J Mar Sci Technol (2008) 13:127–137 DOI 10.1007/s00773-007-0262-9
ORIGINAL ARTICLE
Modeling nonlinear roll damping with a self-consistent, strongly nonlinear ship motion model Ray-Qing Lin · Weijia Kuang
Received: February 5, 2007 / Accepted: November 6, 2007 © JASNAOE 2008
Abstract Appropriate modeling of roll damping is one of the key issues in accurately predicting ship roll motion. The difficulties in modeling roll damping arise from the nonlinear nature of the phenomena. In this study, we report a new effort in modeling the bilge keel roll damping effect based on the blocking mechanisms of an object in the potential flow. This effect can be implemented as a component of appropriate ship motion models. We used our digital, self-consistent, ship experimental 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 of numerical model results (the results were compared with ship experiment data from the Naval Surface Warfare Center, Carderock Division, Maneuvering and Seakeeping Facility). Key words Roll motion · Damping · Numerical model
Introduction Roll 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. Lin David Taylor Model Basin, Naval Surface Warfare Center, Carderock Division (NSWCCD), 9500 McArthur Blvd., West Bethesda, Maryland 20817-5700, USA e-mail:
[email protected] W. Kuang NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
to ship bilge keels and appendages. This sensitivity needs special attention, both in modeling corresponding physical processes and in numerical 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. Bryan1 was perhaps one of the first researchers to study the bilge keel’s effect. Hishida2–4 provided a theoretical model of roll damping for ship hulls in simple oscillatory waves. Martin,5 Tanaka,6–9 Kato,10 Moody,11 Motter,12 and Jones13,14 provided experimental results on the effect of bilge keels. Several researchers tried to address the effect of ship speed on roll damping. Later, Yamanouchi,15 Bolton16 tested the damping with finite forward ship speed. Their findings, as well as many others, showed that there are considerable differences between the experimental data and the existing theoretical results. With the linear strip theory, Himeno17 improved the theoretical model with 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 of 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 in the following analysis) the underwater ship volume. Many studies have been made on the nonlinearity of ship motion. For example, Liu et al.18 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 Lin and Yue,19 Lin
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et al.,20 Xue,21 Xue et al.,22 and Liu et al.23 Unfortunately, there are still empirical or linear parameters used in modeling ship motions with six degrees of freedom, e.g., added mass and damping for each degree of motion. Limitations of these parameters can potentially affect the accuracy of nonlinear ship motion models. For example, the added mass and the damping are often determined from the ship’s geometry below the waterline in calm water (Magee and Beck24). However, the ship’s geometry below the waterline when underway in waves is very different from that in calm water and continually changes due to the speed and wave action. Linear approaches are appropriate if the wave amplitudes are small and the ship speed is low. In 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 comprehensive description, we refer the reader to Lin and Segel25 and to Infeld and Rowlands.26 More recently, Lin and Kuang27–29 developed a new nonlinear ship motion model named the digital, selfconsistent, ship experimental laboratory (DiSSEL). In 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. Parameterization 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 from the basic equations of fluid motions. For this purpose we intend to develop a new, dynamically consistent, bilge keel roll damping function. In a general context, a flow passing a solid object is different from that without the object, as can be observed from the different streamlines of the flow, such as those shown in Figs. 1–4 in Landau and Lifshitz.30 This difference is often called the “blocking 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,31 Garner,32 Lin and Chubb,33 and Holton;34 however, in this study, we use the term for 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 with the surrounding flow possesses similar 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
Fig. 1. ONR Tumblehome hull (left) and Flared hull (right)
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Fig. 2a. The roll damping power spectrum for the Tumblehome hull for an initial angle of p/5 b Roll damping power spectrum for the Flared hull for an initial angle is p/5. W, frequency; L, ship length; g, gravity acceleration
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time; this change depends on the interaction as well. The advantage of this approach is enormous: the derived damping function is independent of the ship hull. The hull geometry effect is described explicitly by a variable
in the function, as are other environmental conditions such as the ship motion state, incident waves, and wavebreaking mechanisms. We will use the DiSSEL ship motion model (Lin and Kuang27–29) and experimental data generated in the David Taylor Model Basin Laboratory to test the newly developed roll damping function. The model includes two components, DiSSEL_SW and DiSSEL_SB: the ship–wave interaction is modeled in the DiSSEL_SW component and the ship solid body motion under the influence of the interaction is modeled in the DiSSEL_ SB component. This article is organized as follows: first, we provide a brief overview of the DiSSEL model and then we demonstrate the importance of ship water line variation. Next, the theoretical derivation of the roll damping function is explained, followed by the numerical results and comparison of the model output and experimental data. This is followed by a discussion.
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Fig. 3. Roll damping of the Tumblehome hull with bilge keels. The bilge keel effect can be measured by the amplitude difference between the two adjacent oscillating peaks. Fraude number (Fu) = 0, and wave height (wh) = 0
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Fig. 4a. Tumblehome hull roll motion time series for Fn = 0.066, a wave steepness (ak) of 0.0465, and an incident wavelength/ship length ration (l) of 2.5. Shown are the experimental results (squares), the numerical results with roll damping (solid line) and the numerical results without roll damping (chained line). The bilge keel span was 1.25 m, and the length was 102.67 m. b For the
Time (second)
Flared hull with ak = 0.0565 and other parameters as above. c For the Tumblehome hull with ak = 0.05 and l = 2.25. d For the Flared hull with ak = 0.1285 and l = 2.25. e For the Tumblehome hull with ak = 0.0465 and l = 1.75. f For the Tumblehome hull with ak = 0.055 and l = 3.5
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DiSSEL 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 Lin and Kuang27–29 and Lin et al.35 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).
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 from the ship (the far field). In computational applications, the far-field boundary conditions are defined at a finite distance from the ship (instead of the asymptotic limit x → ∞):
∂ ϕ ∂ ϕe = − usx , η = ηe ; at x = b ∂x ∂x ∇ϕ = ∇ϕ e + (∇ϕ s − ∇ϕ s ) − u s; η = ηe + (ηs − ηs ); at x = c
(6)
(7)
DiSSEL_SW In 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 reference frame are the incompressibility condition: ∇ 2 ϕ ≡ ∇ 2h ϕ +
∂ 2ϕ = 0 for − H ≤ z ≤ η ∂z2
(1)
where ∇h is the horizontal gradient, ϕ is the total velocity potential, and H is the water depth; the dynamic and kinematic boundary conditions at the free surface z = h:
∂ϕ ⎛ 1 p ∂u + ∇ϕ + u s ⎞ ⋅∇ϕ + gη + + s ⋅ x − ν∇ 2h ϕ = 0 ⎠ ∂t ⎝ 2 ρ ∂t
(2)
∂η ∂ϕ + (∇ hη ) ⋅ (∇ h ϕ + u s ) = ∂t ∂z
(3)
where b is the forward boundary, c is the side and aft boundaries, ϕe and he are the velocity potential and the surface elevation of the environmental waves, and ϕs and hs are those associated with the ship (e.g., waves generated by the ship motion and waves arising from ship–environmental wave interactions). The overbar means the spatial average of the quantities. We should point out that the boundary conditions are consistent with mass conservation. In calm water, ϕe and he vanish. The fully nonlinear Eqs. 2 and 3 are solved in the computational domain with a pseudospectral method; however, on the ship boundary, a finite element or a finite difference method is applied. At every time step, physical quantities are transferred via a quasilinear method between the ship boundary grids and the spectral collocation points. This approach can achieve computational efficiencies for arbitrary ship hulls. DiSSEL_SB
where p is the dynamic pressure, r is the fluid density, n is the kinematic viscosity, x is the position vector, us is the ship’s constant velocity, and h is the free surface elevation; and the impenetrable conditions at the bottom of the ocean and at the ship boundary Γ:
∂ϕ + (∇ h H ) ⋅ (∇ h ϕ + u s ) = 0. at z = − H ∂z
(4)
nˆ ⋅ (∇ϕ + ut ) = 0, on Γ
(5)
mship
where nˆ is the unit vector of Γ, and ut is the total ship velocity (translational and rotational motion speed). It should be pointed out here that an effective dissipation is introduced in Eq. 2 to model wave-breaking mechanisms and the effect of subscale flow. It is not the fluid viscosity that vanishes in an incompressible potential flow, as shown in Eq. 1. For mathematics details, we refer the reader to Lin and Kuang.40
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The six-degree-of-freedom ship solid body motion is divided into a three-degree translational motion of the mass center and a three-degree rotation about the mass center. The translational motion equation is written as: dv restore + Dtrans v = Ftrans + Ftrans dt
(8)
where mship is the mass of the ship, v is ship translational velocity, Dtrans is the damping coefficient for the dissipative effect between the ship body (including its appendages) and the surrounding fluid, and Ftrans and Ftrraesntsore are the net force on the ship surface and the net body restoring force, respectively: Ftrans = − ∫ ∫ n ⋅ pds ∑
(9)
J Mar Sci Technol (2008) 13:127–137 restore Ftrans = ( mship − ρwaterVwet )g
131
(10)
where rwater is the water density, Vwet is the underwater ship volume, and g is the gravitational acceleration. mship, rwater and g are constant. The reference frame that moves horizontally with the ship mass center (the reference frame for surface wave calculations). However, Vwet 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: I⋅
dW I + D+R + W × ( I ⋅ W ) + Drotat W = G rotat + G restore rotat dt
(11)
where Drotat is damping coefficient for rotation motions, W is angular velocity, and I is the ship’s moment of inertia relative to the ship mass center xc, I can be written as Iij and equal: I ij =
∫ ∫∫ dV ( x − x ) δ
2 c i ij
− ( x − xc )i ( x − xc ) j
Vship
+ D+R GoI tat is the pressure moment on the ship body and G rroestattore is the restoring moment arising from the buoyancy force. dV is the volume element, dij is the Kroneeker function, Vship is the total volume of the ship. Equations 8 and 11 are nonlinear in nature because the moments and the forces depend on the ship motion. Linearization or parameterization of these quantities leads to wrong answers when nonlinearity becomes important. In DiSSEL, the wet surface moment and the body restoring moment are evaluated at each time step using:
I + D+R G rotat = − ∫ ∫ ds ( x s − xc ) × np ∑
G restore rotat = ( x wet − x c ) × ρwaterVwet g
(12)
(13)
where xwet is the center of buoyancy and Σ is the wetted surface (the three components of the vector are for roll, pitch, and yaw motions, respectively). Underwater ship geometry and ship motion In a linear parameterized model, the underwater ship geometry is assumed to be unchanged in determining the ship responding motion (Magee and Beck24); in reality, the underwater geometry does change according to the ship’s solid body motion. In DiSSEL, such change is incorporated naturally; its effect on ship motion and on
ship–wave interaction can be identified in the numerical results. In this section, we aim to isolate the impact of time-varying underwater ship geometry on ship motion. It is well known that the response of a solid body to external forcing is strongest when the forcing frequency resonates with the natural frequency of the solid body. The ship natural frequencies in water can be determined in a way similar to that for a pendulum: given an initial ship position in calm water which is away from its equilibrium position, it will oscillate (rotate or move vertically) according to the buoyancy force, i.e., the restoring forces in Eqs. 8 and 11. In numerical simulation, the responses are described by a time series of related quantities (e.g., velocity potential or rotating angles). The natural frequency can be obtained from 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 hulls: the Office of Naval Research (ONR) Tumblehome and ONR 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 hull tilts 10° inward (Fig. 1a) and that of Flared hull tilts 10° outward (Fig. 1b). 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 in Fig. 2. From the figure we can observe clearly that the natural frequency of the Tumblehome hull is lower than that of 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 important for correct modeling of ship motion, in 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 transverse metacentric height (GM)). This naturally affects the ship roll 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 roll motions. As described earlier, in the DiSSEL ship motion model, the pressure moment and the roll motion are updated at every time step. Therefore, it is appropriate for testing the roll damping function. This will be discussed in the following section.
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Bilge keel roll damping function Real and experimental ship hulls often include appendages, such as bilge keels and rudders. These appendages produce roll damping that can reduce ship roll motions by as much as 10%–20%; thus it is important that roll damping be implemented into numerical models (Himeno17). To balance numerical accuracy and efficiency, naval architects have developed many simple, parameterized roll damping models. For example, Himeno17 suggested the following linearized roll damping coefficient: Bφ (θ ) = Beθ
(14)
The roll damping function Bf(q˙ ) is linearly proportional to the rotation rate q˙ with a constant damping coefficient Be. He also suggested a more sophisticated coefficient for simple oscillation with a frequency w : Be = B1 +
8 3 ωθ A B2 + ω 2θ A2 B3 3π 4
(15)
where B1,2,3 are constants and qA is the constant roll motion amplitude that is precalculated by his linear strip theory. Obviously, the damping coefficient Be is constant for a given frequency w and qA. For more details, we refer the reader to Himeno.17 In our earlier approach, the roll damping coefficient was determined by the difference in the two adjacent wave peaks (or valleys) of the free roll oscillation (with a finite initial departing angle) in calm water. An example of such oscillation is shown in Fig. 3. However, this approach, although simple and nonlinear in nature (and thus better than linearized results), has its own limitation: it does not account for possible dependencies of the coefficient on roll oscillation frequency. This could create serious problems when ships are not in calm water. In those more realistic situations, ships roll with a wide spectrum of frequencies. The damping coefficient derived from a single (natural) frequency is simply not sufficient. Even if a more realistic damping coefficient could be measured from the amplitude variation in specified incident waves, this approach is very inefficient: such measurement must be carried out for different ship hulls in different environments. This has prompted us to search for a more realistic approach to modeling the roll damping effect. Instead of reinventing the wheel, we intend to use existing methodologies for our attempt. The approach we take is based on the classical analysis of the blocking effect of a solid object in a flow (Landau and Lifshitz30). This approach has been applied to many fluid systems, e.g.,
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an object in stratified fluid (Freund and Meyer31), the wind past two-dimensional terrain (Garner32), and the current past a seamount (Lin and Thomas37). Bilge keels on ship hulls play a role similar to a seamount in the ocean. This leads us to import the methodology for our problem. However, our problem is unique and is different from the seamount case in that the underwater part of the bilge keels varies in time as the ship moves in water. In 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 mathematical formulations are used to derive the damping function. The price paid is that not all nonlinear damping effects are included. To include the fully nonlinear effects of the appendages, one could use a coupled system, such as a nonlinear ship motion model (e.g., DiSSEL) with a fully nonlinear viscous flow model (e.g., RANS). However, this can be computationally very time consuming, or even impractical. The details of the approach are now explained. In general, the roll damping can be evaluated from the moment arising from the pressure acting on the blocking area A¯block: ( G block )i = − ρ ∫∫
Ablock
( r × n )i pds = D (t )
dθi dt
(16)
where D(t) is a time-varying, nonlinear damping function arising from the bilge keel’s blocking effect (the negative sign implies that it will be deducted from the moments acting on the ship body). However, direct evaluation of Eq. 16 requires correct knowledge of the pressure at the surface, which then depends on the specific geometric properties of the bilge keels and the corresponding dynamic state of the system. While feasible, it can be very demanding numerically because very fine numerical grids are necessary to resolve the small-scale processes. To avoid such difficulties, we borrow the following idea from previous studies of blocking mechanisms: the blocking effect can be included by deducting an effective blocking area Ablock (not A¯block in Eq. 16) from the ship wet surface in Σi the integral in Eq. 16 for the moments on the ship body: I + D+R G rotat =− i
∫∫
∑i − Ablock
d Σ( x s − xc ) × np, i = 4
(17)
Now the problem is to determine the effective blocking area Ablock. In our approach, the effective blocking area is approximated as:
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Ablock ( t ) = Wblock ∗ H bk Lbk (t )
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(18)
N
Ai , A
ω → ∑ω i i =0
where Hbk and Lbk are the width of the span and the underwater length of the bilge keel, respectively. Lbk varies with time because part of the bilge keels can be above water when a ship moves. The underwater length Lbk can be evaluated using:
Lbk (t ) =
J 2( t )
∑ δ l R (t )
j = J1( t )
j
j
(19)
where J1 and J2 are the two time-varying end grid points of the underwater bilge keel, dlj is the length segment at grid point j, and Rj(t) is the underwater percentage of dlj. The time variation of the quantities depends on the ship’s motion and surrounding waves. Wblock is a dynamic factor used to describe other geometric and dynamic effects, such as the effective blocking area width (depending on Hbk and the complexity of flow near the ship boundary). There is no existing result for modeling Wblock. Therefore, we start from physical intuition and the typical approaches of 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: Wblock = c1θ mω
(20)
where qm is the rolling angle magnitude and w is the rolling frequency. On the other hand, general multiscale analysis (Bender and Orszag38) demonstrates that nonlinear effects can be modeled by adding higher-order terms (as the power 2n) to Eq. 20: Wblock = c1(θ mω ) 1 + a1(θ mω )2 + a2 (θ mω )4 + a3(θ mω )6 + . . . K
= ∑ c2 k +1(θ mω )2 k +1
(21)
k =0
up to some given truncation order K. Because roll motions are in general more complicated than a simple oscillation, there is a wide spectrum of oscillating frequencies. These frequencies are often related to those of the external forcing. In our application, the external forcing is provided by incident waves. To properly account for the contributions from individual oscillating modes, we replace the single frequency w in Eq. 21 with a weighted distribution over the incident wave frequency domain:
(22)
where A is the total amplitude of the incident waves, Ai are those of individual modes with the frequencies wi, and N is the number of individual modes of the incident waves. In our analysis, we chose K = 3. Obviously, K can be chosen differently in other applications. The principle is that K should be larger for stronger nonlinearity; however, there is no established theory for selecting an optimal K. The coefficients c2k+1 in the expansion (Eq. 21) depend on the nonlinearity of the physical problem (e.g., the nonlinear equations of the system). The size of the coefficients decreases as k increases. From Eq. 21, we can observe that the coefficient c2k+1 describes the nonlinear effect of mode (qw)2k+1, which is proportional to a2k+1 (a is the typical magnitude of qw). However, we set the coefficients to be normalized in our study, i.e., ∞
∑c k =0
2 k +1
=1
(23)
so that the magnitude variation is only described by qw (similar to the scaling approaches). There are many choices for satisfying the above normalization condition. The simple choice used in this study is: c2 k +1 =
1 , 2 k +1
(24)
So that: ∞
∑c k =0
2 k +1
∞
1 = 1. k +1 2 k =0
=∑
The formulation in Eqs. 18 and 24 of Ablock is based on very simple mathematical and physical considerations: it does not depend on any specific assumption on properties of the ship or the environment. The goal is to develop a damping model applicable to arbitrary ships in arbitrary environments. The coefficients c2k+1 are set constant for the normalization, i.e., the summation of the coefficients is unity. The rolling angle qm in 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 it depends on many characteristics, qm must be determined by solving the coupled ship motion–surface wave system. This is exactly what we have been doing in our numerical
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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-offreedom, ship solid body motions. The details can be summarized as follows: first, an initial state is estimated from the peak of the incident wave (in the ship coordinates). Then the full system (without the bilge keels) is solved from the initial state via numerical time integration. The simulation stops when the system reaches a well-developed, dynamically stable state. The rolling angle amplitude qm is then determined from the stable state. The effective blocking area Ablock is in general larger than the real blocking area A¯block. In our model, the additional area is added to the downstream side of the bilge keels. Since the roll motion is oscillating with time, the downstream is also oscillating with time. Both the size and the location of Ablock are defined by Eq. 21. The expansion given in Eq. 21 is consistent in format with other approaches in modeling solid body motion with damping effects (Kreyszig39). Finally, we want to point out that roll damping is strongly associated with the time-varying underwater geometry. For example, Wblock decreases as the bilge keels rise above the water (an extreme case is that Wblock = 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 roll motions of two ship hulls: the ONR 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 different above-water-line geometries (in calm water), and experimental data is available at the David Taylor Model Basin Laboratory ready for benchmarking our numerical results. We focused on two problems: the impact of the ship hull geometry above the calm water line on roll motion and the blocking effect of the bilge keel. In 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 front of the center of the bow of the moving ship. The incident waves measured were not contaminated by the ship-motiongenerated waves or the ship response motions. Therefore they were used as the initial conditions for the waves in the numerical simulations. Given the initial waves, the initial roll angle can be calculated directly from the constraint that the restoring moment resulting from the roll
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angle completely offsets the driving moment from the initial waves. We considered six different cases for Tumblehome and Flared hulls in regular beam seas, all of which have been studied experimentally in the past. The numerical results were compared with the experimental results and are shown in Figs. 4a–f, which show the time series of the DiSSEL results with roll damping, the experimental results, and the DiSSEL results without the roll damping for different normalized incident wavelengths l (scaled by the ship length, 154 m) and wave steepness, ak. From the figures one can observe clearly that the DiSSEL results with the roll damping agree very well with the experimental data in all cases. The agreement in all six cases suggests strongly that the nonlinear roll damping method described in Eqs. 16–24 is generic, i.e., it is applicable to various ship hulls (Tumblehome and Flared hulls in these cases) in various incident waves (six different waves in this study). It should be pointed out that, as shown in 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. If a regular incident wave has a frequency different from 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 incident wave frequency is to the natural frequency, the stronger the interactions, and thus the more irregular the ship roll motion. In the case shown in Fig. 4f, the two frequencies are so different that the interactions are weak. Therefore the roll 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 roll damping. This effect is more significant when the nonlinear interactions are stronger. In addition, it also depends on incident wavelength and amplitude, as shown in Eq. 21. To compare the mean properties of the numerical results and the experimental data over the entire time period, the typical amplitudes of the all cases were estimated by the root mean square (rms) of the peak amplitudes in the roll angle time series. The results for Hbk = 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 with the experimental data. From Fig. 5 we also note that the ONR Tumblehome hull roll angles are much larger than those of the ONR
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Flared hull near resonance. The difference is largest when l 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 in the ship hull geometry. To better understand its physics, we consider a simple free ship roll motion: the ship is set initially at a small roll angle q0 away from its equilibrium position. It will then oscillate freely with its own natural frequency wnature, governed by the following simplified equation: 2 I11, ω nature θ = mship gRrollθ
(25)
where Rroll is the restoring rolling arms of the ship hull. Therefore, Rroll =
I11, mship g
2 ω nature
(26)
4.5 4 Roll Angle/Ka
3.5 3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2
2.5
3
3.5
4
λ
Fig. 5. Normalized roll motion angle for different values of l with Fn = 0.066. The solid line shows the numerical results for the Tumblehome hull and the dashed line the numerical results for the Flared hull. The experimental data for the two hulls are shown as squares and circles, respectively. The measurement error is 0.05, which is too small to be plotted (i.e., it smaller than symbols representing each experimental data point). Ka, wave steepress
In other words, given the same moment of inertia I1,1, larger natural frequencies lead to a longer restoring arm. Since, from the experimental data, the inertial moment I1,1 and the total ship mass mship, are the same for the Tumblehome and the Flared hulls, and since the Flared hull has a larger natural frequency (the normalized frequency of the Flared hull is 2.8, while that of the Tumblehome is 1.8, see Fig. 2), the Flared hull has a longer restoring arm Rroll 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 in 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 ϕ) with respect to the (discrete) expansion wave numbers (kx, ky) in the (x, y) axes. To show the convergence, we plot in Fig. 6 the distributions of the spectral coefficients of the velocity potential ϕ (for the solution in Fig. 4c at t = 23 s) in the (kx, ky) spectral space. Figure 6a shows the distributions of ⎜ϕ ⎜ in ky for different kx, while in Fig. 6b are the distributions in kx for different ky. In the figures one can see that the spectral coefficient decreases by more than 10 orders of magnitude from small wave numbers (kx, ky) to the maximum wave numbers in the expansion. For example, the coefficient for the wave numbers (kx, ky) = (0.0164, 0.0164) is 12 orders of magnitude greater than that for (kx, ky) = (1.0308, 1.0308). The decrease is monotonic, except at very large values of kx, but in 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.
2 0 -1.5
-1
-0.5
-2
0
-4
0.5
1
1.5
2
2 Kx=0.0164 Kx=0.3436 Kx=0.6709 Kx=1.0308
-6 -8 -10
0.5
1
-4 -6 -8 -10
-12
-12
1.5
Ky=0.0164 Ky=0.3436 Ky=0.6709 Ky=1.0308 Ky=-0.0164 Ky=-0.3436 Ky=-0.6709 Ky=-1.0308
-14
-14
a
0 -2 0 log |phi|
log |phi|
-2
Ky
Fig. 6. The distribution of log10 ⎜ϕ ⎢ (ϕ is the velocity potential) in the spectral space (kx, ky) for the solution at t = 23 s in Fig. 4c. The distributions in the wave number ky for several wave numbers kx
b
Kx
are shown in a and the distributions in kx for several ky are shown in b
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136
Discussion In this article we discussed our approach to modeling the dissipative effect of bilge keels on ship roll motions. Our approach was based on the blocking theory of fluid mechanics, which describes the resistance of a solid object in a flow. Instead of deriving a damping function directly from the basic equations, we employed a multiscale approach with basic physical intuition to formulate the damping function. Thus our approach is quasianalytical in nature. We implemented the new damping function in the DiSSEL ship motion model and tested it numerically with ONR Tumblehome and Flared hulls for different incident waves. Our numerical results agree well with those from 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 interaction, the more significant the damping effect will be. This damping effect depends also on incident waves and ship hull forms, as shown in Figs. 4 and 5. We also demonstrated that the ship geometry above the calm water line is important in correctly modeling ship roll motion, partly because this part of the geometry is critical in determining the natural frequency of the ship, as shown in Fig. 2. Its importance in determining the forces on ship bodies has been reported by the authors elsewhere (Lin and Kuang40). It 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. In the traditional approach (Eqs. 14, 15), an empirical coefficient Be is directly introduced, and a linear damping term Be dq/dt is added directly into the dynamic equation governing the ship roll motion. This empirical coefficient does include partial nonlinear effects, e.g., as a nonlinear function of roll angular velocity. But this approach has an intrinsic deficiency in that modeling can only be carried out if 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. It cannot be used for situations different from the tested cases, and thus cannot be used for ship design. In this study, the blocking moment Gblock is added to the system. This approach is more generic. Not only can the nonlinear effect be well modeled by the moment Gblock, the algorithm itself does not depend on the model environment (e.g., ship hull geometries, incident waves, and ship speed). However, the moment Gblock depends on the ship hull geometry (via the pressure moment integrated over the blocking area), the incident waves, ship
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J Mar Sci Technol (2008) 13:127–137
motions (in very complex manners), the lift force, and kinematic dissipations; we can expect from Eq. 14 that the coefficient Be (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 for ship design. As pointed out, Eqs. 23 and 24 represent only one of many choices for the coefficients c2i+1. Different definitions could be investigated for better modeling of the roll damping effect. Therefore, it would be very interesting to examine the sensitivity of the damping coefficient to the model environment, and the implications of such sensitivity to the numerical model results. Acknowledgments. R. Lin’s work is supported by grants from the David Taylor Model Basin, Carderock Division, Naval Surface Warfare Center Independent Laboratory In-House Research (ILIR) program administered by Dr. John Barkyoumb. W. Kuang is supported by the NASA Earth Surface and Interior program, the NASA Mars Fundamental Research program, and by the NSF Math/Geophysics program. We thank Mr. Mike Davis, Mr. David Warden, and Mr. Terry Applebee for many useful comments and suggestions.
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