3D NUMERICAL STUDY OF WALL EFFECTS ON VORTEX SHEDDING IN

SUBMARINE PIPELINES

Israel Marino Navas

Armando Blanco

Boris Bossio

Euro Casanova

isra.marino91@gmail.com

ajblanco@usb.ve

bossiob@usb.ve

ecasanov@usb.ve

Departamento de Mecánica, UniversidadSimón Bolívar, Caracas, Venezuela.

Abstract.Nowadays, worldwide energy requirements continue to rise. Annual human population

growth rate has become a good motivation for scientists around the world on finding new ways

for reducing energy requirements by trying to optimize energy generation, transportation and

consumption. Consequently, the human race has seen the need to search all possible energy

sources among which are the oil and gas, usually found offshore. Drilling rigs and production

transport systems are needed to move oil and gas extracted from offshore reservoirs to land

facilities, where those hydrocarbons usually go through several processes in order to be ready to

be used as chemical energy sources in cities and industries.

Submarine pipelines are considered one of the most effective and cost-efficient means of

transportation for moving oil and gas from offshore reservoirs to land. Frequently, long

distances should be covered, since several offshore reservoirs are located far from coastlines.

Submarine pipelines are subject to experience both mechanical and chemical stress conditions.

Specifically, inner and outer flows induce those pipelines to vibrate, jeopardizing their useful

lifespan due to the material fatigue.

Although many advances have been made in the numerical simulation of the flow around

submarine pipelines, most of the studies have considered simplifications such as modeling flow

around a two-dimensional structure.

This paper aims to make three dimensional (3D) simulations for studying the effect of marine

floor (i.e. wall effect) on the frequency of detachment of von Kármán’s vortexes that develop

downstream the pipeline. The numerical simulations were performed using the commercial

software CFX ™ which has been previously validated in cases involving two dimensional

(2D)studies. This paper presents a comparison among numerical predictions on vortex-shedding

frequencies between 3D and 2D models. Both models are subjected to laminar flow regimes with

Reynolds numbers 100 and 300, four different pipeline-seabed distances i.e. 7.5D, 2D, 0.75D

and 0.25D, D being the pipe diameter, and in the case of the 3D model, two different pipe

lengths (100 and 150 m). Numerical results show that the lift coefficient and the Strouhal number

increase as the pipeline approaches the seabed. Additionally, it is shown that vortex shedding

behavior depends on the considered section along the pipeline for the 3D model, this feature

been ignored by the 2D model. Finally, it was found that in the case where the pipe is subjected

to gravity loads perpendicular to free flow conditions (i.e. wall effects not considered), the Root

Mean Square (RMS) lift coefficient value is not zero. This finding shows that when the

deformation of the pipe by gravity is considered, flow asymmetry induced by the actual 3D

pipeline profile gives origin to a pressure gradient, which indicates that the pipe is constantly

loaded in cross-flow direction, thus altering its equilibrium position.

It is concluded that the two-dimensional simplification made for this phenomenon could be

inaccurate depending on the flow parameters and pipe configuration.

Keywords: Submarine pipelines, Vortex-shedding, Lift coefficient, Strouhal number.

Resumen.Actualmente, los requerimientos de energía a nivel mundial siguen aumentando. La tasa

de crecimiento poblacional se ha convertido en una motivación para los científicos alrededor

del mundo para encontrar nuevas formas dereducir los requerimientos energéticos al intentar

optimizar la generación, transporte y consumo de la energía.Debido a esto, el hombre se ha

visto en la necesidad de buscar todas las fuentes posibles de energía entre las cuales se

encuentran los hidrocarburos como el petróleo y gas, sustancias que usualmente son

encontrados costa afuera. Para mover el petróleo y el gas extraídos costa afuera hasta tierra

firme son necesarios diversos sistemas de transporte, ya que es generalmente en tierra firme

donde son procesados para ser utilizados como fuente de energía química en ciudades e

industrias.

Las tuberías submarinas destacanentre los medios de transporte más efectivos para trasladar el

petróleo y el gas desde los yacimientos costa afuera hasta tierra firme. Frecuentemente, estas

reservas de gas y petróleo se encuentran muy alejadas de la costa, razón por la cual las tuberías

deben cubrir un largo trayecto. Estas tuberías submarinas están expuestas a condiciones

mecánicas y químicas críticas; específicamente, los flujos internos y externos inducen

vibraciones mecánicas en la tubería, comprometiendo así la vida útil de la estructura debido a

la fatiga de los materiales que la componen.

A pesar que se han hecho grandes avances en las simulaciones numéricas en los estudios de

flujo de fluidos alrededor de las tuberías submarinas, la mayoría de los estudios han

considerado simplificaciones tales como el modelaje del flujo alrededor de estructuras

bidimensionales.

Esta investigación se en estudiar el efecto del lecho marino (efectos de pared) en la frecuencia

de desprendimiento de vórtices de von Kármán que se desarrollan aguas abajo de la tubería,

con un planteamiento tridimensional.Las simulaciones numéricas fueron realizadas utilizando el

software comercial CFX ™, el cual ha sido previamente validado en estudios bi-dimensionales.

Este trabajo presenta una comparación entre la frecuencia de desprendimiento de vórtices en

modelos 3D y 2D. Ambos modelos fueron estudiados en régimen laminar con números de

Reynolds iguales a 100 y 300, cuatro holgurasentre el cilindro y la pared: 7.5D, 2D, 0.75D y

0.25D-donde D representa el diámetro de la tubería- y en el caso del modelo 3D, dos longitudes

de tubería (100 y 150 m). Los resultados muestran que el coeficiente de sustentación y el número

de Strouhal aumentan a medida que la tubería se aproxima al lecho marino. Adicionalmente, se

muestra que el comportamiento tridimensional del desprendimiento de vórtices no es tomado en

cuenta cuando se realiza la simplificación 2D. Finalmente, se observó que en el caso donde la

tubería es deformada por efecto de la gravedad y no se consideran los efectos a pared, el valor

RMS (“Root Mean Square”) del coeficiente de sustentación no se anula. Este hallazgo muestra

que cuando la tubería es deformada por efecto de la gravedad, la asimetría del flujo inducida

por el perfil 3D de la tubería, da origen a un gradiente de presiones tal, que resulta en una

carga perenne en la estructura en la dirección perpendicular al flujo, alterando así su posición

de equilibrio estático.

Se concluye que la simplificación bi-dimensional hecha para este fenómeno puede ser poco

precisa dependiendo de los parámetros del flujo y de la configuración de la tubería.

Palabras Clave:Tuberías submarinas, Desprendimiento de vórtices, Coeficiente de sustentación,

Número de Strouhal.

1. INTRODUCTION

Vortex shedding is a phenomenon that has been highly studied since the early 1900s, both

analytically and experimentally. Nowadays, this topic is taken into account at the design stage of

many engineering applications, such as heat exchangers, nuclear reactors and even in the

climatological field. When studying vortex shedding, the fluid–structure interaction is analyzed,

and it has been found that this phenomenon potentially may compromise the mechanical

integrity of the structure and could cause it to fail, as the case of the Tacoma Narrows bridge,

which collapsed by high vibration amplitudes caused by the wind, in 1940.

Vortex-Induced Vibration (VIV) in submarine pipelines is a case of special interest because

these structures are mechanically affected by external and internal flows. Submarine pipelines

are frequently used in the oil industry for transporting oil and gas from offshore facilities to land.

Those pipelines are supported at the bottom of the sea which has, in some areas, an irregular

topology. For this reason, the supports of the pipes are, sometimes, separated by very long

distances (i.e. around 80-100m), being this support condition the most harmful one, as it

potentially allows the structure to vibrate excessively due to external flow.

The submarine pipelines are frequently modeled as circular cylinders. It is widely known

that the detachment of the vortices occurs when the Reynolds number (Re) value, based on the

cylinder diameter (D), is greater than 40 [1]. Beyond this regime, different flow patterns and

characteristics of the wake downstream of the cylinder are present. If a plane is inserted near the

cylinder, the phenomenon of vortex shedding is affected in structure, stability and frequency of

detachment. This fact has been shown through numerous experimental works (e.g. Taneda, 1965

[2];Göktun, 1975 [3];Bearman&Zdravkovich, 1978 [4];Angriliet al, 1984 [5]). However, none of

these studies takes into account the deformation of the cylinder due to gravity and the effect of

the difference of the gap (G) between the cylinder and the wall in the vortex shedding.

In the simulation field, several studies have been conducted on this phenomenon (e.g. Lei et

al, 2000 [6];Dettner&Peric, 2006 [7];Rajaniet al, 2009 [8]; Medina, 2010 [9]; and Rodríguez,

2012 [1[10]]) recreating accurately and precisely the vortex shedding effect on the

structure,using different codes and methods. Specifically, using CFXTM

, both Medina and

Rodríguez validated their models for analyzing vortex shedding effect on structures, in cases

where the structure was close to a wall, considering laminar flows. However, the cited numerical

analyzes have been simplified modeling a 2D domain. The present study focuses on whether this

simplification is valid for those cases where the ratio G/D is variable along a 3D structure, as is

the case of submarine pipelines which are deformed by the effect of gravity.

2. PHYSICAL MODEL

An air-filled structural steel pipe immersed in water is considered here. This submarine

pipeline has deflected due to gravity effects, therefore, there are different gaps G(Z) along its

longitude between it and the seabed. In order to engage in a short description of the VIV in a

cylinder, the reader must keep in mind that when the fluid encounters an object in its path, the

velocity of the fluid particle along the contact surface is zero causing a velocity gradient near the

object forming a boundary layer. In the case of the cylinder, the formed boundary layer would try

to surround it and at some point, the boundary layer will separate from the cylinder's surface, due

to its excessive curvature. This situation will promote the formation of vortexes. Vortexes are not

frequently formed symmetrically from the cylinder's symmetry plane and this is the cause of

different lift forces at the two sides of the cylinder; this originates a resultant global lift force

acting on the cylinder, inducing structure displacements in the transversal direction to the fluid

flow. The generated transversal movement changes the vortexes formation nature in a way that

vibration amplitude is limited, in most cases. At low Reynolds’s number values, current lines are

expected to be symmetrical, as long as predicted by potential flow theory.

Figure 1-Sketch of the deformed pipe near the seabed and being exposed to an external flow

Figure 1shows a submarine pipeline subjected to amarine current. In the figure it is shown

the velocity profile of the marine current approaching the pipe, where Uo is the free flow speed

of the marine current. Downstream, vortexesare detached, which will result in a harmonic

excitation on the pipe in the cross-flow direction. This study focuses on studying the effect of the

gap (G) between the bottom of the pipe and seabed and marine current velocity values on the

frequency of detachment of the vortices, fs. Different marine current velocity values will be

considered through different Reynolds’s number values.

Strouhal’s number is a dimensionless parameter used in problems of oscillatory and

transitory flux, which represents physically the relation between the inertial forces caused by the

flux instability and the inertial forces caused by the change of velocity from a point to another in

the fluid field[1]. Strouhal’s number is defined as:

(1)

Where,

S is the Strouhal’s number.

fs is the vortex shedding frequency

D is the pipe diameter

Uo is the free-stream velocity

It is well known that the vortex shedding frequency fs is the same that the lift force

frequency and two times the drag force frequency [1].

Fluid dynamics was modeled by transient Navier-Stokes equations for Newtonian and

incompressible fluids:

(2)

(3)

Where,

ρis the fluid density.

V is the velocity vector.

g is the body force vector.

p is the pressure.

μis the dynamic viscosity of the fluid.

The energy equation isnot considered because the system is isothermal and there is no mass or

heat transfer involved.

3. NUMERICAL MODEL

ANSYS CFX software is a high-performance, general purpose fluid dynamics program that

has been applied to solve wide-ranging fluid flow problems.This computational fluid dynamics

software was used for solving the problem in question.This program applies the finite volume

method to solve the equations shown previously, using an implicit method which has a second

order accuracy in both time and space.

The considered boundary conditions are shown in the Fig. 2.The set initial conditions were

obtained from a steady-state simulation using both the same domain and boundary

conditions.The“upwind” advection scheme was set for optimizing the computational required

time to get the results.

Figure 2- Boundary conditions used in the domain

The opening condition (definedby pressure value) was used to allow the recirculation of the flow

in the outlet surface and a velocity definition based on the Reynolds number was used in the inlet

condition.

4. CASES OF STUDY

On this paper, the effect of marine floor (i.e. wall effect) on the frequency of detachment of

von Kármán’s vortexes that develop downstream the pipeline is studied. The obtained 3D results

were compared with the ones from the 2D case studied by Medina [9], in which the“wall effect”

in a 2D cylinder under laminar flow was studied. Four different gaps (G) denoted as: G1=0,25D,

G2=0,75D, G3=2D and G4= 7,5D were considered. The last case was defined as “no-borders”

(NB) because the wall effect at this distance is indeed neglected. It is important to bear in mind

that one of the mainfocuses and motivations of this paper is to compare the 2D simplification

results [9] with the present full model (3D) results, hence,the cases of study of this work were the

same that Medina [9]considered. In order to add a 3D parameter, two pipeline length (L)

wasconsidered as well.Figure 3shows a diagram of the cases studiedon this work.

Figure 3-Cases of study

5. RESULTS

A sweeping method was used to build the discrete geometry around the deformed cylinder.

Therewere considered hexahedral elements witha second order form function in order to be able

to achieve better accuracy in the interpolation of the solution between the faces of the cells. The

block around the cylinder had five layers of prismatic elements and a first element size of 0,425

mm. These parameters were validated by Medina [9], whosuccessfully represents the vortex

detachment of a 2D cylinder. For the case of a 3D domain it is necessary to estimate the number

of cells in the longitudinal axis of the pipe to capture with precision the physics of the problem.

Table 1 showsthe mesh sensitivity analysis for the case of the present study were a high

numerical precision was required (i.e.Re=300, L=150 m, G/D=0,25).CL and CD represent the lift

and drag coefficients, respectively and S is the Strouhal’s number

Table 1-Mesh sensitivity study

Number of divisions CL CD S

20 0,4000 0,1755

40 0,0667 0,4190 0,1873

80 0,0714 0,4286 0,2435

These results show that the mesh criterion convergence has not been reached. However in

Simón Bolívar University's facilities there were no computer that had the capacity to perform a

simulation with a partition of 120 cells in the longitudinal direction of the cylinder. Additionally,

80 divisions is also a challenging goal to achieve because the time required for running all the

study cases with this partition would exceeded the available time to write this work, for this

reason a mesh with only 40 division was employed.

The time step used in the present work was determined by means of the Courant number

which was fixed at a value of 1 for a more accurate solution.Theresults shown in this sectionwere

selected at a certain time, where the RMS valuesand amplitudesof the drag and lift forcesdon't

varied significantly.In order to be able to compare these results with the ones obtained by

Medina [9](Re=300),a monitor point was defined at a distance near the pipe (y/D=1, x/D=1) at

the mid plane of the longitudinal direction.This was made to be able to compare the frequency

value of the vortices detachment in a plane of the domain with the 2D simplification. The tables

2 & 3 show the results obtained at a flow regime of Re=100 and tables 4 & 5 show the results

obtained at Re=300.

As expected, the lift coefficient value increases as the cylinder gets closer to the wall. The

wall effect produce an asymmetry in the flow around the pipe, causing a rise in the pressure field

in the cylinder’s bottom, then, the resultant perpendicular force increases as well. Additionally, it

is observed that for all the flux regimes and pipe lengths, the lift force is not equal to zero in the

NB case. This is caused by the flow asymmetry that is formed around the deformed structure.

The catenary form of the pipe produces a higher pressure distribution in the pipe bottom,

resulting in a persistent force that has to be taken into account because it changes the equilibrium

position of the structure.

To discard a possible numerical error on this affirmation, the simulation of a straight

cylinder was done, considering the same flux regime, pipe-length and gap (NB) conditions. The

resultant lift coefficient for this case was 7,5x10-5

which is a negligible value in comparison with

the one obtained in the simulation of the deformed cylinder. Figure 4 shows the streamlines for

these two cases.

Table 2-Results obtained for Re=100, L=100 m

Seabed Gap, G CL CD S

NB 0,5387 0,1660

2D 0,0408 0,6135 0,2080

0,75D 0,0407 0,6510 0,1750

0,25D 0,0701 0,6211 0,1969

Table 3-Results obtained for Re=100, L=150 m

Seabed Gap, G CL CD S

NB 0,8032 0,1625

2D 0,0475 0,8485 0,1618

0,75D 0,0582 0,8990 0,1796

0,25D 0,0665 0,8770 0,1778

Table 4-Results obtained for Re=300, L=100 m

Seabed Gap, G CL CD S S*

NB 0,4205 0,1805 0,1791

2D 0,1257 0,4633 0,1958 0,1998

0,75D 0,1582 0,4866 0,2040 0,2046

0,25D 0,0701 0,4453 0,2028 0,1694

Note: S* correspond to the Strouhal’s number calculated using the oscillation frequency of

the velocity in the monitor point near the pipe in the mid-plane of the structure.

Table 5-Results obtained for Re=300, L=150 m

Seabed Gap, G CL CD S S*

NB 0,3967 0,1800 0,1752

2D 0,0726 0,3711 0,1738 0,1951

0,75D 0,1177 0,4333 0,1870 0,1898

0,25D 0,0655 0,4281 0,1871 0,1812

In the tables 4 and 5 it is shown that the difference between the Strouhal’s number value

measured using the lift force frequency and the one that is calculated using the monitor point

velocity near the cylinder, is very low (i.e. <5%) for most cases. This proves that the deflection

of the pipe causes a three-dimensional behavior in the flow and that the shedding frequency

depends on the entire flow field.

Figure 4- Dimensionless velocity (U/Uo) streamlines around the pipeline.

(a) straight cylinder (b) deformed cylinder.

The results shown in Figs. 5 to 7 suggest that the 2D simplification doesn’t reproduce

accurately the detachment of vortexes in a deform cylinder who is near a plane, because the three

dimensionality of the vortex street it’s not considered in the 2D model. Fig 8 shows this

phenomenon graphically.

Figure 5-Lift coefficient comparison with Medina [9] (Re=300)

Figure 6-Drag coefficient comparison with Medina [9] (Re=300)

Figure 7-Strouhal's number comparison with Medina [9] (Re=300)

Figure 8-Dimensionless velocity (U/Uo) streamlines around the pipeline

(Re=300 and L=100 m). (a) G/D=NB; (b) G/D=2; (c) G/D=0,75; (d) G/D=0,25

Figure 9-Dimensionless velocity (U/Uo) contour plot in the mid-plane of the pipeline

(Re=300 and L=100 m). (a) G/D=NB. (b) G/D=2. (c) G/D=0,75. (d) G/D=0,25

Figure 8illustrates the three-dimensionality of the flow previously mentioned which show

that the vortex street, that is formed downstream of the pipe, has different frequencies of

detachment. This delayin frequency becomesmore evident as the gap between the cylinder and

the seabed reduces, causing a 3D curvature in the vortex street. Fig 8d shows that for the case

where the gap is smaller than 0,3D, the vortex shedding is suppressed. This behavior, also shown

in Fig. 9d,has beenreportedsome authors [1], [2], [3], [6], [9].

6. CONCLUSIONS

A 3D Numerical study of wall effects on vortex shedding in submarine pipelines was

conducted in this work. The effect of marine floor on the frequency of detachment of von

Kármán’s vortexes, that develop downstream the pipeline, was analyzed throughnumerical

results,showing that the lift coefficient and the Strouhal number increase as the pipeline

approaches the seabed. Results, also showed that vortex shedding behavior depended on the

considered section along the pipeline for the 3D model, being this feature ignored by 2D models.

Finally, it was found that in the case where the pipe is subjected to gravity loads perpendicular

toits longitudinal axis, and no wall effect is considered (i.e. NB condition) the RMS lift

coefficient value is not zero. This finding shows that when the deformation of the pipe due to

gravity is considered, flow asymmetry induced by the actual 3D pipeline profile gives origin to a

pressure gradient, which indicates that the pipe is constantly loaded in cross-flow direction, thus

modifying its equilibrium position. As general conclusions, it may be stated that:

For cases were the wall-effect and the deformation of the pipe due to gravity or any body-

force is important, 2D simplification is not recommended.

The three-dimensionality of the deformed pipe causes a variation in the near-wall vortex

shedding frequency in all the pipeline length.

When the deformation of the pipeline is important, it is necessary to take into account the

force in the cross-flow direction which changes the equilibrium position of the structure.

7. REFERENCES

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[3]. Göktun, S.,The drag and lift characteristics of a cylinder placed near a plane surface.

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[4]. Bearman, P.&Zdravkovich, M., Flow around a circular cylinder near a plane boundary.

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[5]. Angrilli, F., Bergamaschi, S., &Cossalter, V., Investigation of wall induced

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[7]. Dettmer, W., &Peric, D., A computational framework for fluid-rigid body interaction:

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[8]. Rajani, B., Kandasamy, A., &Majumdar, S., Numerical simulation of laminar flow past

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[9]. Medina, M., Estudio numérico del efecto de paredes en la frecuencia de

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