Modeling nonlinear roll damping with a self consistent, strongly nonlinear ship motion model

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J M a r Sci Technol (2008) 13:127-137 D O I 10.1007/s00773-007-0262-9

O R I G I N A L A R T I C L E

Modeling nonlinear roll damping with a self-consistent, strongly

nonlinear ship motion model

Ray-Qing Lin • Weijia Kuang

Received: F e b r u a r y 5, 2007 / Accepted: N o v e m b e r 6, 2007 © J A S N A G E 2008

Abstract Appropriate modeling of roll damping is one

of the key issues i n accurately predicting ship roll motion. The difficulties i n modeling roll damping arise f r o m the nonlinear nature of the phenomena. I n this study, we report a new effort in modeling the bilge keel roll damping effect based on the blocking mechanisms o f an object in the potential flow. This effect can be imple-mented as a component of appropriate ship motion models. We used our digital, self-consistent, ship experi-mental laboratory (DiSSEL) ship motion model to test its effectiveness in predicting ship roll motion. Our numerical experiment demonstrated clearly that the implementation of this roll damping component improves significantly the accuracy o f numerical model results (the results were compared w i t h ship experiment data f r o m the Naval Surface Warfare Center, Carderock Division, Maneuvering and Seakeeping Facility).

Key words Roll motion • Damping • Numerical model

Introduction

R o l l motion is one of the most important ship responses to waves and it is very difficult to predict due to the complexity of ship-wave interactions and its sensitivity

R . - Q . L i n

D a v i d T a y l o r M o d e l Basin, N a v a l Surface W a r f a r e Center, C a r d e r o c k D i v i s i o n ( N S W C C D ) , 9500 M c A r t h u r B l v d . , West Bethesda, M a r y l a n d 20817-5700, U S A

e-mail: R a y . L i n @ n a v y . m i l W . K u a n g

N A S A G o d d a r d Space F l i g h t Center, Greenbelt, M a r y l a n d 20771, U S A

to ship bilge keels and appendages. This sensitivity needs special attention, both in modehng corresponding physi-cal processes and in numeriphysi-cal treatment. One important process is the roll damping. Without an appropriate roll damping model, accurate prediction of ship roll motion is impossible.

There have been several theoretical and experimental attempts to model roll damping in the past. Bryan' was perhaps one of the first researchers to study the bilge keel's effect. Hishida^"* provided a theoretical model o f roll damping f o r ship hulls in simple oscillatory waves. Martin,^ T a n a k a , " Kato,'" M o o d y , " Motter,'^ and J o n e s ' p r o v i d e d experimental results on the effect of büge keels. Several researchers tried to address the effect of ship speed on roll damping. Later, Yamanouchi,'^ Bolton'^ tested the damping with finite f o r w a r d ship speed. Their findings, as well as many others, showed that there are considerable differences between the experimental data and the existing theoretical results. W i t h the linear strip theory, Himeno'^ improved the theoretical model w i t h his roll damping coefficients, which have, to date, been widely used. However, as Himeno himself acknowledged, we still have not fully understood the roll damping process.

One reason that hinders our understanding o f the roll damping effect is the nonlinear interaction between the ship body and surrounding fluid (including waves). This nonlinear interaction affects not only ship motion, but also (more importantly, as we shall demonstrate i n the following analysis) the underwater ship volume.

M a n y studies have been made on the nonlinearity o f ship motion. F o r example, L i u et al.'^ used a high-order spectral method to study nonlinear interactions between the ship and the water surface. Their approach was further improved and extended by L i n and Y u e , " L i n

128 J M a r Sci T e c h n o l (2008) 13:127-137

et al.,'" X u e r ' Xue et al./^ and L i u et a l . " Unfortunately, there are still empirical or linear parameters used in modehng ship motions w h h six degrees o f freedom, e.g., added mass and damping for each degree of motion. Limitations o f these parameters can potentially affect the accuracy of nonlinear ship motion models. For example, the added mass and the damping are often determined f r o m the ship's geometry below the waterline in calm water (Magee and Beck^"'). However, the ship's geometry below the waterline when underway in waves is very different f r o m that i n calm water and continually changes due to the speed and wave action.

Linear approaches are appropriate i f the wave ampli-tudes are small and the ship speed is low. I n this case, nonlinear interactions among the waves are of higher order effects, and thus the linear superposition of the waves is sufficient for describing the ship motion. However, when the nonlinear interactions are strong, i.e., significant compared to the linear terms, such linear superposition is no longer accurate. For a comprehen-sive description, we refer the reader to L i n and Segel"^ and to Infeld and Rowlands."*

More recently, L i n and Kuang""^' developed a new nonlinear ship motion model named the digital, self-consistent, ship experimental laboratory (DiSSEL). I n this model, a hybrid algorithm based on spectral, finite difference, and finite element methods is used to solve

the fundamental nonlinear equations that govern the dynamics of surface waves and ship motions. Parameter-ization that had been assumed in the previous studies on ship motion modeling has been minimized in this model.

One solution to reduce the effect of the empirical or linear parameters is to derive a better model f r o m the basic equations o f fluid motions. For this purpose we intend to develop a new, dynamically consistent, bilge keel roU damping function. I n a general context, a flow passing a solid object is different f r o m that without the object, as can be observed f r o m the different streamlines of the flow, such as those shown in Figs. 1-4 i n Landau and Lifshitz.^" This difference is often called the "block-ing effect" of the object on flow by geophysicists. There is a long history of studying blocking mechanisms in a wide variety of fluid systems, e.g., Freund and Meyer," Garner,^^ L i n and Chubb," and Holton;^'' however, i n this study, we use the term f o r the net pressure force on the surface, instead. Our approach is based on the studies of the blocking mechanisms of an object in potential flow.

The interaction of a ship body w i t h the surrounding flow possesses simflar physical characteristics to the flow passing an object. However, our problem is much more challenging, because the underwater ship geometry, i.e., the solid object interacting with the flow, changes over

F i g . 1. O N R T u m b l e h o m e h u l l (left) a n d F l a r e d h u l l {right) 0.2 f 0.15 I -I 0.05 0 0.12' 0.1 0.08 c Q 0.06 >. Dl 0.04-m 0,02 0 W'(Ug)"0.5 W(L;g)"l).5

F i g . 2a. T h e r o l l damping power spectrum f o r the T u m b l e h o m e h u l l f o r an i n i t i a l angle o f nl5 b R o l l d a m p i n g power spectrum f o r the F l a r e d h u l l f o r a n i n i t i a l angle is TclS. W, frequency; L, ship length; g, gravity acceleration

J M a r Sci T e c l m o l (2008) 13:127-137 129

time; this change depends on the interaction as weU. The advantage of this approach is enormous: the derived damping function is independent of the ship hull. The huU geometry effect is described explichly by a variable

Time

F i g . 3. R o l l d a m p i n g o f t h e T u m b l e h o m e h u l l w i t h bilge keels. The bilge keel effect can be measured b y the a m p l i t u d e difference between the t w o adjacent oscillating peaks. F r a u d e number ( F u ) = 0, a n d wave height ( w h ) = 0

15

Time {second)

F i g . 4a. T u m b l e h o m e h u l l r o l l m o t i o n time series f o r Fn = 0.066, a wave steepness (a/c) o f 0.0465, a n d an incident wavelength/ship length r a t i o n (X) o f 2.5. S h o w n are the experimental results

(squares), the numerical results w i t h r o l l d a m p i n g (solid line) and

the numerical results w i t h o u t r o l l d a m p i n g (chained line). The bilge keel span was 1.25 m , and the length was 102.67 m . b F o r the

in the function, as are other environmental conditions such as the ship motion state, incident waves, and wave-breaking mechanisms.

We will use the DiSSEL ship motion model ( L i n and ICuang""^') and experimental data generated i n the David Taylor Model Basin Laboratory to test the newly developed roh damping function. The model includes two components, DiSSEL_SW and DiSSEL_SB: the ship-wave interaction is modeled i n the DiSSEL_SW component and the ship solid body motion under the influence of the interaction is modeled i n the DiSSEL_ SB component.

This article is organized as fohows: first, we provide a brief overview of the DiSSEL model and then we dem-onstrate the importance o f ship water hne variation. Next, the theoretical derivation of the roll damping func-tion is explained, followed by the numerical results and comparison of the model output and experimental data. This is followed by a discussion.

Time (second)

F l a r e d h u l l w i t h ak = 0.0565 and other parameters as above, c F o r the T u m b l e h o m e h u l l w i t h afc = 0.05 and A = 2.25. d F o r the F l a r e d h u l l w i t h a/c = 0.1285 and A = 2.25. e F o r the T u m b l e h o m e h u l l w i t h a/c = 0.0465 and A = 1.75. f F o r the T u m b l e h o m e h u l l w i t h a/c = 0.055 and A = 3 . 5

130 J M a r Sci T e c l i n o l (2008) 13:127-137

D i S S E L overview

To contain the length of this article, we provide only a brief overview of the DiSSEL ship model used in our study for testing the roll damping function. For the details of the model, we refer the reader to L i n and K u a n g " - ' ' and L i n et a l . ' '

The DiSSEL model consists of two components: a fully nonlinear ship-wave interaction model (DiSSEL_ SW) and a coupled, six-degree-of-freedom, ship solid body motion model (DiSSEL_SB).

DiSSEL_SW

I n this model, the physical processes are described in a reference frame moving horizontally with the mass center of the ship. The corresponding equations in this refer-ence frame are the incompressibility condition:

0 for -H<z<ri

(1)

where V,, is the horizontal gradient, cp is the total velocity potential, and H is the water depth; the dynamic and kinematic boundary conditions at the free surface

2

=77:

^ + riv(p + u Yv(p + ^77 + ^ + ^ - x - v V > = 0 (2)

at \2 ) p dt

| + (V,,).(V,<p + u J = | (3)

where p is the dynamic pressure, p is the fluid density, v is the kinematic viscosity, x is the position vector, is the ship's constant velocity, and r\ is the free surface elevation; and the impenetrable conditions at the bottom of the ocean and at the ship boundary F:

+ (V;,/^)-(V„(p + u J = 0. at -H

ii-(V(p + u,) = 0, on F

(4)

(5) where n is the unit vector of F, and u, is the total ship velocity (translational and rotational motion speed). I t should be pointed out here that an effective dissipation is introduced in Eq. 2 to model wave-breaking mecha-nisms and the effect of subscale flow. I t is not the fluid viscosity that vanishes i n an incompressible potential flow, as shown in Eq. 1. For mathematics detaüs, we refer the reader to L i n and Kuang.'"'

The equations are solved with the radiation boundary conditions to ensure that the ship-generated waves are left behind the ship. They are implemented as open boundary conditions away f r o m the ship (the far field). I n computational applications, the far-field boundary conditions are defined at a finite distance f r o m the ship (instead of the asymptotic limit x 00):

3 ^ = - f ^ - « . v > n = ^e\ at x = b dx dx Vcp = V(p, + ( V ( p , - V ( p J - u , ; '? = ??. + 0?.-??J; at x = c (6) (7) where b is the forward boundary, c is the side and aft boundaries, 9, and 77, are the velocity potential and the surface elevation of the environmental waves, and cp, and rj, are those associated w i t h the ship (e.g., waves generated by the ship motion and waves arising f r o m ship-environmental wave interactions). The overbar means the spatial average of the quantities. We should point out that the boundary conditions are consistent w i t h mass conservation. I n calm water, (p^ and vanish.

The fully nonlinear Eqs. 2 and 3 are solved in the computational domain with a pseudospectral method; liowever, on the ship boundary, a finite element or a finite difference method is applied. A t every time step, physical quantities are transferred via a quasilinear method between the ship boundary grids and the spec-tral collocation points. This approach can achieve com-putational efficiencies f o r arbitrary ship hulls.

DiSSEL_SB

The six-degree-of-freedom ship solid body motion is divided into a three-degree translational motion o f the mass center and a three-degree rotation about the mass center.

The translational motion equation is written as:

trans Iraiis (8)

where m^,,,^ is the mass o f the ship, v is ship translational velocity, £),,„,„ is the damping coefficient for the dissipa-tive effect between the ship body (including its append-ages) and the surrounding fluid, and F„.„,„ and F,;:",'°''" are the net force on the ship surface and the net body restor-ing force, respectively:

- j n • pds (9)

J M a r Sci T e c l i n o l (2008) 13:127-137 131

^:^r=("^s,,-P.a,e,KJS (10)

where p,„„„. is the water density, is the underwater ship volume, and g is the gravitational acceleration, m,,,!^,

p„,„,j,. and g are constant. The reference frame that moves

horizontally with the ship mass center (the reference frame f o r surface wave calculations). However, F„,,, is time-varying and depends on the nonlinear process described in part by Eqs. 2 and 3.

The rotational motion equation (Liouville equation) is written as:

where is damping coefficient for rotation motions,

Q is angular velocity, and I is the ship's moment of

inertia relative to the ship mass center x„ I can be written as and equal:

= ƒ J ƒ r/F[(x - X J,'5, - (X - X J,(x - x

T'f,^,*'^ is the pressure moment on the ship body and V'^l'^ is the restoring moment arising f r o m the buoyancy force. dV is the volume element, 5jj is the Kroneeker function,

F,,„^ is the total volume o f the ship. Equations 8 and 11 are nonlinear i n nature because the moments and the forces depend on the ship motion. Linearization or parameterization o f these quantities leads to wrong answers when nonhnearity becomes important.

I n DiSSEL, the wet surface moment and the body restoring moment are evaluated at each time step using:

r ™ = - l H x . - x J x n ; ; (12)

r : : r = ( x , , . , - x j x p , , , „ , x . g (13)

where x,,,^,, is the center of buoyancy and S is the wetted surface (the three components o f the vector are f o r roh, pitch, and yaw motions, respectively).

Underwater ship geometry and ship motion

I n a linear parameterized model, the underwater ship geometry is assumed to be unchanged i n determining the ship responding motion (Magee and Beck'*); i n reality, the underwater geometry does change according to the ship's solid body motion. I n DiSSEL, such change is incorporated naturally; its effect on ship motion and on

ship-wave interaction can be identified i n the numerical results. I n this section, we aim to isolate the impact of time-varying underwater ship geometry on ship motion. It is well known that the response o f a sohd body to external forcing is strongest when the forcing fre-quency resonates with the natural frefre-quency of the solid body. The ship natural frequencies i n water can be deter-mined i n a way similar to that for a pendulum: given an initial ship position i n calm water which is away f r o m its equilibrium position, i t will oscillate (rotate or move vertically) according to the buoyancy force, i.e., the restoring forces i n Eqs. 8 and 11. I n numerical simula-tion, the responses are described by a time series of related quantities (e.g., velocity potential or rotating angles). The natural frequency can be obtained f r o m the time series via the Fourier transform.

The whole ship geometry, not only the part below the water line, is required to correctly determine the natural frequencies of the ship under consideration. To demonstrate this numerically, we selected two ship huUs: the Office of Naval Research (ONR) Tumble-home and O N R Flared hulls. The geometries of the hulls under the calm water line are identical, but their geometries above water are different: the sidewall of the Tumblehome huU tihs 10° inward (Fig. la) and that of Flared hull tilts 10° outward (Fig. l b ) . Linearized ship motion models would yield the same natural frequency because the under-water-line geometries are identical. However, our model results indicate otherwise. The natural roll frequencies of the two hulls are shown i n Fig. 2. F r o m the figure we can observe clearly that the natural frequency of the Tumblehome hull is lower than that o f the Flared hull.

Without correct natural frequencies, one would not be able to determine correctly the ship responses to external forcing on the ship. Therefore, our numerical results show that the complete ship geometry is impor-tant f o r correct modeling of ship motion, i n particular near the resonant conditions. There are several physical reasons for the differences between the roll damping of the Tumblehome hull and of the Flared hull. First, the two hulls have different roll stability (related to trans-verse metacentric height (GM)). This naturally affects the ship roh motion. Another reason is that the ship underwater geometries (wetted surface) are different, except when in calm water and at rest. Consequently the pressure moment on the different wetted surfaces will be different, thus driving different roh motions.

As described earher, in the DiSSEL ship motion model, the pressure moment and the roh motion are updated at every time step. Therefore, it is appropriate for testing the roll damping function. This will be dis-cvissed in the following section.

132 J M a r Sci Technol (2008) 13:127-137

Bilge keel roll damping function

Real and experimental ship hulls often include append-ages, such as bilge keels and rudders. These appendages produce roll damping that can reduce sliip roU motions by as much as 10%-20%; thus it is important that roll damping be implemented into numerical models (Himeno").

To balance numerical accuracy and efficiency, naval architects have developed many simple, parameterized roU damping models. For example, H i m e n o " suggested the following linearized roll damping coefficient:

B,{d) = Bfi (14) The roll damping function 5 / Ö ) is linearly proportional

to the rotation rate 9 with a constant damping cient B^. He also suggested a more sophisticated coeffi-cient f o r simple oscillation with a frequency co:

B^ = B,+ —(o9^B, + -a'B^B, (15)

371 4

where 5 , 2,3 are constants and 0^ is the constant r o h motion amphtude that is precalculated by his linear strip theory. Obviously, the damping coefficient 5^ is constant for a given frequency (o and 9^. For more details, we refer the reader to Himeno."

I n our earlier approach, the roll damping coefficient was determined by the difference i n the two adjacent wave peaks (or valleys) of the free roll oscillation (with a fiifite initial departing angle) in calm water. A n example of such oscillation is shown i n Fig. 3. However, this approach, ahhough simple and nonlinear in nature (and thus better than linearized results), has its own limita-tion: it does not account f o r possible dependencies of the coefficient on roll oscillation frequency. This could create serious problems when ships are not in calm water. I n those more realistic situations, ships roll w i t h a wide spectrum of frequencies. The damping coefficient derived f r o m a single (natural) frequency is simply not sufficient. Even i f a more realistic damping coefficient could be lueasured f r o m the amphtude variation in specified inci-dent waves, this approach is very inefficient: such mea-surement must be carried out f o r different ship hulls in different environments.

This has prompted us to search f o r a more realistic approach to modeling the roh damping effect. Instead of reinventing the wheel, we intend to use existing meth-odologies for our attempt. The approach we take is based on the classical analysis o f the blocking effect of a solid object i n a flow (Landau and Lifshitz'"). This approach has been applied to many fluid systems, e.g..

an object in stratified fluid (Freund and Meyer''), the wind past two-dimensional terrain (Garner"), and the current past a seamount ( L i n and Thomas"). Bilge keels on ship hufls play a role similar to a seamount i n the ocean. This leads us to import the methodology f o r our problem. However, our problem is unique and is differ-ent f r o m the seamount case i n that the underwater part of the bilge keels varies in time as the ship moves i n water. I n addition, we do not intend to solve the problem directly via numerical modeling (using the fundamental equations of the fluid mechanics). Instead, we intend to use a quasi-analytical approach: certain simplified math-ematical formulations are used to derive the damping function. The price paid is that not all nonlinear damping effects are included. To include the fully noiflinear effects of the appendages, one could use a coupled system, such as a nonlinear ship motion model (e.g., DiSSEL) w i t h a fully nonlinear viscous flow model (e.g., R A N S ) . However, this can be computationally very time con-suming, or even impractical.

The details of the approach are now explained. I n general, the roll damping can be evaluated f r o m the moment arising f r o m the pressure acting on the blocking area A,,,,,:

d9

iX„,,,\=-p\\ (rxnXpds = D(t)-^ (16)

where D(t) is a time-varying, nonlinear damping func-tion arising f r o m the bilge keel's blocking effect (the negative sign implies that it will be deducted f r o m the moments acting on the ship body). However, direct evah nation o f Eq. 16 requires correct knowledge o f the pres-sure at the surface, which then depends on the specific geometric properties of the bilge keels and the corre-sponding dynamic state o f the system. Whfle feasible, it can be very demanding numerically because very flne numerical grids are necessary to resolve the small-scale processes.

To avoid such difficulties, we borrow the following idea f r o m previous studies of blocking mechanisms: the blocking effect can be included by deducting an effective blocking area ^jfo^i (not A/^i^^,, in Eq. 16) f r o m the ship wet surface in E,- the integral i n Eq. 16 f o r the moments on the ship body:

r ^ : f , : ^ = - j j d^ix^-xjxnp, i = 4 (17)

N o w the problem is to determine the effective blocking area Aj^j^^;..

I n our approach, the effective blocking area is approx-imated as:

J M a r Sci T e c h n o l (2008) 13:127-137 133

(18)

where H^^ and L,,i. are the w i d t h of the span and the underwater length of the bilge keel, respectively. L,,,. varies with time because part of the bilge keels can be above water when a ship moves. The underwater length Li,,, can be evaluated using:

.'2(0

(19)

where / , and / j are the two time-varying end grid points of the underwater bilge keel, Slj is the length segment at grid point j, and Rj{t) is the underwater percentage of Slj. The time variation o f the quantities depends on the ship's motion and surrounding waves. I'Fjfo.j, is a dynamic factor used to describe other geometric and dynamic effects, such as the effective blocking area width (depend-ing on i / j i - and the complexity of flow near the ship boundary).

There is no existing result for modeling l-F^fo^-. There-fore, we start f r o m physical intuition and the typical approaches o f nonlinear theory. Intuitively, one could argue that the larger the rolling angle and the faster the oscillation, the larger the effective blocking area. This implies that, in the simplest form:

0} X ® , —

r=0 •A

(22)

where A is the total amplitude of the incident waves, Aj are those o f individual modes with the frequencies to,-, and A''is the number of individual modes o f the incident waves.

I n our analysis, we chose K=3. Obviously, Kcan be chosen differently in other apphcations. The principle is that K should be larger for stronger nonhnearity; however, there is no established theory for selecting an optimal K. The coefficients C2j.+, in the expansion (Eq. 21) depend on the nonhnearity of the physical problem (e.g., the nonlinear equations of the system). The size of the coefficients decreases as k increases.

F r o m Eq. 21, we can observe that the coefficient 02^+1 describes the nonlinear effect of mode (0©)'*"^', which is proportional to (f*'^' ( a i s the typical magnitude of 9có). However, we set the coefficients to be normalized in our study, i.e..

T (23)

so that the magnitude variation is only described by öco (similar to the scaling approaches). There are many choices f o r satisfying the above normalization condhion. The simple choice used in this study is:

1

(20) C2,,, = - ^ , (24)

where ö„, is the rolling angle magnitude and co is the rolling frequency. On the other hand, general multiscale analysis (Bender and Orszag'*) demonstrates that non-linear effects can be modeled by adding higher-order terms (as the power 2n) to Eq. 20:

W„„ck = c,(0„,«)Ll + «,(Ö„,ffl)' -t-«2(0,„ö))* +«3(0,„G))* + . . .J

= i c 2 , „ ( ö , „ « ) ' ^ - ' (21)

/f=0

up to some given truncation order K. Because roll motions are i n general more complicated than a simple oscillation, there is a wide spectrum of oscillating fre-quencies. These frequencies are often related to those o f the external forcing. I n our application, the external forcing is provided by incident waves. To properly account f o r the contributions f r o m individual oscillating modes, we replace the single frequency a in Eq. 21 w i t h a weighted distribution over the incident wave frequency domain:

So that:

A-=0 k=0 ^

The formulation i n Eqs. 18 and 24 of is based on very simple mathematical and physical consider-ations: it does not depend on any specific assumption on properties o f the ship or the environment. The goal is to develop a damping model applicable to arbitrary ships in arbitrary environments. The coefficients Cj^.^, are set constant f o r the normahzation, i.e., the summation of the coefficients is unity. The rolling angle 0„, i n Eq. 21 depends on the ship body characteristics, the ship motion state (e.g., speed, rotation), and the environment (e.g., incident waves, ship motion-generated waves, kinematic dissipation, and wave-breaking processes).

We should also point out again that, since i t depends on many characteristics, 0„, must be determined by solving the coupled ship motion-surface wave system. This is exactly what we have been doing in our numerical

134 J M a r Sci T e c h n o l (2008) 13:127-137

simulation, i.e., via DiSSEL_SW (Eqs. 1-7), which models fully nonlinear ship-wave interactions, and DiSSEL_SB (Eqs. 8-13), which models six-degrees-of-freedom, ship sohd body motions. The detahs can be summarized as follows: first, an inhial state is estimated f r o m the peak of the incident wave (in the ship coordi-nates). Then the f u l l system (without the büge keels) is solved f r o m the initial state via numerical time integra-tion. The simulation stops when the system reaches a weh-developed, dynamically stable state. The rolling angle amplitude 9„, is then determined f r o m the stable state.

The effective blocking area ^„„^^ is i n general larger than the real blocking area ^"jfo^,. I n our model, the

addi-tional area is added to the downstream side of the bUge keels. Since the roll motion is oscillating with time, the downstream is also oschlating with time. Both the size and the location o f ^ j f o ^ i are defined by Eq. 21. The

expansion given in Eq. 21 is consistent i n format w i t h other approaches in modeling solid body motion w i t h damping effects (Kreyszig'').

FinaUy, we want to point out that roll damping is strongly associated with the time-varying underwater geometry. For example, fFifec^ decreases as the bilge keels rise above the water (an extreme case is that = 0 when the bilge keels are completely above water).

Numerical results

To examine the damping function model described above, we used DiSSEL to study the roh motions of two ship hulls: the O N R Tumblehome and Flared hulls. The main reasons for selecting these two hulls are that the two hulls have same geometries under water line, but they have dif-ferent above-water-line geometries (in calm water), and experimental data is available at the David Taylor Model Basin Laboratory ready for benchmarking our numerical resuhs. We focused on two problems: the impact of the ship huh geometry above the calm water line on roll motion and the blocking effect of the bilge keel.

I n all numerical simulations, the initial conditions were determined by both the experimental data and the dynamic balancing constraints imposed on the initial states. For example, the time series of the waves were measured at the location right in f r o n t o f the center of the bow of the moving ship. The incident waves measured were not contaminated by the ship-motion-generated waves or the ship response motions. Therefore they were used as the initial conditions for the waves i n the numerical simulations. Given the initial waves, the initial roh angle can be calculated directly f r o m the con-straint that the restoring moment resulting f r o m the r o l l

angle completely offsets the driving moment f r o m the initial waves.

We considered six different cases for Tumblehome and Flared hulls i n regular beam seas, all o f which have been studied experimentally i n the past. The numerical results were compared with the experimental results and are shown i n Figs. 4a-f, which show the time series o f the DiSSEL resuhs with roh damping, the experimental results, and the DiSSEL resuhs without the roh damping for different normalized incident wavelengths A (scaled by the ship length, 154 m) and wave steepness, ak. F r o m the figures one can observe clearly that the DiSSEL results with the roll damping agree very well with the experiinental data in all cases. The agreement i n ah six cases suggests strongly that the nonlinear roll damping method described i n Eqs. 16-24 is generic, i.e., it is appli-cable to various ship hulls (Tumblehome and Flared hulls i n these cases) in various incident waves (six differ-ent waves i n this study).

It should be pointed out that, as shown i n the figures, both experimental data and numerical simulation results demonstrate that roll motion can be irregular, even though the incident waves (in all cases) are regular. This irregular roll motion can be well explained by ship-wave interactions. Given a ship hull, its natural frequency depends only on the ship structure. I f a regular incident wave has a frequency different f r o m the ship natural frequency, the nonlinear interactions between ship body and the incident wave and between the ship-generated waves and the incident waves will produce moments varying with different frequencies. The closer the inci-dent wave frequency is to the natural frequency, the stronger the interactions, and thus the more irregular the ship roh motion. I n the case shown i n Fig. 4f, the two frequencies are so different that the interactions are weak. Therefore the roh motion is nearly regular.

The bilge roll damping effect can be very large: as shown in the figures, the amplitudes without the roll damping can be approximately one-third larger than those with the roh damping. This effect is more signifi-cant when the nonlinear interactions are stronger. I n addition, i t also depends on incident wavelength and amphtude, as shown i n Eq. 21.

To compare the mean properties of the numerical resuhs and the experimental data over the entire time period, the typical amplitudes o f the all cases were esti-mated by the root mean square (rms) of the peak ampli-tudes i n the roh angle time series. The results f o r H,,,, = 1.25 m and Fn = 0.066 are summarized in Fig. 5. From the figure one can observe clearly that the model results agree well w i t h the experimental data.

From Fig. 5 we also note that the O N R Tumblehome hull roll angles are much larger than those o f the O N R

J M a r Sci Technol (2008) 13:127-137 135

Flared hull near resonance. The difference is largest when X equals the resonant value: the roll angle of the Tumblehome hull is nearly double that of the flared hull. This is due to the differences i n the ship hull geometry. To better understand its physics, we consider a simple free ship roll motion: the ship is set initially at a smah roll angle away f r o m hs equilibrium position. I t w i h then oscillate freely with its own natural frequency a)„„,„r„

governed by the following simplified equation:

hplo,urf> = m,,^gKuQ (25) where is the restoring rolling arms of the ship hull.

Therefore,

0 0.5 1 1.5 2 2.5 3 3.5 4

F i g . 5. N o r m a l i z e d r o l l m o t i o n angle f o r d i f f e r e n t values o f X w i t h

Fn = 0.066. The solid line shows the n u m e r i c a l results f o r the

T u m b l e h o m e h u l l and the daslied line the n u m e r i c a l results f o r the Flared h u l l . The experimental data f o r the t w o hulls are s h o w n as

squares and circles, respectively. T h e measurement error is 0.05,

w h i c h is too small to be p l o t t e d (i.e., i t smaller t h a n symbols rep-resenting each experimental data p o i n t ) . Ka, wave steepress

I n Other words, given the same moment of inertia larger natural frequencies lead to a longer restoring arm. Since, f r o m the experimental data, the inertial moment and the total ship mass w,;,,^, are the same f o r the Tumblehome and the Flared hulls, and since the Flared hull has a larger natural frequency (the normalized frequency of the Flared huh is 2.8, while that of the Tumblehome is 1.8, see Fig. 2), the Flared hull has a longer restoring arm and is therefore more stable than the Tumblehome hull. Consequently, its maximum rolling angle is smaller. However, the actual maximum rolling angle is determined by the nonlinear relationship shown i n Eq. 11, not the linearized approximation given in Eq. 25, particularly near resonance.

The convergence of the numerical solutions can be tested by the distribution of the spectral coefficients of the physical variables (e.g., the velocity potential tp) with respect to the (discrete) expansion wave numbers (/c,., /c^,) in the (x, y) axes. To show the convergence, we plot in Fig. 6 the distributions of the spectral coefficients of the velocity potential cp (for the solution in Fig. 4c at ? = 23 s) in the (k„ k^) spectral space. Figure 6a shows the distri-butions o f |(p| i n ky for different k„ while in Fig. 6b are the distributions in /c,. for different ky. I n the figures one can see that the spectral coefficient decreases by more than 10 orders of magnitude f r o m smah wave numbers Qc^, k^) to the maximum wave numbers i n the expansion. For example, the coefficient for the wave numbers (/c^., /c^,) = (0.0164, 0.0164) is 12 orders of magnitude greater than that for (/c,, /c^,) = (1.0308, 1.0308). The decrease is monotonic, except at very large values o f /c„ but i n the latter case, the magnitude is very small and is beyond the accuracy of the computing systems. Therefore, the convergence of the numerical solutions is very satisfactory. Kx=0.0164 Kx=0.3436 Kx=0.6709 Kx=1.0308 -10 12 -14 Kx -Ky=0.0164 -Ky=0.3436 Ky=0.6709 -Ky=1.0308 Ky=-0.0164 Ky=-0,3436 Ky=-0.6709 -Ky=-1.0308

F i g . 6. The d i s t r i b u t i o n o f log|o|(pl (tp is the velocity potential) i n the spectral space (/c„ /c,) f o r the s o l u t i o n at r = 23 s i n F i g . 4c. T h e distributions i n the wave n u m b e r k,. f o r several wave numbers /c.

are shown i n a and the d i s t r i b u t i o n s i n /c^. f o r several are s h o w n i n b

136 J M a r Sci T e c h n o l (2008) 13:127-137

Discussion

I n this article we discussed our approach to modeling the dissipative effect of bilge keels on ship roh motions. Our approach was based on the blocking theory of fluid mechanics, which describes the resistance o f a solid object i n a flow. Instead of deriving a damping function directly f r o m the basic equations, we employed a multi-scale approach with basic physical intuition to formulate the damping function. Thus our approach is quasi-analytical in nature.

We implemented the new damping function in the DiSSEL ship motion model and tested i t numerically w i t h O N R Tumblehome and Flared hulls for different incident waves. Our numerical resuhs agree well w i t h those f r o m experiments done at the David Taylor Model Basin Laboratory. Furthermore, our results show that the bilge keel's blocking effect can damp the ship roll motion significantly. The stronger the nonlinear interac-tion, the more significant the damping effect will be. This damping effect depends also on incident waves and ship huh forms, as shown in Figs. 4 and 5.

We also demonstrated that the ship geometry above the calm water line is important i n correctly modeling ship roll motion, partly because this part o f the geometry is critical in determining the natural frequency of the ship, as shown i n Fig. 2. Its importance in determining the forces on ship bodies has been reported by the authors elsewhere ( L i n and Kuang'"').

I t is very interesting to discuss the differences between our approach and the traditional approach (Eqs. 14-24) to including the roll damping effect into ship motion models. I n the traditional approach (Eqs. 14, 15), an empirical coefficient is directly introduced, and a linear damping term 5, dO/dt is added directly into the dynamic equation governing the ship roll motion. This empirical coefficient does include partial nonlinear effects, e.g., as anonhnear function of roh angular veloc-ity. But this approach has an intrinsic deficiency in that modeling can only be carried out i f experimental data for the ship hulls in the given environment are available. This implies that such numerical modeling could only be used to verify the experiment. I t cannot be used for situ-ations different f r o m the tested cases, and thus caimot be used for ship design.

I n this study, the blocking moment T,,,„,.^. is added to the system. This approach is more generic. N o t only can the nonlinear effect be well modeled by the moment

r ^ f o j , , the algorithm itself does not depend on the model

environment (e.g., ship hull geometries, incident waves, and ship speed). However, the moment Fdepends on the ship hull geometry (via the pressure moment inte-grated over the blocking area), the incident waves, ship

motions (in very complex manners), the l i f t force, and kinematic dissipations; we can expect f r o m Eq. 14 that the coefficient B^ (D in the equation) also depends on the model environment. The advantage of this approach is that all modeling can be carried out without any prior experimental results, and thus is ideal f o r ship design.

As pointed out, Eqs. 23 and 24 represent only one of many choices for the coefficients C2i+i. Different

defini-tions could be investigated for better modeling of the roll damping effect. Therefore, it would be very interesting to e x a i T u n e the sensitivity o f the damping coefficient to the model environment, and the implications of such sensitivity to the numerical model results.

Acknowledgments. R. L i n ' s w o r k is supported by grants f r o m the

D a v i d T a y l o r M o d e l Basin, Carderock D i v i s i o n , N a v a l Surface W a r f a r e Center Independent L a b o r a t o r y I n - H o u s e Research ( I L I R ) p r o g r a m administered b y D r . J o h n B a r k y o u m b . W . K u a n g is supported by the N A S A E a r t h Surface a n d I n t e r i o r p r o g r a m , the N A S A M a r s F u n d a m e n t a l Research p r o g r a m , and b y the N S F Math/Geophysics p r o g r a m . W e thank M r . M i k e Davis, M r , D a v i d W a r d e n , and M r , T e r r y Applebee f o r m a n y useful coimnents a n d suggestions.

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