NS3 - DHI Capabilities in Computational Fluid Dynamic · PDF fileDHI Capabilities in...

Technical NoteFebruary 2008

DHI Capabilities in Computational Fluid Dynamics

NS3

DHI NS3 Description/gps/edc/lay/ybr/pot/manualer –02/08 i DHI

CONTENTS

1 COMPUTATIONAL FLUID DYNAMICS WITH NS3.......................................................1 1.1 Why NS3?.....................................................................................................................2

2 NS3 APPLICATIONS ....................................................................................................3 2.1 Labyrinth Weir ...............................................................................................................3 2.2 Computational Fluid Dynamics (CFD) Simulation of Power Plant Cooling Water

Intake ............................................................................................................................5 2.3 Hydro Power Station Headrace and Tailrace Hydraulic Transient Modelling .................6 2.4 Road Culvert Design Assessment .................................................................................7 2.5 Combined Sewer Overflow............................................................................................8 2.6 Velocity Cap on Deep Water Intake...............................................................................9 2.7 Numerical Analysis of Forces on a Tank’s Walls during Failure...................................10 2.8 Simulation of Wave and Current-induced Forces on a Windmill Foundation................11 2.9 Wave Run-up on Windmill Foundation ........................................................................12 2.10 Seabed Pipeline Stability – Pluto LNG Field, North West Shelf ...................................13 2.11 Reservoir Sedimentation Study ...................................................................................13 2.12 Analysis of Density Current Effects at a Stormwater Outfall ........................................15 2.13 Vortex-induced Vibrations of Risers and Pipelines ......................................................16 2.14 Wave Interactions with Oil Platform Decks ..................................................................17 2.15 Sediment Transport in the Coastal Zone .....................................................................17 2.16 Backfilling of a Trench in Oscillatory Flow....................................................................18

3 REFERENCES............................................................................................................20

DHI NS3 Description/gps/edc/lay/ybr/pot/manualer – 02/08 1 DHI

1 COMPUTATIONAL FLUID DYNAMICS WITH NS3

In 1994, DHI committed to an intensive research effort to establish a world class capa-bility in three-dimensional (3D) flow modelling. The result of this continuing, decade long commitment is the advanced numerical Navier-Stokes solver, NS3. NS3 has been specifically developed for computation of 3D flows and sediment transport with a focus on an advanced free-surface description using adaptive grid technology. NS3 features: • A flow-adaptive curvilinear grid that allows for moving internal and external

boundaries; • Volume of fluid representation of free surfaces enabling representation of complex

air/fluid surface interactions; • Multi-block grid formulation which makes it possible to describe complex model

arrangements; and • Advanced turbulence models including kε, kω and LES. NS3 has a considerable track record when it comes to successful investigation of real-world problems. The system has been applied to solve a broad spectrum of applications including: • Spillway capacity for complex weir and spillway structures; • Water surface drawdown and set-up in hydropower station headrace and tailrace

channels under emergency shutdown and rapid start-up conditions; • Forces and moments exerted on structures under the combination of currents and

non-linear waves; • Sedimentation in waves and currents; • Wave-breaking and associated sediment transport in the surf-zone; • Sediment transport near reflective structures; • Self-induced vibrations of free spanning pipelines in currents and wave-induced

flow; • Free-surface waves around piers; • Flow through closed structures such as combined sewer overflows. Interpretation of 3D model results can be complicated. NS3 has a range of sophisticated presentation tools to aid in the presentation and interpretation of results including: • 1D point time series; • 2D slice plots colour coded; • Vector plots; • Statistical analysis of 1D, 2D and 3D results; and • 2D slice and 3D animations; • Streamlines; and • Determination of force and overturning moments.


1.1 Why NS3?

NS3 has a range of key features that differentiate DHI’s CFD capability from our com-petitors. Key advantages of NS3 are: • NS3 is DHI’s own proprietary code which we have developed from “the ground

up”. DHI is continually broadening and refining the code on real world applica-tions. As custodians of the code, we are able to adapt and extend the code on an “as needs” basis project to project;

• The adaptive grid applied in NS3 gives the code a unique ability to calculate the response of complex flow patterns and forces on structures that may move, sway or flex under the influence of the incident flow;

• Adaptive grid capabilities also enable the code to calculate advanced morphological bed response to sediment erosion in complex flow situations;

• The code has recently been extended to enable additional flow processes to be represented including: • temperature exchange in complex flow patterns; • stratified flow and density driven currents; • air-driven flow and flows with high sediment concentration (density altering);

• As part of the DHI Software family, NS3 applications can be used in combination with our other well known software systems such as MIKE 21 and LITPACK.

Examples illustrating the advanced application of NS3 are provided below.


2 NS3 APPLICATIONS

2.1 Labyrinth Weir

The objective of the study was to investigate the performance of a proposed labyrinth weir connected to a spillway on a planned dam in Ecuador.

Four main tasks were identified: 1. Performance of the labyrinth weir itself 2. Flow pattern of the flow upstream of the weir 3. Flow down the spillway 4. Assessment of the potential risk of erosion

The labyrinth weir was modelled in DHI’s three-dimensional (3D) numerical model, NS3. Two sections of the weir were included in the set-up. The flow approaching the upstream of the dam/weir was modelled by MIKE 21 HD model. The flow down the spillway was modelled with a 2D model, which included a free surface model and a k-ω turbulence model. Based on the results from this model, potential risk of erosion was evaluated.

Figure 2.1 shows a sketch of the proposed labyrinth weir. Using a labyrinth weir instead of a straight weir lowers the necessary height upstream of the weir for the same discharge.

Figure 2.1 Sketch of the proposed labyrinth weir


Theoretical expression of the performance of a labyrinth weir has been presented in Tullis (1995). The formulas are based on extensive physical experiments. However, the performance can be lowered by for instance nappe interference. Nappe interference occurs when the flow over the weir from two sides interacts too much. One of the main objectives was to test whether the weir performed as expected.

Flow over the labyrinth weir is shown in Figure 2.2, and in Figure 2.3 a comparison of the predicted discharge is compared to the Tullis et al (1995) equations. The analyses showed that the proposed labyrinth weir performed as expected.

Figure 2.2 The flow over two sections of the labyrinth weir in the design case. The flow approaches the weir with a speed of 3.1m/s. The height of the side walls is 4m.

Figure 2.3 Comparison between the NS3 results and the formula given in Tullis et al (1995). The left panel shows the comparison between the discharge per section based on the theory by Tullis et al (1995), and the left panel the estimated discharge coefficient.


2.2 Computational Fluid Dynamics (CFD) Simulation of Power Plant Cooling Water Intake

Figure 2.4 Example of NS3 grid

The purpose of this project was the simulation of power plant cooling water intake that involves a combination of free surface flow and internal pipe flow modelling.

The modelled power plant is located close to a coastline and applies seawater as the source for cooling water. Due to sedimentation issues at the existing intake, it has been decided to move the intake head approximately 250m offshore, which implies that the pumping house and chambers are to be redesigned. The NS3 system has been applied to model the flow in the new pump structure including evaluation of head loos through the new arrangement of pipes, free surface and resonance investigation and sediment transport and deposition in the pumping house.

The entire system from the intake head to the pumping house has been modelled in the NS3, where the total number of computational cells is approximately 500.000. Beside the global model, a number of detailed models were also developed in order to investi-gate local flow patterns in details, for example flow and pressure in specific parts of the system, ie 90o bends between two pipes.

The NS3 simulations did reveal that with the original design surface elevation in the pumping house would be very close to a critical level, during pump start, and as a result layout of the pipes had to be redesigned in order to minimise the head loss and obtain an acceptable water level in the pumping house.


2.3 Hydro Power Station Headrace and Tailrace Hydraulic Transient Modelling

The proposed up-rating of the turbines in Snowy Hydro’s Tumut 3 Power Station required an assessment of the likely impacts to operational limits in the headrace and tailrace channels during station load acceptance and load rejection. The primary con-cerns were air entrainment at the intake structure during large load acceptance and back-surge impact from tailrace surging during rapid load reduction.

The operational constraints of the up-rated station were investigated in detail using NS3. In particular, the specific issue of transient propagation in the headrace and tailrace channels under existing and up-rated operating conditions was modelled.

Figure 2.5 Tumut 3 Power Station Headrace channel

NS3 model application involved the development of two separate model domains – one each for the headrace and tailrace channels. The developed models were successfully validated against transients measured during trials on site at the power station. The validated models were then used to predict transient surge levels and flow patterns for 14 various flow scenarios.


Figure 2.6 Tumut 3 Power Station Tailrace channel. NS3 model results showing undular hydraulic jump

2.4 Road Culvert Design Assessment

Climate conditions in Thule in northern Greenland are extreme. The outlet design of an embankment with dual purposes of flow storage and road embankment was investigated using NS3.

Flow conditions for normal outlet operation and overtopping of the embankment were investigated. A computational grid was developed encompassing one representative culvert from the twelve culvert array. NS3 was used to calculate the stationary surface and pressure across the embankment and the flow speed profile both through and over the embankment crest.

Figure 2.7 Road Culvert Model – Thule, Greenland


2.5 Combined Sewer Overflow

During periods of high rainfall, overflows from combined sewer systems are often man-aged by structures known as Combined Sewer Overflows (CSOs). Under normal flow conditions, sewer flows bypass the CSO and continue to the treatment plant. However, when an increased stormwater volume enters the system, the additional flow may cause the sewers to surcharge. In order to avoid surcharging, relief to the flow pressure in the sewer may be provided by strategically located CSO’s, which may either overflow into a stream or estuary or into a controlled storage. The design of functional CSO geome-tries can be constrained by space requirements resulting in a requirement for an original design for each location. The efficiency of complex CSO geometries can be confirmed using CFD modelling.

The following shows the geometry, the grid, the flow through a full CSO and the flow through a partly filled CSO. In the case with the partly filled CSO, the dynamics of the free surface is modelled with an advanced VOF-method. The inflow velocity is of the order of 1m/s. The speed under the sluice gate is up to 12m/s.

Figure 2.8 The geometry and the block-structured grid for the closed CSO.

Figure 2.9 Example of internal flow with a full CSO (left), and a partly filled CSO (right)


2.6 Velocity Cap on Deep Water Intake

A velocity cap is a device that is placed over vertical inlets on offshore intakes. The functional design of the device is aimed at transforming the vertical flow field devel-oped by an uncovered inlet into a more horizontally aligned flow field. The main justi-fication for these types of flow altering intake devices is that fish are more readily able to avoid rapid changes in horizontal flow fields, thereby reducing the likelihood of fish entrainment by the intake.

A proposed design for velocity caps for a series of flow intakes has been tested for a proposed power plant in South Korea.

The key objective of the analysis was to investigate the hydraulic performance of the submerged intake design. Other issues investigated were: • Investigation of temperature in the intake water • Sediment intrusion • Interference of the flow between neighbour intake caps The intake structures are situated at depth of approximately 23m. The intake flow rate is steady state in the short term, but varies depending on season and operation of the plant.

Figure 2.10 The left panel shows the generic layout of a velocity cap. On the right panel, the velocity cap and streamlines placed 0.1m above the bed upstream of the velocity cap are shown.

3D flow structure, possible sediment intake and the temperature of flows in the intake were assessed using NS3 for various intake flow rates and different ambient current conditions. An assessment of the required spacing between the velocity caps was also made.


Figure 2.11 Longitudinal section of the flow towards and passing the velocity cap. The intake flow rate is

12.5m/s and the ambient current flow speed is 0.25m/s

2.7 Numerical Analysis of Forces on a Tank’s Walls during Failure

The assessment of the possibility of catastrophic structural failure of an alcohol holding tank as a result of a minor fracture of the tank wall was assessed with the assistance of NS3. The study focus was to assess the flow patterns and flow-generated forces on the tank side walls during tank failure.

Two different failure types were considered in the study: • the first failure considered was a hole of approximately 0.55m x 0.55m centered 1m

above the base of the tank • the second failure considered was s a long crack in the tank of approximately 0.25m

height above the tank base and extending half way around the tanks circumference Two different tank sizes were tested; a large tank with a 30m diameter and 18m high; and a smaller tank 9m high with a 12m diameter.

Figure 2.12 NS3 model assessing flow-induced forces in the walls of an alcohol filled holding tank.


2.8 Simulation of Wave and Current-induced Forces on a Windmill Foundation

Figure 2.13 Windmill foundation

The objective of this study was to assess the wave and current-generated load on a windmill foundation. The lower part of the foundation is formed as a truncated cone with a bed diameter of approximately 16m, see Figure 2.13, while the diameter of the foundation shaft is approximately 5m. The windmill will be located at water depth of roughly 25m, and the forces have been evaluated for a matrix of different wave and current conditions. Wave conditions did change from a water height of 6m to 12m, the period from 6.5s to 11.4s and the current from 1.0m/s to 1.2m/s. A 5th order Stokes the-ory was applied.

For these different conditions, the NS3 was applied in order to simulate the pressure distribution around the windmill, and the resulting forces were obtained by integration of the calculated pressure distribution.

The suitability and validity of the applied approach is documented in different refer-ences. A number of test cases have been performed in Christensen (2005b) and Chris-tensen (2005a), with approximately the same resolution as in the set-up in this study. In general, the wave run-up was more accurately predicted using CFD when compared to results from second order boundary integral methods.

In Christensen (2005a), a comparison between measured and computed horizontal forces showed a very good agreement. A study with irregular wave interaction with a gravity based structure has been reported in Bredmose (2006), where a good agreement between CFD and physical experiments was found for the maximum wave force and overturning moment.


2.9 Wave Run-up on Windmill Foundation

Figure 2.14 Windmill foundation

During recent years, numerous offshore wind farms have been constructed. One of these is situated at Horns Rev, a reef just west of Denmark. Observations and meas-urements from the park have clearly shown that the wave run-up can be quite signifi-cant, and the objective of the project was to evaluate the impact of wave run-up under extreme wave conditions.

The in-house CFD software, NS3, was applied to study the run-up on the structure for a number of layouts. Especially the state of breaking may have a large influence on the run-up. The figure shows a case with scour protection. This enhances the run-up, espe-cially for short waves. The run-up was found to be much larger than the run-up found from conventional potential diffraction theories.


2.10 Seabed Pipeline Stability – Pluto LNG Field, North West Shelf

Figure 2.15 Assessment of flow kinematics in a seabed trench using NS3

The Pluto pipeline will transport gas from the Pluto field on the North West Shelf off Western Australia to the Pluto onshore LNG Plant. The gas is transported in a 200km long 36” trunk line from the Production Platform to the Onshore Plant. In order to secure stability of the pipeline, a design has been developed where the pipeline is laid at the bottom of a trench on the seabed.

The stability of the pipeline design was assessed using NS3. The main objectives of the investigation were to determine the reduction in flow kinematics using a wide trench compared to flow conditions on a flat seabed by establishing the relation between wave-current parameters and the trench geometry.

The reduction in hydrodynamic forces on the pipeline as a consequence of the reduction in flow kinematics at the pipeline was also assessed.

NS3 was used to simulate combined wave-induced forces and ocean currents moving over the trench. The figure above demonstrates how the flow kinematics can be influ-enced by the trench resulting in lower velocities and a recirculation slip zone.

2.11 Reservoir Sedimentation Study

The rate of sedimentation is an important consideration for the long-term viability and maintenance of some reservoirs. DHI were engaged to assess sedimentation issues in the Baglihar Reservoir in India.


The scope of the study was to: 1. Determine the development of the bed profile for a time period of 100 years 2. Investigate the development and long-term morphological stability of the bed level

in the vicinity of the sluice gates 3. Study the detailed flow and sediment transport patterns in the vicinity of the sluice

gates, and determine initial scour rates.

The above issues were addressed by using two types of mathematical models: one-dimensional (1D) and three-dimensional (3D). The 1D model (MIKE 11) was applied to address the longer time scale issues, while a 3D model (NS3) was applied to deter-mine the detailed sediment transport patterns in the vicinity of the sluice gates.

Figure 2.16 NS3 model to assist in the assessment of long-term sedimentation trends in a reservoir

A part of the study was devoted to the development of a special facility, which ampli-fies the computed bed shear stresses in MIKE 11. This was necessary for accurate description of the bed development in the vicinity of the sluice gates. The facility used the bed shear stresses computed by the NS3 model, as these were considered more accu-rate than the 1D approach on its own.


2.12 Analysis of Density Current Effects at a Stormwater Outfall

Figure 2.17 Assessment of cooling water outlet jet forces on a ship berth

An outlet for process water and storm water discharge was proposed for a location in a harbour close to a berthing facility for an oil tanker. The additional force from the impact of the discharge water on the ship hull was determined using DHI’s in-house CFD code, NS3.

The combination of process water and storm water was anticipated to be warmer and have a lower salt concentration (less dense) than the ambient harbour water. The den-sity effects needed to be taken into account. These density differences resulted in a two-layer flow, where the less dense outfall water flows over of the ambient harbour water body.

Modelling in NS3 showed that the nature of this flow is quite different from the flow patterns that result when density effects are not accounted for. Forces on the ship’s hull were approximately doubled when density effects were taken into account.

Force-reducing scenarios were then considered. Various angles of the outlet channel were tested and a lowering of the bed in the outlet channel was proposed. As the upstream level of the channel bed could not be lowered, a depth transition was required. A concept design for the required transition was made to ensure that the flow at the outlet was uniform and free of surface waves.


The NS3 was developed to model the 3D flow in the outlet water channel and the immediate area around the ship. Density effects were included in the model. Results with and without density effects were produced and compared.

Production runs for three different angles of the outlet channel were made. The effect of having a free outfall of the cooling water was determined and led to the recommenda-tion of lowering the outlet level. A desk study to determine a concept design for this depth transition was made.

2.13 Vortex-induced Vibrations of Risers and Pipelines

NS3 has proved to be an accurate and cost effective means of analysing the forces and degrees of movement of pipelines and risers in complex flow fields. DHI’s NS3 code has been configured to have an adaptive model mesh that allows pipeline and riser structures to flex and move interactively with the applied flow field.

Along with the solution of description of the flow patterns around the structure, at each model time step, NS3 is able to dynamically calculate: • integration of fluid forces on the pipeline • time integration of the body acceleration to give instantaneous body velocity, and • time integration of body velocity to yield instantaneous body position. The integration of the equations of motion of the body can be performed by any numerical method for solving ordinary differential equations. However, since the acceleration of the pipeline implies acceleration of a significant amount of the surround-ing water (the added mass), the structure body dynamics are tightly coupled to the fluid dynamics.

Figure 2.18 Example of a time sequence of vortex-induced vibrations of two cylinders due to a current.

The flow is directed from left to right.

The NS3 programme has been applied to investigate vortex-induced vibrations of single risers, multiple risers and pipelines. The presence and influence of neighbouring struc-tures and/or the bathymetry on a subject structure can be accounted for in the analysis and has been specifically accounted for in the NS3 code.


2.14 Wave Interactions with Oil Platform Decks

Wave-induced forces on offshore platform decks have become an important issue in the design and lifetime re-assessment of older platforms. Due to seabed subsidence, the lower deck of an offshore platform may be subject to increasing wave impact during the most severe storms. Until now, the problem has primarily been studied experimentally. However, by using the VOF technique to handle the free surface, it is now possible to study numerically the complex flow of a wave hitting one or more structural members of a platform deck. DHI has performed a large number of numerical runs, which have been validated against tests carried out in the large wave flume at the Coastal Research Centre in Hannover, Germany.

Figure 2.19 Free surface impact on two vertical plates simulated by NS3 and measured in a large wave flume.

2.15 Sediment Transport in the Coastal Zone

NS3 has also been extensively applied for the detailed study of sediment transport under spilling breakers in the surf zone. The VOF-method was applied to simulate the free surface. A k-ε turbulence model was used for the production, transport and dissipation of turbulent kinetic energy. This was combined with a model for the sediment transport.


Figure 2.20 The top panel shows the shape of the breaking waves in the surf zone, the second indicates the turbulence intensities and the lowest the sediment concentrations under breaking waves.

2.16 Backfilling of a Trench in Oscillatory Flow

By means of detailed NS3 modelling, the processes responsible for the morphological evolution of small-scale excavations exposed to wave-induced flow can be assessed. The problem is relevant when assessing rates of sedimentation of trenches dredged for the purpose of laying out pipelines or cables in the marine environment.

Figure 2.21 Definition sketch and morphological evolution of trench exposed to waves.


The action of waves is simulated by an oscillating pressure gradient imposed on the horizontal momentum equation. The sediment transport is composed of bed and suspended load taking into account the effect of gravity on the former. The convection-diffusion concept is applied to calculate the distribution of suspended sediment concen-tration. Intra-period flow and sediment transport processes are resolved, and the effect of these on the backfilling is quantified by a parameter defined by the ratio between the amplitude of oscillation of the flow and the width of the trench. The work is described in Jensen and Fredsøe (2001).


3 REFERENCES

Bredmose, H, Skourup, J, Hansen, EA, Christensen, ED, Pedersen, LM, and Mitzlaft, A

(2006): “Numerical reproduction of extreme wave loads on a gravity wind turbine foundation”. Proc. of OMAE 25th Int. Conf. on Offshore Mechanics and Arctic Eng., 4-9 June 2006, Hamburg, Germany.

Buxbom, IP, Fredsøe, J, Sumer, BM, Conley, DC and Christensen, ED (2003): “Large

eddy simulation of turbulent wave boundary layer subject to constant ventilation”. In Coastal Sediments 03, 18-23 May 2003. Sheraton Sand Key Resort, Clearwater Beach, Florida.

Christensen, ED, Bredmose, H, and Hansen, EA (2005a): ”Extreme wave forces and

wave run-up on offshore wind-turbine foundations”, In Proc. of Copenhagen Off-shore Wind Conference, 10 pages.

Christensen, ED, and Hansen, EA (2005b): ”Extreme wave run-up on offshore wind-

turbine foundations”, In Proc. of Int. Conf. on Computational Methods in Marine Engineering (MARINE 2005), Oslo, Norway, 27-29 June 2005, pp 293-302.

Christensen, ED, Zanuttigh, B and Zyserman, J (2003): “Validation of numerical

models against laboratory measurements of waves and currents around low-crested structures”. In Coastal Structures 03, 26-29 August 2003, Portland, Oregon.

Christensen, ED, D-J Waltra and N Emerat (2002): ”Vertical variation of the flow

across the surf zone”, Coastal Engineering, Vol 45, No 3-4, pp 169-198. Christensen, ED, Jensen, JH, and Mayer, S (2000): “Sediment transport under breaking

waves”. Proc. 27th, ICCE00, Sydney, Australia. Emarat, N, Christensen, ED, Forehand, DIM and Mayer, S (2000): "A study of plunging

breaker mechanics by PIV measurements and a Navier-Stokes solver", In Proc. of the 27th Int. Conf. Coastal Eng., ASCE, Vol 1, pp 891-901, Sydney, Australia.

Hansen, EA and Meyer, S (2001): “A numerical model for current-induced vibrations of

multiple risers”. In Proc. of the 20th Conference on Offshore Mechanics and Artic Engineering, OMAE’01, Rio de Janeiro, Brazil.

Hansen, EA, Bryndum, M, Mørk, K, Verley, R, Sortland, L, and Nes, H: “Vibrations of

a free spanning pipeline located in the vicinity of a trench”. In Proc. of the 20th Conference on Offshore Mechanics and Artic Engineering, OMAE’01, Rio de Janeiro, Brazil.

Jensen, JH and Fredsøe, J (2001): “Sediment transport and Backfilling of Trenches in

oscillatory Flow”. Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Sep-Oct 2001, pp 272-281.


Jensen, JH, Madsen, EØ, and Fredsøe, J (1999): ”Oblique flow over dredged channels.

II: Sediment Transport and Morphology”. J. Hyd. Engrg, ASCE, 125(11), pp 1190-1198.

Jensen, JH, Madsen, EØ, and Fredsøe, J (1999a): “Oblique flow over dredged channels.

Part I: Flow Description. Journal of Hydraulic Engineering, ASCE, 125(11), pp 1181-1189.

Kawamura T, Mayer, S, Garapon, A, Sørensen, LS (2001): “Large Eddy Simulation of

the Flow past a free-surface piercing circular Cylinder”. In: Transactions of the ASME. Journal of Fluids Engineering, Vol 124, Issue 1, pp 91-101.

Mayer, S, Nielsen, KB, and Hansen, EA (2005): “Numerical prediction of wave impact

loads on multiple rectangular beams”, Coastal Engineering Journal, Vol 45, No 1, pp 41-65.

Mayer, S, Garapon, A and Sørensen, LS (1998): “A fractional step method for unsteady

free-surface flow with application to non-linear wave dynamics”. Intl. Journal for Numerical Methods in Fluids, Vol 28, No 2, pp 293-315.

Nielsen, KB, and Mayer, S (2004): “Numerical prediction of green water incidents”,

Ocean Engineering, Vol 31, pp 363-399. Nielsen KB and Mayer, S (2001a): “VOF simulations of green water load problems”. 4th

Numerical Towing Tank Symposium, September, Hamburg, Germany, 2001. Nielsen, KB, Mayer, S (2001b): “Numerical simulation of tank sloshing”, SRI-TUHH

mini-Workshop on Numerical Simulation of Two-Phase Flows, Tokyo, Japan. Tullis, JP, Amanian, N, and Waldron, D (1995): “Design of labyrinth spillways”, Jour-

nal of Hydraulic Engineering, Vol 121, No 3, pp 247-255.

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