Vortex-induced vibrations on flexible cylindrical structures coupled with non-linear oscillators

(1)

Proceedings of the ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering OMAE2009 May 31 - June 5, 2009, Honolulu, Hawaii, USA

DeIft University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg 2

2628 CD Deift

Phone: +31 (0)15 2786873 E-mail: p.w.deheer@tudelft.nI

VORTEX-INDUCED VIBRATIONS ON FLEXIBLE CYLINDRICAL STRUCTURES

COUPLED WITH NON-LINEAR OSCILLATORS

Guiiherme F. Rosetti1'3 (cuilheriiie.feitosa@tpn.usp.br)

Kazuo Nishimoto1 (knishimo @?usp.br)

ABSTRACT

The recent escalade of the oil prices encourages the search and exploration of new oil fields. This represents a challenge to engineers, due to more difficult conditions of operation in harsh

environments and deeper reservoirs. The offshore industry

faces, in the edge of technology with new necessities and

limiting conditions imposed by the environment, an increase in

the cost of production. It is, therefore, of vital importance to

have the equipments operating at the most optimized conditions in order to reduce these costs.

VIV software developed in the frequency domain was successful in designing risers and pipelines using large safety

factors and maldng conservative assumptions. These tools only predict single-mode vibrations. In this perspective, the present

paper describes the results obtained from a new time-domain code developed to assess the vortex-induced vibrations of a long flexible cylinder. A time-domain analysis was chosen because this suits the problem well, since it is able to predict and calculate different modes of vibrations. In the model, a cylinder is divided into elements that can be exposed to an

arbitrary current profile. Each of these elements is free to

oscillate parallel and transversely to the flow, and is coupled to

a pair of van der Pol's wake oscillators. This simulates the

Jaap de Wilde2

_I

(j dew i ide @ marin. nl)

Department of Naval Architecture and Ocean Engineering Escala Politécnica - University of São Paulo

São Paulo, SP, Brazil 2

MARIN

Wageningen, The Netherlands 3ANP

(Brazilian Petroleum Agency)

OMAE2009-79022

vortex shedding and, therefore, the fluctuating nature of drag

and lift coefficient during the occurrence of VIV. The governing

equations are solved by 4th-Order Runge-Kutta schemes in time domain. The new time-domain model is compared with

small scale model test data from benchmarking. 1. INTRODUCTION

There has been a considerable effort in the field of VIV to develop codes capable of predicting the main features of this phenomenon. Recent advances in computer technology have

made it possible to solve the Navier-Stokes equations

associated with structural models in some applicable situations,

such as some VIV cases. This approach is, however, one that

consumes a lot of time and computer resources; therefore, it is

interesting to search for a more efficient solution in terms of

computer usage.

One such solution is presented in this paper, in the form of a time-domain code in which the structural model is coupled with a wake oscillator of the van der Pol type. The beam equation is solved by spectral analysis in which the response is regarded as

a series expansion of eigenfunctions multiplied by

(2)

Time-domain analysis stands as an interesting approach to resolve an important aspect of the present application, namely,

multi-mode response, which might be important in situations

such a.s long risers subjected to ocean currents.

In

this model, the body is divided into a number of

segments and each of them has its own pair of wake oscillators. This enables the program to account for space varying current

velocity

and body diameter,

increasing its capability of simulating realistic situations. The tension is also calculated for

any particular instant through axial elongation. The coupling between transverse and parallel vibrations (which will be

referred to as cross-flow and inline, respectively) is performed through the tension.

Validation is performed with the

data obtained from

Chaplin, J.R. et al. (2005a) and from Chaplin, J.R. et al.

(2005b) in experiments with risers under sheared current. 2. FORMULATION OF THE PROBLEM

The physical system considered in the present work is a

flexible cylindrical structure modeling a riser. This system will be subjected to an arbitrary current profile, which is variable in time, modeling the ocean current. The body is free to oscillate

parallel and transversely to the current flow direction (inline

and cross-flow).

The equation describing the motion of the body is

at at ax4 ax ax at

--(M?±)+

EI---(T.-.)= FRL

(1)

The x-axis is coincident with the cylinder axis, the y-axis is parallel to the flow and z-axis is perpendicular to the flow. The deflection is expressed by the complex number

r=r,. + ir. (2)

The equation is valid for O <x < L. In x= 0 and in x= L, the body is pinned.

The mass is composed by structural and added mass, with

the potential added mass coefficient considered equal to unity.

M=m,+fC,pD2

(3)

Following Blevins, R. (1990) and Facchinetti, M.L. et al. (2004), the damping, which is a part of the forcing term, is

composed by structural and hydrodynamic.

R=pD22+2M,

(4)

The structural damping includes the term _{d' which is given} beforehand and depends on the structure itself. The

hydrodynamic damping includes y, which is a stall parameter

associated with the drag component in the direction of the cross

motion. It is responsible for withdrawing energy from the

system. For simplicity, it will be taken as constant during the

simulations. Following Fumes, G. et al. (2007), the frequency

2

will be taken as

the natural frequency in still water,

calculated as

N2=\J+ C,

m pD2

The right-hand side of equation (I) is the hydrodynamic

force including fluid damping. The first is modeled as proportional to the square of current velocity. This force term includes viscous and pressure fractions and will be divided in

lift and drag portions. where drag force acts in the same

direction as y-axis and lift force acts in the same direction as z-axis. Therefore, the equation for the hydrodynamic force on the right-hand side becomes

F

=!

PV2D(iCL +CD) (6)

As outlined before, the structural model is represented by equation (I). The variable r represents the deflection of the riser in a particular moment and at a particular position in length. This deflection is represented by a complex vector in

which the real and imaginary axes represent, respectively, inline and cross-flow motions.

From equation (6),

it can be noticed that the force is

calculated using the drag and lift coefficients. In the structural

model, the

different directions are accounted for through

complex representation. In the wake oscillator, however, such representation is not used. Therefore, to represent the different

directions of motion a pair of oscillators is used for each

segment of the riser. By making use of Van der Pol's equations

and introducing a forcing term proportional to body acceleration, it arrives to the set of equations

.+ej(q2_l)+q

=A-a2q.

aq.

a2r

=

A---(7)

The variables q

and q are the reduced lift and drag

coefficients, related to the "real" coefficients through

C1=GLI

j

(8)

C0=C0(I+K1q2)+Ç).2.. (9)

The lift oscillator is widely used in this form since Hartlen

et al. (1970). However, the drag oscillator is an improvement

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proposed b) Fumes, G. et al. (2007). It differs from the model

proposed by Iwan, W. (1970) and by Blevins, R. (1990)

because the former has a very clear physical significance

whereas the last makes use of variables which are not so clear. The Strouhal frequency is given by

=2.irS-D

The choice of using Strouhal frequency in

the wake

oscillator is made to reproduce the behavior of the structure subjected to increasing current velocity. In experiments, it is observed that, up to a velocity value, the shedding frequency

follows the Strouhal relation. At this point, when the shedding

frequency gets close enough to some natural frequency of the

body, it seems that the shedding frequency remains the same as

this natural frequency for a range of velocity. Further, after

some other value of velocity, the shedding frequency returns to the Strouhal relation or shifts to the next natural frequency. This is known as lock-in.

It can also be noted that, in equations (7), the frequency for cross-flow oscillator is half the frequency for inline oscillator.

This adaptation

was made because

it was observed in experiments that the frequency of inline vibrations is the double of the cross-flow vibrations.

3. FOURIER SPECTRAL METHOD

In this section, a few comments will be made on how to solve partial differential equations. such as equation (1). The basic idea is to assume that the unknown function u(x) can be

approximated by a linear combination of N+1 "basis functions"

(x):

u(x)=uN(x)=aI,Lç(x)

(11)

When this series expansion is substituted in the equation

Lu=f(x)

(12)

where L is the differential operator, the result is the "residual

function" defined by

R=LUNf

(13)

The goal is to choose appropriate basis functions and the

coefficients [a} so as to minimize this residual function. Fourier series

In many problems in engineering analysis, it is necessary to

work with periodic functions. Some of the simplest periodic

functions are trigonometric. Since those functions are very easy

to manipulate, it is worth exploring the possibility of working

with them to try to represent an arbitrary periodic function in a

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series of trigonometric functions. One such expansion is known as Fourier series.

Orthogonal sets of functions

Consider a set of functions

_r(x)(r

= 1,2, ...) defined over the interval O x 2rr. If the functions are such that for any two distinct functions

then the set I)r(X) is said to be orthogonal in the interval of

length 2ir. In addition, if the functions r(X)also satisfy

f1Ç(x)2dX=l,r=l,2,... (15)

then, the set is orthonormal. Hence, for an orthonormal set of

functions

J(x)(xx =

r, s =1, 2,... (16)

where 6rS is the Kronecker delta. It is easy to verify that the set

I

sinx cosi sin2x cos2x

(14)

(17)

is orthogonal.

If there is a set of coefficients c,r = 1,2...not all equal to

zero, for which

(18)

then, the set

ir(x)(r

= 1,2,...)is linearly dependent and does not constitute a set of basis functions. If otherwise, then the set is linearly independent and the set may constitute a set of basis

functions.

Trigonometric series

An orthonormal set of functions i(x)(r

= 1,2,...) is

complete if the series composed by the basis functions can approximate a piecewise continuous function f(x) with any degree of precision. The set (17) is complete in the interval

O x 2m. Therefore, any such

function 1(x)

can be

approximated in the interval by the Fourier series:

(4)

ao

=i$f(x)dr

=

'2 f(x) cos(rx)dr, r

= 1,2,... (20)

2f(x)

sin(rx)dr, r = 1,2,... 2r0

Equation (19) with coefficients in (20) is known as Fourier

series.

Choice of basis functions

A fundamental question in the present method is what set of

basis functions to choose. Some features are desirable for this set of functions: (i) easy computation (ii) rapid convergence and (iii) that any solution can be represented by choosing

enough basis functions.

In the present problem, the choice of basis does not show big complication. By looking at the boundary conditions, one

can notice that the solution should be spatially periodic.

Therefore, as each of the basis functions should automatically

satisfy the boundary conditions, the sines and cosines of a

Fourier series seem suitable as basis. Further in the text,

through the eigenvalue problem, it can be seen that Fourier sines expansion arises naturally as a set of basis functions for

such problem.

Although it is possible to treat the time coordinate spectrally, the method will be applied only in the spatial dependence.

Then, the time dependence can be marched forward, from one time level to another. This is much cheaper in computer usage than solving the equations simultaneously in space-time.

The space

spectral discretization reduces the partial

differential equation to a set of ordinary differential equations

in time, which can be integrated by some numerical scheme. In the present case, this is done by a 4th-order Runge-Kutta solver implemented in the code.

4. SOLUTION TO THE EQUATIONS

The previous sections presented a physical description of

the problem. In a general way, there is a dynamic equation that

models the mechanical response of the structure and this is

coupled

to a wake oscillator that models the near wake

dynamics.

Now, it is necessary to find out how to solve the equations

and how the integration between the two "cores" is done.

Firstly, the procedure on how to solve the equations is shown.

Equation (1) is a partial differential equation that presents

derivatives with respect to time and space. Instead of choosing

a finite difference method, a spectral method is preferred, as

outlined before. This spectral method is based on regarding the

where the sets of coefficients a and br

are the Fourier

coefficients:

response as a composition of the system basis functions

multiplied by time-dependent coefficients. The representation to these words is

77(t) dt2 - ML() d2 ML4() d4

Then, introducing A, a real constant number (assuming that i(t) is real), equation (26) leads to:

(26)

r(x, t) = Ø, (x)tí,, (t) (21)

For the sake of simplicity, some manipulation is done in

equation (1). The axial coordinate was normalized by .Ç = x/L, the mass was considered constant over time and the tension was

considered constant over the length. The last assumption is acceptable as a simplified representation of reality, but some

error is incorporated in the results because in real situations, the tension is not constant over the span.

Therefore, it arrives

a2r

EI a4r

T 2r F

R Jr

+ML - MI]

M M

This equation must satisfy the following initial and

boundary conditions

r(0)=0

r(l)=0

dr(e)

₌₀

d2r()

=0

d2

It is worth looking at the eigenvalue problem associated

with this equation

Elgerivalue problem

In order to solve the eigenvalue problem associated, the

following equation is written

or

EI or

T or

(24)

In this case, the solution becomes separable,

r(, r) = t()i7(r)

(25)

where c1(e) represents the general body configuration and ì7(t)

indicates the kind of motion executed with time. Introducing

equation (25) into equation (24)

(5)

Considering a solution of equation (27) in the exponential

form:

and the solutions are:

(31) If A is a negative number, then the two solutions would be real, equal in magnitude, one positive and one negative, which means that one increases and the other decreases exponentially

with time. However, this would is inconsistent with a conservative system, which is the case. Hence, the parameter A must be a positive constant. Letting A = cuz, where co is a real number, equation (31) yields:

S1 = ¡û)

(32) s2 =

From equation (28) and equation (32):

T d24()

El d4P()

+(t)w2 =0

ML2 d2

ML4 d4

The general solution is:

= C1 sin(ß) + C2 cos(ß) + C3 sinh(ß) + C4 cosh(ß)

Making use of the following boundary conditions:

d2cI)

_-0

d2()

=

d2

_{_,}

It can be concluded that C2 = C3 = C4 = 0, and the solution reduces to sin(ß) = 0 (36) Then

fi

=n,r,n=l,2,3...

(37) d2ï7(t) + 2,7(t) = O dr2 T d24 EI d4

-2

ML() d2 ML4() d4

From equation (36) and equation (37), the normal modes

are:

() = Ç sin(n2r)

(38)

and the natural frequencies are:

T , , El

w=--n,r+

_ML

nr

ML4

Turning the attention to equation (22), substituting the

expansion presented in equation (21), multiplying through by

then integrating over the domain, yields d2 r1 ₂

Rd,bf12

dt2 J0

EJ

+-»f --d

T d

L

q5fld

The choice of the basis functions is arbitrary. Since the boundary conditions demand the solution

to be spatially

periodic, Fourier series is a good choice. In fact, the normal

modes are sine functions, so a basis composed by sine functions

(33) as in a Fourier sine expansion will be used. In spite that the

structure may not, in fact, present orthogonal modes, employing sine functions is an acceptable approximation.

Using sine functions as basis

= sin(n,r)

equation (21) becomes

(39)

(40)

(41)

It is possible to evaluate the term

(42)

It is straightforward to obtain from equation (21)

y,

=2frsin(n7r)d

(43)

Naming

=

Çrsinnirct

(44)

(29)

Therefore, the normal modes for this problem are sine

functions. It yields:

s2 +2 = O (30)

(6)

Therefore, the procedure is to solve equation (50)

for a

certain number of modes and reconstruct the response through equation (45). This is represented in a matrix equation as

2a---+ rçt= H(t)

dr2 dt in which O ;-

¡F sin(ir)d

_!_

_{¡F sin(2,r)d}

-- fFsin(ivr)d

M0 R

2a

, El T ₂ ₂ (O n2r+ fl7t '"' ML4 ML

How to calculate the force

The term on the right-hand side of equation (51) is the

hydrodynamic force due to viscous and pressure effects on the structure. This is modeled by

F=1_{PV2D(iCL +CD)} ₍₅₅₎

The lift and drag coefficients come from the wake oscillator as described before.

The model is able to account for non-uniform current. This is made by dividing the length into N segments bounded by the coordinates

1'2...N The velocity is approximated as a

piecewise linear function for each interval and the Strouhal

frequency is also calculated for each interval using an average of the velocity and a prescribed profile of the Strouhal number.

The velocity for the interval {k k+1] is calculated as

V(,t)

= a(t)+b(r)

(56)

r= 2

q, sin(n,r)

(45)

The tension is calculated at an arbitrary instant by making

use of

T=1)+

S L

L

L (46)

Where the span "5L" can he calculated by the arc length

expression

I I

dr,

SL =

LJll +--(--

]d (47)

Making use of equations

(46) and (47), the natural

frequencies of the structure can be updated at an arbitrary

moment.

The bending moment can also be evaluated at an arbitrary

instant by making use of

Moment=EI

d2

d2r

(48)

Going back to equation (40) and using the fact that the basis functions are now determined, the third and fourth terms can be

evaluated:

'fø.

fø

d=_.n2z2

where the other terms arising from the integration yield zero. Finally, equation (40) becomes:

+n

R +fl4Tr4çoEl

+Tn27r2ç0

1 r

= Fsin(nrr)d

This equation is valid for each of the modes and, in this

code, is solved by Runge-Kutta 4th order scheme.

It is important to notice that the modes are uncoupled and

each of the equations can be solved independently but, at each arbitrary instant, the response is given by equation (45).

At this point,

it

is also important to notice that

it is

necessary to truncate the expansion of terms to a certain

number of modes.

(I

q,2 and H(i)= with a1 O O O

(7)

It is important to notice that the expression will be used as it

is in equation (58) into the right-hand side of equation (51)

simultaneously for each and every mode number. 5. RESULTS

In this section, the program is run with sets of data to check if it delivers realistic results.

The data run is from the experiments performed by Chaplin,

et al. (2005a) and Chaplin, et al. (2005b). In the experiments reported in that work, measurements were performed of the

inline and cross-flow motions of a vertical model riser in

stepped current. The riser was 1312m long and 28mm in

diameter. The testing conditions were such that the lower 45%

of its length was subjected to current speeds of up to lmIs,

while the upper part was in still water (Figure 1).

The test matrix was designed such that the response could

be analyzed under different physical characteristics and current

velocities.

Firstly, the standard set of parameters will be used as

presented in Table 1, Case 1. This parameter set is called

standard because with this set, good agreement with the results was achieved.

The model was also run with different velocities and

pre-tension and the results were compared with the benchmark. In this paper, two other conditions are presented, Case 2 and Case 3.

45e, of

th-length

Case

i

The first twenty natural frequencies were calculated for the

riser with such characteristics. The first twenty modes were included in the calculations and this is enough since, for this

case, the importance of the modes decrease both for inline and cross-flow very rapidly after the 10th mode. Those frequencies are presented in Table 2.

'i

Figure 1: Current profile

Table 1:Testmatrix for calibration

Case i Case 2 Case 3

Properties

External diameter(m) 0.028 0.028 0.028

Internal diameter(m) 0.023 0.023 0.023

Lenqth(m) 13.12 13.12 13.12

E(N/m2) i .82E+09 1 .82E+09 i .82E+09

l(m4) 1.64E-08 1.64E-08 1.64E-08

A(m2) 2.00E-04 2.00E-04 2.00E-04

El 2.99E+01 2.99E01 2.99E01

EA 3.65E+05 3.65E-i-05 3.65E-i-05

Mass(kq/m) 1.85 1.85 1.85 Structural damping 0.33% 0.33% 0.33% Hydrodynamic damping 15% 15% 15% Tension(N) 984 457 1002 Volocity)m/s) 0,65(45%) 0.31(45%) 0.95(45%) Modes 20 20 20 Segments 100 100 100 Time step 0.02 0.02 0.02 Az 96 96 96 Ay 12 12 12 0.15 0.15 0.15 0.30 0.30 0.30 y 0.15 0.15 0.15 Strouhal frequency 0.17 0.17 0.17

and its average in the interval is

k

=!_'(a+b)d°

(57)

Furthermore, each of the N segments has its own pair of

oscillators and, therefore, its own lift and drag coefficients for each computation step. Both are approximated in the interval as piecewise constant functions. This solution makes it possible to

solve the oscillator equations as total differential equations. Such approach shows to be acceptable since experimental studies indicate that shedding process is dependent on local vibration amplitude and nearby elements do not seem to have

appreciable importance.

Therefore, the expression for the forcing is

(8)

Time series

The next plots present the time series of displacement in different points of the riser. The positions shown are 656m, 328m, li 48m and 082m.

Riser elevation - 6.56to (50% of the length)

[

A /

/\ f\A /'\Mb

Riser elevation - 328m (25% of the length)

Table 2: Natural freouencies of the rise

Riser elevation - 11 48m (87.5% of the length)

V\J\/\/VWNW

Riser elevation - O.82m(6.25% of the length)

AMA/í\MJVVV\\f\/\/ \MA/\7

O0 201 202 203 204 205

Figure 2: Cross-flow displacements in different positions of the riser

Rioer elevation . 656m (50% of the hser length(

Riser elevation . 328m (25%of the riser length)

W\N\NVVVVVVW\RAA1vWI\AMMJ\

Riser elevation- f1.480, (87.5% of the riser length)

AfVfv'VJVVV'fI\J'A/V'JVV\AAtí'/\J'\A,VV\A

Riser elevation- 082m (6.25% of the riser length(

NN\IW NWWI\J

201 202 203 204 205

Figure 3 Inline displacements in different positions of the riser

It can he noticed that the frequency of oscillation is the

same for cross-flow in all the points and also for inline, for all

the points. Riser shapes

It is clear that, for cross-flow vibration, the 5th mode is

predominant and, for inline, the 9th mode is the most important.

It is, however, evident that there is an important contribution from other modes. With respect to amplitudes, the cross-flow

and inline vibrations show to be close to the values presented in the experiments, although, the mean inline deflection is slightly bigger in the present model.

Cross-f ow shapev e-0.1 0.2 0.3 0.4 5.5 s-7 s-e 2.e -08 ra/D

Figure 4: Riser shape for cross-flow vibrations

Mode number Air Still water

1 0.76 066 H z 2 1.52 1.32 3 2.29 1.98 4 3.08 2.66 5 3.88 3.35 6 4.69 4.05 7 5.53 4.78 8 6.40 5.53 9 7.30 6.30 10 8.22 7.10 11 9.19 12 10.19 8.80 13 11.23 9.69 14 12.31 10.63 15 13.43 11.60 16 14.60 12.61 17 15.82 13.66 18 17.09 14.75 19 18.40 15.89 20 19.77 17.07 o -i o 8.1 7.5 4.1 3.5 6.75 6.15

(9)

0.1-0.2 0.3-0.4 0.5 0.6-0.7 0.6 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Mean inline deflection

3 4 5 6 7

ry/D

Figure 5: Mean inline deflection

Inline deflection measured from the mean

Figure 6: Riser shape for inline vibrations

It is interesting to see the contribution of each mode to the

results. This is presented in Figure 7 and Figure 8. These plots show the contribution of the modes for one particular time step.

Contribution of each mode to cross.fiow response

Mode number

Figure 7: Contribution of each mode to cross-flow response

-0.02 -0,04 -0.06

2 4 6 8 10 12 14 16 18 20

Mode number

Figure 8: Contribution of each mode to inline response Trajectories

The next plots present the riser trajectory at the midpoint, at 853m and at 105m during 2s as of 200s. The so-called "eight-figure" is easily recognizable in such plots.

T1n$olmy nl ll1 r6m rnldnI

Contribution of the each mode to inline response

0.ti 0.0e 0 06 0) 0.04 E 0, 5 0.02 0, 0 06 -o. ry/7

Figure 9: Eight-figure at riser midpoint

Treocloy ol the roer OIS 53m

9.5 916 9.65 9.7 9.75 9.6 9.90 n'O Figure 10: Eight-figure at 853m 0.25- 0.2-E 0.15 a E 0.1-E 0.05 -0.05 0.3 8 10 12 14 16 18 20 .0.3 .02 -0.1 0 O 0.2 0.3

(10)

Case 2

0.3- IroIdy 01 0,0 rr 0110.5,0 0.2 0.1 .0.0 .0.3 'otOO 8.24 6.26 0.26 0.3 8.32 8.34 8.36 8.36 84 'Y/o

Figure 11: Eight-figure at 1O.5m

riser with such characteristics. The first twenty modes were included in the calculations and this is enough since, for this

Table 3: Natural freauencies of the riser

Riser shapes 0.1 0.2 0,3 04 0.5 0.6 0.7 0.0 0.9 0. 0. o. 0. 0. o. 0. 0. 0. Cross-flow shapes

.04 03 .02 .0,1 0 0.1 0,2 0,3 0.4

rylD

1 0.52 0.45 H z 2 1.04 0.90 3 1.58 1.36 4 2.13 1.84 5 2.70 2.34 6 3.31 2.85 7 3.94 3.40 8 4.61 3.98 9 5.32 4.59 10 6.07 5.24 11 6.86 5.92 12 7.70 6.65 13 8.59 7.42 14 9.54 8.24 15 10.54 9.10 16 11.59 10.00 17 12.69 10.96 18 13.86 11.97 19 15.08 13.02 20 16.36 14.12 0.5 15 ro/O

Mean inline deflecOon

0.1 0.2 2.3 04 03 0.6 0.7 0.8 0.9 3 4 5 ry/D 9 lo

(11)

In this case, it is clear that, for cross-flow vibration, the 4th mode is predominant and, for inline, the 7th mode is the most

important. It is, however, evident that there is an important

contribution from other modes.

With respect to amplitudes, the cross-flow and inline

vibrations show to be close to the values presented in the

experiments.

Case 3

riser with such characteristics and this is enough as, for this

Table 4: Natural freouencies of the riser

Riser shapes Cross-flow shapes o 0.1 0.7 0.0 0.4 0.5 05 0.7 05 0.9 -15 -1 .0.5 0 0.5 rzD

15 0.2- 03- 0.4- 0.8- 0.7- 0-B- 0_9-o 0-1 0_7 0.) 0.4 02 o-E 0.7 0-B 0.9 s-1 0.2 0_3 04 05 0-e 07 0-B 0,8 -lt4 -0.3 -0.2 -0,1

00.1

0.2 03 04 r/D

Figure 18: Riser shape for inline vibrations in one instant

Mean inline deflection

8 12 14 16 18 20

ryJD

-lt 4 -0.3 .0.2 .0.1 0 0.1 0.2 0.3 04 ry/D

For a more clear visualization, the riser shape for inline

vibration is shown in a new plot with only one instant plotted,

in Figure 18.

1 0.77 0.66 H 2 1.54 1.33 3 2.32 2.00 4 3.11 2.68 5 3.91 3.38 6 4.73 4.09 7 5.58 4.82 8 6.45 5.57 9 7.35 6.35 10 8.29 7.16 11 9.26 7.99 12 10.26 8.86 13 11.30 9.76 14 12.39 10.70 15 13.52 11.67 16 14.69 12.69 17 15.92 13.74 18 17.19 14.84 19 18.51 15.98 20 19.88 17.17

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In this case, for cross-flow vibration, the 7th mode is

predominant and, for inline, the

12th mode is

the most

important.

With respect to amplitudes,

the cross-flow and inline

vibrations show to be close to the values presented in the

experiments. Although, the inline mean deflection calculated in the model is a higher value than the one in the experiments. 6. CONCLUSIONS AND DISCUSSION

The model was developed during the authors internship period at MARIN and is in continuous development, as the

work herein presented is the initial stage of the investigation.

The model developed approaches the VIV problem in a manner largely done in the field of VIV studies, which is to

associate a wake oscillator to a structural model. The first

models the near wake dynamics and the last, the dynamics of

risers or pipelines. In this case, the wake oscillator is modeled as a van der Pol equation. The structural model is a beam with bending and axial stiffness with both ends clamped.

The model is able to evaluate motions in cross-flow and

inline directions as well as velocities, accelerations, trajectory

and tension. The present model differs from some industrial

software fundamentally because the present one is able to

model not only cross-flow vibrations but also inline modeling. However, it is important to reinforce that this is the initial stage

of the research and that the modeling is being constantly

revised aiming the best representation of reality.

Since this is a time-domain code, it is possible to evaluate

multi-mode response. This is a desirable feature when

analyzing the behavior of a deep-water riser,

since it is

observed that in the real situation multi-mode vibration is common. In the present case, the results showed that more than one mode was present. However, at this stage, further analysis must be performed to confirm whether multi-mode vibration is

present at high mode or time-sharing of frequencies occurs. Swithenbank, S. (2007) presented an important study on this

matter and showed that, at high mode vibrations, time-sharing

response do happen. The results obtained from the present model shall be analyzed following the approach suggested in

Swithenbank, S. (2007).

To solve the partial differential equation that models the structure a spectral method was used in which the response is considered as a series of eigenfunctions multiplied by time-dependent coefficients. In this case, the eigenfunctions were

chosen to be sines. With this model it is possible to calculate as

many modes as necessary in order to represent high mode

vibrations. However, as the basis functions are simple sines, no

imaginary components of mode shapes are present due to the

simplicity of the eigenfunctions. In this sense, the

implementation of BesseYs functions or the approach suggested in Fumes, G. (2000) would be a necessary improvement.

The differential equations arising from dynamic model and wake oscillator are solved by a 4th order Runge-Kutta scheme implemented in the code.

The sequence of validation was designed to have different

testing conditions as done in the work by Chaplin at al. (2005a)

and by Chaplin et al. (2005b), which is considered to be a good benchmark for VIV. Three of the testing conditions used in such

work (changing flow velocity and tension) were used to be

tested in the model with results showing to be satisfactory. As this is a preliminary stage of validation of the present work, the model has not been compared with other experimental results

with risers of greater aspect ratios. In future steps of the

research, further validation will be performed. NOM ENCLATU R E

Ey Damping coefficient for inline oscillator

Damping coefficient for cross-flow oscillator ç!1 Strouhal frequency [radis]

A Coupling amplification drag parameter Coupling amplification lift parameter Co Drag coefficient for fixed cylinder C0 Drag coefficient

CL Lift coefficient

CLO Vortex shedding lift coefficient

Ca Potential added mass coefficient

Ci0 Vortex shedding drag coefficient R1 Fluid damping [kg/sl

R Structural damping [kg/sl

mf Potential added mass [kg] Structural mass [kg] qy Reduced drag coefficient q Reduced lift coefficient

D Diameter[m]

K, Coupling parameter

L Immersed length [m]

s

Strouhal number

V Flow velocity [mis] Inline displacement [ml

r

Cross-flow displacement [m]

y Damping coefficient

çd Structural damping coefficient

p Fluid density [kglm3]

Vr Reduced velocity

ACKNOWLEDGMENTS

The first author is indebted to MARIN, University of São Paulo and Prof. Dr. A.L.C. Fujarra.

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