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Total wave force on cylinder considering free surface fluctuation
Applied Ocean Research 18 (1996) 37-43 Copyright 8 1996 Elsevier Science Limited PII:
ELSEVIER
Printed in Great Britain All rights reserved 0141-1187/96/$15.00
SOl41-1187(96)00013-2
Total wave force on cylinder considering free surface fluctuation C. C. Tung Box 7908, North Carolina State University, Raleigh, NC 27495, USA
(Received 4 March 1996) Past studies have shown that, in a wave field, fluctuation of the water surface has a profound influence on the wave induced force on objects placed close to the still water level. A recent study examined the effects of free surface fluctuation on the total wave force on a rigid vertical cylinder of small diameter extending from sea floor to beyond the still water level. The waves were considered long-crested, stationary and linear with a narrow-band spectrum and the wave force was evaluated using the Morison formula. In this study, the waves are treated as second order nonlinear Stokes waves with narrow-band characteristics. The effects of wave nonlinearity and free surface fluctuation on the mean and standard deviation of total wave force on a rigid vertical cylinder are examined. Copyright 0 1996 Elsevier Science Limited Keywords: waves, nonlinear waves, wave force, vertical cylinder, random waves, narrow-band, surface fluctuation.
NOMENCLATURE
FI
The following symbols are used in this paper: = function defined in eqn (47); AW = amplitude of vl; a = linear component and second order al, a2 correction of horizontal fluid particle acceleration defined in eqns (13) and (14); = function defmed in eqn (62); W) = function defined in eqn (48); W = drag and inertial force coefficients, CD> cm respectively; = coefficient given in eqn (39); C D = diameter of cylinder;
fx9
t),
@I@,
4
EL.1 F(t),
=
-
element wave force [see eqn (17)], element drag force [see eqn (19)] and element inertial force [see eqn (18)], respectively; expected value of the random quantity enclosed in the brackets;
FD@)v
total wave force [see eqn (20)], total drag force [see eqn (22)] and total inertial force [see eqn (21)], respectively;
F&I
=
FD
= Fn/
( -&K&g),
non-dimensionalized density function
inertial
of U;
= joint probability density function of $ and W; = function defined in eqn (4); = gravitational acceleration; ir, g2, g3, g4 = functions defined in eqns (lo), (ll), (15), and (16), respectively; = Heaviside unit step function; W) h = water depth (see Fig. 1); = quantities appearing in eqns (19) and (18) KD, Km related to drag and inertial forces, respectively; k = wave-number of monochromatic wave and characteristic wave-number of narrow-band waves; L = cylinder length (see Fig. 1); = function defined in eqn (59); QC9 = quantity defined in eqn (59); Iz s = vertical coordinate (see Fig. 1); t = time; = horizontal fluid particle velocity [see eqn (7)]; u(s, 0 = linear component and second order correcu19 u2 tion of U(S, t) defined in eqns (8) and (9), respectively; = horizontal fluid particle acceleration [see eqn +, t)
f,,&*) GP)
@(s, t), flD(.%
= FI/(K,ag), force; = probability
non-dimensionalized
WY;
drag force; 37
C. C. Tung
- auxiliary random variable [see eqn (SO)]; = horizontal axis (see Fig. 1); = dummy variable; = function defined in eqn (58); = random variables defined in eqns (26) and (27), respectively; = quantity defined in eqn (58); = kk, water depth parameter; = quantity defined in eqn (34); = cylinder length parameter defined in eqn (35); = quantity defined in eqn (33); = quantity defined in eqn (37); = non-dimensionalized wave elevation defined in eqn (25); = cylinder protrusion parameter defined in eqn (36); = wave elevation given in eqn (1); = linear component and second order correction of r) defined in eqns (2) and (3), respectively; = quantity defined in eqn (23); = significant wave slope [see eqn (29)]; = water density; = standard deviations of nl,q and u, respectively; = phase angle [see eqn (5)]; = phase function [see eqn (5)]; and = frequency of monochromatic wave and characteristic frequency of narrow-band waves.
1 INTRODUCTION In a wave field, due to free surface fluctuation, an object placed in the vicinity of the still water level experiences wave induced forces only intermittently. It may be anticipated, therefore, that the free surface fluctuation phenomenon would be an important consideration in the design of a floating structure. On the other hand, for a vertical structure extending the entire water depth, the relevant design quantity would be the sum of all the wave forces along the length of the structure. It is then natural to inquire whether the free surface fluctuation phenomenon would have any effect on the total wave force. This problem was addressed in a recent paper.’ Considering the waves as u&directional, stationary and linear with a narrow-band spectrum, the mean and standard deviation of the total wave force were obtained. It was shown that, in general, free surface fluctuation affects total wave force less in deep water than in shallow water. A question which has frequently been raised in connection with wave force evaluation concerns wave nonlinearity. In this paper, the effects of wave nonlinearity and free surface fluctuation on the mean and standard deviation of total wave force are examined.
2 FORMULATION Consider a rigid vertical circular cylinder of diameter D and length L, placed in water of depth h, assumed to be always less than L (see Fig. 1). In a uni-directional sea of iinite depth, the surface displacement q(x, t) of second order Stokes wave propagating in the x-direction, measured upward from the still water level, can be expressed as2 ?(-%t) = ‘I1
+ 72
where qr= acosx and a2k
)72= ~(oL)cos2x
G(a)-cothn(l+
&).
Here, t = time, ql = the linear component of 7, r12- the second order nonlinear correction, a - the amplitude of 71 and x=kx-wt+b
(5)
x = the phase function, 9 = the phase angle, w - the frequency of ql, k = the corresponding wave number related to wave frequency w by the dispersive relation w2= gktanha!
(6)
g = gravitational acceleration and (II( - kk) = water depth parameter. The surface displacement of a random wave train with narrow-band spectral characteristics, to the second order, also takes the form of eqn (1).3-5 The amplitude, a, of ql, is a Rayleigh distributed random variable and the phase angle $I is a uniformly distributed random variable; they are statistically independent and are slowly varying. The quantity w is interpreted as characteristic frequency and k is the corresponding wave-number of the narrow-band process ql. The wave induced fluid force on the cylinder, placed at x - 0, is evaluated using the Morison formula for which the horizontal fluid particle velocity and acceleration at x - 0 are required. The horizontal component of fluid particle
Fig. 1. Schematic diagram of cylinder in wave field.
Total wave force on cylinder consiaizring free surface jluctuation velocity is2
39
and
u(s, t) - Ul + I42
(7)
where UI= wmSx
@I
is the linear component and
Here and hereafter, the arguments s and t may be omitted. The upper limit in these integrals is x- (h + r#z(L - (h + 7)) +LH((h + ?r)-L)
a2k UZ'F.2~S2X
is the second order correction and, from eqn (5), x = -wt+$. The quantity s is the vertical coordinate (see Fig. l), COShkS W~wsinha,
(10)
where I-Z(.) is the Heaviside step function. Equation (23) simply states that, at any time, if the water surface covers the entire length of the cylinder, X = L; otherwise, X = h+v. From eqns (l)-(3), the mean value of 7 is seen to be equal to zero, and the standard deviation is a,, = a(1 + k22G2)“2
and 3 cosh2k.s gz-p$--&q.
(11)
The horizontal component of fluid particle acceleration can be shown to be given by
l&t)=
du(s, t) dt=
du(s, t) -+u at
au,
lax-4
(25)
and the non-dimensional quantities
where al-
(24)
where the argument ar of G(o) is omitted for brevity and u is the standard deviation of the normally distributed linear wave component ql. To the second order, a7 = Q. The surface displacement is normalized as s-rile
+a2
(23)
Z,+sX
(26)
(13)
g3amSx
and a2k a2 = g4 2in2x g3 =
(14) (15)
%l
and g4-02
12cosh2k.r cosh2ks --. SiIlh2CY sinh2a! >
(16)
t)
flI(S, t) -Kn
dub,Oh dt
(18)
2~5:= uk.
F,(t) -J0xMI
where
u-u&
(19)
is the element drag force, K. = C&D/2, and CD - the drag force coefficient. The total wave force on the cylinder is given by A dFd.s - F,(t) + F,(t) F(t) (20) J0
(28)
(29)
The quantity [ is referred to as the significant wave slope! Expressed in terms of Zr and Z2, the horizontal fluid particle velocity is
is the element inertial force, K,,, - CdxD2/4, p - water density, C,,, - inertial force coefficient, and dF&, t) -K&4(s, t)lu(s, t)lds
+ ;2*c(z:-z:)
where
(17)
where
(27)
are introduced. Here, Z1 and Z2, are statistically independent normal random variables with zero mean value and unit standard deviation. Using the trigonometric identity cos 2x = cos’x-sin2x, one gets s-z,
The wave force acting on an element of the cylinder of length d.r is @(s, t) - dFl(s, t) + mi+,
Z, = >inx
z1+ *u(z:-z:)] [
and the horizontal fluid particle acceleration is (31)
(21)
3 TOTAL WAVE FORCE From eqns (21), (18), (31), (15) and (16), it may be verified
C. C. Tung
40
4 MEANANDMEANSQUAREVALUESOFFIAND FD
statistical moments of F, and FD in eqns (32) and (41), respectively, can not be achieved in closed form because of the form of the functions relating these quantities with Zr and Z,. Close examination of the quantity 8 in eqn (33) reveals that, while (Y’ is finite, A is of the order of the small perturbation parameter 4 ( < 0.02, see Huang and Long6). Thus, to order 5,
The
x sinh2/3-p
(32)
where /3=cr’+A
(33)
cY’=cr+(li-cY)H(s-T)
(34)
sir&/3A sinha’ + Acosho’
(42)
ii==kL( >cY)
(35)
sin20 s sinh2ar’ + 2Acosh2cr’
(43)
is the cylinder length parameter,
and
G= (L - h)/cr = (6 - cr)/ka - (5 - ar)/2a5
(36) is a measure of the degree of protrusion of the cylinder above the still water level, and A = (2r[)31(T -s).
(37) Thus, the total inertial force F, is a nonlinear function of the random variables Zr and Z,; s as a function of Z1 and Zr is given in eqn (28). Examination of eqns (22), (19) and (30) shows that the total drag force FD can not be obtained as straight-forwardly as in the case of F,. To simplify the calculation, the element drag force in eqn (19) is statistically linearized. That is, it is assumed that @D(S, r) - cu(r, 0
cosh2/3 6 cosh2o’ + 2Asinh2o’.
By substituting the above relations into eqns (32) and (41), and retaining terms only to the order of 5,
GmG FM = coshar
+ &(sinh2o’(
&-
F,(t)
=
4 K+%
z
sti2a
x
ZAP’)
2Z131(T
-
E[u21ul] =J
u21ulf,(u)du
-m
- *,g1up G
to the order of ,$ so that 4
(45)
+ (294
s)cosh2a’ +
(39)
The symbol Ep] denotes expected value of the random quantity enclosed in the brackets. The quantity E[u’] = & = gf> is the variance of the zero mean process u. To obtain E[u21uI], the probability density function f”(.) of u is required. This may be derived from eqn (30) by the standard method of transformation of random variables.’ Upon obtaining fue) following the procedure of Huang et aZ.,’ it may be shown that
i) -a’)]}
and
where, by the method of least squares,
c=2?r J-g
sinhar’+ (2?r[) dZ( S - s)coshcr’
(38)
E[u2iut] C-E[U21.
(44)
(z,z
-222)
C(a’)
sinh3a A(d)
(46) where Ned-~+
sinh2y (47)
2
and C(y) =A(y)cosh2y.
(48)
In both eqn (45) and eqn (46), (II’given by eqn (34) is a function of s which is, in turn, a function of Zr and Zr given by eqn (28). The determination of the expected values of FI, FD, Ff and Fi from the above equations involves that of terms such as E[Z2 sinh a’], say, which may be carried out by first introducing an auxiliary random variable. Thus, from eqn (28),
lU* S -zr + $2*E)(z:
Using eqn (38) together with eqn (22) and eqn (30),
(49)
By letting
F,(t) -
w-z2
(41) Again, Fe(t) is a nonlinear function of Zr and Z,.
-zz”).
(50)
the joint probability density function f%,,,@,*)may be obtained by the standard method of transformation of random variables.’ Following the procedure given in
Total wave force on cylinder considering free surface fructuation Huang et al,’ to the order of t, 21-s-
32*5)(&w2)
(51)
41
ml[F;] =(&h&)‘& A*(a) + (2 + D) [A’(C) -A’@)]
z2=w
(52)
and the joint probability density function of s and w is fs,Js,w)=
21;[‘-(2*Z)+-
;+
i +(2x@
-4(2+T2)B(o)+2(1+e2) [
‘(;))-szo) II (61)
$)] where
xeap[-i(52+n?)].
(53)
B(y) =A(y)cosh2y.
(62)
In this way, E[Z2sinhcr’]=E[Z,H(T - s)]sinhar +E[z2H(s - T)]sinhE (54) by virtue of eqn (34) and, using eqns (52) and (53)
5 RESULTS By defining
E[ZzH(S -s)] -E[wH(T -s)]
Similarly, E[Z,H(s - S)] = 0. Without further providing the details of the derivation, the following results are given:
JWII=0 ‘%I-
(56)
4 2x J-K
Z+(2r[)(--+l--_)cotha o&
+ (2?rt) Gd G2 - 111 Ate) 2sinh2cr
441
where
Z-Z(T) -&xP(-2) and &QW-
Ji;z(y)dy
(59)
(see Abramowitz and Stegun’). The mean square values of FI and FD are
I E[F;]=-
(K,,~g)2 cosh’ar
and FI = FI/(&ag), results of E[F], E[@] and E[$‘i ] are obtained. These expected values are plotted as functions of (Yfor sea state parameter [ = 0.015 and cylinder protrusion parameter 5: = 0.0 and 3.0. To examine the effects of wave nonlinearity and free surface fluctuation, these expected values are computed for the following four cases: (1) case NF considers both wave nonlinearity and free surface fluctuation, (2) case LF neglects wave nonlinearity but includes free surface fluctuation, (3) cases N and L, respectively, consider nonlinear and linear waves but neglect free surface fluctuation. From Fig. 2, the coincidence of the curves NF and LF and the coincidence of the curves N and L show that wave nonlinearity has no effect on E[F] whatsoever, regardless of the value of CY.Comparison of the curves NF(S -3.0 and T-0.0) with the curve N [and the curves LF(S -3.0 and S v 0.0) with the curve L] shows that free surface fluctuation affects the value of E[p], only moderately in deep water (of the order of 5) but signitkantly in shallower water (cz < 1). Figure 3 gives E[@. Again, wave nonlinearity has no effect on E[$J, but free surface fluctuation does affect E[Ff], moderately when T-0.0, but only negligibly when T = 3.0.
sinh2a + Q(sinh26 - sinh2a) -(2x5)2
sinh2cr + $3 - g2) (sinh2& -sinh2cr) [
+(2*I)i[(&-~)($$$nh%-sinh2o)-(~-cz)]
1
(60)
C. C. Tung
42
E[m 1.0 ,
0.5-
2.E[p]as function of 01.
Fig.
Figure 4 gives E[Fi]. To study the effect of wave nonlinearity, compare the curves NF and LF for T = 0.0 Since these curves coincide, wave nonlinearity has no influence on E[Fi] in this case. However, comparison of the curves NF and LF for G- 3.0 shows that wave nonlinearity becomes rather important as water depth decreases. The effect of free surface fluctuation is seen by comparing the curves NF@ - 0.0 and 5:- 3.0) and that of N [and the curves LF(S=3.0 and T-0.0) and that of L]. The effect of free surface fluctuation becomes pronounced when water depth decreases. It is mentioned here that neither the Stokes wave nor the
i
linear wave gives a good description of waves in shallow water. The aforementioned observations, while still true, should be viewed in that light.
6 CONCLUSIONS Based on the results of this study, it may be concluded that 1. wave nonlinearity has no effect on E[F], 2. free surface fluctuation affects E[F] moderately in deep water but significantly in shallow water,
a
i
Irig. 3. E@] aa function of oz.
Total wave force on cylinder considering free sur$ace fluctuation
’ LF 6347
‘t-
43
NF (&.o)
l.O-
OS-
0.0 0
I 1
a
I 2
Fig. 4. E[Fi] as function of a.
3. wave nonlinearity only affects E[Fi] in shallow water, 4. wave nonlinearity does not affect I!#], 5. free surface fluctuation is an important consideration in calculating IT&,] especially in shallow water, but 6. free surface fluctuation plays an insignificant role in affecting E[FF].
REFERENCES 1. Tung, C.C., Effects of free surface fluctuations on total wave force on cylinder. J. Engng Me& AXE, 121(1995) 274-280.
Sarpkaya, T. & Isaacson, M., Mechanics of Wave Forces on Offshore Structures, Vaa Nostrand Reinhold, New York, 1981. Tayfun, M.A., Narrow-band nonlinear sea waves. J. Geophysicul Rex, 85 (1980) 1548-1552.
Tayfun, MA., On narrow-band representation of ocean waves, 1: Theory. J. Geophysical Res., 91 (1986) 7743-7752. Huang, N.E., Long, S.R., Tung, CC., Yuan, Yeli & Bliven, L.F., A non-gaussian statistical model for surface elevation of nonlinear random wave fields. J. Geophysical Res., 88 (1983) 7597-7606. 6. Huang, N.E. & Long, S.R., An experimental study of tbe surface elevation probability distribution and statistics of wind generated waves. J. Fluid Mech., 101 (1980) 179-200. 7. Papoulis, A., Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, 1984. 8. Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions, Dover, New York, 1964.
ELSEVIER
Printed in Great Britain All rights reserved 0141-1187/96/$15.00
SOl41-1187(96)00013-2
Total wave force on cylinder considering free surface fluctuation C. C. Tung Box 7908, North Carolina State University, Raleigh, NC 27495, USA
(Received 4 March 1996) Past studies have shown that, in a wave field, fluctuation of the water surface has a profound influence on the wave induced force on objects placed close to the still water level. A recent study examined the effects of free surface fluctuation on the total wave force on a rigid vertical cylinder of small diameter extending from sea floor to beyond the still water level. The waves were considered long-crested, stationary and linear with a narrow-band spectrum and the wave force was evaluated using the Morison formula. In this study, the waves are treated as second order nonlinear Stokes waves with narrow-band characteristics. The effects of wave nonlinearity and free surface fluctuation on the mean and standard deviation of total wave force on a rigid vertical cylinder are examined. Copyright 0 1996 Elsevier Science Limited Keywords: waves, nonlinear waves, wave force, vertical cylinder, random waves, narrow-band, surface fluctuation.
NOMENCLATURE
FI
The following symbols are used in this paper: = function defined in eqn (47); AW = amplitude of vl; a = linear component and second order al, a2 correction of horizontal fluid particle acceleration defined in eqns (13) and (14); = function defmed in eqn (62); W) = function defined in eqn (48); W = drag and inertial force coefficients, CD> cm respectively; = coefficient given in eqn (39); C D = diameter of cylinder;
fx9
t),
@I@,
4
EL.1 F(t),
=
-
element wave force [see eqn (17)], element drag force [see eqn (19)] and element inertial force [see eqn (18)], respectively; expected value of the random quantity enclosed in the brackets;
FD@)v
total wave force [see eqn (20)], total drag force [see eqn (22)] and total inertial force [see eqn (21)], respectively;
F&I
=
FD
= Fn/
( -&K&g),
non-dimensionalized density function
inertial
of U;
= joint probability density function of $ and W; = function defined in eqn (4); = gravitational acceleration; ir, g2, g3, g4 = functions defined in eqns (lo), (ll), (15), and (16), respectively; = Heaviside unit step function; W) h = water depth (see Fig. 1); = quantities appearing in eqns (19) and (18) KD, Km related to drag and inertial forces, respectively; k = wave-number of monochromatic wave and characteristic wave-number of narrow-band waves; L = cylinder length (see Fig. 1); = function defined in eqn (59); QC9 = quantity defined in eqn (59); Iz s = vertical coordinate (see Fig. 1); t = time; = horizontal fluid particle velocity [see eqn (7)]; u(s, 0 = linear component and second order correcu19 u2 tion of U(S, t) defined in eqns (8) and (9), respectively; = horizontal fluid particle acceleration [see eqn +, t)
f,,&*) GP)
@(s, t), flD(.%
= FI/(K,ag), force; = probability
non-dimensionalized
WY;
drag force; 37
C. C. Tung
- auxiliary random variable [see eqn (SO)]; = horizontal axis (see Fig. 1); = dummy variable; = function defined in eqn (58); = random variables defined in eqns (26) and (27), respectively; = quantity defined in eqn (58); = kk, water depth parameter; = quantity defined in eqn (34); = cylinder length parameter defined in eqn (35); = quantity defined in eqn (33); = quantity defined in eqn (37); = non-dimensionalized wave elevation defined in eqn (25); = cylinder protrusion parameter defined in eqn (36); = wave elevation given in eqn (1); = linear component and second order correction of r) defined in eqns (2) and (3), respectively; = quantity defined in eqn (23); = significant wave slope [see eqn (29)]; = water density; = standard deviations of nl,q and u, respectively; = phase angle [see eqn (5)]; = phase function [see eqn (5)]; and = frequency of monochromatic wave and characteristic frequency of narrow-band waves.
1 INTRODUCTION In a wave field, due to free surface fluctuation, an object placed in the vicinity of the still water level experiences wave induced forces only intermittently. It may be anticipated, therefore, that the free surface fluctuation phenomenon would be an important consideration in the design of a floating structure. On the other hand, for a vertical structure extending the entire water depth, the relevant design quantity would be the sum of all the wave forces along the length of the structure. It is then natural to inquire whether the free surface fluctuation phenomenon would have any effect on the total wave force. This problem was addressed in a recent paper.’ Considering the waves as u&directional, stationary and linear with a narrow-band spectrum, the mean and standard deviation of the total wave force were obtained. It was shown that, in general, free surface fluctuation affects total wave force less in deep water than in shallow water. A question which has frequently been raised in connection with wave force evaluation concerns wave nonlinearity. In this paper, the effects of wave nonlinearity and free surface fluctuation on the mean and standard deviation of total wave force are examined.
2 FORMULATION Consider a rigid vertical circular cylinder of diameter D and length L, placed in water of depth h, assumed to be always less than L (see Fig. 1). In a uni-directional sea of iinite depth, the surface displacement q(x, t) of second order Stokes wave propagating in the x-direction, measured upward from the still water level, can be expressed as2 ?(-%t) = ‘I1
+ 72
where qr= acosx and a2k
)72= ~(oL)cos2x
G(a)-cothn(l+
&).
Here, t = time, ql = the linear component of 7, r12- the second order nonlinear correction, a - the amplitude of 71 and x=kx-wt+b
(5)
x = the phase function, 9 = the phase angle, w - the frequency of ql, k = the corresponding wave number related to wave frequency w by the dispersive relation w2= gktanha!
(6)
g = gravitational acceleration and (II( - kk) = water depth parameter. The surface displacement of a random wave train with narrow-band spectral characteristics, to the second order, also takes the form of eqn (1).3-5 The amplitude, a, of ql, is a Rayleigh distributed random variable and the phase angle $I is a uniformly distributed random variable; they are statistically independent and are slowly varying. The quantity w is interpreted as characteristic frequency and k is the corresponding wave-number of the narrow-band process ql. The wave induced fluid force on the cylinder, placed at x - 0, is evaluated using the Morison formula for which the horizontal fluid particle velocity and acceleration at x - 0 are required. The horizontal component of fluid particle
Fig. 1. Schematic diagram of cylinder in wave field.
Total wave force on cylinder consiaizring free surface jluctuation velocity is2
39
and
u(s, t) - Ul + I42
(7)
where UI= wmSx
@I
is the linear component and
Here and hereafter, the arguments s and t may be omitted. The upper limit in these integrals is x- (h + r#z(L - (h + 7)) +LH((h + ?r)-L)
a2k UZ'F.2~S2X
is the second order correction and, from eqn (5), x = -wt+$. The quantity s is the vertical coordinate (see Fig. l), COShkS W~wsinha,
(10)
where I-Z(.) is the Heaviside step function. Equation (23) simply states that, at any time, if the water surface covers the entire length of the cylinder, X = L; otherwise, X = h+v. From eqns (l)-(3), the mean value of 7 is seen to be equal to zero, and the standard deviation is a,, = a(1 + k22G2)“2
and 3 cosh2k.s gz-p$--&q.
(11)
The horizontal component of fluid particle acceleration can be shown to be given by
l&t)=
du(s, t) dt=
du(s, t) -+u at
au,
lax-4
(25)
and the non-dimensional quantities
where al-
(24)
where the argument ar of G(o) is omitted for brevity and u is the standard deviation of the normally distributed linear wave component ql. To the second order, a7 = Q. The surface displacement is normalized as s-rile
+a2
(23)
Z,+sX
(26)
(13)
g3amSx
and a2k a2 = g4 2in2x g3 =
(14) (15)
%l
and g4-02
12cosh2k.r cosh2ks --. SiIlh2CY sinh2a! >
(16)
t)
flI(S, t) -Kn
dub,Oh dt
(18)
2~5:= uk.
F,(t) -J0xMI
where
u-u&
(19)
is the element drag force, K. = C&D/2, and CD - the drag force coefficient. The total wave force on the cylinder is given by A dFd.s - F,(t) + F,(t) F(t) (20) J0
(28)
(29)
The quantity [ is referred to as the significant wave slope! Expressed in terms of Zr and Z2, the horizontal fluid particle velocity is
is the element inertial force, K,,, - CdxD2/4, p - water density, C,,, - inertial force coefficient, and dF&, t) -K&4(s, t)lu(s, t)lds
+ ;2*c(z:-z:)
where
(17)
where
(27)
are introduced. Here, Z1 and Z2, are statistically independent normal random variables with zero mean value and unit standard deviation. Using the trigonometric identity cos 2x = cos’x-sin2x, one gets s-z,
The wave force acting on an element of the cylinder of length d.r is @(s, t) - dFl(s, t) + mi+,
Z, = >inx
z1+ *u(z:-z:)] [
and the horizontal fluid particle acceleration is (31)
(21)
3 TOTAL WAVE FORCE From eqns (21), (18), (31), (15) and (16), it may be verified
C. C. Tung
40
4 MEANANDMEANSQUAREVALUESOFFIAND FD
statistical moments of F, and FD in eqns (32) and (41), respectively, can not be achieved in closed form because of the form of the functions relating these quantities with Zr and Z,. Close examination of the quantity 8 in eqn (33) reveals that, while (Y’ is finite, A is of the order of the small perturbation parameter 4 ( < 0.02, see Huang and Long6). Thus, to order 5,
The
x sinh2/3-p
(32)
where /3=cr’+A
(33)
cY’=cr+(li-cY)H(s-T)
(34)
sir&/3A sinha’ + Acosho’
(42)
ii==kL( >cY)
(35)
sin20 s sinh2ar’ + 2Acosh2cr’
(43)
is the cylinder length parameter,
and
G= (L - h)/cr = (6 - cr)/ka - (5 - ar)/2a5
(36) is a measure of the degree of protrusion of the cylinder above the still water level, and A = (2r[)31(T -s).
(37) Thus, the total inertial force F, is a nonlinear function of the random variables Zr and Z,; s as a function of Z1 and Zr is given in eqn (28). Examination of eqns (22), (19) and (30) shows that the total drag force FD can not be obtained as straight-forwardly as in the case of F,. To simplify the calculation, the element drag force in eqn (19) is statistically linearized. That is, it is assumed that @D(S, r) - cu(r, 0
cosh2/3 6 cosh2o’ + 2Asinh2o’.
By substituting the above relations into eqns (32) and (41), and retaining terms only to the order of 5,
GmG FM = coshar
+ &(sinh2o’(
&-
F,(t)
=
4 K+%
z
sti2a
x
ZAP’)
2Z131(T
-
E[u21ul] =J
u21ulf,(u)du
-m
- *,g1up G
to the order of ,$ so that 4
(45)
+ (294
s)cosh2a’ +
(39)
The symbol Ep] denotes expected value of the random quantity enclosed in the brackets. The quantity E[u’] = & = gf> is the variance of the zero mean process u. To obtain E[u21uI], the probability density function f”(.) of u is required. This may be derived from eqn (30) by the standard method of transformation of random variables.’ Upon obtaining fue) following the procedure of Huang et aZ.,’ it may be shown that
i) -a’)]}
and
where, by the method of least squares,
c=2?r J-g
sinhar’+ (2?r[) dZ( S - s)coshcr’
(38)
E[u2iut] C-E[U21.
(44)
(z,z
-222)
C(a’)
sinh3a A(d)
(46) where Ned-~+
sinh2y (47)
2
and C(y) =A(y)cosh2y.
(48)
In both eqn (45) and eqn (46), (II’given by eqn (34) is a function of s which is, in turn, a function of Zr and Zr given by eqn (28). The determination of the expected values of FI, FD, Ff and Fi from the above equations involves that of terms such as E[Z2 sinh a’], say, which may be carried out by first introducing an auxiliary random variable. Thus, from eqn (28),
lU* S -zr + $2*E)(z:
Using eqn (38) together with eqn (22) and eqn (30),
(49)
By letting
F,(t) -
w-z2
(41) Again, Fe(t) is a nonlinear function of Zr and Z,.
-zz”).
(50)
the joint probability density function f%,,,@,*)may be obtained by the standard method of transformation of random variables.’ Following the procedure given in
Total wave force on cylinder considering free surface fructuation Huang et al,’ to the order of t, 21-s-
32*5)(&w2)
(51)
41
ml[F;] =(&h&)‘& A*(a) + (2 + D) [A’(C) -A’@)]
z2=w
(52)
and the joint probability density function of s and w is fs,Js,w)=
21;[‘-(2*Z)+-
;+
i +(2x@
-4(2+T2)B(o)+2(1+e2) [
‘(;))-szo) II (61)
$)] where
xeap[-i(52+n?)].
(53)
B(y) =A(y)cosh2y.
(62)
In this way, E[Z2sinhcr’]=E[Z,H(T - s)]sinhar +E[z2H(s - T)]sinhE (54) by virtue of eqn (34) and, using eqns (52) and (53)
5 RESULTS By defining
E[ZzH(S -s)] -E[wH(T -s)]
Similarly, E[Z,H(s - S)] = 0. Without further providing the details of the derivation, the following results are given:
JWII=0 ‘%I-
(56)
4 2x J-K
Z+(2r[)(--+l--_)cotha o&
+ (2?rt) Gd G2 - 111 Ate) 2sinh2cr
441
where
Z-Z(T) -&xP(-2) and &QW-
Ji;z(y)dy
(59)
(see Abramowitz and Stegun’). The mean square values of FI and FD are
I E[F;]=-
(K,,~g)2 cosh’ar
and FI = FI/(&ag), results of E[F], E[@] and E[$‘i ] are obtained. These expected values are plotted as functions of (Yfor sea state parameter [ = 0.015 and cylinder protrusion parameter 5: = 0.0 and 3.0. To examine the effects of wave nonlinearity and free surface fluctuation, these expected values are computed for the following four cases: (1) case NF considers both wave nonlinearity and free surface fluctuation, (2) case LF neglects wave nonlinearity but includes free surface fluctuation, (3) cases N and L, respectively, consider nonlinear and linear waves but neglect free surface fluctuation. From Fig. 2, the coincidence of the curves NF and LF and the coincidence of the curves N and L show that wave nonlinearity has no effect on E[F] whatsoever, regardless of the value of CY.Comparison of the curves NF(S -3.0 and T-0.0) with the curve N [and the curves LF(S -3.0 and S v 0.0) with the curve L] shows that free surface fluctuation affects the value of E[p], only moderately in deep water (of the order of 5) but signitkantly in shallower water (cz < 1). Figure 3 gives E[@. Again, wave nonlinearity has no effect on E[$J, but free surface fluctuation does affect E[Ff], moderately when T-0.0, but only negligibly when T = 3.0.
sinh2a + Q(sinh26 - sinh2a) -(2x5)2
sinh2cr + $3 - g2) (sinh2& -sinh2cr) [
+(2*I)i[(&-~)($$$nh%-sinh2o)-(~-cz)]
1
(60)
C. C. Tung
42
E[m 1.0 ,
0.5-
2.E[p]as function of 01.
Fig.
Figure 4 gives E[Fi]. To study the effect of wave nonlinearity, compare the curves NF and LF for T = 0.0 Since these curves coincide, wave nonlinearity has no influence on E[Fi] in this case. However, comparison of the curves NF and LF for G- 3.0 shows that wave nonlinearity becomes rather important as water depth decreases. The effect of free surface fluctuation is seen by comparing the curves NF@ - 0.0 and 5:- 3.0) and that of N [and the curves LF(S=3.0 and T-0.0) and that of L]. The effect of free surface fluctuation becomes pronounced when water depth decreases. It is mentioned here that neither the Stokes wave nor the
i
linear wave gives a good description of waves in shallow water. The aforementioned observations, while still true, should be viewed in that light.
6 CONCLUSIONS Based on the results of this study, it may be concluded that 1. wave nonlinearity has no effect on E[F], 2. free surface fluctuation affects E[F] moderately in deep water but significantly in shallow water,
a
i
Irig. 3. E@] aa function of oz.
Total wave force on cylinder considering free sur$ace fluctuation
’ LF 6347
‘t-
43
NF (&.o)
l.O-
OS-
0.0 0
I 1
a
I 2
Fig. 4. E[Fi] as function of a.
3. wave nonlinearity only affects E[Fi] in shallow water, 4. wave nonlinearity does not affect I!#], 5. free surface fluctuation is an important consideration in calculating IT&,] especially in shallow water, but 6. free surface fluctuation plays an insignificant role in affecting E[FF].
REFERENCES 1. Tung, C.C., Effects of free surface fluctuations on total wave force on cylinder. J. Engng Me& AXE, 121(1995) 274-280.
Sarpkaya, T. & Isaacson, M., Mechanics of Wave Forces on Offshore Structures, Vaa Nostrand Reinhold, New York, 1981. Tayfun, M.A., Narrow-band nonlinear sea waves. J. Geophysicul Rex, 85 (1980) 1548-1552.
Tayfun, MA., On narrow-band representation of ocean waves, 1: Theory. J. Geophysical Res., 91 (1986) 7743-7752. Huang, N.E., Long, S.R., Tung, CC., Yuan, Yeli & Bliven, L.F., A non-gaussian statistical model for surface elevation of nonlinear random wave fields. J. Geophysical Res., 88 (1983) 7597-7606. 6. Huang, N.E. & Long, S.R., An experimental study of tbe surface elevation probability distribution and statistics of wind generated waves. J. Fluid Mech., 101 (1980) 179-200. 7. Papoulis, A., Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, 1984. 8. Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions, Dover, New York, 1964.