(1)
Xiping Yu and Zengnan Dong
Department of Hydraulic Engineering, Tsinghua University, Beijing, China
Abstract: A numerical method for direct computation of wave interactions with a submerged plate is described. Both solid and porous plates are considered. The solid plate is treated as a special case of the porous plate with vanishing porosity. The governing equations of the problem are the continuity equation and the Navier-Stokes equations formulated for flows in porous media but reduce to represent ordinary flows by properly determining the porous parameters. Decoupling of the velocity and the pressure in the numerical scheme follows the classical SMAC method. k-model is employed to deal with the turbulence-induced diffusion of momentum. The VOF function is introduced to describe the free surface. The free-surface capturing in numerical computations is realized through solving the advection equation for the VOF function by donor-acceptor method. Finite volume method is employed for spatial descretization. The computational results demonstrate the existence of downward and upward jets at the downwave end of the plate at particular phases. Formation of clockwise vortex right below the downwave end of the plate is also identified. The plate porosity is found to have negative effects on wave control.
Keywords: water-waves, maritime structure, numerical method
A pile-supported plate submerged a certain depth under the sea surface has been proposed as a structure type for mitigating severe wave attack on coast, i.e., for nearshore wave control since 1970’s. In comparison with other types of structures that can be applied for the same purpose, the submerged plate has a few notable features. It is less dependent on the geotechnical conditions of the sea bottom where the structure is to be installed; its cost performance is high, particularly in relatively deep waters; it adds no negative effects to the marine scenery and will never create a dead water zone where great efforts may be necessary to keep the water quality.
Many research efforts have been directed to the development of effective methods for evaluating the functional performance of, and the hydrodynamic forces on the submerged plate. Hattori and Matsumoto (1977) proposed a patching method for the analysis of wave reflection and transmission based on long wave approximations. Yu et al. (1989) studied short waves with the same method and obtained analytic expressions for wave reflection and transmission coefficients of the plate. Siew and Hurley (1977) worked out an analytic solution on long wave interactions with the submerged plate based on the method of matched asymptotic solutions. Patarapanich (1984) obtained a solution that can be applied to relatively short waves, but it was proved that the solution is a special form of Yu et al. (1989) (Yu and Chwang, 1994). Patarapanich (1984) also investigated the conditions at which minimal transmission occurs. A subsequent study showed that minimal transmission could also be predicted with a semi-empirical formula (Yu et al., 1991a).
In the aforementioned studies, evanescent wave modes that represent the effects of the edges of the plate were omitted. Previous studies that take into account of the evanescent waves fall into two categories. For a horizontal plate, the methods based on patching eigensolutions of the Laplace equation in rectangular domains were shown to be efficient. Yoshida et al. (1990) studied the collocation method. Yu (1995) employed Galerkin method. A comparison of the various patching methods was also carried out (Yu, 1995). Under general situations, particularly when the plate inclination is not small, the boundary integral equation method has been widely preferred (Liu and Abbaspour, 1982; Yu et al., 1991b; Parsons and Martin, 1992).
It has been found that the performance of the submerged plate for wave control, under known conditions of incidence and local water depth, mainly depends on the relative length and the submergence of the plate. Porosity of the plate is also an important factor. Influences of wave nonlinearity, plate inclination within a certain range and boundary layer effects were found to be secondary. The condition under which the reflection coefficient becomes maximal and the transmission coefficient minimal was realized to be when the length of the plate is a little greater than a quarter of the incident wavelength.
In this study, a numerical method for direct computation of wave interactions with the submerged plate is described. Both solid and porous plates will be considered. The solid plate is to be treated as a special case of the porous plate with vanishing porosity. The continuity equation and the Navier-Stokes equations formulated for flows in porous media will be employed as the basic equations. The SMAC method will be applied for numerical computation. The VOF function will be introduced to describe the free surface. The governing equation of the VOF function will be solved by the donor-acceptor method. Finite volume method will be employed for spatial descretization.
Assuming the fluid being incompressible, we have
the following governing equations for the fluid flow of the present interest:
(1)
(2)
where u is the velocity vector, p is the dynamic pressure, E is the rate of strain tensor, r is the fluid density, and ne is the eddy viscosity. By invoking the k-e turbulence model, the eddy viscosity can be expressed by
(3)
where k is the turbulent kinetic energy, e is the dissipation rate of k,
and
is a constant. The standard value of
is 0.09. k and e
are known to satisfy the following
advection-diffusion equations:
(4)
(5)
where
,
,
. The constants may be taken as
,
,
,
.
An extended form of Eqs. (1), (2), (4) and (5) that is applicable to flows within porous media has been derived as
(6)
(7)
(8)
(9)
where g is the porosity of the medium,
is an inertia coefficient representing the inertia effects of the porous
skeleton on the fluid motion (CM
is the added-mass coefficient of the porous skeleton);
is the resistance added to the flow
by the porous medium (CD is
the resistance coefficient and d
represents the geometric dimension of the porous skeleton). At
, Eqs. (6), (7), (8) and (9) reduce to Eqs. (1), (2), (4) and (5), respectively.
It is also worthwhile to mention that, at
,
, and
, the porous medium becomes actually solid.
In this study we deal with Eqs. (6)- (9). By properly choosing the porous parameters, it is expected that problems with arbitrary bottom topography and various kind of structures can be accurately represented following a proper determination of the porous parameters.
For numerical convenience, the volume of fluid (VOF) function F is introduced to express the kinematic free surface boundary condition. F is defined by the volumetric percentage of the fluid in an infinitesimal space at any instance and is governed by
(10)
The dynamic free surface condition can be written as
(11)
where h is the vertical displacement of the free surface from the datum and pa is the atmospheric pressure.
The numerical method we employ to solve (6) –
(9) is essentially SMAC, originally proposed by Amsden and Harlow (1970).
Specifically, we use the following explicit scheme to predict the velocity
vector u at the computational step,
i.e., at
(
is the time increment):
(12)
where
is the predicted velocity,
, and the superscript n indicates the
values at the known time step
. By invoking (12), the continuity equation (6) yields
(13)
where
is a variable introduced to
represent pressure increment from the known to the computational time step. As y is solved, the velocity and the pressure at the computational time
step can then be computed by
(14)
(15)
We apply an asymmetric staggered mesh for spatial discretization. Second-order finite volume method is adopted for the advection-diffusion equations (12), (8), and (9). Equation (13) is discretized with the classical five-point scheme and the resulting finite difference equations are solved by the method of conjugate gradient.
The kinematic free surface boundary condition (10) is discretized with the donor-acceptor method so that the discontinuity of the VOF function, from which the free surface is identified, can be kept traceable.
We apply the method developed above to the study of wave motion around a submerged plate. As sketched in Figure 1, the computational domain is 900m in length and 25m in height. The still water depth is 20m. The left boundary is the wave-making boundary where the velocity is given according to Stokes wave theory. The incident wave is assumed to be 8s in period and 1m in height. The length of the plate is assumed to be 25m. This results in a relative plate length to the incident wavelength being approximately but a little greater than 0.25, very close to the condition at which minimal transmission occurs. The submerged depth of the plate is 4m. For numerical computations, the mesh size is 1m in the horizontal direction and 0.25m in the vertical direction. The time increment is 0.05s.
Fig. 1 Definition sketch of the computational domain
Demonstrated in Figure 2 is the computed water surface elevation at the position 100m separated from the wave generation boundary (the solid line), in comparison with the result given by Stokes wave theory (the dot line) without phase alignment. The difference between the two profiles is seen to be insignificant except for the transient stage. This implies that the computational results are accurate.
Fig. 2 Comparison of computed water surface elevation with theory
Fig. 3 shows the variation of the wave-induced flow around the
submerged plate within one wave period at the steady stage. The displayed area
is 125m in the horizontal direction and covers the whole water depth. Figure 4
is a local magnification of Figure 3 to view the details of the flow near the
downwave end of the plate. It can be noted that, in association with the U-turn
of flows at the downwave end of the plate, downward and upward jets are
generated at
and
, respectively, corresponding to the instances when the wave crest and the wave
trough passing the relevant cross-section. Formation of a
Fig. 3 Wave-induced flow around the submerged plate
clockwise vortex right
below the downwave end of the plate at
is also evident. These phenomena
seem to be important features of the wave-induced flow as the condition of
maximal reflection and minimal transmission is approximately satisfied.
Fig. 4 Details of flow at the downwave end of the submerged plate
Figure 5 shows the flow
around a porous plate. The porous parameters of the plate are given by
,
, and
, while the geometric dimension of the porous skeleton is assumed to be equal to
the vertical mesh size. By a comparison with Figure 3, we note that the wave
height behind the porous plate is significantly larger, although the porous
plate is dissipative while the solid plate does not have such a mechanism. This
confirms that the porosity of the plate has negative effects on wave control. It
is also obvious that nonlinear effects of the plate on the wave profile are very
much reduced as a solid plate is replaced by a porous one.
Fig. 5 Wave-induced flow around the
submerged porous plate
We developed a numerical method for direct computation of wave interactions with a submerged plate. Wave transformation by both solid and porous plates was studied. The solid plate was treated as a special case of the porous plate with vanishing porosity. We used the continuity equation and the Navier-Stokes equations formulated for flows in porous media as the governing equations. The SMAC method was applied in the numerical computations. Turbulence-induced diffusion of momentum was dealt with based on the k-e model. We introduced the VOF function to describe the free surface. The VOF function satisfies the advection equation and was numerically solved by the donor-acceptor method. Finite volume method was employed for spatial descretization. From the computational results we noted that downward and upward jets are generated at the downwave end of the plate when the wave crest and the wave trough passing this position. Existence of a clockwise vortex right below the downwave end of the plate was also identified. The plate porosity was found to have negative effects on wave control.
Acknowledgement
This study is supported by Ministry of Education, People’s Republic of China under the grant for core faculty members in universities and the grant for scholars returned from overseas. Cooperation with Professor Masahiko Isobe, University of Tokyo and Dr. Shigeo Takahashi, Port and Harbor Research Institute, Ministry of Transportation, Japan during the development of the computer program is very much appreciated.
References
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