On the added resistance of a SWATH ship advancing in head waves

I1eke1weg 2,2828 CD Deift T&018.788873 Fac O18 781838

4..

On the. Added Resistance. ofYWATH _{ship Advancina in Head. Waves}

By

Ming-Chunq Fang*, Wei-June Shyu**

*prOfessor

**Graduate student

Department of Naval Architecture _{and. Marine Engineering} tional Cheng Kung University

V

ABSTRACT

A technique for predicting the added resistance for a SWATH ship advancing in head waves is presented in the paper. The

nonlinear hydrodynamic force theory and the strip theory with viscous damping correction are used . Basically the nonlinear theory used in the paper originates from that of single ship and the hydrodynainic interactions between two hull bodies are

included in the calculations. _{Three basic assumptions are made:} (1) the hull body is a "weak scatterer" (2) the second order potential is small and can be neglected (3) the ship motion is based on the linear strip theory but includes the viscous damping effect. According to the present study, it shows that the

previously developed nonlinear theory for single ship may be applied to predict the added resistance of a seagoing SWATH _ship. Hopefully the technique developed here may offer some useful

a₀ :. viscous-lift coefficient

vertical velocity of the fluid induced by incident wave

b : indication of body b

b(x): tranverse. distance from the x-axis to the i4dpoit of the beam of one hull

.restoingfore per unit displacement or maximum beam of submerged cross sectin of one hull

c :. contour- of body

:. cross-flow drag coeffceint

d1 (x): depth to the maximum breadth point at cross section

F : second-order force

F : mean second-order force

Fk : wave exciting force in it mode direction

:

Frouae number (U//L

g : gravitational acce1era.ton G :

pulsating oce function

k : indicating the diretion of force

characteristic

length

indicating mode of motion total mass of the ship

M :. added .mss

Q : source intensity

Sm : displacement of response motion of mode m of ship SB the submerged surface Of the body

: a control surface in the far-field

T : the oscillation period (2.iT/)

U : ship speed

: angle of segment Of contour with

respect to y-axis

: wave length

angIe of wave -incidence wave number

wave number based on encounter frequency water density

: resultant velocity potential

the potential due to the body isturbance

incident and-diffraction v.e1oci.t _{potentials respectively}

radiation velocity potential

R

wave frequence

INTRODUCTION

The technique for predicting the total ship resistance in the calm water has been developed _{quite. well, either by experiment or} by thery. Generally the _{horsepower needed for a ship can be well} estimated by considering _{the friction, wave-making viscous,}

air and appendages resistances. _{However, while the ship is advancing}

in waves the ship engine _{power output will be reduced, due to the}

Ship motion. Traditionally he _{power required to attain a certain speed} in a seaway has been determined from the still water perforEance of the vessel after making an allowance of 15 to 30% for wind and

waves [1].

However, the analytical mthods _{and experimental techniques}

for determining. the _{added resistance Of ship in a seaway has been} developed quite well and can be used for design purpose. Havelock [2] developed a theory tO estimate the added resistance with the right order although it is not so accurate. The refined _theories were made by Maruc [3] 'and Gerritsma and Beukelman [4] _{and have} been shown to give better. results. A different but fairly accurate theory was also developed by Strom-Tejsen[5]. Ankudinow [6] used the sane manners as did Havelock but includes _{the diffraction}

effects to calculate the added resistance.salvesen [7] derived the added resistance of a ship in a way which closely followed _Newman's derivation for submerged bodies(8]. Fang[9] followed the same weak scatterer assumptjo _{as Salvesen[7] did but different diffraction} expression and better: 'results were obtained.

The theories for predicting the motions of a SWATH ship have been well developed recently, e.g. [1O][11][12](13]. _{However it} seems no authors have done any analysis for the added _{resistance of} a SWATH ship advancing in waves. Similar to the design _{for the}

single inonohufl ship, _{the added resistance is also an important} factor for estimating the power of a _{SWATH ship advancing in seaway} in the design stage. Therefore it will _{be quite helpful to develop} the method for predicting the added resistance _{of the SWATH ship.in} the present paper the author uses the _{nonlinear theory [9]}

incorporated the well developed SWATH ship 'notion prediction _method (13] to calculate the added resistance for a SWATH ship advancing in waves. The viscous _{damping effect[].4] is}

also considered to improve the motion prediction and the results are quite

satisfactory.The corresponding theory is described in the following sections..

'V

EQUATION OF SWATH MOTION IN HEAD SEA

The motion of a SWATH _{advancing in waves are described by the}

inertial coordinate, O-XYZ, and body _{systems, o-xyz, as shown in} Figures 1 and 2. The body-fixed system coincides with inertial system at time=O and moves along the positive X-axis at speed U.

The ship motion in response to head wave, i.e. 1-' =1800 is

calculated by the following linear _{coupled equations,}

rn35

- M)- 1N

In the eq-uatiorjs, M, Nwc and B _{represent the added mass, damping}

and restoring force respectively. Sm is the motion response and Fc is the exciting force.

Because of the disturbance between two hull bodies, the hydrodynpjc coefficients mentioned _{above must be solved by the} boundary conditions including _{the hydrodynainic interactions. The} corresponding potentials can be _{expressed in the following form,}

D (x.,v.,z.,z) _{= D}

(y:,z1)et)

with

ÔD

(y,

(n,

)G(y4,z47n,ç4,v)dc

7 where .i,j =a or b, Q is the source intensity and the Green function G is the two-dimensional pulsating source potential per unit intensity at the point in the lower half of the the y-z plane. The c represents the hull section contour below the calm water surface. VISCOUS DANPING COEFTICIENTS Due to the small waterplane, the SWATH ship generally does not generate large waves when it oscillates in the vertical plane modes. Therefore the wavexnaking damping will be relatively small and the viscous effects may contribute significantly to the damping especially in the neighborhood of motion resonance which should be carefully considered. A technique [14] to determine the supplemental viscous damping according to Thwaites[15J is used here to improve the motions prediction.The corresponding formulas for damping coefficients. restoring forces and the wave-exciting coefficients due to the viscous effect are shown in the Appendix A. SECOND-ORDER STEADY FORCE OF ADDED RESISTANCE The techniques for predicting the added resistance of monohufl ship [7][9] are used here for SWATH ship. Two basic assumptions are made: (1) the twin-hull bodies are considered as weak scatterers (2) the second-order potential is neglected. Consequently the second-order force for a SWATH ship in waves can be written as

F

= - p 55

L.

_an

In order to obtain the mean value of the forces, we take time average of the force in eq.(4). The

time-average function F is defined by

=.! ft+T

_TJ

_(t)

(4)

and using the momentum theory, _{the steady-state second-order force} can be written as

= Re _(P3i.

_{7_ds}}

*

-jux

where _{denoted the complex conjugate of 'p1e}

. If we only

consider the horizontal component in head wave, i.e. 14 =180°, then

the mean second-order force is obtained by

= -Re (

PVofJ3

(7)

where

fl35

D

(8)

The potentials _and rj in eq..(8) have included the

interaction effect between two hulls in the calculation of the potentials, i.e eq.(3).

Substituting eq.(8) into eq.(7), _{the added resistance term} can be obtained Assume = -Re

7PUJf

j-J

---)

_Cpv

znfI

_th

=-Re

jff

)s

-

3"

then the added resistance _is - C

---,

=1

(F-- F

--

_{ni3, 5}

The derivations of the eqs.(1O), (11) and (12) are discussed one by one in the followings:

(A) For : [i.e. eq.(10)]

-±ri

Zn

(14)

and

and

ac. (x,y,z)

raax = m"° (y,z)

the eq. (15) can be _{rewritten as}

-Re t

_-PVS

fI

ifs

= -Re 5rn(F

) }

where _{= Froude-Krylov force}

(B) For F _{[i.e. eq.(11)J}

a

*

an =

-n

in which n1 _{<< n3 and is neglected. Substitute}

eq.(19) into eq.(11), we obtain = -Re

C.v2 sJIe

(-in1

Applying the following Stoke's Theorem

= (16)

(17)

(f

2

_{=-Re(-pv ZS5}

(x-)o

_{_ dS}}

2o

--B

(C) For F3 [i.e. eq.(12)]

The boundary condition for diffraction problem is

D

3fl an

The second term of the right side of eq.(12) can easily be shown to be zero by applying _{eq.(22), therefore}

- r

= -Re _{u0 dS:} (23)

Substituting eq.(19) into eq.(23), we obtain

= -Re z

(24)

where 'D should be finally _{kept in three dimensional pattern} although it is replaced by a two-dimensional one, i.e. the

-i\I X

RESULTS AND DISCUSSION

For verifying the validity of the present theoretical

technique, an stretched ssp was selected as the experimental model. The principal dimensions of the model are shown in Table 1. The model scale ratio is 1:17.63. Tests are carried out in regular waves in head waves.The test arrangement and the configuration of the model are shown in Figure _{3-. Three major speeds are considered,} i.e. Fr.=0.073, 0.163 and 0.196. For comparison only the numerical calculations for head waves , i.e.1.i =1800, are made in the _paper.

Figure 4-9 show the heave and pitch for SSP model running in head waves with the corresponding speeds, Fr.= 0.073, 0.163 and

O.l96.From these figures we can see that the theoretical results agree with the experimental data fairly well if the viscous damping is considered. In Figures 4 and 5 the theory without viscous

damping shows the peak values around A

_IL=1.65

generally agree with the experimental data well. Therefore the viscous damping indeed plays _{an important role for the SWATH}

motion. For higher _{speed in Figures 6-9, the same condition can be} seen. Although there is still little discrepancy appear in the longer waves for Fr.=O.].96.in Figures 8-9, the general _comparison with the experimental data are regarded to be _{very satisfactory if} the viscous damping effect is considered.

13

The added resistance for SSP model running in head,waves with different speed is shown in Figures _{10-12. The experimental values} are obtained by integrating over a full number of wave periods from the measured data of _{added resistance. Figure 10 shows the added} resistance with Fr.=0.073. The comparisons between theory and experiment are generally good. The peak values around A/L=1.5 may be due to the large pitch motion if Figure 5 are referred The

viscous damping effect also shows its effect on the peak values and no significant effect on the rest region. _{There seems to have}

another peak in the short wave around A/L=0.5 for both theory and experiment. However the peak seems not due to the motion effect, therefore the viscous damping has _{no obvious effect on this region.} In Figure 11, the added resistance _{of the SSP model with Fr.=0.163} is shown. Again the agreement between theory and experiment is

good. In theory there are two peaks around A/L=0.75 and 2.0 and the latter again seems due to the pitch motion resonance, i.e. refer to Figure 7. The viscous damping effect is similar to the case of low speed, Fr.=0.073, stated in the above. Similarly the added

resistance is also large even the motions are small, i.e. XIL=0.75. Figure 12 shows the results for added resistance of the model with Fr.=0.196. The similar discussion to Figure 11 can be applied in this case. However the peak value of the experiment around AlL =2.3 seems higher than the theoretical prediction value with viscous damping effect. It may be due _{to some erroneous measured .data or} other uncertain factors. But the trend for both theory and

CONLUS IONS

Several analyses for the added resistance of the SSP model running in head waves have been carried out in the paper. Some experimental data are also obtained in NCKUDNAME towing tank and used to verify the validity of the present theory. From the above comparison and discussion, some conclusions can be drawn as below:

The present theory, i.e. with weak scatterer assumption, may be still regarded as a useful tool to predict the added resistance for a SWATH _{ship running in head waves generally according to}

present analysis.

The viscous damping indeed plays an important role in the prediction of SWATH motion and added resistance especially at

resonance region.

Generally the pitch motion resonance causes large added resistance in head waves from the present study.

In shorter waves, there also generally exist added resistance even the motion is small and should be carefully concerned. Although the present study shows that the present theoretical results may be regarded to be reasonable for predicting the added resistance of the SSP model, _{more studies especially for} the other SWATH _{models are recoimnended to be handled in the}

REFERENCES.

Bhattacharyya, Rameswar, et. al., _{"Dynain-ics of Marine} Vehicles", John Wiley. & Sons, _{New York, 1972.}

Havelock, T. H., "Drifting Force _{on a Ship Among Waves",} Philosophical Magazine, Vol.33, 1942, _pp.467-475.

Maruo, H., "The Excess Resistance of _{a Ship in Rough Seas"} ,Internatjonal Shipbuilding Progress, _{Vol.4, 1957, pp.337-345.} Gérritsma, J. and Beukelman, W., _{"Analysis of the}

Res-istance Increase ifl Waves of _{a Fast Cargo Ship", International} Shipbuilding Progress, Vol.19, _{1972, pp.285-293.}

Strom-Tejsen, J., Yeh, H. Y. _{H.., and Moran, D. D., "Added} Resistance in Waves", SNAME _{Transactions., Vol.81, 1973,}

pp. 109-143.

Ankudjnov, V. K., "The added Resistance of a moving Ship in Waves", International Shipbuilding _{Progress, Vol.19. 1972.}

Salvesen, N.., _{"Second-Order Steady State Forces and Moments}

on Surface Ships in Oblique Regular _{Waves," International Sympos.} on the Dynamics of Marine Vehicles _{and Structures in Waves,}

Univ. College, London, _{1974, pp.212-226.}

Newman, J. N., "The Second-Order time-average vertical force on

a Submerged bodies moving beneath _{a regular wave system," 1970,} unpublished.

Fang, M.-c., "Second-Order Steady Forces on a Ship Advancing in Waves," Internatjon1 Shipbuilding _{Progress, (To be published).} Lee, .C. M., "Theoretical prediot-ion of Motion of

Small-Waterpiane-Area Twin-Hull (SWATH) Ships _{ih Waves," DTNSRDC Report} 76-0046, David W.. Taylor Naval Ship Research and Development Center, Bethesda., Nd., 1976.

11. Hong, y. S., "Improvements in the _{Prediction of Heave and}

Pitch Motion for SWATH Ships," _{DTNSRDC Departmental Report} SRD-092802; David W. Taylor Naval Ship Research and Development Center, Bethesda, Md., 1980.

Hong, Y. S., "Heave and Pitch Motions _{of SWATH Ships,"} Journal of Ship Research, Vol. 30, No. _{1, 1986, pp.12-25.} Fang, M.-C., "The Motions of SWATH _{Ships in Waves,"}

Journal of Ship Research, Vol.32, _{No.4, 1988, pp.238-245.} Fang, M.-C. and Shyu, W.-J., _{"Hydrodynamjc Pressure}

Distribution on Hull Surfaces of a Seagoing _{SWATH Ship" NSC 80-0403-E006-05,} National Science Council, _{Republic of China, 1992}

Thwajteg, B. (Editor),

"Incompressible Aerodynamics," Oxford University press, 1960, pp.405-421.

ACKNOWLEDGMENTS

The authors wish to express the deep appreciation to National Science Council for their financial support under Contract No. 78-0403-E006-03. The acknowledgment is also extended to Mr. W.-Y. Shu for his help to handle the experimental operation.

TABLE 1. Principal _{dimensions of models}

Stretched

SSP

Displacement (long ton) ₆₁₈

Characteristic length L (in)

₄₅₅

Length of main hull (in) _46.3

Beam of each hull at waterline (m) _3.4 Draft at midship (m) _5.0

Maximum _{diameter of main hull (in)}

3.4 Hull spacing (in) _12.2

Longitudinal _center of gravity aft of main hull

ne

21.9 Vertical center

of

Longitudinal GM (in) _18.7

Transverse GM (in) _1.5 Radius of gyration for pitch (in) _12.4 Radius of gyration for roll (in) _5.6

Waterplane _area (in2)

- 68.4

Length of _{Strut 31.5} Strut _gap (in)

1 0.8

Y,y

Figure 1. Coordinate _{systems and plan view}

Figure 2. Coordinate system and section view

potential meter

modular force

gauge

Figure 3. The set-up of experimental model

wave

2.50

-S3/a

-2.00

₌

1.50-1.00

₌

0.50

₌

Theory without viscous damping effect

Theory with viscous

_{damping effect}

0

_{Experiment ( Fr}

_{= 0.070 - 0.078 )}

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

0.00

1.00

_2.00

_3.00

_4.00

_5.00

6.00

Figure

The Heave Motion

of SSP in Head Sea

at Fr

2.00

1.00

-0.00.

0.00

0

Theory without viscous

_{dar.ping effect}

Theo.ry with viscous

_{damping effect}

Experiment ( Fr

_{= 0.070 - 0.078 )}

1.00

_2.0:0

_3QQ

II

I I I I I I I I I I _j I I I F F

4.00

5.0.0

Figure 5

_{The Pitch Motion}

_{of SSP in Head}

_{Sea at Fr}

= Q;073

6.00

A/L

3.00

S3/a

_{Theory without}

Viscous damping effect

Theory with viscous

_{damping effect}

a

_{Experiment ( Fr}

_{= 0.158 - 0.169 )}

I I I I I I I I I I I I I I I I I I

J

I I I I I I I I I I I I I I I I I I I I I I I I I

0.00

1.00

_2.00

_3.00

_4.00

_5.00

_6.00

Figure

The I-leave Motion of SSP

in Head .Sea at Fr

= 0.163

2.00

1.00

-0

4.00

-3.00

-2.00

-1.00

₌

0.00

Figure

Theory without viscous damping effect

Theory with viscous

_{damping effect.}

a

_{Experiment ( Fr}

_{= 0.158 - 0.169 )}

*.,1.-.

0

II

II

1.00

_2.00

_3.00

_4.00

5.00

_6.00

7

The Pitch Motion

of SSP in Head Sea

_{at Fr}

3.00'

s/a

1.0:0

-2.00

Theory without viscous

_{damping effect}

Theory with viscous

_{damping effect}

Q Experiment ( Fr = 0.190 - 0.204

0.00

1.00

_2.00

_3.00

_4.00

_5.00

_6.00

X/L

5/

S/av0

4.00

2.00

-0.00

0.00

Theory without viscous

_{damping effect}

Theory with viscous damping

_effect

o

_{Experiment (}

_Fr

_{= 0.190}

_{0.204 )}

0

IIIIIIIIIIIIII-.-.-.1-IlI,I1IIIIIIIIIIIIIIIlIIIIIIIIIII

1.00

2.00

3.00

_4.00

_5.00

_6.00

8.00

₌

2.00

0

Theory without, viscous damping effect.

Theory wit.h viscous damping

_effect,

a

_{Experiment. ( Fr}

_0.070

_{- 0.078}

0.00

0.00

_1.00

_2.00

_.3.00

_4.00

II

TI1IIIIIIIII1

5.00

Figure 10 The Added Resist,ance of 5SF in Head

_{Sea at Fr}

_0.073

6.00

A/L

4.00

2.00

-0.00

0.00

1.00

_2.00

_3.00

II Iii IIIIIIIIIIIIIIIIII.IIIIII

_4.00

II

5.:00

6.00

12.00

4O0

-0.00

111111

0.00

Figure

0

0

'tIIlIltIIEiip i-i-I

1.00

2.00

Theory wfthout viscous damping

_effect

Theory with viscous damping

_effect

o

Experiment ( Fr

0.190

- 0.204 )

0

III

6.00

The Added Resistance of

_{SSP in Head Sea at}

_Fr

_0.196

k/L

III

JI II

=ipa0UwafB e"°

APPENDIX A

The damping coefficients, restoring forces and wave exciting forces. due to the viscous effect:

=

paoUf B dx p-1- CDfZ!Q B. dx

=

-paoUfxB. dx

_-

P---CDf

4

xB. dx

=

-pa0 UfB dx -

=

pa0

Ufx2 B dx

=pa0U2fB. dx

=-pa0U2fxB dx

dx +ip-CDUQafB e0!

3r

= -

ipa0 Ucii afxB e

'° -

_-

_{ipi_CDwQafxB}

e",dx

L

where

ZO=Z

Zth

S3 -xS5 -a(x,-b(x),-dj(x))

'q77

fC(

H.

7o7Li

/L

Cr.

(7