Hull component interaction and scaling for TLP hydrodynamic coefficients

Ocean Engineering 29 (2002) 513–532

Hull component interaction and scaling for TLP hydrodynamic coefficients James J. O’Kane a, Armin W. Troesch

b,*

, Krish P. Thiagarajan

c

a United States Coast Guard, Naval Engineering Support Unit, Boston, Massachusetts, USA Department of Naval Architecture and Marine Engineering, University of Michigan, 2600 Draper Road, Ann Arbor, Michigan 48109-2145, USA Centre for Oil and Gas Engineering, The University of Western Australia, Nedlands, Western Australia

b

c

Received 6 September 2000; accepted 29 December 2000

Abstract The purpose of this paper is to validate a new method that can be used by offshore platform designers to estimate the added mass and hydrodynamic damping coefficients of potential Tension Leg Platform hull configurations. These coefficients are critical to the determination of the platform response particularly to high frequency motions in heave caused by sum-frequency wave forcing i.e. “springing”. Previous research has developed the means by which offshore platform designers can extrapolate anticipated full-scale hydrodynamic coefficients based on the response of individual model scale component shapes. The work presented here further evaluates the component scaling laws for a single vertical cylinder and quantifies the effects due to hydrodynamic interaction. Hydrodynamic interaction effects are established through a direct comparison between the superposition of individual hull component coefficients and those evaluated directly from complete hull configuration models. The basis of this comparison is established by the experimental evaluation of the hydrodynamic coefficients for individual hull components as well as partial and complete platform models. The results indicate that hydrodynamic interaction effects between components are small in heave, and validate component scaling and superposition as an effective means for added mass and damping coefficient estimation of prototype platforms. It is found that the dependency of damping ratio with KC for a TLP is almost identical to that of a single column, thus offering a scaling methodology for prototype damping ratio values.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Tension leg platform; Springing; Heave hydrodynamic damping; Form drag; Friction drag; Prototype scaling; KC number dependence

* Corresponding author. Tel.: +1-734-763-6644; fax: +1-743-936-8820. E-mail address: [email protected] (A. W. Troesch). 0029-8018/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 1 ) 0 0 0 3 9 - 7

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Nomenclature A w B Bh Bv B⬘ Cd Cm Ca D Dh Dv F KC KCcr Lh Lv S T U f m u a1,2 b m r n

amplitude of oscillation angular velocity generic linear damping coefficient linear damping coefficient for horizontal members linear damping coefficient for vertical members non-dimensional damping coefficient drag coefficient of Morison’s Equation inertia coefficient of Morison’s Equation or combined hydrodynamic and platform inertia coefficient for oscillating bodies added mass coefficient for oscillating bodies diameter of cylinder component diameter of horizontal cylinder component (pontoon) diameter of vertical cylinder component (column) inline hydrodynamic force on cylinder given by Morison’s Equation Keulegan–Carpenter Number, 2pA/D KC critical length of horizontal cylinder length of vertical cylinder (T ) component shape area normal to fluid flow draft of vertical cylinder component (Lv) cylinder component velocity frequency of oscillation (Hz) in heave displaced mass flow velocity offset and slope respectively of B⬘-KC curve Beta-frequency parameter, D2f/n dynamic viscosity water density kinematic viscosity

1. Introduction Hydrodynamic model testing using scaled geometrically similar models or “geosims” is a vital component in the design process of novel deep water structures. Model testing in an ocean basin provides visual proof of the operability of a platform, along with providing key information such as Response Amplitude Operators in different modes of motion. It has been widely agreed (see e.g. Chakrabarti, 1994, § 1.4) that a scale of 1:100 is the upper limit on the size of models that can realistically reproduce a prototype environment in a laboratory. However, with present-day deep-water

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production exceeding depths of 1000 m, and existing facilities of the order of 10 m water depth,1 using scales larger than 1:100 seems unrealistic. Research has thus been recently focussed on obtaining reliable predictions from smaller scale models of up to 1:250 (Stansberg, 2000). An alternative is to develop new scaling strategies which can use limited model testing information with larger models (⬎1:100) and obtain predictions for prototype. One such strategy explored by the present authors in an earlier paper (Thiagarajan and Troesch, 1994), in particular applies to estimating hydrodynamic coefficients of a Tension Leg Platform (TLP) in heave. The premise to this strategy is that there exists significant amounts of research dedicated to understanding the hydrodynamic coefficients for TLP components such as vertical and horizontal cylinders. Although, only recently has research been completed that models the actual range of the amplitude and frequency experienced by a TLP during springing. The “component scaling” methodology (Fig. 1) of Thiagarajan and Troesch (1994) evaluated the TLP added mass and damping by separately scaling the vertical and horizontal components of the hull and then adding them together. This method neglects any effects of interaction between the components by assuming that these effects are typically of higher order and therefore expected to be small in magnitude. The present paper investigates the validity of this assumption by experimentally quantifying the effects of component interaction on the added mass and damping coefficients. Model tests are conducted on progressively complex models of TLP and the relationship between measured coefficients and geometry is explored. Interaction effects between pontoon and columns are investigated.

2. Background and theoretical considerations Arguably, the most significant motion related to TLP design and response is highfrequency motion in heave. Although most incident wave periods are longer than TLP heave natural periods (2–4 s), these vertical plane motions are important since non-linear effects at sum-frequencies cause high frequency vertical excitation (Bar-

Fig. 1.

Description of component superposition methodology.

1 The deepest model basin in the world presently is the MARIN ocean basin at Netherlands at 10.5 m depth.

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Avi and Benaroya, 1997). This non-linearity is due to the wave forces varying with the square of the incident wave heights, the variable wetting of the TLP columns, and the effect of the velocity squared term in the Bernoulli equation. This loading condition can lead to a resonant steady state response phenomenon termed “springing”. Springing vibration has been associated with several serious problems including structural fatigue in the tendons, discomfort for the crew, and safety concerns for the production facility. The magnitude of the response for a compliant structure depends critically on the added mass and damping (Bearman and Russell, 1996). Damping is unimportant in off-resonant conditions, since the response is primarily due to a function of the system inertia, system stiffness, and external excitation. However, at resonance the large stiffness and inertia forces cancel, and the response is dictated solely by the ratio of the excitation to damping (e.g. Troesch and Kim, 1991). Therefore, an accurate estimation of the responses at these frequencies requires a detailed understanding of the damping levels. Potential sources of damping include wave radiation damping, viscous damping, column footing damping, soil damping, and structural damping (BarAvi and Benaroya, 1997). This paper focuses on the hydrodynamic damping arising from the viscous flow around a TLP hull. The various components of hydrodynamic damping for bodies such as TLP’s are due to the following: 1. Time dependent boundary layer flows, 2. Flow separation, vortex formation and shedding, and 3. Free surface effects such as wave generation and diffraction. It has been noted (e.g. Thiagarajan and Troesch, 1994; Bar-Avi and Benaroya, 1997) that free surface effects have a relatively minor effect on springing damping and will not be considered in this paper. The two main components are influenced by several factors including surface roughness, incident currents, and waves. The resulting damping forces are surface integrations of both shear stress and normal pressures about the hull. Since a typical TLP hull is comprised of both vertical and horizontal cylinders each with dissimilar cross-sectional properties, separate flow properties and scaling laws coexist (Thiagarajan, 1993). Therefore, the total flow description about a vertically oscillating TLP hull will have to be considered as two separate flows: 1. Cylinders in an oscillatory cross flow (the case of the TLP pontoon) 2. Cylinders in an oscillatory axial flow (the case of the TLP column). Added mass, the other hydrodynamic component resulting from oscillatory motion, is defined as the quotient of the additional force required producing the fluid particle accelerations divided by the acceleration of the body. This fluid-mass transport induces a higher effective force requirement for the body acceleration in the fluid and lowers the natural frequency of the body. In general, added mass coefficients depend on parameters characterizing the history of the motion, time, type

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and direction of motion, Reynolds Number, and proximity effects (Sarpkaya and Isaacson, 1981). Much like damping, added mass values in oscillatory flows for the vertical and horizontal members differ based on the separately exposed cross sectional areas perpendicular to the flow. Therefore, the component damping and added mass coefficients determined through both experimental measurement and theoretical calculation in this work will be evaluated separately and superimposed for comparison with the measured values of complete hull forms. The frequency and amplitude of oscillation for circular cylinders are characterized in non-dimensional form by the Keulegan–Carpenter number (KC) and “frequency parameter” (b). These are defined as follows: KC⫽ b⫽

2pA D

D2f v

(1) (2)

where symbols are defined in the nomenclature section of this paper. Originally, the KC number was defined as the amplitude of incident oscillatory fluid motion relative to the cylinder size (Keulegan and Carpenter, 1958). The magnitude of KC indicates the relative importance of drag and inertia forces (Sarpkaya and Isaacson, 1981). The product of KC and b equals the respective Reynolds number, Re, characterizing the flow based on the flow oscillation velocity. The expression for modeling the inline hydrodynamic damping and inertia forces on a circular cylinder in such an oscillatory flow perpendicular to the cylinder axis is given by Morison’s Equation (Morison et al., 1950): 1 dU F 1 ⫽ rDCd|U|U⫹ prD2Cm Lh 2 4 dt

(3)

The force is divided into two parts. The first term is the drag force per unit length in phase with the square of the fluid velocity, and the second term is the inertia force component per unit length in phase with the acceleration. Both the inertia and drag coefficient may be estimated experimentally using either the method of least squares, or Fourier averaging (e.g. Sarpkaya and Isaacson, 1981). A similar expression can be used to describe the sectional forces acting on oscillating twodimensional bodies where Cm is replaced by Ca, added mass coefficient. For a circular cylinder Cm=1+Ca. In the limit of small KC values, Morison’s drag coefficient may not be the most appropriate form to represent hydrodynamic damping. As KC goes to zero, the drag coefficient varies inversely with velocity, leading to drag coefficients approaching infinity, at least in principle. And although the primary source of heave excitation is due to nonlinear wave forces, the response of the platform can be effectively modeled as a linear dynamic phenomenon using an equivalent linear damping ratio. For this paper, then, Morison’s nonlinear drag force term will therefore be rewritten in an equivalent linear version, which is accomplished by approximating the

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cosq|cosq| term of the drag force by the first term of an equivalent Fourier series equal to 8/3pcosq, where q=wt (e.g. Sarpkaya and Isaacson, 1981). The drag force may be alternately represented by a linearized equation given by Fdh⫽BhU

(5)

where Bh is the linear damping coefficient, which can be related to Cd by Bh⫽

4 mC bKCLh 3p d

(6)

The subscript “h” is used to denote coefficients and forces acting on horizontal members, e.g. the pontoons of a TLP. Since the damping for a horizontal cylinder was not individually tested in this work, Eq. (6) is used to calculate the pontoon’s damping coefficients for small KC values. These are then used in the superposition calculation for comparison with damping values measured for the complete hull form. The drag coefficient values used in the above expression to determine the corresponding damping coefficients are calculated based on experimentally validated expressions from previous research as presented below. In the analysis of the results, the linear damping coefficient B is normalized to give a non-dimensional damping ratio B⬘⫽

B 2mw

(7)

The critical damping coefficient, z or Z, both used by Thiagarajan and Troesch (1994, 1998), respectively), is similar to the expression for B⬘ except that the frequency of oscillation is replaced by the natural frequency of the system and the mass includes the added mass, i.e. z⫽Z⫽

B 2m(1+Ca)wn

(8)

The use of B⬘ facilitates the inclusion of off-resonant frequencies in the analysis of relative damping. 2.1. Hydrodynamics of horizontal cylinders Cylinders in an oscillatory cross flow can be described by several flow classifications that have been observed through numerous experimental investigations. All of the flow classifications are governed primarily by KC; however, a weaker dependency also exists on Reynolds number (Bearman et al., 1985). When KC is small, the oscillatory boundary layer is attached, two dimensional, and laminar. In this case of two-dimensional laminar flow, Wang (1968) extended Stokes (1851) solution of expressions for the drag and inertia coefficients to obtain

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Cd⫽

3p3 [(pb)−0.5⫹(pb)−1⫺0.25(pb)−1.5] 2KC

Cm⫽2⫹4(pb)−0.5⫹(pb)−1.5

519

(9) (10)

Several experiments have been conducted investigating hydrodynamic damping of horizontal cylinders at different ranges of KC and b, including Sarpkaya (1986), Otter (1990), Troesch and Kim (1991), Bearman and Russell (1996) and Chaplin and Subbiah (1998). In each case the drag coefficient was consistently inversely proportional to KC as in theory. Agreement with Wang’s (1968) theory occurs only below a particular critical value of KC, called KCcr, after which Cd values step upwards and then continue along the previous trend slope at a higher offset. Beyond KCcr, the measured drag coefficients of each test consistently exceeded that predicted by Eq. (9) by ratios ranging from 1 to 5. The step and offset change in Cd are attributed to characteristic changes in the flow from laminar to an eventually turbulent boundary layer. Typically this transition begins at KCcr. An expression for KCcr is given by Hall (1984): KCcr⫽5.778b−0.25(1⫹0.205b−0.25)

(11)

As suggested by Eq. (11), the KC value at which laminar-to-turbulent transition is initiated decreases with increasing b. A fully turbulent boundary layer is expected to occur at a Reynolds number based on amplitude of oscillation of about 104–105. Two particular works, one by Bearman and Russell (1996) and one by Chaplin and Subbiah (1998), have derived consistent values, of 2 and 2.2 respectively, for the ratio of increase of Cd for KC ⬎ KCcr. The reliability of the values derived is enhanced by the fact that the experimental techniques in the two sets of experiments were different from both each other and the remaining works noted. Therefore, in the present paper, a recently developed expression by Chaplin (2000) given as Cd⫽

55

冑bKC

⫹0.08KC

(12)

is used as the theoretical expression to calculate Cd for the pontoons in the range KC ⬎ KCcr. 2.2. Hydrodynamics of vertical cylinders Considering cylinders in oscillatory axial flow, e.g. the vertical members of a TLP, the two major components of hydrodynamic damping are (1) friction drag due to viscous shear stress on the surface, and (2) form drag due to separated flow at the bottom of the cylinder. Contributions from other heave damping sources such as wave generation and diffraction are usually insignificant for typically deep draft TLP’s at very low amplitudes of oscillation (Thiagarajan and Troesch, 1994; Huse and Utnes, 1994). These authors showed experimentally that the critical damping

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ratio of a cylinder undergoing heave oscillations, is a linear combination of form and friction drag. Based on these previous experiments for a surface piercing vertical cylinder and previous research on totally submerged horizontal cylinders (e.g. Moe and Verley, 1980 or Sarpkaya and Isaacson, 1981), the system damping plotted versus KC is effectively modeled as a straight line with a nonzero offset. The offset is identified as an effect of friction drag and the slope is due to form drag. Form drag for the vertical cylinders was found to be a much larger component of the total drag force even for low KC values (Thiagarajan and Troesch, 1994 or Lake et al., 2000). Based on these conclusions the following expressions were derived B⬘v⫽B⬘frictionv⫹B⬘formvKC⬅a1v⫹a2vKC

(13)

where Dv B⬘frictionv⬀b−0.5 and B⬘formv⬀ T

(14)

The subscript “v” is used to denote coefficients and forces acting on the vertical members of a TLP. Huse and Utnes (1994), following momentum theory considerations for very small values of KC, have argued that form drag becomes independent of KC in the limit of KC→0. This conclusion was numerically validated by Tao et al. (2000) for KC values below 0.01 and consequently suggests a lower bound on the validity of Eq. (13). The behavior of form drag at very low KC thus can affect the offset value of a linear fit to data. It can however be argued that the offset part of the form drag is considerably less significant compared to the slope part for most of the KC range of interest (0.1–1.0).

3. Modeling and experimental set-up Several simplifying assumptions are made to experimentally model the dynamic behavior of a Tension Leg Platform. First, the six degrees of freedom are decoupled, since heave is the selected motion of interest associated with springing. Second, the response of the model is assumed periodic since the motion characterizes a resonant condition. Third, the production facility, drilling equipment, and hull are modeled as a single rigid body. Lastly, the dynamics of the risers and tendons were not considered but must normally be included when calculating full-scale estimates. Three models were constructed: a single vertical cylinder model (1 Leg), two vertical cylinders connected by a single circular pontoon (2 Leg), and four vertical cylinders all connected by circular pontoons (4 Leg). The models as shown in Fig. 2 are 1:80 scaled models of the Jolliet TLP in the Gulf of Mexico. All of the vertical cylinders and pontoons are the same dimensions respectively between models and are merely added together to create the different configurations. All models are constructed out of aluminum tubing with the members welded together and the welds filed smooth. The bottoms of the vertical cylinders are capped and welded from the

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Fig. 2. 1 Leg, 2 Leg, and 4 Leg models used in experiments.

inside, leaving the topside open for ballast. Also, an aluminum frame is welded to the tops of the vertical columns to provide support to the model and structure for load cell and forcing mechanism attachment (see Fig. 3 for the schematic of the basic model dimensions and Table 1 for model component particulars). Experiments were conducted at the Marine Hydrodynamics Laboratory at the University of Michigan. The particulars of the towing tank facility are given in Table 1. All model tests were conducted from the tow tank carriage platform. These tests followed the experimental techniques given by Lake (1999). Damping and added mass are calculated by measuring the response to various frequencies and amplitudes of forced oscillation in heave. The models were attached via load cells to a Vertical Motion Mechanism (VMM), which created the forced oscillation at prescribed amplitudes and frequencies. The VMM has a fore and aft unit each with a dynamic drive system consisting of a ballscrew-type shaft fed through an electric motor capable of driving a stroke of ±0.102 m (4 inches) at a

Fig. 3.

Schematic of basic model dimensions

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Table 1 Model testing particulars Surface area of water tank Water depth Diameter of vertical cylinder Draft of vertical cylinder Overall length of vertical cylinder Diameter of horizontal cylinder Length of horizontal cylinder 1 Leg model Displaced mass Natural frequency 2 Leg model Displaced mass Natural frequency 4 Leg model Displaced mass Natural frequency Amplitudes of oscillation KC range b values

100.5 m × 6.7 m (330 ft × 22 ft) 3.0 m (10 ft) 0.1524 m (0.5 ft) 0.3018 m (0.99 ft) 0.5608 m (1.84 ft) 0.0876 m (0.2875 ft) 0.381 m (1.25 ft) 5.51 kg (0.3765 slugs) 0.7565 Hz 13.31 kg (0.9101 slugs) 0.7101 Hz 31.21 kg (2.134 slugs) 0.6163 Hz 3.05–61.0 cm (0.1–2.0 in) 0.2–1.8 17 979, 20 202

rate of ±0.076 m/s (3.0 in/s). The bearing sets of each unit are arranged so that only vertical loads are transmitted to the load cell attached through each unit, which removes unwanted side forces and moments. This enables the VMM to produce a uniform vertical motion with negligible motion in the other five degrees of freedom. Test measurements include force from a single load cell, displacement from both a linear variable displacement transducer and potentiometer, acceleration from an accelerometer, and the input signal used to drive the VMM. Data was filtered and recorded at 100 points/s for over 16 cycles. This data was then Fourier analyzed to identify the magnitude and phase of the force and displacement output signals, which are required to calculate the added mass and damping coefficients using the formulas derived from the system equation of motion. Each model’s natural frequency was estimated, experimentally identified, and tested. Experimental identification was required due to the initially unknown influence of the added mass coefficient on the system natural frequency. Resonance is observed when the phase difference between the measured force in heave and model displacement equals 90°. In addition, four common non-resonant frequencies were used for inter-model comparison over a common range of amplitudes as identified in Table 1. The frequencies used for comparison include 0.7565 Hz and 0.85 Hz corresponding to b values of 17979 and 20202, respectively.

4. Results and discussion The results presented address both the issues surrounding particular scaling dependencies of a single vertical cylinder as well as the hydrodynamic interaction effects

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pertaining to component superposition of inertia and damping coefficients. Furthermore, accurate damping and added mass coefficients were successfully recovered at both the natural frequency and off-resonant conditions for all three model configurations. Thiagarajan and Troesch (1994) noted straightforward estimation of damping to be less accurate in off-resonant conditions due to the inertia dominance of the total force. However with accurate Fourier component phase resolution of the time histories, repeatable results of damping coefficients for “neighboring” non-resonant frequencies were possible. (The natural frequency condition for the vertical cylinder (1 Leg model) as shown in Fig. 4, corresponds to b=17979.) This finding was consistent for all three of the models tested and illustrates the need for precise estimates of both the Fourier magnitudes and phases. 4.1. Scaling dependencies of a single vertical column Based on the 1 Leg model experiments, the non-dimensional damping coefficient (B⬘) plotted versus KC (i.e. Fig. 4) is almost a straight line with a non-zero offset. This finding is consistent with previous works by Thiagarajan and Troesch (1994) and Huse and Utnes (1994); extrapolated) for a vertical cylinder with KC greater than 0.1. The offset is primarily attributed to friction drag, and to a lesser extent on form drag. Friction drag is due to the viscous tangential stresses acting along the sides of the cylinder and scales according to the theory of laminar, oscillatory flow over an infinite flat plate. Therefore as shown by Eq. (14), the friction drag contribution to damping is proportional to b⫺0.5 and should effectively become smaller with higher b values. However as shown in Fig. 4 the offset of linear fits to the present experimental data increases rather than decreases with the small increase in b from 17 979 to 20 202. This trend is the opposite of that predicted by Eq. (14), but appears to be within the scatter of data shown in the figure. Comparing the

Fig. 4. Nondimensional damping ratio (B⬘) of single vertical cylinder vs. KC.

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average of these two offsets with the previously referenced works shows the correct b⫺0.5 dependence expected as the b variation become large. When the actual numerical values of the offsets are examined, it is found that the current offsets are twice the numerical value of the offset anticipated when scaled from the results of both Thiagarajan and Troesch (1994) at b =89 236 and Huse and Utnes (1994) at b =959 123, see Table 2. The high offset of the present data may be attributed to a disproportionate contribution of form drag to the offset value, since the models used here were on a significantly smaller scale than the previous works. As previously mentioned, the slope of the non-dimensional damping ratio lines are due to form drag caused by vortex formation and shedding along the bottom edge of the cylinder. The slope is therefore scaled proportional to Dv/T (Eq. (14)). When compared to Thiagarajan and Troesch (1994), the current values for the slopes differ less than 9.0 percent from the values anticipated by the scaled value. A comparison was not formed with a slope extrapolated from Huse and Utnes (1994), because there is insufficient information for the KC range of interest. Overall, the results closely follow the suggested form drag scaling law, the primary contributor to the slope of the system damping curve. 4.2. Hydrodynamic interaction effects in coefficient superposition Interaction effects on both the normalized added mass and damping coefficients are identified by comparing the values calculated using the superposition of individual components versus the actual values of the complete hull form. Since the diameters of the columns and pontoons are different (0.1542 m and 0.0876 m respectively), the two member types are represented by different KC and b values when calculating the individual component coefficients. The respective component dimensional coefficients are first found for a given frequency and amplitude of motion and then added together to determine the value for the complete hull form. The final composite nondimensional coefficients are then characterized solely by the KC and b corresponding to the vertical column. These values are compared directly with the actual coefficients of the complete 2 Leg and 4 Leg models (see Fig. 2), both again represented by KC and b numbers based on the diameter of the vertical column. For this work, the hydrodynamic inertia (added mass) will be combined with the Table 2 Beta dependence summary Previous ref.

Thiagarajan & Troesch (Beta =89236) Huse et al. (Beta=959123)

Extrapolated and current work Beta

Anticipated offset

Actual offset

Difference (%)

17 979 20 202 17 979 20 202

0.0076 0.0071 0.0073 0.0069

0.0115 0.0146 0.0115 0.0146

51% 105% 58% 111%

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Fig. 5.

525

Inertia coefficient: 2 Leg—Cm superposition vs Cm actual.

structure’s or component’s displaced mass since the dynamics of the TLP are governed by their sum, i.e. Cm. The inertia coefficient comparison between component superposition and the 2 Leg and 4 Leg hull shapes revealed only slight differences as shown in Figs. 5 and 6. These figures only include the plots for the 2 Leg and 4 Leg models at b =17 979, since Cm is b independent at this range of KC (as shown in Table 3). The differences between the superposition estimates and measured 2 Leg and 4 Leg model results are on the order of 16–17% for the added mass normalized by the system mass (i.e. Ca), or 4–6% of the mean Cm superposition value respectively. The agreement is thus quite reasonable. The superposition calculation of the damping coefficients includes a combination of the single vertical cylinder (1 Leg model) results from this work and the recently

Fig. 6. Inertia coefficient: 4 Leg—Cm Superposition vs Cm actual.

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Table 3 Averaged Cm values Measured values

1-leg 2-leg 4-leg

Predicted values

b =17979

b =20202

b =17979

b =20202

1.185 1.371 1.514

1.188 1.383 1.523

1.330 1.434

1.321 1.435

expanded expressions for a horizontal cylinder, the pontoon component, as given by Eqs. (9) and (12). As previously mentioned, these two different expressions (Eqs. (9) and (12)) are needed to recover the drag coefficient since the characteristic flow changes from laminar to turbulent flow. For a b of the order of 18 000, the laminarturbulent transition occurs at a KC number of approximately 0.39. A jump in the value of the pontoon damping coefficient is observed at this KC number (Figs. 7 and 8). Once the coefficient of drag is calculated, the pontoon’s damping coefficient can be determined using Eq. (6). (Note that the pontoons’ b and KC numbers must be converted to the TLP’s b and KC numbers based upon the diameter of the vertical column.) The non-dimensional damping comparison between component superposition and the 2 Leg and 4 Leg experimental data shows remarkable agreement as evidenced in Figs. 7 and 8. In the case of the 2 Leg configuration, the superposition and direct estimation are essentially equal to within experimental scatter. Likewise, the 4 Leg configuration coefficient estimates for each method are similar as shown in Fig. 8. Linear curve fits to the two sets of data in Fig. 8, “B⬘ Superposition” and “B⬘ Actual”, show that the offsets differ by 13% and the slopes by 8%.

Fig. 7.

Nondimensional damping ratio: 2 Leg—B⬘ superposition vs B⬘ actual.

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Fig. 8.

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Nondimensional damping ratio: 4 Leg—B⬘ superposition vs B⬘ actual.

Figs. 7 and 8 also show the individual contributions from the columns and pontoons to the total damping coefficient. For b of the order of 20 000, the pontoons contribute approximately 45% of the total system damping for KC values near 0.5 and reduce to 33% for KC values near 1.0. For larger b and small KC values, the pontoons’ contribution will diminish significantly as the next section on prototype scaling demonstrates. To summarize, the results presented in Figs. 5–8 depict component superposition to be an effective means of estimating platform heave inertia and damping coefficients, at least for the b and KC ranges considered. The hydrodynamic interaction effects on the effective individual component inertia and damping coefficients are visibly small and therefore justify superposition as a means of achieving accurate engineering estimates.

5. Efficacy of B⬘ as a scaling parameter and example calculation This section addresses the issue of scaling experimental results from model size, with a b value of order 20 000, to prototype, with a b value of order 90 000 000. Consider, for example, a model of the Jolliet TLP as represented by the 4 Leg model shown in Fig. 2. The damping coefficient for the model at a b value of 17 979 is given in Fig. 8. Since the various components (i.e. vertical cylinders and pontoons) follow different scaling laws, a direct extrapolation of the model damping results is not possible. To estimate the damping coefficient for any scale, we combine the various component damping contributions. These are given by Eqs. (6) and (12) for the pontoons (valid for KC beyond turbulent transition) and Eqs. (13) and (14) for the vertical cylinders. The total dimensional system damping, B, is then given by

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B =Bh+Bv

冋

=4

4 mC b KC L +2mvw(a1v+a1vKCv) 3p d h h h

⫽4

冤冢

册

冣

4 55 +0.08KC2h bhLh+mbvp2Lv(a1v+a2vKCv) m 3p 冑bh

冥

(15)

Here the subscripts “v” and “h” are used to denote coefficients and dimensions associated with the vertical columns and horizontal cylinders, respectively. To better illustrate the b dependence of Eq. (15), we use Eq. (14) to re-define the offset of the vertical cylinder as a1v⫽a⬘1v/冑bv

(16)

The non dimensional damping coefficient can then be expressed as B⬘ =B/2mw

冤

冉

冊

冢

冣冥

a⬘1v D2vrLv 4 Lh 55 Dh +0.08KC2 +p +a KC = 2 m 3p Lv 冑b Dv 冑b 2 v

(17)

where m is the total TLP displaced mass and a⬘1v is a constant, independent of b and KC. The subscripts on b and KC have been dropped with the understanding that both coefficients are now based on the diameter of the vertical columns. For the present example, the offset and slope of the vertical columns can be determined by taking the average of the slope and offset curves (b values of 17 979 and 20 202) shown in Fig. 5. That is a⬘1v=1.81, and a2v=0.0174. Eq. (17) has the form of a quadratic equation in KC, with the offset a function of b⫺0.5. The offset is due to both TLP components, the pontoons and the vertical columns, while the linear term is due solely to the vertical columns and the quadratic term is due solely to the pontoons. To understand the contributions of the various TLP components and to extrapolate TLP model damping to full scale, we write the numerical form of Eq. (17) for two values of b:19 000 (an average model scale b) and 90 000 000 (a typical prototype value of b). The equations become: Model b=19 000 B⬘=(0.0088+0.0093)+0.0123KC+0.0031KC2

(18)

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and Prototype b=90 000 000

(19)

B⬘=(0.0001+0.0001)+0.0123KC+0.0031KC2.

The first and second terms in the offset pairs of Eqs. (18) and (19) represent the contributions from the pontoons and vertical columns respectively. The applications of Eqs. (18) and (19) are given in Table 4. Table 4 shows the non dimensional damping coefficients and ratios of pontoon damping to total TLP damping for various KC values. The following observations may be made: 5.1. Damping offset for large b Since the offset of the damping curve is modeled by a b⫺0.5 behavior, the offset does not significantly contribute to the full scale damping until the KC number is of the order 0.01 or smaller. However, as noted in the discussion following Eq. (14), Huse and Utnes (1994) and Tao et al. (2000) suggest that form drag becomes independent of KC in the limit of KC→0. This places a practical limit on the method restricting the above equations to KC values greater than 0.01. 5.2. Contribution of the pontoons to the total system damping The pontoons clearly add a disproportionate amount of damping in model scale. For values of 0.1ⱕKCⱕ0.5, the model scale pontoons contribute an average of 42% to the total model scale damping while the full scale pontoons contribute only 11% to the full scale damping. It is not until the KC numbers approach a value of 2.0, that the pontoons have comparable influence on the total system damping for model and full scale.

Table 4 Model and prototype total damping and percent of pontoon contribution to total damping b=19 000

b=90 000 000

KC

B⬘

% pontoon to total B⬘

% pontoon to total

0.1 0.3 0.5 1.0 1.5 2.0

0.0193 0.0220 0.0249 0.0334 0.0434 0.0549

45.7% 41.2% 38.3% 35.5% 36.2% 38.6%

10.4% 9.6% 12.5% 20.5% 27.5% 33.4%

0.0015 0.0042 0.0072 0.0156 0.0256 0.0371

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5.3. Critical damping ratio for model and full scale geosims While the results for the system damping given in this paper are expressed as B⬘, the critical damping ratio, typically denoted as z or Z (see Eq. (8)), may be determined by including the added mass and interpreting the frequency of oscillation as the natural frequency of the system. For example, by replacing (1+Ca) in Eq. (8) with Cm read from Fig. 6 or Table 3, the values for B⬘ in Table 4 can be used to determine z for system natural frequencies corresponding to b values of 19 000 or 90 000 000. From Table 4, a KC value of 0.1 gives critical damping ratios of 1.30% and 0.10% for b values of 19 000 or 90 000 000, respectively. In other words, the model TLP has thirteen times the critical damping that the prototype would have for the same KC value, exclusive of appendages, mooring lines, and risers. This difference decreases as KC numbers increase, e.g. KC =0.3 corresponds to critical damping ratios of 1.50% and 0.28% for model and prototype, respectively. At this KC value, the model critical damping is only five times that of the prototype’s. The KC =0.1, the prototype value compares reasonably well with the Jolliet TLP value published by Thiagarajan and Troesch (1994) of 0.11%.

6. Conclusions The objective of this research was to further evaluate the scaling dependence of a single vertical cylinder as well as to identify the effects of interaction on the heave added mass and heave damping estimation of a TLP. Experiments were conducted measuring the hydrodynamic response of a single vertical cylinder component, 2 Leg, and 4 Leg TLP models subject to oscillatory heave excitation. The response for the individual horizontal pontoon component was derived from recently expanded theoretical expressions based on experimental observations (Bearman and Russell (1996), Chaplin and Subbiah (1998) and Chaplin (2000)). The vertical column nondimensional damping values were compared with previous works (Thiagarajan and Troesch (1994, 1998)) and the scaling laws associated with the offset and slope of the curves evaluated. The determination of interaction effects involved the comparison of measured added mass and damping coefficients calculated from the superposition of individual hull components versus the measured response of the 2 Leg and 4 Leg hull configurations. The data presented here for the composite TLP have shown unequivocally the nonlinear trend of B⬘ with KC. Quadratic behavior, due in part to the horizontal pontoon, have shown to be a better explanation of the trend than linear fits for values of KC greater than 0.01. On one hand, we have the trend of flattening of the curve as KC→0, while on the other hand, we have a nonlinear trend at KC ⬎1, indicating as mentioned before some complex vortex interactions. Results indicate that the offset of the relative system damping curve for a single cylinder may further depend on form drag as a function of model scale rather than solely on the b dependence due to friction drag. Form drag contributions to the relative damping matched the expected values scaled based on the diameter to draft

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dependence. Inertia coefficients estimated using superposition were slightly lower and therefore more conservative than those measured for the complete hull configurations. In summary, component scaling is an effective method by which platform designers can estimate the hydrodynamic response of a prototype hull in heave. The opinions, equations, and conclusions found herein are not those of the Department of Transportation or of the United States Coast Guard.

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