### Defft University of

_{Technology}

### 09 ME 1988

Ship Hydrornechan.ics LaboratoryMekelweg 2 _{2628 CD DELFT}
The Netherl.,)nds

WAVE INDUCED MEAN SHIFTS IN VERTICAL ABSOLUTE AND RELATIVE MOTIONS

by

### John

O'DeaDavid W. Taylor Naval Ship Research and Development Center Bethesda, Maryland 20084

Abstract

Ship slamming and deck wetness in waves are strongly affected by the
mean draft or mean freeboard at a given location _{on a ship.} _{It is}

well known. that sinkage, trim and bow waveprofile can change the local
draft and freeboard significantly. _{Calculations of these effects}

are normally made for the case of steady forward motion in calm'water. However, the mean values of these quantities may be changed when a

ship is advancing into waves. _{This paper presents experimental}
measurements of this effect, which are compared to predictions _{based}
on a second-order approach similar .to that used to calculate _{a.dded}
resistance or drift forces.'

Introduction

'One

### of

_{the significant problems in the seakeeping field}

_{is the }

pre-diction of the occurrence and severity Of deck wetness and bottom
slamming. _{These phenomena are strongly influenced by the frequency}
and degree to which the local free surface either rises above the
.deck or falls below the keel, respectively. _{Both are examples of the}
need to accurately predict vertical relative motions in waves near

the bow.

It has been recognized that relative motion cannot be calculated
simply as the difference between the local absolute vertical motion
and the wave elevation due to the local undisturbed incident wave
potential. _{Such a method usually leads to underprediction of relative}
motion at the bow. _{The discrepancy has usually been ascribed} _{to a}
"dynamic swell-up" of wave elevation near the bow. In fact, the
dis-crepancy in the oscillatory part can be attributed to several _{causes,}
including diffraction of the incident wave and radiation of waves by
the oscillating hull. _{These components have been investigated by}
Beckl and Lee.'

It is also recognized that the mean freeboard and draft are critical in predicting the frequency of occurrence of wetness and slamming,

, respectively, in random seas.

Within the usual framework of
assump-tions, i.e., linear motion _{responses, and narrow band Gaussian random}
processes for both waves and responses, the frequency with which _{a}
mean level is exceeded is an exponentially decreasing function of the

square of the mean level. _{Thus, it is extremely important to be able}
to accurately estimate the mean freeboard or draft. When a ship is

moving at forward speed, it. assumes a. certain dynamic trim and bow wave profile, which

### will

'change:thamean freeboard and draft. These components have been eatiMated by.Lee et al.3The current state of the art may be thus 'summarized as follows: The oscillatory relative motion is calculated as the difference between the absolute vertical motion (a function of the rigid body motions) at any point, and the actual local oscillatory free surface including

incident, diffracted and radiated waves. The sinkage, trim and bow wave profile due to steady forward speed in calm water are then used to correct the static freeboard or draft at that particular location. Finally, random process and linear system theory are-used to predict the severity of exceedence of relative. motion beyond. the mean free-board and draft.

One aspect of the overall problem, which does not appear to have received previous attention is the question of whether the mean sinkage, trim and wave profile are the same for a ship when running in incident waves, as when running in calm water at the same speed. Some recent experimental data obtained at DTNSRDC has indicated that there is in fact an additional shift when running in waves, parti-cularly for trim or mean pitch angle. This results in a mean free-board or draft which is different from that found for the calm water

case. Although the shifts are quite small compared to the amplitudes of oscillatory quantities, it is felt that they may still be signifi-cant because of the very strong exponential dependence of exceedence events on the mean level to be exceeded.

The mean shifts in trim and sinkage are analogous to the mean values of horizontal force in waves in the axial and transverse directions, usually referred to as added resistance and drift force. Such forces arise due to quadratic interactions between various components of the

potential flow. When the linear forces and motions are sinusoidal, the quadratic components give rise to non-zero mean forces and to second harmonics. When the first-order problem consists of more than one harmonic, quadratic components will also appear at the sum and difference frequencies. This latter effect is the source of low-frequency drifting motions of ships. Only the mean values in single-frequency regular head waves will be considered here.

Hydrodynamic Formulation

The problem is approached as a free-surface potential flow problem, in which the various boundary conditions are expanded in perturbation series. In order to allow the possibility of a non-zero mean value, the perturbation expansion must be carried to second order. On the other hand, the desire to obtain only the mean pressures, forces and moments on the hull obviates the necessity to solve the second-order potential, which would contribute only oscillatory. components.

In-stead, it is only necessary to carefully evaluate all the terms which are products of the first order potential and its derivatives. It is assumed that the potential flow of a ship advancing and oscil-lating

### in

head waves (see Figure 1for coordinates) can be given as:### FIGURE 1

### COORDINATE SYSTEM

The first and second terms represent the uniform forward motion of the ship, while the terms in the brackets represent the oscillatory components (the real part of all complex quantities is assumed). The (x,y,z) coordinates move with the mean forward velocity of the ship, but the x, y axes remain in the undisturbed free surface. The term 00 is the undisturbed incident wave potential,

g .

= A ekz-Ikx.e

4)0

where w.and k are the wave, frequency and wave number, satisfying

U =

### we -

kU in deep water; and is the wave amplitude. The termOd is the diffraction potential,

### satisfying the

linearized free surface Condition and the kinematic'condition.on the hull,30a/3n = -300/3n, while the terms

### 03,

05 are the heave and pitch forced oscillation potentials satisfying305

### 7g7 =

### -iwenz

and### 7Ft

### =xnz

_{+ TJnz}

and

nz is the vertical component of the unit normal to the hull sur-face.

By. using the assumptions of strip theory, the problem of solving for the three-dimensional potentials

### 0d, 03

and 05 is reduced to thesolution of a set of two-dimensionalproblems-along the length of the ship. If IP is a two-dimensional potential at a ship section at x, satisfying,

(2) t

w 2

1.1) = 0 on z = 0

Z g

and Wan =.-iwenz on-the section contour C(x), then according to.. strip theory, the heave and pitch potentials are given as:

03 = j(6 tlidx

dx +

171Pdx

(3)

The diffraction potential is obtained in a similar way from a two-dimensional potential satisfying

_{alVan =}

-401'3n on C(x).
The linearized oscillatory pressure from Bernoulli's equation (neglecting the hydrostatic component, the effect of which is in-cluded separately in the equations of motion as hydrostatic coef-ficients Cij)

### is

P = -P (,7-a

### - u

-57-c)[### +

4)d + 4)3113Integration of the pressure on the undisturbed immersed hull surface results in forces and moments which are the exciting force and moment in the case of

### (0

_{+d}

), and in the case of 03 and ### 05,

when resolved.o

into components in phase with velocity and acceleration, give the familiar added mass and damping terms of the equations of motion for strip theory (Salvesen, Tuck and Faltinsen4):

A33 =.1r. a33dx, B33

### =f

b33dx Xa 3dx -U2 B33 we Xb dx + UA33 V xa33dx +_{B33}we A35 B35 A53

_{-f}

-iw t
e e. (4).
### [-W2A-1WeB

### e53

53 U2 A5### =f

x a53dx + --n A33 we` B55### =Jr

x2b33dx + --T.12 y B33 we F3### =je

### (3

113)dx F5 = x(f3 + 1,3)dx + (5)where a b33 are the two-diMensional added mass and damping obtained from IP, f,1 is the sectional FroUde-Ktiloff excitation, from the intident potential ci)0, and h3. isthe sectional diffraction force. The linear coupled equations for the Complex amplitudes

n3' n5 are .then given as:

2 2

E7we(11433)e11334-C331n5 E-(4eA3 e 35 35 5 3

B +C in = F

fT_LA m

### in

+ 2 = F5

.where.C1.4 are .hydrostatic.coefficients.calculated from the ship's waterplene shape. The actual heave and pitch motions are given by

Re[n3eiwe;] and ReN5e-iwet], respectively.

Second-Order Calculation- s

The first-order problem of calculating the rigid body oscillatory motions requires the solution of potentials which satisfy the linear

free surface condition on z = 0 and a prescribed normal velocity condition on the undisturbed hull surface. In contrast, when the

perturbation expansion is continued to quadratic terms, several com-plications arise. First, a second order potential exists, which must satisfy a considerably more complex free surface condition, and which will have a harmonic component at twice the frequency of the linear problem. Also, the normal velocity condition for this potential must be applied on the actual moving hull surface, consistently to second order. Fortunately, this potential does not contribute to mean pres-sures and will not be further considered here.

Another complication is that the pressure must also be calculated up to second order. This includes the velocity squared term of Bernoulli's equation evaluated on the Undisturbed hull surface and a component due

to the expansion of the first order pressure onto the moving hull sur-face, rather than the undisturbed surface:

(2)

### =

### -p =

(1) (1) (2)### z +

### N1)-14x(1)]

+ [11V0 VO### +0t

### (op.)..0 (

))0(1)### vo(1)

/ tyIn both equations (7) and (8), the quantity of interest is the mean value. Since the only contribution of the second-order potential is

through Its time or axial derivative, it. make's no contribution to the mean value of the expressions. On the other. hand, mean values as well as second harmonics do arise from products of first order oscillatory potentials and their derivatives. That is,

-iw t -iw t

### Re[Ie

### e ] R+

e e### 1hRe

-### r,

### l

### ,*

### , 0

e-21we].### rsY2

(.9)where the asterisk superscript denotes the complex conjugate, and the first term in the brackets contributes the mean value.

Second Order, Two-Dimensional

### Case

It is.informative to consider next the case of a two-dimensional cylinder oscillating vertically in a sinusoidal wave field. This problem has been considered before by Potash5 and Lee6 for forced oscillation in calm water, and by Soding7 and Papanikolaou and

-14(2) (1)

### (01)_

2(c1))]_{(7)}

(1)

Where n isthe local first-order vertical displacement due to heave and pitch, 11(1):=

### n3 7

20.5, and the. superscripts (1), (2) indi-cate first-. and second-order quantities.Finally, there is a second-order term, evaluated at the intersection of the hull and free-surface, which is a force term arising essen-tially from the integration of a first-order pressure term over a unit of area which is the first order deviation of the hull-free surface intersection from the undisturbed location. If a hull is wall-sided at the waterline, this term will have no effect on the vertical force, since the vertical component of the normal would be

zero. On the other hand, when a hull has flare at the waterline this component makes a significant contribution to the total mean vertical force. It should be noticed that this component will make a signifi-cant contribution to the mean horizontal (added resistance or drift) forces, whether or not the hull has flare.

The second-order displacement of the free surface (ignoring the wave component due to the steady forward disturbance of the hull), is given by

(1)]

### 111117----

8Nowacki for free floating bodies in waves. Let a two-dimensional cylinder be Oscillating sinusoidelly in heave, with its motion given

by z= Re[n3e-iwt], either in calm water or in the presence of inci-dent (and diffracted) waves. The mean second-order pressure is, from equations (7) and (9), --(2)

_{p Lr,(1)}

### +

,(1)* . .(1) (1)* + 2i P -### T

T1 Tyevaluated on the undisturbed cylinder surface. The integration of this pressure on the undisturbed immersed section results in one compOnent of the total vertical force. An additional component must

be considered at the intersection ofthe free surface,. when the Cylinder is'not well-Sided at the free surface.

The geometry of the intersection is shown in Figure 2.. The integra-tion of the second-order pressure (see equaintegra-tion (10)) On the

undis-turbed cylinder surface Is equivalent to the integration on the actual instantaneous surface, correct to the second order, up tO the point,z = nl. However, the integration of the-first-order pressure, including tfie hydrostatic term, from .z = n3 to z = where 4 is the first-order wave.: eleVition; also results inl:a second-order contribu-tion;to the .vertical force if the intersection is not vertical

### (a 0

7r/2):-fink-ads -r13

### sum

_{=}

_{147}

### (

2t### + gn)y

3. tan..4_{(11)}

### dz

The mean force component. when the potential is harmonic is thus:

P[Re

n3cl)*1

### 14k4/2

gl3/2)]/ tan a (12)

This term, when evaluated at the intersection on both sides of the cylinder and combined with the integration of the pressure (see equation (10)) on the undisturbed cylinder surface, results in the

mean vertical force. F.

Sample calculations have been made for a typical ship section, shown in Figure 3. Its motion has been restricted to the heave degree of freedom only for illustrative purposes, although it is recognized that the results for the second-order quantities would be different if roll and sway were included. The first-order heave transfer func-tion is shown in Figure 4. It is interesting to note that the Lewis-form approximation to this section, also shown in Figure 2, has

virtually the same first-order properties as the actual section. However, the second-order properties are quite different. Contribu-tions to the mean vertical force for the two shapes are shown in Figure 5 for two cases: pure heave in calm water, and incident wave excitation on a fixed cylinder. In both cases, there are significant differences in the forces between the actual shape and its Lewis form approximation. The difference is caused primarily by the "inter-section" term (see Equation (12)) for the actual cylinder. Since Lewis forms are by definition wall-sided, this term does not produce a force on a Lewis form.

### UNDISTURBED POSITION

### INSTANTANEOUS POSITION

### FIGURE 2

### GEOMETRY OF HULL-FREE SURFACE INTERSECTION

### FIGURE 3

1 .0

### -90°

MIMED### 0.5

### 1.0

### 1.5

### 2.0

### ire

### FIGURE 4

### MAGNITUDE AND PHASE OF

### CYLINDER HEAVE MOTION

The mean vertical force on the cylinder heaving in waves is shown in Figure 6. Since this force is a result of nonlinear interaction among potentials, the force cannot be calculated from a superposi-tion of the diffracsuperposi-tion and forced mosuperposi-tion components of Figure 5. In particular, for a given amplitude of incident wave and heave motion, the mean force will vary if the phase angle between the incident wave and motion changes. This effect is also shown in Figure 6, where the curve labeled 00 is the curve for the freely floating heave motion of Figure 4, and the other curves represent successive shifts of 900, 1800, and 2700. These curves are shown to illustrate the importance of the phase relationship, since in the case of estimating the vertical force on a three-dimensional hull through the use of strip theory, the phase between the vertical motion at a given station and the local incident wave is governed by

the three-dimensional rigid body dynamics and can be quite different from the phase based on simple two-dimensional calculations.

113

### 0.25

### 0

### pgA2

### 0.50

### 0.75

### 0.25

### -0.25

### ACTUAL SECTION

### LEWIS FORM APPROXIMATION

## //"°

OM=### 270°

### 90°

### 2.0

### fz

### -0.25

### P9tA2

### 2.0

1.5### 0.5

1.0 7rB### FIGURE 5

### COMPARISON OF MEAN VERTICAL FORCE

### COMPONENTS FOR LEWIS FORM AND ACTUAL SECTION

### 0.50

### 0.5

1.0### 1.5

.TrEi

X

### FIGURE 6

### MEAN VERTICAL FORCE (COMBINED HEAVE AND DIFFRACTION)

### INCLUDING EFFECT OF PHASE SHIFT

The final quantity calculated for the two-dimensional example is the mean wave elevation at the sides of the cylinder, shown in Figure 7.

The mean wave elevation, from Equations (8) and (9), is:

### k2001

(13)

This calculation is shown for the free heaving cylinder, with the incident wave approaching from the left side. The mean shift is everywhere positive, indicating an effective reduction in mean free-board for the cylinder. As wave length becomes shorter, the mean shift on the right side approaches zero, indicating a sheltering effect, while the nondimensional mean shift on the left approaches a value of 1.0, which is the value for a plane progressive wave re-flected from an infinitely deep vertical wall.

### 1.0

k4A2

### 0.5

### 0.5

### 1.0

### 1.5

### 2.0

### irB

### FIGURE 7

### MEAN WAVE ELEVATION ON HEAVING CYLINDER

### IN WAVES (WAVES APPROACHING FROM LEFT)

Three-Dimensional Case

The calculations previously illustrated for a two-dimensional

cylinder, may also be carried out for three-dimensional ships using strip theory. This has been done for a case in which experimental model-scale results were available, showing a noticeable shift in waves. The body plan of this ship is shown in Figure 8. As can be

seen, there is considerable flare at the waterline in both the fore-body and afterfore-body, and the effect of this flare will be illustrated in the calculations below.

### FIGURE 8

### HULL BODY PLAN

The first-order heave ancLpitch Motions,' calculated by normal strip theory (Reference 4), are shown in

### Figure 9.

These results### are

inline with previous calculations for this

### hull;

however,. It is known (O'Dea and Jones9.) that the 'heave motion forYthis hull is signifi-cantly overpredicted, apparently caused by errors in the calculation of cross-coupling coefficients between heave and pith..The mean shifts in sinkage and trim, caused by interactions of the wave potential and the forced oscillation potential, have been

cal-culated by integrating the second-order pressures to give the mean heave force and trim moments:

.ir Fz

3 .

### -.1rezdk

where Fz is the mean vertical sectional force obtained by the
combina-tion of integrating second order pressures around the contour and
evaluating the "intersection" correction, Equation (11). These mean
loads may then be used together with the hydrostatic coefficients _{to}
determine the mean sinkage and trim. The calculated mean shifts in
heave and pitch are presented in Figures 10 and 11, together with
experimentally measured values. In general, the agreement is good
for heave, but the calculations underpredict the mean pitch. In
consideration of the small magnitude of the measured shifts in mean
pitch and heave (on the order of a few tenths of a degree and a few
millimeters, respectively), the overall correlation between prediction
and measurement is considered reasonable. _{It is interesting that}
-both the calculations and measurements indicate a rise in the center

of gravity, compared to its location while advancing in calm water, and both predictions and measurements indicate a shift in trim toward a bow up condition. Both trends will tend to increase mean free-board forward.

### 1.5

### 1.0

### 0.5

### X/L

X/L### FIGURE 9

### FIRST-ORDER HEAVE AND PITCH RESPONSE

1

### 2

### 3

-1.0 1.5 -713

### kA2 1.0

.5### PREDICTED

### 0 0 0 0 MEASURED

### X/I-FIGURE 10

### COMPARISON OF PREDICTED

### AND MEASURED MEAN PITCH SHIFT

1 2

### FIGURE 11

### COMPARISON OF PREDICTED

Figure 12 shows the change in mean wave elevation at a point 10 per-cent of the length from the bow. The experimental data, which were

taken with a relative motion probe mounted to the hull, have been corrected for the shift in mean absolute motion at that station,

associated with the mean values of heave and pitch. Both predicttions and measurements show a strong variation over the wavelength range

considered, including both positive and negative values, and the correlation is not very good campared to the motions. A possible cause of the discrepancy is a possible interaction between the mean wave caused by oscillatory quantities, and the mean wave generated

by the steady forward motion.

.k4 A2

### 00

### a---.PREDICTED

### 0 0 0 MEASURED

### k/L

### FIGURE 12

### PREDICTED AND MEASURED

### SHIFT IN MEAN WAVE LEVEL

### AT Fn = 0.3, x/L = 0.4

A comparison of the components of the vertical mean loads is shown in Figure 13. Three components are shown. The first is from the

integration of the (-1gpv2) term of Bernoulli's Equation (7) on the undisturbed hull. The second is from the Taylor expansion of the first-order pressure onto the moving hull surface. The third term is from the evaluation of the pressure integration at the intersection of hull and free surface at the waterline (Equation 12). As can be seen, the last term is daminant for a hull with flared sections. In fact, if this hull were wall-sided, the net mean vertical force would be nearly zero, and the mean pitch moment would be in the bow down direction.

### F(x)

### pg e,

4 2 -2### -0.5

### x/L

### 0.5

### FIGURE 13

### COMPONENTS OF MEAN

### VERTICAL FORCE

### Fn = 0.3, X/L = 1.0

ConclusionsA method has been developed to compute the mean shifts in pitch and heave, caused by interaction of a ship with incident waves. The

effect of flare at the waterline has a strong effect on these computa-tions. The overall effect is to indicate a change in mean freeboard and draft at the bow, when running in waves. This is expected to have an influence on the occurrence of slamming or deck wetness in random

seas.

Acknowledgement

This work was sponsored by the Naval Sea Systems Command under the General Hydromechanics Research Program administered by the David W. Taylor Naval Ship Research and Development Center.

/-\

References

Beck, Robert F., "Relative Motion Components for a Mathematical Form in Regular Waves," 14th ONR Symposium on Naval Hydrodynamics, Ann Arbor, Michigan, 1982.

Lee, Choung M., "Computation of Relative Motion of Ships to Waves," DTNSRDC Report 82-019, March 1982.

Lee, Choung M., John F. O'Dea and William G. Meyers, "Prediction of Relative Motion of Ships in Waves," .14th ONR Symposium on Naval Hydrodynamics, Ann Arbor, Michigan, 1982.

Salvesen, Nils, E.O. Tuck and Odd Faitinsen "Ship Motions and Sea Loads," Transactions SNAME, Vol. 78, 1970.

Potash; Roger ,L., "Second Order Theory of Oscillating Cylinders,' UniVersity of California, Berkeley Report NA10-3, June 1970. Lee,-Choung M., "The Second-Order Theory of Heaving Cylinders in a .Free Surface," Journal of Ship Research, December 1968.

Soding, H., "Second-Order Forces on Oscillating Cylinders in Waves," Schiffstechnik, Bd. 23, 1976.

Papanikolaou, A. and H. Nowackl,_"Second-Order Theory of Oscillat-ing Cylinders in a Regular Steep Wave," 13th ONR Symposium on Naval Hydrodynamics, Tokyo, Japan,

1980-O'Dea, John F. and Harry Jones, "Absolute and Relative. Notion Measurements on a Model of a:High-Speed Containership," 20th American Towing Tank Conference, August 1983.