Aerodynamic Response of Offshore Spar · PDF fileThe fundamental frequency of spar platform is designed to. INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1, No 3, 2010

© Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN 0976 – 4399

327

Aerodynamic Response of Offshore Spar Platforms

Mohd Umair 1 and A. K. Jain 2

1 – Research Scholar, Department of Civil Engineering, IIT Delhi, New Delhi – 16. , & Assistant Professor, Department of Civil Engineering, Jamia Millia Islamia, New Delhi25.

2 – Professor, Department of Civil Engineering, IIT Delhi, New Delhi – 16.

[email protected]

ABSTRACT

Offshore Spar platforms are the floating platforms and behave as moored structures in horizontal as well as vertical planes in deep oceans. Under the influence of wind, superstructure of the spar is subjected to aerodynamic drag forces in the windward direction and the underwater portion is exposed to random waves and currents. Spar motions have time periods falling in the wind excitation frequency range due to its compliant nature. Dynamic response analysis of spar platforms to a low frequency wind forces with random waves is presented in the paper. Nonlinear effects due to variable submergence of the structure, hydrodynamic forces and cable tension variation are also considered. The random sea state is characterized by PiersonMoskowitz sea surface elevation spectrum. The wave forces on the elements of the structure are calculated using Morison’s equation with Airy’s linear wave theory ignoring diffraction effects. The fluctuating wind has been estimated using Emil Simiu’s wind velocity spectrum for offshore structures. The nonlinear equation of motion is solved in the time domain by numerical integration procedure. The wind structure interactions along with the influence of various other parameters on the structure’s response are investigated. Results indicate that the low frequency wind forces alter the response of the spar to a considerable extant which otherwise is not enhanced in the presence of random waves alone. The mean wind modifies the mean position of the surge response to the positive side, causing an offset. Apart from the prominent peak occurring at the natural frequency, other peaks also appear showing the superharmonic responses.

Keywords: Aerodynamic, spar, floating, mooring, power spectral density function.

1. Introduction

Offshore spar platforms are the floating compliant structures which are tied to the sea bed by the flexible cables. They comply in the direction of wind and wind driven waves when exerted by these forces. Being the structures of high importance level their dynamic analysis is quite important as their failure may cause a large destruction both monitory as well as in terms of human life. The fundamental frequency of spar platform is designed to



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be well below the frequency range of waves and therefore it is very low. As a result spar platforms are susceptible to low frequency wind forces. Fig.1 shows the various degrees of freedom of spar platform. As presented by Ochi and Shin (1988), fluctuations in the wind velocity over the open ocean are quite significant and may have large offset on the response of the platform. This is particularly true because wind may have substantial energy at low frequencies where spar is susceptible to resonant vibrations.

Figure 1: Degreesoffreedom of Spar platform

2. Modeling of Spar for Wind Forces

Figure 2(a) shows the plan at hull of the spar taken for the study. The schematic diagram of the spar exposed to wind and wave is shown in Fig. 2(b). Under the influence of wind, the super structure of the spar is subjected to aerodynamic drag force in the windward direction. The wind fluctuations are modeled as described by Kareem (1983). The Power Spectral Density Function provides information on the energy content of the fluctuating wind with frequency. The wind induced response of the structure is mainly due to drag and lift forces in the direction parallel and normal to the wind flow, respectively. The analysis due to wind forces is carried out by decomposing the wind velocity component into its mean and fluctuating components. Out of all the sources of aerodynamic oscillations, spars, like other compliant structures are susceptible to excessive responses in surge mode (Simiu and Stefan (1984)). Horizontal as well as part of vertical cylinder above the water level obstruct the wind in surge direction and hence they are replaced by their equivalent projected areas. The underwater portion is exposed to random waves and current and the superstructure is exposed to wind forces.

2.1 Dynamic Analysis

The equation of motion is assembled as below:

) , , , ( ] [ ] [ ] [ . .. . ..

t X X X F X K X C X M = + + (1) The mass matrix [M] is diagonal in nature and constant. The added mass, Ma due to the water surrounding the structural members and arising from the modified Morison



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equation has been considered up to the mean sea level only. The presence of offdiagonal terms in mass matrix indicates the contribution of added mass due to hydrodynamic loading. The coefficients of the stiffness matrix [K] of the spar are derived from the first principles (Chandrasekaran and Jain (2002)). These coefficients have nonlinear terms due to cosine, sine, square root and squared terms of displacements. The change in mooring line tension due to movement of the spar makes the stiffness matrix response dependent and also affects the buoyancy. Hence [K] is not constant for all values of time instants but the coefficients are replaced by new values computed at each time instant depending upon the response value at that time instance. The damping matrix [C] is assumed to be proportional only to the initial values of [K] and [M].

Figure 2: (a) Plan of the hull of spar platform (b) Schematic Diagram of coupled spar platform subjected to wind

and wave forces.

2.2 Wave Forces

As the random process is considered to be linear superposition of large number independent waves, its distribution becomes Gaussian. In the present simulation



330

procedure, waves are assumed to be stationary, homogeneous and ergodic in statistical sense. The modified PeirsonMoskowitz (PM) spectrum model representing the sea state is given by:

−

=

− − 4 5

2

2

2 1 exp

2 8 ) (

π ω

π π ω

π ω ηη

Tz Tz Tz Hs S

(2) The wave super position technique is adopted based on the approach suggested by Goda (1970) for the simulation of sea surface elevation. The component frequencies determined are noncorrelating. The random selection of component frequency is repeated for each run of sea surface elevation spectrum. The simulation is carried out with 245 component waves. The length of the simulated wave record is controlled so that about 2000 data points are generated in one run. Record length of 1024 s is chosen which

gives 2048 (2 11 ) data points. The horizontal water particle velocity ) , ( .

t x u and the

vertical water particle velocity ) , ( .

t x v are given as:

)) ( sinh( ) cosh( ) cos( ) , (

1

.

η φ ω ω

+ + − Σ =

= d k y k t x k A t x u

i

i i i i i i

K

i (3)

)) ( sinh( ) sinh(

) sin( ) , ( 1

.

η φ ω ω

+ + − Σ =

= d k y k

t x k A t x v i

i i i i i i

K

i (4)

The horizontal water particle acceleration ) , ( ..

t x u and vertical water particle acceleration

) , ( ..

t x v are given as:

)) ( sinh( ) cosh(

) sin( ) , ( 2

1

..

η φ ω ω

+ + − Σ =

= d k y k

t x k A t x u i

i i i i i i

K

i (5)

)) ( sinh( ) sinh(

) cos( ) , ( 2

1

..

η φ ω ω

+ + − Σ =

= d k y k

t x k A t x v i

i i i i i i

K

i (6)

Where ki is the i th component wave number, y is the vertical distance at which the wave kinematics is calculated, d is the water depth and η is the sea surface elevation. Water particle kinematics is calculated using Airy’s linear wave theory along with the stretching modifications suggested by Hogben. The wave forces are computed using Morison’s equation, which is given as:

f =0.5ρwCdD| . u

. x + c u

.

|( . u

. x + c u

.

) + 0.25 π D 2 ρwCm

.. u ± 0.25 π D 2 [Cm1] ρw

.. x (7)

The equation of motion is solved in time domain using numerical integration technique, incorporating all the nonlinearities caused in the hydrodynamic loading and those caused due to structural stiffness.



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2.3 Wind Forces

The direct wind pressure on the superstructure of spar causes translational surge force and moments in pitch and yaw directions. Due to the coupled nature of heave degree of freedom with surge, pitch and yaw degrees of freedom, the heave motion is also activated. This coupled wind force also influences the mooring tension fluctuations. The basic expression for the wind induced force is given as the drag per unit area, projected on a plane, normal to the mean wind velocity (Simiu and Stefan (1984)) is given by:

f(y,z,t)=0.5ρaCa(y,z)Aa[ . u (y,z,t)

. x (t)] 2 (8)

where, f(y,z,t) is the force per unit area and is a function of space (y,z) and time (t).

ρa is the air density,

Aa is the exposed area in surge direction ,

Ca(y,z) is the force coefficient at elevation z and at horizontal coordinate y, . u (y,z,t) is

the wind velocity in the surge direction varying with time, . x (t) is the structural velocity

in the surge direction. It is assumed that directions of wind and surge motions are collinear. The wind velocity is expressed as:

. u (y,z,t) =

. u +

. u ’ (y,z,t) (9)

Where, . u is the mean wind velocity at 10 m above mean sea level,

. u ’(y,z,t) is the

fluctuating wind velocity. Hence the force due to wind in n segments above the p th segment i.e. starting from MSL, shall be expressed as:

−

− + Σ = 2 . . . )] ( ) , , ( ' [ ) , , ( t x t z y u u C t z y F p

N

p (10)

ap p a p A C C ρ 5 . 0 = −

(11)

Where Aap is the projected area of the p th segment, Cp is the aerodynamic force coefficient for the p th segment and N is the total number of segments. The mean wind velocity using the logarithmic law is as follows:



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0

0

ln

ln ) ( ) (

z z z z

z u z u ref

ref =

(12) Where zref is the reference elevation usually taken as 10 m above mean sea level (MSL),

z0 is the roughness length, ) (z u is the mean velocity at elevation of the vertical coordinate z. Specifying the value of Cp with roughness length z0 over the sea surface as:

2

0

10 ln

=

z

K C p

(13)

Where K is the Von Karman constant (k=0.4), the equation 12 gives a conservative estimate of the mean wind velocity component, due to the presence of water wave as compared to flow over a rigid surface. The random variation of the wind velocity fluctuation is assumed to be constant all through the projected area. This fluctuating velocity component is estimated by the Fourier synthesis of wind spectrum suited to the offshore environment (Simiu and Stefan (1984)).

2.4 Emil Simiu’s sea site Spectrum

Kareem and Datton (1982), Kareem et al (1986) and Kareem (1980) have proposed wind spectra to describe the longitudinal wind velocity fluctuations in Eq. (10). As the deep water compliant type platforms like spar have the frequencies of interest at very low values, the ordinates of this spectrum is maximum as shown in Fig.5 Shape of the spectrum in the very low frequency range has very little impact on the designed of land based fixed structures, while it significantly influences the design of compliant offshore structures like spars. Expression for the fluctuating wind velocity spectrum by Simiu and Stefan (1984) is given below:

≥ =

+ + =

≤ + + =

− ) ( 26 . 0

) (

) 0 ( ) , (

3 2

2 2 2 2

3 1

2 1 1 2

*

s

s m

m

f f f

f f f f b f a c

f f f d f b f a u

n z nSu

p p

p

(14) In which,



333

0

* 10 ln

) 10 (

z

Ku u =

(15)

z L

a u β 4 1 =

(16) 3 2

1 26 . 0 − = s f β (17)

) (z u nz f =

(18) Where, f is a nondimensional frequency.

( ) ( ) m

s m s s m s m s m s m

m

s m

f f

f f f f f f f f f f

f f

f a b

ln ) 2 ( ) ( 2 2 1

6 5

ln 3 7

3 1

2 2 2

1 1

2

− + − + − + −

−

+ +

= β β

(19)

m f b a 2 2 2 − = (20)

− + − = 2

2 1 1

3 1 ) ( 2

2 s m

m

m

f f b f a

f d β

(21)

1 1

1 5 . 1 2

d f f a

b m m

− − = (22)

2 2 2 1 2 s f b f a c − − = β (23)

In the above expressions, z0 is taken as 0.001266 m and Lu is taken as 180 m. The other wind characteristics are given in Table 4. The values for mean wind velocity are taken as 35 m/s, 40 m/s and 45 m/s and the corresponding values of u* are 1.56 m/s, 1.76 m/sand 2.01 m/s, respectively in the numerical study. Spar has quite low frequency for which the corresponding wind velocity spectral ordinate is the highest. Fluctuation wind represented by Emil Simiu’s spectrum is simulated using MonteCarlo procedure.

2.5 Wind and Wave Forces

( )

=

61

51

41

31

21

11

F F F F F F

t F

(24) Where,

F11 = Force in the surge degreeoffreedom due to wind and wave. F21 = Force in the sway DOF due to wind and wave. F31 = Force in the heave DOF due to wind and wave. F41 = Force in the roll DOF due to wind and wave.



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= Moments of the above coefficients (i.e. F11, F21 and F31) about xaxis. F51 = Force in the pitch DOF due to wind and wave.

= Moments of the above coefficients (i.e. F11, F21 and F31) about yaxis. F61 = Force in the yaw DOF due to wind and wave.

=Moments of the above coefficients (i.e. F11, F21 and F31) about zaxis.

The equation of motion is solved by using numerical integration technique, presented by Bathe and Wilson (1976), incorporating all the nonlinearities. Since the assembled equation of motion is coupled and nonlinear, a time domain analysis is employed to calculate the time history response. At each step, the restoring force matrix is updated to take into account the change in the cable tension. The solution procedure is iterative due to the nonlinearities and solved by using Newmark’s Beta method. The numbers of wave components are taken as 245 and the record length is 1024 seconds. The time step is taken as 0.5 s. The general approach to solve such equation of motion is through integration of acceleration and velocity in time domain.

3. Numerical Studies and Discussions

Numerical studies for evaluating the coupled responses of spar platform under various combinations of hydrodynamic and aerodynamic loading have been carried out. The geometrical properties of the spar are given in Table 1. Table 2 shows the natural time periods of the spar and Table 3 shows the random wave data and the hydrodynamic coefficients used in the analysis. Table 4 shows the wind characteristics used in the study.

Table 1: Geometric properties of the spar platform

Degreeoffreedom Time Period (sec) Surge 214.47 Sway 174.69 Heave 34.42 Roll 50.84 Pitch 50.84 Yaw 72.81

Table 2: Natural Time Periods of the coupled spar platform

Data Description Weight of structure (W) 2.6*10 6 KN Height of spar cylinder (hs) 216.4 m Radius of spar cylinder R 20.26 m Distance of C.G to buoyancy (h1) 6.67 m Distance of C.G to keel (hcg) 92.4 m Distance of C.G to fairlead attachment point (h2)

0.2 m

Initial draft of spar (hd) at rest 198.12 m



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Table 3: Wave data and hydrodynamic coefficients

Mass density of sea water, (ρw) = 1049.95 Kg/m 3 Hydrodynamic Drag coefficient, CD = 0.6 Hydrodynamic Mass coefficient, CM = 1.5 Structural Damping, ξ = 5٪ Random Sea spectrum (PeirsonMoskowitz) Case 1 Significant wave height, Hs = 15m

Zero upcrossing period, Tz = 15s Case 2 Significant wave height, Hs = 12m

Zero upcrossing period, Tz = 15s Case 3 Significant wave height, Hs = 10m

Zero upcrossing period, Tz = 10s

Table 4: Wind characteristics

Force coefficient, Cp = 0.002 Mean Wind Velocity = 35m/s, 40m/s, 45m/s Reference elevation, z = 10 m Total equivalent area of the super structure of Spar platform = 2572 m 2 Distance of AC from mean sea level = 58.34 m Decay coefficient = 16.0 Emil Simiu’s seasite wind velocity spectrum constants: (β=6.0, fm=0.07, fs=0.21)

3.1 Coupled Surge Response of Spar Platform

3.1.1 Effect of wind on coupled surge response under random waves

The large wind exposed area of the spar deck structure attracts wind forces in the surge degreeoffreedom. The spar platform model is exposed to three cases of environmental loadings visàvis random wave of Hs=15m and Tz=15s acting along with wind having a mean wind velocity of 45m/s, random wave of Hs=12m and Tz=15s acting along with wind having a mean wind velocity of 40m/s and random wave of Hs=10m and Tz=10s acting along with wind having a mean wind velocity of 35m/s. The maximum +ve and minimum –ve values from the time history of coupled surge response under random waves and wind with variable submergence is given in Table 5. By comparing the values of coupled response, it is seen that the wind velocity of 45m/s with the random wave height of Hs=15m, Tz=15s increases the response by 35.7%, the wind velocity of 40m/s with the random wave height of Hs=12m, Tz=15s increases the response by 31.63% and the wind velocity of 35m/s with the random wave height of Hs=10m, Tz=10s increases the response by 26.3%. Hence it may be concluded that the consideration of wind along



336

with random waves acting on spar platform increases the coupled surge response significantly. Fig. 3 shows the PSDF plot of the coupled surge response under random waves and wind with variable submergence. The first peak occurs nearer to the natural frequency of the surge degreeoffreedom of the spar platform. The second peak occurs nearer to the natural frequency of the random wave.

Table 5: Coupled Surge response of spar platform with variable submergence

Fig.3 Coupled Surge Response due to Random Wave (15m15s) along

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Frequency (rad/sec)

PSD sur

ge (m

2 sec

/rad)

wind+wave

only wave

Figure 3: Coupled Surge response with variable submergence

Maximum values (m)

Minimum values (m)

Hs Tz

Mean wind velocity (m/s)

Wave + wind

Only wave

Wave + wind

Only wave

15m15s 45 16.93 12.08 1.47 19.54 12m15s 40 13.65 10.37 1.06 13.83 10m10s 35 10.65 8.43 0.34 08.20



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3.1.2 Effect of variable submergence on the coupled surge response under wind and random waves

The maximum +ve and minimum –ve values of coupled surge response under random waves and wind forces without variable submergence are taken and presented in Table 6. By comparing the maximum +ve values that obtained in the presence of wave and wind with that obtained under wave only, it is observed that surge response increases by 45.3% for the random wave field of Hs= 15m, Tz=15s along with the mean wind velocity of 45m/s, increased by 30.5% for the random wave field of Hs= 12m, Tz=15s along with the mean wind velocity of 40m/s, increased by 21.7% for the random wave field of Hs= 10m, Tz=10s along with the mean wind velocity of 35m/s. The above data clearly prove that wind plays an important role in altering the coupled surge response of spar platform. The presence of wind acting along with the random wave field (without variable submergence) reduces the effect of variable submergence in the coupled surge response. As we compare the coupled surge responses obtained in the presence of wind and wave, as presented in Table.5 and Table.6, it is observed that in the presence of variable submergence the coupled surge response is increased by 14.9% for the random wave field of Hs= 15m, Tz=15s along with the mean wind velocity of 45m/s, increased by 19.95% for the random wave field of Hs= 12m, Tz=15s along with the mean wind velocity of 40m/s, increased by 14.4% for the random wave field of Hs= 10m, Tz=10s along with the mean wind velocity of 35m/s. Hence it is clear that under the presence of wind along with random waves, variable submergence markedly increases the coupled surge response.

Table 6: Coupled Surge response of spar platform without variable submergence

Maximum values (m)

Minimum values (m)

Hs Tz


Wave +

wind

Only wave

Wave +

wind

Only wave

15m15s 45 14.73 10.20 1.31 9.92 12m15s 40 11.38 8.72 1.02 9.27 10m10s 35 9.31 7.65 0.06 7.20

3.2 Coupled Sway Response of Spar Platform

3.2.1 Effect of wind on coupled Sway response under random waves

The maximum +ve and minimum ve values from the time history of coupled sway response under wind and random waves with variable submergence are given in Table 7. By comparing the values of response, it is seen that the coupled sway response is nearly negligible under random waves only than its value under wind and random waves. Coupled sway response under all wind speeds (45, 40 and 35 m/s) and random waves of



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all Hs,Tz (15,15; 12,15 and 10,10) acquires the values of the order of 89 meters whereas under only random waves of all the three Hs,Tz stated above, it is of the order of only 0.02 meters. From the above data it is observed that it is necessary to analyse the sway response of spar platform under both wind and random waves to obtain real responses taking place in the ocean environment.

Table7: Coupled Sway response of spar platform with variable submergence

Maximum values(m)

Minimum values (m)

Hs Tz


Wave +

wind

Only wave

Wave +

wind

Only wave

15m15s 45 9.995 0.025 0.002 0.02 12m15s 40 9.180 0.024 0.0015 0.03 10m10s 35 8.230 0.024 0.00086 0.05

Fig. 4 shows the PSDF plots of the coupled sway responses under wind and random waves and under random waves only with variable submergence. There is only one peak that occurs nearer to the natural frequency of the sway degreeoffreedom of the spar platform.

0

500

1000

1500

2000

2500

0 0.1 0.2 0.3 0.4

Frequency(rad/sec)

PSD sway (m

2 sec/rad)

wind+wave

only wave

Figure 4: Coupled Sway response with variable submergence

3.2.2 Effect of variable submergence on the coupled sway response under wind and random waves

The maximum +ve and minimum ve values of coupled sway response under random waves and wind forces (with and without variable submergence) are taken and presented



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in Table 7 and 8 respectively. By comparing the maximum +ve values that obtained in the presence of wave and wind with that obtained under wave only, it is observed that sway response increases drastically i.e. from just 0.025 meters to about 10.0 meters with variable submergence and from 0.024 meters to 7.3 meters without variable submergence when we consider wind with velocity of 45 m/s along with the random wave field of Hs= 15m, Tz=15s, the response increased from just 0.024 meters to 9.2 meters with variable submergence and from 0.025 meters to 6.76 meters without variable submergence when we consider wind with mean wind velocity of 40 m/s along with the random wave field of Hs= 12m, Tz=15s and it increased from just 0.024 meters to 8.23 meters with variable submergence and from 0.024 meters to 6.0 meters without variable submergence when we consider wind with mean wind velocity of 35 m/s along with the random wave field of Hs= 10m, Tz=10s.

Table 8: Coupled Sway response of spar platform without variable submergence

Maximum values (m)

Minimum values(m)

Hs Tz

Mean wind velocity (m/s) Wave

+ wind

Only waves

Wave +

wind

Only waves

15m15s 45 7.328 0.025 4.5e3 0.06 12m15s 40 6.759 0.025 3.6e3 0.04 10m10s 35 5.997 0.024 2.7e3 0.02

The above data clearly proves that wind plays an important role in altering the coupled sway response of spar platform. The presence of wind acting along with the random wave field (without variable submergence) reduces the effect of variable submergence in the coupled sway response. As we compare the coupled sway response obtained in the presence of wind and wave, as given in Table 7 and Table 8, it is observed that in the presence of variable submergence the coupled sway response is increased marginally under waves only. When we consider wind with mean wind velocities of 35, 40 and 45 m/s along with random waves of Hs,Tz as 15,15; 12,15 and 10,10 respectively, the sway responses increase from 30% to 40% in the presence of variable submergence which is indeed a significant enhancement necessary to be considered in the analysis. From the above factual figures it is clear that the consideration of wind is a mandatory for analyzing the spar so as to obtain the responses as near as possible to the responses really taking place in the ocean environment.

3.3 Coupled Heave Response of Spar Platform

3.3.1 Effect of wind on coupled Heave response under random waves



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The coupled heave responses under wind and waves are presented in Table 9 and Table 10. Comparing the maximum +ve values of the heave responses obtained under wind and waves with that obtained under waves only, it is observed that the presence of wind velocity of 45m/s with random waves of Hs=15m, Tz=15 s has increased the response by about 3.5 times, the presence of wind velocity of 40m/s with random waves of Hs=12m, Tz=15 s has again increased the response by about 3.5 times and the presence of wind velocity of 35m/s with random waves of Hs=10m, Tz=10 s has increased the response about 8 times. Hence it may be concluded that the consideration of wind along with random waves acting on spar platform increases the coupled heave response manifolds and therefore is necessary.

Table 9: Coupled Heave response of spar platform with variable submergence

Fig. 5 shows the PSDF plots of the coupled heave response under random waves and wind, and only random waves with variable submergence. The peak occurs in the vicinity of the natural frequency of the heave degreeoffreedom of the spar platform.

3.3.2 Effect of variable submergence on Coupled Heave response under wind and random waves

The maximum +ve and minimum –ve values of the coupled heave responses under random waves and wind (without variable submergence), are given in Table 10, whereas the maximum +ve and minimum –ve values of the coupled heave responses under random waves (with variable submergence) and wind, are given in Table 9. As we compare the coupled heave responses obtained in the presence of wind and wave, it is observed that in the presence of variable submergence the heave response is increased marginally under waves only. When we consider wind with mean wind velocity from 35 to 45 m/s along with random waves of Hs;Tz as 15;15,12;15 and 10;10 respectively, the heave responses obtained when variable submergence is considered is increased from 30 to 40% in comparison with the heave responses obtained without considering variable submergence.

Maximum values (m)

Minimum values (m)

Hs Tz


+ wind

Only waves

Wave + wind

Only waves

15m15s 45 1.891 0.505 7.72 7.64 12m15s 40 1.532 0.456 6.87 6.31 10m10s 35 1.291 0.154 5.83 5.36



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0

5

10

15

20

25

30

0 0.5 1 1.5 2

Frequency(rad/sec)

PSD hea

ve (m

2 sec

/rad)

wind+wave

wave only

Figure 5: Coupled Heave response with variable submergence

Table 10: Coupled Heave response of spar platform without variable submergence

Maximum values (m)

Minimum values(m) Hs Tz


Wave + wind

Only waves

Wave + wind

Only waves

15m15s 45 1.40 0.405 7.67 7.25 12m15s 40 1.29 0.353 6.42 6.12 10m10s 35 0.853 0.101 5.46 5.17

3.4 Coupled Pitch Response of Spar Platform

3.4.1 Effect of wind on coupled Pitch response under random waves

The coupled pitch response under the random waves and wind with variable submergence is given in Table 11. By comparing the maximum positive values for the cases of wave + wind and for only waves, it is observed that the presence of wind velocity of 45 m/s with the random wave of Hs=15m and Tz=15s has increased the response by 8%, the presence of wind velocity of 40 m/s with the random wave of Hs=12m and Tz=15s has increased the response by 23%, the presence of wind velocity of 35 m/s with the random wave of Hs=10m and Tz=10s has increased the response by 15%. On the basis of results it can be



342

seen that the presence of wind significantly increases the coupled pitch response of the spar platform.

Table 11: Coupled Pitch response of spar platform with variable submergence

Maximum values (m)

Minimum values (m)

Hs Tz


+ wind

Only waves

Wave +

wind

Only waves

15m15s 45 0.138 0.116 0.141 0.08 12m15s 40 0.108 0.082 0.103 0.06 10m10s 35 0.095 0.071 0.098 0.05

Figure 6 shows the PSDF plots of the coupled pitch response under random waves and wind with variable submergence. The second peak occurs in the near vicinity of the peak frequency of the PM spectrum. The first peak occurs at a very low frequency corresponding to the peak frequency of the wind velocity spectrum showing the influence wind in the peak response.

0

0.01

0.02

0.03

0.04

0.05

0 0.2 0.4 0.6

Frequency(rad/sec)

PSD pitch (m

2 sec/rad)

wind+wave

wave only

Figure 6: Coupled Pitch response with variable submergence

3.1.2 Effect of variable submergence on Coupled Pitch response under wind and random waves

The maximum positive and minimum negative values of the coupled pitch response under the random waves (without variable submergence) and wind are given in Table 12. Comparing the maximum positive values that obtained under the presence of random waves and wind with that of random waves only, it is observed that the pitch response is increased by 14.5% for random field of Hs=15m and Tz=15s along with mean wind velocity of 45 m/s, increased by 27.7% for random field of Hs=12m and Tz=15s along



343

with mean wind velocity of 40 m/s, increased by 29.5% for random field of Hs=10m and Tz=10s along with mean wind velocity of 35 m/s.

Table 12: Coupled Pitch response of spar platform without variable submergence

Maximum values (m)

Minimum values (m)

Hs Tz


+ wind

Only waves

Wave +

wind

Only waves

15m15s 45 0.127 0.112 0.103 0.07 12m15s 40 0.095 0.076 0.095 0.05 10m10s 35 0.089 0.065 0.081 0.04

On the basis of results it can be seen that the presence of wind significantly increases the coupled pitch response of the spar platform under the absence of variable submergence. The presence of wind acting along with random wave field (without variable submergence) reduces the effect of variable submergence in the coupled pitch response. From Fig. 5, it is observed that the wind contribution remains almost same in the PSDF with and without variable submergence and only wave contribution gets altered. Comparing the coupled pitch response that obtained under the presence of random waves and wind, as given in Table 11 and Table 12, it is observed that the presence of variable submergence has increased the response by 14.8% (for random wave loading of Hs=15m and Tz=15s), increased the response by 17.2% (for random wave loading of Hs=12m and Tz=15s), increased the response by 7% (for random wave loading of Hs=10m and Tz=10s), On the basis of results it can be seen that under the presence of wind along with random waves, variable submergence significantly increases the coupled pitch response of the spar platform.

4. Conclusions

Dynamic response analysis is carried out for the spar platform under wind and random waves in time domain. Emil Simiu’s wind velocity spectrum and PM sea surface elevation spectrum are used. The major nonlinearities are incorporated and their respective influences on the response are studied. Wind plays an important role on the response in various degreesoffreedom. The coupling of various degreesoffreedom through stiffness matrix plays a vital role in influencing the response in some degreesof freedom. Time histories for various results are developed until the effect of transients diminishes. Power Spectral Density Function (PSDF) gives an idea of energy content of response. Based on the numerical study of spar platform, the following main conclusions are drawn:



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1. Aerodynamic surge response of the platform consists of a nonzero mean due to mean wind and the superimposed fluctuating surge response caused by the wind gustiness about the displaced position. The mean wind modifies the mean position of the surge response to the positive side, causing an offset. It oscillates about this offset position under the wind and random wave loading.

2. The sway response is significant manifestation of windinduced vibration. It is negligible when the spar platform is analyzed under random waves only whereas it is significant when wind is also considered along with random waves. It is mainly due to the fluctuating component of the wind.

3. Apart from surge and sway, the heave and pitch responses are also enhanced considerably which would not have been computed without considering the wind along with waves.

5. References

1. Bathe, K.J. and Wilson, E.L.(1976),” Numerical methods in finite element analysis”, Prentice Hall International Inc, Englewood Cliffs, USA.

2. Chakarabarti, S. K. (1971),”Discussion of dynamic response of single point moorings in deep water”, Journal of waterway, Harbor and Coastal Engineering Division, ASCE, Vol. 97, No. WN3, pp. 558590.

3. Chandrasekaran, S and Jain, A.K.(2002),”Dynamic behavior of square and triangular offshore Tension Leg Platforms under regular wave loads”, Ocean Engineering, Elsevier Science, Vol. 29, No.3, pp. 279313.

4. Chandrasekaran, S and Jain, A.K.(2002),” Triangular configuration TLP behavior under random sea wave loads”, Ocean Engineering, Elsevier Science, Vol. 29, Issue 15, pp. 18951928.

5. Goda, Y. (1970),”Numerical experiments on wave statistics with spectral simulation”, Report of the Port and harbor research institute, Japan, Vol. 9, No. 3, pp. 1618.

6. Kareem, A. (1980),”Dynamic effects of wind on offshore structures”, Proc. of Offshore Technology Conference, No. 3764, pp. 235246.

7. Kareem, A. and Datton, C. (1982),”Dynamic effects of wind on TLP”, Proc. of Offshore Technology Conference, No. 4229, pp. 749757.

8. Kareem, A. (1983),”Nonlinear dynamic analysis of compliant offshore platforms subjected to fluctuating winds”, Journal of wind engineering and industrial aerodynamics, Vol. 14, pp. 345356.



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9. Kareem, A., Lu, P.C., Finnigan, T.D. and Liu, S.L.V. (1986),”A wind tunnel investigation of aerodynamic loads on a typical TLP”, Proc. of Offshore Technology Conference, No. 5173, pp. 187197.

10. Ochi, M.K., and Shin, Y.S. (1988),”Wind turbulent spectra for design consideration of offshore structures”, Proc. of 20 th Annual Offshore Technology Conference, No. 5736, pp. 461467.

11. Simiu, E. and Leigh, S.D. (1983),”Turbulent wind effects on TLP surge”, National Bureau of Standards, Science series 151.

12. Simiu, E. and Stephen, L.D.(1984),” Turbulent wind and TLP surge”, Journal of Structural Engineering, ASCE, Vol. 110, No. 4, Paper No. 18672, pp. 785802.

Aerodynamic Response of Offshore Spar · PDF fileThe fundamental frequency of spar platform is designed to. INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1

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