ChenXB-ICHD06

7/30/2019 ChenXB-ICHD06

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The set-down in the second-order Stokes waves

Xiao-Bo Chen

Research Department, Bureau Veritas, Paris La Defense, France

Fax: +33-1-4291.3395 Email: [email protected]

ABSTRACT

Under the assumption of small wave steepness, the classical Stokes expansion is used to analysisregular and bichromatic waves upto the second order. An additional set-down term is derived byconsidering the limit of second-order bichromatic waves of two wave frequencies when one wavefrequency tends to another and its equivalence to the second-order regular waves. It is shown thatthe resulting contribution to set-down is much more significant than the classical Stokes term.Some discussions and conclusions are given at the end of the paper.

1 INTRODUCTION

Based on the power series of the wave steepness ka which is assumed to be small (ka 1), Stokes(1847) gave the nonlinear solution for regular wave trains in deep water and then extended to finitewaterdepth. The largely used form of the Stokes waves up to the second order is written as :

= a sin(kxt +) ka2

2

3tanh2(kh)

2 tanh3(kh)

cos(2kx2t+2)

ka2

2 sinh(2kh)(1)

in which (a,k,,,h) stand for wave amplitude, wavenumber, wave frequency, phase and water-depth, respectively. The first term on the right hand side of (1) is the first-order Stokes wavesalso called as Airys waves. The second term is the second-order correction which makes the crestof Airys waves sharper and the trough flatter.

The third term

D = ka2

2 sinh(2kh)(2)

is a negative constant called the set-down which represents the mean level in regular Stokes waves.

There is an extra arbitrary term involved in the Bernoulli equation related to the need to specifyan initial condition. By assuming that the mean level in regular waves is the same as that of acalm sea by Molin (2002), the value of is simply determined as the magnitude of the third termbut with the opposite sign, i.e. = D.

The so-defined regular Stokes waves of the second order have two issues. One concerning itsvalidity in describing free-surface elevation especially in shallow water (kh 1), is solved by therequirement that the ratio between the magnitude of the second term and that of the first term issmall :

ka[3tanh2(kh)]

4 tanh3(kh) 1 or ka/(kh)3 1 for kh 0 (3)

in agreement with the analysis by Ursell (1953). The value ka/(kh)3 is then called as Ursells

parameter.


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The other issue concerns the inconsistence of the global set-down in regular waves and in bichro-matic waves. Considering two regular waves with frequencies 1 and 2, unit amplitude and thesame initial phase, the set-down in the limit as 1 2 is not equal to that of a regular wave ofthe same frequency with the amplitude doubled. The problem was raised by Eatock Taylor (1989)but the controversy has persisted or totally ignored as in Dalzell (1999).

In this paper, we start with the analysis on the second-order velocity potential in both regular andbichromatic waves. By making the limit of bichromatic wave of two frequencies when one tends toanther and considering the equivalence to the regular wave, an additional set-down term is derivedso that a consistent formulation of second-order Stokes waves is developed.

2 SECOND-ORDER VELOCITY POTENTIAL

The velocity potential describing regular and bichromatic waves satisfies the Laplace equation,a no-inflow condition on the sea bed and a nonlinear boundary condition on the free surface.Following Stokes expansion, the velocity potential is written as

= (1) + (2) + (4)

The first-order potential (1) is simple as it satisfies a homogeneous condition at the mean freesurface while the second-order potential (2) is conditioned by :

g(2)zz + (2)tt = Q (5a)

with

Q = 2(1)(1)t + (

(1)zz +

(1)ztt/g)

(1)t (5b)

at z =0 with the non-homogeneous term Q in quadratic function of the first-order potential.

2.1 Potential of regular waves

The velocity potential of Airys waves is well known and expressed by the real part of a complexfunction :

(1) = {(1) eit} (6a)

with

(1) = ag

cosh[k(z+h)]

cosh(kh)eikx+i (6b)

in which (, k) satisfy the dispersion relation 2 = gk tanh(kh) with g the acceleration of gravity.

Introducing the first-order potential (1) of regular waves into the second equation of (5b), we have

Q = {qei2t} (7a)

with

q =i3a23

2 sinh2(kh)ei2kx+i2 (7b)

The second-order potential, which satisfies the Laplace equation and the free-surface condition (5a)can be expressed as :

(2) = {(2)ei2t} Cg t (8a)

with

(2)

=

i3a2

8 sinh4(kh) cosh[2k(z +h)]ei2kx+i2

(8b)


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in which the term (Cgt) is the missing term in the original Stokes waves with g the accelerationof gravity and C a constant to be determined and (2) is called double-frequency potential.

2.2 Potential of bichromatic waves

The first-order velocity potential of bichromatic waves can be written as:

(1) = {(1)1 e

i1t + (1)2 e

i2t} (9a)

in which (1)1 and (1)2 are defined by the same expression as (6b) in which we replace (, k) by

(1, k1) and (2, k2), respectively. Introducing (9a) into the second equation of (5a), we have

Q = {q1ei21t} + {q2e

i22t} + {q+ei+t} + {qe

it} (9b)

where q1 and q2 are defined by the same formulation as q given by (7a). The terms q associatedwith are given by

q+ = ia1a2g2A+e

ik+x+i+ (10a)

and

q = ia1a2g2Ae

ikx+i (10b)

with

= 1 2 , k= k1 k2 ,

= 1 2

and

A+ =1+2

12k1k2(T121)

1

2

k21/1

cosh2(k1h)+

k22/2

cosh2(k2h)

(11a)

A =12

12

k1k2(T12+1)+1

2

k21/1

cosh2

(k1h)

k22/2

cosh2

(k2h) (11b)

in which T12 = tanh(k1h) tanh(k2h). Accordingly, the second-order velocity potential in bichro-matic waves is expressed as :

(2) = {(2)1 e

i21t} + {(2)2 e

i22t} (C1+C2)gt

+ {(2)+ e

i(1+2)t} + {(2) e

i(12)t} (12a)

where (2)1 and (2)2 are given by the same formulation as (8a) for

(2), while (2)+ and (2) are

given by :

(2)+ = q+

cosh[k+(z +h)]/ cosh(k+h)

gk+

tanh(k+

h) (+

)2

(12b)

and

(2) = q

cosh[k(z+h)]/ cosh(kh)

gk tanh(kh) ()2(12c)

The potentials (2)+ and (2) are usually called sum-frequency (or high-frequency) potential and

difference-frequency (or low-frequency) potential, respectively.

2.3 Determination of the constant

To determine the constant C in (8a) and (C1, C2) in (12a), we perform 1 2 and 12 at thesame time. The first-order potential of bichromatic waves (9a) becomes the potential of regular

waves (6a) with the amplitude doubled. In order to keep the consistence, the components at double


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frequency and the set-down of the second-order potential in bichromatic waves are quadruple ofthat in regular waves, i.e. :

(2)1 +

(2)2 +

(2)+ 4

(2) (13)

and

(C1 + C2)gt + {(2) e

i(12)t} 4(Cgt) (14)

The first limit (13) is satisfied as can be easily checked. The second limit (14) gives the constant :

C = lim0

{(2) e

it}/(2gt) = ka2

4

4S+1 tanh2(kh)

4S2kh tanh(kh)

(15a)

with

S =sinh(2kh)

2kh+sinh(2kh)(15b)

since C1 C2 C for 0.

3 SECOND-ORDER STOKES WAVES

The free-surface elevation is expanded in the same way as the velocity potential :

= (1) + (2) + (16a)

in which the first-order and second-order components (1) and (2) are defined by :

(1) = (1)t /g (16b)

and

(2) =

(1)t (1)zt /g (1)(1)/2 (2)t

/g (16c)

Introducing the first-order potential of regular waves (6a) into the first equation of (16c), we getthe first term of the Stokes waves (1). From (16c), we observe that the second-order wave (2)

is composed of two parts. One part is quadratic products of the first-order potential representedby the first two terms in (16c). The second part is related directly to the second-order potentialpresented by the third term.

3.1 Regular waves

By introducing (6a) and (8a) of the first- and second-order potentials of regular waves into (16c),we obtain :

(2) = a2A cos(2kx 2t +2) a2(D + C) (17a)

with

A = k

2

3tanh2(kh)

2 tanh3(kh)

(17b)

D = D/a2 =k/2

sinh(2kh)(17c)

and

C = C/a

2

=

k

44S+1 tanh

2(kh)

4S2kh tanh(kh)

(17d)


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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

C (kh)3/k

D (kh)/k

kh

Figure 1: Additional set-down term C (17d) and classical set-down term D (17c)

The first term in (17a) is the double-frequency waves. The second term is the set down. The termD is the original one in (1) contributed by the first-order potential while C, which is missing in theclassical Stokes waves (1) and present in (8a) and (15a), comes from the second-order potential.

It can be shown that :

C (3/4)k/(kh)3 for kh 0 (18a)

and

C k/(4kh1) for kh (18b)

which is more significant than the original set-down in Stokes waves (1) :

D =k/2

sinh(2kh)

k/(4kh) for kh 0 (18c)

and

D ke2kh for kh (18d)

The variation ofC(kh) and D(kh) is presented on the left part of Figure 1 against the wavenumberkh. The values of both C(kh) and D(kh) in the figure are multiplied by (kh)3/k and (kh)/k,respectively.

3.2 Bichromatic waves

By introducing (9a) and (12a) of the first- and second-order potentials in bichromatic waves into

(16c), we get :

(2) = (2)1 + (2)2 +

(2)+12 +

(2)12 (19)

(2)1 is the second-order wave associated with the double-frequency 21 and the two set-down

terms given by the same formulation as (17a) in which we replace ( a,,k,) by (a1, 1, k1, 1).

The formulation (17a) is applicable as well to (2)2 by replacing (a,,k,) by (a2, 2, k2, 2). The

new components (2)+ and

(2) are given by :

(2)+12 = a1a2H12 cos(k

+x +t + +) (20a)

and

(2)12 = a1a2B12 cos(kx t + ) (20b)


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with H12 and B12 defined by

H12 = 1

2g

12(1 T

112 ) +

21 +

22 +

2g2+A+gk+ tanh(k+h)(+)2

(20c)

B12 = 1

2g

12(1 + T

112 )

21

22 +

2g2Agk tanh(kh)()2

(20d)

in which the notations (k, , ) = (k1k2, 12, 12), T12 = tanh(k1h) tanh(k2h) and Adefined in (11) are used.

3.3 Irregular waves

In irregular waves, the free-surface elevation of the first order is represented by the Fourier series :

(1) =Nj=1

aj sin(kjx jt + j) with aj =

2S(j)j (21)

for a sea state represented by its energy spectrum S(). The second-order waves :

(2) =Nj=1

(2)j +Nj=2

j1k=1

(2)+kj +Nj=2

j1k=1

(2)kj

which can be further written explicitly :

(2) = Nj=1

a2j(Dj + Cj) +Nj=1

a2jAj cos(2kjx2jt+2j)

+

Nj=2

j1k=1

akajBkj cos[(kkkj)x(kj)t + kj]

+Nj=2

j1k=1

akajHkj cos[(kk+kj)x(k+j)t + k+j ] (22)

in which the coefficients (Dj , Cj , Aj) are defined in (17b) while Bkj and Hkj by (20d) and (20c),respectively.

4 DISCUSSIONS AND CONCLUSIONS

By an analysis of Stokes waves up to the second order in regular and bichromatic waves, theset-down is found to be the sum of the original one D and the new one C. The component D isdependent on the first-order wave field while C is contributed by the second-order velocity potentialwhich seems to have been missed. Furthermore, the component C is much more significant than

D as shown on Figure 1. Different wave components of bichromatic waves of (1, 2)=(0.53, 0.47)rad/s with an equal amplitude (a1= a2 =0.8m) and zero phase in water of depth (h =15m) at theinstant t = 0 are depicted on Figure 2 against x. The free-surface elevation of (1) is representedby the dot-dashed line. The low-frequency component is illustrated by the dotted line while theset-down component C= 0.152 (much larger than D = 0.016) by the dashed line. The totalfree-surface elevation including also the double-frequency and high-frequency components whichare not shown for the sake of clarity, is traced by the solid line.

This new component C of the set-down in the second-order Stokes waves can also be derived ina regular wave if the method of multiple scales performed in Davey & Stewartson (1974) and Mei(1989) is applied. This finding is hoped to fill the gap in the consistent simulation of second-orderStokes waves in the water of small depth wherever the Stokes theory is still applicable, i.e. thecases when the Ursells parameter is small. Although this set-down component does not contribute

to the horizontal components of low-frequency wave loads, the vertical components of wave loads


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-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-600 -400 -200 0 200 400 600

(1) + (2)

D + C

(2)12

(1)

x

Figure 2: Different wave components of bichromatic waves at t = 0

are much affected. The mean position of a floating structure is pulled down so that the clearancebetween the structures bottom and sea bed (one of design criteria) is reduced.

This effect appears to have been widely overlooked in usual second order analyses. As pointed outin Eatock Taylor (1989), it is necessary to take account of both set-down terms to maintain theconsistency between the second-order mean vertical force in a regular wave and the low-frequencyforce in bichromatic waves. A discontinuity should appear on either side of the diagonal of thequadratic transfer function for second-order vertical forces if the additional set-down term C isomitted.

ACKNOWLEDGMENTS

The author would like to thank Dr. Jerry Huang (ExxonMobil), Dr. Marc Prevosto (IFRE-MER) and Prof. Eatock Taylor (Oxford University) for their constructive inputs, discussions andcomments on the subject.

REFERENCES

Dalzell J.F. (1999) A note on finite depth second-order wave-wave interactions. Applied OceanResearch 21, 105-11.

Davey A. & Stewartson K. (1974) On three-dimensional packets of surface waves. Proc. R. Soc.London A 338, 101-10.

Eatock Taylor R. (1989) Is there an inconsistency in the treatment of low frequency second ordervertical forces ? Proc. 4th IWWWFB, 55-9.

Mei C.C. (1989) The applied dynamics of ocean surface waves. World Scientific.

Molin B. (2002) Hydrodynamique des structures offshore. Editions Technip.

Stokes G.G. (1847) On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441-55.

Ursell F. (1953) The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc.49, 685-94.

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