Numerical prediction of performance of submerged breakwaters

Ocean Engineering 32 (2005) 1235–1246 www.elsevier.com/locate/oceaneng

Numerical prediction of performance of submerged breakwaters A. Chiranjeevi Rambabu*, J.S. Mani Department of Ocean Engineering, Indian Institute of Technology, Madras, Chennai 600 036, India Received 1 October 2003; accepted 25 October 2004 Available online 1 April 2005

Abstract The results of a numerical model study on the transmission characteristics of a submerged breakwater are presented. Study aimed to determine the effect of depth of submergence, crest width, initial wave conditions and material properties on the transmission characteristics of the submerged breakwater. The results highlight the optimum crest width of the breakwater and optimum clear spacing between two breakwaters. A submerged permeable breakwater with ds/dZ0.5, pZ0.3 and fZ1.0, reduces the transmission coefficient by about 10% than the impermeable breakwater. The results indicates an optimum width ratio of B/dZ0.75 for achieving minimum transmission. By restricting the effective width ratio of the series of breakwaters to 0.75, studies were conducted to determine the effect of clear spacing between breakwaters on transmission coefficient, suggesting an optimum clear spacing of w/bZ2.00 to obtain Kt below 0.6. q 2005 Elsevier Ltd. All rights reserved. Keywords: Depth of submergence; Crest width; Submerged breakwater; Porosity; Friction; Transmission coefficient

1. Introduction With the developmental activities, the dynamic equilibrium of the coastal region is disturbed resulting in coastal erosion and accretion. Coastal erosion is a severe problem * Corresponding author. Fax: C91 44 2352545. E-mail addresses: [email protected] (A.C. Rambabu), [email protected] (J.S. Mani). 0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2004.10.023

1236

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

Nomenclature Hi T g d ds hs B p f w b Kt

incident wave height wave period acceleration due to gravity water depth depth of submergence height of the structure crest width porosity friction coefficient clear spacing crest of series of breakwater transmission coefficient

worldwide, threatening the coastal properties, degradation of valuable land and natural resources, disruption to fishing, shipping and tourism. The development of coastal facilities has necessitated proper management of the sea front warranting for construction of coastal protective structures. The choice of the structure would depend on the wave environment and the morphology of the coastal region. Shore-parallel, detached breakwaters have succeeded in a number of coasts to mitigate the erosion. They provide good shelter for the leeside, but unfortunately their emerged crests obstruct the aesthetic view of the sea. Furthermore, a significant reflected wave is likely to create problems of scour on the seabed in front of the structure. To solve this problem, a submerged breakwater is developed as an alternative means. The main function of these breakwaters is to protect the seaward area from the severe wave actions, by attenuating the wave pass over the structure. Submerged breakwaters absorb some of incoming wave energy by causing the waves to break prematurely, thus diminishing the transmitted wave energy. These structures trap the sediment entering into the protected zone, on the leeside of breakwater, enhances to chance of building up of the beach. The main parameters used to describe the general geometry of a submerged breakwater are shown in Fig. 1(a). Fig. 1(b) shows series of submerged breakwaters separated by a clear spacing (w). The studies conducted by Dick and Brebner (1968) on solid and permeable type submerged breakwaters indicate that these breakwaters with near zero submergence are capable of reducing the incident wave energy by 50%. Raman et al. (1977) reported on the damping action of rectangular and rigid vertical submerged barriers and expressed the transmission coefficients in terms of the transmitted energy to the total power of the incident wave. Further it is stated that in case of rectangular submerged barrier, the top width of the barrier plays an important role in controlling transmission coefficient. The test on wave transmission and reflection characteristics of laboratory breakwater conducted by Seelig (1980) indicate that both the depth of submergence and top width are important in determining the performance of the breakwater and it is suggested that for near zero submergence the submerged breakwater is efficient in reducing the transmission.

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

(a)

1237 Z

Incident wave

Transmitted wave

Coastline

X ds

B 1.5

d

1.5 1

1

hs

Z

(b) Coastline

Transmitted wave

Incident wave X b

b 1.5 1

1.5

1.5 1

1

ds 1.5

d 1

hs

w

Fig. 1. (a) Definition sketch of submerged breakwater. (b) Definition sketch of series of submerged breakwater.

Kobayashi and Wurjanto (1989) presented a numerical approach based on finite amplitude shallow water equation for determination of wave transmission over submerged breakwater. Design equations for wave transmission over a submerged breakwater are detailed by Rojanakamthorn et al. (1989) based on mild slope equation and the result suggests that when the height of the submerged breakwater is 50% of the total depth, the transmission coefficient varies between 0.4 and 0.7. In this paper, the results of a numerical model study to determine the transmission characteristics of the submerged breakwater by varying depths of submergence, crest widths, material properties and clear spacing under a range of regular wave conditions are presented. In the formulation of the scheme, the boundary-value problem is restricted to two dimensions, and the solution is written in terms of velocity potential (f(x, z)) of the waves, which satisfies Laplace’s equation. The scattered portion of velocity potential of the wave is solved numerically based on Green’s formulation. The results are compared with those reported in the literature (Abdul Khader and Rai, 1980; Hall and Seabrook, 1998), wherein the authors have conducted experimental studies on the submerged breakwater. An agreement between experimental and present studies validates the numerical model.

2. Numerical model The problem is restricted to two dimensions, and the usual assumptions of incompressible, irrotational flow are made with in the framework of linear wave theory.

1238

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

The solutions thus can be written in terms of a velocity potential, f(x, z) which satisfies Laplace’s equation and is a harmonic function of the time, t. The velocity potential is given as fðx; zÞ Z f0 ðx; zÞ C fs ðx; zÞ f0 ðx; zÞ Z

gH cosh kðz C dÞ ikx e 2s cosh kd

(1) (2)

where f0(x, z) is the incident wave potential, fs(x, z) scattered wave potential, x the coordinate in the direction of the wave, and z the vertical coordinate measured positive upward from the still water level. Under the assumptions of the linear theory, f(x, z) will satisfy the differential equation V2 fðx; zÞ Z 0

(3)

The solution for fs(x, z) is obtained by the method of integral equations using the source function in two dimensions. The function satisfies boundary conditions except at the source point and is given in complex form as follows Gðx; z; x; hÞ Z gðx; z; x; hÞeKist

(4)

where G(x, z; x, h) is the Green’s function with gðx; z; x; hÞ Z g1 ðx; z; x; hÞ C ig2 ðx; z; x; hÞ g1 Z

1 r 1 r log C log 2 2p d 2p d  ð  1 N k C n eKkd cosh kðd C zÞcos kðx K xÞ eKkd C K dk p 0 k k sinh kd K n cosh kd k

g2 Z g0 cos kðx K xÞ

(5)

(6)

(7)

The arguments x and h are the coordinates of the source point and g0 Z

v cosh kðd C hÞcosh kðd C zÞ k vd C sinh2 kd

(8)

The function g1 may also be written in series form as g1 Z g0 sin kjx K xj K

X 1 m2i C v2 cos mi ðz C dÞcos mi ðh C dÞeKmi jxKxj 2 2 m m d C v d K v i i iZ1 (9)

where mi iR1 are positive real roots of the equation m tan(md)CnZ0. This expression of g1 can be evaluated numerically much more efficiently than Eq. (6), when jxKxj is large.

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

1239

By the application of Green’s theorem in two dimensions to fs and g, a representation for fs in terms of its value on the object is obtained  ð vf vg (10) fs ðx; zÞ Z Kgðx; z; x; hÞ 0 ðxxhÞ K fs ðxxhÞ ðx; z; x; hÞ ds vn vn where use has been made of the object surface condition ðvfs =vnZKvf0 =vnÞ. Thus, if the scattered potential on the surface of the object is known the scattered potential at any point in the wave field may be evaluated. In the case of permeable breakwater use has been made on the surface condition (fsZ (sKif)f0)   ÐT Ð 2 3 p2 v p3 C f p ffiffiffiffi jqj jqj þ V dt dV Kp Kp 1 Kp 1 0 Cm f ¼ S¼1þ ÐT Ð 2 p u 0 Vpjqj dt dV 2.1. The potential on the object We now reapply Green’s theorem to fs and g, letting the point (x, z) in Eq. (10), the equation becomes ð ð 1 vg vf f ðx; zÞ C ðx; z; x; hÞfs ðx; hÞvs Z K gðx; z; x; hÞ 0 ðx; hÞds (11) 2 s vn vn This equation is solved numerically by dividing the surface of the object in to small segments and solving for fs(x, z) at the center of these segments. 2.2. Numerical computation For the numerical solution of fs, Eq. (11) is replaced by a sequence of algebraic equations. The contour describing the two-dimensional object is divided in to N segments and the center of the segments is described by the coordinates xi,hi (iZ1,2,3,.N). Then the algebraic equations take the form (Naftzger and Chakrabarti, 1979) N N X X 1 vg vf fs ðxi ;hi ÞK ðxi ; hi ; xj ; hj Þfs ðxj ;hj ÞðDsÞj Z gðxi ; hi ; xj ; hj Þ 0 ðxj ;hj ÞðDsÞj 2 vn vn jZ1 jZ1

which may be equivalently written as N X

Aij Xj Z Bj

jZ1

in which Aij is a complex matrix given by 1 vg Aij Z dij K ðxi ;hi ; xj ; hj ÞðDsÞj 2 vn

1240

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

dij is the Kronecker delta, Xj are the unknown scattered potentials at the center of the segment, and Bj is obtained by the Gauss-Jordan pivotal elimination technique. Once fs (xi, hi) is known, the total potential is computed from Eq. (1).

3. Results 3.1. Wave transmission of impermeable trapezoidal submerged breakwater † Variation of Kt with Hi/gT2 for constant water depth and different values of relative depth of submergence ds/d (0.5, 0.625, and 0.75) is shown in Fig. 2. The results suggest that for ds/d!0.625 and Hi/gT2O0.006, the breakwater is capable of reducing incident wave by about 60%. † Effect of crest width of the structure on Kt is shown in Fig. 3. The results shows that increase in Hi/gT2 from 0.0014 to 0.0081 the Kt decrease by about 45% with increase in B/d from 0.25 to 1.25. However, there is no appreciable change in Kt observed with increase in B/d from 0.75 to 1.25, suggesting an optimum width ratio of B/dZ0.75. † The results obtained based on the numerical study (Fig. 4) suggest that the Kt decreases by 50% with an increase in hs/d from 0.4 to 0.95. Comparison of the results with experimental studies (Abdul Khader and Rai, 1980) shows a good agreement.

constant water depth

ds/d=0.5 ds/d=0.625 1.00

ds/d=0.75

0.90 0.80 0.70

Kt

0.60 0.50 0.40 0.30 0.20

Hi

d

Ht B

ds hs

0.10 0.00 0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140

Hi/gT 2

Fig. 2. Variation of Kt with Hi/gT2 for constant water depth.

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

1241

1.00 0.90 constant water depth

Kt

0.80

Hi/gT2=0.0081

0.70

Hi/gT2=0.0041

0.60

Hi/gT2=0.0031

0.50

Hi/gT2=0.002

0.40

Hi/gT2=0.0014

0.30

Hi

Ht B

d

0.20

ds hs

0.10 0.00 0.00

0.25

0.50

0.75

1.00

1.25

B/d Fig. 3. Effect of top width of submerged breakwater on Kt.

Hi/Li=0.03620 d/Li=0.1189 B/Li=0.0973 Hi/d=0.304

1.00

0.80

Hi

0.60

Kt

d

Ht B

ds hs

0.40

literature (Abdul Khader and Rai , 1980) present

0.20

0.00 0.00

0.20

0.40

0.60

0.80

1.00

hs/d Fig. 4. Comparison of present results with available literature (Abdul Khader and Rai, 1980) for impermeable submerged breakwater.

1242

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246 0.80

Ht

Hi

water depth constant p=0; f=0

b

d

Hi/gT2=0.0025 Hi/gT2=0.0081

b

ds hs

w

Kt

0.70

0.60

0.50 0.00

0.25

0.50

0.75

1.00

1.25

1.50

w/B Fig. 5. Effect of clear spacing between breakwaters on Kt.

† By restricting the effective width ratio of the series of breakwater to 0.75, studies were conducted to determine the effect of clear spacing between breakwaters on transmission coefficient. Fig. 5 show the results, suggesting an optimum clear spacing of w/bZ2.00. 3.2. Wave transmission of permeable trapezoidal submerged breakwater (pZ0.3, fZ1) † Variation of Kt with Hi/gT2 for constant water depth and different values of relative depth of submergence ds/d (0.5, 0.625, and 0.75) is shown in Fig. 6. The results suggest that for ds/d!0.625 and Hi/gT2O0.006, the breakwater is capable of reducing incident wave by about 55%. † Fig. 7 shows the variation of Kt with Hi/gT2 for constant water depth (and ds/dZ0.5) with and without porosity and friction. The results suggest that transmission coefficient (Kt) reduces by about 10% due to the effect of porosity (pZ0.3) and friction (fZ1) for the range of Hi/gT2 considered in the present study. † The results obtained based on the numerical study (Fig. 8) suggest that the Kt decreases by 50% with an increase in hs/d from 0.4 to 0.95. Comparison of the results with experimental studies (Hall and Seabrook, 1998) shows a good agreement. 3.3. Wave transmission of rectangular submerged breakwater † Numerical model studies were conducted (by setting side slopes of the trapezoidal submerged breakwater to zero) to study the performance of rectangular submerged

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

1243

constant water depth p=0.3 ; f=1 ds/d=0.5 ds/d=0.625

1.00

ds/d=0.75 0.90 0.80 0.70

Kt

0.60 0.50 0.40 Hi

Ht

0.30 d

0.20

B

ds hs

0.10 0.00 0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 Hi/gT2 Fig. 6. Variation of Kt with Hi/gT2 for constant water depth.

breakwater in controlling wave transmission. Fig. 9 shows the comparison of numerical model results with those reported in the literature (Twu et al., 2001; Lee and Liu, 1995). Comparison indicates that the present model overestimates the Kt by 12% due to the fact that the porosity and friction aspects are not considered in the present case. † Results of the studies for the present case, incorporating porosity and friction are shown in Fig. 10. Comparison of the results with those reported in the literature (Twu et al., 2001; Lee and Liu, 1995) indicates reasonable agreement.

4. Conclusions † The effect of relative depth of submergence on Kt, suggesting that for ds/d!0.625 and Hi/gT2O0.006, the breakwater is capable of reducing incident wave by about 60%. † Effect of crest width of structure on Kt, indicates an optimum width ratio of B/dZ0.75. † The effect of clear spacing between breakwaters on Kt, indicates an optimum clear spacing of w/bZ2.00. † The studies on series of breakwater indicate that the clear spacing between breakwaters has no much influence on controlling transmission.

1244

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

1.00 0.90 water depth constant

0.80

ds/d=0.5 p=0.3; f=1

0.70

p=0; f=0

Kt

0.60 0.50 0.40 Hi

0.30

Ht B

d

0.20

ds hs

0.10 0.00 0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 Hi/gT2 Fig. 7. Variation of Kt with Hi/gT2 (with and without porosity and friction).

literature(Hall and Seabrook, 1998)

1.00

present

0.80

Kt

0.60 Hi

d

0.40

Ht B

ds hs

0.20

0.10

0.20

0.30

0.40

0.50

0.60

hs/d Fig. 8. Comparison of present results with available literature (Hall and Seabrook, 1998) for permeable submerged breakwater (pZ0.3, fZ1).

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

1245

1.00 B/d=1; ds/d=0.2

0.90 0.80 0.70

Kt

0.60

B

ds

0.50

hs

d

0.40 0.30

present (p=0; f=0)

0.20

Twu et al., 2001 (p=0.3; f=1)

0.10

Lee and Liu, 1995 (p=0.3; f=1)

0.00 0.00

0.50

1.00

1.50

2.00

2.50

3.00

kd Fig. 9. Comparison of present results with available literature (Twu et al., 2001; Lee and Liu, 1995).

Fig. 10. Comparison of present results with available literature (Twu et al., 2001; Lee and Liu, 1995).

1246

A.C. Rambabu, J.S. Mani / Ocean Engineering 32 (2005) 1235–1246

† Effect of porosity (pZ0.3) and friction (fZ1.00) on Kt, indicates that for ds/dO0.5 it have less influence in controlling Kt.

References Abdul Khader, M.H., Rai, S.P., 1980. A study of submerged breakwaters. Journal of Hydraulic Research 18, 113–121. Dick, T.M., Brebner, A., 1968. Solid and permeable submerged breakwaters. Proceedings of the 11th International Conference on Coastal Engineering 1968; 1141–1158. Hall, K.R., Seabrook, S.R., 1998. Design equation for transmission at submerged rubble mound breakwaters. Journal of Coastal Research 26, 102–106. Kobayashi, N., Wurjanto, A., 1989. Wave transmission over submerged breakwaters. Journal of Waterway, Port, Coastal and Ocean Engineering 115, 662–680. Lee, J.K., Liu, C.C., 1995. A new solution of waves passing a submerged porous structure. Proceedings of the 17th International Conference on Coastal Engineering 1995; 593–606. Naftzger, R.A., Chakrabarti, S.R., 1979. Scattering of waves by two-dimensional circular obstacle in finite water depths. Journal of Ship Research 23, 32–42. Raman, H., Shankar, J., Dattatri, J., 1977. Submerged breakwaters. Central Board of Irrigation and Power Journal 34, 205–212. Rojanakamthorn, S., Isobe, M., Watanabe, A., 1989. Design equation for transmission at submerged rubble mound breakwaters. Coastal Engineering in Japan 32, 209–234. Seelig, W.N., 1980. Two-dimensional tests of wave transmission and reflection characteristics of laboratory breakwaters. Tech. Rept. No. 80-1, US Army Coast. Engrg. Res. Ctr., Fort Belvoir, VA. Twu, S.W., Liu, C.C., Hsu, W.H., 2001. Wave damping characteristics of deeply submerged breakwater. Journal of Waterway, Port, Coastal and Ocean Engineering 127, 97–105.