Fully three-dimensional ship seakeeping computations with a surge-corrected Rankine panel method

J M a r s . Techno, (1998) 3 . 4 - 1 0 1

^

6 SCieilCe

and Technology

Fully three-dimensional ship seakeeping computations with a

surge-corrected Rankine panel method

V O L K E R B E R T R A M * and G E R H A R D T H I A R T

Department of Mechanical Engineering, University of Stellenbosch, 7600 Stellenbosch, South Africa

Abstract: A 3-d seakeeping code uses first-order Rankine

panels witti special numerical integration on the ship's hull and Rankine point source clusters above the free surface. The code computes the motions of the ship i n regular waves of small height (Knearized). The steady flow is captured without simplification by solving the f u l l y nonlinear wave-resistance problem first. A special treatment of the surge m o t i o n con-siders the influence of periodic quantities on thrust and resis-tance, and improves surge m o t i o n predictions. Radiation and open-boundary conditions are enforced by staggered grids. Results f o r the I T T C standard test case S-175 containership agree well with experiments except f o r very long waves. The importance of capturing the three-dimensional steady flow contributions is also demonstrated.

Key words: seakeeping, panel method, containership, surge

correction

Introduction

I n their 1950 milestone paper,^ Weinblum and St. Denis said: "The present hydrodynamic methods used in studying problems of seaworthiness are based largely on the powerful concept of sources and sinks." Almost half a century later, this statement still describes the state of the art. O f course, today's methods are far more powerful, w i t h the main advances being i n capturing nonlinearities and the three dimensionality of the flow. We present here a " f u l l y " three-dimensional Rankine panel method, capturing both the steady and the

time-Address correspondence to: V . Bertram, T U H H , Lammersieth

90, 22305 Hamburg, Germany

Received f o r publication on Oct. 6,1997; accepted on Jan. 28, 1998

* Visiting scientist

harmonic potentials three-dimensionally. For a recent survey of Rankine panel methods for forward-speed seakeeping, we refer to Bertram and Yasukawa.^ Our method captures all forward-speed effects, so i n addi-tion to the change in encounter frequency, we capture: dynamic t r i m and sinkage;

steady wave profile (average wetted surface) and gener-ally the steady wave elevation on the free surface; local steady flow field.

Pliysical model

We consider a ship w i t h average speed J7 i n a regular wave of smafl amphtude h. The boundary conditions w i f l be linearized w i t h respect to h (and all other related time-harmonic quantities). We refer to Bertram^for an extensive derivation of the boundary conditions.

The fundamental differential equation for the as-sumed ideal flow is Laplace's equation, which can be interpreted as describing conservation of mass. I n addi-tion, we formulate the following boundary conditions: 1. water does not penetrate the ship hull;

2. water does not penetrate the free water surface; 3. there is atmospheric pressure on the free surface; 4. there is undisturbed flow far away f r o m the ship; 5. waves created by the ship propagate away f r o m the

ship; for T > 0.25 these waves are limited to a sector downstream;

6. waves created by the ship must leave an artificial boundary of the computational domain without reflection;

7. the forces on the ship result i n periodic motions. (We assume that the time-averaged added resistance is compensated by increased propulsion forces, i.e., the average speed remains constant.)

The radiation condition (5) deserves a more detailed discussion. T o date, our method is limited to cases

where the parameter T = (O.UIg is larger than 0.25. co, is the encounter frequency, g = 9.81 m/s^. While a number of techniques exist^ to extend Rankine panel methods to arbitrary T, these require considerably higher computa-tional e f f o r t and wih be the subject of a separate future investigation. For the test case selected here, as f o r most ships, a restriction to T > 0.25 excludes only fohOwing waves (Fig. 1). For r > 0.25, the radiation condition is analogous to the steady wave resistance problem in shallow water f o r undercritical depth Froude numbers. The numerical "shifting" or "staggered g r i d " technique, developed originally f o r the steady wave-resistance case, can be adapted without problems to the time-harmonic problem" and also automatically fulfills the open-boundary condition (6).

The problem is formulated i n right-handed Cartesian coordinate systems. The inertial Oxyz system moves uniformly with velocity U. x points forward and z points downward. The Oxyz system is fixed at the ship and follows its motions. When the ship is at the rest position, X, y, and z coincide with x, y, z. The angle of encounter jJL between the body and the incident wave is defined such that jl = 180° denotes a head sea and ji = 90° denotes a sea f r o m the starboard.

The ship has six degrees of freedom f o r rigid body motion, expressed in the motion vector ü = { M J , H J , U^Y and the rotational motion vector « = {«4, « 5 , u^Y-{«i, «21

«1 is the surge motion of O i n the x-direction, relative to O;

«2 is the sway motion of O in the y-direction, relative to

O;

H3 is the heave motion of O i n the z-direction, relative to O;

« 4 is the angle of roll = angle of rotation around the 'x-axis;

is the angle of pitch = angle of rotation around the j - a x i s ;

Wg is the angle of yaw - angle of rotation around the z-axis. T = 0.25 '

/

/

/

Fig. 1. The parameter T= öJ.Wg depends on the angle and

wavelength of the incident wave. Only f o r following and very long quartering waves does T become smaller than 0.25

A l l motions are assumed to be of first-order small. A perturbation formulation for the potential is used omitting higher-order terms,

0? = 0W+0W ₍₁₎

where 0'"' is the part of the potential independent of the wave amphtude h. I t is the solution of the steady wave-resistance problem. is proportional to h and accounts (linearized) f o r the contributions of the sea-way. Higher-order (seaway) terms are neglected.

We describe the elevation of the free surface ^ i n a similar f o r m as the potential, where we exphcitiy specify that quantities are time-harmonic i n O), (solution i n the frequency domain):

4>t(x, y, z; f ) = 0W(x, z]

+ ^ W ( x , y, z; ^) = 0(")(x, y, z]

+ Re(4>^'\x, y, z)e"'''

C ' ( x , y; ^) = f W ( x , + y; t)

= i;^'\x,y) + Re[^^'\x, yy--'

(2)

(3)

The symbol """ denotes the generally complex ampli-tude of a time-harmonic quantity.

The harmonic potential is decomposed into the potential of the incident wave f \ the diffraction poten-tial ^ , and six radiation potenpoten-tials

6

- h ^ ^ ' i i . (4)

t=i

I t is convenient to divide </)"' and <p'' into symmetrical and antisymmetrical parts to take advantage of the (usual) geometrical symmetry:

0'"(x, y, z) + - y , z)

(5)

(6) The steady potential can be computed by a " f u l l y nonlinear" wave resistance program which also yields second derivatives of the potential using higher-order panels on the huU.' The second derivatives of the poten-tial i n the x-direction are limited to the maximum value at 80% of the ship's length in the aftbody to avoid unrealistically high values at the stern, reflecting to some extent the effect of viscosity in reality.

96 V . Bertram and G. Thiart: Surge-corrected 3-d seakeeping computations Tlie linearized potential of the incident wave on

water of infinite depth is expressed i n the inertial system

(

=Re ^S^'^ „-ik(x cos fi~y sia tiyicz iai^l

(7)

where co = is the frequency of the incident wave,

co^= {co - kU cos fl I is the frequency of encounter, and k is the wave number.

So the remaining unknowns are the diffraction and (unit motion) radiation potentials. These are deter-mined by solving the Laplace equation subject to the boundary conditions given below.

A t the average free surface {z = ( - £ 0 2 + 5 r £ » , ) ^ W

+ V0W(V0WV)V0W = O (8) This equation corresponds to E q . 3-23 of Newman.*^

a = (V0('')V)V</)(°) is the steady particle acceleration, a^ = d - { 0 , 0, g}\ B = -(l/af)i(V0(«)fl«).

Let the ship's hull be defined in the ship-fixed sys-tem by 5(1) = 0. Then after introducing the m terms

m = {fN)V(f°\ the boundary condition at S{x) = 0

becomes

(9)

This equation corresponds to E q . 3-30 of Newman.** I n Eqs. 2 and 4, the total velocity potential has been decomposed:

|- ^ ^ ^ 6 ^

0' = 0 ( ° ' +Re + +tpd-"

'=1 ; J (10)

This equation contains eight unknown potentials (0''-", and the 0')- These potentials are solved i n a Rank-ine panel method which uses first-order panels (plane and constant strength) on the hull (up to a height above the average wetted surface) and around the ship. The Laplace equation and the decay condhon (4) are then automatically fulfihed.

For the diffraction problem, all and «; are set to zero. For a radiation problem, the relevant m o t i o n am-plitude is set to 1. A l l other motion amam-plitudes, f \ and 0'' are set to zero. Then the boundary conditions at the free surface (Eq. 8) and the hull condition (Eq. 9) are

fulfilled in a collocation scheme. The cohocation scheme forms eight systems of linear equations in the unknown source strengths. The four symmetrical (and also the four antisymmetrical) systems of equations share the same coefficient matrix with only the right-hand side being different. A h four cases are solved simultaneously using Gauss elimination. Then the com-putation of all potentials and their derivatives at aU collocation points is straightforward. However, f o r the total potential, the so-far unknown motion amplitudes stih need to be determined.

The expressions f o r this final step are derived i n prin-ciple f r o m "force = mass x acceleration" to

m\Li + axx +

---ax

ix_g X ïi j -I- / a = -Xg xiaxG

+ J i

- p{üas + a[xxas

)jj

(x x «) dS

The unsteady pressure is decomposed into contribu-tions f r o m the incident wave, diffraction, and radiation,

^ p + p<i,s + p U'^ + p d ^ + ^ p ' U i ₍₁₂₎

where each unsteady pressure component on the right-hand side is computed i n the linearized f o r m as

^y<pi°)v^i)' (13)

where G= (0, 0, jngY is the ship's weight, in the dis-placement mass,Xg is the center of gravity, and / is the matrix of mass inertia moments with respect to the origin of the ship-fixed system. Equation 11 yields a system of linear equations f o r u, ( i = 1 , . . . , 6) which is quickly solved by Gauss elimination.

Corrections for surge

So far we have considered only the potential flow influ-ences. As well as viscosity, the resistance and propulsion characteristics and autopilots (trying to keep the ship on a constant course and possibly at a constant speed or r.p.m.) introduce additional restoring and damping i n -fluences. We now present an ad hoc approach to correct surge for the propulsion characteristics.

The thrust and resistance forces acting on the ship are affected by the motion. One could include thrust and resistance vectors similar to the weight vector G to ac-count for all motion. However, the main effect comes f r o m surge motions i n long waves, and this allows a

somewhat simpler treatment. Surge motions change the longitudinal velocity of the ship. The correspondingly changed resistance and propulsion characteristics of the ship w i l l induce considerable damping of surge motions, especially for long waves. Also the local orbital velocity of the waves may have some influence. I f these effects are neglected, surge motions may be overestimated by the computation. The thrust T,, depends on the instanta-neous speed of advance of the propeUer v„:

v„ = ( l -

w){u

<t>i

= i V o + A v „ (14) where v„o = (1 - w)U is the speed of advance of the

propeUer at steady speed U, w is the Taylor wake fraction, and the location of the propefler is x^. For long waves, the contribution of the incident wave w i f l dominate and the diffraction part may be ne-glected.' T,, is developed in a linear Taylor expansion around v„o:

r „ » T „ ( v , o ) + A v „ r , : ( v „ , ) , (15) The calm-water resistance R at an instantaneous

speed is similarly developed i n a Taylor expansion around U\

R ^ R ( U ) + AUR'(U) = R(U) + (Ü,-V,,^)R'(U] (16)

V = (17)

S} ' , ( 0 )

where v^^f approximates the influence of the orbital ve-locity averaging over the wetted surface of the ship. For short waves, this wifl lead automatically to negligible values, leaving contributions only f o r long waves. Thrust and resistance at calm-water speed cancel out, yielding

(l-t)T,-R = ((l-t)(l-w)T,:-R')u,

- ( 1 - t)(l - w)T,:(^^' (x^) + 0i {x^)) + v,,R' (18) This expression is added as a correction on the

right-hand side o f t h e first component of vector Eq. 11. The lij t e r m can be interpreted as surge damping; the remain-ing terms contribute to the excitremain-ing surge force. T,', R', thrust deduction fraction t, and w are taken f r o m model tests o r — i f these are not avaflable—approximated by empirical formulas.^ We used a resistance estimation following Guldhammer-Harvald in our results. Numerical tests f o r our case, however, showed that the inclusion of the last two terms in E q . 18 always affects the resuhs by less than 1 % and may therefore be omitted.

A p p l i c a t i o n to the S-175 containership

The S-175 containership (Table 1) was chosen as a test case because i t is one of the recommended I T T C test cases f o r seakeeping, and f o r this ship there are also relatively extensive experimental data f o r motions i n obhque waves.'-^" The tests were carried out at various times (and possibly f o r various unrelated purposes) in four Japanese towing tanks (the Ship Research Institute, N i p p o n Kokan, Ishikawajima-Harima Heavy Industries, and Sumitomo Heavy Industries). Owing to the voluntary nature of the work, the coverage of the specified matrix of headings and responses is uneven. The differences between testing estabhshments were usually relatively small, so that no distinction between the various sources of data has been made here. A l l cases were computed f o r the design condition w i t h

F„ = 0.275.

We approximate the propulsion data i n Table 1, based on the propefler data of Z h o u . '

Steady flow computation

Figure 2 shows the h u l l grid w i t h 631 elements. The h u l l was modified in the aft region by integrating the rudder into the huU. For symmetric motions, this w i l l have only a negligible effect, but f o r antisymmetric motions it should capture the physics better than omitting the r u d -der totally. A better modelling of this region would require the inclusion of viscous effects and the propeller action as well, which is beyond the scope of our current abihties.

I n a first step, the nonlinear wave-resistance problem was solved to determine the steady potential and its derivatives. The structured grid on the free surface used 1430 elements. The steady computation converged

rap-Table 1. Principal particulars of S-175 containership

Length B 175.00 m W i d t h B 25.40 m D r a f t A P T 9.50 m D r a f t FP 9.50 m Block coeff. 0.5716 Wetted surface 5540 m 2 before L^p/l Zg below C W L 2.48 m before L^p/l Zg below C W L -0.02 m Mass m Radius of inertia 8.33 m Radius of inertia _K 42.00 m Radius of inertia _K 42.00 m Radius of inertia 0.00 m Resistance slope R' 500000 Ns/m

Reduced thrust slope (1 - 0 ( 1 - - 1 5 0 0 0 0 Ns/m

Propeller location - 8 7 . 5 m

98 V . Bertram and G. Ttiiart: Surge-corrected 3-d seakeeping computations idly, reducing the error on the free surface to 0.03% of

the hnear solution within three iterative steps. Sinkage was 0.0025 L^^ and trimT).00045aft, which appears plau-sible, although no experimental data are available f o r comparison. The wave pattern appeared plausible with-out any noticable reflections at the open boundary (Fig. 3). Overafl, the result of the steady computation gave no indication of undue errors.

Sea keep ing compu ta tions

The same discretized hufl model was used f o r the seakeeping computations. The grid on the free surface had about 1300 elements (on the starboard half only). Figure 4 shows the response amplitude operators f o r motions. The results of our panel method generally agree well with experiments for all motions and all fre-quencies, in both absolute values and phases. Only for higher frequencies do experimental and computed phases differ, sometimes considerably. However, in these cases the absolute values are almost zero, and in this case the phase is insignificant.

The heave motion f o r low frequency shows a slight inflection, but is stifl close to the experimental values. We suspect that this may be induced by discretization errors. Other ships, hke a Series-60 of similar block coefficient, did not exhibit this behavior.

The computed surge motions for low frequencies are somewhat higher than those measured. The reason is unclear, but could lie i n nonlinear effects or margins of errors i n the experiments.

For comparison, we also show results for the same grids, but with the classical steady-flow approximation.

i.e., no trim and sinkage, flat free surface, uniform flow, and integration only to the calm-waterline. This ap-proximation yields differences in the heave and phch motions of up to 20% for medium wave lengths, which are simflar to recent M I T results" with a time-domain code that captures the higher-order seakeeping butions, but only approximates the steady flow contri-butions. The results for the two computational methods f o r very long and short waves show better agreement. Similar effects were observed f o r a Series-60 ship."^" These pubhcations also demonstrate that the local pres-sures i n the bow change by up to 20% between a " f u l l y 3-d" solution and a 3-d seakeeping solution based on the classical approximation of the steady flow compo-nents. The surge correction has a significant effect only f o r long waves (low frequencies). This effect is naturally most pronounced for the surge motions, where it re-duces the surge motion f o r the longest wave by 20%, but due to cross-coupling a smafler effect is also appar-ent f o r pitch motions.

There are several steady free-surface effects which in sum create these differences. One of these effects can be incorporated relatively simply i n all seakeeping meth-ods, i.e., dynamic t r i m and sinkage. The dynamic t r i m and sinkage may be computed by any Rankine panel method or estimated by simpler methods. Then for the resulting geometry a Green function method grid or a strip method grid may be created. T o illustrate the influ-ence of just the effect of trim and sinkage, the computa-tions were repeated with t r i m and sinkage suppressed in the steady computations. Figure 5 compares the results. The influence, on heave is negligible i n this case except f o r one rather low frequency. However, f o r

Fig. 2. Discretization of a S-175 container

1.0 • -90 .3 4 ^^/L/g 1.0 1 0.5 \Ü3\/h 90 -90 tl.Y 1.(1 0.5 ^ [h\/kh 90 0 - 9 0 4 ui^/Lfg

Fig. 4. Response amplitude operators f o r S-175, \i = 180°, F„ = 0.275. • , experiment; o, Rankine panel method ( R P M ) w i t h all forward-speed effects; *, R P M w i t h classical u n i f o r m flow approximation; +, R P M without surge cor-rection. The classical approximation introduces differences of up to 20% i n the medium f r e -quency (medium wavelength) region

1.0 0.5 \Ü3\/h 90-1 0 -90-1 3 4 w^yL/g 1.0 0.,5

\ü,\/kh

Fig. 5. Response amplitude operators f o r S-175, fl = 180°, F„ = 0.275. • , experiment; o, R P M w i t h all forward-speed effects; *, R P M w i t h trim and sinkage suppressed

pitch in medium to long waves, about half of the discrepancy between the results, considering all forward-speed effects and the classical approximation, can apparently be attributed to not capturing t r i m and sinkage.

The results f o r oblique waves (ji = 150°) show a simi-larly good agreement f o r the symmetric m o t i o n modes

(Fig. 6). For the antisymmetric modes, computation and experiment agree well f o r rolling. For very long waves, the response amplitude operator rises sharply. While

100 V . Bertram and G. Thiart: Surge-corrected 3-d seakeeping computations 1.0 \uMh 90 -90 4 w i / i / s o o o o 1,0 0.5 ^ \Ü2\lh 1 2 3 4 wyjLlg 901 -90 1.0 - 0.5-901 -90-1 2 3 4 i^^/Lfg 2.0 1.0 90 0 -90 I O • O p O -1 2 4 W T / L / a 1.0-0.5 1 90- -90-1 2 3 4 w ^ L / g 1.0 0.5 kh 3 4 ujyjLjg 90 0 -90

Fig. 6. Response amplitude operators f o r S-175, [I = 150°, F„ = 0.275. • , experiment; o, R P M w i t h all forward-speed effects

there is no experimental data f o r the lowest frequency (longest wave) computed, the computed value of 12 is unreahstically high. This is attributed to the neglect of viscous damping and nonhnear effects, and w i l l be sub-ject to further investigation. Yaw and sway look plau-sible except f o r a similar sharp increase f o r the longest wave.

C o n c l u s i o n

As far as the results obtained so far ahow any con-clusions, capturing the steady flow i n seakeeping computations apparently increases the accuracy of predictions considerably. The agreement w i t h experi-ments seems good, at least f o r slender ships, with

the exception of resonance regions f o r the antisymmet-ric modes of motion. The investigated surge correction is easy to implement, especially i f the last two terms, which were found to contribute less than 1 % i n numerical tests, are omitted. Stih many open questions remain. System-atic grid refinement tests were performed. W h i l e the investigated grids were close to the computer capacity available to us (EWS), and the grids were fine enough f o r what is today's standard i n steady computations, we cannot exclude the possibihty that discretization errors may still play a role. M o r e important is that viscous effects are not yet included f o r the antisymmetric modes, and that the method is stih hmited to T > 0.25. M u c h more research and development work w i h be necessary before the method can be regarded as a practical tool. I n the meantime, it allows numerical study of the effects of widely adapted simplifications, and some of the techniques presented can be included i n other approaches such as strip methods or Green-func-tion methods.

R e f e r e n c e s

1. Weinblum G, St Denis M (1950) On the motions of ships at sea. SNAME Trans 58:184-248

2. Bertram V, YasUkawa H (1996) Rankine source methods for seakeeping problems. Jahrbuch Schiffbautechn. Gesellschaft, Springer, Berlin Heidelberg, pp 411^25

3. Bertram V (1996) A 3-d Rankine panel method to compute added resistance of ships. IfS Report 566, University of Hamburg 4. Bertram V (1990) Fulfilling open-boundary and radiation

condi-tion in free-surface problems using Rankine sources. Ship Technol Res 37(2):47-52

5. Hughes M , Bertram V (1995) A higher-order panel method for steady 3-d free-surface flows. IfS Report 558, University of Hamburg

6. Newman JN (1978) The theory of ship motions. Adv Appl Mech 18:222-283

7. Zhou Y (1989) Bestimmung der im Seegang zusatzlich erforderlichen Antriebsleistung von Schiffen. IfS Report 490, University of Hamburg

8. Schneekluth H (1988) Hydrodynamik zum Schiffsentwurf. Koehler, Hamburg

9. ITTC (1978) Seakeeping committee report. 15th International Towing Tank Conference, The Hague, pp 55-114

10. ITTC (1981) Seakeeping committee report. 16th International Towing Tank Conference, Leningrad, pp 185-247

11. Huang Y, Sclavounos P (1997) Nonlinear ship wave simulations by a Rankine panel method. 12th International Workshop Water Waves and Floating Bodies, Marseilles

12. Bertram V (1997) Vergleich verschiedener 3D-Verfahren zur Berechnung des Seeverhaltens von Schiffen. Jahrbuch Schiff-bautechn. Gesellschaft, Springer, Berlin Heidelberg

13. Bertram V (1997) The influence of the steady flow in seakeeping computations. 5th International Symposium Nonlinear and Free-Surface Flows, Hiroshima, pp 29-32

14. Bertram V (1998) Numerical investigation of steady flow effects in 3-d seakeeping computations. 22nd Symposium Naval Hydro-dynamics, Washington