Title: Using strip theory to model vibrations in offshore risers

Author 1, corresponding author

Seyed Hossein Madani MSc, PhD

Research Fellow, Department of Mechanical, Aerospace and Civil engineering, Brunel University

London, UK Email: Hossein.Madani@brunel.ac.uk ; Tel: (+44)01895266383

Author 2

Jan Wissink MSc, PhD

Senior Lecturer, Department of Mechanical, Aerospace and Civil engineering, Brunel University

London, UK

Author 3

Hamid Bahai MSc, PhD

Professor, Head of Mechanical, Aerospace and Civil engineering Department, Brunel University

London, UK

ABSTRACT

We present in this paper an algorithm for the coupled time-domain solution of governing equations of

flow and those of the structure using the strip theory. We adopt this approach for numerical

simulation of the multi-mode response of a flexible long riser to Vortex-Induced Vibrations (VIV). In

this research, various models which have been used to solve the riser problem are briefly studied and

compared. The work reported here initially includes calculation of eigenvalues and eigenmodes of the

riser, the results of which are subsequently used to predict the response of the riser based on a modal

analysis. The 2D flow simulations used in the strip theory are validated using experimental and

numerical results presented in the literature. The Root Mean Square (RMS) of the amplitude of the

riser movement in the cross flow direction for various top tensions is studied and compared with

similar cases reported in the literature.

Key words: Offshore industry, Computational mechanics, Mathematical modelling.

List of notation:

L, H Length / Height of Riser (m)D Riser Diameter (m)E Young modules (N/m2)I Moment of inertia (kg.m2)u Cross flow displacement (m)w Axial displacement (m)y Cross flow directionz Axial directionx Flow directionTtop Riser top tension (N)t Time (s)A Riser cross section (m2)F Forces (N)V Flow velocity (vector) (m/s)P Pressure (N/m2)U∞ Flow velocity (m/s)m Mass of the riser per length (kg/m)Cl Lift coefficientCaxial Axial force coefficientm* Mass ratiour Flow velocity- radial (m/s)uφ Flow velocity- circumferential (m/s)r Position vector (m)Vr Reduced velocityAr Reduced areaM Mass matrixK Stiffness matrixC Damping matrixα, β Rayleigh damping coefficientq Distributed load (kg/m)

1 Introduction

Marine risers typically are flexible, long pipes with a circular cross section. These pipes are used in

the offshore industry to transfer fluid from the bottom of the sea to the platforms and vice versa. Top

Tensioned Risers (TTRs) and Steel Catenary Risers (SCRs) are among the most common and

traditional flexible risers. The former is normally vertical while the latter is highly curved and flexible

in a catenary shape connecting a sub-sea pipeline to a floating production structure. Recently,

composite risers have drawn significant attention due to their relatively low weight and low costs,

however, the design and analysis of the composite risers are challenging especially when the

durability in harsh environments is involved (Tan, Chen et al. 2015). For applications with large

physical domains and large body displacements it is of vital importance to use an accurate and

computationally affordable numerical model (Mittal, Iaccarino 2005). Despite the considerable

research efforts on Vortex-Induced Vibration during the last decades, the prediction of the load and

response is still subject to a considerable amount of uncertainty. Therefore, high safety factors of

between 10 and 20 are required in the design process (Wang, Fu et al. 2015). CFD analysis of off-

shore engineering applications is becoming increasingly popular (Oakley, Constantinides et al. 2005,

Constantinides, Oakley 2006, Constantinides, Oakley et al. 2007, Holmes, Oakley et al. 2008, Tan,

Chen et al. 2015). However, due to the size of the problem which leads to a high computational

demand, a full 3D VIV simulation of the interaction between the flow and the riser is unfeasible.

Hence, simulations rely (and for the foreseeable future will continue to rely) on simplifying

assumptions (Chaplin, Bearman et al. 2005) and simulations are necessary to reduce the cost and

increase the scope of expensive testing programs.

There are three main approaches to overcome this issue. In the first approach experimental data is

interpolated to represent the hydrodynamic forces on the riser so that there is no need to solve the

fluid flow and only the structural equations need to be solved. An example of this method is VICoMo

Code (Gopalkrishnan 1993). Alternatively, semi-empirical data from Morrison’s equation is used to

predict the hydrodynamic forces (Lyons, Vandiver et al. 2003, Tan, Chen et al. 2015). The SHEAR7

code, which is mainly used in industry, is based on this method. In the third approach a CFD based

model is coupled with the structural equations to simultaneously simulate the Fluid-Structure-

interaction (Willden, Graham 2005). Chaplin has made a remarkable effort to compare 11 codes based

on these methods with experimental data (Chaplin, Bearman et al. 2005).

In the present research, methods presented in the literature in which hydrodynamic forces are

calculated to model the flexible riser problem are studied. Also an algorithm based on 2D strip theory

and well-known flow and structural methods is presented which has the potential for further

development into a 3D strip model. In addition, the simulation results for the 2D strips are compared

with the experimental and numerical results presented in the literature and the modal analysis of the

riser using various top tensions is studied and compared with ANSYS. The RMS of the amplitude of

oscillation is presented for various riser tensions.

2 Various CFD based approaches

(Schulz, Meling 2004) and (Willden, Graham 2005) introduced a strip-theory numerical method to

model long, flexible risers. The hydrodynamic forces at each 2D strip are interpolated to obtain the

overall loading along the span of the riser. This loading is then used to integrate forward a single time-

step in the riser equations of motion to obtain an updated riser displacement profile. Closure of the

coupled flow-structure method is achieved by updating the riser displacements for each of the

corresponding hydrodynamic strips in the next time-step integration.

In another attempt, a scaled three dimensional CFD model with high element aspect ratio in the axial

direction is studied using the AcuSolveTM package which is a finite element Navier-Stokes solver

based on a Galerkin/Least squares formulation. To minimize the number of mesh points in the

discretisation, the mesh around the riser is coupled to the riser displacement so that at all axial

positions the riser cross section is in the middle of grid (Constantinides, Oakley et al. 2007). In these

studies a simple linear structural vibration analysis is used in which sinusoidal eigenmodes are

assumed and the displacements of the riser are a linear summation of the modal amplitudes times the

corresponding eigenvectors (Constantinides, Oakley et al. 2007, Holmes, Oakley et al. 2008,

Constantinides, Oakley 2006). It is claimed that the model was able to simulate properly the effects of

three-dimensional structures such as strakes, buoyancy modules and catenary riser shapes. The

simulation was benchmarked against laboratory and offshore experiments.

(Huang, Chen et al. 2010) proposed a scaled riser model with a high aspect ratio of about ten times the

pipe’s diameter (10D) in the axial direction to limit the computational costs, which resulted in a poor

resolution of the flow in the axial direction. The simulations were carried out using the Finite Analytic

Navier-Stokes solver, FANS, code (Huang, Chen et al. 2007). The riser is modelled as a beam with

top tension and solved explicitly using beam theory. It is shown that the cross flow VIV in the riser

top section is not similar to that of the bottom section. One end was found to have considerably higher

cross flow vibrations than the other. It was concluded that the presented CFD approach was able to

provide reasonable results and is suitable for 3D riser VIV analysis in deep water and complex current

conditions. Also it was possible to predict similar dominant modes and amplitudes as observed in the

experiment performed by Marintek, Trondheim, Norway, and donated by ExxonMobil URC,

Houston, TX, USA.

(Wu, Zhu et al. 2015) used a scaled 3D model to analyse the loads and response of a flexible riser in a

wave-current environment. The loads and responses were calculated using the CFD module System

Coupling in the software package ANSYS14.5. The results showed that the vibration equilibrium

location of the riser offsets in the direction of the current when wave and current are aligned. In this

case, the response of the riser was larger than when having only a wave and no current. The vibration

amplitude was found to increase with the current. Results opposite to the above were found when

wave and current had opposite directions.

Table 1: CFD based model presented in the literature to model the flexible riser problem

Model Riser L/D Riser Type Other important aspectsStrip Theory (Schulz, Meling 2004)

………… ……… 2D strips, FVM, RANS Loose coupling in time domain Uniform and shear current Single and array risers Sensitivity study to the number of the strip

Strip Theory (Willden, Graham 2005)

1544(500/0.3239)

TTR,SCR 2D strips, LES, velocity vortices modelUniform and stepped current profileSuggest 6 to 7 strips per half wave lengthEuler-Bernoulli Beam model

3D CFD, high aspect radio in axial direction (Holmes, Oakley et al. 2006, Constantinides, Oakley 2006, Constantinides, Oakley et al. 2007)

1400(38/0.027)4000(147/0.035)

TTR,SCR High element aspect ratio in riser axial direction, highly sparse mesh

Simple structural modelUniform and linear shear flow Include the effects of Strakes and

buoyancy modules3D CFD, high aspect radio in axial direction (Huang, Chen et al. 2010)

482(10/0.0207)

TTR 817N High element aspect ratio in riser axial direction.

RANS in conjunction with LESUniform flowPinned supports riser

3D CFD (Wu, Zhu et al. 2015)

…………. a pipe FVM, RANSWave- Current profile 3D FEM for structure using ANSYS 14.5

In the present research an affordable computational model based on strip theory is developed to

simulate the behaviour of a flexible riser that is exposed to unsteady hydrodynamic forces caused by

vortex shedding. Primitive parameters are used to solve the two dimensional Navier-Stokes equations

using cylindrical coordinates (Verzicco, Orlandi 1996). The moving frame of reference (Li, Sherwin

et al. 2002) is attached to the riser at each 2D plane to be able to use an efficient fixed mesh to

simulate the fluid-structure interaction with large displacement. A modal analysis of the riser is

conducted to be able to define structural parameters that might lead to lock-in phenomena. The

dynamic response of the riser is computed in the time domain with a finite element structural model

based on the Euler–Bernoulli beam theory (Patel, Witz 2013).

3 Physics of the problem

Riser pipes typically have a length, L, of about a few hundred meters with an outer diameter, D, less

than a meter which gives an L/D ratio on the order of O(103). Risers are normally exposed to a current

with a maximum of 2m/s and a current profile that can vary with depth as well as waves. The

Reynolds number of the flow typically is on the order of O(105) to O(106). At very high depths, the

riser pipes become longer and more flexible and can be exposed to a higher vibrational mode (above

40th) (Willden, Graham 2005). Especially when the frequency of the vortex shedding of the flow

behind the riser coincides with the natural frequency of the structure, VIV phenomena might occur.

In this research, the riser dynamics in the presence of low Reynolds number flow is investigated so

that 2D domain strips can be used without turbulence modelling. The structure characteristics are

chosen such that the natural frequencies of the riser are in the range where VIV is expected to happen.

4 Numerical model

Here, the motion of a point on the riser in the cross-flow and axial directions is considered. The axial

motion is assumed to be locally tangential to the riser. The motion normal to the axis of the riser is

calculated based on bending and rotation about an axis perpendicular to the plane of the translation.

The stiffness in the transverse direction (normal to the plane spanned by the axis of the riser and the

flow direction) includes bending stiffness and geometrical stiffness. The geometrical stiffness arises

from moments generated by the axial forces.

In the case of small displacements (due to VIV of a riser), the geometrical stiffness contribution can

be ignored. However, and initial stress contribution cannot be neglected if there is a distribution of

stress in the equilibrium configuration. In addition, because of using linearization, the axial stress

dependency upon the bending motions can be ignored.

Therefore, before linearization it is necessary to determine the static configuration and axial stress

distribution about which to linearize. In the case of a straight and vertical riser the static configuration

and equilibrium axial stress are known before hand, and axial deformations due to the buoyancy and

gravity are neglected. Also, the orthogonal mode of vibration can be studied independently. Note that

in this study, the displacement in the flow direction is ignored.

4.1 Governing equations – structural

The governing equations of motion for a slender, linearly elastic pipe can be determined by

considering a short segment of the pipe in local coordinates, where u, w are the local displacements in

in the y and z directions, respectively (, left), where z is the axial direction and the y direction is

transverse to the flow direction x. The origin of the linearized riser structural dynamic model is

located at the bottom of the riser.

Using a linearized assumption, the motions in the orthogonal directions are solved independently.

However, the riser’s local displacements will be coupled through the enforcement of displacement

compatibility at the nodes of the elements in the finite element discretisation.

Using Bernoulli-Euler beam theory and Newton’s second law the equation of motion in the transverse

direction is established (Appendix). It is assumed that the hydrodynamic forces act perpendicular to

the axis of the riser and the weight of the segment and buoyancy forces act in the axial direction (,

Right). The equation for the transverse motion of a point on the riser reads:

EI ∂4 u∂ z4 −

∂∂ z (T ∂ u

∂ z )+m ∂2u∂ t2 =F transverse

(1)

In this equation, t is time, F transverse is the lift force per unit length of the riser which is calculated from

the hydrodynamic forces. AlsoE, I and m are Young’s modulus, moment of inertia of the cross

section and mass per unit length of the riser, respectively. In equation (1), T is the axial tension which

varies along the axial direction and is defined by the following equation:

T ( z )=T top−mg ( H−z )+ πρ D2

4( H−z ) g (2)

In this equation, the origin is located at the bottom (foot) of the riser, the height of the riser is H and

Ttop is the initial tension at the top. The last term in equation (2) is the buoyancy force, where it is

assumed that the entire riser is submerged in a fluid of density, ρ, and g is the gravitational

acceleration.

In the axial direction, by neglecting shear deformation and assuming a linear material with constant

cross section and Young’s modulus, the equation of motion for a point on the riser can be written

using Newton’s second laws in axial direction (Appendix). The linearization of the axial stress

removes its dependency on bending motions, so that the equation of motion in the axial direction

becomes:

−EA ∂2 w∂ z2 +m ∂2 w

∂t2 =Faxial(3)

In the above equation, A is the riser cross section and Faxial is the distributed axial forces per unit

length of the riser which includes the weight of the riser and buoyance forces.

4.2 Governing equations – fluid flow

The governing equations for an unsteady, incompressible fluid flow in vector form are given by the

Navier–Stokes equations which read:

ρ( ∂ V∂t

+V .∇V )=−∇ p+μ∇2V + f(4)

∇ .V =0 (5)

In the above equations, μ is the dynamic viscosity and f is the body force. The governing equations

are rewritten in cylindrical coordinates and subsequently discretised using a staggered arrangement of

variables to guarantee a strong coupling between the velocity and pressure. Second order central

difference methods are used to discretise the spatial terms in the governing equations. The fractional

step method, finally, is used to integrate the Navier-Stokes equations in time. In the predictor stage, a

velocity field is predicted by integrating the convective and diffusive terms using a 3 rd order Runge-

Kutta method. In the corrector step, the Pressure Poisson Equation is solved to enforce conservation

of mass through the pressure field using the Strongly Implicit Procedure (SIP solver). The origin of

each 2D fluid plane is attached to the riser cross section for each 2D strip and the Navier-Stokes

equations are derived in the moving frame of references. Therefore it is possible to simulate structural

displacements without any mesh deformation as at the moving boundary the velocities of the fluid

flow and the structure are the same.

4.3 Non-dimensionalisation

In order to couple the equations of structural motion to the flow governing equations, the equations

are non-dimensionalised using the same parameters.

t '=tU∞

D;m¿=

ms

mf=

ms

πρ D 2

4

;u'= uD

; ∂ u'

∂t= 1

U ∞

∂u∂ t

;

CL=F transverse

12

ρU ∞2 D

;Caxial=Faxial

12

ρ U∞2 D

(6)

In the above equation, ms is the mass of the solid per unit length of the riser, mf is the mass of the

displaced fluid per length of the riser, m¿ is the mass ratio, U∞ is the free flow velocity, CL is the lift

coefficient and Caxial is the non-dimensional force in axial direction. t ',u' are non-dimensional time

and velocity, respectively. The reduced velocity,V r, is defined based on the first mode of the in vacuo

natural frequency, f beam1 , of an equivalent straight beam which has simple support ends and no axial

tension and is defined below:

f beam1 =

12 π ( π

L )2√ EI

m(7)

V r=U∞

f beam1 D

=2 π U ∞

D ( Lπ )

2

( mEI )

12

(8)

The non-dimensional equations of structural motion in the transverse and axial directions then read:

4 ( L/ D )4

( π V r )2 [ ∂4 u∂ z4 −

∂∂ z (T r

∂u∂ z )]+ ∂2u

∂ t2 =C L

(m¿ π2 )

(9)

−Ar4 (L /D )4

( π V r )2∂2 w∂ z2 + ∂2w

∂t 2 =Caxial

( m¿ π2 )

(10)

In the above equations, a reduced tension,T r, and a reduced area, Ar, are introduced to non-

dimensionalise the axial tension and cross sectional area to simplify the structural equations of motion

(Appendix). The non-dimensional form of the fluid flow equations in cylindrical coordinates reads:

∂ur

∂ t+ur

∂ur

∂ r+

uφ

r∂ ur

∂ φ−

uφ2

r=−∂ p

∂r+ 1

ℜ [ 1r

∂∂r (r ∂ ur

∂r )+ 1r2

∂2 ur

∂ φ2 −ur

r2 −2r2

∂ uφ

∂ φ ]−a ysolidsin ( φ )(11)

∂uφ

∂ t+ur

∂ uφ

∂ r+

uφ

r∂uφ

∂ φ−

ur uφ

r=−1

r∂ p∂ φ

+ 1ℜ [1

r∂

∂ r (r∂ uφ

∂r )+ 1r2

∂2uφ

∂ φ2 −uφ

r2 + 2r2

∂ ur

∂ φ ]−aysolid sin (φ )(12)

∂∂ r (r ur )+

∂ uφ

∂ φ=0

(13)

In the above equation, ur and uφ are velocities in radial, r , and circumferential, φ, directions,

respectively. a ysolid is the solid acceleration in the cross flow direction and ℜ is the Reynolds number.

4.4 Numerical approximation – Structural

The Galerkin weighted residual method is used to derive the numerical approximation of the

structural equations of motion in local coordinates for each element. Because the axial and bending

motions are orthogonal, they can be solved independently as uncoupled equations.

The pipe’s displacement in the transverse direction to the flow can be expressed in terms of shape

functions and nodal displacements as:

ue (ze , t)=∑m=1

M ye

ume (t ) N ym

e (ze),(14)

Where, the transverse displacement ue (ze , t), is calculated using the displacement at every nodal

point (M ye points) of that element, um

e ( t ) , multiplied by the predefined element local shape functions,

N yme (ze). ze is the local coordinate for a general element in the axial direction.

The Galerkin weighted residual approach tries to minimise the error (residual) that occurs in

approximating the solution. The residual is minimized by integrating over the element using suitable

shape functions as the weighting functions (see Smith, Griffiths 1988).

The discretised equations of motion for the transverse and axial directions can be expressed as a

single matrix equation.

M ∂2 r∂ t2 +C ∂r

∂ t+K r=F

(15)

In the above equation M is the mass matrix, K is the stiffness matrix (bending stiffness + geometric

stiffness),C is the damping matrix and F is the force matrix which is calculated based on non-

dimensional parameters (Appendix). For the damped vibration it is assumed that the un-damped mode

shapes are orthogonal with respect to the damping matrix if C is taken to be a linear combination of M

and K.

C=αM+ βK (16)

Where the scalars α and β are the so called ‘Rayleigh damping’ coefficients (Smith, Griffiths 1988).

5 Interpolating the force from the flow to the structure

For each 2D strip the distributed load over a unit length part of the riser is integrated using flow

variables adjacent to the cylinder boundary. As the locations of the 2D strips do not necessarily

coincide with the structural elements, an interpolation method is needed to find the distributed force

over each element. After the interpolation, the distributed load over the structural elements is known.

A minimum of 5 to 6 strips are needed to simulate a full oscillation wave length. Therefore, to be able

to model the 10th natural mode of a riser at least 50-60 strips are necessary. In Figure 4, the locations

of the strips are identified by S1 to Sp where “p” is the number of processors (each 2D strip is

calculated using a separate processor). The distributed load needs to be replaced by an equivalent set

of nodal loads. This can be done by using Przemieniecki’s method. In this case, the load can be

assumed as the negative of the end reactions and moments that would apply if each element was fully

supported (removing all degrees of freedom) at both ends (Przemieniecki 1985).

By assuming linear distributed forces over each element, the equivalent forces and moments at both

ends of each element are calculated from equations (17-(20) and shown in Figure 4.

FL=L e qL

2+

Le(qR−qL)6

(17)

FR=Le qL

2+

Le (qR−qL)3

(18)

M L=[ Le2qL

12+

Le2 (qR−qL)

30 ] (19)

M R=−[ Le2qL

12+

Le2 (qR−qL)

20 ] (20)

In the above equations it is assumed that the distributed loads are vectors and include the sign, in

addition the moments in the counter clock wise direction are positive.

6 Solution algorithm

At the start of the simulation the geometry of the structure is defined once and the locations of the 2D

strips are stored. The flow governing equations for each 2D strip are solved on separate processors

and the interpolations and structural equations are solved and integrated in time on the root processor.

The structural equation of motion (15) is expressed in state space form and the Crank-Nicolson

scheme is used for the temporal integration:

rin+1−ri

n

∆ t=1

2 ( ∂ rin+1

∂ t+

∂r in

∂ t ) (21)

M

∂r in+1

∂ t−

∂r in

∂ t∆ t

+C∂ r i

n+1

∂ t+K ri

n+1=Fn

(22)

The overall procedure for advancing the solution from the tn to tn+1 can be summarized as follows:

1. Use the solution (Fn , rin ,

∂rin

∂ t) at tn

as initial value to calculate the location (rin+1),

velocity and acceleration of the structural nodes at time tn+1 by rearranging equations

(21), (22).

2. Interpolate the location, velocity and acceleration of the structure (using cubic splines)

at tn+1 and distribute these values to each 2D flow level.

3. Solve the flow governing equations (11-(13) to advance the pressure and velocity

fields to tn+1 for each 2D flow strip using the new position of the structure.

4. Calculate hydrodynamic forces at tn+1 for each 2D flow strip.

5. Interpolate the forces from each 2D flow level (using cubic splines) to calculate the

overall hydrodynamic force distribution along the riser.

6. Check if the structural solution reached steady state for the structural response

otherwise continues to advance in time by going back to the first step.

7 Results

As mentioned earlier, the hydrodynamic forces are calculated in 2D strips at various levels along the

riser axis while the connection between these hydrodynamic forces happens through the structure’s

response. In this section some of the VIV results related to the simulation of these two dimensional

strips are presented and compared with literature. In addition, the modal analysis of the structure is

presented to investigate the modes that are more likely to be excited by vortex shedding and to explain

some of the structural parameters’ values. In the final part of this section, the results obtained by

applying strip theory to the riser problem are presented and discussed.

7.1 Fluid flow at 2D strips

The numerical simulation of flow past a moving structure is one of the most challenging problems in

computational mechanics and numerous researches have been carried out in this area. Flow over a

circular cylinder has been used as a bench mark to compare methods in the past decades. However,

the reported results do not all agree and some of the discrepancies are highlighted in this section. In

this paper, the flow around the cylinder is simulated using cylindrical coordinates. The origin of the

coordinates is attached to the cylinder and the governing equations are solved in the moving frame of

references. The relative velocity is used to avoid expensive mesh regeneration while modelling the

moving cylinder. In this section, a cylinder with one degree of freedom in the cross-flow direction is

simulated for low Reynolds numbers, 90 < Re < 140, and reduced velocities 5 < V r < 8 to verify the

accuracy of the code (that will be used in combination with the strip theory) by comparing it to the

results presented in the literature. In this simulation, the mass ratio and damping ratio are m *=150 and

ξ=0.0012 respectively.

Table 2 shows that the maximum oscillation amplitude with respect to cylinder diameter found here

compares well with the results obtained using other numerical methods and experiments. For instance,

some of the numerical methods predicted an oscillation amplitude above 0.3 at Re=100, while in the

experimental results the amplitude of oscillation is low. This low amplitude is correctly predicted in

the present simulations and also by (Li, Sherwin et al. 2002), who also use a moving frame of

reference to model the moving boundaries. Figure 5 shows the development of the oscillation

amplitude and the lift coefficient at Re=105 which matches well with the results presented in

literature, showing an amplitude of oscillation of A/D=0.382.

Table 2: Maximum oscillation amplitude respect to diameter as a function of reduced velocity and Reynolds numberReynolds number 90 95 100 105 110 120 130 140

Reduced velocity 2.02 5.30 5.58 5.86 6.14 6.70 7.26 7.81

Amplitude found here 0.002 0.0034 0.008 0.38

3

0.339 0.008 0.004 0.004

(Anagnostopoulos, Bearman

1992), Experimental

------ 0.015 0.015 0.52 0.51 0.4 0.2 0.002

(Kara, Stoesser et al. 2015)

CaseA

0.005 0.007 0.36 0.30 0.25 0.005 0.005 0.007

(Kara, Stoesser et al. 2015)

CaseB

0.002 0.007 0.42 0.38 0.35 0.005 0.005 0.007

(Li, Sherwin et al. 2002) ------ ----- 0.01 ----- 0.25 0.20 0.005 -----

(Yang, Preidikman et al. 2008) 0.002 0.42 0.38 0.35 0.30 0.005 ------ 0.002

7.2 Modal analysis

As previously stated, altering the riser characteristics, including top tension, mass ratio, size and

material, will change the natural frequency of the structure and hence change the possibility and range

of lock-in phenomena. To be able to study and observe vortex-induced vibration in a riser, the

frequency of vortex shedding (Strouhal frequency) and the natural frequency of the structure should

be close to each other. This has been studied in three cases (). In the first scenario both the top and the

bottom of the riser are assumed to be simply supported (with only a rotational degree of freedom). In

the second case, a top tension is applied in the axial direction and the degree of freedom in the

transverse direction is removed. In the third scenario, tension is applied only in the axial direction and

the riser is able to move in the transverse direction. Figure 6 shows the natural frequency of a riser

based on mode number for a simply supported riser and for a riser under top tension. In this figure

modal analysis results obtained by the in-house code are validated using ANSYS software. The effect

of top tension on the riser natural frequency is shown in Figure 7. Here, the riser natural frequency, fn,

is non-dimensionalised by the vortex shedding natural frequency, f s. It is expected that for cases with

fn/fs > 1, the riser will be excited at the vortex shedding frequency.

Table 3: Riser characteristics

Length D (outer) D (inner) Density Density

fluid

Young

modules

Top tension

500m 0.324m 0.302m 8750kg/m2 1025kg/m2 200GPa 2.285E5 N

7.3 Strip theory results

The riser VIV responses in a uniform cross flow of U∞=1 m/s are analysed. The present study focuses

on the cross flow VIV results, which are more important due to the higher fluctuation amplitude in the

lift coefficient as compared to the drag coefficient (in-line to the flow direction). The riser is chosen

such that the weight of riser and the buoyancy forces cancel each other (Table 3), so that the tension

in the riser is constant and equal to the riser top tension. The simulations start from a straight riser. In

the beginning of the simulation the riser is observed to deflect to one side until the restoring force is

large enough to overcome the lift coefficient. The simulations are carried out for 650 non-dimensional

time units, as normally the simulations reach a steady state after 500 time units. Figure 8 presents the

time evolution of the middle of a riser. The simulation results show a non-uniform deflection along

the riser which is due to the top tension, weight of riser and buoyancy forces. The amplitude of the

maximum oscillation is predicted to be about 0.4D due to VIV which is in reasonable agreement with

the values found in the literature (Huang, Chen et al. 2010). The root mean square (RMS) of the riser

response is shown in Figure 9. The power spectrum amplitude of the RMS graph shows that the first

six modes of the oscillation are activated in the oscillation. Also the third mode of oscillation is found

to match the vortex shedding frequency. To be able to predict and capture higher frequencies of

oscillation, the number of 2D strips would need to be increased. Future work will focus on a detailed

analysis of the dependency of the results on the mass ratio and the tension.

8 Summary and conclusion

In this study, various methods found in the literature that were used to solve the flexible riser problem

are classified and the CFD based methods are briefly discussed. A full 3D simulation of the riser

problem is nearly impossible due to the huge size of the problem. To circumvent this, simplifying

assumptions need to be employed. Three main approaches were found in the literature: 1)

Determining the forces on the riser by calculating 2D flow in several strips located along the riser.

The hydrodynamic forces are subsequently coupled through the structural movement, (strip theory) 2)

Applying a scaled model, which may give rise to problems when extending it to a full scale model

with multiple modal frequencies. 3) The application of large-sized elements with a very high aspect

ratio in the axial riser direction to decrease the computational time. This method may compromise the

accuracy of the results especially near the riser wall.

In this paper, an in-house code was used to develop a strip theory model using well-established

structural and flow solvers from the literature (partition approach). This method has the potential to

replace the 2D strips by 3D strips as well as to allow for deformation of the riser’s cross-section. The

2D flow solver used in the 2D strips is validated using experimental and numerical results from the

literature. Modal analysis is used to define the proper characteristics for the structure, to be able to

predict possible lock-in phenomena and to define a proper top tension in the riser problem. All

simulations were performed at low Reynolds numbers to be able to ignore three-dimensional effects

of vortex shedding. In future, the code will be expanded by replacing the 2D strips by 3D strips and

by allowing hydrodynamic forces in the axial direction.

9 Acknowledgement

The funding provided by EPSRC (grant number EP/K034243/1) and BP for this research is gratefully

acknowledged.

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Appendix

The assumption to derive and discretise the structural governing equations is presented in this section.

Bernoulli-Euler beam theory and Newton’s second law is used to write the equation of motion. It is

assumed that the hydrodynamic forces are act perpendicular to the axis of the riser and the weight of

the segment and buoyancy forces are act in the axial direction (, Right). The Fz and Fy are the external

loads per unit of length of the segment in that figure.

The flexural rigidity, EI, and the mass per unit length of riser, m are taken to be constant, however the

axial tension, T, in the riser is not constant and is changing according to the height of the riser.

Therefore the equation of the transverse motion for a point on the riser reads:

EI ∂4 u∂ z4 −

∂∂ z (T ∂ u

∂ z )+m ∂2u∂ t2 =F transverse

(1)

In this equation u is the displacement in the transverse direction (to the flow direction), y, and z is the

coordinate in the axial direction of the riser (see Smith, Griffiths 1988). T, the axial tension at each

point along the axial direction, z, can be formulated based on the height of the riser, H, the top initial

tension, Ttop, the riser mass and buoyancy forces.

T ( z )=T top−mg ( H−z )+ πρ D2

4( H−Z ) g

(2)

The last sentence in this equation is the buoyancy force and it is assumed that the whole of the riser

has been submerged in the fluid.

In the axial direction, by neglecting shear deformation and assuming linear material, the equation of

motion for a point on the riser can be written using Newton’s second laws in axial direction ( ). The

linearization of axial stress removes its dependency upon bending motions. Therefore the equation of

motion in the axial direction can be written as:

−EA ∂2 w∂ z2 +m ∂2 w

∂t2 =Faxial(3)

In order to couple equations of structural motion to the fluid governing equations the problem

governing equations are nondimensionalised with the similar parameters (equations (6),(7),(8)).

In addition a reduced tension is introduced to non-dimensionalise the axial tension based on beam

flexural rigidity to be able to simplify the transverse equation of motion.

T r ( z ' )=T (z' ) D2

EIA reduced area, Ar and reduced gravity acceleration is defined as bellow.

Ar=A D 2

I;gr=g D

U∞2

Therefore the axial tension in non-dimensional form can be written as bellow:

T r ( z ' )=T r top−(π V r )2

4 ( L/ D )4gr ( H '−z' )(1− 1

m¿ )The non-dimensional equations of structural motion in transverse and axial direction become

equations (9)(10).

The Galerkin weighted residual method is used to drive the numerical approximation of the above

equations of motion in local coordinates for each element in which the axial and bending motions are

orthogonal. Therefore they can be solved independently as uncoupled equations.

The discretised equations of motion for the transverse motion in y direction and for the axial direction

in z direction can be express as a single equation in matrix form.

M ∂2 r∂ t2 +C ∂r

∂ t+K r=F

(15)

In the above equation M is the mass matrix:

Le [ 13 0 0

156420

¿4 Le

2

420

16

0 0

0 54420

−13 Le

420

013 Le

420−3 Le

2

420

Symmetrical

13

0 0

156420

¿4 Le

2

420

]And K is the stiffness matrix (bending stiffness + geometric stiffness):

4 ( L/ D )4

( π V r )2 {[ A r

Le0 0

12Le

3 ¿ 4Le

−A r

Le0 0

0 −12Le

36

Le2

0 −6Le

22Le

Symmetrical

A r

Le0 0

12Le

3 ¿4Le

]−T r

30 [ 0 0 036Le

¿ 4 L

0 0 0

0 −36Le

3

0 −3 −Le

symmetrical0 0 0

36Le

¿ 4 Le]}

And F is the force matrix which is calculated based on non-dimensional parameters.

Le

60m¿ π

2 {20 Caxial1+10Caxial 2

21C L1+9CL2

3Le CL1+2 Le CL2

10 Caxial1+20Caxial 2

9CL1+21 CL2

−2Le C L1−3 Le CL 2

}WhereCaxial1, Caxial2, CL1, CL 2 are force coefficient in the axial and transverse direction at two side of

an element with linear distribution between two side.

Figure 1: Structural element, Left) directions and displacements, Right) forces and moments

Figure 2: Left) schematic of a strip theory model for a Top Tensioned Riser. Right) definition of the fluid domain for

each 2D flow strip

Figure 3: Configuration of the riser supports

(a) (b) (c)

Ttop Ttop

H

Z

yz

x

Δu FyΔz Fz

y

z

Figure 4: Interpolating the force from the flow strips (Shown by S1 to Sp) to the nodes (shown by N1 to

Nn+1).

Figure 5: Development of amplitude of oscillation and lift coefficient in time, Re=105 and Vr=5.86

FRnFRn-1FRn-2FR4FR3FR2FR1 FLn

FLn-1FLn-2FL4FL3FL2FL1

MRnMRn-1MRn-2MR4MR3MR2MR1 MLnMLn-1MLn-2ML4ML3ML2ML1

qRnqRn-1qRn-2

qR4qR3qR2qR1 qLnqLn-1qLn-2

qL4qL3qL2qL1

enen-1en-2e4e3e2e1

Sp-1S3

N+1N1 NnNn-2Nn-3N5N4N3N2S2 S4 Sp-2S1 Sp

Figure 6: Riser modal analysis – validation using commercial software (ANSYS)

Figure 7: Riser modal analysis – effect of riser top tension on natural frequency

Figure 8: Oscillation amplitude of the middle of a riser under tension T=8.285E5 and uniform current

Figure 9: Root Mean Square (RMS) of the riser oscillation between time= 500 and 620