Flow+Field+Analysis+of+Submerged+Horizontal+Plate+.pdf

China Ocean Eng., Vol. 27, No. 6, pp. 821 – 828 2013 Chinese Ocean Engineering Society DOI 10.1007/s13344-013-0067-z, ISSN 0890-5487 Flow Field Analysis of Submerged Horizontal Plate Type Breakwater* ZHANG Zhi-qiang 张志强a, b, LUAN Mao-tian 栾茂田a, b and WANG Ke 王 科c, 1 a State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China b Institute of Geotechnical Engineering, Dalian University of Technology, Dalian 116024, China c State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China Received 15 May 2012; received revised 2 November 2012; accepted 10 March 2013 ABSTRACT Submerged horizontal plate can be considered as a new concept breakwater. In order to reveal the wave elimination mechanism of this type breakwater, boundary element is utilized to investigate the velocity field around plate carefully. The flow field analysis shows that the interaction between incident wave and reverse flow caused by submerged plate will lead to the ation of wave elimination area around both sides of the plate. The velocity magnitude of flow field has been reduced and this is the main reason of wave elimination. Key words boundary element ; plate type breakwater; flow field analysis 1. Introduction Owing to the capable of maintaining water exchange, the floating breakwater has important practical engineering interests in the coastal erosion area, aquaculture area, and the peripheral protection area of deep-sea plats. Therefore, the study of its hydrodynamic characteristics has aroused increasing attention in recent years. The main design concept of this type breakwater is the attenuation of part of the wave energy, which can keep the huge wave forces from acting directly on the marine structures. Besides, the free water exchanges behind the breakwater can keep the sea water clean and marine ecosystem uninfluenced. Many mathematical derivations and experiments have been done in the study of the hydrodynamic characteristics of floating breakwater. Parson and Martin 1992 established a high order singular integral equation, and studied the diffraction of the submerged horizontal plate by solving the discontinuous velocity potential on its both sides. Neelamani and Reddy 1992, Yu and Chwang 1993 solved the diffraction problem of the horizontal plate in finite water depth using the eigenvalue approximation in finite region. Hu et al. 2002 presented a two-dimensional analytical solution to study the reflection and transmission of linear water waves propagating past a submerged * This research is supported by the Fundamental Research Funds for the Dalian University of Technology Grant No. DUT10LK43 and the National Key Basic Research Program of China Grant No. 2013CB036101. 1 Corresponding author. E-mail kwang ZHANG Zhi-qiang et al. / China Ocean Eng., 276, 2013, 821 − 828 822 horizontal plate and through a vertical porous wall. Usha and Gayathri 2005 considered the scattering of surface waves by a submerged, horizontal plate or disc, by using eigenfunction expansions within the finite domain. Zheng et al. 2007 studied the hydrodynamic coefficients and wave exciting forces for long horizontal rectangular and circular structures by boundary element . Dong et al. 2008 studied wave transmission coefficients of floating breakwaters by experiment. Liu et al. 2009 investigated the hydrodynamic perance of a submerged two-layer horizontal plate by the matched eigenfunction expansion . However, up to now, no clear and direct elaboration has been made about wave elimination mechanism from the viewpoint of flow field analysis. Therefore, the present study adopts the boundary element to solve the diffraction problems of the submerged horizontal plate and obtain the distribution of velocity field around the plate. The numerical result confirms that this is efficient to the analysis of the velocity field around submerged horizontal plate. In Section 2, mathematical models and numerical scheme concerning the velocity field are elaborated. Section 3 provides the numerical results and discussion. Conclusion is drawn in Section 4. 2. Coordinate System and Theoretical ulas 2.1 Governing Equation and Boundary Conditions Fig. 1 shows the definition of Cartesian coordinate system oxyand plate location. This coordinate system is stationary and related to the undisturbed position of the free surface. The origin o is located on free surface, positive x-axis is from left to right and y-axis is positive upward. The incident wave is propagating from right to left along x-axis. Besides, plate length is 2Ba, plate thickness is TT and plate submergence is S H. It is also supposed that the plate is rigid and thin, and normal vector is positively pointing out of the fluid domain. Fig. 1. Calculation sketch of the submerged horizontal plate. The fluid is assumed to be non-viscous, incompressible and irrotational. Supposing the motion of the object is harmonic oscillation, then, the fluid velocity can be expressed by the gradient of velocity potential Φ and Φ can be defined as i , ; Re[ , e] t x y tx y    , where i Re[ , e] t x y    denotes the real part,  is the frequency of incident wave, t is time, i1, and  is the spatial complex velocity potential irrelevant to time. The velocity potential  satisfies Laplace equation and related boundary conditions Faltinsen, 1991; Wang et al., 2011a. ZHANG Zhi-qiang et al. / China Ocean Eng., 276, 2013, 821 − 828 823 Based on linear assumption, complex velocity potential  can be decomposed as follows 3 ID 1 i ii i x    , 1 where i x is the motion amplitude of the object, I  is the incident potential, D  is the diffraction potential, SID  is the scattering potential and i  is the radiation potential for sway, heave, and roll, respectively. The boundary condition for diffraction and radiation potentials is DI 123 , ,, i n nn nnn     , 2 where, 123 ,, and n nn are the components of normal vector on the object surface and 1x nn, 2y nn, 300 xy nyy nxx n, 00 ,xy being the rotating center of the object. The incident wave potential I  can be expressed as i I i e KyKx gA     , 3 where,  is the wave circular frequency, 2 K g   is the wave number, g is the acceleration of gravity, and A is the amplitude of incident wave. 2.2 Boundary Integral Equation The hydrodynamic of submerged horizontal plate can be solved by establishing integral equation on the object surface with Green’s theorem. Following is the boundary integral equation about i  and D  , d ,d SS G P QQ CPPsG P Qs nn       , 4 where D or i P, , P x y is field point, ,Q  is source point, ,G P Q is Green’s function, and C is solid angle. ,G P Q can be expressed as follows Wang et al., 2011b  12C ,lglg2G P QrrI. 5 where  2222 22 12 , rxyrxy; 6  C 0 0 ecos[ ] lim i d u y u x I uK u            . 7 2.3 Flow Field Velocity If  represents scattering potential S , radiation potential R123 ,,    and total velocity potential  respectively, the velocity of flow field , u v induced by  can be obtained through the following equation ZHANG Zhi-qiang et al. / China Ocean Eng., 276, 2013, 821 − 828 824 , uv xy    , 8 where p at any point p can be solved by source point qas follows d , d Gq pqsG p qs nn        . 9 Supposing the fluid domain is discritized into finite elements, the velocity of any point in the fluid domain can be calculated as    1T i i i N u J vN                   , 10 where [ ]J is Jacobian determinant, i N is shape function, and and  are local coordinates. 3. Flow Field Analysis For further revealment of the wave elimination mechanism of plate type breakwater, this section explores the velocity of water particles in the whole process of wave elimination. In the calculation, the most effective wave condition is chosen when the plate location is S 0.05 mH, T 0.005 mT and 0.4 mBWang et al., 2011a. Besides, three kinds of incident waves / 20.4, 0.8, 1.2KB are selected and the amplitude of incident wave is 0.02 m. The sway velocity field is small due to the plate thickness, the result is not discussed here. In following results, figures ‘a’ and ‘b’ are the velocity field of radiation potential 2  and 3 , respectively; figure ‘c’ is the velocity field of scattering potential S ; and figure ‘d’ is the velocity field of whole potential . Figure ‘e’ is also the velocity field of potential , but the calculation area is extended to domain 2.0 m, 2.0 m in order to display the velocity field far away from the plate. The above mentioned results are all corresponding to the waves in crest time. 3.1 Velocity Field at/ 20.4KB Fig. 2 shows the calculated results of velocity field for submerged horizontal plate when / 20.4KB, where the transmission coefficient is 0.3 Wang et al., 2011a and the heave wave exciting force is the largest Wang et al., 2011b. Since plate is symmetric about y-axis, the generated radiation heave flow field 22 ,u v is symmetric and roll flow field 33 ,u v is anti-symmetric. For the roll motion, the flow directions of water below the plate and above the plate are opposite, and the velocity amplitude below the plate is smaller than that above the plate. That is the characteristic of roll motion. When the plate heaves, water particles around the plate also reciprocate vertically at the same time. Since the plate is thin and close to the free surface, the plate length is decisive for the change of velocity field. It can be seen from Fig. 2d that water particle above the plate ZHANG Zhi-qiang et al. / China Ocean Eng., 276, 2013, 821 − 828 825 at crest time is separated into two parts and flow horizontally to the fluid field with opposite direction. This wave-plate interaction repeats reciprocally. The interaction between reverse flow above the plate and the incident wave leads to the ation of wave elimination area in the head-sea of the plate, where fluid velocity changes dramatically. From Fig. 2e we can see that the velocity magnitude in head sea is much larger than that behind the plate. As a result, the reverse flow caused by submerged plate should be the main reason of wave elimination. Fig. 2. Velocity field for submerged horizontal plate at / 20.4KB. ZHANG Zhi-qiang et al. / China Ocean Eng., 276, 2013, 821 − 828 826 3.2 Velocity Field at / 20.8KB Because the curve of transmitted coefficient is symmetric around/20.6KBWang et al., 2011a, at/20.8KB, transmission coefficient is also about 0.3. That means if we want to obtain the same wave transmitted coefficient, a smaller plate length should be chosen. Fig. 3 shows that, at/20.8KB, the velocity field around the plate is almost similar to that at /20.4KB, but the velocity magnitude and the wave elimination area are smaller. Fig. 3. Velocity field for submerged horizontal plate at / 20.8KB. ZHANG Zhi-qiang et al. / China Ocean Eng., 276, 2013, 821 − 828 827 3.3 Velocity Field at/ 21.2KB It can be seen from Figs. 4a and 4b that the variations of heave and roll velocity field is also similar to those at / 20.4 and 0.8KB. But different from / 20.4 and 0.8KB, we can see from Fig. 4e that the velocity field around the plate within one wave length is little disturbed and wave elimination is not effective. That is mainly because the shallow water above the plate does not generate strong horizontal flow. Fig. 4. Velocity field for submerged horizontal plate at/ 21.2KB. ZHANG Zhi-qiang et al. / China Ocean Eng., 276, 2013, 821 − 828 828 4. Conclusions This study explores the flow field distribution around submerged horizontal plate under different wave conditions by use of boundary element . It is found that 1 The heave flow field is symmetric, and the roll flow field is anti-symmetric. Because of the existence of submerged plate, the incident flow field is disturbed by the plate radiation and diffraction. 2 The shallow water effect will generate reverse current above plate and flow horizontally to the edges of the plate. The interaction between reverse flow and incident wave leads to the ation of wave elimination area around both sides of the plate, where fluid velocity changes dramatically. As a result, the velocity magnitude has been reduced and this is the main reason of wave elimination. References Dong, G. H., Zheng, Y. N., Li, Y. C., Teng, B. and Lin, D. F., 2008. Experiments on wave transmission coefficients of floating breakwaters, Ocean Eng., 358–9 931–938. Faltinsen, O. M., 1991. Sea Loads on Ships and Offshore Structures, Cambridge University Press, Cambridge, UK. Hu, H., Wang, K. H. and Williams, A. N., 2002. Wave motion over a breakwater system of a horizontal plate and a vertical porous wall, Ocean Eng., 294 373–386. Liu, Y., Li, Y. C. and Teng, B., 2009. Wave motion over two submerged layers of horizontal thick plates, Journal of Hydrodynamics, Ser. B, 214 453–462. Neelamani, S. and Reddy, M. S., 1992. Wave transmission and reflection characteristics of a rigid surface and submerged horizontal plate, Ocean Eng., 194 327–341. Parson, N. F. and Martin, P. A., 1992. Scattering of water waves by submerged plate using hyper-singular integral equations, Appl. Ocean Res., 145 313–321. Usha, R. and Gayathri, T., 2005. Wave motion over a twin-plate breakwater, Ocean Eng., 328−9 1054–1072. Wang, K., Zhang, Z. Q. and Xu, W., 2011a. Transmitted and reflected coefficients for horizontal or vertical plate type breakwater, China Ocean Eng., 252 285–294. Wang, K., Zhang, X. and Gao, X., 2011b. Study on scattering wave force of horizontal and vertical plate type breakwater. China Ocean Eng., 254 699–708. Yu, X. P. and Chwang, A. T., 1993. Analysis of wave scattering by submerged circular disk, J. Eng. Mech., 1199 1804–1817. Zheng, Y. H., Shen, Y. M. and Ng, C. O., 2007. Effective boundary element for the interaction of oblique waves with long prismatic structures in water of finite depth, Ocean Eng., 355−6 494–502.