Analytical solution for the linear wave diffraction by a uniform vertical cylinder with an arbitrary smooth cross-section

Ocean Engineering 126 (2016) 163–175

Contents lists available at ScienceDirect

Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Review

Analytical solution for the linear wave diffraction by a uniform vertical cylinder with an arbitrary smooth cross-section Jiabin Liu, Anxin Guo n, Hui Li Ministry-of-Education Key Laboratory of Structural Dynamic Behavior and Control, School of Civil Engineering, Harbin Institute of Technology, Harbin, China

art ic l e i nf o

a b s t r a c t

Article history: Received 30 January 2016 Received in revised form 15 June 2016 Accepted 6 September 2016

An analytical method is proposed to investigate the wave diffraction of linear waves with a uniform, bottom-mounted cylinder with an arbitrary smooth cross-section. Based on the condition that the radius function of the cylinder surface can be expanded into a Fourier series, the linear diffraction theory is extended to solve the diffraction problem of linear waves in such large-scale structures. The present method is first validated using a uniform vertical cylinder with cosine-type radial perturbations. Then, the wave diffraction, wave force and wave run-up are investigated for such structures under wave attacks with different rotation angles. Finally, this method is further extended to a practical engineering application in a quasi-ellipse caisson foundation for a cross-strait bridge pylon. The results show that the method that we have developed can be effectively used for predicting the wave force and wave run-up of large-scale cylinders with arbitrary smooth cross-sections considering the wave diffraction effects. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Wave diffraction Cylinder Arbitrary smooth cross-section Analytical solution Wave force Wave run-up

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation and case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Case study 1: cylinders with cosine-type perturbation cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Case study 2: a quasi-ellipse caisson foundation of the bridge pylon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction In offshore engineering, an ocean wave is a major load that threatens the safety of structures in a marine environment. For a large-scale body, quantitative understanding of the effects of diffraction due to wave-structure interactions is of primary importance to determine the wave force acting on the coastal structures when subjected to ocean wave actions. Linear diffraction theory is a common method to theoretically analyze the interaction of a linear wave with a cylinder based on the potential theory. An analytical solution to the interaction between linear n

Corresponding author. E-mail address: [email protected] (A. Guo).

http://dx.doi.org/10.1016/j.oceaneng.2016.09.010 0029-8018/& 2016 Elsevier Ltd. All rights reserved.

163 164 167 167 171 173 174 174 174

waves and a bottom-fixed vertical circular was initially proposed by Havelock (1940) for the deep-water case. Later, this theory was extended by MacCamy and Fuchs (1954) to a finite water depth. The experimental study of Chakarabarti and Tam (1975) demonstrated that the linear diffraction theory was reasonably accurate to predict the wave force on a circular cylinder if 2A /h ≤ 0.25 ( A is the wave amplitude, and h is the water depth) and 0 ≤ ka0 ≤ 3 (k is the wave number and a0 is the cylinder radius). Nevertheless, the linear diffraction theory is no longer suitable for calculating the wave action of a large body under a strong nonlinear wave. Therefore, many studies were also conducted by researchers to achieve an exact estimation of the wave force on a circular cylinder under nonlinear wave actions (Chau and Taylor, 1992; Lighthill, 1979; Malenica and Molin, 1995; Molin, 1979; Newman, 1996).

164

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

Except for a circular cylinder, the wave-structure interaction problem of a bottom-fixed cylinder with some other specific geometric shapes of the cross-section was also addressed by researchers to obtain an analytical solution. Chen and Mei (1973) presented an exact solution of wave forces acting on an elliptical cylinder by the Mathieu function in elliptic cylindrical coordinates. By this method, the complete solution is very complex due to the requirement of calculating the infinite series of the Mathieu function. To reduce the computational work, Williams (1985) developed two alternative methods for solving the same problem. For the numerical methods to simulate wave loading on largescale objects with arbitrary shapes, Green's function plays an important role for the analysis of this class of problems. Recently, a detailed history and discussion of Green's function were presented by Duffy (2015). The original concept of Green's function came from classical electrostatics, and this concept enjoyed great success in the classic field of an irrotational water wave. Cauchy and Poisson first applied Green's function to solve the two-dimensional problem of the water wave surface in the nineteenth century. Later, Green's function was studied extensively during the 1940s and early 1950s, and several alternative integral representations were given, as reviewed by Wehausen and Laitone (1960). For the wave problem of the square caisson, Isaacson (1978) developed a method by assuming a distribution of vertical line wave sources over the submerged body surface. Mansour et al. (2002) presented two methods to analyze the wave diffraction of linear waves by a uniform vertical cylinder with cosine-type radial perturbations. In this study, an analytical solution based on perturbation theory was developed for small perturbation amplitudes of the circular cross-section. Nevertheless, a boundary element solution, which is similar to the solution of Isaacson (1978), was also presented in that study for the case of no restriction on the magnitude of the perturbation amplitude based on Green's theorem. Furthermore, to overcome the defect that the wave source is not effective for calculating the hydrodynamic forces when the cylinder is oscillating, the local disturbance source was introduced by Miao and Liu (1990) and Miao et al. (1993) to solve the hydrodynamic forces acting on a single cylinder with an arbitrary cross-section vibration in still waters. In the study of Ghalayini and Williams (1989), the first-order potential on a vertical cylinder with arbitrary cross-section, expressed in terms of eigenfunction expansions, was calculated by using Green's function. In this study, the second-order wave force was also calculated by an efficient numerical technique. For the wave loading acting on the arbitrary shape, some researchers also addressed the numerical method to solve this type of problem. Methods such as the finite element method (Shankar et al., 1984), boundary element method (Au and Brebbia, 1983; Zhu and Moule, 1994). Recently, Tao et al. (2007) used the scaled boundary finite element method, which is a semianalytical method developed in the elasto-statics and elastodynamics areas, to solve the boundary-value problem composed of short-crested waves diffracted by a vertical circular cylinder. This method was also utilized by Song et al. (2010) to analyze the water wave interaction with multiple cylinders of arbitrary shape. The analyzed results indicated that this method has a great advantage in treating the cylinders with prismatic surface. Naserizadeh et al. (2011) developed a BEM-FDM technique to solve the modified mild slope equation by using the combination of the boundary element method (BEM) and the finite difference method (FDM). The main idea of this method was to utilize BEM in the exterior domain with constant depth and FDM in the interior domain with variable depth. The refraction and diffraction problem of waves from submerged bottom mounted obstacles was analyzed and compared well with experimental measurements. Focused on the wave-power farm, McNatt et al. (2013) developed a new method

for computing the cylindrical wave-field coefficients for an arbitrary geometry. In this study, the Fourier transform and the orthogonality property of the depth dependence was employed, and the circular-cylindrical section of the wave field was computed with the boundary-element-method solver. With a current trend of more cross-strait bridges being built in deeper waters (Feng, 2013), the structural safety of coastal bridges in a marine environments, especially for the long-span navigation bridges with a large-size foundations, becomes more and more important when the bridge is subjected to wave loading and the combined action of waves and other types of natural hazards. For most coastal bridges, some specific geometrical shapes rather than a circular cylinder (such as a quasi-ellipse caisson) have been selected for the foundation of the pylons. Understanding the wavestructure interaction mechanism and establishing an accurate method to predict the wave loading acting on such types of foundations are important issues for researchers and engineers. However, most of the analytical studies on the wave-structure interaction on a bottom-fixed cylinder mainly focus on a crosssection with circular and elliptical shapes. In this article, the linear diffraction theory is extended to solve the wave force and wave run-up on a bottom-fixed uniform cylinder with an arbitrary smooth cross-section in which the radius function can be expanded into a Fourier series. The main contents of this study are organized as follows. First, the definition of the physical problem and the mathematical derivation of the analytical solution of the scattered-wave potential is presented in Section 2. Then, the proposed method is validated by the vertical uniform cylinder with a noncircular section, and the comparative results are introduced in Section 3.1. Focused on the circular cylinder with cosine-type perturbations, the wave diffraction, wave force and wave run-up are investigated and discussed in Section 3.2 considering the effects of rotation angle, shape perturbation and wave number. This method is further extended to a practical engineering application with a quasi-ellipse caisson foundation of a cross-strait bridge pylon in Section 3.3. The main findings of the present work are summarized in the final section.

2. Mathematical formulation Fig. 1 shows the schematic diagram of the wave diffraction around a uniform surface-piercing cylinder, which is assumed to be rigid and mounted at the bottom of the seedbed. In the analysis, the origin of the coordinate system is set inside the cross section at the still water level (SWL). In polar coordinates, r and θ are defined in the horizontal plane, and the z -axis is perpendicular to the SWL and positive upward. Under the action of gravity, the water wave is assumed to be an ideal fluid with incompressible, inviscid and irrotational characteristics. Based on these conditions, the total velocity potential of the fluid, Φ(r , θ , z, t ), can be written in a complex form as

Φ(r , θ , z, t ) = ⎡⎣ ϕI (r , θ , z ) + ϕD(r , θ , z )⎤⎦e−iωt

(1)

in which ϕI (r , θ , z ) and ϕD(r , θ , z ) are the spatial velocity potential of the incident wave and the scattered wave, respectively; ω is the circular frequency of incident wave. In polar coordinates, the velocity potential of the incident wave can be expressed as

ϕI ( r , θ , z ) = − i

Ag cosh k(z + h) ω cosh kh

∞

∑ m=0

εmi mJm (kr )cos mθ

(2)

in which g is the gravity acceleration; A is the wave amplitude; k is the wave number, which is related to the wave frequency through the dispersion equation ω2 = gk tanh(kh); h is the water depth; and

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

165

Fig. 1. Definition of the problem.

Jm ( ⋅) is the first-kind Bessel function with m order. In this equation, εm = 1 in the case of m = 0; otherwise, εm = 2. The velocity potential of the fluid should satisfy Laplace's equation and the corresponding boundary conditions. Herein, the governing equation and boundary conditions for the scatteredwave potential are summarized as follows (Molin, 2002; Sarpkaya, 2010): Laplace's equation:

∇2ϕD = 0

(3)

2

ω − ϕ =0 ∂z g D

∂ϕD

=0

∂z

Cylinder surface condition:

→ → ∇ϕD⋅ ns = − ∇ϕI ⋅ ns

(6)

Furthermore, the scattered wave should also satisfy the Sommerfeld radiation condition at the far field:

⎛ ∂ϕ ⎞ lim r ⎜ D − ikϕD⎟ = 0 r →∞ ⎝ ∂r ⎠

(7)

nr

cos nr θ + bnr sin nr θ

∞

∑

S(r , θ ) = r −

)

(10)

(a

nr

cos nr θ + bnr sin nr θ

)

nr = 0

(11)

With the above surface function, the partial differential ∂S/∂θ in Eq. (9) can be obtained as ∞

∑

(

nr a nr sin nr θ − bnr cos nr θ

)

nr = 0

(12)

By using the body-surface condition, we can obtain

(

(5)

(a

The surface function, S , can also be given by

(4)

Seabed condition ( z = − h):

∑ nr = 0

∂S = ∂θ

Free surface condition ( z = 0):

∂ϕD

∞

r( θ) =

∂ϕD ∂r

+

∂ϕ 1 ∂S ∂ϕD 1 ∂S ∂ϕI ) =−( I + 2 ) ∂r r ∂θ ∂θ S = 0 r 2 ∂θ ∂θ S = 0

(13)

Because the coordinate origin is located at the inside of the cross-section of the cylinder, there is an implied condition that the function r (θ ) > 0. Substituting Eqs. (2) and (8) into Eq. (13), the body condition becomes ∞

∑

′ (kr )k + mSθ′A m Hm(kr ))cos mθ + εmi m(r 2BmHm

m=0 ∞

∑

εmi m(r 2A m H′′m (kr )k − mSθ′BmHm(kr ))sin mθ

m=0 ∞

Similar as the solution of MacCamy and Fuchs (1954), the velocity potential of the scattered wave can be calculated by using the Hankel function as follows

ϕD( r , θ , z ) =−i

Ag cosh k( z + h) ω cosh kh

∞

∑

εmi mHm( kr )( Bm cos mθ + A m sin mθ )

m=0

=−

∑

′ (kr )k cos mθ − mSθ′Jm (kr )sin mθ ) εmi m(r 2Jm

m=0

where Sθ′ is a simplifying expression of the partial differential ∂S/∂θ . To determine the coefficients Am and Bm in the above equation, the Hankel and Bessel functions and their derivation are also expanded into a Fourier series as follows. ∞

(8)

Hm(kr (θ )) =

∑ ( cmn cos nθ + dmn sin nθ ) n= 0

in which Hm( ⋅) is the first-kind Hankel function with m order; Bm and Am are the unknown coefficients that should be solved by using the cylinder surface condition. In polar coordinates, the → normal vector, ns , in Eq. (6) is given by

→ ns =

1 1+

1 ∂S ( r ∂θ )2

⎛ 1 ∂S ⎞ ⎜1 0⎟ ⎝ r ∂θ ⎠

(14)

(15)

∞

Jm (kr (θ )) =

∑ ( emn cos nθ + fmn sin nθ) n= 0

(16)

∞

′ (kr (θ )) = Hm

n= 0

(9)

In this study, it is assumed that the radius function can be expanded into the Fourier series based on the assumption of the smooth cylinder surface. Therefore, the radius function r (θ ) of the cylinder surface can be written as

∑ ( c˜mn cos nθ + d˜mn sin nθ)

(17)

∞

′ (kr (θ )) = Jm

∑ ( e˜mn cos nθ + f˜mn sin nθ) n= 0

(18)

To simplify the following derivation, r 2(θ ) is also expanded into

166

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

the Fourier series as

∞

−

m=0

∞

r 2( θ ) =

εmi m(r 2J′m (kr )k cos mθ − mS′θ Jm (kr )sin mθ )

∑

∑

( a˜

nr

cos nr θ + b˜ nr sin nr θ

)

(19)

nr = 0

The coefficients b0 , b˜0 , dm0 , d˜ m0 , fm0 and f˜m0 in the corresponding equations are equal to zero. Substituting Eqs. (15–19) into Eq. (14), the left terms of Eq. (14) can be organized according to the sinusoidal and cosinoidal functions as follows.

∞

=−

∞

∑ ∑ m = 0 nr = 0

∞

∑

′ (kr )k + mSθ′A m Hm(kr ))cos mθ + εmi m(r 2BmHm

m=0 ∞

∑

′ (kr )k − mSθ′BmHm(kr ))sin mθ εmi m(r 2A m Hm

⎫ ⎧ ⎡ 1 ⎤T ⎪ ⎪ ⎢ ζnnrm ⎥ ⎪ ⎢ 2 ⎥ ⎡ cos(n + n + m)θ ⎤⎪ r ⎪ ⎢ ζnnrm ⎥ ⎢ ⎥⎪ cos(n + nr − m)θ ⎥⎪ ⎪⎢ ⎥ ⎢ 3 ⎪ ⎢ ζnnrm ⎥ ⎢ cos(n − n + m)θ ⎥⎪ r ⎪⎢ ⎥⎪ ⎢ ∞ ⎪ ζ 4 ⎥ ⎢ cos(n − nr − m)θ ⎥⎪ ⎪ ⎪ m ⎢ nnr m ⎥ ∑ εmi ⎨ ⎢ 5 ⎥ ⎢ sin(n + n + m)θ ⎥⎬ r ⎪ ζnn m ⎢ ⎥⎪ n= 0 ⎪ ⎢ r ⎥ ⎢ sin(n + n − m)θ ⎥⎪ ⎥ ⎢ r 6 ⎪ ζnn m ⎢ ⎥⎪ ⎪ ⎢ r ⎥ ⎢ sin(n − nr + m)θ ⎥⎪ ⎥ ⎢ 7 ⎪ ζ ⎥⎪ ⎢ ⎪ ⎢ nnrm ⎥ ⎣ sin(n − nr − m)θ ⎦⎪ ⎪ ⎪⎢ζ8 ⎥ ⎪ ⎪ ⎣ nnrm ⎦ ⎭ ⎩

(22)

m=0

∞

=

∞

∑ ∑ m = 0 nr = 0

⎧⎡ 1 ⎪ ⎢ ξnnrmBm ⎪⎢ 2 ⎪ ⎢ ξnnrmBm ⎪⎢ 3 B ⎪ ⎢ ξnn rm m ⎪⎢ 4 ∞ ⎪ ⎪ ξnn mBm ∑ εmim⎨ ⎢⎢ 5 r ⎪ ξnn mBm n= 0 ⎪⎢ r 6 ⎪ ⎢ ξnn mBm ⎪⎢ r ⎪⎢ξ7 B ⎪ ⎢ nnrm m ⎪⎢ξ8 B ⎪ ⎩ ⎣ nnrm m

i in which ξnn rm

1 4 1 2 = ξnn rm 4 1 3 = ξnn rm 4 1 4 = ξnn rm 4 1 = ξnn rm

( ka˜ ( ka˜ ( ka˜

T

5 A ⎤ − ξnn rm m ⎥ 6 A ⎥ + ξnn rm m ⎥ ⎥ 7 A − ξnn rm m ⎥ ⎥ 8 A + ξnn rm m ⎥ ⎥ 1 A + ξnn rm m ⎥ ⎥ 2 A − ξnn rm m ⎥ ⎥ 3 A + ξnn rm m ⎥ ⎥ 4 A − ξnn rm m ⎦

⎫ ⎪ ⎡ cos(n + nr + m)θ ⎤⎪ ⎥⎪ ⎢ ⎢ cos(n + nr − m)θ ⎥⎪ ⎢ cos(n − nr + m)θ ⎥⎪ ⎥⎪ ⎢ ⎪ ⎢ cos(n − nr − m)θ ⎥⎪ ⎢ sin(n + n + m)θ ⎥⎬ r ⎥⎪ ⎢ ⎢ sin(n + nr − m)θ ⎥⎪ ⎥⎪ ⎢ ⎢ sin(n − nr + m)θ ⎥⎪ ⎢⎣ sin(n − n − m)θ ⎥⎦⎪ r ⎪ ⎪ ⎪ ⎭ (20)

(i = 1, 2, … , 8) are written as

4 ζnn = rm 5 ζnn = rm 6 ζnn = rm

˜mn nr e

− kb˜ nr f˜mn + mnr a nr emn − mnr bnr fmn

˜mn nr e

− kb˜ nr f˜mn − mnr a nr emn + mnr bnr fmn

˜mn nr e

+ kb˜ nr f˜mn − mnr a nr emn − mnr bnr fmn

( ka˜ ( ka˜ ( ka˜ ( ka˜ ( ka˜

) ) )

˜mn nr e

+ kb˜ nr f˜mn + mnr a nr emn + mnr bnr fmn

nr fmn

˜

+ kb˜ nr e˜mn + mnr a nr fmn + mnr bnr emn

nr fmn

˜

+ kb˜ nr e˜mn − mnr a nr fmn − mnr bnr emn

nr fmn

˜

− kb˜ nr e˜mn − mnr a nr fmn + mnr bnr emn

˜

− kb˜ nr e˜mn + mnr a nr fmn − mnr bnr emn

) ) ) ) )

− kb˜ nr d˜ mn + mnr a nr cmn − mnr bnr dmn

8 ζnn = rm

nr cmn

˜

− kb˜ nr d˜ mn − mnr a nr cmn + mnr bnr dmn

Substituting Eqs. (20) and (22) into Eq. (14), we can obtain a set of equations

nr cmn

+ kb˜ nr d˜ mn − mnr a nr cmn − mnr bnr dmn

⎧ L C 0(B0, A 0 , B1, A1, ⋯)cos(0θ ) = RC 0cos(0θ ) ⎪ L (B , A , B , A , ⋯)sin(0θ ) = RS 0 sin(0θ ) ⎪ ⎪ S0 0 0 1 1 ⎨ L C1(B0, A 0 , B1, A1, ⋯)cos(1θ ) = RC1cos(1θ ) ⎪ ⎪ L S1(B0, A 0 , B1, A1, ⋯)sin(1θ ) = RS1 sin(1θ ) ⎪ ⎩⋮

( ka˜

˜

nr cmn

˜

) ) )

+ kb˜ nr d˜ mn + mnr a nrcmn

(

1 ka˜ nr d˜ mn + kb˜ nr c˜ mn − mnr a nrdmn 4

(

)

− mnr bnr cmn

NC = J

1 = ka˜ nr d˜ mn − kb˜ nr c˜ mn − mnr a nrdmn 4

( (

Similarly, the right term of Eq. (14) can also be written as

(24)

(25)

⎡ N11 N12 ⋯⎤ ⎥ ⎢ N = ⎢ N21 N22 ⋯⎥ ⎢⎣ ⋮ ⋮ ⋱⎥⎦

1 = ka˜ nr d˜ mn − kb˜ nr c˜ mn + mnr a nrdmn 4

)

(23)

in which

)

+ mnr bnr cmn

− mnr bnr cmn

nr fmn

in which LCi(B0, A0 , B1, A1, ⋯) and LSi(B0, A0 , B1, A1, ⋯) are functions related to cos(iθ ) and sin(iθ ) with undetermined coefficients of Ai and Bi . RCi and RSi are the coefficients of cos(iθ ) and sin(iθ ) from Eq. (22). The set of equations can also be written in the form of matrix equation as

)

+ mnr bnr cmn

8 ξnn rm

3 ζnn = rm

( ka˜ ( ka˜ ( ka˜

˜

)

7 ξnn rm

2 ζnn = rm

1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4

(i = 1, 2, … , 8) are given by

nr cmn

1 = ka˜ nr d˜ mn + kb˜ nr c˜ mn + mnr a nrdmn 4

6 ξnn = rm

1 ζnn = rm

7 ζnn = rm

+ mnr bnr dmn 5 ξnn rm

in which

i ζnn rm

(26)

(21) T C = ⎡⎣ C 0 C1 ⋯⎤⎦ , T J = ⎡⎣ J0 J1 ⋯⎤⎦ ,

Ci = ⎡⎣ Bi Ai ⎤⎦

(27)

Ji = ⎡⎣ RCi RSi ⎤⎦

(28)

Actually, the size of matrix N is infinite and the coefficients Bm and Am are dependent upon each other, except in the case of a circular section. For a circular cylinder, the matrix N is simplified to a diagonal matrix, and the solution of Eq. (25) can be obtained

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

as

Bm = −

J′ (ka 0) e˜m0 =− m , ′ (ka 0) c˜ m0 Hm

Am = 0

)

(

→ nS )⋅ nX r (θ )dθ dz ∫−h ∫S = 0 P( −→

F¯Y =

→ nS )⋅ nY r (θ )dθ dz ∫−h ∫S = 0 P( −→

(33)

0

(34)

→ → in which nX and nY are unit vectors in the x− and y−directions, respectively. In the following equation, the maximum dimensionless hydrodynamic force defined in Mansour et al. (2002) is adopted in the analysis

FX =

F¯X − max 2ρgAπa 02

; FY =

F¯Y − max 2ρgAπa 02

(35)

in which a0 is the coefficient of cos( 0θ ) in the Eq. (10). The maximum dimensionless wave run-up can also be defined as

R = Δmax /2A

(36)

in which Δmax is the maximum wave run-up around the cylinder surface. In the linear wave theory, the dimensionless wave forces defined in Eq. (35) are affected by the water depth, which can be seen from Eqs. (33) and (34) by substituting Eq. (32) into them. In other word, the wave forces are dependent on h/a0 from the

(

)

function of tanh ( ka0)⋅( h/a0) in the velocity potential. As for the dimensionless wave run-up defined in Eq. (36), it is not affected by the water depth.

)

complex conjugation transpose of the matrix). Generally, for a specific matrix N , the Moore–Penrose pseudoinverse matrix N+ s can be obtained by using the MATLAB software package. Then, the solution of Eq. (25) is given by +

C=N J

(30)

Once the matrix C is obtained, the velocity potential of the incident and scattered waves in the computation domain can be calculated by substituting the coefficients Ai and Bi into Eq. (8). With the velocity potential, the free surface elevation can be computed by

⎧ ⎪ 1 ∂Φ η( r , θ , t ) = Re⎨ − ⎪ ⎩ g ∂t

0

F¯X = (29)

From the above equation, it can be found that the solution of this case obtained from the present method in this study is equal to that of MacCamy and Fuchs (1954). For noncircular cross-section, the matrix could be truncated at a specific order to balance the accuracy and computer efficiency because the values of Bm⋅Hm( k⋅ min(r (θ ))) and Am ⋅Hm( k⋅ min(r (θ ))) become very small when the order of m is large. In this case, if the matrix is truncated at M-order, the matrix dimensions of N and J become ( 2M + 2) × ( 2M + 2) and ( 2M + 2) × 1, respectively. In the present method, the matrix N is singular in some cases, such as a circular cylinder. Therefore, the inverse matrix of N does not exist in such cases. Moreover, the modulus of the Hankel function Hm(⋅) increases sharply with the increase of order. Therefore, the condition number of the matrix N becomes much larger with the increase of m, resulting in the ill-posed problem of Eq. (25). To assure the solution of Eq. (25), it should find a solution to make the Euclidean norm ‖NC − J‖ reaching its least value. In this study, the Moore–Penrose pseudoinverse, which was proposed by Moore (1920) and Penrose (1955, 1956), is utilized as the tool to deal with the least squares solution of the linear systems. For the Moore–Penrose pseudoinverse matrix N+ , it should satisfy the following four criteria: (1) NN+N = N ; (2) N+NN+ = N+; (3) NN+ * = NN+ ; and (4) N+N * = N+N (the superscript * means the

(

167

⎫ ⎬ ⎪ z= 0 ⎭ ⎪

(31)

The pressure in the fluid domain is given by

P ( r , θ , z, t ) = − ρgz − ρ⋅Re

{ } ∂Φ ∂t

(32)

With the pressure applied on the cylinder, as shown in Fig. 2, the wave force acting on the cylinder along the x− and y−axes are then computed by

3. Validation and case study 3.1. Validation The bottom-mounted uniform vertical cylinder with cosinetype radial perturbations is used for validating the present method. The geometric surface of such a cylinder is expressed by the radius function of the cross section as

r (θ ) = a 0(1 + ε cos nr θ )

(37)

in which a0 is the radius of circular section and ε is the perturbation value of the radius. With the defined radius function, the cross-sectional geometric shape of the cylinder is varied with the increase of the parameters nr and ε . As an example, Fig. 3 shows five types of cross-section under the condition of ε = 0.05 and increasing nr . An analytical study based on the perturbation theory and a numerical study based on Green's theory were investigated by Mansour et al. (2002) to determine the wave action on such types of cylinders subjected to a train of regular waves. The results of the maximum dimensionless wave force and wave run-up by the

Fig. 2. Definition of the rotation angle and the pressure vector on the cylindrical surface.

168

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

Fig. 3. Cross-section with different value of nr (ε¼ 0.05).

Fig. 4. Comparative results of a cylinder with ε = 0.05 and nr = 2: (a) maximum dimensionless wave force in the x-direction; and (b) maximum dimensionless wave run-up.

Fig. 5. Comparative results of a cylinder with ε = 0.05 and nr = 3: (a) maximum dimensionless wave force in the x-direction; and (b) maximum dimensionless wave run-up.

Fig. 6. Comparative results of a cylinder with ε = 0.05 and nr = 4 : (a) maximum dimensionless wave force in the x-direction; and (b) maximum dimensionless wave run-up.

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

169

Fig. 7. Comparative results of the cylinder with ε = 0.1 and nr = 3 and 4: (a) maximum dimensionless wave force in the x-direction; and (b) maximum dimensionless wave run-up.

Fig. 8. Effects of the truncated error for the cylinder with ε = 0.05 and nr = 3. (a) Maximum dimensionless wave force in the x-direction; and (b) maximum dimensionless wave run-up.

Fig. 9. Elliptical cross-section and the Fourier expansion: (a) schematic diagram of the cross-section; and (b) Fourier expansion of the cross-section.

numerical methods is adopted herein for validating the present method. In this study, the value of h/a0 equaling to 1.16 is adopted in the following analysis. Figs. 4–6 show the comparative results for the structure with ε = 0.05 and nr = 2, 3, 4 , respectively. The analytical method developed by Mansour is valid for the perturbation value of ε within the range of (0, 0.05]. Beyond this range, Mansour suggested using the numerical method. As shown in these figures, the maximum dimensionless wave force is increased with the increase of ka0 and

then decreased. The analytical solution of the Mansour method can effectively match the numerical simulation. However, when ka0 > 2.5, the discrepancy becomes relatively large, especially for the case of nr = 4 . For the wave run-up, the analytical solutions of Mansour exhibit some large errors compared to the numerical results when ka0 becomes large. However, the comparison shows that the analytical results of the present method are in good agreement with the numerical results. For a larger value of ε , comparisons were also conducted to

170

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

Fig. 10. Comparative results of the cylinder with elliptical cross-section: (a) maximum dimensionless wave force in the x-direction; and (b) maximum dimensionless wave run-up.

Fig. 11. Wave run-up around the cylinders with various rotational angles ( ε = 0.05, ka0 = 2): (a) nr = 2; (b) nr = 3; (c) nr = 4 ; and (d) nr = 5.

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

171

proposed method, the maximum dimensionless wave force in the x-direction and the wave run-up are depicted in Fig. 10 under the wave incident angle of 0o and 90o . For comparison, the numerical simulation results obtained from the open source code NEMOH (Babarit and Delhommeau, 2015), which is a software code for solving the linear hydrodynamic problem of offshore structures, are also presented in this figure. It is seen from the comparison that the analysis results obtained from the present method shows a good coincidence with the numerical results, which also validates the effectiveness and accuracy of the developed method for solving the linear wave diffraction problem with a complex crosssection. 3.2. Case study 1: cylinders with cosine-type perturbation cross sections

Fig. 12. Dimensionless wave force versus the rotation angle ( ε = 0.05, ka0 = 2): (a) x-direction; and (b) y-direction.

investigate the effectiveness of the present method with ε = 0.1 and nr = 3, 4 . Fig. 7 shows that the calculated wave forces acting on the cylinders and the wave run-up obtained from the present method also coincide well with the numerical results, which demonstrates the accuracy and effectiveness of the present method. For the present method, the effects of the truncation error are analyzed in the following. Fig. 8 shows the dimensionless wave forces and wave run-up for the cylinder with nr = 3 and ε = 0.05. From the comparative results, the wave force can achieve an accuracy result when the truncated order is selected as 3. However, the effect of the truncation error is large for the wave run-up with the same truncated order, especially for ka0 in the range of [2,4]. When M ≥ 9, the wave run-up can achieve accurate results by using the present method. Furthermore, the appropriate truncated order is also dependent on the shape of the cylindrical cross-section. For uniformity, a truncated order of 20 is used for the analysis of the cosine-type section. The second example is to solve the wave diffraction by an elliptical cylinder. Fig. 9(a) shows the schematic diagram of the elliptical cross-section. The dimensions of y0 : x0 = 0.8 and h: x0 = 2 were considered in this study. With this given configuration, the elliptical cross-section can be represented well by a Fourier series with a truncation order of 6, as shown in Fig. 9(b). By using the

In the following discussion, the wave run-up and wave force of the above cylinders with cosine-type perturbation cross-sections is analyzed considering the effects of the rotation angle, as defined in Fig. 2. As an example, the parameters of ε and ka0 are set as 0.05 and 2, respectively. Because the geometric shape of the cylinder in the cases of nr = 1 is approximately similar to a circle, the effects of the rotation angle are very small for such a geometrical shape. Therefore, the following analysis focused mainly on the cases of 2 ≤ nr ≤ 5. Fig. 11 shows the wave run-up profiles around the cylinders under various rotational angles. The effects of the rotation angle are very small for the case of nr = 2 because the configuration of the cross-section is close to the circle. In this case, a very slight variation of the wave run-up curves versus the rotation angle can be observed from Fig. 11(a) for the whole circle. For all of the geometrical shapes, the maximum wave run-up often occurs somewhere upstream. With the increase of nr , the variation of the wave run-up of the upstream portion of the circle becomes more obvious and is significantly affected by the more complex geometrical shape of the cylindrical surface, as shown in Fig. 11(b)-(d). From the analysis results, the wave run-up at the downstream part of the cylinder, especially at the specific location near θ ¼0° (or θ ¼ 360°), is not sensitive to the geometry of the cross-section. Fig. 12 shows the maximum dimensionless wave force in the xand y-direction versus the rotational angle in one period. The maximum wave force for the case of nr = 1 is almost equal to the maximum wave force of the circular cylinder, which can be theoretically proven to be 0.275 and 0 for FX and FY , respectively. The analysis results show that the effect of the rotation angle is more obvious for the cylinders with nr = 3, although the wave run-up shown in Fig. 11(b) is less influenced by the rotation angles compared with Fig. 11c and d. When nr ≥ 3, the fluctuation varies with rotation angle decrease and the wave force in both directions

Fig. 13. Effects of ka0 for the cylinder with ε = 0.05 and nr = 3: (a) maximum dimensionless wave force in the x-direction; (b) maximum dimensionless wave run-up.

172

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

Fig. 14. Wave run-up around the cylinder with various ka0 ( ε = 0.05, α = 0o ): (a) nr = 2; (b) nr = 3; (c) nr = 4 ; and (d) nr = 5.

Fig. 15. Schematic diagram of the caisson foundation.

closes gradually to the circular case. Parameter ka0 represents the relative dimension between the wave length and the geometrical size of the cross-section of the

cylinder. The diffraction effects of the wave action are not significant as ka0 < 0.63 for the small-scale structures. With the increase in the cylindrical size, the effects of the wave diffraction

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

Fig. 16. Fourier expansion for the cross-section. Table 1 Wave properties of the extreme sea states. Sea state

T (s)

H (m)

h/L

H /L

25-year return period 100-year return period Long-period surge wave (case 1) Long-period surge wave (case 2)

8.7 9.6 15 30

7.1 8.6 10.1 12

0.400 0.334 0.169 0.075

0.061 0.062 0.037 0.019

Table 2 Results of the max inline force and max wave run-up.

α

T (s)

ka0

F¯X − max (MN)

FX

Δmax (m)

R

0°

8.7 9.6 15 30 8.7 9.6 15 30

1.92 1.60 0.81 0.36 1.92 1.60 0.81 0.36

94.4 141.2 296.2 244.0 63.8 100.9 217.2 187.7

0.34 0.42 0.75 0.52 0.23 0.30 0.55 0.40

7.46 8.77 9.39 7.56 6.60 7.83 8.08 7.20

1.05 1.02 0.93 0.63 0.93 0.91 0.80 0.60

90°

should be considered for the calculation of the wave force due to the obvious effects of the scattered waves. Fig. 13 shows the maximum dimensionless wave force and wave run-up of the cylinder with ε = 0.05 and nr = 3 versus ka0 and rotation angle α . The effects of the rotation angle α are negligible on the wave force FX when ka0 is less than 0.75. With the further increase of ka0, the influence of α on the wave force becomes obvious. Following the increase of the ka0, the wave force exhibits a tendency to decrease

173

with the increase of the rotation angle α . For the wave run-up, a similar tendency can also be found in Fig. 13(b). The rotation angle α has a negligible effect on the maximum wave run-up when ka0 is smaller than 0.5. Then, an obvious difference of the maximum wave run-up can be observed with the change of α . Compared to a flat surface, a smooth corner in the upstream would cause less wave action. It is therefore easy to understand why the wave force and wave run-up are smaller than the value of other rotation angles as α = 60o . Fig. 14 shows the simulation results of the wave run-up for the condition of ε = 0.05 and α = 0° with various values of ka0 . The wave run-up is obviously affected by ka0 upstream of the cylinder. For the cylinder with nr = 2, the dimensionless wave run-up upstream is slightly increased with the increase in ka0 . When the geometry of the cylinder becomes complex with a large nr , the effects of the parameter ka0 also become complex. For example, in the case of nr = 4 , the maximum dimensionless wave run-up occurs at the location with angles of 123o and 237o. Downstream of the cylinder, the effects of ka0 on the wave run-up also become relatively obvious. Furthermore, the downstream wave run-up is not sensitive to the geometrical features of the cylinder if ka0 is the same, which is similar to the conclusion obtained from Fig. 11. 3.3. Case study 2: a quasi-ellipse caisson foundation of the bridge pylon Large-scale caisson foundation is a typical component for supporting the tall bridge tower of the long-span bridge structure in deep water due to the advantage of large stiffness and easy construction. In deep water, the caisson foundation exhibits obvious wave diffraction characteristics when subjected to wave attack. Establishing a simple method to estimate the wave forces acting on the caisson foundation is an important issue for engineers to assure the safety of the structures under wave action. As an example, the wave force and wave run-up of a quasiellipse caisson foundation of a cross-strait bridge pylon are analyzed by using the present method in this subsection. The schematic diagram of the foundation is shown in Fig. 15. The caisson foundation has an approximately elliptical section with dimensions of 60 m  80 m, combining a central rectangle of 20 m  60 m and two external half circles with radii of 30 m. To employ the present method, the radius function of the quasi-ellipse cross section of the caisson foundation is first expanded into a Fourier series. As an example, Eq. (38) gives the formula of the radius function in the case of α = 0o . Fig. 16 shows the comparison of the actual shape of the curve of the cylinder surface and the approximate fitting curves by the Fourier series with the truncated order of 2 and 14, respectively. From the figure, the expanded Fourier series can be effectively used to represent

Fig. 17. Wave run-up profiles around the cylinder surface: (a) α = 0o ; and (b) α = 90o .

174

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

the quasi-ellipse cross section of the caisson foundation with an order of 14.

r (θ ) α= 0o

⎧ R0 for 0 ≤ θ < arctan(L/R 0) ⎪ ⎪ cos(θ ) ⎪ 2L sin(θ ) + 4R 02 − 2L2(1 + cos(2θ )) ⎪ ⎪ =⎨ 2 ⎪ ( for arctan L /R 0) ≤ θ ≤ π /2 ⎪ ⎪ r (π − θ ) for π /2 ≤ θ < π ⎪ ⎪ ⎩ r (θ − π ) for π ≤ θ ≤ 2π

(38)

The water depth, h, at the construction site of the bridge is 46.64 m. The extreme sea states with a return period of 25 and 100 years are used for the structure design according to the meteorological and hydrological data. Furthermore, surge waves with a long period of 15 s and 30 s are also considered for the foundation. Table 1 lists the wave properties of the corresponding sea states, including the wave height, H ; the wave period, T ; and ratios of h/L and H /L . The values of h/L show that all of the extreme waves are in the region of intermediate water depth. Based on the wave properties and the geometric configuration of the foundation, the inline wave force acting on the cylinder and the wave run-up are calculated by using the present method. Considering the space limitation, only the analysis results for the structure with a rotation angle of 0° and 90° are given in this study. As shown in Table 2, the inline wave force acting on the foundation is found to be 94.4 MN with a dimensionless wave force of 0.34 for the case of α = 0o and T = 8.7s. For the sea state with a 100-year return period, the dimensioned and dimensionless wave forces are increased to 141.2 MN and 0.42, respectively. The wave run-up achieves the maximum dimension value of 8.77 m for the case of the 100-year return period, although the dimensionless value is relatively smaller than that of the 25-year return period. For the surge wave with long periods, the wave force that is obtained is significantly larger than the other cases. When the rotation angle between the pylon foundation and the incident wave is changed to 90°, both the wave force and the wave run-up are smaller than the wave force and the wave run-up at α = 0o for the same sea states. To more obviously investigate the wave run-up around the pylon foundation, Fig. 17 shows the profiles of the maximum dimensionless wave run-up under four sea states. Significant differences in the wave run-up around the pylon foundation is observed for the sea state of the 25-year return period with a short wave period. With the increase of the wave period and the wavelength, less wave disturbance is found from the wave run-up profile, especially for the surge wave with a period of 30 s From the above analysis, it can be found that the wave force and wave run-up of the caisson foundation with noncircular crosssection can be calculated by using the present method. With this method, it can also obtain the surface pressure of the structures and wave surface elevation. From the engineering point of view, the method developed in this study provides a relative simple and easy way for structural design compared with the common numerical method.

4. Conclusions An analytical method is presented in this study to investigate the linear wave diffraction around a uniform vertical cylinder with an arbitrary smooth cross-section. Based on the assumption of the radial function of the cylinder surface being expanded into a Fourier series, the unknown coefficients for the scattered-wave potential can be solved by a linear equation set. The analytical

method is validated by a uniform vertical cylinder with cosinetype radial perturbations. The wave-structure interaction with such types of cylinders and the practical engineering application with a quasi-ellipse caisson are also comprehensively investigated by using the proposed method. The following conclusions can be drawn from this study: (1) The analytical method presented is capable of predicting the wave force and wave run-up, considering the wave diffraction effects, for the vertical uniform cylinder with an arbitrary smooth cross section. Compared to the numerical method, the analytical solution is simple and accurate for the estimation of the wave force for the structure design. (2) From the comparable results for the cylinder with cosine-type radial perturbations, this approach has a higher accuracy for the prediction of the wave force and wave run-up with large ka0. Furthermore, because the cylinder surface is represented by a Fourier series expansion, this approach can be effectively implemented to solve the wave force and wave run-up of such a structure considering the effects of the rotation angle between the structure and the incident wave on the wave diffraction. (3) For the large-scale quasi-ellipse caisson foundation bridge pylon, the long-period surge waves induced a larger wave force on the structure. The effects of such waves should be considered in bridge design.

Acknowledgments The authors would like to acknowledge the financial support from the National Key Technology R&D Program of China (2014BAL05B02), the National Natural Science Foundation of China (51222808), the Fundamental Research Funds for the Central Universities of China (HIT.BRETIV.201320).

References Au, M., Brebbia, C., 1983. Diffraction of water waves for vertical cylinders using boundary elements. Appl. Math. Model 7 (2), 106–114. Babarit, A., Delhommeau, G., 2015. Theoretical and numerical aspects of the open source bem solver nemoh. In: 11th European Wave and Tidal Energy Conference, Nantes, France. Chakarabarti, S., Tam, A., 1975. Interaction of waves with large vertical cylinder. J. Ship Res. 19 (1), 23–33. Chau, F., Taylor, R.E., 1992. Second-order wave diffraction by a vertical cylinder. J. Fluid Mech. 240, 571–599. Chen, H., Mei, C., 1973. Wave forces on a stationary platform of elliptical shape. J. Ship Res. 17 (2), 61–71. Duffy, D.G., 2015. Green's Functions with Applications. CRC Press, Boca Raton, USA. Feng, M., 2013. Modern bridges in china. Struct. Infrastruct. Eng. 10 (4), 429–442. Ghalayini, S.A., Williams, A.N., 1989. Nonlinear wave forces on vertical cylinders of arbitrary cross-section. J. Waterw. Port Coast Ocean Eng. 115 (6), 809–830. Havelock, T., 1940. The pressure of water waves on a fixed obstacle. Proc. R. Soc., Lond., Ser. A 175, 409–421. Isaacson, M., 1978. Vertical cylinders of arbitrary section in waves. J. Waterw. Port. Coast Ocean Div. 104 (3), 309–324. Lighthill, J., 1979. Waves and hydrodynamic loading. In: Proceedings of the Second International Conference on the Behaviour of Off-Shore Structures, Imperial College, London, England, pp. 1–40. MacCamy R., Fuchs R., 1954. Wave Forces on Piles: a Diffraction Theory, Technical Memo., US Army Board, US Army Corp. Eng. Malenica, Š., Molin, B., 1995. Third-harmonic wave diffraction by a vertical cylinder. J. Fluid Mech. 302, 203–229. Mansour, A.M., Williams, A.N., Wang, K.H., 2002. The diffraction of linear waves by a uniform vertical cylinder with cosine-type radial perturbations. Ocean Eng. 29 (3), 239–259. McNatt, J.C., Venugopal, V., Forehand, D., 2013. The cylindrical wave field of wave energy converters. Int. J. Mar. Energy 3–4, e26–e39. Miao, G., Liu, Y., 1990. Hydrodynamic forces for large vertical cylinders of arbitrary section. China Ocean Eng. 4 (2), 145–158. Miao, G., Liu, Y., Mi, Z., 1993. Computation of hydrodynamic forces on vibrating

J. Liu et al. / Ocean Engineering 126 (2016) 163–175

multiple large vertical cylinders of arbitrary section. Mar. Struct. 6 (2), 279–294. Molin, B., 1979. Second-order diffraction loads upon three-dimensional bodies. Appl. Ocean Res. 1 (4), 197–202. Molin, B., 2002. Hydrodynamique Des Structures Offshore. Editions Technip, Paris, France. Moore, E.H., 1920. On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26, 394–395. Naserizadeh, R., Bingham, H.B., Noorzad, A., 2011. A coupled boundary elementfinite difference solution of the elliptic modified mild slope equation. Eng. Anal. Bound. Elem. 35 (1), 25–33. Newman, J.N., 1996. The second-order wave force on a vertical cylinder. J. Fluid Mech. 320, 417–443. Penrose, R., 1955. A generalized inverse for matrices In: Proceedings of the Cambridge Philosophical Society. Cambridge University Press, 51, vol. 3, pp. 406– 413. Penrose, R., 1956. On best approximate solutions of linear matrix equations. In: Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge

175

University Press, 52, vol. 01, pp. 17–19. Sarpkaya, T., 2010. Wave Forces on Offshore Structures. Cambridge University Press, New York, USA. Shankar, N.J., Balendra, T., Soon, C.E., 1984. Wave loads on large vertical cylinders: a design method. Ocean Eng. 11 (1), 65–85. Song, H., Tao, L., Chakrabarti, S., 2010. Modelling of water wave interaction with multiple cylinders of arbitrary shape. J. Comput. Phys. 229 (5), 1498–1513. Tao, L., Song, H., Chakrabarti, S., 2007. Scaled boundary fem solution of shortcrested wave diffraction by a vertical cylinder. Comput. Methods Appl. Mech. ENG 197 (1–4), 232–242. Wehausen, J.V., Laitone, E.V., 1960. Surface Waves. Springer, Berlin, Heidelberg. Williams, A.N., 1985. Wave-forces on an elliptic cylinder. J. Waterw. Port Coast Ocean Eng. 111 (2), 433–449. Zhu, S.P., Moule, G., 1994. Numerical-calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-section. Ocean Eng. 21 (7), 645–662.