ipc2012 90225 computational fluid dynamics

1 Copyright © 2012 by ASME

Proceedings of the 9th International Pipeline Conference IPC2012

September 24 – 28, 2012, Calgary, Alberta, Canada

IPC2012- 90225

ADDED VALUE OF COMPUTATIONAL FLUID DYNAMICS IN OFFSHORE PIPELINE DESIGN

F. Van den Abeele

OCAS N.V. Zelzate, Belgium

J. Vande Voorde ArcelorMittal Global R&D Ghent

Zelzate, Belgium

F. Kara Cranfield University

Cranfield, UK

ABSTRACT

The advent of high performance computing systems has unlocked the promising potential of computational fluid dynamics (CFD) to assist in the design, calculation and optimisation of engineering structures. Indeed, CFD provides a powerful and efficient means of evaluating flow mechanics and predicting hydrodynamic loads on offshore structures. In this paper, the added value of CFD in offshore pipeline design is demonstrated with two practical case studies: stability of subsea pipelines, and vortex induced vibrations in marine risers.

First, the stability of offshore pipelines in close proximity

to the seabed is studied. In traditional offshore pipeline design, the on-bottom stability is governed by the Morison’s equations. According to this set of equations, offshore pipelines are designed to satisfy two stability conditions: the submerged weight of the pipe has to be greater than the lift force, and the horizontal friction force should exceed the combined drag and inertia forces. It is common practice to use fixed hydrodynamic coefficients (drag, lift and inertia) to calculate the pipeline stability, based on the assumption that the pipeline is either trenched or in contact with the seabed. However, due to uneven seabed topology and/or scouring, a gap may exist between the pipe and the seafloor. In such a case, the force coefficients not only depend on the relative gap between the pipe and the seabed. Moreover, in unsteady oscillatory flow (induced by waves), the time-dependent laminar or turbulent characteristics of the boundary layers become important. In this paper, a CFD model is presented to evaluate lift, drag and inertia forces exerted on subsea pipelines to reveal the effect of boundary proximity.

In the second application, turbulence modelling is applied

to predict vortex induced vibrations (VIV) in multiple marine risers. One of the most important design requirements for marine risers in (ultra)deep water is to limit the fatigue damage caused by VIV. Even moderate currents can induce vortex shedding, at a rate determined by the flow velocity. Each time a vortex sheds, a force is generated in both the in-line and cross-

flow direction, causing an oscillatory multi-mode vibration. Vortex induced vibration (VIV) can give rise to cyclic stresses that might cause fatigue failure. For floating production platforms in particular, there is a risk of interference between adjacent production or export risers, or possibly between other combinations of tendons, drilling risers and production risers. Numerical simulations of fluid flow and vortex shedding allow calculating the optimum spacing between multiple marine risers in tandem arrangement.

COMPUTATIONAL FLUID DYNAMICS Computational fluid dynamics (CFD) is an emerging field

of research with promising potential, and has been receiving more and more attention from the offshore in the oil and gas industry the last few years. A CFD solver typically uses a generalized version of the Navier Stokes equations, solving for the velocity field and the pressure . In its most general form this set of equations comprises an energy balance, a continuity equation expressing the conservation of mass, and an equation for the conservation of momentum [4]. Assuming a constant (seawater), the Navier Stokes equations reduce to the formulation of incompressible Newtonian flow:

∙ ∙

∙ 0

In general, these equations are solved for the three-

dimensional flow field. In the present formulation, the velocity field is taken as two-dimensional, i.e. , .

When the fluid flows past a fixed cylinder like a marine

riser or an offshore pipeline, a region of disturbed flow is formed, like schematically shown on Figure 1. In this simulation of laminar flow, the free stream velocity is shown in green. Lower velocities are depicted in blue, whereas yellow indicates values higher than the stream velocity.


Figure 1: Regions of disturbed flow

Evidently, the velocity varies in terms of magnitude,

direction and time, and four regions can be distinguished: 1. The retarded flow is a narrow region in front of the

cylinder, where the local (time-averaged) velocity is lower than the free stream velocity

2. Two boundary layers attached to the surface of the cylinder

3. Two sideway regions where the local (time-averaged) velocity is higher than the free stream velocity

4. The wake, which is the downstream region of separated flow where the local (time-averaged) velocity is less than the free stream velocity

The fluid flow around a circular cylinder, like shown on

Figure 1, is a well known and documented [5-7] problem in computational fluid dynamics, and often used as a benchmark for CFD solvers [8]. The flow pattern in the wake of the cylinder is primarily governed by the Reynolds number

⁄ which expresses the ratio of inertia forces to viscous forces, with the fluid flow velocity, the total outer diameter, and the kinematic viscosity .

A detailed analysis of the different flow regimes around

subsea structures can be found in [9-10]. In summary, the regimes of fluid flow across a smooth subsea structure can be divided in

Unseparated flow for very low ( 5) Reynolds numbers

The regime for 5 40 , where a pair of Föppl vortices develop in the wake

The transition range (150 300) from laminar flow to turbulence

The regime where the vortex street is fully turbulent (300 3 ∙ 10 )

For even higher numbers 3 ∙ 10 3 ∙ 10 , the laminar boundary layer undergoes turbulent transition, and the wake will be narrower and disorganized

At very high Reynolds numbers ( 3 ∙ 10 ), re-establishment of a turbulent vortex street occurs

For the range of Reynolds numbers relevant to offshore pipeline engineering, the flow is fully turbulent, and it becomes increasingly difficult –if not impossible- to predict the transient flow behavior with a laminar solver for the Navier Stokes equations. The possible options for CFD simulations at very high Reynolds numbers are:

Direct Numerical Simulation (DNS), which solves the Navier Stokes equations for the pressure and the velocity components in a time-dependent domain. This approach requires a very fine mesh size and very small time steps to resolve the smallest eddies and capture the fluctuations in the turbulent flow [03]. As a result, this approach is not economically feasible for pipeline design.

Large Eddy Simulation (LES), where large turbulent eddies are computed in a time-dependent simulation, whereas small eddies are predicted with a compact model. Indeed, smaller eddies have an isotropic (and hence more universal) behavior, but larger eddies in the turbulent flow tend to be anisotropic, and their behavior is directly influenced by the problem geometry. The viability and accuracy of Large Eddy Simulation for complex turbulent flows at high Reynolds numbers is investigated in [11], but has proven to be not feasible for full 3D analysis of offshore structures [12].

Reynolds Averaged Navier Stokes (RANS) turbulence

model. In the RANS approach, all flow characteristics are decomposed as the sum of a steady (mean) value and a fluctuating term. This decomposition gives rise to a Reynolds stress tensor, which adds six unknowns to the system of equations. As a result, turbulence models are required to provide additional transport equations to close the system [13]. In this paper, the Spalart-Allmaras Turbulence model is applied to study the stability of offshore pipelines close to the seabed, and an enhanced model is used to simulate vortex induced vibrations in multiple marine risers.

ON BOTTOM STABILITY OF OFFSHORE PIPELINES In traditional offshore pipeline design, the on-bottom

stability of submarine pipelines is governed by the semi-empirical Morison’s equations [14]. Assuming that waves are approaching the pipeline with a velocity and at an angle , and the current with steady velocity is approaching at an angle , the fluid flow will impose a lift force

12

(1)

and a drag force (2)


12 | |

where and are the lift and drag coefficients respectively. In addition, the wave induced acceleration gives rise to an inertia force

4

(3)

with the inertia coefficient. The Morison’s equations show that the drag and lift forces are proportional to the square of the fluid particle velocity, and that the inertia force is directly proportional to the fluid particle acceleration. The drag force acts in a direction parallel to the fluid flow, while the lift force is generally upwards (i.e. normal to the seabed). The inertia force acts in the direction of the flow or against it, depending on whether the flow is accelerating or decelerating.

The Morison’s equations are used to determine the appropriate thickness of a concrete weight coating to ensure offshore pipeline stability. The pipeline stability condition is considered to be satisfied when the forces that resist the pipeline displacement are greater than the forces that tend to displace it. As a result, the pipeline is stable when the submerged weight of the pipe is greater than the lift force in vertical direction:

(4) with the weight of the pipe, coatings and contents, the buoyancy forces acting on the pipe, and an appropriate safety factor [15]. At the same time, the horizontal friction force has to remain greater than the combined drag and inertia forces:

(5) where is the coefficient of friction between the pipe and the seabed. Self-weight of the pipe (and its contents) is generally not sufficient to satisfy these criteria. In order to achieve stability, subsea pipelines are coated on the outside with high density concrete. The required thickness of the concrete coating is determined by an iterative procedure [16] such that the above criteria (4)-(5) are satisfied for the most severe load combinations. It should be noted that the absolute lateral stability approach, proposed in equation (5), results in a heavy pipe, and so is used only for special cases [15]. When the pipeline is sitting on the seabed, the hydrodynamic coefficients are frequently fixed [17] to 0.9, 0.7 and

3.29. Obviously, these values depend on the roughness of the pipe, and various guidelines and codes may recommend different values. In addition, the hydrodynamic coefficients depend on both the Reynolds number and the Keulegan-Carpenter number [18]

with the wave period. In addition, the value for , and is dependent on the position of the pipe as well. If the pipeline is sitting on the seabed –which is always intended by design- the hydrodynamic coefficients will be significantly different from those for pipeline spans with a gap between the pipe and the seabed, or for partially buried pipes [19].

Sarpkaya has performed experimental research [19] to study the hydrodynamic forces on cylinders placed at various distances from the bottom of a U-shaped water tunnel. The main conclusions from these investigations read

The hydrodynamic coefficients are functions of the

Reynolds number , the Keulegan-Carpenter number , the gap between the pipe and the seabed and the

depth of penetration of the viscous wave or the boundary layer thickness:

, , , , , (6)

The effect of the boundary layer or the penetration

depth of the viscous wave is small, provided that the boundary layer remains laminar. Boundary layer effects are ignored for 0.1⁄ . For turbulent oscillatory boundary layers, the characteristics of the wall jet and separation over the cylinder may be significantly affected.

The drag and inertia coefficients for the in-line force acting on the cylinder are increased by the presence of the wall. This increase is most evident in the range of

0.5⁄ .

The proximity of the wall helps to decouple the frequency of oscillations in the top and bottom shear layers. This decoupling effect prevents the occurrence of regular vortex shedding for small values of ⁄ .

The transverse force towards the wall is relatively small and fairly independent of ⁄ . The transverse force away from the wall is quite large and dependent on ⁄ , particularly in the range 0 0.5⁄ .

The use of the Morison’s equations to decompose the in-line force into two components is a sound approach. The lumping of the entire in-line force into a single coefficient is not justified, and obscures flow mechanics.

As already indicated in [20], computational fluid dynamics proves to be a powerful tool to study the stability of offshore pipelines.


Numerical simulations can shed a brighter light on the effects of boundary proximity, and enable a more profound understanding of the experimental insights developed by Sarpkaya. To study the influence of boundary proximity on the evolution of the hydrodynamic coefficients, a CFD model was constructed with a fixed pipeline with diameter . The pipeline exhibits a gap with the seabed and was subjected to a fluid flow velocity . By changing the velocity (and hence the Reynolds number) and the gap , the experimentally observed relations (6) can be calculated.

The simulations were performed at Reynolds numbers in

the range of 1 ∙ 10 1 ∙ 10 , which requires turbulence modeling to accurately capture the fluid flow patterns. For the simulations, presented in this paper, the Spalart-Allmaras one-equation turbulence model has been applied. This RANS model, originally developed for aerodynamic flow [21], solves for the undamped viscosity , whereas the turbulent eddy viscosity can be computed from . A detailed review of the Spalart-Allmaras turbulence model and its (mostly aeronautical) applications can be found in [22]. The values for the constants, used in this paper, are listed in Table 1.

Table 1: Values for the Spalart-Allmaras model constants

0.13555 0.622 7.1

0.3 2 7.1 0.41

On Figure 2, the CFD model was used to predict the

variation of the drag coefficient as a function of seabed proximity ⁄ . As can be seen, the drag coefficient for a specific Reynolds number will change from a relatively high value when the pipe is close to the seabed ( ≪ 1⁄ ) to the free stream value when 1⁄ .

Figure 2: Drag coefficient as function of seabed proximity

In Figure 3, streamlines of the flow and contour plots of

the pressure are plotted for 2 ∙ 10 . In these plots, it is clear that the wake behind the pipe changes from primarily a blunt body on the seabed ( 0.1⁄ ) to a free stream cylinder with increasing distance from the seabed. The shape

and the wake behind the pipe changes accordingly, which in turn strongly influences the drag (and drag coefficient) of the pipe. The distinct advantage of computational fluid dynamics over experimental testing is that a vast amount of flow data is available to analyze the physical phenomena that are taking place. In addition, it is straightforward to assess the effects of modifications, and hence quickly optimize the solution.

0.1⁄

0.2⁄

0.5⁄

1⁄

Figure 3: Flow patterns as function of seabed proximity


More details on the use of computational fluid dynamics to

study on-bottom stability of offshore pipelines in close proximity to the seabed can be found in [20]. In the next section, the merits of CFD in predicting vortex induced vibrations in multiple marine risers exhibiting wake interference are briefly highlighted.

PREDICTION OF VORTEX INDUCED VIBRATIONS During the design of floating production platforms in

deepwater, it has been recognized [23] that there is a risk of interference between adjacent production or export risers, or possibly between other combinations of tendons, drilling risers and production risers. The consequences of most concern are the possible increase in fatigue damage due to vortex induced vibrations (VIV), and the likelihood of contact between adjacent risers.

A large body of work has been published addressing

measurement, modeling and analysis of marine risers in tandem arrangement [24]. A careful review of flow interference between two circular cylinders in various arrangements has been presented by Zdravkovich [25-26], including an extensive list of references on this subject. He has also introduced a classification of flow regimes around two circular cylinders, depending on their relative position.

Different studies for the tandem arrangement of two

adjacent risers [23,27-29] have shown that the changes in drag, lift and vortex shedding are not continuous. Instead, an abrupt change for all flow characteristics is observed at a critical spacing between the risers. An exhaustive description on proximity effects and wake interference can be found in [30], and a comprehensive summary of VIV in tandem risers is provided in [31]. Recent research results have been published in a.o. [31-33].

In this paper, the published data on riser interference tests for flexible tubulars [23] will be used as experimental validation. To simulate these experiments, a 2D CFD model is constructed, assuming fixed rigid cylinders with an outer diameter of 114.3 mm. The simulation setup, with a grid of 50 by 15 , is shown on Figure 4.

Figure 4: Simulation setup to study wake interference

For the simulations of fluid flow around marine risers in

tandem arrangement, the computational grid comprises some 250 000 cells. Depending on the end spacing, the dimensionless wall distance is in the range of

20 yρ u y

μ30 (7)

with the distance to the nearest wall, and the friction velocity defined by

u τρ

(8)

where is the average wall shear stress. As long as (7) is satisfied, the problem is well conditioned. However, for simulations requiring a high amount of vorticity and flow separation, the Spalart-Allmaras turbulence model used in the previous section did not yield reliable results. Indeed, the calculations failed when negative pressure gradients occurred. Hence, this turbulence model is not appropriate to predict vortex induced vibrations in marine risers.

To mitigate the problems associated with the Spalart-

Allmaras model, the turbulence model was selected to simulate wake interference in adjacent marine risers. This model is frequently used to model turbulent flow, and was identified by [11] as the most appropriate RANS model to predict vortex induced vibrations in marine risers for Reynolds numbers up to 10 .

The turbulence model [35, 36] is a two equation

model, providing a transport equation for the kinetic energy and an additional expression for the viscous dissipation rate .

Table 2: Values for the k- model constants

1.35 1.80

1.0 1.3

The values for the model constants are listed in Table 2. This standard model is widely used in computational fluid dynamics, and was adopted by [11,37] to predict vortex shedding around circular cylinders at high Reynolds numbers ( 10 ). The model performs quite well for boundary layer flows, but is less accurate for risers in which a high mean shear rate is present or massive separation occurs (which could be expected for risers in tandem arrangement). In these cases, the eddy viscosity is over-predicted by the standard formulation. Moreover, the dissipation rate equation does not always give the appropriate length scale for turbulence.


To improve the ability of the standard model to predict complex turbulent flows, an enhanced eddy viscosity model is proposed in [38]. This model consists of a new formulation for the viscous dissipation rate based on the dynamic equation of the mean square vorticity fluctuation at large turbulent Reynolds numbers. In addition, a new eddy viscosity formulation is introduced based on the positivity of the normal Reynolds stresses and the Schwarz’ inequality for turbulent shear stresses [38]. The model constants, calibrated in [38], are listed in Table 3.

Table 3: Values for the enhanced k- model constants

max 0.43,⁄

5 ⁄ 1.90

1.0 1.2

On Figure 5, the turbulent eddy viscosity is shown for very high ( 2.5 ∙ 10 ) Reynolds numbers, clearly indicating that this enhanced eddy viscosity model is capable of simulating a turbulent wake with significant separation.

Figure 5: Distribution of turbulent eddy viscosity

Figure 6: Tandem risers with different end spacing


The parametric approach, suggested in Figure 4, enables the investigation of risers in staggered arrangements as well, for 0. In this paper, we focus on risers in tandem arrangement

( 0 with different end spacings 2 ⁄ 6. It has been shown experimentally [26-28] that there is strong interference between two cylinders in tandem arrangement for spacing ratios with ⁄ 3.5. At a spacing ⁄ 3.5, a sudden change of the flow pattern in the gap between the adjacent risers is observed.

On Figure 6, the influence of the end spacing on the fluid

flow pattern in the wake of the tandem risers is shown for a Reynolds number 10 , i.e. the two-bubble regime of the transition in the boundary layers. These simulation results indeed endorse the experimental observations of Allen [23], Zdravkovich [26] and King [27]:

For small end spacing ( ⁄ 3), vortex shedding

only occurs in the wake of the downstream riser: the free shear layers which separate from the upstream riser are permanently re-attached to the downstream riser. In [39], Zdravkovich refers to this type of wake interference as quasi-steady re-attachment.

When increasing the gap ( ⁄ 3) between both risers, a turbulent vortex street appears in the wake of both the upstream and the downstream riser. The vortices shed by the upstream riser coalesce with the vortex street of the downstream riser, and binary eddy streets are observed. It can be clearly seen that there is no re-attachment of the free shear layers separated from the upstream riser to the downstream one.

Drag coefficient data [26, 28] shows that the upstream riser takes the brunt of the burden, and that the downstream riser has little or no effect on the upstream one. For different values of spacing ⁄ , the drag coefficient is shown on Figure 7.

Figure 7: Drag coefficients at Re = 105

Apparently, the drag coefficient on the upstream riser is not

significantly influenced by the downstream one, but a significant change in drag is observed on the downstream cylinder for ⁄ 3.

In [23], drag coefficients are measured on risers in tandem arrangement with increasing end spacing for Reynolds numbers from 1 ∙ 10 up to 2.5 ∙ 10 . On Figure 8, for instance, the measured drag coefficients for both upstream and downstream riser are shown for a spacing ⁄ 3. The drag coefficients, predicted by the CFD simulations at 1 ∙ 10 , are indicated as well.

Figure 8: Drag coefficients for L = 3D [23]

Figure 8 shows that for the upstream cylinder, the drag

crisis occurs somewhat earlier (i.e. at a lower Reynolds number) than traditional measurements of this phenomenon [27, 28], which could be attributed to the combined effects of free-stream turbulence and cylinder displacement. The combination of an early drag crisis on the upstream riser and large displacements of the downstream riser produces a larger total drag force on the downstream riser for 1.7 ∙ 10 . More details on the effect of end spacing on drag coefficients and transverse displacements of tandem risers can be found in [23]. The results, presented here, indicate that computational fluid dynamics can indeed contribute to deepwater risers design.

CONCLUSIONS In this paper, the added value of computational fluid

dynamics (CFD) was demonstrated by means of two case studies. First, the stability of offshore pipelines in close proximity to the seabed was studied. The main conclusion from that investigation is that for a given Reynolds number, the drag coefficient changes from a relatively high value (close to the seabed) to the free stream value ( 1⁄ ). The flow in the wake of the pipe changes from primarily a blunt body on the seabed to a free stream cylinder with increasing distance to the seabed.

In the second application, turbulence modelling is applied to predict vortex induced vibrations (VIV) in multiple marine risers. The most striking observations and conclusions from that study read


Given the high Reynolds numbers involved in deep water riser design 10 10 , turbulence modeling is required to capture vortex shedding. The enhanced model, proposed in [38], proved to be the most appropriate RANS model to predict VIV.

For two risers in tandem arrangement, there is a sudden change in flow characteristics for a critical end spacing ⁄ 3.5. The upstream riser takes most of the burden,

while the drag coefficient on the downstream riser is lower at 1.7 ∙ 10 .

For low Reynolds numbers, there is little effect of end

spacing on the drag coefficients and displacements, whereas the effect of end spacing is obvious and distinct for 1.7 ∙ 10 .

Both case studies prove that computational fluid dynamics provides a powerful means of evaluating flow mechanics and optimize the design. Hence, CFD can indeed provide added value in ensuring offshore pipeline stability and supporting the design of marine risers in deep water.

REFERENCES [1] Dixon M. and Charlesworth D., Application of CFD for

Vortex Induced Vibration Analysis of Marine Risers in Projects, Proceedings of the Offshore Technology Conference, OTC 18348 (2006)

[2] Huang K., Chen H.C. and Chen C.R., Deepwater Risers VIV Assessment by Using a Time Domain Simulation Approach, Proceedings of the Offshore Technology Conference, OTC 18769 (2007)

[3] Mainçon P. and Larsen C.M., Towards a Time Domain Finite Element Analysis of Vortex Induced Vibration of Risers with Staggered Buoyancy, Proceedings of the 30th ASME Conference on Ocean, Offshore and Arctic Engineering, OMAE2011-49046 (2011)

[4] Batchelor G.K., An Introduction to Fluid Dynamics, Cambridge University Press (1967)

[5] Norberg C., Flow Around a Circular Cylinder: Aspects of Fluctuating Lift, Journal of Fluids and Structures, vol. 15, pp. 459-469 (2001)

[6] Rocchi D. and Zasso A., Vortex Shedding from a Circular Cylinder in a Smooth and Wired Configuration: Comparison between 3D Large Eddy Simulation and Experimental Analysis, Journal of Wind Engineering and Industrial Aerodynamics, vol. 910, pp. 475-489 (2002)

[7] Norberg C., Fluctuating Lift on a Circular Cylinder: Review and New Measurements, Journal of Fluids and Structures, vol. 15, pp. 57-96 (2003)

[8] Schäfer M. and Turek S., Benchmark Computations of Laminar Flow Around a Cylinder, Flow Simulation with High Performance Computers II, vol. 52, pp. 547-566 (1996)

[9] Sumer B.M. and Fredsoe J., Hydrodynamics around Cylindrical Structures, World Scientific, Signapore (1999)

[10] Gourmel X., Numerical Simulation of the Flow over Smooth and Straked Riser at High Reynolds Numbers, M.Sc. Thesis, Cranfield University, School of Applied Sciences, Offshore and Ocean Technology (2010)

[11] Catalano P., Wang M., Iaccarino G. and Moin P., Numerical Simulation of the Flow Around a Circular Cylinder at High Reynolds Numbers, International Journal of Heat and Fluid Flow, vol. 24, pp. 463-469 (2003)

[12] Constantinides Y. and Oakley O., Numerical Prediction of Bare and Straked Cylinder Vortex Induced Vibrations, Proceedings of the 25th International Conference on Offshore Mechanics and Arctic Engineering, OMAE2006-92334, Hamburg, Germany (2006)

[13] Wilcox D.C., Turbulence Modelling for Computational Fluid Dynamics, Second Edition, DCW Industries (1998)

[14] Morison J.R., O’ Brien M.P., Johnson J.W. and Schaaf S.A., The Forces Exerted by Surface Waves on Piles, Journal of Petroleum Technology, vol. 189, pp. 149-154 (1950)

[15] Det Norske Veritas, On Bottom Stability Design of Submarine Pipelines, DNV-RP-F109 (2007)

[16] Palmer A.C. and King R.A., Subsea Pipeline Engineering, Penwell, Tulsa, Okla, 572 pp. (2004)

[17] Mohitpour M., Golshan H. And Murray A., Pipeline Design and Construction, A Practical Approach, Third Edition, ASME (2007)

[18] Olbjorn E.H., Some Submarine Pipeline Engineering Problems, Submarine Pipeline Engineering

[19] Sarpkaya T. and Isaacson M., Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, 651 pp. (1981)

[20] Van den Abeele F. and Vande Voorde J., Stability of Offshore Pipelines in Close Proximity to the Seabed, Proceedings of the 6th Pipeline Technology Conference, Hannover, Germany (2006)

[21] Spalart P.R. and Allmaras S.R., A One Equation Turbulence Model for Aerodynamic Flows, Recherche Aerospatiale, vol. 1, pp. 5-21 (1994)

[22] Javaherchi T., Review of Spalart-Allmaras Turbulence Model and its Modifications (2010)

[23] Allen D.W., Henning D.L. and Lee L., Riser Interference Tests on Flexible Tubulars at Prototype Reynolds Numbers, Proceedings of the Offshore Technology Conference, OTC 17290 (2005)

[24] Kondo N., Three Dimensional Analysis for Vortex Induced Vibrations of an Upstream Cylinder in Two Tandem Circular Cylinders, Proceedings of the 30th ASME Conference on Ocean, Offshore and Arctic Engineering, OMAE2011-49655 (2011)

[25] Zdravkovich M.M., Review of Flow Interference between Two Circular Cylinders in Various Arrangements, Journal of Fluids Engineering, vol. 99 (4), pp. 618-633 (1977)

[26] Zdravkovich M.M., Flow Induced Oscillations of Two Interfering Circular Cylinders, Journal of Sound and Vibration, vol. 101(4), pp. 511-521 (1985)


[27] King R., Wake Interaction Experiments with Two Flexible Circular Cylinders in Flowing Water, Journal of Sound and Vibration, vol. 45(2), pp. 559-583 (1976)

[28] Zhang H. and Melbourne W.H., Interference between Two Cylinders in Tandem in Turbulent Flow, Journal of Wind Engineering and Industrial Aerodynamics, vol. 41-44, pp. 589-600 (1992)

[29] Allen D.W. and Henning D.L., Vortex Induced Vibration Current Tank Tests of Two Equal Diameter Cylinders in Tandem, Journal of Fluids and Structures, vol. 17, pp. 767-781 (2003)

[30] Mittal S. And Kumar V., Flow Induced Oscillations of Two Cylinders in Tandem and Staggered Arrangements, Journal of Fluids and Structures, vol. 15, pp. 717-736 (2001)

[31] Lestang Parade E., Dual String Risers – Local Behaviour, M.Sc. Thesis, Cranfield University, School of Applied Sciences, Offshore and Ocean Technology (2010)

[32] Li L., Fu S., Yang J., Ren T. and Wang X., Experimental Investigation of Vortex Induced Vibration of Risers with Staggered Buoyancy, Proceedings of the 30th ASME Conference on Ocean, Offshore and Arctic Engineering, OMAE2011-49046 (2011)

[33] Corson D., Cosgrove S., Hays P.R., Constantinides Y., Oakley O., Mukundan H. And Leung M., CFD Hydrodynamic Databases for Wake Interference Assessment, Proceedings of the 30th ASME Conference on Ocean, Offshore and Arctic Engineering, OMAE2011-49407 (2011)

[34] Gao Y., Yu D., Wang X. And Tan S.K., Numerical Study of an Oscillating Cylinder in the Wake of an Upstream Larger Cylinder, Proceedings of the 30th ASME Conference on Ocean, Offshore and Arctic Engineering, OMAE2011-49126 (2011)

[35] Jones W.P. and Launder B.E, Prediction of Laminarization with a Two Equation Model of Turbulence, International Journal of Heat and Mass Transfer, vol. 15 (1972)

[36] Chien K.Y., Predictions of Channel and Boundary Layer Flows with a Low Reynolds Number Turbulence Model, AIAA Journal, vol. 20(1), pp. 33-38 (1982)

[37] Majumdar S. and Rodi W., Numerical Calculation of Turbulent Flow Past a Circular Cylinder, Proceedings of the 7th Turbulent Shear Flow Symposium, pp. 3.13-25, Stanford, USA (1985)

[38] Shih T.H., Liou W.W., Shabbir A., Yang Z. And Zhu J., A New k- Eddy Viscosity Model for High Reynolds Number Turbulent Flows, Computer Fluids vol. 24(3), pp. 227-238 (1995)

[39] Zdravkovich M.M., Flow Around Circular Cylinders – Applications, Oxford University Press (2003)

ipc2012 90225 computational fluid dynamics

Documents