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OCR for page 513

Interaction between Current, Waves and Marine Structures
R. Zhao and 0. M. Faltinsen
Norwegian Institute of Technology
Trondheim, Norway
Abstract
A theoretical method to analyse current wave
interaction on large-volume marine structures
is presented. It is shown how to circumvent
the problems associated with the mj-terms.
Numerical results for mean wave forces on
floating vertical cylinders are discussed.
The results are based on using both direct
pressure integration and conservation of
momentum.
Introduction
Zhao & Faltinsen [9] have presented a two-
dimensional theory that hydrodynamically ana-
lyses the combined effect of waves and
current on two-dimensional floating struc-
tures. Zhao et al. [10] generalized the
theory to three-dimensional flow. A
hemisphere was analysed and satisfactory
agreement between numerical and experimental
prediction of linear and mean wave forces was
documented.
The theory is based on matching a local solu-
tion to a far-field solution. In the far-
field the waves "ride" on the undisturbed
current velocity, while in the near-field the
waves "ride" on the local steady flow. The
theoretical solution for the velocity poten-
tial is expressed as a series expansion in
the wave amplitude (a and the current velo-
city U. The problem is solved to first order
in (a and first order in U. It is assumed
that the wave slopes of different wave
systems and the Froude number are asymp-
totically small. In the free surface con-
dition and the body boundary condition the
interactions with the steady motion potential
are taken care of. In addition a radiation
condition is specified. In the numerical
solution a boundary element method based on
Green's second identity is incorporated. The
far-field solution is represented by a sum of
multipoles (including sources) with singu-
larities inside the body. The multipoles
satisfy the radiation condition and the far-
513
field free surface condition. For a general
body several singularity points are used for
the multipoles inside the body. The coef-
ficients in the multipole expansion are
determined by matching the local- and far-
field solution.
In the main text we will outline the theory
in more detail. We will focus our attention
on the m -terms in the body boundary con-
dition. The mj-terms are due to interaction
between the current and the oscillatory fluid
motion. The same terms occur when ship
motions at forward speed are analysed (Newman
[8]. Numerical inaccuracies and unphysical
effects of these terms can cause large errors
in the numerical prediction, in particular
for a body surface with sharp corners or
local high curvature. It is shown numerically
and analytically how problems with the mj-
terms can be avoided.
Numerical results for horizontal and vertical
mean wave forces on floating vertical cylin-
ders of finite length are presented. Both a
direct pressure integration method and a
method based on conservation of momentum are
used. Convergence of the results are docu-
mented as a function of number of panels
approximating the body surface, the free sur-
face and the control surface. It is
demonstrated that special care has to be
shown in modelling the cylinder surface
around sharp corners. This is particularly
true when the direct pressure integration
method is used.
Theory
Consider a structure in uniform current and
regular incident waves with small amplitude
in deep water. The structure is restrained
from drifting, but is free to oscillate har-
monically in six degrees of freedom. We
choose a right-handed coordinate system
(x,y,z) fixed in space. z = 0 is in the mean
free surface, positive z-axis is upwards and
goes through the centre of gravity of the
structure when the body is at rest (see Fig.
1). Surge (~1), sway (~2) and heave (~3) are

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~3
- _=
with respect to (a, ¢2 is proportional to
(a. Far away from the body
'~2 As - U(x case + y sing)
~1
Fig. 1 Coordinate system and sign convention for translatory and
angular displacements.
the translatory displacements of the body in
the x-, y- and z-direction referred to the
origin of the (x,y,z) coordinate system, when
the body is at rest. Roll (~4), pitch (~5)
and yaw (~6) denote the angular displacements
about the x-,y- and z-axis, respectively.
Consider the fluid to be incompressible and
the fluid motion irrotational so that there
exists a velocity potential ~ which satisfies
Laplace equation.
v2~= 0
(1)
In reality the flow is always rotational in a
boundary layer close to the body. In addition
the flow may separate from the body and inva-
lidate a potential flow description also in
parts of the flow outside the boundary layer.
This depends for instance on the shape of
body, the Reynolds number, the Keulegan-
Carpenter number (KC), a non-dimensionalized
frequency of oscillation, the roughness ratio
and the ratio between the current velocity U
and the maximum horizontal oscillatory
ambient fluid velocity UM in the current
direction. Obviously the flow will always
separate from a body with sharp corners in
any type of ambient flow. This is also true
for the flow around any blunt-sharped marine
structures in current without waves. However,
when the free stream velocity along the
current changes direction with time, i.e.
U/UM < 1, the flow around bodies with curved
surfaces will not separate for small KC-
numbers (Zhao et al. [10]).
The potential flow solution will be written
as a series expansion in the wave amplitude
(a of incident waves. It is assumed that (a,
the body motion and the steepness of the dif-
ferent wave systems are asymptotically small.
We write
~ = As + ¢1 + ¢2 (2)
where As is independent of (a, ¢1 is linear
Here a is the angle between the current
direction and the x-axis. We assume that the
current velocity U is small and solve the
problem correctly to O(U). A consequence of
neglecting terms of o(U2) is that we disre-
gard the effects of the steady wave system
generated by the current flow past the body.
In practice this is expected to be a good
approximation. On the free surface, As
satisfies the rigid free surface condition,
'.e.
a¢S
- = 0 on z = 0 (4)
On the mean position of the body surface SB,
As satisfies a zero-normal velocity con-
dition, i.e.
a¢S
an = 0 on SB (5)
In general a numerical method has to be used
to find As In our case we used Hess &
Smith's method [2]. This is based on distri-
buting sources over SB.
Due to linearity we can decompose the first-
order velocity potential ¢1 into separate
components due to the rigid-body motions ski
the incident waves and a diffraction poten-
tial theist. We can write
¢1 = woe t + Ageist + ~
ok (6)
k=1
The incident wave potential can be written as
At
tOe =
9
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By using the dynamic and kinematic free sur-
face conditions, it follows that Ski k = 1,6
satisfy correctly to O(U)
a2¢S a2¢S
k sieves V.k ~ id ~ 2 ~ - ]
atk
+ gaz = 0, on z = 0
all)
Also (~0~67)e;~t satisfies equation (10). Far
away from the body equation (10) becomes
2 ask ask
ok ~ 2ieU (cos a a + sin a ay )
O. on z = 0
(11)
Equation (10) expresses the fact that the
waves interact with the local steady flow As
around the body. Equation ( 11 ) resembles the
classical free-surface condition with forward
speed, which for a = 0 can be written as
2 ark 2 amok
ax
ask
gaz = 0 , on z = 0
(12)
The difference between equation (11) and (12)
is that terms of o(U2) are neglected. Zhao &
Faltinsen [9] found for two-dimensional flow
that it is appropriate to use equation (11)
when T = WU/9 ~ ~ O. 15. In the three-dimen-
sional flow case we expect a similar limita-
tion. An example on calculations of the real
part of the velocity potential due to a har-
monically oscillating source satisfying
either free surface condition (11) or (12)
is shown in Fig. 2. The curves in the upper
half plane correspond to calculations with
equation (11). The T-value is 0.1,
s ~/9 = 0.6773 where zs is the z-
coordinate of the source point. The calcula-
tions in Fig. 2 are for points on the mean
free surface. We note a small phase dif-
ference between the wave systems when con-
dition (11) or (12) is used. This is more
evident on the upstream side of the source.
The amplitudes are in good agreement. The
consequence of neglecting terms of o(U2) in
the free surface condition is that fewer wave
systems occur far away from the body.
However, the consequence of this is of no
practical importance for small T-values below
0.15. A further simplification of the source
potential G is sometimes used. One writes
G = G. ~ T aG | +
U
Fig. 2 Calculated values of the real part of the Green's function on the
free surface. In the upper half part of figure equation (11)
is used. In the lower half part the classical free surface con-
dition (12) is used (T ' 0.1, me; ~ 0.6773)
( The ca l cu l a t i ons i n the l over ha l f p l ane has been prov i deaf by
J. Hoff).
This simplification can lead to large errors
at some distance from the source point. This
is illustrated in Fig. 3. The conditions are
the same as in Fig. 2. The curves in the
upper half plane correspond to that equation
(13) has been used, while the curves in the
lower half plane corresponds to that the free
surface conditions (11) has been used.
Fig. 3 Calculated values of the real part of the Green's function on the
free surface. In ache upper half part of the figure equation (13)
has been used. In the lower half part free surface condition (11}
has been use ~ T ~ 0.1, ~j - O.6773 ~ .
515