HIGH MODE NUMBER VIV EXPERIMENTS

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1 IUTAM Symposium On Integrated Modelg of Fully Coupled Fluid-Structure Interactions Usg Analysis, Computations, and Experiments, 1 June-6 June 003, Kluwer Academic Publishers, Dordrecht. J. KIM ANDIE, HAYDEN MACOO Invited Plenary Presentation HIGH MODE NUMBE I EXPEIMENTS Abstract. A simple equation is presented which provides the imum achievable mode number for a flexible cylder, towed by the top end with a weight at the bottom end. The imum achievable mode number, while towg still water, is shown to depend primarily on mass ratio, length to diameter ratio and the imum allowable angle of departure from vertical at the top end. Modal overlap lock- regions is shown to depend strongly on mass ratio uniform flow, but not sheared flow. The reduced velocity bandwidth parameter is troduced to quantify the extent of lock- regions sheared flow. Two shear parameters are shown to be useful characterizg low and very high mode number response. Fite length lock- regions are described for cylders with fite length dynamic properties. 1. INTODUCTION As offshore oil and gas production pushes to deeper water, drillg, production and export risers as well as TP tendons have become critical elements the design of these facilities. All of these long cylders are susceptible to significant vortexduced vibration. Gulf of Mexico loop currents and detached eddies are the source of considerable concern to engeers havg to design for fatigue resistance to I. The prediction of I is currently based on data from relatively short cylders tested laboratory environments. The most successful I prediction programs for long cyldrical mare structures are empirically based and make use of the data from these laboratory models. Many assumptions are required to extend the simplified experimental results to the prediction of long cylders ocean currents. For the most part the prediction programs are tentionally quite conservative, sometimes resultg the unnecessary use of I suppression. Model tests with long flexible cylders, respondg at high mode number are highly desirable and are needed to help validate and fe-tune the prediction programs. There are many difficult challenges faced conductg experiments at high mode numbers. The challenges range from the design of the experimental apparatus, to the identification of the key fluid and structural parameters essential to the understandg of the model test results. This paper deals with some of the challenges posed the experimental design and then addresses the important physical parameters that help one to understand the response of a long cyldrical structure to I. By carefully examg the dynamics of the expected behavior, several new parameters have been formulated. These parameters make it easier to understand the behavior of long cylders shear flows.

2 J. KIM ANDIE, HAYDEN MACOO. EXPEIMENT DESIGN.1 Objectives High mode number experiments are expected to address a variety of questions, cludg: Does lock- occur at high mode number? How much I suppression coverage is needed over the length of a riser to prevent high damage rate I events? What is the probability of sgle versus multi-mode response? What is the relationship between -le and cross-flow response? A Phase I testg program, sponsored by an dustry consortium of companies, known as Deepstar, is beg planned for the deep waters of ake Seneca Upper New York State. These tests will serve as a proof of concept opportunity to test the sensors, the data acquisition system and the testg methods. The model will be towed behd a boat a near vertical configuration with a large weight hangg off the bottom. The test matrix will address some of the issues mentioned the questions above. Both uniform and sheared flow tests will be attempted. Phase II testg is beg discussed for a real, sheared current profile the offshore environment, such as the oop or the Gulfstream. The offshore environment troduces challenges and conditions beyond our control, such as the presence of surface waves.. Maximum mode number In order to vestigate the effects of high mode number I with model testg it is desired to design the model so that relatively high mode numbers are achievable. By performg some simple rearrangements of formula with approximations, the mode number, n, can be estimated givg sights to the important physical parameters required for the model. For long cylders where the tension domates the bendg stiffness effects determg the natural frequencies, the natural frequencies may be approximated by f n n T m (1) T where and T are the length and average tension, while m T is the mass per unit length cludg added mass. For the lake experiments the tension is nearly constant along the length. The highest sheddg frequency, f v, is related to the highest velocity, U, the diameter, D and the Strouhal number, St as follows:

3 HIGH MODE NUMBE I EXPEIMENTS 3 f v, U St D () The frequency of the highest mode number possibly excited will be approximately the same as the highest excitation frequency. Equatg the frequencies and rearrangg gives: U mt n St D T. (3) In the case of a towed riser model, the horizontal component of the towg force required at the top end of the riser is equal to the total drag force, and is given by F s( ) D T (4) The total drag force, F D, is the accumulation of local drag forces tegrated over the length of the cylder. Solvg for T yields T FD s D f z z0 1 C D U dz s (5) Assumg the tension is approximately constant requires that be small. By squarg expression (3) and substitutg for T from (5) yields an expression for n, the highest achievable mode number for the specified conditions. n St U s Ca f D (6) s CD U If T is not approximately constant, then Equation (6) is a lower bound. The expression U is the spatially averaged square of the velocity, U z, over the length of the cable. s and f are the densities of the structure and fluid, while Ca is the added mass coefficient. Expression (6) has several different parameter groups. For uniform flows the ratio of the imum velocity to the average velocity equals one, for sheared flows

4 4 J. KIM ANDIE, HAYDEN MACOO the ratio is higher and thus the imum mode number achievable is higher. The equation also reveals that the imum mode number creases with top angle, mass ratio and aspect ratio. It may be desirable to limit the top angle so that the cident flow remas close to perpendicular to the pipe axis. The mass ratio should be near to realistic prototype risers. The easiest parameter to systematically vary is limited to the aspect ratio. arger aspect ratios lead to higher respondg mode numbers. For the proposed experimental model the aspect ratio is about By limitg the top angle to 0 degrees, the achievable cross-flow mode number, as estimated from Equation (6) is 18. This is for a cylder length of approximately 47 feet (130m), a diameter of 1.5 ches (0.0318m), a specific gravity of 1.5, an S t of 0.17, C D a of.0, and for a Ca of 1.0. In-le frequency content is approximately a factor of two greater than the domant cross-flow frequencies. For a tension domated system this means that if the cross-flow imum mode number is 18 then the imum -le mode will be approximately Mode esolvg -le and cross-flow components Maximizg the aspect ratio naturally results choosg as small a diameter as practical. With a small diameter, the ability to stall struments becomes more difficult. Additionally smaller diameters have lower torsional stiffness. One of the objectives of the testg program is to be able to resolve both -le and cross-flow motion components at every location that there is a sensor. At each sensor one must also know the orientation. This requires that the torsional stiffness be kept reasonably high. The torsional stiffness K is given by GJ K (7) where G is the shear modulus and J is the polar area moment of ertia. The proposed model is Alumum tubg (6061) with a 1.5 ch O.D. x 0.10 ch wall thickness. The length will be 130 m for the Phase I tests and approximately 400 m for the Phase II tests. The respective values of torsional stiffness are 9.1 and 3.0 N-m per radian. This is adequate for the shorter model, but might be a problem at longer lengths..4 The Number of required Sensors A current topic under evaluation is the number of sensors required to describe the vibration space and time. If the mode shapes are approximately susoidal shape, then a spatial Nyquist criterion would suggest a mimum of two measurement pots per wavelength. The number of wavelengths each mode shape is n/. Therefore, a rough estimate of the number of required sensors is the imum mode number. In the example computation Section. that number is 36 biaxial sensors for the 130-meter long model. For a fixed imum top towg

5 HIGH MODE NUMBE I EXPEIMENTS 5 angle the number of required sensors creases approximately proportion to the square root of the length. The large number of sensors required to resolve the response at high mode numbers is both a fancial and technical challenge. Current technology provides a choice among stra sensors, accelerometers and angular rate sensors. The stallation of a large number of sensors requires a serial, digital, data transfer. Otherwise there would be too many conductors runng down the center of the pipe. Additionally if there is too much formation to be passed along the serial connection, as would happen with a high samplg rate and high number of sensors, then the formation must be stored at each sensor location. Synchronous data acquisition among all sensors is extremely important if modal analysis is to be performed. For this reason a common trigger has to be designed to the acquisition system so that the sensors sample together. 3. DIMENSIONESS PAAMETES IMPOTANT TO THE PEDICTION OF OTEX-INDUCED IBATION OF ONG, FEXIBE CYINDES IN OCEAN CUENTS The headg above is the title from a paper (andiver, 1993) which discusses several important parameters cludg the reduced dampg or mass-dampg parameter, mass ratio, shear fraction, /, the number of modes with the shear bandwidth N s, and the wave propagation parameter n n. Some additional discussion and some new parameters follow. 3.1 The educed elocity Bandwidth Also troduced that paper is the concept of the lock- bandwidth of the wake. This is a measure of the ability of the wake to synchronize with the motion of a vibratg cylder a sheared flow. It is based on the concept that at a specific vibration frequency and amplitude there is a flow velocity,, which is ideally suited to supportg lock-. Furthermore, this ideal flow velocity is centered a range of flow velocities which make up the lock- region. At this center velocity an ideal reduced velocity may be defed as c c f D c (8) v It is important to note that this reduced velocity is defed terms of the vibration frequency, and not any fixed natural frequency. In the presence of a shear, the velocity varies the axial direction along the cylder both directions from this most favorable position. ock- or wake

6 6 J. KIM ANDIE, HAYDEN MACOO synchronization is able to persist over a limited range of velocity values, which is defed as. This variation when divided by the center velocity provides a defition of the lock- bandwidth, d. d c U, c, (9) The second part of this equation is terms of the upper and lower reduced velocities correspondg to the limits of wake synchronization. This lock- bandwidth parameter has also been referred to as the reduced velocity bandwidth the user guide to the response prediction program SHEA7 (andiver et al, 00). This is a parameter used design computations to predict the extent of a potential lock- region. A commonly prescribed design value for this number is 0.4. This specific value is to be terpreted as meang lock- may exist a sheared flow over a range of flow velocities, which may vary +0% around the most favorable velocity. Figure 1 is a conceptual sketch borrowed with permission from Prof. Michael Triantafyllou of MIT (003). It shows how the lock- bandwidth and the strength of the excitation might vary with reduced velocity, defed terms of the measured vibration frequency and cylder vibration amplitude. Outside of the positive energy region the lift coefficient will be negative and will produce hydrodynamic dampg. Outside of the wake capture region, the wake will not be correlated to the motion of the cylder. Experimentally such formation has been obtaed by drivg D rigid cylders, at constant frequency and amplitude a constant speed uniform flow. A very useful dimensionless frequency has been described by (Govardhan and Williamson, 000). It is fv f vo where fvis the vibration frequency and f vo is the stationary cylder Strouhal frequency associated with the ideal center velocity, c, and is given by f S D t c vo (10) If f U and f are the upper and lower bounds of the lock- range, then an experimentally determed estimate of the lock- bandwidth is given by d f f f U (11) fvo fvo

7 HIGH MODE NUMBE I EXPEIMENTS 7 Figure 1. Wake capture range terms of A/D and reduced velocity. From Triantafyllou, 003. This way of expressg the bandwidth has the advantage that S and therefore t fvo may be allowed to be a function of eynolds number. Equations 9 and 11 are equivalent if with a lock- region the spatial velocity variation along a vibratg cylder a sheared flow is equivalent to a slowly changg frequency variation over time a uniform flow. This cannot be strictly true, but there is considerable experimental evidence to suggest that, as an engeerg model of the real world, it is adequate for use designg risers to resist I. In section 3.3 the role of reduced velocity bandwidth determg response of a flexible cylder sheared flow is addressed. Before engagg that discussion it is important to address the role of added mass, because it is often not correctly understood. 3. The ole of Added Mass Uniform Flow The reduced velocity bandwidth discussion above made no reference to added mass. The effect of added mass on lock- of sprg-mounted cylders uniform flow is well understood. (andiver, 1993) describes it quite carefully. In brief, the most significant aspect is that a uniform flow added mass decreases dramatically as the reduced velocity is creased through the lock- range. A decrease added mass results an crease the natural frequency of the cylder. A very common form of the reduced velocity is one defed terms of a fixed frequency, such as the natural frequency vacuo or still water. The symbol used for this form this paper is n. A plot of response A/D versus n, for a flexible cylder vibratg first mode with pned ends is shown Figure a.

8 8 J. KIM ANDIE, HAYDEN MACOO Figure a. MS response/d versus reduced velocity, n, based on the natural frequency air. First and second mode cross-flow vibration of a flexible cylder From a 1981 experiment conducted by Exxon Production esearch. Published with permission from ExxonMobil. Figure b. ibration frequency versus reduced for the 1981 EP experiment on a flexible cylder water. n defed usg the f n air. Published with permission from ExxonMobil. This is data from an experiment conducted by Matt Greer of Exxon Production esearch 1981 at the Skibstekmisk aboratorium Denmark. Ten hollow alumum segments (outside diameter=0.1m, side diameter=0.11m) were

9 HIGH MODE NUMBE I EXPEIMENTS 9 mounted on a central tensioned steel rod. Plastic spheres were placed side the segments to achieve the desired mass ratio, when flooded. The model was 9.93 meters length and had an external diameter of 0.10 meters. The composite cylder had a specific gravity of 1.0 and a mass per unit length, cludg trapped water, but not cludg added mass of 11.3 kg/m. The cylder was tensioned between struts beneath a towg tank carriage. The dampg ratio of the first mode air was approximately 1.0 to 1.5% of critical and the natural frequency air was Hz. The tension was 16.7 kn. The tension creased by as much as 8% due to drag force durg the tests. The value of f n used to compute n Figures a and b cluded the variations caused by tension variation, but does not clude the variation natural frequency caused by added mass variation. When reduced velocity is defed terms of a fixed natural frequency such as the natural frequency vacuum, air or still water, the extent of the lock- range depends on the mass ratio. When this defition of reduced velocity is used, low mass ratio cylders have a much greater value of the upper limit of the reduced velocity range than high mass ratio cylders. This is quite evident Figure a. The upper end of the lock- range is 9.5. Williamson has shown that the imum upper limit value of is given by n n, u 9.5 m* C a m*.54 (1) where the mass ratio, m*, is defed as m* m D 4 (13) f The upper limit of the reduced velocity Figure, for a C a of 0.0, is predicted by equation (1) to be The actual upper bound shown the figure is approximately 9.5. The reason the upper bound is not achieved is that the second mode of vibration asserts control of the wake synchronization. The second mode amplitude is also shown Figure a. The second mode s potential lock- range overlaps that of the first, and for reasons not yet fully comprehended, the second mode becomes domant at this particular reduced velocity. This phenomenon is central to the behavior of high mode number response. For typical risers with mass ratios of 1.0 to 3.0, lock- regions will always overlap at high mode number. Describg high mode number response under controlled uniform flow conditions is a high priority objective for the experiments and understandg mode to mode transitions is a particular focus. Thus far two kds of reduced velocity bandwidth have been described. n is based on a fixed natural frequency and is relevant to determg the lock- bandwidth and extent of modal overlap for uniform flow conditions only. As the flow speed changes with time the natural frequency changes with the change added mass. Equation (1) reveals a significant disadvantage of usg,n. The

10 10 J. KIM ANDIE, HAYDEN MACOO values depend on the C a value the user prescribes. Usg an added mass coefficient of 0 or 1 to defe the natural frequency used to compute,n shifts the value., which depends on the actual vibration frequency, has a bandwidth defed as d. This bandwidth will be observed a uniform flow experiment which the cylder is forced to vibrate over a range of frequencies and it will also determe the range of the lock- region on a cylder a sheared flow. In uniform or sheared flow reduced velocity defed terms of the actual vibration frequency does not change with added mass, added mass coefficient or natural frequency. This topic is expanded upon the next section. 3.3 The ole of educed elocity Bandwidth The reason that low mass ratio cylders have such a broad lock- range uniform flows is that the natural frequency creases with flow speed. This is because the fluid added mass decreases as the flow speed and reduced velocity crease. If the mass ratio is less than 0.54, there is no upper bound of lock--reduced velocity (Govardhan and Williamson, 000). This is also evident from Equation (1). If the reduced velocity is defed terms of the actual observed vibration frequency,, then the dependence on mass ratio is removed. f v U (14) fd v When plotted this way the bandwidth of the lock- range is a function only of the ability of the wake to synchronize with the motion of the cylder, and is not a function of added mass. Plotted Figure b is n versus the ratio of the actual vibration frequency to the natural frequency air for the same experiment that produced the data Figure a. In Figure a the upper and lower bounds of the lock- range terms of n for Mode 1 are 3 and 9.5. The same range terms of defed Eq. (14) is 5. and 9.5. This is a variation of +30% around a center value of Alternatively it could be said that the reduced velocity bandwidth, d, was 0.6. If we defe the ideal reduced velocity for the occurrence of lock- as the verse of the Strouhal number, S t 1 e f c D (15) vo then a reduced velocity lock- bandwidth may be defed as d f f U,, U (16) i, fvo

11 HIGH MODE NUMBE I EXPEIMENTS 11 which is another way of arrivg at Equation (11). Two phenomena fluence the range of flow speeds over which a natural mode of response of a flexible cylder may exhibit lock-. One is added mass variation and the other is the ability of the wake formation process to synchronize with the motion of the cylder. Both may affect the range of flow speeds that defe the lock- region, but at high mode number sheared flow, it is only the ability of the wake to synchronize with the motion of the cylder that is important. The prciple focus of this paper is to understand the vibration of flexible cylders sheared flow. It is helpful to pose the problem as a question. Over what spatial variation flow speed is the wake able to synchronize with the vibration of a cylder at one of its natural frequencies? Figure 3 shows a lear sheared flow. Mode n of a riser exposed to the flow is likely to experience flow-duced excitation at its natural frequency regions where the flow speed will give rise to cross-flow lift forces whose frequencies are at or near the natural frequency of the mode. Such a region is shown the figure. The region is characterized by upper and lower velocities which defe the limits of the wake synchronization region. Between these limits it is assumed that the wake is synchronized with the cylder motion. The length of this region can be described terms of a reduced velocity bandwidth as given Equation (9). dr U (17) i c The extent of the lock- region a sheared flow is not dependent on mass ratio or added mass. It is prcipally dependent on the ability of the wake to synchronize with the cylder vibration. It is not dependent on added mass variation because that effect would require the natural frequency to vary with position along the riser, whereas there can be only one natural frequency for each mode a given sheared profile. In a shear the added mass does vary with reduced velocity all along the riser, but at steady state the modal mass cludg added mass is a constant, given by (18) M m x m x x dx n a n o where m(x) is the riser material mass per unit length cludg contents and m a x is the added mass/length. The added mass distribution along the riser for a given shear profile and vibration frequency is fixed and so also is the modal mass and the associated natural frequency. If the sheared flow profile were to change, for example as with a tidal variation, the position of the power region would move for each mode. This would cause the added mass distribution for each mode on the riser to vary slowly because outside of the lock- region the added mass at high and low reduced velocities

12 1 J. KIM ANDIE, HAYDEN MACOO current speed U U C fn D St U U U C U d U U U U C Figure 3. Power- region for mode 'n' a lear sheared flow. changes slowly. Therefore the modal mass and the natural frequency of each mode changes slowly with cremental changes the sheared profile. Added mass effects on the natural frequencies and therefore on lock- behavior are only associated with actual changes the flow profile. For a steady state sheared profile the extent of power- regions is governed only by the wake synchronization bandwidth as quantified the parameter d. When reviewg drillg riser data, this author has found that as the velocity profile changes slowly with time, only the first two or three modes show significant variation natural frequency due to added mass variation. The first mode shows the most variation, followed by the second and then others descendg order. It is predicted that higher mode number response sheared flows will show little variation natural frequency due to variation added mass. 3.4 Shear Parameters and Correlation ength Another way to thk about the power- region defed by the bandwidth d is as the length over which vortex-duced lift forces are correlated with the vibration of

13 HIGH MODE NUMBE I EXPEIMENTS 13 the cylder at its natural frequency. This length is assigned the symbol lear shear the change velocity over a length When X X is given by. In a d X (19) dx is the variation of velocity over the wakesynchronized power- region. Dividg both sides of Equation (19) by, the center velocity for the lock- region, provides a new expression for of the shear gradient. is the length, then d c terms d d (0) dx c c Solvg this equation for yields d 1 d c dx (1) A useful expression for vibration of a riser, which exhibits standg wave vibration over its entire length, is the ratio of the power- length to the total length. Equation (1) is easily modified to yield such an expression. d d c dx () The quantity square brackets is a shear parameter, which is conceptually useful when considerg the structural dynamic response prediction problem. For example, consider a lear velocity profile specified by then x x (3)

14 14 J. KIM ANDIE, HAYDEN MACOO d dx (4) and from Equation () d c c d (5) This tells us that the fraction of the length occupied by the power- region is larger at higher flow velocities. This expression fails as the center velocity approaches because the lock- region cannot extend above and will be only half as long as predicted by Equation (5) when c =. In summary Equation (5) is a conceptually useful tool understandg the relationship between shear and the correlation length which is available to excite each mode of vibration. It is probably true that the reduced velocity lock- bandwidth, d, is itself a function of the shear gradient, but that level of understandg must await future experimental research. 3.5 educed Dampg Parameter The mass-dampg or reduced dampg parameter as defed for uniform flows is given by S g r m S U s f 8 t fd (6) The expression on the right is from Griff (1998) and that on the left is from andiver (1993). r is the structural dampg constant per unit length, is the vibration frequency, U is the flow velocity, is the structural dampg ratio, and is the fluid density. The expression on the right is shown andiver (1993) to f reduce to that on the left. It is well known that this parameter may be used to predict resonant response A/D. esponse decreases with creasg values of S. At values less than about 0.1, cross-flow I reaches limit cycle amplitudes of one to two diameters dependg on the mode shape. This parameter has been extended by andiver (00) to the sheared flow case, and may be written as s g

15 HIGH MODE NUMBE I EXPEIMENTS 15 S u n n fu where n is the modal dampg, n is the natural frequency, and is the power length as defed earlier. andiver (00) may be checked for further details of the derivation and examples of its use. 3.6 The esponse of Dynamically Infite Cylders Infite cylders are defed andiver (1993) as ones which vibration waves excited one region die out due to structural and hydrodynamic dampg before reachg boundaries. Thus, standg waves do not domate the response as with low mode number short cylders. andiver (1993) provides a parameter, n, for assessg if a cylder is likely to behave as of fite length with travellg waves or of short length with standg wave mode shapes. n is the mode number and is the modal dampg ratio. Above n 1 n n (7) the vibration approaches that of an fite cylder. In a recent doctoral dissertation by Jung Chi iao (001), a theoretical solution is presented for an fite cylder excited over a fite region by an harmonic force applied a standg wave pattern. This is shown Figure 4. The excitation is of x the form f xt, Fs cost and is applied over a fite o length,. If it is assumed that the force is due to cross-flow I, expressed as n F o may be 1 Fo fu DC (8 C is the local lift coefficient, which is A/D and reduced velocity dependent. iao s solution is shown Figure 4. With the power- region a standg wave appears. Outside of the power- region the solutions are travelg waves, which decay due to structural and hydrodynamic dampg as they travel away from the power- region. The imum amplitude of the standg wave at the center of the excitation region is given by iao as A x C U D r 0 f N 1e (9)

16 16 J. KIM ANDIE, HAYDEN MACOO N and is the wave length of vibration waves the cylder where with frequency. The dampg,, is the structural dampg only side of the power- region and is assumed to be reasonably small, less than This equation tells us that given a sufficiently long power- region N imum amplitude standg waves may be achieved. excitation Axial coordate, X Figure 4. Harmonic excitation a fite length power- region on an fitely long cylder. The wavelength of the excitation matches that of the standg waves the cylder at that frequency. response amplitude Travelg wave region Standg waves Travelg wave region 0 0 Axial coordate, x Figure 5. Magnitude of the response of the fitely long cylder, given the excitation shown Figure 4. The lift coefficient is of course dependent on many factors cludg reduced velocity and AD. If the power- region is not long enough for the waves to reach the imum value, Equation 7(9) tells us that the imum amplitude response the power- region will be smaller.

17 HIGH MODE NUMBE I EXPEIMENTS 17 Such behavior is unlikely to be seen on typical offshore petroleum dustry risers. However, it is very likely to occur on cables used to tow heavily weighted deep survey vehicles behd ocean vessels. The tow cable typically enters the water almost horizontally due to total system drag as given Equation (4) but then curves downward, eventually becomg nearly vertical at the survey vehicle. The survey vehicle serves as a reflectg boundary condition. The near vertical region of the cable near the vehicle will have passg by it a nearly uniform flow crossg the cable perpendicular to its axis. This flow will excite standg wave I near the boundary. This is a semi-fite cable problem. The waves generated a power- region of length, startg at the attachment pot on the vehicle are equivalent to those that would be generated a power- region twice as long on a cable extendg to fity both directions. Half of the wave energy travels each direction for the fite cable. By symmetry arguments only half the power- length is needed to supply the wave energy travelg only one direction the semi-fite case. Waves generated the power- regions propagate along the cable never to return. At any equilibrium amplitude the power gog to creatg waves the power- region is equal to the sum of the power lost to dampg the power- region and the power radiated away from the power- region as travelg waves. There is a convenient expression for N terms of the local velocity gradient. As before may be expressed as d 1 d dx (30) sce N, then N dr d dx (31) From this expression it is easy to see that the stronger the shear, the shorter the power- region as measured wave lengths. From Equation (9) it is easy to see that as N creases so does the response amplitude the power- region. d is a very useful dimensionless parameter which expresses the shear gradient dx a way which is significant to structural dynamic response on very long cylders. There is one situation applicable to offshore risers where the previous results may be appropriate. Consider the example of a very long riser which is covered by

18 18 J. KIM ANDIE, HAYDEN MACOO fairgs over most of its length, except for one bare section. How long must the bare section be order for I to result significant response amplitudes the bare region? An approximate answer is given by Equation (9). The faired region of the riser will have significant hydrodynamic dampg for any travelg waves created the region without fairgs. Due to the large dampg the faired region the riser will behave as if it were of fite length. No standg waves will occur outside of the bare region. As before a bare region at one end of a riser of length is equivalent to a region of length found the middle of an fite cable. 4. CONCUSIONS The purpose of this paper was to identify areas of weakness our understandg of high mode number I, describe experiments that might help resolve some of the questions, and identify dimensionless parameters which might help the planng of experiments and help us to better understand the results. Four particularly useful parameters have been discussed: a reduced velocity bandwidth parameter, a reduced dampg parameter appropriate for sheared flows and, fally, two shear parameters which have significance structural dynamic response prediction. A simple formula has been provided for estimatg the imum achievable mode number. It shows that /D, mass ratio and top towg angle are the key parameters that control imum mode number. 5. EFEENCES Govardhan,. and Williamson, C.H.K. (000). Modes of vortex formation and frequency response for a freely-vibratg cylder. Journal of Fluid Mechanics, 40, Griff, O. M. and amberg, S. E. (198). Some recent studies of vortex sheddg with applications to mare tubulars and risers. ASME Journal of Energy esources Technology, ol. 104, pp iao, J. C., ortex-induced ibration of Slender Structures Unsteady Flow, Doctoral dissertation, MIT Department of Mechanical Engeerg, February 00, J.K. andiver, Supervisor. Triantafyllou, M.S., I of Slender Structures Shear Flow, IUTAM Symposium on Flow-Structure Interactions, June 003, New Brunswick, NJ andiver, J. K.. (1993). Dimensionless Parameters Important to the Prediction of ortex-induced ibration of ong, Flexible Cylders Ocean Currents. Journal of Fluids and Structures, ol. 7, No. 5., pp andiver, J. K. (00). A Universal educed Dampg Parameter for Prediction of ortex-induced ibration. Proceedgs of the 0 th International Conference on Offshore Mechanics and Arctic Engeerg, OMAE 00, Oslo, Norway. andiver, J. K. et al, (003). SHEA7 User Guide, MIT Department of Ocean Engeerg.