A three-dimensional panel method for motions and loads of ships with forward speed

Journal for Research in Shipbuilding and Related Subjects

SHIP TECHNOLOGY RESEARCH/SCHIFFSTHMNIK vm founded by K. Wendel ih 1952. It is edited by H. Söding and V. Bertram in cdlaboration with ex^its from univeisities and model basins in Berlin, Duisbocrg and Hamburg, from

Germa-nischer Lloyd and other researdi-orienKd organizations in Germany.

Papers and discussions proposal for publication should be sent to Prof. Dr.-Ing. H. Söding, Institut für Schifflau, Lammer-sieth 90^ 2000 Hamburg 60, Gemmny.

W . 39 • No. 4 • November 1992 TECHNISCHE UNIVERSITBI Laboratoflum voor $cheepshydromechantea ' Archief Mekefweg 2628 CD Dolft TflU 016 - 7BS37S - fexi 018 • 781838 Apostolos D. Papanikplapü, Thomas E . Schellin

A Three-Dimensional Panel M e t h o d for M o t i o n s and Loads o f Ships w i t h Forward Speed

Ship Technology Research 39 (1992), 147-156

A new three-dimensional pand method efficiently calculates motions and wave induced loads of arbitrarily shaped but slender single-hull displacement ships with forward speed in regular waves. Pulsating three-dimensional sources are distributed on the wetted hull surface. The total vdodty p o t ^ t i a l Is assumed to comprise a steady, speed-independent part and £m osdllatory, speed-corrective part. A linear theory solves the correspond-ing boundary value problem on the basis of the zero-speed Green function. Numerical evaluation of surface integrals containing derivatives of the vdocity potential yidd hy-drodynamic pressures. A n additional term in the roll damping coeffident accounts for viscous resistance to roll. Restdts for the I T T C cbntainership S7-175 compare favorably with measurements.

Keywords: pand method, ship motion, load, wave, speed, containership, seakeeping

A Three-Dimensional Panel Method

for Motions and Loads of Ships with Forward Speed

Apostolos D . Papanikolaou, Dept. N A & M E , National Technical Umversity Atbeas Thomas E . Schellin, Germanischer Lloyd ^

1. M a t h e m a t i c a l Forn^ulation

We considered a ship advancing at constant mean forward speed U in regular sinusoidal waves of small amplitude. The ship's heading was defined by angle ß measured between the direction of Ü and the direction of wave propagation {ß — 0° r^esented following waves,

ß = 90° beam Waves from starboard, and ß = 180° head waves). The wave frequency was

rdated to the ship's frequency of encounter to by

u=^\wo-Ukcosß\ (1)

with k = 2r/\, where fc is wave niunber and A wave length.

The resulting osdUatory motions of the ship were asisûmed Ùneax ^ d harmonic. A n or-thogonal coordinate system x^y^z was fixed With respect to the mean position of the ship, with z vertically upward throùgh the center of gravity of the ship, x in the direction of forward motion, and the origin 0 in the plane of the undisturbed free stirface. The ship was assumed to oscillate as a rigid body in six degrees of freedom with (complex) amplitudes £i (i ~ 1,2, ...6), where % = I,2,3,4i5 and 6 refer to surge, sway, heave, roll, pitch and yaw, respectivdy. The ship was assumed slender in the s^se of slender body theory, i.e., derivatives in the transvörse direction Were assumed large compared to derivatives in the longitudinal dltection.

Using subscript notation, the six linear coupled diiferential equations of motion were 6

X ; [ ^ ' ( M i f c + ^fc) - ju; Bik + Cik] ffc = fi c-^"*. t = 1,2, ...6 (2)

where Mîh are components of the generalized mass matrix of the ship, Aih and Bik a^e, respec-tivelyy added mass and damping coeffidents, dk are hydrostatic restoring force coefficients,

fi are complex amplitudes of wave exdting forces and moments, and j is the imaginary unit

assodated with a harmonic time function. Only the real part is taken in expressions involving a factor e'^"^. Amplitudes / i , ƒ 2 , ƒ 3 refer to surge, sway, heave exciting forces and amplitudes ƒ 4 , /B» fs to roll, pitch, yaw exdting moments, respectivdy.

We assumed potential flow. The vdodty potential ^(x,y, z;t) was separated into a time-independent steady contribution due to the ship's speed Ü and a tinie-dep€xidant part assod-ated with the inddent wave system and the unsteady body motions:

<f>{x,y,z;t) ^[-Ux-\. (^,(a:,y,z)] -f- ^(x,y,z)e-^*^ (3)

Here —Ux + is the steady contribution <f>T the cpmplex ampUtude of the unsteady potential.

The total vdocity p o t ^ t i a l ^ had to satisfy the hull surface boundary condition (i.e., rigid wall condition for the osdllating hull surface) and the.free surface boundary condition (i.e., zero pressure differential on the unknown free surface). We linearized t h ^ boundary conditioos,

Salvesen et al. (1970): we assumed that the slender hull geometry leads to smaU values of

the steady perturbation potential (f>, and its derivatives. For small osdUatory motions the potential 4>T and its derivatives axe small. Under these conditions we linearized the problem

^PO Box 11 16 06. 2000 Hamburg II, Germany

by disregaxcËng higher order terms i n 0« and 4>T ^ d t^ms containing thdr cross products. We decooïposed the amplitude of the time-dependent part of the potential associated with the inddent wave system and the tinsteady body motions:

fi

<h' = <h-^<h + 'E,(i<k (4)

where ^ is the inddent wave potential, 4*7 the diffracted wave potential, and ^ the contribution to the vdodty potential from the tth mode of motipn.

Besides the known wave potential aU pptentials of ^ in (4) had to satisfy the Laplace equation in the fluid domain, a radiation cpndition at infinity, and a rigid waU cbnditiion on the sea bpttom. The difiracted potential ^ also had to satisfy the hûÛ surface (body) boundary condition at mean position for a point x , ^ , z, on surface 5

7tV4>7 - (5)

and the free surÊice condition on the stiU water surface aX z = 0

[(jw -f- Ud/dx)^ + g ô/dz]<h = O (6) On the hull at its mean position, the osciUatory radiation potentials had to satisfy the condition

rt^.4>i - - j w n j + Umi (7)

and, on the still water surface at z = 0, the condition

+ UBjdxf -h gdldz]<i>i = 0 (8)

where the generalized normal vector is

rt = (ni,n2,«3)^ and f x Ä = {n^^n^^nef (9) mth TI denoting the outward unit normial vectpr and f thê position vector frbm the prigin of

the coordinate system. In (7), TTK = 0 for t = 1,2,3,4 while ms = m and m^ = -n2. Body boundary cpndition (7) was simpUfied by dividing the oscillatory (radiation) potentials into two parts:

A = 4>? -

g^f

where is speed-independent. This resulted in the additional hull conditions

ÄV^S* = -jùjUi and n V ^ f = -jùjm^ (11) Potentials and had to satisfy the Laplace equation, the free surface condition (8), the

boundary conditions at infinity and, for the case of fizdte depth, the rigid wall condition on the ocean bottom. Hull conditions (11) and ^ = 0 for i = 1,2^3,4 and mg = ns led to

^ = ^ a n d ^ ^ = ^S5 (12) The radiation pot«itial components w^e expressed in terms of the speed-independent part of

the pptential:

' <l>i = ^? f o r i =1,2,3,4 (13)

<k = <l^-j^4 (14) <k = 4 + ^ 4 (15) 148

where 4*° for i = 1,2, ....6 had to satisfy, on the hull at mean ppsition, the condition

fiV<l>^ =-jumi (16)

and, on the stiU water surface at z = 0, the condition

[ ( j o ; -\- Udldxf -I- ffÔ/Ôz]^? = 0 (17) For a; <C UÔ/Ôx the depend^ce of tpj and <f>iOzi U disappeared. This condition was met at

high encounter frequency a;, at low forward speed Uy and/or for a dender body where the rate pf lengthwise variation of hydrodynamic pressures are small rdatiVe to the rate of girthwise variation. Even at low frequendes of ^counter, errors introduced in computed motions tended to be small because at low frequendes forces are dpminated by hydrostatics not affected by this cpndition.

2. Hydrodynamic Response

Hydrodynamic pressure p was calculated by the Bernpulli equation for unsteady flow:

P - ' p i ^ + h^'^l' + Sz) (18)

Expanding this presisure in a Taylor series about the undisturbed ppsitipn pf the hull, liiaearidng by neglecting h i ^ e r than first-order terms and ignoring steady pressure terms, we obtained hydrodynamic pressure on the ship's surface at its undisturbed position:

p = /> {jw + Ua/dx) <he-''^ - P9{i3 + - iBx)e-^'^ (19) for a point defined by coordinates ar,y, z. The last term in (19) represents buoyancy dependant

restoring forces and mpments. For sake of brevity, these terms axe pmitted ih the following expresdons of hydrodynamic forces and moments. Integration of pressure (19) over the huU surface yidded hydrodynamic force and momcùat amplitudes:

= - J j n i p d s c f t S i - - p j J UiiJu -f- Ud/dx)4>T ds (20)

5 S

where $ represents the ship's surface at its mesm position, while i f j , j?3, are force com-ponents in the x.,yjZ directions and B^^ 3$^ SQ moments about the z , y , z axes, respectivdy. Based on (4), hydrodynamic forces and moments were subdivided into waye exdting forces and moments Fi and forces and moments of hydrodynamic response to the six degrees of freedom ship motions Gii

Si = Fi-j-Gi (21) Fi = ~pJJniUi^-\-Ud/dx)(4>Q + 4>7)ds (22)

s

Gi = - p f f n,(jw -H Udldx) ^ ds ='£ ^^Tik (23)

- ' s k=i k=i

Here Tik, the hydrodynamic force/moment in direction i due to a unit amplitude motion in direction fc, is

Tik = -pJJ ^U*^ + Ud/dx}<t>k ds = -pjuj j j ni(t>k ds-pU j j nid4>k/dx ds (24)

Hydrodynamic response forces and moments Tik were separated into thdr real and ima^nary parts:

Tik = A * + iM^Bih (25)

The equations of motion were (see also (2))

2^[-w2(Jlfi;b + Aik) - jojBik + Cik](k = Fi for i = ly2, ...6 (26)

Hydrostatic restoring force coeffidents Cik resulted from the buoyancy term in the pressure equation (19). The problem left now was to determine added mass and damping coeffidents iiifc, Bih and wave exdting forces and mpments f^. To evaluite hydrodyiutmic cpeffidents, we detennined the complex value of Tik from (24), where the speed-independent part is given by

T?k = -P3<^ J jni<t>kds (27) s

and the speedrcorrective part is defined by

TYk ^'P^ j ƒ -^Hkldx ds (28) s

To evaluate the surface integral on the RHS of (28), Salvesen ei <U, (1970) used a variant pf Stokes theorem according to Ogilvie and Tuck (1969), ^ving

ƒ ƒ mdtpk/dx ds = J J mi<f>k ds- j ni4>k dl (29)

5 s c

where differential length dl of the line integral term is taken along waterline C of the ship. In thdr strip theo^, Salvesen et al. (1970) ignpred this line integral, assuming the angle between waterline and x-axis tp be small. We retained this line integral by directly evaluating surface integral (29) in (28) frpm ayailable derivatives of three-dimensional pptentiid fonctions. Surface integrals to be dealt with were given by (27) and T^l given by

Tg

=

= U{~pJ jnid4Pkldxds) + U \ p ^ j fnid4>^/dxds) (31) S s

Hydrodynamic respoiüe coeffidents were derived from Tik hy separation into real and imaginary parts:

Bik = ^ + UB}i' + U^Bg'

where 4?* = IU(lS)/a,», = M î S ) / a < (32) (33) (34) (35) (36)

3. Viscous Roll Damping

Following Bimenp (1981), we computed viscous roÜ damping as the sum of four effects-huU friction, eddy maîûng redstance, normal force damping of bilge keds, and effects-huU presstire

damping pf bilge keels. These effects, which are npnlinear with respect to the roU vdodty (4, were introduced in the equations of motions as the quasi-linear term

BÏ4 = BIIA (37)

also referred to as the equivalent linearized viscous roU damping coeffident. The total viscous roll damping coeffident B^ depended on frequency of encounter, bilge ked dimensions, hull geometry, and ship speed. This coeffident was added to the linear (pptential) rpU dainpmg coeffident accounting fbr radiated waves due to roU osciUations.

Thé (maximum) roU vdodty ^4 had to be estimated b^ore computing B^^. We estimated the roU vdodty by first solving the equations of mption without the viscous roU damping term and then resolving them with this term, using the previoudy obtained motions. This iteration process was repeated until a reasonable convergence of the motions was reached. For further details regarding this approximate equiUnearization method see Óstergaard and Schellin

(1977) . The total viscous roU damping coeffident

5 ? = S F + + BBKN + BBKH (38)

comprised four components cprresponding to the four effects m^tipned abpve. (We did not cpndder effects pf appendages except for rudder and bUge keds. The rudder was assumed induded in the huU configuration.)

Tó caltulate friction damping, we used the formula of Tamiya and Komùrà (1972), which is based on an analysis of three-dimensional boundary layers of dongated spheroids In roU motion:

BF = BFO{1 + ^-lU/ufL) (39)

where Bpo represents friction damping at zero speed and L is ship length between perpendic-ulars.

To estimate damping due tp eddy making, we used the empirical formula of Ikeda et al.

(1978) , which is based on the eddy damping of arbitrary ship forms:

BE = 5Bo(0.04a;2iV(^7^ + 0.Q4co^L^)) (40)

where £^0 denptes eddy damping at zerp ship speed.

Bilge ked damping indudes nPt pnly the damping pf bilge keds t h ^ s d v e s , but also aU interaction effects between bUge keds, huU, and waves. We used the formula of Yûasa et al.

(1979) , which is based on the approximation of low aspect rationings to the case of bilge keds:

BBKN = BBKNO + '^b%K-^BK^f2 (41)

where BBKNO stands for normal force damping at zero forward speed, hg^ is bilge ked width, and TBK is distance from the ship's center of gravity to the bilge keds.

H1ÜI pressure damping of bUge keels was calculated using a formula of Jheda et a/. (1978),

established for damping due to the huU pressure difference with and without the presence of bilge keels:

BBKH - -^P^i^T^BK^^flkn^BKH (42)

where D is huU draft, fsKH ^ empirical c o r d e n t of vdodty incrment at the bUge drde, and IBKH <m integral of pressure difference taken around the girth of the ship. HuU pressure damping was assumed independent of ship speed.

4. E x c i t i n g Forces a n d M o m e n t s

Wave exdting forces and moments (22) were separated into the inddent wave part F^ and the diffracted part F f :

F i = ƒ/ -I- i ï f = -/) ƒ ƒ n.(iw Udldx)<h <^^'P J J ^O'w + ^d/dx)<f>7 ds (43)

s $

The potential for the inddent wave was

cosh fca

where a^, is wave amplitude. The wave frequency too is rdated to water depth d by the Unear dispersion rdationship: = ^fctanh(fcd). A t time ^ = 0 the wave crest is located at 2 = 0 (cosine wave at midships).

Substituting this wave potential 4>o into (43) yidded the Froude-Krylov forces and moments:

Fi ~ -p j j ni(jw + jUk co^ß)4>Q ds = - jpuj^ j ƒ ds (45) s s

The di£&action parts of wave excitation were expressed i n terms of surface integrals:

= i f » + f f = -jpu j jni<hds=-pU j I nid4njdx ds (46) 5. W a v e Induced Loads

We let Vi be the generalized dynamic force or moment at positipn x along the ship's longi-tudinal ajds. Indices i= 1,2,3 refer to fprce comppnents in the x-,y~,z- directipus; indices i = 4,5,6 to moments about the y - , z - directions. We obtiwned this generaUzed dynamic force by subtracting the hydrodynamic pressure force Pi from the inertia force ƒ» acting on the portion ofthe ship's huÜ forward of podtion x:

Vi^Ii- Pi (47) mihPi^F^ + Gt-^Rt (48)

where i ^ * is wave exdting force or moment, G* is hydrodynamic response force or moment (due to body motions), and R* is hydrostatic restoring force or moment. Inertia forces and mpments were the product of sectional masses and sectional accderations:

A = ƒTn(ël + ^Gê5)rf^

h = / m ( 6 + ^ 6 - Z G C 4 ) d f

/3 = Jm(is'^U)d^

A = JlixÜ-^.^G(^2-\-^ü'ZG(4)]d^ is =

- ƒ

/é =

ƒ

Here m = m({) is sectional mass per unit length of the ship, ZQH) is vertical positipn of the center of gravity of the sectional mass, and i^d) is sectional mass moment of inertia about the ar-axis. Int^ations are over the length of the ship forward of the cross section at podtion i .

152 SchiÉfetechnik Bd. 39 — 1992 / Ship Techhblpgy Research VbL 39 — 1992

(49) (50) (51) (52) (53) (54)

Wave exdting forces and mpments Pver the pprtion of the ship forward of the section bdng conddered were obtained from (22) by integrating over the htdl surface 5* forward of podtibn

x:

F : ^ - P J

J

Fi - - jpu^o J j ni4>ods- jpujQ j j ni<p7ds-pU j j nid<hlàxds (56) s* s*

Similarly, hydrodynamic response forces and moments over the ship forward ofthe cross section being considered were obtain«i from (23):

<?: = 'pjjni(3UJ^Üdldx)(^ik<Mds (57) S* *!=1

= E a 2 7 * = E & ( ï a ' + 2'S') (58)

with the speed-independent part T^* ~ P j j ^^fc (59) and the speed-corrective part Tg* = Pj^ j j n t | ^ ds - pS~ j j 45(60)

which was evaluated in analogy to (31):

Hydrostatic restoring forces and moments over the ship forward of podtion x were

Ä 3 = - ƒ ƒ nzp, ds (62) 5*

Ä 4 = - ƒ ƒ (ns - zn2)ps ds 4- g j mzaU (63)

Ä 6 = ƒ {{i - x)ns - zni)p, ds + g J J mza^s d^ (64)

s* 5*

where Pt = -pg(iz + i^y - i^z) is the increase of hydrostatic pressure at pdiat x,y,z on the hnU surface due to body uLOtions ( j . The dpuble integration is over the hull surface forward of podtion x\ the single integration, over the length of the ship forward of podtion x.

6. Results and Discussion

Ship mptipns and wave induced Ipads were calculated for the ITTC containership 87-175 in infinite water depth at a Froude number of 0.275. The wetted surface of the ship's huU was discretized by 190 surface dem^ents, Fig.1. Prindpal particulars of the ship are found in ITTC

(1983). We assumed bilge keds of 87.5 m length and 0.4 m width were fitted. BÛge radius was

2.92 m. Maximum and mean distances from the ship's center of gravity to the bUge keels Were taken to be 14.83 m and 13.92 m, respectivefy.

Wave directions were varied betwe^ fóÜowing seas and head seas at 30° intervals. Modd test measurements validated the calculations, Figs.2 to 5. For motions i n head waves, Fig.2,

phase angles are also given. Predicted responses compared favourably with measurements fox all wave headings and wave frequendes.. Motions agreed more dosdy than wave induced forces. Near heave resonance, calculated heave ampUtudes were larger than measured heave amplitudes, Figs.2d and 3a, due to the absence of viscous heave dainping i n the theory. Near roll resonance, no measurements were available for comparison, Fig.3b. Because vertical shear forces tend to apprpach tlidr minima at roidships, Fig.5a shows, as expected, larger differences at nondimendonal frequendes of around three. No measurements were available for lateral shear forces at midships, Fig.Sd.

To compare our results with those obtained by the strip theory method of Salvesen et al.

(1970), we extracted from ITTC (198$) the cprresponding ship motions and wave loads and

induded them in Figs.2 to 5. Differences were rdativdy smiall except for vertical shear force and vertical b i d i n g moment at midships (Figs.4 and 5a,b). Unfortunatdy, the large spread of the measured data for these wave loads makes it imposdble to demonstrate that predictiôns with our panel method were more accurate. Our rdativdy coarse discretization may have caused inaccurades i n calculated responses, especiaUy at this high ship speed.

Reference

HIMENO, Y. (1981), Prediction of ship r d l damping - state of the art, Rep. 239, Dept . NA&ME, Univ. of Michigan

IKEDA, v . , HIMENO, Y. aad TANAKA, N. (1978), Components of roll damping of ship at forward speed, Trans. JSNA 143, (in Japanese)

ITTC (1983), Summary of resûlU obtained with computet progiams to predict ship motions in six degrees of freedom and related responses, 15th and 16th ITTC, Seakeeping Comm. Comp. Siudy on Ship Motions Program 1976 -1981,

ÖSTERGAARD, C. and SCHELLIN, T.E. (1977), On the treatment of viscous effects in the analysis of ocean platforms, Schiff und Hafen 29/4, pp. 343-349

OGILVIE, T. F. and TUCK, E. O. (1969), A rationd strip theory for ship motions - part I, Rep. 013, Dept. NA&ME., Univ. óf Michigan

SALVESEN, N., TUCK, E. O. and FALTINSEN, 0. (1970). Ship motions and sea loads, TVans. SNAME 78, pp. 250-287

TAMIYA. M. and KOMURA, T. (1972), Topics on ship rolUng characteristics with advance speed, Tcans. JSNA 132 (in Japanese)

YUASA, K., FUJINO, M. and MOTORA, S. (1979), New approach to hydrodynamic forces on oscil-lating krw a^ect ratio wings, Ttans. JSNA 144, (in Japanese)

Fijg. 1. Surface discretization of ITTC ship S7 - 175

154

(a) SURGE PHASE ANGLE (0) HEAVE PHASE ANGLE 0 - -180- -36 0- -180- -360- -540--180-^ -720H

(e) PITCH PHASEANGL£

a w m SURGE

15H

loH

O H

(d) HEAVE (0 PTTCH

Rg. 2 (TTC ship S7 -175 in regular lieâd waves ( ß = 180"* ) :

transfer functions and phase angles of surge,

heave and pitch motions; Fn = 0.275

( • measured, —calculated 3D,cateuiated strip theory )

1.5H (b) ROLL (a HEAVE K BW (Ç) PITCH asH 15-H uoH 05 H

Rg. 3 ITTC ship S7 -175 in regular bow quartering waves ( ß = 150° ) :

transfer functions of heave,róll and pitch motions; Fp - 0,275

( • measured, —calculated 3D, - caicuiated strip theory )

(a) VERTICAL SHEAR f=bRCE AT® pgLBaw 003^ (b) VERTIGALBENDINGHOMBITAT® Vs 4 (BVLJÓ

Rg. 4 ITTC ship S7 -175 in regular head waves ( ß = 180» ) •

transfer functions of vertical shear force and vertical

tending moment at midships; Fn = 0.275

( • measuried, -calculated 3D.-- - calculated strip theory )

(a) VERnCALSHEARFORCEAT®

(d) LATERAL SHEAR AT®

Q03^

002-^

aoH

4 ( DT Ï ^ m VERTICAL BENDING MOMENT AT®

pgLBaw

Q02H

4 fûJÏ^

(e) U T E R A L BENDINiS MOMENT AT®

PSL^Baw 002-^ _pgL^Baw (o-fÖÏ

ojïoioH

Rg.5

ITTC sftip S7 -175 in regular

bow quartering waves ( ß

s

150'' ) :

trar^fer functions of vertical and

lateral shear forces, vertical and

lateral bending moments,

and torsional moment al midships;

Fn = 0,275

( • measured, —calculated 3D,

- - - caicuiated strip theory )

Schif&technik Bd. 39 - 1992 / Ship Ifechnology Researdi Vol. 39 - 1992