Ship motions and sea loads in restricted water depth

I

23 SEP. 19O

ARCHIE.F

Ocean Engng. Vol. 6, pp. 557-569.

Pergamon Press Ltd. 1979. Printed in Great Britain

SHIP MOTIONS AND SEA LOADS IN RESTRICTED

WATER DEPTFL

POUL ANDERSEN

Assistant Professor, Department of Ocean Engineering, Technical University of Denmark, Lyngby, Denmark

AbstractThe influence of the sea bottom on ship motions and sea loads is examined. It is described how to calculate the vertical motions and loads for a ship with non-zero

forward speed in regular waves by use of strip theory and fluid finite element method. Results of such calculations are shown. The effects of shallow water are significant as is seen from

several figures.

NOTATION

B breadth

F functional

V

Froude depth number

J dividing line

L ship length

length between perpendiculars

Al bending moment N normal vector of surface

N, N components of surface normal vector N shape function

T draught, shear force

V ship speed

a b constants

a3 added mass coefficient b3 damping coefficient

f

Froude-Kryloff force coefficient h*3 diffraction force coefficient h water depth

k wave number n10, ni, cigenralues

motion amplitude ordinate of dividing line J

z heave amplitude

A displacement, Laplace operator

a wave amplitude heading angle block coefficient o pitch amplitude 0, elgenfunctions A wave length p potentials

nodal point value of potential coefficient of eigenfunctions

ro frequency of encounter, wave frequency, angular p water density

557

Lab.

v.

Scheepsbouwkunde

Technische Hogeschool

INTRODUCTION

Wiiri THE increasing offshore activity and the large ships which have come into operation

during the last decade the effect of restricted waters on ship motions and sea loads becomes

still more important.

The interest in the shallow water and restricted water cases can be seen in view of the efforts exercised to solve these problems. An extensive examination of most aspects of the problem has been carried out by Huuska (1976) in order to evaluate underkeel clearances in Finnish waterways. Calculations of the hydrodynamic forces and motions of a ship, oscillating about a state of rest in 6 degrees of freedom in response to excitation by a harmonic wave in shallow water have been performed by van Oortmerssen (1976). He linearizes the problem of finding the velocity potential around the ship's hull with zero for.vard speed describing the flow by a distribution of three-dimensional sources. His results from calculations show good agreement with experimental ones. The problem has also been under theoretical investigation by Tuck (1970). An important aspect is the influence of forward speed. The reasoning is that squat, i.e. increased draught of ship because of suction between bottoms of ship and sea, must be combined with the motions. An examination of the influence of speed has been made by Hooft (1974) for water depth greater than 1.875 times the draught of the ship. In the theoretical calculations of this work

it has been assumed that the method of strip theory which is applicable for the determination of ship motions and sea loads in infinite water depth is also valid for the vertical motions in

restricted water. This assumption has been verified by Hooft by elaborate model

experi-mentation (1974).

That the strip theory is valid for the case of finite water depth makes it worthwhile to consider the influence of water depth on the hydrodynamic properties of two-dimensional

sections. A Green's function method by 1-b and Harten (1975) is very general and can even

take a quay into account. Svendsen (1968) has calculated forces induced on a rectangular

cylinder by a forced heave motion with draughtdepth ratio close to unity. Hisexpression for the pressure at the bottom ofthe section contains a term. proportional to the square of the heave

velocity, i.e. a non-linear term. Svendsen (1977) has also analysed the wave induced heave

motion of a rectangular section, this time under the assumption of small motion amplitudes

compared with the underkeel clearance. This assumption implies linearity in the motion.

Newton (1975) has used the finite element method to compute the added mass and damping

of a two-dimensional section. An improved version of this method making use of eigen-function expansions has been given and applied with successful result by Bai and Yeung

(1974).

To give a more complete description of the problems of ships in shallow and restricted

waters squat and waves will be described in the following sections. A further analysis will be

made on the calculation of ship response amplitude operators and results will be presented in later sections.

SQUAT

Squat, i.e. mean sinkage because of suction between bottoms of ship and sea, as well as changes in trim have been dealt with both theoretically and experimentally. The theoretical

evaluation given by Tuck and Taylor (1970) seems to give good results as compared with experiments carried out by Hooft (1974).

For a channel of infinite width, i.e. a sea of restricted depth only, Huuska (1976) has

gi' wh resm sm Eqi Rcs 558 POUL ANDERSEN

Ship motions and sea loads in restricted water depth 559

given a formula which in a simplified form is given as

iT

B2.4Ph

T

(1

-where Fflh is the Froude number based on water depth Ii V

Fh

-FIG. 1.

For the ship given in Fig. 1 and Table I the squat has been calculated using Equation (1). Results are shown in Table 2.

TABLE 1. MAIN PARTICULARS OF SHIP OF Fio. 1

-/g h

(1)

The formula is applicable for ships of normal dimensions. It is based on a collection of results from several model experiments. The scattering as Huuska points out is about ± 25%. Equation (1) is of course only valid for F,,, < 1. In this region the trim is rather small. For a more complicated bottom topography such as a fairway or inclined bottom, Equation (1) cannot be used.

L 158.5Gm B 23.2Gm T, even keel 7.75 m A 18,000t LCG 0.002 L abaft midships Radius of gyration 0.315 L WAVES

When dealing with wave induced motions of a ship in shallow and restricted waters it seems appropriate first to describe briefly the waves. Descriptions of such waves may be found in textbooks on coastal hydrodynamics, e.g. Svcndsen and Jonsson (1976).

The waves of significant interest in this connection according to Sjöfartsstyrelsen (1968) p, a is as ip s. th in 1-to al ar or ye ye es n. rig n-. rig ed be ed as :al ith tas

I

TIT

TABLE 2. RELATIVE SQUAT. PROBABLE TOUCH OF BOTTOM

ER V 20 KNOTS, lilT = 1.5

are generated in a deep water area and then propagate into areas of restricted depth. Hereby the wave velocities and the waves become shorter, the crests steeper while the troughs become

longer and flatter. These are more severe for the longer waves. Further, because of the

steeper crests breaking of waves will occur more frequently. This seems to indicate that the normal statistical description of the sea by means of sea spectra is inaccurate because of the linearization which cannot take into account effects of steep crests and fiat troughs. However, spectral description of the sea seems to be the only technique which is now applicable in practice. Following a method given in Sjofartsstyrelscn (1968) the spectrum

of the given sea area assuming infinite depth, e.g. the ITTC-spectrurn, with correct duration

and fetch of wind, may be modified for finite depth. This may be done by multiplying the spectrum with the square of two coefficients taking into account effects of shoaling and refraction. The shoaling coefficient which is a function of wve frequency and water depth

is the ratio of wave heights in water of finite and in6nite depths due to simple shoaling, i.e.

propagation of waves from one water depth to another with the wave fronts parallel to the

bottom contour. Calculation of the shoaling coefficient may be carried out with theformulae ofSvendsen and Jonsson (1976). This reference also describes the evaluation of the refraction

coefficient. This coefficient which is a function of the wave frequency is the ratio of wave heights of finite and infinite water depths and it takes into account the influence of the deformation of wave fronts, called refraction.

MOTIONS IN WAVES

With the sea spectrum known the task left is to obtain the response amplitude operator

for the ship. This may be done either by experiments or by theoretical calculations. Because

of the finite water depth experiments must be expected to be extremely. time-consuming and costly which turns the interest towards a theoretical approach. As mentioned in the introduction to this paper the strip theory is valid also for the finite depth case at least for the vertical motions of the ship. This provided that the draughtdepth ratio is not close to

unity where non-linear effects will occur and where the flow near the ends of the ship tends

to be more three-dimensional.

For the results presented later in this paper the strip theory due to Salvesen et al. (1970)

has been used for calculating the wave induced motions, i.e. heave and pitch, and loads, to heave and pitch corresponding vertical bending moment and shear force. The two-dimen-sional sectional properties, i.e. two-dimentwo-dimen-sional added mass and damping and exciting

forces, the latter consisting of a FroudeKryloff and a diffraction term, have been calculated

using a finite element method.

V, knots 0 10 20 aD 0 0 0 hIT: 3 0 0027 0.137 1.5 0 0.057 0.745 560 PouL ANDERSEN con. is a Wat flui spec has a tioij And bout abo into matc bo UT whei the F (p

+-4,

0 (2) Y

L4- 0

q so e'°' '(p

-FIG. 2. Definition of problem.

The problem of a heaving, infinitely long cylinder has been outlined in Fig. 2 as a boundary value problem in the potential p. There is symmetry in motion and geometry about the centerline (z-axis) and the free surface is at z = 0 (j-axis). The region is divided into an inner (I) and an outer (2) and the potentials of each of the regions p1 and (7 must match on the boundary line J. Calling the area of the inner region S1 surrounded by the boundary ! which does not include J and generalizing the boundary conditions on 1 to

+ ap + b = 0,

where a and b are appropriate constants, it can be shown that a functional which contains the boundary value problem above when made stationary is

F(p1, p) =

+

(1)2)ds

±

_{5 (} ap21 ± bp1)dl On J: (p1 =

+ 5

((pr - tp2)

dl.

y (2)

Ship motions and sea loads in restricted ater depth 561

1. Finite elenient method

The finite element method has been widely used in structural analysis but it has only come into use in the field of hydrodynamics in recent years. Because of this andbecause it

is a very computer-oriented method it was decided to use this method on the restricted water problem thus gaining experience in the applicability of the method on this particular

fluid flow problem.

A thorough description of the finite element method is given by Zienkiewicz (1971). A

special version given by Bai and Yeung (.1974), called "the localized finite element method", has been used. A short description of the method is given below. A more thorough descrip-tion including details regarding elements, strip equadescrip-tion etc. and further results are given in Andersen (1978).

This is, however, only the case if the potential of the outer region P2 fulfils the field and the

boundary equations, not including the conditions on J, and if it contains no constants. This

functional and a proof of its validity has been given by Bai and Yeung (1974).

The inner region is divided into elements with corresponding nodal points as usual in the finite element method. Also shape functions are introduced by writing the potential in the form given below as a linear sum of shape functions and nodal point values of the potential, the determination of which will be shown later.

(p1=Z N(y,z)q1.

(3)

i= I

Similarly, the potential of the outer region may be written as a sum 'Pt -I

=

_{'Pj 0 (y, z)} (4)

1=0

The functions 0 must satisfy the field and the bottom and free surface boundary conditions.

Using the method of separation of variables the eigenfunctions of the problem can be evaluated satisfying the given conditions. p21 of Equation (4) are unknown coefficients to

be determined later. Thus 01 are given as

0 = e

im, (YaY)cosh in0 (h--z)

cosh ni0h e nt,(y0_y) cos n,1 (hz) ,,z e

'

cos in1 (hz), where (02

rn0 tanh ni0h =

-g (02 rn1 tan

= - -

J = 1,.. .,inl.

g

In order to solve the problem, i.e. to find the coefficients Pn, P2J, Equations (3) and(4)

are inserted into Equation (2). Since F contains the solution to the problem in the point

where it is stationary it is differentiated with respect to the variables, Pu and (p21, and each differential coefficient is put equal to zero. This gives the following system of linear equations:

(5) Pt

(7±A)(p+

j=1 n

tnI

E B*11p1 +

Ap21

1=1 j=0

B*(p23+B1=0

1=0

= 0

i = 0,. .

i==l,...,n,

., rnI

(6) wh is ti for arni eler 562 POUL ANDERSEN wF Tb be 01) In mc J 3

where

= - co

e

Ship motions and sea loads in restricted water depth

Krf (+1)ds

cy

bz cz

A=faNiNjd1

Bj==J bN1dl,

A*..r= 1 80. 80.

JJO

+01d1

i=l,...,n

The integrals may be evaluatedelement by element and summed up over the entire region.

A*u can be evaluated analytically.

If a normal finite element approach was to be followed the wholeregion in Fig. 2 should

be divided into elements. A radiationcondition such as outgoing waves should be applied on the boundary line J which should be moved as far to the right as practically possible. In Equation (6) the terms A*u and B*11 should be omitted while A11 and B1 should be modified to take into account the radiation boundary condition.

\Vhen the potential has been found the added mass a3, dampingb3 and exciting forces

f

and h*3 may be found as

- hub3 = - pie) f N p dl

cx

= g

e ikysin 3 - ikysin 3 cosh

(k (hz))

_Ndl,

cosh k/i

cosh (k (hz))

_{(iNlauh (k( h-z)) - -. Nsin}

pdl

cosh kh

563

(7)

where C is the section contour and is the heading angle (3 l80 for head seas). (D is the wave frequency, e) the frequency of encounter. The Froude-Kryloff and diffraction force terms f*3 and /1*3 contain a factor p; p being thedensity of water and the wave

amplitude.

In Table 3 some results are presented from calculations with both the normal finite

element method and the localized finite element method. Theresults of Lee (1976) with which

1 80.

564

*Fjfle mesh.

Poui ANDERSEN

TABLE 3. RESULTS OF CALCULATIONS BY STANDARD FINITE ELEMENT METHOD (FEM) AND LOCALIZED FINITE

ELEMENT METHOD (LOCA. FEM) COMPARED Will! RESULTS OF LEE (1976). TuE CALCULATIONS ARE CARRIED OUT FOR DEEP VATER FOR A RECTANGULAR SECTION WITH BIT = S/it

- a3 ..

bA

A: sectional area B: sectional breadth

f* + /1*31 2gB/2 T: sectional draught

the finite element results are compared are obtained by the method of source distribution.

The mesh used by the normal finite element method has been constructed with a horizontal

distance between nodal points of 1/10 of a wave length while the radiation condition is applied 3 wavelengths from the centreline for a relative wavelength 2./B/2 = 10. Near the section the mesh has been made finer. The notation fine mesh indicates that for these

calculations the mesh has been much refined near the section while further away the standard

mesh and the fine mesh are identical. Linear shape functions have been used and the same meshes have been used for all calculations despite changes in wavelength. The meshes are

thus somewhat coarser than recommended by Newton (1975) and the accuracy of the results must be seen in viev of this. The mesh for the localized finite element method has the same

subdivision near the section as the standard mesh while of course in the outer region the potential has been described by the eigenfunction expansion.

The better results obtained by the localized finite element method compared to those of the standard finite element method arc most likely due to the better description of the

potential far away from the section. The advantage of the localized finite element method is

further demonstrated by Table 4 from which it can be seen that the better results are

obtained with considerably less computational effort.

During other calculations with the normal and with the localized finite clement method for other genlnetries and water depths no numerical problems have been encountered. This leads to the conclusion that the localized finite element method is very suitable, if not

efficient mathematically.

2. Results

For the ship of Fig. I and with main particulars of Table 1 a series of calculations has

* )./B/2

f,

Lee (1976) FEM Loca. FEM Lee (1976) FEM Loca. FEM Lee (1976) FEM Loca. FEM 0.7167 60 1.64 1.343 1.576 1.65 2.207 1.474 0.80 0.815 0.810 1.0136 30 1.24 1.124 1.219 1.23 1.354 1.212 0.69 0.716 0.691 1.2414 20 1.08 1.015 1.047 0.95 0.992 0.957 0.61 0.627 0.615 1.4339 15 1.01 0.964 0.974 0.75 0.766 0.755 0.54 0.552 0.546 J4339* _{15 1.01 0.987}

-

_{0.75 0.752}

-

0.54 0.548

-1.6026 12 0.99 0.945 0.949 0.59 0.606 0.598 0.48 0.488 0.484 1.7556 10 0.99 0.942 0.949 0.46 0.484 0.475 0.43 0.441 0.432 1.7556* 10 0.99 0.96S

-

0.46 0.473

-

0.43 0.434

-1.9627 8 1.02 0.968 0.974 0.33 0.324 0.337 0.36 0.360 0.365 2.2664 6 1.09 1.047 1.038 0.18 0.181 0.192 0.27 0.256 0.276 2.7757 4 1.20 1.146 1.156 0.06 0.062 0.064 0.16 0.159 0.160 2.7757* ₄ 1.20 1.181 0.06 0.058

-

0.16 0.167

-3.9255 2 1.37 1.283 1.311 0 0.031 0.003 0.03 0.032 0.032

I

FEM, standard mesh 571

-

350 K 72 s

FEM, fine mesh 939

-

850 K 178 s

Localized FEM 194 20 250 K 16 s I.0 N 0.5 N 0.5 Heave , V 0 kn Deep h/T' 3 '.5 _h/Tl.5 0.5 _X/L Heave, V 2Okn Fio. 3 1.0 1.5 Deep h/T' 3 h/T' .5

Ship motions and sea loads in restricted waler depth 565

TABLE 4. COMPUTER RESOURCES BY DIFFERENT METHODS FOR A REPRESENTATIVE WAVE FREQUENCY. CALCULATIONS ARE CARRIED OUT ON AN IBM 370/165

COMPUTER

0 0.5 _X/L

Heave, V IOkn

No. of

No. of

eigen-Method nodes function Storage CPU-time

1.0 1.5 '.5 1.0 0 .5 1.0 0.5 0 0.5 _{X/ L} 1.0 I .5

I.5 1.0 0.5 0 1.5 1.0 0.5 0 I .5 a:, 0.5 Pitch, V 0 kn Deep

h/T3

h/TI5

/

/

,. 05 1.0 1.5 X/ L Pitch, VlO kn Deep h/T3 h/Tl.5

//

I

-._."

I I I 0.5 1.0 1.5 X/L Pitch V 20 kn _Deep h/T3 h/T'I.

-/

/

.1'

7/

I. I

//

I,

I 0 05 1.0 1.5 X/L F[G. 4

been carried out for combinations of speed and water depth as in Table 2. Only head seas have been considered. Response amplitude operators for heave, pitch, vertical bending

moment at midship section and shear force at L/4 forward of midship section have been calculated. The results are shown in Figs. 3-6 from which the effects of water depth can

be seen. Phase angles have not been plotted since they hardly vary with varying water depth.

From the figures it can be seen that the water depth has a significant influence on the

response of the ship. Generally, the peak values of the response amplitude operators for the

motions, i.e. heave and pitch, are decreased while those for the loads, i.e. vertical bending

mo -pro spec for

X/L

Shear force, sect. 5, Vl0 kn

/

N

moment and shear force, are increased for decreasing water depth. This effect is more pronounced with the increase of ship speed. For constant water depth an increase in ship

speed will increase the maxima of the response amplitude operators as can also be observed

for deep water.

CONCLUSION

The behaviour of a ship in waters of restricted depth is strongly influenced by the

water depth and the ship speed and also by combination of the two. A complete examination

Deep 1.5 h/TX 3 h/Tj .5 "--.5.. 1.0 '5..

/

/

/ /

/

/

0.5 0 0.5 .0 '.5 X/ L

Shear force, sect. 5, V- 20

I.5

'

N Deep h/T-3

--- h/TI.5

oa .5 1.0 0.5 0

Ship motions and sea loads

Shear force, sect.

/

'I

in restricted water depth 567

5, V 0 kn Deep

h/T3

h/T'I.5 ----S 0.5 1.0 .5 C 0 -J 1,0

I-a

0.5 0 1.5 I ( 0.5 1.0 X/L FIG. 5

.-,

568 POUL ANDERSEN

Bending moment, amidships, V0 kn B

H Deep h/T3 H 0

_J5

H N

/

S 0 05 1.0 .5 X/L S

Bending moment, amidships V kn S\

s' s' Deep h/T 3 h/T'l.5 N₀ S

J5

Tt Zi 0 0.5 >'/L .0 .5

Bending moment, amidships, V' 20 kn

Deep h/T3 h/TI.5 N

7

0

_-J5

_N. a 0 0.5 _X/L 1.0 1.5 FIG. 6

of this problem should thus include the effect of speed on the wave induced motions and loads. For such an examination, at least for the vertical motions, the strip theory is quite useful. The finite element method is very suitable for calculating the corresponding

two-dimensional properties.

REFERENCES

ANDERSEN, P. 1978. Bolgeinducerede bevaegeiser og helastningcr for skib pa laegt vand (Wave induced

motions and loads on a ship in shallow water). Dept. of Ocean Engineering, Tech. Univ. of Denmark. (In Danish.)

Ship motions and sea loads in restricted water depth 569

BAt, K. 3. and YEUNG, R. W. 1974. Numerical solutions to free-surface flow problems. Proc. ofthe 10th ONR Symposium.

Ho, R-T. and 1-IARTEN, A. 1975. Green's function technique for solution of floating body problems. Proc. Civil Engineering in the Oceans II!, Newark.

Hoon, J. P. 1974. The behaviour of a ship in head waves at restricted water depths.hit. Shipbldg Prog. 21,

No. 244.

HUUSKA, 0. 1976. On the evaluation of underkeel clearances in Finnish waterways. Report No. 9, Ship Hydrodynamics Laboratory, Helsinki Univ. of Tech.

LEE, C. M. 1976. Motion characteristics of floating bodies. J.Ship Rcs. 20.No. 4.

NEWTON, R. E. 1975.Finite cle,nent analysisoftwo-dimensional added mass and damping. Finite elements in

fluids. Edited by Gallagher, Oden. Taylor, Zienkiewicz. Wiley.

VAN OORTMERSSEN, Cr. 1976. The motions of a ship in shallow water. Ocean Engng 3, No. 4.

SALVESEN, N., TUCK, E. 0. and FALTINSEN, 0. 1970. Ship motions and sea loads.Trans. Soc. nm. Archit. mar. Engrs78.

SJöEARTSSTYRELSEN. 1968. Farleders Djupmarginaler (Underkeel clearances of fairways). Stockholm.

SVENDSEN, I. A. 1968. On the forces induced on a rectangular cylinder by a forced heave motion with

draughtdepth ratio close to unity. J. IJydr. Res.6, No. 4.

SVENDSEN, I. A., MADSEN, P. A. and MICE1AELSEN, C. 1977. Wave-induced heave motion of a ship in shallow

water. Progress Report 43, 1SVA, Tech. Univ. of Denmark.

SVENDSEN, I. A. and JoNssoN, I. U. 1976. Hydrodynamics of coastal regions. Den private Ingeniorfond, Tech. Univ. of Denmark.

TUCK, E. 0. 1970. Ship motions in shallow water. J. ship Res. 14,No. 4.

TUCK, E. 0. and TAYLOR. P. J. 1970. Shallow water problems in ship hydrodynamics. Proc._ofthe 8h ONR

Symposium.

ZIENKIEWICZ, 0. C. 1971. The finite element method in engineering science (2nd edition). McGrawHill,