Flow and flow control modeling for a drilling riser system with auxiliary lines

Ocean Engineering 123 (2016) 204–222

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Flow and flow control modeling for a drilling riser system with auxiliary lines Wenbo Wu a, Jiasong Wang a,n, Shiquan Jiang b, Liangbin Xu b, Leixiang Sheng b a b

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Research Institute of CNOOC, Beijing 100028, China

art ic l e i nf o

a b s t r a c t

Article history: Received 29 September 2015 Received in revised form 5 April 2016 Accepted 28 June 2016

In this numerical study, it was mainly intended to investigate the flow and flow control for a practical drilling riser system with auxiliary lines at various incidence angles. The flow past the riser system was simulated through solving the Reynolds averaged Navier–Stokes equations and k − ω turbulence model. The cases about fluid flow past single cylinder, two staggered cylinders and eight even-distributed auxiliary lines around main line were simulated first, and the results match well with previous researches. The Reynolds number (Re) used here was 35,000, which belongs to the subcritical Reynolds region, and the flow in this Reynolds region can shows the typical flow characteristics in practical drilling operator. The effects of the incidence angle, which was in the range of 0–360° and at 30° intervals, on the drag and lift coefficients, the pressure distribution around the main line, the vortex shedding frequencies and the flow structure were investigated. The mean and RMS values of the drag and lift coefficients of the main line vary with the incidence angle irregularly due to the complexity of the geometry of the riser system. The flow patterns can be classified into six types based on the time-averaged streamlines. The effects of the auxiliary lines on mean force coefficients of the main line in different flow patterns were discussed in detail based on the distributions of the pressure on the main line. The auxiliary lines can suppress the vortex shedding on the main line at all incidence angles. Especially, in the CVP (clamped vortices pair, α = 210° and α = 330°) pattern, the vortex shedding behind the main line is suppressed most effectively by multiple downstream auxiliary lines. The mean value of the drag coefficient is just about 45.5% of that of the single cylinder, and the RMS value of the lift coefficient is about 1.5% of that of the single cylinder. This is the favorable condition for drilling operation. The worst effect of suppression occurs in the SLA (shear layers afflux, α = 90°) pattern, and the RMS value of the lift coefficient is about 76.8% of that of the single cylinder. Due to the interaction between the vortices shed from different lines in riser system, the frequencies of the lift coefficients of the lines in riser system change largely, and some higher frequencies appear in the lift coefficients of the main line. As α = 60° and α = 120°, the Strouhal number of the main line is about 0.75. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Flow control Riser system Auxiliary line Vortex shedding Drag and lift coefficients Numerical simulation

1. Introduction During the development of offshore oil and gas, drilling risers are immerged in deep sea. The flow analysis and flow control for risers are concerned by researchers and engineers to prolong the service life of the risers and optimize the drilling operating environment. Roughly two flow control methods are known with respect to energy expenditure, active method and passive method. In the active method, such as surface heating (Lu et al., 2012), blowing and/or suction (Sohankar et al., 2015), external energy should be input into the system. While the passive method is n

Corresponding author. E-mail address: [email protected] (J. Wang).

http://dx.doi.org/10.1016/j.oceaneng.2016.06.043 0029-8018/& 2016 Elsevier Ltd. All rights reserved.

concerning about the geometrical modification of the flow system without any additional energy consumption, such as splitter plate (Qiu et al., 2014, Gu et al., 2012b), fairing (Assi et al., 2011,Wang and Zheng, 2015), helical strakes (Gómez et al., 2013) and control rod (Firat et al., 2015, Lee et al., 2004) can be used to change the flow structures around the cylinder. The auxiliary lines located close to the main line of the drilling risers are necessary to achieve special functions for drilling operation, and can also be expected to control the flow around the risers to some extent, which can be classified into the passive method. Due to the lack of the knowledge about the influences of the auxiliary lines on the main line, the current design practice of the risers simplifies the riser system with auxiliary lines as a single cylinder, which cannot reveal the real flow and fluid force characteristics. The issue about flow past drilling riser system relates to flow around multiple cylinders with

W. Wu et al. / Ocean Engineering 123 (2016) 204–222

different diameters, which has been scarcely studied up to now. Flow around cylinders is a common phenomenon in engineering practice. A lot of researchers have paid considerable attentions to this issue, and many important insights into this problem have been achieved. In the early researches, the main point was focusing on flow around a single cylinder, both experimentally and numerically, for several decades, as reflected in the reviews of Williamson (1996) and Norberg (2003). Although the geometry is simple, the interesting flow features are abundant. Fung (1960) and Roshko (1961) investigated the flow around a single circular cylinder at supercritical Reynolds numbers and high Reynolds numbers respectively. The relationship between the lift coefficient, the drag coefficient, the Strouhal number and the Reynolds number were summarized systematically. Up to now, the flow past two cylinders has been studied extensively, both experimentally and numerically, for several decades, as reflected in the reviews of Zdravkovich (1977) and Sumner (2010). The flow around a group of structures is different from and more complicated than that around a single structure. In the issue about the flow past a group of cylinders, the main parameters are the Reynolds number and the configuration of the cylinders. Spivac (1946) studied the flow past two parallel cylinders at different gap ratios. It was found that the flow around the two parallel cylinders is characterized by a single frequency. Wang and Zhou (2005) investigated the flow behind two side-by-side circular cylinder experimentally based on laser-illuminated flow-visualization. According to different gap ratios, the flow was classified into three regimes: single street, asymmetrical flow and two coupled flow. It was found that the flow structure and its downstream evolution are closely linked to the phase relationship between the gap vortices in the wide wake and in the narrow wake. Meneghini and Saltara (2001) observed sub-harmonics at the gap L ¼3D, and the power-spectrum estimation of the lift coefficient shows a very distinctive peak at the Strouhal frequency. Gu and Sun (1999) investigated the interference between two parallel circular cylinders arranged in staggered configurations at high subcritical Reynolds number using a wind–tunnel, three pressure-distribution patterns on the downstream cylinder and two switching processes were observed at different incidence angles. The corresponding flow patterns were classified as wake, shear layer and neighborhood interference. Sumner et al. (2000, 2003) and Li and Sumner (2009) found that the behavior of the Strouhal number could also be broadly classified according to pitch ratio into three types: closely spaced configurations, moderately spaced configurations and widely spaced configurations. The small changes in the incidence angle (wind direction) can lead to marked changes in the vortex shedding (dominant) frequencies. Zhou et al. (2009) investigated the effect of the Reynolds number on the wake of two cylinders arranged in staggered configurations based on the measured/reported Strouhal number and the flow structure. The St-Re relationship was classified into four distinct types. Each is linked to distinct initial conditions, namely, interactions between the four shear layers around the cylinders. It should be noted that, the researches mentioned above were based on two cylinders with the equal diameter. Zhao et al. (2005, 2007) investigated the viscous flow and turbulent flow past two circular cylinders with different diameters numerically using a finite element method. The effect of the gap ratio and the position angle of smaller cylinder on drag coefficient, lift coefficient and vortex shedding frequency were discussed detailedly. Besides the work listed above, more investigations (Hu and Zhou, 2008a, 2008b; Zhang et al., 2006; Mittal et al., 1997) about the flow past two cylinders arranged in different configurations can be found. However, the work about the flow past more cylinders is scarce. Sun and Gu (1995) studied the flow past two and more cylinders and pointed out that, the interference of flow

205

around groups of structures, or simply the group-effect, is important and essential both from viewpoints of fundamental research in fluid dynamics and engineering applications. Lam and Fang (1995) studied the static pressure distribution on four cylinders arranged in a square configuration subjected to cross flow for spacing ratios ranging from 1.26 to 5.8 and incidence angles ranging from 0–45°. It was found that a cluster of cylinders would result in a reduction in drag coefficient and hence a reduction in total drag. Sewatkar et al. (2012) studied the flow around six inline square cylinders numerically and experimentally, four basic flow regimes were initially proposed as a function of spacing ratio for Re ¼100. Gu et al. (2012a) and Zhao et al. (2012) studied the effects of the auxiliary lines on the vortex shedding on the main line experimentally and numerically respectively based on an ideal riser model. It was found that, the auxiliary lines can suppress the vortex shedding, and the effect of suppression is closely related to the arrangement of the auxiliary lines and the incidence angle of the flow. Lu et al. (2014) studied the laminar flow past a circular cylinder with multiple equally distributed small-diameter control rods numerically. The effects of rod-to-cylinder spacing ratio, rod to cylinder diameter ratio, Reynolds number, number of control rods and angle of attack on the hydrodynamics of the main circular cylinder were investigated. Four different flow regimes were identified based on the mechanism of lift and drag reduction. In this paper, the flow analysis and flow control for a real drilling riser system with six auxiliary lines were studied numerically. The drilling riser system was modeled as seven circular cylinders with different diameters, it was expected that the flow structures were more complex than previous researches mentioned above. It should be noted that the work in this paper was based on a drilling system currently used in offshore oil industry, and the geometry of the system was fixed. The aims of this study were to investigate and evaluate the effects of the auxiliary lines on suppression of the vortex shedding behind the main line at different incidence angles, and to study the new phenomenon that appears in complex cylinder configuration (multi-cylinders with different diameters). The flow structure around the multiple cylinders was validated based on the insights of the flow past single cylinder, two staggered cylinders and eight even-distributed auxiliary lines around the main line. The effects of the incidence angle on flow structure, hydrodynamic force and vortex shedding frequency at Re¼35,000 were studied systematically. Based on the diameter of the main line, the Reynolds number investigated in this paper was selected to be 35000, which was based on following considerations. First, Re¼35,000 belongs to a subcritical Reynolds region, and the flow in this Reynolds region can shows the typical flow characteristics in practical drilling operator. Second, the first natural frequency of the drilling riser with the length of 1000 m may be estimated about 0.0244 Hz, and the reduced velocity (Ur ¼ UD/ν, U is the free-stream velocity, D is the diameter of the cylinder, ν is kinematic viscosity of the fluid) corresponding to this Reynolds number mentioned above is about 5, which is a typical value that is cared about in vortex induced vibration (VIV) research. The outline of this paper was arranged as the followings. In Section 2, the problem description of the flow past the riser system was given. Some parameters used to describe the issue were defined. In Section 3, the governing equations and the turbulence model were introduced. In Section 4, the model and numerical method were validated by comparing with the classic researches on flow past single cylinder, two identical staggered cylinders and riser with eight even-distributed auxiliary lines. The computational grid had also been tested using four different meshes. The simulation results were presented and discussed in Section 5. In the final section, some conclusions were drawn.

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2. Problem description

Table 1 The diameters of lines and geometrical parameters in the riser system.

Fig. 1 is the sketch of the cross section of a real drilling riser system with auxiliary lines. In the riser system, the auxiliary lines have different diameters and distribute asymmetrically around the main line. It was expected that, the flow past this complex structure was complicated, which included the characteristics of flow past multi-cylinders, staggered cylinders and cylinders with different diameters. The complicated flow was investigated based on the different incidence angles. In Fig. 1, α represents the incidence angle of the flow, θ is the circumferential angle of the main line, U is the free stream velocity of the flow. The diameter of the main line is represented as D, the diameters of the auxiliary lines in the riser system and the geometrical parameters shown in Fig. 1 are listed in Table 1.

3. Governing equations and computational model In this paper, OpenFOAM was used to compute the flow past the riser system. The finite volume method is used in the OpenFOAM, which is based on the integration of the fluid variables on the control volumes (OpenCFD Ltd produce OpenFOAM TM, 2009). The analytical form of the governing equations of unsteady flow of viscous incompressible fluid in Cartesian coordinates can be expressed as follows:

∂Ui =0 ∂xi

C&K line Hydraulic line Booster line Chemical line l1 l2 l3 l4 l5 l6 l7 l8

where μt is the turbulence viscosity, δ is the Kronecker tensor operator, k is the turbulent kinetic energy, ω is the turbulence frequency. The transport equations for k and ω for turbulent flows can be expressed as follows (Wilcox, 1998)

μ ⎞ ∂k ⎤ ∂ ⎡⎛ ⎥ + Pk − β*ρkω ⎢ ⎜μ + t ⎟ σ k ⎠ ∂xi ⎦ ∂xi ⎣ ⎝

(5)

∂ ( ρωUi ) ∂( ρω) μ ⎞ ∂k ⎤ ∂ ⎡⎛ + = ⎥ + Pω − β1ρω2 ⎢ ⎜μ + t ⎟ ∂t σω ⎠ ∂xi ⎦ ∂xi ∂xi ⎣ ⎝

(6)

∂ ( ρk ) ∂t

+

∂ ( ρkUi ) ∂xi

∂τij ∂ ∂ 2Ui ∂p ∂Ui +ρ − + Uj Ui ) = μ ( ∂t ∂xi ∂xj ∂xj ∂xi ∂xj

(2)

where μ is the molecular dynamic viscosity, U is the time-average velocity vector of the fluid, τ is the Reynolds stress tensor, p is the pressure, ρ is the density of the fluid. k − ω model was used here, the Reynolds stresses were computed in two-equation models with the Boussinesq expression

⎛ ∂U ∂Uj ⎞ 2 ⎟ − ρkδij τij = μt ⎜ i + ∂xi ⎠ 3 ⎝ ∂xj

μt =

ρk ω

=

(1)

Pk = 2μt Sij Sij − ρ

0.285D 0.119D 0.238D 0.125D 0.973D 0.12D 0.212D 0.902D 0.928D 0.857D 0.154D 0.163D

(3)

(4)

2 ∂Ui ρk δij 3 ∂xj

⎛ 2 ∂U ⎞ Pω = γ1 ⎜ 2ρSij Sij − ρω i δij ⎟ 3 ∂xj ⎠ ⎝

(7)

(8)

where S is the average strain rate tensor, the model constants are as follows. σ k = 2.0, σω = 2.0, γ1 = 0.553, β1 = 0.075, β* = 0.09. A second-order Gauss integration scheme with a linear interpolation for the face-centered value of the unknown was used for the divergence, gradient, and Laplacian terms in the governing equations. The second-order backward Euler method was adopted for time integration. Thus, the numerical discretization scheme gives second order accuracy in space and time. The pimpleFoam solver in the OpenFOAM was employed to solve the dynamic equations.

4. Validation 4.1. Comparison with others In order to validate the computational method in this paper, the results of flow past single circular cylinder, staggered cylinders and risers with eight even-distributed auxiliary lines computed with OpenFOAM were compared with the results of other researchers. In this paper, the drag coefficient ( Cd ), lift coefficient ( Cl ) and Strouhal number ( St ) were defined as

Cd =

Fig. 1. Sketch of the cross section of a real drilling riser with auxiliary lines.

2Fy f D 2Fx , Cl = , St = s ρU 2D ρU 2D U

(9)

where Fx and Fy are the fluid force (on the unit length) components in the in-line direction and the cross-flow direction respectively. ρ, D and U are the same parameters as mentioned above. fs is the

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Table 2 Comparison with other papers. Re

Source

Cd

Cl

St

1000

Braza et al. (1986) Mittal and Kumar (2001) Labbe and Wilson (2007) Wang et al. (2010) Present study

1.15 1.46 1.20 1.14 1.35

1.35 0.72 0.54 1.07

0.22 0.22 0.19 0.21

Norberg (1987) Wang et al. (2010) Present study

0.99 1.06 1.15

0.60 0.97

0.22 0.20 0.19

Dong and Karniadakis (2005) Gopalkrishnan (1993) Present study

1.14 1.19 1.03

0.43 0.38 0.86

0.20 0.19 0.20

3900

10,000

frequency of the vortex shedding, which was calculated by the FFT of the lift coefficient. Further, the mean drag coefficient (Cd ), mean lift coefficient (Cl ) and RMS values of the drag and lift coefficients ( Cd, rms and Cl, rms ) were defined as

Cd =

1 N

N

∑ Cd, Cl = n= 1

1 N

N

∑ Cl

(10)

n= 1

Strouhal number were present. In this paper, a simulation was carried to valid the computational model. Fig. 2a shows the configuration of the two cylinders, P is the center-to-center distance, D is the diameter of the cylinder, U is the velocity of the free flow, α is the angle between the center-to-center line and the oncoming flow. Fig. 2b shows the computation domain used in this paper, 55D in the streamline direction and 30D in the cross flow direction. The center of the upstream cylinder was located 15D downstream from the left side and at the centerline in the cross flow direction. The left side was defined as velocity-inlet boundary, the top and bottom sides were defined as symmetry boundary, the right side was defined as outflow boundary. The Reynolds number of the case simulated here is 3.2  104, and P/D ¼2. Fig. 3 shows the mesh around the staggered cylinders. The unstructured grid were used to discrete the computation domain. The body fitted anisotropic meshes were applied to both cylinders, and yþ is less than 5 for both cylinders. The numbers of the circumferential nodes on both cylinders are 160. The time step is 2  10  4 s, which can make the solutions stable in time, and the time integration errors are found insignificant. The calculation was initialized by the variables at the velocityinlet boundary. The velocity at the inlet boundary is a constant value U, the turbulence variables of the k − ω turbulence model were calculated using turbulence intensity and eddy viscosity ratio as follows.

k= Cd, rms =

1 N

N

2

∑ ( Cd − Cd ) n= 1

, Cl, rms =

1 N

N

3 ( UI )2 2

(12)

2

∑ ( Cl − Cl ) n= 1

207

(11)

ω= where N is the number of the data in integral multiple periods (the period is the reciprocal of the fs ). To reduce the error, the data in twenty periods were used in this paper. 4.1.1. Case1. Single cylinder The comparisons of single circular cylinder are listed in Table 2. From the comparison, it can be found that Cd and St agree well with other results. But Cl is different from other results, and the results of others are also differing largely from each other. The multitudes of those differences were arisen from the experimental uncertainties, using various boundary conditions and grid structures, the effects of blockage, etc. Therefore, the computations in this paper are reasonable. 4.1.2. Case2. Two staggered cylinders In the research of Sumner and Richards (2003), two staggered circular cylinders with the same diameter were located in a low speed wind tunnel. The drag coefficient, the lift coefficient and the

−1 ρk ⎛ μt ⎞ ⎜ ⎟ μ ⎝ μ⎠

(13)

where the turbulence intensity is 0.5%, the Eddy viscosity ratio is 10. Kim et al. (2008) also computed this case using LES method, but only results of α = 0° ∼ 15° were presented. The comparisons of the present result with those of Sumner and Richards (2003) and Kim et al. (2008) are shown in Fig. 4. It can be found that, the drag coefficient matches well with the results in Sumner and Richards (2003) and Kim et al. (2008). The lift coefficient does not match the result of Sumner and Richards (2003) when the incidence angle is 10°, but it matches well with the result of Kim et al. (2008). The Strouhal numbers computed in this paper are little larger than those in Sumner and Richards (2003) at low branch as α > 50°, but the major characteristics and the tendency are the same. The results at the upper branch match well with those of Sumner and Richards (2003). The difference in these results may be due to that the turbulent features cannot be set as the same as the experiment at the inlet boundary or the roughness of

Fig. 2. The staggered configuration of the two cylinders and the computation domain.

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Fig. 3. The mesh around the cylinders.

the cylinders in experiment cannot be ignored for the strong interference of the flow between cylinders. The variation of the lift coefficient is abrupt at α = 10°, which is sensitive to the parameters mentioned above. So the present method can be taken as accurate enough to be used in the following simulation. 4.1.3. Case3. Eight auxiliary lines around cylinder Zhao et al. (2012) had studied the effects of the auxiliary lines on the vortex shedding behind the main line experimentally and numerically based on an ideal riser model. In this paper, the flow past the riser with eight auxiliary lines was simulated at Re¼3  104. Fig. 5a shows the configuration of the cylinders, δ is

the space between the main line and the auxiliary line, d is the diameter of the auxiliary line, D is the diameter of the main line, α is the angle expressing the direction of the oncoming flow, U is the velocity of the free stream. Fig. 5b shows the mesh around the cylinders. The unstructured grids were used to discrete the computation domain. The body fitted anisotropic meshes were applied to all cylinders, and y þ is less than 5 for all cylinders. The number of the circumferential nodes on the main line is 310. The numbers of the circumferential nodes on auxiliary lines are 80. The time step is 0.0006 s. The k − ω turbulence model was used, and the turbulence intensity is 0.1%, the Eddy viscosity ratio is 1. Fig. 6 shows both the results of Zhao et al. (2012) and this paper. It can be found that the auxiliary lines actually reduce the drag and lift coefficients on the main line. The Strouhal numbers on the main line are also reduced by the auxiliary lines. As α = 0°, the result in this paper is almost equal to that in Zhao et al. (2012). For the two remaining angles, the results in present study can match well with the numerical or experiment results in Zhao et al. (2012). So the present method can be taken as accurate enough to simulate the flow around multiple cylinders. 4.2. Grid testing A rectangular domain was used in present work, 65D in the streamline direction and 30D in the cross flow direction. The center of the main line was located 15D downstream from the left

Fig. 4. Comparison of the mean values of lift and drag coefficients of the downstream cylinder and Strouhal number with the results of Sumner and Richards (2003) and Kim et al. (2008) ( Re = 3.2 × 104 , P/D ¼2).

W. Wu et al. / Ocean Engineering 123 (2016) 204–222

209

Fig. 5. The configuration of the cylinders and the mesh around the cylinders.

Fig. 6. Comparison of the mean values of the drag coefficients and the RMS value of the lift coefficients and Strouhal numbers on the main line with the results of Zhao et al. (2012).

side and at the centerline in the cross flow direction. The left side was defined as velocity-inlet boundary, the top and bottom sides were defined as symmetry boundary, the right side was defined as free-stream boundary, as shown in Fig. 7a. The computational domain was meshed with the unstructured grids. The mesh near the cylinders is shown in Fig. 7b, body fitted anisotropic meshes were applied to all of the cylinders in the riser system, and y þ is less than 5 for all of the cylinders, the number of the circumferential nodes on the main line is 180, and the numbers of the circumferential nodes on the auxiliary lines are 140. The time step is

0.02 s. The k − ω turbulence model was used, and the turbulence intensity is 0.1%, the Eddy viscosity ratio is 10. To verify the influence of the computational grid on the numerical results, four different grids were used for the flow past the riser system at Re¼3.5  104 while incidence angle is 0°. The grids tested have 64,000, 72,000, 82,000 and 115,000 elements (called coarsest, coarse, fine and finest grids respectively). As shown in Table 3, the mean drag and lift coefficients on all lines in the riser system were summarized, and the dominant frequencies of the lift coefficients on the main line were also presented for these

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Fig. 7. Computational domain and Grid generated around the cylinders.

Table 3 Mean drag and lift coefficients acted on lines in the riser bundle, and dominant frequency for three different grid refinements and Re¼ 35,000, m – main line, CK1 – C&K line 1, CK2 – C&K line 2, H1 - Hydraulic line 1, H2 - Hydraulic line 2, B - Booster line, C - Chemical line. Results m

Coarsest grid

Coarse grid

Fine grid

Finest grid

Cd

0.59

0.59

0.59

0.58

m Cl CK1 Cd CK1 Cl CK2 Cd CK2 Cl H1 Cd H1 Cl H2 Cd H2 Cl B Cd B Cl C Cd C Cl

0.038

0.035

0.037

0.038

f

0.28

0.28

0.27

0.27

0.0020

0.0017

0.0020

0.0020

1.19

1.17

1.17

1.17

0.067

0.060

0.065

0.066

1.21

1.22

1.21

1.21

0.0012

0.0011

0.0012

0.0013

0.64

0.63

0.62

0.62

0.0022

0.0022

0.0023

0.0023

1.03

1.04

1.04

1.04

0.034

0.030

0.032

0.033

0.97

0.98

0.97

0.97

0.0013

0.0012

0.0013

0.0013

0.092

0.090

0.092

0.092

different grid refinements. Little difference among the results for these meshes was observed, namely, the effect of the grid refinement can be ignored. Here, the fine mesh, 82,000 elements was used for all cases.

5. Results and discussions In this section, the results and discussions about this simulation were given from static (average) variations to dynamic variations. The flow patterns were classified into six types in Subsection 5.1 based on time-averaged streamlines, and the characteristic of every type of flow pattern was described in detail. In Subsection 5.2, the mean drag and lift coefficients of the main line and auxiliary lines at different incidence angles were discussed. To express the variation of the mean force coefficients on the main line, the timeaveraged pressure distributions on the main line were shown and discussed in Subsection 5.3. The RMS values of the force coefficients were discussed in Subsection 5.4, and the variation of the RMS force coefficients with the incidence angle was expressed based on instantaneous vorticity contours in Subsection 5.5. In Subsection 5.6, the vortex shedding frequencies of the main line and auxiliary lines were discussed.

5.1. Flow patterns based on the time-averaged streamlines Considering the complex geometry of the riser system, the incidence angle cannot express the effect of all auxiliary lines on the main line adequately, the configuration of the riser system at different incidence angles are given in Fig. 8. Because the configuration of the riser system is not symmetric about any line, the flow past the riser system at any incidence angle among 0–360° is a distinctive problem. In this paper, the incidence angles were picked from 0–330°, with the interval being 30°. According to the time-averaged streamlines at different angles shown in Fig. 8, six different types of flow patterns can be identified, as shown in Table 4. As α = 60° and α = 120°, the shear layers of the main line separate alone. Neither the shear layers of the auxiliary lines attach to the main line, nor do the shear layers of the main line attach to the auxiliary lines. This type of flow pattern is referred to as shear layers separation alone (SLSA) pattern. As α = 30°, α = 150° and α = 300°, a single vortex form behind the main line because of the influence of the auxiliary lines behind the main line. For this type of flow structure, the shear layers separated from one side (top side or bottom side) of the main line form a single vortex, while the vortex formed at the other side is suppressed. Briefly, this type of flow pattern can be called as single vortex forming (SVF) pattern. As α = 210° and α = 330°, the shear layers separated from the main line are clamped in a small domain by the downstream auxiliary lines, and a pair of vortices with different sizes form behind the main line. This type of flow pattern can be called as clamped vortices pair (CVP) pattern. As α = 0°and α = 180°, the shear layers of the upstream auxiliary lines attach to the main line. The downstream auxiliary lines locating in the shear layers separated from the main line and split the vortices behind the main line into many smaller vortices. The shear layers of the auxiliary lines locating side by side with the main line do not interact with the shear layers of the main line directly, and these auxiliary lines just affect the blockage ratio around the main line. In this type of flow pattern, the separation of the shear layers of the main line is affected by the upstream and downstream auxiliary lines. It can be called as shear layers afflux and vortices splitting (SLAVS) pattern. As α = 90°, the shear layers of the upstream auxiliary lines attach to the main line, and the shear layers of the main line shed freely. In this type of pattern, the separation of the shear layers of the main line is just affected by the upstream auxiliary lines. It can be called as shear layers afflux (SLA) pattern. As α = 240° and α = 270°, no shear layer of the auxiliary lines attaches to the main line, and the vortices behind the main line are split into many smaller vortices by the downstream auxiliary lines. This type of flow pattern can be referred to as vortices splitting (VS) pattern.

W. Wu et al. / Ocean Engineering 123 (2016) 204–222

Fig. 8. The configurations and time-averaged streamlines of riser system at different incidence angles.

211

212

W. Wu et al. / Ocean Engineering 123 (2016) 204–222

Table 4 The list of flow patterns and corresponding incidence angles. Abbreviation Full meaning

Incidence angle

SLA SLAVS

α = 90 ° α = 0°, α = 180 °

Shear layers afflux Shear layers afflux and vortices splitting Shear layers separation alone Clamped vortices pair Single vortex forming Vortices splitting

SLSA CVP SVF VS

α = 60 °, α = 120 ° α = 210 °, α = 330 ° α = 30 °, α = 150 °, α = 300 ° α = 240 °, α = 270 °

5.2. Mean values of the drag and lift coefficients The mean values of the force coefficients of the main line and auxiliary lines were computed in the following form: i

Cj =

1 N

N

∑ C ij, n n= 1

(14)

where i can be referred as m, CK1, CK2, H1, H2, B and C corresponding to main line, C&K line 1, C&K line 2, Hydraulic line 1, Hydraulic line 2, Booster line and Chemical line in the riser system, j refers as d and l corresponding to drag and lift coefficients, other parameters are the same as those in Eq. (10). m m Cd and Cl vary with the incidence angle irregularly, as shown m in Fig. 9. Cd is positive in all incidence angles, and smaller than that of the single cylinder with the same diameter as the main line (expressed as single cylinder in the following contents) in most m angles except 270°. Cl oscillates between positive value and negative value, and approaches to zero as the incidence angle increasing. Considering the flow patterns mentioned in previous m m section, Cd and Cl can be classified using the flow patterns. The m minimum value of Cd appears in the CVP pattern (about 45.5% of that of the single cylinder as α = 210°), and the maximum value appears in the VS pattern (larger than that of the single cylinder as α = 270°). It can be observed from Fig. 11f that the base pressure of the main line is slightly affected by the auxiliary lines as α = 270°, and the zone of positive pressure is enlarged by the side auxiliary lines, which increases the mean drag coefficient. The base pressures of the main line are magnified by the side auxiliary lines in the CVP pattern, as shown in Fig.11d, so the mean drag coefficients m are largely reduced. For Cl , the maximum deviation from zero appears in SLSA pattern, and the minimum deviation from zero appears in the SLA pattern. This is due to that the geometry is largely asymmetric in SLSA pattern and is almost symmetric in SLA

pattern. Fig. 10 shows the mean values of drag and lift coefficients on the auxiliary lines. In the geometry of the riser system, the auxiliary lines are divided into three pairs, and the auxiliary lines are located closely to each other for every pair. For the convenience of the comparison, the mean values of the force coefficients were plotted together for every pair of auxiliary lines in Fig. 10. It can be found that, the values of the adjacent auxiliary lines are slightly different at most locations. Large difference is observed as the auxiliary lines pair is almost side by side with the main line, and the difference is even larger when the larger auxiliary line is located upstream of the smaller one. The main reason is that the shear layers of the upstream one of the auxiliary lines pair would attach to the downstream one, and the gap is occupied by the revolving flow, as shown in Fig. 8. The downstream auxiliary line is not exposed in the free stream, and the mean drag coefficient on that auxiliary line is negative, which results in the large difference in the mean drag coefficients. The vortex shedding of the auxiliary lines pair is similar to that of a single cylinder, and the vortices just shed from larger auxiliary line, as shown in Fig. 14, which results in the large difference in the mean lift coefficients. This phenomenon can also be found in the study of Zhao et al. (2005). 5.3. Pressure distribution To understand the variation of the mean values of the drag and lift coefficients on the main line with the incidence angle clearly, the time-average pressure coefficient distribution along the perimeter of the main line was studied, as shown in Fig. 11. For the purpose of the comparison, the pressure distribution of the single cylinder was also included in the figure. The pressure coefficient Cp was defined as

Cp =

2 ( p − p0 ) ρU 2

(15)

where p0 is the pressure at a faraway location upstream of the riser system. In the SLA and SLAVS patterns, the shear layers of the upstream auxiliary lines attach to the main line, Cp of the main line is smaller than that of the single cylinder before the point where the shear layers of the auxiliary lines afflux into (as shown in Fig. 8a, d and g). Behind the point, Cp of the main line changes suddenly and is larger than that of the single cylinder because of the influence of the side auxiliary lines. Sakamoto and Haniu (1994) indicated that, the small cylinder located in the front edge of a large cylinder can

Fig. 9. The mean values of the drag and lift coefficients on the main line.

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213

Fig. 10. The mean values of the drag and lift coefficients on the auxiliary lines.

cause the displacement of the separation point alone the large cylinder, and this can also be seen in Fig. 11a and b. In the SLSA and CVP pattern, Cp along the upstream part of the main line is similar to that of the single cylinder, as shown in Fig. 11c and d. But Cp of the main line is larger than that of the single cylinder for the downstream part, because the separation of the shear layers is suppressed by the downstream auxiliary lines, as shown in Fig. 8c, e, h and l. In the SVF pattern, Cp of the main line is slightly different

from that of the single cylinder along the side where the vortices form (Fig. 8a, f and k). However, the large difference is observed on the side where the vortex forming is suppressed. The separation point moves upstream, and Cp is larger than that of the single cylinder on the rear edge, as shown in Fig. 11e. In VS pattern, the auxiliary lines located in the shear layers separated from the main line, as shown in Fig. 8i and j, have little influence on the pressure distribution along the main line. The pressure distribution is

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Fig. 11. The time-averaged pressure distribution on the main line.

similar to that in the SLSA pattern as α = 240°, and is just affected by the side auxiliary lines as α = 270°, as shown in Fig. 11f. Zhao et al. (2005) had studied the pressure distribution along the cylinders based on two staggered cylinders with different diameters, all the variation of the pressure distribution mentioned here can be verified with the results in their research. It can also be observed that the pressure changes more slowly at the minimum as

two pairs of auxiliary lines are located in one side of the main line, as shown in Fig. 11c and f. 5.4. RMS values of the drag and lift coefficients In the flow past the riser system, the vortices are shed from the main line and auxiliary lines alternately, and the fluctuating forces

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215

Fig. 12. The RMS values of the drag and lift coefficients on the main line.

are acted on the lines. The RMS values of the force coefficients mentioned in this section were computed in the following form

C ij, rms =

1 N

N

∑ n= 1

(C

i j, n

i 2

− Cj

)

(16)

where i and j have the same definitions as those mentioned in Eq. i

(14), C¯ j is the mean value of the force coefficient, other parameters are the same as those in the Eq. (10). Fig. 12 shows the RMS values of the drag and lift coefficients on the main line. It can be found that Cdm, rms are larger as α ≤ 180° except α = 30° and are smaller as α > 180° than that of the single cylinder. Overall, Cdm, rms are very small for all cases. The differences between the results (Cdm, rms ) of the cases that α ≤ 180° and α > 180° are mainly due to the asymmetry of the Booster line and Chemical line. As α > 180°, the auxiliary lines pair (Chemical line and Booster line) locate downstream of the main line, and the shear layers of the main line are stretched, as shown in Fig. 14. The shedding of the vortices from the main line is farther downstream at α > 180°, and the force coefficients are smaller than those in the case α ≤ 180°. Clm are smaller than that of the single cylinder in , rms all incidence angles, which can be taken as a favorable condition to suppress the vortex shedding or control VIV of the risers. In the CVP pattern, Clm are just about 1.5% of that of the single cylinder. , rms It can be observed in Fig. 14h and l that the vortex shedding of the main line is delayed by the auxiliary lines in CVP pattern. The shear layers of the main line merge to the shear layers of the auxiliary lines, and no obvious vortex can be observed in the near wake, which reduces the RMS force coefficients of the main line largely. Fig. 13 shows the RMS values of the drag and lift coefficients on the auxiliary lines. All of the auxiliary lines pairs located closely have the same variation tendency with the incidence angle. As the auxiliary lines pair is almost located directly in the rear of the main line, the shear layers separated from the main line have strong interference on the auxiliary lines pair, and the RMS values of the force coefficients on the auxiliary lines pair are large. In this situation, the direction of the flow past the auxiliary lines pair is changed by the revolving fluid behind the main line, and the auxiliary lines pair acts like two tandem cylinders. The reattachment of the shear layers of one auxiliary line to the other can be observed (as shown in Fig. 14a, g and j), which results in a large difference between the RMS values of force coefficients on the auxiliary lines pair. As the auxiliary lines pair is almost located side by side with the main line, the velocity of the flow past the

auxiliary lines pair is enlarged by the increasing blocking ratio, and it is expected that the RMS values of the force coefficients should be increased. However, the shear layers of the auxiliary lines pair are stretched and stable as the larger auxiliary line locates upstream of the smaller one (as shown in Fig. 14), and the large RMS values of the force coefficients only appear in the case that the larger auxiliary line locate downstream of the smaller one. 5.5. Instantaneous vorticity contours Fig. 14 shows the instantaneous vorticity contours of the riser system at different incidence angles. In the SLA pattern ( α = 90°), the vortex cannot shed from the upstream auxiliary lines, and the shear layers of the upstream auxiliary lines attach to the main line. Three vortex streets were observed in the near wake. After vigorous interaction in these vortex streets, a single vortex street is formed in the far wake, as shown in Fig. 14d. It can be seen that, the vortices shed from the auxiliary lines are weaker than those shed from the main line, and the directions of the vortices in the far wake are the same as the directions of the vortices shed from the main line. In the SLAVS pattern ( α = 0° and α = 180°), the vortex still cannot shed from the upstream auxiliary lines. The downstream auxiliary lines immerse in the shear layers of the main line, and no vortex shed from them can be observed, as shown in Fig. 14a and g. As α = 0°, the vortices shed from the low side of the main line and the vortices shed from the top side of the side auxiliary lines cancel out each other, and disappear quickly. So, only the vortices shed from the free-stream sides of the riser system can be observed. As α = 180°, the vortices shed from the side auxiliary lines merge to the upper row vortices shed from the main line, and a vortex with irregular shape is formed. In the SLSA pattern ( α = 60° and α = 120°), both the positive and negative vortices of the main line are squeezed in a vortex area (positive or negative), and the vortices with anti-direction disappear quickly. No positive vortices shed from the main line is observed as α = 60°, only the negative vortices at the top side shed and merge to the vortices shed from the upstream auxiliary lines, as shown in Fig. 14c and e. In the CVP pattern ( α = 210° and α = 330°), the vortices shed from the main line interact with the vortices shed from the auxiliary lines in the near wake, and quickly form a single sign vortex area (positive or negative), then shed alternately, as shown in Fig. 14h and l. In the SVF pattern ( α = 30°, α = 150° and α = 300°), the vortex shedding is suppressed on one side of the main line by the downstream auxiliary lines. As α = 30° and α = 150°, the shear layer separated from one side of the main line

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Fig. 13. The RMS values of the drag and lift coefficients on the auxiliary lines.

and the shear layers separated from the side auxiliary lines blend together, then roll up to forming one vortex street with the vortices shed from other auxiliary lines, as shown in Fig. 14b and f. As α = 300°, the vortices forming behind the main line and the auxiliary lines are comparative, the vortices shed from the main line disappear quickly. Only the vortices shed from the auxiliary lines can be observed, as shown in Fig. 14k. In the VSP pattern ( α = 240° and α = 270°), the downstream auxiliary lines are

immersed in the shear layers separated from the main line. As α = 240°, the shear layers separated from the main line are squeezed in a negative vortex area, and the positive vortices disappear quickly. Then, the above blend vortices shed in the far wake, and form a single vortex street, as shown in Fig. 14i. As α = 270° (see Fig. 14j), three vortex streets are observed in the near wake, which is similar to the flow structure in the SLA pattern. However, the downstream auxiliary lines make the vortices

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217

Fig. 14. The vorticity contours of the riser system at different incidence angles.

behind the main line longer and narrower than those in the SLA pattern, and this is the reason why the RMS values of the lift coefficients are smaller than those in the SLA pattern (Fig. 12b). It can be found that the shear layers of the main line are

squeezed together in CVP pattern, which results in a strong interference between the vortices with different directions shed from the main line, as shown in Fig. 14h and l. These vortices are canceled out quickly in the near wake, and the RMS values of the

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Fig. 15. The instantaneous vorticity contours of the riser system at α = 60°.

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Fig. 16. The instantaneous vorticity contours of the riser system at α = 210° .

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numbers of different lines in riser system. As α = 90°, the vortex can shed from the C&K line and main line (Fig. 14d), two Strouhal numbers can be observed. As α = 150°, the vortex can shed from the Booster line, Hydraulic line 1 and the main line (Fig. 14f), and three Strouhal numbers can be observed. As α = 180°, the vortex can shed from the Booster line and the main line (Fig. 14g), and two Strouhal numbers can be observed. In other cases, the shear layers separated from the main line and the auxiliary lines blend together before rolling up to form one vortex (as shown in Fig. 14), so only one Strouhal number dominate in these cases. Fig. 19 shows the histories of the lift coefficients of the main line at α = 60°, α = 210°, α = 240° and α = 300°. They present different amplitude values and shedding frequencies.

6. Conclusions Fig. 17. The Strouhal numbers of the main line.

lift coefficients of the main line are reduced largely, as shown in Fig. 12b. Figs. 15 and 16 show the instantaneous vorticity contours at α = 60° and α = 210° respectively. The RMS lift coefficients are small at these two angles, but different flow patterns are observed (as shown in Table 4). As α = 60° (Fig. 15), the shear layers A and B separated from the main line do not shed alternately. After 9T/15, the shear layer A develops into a single vortex C and disappears quickly. The shear layer B changes little in the whole period. As α = 210° (Fig. 16), the shear layers A and B separated from the main line interact strongly, and form a single vortex C together. This vortex disappears quickly. 5.6. Strouhal number based on the fluctuating lift force The Strouhal numbers shown in Figs. 17 and 18 are based on the diameter of the main line. Fig. 17 shows the Strouhal numbers of the main line at different incidence angles. As α = 60°, α = 120°, α = 210° and α = 240°, the shear layers separated from the main line are squeezed together and the vortices forming behind the main line are canceled out quickly (as shown in Fig. 14c, e, h and l). The vortices are mainly shed from the auxiliary lines, and the Strouhal number in these cases is much larger than that of single cylinder. In other cases, the vortices shed from the main line are stronger than those shed from the auxiliary lines, as shown in Fig. 14. It can be found that the dominant Strouhal numbers are smaller than that of the single cylinder. Fig. 18 shows the Strouhal

Fig. 18. The Strouhal numbers of lines in riser system.

Flow analysis and flow control for a real riser system with six auxiliary lines used for ocean drilling engineering were numerically studied in this paper. Three types of traditional experimental examples were used to validate the present method. The effects of the incidence angle, which was in the range of 0–360° and at 30° intervals, on the drag and lift coefficients, the pressure distribution around the main line, the vortex shedding frequencies and the flow structure were investigated. Some conclusions can be summarized as follows: 1. The flow patterns were classified into six types: SLA, SLAVS, SLSA, CVP, SVF and VS based on the varying incidence angles. In the same flow pattern, the main characteristics of the flow are similar. 2. The auxiliary lines can suppress the vortex shedding in all cases, but the effect of the suppression depends on the incidence angle. The mean value of the drag coefficient on the main line is smaller than that on the single cylinder at all oncoming angles expect α = 270°, and the RMS values of the lift coefficients on the main line are largely reduced by the auxiliary lines. 3. In the SLSA and CVP pattern, the vortex shedding from the main line is suppressed effectively by the auxiliary lines, so the RMS values of the lift coefficients of the main line are reduced largely. Especially, in the CVP pattern, the vortices shed behind the main line are squeezed together by multiple downstream auxiliary lines, and cancel out each other quickly because of the strong interaction between the vortices at anti direction (which was not observed in the case that one control rod was located around the cylinder, such as the studies of Zhao et al. (2005, 2007) and Firat et al. (2015)). Both the mean and RMS values of the drag and lift coefficients are reduced effectively. The mean value of the drag coefficient is just about 45.5% of that of the single cylinder, and the RMS value of the lift coefficient is about 1.5% of that of the single cylinder. In this condition, the drilling operation can be easier and safer. The worst effect of the suppression occurs in the SLA pattern, and the RMS value of the lift coefficient is about 76.8% of that of the single cylinder. 4. Three vortex streets were observed in the near wake in some incidence angles, which is different from the flow past two cylinders (Sumner et al.,2000, 2003; Hu and Zhou, 2008a). The drag and lift coefficients of the main line have been increased in these cases, such as the cases α = 90° and α = 270°. 5. The auxiliary lines can change the frequency of vortex shedding from the main line, and some larger Strouhal numbers are gotten in some cases. As α = 60°, α = 120°, α = 210° and α = 240°, the vortices shed from the auxiliary lines are stronger than the vortices shed from the main line, and the dominate frequencies are based on the auxiliary lines. As α = 60° and α = 120°, the Strouhal number of the main line is about 0.75.

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221

Fig. 19. Time histories of the lift coefficients of the main line at some angles.

Acknowledgment The authors are grateful to the support from the National Natural Science Foundation of China (Grant nos.11372188, 51490674) and the National Basic Research Program of China (973 Program) (Grant no. 2015CB251203).

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