell that has no empty neighbors. Both of these situations are likely to occur repeatedly in a free surface fluid flow simulation, but neither situation has been explicitly taken into account in previous marker and cell methods. Numerical simulation results obtained both with and without the use of micro cells are compared with each other and with theoretical solutions to demonstrate the capabilities and validity of the new method.

Advanced CFD Simulations of free-surface flows around modern sailing yachts using a newly developed openFOAM solver

2016 ◽  
Author(s):  
Janek Meyer ◽  
Hannes Renzsch ◽  
Kai Graf ◽  
Thomas Slawig
Keyword(s):  
Free Surface ◽  
Large Scale ◽  
Breaking Waves ◽  
Free Surface Flows ◽  
Body Motion ◽  
Computational Time ◽  
Efficient Computation ◽  
Surface Flows ◽  
Scale Free ◽  
Wide Range

While plain vanilla OpenFOAM has strong capabilities with regards to quite a few typical CFD-tasks, some problems actually require additional bespoke solvers and numerics for efficient computation of high-quality results. One of the fields requiring these additions is the computation of large-scale free-surface flows as found e.g. in naval architecture. This holds especially for the flow around typical modern yacht hulls, often planing, sometimes with surface-piercing appendages. Particular challenges include, but are not limited to, breaking waves, sharpness of interface, numerical ventilation (aka streaking) and a wide range of flow phenomenon scales. A new OF-based application including newly implemented discretization schemes, gradient computation and rigid body motion computation is described. In the following the new code will be validated against published experimental data; the effect on accuracy, computational time and solver stability will be shown by comparison to standard OF-solvers (interFoam / interDyMFoam) and Star CCM+. The code’s capabilities to simulate complex “real-world” flows are shown on a well-known racing yacht design.

Distribution of Wave Impact Forces From Breaking and Non-Breaking Waves

2009 ◽  
Author(s):  
Anne M. Fullerton ◽  
Thomas C. Fu ◽  
Edward S. Ammeen
Keyword(s):  
Free Surface ◽  
Wave Height ◽  
Breaking Waves ◽  
Qualitative Assessment ◽  
Total Force ◽  
Impact Loads ◽  
Wave Impact ◽  
Data Set ◽  
Wave Characteristics ◽  
Wide Range

Impact loads from waves on vessels and coastal structures are highly complex and may involve wave breaking, making these changes difficult to estimate numerically or empirically. Results from previous experiments have shown a wide range of forces and pressures measured from breaking and non-breaking waves, with no clear trend between wave characteristics and the localized forces and pressures that they generate. In 2008, a canonical breaking wave impact data set was obtained at the Naval Surface Warfare Center, Carderock Division, by measuring the distribution of impact pressures of incident non-breaking and breaking waves on one face of a cube. The effects of wave height, wavelength, face orientation, face angle, and submergence depth were investigated. A limited number of runs were made at low forward speeds, ranging from about 0.5 to 2 knots (0.26 to 1.03 m/s). The measurement cube was outfitted with a removable instrumented plate measuring 1 ft2 (0.09 m2), and the wave heights tested ranged from 8–14 inches (20.3 to 35.6 cm). The instrumented plate had 9 slam panels of varying sizes made from polyvinyl chloride (PVC) and 11 pressure gages; this data was collected at 5 kHz to capture the dynamic response of the gages and panels and fully resolve the shapes of the impacts. A Kistler gage was used to measure the total force averaged over the cube face. A bottom mounted acoustic Doppler current profiler (ADCP) was used to obtain measurements of velocity through the water column to provide incoming velocity boundary conditions. A Light Detecting and Ranging (LiDAR) system was also used above the basin to obtain a surface mapping of the free surface over a distance of approximately 15 feet (4.6 m). Additional point measurements of the free surface were made using acoustic distance sensors. Standard and high-speed video cameras were used to capture a qualitative assessment of the impacts. Impact loads on the plate tend to increase with wave height, as well as with plate inclination toward incoming waves. Further trends of the pressures and forces with wave characteristics, cube orientation, draft and face angle are investigated and presented in this paper, and are also compared with previous test results.

scholarly journals Free-surface flow simulations for discharge-based operation of hydraulic structure gates

2013 ◽  
Vol 16 (1) ◽  
pp. 189-206 ◽  
Author(s):  
C. D. Erdbrink ◽  
V. V. Krzhizhanovskaya ◽  
P. M. A. Sloot
Keyword(s):  
Free Surface ◽  
Hydraulic Structure ◽  
Flow Characteristics ◽  
Surface Flow ◽  
Water Levels ◽  
Surface Model ◽  
Local Flow ◽  
Gate Operation ◽  
Flow Equations ◽  
Flow Simulations

We combine non-hydrostatic flow simulations of the free surface with a discharge model based on elementary gate flow equations for decision support in the operation of hydraulic structure gates. A water level-based gate control used in most of today's general practice does not take into account the fact that gate operation scenarios producing similar total discharged volumes and similar water levels may have different local flow characteristics. Accurate and timely prediction of local flow conditions around hydraulic gates is important for several aspects of structure management: ecology, scour, flow-induced gate vibrations and waterway navigation. The modelling approach is described and tested for a multi-gate sluice structure regulating discharge from a river to the sea. The number of opened gates is varied and the discharge is stabilized with automated control by varying gate openings. The free-surface model was validated for discharge showing a correlation coefficient of 0.994 compared to experimental data. Additionally, we show the analysis of computational fluid dynamics (CFD) results for evaluating bed stability and gate vibrations.

scholarly journals A Second Order Weighted Numerical Scheme on Nonuniform Meshes for Convection Diffusion Parabolic Problems

2020 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Time Derivative ◽  
Perturbation Parameter ◽  
Rectangular Domain ◽  
Shishkin Mesh ◽  
Second Order ◽  
Parabolic Problems ◽  
Uniform Mesh ◽  
Central Difference ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

In this article, a singularly perturbed parabolic convection-diffusion equation on a rectangular domain is considered. The solution of the problem possesses regular boundary layer which appears in the spatial variable. To discretize the time derivative, we use two type of schemes, first the implicit Euler scheme and second the implicit trapezoidal scheme on a uniform mesh. For approximating the spatial derivatives, we use the monotone hybrid scheme, which is a combination of midpoint upwind scheme and central difference scheme with variable weights on Shishkin-type meshes (standard Shishkin mesh, Bakhvalov-Shishkin mesh and modified Bakhvalov-Shishkin mesh). We prove that both numerical schemes converge uniformly with respect to the perturbation parameter and are of second order accurate. Thomas algorithm is used to solve the tri-diagonal system. Finally, to support the theoretical results, we present a numerical experiment by using the proposed methods.

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