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2 Proceedings of the th International Conference on Ocean, Offshore and Arctic Engineering OMAE May - June,, St. John s, Newfoundland and Labrador, Canada OMAE- HYDRODYNAMIC COEFFICIENT MAPS FOR RISER INTERFERENCE ANALYSIS Suneel Patel H Offshore Inc. Shankar Sundararaman H Offshore Inc. Pete Padelopoulos H Offshore Inc. Kamaldev Raghavan Chevron Energy Technology Company Metin Karayaka Chevron Energy Technology Company Houston, TX, U.S.A Paul Hays Chevron Energy Technology Company Yiannis Constantinides Chevron Energy Technology Company ABSTRACT Riser wake interference analysis is conducted based on analytical / semi-empirical models such as Blevins and Huse s models. These models are used for modeling the reduction in particle flow velocity due to the presence of a cylindrical object upstream in the flow path. However, these models are often too conservative and accurate only for circular cylinders. Many top tensioned risers (TTRs) use vortex induced vibration (VIV) suppression devices such as strakes or fairings. There is a need for alternate methods to obtain drag and lift coefficient datasets for circular cylinders with strakes and fairings. Two such approaches are to obtain data from Computational Fluid Dynamics (CFD) simulations or from experimental large-scale model test data. Interpolation and/or extrapolation methods are needed to obtain additional data points for global riser finite element analysis. This paper presents a methodology to obtain hydrodynamic coefficients for TTRs with VIV suppression devices. The proposed methodology uses a combination of empirical formulas based on Blevins model and numerical interpolation techniques along with experimental tow tank test data and CFD analysis. The resulting data is then input as user-defined drag/lift coefficients into a global riser finite element analysis to obtain a more realistic riser system response. INTRODUCTION Analytical and semi-empirical models such as Blevins and Huse s are used for modeling the reduction in current velocity due to the presence of a cylindrical object upstream in the flow path. However, these models are often too conservative and accurate only for circular cylinders. With the presence of VIV suppression devices on both the upstream and downstream risers, the risers no longer react as circular cylinders. Additionally, existing models found in industry guidelines are based on approximate theoretical models of bare cylinder wake and nominally checked against small scale tests at low Reynolds numbers. In actual conditions, the Reynolds number is sufficiently high for risers fitted with VIV suppression devices, []. Matrices of hydrodynamic drag and lift coefficients are required for the user-defined wake interference capabilities in global riser finite element analysis software. These drag coefficients/drag coefficient factors and lift coefficients vary as a function of the distance of the downstream riser from the upstream riser. This variation in drag and lift coefficients due to the wake effects induced by the upstream riser depend on a number of factors including the distance between risers, drag diameter, geometry of the riser (bare, fairing, strake), current velocity, current direction and Reynolds number. Copyright by ASME

3 Wake Interference Analysis Wake interference studies between riser pairs are carried out by assessing a number of factors related to the flow regime, and the structure. These parameters also include effects of any VIV suppression devices. The interference between two risers is evaluated by considering a conservative but realistic wake model in which the flow behind the upstream cylinder is modified resulting in reduced velocity downstream. The downstream structure drag and lift coefficients used to model this are expressed in terms of the upstream drag diameter because of the geometric similarity of the wake field behind the upstream structure. A simplified riser schematic in profile view of the upstream-downstream riser pair and the current profile is illustrated in Figure. theory and, F D and F L denote the drag/lift force per unit length for the downstream riser, U is the undisturbed current speed, D u is the upstream drag diameter and ρ is the fluid density. As illustrated in Figure, the upstream cylinder is at (X u, Y u ), and the downstream cylinder is at (X d, Y d ). The in-line (L) and transverse (T) distances can be calculated using Equation and Equation, respectively: L = dx cos α + dy sin α () T = dx sin α + dy cos α () where α is the current direction angle, dx=x d -X u and dy=y d - Y u. Current Profile Upstream Riser Downstream Riser dx Downstream Riser (X d,y d) T L dy Upstream Riser (X u,y u) Current Direction Y α Global Axes Current Direction X (α=) Figure Upstream-Downstream Riser Pair with Current The plan view of the upstream-downstream riser pair is illustrated in Figure. The drag and lift coefficients are defined in terms of functions of non-dimensional distances, u and T/D u, where L and T denote the in-line and transverse distances between the centerlines of the downstream and upstream cylinders, and D u is the drag diameter of the upstream cylinder. When the downstream cylinder is within the wake generated by the upstream cylinder, it experiences a reduced drag force due to reduced mean current velocities in the wake and a lift force as a result of varying pressures across the cylinder. The drag and lift coefficients for the downstream cylinder are calculated as functions of its position in the wake as given in Equation and Equation, respectively: Figure Cylinder Model Plan View with Current When considering the in-line and transverse distances between two adjacent risers, there are three key wake regions as defined in Figure. The three key regions are the suction region, mid-range region and the far-field region. The suction region corresponds to low in-line and transverse distances. In the suction region, the drag and lift coefficient factors are reduced so adjacent risers move closer to each other. The midrange region corresponds to in-line and transverse distances that are typically experienced during -year and -year return period environments. The far field region corresponds to in-line and transverse distances that are great enough to produce little change in the nominal drag and lift coefficients. C d = C d λ d (L D u, T D u ) = F D ( ρu D u ) () C L = λ L (L D u, T D u ) = F L ( ρu D u ) () where, C d denotes the undisturbed drag coefficients for the downstream cylinder in the absence of wake effects; λ d and λ L are the coefficient factors calculated from the appropriate wake Copyright by ASME

4 where, a, and a are Blevins constants; and the other terms are defined above. In the typical Blevins formulation for bare cylinders, a =, and a =.. The drag coefficient is then described by Equation 9: C D = C d (V d V d ) (9) The mean lift coefficient is given by Equation : C L = a dc D = d(t/d d ) a ( TC dd d ) ( C dd u ) / ( LC u D u L a ( C dd u ) / exp ( a T ) exp ( a T )) () L C d D u L C d D u L where, T ; a is the Blevins lift constant; D d is the downstream drag diameter; and, C u is the nominal upstream drag coefficient. In the typical Blevins formulation for bare cylinders, a =-.6. The lift coefficient is anti-symmetric about the wake and its sign changes for T<. Figure Key Wake Regions Huse s Formulation Huse s formulation, [], is an analytical approach used to model downstream velocity reduction due to wake by calculating drag forces with the absence of lift forces. The downstream velocity at a coordinate location (L, T) due to the presence of the upstream cylinder is given by Equation : V d = V d k V u ( C dd u x s ) / exp ( k ( T b ) ) () x s = L + D u /C d (6) b = k (C d D u x s ) / (7) where, V d and V u denote the undisturbed downstream and upstream current velocities; k, k and k are Huse s constants; x s and b are defined in Equation 6 and Equation 7, respectively, and the other terms are defined above. In the typical Huse s formulation for bare cylinders, k =, k =. and k =.69. Blevins Formulation Blevins formulation, [], is an analytical approach used to model both downstream velocity reduction along with wake lift forces. The downstream velocity at a coordinate location (L,T) due to the presence of the upstream cylinder is given in Equation 8: V d = V d a V u ( C dd u ) / exp ( a T ) (8) L C d D u L The lift coefficient is dependent on the downstream drag diameter. This approach of using lift and drag coefficients is valid for cylinder-to-cylinder distances of greater than - times the diameter of the upstream cylinder. At distances less than this value, the interaction becomes more complex with suction as well as drag forces coming into play. Hydrodynamic Maps To develop hydrodynamic coefficient maps for the key wake regions, shown in Figure, experimental testing was conducted to account for the higher Reynolds number along with non-circular VIV suppression devices. CFD simulations were conducted to account for data that was not available from the experiments. Global analysis software packages such as FLEXCOM-D and Orcaflex, [] [6], allow for use of user-defined wake parameters. These user-defined wake parameters have the ability to capture the three critical wake regions for non-circular cylinders. Therefore, hydrodynamic coefficient maps are needed to accurately capture the wake responses of riser systems with VIV suppression devices. This paper discusses the development of hydrodynamic coefficient maps from the discussion presented in Constantinides et al, []. The full experimental load case matrix is presented in Table and Table. The properties of the VIV suppression devices corresponding to the cases discussed in this paper are presented in Table. In addition to the experimental tests for case and case, additional CFD analysis was conducted. The hydrodynamic maps for case and case are further discussed below. Copyright by ASME

5 Case No. Table Load Case Matrix for Study, [] Upstream Downstream Riser Pipe Riser Pipe Riser Pipe with. Riser Pipe with ADFS Fairings (9 Span) Riser Pipe with ADFS Fairings (7 Span). Riser Pipe with. Riser Pipe with Bare Riser Pipe 6. Riser Pipe with. Riser Pipe with Riser Pipe with ADFS Fairings (9 Span) Riser Pipe with ADFS Fairings (7 Span). Riser Pipe with. Riser Pipe with Table Load Case Matrix Schematics Case No. Schematic 6 Table VIV Suppression Devices for Case Study, [] Component S/D () Pitch ADFS Fairings (7 Span). N/A ADFS Fairings (9 Span). N/A N/A 7D / Denotes span/diameter ratio. DRAG COEFFICIENT DATA FROM LAB EXPERIMENTS AND CFD SIMULATIONS To understand the effect of vortices being shed from an upstream riser with non-cylindrical VIV suppression devices on a downstream riser as a function of the spacing between them and their offset relative to the flow direction, experimental testing was conducted at the Institute for Ocean Technology with the assistance of Oceanic Consulting Corporation. Additional details regarding the experimental testing are provided in Constantinides et al, []. A sequence of -dimensional, transient CFD simulations of a pair of cylinders was performed for certain load cases; each containing a variation of cylinder position, current velocity, cylinder geometries and VIV mitigation devices. The load cases and VIV suppression devices considered in the CFD simulations were case and case (as defined in Table and Table ). The simulations were performed at full scale, prototypic Reynolds Numbers (Re ADFS-9 x 6, Re ADFS-7 9x, Re Strake 6x ). To accurately produce lift and drag values for risers, a transient simulation approach and a high resolution turbulence model to resolve the wake vortex structures is considered. Using this approach, a portion of the riser is modeled as it is subjected to a constant undersea current. The simulation proceeds in a time accurate fashion, resolving the - dimensional vortices that are created in the wake, and tracking the propagation downstream. The simulations are run for - shedding cycles of the riser to acquire statistics about the lift and drag as well as the time averaged velocity field in the wake. INTERPOLATION METHODOLOGY Experimental and CFD simulation data (if necessary) are interpolated to obtain values for various distances from the centerline of the upstream cylinder. A matrix of values is generated in terms of ratios of longitudinal distance over drag diameter ( u ) and transverse distance over drag diameter (T/D u ). Three types of interpolation techniques are used:. Linear interpolation;. Blevins function fit in conjunction with piecewise spline interpolation;. Piecewise Hermite interpolating polynomial. Linear interpolation and linear extrapolation ensures a straight line fit over each data interval. Spline interpolation uses low-order polynomial fits in each of the data intervals and ensures that they fit smoothly (i.e., splines have continuous second derivatives). Hermite interpolating polynomials also make use of polynomials fits, but the coefficients are instead obtained using a recursive division process. Typically, spline interpolation is more accurate than Hermite interpolating polynomials if the data are values of a smooth function. However, Hermite interpolating polynomials have no Copyright by ASME

6 Cd-downstream/Cd-nominal Cd-downstream/Cd-nominal Cd-downstream/Cd-nominal overshoots and have fewer oscillations if the data is not smooth (non-monotonic). Additionally, the Blevins function is used to fit a bulk of the data points along the transverse direction. The Blevins function is also incorporated in a piece-wise sense; i.e., different sets of Blevins coefficients are used to fit pairs of data points. The Blevins coefficients used to generate the tables are based on the nominal drag coefficients and diameters used in the respective CFD simulation/laboratory experiment. Two sets of Blevins functions are used, depending on whether the drag coefficients or lift coefficients are generated. Approach to Extract Drag Coefficients A four step process is used to extract the drag coefficients for all cases presented in Table and Table. The steps to generate the drag coefficients are provided using case as an example. The contour plots of drag coefficient ratios are provided for case and case to show how these cases differ. The steps used to generate the drag coefficients are as follows:. The drag coefficient ratios are fitted along the in-line longitudinal direction as shown in Figure for case. A piecewise cubic Hermite interpolating polynomial is used to interpolate or extrapolate data based on each set of experimental/cfd simulation data points.. The Blevins function is used to generate data and fit the data along the transverse direction (, T /D u, T /D u, T n /D u ), where T n /D u refers to the ratio of the transverse distance of the nth data point to the upstream riser drag diameter. This fit is carried out along each set of data points obtained in the in-line longitudinal direction (, L /D u, L /D u, L m /D u ), where L m /D u refers to the ratio of the longitudinal distance of the mth data point to the upstream riser drag diameter. Wherever the Blevins fit does not pass through all the data points (typically in the suction region), a combination cubic spline and piecewise cubic Hermite interpolating polynomial fit is used to ensure that the final fit passes through all the data points obtained. Step is illustrated in Figure.. The drag coefficient ratios are then fitted along the inline longitudinal direction (, L /D u, L /D u, etc.) as illustrated in Figure 6. A combination of linear interpolation (in the suction region) and piecewise cubic Hermite interpolating polynomial (when the distance between the two risers is great enough) is used to extrapolate the remaining data points along the longitudinal direction (, L /D u, L /D u, etc.).. Steps - are repeated along every longitudinal and transverse line of interest to generate a contour plot. This data, including the illustrations in Figure 7 and Figure 8, is generated with a longitudinal spacing of. between L=Du and L=Du,. between L=Du and L=Du, and. greater than L=Du and a transverse spacing of. between T= and T=Du, and. greater than T= Du. This spacing is adopted because most global riser analysis software, such as FLEXCOM-D and Orcaflex, [] [6], require fine spacing when the two cylinders are in close proximity. -. -upstream Blevins Eq. Hermite Polynomial Fit Huse Eq. Figure Case ; Drag Coefficient Ratio Fitted along Longitudinal Direction (T/D u =) (Step ) Piecewise Blevins fit Piecewise cubic spline+cubic Hermite interpolating polynomial fit (Suction Region) T/D-upstream Blevins Eq. Blevins Eq./Spline Fit Huse Eq. Figure Case ; Drag Coefficient Ratio Fitted along Transverse Direction ( u =) (Step ) Piecewise Cubic Hermite Interpolating Polynomial Fit -. -upstream Blevins Eq. Linear Fit (Suction Region) Hermite Polynomial/Linear Fit Huse Eq. Figure 6 Case ; Drag Coefficient Ratio Fitted along Longitudinal Direction (T/D u =) (Step ) Copyright by ASME

7 T/D T/D / Contour lines are defined as C d, / C d. Figure 7 Case ; Contour Plot of Drag Coefficient Ratios (Step ) Drag Coefficient Ratio (Non-Buoyant Fairing-Strake) / Contour lines are defined as C d, / C d. Figure 8 Case ; Contour Plot of Drag Coefficient Ratios (Step ) When comparing the results of the drag coefficients for case and case, both cases do not differ significantly in the suction region (where T/D u ranges from - to and u is less than ). However, the actual in-line and transverse distances are much higher for case as the upstream outer diameter is much smaller than that of case. All other load cases provided in Table and Table follow a similar trend. Approach to Extract Lift Coefficients A similar approach is used to generate lift coefficients. The start point for generating the lift coefficient values are the corresponding drag coefficient factors. Then the mean lift coefficients are obtained by suitably solving a modified form of the Blevins equation. This equation is expressed as: C l a a C d C d ( C d C d ) ( Tc dd d Lc u D u ) ; T, L > () where a is obtained from the Blevins function fit for the drag coefficients, and C d /C d is the drag coefficient factor obtained from the extraction of the drag coefficients. This process is done piecewise (i.e., for every pair of experimental/cfd simulation data points). Lift coefficient data should ideally be extracted from experimental or CFD datasets. However, lift coefficient sets may not be available from experiments or CFD analysis. Two different methods can be adopted, depending on the availability of lift coefficient data. These two methods are as follows:. Lift coefficients are obtained from the CFD simulation datasets. The modified Blevins equation is then solved to generate the constant, a and the values obtained are used to generate data and fit the data along the transverse direction (, T /D u, T /D u, etc.).. No lift coefficients are provided from the experimental datasets. Although lift data is not available, lift data needs to be generated because the absence of lift from riser interference analysis results in a less conservative result. Consequently, Blevins function lift coefficient value applicable to circular cylinders of a =-.6 is used in the above expression (Equation ) to generate the lift coefficients and fit the data along the transverse direction (, T /D u, T /D u, etc.). If the lift coefficient values exceed., the coefficients are scaled to an absolute maximum of.. This is because the value of a =-.6 is empirical and generally true for smooth cylinders and must be suitably tuned in cases where the interacting structures are not smooth cylinders. Since this kind of empirical tuning would require experimental testing for all pairs, an alternative is choosing a reasonable upper bound for the lift coefficients. Blevins shows maximum lift coefficients restricted to., []. This serves as a justification for setting the upper bound for C l to. and scaling them suitably in cases where the values are exceeded. The lift coefficients are then fitted along the in-line longitudinal direction (, L /D u, L /D u, etc.) for intermediate values of u by using a piecewise cubic Hermite interpolating polynomial fit. The lift coefficient contour plots using method, directly obtained from CFD simulation, are shown in Figure 9 and Figure for case and case, respectively. The lift coefficient contour plots using method are shown in Figure and Figure for case and case, respectively. In this instance, the modified Blevins equation is used to generate the constant, a, and the values obtained are used to generate the lift coefficient curves and then scaled. The non- 6 Copyright by ASME

8 T/D T/D. -.. T/D T/D Lift Coefficient (Drag Extrap.) (Buoyant Fairing-Strake) linear interpolation procedure outlined is used to obtain coefficients at multiple points within and outside the range specified. The interpolated drag coefficients are shown using contours and the data points from which the contours are obtained are shown using red filled circles CFD Data Figure 9 Case ; Contour Plot of Lift Coefficients Obtained Using Lift Coefficient Data (Method ) Figure Case ; Contour Plot of Lift Coefficients Obtained Using Lift Coefficient Data (Method ) Lift Coefficient (Slick Fairing-Strake). -. CFD Data Figure Case ; Contour Plot of Lift Coefficients Obtained Using Drag Coefficient Data (Method ) Figure Case ; Contour Plot of Lift Coefficients Obtained Using Drag Coefficient Data (Method ) For the example cases presented considering riser pipe with ADFS fairings upstream to riser pipe downstream with strakes, the lift contour plots generated from the two methods yield quite different results. The difference can be attributed to the fact that a Blevins fit is used to obtain lift coefficients from drag data. While the lift forces are concentrated between values of to and T/D values of to +./-. for contours generated using CFD data points, the lift coefficients extrapolated from experimental drag coefficients have a wider spread Lift Coefficient (Drag Extrap.) (Non-Buoyant Fairing-Strake) CONCLUSIONS When considering adjacent risers with VIV suppression devices, drag coefficients for both riser pairs can be derived using experimental test data and CFD simulations with the use of linear interpolation, piecewise spline interpolation and a Piecewise Hermite interpolating polynomial. For cases considering u > and T/Du =, shown in Figure, the drag coefficient ratios for all the experimental data and the corresponding data fit are higher than the corresponding Blevins and Huse s drag coefficient ratios. The reduction in 7 Copyright by ASME

9 drag coefficient will affect the clearance between adjacent riser pairs. Lift coefficients can also be derived from either drag data or CFD simulations using the modified Blevins equation provided by Equation. As the validity of lift coefficients generated from either method is somewhat debatable, the more conservative of the two techniques, evaluated on a case by case basis, should be used when evaluating the global riser system response. The resulting drag and lift coefficients are then input as user-defined drag and lift coefficients into a global riser finite element analysis to obtain a more realistic riser system response. Further discussion regarding the use of inputs to userdefined drag and lift coefficients and its effect on riser-to-riser clearance is provided by Sundararaman et al, []. NOMENCLATURE ADFS AIMS Dual Fin Splitter CFD Computation Fluid Dynamics TTR Top Tensioned Riser VIV Vortex Induced Vibration REFERENCES [] Huse,.E., - Experimental Investigation of Deep Sea Riser Interaction ; OTC996-87; May 996. [] Blevins, R.D. Forces on and Stability of a Cylinder in a Wake ; J. OMAE, 7, 9-,. [] Constantinides, Y., Raghavan, K., Karayaka, M., Spencer, D., - Tandem Riser Hydrodynamic Tests at Prototype Reynolds Number, OMAE-9,. [] Sundararaman, S., Saldana, D., Patel, S., Andrew, B., Padelopoulos, P., Karayaka, M., Raghavan, K., Hays, P., - Interference Assessment between Top Tensioned Risers for Tension Leg Platform Using a Simplified Screening Approach, OMAE-66,. [] Marine Computational Services (MCS) FLEXCOM-D Three-Dimensional Nonlinear Time Domain Offshore Analysis Software. Version 7.,. [6] Orcina Ltd. ORCAFLEX User Manual. Version 9.a, 9. 8 Copyright by ASME