BARRIER. Journal of Hydrodynamics, Ser. B,1 (2001),2833 China Ocean Press, Beijing  Printed in China


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1 28 Journal of Hydrodynamics, Ser. B,1 (2001),2833 China Ocean Press, Beijing  Printed in China WAVE TRANSMISSION AND BARRIER REFIECTION D UE TO A THIN VERTICAL Ξ Gao Xue2ping Hydraulics Instit ute, School of Civil Engineering, Tianjin U niversity, Tianjin300072, China Nakamura Takayuki, Inouchi Kunimitsu Ehime University, Matsuyama , Japan Kakinuma Tadao Nagasaki Comprehensive U niversity, Nagasaki , J apan (Received Oct ) ABSTRACT: A numerical method, the boundary fitted coordinate method ( BFC ), was used to investigate the transmission and reflection of water waves due to a rigid thin vertical barrier descending from the water surface to a depth, i. e., a curtain2wall type breakwater. A comparison between the present computed results and previous experimental and analytical results was carried out which verifies the prediction of the BFC method. Wave transmission and reflection due to the barrier were computed, and the transmission and refiection coefficients were given in a figure. KEY WORDS : wave transmission, wave reflection, vertical barrier, boundary fitted coordinate method thin 1. INTROD UCTION A t rain of surface water waves incident f rom infinity on an obstacle present in water, experiences t ransmission and reflection by and over or below t he obstacle. The obstacle may be in t he form of a rigid t hin vertical barrier. The rigid t hin barrier descending from the water surface to a depth, which is also called curtain2wall type breakwater, is one type of t hin vertical barriers. Numerous st udies for t his type barrier have been conducted experimentally, t heoretically and numerically. Wiegel (1960) devised a semi2empirical solution for t he t ransmission coefficient s in intermediate and shallow water conditions based on energy flux. Ursell ( 1974 ) developed an analytical solution for the transmission coefficients in deep water condition. Morigira et al., ( 1960 ) examined t he characterstics of wave t ransmission and wave force acting on t he st ruct ure experimentally. Liu et al., ( 1982 ) st udied t he t ransmission and reflection coefficient s of bot h t he vertical and inclined t hin breakwater by t he boundary integral equation met hod ( B IEM ). Theoretical solution was obtained by eigenf unction expansion met hod for t he t ransmission and reflection coefficient s by Losada et al., ( 1992). Mandal et al., ( 1994 ) st udied t he reflection and t ransmission coefficient s by an appropriate one2term Galerkin approximation. In t he present st udy, a numerical met hod, t he boundary fitted coordinate method (BFC), is used to study the transmission and reflection due to the rigid thin vertical barrier descending from the water surface to a depth (curtain2wall type breakwater). With the BFC method, which is suitable for free surface flow problems, one t ransforms a p hysical domain into a rectangular domain using a suitable t ransform and solves the governing equations (the N2S equation and continuity equation ) by finite difference approximation. Moreover, satisfying accurately t he complex p hysical boundaries is t he greatest advantage of t he numerical met hod. The FF T met hod toget her wit h t he resolution technique presented by Goda and Suzuki ( 1976 ) are used to calculate t he incident, reflected and t ransmitted wave height s. Some computed result s are compared wit h previous analytical solutions and experimental data to verify t he prediction of the BFC method. Wave transmission and reflection due to the barrier are computed, and the transmission and reflection coefficient s for various cases are given in a figure. 2. BOUNDARY FITTED COORDINATE METHOD The p hysical domain is t ransformed into t he rectangular domain representing accurately t he complex p hysical boundary ( Fig. 1 ). Governing equations and boundary conditions are also t ransformed. In t he rectangular domain, t he t ransformed governing equations are solved by t he finite difference met hod. Ξ This is supported by the Returned Student Foundation of State Education Ministry of China
2 29 where w 1 = w  s u, w 2 = w 15s 5 (7) = s 5 2 s s s 5 s 5 (8) Fig. 1 Coordinate transformation ( a) Physical domain and (b) rectangular domain 2. 1 Governing equations Wave motion is governed by t he continuity equation and Navier2Stokes equations. In two2 dimensional Cartesian coordinates, t hese equations are expressed as 5 x + 5 z 5 t 5 t = 0 (1) + u 5 x + w 5 z =  5 P 5 x + 52 u 5 x u 5 z 2 + g x + u 5 x + w 5 z (2) =  5 P 5 z + 52 w 5 x w 5 z 2 + g z (3) and s = s ( x, t) is the surface elevation ( Fig. 1), s = 5s/ 5, s = 5 2 s/ Boundary conditions Free surface conditions. The f ree surface conditions include kinematic and dynamic boundary conditions. In t he p hysical domain, t he kinematic condition is expressed as 5s 5 t + u 5s 5 x = w (9) By considering the normal stress and tangential stress at t he f ree surface, t he dynamic boundary conditions are expressed as ( Kawamura and Hijikata, 1995) : P = s x x (1 + s 2 x ) 3/ v s x 5 z + s 2 x 5 x  s x 5 z + 5 x (10) where ( x, z) is the physical domain coordinate ; t is time ; u and w are t he corresponding component s of 5 z + 5 x  2 s x s x the velocity in the x and z directions ; P = p/ ( pis the pressure, is the density of water) ; g x and g z are t he component s of unit body force in t he x and z directions respectively ; is the kinematic viscosity coefficient. By a suitable t ransformation ( see Kawamura and Hijikata 1995 ; Gao et al., 1998), t he governing equations in t he p hysical domain of t he coordinate system ( x, z, t) as above are transformed into the rectangular domain of the coordinate system (,, ) t he dynamic conditions as follows : P =  s s 5 5 ( su) ( w 1) = 0 (4) 5 + u 5 + w 1 2 s 5 =  5 P 5  s 5 P s 5 + u + g x (5) 5 + u 5 + w 1 2 s 5 = P s 5 + w + g z (6) 5 x  5 z = 0 (11) where is the surface tension, and s x = 5s/ 5 x, s x x = 5 2 s/ 5 x 2. In the rectangular domain, the surface conditions are reduced to t he kinematic condition : 5s 5 = w 1 ; at = 1 (12) 5  s 1 s (13) 1 s 5 = s s 5 at = 1 (14) Rigid boundary conditions. A rigid boundary may be either of two types ; non2slip or free slip. Here the non2slip condition is utilized, that is, in the physical domain, t he normal and tangential velocities at t he
3 30 rigid boundary are zero. Therefore, in t he rectangular domain, t he rigid boundary conditions are expressed as at = 0, u = w = 0 (15) 2. 3 Finite dif f erence approxi m ation The governing equations (4) (6), together with boundary conditions (12) (15), are solved by t he finite difference approximation. A staggered mesh system is used, i. e., velocities are defined at cell boundaries while pressure is defined at cell center. Forward differencing in time and centered differencing in space are utilized in t he finite difference approximation. 3. METHOD OF COMPUTATION AND VERIFICATION 3. 1 Com putational method For t he p hysical domain ( Fig. 2 ), given t he incident boundary and open boundary conditions, t he wave motion is calculated by t he boundary fitted coordinate met hod described as above. A train of the second2order Stokes waves is given at t he incident boundary. The Sommerfeld radiation condition is utilized at t he open boundary. For the initial condition, the fluid is assumed to be entirely at rest, and the physical domain is divided into cells, the size of a cell in the x direction is about x = L / 50 ( L is incident wavelength), the size of a cell in the z direction is about z = h/ 25, where h is still water depth, and z will change as time continues. The time step t = s is taken in this paper simply. Fig. 2 A thin vertical barrier descending from the water surface to a depth and coordinate system 3. 2 Com putation of incident, ref lected and t ransmitted w ave heights It is well known that, in a wave flume, waves generated by a wave paddle propagate forward in the flume and are reflected by a test st ruct ure ; t he reflected waves propagate back to the wave paddle and are reflected ; t he reflected waves propagate forward again and t he process is repeated until t he multi2 reflected waves are fully attenuated ; then the value of the incident wave height may be changed. Like a wave flume, because of the reflection due to a barrier in t he p hysical domain, when waves a stable state, the value of incident wave height may be different from the one given initially. Therefore, it is necessary to determine t he incident wave height when waves reach a equilibrium state. In t his st udy, t he FF T met hod toget her wit h t he resolution technique presented by Goda and Suzuki ( 1976) are utilized to determine t he incident wave height and resolve t he reflected and t ransmitted wave height s. The specific procedures for t he calculation can be summarized as follows. The first step is to record t he time historys of wave surface elevations. A two wave system is utilized for recording wave surfaces at different locations along t he wave propagation direction. The spacing of t he two wave gages has to be determined based on t he fundamental frequency f 1 ( = 1/ T) and the frequency of the highest harmonic f n ( = 1/ n T), which might be t he t hird or fourt h harmonic f requency. It is suggested that the spacing of the wave gages should be greater than L 1 and less than L n, where L 1 and L n are the wavelengths corresponding to the fundamental frequency f 1 and the frequency of the highest harmonic f n, respectively. The second step is to calculate t he Fourier components, i. e., the amplitudes of sine and cosine terms, by the FFT method from the time history of wave surface elevations recorded. The t hird step is to calculate t he amplit udes corresponding to different order harmonic f requencies by the formula presented by Goda and Suzuki (1976). The fourt h step is to calculate t he incident and reflected wave heights by summing the energies of all frequency components from f 1 of f n. The energies of resolved incident and reflected waves, E in and E rf, are as follows : E in E rf = 1 2 g f n ( a in ) 2 i (16) i = f 1 = 1 2 g f n ( a rf ) 2 i (17) i = f 1 where a in and a rf, are the amplitudes of incident and reflected waves, respectively corrsponding to different order harmonic f requencies. Converting the energies, E in and E rf, to the so2 called representative wave height s ( incident and reflected wave heights), H i and H r, leads to
4 31 H i = 2 H r = 2 2 E in g 2 E rf g (18) (19) Transmitted wave height is also calculated by t he same energy consideration. According to t he procedure described above, incident, reflected and transmitted wave heights due to t he barrier ( curtain2wall type breakwater ) are calculated by the present BFC method ( Fig. 3). It is shown that the wave heights (incident, reflected and t ransmitted wave height s) along t he wave propagation are almost t he same, which means t hat t he present numerical met hod is valid for t he steady state calculations. After knowing these wave heights, the reflection and t ransmission coefficient s can be calculated as follows : behind a barrier were conducted in a 26m long, 1m wide and 1. 2m deep wave flume. Fig. 4 shows a comparison between computed and experimental wave surface elevations. It is shown t hat t he present met hod is good for t he predictions of wave deformations around a barrier. C r = H r H i (20) C t = H t H i (21) where H i is the incident wave height ; H r is the reflected wave height ; and H t is the transmitted wave height. Fig. 4 Comparison of wave surface elevations by a curtain2 wall breakwater. a,b,c : at 5. 82m, 3. 88m, 1. 94m in front of the curtain2wall breakwater. d : at 1. 94m behind the curtain2wall breakwater. Case : h = 0. 75m, H = m, T = s, d = 0. 21m Fig. 3 Wave heights near a curtain2wall breakwater. Case : h = 0. 75m, H = m, T = s, d = 0. 21m 3. 3 Com parisons of w ave surf ace elevations A numerical computation flume of 13. 2L ( L is the wavelength), where a barrier is located at 10. 2L from the incident boundary, is set up for the study by the present method. Calculations show that the waves are stable (see Fig. 3). For the case that there is a barrier in the physical domain, laboratory experiment s for measuring wave surface elevations at selected locations in f ront of and 3. 4 Com parisons of t ransmission coef f icients Transmission coefficient s computed by t he present met hod are compared wit h experimental data, numerical result s and analytical solutions published by researchers, including Wiegel ( 1960), Liu et al., (1982), Ursell ( 1974) and Losada et al., ( 1992). Figs. 5 ( a), 5 ( b) and 5 ( c) show the transmission coefficient s plotted versus d/ h for different relative water dept hs, kh = , and , respectively. The present computations are in good agreement wit h t he experimental data. 4. COMPUTED RESULTS AND DISCUSSION The barrier descending from the water surface to a dept h (curtain2wall type breakwater) is considered in
5 32 Fig. 5 (a) Comparison of transmission coefficients ( kh = ) Fig. 5 (c) Comparison of transmission coefficients ( kh = ) ts, C r, increase while transmission coefficients, C t, decrease. Moreover, as t he relative water dept h kh increases, the variations of reflection coefficients C r and transmission coefficients C t become smaller. This is because wave disturbance is confined within a layer near t he wave surface as t he relative water dept h kh increases. For t he fixed values of kh, as d/ h increases, C r increase and C t decrease. Table 1 Computational conditions relative water depth kh wave period T (s) incident wave height H (m) d/ h (refer Fig. 2) Fig. 5 (b) Comparison of transmission coefficients ( kh = ) t he present paper, Computational conditions are shown in Table W ave t ransmission and ref lection For different wave conditions and barrier sizes, extensive numerical computations have been carried out by t he present B FC met hod to investigate t he wave t ransmission and reflection due to t he t hin vertical barrier. Fig. 6 shows t he wave refection and transmission coefficients due to the thin vertical barrier descending f rom t he water surface to a dept h. It is observed that for fixed values of d/ h, as the relative water dept h kh increases, reflection coefficien 5. CONCL USIONS The main conclusions f rom t his st udy are summarized as follows. Comparisons between t he computed result s and previous analytical and experimental result s have been made. The t ransmission coefficient s due to a barrier descending from the water surface to a depth computed by t he boundary fitted coordinate met hod ( B FC ) shows good agreement wit h previous experimental data. A thin barrier descending from the water surface to a depth is investigated. For the fixed values of d/ h, as t he relative water dept h kh increases, reflection coefficients C r increase while transmission coefficients
6 33 REFERENCES Fig. 6 Wave reflection and transmission coefficients due to a curtain2wall breakwater C t decrease. Moreover, as the relative water depth kh increases, the variations of reflection coefficients C r and transmission coefficients C t become smaller. Wave t ransmission and reflection coefficient s due to t he barrier for various cases are given in a figure. 1. Gao Xueping, Inouchi K. and Kakinuma T., 1998 : Wave Motions Around Different Submerged Structures, Acta Oceanologica Sinica, 17(3), Goda Y. and Suzuki Y., 1976 : Estimation of Incident and Reflected Waves in Random Wave Experiments, Proc. 15th Coastal Engineering Conf, ASCE, 1, Honolulu, Kawamura H. and Hijikata K., 1995 : Simulation of Heat and Flow Maruzen Co. Ltd., Tokyo, (in Japanese) 4. Losada I. J., Losada M. A. and Roldan A. J., 1992 : Propagation of Oblique Incident Waves Past Rigid Vertical Thin Barriers. Appl. Ocean Res., 14, Liu P. L2F. and Abbaspour M., 1982 : Wave Scattering by a Rigid Thin barrier. J. Waterways, Port, Coastal and Ocean Div., ASCE, 108(WW4), Mandal B. N. and Dolai D. P., 1994 : Oblique Water Waves Diffraction by Thin Vertical Barriers in Water of Uniform Finite Depth. Appl. Ocean Res., 16, Morihira M., Kakizaki S. and Goda Y., 1960 : Experimental Investigation of a Curtain2Wall Breakwater, Report of Port and Harbour Research Institute, Japan, 3 (1). 8. Ursell F., 1974 : The Effect of a Fixed Vertical Barrier on Surface Waves in Deep Water, Proceedings of the Cambridge Philosophical Society, 43, Part 3, Wiegel R. L., 1960 : Transmission of Waves Past a Rigid Vertical Thin Barrier, J. Waterways and Harbors Div., ASCE, 86 (WW1), 112.
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