NBS BUILDING SCIENCE SERIES 151

Turbulent Wind Effects

on Tension Leg Platform Surge

U.S. DEPARTMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS
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NBS BUILDING SCIENCE SERIES 151

Turbulent Wind Effects

on Tension Leg Platform Surge

Emil Simiu

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

and

Stefan D. Leigh

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

Sponsored by:

Minerals Management Service

United States Department of the Interior

Reston, VA 22091

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued March 1983
Library of Congress Catalog Card Number: 83-600507

National Bureau of Standards Building Science Series 151

Natl. Bur. Stand. (U.S.). Bldg. Sci. Ser. 151, 46 pages (Mar. 1983)

CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1983

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $4.75

(Add 25 percent for other than U.S. mailing)
ABSTRACT

A procedure Is presented for estimating surge response to turbulent wind in the

presence of current and waves. The procedure accounts for the nonlinearity of

the hydrodynamic forces with respect to surge and for the coupling of aero- dynamic and hydrodynamic effects. It is shown that current wind spectra do not

model correctly the wind speed fluctuations at very low frequencies and an alternative model of the wind spectrum, consistent with fundamental principles,

is presented. The equation of surge motion under turbulent wind in the presence

of current and waves is solved for typical tension leg platforms, and it is

shown that under extreme wave conditions the damping provided by the hydrodynamic forces precludes the occurrence of significant wind-induced resonant

amplification effects even if the drag coefficient in the Morison equation is

very small (e.g., C^j = 0.1). It is verified that for the platforms being

investigated the use of a one-minute wind speed to represent the effect of the

mean wind and of the turbulent wind fluctuations is acceptable for the purpose

of estimating peak surge response.

Key Words: compliant platforms; ocean engineering; offshore platforms;

structural engineering; tension leg platforms; turbulence; waves;

wind loads.

iii
ACKNOWLEDGMENTS

The cover picture is excerpted from "Hutton TLP Vessel - Structural

Configuration and Design Features" by Norman Ellis and Jeffery Howard Tetlow

of Conoco (U.K.) Ltd., Eraser Anderson of Scott Lithgow Ltd., and Alan Lindsay

Woodhead of Brown and Root (U.K.) Ltd., paper OTC 4427, Proceedings , 1982

Offshore Technology Conference, May 3-6, 1982, Houston, TX, Vol. 4, pp. 557-571.

The writers would like to acknowledge the use of documentation on tension

leg platform designs kindly provided by Mr. G. Sebastiani, Manager, Research

Department, Tecnomare S.p.A., Venice, Italy.

The writers would also like to express their apppreciat ion to Ms. Mary Ramsburg

for her capable typing effort, and to Ms. Jennie Covahey for her editorial

assistance.

iv
TABLE OF CONTENTS

ABSTRACT ill

ACKNOWLEDGMENTS Iv

LIST OF SYMBOLS vi

LIST OF FIGURES x LIST OF TABLES xi

1. INTRODUCTION I

2. WIND LOADS 3

2.1 Basic Expressions 3 2.2 Mean Wind Speeds 3 2.3 Fluctuating Wind Speeds: Critique of NBC Spectrum 4 2.4 Fluctuating Wind Speeds: Proposed Spectrum 6

2.5 Fluctuating Wind Speeds: Spatial Coherence 6 2.6 Mean Wind Loads 10

2.7 Fluctuating Wind Loads 10

3. Hydrodynamic Loads 15

4. Restoring Force 17

5. Numerical Estimates 19

5.1 Equation of Surge Motion 19

5.2 Assumed Characteristics of Environment and of Platforms 19

5.3 Linear vs. Nonlinear Restoring Force 21

5.4 Nominal Damping Ratio of Pseudo-Linear System Equivalent to

Equation 38 24

5.5 Peak Surge Response Under Turbulent Winds 27

5.6 Surge Response Estimated Under Various Simplifying Assumptions . 27

5.7 Sensitivity of Surge Response to Changes in Values of Wind

Environment and Aerodynamic Parameters 30

6. SUMMARY AND CONCLUSIONS 33

REFERENCES 35

APPENDIX - Expression for the Spectrum of the Longitudinal Velocity

Fluctuations 39

V
LIST OF SYMBOLS

A = surge added mass

= projection of above-water part of platform on plane normal to wind

speed

Ap = elemental area of submerged structure projected on a plane normal

ij to direction of current and wave motion

b = width of main deck

B = surge wave-radiation damping coefficient

= overall aerodynamic drag coefficient defined by Eq. 17

^C'^m ~ drag inertia coefficient in Morison's equation, respectively

^Dsea ~ drag coefficient (Eq. 5)

^d..>^m.. ~ drag, inertia coefficient in Morison's equation corresponding to

"J volume , respectively

CjjL ~ downdraw coefficient (Eq. 37)

Coh = square root of coherence function (Eq. 15)

Cp = pressure coefficient (Eq. 1)

Cy,C2 = exponential decay coefficients (Eq. 16)

D, D(, = diameter of immersed body or component, diameter of buoyant columns,

respectively

E = function defined by Eq. 29

f, f^i, fg = nondimensional frequency, nondimensional frequency at which product

nSy is maximum, nondimensional frequency above which Eq. Ale

is valid

' ^fi» ^onn » T-» ~ total hydrodynaraic load, total wave-induced i L C vj LI y L LI U y *- V exciting force, equivalent force defined by Eq, 23, total wind load,

fluctuating part of wind load acting on body at rest, total

hydrodynamic viscous force

g = acceleration of gravity

H = wave height

J = reduction factor defined by Eq. 28

vi
LIST OF SYMBOLS (Continued)

k = von Karman's constant

= wave number given by Eq. 34

K = Keulegan-Carpenter number

= nominal length of tethers

L = wave length

Ly = integral scale of longitudinal wind velocity fluctuations (in

direction of mean wind speed)

M = total mass of platform

= point in space

Max(xjQgx) ~ sample maximum value of Xj^^^

n, n^j = frequency

p = wind pressure

R = restoring force

Re = Reynolds number

R^ = autocovariance function of longitudinal velocity fluctuations

s(xjQ3x) ~ sample standard deviation of x^iax

Sj, ,Sp = spectrum of f' , of f' , respectively

equ,r u,r ^ »

r ^u'^u u »^u u

~ spectrum, cross-spectrum, cospectrum of longitudinal velocity 12 12 fluctuations, respectively

^ueq ~ spectrum of equivalent wind speed fluctuations, defined by Eq. 21

t = time

T = total tension in tethers at x = 0

Tf = period of harmonic relative fluid-body motion

= nominal natural period of surge motion

= wave period

vii
LIST OF SYMBOLS (Continued)

I t t u,u ,Ugq,Ugqj = longitudinal wind velocity, longitudinal velocity fluctuation,

equivalent wind speed fluctuation, component of equivalent wind speed

fluctuation

u^ = friction velocity

v^ = current velocity

Vj^j = horizontal wave-induced velocity at coordinates Zj^,Xj

V = amplitude of relative fluid-body velocity

Vj^j = elemental submerged volume

X = platform surge displacement

^max ~ peak surge displacement

^umax ~ amplitude of contribution to surge of a harmonic force (Eq. 14)

X = horizontal coordinate

y,yj^ = horizontal coordinate, horizontal coordinate of point M^, respectively

Yj^ = arbitrary coordinate

z = vertical coordinate above mean water level

Zj^j = vertical distance from free surface to center of volume V-j^j

Zq = roughness length

3 = coefficient defining mean square value of turbulent fluctuation in

terms of friction velocity (Eq. 14)

A = denotes increment of a quantity

5 = nominal damping ratio

V = dynamic viscosity

Pa'Pw ~ air, water density, respectively

T = time lag

(|) = arbitrary function

<j)j = phase angle

viii
LIST OF SYMBOLS (Continued)

denotes differentiation with respect to ti

denotes mean value

denotes fluctuating part

ix
LIST OF FIGURES

Page

Figure 1. Spectra of Longitudinal Velocity Fluctuations 7

Figure 2. Spectra of Longitudinal Velocity Fluctuations 8

Figure 3. Spectra of Longitudinal Velocity Fluctuations 9

Figure A. Integration Domain and Transformation of Variables 12

Figure 5. Geometry of Tension Leg Platform 20

Figure 6. Surge Response of Tension Leg Platform with Nonlinear

Restoring Force 22

Figure 7. Surge Response of Tension Leg Platform with Linear Restoring

Force 23

Figure 8. Surge Response of Tension Leg Platform to Harmonic Wind Load

in the Presence of Current and Waves, Cjj, = 1,8, C^j = 0.8 26

Figure 9, Surge Response of Tension Leg Platform to Turbulent Wind

Load in the Presence of Current and Waves 28

.1

X
LIST OF TABLES

Pa^e

Table 1. Assumed Current Speeds at Various Depths 19

Table 2. Estimated_Nomlnal Damping Ratio, ^, and Mean and Peak Surge

Response x and Xpi^ , in Meters 25

Table 3, Statistics of Peak Surge Response, x^j^^j Under Turbulent Winds

(in meters) 27

Table 4, Estimates of Surge Response Corresponding to Various

Deterministic Loading Assumptions (in meters) 29

Table 5. Dependence of J''"/^(n) Upon Exponential Decay Coefficient, C^ . . 31

Table 6. Dependence of u(z) Upon Zq (u(35 m) = 45 m/s) 31

xi

1 . INTRODUCTION

Wind effects on tension leg platform (TLP) surge response can be divided into

two categories: the wind-induced steady drift, due in practice solely to the mean wind speed, and oscillatory motions induced by the turbulent wind speed

fluctuations.

If the requisite aerodynamic information is available the estimation of the

wind-induced steady drift is straightforward. However, the estimation of the

wind-induced oscillation raises a number of problems that merit careful investigation. It has been stated [24] that "determining the response to wind is

possibly of greater importance to the design of compliant structures than the

wave and current aspects. Wind has a spectrum which has its peak near the

structure's natural frequency in surge. In many instances, the movements

derived from varying wind will be greater than those for the wave drift."

Investigations into wind-induced oscillatory compliant platform motions have

been reported recently in references 6 and 28. However, these investigations

do not take into account nonlinearities due to the hydrodynamic viscous forces,

and assume instead that the response to wind is described by a system with

proportional damping, the damping ratio being of the order of 5 percent. It

appears that no research has been reported that takes into account explicitly

the turbulent nature of wind as reflected by spectral and cross-spectral information, while considering simultaneously the nonlinear effects of current and waves. Rather, estimates of the effect of wind speed fluctuations in the

presence of current and waves have been advanced that assume a constant

(1-minute wind) or harmonic loading in lieu of the actual turbulent wind load

[11, 17, 19]. The extent to which these simplified representations are acceptable has not yet been established in the literature. This was noted in a recent evaluation of the Hutton TLP response to environmental loads, which

states: "Wind gusts are typically broad-banded and may contain energy which

could excite surge motions at the natural period. These would be controlled by

surge damping. Theoretical and experimental research is required to clarify

the importance of this matter" [11]. The investigation presented here was undertaken in response to this need.

In this paper the surge response is estimated by solving the equation of surge

motion in the time domain. The forces represented in this equation consist

of: the forces of inertia, the external forces (hydrodynamic and aerodynamic),

and the internal forces (restoring, and damping due to internal friction within

the structure).

In the following sections the models used in this investigation for the external

and internal forces are described, and the equation of surge motion is solved

for TLP's with specified characteristics in specified wave, current, and wind

environments. To gain insights into the behavior of the platform viewed as a hydrodynamically damped system, the equation of surge motion is also solved in

the idealized case where the fluctuating wind load is represented by a harmonic

function. Numerical results are presented, and their sensitivity to changes

in the values of various parameters is assessed. Current simplified wind

loading models are evaluated in light of these results.

1

2. WIND LOADS

2.1 BASIC EXPRESSIONS

Like the hydrodynamlc loads, the wind loads consist of a component due to the

presence of viscosity and the consequent flow separation, and an inertial

component associated with the relative fluid-body accelerations. However, it can be verified that the inertial component is about two orders of magnitude

smaller than the component due to flow separation, and can therefore be

neglected in practical applications.

To estimate the wind-induced drag force it is assumed that the elemental drag

force per unit of area projected on a plane, P, normal to the mean wind speed

can be written as

p(y,z,t) = i P„ C (y,z)[u(y,z,t) - x(t)]2 (1)

2 d P

where Pa = air density, Cp(y,z) = pressure coefficient at elevation z and

horizontal coordinate y in the plane P, t = time, x = surge displacement, the

dot denotes differentiation with respect to time, and u(y,z,t) = wind speed

upwind of the structure in the direction of the mean wind. It is assumed that

the directions of the mean wind and of the surge motion coincide. The speed,

u(y,z,t), can be expressed as:

u(y,z,t) = u(z) + u'(y,z,t) (2)

where the bar and the prime indicate mean value and fluctuating part,

respectively. The total wind-induced drag force is

Fu(t) = /a P(y,z,t) dy dz (3) a

where A^ = projection of above-water part of the platform on a plane normal to

the mean wind speed.

It is seen that a prerequisite for the modeling of the elemental and total drag

forces is the modeling of the wind speed, u(y,z,t),

2.2 MEAN WIND SPEEDS

The mean wind speed can be modeled by the equation:

z In

u(z) = u(Zj-gf)

z.

JL. (4)

in

z ref

where Zj-gf = reference elevation (usually in meteorological practice Zj-gf

10 m), and Zq = roughness length.

3
Information on the roughness length, Zq, over the ocean is commonly provided byspecifying the value of the sea drag coefficient, defined as

%sea = [k/MlO/z^)]2 (5)

where Zq is expressed in meters and k = von Karm^n constant (k 0.4),

Measurements have shown that the sea drag coefficient increases with wind

speed. This dependence is not completely understood; however, on the basis of

a large number of measurements, the following empirical relations were proposed

for the range 4 < u(10) < 21 m/s:

%sea = 5.1 X 10"^ [^TOO)"]^*^^ (6)

or

^Dsea = ^^'^ 1^-5 + '^^(JOT] (7)

where u(10) is expressed in m/s [25], For values of about u(10) > 20 m/s the

variation of Cpggg with wind speed appears to be insufficiently documented.

According to reference 29, for such values Cj)gea is constant. On the other

hand, results of limited studies summarized in reference 5 would suggest that

for about u(10) > 20 m/s Cpgea can be expressed as an average of the values

given by equations 6 and 7,

It is shown subsequently that uncertainties with respect to the actual value of

^Dsea of little consequence in the estimation of the total surge response; errors in the estimation of Cpgg^ of the order of 50 percent result in differences in the calculated total surge of less than 5 percent. Note that, even if

the actual value of Zq were known, equation 4 should not be regarded as "exact,"

particularly near the mean water surface. Indeed, it is argued in reference 9

that owing to the presence of waves "wind profiles are distorted to lower wind

speeds compared to flow above a rigid surface," The use of equation 4 in

structural calculations is therefore likely to be conservative,

2,3 FLUCTUATING WIND SPEEDS; CRITIQUE OF NBC SPECTRUM

For design purposes, it is necessary to describe the longitudinal wind speed

fluctuations, u', in equation 2 in terms of their spectra and cospectra [23,

p, 446],

The expression for the spectrum of the longitudinal velocity fluctuations used

in the National Building Code of Canada (NBC) [12] and the American National

Standard ANSI A58,l - 1972 [1] is

NBC

^

^ 2 [l,200/u(10)]^n

{l+[1200n/u(10)]2}

where n = frequency, u(10) is expressed in m/s, and the friction velocity, u*

is defined as

4
= k u(lQ)

(9)

Equation 8 is represented in figure 1 for u =1.76 m/s and Cj)gea = 0.002

* (zq = 0.0013 m). It is seen that according to equation 8 the ordinates of the

spectrum at frequencies of interest in deep water platform design (i.e.,

n = 0.01 Hz) are quite small. However, this is actually not the case. As

pointed out by Luraley and Panofsky [10], the assumption inherent in equation 8

to the effect that the spectrum vanishes at n = 0 is incorrect. This can be

shown by considering the relationship between the spectrum and the

autocovariance function, Ru(t):

Su(n) = 4 / Ru(t) cos 2TrnTdT (10)

0

For n = 0:

Su(0) = 4 / Ru(t) dT (11)

0

The function Ru(t) decreases with increasing t from Ry(0) = u' to Ru(t) - 0

for large values of t. Defining the integral length scale of the longitudinal

turbulence as

L„ =

/ Ru(t) dT

.2 0

(12)

it follows immediately from equations 11 and 12 that

Su(0) = 4 Lu (13)

Information on and its variation in the atmosphere cannot be obtained from

wind tunnel measurements, owing to the difficulty of modeling the mesometeorological features that presumably control the lower frequency portions of the

spectra [13]. However, values of obtained from numerous full-scale measurements are listed in the literature, and in particular in reference 2. These

show that Lu Increases with height above ground, and that it also increases

if the surface roughness decreases. From these measurements it may be inferred

roughly that at elevations over water of about 20 to 60 m, which are of primary

interest in platform design, 100 m < < 240 m. Even larger variations

(60 m < < 450 m) were reported by Shiotani [22] (see also reference 23,

p. 55). The effect of such variations upon the magnitude of surge response

is examined subsequently.

5
2.4 FLUCTUATING WIND SPEEDS: PROPOSED SPECTRUM

No expression for the wind spectrum compatible with equation 13 has been

developed so far in the literature primarily because the shape of the spectrum

in the very low frequency range has little effect on the design of land-based

structures or fixed offshore platforms [23]. To develop such an expression

the following conditions, in addition to equation 13, must be satisfied (for

details, see reference 23, pp. 53-55):

1. In the inert ial subrange (f > fg), where the nondimensional frequency

f = nz/u(z), and fg = 0.2, Kolmogorov's first and second hypothesis

apply, and the energy production is approximately balanced by the energy

dissipation.

2. The product nS^Cn) is a maximum (i.e., the derivative of the function

nSu(n) vanishes) at some nondimensional frequency, f^ < fg. Measurements

at elevations of interest in platform design suggest that f^jj - 0.05 to 0.09.

3. The area under the spectral curve is

where u is defined by equation 9 and 3 - 6.0.

An expression for the spectrum derived from these conditions and Eq. 13 is

given in the appendix (equation Al). This exp ression is plotted in figure 1

for k = 0.4, Zq = 0.001266 m, z = 35 m, u(35) = 45 m/s (u = 1.76 m/s), 3 =6.0,

fg = 0.2, fm = 0.07, and = 180 m. For the same values of the parameters k

through f g , the expression for the spectrum is also plotted in figure 2 for

= 0.07 and = 100 m, 180 m, and 240 m, and in figure 3 for = 180 m

and fjn = 0.05, 0.07, and 0.09. It is seen from figure 3 that the influence

of the nondimensional frequency f^^ on the shape of the spectrum is weak.

2.5 FLUCTUATING WIND SPEEDS: SPATIAL COHERENCE

It is a characteristic of turbulent flows that the velocity fluctuations at any

two points separated by some distance, d, do not exhibit perfect mutual coherence. Owing to imperfect coherence such fluctuations do not attain their maxima

or minima at the same time, and the resultant wind-induced fluctuating loads are lower than would be the case if perfect spatial coherence were assumed.

Mathematically, the imperfect coherence is reflected in the expression for the

cross-spectrum of the velocity fluctuations, S^ ^ (n), whose absolute value

can be written as: 12

where Su 12(n) and S^ (n) = spectra of the velocity fluctuations, u*12and u', at

points M]^ and M2, respectively, and Coh (M]^, M2, n) is referred to as the

u'2 B u2 (14)

*

(15)

6
(s/ziu) MnUiOBdS

7
(s/zuj) i/yndi03dS

8
9
square root of the coherence function. Coh (M]^, M2, n) is unity for = M2

and decreases with increasing distance between M]^ and M2. For any given M]^ and

M2 the coherence decreases with increasing n. An expression similar to

equation 15 can be written for the cospectrum (i.e., the real part of the

f t c cross-spectrum) of u and u , S (n). Measurements have shown that if M-, and

1 2 "l"2 ^ M2 are contained in a plane normal to the mean velocity, it is possible to

write [3]:

1/2

C 1/2 1/2 -[C2(y -y )2+c2(z -z )2] n V2^"^ ^ '-1 W""^ exp—Z_l_2

z

1 2 (16)

where Cy, C2 are empirical factors referred to as exponential decay

coefficients. Commonly accepted values are Cy = 10, C^ = 16 [28]. However,

actual values as obtained from experiments vary widely [8, 22, 23]. The effect

of this variation on the calculated surge response is examined subsequently.

Equation 16 can be modified to include the effect of longitudinal separation

(in the direction of the mean wind) as well. However, it follows from information presented in reference 7 that this effect is negligible as far as fluctuating aerodynamic loads on offshore structures are concerned.

2.6 MEAN WIND LOADS

It can be verified that the mean square values of u and x and the mean value

of the product u'x are small compared to the square of u. It then follows

from equations 1, 2, and 3 that the mean drag load can be written as

where the overall aerodynamic drag coefficient is

Ca =

/. C_(y,z) ^ITiT^ dydz (18)

and = elevation of aerodynamic center of above-water part of the platform.

2.7 FLUCTUATING WIND LOADS

From equation 1, 2, 3, 17, and 18 it follows that the fluctuating part of the

wind drag load that would act on the platform at rest (i.e., with x = 0) is

^u r(t) = Pa /a ^0(7,2) u(z) u'(z,t) dy dz (19)

10
where the subscript r refers to the fact that the platform is at rest. The

Fourier transform of the autocovariance function of F _(t) yields: u , r

^Fu r^""^ = Pa^ / / Cp(yi'Zi) Cp(y2.Z2) uCz^^) u(z2)

a a

C

Su^U2^yi'y2>zi»Z2) dyidy2dzidz2 (20;

C

where ^ is given by equation 16. The spectrum Sp (n) can be estimated

12 u, r numerically by assuming Cp(yj^,Zj^) - (i=l,2) and using equations 4, 16, and Al

(see appendix). An equivalent wind fluctuation spectrum can then be defined as

Sp (n)

^u,eq = "'^ (21)

[p C A u(z )]2

a a a a

From Su^eq(n) it^is possible to generate by Monte Carlo simulation realizations

of the process u (t): eq

Ueq(t) = I u^qj cos(2TTnjt + <^.) (22)

j

In equation 22 the phase angle, , is generated by random sampling from a uniform

distribution in the interval 0 < < Iv.

Let the spectrum of the force, F^^^ j.(t), defined as

t Fequ,r(t) = Pa ^^^e) "eq^^) (23)

be denoted by Sp (n). Clearly,

equ,r

Sp (n) = Sp (n) (24)

equ,r u,r

Thus, Ugq(t) can be viewed as an equivalent wind speed fluctuation which is

perfectly coherent over the area A^ and whose effect upon the structure at rest

is the same as that of the actual fluctuating wind field.

The total wind load acting on the platform can thus be expressed as

^u(t) = I

Pa ^a ^a ^Iq^^^ " ^ ^^^l' ^^5)

11
Numerical calculations have shown that if the difference between the elevation

of the helideck (or the top of the crew quarters) and the underside of the

lower deck in a typical drilling and production platform is less than about twothirds to three quarters of the width of the main deck, the term C^^ (z-^-Z2)^ of

equation 16 can be neglected when evaluating the integral of equation 20. This

is a consequence of the fact that is smaller than Cy by a factor of about 1.5.

The approximation inherent in the neglect of cj^ (z-^-Z2)^ is slightly conservative from a structural engineering point of view (though insignificantly so).

Noting, then, that for any arbitrary function, cj),

1 1 1

/ / <i>(|Yi-Y2|) dYidY2 = / Ht) (1-t) dt (26)

0 0 0

(figure 4), and assuming Cp(yi,Zi) - C^, uiz^) ^ uTz^T, and Su^(n) Su(za,n),

(i=l, 2), it follows after some algebra from equations 20, 16, and 21 that

Su,eq(") =^ Su(Za,n) J(n) (27)

12
where 8^(23,11) = spectrum of longitudinal velocity fluctuations at elevation Zg

given by equations Al, and J(n) = reduction factor accounting for the imperfect

coherence among the fluctuating wind pressures at different points of the

platform, given by the expression

J(n) = -1 {-exp(-E) + (l-l)[exp(-E)-l]l (28)

E E

E = Cy b (29)

In equation 29 b = width of main deck. Equation 27 can be used in lieu of

equations 20 and 21 for the Monte Carlo simulation of the equivalent velocity

fluctuations u^gCt) (see equation 22) needed in the expression for the

total wind load acting on the platform, F,j(t).

13

3. HYDRODYNAMIC LOADS

The total hydrodynamic load, F|^, is written in the form:

Fh = + - Ax - Bx (30)

where = total hydrodynamic viscous force, Fg = total wave-induced exciting

force, A = surge added mass, and B = surge wave-radiation damping coefficient.

Following reference 17, it is assumed for convenience that the wave motion is

monochromatic, hence the absence of second-order drift forces in equation 30

[16]. It is assumed in addition that B = 0 since the radiation damping at low

frequencies is negligible (17, 26).

The total wave-induced exciting force and the surge added mass can be estimated

numerically on the basis of potential theory. Alternatively, these two terras

may be assumed to be given by the inertia component of the Morison equation,

i.e.

,

A - EE Vi-j(Cn, - 1) (31)

ij ij

, 3v . = _ • 9v . .

Fe Pw ^ij . {-il + + - X] (32)

ij ^ ij 3t ax

[18, p. 31], where = water density, V^j = elemental volume of submerged

structure, Cj^ = surge inertia coefficient corresponding to Vj^-j , X = horizontal

ij

distance from some arbitrary origin to center of Vj^^ along direction parallel

to surge motion, Vj^ and v^j = current velocity and horizontal particle velocity

due to wave motion, respectively, at center of j • Equations 31 and 32 may be

employed if for the component being considered the ratio of diameter to wave

length, D/L < 0.2 [18, p. 283]. Since for T^ 15 sec, L = g T2/2Tr - 350 m

[18, p. 283], where T^ = wave period, and g = acceleration of gravity, it

follows that for members of typical TLP structures, for which D < 20 m or so,

the use of equations 31 and 32 is acceptable if three-dimensional flow effects

are not taken into account. The wave motion is assumed to be described by deep

water linear theory, so that

-kz^

Vij =IH e cos (k^X^ - 2lt) (33) w w

where H = wave height , k^ = wave number given by

k„=i.(|l)' (34)

[18, p. 157]. It is recalled that the directions of the mean wind, current,

and wave propagation are assumed to coincide.

15
The total hydrodynamic viscous load is assumed to be given by the viscous

component of Morison's equation

• • = 0.5 (V EE Cd Ap _ |vi + Vij - x| [vi + Vij-x] (35)

ij ij ij

where Ap = area of elemental volume V-[-; projected on a plane normal to the

^ij

direction of the current, and Ca = drag coefficient corresponding to Ar. ij ^ij

If the relative motion of the body with respect to the fluid is harmonic,

the drag and inertia coefficients in Morison's equation can be determined on

the basis of experimental results as functions of local oscillatory Reynolds

number. Re = 2TrD^/(vT£), Keulegan-Carpenter number, K = V T^^/D, and

relative body surface roughness, where D = diameter of body, v = kinematic

viscosity, V and Tf = amplitude and period of relative fluid-body velocity.

However, actual relative fluid-body motions are not harmonic. This introduces

uncertainties in the determination of the drag and inertia coefficients even

if experimental information for harmonic relative motions were available in

terms of Re and K. Unfortunately, such information is not available for the

small K numbers (of the order of 2) and large Reynolds numbers (of the order

of 10^) of interest in TLP design. For this reason calculations will be carried

out for various sets of values C^j, Cm, and the sensitivity of the results to

changes in these values will be investigated.

16
4. RESTORING FORCE

The restoring force, R, at any Instant, t, is equal to the horizontal

projection of the resultant of the tether tensions at that instant, i.e.,

R(t) - - (T+ AT) r— - (36)

^n + ^^n

where T = initial pretensioning force, AT = incremental tension due to surge

motion, = nominal length of tethers at x = 0 Ail^ = incremental length, and

T + ^I_+c [1 - / l-(x/£)2] (37)

Si + M I NL

n n n where Cjjl = downdraw coefficient, equal to the weight of water displaced as the

draft is increased by a unit length [17].

17

5. NUMERICAL ESTIMATES

5.1 EQUATION OF SURGE MOTION

Numerical estimates of surge motion are obtained by solving the equation of

surge motion for structures with specified characteristics in specified wind,

current, and wave environments. The equation of surge motion can be written a

(M + A)x + [I_ + Cf^ (1 - / l-(x/£)2)] X =

0.5 EE C<i Ap I vi + vi-j - X I [vi + vj^^ - x]

. ij iJ ij _

viscous hydrodynamic force, FyCt) [equation 35]

3v. . ij m

ij 3t ij

9v,

x] —1

ax

wave-induced exciting force, Fg(t) [equation 32]

+ 0.5 p^ A^ t^Cz^) + E Ugq^j^ cos(2Trnj^t + (j)^) - x(t)]

wind force, F^Ct) [equations 25 and 22] (38

The equation of motion is solved by using the PORT Ordinary Differential

Equations Solution Subroutine [4]. For details on computational aspects of

differential equation solutions, see references 20 and 21.

5.2 ASSUMED CHARACTERISTICS OF ENVIRONMENT AND OF PLATFORMS

The assumed characteristics of the environment and of the platforms are listed

below.

(a) The current speed , v-^, varies linearly between the values given in table

Table 1. Assumed Current Speeds at Various Depths Zj^

Z£ (meters) v± (meters/second)

0 1.40

35 0.90

300 0.30

550 0.15

19
(b) The horizontal velocities, v-j^-s , associated with the wave motion, are given

by equation 33 in which H = li m and = 15 sec.

(c) The mean wind speed is specified as uCz^) = 45 m/s, unless otherwise

stated,

(d) The equivalent wind speed fluctuation components , ^pqj^ » obtained

by Monte Carlo simulation from the spectrum S^^gq(n), given by equation 27,

unless otherwise stated. In equation 27, the spectrum of the longitudinal

velocity fluctuations, S^Cn), is given by equation Al and A2, where the

friction velocity, u , is obtained from equations 9 and 5, and where,

* unless otherwise stated, it is assumed Cpgg^ = 0.002, L^ = 180 m, 3 = 6.0

(equation 14), f^j = 0.07, and fg = 0,2. The experimental decay coefficient

in the function J(n) of equation 27 (see also equations 28 and 29) is

Cy = 16, However, values of Cpsg^, L^, 3, fni» f s > ^'^^ ^y different from

those just listed are considered in subsequent sensitivity studies,

(e) Geometric characteristics of the platforms are shown in figure 5. Unless

otherwise stated, the nominal length of the tethers is assumed to be

= 590 ra, corresponding to about 600 m depth of water. The case

= 150 m is also considered where noted.

67m

— n —

32ni

\< ^

1

1

1

•4 > — Diam.=18m

Diam.=9in

—n —

Depth: ^ 600m

Figure 5. Geometry of tension leg platform

20
(f) Mechanical characteristics of the platforms are specified as follows: mass

of platform, M = 34.3 x 10^ kg, total tension In legs, T = 1.56 x 10^ kN.

From figure 5 It follows that = 4(itD^/4) = 1.03 x 10 kN/m, where

= diameter of buoyant columns. Note that the values of M and T and the

geometric characteristics of the platform shown In figure 5 are similar to

those assumed In reference 17. This choice was aimed at facilitating com- parisons with results obtained therein. Damping due to internal friction

within the structure Is assumed to be negligible.

(g) Hydrodynamlc characteristics of the structure consist of the Morison

equation coefficients. Unless otherwise stated, these are specified as

= 1.8, C(j = 0.6. (Note that C^, = 1.8 yields the surge added mass, A,

obtained from calculations based on potential theory in reference 17.) In

addition, the sets (a) Cm = 1.9, = 0.1, (b) = 1.9, = 0.2,

(c) = 1.8, C(j = 0.8, and (d) = 1.3, C^j = 1.2 (assumed to correspond

to the case of submerged surfaces roughened by blofouling) are considered

in sensitivity studies. Note that low C^j values have been suggested for

small K numbers in low Reynolds number flow [26]. Corresponding values in

the high Reynolds number regime do not appear to be available in the

literature.

The nominal natural period of the surge motion. In seconds, is given by

(M + A)

T = 2TT [ EL] (39) n "p

where i^i ~ nominal length of tethers. For the platforms with ^.^i ~ ^

and Jlfi = 150 ra, if = 1.8, = 103 sec (as in reference 17) and

Tj^ = 52 sec, respectively.

(h) Aerodynamic characteristics of the structure consist of the product 0.5 Pg

C^A^ (equations 17 and 19), and the elevation of the aerodynamic center, z^,

The values specified for these parameters are 0.5 p^ C^ A^ = 2,700 kg/ra, as in reference 17, and = 50 m. In addition, the value = 35 m is

used to investigate the influence of Zg upon the calculated surge.

5.3 LINEAR VS. NONLINEAR RESTORING FORCE

To assess the effect of neglecting the nonllnearlty of the restoring force,

solutions of equation 38 were obtained In which, all other parameters being the

same, the cases C-^ = 1.03 x 10^ kN/m and = 0 were considered. In both

cases it was assumed that ^gq(t) (see equation 22) Is given by the harmonic

function

u' (t) = 5.9 cos t (40)

Tn

in meters per second, where = 103 sec. It is seen from figures 6 and 7 that

if the restoring force is assumed to be linear i.e., if it is assumed Cjjl ~ 0>

the surge response is amplified with respect to the nonlinear case by about 7

21
22
23
percent. This is comparable with results reported in reference 17. Once the

approximate magnitude of these differences was established, all estimates of

the surge response reported in reference 17 were based on the assumption that

the restoring force is linear. To facilitate comparisons with the results of

reference 17, the same assumption is used henceforth in the present work.

5.4 NOMINAL DAMPING RATIO OF PSEUDO-LINEAR SYSTEM EQUIVALENT TO EQUATION 38

It was indicated previously that the estimation of the effect of turbulent

wind on surge was attempted in the literature on the basis of the assumption

that the equation of surge motion represents a linear system with a viscous

damping terra characterized by a nominal damping ratio, ^. The effect of this

terra is postulated to be equivalent to the damping effect of the hydrodynamic

viscous force. This approach was used, for instance, in references 6 and 28,

where calculations are presented based on the assumption that ^ is of the

order of 5 percent.

Such an approach is acceptable if the order of magnitude of the nominal

damping ratio is consistent with the hydrodynamic behavior of the system. Calculations are now presented that illustrate how such nominal damping ratios can

be estimated.

t It is assumed that Ug^Ct) (see equation 22) is given by the harmonic function

• - ' 2_Tf Tn where T^^ = natural period in surge, and that the systera under consideration is

linear with mass M + A, natural period T^, and damping ratio The amplitude of the contribution to the surge response of a harmonic force A^ uCz^)

u„„i cos 2Trnt is denoted by x,„^^, and is given by the relation: eqi Umax' ^ •'

Pa u(z^) u^^^

(M+A)(2^)^ {[l-(nTn)^]^ + (25nTn)2}^^^

n

(42)

The nominal damping ratio, is estiraated from equation 42 by equating x^^^^

and the contribution to the surge response of the force A^ ^(^3)

"equl ^® obtained by solving equation 38. By substituting 1/T^

T^n

for n in equation 42, it follows that

oaaa p C A u(z^a') u' 1 eql

C = f -—L, (43)

X (M+A) (2l)2 Umax

n Calculated values of C are shown in table 2 for the platforra described

previously with norainal tether lengths = 590 m and 150 m. These values are

24
based on the assumption uCz^) = 40 m/s , u'gq^ = 4 m/s. For example, In the case ~ 590 m, Cj^ = 1.8, = 0.8, equation 38 yields x^^gJ^ 2.5 ra (i.e.,

half the double amplitude of the low frequency oscillation in figure 8). Since

M+A = 71.1 X 10^ kg, T„ = 103 sec, and 0.5 = 2,700 kg/ra, the estimated LI cl d cL nominal damping ratio in this case is ^ = 0.65. Results are given in table 2 for each of the five sets of values Cj^, C^j listed previously. Also shown in

table 2 are calculated values of the steady and peak surge response in meters.

It is seen that as the damping coefficient, C(j, increases, both the steady and

the peak response increase, even though the contribution to the response due to

wind speed fluctuations decreases as a result of the stronger hydrodynaraic

damping.

It is seen in table 2 that the estimated damping ratios are considerably

larger than assumed, e.g., in references 6 and 28. Large values of ^ indicate

that the damping inherent in the viscous hydrodynamic forces is sufficiently

strong to preclude the occurrence of significant resonant amplification effects

under fluctuating wind loads. Note that results similar to those of table 2 are obtained from the surge response calculations reported in reference 17.

Table 2. Estimated Nominal Damping Ratio, ^, and Mean and Peak Surge

Response x and Xp]^, in Meters

^n = 590 meters ^n = 150 meters

Cm Cd

X

meters

Xpk

meters

X

meters

Xpk

meters

1.9 0.1 0.25 25. 36.0 0.10 6.0 15.5

1.9 0.2 0.30 28.5 38.0 0.15 7.0 16.0

1.8 0.6 0.55 34. 43.0 0.20 9.0 17.0

1.8 0.8 0.65 37. 44.5 0.30 10.0 18.0

1.3 1.2 0.80 46. 56.0 N.C.a N.C.a N.C.a

3 N.C. = not calculated

The results of table 2 suggest that, unless biofouling effects are significant.

It is reasonable to assume for the purpose of estimating peak surge response

that C(j = 0.6, Cjjj = 1.8. This set of values is used subsequently in this

paper for the estimation of surge response under turbulent wind loads.

25

5.5 PEAK SURGE RESPONSE UNDER TURBULENT WINDS

The equation of surge motion (equation 38), with the environmental, geometric,

mechanical, hydrodynamic , and aerodynamic parameters specified previously, was solved for fifty different realizations of the random process u'gq(t)

(equation 22)_in each of the following cases: = 590 m and = 150 m. The

sample mean, Xj^^x* the sample standard deviation, s(x^3x)> 3"*^ the sample maximum Max(xjj,3x)> of the peak surge response, x^^j ^'^^ listed in line (1) of

table 3 (corresponding to L^ = 180 m). A typical solution of equation 38 is

shown in figure 9 for the platform with = 590 m (T^ = 103 sec), and

Lu = 180 ra.

Table 3. Statistics of Peak Surge Response, yi^g^y^y Under Turbulent

Winds (in meters)

^n = 590 m (Tn == 103 sec) ^n = 150 m (Tn = 52 sec)

^ ( ^raax

)

Max(xniax) ^ (^raax

)

^raax ^max Max(xniax)

(1) Lu = 180 m 48.7 1.19 51.4 18.3 0.40 19.7

(2) Lu = 100 m 48.1 0.64 49.9 18.2 0.44 19.5

(3) Lu = 240 m 48.9 1.13 51.7 18.5 0.50 19.8

5.6 SURGE RESPONSE ESTIMATED UNDER VARIOUS SIMPLIFYING ASSUMPTIONS

It is of interest to estimate the surge response that would occur if the

platform were subjected to forces induced by:

Case A: Mean wind alone

Case B: Current and waves alone

Case C: Current, waves, and mean wind alone (no wind speed fluctuations)

Case D: Current, waves , mean wind , and harmonic wind speed fluctuations

Ueq(t) = 0.1 M{Zg) cos (2Trt/T^)

Case E: Current, waves, mean wind, and harmonic wind speed fluctuations,

u^n(t) = 1.4l/3u cos2ti t/T„) [recall that /gu = r.m.s.

* " * of turbulent wind speed fluctuations - see equation 14].

Case F: Current, waves, and steady (1-minute) wind speed [No wind speed

fluctuations]. (Note that 1-minute speed - 1.24 u(za), see reference 23, p. 62.)

27
<i:

I—

I

3

Ht—

I

1—I—I—

r

CD

too

o

CO

•H

c

-H

O

4-1 S 03 O

14-1

Oi 4-1 bO 0)

(U U

r^ !-i

o

•rH U-(

Cfi O

c

tU 0)

4-> O

o w

0)

0) S-i CO P- do cu o- ^

«! 4-)

!-i C

•H bOU n3 S O

CO iH

28
The results of the calculations are shown in table 4, which lists for each

of these cases the peak surge, as well as the steady surge, the amplitude of

the wave-induced fluctuating surge, and the amplitude of the wind-induced

fluctuating surge.

The results of tables 3 and 4 show that:

- The contribution of the mean wind to the peak surge response is about

40 percent and 25 percent for the platforms with ~ 590 m, and

= 150 m, respectively (case A, table 4, versus Max(mj^a5j), line (1),

table 3).

- The contribution of the wind speed fluctuations to the peak surge

response is about 12 percent for both values of £^ (case C, table 4, versus Max(xjjjg-j^ ), line (1), table 3).

- The contribution of current and waves to the peak surge response is

about 50 percent and 60 percent for the platforms with = 590 ra and

^j^ = 150 m, respectively (case B, table 4, versus Max(xn,ax)» table 3).

- Representing the turbulent wind speed fluctuations by a harmonic

fluctuation with amplitude equal to one-tenth of the mean wind speed,

as done in reference 17, (case D), results in an underestimation of peak

surge response by 5 percent to 10 percent. Calculations were carried

out which showed that, in order for harmonic and turbulent wind speed

fluctuations to be equivalent in terras of surge response, the amplitude

of the harmonic wind speed fluctuations should be about twice the r.ra.s. of the turbulent wind fluctuations.

Table 4. Estimates of Surge Response Corresponding to Various

Deterministic Loading Assumptions (In meters)

Case

^n = 590 m ^n = 150 m

Steady

WaveInduced

Fluct.

WindInduced

Fluct.

Total

(peak) Steady

WaveInduced

Fluct.

WindInduced

Fluct.

Total

(peak)

A 20.5 20.5 5.3 5.3

B 18.0 6.5 24.5 4.5 7.5 12.0

C 39.0 6.0 45.0 10.0 7.5 17.5

D 39.0 5.0 5.0 49.0 10.0 6.0 2.0 18.0

E 39.0 5.0 5.5 49.5 10.0 6.0 2.5 18.5

F 50.0 6.0 56.0 13.5 6.5 20.0

29
- Accounting for the contribution of the wind speed fluctuations to the

peak surge response by assuming UgqCt) = 0 and replacing uCz^) In

equation 25 by the one-minute wind speed, as done in references 11 and

18 (case F), results in an overestimatlon of the total peak surge by

about 10 percent for the platform with = 590 m, and 1 percent for

the platform with = 150 m.

5.7 SENSITIVITY OF SURGE RESPONSE TO CHANGES IN VALUES OF WIND ENVIRONMENT AND

AERODYNAMIC PARAMETERS

This section examines the effect upon the surge response of changes in the

values of the wind environment parameters L^, fjj,, fg, and Cy, and Cpgg^

(or Zq - see equation 5), and of the aerodynamic parameter z^.

The effect of the magnitude of the integral turbulence scale, L^, is

investigated by estimating the surge response under the same assumptions that

led to the results of line (1) of table 3, except that in lieu of = 180 m

the values = 100 m and = 240 m are used. Again, for each of the cases

= 590 and Jl^ ~ 150 m, equation 38 was solved for fifty different realizations of the random process Ugq(t). The results are given in lines (2) and

(3) of table 3. It is seen that the statistics of the peak response are not

affected significantly by L^; the sample maxima, Max(xnj3x)» differ in both

cases by 3.5 percent or less when the integral scale of turbulence increases

by a factor of 2.4.

An inspection of figure 2 and 3 shows that the influence upon the spectral

shape of the nondimensional frequency, f^, at which nSy(n) is maximum (see

equation Al), is considerably weaker than the influence of L^. Hence, it can

be expected that the influence of f^j on the surge response is minimal. Solutions of equation 38 in which the values f^ = 0.04 and f^ = 0.10, in lieu of

fjn = 0.07, were used, all other parameters being unchanged, showed that this is

indeed the case.

The nondimensional frequency fg (see equation Al) affects the higher frequency

portion of the spectrum, which has no perceptible influence on the surge

response. Hence, even large variations of fg (between 0.1 and 1, say) are inconsequential as far as their effect on surge is concerned.

It is well known that the average size of the longitudinal turbulent eddies

in the atmosphere is considerably less in the transverse than in the longitudina

direction [22, 23]. For this reason < Cy, the ratio C2/Cy being about 0.6

or less. It was pointed out previously that for typical platform shapes this

results in the effect of being negligible with respect to the effect of Cy

in equation 16 and 20, so that equation 27 through 29 may be used in lieu of

equation 21. If the value Cy = 24 in lieu of Cy = 16 is used, the values of

jl/2(n) in and near the range of frequencies where the energy of the wind

speed fluctuations is large (0 < n < 0. 5 or so - see figure 2) decrease as shown in table 5.

30
Table 5. Dependence of J-'-''^(n) Upon Exponential Decay Coefficient, C

n 0.01 0.02 0.05 0. 1

16, n) 0.95 0.91 0.79 0. 68

24, n) 0.93 0.87 0.73 0. 59

It is seen that the influence of large changes in the value of C upon J^^^(n)

in this frequency range is on the average of a few percent. It follows that

the influence of such changes on the peak surge response is negligible for

practical purposes.

The choice of value t-^psea effects both the mean wind profile and the spectrum

of the longitudinal velocity fluctuations. To Cpgg^ = 0.002 and 0.003, for

example, there correspond values Zq = 0.0013 ra and 0.0067 m, respectively

(equation 5). Based on equation 4, the respective calculat ed mean speeds at

z = 20 m and z = 50 m above mean water level, given that u(35 m) = 45 m/s, are listed in table 6.

Table 6. Dependence of u(z) Upon Zq (u(35m) = 45 m/s)

Elevation, z 20 m 50 m

u(z) for Zq = 0. 0013 m 42.4 m/

46.5 m/s

u(z) for Zq = 0. 0067 m 42.0 m/ s 46.8 m/s

The influence of Cpgg^ is more significant in the case of the spectra,

S (n), which are proportional to u^ (equations Al). To differences between

values of C^gg^ of the order of 50 percent there correspond differences in

the magnitude of S^^'^(n) of about 20 percent. Ov;ing to nonlinearity effects,

the differences between the corresponding contributions of the wind speed

fluctuations to the surge response are somewhat less. Since these contributions were previously estimated to be of about 12 percent of the peak surge

response, differences between C-qqqq^ values of the order of 50 percent result

in differences between the corresponding values of the peak surge response of

less than 3 percent.

To summarize, although the uncertainties with respect to the parameters

defining the wind environmental are large, the effects of these uncertainties

upon the peak response translate in all cases into differences of the order of

a few percent.

31
The aerodynamic parameter used in the calculations presented so far was specified as = 50 m, as indicated previously. Lower values, e.g. z^ = 35 m,

may be encountered in typical practical situations. Calculations based on the assumption = 35 ra, all other parameters being unchanged (including the speed

uCz^) = 45 m/s), have shown that the peak surge response is smaller by less

than 2 percent than the corresponding response based on the value z^ = 50 m.

32
6. SUMMARY AND CONCLUSIONS

This paper presents a procedure for estimating surge response to the action

of turbulent winds in the presence of current and waves. The procedure accounts

for the nonlinearity of the hydrodynaraic forces and the coupling of aerodynamic

and hydrodynamic effects.

It is demonstrated that, unlike wave spectra, wind spectra peak at zero frequency, where they are proportional to the integral scale of turbulence.

An expression for the wind spectrum consistent with this requirement is derived,

and a critique is presented of current expressions in which this requirement

is violated. It is shown that such expressions may lead to the gross underestimation of wind speed fluctuation components with frequencies comparable to the

natural frequencies of compliant platforms.

A simple expression is derived that accounts for the imperfect coherence of

fluctuating wind pressures acting at different points on the surface of the

platform.

For illustrative purposes the procedure for estimating surge response is applied

to typical tension leg platforms (TLP's) for water depths of about 600 m and

150 m. Solutions of the equation of surge motion are obtained, which suggest

that for these platforms the hydrodynamic damping is sufficiently large to

preclude the occurrence of significant resonant amplification effects due to

wind loads, even if the assumed drag coefficients in Morison's equation are as low as C(j = 0.1. Equivalent linear damping ratios with respect to wind effects

were estimated to be of the order of 25 percent to 55 percent corresponding to

C(j = 0.1 and C(j = 0.6, respectively, in 600 m deep waters, and 10 percent to

20 percent for = 0.1 and C^j = 0.6, respectively, in 150 m deep waters. The

calculations show that as the viscous damping due to hydrodynamic effects

increases, the peak total surge response also increases, even though the contribution of the fluctuating wind effects to the total surge is reduced. The use

of a Morison equation drag coefficient C^j = 0,6 appears to be reasonable for

the purpose of estimating peak total surge.

Parameters defining the wind environment are reviewed, and it is shown that

uncertainties with respect to the actual values of these parameters have little

effect on the estimated peak surge response. However, uncertainties with

respect to the sea surface drag coefficient (or equivalently , to the roughness

length parameter for the sea surface) may cause differences of the order of

20 percent in the estimated contribution of the wind load fluctuations to the

surge response.

For the platforms considered in this work, and under the assumption that the

drag coefficient in Morison's equation is C^j = 0.6, a fictitious harmonic wind

speed fluctuation and the actual turbulent wind speed fluctuations are equivalent from the standpoint of their contribution to the total surge if the amplitude of the harmonic fluctuations is equal to about twice the r.m.s. of the

turbulent fluctuations. Finally, it is concluded that, under the same assumption, the use of a one-minute wind speed to represent the effect of the mean

wind and of the turbulent wind fluctuations is acceptable for the purpose of

estimating peak surge response.

33

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35
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,

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36
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37

APPENDIX: EXPRESSION FOR THE SPECTRUM OF THE LONGITUDINAL VELOCITY FLUCTUATIONS

n Sy(z,ii) C2 + 32-^ ^^2^

0.26 f

2/3

— in

<

f < fs

where u is given by equations 9 and 4, f = nz/u(z), and

(Ala)

(Alb)

(Ale)

^1

=

h

4 L^ 3

= 0.26 f, -2/3

bo =

f^V^s) -^|(fm-fs) + 2f^(fs-fm) + ^s s'^fn.)^^

.fs

(A2a)

(A2b)

(A2c)

a2 = -2 b2 f m

di = ^ [-^ -

3i + b2 (f^ - fs)^

:3 2 m

(A2d)

(A2e)

bi =

^^2 =

-a- 2f

-1.5 f^d

m

m°l

^2fs -^2fs

(A2f)

(A2g)

U^. GOVERNMENT PRINTING OFFICE . 1983 0—380-997 (908)

39
NBS-n4A (REV. 2-8C)

j U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publ ication Date BIBLIOGRAPHIC DATA

REPORT NO.

March 1983

1 SHEET (See /nstruct/onsj NBS BSS 151

4. TITLE AND SUBTITLE

Turbulent Wind Effects on Tension Leg Platform Surge

5. AUTHOR(S)

Emll Simiu and Stefan D. Leigh

6. PERFORMING ORGANIZATION (If joint or other than NBS. see /nstruct/onsj 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State, ZIP)

Minerals Management Service

U.S. Department of the Interior

Reston, VA 22091

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 83-600507

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached. H. ABSTRACT (A 200-word or /ess factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

A procedure is presented for estimating surge response to turbulent wind in the

presence of current and waves. The procedure accounts for the nonlinearity of

the hydrodynamic forces with respect to surge and for the coupling of aerodynamic and hydrodynamic effects. It is shown that current wind spectra do not

model correctly the wind speed fluctuations at very low frequencies and an alternative model of the wind spectrum, consistent with fundamental principles,

is presented. The equation of surge motion under turbulent wind in the presence

of current and waves is solved for typical tension leg platforms, and it is

shown that under extreme wave conditions the damping provided by the hydrodynamic forces precludes the occurrence of significant wind-induced resonant

amplification effects even if the drag coefficient in the Morison equation is

very small (e.g., C^j = 0.1). It is verified that for the platforms being

investigated the use of a one-minute wind speed to represent the effect of the

mean wind and of the turbulent wind fluctuations is acceptable for the purpose

of estimating peak surge response.

12. KEY WORDS (Six to twelve entries; o/phabet/co/ order; capitalize only proper names; and separate key words by semicolons)

compliant platforms; ocean engineering; offshore platforms; structural engineering;

tension leg platforms; turbulence; waves; wind loads.

13. AVAILABILITY

[3c] Unlimited

I I For Official Distribution. Do Not Release to NTIS

fx] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

46

15. Price

$4.75

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