Introduction

Comparisons between simulation predictions with experimental observations are indis-pensable to validate the derived mathematical models and adopted methods of solution.

The derived 3-D motion equations are primarily concerned with internal flow, the struc-tural nonlinearity, the coupling of the in-line (IL) and cross-flow (CF) motions and the fact that displacements vary along the span. Unfortunately, there is few good quality measurement of the response of a fluid-conveying cantilevered pipe in literature.

Before going on our simulation studies, it is necessary to make it clear about the calcu-lations of natural frequencies and damping ratio of a cylinder system immersed in water.

Two experiments performed by former researchers are explained in detail which will be utilized in the following sections.

Natural frequency

The natural frequenciesω_n(n= 1,2· · ·) of a free hanging pipe are dependent on length, mass, bending stiffness, tension including self-weight effect, and boundary conditions.

There is no exact analytical formula if tension and bending stiffness are both taken into account. Sch¨afer and Oberpfaffenhofen [140] determined an approximate analytical closed-form solutions by use of the Ritz-Galerkin method with gravity-free beam eigen-functions in the series expansion for the free vibrations of a gravity-load clamped-free Euler-Bernoulli beam. Decay tests can be conducted to calculate the natural frequencies experimentally for scaled models. Depending on whether they are tension dominated or

not, the cylinders exhibit dynamic characteristics similar to tensioned strings or unten-sioned beams.

Neglecting gravity-induced tension, a clamped-free pipe can be modelled as a can-tilevered beam, the natural frequencies can be calculated by formula:

ωbn=

2n−1 2 π+en

2 1 L²

rEI ms

(e1= 0.3042, e2=−0.018,· · ·; n= 1,2· · ·), (4.1) where

ms=m+Mi+ma. (4.2)

m, Mi and ma are the masses of the pipe, internal fluid, added mass due to the sur-rounding water per unit of length.

Neglecting bending stiffness, a flexible long clamped-free pipe can be modelled as a hanging string. Referring to [88], the natural frequencies can be calculated by the fol-lowing formula based on the first Bessel Function:

ωsn= 1 2√

Lj0,n

s gβ1

1 +β₂ (j0,1 = 2.405, j0,2= 5.52 · · ·;n= 1,2· · ·), (4.3) whereβ1= (m+Mi−Me)/m,β2= (Mi+ma)/m.

When there is an end-mass at the bottom end, if a flexible long clamped-free pipe is modelled as a hanging string, the natural frequencies can be calculated based on the Bessel Functions

1 αωn

q_m

eg µ

h C1J1

2ωn

qmeg µ

α

+C2Y1

2ωn

qmeg µ

α i

−1 g h

C1J0

2ωn

q_m

eg µ

α

+C2Y0

2ωn

q_m

eg µ

α i

= 0, (4.4)

where µ= (m−M_e+M_i)g,α =p

µ/(m+m_e). J₁ is the first kind of Bessel funciton of order one. Y0 is the second kind of Bessel function of order zero.

Tension can be dominant and equation (4.2) has been validated by a scaled model and a real drilling riser model by Park, et al. in [88]. Equation (4.3) has been validated by a cantilever string with an end-mass by decay tests at System Planning Lab. in Kyushu University.

Structural damping

As a mass-spring-damper oscillator, the equation of motion is given as:

my¨+cy˙+ky=F , (4.5)

where the system’s mass is m, damping coefficient is c, and spring constant is k, y is the displacement, F is the exerted force. The fundamental nature frequency ω1 can be calculated by

ω₁ = rk

m, (4.6)

and the critical damping of the first natural mode c_r is c_r = 2√

km= 2mω₁. (4.7)

Referring to [89], the damping ratio ζ of a cylinder system subject to VIV is the ratio of the structural damping and critical damping in water:

ζ = c

c_r = c 2p

k(m+ma) = c

2(m+m_a)ω₁. (4.8)

Exp. 1 employing a cantilevered cylinder

Pesce and Fujarra [141] performed an experiment employing a 3-m cantilevered cylin-der. The cylinder system was comprised of an inner circular solid rod which was rigidly attached to an external pipe by means of equally spaced annular rings. It was free to move in both the IL and CF directions. The parameters of the cylinder system is given in Table 4.1. The external fluid velocity range excited only the first mode of vibration.

Using the data from in Table 4.1 and employing formula equation (4.1), the first nat-ural frequency can be calculated ωb1 = 11.8842 rad/s (f_b1^∗ = 1.8914 Hz) while us-ing formula equation (4.2), the first natural frequency is achieved ω_s1 = 1.1569 rad/s (f_s1^∗ = 0.1841 Hz) which can be negligible. It agrees well with f1 = 1.8915 Hz from the decay tests and validates the formula equation (4.1).

Making use of equation (4.7), the first mode structural damping coefficient in water

Table 4.1: Physical parameters

Length of the cylinder system,L 3 m

External diameter, D_o 31.75 mm

The measured bending stiffness, (EI)_eq 2568.4 Nm² 1st mode, total damping coefficient measured in water,ζ_1w 1.25%

1st mode, structural damping coefficient measured in water, ζ₁^s_w 0.46%

1st mode, structural damping coefficient measured in air,ζ₁^s_air 0.57%

1st mode, nature frequency measured in still water,f1w 1.8915 Hz 1st mode, nature frequency measured in air,f1air 2.313 Hz

Structural mass,mstruct 1.568 kg/m

Wiring mass, mwires 0.300 kg/m

Total system mass,msys 1.868 kg/m

1st mode, added mass,m_a 0.927 kg/m

1st mode, added mass coefficient,C_a=m_a/(ρ_eπD_o²/4) 1.17

c per unit length can be calculated by

c = 2(m_sys+m_a)ω_1wζ₁^s_w/L

= 4π(msys+ma)f1wζ₁^s_w/L

= 0.1262 Ns/m², (4.9)

whereω1w is the angular natural frequency. The Strouhal numberS was assumed to be 0.208 over the range of Reynolds numberRe= 1000−2500 referring to [113]. The vortex shedding frequencyf_s and the corresponding angular frequency Ω_scan be calculated by

f_s=Sv/D_o, Ω_s = 2πf_s. (4.10)

The reduced velocity can be calculated by

Vr=v/(f1wDo), (4.11)

which is not the real reduced velocity because f1w is the natural frequency measured in still water. The parameters used in the simulations are given in Table 4.2.

Table 4.2: Input data in simulations

Length of the pipe,L 3 m

The bending stiffness,EI 2568.4 Nm²

Structural damping coefficient, c 0.1262 Ns/m² The mass of the pipe per unit length,m 1.868 kg/m Density of external fluid, ρ_e 1000 kg/m³ Mass ratio,m^∗=m/(ρeπD_o²/4) 2.36

Added mass coefficient, Ca=ma/(ρeπD_o²/4) 1.17

The Strouhal number,S 0.208

External diameter of the pipe, Do 31.75 mm Inner diameter of the pipe, Di 0 mm Reference drag coefficient, C_D0 0.2 Reference lift coefficient,C_L0 0.3

Internal flow velocity, U 0 m/s

Added force due to a nozzle, ∆P_L 0P_a Exp. 2 employing a mass-spring-damper model

Stappenbelt et al. [130] conducted experiments in the towing tank at the University of Western Australia’s Laboratory to examine the single (CF motion only) and two-degree of freedom (2-DOF) VIV response of an elastically mounted rigid cylinder subject to steady, uniform current conditions. The apparatus illustrated in Figure 4.1 consisted

Figure 4.1: Schematic of the experimental set-up by Stappenbelt et al. [132]

of a towing carriage and a parallel linkage mechanism capable of translation motion in both IL and CF directions. The motion was calculated by measuring the spring force via calibrated load cells. Apparently, this device is based on a mass-spring-damper model.

The damping ratioζ is around 0.6%. The mechanism employed ensured identical mass ratios and natural frequencies in both directions. The mass ratio range covered was 2.36 to 12.96, achieved by the addition of lumped masses to the plate attaching the cylinder test section to the parallel linkage. Part of the natural frequencies f_n (angular natural frequencyωn) in still water are given in Table 4.3.

Table 4.3: Natural frequencies in still water at different mass ratios m^∗ m^∗ζ fn(Hz) ωn(rad/s)

2.36 0.014 1.711 10.75 3.68 0.022 1.502 9.44 5.19 0.031 1.359 8.54 ... ... ... ...