P.Bearman, M.Russell (Imperial College of Science, Technology and Medicine, United Kingdom)

A static water tank to test oscillating cylinders and scale models of offshore structures has been constructed. The main purpose of the tank, which is approximately a cube of side 2.4m, is to provide a facility to measure the viscous contribution to the hydrodynamic damping of models. Models are mounted from a pendulum suspension system and the damping is measured following the deflection and release of the pendulum. Circular cylinders with diameters of 150mm and 312mm have been tested and the damping measured up to values of the viscous scale parameter β of about 6×10^{4}. The processed data is presented as a variation of drag coefficient with Keulegan Carpenter number for different values of β. The KC range studied is from about 0,003 to 3. Measurements have also been made using a square cross-section cylinder of side 300mm. Some preliminary results are presented for an approximately 1/100 scale model of a TLP hull. The form of the experimental results is explained by considering the drag to be composed of a boundary layer and a vortex component.

Wind, waves and currents induce loading on tethered floating structures and this loading can lead to both a mean displacement and an oscillatory response. In the case of waves the resulting response may be at the main wave frequency and at frequencies substantially higher and lower than the wave frequency. For a tension leg platform (TLP) the high frequency loading can lead to the phenomenon known as springing and the low frequency loading is responsible for the slow drift oscillations of floating systems. Accurate estimation of the responses in these frequency ranges depends on many factors but it is particularly important to have a reliable evaluation of the hydrodynamic loading and precise knowledge of damping levels. The hydrodynamic damping arises from a number of sources and includes wave-drift damping of the hull which is related to wave radiation and diffraction and can be predicted using irrotational flow theory. Other forms of damping include mooring line damping, damping from the riser array, tether damping in the case of TLPs and additional damping from the hull arising as a consequence of the viscous nature of water.

In this paper the term “hydrodynamic damping” refers to the viscous contribution to the damping arising from flow around the hull of a TLP which typically may be constructed from circular and rectangular cross-section members. In the design process the hydrodynamic loading due to waves is usually considered separately from the damping and for large structures the loading is derived from ideal flow theory. In the case of large volume structures, viscosity is likely to have only a small effect on the total hydrodynamic loading. This implies that the loading on the structure is inertia dominated and that the influence of boundary layers, flow separation and vortices are all very small. However, for a compliant

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Measurements of Hydrodynamic Damping of Bluff Bodies with Application to the Prediction of Viscous Damping of TLP Hulls
P.Bearman, M.Russell (Imperial College of Science, Technology and Medicine, United Kingdom)
SUMMARY
A static water tank to test oscillating cylinders and scale models of offshore structures has been constructed. The main purpose of the tank, which is approximately a cube of side 2.4m, is to provide a facility to measure the viscous contribution to the hydrodynamic damping of models. Models are mounted from a pendulum suspension system and the damping is measured following the deflection and release of the pendulum. Circular cylinders with diameters of 150mm and 312mm have been tested and the damping measured up to values of the viscous scale parameter β of about 6×104. The processed data is presented as a variation of drag coefficient with Keulegan Carpenter number for different values of β. The KC range studied is from about 0,003 to 3. Measurements have also been made using a square cross-section cylinder of side 300mm. Some preliminary results are presented for an approximately 1/100 scale model of a TLP hull. The form of the experimental results is explained by considering the drag to be composed of a boundary layer and a vortex component.
INTRODUCTION
Wind, waves and currents induce loading on tethered floating structures and this loading can lead to both a mean displacement and an oscillatory response. In the case of waves the resulting response may be at the main wave frequency and at frequencies substantially higher and lower than the wave frequency. For a tension leg platform (TLP) the high frequency loading can lead to the phenomenon known as springing and the low frequency loading is responsible for the slow drift oscillations of floating systems. Accurate estimation of the responses in these frequency ranges depends on many factors but it is particularly important to have a reliable evaluation of the hydrodynamic loading and precise knowledge of damping levels. The hydrodynamic damping arises from a number of sources and includes wave-drift damping of the hull which is related to wave radiation and diffraction and can be predicted using irrotational flow theory. Other forms of damping include mooring line damping, damping from the riser array, tether damping in the case of TLPs and additional damping from the hull arising as a consequence of the viscous nature of water.
In this paper the term “hydrodynamic damping” refers to the viscous contribution to the damping arising from flow around the hull of a TLP which typically may be constructed from circular and rectangular cross-section members. In the design process the hydrodynamic loading due to waves is usually considered separately from the damping and for large structures the loading is derived from ideal flow theory. In the case of large volume structures, viscosity is likely to have only a small effect on the total hydrodynamic loading. This implies that the loading on the structure is inertia dominated and that the influence of boundary layers, flow separation and vortices are all very small. However, for a compliant

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structure the magnitude of the response depends critically on the damping level and viscous effects may make a significant contribution to this damping. Estimates of damping coefficients are required to predict both surge and heave motions. Here we will present some direct measurements of the viscous damping of components of TLP hulls obtained by displacing them and recording their decay in still water. By appropriately combining these it should be possible to approximate the damping of a complete hull. It is intended to check the accuracy of this procedure by also measuring the damping of a model of a TLP hull.
In reality, of course, the fluid is always viscous and the damping is just that part of the total loading that happens to be in phase with the structure velocity. When a structure is excited into oscillation by wave forces is it permissible to predict the response using damping levels obtained in still water? Sarpkaya (1) has expressed a similar concern about this approach and states “damping is used to lump into one parameter our inability to solve the fluid-structure interaction problem”. However, until we are able to solve satisfactorily this interaction problem designers will need estimates of damping. Hence damping values, expressed in terms of drag coefficients, will be presented in this paper.
When considering the damping of TLP hulls in real seas the relative motion between the water and the structure is considerably more complex than the harmonic motion considered in simple decay tests. The relative flow has three components which in the general case are not collinear. These motions are due to waves, currents and the response of the structure. The wave motion may excite response of the structure in three distinct frequency ranges appropriate to slow drift response, wave response and springing response. There is evidence available to suggest that damping due to a current and waves, together with response in surge, can be dealt with by the relative motion form of Morison's equation, provided there is no resonance with vortex shedding. The heave response in the springing mode is characterised by relatively high frequencies and very small motions and it may not be appropriate to lump this motion together with the others into Morison's equation. One possibility is that it may act independently within a much slower varying velocity field. It is clear that there are many outstanding questions surrounding the concept of hydrodynamic damping. However, the study of viscous effects for small amplitude oscillatory motion is in itself an interesting subject and some new results, and perhaps some fresh understanding, may result from our experiments.
Assuming that the fluid loading on an oscillating body can be described by Morison's equation then the hydrodynamic damping is related to the drag term in this equation. It can be shown, see for example Bearman and Mackwood (2), that the logarithmic decrement of damping, δ, is related to the drag coefficient of a body, CD, through the relationship:
δ=2ρD2.KC.CD/3πm. (1)
In this expression ρ is water density, D a length scale used in CD (in the case of a circular cylinder it would normally be the diameter), KC is Keulegan Carpenter number and m is the effective mass per unit length of the body. KC is defined as UT/D, where U is the maximum velocity of the body relative to the water during a cycle and T is the period of oscillation. For harmonic motion KC can be defined as 2πA/D, where A is the amplitude of oscillation.
The viscous drag coefficient is composed of a skin friction component and a component related to the pressure force on the body. For circular cylinders, at KC values of order unity or less the contributions from pressure and skin friction are of a similar magnitude. However, at higher KC numbers separation occurs and the drag coefficient, and hence damping, is then dominated by the pressure component. Mooring lines, tethers and risers experience large motions relative to their diameters and hence their KC numbers are also large and in a range where there is considerable data available on drag coefficients. Apart from large amplitude slow drift oscillations, the flow around hulls is characterised by small KC numbers and is in a regime where there is sparse information on CD values that can be applied with confidence to full scale structures.
In the case of TLP hulls viscous damping arises from flow about the vertical columns and from the pontoon. The columns are usually circular whereas the pontoon may be constructed from square or rectangular section with various degrees of corner rounding. The damping level is dependent on characteristics of the boundary layer flow and on whether separation occurs from the circular members

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and from the corners of the pontoon. In addition to KC, an important parameter in determining the damping level is the ß parameter, where ß=D2/υT. In the case of slow drift oscillations of full scale structures CD values are required that are appropriate for the resulting high values of ß.
At low KC the oscillatory flow around a circular cylinder develops a three-dimensional instability, known as the Honji (3) instability. This takes the form of a regular array of vortex structures along the cylinder span, the wavelength of which reduces with increasing ß. This instability appears to be part of the transition process whereby the boundary layer flow changes from a laminar to a turbulent state. Experiments show the instability to be present over a wide range of ß and it is known to exist up to ß values of at least 5×104. The generation of the vortex structures extracts energy from the flow and the damping levels are higher than those given by the analytical solution for laminar attached flow. The transitional nature of the flow at low KC makes it particularly difficult to predict.
Much of the previous work on viscous damping has been carried out for circular cylinders. Bearman and Mackwood (2) mounted circular cylinders on a pendulum suspension system and measured damping by recording the amplitude of decaying oscillations. The results are presented in the form of CD versus KC and typically the KC numbers range from 0.1 to 3 and ß values are up to 3×104. A similar technique has been used by Otter (4), (5) and he presents two sets of CD values for KC up to around 2 and for ß=5.47×104. There is substantial disagreement between the two sets which he ascribed to sloshing in his tank. Sarpkaya (6) has made direct measurements of CD at low KC in a U tube for values of ß up to 1.1×104. Direct measurement of CD at very low KC is extremely difficult because the fluid loading is dominated by the inertia component. For example, at KC=1 and for ß=3×104 the maximum drag load in a cycle is only about 2% of the maximum inertia load. At KC=0.1 it drops to about 1%. One of the main reasons why viscous damping at low KC remains so uncertain is because it is caused by such a small fraction of the total hydrodynamic loading.
The values of KC and β to be modelled depend on the mode of response being considered and the dimensions of the structure. A diagram of a TLP hull is shown in figure 1 and it has circular
Figure 1 A typical TLP hull below the still water level
columns and the pontoon is constructed using a square cross section. Data appropriate to such a structure is used to estimate typical values of the flow parameters. Basing parameters on the length of a side of the cross section of the square pontoons, 8.5m; in surge β may be up to about 1.5×106 and KC up to 15 or 20. On the other hand, for the high frequency springing motion in heave β is likely to be about 2.5×107 and KC only around 0.004. For large values of KC and β, drag coefficients can be estimated from existing data for circular cylinders and, because of the relatively smaller amount of data, perhaps less reliably for square sections. Bearman et al (7) have measured the CD, for KC numbers down to about 1.5, of square-section cylinders with various degrees of corner rounding. However, the highest ß value in these experiments is limited to 432. Little or no information seems to exist on the viscous damping contribution due to free ends or that due to intersections between members, such as occurs for TLPs between the columns and the pontoon.
The present investigation is aimed at providing the viscous contribution to the hydrodynamic damping for small amplitude motions, i.e. KCs of order unity and less. In order to be able to generate data that will be applicable to design, drag coefficients at as large a value of β as possible are required. β can be increased by increasing D, reducing T and using a fluid with a smaller kinematic viscosity. Using as large a linear scale as possible requires as big a tank as possible in order to keep the blockage of the model to a reasonable limit. For the present study a new tank 2.4m×2.4m and 2.4m deep was constructed. There is no cheap, safe and easily available fluid that has a substantially

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lower kinematic viscosity than water and so water was chosen for this investigation. By mounting models from a pendulum arrangement and then increasing the stiffness by the addition of springs the time period can be reduced. A number of bluff sections have been tested. These include circular cylinders with diameters of 150mm and 312mm and a square section with sides of 300mm. In the case of the square cylinder, damping has been measured for a range of angles of incidence to the ambient relative flow. Using models with widths of about 300mm, β values up to 6×104 could be obtained. The KC range is then from around 0.001 to 2 or 3.
EXPERIMENTAL ARRANGEMENT
A water tank was constructed using prefabricated GRP panels supplied by BTR Hydroglas, that are available in sizes of 1.22m×1.22m and 0.61m×1.22m. The panels are bolted together using a bitumastic sealant between the flanges to provide a water-tight joint. Ten large and twenty small panels were purchased to permit a cube of side length 2.4m to be constructed. Visual access to the tank was required to allow flow visualisation techniques to be used and four of the large panels were converted into window panels. This was achieved by cutting out a section of the panels, (dimensions 0.91m×0.91m) and bolting a sheet of clear acrylic (1.22m×1.22m by 15mm thick) to the inside, using a bitumastic sealant to provide a water tight joint. When not in use the windows were covered to prevent light affecting the water. To support the tank an external steel corsetry was constructed from rectangular box sections. Steel channel was attached to the flanges of the uppermost panels, which then supported a steel framework spanning the tank. The steel framework in turn supported a decking of marine quality plywood panels which almost completely covered the tank, preventing light and dust particles from entering the water. It also provides a working platform on top of the tank.
An acrylic cover was installed to prevent free surface waves from developing and affecting the damping measurements when models were oscillating. The cover was suspended from the sides of the tank, 5cms below the water level. A section was removable to allow models to be inserted and within this piece a further removable section was fabricated to fit around the model support system to complete the cover.
Figure 2 Cylinder and pendulum support system
In order to support the models a larger version of the pendulum arrangement used by Bearman and Mackwood (2) was constructed. It is shown in figure 2 and consists of an ‘A frame' structure secured to the steel uprights of the tank. The arms of the pendulum were constructed from

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aluminium lengths of 2m×50mm×50mm which are held by the steel frame at their top end. At the bottom end they support a second steel frame to which models are attached. As shown in figure 2, the attachment of these arms is by spring steel flexures. Models are supported from the lower frame via a streamlined, stainless steel strut. Springs of different stiffness are attached to the lower frame of the pendulum to increase the natural frequency of oscillation of the pendulum.
Bearman and Mackwood (20) had tested models in the horizontal plane, supported by a streamlined strut at each end. By now mounting the cylinders vertically only a single strut is required and the correction to the data needed to take account of the additional damping caused by the support arrangement should be less. The streamlined strut supported models at a depth of 300mm below the water level. A steel plate was welded to the lower end of the strut and steel rods then bolted to it. These rods passed through the models and were attached to a plate at the bottom of the models. This then held the hollow cylinders in compression and transferred the loading through the model and through the strut to the pendulum. The models were made water tight by using a silicon sealant on all the joints. A diagram of a model is shown in figure 3. To ensure the models were almost neutrally buoyant lead weights were bolted to steel plates inside the models. Various length models could be accommodated, by the use of different length rods, and various diameter models, by the addition of different fittings to the plate attached to the strut. Various length circular cross-section models with diameters of 150mm and 312mm, along with a square cross-section model of 300mm across the flats, were constructed and tested.
Two methods were employed to sense the amplitude of the pendulum. In the first method used, strain gauges were attached to the flexures. However, the signal became noisy as the amplitude of oscillation became small and hence the results for low KC were unreliable. In order to obtain accurate decay data at very low KC, a non-contacting displacement transducer was used which could be placed so as to pick up the displacement at various distances along the pendulum arms. Both methods had to be carefully calibrated in order to calculate KC and to be able to obtain accurate CD results from decay records. Data acquisition and analysis was carried out using a PC-based system. The system was programmed to provide a sampling frequency of 50Hz for the analogue to digital conversion of the signals from the displacement gauges. In addition, the PC provides a digital to analogue signal to trigger the release of the models. A pair of electromagnets hold the pendulum, at a known displacement, prior to its release by the signal from the PC.
Figure 3 Details of model construction
EXPERIMENTAL METHOD
While the experimental method is basically very simple, accurate results can only be obtained by carrying out the various stages as carefully as possible. The main difficulty is separating the viscous damping of the model from other sources of damping such as the structural damping of the pendulum, the viscous damping of the support strut and the contribution from end conditions, such as end plates. Tests carried out without a model and strut, but with mass added to the pendulum to

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represent the mass of the model, showed the structural damping to be extremely small and much smaller than any contribution arising from hydrodynamic effects.
Two techniques were used to correct data for the effects of the additional damping that arises from end plates and from the way models were mounted. In the first method the additional components of damping are measured in a separate experiment with the model removed but with first the strut and then the strut and an end plate in place. Mass is added to the pendulum in order to ensure that the frequency of oscillation is the same, so that the value of β will be maintained. This method is referred to as the plate subtraction technique. The second method involves testing two lengths of model at the same value of β. Provided the tests are conducted at the same frequency, then the structural damping of the pendulum and the viscous damping due to the end plates and strut should be identical for similar KC numbers. Therefore subtracting the product of the drag coefficient and the length for the short model from that for the long model and by dividing this result by the difference between the two lengths provides the drag coefficient for the cylinder. This method is referred to as long-short.
Once the log decrement due to just the hydrodynamic damping of the model is known as a function of amplitude then the corresponding drag coefficient can be calculated as a function of Keulegan Carpenter number, using equation (1). In order to obtain CD the effective mass of the model and suspension system, including the added mass of the water, has to be known. This is determined by displacing the model a known distance and measuring the force required to hold the model in place. Once the stiffness and frequency of oscillation of the system are known, the effective mass can be determined. This experiment also enables the calibration of the strain gauges to be verified.
EXPERIMENTAL RESULTS AND DISCUSSION
When using a pendulum suspension system the model experiences a small change in its vertical position as it oscillates back and forth. This small vertical displacement may have some influence on the flow and the effect may be different if a cylinder is mounted vertically rather than horizontally. In order to determine if this has a significant effect on the results experiments were carried out on a 150mm diameter circular cylinder. A model of length 586mm, fitted with 234mm diameter end plates, was tested horizontally using a double strut system and vertically employing a single strut. In separate experiments, the damping due to the struts, end plates and the structural damping was measured, for the two orientations, at similar β values. Obtaining drag coefficients by subtracting the end plate, strut and structural damping from each set of data indicated a good degree of agreement between the two orientations. Figure 4 shows drag coefficient versus KC for β=16,538 and it can be seen that there is reasonable agreement between the results for the horizontal and vertical cylinder. The figure also shows the large scatter obtained at low KC due to noise from the strain gauges.
Figure 4 CD versus KC for a circular cylinder with β=16,538. x, vertical model; +, horizintal cylinder; ——, Wang.
The straight line in figures 4 represents results from a theoretical analysis due originally to Stokes (8), and which was extended later by Wang (9). These theoretical results, which will be referred to here as the Wang results, apply to two-dimensional, laminar attached flow about an oscillating circular cylinder. The theory gives rise to the following expression for the drag coefficient:
CD=3π3/2 KC (πβ)1/2. (2)
Alternatively, this can be written as:
CD KC β1/2=26.24. (3)
As KC is increased from very small values the above theory will break down due to two reasons: firstly the flow may separate and additional drag will be

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created by the generation of vortices and secondly the flow may become three-dimensional due to the onset of instabilities. When the flow becomes three-dimensional, even though it may remain attached, the drag coefficient rises above the Wang value, as shown by Sarpkaya (6). This is assumed to be due to the generation of Honji (3) vortices. Hall (10) studied the stability of oscillating cylinder flow and found that there is a critical KC value, KC*, above which three-dimensional instabilities are amplified. For high β values this KC is given approximately by:
KC*=5.778/β1/4. (4)
Over the β range studied in this investigation, KC* varies between 0.58 and 0.37. The results plotted in figure 4 are typical of all the ones we obtained and show that drag coefficient values are generally greater than those predicted using equation (2), even for KC values below the critical values given by equation (4). It is not clear why this difference occurs below KC* but perhaps, with decaying motion, three dimensional boundary layer flow established at a higher KC value remains to influence the flow as KC falls. However, it is apparent that at low KC the drag coefficient and KC number are related by the expression:
CD KC=A (5)
where A depends on β.
The experiments with the 150mm model established that the orientation of the cylinder had little effect on the results and that drag coefficient values similar to those reported by Bearman and Mackwood (2) could be obtained. Hence it was decided to move on to a 312mm diameter circular cylinder model. Using this larger diameter model, tests were first carried out to determine if the amplitude of release has any effect on the subsequent decay data. A model of length 1214mm and diameter 312mm, with 392mm diameter end plates, was tested at various β values. The tests consisted of releasing the model from different amplitudes and comparing the variation of drag coefficient with KC number, where the drag coefficient has been calculated without any correction for the drag of the strut and end plates. Results are shown plotted in figures 5 for β=61022 and indicate that the drag coefficients are higher than those obtained from continuous decay tests for the first few cycles after release. Drag coefficients gathered from the first three or four oscillation cycles are generally neglected from further presentations of results..
Figure 5 CD versus KC for a circular cylinder with β=61,022; effect of release Keulegan Carpenter number with release from KC=0.44 to 3.4.
Circular cylinder models of diameter 312mm and lengths of 1214mm and 620mm, with 392mm end plates, were tested vertically using a single strut for a range of β values between 18,000 and 61,000. The drag coefficients were first obtained from these tests by subtracting the difference in the damping between the long and the short models. Next the drag component arising from the strut, end plates and the structural damping of the pendulum was determined separately. This was achieved by testing a single 392mm diameter plate, suspended from a wooden strut with similar dimensions to those of the stainless steel strut used for supporting the models. The tests were carried out at similar β values to those of the main experiments. The resulting drag coefficients, which are referred to as the tare drag coefficients, were then subtracted from those obtained for the long and short models individually. Results for β=61,022 are shown in figure 6. The figure shows the original long and short cylinder drag coefficients, the drag coefficients obtained by subtracting the long and short cylinder data and the drag coefficients for the long and short cylinders obtained by subtracting the tare drag. The Wang prediction of drag coefficient is also plotted. By comparing the three different estimates of drag coefficient obtained from the experiments it is clear that there are differences. In each case the CD for the short cylinder is larger than the CD for the long cylinder, which in turn is larger than the CD obtained by subtracting the long and short cylinder data. This suggests that a larger tare drag should

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Figure 6 CD versus KC for a 312mm diameter circular cylinder with β=61,022 and with 392mm end plates. ◇, long cylinder; ☐, short cylinder; x, long minus short; - long minus plate; —, short minus plate
have been subtracted from the total drag and that the effect of the ends is greater than assumed.
In order to try to identify where the extra component of drag might be coming from some flow visualisation was carried out using the electrolytic precipitation method. Fine white particles are emitted from the model surface into the flow. By observing the flow near one end of the cylinder it became evident that pairs of vortices were being shed from the edge of the end plate. Also close to the end plate there was a region of three dimensional flow on the cylinder as a vortex formed in the corner between the end plate and the cylinder. This vortex appeared to be spilling out past the end plate and causing the shedding of the vortex pairs. The diameter of the end plates was now increased to 700mm and the visualisation repeated. The shedding of vortex pairs no longer occurred and the flow appeared much more nearly two dimensional near an end plate. Encouraged by these findings it was decided to repeat the earlier measurements using the larger end plates.
The two different length 312mm diameter models were tested again with the larger end plates (700mm) for a range of β values from 20,000 to 58,000. A 700mm end plate was also tested at similar β values to those of the cylinder measurements. A comparison of drag coefficients obtained by the plate subtraction and long—short methods is shown in figure 7 for β=20,526. Although there are still some differences between the three sets of data, the spread is substantially reduced compared to that obtained with the smaller end plates. The results with the smaller end plates show that end conditions are important, even for a cylinder in oscillatory flow at low KC.
It is apparent for all the circular cylinder data, when plotted in a log log form, that as KC reduces so the drag coefficient follows a line parallel to the theoretical values of Wang. This confirms that at low KC, for a particular β value, the product of drag coefficient and Keulegan Carpenter number tends to a constant. Also the ratio of the measured drag coefficient to the Wang drag coefficient, C D/CDW, is greater than unity. Typically this ratio seems to take a value between about 1.5 and 2.5. A further feature to observe from the circular cylinder CD plots is that above a KC value of about 0.5 the data deviates from the straight line form given by equation (5). Perhaps this marks the first appearance

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Figure 7 CD versus KC for a 312mm diameter circular cylinder with β=20,526 and with 700mm end plates. +, long minus plate; -, short minus platex, ☐, long minus short; ——, Wang.
of separation which, as KC increases further, leads to vortex shedding and a rising drag coefficient. The minimum CD in our experiments seems to occur for KC values between about 2 and 3.
Bearman et al (11) have suggested that the drag of a bluff body in oscillatory flow can be considered as the sum of a boundary layer component and a vortex component. Further it is argued that at low KC the vortex drag coefficient for a circular cylinder should increase linearly with KC. Figure 8, taken from Bearman et al (11), shows the vortex drag component, CDV, versus KC for low values of β of about 1000. Also plotted on the same figure is an estimate of CDV obtained from large scale oscillatory flow and wave experiments carried out at SSPA and DHL respectively, for β of order 100,000. This gives
CDV=0.08 KC. (6)
The boundary layer contribution, CDBL, takes the form predicted by Wang but our experiments show that approximately,
CDBL=2 CDW.(7)
Hence we can express the variation of CD with KC as follows:
CD=52.48/(KC β1/2)+0.08 KC. (8)
The prediction obtained using equation (8) is plotted in figure 9 for β=34,946 and shows good agreement with the experimental results.
CD values for the square section cylinder model, with sides of length 300mm, are shown plotted in figure 10 against KC for β=8, 14, 28, 39 and 48×103. These results have been obtained using the plate subtraction method. Compared to the circular cylinder drag coefficient results there is considerably less scatter at very low KC. However, this is not related to the flow but is due to the fact that the displacement transducer was used for amplitudes corresponding to KC values less than about 0.1.

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Figure 8 Circular cylinder vortex drag coefficient. Symbols for β=(O) 1,000; ——, β=(O) 100,000.
Using an analysis similar to that of Wang (9), Bearman et al (11) showed that the drag coefficient for attached laminar flow on a square cylinder is given by
CD KC β1/2=38.18. (9)
The predictions obtained using equation (9) are plotted in figure 10 and they reproduce the correct form of the β dependence but, just as in the case of the circular cylinder, the CD is under estimated. In this case the measurements are 50% to 60% higher and for the square section the equivalent of equation (7) is roughly
CD KC β1/2=60. (10)
Bearman et al (11) also estimate the vortex-related drag for a square section cylinder using a simplified isolated edge, vortex analysis. This predicts a CD which is independent of KC and β, and takes the value 5.88. Inspection of figure 10 shows that as KC approaches unity the CD values do lose their β dependence, as predicted, but the asymptotic value of CD is nearer 2.5 that 5.88. The vortex analysis assumes that each edge of the square behaves independently and this may well be too idealised and lead to an excessively high prediction of drag. To obtain a more accurate result a full interactive flow field calculation would need to be carried out. Bearman et al (11) found that the isolated edge analysis also overestimated their drag coefficient measurements for β=231.
Following the method used to describe the circular cylinder drag coefficient variation with KC and β, an equivalent expression for the square section is given by:
CD=60/(KC β1/2)+2.5 (11)
Equation (11) is plotted in figure 11 and is seen to provide a good representation of the data. The drag of a square section is dominated by the generation of vortices from its sharp corners at all but extremely low KC values. At low enough KC the separated flow regions will be so localised near the edges that the drag contribution from vortices will be small and viscous damping will be due mainly to boundary layer drag arising from the regions of attached flow over the body.
The TLP hull shown in figure 1 was constructed to a scale of 1:113. The design allows the hull to be tested only in surge but the model can be oriented in various directions to the relative flow, in a similar manner to the square model. Preliminary drag coefficient results are shown plotted in figure 12 versus KC, for 0° and 45° orientations and for β=19,000. In this case β and KC are based on the diameter of the TLP columns. The results have not been corrected for the effect of tare drag and it should be noted that the area used in the drag coefficient is the same for both orientations. At very low KC the inverse relationship between CD and KC is similar to that observed for the circle and square measured separately. Near KC=1 the vortex drag of the square section pontoon seems to dominate the drag coefficient. It is interesting to note that the damping of the TLP seems to be unaffected by its orientation. However, these are preliminary results and they need to be carefully checked.
CONCLUSIONS
Experiments have been carried out to determine the viscous damping of a circular and a square cross section cylinder by mounting the sections on a pendulum and observing the decay of oscillations. Circular cylinders with diameters of 150mm and 312mmm have been tested as well as a square section of side 300mm. The data is presented

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Figure 9 CD versus KC for a circular cylinder with β=34,946; ——, prediction using equation (8).
Figure 10 CD versus KC for a square section cylinder; ○, β=40,000; Δ, 39,000; x, 28,000; ◇, 14,000; ☐, 8,000; ——, prediction using equation (9) with β=48,000; -----, prediction using equation (9) with β=8,000.

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Figure 11 CD versus KC for a square section cylinder; ○, β=40,000; Δ, 39,000; x, 28,000; ◇, 14,000; □, 8,000; ——, prediction using equation (11) with β=48,000; -----, prediction using equation (11) with β=8,000.
Figure 12 Unconnected drag coefficient versus KC for TLP model with β=19,000; KC and β based on column diameter. x, zero degrees; ◇, 45 degrees.

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as a variation of drag coefficient with Keulegan Carpenter number for different β values. The KC range tested was from about 0.003 to 3 and the maximum β was about 60,000. Two methods have been used to correct the results for the effect of tare drag. It is found that end conditions are important at low KC and large diameter end plates were required to give approximately two dimensional flow. Some preliminary results have been presented for a model of a TLP hull.
At low KC all results indicate that the product of CD and KC tends to a constant, as predicted by laminar flow theory. However, the level of CD is higher than that predicted by the theory at all KC values. By considering the drag to be composed of a boundary layer and a vortex component, relationships are proposed for the variation of CD with KC and β. These are shown to give a reasonable fit to the experimental data for the complete KC range examined. It remains for these relationships to be applied to predict the damping of the complete TLP hull. The TLP results show the same inverse relationship between CD and KC at low KC and CD appears to be dominated by the drag of the square section pontoon at higher KC values. The damping levels measured for the TLP at 0 ° and 45° incidence are similar.
ACKNOWLEDGEMENT
This research is sponsored by the Marine Technology Directorate Ltd and is funded by EPSRC and the Offshore Industry. It forms part of a managed programme of research entitled Uncertainties in Loads of Offshore Structures.
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