irregular sea states. To calculate these loads, the local relative fluid velocities acting normal to the bilge keel are combined with a KC dependent drag coefficient.
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Proceedings of the Twenty-second (2012) International Offshore and Polar Engineering Conference Rhodes, Greece, June 17–22, 2012 Copyright © 2012 by the International Society of Offshore and Polar Engineers (ISOPE) ISBN 978-1-880653-94–4 (Set); ISSN 1098-6189 (Set)
Forces on Bilge Keels in Regular and Irregular Oscillating Flow
Riaan van ‘t Veer
Alessio Pistidda
Arjen Koop
SBM Offshore Schiedam, The Netherlands
SBM Offshore Schiedam, The Netherlands
MARIN Wageningen, The Netherlands
ABSTRACT
To assess the roll motions the (non-linear) roll damping contribution from the appendages is to be quantified. For a spread moored system with a submerged lower riser balcony of significant dimensions, the influence of the riser balcony has to be included in the damping evaluation (Van „t Veer et al. (2011)). On FPSOs relatively large bilge keels are found necessary to achieve acceptable roll motion amplitudes in extreme design conditions. Such large appendages require proper structural design evaluation, which is basically lacking in current offshore class regulations or recommended practices.
This paper presents a validated methodology to calculate the oscillatory loads on bilge keels of ships operating at zero forward speed in irregular sea states. To calculate these loads, the local relative fluid velocities acting normal to the bilge keel are combined with a KC dependent drag coefficient. The local relative velocity to the bilge keel is obtained from 3D potential flow calculation which implies that the hull geometry and bilge keel location are incorporated and that all wave velocities (from radiation, diffraction and the incident wave) are accounted for. The KC dependent drag coefficient of the bilge keel geometry is calculated by 2D CFD simulations in harmonic flow oscillations utilizing a rectangular fluid domain. This method is verified to experimental data from the literature. With the present approach it is possible to quantify the ultimate load on the bilge keel in design extreme conditions and to obtain the long term load distribution necessary for fatigue analysis. It respects the vessel heading and sea state parameters. Model tests for several FPSO vessels have been used to validate and calibrate the methodology. The calculation method is currently further evaluated and applied for structural design analysis on bilge keels by SBM Offshore.
The most extensive roll damping research and a resulting empirical roll damping prediction method has been reported some 35 years ago by Ikeda et al. (1976) and Himeno (1981). It has become the ITTC recommended procedure for numerical assessment of roll damping in absence of experiments (ITTC (2011)). From this research it is known that the roll damping from bilge keels at zero speed can be attributed to two equally important contributions: the normal drag force on the bilge keel and a roll damping moment due to a pressure change on the hull. In the method by Ikeda et al. empirical coefficients derived from model test data are provided to calculate the bilge keels damping. However, present FPSO vessels were not included in this research.
KEY WORDS: Bilge keel loads; Oscillating flow; Model tests; CFD
The methodology presented in this paper only concerns the normal bilge keel (drag) force and it is not a roll damping prediction method. The objective is to formulate a methodology to calculate the loads on the bilge keels in irregular sea states, to allow structural verification of the bilge keel design. The vessel response is a known input, for example from sea keeping model tests. The method itself is further based on first-principle application of 3D potential flow, accounting for the bilge keel location as well as irregular sea state properties and vessel heading. This generalises the approach to any vessel and does not restrict the methodology to FPSOs bilge keels.
INTRODUCTION FPSO vessels are designed to operate continuously without interruption for about 20 to 25 years. The FPSO mooring system can be either a spread mooring system or a single point mooring. A single point mooring system is an option that SBM had already applied on a crude oil Floating Storage and Offloading (FSO) platform in the early 1970s. Since then, it has become the common way to provide station keeping capacity to F(P)SOs. Nowadays most single point moorings are turrets. The turret mooring allows the vessel to freely weathervane and so to show the least resistance to the combined forces of wind, waves, and current. This minimizes the exposure to beam seas. But for various reasons the preferred FPSO mooring might be a spread moored system. In that case, the probability of beam seas is defined by the selected vessel compass heading in combination with the distribution of wave environments at site location.
This paper starts with a discussion of the flow phenomena associated with a (flat) plate in a perpendicular oscillating flow. Experimental data for a flat plate bilge keel are discussed and compared to new executed Computational Fluid Dynamics (CFD) calculations. The paper continues with a description of the developed bilge keel load prediction model. Validation of the model is presented by means of measured loads in regular and irregular sea states during FPSO model tests.
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FORCES ON A PLATE IN A NORMAL FLOW
CD
Oscillating Flow Forces
.5 DCDU t | U t | CM A0U t
U mT 2h
1
2.4 45
U mT h
1
2.4
(2)
The velocity amplitude Um at the bilge keel is formulated by Ikeda et al. (1976) in terms of the local velocity induced by global motions. When the bilge keel is located at a distance r from the rotation centre, and the harmonic motion has amplitude φm and frequency ω, the velocity amplitude is given by Um=f(rωφm). The KC number can be written as KC=UmT/(2h)=πrfφm/h. Here f is an increment factor used to account for the hull geometry (bilge radius). It is defined by the sectional area coefficient σ and given by f=1+0.3exp(-160(1-σ)). The limiting factor is herein 1.3 for a square section.
The Morison model is the most often applied force model to describe the load on fixed or moving objects in a varying flow field. It originates from the publication by Morison et al (1950) on cylinders although, as referred by Keulegan and Carpenter (1958), its basis is already put forward 100 years before by Stokes (1850). It has been successfully applied on flat plates by Keulegan and Carpenter (1958). The Morison equation reads, for the force per unit length on a fixed object in an oscillatory flow described by U(t) as:
F t
22.5
In Figure 1 the drag coefficients are given for free and wall-bounded rectangular flat plates, as obtained in experiments by Keulegan and Carpenter (1958), Radjanovic (1962) and Sarpkaya and O‟Keefe (1996). Radjanovic used a pendulum apparatus with a free plate and derived the drag coefficient from the decaying motion. Keulegan used a fixed free plate in a wave tank, while Sarpkaya used a fixed plate in a U-tank (mounted and free). All experiments used different plate dimensions and experimental set-up, but the drag coefficients agree well between the different methodologies. Included in Figure 1 is the trend line from Ikeda according equation (2) (solid line), and a trend line based on the given empirical coefficients and KC=UmT/h, that is CD=22.5/KC+2.4 (dashed line). It is interesting to see (and perhaps coincidental) that Ikeda‟s formulation with a KC number based on twice the bilge keel height (equation (2)), agrees well with the wallbounded plate drag coefficients which were not known at the time Ikeda fitted his expression to the free plate data values (replacing the KC definition).
(1)
Where D is the flow diameter of the object, A0 is the added mass reference area, usually taken as an enclosed circular area A0=πD2/4. The inertia coefficient CM is related to the added mass coefficient by CM=1+CA and is considered to be constant throughout an oscillating cycle, as is the drag coefficient CD. It is known that the Morison equation (1) has its limitations in terms of describing the exact force history as well as the maximum load peak. In fact, equation (1) is a two-coefficient Fourier fit, assuming that higher order coefficients can be neglected. Based on measured forces on fixed plates and cylinders located in the middle of a wave tank, Keulegan and Carpenter (1958) reported that for plates the higher order terms are more important than they are for cylinders. Using the data from their paper it can be calculated that for KC=2, 12 and 40, the force peak is under-predicted by the Morison equation by about 2%, 12% and 7%, respectively. Such difference might be important to correct for. To capture the variation in the measured drag and inertia coefficients, Keulegan and Carpenter (1958) introduced the flow parameter UmT/D, later denoted by the Keulegan-Carpenter number KC. Here, Um is the harmonic flow oscillatory amplitude far away from the flow obstruction, T is the oscillation period and D is the dimension of the body normal to the flow. For the free plate, D was set equal to the height h of the plate, thus leading to KC=UmT/h. For harmonic motions, X(t)=Xmsin(ωt), the KC number can be expressed as a ratio of the motion amplitude Xm and the object dimension parameter, leading to KC=2π(Xm/h). Sarpkaya an O‟Keefe (1996) extended the work by Keulegan and Carpenter (1958) by measuring the force on wall-bounded plates in a U-shape water tunnel. Sarpkaya an O‟Keefe (1996) present the free plate and wall-bounded plate inertia coefficients based on the same reference area, being the fictitious cylinder enclosed by the plate height. Thus, for the wall-bounded plate with height h the applied reference area is as well A0=πh2/4; a fictitious circle completely above the wall enclosing the plate. As mentioned Sarpkaya an O‟Keefe (1996), if the wall acts as a symmetry plane for the wall-bounded plate configuration, half of the enclosed fictitious cylinder with radius h might be more applicable as reference area, that is A0=πh2/2.
Figure 1: Drag coefficient as function of KC number. Apart from the drag force due to the flow velocities, the flow accelerations will lead to an inertia force on the bilge keel. The inertia coefficients for a free- and wall-bounded plate, as measured in different experiments, are presented in Figure 2. For reference purpose the inertia coefficients from Keulegan and Carpenter (1958) for the (free) cylinder are shown as well, which converge to 2 for low KC numbers (inertia dominated force). The lowest value for the plate is CM=1.35 at KC=1.7. The inertia coefficient for the wall-bounded plate at low KC converges to a value close to one.
The wall symmetry argument is used by Ikeda et al. (1976) to formulate the Keulegan-Carpenter number as KC=UmT/(2h) in the formulation for bilge keel drag coefficients. A formulation for the drag coefficient was derived from a fit on the free plate experimental data of Keulegan and Carpenter (1958) and Shih and Buchanan (1971) in the range 4
