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\ and 20 = 2*2 gives 1 / g3 \
23i0
3 3x 3x
9i0 3 \ / 3z 3 z / \
23i0
320 3x 3X
3i0 320 3Z
3Z
= 11 B\ 2B2 {(1 + 4 cosh 2
cosh(4
and where i ^ V--1- The majority of the products of derivatives in the nonlinear terms in (m+\)FLHs, {m+\)FRHS, (m+i)H and ( m + i)n may be replaced for m > 0 by algebraic substitutions of Eqs. (6.22f-u); and this algebraic substitution strategy eliminates the tedious pedestrian tactic of term by term differentiation as the number of terms grow with the nonlinear order m (vide., Fenton, 1985, p. 220, or Mei, 1989, Chapter 12.2.1). Linear Solution €° The nonlinear free-surface \F and \H and pressure 1 n terms in Eqs. (6.21) are i F = 0, i # = 0, i n = 0, (6.23a,b,c) so that the linear solutions from Eqs. (6.22) are {cp
= !! = —ifii coshq(z + l)sinx, Xi = 0,
\rj = ifi = \A\ cosx, (6.23d,e) (6.23f)
333
Nonlinear Wave Theories
provided that the dimensionless linear dispersion equation from Eq. (6.21c) for i F = m = 0 and n = 1 is \C2=tanhq.
(6.23 g)
Recall from strategy #4 in the General Strategy for Solving the BVP in Eqs. (6.21) that non-zero corrections to the dimensionless wave celerity ( m+ i)C in Eq. (6.20d) are computed only at odd perturbation orders m + 1 = 1,3,5,..., etc. A zero-mean free-surface displacement may be computed from
if71 (v) = — /
rn
i
r)dx - — /
{m+\)i;ndx
= 0,
(6.23h)
so that substituting Eqs. (6.23b,d,e) into Eq. (6.2Id) for m — 0 and n = 1 gives i<2 = 0. (6.23i) Recall from strategy #4 in the General Strategy for Solving the BVP in Eqs. (6.21) that non-zero corrections to the Bernoulli constant (m+\)Q in Eq.(6.20a)arecomputedonlyattheet;en/>ertarfearionon3fersm-|-l = 2 , 4 , . . . , etc. The linear homogeneous coefficient i B\ may be computed by substituting Eqs. (6.23b,d,e,i) into the initial condition Eq. (6.21 d) for i A \ = 1 according to lAi = l,
iBx = (XC cafoq)-1
=-}—
(6.23j,k)
sinhg by Eq. (6.23g) so that cosh<7(z + 1) ~ ^sinx, (6.231) sinh^ IV = i£i = cosx. (6.23m) The linear dynamic pressure i p may be computed by substituting Eqs. (6.23g,l) into Eq. (6.2le) according to 3i0 cosh[^(z + l)] cosh[<7(z + l)] IP = -lC-r— = cosx = it]. (6.23n) ox cosh q cosh q Second-Order Solution e 1 The nonlinear free-surface terms iF in Eq. (6.21c) are 1 0 = 1
2 2F
= 2FLHS + 2^Rf/5 = X l 2 , 7 " s i n " x ' n=l
z = 0,
(6.24a)
334
Waves and Wave Forces on Coastal and Ocean Structures
where substituting algebraically for ^FLHS the odd-order derivative from Eqs. (6.22e,f,h) in term {2} below with l = Q,i = j = k — n — \,\r}= £1 from Eq. (6.23e) and \
on the LHS of the CKDFSBC above was identical to the linear velocity potential
and only measurements along the constant unit vectors Pi (t) and P2 (t) are required to determine if Ws (Ty) and WS(TY) intersect transversely (vide., Eqs. (6.197)). Wiggins (1988, Fig. 4.1.6) illustrates the intersection of Ws(Mya) and Wu(Mya) with an homoclinic plane that is spanned by N^> from Eq. (6.207) and the constant unit vectors in the Pi (?) and the P2W directions denoted as Pi (t) and Pi{t), respectively. It is not possible to sketch an analogous figure for the perturbed dissipative system a > 0 and y > 0 because the dimensionality of this system is n = l,m = 2,i = 2 (vide., Wiggins, R 2n x Mm x Tl, 1988, p. 362). Melnikov Integral (a > 0, y > 0) The distance between W\Tya(Q, 0)) and Wu(Tyoi(0,0)) at any point 3> e H may be computed from (Wiggins, 1988, (4.1.47), p. 365) 3/ 3$ — = -T7+ — / + — / - ^ - = 0 , Dt Ot dx ox ay dy dz (x) = 0, +oo [ dz
M«2l(0)) f Antigq-pgp =„
11
A !™
(Xfl , Pfl
+ QigPl + Q2gP2K*(m(t);y Pl
im
-Ql(ci0(PuP2),P0(P2Y,Y = 0)ftng W ('y'Y
= 0)dt = (»dt
,
T CX)
0
-Q2(qolPl,P2).Po(Pz);Y = 0 ) / ^ n ^ ( * ( ° . ) ( r ) ; y =0)dt
(6.208a)
483
Nonlinear Wave Theories
where the hyperbolic saddle points qo(P\ (0), /MO)) and po(^2(0)) are given by Eqs. (6.180c,a) and where * ( 0 ' 0 ) (O = {qh(t),ph(t), Qih(t), G2*(0}(P,(0 = 0,^(0 = 0)
(6.208b)
is a heteroclinic separatrix of the unperturbed system a = y = 0 in Eq. (6.193) where Pi (/) = Piit) = 0 that corresponds to the hyperbolic fixed point of the averaged vector field on the perturbed hyperbolic locally invariant manifold Mya in Eq. (6.203). Because Eq. (6.180d) is valid on the unperturbed normally hyperbolic invariant manifold Min Eq. (6.181), then Pi(t) = P2(t) = 0 is a point on M provided that a\ = eQ,, (6.209a) that is the lowest point on the unperturbed heteroclinic manifold shown in Fig. 6.18. Bowline et al. (1999, p.39) identified this fixed point as a point of weak chaos for cross waves without surface tension or dissipation. Substituting Eq. (6.168c) for eQ and a\ into Eq. (6.209a) and solving for /3 yields
'44
l-All-JL
-11/2
(6.209b)
where
A
= ?7R-
(6 209c)
-
Expanding /3 by the binomial expansion (Chapter 2.3.4) gives, approximately, P - \ (l - ~|) •
(6-209d)
The Melnikov integral M(gi(0)) in Eqs. (6.208) represents (to 0(y2)) the distance between Ws(TYa(0,0)) and W"(T ya (0,0)) at any point $> e H along the one-dimensional vector space Np defined by Eq. (6.207). In order to compute the integral in Eq. (6.208a), only the perturbed dissipative vector field a > 0 and y > 0 in Eqs. (6.202) and the trajectories along the unperturbed heteroclinic manifold H in Eq. (6.193) are required. Computing the Melnikov integral in Eq. (6.208a) on the Pi(t) = Pi(t) = 0 energy level and evaluating the improper integrals as limit of discrete time sequences leads to M(<2i(0)) being identically equal to zero. This implies that the Melnikov
484
Waves and Wave Forces on Coastal and Ocean Structures
method fails to provide the necessary condition for this dissipative system to be chaotic. Liapunov Characteristic Exponents In contrast to Hamiltonian (conservative or non-dissipative a — 0) systems in which the phase-space volume is conserved by the Liouville theorem (Verhulst, 1990), dissipative systems are characterized by continued contraction of the phase-space volume with time t. Dissipative systems are characterized by the attraction of all trajectories passing through a certain domain towards an invariant surface or an attractor of lower dimensionality than the original space. If the dissipative system parameter is changed, then the motion on the attractor may also change from regular (i.e. sink or limit cycle) to chaotic (i.e. strange attractor). Rates of divergence or convergence of trajectories may be computed by Liapunov characteristic exponents. One positive Liapunov characteristic exponent is a strong indicator of chaotic motions (Rasband, 1990). A strange attractor must have at least one negative, one zero, and one positive Liapunov characteristic exponents (Parker and Chua, 1989). For the six-dimensional phase space defined in Eq. (6.178g); there are six real exponents that may be ordered as AH > M2 > M3 > M4 > M5 > M6,
(6.210)
where fi\ is the largest Liapunov characteristic exponent and where one of the remaining five exponents represents the direction along the perturbed dissipative flow and must be equal to zero. For the six-dimensional phase space defined in Eq. (6.178g), there are ten distinct strange attractors with the following spectral signs of the Liapunov characteristic exponents: (+,+,+,+,0,-), (+,+,0,0,0,-), (+,0,0,0,0,-),
(+,+,+,0,0,-), (+,+,0,0,-,-), (+,0,0,0,-,-),
(+,+,+,0,-,-), (+,+,0,-,-,-), (+,0,0,-,-,-),
l
"
}
(+,0,-,-,-,-). Numerical Calculation of /ii For chaos to exist, only the largest Liapunov characteristic exponent fi\ is required in order to determine whether nearby trajectories diverge (fi\ > 0) or converge (fii < 0) on the average (Moon, 1992). The perturbed dissipative vector field a > 0 and y > 0 in Eqs. (6.202) is integrated numerically by the fourth-order Runge-Kutta method with a time step At = 0.02 sec to
485
Nonlinear Wave Theories
Trajectory
Fig. 6.21. Numerical calculation of the largest Liapunov characteristic exponent (after Benettin et al, 1976) where do — 1, x — f, y = i/r + 8f, At — 0.1 seconds.
determine a reference trajectory i/r(t) in the six-dimensional phase space in Eq. (6.178g). The largest Liapunov characteristic exponent /JL\ is calculated from the first variation of the perturbed dissipative vector field a > 0 and y > 0 in Eqs (6.202) according to
/ Sq(t) \ Sp(t) SPl(t) SP2(t) SQl(t) \SQ2it))
dq(t) dq(t) dp(t) dq{t) 3Pl(f) dq(t)
dq(t) 8p(t) dp(t) dp(t) dP\(t) dp(t)
dP2
dP2
dq(t)
dp(t)
9Gi(f) dq(t)
32l(0 dp(t) dQliO dP(t)
dQlit) dq(t)
dq(0 dPl(t) dp(t) dPlit) 3Pi(0 dPi(t) dP2(t)
dq(t) dP2(t) dp(t) dP2(t) dPl(t) dP2(t) dP2(t) dP2(t)
9P\(t) dQ\(t) dPi(t)
dQlit) dP2it)
322(0 dPi(t)
3fi 2 (0 dP2(t)
dqit) dQlit) dp{t) dQlit) dPlit) dQlit) dPlit) dQlit) dQlit) dQlit) dQlit) dQx{t)
dqit) dQlit) dp(t) dQlit) dPlit) dQlit) dP2it) dQlit) dQlit) dQlit) dQlit) dQ2(t)
( Sqit) \ Sp(t) SPiit) SPlit) SQiit) \SQiit)/
(6.212)
Figure 6.21 illustrates a reference trajectory x//(t) and a nearby trajectory VKO + W(t) with initial conditions fo(t = 0) and f0(t - 0) + 8\J/0(t = 0), respectively, that evolve with time yielding the tangent vector 8xjr {^Q, t)'m the six-dimensional phase space defined by Eq. (6.178g) with an Euclidean norm
d[0 = ll^(^o,OII,
(6.213)
where the symbol d(t) is not to be confused with the ordinary derivative symbol dt. A renormalization procedure due to Benettin et al, (1976) and applied by Umeki and Kambi (1989) for parametrically excited surface waves is adopted
Waves and Wave Forces on Coastal and Ocean Structures
e 0.8 " 0.6 " Chaotic Motion 0.4 -
u,>o
0.2 "
1 0.2
1 0.4
-i
1 O.i
0.6
Fig. 6.22. Chaos diagram for the regions in the v, e parameter space where chaotic motion may exists (Hudspeth, et al., 2005).
to avoid overflows and other computation errors that result from the exponential growth of the Euclidean norm d(t). For computational convenience, the initial norm is chosen to be unity and H-ifr is renormalized to a norm of unity every At = 0.1 seconds. The values for Euclidean norm d(t) in Eq. (6.213) are determined iteratively with a total number of time steps N = 500. The Liapunov restart time step is NAt = 50 sec. The largest Liapunov exponent is (Lichtenberg and Lieberman, 1992, p. 315) 1 N ui(if,Sxl/) = lim >hW;(0A/_>.™ MNAt At i—t N->oo
(6.214)
(=1
The following initial conditions: ^o = {0,0,0.1,0,0,0}, Wo = {0,1,0,0,0,0},
(6.215a) (6.215b)
and the following numerical values for the dimensionless parameters: y = 0.25, X,. = 0.5, | =24TT, / I = 1, 0 = 0.45, r =0.016, b = n are selected
487
Nonlinear Wave Theories
for the numerical computations. For twenty different values of the dimensionless damping parameter 0 < v < 1 and twenty different values of the dimensionless Floquet parametric forcing parameter 0 < e < 1, the largest Liapunov characteristic exponents are calculated and a chaos diagram is constructed in Fig. 6.22 for the positive values of the largest Liapunov exponents. This chaos diagram is useful for searching for those regions of the e — v parameter space where chaotic motions may exist. In the e — v parameter space in Fig. 6.22, chaotic motions may occur inside the shaded region shown and regular motions occur outside of the shaded region.
A. Nonautonomous Canonical Transformations by the Generalized Herglotz Algorithm (GHA) In a canonical transformation from the set (q, p) to the set (Q, P), the dynamical state of the system may be defined in terms of the new set of variables (Q, P). Because the canonical transformation destroys the distinction between generalized coordinates and conjugate momenta, the set of new variables (Q, P) may no longer be restricted to sets that define the configuration of the system as did the original set of generalized coordinates q (Desloge, 1982; Guenther and Schwerdtfeger, 1985; and Guenther et al., 1996). The extension of the Herglotz algorithm to nonautonomous dynamical systems (GHA) significantly reduces the effort required to suspend a nonautonomous Hamiltonian component (Orum et ah, 2000 and Hudspeth et al., 2005). Canonical Transformations: Definition and Properties A time-dependent transformation (q, p) =>• (Q, P) where Qi = Qi (q, P, 0,
Pi = Pi (q, P, 0 ,
i = l,2,.-.,N
(A. la,b)
is canonical if and only if there exists a generating function F(u,U, t) such that (Desloge, 1982, p. 794) N
dF(u, U, 0 = ^2(vt dui - Vt dUi), !=1
(A.2a)
488
Waves and Wave Forces on Coastal and Ocean Structures
where d2F du\dU\
32F duidllj
82F duNdUi
d2F du\dUN 7^0,
(A.2b)
32F
and where the combinations of 2N new and old variables are Ui,Vi
=
\
Ui,Vt =
\Pi,-9i,
Qi,Pi
i =
l,2,...,N.
Pi,-Qi,
(A.2c,d) (A.2e,f)
Equations (A.2) represent a generalized definition for a canonical transformation (Desloge, 1982). Poisson Brackets The Poisson bracket for any two members Rj, Rk are of the transformed set of 2N variables Q = Q\,..., QN and P = P\,...,P^ for the transformation inEq. (A.l) is N
r
dRjdRk
lRj,Rklw = ^ Y, |_ dqi dpi
dRjdRt dpi dqi
(A.3)
and the transformation by Eq. (A.l) is canonical if and only if the Poisson brackets satisfy IQi, Cjlqp = 0,
IQi, Pjlqp = SU,
IPi, Pjlqp = 0,
(A.4a-c)
where Sjj is the Kronecker delta (vide., Chapter 2.2.3). Generalized Herglotz Algorithm (GHA) The Herglotz algorithm for autonomous dynamical systems (Guenther et ah, 1996) may be generalized by: (1) including time t as a parameter, and (2) defining a generating function by Eq. (A.2a) with a non-zero determinant of second derivatives with respect to u, U defined in Eq. (A.2b). The GHA transforms a set of 2N variables (u, v) to a set of 2N new variables (U, V) by choosing
489
Nonlinear Wave Theories
N new variables £/, and then computing the remaining N new variables V, uniquely from the chosen [/,• so that the transformation (u, v) ->• (U, V) is canonical. The old («,-, u,-) and new (£/,-, V,) variables may be any of the four combinations in Eqs. (A.2c-f). The transformation (u, v) -> (U, V) may be shown to be canonical by the Poisson bracket conditions equations (A.4a-c). There are two types of the GHA. Type I equates N new variables to N old variables [/,• (u, v, t); and Type II equates N old variables to N new variables m — ui (U, V, t). In both types, however, the N new variables are to be chosen. GHA: Type I Equate N new variables to N old variables Ui =
Ui(u,\,t),
i = l,2,-
,N,
(A.5a)
,N
(A.5b)
such that the Poisson bracket in Eq. (A. 3) satisfies
m,UjJw
= 0,
i,j = 1,2,
and the determinant dUi dUi
dUi dvN #0.
dVj
dUN
(A.5c)
dUN dvN
Equate N Herglotz auxiliary functions X, (u, v) to the absolute value of the ratios of the Af old variables in either of the following ratio forms: Xi=Xi(u,v)
=
i =
l,2,-..,N,
(A.5d) (A.5e)
such that the Jacobian of the chosen N new variables U and the Af Herglotz auxiliary functions X are nonzero; i.e. 9(U,X) #0. 3(u,v)
(A.5f)
490
Waves and Wave Forces on Coastal and Ocean Structures
Solve Eq. (A.5d or e) for vi = u;(u,X), substitute f,(u,X) into Eq. (A.5a) and invert Eq. (A.5a) to obtain Ui=Ui(V,X,t),
i = l,2,-..,N.
(A.5g)
Compute the generating function F(u, U, t) in Eq. (A.2a) from N
du
dF(\\{\J, X, t), U, t) = J^^i
V dlJ
i -
i
( A - 5h >
i)
i= l
by expanding to obtain N
E
/dF
N
dXi + (
N
dF
dUt
\
3 V
^ 3 V
dUi
j=\
'
dx
\
= E E J^Yt i + E J w i=i y = i
'
-
VidUi
)-
^ A - 5i )
/
Equate the coefficients of like differentials in Eq. (A.5i) and compute: (1) the generating function F(u, U, t) from the integral N
pXi
N
a
V
F(U, u, o = E / E J ^rdx'i + C(U> 0> and 1=1
7=1
(A 5j)
-
'
(2) the remaining N new variables Vi (u, U, t) from
E
duidF J-^T--Z7T>
V
i = h2,...,N,
(A.5k)
y=i
where the integration constant C(\J,t) in Eq. (A.5j) is an arbitrary additive function of both time and U that may be neglected. The 2N old («,, i;,-) and new ([/,, Vi) variables may be any of the four combinations in Eqs. (A.2c-f). The transformation given by Eq. (A.5a) and Eq. (A.5k) may be shown to be canonical by the Poisson bracket equations (A.4a-c). In order to compute the transformed Hamiltonian in terms of the new variables (U,V), the inverse canonical transformation (u(\J,V,t), v(U,V,t)) must be computed and the new Hamiltonian K(Q, P, t) is given by K(Q, P, t) = tf(q(Q, P, t), p(Q, P, t), t) +
dF(u,\J,t) \' ', at
(A.51)
491
Nonlinear Wave Theories
where F(u,V,t) is the generating function in Eq. (A.5j) for the canonical transformation (U(u, v, t), V(u, v, t)). GHA: Type II Equate N old variables to N new variables ut = m(lJ,V,t),
i = 1,2,..., AT,
(A.6a)
such that the Poisson bracket Eq. (A.3) satisfies EM/.M/IUV
= 0,
i,j = 1,2, ...,N,
(A.6b)
and the determinant dVi
dvN
dui
(A.6c)
^0.
dVi
du N dVN
8UN
dVi
Equate A^ Herglotz auxiliary functions X, (U, V) to the absolute values of the ratios of the new variables in either of the following forms: Xi = Xi(U,\)
=
\Ui/Vi\ Wi/Ui\,
i =
l,2,..-,N,
(A.6d) (A.6e)
such that the Jacobian of the N old variables u and the Af auxiliary functions X is nonzero; i.e. 3(u, X) #0. (A.6f) 9(U,V) Solve Eq. (A.6d or e) for Vt = V/(U, V), substitute V,(U,X) into Eq. (A.6a) and invert Eq. (A.6a) to obtain Ui = Ui(u,X,t),
i=
l,2,...,N.
(A.6g)
Compute the generating function F(u, U, t) in Eq. (A.2a) from N
JF(u,U(u, X, 0,0 = J2iVi i=i
dUi
~ Vi
dui)
(A.6h)
492
Waves and Wave Forces on Coastal and Ocean
Structures
by expanding to obtain N
x^{
dF
E
i^rdXi + ir-du>
*—' \ aX; (=1
dF dui
V
dXi +
N
E ( E J jY
E VJ °irdu' - u«du' I •
i=l \ y = i
7=1
'
(A-6i>
Equate the coefficients of like differentials in Eq. (A.6i) and compute: (1) the generating function F(u, U, t) from the integral N
CXi
F(U,u, o = E / ;' = 1
N
dUV
E J^hdx'i+C(u'tX 7= 1
and
(A 6j)
-
'
(2) the remaining N old variables v,- (U, V, t) from vi(V,V,t)
J^ dUj = YiVj—±-—,
dF
i = l,2,...,N,
(A.6k)
7=1
where the integration constant C(u,f) in Eq. (A.6j) is an arbitrary additive function of both time and u that may be neglected. The 2N old («,, u,-) and new (£/,-, V,) variables may be any of the four combinations in Eqs. (A.2c-f). The transformation given by Eq. (A.6a) and Eq. (A.6k) may be shown to be canonical by the Poisson bracket equations (A.4a-c). The transformed Hamiltonian A^(Q, P, t) in terms of the new variables is given by K(Q, P, t) = H(q(Q, P, 0 , P(Q, P, 0, 0
-ir^'
at
where F(u,U,f) is the generating function for the canonical transformation (u(U,V,0,v(U,V,0).
6.9. Problems 6.1.
Applying the Dean stream function coefficients from the Dean Stream Function Tables (1974), compute the dimensionless water surface elevation r)' and horizontal water particle velocity u' for the following
493
Nonlinear Wave Theories
conditions and compare with the tabulated values from the Dean Stream Function Tables (1974): Dean * Tables (1974) Case 10D
s—z+ h h 0.9 0.9 0.9
0 (deg) 130 140 130
11
H
- 0.294
, u =
u H/T
- 0.625
in which
v'm = |r - ^ !>(«)sinh I (^r) ^ +ti}cosnd> u'{6) = - 22 X(n) I
1 cosh j
cos nd.
Case 10D: 3rd Order Stream Function Wave Theory Definitions: XQ = (g/2n)T2 is the deep-water wavelength calculated from linear wave theory, H is the wave height, g is the gravitational constant, T is the wave period, X(n) is the nth stream function coefficient, X is the nonlinear wavelength, *,, is the value of stream function on the free surface r/(9). Wave parameters: H/X0 = 0.170401, h/X0 = 1.999993, H/h 0.085201, A./A.0 = 1-222070, V^/igHT) = -0.015407,
6.2.
=
Dimensionless stream function coefficients: X(l)/(HTg) = -0.488993 - 0 5 , X(2)/(HTg) = -0.863825 - 11, X(3)/(HTg) = - 0 . 6 3 8 3 0 4 - 16. Verify that Eqs. (6.82 and 6.83) satisfy exactly the kinematic free surface boundary condition given by Eq. (6.81c).
Chapter 7
Deterministic Dynamics of Small Solid Bodies
7.1. Introduction Engineering analyses generally begin with an observation of the principles of physics that govern a physical phenomena. After these principles of physics have been carefully identified, a mathematical model may be developed that approximates these principles of physics. The careful selection of a mathematical model is always done after the physics have been carefully evaluated. It is crucial that a coastal or ocean engineer never juxtapose the sequence of these two elements in the engineering design and analysis. In this spirit, an observer standing on the boat landing near one of the corner jacket legs of a pile-founded steel jacket structure would observe visually that this leg has no sensible effect on the incident wave field. Under careful observation, especially in the case of long-crested waves, the observer could see or not see the following two major physical phenomena: Seen:
The observer would see flow separation around the slender jacket leg that results in vortex shedding and wake turbulence on the down-wave side of the jacket leg. In addition, some slight decrease in the water surface elevation may be seen in this down-wave, turbulent-wake zone. Not Seen: The observer would not see the incident wave modified significantly by the slender jacket leg. Additionally, if this platform were free to respond with dynamic motions such as in a small, semi-submersible buoy, no radiated waves would be observed radiating away from the slender member or buoy. 495
496
Waves and Wave Forces on Coastal and Ocean Structures
In contrast to what is seen or not seen in the case of a large body (vide., Chapter 8), the effects of flow separation and wake turbulence require that real fluid effects must be considered in the selection of a mathematical model to approximate the wave-induced pressure forces on a small body. Physically, viscous and turbulent stresses dissipate energy. As is the case with every mathematical model for dissipative stress tensors, empirical coefficients are required. This would be the case if it would be possible to model analytically the flow field around the small body. Just as Newton's law of viscosity requires an analytical model of the flow field (the gradient of the flow field dll/dy) plus an empirically determined coefficient (fi = dynamic viscosity), the mathematical model for the wave-induced pressure forces on a small body represented by the Morison equation requires an analytical model for the flow field (wave field kinematics) and empirically determined coefficients (Cd and Cm). Figure 7.1 illustrates the physics of a fixed and oscillating small member structure. The question posed in Fig. 7.1 is whether or not the inertial Cm and
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c
)
Dynamic Rasponsa Jn Portion
MtKli M o d a l
FT = Cmu, +CdLu-
(Cm - 1) X- CdL X
Fig. 7.1. Small body dynamic response.
497
Deterministic Dynamics of Small Solid Bodies
kA
2b <—»
kA>l;kA/(kh)3>l
2b V7.
Deep-Water
2b
2b 7J^
Z7
Finite-Depth
Fig. 7.2. Comparisons of small and large bodies in deep- and finite-depth water.
linearized drag Cdi coefficients for the exciting force on a fixed body are the same as the added mass Ca = Cm — 1 and viscous damping Cm coefficients for the same body oscillating in otherwise still water. In contrast to Fig. 8.1, the small body in Fig. 7.1 that is oscillating in otherwise still water does not generate radiated waves; but, rather only local vortices due to flow separation around the body. Figure 7.2 illustrates these two extreme possibilities as well as the possibility of a complex structure that may simultaneously include both sizes of members and, consequently, both effects. Because there is at present no single unified wave force field theory for computing the entire spectrum of wave force possibilities between these two extremes, only the effects based on the two extreme member sizes will be reviewed. The first effect to be reviewed is the small body or Froude-Kriloff theory where the size of the structure makes no sensible modification to the incident wave field (Chapter 7). The second effect to be reviewed is the large body or diffraction theory that also includes the possibility of wave radiation for compliant or for semi-constrained floating bodies (Chapter 8). Figure 7.3 illustrates the three methods that are applied to estimate each of these two extreme loads. Morison, et al. (1950) introduced a semi-intuitive equation to compute wave forces on fixed immersed objects having characteristic dimensions that are small compared to the wavelength of the incident wave. This equation has since been extended to the three-dimensional analyses of arbitrarily-oriented small members on offshore structures and to the relative-motion of small members. Sarpkaya and Isaacson (Chapter 3, 1981) give a detailed analysis and history of the Morison equation. Mei (1989) gives a concise but
Waves and Wave Forces on Coastal and Ocean Structures
498
METHODS FOR WAVE LOAD PREDICTIONS
ANALYTICAL
FROUDE-XRILOFF HYPOTHESIS
DIFFRACTION THBORI
DIMENSIONAL AKALIBES
NAVIER-SrOXES EQUATION
|
BERNOULLI EQUATION
| B . I . E . M . | | FBOUDE-KBILOFF | | M F F R A C H o i r ~ | MORISON H ) .
STRIP THEORT
VSl
SPECIAL CBOHETRZ
m
U R E A S THEORY KIHHATIOS
NONLINEAR THEORY KINEMATICS
URSELL HULTHOLE | CREEK'S FUNCTION
NUMERICAL STREAM FUNCTION
VARIATIONAL METHOD STOKSS PERTURBATIONS
CONFORMAL TRANSFORMATIONS
Fig. 7.3. Three methods for computing loads on coastal and ocean structures.
cogent argument applying classical viscous flow theory to demonstrate why the Keulegan-Carpenter parameter (or period parameter) K = UmT/D is important in separated flow. The Morison equation for the horizontal force per unit length on a fixed small body is approximated by dFx(z,t) = dFm(z,t)
+ dFd(z,t),
(7.1)
where dFx{z,t) = the wave-induced, hydrodynamic, horizontal pressure force per unit length acting normal to the axis of the small body; dFm(z, t) = the inertia component; and dFd(z, t) = drag component.
7.2. Small Body Hypothesis (Morison Equation) In order to appreciate the small body hypothesis in the Morison equation, it is helpful to compare the hydrodynamic pressure force on a totally porous pile that is composed completely of the same fluid. Newton's second law of motion may be applied to determine the time rate of change of momentum on this fluid pile in a well-known integral form that is derived in elementary texts on the fundamental mechanics of fluid for unsteady flows (e.g., Schlichting, 1979). A finite control volume may be applied to demonstrate that Newton's second law for the time rate of change of momentum M(t) in an accelerating
499
Deterministic Dynamics of Small Solid Bodies
fluid is dF(t) =
dM(t) dt '
(7.2a)
where dF(t) = sum of external surface and body forces; viz., dF(t) = dFN(t) + dFs(t) + dFB(t),
(7.2b)
where dF^it) = normal surface force; dFs(t) = tangential surface shear stress force; and dFs (f) = body force. The total force in Eq. (7.2b) is equal to the time rate of change of momentum within the control volume plus the transport of momentum or flux of momentum across the control volume boundaries (i.e., the inertia of the Eulerian fluid field). The finite control volume shown in Fig. 7.4 with an outward pointing surface normal vector dS = ndS contains a mass of fluid in the two finite control volumes Vol. 1 + Vol. 2 at time = t. The same mass at time = t + At is in Vol. 2 + Vol. 3. The fluid momentum is given by M(t)
•E
pu(x,t)dV(x),
v r >
dFx \
dz
ItTfl
$
= &
Fig. 7.4. Finite control volume for a "water pile".
(7.3)
500
Waves and Wave Forces on Coastal and Ocean Structures
and for a finite control volume, the time rate of change momentum from Fig. 7.4 is dM(t) dt
=
JM2 (0 + M3 (t) [ {+^
JMj (t) + M2 (t)}
At
At
hm At^o
(0
Adding and subtracting the quantity {M\(t)}(t+At)/At terms in Eq. (7.4) gives {Afl(0 + M2(t)}{t+At)
(7.4)
and rearranging
- {Mi(0 +M 2 (01(f) At
dM(t)
=
dt
hm At^o
+
{M3(t)}{t+At)
~ {A*i(0}(f+A0
At
-if
(7.5) The first term [•]£ on the RHS in Eq. (7.5) yields the local time rate of change of momentum in the fixed control volume given by lim
{Mi(t) +M2(t)}{t+At)
- {Mi(0 + M 2 ( 0 } ( 0
dt
At
At^O
dM(t)
Substituting for Af (?) from Eq. (7.3) yields dM(t)
*7r = 7,llh^-<)dV&-
(76a)
The second term on RHS in Eq. (7.5) [ • ] / yields the transport of momentum (or flux of momentum) out of the fixed control volume given by lim Af-+0
{A/ 3 (0}(t+A0 At
{Afi(0}(r+A0 At
= (p pu(x,t)[u(x,t)
•n]dS(x).
-1/ (7.6b)
Substituting Eqs. (7.6a and 7.6b) into Eq. (7.2) gives dF(t)
dM(t) dt = —
pu(x,t)dV(x)+
j> pu(x,t)[ii(x,t)»n]dS(x)
(7.7)
501
Deterministic Dynamics of Small Solid Bodies
that relates the time rate of change of momentum of a fluid particle in the Lagrangian coordinate axes to the fluid inertia in the Eulerian coordinate axes. Now, if a slender vertical circular pile were placed within the fluid control volume Vol. 1 + Vol. 2 in Fig. 7.4 and if the velocity distribution could be determined both within the control volume and across the surface of the finite control volume, then the total effect of the sum of all the external surface forces (normal and tangential) and body forces could be computed from the inertia of the fluid by Eq. (7.7). Because a closed-form analytical solution is not known for the exact velocity distribution around a slender cylinder in a real viscous fluid because of flow separation and wake turbulence in unsteady fluid flows, it is necessary to apply approximate methods to estimate the integrals in Eq. (7.7). If the slender cylinder radius is small compared to the incident wavelength (b/k <£. 1), one approximate method for estimating these integrals would be to expand the integrands in a Taylor series about the vertical centerline of the slender cylinder. Retaining only the linear terms in this series expansion would be equivalent to estimating the wave particle kinematics at the centerline (i.e., xo, yo, Zi) of the slender cylinder as though the cylinder were not present. For the horizontal force component in the positive x direction, Eq. (7.7) reduces to
<1FX(XQ + Ax, jo + Ay,
n,t)
9 + —idFx(xo, ox
«s dFx(x0,yo,Zi,t)
+ — [dFx(x0,yo,Zi,t)]Ay dy ^ ~dt / / /
y0,Zi,t)]Ax
+ 0(Ax)z
Pu(xo>yo,Zi,t)dV(x)
+ (f> pu(x0,yo,Zi,t)u(x0,yo,Zi,t) Jcs du(x0,yo,Zi,t) P / / / dV& dV(x) Jt
•hdS(x)+
0(Ax)
•Iffcv >
+ pu(xo,y0,Zi,t)
(t u(x0,yo,Zi, t) • ndS(x) + 0(Ax) Ics
502
Waves and Wave Forces on Coastal and Ocean Structures
Ifu(xo,yo,Zi,t)
• « ~ \u(xo,yo,Zi,t)\cos9,
/// dV(x) = 7Tb2dzr, JJJcv
and if
(b dS(x) = n2bdzi Jcs
for the z'th circular disk of height dzi in Fig. 7.4; then, dFx(x0,y0,Zi,t)
«s (prtb2)
' ' " at
dzi +
p(7T2b)u(x0,yo,Zi,t)\u(x0,yo,Zi,t)\dzi.
(7.8) In order to obtain an equality in Eq. (7.8), two empirically determined coefficients Cm and Cd/4jt reduce Eq. (7.8) to dFx(xo,yo,Zi,t) dzi 2
du(xQ,yo,Zi,t)
= Cm{pTtb )
1 h -Q(p^)«(xo,jo,Z(,OI"Uo,) ; o,Z(,OI(7.9a)
Examining carefully the sum of the external forces in Vol. 1 in Fig. 7.4 illustrates that the momentum transport theorem is equal to both the skin (friction) and form (wake) drag forces. The total resultant force in Eq. (7.2b) includes: (1) a normal total pressure force p(b,6,zi)dS and (2) a tangential shear force x{b,6,Zi)dS. The total force per unit length in the direction of wave travel in the x coordinate direction shown in Fig. 7.4 is given by (Dean andHarleman, 1966) dFx(xo,yo,Zj,t) dzi = j -/
P(b,0,zi,t)bcos6d6-
= {-/
lPo(zi) + 8p(b,0,Zi,t)]bcos6d6-
U
2it
/
r(b,6,Zi,t)b
sin0d9\
r(b,e,Zi,t)bsm9de\
p2n
Sp(b,9,Zi,t)cos6d6-
I
I
T(b,9,zi,t)sin9d9\.
(7.9b)
503
Deterministic Dynamics of Small Solid Bodies
Equating Eq. (7.9a) to Eq. (7.9b) demonstrates that the Morison equation (7.2) is equal to both the dynamic pressure force 8p(b,9,Zi)dS and tangential shear force r(b, 6, Zi)dS but not to the hydrostatic (body force) pressure force po(Zi)dS. Designs of slender body structures by the Morison equation must account for the collapse of the slender body due to the hydrostatic pressure force separately from the hydrodynamic pressure forces in Eq. (7.9b) computed by the Morison equation. Comparing Eq. (7.9a) with Eq. (7.9b), it may be observed that the hydrodynamic pressure forces on a slender circular cylinder computed by the Morison equation may be obtained from Eq. (7.7) by replacing the first volume integral on RHS with an inertia force dFm(z,i,t); and by replacing the second surface integral on RHS with a drag force dFd(zi,t). Assuming that the fluid is incompressible, Dp/Dt = 0 and that the small body hypothesis is valid, Eq. (7.7) may be written as dFx{z,t) = dFm(zi,t)+dFd(Zi,t),
(7.10a)
where for a right circular cylinder of diameter D = 2b; AV < r ( 7tL>2\ du(xo,yo,Zj,t) j t\ dFm(x0,yo,Zi,t) = Cm I p—— I — dzt,
rn,nU\ (7.10b)
dFd(xo,yo,Zi,t)
(7.10c)
= -Cd(pD)u(xo,yo,Zi,t)\u(xo,yo,Zi,t)\dzi.
Identification from load data and selection of the force coefficients Cm and Cd are deferred until Sec. 7.6. The Morison equation (7.2) for a fixed slender body may be extended heuristically to a dynamically responding slender body by the modified Wave Force Equation (WFE, relative motion generalization of the Morison equation, American Petroleum Institute API RP2A, 1987) that is given by AT? i *\ ,du(zi,t) dFm(zi,t) = pVi———
fdu(zi,t) + (Ca)pVi I ——
dFd{zi,t) = -CdpAi[u(zi,t)
- Xi(t)]\u(Zi,t)
» \ Xiit) I , - Xi(t)\,
(7.11a) (7.11b)
where Xt(t) = horizontal Lagrangian motion of the ith element of a slender body and where the over dots denote ordinary temperal derivatives of Lagrangian solid body motions (vide., Eqs. (3.48) in Chapter 3.5). Applications of a linearized form of the WFE in Eq. (7.11) are reviewed in Sec. 7.8
504
Waves and Wave Forces on Coastal and Ocean Structures
for single-degree-of-freedom (SDOF) articulated tower and for an uncoupled three degrees-of-freedom (MDOF) semi-immersed moored pontoon made of rigidly-connected double horizontal circular cylinders. The Morison equation (7.10) may be integrated for a total depth-integrated force on a vertical circular cylinder by applying linear wave theory kinematics (vide., Sec. 4.4). The horizontal water particle velocity and local acceleration at the vertical centerline of a circular cylinder at x = 0 are given by u(z,t) =
H gTcoshk(z 2 A.
+ h)
cosh kh
2nt cos
Hgn cosh k(z + h) (2nt cosh kh sin
du(z,t) dt
(7.12a) (7.12b)
V T
Substituting Eqs. (7.12) into Eq. (7.10a); assuming that the force coefficients Cm and Cd are constants; and integrating over the still-water-depth —h
ynD HKmsm[
4
r
Cd
\ T •}
/2nt
+ ^-yDH2Kdcosl Km =
—
2nt cos ——
VT
tanhkh
(7.13a) (7.13b)
2kh d = a 1+ sinh 2kh
(7.13c)
K
An example of the variations of Fx(t), Fm{t) and Fd(t) over one wave cycle for the following conditions: H = 34 ft, D = 4 ft, is illustrated in Fig. 7.5.
T = 12 s,
A = 8 5 ft Cd = 0J
Deterministic Dynamics of Small Solid Bodies
505
_3 000 o t T nTTlTrTTI111TtrtTrTTr1llTITtTTTTTI -180 -120 -60 0 60 120 180 S = 2m/T Fig. 7.5. Morison equation total and component depth-integrated forces on a vertical circular cylinder.
7.3. Drag dFd and Inertia dFm Forces An application that requires some knowledge of wave kinematics and dynamics is the response of small bodies to wave-induced velocities (kinematics) and pressures (dynamics). The adjective small restricts this analysis to body dimensions that are a small fraction of both the wavelength (b/X.
506
Waves and Wave Forces on Coastal and Ocean Structures
U(t)
WAKE
kWWWM
Fig. 7.6. Three types of elliptical cylinders.
ideal fluids may be applied to estimate these forces in unsteady flows. These potential flow estimates are also modified by real fluid effects. Drag Forces dFd: Two general types of drag forces are encountered; viz. (1) skin (friction) drag and (2) form (wake) drag. Figure 7.6 illustrates this general case for an elliptical cylinder of various ratios of the semi-minor to the semi-major axes. For an elliptical cylinder with a zero semi-minor axis (i.e., horizontal flat plate), only skin (friction) drag occurs. For equal semiminor and semi-major axes (i.e., a horizontal circular cylinder), both skin and form drag occurs. For an elliptical cylinder with a zero semi-major axis (i.e., a vertical flat plate), only form (wake) drag occurs. Form drag is associated with the stagnation pressure caused by a body blocking local velocities. Consider the verticalflatplate oriented perpendicular to the horizontal flow in Fig. 7.7. The pressure rise from point 1 to point 2 in Fig. 7.7 may be estimated from the unsteady form of the Bernoulli equation (Eq. (3.46) in Chapter 3.4) for horizontal flow by 30 at
p p
U2 2
•
Q(t)
d4> "dt
uz + - + — + gz P
J2 P 2 that for steady flow (d(»)/dt = 0) at the same vertical elevation (zi = n) reduces to P2 ~ Pi
ul
Ap
P where Ap = the local rise in pressure at the body due to the flow blockage effect and U\ = steady free-stream velocity. The force may be estimated
Deterministic Dynamics of Small Solid Bodies
507
•^ i
U(t)
3T
Fig. 7.7. Form (wake) drag past a vertical flat plate.
by integrating the pressure over the frontal area only of the body normal to the flow
r uf Fx « / pdA a pA-£that may be made into an equality by defining Cd to be the required constant of proportionality; i.e., Fd = \cd{pA)U\,
(7.14)
where Cd = dimensionless drag coefficient that is a function of the body shape and turbulence state that may be determined by the Reynolds parameter R from Chapter 3.6; p = fluid mass density; U\ = horizontal steady velocity; A = frontal area perpendicular to U\. Although the drag coefficient Cd varies with the Reynolds parameter R, fully turbulent values are approximately 1.0 for a cylinder and 2.0 for a sphere. The Reynolds parameter dependence for a circular cylinder in steady flow (cable, pile, etc.) is shown in Figs. 7.8 and 7.9. Equation (7.14) must be modified for oscillatory flows by Fd(z,t) = -Cd(pA)u(zi,t)\u(z,t)\
(7.15)
to insure that Fd(z, t) acts in the same direction as u(z, t) for oscillatory flows.
508
Waves and Wave Forces on Coastal and Ocean Structures
III III
1 1 1 1 III
D[mmJ •> 0.05, • 0.1 • RJ • 1.0 H83SWC6 " 3 0 if • 7J Wieselsberger » *ZJ> e 60.0 • 300.0 ---Thcotydueto lamb
,
_--
s \
L-
Fig. 7.8. Drag coefficient for circular cylinders in steady flow (Schlichting, 1979).
I
\ Co
s %
* f!S^ \
®
•
Uehster
® ' • "
v
"^
"V
MAX NJJ a.
*" *=»,
\ \ »<:
«r»' * e V ?
4 6
-e
V2 ' V
?
*£V* *SV?
4S
V2
*eV
e_B3 *~ v
Fig. 7.9. Steady-state drag coefficient, Cds, for a smooth cylinder in the transition and transcritical regimes (Schlichting, 1979).
7.3.1. Inertia Forces Inertia forces are the body reaction to a fluid accelerating in the vicinity of the body. The volume of fluid accelerated is proportional to the volume of the body. The constant of proportionality is Cm where Cm = an inertia mass
509
Deterministic Dynamics of Small Solid Bodies
coefficient that integrates the effects of all local fluid accelerations in terms of (1) the displaced fluid volume V and (2) the approach acceleration 3 U/dt; i.e., Fm(z,t) = CmPVd-^^-. at
(7.16)
The inertia coefficient Cm may be computed from potential theory for flow around bodies having simple geometric shapes (e.g., cylinders, spheres, bodies of revolution) by solving for a scalar velocity potential O from Laplace's equation constrained by the boundary conditions; solving for the pressure p from <E> by the Bernoulli equation; integrating p around the body surface to obtain Fm; then dividing by pVdU/dt to obtain Cm. For example, Cm = 1.5 for a sphere and Cm = 2.0 for a circular cylinder. These values are exact and are not determined experimentally. However, they need to be adjusted empirically to account for real fluid effects that are not included in the potential flow solution.
7.4. Comparison Between a Fixed Cylinder in Accelerating Flow and an Accelerating Cylinder in Still Fluid To illustrate the concepts of an added mass coefficient Ca and an inertia coefficient Cm — 1 + Ca, consider the following two examples: (i) an infinitely long circular cylinder accelerating in an unbounded still fluid (i.e., no free surface or other free or fixed boundaries); and (ii) the acceleration of an unbounded fluid (i.e., no free surface or other free or fixed boundaries) past an infinitely long fixed circular cylinder. In both of these examples, the infinitely long circular cylinder may be either vertical (x = r cos 9; y = r sin 6) or horizontal (x = r cos 9; z = r sin 9) in a 3D Cartesian coordinate system with the z axis positive up. The velocity vector is defined by the negative gradient of a scalar velocity potential according to
-
,
,
^
q = {ur,ue} = -V
f9*-
19$
-
—er + -—e# [ dr r d9
(7.17)
510
Waves and Wave Forces on Coastal and Ocean Structures
7.4.1. Accelerating Cylinder in Still Fluid For the horizontal circular cylinder in Fig. 7.10 in an unbounded otherwise still fluid where x = r cos 0 and z = r sin 0 a scalar velocity potential
0
(7.18)
b2 *(r,0,f) = -U(t)— cos<9,
(7.19)
rdr\dr is given by (Lamb, 1932, pp. 62-65)
where the cylinder is accelerating in the negative x direction. The total pressure is given by the unsteady form of the Bernoulli Eq. (3.46) in Chapter 3.4 p p
[a
l
[ dt
2
2
na<&\2"
[(£) + [r
gz +
86 )
Q(t)
(7.20)
The x component of the pressure force per unit length may be determined from -In
dFx = - f
pbcos9d9;
r = b.
(7.21)
Jo
Substituting Eq. (7.19) into Eq. (7.20) gives the total pressure on the cylinder r = b as P_ P
b
dU U2 n cosO -\ dt 2
gz +
Q(t)
(7.22)
Substituting Eq. (7.22) into Eq. (7.21) and integrating gives ,dU ,dU dFx = Capnb — = onb — dt dt
(7.23)
that gives an added mass coefficient Ca — 1.0. 7.4.2. Fixed Cylinder in an Accelerating
Flow
For a fixed cylinder in an accelerating unbounded fluid domain, the solution to Eq. (7.18) must be modified to account for the accelerating fluid in the
511
Deterministic Dynamics of Small Solid Bodies
Fig. 7.10. Cylinder accelerating in still fluid.
positive x direction according to Q(r,0,t) = -U(t)r{l
+
cos 9.
(7.24)
The total pressure on the cylinder r = b from Eq. (7.20) becomes 7 . 9 .
dU
£
b—cosO
P Pm
Pv
P
P
+ U2 sin2 6>| - gz + Q(t)
(7.25a)
(7.25b) + — + Q(t) p that gives both an inertia and drag component of pressure. Substituting Eq. (7.25a) into Eq. (7.21) and integrating gives ,dU
dFx = CmpTtbL'—dt
= IpTtb2
dU — dt
(7.26)
that gives an inertia coefficient Cm = 2.0. The inertia coefficient may be expressed as t-'/n — 1 "T C a .
(7.27)
From Eqs. (7.23 and 7.26), there is no net horizontal drag component of the force per unit length on a cylinder in an ideal fluid (d'Alembert's paradox, Milne-Thompson, 1968). This is a consequence of neglecting real fluid effects that result in flow separation and the formation of a wake. From Eqs. (7.25), the dynamic pressure in brackets {•} may be expressed as a linear sum according to PD - Pm + Pv
(7.28)
512
Waves and Wave Forces on Coastal and Ocean Structures
Fig. 7.11. Comparison between ideal and real fluid flow around a circular cylinder (R 2.0 x 105).
The dynamic pressure distribution around a circular cylinder in steady flow (i.e., dU/dt = 0) from Eqs. (7.21 and 7.20) with r = oo is Cn
Poo 2
pU /2
1 -4sin20,
(7.29)
where Cp = pressure coefficient (cf., the inverse of the Euler parameter in Table 3.2 in Chapter 3.6). Figure 7.11 compares Eq. (7.29) for a real and ideal fluid flow around a circular cylinder for a Reynolds parameter R = 2.0 x 105. To help interpret the difference between accelerating a cylinder in a still fluid from accelerating the fluid past a fixed cylinder, it is necessary to consider the momentum equation. The buoyant force in a static fluid may be computed from dz
+ y =0
(7.30a)
for a fluid volume V by
y(V) = FB =
-[d^-\v.
(7.30b)
The x-component of the horizontal momentum in an incompressible fluid from Euler's equation (Eq. (3.41) in Chapter 3.5) is 9
Du Dt
dx
(7.31)
513
Deterministic Dynamics of Small Solid Bodies
or for a circular cylindrical volume of fluid of unit height (d V = nb2) •) du
T
dp
dFm = P(7tb2)— = -{itb2)-f. 6t ox
(7.32)
Comparing Eq. (7.32) with Eq. (7.23) demonstrates that the inertia coefficient includes the effect of the pressure gradient in an accelerating fluid. Establishing that oscillating a cylinder in a still fluid is kinematically similar to oscillating a fluid past a fixed cylinder is most easily demonstrated by a noninertial moving coordinate axes attached to the center of a horizontal circular cylinder (Garrison, 1980 and 1990). The total derivative of the fluidvelocity vector that appears in the Navier-Stokes equations may be written in terms of moving coordinates attached to the oscillating cylinder, that oscillates horizontally with instantaneous velocity U as Dq Dq — = Dt Dt
dU ex, dt
v(7.33)
'
where Dq/Dt denotes the total derivative operator defined in moving coordinates where q denotes the fluid-velocity vector measured in moving coordinates. The vector gradient operator V(«) in Chapter 2.2.7 is the same in either the moving- or fixed-coordinate system; so that the complete boundary value problem written in moving coordinates is given by (Garrison, 1980 and 1990) -7-e x + - 7 = dt Dt p where v = kinematic fluid viscosity where V»qr = 0,
q =—Uex, x -* oo,
+ W2~q,
q = 0 on r = b.
(7.34)
(7.35a-c)
If the equivalent boundary value problem is formulated for oscillation of the fluid with instantaneous velocity £/ex about a.fixedcylinder, the result is Dq _
=
-WP _
+
7^ „V>,,
(736)
where V » ^ = 0,
q = -Uex,
x -» oo,
q = 0 on r = b.
(7.37a-c)
514
Waves and Wave Forces on Coastal and Ocean Structures
At an infinite distance x —>• +00 from the cylinder, the viscous and convective acceleration terms inEq.(7.36) vanish leaving only the following x component from Eq. (7.36): dU 1 3Poo ~r = ^> (7-38) v dt p dx ' where Poo denotes the pressure in the oscillatory uniform external flow. This pressure field Poo = 0 in Eqs. (7.34 and 7.35), but dPoo/dx is needed to accelerate the flow in Eqs. (7.36 and 7.37) so that P = Poo + P*, (7.39) where P* denotes the pressure due to the disturbance caused by the cylinder; and Poo = pressure in the absence of the cylinder. Substituting Eqs. (7.38 and 7.39) into Eq. (7.36) gives
515
Deterministic Dynamics of Small Solid Bodies
still water is kinematically identical to oscillating water past a fixed cylinder. The significance of the kinematic similarity between these two experiments is that data measured from oscillating cylinders in a still fluid are equally valid for wave force computations as are data measured from a fixed cylinder in an oscillating stream. This is important in that the oscillation of a cylinder in still water as a testing method has some advantages over fixed-period apparatuses such as oscillating U-rubes or variable-period wave tanks where the Reynolds parameter and the displacement ratio for a fixed cylinder size are coupled. Specifically, oscillating a cylinder in still water usually has the versatility that allows independent control over both the amplitude and frequency of oscillation. This allows independent variation of both the Reynolds parameter R and Keulegan-Carpenter parameter K (or amplitude ratio or period parameter G). In addition, when the cylinder is oscillated in still water, the horizontal pressure gradient term dPoo/dx that contributes 1.0 to the inertia coefficient is eliminated and, accordingly, so is the requirement to subtract from the measured force the force due to the pressure gradient that accelerates the free stream flow (Garrison, 1980 and 1990).
7.5. Maximum Static-Equivalent Force/Moment (Fixed-Free Beam) For constant values of Cd and Cm in Eqs. (7.10) (i.e., no dependence on R or K through depth z), a particularly useful dimensionless form for the inertia dFm (xo, yo, Zi, t) and drag dF^(xo, yo, Zi, t) force components may be obtained for vertical circular cylinders. The total integrated force/moment loads in the static-equivalent fixed-free cantilevered beam analogy shown in Fig. 7.12 where the distributed hydrodynamic wave-induced pressure loading is replaced by a total depth-integrated force Fx (r) and a moment arm I where the dimensionless time x = 2nt/f. In a 2-D non-inertial moving coordinate system {xm,z}, dimensionless variables may be scaled by dimensional variables (denoted by a tilde ~) according to: 2nxm ~ ~ § = —~— = k(xfJ - Ct), X r)(6) = —z-, H
u(%,st) =
~
2nt x = —^, T ~
(H/T)
9
s z+h s = - = —=—, h h Dx
(H/T2)
516
Waves and Wave Forces on Coastal and Ocean Structures
^a£ ^^.y
M(x) "777777777777777777777777777777,
Fig. 7.12. Static-equivalent fixed-free cantilevered beam.
and the dimensionless total depth-integrated inertia force/moment for x/ = 0 are given by
Fm(t) =
Fm(cbi)
Cm (pnfr/A)
(Hh/f*)
-L
Jo
(7.41a)
DT
r\+Hr,{z)fh
Cm (pwD 2 /4) (Hh2/f^
D M ( T > 5 )
ds,
Jo
Mm{&t)
Mm(r) =
\+Hr,(x)/h
sDu(r,s)
Dx
ds, (7.41b)
and the dimensionless total depth-integrated drag force/moment for x/ = 0 are given by
Fd(r) =
Fd{S>t)
Cd~pb~h/2 (H/ty Md(x) =
Md{(bt) 2
CdpDh /2
(Hit)'
-L
\+Hrt(x)/h
u{x,s)\u(x, s)\ds,
(7.41c)
su(x,s)\u(x,s)\ds,
(7.41d)
l+Hr,(r)/h
=1
517
Deterministic Dynamics of Small Solid Bodies
where xf = 0. A dimensionless total depth-integrated force/moment may be defined by a linear combination of Eqs. (7.41a-d) as Fx{t)
Cm7tD
Fx(a>t)
=
_
Cd (pbh/2)
_\2
My {cot)
My{x)
Fm{x) + Fd{x),
Cm TTD
2
Mm{x) + Md{x),
(7.41f)
Cd 2H
2
cd (pbh /i)
(7.41e)
Cd 2H
(H/T)
(H/T)
where nD/2H = TT2/2K and K = Keulegan-Carpenter parameter defined by Eq. (7.72). Dean (1974) tabulated Eqs. (7.41) applying his nonlinear stream function wave theory (vide., Chapter 6.5) (Dean, 1965). The solutions in dimensional moving coordinate axes {um, z\ for a dimensional streamfunction ^(^,s) and water surface elevation ??(§) are given by Eqs. (6.82 and 6.83) in Chapter 6.5. The dimensional horizontal water particle velocity u{i;,s) and acceleration Du{%,s)/Dt are given by {vide., Dean (1965); Rienecker and Fenton (1981) or Huang and Hudspeth (1984)) N+2
2ir{n — 3)s
-J^— ) V {n - 3 ) i ( n ) c o s h
.
cos((n-3)£),
X{3) (7.42a)
Du(M —
=
Dt
W{£,S)
X^'
—
'
T
du{%,s)
—
\X{3)J
h W{1;,S)
9«(g,£)
3§ (7.42b)
and the dimensional vertical water particle velocity w{%,s) and acceleration Dw{%,s)/Dt are given by N+2
w($,s)
= ( - = ^ - ) J2^n .X{3)/ n=4
3
27z{n — 3)s
)*(ra)sinh
.
X{3)
sin {{n - 3 ) £ ) ,
. (7.42c)
DwGJ) z
Dt
~(b~, = I w{t,s)
X{3)\( —
2JV —
\dw{$)
I "
T
\X{3)J
3§
dw{%,5) h w{£,s)
ds (7.42d)
518
Waves and Wave Forces on Coastal and Ocean Structures
The dimensionless integrals for Fm{x), Fd(t), Mm{x) and Md{x) are tabulated by Dean (1974) and may be applied to quickly evaluate the total dimensionless integrated loads on the vertical circular members of a spaceframe structure for dimensionless combinations of H, f and h if the inertia and drag coefficients Cm and Cd are assumed to be constants that are independent of the R and K parameters over depth. The load components may be plotted on scaled transparent paper and the space-frame structure may be loaded graphically by a wave progressing at an angle a to the structure as shown in Fig. 7.13 (Dean, 1972). The dimensionless forces and moments computed by the nonlinear Dean stream function theory may be compared with the dimensionless forces and
PLATFORM SUPPORX PILING,
0 DISTANCE (FT.)
40
80
120
Fig. 7.13. Graphical force loading on a pile supported offshore platform from the Dean Stream Function Tables (Dean, 1972 and 1974).
519
Deterministic Dynamics of Small Solid Bodies
moments computed with LWT kinematics with constant values for Cm and Cd- For a dimensional horizontal water particle velocity u(xf,z,i) given by Eq. (4.37b) in Chapter 4.4 and the local water particle acceleration computed by du(xf,z,t)/dt, the total depth-integrated inertia force Fm(r) may be computed for x/ = 0 from
Fm(r) =
kh
Fm(o)t)
L
~2~
Cmfn^H tanh kh
1
cosh.(khs) cosh kh
sinr
X — sm T. 2X0
sin T = 2
(7.43a)
where y = pg; Xo = gf2/2n = deepwater wavelength; and the wavelength X must be computed from the LWT dispersion equation according to cb2h
k$h =
(7.43b)
khtanhkh.
The total depth-integrated drag force Fd (r) may be computed for x/ = 0 from
Fd(t)
=
Fd(cbt)
gh
cdyDH2 2
4^2
cosh (khs) ^^ds JO cosh kh
gh[koh + (kh sech kh)2 COST COST
2(2Ckh)2
cost cost (7.43c)
where the wave celerity C = cio/k = X/f. The total depth-integrated inertia moment Mm{x) may be computed for Xf = 0 from
Mm(x) =
Mm(ut)
_
kh
2 cosh kh _/ ' Jo sech kh + koh — 1
Cmyn^-Hh
2kh
SUIT.
s cosh(khs)ds sinr (7.43d)
520
Waves and Wave Forces on Coastal and Ocean Structures
The total depth-integrated drag moment Md(x) may be computed for Xf = 0 from 2
Md(x) =
Md(cbt) 2
CdyDH h 2
ACkh
L
2C
scosh (khs) Y^—as cosh kh
[(1 + (khf)sech2kh
COST COST
+ k0h - 1] COST|
COST|.
(7.43e)
7.6. Parametric Dependency of Force Coefficients Cm and Ca In general, the Morison equation (7.1) is applicable to wave loadings where: (i) the wavelength is large with respect to a characteristic body dimension; i.e., b/k <3C 1; and (ii) the characteristic body dimension is a small fraction of the water depth;
i.e.,b/h « 1. The small body condition (b/k « 1 ) assures that the incident wave kinematics and dynamics are not modified (diffracted or scattered) by interacting with the small body. 7.6.1. Relative Importance ofdFm (zi, t) and dFj (zi, t) The Morison equation (7.10a) quantifies both ideal and real fluid forces through the inertia dFm{zi,t) and drag dFd(zi,t) terms, respectively. In general, both of these force components contribute to the total force on a small body. The relative contribution of each force component may be estimated from Eqs. (7.10a-c) for the x component by dFx(zi,t) = dFd(zi,t) = dFd(zi,t) 1 +
fdFm(zj,t)
\dFd(zi,t)J] L,m ~C~d
Ttb
ut(xo,yo,Zi,t) u(x0,yo,Zi,t)\u(xo,yo,Zi,t)\/
Deterministic Dynamics of Small Solid Bodies
521
where the subscript notation (»)f denotes partial temporal differentiation of the Eulerian velocity field according to ut(xo,yo,Zi,t)
-
du(xo,yo,Zj,t) dt
Linear wave theory scales give H u <x um(zi) oc —cof(zi), 2
du H , — <x -umco a - — a r / f o ) > at I
f(zi) = cosech kh coshk(zi +h),
a> = —
and consequently dFx (zi, t) reduces to 21
dFx{zt,t)
=dFd(zt,t) 1 - | ^ ) Cd) K
(7.44a) 7t2b
1 f(zt)\CdJ\
= dFd(zt,t) 1
H
(7.44b)
where the Keulegan-Carpenter parameter is defined by um(zi)T (7.44c) D Figure 7.14 (Garrison, 1978b) demonstrates the regions of validity for Eqs. (7.44) by identifying the regions where viscous and diffraction effects K =
4.0 "' I 1 '
3.0 •\ tl
HI
p
UU \
§2.0 Hi
C
\ \ \ \
\ diffraction \^_/^effects
.x V
/ \ viscous I f effects
1.0//2X/X/>// / / / / / / / I
0
I
•• E&fFRACTION THEORY • • • _^ i - - 1- —— i
0.1
0.2 D/X
0.3
0.4
Fig. 7.14. Parametric regions of relative importance of drag and inertia force components on a vertical pile (Garrison, 1978b).
Waves and Wave Forces on Coastal and Ocean Structures
522
(U
H
'5 '10 X/D
TTOO
' 1000*
Fig. 7.15. Hydrodynamic loading regimes for small bodies.
dominate the physics of the wave-induced hydrodynamic loadings. Figure 7.15 demonstrates the region of validity of the Morison equation in deep-water where the limiting steepness H/X — 1/7 is imposed. The full Morison equation (i.e., including both inertial and drag force components) is valid in deep-water when 1.0 < H/D < 10 and X/D > 20. 7.6.2. Computing the Force Coefficients Cm and Cj In order to more clearly understand the parametric dependency of the empirical force coefficients Cm and Q , it is important to note that there have been four basic types of laboratory tests plus field tests in the ocean; and that there have been three basic types of analyses applied to obtain these coefficients. These empirical evaluation methods are summarized in Table 7.1. Figure 7.16 from Garrison (1982) compares portions of the results for the drag Q and inertia Cm coefficients from the U-tube tests of Sarpkaya (1976); from the oscillating cylinder tests of Yamamoto and Nath (1976) and from Garrison, et al. (1977c). The Reynolds parameter range shown extends beyond the transition region and it appears that the post-critical region for these oscillatory flows lies well beyond R = 106 for a smooth cylinder. Sarpkaya (1977) analysed cylinders ranging in diameter between 2.0 < D < 6.5 inches. Yamamoto and Nath (1976) oscillated horizontally a cylinder with D = 12 inches in a wave channel filled to a depth of 13.0 feet. Garrison, et al. (1977c)
523
Deterministic Dynamics of Small Solid Bodies Table 7.1. Summary of 4 types of tests and 3 types of empirical evaluation methods. Test Type 1. LABORATORY i) Progressive Wave ii) Standing Wave Node iii) Oscillating U-tube iv) Oscillating Cyl Oscillating Cyl Oscillating Cyl 2. OCEAN i) Wave Project I Wave Project II ii) Ocean Test Structure (OTS)
Kinematics
Analysis
Theoretical
Phase
Morison etal. (1950)
Measured
Fourier
Measured
Fourier
Measured
Least Sqs
Theoretical Theoretical
Fourier Fourier
Keulegan and Carpenter (1958) Sarpkaya (1975, 1976, 1977) Yamamoto and Nath (1976) Garrison et al. (1977c) Rodenbush and Gutierrez (1983)
Theoretical
Least Sqs
Theoretical
Phase
Theoretical and Measured
Least Sqs and Phase
References
Dean and Aagaard (1969) Kim and Hubbard (1975) OCEAN IV Proceedings (ASCE, 1979)
Sarpkaya(1976) Yamamoto & Nath (1976) Garrison et al (1977c)
-' Roshko (1960)j steady flow 0.4
0.6
HD
2.0
4.0
6.0
ta
20.
4a
R x 10~5
Fig. 7.16. Drag and inertia coefficients in oscillatory flow past smooth circular cylinders (Garrison, 1982).
524
Waves and Wave Forces on Coastal and Ocean Structures
oscillated horizontally cylinders of D = 2.0 and 3.0 inches in a channel 2.0 feet wide, 16.0 feet long and 42 inches deep. Although exact correlation does not exist between results of the three independent laboratory experiments shown in Fig. 7.16, it is noteworthy that the trends of the data are all very similar and the agreement is actually as good as exists for equivalent steady flow experiments. However, a certain amount of disagreement should be expected in the transition region because the drag coefficient is fairly sensitive to free stream turbulence (Garrison, 1982). Some variations no doubt existed in the rate of dissipation of the turbulence level and vibration of the test cylinder. At the high Reynolds parameters the disagreement appears to decrease and the results appear to converge with the asymptotic values of Cd increasing with decreasing values of the displacement ratio 2A/D (Garrison, 1982) where A — wave amplitude = H/2. It is significant that relatively good agreement exists between these different experiments. This may have been unexpected compared to the large degree of scatter that characterizes the drag and inertia coefficient data obtained from ocean test platforms and from small-scale wave tank test (Heideman, et al., 1979). Because these results cover the transition region where the coefficients are sensitive to a number of different effects, it appears that a correct answer does not exist from the viewpoint of practical applications. However, the postcritical Reynolds parameter region is generally of more interest in design and, although data in this range for smooth cylinders is still somewhat lacking, it appears that the data in Fig. 7.16 do tend to converge at high Reynolds parameters. 7.6.3. Methods of Analyses Three methods have been applied to obtain the force coefficients Cm and Q from measured force data by minimizing the error between the measured force and the predicted force. These three methods are: (i) Phase method (ii) Least-squares method for both continuous and discrete time series (iii) Fourier analyses method Each of these three methods are reviewed briefly. If the cylinder is horizontal instead of vertical (such as in an oscillating U-tube or at the node of a
Deterministic Dynamics of Small Solid Bodies
525
standing wave), the horizontal force component dFx(zi,6) is the total force per horizontal unit length at vertical elevation z,. (i) Phase Method: Evaluating Cm and Cd from measured force traces depends on the intended application of the values. For maximum wave forces, it appears reasonable to reduce the data in a manner similar to the method applied by Morison, et al. (1950) and by Kim and Hibbard (1975). The drag coefficient Cd was computed from the phases of the measured force trace where the horizontal velocity was a local maximum and the total force was mostly drag. The inertia coefficient Cm was computed from the phases of the measured force trace where the horizontal acceleration was a local maximum and the total force was mostly inertia. In order to estimate the phase of the wave where the force predicted by the Morison equation is a maximum, the dimensional wave kinematics may be approximated for simple harmonic motion from linear wave theory by u{z,9) = um(z)cos9,
ut(z,0) = —coum(z) sinO
(7.45a,b)
Aco cosh k{z + h) 7-r-r. (7.45c) sinh kh where A = wave amplitude; and 6 = cot. The predicted dimensional waveinduced pressure force per unit length may be expressed by the Morison equation for simple harmonic data as um(z) =
dFx(z,0) = dFm(z,e) + dFd(z,e) -dFm (Z) sin 6 + dFd (z) cos 6 | cos 0 |
(7.46a)
where dFm(z) = Cm (p^-
J coum{z),
dFd{z) = -CdPDu2m (z).
(7.46b,c)
The phase at the time of the maximum force occurs near 0 = 0° and may be determined from (Morison, et al., 1950) ddFx(z,0) ^ r ^ = 0,
6 = 6max
(7.47a,b)
Waves and Wave Forces on Coastal and Ocean Structures
526
that requires that cos9max[dFm(z) + 2dfd(z) sin# max ] = 0.
(7.48)
The two possible solutions to Eq. (7.48) may be computed from cos #max = 0;
and
[dFm (z) + 2dFd(z) sin 0max] = 0
giving the following two possibilities: it
(i) #max = (2n + 1)—,
n > 0 and integer
r-\ • a dFm(z) (u) sm# max = 2dFd(z) that imply that either:
f ~ cos6»max = J 1 - sin i
(i) the maximum force is inertia-dominated and the Morison formulation is not valid
2
y/2
dFx (z, (In + 1 ) | ) = dFm{z)
(7.49a)
or (ii) the maximum force is drag-dominated and given by, approximately
dFx(z,0m!a)
=dFd(z)
1 +
/dFm(z)Y 4\dFd(z)J
(7.49b)
527
Deterministic Dynamics of Small Solid Bodies
The dimensionless force ratio dFd(z) = W dFm(z)
(7.49c)
appears repeatedly in the analyses of wave force data and is called the O'Brien parameter W in recognition of his pioneering contributions to coastal and ocean engineering. Least-Squares Method An alternative to the phase method that is readily adaptable to strictly periodic fluid or structure motion with period T = 2n/to is to apply a least-squares fit of the measured force trace to a computed Morison equation over a complete cycle of the forcing. For periodic motion past a fixed vertical circular cylinder of diameter D, a mean square error between the measured force dFj(z, 9) and the computed force dFx(z, 6) may be defined by e 2 = ([dFT(z,e) -dFx(z,8)]2)
(7.50a)
where the temporal averaging operator ((•)) is defined for simple harmonic data by ((•)> = ^~ f \*)d0
(7.50b)
and where dFj (z, 9) = measured force per unit length; 9 = cot; and the computed force per unit length dFx{z, 9) is computed from the Morison equation (7.46) by dFx(z,6) = dFm(z,6) + dFd(z,9)
(7.51)
where dFm(z, 9) = an inertia force component and dFd(z, 9) = a drag force component. Minimizing Eq. (7.50a) with respect to Cm and Cd in a best least-squares sense requires that de2 dCm
n
0,
de2 — dC- = 0. d
(7.52a,b)
528
Waves and Wave Forces on Coastal and Ocean Structures
For simple harmonic motions given by Eqs. (7.46), Eqs. (7.52) give the following formulas for Cm and Q : 2
« dFT{z,9)
. sm9d9, pDuliz)
(7.53a)
2
Cd =
* dFT(z,9)
V w Jo
pDul(z)
cos 9\ cos 9\d9,
(7.53b)
where the Keulegan-Carpenter parameter K is defined by um{z)T
(7.53c) D The coefficients Cm and Cd are functions of the vertical location z. This vertical dependence is reflected in the parameters K and R through the amplitude of the velocity um{z). K
Fourier Analysis Method Keulegan and Carpenter (1958) analyzed the standing wave-horizontal cylinder system shown in Fig. 7.17 and applied a Fourier analysis of the measured force data to obtain the following expansion for the dimensionless predicted, wave-induced force per unit length: dFx(z,9) pDum(z)
= YJ[A2n(z)cos(2n9)
+ A2n+i(z)cos[(2n + \)9]
n=0
+B2n (z) sin(2n6) + B2n+i (z) sin[(2n + 1)9]],
(7.54a)
Fig. 7.17. Definition sketch for Keulegan and Carpenter experiments (1958).
529
Deterministic Dynamics of Small Solid Bodies
where the coefficients Aj (z) and Bj (z) are again functions of the vertical location z. These coefficients may be computed in a best least squares sense by substituting Eq. (7.54a) into Eq. (7.50a) and minimizing according to 3e 2 dA2n
0,
3^ 2 dA2n+\
3e 2 n = 0, 3B2n
n
= 0,
de2 n — = 0 d#2n+l
lncA^
, (7.54b-e)
that give -2;r
A2n{z) = I" A2n+i(z)
* . 1 f ^
d F
^ \
cos(2n9)d9,
zjr 1 f*2n dFT(z 9) = - / „ ' cos((2n + l)0)
"2» dFT(z,9) . " ' ' K?"' sin(2n9)d9, pDuUz) 7T Jo -n-''~s
B2n(z) = - [
1 C2n dFriz 9) Bin+liz) = - / „ ' { sin((2n + \)9)d9.
(7.55a) (7.55b) (7.55c) (7.55d)
The vertical dependence on z is again reflected in the parameters K and if through the amplitude of the velocity um{z). If the dimensional measured force per unit length dFj(z, 9) is assumed to be given by Eqs. (7.46), then Eqs. (7.55) reduce to Azn(z) = 0,
A2n+i(z) =
4(-P"+1 Cd_ ,— -, r o -3 n [8n + 1 2 n 2 - 2 n - 3 ] '
fl2»(z) = 0,
B2n+1(z) = -Cm"Jn0,
(7.56a,b)
(7.56c,d)
where 5,7 = Rronecker delta function (vide., Chapter 2.2.3). Substituting Eq. (7.56d) with n = 0 into Eq. (7.55d) gives Cm =
2K f2n T /
dFT(z,9) ' V , \in 9d9,
(7.57a)
and substituting Eq. (7.56b) with n = 0 into Eq. (7.55b) gives 3 /,2?r d = -A\ 4;0
C
dFT(z,9) *\, cos 9 d9. pDuUz)
(7.57b)
530
Waves and Wave Forces on Coastal and Ocean Structures
1.6.4. Linearized Drag Force Both the least-squares and Fourier methods have been applied to provide a better fit between the Morison equation and a measured force trace. However, it is of equal importance to be able to compute the added mass and damping coefficients associated with the motion of compliant bodies. In the case of damping, it appears to be less important to accurately represent the exact temporal variation ofthe force. Rather, it appears to be more important to accurately represent the energy or work done during a periodic cycle of the motion. This concept is called the Lorentz (1926) method of equivalent work (Ippen 1966, Chapter 10.4). If the work done by the measured force is equated to that done by the velocity-squared term in the Morison equation dFd(z, 9), a definition for Cdi is obtained that is appropriate to damping of compliant bodies. The result is identical to the Fourier method; i.e., the Fourier average drag coefficient Cdi is defined in such a way as to preserve the energy dissipated over a complete periodic cycle of the motion (Garrison, 1982). The linearized drag force is equivalent to assuming that the quadratic drag force in Eq. (7.46a) may be replaced by a linearized form of the drag force by applying the LWT kinematics in Eq. (7.45a-c) according to dFdt(z,9) = dfdi{z) cos6,
(7.58a)
where dfdl(z) = Cdi^-um(z)
(7.58b)
and Cdi = a linear drag coefficient that has the dimensions of velocity um [Length/Time]. The time average rate of work (WL) done by a linear drag force dFdi(z, 9) may be equated to the time average rate of work ( WNL) done by the nonlinear drag force dFd(z,9). The time average rate of work done by the linearized drag force dFdi(z, 9) over one wave cycle is "27T
Pi = (n't) = j - /
dFdl(z,e)u{z,9)de
CdWD 2 f2* 2 cos^ 9d9 -«m(z) / An JO CdlpD
2
—-.—u m {z).
(7.59a)
531
Deterministic Dynamics of Small Solid Bodies
The time average rate of work done by the nonlinear drag force dFd (z,9) over one wave cycle is 1 f2jT PNL = {WNL) = — / 2TT JO
dFd(z,6)u{z,6)dd
CdpD 1 3 f ^ „2z cos 6>|cos#|d6» An -um(z) I Jo 2CdpD 3 = —: um(z). in
(7.59b)
Equating the average linear power P^ to the average nonlinear power P ^ L yields g Cdi = -z—CdUm(z). in
(7.60)
The linearized drag force given by Eqs. (7.58a,b) may be equated to the nondimensional A\{n = 0) coefficient term in Eq. (7.56b) for n = 0; i.e., dFdi(z,0) = PDu2m(z)AiU) (4C u (z)\ = pD I d m
cose \um(z)cos6
(7.61)
by Eq. (7.56b) for n = 0. Equating Eq. (7.61) to the linearized drag force in Eqs. (7.58a,b) also gives Eq. (7.60) proving that the Lorentz method of equivalent work (Ippen, 1966) is equal to the first term in a Fourier series expansion. The Fourier definition for Cd appears to be the most appropriate to apply to compliant bodies while the least-squares definition appears to be the most appropriate for representing the temporal variation of the wave force (Garrison, 1982). Garrison (1982) has compared the values of Q computed by both of these methods for three types of the four laboratory tests listed in Table 7.1. Garrison (1980 and 1982) demonstrates that the Fourier method gives slightly higher values than the least-squares method; however, it does not appear to be significant to distinguish between Cd coefficients computed by these two methods for practical design purposes.
532
Waves and Wave Forces on Coastal and Ocean Structures
7.6.5. Laboratory U-Tube Data Sarpkaya (1975) applied Fourier analysis to data from the oscillating U-tube shown in Fig. 7.18 to obtain drag Cd and inertia Cm coefficients on horizontal cylinders in oscillating flow without a free surface. He concluded that these two force coefficients were reasonably well-correlated with the period or Keulegan-Carpenter parameter K = UmT/D; but that plots "versus Reynolds number (Re = UmD/v) have shown that there is absolutely no correlation between the said parameters and the Reynolds number" (Sarpkaya, 1975, p. 35). Miller (1975) replotted both the original standing wave data from Keulegan and Carpenter (1958) and the oscillating U-tube data from Sarpkaya (1975) and demonstrated that both of these data demonstrate a clear Reynolds parameter dependence. Sarpkaya (1977) then re-examined both the original Keulegan and Carpenter data (195 8) as well as his own data (1975) and determined that a frequency parameter f5 = D2/vT and roughness Reynolds parameter R€ = ew m /v where e = characteristic length of the cylinder roughness could explain the earlier incorrect interpretation regarding the total lack of dependence on Reynolds parameter. The frequency parameter f) = D2/vT represents a ratio of the Reynolds parameter R to the Keulegan-Carpenter parameter K. This ratio may be computed from Eq. (3.58) in Chapter 3.6 by dimensional analysis. If the Reynolds parameter if = pb^fgbl^ is divided by the dimensionless parameter T^/g/b in Eq. (3.58) where the time was scaled by gravity g, then the frequency parameter is given by fi = R/K = b2/vT.
_v_ -VJ
Y^
5Z. / / / / / / / / / / / A
\ 777777777777777777
Jy
Fig. 7.18. Definition sketch for oscillating U-tube.
Deterministic
Dynamics of Small Solid Bodies
533
In U-tube laboratory tests, oscillatory flow past test cylinders have been obtained by sinusoidal current flow represented by u{6) = Um cos# where Um = Hn/T where H = 2 A that is twice the amplitude of the U-tube oscillation shown in Fig. 7.18. The period or Keulegan-Carpenter parameter K is a measure of the wake effects in oscillatory flows. The Keulegan-Carpenter parameter differs from the relative amplitude parameter by TT, a constant for harmonic laboratory fluid flows or sinusoidally forced body motions in still water. In the supercritical Reynolds parameter region R > 106, the drag Q and inertia Cm coefficients are sensitive to wake effects that depend on the Keulegan-Carpenter parameter K (or relative displacement or period parameter H/D) and to the relative roughness e/D as shown in Figs. 7.19 and 7.20 (Garrison, 1980 and 1982). For smooth cylinders in steady flows, the drag coefficient Q = 0.6 in Fig. 7.19; and the effect of relative roughness e/D is to increase the drag coefficient with increasing relative roughness. The additional increase in the drag coefficient Cn in the supercritical Reynolds parameter region is due to the wake effects that correlate with the amplitude parameter H/D. As the wake is washed back and forth over the cylinder in oscillatory flows, values for the drag coefficient are elevated over the values for roughened cylinders in steady flow with the more elevated values for the drag 2.4
' ' ""I ' ' ' ' ""I Typical in-service roughness
2.2
-<
• -
2.0
HID = 6.4
1.8 1.6 -
U-Tube i Sarpkaya" (1976)
1.4 1.2 1.0
r o u g h n e s s ^ l ^ y , rough, 2A/D = oo
0.8
6ffect
0.6
TTT77T7-T
' steady, smooth-*
0.4 0.2
e/D
0.0
10 s
10 4
Fig. 7.19. Effects of period parameter H/D (Garrison, 1980 and 1982).
io- 3
= 2A/D
lO"2
on drag coefficient Q
IO"1
forR
> 10 6
534
Waves and Wave Forces on Coastal and Ocean Structures
CO 2.0 1.8
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1C
Fig. 7.20. Effects of period parameter H/D = 2A/D on inertia coefficient Cm for/? > 106 (Garrison, 1980 and 1982).
coefficient occurring for relative small values of the amplitude parameter H/D. However, as the amplitude parameter H/D increases, the oscillatory flow or cylinder motion appear more like steady flow conditions and the drag coefficient values reduce toward the steady flow values (Garrison, 1980 and 1982). The effects of the relative roughness parameter e/D and the relative amplitude parameter H/D on the inertia coefficient Cm are compared in Fig. 7.20 to a smooth cylinder where Cm = 2.0 (cf., Eq. (7.26) in Sec. 7.4) in the supercritical Reynolds parameter region R > 106. In contrast to the wake effects on the drag coefficient Cd illustrated in Fig. 7.19, wake effects decrease the values of the inertia coefficient Cm from the smooth cylinder values. Again, wake effects are most pronounced at relative small values of the amplitude parameter H/D; and the wake effects are reduced as these values increase. The highest value of fi measured in the Sarpkaya data (1977) for the force coefficients Cd and Cm appears to be approximately fi = 8370. A representative experiment measuring wave forces on a vertical circular cylinder in the O.H. Hinsdale-Wave Research Laboratory (OHH-WRL) at Oregon State University, USA (vide., Fig. 5.1 in Chapter 5.1) would be for a cylinder diameter D = 1.0 ft., a wave height H = 3.0 ft. and a wave period T = 3.5 sec giving a /J = 30 000. A representative design specification for a tubular member on a jacket-type offshore platform may require a diameter D = 13 ft ^ 4^- ft, a wave height H = 48 ft and a wave period T = 13 s ^ An s.ForUm & Hn/T, P = 1.3 x 106. Because none of the /? values from the oscillating U-tube experiments are of these magnitudes, the graphs for the drag and inertia coefficients from Sarpkaya (1977) are not included in this review.
535
Deterministic Dynamics of Small Solid Bodies
7.6.6. Ocean Wave Data Dean and Aagaard (1970) analyzed measured wave pressure forces on two prototype offshore oil platforms in the Gulf of Mexico from Wave Projects I and II. Force coefficients Q and Cm were computed from the data by a least-squares regression algorithm applying theoretical nonlinear water particle kinematics computed by the theoretical stream function theory (vide., Chapter 6.5). They found that the drag coefficient varied with Reynolds parameter and that the inertia coefficient was essentially a constant. The Dean and Aagaard regression analysis illustrates the parametric dependence of the coefficients and explains some of the scatter observed by other investigators. Dean and Aagaard (1970) discretized continuous force records by tn — n A t and computed Cd and Cm from a least-squares fit to the discretized measured forces per unit area (psf) dFj (z,ri)by minimizing the following mean square error for discrete time series: 1 N e = -J2idFT(z,n) 2
- dFx(z,n)f
(7.62)
n=\
where dFj{z,n) = discretized measured force per unit area (psf) and dFx (z, n) = discretized predicted force per unit area (psf) given by / TCD\ Du(z,n) /p\ dFx(z,n) = Cm I p—J ^ + Cd ( | ) u{z,n)\u{z,n)\
(7.63)
where the Stokes material derivative operator D(»)/Dt is defined in Chapter 2.2.10. Note that both the measured dFj (z,n) and predicted dFx (z,n) pressure forces have the dimensions of pressure = [Mass]/[Length]/[Time]2. Minimizing the mean square error Eq. (7.63) by Eqs. (7.52a, b) yields
CmAN + —CdBN
=
TtD
Cm—BN TtD
+ ——CdbN (nD)z
-FiiN,
(7.64a)
pnD
=
-F2,N, pnD
(7.64b)
536
Waves and Wave Forces on Coastal and Ocean Structures
where N
\\ 2
/ r> I
AN=
2S\—~j)t—J
N
'
BN =
Z-,—^—u(z,n)\u(z,n)\,
N
bN = y^M4(z,w), n=l AT
Du(
Fl,N = ^2dFT(z,n)——j—, Dt
,
N
F2,N =
n—\
^dFT(z,n)u(z,n)\u(z,n)\. n=\
Solving Eqs. (7.64a, b) by Cramer's rule gives the following formulas for the force coefficients: r m
— FI>NDN ~ F2,NBN ~ p7t(D/4)DET(Cm,Cd)'
d
_ F2,NA„ - F\yNBN ~ (p/2)DET(Cm,CdY
(
' °'
where DET(Cm, Cd) = ANDN
2
- BZN.
(7.64e)
Dean, etal.{\ 91 A) reanalyzed the Wave Projects I and II data from Dean and Aagaard (1970) and their results for the inertia coefficient Cm are illustrated in Fig. 7.21 and for the drag coefficient Cd in Fig. 7.22. Because the direction of wave advance varied azimuthally around the vertical piles on the platforms in Wave Projects I and II, an inline direction of wave travel over the crest portion only of each wave profile analyzed was computed in a best-least-squares sense and an in-line wave force computed with the in-line angle for each wave crest time sequence analyzed. A resultant wave force was also computed over each wave crest time sequence analyzed. A comparison between the inline drag force coefficients and the resultant drag force coefficients is illustrated in Fig. 7.23.
537
Deterministic Dynamics of Small Solid Bodies
2J0 T V
.5 ?
08
?»Ul + •
*
AO
• a
IJO
X
• X
D
J,
V
0.6
-mO.4
0.2
0.1IXI0 4
4
8 IXIO5
6
2 Reynolds
4 6 No.R
D/A Symbol (Ft.)
4
8 1X10*
UF Symbol
Resultant
2*
o
•
©
t
'!
&
•V
+
I *
•
0
#
•
•
X
TV
It
w a
4*
371*
* f
Resultant
8 IXI0 T
Wave Project
Inline
Inline
6
Water Depth ~ 3 3 Ft. Water Deplh~ICO Ft.
Fig. 7.21. Comparison of inertia force coefficients Cm from Wave Projects I and II (Dean, et al., 1974 and Dean and Aagaard, 1970). 2.0
*
i.o
*%< x*
0.8 0.6
i r? »
va DO
"<*0.4
0.2
6
8 1X10°
2 Reynolds
DA Symbol (Ft.) 2* 3* 4* 3.7.t
Inline
Resultant
o
• *
A V
o
* f
4 6 No./}
8 IXIO"
UF Symbol Inline
Resultant
o
t
T
+ o
. •
•
X
it
4
6
8 1X10'
Wave Project i *
it
Wfaler Depth ~ 33 Ft. Water Depth ~ 100 Ft.
Fig. 7.22. Comparison of drag force coefficients C^ from Wave Projects I and II (Dean, et al., 1974 and Dean and Aagaard, 1970).
538
Waves and Wave Forces on Coastal and Ocean Structures
;
—r
i /^Resultant
J
~^J — —
- Inlin
IXIO4
2
4
6
8 IXIO*
2 Reynolds
4 6 No. A
8 IXIO*
•
•
2
4
6
8 IXH3T
Fig. 7.23. Comparison of inline and resultant drag force coefficients Q from Wave Projects I and II (Dean, et ah, 1974 and Dean and Aagaard, 1970).
7.7. The Dean Eccentricity Parameter and Data Condition Much effort has been directed toward resolving the parametric dependency of the two empirical force coefficients Cm and Cd that are applied to estimate the wave-induced pressure loads on small members by the Morison equation (cf, Sarpkaya and Isaacson, 1981 or Chakrabarti, 1987). The two most commonly applied parameters are the Reynolds parameter R = um D/v and the Keulegan-Carpenter parameter K = um T/D where um = amplitude of the horizontal water particle velocity. However, only Dean (1976) appears to have recognized the importance of the condition of the data when identifying these two empirical force coefficients in any parameter identification algorithm. Although the error ellipse concept was originally proposed by Dean (1976) to demonstrate geometrically the condition of data for identifying Cm and Q ; his original development did not demonstrate that the alignment of the axes of the error ellipse depends explicitly on either J? or K. Because the data are relatively better-conditioned for identifying the empirical force coefficient on the axis that is parallel to the semi-minor axis of the error ellipse, it is essential to be able to demonstrate that the alignment of the axes of the error ellipse depends explicitly on either R or K or, preferably, both of these parameters. It is possible to connect the parametric dependency of Cm and Cd with the Keulegan-Carpenter parameter K by the condition of the data through the Dean eccentricity parameter E as illustrated in Fig. 7.24.
Deterministic Dynamics of Small Solid Bodies
539
? CONNECTION
Fig. 7.24. Connection between parametric dependency of Cm and C^ and the data condition (i.e., orientation of error ellipse axes).
It is possible to demonstrate that, for data with kinematics that are simple harmonic, the Dean eccentricity parameter E is proportional to the Keulegan-Carpenter parameter K. Thus, the Dean eccentricity parameter E provides an explicit measure of the parametric dependency of the alignment of the semi-minor axis of the error ellipse on the parameter K. Specifically, when E < 1.0, then K < 11.40 and the semi-minor axis of the error ellipse is parallel to the Cm axis and the data are relatively better conditioned for computing Cm. Conversely, when E > 1.0, then K > 11.40 and the semi-minor axis is parallel to the Cd axis and the data are relatively better conditioned for computing Cd- When E = 1.0, then K = 11.40 and the error ellipse is a circle with zero eccentricity. It is interesting to note that a value of K = 11.40 is approximately the value of K where the peak in Cd and the trough in Cm occur in the replotted Keulegan-Carpenter data (vide., Fig. 7.26, Sarpkaya and Isaacson, 1981 and Chakrabarti, 1987) and, also, where the amplitude of the transverse lift force in the Hayashi and Takenouchi (1979) data is completely stable (vide., Fig. 7.27). Dean postulated an algorithm that is capable of including the effects of errors in the amplitude/phase of the wave kinematics on the estimates of Cm and Cd (Hudspeth, 1974). The Dean error ellipse methodology (1976) may be compared geometrically with his earlier amplitude/phase error methodology (Hudspeth, etal., 1988). In addition to demonstrating geometrically the importance of the condition of the data, the amplitude/phase error methodology also
540
Waves and Wave Forces on Coastal and Ocean Structures
(Cml
IT
Fig. 7.25. Dean error ellipses for determining the condition of wave force data to compute force coefficients (Dean, 1976).
demonstrates the importance of errors in the measured amplitude/phase of the wave kinematics. In contrast to the Dean error ellipses illustrated in Fig. 7.25, the amplitude/phase error methodology demonstrates geometrically the parametric dependency of Cm and Cd on the parameter K (or 1J) by the magnitude of the slopes of the contours of the dimensionless O'Brien force ratios W = \dFd\/\dFm | that pass through a zero error in phase at cox — 0. The advantage of the amplitude/phase error methodology is that the infinite suite of separate graphs shown in Fig. 7.25 that are required by the Dean error ellipse methodology for each constant value of the O'Brien force ratio W = \dFd\/\dFm\ may be replaced by a single graph with contours of constant values of W. The effect of W on the alignment of the Cm and Cd error ellipse axes in the Dean error ellipse methodology may be demonstrated by the slopes of the contours of W that pass through a zero error in phase. Comparisons with synthetically phaseshifted laboratory data for E ^ 1.0 (or equivalently, K ^ 11.4) are excellent for phase shifts in the range of \u>x\ < n/S (Hudspeth, et al., 1988). A definition for the term data condition may be appropriate here. Data condition is defined as the ability of a least-squares algorithm to locate a global minimum on an error surface for given wave kinematic/force data (vide., Marquardt, 1963). It is, of course, related to the numerical condition number of a least-squares error matrix (vide., Atkinson, 1989). Matrix condition numbers are computed by four standard measures for least-squares error analysis of the Morison equation. The matrix condition number is identically equal to unity whenK = 13.16 andE = 1.15.
541
Deterministic Dynamics of Small Solid Bodies
Because the Morison equation represents the inertia of the fluid (Sec. 7.1 and Sec. 7.2), it does not contain an explicit constitutive relationship for the viscous stress tensor. Therefore, it is not possible to demonstrate a similar explicit dependency of the alignment of the axes on the Reynolds parameter/?. 7.7.1. Dean Error Ellipse and Eccentricity Parameter E (Geometric) The mean square error e2 between the true force per unit length (denoted by upper case unprimed letters) dFj{z,9) and the computed force per unit length (denoted by lower case primed letters) df'x (z,0) may be estimated from Eq. (7.50a) in Sec. 7.6.3 where the temporal averaging operator (•) is defined for simple harmonic data by Eq. (7.50b) in Sec. 7.6.3 and where 9 = 2nt/T. The true force per unit length is assumed to be represented exactly by the two-term Morison equation (7.1) and is given by dFT(z,9) = dFm(z,e)+dFd(z,e)
= KmUt{z,d) +
KdU(z,9)\U(z,9)\ (7.65)
and the computed force per unit length is given by dF^z,9)^dF^(z,9)+dFd(z,9)
= k'mut(z,9) +
k'du(z,9)\u(z,9)\, (7.66)
where (»)f = 3 (•)/dt= partial temporal derivative of an Eulerian velocity field and where the true and computed generalized inertia and drag coefficients, are, respectively Km = Cm ( — - — I,
4
r
C = C(^),
Kd = Cd\ - r - I,
"
" 2
4-ci(f).
where Cm and Cd are the true coefficients and C'm and C'd are the computed coefficients. The computed inertia and drag coefficients are denoted by superscript primes (•)' in order to distinguish them from the true coefficients that are unprimed.
542
Waves and Wave Forces on Coastal and Ocean Structures
Substituting the computed force in Eq. (7.66) into Eq. (7.50a) and expanding yields e2 = [ ( / > D / 2 ) 2 ( M ( Z , 0 ) 4 ) ] X2 + [(pnD2/4)2(ut(z,9)2)]
Y2
2[(pD/2)(P7tD2/4){u(z,e)\u(z,e)\ut(z,e))'jXY
+
-2[(pD/2)(FT(z,9)u(z,9)\u(z,6)\)]X - 2 [(p7zD2/4)(FT(z,8)ut(z,9))~]
Y
+ (FT(z,9)2},
(7.67)
whereX = C^and7 = C^. Equation (7.67) is a conic section equation for an ellipse (Dean and Daliymple, 1991 or Thomas, 1965) with an origin that has been translated and rotated; i.e., (aX)2 + 2HXY + (PY)2 - 2GX - 2JY + C = 0.
(7.68)
The coordinates of the translated and rotated origin Xo = (C^)mjn and yo = (C'm)mm may be computed from the data according to (ut(z,9)2){dFT(z,9)u(z,9)\u(z,9)\)
\
-(u(z,9)\u(z,9)\ut(z,9))(dFT(z,9)ut(z,9))J
v
•^0 — (C^)min —
(p D/2) DET[X0, Y0] (7.69a) 4
lu(z,9) )(dFT(z,8)ut(z,8)} -(u(z,9)\u(z,9)\ut(z,9))(dFT(z,9Mz,9)\u(z,9)\)/ ^0 — (^m)min —
(p7zD2/4)DET[Xo,Y0] (7.69b)
DET[XQ,Y0]
= (u\z,9))(ut{z,9)2)
- (u(z,e)\u(z,9)\ut(z,9))2,
(7.69c)
and the angle of rotation may be computed from only the kinematics by co. 20 = ( " f a < » 2 > - ( ^ > 2 ( - f o ^ ) . JrD(ii(z,9)|n(z,0)|ii,(z,»)>
(7.69d)
543
Deterministic Dynamics of Small Solid Bodies
If the true and computed kinematics are simple harmonic oscillations that are given by U(z,9) = Um(z)cos(9),
Ut(z,9) = -Um(z)cosm(9),
(7.70a,b)
u(z,6) = um(z)cos(6),
ut(z,d) =-um(z)eosm(6),
(7.70c,d)
then the time averaged quantities (•) that are required in Eq. (7.67) are given by / /
m4\
(u(z,9y)
3
"m(z)
/
= —-—,
,
m
(coum(z))2
2\
(ut(z,9n
=
,
{u(z,e)\u(z,9)\ut(z,e))
(7.70e,f)
=0
(7.70g)
and Eqs. (7.67 and 7.69a-d) reduce to e2 = l(PD/2)2(3um(z)4/8)]X2
l(P7t2D/4)2(2um(z)4)/K2jY2
+
-2[(pD/2){dFT(z,0Mz,9)\u(z,e)\)]X - 2[(P7TD2/4)(dFT(z,9)ut(z,9))]Y ,r,. 0 = (Q) m in =
+ (dFT(z,9)2),
l6(dFT(z,9)u(z,9)\u(z,9)\)
v X
,
4
2K2(dFT(z,9)ut(z,9))
Y
0 = (Cm)min
(7.71a)
=
5
2
47^
'
(7.71b) (7-71c)
n pu^(z) where the Keulegan-Carpenter parameter K is defined for simple harmonic kinematics by K = ^ f l .
(7.72)
Equations (7.71a-c) demonstrate the parametric dependency of the translation of the coordinates of the origin (XQ, YO) on K for simple harmonic kinematics. Equations (7.71a-c) are similar to those given by Dean and Dalrymple [1991, Eq. (8.47), p. 226] but that are not dimensionally homogeneous and that do not demonstrate an explicit parametric dependency on K. In order to demonstrate explicitly the parametric dependency of the eccentricity of the error ellipse on the parameter K for simple harmonic kinematics,
544
Waves and Wave Forces on Coastal and Ocean Structures
note that in Eq. (7.68) that: a2 = (pD/2)2(u(z,0)4) R2 =
p2 = (P7zD2/4)2(ut(z,9)2)
> 0,
> 0,
(S/3){dFT(z,9)u(z,9)\u(z,9)\)2/u4m(z) + 2{dFT{z,9)ut{z,9))2/{coum{z))2
+ e2 -
(dF2(z,9))
and that H = 0 in Eq. (7.68). This implies (Thomas, 1965) that the translated axes of the error ellipse are not rotated and are parallel to the Cartesian axes X = Cd and Y — Cm as illustrated in Fig. 7.25. Dean and Dalrymple (1991) demonstrate that it is more illustrative for the case of simple harmonic data to write the conic section equation by completing the square of Eq. (7.68) in the following manner:
(X-Xp)2
-WaT
|
+
(F-FQ)2
-(KW- = h
(7 73a)
"
The eccentricity of the error ellipse may be defined by the Dean eccentricity parameter E given by E2 = (a/p)2 = 3(K/2n2)2.
(7.73b)
The eccentricity determines geometrically the condition of the data for identifying C'd and C'm and may be evaluated from the ratio a/p. For K < lit21V3 = 11.40 the eccentricity parameter is E2 < 1.0; and R/a identifies the semi-minor axes parallel to the Y(=C'm) axis and R/fi identifies the semi-major axis parallel to the X{=C'd) axis. For K > 2n2/*j3 = 11.40 the eccentricity parameter is E2 > 1.0; and R/a identifies the semi-major axis parallel to the Y(=C'm) axis and R/fi identifies the semi-minor axis parallel to the X(=C'd) axis. The data are relatively better conditioned for identifying the force coefficient on the axis that is parallel to the semi-minor axis as illustrated in Fig. 7.26. The eccentricity e2 of the error ellipse determines geometrically the condition of the data for identifying C'd and C'm and may be computed from (Thomas, 1965) +2
a2 = l0_E(±)\
>
E<10
> «/0 < 1-0, E > 1.0, a/p > 1.0.
(7.74a) (7.74b)
545
Deterministic Dynamics of Small Solid Bodies
R-15,000
100 K
K = ll.4
Kzll.4
E
E = I.O
E>I.O
C,
1.0 0.5
-R= 25,000 - / "-R= 20,000 R-15,000 E
.4
_ i —
10
20
, 1 ) 1 1 1
100
K
Fig. 7.26. Replotted Keulegan-Carpenter data (Hudspeth, 1991).
The parametric dependency of the eccentricity of the error ellipse and the alignment of the axes are now shown to depend explicitly on the Keulegan-Carpenter parameter K by the Dean eccentricity parameter E defined in Eq. (7.73b). There appear to be at least two sets of data where the significance of the Dean eccentricity parameter is obvious. The first data set is the well-known replotted Keulegan-Carpenter force data for a horizontal circular cylinder (vide., Sarpkaya and Isaacson, 1981 and Chakrabarti, 1987). The second data set are the transverse lift forces on circular cylinders reported by Hayashi and Takenouchi (1979). Figure 7.26 (cf., Sarpkaya and Isaacson, 1981 and Chakrabarti, 1987) demonstrates that the peak in the Q graph and the trough in the Cm graph of the replotted Keulegan-Carpenter data occur approximately at a Dean eccentricity parameter of unity or K = 11.40. A Dean eccentricity parameter of unity identifies the value of A" where the eccentricity of the error ellipse is zero and the semi-major and semi-minor axes of the error ellipse are equal (i.e., the error ellipse is a circle). Data for values of A" < 11.40 are relatively well-conditioned for identifying Cm (i.e., the semi-minor axis
546
Waves and Wave Forces on Coastal and Ocean Structures
is parallel to the Cm axis for E < 1.0); while data with values of K > 11.40 are relatively well-conditioned for identifying Cd (i.e., the semi-minor axis is parallel to the Cd axis forE > 1). It is obvious in Fig. 7.26a that the values of Cm all collapse onto a single line for E < 1.0 (or K < 11.40). However, this is not the case for E > 1.0 (or K > 11.40). Conversely, in Fig. 7.26b, there is less correlation for Q for E < 1.0 (or K < 11.40); but the data are relatively better correlated for E > 1.0 (or K > 11.40). The Dean eccentricity parameter E identifies the peak in Cd and the trough in Cm with data that are equally well-conditioned for identifying both of these two coefficients. It also delineates the two regions where the Reynolds parameter R must also be considered in addition to K. Figure 7.27 (Hayashi and Takenouchi, 1979) illustrates the transverse lift forces on a vertical circular cylinder under periodic wave excitation for various values of the Keulegan-Carpenter parameter K. The irregular behavior of the transverse lift force is evident for all values of the Keulegan-Carpenter parameter except for K = 11.8. At K = 11.8, the amplitude of the transverse lift force is completely stable. This could be explained by the error ellipse
Run No. (1)
W*H^|jh^^
RMSKC = 5.2 ,\^www-y\vWM\iviw
(3)
"iMv^^:.,;^..'.*^,,.;, .,,.^ f ,.,. J l ,,
(4)
RMS KC = 10.1
RMSKC=12
(5)
(6)
RTvlS KC = 6 2
RMS KC= 18.2
j*i§i^^
RMSKC=11.8
20 sec Fig. 7.27. Transverse lift forces on vertical circular cylinders. Note the completely stable behavior when K(= KC) = 11.8 (Hayashi and Takenouchi, 1979).
547
Deterministic Dynamics of Small Solid Bodies
being a circle at approximately K = 11.8. Other possibilities exist and the dependence on R requires further analyses. In order to determine the suitability of the data to resolve the force coefficients Cm and Cd, changes in the minimum values of these coefficients for a given mean square error must be minimized from the following Taylor series expansion (vide., Chapter 2.3.3):
e2 = (e2)ffiin + ^r(SCd) + ^(SCm) otd
+
^hL
oCm
+
JCJ-2L
+ J"
dCdaLm
(SCd)(8Cm)
+m
( 5)
-
"
The slope of the error surface at the minimum is zero; therefore, Eq. (7.75) reduces to 9 2 e 2 ,o„ w*„ , d2€2(8Cd)2 €2 = (e2)min + ——— x (8Cd)(8Cmm) + \ - 2 x ' ' dCddCm ' dC 2
, +
d2€2(8Cm)2 8Cm 2
For Cm = constant, it may be shown that this minimization of the changes in the minimum values gives (Dean, 1976)
8cd = ~y and for Cd = constant (Dean, 1976)
2 JJ -
(^
J(ut(z,6)2)
P*D
The ratio of the axes of the error ellipse may be computed from E by
8Cd
TTD]) (ut(z,0)2)'
K
"
;
For simple harmonic oscillations, Eq. (7.76) reduces to „
V3.fi:
Cm V3 \dFd\
where \dFd\ = Cdpu2n(z)/2 harmonic motions.
Cm V3
and \dFm\ — CmpjtDcoum(z)/4
for simple
548
Waves and Wave Forces on Coastal and Ocean Structures
Table 7.2. Suitability criteria for determining Cd and Cm (Dean, 1976). E
<0.26 0.26-4.0 >4.0
K
<3 3-46 >46
Relatively well-conditioned to determine Cm Cd and Cm
cd
Equation (7.77a) demonstrates that data are relatively well-conditioned to determine Cm for small values of K and are relatively well-conditioned to determine Q for large values of A". The error surface for simple harmonic oscillations in dimensionless form is e2 - (c2)mm
\fdV
Q/S)(8Cd/Cdy
2
+
(P^-ml^m)
2W1
(7.78)
Dean (1976) gives criteria for estimating the suitability of the data for determining Cd and Cm that are summarized in Table 7.2. 7.7.2. Amplitude/Phase Method
(Geometric)
The two-term Morison equation coefficients Q and Cm for a cylinder in waves depend on the correct measurement of amplitudes and phase shifts between the ambient wave kinematics and the measured force. Hudspeth, et at. (1988) apply a regression analysis to develop an algorithm that illustrates how these force coefficients will change if incorrect amplitudes or phase shifts are introduced in the analysis due to: 1) errors in the data acquisition; 2) numerical data reduction techniques; or 3) natural variations due to vortex shedding. The algorithm assumes that the two-term Morison equation models the measured forces exactly and that linear wave theory models the wave kinematics exactly. A least-squares analysis of the time-averaged, mean square error between measured and predicted forces is applied. Dimensionless variations in the force coefficients are shown to depend on two dimensionless parameters: 1) a dimensionless force amplitude ratio W (the O'Brien parameter
549
Deterministic Dynamics of Small Solid Bodies
Eq. (7.49c)) that is proportional to the Dean eccentricity parameter E and to the Keulegan-Carpenter parameter K); and 2) a dimensionless velocity amplitude ratio V(z) that is a function of the vertical elevation in the water column z. Their algorithm combines both the effects of data condition and of the wave amplitude/phase that complements the earlier development by Dean (1976). Good agreement is obtained with laboratory data of wave forces on a vertical, sand-roughened cylinder wherein the force measurements were purposefully phase shifted with respect to the wave phase in small increments up to ±33.8 degrees (±3n/l6 radians). Variability observed in the values of Cd and Cm may be due to several causes: viz., the accuracy of the two-term Morison equation; incorrect estimates or measurements of the wave kinematics; the influence from unknown roughness; measurement errors; poor condition of the data; wake encounter effects; or the inadvertent introduction of erroneous amplitudes or phase shifts into the data acquisition or the numerical analysis. There are several possible causes for a phase shift error. For example, there may be a spatial separation between the wave profile recorder, the current meter (if used), and the pile where the force is measured (vide., Fig. 7.28). The electronic or numerical filtering of data signals may introduce both a phase shift and an amplitude distortion. The sequential sampling of multiple data channels by analog-to-digital recorders introduces a small phase shift. If these potential amplitude and phase shift errors and the conditioning of the data for parameter estimation are not considered appropriately, variations in the values of the force coefficients will result. •
Wave staff
Instrumented pile
D Currentmeter
ggg •*- Force transducer
Fig. 7.28. Typical wave and wave force measurement configuration.
550
Waves and Wave Forces on Coastal and Ocean
Structures
Figure 7.28 illustrates a typical experimental configuration. The wave staff and current meter (that may not be superimposed as shown) are located at the origin, while the pile is located at some distance from the origin. A phase shift in the measurements will result from this spacing and must be taken into account. The true force is assumed to be represented exactly by the two-term Morison equation given by (Eq. 7.65). An erroneous phase shift cox between the computed force and the computed kinematics is denoted by 0 = to it + T) in Eqs. (7.70c,d). Minimizing the mean square error between Eqs. (7.65 and 7.66) by Eqs. (7.50) with respect to the computed coefficients (denoted by superscript primes) according to 0,
dk'm
dk'
(7.79a,b)
0
yields the following two equations: -Kd{U\U\ut)
- Km{Utut)+k'd{u\u\ut)
+ k'm(u2t) = 0,
-Kd(Uu\U\ \u\) - Km (Utu\u\) + k'd{u4) + k'm{u, u\u\) = 0,
(7.80a) (7.80b)
where the true and computed kinematics are defined in Eqs. (7.70). Equations (7.80) may be rearranged to give a dimensionless inertia coefficient ratio em and a dimensionless drag coefficient ratio e^ defined by the following: €/n —
c
Ca 2 (U\U\ut) + (Utut) Cm7tD (M2)
c
(Uu\U\ \u\) +
(7.81a)
(Cm/Ca)(7TD/2)(Utu\u\
(7.81b) (u4) Cd It is not a trivial task to evaluate some of the integrals in Eqs. (7.81a, b) that require absolute values of elementary transcendental functions (Hudspeth, et al., 1988). Both negative and positive phase shifts must be considered. The dimensionless inertia coefficient ratio em is given by *d
em = V(z) cos(
sin(<wr) 5TT
\(OX\ < 7T
(7.82a)
551
Deterministic Dynamics of Small Solid Bodies
and the dimensionless drag coefficient ratio ed by 1
T
ed = — V2(z)[2n + 3sm(\2(or\)-2\2a)x\ 3n + ^ - sin(|2a>T|)], 5W
- (|2WT| - 7r)cos(|2«r|)
\2cox\ < it,
(7.82b)
where the dimensionless velocity amplitude ratio V(z), the dimensionless O'Brien force amplitude ratio W, and dimensionless Keulegan-Carpenter parameter K, are defined by , Um(z) V(z) = — — , T//
W=-
Cd K j '
K
Um(z)T = —Fi—•
(7.83a-c)
The O'Brien force ratio W is the ratio between the true drag force and the true inertia force (vide., Eq. (7.49c) in Sec. 7.6.3). The magnitude of W may be evaluated to determine the condition of the data to estimate the force coefficients because it is directly proportional to the Dean (1976) eccentricity parameter E; i.e.,
where the Dean eccentricity parameter E is defined in Eq. (7.73b). The dimensionless force coefficient ratios defined by Eqs. (7.82a-d) incorporate not only the effects of amplitude/phase shift errors but also the condition of the data for estimating the force coefficients through the parametric dependency on the Dean eccentricity parameter E. The parametric dependency on the two dimensionless parameters V (z) and W (or E and K) are evaluated separately. Figure 7.29 illustrates the parametric dependency of the dimensionless drag coefficient ratio €d on the dimensionless force amplitude ratio W (or E or K) for a constant dimensionless velocity amplitude ratio V(z) = 1.0 (i.e., the computed velocity amplitude equals the true ambient velocity amplitude). For relatively large values of W(> 4.0), €d is not sensitive to the magnitude of the phase shift near the origin \cor\ = 0 . Relatively large values of W (or E or K) imply that the data are drag-dominated and are relatively well-conditioned for
552
Waves and Wave Forces on Coastal and Ocean Structures
COT ( r a d i a n s ) Fig. 7.29. Parametric dependency of dimensionless drag coefficient ratio on O'Brien force ratio W for V(z) = 1.0.
determining the drag coefficient C'd. Note that if Cd ^ 0.9 and Cm ~ 2.0, then*" ~ 22W. For relatively small values of W(< 0.1), u is very sensitive to the magnitude of the phase shift near the origin. This implies that for small values of W{< 0.1) (or K < 2.2), the data are relatively ill-conditioned for determining the drag coefficient C'd. The slope Sd of the dimensionless drag coefficient ratio ed near the origin provides additional insight into the condition of the data for estimating C'd and will be examined in detail later. Figure 7.30 illustrates the parametric dependency of the dimensionless inertia coefficient ratio em on the dimensionless force amplitude ratio W (orE or K) for a constant velocity amplitude ratio V(z) = 1.0. For relatively small values of W(< 0.1), em is not very sensitive to the magnitude of the phase shift near the origin \a>x | ~ 0. Relatively small values of W (or E or K) imply that the data are inertia-dominated and are relatively well-conditioned for determining the inertia coefficient C'm. For relatively large values of W(> 4.0), em is very sensitive to the magnitude of the phase shift near the origin. This implies that for large values of W(> 4.0 or K > 88), the data are relatively ill-conditioned for determining the inertia coefficient C'm. The slope Sm of the dimensionless inertia coefficient ratio €m near the origin provides additional insight into the condition of the data for estimating C'm and will be examined in detail later.
553
Deterministic Dynamics of Small Solid Bodies
COT (radians) Fig. 7.30. Parametric dependency of dimensionless inertia coefficient ratio on O'Brien force W ratio for V(z) = 1.0.
For small phase shifts (\cor\ ~ 0), values for Eqs. (7.82a, b) are given by the following approximations:
V{l-|'£)(«r>
(7.84a) (7.84b)
Q
Differentiating Eqs. (7.84a, b) with respect to COT gives the following formulas for the slopes: dem *Jm —
d(cor)
nWV\ wv\ 3TT J '
_= Sd
da Jo_ d(coz)
/64\ ,9^
/V 2 N F
) .
(7-85a,b)
Figure 7.31 illustrates that the slope Sm is negative near the origin, independent of cot and proportional to the product V(z)W. This confirms the earlier observation that for inertia-dominated data {W < 0.1) changes in em are relatively small and nearly independent of the phase shift cox near the origin. For the data that are ill-conditioned for determining the inertia coefficient (W > 4.0), the slope Sm and the changes in em are relatively larger near the origin. Figure 7.32 illustrates that the slope Sd is positive near the origin, independent of cor, and proportional to the ratio V2(z)/W. This confirms the earlier
554
Waves and Wave Forces on Coastal and Ocean Structures
TT "4"
-rr
n
"T
n
TT 8
4
COT (radians) Fig. 7.31. Parametric dependence of slope 5 m on the ellipse eccentricity E near the origin.
IT
Tr
4
" 8
n
jr
JT
8
4
OJT (radians) Fig. 7.32. Parametric dependence of slope Sd on the ellipse eccentricity E near the origin.
observation that for drag-dominated data (W > 4.0) changes in Q are relatively small and nearly independent of the phase shift near the origin. For data that are ill-conditioned for determining the drag coefficient (W < 0.1), the slope Sd and the changes in Q are relatively larger near the origin. Figure 7.33 compares em and Q with synthetically phase-shifted laboratory data on a vertical circular cylinder measured in the O.H. Hinsdale-Wave
555
Deterministic Dynamics of Small Solid Bodies
_| [ _ TT 4
1
1 _TT 8
,
L —, O
COT (radians)
, TT 8
,
1 TT 4
Fig. 7.33. Comparison of em ande^ with synthetically phase-shifted laboratory data (Hudspeth,
era/., 1988).
Research Laboratory at Oregon State University (Hudspeth, et ah, 1988). These synthetically phase-shifted data are for both a relatively small value of K = 7.9 (E < 1.0) and a relatively large value of* = 23.3 (E > 1.0).
7.7.3. Matrix Condition Numbers
(Numerical)
The Dean error ellipse methodology (Sec 7.7.1) and the amplitude/phase error methodology (Sec 7.7.2) provide geometric interpretations of the condition of the wave kinematic data to identify the drag and inertia coefficient C'd and C'm. Because both of these methods are derived from a least square error, standard techniques from error analyses are available to determine matrix condition numbers (Atkinson, 1989). These matrix condition numbers provide numerical measures of the sensitivity of the computed empirical force coefficients to small perturbations in the wave kinematic/force data. These numerical measures of the condition of the data may again be related to the two geometric methodologies by the Dean eccentricity parameter E. Minimizing the mean squared error defined in Eq. (7.50a) with respect to C'd and C'm gives the following matrix equation: AX = B
(7.86)
556
Waves and Wave Forces on Coastal and Ocean Structures
"^ •
~**W
RUN 131 ' K= 30.8
RUN 129' K= 28.76
VTVAJ RUN 133 = K = 17.18
RUN 145= K= 13.02
Fig. 7.34. In-line and transverse lift forces on a vertical circular cylinder forced to oscillate in still water (Maull and Milliner, 1979).
where the scaled matrices in Eq. (7.86) are given by A=
3K
V
0
1
\in °1 0
X =
1
c
(7.87a)
(7.87b) (7.87c)
LWJ
(Fu Ut t) B = ¥\}f(Fu\u\) \.].
(7.87d)
Matrix A is Hermitian and unitary. It becomes a unit matrix with matrix condition numbers identically equal to unity when A" = 4TT 2 /3 = 13.16 and E = 2/y/3 = l.5. The transverse lift force on a vertical circular cylinder forced to oscillate in still water measured by Maull and Milliner (1979) are shown in Fig. 7.34. The transverse lift forces are stable and repeat exactly only when K = 13.02 for RUN 145. The four standard measures of the condition number of the error matrix A defined in (Eq. 7.87a) are summarized in Table 7.3. The matrix condition number when K = 11.40 or E— 1.0 are also tabulated in column 3 of Table 7.3.
557
Deterministic Dynamics of Small Solid Bodies
Table 7.3. Summary of matrix condition numbers. Matrix condition number (Atkinson, 1989)
K= 13.16 £=1.15 (2)
# = 11.40 E= 1.0 (3)
1.0
1.15
1.0
K < 13.16 E < 1.15 (4)
K > 13.16 E> 1.0 (5)
(1) Cond (A)i
4JTZ _
2
7
3K
£V3
2
IK ~ £\/3"
4;r
1.15
4 ^ _ £A/3 3tf
3A"
1.15
4*f
Cond (A) c Cond (A) 2 Cond (A)*
1.0
=
3# ~ £ v ^
4JT
3tf 4jr
£V3
2
£V3
2
The four standard matrix condition numbers in column 1 of Table 7.3 are defined as follows (Atkinson, 1989): Cond(A)i = Cond(A)oo Max | A. |
Cond(A)2 =
kecr[A x A] Min \k\
lAI.IA" - i i
1/2
" Max W '
Cond(A*) =
kea[A] Min \\\
Xeo[A x A] where | • | = a matrix norm; A - 1 = matrix inverse; A. = eigenvalue of the matrix;CT[»]= spectral radius of the matrix ([•]); and A* = complex conjugate transpose (Atkinson, 1989). Summary The condition of wave kinematic/force data to identify the empirical force coefficients Cd and Cm in the Morison wave force equation for small bodies may be evaluated by three methods; two geometric (Sees. 7.7.1. and 7.7.2) and one numerical (Sec. 7.7.3). The two geometric methods are the Dean error ellipse methodology and the amplitude/phase error methodology. The Dean error ellipse demonstrates geometrically the condition of the data by the alignment of the Cd and Cm axes of the error ellipse. A separate error ellipse is required for each value of the O'Brien force ratio W as shown in Fig. 7.35. The amplitude/phase methodology demonstrates geometrically the condition
558
Waves and Wave Forces on Coastal and Ocean Structures
e
2>el / " ~ \
E >I.O IT > 1.0
E< 1.0 w< 1.0
Cm 63 >6|
'CmW
6£0
'^d'min
I
* 'cd'min Fig. 7.35. Suite of Dean error ellipses required for each value of W.
of the data by the magnitude of the slopes of contours of the force coefficient ratios W passing through a zero phase error cox = 0 as shown in Figs. 7.31 and 7.32. Each of the separate graphs in Fig. 7.35 for each separate value of W required by the Dean error ellipse methodology may be replaced by a single graph with contours of W as shown in Figs. 7.29-7.32. Both of these two error methodologies may be related to the Keulegan-Carpenter parameter K by the Dean eccentricity parameter E. The Dean eccentricity parameter E = */3K/(2TT2) provides a geometric measure of the condition of wave force data on circular members for estimating the force coefficients C'd and C'm. Two sets of data appear to illustrate the physical significance of the Dean eccentricity parameter. The variability in Cm for E > 1.0 (or K > 11.40) and in Cd for E < 1.0 (or K < 11.40) in the replotted Keulegan-Carpenter data in Fig. 7.26 may be explained by dividing the data into two parts determined by a Dean eccentricity parameter equal to unity (E = 1). The unusual stable behavior of the amplitude of the transverse lift force in the Hayashi-Takenouchi (1979) wave force data in Fig. 7.27 may possibly be explained by a Dean eccentricity parameter of unity. The axes of the Dean error ellipse are shown to be parallel to the Cm and Cd axes for simple harmonic kinematics (i.e., (u\u\ut) = 0). The Dean eccentricity parameter E may be incorporated into an error analysis that also includes errors in the amplitudes/phases of the kinematics. Comparisons shown in Fig. 7.33 with synthetically phase-shifted laboratory data are quite good for phase-shifts \cox\ < T T / 8 .
Deterministic Dynamics of Small Solid Bodies
559
Four measures from standard matrix error analyses may be computed to determine the matrix condition numbers for the least square error between the measured and predicted wave forces. Each of the four error matrix condition numbers are identically equal to unity when K = 13.16 or when E = 1.15. The only stable transverse lift force in the Maull and Milliner (1979) data shown in Fig. 7.34 occurs when K = 13.02. The matrix condition numbers tabulated in Table 7.3 are equal to 1.15 when K = 11.40 and when E -\.Q. The Dean eccentricity parameter E may be applied to compare each of the three methods to evaluate the condition of the wave kinematic/force data to identify the force coefficients Cj and Cm. It also connects each of the methods to the Keulegan-Carpenter parameter K. To illustrate the connection of the eccentricity E to design conditions, a typical design specification for a member on a jacket-type offshore platform may require a pile diameter D = 13 ft «s 4TV ft (4 m) for a design wave height H = 48 ft (15 m) and a wave period T = 13 s « 4n s. For Umax « HTT/T, thenA" = 12andi? = 1.5xl0 7 and/J = 1.3 x 106. There are very few data sets available for Cm and Cd in the narrow interval between 11.40 < K < 13.16 with correspondingly high values of the Reynolds R or frequency fi parameters.
7.8. Modified Wave Force Equation (WFE, Relative Motion Morison Equation) The linearized modified Wave Force .Equation (WFE, relative motion Morison equation, American Petroleum Institute, 1987) is applied to two types of structures that consist of small members. The first type is a vertical circular cylinder that is an articulated tower and that oscillates about a base that is connected to the ocean bottom. This is a single-degree-of-freedom oscillator (SDOF) that responds to a wave induced, relative motion moment load M$(t, ao, ©5, 6)5). The second type consists of two semi-immersed horizontal circular cylinders that are tautly moored to the ocean bottom, connected on each end by rigid horizontal members, and floats horizontally as a pontoon system. This is a three (multiple)-degree-of-freedom oscillator (MDOF) that responds to two wave induced, relative motion force loads F2(x{,Zi,t,ao,X2,X2) and Fi(xi,Zi,t,ao,Xi,Xi) and a wave induced, relative motion moment load Ms(xi,Zi,t, ao, ©5, ©5). These two examples illustrate both translational and rotational dynamical systems of small members. The goal of both of these
560
Waves and Wave Forces on Coastal and Ocean Structures
two examples is to solve the following dimensionless dynamic equations of motion in the y'th mode of oscillation/rotation:
Xj(r) + 2!ja>JhXj(T) + &2jhXj{x) = SstjS>2Jh J ^ J (T -ft),
j = 1,2,3, (7.88a-c)
&j(r) + 2Cj5>jh®j(T) + a>2jh®j(r) = AstjS>2h {**| (r -
ft),
j = 4,5,6, (7.88d-f)
that may be readily solved by the well-known method of undetermined coefficients (vide., Chapter 2.5.6) and where the overdots (•) denote ordinary temporal derivatives of a Lagrangian solid body; i.e., d (i) = —,
di'
.. _ d2 (i) (#) = - j .
~ dJi
(7.89a,b)
The linearized WFE separates the wave induced relative motion loads into two components: viz., 1) an added mass and a fluid viscous damping component that are proportional to the Lagrangian solid body acceleration Xj(z,aj)/@j(T,<Xj) and velocity Xj(r,aj)/&j(r,aj) in the ;'th mode, respectively; and 2) a wave induced exciting force/moment load F?(t, ceo, Pj)/M?(t, ao> Pj) m the jth coordinate direction on a fixed body. Because this linear decomposition of the WFE loads results in a wave induced hydrodynamic force/moment load on a fixed body, fixed bodies are not analyzed separately because fixed bodies experience the same exciting force/moment loads (correct to the linear approximation) as bodies that respond dynamically! The damping ratio fy in the jth mode of oscillation/rotation is due entirely to the fluid hydrodynamic force/moment from the linearized WFE. Once the dimensionless dynamic equations of motion have been derived, the fundamental problem solution technique is an application of the method of undetermined coefficients (vide., Chapter 2.5.6) given by the following sequence: (i) Compute the amplitude F?/M? and phase fij of the jth mode exciting force/moment load on a fixed body by the linearized WFE
561
Deterministic Dynamics of Small Solid Bodies
according to:
Ff(t,a0,Pj) Mf(t,a0,Pj)
•ao±Pj).
(7.90a) (7.90b)
(ii) Select the form of the solution for the dynamic response in the jth mode of oscillation/rotation by the method of undetermined coefficients in Chapter 2.5.6 from the form of the family of the exciting force/moment load F?(t, ao, f5j)/MJ(t, ao, fij) according to [Xj(f,aj)\ _ Uj 1 (sin)
{0,(^ 7 .)J-i^Hcosl
_
(ft;f
(7.91a)
ao±
^'
(7.91b)
where ao = phase of the free surface wave profile r](x,t,oto) and the choice of {sin/cos} depends on the family of the exciting force/moment loads in Eqs. (7.90). (iii) Compute the dimensionless natural frequency for the jth mode with hydrodynamic effects (viz., added mass) a>jnfromthe stiffness Kj /KJ and mass/mass moment of inertia m/Ij by lKj/(m -2 °>jh =
w
-2
+ Ma)
(7.92a) )
'
[ Kj/Vj + Ia) J
(7.92b) where Ma/Ia = the dimensional added mass/mass moment of inertia from the linearized WFE. Compute the dimensionless damping ratio £j from the dimensional damping coefficient Cj from the linearized WFE and critical damping Cjcr by
0=
-jcr
y/Kj{m
t-' jcr — -^
I V^^J
+ M_a)\ +
la) V
(7.93a,b)
(iv) Compute the static displacements 8stj/Astj from the amplitudes of the exciting force/moment load ^ ^ / M ? and stiffnesses KJ/KJ (vide., Chapter 2.5.6) by (7.94a) •
=
'
[ Kj
)
(7.94b)
562
Waves and Wave Forces on Coastal and Ocean Structures
and dynamic magnification factor D (&>;%) (vide., Chapter 2.5.6) from DOSJfc) = L(l - 5>~]l? + Qtjl&jhfr1'2.
(7.95)
(v) Compute the dimensionless amplitudes |§/|/|£27-| of the y'th mode of oscillation/rotation from the static displacement 8st/Ast. (vide., Chapter 2.5.6) and dynamic magnification factor D \(o~^) by l£/l 1
TW~-2N f &stt \
(7.96a) (7.96b)
(vi) Compute the phase angel aj in Eqs. (7.91a,b) from the family of the exciting force/moment solutions by the method of undetermined coefficients (vide., Chapter 2.5.6) by first defining tj = arctan ( /
__, J .
(7.97a)
For the family of exciting force/moment given by Eq. (7.90a) with the response solution given by Eq. (7.91a) and with the same choice of ± sign in each equation, the phase angle aj is given by a;3 = arctan
/ tan 8 ± tan e \ . VlTtanetanB/
v(7.97b,c)
'
For the family of exciting force/moment given by Eq. (7.90b) with the response solution given by Eq. (7.91b) and with the same choice of ± sign in each equation, the phase angle aj is given by a,1 = arctan
/ tan B :p tan e \ }. Vl±tanetanB/
v(7.97d,e)
"
The following two examples in Sec. 7.8.1 and 7.8.2 demonstrate this problem solution technique outlined above in Steps (i)-(vi) by the method of undetermined coefficients (vide., Chapter 2.5.6) for both SDOF and MDOF dynamical systems.
563
Deterministic Dynamics of Small Solid Bodies
Fig. 7.36. Definition sketch for articulated SDOF tower.
7.8.1. Articulated Circular Cylindrical Tower: SDOF System The dynamic equation for the SDOF motion in the pitch mode response rotating about a bottom-fixed base located at s = z + h of the articulated tower shown in Fig. 7.36 is /5©s(0 = [-FBhB
+ Wbhb + Wghg + Wphp]&s(jt) +
= -K*5e5(t)+
M5(t,ao,®5,®s) (7.98a)
M5(t,a,&5,G5),
where the over dots represent ordinary temporal derivatives of a Lagrangian tower according to ..
d2& dtz
.
d@ dt
(7.98b,c)
where I5 = total mass moment of inertia about the bottom-fixed base; Fg = buoyantforce(=pg7rD 2 /i/4); Wb = total weight of ballast; Wg = total weight of the tower material; Wp = total lumped-mass weight of the platform superstructure; M5(t,ocQ,@5,®5) = wave-induced hydrodynamic load moment computed by the linearized WFE where ao = an arbitrary phase angle for the free surface profile r)(x,t,ao); K^5 = rotational stiffness due to excess buoyancy that is given by K;5
= Wgh
El
m-imymm-m (7.99a)
564
Waves and Wave Forces on Coastal and Ocean Structures
and where 0 5 (O = |fi 5 | cosioot - a0 - a5) - RQ{Q5 exp ~i(cot - ao)}.
(7.99b)
The contribution from each individual moment in the term in square brackets [•] in Eqs. (7.99a) to the total restoring moment K^5®s(t) about the point of rotation with coordinates {XR,SR} = {0,0} may be computed from the geometry shown in Fig. 7.36 according to J^rRj
xFj
= - £ ( X - XR)j(±Fj)ey,
j
j = b, B,g,p,
(7.99c)
J
where each moment arm (X — XR)J may be approximated for small rotations ©5(f) from the vertical equilibrium geometry shown in Fig. 7.36 by (X - XR)j = hj tan &s(t) « hj&s(t).
(7.99d)
The total mass moment of inertia I5 for rotations about the base of the articulated tower located at the ocean bottom at s = z + h = 0 is a linear combination of the individual mass moments of inertia about the bottom-fixed base from the ballast component l\,\ from the structural tower component Ig; and from the lumped mass platform component Ip: h = h + Ig + IP,
(7.100a)
where the mass moment of inertia about the center of gravity IJCG for each y'th structural component may be computed from non-inertial axes x'j located at the center of gravity of each 7'th structural component from IJCG
= /
rfdnij
Jm i
PjrfdVj;
rf = xf + yf,
dVj = r'jdcpdr'jds,
- / / / . (7.100b-d) where the homogeneous mass density pj for the y'th structural component may be computed from WPj = -rjr, (7.100e) where Wj — the total weight of the jth structural component and where the vertical integration over ds is over the total length Lj of each yth structural
565
Deterministic Dynamics of Small Solid Bodies
component at equilibrium ©5(0). The mass moment of inertia Ij about the point of rotation at s = X = Y = 0 for the y'th structural component in Eq. (7.100a) may be computed by the parallel axis theorem according to
/,- = IJCG + h2j^ = fffWj PF?dv + ^ 5 y ^ / gn{D>l/4Lj)J0 W D2
, u2^j
28 4
+
J
/ Jo
_u2Wj
= hi
^
+ h1:^J 8
/ r'fd^drjds J Jo J 1 +
1 /D 8 \h
j = b,g,p.
(7.100f)
Because the platform super structure is idealized as a lumped mass with no size (i.e., D = 0), Eq. (7.100f) reduces to h2pWp
h=
(7.100g)
8
Substituting Eqs. (7.100f, g) into Eq. (7.100a) and neglecting the D/hj terms gives for the total mass moment of inertia / 5 = [Wbh\ + Wgh] + Wph2p]/g Wgh? 8
h,
+
(7.101)
where the heights hj from the fixed base bottom at 5 = X = Y = 0 are illustrated in Fig. 7.36. The wave-induced, hydrodynamic load moment M${t, ao, ©5, ©5) may be approximated by the linearized modified wave force equation (WFE, relative motion generalization of the Morison equation) according to M it „ & & \
I f0hdMm(s,t,ao,&5) +JQ dMdi(s,t,a0,®5) [ +Jo dMFK(s,t,a0)
I J (7.102)
where the relative motion differential inertial component dMm (s, t, ao, ©5) is dMm(s,t,a0,&5)
= Ca(p7TD2/4)s[ut(s,t,ao) - s@5(t,a0)]ds,
(7.103a)
566
Waves and Wave Forces on Coastal and Ocean Structures
where the temporal partial derivative of an Eulerian field variable is denoted by a subscript according to 3(.) (•)t
=
dt '
and where the relative motion differential linearized drag component dMdi(s,t,a0, ©5) is dMdi(s,t,a0,&5)
= CaiipD/2)s[u(s,t,ao)
- s@5(t,ao)]ds,
(7.103b)
and the Froude-Kriloff component dMp# (s,t,ao) is dMpK(s,t,ao) = (picD /4) sut(s,t,ao)ds
(7.103c)
where the Froude-Kirloff component is the inertia of the fluid in the absence of the body (vide., Chapter 3.3.1). Assuming that linear wave theory is valid and that the free surface displacement is given by H
\H
rj(x, t, ao) = ~r COS(KX — cot + ao) = Re | — exp i(kx
— cot + ceo)
(7.104a) where the arbitrary phase angle «o m Eq. (7.104a) is introduced for data analyses by Finite Fourier Transform (FFT) algorithms (vide., Chapter 9.2), then the horizontal components of the wave-induced velocity u(s,t, ao) and acceleration ut (s, t, ao) may be expressed by Hco cosh ks u(s, t, ao) =
2 sinh kh COS(KX — cot + ao) Hco cosh ks expi(kx — cot + ao) = Re 2 sinh kh — RG{UM(S) exp i(kx
— cot +
ao)}
(7.104b)
and by Hco2 cosh ks . ut(s,t, ao) = — sin(o: — cot + ao) 2 sinh kh Hco2 cosh ks = Re • exp i (kx — cot + ao) 2 sinh kh Re{—icoUM(s) exp i(kx — cot + ao)},
(7.104c)
567
Deterministic Dynamics of Small Solid Bodies
where the depth dependent velocity amplitude Um (s) from linear wave theory is given by Hcocoshks ,„,„..,, UM(s) = -^r^-rvT(7.104d) 2 sinh kh If the slope of the instantaneous water surface profile r\x (t, ao) and the temporal derivatives of the slope of the instantaneous water surface profile evaluated at x = 0 are given by r]x(t,ao) = sm(cot — ao) = Re ji—— exp — i(cot — ao)} (7.105a) Vxt(t,cio) =
Vxtt (t, ao) =
cos(cot — ao) = Re I —— exp —i(a>t — ao)} (7.105b) kHco2 f Ww2 , 1 — sm(cot - a 0 ) = Re j -i —-— exp -i(cot - a 0 ) | (7.105c)
then the wave-induced water particle kinematics in Eqs. (7.104a, b) may be defined in terms of the temporal derivatives of the slope of the free surface profile at x = 0 according to Hu> cosh ks 1
{
{
————- exp-i (cot -ao)\ = 2 sinh kh J Hco2 cosh ks -i—
2
hV(s)r)xt(t,ao), (7.106a)
1
. exp -i(cot - a 0 ) } = smh A:rt J
h^(s)r)xtt(t,a.o),
(7.106b) where the dimensionless depth dependency is cosh ks kh sinh Krt The relative motion differential component of the inertial dMm (s,t, ao, ®s) and of the Froude-Kriloff dMpK (s,t, ao) moments may be combined with the second temporal derivative of the slope of the water surface profile x]xtt (t, ao) according to dMm(s,t,a0,@5)+dMFK(s,t,ao) = (P7tD2h/4)(s/h) [Cmh^(s){r]xtt(t,ao)}
- Cas{®5(t,a0)}]ds,
(7.107a)
568
Waves and Wave Forces on Coastal and Ocean Structures
that may be integrated over the SWL depth h in Eq. (7.102) if both Cm and Ca are constants to obtain Mm(t,a0,®5)
+MFK(t,a0)
/•Ffl/i 2 \ f"
fh
= / {dMm(s,t,a0, Jo
„,,?/,
3C
2 l
,, ,
1
1
©5)
+dMFK(s,t,a0)}
cosechMX
= (-IT j [ ^- ( ~ ^ r + - £ - ) x{rixtt(t,ao)}-Ca{G5(t,a0)}
,
(7.107b)
where Cm = 1 + Cfl,
FB = ynD2h/4.
(7.107c,d)
The linearized drag coefficient Cdi{s) is a function of the depth 5 and is related to the quadratic drag coefficient Cd in Eq. (7.60) in Sec. 7.6.4 by f27T 2 Cdi(s) — / cos (a)t — ao)\ cos(a>t — ao)\d(o)t) n Jo SCdUM(s) (4Ha>\ = \ = Cdkh — - *(*) (7.108) CdUM(s)
that has the dimensions [Length/Time] of the velocity Um(s) in Eq.(7.104d). The differential component of the linearized relative motion drag moment dMdi(s, t, ao, ©5) may also be combined with the first temporal derivative of the dimensionless wave slope r)xt(t,ao) according to dMdi (s,t,UQ,®5) (pD\ = Cd l^-J
{4Hkhco\
r ?
•
1
( — 3 — ) (sh) [* 2 (j){ifc,(f,a 0 )} - ^(5){© 5 a,Qfo)}J ds (7.109a)
and may be integrated over the SWL depth h if Cd is a constant to obtain fh Mji(t,ao,&s) ^ ^
= I Jo
dMdi(s,t,ao,®5)
(H/D)(ko/k)(FBh/3) a>{Tckh)2
(2kh (kh + sinh 2kh) + 1 - cosh 2kh \ . ,2,, V sinh kh / -S[kh(kh 2(k/k0))+2]{®5(t,a0)}
{lxt(t,a0)}
(7.109b)
569
Deterministic Dynamics of Small Solid Bodies
The dimensional dynamic equation of motion given by Eqs. (7.98a-c) may be scaled by the following dimensionless variables (with tildes ~ except for a dimensionless time r): T
= cot, x = kx, ®5(r) — (2h/H)®5(t,a0),
fj(r,ao) =
rj(t,ao)/(H/2).
Dimensionless temporal and spatial derivatives are now given by m(r,a0)
= nx(t,a0)/(kH/2),
fjxzir, «o) = r)xt(t,a0)/(kHco/2), 2
ilxxr(r,c(o) = r,xtt(t,ot0)/(kHco /2),
05(T) =
®5(t,a0)/(H/2h),
05(r) =
®5(t,a0)/(Hco/2h),
05(r) =
®5(t,a0)/(Hco2/2h).
Equations (7.98) may now be scaled by dividing by a buoyant moment = (FBh/3)(koH/2) to obtain the following dimensionless equation of motion: L/5 + C a J© 5 (r,ao) + Bv®5(t,a0)
+ K5®5(i,a0)
= M 5 £ (r,a 0 ,&) =
Km{fjxtt(r,ao)} +
Cddfjxtir,ao)}, (7.110a)
where M$ (r, ao, fis) = exciting moment on a fixed tower. The dimensionless mass moment of inertia is given by
<%)m+my<m)\ <•»*> FB.
the dimensionless added mass by (7.110c) the dimensionless viscous damping moment coefficient by Bv = Cd
(tor)
kh(kh-2(k/ko))+2 ,
(7.110d)
(Ttkh)3
and the dimensionless rotational stiffness moment coefficient by ^5
=
Mi^KF \)
w
g
B
~(FB\(hB\
(Wb\ fhb
l\wj\h J \wg)
m -m
U
(7.110e)
570
Waves and Wave Forces on Coastal and Ocean Structures
The dimensionless inertia moment coefficient is Km =
(kh)(k0h)
[koh — 1 + sech kh],
(7.110f)
and the dimensionless linear drag moment coefficient is '2kh(kh + sinhlkh) + 1 - cosh2A:/z Cdi = Cd ( —
(nkh)2 sinh2 kh
(7.110g)
Figure 7.37 illustrates the parametric dependencies on the dimensionless deepwater wave frequency koh = co2h/g of the dimensionless viscous damping moment coefficient B1' B" =
Bv Cd(H/DY
(7.11 Oh)
of the dimensionless inertia moment coefficient K'„ K
m =
(7.110i)
-fT>
and of the dimensionless linear drag moment coefficient Cidi
(7.110J)
r" -dl —C (H/DY d 100
0.01 0.01
0.1
1
10
kjl
Fig. 7.37. Dimensionless hydrodynamic coefficients for an articulated SDOF tower.
571
Deterministic Dynamics of Small Solid Bodies
The solution kh to the following dimensional linear dispersion equation is also shown: co2h
(7.110k)
= koh = khtanfakh.
The dimensionless dynamic equation of motion in Eq. (7.110a) may now be expressed in the following dimensionless form (cf, Eqs. (2.99) in Chapter 2.5.3) for a SDOF damped harmonic oscillator: 0 5 ( r , a o ) + 2£ 5 <»5A©5(T, «O) + a>ih@s(T,a0) = Mf(T,a 0 ,ft),
(7.111)
where the dimensionless natural frequencies <wo, /o and period To without hydrodynamic added mass are given by
where the dimensionless critical damping Bcr and damping ratio £5 are given by
Bcr-2y/K5(I5
+ Ca),
^5 =
Bv ^ , Brr
(7.112d,e)
and where the dimensionless frequency ratios a>$h, fs„ and period fsn with hydrodynamic added mass are given by (cf., Eqs. (2.99) in Chapter 2.5.3) u\h
K5
a>5h
(h + Ca)
(JO''
T Tsh
— fsh>
(7.112f-i)
where Eq. (7.112g) for fen is the inverse of the traditional definition for the dimensionless frequency ratio (vide., Chapter 2.5.3 or Clough and Penzien, 1975 p. 53, Eqs. (4-8)). The dimensionless exciting load moment on a fixed tower is given by M5£(r,a0,#0 =
h + Ca
= M5 cos (T — ao + P5),
(7.113a)
572
Waves and Wave Forces on Coastal and Ocean
Structures
where the dimensionless exciting moment amplitude M^ and phase ^5 in Eq. (7.113a) are
- F J (Km)2 + (Q/)2 Mf = 1 = , h + Ca
05 =- arctan I -z^\Cdi
(7.113b,c)
The solution to Eq. (7.110a) by the method of undetermined coefficients (vide., Chapter 2.5.6) is assumed to be a near-surface slope follower r\x given by the real part ofEq. (7.99b) as 05(r,ao) = ^5 cos(r — ao — as),
(7.114a)
where Q5 = D [(S)5hr2]
A« 5 ,
Ast5 = ^ - ,
Si5h
D [(^5/z)-2] = [ (1 - &jfy + as5/u5hy
(
(7.114b,c)
1/2
(7.114d)
( €5 = arctan I \ 1 - a>.5h-2 , (7.114e,f) where the signs in Eq. (7.114e) are changed from the signs in Eq. (2.101c) in Chapter 2.5.3 because the sign of the forcing phase angle ^5 in Eq. (7.113a) is changed from the sign of the forcing phase angle fi in Eq. (2.99a) in Chapter 2.5.3. The dimensionless damped natural frequencies a>5h,fsd and period 7 ^ are .,2 5d_
/ T \ 2
^
\ + Ju /
*h = ^T = (^-)
tan65 — tanfis \ , l+tane5tanft/
/ i \ 2
=^d-?52)-
=fsd = (^-)
(7.H4g-k)
\ •« 5 d /
The dynamic magnification factor DL(ft>5/,)~2J and phase angle €5 are shown in Fig. 7.38. The dimensional (without tilde ~) natural frequencies are easily shown to be col = a?w\,
co\h = to2 &\h,
a>\d = co2 d)25d.
(7.115a-c)
The dimensional (without tilde ~) inertia, damping and stiffness are /s =
'FBh^ 3g
k;Bv = (^\
k0hBv;K5 = H^kohKs
(7.116a-c)
573
Deterministic Dynamics of Small Solid Bodies
(a)
>J3 P
(b) 18° 160 140 120
10
^ ° f» 80
|c. =«h-* _/^^r===== [771 1MJ I—i
MJ
"li/^^^^^^^^ ^ **\i3\ -|oT|
60 40 20 0
Hi.o [ 0.5
1.5
2.5
w"« = r,,, IT Fig. 7.38. Parametric dependencies on the dimensionless damping ratio £5 of the a) dimensionless dynamic magnification factor D[ftW ~\ and b) relative phase angle 65.
7.8.2. Two Semi-Immersed Horizontal Circular Cylinders: MDOF System A single vertical circular cylinder is analyzed in Sec. 7.8.1 as a SDOF articulated tower that is constrained to rotate about a bottom anchored base in the x-z plane in response to a wave induced moment load. Here two circular cylinders are rigidly connected at their centers as shown in Fig. 7.39 and floated horizontally as a tautly-moored semi-immersed pontoon system. For normally incident surface gravity waves and a collinear current Uc, this system will oscillate as an uncoupled 3 degrees-of-freedom (MDOF) oscillator in the sway X2O), heave X3O) and pitch &s(t) modes of oscillation (vide., Chapter 8.1 for the definitions of these modes of oscillation). The pontoon system is constrained by two taut mooring lines with equal tension magnitude
574
Waves and Wave Forces on Coastal and Ocean Structures
Fig. 7.39. Two semi immersed, tautly moored horizontal circular cylinders of length B.
Fig. 7.40. Locations of the five separate coordinate axes on the MDOF pontoon system.
P in each mooring that are attached to the rigid connecting members "a-c-b" at point "c" at each end as shown in Fig. 7.39. The two horizontal circular cylinders, each of equal length B, are connected at their centers "a" and "b" at each end with a rigid connector "a-c-b" of length x as shown. The two pontoons float horizontally under the action of a steady current Uc only with two rigid connections "a-c-b" at each end of the pontoons that are located at the still water level (SWL). This MDOF application illustrates the static equilibrium conditions required between the Archimedes buoyant forces and external mooring constraints (vide., Chapter 8.2 where this static equilibrium constraint is generalized in the dynamic boundary conditions on a 6 DOF large Lagrangian solid body). The five separate sets of coordinate axes shown in Fig. 7.40 at the static horizontal equilibrium position of the semi immersed, double cylinder pontoon system under the action of a steady current only are required in order to analyze the dynamics of this relatively simple 3 degrees-of-freedom
Deterministic Dynamics of Small Solid Bodies
575
(MDOF) oscillating system. The first axes set is an inertial coordinate system x — {x,y,z} that is located at time t = 0 at the attachment point "c" of the two taut mooring lines with the vertical z axis positive up and the x-y axes in the horizontal plane at the SWL at z = 0. The second axes set is a body-fixed, non-inertial, moving coordinate systemic' = {x',y',z'} that is also located at time t — 0 at the attachment point "c" of the two taut mooring lines with the vertical z' axis positive up and the x'-y' axes in the horizontal plane at the SWL at z' = 0. The third axes set is also a body-fixed, non-inertial, moving coordinate system*^ = WR,y'R,z'R} for the point of angular rotation and is also located at the attachment point "c" of the two taut mooring lines with the vertical z'R axis positive up and the x'R-y'R axes in the horizontal plane at the SWL at z'R = 0 at time t = 0. Both of these latter two body-fixed, non-inertial, moving coordinate systems x' = {x',y',z'} and x'~ = [x'R,yR,zR} translate in linear momentum, but neither the point of translation x' = {x' = 0, y' = 0, z' = 0} nor the point of rotation xR = {XR = 0, y« = 0, ZR = 0} will also translate under angular rotations. The contributions to translational momenta from angular momenta is generalized in Chapter 8 to allow for additional translational displacements due to angular rotations when the origin of the translational axes x' = {x' = 0,y' = 0, z' = 0} and the origin of the rotational axes XR = {XR = 0, yR = 0, ZR = 0} do not coincide at t = 0. However, for this relatively simple two-dimensional planar 3 degrees-of-freedom oscillator, no additional translations due to angular rotations are required because the point of translation located at x' — [x' = 0, y' — 0, z' = 0} is identical to the point of rotation located at xR = {x'R, y'R, z'R); and, consequently, the vector r' between the point of translation x' and the point of rotation x'R is identically equal to zero; i.e., r' = x' — xR = 0. Finally, the fourth and fifth axes sets are also non-inertial, body-fixed, moving coordinate systems that are located at the center "a" of the large cylinder^ = [x'L, y'L,z'L] and at the center "b" of the small cylinder x's = {x's, y's, z's], respectively, with the vertical z/ axes positive up and the Xj'-y/ axes (j = L or S) in the horizontal plane at the SWL at z/ = 0 at time / = 0 for both coordinate systems. These latter two axes sets are required in order to compute the polar mass moments of inertia for each of the two cylinders by the parallel axis theorem.
576
Waves and Wave Forces on Coastal and Ocean Structures
The stiffnesses of each of the two mooring lines Kj in the vertical (j = z = 3) and in the horizontal (j = x = 2) modes are
K2 K3
sin2 8^ (2AE -P) cos2 8t
+P
(7.117a) (7.117b)
where 8i = the angle between each taut mooring line and the vertical z axis (vide., Fig. 7.41); P = the magnitude of the tension that is equal in each mooring line; I = total length of each taut mooring line; E = linear modulus of elasticity of the mooring lines; and A = cross sectional area of each mooring line. The large cylinder "a" weighs WL with a diameter Di; and the small cylinder "b" weighs Ws with a diameter D$.
Static position with only a steady current Uc The static horizontal equilibrium position of the semi-immersed, tautlymoored double cylinder pontoon system under the action of a normally incident steady current only is shown in Fig. 7.41. The origin of the inertial Cartesian coordinate system x — [x — y = z = 0} is located at the taut mooring line attachment point "c" with the x-y axes horizontal along the SWL at z — 0 at the static equilibrium position shown at time t = 0. The two mooring lines are attached to the rigid connector "a-c-b" at point "c" that may be computed from the requirement that the rigid connector "a-c-b" be horizontal at the SWL at
Fig. 7.41. Static free-body-diagram for only a normally incident steady current Uc.
577
Deterministic Dynamics of Small Solid Bodies
static equilibrium by summing the static moments about "c" according to
= (X - x0)Ws - x0WL - (x - XQ)FBS + XQFBL
(%)M?K
(7.118a)
where x and xo are defined in Fig. 7.39. Solving for a dimensionless distance xo/x f° r t n e attachment point "c" gives *o
<«w*>-(£)>-(?)
F
£lL
—.
(7.118b)
X WLFBL + Ws - FBs A numerical value for the circular cylinder "mass densities p" may be defined by 4Wi (7.119a) for j = L, S, Pi TtgD)B for the Archimedes buoyant forces FBj by jtD2- B FBJ=P8—^~
for j = L,S,
(7.119b)
and for the drag forces FQ for a normally incident steady current by FCj=CdjjDjBU?
for j = L,S,
(7.119c)
where C^. = drag coefficient for a normally incident steady current and may have different values for each cylinder because of their parametric dependencies (vide., Sec. 7.6). Substituting Eqs. (7.119a-c) into Eqs. (7.118a, b) gives xo
«*-'>(&)'-%(%)('+% (a?
X
(25 L - 1) + (2SS - 1) ( g )
(7.120)
2
where Sj = Pjg/pg = Yj/y = numerical value for the "specific gravity" of the j'th cylinder. For the special cases when In
\gxJ
CdL \DL)
« (25 s
(2SS - 1) * (2SL - 1),
1)
m
(7.121a) (7.121b)
578
Waves and Wave Forces on Coastal and Ocean Structures
then a very simple approximation for Eq. (7.120) is given by (Ds/DL)2
XQ
-r. (7.122) X 1 + (DS/DL)2 The equal tensions P in each of the two taut mooring lines may be computed from the following static equilibrium equations: Yl F* = FCL + Fcs ~ IP sin h = 0, (7.123a) Y^ Fz = - WL + FBL - Ws + FBs - IP cos h = 0,
(7.123b)
that may be solved for P and 8 values given by
(l-2SL)
7iD2LB
(£)'<'"
+
Pg
+
2CdL ( U, Uc\(l gDL)\
•
2SS)
CdsDsV CdLDL)_
*L (Ui U i | C"SDS\ n \gDL) y ^ cdL DL)
(7.123c)
2C
hi = arctan
(7.123d)
(l-25 L ) + ( g ) (1-2S 5 ) For the special cases when 5s
C
ds
(7.123e,f)
C
dL
then Eqs. (7.123c, d) reduce to P P8—§
TTDIB
-i2
(1 - 2SL)
2 QL
+L
it
8i — arctan
I
1+
U
C
\gDLJ
El DS
1+ V DL
(7.123g)
(7.123h)
a-25 L )(i+(gy
579
Deterministic Dynamics of Small Solid Bodies
Dynamic response an The wave-induced water particle velocity q(x,z,t,ao) d dynamic pressure po(x,z,t,ao) may be computed from a scalar velocity potential $>(x,z,t,cto)by q(x,z,t,ao)
= -VQ>(x,z,t,ao),
po{x,z,t,ao)
= p
d$>(x,z,t,ao) dt
(7.124a,b) where ao = an arbitrary phase angle for the free surface profile r](x,t,ao). A solution to the linear wave BVP in Chapter 4.3 is given by ®(x,z,t,ao)
= [ -^- ) V(kz)sm(a>t - a 0 - kx),
(7.125a)
where
V(kz)
exp(fcoz), coshpUi(l + f)l ooshkh
'
k0h>7T
(7.125b)
k0h
( 7 - 125c >
provided that co2h g
\koh, = koh = \ ikhtmhkh,
koh > TZ
(7.125d) (7.125e)
koh < n
The free surface displacement r)(x,t,ao) for linear waves that are collinear with the steady current normally incident to the pontoon system as shown in Fig. 7.39 may be determined from t](x,t,ao) =
1 d$>(x,0,t,a0) , g dt
z= 0
(H\ = 1 — 1 cos(
(7.126a)
580
Waves and Wave Forces on Coastal and Ocean Structures
the horizontal [vertical] water particle velocity u(x, z, t, ao) [w(x, z, t, ao)] by 9<J>(x, z, t, ao) u(x,z, t,ao) dx 9$(x, z, t, ao) w(x,z,t,ao) dz (Hgk\\ * (kz) cos(a>t — ao — kx) ^kz(kz) sm(cot — ao — kx) \2co (%)*(kz)ri(x,t,ao) {0)^kz{kz)m{x,t,ao)
"I-
(7.126b) (7.126c)
|'
UM(kz) cos(cot - a 0 - kx) 1 -Wntikz) sin(cot - a 0 - kx) J '
(7.126d) (7.126e)
where the amplitudes Uniikz) [W^fe)] of the horizontal [vertical] velocities are ,'Hco\/kh\ /Ho) v UM(kz) = — — *(**)> W (*z) = M ~' 2 y \kohJ ~" '»^ \ 2 J \kohJ (7.126f,g) and the local horizontal [vertical] water particle acceleration « ; (x,z,/,ao) [w f (x,z,?,ao)] are ut(x,z,t,ao)\ wt(x,z,t,a0)\
_ ~
(Hsk\\ y(kz)sin(cot-ao-kx) V 2 / {*a(^)cos(&)r - ao (%)*(kz)rit(x,t,ao) (^)^kz(kz)mt(x,t,a0)
{
UM(kz)co sin(a>? — ao — kx) WM(kz)(ocos{a>t — ao — kx)
1 kx)\ (7.126h) (7.126i) (7.126J) (7.126k)
where the dimensionless derivative ^kz (kz) in Eq. (7.16c, g and i) is given by (7.1261) exp(koz), koh > jr *kz(kz) = sinh[kh(l + (z/h))] (7.126m) koh < 7i cosh kh and where the subscripts t and kz denote partial derivatives of Eulerian fields; viz., 3W 9(«) (•)t (•)kz = 9? ' 9*z'
581
Deterministic Dynamics of Small Solid Bodies
Fig. 7.42. Sway mode Xj 0) free-body-diagram for coUinear, normally incident steady current and waves.
Sway mode Xi(i) For coUinear and normally incident steady current and waves shown in Fig. 7.42 the horizontal sway mode response X2U) may be computed from WL
1+
DsY SL
d2X2(t) dt2
DL)
= - 2 ^ 2 * 2 ( 0 + ^2 lxitZi,t,ao,
2
,
I,
(7.127)
where F2(xj,Zj,t,ao,d2X2(t)/dt2,dX2(t)/dt) = wave-induced hydrodynamic sway force load computed by the linearized WFE. The horizontal sway stiffness coefficient K2 is computed from the two taut mooring line stiffnesses Kx — K2 in Eqs. (7.117a, b). For normally incident waves, the wave-induced hydrodynamic sway force load may be computed from the linearized WFE (relative motion Morison equation with a linearized drag force) applied separately to each cylinder according to F2(xj,Zj,t,ao,d2X2(t)/dt2,dX2(t)/dt) = F2L (XL,ZL, t, a0, d2X2(t)/dt2, 2
2
+ F2S (xs, zs, t, a, d X2(t)/dt , L
=E
(C2mj
dX2{t)/dt) dX2(t)/dt)
DP ( K~T-) \utj(xj,zj,t,a0)
+ C2dijP ( ^ ) [uj(xj,zj,t,ao) (nD2.B\
+ P I —f— I utj
(XJ,zj,t,ao)
-
d
^P]
^ f ]
(7.128)
582
Waves and Wave Forces on Coastal and Ocean Structures
where C2mj {C2dij) = inertia coefficient (linearized drag coefficient that has the dimensions of velocity; vide., Eq. (7.60) in Sec. 7.6) for the y'th cylinder. The force coefficients are assumed to be different for each mode of oscillation and cylinder size because of their parametric dependencies (vide., Sec. 7.6) and interference or sheltering effects. Substituting Eqs. (7.126d,f and j) into Eq. (7.128) with zj = -Dj/4; approximating that V(-kDj/4) ^ 1 in Eqs. (7.125b or c); and separating terms that are proportional to only the pontoon motion from terms that are proportional to only the fluid motion gives
M
d2X2(t)
c
dX2(t)
> - ^ r - + > — + »«**« = F2L(t,x,xo,a0) = 3r2E(x,xo)cos(cot = ^(X^xo)
F2Es(t,x,x0,a0)
+
-ao + fi2)
[cos#2 cos(<*>/ — ao) — sin /% sm(cot — ao)] >
(7.129)
where K2 for each mooring line is defined in Eq. (7.117a) and where F2E(t,x,xo,ao) = the wave-induced, exciting sway force load on the fixed pontoon system that is a linear combination of the sway mode force load on the large cylinder F2E (t, x, xo, ceo) and the sway mode force load on the small cylinder F2E (t,x,xo,ao)- Scaling Eq. (7.129) by the following dimensional variables (tildes ~ denote dimensionless variables except for dimensionless time r and damping ratio £2):
(Ot == x,
dX2(t) dt
(Hco\ dX2(r) ~\ 2 ) dt '
&--= -^-,
c2l
(7.130a,b)
X2(t) = 1 — 1 X2(z), d2X2(t) dt2
C2cr = 2y/2K2M2;
{—)-*?-'
(7 13 C d)
- °'
(7.130e,f)
Deterministic
/ ^ / i v2
~2
?St2 ^Jf 2
=
583
Dynamics of Small Solid Bodies
(f)
2^2
#2
M2
M 2
a>2d = [ — ) = J l - & a>2fc
(7.130g,h)
J-2
(7.130i)
(^2//)'
where the natural frequency with hydrodynamic added mass a>2h is defined in Eqs. (7.112f-i) and the damped natural frequency a>2d is defined in Eqs. (7.114g-k). For ZJ = -Dj/4 and approximating ^(-kDt/4) « 1 in E Eqs. (7.125b or c), the dimensionless magnitude !F2 and phase angle fij of the normally incident wave induced hydrodynamic sway exciting force load are ^ (x.*o) =
y 2 £ (x.*o)
p 0*0 (¥)(&)
C 2 m L cosfcto + C 2 m s ( c f ' ) COS&(/ - x0) +
(*?) (A) (*W \CuL Sinkx0 ~ C2ds(£t) ™Hx - *0))_
(7.130J)
C2„,L sin kx0 - C2ms ( jfc ) sin *(x - * 0 )
( $ ) (A) ( W (CMt C°S^° + C2* (^f) C°S*(X - *o)). C2mL coskx0 + C2ms y^-j
c
/S2 — arctan
sin
cosk(x
-x0)
CM
+ G&) ( 4 : ) ( $ 0 ( ^ **° - S ( I £ ) sin*(x - *<»)_ C 2 m L sintoo- C2ms (75^) sin*(x - *o) " ( $ ) ( ^ l ) ( $ ) (C2"L
cos
^ 0 + C 2 , s ( £ * ) cos k(X - x0))
(7.130k) A dimensionless total cylinder and added mass M2 may be computed from M2 =
M2 pit DIB
2SL
+ (Clm, - 1) + (C2ims
SL
\DL)
1) El
(7.1301)
DL
where Sj — Pjg/pg = Yj/Y = numerical value forthe "specific gravity'' of the y'th cylinder. A dimensionless hydrodynamic viscous damping coefficient C2
584
Waves and Wave Forces on Coastal and Ocean Structures
may be computed from
C2 =
C2
p (H/2T) (4BDL/3)
(kh/k0h)
= CMi(l + g ^ ) .
(7.130m)
The dimensionless horizontal sway mode equation of motion is now given by (cf, Eq. (2.99a) in Chapter 2.5.3)
^+2^2*^ + ^
X2(r) - 8st26>2h COS(T - a 0 + &).
(7.131)
Applying the method of undetermined coefficients in Chapter 2.5.6, the dimensionless sway mode displacement X 2 (r) may be assumed to be given by X 2 (r)
= \h\ COS(T - ao +
(7.132a)
a 2 ),
where the dimensionless sway amplitude |f2| and dynamic magnification factor T>[(a)2h)~2] are 1/2 2
\h\ = D[(& 2 / ! )- ]^ 2 ,
2
T)[(w2hr ]
=
1 - (2A)~
-I
\(02h/
(7.132b,c) and the sway mode phase angle a 2 is tan ^2 — tan e2 _ 1 + tan €2 tan y$2 (2£2/<w2/,)
a 2 = arctan tane 2 =
- 22]i • [1 - {&2h)-
(7.132d) (7.132e)
The dynamic magnification factor D [(w2/!) 2 J and phase angle e are shown in Fig. 7.38 in Sec. 7.8.1.
Deterministic Dynamics of Small Solid Bodies
585
Fig. 7.43. Heave mode XT, (t) free-body-diagram for collinear, normally incident steady current and waves.
Heave mode X3 (t) For normally incident and collinear steady current and waves shown in Fig. 7.43, the vertical heave mode response X3(t) may be computed from WL
g
1+
mm
d2X3(t) dt2
V ^ - L p f
TMAt
^*3(0
= -K3X3O) + F 3 lxj,-Di/4,t,a,
dX3(t)\
,—
I,
(7.133)
where F-i(xj,—Dj/A,t,ao,d2X3(t)/dt2,dXi,{t)/dt) = wave-induced hydrodynamic heave force load computed by the linearized WFE. The vertical heave stiffness coefficient K3 is a linear combination of the water plane stiffAi nesses K3 from each cylinder and of the taut mooring line stiffness in each of the two taut mooring lines K3 from Eq. (7.117b) that may be expressed as AT3 = K^L + K*s + 2K3
= pgBDL(l
+
^\+2K3
(7.134a)
and a dimensionless vertical heave stiffness coefficient £3
K3 =
*3
2K3
i+
fpgBDL(i + %y 2/T 3
(7.134b)
When the total dimensional stiffness from both of the two taut mooring lines 2K3 is large compared to the water plane area or, equivalently, when &i «» 0
586
Waves and Wave Forces on Coastal and Ocean Structures
(that is frequently true for small pontoon bodies), then KT, « 2K$. For normally incident waves, the wave-induced hydrodynamic heave force load may be computed from the linearized WFE (relative motion Morison equation with a linearized drag force) applied separately to each cylinder according to F2(xj,-Dj/4,t,a0,d2X3(t)/dt2,dX^t)/dt) =
\F2L(xL,-DL/4,t,a0,d2X3(t)/dt2,dX3(t)/dt) 1 1 +F.35 (xs, -Ds/4, t, a0, d2Xj (t)/dt2, dX3 (t)/dt) J L
(Csmj - DP (xD2B/s)
-E
[wtj(xj,Zj,t,a0)-d2X3(t)/dt2]
+ C3dijP {DjB/2) [wj(xj,zj,t,a0) + p (nD2B/8)
-
dX3(t)/dt]
wtj (XJ, ZJ, t,a0) (7.135)
where Csmj {C^dij) = inertia coefficient (linearized drag coefficient) that has the dimensions of velocity; vide., Eq. (7.60) in Sec. 7.6) for the jth cylinder. The force coefficients are assumed to be different for each mode of oscillation and cylinder size because of their parametric dependencies (vide., Sec. 7.6) and interference or sheltering effects. Substituting Eqs. (7.126e, g and k) into Eq. (7.135) with ZJ = -Dj/4; approximating that Wkz(-kDj/4) « 1 in Eq. (7.1261) or «s tanhkh in Eq. (7.126m); and separating terms that are proportional to only the pontoon motions from terms that are proportional to only the fluid motion gives d2X3(t) =
dXi(t)
Ff(t,X,xo,ao)
= PfL (t, X, XQ, a 0 ) + Fis (f> *' x ° ' a ° ) = £3 £ (x,*o) cos(cot - ao - £3) = ^E(X> xo)[sin /% sm(cot — ao) + cos $3 cos(a>t — ao)],
(7.136)
where F3£ (?, x, *o, «o) = the wave-induced, exciting heave force load on the fixed pontoon system that is a linear combination of the heave mode force load on the large cylinder FfL(t,x,xo,<xo) and the heave mode force load on the small cylinder Ffs(t, X,XO,UQ). Scaling Eq. (7.136) by the following
587
Deterministic Dynamics of Small Solid Bodies
dimensional variables (tildes ~ denote dimensionless variables except for dimensionless time x and damping ratio £3): cot = x,
d2X3(t) dt2
Hco\ dX3(x) dx
dX3(t) dt
C
3 fc C , C3cr = 3cr
~2
(7.137a,b)
X3(t) = ( y ) X3(r),
d2X3(x) -, (7.137c,d) dx2 (7.137e,f)
2^/K3M3,
/ ^3/1
K3
K3
1
M3
M3W
<w /
(Hco2\
,
<^3rf\
co3d = [ —^- ) = ^ / l - f32, <W3ft/
(7.137g,h) Stf3 =
£ 5^ •^3
°,S*3
(H/2)
(7.137i)
(K3H)
where the natural frequency with hydrodynamic added mass u>2h is defined in Eqs. (7.112f-i) and the damped natural frequency coja is defined in Eqs. (7.114g-k). For zi = - A / 4 and assuming that tykz(-kDi/4) % 1 in Eq. (7.1261) or « tanhfc/r in Eq. (7.126m), the dimensionless magnitude !F^ and phase angle /% of the normally incident wave induced hydrodynamic exciting heave load are y3£(x,*o)
rf(x,*o)
Hco2] C3mLsmkx0-Cims(%l) C
- {&) (A) \k%) { ^L
cos
smk(x-x0)
^0 + C3ds (£*) cosk(x - , 0 )) .
C 3 m L coskx0 - Cims ( - ^ j cosfe(x - *o)
-
+
( $ ) {£) ( $ ) ( C ^ Sinfa0 + C3ds (§*) sin<:(x - *„)) (7.137J) - C 3 m ^ sintao - C3m^ (-p^-j sinfe(x - *o) C Cosfa C3rf cos
/J3 = arctan
' (fa?) ( # ) (ffi) ( ^
°+
5 ( jfc) *(* ~ *o))
C 3 m i coskx0 - C 3 m s ( - ^ - j cosA(x - *o)
• ( $ ) (BT) ( $ 0 ( C ^ s i ^ 0 - C3,s (£*) sin*(x - *„)) (7.137k)
588
Waves and Wave Forces on Coastal and Ocean Structures
A dimensionless total cylinder and added mass M 3 may be computed from M3 =
M3
2SL
(PTTD2LB/S)
SL
+ (C3mL
\DL)
1) + (C3,ms
1)
:
m
,
(7.1371)
where Sj — Pjg/pg = Yj/y = numerical value for the "specific gravity" of the jth cylinder. A dimensionless hydrodynamic viscous damping coefficient C3 may be computed from C3 =
C3
p (H/2T) (&BDL/3) (kh/k0h)
= CM,
1 +
(7.137m) CM,
DL
The dimensionless vertical heave mode equation of motion is now given by (cf, (2.99a) in Chapter 2.5.3)
H?
+2 h
^ Tx
+
3h X3(r) = 8st3 h cos(T
'
^
~ a°+ ^3)'
(7 138)
'
Applying the method of undetermined coefficients in Chapter 2.5.6, the dimensionless heave mode displacement X 3 (T) is assumed to be given by *3 (T) = \h I COS(T - a0 +
(7.139a)
a?),
where the dimensionless heave amplitude |f3| and dynamic magnification factor D [(<53/;)~2] are 1/2
2
2
If31 = D [(^h)- ] ht3, D [(u3h)- ] =
1 - (mhY
\mhj
J
(7.139b,c) and the heave mode phase angle a 3 is tan yS3 — tan e3 a 3 = arctan _ 1 + tan e tan /3 _ 3 3
tane 3 =
?
^.
[i-{mh)-2]
(7.139d,e)
589
Deterministic Dynamics of Small Solid Bodies
The dynamic magnification factor D [(w^) in Fig. 7.38 in Sec. 7.8.1.
2
] and phase angle 63 are shown
Roll mode &s(t) about the mooring line attachment point "c" Because the point of translation "c" for sway and heave motions is the same as the point of rotation "c" for angular roll modes, the translational dynamics are simplified significantly (vide., Chapter 8 for applications where this simplification is not valid). For normally incident and collinear steady current and waves, a relatively simple approximation for the roll mode response ®s(t) about point "c" may be computed from the moments induced by the horizontal, relative motion sway mode load F2(xj,—Di/4,t,ao, d2X2 (t)/dt2, dX2(t)/dt) and by the vertical, relative motion heave mode load 2 2 F^{XJ, 0, t, oto,d Xi(t)/dt , dX^(t)/dt). In this approximation, however, the moment load may be reduced to only a moment due to the exciting sway mode load on a fixed pontoon system F^ (t, x, xo, ao) because the horizontal, relative motion of each cylinder Sj(t) may be shown to be of second order (i.e., st(t) a @5(0> where Sj(t) is defined in Fig. 7.45), and therefore, may be neglected for small amplitude motions. However, this will not be the case for the moment from the vertical, relative motion heave mode force load. For normally incident and collinear steady current and waves, the roll mode response @5(0 about "c" may be computed from dzG5(t) ' dt2
= -K5@5(t) + M5
__^2,_±2),
(7 ,40,
where M5(xj,Zj,t,ao,d2&5(t)/dt2,d@5(t)/dt) = wave-induced, relative motion hydrodynamic moment load on the jth pontoon that may be computed by the linearized WFE. This moment load is linear combination of the waveinduced, exciting sway force load on thefixedpontoon system F^ (t, x, xo, ao) from Eq. (7.129) and from the vertical, relative motion heave mode force Ff(t,x,xo,ao) fromEq. (7.136). The total polar mass moment of inertia 15 = a linear sum of the polar mass moments of inertia for each cylinder Im (j = L or S) that may be computed by the parallel axis theorem about the horizontal y-axis through the mooring
590
Waves and Wave Forces on Coastal and Ocean Structures
point "c" in the x-z plane according to
h = iic+A ( Y)
+7
& + {x~Xof ( y ) '
(7 141a)
-
where IJ^ = polar mass moment about the center of gravity "C.G." in the x'j-z'j plane of the ;'th cylinder. If each of the 7'th cylinders is an homogeneous circular cylinder of diameter Dj and wall thickness dj, then IJ^ may be computed from
"&>m-<®
(djD]B'
(7.141b)
A solid homogeneous circular cylinder may be computed from Eq. (7.141b) by substituting dj = Dj/2. Because the pontoon system is rolling about the point "c" that is the point of attachment of the two taut mooring lines, the roll stiffness coefficient K5 is not a function of the taut mooring line stiffnesses but is solely a function of the moments from the vertical heave mode water plane stiffnesses K^1 of each cylinder and may be expressed as K5S5(t) = x0K*LZL(t)
+ (X-
x0)K*sZs(t),
(7.142a) A •
where the vertical heave mode water plane stiffnesses K^J of each cylinder j = L or S are given in Eq. (7.134a). Because xo ^ x/2, the vertical displacements Zj (?) for each cylinder are not equal. The unequal vertical displacements Zj(t) for each cylinder j = L or S may be converted to angular roll rotations ®s(t) from the geometry shown in Fig. 7.44 (cf., Sec. 7.8.1) according to sin 0 5 (O =
7
^
L
Xn
= - ^ Y — Xn
« 6 5 ( 0 + O (&l(t)) . V
(7.142b,c)
/
Substituting Eqs. (7.142b, c) into Eq. (7.142a) yields K5@5(t) = [X%K}L
+ (x - xo)2 K*s] 0 5 (O.
(7.143)
For normally incident and collinear steady current and waves, the waveinduced, relative motionroll mode moment load M5 (xy-, ZJ , f, an, d2@s (t)/dt2,
591
Deterministic Dynamics of Small Solid Bodies
Fig. 7.44. Heave-roll modes geometry.
d@5(t)/dt) may be derived from a linear combination of the moments about "c" that are induced by the horizontal, relative motion sway mode force load F2(xj,-Dj/4,t,ao,d2X2(t)/dt2,dX2(t)/dt) that is applied at the location Zj = —Dj/4 and by the vertical, relative motion heave mode force load F3 (XJ, 0, t, <XQ, d2X-$(t)/dt2, dX3(t)/dt) that is applied at z = 0 according to M5(xj,zj,t,a0,d2®5(t)/dt2,d@5(t)/dt) =
M52 (xj,-Dj/4,t,a0,d2&5(t)/dt2,d@5(t))} + M53(xj,0,t,ao,d2@5(t)/dt2,de5(t)) j (-xo,^,t,ao,d2X2(t)/dt2,dX2(t)/dt^
~^F2L ~^F2s
[x
-xQ,^,t,aQ,d2X2{t)/dt2,dX2{t)/dt)
, ( xoF3L(-x0,O,t,ao,d2X3(t)/dt2,dX3(t)/dt) + {-(X-xo)F3s(x-xo,0,t,ao,d2X3(t)/dt2>dX3(t)/dt)l
1 K
'
'
The relative motion sway-roll mode moment MS2(XJ,—DJ/4, t, ao, d2&5(t)/dt2, d@5(t)/dt) about the point of rotation "c" from the horizontal sway loads in Fig. 7.42 is a negative moment (counterclockwise); and the relative motion heave-roll mode moment load Ms3(xi,0,t,ao, d2@${t)/dt2,d®s{t)/dt) about the point of rotation "c" on both the small and large cylinders from the vertical heave loads in Fig. 7.44 are positive moments. In order to demonstrate that the heave-roll mode moments on both cylinders in Fig. 7.44 are positive, the moments due to the relative velocities
592
Waves and Wave Forces on Coastal and Ocean Structures
and accelerations may be computed from
I
dZL(t)
(d2ZL(t)\)
dt \ dt2 ) I = wL(t)(wtL(t)) \
-xo
0 0
0 tdZiXt) \ dt
(d2ZL{t)\ dt2 )
\
{ wL(t)(wtL(t)) rfZifr) /rf2zL(p^
dt2
dt
= +XQ
nh.
wL(t)(wtL(t))
rs x
dZs(t) dt
w
ws(t)(wtJt))
ex (X-xo) 0
J
0
ez 0
0
' dZs{t) dt \
(d2Zs(t)\ dt2 )
ws{t){wtJt)) 2 • dZs(t) (dI*Z2 s(t)\\
+ (x-xo)>
\ dt2 ) 5V ws(t)(wts(t)) J dt
The relative motion sway-roll mode moment load d2@5(t)/dt2,d@5(t)/dt) is
Ms2(xi,—Di/4,t,ao,
M52{xj,-Dj/A,t,a(i,d2®s{t)/dt2,d®s{t)/dt\
A
(7.145)
The relative motion sway-roll mode moment Af52(JC,-, — D,/4,?,ao, d2&5(t)/dt2, d@5(t)/dt) in Eq. (7.145) due to the horizontal, relative motion sway mode force load on each y'th cylinder F2(xj,0,t,ao,d2Sj(t)/dt2, dsj{t)/dt) in Eq. (7.128) may be approximated by a moment load that is due only to the sway mode exciting force load on a fixed pontoon system F2~(t,x,xo,(Xo) from Eq. (7.129) and Eqs. (7.130J, k) by an analysis of the geometry illustrated in Fig. 7.45. Similar to the heave-roll mode geometry illustrated in Fig. 7.44, the horizontal displacements Sj(t) for each cylinder
593
Deterministic Dynamics of Small Solid Bodies
z,{t),
S+Jz
*i (0 . ©5 (<) \*M5
(t) *,»
Fig. 7.45. Sway-heave-roll modes geometry.
j = L or 5 in Fig. 7.45 may be converted to the angular roll rotation ©5(0 from the geometry shown in Fig. 7.45. For the large cylinder DL, this approximation is sL(t) w
1
=cos©5(0~l
XQ
sL(t) XQ
©2(0 x~ +
2! ©2(0 2!
(7.146a)
and for the small cylinder D$ 1 -
ssit) X
= COS © 5 ( 0 ^ 1
-XQ
ssit) X -*o
©jKo +
2! ®i(t) 2!
•••
(7.146b)
so that
«(.> = (f)e3w. ^ ^ = (?) rfei<0 d? d2sL(t) dt2 ss(.t) =
=(?)
d2Q2(t) 5 Jr 2
2
(7.147c) X-x
® 5 (0,
<*2*s(0 J?
(x0^ 2/
X - x o \ c / 2 0 2»(5
(7.147a,b)
0
\ d©|(0
(7.147d,e) (7.147f)
Waves and Wave Forces on Coastal and Ocean Structures
594
Because of the small amplitude motion assumption, higher order terms of 0{(d\(t)) are neglected. Consequently, the relative motion sway mode moment load in Eq. (7.145) reduces to only the sway-roll mode exciting moment load on a fixed pontoon system Mf2(t, X>*o, «o) given by
Mf2(t,a0)
= M%2L(t,a0) + M%2s(t,ao) = Mf 2 (x,*o) sin(cot -a0-
fcl)
— Mf2(x>*o) [cos ^52 sin (art - ceo) - sin £52 cos(otf - «o)], (7.148)
where the dimensionless moment magnitude M^2(X> x o) is
Mf2(x,*o):
Mf2(x,*o)
*(*)(¥)(&) C2mL sinkx0 - C2m$ ( ^ - ) sink(x - x0)
+
_ (3)
( $ ) (C"L
(A)
COskx
° + C2ds(%t)2
C2mLcoskxo + C2ms(-^) • ( 3 ^ ) (A)
cosHx -xo)
cosk(x-xo)
( $ ) (C2
sm«X - *o))
(7.149a) and phase angle /S52 are
C2mL sin/jx0 - C2ms(j^ ,852 = arctan
sin*(x - x0)
• (£) (A) ( $ ) {C"L COS^0 + C™S (% f OP*** - *<>>) C2mLcoskx0 + C2msyifi) +
cosk(x-x0)
{&) (A) ( $ ) ( C ^ sin^0 - C2ds (ft f Sink(X - xo)) (7.149b)
595
Deterministic Dynamics of Small Solid Bodies
The relative motion heave-roll mode moment load in Eq. (7.144) is M5i(xi,0,t,a0,d2e5(t)/dt2,d@5(t)/dt)
' V^, -1) ("?**) (u,,,-*™) - L X ~xo)
+cus{Zl){pWM<*?)DsB){«>s
d
T) (7.150)
The relative motion heave-roll mode moment load in Eq. (7.150) that is due to the vertical, relative motion heave mode force load on each jth cylinder in Eq. (7.135) may not be simplified to a moment load that is due solely to the heave mode exciting force load on a fixed pontoon system F^(t,x,xo, ao). In addition, some care must be exercised with the relative motion moment load due to the vertical heave mode force load on the small cylinder because this moment load is negative and would result in a negative added mass and viscous damping coefficients if the temporal derivatives of the vertical displacement Zs(t) in Eq. (7.150) are not converted to the angular roll rotation ©5(?) correctly from the geometry shown in Fig. 7.44. For positive roll rotations ©5(0 (clockwise rotations) in Fig. 7.44, the vertical displacement of the large cylinder Zi(t) is positive (up); but the vertical displacement of the small cylinder Zs (t) is negative (down). However, both moments have already been demonstrated to be positive moments following Eq. (7.144). The temporal derivatives of the vertical displacements Zj(t) for each jth cylinder in Eq. (7.150) may be converted to angular roll rotations ©5(f) from the geometry shown in Fig. 7.44 by Eqs. (7.142b, c). Substituting Eqs. (7.126e-g) into Eq. (7.150) with ZJ = 0; approximating that tytz(0) = 1 in Eq. (7.1261) or = tanh kh in Eq. (7.126m); and then separating terms that are proportional to only the pontoon motions from terms that are proportional to only the fluid
596
Waves and Wave Forces on Coastal and Ocean
Structures
motions gives for the heave-roll mode exciting moment load M^3(t, x,x0,a0)
= M^L(t,x,x0,ao)
-
Ms3s(t,x,x0,ao)
= Mf 3 (x,*o) sin(cot -ao- £53) = Mf3(x,xo)[cos^53 sin(cot - ao) - sin £53 cos(tot - ao)], (7.151) where the dimensionless moment magnitude M^3 (x, XQ) and phase angle ^53 of the normally incident wave induced hydrodynamic exciting heave-roll moment load are Mf30(,xo) =
M^3(x,-*o) 1
( I 4 £ ) ff^'o)
+(&) (A) K •**» + C 3 4t) ( ^ ) sin*(* -xo)) +
- ( $ ) (&) K cos **° - C ^(^) (^r)cos*(* ^ 0) )
(7.152a)
C 3 m L c o s ^ 0 - C 3 m s ( ^ f ) ( 2 ^ f i ) c o s t ( x -XQ) C sinfa C
%) {A) ( ^L
/S53 = arctan
0 + Us (jfc) ( ^ ) ™*<X ~ XQ))
CimL sinkx0 - C3ms ( ^ - j ( ^ ^ T ) " & ) (A)
K
COS
sin
* ( * ~ *<>)
^0 - C3rf5 ( £ * ) ( * i f ) eos.(x - ,0)) J I
(7.152b) The wave-induced roll mode exciting moment on a fixed pontoon system Mf(t, x,XQ,a(j) may now be written as a linear combination of Eqs. (7.148 and 7.151) as M5 =
(t,x,xo,a0) M
? 5^( ' X, XQ, ao) + M5£3 (r, x, ^0, «o) 52 (t,X,x<
= (Mf 2 (x,*o)cos/352+ Mf 3 (x,x 0 )cos^53)sin(tt.? - ao) - (Mf 2 (x,x 0 )sin y 6 52 + Mf 3 (x,x 0 ) siny653) cos(a.f - a 0 ) = Mf (x,*o) sin(&;? - ao - ft) = Mf (x,xo)[cosft sin(ft>? — ao) — sinft cos(atf - ao)],
(7.153a)
597
Deterministic Dynamics of Small Solid Bodies
where
Mf(x,*0) =
1/2
[Mf 2 (x,xo)cos^52 + Mf3(x,xo)cosj653]2 + [Mf 2 (/,x 0 ) sin ^52 + Mf 3 (x,x 0 ) s i n ^53] 2
(7.153b) l$5 — arctan
[Mf2(x,x0)sinj652 + Mf3(x,A:o)sin;653] [Mf 2 (x,*o) c o s f e + Mf 3 (x,x 0 ) cos £53] (7.153c)
Separating terms in the relative motion roll mode moment in Eq. (7.144) that are proportional to only the pontoon roll mode rotations ©5 (t) from terms that are proportional to only the fluid kinematics transforms Eq. (7.140) to d2G5(t) dt2
d@5(t) dt
F
(7.154)
Because the pontoon system may be sited in deep-water conditions, scaling the roll mode rotations ®s(t) by the water depth h as was done for the articulated tower in Sec. 7.8.1 is not appropriate and, instead, will now be scaled by the following variables (tildes ~ denote dimensionless variables except for dimensionless time x and damping ratio £5): x = cot, © 5 ( T ) =
£L
d@5(x) dx
DL\d@s(t) Hco) dt
d2@5(x) dx2
'
C5 & = --
(7.155a,b)
©5(0,
H
DL \ d2@5(t) dt2 Hco1
(7.155c,d) (7.155e,f)
C5cr = 2Jic5l5,
*~-5cr ~2
(U>5h\2
K5
K5
h
hco'
A,(J =
Mf
«>5d
<5,
(7.155g,h)
\C05h/
(7.155i)
*5
Scaling Eq. (7.154) by the same scaling as Mf3 in Eq. (7.152a) (viz., p7tD2^BHco2xo/l6) gives for the dimensionless pontoon cylinder polar mass and added mass (from the heave-roll moment load Eq. (7.150)) moments of
598
Waves and Wave Forces on Coastal and Ocean Structures
inertia I5
h=
h (pnD\BxQ/16)
(pnDlLo/16) + 2(C3 ms
+ 2(C3mL
-»m
" 1} \lk (X -
XQ)2
(7.155J)
DLXQ
from Eqs. (7.141a and 7.150); for the dimensionless viscous damping coefficient C$ from the heave-roll moment load Eq. (7.150) C5 =
C5 (p7t2DlBx0/ST) 32 3n2
H
XQ
~D~L
~D~L
El
C^dr + C-ids I "=—
XQ
xo (7.155k)
for the dimensionless rotational stiffness £5 K5
(PTTDIBQ)2X0/16)
(7.1551)
and for the amplitude of the dimensionless roll mode exciting moment load M5£(X,*o) Mf(X.Jco)
Mf(x,x0) (p7TD2LBHco2xo/l6) [ M f 2 ( X , x o ) ( ^ ) ( g ) c o s ^ 2 + M 5 £ 3(X^o)cos^ 3 ] 2 + [ f i * ( X ,* 0 ) ( $ ) ( g ) s i n f e +M 5 £ 3 (x,xo)sin^ 3 ]^ (7.155m)
599
Deterministic Dynamics of Small Solid Bodies
The dimensionless roll mode equation of motion Eq. (7.154) is (cf, Eq. (2.99a) in Chapter 2.5.3) ^ 2 + 2^S)5h'd^
+ &
5h
0 5 ( T )=
*«A
sin T
( - "o - ft). (7.156)
Applying the method of undetermined coefficients in Chapter 2.5.6, the dimensionless roll mode rotation ©s(r) is assumed to be given by ©5(r) = |£25|sin(r - ao -as),
(7.157a)
where the dimensionless roll amplitude | Q51 and dynamic magnification factor D[(<S5/i)"2] are • 1/2 2
l«51 = D [ ( & 5 A ) " ] A , / 5 ^ ,
2
D[(5 5 *)" ] =
2 2
[1 - (&5hr ]
+, _
.
<»5h/
(7.157b,c) and roll mode phase angle 0:5 is tan Ps — tan €5 1 + tan 65 tan $5 _
,„ , ._ , . =-. (7.157d,e) [1 - ( ^ ) - 2 ] 2 The dynamic magnification factor D[(a)5n) ] and phase angle €5 are shown in Fig. 7.38 in Sec. 7.8.1. tanas =
(2ft/<W5ft)
tanes =
7.9. Transverse Forces on Bluff Solid Bodies The bluff solid body shown in Fig. 7.46 is subjected to three forces; viz., an in-line hydrodynamic force per unit length dfx(t) that may be due to either a steady flow Uc or an oscillatory flow u(t) or both; a transverse force per unit length dfj(t) that is due to viscous forces and eddy shedding and a resisting force per unit length dfR (t) that is the dynamic response of the body due to the alternate shedding of vortices. The fluid viscosity [i causes a velocity defect near the bluff body boundary that results in the flow in the boundary layer coming to rest and forming the separation points and a wake as illustrated in Fig. 7.46. In the wake region between the lines formed at the two separation points, vortices are formed and shed from the bluff solid body. In steady flow,
600
Waves and Wave Forces on Coastal and Ocean Structures
dfAt)
separation point
Uc,u(t)
JKy
'
point
Fig. 7.46. Definition sketch for the transverse force dfj, in-line hydrodynamic force dfx and resisting force dfn loading a bluff solid body.
these vortices are shed alternately from both sides of the body and create an oscillatory force transverse to the direction of the flow shown in Fig. 7.46 that is called a transverse force dfy (t).In unsteady flow, the same vortex formation due to flow separation in the boundary layer occurs; but now this phenomenon is much more complicated because the location of the wake now oscillates from the up-wave side of the body to the down-wave side over a wave period T as the wave-induced hydrodynamic pressure oscillates from the up-wave side of the body to the down-wave side of the body (vide., Chapter 4.5). A semiconstrained bluff body also induces additional dynamic pressures on the fluid as it responds dynamically to the vortex-shedding forces. The force that is due to the dynamic response of a bluff body to the vortex-induced forces is called a resisting force dfR(t). Sarpkaya (1979 and 1989) reviews in detail the computational methods available for analyzing vortices and vortex-induced motions. Keulegan and Carpenter (1958) identify the location of the oscillating separation points illustrated in Fig. 7.46 as a cause of additional difficulty in estimating wave-induced pressure forces on small bluff bodies. They introduce a physical interpretation for their periodparameter (or the Keulegan-Carpenter parameter K = Um T/D in Eq. (7.44c) in Sec. 7.6.1) that is based on the length of travel of a fluid particle past a small bluff body during one wave period T. Specifically, the length of travel of a fluid particle past a small bluff body during exactly one-half of a wave cycle T/2 provides great insight into the stability of eddy shedding in oscillatory flow of viscous fluids. A ratio between the wave period T and the duration of the shedding of a single eddy Ts may
601
Deterministic Dynamics of Small Solid Bodies
be computed from the following product between the Strouhal parameter St from Chapter 3.6 and the Keulegan-Carpenter or period parameter K:
When the ratio T/Ts = 2 in Eq. (7.158), only one single eddy will be formed and separated during exactly one-half of a wave period T. For values of the Strouhal parameter in the range 0.18 < St < 0.2 (Dean and Harleman, 1966) and for T/Ts = 2, then the Keulegan-Carpenter or period parameter K « 11. This value is a critical value that is associated with an eccentricity of the Dean error ellipse of unity (vide., Sec. 7.7.1) and implies that the wave force data are equally well-conditioned for identifying both of the force coefficients Cm and Cd- In the replotted Keulegan-Carpenter data shown in Fig. 7.26, the minimum value for the inertia force coefficient Cm occurs at a value of the Keulegan-Carpenter parameter K « 12.5 and only one single eddy was observed to form and be shed during the experiments (Keulegan and Carpenter, 1958, p. 439, Fig. 29b). The critical values of the Keulegan-Carpenter parameter between 11.40 < K < 13.16 may be connected to the condition of the data when the inertiald/ m and the drag dfd force components per unit length contribute equally to the mean square error between the measured force data and the forces computed by the Morison equation on small piles (vide., Sec. 7.7). Additional experimental data that support the existence of stable transverse forces for critical values of the Keulegan-Carpenter parameter between 11.40 < K < 13.16 are given by Hayashi and Takenouchi (1979) in Fig. 7.27 in Sec. 7.7.1 and by Maull and Milliner (1979) in Fig. 7.34 in Sec. 7.7.3. The physical interpretation that the Keulegan-Carpenter parameter K is a measure of the length of travel of a fluid particle past a small bluff body during one-half of a wave period T is apparently related to the wake effect that may explain the elevation of values of the drag coefficient Cd in the supercritical region where the Reynolds parameter R > 106 in Fig. 7.19 in Sec. 7.6.5 (Garrison, 1980 and 1982). Garrison (1980) defines a relative amplitude parameter by 2D
D'
(7.159)
602
Waves and Wave Forces on Coastal and Ocean Structures
where 2 A = H and that is proportional to the Keulegan-Carpenter parameter K for simple harmonic flows when UmT (Hn\ T H K=-^—^[—r\= —n=2nG. D \ TJD D
(7.160) '
v
Garrison (1980) demonstrates that the supercritical region where R > 106 may be separated into two effects: viz. 1) an effect due to pile roughness e/D and 2) an effect due to vortex wakes that correlate with the relative amplitude parameter G in Eq. (7.159) (vide., Sec. 7.6.5). In addition to the Keulegan-Carpenter experiments, transverse forces have been measured in two very different types of experiments. Hayashi and Takenouchi (1979 and 1985) measure transverse forces on vertical circular cylinders in progressive waves. Maull and Milliner (1979) measure transverse forces on a vertical circular cylinder forced to oscillate in still water. Hayashi and Takenouchi (1979) measure the transverse forces on fixed vertical circular cylinders in progressive laboratory wave as shown in Fig. 7.27 in Sec. 7.7.1. All of these transverse forces are all irregular except when the Keulegan-Carpenter parameter K= 11.8 (vide., Sec. 7.7 for critical values of K) where the transverse force is stable and periodic. In addition, their data records indicate that there are exactly two vortices shed over each wave period T. Hayashi and Takenouchi (1985) also reproduce stable two-vortex transverse forces for two values of the Keulegan-Carpenter parameter K — 10.5 and K = 11.2 in experiments with a 4 cm diameter cylinder at higher values of the Reynolds parameter R than the values that are illustrated in Fig. 7.27. Maull and Milliner (1979) measure the transverse forces on a vertical circular cylinder forced to oscillate in still water as shown in Fig. 7.34 in Sec. 7.7.3. The oscillograph traces for each run shown in Fig. 7.34 represent 15 continuous cycles so that each upper trace measured at each Keulegan-Carpenter parameter K represent 5 repetitions superimposed from each of the 3 cycle recordings. The one-vortex stable transverse force measured for K — 13.02 (vide., Sec. 7.7 for critical values of K) demonstrates remarkable stability. The conclusion from these experiments for transverse forces is that when the values for the Keulegan-Carpenter parameter K are in the critical range 11.40 < K < 13.16, then the transverse forces are stable and periodic. As
603
Deterministic Dynamics of Small Solid Bodies
a consequence of this periodicity, Fourier series approximations with deterministic force coefficients for the transverse forces on small bodies may be appropriate. For values of the Keulegan-Carpenter parameter K that are not in the critical range, the transverse forces are irregular and non-periodic. Consequently, only Fourier series transverse force models are reviewed. A transverse force per unit length dfy is given by dMO =
l
-CLpAu2{t),
(7.161)
where CL = dimensionless lift coefficient, p = fluid mass density, A = cross-sectional area of a body perpendicular to the in-line flow, and u(t) = a characteristic in-line flow velocity that is perpendicular to the direction of the transverse force (e.g., a maximum amplitude of a time-dependent in-line flow, an instantaneous value of a time-dependent in-line flow, a steady flow amplitude, etc.). An alternative to Eq. (7.161) for a transverse force per unit length assumes a velocity w(t) that is in-line with the direction of the transverse force and perpendicular to the direction of the in-line flow velocity u(t) in Eq. (7.161); viz., dMO = -CLpAw2(t).
(7.162)
The transverse force per unit length on a bluff body may be approximated by dMt)
= dMt)
- dMO,
(7-163)
where dfj{t) = a time-dependent transverse force per unit length, dft(t) = a time-dependent lift force per unit length and dfR(t) = a time-dependent resisting force per unit length. Chakrabarti (Chapter 6.3, 1987) proposes an alternative to the linear combination of a transverse and resisting forces in Eq. (7.163) that is a relative motion transverse force model given by dMO = \cLpA(.w{t)
- y(0)\w(t)
- y(0\,
(7.164)
where y(t) = transverse velocity of the oscillating bluff body in Fig. 7.46 and the overdot denotes ordinary temporal derivative of a Lagrangian solid body;
604
Waves and Wave Forces on Coastal and Ocean Structures
viz., y(t) = dy(t)/dt. Because the fluid velocity w(t) that is transverse to the wave direction must be known in order to apply Eq. (7.163), the model in Eq. (7.164) is difficult to apply with wave forces. Relative velocities could also be applied in Eq. (7.163). In unsteady flows, vortex shedding from bluff bodies becomes very complicated and the transverse force is irregular and random. Because of this random behavior, the transverse force per unit length may be expanded in a Fourier series (vide., Sec. 7.6.3 and Chapter 9.2) with frequency dependent lift coefficient Cm according to (Chakrabarti, 1987) / r ( 0 = ^ ^ " m a x J2 n
C t n C0S M
( nt
+ €n),
(7.165)
where umax = the amplitude of the periodic wave velocity that is perpendicular to the transverse force. A maximum Cz,max and a root-mean-square Cirms lift coefficients may be computed from the maximum fTmax and root-mean-square fTrms transverse forces, respectively, from Chmax =
•^jTmax • 7 ,
pAu*
„ CLTMS =
^JTr . 9
pAu{
•
(7.166a-b)
A model for a one-half cycle transverse force per unit length proposed by Chakrabarti (1987) is given by 1 T frit) = -C L pAw m a x cos — (1 -cosa>0 + * sin,2z o;/, 2'
(7.167)
where ^ = a phase angle that depends on the history of the transverse flow. Transverse forces per unit length on marine risers may be modeled with an amplification ratio according to dfrit) = l-CLPAu2mw, (~\
cos(cot - * ) .
(7.168)
Finally, a relatively simple transverse force per unit length that is periodic and that has been applied to tension leg platform tethers is given by dfT(t) = ^CLpAulmcos((ot).
(7.169)
605
Deterministic Dynamics of Small Solid Bodies
7.10. Stability of Marine Pipelines The algorithm derived by Huang and Hudspeth (1982) to determine the lateral static stability of bottom-laid pipelines under finite-amplitude deterministic waves is reviewed. The dynamic behavior of deep ocean bottom-laid pipelines is analyzed by Zimmerman, et al. (1986). An unsteady, wave-induced oscillatory flow u(x,y,z,t) combined with a steady, uniform current Uc exerts hydrodynamic pressure forces on a pipeline laid on a rigid, impermeable bottom that may be expressed by lateral and vertical force components as shown in Fig. 7.47. The dynamic response of a pipeline or cable protector to these hydrodynamic forces may be divided into the two categories defined by Eagleson and Dean (1959) for sediment motion under waves; viz., incipient motion and established motion. Only the case of incipient motion will be considered for pipeline stabilization. The incipient motion criterion for the pipeline requires that the x-component of the hydrodynamic force Fpx in Fig. 7.47 be less than the sum of the two resisting forces represented by the tensile strength of the pipeline FT and the Coulomb friction force Fp. This stable equilibrium condition may be expressed as FP • 2 r
x \FT\l\FF\WF»ex < 1.
W+
(7.170)
Neglecting the ratio \Fr \/\Fp | in Eq. (7.170) results in a conservative lateral stability criterion given by FPX
=
FFX
dFm+dFd
< h
(?
i?i)
FFx
where the x -component of the lateral inertia dFm and drag dFd forces per unit length on the pipeline may be computed from the Morison equation Eqs. (7.10a-c) evaluated at the bottom z = —h{y) according to (TTD2\
dFm = Cmp
——
ut cos #00,
dFd = '-CdpD^u1 + U'£+ 2uUc cos0(y) [u cos6»(y) + UC],
(7.172a) (7.172b)
Waves and Wave Forces on Coastal and Ocean Structures
Fig. 7.47. Definition sketch for bottom-laid pipeline stability (Huang and Hudspeth, 1982).
where the horizontal {x, y} fluid velocity vector q evaluated at z = —h(y) is q = (ucosO + Uc)ex + u sin6ey;
|^| 2 = u2 + t/£
+2uUccosO, (7.172c,d)
where Q = drag force coefficient; Cm = inertia force coefficient; D = pipeline diameter; and u and ut — instantaneous wave-induced horizontal water particle velocity and local acceleration, respectively, evaluated at z = —h(y); p = mass density of water; and 6(y) = local incident wave angle in Fig. 7.47.
607
Deterministic Dynamics of Small Solid Bodies
The friction force per unit length of pipe Fp may be computed from Coulomb's law of friction with a coefficient n by F F = ^(|Fiv|-|i^l)cos£2x,
(7.173a)
where fi = beach slope; and where 71D2
(7.173b)
FN = PgiYs ~ 1) FL = -CLpDyJu2
+ UC + 2uUc cos6>(y) [u cos6>(y) + Uc] ez, (7.173c)
where the specific gravity of pipe ys = ps/p',ps = mass density of pipe; g — gravitation constant; and CL = lift force coefficient. The application to design conditions of the lateral stability criterion Eq. (7.171), where both the inertia dFm and drag dFd forces per unit length are included, may be reduced to Cd l+li[-^-)cosp\ Cd
yju2 + UC + 2uUc cosG(y) [u cos 9{y) + Uc]
TtD
+ Cm ( — ) ut cos6{y) < ixg(y.• s - l )
—
cos/3.
(7.174)
Dimensionless horizontal water particle velocity and acceleration (denoted by tildes ~) u and ut evaluated at the bottom z = —h{y) and a dimensionless steady, uniform horizontal current Uc are defined by H(y)/T'
ut ut = H(y)/T2' '"
Uc UC = ^ H(y)/T
(7.175a-c)
that reduces Eq. (7.174) to the following dimensionless form: H(y)
D < V(Ys ~ l)?r — cos/S, x [ ^ + cos 0(y)\+Cm^^ut
cos 6(y) (7.176)
where the dimensionless force coefficient ratio G is G = 1 +ii\
— )cos/3,
(7.177)
608
Waves and Wave Forces on Coastal and Ocean Structures
H = friction coefficient and XQ = gT2/2n = the linear theory wavelength in deep-water given by Eq. (4.24h) in Chapter 4.3. For a bottom-laid pipeline with a diameter that is small compared to a wavelength (D/X <^ 1), the dimensionless lateral water particle bottom velocity and acceleration u\j and Ubt computed at the center of the pipeline (i.e., z = —h(y)[l — D/2h(y)] « —h (y) in Fig. 7.47 may be computed to represent the water particle kinematics at the bottom at z — —h(y) in the Morison equations (7.10a-c). For the special case where Uc — 0, Eq. (7.176) may be reduced further to PB < Ps,
(7.178)
where the dimensionless wave force acting on the pipeline PB is PB = ul + T^Gubt
(7.179a)
and the dimensionless pipeline stability parameter Ps is t Ps = -P11 — _ -T, 2 {(#(v)Ao) QGcos0Oo}'
(7.179b)
where G is a dimensionless force ratio defined by the following: Cm G =
D (7.179c)
Cd[l+n(CL/Cd)co*p]H(y) that is a measure of the relative contributions between the inertia force component dFm and the drag force dFd component. The friction resistance between a pipeline and a rigid, impermeable bottom depends on the following factors (Anand and Agarwal, 1980): 1) the surface coating of the pipeline; 2) the benthic material (viz., the geotechnical properties, the angle of the internal friction, the cohesion of the benthic material, and the degree of saturation); 3) the contact area between the pipeline and the bottom; and 4) the direction of the pipeline motion. Potynody (1961) estimates values for the friction coefficient (x between various soils and pipe materials by conducting direct shear tests where the contact surface between the pipe materials and soils was horizontal. Valent (1979)
609
Deterministic Dynamics of Small Solid Bodies
Table 7.4. Summary of friction coefficient from Potynody (1961) and Valent (1979). Pipeline Material Smooth steel
Soil Material
M
Sand Dense cohesionless silt Clay
0.64 0.68 0.50
Coralline sand Oolitic sand Foram sand-silt Rough steel
Sand Dense cohesionless silt Clay Coralline sand Oolitic sand Foram sand-silt
Smooth concrete
Sand Dense cohesionless silt Clay Coralline sand Oolitic sand Foram sand-silt
Grained concrete
Rough concrete
0.17-0.18 0.31 0.37 0.80 0.75 0.80 0.55 0.50-0.51 0.66 0.80 0.87 1.00 0.56 0.52-0.54 0.67
Reference Potynody (1961)
Valent (1979)
Potynody (1961)
Valent (1979)
Potynody (1961)
Valent (1979)
Sand Dense cohesionless silt Clay
0.88 0.96 1.00
Potynody (1961)
Sand Dense cohesionless silt Clay
0.90 1.00 1.00
Potynody (1961)
Coralline sand Oolitic sand
0.59 0.57
Valent (1979)
also computes experimental values for the friction coefficient /x for pipelines made of steel and of concrete by conducting direct shear tests using three different beach sand materials and two different building materials. Both the Potynody (1961) and Valent (1979) tests results are summarized in Table 7.4.
610
Waves and Wave Forces on Coastal and Ocean Structures
Coulomb friction coefficients for concrete-coated pipelines on both sandy and silty soils are reported by Lyons (1973) and by Anand and Agarwal (1980) using both model and prototype pulling tests. There are no laboratory or field measurements for the friction coefficient for pipeline materials on an impermeable rock bottom. For design purposes (if no accurate site survey data are available), a value of /u. = 0.33 for the friction coefficient is recommended for stone masonry and wet undisturbed ground (Eshbach, 1974). The maximum dimensionless transverse hydrodynamic force exerted on a bottom-laid pipe may be divided into three distinct regions on a dimensionless relative displacement H/D relative water depth h/Xo dissection plane where the linear wave theory deep-water wavelength ko = gT2/2TZ by Eq. (4.24h) in Chapter 4.3. These three regions are determined from values of the ratio of the amplitude of the drag force \dFd\ to the amplitude of the inertia force \dFm | computed by linear wave theory from \dFd\ = (Cd\ W = ^\dF \ = \C (^L)) m m
H_ = (Cd\
K_
— = \C ( 7^) "J. TTD m) TT<
(7180>
where the Keulegan-Carpenter parameter K — Hn/D for linear wave theory kinematics (Eq. (7.44c) in Sec. 7.6.1). For design purposes, two values of W may be shown on this dissection plane for W — 20 (drag force component dominates) and for W — 0.05 (inertia force component dominates). Region A in the upper-left corner of Fig. 7.48 applies to design conditions where the drag force dominates over the inertia force. In general, steep waves in shallow water create the conditions encountered in this region. In Region C in the lower right corner of Fig. 7.48, the situation is reversed. In Region B, both the drag force and inertia force components should be considered for design. The wave height H(y) in Eq. (7.176) may be expressed in terms of the deep-water swell height Ho, a refraction coefficient Kr(y), and a shoaling coefficient Ks(y) by the following
H(y) = Kr(y)Ks(y)H0,
Kr(y) = Jjiy),
K^l(y) = J 2 n ( y ) ^
Deterministic Dynamics of Small Solid Bodies
611
1000
Fig. 7.48. Parametric regions of importance for ratio of drag/inertia force components (Huang and Hudspeth, 1982).
where Eq. (4.60b) in Chapter 4.5 gives n(y) = - [1 +
2kh(y)cosech2kh(y)]
and bo and b{y) = the horizontal distances between wave orthogonals in deepwater and at a specific pipeline location y, respectively, shown in Fig. 7.47 and Fig. (4.12) in Chapter 4.6. Substituting for u, ut and H(y) from linear wave theory and invoking the small argument approximation z ~ —h(y), the general stability criterion given by Eq. (7.176) may now be expressed by: Kr(y)Ks(y)
Kr(y)Ks(y) ( § )
CdG
cosh kh(y)
cos 2 cot
y 1 + (Uc7t~l sinh[fc/i(;y)]cos_1 cot)2 + 2Ucn~1 sinhlMOOlcos -1 cotcosO(y) x[Ucsinh[kh(y)]cos •C,
l
cot + n cos6(y)]
-Hi) cosh lkh(y) smcot cos0(y) J™?'
(7.181)
Waves and Wave Forces on Coastal and Ocean Structures
612
Fig. 7.49. Contours of dimensionless wave force on a pipeline P# computed by Dean stream function wave theory (1965) for G = 5 (Huang and Hudspeth, 1982).
The maximum value of the LHS of Eq. (7.181) for the special case in that Uc = 0 is given by CmnD sinh[kh(y)] V
[Kr(y)Ks(y)cosech[kh(y)]y < fi(ys - 1)
Ho)
2Kr(y)Ks(y)HoCdGj \H0
(QG^cosjS
cos^(v) (7.182)
and defines the lateral stability criterion for linear wave theory kinematics with UC = 0. Because of the infinite number of wave-current combinations that are possible, Huang and Hudspeth (1982) applied Eq. (7.178) to represent the lateral stability for a bottom-laid pipeline under a shoaling, finite-amplitude wave without a current Uc- The local values for the wave height H(y) and wave direction 0{y) at a specific location may be determined iteratively following the procedures proposed by Dean (1974) or directly from site survey data. The maximum values of the dimensionless wave force PB given by Eq. (7.179a) are computed by the Dean stream function kinematics and contoured on a wave height parameter H/T2 versus water depth parameter h/T2 dissection plane. Figure 7.49 illustrates the parametric dependency of contours of the dimensionless wave force PB for four wave height H to breaking wave height
613
Deterministic Dynamics of Small Solid Bodies
HB ratios H/HB (viz., A = 25% breaking, B = 50% breaking, C = 75% breaking and D = 100% breaking in Fig. 6.10 in Chapter 6.5) when the dimensionless force ratio G equal to 5.0. The dimensionless relative errors for PB computed by the stream function wave theory compared to the linear wave theory are summarized by Huang and Hudspeth (1982) for the 40 Dean stream function wave cases. The values of the dimensionless force ratio G may be calculated for the following three design conditions: 1) /x = 0.33 (hard-rock), 2)11 = 0.55 (sand), and 3) //, = 1.0 (silty-soil), with zero-slope (cos ft = 1.0). The results are presented in Figs. 7.50-7.52 as contours of the dimensionless force ratio G on a dimensionless relative displacement H/D versus dimensionless relative water depth h/Xo dissection plane. These results provide an approximate range for values of the dimensionless force ratio G that may be applied as design aids for estimating the lateral stability of a bottom-laid pipeline. In conditions where bottom-laid pipelines are subjected to shoaling finiteamplitude waves, it is necessary to estimate the maximum water depth where lateral stability protection is required. Between the shoreline and this maximum depth offshore, it is necessary to either bury a pipeline or to use other means of stabilization in the case of a rigid, impermeable beach. This maximum depth may be determined from the dimensionless design curves of PB
,o 2
H D 10°
K Fig. 7.50. Dimensionless force ratio G computed from linear wave theory for /J, — 0.33 (rock) and p = 0° (Huang and Hudspeth, 1982).
Waves and Wave Forces on Coastal and Ocean Structures
614
Fig. 7.51. Dimensionless force ratio G computed from linear wave theory for ^ = 0.55 (sand) and yS = 0° (Huang and Hudspeth, 1982). 10 Os"
1
10
H D 10°
' S /
..-1 10
2 5
/ ^ -2 10
/,
-1 10
Fig. 7.52. Dimensionless force ratio G computed from linear wave theory for fi = 1.0 (siltysand) and p = 0° (Huang and Hudspeth, 1982).
from Huang and Hudspeth (1982) by the following procedure: (i) Estimate the design wave height H(v) wave period T and the direction of wave propagation 9(y) along the proposed pipeline route, (ii) Compute Kr(y) and Ks(y) along the proposed pipeline route, (iii) Determine the local wave height H (v) and wave direction 9 (v) along the proposed route. (Steps (ii) and (iii) may be omitted if site survey data are available that give both local wave height and direction.)
615
Deterministic Dynamics of Small Solid Bodies
(iv) Beginning seaward of the breaker zone determined in Steps (ii) and (iii), compute the wave height parameter H/T2, depth parameter h/T2, and the dimensionless force ration G either by applying Eq. (7.179c) or Figs. 7.50-7.52. (v) Estimate the value of the dimensionless force PB from the design curves given by Huang and Hudspeth (1982) (e.g., Fig. 7.49) by entering a vertical line through the design depth parameter h/T2 and a horizontal line through the design wave height parameter H/T2. The value of the dimensionless force PB may then be estimated at the intersection of these two lines using the values from the adjacent contours of constant PB • Numerical interpolation may be used for values of G that differ from the three values selected by Huang and Hudspeth (1982). (vi) Compute the dimensionless pipeline stability parameter Ps from Eq. (7.179b). (vii) If PB > Ps then the pipeline at this depth must be stabilized, (viii) Repeat Steps (iv) to (vii) moving seaward along the proposed route until PB < Ps- The maximum depth for pipeline stabilization is at this location.
7.11. Problems 7.1.
Given the following properties for the articulated tower analyzed in Sec. 7.8.1: h = 135 m; D = 4.5 m; Lp = Lg = 150 m; Mg = 880,000 kg; Wp/Wg = 0.2; Wb/Wg = 0.5; hb/h = 0.45; Lg/h = 1.111; Lt,/h = 1.0; Lp/h = 1.111 and the following wave conditions: H(m) T(sec)
7.5/10/15/20/30 9.8
15/20/30 13.9
10/30 17.4
a) Compute the natural undamped period To and frequency /o in air in the pitch 05 mode about the base at z = — h. b) Select an appropriate hydrodynamic load model for the waveinduced hydrodynamic pressure loading and justify your selection.
616
Waves and Wave Forces on Coastal and Ocean Structures
c) Compute the undamped Tn and damped Td natural periods in pitch 05 about the base including hydrodynamic effects. d) Compute the following dynamic response constants: Ast = rads; £2s = rads; 0:5 = rads e) Compute the damping ratio £5. 7.2. Assume that the measured force/unit length on a small vertical circular cylinder may be written as dFr(z,t) = —dfm(z) sin cot + dfd(z)coscot\ cos cot \ Apply Fourier analyses to show that if the Morison equation is given by /pitD2\
dFx(z,t) = -Cm I —-— I um(z)co sin cot D 2 + CdP-rum(z) coscot\ coscot\ then the Morison equation force coefficients Cm and Cd may be computed from r
4dfm(z) pit D2coum{z)'
_
2dfd{z) pDu^iz)
7.3.
Applying linear wave theory kinematics from Chapter 4.4, compute the total force on a vertical circular pile for Q = 1.0; Cm = 1.5; D = 3.0 ft; H = 30 ft; T = 10 sec;h = 550 ft(F max = 13.2kips). 7.4. Assume that the inertia and drag coefficients in the Morison equation areC m = 1.40 and Cd = 1.05. Apply the Morison equation with linear wave theory kinematics from Chapter 4.4 and determine horizontal and vertical forces on the pontoon shown in Fig. 7.39 in Sec. 7.8.2 for h = 50ft;// = 10ft; andT = 12 sec. Plot these forces as a function of time for one wave period T and estimate the maximum values from these plots. 7.5. A vertical circular cylinder of D = 4 ft is located in h = 100 ft of water. a) what is the Froude parameter Fr and Reynolds parameter R for the maximum velocity of a wave with a height H = 42 ft and a period T = 13 sec by linear wave theory kinematics?
617
Deterministic Dynamics of Small Solid Bodies
b) compute the maximum force the pile by the Morison equation with linear wave theory kinematics from Chapter 4.4. 7.6.
Two horizontal cylinders, each 40 ft long, are connected at each end with a rigid connector as shown in Fig. 7.39 in Sec. 7.8.2. Assume that the weight ofthe connector is included in the cylinder weights. The large cylinder diameter is DL = 4 ft and weighs 5,220 lbs; the small cylinder diameter is Ds — 2 ft and weighs 1,350 lbs; / = 11 ft. The water depth is h = 80 ft; wave height H = 4 ft; wave period T — 4.1 seconds; uniform current is Uc = 1.0 knot. Compute the following: a) with the current only acting, determine the length I of the mooring line at each end and the rigid connector distance xo so that the rigid connector line a-c-b will be horizontal at the still water surface. Each of the two mooring lines has a cross-sectional area of .44 in2. The linear modulus of elasticity E of the mooring is 20 x 106 psi. Assume that the ultimate breaking strength is very large. The stiffnesses in x (horizontal) and z (vertical) per mooring line are given by Eqs. (7.117) in Sec. 7.8.2. b) compute the dimensionless dynamic amplification factor D [a)j£] for sway j = 2 and heave j = 3 about the steady state position with current. In addition, compute the dynamic magnification factor D[<w -h ] for roll j = 5 about the points, a, b, and c. c) tabulate each of the quantities in the following table: MODE SWAY (j = 2) HEAVE O" = 3) ROLL (J = 5)
7.7.
^(Mf)
Pj
«>jh
*stj
O l^-KI^I)
a
j
Compare the general quadric error surface given by the equation (7.68) in Sec. 7.7.1 (aX)2 + 2HXY + OSy)2 - 2GX - 2JY + C = 0. with the least-squares equation (7.67) for Cd and Cm given by A
(CdP\2
. 4,^ (r
x (u\u\ut) - 2 (^\
P*D\2
2
(CdP\
(CmPitD\
(FTu\u\) - 2 (CmP*D\ {FTUt)
+
{F2}
618
Waves and Wave Forces on Coastal and Ocean Structures
where Cd = X and Cm = Y. Compute the equations (7.69a-c) for the origin (Xo, Fo) of the translated axes for Q and Cm if the equation for the translated origin is derived from the simultaneous solution to HXQ + p2Y0 -J
= 0,
a2X0 + HY0 -G
=0
Compute Eq. (7.69d) for the rotation of the Cd and Cm axes. 7.8. Verify that the following phase method Eq. (7.49b) is correct: dFx(z,9max)
= dfd(z) 1 +
1 (dfrniXyV 4\dfd(z)J
(7.49b)
7.9. Verify that the following least squares Eqs. (7.53) are correct.
2
Cd =
*
\M) Jo
°2* dFT(z,e) sin 9 dO, o pDum{z)
(7.53a)
dFT(z,9) cos 9\cos0\d9 pDum(z)
(7.53b)
7.10. Verify that the following Fourier Method Eqs. (7.55) are correct K2
—If
Mn(z) =
* dFT(z,6) cos(2n9)d0, pDum(z)
(l+8„o)7t]Jo • Sn2o)n J Jo ' * dFT(z,9) A2n+\(z) cos((2« + \)9)d9, n Jo pDum(z) 271
B2 n(z)
= - f0 •x Jo
B2n+l(z)
7.11.
=-f
In
0
dFT(z,9) sin(2n9)d9, pDum(z)
dFT(z,9) pF>um(z)
sin((2n + \)9)d9.
(7.55a) (7.55b) (7.55c) (7.55d)
7T Jo
Verify that the following linearized drag coefficient Cdi in Eq. (7.60) is correct: Cdi =
—CdUm(z) 571
(7.60)
Chapter 8
Deterministic Dynamics of Large Solid Bodies
8.1. Dynamic Response of Large Bodies: An Overview The importance of focusing on the principles of physics before selecting mathematical models to analyze the dynamic response of coastal and ocean structures is highlighted in Chapter 7.1 and will not be repeated here. In contrast to the dynamics of small solid bodies, the effect of large bodies on an incident wave field is not negligible. Carefully observing what is seen or not seen when long-crested surface gravity waves interact with large solid bodies identifies the following two major physical phenomena: Seen:
Waves may be observed radiating away from large, semiimmersed solid bodies and the incident wave may be modified significantly. Not Seen: When the solid body is sufficiently large compared to the incident wave height and wavelength, no wakes due to flow separation may be observed. Figure 8.1 illustrates the linear superposition of an exciting force on a. fixed body plus the restoring force (wavemaker theory) on a large solid body. In contrast to Fig. 7.1 in Chapter 7.1, the motion of the large body in otherwise still water generates radiated waves (wavemaker theory). Also in contrast to Fig. 7.1 in Chapter 7.1, the added mass Hjk and radiation damping kj^ coefficients from the radiated force are not equal to the exciting force coefficients C,f and C,f. 619
Waves and Wave Forces on Coastal and Ocean
PHYSICS
Structures
LARGE BODY DYNAMICS SCATTERED & RADIATED WAVES v X T L O W SEPARATION IDEAL & INVISCID POTENTIAL THEORY
SEEN NOT SEEN FLUID SOLUTION EXCITING FORCE on a FIXED BODY
+
RADIATED FORCE (WAVEMAKER THEORY)
•^s\r il
I
II DYNAMIC RESPONSE LINEAR SUPERPOSITION
MATH MODEL
F(t) = FE(t) + F'(t) = CEuu,(t) + £EU u(t) - aik u,(t) - Xik u(t)
Fig. 8.1. Large body dynamic response.
When no flow separation occurs, real fluid effects due to fluid viscosity may be neglected and a potential theory for an ideal fluid may be applied to compute the wave loads. A boundary value problem (B VP) for the motion of an ideal fluid may be solved to account for this type of wave-structure interaction. Because both the radiation conditions and the fundamental singular solutions to Laplace's equation (viz., Green's functions) are different for 2D and for 3D geometry, 2D solid bodies must be distinguished from 3D solid bodies. The solid body boundary between an ideal fluid and a solid body requires special care. Because this boundary isfree, both kinematic and dynamic boundary conditions are required. Because the fluid motion is an Eulerian field with time and space as independent variables and the solid body is a Lagrangian body with time and the body as independent variables, these kinematic and dynamic boundary conditions will be different from the free surface boundary that separates two Eulerian fields; viz., air and water (cf, Chapter 4.2). The boundary between the fluid and solid body separates an Eulerian field from a Lagrangian body and these boundary conditions are somewhat more complicated.
Deterministic Dynamics of Large Solid Bodies
621
Fig. 8.2. (a) Definition sketch for semi-immersed MDOF Lagrangian solid body, (b) Locations of the origins of the inertial and non-inertial coordinate systems.
The solid body motion of the semi-immersed body shown in Figs. 8.2 may be described in Lagrangian coordinates by three translational displacements [X\(t)= surge; X2(t)— sway; X^(t) = heave] and three rotational displacements [04(f) = roll; ©5(0 = pitch; and ©6(0 =yaw]. The direction of the unit normal to the solid body is ambiguous and must be defined as either into or out of the solid body surface. The unit normal vector in Fig. 8.3 points out of the body and into the fluid and is defined by n = n\e\ + «2
(8.1)
with direction cosines defined by n\ = cos(0i,*'),
«2 = cos (02,/),
«3 = cos(03,z'),
(8.2a,b,c)
where 9j = angle between the unit normal vector n and the unit basis vector ej. The direction cosines nj defined by Eqs. (8.2) are dimensionless. Pseudodirection cosines for the three rotational modes about an arbitrary point of rotation R in Figs. 8.2 in the body-fixed, non-inertia coordinates that are defined by the coordinate vector XR = {x'R,y'R,zR}
(8.3a)
and by an arbitrary point of translation C in Figs. 8.2 that is defined in bodyfixed, non-inertial coordinates by the coordinate vector xc = {x'c,y'c,z'c}.
(8.3b)
622
Waves and Wave Forces on Coastal and Ocean Structures
Fig. 8.3. Unit normal points out of the Lagrangian solid body and into the fluid.
The pseudo-direction cosines are defined by the following: rc x n = nAe\ + n5ez + n6ei,
(8.4a)
r' — v ' - v '
(8.4b)
where from Fig. 8.2(a)
and » ' = Sjkm rj nk em,
j,k,m
(8.4c)
- 1,2,3,
where Sjkm = the Levi-Civita permutation symbol (vide., Chapter 2.2.4); and r
'd =x'c~
X
n
4 = {rc2"3 - r'c^m),
(8.5a,b)
r
C2 =y'c-
y'R>
" 5 = {rC3wl -
C! W 3},
(8.5c,d)
{r'Cln2-r'C2ni}.
(8.5e,f)
r
'R>
C3=Zc-4.
n'6 =
r
The pseudo-direction cosines n'- defined by Eqs. (8.5b,d,f) have the dimensions of [Length].
Deterministic Dynamics of Large Solid Bodies
623
The fixed inertial x axis is defined by the longitudinal axis (or generator) of the body and not by the direction of wave propagation. These motions are defined by the body coordinate axis and not by the wave-field coordinate axis. The three translational modes of oscillation for a Multiple-Degree-OfFreedom oscillator (MDOF) are given by X(t)=X(t)+R(t),
R(t) = 0(t)xr'c,
(8.6a,b)
where r'c in Figs. 8.2 is defined by Eq. (8.4b). The displacement R(t) is time dependent and defined by Eq. (8.6b), whileic^ is a time independent coordinate vector in the body-fixed, non-inertial coordinates defined by Eq. (8.3a). For strictly periodic solid body motions with period T = 2n/co, the harmonic time dependence may be given by exp — (icot). The complex-valued, scalar components of the Lagrangian displacement, velocity and acceleration of the solid body are given by the real parts of Xj(t) = Re{i%j exp - icot} = |£/ | sin(cot - ctj), ®j(t) = Re{i£2j exp - icot} = \Qj\ sin(cot - ctj),
j = 1,2,3, j = 4,5,6,
Xj (t) = Re{co%j exp - icot} = co\i-j | cos(cot -ctj),
j = 1,2,3,
®j(t) = Re{coQj exp — icot} = co\£2j\cos(cot — ctj),
(8.7c)
(8.7e)
j = 1,2,3,
(8.7f)
®j(t) = Re{-ico2Qj exp - icot} = -co2\Qj\ sin(cot - ctj), = Re{-i<w©;(0},
(8.7b)
j = 4,5,6, (8.7d)
Xj(t) = Re{-ico2t-j exp - icot} = -co2\i-j\sin(cot - ctj), = Re{-icoXj(t)},
(8.7a)
j = 4,5,6,
(8.7g) (8.7h)
where the over dots represent ordinary Lagrangian temporal derivatives according to , ,
x
dXj(t)
.
d®,(t)
where the complex-valued amplitudes of the translational l-j (j = 1,2,3) and the rotational modes £2j (j = 4,5,6) are defined by S; = IfyI expiaj,
j = 1,2,3,
Slj = \Slj\ expictj,
j = 4,5,6. (8.7k,l)
624
Waves and Wave Forces on Coastal and Ocean Structures
The free boundary between the Eulerian fluid and the Lagrangian solid body may be described by a Stokes material surface S(x,y,z,t) (vide., Chapter 2.2.10) that requires two boundary conditions; viz., 1) an Eulerian .Kinematic 5ody .Boundary Condition (KBBC) for an ideal fluid; and 2) a Lagrangian Dynamic .Boundary Condition (DBC). It is important to focus on the differences in the boundary conditions for 1) a free boundary between a Lagrangian solid body and an Eulerian fluid field; and 2) a free boundary between two Eulerian fluid fields such as water and air (vide., Eqs. (4.5e-h) in Chapter 4.2). The fundamental fluid unknowns for the flow of an incompressible fluid are the velocity (a vector q) and pressure (a scalar p). Two boundary conditions are required (just as in the case of a deformable free surface fluid boundary between two Eulerian fluid fields) because the solid boundary isfree to respond to dynamic pressure loads; i.e., the motion is not prescribed (vide., John, 1949 and 1950). Free boundaries between either two Eulerian fluid fields or between an Eulerian fluid field and a Lagrangian solid body require both a .Kinematic (KBC) and a Dynamic (DBC) .Boundary Condition. The principal distinction between the Lagrangian solid body boundary S(x, y, z, t) and the Eulerian free surface fluid field boundary F(x, y, z, t) is that the Lagrangian solid body free boundary S(x, y, z, t) is a non-deforming solid boundary between a Lagrangian solid body and an Eulerian fluid field. In contrast, the Eulerian fluid field free surface boundary F(x, y, z, t) is a deformable boundary between two Eulerian fluid fields. The Problem Solution Technique (PST) for linearly decomposing the boundary value problem for a scalar velocity potential <$> (x, y, z, t) for the Eulerian fluid field motion into three separate boundary value problems is easily demonstrated and motivated by a close examination of both the .Kinematic (KBC) and Dynamic (DBC) .Boundary Conditions on the Lagrangian solid body free boundary S(x, y, z, t). Free boundaries between an Eulerian fluid field and either another deformable Eulerian fluid field or a non-deforming Lagrangian solid body require both a .Kinematic (KBC) and a Dynamic (DBC) boundary condition because the boundaries are free and because there are two fundamental fluid unknowns for an incompressible fluid; viz., velocity and pressure. The exact forms of these two boundary conditions depend on 1) whether or not the fluid is a real or an ideal fluid; and 2) whether or not the^ree boundary is a nondeforming boundary between a Lagrangian solid body and an Eulerian fluid field or a deformable boundary between two Eulerian fluid fields.
Deterministic
625
Dynamics of Large Solid Bodies
Because the Lagrangian solid body is assumed to be a dimensionally large body with respect to the dimensions of both the wavelength X and wave height H, no fluid flow separation will occur around the Lagrangian solid body; and, consequently, a potential theory solution for an ideal, inviscid and incompressible fluid may be computed. The fundamental fluid unknowns of velocity q(x, y, z, t) and dynamic pressure p(x, y, z, t) may be computed from a scalar velocity potential <£>(x, y, z, t) according to the following: q(x,y,z,t)
= {u(x,y,z,t),v(x,y,z,t),w(x,y,z,t)}
=
-VQ(x,y,z,t), (8.8a)
p(x,y,z,t)
= P
dQ(x,y,z,t) at
•
(8.8b)
From physical observations, it is easy to motivate a linear decomposition of the scalar velocity potential <£(x, y,z,t) into an incident wave potential ^wi*, y, z, t) and a body disturbance potential <&b(x, y, z, t) that is due to the presence of the solid body. This linear decomposition is valid only for small oscillations about an equilibrium position of the solid body QS(X, y, z, 0). The .Kinematic Body boundary Condition (KBBC) for an ideal (incompressible, inviscid) fluid is given by the following single scalar equation that is evaluated at the equilibrium position oS(x, y, z, 0): -V4> • n=X»ii
+
&»n',
- ( V ^ u , + V0>fe) • n=X»n
+
Q»n',
— (w + (S>b)+X*n + ®»n'= 0; on 0S(x,y,z,0), (8.9a) an where the six solid body Lagrangian velocities are defined by Eqs. (8.7c, d). The scalar kinematic boundary condition transforms the solid body Lagrangian velocities X(t) and 0(?) into Eulerian fields through vector dot products with the spatially dependent unit normals « and n' to the Lagrangian solid body 0S(x,y,z,0). Because the free boundary is a boundary between an non-deforming Lagrangian solid body and an Eulerian fluid field, the Dynamic boundary Condition (DBC) is not expressed in terms of the local Eulerian fluid field pressure as in the case of the deformable free surface boundary in the DFSBC in Eq. (4.5f) in Chapter 4.2. Instead, the Dynamic 5ody boundary Condition
626
Waves and Wave Forces on Coastal and Ocean Structures
(DBBC) is expressed in terms of the integrated fluid pressure force/moment through the following vector equations of Lagrangian solid body motions evaluated at the equilibrium position QS{X, y, z, 0):
m — _^_
'
iX(t) —— — —e
. iO(0 ,
J
J
• +
•
—— —
£lifr;(0 j
(8.9b) (8.9c) (8.9d) ~(8~9e)~ (8.9f) (8-9g)
where e = kA = dimensionless perturbation scale for the smallness of the time-dependent motions about equilibrium QS{X, y,z,0); and where both the time-dependent (subscript = 1 ) and time-independent (subscript = 0) forces/moments are derived in detail in Sec. 8.2. The notation in Eqs. (8.9b-g) and in all subsequent equations for linear and angular momenta is an implied stacked notation with the three translational momenta stacked vertically above the three angular momenta and separated by a horizontal dashed line [ ];i.e.,
lX(t)
|
10(0 J
iXi(0 1*2(0 1*3(0 104(0 105(0 106(0
The forces/moments on the RHS of Eqs. (8.9b-g) include: 1) stiffnesses due to buoyancy, body weight and water plane surface areas; 2) external mooring constraints; 3) the exciting forces/moments on a fixed, still body; and 4) the restoring forces/moments on a body oscillating in otherwise still-fluid (i.e., the wave maker problem in Chapter 5.). The Lagrangian vector dynamic boundary conditions in Eqs. (8.9b-g) transform the Eulerian dynamic pressure field in Eq. (8.8b) and hydrostatic pressure ps = —yz into a time-independent and a time-dependent Lagrangian solid body force/moment by integration over the immersed surface area of the Lagrangian solid body at the equilibrium position 0S(x,y,z,0).
Deterministic
Dynamics of Large Solid Bodies
627
The scalar Eulerian .Kinematic fiody boundary Condition (KBBC) Eq. (8.9a) and the Lagrangian vector Dynamic .Body .Boundary Conditions (DBBC) given by Eqs. (8.9b-g) represent seven scalar equations in seven unknowns; viz.,
(8.10)
where <£>s (x, y, z, 0 = scattered wave potential and <£>r (x, y, z, 0 = radiated wave potential. Substituting Eq. (8.10) into Eq. (8.9a) gives fd
on
0S(x,y,z,0),
(8.11) that gives a diffraction boundary value problem for <&D(x,y,z,t) = $>w(x,y,z,t)
+
$>s(x,y,z,t)
= Re{ A((j)w (x, y, z) + (/>s(x, y, z)) exp - icot}
(8.12a)
with the following homogeneous .Kinematic .Body .Boundary Condition (KBBC) at the equilibrium position oS(x, y, z, 0): ^ = 0 , an
on
0S(x,y,z,0)
(8.12b)
628
Waves and Wave Forces on Coastal and Ocean Structures
or, equivalently, the following inhomogeneous kinematic 5ody boundary Condition (KBBC) for the scattered potential <&s(x, y, z, t): a^
=
an
_ 3 ^
o n
z>0)_
o 5 (
( 8 1 2 C )
an
Because of the identically equal sign [=] in Eq. (8.11), the second term in brackets {•} on the left-hand-side of Eq. (8.11) may also be linearly decomposed into six separate .Kinematic 5ody .Boundary Conditions (KBBC) by the following linear superposition for <&r (x, y, z, t): 6
<$>r(x,y,z,t) = ^ l > y ( x , j , z , 0 7=1
3
(8.13) v
6
'
= ] T Xj (t)4>j (x, y, Z) + J2 ®j (*)4>j (*, y, z) 7=1
7=4
and substituted into the second term in brackets {•} in Eq. (8.11) to obtain
*l(0 {^7 +nx\+ ±2{t) {^T + m\ + *m \lt+n3} +
@4(t)\^1+nf4\+@5(t)\^p-+n'5 [an
J
+ ©6(0{^+<}=0,
[ dn
on
0S(x,y,z,0).
(8.14a-f)
By factoring explicitly the unknown amplitudes of the six Lagrangian solid body velocities Xj(t) and ®j(t) for each of the unknown radiated (or wavemaker) potentials y (x, y, z, t), six inhomogeneous kinematic .Body .Boundary Conditions (KBBC) are obtained that depend only on the equilibrium position of the solid body oS(x,y,z, 0) and not on the amplitude of the motion of the solid body (i.e., d(j)\/an = — n\, dfo/dn = — nj, etc.)! Of course, these unknown amplitudes must still be determined; and they are readily computed from the Dynamic .Boundary Conditions (DBC) in Eqs. (8.9b-g). In the linear diffraction theory, the interaction of a fixed Lagrangian solid body with the incident wave field generates a scattered wave. If a semi-immersed body is freely floating or semi-constrained, the response also generates radiated waves. Because of linear superposition, the total fluid
629
Deterministic Dynamics of Large Solid Bodies
motion resulting from this wave-structure interaction is assumed to be a linear combination of the incident wave <&w(x, y,z,t) plus a scattered wave <£>s(x,y,z,t) plus six individual radiated waves each resulting from one of the six modes of solid body motions from <&r(x,y,z,t). Because the incident wave and the resulting Lagrangian solid body motions are assumed to be simple harmonic in time (viz., oc exp— icot), each of these three uppercase scalar velocity potentials may now be defined by the real part of a linear sum of complex-valued, time dependent amplitudes and lower-case spatially dependent velocity potentials by Eqs. (8.12a and 8.13) according to <&(*, y, z, t)= <&D(x, y, z, t) + ®r(x, y, z, t) I" A[cj)w(x,y,z) +
fa(x,y,z)]
3
~]
6
+co^2^j
~ L
•/=!
XP
lC
°
'
J
7=4
(8.15) where each of the lower-case scalar velocity potentials fa (x,y,z) are functions only of the spatial coordinates (x,y,z); and where the amplitudes of each velocity potential A, %j and Qj are complex-valued amplitudes of the incident wave and the translational and rotational Lagrangian solid body displacements, respectively. The boundary value problems for each of the seven unknown lower-case scalar potentials fa (x,y,z) plus the known incident wave potential are identical with the exception of: 1) the inhomogeneous kinematic 5ody boundary Condition (KBBC) on the Lagrangian solid body and 2) a radiation condition to be imposed at large horizontal distances from the equilibrium position of the Lagrangian solid body oS(x, y, z, 0). The governing field equation for each of the lower-case scalar velocity potential fa (x, y, z) is Laplace's equation (2.12a) in Chapter 2.2.9 for the conservation of mass of an ideal (incompressible and inviscid) fluid in irrotational flow; i.e., V2fa=0,
|x|
\y\
-h
(8.16)
where the scalar Laplacian operator in Eq. (2.12a) in Chapter 2.2.9 is defined by V 2 (.) = 3 2 (.)/3x 2 + 9 2 (*)/3y 2 + 3 2 (.)/9z 2 . Each scalar potential fa(x,y,z) must satisfy a .Kinematic boundary Condition (KBC) on the horizontal impermeable bottom given by a Stokes material
630
Waves and Wave Forces on Coastal and Ocean Structures
surface B = z + h = 0 according to DB D(z + h) = - V 0 ; • n = V(pj • e3 = 0 Dt Dt that reduces to
(8.17a)
^p- = 0, | J C | < O O , | y | < o o , z = -A (8.17b) oz for a rigid, impermeable, horizontal seabed. Each potential j must satisfy the linearized Combined kinematic and Dynamic Free Surface boundary Condition (CKDFSBC) given by -P-- koj = 0, \x\
+ ®(t)»n'
= 0,
on
0S(x,y,z,0),
that yields seven inhomogeneous kinematic body boundary conditions (KBBC). Again, this scalar kinematic 5ody boundary Condition (KBBC) converts the Lagrangian motions of the solid body X(t) and 0 ( 0 to Eulerian fields through the vector dot product with the spatially dependent unit normals it and n' to the equilibrium position QS{X, y, z, 0) of the Lagrangian solid body. For the diffraction of waves by a fixed solid body, the inhomogeneous kinematic .Body boundary Condition (KBBC) for s may be computed from V ( 0 U , + 0 J ) « » = O,
on
0S(x,y,z,0)
(8.19a)
or d
30u
,
on
0S(x,y,z,0)
(8.19b)
on on For the radiation of waves by a solid body moving in otherwise still fluid (i.e., the wavemaker boundary value problem from Chapter 5), the inhomogeneous linear kinematic 5ody boundary Condition (KBBC) becomes „ .
—^
3
= HXJ^ni 7=1
6 +
H®}(t)n'j> j=4
on
oS(x,y,z,0),
(8.20)
631
Deterministic Dynamics of Large Solid Bodies
that will yield an inhomogeneous .Kinematic Body boundary Condition (KBBC) for each of the six radiated wave potentials 4>j(x,y,z). Substituting Eqs. (8.7c,d) into Eq. (8.13) provides a separate radiated wave potential for each jth mode of oscillation according to
3 =
CO
6
] T %j
Qj(/)j (x, y, z)exp — icot.
j=4
(8.21a) Substituting Eqs. (8.7c,d) into Eq. (8.20) and equating to the normal derivative of Eq. (8.21a) for each of the potentials cj>j (x, y, z) to the negative of the unit normal results in
Wj
\-njlj
= 1,2,3 1
1 7 = {-»$;;• = 4,5,6 J
^ :
°
n
o*(W,0).
(8.21b) (g21c)
By factoring explicitly, a priori, the unknown, complex-valued amplitudes of the velocities of the Lagrangian solid body defined by Eq. (8.7c,d), the inhomogeneous .Kinematic Body Boundary Condition (KBBC) for the radiated wave potentials requires knowing only the equilibrium geometry QS(X, y, z, 0) of the Lagrangian solid body defined by the direction cosines rij and pseudo-direction cosines n'-. In principle, the unknown amplitudes have been assumed symbolically to be known and the computation of their numerical values is deferred until the Lagrangian Dynamic Boundary Conditions (DBC) Eqs. (8.9b-g) are solved for these Lagrangian solid body motions. For the translational motions denoted by j = 1,2,3, the dimensions of
632
Waves and Wave Forces on Coastal and Ocean Structures
required to account for local pressure effects near the solid body (vide., the wave maker evanescent eigenmodes in Chapter 5.2). The radiation condition eliminates both incoming wave energy and unbounded non-wave, spatially transient solutions. This kinematic Radiation boundary Condition (KRBC) depends on the choice of the time dependence exp ±icot. Two-dimensional surface waves behave like exp ± [sgn(x)]?' Kx; while three-dimensional waves behave like [exp±iKr]/^J~Kr where the free-surface coordinates r 2 = x2 + y 2 > 0. Accordingly, the .Kinematic Radiation boundary Condition (KRBC) at large horizontal distances from the Lagrangian solid body must be (Stoker, 1957) lim (Kr)^-^2 M=>oo
\ ~ ± [8n2 + Snl sgn(x)] IK) 0, = 0, I on J
(8.22)
where n = the number of horizontal surface coordinates; i.e., n = 1 for 2D waves; and n — 2 for 3D waves. The choice of the ± sign is determined totally by the choice of the sign of the argument of exp ±icot. The premultiplication in Eq. (8.22) by */Kr in the case of 3D waves (n = 2) is required in order to insure that only the coefficients in the solution are applied to satisfy the radiation condition and not the radially decaying behavior of the solution given by r _ 1 / / 2 .
Summary The complicated irrotational motion of an ideal fluid that results from the interaction of a semi-immersed Lagrangian solid body with surface gravity waves may be linearly decomposed into three less complicated boundary value problems (BVP) by invoking the following linear superposition. First, the Lagrangian solid body is completely removed from the fluid and a boundary value problem for the incident wave potential (pw(x,y,z) is obtained. Engineering design specifications are given by the wave height H = 2A, the wave period T, and the water depth h and allow this potential to be completely quantified (vide., Chapter 4.2). Second, a fixed Lagrangian solid body is placed in the incident wave field and a scattered wave potential
Deterministic Dynamics of Large Solid Bodies
633
Condition (KBBC) on the Lagrangian solid body given by the normal gradient of the incident wave potential having a known complex-valued amplitude. Third, the Lagrangian solid body is oscillated in otherwise still fluid (i.e., the wavemaker boundary value problem in Chapter 5). The complex-valued amplitudes of oscillations are assumed to be known symbolically; and six radiated wave potentials
=
[u(x,y,z,t),v(x,y,z,t),w(x,y,z,t)}
= -VQ(x,y,z,t)
(8.23a)
634
Waves and Wave Forces on Coastal and Ocean Structures
and the dynamic fluid pressure from the linearized Bernoulli equation d
The three boundary Falue Problems (BVP) for each of the potentials 4>„,(x, y, z, t), <$>s (x, y, z, t), and $>r (x, y, z, t) may be summarized as follows: <&(x,y,z,t) = <S>w(x,y,z,t) +
3
6
7=1
7=4
exp — icot (8.24a)
Laplace's equation 3
2
2
coJ^^V2(t)j{x,y,z)
A[W 4>w{x,y,z) + y 4>s{x,y,z)'\ +
7=1 6
+ a ) ^ ^ 7 ' V 2 0 ; - ( x , ) ' , z ) = 0;
\x\ < oo,
\y\ < oo,
— h < z < 0.
7=4
(8.24b) Combined kinematic and dynamic free-surface boundary condition (CKDFSBC) 3
A[Lf((f>w) + Lf((/)s)] + coJ2^Lf((l)j) 7=1 6
+ o^QjLfi^j)
= 0,
|JC| < oo,
\y\ < oo,
z = 0,
(8.24c)
7=4
where the linear free-surface operator £ / ( • ) is defined by L / ( . ) = [3/3z - *<>](•).
(8-24d)
Bottom boundary condition (BBC) d&w dj>s_ + CO dz dz
3
907 I>^dz + £ V dz 7= 1
x\ < oo,
~~
\y\ < oo,
7=4
z = —/?.
= 0,
(8.24e)
635
Deterministic Dynamics of Large Solid Bodies
Eulerian kinematic body boundary condition (KBBC) d*w dn
9<*>* dn
+
dn
+ X(t)»n + @(t)»ri = 0 on 0S.
(8.24f)
Kinematic radiation boundary condition (KRBC) 3
6
(8.24g) where the deep-water wave number &o = (*)2/g and the linear radiation condition operator Loo(») is defined by Loo(.) = , lim (Kmr)(n-l)'2
\ — ± [8n2 + 8nl sgn(x)]K„ (•), (8.24h)
where n = 1 for 2D waves; n = 2 for 3D waves; and £"i = — ik because the time dependence is chosen to be exp — icot; and the Lagrangian dynamic body boundary condition (DBBC)
m — _/*_
•
(8.24i) (8-24J) (8.24k)
lX(t) — — — = e
. 16(0
£iM„(f)
+
Eo^«
(8.241) (8.24m) (8.24n)
where both the time-dependent 0(e=kA) and time-independent 0(€°) forces/moments are derived in detail in Sec. 8.2. These forces/moments include the following: 1) stiffnesses due to buoyancy, weight and water plane surface area (vide., Eqs. (8.53c-h) in Sec. 8.2); 2) external constraints (vide., Eqs. (8.55a-d) in Sec. 8.2); 3) the exciting forces/moments on a still body (vide., Eq. (8.43) in Sec. 8.2); and 4) the restoring forces/moments on a body oscillating in otherwise still-fluid (i.e., the wave maker boundary value problem and vide., Eqs. (8.44) in Sec. 8.2). The Lagrangian vector dynamic body boundary conditions in Eqs. (8.24i-n) transform the Eulerian dynamic and static pressure fields into time-dependent and time-independent, respectively, Lagrangian solid body forces/moments by integration of the total
636
Waves and Wave Forces on Coastal and Ocean Structures
fluid pressure field over the immersed surface area of the equilibrium position QS(X, y, z, 0) of the Lagrangian solid body. Linear superposition requires the identically equal [=] notation in the kinematic 5ody boundary Condition (KBBC) Eq. (8.24f). Both the scattered 4>s{x,y,z) and radiated
8.2. Linearized MDOF Large Solid Body Dynamics A mathematically consistent theory for evaluating the large solid body Lagrangian motions of the semi-immersed large solid body shown in Fig. 8.4
Fig. 8.4. Free body diagram for large solid body.
Deterministic Dynamics of Large Solid Bodies
637
was developed by John (1949, 1950). Extensions of this theory are given by Peters and Stoker (1957), Stoker (1957), Wehausen (1960, 1971), Newman (1961, 1977) and Mei (1989). The extensions by Newman (1961) include three separate perturbation parameters that represent the orders of: 1) the wave height/wavelength ratio (H/k); 2) the beam/wavelength ratio; and 3) the magnitude of the oscillations of the body. This less ambitious summary follows John (1949, 1950) with the exception of the orientation of the vertical z axis instead of the y axis; the negative gradient of the scalar velocity potential to recover the fluid velocity q = — V4>; and the addition of zero-order external constraints (Mei, 1989). There are two well-known spring stiffness constants for the hydrostatic stiffness due to the change in water plane area for semi-immersed large solid bodies. One of these two spring constants is the vertical heave mode X$(t) linear spring given by the well-known formula K& = pgA0, where AQ = the water plane area at equilibrium (e.g., AQ = nD2/4 for a vertical circular cylinder of diameter D). The second spring constant is the pitch mode ©5 (t) rotational spring constant for rotations about the origin of the body-fixed coordinates x' given by the well-known formula
where the second moment of area may be computed from
0A=
Jj(x'-x'R)2dx'dy', Ao
where 1^ , = rrD4/64 for a vertical circular cylinder and Ao = water plane r r
i i
area at equilibrium (i.e., t = 0). When these two well-known spring constants are derived from the fluid pressure given by Eq. (8.47) below, it is then possible to compare the rotational spring constant due to excess buoyancy at equilibrium K$5 with the spring constant due to changes in water plane area at equilibrium A55 for semi-immersed solid bodies (cf, the articulated tower in Chapter 7.8.1 where K^5 is neglected compared to K^5).
638
Waves and Wave Forces on Coastal and Ocean Structures
8.2.1. Kinematic Body Boundary Conditions (KBBC) The instantaneous wetted-surface of the solid body shown in Fig. 8.4 may be described by a Stokes material surface S(x, y, z, t) given by S(x,y,z,t)
= z-f(x,y,t)
= 0.
(8.25)
The solid body kinematic .Sody boundary Condition (KBBC) may be computed from the Stokes material derivative of the Stokes material surface (vide., Eq. (2.13a) in Chapter 2.2.10) according to DS of d
onS(x,y,z,t).
(8.26a)
The unit normal surface vector may also be determined from Eq. (8.25) by n= -^— =
±|VSI
dx
dy
2
(8.26b)
2
^[m +m +i'_
where the ambiguous sign of the modulus of the gradient vector ± | V 51 must be determined from whether the unit normal points out of the surface and into the fluid or into the surface and out of the fluid (Hildebrand, 1976, Chapter 6.12). The perturbation procedure involves expanding the scalar velocity potential <£>(x,y,z,t); the translational displacements X(t) of the center of mass x'c (x'c,y'c,z'c) of the solid body; the rotational displacements 0 ( 0 about an arbitrary point of rotation xR(xR,yR,zR); the external loads F(t) and M{t); and the Stokes material surface S{x,y,z, t) in a power series ordered by a perturbation parameter given by the wave amplitude/wavelength ratio e — kA = 2nA/X > 0 (cf., Chapter 6). Simple coordinate transformations relate the fixed inertial coordinate system x to the moving body-fixed, non-inertial coordinate system x' where x = x' at t = 0. Note that x may be a function of time; but x' is never a function of time. Solidbody rotations in the moving body-fixed, non-inertial axes x' that coincide with the fixed inertial axes x at t = 0 may be about any arbitrary point of rotation xR(x'R, y'R,z'R) shown in Figs. 8.2 at point R. Substituting the linearized Bernoulli equation for the total fluid pressure (i.e., static plus dynamic) into the body surface integrals evaluated over the equilibrium wetted-surface area of the immersed body o^(x, y, z, 0) yields: 1) zero-order
639
Deterministic Dynamics of Large Solid Bodies
e° equilibrium equations for the Archimedes buoyant force/moment F /M and 2) a first-order e 1 linearized dynamic pressure force/moment components Fp /Mp. The Archimedes buoyant force/moment F /M may be combined with the body weight force/moment F /M to give the hydrostatic pressure -> k
-> k
force/moment stiffness components F /M . Equality between two different forms of the kinematic boundary condition for an ideal fluid on the solid body boundary may also be demonstrated (Mei, 1989). The perturbation expansions are given by (cf. Chapter 6.2) $>(x,y,z,t)
+ e22®(x,y,z,t)
= €i®(x,y,z,t)
+•••
= JV„
(8.27a)
n=\
X(t) = €lX(t) + €22X(t) +-.. = J2 €\X(t)
(8.27b)
n=\
0 ( 0 = e i 0 ( O +e22®(t)
+--- = ^€nn@(t)
(8.27c)
n=\
z = f(x,y,t)
= of(x,y)
+€if(x,y,t)
+•••=
^2ennf(x,y,t)
n=\
(8.27d) P(x,y,z,t)
= op(z) +e\p{x,y,z,t) = ^j£nnP(x,y,z,t)
+€22p{x,y,z,t)
-\ (8.27e)
n=0
that imply that there is nofluidmotion for e = Oandop(z) = hydrostatic pressure. The Stokes material surface S(x, y, z, t) and the pressure P(x, y, z, t) all contain zero-order equilibrium components that are independent of time; viz., of(x, y) andop(z). The complex-valued scalar velocity potentials <&(x, y, z, t) represent only wave-induced fluid motions and, therefore, do not contain zero-order e°, steady-state, time-independent components. The coordinates in Fig. 8.2a to an arbitrary point of translation xc(xc,yc,zc) iQ the wave inertial coordinate system fixed at the still water level equilibrium position are equal to the moving, body-fixed non-inertial coordinates xc{x'c,y'c,z'c) at t = 0 (John, 1949). An arbitrary point of rotation expressed in the moving body-fixed non-inertial coordinates x'R(x'R,y'R,z'R) in Fig. 8.2a need not be
640
Waves and Wave Forces on Coastal and Ocean Structures
Fig. 8.5. Translation and rotation of a large solid body.
located at the body center of mass x'c(x'c, y'c,z'c) (cf., the articulated tower Chapter 7.8 and waves in a 2D channel Chapter 5.2). Again, x = x' at t = 0; and x may be time-dependent but 3c' is always time-independent. An expression for \f(x,y,t) in terms of the static equilibrium position of(x,y) is required in order to proceed further with the linearized problem (John, 1949). In order to obtain this required expression for i / (x, y, t) in terms of of(x, y), the moving body-fixed, non-inertial coordinate system x must be related to the fixed, wave inertial coordinate systems; and the Stokes material surface S(x, y, z, t) must be expanded in a Taylor series about the equilibrium position oS(x, y, z, 0) of the solid body in order to be able to equate terms of 0(e). It is important to remember that the goal of the following derivation is to replace the time-dependent location of the Lagrangian body at i / ( x , y, t) with the time-independent location at equilibrium at of(x, y). The coordinate transformations between the fixed, wave inertial coordinate system fixed at the still water level at the body equilibrium position x and the moving, body-fixed non-inertial coordinate system x' may be obtained by a linear combination of pure translation without rotation x(t) — x' + e\X(t) + 0(e 2 ) (vide., Fig. 8.5) plus pure rotation without translationR(t) = ei®(t) xr' + 0(e2) about an arbitrary point R in the moving body-fixed, non-inertial coordinates x' that is aligned with the fixed inertial axes x at t = 0 according to
x(t) =x' + €[iX(t) + 10(0 x (x'-x'R)] = 3c' + 6 i x ( 0 + O(6 2 ),
+ 0(e 2 ) (8.28a)
641
Deterministic Dynamics of Large Solid Bodies
that may be rearranged to X{t)-X'
= 6lX(0 + O(6 2 ) + 0(e2)
= e [ i £ ( 0 + !0(f) x (x'-xR)]
(8.28b)
with the following scalar components: x'-x(t)=
- e [ i X i ( 0 + i® 5 (0(z' - z'R) - i®6(t)(y'-
y'R)],
(8.29a)
y' - y(t) = - e [ i X 2 ( 0 + i©6(0(*' - 4 ) ~ i©4(0(z' - z'R)],
(8.29b)
z' - z(0 = - e [ i X 3 ( 0 + i © 4 ( 0 ( / - 4 ) - i®s(0(*' - 4 ) ]
(8-29c)
because 3c = x correct to 0(e). At equilibrium or, equivalently, at time t = 0, z(0) = z' and the Stokes material surface becomes oS(x,y,z,0)
=
z-of(x,y)=
(8.30)
z / - o / ( x / , / ) = 0. The Stokes material surface may be expanded in a Taylor series about the equilibrium body position o5(x, y, z, 0) by first expanding f(x,y, t) by Eq. (8.27d) to obtain S(x,y,z,t)
= z-
f(x,y,t)
« [z' - of(x',y')]
- eif(x,y,t)
+ 0(e2) + •••
(8.31a)
and then expanding S(x, y, z, t) by S(x,y,z, t)*0S
+ - V ( z ' - z(t)) + 1rr(x'dz ox' 6 nS , + ^r(y'y(t)) + • • •
x(t))
dy' *(z'-0f(x',y')) ^~f
dx'
+
(z'-z(t))
(x - x(t))
—
(y - y(t))
+ ••• .
dy' (8.31b)
642
Waves and Wave Forces on Coastal and Ocean Structures
Substituting Eqs. (8.29a, b) into Eq. (8.31b) and equating 0(e) terms gives the following first-order approximation for i f(x,y, t) in terms of of(x, y) (John, 1949, p. 24, Eq. (2.2.4)): e1 :
lf(x,y,t)
= iX3(t) +
l®4(t)(y'-y'R)-l®5(t)(x'-x'R)
- - ^ - { i ^ i W + i0 5 (O(z' - z'R) - 1 © 6 ( 0 ( / - y'R)) ri
-f
- -jriiXjit)
+1©6(0(*' - 4 ) - i©4(0(z' - z'R)}
= 1X3(0 - ofxiXiiO - ofyiX2(t)-
(8-32)
Substituting the temporal derivative of Eq. (8.32) into the .Kinematic .Body .Boundary Condition (KBBC) given by Eq. (8.26a) yields, correct to 0(e) dif(x,y,t) dt
=
diQdpf dx dx'
|
d^dpf dy dy'
3iO dz
= jX 3 (0 + i ® 4 ( 0 ( / - y'R) - i® 5 (0(*' - x'R) - -2L{xXi(t)
+ i©5(0(z' - zR) - i © 6 ( 0 ( / - y'R)}
~ - ^ 7 { l X 2 ( 0 + i© 6 (0(*' - 4 ) - 1©4(0(Z' - ZR)} ay (8.33) or
3$
A
— +X»n an
A
+ @»ii' = 0,
on
0 5(x,v,z,0),
(8.34)
that readily demonstrates the well-known equality between the .Kinematic .Body .Boundary Condition (KBBC) for 1) a boundary between a solid body and an ideal fluid given by a Stokes material derivative (vide., Chapter 2.2.10) and 2) the continuity of the normal component of the velocity of a solid Lagrangian body and the normal component of the velocity an ideal fluid (Dean and
643
Deterministic Dynamics of Large Solid Bodies
Dalrymple, 1991 orMei, 1989) i.e., 3
A
A
DS
\-X»n + Q»n
= 0=
,
on
0S(x,y,z,0),
(8.35a)
where the unit normal on oS(x, y, 0) is computed from Eq. (8.26b) by n=
H>/*,-o/y,l) 2
.
(8.35b)
2
- A Ao/, + o/v + l]
The scalar kinematic Body Boundary Condition (KBBC) in Eqs. (8.35) transforms the large Lagrangian solid body motions X{t) and 0 (t) to Eulerian fields by the vector dot product with the spatially-dependent unit normals and the body motions are now functions of both time and space in the KBBC.
8.2.2. Dynamic Body Boundary Condition (DBBC) Because the semi-immersed body in Fig 8.6 is free to respond to the wave induced pressure loadings, a dynamic boundary condition must also be prescribed in addition to the kinematic Body Boundary Condition (KBBC) given by Eq. (8.34). The three translational displacements \X{t) may be referenced to any arbitrary differential point of mass dm in the body shown in Fig. 8.6. The Lagrangian linear and angular momentum principles for a solid body may be
Fig. 8.6. Locations of 2 axes: 1) Inertial {x, y, z} and 2) Non-inertial {x', y', z'
644
Waves and Wave Forces on Coastal and Ocean
Structures
written as n
(8.36a)
x(t)dm = '(Ox where r{t) = (x-xR)
+ eiR(t) = r'(t) + erR(t) +
0(e2)
(8.36b)
ic(0 =x' + ehX(t) + iR(t)] = ehX(t) + i © ( 0 x r'] + 0(e 2 ); (8.36c) x(t) = eiX(0 = ehX(t) + i 0 (O x r'] + 0(e 2 )
(8.36d)
The notation in Eq. (8.36a) and in all subsequent equations for linear and angular momenta is an implied stacked notation with the three translational momenta stacked vertically above the three angular momenta and separated by a horizontal dashed line [ ]; i.e.,
n
EF2„(o n
£iW)l
X>„(0 (8.37a) n
I>2„(0 X>3Bc) . n
Careful consideration must be given to r(t) when transforming from linear momentum to angular momentum. In Fig. 8.6, r(t) may be seen to be the (8.37b) r(t)=? + €iR(t). vector sum of static r' and dynamic R(t) components according to
645
Deterministic Dynamics of Large Solid Bodies
Large Solid Body Lagrangian Inertia (Crandall et al., 1968, Ch. 3) The Lagrangian inertia of an homogeneous large solid body correct to O (e) becomes
d dt / / /
x(t)dm .'(Ox \X(t)dm
- / / /
r x (?) x r'dm + 0(e2)
"l[l{y.¥ m\X(t) Sijkm(r'c)j
\Xk{t)ei/
( + €
Sijkm(r'c)k
lO(0 Iff \r'.r'\dm-
i@j+3(t)ei
i&(t) • ffjV
r'dm )
(
m\X{t)ei
= e \msijk(r'c)j
(
\Xk{t)ei
msijk(r'c)k i0 7 + 3 (Oe; __ \,i,j,k / r v.i ©,-+3 ( 0 5 , - / ^ 1 ©j+3 (0?«
= 1,2,3,
(8.38)
where e = &A = the perturbation parameter; Sijk = the Levi-Civita permutation symbol (vide., Eq. (2.5) in Chapter 2.2.4); and repeated subscripts imply tensor summation. The total mass m and center of mass x'c of the homogeneous
646
Waves and Wave Forces on Coastal and Ocean Structures
large Lagrangian solid b o d y are defined b y III
dm = m,
x'c = m~ III
vm
x dm,
(8.39a,b)
vm
w h e r e Vm is the total v o l u m e of the h o m o g e n e o u s large solid body. T h e first(translational inertia) a n d second- (rotational inertia) m o m e n t s o f total m a s s v o l u m e for a large h o m o g e n e o u s solid b o d y are defined b y / / / r'jdm = \ \ \ (x' -x'R)jdm vm
= m(x'c -x'R)j
=m(r'c)j,
j = 1,2,3
vm (8.39c) (rc)i =x'c
-x'R,
(rc)i = y'c - y'R,
(rch = z'c - z R (8.39d-f)
'rt = Iff ^'^ = fff(X' ~X'R)i{X' ~X*)idm vm
vm 2
[(x' - x'R) + (y' - y'R)2 + (z' - z'R)2] dm
= jjf vm r r
l l
V2
r r
33
where I™, = mass m o m e n t o f inertia (Crandall et ah, 1968, p . 172) a n d where qr,
= jjf
r'/jdm = Jjj(x'
vm
x'R)jdm
vm
= I™x, -m(xR)i(x'c)j i,j
- x'R)i(x' -
= 1,2,3
-m(x'R)j(x'c)i
+m(xR)i(x'R)j (8.39h)
w h e r e I"], — products of mass m o m e n t s inertia (Crandall etal., 1968, p . 172); and where Eq. (8.39h) is a convoluted form o f the parallel axis theorem for a large solid h o m o g e n e o u s body. For a non-inertial axis x" fixed to the large solid h o m o g e n e o u s b o d y at the center of mass x'c, the parallel axis theorem
647
Deterministic Dynamics of Large Solid Bodies
for rotations about the body-fixed non-inertial coordinates xR is given by ==
x jll
v Q == (x
v
x D)
\^c
x"x'-dm = J I fix' - x'c)iix' -
vm
^ R)
x'c)jdm
vm =
Qx'. ~ m(xc">i(x'ch>
iJ = 1 ' 2 ' 3 -
External Forces/Moments F/M The total sum of the external forces/moments F/M may be evaluated by a variety of methods; but the method that most easily illustrates the spring constants K^ and K^5 is of primary interest. The most common method is to compute these spring constants from the total fluid pressure from Eq. (8.27e); i.e., hydrostatic opiz) plus dynamic eipix, y, z, t). The total sum of the external forces/moments include the following vectors: r n •
=
-P
F it)
+ •
'
J^Mnit)
' Pw '
1
( ->C • +
•
-» w M it)
P
M it)
1
F it) MCit)
where the superscript P = total Pressure forces/moments; the superscript W = total fTeight forces/moments and the superscript C = total mooring and anchoring Constraints. These forces/moments may be expanded and combined to obtain f - F
f ~R
1
F it)
F \t)
n •
=
• +
•
E
M it)
R
M it)
^
\PKit)} +| K
[M it)
f ->C
1
F it) • +
•
MCit)
n
where the superscript E = Exciting dynamic pressure force/moment on a fixed body; the superscript R = the hydrodynamic Restoring force/moment on a large Lagrangian solid body oscillating in still water (i.e., the wavemaker BVP).
648
Waves and Wave Forces on Coastal and Ocean Structures
Total Fluid Pressure Forces/Moments K(t)/M(t) The total fluid pressure loads may be computed from /
FP(t)
fjp(x,y,z,t) P
M {t)
I , \
\
1
hdS = -e
^ r(f)x /
oS
(
I
\p(x,y,z,t)
QS
ndS
V
rx
)
\
1
hdS
//OPCZ) y(f'
Fp(t)
+
€lX(t)+€1R(t))x/
F\t) • +
p
M (t)
(8.40)
•
b
M {t)
where o S is the wetted surface of the large solid body at equilibrium or at t = 0; and where the minus sign is required because the unit normal n is directed out of the body and into the fluid. This method includes the body force/moment ^b
- b
loads F (t)/M (t) through the hydrostatic pressure component op(z). The total fluid pressure is computed from the linearized Bernoulli equation and includes both hydrostatic e° and dynamic pressure components correct to the first-order 0(e); i.e., P(x,y,z,t)
= op(z) + €ip(x,y,z,t) = -pgz
+ ep—-
at
= -pg[of(x,y)
+ 0(e )
+ 0(e2)
+ €if(x,y,t)]
(8.41a) + €P-^-
+ 0(e2)
(8.41b)
at
by Eqs. (8.27d, e). The zero-order hydrostatic component op(z) represents a body force/moment contribution to the total sum of the external forces/moments. It is this body force component that will eventually contribute to the hydrostatic spring constants in Eq. (8.50a) below; viz., K&
and
K& + KL
649
Deterministic Dynamics of Large Solid Bodies
Dynamic Pressure Force/Moment The dynamic pressure force/moment loads may be computed by substituting Eq. (8.24a) for I<J>(JC, y, z, t) from Sec. 8.1 into Eqs. (8.41a, b) for \p(x, y, z, t) and then by substituting into the first curly bracket term {•} on the RHS of Eq. (8.40) to obtain (Fp{t)\
(
1
\ ndS
= -Re \MP{t))
y rx J
0S
(
1
\ wrfS
= -Re
V rx / /
1
\
ipoj / / A[4>w +4>s]
= Re
ndS
exp — icot
oS 3
6
/ , ?m0m + 2 J £lm4>ir, m=\ m=4
+ Re
ndS
i pay
/ /
3
exp — ittrf
6
/ , Hm4>m + 2 J ^fflfc .m=l
( J\t)
m=4
\ (8.42a-f)
+ 6
\iM (t)J -> E
—E
where i F {t)/\M
{t) = the .Exciting force/moment on afixedbody given by /
(i^v) = eRe £
\,M (f)y
1 \
'W/'
ipoj I I A[
ndS exp — ('a;/
(8.43a) (8.43b) (8.43c) = eRe
d S e x p — icot
i/ooj / / A[0„, +4>s] " / / •
oS E
\Mj (t)
V«/
(8.43d) (8.43e) (8.43f)
650
Waves and Wave Forces on Coastal and Ocean
Structures
The \F (t)/\M (t) = the Restoring force/moment on a large solid Lagragian body oscillating in otherwise still fluid (i.e., the wavemaker BVP in Chapter 5). Substituting Eqs. (8.2 lb,c) from Sec. 8.1 for each of the j unit nor-* R
-* R
mals and Eqs. (8.7c,d) for the velocities into Eqs. (8.42) for \F (t)/\M and expanding each
(8.44a)
0 7 = R e { 0 7 } + nm{0 7 }, -* R
(t)
-* R
the Restoring force/moment \F (t)/\M
{t) may be expressed as
xFfit)
\Mf{t)
3
J2ixm(t)fj (Re{^ }
+
m
6
iIm{4>m})dS
OS
3«
+ X ) i©m(0 [J(*B{4>m} + i \m{4>m})dS m=4
= eRe
icop
>J = 3
J2iXm(t)jj (Re{4> } + ilm{
m
oS
J2 i®«<') /7(Re{0 m } + i lm{4>m})dS m=4
dn
oS
(8.44b-g)
651
Deterministic Dynamics of Large Solid Bodies
where the two integral terms in the elements of each of the two 3 x 6 matrices in the RHS of Eq. (8.44b) represent typical matrix elements. Substituting the Lagrangian ordinary temporal derivatives of Eqs. (8.7f,h) given by iXj(t) = -ia>iXj(t),
j = 1,2,3,
l©j(t) =-ia>i®j(t),
j =4,5,6
into Eq. (8.44b-g) gives
iFfit) J =
\Mf{t)
3
E
j j Re{
,
m=l
E m=4
co j j \m{cj>m}d-^-dS
\Xm{t) +
j j Re{
LloS
l©m(0 +
j
\Xm{t)
oS
co Jj
Iml^^dS l©m(0
I o5
-epRe 3
E
//
Re{c/>m}—J-dS \Xm{t) + an
co Jj' \m{4>m}d-^-dS
\Xm(t)
m=l
E
jJ' Re{4>m)d-p-dS on
l©m(0 +
co J j'lm{<M^S l©m(0
m=4
(8.44h-m)
652
Waves and Wave Forces on Coastal and Ocean Structures
that may be reduced to
/ , i^mj 1 Xm (t) + ^mj \ Xm (t)] m=l
+ 6
m (') + ^mj 1 ®m (01
lF/(0
• m=4
= -Re
lMf{t) / , [Mm; 1 ^m (f) + A.my-1 X m (f)] m= l
+ 6
XI [iJ-mj 1 ®m (0 + \mj i ©m (/)] -m=4
(8.44n-s)
where the added mass coefficients iJLmj and the radiation damping coefficients kmj in Eq. (8.44n-s) may be computed from
fimj = P if o5
RelM-^-dS;
Xmj = pco jj
Im {
(8.45a,b)
oS
where, again, the square bracket terms [•] in the elements in each of the two 3 x 6 matrices in the RHS of Eqs. (8.44n-s) represent typical matrix elements. The restoring force/moment matrix in Eqs. (8.44) may be linearly decomposed into two matrices; viz., a matrix for the accelerations Xm(t)/@m(t) and the added mass fimj and a matrix for the velocities Xm (t)/®m (t) and the
653
Deterministic Dynamics of Large Solid Bodies
radiation damping Xm; coefficients according to
\Ff{t)
\Mf{t) •
xFfif)
•
1
lF|(0
2
•
•
• - eRe
-eRe
3
e
,
J=
-
•
•
4
lMf(t)
\M)(t)
5
•
•
6
3
6
^ Mm; 1 *m (t) + Y^ Hmj m=\ m=4
\®m(t)
= -eRe 3
6
XI V-mj\Xm(t) + ^2 Vmj\®m(t) m=\
3
m=4
6
/ , -^mj 1 ^m (t)+Y^kmjl®m(t) m=\
m=4
1 2 3
(8.45c) (8.45d) (8.45e)
-eRe 4
22 Xmj 1 Xm 0) + 22 ^mJ l ®m ('} m=\
m=A
5 6
(8.45f) (8.45g) (8.45h)
654
Waves and Wave Forces on Coastal and Ocean Structures
The added mass matrix elements in Eqs. (8.45) and the radiation damping matrix elements in Eqs. (8.45) may be transposed to the LHS of the dynamic equation of motion and coupled into the mass matrix and the damping matrix. Hydrodynamic Stiffness Fb(t)/Mb(t) Substituting the hydrostatic pressure component op(z) Eq. (8.41a) into the third integral in Eq. (8.40) yields
P\t) M\t)
oS
—pgz from
1
= P8
ff-
=
ndS. v
(8.46)
{F + eiA"(0+ *!*(*)}x
Substituting Eq. (8.31a) for z into Eq. (8.46) gives F (0 = pg
z
ndS -epg
M\t)
l f(x, y,t)\
r x -*Pg
r x
ffof(,x',y')l JJ
\hdS
\ndS. x
ix(0x
The gradient Theorem in Eq. (2.122b) in Chapter 2.6 converts the surface integral over dS to a volume integral over d V according to
llFidS = lll*FdVoS
Vo
where Vz = g3 (Hildebrand, 1976, Chapter 6.12); where Vo = submerged volume of the solid body at equilibrium; and where the horizontal projection dxdy of the wetted surface may be computed from dx dy = n-}d S =
dS
y/ofi + of? + l'
Deterministic
655
Dynamics of Large Solid Bodies
so that Eq. (8.46) becomes
F\t) M\t)
= M [[[I
\dV-epg [fifl
~€p8fff\J J
l
| hdxdy
)dv'
Vvx(z;x(0)/
where QA = the water plane area at equilibrium. Collecting order e terms for bodies that are symmetric at the still-water-level (SWL) gives F\t)
pgVoh
M\t)
yPgV0[(y' - y'R)ex + (*' - x'R)e2] (
[OA1X3(O + / r | l 0 4 ( O - / r | l 0 5 W ] ^
+ £pg
I$lX3(t) + I * 104(O - / I , 105(O / l x 3 ( 0 + /;l,i©4(0 - / l , 105(0
+
\
+ epg
6 (o-/Vi0 4 (ol2i + kVi0 L l 3 J r
r
kVi0 5 (o-/Vi0 6 (o L
r
3
r
2
+ epg _-VoiX2(Oei + VoiXi(0«2_
^
656
Waves and Wave Forces on Coastal and Ocean Structures
and combining zero-order e° and first-order e 1 terms gives: xF\t)
OF"
+ €
\M\t)
oM"
pgVoh pgVo [(y' - y'R)h + (*' -
x'R)e2\
- [oA iX 3 (0 + 7^184(0 - /^105(O]«3 + epg - fa 1*3(0 + fa . + 7?) 1©4(0 - 7* 105(0 - # 1 0 6 ( 0 " V iX (/)l e 0 2 X L r2
V r2r2
r
r r
3/
l 2
r
l
J
+ [ # i X 3 ( 0 + / 4 , i ©4(0 - fa, + iY) 105(0 + 7Vi06(O - v 0 iXi(ol h L rl
r
2rl
V rlrl
r
3/
r
2
J
•
(8.47) where # = / Y (*' - 4 ) , . r f * ' ^ ; J J0A
# r , = / / " (*' ' J
4),-(JC'
-
x'R)jdx[dx'2
J J0A
(8.48a,b) IVr, = \ \ \ (*' - 4)« dV = V0(x'B - x'R)i,
(8.48c,d)
and where x'B is the vector coordinate to the center of buoyancy B in Fig. 8.4. -w ^w Body Weight Force/Moment F /M The body weight force/moment due to gravity may be computed from ™ge3
- w M (0
p'c+eiX(0]
oF
-mge 3
»* oM
w
- w \M (0
mg(yc ~ y'R^i ~ mS(x'c - x'R)e2 +6
- m g [ i © 6 ( 0 ( 4 - X'R) ~ i®t(0(Zc - Z'R) " i^2(0]2i +mgh®5(t)(z'c - z'R) - i®6(t)(y'c - y'R) + iXi(0]? 2
(8.49)
657
Deterministic Dynamics of Large Solid Bodies
where (vide., Fig. 8.4) r'c = (xc ~ X'R) «i + (y'c ~ y'lt) *2 + (z'c - 4 ) & Stiffness due to Buoyancy and Weight The stiffness due to buoyancy and weight may be written as FK{t)
i
vb 1
=1
b
\ J\t) |
• +€
•
I oM
K
M (t)
• +
+e
•
.M W (0
\M\t) \FK(t) '
oFK •+ €
\MK{t)
oMK where o ^ K
(-mg + pgV0)h
1
[-mg (y'c - y'R) + pgV0 (y'c - y'R)] ex + [-mg (x'c - x'R) + pgV0 [x'c - x'R)] e2
(8.50a)
and where Pg [ A O I X 3 ( 0 + 7.|i© 4 (0 -
i^(0
> 8 ifrXrt)
+ [p g ( / £ , + 7rV) - mg (z'c - z'R)\ 0 4 ( 0
-pg I*Ai05(O lMK(t)
• '2
Irli@5(t)]e3
- [pg /rV - mg ( 4 - jc^)] 106(0
ei
p g / . | i X 3 ( 0 + PgI^r[ 104(0
[ p « ( / r | r ; +/ r V) " m g (Z'c- Z'«)] 105(0
-[pg/rV-mg(^-^)]i©6(0 (8.50b) External Constraining Forces/Moments F (f)/M (t) The total external constraining forces/moments represent both static and dynamic external constraints such as mooring line guys, tension leg tethers, anchors, etc. These external constraining forces/moments may also be
658
Waves and Wave Forces on Coastal and Ocean Structures
expanded in a perturbation series according to F (t) = 0F +€rF
(0 + f V w
+ -
= £ f " / ( 0 ,
(8.51a)
n=0
MC(t) = 0MC + eiM C (0 + e22MC(t) + • • • = J ] e"„M C (0.
(8.51b)
n=0 -*c
•* c
Zero-order, time independent constraints o-F /o-^ are required, for example, in a tension leg platform that has excess buoyancy (vide., the articulated tower in Chapter 7.8.1 and the two semi-immersed horizontal pontoons in Chapter 7.8.2 in Chapter 7.8). Collecting terms only to 0(e) gives oFC
FC(t) MC(t)
C
+ [?; x 0FC]
0M
iFC(t) (8.51c)
+€ C
C
C
iM (f) + [r'e x lJF (f)] + [ix(0 x 0F ] lFC(t)
oF oM°
•+•
(8.5 Id)
liMfw + iM^OJ
where C
= 0MC + [r'e x 0 F C ]
(8.51e)
MCx(t) = e\MC{t) + [r'e x e , F C « ]
(8.51f)
M°2{t) = [ei*(0 x 0 J F C ] + [ei*(0 x <>FC]
(8.51g)
0M
and r'e is(are) the position vector(s) from the point of rotation R in Fig. 8.4 to the point(s) where the external constraint(s) is(are) applied. If the point of rotation J?
Deterministic Dynamics of Large Solid Bodies
659
in Fig. 8.4 is the only single point of external constraint r'e in Fig. 8.4, then MC2{t) = [€lX(t) x 0 F C ] + [ei*(f) x 0 F C ] •
(8.5 lh)
Summary The Lagrangian dynamic equations of motion for a homogeneous large solid body are given by ~m~ _M_
•
iX(0 —
~K~ +€ _K
.iO(0. oF'
f l
— \+ 0M J f
r
_
\X(t)' — l&(t) r
->C i
-> F
l
,^(0
1^(0 — 0M -C
+ €|
iFC(t)
• — e
£
C
[iM (0. ~\
+e•
•
R
iM (t) (8.52a-c)
J
.lMfa ) + \M £(0.
(8.52d-f)
where the hydrodynamic stiffnesses in Eq. (8.50b) have been transposed to the LHS in Eqs. (8.52) and where the 36 element inertia matrix (Crandall et al., 1968) is given by
L.AU
where the first three rows of the mass matrix are
[m] =
m 0 0
0 0 0 m 0 —w(zc —z'fl) 0 m m (y'c — y'R)
m (z'c - z'R) -m (y'c - y'R)~ 0 m (x^ — x^) —m (x'c — x'R) 0 (8.53a)
Waves and Wave Forces on Coastal and Ocean Structures
660
and the last three rows of the mass moment of inertia are
0
-m(z'c-z'R) m (y'c - y'R)
m(z'c-z'R) m
- (y'c-y'i()
_
(4
-Ijj •2
-m{x'c-x'R)
0
m
\i"!, +ir r V V2
-Im r'r' r r
r r
V Vl
0
4)
I
3 3/
~'jj
3 2
W l
+H , (8.53b)
The corresponding 36 element stiffness matrix is given by "K" _K _
where the first three rows of the stiffness matrix are 0 0 0 0 0 0 0 0 0 0 [K] = 0 0 pgtf -pgrf ^33
0 0 0
(8.53c)
and the last three rows are M =
0
0^ 3
C
pgI$-0F% r 2
0
.+0^(4-4)-
-oF${y'e-y'R)\ r
04
r/4 r,r 2
-pgrf+oFC
~l
^ 5 5 + *55
-0Ff(x'e-x'R)
_-of 3 C (4-4).
tf
-off
0
-PS? +mg {x'c - x'R)
o^ & -4)
z
0^2 ( «
Z
~ 'R)
_+oFC (x'e-x'R)
r -p8V
2
+mg b e - y'R) c MFc,{y'e-y' R)
"- 0 f 2 (4-4) .-off ( 4 - 4 ) (8.53d)
and where the water plane stiffness elements in Eqs. (8.53c, d) are given by K^ = pg0A, K55 =P80r"
K& = pgl^,
KlA = Kv55=pgll-mg{z'c-z'R),
(8.53e,f) (8.53g,h)
661
Deterministic Dynamics of Large Solid Bodies -»E
-* E
where I"}, is defined in Eqs. (8.39g,h); \F (t)/\M (t) are defined in Eqs. (8.43); \FR {t) / XMR {t) are defined in Eqs. (8.44); and 1$, 1^, and /,? are defined in Eqs. (8.48). If the center of mass x'c and the center of rotation xR coincide {i.e., r'c = xc — xR = 0), both the mass and stiffness matrices are simplified significantly. Static Equilibrium Applying Archimedes buoyant principle, the weight of the large Lagrangian solid body plus the displaced weight of the fluid may be equated to the residual of the time-independent external constraints at static equilibrium (e°) to obtain
P8 z
ff
r li l
" 1 " 1
oF mge-i =
ndS-
/x.
(8.54) C C 0M + [P'e x 0F ]
that gives the following: Forces h • PgVo -mg = 0F3 ,
(8.55a)
Moments
h:pgll
- (y'c ~ y'R) mg = 0Mf+ [(y'c - y'R) 0 F 3 C - (z'c - z'R) 0 F 2 C ] , (8.55b)
h: -Pgtf
+ {x'c - x'R) mg = 0M2C + [(z'c - z'R) 0 / f - (x'c - x'R) 0 F 3 C ] , (8.55c)
h: 0 + 0 = 0M3C + [(x'c - x'R) o / f - (y'c - y'R) 0Ff] .
(8.55d)
-* c -c If there are no external constraints [oM = 0 and o^ = 0] and if x'c — xR = 0 the stiffness matrix is simplified further.
662
Waves and Wave Forces on Coastal and Ocean Structures
Two-Dimensional Applications Many applications do not require solving the Lagrangian dynamic equations of motion for afloatingbody in three dimensions (i.e., six degrees offreedom).An example is an asymmetric, semi-immersed body of revolution (Isaacson, 1982 or Hudspeth et al., 1994). In the two-dimensional case, translational motions are restricted to surge X\(t) and heave X^it) only, while the rotational motion is restricted to pitch &s(t) only. Because the equations of motion in Eqs. (8.52) for the three-dimensional case are written in matrix form, the conversion to the two-dimensional case is very simple. Because the sway X2 (t), roll ©4(f) and yaw @6(0 modes are neglected in the 2D case, the rows and columns associated with these motions in the matrices Eqs. (8.29) may be omitted. The reduced mass and stiffness matrices for 2D dynamics are m (z'c - z'R) -m (x'c - x'R)
0 m
m [m'] = 0 m (z'c - z'R)
(8.56a)
0 -pgIA
K
yi
[k'] = A
-PgI
K?5 + K} -oFf(x'e-x'R) l-oF3c(z'e-z'R)j]
(8.56b)
and may be simplified further if the point of translation xc coincides with the point of rotation*^.
8.3. Froude-Kriloff Approximations for Potential Theory The Froude-Kriloff hypothesis for estimating the wave-induced hydrodynamic pressure loads on small bodies formed the basis for the Morison equation force/moment loadings in Chapter 7. The linear decomposition of a 3D scalar velocity potential into an incident <$>w(x, y, z, t), a scattered ®s(x, y, z, t), and a radiated <$>r(x, y,z, t) wave potential for estimating the wave-induced loadings on large bodies also yields a Froude-Kriloff loading. In this case, the hydrodynamic pressure field induced by only the incident wave potential Q>w(x, y,z, t) yields a Froude-Kriloff load. This approximation forms the first estimate for the wave-induced loadings on large bodies; and may be applied to calibrate the magnitude of the diffraction effects on large fixed bodies. The scattered wave
663
Deterministic Dynamics of Large Solid Bodies
potential Q>s(x,y,z,t) results in a first-order correction to the hydrodynamic pressure field induced by the incident wave. If the diffracted wave load on a large fixed body is normalized by the Froude-Kriloff load, the magnitude of the scattered wave load is easily quantified. Because of this, all diffracted wave loads on largefixedbodies may be normalized by the Froude-Kriloff load due to only the incident wave pressure field. Examples in both Cartesian and circular cylindrical coordinates are reviewed.
8.3.1. Froude-Kriloff Load in 2-D Cartesian Coordinates The relatively large three-dimensional body shown in Fig. 8.7 is submerged with the depth of the center of gravity of the body located at z = —d. The effects of flow separation around the sharp corners are neglected. A normally incident, 2-D surface gravity wave in the absence of the body may be described by a scalar velocity potential given by &w(x,z,t) = Re{Acbw(x,z)exp-i(cot = Re \i
+ v)}
cosh kh(\ +z/h) exp i (kx — cot — v) > cosh kh coshfc/?(l + z/h) sin(kx -cot — v), (8.57) cosh kh
where the phase angle v in Eq. (8.57) is introduced for data analyses by Finite Fourier Transform (FFT) algorithms (vide., Chapter 9.2). The wave-induced fluid
nH
I
J f
JVV\.V*\.V*VJ
x SWLX7
1
~t'"7
IWWWW\
Fig. 8.7. Definition sketch for 2-D body in Cartesian coordinates.
664
Waves and Wave Forces on Coastal and Ocean Structures
velocity is computed from the negative gradient of a scalar potential according to q(x,z,t) =
-V®w(x,z,t)
The total linearized fluid pressure (viz., the hydrostatic ps(z) plus hydrodynamic p(x, z, t) in Eq. (4.47c) in Chapter 4.5) is computed from the linearized Bernoulli equation by P(x,z,t) = ps{z) + p(x,z,t) = -pgz + p at
f f coshkh(l+z/h) = — PS\z — Re A — expi(kx — cot — v) [ |_ cosh kh (8.58a) = -pg\z-A [
, ' cosh kh
cos(kx-cot-v)
.
(8.58b)
J
The net pressure loading on the body may be estimated from F(t) = - IJPhdS,
(8.59a)
where the unit normal n is directed out of the body surface and into the fluid and oS is the wetted surface area of the body in Fig. 8.7 at equilibrium at t = 0. The horizontal components of the surface area vector n\ dS in Eq. (8.59a) in the horizontal ±x -directions are nidS = ±dzdyei-
(8.59b)
Substituting Eq. (8.58b) and Eq.(8.59b) into Eq. (8.59a) for the horizontal components of the net hydrodynamic pressure load and integrating over the total vertical surfaces of the body in the horizontal ±x -directions gives the following dimensionless horizontal load component: rr^
Fl(T)
Spg(WBD)(A/h) sin kB sinh kD [1 = ko h sinh kd [1 — — (kh/koh) (kh/koh) coth coth kd] kd] sin(r sin(r + v), kD kB (8.60)
665
Deterministic Dynamics of Large Solid Bodies
where the deep-water wave number ko — co2/g and x = cot. The dimensionless horizontal load in Eq. (8.60) may be verified from the horizontal component of the Euler vector equation (3.41) in Chapter 3.3 that is given by du(x,z,t) dt
dFi(x,z,t),
(8.61a)
where u(x,z,t) =
d
=
COS(KJC — co t — v)
co
(8.61b)
cosh kh
and where du coshkh(l+ z/h) — = Agk sm(kx — cot — v). (8.61c) dt cosakh In order to obtain the net horizontal component, Eqs. (8.61) must be integrated over the fluid volume displaced by the large body at equilibrium Vo in Fig. 8.7 according to Fl(0 = fffdFM = / / / p3-^±dV, (8.62) Vo
Vo
where d V = dxdydz. The dimensionless load from Eq. (8.62) may be verified to be equal to Eq. (8.60). For body dimensions that are small compared to the wavelength A. (i.e., kB, kd and kD « ; 1), Eq. (8.60) reduces to FI(T) « kd(h/d)[k0d - l]sin(r + v).
(8.63)
For a semi-immersed body where d = D, Eq. (8.60) reduces to Fi(r) =
kh sinfcZ? / cosh2fcD - 1 2kD kB \
kh \ smh2fcD sin(r + v), koh )
(8.64a)
and for body dimensions that are small compared to the wavelength X, Eq. (8.64a) may be further reduced to Fi(r) « -kh(k/ko) sin(r + v).
(8.64b)
666
Waves and Wave Forces on Coastal and Ocean Structures
The vertical components of the surface area vector n^dS in Eqs. (8.59a, b) in the vertical indirections are h$dS = ±dxdyes.
(8.65a)
Substituting Eqs. (8.58b and 8.65a) into Eq. (8.59a) for the vertical components of the net hydrodynamic pressure load and integrating over the total horizontal surfaces of the body in the vertical ±z-directions gives the following dimensionless vertical load component: F 3 (T) =
F3(T)
Sy(WBD)(A/h) h sm kB sinh&D = h *o« suiakd [(k/ko) — cotakd] cos(r + v), A kB kD (8.65b)
where h/A = dimensionless Archimedes buoyant load (cf. Eq. (3.47b) in Chapter 3.5). Again, Eq. (8.65b) may be verified from the vertical component of the Euler vector equation (3.41) in Chapter 3.3 as dw(x,z, t) = dF3(x,z,t)-Y, (8.66) ' dt where y = pg and where d
COS(KX — cot — v).
dt cosh kh Integrating Eq. (8.66) over the fluid volume displaced b y the large body at equilibrium according to t) y dV (8.67) F3(,) = ///^3fe Z ,.) = / / / dw(x,z, dt vb v0 that verifies that the dimensionless load from Eq. (8.67) is equal to Eq. (8.65b).
667
Deterministic Dynamics of Large Solid Bodies
For body dimensions that are small compared to a wavelength k (i.e., kB, kd and kD < 1), Eq. (8.65b) reduces to h F 3 (r) « — + kh(kd - k0/k) COS(T + v). A
(8.68)
For a semi-immersed body where d — D, then Eq. (8.65b) reduces to h kh sinkB / , „, , fco^ . , ^, ^ , F 3 (r) = - + cosh2fcD - 1 - — sinh2£D COS(T + v) A 2kD kB \ kh ' (8.69a) and for body dimensions that are small compared to the wavelength k, Eq. (8.69a) reduces to F^ ( t ) «*
h A
(8.69b)
Ko/Z COS(T + V).
8.3.2. Froude-Kriloff Load in Circular Cylindrical Coordinates The relatively large, three dimensional vertical right circular cylinder shown in Fig. 8.8 is submerged with the location of the center of gravity C. G. of the body located at z = —dcG and the point of rotation located at R. Fluid flow separation effects around the bottom are neglected. In order to integrate the incident wave field potential around the circumference of the cylinder, the Cartesian coordinates for the horizontal plane (x, v) must be transformed into the circular
.OgTl
SWL
y
Fig. 8.8. Definition sketch for 3D body in circular cylindrical coordinates.
668
Waves and Wave Forces on Coastal and Ocean Structures
cylindrical coordinates (r, 9) by x = r cos 9 and y = r sin# and the outward pointing unit normal in circular cylindrical coordinates by n = cos 9e\ + sin 9Si. If £ = exp/(# + 7r/2) = exp(/^/2) expi9 = iexpi9 and f = kr, the generating function for Bessel functions Jm(t;) is given by (vide., Eq. (2.57a) in Chapter 2.4.3.1 or Hildebrand, 1976, p. 177)
exp[?(f - l/£)/2] = £ ^ ( 0 that gives the following Jacobi-Anger circular cylindrical coordinate expansion from Eqs. (2.57) in Chapter 2.4.3.1: gi8
e -(6
+
exp ikx = exp ikr cos 9 = exp ikr = ^ « ' m expO'm 9)Jm(kr)
'jfcr = exp
= ^ ( 2 - 5 m0 )J m
T
(*"-£):
cosm9Jm(kr),
(8.70)
m=0
where /«,(•) = Bessel function of the first kind of order m in Eqs. (2.46) in Chapter 2.4.3 and <5mo = the Kronecker delta function in Eq. (2.2) in Chapter 2.2.3. The incident wave potential defined by Eq. (8.57) in Cartesian coordinates may now be expressed in circular cylindrical coordinates by ®w(x,y,z,t)
= ®w(r,9,z,t)
= Re{A<)>w(r,9,z)exp-i(cot
.Ag coshfc/i(l + z/h) expi(kx = Re \i u> cosh kh = Re i
Ag coshM(l + z/h) co
cosh kh
Ag cosh.kh(\ + z/h) co
cosh kh
+ v)}
— cot — v)
exp i (kr cos 9 — cot — v) 00
m+l exp(im9)Jm(kr) Re ^2 i
exp - iicot + v) \
—oo
J
(8.71a) Ag coshfc/i(l + z/h) co
cosh kh (
00
x Re | £ ] ( 2 - Sm0)im+l lm=0
cosm9Jm(kr)
exp - i(cot + v)\.
(8.71b)
669
Deterministic Dynamics of Large Solid Bodies
The total linearized pressure is now given by Eq. (4.47c) in Chapter 4.5 as P(r,8,z,t)
= ps(z) +
p(r,6,z,t)
d
coshkh(l+z/h) cosh kh
m x R e y~] i sxp(im6)Jm(kr)
exp — i(wt + v) (8.72a)
z-A = ~Pg
xRe
coshkh(\+z/h) cosh kh
y ^ ( 2 - Smo)im cosm9Jm(kr) exp — i(cot + v) jn=0
(8.72b) The horizontal component of the surface area vector in Eq. (8.59b) in the xdirection is (8.72c)
«i dS = b cos 9d0 dz ei.
Substituting Eqs. (8.72b, c) into Eq. (8.59a) for the horizontal component of the net linearized pressure load yields the following dimensionless horizontal load component: ^l(T)=
*l(T)
pgnb2d(A/h) 2koh Ji(kb) (kd)(kb)
kh 1 — cosh kd + ( koh
) sinh kd sin(r + v), (8.73a)
where the deep-water wave number ko = co2/g and x = cot. For bodies that are small compared to a wavelength k,kb <§; 1, J\ (kb) ~ (kb)/2 (vide., Eq. (2.58a) in Chapter 2.4.3.2), Eq. (8.73a) reduces to Fi(r)
koh
Id
1 — cosh kd +
kh\
. sinh £ J sin(r + v). si
(8.73b)
koh J
For comparison with the MacCamy-Fuchs (1954) dimensional diffraction load on a full-draft vertical circular cylinder (d = h), the dimensional and dimensionless Froude-Kriloff load on a full-draft vertical circular cylinder where the
670
Waves and Wave Forces on Coastal and Ocean Structures
draft d = h from Eq. (8.73a) is Fi(r) = (
) J^kb)
kbkh
sin(T
+v>
( 8 - 73c >
knh J\(kb) Fl(T) =
Th ~lcir
(T + v)
(8-73d)
"
Similarly, the vertical component of the horizontal surface area vector in Eq. (8.59a) is n3dS = ±rdrdG e3-
(8.74a)
Substituting Eq. (8.72b) and Eq. (8.47a) into Eq. (8.59a) for the vertical component of the net linearized pressure load yields the following dimensionless total vertical load component on the bottom of the cylinder at z = —d: F3(z) =
p gn
_ h_ ~ A
+
b2d(A/h) 2(k0h)Ji(kb)
/
kh\
(kb){kd)
cosh k d — sinh k d cos(r + v). (8.74b)
For bodies that are small compared to a wavelength A. so that tt « Mkb) % (kb)/2 and Eq. (8.74b) reduces to h kh ( fk0h\ \ FI(T) « - H coshkd - - ^ - smhkd COS(T + v). A kd \ \kh J J
1, then
(8.74c)
If the draft d is also small compared to a wavelength A so that kd <£ 1, then Eq. (8.74c) further reduces to h F
3
I' kh
\
( T ) « - + ( — - * o A j c o s ( T + v).
(8.74d)
The moment about the point of rotation R in Fig. 8.8 may be computed from MR(t)
= - / // p(r,9,z,t)l p(r,9,z,>
x ndS,
(8.75a)
where d S for the vertical boundary of a vertical circular cylinder is given by dS = bd6dz.
(8.75b)
Deterministic Dynamics of Large Solid Bodies
671
If the point of rotation R in Fig. 8.8 is located on the vertical axis of symmetry z at r = 0, then the pseudo-direction cosines i x n may be computed from e\ I x n = b cos cos 9
e% b sin 0 sin 0
ej, z + dR 0
= - [ ( z + dfl) sin#]2i + [(z + d«) cos 6>]e2-
(8.75c)
Substituting Eq. (8.72b) and Eqs. (8.75b, c) into Eq. (8.75a) yields only one moment component in the y, ei coordinate direction that is due only to the dynamic pressure component p(r, 0, z, t) in Eq. (8.72b) and that is given by
Afs(f) =
pgAlnbh2 J\{kb) (kh)2 kdR tanh kh — 1 + k (d — dR) cosh kd[tanh kh — tanh kd] sin(o)t + v). (8.75d) + cosh kd(l — tanh kd tanh kh)
A dimensionless moment Ms(r) may be computed from M5(r) M5(r) pgnb d2 (A/h) 2
2khJi(kb) kbikd)2
kdR tanh kh — 1 + k(d — dR) cosh kd[tanh kh — tanh k d] sin(r + v). + cosh kd{\ — tanh &d tanh kh) (8.75e)
If the point of rotation R is located at the draft depth, then dR = d and Eq. (8.75e) reduces to M5(T)
=
2khJi{kb) kb(kd)2 x [kdtauhkh
— 1 + coshfcd(l — tanhfcrftanh kh)] sin(r + v). (8.75f)
672
Waves and Wave Forces on Coastal and Ocean Structures
For comparison with the MacCamy-Fuchs (1954) diffraction moment about the bottom dR = h on a full-draft vertical circular cylinder, the dimensional and dimensionless Froude-Kriloff moments on a full-draft vertical circular cylinder from Eqs. (8.75d, f) are MsW =
pgA2nbh2 J\(kb) ,,,,2 [k0h - 1 + sech kh] sin(cot + v),
(8.75g)
\KTI)
M5(r) =
2Jx(kb) , , , [k0h - 1 + sech kh] sin(r + v). kbkh
(8.75h)
Finally, if the cylinder radius b is small compared to the wavelength k then tt« 1 and J\ «« (kb)/2, and Eqs. (8.75g, h) reduce to pgAnb2 h M5(t) «s — [k0h - 1 + sech kh] sm(cot + v), (kh) [koh — 1 + sech kh] M5(r) « sin(r + v), 7r^-2 (kh)
(8.75i) (8.75J)
where Eq. (8.75j) is independent of the small cylinder radius b.
8.4. Diffraction by a Full-Draft Vertical Circular Cylinder There is a very simple closed-form solution for the diffraction of monochromatic simple harmonic linear waves incident on a vertical circular cylinder of constant diameter that extends over the entire fluid depth d = h in Fig. 8.8 in Sec. 8.3. The reason that the solution is so simple is because the water particle velocity in the radial direction r computed from the incident wave may be completely cancelled by a scattered propagating wave from the homogeneous diffracted wave kinematic body boundary condition (KBBC) on the cylinder by Eq. (8.12b) and, consequently, no evanescent eigenmodes are required to satisfy exactly the kinematic body boundary condition on the vertical cylinder. The theory is credited to MacCamy and Fuchs (1954) and there are also additional solutions for circular cylinders by potential theory that are given by Morse and Ingard (1968), by Mei (1989) and by Ingard (1990). The diffraction boundary Falue Problem (BVP) may be obtained from Eqs. (8.24a-n) in Sec. 8.1. A dimensional scalar diffraction velocity potential Q>D(r,6,z,t) for the full-draft circular cylinder shown in Fig. 8.8 with dR = d = h and a wave
673
Deterministic Dynamics of Large Solid Bodies
period T = 2n/co may be defined by a linear combination of the real parts of a linear incident wave potential
= Re[<j>D(r,9,z) exp - i(cot + v)} = Re{[0 r a (r,0,z) + & ( r , 0 , z ) ] e x p - i ( < w f + v)},
(8.76)
where the phase angle v in Eq. (8.76) is introduced for data analyses by Finite Fourier rransform (FFT) algorithms (vide., Chapter 9.2). The fluid motion may be obtained from the negative gradient of a scalar diffraction velocity potential ®D(r, 9, z, t) according to q(r,6,z,t)
= ur(r,0,z,t)er
+ ud(r,6,z,t)eg
= -VQD(r,0,z,t)
+ w(r,0,z,
t)e3
=Rs{-V4>D(r,0,z)exp-i(cot
+ v)}
= R e { - [ v 0 t u ( r , 0 , z ) + V 0 J ( r , 0 , z ) ] e x p - i ( a > f + v)J, (8.77a) where the three-dimensional gradient operator in circular cylindrical coordinates given by Eq. (2.10b) in Chapter 2.2 with ez = e3 is f 3 .,
vw
=| ^
/ 1 \ 3 _
+
(7)w*
3^1
+
ta«|w-
<8 77b)
'
The total linear pressure field P(r, 6, z, t) may be computed from the unsteady Bernoulli equation in circular cylindrical coordinates with the incident wave pressure given by Eqs. (8.72a, b) in Sec. 8.3 according to P(r,e,z,t)
= p(r,9,z,t) = P
+ Ps(z)
d
Pgz
at
= Re{-ipco
+ v)} - pgz
+ s(r,9,z)]exp — i(a>t + v)} -
pgz. (8.77c)
674
Waves and Wave Forces on Coastal and Ocean Structures
The free surface elevation r](r,9,t) may be computed from Eq. (5.71f) in Chapter 5.3 according to i
d
rj(r,9,t) =
, g
z= 0
dt CO
= Re{-i—4>D(r,8,z)exp-i(cot + v)}, z = 0 g CO = Rel-i-[(/)w(r,0,z) +
z = 0. (8.77d)
The incident wave potential (pw(r,9,z) may be expressed by the FroudeKriloff velocity potential from Eqs. (8.71a,b) in Sec. 8.3 with the Jacobi-Anger expansionfromEqs.(2.57) in Chapter 2.4.3.1 in circular cylindrical coordinates as .Ag cosh k(z + h) (pw(x,z) = i ———expikx, co cosh k h
(8.77e)
A c a ^ . Ag COSh k(z + h) 4>w(r,6,z) = i —— expikr cosd co cosh k h oo
Agcoshk(z -^—^/ + h) ^m + 1,expO'm0)./ m+1 m (fcr) co cosh -oo oo
Ag cosh k(z + h) <^ +1 — — > (2-yim+lcosfflfl4(^), co cosh K « m=0 •*—'
(8.71a) ,.,.., (8.71b)
where 5mo = Kronecker delta function from Eq. (2.2) in Chapter 2.2.3. The BVP for the scattered velocity potential (f>s (r, 0, z) is (vide., Chapter 5.3) V2
r > b, 0 < 9 <2JT,
- h < z < 0,
d
(8.78a) (8.78b) (8.78c)
675
Deterministic Dynamics of Large Solid Bodies
lim
i Kn
\
d
d(j)w(r,0,z) dr
dr
r = b, 0 < 9 <2n, h < z < 0.
(8.78d)
(8.78e)
The solution to the BVP in Eqs. (8.78) may be computed from an orthonormal eigenfunction expansion from the following amplitude modulated circular wavemaker solution in Eqs. (5.86a and 5.79c) in Chapter 5.3: 00
00
(8.79a) where the orthonormal eigenfunctions are (vide., Eqs. (5.78) in Chapter 5.3) cosh Kn(z + h) n = 1,2,3,..., (8.79b) *„(Kn,z) = Nn where the propagating eigenvalue A^ = k for n — 1 and where the evanescent eigenvalues Kn = iKn for n > 2 so that the evanescent eigenmodes are defined from Eq. (8.79b) as Hi l
\
COS Kn(Z
+ h)
(8.79c) rT , n>2, N„ and where the dimensionless normalizing constants Nn are computed from *«(Kn,z) =
tf-/
"0
»*>*.* (i+ £)<((£)
2 k h + sinh 2k h 2Knh + sm2Knh AKnh
n=1 ,
(8.79d)
n > 2 and integer (8.79e)
provided that the eigenvalues k and Kn for n > 2 and integer are computed from koh =
\khtanhkh, n= \ —Knh tanKnh for n > 2,
(8.79f) (8.79g)
676
Waves and Wave Forces on Coastal and Ocean Structures
where ko = co2/g = deep-water wave number. The complex-valued Bessel functions of the third kind or, equivalently, Hankel functions of the first kind H„ (Knr) may be defined as (vide., Eq. (2.49a) in Chapter 2.4.3 or Hildebrand, 1976, p. 146, Eq. 92) H£\Knr)
=
Jm(Knr)+iYm(Knr).
(8.79h)
The orthonormal eigenfunction solution in Eq. (8.79a) satisfies exactly Eqs. (8.78a-d) from the BVP. The unknown coefficients Cmn may be computed in a best-least-squares sense from the inhomogeneous kinematic boundary condition in Eq. (8.78e). Substituting Eq. (8.71b) for w(r,6,z) and Eq. (8.79a) for (ps(r,9,z) into Eq. (8.78e), equating terms of cosm0 and integrating over the dimensionless interval of orthogonality — 1 < z/h < 0 yields *~-mn —
Ag Ni co cosh kh
on\
dJm(kb)/d(kb) dH^\kb)/d{kb)'
(8.80a)
where the derivatives ofthe Bessel functions d(»)/d (kb) are evaluated at the cylinder boundary r = b and are dimensionless derivatives with respect to the argument kb of the Bessel functions. The diffraction velocity potential in Eq. (8.76) is now given by <J>o(r,6,z, t) = ~Re{(pD(r,6,z) exp - i(cot + v)} = Re{[4>w(r, 0, z) +
Ag coshfc(z + h) £](2-S m o)/ m + 1 cos(m0) co cosh kh m=0 Jm(kr) -
dJm(kb)/d(kb) dHJn\kb)/d(kb)
(8.80c)
l
Hi \kr)
Because the vertical circular cylinder of constant diameter lb extends over the entire fluid depth (or, equivalently, the total dimensionless interval of orthogonality — 1 < z/h < 0), only the propagating eigenmode W\(k,z) for n = 1 is required in order to satisfy the zero normal flow kinematic boundary condition in Eq. (8.78e).
671
Deterministic Dynamics ofLarge Solid Bodies The dynamic pressure p(r,9,z,t) in Eq. (8.80c) by Eq. (8.77c) is
p(r,0,z,t)
computed from the diffraction potential
= Re{-( pco^oir ,9, z) exp - i (cot + v)} PSA
coshfc(z + h) s-^ TT^TTT. Z ^ cosh kh
2
~ 8>"0)l COS (mG)
m=0
= Re
Jm(kr)
dJm(kb)/d(kb) dH^\kb)/d(kb)
H$\kr)
exp — / (cot + v) (8.80d)
and the free surface profile r\(r,B, t) in Eq. (8.77d) for the total diffracted wave is r)(r,G,t) =
1 3<& p (r,9,z,Q , R Bt
= Re{-i-(f>D(r,8,0)
z= 0 exp - i(cot + v)},
z-0
^ ( 2 - 5 m 0 ) « m cos (mO) = A Re
m=0
Jm(kr)
dJm(kb)/d(kb) l
//(1)
H^(kr)
dHi \kb)/d(kb)
m
exp — i(cot + v) (8.80e)
where, again, the derivatives of the Bessel functions d(»)/d(kb) are evaluated at the cylinder boundary r = b and are dimensionless derivatives with respect to the argument kb of the Bessel functions. The series in Eq. (8.80e) may be interpreted as a normalized wave profile and applied to compute the wave runup around the circumference of the cylinder (MacCamy and Fuchs, 1954 and Mei, 1989). Isaacson (1978b) extended the relatively simple approximation to Eq. (8.80e) given by MacCamy and Fuchs (1954) for the wave runup on a small cylinder to
V(b,6)
= Aj\
+ (kb)2 2 c o s ^ + l n ( | ) +
( ^ i
(8.80f)
where yE = 0.5772157 = Euler's constant from Eq. (2.41e) in Chapter 2.4.2.
678
Waves and Wave Forces on Coastal and Ocean Structures
Horizontal Pressure Force The horizontal pressure force in the +x direction of the incident wave propagation may be computed by substituting Eq. (8.80d) into Eq. (8.59a) in Sec. 8.3 and obtaining
Fl(t) = -hJ
bdO I pgAhb cosh kh
= Re
p(b,e,^,t\cosOd(^\ Wm(kb)
f
x ;
=Re
exp - i(wt + y) ^
dH^\kb)/d(kb)
cos mO cos Ode /
Jo pgAnhbkoh 2Sml
w
(2 - Smo)i"
m= 0
cosh(kh(l+z/h)d
W(kb)
i exp — i {cut + v)
dH$\kb)/d{kb)
(8.81a)
where the dimensionless Wronskian W(kb) is
Wm(kb) = Jmikb)
dH^ikb) dikr)
-Hml\kb)
dJm(kb)
2i
d{kb)
Ttkb
(8.81b)
that is independent of the 0m mode m; and where the dimensionless derivative of the Hankel function is dH%] (kb)/d(kb)
1 dH^\kb)/d{kb)
exp - ie„ Rmikb)
dHml\kb)/d{kb)
Rmikb) = ^[dJm{kb)/d{kb)f em = arctan
+ [dYm(kb)/d(kb)]2,
~dYm(kb)/d{kb)' dJm{kb)/d{kb)
(8.81c)
(8.81d) (8.81e)
r(2)/ where ffj (kb) = Hankel function of the second kind in Eq. (2.49b) in Chapter 2.4.3. The dependence of the diffracted forces and moments on the coefficient \/{kbR\(kb)) is shown in Fig. 8.9. Substituting Eqs. (8.81b, e) into Eq. (8.81a) reduces the horizontal force F\{t) in the +x direction of the
679
Deterministic Dynamics of Large Solid Bodies
incident wave propagation for m = 1 to
F i W =(
ApgAhbkph \
U(wWcos(
+v 61)
~
= C(W?) cos (cot + v — ei).
(8.82a) (8.82b)
In order to evaluate the contributions of the scattered potential to the total horizontal force, the amplitude of the diffraction force coefficient C(kb) in Eq. (8.82b) may be scaled by the amplitude of the Froude-Kriloff force in Eq. (8.73c) in Sec. 8.3 to obtain C{kb) =
C(kb) (2pg7Tb2Ak0h/kbkh)
Jx {kb)
nkbRi (kb)Ji (kb)
(8.82c)
that is illustrated in Fig. 8.9. The diffraction force amplitude C(kb) in Eq. (8.82b) may then be computed by simply multiplying the dimensionless diffraction force amplitude from Eq. (8.82c) that is illustrated in Fig. 8.9 by the Froude-Kriloff force amplitude in Eq. (8.73c) in Sec. 8.3. There are two significant points that are illustrated in Fig. 8.9. First, the value of the amplitude of the diffraction force Cikb) in Eq. (8.82b) is approximately = 2 for kb -> 0 that is equivalent to the inertia coefficient Cm in Eq. (7.16) in Chapter 7.3 for a circular cylinder. Second, the values of the amplitude of the dimensionless diffraction force amplitude C(kb) inEq. (8.82c) are approximately equal to unity between values of 1.8 < kb < 2.5 in Fig. 8.9; and this implies that the diffraction effects are negligible in this restricted range of values for kb.
Fig. 8.9. Parametric dependence of the MacCamy-Fuchs diffraction force coefficients and phase shift.
680
Waves and Wave Forces on Coastal and Ocean Structures
In contrast to the dimensionless Froude-Kriloff horizontal force in Eq. (8.73c) in Sec. 8.3, the diffraction force in Eq. (8.82a) is phase-shifted from the incident wave phase by the phase angle ei that is defined in Eq. (8.81e). Consequently, the diffraction force per unit length may be compared with the linearized wave force per unit length on small bodies where the linearized drag force per unit length is given by Eq. (7.60) in Chapter 7.6.4. The horizontal water particle velocity of the incident wave evaluated at the vertical centerline of the vertical circular cylinder at x = r = 0 may be computed from Eq. (8.71b) in Sec. 8.3.2 by u(z,t) = Re {
exp — i(cot + v) r=0 dr Agkcoshk(z + h) = cos(oot + v), co cosh kh
(8.83a)
and the horizontal water particle local acceleration of the incident wave evaluated at the centerline of the vertical circular cylinder at x = r = 0 may be computed from Eq. (8.83a) by ut(z,t) =
du(z,t) dt
+ h) = -AgkCOShkiz (8.83b) Sm(cot + v). cosh kh An inertial and a linearized drag diffraction force per unit length may be expressed in a form following Eq. (7.23) in Chapter 7.4 as dFx(
f't) = pnb2 (CMut(z, t) + CDLu(z, t)), (8.83c) dz where CM = an inertia coefficient for a diffraction force per unit length and CDL = a linear drag force coefficient that is due to radiation damping by the scattered wave and not to viscous fluid damping as in Chapter 7. The complex-valued amplitude of the diffraction force per unit length df\(b,z) may be computed by modifying Eq. (8.81a) to dfdb,z) dz
= ipoobf Jo An
-pgA2nb
(pD(b,6,z)cosdd9 , coshk(z + h) — Jm(kb) coshra
ApgAb cosh&(z + . kbR\{kb) coshM
exp - iei
dJm(kb)/d{kb) (1) " HX>(kb) dHln\kb)/d{kb) (8.84a)
681
Deterministic Dynamics of Large Solid Bodies
and the total diffraction force per unit length dF\ (b, z, t) is given by dFx(b,z,t) dz
= Re {I
{
\df\{b,z) exp —i (cot + v + ei) dz 4pgAb coshk(z + h) .,
1
—^-2 — exp -i (cot + v + ei) F kbR\(kb) cosh kh J ApgAb cosh£(z + h) _ . / r kbRi(kb) cosh kh {cos ei cos(cot + v) — sin ei sinfwf(8.84b) + v). Equating Eq. (8.84b) to Eq. (8.83c) and then equating the coefficients of the sin(») and cos(«) terms gives the following force coefficients (Mei, 1989):
4
sin.,
=4<,YIm,
2
n(kb) Ri(kb) =
4co
cos 6i
(kb)2R2(kb)
n
4codMkb)/dm
=
2
n (kb) Ri(kb)
TZ
(kb)2R2(kb)
Again, the analytically-computed force coefficients CM and CDL in Eqs. (8.84c, d) must be distinguished from the empirically-computed inertia Cm and linear drag Cdi coefficients in Chapter 7. In addition, the viscous drag coefficients Cdi are positive-definite, but the radiation damping coefficients CDL are both positive and negative.
Pressure Moment about the Bottom The hydrodynamic pressure moment about the vertical axis of symmetry at r = 0 and z — —h may be computed by substituting Eq. (8.80d) and Eqs. (8.75b, c) into Eq. (8.75a) from Sec. 8.3 with the vertical point of rotation evaluated at J« = h and obtaining
M5(t) = - h 2 f
n
bdO j
p(b,9,^,t\(l
ApgAh2b Ms(t) = ,,,,,.2 2 n /i TN [koh ~ * + kb(kh) Rx(kb)
secn
+ ^1
c o s
^\e2dU (o)t + v — e\)
= C(kb) [k0h - 1 + sech kh] cos (cot + v - e i ) .
(8.85)
682
Waves and Wave Forces on Coastal and Ocean Structures
In order to evaluate the contributions from the scattered potential to the total moment, the amplitude of the diffraction moment coefficient C(kb)[koh — 1 + sechkh] in Eq. (8.85) may be scaled by the amplitude of the Froude-Kriloff moment in Eq. (8.75g) in Sec. 8.3 to obtain, again, Eq. (8.82c); i.e., C(kb) =
C(kb)[k0h- I+ sech kh] (pgA27tbh2Ji(kb)/(kh)2) [k0h - 1 + sechkh]
nkbRi(kb)Ji(kb)' (8.82c)
8.5. Reciprocity Relationships Haskind's Theorem for Wave Forces on Fixed Semi-Immersed Lagrangian Bodies The total 3D velocity potential <£>(x,y,z,t) for the fluid motion with a 3D, semi-immersed Lagrangian solid body may be represented by a diffraction wave potential <&D(X, y, z, t) that is a linear combination of an incident wave potential ®w(x, y, z, t) plus a scattered wave potential <&s(x, y, z, t)(<$>D = 4>w + $?) pl u s a six degrees of freedom radiated wave potential <S>r (x, y, z, t) from Eq. (8.24a) in Sec. 8.1 according to &(x,y,z,t)
= <S>D(x,y,z,t) + <S>r(x,y,z,t) Aw[
= Re
3
+ico ^
6
%j
7=1
&j
7=4
(8.24a) where a> = 2n/T. Each of the spatial velocity potentials
\x\
\y\
-h < z < 0, (8.24b)
|x|
\y\ < oo,
z = —h,
|JC|
\y\ < oo,
z = 0,
(8.24e) (8.24c)
Deterministic Dynamics of Large Solid Bodies
683
where the deep-water wave number ko = co2/g. The scattered potential
= 0,
on the body equilibrium surface o 5
(8.86)
or d
d
(8.87) on the body equilibrium surface QS. dn dn For the radiated O r BVP, the six components of the Lagrangian solid body velocities of the forced oscillations (wavemaker) may be generalized as in Eqs. (8.7c, d) to CO
l©;(0
'b Qi
1,2,3 4,5,6
exp — icot
(8.88a-c) (8.88d-f)
The radiated wave velocity potential cbj may be determined from the kinematic boundary condition on the oscillating Lagrangian solid body boundary oS in six degrees-of-freedom from d
= Ke\co 'Sj
exp — icot:
IPJ J
nj(x,y,z) n'j(x,y,z)
1,2,3 J =
_4,5,6J (8.89a-f)
Each of the six radiated wave velocity potentials cpj (JC, v, z) in Eq. (8.24a) may be computed from the direction itj{x, y, z) or the pseudo-direction n'- (x, y, z) cosines for a Lagrangian solid body oscillating with six degrees-of-freedom (MDOF) according to d
-nj(x,y,z)\ -n'j(x,y,z)\
J =
1,2,3 4,5,6
(8.90a-f)
n\ = cos(9\,x),
surge,
(8.90g)
«2 = cos(^2, y),
sway,
(8.90h)
«3 = cos(#3, z),
heave,
(8.90i)
- z cos(6»2, y),
roll,
(8.90J)
n'5 = zcos(9\,x)
— x cos(^3,z),
pitch,
(8.90k)
n'6 = xcos(02,y)
— ycos(9i,x),
yaw,
(8.901)
n'4 = y
COS(9T„Z)
684
Waves and Wave Forces on Coastal and Ocean Structures
where Oj = angle between the unit normal n and the basis vector gy (vide., Fig. 8.3 in Sec. 8.1). The exciting forces/moments on a fixed solid body may be computed from a surface integral over the immersed solid body at its equilibrium position oS and the hydrodynamic pressure p{x, y, z, t); i.e.,
F/(0 Mf(t)
dS,
I =
1,2,3 4,5,6
(8.91a-c) (8.91d-f)
where the hydrodynamic pressure may be computedfromthe linearized Bernoulli equation by p(x,y,z,t)
=p
d<&l(x,y,z,t) dt
= Re{—picoA^e(x,y,z)
exp — icot} ,
I = w,s, 1,2,... ,6, (8.92)
where Ag = amplitude of the Ith radiated velocity potential <$>i(x, y, z, t). An integral for the hydrodynamic pressure force/moment on a fixed body QS that is independent of the scattered velocity potential cps may be computed from the following volume integral that relates the scattered wave potential
/// VsS/2(t)i ~
^^1dv=°-
(8.93)
Integrating once by parts and substituting Eq. (8.24b) gives: d
/ / 35
d(ps
dS = 0,
(8.94)
where the surface integral is to be evaluated over the following total surface dB that bounds the total fluid volume V shown in Fig. 8.10: dB = Sf + oS + Soo + S-n
(8.95a)
Deterministic Dynamics of Large Solid Bodies
685
Fig. 8.10. Surfaces Sj bounding fluid domain V.
and the integral over the total surface 3 B in Fig. 8.10 may be linearly decomposed into the following sum of integrals:
jj (.)dS = jj (.)z=0 dS + jj (*)dS + jj (.)r^oo dS dB
Sf
Soc.
QS
+ jj(.)z=_hdS
= 0,
(8.95b)
S-h
where the horizontal coordinates r 2 = x2 + y2 and the integrand (•) in each integral is
(•) = (Ps-
0/^—•
(8.95c)
on dn The integral across the horizontal, impermeable bottom boundary S-h vanishes by the bottom boundary condition (BBC) in Eq. (8.24e); so that, accordingly dQj d
//I
dZ
dz
\z=_h
Similarly, the surface integral across the free surface Sf also vanishes because the combined kinematic and dynamic free surface boundary condition (CKDFSBC) given by Eq. (8.24c) requires that d(ps ko
//I
686
Waves and Wave Forces on Coastal and Ocean Structures
so that Eq. (8.94) now reduces to
jfwiS = -//(.),
,dS.
Finally, the integrand on the vertical boundary in the far-field integral at infinity Soo vanishes due to the kinematic radiation boundary condition (KRBC) from Eq. (8.24g) in Sec. 8.1 for n = 2 and for a temporal dependency of exp — /cot; i.e., lim sfkr \ — r-*oo
ik | ^ = 0
\ dr
so that
//{
d
d
dS = 0
Consequently, Eq. (8.94) now reduces to
//{
84>j 4>s-r, on
d
(h-stds=Jf^ds-
(8.96)
The exciting hydrodynamic pressure forces/moments from the (fixed body) diffraction BVP for $D(JC, y, z, t) may be computed from the following surface integral over the fixed body surface boundary QS:
[Mf(f)J-
Re
ipcoAw exp— icot
/ /
{4>w +
[1,2,3,1 [4,5,6 J
\dS
(8.97a-f) The direction cosines m and the pseudo-direction cosines n't are equivalent to the normal derivatives of the radiated wave potential dcpj/dn from Eqs. (8.21 b, c). Substituting fromEqs. (8.21b, c)forthedirectionorpseudo-directioncosinesyields ME(t)\
=i (aAw
P
I
Ww+M-^-dSexp-icot
OS
ipcoAu
exp — icot,
I ••
f 1,2,31 |4,5,6
687
Deterministic Dynamics of Large Solid Bodies
Substituting Eq. (8.96) and the kinematic body boundary condition for the diffraction BVP given by Eq. (8.87) into the second integral gives Flit) Mf(t)
\ =ipcoAw
= iAwTl
II
exp — (icot),
fa
I =
dfao dS exp- - (icot) dn [1,2,31 I4,5,6j )
(8.98a-c) (8.98d-f)
where the generalized exciting force/moment amplitude per unit incident wave amplitude Tl may be computed from the integral
= p(
d(j>t d
n
°ff
/ / / Kv2<^ _4>jV24>*,}dv = j j U J ^ -
dB
where the integral over the entire boundary surfaces dB shown in Fig. 8.10 that bounds the fluid domain V is again given by Eq. (8.95b). Again, the integrals over S-h and Sf vanish by the BBC (Eq. (8.24e)) and the CKDFSBC (Eq. (8.24c)) giving:
//{^-^}'—//{^-^L«The surface integral over QS may now be replaced by an asymptotic surface integral according to •FtHt) \Mf(t)l~
=
—ipcoAy lHW w
"
= —ipcoAwTl
/ /
dfa
exp — (icot),
dcj)w fa-rdn
£ =
1,2,3 4,5,6
dS
exp — (icot)
(8.100a-c) (8.100d-f)
688
Waves and Wave Forces on Coastal and Ocean Structures
that is a remarkable result! The magnitude of the exciting force/moment on a fixed body may be computed without solving for the scattered potential 4>s or integrating over the body surface QS. The radiated wave potentials
8.6. Green's Functions and Fredholm Integral Equations There are three fundamental methods commonly applied to evaluate numerically the wave-induced pressure forces on fixed and floating large solid structures: 1) the Ursell source method (Ursell, 1950), 2) the Schwinger variational method (Black and Mei, 1969 and Black et ah, 1971), and 3) the Fredholm integral equation method (John, 1949 and 1950). Only the Fredholm integral equation method is reviewed in detail because the Ursell source method and the Schwinger variational method are generally restricted to special structural and fluid geometries that are separable in one of the 13 separable coordinate systems (vide., Morse and Feshbach, Chapter 5, 1953). This review of Green's functions for surface gravity waves follows John (1949 and 1950) and may also be found in Mei (1989). Alternative derivations may be found in Stoker (1957, Chapter 9), Wehausen and Laitone (1960) and Newman (1977). An excellent application of singular integral equations that is relatively easy to understand and that is applicable to surface gravity waves is given by Lamb (1932, Sec. 240-244). A very robust analyses of Green's functions with many physical applications is given by Courant and Hilbert (1966, Vol I and 1962, Vol II) and by Stakgold (1979). John (1950) derives four forms for Green's functions for surface gravity waves in both two-dimensions (2D) and in three-dimensions (3D). These eight forms of Green's functions for surface gravity waves may be applied as resolvent kernels in four forms of Fredholm integral equations (Guenther and Lee, 1996, Chapter 7-3). Green's functions and Fredholm integral equations may be considered to be analogous for boundary value problems to the impulse response function and the Duhamel convolution integral for initial value problems (vide., Chapter 2.5.3). Morse and Feshbach (1953, Chapter 7) derive the fundamental singular solutions for the Laplace equation in 2D and in 3D in terms of the position vector Rj(x,xo) shown in Fig. 8.11 where j = 2 for 2D and 7 = 3 for 3D
689
Deterministic Dynamics of Large Solid Bodies
P
o(*o)^-pP(x)
Fig. 8.11. Observer PQ(X, y, z) and source P(XQ, yo, ZQ) point vector definitions in 3D.
position vectors from the source point Po(*o) to the observer point P(x) and where y — yo = 0 for j = 2. The four fundamental Green's functions for 2D and for 3D are reviewed first; and then the four forms for the Fredholm integrals are reviewed.
Green's Functions in 2D (y — jo = 0) The 2D position vector ^2(^,^0) in Fig. 8.11 is given by (8.101a)
/?2(£,£o) = Rx2(x) - J?o2(*o) x = xe\ + zey, R2(x,xti)
xo = xoe\ + zo?3
= (x - xo)e! + (z -
zo)h
Jf 2 (x,*o)| = [(* - xo)2 + (z - zo) 2 ]
(8.101b,c) (8.101d)
1/2
(8.101e)
Morse and Feshbach (1953) derive the fundamental singular solution for the 2D Laplace's equation from the flux of fluid across the arc 0:2 of the circumference of a small disk of radius e in the vertical x-z plane according to 9G2(£,£o) 8r
dSe =
dG2(x,x0) dr
rd9 = o?2,
(8.102a)
where r = radial coordinate in the non-inertial circular cylindrical coordinate axis located at the center of the small disk of radius e. Integrating around the arc a2
690
Waves and Wave Forces on Coastal and Ocean Structures
of the circumference of the disk of radius e yields a2G2(x,xo) = ct2 In |J?2(x,3co)| G2{X,XQ)
(8.102b)
= In |-R2(*,*o)l
(8.102c)
where a2 = the arc on the circumference of the small disk of radius € in 2D. John (1950, p. 100) derives the following 2D integral form of the Green's function for deep-water h -> oo: G2(x,x0) = ]n(\R2(x,xo)\) +
ko + k
-k
cos (k{x — XQ) ) — exp
exp (k(z + zo))
&_
(8.103a)
V where ko = co2/g = deep-water wave number; and where the integral j> (•) denotes a Cauchy principal value integral (Morse and Feshbach, 1953, Chapter 9.2, and Hildebrand, 1976, Chapter 10.6). The corresponding 2D integral equation for finite water depth is (John, 1950, Eq. (A25), p. 99) ln(j?2(*,ico)h +ln^« 2 (Jc,3c 0 )h G2(x,xo) =
f°°
-2lnh•
cosh[k(z+h)]cosh[ic(z0+h)]cos\k\x-xo\\ ko cosh kh — k sinh kh
+
+ exp(kh)
dk k
(8.103b) where the position vector singularity ^2(^,^0) that is reflected below the horizontal bottom at z — — h is defined by R2(x,xo) = (x - xo)ei + (z + 2h + zo)h
(8.103c)
\R2(x,x0)\ = [(JC - * 0 ) 2 + (z + 2h + zo)2]
(8.103d)
and where the linear combination of the two logarithmic terms in Eq. (8.103b) satisfy exactly the kinematic bottom boundary condition (KBC) for 2D surface gravity waves that requires no flow across a horizontal impermeable bottom boundary given by Eq. (8.17b) in Sec. 8.1.
691
Deterministic Dynamics of Large Solid Bodies
Finally, John (1950, Eq. (A2), p. 100) expands the 2D Green's function in the following orthonormal eigenseries ty„(Kn,z/h) (cf, Eqs. (5.30) in Chapter 5.2): G2(x,h)
ij^(KnhrlVn(Kn,z/h)Vn(KH,zo/h)expi(Kn\x-xo\),
= 1
„=i (8.103e)
where *n(Kn,Z/h)
COShK 1+Z/h)
^
;
„ = 1,2,...
2kh+«&2kht
r° Nl = /
=
2
cosh Knh{\ + z/h)d(z/h)
where K\ =k;Kn
=
l
(8.103 g)
->
(8103h)
n =
2lc h
" +™2lc"h,
(8.103f)
n 2
= iKn for n > 2; and provided that
= koh = kh tarih kh =—Knh tan Knh;
n>2.
(8.103 i, j)
g
Green's Functions in 3D] (y—yo ^ 0) The 3D position vector ^3(3c,ico) in Fig. 8.11 is given by Rl(5i,xo) = % 3 ( x ) -i?o 3 tfo) x = xei + yh + zey, R3(x,x0)
= (x-
(8.104a)
*o = ^o?i + yoh + zoh
(8.104b, c)
xojei + (y - yo)h + (z - zo)h
(8.104d)
l*3(*,*o)l = [(* - *o) 2 + (v - yo)2 + (z-
zo) 2 ] 1 / 2
(8.104e)
Morse and Feshbach (1953) derive the fundamental singular solution for the 3D Laplace's equation from the flux of fluid across the sector 0:3 of the surface of a small sphere of radius e in 3D according to 3G3(3c,3c0)
3G3(Jc,Jc0) \r=e dSe =
dr
dr
2 • , . , ,Q \r=e r smfdfdO =013,
, c ,n* x (8.105a)
692
Waves and Wave Forces on Coastal and Ocean Structures
where r = radial coordinate in the non-inertial spherical coordinate system located at the center of the small sphere of radius e. Integrating around the section a3 of the surface of the small sphere of radius e yields (8.105b) R3(X,X0)
G3(x,*o)
1
(8.105c)
/-* -* ' l^3(*,*0)l
where a3 = section of the surface of the small sphere of radius e. John (1950, Eq. (A24), p. 98) derives the following 3D integral form of the Green's function for deep-water h ->• oo:
G3(x,x0) =
_
/»0O
+J
1 \R3(x,xo)\
[(tS)
exp
fa
+ Zo)
) J° [* (\*x,y(Xx,y,xo*j) |]] dk~ (8.106a)
where /o(«) = Bessel function of the first kind of order 0 (vide., Eqs. (2.46) in Chapter 2.4.3) and where the horizontal position vector Rxy (•, •) in the argument of the Bessel function is (John, 1950, p. 93) Rxy(Xxy,xoXj) = (x- xo)et + (y- yo)e2,
\RxAx^'x^)\
= [(* - xo)2 + (y- yo)]
(8.106b)
•
(8.106c)
The corresponding integral for finite depth is (John, 1950, Eq. (A9), p. 93) G3(x,x0) \R3(x,x0)\
+— \R3(x,x0)\
/•oo (^o + k) exp -kh cosh[ic(z + h)]cosh[fc(z0 + h)]Jo (k\Rxj>(xXty,xo )|) +2
(8.106d) where the reflected position vector singularity R3(x,x§) below the horizontal bottom at z — — h is R3(i,x0) = (x- x0)h + (y- yo)h + (z + 2h + zo)h, \h(x,x0)\
= [(x - xo)2 + (y - y0)2 + {z + 2h + z 0 ) 2 ] 1 / 2
(8.106e) (8.106f)
693
Deterministic Dynamics of Large Solid Bodies
and where the linear combination of the modulus of the two inverse position vector terms in Eq. (8.106d) satisfy exactly the kinematic bottom boundary condition (KBC) in 3D that requires no flow across a horizontal impermeable bottom boundary given by Eq. (8.17b) in Sec. 8.1. Finally, John (1950, Eq. (A16), p. 95) expands the 3D Green's function in the following orthonormal eigenseries tyn(Kn,z/h) (vide., Eqs. (5.30) in Chapter 5.2): 00
G3(x,x0) = i |
£(tf B Ar 1 * B (tfi.,z/A)*»(tf»,zo/fc)
x #0(1) (Kn [|jkj, (xXj, *o,,,)|]) ,
(8.106g)
where K\ = k; Kn = iic„ for n > 2; *„(•) are given by Eqs. (8.103f-j); H^\») = Hankel function of the first kind of order zero (vide., Eq. (2.64a) in Chapter 2.4.3.2); and provided that = k0h = khtaahkh = —Knhtan.Knh ,
n > 2,
(8.106h,i)
8 and where (vide., Eq. (2.54) in Chapter 2.4.3) r;r//0(1) (iKn Rxy (xX)y, x 0xj ,)|j = 2 K0 [Kn | ^ j , (xXJh ioX(V) J ,
n > 2, (8.106J)
where Ko(») = the modified Bessel function of the second kind of order zero (or Kelvin function, vide., Eq. (2.54) in Chapter 2.4.3). Loads on vertical, axisymmetric 3D bodies may be computed from 2D vertical plane integrals with an axisymmetric Green's function (Fenton, 1978 and Isaacson, 1982). Hudspeth et al. (1994) discuss in detail the convergence criteria for axisymmetric Green's function series. Fredholm Integral Equations Lamb (1932, Sees. 57-5 8) provides an excellent introduction to singular Fredholm integral equations that is reviewed below. More detailed derivations are given by Morse and Feshbach (1953, Chapter 8) and by Courant and Hilbert (1962, Vol II, Chapter IV). A treatment that is directed specifically toward engineers and physicists is given by Hildebrand (1965, Chapter 3). The Fredholm integral
694
Waves and Wave Forces on Coastal and Ocean Structures
equations reviewed here are boundary surface integrals over all or only part of the boundary surfaces dB that bound the entire fluid domain in either 2D or 3D as illustrated for a 3D domain in Fig. 8.11. The portions of the fluid boundary that are required to compute a scalar velocity potential
xeD
V2 G (x,x0) = amS(x - x0)S(y - y0)S(z - z 0 ),
(8.107a) x, x0 e D.
(8.107b)
where m = 2 or 3. Substituting Eqs. (8.107) into Green's second identity (vide., Eq. (2.123c) in Chapter 2.6) yields
V
dB
A, ff(JG rd(t>\^ = G + Jj{+-to- to)dS'
am
m=
J2for2£> l3for3D,
dB
(8.107c) where am =fractionof an arc on a circular disk about the singular point at x = xo for m = 2 in 2D or the fraction of the surface of a sphere about the singular point at x = xo for m = 3 in 3D. Equation (8.107c) is a Fredholm integral equation of the first kind or homogeneous Fredholm integral equation of the second kind (Guenther and Lee, 1996, Chapter 7-3; Hildebrand, 1965, Chapter 3.1; or Morse and Feshbach, 1953, Chapter 8.3). When the fundamental Green's function from Eq. (8.102c) in 2D or from Eq. (8.105c) in 3D or the deep-water integral Green's function Eq. (8.103a) in 2D or from Eq. (8.106a) in 3D is substituted into Eq. (8.107c), the integral must be computed over the entire boundary surfaces dB because the fundamental Green's function does not satisfy any boundary conditions except for the deepwater integrals where the integrals vanish as z ->• — oo. Black (1975a, b) applies Eq. (8.107c) to compute wave forces on large solid bodies.
695
Deterministic Dynamics of Large Solid Bodies
When the combination of reflected fundamental Green's functions plus an integral over all wave numbers k in Eq. (8.103b) in 2D and Eq. (8.106d) in 3D are substituted into Eq. (8.107c), the surface integral over the fluid boundary dB reduces to an integral over only the immersed surface of a solid body oS that is illustrated in Fig. 8.10. Finally, when the Green's function that is constructed from the orthonormal eigenfunction expansion in Eq. (8.103 e) in 2D or Eq. (8.106g) in 3D are substituted into Eq. (8.107c) as degenerate kernels (Guenther and Lee, 1996, Chapter 7-5), the surface integral reduces to an integral over only the immersed surface of a solid body oS that is illustrated in Fig. 8.10. The integral in Eq. (107c) is not a very efficient integral for computing wave forces on solid bodies. There are two other Fredholm integral equations that require computingfirstan auxiliary function from a Fredholm integral equation of the second kind and then computing the scalar velocity potential from a Fredholm integral equation of the first kind. The derivation that is reviewed here follows Lamb (1932, Sees. 57 and 58). The surface boundary integral over the entire fluid boundary 3 B in Eq. (8.107c) may be reduced to only a surface integral over the immersed equilibrium surface of the solid body QS by requiring that the Green's function satisfy certain boundary conditions on all of the other surface boundaries illustrated in Fig. 8.10. The surface boundary integral over dB may be linearly decomposed into separate surface boundary integrals over each of the four boundary surfaces identified in Fig. 8.10; viz.,
dB
dB
= [[(•)dS + dS+ [[(•)dS+ S-h
Soo
ff(»)dS+ Sf
ff(»)dS.
(8.107d)
0^
The boundary value problems (BVP) for both scattered <j>s and radiated <pr wave potentials are prescribed in Eqs. (8.24) in Sec. 8.1. The boundary conditions for both wave potentials are identical on three of the four surface boundaries in Eq. (8.107d) except for the inhomogeneous Neumann boundary conditions (Hildebrand, 1976, Chapter 9.2) that are prescribed on the boundary surface of the immersed solid body oS. If the Green's function G (X,XQ) is constructed so that it satisfies the same boundary conditions on the three boundary surfaces
696
Waves and Wave Forces on Coastal and Ocean Structures
that both
am
(8.107e)
oS
that may be transformed into a Fredholm integral equation of the second kind when the inhomogeneous Neumann boundary condition Eq. (8.24f) in Sec. 8.1 is substituted for dfa/dn. Lamb (1932, Sees. 57 and 58) derives formulas for two auxiliary functions from Eq. (8.107e) that are more computationally efficient. It is convenient to represent the inhomogeneous Neumann boundary conditions in Eq. (8.24f) on the immersed solid body boundary surface oS by the following generic notation:
dn
=
Uj=\
- ^ f , diffraction, ; = s, rij oxn'j, radiation, ;' = 1,2,3,4,5,6.
(8.107f) (8.107g)
The BVP may be separated into two domains; viz., an exterior problem that is defined inside the immersed solid body QS and exterior to the fluid domain V; and an interior problem that is defined inside of the fluid domain V and exterior to the immersed solid body QS. The potential for the exterior problem is defined asty(x) and the potential for the interior problem as
am
(8.108a)
oS
- / / (
an
dn
where the normal derivatives to the immersed solid body surface o S are defined by
M_-»W. dn
dn
(8,08c)
697
Deterministic Dynamics of Large Solid Bodies
Adding Eq. (8.108a) and Eq. (8.108b) and substituting Eq. (8.108c) yields (Lamb, 1932, Sees. 57 and 58) otm
G (^ V9"
+ ^)dS on J
+ [[ (4, - ir^dS. on JJ0s
(8.108d)
Lamb (1932) identifies two possibilities for computing the potential <j> from (8.108d) that depend on whether the boundary conditions on the surface boundary are inhomogeneous Dirichlet or Neumann conditions (vide., Table 2.3 in Chapter 2.5.1). Each of these two possibilities correspond physically to whether the tangential velocities or the normal velocities are continuous on the body surface. Continuous tangential fluid velocities: Source Density a (Lamb, 1932) If the tangential fluid velocities are continuous on oS, then \jr =
where a = the first auxiliary function. The source density method requires computing numerical solutions to two Fredholm integral equations that are singular integral equations. First, the source density auxiliary function a must be computed from the inhomogeneous Neumann boundary condition on oS given by Eq. (8.24f) / dG umOj -
= Uj,
(8.109b)
that is a Fredholm integral equation of the second kind and where § denotes a Cauchy principal value integral (Hildebrand, 1976, Chapter 10.15). Once the auxiliary source density function a is computedfromEq. (8.109b), then the scalar velocity potential may be computed from Eq. (8.108d) with
j = s, 1,2,3,4,5,6.
(8.109c)
698
Waves and Wave Forces on Coastal and Ocean Structures
Continuous Normal Fluid Velocities: Dipole Source Density /A (Lamb, 1932) If the normal fluid velocities are continuous on QS, then an
an
an
an
by Eq. (8.108c) and the first surface integral in Eq. (8.108d) vanishes. Consequently, the tangential fluid velocities are discontinuous; and the resulting fluid motion is caused by a surface distribution of dipole source densities /A that may be defined by fji, = ^-f,
(8.110b)
where fi = the second auxiliary function. The dipole source density method also requires computing numerical solutions to two Fredholm integral equations that are singular integral equations. First, the dipole source density auxiliary function fx must be computed from the inhomogeneous Neumann boundary condition on 0S given by Eq. (8.24f) / a2G ocmH-j -
(8.110c)
that is a Fredholm integral equation of the second kind and where
Deterministic Dynamics of Large Solid Bodies
699
8.6.1. Orthonormal Eigenfunction Expansion of Green's Function for 2D Wavemaker The 2D wavemaker theory reviewed in Chapter 5.2 focuses on the connection between the wavemaker boundary value problem (BVP) and the radiated wave potential given by Eqs. (8.21) in Sec.8.2 for dynamically responding large Lagrangian solid bodies. Consequently, it is appropriate at this point to construct a Green's function for a planar wavemaker in a 2D channel with the same orthonormal eigenfunctions given by Eq. (5.30) that are derived in Chapter 5.2 from the wavemaker BVP. In order to do this, it is first necessary to construct a BVP for a Green's function in a 2D wave channel. However, at this point it may be instructive to identify an analogy between the Green's function G(X,XQ) as a resolvent kernel in a Fredholm integral equation for a BVP and the convolution kernel h (x, £) (or impulse response function) in the Duhamel convolution integral for an initial value problem (vide., Chapter 2.5.3). The convolution kernel is constructed from the homogeneous solutions to the governing momentum equation for a damped harmonic oscillator in Eq. (2.92a) in Chapter 2.5.3; while the Green's function may be constructed from the homogeneous boundary conditions in a BVP. If the structural properties of a Lagrangian oscillator remain unchanged, the solution to a different loading requires only convolving the convolution kernel with the new loading because the convolution kernel is unchanged. Analogously, if the boundary geometry remains unchanged for a BVP, a solution for a velocity potential for new inhomogeneous boundary conditions may be computed simply by convolving the Green's function with the new boundary conditions in a Fredholm integral equation. The BVP given by Eqs. (8.24b-h) in Sec. 8.1 for either the scattered (f>s or the radiated (pr velocity potential consists of a homogeneous field equation given by Laplace's equation (8.24b) with an inhomogeneous boundary condition given by Eq. (8.24f) on the boundary of the Lagrangian solid body at the equilibrium position oS. The BVP for the Green's function may be constructed from Green's second identity in Eq. (2.123c) in Chapter 2.6. The result is that the BVP for the scalar velocity potential with an homogeneous second-order partial differential equation (Laplace's equation) with inhomogeneous boundary conditions is traded for a BVP for the Green's function with an inhomogeneous second-order partial differential equation (Poison's equation) with homogeneous boundary conditions.
700
Waves and Wave Forces on Coastal and Ocean Structures
It may not be easier to construct a solution for the Green's function BVP than for velocity potential BVP, but once the Green's function is constructed for a particular fluid domain, a solution for the velocity potential with different inhomogeneous boundary conditions on the body boundary may be computed simply by convolving the Green's function with the new boundary conditions in a Fredholm integral equation. Specifically, the focus of the Green's function BVP is on the boundary condition on the wavemaker surface boundary at QS and may compared by V2G = a2S(x - x0)S(z - zo), 3dG G dn - o ,
where U is given by Eq. (5.7e) in Chapter 5.2. The homogeneous Laplace's equation with an inhomogeneous Neumann boundary condition U on the wavemaker surface boundary for the wavemaker BVP is traded for the inhomogeneous Poison's equation with an homogeneous Neumann boundary condition on the wavemaker surface boundary (Courant and Hilbert, 1962, Vol II, Chapter IV or Stakgold, 1979, Chapter 3). The BVP for the Green' function for a wavemaker begins with the governing field equations that is Laplace's equation (8.107a) for the velocity potential and Poison's equation (8.107b) for the Green's function. Substituting Eqs. (8.107) into Green's second identity in Eq. (2.123c) in Chapter 2.6 and expanding the boundary dB over the entire 2D or 3D boundary of the fluid domain leads to Eq. (8.107d) that is given by
a2
= f
on J
JdB
(«)rf€(jc0) + I (*)^(*o) + f
+ f Wdlixo),
(m)d£(x9) (8.107d)
where the dimensions of a2 = [Length] and where the line integrals are integrated over the JCO coordinates. The procedure for constructing the boundary conditions for the Green's function BVP discussed in Sec. 8.6 may now be demonstrated from the four boundary surfaces by substituting the boundary conditions for the velocity potential into the four boundary surfaces on the RHS of Eq. (8.107d).
701
Deterministic Dynamics of Large Solid Bodies
The boundary conditions that are required for the velocity potential are given in the wavemaker BVP in Eqs. (5.7a-e) in Chapter 5.2. The first surface boundary on the RHS of Eq. (8.107d) is integrated across a horizontal, impermeable bottom boundary S-h that is given by /
Ud-^--Gdf)dl{x*)=
f °°4>(^-)
dx0
(8.111a)
by the bottom boundary condition in Eq. (5.7b) in Chapter 5.2. In order to make this integral vanish, the bottom boundary condition for the wavemaker Green's function BVP must be dG — = 0 , 0 < x < + o o , z = -h. (8.111b) dz The second surface boundary integral on the RHS of Eq. (8.107d) is integrated across the free surface boundary Sf that is given by f ( dG d4>\ J U—-G-£\dl(xo)
,
f+0° =]
(dG \
dx0
(8.111c)
by the combined kinematic and dynamic free surface boundary condition in Eq. (5.7c) in Chapter 5.2 where the deep-water wave number is ko = co2/g. In order to make this integral vanish, the combined kinematic and dynamic free surface boundary condition for the wavemaker Green's function BVP must be k0G = 0,
0<x<+oo,z
= 0.
(8.111d)
dz The third surface boundary integral on the RHS of Eq. (8.107d) is integrated across the vertical surface boundary S^ as x ->• +oo that is given by / U^- - G^-) JSoo\ dx dxJx^+00
dl(x0) = f 4>(^-~ J_h \dx
iKnC)
dzo /^+oo (8.1 lie)
702
Waves and Wave Forces on Coastal and Ocean Structures
by the radiation boundary condition in Eq. (5.7d) in Chapter 5.2. The minus sign in Eq. (8.1 lie) corresponds to the assumed negative sign for the temporal dependency of the wavemaker potential
iKnG\=0, J
x -+ +oo, -h < z < 0.
(8.111f)
The fourth and last integral in Eq. (8.107d) is integrated across the wavemaker surface boundary at equilibrium oS that is given by / ( V ^ - G ^ W i c o ) = / U^--Gu] J0S\ dx 8xJ J_h\ dx
Jx
dzo
(8-lllg)
=0
by the kinematic wavemaker boundary condition in Eq. (5.7e) in Chapter 5.2. In order to convert this integral into a Fredholm integral equation for the wavemaker potential, the kinematic wavemaker boundary condition for the wavemaker Green's function must be 9G
„ — = 0, dx and Eq. (8.107d) reduces to
(8.111h)
x = 0, -h < z < 0
a2(p --= - /
(8.1 Hi)
GUdzo-
J-h The BVP for the wavemaker Green's function is now complete and may be compared with the wavemaker BVP in Eqs. (5.7a-e) in Chapter 5.2; i.e., = 0;
V2G = -^8(x-x0)8 h
(-- * h)' \h
d
k0G = 0;
\-h
| 0 < x < +oo, 0 < x < +oo, z= 0
(8.112a,b) (8.112c,d) (8.112e,f)
Deterministic
703
Dynamics of Large Solid Bodies
lim I — -iKn^>\=0; x-»+oo I dz
lim
I
*->-+oo
d(p _ dx
dG dz
iKnG
dG ' dx
[-h < z
x —> + 0 0 ,
= 0;
-h < z < 0, (8.112g,h) (8.112ij)
The alternative to the Poison's equation (8.112b) for the wavemaker Green's function BVP is to solve an homogeneous Laplace's equation for the Green's function and specify a jump and a continuity condition (Stakgold, 1979). Because the boundary conditions for G in Eqs. (8.112d, f) are homogeneous and are identical to the boundary condition for (/> in Eqs. (8.112c, e), the solution for G may be expanded in the orthonormal eigenseries tyn(Kn,z/h) from Eq. (5.30a) in Chapter 5.2 and the jump and continuity conditions may be determined by integrating Eq. (8.112b). The wavemaker Green's function G(x, z : xo, zo) may be expanded in the following orthonormal eigenseries: G(x,z : x0, zo) = YlXr>(x
:x
°> zo)^n(Kn,z/h),
(8.113)
n
where the orthonormal eigenseries tyn(Kn,z/h) from Eq. (5.30a) in Chapter 5.2 is given by Vn{Kn,z/h)
=
cosh Knh{\
+z/h)
n=
l,2,...,
(5.30a)
where the nondimensional normalizing constant Nn is Nj = I cosh2 Knh{\ + z/h)d(z/h) =
2kh+smh2kh 4kh ' 2K„h+sm.2icnh 4K „h '
n = 1,
(5.30b,c)
provided that ^1 = k for n = 1, Kn = IK„ for n > 2, and that the eigenvalues are computed from koh =
khtanhkh, -Knh tan Knh,
where k0h = co2h/g (John, 1949 and 1950).
n=1 n > 2,
(5.30d) (5.30e)
704
Waves and Wave Forces on Coastal and Ocean Structures
Substituting Eq. (8.113) into Poison's equation (8.112b) yields V2G(x, z : XQ, zo) = -rS(x j
~2
~ x°^8 {j, ~ ~h) '
+ K;Xn(x : xo, zo) | Vn(Kn,z/h)
*
n
'
= ^(x-x
0
)*g-^).
(8.114a)
Multiply both sides of Eq. (8.114a) by a member of the orthonormal eigenseries ^m{Km,z/h) and integrate over the dimensionless interval of orthogonality - 1 < zjh < 0 to obtain d Xn(x:x0,zo) dxz
+ K2
x
.^
o^S(x _ h
zo) =
Xo)yn(KnfZo/h)
(8 114b)
Define the solutions for x < xo and for x > xo by the following symbols: X„ forO < x < xo,
X„ forxo < A: < oo.
(8.114c,d)
Integrating Eq. (8.114b) with respect to x, yields the following two ODE BVP's in the x coordinate: a An yx . * 0 . « w
,
K ! V
< , .
xo, zo) = 0,
(8.114e)
*o, zo) = 0,
(8.114f)
dx2
8>X>(x:2X0,zo)+K2K(x dX<(x:x0,zo)=^ dx lim *->-+oo
x = Q^
to-""
Z;CI
(8.114g) xo, zo) = 0.
(8.114h)
The required jump and continuity conditions, respectively, may be obtained by integrating Eq. (1.114b) once to obtain dX>(x : xo, zo) —-— 6x
dX<(x:x0, — ax
zo)
«21I( . „ ... = —Wn(Kn,zo/h), n
X>(x:x0,zo)-X<(x:xo,zo)
= 0,
x = x0,
/on>r\ (8.114i)
x = x0. (8.114J)
The solution to Eqs. (8.114e, g) is X<(x : XQ, zo) = an cosKnx,
(8.115a)
705
Deterministic Dynamics of Large Solid Bodies
and the solution to Eqs. (8.114f, h) is X>(x : xo, zo) = An expiKnx.
(8.115b)
The two coefficients an and An may be determined by substituting Eqs. (8.115a, b) into the jump condition in Eq. (8.114i) and the continuity condition in Eq. (8.114j), respectively, according to an co$K„xo — An expiKnxo -anKn
= 0,
s i n £ „ * 0 - iAnKn expiK„x0
(continuity),
(8.115c)
= — V„(Kn,zo/h), h
(jump). (8.115d)
The determinant of Eqs. (8.115c, d) is the Wronskian of Eqs. (8.115a, b) and is equal to —/. The solutions for the coefficients an and An from Eqs. (8.115c, d) are .a2^n{Kn,zo/h) an = i — Knh
exp i Knxo,
An = i
.a2^n{Kn,zo/h) — cos Knxo Knh (8.115e,f)
and the 2D planar wavemaker Green's function is G(x, z : xo, zo) .^n(^-n,<.0/n)^nK^n,<-/"J .Vn(K n,Z0/h)Vn(Kn,Z/h) Knh i nh ^.Vn(Kn,zo/h)Vn(KKn,z/h) a2 > i —— Knh
E
.„
„
exp i KnXQ cos Knx,
x < xo,
exp i Knx cos K„xo,
x > xo, (8.115g,h)
provided that ^ i = k;Kn = iicn and their values are computed from Eqs. (5.30d, e). The wavemaker potential may now be computed from Eq. (8.111 i) by ,o
c,z) = -— I G(x,z:0,zoW(zo)dzoG(x,z : 0,
(8.116a)
The critical element now for computing Eq. (8.116a) is to determine which of the two Green's functions in Eqs. (8.115g, h) is to be substituted into the integral. Because the Green's functions are symmetric with respect to z and zo, the decision
706
Waves and Wave Forces on Coastal and Ocean Structures
rests with whether x is greater than or less than XQ. Because xo = 0, then x > x$ and Eq. (8.115h) is the correct choice and Eq. (8.116a) reduces to
expiKnxf
Vn(Kn,z0/h)U(z0)d
(^) (8.116b)
The dimensional integral in Eq. (8.116b) is identical to the dimensionless integral Eq. (5.40) in Chapter 5.2 that computes the wavemaker potential coefficients Cn. Again, in an analogy to the convolution kernel in the Duhamel convolution integral for initial value problems in Chapter 2.5.3, the degenerate Green's functions in Eqs. (8.115g, h) may be constructed from a BVP with homogeneous boundary conditions. Once the Green's function is constructed for a given fluid domain, the coefficients for either the scattered or radiated velocity potentials may be easily computed from an integral over the equilibrium surface of the solid body QS for any new values for U.
8.7. Wave Loads Computed by the FEM The geometries of many ocean and coastal structures are not separable in one of the thirteen separable coordinate systems (Morse and Feshbach, 1953, Chapter 5). Consequently, the hydrodynamic wave pressure loads on these structures must be computed by numerical solutions of integral equations. The two most commonly applied numerical methods are the boundary integral Element Method (BIEM) and the finite Element Method (FEM). An example of the BIEM is given by Hudspeth et al. (1994) who compute numerically wave loads on vertically axisymmetric bodies from a Fredholm integral equation with a resolvent kernel that is an orthonormal eigenfunction expansion of an axisymmetric Green's function. Only the FEM is reviewed here. The fundamentals of the FEM are given by Zienkiewicz (1967), inter alios. Analyses by the FEM for wave interference between two horizontal floating circular cylinders in oblique seas is given by Leonard et al. (1983); for wave interference between multiple vertical surface piercing cylinders of arbitrary cross-section shapes with radiation boundary dampers is given by Huang et al. (1985a); and for 3D wave interference between multiple permeable and nonpermeable semi-immersed structures of arbitrary cross-sections with radiation
707
Deterministic Dynamics of Large Solid Bodies
boundary dampers is given by Huang et al. (1985b). These FEM analyses are reviewed below. Because the matrix equation for the dynamic response of a single solid body in six degrees-of-freedom given by Eqs. (8.52) must now be replaced by matrices for NB bodies, it is notationally convenient to assign all six degrees-of-freedom motions and the forces and moments to single variables instead of the two variables previously assigned to the translational and rotational modes and to the forces and moments in Sec. 8.2. The single subscripted variable assigned to the motion modes of the rath body is defined by R e f e e x p - ^ } = (Zj'"W' 1
'
j =
123
j
=4,5,6,
\njn{t),
'''
(8.117a)
where co = 2n/T, where T = wave period and where the complex-valued amplitude £,-„ is defined by £yi, = l£/i,|expi« / l I ,
7 = 1,2,3,4,5,6.
(8.117b)
The Exciting (E) and Restoring (R) forces and moments on the nth body are defined by
where the complex-valued amplitude / • ffnR)
= \ffnR)\
«P»"ft».
is defined by 7 = 1,2,3,4,5,6.
(8.117d)
The three boundary Falue Problems (BVP) for the incident wave potential ^w (x, y, z, t), for the scattered wave potential <E\ (x, y, z, t) and for the radiated wave potential <$>r (x, y, z, t) that are summarized in Sec. 8.1 must now be modified for multiple NB body interactions. The total dimensional scalar velocity potentials for the diffraction wave potential & (x, y, z, t) and for the radiated wave potential <J>r(x, y, z, t) in Eq. (8.15) in Sec. 8.1 are now given by
+
A\
L +££fi£$=i^B(*,:v,z)
J
j (8.118)
708
Waves and Wave Forces on Coastal and Ocean Structures
^ wi\JC\?A
ds,FS n*
Fig. 8.12. Definition sketch of Ng large solid bodies in 3D seas.
where NB = number of large bodies and where each of the lower-case scalar velocity potentials 4>i(x,y, z) for I = w,sn or jn are functions only of the spatial coordinates {x, y, z). For an incident linear wave propagating at an arbitrary oblique angle of incidence a in Fig. 8.12, the spatial incident wave potential 4>w (x, y, z) in Eq. (8.118) in given by cosh k (z + h) 4>w (x, y, z) = — ——— exp / (kx cos a + ky sin a), KQ cosh kh
(8.119a)
where the deep-water wave number &o = &>2 /g and the wave number k = 2n/k is the solution to the linear dispersion equation koh = kh tanh kh
(8.119b)
and where X = wavelength. Each of the dimensional scalar velocity potentials (pi{x,y,z) in Eq. (8.118) must be a solution to Laplace's equation (2.12a) in Chapter 2.2.9 that is given by V 4>i(x,y,z) = A
d24>e(x,y,z) dz2
I = w,sn, jn (8.120a)
and to the following boundary conditions for I = w, sn, jn:
dz — dz
= 0,
kofa = 0,
=
(8.120b)
z = 0.
(8.120c)
z
709
Deterministic Dynamics of Large Solid Bodies
The scattered and radiated scalar velocity potentials must satisfy a radiation condition given by lim Vkr — r^oo [ dr
[ik\
— )(j)i 2r J
= 0,
I =
sn,jn
(8.120d)
where the magnitude of the horizontal position vector is r 2 = x2+y2 inFig. 8.12. The body boundary conditions for the scattered and radiated scalar velocity potentials are ^
+
^ L
dnn
=
0on305„,
d(pjn = rijn ondoSn, dnn \L
"nm)
d
l,2,...,NB
(8.120e)
n,m — 1 , 2 , . . . ,NB
(8.120f)
n=
n,m — 1,2,
...,NB
(8.120g)
where Snm = Kronecker delta function (vide., Eq. (2.2) in Chapter (2.2.3)); and where itjn = the jth direction and pseudo-direction cosines for the nth body that are defined by Eqs. (8.1 and 8.4) in Sec. 8.1. The hydrodynamic pressure field may be computed from
p(x,y,z,t)
=Re
poo L
NB
6
n=\
j=\
exp — icot
(8.121)
where p = fluid mass density. The hydrodynamic exciting load on the fixed nth solid body in the j t h degree of freedom may be computed from R*{/£exp-iarf}
1 = Re poo A
/ / {fiw + ^fak)
\j =
l,...,6,
\n =
l,...,NB,
njndS exp — icot
(8.122a)
710
Waves and Wave Forces on Coastal and Ocean Structures
where 3o5„ = the wetted surface area of the nth solid body and where ff„ = the complex-valued hydrodynamic exciting load coefficient on the nth body in the y'th degree-of-freedom. The hydrodynamic restoring load on the nth solid body in the yth degree-of-freedom may be computed from NB
2
Re{/£exp-ia)f}=Re
6
pco ^2 ^
. .
Hjn / / (pmknjndSn exp - it doS„
f NB 6
(8.122b) where Ffnmk = the complex-value hydrodynamic restoring load coefficient on the nth body in the jth degree-of-freedom due to the motion of the &th body in the mth degree-of-freedom. The restoring load coefficient in Eq. (8.122b) may be separated into real and imaginary components according to F
fnmk
= P™2 / /
+ io)Xjnmk,
(8.122c)
doSn
where \Xjnmk = the added mass coefficient and'kjnmk= the radiation damping coefficient for the nth body in the y'th degree-of-freedom. Applying Green's second identity in Eq. (2.123c) in Chapter 2.6 to the radiation potential yields the following symmetric identities: r
_
R
_
pR
inmk ~ tmkjn^
_
\n,k=\,...,NB,
l^jnmk — H-mkjn, *-jnmk — A-mkjn,
i
[J,m=l,...,6. (8.122d,e) The dynamic responses of each of the NB bodies may be computed from the solutions to the following simultaneous equations of motions: -a)2(mjnmk + fijnmk)-ia)kjnmk
I
6
El
m=l
+ Efcf 1 0 - &kn) [-(02Hjnmk
= ffn,
\
i =
\j =
\'-'
6
'
l,...,NB,
+Ffnmk + Ffnmkj^jn
- lCoXj„mk + Ffnmk + Ffnmk\ $mk
(8-123)
711
Deterministic Dynamics of Large Solid Bodies
where Ffnmk = the hydrostatic stiffness coefficients in the stiffness force/moment loads Eqs. (8.50) in Sec. 8.2 and F
FEM Algorithm Solutions to the 3D BVP's for the scattered and the radiated scalar velocity wave potentials may be computed from the standard FEM. Because boundary dampers are applied in the FEM algorithm, the kinematic radiation boundary condition (KRBC) in Eq. (8.120d) may be applied at only moderate distances from the structures; and all of the closely spaced NB structures are enclosed within this reduced radiation boundary. Radiation boundary dampers, Bettess and Zienkiewicz (1977) The .Kinematic .Radiation boundary Condition (KRBC) in Eq. (8.120d) is applied at a finite distance rp according to
"^\^~(ik~2~r~)(pl}
= 0, l=zsnJn
-
(8 124)
-
Kinematic fictitious bottom boundary condition In order to avoid discretizing the fluid domain below the depth where wave dynamics and kinematics are negligible in deep-water applications, the kinematic bottom boundary condition in Eq. (8.120b) may be modified for the scattered and the radiated wave potentials to the following: d dz
ktanhk(z + h)
tn
= 0,
i = s,j,
(8.120b')
that may be applied at afictitiousbottom z = —hf where hf < h and that reduces to Eq. (8.120b) when z = -h. The FEM solutions may be computed from variational functional that are integrals over the entire fluid domain V, over the free surface boundary dSps, over the radiation boundary surface dSrD, over the bottom surface boundary dSh and over the NB body surface boundaries doS„. The FEM functional for
712
Waves and Wave Forces on Coastal and Ocean Structures
the scattered velocity potential BVP is given by
L
V
dSFs
4>2sdS - - / / ktemhk(z + h)
n(&) =
dS/,
dSr[)
+ :Ln l
- JJ
-<j)sdS
dnn
(8.125a)
and for the radiated velocity potential BVP by
Uff\(°th(^hmV-lff^ SSFS
n (*;„)
-\ fj («'* - ^ j ) 4>)ndS - l- fjkUahkiz + h)4>%dS dSrD
- I
3 S/,
These functionals are discretized into finite elements; and the unknown potentials are expanded in each finite element in a set of nodal unknowns tye and prescribed shape functions N" where q = \,...,NE where Ng = the total number of nodal points in each finite element. A typical element may be denoted by the superscript e. Expanding the functionals in Eqs. (8.125) in the nodal parameters ifr* gives the following combinations of discrete
713
Deterministic Dynamics of Large Solid Bodies
functionals:
n(&) =ni(0,) - n2(&) - n3(&) - n4(&) + n5(&) N
NV
NFS
q=\
q=\
rD
n
= J2 nf (&) - J2 K&) - E n 3^) q=\
N
hf
NB
NNB
q=\
n=\
q=\
(8.126a)
u{(j)jn) =nl{(t>jn) - n2(4>jn) - n 3 ( ^ „ ) - n4(7„) - n5(
NFS
"rD
q=\
q=\
J= l
Nh
f
NB
n
NNB
-E 4<w-EEn5(u n—\
7=1
(8.126b)
o=l
where Afy = the total number of finite elements in the fluid domain V; Nps = total number of surface area elements on the free surface boundary; NrD = the total number of damper area elements on the radiation boundary at ro', Nhf = total number of surface elements on the bottom boundary at z = — h or on the fictitious bottom boundary in deep-water at z = — ft/ and NBE = total number of surface area elements on each of the NB structural boundary surfaces. Each of the five functionals for the scattered velocity potential 4>s{x,y,z) in Eq. (8.126a) is minimized with respect to the nodal parameters tye over the boundaries of each element according to
anf(&) ab
-III
dx
dx
dy
dy
dz
dz
VadV (8.127a)
9n|(&) ab
-SI
dS%
k0NeaNebradS
(8.127b)
714
Waves and Wave Forces on Coastal and Ocean Structures
(8.127c) dSS
rD
+ h)NeaNebfeadS
= ffktanhk(z
(8.127d)
3 st
II —K dS
(8.127e)
a
J Je dnB 3oS„
and for the five almost identical functionals for the radiated velocity potential
HI
ab
-IS!
dx
dx
dy
dy
dz
dz
VadV (8.128a)
= ff' k0NeaNebradS
(8.128b)
as*FS
= ab
.
are
ff(ik-£)NMr°dS
*n L -If
k tanh k(z + h)NeaNebfeadS
(8.128c)
(8.128d)
dSi
dne5(
Hi
ab
= / / * *bNebnjndS.
(8.128e)
doS<
The functionals in Eqs. (8.127a-d) for the scattered potential are identical to the functional in Eqs. (8.128a-d) for the radiation potentials; and only the inhomogeneous kinematic boundary conditions in Eqs. (8.127e and 8.128e) are distinct. The minimization and assemblage of the functional derivatives in Eqs. (8.127 and 8.128) follow the standard porcedures that are outlined
Deterministic Dynamics of Large Solid Bodies
715
in standard FEM texts and references (e.g., Zienkiewicz, 1967). The system matrices for the scattered and the radiation potentials, respectively, are given by [A]{<M = {Us},
[A]{
(8.129a,b)
The system matrix [A] in Eqs. (8.129) is symmetric and banded and is identical for both BVP's. Consequently, this system matrix need be assembled only once for both BVP's. However, this system matrix may be very large for 3D BVP's; and a very efficient Gauss elimination solution method with block form that uses secondary computer storage (Wilson et al, 1974) may be required for some computer systems. Semi-immersed discus buoy Huang, et al. (1985b) computed the dimensionless surge amplitude 2|£i|///; the dimensionless heave amplitude 2|§ 3 |/# and the dimensionless pitch amplitude |£5 \/{H/2b) for a semi-immersed discus buoy by the FEM algorithm given above and compared these three dimensionless motion amplitudes with data from Hoffman et al. (1973) and with the three dimensionless motion amplitudes computed from the BIEM by a 3D source Green's function (Garrison, 1975) and by an orthonormal eigenfunction expansion of an axisymmetric Green's function (Hudspeth et al, 1994). In the FEM algorithm, the fluid domain is discretized into 3D rectangular prism volume elements and 2D
1 .1875b
150.
±
TZ .225b
375b •
175b
h,
fictitious bottom
Fig. 8.13. Dimensions and FEM mesh for a semi-immersed discus buoy where b (h/b = oo;hf/b = 1.0; h\jb = 0.5) (Huang et al., 1985b).
2.0 ft
716
Waves and Wave Forces on Coastal and Ocean Structures
2.0
1.5
1.0
0.5
0. 0. 0.5 1.0
1.5 2.0 2.5
3.0 3.5
Fig. 8.14. Dimensionless dynamic response amplitudes ofa semi-immersed discus buoy (h/b = oo;hf/b = \.0;}n/b = 0.5)(— : 2 | £ 3 | / # & \!-5\/(H/2b) BIEM 3D source Green's function, Garrison, 1975; • & • : 2|f 3 |/ff, & \$5\/(H/2b) experimental data, Hoffman etal, 1973; + , • & • : 2II] \/H,2\h\/H, & \^\/(H/2b) BIEM axisymmetric Green's function, Hudspeth etal, 1994; A, o&D: 2 | ^ \/H,2\^\/H, & \^\/{H/2b) FEM, Huang et al, 1985b).
quadrilateral area elements. Two isoparametric finite elements with quadratic shape functions are applied, viz., a 20-noded prism and an 8-noded quadrilateral. The FEM mesh and buoy geometry are illustrated in Fig. 8.13 where the radius of the discus buoy b = 2.0 ft, the weight mg = 198.7 lb, and the polar mass moment of inertia / = 7.20434 slug-ft2. Cylindrical radiation dampers were applied at a distance rD = 1.3b and a fictitious bottom athf = b where b = radius of the discus buoy. Comparisons with data and the two BIEM algorithms are illustrated in Fig. 8.14.
8.8. Problems 8.1
8.2
Derive a Green's function for each of the two circular wavemakers analyzed in Chapter 5.3 by orthonormal eigenfunction expansion method applied in Sec. 8.6.1. Compute the maximum horizontal force and overturning moment about the bottom on a full depth vertical circular storage tank of diameter
111
Deterministic Dynamics of Large Solid Bodies
8.3
D = 300 ft in a water depth of h = 300 ft from a progressive wave with H = 45 ft and period 7 = 15 sec. The dimensionless roll mode equation of motion for a semi-immersed Lagrangian solid body is ' d2 _ d ~-j~2 +2^ + 2i; + <*4 [ 0 5 ( r ) = A5cojh sin(r - a0 - fa) 5co 5a>5h^r; 5h-rL dx dx where r — cot. Applying the method of undetermined coefficients from Chapter 2.5.6, the dimensionless roll mode rotation @5(r) is assumed to be given by @5(r) = | £251 sin(r - aQ - as). Prove that the dimensionless amplitude Q5 and dynamic magnification factor D [((Ss/,)-2] are \Q5\=B^(co5hr2'JA5co25h;
2
2 2
D[(^r ] = |[i- ( ^r ]
+
(gy
-1/2
and that the sway mode phase angle a$ is tan 0:5 = 8.4
tan /S5 + tan e$ 1 - tane 5 tan£5 J '
tan 65 =
(2£5/<W5fc) [l - (co5h)~2]'
Prove that the added mass coefficient fiji and radiation damping coefficient kji are given by
ff
d
ff
30;
\Ljt = P
8.5
Re{(f)j}—-dS; kjt = pco / / lm{<j)j}—dS. JJ0s dn JJ0s dn In the articulated tower analysis in Chapter 7.8.1, the pitch mode freesurface spring constant K$5 was neglected compared to the pitch mode excess buoyancy spring K^5. Compare the ratio K^5/K^5 of the pitch mode free-surface spring constant to the pitch mode excess buoyancy spring for a large homogenous circular cylinder fixed to the bottom that are given by
*55 = Pgrfy, 5 < 4 = ^ 5 = S^ ~ mg(z'B - l'R)
Chapter 9
Real Ocean Waves
9.1. Introduction Kinsman (1965, p. 386) is careful to point out that a valid specification of real ocean waves must integrate the following three concepts: 1) Fourier and spectral analyses of random processes, 2) probability theories applied to stochastic processes, and 3) hydrodynamics. Techniques from the first two concepts that do not depend on the physics of the hydrodynamic processes are available for analyzing real ocean waves. However, only those techniques from concepts that may be related rigorously to the physics of the hydrodynamics of real ocean waves are reviewed. The theoretical techniques reviewed are applicable to stationary ergodic processes and are limited strictly to short term statics. Isaacson and MacKenzie (1981) give an excellent review of long term statistical and probabilistic techniques applied to real ocean waves. The significance of the stationary ergodic hypothesis is that the ensemble average E[x(t\)] at the same time t\ shown in Fig. 9.1 of an infinite number of finite length time series x\(t\),X2(t\),x-s(t\), ,*oo(^l) is equivalent to the temporal average over all times shown in Fig. 9.2 of an infinitely long single time series X\(t\),X\(t2),Xl{tl),....,X\(tooY,i-S;
E[x(tn)]=
lim — I TR^OO
1R
719
JO
R
xi(t)dt.
(9.1)
720
Waves and Wave Forces on Coastal and Ocean Structures
Xjrft)
-A/l/yAA^JjV/ x3(0
x2(t)
-X1
xtf)
'
!/ *
/
/
Fig. 9.1. Two ensemble averages of finite length records *,- (t) at times t\ and ti (Bendat and Piersol, 1986).
Xrft)
Fig. 9.2. Temporal average of a single time series x\ it) of infinite length.
9.2. Fourier Analyses Definitions of Fourier coefficients Fourier coefficients are defined separately for deterministic and for nondeterministic (or, equivalently, random) analyses.
721
Real Ocean Waves
Deterministic: Fourier coefficients are those coefficients that provide the best least squares fit to the data. Non-Deterministic: Fourier coefficients are those coefficients that explain the contribution by each frequency to the total variance of a random process. When both time and frequency are continuous, the Fourier transform pairs are given by the following integrals:
r+0°
i
r)(t) = —f==. 'In j — /oo J—c +
i
F(a))exp±(i(ot)d(D,
(9.2a)
t](t)exp^(icot)dt,
(9.2b)
r °°
F(a>) = - = V2TT J—OO
where u> = 2nf = radian frequency; where F(co) = Fourier transform of the time series r){t); and where the plus + sign in Eq. (9.2a) must be paired with the negative — sign in Eq. (9.2b). Both choices for the ± signs in the arguments of the exponential functions exp(«) in Eqs. (9.2) may be found in the literature; and the placement of the normalizing constant 2n is also arbitrary (Lighthill, 1964). If the frequency / is given in Hertz, the normalizing constant 2n does not appear and Eqs. (9.2) reduce to
+00
/
F(f)exV±(i2nft)df,
(9.3a)
r,(t)expT(i2nft)dt.
(9.3b)
-00
/+0O -00
Data records that are continuous in time are termed time series; and data records that are digitized to discrete values of time are termed time sequences. Modern Fourier analyses employ discrete/inite Fourier transform (FFT) algorithms that are designed to take advantage of high speed digital computers. Discretization of Eqs. (9.3) requires a finite record length TR and discrete
722
Waves and Wave Forces on Coastal and Ocean Structures
values of both time and frequency; i.e.; t ->• t„ = nAt,
f ->• fm=
mAf,
co -» com = 2nfm =
InmAf, (9.4 a-c)
where At(=T[t/N) and Af(=l/NAt) are constant fixed temporal and frequency intervals, respectively, of N total discrete values of time and frequency. A discrete Fourier transform pair may be approximated from Eqs. (9.2) by N-l
rj(n) = A / ^
F(ra)exp ± {ilizrimAf At),
n = 0,1,2,... ,N - 1,
m=0
(9.5a) N-l
F(m) = At y ^ rj(n)exp ^(UnnmAtAf),
m = 0,1,2,.. .,N — 1,
n=0
(9.5b) where the total number of discrete time and frequency values are equal to ./V; and where the FFT coefficients F(m) are complex-valued quantities. The mean value of the discrete time sequence r\ (n) is given by the real-valued FFT coefficient ^(0); and the FFT coefficient F(N/2) is also real-valued at the Nyquist or folding frequency fy = (N/2)Af in Eq. (9.5a). The discrete positivedefinite frequencies fm > 0 in Eqs. (9.2 and 9.3) are represented by the indices 1 < m < N/2 — 1; while the discrete negative-definite frequencies fm < 0 in Eqs. (9.2 and 9.3) have the indices N/2 + 1 < m < N - 1. The complex-valued FFT coefficients F(m) at these discrete negative-definite frequencies are the complex conjugate values (denoted by superscript asterisks *) of the FFT coefficients at the corresponding positive-definite frequencies; i.e., F(N -m)
= F*(m),
N 1 < m < — - 1.
(9.5c)
Both of the summations of the series of discrete values in Eqs. (9.5) begin with a zero index n = m = 0 that are compatible with modern digital computer algorithms. However, older FORTRAN versions of the discrete algorithm
723
Real Ocean Waves
Eqs. (9.5) did not permit zero indices and Eqs. (9.5) were given by only positive-definite indices m, n > 0 according to N
rj(n) = A / ^ F ( m ) e x p ± ( / 2 ; r ( M - l)(m - 1 ) A / A 0 , n =
l,2,...,N (9.6a)
N
F(m) = At^t](n)exp
=F(/2jr(n - \)(m - \)AtAf),
m = 1,2,...
,N.
(9.6b) The mean value of r)(n) in Eqs. (9.6) is given now by the real-valued FFT coefficient F(l); and the real-valued coefficient F(N/2 + 1) is now at the Nyquist or folding frequency /N = (N/2 + 1)A/. The discrete positivedefinite frequencies fm > 0 in Eq. (9.5a) now have the indices 2 <m< N/2 in Eq. (9.6a); while the discrete negative-definite frequencies fm < OinEq. (9.5a) now have the indices N/2 + 2 < m < N in Eq. (9.6a). The complex-valued FFT coefficients F(m) at these discrete negative-definite frequencies are the complex conjugate values of the FFT coefficients at the positive-definite frequencies; i.e., F(N + 2-m)
= F*(m),
2 < m < —.
(9.6c)
A normalizing constant CN appears in all modern FFT algorithms that is related to the constant discrete intervals At and A / . Just as there is no standard convention regarding the normalizing constant 2n in the transform pair in Eqs. (9.2) for continuous time series (Lighthill, 1964), there is also no standard convention for where to place the normalizing constant CN in the FFT transform pairs in Eqs. (9.5) or Eqs. (9.6) for discrete time sequences; and it must be determined uniquely for each FFT algorithm before applying them. To illustrate how this normalizing constant CN is related to A? and A / , consider the geometric sequence
sN = i + z + z2 + N-l
«=0
3 z
+ ... +
N 1 z
-
724
Waves and Wave Forces on Coastal and Ocean
Structures
that may be reduced to the following compact result: -;\-zN
,
z#l
, N If z is a complex-valued variable given by
z= l
SN
—
\-z
(9.7b) (9.7c)
z = exp ± {i2n(m — m)/N),
(9.8)
where m and m are integer constants; then Eq. (9.7a) becomes N-l
SN = ^
exp[±i2jr(#n -
m)/Nf
(9.9a)
1 — exp ± \i2n{m — m)\ 1 — exp ± [ilTz{m — m)/N]
(9.9b)
n=0
and Eq. (9.7b) is SN
=
N, 0,
m — m = 0, 0 < m — m < N — 1.
(9.9c)
The normalizing constant CN may be determined by substituting Eq. (9.5a) into Eq. (9.5b) with a careful change of dummy indices from m to m and obtaining N-l
N-l
F(m) = AtJ2
A / y ^ F(m) exp ± (ilnnmAf
At)
n=0
x exp == | {ilnnmAf N-l
At) (N-l
= A?A/ ] T F(m) | ] T exp ± (ilnim m=0 N-l
m)nAfAt)
ln=0 N-l
= AtAf J^ F^) m=0
J]{exp
±
(i2n(™ ~ m)AfAt)f.
(9.9d)
n=0
If AtAf = N 1; then the term in curly brackets {•}" = N (5^m by Eqs. (9.9b, c and 9.5b) reduces to F(m) = ( A f A / ) F ( m ) { A ^ m )
(9.9e)
725
Real Ocean Waves
that is possible only if At • Af • N = I, so that Af A / = i
(9.10)
= CN
and the normalizing constant CN is inversely proportional to N. For example, the FFT algorithm in the symbolic software MATHEMATICA™ employs the finite Fourier transform pair Eqs. (9.6), pre-multiplies the complex-valued FFT coefficients F(m) by a normalizing constant CN = s/N and selects the minus (—) sign for the exponential function in Eq. (9.6a) and the positive sign (+) for the exponential function in Eqs. (9.6b). With this convention, Eqs. (9.6) become N
r](n)= ] [ ^ f l ( m ) e x p - ( i 2 j r ( n - l)(m - \)/N);
n=
l,2,...,N, (9.11a)
N
B(m) = */NF(m) = ^
rj(n) exp+(/27r(n - l)(ra -
m = l,2,...,N.
l)/N); (9.11b)
Because Eq. (9.11a) applies the minus (—) sign for the argument in the exponential function exp(«), the complex-valued FFT coefficients B(m) are expressed with a negative (—) sign for the phase by B(m) = \B(m)\ exp - ia(m),
(9.11c)
where a(m) = phase angle at the discrete frequency mAf. Figure 9.3 illustrates the frequency domain representation of the amplitudes (moduli) \B(m)\ in Eq. (9.11c). To illustrate a simple numerical test that may be applied to determine where a particular FFT algorithm places the normalizing constant CN, synthesize the following time sequence that consists of a non-zero mean F(l); first F(2) and fourth F(5) harmonic components with positive and negative phase
Waves and Wave Forces on Coastal and Ocean Structures
726
\B(m)\ ,
I-J I i I U,J_I
I .
t i > i s 1 e • \i|J«J.!!« "•,M,-mHH"'' "
fff
Fig. 9.3. Frequency domain representation of the amplitudes of the FFT coefficients \B{m)\. 11 0.5 , 0
1 i
"-•
"sT -l .
-2 -
1
1
1
1
i
r J! 1 • L I • 1I
i
r
i
i
T
r
J1 1 L I »-H I
•
.
j
r
•
'
J1 • 4_ ! 11 L » L I I -i i I I
i
*
i
i
4
7
10
13
1
n Fig. 9.4. Discrete nondimensional time sequence for N = 16.
angles a(2) and a(5), respectively; and a total number of discrete sequences N = 16 = 2 4 (NOTE: «/w«ys select N = 22M where 2M must always be an even integer so that if the FFT algorithm applies the square root J convention for the normalizing constant CN then the normalizing constant CM will always be a rational number). Consider the discrete time sequence
3
(Inn
TT
+
1
(litAn
C S
n
2 ° U^~4
(9.12)
that is illustrated in Fig. 9.4. A program for synthesizing the normalized discrete time sequence in Eq. (9.12) by the FFT algorithm in MATHEMATICA™ is listed below. The amplitudes \B(m)\ of the complex-valued FFT coefficients are illustrated in Fig. 9.5; and the phases aim) of the complex-valued FFT coefficients are illustrated in Fig. 9.6. Note the change in sign of the phases a(m) in Fig. 9.6 as a result of the sign convention defined for the
727
Real Ocean Waves 5 4 ~^~ 3 i i
-r-i
1 0 7
10
13
16
m Fig. 9.5. Amplitudes of the FFT coefficients for the discrete time sequence r)(ri) in Fig. 9.4.
1
4
Fig. 9.6. FFT phase angles for the discrete time sequence shown in Fig. 9.4.
FFT coefficients in Eq. (9.11c). Table 9.1 lists both the expected amplitudes \F(m) | of the Fourier coefficients without regard for the normalizing constant CN and the amplitudes \B(m)\ from the program MATHEMATICA™. By dividing the values of the amplitudes \B(m)\ from MATHEMATICA™ by the expected amplitudes \F{m)\ of the Fourier coefficients in Table 9.1, it is easy to obtain the normalizing constant CV = V16 = 4. Note that in Table 9.1 that the negative-definite frequencies in the FFT algorithm are stored in the discrete frequencies identified by the indices N/2 + 2 < m < Af. Accordingly, the complex-valued FFT coefficients B(m) for the negative-definite frequencies with indices N/2 + 2 < m < JV are the complex conjugate values B*(m)
728
Waves and Wave Forces on Coastal and Ocean Structures
Table 9.1. Determination of normalizing constant Qy and phase angles a (m) from MATHEMATICA™ FFT algorithm for N = 16 = 4 2 {terms in (•) are the amplitudes and phases of the complex-conjugate FFT values}.
m
\F(m)\
1 2(16) 3(15) 4(14) 5(13) 6(12) 7(11) 8(10) 9
1.0 0.75 (0.75) 0(0) 0(0) 0.25 (0.25) 0(0) 0(0) 0(0) 0
\B(m)\ =
CN\F(m)\
4.0 3.0(3.0) 0(0) 0(0) 1.0(1.0) 0(0) 0(0) 0(0) 0
a(m) (Radians)
CN = JN 4.0 4.0 (4.0)
-(-) -(-) 4.0 (4.0)
-(-) -(-) -(-) -
n -JT/4(+TC/4)
0(0) 0(0) +jr/4(-7r/4) 0(0) 0(0) 0(0) 0(0)
of the positive-definite frequencies with indices 2 < m < N/2; i.e., B(N + 2-m)
= B*(m),
N 2 < m < —.
(9.13a)
The mean of the time sequence is stored in the first real-valued FFT coefficient B(\); and the Nyquist or folding frequency is fy = (N/2)Af and is stored in the real-valued FFT coefficient B (N/2 +1) for 1 < m < N when the positivedefinite index notation of Eqs. (9.6) are applied. Both the mean B(l) and the complex-valued FFT amplitude at the Nyquist or folding frequency B{N/2 + 1) are real-valued because, for the Nyquist frequency at m — N/2 + 1, the complex-valued FFT coefficients B*(N/2 + 1) = B(N/2 + 1) by Eq. (9.13a). For FFT algorithms that employ the positive semi-definite notation for 0 < m < N — 1 given by Eqs. (9.5), the complex-valued FFT coefficients F(m) for the negative frequencies with indices N/2 + 1 < m < N are the complex conjugate values F*(m) of the positive-definite frequencies with indices 1 < m < N/2 — 1 given by F(N -m)
= F*(m),
N 2 <m < —- 1
(9.13b)
Real Ocean Waves
729
and the Nyquist frequency is /N = (N/2)Af. Again, both the mean F(0) and the complex-valued amplitude at the Nyquist frequency F(N/2) are both realvalued, because for the Nyquist frequency m = N/2, F*(N/2) = F(N-N/2) by Eq. (9.13b). The algorithm from the software package MATHEMATICA™ is not optimized and is dated. It is intended only to illustrate one of many possible algorithms that may be applied to determine the normalizing constant CN and the sign of the phase angles a(m) for complex-valued FFT coefficients. Some of the commands illustrated in the program may need to be changed for later versions of MATHEMATICA™. (* PROGRAM DEBUGFFT.MTH FOR DEBUGGING FFT MATHEMATICA *) (* RULES*) TagReal[x_] := (x/: Re[x] = x; x/: Im[x] = 0;) Unprotect[Arg]; Arg[0.0] :=0.; Arg[0] :=0;Protect[Arg]; els :=Run["cls"] (* TIME SEQUENCE w/MEAN AND 2 HARMONICS *) fn :=N[m + al Cos[2 Pi n/npts - Pi/pl] + a2 Cos[8 Pi n/npts - Pi /p2]] f=Table[fn/.{al->1.5,a2->0.5,npts->16,pl->-4.,p2->+4.,m->-l.},{n,0,15}] f=Chop[fJ (* SEQUENCE OF INTEGERS FOR N = 16 *) t=Table[t,{t,0,15}];tf=Table[{t[[i]],f[[i]]},{i,l,16}] tsplot=ListPlot[tf,Frame->True,FrameLabel-> {"n dt","f(n dt)"}, FrameTicks-> {t,Automatic,t,Automatic}, GridLines->Automatic, Prolog->AbsolutePointSize[10]] Display ["ts.plt",tsplot]; (* COMPUTE FFT COEFFICIENTS B(m) *) bnX2=Fourier[fJ;bn=Chop[bnX2];absbn=Abs[bn]; tbn=Table[{t[[i]],Abs[bn[[i]]]},{i,l,16}] bnplot=ListPlot[tbn,Frame->True,FrameLabel-> {"m df","F(m)"}, FrameTicks->{t,Automatic,t,Automatic},GridLines-> Automatic, Prolog- > AbsolutePointSize[ 10]] Display ["bn.plt",bnplot]; phase = N[Table[Arg[bn[[i]]],{i,l,16}]];tph=Table[{t[[i]],Phase[[i]]},{i,l,16}] tphplot=ListPlot[tph,Frame->True,FrameLabel-> {"m df","phase(m)"}, FrameTicks-> {t,Automatic,t,Automatic}, GridLines->Automatic, PlotRange->All,Prolog->AbsolutePointSize[10]] Display ["tph.plt",tphplot]; finv=InverseFourier[bn];finv=Chop[finv]
730
Waves and Wave Forces on Coastal and Ocean Structures
SameQ[finv==f] (* OUTPUT FILE TO BE IMPORTED TO SPREADSHEET "QPRO" *) SetDirectory["C:\lfn\"]; z=Table[ {t[[i]],f[[i]],absbn[[i]],phase[[i]]}, {i, 1,16}] ColumnForm[z];»debugfft.prn
9.3. Ocean Wave Spectra Spectral representations of stationary ergodic random seas are required for engineering analyses of random ocean waves. Several two-parameter theoretical spectral models have been applied to engineering applications along with five- and six-parameter spectral models. The parameters applied in the original derivations of these spectral models are varied. Among the most commonly applied parameters are wind speed Uw, fetch length F, significant wave height Hs and period Ts. For those two-parameter spectral densities, it is convenient for engineering design to replace the original parameters with the zeroeth spectral moment mo (= to the variance a2 of the time series or the area under the spectrum) and the frequency of the spectral peak /o or COQ. The mean (= p,\), the mean square value (= \xi), the variance (= a% = mo) and the standard deviation (= <JX) of a continuous time series may be computed from a stationary ergodic random process by either an ensemble average of an infinite number of finite length records (vide., Fig. 9.1) or from a temporal average of a single record of infinite length TR (vide., Fig. 9.2) or from probability moments /x„(«) by the following (Bendat and Piersol, 1986): +oo
/
=
xnp{x)dx
-00
lim — [ *xn(t)dt, TR-^oo TR JQ
n = l,2,3,...,
(9.14a)
+oo
/ 1
=
f
TR
(X -
pLX)2p(x)dx
-00
->
9
lim — / (x(t) - n\Ydt TR^oo TR JO
= 112- p\,
(9.14b)
731
Real Ocean Waves
where £[•] = an ensemble averaging operator defined in Eq. (9.1); p(») = the probability density function (pdf) for the random variable (•); and 7> = a temporal record that is infinitely long in Eq. (9.1). Data are frequently normalized by subtracting the mean \x\ and then dividing by the standard deviation ax of x(t) in order to obtain the following zero-mean, unit variance data record: =
*(O-MIW
(914c)
Ox
The Wiener-Khinchine Fourier transform pair is similar to the Fourier series transform pair from Eqs. (9.2) but relates the covariance function Cxy(x) to the two-sided spectral density function Gxy(co) (Bendat and Piersol, 1986) according to 1 r+°° Cxy(r) = —— / Gxy(oo)exp ±(ioor)dco, V2TT J—oo
i Gxy(co) = —=
(9.15a)
r+°° I
Cxy(r)exTp^(ia)T)dr,
(9.15b)
V2TT J-OO
where Gxy (co) = a complex-valued, two-sided cross-spectral density function that may be expressed as Gx