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Linearization methods and the influence of current on the nonlinear hydrodynamic drag force
Linearization methods and the influence of current on the nonlinear hydrodynamic drag force OVE T. GUDMESTAD Senior Engineer, Statoil, Stavanger, Norway JEROME J. CONNOR Professor of 6)'vil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Key Words: Linearization of relative velocity terms, hydrodynamic damping, nonlinear expansion of the force, influence of current, sinusoidal design wave and stationary Gaussian wave excitation.
INTRODUCTION With the trend toward deployment of conventional fixed offshore structures in deeper water, the question of dynamic amplification becomes more important since the fundamental natural frequency, 6oN, of deep water structures is closer to the dominant wave frequency. For piled steel platforms in 150 m water depth, typical value of WN are 3-4 rad/s. Similar platforms in 300 m water depth have fundamental natural frequencies in the region of 1.5 rad/s, which is approximately three times the peak frequency, coo, of the wave excitation. Morison's formula is generally applied to evaluate the hydrodynamic forcing, and an equivalent linearization technique 1 is used to linearize the forcing function in the frequency domain. Thus, terms emerging from nonlinear drag force having frequencies 2wp, 36Op, etc. are not accounted for in a frequency domain analysis.2 However, for deep water structures, these terms may increase the force spectral density near the fundamental natural period of the structure and thereby increase the response of the structure considerably, a, 4 This paper is concerned with procedures for linearizing the relative motion term in the forcing function. As a first step, the approach proposed by Blevins s for a sinusoidal wave is extended here to a general stationary Gaussian sea. It is shown that the procedure leads to lower values for hydrodynamic damping both for a sinusoidal wave as well as for a Gaussian sea state. A combination of sinusoidal waves and a current U is also examined. The effect of including a realistic North Sea current when calculating the nonlinear drag force terms is illustrated. Procedures appropriate for frequency domain analysis, allowing for a current coupled with a stationary Gaussian sea state, are considered next. An expansion of the drag force derived from a least square approximation is used to generate the spectral formulation of the force. It is shown that this method yields the same results as obtained by Borgman 6'7 directly from the autocorrelation function and cross-correlaion function. However, the expansion approach is less complex and the physics of the problem is more easily understood since the effects of the nonlinear forcing terms are evaluated directly.
Received November 1981. Discussion closes December 1983.
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Applied Ocean Research, 1983, Vol. 5, No. 4
An interesting aspect of this study is the application to realistic North Sea environmental conditions defined by a Jonswap spectrum and a current profde typical of that area. A narrow-banded white noise wave velocity spectrum is also investigated. The inclusion of current in the examples is believed to represent a new application of nonlinear spectral density calculations.
LINEARIZATION OF THE HYDRODYNAMIC DRAG FORCE General expression For short-term intervals, of the order of a few hours, the water surface elevation ~(t) at a fixed location in the sea can be approximated as a stationary process. In this study, r/(t) is considered to be a non-zero mean, Gaussian, stationary process and is represented by a linear summation of an infinite number of sinusoids with phase angles randomly distributed between 0 and 2rr. The waves are assumed to be unidirectional and straight crested, and linear wave theory is used to relate water particle motion at any depth to the fluctuating surface elevation. It is also assumed that the total hydrodynamic force per unit length acting on a cylinder of diameter D is given by Morison's equation: P(t) = p (CM -- 1) 47r D 2(u(t) -- 5(t)) + p ~_D2//(t)
+ ½pCDDIf~(t) -- zS(t)l(u(t) -- zS(t))
(1)
where CM, Co are the inertia and drag coefficients consistent with the flow and structural response; ti,/J are the water particle velocity and accelerations; v is the displacement of the structure. One of the uncertainties influencing the response is associated with the values for CM and CD which exhibit a significant variation with Reynolds number (Re), Keulegan-Carpenter number (K) and relative roughness (k/D). However, since the interest here is on different linearization techniques for the nonlinear term in equation (1), CM, CD, are considered to be constants.
Linearized force expression Unfortunately, P(t) is nonlinear with respect to the drag term. When the wave amplitude is much greater than the maximum displacement of the structure: rl(t) ~ v(t)
(2)
0309~1708183/040184-11 $02.00 © 1983 CML Publications
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T Gudmestad and J. J. Connor
[_
the following linearization s can be employed:
1(0
/4+1
0
2
It i ( t ) - z~(t)l( t i ( t ) - z~(t)) = Iti(t)- r)(t)lfi(t)- Iti(t)- ~)(t)l ~)(t) It~(t)lti(t)-I ti(t)l ~)(t)
(3)
Two approximations have been introduced: 1. Setting lit(t)--i;(t)lft(t)= Iti(t)lti(t) introduces an error in the forcing term which diminishes as the amplitude of the flow velocity becomes much greater than the velocity of the structure. 2. The term Ift(t)--i~(t)li)(t) represents fluid damping of the structure. Taking lu(t)--i)(t)li~(t)= Ifi(t)l~)(t) introduces an error in the estimate of damping which also diminishes as the amplitude of the flow velocity becomes greater than the velocity of the structure. For the design wave condition, the dominant wave frequency is usually much smaller than the natural frequency of the structure, and (2) is quite reasonable since the structural response is quasi-static.
Estimate of Ifi(t)ltJ(t) An approximation for It~(t)lfi(t) can be generated by expanding in a power series: Ifi(t)lt~(t) ~ Co + el/~(t) 4- c 2 (tJ(t)) 2 + . . . + Cn(fi(t)) n (4) and applying a weighted residual procedure to evaluate the coefficients. Since u(t) is a random variable, the weighted residual, R, is the expected value of the weighted error measure, we: e = It~(t)la(t)--Co --cl u(t)--c2 (ti(t))2--...--Cn(U(t)) n
R=E[co]e]=O
j=O, 1,2 . . . . . n
(5)
where co] are weighting functions. Taking col = (it( t))] corresponds to a least square approximation and leads to the following equations:
+
2
+
0
2
tow(0] ]
x = a(t)/on(t)
10loa(t) qb = 2 x ~ n
J
exp [-(t=/2)] dt = 2 erf(O/oa(t) )
0
~=~
exp - 2
Equation (7) reduces ,to Borgman's 7 result when O is set equal to 0, The method can also be extended to the case where the excitation is. not Gaussian. One replaces the Gaussian probability density function (p.d.f.), with the actual p.d.f, for the excitation in equation (6).
Estimate of Iti(t)lz)(t) The fluid damping coefficient is approximated by Iti(t)l. In the case of a Gaussian excitation with mean U, the most accurate least square estimates are: Iti(t)l~
__
qb+2¢
°a (0 {2~b} + {cb} x
1o- (±t+
c o E [ # ] + c,E[u j+l] + . . . + c,E[ui+,]
(8)
= E[Itilui +1]
+,~
x/%oa(0 o~(t)
] ida
(6)
Results for up to fourth order expansions are listed below. Details pertaining to the evaluation of the expectation terms are presented in the Appendix. lu(t)lti(t)__ Ixlx ~ 2
--
~--2
4~
oa(t)
~b + {4~b}x + {~} x 2
~ {-~3 (o~(t))a ¢} +{ 2 [(o~t,~+l]~} x +
~-2
The "constant' approximation is used in this analysis. For zero mean excitation, it corresponds to that which would be generated by a steady flow of velocity ~ aa (t)Blevins s has shown that this approximation for fluid damping yields good agreement with the 'exact' nonlinear solution of a one degree of freedom system subjected to a sinusoidal design wave when the structural displacement is much less than the fluid displacement. For large structural displacement, numerical integration in the time domain is required in order to account for the nonlinear damping in an accurate way.
Comparison with other methods An equivalent linearization technique proposed initially by Malhotra and Penzien 1 for the zero mean case and later extended by Wu a to allow for current is based on the linear approximation:
I \oa(t)! --2
(7)
where:
~ x2+{-~}x a
Ift(t)--i)(t)l(ft(t)--i;(t)) ~ ao +al(fi(t)--i)(t))
(9)
The approach presented in this paper works with separate expansions for the coefficients of tJ(t) and z)(t), and the linearized version has the following form:
Iti(t)--i)(t)l(ffft)--iJ(t)) ,~ Iti(t)lti(t)- It~(t)lT)(t) bo + blu(t)--Coi)(t) Applied Ocean Research, 1983, Vol. 5, No. 4
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185
Linearization methods and influence of current on nonlinear hydrodynam& drag force: O. T. Gudmestad and J. J. Connor where bo, b~ are defined in (7) and Co is given by (8). Wu 8 assumes/~(t) is Gaussian when evaluating the expectation of the weighted residual. As will be shown later, the nonlinear drag force represents a non-Gaussian excitation, and therefore the response is not Gaussian. One finds that ao = bo, a l = b b and co=bl/2. The hydrodynamic damping coefficients differ by a factor of two. Time history simulation may resolve this issue.
ESTIMATE OF NONLINEAR DRAG FORCE WITH SINUSOIDAL WAVE LOADING AND CURRENT
Linear wave theory Before considering random sea-states, the extreme force calculation for a steady-state sinusoidal wave is first reviewed. The free surface elevation r/(x, t) is given by: n(x, t) = A cos (kx -- cot)
(11)
where A is the surface wave amplitude and 0 = k x - c o t is the wave phase. In linear gravity wave theory, the wave number k and frequency co are connected by the dispersion relation:
co2 = kg tanh kh
(12)
where g is the acceleration of gravity and h is the water depth. Also, the water particle velocity it(x, t) at depth --z is determined by superimposing the current and wave induced components: i t ( t ) = U+AGcosincot = U+Urn sin cot
(13)
where 0 is the steady current and G defines the variation with depth:
G = coshk(x+h)/sinhkh
--h
(14)
Linearization o f nonlinear drag force When ( i < Urn, the Fourier expansion for the drag lit(t)lit(t) has the form: ld(t)lit(t) = -
U= +
n +--
2V+4UUrn
2i
~
2 (sinV(2--j)
)
Utide(Z): Vtide(Z=o)(h+ztl'7 t h /
{
\
0
ho I
(19) for 0 t> z >1 --ho for z < --ho
C1
2.0
JJ cosScot
4 >-cossV l
n i : l , 3 , s ....
- vL
0---- Utide(Z) + Uwind(Z)
cos V(1 + ] ) ]
5+j
~sinV(1--j)
(18)
where ci(i = 0, 1,2 . . . . ) are functions of (l/Urn. The significance of current relative to the wave excitation is shown in Fig. 1 which contains plots of ci(i = O, 1, . . . . 4) vs. O/urn. A s t h e currentincrease-sw~(fi- respec{ to Urn, the forcing terms with frequencies co, 26o become more important while the terms with frequencies 360 and 5co approach zero. The magnitudes of the forcing terms with higher frequencies decrease with increasing j. However, these forces can excite the natural frequency of the system and it may therefore be of importance to keep them in the calculation o f the system response. This is especially true for flexible structures in very deep water. As an example, the Cognac platform installed in 1025 ft of water has a natural frequency ~ 1.3 rad/s (Fig. 2). Typical hurricanes in the area give dominant wave frequencies ~ 0.5 rad/s. 9 Thus, the natural frequency is ~ three times the dominant wave frequency. The relative importance of current versus the wave velocity depends upon the depth and should be considered when analyzing a particular structural member. Both the current 0 and the wave velocity amplitude Urn have a maximum at the surface. However, Urn drops exponentially with depth as given by (14) while 0 decreases much slower with depth. Det norske Veritas m proposes the following variation of 0:
Uwind (z) =
-- sinj V
+
=+
+ 40Urn\
+c3sin3cot+c4 c o s 4 w t ...}
sin2V
sinV(2+])i 1
-u;.
E
Ifi(t)lit(t) = U2m{Co + c1 sin cot + c2 cos 2cot
2 I ]
(cos V(l --j) + 4UUm \ -17j
1
~m
2
02+
7/'/=2,4, 6 ....
+-
cosV--
Equation (15) specialized for pure sinusoidal flow reduces to the expansion obtained by Blevins. s As a point of interest, the least square approximation based on the probability density function for a sinusoidal wave gives an identical result. Equation (15) can be written as:
sin
~
V(1 + j ) ] 1-+i
]
V(2 +i)'~l cosy(z-i) v cos gUs ljsinScot 2-7
(15)
, 0"
i
i
I iJ i.
where : sin V = U/Urn
(16)
If O ~> Urn, it(t) > 0 for all t and it follows that: lit(t) lit(t) = it2(t) = 02 + 2UUrn sin cot + U2m sin 2 cot
186
Applied Ocean R&earch, 1983, Vol. 5, No. 4
(17)
Figure 1. Plot o f Ci VS (i/Urn showing relative importance o f current in the expansion of lit(t)]fi(t) in the case of a sinusoidal wave
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor where ho = reference depth for wind generated current = 50 m, and z = distance from still water level, positive upward. Using a typical design wave for the North Sea: u
500
H = 2A = 30 m T = 14s H s = 40.3
400
ft.
X = 306m using linear (Airy) wave theory; and evaluating (13) leads to: 300
< w
Umax = 6.732
U
k
I 20O
% r~
100
0.5
1.0
2.0
I .5
RAD/SEC
Figure 2. Natural frequency and wave spectrum for the Cognac Platform in the Gulf of Mexico. From Sterling et al.9
cosh k(z + h) sinh kh
- Urn
(20)
The variation of lima x with z for two water depths, 150 m and 300 m, is shown in Figure 3. These depths are typical for petroleum fields under development and deep water findings in the North Sea, respectively. Also plotted is the variation of current 0 using (19) and typical data from the Norwegian Continental Shelf: Utiae(z = 0) = 0.5 m/s and Owind(Z = 0) = 0.5 m/s} ° The current velocity in the wave crest above the still water level is chosen equal to the velocity at the still water level, a° Lastly, the variation of U/Urn is generated. The forcing function is proportional to U~m (see (18)). Due to the slow variation of the resonant terms with U/Urn (Fig. 1), each resonant term will have its maximum at the surface. Note that c2 is of the same order o f magnitude as co for this example. Treatment of the damping term proceeds in a similar manner. Assuming linear total damping corresponds to retaining only the first term in the Fourier expansion of
It~(t) l: .15
2 [fi(t)l = - ( U V + Um cosV) 7r
Crest° -15
where V is defined by (16). For zero mean wave excitation, (21) reduces to:
-30
~
-60
~
Figure ~a
A
Plot_o~
2 Iti(t)l = - Urn ~r
-
u m, u
c~ "~
-90
-12
-15I
Figure 3[a).
+15 Crest o
i~--
/_! ~-
" 0,5
The total damping (fluid damping +structural damping) for a single degree of freedom system has been studied by Blevins. s Correct representation of the fluid damping term is essential in order to obtain an accurate solution for forcing near the resonant frequencies.
~/U. (bottom scale)
]I
I 1,5
I 2
]L
U(m/s)
and U/U m
Plot o f U m, Oand U/Umfor150m w.d. 1
2
3
4
5
6
7
8
(22)
9
10
Structural response Our starting point is the equation for the response v(t) of a spring supported structure of mass mo in an oscillating flow with mean 0: mo~)(t) + 2mo ~'o 6Ooz~(t) + kv(t) = e(t)
-12{
g -IB(
7[
-24C
m =mo+p(CM--1)-D 2 4
;
Figure 3(b ).
(23)
where ~o = structural damping, k = linear stiffness, and P(t) is considered to have the same form as (1). The value, U/Urn = 0.15, is used here since it is representative of North Sea design wave conditions. Applying the linearized force expansion specialized for a sinusoidal wave (15), (21), transferring the added mass and hydrodynamic damping terms to the left side and introducing:
-6o
-30(
(21)
I • o,s
I
l,s
12
~ ~Cm/s) and 6/Um
Plot of Urn, U, U/U,,for 3OOm w.d.
~o x/1 + pA(CM -- 1)/mo
Applied Ocean Research, 1983, Vol. 5, No. 4
187
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor
(oo+]( Vm1
+ 0.159 \ - - - ~ / [ ~ N D ! Co
(24) Nonlinear
I ~
7[
A =-D 2 4
]I tu
I
+ ......... I
/~
N :°'~)5
1%(N
= k/m ~4
leads to:
m~(t) + 2m~wcoNO(t) + k~v(t)
/,I,
%
+
1 2 o +C 1 sin cot = pACMUrncocosco t + ~PCDDUm{c
0.05 LinearSoIution
,I
o
!j.
If
', u
=
/
\\,\
/i
+ c 2 cos2cot + c3 sin 3cot +c4 cos4cot +Cs sin 5cot + ... }
(25)
where: Co = 0.192, cx = 0.877, e2 = --0.126, ca = - - 0 . 1 6 0 , c4 =--0.034, Cs = - - 0 . 0 1 9 After expanding the displacement v(t) in Fourier series, the solution is found to be: - - = rl D
o
0.5
1,0
1.5
"N
Figure 4. Typical dynamic in-line response of a spring supported cylinder to an oscillating flow with mean 0 = O.15 Urn. (Two different values of damping, ~N, are chosen.)
IDll cos (cot--C1
+ r2 {Co + cl IDd sin ( c o t - ¢ 0 + c2 ID2I cos (2cot-4~2) + ca ID31 sin (3cot --¢3) + c, ID4I cos (4cot -- ¢4) + cs [D4I sin (5 cot -- Cs) }
(26) where: 1
I,
(,7)
60N/
quency, the structure resonates with a component of the fluid force. The resonant motion increases with decreasing damping. For low values of co[coN the response of the structure approaches the static response. For co/coN>> 1, the forces cannot transfer energy into the structure and the response approaches zero. Figure 4 clearly demonstrates that for O < Urn the resonances 2co and 3o2 are very important for the range of damping values reported for offshore structures. 13 The resonances 4co and 5co may be important for very low damping values.
\CON l
C0S~b/: d[l __(~N/112 + (2~.N~_~)2 r,
=
(28)
CM [ W / [-~P (29)
1 ( / ) O 2 / ( Um ~2
= 7 t G ] t-----N-6] ca For a slender offshore structure it is easily shown that rl "¢r2. The frequencies contained in the response are reasonably well separated and the maximum value of the oscillating part of v(t)/D can thus be approximated by: D lrnax r2
d2 =1
Wave theory and wave excitation In the case of a general stationary Gaussian sea state, the orbital wave velocity and acceleration terms are expressed as functions of the spectral density function Snn(co) of the free surface elevation ~(t). Taking Snn(co) to be a one sided spectrum, and considering only linear wave theory, one obtains: u(t) = f x/2Snn(co ) dco coG cos(--cot+kx+ q;) (31) 0
IDilZc}
(30)
The response is expressed as the ratio of the dynamic response (30), to the response produced by the maximum fluid force applied statically to the structure. In the case of a current /7, the maximum of the fluid force can be calculated according to the approach presented by Moe and Crandall.12 Figure 4 shows a typical dynamic response of a spring supported structure as a function of the ratio of oscillating flow frequency to the natural frequency of the structure. When the frequency of the wave, co, approaches the natural frequency of the structure or a submultiple of that fie-
188
ESTIMATE OF NONLINEAR DRAG FORCE WITH GENERAL STATIONARY GAUSSIAN WAVE AND CURRENT
Applied Ocean Research, 1983, Vol. 5, No. 4
7
//(t) = J ~/2Snn(co ) dw co2G sin (--cot + kx + ~) (32) O
where ~O is a random phase angle, uniformly distributed between 0 and 27r and G is the depth dependent factor defined by (14). Equation (1) is now rearranged so that the response dependent terms are separated from the flow dependent terms. Introducing: ;7
Ma = O -~ (CM -- 1) D 2
(33)
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor 1
2
Ca= g p C D D ( ~ o f i ( t ) )
(34)
and applying the linearization approximation, (3), results in:
P(t) =
4 CMD
ii(t)
(½PCDD)Iti(t)lti(t)
+
-- Mai)( t ) -- CigfJ(t )
Figure 5 compares pFig(Fig) as a function ofz = Fig/kigo~(t). There is a significant difference between the p.d.f, for the exact force and the p.d.f, for the linearized drag force in the high force range. Hence, the Gaussian approximation does not predict the extreme values with good accuTacy.4 The probability density for the cubic approximation to Fig(t ) is found by using (41).
(35)
1
F(t) = Fm(t ) + Fig(t)
1
pyig(Fd) -- 4kigog(t ) ~
The terms MaiJ(t ) and CigiJ(t) are called the added mass and hydrodynamic-drag damping terms, respectively. The flow dependent part of the force can thus be divided into an inertia force term and a drag force term:
{(b + ~b-~-g]) 1/a
-- (b -- ~
exp (a)
1 ) 1/3 }
(46)
where
(36)
_
Fig(t)
£
where
Fro(t) = P 4 CMD
ii(t) = kmii(t )
Fa(t) = (~oCDD) Iti(t)lti(t) = kalifft)lti(t)
a ={1 --½[(b + X/~ + 1)2/3 + (b -- bx/'bT-g1)2/3] } (47)
(37) (38)
The resulting p.d.f, is also shown in Fig. 5. This approximation gives very close agreement with the exact result, both for the p.ckf, and the variance:
Linearization of nonlinear drag force Comparison studies for a single sinusoidal wave indicated that a cubic expansion for the drag force term is a very good approximation. In order to establish the appropriate expansion for the case of a random sea state, it is necessary to examine the probability density function for the force. When the current U is neglected, ti(t) is a zero mean Gaussian process with probability density: {_ 1 ti2(t) I exp - 2- _-22--P~(ti) = X~ofi(t) oa(t) J 1
(39)
Since the exact drag force, (38), is a monotone function, the inverse function exists.
[ F~a[ka;
Fig >t 0
(40)
ti(t) = I - Fa',/P~lk-Sd; F a < 0
o2FIg=E[F5 ]=_28 4 4 3rr kig2 oa(O ~ 2-9708kig2 oa(o
(48)
which is only 1% smaller than the exact variance, (43). As for no current, the cubic approximation will probably give a good approximation to the drag force term when the current is included. Current is introduced by writing: ti(t) = 0 + tig(t )
(49)
where tig(t) is a Gaussian zero-mean process. Substituting for ti(t) in (38), the drag force can be represented as a sum of two components:
Fd(t ) : ff d(t) + Fd(t)dynami c
(50)
where fig(t) represents the static force due to current and Fig(t)dynami c contains the time varying terms. Their form corresponding to a cubic expansion is:
Using (40) and the definition equation:
fiig(t)
dti(Fa)
Pyig(Fig) = Pu(ti(Fig)) - - dFig
= kigo~(t)
+ ~a.2
(41)
leads to the 'exact' expressions:
1
]-k-a
[ Ifigl/
2 \
PFig(Fig) = 2 Vr2Ou(Okig d--~aaaexp t - [ ~ - a 1 / 2 ° a ( t ) ) (42)
and
o~ig
=
e [ F S ] = 3kigou(t) ~ 4
(43)
When Fa(t ) is approximated by a linear expansion in (41), the probability density function reduces to the Gaussian form:
pFa(Fig)-4kig 2 exp -(Lk~a)/(---~oti ( Ou(O
v
% 5
-1 1o
c~ >.
k
- "2 =
i0
% ,%
2
, b
% % 10_3
(44)
%
c~
This is to be expected since a linear transformation of a Gaussian process gives another Gaussian process. The variance of the linear approximfition is:
°2Fa = E[FS] = [ k 5 og(t)
v "o
(45)
lr
which is 15% smaller than the exact variance given by (43).
-4 10
1
2
3
4
5
~
7 2
Z:Fd/(kd Ju(t) )
Figure 5. Probability densities of drag force term. (a) Probability density of exact drag force term. (b) Probability density of linear approximation to drag force term. (c) Probability density of cubic approximation to drag force term. Curves (a) and (b ) after Smith 4 Applied Ocean Research, 1983, Vol. 5, NO. 4
189
Linearization methods and influence of current on nonlinear hydrodynamic drag force." O. T. Gudmestad and J. J. Connor
t
od(t)]
+ {qb} x 2 + {]~b} x3[
(51)
Ug(t) Od(t)
The quartic approximation is generated by adding a fourth degree term to (51):
Fa(t)
equation (51)4
=
[ 1 -U ~ ] [ 6 Od(t) !
X4
which can be shown to have the same form as for a cubic approximation with an additional term quartic in Raa(r ). (See Borgman's result. 6) Therefore, it follows that expanding the drag force term using a least square fit and then calculating the autocorrelation function yields the same result as calculating the autocorrelation function directly. However, the expansion approach is more convenient for incorporating the current. The spectral density function of the force is defined as the Fourier transform of (57) (see also Borgman6).
"(O 2
(52)
l
Spectral density function o f the force
+4 •
The general cross spectral density function for the case with current U was established by Borgman. 6 It will be shown here that the expansion approach is more convenient to determine this function. In general, the spectral density function of the force is given by the Fourier transform of the autocorrelation function which is determined as follows: Let X = g ( F ) be a monotone function and let its inverse function be F = if(X). If F is a random variable of a zero mean Gaussian process, the autocorrelation function of X(t) is related to the autocorrelation function o f F ( t ) by:
Rxx(7) = f 1 b 2 R~F(r_____) )
+2~
O 2 2 Sad(~)loa(t)
q_ 2{(I)}2[Sdd((.O)]*2 /Ou(t) 4
+ ~{~} 8 2[Sdd(co)],3/ad(t) 6 ~2 0 z +3_Il(od(,))~lI [Sdd(~o)],45 /od(0)
+ k~S.~(~)
(58)
where the convolutions of Sdd(co) are given by:
(53)
[Sdd(6o)] *n = ~ [Sua(g)]*(n-')Sdd(Oo--g) dg (59)
(54)
The expression (Sdd(ao)) *~ is understood to mean S~d(co). For no current, (58) reduces to: 8 Spp(~o) = - kdOd(t) 2 4 [Sad(CO)/oa(t) =
n=0 n r n R~F(O) where oo
1
___f2
bn - x / ~ o f
f g ( f ) exp (-~o~)Hn ( ~ ) d f
7r
_t_ 1
6 g [Sad(co)] * 3 /oa(t)] + k2mSr,n(co)
and Hn(x ) are Hermite polynomials:
Hn(x) = (--1)n exp
(x2~ dn [~
,2
, xp
/.____]ix 2
(55)
Their series expansion has the form:
Hn(x) = x n
n(n--1) xn_2 + n(n -- I) (n -- 2) (n -- 3) x n-4 2.1!
22.2!
n(n -- 1) (n -- 2) (n -- 3) (n -- 4) (n -- 5)
X n-6
(60)
Figure 6 shows values of the coefficients of (58) as functions of U/od(t) for U/oa(t)< t. Note that the same trend was observed in Fig. 1. In order to proceed with example calculations of spec2 0 has tral density functions of the force, the variance oa( to be calculated. For a zero mean wave, the variance is given by: 1 M 2 2 Z -AnCOnD -2 2--2 n (61) Od(t) :
23 .3!
n~=l
(56) A general reference for this derivation is Smith.4 The drag force corresponding to the quartic approximation is given by (52) which is a sum of monotone functions. Also, the random variable ftg(t)/od(t ) is the random variable of a zero-mean Gaussian process. Then, including the inertia, force term from (37), the autocorrelation function is given by:
RFte(r) : kSo4(t) +4{~
¢b
+ 1 + 20
,c i
~]it} ,,g 4. ,) %:,
~(0] ~
o ~ ( t ) + 2 ~ } Rda(r)/oZ(t)
+ 2 {~b}2 Rad(r)/Od(t) 2 4 3 6 + 8~(~} 2 Rdd(r)/Od(t )
+-
3~
4 8 2 ) ~ Rdd(r)/%(O + kmR~.(r
(57)
190
Applied Ocean Research, 1983, Vol. 5, No. 4
Figure 6. Values of coefficients of ci vs O/Od¢O showing relative importance o f current in the case of General Stationary Gaussian wave
Linearization methods and influence of current on nonlinear hydrodynamic drag force." O. T. Gudmestad and J. J. Connor Since 0 is a static current, the variance deffmed by (61) also applies when the" current Uis present.
SFdFd/10% N2-S
11 I~ o
Example 1 Spectral density function for drag force in the narrow banded case. As an example, a narrow banded white noise wave velocity is considered. The spectral density function for the wave velocity is given by:
~: Linear drag force spectrum FdFd( ) 2.10 8
b: Nonlineor drag for¢~ spectrum, ~o ~:urrcnt, U=U iO 8
with current, IJ=l.Om/s
5.10 7
So for --cop -- - - < co < --COp + -2 2 S~(CO)= /
co < COp+ _A A
and COp_AA2 _<
(62)
2
7
0 elsewhere
1° 7
For extreme sea states, typically Tp ~ 14 s and aa(t) = 1.5 m/s. A is chosen equal to 0.1125 rad/s giving Sa(CO)= 10 m2/s. 14 For North Sea conditions 0 ~ 1 m/s. Thus, U[oa(t) 0.667 and • ~ 0.426, ~ ~ 0.3194 (equation (7)). Choosing CD = 1.4, 0 = 1025 kg/m a and D = 1 m, gives for the onesided spectral density function for drag force:
,, ~'1°° ~ t
0
+ 0.492 [Sa(co)]*2/a~(O
\ " ......-.
Ill
1°6
SFaFa(CO) = 2.317 000. (2.933 + 3.753 Su(w )
0.2
~ 0.4
zw 0.6
0.8
1.0
1.2
J~P 1.4
1,6
q~ 1.8
2.0 rad/sec
Figure 8. Spectral density function for drag force in the case of a Jonswap state state, cop=O.45rad/s, 3'=7, a = 0.015
+ 0.271 [Sa(co)]*a/o~(t) + 0.030 [Sa(CO)l*'/a6(t)
(63)
A plot of the one-sided spectral density function for the drag force corresponding to a narrow banded white noise wave velocity is shown in Fig. 7, Peaks are observed at two and three times the peak frequency of the wave excitation. Note that the energy at four times the peak frequency is only "~5% of the energy at 3cop. The figure shows that there is generally a significant increase in the spectral density function when the current
:i
c: Nonlinear drag forcc~ spectrum,
is included. However, at the 3cop resonance point, the spectral density function is less when the current is included (cf. Fig. 6).
Example 2 Spectral density function for drag force when the sea state can be described by a Jonswap spectrum. The Jonswap spectrum for surface elevation has the form: 5 (__~_~t-4]. 7 ( _ ( c o _ cop)2] Snn(~°)=°~g2w-Sexp[--4\copl j exp[ 2o-~w~ 1
{h)) (N2-S)
(64)
~
where the spectral parameters according to the Norwegian Petroleum Directorate Is are determined by:
Including current NO current, 0=0
[ 0.07 for co ~< cop - a = t 0.09 for co > cop The peak period Tp is varied within the period interval 13 s < Tp < 20 s The peakedness parameter 3' is chosen between 1 and 7 The a parameter is chosen so that:
-i0 ~
-
1(
-
~q 5.1(
-
/
oo 4
4 j x/Snn(f ) d e = Hs 106,6(0)
o
wp=O. 45
2~p~O. 90
3~m= 1.35
~ (tad/s )
3A
Figure Z Dragforce spectral density function for narrow band white noise wave velocity spectrum. (Note: Energy at 4cop ~ 5% o f energy at 3cop)
where H s is the significant wave height.* The most peaked wave spectrum will give the highest values for the resonant peaks at 2cop, 3cop in the force spectrum. 4 It is therefore of special interest to study the case 3' = 7. Figure 8 gives the spectral density function for * Possible change in surface elevation spectrum due to a current is not accounted for.16
Applied Ocean Research, 1983, Vol. 5, No. 4
191
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor drag force (58) with krn- 0) in the case of a design sea state; and a typical current : Hmax = 30 m, duration 6 h
Tp = 14s U = 1.0m/s Note that:
co2G~S~(co)
S~(co) = O2a(O) +
where Snn(co ) and G are given by (64) and (14) respectively, and 8(0) is the Kronecker delta function.
Cross spectral density o f wave forces The cross spectral density of wave forces is needed when the response of a structure with more than one degree of freedom is to be calculated, in general, the cross spectral density of wave forces is the Fourier transform of the crosscorrelation between:
F1 (t) = ka, lul(t)lu~ (t) + kmfi~ (t) and
(65)
F2 (t) = ka2 lu2(t)lu2 (t) + km2~t2(t) and can be found in a similar way as followed for the spectral density. The resulting spectral density is:
Sp
I
I(
'l
L~Oa2(t):
2
2
8
3 3 x [S<,i~(co)] ~ 3 /(o,i,(Oo,i2(O)
3 to~.(t)
oa~(t)
\oaf(t): + krn lkrn2. Sti,iifco )
(66)
Borgman 6 assumed that the coefficient of Sa,u~(co) was equal to zero. Recent research 17 shows that ka and krn cannot be considered as constants. Further, the variance Ou(t) depends upon the depth of the member (see equation (61)). Therefore, it is more consistent to include the term Sa,u2(w ) in the cross spectral density of the wave force.
SUMMARY
AND
CONCLUSIONS
The main conclusions of this work are itemized below. 1. The proposed linearization of relative velocity terms Iti(t) -- b(t)l (fi(t) -- b(t)) leads to a hydrodynamic damping which is half that obtained by other linearization methods}
192
A broadening of the force spectra are observed, thus increasing the variance aa(t) of the flow. This broadening increases with increasing current, U. - The value of the spectra at the peak frequency, cop, of the wave excitation increase substantially with increasing value of U/oa(t). An increase of about 20% is found for North Sea design conditions. - Peaks are observed at frequencies 2cop and 3cop. The peak at 2cop is introduced because of the current O and vanishes if O = 0. The peak at 3cop is reduced with increasing U/oa(0. Another peak at 4cop is observed in the case of a current. However, this peak is very small for North Sea design currents. A cubic approximation to the force will suffice in most cases. Conventional piled deep water structures in 250400 m have fundamental natural frequencies in the range 2cop-4cop. Thus the nonlinear forcing terms will be o f importance in calculating the response of these structures. - The energy at higher frequencies is increased by the nonlinearity. An increase of an order of magnitude is found for co > 10cop in the case of typical North Sea design conditions. For lower sea states representing day-to-day loading of the structure, the effect of the nonlinear terms should be included in fatigue analysis. At low frequencies, nonlinearities introduce wave force energy which increases with increasing U/oa(o, even if the water particle velocity has no energy in this region. This energy may be of importance in the calculation of the response of very flexible structures such as guyed towers proposed for deep water. -
x Sc,,a~(co)/(oa,(Ooc,=(O) + 2{'1} {'2} *2
2. For a sinusoidal design wave with frequency co, the nonlinear terms of an expansion of the drag force [u(t)lfi(t) represent forcing functions with frequencies/'co, ] = 2, 3, 4, 5, etc. The odd terms emerge from the sinusoidal design wave which has a particle fluid velocity amplitude equal to Urn, while the even terms are introduced in the case of a current O > 0. The forcing terms with frequency co and 26o increase with increasing U/Urn while the forcing terms with frequency 36o and 56o decrease with increasing U/Urn. The forcing term with frequency 46o increases with U/Um up to U]Um ~ 0 . 5 and then decreases. For typical North Sea design conditions, U/Um ~ 0.15 at the free surface and increases with depth. For U/Urn > 1, u ( t ) > 0 for all t and there are no forcing terms with frequency larger than 260. The resonances at 2co and 3oo are shown to be important for calculating response of structures or structural members with natural frequency in that range. The resonances at 46o and 56o will be important for very lightly damped structures only. Thus, for a structure with fundamental natural frequency coN in the range co-5co, the nonlinear terms of the drag force should be retained. The calculated response could be much larger than the linear response, depending on the damping of the structure. 3. For stationary Gaussian wave excitation, it is shown that the Gaussian approximation to the drag force does not predict the extreme values of the force with good accuracy while a cubic approximation gives a very close prediction. The force spectral density function and cross-correlation spectrum have been studied. The effects of the nonlinear terms of an expansion of drag force ]u(t)lu(t) are as follows:
Applied Ocean Research, 1983, Vol. 5, No. 4
4. The inclusion of the current is shown to be important for typical North Sea velocity conditions.
Linearization methods and influence o f current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor ACKNOWLEDGEMENTS The first author was partially supported b y the Norwegian Council for Scientific and Industrial Research through NATO Science Fellowship Grant 1322. Intevep, SA, Venezuela, provided partial support for the second author. In addition, the authors are most appreciative of the valuable suggestions and stimulating discussions with S. Shyam Sunder and Enrique Laya.
density function, p(x), has the form:
1
p(x) = ---~
exp [ - ( x -m)2/2o21
(A1)
oV~zr Noting the following relation: d
(x - - m )
dx p - p ' =
--o 2
(A2)
P
allows one to write:
xp = --aSp ' + mp
REFERENCES 1 Malhotra, A. K. and Penzien, J. Nondeterministic analysis of offshore structures, J. Eng. Mech. Div., ASCE 1970, 96 (EM6), 985 2 Sigbjomsen, R., Bell, K. and Holand, I. Dynamic response of framed and gravity structures to waves, in Numerical Methods in Offshore Engineering, Zienkiewcz, O. E., Lewis, R. W. and Stagg, K. G. (eds.), John Wiley & Sons, Chapter 8, pp. 245280, 1978 3 Mes, M. J. New studies improve wave force spectral calculations, The Oil and Gas Journal, April 1978 4 Smith, E. On nonlinear random v~ration, PhD Thesis, Division of Structural Mechanics, The Norwegian Institute of Technology, Trondheim, 1978 5 Blevins, R. D. Flow Induced Vibration, Van Nostrand Reinhold Company, New York, 1977 6 Borgman, L. E. Random hydrodynamic forces on objects, Ann. Math. Statistics 1967, 38, 37 7 Borgman, L. E. Ocean wave simulation for engineering design, J. Waterways and Harbours Division, ASCE 1969, 95 (WW4), 557 8 Wu, S. C. The effects of current on dynamic response of offshore platforms, Offshore Technology Conference, Houston, May 1976, II. Paper No. 2540, pp. 187-196 9 Sterling, G. H., Cox, B. E. and Warrington, R. M. Design of the Cognac Platform for 1025 feet water depth, Gulf of Mexico, Offshore Technology Conference, Houston, May 1979, II, Paper No. OTC 3494, pp. 1185-1198 10 Det norske Veritas (DnV) Rules for the design, construction and inspection of offshore structures, Appendix A, Environmental conditions, Oslo, 1977 11 Moan, T., Syvertsen, K. and Haver, S. Stochastic dynamic response analysis of gravity platforms, Report SKIM-33, Division of Ship Structures, The Norwegian Institute of Technology, Trondheim, 1976 12 Moe, G. and Crandall, S. H. Extremes of Morrison-type wave loading on a single pile, J. Mech. Design, ASME 1977, No. 77-DET-82 13 Vandiver, J. K. and Campbell, R. B. Estimation of natural frequencies and damping ratios of three similar offshore platforms using maximum entropy spectral analysis, ASCE Spring Convention, Boston, Mass, April 1979 14 Moe, G. Stochastic dynamic analysis of jacket type platforms; nonlinear drag effects and effective nodal areas, International
Research Seminar on Safety of Structures under Dynamic Loading, Trondheim, July 1977 15 Norwegian Petroleum Directorate. Environmental Loads, NonMandatory Supplement to Regulations for the Structural Design of Fixed Structures on the Norwegian Continental ShelJ~ Stavanger, 1977 Huang, N. E., Chen, D. T. and Tung, C.C. Interactions between steady non-uniform currents and gravity waves with applications for current measurements, J. Phys. Oceanog. 1972, 2, 420 17 Sarpkaya, T., Collins, N. J. and Evan, S. R. Wave forces on rough-walled cylinders at high Reynolds numbers, Offshore Technology Conference, Houston, May 1977, II1, Paper No. OTC 2901, pp. 175-184 18 Papoulis, A. Probability, Random Variables, and Stochastic Processes, McGraw-Hill,Inc., 1965
(A3)
Equation (A3) is useful for evaluating the expected value of a power o f x . One can express E [ x n] in terms of lower order terms with the recurrence relation:
E[xn]=
f --
xnp(x) dx oo
x n - a p dx + m f
----o2f --oo
x n - t p dx
--oo
= o 2 ( n - - 1 ) E [ x n-2 ] + m E [ x n-l] n = 3, 4 , . . .
(A4)
Results up to n = 6 are listed below for reference.
E[x] = m E [ x 2] = a2 + m 2 E [ x 3 ] = m ( 3 o 2 + m 2) (A5)
E[x 4] = 3o 4 + 6o2m 2 + m 4 E[x s ] = m(15(r 4 + 10o2m 2 + m 4) E[x 6] = 15o 6 + 45oam ~ + 15o2m 4 + m 6 When m = 0, (A4) reduces to (see Papoulis): TM
E[x n] = 1 × 3 x ... × ( n - - D o n = 0
n even (A6)
n odd
The right-hand side of (A6) involves the tion o f the expected value of [xlx n. Letting
evalua-
y = x-- m
(A7)
allows one to write:
16
- - m
f
--
( - - y - - m ) ( y + m)n f ( y ) dy oa
o
+ f
(y+m)(y+m)nf(y)
dy
- - m
APPENDIX Consider a normally distributed random variable, x, with mean m, and variance o. By definition, the probability
+ i ( y + m)n + i f ( y ) dy o
Applied Ocean Research, 1983, Vol. 5, No. 4
(A8)
193
Linearization methods and influence of current on nonlinear hydrodynamic drag force." O. T. Gudmestad and J. J. Connor
bl = erf(mlo) = ~
1
f(Y) = o---X/~ exp (--y2/2o2)
f --oo
exp (-y2/2) dy
o
1
Rearranging the limits on certain integrals leads to: -]-oo
1/
m/o
where
b2 = ~ -
exp [--m2/2o 2]
oo
[ x [ x n p ( : ~ ) d x = f { ( y + m ) n+l-- ( - - y + m ) n+l} f ( y ) dy
~E[Ixl]
=
mbx + ob2
~E[lxlx] = b1(o 2 + m 2) + b2mo
0
~E[Ix[x 2] = bl [m(m2 + 302)] + b210(m2+ 2o2)] m
+
2f
(A9)
(m--y)n*lf(y) dy
o
I / f [IX[X3 ] = b l [ m 4 q - 6m202q - 304 ]
+
b2[mo(m2+ 502)]
~E[Ixlx 4] = bl[m(m 4 + 10m2o2 + 15o4)] + b2[o(m 4 + 9m202 + 804)]
Results for n up to 4 are presented below.
Sponsoredby the InternationalSocietyforComputationalMethodsin Engineering
2nd International Conference on
Computational Methods and Expgrimental Measurements On board the liner the Queen Elizabeth II June 27th - July 2nd 1984 New York to Southampton
In view of the increasing interest for interaction between numerical and experimental approaches and the success of the 1st International Conference on Numerical Methods and Experimental Measurements, the Conference is to be reconvened in 1984. The primary aim of the Conference is to provide a forum for presentation and exchange of innovative approaches in the fields of numerical methods and experimental studies, with emphasis on their interaction and application in engineering problems.
The Queen Elizabeth II provides an ideal environment for an international meeting and one that is equally convenient for USA and European researchers. The QE II has excellent conference facilities and offers an environment propitious to the interchange of ideas and close contact between participants?
194
• • • •
• •
Experimental versus analytical or numerical models Interaction of computer codes and experimental models Material property characterization through numerical models and experimental prototypes Computer interaction and/or control of real time experiments, interface with computational models and calibration of mathematical models Real time simulations Microprocessor implementation for data acquisition and processing phases
• Structural Applications and Fracture Mechanics • Geomechanics and Soil Dynamics • Heat Transfer • Fluid Dynamics, including Geophysical Applications
Applied Ocean Research, 1983, 1Iol. 5, No. 4
(A10)
• Fluid Structure Interaction • Water Resources and Sediment Transport • System Identification • Material Characterization • Data Identification
Enquiries regarding the conference should be addressed to one of the Conference Directors Dr. C.A. Brebbia Computational Mechanics Centre Ashurst Lodge Ashurst Southampton SO4 2AA AA ENGLAND. Tel: 042 129 3223 Dr. G.A. Keramidas
Naval Research Laboratory (Code 5841) Washington, DC 20375, U.S.A. Tel: 202 767 3389
INTRODUCTION With the trend toward deployment of conventional fixed offshore structures in deeper water, the question of dynamic amplification becomes more important since the fundamental natural frequency, 6oN, of deep water structures is closer to the dominant wave frequency. For piled steel platforms in 150 m water depth, typical value of WN are 3-4 rad/s. Similar platforms in 300 m water depth have fundamental natural frequencies in the region of 1.5 rad/s, which is approximately three times the peak frequency, coo, of the wave excitation. Morison's formula is generally applied to evaluate the hydrodynamic forcing, and an equivalent linearization technique 1 is used to linearize the forcing function in the frequency domain. Thus, terms emerging from nonlinear drag force having frequencies 2wp, 36Op, etc. are not accounted for in a frequency domain analysis.2 However, for deep water structures, these terms may increase the force spectral density near the fundamental natural period of the structure and thereby increase the response of the structure considerably, a, 4 This paper is concerned with procedures for linearizing the relative motion term in the forcing function. As a first step, the approach proposed by Blevins s for a sinusoidal wave is extended here to a general stationary Gaussian sea. It is shown that the procedure leads to lower values for hydrodynamic damping both for a sinusoidal wave as well as for a Gaussian sea state. A combination of sinusoidal waves and a current U is also examined. The effect of including a realistic North Sea current when calculating the nonlinear drag force terms is illustrated. Procedures appropriate for frequency domain analysis, allowing for a current coupled with a stationary Gaussian sea state, are considered next. An expansion of the drag force derived from a least square approximation is used to generate the spectral formulation of the force. It is shown that this method yields the same results as obtained by Borgman 6'7 directly from the autocorrelation function and cross-correlaion function. However, the expansion approach is less complex and the physics of the problem is more easily understood since the effects of the nonlinear forcing terms are evaluated directly.
Received November 1981. Discussion closes December 1983.
184
Applied Ocean Research, 1983, Vol. 5, No. 4
An interesting aspect of this study is the application to realistic North Sea environmental conditions defined by a Jonswap spectrum and a current profde typical of that area. A narrow-banded white noise wave velocity spectrum is also investigated. The inclusion of current in the examples is believed to represent a new application of nonlinear spectral density calculations.
LINEARIZATION OF THE HYDRODYNAMIC DRAG FORCE General expression For short-term intervals, of the order of a few hours, the water surface elevation ~(t) at a fixed location in the sea can be approximated as a stationary process. In this study, r/(t) is considered to be a non-zero mean, Gaussian, stationary process and is represented by a linear summation of an infinite number of sinusoids with phase angles randomly distributed between 0 and 2rr. The waves are assumed to be unidirectional and straight crested, and linear wave theory is used to relate water particle motion at any depth to the fluctuating surface elevation. It is also assumed that the total hydrodynamic force per unit length acting on a cylinder of diameter D is given by Morison's equation: P(t) = p (CM -- 1) 47r D 2(u(t) -- 5(t)) + p ~_D2//(t)
+ ½pCDDIf~(t) -- zS(t)l(u(t) -- zS(t))
(1)
where CM, Co are the inertia and drag coefficients consistent with the flow and structural response; ti,/J are the water particle velocity and accelerations; v is the displacement of the structure. One of the uncertainties influencing the response is associated with the values for CM and CD which exhibit a significant variation with Reynolds number (Re), Keulegan-Carpenter number (K) and relative roughness (k/D). However, since the interest here is on different linearization techniques for the nonlinear term in equation (1), CM, CD, are considered to be constants.
Linearized force expression Unfortunately, P(t) is nonlinear with respect to the drag term. When the wave amplitude is much greater than the maximum displacement of the structure: rl(t) ~ v(t)
(2)
0309~1708183/040184-11 $02.00 © 1983 CML Publications
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T Gudmestad and J. J. Connor
[_
the following linearization s can be employed:
1(0
/4+1
0
2
It i ( t ) - z~(t)l( t i ( t ) - z~(t)) = Iti(t)- r)(t)lfi(t)- Iti(t)- ~)(t)l ~)(t) It~(t)lti(t)-I ti(t)l ~)(t)
(3)
Two approximations have been introduced: 1. Setting lit(t)--i;(t)lft(t)= Iti(t)lti(t) introduces an error in the forcing term which diminishes as the amplitude of the flow velocity becomes much greater than the velocity of the structure. 2. The term Ift(t)--i~(t)li)(t) represents fluid damping of the structure. Taking lu(t)--i)(t)li~(t)= Ifi(t)l~)(t) introduces an error in the estimate of damping which also diminishes as the amplitude of the flow velocity becomes greater than the velocity of the structure. For the design wave condition, the dominant wave frequency is usually much smaller than the natural frequency of the structure, and (2) is quite reasonable since the structural response is quasi-static.
Estimate of Ifi(t)ltJ(t) An approximation for It~(t)lfi(t) can be generated by expanding in a power series: Ifi(t)lt~(t) ~ Co + el/~(t) 4- c 2 (tJ(t)) 2 + . . . + Cn(fi(t)) n (4) and applying a weighted residual procedure to evaluate the coefficients. Since u(t) is a random variable, the weighted residual, R, is the expected value of the weighted error measure, we: e = It~(t)la(t)--Co --cl u(t)--c2 (ti(t))2--...--Cn(U(t)) n
R=E[co]e]=O
j=O, 1,2 . . . . . n
(5)
where co] are weighting functions. Taking col = (it( t))] corresponds to a least square approximation and leads to the following equations:
+
2
+
0
2
tow(0] ]
x = a(t)/on(t)
10loa(t) qb = 2 x ~ n
J
exp [-(t=/2)] dt = 2 erf(O/oa(t) )
0
~=~
exp - 2
Equation (7) reduces ,to Borgman's 7 result when O is set equal to 0, The method can also be extended to the case where the excitation is. not Gaussian. One replaces the Gaussian probability density function (p.d.f.), with the actual p.d.f, for the excitation in equation (6).
Estimate of Iti(t)lz)(t) The fluid damping coefficient is approximated by Iti(t)l. In the case of a Gaussian excitation with mean U, the most accurate least square estimates are: Iti(t)l~
__
qb+2¢
°a (0 {2~b} + {cb} x
1o- (±t+
c o E [ # ] + c,E[u j+l] + . . . + c,E[ui+,]
(8)
= E[Itilui +1]
+,~
x/%oa(0 o~(t)
] ida
(6)
Results for up to fourth order expansions are listed below. Details pertaining to the evaluation of the expectation terms are presented in the Appendix. lu(t)lti(t)__ Ixlx ~ 2
--
~--2
4~
oa(t)
~b + {4~b}x + {~} x 2
~ {-~3 (o~(t))a ¢} +{ 2 [(o~t,~+l]~} x +
~-2
The "constant' approximation is used in this analysis. For zero mean excitation, it corresponds to that which would be generated by a steady flow of velocity ~ aa (t)Blevins s has shown that this approximation for fluid damping yields good agreement with the 'exact' nonlinear solution of a one degree of freedom system subjected to a sinusoidal design wave when the structural displacement is much less than the fluid displacement. For large structural displacement, numerical integration in the time domain is required in order to account for the nonlinear damping in an accurate way.
Comparison with other methods An equivalent linearization technique proposed initially by Malhotra and Penzien 1 for the zero mean case and later extended by Wu a to allow for current is based on the linear approximation:
I \oa(t)! --2
(7)
where:
~ x2+{-~}x a
Ift(t)--i)(t)l(ft(t)--i;(t)) ~ ao +al(fi(t)--i)(t))
(9)
The approach presented in this paper works with separate expansions for the coefficients of tJ(t) and z)(t), and the linearized version has the following form:
Iti(t)--i)(t)l(ffft)--iJ(t)) ,~ Iti(t)lti(t)- It~(t)lT)(t) bo + blu(t)--Coi)(t) Applied Ocean Research, 1983, Vol. 5, No. 4
(10)
185
Linearization methods and influence of current on nonlinear hydrodynam& drag force: O. T. Gudmestad and J. J. Connor where bo, b~ are defined in (7) and Co is given by (8). Wu 8 assumes/~(t) is Gaussian when evaluating the expectation of the weighted residual. As will be shown later, the nonlinear drag force represents a non-Gaussian excitation, and therefore the response is not Gaussian. One finds that ao = bo, a l = b b and co=bl/2. The hydrodynamic damping coefficients differ by a factor of two. Time history simulation may resolve this issue.
ESTIMATE OF NONLINEAR DRAG FORCE WITH SINUSOIDAL WAVE LOADING AND CURRENT
Linear wave theory Before considering random sea-states, the extreme force calculation for a steady-state sinusoidal wave is first reviewed. The free surface elevation r/(x, t) is given by: n(x, t) = A cos (kx -- cot)
(11)
where A is the surface wave amplitude and 0 = k x - c o t is the wave phase. In linear gravity wave theory, the wave number k and frequency co are connected by the dispersion relation:
co2 = kg tanh kh
(12)
where g is the acceleration of gravity and h is the water depth. Also, the water particle velocity it(x, t) at depth --z is determined by superimposing the current and wave induced components: i t ( t ) = U+AGcosincot = U+Urn sin cot
(13)
where 0 is the steady current and G defines the variation with depth:
G = coshk(x+h)/sinhkh
--h
(14)
Linearization o f nonlinear drag force When ( i < Urn, the Fourier expansion for the drag lit(t)lit(t) has the form: ld(t)lit(t) = -
U= +
n +--
2V+4UUrn
2i
~
2 (sinV(2--j)
)
Utide(Z): Vtide(Z=o)(h+ztl'7 t h /
{
\
0
ho I
(19) for 0 t> z >1 --ho for z < --ho
C1
2.0
JJ cosScot
4 >-cossV l
n i : l , 3 , s ....
- vL
0---- Utide(Z) + Uwind(Z)
cos V(1 + ] ) ]
5+j
~sinV(1--j)
(18)
where ci(i = 0, 1,2 . . . . ) are functions of (l/Urn. The significance of current relative to the wave excitation is shown in Fig. 1 which contains plots of ci(i = O, 1, . . . . 4) vs. O/urn. A s t h e currentincrease-sw~(fi- respec{ to Urn, the forcing terms with frequencies co, 26o become more important while the terms with frequencies 360 and 5co approach zero. The magnitudes of the forcing terms with higher frequencies decrease with increasing j. However, these forces can excite the natural frequency of the system and it may therefore be of importance to keep them in the calculation o f the system response. This is especially true for flexible structures in very deep water. As an example, the Cognac platform installed in 1025 ft of water has a natural frequency ~ 1.3 rad/s (Fig. 2). Typical hurricanes in the area give dominant wave frequencies ~ 0.5 rad/s. 9 Thus, the natural frequency is ~ three times the dominant wave frequency. The relative importance of current versus the wave velocity depends upon the depth and should be considered when analyzing a particular structural member. Both the current 0 and the wave velocity amplitude Urn have a maximum at the surface. However, Urn drops exponentially with depth as given by (14) while 0 decreases much slower with depth. Det norske Veritas m proposes the following variation of 0:
Uwind (z) =
-- sinj V
+
=+
+ 40Urn\
+c3sin3cot+c4 c o s 4 w t ...}
sin2V
sinV(2+])i 1
-u;.
E
Ifi(t)lit(t) = U2m{Co + c1 sin cot + c2 cos 2cot
2 I ]
(cos V(l --j) + 4UUm \ -17j
1
~m
2
02+
7/'/=2,4, 6 ....
+-
cosV--
Equation (15) specialized for pure sinusoidal flow reduces to the expansion obtained by Blevins. s As a point of interest, the least square approximation based on the probability density function for a sinusoidal wave gives an identical result. Equation (15) can be written as:
sin
~
V(1 + j ) ] 1-+i
]
V(2 +i)'~l cosy(z-i) v cos gUs ljsinScot 2-7
(15)
, 0"
i
i
I iJ i.
where : sin V = U/Urn
(16)
If O ~> Urn, it(t) > 0 for all t and it follows that: lit(t) lit(t) = it2(t) = 02 + 2UUrn sin cot + U2m sin 2 cot
186
Applied Ocean R&earch, 1983, Vol. 5, No. 4
(17)
Figure 1. Plot o f Ci VS (i/Urn showing relative importance o f current in the expansion of lit(t)]fi(t) in the case of a sinusoidal wave
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor where ho = reference depth for wind generated current = 50 m, and z = distance from still water level, positive upward. Using a typical design wave for the North Sea: u
500
H = 2A = 30 m T = 14s H s = 40.3
400
ft.
X = 306m using linear (Airy) wave theory; and evaluating (13) leads to: 300
< w
Umax = 6.732
U
k
I 20O
% r~
100
0.5
1.0
2.0
I .5
RAD/SEC
Figure 2. Natural frequency and wave spectrum for the Cognac Platform in the Gulf of Mexico. From Sterling et al.9
cosh k(z + h) sinh kh
- Urn
(20)
The variation of lima x with z for two water depths, 150 m and 300 m, is shown in Figure 3. These depths are typical for petroleum fields under development and deep water findings in the North Sea, respectively. Also plotted is the variation of current 0 using (19) and typical data from the Norwegian Continental Shelf: Utiae(z = 0) = 0.5 m/s and Owind(Z = 0) = 0.5 m/s} ° The current velocity in the wave crest above the still water level is chosen equal to the velocity at the still water level, a° Lastly, the variation of U/Urn is generated. The forcing function is proportional to U~m (see (18)). Due to the slow variation of the resonant terms with U/Urn (Fig. 1), each resonant term will have its maximum at the surface. Note that c2 is of the same order o f magnitude as co for this example. Treatment of the damping term proceeds in a similar manner. Assuming linear total damping corresponds to retaining only the first term in the Fourier expansion of
It~(t) l: .15
2 [fi(t)l = - ( U V + Um cosV) 7r
Crest° -15
where V is defined by (16). For zero mean wave excitation, (21) reduces to:
-30
~
-60
~
Figure ~a
A
Plot_o~
2 Iti(t)l = - Urn ~r
-
u m, u
c~ "~
-90
-12
-15I
Figure 3[a).
+15 Crest o
i~--
/_! ~-
" 0,5
The total damping (fluid damping +structural damping) for a single degree of freedom system has been studied by Blevins. s Correct representation of the fluid damping term is essential in order to obtain an accurate solution for forcing near the resonant frequencies.
~/U. (bottom scale)
]I
I 1,5
I 2
]L
U(m/s)
and U/U m
Plot o f U m, Oand U/Umfor150m w.d. 1
2
3
4
5
6
7
8
(22)
9
10
Structural response Our starting point is the equation for the response v(t) of a spring supported structure of mass mo in an oscillating flow with mean 0: mo~)(t) + 2mo ~'o 6Ooz~(t) + kv(t) = e(t)
-12{
g -IB(
7[
-24C
m =mo+p(CM--1)-D 2 4
;
Figure 3(b ).
(23)
where ~o = structural damping, k = linear stiffness, and P(t) is considered to have the same form as (1). The value, U/Urn = 0.15, is used here since it is representative of North Sea design wave conditions. Applying the linearized force expansion specialized for a sinusoidal wave (15), (21), transferring the added mass and hydrodynamic damping terms to the left side and introducing:
-6o
-30(
(21)
I • o,s
I
l,s
12
~ ~Cm/s) and 6/Um
Plot of Urn, U, U/U,,for 3OOm w.d.
~o x/1 + pA(CM -- 1)/mo
Applied Ocean Research, 1983, Vol. 5, No. 4
187
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor
(oo+]( Vm1
+ 0.159 \ - - - ~ / [ ~ N D ! Co
(24) Nonlinear
I ~
7[
A =-D 2 4
]I tu
I
+ ......... I
/~
N :°'~)5
1%(N
= k/m ~4
leads to:
m~(t) + 2m~wcoNO(t) + k~v(t)
/,I,
%
+
1 2 o +C 1 sin cot = pACMUrncocosco t + ~PCDDUm{c
0.05 LinearSoIution
,I
o
!j.
If
', u
=
/
\\,\
/i
+ c 2 cos2cot + c3 sin 3cot +c4 cos4cot +Cs sin 5cot + ... }
(25)
where: Co = 0.192, cx = 0.877, e2 = --0.126, ca = - - 0 . 1 6 0 , c4 =--0.034, Cs = - - 0 . 0 1 9 After expanding the displacement v(t) in Fourier series, the solution is found to be: - - = rl D
o
0.5
1,0
1.5
"N
Figure 4. Typical dynamic in-line response of a spring supported cylinder to an oscillating flow with mean 0 = O.15 Urn. (Two different values of damping, ~N, are chosen.)
IDll cos (cot--C1
+ r2 {Co + cl IDd sin ( c o t - ¢ 0 + c2 ID2I cos (2cot-4~2) + ca ID31 sin (3cot --¢3) + c, ID4I cos (4cot -- ¢4) + cs [D4I sin (5 cot -- Cs) }
(26) where: 1
I,
(,7)
60N/
quency, the structure resonates with a component of the fluid force. The resonant motion increases with decreasing damping. For low values of co[coN the response of the structure approaches the static response. For co/coN>> 1, the forces cannot transfer energy into the structure and the response approaches zero. Figure 4 clearly demonstrates that for O < Urn the resonances 2co and 3o2 are very important for the range of damping values reported for offshore structures. 13 The resonances 4co and 5co may be important for very low damping values.
\CON l
C0S~b/: d[l __(~N/112 + (2~.N~_~)2 r,
=
(28)
CM [ W / [-~P (29)
1 ( / ) O 2 / ( Um ~2
= 7 t G ] t-----N-6] ca For a slender offshore structure it is easily shown that rl "¢r2. The frequencies contained in the response are reasonably well separated and the maximum value of the oscillating part of v(t)/D can thus be approximated by: D lrnax r2
d2 =1
Wave theory and wave excitation In the case of a general stationary Gaussian sea state, the orbital wave velocity and acceleration terms are expressed as functions of the spectral density function Snn(co) of the free surface elevation ~(t). Taking Snn(co) to be a one sided spectrum, and considering only linear wave theory, one obtains: u(t) = f x/2Snn(co ) dco coG cos(--cot+kx+ q;) (31) 0
IDilZc}
(30)
The response is expressed as the ratio of the dynamic response (30), to the response produced by the maximum fluid force applied statically to the structure. In the case of a current /7, the maximum of the fluid force can be calculated according to the approach presented by Moe and Crandall.12 Figure 4 shows a typical dynamic response of a spring supported structure as a function of the ratio of oscillating flow frequency to the natural frequency of the structure. When the frequency of the wave, co, approaches the natural frequency of the structure or a submultiple of that fie-
188
ESTIMATE OF NONLINEAR DRAG FORCE WITH GENERAL STATIONARY GAUSSIAN WAVE AND CURRENT
Applied Ocean Research, 1983, Vol. 5, No. 4
7
//(t) = J ~/2Snn(co ) dw co2G sin (--cot + kx + ~) (32) O
where ~O is a random phase angle, uniformly distributed between 0 and 27r and G is the depth dependent factor defined by (14). Equation (1) is now rearranged so that the response dependent terms are separated from the flow dependent terms. Introducing: ;7
Ma = O -~ (CM -- 1) D 2
(33)
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor 1
2
Ca= g p C D D ( ~ o f i ( t ) )
(34)
and applying the linearization approximation, (3), results in:
P(t) =
4 CMD
ii(t)
(½PCDD)Iti(t)lti(t)
+
-- Mai)( t ) -- CigfJ(t )
Figure 5 compares pFig(Fig) as a function ofz = Fig/kigo~(t). There is a significant difference between the p.d.f, for the exact force and the p.d.f, for the linearized drag force in the high force range. Hence, the Gaussian approximation does not predict the extreme values with good accuTacy.4 The probability density for the cubic approximation to Fig(t ) is found by using (41).
(35)
1
F(t) = Fm(t ) + Fig(t)
1
pyig(Fd) -- 4kigog(t ) ~
The terms MaiJ(t ) and CigiJ(t) are called the added mass and hydrodynamic-drag damping terms, respectively. The flow dependent part of the force can thus be divided into an inertia force term and a drag force term:
{(b + ~b-~-g]) 1/a
-- (b -- ~
exp (a)
1 ) 1/3 }
(46)
where
(36)
_
Fig(t)
£
where
Fro(t) = P 4 CMD
ii(t) = kmii(t )
Fa(t) = (~oCDD) Iti(t)lti(t) = kalifft)lti(t)
a ={1 --½[(b + X/~ + 1)2/3 + (b -- bx/'bT-g1)2/3] } (47)
(37) (38)
The resulting p.d.f, is also shown in Fig. 5. This approximation gives very close agreement with the exact result, both for the p.ckf, and the variance:
Linearization of nonlinear drag force Comparison studies for a single sinusoidal wave indicated that a cubic expansion for the drag force term is a very good approximation. In order to establish the appropriate expansion for the case of a random sea state, it is necessary to examine the probability density function for the force. When the current U is neglected, ti(t) is a zero mean Gaussian process with probability density: {_ 1 ti2(t) I exp - 2- _-22--P~(ti) = X~ofi(t) oa(t) J 1
(39)
Since the exact drag force, (38), is a monotone function, the inverse function exists.
[ F~a[ka;
Fig >t 0
(40)
ti(t) = I - Fa',/P~lk-Sd; F a < 0
o2FIg=E[F5 ]=_28 4 4 3rr kig2 oa(O ~ 2-9708kig2 oa(o
(48)
which is only 1% smaller than the exact variance, (43). As for no current, the cubic approximation will probably give a good approximation to the drag force term when the current is included. Current is introduced by writing: ti(t) = 0 + tig(t )
(49)
where tig(t) is a Gaussian zero-mean process. Substituting for ti(t) in (38), the drag force can be represented as a sum of two components:
Fd(t ) : ff d(t) + Fd(t)dynami c
(50)
where fig(t) represents the static force due to current and Fig(t)dynami c contains the time varying terms. Their form corresponding to a cubic expansion is:
Using (40) and the definition equation:
fiig(t)
dti(Fa)
Pyig(Fig) = Pu(ti(Fig)) - - dFig
= kigo~(t)
+ ~a.2
(41)
leads to the 'exact' expressions:
1
]-k-a
[ Ifigl/
2 \
PFig(Fig) = 2 Vr2Ou(Okig d--~aaaexp t - [ ~ - a 1 / 2 ° a ( t ) ) (42)
and
o~ig
=
e [ F S ] = 3kigou(t) ~ 4
(43)
When Fa(t ) is approximated by a linear expansion in (41), the probability density function reduces to the Gaussian form:
pFa(Fig)-4kig 2 exp -(Lk~a)/(---~oti ( Ou(O
v
% 5
-1 1o
c~ >.
k
- "2 =
i0
% ,%
2
, b
% % 10_3
(44)
%
c~
This is to be expected since a linear transformation of a Gaussian process gives another Gaussian process. The variance of the linear approximfition is:
°2Fa = E[FS] = [ k 5 og(t)
v "o
(45)
lr
which is 15% smaller than the exact variance given by (43).
-4 10
1
2
3
4
5
~
7 2
Z:Fd/(kd Ju(t) )
Figure 5. Probability densities of drag force term. (a) Probability density of exact drag force term. (b) Probability density of linear approximation to drag force term. (c) Probability density of cubic approximation to drag force term. Curves (a) and (b ) after Smith 4 Applied Ocean Research, 1983, Vol. 5, NO. 4
189
Linearization methods and influence of current on nonlinear hydrodynamic drag force." O. T. Gudmestad and J. J. Connor
t
od(t)]
+ {qb} x 2 + {]~b} x3[
(51)
Ug(t) Od(t)
The quartic approximation is generated by adding a fourth degree term to (51):
Fa(t)
equation (51)4
=
[ 1 -U ~ ] [ 6 Od(t) !
X4
which can be shown to have the same form as for a cubic approximation with an additional term quartic in Raa(r ). (See Borgman's result. 6) Therefore, it follows that expanding the drag force term using a least square fit and then calculating the autocorrelation function yields the same result as calculating the autocorrelation function directly. However, the expansion approach is more convenient for incorporating the current. The spectral density function of the force is defined as the Fourier transform of (57) (see also Borgman6).
"(O 2
(52)
l
Spectral density function o f the force
+4 •
The general cross spectral density function for the case with current U was established by Borgman. 6 It will be shown here that the expansion approach is more convenient to determine this function. In general, the spectral density function of the force is given by the Fourier transform of the autocorrelation function which is determined as follows: Let X = g ( F ) be a monotone function and let its inverse function be F = if(X). If F is a random variable of a zero mean Gaussian process, the autocorrelation function of X(t) is related to the autocorrelation function o f F ( t ) by:
Rxx(7) = f 1 b 2 R~F(r_____) )
+2~
O 2 2 Sad(~)loa(t)
q_ 2{(I)}2[Sdd((.O)]*2 /Ou(t) 4
+ ~{~} 8 2[Sdd(co)],3/ad(t) 6 ~2 0 z +3_Il(od(,))~lI [Sdd(~o)],45 /od(0)
+ k~S.~(~)
(58)
where the convolutions of Sdd(co) are given by:
(53)
[Sdd(6o)] *n = ~ [Sua(g)]*(n-')Sdd(Oo--g) dg (59)
(54)
The expression (Sdd(ao)) *~ is understood to mean S~d(co). For no current, (58) reduces to: 8 Spp(~o) = - kdOd(t) 2 4 [Sad(CO)/oa(t) =
n=0 n r n R~F(O) where oo
1
___f2
bn - x / ~ o f
f g ( f ) exp (-~o~)Hn ( ~ ) d f
7r
_t_ 1
6 g [Sad(co)] * 3 /oa(t)] + k2mSr,n(co)
and Hn(x ) are Hermite polynomials:
Hn(x) = (--1)n exp
(x2~ dn [~
,2
, xp
/.____]ix 2
(55)
Their series expansion has the form:
Hn(x) = x n
n(n--1) xn_2 + n(n -- I) (n -- 2) (n -- 3) x n-4 2.1!
22.2!
n(n -- 1) (n -- 2) (n -- 3) (n -- 4) (n -- 5)
X n-6
(60)
Figure 6 shows values of the coefficients of (58) as functions of U/od(t) for U/oa(t)< t. Note that the same trend was observed in Fig. 1. In order to proceed with example calculations of spec2 0 has tral density functions of the force, the variance oa( to be calculated. For a zero mean wave, the variance is given by: 1 M 2 2 Z -AnCOnD -2 2--2 n (61) Od(t) :
23 .3!
n~=l
(56) A general reference for this derivation is Smith.4 The drag force corresponding to the quartic approximation is given by (52) which is a sum of monotone functions. Also, the random variable ftg(t)/od(t ) is the random variable of a zero-mean Gaussian process. Then, including the inertia, force term from (37), the autocorrelation function is given by:
RFte(r) : kSo4(t) +4{~
¢b
+ 1 + 20
,c i
~]it} ,,g 4. ,) %:,
~(0] ~
o ~ ( t ) + 2 ~ } Rda(r)/oZ(t)
+ 2 {~b}2 Rad(r)/Od(t) 2 4 3 6 + 8~(~} 2 Rdd(r)/Od(t )
+-
3~
4 8 2 ) ~ Rdd(r)/%(O + kmR~.(r
(57)
190
Applied Ocean Research, 1983, Vol. 5, No. 4
Figure 6. Values of coefficients of ci vs O/Od¢O showing relative importance o f current in the case of General Stationary Gaussian wave
Linearization methods and influence of current on nonlinear hydrodynamic drag force." O. T. Gudmestad and J. J. Connor Since 0 is a static current, the variance deffmed by (61) also applies when the" current Uis present.
SFdFd/10% N2-S
11 I~ o
Example 1 Spectral density function for drag force in the narrow banded case. As an example, a narrow banded white noise wave velocity is considered. The spectral density function for the wave velocity is given by:
~: Linear drag force spectrum FdFd( ) 2.10 8
b: Nonlineor drag for¢~ spectrum, ~o ~:urrcnt, U=U iO 8
with current, IJ=l.Om/s
5.10 7
So for --cop -- - - < co < --COp + -2 2 S~(CO)= /
co < COp+ _A A
and COp_AA2 _<
(62)
2
7
0 elsewhere
1° 7
For extreme sea states, typically Tp ~ 14 s and aa(t) = 1.5 m/s. A is chosen equal to 0.1125 rad/s giving Sa(CO)= 10 m2/s. 14 For North Sea conditions 0 ~ 1 m/s. Thus, U[oa(t) 0.667 and • ~ 0.426, ~ ~ 0.3194 (equation (7)). Choosing CD = 1.4, 0 = 1025 kg/m a and D = 1 m, gives for the onesided spectral density function for drag force:
,, ~'1°° ~ t
0
+ 0.492 [Sa(co)]*2/a~(O
\ " ......-.
Ill
1°6
SFaFa(CO) = 2.317 000. (2.933 + 3.753 Su(w )
0.2
~ 0.4
zw 0.6
0.8
1.0
1.2
J~P 1.4
1,6
q~ 1.8
2.0 rad/sec
Figure 8. Spectral density function for drag force in the case of a Jonswap state state, cop=O.45rad/s, 3'=7, a = 0.015
+ 0.271 [Sa(co)]*a/o~(t) + 0.030 [Sa(CO)l*'/a6(t)
(63)
A plot of the one-sided spectral density function for the drag force corresponding to a narrow banded white noise wave velocity is shown in Fig. 7, Peaks are observed at two and three times the peak frequency of the wave excitation. Note that the energy at four times the peak frequency is only "~5% of the energy at 3cop. The figure shows that there is generally a significant increase in the spectral density function when the current
:i
c: Nonlinear drag forcc~ spectrum,
is included. However, at the 3cop resonance point, the spectral density function is less when the current is included (cf. Fig. 6).
Example 2 Spectral density function for drag force when the sea state can be described by a Jonswap spectrum. The Jonswap spectrum for surface elevation has the form: 5 (__~_~t-4]. 7 ( _ ( c o _ cop)2] Snn(~°)=°~g2w-Sexp[--4\copl j exp[ 2o-~w~ 1
{h)) (N2-S)
(64)
~
where the spectral parameters according to the Norwegian Petroleum Directorate Is are determined by:
Including current NO current, 0=0
[ 0.07 for co ~< cop - a = t 0.09 for co > cop The peak period Tp is varied within the period interval 13 s < Tp < 20 s The peakedness parameter 3' is chosen between 1 and 7 The a parameter is chosen so that:
-i0 ~
-
1(
-
~q 5.1(
-
/
oo 4
4 j x/Snn(f ) d e = Hs 106,6(0)
o
wp=O. 45
2~p~O. 90
3~m= 1.35
~ (tad/s )
3A
Figure Z Dragforce spectral density function for narrow band white noise wave velocity spectrum. (Note: Energy at 4cop ~ 5% o f energy at 3cop)
where H s is the significant wave height.* The most peaked wave spectrum will give the highest values for the resonant peaks at 2cop, 3cop in the force spectrum. 4 It is therefore of special interest to study the case 3' = 7. Figure 8 gives the spectral density function for * Possible change in surface elevation spectrum due to a current is not accounted for.16
Applied Ocean Research, 1983, Vol. 5, No. 4
191
Linearization methods and influence of current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor drag force (58) with krn- 0) in the case of a design sea state; and a typical current : Hmax = 30 m, duration 6 h
Tp = 14s U = 1.0m/s Note that:
co2G~S~(co)
S~(co) = O2a(O) +
where Snn(co ) and G are given by (64) and (14) respectively, and 8(0) is the Kronecker delta function.
Cross spectral density o f wave forces The cross spectral density of wave forces is needed when the response of a structure with more than one degree of freedom is to be calculated, in general, the cross spectral density of wave forces is the Fourier transform of the crosscorrelation between:
F1 (t) = ka, lul(t)lu~ (t) + kmfi~ (t) and
(65)
F2 (t) = ka2 lu2(t)lu2 (t) + km2~t2(t) and can be found in a similar way as followed for the spectral density. The resulting spectral density is:
Sp
I
I(
'l
L~Oa2(t):
2
2
8
3 3 x [S<,i~(co)] ~ 3 /(o,i,(Oo,i2(O)
3 to~.(t)
oa~(t)
\oaf(t): + krn lkrn2. Sti,iifco )
(66)
Borgman 6 assumed that the coefficient of Sa,u~(co) was equal to zero. Recent research 17 shows that ka and krn cannot be considered as constants. Further, the variance Ou(t) depends upon the depth of the member (see equation (61)). Therefore, it is more consistent to include the term Sa,u2(w ) in the cross spectral density of the wave force.
SUMMARY
AND
CONCLUSIONS
The main conclusions of this work are itemized below. 1. The proposed linearization of relative velocity terms Iti(t) -- b(t)l (fi(t) -- b(t)) leads to a hydrodynamic damping which is half that obtained by other linearization methods}
192
A broadening of the force spectra are observed, thus increasing the variance aa(t) of the flow. This broadening increases with increasing current, U. - The value of the spectra at the peak frequency, cop, of the wave excitation increase substantially with increasing value of U/oa(t). An increase of about 20% is found for North Sea design conditions. - Peaks are observed at frequencies 2cop and 3cop. The peak at 2cop is introduced because of the current O and vanishes if O = 0. The peak at 3cop is reduced with increasing U/oa(0. Another peak at 4cop is observed in the case of a current. However, this peak is very small for North Sea design currents. A cubic approximation to the force will suffice in most cases. Conventional piled deep water structures in 250400 m have fundamental natural frequencies in the range 2cop-4cop. Thus the nonlinear forcing terms will be o f importance in calculating the response of these structures. - The energy at higher frequencies is increased by the nonlinearity. An increase of an order of magnitude is found for co > 10cop in the case of typical North Sea design conditions. For lower sea states representing day-to-day loading of the structure, the effect of the nonlinear terms should be included in fatigue analysis. At low frequencies, nonlinearities introduce wave force energy which increases with increasing U/oa(o, even if the water particle velocity has no energy in this region. This energy may be of importance in the calculation of the response of very flexible structures such as guyed towers proposed for deep water. -
x Sc,,a~(co)/(oa,(Ooc,=(O) + 2{'1} {'2} *2
2. For a sinusoidal design wave with frequency co, the nonlinear terms of an expansion of the drag force [u(t)lfi(t) represent forcing functions with frequencies/'co, ] = 2, 3, 4, 5, etc. The odd terms emerge from the sinusoidal design wave which has a particle fluid velocity amplitude equal to Urn, while the even terms are introduced in the case of a current O > 0. The forcing terms with frequency co and 26o increase with increasing U/Urn while the forcing terms with frequency 36o and 56o decrease with increasing U/Urn. The forcing term with frequency 46o increases with U/Um up to U]Um ~ 0 . 5 and then decreases. For typical North Sea design conditions, U/Um ~ 0.15 at the free surface and increases with depth. For U/Urn > 1, u ( t ) > 0 for all t and there are no forcing terms with frequency larger than 260. The resonances at 2co and 3oo are shown to be important for calculating response of structures or structural members with natural frequency in that range. The resonances at 46o and 56o will be important for very lightly damped structures only. Thus, for a structure with fundamental natural frequency coN in the range co-5co, the nonlinear terms of the drag force should be retained. The calculated response could be much larger than the linear response, depending on the damping of the structure. 3. For stationary Gaussian wave excitation, it is shown that the Gaussian approximation to the drag force does not predict the extreme values of the force with good accuracy while a cubic approximation gives a very close prediction. The force spectral density function and cross-correlation spectrum have been studied. The effects of the nonlinear terms of an expansion of drag force ]u(t)lu(t) are as follows:
Applied Ocean Research, 1983, Vol. 5, No. 4
4. The inclusion of the current is shown to be important for typical North Sea velocity conditions.
Linearization methods and influence o f current on nonlinear hydrodynamic drag force: O. T. Gudmestad and J. J. Connor ACKNOWLEDGEMENTS The first author was partially supported b y the Norwegian Council for Scientific and Industrial Research through NATO Science Fellowship Grant 1322. Intevep, SA, Venezuela, provided partial support for the second author. In addition, the authors are most appreciative of the valuable suggestions and stimulating discussions with S. Shyam Sunder and Enrique Laya.
density function, p(x), has the form:
1
p(x) = ---~
exp [ - ( x -m)2/2o21
(A1)
oV~zr Noting the following relation: d
(x - - m )
dx p - p ' =
--o 2
(A2)
P
allows one to write:
xp = --aSp ' + mp
REFERENCES 1 Malhotra, A. K. and Penzien, J. Nondeterministic analysis of offshore structures, J. Eng. Mech. Div., ASCE 1970, 96 (EM6), 985 2 Sigbjomsen, R., Bell, K. and Holand, I. Dynamic response of framed and gravity structures to waves, in Numerical Methods in Offshore Engineering, Zienkiewcz, O. E., Lewis, R. W. and Stagg, K. G. (eds.), John Wiley & Sons, Chapter 8, pp. 245280, 1978 3 Mes, M. J. New studies improve wave force spectral calculations, The Oil and Gas Journal, April 1978 4 Smith, E. On nonlinear random v~ration, PhD Thesis, Division of Structural Mechanics, The Norwegian Institute of Technology, Trondheim, 1978 5 Blevins, R. D. Flow Induced Vibration, Van Nostrand Reinhold Company, New York, 1977 6 Borgman, L. E. Random hydrodynamic forces on objects, Ann. Math. Statistics 1967, 38, 37 7 Borgman, L. E. Ocean wave simulation for engineering design, J. Waterways and Harbours Division, ASCE 1969, 95 (WW4), 557 8 Wu, S. C. The effects of current on dynamic response of offshore platforms, Offshore Technology Conference, Houston, May 1976, II. Paper No. 2540, pp. 187-196 9 Sterling, G. H., Cox, B. E. and Warrington, R. M. Design of the Cognac Platform for 1025 feet water depth, Gulf of Mexico, Offshore Technology Conference, Houston, May 1979, II, Paper No. OTC 3494, pp. 1185-1198 10 Det norske Veritas (DnV) Rules for the design, construction and inspection of offshore structures, Appendix A, Environmental conditions, Oslo, 1977 11 Moan, T., Syvertsen, K. and Haver, S. Stochastic dynamic response analysis of gravity platforms, Report SKIM-33, Division of Ship Structures, The Norwegian Institute of Technology, Trondheim, 1976 12 Moe, G. and Crandall, S. H. Extremes of Morrison-type wave loading on a single pile, J. Mech. Design, ASME 1977, No. 77-DET-82 13 Vandiver, J. K. and Campbell, R. B. Estimation of natural frequencies and damping ratios of three similar offshore platforms using maximum entropy spectral analysis, ASCE Spring Convention, Boston, Mass, April 1979 14 Moe, G. Stochastic dynamic analysis of jacket type platforms; nonlinear drag effects and effective nodal areas, International
Research Seminar on Safety of Structures under Dynamic Loading, Trondheim, July 1977 15 Norwegian Petroleum Directorate. Environmental Loads, NonMandatory Supplement to Regulations for the Structural Design of Fixed Structures on the Norwegian Continental ShelJ~ Stavanger, 1977 Huang, N. E., Chen, D. T. and Tung, C.C. Interactions between steady non-uniform currents and gravity waves with applications for current measurements, J. Phys. Oceanog. 1972, 2, 420 17 Sarpkaya, T., Collins, N. J. and Evan, S. R. Wave forces on rough-walled cylinders at high Reynolds numbers, Offshore Technology Conference, Houston, May 1977, II1, Paper No. OTC 2901, pp. 175-184 18 Papoulis, A. Probability, Random Variables, and Stochastic Processes, McGraw-Hill,Inc., 1965
(A3)
Equation (A3) is useful for evaluating the expected value of a power o f x . One can express E [ x n] in terms of lower order terms with the recurrence relation:
E[xn]=
f --
xnp(x) dx oo
x n - a p dx + m f
----o2f --oo
x n - t p dx
--oo
= o 2 ( n - - 1 ) E [ x n-2 ] + m E [ x n-l] n = 3, 4 , . . .
(A4)
Results up to n = 6 are listed below for reference.
E[x] = m E [ x 2] = a2 + m 2 E [ x 3 ] = m ( 3 o 2 + m 2) (A5)
E[x 4] = 3o 4 + 6o2m 2 + m 4 E[x s ] = m(15(r 4 + 10o2m 2 + m 4) E[x 6] = 15o 6 + 45oam ~ + 15o2m 4 + m 6 When m = 0, (A4) reduces to (see Papoulis): TM
E[x n] = 1 × 3 x ... × ( n - - D o n = 0
n even (A6)
n odd
The right-hand side of (A6) involves the tion o f the expected value of [xlx n. Letting
evalua-
y = x-- m
(A7)
allows one to write:
16
- - m
f
--
( - - y - - m ) ( y + m)n f ( y ) dy oa
o
+ f
(y+m)(y+m)nf(y)
dy
- - m
APPENDIX Consider a normally distributed random variable, x, with mean m, and variance o. By definition, the probability
+ i ( y + m)n + i f ( y ) dy o
Applied Ocean Research, 1983, Vol. 5, No. 4
(A8)
193
Linearization methods and influence of current on nonlinear hydrodynamic drag force." O. T. Gudmestad and J. J. Connor
bl = erf(mlo) = ~
1
f(Y) = o---X/~ exp (--y2/2o2)
f --oo
exp (-y2/2) dy
o
1
Rearranging the limits on certain integrals leads to: -]-oo
1/
m/o
where
b2 = ~ -
exp [--m2/2o 2]
oo
[ x [ x n p ( : ~ ) d x = f { ( y + m ) n+l-- ( - - y + m ) n+l} f ( y ) dy
~E[Ixl]
=
mbx + ob2
~E[lxlx] = b1(o 2 + m 2) + b2mo
0
~E[Ix[x 2] = bl [m(m2 + 302)] + b210(m2+ 2o2)] m
+
2f
(A9)
(m--y)n*lf(y) dy
o
I / f [IX[X3 ] = b l [ m 4 q - 6m202q - 304 ]
+
b2[mo(m2+ 502)]
~E[Ixlx 4] = bl[m(m 4 + 10m2o2 + 15o4)] + b2[o(m 4 + 9m202 + 804)]
Results for n up to 4 are presented below.
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2nd International Conference on
Computational Methods and Expgrimental Measurements On board the liner the Queen Elizabeth II June 27th - July 2nd 1984 New York to Southampton
In view of the increasing interest for interaction between numerical and experimental approaches and the success of the 1st International Conference on Numerical Methods and Experimental Measurements, the Conference is to be reconvened in 1984. The primary aim of the Conference is to provide a forum for presentation and exchange of innovative approaches in the fields of numerical methods and experimental studies, with emphasis on their interaction and application in engineering problems.
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194
• • • •
• •
Experimental versus analytical or numerical models Interaction of computer codes and experimental models Material property characterization through numerical models and experimental prototypes Computer interaction and/or control of real time experiments, interface with computational models and calibration of mathematical models Real time simulations Microprocessor implementation for data acquisition and processing phases
• Structural Applications and Fracture Mechanics • Geomechanics and Soil Dynamics • Heat Transfer • Fluid Dynamics, including Geophysical Applications
Applied Ocean Research, 1983, 1Iol. 5, No. 4
(A10)
• Fluid Structure Interaction • Water Resources and Sediment Transport • System Identification • Material Characterization • Data Identification
Enquiries regarding the conference should be addressed to one of the Conference Directors Dr. C.A. Brebbia Computational Mechanics Centre Ashurst Lodge Ashurst Southampton SO4 2AA AA ENGLAND. Tel: 042 129 3223 Dr. G.A. Keramidas
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