and velcoity potential f
is defined as
,where
is gradient operator. It can be shown, the velocity potential satisfies the
Laplace Equation,
Bang-Fuh Chen1 and
Pei-Hong Chen2
1. Professor, 2. Graduate student
Department of Marine Environment and Engineering
National Sun Yat-sen University
Kaohsiung, Taiwan 804
E-mail: chenbf@mail.nsysu.edu.t65
fax:886-7-5255065
Abstract: A time-independent finite-difference method is developed to solve for the fully nonlinear waves past a submerged or floating breakwaters. The wave reducing effect of various combinations of height and length of breakwater are compared and discussed. The effects of surge and heave motions of floating breakwater are studied. The developed finite-difference method can be easily extended to analyze the problems including coupled surge-heave-pitch motions of floating breakwater and wave-structure interaction.
Keywords:
fully nonlinear analysis,
submerged breakwater, floating breakwater
The
tension leg platform, (TLP) has been widely used in ocean engineering, such as
floating oil tank, floating airport and aqua-culture etc. The TLP can also be
used as a floating breakwater to reduce wave height as well as wave forces and
then protect coastal bank against assaulting wave. The tension leg floating
structure is consist of pre-tensioned wire and floating structure. As the wave
action, the floating structure is forced to moving, the motion of the floating
structure will generate wave and change the flow field around structure. The new
flow field would affect the motion of floating structure that will change the
flow field again. The whole process is coupled and the fluid-structure
interaction should be considered in studying the real fluid dynamic phenomenon
of floating structure under real marine time environment.
The
related studies have been made as early as 1969, Black et al. studied the
scattering of wave passing fixed floating and submerged breakwater. They also
discussed the reflection and transition behavior for various water depth and
wave lengths. Black and Mei (1971) used Schwinger¡¯s variational formulation
and Haskind¡¯s theorem study radiation problem and the effects of heave, sway
and roll motions on flow field. Considering surge, heave and pitch motion of
floating body, Yamamoto (1982) used Green¡¯s function and potential function
model the flow filed and studied the flow behavior and corresponding wave forces
on floating body. Their numerical results were validated by experimental
visualization. However, all the above analyses are linear or weakly nonlinear
and the moving boundaries were neglected in the analyses. In order to make a
fully nonlinear and complete 2-dimensional analysis, a time-independent
finite-difference method was developed to study the dynamic responses of
floating breakwater under wave force. The coupled surge, heave and pitch motion
of floating breakwater and the wave-structure interaction are all included in
the analysis. The numerical results were validated by several bench mark studies
and existing reporting results. The wave reducing effect of both submerged and
floating breakwaters were analyzed and discussed.
In the present study, the incident wave is
generated by a piston type wave maker. An incompressible and invicid flow is
considered and the flow is assumed to be irrotational and the velocity potential
can be used to reduce the equations of motion of fluid as a Laplace equation.
The definition sketch of coupled surge, heave and pitch motion under wave action
is shown in Fig. 1. The relationship between velocity
and velcoity potential f
is defined as
,where
is gradient operator. It can be shown, the velocity potential satisfies the
Laplace Equation,
(1)
and also satisfies the Bernoulli¡¯s Equation as
(2)
where p is pressure, r
is fluid density, g is acceleration of gravity. The stroke of the wave maker is
given as S(z,t)=So/2 cos(st),
s=2p/T,
T is the period of wave maker. Thus the horizontal velocity at the wave maker is
(3)
The free surface condition includes kinematic free surface
boundary condition (F.S.K.B.C)
(4)
where
is the elevation of the free
surface and dynamic free surface boundary condition (F.S. D.B.C)
(5)
where u and v are the velocity components in x and z
directions. At sea bed, the impermeable bed is assumed and the normal velocity
at seabed is zero. The Sommerfeld¡¯s radiation condition is used at the
downstream truncate boundary, i.e.,
(6)
where
is wave speed. At the solid surface
of floating body, the fluid velocity normal to the body is equal to the normal
velocity of the floating body.
As
floating body is set to move, the gravity center of the body moves from its
origin position (xo,yo) to (xt,yt).
The fluid velocity normal to floating body is equal to the velocity of the body.
The new position of center of the body and corresponding velocity can be
evaluated by solving equations of motion of the floating body and can be found
in Mei (1983).
In
order to remove the time dependent free surface and moving boundary of the
floating body, proper coordinate transformations are used (Chen, 1994). The
whole computational domain is separated into three parts, as shown in Fig. 1. In
the first system (x1¨Cy1), the right boundary shape is
described by function x1= b1(y1,t), the free
surface profile is indicated by y1= h1(x1,t).
In the second part, (x2¨Cy2), the left and right boundary
are described by x2= b2(y2,t) and x2=
b3(y2,t) respectively. The bottom shape of the floating
body is indicated by y2= h2(x2,t). And in the
third part (x3¨Cy3), the left bound is described by x3=
b4(y3,t), and the corresponding free surface profile is
represented by y3= h3(x3,t). The following
mapping functions are used to transform the irregular computational domain of
each part to a rectangular one:
The
first part:
;
(7)
The second part:
;
(8)
and the third part
;
(9)
By above
transformation, the left boundary will be mapped onto
, right boundary to
, upper boundary to
and bottom boundary to
. Thus all the computational domains are mapped onto a rectangular shape, which
is time-independent. The coordinates
can be further transformed so that the layer near the free surface and
boundaries will be stretched by the proper exponential functions (Chen, 1997).
The variables used in the analysis are normalized as following dimensionless
parameters.
According
to aforementioned transformation and dimensionless variable, the equations and
boundary condition can be rewritten by chain rule and are omitted in the
manuscript. Although the transformed equations are more complicated than their
origin forms, the moving boundary and irregular boundary shapes are mapped onto
a rectangular one and the computational domain is time-independent throughout
the analysis. The developed finite-difference method by Chen is used in the
present analysis (Chen 1994, Chen 1997). Basically, the fluid flow is solved in
a rectangular mesh network of the transformed domain. The staggered grid system
is used in the analysis. The Crank-Nicholson iteration scheme and Gauss-Seidel
Point successive over-relaxation iteration procedures are used in the analysis,
detailed solving procedures can be found in (Chen 1997).
Since the time limit, the results presented in this section are for submerged and floating breakwaters. The separated surge and heave motions of floating breakwater are also studied, while the coupled surge-heave-pitch motion and related fluid-structure interaction are not included in the present report. In the X-Z computational domain, the mesh size DX and DZ are chosen by parametric tests, as shown in Fig. 2, and the results for M = 140 and N=30 are suitable for numerical accuracy and are used in the present analysis. In order to control the mesh size near the free surface and solid boundary, the mesh size is further stretched by setting ak=1 (k=1,2,3) and bk=0.5 (k=1,2,3) to give finer mesh size distribution (Chen, 1997).
The accuracy of the present analysis is validated by made a bench mark study. A reported numerical analysis by Hwang et al. is made first. In this case, the undisturbed water depth is d0=0.376m, the height of submerged breakwater is 0.7 d0, the stroke of piston type wave maker is 3.6cm, the exciting period is 1.85 sec. Fig. 3 shows the comparison between the present numerical results and those of Hwang et al. The present numerical results agree well with those of Hwang with minor difference for the viscous effect was included in Hwang¡¯s analysis.
In the present section, same parameters used in the previous section are used, except that finite length of the submerged breakwater is assumed. To discuss the height and length effects on wave passing submerged breakwater, various combinations of height bH and bL of submerged breakwater are assumed. Fig. 4(a), (b) and (c) are the surface profile at T = 10.77sec for bL=2m, 4m and 6m respectively. Three heights values are assumed, bH=0.3d0, 0.5d0 and 0.6d0. Figure 4 focuses on the effects on wave reducing, as can be seen in the figure, the wave height of waves in the region of breakwater will increase as the height of submerged breakwater increases. But when waves past breakwater, the wave height will decrease as the height of submerged breakwater increases. Comparing Fig. 4(a) to (c), as can be expected, the larger the length of breakwater, the reducing effect is also larger.
Various depth and length of floating breakwater are assumed, the floating breakwater is assumed motionless. The basic parameters are same as previous section, but three depth values are assumed, bd=0.3d0, 0.5d0 and 0.6d0 with bL=2m, 4m and 6m. Fig. 5 plots the surface profile of wave passing floating breakwater, at T = 10.77 sec., as can be seen in the figure, the wave reducing effects of all three depths with constant length are about the same and significant reducing effects are presented when the length of breakwater increases. However, the stability of the floating breakwater is another factor that should be considered to determine the optimal dimension of the breakwater. It can be determined when the coupled surge-heave-pitch motion and fluid-structure interaction are included in the analysis and will be done in the near future.
For future study of the coupled surge-heave-pitch motion and fluid-structure interaction, the effects of separated surge, heave and pitch motions of floating breakwater are studied in the present section. For surge motion effect, the floating breakwater is assumed to be moving harmonically with a constant amplitude but various oscillating frequencies. The amplitude is 3.6cm, T = 1.85sec, 2T and 0.5T are given. Figure 6 depicts the free surface profile of waves before and behind the breakwater. As can be seen in the figure, the surge motion of the breakwater will generate waves and distorts the wave profile before the breakwater, and increase the wave height of the wave behind the breakwater. That is, the breakwater with surge motion behaves as a wave generator, and the wave behind the breakwater is dominated by the surge motion of the floating breakwater. Fig. 7 are corresponding results for heave motion of breakwater, the oscillating speed is constant with two exciting oscillating periods, T = 1.85sec and T = 3.7sec. The wave heights before and behind the breakwater are larger than those of fixed floating breakwater. But the wave heights of both cases, T= 1.85sec and T= 3.7sec are about the same, the wave height behind breakwater of T = 1.85sec is slightly larger than that of T = 3.7sec.
The present time-independent finite difference method can be successfully used to simulate the fully nonlinear waves past a submerged or floating breakwater. The breakwater can be fixed or set in motion. For submerged breakwater, as the length of breakwater increases, so does the reducing effect on wave height. While, for floating breakwater, the wave reducing effects of three depths with constant length of breakwater are about the same. However, the stability of the floating breakwater is another factor that should be considered to determine the optimal dimension of the breakwater. As breakwater set to motion in x and z directions, the floating breakwater becomes a wave generator. For surge motion with constant oscillating displacement, the waves behind the breakwater increase as exciting frequency increases. Similar behavior exists in heave motion of the breakwater, the wave height behind breakwater of smaller exciting period is slightly larger than that larger exciting period. The present finite-difference scheme can be easily used to simulate the couple-surge-heave-pitch motion and fluid-breakwater interaction.
Reference
[1] Black, J. L. and Mei, C. C., ¡°Scattering of surface waves by rectangular obstacles in water of finite depth¡±, J. Fluid Mechanics, Vol. 38, pp. 499-511, 1969.
[2] Black, J. L., Mei, C. C. and Bray, M. C. G., ¡°Radiation and scattering of water waves by rigid bodies¡±, J. Fluid Mechanics, Vol. 46, pp. 151-164, 1971.
[3] Chen, B. F., ¡°Nonlinear hydrodynamic pressure by earthquake on dam faces with arbitrary reservoir shapes¡±, J. Hyd. Res., 32(3), pp. 401-413, 1994.
[4] Chen, B. F., ¡°3D nonlinear hydrodynamic analysis of vertical cylinder during earthquakes. I:rigid motion¡±, J. Engrg. Mech., 123(5), pp. 458-465, 1997.
[5] Mei, C.C., The Applied Dynamics of Ocean Surface, Wiley, New York, 1983.
[6] Yamamoto, T., ¡°Morred floating breakwater response to regular and irregular waves¡±, In Dynamics Analysis of offshore Structures, edited by Kirk, C. L. CML, Southampton, pp. 114-123, 1982.
Fig. 1 The definition sketch and coordinate system
Fig. 2 Free surface profile of (a) various M with N=30, (b) Various N with M=140.
Fig. 3 The comparison of surface profile of waves over an infinte long submerged breakwater, solid line: present results, dash-line: Hwang’s viscous analysis.
Fig. 4 The surface profile of wave over a submerged breakwater with finite eight and length.
Fig.5 The surface profile of wave over a floating breakwater with finite height and length.
Fig. 6 The surface profile of wave past a floating breakwater with surge motion.
Fig. 7 The surface profile of wave past a floating breakwater with heave motion.