Drift forces and slowly varying forces on ships and offshore structures in waves

24 JVLJi9j8Lft_r:

RCH1EP

Drift Forces and Slowly Varying Forces on

Offshore Structures in Waves

by

Q.M. Faltinsen, Professor

Norwegian Institute of Technology, Trondheim A.E. Løken, Research Engineer,

Research Division, Det norske Veritas, Oslo.

1. Abstract

A suriey of numerical methods used to calculate drift forces and moments on -ships or other marine structures in regular and uregular waves is presented. Methods to calculate slow drift excitation forces in waves are also included. Results for drift forces and moments for different objects in beam, head and oblique sea are given. Eight different methods are used. Values for second-order difference frequency force due to presence of two simultaneous regular wave components are presented.

2. Introduction

Slow drift oscillations of a moored structure in irregular waves may be an important problem. The large horizon-tal excursions that occur can cause large forces in anchor lines and limitations in drilling operation. The phenome-non is commonly seen in model tests.

Hsu and Blenkarn (1) have given a simple explanation of the phenomena. They imagine the irregular wave sys-tem divided into approximate regular wave parts. In each regular wave part the structure will experience a constant horizontal drift force (and yaw moment). This is illu-strated in figure 1 where the drift force in each "regular

Fig. 1: Example of drift force obtained from wave record.

wave part" is indicated by an arrow. In this way a slowly varying excitation force is obtained. The magnitude is not large, but if the mean period is close to a natural pericd in yaw, sway or surge, a significant amplification may occur due, to small damping in the system.

The drift force in regular waves is the important

building brick in Hsu and Blenkarn's analysis of the slowly-varying drift

force. The same is true in the

approach by Remery and Hermans (2) and Newman (3). Various theories exist to predict drift forces in regular waves. For a ship in regular beam sea waves one may use the Maruo's formula (Maruo (4)). For a ship in regular

Norwegfan Maritime Re.earch No. 1/1978 2

v.

c eepsouw on

Technische Hogeschool

;hipsan'

2

,.3.

9

head sea waves the method of Gerritsma and Beukelxnan (5) and Maruo (6) seems to give good results in many cases. For the oblique-wave case Maruo (4) derived a theory for the drift force, but his work does not include numerical results for the oblique sea case. Newman (7) used a different approach than Maruo and, for the zero-speed case, derived the drift-yaw moment and re-derived the -force results of Maruo. His numerical results were based on slender.body theory combined with long wave length assumption. The drift

force and moment in

oblique sea were also studied by Faltinsen and Loken (8). The first order potential was found by using strip theory and solving two-dimensional Helmholtz equation problems for the diffraction potential and two.dimen-sional Laplace equation problems for the forced motion potentials. The drift force and moment are obtained by Newman's formula (Newman (7)). Salvesen (9) derived a result where the final results were expressed in the frame-work of the shipmotion strip-theory. He showed partial agreement between theory and experiment for the drift force but discrepancy between theory and experiment for the drift moment. Kaplan and Sargent (10) proposed to use the formula of Gerritsma and Beukelman (5) also for oblique sea. This is a simplification since the influ-ence of roll, sway and yaw motion is neglected. However, the formula is simple and may be a practical tool to give a rough estimate of drift forces and moments on a ship in regular oblique sea waves. Kim and Chou (11) have proposed a method where the first order potential is obtained by strip theory and solving 2-D Laplace equation problems for the cross-section of the ship. For a large-volume structure of any shape Faltinsen and Michelsen (12) presented a method based on three-dimensional source technique and a generalization of the formula of Newman (7). The method is applicable for any wave direction and finite water depth. It shows good agreement with experiments. Pinkster and van Oortniers-sen (13) preOortniers-sented a method to calculate mean drift

forces and moments on an offshore structure. The

method is based on integration of second-order pressure along the body. The assumptions are the same as the Faltinsen and Michelsen method (12). The drift force on a ship in regular beam sea waves and in the vicinity of another structure has been theoretically determined by Ohkusu (14). For certain frequencies and for a given distance between ship and structure he predicted a nega-tive drift force. This causes a slowly varying oscillation even in regular waves. It should be kept in mind that all methods presented above are based on potential theory. Viscous effects may be the most important contribution to mean wave forces on semi-submersibles and other small-volume structures. Huse (15) has pointed out that viscous effects may create a negative drift force on semi-submersibles in beam sea. Pijfers and Brink (16) calcu-lated the drift forces for two semisubmersibles due to different current velocities and mass transport velocity of waves based on second-order incoming waves, harmo-mc wave and structure motions. The results from their method show quite high values and should be used with care since no experimental or full-scale measurements are available at the present time that justify the values. All the results mentioned above are based on waves of moderate wave height. Waves close to breaking may change the results significantly (Longuet-Higgins (17)). The

Hsu and Blenkarn approach or Newman's

approach (3) of calculating slow drift oscillations is a simplification. They disregard several non-linear

inter-action terms between wave and structure. Kim and

Breslin (18) have presented a different approach than Hsu and Blenkarn. They calculate second-order transfer funç-tions by generalizing the procedure by Salvesen (9) for mean forces and moments..

Faltinsen (19) (see also Faltinsen and Loken (8)) presented a new method to calculate slow drift oscilla-tions of a ship in irregular beam sea waves. The hydro-dynamic boundary value problem was formulated and solved correctly to second order in wave amplitude. The first-order problem is the well-known linear ship motion problem, and the second-order problem con-tains the necessary slow drift excitation forces. The second-order potential satisfied Laplace equation with inhomogenous boundary conditions on the free surface and the body boundary. Green's theorem was used to derive a formula for the drift force and slowly-varying horizontal force.

The models for slow drift used in this report do not include the response to the slow drift excitation force and moment. In comparing with model tests and full-scale tests, such a model has to be set up. One way of doing this is to use a conventional mass-damping-spring system. The effect of the mooring is calculated in a quasi-static way by using non-linear characteristics, for the mooring force. The damping of the system is usually due to viscous effects. When model tests for slow drift oscillations are done in irregular waves, the second-order transfer function can be estimated by means of higher order spectra. This requires quite a long testing period to get reliable results. Further transient factors may affect the results over a substantial time due to small damping in the system. Another way of conducting model testing would be to do tests with two simultaneous regular wave components of different frequencies and systematically

vary the frequencies. In this way the necessary second-order excitation force or moment component FiJc and F1S (see equation 13) can be determined directly. Due to the small quantities involved it might be difficult to perform the tests. When comparing with full-scale tests, the results will be obscured by other 'physical effects such as effect from current, ivind and three-dimension-' ality of the wave system. It is considered important to

measure the incident undisturbed wave 'system to try to detect, slow drift excitation mechanisms that might not be in the theoretical model.

In the following text eight different numerical methods will be discussed in more detail. Their appli-cabiities and limitations will be stressed. Numerical comparative 'tests will be presented and discussed.

3. Numerical Methods '

In the numerical calculations to be presented in this report eight different methods will be used. The assump-tions and limitaassump-tions of the methods are discussed. All the methods are based on potential theory and the pro-blems are solved as a perturbation problem where the amplitudes of oscillation of the fluid ancLthe hody are assumed to be small compared to body dimensions and the wave length. The methods are not expected to be valid for waves close to breaking, but for how large waves they are valid is difficult to say a' priori. This is a question of experience. Further the methods are not valid for small-volume structures,where viscous effects might be important.

a Faltinsen and Michelsen's Method (12)

This method is applicable for any structure, wave head-ing and water depth. It is assumed that the sea floor can be approximated by a horizontal rigid plane. The first-order approximation of the incident waves 'are regular sinusoidal waves of the Airy type. The structure has zero mean forward speed and 'there is no current. In principle it is possible to make calculations for several structures present in the fluid at the same time. The basis for the method is the following expressions for drift

force components F, F

and yaw drift moment com-ponents M.

F = - f f [ p cos e + p Vr (Vr COS 0 - V0 5ifl 0)] rd 0 dz S00

Fy=-ffEPsiflO+PVr(VrsiflO+Y000SO)]rd0dZ

S00

Mz=_pffVryOr2 dOdz

(3)

SQ0

where the bars denote time average 'and the rntegration is. over the surface Sth of a vertical cylinder of large radius r, that is extending from the free surface down to the:

Norwegiw, Mgrftme Resec.rch No. 1/1978

horizontal sea floor z = -h. Further p is the pressure and p the massdensity of fluid. To define, the other.quantities it is necessary to define the coordinate system (x, y, z) which is a right-handed co-ordinate system fixed with respect to the mean position.of the body, with positive z vertically upwards. through the centre of gravity of the body and the origin in the plane of the undisturbed free surface. The coordinate system is shown in the case of a ship in figure 2. Let us now return to the explanation of the quantities in equations 1-3. F and F are the x and y components of the horizontal drift force and M

is the drift moment about the z axis. We have used

(r, 0, z) as cylindrical polar co-ordinates with x = r

cos 0 and y = r sin 0.

1r and V0 are the radial and tangential velocity components, respectively.

x,Y

.1Y

xJ

0 2 6 6 8 10 12 14 16 lB 20

l.

Fig. 2: Tanker - displacement of 155,000 tons

The expressions are exact within potential theory and are derived using conservation of momentum and energy in the fluid. In Faltinsen and Michelsen (12) the express. 'ions are further approximated and formulae that are correct to second order in wave amplitude are derived. In these formulae it is necessary to know only the first-order quantities of the physical variables in the fluid. The necessary first-order quantities are calculated using a three-dimensional sink-source technique which is a powerful method to evaluate wave loads on fixed as well as free-floating large-volume objects. In the computer program available at Det norske Veritas it is possible to use up to 500 elements to approximate the wetted body surface. For small wave lengths compared to character

Norwegian Maritime Research

No. 1/1978

NV 459 COMPUTER MODEL -LOADED

CONDITION

READING

ANGLE OF \wAv ES

istic structural dimensions, and for complicated struc-tures 500 elements might be too few. But in normal cases it is a sufficient number of elements. The accuracy of the method is expected to be very good. A drawback of the method is long computer time, increasing at least as the square of the number of surface elements. In the lower wave length range irregular frequencies may occur, which means there is no solution by the method. This problem can be circumvented (Sayer and Ursell (20)) but in normal cases it does not seem to represent any problem.

b. Newman's Long Wave Length Formula (7)

Newman's long wave length formula is based on the following formulae for horizontal drift force components and yaw moments

k2 2ir

F=--_f

H(0) 2cos OdO

+

PWac05 13

Re{HØT+$)}

pk2 2ir

F'=.f H(0)

OdO 0

+ 4 PWa Sifl

13 Re { H (r 2ir

M = -

Im f H* (0) H' (9)dO

0

+

PCa)a Im{H' (2r+13)}

rd H01

where H'(ir+13) is to be interpreted as L '

_Jo

'zr+13

Expressions (4), (5) and (6) are slightly different' from Newman's expressions (7) due to different expressions for the incident wave potential. Further k is the wave number, w is the circular frequency of oscillation, _athe wave amplitude and 13 the angle between the x axis and the propagation direction of the incident waves. 13 = 00 corresponds to head sea and 13 = 900 corresponds to beam sea. The structure has zero mean forward speed. The asterisk denotes the complex conjugate and Re and Im the real and imaginary part respectively. H (0) is the Kochin function which is given by

(7) a

ffds (----Ø8--)exp(kz+ikxcos 0+ikysin 0)

SB (4) (5) (6)

Here i is the complex unit and 'B e

it is the velocity

As long as the heading is not close to head sea the

potential due to the presence of the structure. That

method is expected to be valid for a broad range of wave means we may write the total velocity potential as lengths.

(i exp (kz+ikxcos 3+ikysin

13)+B)e_t

where the integration is over the mean body surface SB. Further a/an is the derivative along the normal vector of the body surface (positive normal direction is out of the fluid). The expressions above are correct to second order in wave amplitude. Infinite water depth is assumed and the first order approximation of the incident waves are assumed to be regular sinusoidal waves of the Airy type. Newman makes further approximations in his calcu,-lation. He assumes long wave length slender body theory

and applies the formula to ships. The slender body

assumption implies that the cross-dimensions of the ship are small compared to the length of the ship. It is there-fore questionable to apply the formula to a ship or a barge with a small length-to-beam ratio. Due to the long wave length assumption the theory should be used with great care for wave lengths much smaller than the ship length.

c.. Newman-He/mholtz Formula (8)

The basis for the Newman.Helmholtz formula is the same as Newman's long wave length formula. The differ-ence from the Newman long wave length formula is in

the calculation of the velocity potential

B in the Kochin function. The forced motion part of the potential is calculated using strip theory. The flows around each cross-section of the ship are assumed to be independent of each other and calculated by solving two-dimensional Laplace equations with the classical linear free surface

condition, a radiation condition and a linear

body-boundary condition dependent on the oscillation mode. The water depth is assumed to be infinite. Either the Frank Closefit method (Frank (21)) or the Lewis form method (Tasai (22)) is used in the calculation. In the

Frank Closefit method the wetted body surface is

- described by offset points, usually a total of 15 points. This method is applicable to a large family of cross-section forms. Exceptions might be cases where the body surface intersects the free surface at a small beam, draught and area. The method is applicable for normal ship forms, but is questionable for instance for bulbous bows. For more details see Van Kerczek and Tuck (23). The velocity potential due to the presence of the body when the body is restrained from oscillating (i.e.

the diffraction potential) is calculated by strip theory and solving two-dimensional Helmholtz equation pro-blems with proper free-surface conditions. The water depth is assumed to be infinite (Troesch (24)). In the beam sea case the Helmholtz equation reduces to a two-dimensional Laplace equation. In the head-sea case the solution breaks down. The reason is the strip theory assumption, which neglects hydrodynamic interaction between the cross-sectiOns. The solution could be done in the way described by Faltinsen (25) and Maruo (26).

5

Maruo's Beam Sea Formula (4)

According to Maruo the drift force in beam sea on a slender ship can be written as

F =

hA-i2 dx

(8)

where the integration is over the ship length and IA 1 is

the amplitude of reflected wave for each crors-section. The reflected wave is due both to the swaying, heaving and rolling of the ship and the diffraction effect of the restrained ship. Both the forced motion potentials and the diffraction potential are in this paper calculated by Lewis form technique (Faltinsen (27)). Maruo's formula has been derived using the equations of conservation of momentum and energy. The formula is correct to second order in wave amplitude. The first-order approximation of the incident waves are regular sinusoidal waves of the Airy type. The ship has zero mean forward speed and there is no current. Infinite water depth is assumed.

Kim and C/iou's Formula (11)

Kim and Chou use Maruo's beam sea formula for oblique sea. Kim and Chou find the reflected wave amplitude by. solving 2-D Laplace equation problems along the ship by means of Frank Closefit method. The rational expla-nation for doing this is questionable.

f

Salvesen's Formula

Salvesen (9) starts out with a formula he claims is

accurate within potential theory. As pointed out by Faltinsen (28) one term is missing in his expression. This term is important in the calculation of vertical force, but its importance in the calculation of horizontal force has not yet been examined. Salvesen assumes that the poten-tial du to the presence of the body is much smaller than the incident wave potential. This-assumption-makes his procedure questionable for wave lengths shqiethan the ship length. Salvesen assumes that the first order approxi-mation of the incident wave,s are regular sinusoidal waves of the Airy type. The water depth is infmite. The calcu-lations may be performed fbr any wave heading and non-zero forward speed.

g. Geffftrna and Beuke/man's Formula (5)

The formuia, which is derived for head sea, is a simple expression that gives good results in many cases. The rational basis for the formula is somewhat vague as it is based on the relative motion hypothesis. The formula is valid for inifinite water depth and non-zero mean

for-Norwegian Maritime Research No. 1/1978

ward speed of the ship. The first-order approximations of the incident waves are assumed to be regular sinusoi-dal waves of the Airy type. In the zero mean forward speed case the formula can be written as

F,=&-fb33V2

where b33 is the sectional two-dimensional damping coefficient in heave and Vza the amplitude of a kind of relative vertical velocity V along the ship. It is written as

di73 di75

d*

Vz-j-- _x.i__.ar

where .4- means the time derivative and i and i are the heave-and pitch oscillations. Further

*

k°

I

="I--

I

yedz)

where is the undisturbed incident wave profile as a function of x, Yw the sectional half-breadth and d the sectional draught.

h. Faltinsen's Formula (19)

Faltinsen assumes beam sea incident waves on a slender ship. The first order approximation - of the incident waves may be a finite or infinite number of regular

sinus-oidal wave components of the Airy type. The first

approximation of the urcident velocity potential

is written as

N-gA1

=

--e

k z

_{sin (k1y - t + e1) (12)}

(ie. long-crested waves)

where g is acceleration of gravity, cj the circular fre-quency of oscillation of wave component No. i, and

(10)

c1,i2

ki=g

The a-nplitudes A1 of the wave components may be given by a wave spectrum and e1 may be considered as random phase angles. The water depth is infinite. The ship has zero mean forward speed and there is no current. The hydrodynamic problem is solved correctly to second order in wave amplitude. The expression for the slow drift excitation force and the mean drift force is written as

Norwegian Maritime Research No. 1/1978

FHSV

=

AA [F1Ccos (c-c,.)t+

FijSsin(cA,j_,i)t]

(13) The mean drift force is easily obtained from equation (13)as

FHSV= . Al2F11C (14)

1=1

l'his formula says that the mean drift force in irregular waves may be found by fIrst dividing the irregular sea into regular wave components, then fmding the drift force in each regular wave component and adding those

drift force components to obtain the drift force

in irregular sea. This is common procedure (Gerritsma et. al.

_{(29)). In the calculation of slow drift excitation}

forces the procedure is not so simple. Non-linear hydro-dynamic interaction between the wave components has to be considered. This is expressed in terms for F1.0 and FiJS, which is a function of only wave componens Nos. i and j. A degree of ambiguity exists in the coefficients F1iC and F1S when_{i j. We could for example impose} the restriction that F1f-' and FS are equal to zero when i > j. Another possibility is to require that FiiC = FJiC and ijS =

-

FS when j * i. This will be done here. FC and F1S depend on the phase angles e and ej, but we could rewrite equation (13) in terms of second-order transfer functions.

The calculation of FijC and FjS is quite complicated. For details we refer to Faltinsen (19). It should be noted that Faltinsen used Lewis form representation of the cross-sections of the ship. But the procedure may be easily generalized to unconventional ship forms.

4. Numerical Calculations

4.1

Description of Models used

Objects of different forms were used in the numerical

computations. One of these was a cylinder with L x B x d = 120 m x 20 m x 10 m, (L is length, B is beam at water level and d is draught) roll radius of gyration 6 m and centre of gravity at the mean position of the free surface. Further we used Series 60 ships (see Todd (30)), a loaded tanker (see Table 1 and Fig 2), a vertical circu-lar cylinder (see Fig. 17) and a box with 2 draughts (see Table 2). The parameters k, k and k in table 2 are the radii of gyration in roll, pitch and yaw, respectively with respect ot the (x, y, z) coordinate system. Further (xa, ya, z) are the coordinates of the centre of gravity C.G.

4.2 Results from Numerical Calculations Horizontal Cylinder

Figure 3 show drift forces on the horizontal circular

Table 1: Main Particulars

of

130,000

dwt

Tanker Length between perpendiculars m 285.60 Beam m 46.71 Depth m 20.35 Draught fore m 13.82 Draught aft m 13.82 Draught mean m 13.82 Displacement tons 154980.00 Cent.re of gravity, longitud. from m

+

6.46 above gravity of Centre

baseline m 11.03 height Metacentric m 8.97 Pitch/yaw radius of gyration m 71.40 Roll radius of gyration m 16.35 Natural pitch period s 9.80 Natural roll period s 12.80 Note: Centre

of

gravity, longitud. from

+

means

forward

of

or

station 10. Table 2: Geometrical Data

for

Floating Box, L

=90

m,

8=90

m

0.60

0.5

0.40

0.30 0.20 0.10 Draft=40m Draft=20m.

p.g.2.L

ASYMPTOTIC SOLUT ION 'I' 0.5 1.0 regular waves. The results are presented as horizontal drift force non-dimensionalized by

pg2

L as a func- tion of X/L where A is the wave length of the incident waves. The asymptotic value for small SJL is 0.5 which follows from equation (8) and the fact that the wave is totally reflected for small wave lengths. Both results for restrained ship and free-floating ship are presented. We note a marked influence of the body motions. The Marou's beam sea formula, the Newman-Helmholtz formula, and the Faltinsen's formula are quite different in form, but they are all correct to second order in wave amplitude. We note that the Newman-Helmholtz formula may give somewhat higher values than 0.5 for the non-dimensionalized force. According to Maruo's beam sea formula equation (8) this corresponds to a higher reflected wave amplitude than the incident wave amplitude. This is unphysical and the results must be due to numerical approximations. The values calculated by Salvesen's formula are con- siderably lower than values calculated by the other three methods mentioned above. But it should be noted that Salvesen used the assumption that the velocity potential due to the presence of the body ØB is much smaller than the incident wave potential

t.

This i questionable for small wave lengths. Asymptotically for small AlL the amplitude of

'B

is equal to the amplitude of cf for the horizontal cylinder in beam waves. This explains the general tendency that Salvesen's method diverges more and more from the other three methods for decreasing wave lengths. o MARUO S BEAM SEA FORMULA (CYLINDER RESTRAINED) MARUOS BEAN SEA FORMULA (FREE MOTION) A NEWMAN-BELMHOLTZ FORMULA (CYLINDER RESTRAINED) NEeiAN-HELMHOLTZ FORMULA (FREE MOTION) X SALVESEN FORMULA (FREE MOTION) a FALTINSEN FORMULA (FREE MOTION) 1.5

AlL

Fig.

3:

Drift

force on circular cylinder

in

beam sea. (L

x

B

x

d

= 100 m

x

20

m

x

10 m) 7 Norwegian Maritime Research 1/1978 No. C G,(X, YG' ZG) 0, 0, 10.62 m 0, 0. 8.82 m 33.04 m 37.32

m

m 33.30 m 32.09 k Series 60 Ships k 32.92 m 40.08 m Calculations of drift forces on a Series 60 ship (CB 0.6) in beam sea regular waves are presented in figure 4. Both PARAMETER UN IT

0.5 0.4 0.3 0.2 o.I F p. gç L £

.

0.5 1.0 ISA/I.

60 CB6:

No.1 --MARUOS FORMULA

No.2 U KIM AND CHOUSFORMULA

-G-MARUQ5 FORMULA

OIFERENT VISCOUS

.IDAMPING IN ROLL.

No.4-0-KIM AND CHOUS FORMULA No.5 LALANGAS EXPERIMENT

No.6 NEWMAN -HELM HOLTZ FORMULA

SERIES 60 CB.O.8:.

No. 7-v-MARUOS FORMULA }FIXEO

SERIES 60 C80.7:

No. 8 £ OGAWA EXPERIMENT)FIXED SERIES

Fig. 4: Drift force in beam seia series 60 C8 =

0.6 (0.7,0.8).

Maruo's beam sea formula and Newman.Helmholtz' for-mula are used. The values are in close agreement except in the vicinity of X/L = 0.9 which is close to resonance in roll. This may be explained in the following way:.The two formulae are derived using conservation of energy assuming no energy loss due to viscous effects. But

Norwegian Marhime Research No. 1/1978

'-0

FIXED FREE MOTIONS

0.5

1.0 1.5

X/L

Fig. 6: Drift force on a series 60 ship, CB = a 6 heading angle 60°.

8

viscous effects are significant in the calculation of roll around resonance. See Salvesen, Tuck and Faltinsen (31), which is the method used to calculate first-order poten-tial.

The results obtained by Kim and Chou (11) and

experiments by Ogawa and Lalangas referred to in (32) are also given in Fig. 4.

The results given by Kim and Chou show wide dis-agreement for a free-floating ship compared with the two formulae of Maruo and Newman-Helmholtz and experiments by Lalangas.

The tendency of the results when the wave length increases represented by curve No. 4 in Fig. 4 seems not to be correct when compared with the results for the res-trained ship (curve No. 2). The curve for the free-floating ship should rather approach zero for long wave lengths than the value for restrained ship, as is the case in the

0.1

-.0.1

-0.2

I

NEWMAN - HELMHOLTZ FORMULA

(FREE MOTIONS)

A NEwMAN'S LONG WAVE.

LENGTH FORMULA

(FREE MOTIONS)

Fig. 5: Drift force on a series 60 ship, CB =

0.6

heading angle 60°.

KIM AND C!IOUS FORMULA

(RESTRAINED) C3 0.6 SALVESEN'S FORMULA MARINER HULL 0.61)

(FREE MOTIONS)

NEWMAN'S LONG WAVE LENGTH F0R1ULA (C3 = 0.6)

(FREE MOTIONS.)

NEWMAN-RELMHOLTZ FORMULA (FREE MOTIONS) (CB = 0.6) OGAWAS ECPERIMENTAL VALUES

(SNIP RESTRAINED) (C3 = 0.7) NEWNAN-HELMNOLTZ FORMULA (SHIP RESTP.AINEO) (C3 = 0.6) £

S.-.

F

0

ASYMPTOTIC

SOLUTION

-L 0.3-o

0.2

"I

D t

Ii"

-'

\1

.

I

0.1- _I0

I

work of Kim and Chou. This is a function of the assump-tions that we have made. If we had not assumed the

oscillation amplitudes to be smaller than structural

dimensions, our results would have given drift even in the long wave length range. This can be visualized by looking on a small particle in regular wave field in a wave flume.

Figure 5 presents results for the longitudinal drift-force component in wave heading 60°. The agreement between the Newman-Helmholtz formula and the Newman-long-wave-length formula is

quite poor for

smaller wave length. The same tendency to disagreement is seen in figures 6 to 8. In figure 6 is presented trans-verse drift-force component for wave heading 600 _and

in figure 7 and 8 longitudinal and transverse drift-force components for wave heading 450, _{In figure 6 are also}

analytical results by Kim and Chou (11) and Salvesen (9) in addition to experimental results by Ogawa (33) presented.Ogawa's results are for Series 60, Cb = 0.7, but previous experience indicates that the influence of block

F

-0.06

-0.05

-004

-0.03

-002

-0.01

1-i'

A

05

0'

-.

0.6 0.3 M2

p.gL2.2

-\

I'

I'

I,-- p%t4 I

'

I g

is

c I

-Al

A

i

3s

-I

-I

Al I

0.03W 0.02' 0.01

p.g.ç12

+ 0.01

Fig. 10: Drift moment on a series 60 ship, CB = a 6 heading angIe 60°.

9

NEWMAN - HELMHOLTZ FORMULA

(FREE MOTIONS) 6 NEWMAN'S LONG WAVE

LENGTH FORMULA

(FREE MOTIONS)

0.5 1.0 1.5 IL 2.0

Fig. 8: Drift force on a series 60 ship, CB = a 6 heading angIe 45°.

NEWMAN - HELMHOLTZ FORMULA

(FREE MOTIONS

Norwegian Maritime Research No, 1/1978

0.5 1.0 1.5 AlL

-0.01'

v.

-0.02

-0.03

Fig. 9: Drift moment on a series 60 ship, = a 6 heading angIe 45°.

0.5 1.0 1.5 IL

NEWMAN - HELMHOLTZ FORMULA

(FREE MOTIONS)

I a NEWMAN'S LONG WAVE_{LENGTH FORMULA}

(FREE MOTIONS)

Fig. 7: Drift force on a serfes 60 ship, 8 = heading angle 45°.

0 KIM AND CHOU S FORMULA

(FREE MO'rIONS) C = 0.6

0 NEWMAN-HELMHOLTZ FORMULA

(SHIP RESTRAINED) CB = 0.6 OGAWA'S EXPERIMENTAL RESULTS

(SHIP RESTRAINED) CB = 0.7

A NES1AN'S LONG WAVE LENGTH FORMULA (FREE MOTIONS) CE = 0.6 NEWMAN-HELMHOLTZ FORMULA (-FREE MOTIONS) CE = 0.6 0 SALVESEN'S FORMULA (FREE MOTIONS) MARINER HULL- -(CB 0.61) 1.5

AlL

0.1 ASYMPTOTIC SOLUTION 0.2

\'

0.20 0.15 0.10 0.05

p.g 'y SPECIFIC WEIGHT OF WATER

Fig. 11: Non-dimensiona/ized added resistance in head sea.

coefficient is not significant. We note a good agreement between Newman-Helmholtz formula, Ogawa's experi-mental results and Kim and Chou's formula. Salvesen's results for a Mariner hull with block coefficient 0.61 is significantly smaller than the results by Newman-Helm-holtz formula. The same tendency was seen for the

circular cylinder and can be explained in the same way as for the circular cylinder.

In figures 9 and 10 are presented results for drift-yaw moment on Series 60 ship (Cb = 0.6) for wave headings 450 and 600. We note that the experimental values by Ogawa agree to some degree with Newman-Helmholtz formula and Kim and Chou's formula, while both the results by Newman's long wave length formula and Sal-vesen's formula disagree significantly with the expen-mental values. From the results in figures 4-10 it does not seem easy to present the results as a simple function of wave heading.

Figure 11 shows the results from computations by Det norske Veritas computer program NV428 of non-dimensionalized added resistance coefficients in head sea regular waves. The values are plotted for comparisons with those obtained by Gerritsma and Beukelman (5) by experiments and computations. The computer program NV428 is based on Gerritsma and Beukelman's pro-cedure. The slight difference in results is due to different

procedures to calculate firder potential. In NV428

the procedure by Salveséñ, Tuck and Faltinsen (31) is followed.

The computations and experiments refer to the ship M.V. "S.A. van der Stel" with CB = 0.564. The results are for Froude numbers Fn = 0.20 and F = 0.15. It should be noted that the results are non-dimensionalized with respect to wave height. The main particulars and

Norwegian Maritime Research No. 1/1978

L LENGTH BETWEEN PERPENDICULARS

10

-0

0.5 1.0 1.5

body plan for this ship may also be found in (5). The results from computations given by N. Salvesen in (9) for the "Mariner Hull" with CB = 0.634 for the case of zero speed is also included in Fig. 11.

Loaded tanker

Results for transverse drift-force component on the loaded tanker in beam sea are presented in figure 12. We note that Newman-Helrnholtz formula agrees well with Maruo's beam sea formula except around roll resonance. This has been explained earlier to be due to viscous effects. The results by Faltinsen and Michelsen's formula

are expected to be the most accurate of the three

methods. We note quite good agreement between New-man-Helmholtz formula and Faitinsen-Michelsen formula. Results for longitudinal drift-force component and drift-yaw-moment in beam sea are presented in figure 13. Keeping in mind that the quantities are quite small, we note good agreement between Faltinsen-Michelsen's formula and Newman-Helmholtz formula.

Results for longitudinal and transverse drift-force components as well as drift-yaw moments for wave head-ing 45° are presented in figures 14 and 15. The difference between Newman-Helmholtz formula and

Faltinsen-Michelsen formula is greater than in beam sea.

The Newman-Helmholtz formula breaks down in head sea. The reason is that the solution of the first-order potential by Helmholtz equation breaks down for head sea (Troesch (24)). The results by the method in head sea are presented in figure 16. In the same figure are pre-sented results by Faltinsen and Michelsen's formula and Gerritsma and Beukelman's formula. The agreement between these two methods is not satisfactory.

-J

WAVE HEIGHT WAVE

0 BEAM OF SHIP

LENGTH

-I

S.A. VAN DER STEL CB .0.56k F0.1S S.A. VAN DER STEL CB.0.564 F0.20

p --0-- COMPUTEO BY GERRITSUA I --0--COMPUTED BY GERRITSMA

a

-a-EXPERIMENT - -

a

AEXPERIMENT

-Q.

-

0. a a COMPUTED BY NV28 V COMPUTED BY SALVESEN - U-COMPUTED BY NV42B MARINER HULL CB.0.63L F, .0.0 2- COMPUTED BY NVL28

"S.A. VAN DER STEL" CB .0.564 2'

,0

0

0 I,a%t

I,

\

I-

/1

Vt 1-05 1.0

0.01 -001 N1 p.9. .12 0.5 0.4

0.3

0.2

0.1

F

p.g..L

ASYMPTOTIC

SOLUT ION

o---q

Fig. 12: Drift force on loaded tanker in beam seas.

culated here are quite small and hence more subject to influence from surge motion and the bluntness of the bow which are neglected in Gerritsma and Beukelman's formula. U

0.5

1:0 NEWMAN-HELMHOLTZ FORMULA FREE NOTIONS 6 SHIP RESTRAINED

FALTINSEN AND MICHELSEN FORMULA FREE MOTIONS

oSHIP RESTRAINED

Fig. 13: Drift force and moment on loaded tanker in beam seas.

05

NEWMAN-HELMHOLIZ FORMULA FREE MOTIONS

a SHIP RESTRAINED

ASYMPTOTIC FALTINSEN AND MICHELSEN FORMULA

SOLUTION FREE MOTIONS

a SHIP RESTRAINED _oiL ii 0.12

-1.0 1.5 A/I 0

Fig. 14: Drift force on loaded tanker, heading angle 45°.

Gerritsma and Beuckelman's formula give generally satisfactory results for added resistance of a ship at Froude number different from zero; see for instance figure 11. But it should be noted that the quantities

cal-11

Norwegian Marftime Rezearch No. 1/1978 -001 00. 0. 002 0.10 002

-

S - NEWMAN-HELMHOLTZ FORMULA (FREE MOTIONS) - 1k - NEWMAN-HELMHOLTZ FORMULA (SHIP RESTRAINED>

- V - MARUO S BEAM SEA FORMULA

(FREE MOTIONS)

- 0 - MANUOS BEAN SEA FORMULA

(SHIP RESTRAINED)

-. - FALTINSEN AND MICHELSEN

FORMULA (FREE MOTTIONS)

'0

1.5

2.0

AlL

._-_

.---NEWMAN-HELMHOLTZ FORMULA FREE MOTIONS a SHIP RESTRAINED

FALTINSEN AND MICHELSEN FORM.

FREE MOTIONS o SHIP RESTRAINED

.

S...

0-,

1.0 1.5 A/L

Fig. 15: Drift force and moment on loaded tanker, heading angIe 45°.

DRIFT FORCE ON LOADED TANKER

HEADING ANGLE 0

NEWMAN-HELMHOLTZ FORMULA (FREE MOTIONS)

FALTINSEN AND MICHELSEN FORMULA (FREE MOTIONS)

o GERRITSMA AND BEUKELMAN

FORMULA

1.0 1.5 A/L

Fig. 16: Drift force on loaded tanker, heading angle 0°.

0.03- N1

O.0 0.01

--

FALTINSEN AND MICHELSEN

FORMULA

Vertical Circular Cylinder

Figure 17 shows results of drift force on a vertical cir-cular cylinder in finite water depth and regularwaves. Det norske Veritas computer program NV459, which is based on the Faltinsen and Michelsen method (12), is compared with theoretical and experimental values by van Oortmerssen (34). The cylinder is restrained from oscillating. We note good agreement between. the two methods and the experiments. Only 48 elements were used in the computer program NV459 to describe the wetted body surface. Van Oortmerssen (34) used only velocity square term in the Bernoulli equation in calcu-lating the drift force. It is our experience that this might yield quite erroneous answers in many cases. This has also been pointed out by Pinkster and van Oortmerssen (19). 06 0.5 0.4- 0.3-O.2 0.l 0 0.3 0.4 pg[ 0.5 - COMPUTED (34) MEASURED (34) O NV459, 48ELEMENTS 2a TOTALLY l0 o HEADING ANGLE 0' A HEADING ANGLE 45' O HEADING ANGLE O £ HEADING ANGLE 45

Norwegian Maritime Research No. 1/1978

0.5 . 06 K@

Fig. 17:

Drift force on a vertical circular cylinder. 14 18 PERIOD Tis) NV459 68ELEMENTS, EFFECT OF MOTION INCLUDED NV 459 68 ELEMENTS, EFFECT OF MOTION NOT INCLUDED

Fig. 18: Drift force on floating box (L x B x d

=90 m x 90 m x 20 ml at infinite depth. 12 9. L 0. 0.4 0.2 10 14 18 -PERIOD Tis) --: NV-459 108 ELEMENTS,

EFFECT OF MOTION INCLUDED

: NV-459 108 ELEMENTS

EFFECT OF MOTION NOT INCLUDED

0:

NV-459 48 ELEMENTS,

EFFECT OF MOTION INCLUDED .Ø NV-459 48 ELEMENTS

EFFECTOF MOTION N6T INCLUDED

: EXPERIMENT VALUES

Fig. 19: Drift force on floating box (L x B x d =

90m x 90mx40 ml at infinite depth.

0.6 0.4 0.2 10 14 18 PERIOD Ts) --: NV-I.59 108 ELEMENTS,

EFFECT OF MOTION INCLUDED

NV-459 108 ELEMENTS

EFFECT OF MOTION NOT INCLUDED

0:

NV-/.59 48 ELEMENTS,

EFFECT OF MOTION INCLUDED

0:

NV-459 48 ELEMENTS,

EFFECTOF MOTION NOT INCLUDED

S EXPERIMENT VALUES

Fig. 20: Drift force on floating box (L

x B x d =

9Omx9Omx 40 ml at infinite depth.

Floating Box

Results for drift force on a floating box in regular wives are presented in figures 18, 19, 20. The results are from Faltinsen and Michelsen (12). The main particulars of the box are given in Table 2. The asymptotic values for

ANGLEO°

'P

HEADING S

IT

ASYMPI. ASYMPT VALUE 4S.

---S.

-'S.-. A a' 0. 0.2 0.1 0 HEADING ANGLE :450

I

S

small periods are also indicated. Expressions for these have been derived by Maruo (4).

In our case, for a heading angle = 00, we get when A/L goes to zero

Fx=O.5pg2 L

For both boxes this agrees quite well with our calcula-tions.

For heading angle = 450, we get when AlL goes to

zerO.

V2

₈

P8a

for the drift force. Again the asymptotic values agree very well with our computations.

We may further note that the effect of the motion is

small for the lower periods. For the floating box of

draught 40 m and close to the heave resonance fre-quency, there is strong dependence on the motions. This may be expected smce the maximum immersion of the body is increased due to large relative vertical motion between wave and body. For the large periods there will also be strong dependence on the motions. The free-floating body moves more or less as a water particle for these periods and the drift force is therefore small. The agreement with theoretical and experimental values is quite good. We note that both theory and experiments predict non dimensionalized values higher than 0 5 at heave resonance. In the case of a two-dimensional body in beam sea it is impossible to get higher non-dimension-alized values than 0 5 according to Maruo s formula (8)

Mean Drift FOrces and Moments in irregular Seas

Mean drift forces and moments in irregular sea on the loaded tanker presented in figure.2 have been calculated. The formula by Gerritsma et. al. have been used (see equation (14)) together with Faltinsen and Michelsen regular wave results presented in figures 12, 13, 14, 15 and 16. The calcUlations were based on the Jonswap wave power density spectrum with parameters a=0.0087, P = 3.3 U 0.07, ai, = 0.07. (For definition of quanti-ties see Hasselmann et. al. (35).) The results are presented in figUres 21 and 22 as a function of the mean zero up-crossing period T of the stationary (i.e. short-term predictions) irregular sea. The results are given as

.J/H5 where H is the significant wave height. The

drift force and moment can be obtained from figures 21 and 22 by using the formula

2 2

i 8

where n is 1 for the x and y components of the fOrce and n is 2 for the yaw moment.

Fig. 21 and 22 may be used for other ship lengths then L = 28560 if is plotted as a function of T.\/T.

H5 _L.

The moment in Fig. 22 is negative referred to the co-ordinate system in Fig. 2.

13 0.40 A 0.32 a a A 0.16 0.08

Fig. 21: Wave drift forces in irregular Waves on tanker; XCOMPONENT VCOMPONENT 0 : 0° A 3 :45'

a:90'

a 10 12 16 Tz(s 10 12 14

Fig. 22: Wave drift moments in irregular waves on tanker.

Calculation of Second-Order Difference Frequency Force

The second-order difference frequency fore due to the

action of two simultaneous waves with frequencies c and ,j was calculated. The horizontal circular cylinder

Norwegian Maritime Research No. 1/1978

H5 0 048

described in a preceding section was selected. Beam sea was assumed. The non-dimensionalized wave frequencies

were chosen among the numbers 0.64, 0.79, 0.84, 0.95 and 1.12. The phase angles e1 and e1 were chosen equal

to zero. The results are presente( as

FjC/(pgL) and F1S/(pgL) in Table 3 and 4. FiC and FjS are defined by equation (13). When e1 = e they are the same as second order transfer functions. Newman (3) assumed that F1 could be approximated by F11C and FiS was small when.

C&)j were close to w. Our results indicate that Newman's

approximative way of calculating slow-drift oscillations is reasonable. But a. final conclusion cannot be made before extensive investigations for other cross-sections have been performed. ..

0.64 0.72 0.84 0.95 1.12

Table 3: Numerical Calculatibn of Fjfl/('pgL) for. circular cylinder in beam sea

Table 4: Numerical Calculation of Fj,SftpgL) for circular cylinder in beam sea

5. Conclusions

Eight different numerical methods have been presented. It is shown that the methods used to predict drift forces and moments should be selected with great care, depend-ing on the structural form, the incident wave length and the incident wave heading. Simplified methods may yield quite errçneous answers. It

is shown that the

magnitude of drift forces and moments on a structure. may depend strongly upon wave-induced motions as well

as the structural form. It does not seem possible to

represent drift forces and moments as a simple function of wave heading.

The presented results indicate that Newman's approximative way of calculating slow drift excitation force in irregular waves is reasonable. The final conclusion

Nonveglan Maritime Research No 1/1978

14

can not be made befOre calculations for other structures have been performed.

Acknowledgement

This study has been financially supported by the Royal

Norwegian Council of Scientific and Industrial Research (NTNF) and by Det norske Veritas.

References

Hsu, F.H. and Blenkam, KA.: "Analysis of peak mooring forces caused by slow vessel drift oscilla-tions in random seas", Paper 1159, Offshore

Techno-logy Conference, Houston 1970.

Remery, G.F.M. and Hennans, AJ.: "The slow drift oscillations of a moored object in random seas", Off-shore Technology Conference, Houston 1971. Newman, J.N. "Second order slowly varying forces on vessels in irregular waves", International Sympo-sium on The Dynamics of Marine Vehicles and Structures in Waves, University of College London, April1974.

Maruo, H.: "The drift of a body floating on waves", Journal of Ship Research, December 1960.

Gerritsma, J. and Beukelman, W.: "AnalysiS of the resistance increase in waves of a fast cargo ship", Netherlands Ship Research Centre TNO, Report NO. 169.S, April 1972;

6.- Maruo, H.: "The excess resistance of ship in rough seas", mt. Ship Prog. 1957, 4 (35).

7. Newman, J.N.: "The drift force and moment on ships in waves", Journal of Ship Research, 1967, 11(1).

& Faltinsen, 0. and Loken, A.E.: "Drift forces and slowly varying horizontal forces on a ship in waves"* Presented at Symposium on Applied Mathematics

dedicated to the late Professor Dr.

. R. Timman,. University of Technology Deift, Jan. 1978.

Salvesen,. N.: "Second Order Steady-state Forces and Moments on Surface Ships in Oblique Regular Waves". International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, Univer-sity College London, April 1974.

Kaplan, P. and Sargent T.P.: "Motions of offshore structures as influenced by mooring and positioning systems".. Proceedings of BOSS'76, TrondhCim 1976. Kim, C.H. and Chou, F. "Prediction of Drifting Force and Moment on an ocean platform floating in oblique waves". International Shipbuilding.Progress, 1973.

Faltinsen, 0. and Michelsen, F.C.: "Motions of

Large Structures in Waves at Zero Froude Numbers". Proceedings of the International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London 1974. 0.36 0.22 0.06 . 0.01 0.01 0.32 0.21 0.07 0.02 0.01 0.29 0.24 0.11 0.07 0.06 0.36 0.38 0.24 0.21 0.22 0.47 0.36 0.29 0.32 0.36

-0.16 -0.2

-0.17 -0.07

-0.12 -0.17

-0.13

0.17

-0.04 -0.09

. . 0.13 0.1.7 .0.01

.0.09 0.17

0.2 -0.01 0.04 0.12 0.16 1.12 0.95 0.84 0.72 0.64 c.-l!-Jyg 1.12 0.95 0.84 0.72 0.64

ig

0.64 0.72 0.84 0.95 1.12

Plnkster and van Oortmerssen: "Computation of the first and second order wave forces on Oscillating bodies in regular waves". Proceedings of the Second International Conference on Numerical Ship Hydro. dynamics, University of California, Berkeley,

Sep-tember 1977.

Ohkusu, M.: "Ship Motions in Vicinity of a Struc. ture". Proceedings of BOSS'76, Trondheim, 1976. Huse, E.: "Wave Induced Mean Force on Platforms in Direction Opposite to Wave propagation". Pro. ceedings of Interocean, 1976.

Pijfers, J.G.L. and Brink, A.W.: "Calculated Drift forces of two semisubmersible platform types in regular and irregular Waves". OTC 2977, 1977. Longuet-Higgins, F.R.S.: "The mean forces exerted by waves on floating or submerged bodies with applications to sand bars and wave power machines" Proc. R. Soc. Lond. A 352,463-480, 1977.

Kini, C.H. and Breslin, J.P.: "Prediction of Slow drift oscillations of a moored ship in head. seas". Proceedings of BOSS'76, Trondheim 1976.

Faltinsen, 0.: "Slow drift excitation of a ship in irregular beam sea waves". * Det norske Veritas Report No. 76-083, 1976.

Sayer, P. and Ursell, F.: "Integral-equation methods for calculating the virtual mass in water of finite depth". Proceedings of Second International Con-ference on Numerical Ship Hydrodynamics, Univer-sity of California, Berkeley, September 1977.

Frank, W.: "Oscillation of Cylinders In or Below

the Free Surface of Deep Fluids". Naval. Ship

Research and Development Center, Washington DC, Report237S, 1967.

Tasai, F.: "On the Damping Force and Added Mass of Ships Heaving and Pitching". Journal Zosen Kiokai, Vol. 105, 1959.

Van Kerczek, C. and Tuck, E.O.: "The Represen. tations of Ship Hulls by Conformal Mapping Func. tion". Journal of Ship Research, Vol. 13, No. 4, Dec. 1969.

Troesch, A.W.: "The Diffraction Potential For a Slender Ship Moving Through Oblique Waves". The Department of Naval Architecture and Marine Engi-neering, The University of Michigan, Report No. 176, 1976.

Is

Faltinsen, 0.: "Wave Forces on a Restrained Ship in Head Sea Waves". Det norske Veritas, Publication No. 80, March 1973.

Maruo, H.: "On the Wave Pressure Acting on the Surface of an Elongated Body Fixed in Head Sea". JSNA Japan, Vol. 136, Dec. 1974.

Faltinsen, 0.: "Drift forces on a ship in regular

waves". Detnorske Ventas Report No. 75-4-S. 1975. Faltinsen, 0.: "Numerical Solutions of Transient Nonlinear Free Surface Motion Outside Or Inside Moving

Bodies". Proceedings of Second

Inter-national Conference on Numerical Ship Hydro. dynamics, University of California, Berkeley, Sep. tember 1977.

Gerritsma, J. et

al.: "Propulsion in Reular and

Irregular waves". International Shipbuilding Pro. gress, Vol. 8, 1961.

Todd, F.H.: "Some further experiments on single screw merchant ship forms - Series 60". Trans. SNAME, Vol. 61, 1953.

Salvesen, N., Tuck, E.O. and Faltinsen, 0.: "Ship Motions and Sea Loads". Trans. SNAME, Vol. 78, 1970.

Remery, G.F.M. and Oortmersen, G. van: "The Mean Wave, Wind and Current Forces on Offshore Structures and Their Role in Design of Mooring Systems". OTC 1941, 1973.

Ogawa, A.: "The Drifting Force and Moment on a Ship in Oblique Regular Waves". Publication 31, Delft Shipbuilding Laboratory, l.F.P. Vol. 14, No. 149, Jan. 1967.

Oortmerssen, G. van: "Some Aspects on Very Large Offshore Structures". Proceedings .of the ONR Ninth Symposium on Naval Hydrodynamics, Paris, 1972.

Hasselmann, K. et al.: "Measurements of Wind Wave Growth and Swell during the Joint North Sea Wave Project". Deutsches Hydrographisches Zeitschrift Nr. 12, 1973.

*

Please note that some misprints are present.

Norwegian Maritime Reseanh. No. 1/1978