01 - hydrodynamics - 05 - hydrodynamic analysis for offshore lng terminals chen xiao

7/30/2019 01 - HYDRODYNAMICS - 05 - Hydrodynamic Analysis for Offshore LNG Terminals Chen Xiao

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HYDRODYNAMIC ANALYSIS

FOR OFFSHORE LNG TERMINALS

Xiao-Bo Chen

Research Department, BUREAU VERITAS17bis, Place des Reflets, 92400 Paris La Defense (France)

Email: [email protected]

New developments in the hydrodynamic analysis of second-order wave interaction withthe floating system are presented. They consist of the application of the middle-fieldformulation newly-obtained by Chen (2004) to evaluate not only the drift loads but alsothe low-frequency wave loads, and the adoption of the notation of fairly perfect fluid to

introduce the dissipation in resonant wave kinematics due to complex interaction betweenfloating bodies in offloading operations. The numerical results validated with measure-ments of model tests show that the Newman approximation largely used in practice isnot appropriate in most applications, and that the usual near-field formulation gives re-sults of low-frequency loads with poor convergence. The innovations presented here solvethese issues and provide accurate and efficient computation of full QTF. Furthermore,the dynamic effect of liquid motion in tanks is analyzed and taken into account in thecomputation of global responses.

INTRODUCTION

Recently, there are more and more large floating LNG terminals being developed inremote offshore locations where marine environment can be hostile, in order to distancethemselves from neighbors and minimize permitting issues. As the important part of theLNG system, the terminal can be of a barge type LNG/FPSO including accommoda-tions, gas preconditioning and liquefied plant, a number of storage tanks and offloadingfacilities. It serves also as a support to moor a LNG carrier during offloading operations.The mooring of LNG carrier in side-by-side of the terminal is being considered as thepreferred option. In the design of such mooring system of LNG/FPSO terminals andLNG carriers in deep water or in a zone of shallow water, one key issue is the accu-rate simulation of low-frequency motions of the system to which the second-order waveloading is well known as the main source of excitation. Associated with this issue, themultibody interaction and the dynamic effect of liquid motion in tanks have to be takeninto account in a consistent and efficient way.

The formulations of second-order wave loads are elaborated after this introduction.In particular, the low-frequency load is given by its complete expression including the

contribution of the second-order velocity potential. In the usual way, the low-frequencywave load is expressed by the sum of one part depending on the first-order quantitiesand another contributed by the second-order wave field. The first part being function ofquadratic product of the first-order wave field and responses can be directly evaluatedonce the first-order solution is obtained. The second part can be further decomposed into


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one component depending on the incoming waves and another relating the second-orderdiffracted waves. The indirect method (Molin, 1979) is used to evaluate the second-orderdiffraction load which is formulated by the sum of two integrals of Haskind type, one onthe hull and another on the free surface. The analysis by developing the low-frequencyload into a Taylor series with respect to the frequency () shows that the Haskindintegral on the free surface is of order [()2] or higher so that the approximation offirst-order () is proposed in Chen (1994), one order better than the zeroth-orderNewman approximation largely used in practice.

Amongst the publication on the low-frequency load, more attention is paid on theconstant drift load. In particular, two classes of formulations have b een developed.One is called the near-field formulation derived from the pressure integration on bodys

hull, as in Molin (1979), Pinkster (1981) or Ogilive (1983). Another called the far-fieldformulation in Maruo (1964) and Newman (1967) is obtained by applying the momentumtheorem to the fluid domain. Since the starting points of the two formulations are sodistinct that their appearances are very different especially for floating bodies, additionalterms associated with bodys motion appear in the near-field formulation while the far-field formulation keeps the same form without explicitly involving bodys motions. Theconnection between both formulations has been an interesting issue.

Very recently, an important analysis on the classical near-field formulation has beenrealized in Chen (2004). Based on the use of two variants of Stokess theorems givenin Dai (1998), its shown mathematically that both formulations are indeed equivalent.This theoretical breakthrough brings, in addition, several new formulations. A new near-field formulation is obtained by direct application of the variants of Stokess theorems.It is essentially similar to the classical one with some improvements as terms associated

directly with bodys translations and rotations disappear. Applying the Greens theoremto the domain limited by a control surface, a second new formulation is obtained andinvolves the integrals on the control surface and along its intersection with the meanfree surface. Unlike the formulation given in Ferreira & Lee (1994) obtained by applyingthe momentum theorem and applicable only to the drift loads, this new formulationis absolutely general as it can apply to the high-frequency loads as well as the low-frequency loads, to horizontal load components as well as vertical load components. Aninteresting feature of the formulation concerns the low-frequency wave load for whichthe formulation is largely simplified. In particular, the horizontal components of driftloads involve only a surface integral on the control surface and a line integral along itsintersection with the free surface. This formulation written on the control surface atsome distance from the body is called as the middle-field formulation. It is shown that ithas the same virtue as the far-field formulation to have rapid numerical convergence for

horizontal drift loads. Furthermore, in the case of multiple bodies, the control surfacecan be one surrounding an individual body and the wave loads applied on the surroundedbody are then obtained, while the far-field formulation provides only the sum of waveloads applied on all bodies.

An important application of the developed method is the multibody interaction.The side-by-side situation amplifies the interaction and can yield large kinematics ofwave field in the confined zone. Within the framework of the classical linear potentialtheory, there is not any limit in predicting wave elevations at the free surface whilethe resonant motion in the reality must be largely damped by different mechanisms ofdissipation. Unlike the method developed by Buchner et al. (2001) or that by Newman(2004), we apply directly the authentic equations, presented in Chen (2004), of thefairly perfect fluid involving the energy dissipation via introducing the damping force.The integral equation extended to a limited zone of the free surface is then developed.Numerical examples show that the method is efficient and provide results closer to theexperimental measurements.

The effect of liquid motion in partially-filled tanks of a LNG carrier is taken intoaccount in the seakeeping analysis. Classically, only the hydrostatic effect is taken into


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account by subtracting the corresponding stiffness from the global hydrostatic matrix.This is only valid for very low wave frequencies. The dynamic effect is important es-pecially at a wave frequency close to one resonance frequency of tanks. Under theassumption of linear potential flow, the fluid motion in tanks can be evaluated by solv-ing the boundary value problem involving the same Green function which satisfies thefree surface condition. To approximate the damping effect to liquid motion in tanks, thecondition on tanks wall is modified by introducing a small positive parameter equivalentto a partial reflection of walls. This implies that the main part of dissipation occurs inthe boundary layer. Numerical results show that the approximation is good enough tocapture the major coupling effect of liquid motion with the global motion of vessels.

Finally, some discussions and conclusion on the foregoing analysis are addressed.

Furthermore, an important issue on the set-down in the second-order Stokes waves israised. The inconsistence of the global set-down in regular waves and in bichromaticwaves when one of two frequencies tend to another has been controversy : ones supposethe zero constant set-down and others argue that the limit of a bichromatic wave isnot a regular wave. The analysis in Chen (2005) gives a consistent expression of thesecond-order incoming waves which includes an additional term ignored in the classicalformulation of Stokes waves. This analysis is hoped to be useful in healing the breach.

LOW-FREQUENCY LOAD AND APPROXIMATION

Numerous studies have been devoted to the analysis of second-order wave loads. Anon-exhaustive list includes the classical work by Maruo (1964), Newman (1967, 1974),Molin (1979), Pinkster (1980) and Ogilvie (1983). The general formulation of second-

order wave loads can be obtained by directly integration of the second-order pressureon the hull surface of bodys mean position and the variation of the first-order loadsdue to the first-order motions. The second-order wave load is then composed of onepart dependent on the quadratic product of the first-order quantities and another partcontributed by the second-order potential :

(F, M) = (F1, M1) + (F2, M2) with (F2, M2) =

H

ds (2)t (n, rn) (1)

where F = (Fx, Fy, Fz) stands for the forces, M = (Mx, My, Mz) for the momentsand (2) for the second-order potential. The commonly-used formula of the first part(F1, M1) is given by :

F1 =g

2

d (23)n + H

ds ()2/2+ Xt+Rtn (2a)M1=

g

2

d (23)(rn)+

H

ds

()2/2+ Xt+Rt

(rn)+ tTn

(2b)

in which all quantities are of the first order as for the free-surface elevation, for thevelocity potential, X = T+Rr = (1, 2, 3) for the displacement due to the translationT = (1, 2, 3) and rotation R = (1, 2, 3), and r = (xx0, yy0, zz0) for the positionvector with respect to the reference point (x0, y0, z0) of rotation. In (2), stands forthe intersection of the hull H at its mean position with the mean free surface F(z = 0)which is supposed to be wall-sided. The normal vector n is oriented inwards to the fluid,as already described previously.

The line integral in (2) is the result of the integration of the first-order pressureon the intermittent zone around the waterline. The first term in the hull integral of (2)

comes directly from the convective term in Bernoullis equation while the second term isthe correction of the first-order dynamic pressure with respect to the displacement. Theterm associated with the rotation R takes into account of the variation of the normalvector. Finally, the last term in (2b) is the moment induced by the first-order dynamicpressure applied to translated reference point.


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We consider bichromatic waves associated with frequencies (j , k) in which thefirst-order quantities (, X, R, T) are supposed to be in the same form as the first-ordervelocity potential written by

=

ajjeijt

+

ak

keikt

(3)

with (aj , ak) b eing amplitudes of first-order incoming waves. Introducing the form (3)for all first-order quantities into (2), we obtain different components of the second-orderload associated with different frequencies equal to (2j), (2k), (j+k), 0 and (jk),respectively. Since we are interested here only to the low-frequency load, the componentsassociated with the frequencies (2j), (2k) and (j + k) are ignored. Furthermore,the drift load (at zero frequency) can be obtained by the limit of the low-frequency loadassociated with the frequency (jk) when k tends to j . Without loss of generality,we may write the low-frequency load by :

(F, M) =

ajak(f, m)ei(jk)t

with (f, m)= (f1, m1)+(f20, m20)+(f2D, m2D) (4)

where ak stands for the complex conjugate of ak. This rule to denote the complexconjugate by the over line is applied to all first-order quantities in the following.

The first part (f1, m1) can be directly obtained by (2) while the second part (f2, m2)is decomposed into the component (f20, m20) contributed by the incoming waves andthat (f20, m2D) by the diffracted waves :

(f20, m20) =i(jk)H

ds (2)0 (n, rn) and (f2D, m2D) =i(jk)H

ds (2)D (n, rn)

The second-order incoming velocity potential is written as

(2)0 = iA

g2 cosh(kjkk)(z +h)/ cosh(kjkk)h

g(kjkk) tanh(kjkk)h (jk)2eik

(x cos+y sin) (5)

with A defined by

A =jk

jkkjkk

1 + tanh kjh tanh kkh

+

1

2

k2j/j

cosh2 kjh

k2k/k

cosh2 kkh

The contribution by the second-order diffraction potential can be evaluated by Molinsmethod (1979) :

(f2D, m2D)j = i(jk)

H

ds(2)0

nNH

j + i(jk)

g

F

dsNFj (6)

where j is the additional radiation potential at (jk). The non-homogeneous termsare given by :

NF= i(jk)j

k

P +jP

k

0

ij2g

j(2kz + g

2zz)

k

P + gk2k(1tanh

2 kkh)jP

k

0

+

ik2g

k

(2jz + g2zz)

jP + gk

2j (1tanh

2 kjh)k

Pj

(7a)

2NH= (ikxk

k)(Rjn)(ijx

j+j)(Rkn)(xj)

kn(xk)jn (7b)

in which j,kP = (j,kj,k0 ) stands for the perturbation part, the sum of the diffraction

and radiation potentials.


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In summary, the second-order low-frequency wave load is composed of one partdepending on the first-order quantities and another part on the second-order potential.The second part can be further decomposed into one term of integration of incomingwave pressure (f20, m20), one Haskind integral on the hull (f2H, m2H) and one Haskindintegral over the free surface (f2F, m2F) resulting from the second-order forcing on thefree surface. The sum of last two represents the integration of diffraction wave pressure(f2D, m2D), according to (6). Thus, we may write the lth component of QTF :

F(j , k) = (f1, m1) + (f20, m20)

+ (f2H, m2H) + (f2F, m2F)

(8)

Furthermore, the QTF F(j, k) is assumed to be regular function of (j , k) and aTaylor expansion with respect to = (jk) can be developed :

F(j , k) = F0 (j) + F

1(j) + F

2(j)()

2/2 +

The analysis in Chen (1994) shows that the free-surface Haskind integral ( f2F, m2F)

represented by the second integral on the right side of (6) is of order O[()2] or higherso that an approximation of the low-frequency QTF is proposed. This approximationconsists of keeping all terms of (8) excluding only the free-surface integral (f2F, m2F)

is of order O(), one order higher than the approximation of Newman (1974) whichconsists of using only the drift loads and is qualified as the zeroth-order approximation.Furthermore, the numerical results presented in Chen (1994) using the first-order ap-proximation are in good agreement with experimental measurements on the NKossaFPSO while the wave loads based on Newman approximation are largely underesti-mated. As shown on Figure 1, the low-frequency force in surge at = 0.06 rad/s is

depicted on the left by the solid line for the first-order approximation and by the dashedline derived from the Newman approximation. On the right, the surge response spectraby using the first-order approximation (solid line) and Newman approximation (dashedline) are compared with the experimental measurements represented by circles. The time

Figure 1: Low-frequency load Fx( = 0.06) (left) and Surge response spectrum (right)

simulation of low-frequency motions confirm the Newman approximation gives resultstoo low: RMS being 35% (and extreme values up to 60%) smaller than those of modeltests. Very recently, Newman (2004) confirms that the zeroth-order approximation ispoor as the waterdepth is below 100m. It is further recognized that the approximationincluding the effects of the second-order incoming wave potential, and its diffraction bythe body, but not the part resulting from the second-order forcing on the free surface(as proposed by Chen, 1994) gives much better results.

MIDDLE-FIELD FORMULATION

The pressure-integration formulation (2) to compute the first part of low-frequency loadis called as near-field one since the involved terms are evaluated on the hull and along


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the waterline. In the particular case of k = j , the low-frequency wave load becomesconstant drift load which is contributed only by the first part. Another formulationbase on the momentum theorem for the horizontal drift forces has been developed byMaruo (1960) and extended to the moment around the vertical axis by Newman (1967).This formulation involving first-order wave field in the far field is often called far-fieldformulation and preferable in practice thanks to its better convergence and accuracy. Byperforming a local momentum analysis, Ferreira & Lee (1994) developed a formulationover a control surface surrounding the body to evaluate the constant drift load.

Unlike the previous approach based on the momentum theorem for the drift load,the middle-field formulation for low-frequency load has been developed by Chen (2004).Starting with the near-field formulation and making use of the variants of Stokes theo-

rem given in Dai (1998), we obtain a new near-field formulation :

F1 = g

2

d

2n2(Xn)k

+

2

H

ds

()n+2t(Xn)

(9a)

M1=g

2

d

2(rn)2(Xn)(rk)+

2

H

ds

()2(rn)+2(rt)(Xn)

(9b)

which is essentially similar to (2) with some interesting improvements such as all termswith body motion (T, R) disappear and the term involving the displacement in thewaterline integral gives a contribution only to the vertical components. Applying theGreen theorem in a domain D surrounded by S= H C F with the body hull H atits mean position, a fictitious (control) surface C surrounding the body and the meanfree surface F limited by the intersection of H with z =0 and that c of C with z = 0,

we obtain :

F1= g

d (Xn)k+

H

ds(Xtn)+t(Xn)

F

ds

(z+ t)(zt+/2)k

+g

2

c

d 2n +

2

C

ds

2n()n

(10a)

M1= g

d (Xn)(rk)+

H

ds r(Xtn)+t(Xn)

F

dsr(z+ t)(zt+/2)(rk)+

g

2

c

d 2(rn) +

2

C

ds

n(r)()(rn)

(10b)

The new formulation (10) is absolutely general as it can apply to the high-frequency loadsas well as the low-frequency loads, to horizontal load components as well as vertical loadcomponents. The control surface C can be at a finite distance from the body or onepushed to infinity. In the first case, C may be pushed back to H while in the second case,C may be composed of the surface of a vertical cylinder plus the seabed. Furthermore, inthe case of multiple bodies, the control surface C can be one surrounding an individualbody and (10) gives the wave loads applied on the surrounded body.

An interesting feature of (10) concerns the low-frequency wave load for which theformulation is simplified. It can be easily checked that the values of the hull integraland of the first term in the free-surface integral are of order O(). Furthermore, thewaterline integral as well as the second term in the free-surface integral contribute onlyto the vertical loads including the vertical force Fz1 and moments around the horizontalaxis (Mx1 , M

y1 ). Thus, the horizontal components (F

x1 , F

y1 , M

z1 ) of low-frequency loads


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can be expressed as :

Fx1 =jk

2g

c

d jk

n1+

2

C

ds

jnk

x + k

njx

jk

n1

+ fx1 (11a)

Fy1 =jk

2g

c

d jk

n2+

2

C

ds

jnk

y + k

njy

jk

n2

+ fy1 (11b)

Mz1 =jk

2g

c

d jk

n6+

2

C

ds

jn(xk

yyk

x)+ k

n(xjyy

jx)

jk

n6

+ mz1

(11c)

with the additional terms (fx1 , fy1 , m

z1) given by :

fx1 =

2

H

ds

jnk

x/j k

njx/k

2g

F

ds

jj

k

xkk

jx

(12a)

fy1 =

2

H

ds

jnk

y/j k

njy/k

2g

F

ds

jj

k

ykk

jy

(12b)

mz1 =

2

H

ds

jn(xk

yyk

x)/j k

n(xjyy

jx)/k

2g

F

ds

jj(x

k

yyk

x)kk

(xjyyjx)

(12c)

The formulations (11-12) provide, for the first time, an original way to evaluate thehorizontal components of low-frequency wave loads. The additional terms given by (12)are of order (). If the bodys motion is small (X 0) in waves of small period, the

integral over hull surface is negligible since n = Xtn on H. The integral over the part offree surface is ease and accurate since the velocity potentials are not evaluated at bodyssurface.

In regular waves, the formulation (11) reduces to the first two integrals on thecontrol surface since j = k. The low-frequency loads by (11) becomes the drift loads.If the control surface C is put to infinity, the expression (11) is in agreement withthose by Maruo (1960) and Newman (1967). On the surface C at infinity, asymptoticexpressions of the first-order potential can be used to simplify further the formulationto the single integrals involving the Fourier polar variable. This shows formally that theusual near-field formulation and far-field formulation are indeed equivalent.

The near-field, middle-field and far-field formulations are first compared in the com-putation of second-order drift loads on a LNG terminal of size (LengthWidthDraught

= 350m 50m 15m) moored in water of finite depth (h =75m). The meshes of the hullcomposed of 1490 panels, and the control surfaces C F including the part of free sur-face F are illustrated on the left part of Figure 2. Only the half of the hull (y0) andthat of CF for (y0) are presented in the figure. On the right part of Figure 2, thenon-dimensional values of drift load FyD/(gL/2) with L = 350m in waves of heading=195 are depicted against the wave frequency (). Three meshes composed of 1490,3816 and 7824 panels on the hull surface are used. The results using the near-field andfar-field formulations are represented by the dashed, dot-dashed and solid lines for threemeshes (1490, 3616 and 7824 panels), respectively. The results using the middle-fieldformulation are shown by the symbols of circles (1490 panels), crosses (3616 panels) andsquares (7924 panels). The curves associated with the near-field formulation are sepa-rated for > 0.45 rad/s. This shows that the results using the near-field formulationare not convergent in most part of wave-frequency range. On the other side, the results

obtained by the far-field formulation (dashed, dot-dashed and solid lines) are indistin-guishable on the whole range of wave frequency. The same feature is observed for theresults associated with the middle-field formulation (circles, crosses and squares). Fur-thermore, the results of middle-field formulation are in excellent agreement with thoseof far-field formulation.


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EE

Wave heading

Control surfaces

Terminals hull

x

y

-0.4

-0.3

-0.2

-0.1

0

0.2 0.4 0.6 0.8 1 1.2 1.4

Middle-field and far-field

Near-field

Figure 2: Terminals hull & control surfaces (left) and drift loads FyD (right)

-0.2

-0.1

0

0.1

0.2

0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.2 0.4 0.6 0.8 1 1.2 1.4

Middle-field

Near-field

Near-field

Middle-field

Figure 3: Real (left) and imaginary (right) parts of Fy1 in oblique sea

Now, we consider the low-frequency load Fy1 /(gL/2) at a difference frequency(jk) = 0.04 rad/s in waves of the same heading =195

. The results in complex arepresented on Figure 3 against wave frequencies (k). The real part and imaginary partof Fy1 are depicted respectively on the left and right part of the figure. The results ob-tained by using the near-field formulation are illustrated by the dashed, dot-dashed andsolid lines associated with the meshes of 1490, 3616 and 7924 panels, respectively. Theresults obtained from the middle-field formulation are shown by the symbols of circles,crosses and squares associated with three meshes. Again, we observe that the near-field formulation gives the results with poor precision while the middle-field formulationprovides the results of excellent convergence.

MULTIBODY INTERACTION IN OFFLOADING SYSTEMS

The interaction of multiple bodies includes the mechanical and hydrodynamic interac-tions. The mechanical interaction is defined by the mechanical properties of the con-nection between bodies which depend only on the design and operation procedure. Thehydrodynamic interaction is more complex and requires a complete solution taking ac-count of full interaction between multiple bodies. In some cases such as side-by-sidevessels, the hydrodynamic interaction may annul any motion in the confined zone atsome wave frequencies, or create violent kinematics of wave field at other wave frequen-cies. Particular attention to this resonant phenomena is paid and new method based onthe notion of fairly perfect fluid is developed to take into account the damping mecha-

nism in fluid.A few of publications have been realized recently on the resonant motion of wave

field in the confined zone between two floating bodies. Unlike the resonant response ofbodys motion associated with the balance of inertia and stiffness loads, this resonant


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kinematics of fluid is due to the hydrodynamic interaction - wave kinematics annulledor amplified by the complex scattering between bodies. Within the framework of theclassical linear potential theory, there is not any limit in predicting wave elevations atthe free surface while the resonant motion in the reality must be largely damped bydifferent mechanisms of dissipation. This unrealistic fluid motion magnifies the waveloads on the bodies. To hold the wave motion back to a realistic level, Buchner et al.(2001) developed a method consisting to place a lid on the gap in between the twobodies. The unrealistic wave kinematics is then suppressed. In fact, no wavy elevationis possible under the rigid lid and noticeable perturbation around the ends of the liddue to the diffraction effect can be observed. To make wavy motion allowable on thelid, Newman (2004) renders the lid flexible using a set of basis functions of Chebychev

polynomials. The deformation of the flexible mat (equal to the free-surface elevation) isthen reduced by introducing a damping coefficient.

Unlike above methods using an artificial lid, we apply directly the authentic equa-tions, presented in Chen (2004), of the fairly perfect fluid involving already the energydissipation via introducing the damping force. The dissipation term appears in theboundary condition on the free surface :

z k ik = 0 for P F(z = 0) (13)

with k = 2/g and the dissipation coefficient. Following the analysis in Chen (2004),we have :

(P) =

S

ds (Q)G(P, Q) with S = H F F (14)

and the integral equations to determine the source distribution are :

2(P) +

S

ds (Q)Gn(P, Q) = vn P H (15a)

4(P)

S

ds (Q)Gn(P, Q) = 0 P F (15b)

4(P) + ik

S

ds (Q)G(P, Q) = 0 P F (15c)

The integral equation (15b) on the internal waterplane surface F is necessary to elim-inate the irregular frequencies. The integral equation (15c) is written over entire F.However, we know =0 if =0 from (15c). As we need to apply a non-zero value of only in the zone where the fluid kinematics is susceptible to be violent, the discretization

ofF is limited. A practical way is to mesh the zone between two vessels on which a con-stant or a distribution of varying in space can be applied. The first example concerns

0

1

2

3

4

5

6

7

8

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Num. = 0Num. = 0.016Measurement

Gap meshed as the damping zone

Figure 4: Side-by-side barges & damping zone (left) and wave elevation in the gap (right)

2 side-by-side barges of the same dimension in meter (LBT= 2.470.60.18) with


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mechanical properties (zG= 0.02 and gyration radii=0.187/0.527/0.527) free floating inhead waves. The mesh of barges and the whole gap (0.116 as width) meshed as thedamping zone are presented on the left of Figure 4. The free-surface elevation is mea-sured at the center of the gap. The results of numerical computation with two valuesof parameter = 0 and 0.016 are drawn on the right together with those of measure-ments, against the wave frequency (rad/s). Large free-surface elevations are remarkableat three wave frequencies. The results with = 0 (no damping) are much larger thanthose measured while the results with = 0.016 agree well with the measurements. Bycomparison between the curves of numerical results corresponding to = 0 and 0.016,we see that the damping affects only the values in the range of frequencies around onewhere large elevations occur, as expected.

The second example is the case of a Wigley hull placed side-by-side with a barge,presented in Kashiwagi (2004). Both vessels are of dimension in meter (LBT =20.30.125) and set in beam waves with the two separation distances (S1= 1.097 andS2 = 1.797) between two centerlines of the vessels. The case of S1 = 1.097 is considered

Damping zone

Control surfaces

Figure 5: Side-by-side vessels & damping zone (left) and control surfaces (right)

here. The mesh of two vessels is represented on the left of Figure 5 on which a rectangularzone (damping zone) between the vessels is shown as well. On the damping zone, Thevalues = 0 (no damping) and = 0.016 are applied. On the right of the figure, twoseparate control surfaces surrounding respectively the two vessels are illustrated togetherwith the vessels mesh. The drift loads in the beam sea with the Wigley hull on theweather side are computed and compared with the measurements by Kashiwagi et al.(2004). Two vessels are fixed during model tests. The middle-field formulation is usedand the results are found to be quite close to those by the near-field formulation. Thecomparison of drift loads is presented on Figure 6. The sway drift forces Fy

D

/(gLa20

/4)on the Wigley hull and on the barge are shown against the wavenumber kL/2 on theleft and on the right, respectively. Furthermore, the sway drift forces on the Wigley hullalone (without the barge) and those on the barge alone (without the Wigley hull) arerepresented by the dashed lines. It can be seen on Figure 6 that the numerical resultsfrom the middle-field formulation are in good agreement with measurements, exceptthose around kL/2 4.71 where large values appear. The curves (solid lines) with = 0.016 are very close to those (dot-dashed lines) of =0 (no damping) except aroundkL/2 4.71 where the curves with damping are closer to the model tests, as expected.This shows that the importance of damping effect on the strong interaction between twovessels, and that the utility of the small parameter introduced in the boundary conditionat the free surface. It is remarkable that the sway drift force on the Wigley hull on theweather side of beam waves becomes large negative around kL/2 4.71 while the forceon the barge (on the lee side) keeps the same sign and with large values. The sum offorces on two vessels remains positive in the whole range of wave frequency. This showsagain that the multibody interaction is important and can create large forces of repulsion(and attraction as well) between two bodies. At large wave frequencies, the barge onthe lee side withstands less forces than those when it is alone due to the screen effect of


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-4

-3

-2

-1

0

1

2

3

1 2 3 4 5 6 7 8

-1

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8

M.F. = 0M.F. = 0.016MeasurementWigley hull alone

M.F. =0M.F. =0 .016MeasurementBarge alone

Figure 6: Drift load FyD on Wigley hull (left) and that on the barge (right) in beam sea

the Wigley hull on the weather side. At the limit of infinity frequency, the drift forceson the barge should be nil while those on the Wigley hull tend to the value ( gLa20/2)when it stands alone in beam waves.

EFFECTS OF LIQUID MOTION IN TANKS

The full account of liquid motion in tanks such as sloshing effect is difficult due to thenon-linearity of the phenomena and a time-domain solution of coupling is necessary aspresented in Malenica et al. (2004). It is assumed that the major effect to vessels globalmotion is linear so that a solution in frequency domain is possible. This assumption isindeed valid when the liquid motion in tanks is not violent in the range outside of

resonant sloshing frequency. At low wave frequencies, only the hydrostatic effect isimportant and can be taken into account just by introducing negative values for thenon-zero terms of stiffness matrix :

C44 = gIyy; C

45 = gIxy = C

54 and C

55 = gIxx (16a)

with Ixx,xy,yy are the moments of waterplane with respect to its center.

In general cases especially at a wave frequency close to one of resonance, the liquidmotion induces additional inertia loads and damping if energy dissipation is modeled.In fact, the motion equation of the vessel is modified as :

6

j=1 2(Mkj +Akj +A

kj) i(Bkj +B

kj) + Ckj+C

kjaj = Fk (16b)for k = 1, 2, , 6 and the inertia matrix Mkj associated with the mass distributionexcluding the liquid in tanks. The additional mass matrix Akj is equal to that toconsider the liquid as a solid mass in classical approximation valid for low frequenciesand the damping Bkj =0 in this case.

The linear velocity potential due to forcing oscillations of the tank can be solvedin the same way as the solution of radiation problem for the vessel. The matricesAkj and B

kj can then be obtained for each wave frequency. At low wave frequencies,the contribution of liquid in tanks is nearly like solid mass. When wave frequencyapproaches the resonant frequency, the value of inertia increases rapidly without limitat the resonance. The added-inertia changes the sign when the wave frequency goesacross the tank resonant frequency. This variation of inertia modifies the response of

the barge. Instead of one peak without dynamic effect of liquid motion, there are two :one on the left of the tank resonance and another on the right. This can be explainedby the fact that the inertia is largely amplified when the excitation frequency is close tobut smaller than the first tank resonant frequency, the peak of global response is thenshifted on the left. At the tank resonance, the response is largely reduced due to the


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large value (up to infinity if no damping) of added-inertia. At a higher wave frequency,the large negative values of the inertia due to liquid motion yield a second peak resultantfrom a new balance between the total inertia force and stiffness force of the system.

We consider a LNG carrier of 274m in length, 44.2m in width and 11.58 in draught.The tank No.2 of size (LB = 47.1839.1) and the tank No.4 of size (LB = 41.439.1)with a filling height of 10m are placed at the position 144.55m and 64.25m from theafter perpendicular of LNG, respectively. The bottom of tanks is at the height of 3mfrom the baseline. The mesh of LNG together with the two tanks are illustrated onthe left of Figure 7. Different sets of model tests in irregular waves varying significant

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Num. = 0Num. = 0.02

Test n1Test n2Test n3Test n4Test n5

Figure 7: LNG carrier and two tanks (left) and sway RAOs in beam sea (right)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Num. = 0Num. = 0.02


Num. =0Num. =0 .02


Figure 8: Heave RAOs (left) and roll RAOs (right) in beam sea

height HS from 2.5m to 6m and peak periods TP from 8s to 16s associated with thespectrum of Jonswap type were made with a length scale of 1:50. The results of numericalcomputations with =0 and 0.02 are compared with those of measurements. The RAOof sway motion is presented on the right of figure 7 while those of heave and roll areon the left and right of Figure 8. There is not significant difference between the resultswith = 0 and those with = 0.02 except the peak values of sway and roll are slightlysmaller for = 0.02, as expected. It is shown that the numerical results are in very goodagreement with model tests. Not only the position of peaks in sway and roll motionsof numerical computation coincides with that of measurements, but also the values ofpeaks in two sets of results are in excellent agreement.

DISCUSSION AND CONCLUSION

The low-frequency wave load is composed of one part depending on the first-order quan-tities and another part contributed by the second-order wave field. The zeroth-orderapproximation (Newman, 1974) consists of using only the drift load derived only fromthe first-part of wave load is shown to not be appropriate for most applications. On


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the other side, the first-order approximation (Chen, 1994) gives much better results andis considered to be sufficient for most applications in deep water as well as in water offinite depth.

To evaluate the quadratic transfer function of low-frequency load in bichromaticwaves, the near-field formulation derived from the pressure integration is largely usedand considered to be the only way to go, unlike the constant drift load for which thefar-field formulation based on the momentum theorem is available as well. However,the near-field formulation is reputed by its poor precision and convergence, especiallyfor structures hull with sharp geometrical variations. The method using higher-orderdescription of hull geometry (B-spline patches, for example) was hoped to give betteraccurate results than the lower-order method (constant panels). However, the higher-

order method is more sensitive to the singularities which are present in the velocity fieldat sharp corners. As concluded in Newman & Lee (2001), this sensitivity is manifestedwhen the tangential fluid velocity is computed as in the evaluation of the mean pressureor the low-frequency pressure. As a result, the low-frequency load converges slowly orin the worst cases, it may be non-convergent.

The middle-field formulation newly-obtained in Chen (2004) solves this issue. Itsapplication in the computation of second-order low-frequency loads confirms its im-portant advantages. Firstly, it permits to make the connection between the near-fieldformulation derived from the pressure integration and the far-field formulation based onthe momentum theorem for the constant drift load. Secondly, it accumulates the virtuesof both near-field and far-field formulations, i.e. the excellent precision of far-field for-mulation and the access to the low-frequency wave loads as the near-field formulation.Furthermore, in the case of multiple bodies, the middle-field formulation provides the

drift load as well as the low-frequency load on each individual body while the far-fieldformulation can only give the sum of drift loads on all bodies.

Based on the notion of fairly perfect fluid, the damping to reduce, to a reason-able level by comparing to model tests, the resonant kinematics of wave elevation isapplied via the boundary condition at the free surface. Following the same principle,the boundary condition on bodys hull can be modified as well to include a partial re-flection equivalent to energy dissipation in boundary layer. The new integral equationsare established following these modifications. The applications to the side-by-side multi-body interaction and to the liquid motion in tanks show its soundness and efficiency.It is natural to extend the application to the moonpool issue for which the success canbe envisioned. In spite of these successful applications, the method remains to be anapproximation to the dissipation mechanism - an important and complex aspect of fluidmechanics. The involved parameters need to be determined by comparing to experi-mental measurements or results of elaborated CFD simulations.

The dynamic effect of liquid motion in tanks is represented by the added-massand damping (if a dissipation coefficient is applied on tank walls) terms. These termscan be obtained in a similar way as the solution of radiation problems. The importantcoupling effect on global responses of LNG carriers (or floating terminals) is shown andcompared with experimental measurements. The second-order low-frequency load onLNG carriers/terminals must be much affected by these effects.

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

= a sin(kxt+) ka

2

2

3

tanh

2

(kh)2tanh3(kh)

cos(2kx2t+2) ka

2

/2sinh(2kh) (17)

in which (a,k,,,h) stand for wave amplitude, wavenumber, wave frequency, phase andwaterdepth, respectively. The first term on the right hand side of (17) is the first-order


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Stokes waves also called as Airys waves. The second term is the second-order correctionwhich makes the crest of Airys waves sharper and the trough flatter. The third termis a negative constant called the set-down which represents the mean level in regularStokes waves.

The so-defined regular Stokes waves of the second order have two issues. Oneconcerning its validity in describing free-surface elevation especially in shallow water(kh 1), is solved by the requirement that the ratio between the magnitude of thesecond term and that of the first term is small, i.e. :

ka[3tanh2(kh)]

4tanh3(kh) 1 or ka/(kh)3 1 for kh 0 (18)

in agreement with the analysis in Ursell (1953). The value ka/(kh)3 is often calledas Ursells parameter. The other issue concerns the inconsistence of the global set-down in regular waves and in bichromatic waves. Considering two regular waves withfrequencies j and k, unit amplitude and the same initial phase, the set-down in thelimit as j k is not equal to that of a regular wave of the same frequency with theamplitude doubled.

In the notes of Chen (2005), the second-order problem of wave-wave interactionsis described by the system of classical differential equations. The solution providesthe complete expression of second-order bichromatic waves. By making the limit ofbichromatic waves, an additional term written as :

C = ka2

4 4S+1tanh2(kh)

4S2kh tanh(kh) with S =sinh(2kh)

2kh+sinh(2kh)(19)

is obtained. This term is a negative constant in water of finite depth, more significantthan the existing one ka2/[2 sinh(2kh)], and has been ignored in the classical expres-sion of Stokes waves. Although this set-down component does not contribute to thehorizontal components of low-frequency wave loads, the vertical components of waveloads are much affected. Without this term in the analysis of bichromatic waves, an in-consistent discontinuity would appear on either side of the diagonal of quadratic transferfunction for second-order vertical load. With this term, the mean position of a floatingterminal is pulled down so that the clearance between the structures bottom and seabed (one of design criteria in shallow water) is more reduced. Furthermore, this termmust play a role in the second-order decomposition of real waves measured in the siteinto components of free waves and bound waves. As well as in the third-order analysis,there must exist components associated with this term.

The innovative developments have been realized within the software HydroStar -the hydrodynamic part of the software package VeriSTAR-Offshore of Bureau Veritas.HydroStar has benefited from continuous elaborations, inspirations of most recent the-oretical findings and developments of efficient numerical algorithms. The analysis onthe free-surface Green function of wave diffraction and radiation in water of finite depthleads to the development of powerful algorithms. The removal of irregular frequenciesby the extended integral equation method solves the issue associated with the classicalmethod. The implementation of innovative formulations for the computation of second-order wave loads creates new reliable and practical options. Applications to multibodyinteraction and the dynamic effect of liquid motion in tanks with numerous results ex-tend the range of validity of established formulations and developed algorithms, andenrich the database of HydroStar.

ACKNOWLEDGMENTS

The author would like to thank Dr. Marcos Donato Ferreira (CENPES) for his valuablecomments on the middle-field formulation and Dr. Jerry Huang (ExxonMobil) for hisconstructive inputs and discussions on the set-down issue of Stokes waves.


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REFERENCES

[1] Buchner B., van Dijk A. & de Wilde J. (2001) Numerical multiple-bodysimulation of side-by-side mooring to an FPSO. Proc. 11th ISOPE, Stavanger,343-53.

[2] Chen X.B. (1994) Approximation on the quadratic transfer function of low-frequency loads, Proc. 7th Intl Conf. Behaviour Off. Structures, BOSS94, 2,289-302.

[3] Chen X.B. (2004) New formulations of the second-order wave loads. Rapp. Tech-nique, NT2840/DR/XC, Bureau Veritas, Paris (France).

[4] Chen X.B. (2004) Hydrodynamics in offshore and naval applications - Part I.

Keynote lecture of 6th Intl. Conf. HydroDynamics, Perth (Australia).[5] Chen X.B. (2005) The Set-Down in the Second-Order Stokes Waves, Rapp. Tech-

nique, Bureau Veritas, Paris (France).

[6] Dai Y.S. (1998) Potential flow theory of ship motions in waves in frequency andtime domain. (in Chinese). The Express of the National Defense Industries, Bei-jing (China).

[7] Ferreira M.D. & Lee C.H. (1994) Computation of second-order mean waveforces and moments in multibody interaction, Proc. 7th Intl Conf. Behaviour Off.Structures, BOSS94, 2, 303-13.

[8] Kashiwagi M. (2004) Wave drift forces on two ships in close proximity, Proc.Joint Intl. Conf. OCEANS04 & TECHNO-OCEAN04, Kobe, 578-84.

[9] Malenica S., Zalar M. & Chen X.B. (2003) Dynamic coupling of seakeepingand sloshing, Proc. ISOPE2003, Honolulu.

[10] Maruo H. (1960) The drift of a body floating on waves. J. Ship Res.,4, 1-10.

[11] Molin B. (1979) Second-order diffraction loads upon three-dimensional bodies.App. Ocean Res. 1, 197-202.

[12] Newman J.N. (1967) The drift force and moment on ships in waves. J. Ship Res.,11, 51-60.

[13] Newman J.N. (1974) Second-order, slowly-varying forces on vessels in irregularwaves, Proc. Intl Symp. Dyn. Marine Vehicle & Struc. in Waves, Mech. Engng.Pub., London, 193-97.

[14] Newman J.N. & Lee C.H. (2001) Boundary-element methods in offshore struc-ture analysis, Proc. 20th Intl Conf. Off. Mech. Arc. Engeng, Rio de Janeiro.

[15] Newman J.N. (2004) Progress in wave load computations on offshore structures,Oral presentation at OMAE2004, Vancouver.

[16] Ogilvie T.F. (1983) Second-order hydrodynamic effects on ocean platforms, Proc.Intl Workshop on Ship & Platform Motions, Berkeley.

[17] Pinkster J.A. (1980) Low frequency second order wave exciting forces on float-ing structures. H. Veenman En Zonen B.V. - Wageningen, Wageningen (TheNetherland).

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

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

01 - hydrodynamics - 05 - hydrodynamic analysis for offshore lng terminals chen xiao

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