Proceedings of OMAE ‘01
20 th International Conference on Offshore Mechanics and Arctic Engineering
June 3-8, 2001, Rio de Janeiro, Brazil
OFT 01-1364
Anthi Miliou, Spencer J. Sherwin, J. Michael R. Graham
Department of Aeronautics, Imperial College of Science, Technology and Medicine
London SW7 2BY, U.K. e-mail: a.miliou@ic.ac.uk
, s.sherwin@ic.ac.uk
, m.graham@ic.ac.uk
ABSTRACT
In order to gain a preliminary understanding of the fluid centerline of the body, would fail at the sections where the vortices are not shed normal to the axis of the body. The need dynamics developed past a curved riser pipe, a numerical investigation into the flow past curved cylinders at a Reynolds number of 100 has been performed. To approximate the flow conditions on curved riser pipes, different velocity profiles and flow directions were applied and the corresponding results compared. In addition, the fluid dynamic loading and the wake structures for curved cylinder flows were investigated.
The fully three-dimensional simulations were computed with a spectral/ hp element method. The computational results were compared with experiments undertaken in the towing tank facility of the Department of Aeronautics of Imperial College.
INTRODUCTION
As the need to develop deep-water reservoirs for the exploitation of hydrocarbons below the sea has been increasing, riser pipes are now required to operate in depths up to 2000m.
Catenary risers, which are effectively more flexible riser pipes, are preferred over vertical risers for deep water reservoirs as they can accommodate larger motions of a floating vessel.
While steel catenary risers are normally associated with large radii of curvature, flexible catenary risers can have quite drastic localised deformation. A catenary riser is subject to fluid dynamic loading as a combination of waves and current and may be in tangential contact with the seabed. In this case part of its length lies near horizontal at the seabed being semi- or even fully- buried occasionally.
Understanding the unsteady flow patterns developed behind a deformed catenary riser pipe is a problem that requires investigation in three dimensions. The empirical models
(Morison's equation, Shear 7 [5]) in commercial use are applied on a two-dimensional sectional basis. A sectional approach that assumes two-dimensional flow for a curved pipe, for the cases where the free-stream velocity vector is not normal to the to develop the computational means for modelling and understanding the fluid dynamic loading and the vortex shedding patterns past curved riser pipes is therefore very important. With the motivation stemming from the offshore industry hence, the computational investigations described in the current work have focused on gaining a better understanding of the three-dimensional dynamics of the flow past curved cylindrical pipes and their wake structures.
COMPUTATIONAL METHOD
Computations have been carried out using Nektar [1, 8,
10], a three-dimensional incompressible Navier-Stokes solver, based on the spectral/ hp element method. This spectral method is a high-order discretisation method where one can increase simultaneously the number of elements (h-refinement) and the order, P , of the interpolating polynomials in the domain (prefinement). Unstructured conforming discretisations have been used in the computational domain. For smooth solutions this spectral/ hp element method accounts for exponential convergence.
The mesh generation for all the computational investigations was accomplished with Felisa where a modified advancing layers method [3] is employed near the pipe wall regions and a method based on the advancing front technique
[4] is employed for the rest of the domain. Hybrid meshes were constructed where the boundary layer mesh was composed of prismatic elements growing in the direction of normal to the pipe surface and the interior of the domain was filled with tetrahedra.
RESULTS AND DISCUSSION
The first set of computations has been performed at a
Reynolds number, based on the diameter, of 100 and is aimed at
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investigating the effects of curvature compared to a straight cylinder and the effects of the interaction with the seabed as well as a sheared current profile. The computations included cases of a quarter turn with either a horizontal or a vertical extension, subject to either uniform or sheared flow in directions normal or parallel to the plane of curvature.
Symmetry boundary conditions were assigned to the plane boundaries at the intersection with the curved cylinder.
Specifically, the five cases presented in this paper include:

a quarter turn in uniform flow normal to the plane of curvature

a deformed quarter turn with a vertical addition of two diameters being semi-buried in the ground and subjected to shear normal flow

a deformed quarter turn with a vertical addition of two diameters being semi-buried in an inviscid wall and subjected to shear normal flow

a deformed quarter turn with a vertical addition of two diameters being semi-buried in an inviscid wall and subjected to uniform normal flow

a quarter turn with a horizontal extension of ten diameters in uniform flow parallel to the plane of curvature
The selection of the above cases was done on the basis that a generic shape for a catenary riser would consist effectively of a quarter turn with the vertical extension to the offshore vessel and the horizontal extension lying on the seabed partially- or fully- buried in the ground.
This work represents a fundamental study of the influence of the seabed and of a sheared current profile on a curved cylinder. Since the main focus was to gain an understanding of the fundamental mechanisms of vortex shedding past curved cylinders, a low Reynolds number, i.e. 100, was chosen. At this low Reynolds number the detailed flow structures can be accurately captured numerically and the flow physics of the straight cylinder case is well understood. In future computations, the Reynolds number will be increased so that the development of intrinsic three-dimensionality in the wake can be studied in a progressive manner. a) Quarter turn in uniform normal flow
The flow past a quarter turn with an aspect ratio of 12.5, defined as the ratio of the radius of rotation from the centre of curvature to the cross-section diameter of the cylinder, subject to uniform flow normal to the plane of curvature has been simulated. The diameter of the cylinder was 1 unit. This aspect ratio is in the correct order of magnitude and representative for some flexible catenary riser pipes. Steel catenary riser pipes, on the other hand, exhibit much larger curvature ratios. In terms of the aspect ratio, hence, the results for the flow past curved pipes computed in this work have more relevance to the flexible riser pipes, where the three-dimensional effects are much stronger, as opposed to the steel catenary riser pipes.
The hybrid mesh created for this computation comprised a total of 7464 elements, out of which 790 were the prismatic boundary layer elements.
Figure 1 depicts a cut in the computational domain in the streamwise direction and exhibits symmetry about the diagonal of the region. The flow direction was in the y-direction, effectively into the page and therefore the inlet and outlet planes would be parallel to the page. In the streamwise direction, the distance to the outlet was 15 diameters from the axis of the body and the inlet distance was 5 diameters. Boundaries A and
B include an intersection with the curved pipe and so a symmetry boundary condition was imposed. Boundaries C and
D correspond to free-stream boundaries assigned a uniform velocity boundary condition, V = [0, 1, 0].
Figure 1 : Computational boundaries (not to scale)
The computation was performed originally with 3 modes per edge of each element (P = 2), and was then restarted with 5 modes (P = 4) and finally with 7 modes (P = 6). When the computation was performed with 7 modes, the mesh accounted for 715,456 local degrees of freedom per variable. Blockage for these computational cases was 4.5%.
Figure 2 depicts the force coefficients in the x-, y-, and z- directions. The x- and z- directions represent the horizontal and vertical component of the transverse force respectively and the y-direction is the direction of the drag. The x- and z- forces come out to be of equal amplitude and frequency as expected for an axisymmetric body with symmetry boundary conditions.
As Figure 2 demonstrates, the curved cylinder, unlike the straight cylinder which exhibits a drag force with a frequency equal to twice that of the lift, accounts for a second lower amplitude peak in the drag force as an effect of curvature.
Moreover, the mean in the lift force coefficient is not zero as is the case for a straight two-dimensional cylinder. Unlike the straight two-dimensional cylinder, the two vortices shed per lift cycle do not have to be of equal strength. It should also be mentioned, to show that this result is free from the effect of the resolution of the mesh, that when the polynomial order for the computation was increased from 4 to 6, the lower amplitude peak in the drag force was still apparent.
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in phase along its length at this Reynolds number. In-phase shedding at this Reynolds number would also be expected from a straight cylinder implying that a sectional analysis might still be appropriate for these flow conditions. Figure 4(a) shows the extraction of pressure isocontours at a level of –0.30
ρu 2 , which corresponds to a pressure coefficient of –0.82 (nondimensionalised by the pressure coefficient at stagnation being equal to 1). Figure 4(b), on the other hand, shows the extraction of pressure isocontours at a level of –0.40
ρu 2 projected onto the xy-plane, which corresponds to a pressure coefficient of –1.02.
As Figure 5 depicts, this pressure level is geometrically included in the previous one as it corresponds to higher suction.
Figure 2 : Quarter turn in uniform flow normal to the plane of curvature, Re d
= 100
The power spectral density for the forces in x-, y- and z- directions are shown in Figure 3 . With the blockage in the computational domain being 4.5%, the Strouhal frequency for a quarter turn in uniform normal flow has been found to be equal to 0.175 at a Reynolds number of a 100. The drag force exhibits two frequencies, 0.175 and 0.350. The latter is equal to twice the Strouhal frequency and they are both of approximately the same amplitude in the power spectra.
Figure 4 : Pressure isocontours for a quarter turn in uniform normal flow, Re d
= 100; (a) p = –0.30
ρu 2 , C p
= –0.82;
(b) p = –0.40
ρu 2 , C p
= –1.02, projection in the xy-plane
Figure 3 : Power spectra density for (a) F x
; (b) F y
; (c) F z
Extraction of pressure contours at slices normal to the axis of the curved cylinder in variable locations along the span indicated in-phase shedding as can be observed from Figure 2.
Extraction of pressure isocontours in three-dimensions verified this as the axis of the vortex was found to be parallel to the axis of the cylinder as Figure 4(b) depicts. The ring, being an axisymmetric body, should have indeed been expected to shed
Figure 5 : Pressure contours for a quarter turn in uniform normal flow, Re d
= 100; p = –0.30
ρu 2 and p = –0.40
ρu 2
The root mean square (rms) value of the fluctuating lift coefficient was found to be equal to 0.16

0.01. Graham [7] compiled results produced from a number of two-dimensional codes on the comparison of fluctuating lift coefficients for a circular cylinder at a Reynolds number of 100. Subsequently the data was updated and extended in Bearman [6]. The C
Lrms values covered a range from 0.16 to 0.34 and C
Lrms was found to be very sensitive to detailed aspects of the simulations [6].
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b) Curved riser in contact with the seabed, sheared normal flow
Following the same procedure as for case (a), a quarter turn with a vertical extension of two diameters ending semi-buried in the seabed and subject to sheared normal flow was also considered. This structure had the same aspect ratio, 12.5, as before and the computational domain comprised of 7902 elements, out of which 864 were the prismatic surface elements.
Artificial blending had to be applied at the touch down point of the riser with the seabed to alleviate mesh generation issues. Nevertheless, the resultant geometry has been effectively an even closer representation of a catenary riser being semiburied in the seabed having seabed material adhered to it in the long run of its lifetime in a well.
The bottom plane of the computational domain was represented with a viscous wall as a boundary condition and an exponentially decaying sheared profile was chosen for the computational planes assigned a velocity boundary condition.
The shear profile applied was: v = 1-e
-(z+12.5)
This shear profile resembles that of a boundary layer developing from the seabed. The velocity is equal to one at the top of the riser and equal to zero at the bottom to conform with the wall boundary. The shear profile is such that at 4.6 diameters from the seabed, the velocity achieves 99% of its maximum value.
The computation was performed originally with 3 modes per edge of each element (P=2) and was then restarted at P=4 and P=6. When a computation was performed at sixth polynomial order, the mesh accounted for 760,536 local degrees of freedom per variable. Blockage for these computational cases was roughly in the order of 6%.
Figure 6 depicts the force coefficients in the x-, y-, and z- directions along with the pressure contours at slices normal to the cylinder axis. The x- and z- directions represent the horizontal and vertical component of the transverse force respectively and y- is the direction of the drag. The force plots indicate a modulation in the horizontal and vertical component of the transverse force. Moreover, unlike case(a), the extraction of pressure contours at the slices demonstrates lack of correlation along the span of the riser pipe.
Figure 6 : Curved riser in contact with the seabed.
Shear flow normal to the plane of curvature, Re d
= 100
Furthermore, in order to investigate the wake structures developed by such a type of flow the λ
2
criterion presented by
Jeong & Hussain [2] was used for identifying the vortex cores in the wake. The λ
2
criterion stems from the pressure minimum criterion and dictates that the second largest eigenvalue of
S 2 +Ω 2 (where S and Ω are the symmetric and anti-symmetric components respectively of the velocity gradient tensor) must be negative in order to locate a vortex core. As Figure 7 demonstrates, the vortex cores are no more parallel to the cylinder axis.
In summary, the vortex shedding pattern past this structure exhibited a lack of correlation along the cylinder axis associated with a lower frequency modulation developing in the horizontal and vertical directions of the transverse force and an indication of a potential dislocation occurring as the curved pipe approached the seabed.
Figure 7 : Vortex cores in the wake of a curved riser in contact with the seabed. Shear normal flow, Re d
= 100, λ
2
= -0.1
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c) Curved riser in contact with an inviscid wall, sheared normal flow
A direct comparison between cases (a) and (b) involves the effects of both the sheared profile and the wall boundary.
Intermediate cases were therefore computed whereby the bottom wall was assigned a symmetry boundary condition and thus the effect of the sheared profile alone could be investigated separately from that of the viscous wall. The bottom wall is effectively now an inviscid wall and is therefore acting merely as a plane blocking the flow, but not as a viscous boundary.
The mesh for this computational case is exactly the same as the one for case (b) with the same sheared velocity profile being applied. The force coefficients and the pressure contours are depicted in Figure 8 and a preliminary comparison with those for case (b), as depicted in Figure 6, can be drawn. The force coefficients in general are approximately of the same magnitude as those for case (b). The imposition of a different wall boundary condition, however, has weakened significantly the lower frequency modulation in the horizontal direction of the transverse force, but not in the vertical one. In other words, a sheared profile with a viscous wall exhibits a lower frequency modulation in both the horizontal and vertical directions of the transverse force whereas a sheared profile with an inviscid wall exhibits a much weaker frequency modulation in the horizontal direction of the transverse force than in the vertical one. inviscid wall consists of the immediate vortex tube ending on the cylinder for the viscous wall as opposed to the inviscid wall case where linking of the vortex structures occurs close to the cylinder. In general, the shedding pattern for the inviscid wall case seems to be more complex and the wake more disorganised than that for the viscous wall.
Figure 8 : Curved riser in contact with an inviscid wall.
Shear flow normal to the plane of curvature, Re d
= 100
In order to investigate the wake structures developed past this type of flow, vortex cores at a λ
2
value of –0.1 were extracted following the same criterion and are depicted in
Figure 9. The vortex cores are not parallel to the cylinder axis as has been the case also for the sheared profile with a viscous wall. This emphasises the significance of the sheared profile versus the wall boundary. Nevertheless, comparing Figures 7 and 9, one can claim as a preliminary comparison that in the very near wake the difference between the viscous and the
Figure 9 : Vortex cores in the wake of a curved riser in contact with an inviscid wall. Shear normal flow, Re d
= 100, λ
2
= -0.1
Hence, the significant differences between cases (a) and (b) can be attributed more to the sheared profile rather than to the wall boundary. d) Curved riser in contact with an inviscid wall, uniform cross flow
In order to validate the above argument, a computation was performed with exactly the same mesh as for cases (b) and (c), but now a uniform velocity profile has been applied in the normal to the plane of curvature direction. The wall boundary condition remained a symmetry boundary condition representing an inviscid wall.
The force coefficients for a segment of the force history and pressure contours are depicted in Figure 10. The lower frequency modulation in the horizontal direction of the transverse force has now completely vanished. Moreover, the drag force pattern depicted in Figure 10 exhibits noticeable differences with the drag force coefficients for cases (b) and (c).
At this point, however, it should be emphasised that this case brings us back to case (a), which represented the base case, for comparison. Even though there is a difference in the exact geometry studied (this case ends semi-buried at the seabed as opposed to case (a)), both cases had the same uniform normal flow velocity profile being applied. Specifically, the lower amplitude peak in the drag direction for the quarter turn case discussed as an effect of curvature is now apparent for this case too. However, unlike the case for the quarter turn, there is a low frequency variation. Note that this lower amplitude peak
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phenomenon attributed to the unequal strength of the two vortices did not become clear for cases (b) and (c) where the effect of the sheared velocity profile was dominant.
Figure 10 : Curved riser in contact with an inviscid wall.
Uniform flow normal to the plane of curvature, Re d
= 100
Finally, one can observe going from case (c) to the present configuration, that the modulation in the vertical direction of the transverse force has not vanished as in the horizontal one.
Overall, therefore, one can claim that going progressively from case (b) to (c) and then (d), the horizontal direction is affected before the vertical direction of the transverse force.
Vortex core extraction in the wake of this flow is depicted in Figure 11 employing the same λ
2
criterion. Contrary to cases
(b) and (c), the vortex cores are not inclined to the cylinder axis as the curved pipe approaches the seabed. This verifies the argument developed at the end of the section for case (c) i.e. that the inclination of the vortex cores with respect to the cylinder axis should be attributed to the sheared flow profile primarily rather than to the wall boundary.
Figure 11 : Vortex cores in the wake of a curved riser in contact with an inviscid wall. Uniform normal flow,
Re d
= 100, λ
2
= -0.1 e) Quarter turn with an in-line extension in uniform flow parallel to the plane of curvature
The flow past a quarter turn with an in-line extension of ten diameters subject to uniform flow in parallel direction to the plane of curvature was also computed. The mesh comprised of
10250 elements, out of which 1184 were the prismatic surface elements. The computation was performed originally at second polynomial order and was then restarted at fourth and sixth order. When the computation was performed at sixth polynomial order, the mesh accounted for 993,608 local degrees of freedom per variable. Blockage for these computational cases was 7.2%.
Figure 12 depicts the force coefficients in the x-, y-, and z- directions for a segment of the force history and the pressure contours at slices normal to the cylinder axis. The x-direction is now the direction of the drag and the y- and z- directions represent the horizontal and vertical component of the transverse force respectively. As the pipe bends and becomes aligned with the free-stream, vortex shedding in the near wake becomes much less vigorous.
Figure 12 : Quarter turn with an in-line extension in uniform flow parallel to the plane of curvature, Re d
= 100
Even though C z
from Figure 12 may seem to exhibit an increase, one must look at the whole force history. Figure 13 depicts the convergence to an almost saturated state achieved by increasing the order of the interpolating polynomials from 4 to
6. The force coefficients depicted in Figure 12 correspond to a segment towards the end of the force history depicted in Figure
13 and represent a polynomial order of 6. The transition from order 4 to order 6 occurs at time t = 350.5. Moreover, one
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should note that the fluctuations and the scaling in the axes of the x- and z- directions are significantly smaller compared to those in the y- direction.
As a general comment, one should say that the force coefficients produced for this case behave in a similar manner as those for the standard example of a straight two-dimensional cylinder. As Figure 12 depicts, the deviations from the mean are much smaller for the drag force (x-direction) than for the horizontal component (y-direction) of the transverse force. The frequency of the drag was found to be equal to 0.35 and that of the horizontal component of the transverse force to be equal to
0.175 under the current blockage conditions; hence the similarity with the two-dimensional cylinder which exhibits a drag force with a frequency equal to twice that of the lift force.
Furthermore, in order to investigate the wake structures developed by such a flow the
λ
2
criterion was employed again.
As Figure 14 shows, in the near wake the vortex cores are straight, with their axes normal to the free-stream, but start to distort further downstream in the wake.
Figure 15 depicts a projection of the vortex cores in the xyplane showing a staggered array of vortices in the wake.
Figure 14 : Vortex cores in the wake of a quarter turn with an in-line extension in uniform parallel flow, Re d
= 100, λ
2
= -0.1
Figure 13 : Force history with interpolating polynomials of order 4 and 6
Figure 15: Projection of the vortex cores in the xy-plane
Experimental work was also conducted in the Hydrolab facility of the Department of Aeronautics of Imperial College, in order to compare the computational results with the flow visualisations. A model riser with the same aspect ratio as in the computational cases was towed in a tank [9] at an effective
Reynolds number equal to that of the computations. The flow visualisation images obtained by illuminating dye diffused from the stagnation line of the riser have shown good qualitative agreement with the numerical investigations. Specifically, the flow visualisation image obtained for the wake of case (e) is shown in Figure 16 and can be compared with Figure 14. The bending of the vortices further downstream in the wake is more apparent in the experiments than in the computations where the mesh is less refined far away from the curved pipe.
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Figure 16 : Flow visualisation for a quarter turn with an in-line extension in uniform flow parallel to the plane of curvature,
Re d
= 100
CONCLUSIONS
This paper has presented the fluid dynamic loading and the wake structures developed past curved cylinder flows either in uniform and shear flow or in contact with the solid boundary in directions normal or parallel to the plane of curvature.
The simulations in the direction normal to the plane of curvature aimed at the comparison between the uniform flow past a quarter turn and the more representative case for a catenary riser, i.e. that of a deformed quarter turn with a vertical addition of two diameters ending semi-buried in the seabed and subject to sheared normal flow. The intermediate stages were chosen so that the effects of the sheared profile and the type of the wall boundary could be studied separately.
In the normal flow computations the effect of the shear velocity profile was found to be more significant when compared to that of the boundary and the effect of curvature.
Similar to the case of a straight two-dimensional cylinder, the pressure contours and the identification of vortex cores in the wake have shown that the application of a sheared velocity profile is associated with a lack of correlation along the span of the cylinder and the development of a lower frequency modulation in the horizontal and vertical directions of the transverse force.
Furthermore, the experimental results for the flow in the direction parallel to the plane of curvature have shown good qualitative agreement with the numerical investigations. The quarter turn with a horizontal extension of ten diameters has been found to shed much less vigorously as the pipe bends to become aligned with the free-stream. The force coefficients produced for this case have been found to behave in a similar manner as those for the standard case of a straight twodimensional cylinder.
In summary, the results from the simulations where the flow was normal to the plane of curvature have shown that the sheared inflow was predominant and the structure behaved in a qualitatively similar manner as the straight two-dimensional cylinder. This would support the approximation adopted in a two-dimensional sectional approach. Nevertheless, a twodimensional approach can clearly not capture the threedimensional vortex shedding features. Moreover, the low
Reynolds number of 100 implies relatively strong viscous effects and that we could envisage a stronger influence of the body than at higher Reynolds numbers. Further, the high curvature adopted in this study, which could be considered as an upper limit for catenary riser pipes, would also be more likely to promote three-dimensional body interactions than smaller curvatures. Since the current results have shown that the curved cylinders of this aspect ratio when subjected to flow normal to the plane of curvature behave qualitatively similar to the straight two-dimensional cylinders, we believe that a smaller curvature configuration would not demonstrate any significant difference probably at low Reynolds number.
However, this is not the case for the computational investigations where the flow is parallel to the plane of curvature. The in-plane flow characteristics are significantly different, as the component of the incoming flow vector in the sectional planes is not significant in some sections; therefore , vortex shedding may not occur, even under a two-dimensional argument, in all sections. On-going investigations are therefore focusing on flows where the flow is parallel to the plane of curvature, where the fully three-dimensional computations can provide more insight into the limitations of the sectional approach, as well as increasing the Reynolds number.
ACKNOWLEDGMENTS
The authors would like to thank J.C. Owen and C.J. Hogan for all their help with the experimental part of this work.
Computational resources were provided by the Imperial College
Parallel Computing facilities (Fujitsu AP3000) and the
Computer Services for Academic Research (CSAR) at the
University of Manchester under the EPSRC grant number
GR/R64957.
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