Development of a Calculation Method for Vortex Induced
Vibration of a Long Riser Oscillating at its Upper End
Hidetaka SENGA*1,† Wataru KOTERAYAMA*2
senga@riam.kyushu-u.ac.jp(Received October 31, 2005)
A numerical scheme is developed to simulate three-dimensional dynamics of a flexible riser.
Equations of the riser motion are derived based on ‘Hamilton’s Principle’ and solved using the mode
expansion method. This scheme was verified by comparing with results of forced oscillation experiments. Finally, to validate the accuracy of this numerical scheme, the numerical and experimental
results were compared and showed good agreements.
Key words: Riser, Vortex induced vibration, Irregular motion, Hamilton’s principle, Mode expansion

method1.

Introduction

Risers are used for many purposes such astransporting natural resources from the seabed and lifting cold water for OTEC. They arevery long compared with their cross-area section. The stiffness, therefore relatively decreases and risers show very flexible behavior.
One reason that the dynamics of the riser isso complex is the VIV effect induced by theshedding vortices. Even though the maximum
amplitude of VIV is small compared with its
inline motion, it results in fatigue damage andincreases drag forces for inline motion.
Many researchers have investigated experimentally and numerically the VIV of risers.
For example, Chung1) and Whitney2), and the
textbook of Blevins3) describes a survey of the
VIV.
Experimental studies have provided us
many useful results. Full-scale experiments
were carried out in the Norwegian Deepwater
Program, and Halse4) introduced an overview
of the experimental results.
Some researchers solved the VIV using experimental fluid coefficients (Vandiver5),
Otsuka6), and Etienne7) reported a numerical
scheme combined with the fluid force and finite element method (FEM). However, a numerical scheme, with which we can calculate
the dynamics of such a long flexible riser practicallyandaccurately
enoughin*1 Department of Earth System Science and Technology,
Graduate student*2 Research Institute for Applied Mechanics, Kyushu
University

three-dimension, has not yet been established.
From this view point authors have started to
study on the dynamics of the riser. In this research,wedevelopedaschemeofthree-dimensional dynamics of the riser.
Good agreement between the experimental
and numerical result are seen in comparisons.

2.

Numerical Scheme

The equations of the riser motion are derived
by Hong and Koterayama8) based on ‘Hamilton’s Principle’ as follows:
In X direction{}~ + a ) u&& + zφ&& + X&&(m
X
20+ ( EIu′′)′′ − (Teu′)′ + µgφ2 − QX = 0

In Y direction

{

(1)

}

~ + a ) v&& − zφ&& + Y&&(mY
10+ ( EIv′′)′′ − (Tev′)′ − µgφ1 − QY = FL ( z, t )In Z direction~ {w&& + Z&& }− EA w′′ − µgm0t

− {Q FX − Q DX }φ 2 + {Q FY − Q DY }φ1 − T0δ D ( z ) = 0

(2)

(3)

In φ1 degree of freedom{}~ z 2φ&& − z (v&& +Y&& ) + µg {zφ − v} ⎤
⎡m
1
0
1
⎥
⎢
&
&
&
&
&
&
Iφ1 + ∫ ⎢ + z QFY − aY v&& − zφ1 + Y0 − QDY ⎥ dz = 00 ⎢⎥+ δ ( z )(zR y − vT0 )
⎦
⎣ D
l

{

(

)

}

(4)

In φ2 degree of freedom{}~ z 2φ&& + z (u&& + X&& ) + µg {u + zφ } ⎤
⎡m
2
0
2
⎢
⎥
Iφ&&2 + ∫ ⎢+ z QFX − a X u&& + zφ&&2 + X&&0 − QDX ⎥ dz = 00 ⎢⎥− δ ( z )( zRx − uT0 )
⎣ D
⎦
l

{

(

)

}

(5)

These equations are solved using the mode
expansion method. To express the modal functions with ai(z), satisfying roughly the boundary conditions that top-bottom is simple-free or
fixed-free, and the time component of deflections to X and Y direction with Ui(t), Vi(t), deflection variables are replaced as follows:
For simple support⎛ (2i − 1)π z ⎞ai ( z ) = sin⎜
⎟ , i = 1,..., N
2
L⎠
⎝

(6)

For fixed condition⎛ (2i − 1)π z ⎞ai ( z ) = sin⎜
⎟ , i = 1,..., N
2
L⎠
⎝

(7)

Deflection variablesN⎧⎪u ( z, t ) + zφ2 (t ) = ∑ ai ( z )U i (t )
⎪
i =1
⎨
N
⎪v( z, t ) − zφ (t ) = a ( z )V (t )
∑
1
i
i
⎪⎩
i =1

(8)

As is shown in the textbook written byBlevins, the vortices periodically shed from thecylinder exposed to the free stream flow. The
lift forces caused by such shedding vortices
affect the cylinder. The strength and period of
this lift force are determined by the inline velocity U. Meanwhile, when the top end of a
hanging riser moves, each depth of the riser
moves with its own amplitude and phase according to the top-end motion. Therefore, theinline velocity at each depth U(z) differ from
top to bottom and the lift forces are as follows;
FL ( z , t ) =1
ρU ( z ) 2 DC LY sin (ω s ( z )t )2

(9)Such various lift forces FL(z,t) at each depth
were introduced into the transverse equation.

3.

Model Experiment

A flexible riser model made of polyethylene
and Teflon (PTFE) was used in this experiment.
Its characteristics are shown in Table 1.
Table 1 Characteristics of the riser model

Model length (m)Outer diameter (mm)
Inner diameter (mm)
Mass per length (kg/m)
Young’s Modulus (MPa)
Bottom weight in water (N)Natural frequencies 1stof the model (rad/sec) 2nd
3rd
4th

The natural frequencies of the riser model
shown in Table 1 were calculated analyticallyby solving approximate free vibration equationof the model. The free vibration equation assumes that the bending stiffness effect is smallenough to neglect, and this assumption can be
adopted for rough calculation of natural frequencies for very flexible riser model (Park9)).
The dynamics of the model were measured
by a three-dimensional motion measurement
system using two wing-shaped frames eachholding five built-in CCD cameras (Fig. 1)

6.5
22.5
12.7
0.4
8.847
3.489
0.571
1.308
2.207
2.732

Fig.1

Overview of the measuring system.

The camera C1A ~ C5A and C1B ~ C5B
measured the transverse (Y-Z) motion andinline (X-Z plane) respectively.
A parallel mechanism forced oscillator (Yamaguchi10)) generated irregular motions at the
top end of the model. The gap length between
the water line and the top end of model
was 202 mm, which was 3 percent of the totalmodel length. In the simulation scheme, thisgap was treated as not being affected by fluid
force. In this experiment, other external forces
like wave or current were not generated. Under these conditions, the maximum Reynolds
number was 2.0ｘ103.
We carried out forced oscillation experiments in still water. The top end of the model
was regularly and irregularly oscillated along
the X-axis. The inline irregular motion assumes that the floating structure on which themarine riser is installed is affected by ocean
waves.
Fig. 2 illustrates this measuring flow of theexperiments. For measuring the riser motion
CCD cameras are used, of which arrangements
are shown in Fig.1, forces and moment aredone by using a dynamometer.

Fig.2

4.

Flowchart for the measurement
easurement of the riser motion
and force/moment.

Experimental and Numerical Results

4.1.1

The excitation period is 2.0s, which is theshortest period in experiments, and excitation
amplitude is 100mm under Fixed and Simple
support boundary condition. This condition of
the amplitude and period is very severe for ariser and non-linear effects are expected. The
solid lines in the figures are experimental results and dash-dot lines are simulation results.
The suffix A~E of X are showing the vertical
position of measuring points.
Fig. 4 shows the Configuration profiles of theriser model at AX0 =100mm for fixed and simple
support-1.0

1.0 -1.0

X/AX0

X/AX0

1.0 -1.0

X/AX0

1.0

0

regular oscillation

At first, results of the numerical simulationand experiments regarding to in-line motion
are compared in order to verify the accuracy ofthe developed simulation scheme.
Figs. 3 shows the inline riser motion response of experimental and simulation results.
100

-1

-2

-3
Z (m)

AX0=100mm
Fixed top end-4

Simulation
Experiment

0

-5

-100
100

-6

0
-7

-100
100

T=2.0sFig.4aT=7.0s

T=11.0s

Configuration profiles of the riser (Fixed S.)
Configuration

0
-100
100

-1.0

Simulation
Experiment0

1.0 -1.0

X/AX0

X/AX0

1.0 -1.0

X/AX0

1.0

0

-1

-100
Fig.3atime (s) 20
010Time series of inline motion (Fixed support).

-2

-3

100

Z (m)

AX0=100mm
Simple Support

0-4

Simulation
Experiment

-100
100

-5

0
-6

-100
100

-7

0
-100
100

Simulation

T=2.0s

T=7.0s

T=11.0s

Configuration profiles of the riser (Simple S.)
Configuration

Experiment

0-100
Fig.3b

Fig.4btime (s) 20
010Time series of inline motion (Simple support).

In Figs. 4, we can see the good agreement
between the experimental and numerical results.
Then, to compare the numerical estimationsof the share
share force and moment at the top end of

the model with those of the experimental ones,
time histories of top end forces at excitation
period of 2.0s and AX0=100mm are shown in
Figs. 5.
100.0

Fixed top end
Exp: AX0=100mm

1.0

Exp: AX0=200mm

Sim: AX0=100mm

X0 (mm)Sim: AX0=200mm

0.0-100.012.0

Fz (N)

11.0
10.0

2.0

Simulation

0.0
-2.0

4.0

0.0

Experiment

Fx (N)

0.0
Fig.6b

4.0

8.0

12.0

T (s)

Amplitude of top end moment MY (Fixed support).

My (Nm)0.0

Simple supported-4.0

Exp: AX0=25mm

0Fig.5a2

4

Exp: AX0=50mm

time (s) 86Time histories of top end forces (Fixed support).

Exp: AX0=100mm

2.0

Sim: AX0=25mm
Sim: AX0=50mm

100.0

X0 (mm)Sim: AX0=100mm

1.0

0.0
-100.012.0

Fz (N)

0.0

11.0
10.0

2.0

0.0
Experiment

Fx (N)

Figs.3, 4, 5 and 6 verified the accuracy of thenumerical simulation scheme for inline motion
developed in this study.

-2.0

0

2

4

6

time (s) 8Time histories of top end forces (Fixed support).

In Figs. 5, we can confirm the accuracy of theestimation of share forces and moment.
To show the comparisons of the other experimental conditions, none-dimensional values of the share forces and moment are shownin Figs. 6

Fixed top end
Exp: AX0=100mm
Exp: AX0=200mm

2.0

Sim: AX0=100mm
Sim: AX0=200mm

1.0

0.0

0.0
Fig.6a

4.0
8.0
T (s)
12.0
Amplitude of top end force FX (Simple support).

Simulation

0.0

Fig.5aFig.6c

4.0

8.0

12.0

Amplitude of top end force FX (Fixed support).

4.1.2

Transverse motion

Next, the vortex-induced vibration (transverse motion) is discussed in the case of regular forced oscillation experiments. In theseexperiment, the boundary condition of the
top-bottom of the riser was simple-free.
In order to understand the phenomenon of
VIV simply, we classified all experiments into
two patterns according to the analysis of themeasured transverse motion. One is the casein which a few dominant circular frequencies( ω s) for the transverse motion exist on one
point of the model, and the other is where asingle circular frequency dominates thetransverse motion from top to bottom. For
these two patterns examples, two experiments(T0= 4sec X0=100mm, T0=4sec X0=50mm) wereselected and considered in detail. Figs. 7 show
the experimental and simulation results of thetime histories of X0 and the transverse motion
at the target points A to E (left side) and their
FFT analysis with circular frequency domain(right side) are shown.

-100
20
0
-20

EXP.
SIM.

20
0
-20
20
0
-20
20
0
-20
20
0
-20

20

EXP.
SIM.

20
0
-20
20
0
-20
20
0
-20
20
0
-20
20
0
-20

Fig.7a10 time (sec)

20

y(mm)
100.0 -25.0

0.0

25.0

-1.0

0
10

-2.0

0
10
0
10

-3.0
-4.0
-5.0
-6.0

0
10

-7.0
0

0

2

4
6
(rad/sec)

8

Fig.8aConfiguration profiles of the riser (X0=100mm,
T0= 4sec).

x(mm)-100.0
0.0

50

0.0

y(mm)
100.0 -25.0

0.0

25.0

EXP.
SIM.

-1.0

0
20
10

-2.0

0
20

z(m)

0
-100

0.0

10
0
20
10

-4.0
-5.0

0
20

-6.0

10
0
20

-7.0

10
0

-3.0

0

2

4
6
(rad/sec)

Fig.8b

Configuration profiles of the riser (X0 = 50mm,
T0=4sec).

4.2.1

Irregular oscillation

8

Time histories of transverse motion and its FFT
response (X0=50mm, T0= 4sec).

From the FFT analysis of Fig. 7a and Fig. 7b,
we can confirm that the dominant ωs is theintegral multiples of ω0; this means that the
number of shed vortices during one period offorced oscillation is an even number. For example, ωs /ω0 =1 is the case where a pair ofvortices is shed, and in ωs /ω0 =2, two pair are
shed during one period. Because a pair of vortices is likely stable, ωs is nω0 (n=1,2,3,…) sothat some even–number vortices are shed
during each period of forced oscillation.
Fig. 8 shows configuration profiles of X-Z(inline) and Y-Z (transverse) plane. At the figure X-Z plane, circles show the experimental
results at every T / 8 [sec] and solid lines show
the simulation results. At the figure Y-Z plane,
circles show the measured maximum amplitude at all marked points and solid lines show
the simulation results when the amplitude ismaximum at points A to E

At first, the time history of inline motion
caused by the top-end irregular motion X0 isshown in Figs. 9.
X0 (mm)

100

0

0
10

Time histories of transverse motion and its FFT
response (X0=100mm, T0= 4sec).
YE (mm) YD (mm) YC (mm) YB (mm) YA (mm) X0 (mm)

YE (mm) YD (mm) YC (mm) YB (mm) YA (mm) X0 (mm)10 time (sec)-100.0
0.0

EXP.
SIM.

50

100
0
-100

XE (mm) XD (mm) XC (mm) XB (mm) XA (mm)0

Fig.7a

x(mm)

100

z(m)

0

YE (mm) YD (mm) YC (mm) YB (mm) YA (mm) X0 (mm)

YE (mm) YD (mm) YC (mm) YB (mm) YA (mm) X0 (mm)

100

100
0
-100
100
0
-100

EXP.
SIM.

100
0
-100
100
0
-100
100
0
-100
80

Fig.9atime (sec)120Time histories of inline motion (Mean Period
Tx=4.0sec, Significant Amplitude Xw=200mm).

X0 (mm)

X0 (mm)

100
0
-100

EXP.
SIM.

100
0
-100
100
0
-100

X0 (mm)In Figs. 9, the xA to xE show the deflection at
the selected points A to E, respectively.
The accuracy of this numerical scheme forinline motion of a free hanging riser moving
regularly is already mentioned above andSenga and Koterayama11). As shown in Figs.9,
this numerical scheme also can simulate accurately even if the inline dynamics from top tobottom end of a riser moves irregularly.
Next, the vortex-induced vibration (transverse motion) is discussed in the case of irregular forced oscillation experiments.
The time histories of inline motion X0 andtransverse motion at selected points A to E are
shown in Figs.10 and the FFT responses ofboth results are shown in Figs. 11

20

0
-20
80

-20
0
-20
0
-20
20
0
-20

Exp.

0

2

2

0

0

2

2

0

0

2

2

0

0

2

2

0

0

2

2

0.0

Fig.11a

2.0

4.0

6.0

ω(rad/sec)

8.0

0.0

2.0

4.0

6.0

ω(rad/sec)

8.0

FFT response of transverse motion (Mean Period
Tx=4.0sec, Significant Amplitude Xw=200mm).

20

Exp.

20

0

0

2

2

0

0

2

2

0

0

2

2

0

0

2

2

0

0

2

2

0

Sim.

0

2.0

4.0

ω(rad/sec)

160

Time histories of transverse motion (Mean Period
Tx=4.0sec,
Significant
Amplitude
Xw=200mm).

Sim.

0

0

0.0

time (sec)

140

20

0

YD (mm)20

120

Time histories of transverse motion (Mean Period
Tx=4.0sec,
Significant
Amplitude
Xw=100mm).

20

YC (mm)20

time (sec)100

Fig.10b

YB (mm)0

Fig.10a20

YA (mm)

0
-20

120

0
-20

YE (mm)

YE (mm) YD (mm) YC (mm) YB (mm) YA (mm)

EXP.
SIM.

20

20

140

Time histories of inline motion (Mean Period
Tx=4.0sec, Significant Amplitude Xw=100mm).

100
0
-100

0
-20

X0 (mm)120

20

X0 (mm)time (sec)0-20

YA (mm)

100

20

YB (mm)80

Fig.9b0-20

YC (mm)

100
0
-100

EXP.
SIM.

20

YD (mm)

100
0
-100

YE (mm) YD (mm) YC (mm) YB (mm) YA (mm)

100
0
-100

YE (mm)

XE (mm) XD (mm) XC (mm) XB (mm) XA (mm)

100
0
-100

Fig.11b

6.0

0.0

2.0

4.0

6.0

ω(rad/sec)

FFT response of transverse motion (Mean Period
Tx=4.0sec, Significant Amplitude Xw=100mm).

In Figs. 10, we can see the very similartransverse vibration between the experimental
and numerical results. Even though the vibration amplitude is the almost same, there exist
parts in which both the results are the opposite
motion to the X-axis like 140~160 [sec] of rightfigure. Because the vortex would be sheddingin the opposite direction at such a time in theexperiment and simulation, the transverse
motion became reverse. The direction oftransverse motion is accidental and it is notnatural to consider as deterministic.
Then, in Fig. 11, we can see some peak frequencies bands exist and they are different at
each depth of the model. These regions of peakband are in good agreement with the experimental and numerical results.
To verify the accuracy of the numerical
scheme, three examples of significant amplitudes (Yw) and mean periods (Ty) of each point,
which calculated from the power spectrum oftransverse motions, are compared withexperimental results in Table 2.
Table 2 Examples of Significant Amplitude (Yw) and Mean
Period (Ty) of transverse motion

Case of large amplitude
Top end inline Significant Amp. Xw=150mm
Mean period Tx=4.0 sec
motionTransverseYw [mm]Ty [sec]
Depth
Exp.
Sim.
Exp.
Sim.
A
11.5
8.4
1.93
1.96
B
19.9
14.6
3.53
2.92
C
23.8
19.3
4.11
3.34
D
24.9
17.6
4.01
4.04
E
29.4
25.7
4.44
3.33
Case of small amplitude
Top end inline Significant Amp. Xw=100mm
motionMean period Tx=6.0 sec
TransverseYw [mm]Ty [sec]
Depth
Exp.
Sim.
Exp.
Sim.
A
5.00
4.8
3.07
3.11
B
14.5
14.0
4.39
4.46
C
17.0
14.4
5.27
5.22
D
17.1
15.6
8.45
7.65
E
22.6
23.0
5.57
4.91
Calculation results on the case of small
forced oscillation amplitude showed better
agreements than large forced oscillation case,and this is same tendency as regular forced
oscillation experiments as shown in Figs. 8.

4.

Summary and Conclusions

To analyze the three-dimensional dynamics

of a hanging riser, experiments were carriedout using a flexible riser model. The freehanging riser model was forced to oscillate
regularly and irregularly at its top end in stillwater, and its three dimensional motion wasmeasured by CCD cameras. Through theanalysis of these experimental results and bycomparing the experimental and numericalresults, the following conclusions were obtained.
1) The numerical scheme developed in thisstudy brings us very accurate results ofsurging motion, restoring force and moment regarding to the in-line motion.
2) Regarding to the transverse motion (VIV)
the numerical results agree well with experiments except the case of very large
forced oscillation amplitude and short period.
3) The peak regions of vibration frequenciesof the transverse motion induced by theshedding vortices were in good agreementwith experimental results.
4) In the case of very large amplitude andshort period forced oscillation experiments,
the amplitude of in-line motion decay
quickly toward the bottom, and therefore
the period of the transverse motion arevarious along the riser. This makes transverse motion complex and results in inaccuracy of numerical simulation.
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