Hydrodynamic consideration on added resistance and ship-generated unsteady waves

Delft University of Technology

Ship Hydromechanics laboratory

Ubrary

Mekelweg 2 26282 CD Delft

Phone: +3i (0)15 2786873

E-mail: p.w.deheer@tudelft.nl

Hydrodynamic Consideration on Added Resistance

and Ship-generated Unsteady Waves

Masashi Kashiwagl, Tbkuma Sasakawa and Tbmoki Wakab^rashi

Department of Naval Architecture & Ocean En^eeriiig, Osaka University

2-1 Yamada-oka, ~ Suita, Osaka 56&-0871, japan E-maU: ka8bi@Daoe.eDg.08sJiai-vuac.jp

Abstract

In o r d ^ to investigate wfaich component or psirt of ship-generated unsteady waves is dominant in the added resis-tance, measurements of imsteady w:aves and the subsequent unsteady wave analysis using tfae Fourier trsinsfarm to com» pute tfae added resistance axe carried out for canonicai prob-lems of the wave di&action, tfae foreed osciUatipns in heave

and pitcfa, and the free-response of ship motions in head

waves. With tfaese results, validity of the linear superposi-tion of component waves ^ studied and discussion is made on associated nonlinear effects in tfae wave generaticm and dominant wave components in tfae added resistance.

1. Introduction

Wfaen a.8faip advances in waves, tbye resistance on tfae sfaip increases as compared to tfaat in calm sea. Tfais increase of resistance is called tfae added resistance. Since Maruo's pio-neering wtxck, it faas been well recognized that the dominant component m the added resistance is the one due to gener-ation of unsteady waves and their interaction with incident wave. Notwithstanding a large amount of work so far, details of the iQrdrodynamic relation between the added resistance and sfaip-generated unsteady waves seem to be imclear, be-cause most comparisons faave been made between the total increase in the ship resistance measured by a dynamometer in waves mid tfae calculated value witfa a simplified potential-flow theory. In order to evaluate accurately the amomit of unsteady wave-m£^dng resistance and to underststnd hydro-dynamic relations with ship disturbance waves (for instance, which component or wfaich part of xmsteady waves is dom-inant in the added resistance), it is useful to apply the un-steady wave-pattern analysis proposed by Ofaküsü (1980).

Rfom that viewpoint, Kasbiwagi (2010) sfaowed in tfae 25tfa Worksfaop some measured results of unsteady waves us-ing a modified Wigley model and correspondus-ing computed results by Enhanced Unified Theory (EUT) developed by Kasfaiwagi (1995). In that comparison, a large discrep-ancy was observed near the peak of tfae added resistance where waveinduced ship motions also become large. A n -other prcäoMn^t discrepancy observed in a comparison of tfae wave profile was that short-wavelength components in measured wayes were very small in amplitude as opposed to numerical results by E U T . In order to study possible rea-sons of these discrepancies, additional experiments using tfae same modified Wigley model were newly conducted, mea-suring the unsteady waves for three canonical cases of the difiniction problem, the forced ösciÜätion problem in heave and pitch:, and the mot ion-free problem in incident waves.

by means of largö: number of wave probes. With these mea-stu^d results togethier witfa analytical and numaical studies, discussions are made on tfae validity of lirieax superposition of the diffiraction and radiation waves, and on whidi com-ponent in the unsteady waves is dominant in predicting the added resistance.

2. Added Resistance and Unsteady Waves

We consider a ship advancing at constant forward speed U into a regular incident wave of amphtude A, circular fire-quency UQ. The depth of water is assumed infinite; thus the wayenumber of inddent wave is given by fco = wp/ff» witfa g tfae acceleration due to graivity. Corresponding to the experiment, only the head wave is considCTed, and the analysis is made with a right-hand Cmiiesian coordinate sys-tem O-xyz, with the origin placed at the center of a ship and on tfae imdisturbed firee surface, wfaich translates witfa tfae same constant speed as tfaat of a ship along the positive X-axis. The 2-äxis is poMtive downwaird. Unsteady ship re-sponses and ambient unsteady flow of fluid are assumed to be linear and periodical with circular fi^uency of encounter w = Wo -H koU.

Tfae added r e l a n c e In regular waves can be computed in tenus of tfae Kocfain function which can be associated witfa the Fourier transform of sfaip-generated unsteady waves. Us-ing the Fourier transform witfa respect to x, as sfaown l^^ Kasfaiwagi (2010), the calculation formula for the added re-sistance in head waves cstn be written as follows:

- jfe2 {k + ko)dk (1) where K=-(i^+kU)^ = K + 2 k T + ^

1

For T > 1/4, fca and fc4 become complex and tfae intégra^ tion range in (1) must be treated as continuous for /C2 < i^-C'(Â!,î() i n (1) is tfae Fourier trMisform defined as

X

Fig. 1 Coordinate system and schematic illustration of wave onnponents

whiere Ci^^y) denotes the ship-generated unsteady wave, which is asstraied in the linear theory to be given the linear superposition of scattering wave C7ix,y) and radiär tion waves y) by surge (J = 1), heaye (J = 3) and pitcfa (J = 5) motions, as in tfae following form:

i^{x,y) = AÇ7{x,y) + ^ Xj€jQ{x,y) (6) ^=1,3,5

wfaere Xj is tlie complex arcpfitude in tfae j-th mode of mo-tion and symbol Cj is adopted to express tfae length dimen-sion fór pitch; that is, es = L/2 and c, = 1 for surge and heave.

Here in faead waves UJ — UJO + koU, wfaicfa gives

= | ^ ( - i + v T T 4 ? ) > o ko = '^ = ^ { l + 2T->/rT4f) = -k2 = \k2\

(7)

On tfae otfaer faand, we can prove tfaat tfae relations be-tween the ship's speed U and tfae phase veloaty c of com>-ponent wave kj (J = 1, 3,4) are

0<U < - for fcs-wave - <U < c for fe4-wave

c<V for fci-wave

(8)

Since c/2 is equal to the grou4> velocity witfa whicfa the wave energy is transported, we can undérstEÎnd tfae location of existCTce, tfae relative wavelengtfa, and tfae travelling direc-tion wfaen viewed &om a Bhip moxdug at forward speed U, for eadh of tfae componerit waves kj (j = 1 ~ 4); these are schematically shown in Fig. 1. We note that at r = 1/4, kà = kt, and U becomes equal to the group velocity of wave.

and Bhip-diaturbance wave. Numerical examples of Wi(&) and W2{k) will be shown in the Workshop dùe to paucity of enougfa space in tfals paper.

In order to understand quahtatively the dominant wave components and general characteristics in the Fourier trans-form of tfae wave, let us consider a wave component, prop-agating in the positive z-axis witfa wavenumber kt and am-phtude of the following form:

ax.y) = a ^ p ^ e - ^ ' ' ^

y/\x-x,\ (12)

where a denotes the ampUtude coefficient, x, the starting point of wave existence along a liiie parallel to the x-axis, and u{xg — x) tfae unit step function.

Tfae Fourier transform ctf this wave may be expressed as

C'(fe.y)= r C{x,y)e"'^dx J _ O D

\k-kt{

Therdore it is obvious tfaat the value of Wi(k) becomes large at k'^kt and decays in proportion to l/\k — fc/|.

For larger values of Jfc, W2ik) becomes small at oriier of 0(l/fc) and tfae amplitude coöffident a must be anall in reahty for waves with Ujrge k (siuall wavelength). As aheady noted, kj (j = 1 4) is a root of — k^ = 0, and k + ko =

fc - &2. Hence, dominant wave components in the added

resistance may be relative^ longer waves with smaDer value of fc satisfying k2 < k. We note that if fc2 < fc < 0, tfae wave propagates in the negative x-axis like fc2-wave in Fig. 1, and if 0 < fc, the wave prc^)agates in the positive x-axis Hke

fcs-and fc4-wave8 in Fig. 1.

3. Dominant Components in Added Resistance 4. Experiments

In reaJily, the wavenmnbCT of progressive w a v K geiierated by a ship varies over the integration range shown in (1). In or-der to see which component of progressive waves coiitributes predominantly to the added resistance, w e will check tfae val-ues of tfae integrand of (1), hy rewriting (1) in the form

RAW

= ^ ~f '+JJ-^j^

Here it sfaould be noted tfaat Wa = 0 at fc = fcj (j = 1 4) because of = fc^, and fc -I- fco = fc - fc2 t y (7). Wfara computing the added resistance for the case of forced oscil-lation problem (i.e. the radiation problem), fc + fco in (11) must be replaced simply with fc, because the term related to fco in (11) r^resents interactions between the incident wave

In onler to see the degree of contribution of eadi wave Cji^yy) expressed in Eki- (6) to the added resistance, the ex-periments were conducted for the cases of wave diffraction (wfaere sfaip motions are completely fixed), forced OKÜlation in faeave and pitcfa (where incident waves are absent), and £ree-response of ship motions in head waves (where surge, heave, and pitch are &ee to respond to waves). Measmred in these eiq>eriment8 are priiicipally the added rœistance t y a dynambmetOT and sfaiip-generated unsteady wa.ves using a larger number of wave probes, and also, sfaip motions in the motion-&ee case.

In the unsteady wave analysis, the number of wave probes was increased up to 12 from 6 used in tfae experiment last year to confirm tfae resolution accuracy particularly for afaort-wavelengtfa waves. Those wave probes were positioned with ahnost equal intervals oVer tfae distance of sfaip's move-ment in one period of encounter along a lon^tudinal line parallel to the x-axis (at constant y). Using tfae least-square metfaod in terms of the data measured with 12 wove probes,

the x-direction distribution was obtained of cosine and sine coeffidents in the Fburier-series ^cpansion for the unsteady wave oscillatmg at circular firequency of encounter.

The ship model used in tfae experiments is the same as that used in the previous experiment; th&t is, a modified W i g l ^ model exi»%ssed mathematically as

+ C ' ( i - C ' ) ( i - € Y

. 2 . 2 . z

(14)

wh«% the real dimensions are L = 2.5 m, B = 0.5 m, d = 0.175 m. The gyrational radius hi pitch and tfae center of gravity were set equal to Kyy/L = 0.238 and OG/d = 0.189 (below tfae firee-surface).

The lateral distance of a longitudinal line used for tfae wave measurement froin the centerline of a ship (x-axis) was set eqüïd toy = S/2+ 0.1 m = 0.35 m. The FVoùdenumber was Fn = 0.2 in all measurements.

5. Results and Discussion

Figures 2 tfarougfa 5 sfaow comparisons of tfae added resis-tance for the cases of wave difirEiction (Fig. 2), forced faeave (Fig. 3), forced pitcfa (Fig. 4), and fi^response of sfaip mo-tions in Werves (Fig. 5)- Basically the results measured cU-rectly by a dynamometer are shown by closed circles, tfae results obtained fi-om the unsteady wave analysis are shown by open circles, and computed results by Enhanced Unified Tfaeory (EUT) are shown by the soHd line or otfaer symbols. It is confirmed in numerical computations t^ E U T that the adàed reeöstances computed toectly fix>m the Kodiin func-tion and fi-om the wave-pattern analysis method using com-puted wave profile are virtUEdly the saine.

In Pig. 2, we can observe that tiie results by the unsteady wave analysis are in good agreement with computed ones by EUT, but those are almost half of the value t y the direct measurement irrespective of the wavelength tested. This implies that nonlineai lo>cal waye generation, including wave breaking, may exist in the wave difiractioh problem.

In tfae forced-oscillation problem shown as Figs. 3 and 4, tfae agreement between B U T and the results unsteady wave analysis is generally favorable, and noticeable discrep-ancy ftxim the results by the dkect measurement can be observed in the short-wavelengtfa range espedally in forced pitch oscillation. This discrepancy niay be attributed to nonlinear local wave generation which cannot be explained by a linear potential-flow theory. Howeyer in this short-wavelengtfa range, actual amplitudes of sfaip motions are normally very small. Thus httle effects will arise fiom this discrepancy On the total value of tfae added resistance.

Measured results shown in Fig. 5 for the motion-&ee case are essentially the same as tfaose obtmned in tfae experi-ment one year ago. A large discrepancy can be seen be-tween the results by tfae direct measurement and the wave analysis using measured waves, particularly near the peak around A / L = 1.1. In order to investigate a possible reiason for this discrepancy, the wave profile was computed by the linesir sùperpcxsîtion according to Ëq. (6), using tfae compo-nent waves obtained by the experiments of wave diffiraction (j = 7), forced heave (J = 3), and forced pitdi (j = 5), to-gether with complex ampHtudes of heave and pitch motions measured in the motion-free experiment. (The surge mode is ignored, because the forced oscillation test in siirge was

not conducted.) Then tfae superimposed wave profile was Fourier-transformed and the added resistance was computed firom Eq. (1). Tfae results of added redstance obtained from tfais linear siiperpoHttoD of component waves and complex motion amplitudes are also shown in Fig. 5. It is remarkable tfaat these results become much closer to tfae results by tlie direct measurement and computed by EUT, especially near the peak wfaere ship motions also become large.

Figure 6 provides tbe information on the profiles of scat-tering and radiation waves and on the difference between tfae wave profiles measured by the motion-frOe experiment and obtuned 1^ the superposition without surge motion, for a case of X/L =1.1 (which corresponds approximately to KL = 12.5 at F n = 0.2).

6

ä 2

— M .

ftlUfinVOfl |lllOliPII aOQ Fn=0.20, ß=UO deg.

a Dbretmta: O I K M » Mmm^mtÊ '• m , ^ a 1 ; : • J

» • • • • *

• • •

• •

Fig. 2 Added resistance m the diffraction problem oh modified Wigley model at Fn = 0.2

fieroBd H e m Ovcfltotton Fn=0.20

5 *

<5

•

_•

_•

•

Fig. 3 Added resistance in forœd heave oscillation {X3 = O.Ol m) dn modified Wigley model at Fri = 0.2

Fn=0.20 -1 0.0 < 0 WmvemaffA ^ WUtem^méiKgmPrMVT

•

•

•

•

Pig. 4 Added resistîmce in forced pitch oscillation {X^ = 1.4 deg) on modified Wigley model at F n = 0.2

10

I '

t Vt \ * ml m fa t / * 0 V

•

• /

Pig. 5 Added resistajice in waves (motion firee) on modi-fied Wigley model at F n = 0.2

From tfaese figures, we can see tfaat tfae overall appear-ance of wave profile is very similar between superiniposed and directly measured waves, but a large difference exists near tfae fore-front part of the wave. Tfae source of tfais dif-ferece seems to come from the wave by the forced pitcfa osculation. It'is noteworthy that the forced oscillation tests were performed with relatively small amplitude (JC3 = 0.01 m and = 1.4 deg.) within the range of linear theory being valid. Therefore, when the ampUtùde of ship motions becomes large, hnearity in tfae ampUtude of generated wave may be violated partumlarly near the sfaip's bow due to large pitcfa motion, and as a result, some nonlinear local waves witfa energy dissipation may be generated.

6. Conclusions

By using the üï^teady waves measured in tfae diffi^ction and radiation (heave and pitch only) problems and the com-plex amplitude of wave-induced motions, the unsteady wave corresponding to the one in the motion-free condition was produced t y the Unear superposition. Tbe overall agreement in tfae wave was favorable, but a prominent difference was observed in the fore-firait part of the wave, especially wheoi

the ship motions are large. The added resistance coïnjputed firom tfaia superimposed wave was in better agreement with t±ie directly measured value. We can envisage from tfaese results that, wfaen sfaip motions become large, some

nonlin-local waves may be generated, resulting in a noticeable discrepancy in the added resistance between the results of direct measurémrât and wave analysis.

References

[1] Ohkusu, M (1980). "Added Resistance in Waves in the L i ^ t of Unsteady Wave Pattem Analysis", Froc of 13th Symp on Naval Hydrodynamics, Tokyo, pp. 413-425. [2] Kasfaiwagi, M (1995). Treiüction of Surge and Its Effect

on Added Resistance by Means of the Enfaanced Unified Tfaeory", Thma West-Japan Society of Naval Architects, No. 89, pp. 77-89.

[3] Ka^iiwagi, M (2010). "Prediction of Added I^sistmce by Means of Unsteady Wave-Pattern Analysis", Proc of

25th Int Workshop on Water Waves & Floating Bodies, Harbin, pp. 69-72.

(B) aeanmlBaWkM Fn-O.!, \/L « 1.1, 180 det

at v/(B/2)-1.4H 0 -1 (b) Hm\mrmß»toi\mm rn=o.2.Jïx-i2.5.^=iao*». ümm at u/(B/2 =1,4 0 -1 xAL/2) | a t g / ( f l / 2 ) « 1 . 4 g 0 -1 */(Va) Fn=QX >/i-1.1. (9-180*1-irtV/{B/2)=l.4B 0 -t e) mmtiuoaonnm Fn=0.2. A / t - J l i , g - l M * « . »t;v/(ß/2)-1.4H xAL/2)

Fig. 6 "Wave profiles generated by modified W i g l ^ model at F n = 0.2. (a): Scattering wave tn the difirac-tion problem st X/L = 1.1, (b): Kadiadifirac-tion wave by forced heave oscillation at KL = 12.5 and X3 = O.pl m, (c): Radiation wave by forced pitc^ oscil-lation at KL = 12.5 and Xs = 1-4 deg, (d): Su-perinq)osed wave using the wives of (a)'-(c) and measured complex ampUtudes of heaive and pitcfa at X/L — 1.1, (e): Measured wave i n tbe motion-free condition at X/L = 1.1.