(1)
E.E. Kriezi1,
Th. V. Karambas1,
P. Prinos2
and C.
Koutitas2
1 Research Associate 2 Professor
Department of Civil Engineering, Division of Hydraulics and Environmental Engineering, Aristotle University of Thessaloniki, Thessaloniki, 54006, GREECE
Tel - Fax: +30 31 995689 , E-mail: kkriezi@civil.auth.gr
Abstract: The purpose of the present work is to investigate numerically the effects of floating breakwaters on wave propagation in shallow waters.Two cases are studied: (a) the floating body is fixed with a certain draft and (b) the floating body is allowed to move vertically (heave motion)
A linear dispersion characteristic Boussinesq model is used for the description of the waves before and after the structure. Under the floating body the continuity equation is valid at the edges beneath the structure. The pressure beneath the breakwater is calculated with respect to the wave pressure, calculated from the wave model, at the edges of the structure. For a fixed floating body the pressure is considered to be linear. In the case of heave motion, the equation of motion is applied into the model in addition to the above mentioned equations for calculating the pressure, which has a parabolic function. Computed results are compared satisfactorily against analytical solutions and experimental measurements of other investigators.
Keywords: floating breakwaters, boussinesq equations, transmission, reflection
Floating breakwaters can be used for coastal protection and restoration in a region with limited tidal mean water variation. Their development and withdrawal in different periods of the year offer dynamic opportunities for full control of the hydromorphodynamic conditions in the coastal zone. They can offer a satisfactory -not high - level of wave attenuation and protection from external wave disturbance. The main advantages of a floating breakwater are: (a) Low cost construction independent of water depth and seabed geological condition (b) Easy transportation, essential for temporary facilities and (c) Ecologically advantageous since the sea water exchange beneath the structure is allowed.
From a practical viewpoint, theoretical reasoning and limited model testing has shown that the most important parameters in determining the attenuation of a floating breakwater are: (a) Relative depth, D/d (D is the draught of the structure and d is the water depth) and (b) Relative width, B/L (B is the structure width and L is the wavelength). Waves with higher wave periods (i.e. longer wavelengths) are less affected by the presence of a floating structure than waves with lower periods.
There is a number of performance studies dealing with the hydrodynamic problem of floating breakwaters in deep water. Linear model and analytical solutions have been developed, which describe the full hydrodynamic problem. (Hwang and Tang 1986, Drimmer et al, 1992, Bhatta and Rahman, 1993). A coupled solution for diffraction and body movement is proposed in order to eliminate the error which occurred by the linear approach of the problem (Fugazza and Natale, 1988). Different models have been studied which calculate the forces and the mooring system of a floating breakwater (Takayama and Moroisi,1984, Yeung and Ananthakrishnan, 1992, Isaacson and Bhat, 1994). A limited number of experimental studies exists. (Sutko and Haden, 1974, Tolba, 1998)
In the present study, a simple depth-averaged numerical model is developed. The model is applied in coastal areas and can describe the wave propagation in shallow waters and the interaction between an incident wave and a floating breakwater as well as the wave-wave interaction between the incident and the radiated wave. The energy dissipation mechanism has been simulated and the hydrodynamic pressures and forces on the structure have been tested against analytical solutions of Stoker (1957) and experimental measurements of Tolba (1998).
The main function of a floating breakwater is to attenuate the wave action. Such a structure can not stop all the wave action. The incident wave is partially transmitted, partially reflected and partially dissipated. Energy is dissipated due to damping and friction and through the presence of eddies at the edges of the breakwater. Due to wave energy the object can be put in motion and a radiated wave is produced which is propagated in both direction - offshore and nearshore. The movement of the body is specified in terms of the anchoring which defines the degrees of freedom of the breakwater.
The development of the numerical model it studied step by step in order to understand thoroughly the physical problem. A rectangular breakwater is used which is: (a) rigid and fixed in one position (zero degrees of freedom) and (b) allowed to do a heave motion (one degree of freedom).
For the development of the numerical model a Boussinesq type wave model is used. The model is applied in shallow waters for which d/L<0.3 and mainly can describe low frequency waves (wave period around 6-10 secs). In addition, higher frequency waves (with period less than 4 secs) can be described by the same model. The floating breakwater with its movement produce radiated waves that have frequencies in the same order, as the resonant frequencies of the construction (period should be around 1-2 sec). Thus a dispersive wave model with improved linear characteristics is required in order to describe both low and high frequency waves and their interaction. This wave model can also describe a sequence of physical phenomena that occur in the surf zone.
The equations proposed by Schaffer and Madsen (1995) are adopted which take the following form in one -dimension:
(1)
(2)
where u is the depth-averaged velocity and ζ is the surface elevation. The subscripts t and x indicate derivatives in time and space respectively. The celerity is calculated by the following expression (Schaffer and Madsen, 1995).
(3)
The parameters α, β, and γ are related with the linear dispersion characteristic and take the values –0.305, 0.039 and 0.101respectively.
At the inlet an incident wave condition with free passage of reflected waves is applied. The incident wave surface elevation is given as a periodic sinusoidal compulsion and the reflected is described by the following formula.
(4)
At the outlet of the computational domain a free propagation boundary condition and a sponge layer condition are applied (Larsen and Dancy, 1983) where velocity and elevation at a distance xs from the boundary are divided by a coefficient μ(x):
(5)
where βs
is a constant dependent on the grid points in the xs and defines the
reflection on the boundary.
The floating breakwater is considered as a rectangular rigid body that is partially immersed and fixed. It has zero degree of movement and hence the draught is constant (D) and the width in x axis is B. The length in y axis is considered as infinite (Fig. 1). For simulating the flow beneath the body we adopt the following considerations: (a) The continuity equation (mass conservation) beneath the edges of the breakwater is valid and is applied on the boundaries 1-1 and 2-2 (Fig. 1) and (b) the pressure is varied linearly underneath the floating breakwater. The pressure at the two boundaries of the breakwater 1-1 and 2-2 are calculated from the wave model.
Thereafter, the velocity beneath the object is constant and calculated by the momentum equation applied at the edges of the breakwater
or
(6)
where PR and PL the pressures left and right of the breakwater as they calculated from the following equation.
(7)
The above assumptions are valid when the length, related with the wavelength, is small and as it is deduced from experimental studies (Suko and Haden, 1974, Hwang and Tang 1986) the ratio B/L should be between 0.1-0.4.
A significant amount of energy is dissipated due to eddies that occur in front and behind the breakwater. These are calculated with the use of an artificial eddy viscosity at the right part of the momentum equation (eq. 2) for a length Led, the length of the eddies.
,
(8)
where
is the eddy-viscosity which is a
function of the eddies length and is defined as:
(9)
The length Led is proportional to the draft D (Led = K D). The constant K is a non-dimensional parameter which is given by the following relation:
(10)
where
is the velocity amplitude beneath
the breakwater and T is the wave period. The relation between the rate K and d/L
is given in fig. 2. For the calibration of K comparison with experiments (Tolba
1998) was used. The forces (horizontal
and vertical), applied on the breakwater, are calculated with the integration of
the hydrodynamic pressure on the body.
Fig. 1 Definition Sketch
Fig. 2 Dimensionless number K versus d/L
The heave motion of the structure is introduced in the model with the use of the dynamic equation. The dynamic equation of motion is applied in the center of gravity of the body.
(11)
where m is the mass of the body, hb is the vertical displacement of the structure.
The vertical forces are W, Bu and Rst where W is the weight, Bu the buoyancy and Rst the resistance due to the motion of the body in the water, (Fig.1). The latter is calculated by the following formula:
,
(12)
where S the submerged
body surface,
the vertical velocity of the breakwater, hb the vertical displacement
and Cd the draft coefficient which is considered equal to 2. Buoyancy
is calculated from the pressure variation beneath the body:
(13)
The momentum and continuity equation beneath the breakwater are used in order to calculate the pressure variation beneath the body
(14)
(15)
where,
(16)
and
the depth under the body in the
balance condition and without wave action
The analytical form for the pressure variation, which is deduced from the above equation, is:
(17)
where,
(18)
The function ho
is used instead of
(eq. 7) while hot and hott
are the first and second derivatives respectively. Integration of
eq. (13) gives the buoyancy
force.
(19)
The velocity beneath the breakwater is calculated by the momentum equation with the use of the pressure variation expression.
(20)
(21)
The momentum and continuity equations (eqs. (14) and (15)) are coupled in the numerical solution of the model and give new boundary conditions at the edges of the breakwater in every time step.
For the validation of the prescribed model
computed results are compared against analytical solutions, other numerical
results and experimental measurements. The analytical solution of
Stoker (1957) for semi-submerged body was used (Fig. 3). The transmission
and reflection coefficients are:
, and
respectively, where
, dund = h-D and h is the depth (Stoker 1957, Tolba 1998) (Fig.3).
The analytical solution considers that there is no energy loss and hence the
model was applied with
. It is shown that computed and analytical results for both coefficients are in
close agreement.
In addition, the reflection and transmission coefficients are compared with the experimental data of Tolba, 1998 (Fig. 4). Again, comparison between computed and experimental data is satisfactory for d/L ranging from 0.09 to 0.22 and D/d equal to 1/6.
The computed transmission and reflection coefficient and the amount of energy dissipation versus the wave parameter B/L are shown in Fig. 5. Forces, deduced from the model, are compared with the forces from Drimmer et al (1992) in Fig. 6. The model is shown to underestimate slightly both vertical and horizontal forces.
In
the case of the heave motion several tests have been performed. The reflection
and transmission coefficients of the breakwater have been tested with the
experimental data of Tolba, 1998 (Fig. 7). It is shown that model predictions
agree very well with the experimental
Fig. 3 Computed and analytical Ct and Cr coefficients (fixed breakwater)
Fig. 4 Computed and experimental Ct and Cr coefficients (fixed breakwater)
Fig.5 The effects of relative width on wave characteristics
Fig. 6 Computed horizontal and vertical
values of Ct and
Cr. The variation of the vertical forces with time have been tested
in the case where there are no incident waves and the body is forced to a
sinusoidal vertical motion,
. Computed results are compared with those of Yeung and Ananthakrishnan (1992),
who studied the vertical oscillation of a floating body in a viscous fluid due
to a force perturbation with the Navier-Stokes equations.
Fig. 7 Variation of Ct and Cr with D/d
Fig. 8 Computed forces due to hydrodynamic pressure
The data used for the comparison
are
, dund/B = 0.5 and a/B = 0.1 or 0.2 where
is a non-dimensional
parameter related with the motion frequency. The numerical
model overestimates the amplitude of the forces in this case (Fig. 8). The
simulation of the radiation wave is presented in fig. (9) in the case of zero
incident wave and the breakwater is put in motion due to an initial vertical
perturbation.
Fig. 9 Radiated wave with time for zero incident wave
Fig. 10 Standing and propagated waves (heave motion)
In the case of the interaction of incident wave and a floating breakwater with one degree of freedom (heave motion) Fig. (10) shows the partially standing wave offshore and the partially propagated wave nearshore). The variation of vertical displacement hb with wave (d/L) and body (D/d) characteristics is shown in Figs. (11) and (12) respectively. It is shown that the ratio hb/Hinc decreases with increasing d/L while it increases with increasing D/d. Both variations of hb/Hinc are physically correct but further against experimental measurements is required.
Fig. 11 Variation of vertical displacement of a floating breakwater with d/L
Fig. 12 Variation of vertical displacement of a floating breakwater with D/d
In this study a numerical model has been developed for describing the interaction between waves and a floating breakwater. The model is based on a linear dispersion characteristic Boussinesq model for the description of the waves before and after the structure and is able to calculate the energy dissipation and the hydrodynamic pressures and forces on the breakwater. The following conclusions can be derived from the above study:
(1) For both fixed and moving (heave) breakwater computed transmission and reflection coefficients are shown to agree well with analytical and experimental results odf other investigators.
(2) Computed forces for a fixed breakwater are shown to be underestimated with regard to the ones calculated analytically by other investigators.
(3) The amplitude of forces is overestimated slightly in the case of breakwater forced to a sinusoidal vertical motion with no incident waves.
(4) From a practical viewpoint it is deduced that for a breakwater with geometrical characteristics (D/d = 0.3-0.4 and B/L =0.2-0.4) and wave characteristics d/L<0.3, the transmission coefficient is varied between 0.5 to 0.3 which is an acceptable coefficient related with the function of such a structure.
References
Bhatta D.D., Rahman M. (1993), “Computational methods and experimental measurements to a floating cylinder in waves”, Proc. Computational modelling of free and moving boundary problems, pp. 395-402.
Drimer N., Agnon Y. , Stiassnie, M. (1992), “A hybrid boundary element method of second order wave-body interaction”, Applied Ocean Research, 16,pp. 27-45.
Fugazza M., Natale L. (1988), “Energy losses and floating breakwater response”, Journal of Waterway, Port, Coastal and Ocean Engineering, vol.114, pp. 191-205.
Hwang C., Tang F.L.W., (1986) “Studies on rectangular surface barrier against short waves”, Proc. 20th Int. Conf. on Coastal Engineering, ASCE, pp. 1915 - 1928.
Isaacson M., Bhat S. (1994), “Wave force on an horizontal plate”, International Symposium: waves - physical and numerical modelling , pp. 1184 – 1193.
Larsen J., Dancy H.(1983) “Open boundaries in short waves simulation - A new aproach”, Coastal Engineering vol. 7 pp.285 – 297.
Schäffer H.A., Madsen P.A. (1995) “Further enhancements of Boussinesq-type equations”, Coastal Engineering, 26, pp 1-14.
Stoker J.J., (1957), “Water Waves”, Interscience Publishers Inc., New York.
Sutko A.A. , Haden E.L., (1974), “The Effect of Surge, Heave and Pitch on the Performance of a Floating Breakwater”, Proceedings of Floating Breakwater Conference, Rhode Island, pp 41 -53.
Takayama T., Moroisi K. (1984), “Motion and mooring force of an axisymmetric floating bodies”, Coastal Engineering in Japan, vol. 27, pp. 265-277.
Tolba E.R.A.S. (1998) “Behaviour of floating breakwaters under wave action”, Ph.D thesis, Suez Canal University.
Yeung W., Ananthakrishnan P. (1992) “Oscillation of a floating body in a viscous fluid”, Journal of Engineering Mathematics 26, pp 211-230.