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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 863
On Roll Hydrodynamics of Cylinders Fitted with Bilge Keels
R.W.Yeung1, D.Roddier1, B.Alessandrini2*, L.Gentaz2, and S.-W.Liao1
(1University of California at Berkeley, USA, 2Ecole Centrale de Nantes, France)
ABSTRACT
Bilge keels are commonly used appendages that are effective in reducing roll motion. The complex fluid flow around
such sharp-fin objects in the presence of hull geometry and a free surface has hardly been addressed. Two recently
developed Navier-Stokes Solvers are utilized to examine the forced-motion hydrodynamic properties of rectangular
cylinders fitted with bilge keels. One solver is the Free-Surface Random Vortex Method (FSRVM, Yeung &
Vaidhyanathan, 1994) developed at the University of California at Berkeley, while the other is a Boundary-Fitted Finite-
Difference Method (BFFDM, Alessandrini & Delhommeau, 1995) developed at the Ecole Centrale de Nantes. Cross-
continents collaborations provided the opportunity for the authors to evaluate the consistency of these two different but
equally successful methods. Where available, these theoretical calculations are validated against laboratory measurements
taken from forced-oscillation experiments. The capabilities of such first-principle approaches to predict the added roll
inertia and the roll damping are shown to be excellent. The shortcomings of traditional estimates of roll hydrodynamic
coefficients based on potential-flow calculations are illustrated. The dependence of these coefficients on the parametric
space of frequency, bilge-keel depths, and amplitude of roll is examined and discussed.
KEY WORDS: Bilge keels, roll motion, viscous damping, vorticity, experimental hydrodynamics, vortex methods,
Navier-Stokes equations.
1 INTRODUCTION
Predictions of the roll motion of ships have traditionally relied on the empirical estimates of relevant hydrodynamic
coefficients (Himeno, 1981) since viscosity effects cannot be easily incorporated analytically. The presence of stabilizing
appendages such as bilge keels have made the motion prediction even more difficult (Miller et al., 1974, Cox and Lloyd,
1977). Martin (1958) and Ridjanovic (1962) were among the first to obtain experimental data for a flat plate in oscillating
flows. Even though the mitigation effect of bilge keels on ship motion has been known from the time of William Froude,
who proposed the usage of “bilge pieces” in 1865 and later measured its resistance, very little advances, based on fluid-
mechanics first principles have taken place. The modeling of unsteady viscous forces by sharp-edge fins have been a
difficult one.
Amid the long history of the subject on roll motion, we will mention a few recent references which is not an
exhaustive list. Robinson and Stoddart (1987) and Standing et al. (1992) discussed the effects of damping on a barge's roll
response. Using a single vortex method, Faltinsen and Sortland (1987) investigated the eddy-making damping in slow-
drift motions and showed the importance of bilge-keel depth, especially at low KC (Keulegan-Carpenter) numbers.
Cermelli (1995) observed a very distinctive asymmetric flow pattern associated with oscillating a flat plate near a free
surface. He meticulously investigated the vortical structure of the flow around the tip of the plate using DPIV (Digital
Particle Image Velocimetry) and compared them with the FSRVM (Free-surface Random Vortex Method)
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*Visiting Scholar, University of California at Berkeley

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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 864
of Yeung and Vaidhyanathan (1994). Sarpkaya and O'Keefe (1996) provided related, and similarly interesting features of
the oscillating flows about bilge keels.
Roll damping of ships is not strictly related only to flow separation around the bilges and keels. Of equal importance
is that bilge keels, if present, also generate lift like a low-aspect wing because of the forward speed of the hull. It is
primarily the first aspect that we address in this paper, even though the latter aspect could be effectively modeled by the
first without a full-fledge three-dimensional solution.
An extensive amount of literature exists in motion predictions by inviscid-fluid theories. It is generally assumed that
inviscid-fluid ship-motion theory (Wehausen, 1971) can be largely improved by including viscous effects as ad-hoc
hydrodynamic coefficients. On the other hand there are good reasons to believe that viscous effects could be fully coupled
to the inviscid-fluid motion. The nature of this inviscid and viscous coupling was first examined by (Yeung and Wu,
1991) using a sophisticated Green function, followed by Yeung and Ananthakrishnan (1992), and Gentaz et al. (1997),
using boundary-fitted coordinates methods. A complete understanding of this subject is yet to be achieved. Similar to the
work of Fink and Soh (1974), Braathen (1987) used a vortex tracking method to predict the roll damping of a two-
dimensional body. Korpus and Falzarano (1997) studied the viscous damping caused by bilge keels using a finite analytic
technique to discretize the unsteady Reynolds-averaged Navier-Stokes equations. They used a -ε turbulence model, but
did not consider the effect of the free-surface. Yeung et al. (1996) examined the flow about a rolling plate using FSRVM
and validated their results against their DPIV experiments. Very recently, Yeung et al. (1998) applied the FSRVM to a
rectangular ship-like section oscillating in roll motion. These results were substantiated with new experimental data,
which revealed some discrepancies with the classical measurements reported by Vugts (1968), namely, in the diagonal
terms of the added moment of inertia matrix.
In this paper, bilge keels of various depths are fitted to the corners of a rectangular cylinder (See Fig. 1). The effects
of viscosity, vortex shedding, and the associated fluid inertia and damping will be investigated systematically. Numerical
results based on two rational methods of very different formulations: the FSRVM and a boundary-fitted finite-difference
method (BFFDM), are compared with new experimental measurements. The validity of solving the roll hydrodynamics
problem from first principles can thus be ascertained.
2 THEORETICAL MODELS
2.1 The Free Surface Random Vortex Method
The Free-Surface Random-Vortex Method (FSRVM), as introduced first by Yeung and Vaidhyanathan (1994), has
been successfully validated for the relatively simple geometry of a plate and for a rectangular cylinder (Yeung and
Cermelli, 1998, Yeung et al., 1998). The theoretical formulation allows for arbitrary body shapes. It is therefore a
relatively straight-forward extension to apply the same algorithm to the more complex problem of a rectangular cylinder
fitted with keels around the bilge corners. As this method is relatively new, a brief exposition of the is given here, while
more details are available from Vaidhyanathan (1993) and the three references cited above.
Figure 2 shows a schematic of a floating rectangular cylinder undergoing prescribed roll motions. In indicial
notations, the Cartesian coordinates are designated by (x=x1, y=x2) with the instantaneous angular position of the cylinder
∂ Db denoted by α(t), measured positive counterclockwise. The velocity field is given by (u1, u2). This grid-free method of
FSRVM solves the flow field by decomposing it into irrotational and vortical parts. The irrotational part is solved using a
complex-valued boundary-integral method, utilizing Cauchy's integral theorem for a region bounded by the body, the free-
surface, and the open boundary. The rotational part is described by the vorticity equation for ζ3, where is the
vorticity in the direction normal to the two-dimensional space, and εjkl is the alternating tensor. The vorticity equation is
solved using a fractional step method.
In two-dimensional flow, the governing equations for an incompressible, viscous fluid based on a vor
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 865
ticity and stream-function formulation are given by:
Figure 1: Geometry of cylinders with bilge keels.
Figure 2: Computational domain and definitions for FSRVM.
(1)
(2)
where Dt is the material derivative and v the kinematic viscosity. The velocity components of the flow are given in
terms of ψ by (u1, u2)=(ψ,2,−ψ,1).
In the context of a RVM, the vorticity field is assumed to be a collection of discrete vortices of finite core size
(Chorin, 1973), whose motion is governed successively by a convection step and a diffusion step.
In the first ‘half step', the convection equation of ζ3 is obtained by dropping the right-hand side of Eqn. (2)
(3)
Since vorticity lines are material lines in a two-dimensional flow and the vorticity field is assumed to be an
aggregation of vortices, Eqn. (3) implies that the blobs are convected with the flow velocity at their locations. Thus the
location of the ith vortex blob is given by
(4)
In the second ‘half step', to simulate solution of the diffusion equation,
(5)
every blob is given a “random walk” with a Gaussian distribution of zero mean and standard deviation. The
diffusion step computations are simple whereas the convection-step computations are time consuming. This latter
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difficulty is overcome by an O(N) algorithm (Yeung et al., 1996). The diffusion process is assumed to be sufficiently
local so that effects of free surface is negligible.
To compute the stream function, we let ψ be written as the sum of a vortical part and a homogeneous part,
(6)
where
(7)
ψv is known if the position and strength of each vortex blob is known. For instance, for N blobs, each of circulation Γi,

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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 866
(8)
for any point xj outside of the core of the blob.
It can be seen that only ψh needs to be solved, which must consider all of the conditions on the fluid boundary ∂ D. A
complex potential formulation similar to Grosenbaugh and Yeung (1989) (see also Vinje and Brevig, 1980) is used. Let
the complex potential be defined by where is the (conjugate) velocity potential associated with ψh and
z=x1+ix2. An outline of the solution method for βh follows.
1. At any given time t, Cauchy's integral theorem is applied so that either or ψh is solved on the fluid
boundary when its conjugate part is specified:
(9)
On the body boundary, the no-leak condition can be shown to yield:
(10)
where is the angular velocity of roll.
On the free surface ∂ DF, the kinematic boundary condition for the complex velocity w= u1−iu2 is used to
advance the location of the free surface, while the dynamic condition is used to advance
(11)
(12)
where indicates complex conjugate. The damping function vd in Eqns. (11) and (12) is zero except in the
damping layers −Lx1≥xr on the left and right ends of the free surface, and zo is the initial
location of the lead free-surface node of the layers at t=0 (Israeli and Orszag, 1981, Cointe et al. 1991, Liao
and Roddier, 1998). Because of the use of damping layers, the no-disturbance condition can be used on the
open boundary ∂DΣ,
(13)
2. After βh is solved, the “no-slip” boundary condition on ∂Db needs to be satisfied since βh generates a relative
tangential velocity. Vortices are generated to satisfy the no-slip boundary condition. On ith body panel, the
strength of the vortex is
(14)
where is the relative velocity between the body and fluid, τi is the tangential unit vector and si is the
panel length. Then, the vortex is convected and diffused into the fluid domain according to Eqns. (4) and (5).
3. In order to obtain the forces and moment on the body, we need to solve for βh,t, since is needed in Euler's
integral to obtain the surface pressure. An integral equation similar to Eqn. (9) can be set up in parallel with
are given by Eqns. (12) and (13). On ∂Db,
βh,t. The boundary conditions on ∂ Df and ∂Do of
(15)
4. The new boundary conditions are now available for the next step. Step 1 to step 3 can be repeated till the end
of simulation.
Note that in the absence of ψv, the flow is entirely irrotational. Thus, a fully nonlinear inviscid solution can be
recovered by FSRVM by shutting off the vorticity generation process, with few changes in the solution procedure.
2.2 The Boundary-Fitted Finite-Difference Method
Boundary-fitted coordinate methods for free surface flows have been investigated by Shanks and Thompson (1977)
for a hydrofoil, by Yeung and Ananthakrishnan (1992) for heaving two-dimensional bodies with elaborate grid-control
algorithms of Steinberg and Roache (1986) that were based on variational principles. This present method based on the
RANS equations was developed by Alessandrini and Delhommeau (1995), which has been successfully validated in both
two and three dimensional geometry.
The convective form of Reynolds Averaged Navier-Stokes Equations are written through partial transformation from
Cartesian space (x1, x2) to
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 867
curvilinear space (ξ1, ξ2) fitted to the body and the free surface at each time. Free surface elevation, η(x1, t), the two
Cartesian velocity components (ui), the dynamic pressure (p) including gravitational effects (ρgx2), and turbulent kinetic
energy are the normal dependent unknowns.
Mass conservation is expressed as the classical continuity equation:
(16)
where ai is the contravariant basis. The mean momentum transport equations, written in the moving reference system
attached to the hull are:
(17)
for α ∈ {1, 2}. gij is the contravariant metric tensor, fi the control grid function, the grid velocity which indicates
the displacement of the mesh. veff is the effective viscosity obtained by adding the physical kinematic viscosity to the
turbulence viscosity. Inertia forces due to non-galilean reference (gyration, accelerated translation) are taking into account
in the qi terms.
If a turbulence closure model is adopted, the turbulence term vt needs to be solved. For example, one might use the
classical k-ω model of Wilcox (1988). The inclusion of turbulence effects is being pursued. Here, we only consider the
special case of veff=v, with vt simply taken as zero.
Free-surface boundary conditions are one kinematic condition, two tangential dynamic conditions and one normal
dynamic condition. The kinematic condition, coming from the continuity hypothesis, expresses that the fluid particles of
free surface stay on it:
(18)
bi
where are the bi-dimensional contravariant basis.
Dynamic conditions are given by the continuity of stresses at the free surface. If the pressure is assumed to be
constant above free surface, normal dynamic condition is:
(19)
a1 ǁ 2
where ǁ is the norm of the contraviant vector normal to the surface, and γ is the surface tension coefficient and
R the free-surface medium curvature radius (Ananthakrishnan & Yeung, 1994). Tangential dynamic stress conditions are
simply given by a linear combination of first order velocities derivatives:
Figure 3: Numerical grid for the BFFDM at t/T=4–1/6, 4% keels.
(20)
where a2i is a covariant vector tangential to the free surface.
At each iteration, governing equations and exact nonlinear free surface boundary conditions are discretized in space
and time by second-order finite-difference schemes on a monoblock structured grid that is fitted to the hull and the free
surface. The algorithm of solution involves solving the discrete system of unknowns for the velocities, pressure and free
surface in a fully coupled manner. Transport equations are first linearized and then discretized at each grid node i with
second-order non-centered (unsteady and convection terms) or centered (diffusion term) finite difference schemes. The
pressure gradient is interpolated at grid nodes since the discrete pressure unknown P are defined at the center of the cells.
In order to decrease memory requirements, unknown pseudo-velocities U* are introduced. So, the transport equations for
discrete velocity U at grid node i can be written as:
(21)
Meanwhile, the kinematic boundary condition and the tangential and normal stress conditions on the
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 868
free surface can be manipulated into the following discrete form in time:
(22)
The use of the discrete normal dynamic condition can lead to the following final form of free surface boundary
conditions or velocities:
(23)
where the Mik and f are quantities defined in Alessandrini and Delhommeau (1995). The “tilde” notation indicates
that interpolation to the grid points from cell center is necessary. The pressure equation, in matrix form and utilizing the
pseudo-velocity, is solved by using the Rhie and Chow method (1983) so as to avoid checkerboard oscillations.
(24)
where Πpv, Πpp, and Φp result from the left-hand and right-hand sides of Eqns. (22) and (23), respectively. This full
set of equations is solved at each iteration in a coupled manner using the BI-CGSTAB algorithm of Van Der Vorst
(1992), conditioned with an incomplete LU decomposition.
Figure 3 illustrates the Ø—type mesh fitted around the cylinder at one instant of time.
2.3 Forces and moment
The hydrodynamic forces and moment acting on the body can be calculated according to
(25)
(26)
(27)
(28)
In the case of the FSRVM, the only stress on the body is the pressure which is given by Euler's integral:
(29)
In Yeung and Ananthakrishnan (1992), the shear stress was found to be of secondary importance in strongly
separated flow, the pressure is the dominant dynamic stress contributing to the total fluid force and moment. However,
BFFDM can provide the shear stress as part of the solution procedure, and both components of the stress are used to
compute the force and moment integrals (Eqns. 25–27). The hydrodynamic coefficients will be directly obtained from
these time histories as explained further in Section 3.2.
3 EXPERIMENTAL SETUP AND HYDRODYNAMIC COEFFICIENTS
The experiments were conducted at the Richmond Ship-Model Testing Facility of the University of California at
Berkeley. A 2.54×0.3×0.3 m rectangular acrylic cylinder as shown in Fig. 4 was hinged at the water level by the sides of
the tank. Bilge keels of 1.25 and 1.90 cm were mounted on the bilge corner of the cylinder at a 45° angle. The model
weighted 100 kg and displaced 0.11 m3. The rolling motion was induced by a hydraulic cylinder that can accept a random-
motion input (Random Motion Mechanism, Hodges and Webster, 1986). The cylinder was oscillated from 2.5 to 10 Hz.
Further details on the apparatus are explained in Yeung et al. (1998). In “dry conditions”, the model was oscillated to
obtain its mass moment of inertia (natural period=1.424 secs). The center of gravity was measured by an inclining
experiment. Force transducers placed at the end of the piston rod and just above the hinges provide an analog signal to a
computer. The transducers were calibrated statically. Measured forces and moment were obtained after filtering
instrument noises and vibrations,
3.1 Measured forces and moment
Figure 5 shows a free-body diagram of the system that allows one to analyse the applied forces. Let W be the weight
of the cylinder, m the mass, I0 the mass moment of inertia about O, G the center of gravity at (xG, yG) with |OG|=rG and |
OA|=rA. Let also Xw, Yw and Mw be the forces and moment exerted by the fluid on the cylinder, and X′A and Y′A the forces
exerted by the horizontal driving rod on the cylinder, The subscript o refers to quantities
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 869
measured at the roller bearing O, and the X′ and Y′ refers to the forces in Ax′y′ coordinate system as shown in Fig. 5. Force
transducers are adjusted to zero and calibrated when the cylinder is installed on the hinges and in the upright position.
Their outputs are zeroed when no motion is present even though there may be static loads. When the cylinder is in
motion, the transducer respoonse are proportional to the additional forces present. Therefore, the total force applied to the
cylinder at a hinge can be expressed as
Figure 4: Experimental apparatus and setup.
Figure 5: Free body diagram of the cylinder.
(30)
where F can be either X or Y and the second term ‘m' is proportional to the output voltage of the transducer.
By making use of Eqn. (30) and noting that the F0's are the static terms computable from the geometric properties of
the cylinder, we can arrive at the following equations for the hydrodynamic forces and moment with buoyancy effects
removed:
(31)
(32)
(33)
where θ is the angle about O between the center of gravity G and the vertical direction when the cylinder is at rest,
i.e. θ=|atan(xG/yG)|. Wrod is the weight of the driving rod, ∀ the submerged section area, with the last term of (33)
representing buoyancy effects. We note that even though the measured force carries the dynamic information of the fluid
inertia term, it can be shadowed by the hydrostatics terms unless the body weights cancels it. This can be done by making
the cylinder having a zero metacentric height.
By Fourier decomposing the measured time series, hydrodynamic coefficients are obtained as shown in Section 3.2.
We note in passing that the roll motion of amplitude αo is started smoothly by using a hyperbolic-tangent ramped function
of the form
(34)
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This is also the case in the numerical methods.
3.2 Hydrodynamic coefficients
In linear, potential-flow theory, the hydrodynamic forces and moment are usually written as the sum of inertia and
linear damping terms (Wehausen, 1971):
(35)
(36)
where Xw and Mw are the hydrodynamic sway forces and roll moment as obtained from both the numerical and
experimental simulations. Since the forced roll motion is of the form α(t)=α0 sin(ωt), the linear hydrodynamic
coefficients at t=t0 can be calculated by extracting the Fourier coefficient of the

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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 870
primary frequency over a period T. Hydrodynamic coefficients of the sway force induced by roll motion can be shown to
be:
(37)
(38)
while the diagonal terms are:
(39)
(40)
Earlier, we have completed some experiments for a model without bilge keels Yeung et al. (1998) and compared
some with the reported results of Vugts (1968). Here, for reason of consistency and ease of interpretations, we define the
same normalized coefficients as before:
(41)
(42)
Because the flow is often governed by viscous separation, a nonlinear formulation for the damping could also be
expressed as:
(43)
where λ′66 is the quadratic damping coefficient and is a traditional way of representing the overall flow effects.
However, if the process is assumed to be sinusoidal, equivalent energy dissipation rate would imply the following formula
for λ′66:
(44)
Thus, under the stated assumption, Equation (44) is related to Eqn. (40) by
(45)
The damping coefficient presented in Section 4 will follow Eqn. (40) and the nonlinear coefficient can be deduced
accordingly. However, if both linear and nonlinear damping are present, the uniqueness of each of these coefficients can
be determined only with more information. A formulation for this case can be found in Yeung et al. (1998).
4 RESULTS AND DISCUSSIONS
For a given cylinder geometry and with the assumption of prescribed periodic motion, the problem is characterized
by the roll amplitude αo and the following nondimensional parameters:
(46)
where is the percentage of bilge-keel depth relative to the beam and Re is a characteristic Reynolds number
similar to Yeung et al. (1996). In the FSRVM, the Reynolds number is taken the same as the physical one in the
laboratory, which ranges from 16,000 to 144,000. Typically, the number of vortex blobs used is of O(50,000) though a
larger number can be accommodated. In the BFFDM, the grid size is adapted to the medium Re to avoid remeshing and
consists of 200×200 nodes.
4.1 Inviscid-fluid results
Viscous-flow solutions are emerging because of the increasing power of computers. Inviscid results are still very
useful in providing not only a guideline but also a validation for new numerical methods. The problem of a rectangular
section in roll motion has long been solved by a variety of methods. Yeung (1975) used a Hybrid Integral-Equation
Method, which is very powerful for the case of arbitrary geometry. Boundary-element methods based on various Green
functions is also a possible way for solution. An advantage of the FSRVM is that it yields the inviscid (potential-flow)
solution when the vortical part of the stream function corresponding to Eqn. (8) is not “turned on.” In Figs. 6 to 9, the
inviscid-fluid hydrodynamic coefficients are presented. The solution for a rectangular cross section (with a small bilge
radius, see Fig. 1) is included for reference purposes. These results can be considered as linear results, being independent
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of the roll amplitude. In the actual FSRVM computations, αo is taken as 2.85 degrees. Even though the potential-flow
problems solved were nonlinear, the inviscid calculations were independent of the amplitude of roll for angles less than
10 degrees.
The FSRVM inviscid-fluid results for the “No Keels” case are obtained using a rounded corner similar to the
experimental model where the corner radius was 2% of the beam. Note that the FSRVM for a square-cylinder yielded the
established

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solution.
Figure 7: Added mass of roll into sway, inviscid-fluid
Figure 6: Added moment of inertia, inviscid-fluid solution.
ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS
solution.
fluid solution.
Figure 8: Damping coefficient of roll, inviscid-fluid
Figure 9: Damping coefficient of roll into sway, inviscid-
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 872
solution of Yeung (1975) abd was given in Yeung et al. (1998). As expected, the effect of the keels is to increase the
added moment of inertia and the radiation damping considerably. The location of the peak values, around for the
added inertia and for the damping, remain the same as the cylinder without keels.
4.2 Theory versus experiments— a validation
In Figs. 10 and 11, typical time histories of the hydrodynamic moment at the condition of are shown for both
numerical methods described in Sec. 2 and the experimental measurements. The lower figure presents the average values
of all three set of time series over three or more periods from t/T=0.5 onwards. The agreement is seen to be very good
between FSRVM and BFFDM, both solutions overpredicting slightly the experimental results. Because the phase
difference is minimal between all three curves, we expect the added mass and damping coefficient predicted to be similar,
as will be seen later.
A more detailed comparison of the two methods of solution is shown in Fig. 12. The variation of the local pressure
on the body contour is shown, in the fourth period, as functions of the arc-length parameter s/b measured in a counter-
clockwise direction and with the zero reference point taken at the centerline. Each subfigure corresponds to an instant of
time successively T/6 apart, with the cylinder rotating clockwise. The tips of the left and right bilge keels show up as s/b
approximately equal to −1 and +1, respectively. Results from both numerical methods agree remarkably consistently,
particularly in view of the drastic difference in formulations. Further, since the boundary condition on the free surface are
treated differently, most of the disagreement is expected to occur at values of s/b near the waterline. The pressure
difference between both faces of the keel lead to a jump in the pressure from the side to the bottom of the cylinder. The
presence of the keel is therefore a major contributor to the hydrodynamic roll moment. It should be noted that during the
second half-period which is not shown in Fig. 12, information indentical to those presented in the first half is expected
since there is quasi-symmetry of the solution about the T/2 point. The pressure distribution will have a mirror image about
the point s/b=0 to the first half-period.
The time series for the sway force and roll moment in this study are very similar to the ones presented in Yeung et
al. (1998), except for an increase in amplitude as the keel depth or the amplitude of motion increases. It is more useful,
from the practical standpoint, to present the Fourier averaged values of the force and moment in the form of
hydrodynamic coefficients as defined in Eqns. (37) to (40). In Figs. 13 to 16, the roll added moment of inertia and roll
damping, as obtained experimentally and numerically using FSRVM, are presented for bilge keels whose depths are 4%
and 6% of the beam of the cylinder, and for two amplitudes of roll angle. Results for the rounded-corners geometry can be
found in Yeung et al. (1998). The diagonal terms of the coefficients Figs. 13–16 are grouped separately from the coupling
terms, Figs. 17–20. Numerical results obtained using BFFDM are added to Figs. 13 and 15, when and αo=2.85
for comparison purposes. It is gratifying to see these two very different and independent methods yield very predictions
close to each other. These predictions also tend to agree well, for the most part, with the experimental results. Generally
speaking, FSRVM and BFFDM do agree better in the diagonal terms than the coupling coefficients. In the case of
comparisons with experiments, one should keep in mind that the reliability of the experimental measurements decreases
with an increase in frequency because of vibration of the test apparatus.
Detail examination of this extensive set of data will suggest the following trends.
• An increase in keel depth (4% to 6%) would increase both the added inertia and damping for the entire range of
frequency. This is expected intuitively and from the computations. The inertia measurements indicate otherwise
even though the change is not seen as substantial.
• With bilge-keels size fixed, experiments and real-fluid theory suggest that an increase in roll amplitude lead to a
decrease in the inertia coefficients for both diagonal and off-diagonal terms, or at least up to the largest angle of
5.75° investigated here. Larger roll amplitude yields an appreciably larger damping coefficients.
• The agreement between theory and experi
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history,
Figure 10:
Moment
of roll hydrodynamic moments.
Figure 11: Steady-state averages
Mw(t)
ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS
body
solutions of two methods.
for numerical
Figure 12: Local pressure on the
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 874
ments is better for the diagonal terms than for the coupling terms. Larger discrepancies between theories and
experiments exist in the high-frequency regime, in which experimental measurements are not as reliable.
• Traditional computations based on inviscid-fluid theory significantly over-estimate the inertia coefficients, and
under-estimate the damping effects. Similar observations in this regard were reported in Yeung et al. (1998) for
the same cylinder without any keels.
• The inertia coefficients approach the inviscid-fluid theory at high frequency. This seems consistent with an
analytical viscous-fluid theory by Yeung and Wu (1991), which modeled only diffusion effects.
• In the high frequency range, both experiments and theory are suggesting that the damping are not vanishingly
small as normally expected from inviscid-fluid computations. This can have important design implications.
4.3 Vorticity and flow patterns
Figure 21 details the flow patterns for a period of one oscillation after a steady state is reached. Strong vortices are
created in the vicinity of the aft edge of the bilge keel and are later shed away as the keel moves in the opposite direction.
During every half period, the vortices from the fore keel have a strong tendency to move away sideways1, whereas those
from the aft keel would move downward. The phenomenon is not entirely mirror in image for the keel during each half of
one full period. This slight lack of symmetry was explained by Yeung and Cermelli (1998) as attributable to memory
effects of the starting swing.
In Fig. 22, the vorticity fields obtained using both numerical models are displayed as color-contour plots. The vortex
method predicts larger and more distinct vortices, as well as larger distances from the body once they become separated
from the keel. A stronger dissipation appears to have taken place in the BFFDM. As in most finite-differencing schemes,
numerical (artificial) damping is always present. The grid-free method of FSRVM does not contain such dissipation, thus
preserving the details better. However, FSRVM does not model the boundary layer flow as well as BFFDM.
5 CONCLUSIONS
Experimental and theoretical studies of the forced roll-motion hydrodynamics of a cylinder with bilge keels are
conducted. The theoretical model includes the use of two free-surface Navier-Stokes Solvers (FSRVM & BFFDM).
Overall speaking, the results obtained using the FSRVM method are well validated by the experimental results. In
cases where both numerical solutions are available, very acceptable agreement is observed, at both global and detailed
level. This lends much credence to the present methodologies of predictions.
Added moment of inertia and (equivalently linearized) damping coefficients are presented as functions of frequency
and bilge-keel depth. The computational and experimental results are compared also with those from classical methods
based on an ideal fluid. The computations suggest that an increase in keel depth will increase both the added inertia and
damping. Traditional computations based on potential-flow theory will significantly over-estimate the added inertia
coefficients and under-estimate the damping effects. Real-fluid effects tend to “round” the shape of the body, thus
reducing the effective inertia, whereas flow separation around the keels increases the damping. Similar observations were
given by Yeung et al. (1998) for a cylinder without keels. Details of such flows are easily visualizable from the solutions
presented.
For a given bilge-keel size, an increase in roll amplitude reduces the nondimensional inertia coefficients slightly but
increases the damping coefficients appreciably. The observation applies to both diagonal and off-diagonal terms. Being
investigated is the amplitude dependence at angles larger than the maximum value of 5.75° studied here.
The complete treatment of the free motion of a floating cylinder in waves, with the full effects of viscosity included,
can be found in Roddier et al. (2000). Effects of turbulence on the present solutions are being examined.
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1The fore keel is deined as the more forward keel in the direction of roll motion. Every half period, the fore keel becomes the aft
keel, and vice-versa.

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Figure 14: Added moment of inertia coefficient, αo=5.75°.
Figure 13: Added moment of inertia coefficient, αo=2.85°.
Figure 16: Equivalent linear damping coefficient of roll, αo=5.75°.
Figure 15: Equivalent linear damping coefficient of roll, αo=2.85°.
ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS
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Figure 20: Roll-sway coupled damping coefficient, αo=5.75°.
Figure 19: Roll-sway coupled damping coefficient, αo=2.85°.
Figure 18: Roll-sway coupled added inertia coefficient, ao=5.75°.
Figure 17: Roll-sway coupled added inertia coefficient, αo=2.85°.
ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS
Figure 21: Flow visualization: velocity vectors of the vortex-blob field by FSRVM.
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based on a non-dimensional vorticity defined by
ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS
Figure 22: Vorticity contours for a rolling cylinder with 4% bilge keels at one instant of time, αo=5.75°. The contour scale is
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ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 879
ACKNOWLEDGEMENT
This research reported has been supported in part by the Office of Naval Research and a Shell Foundation Grant of
the US authors and by the French Ministere de la Recherche and the French Ministere de la Defense (DGA) of the French
collaborators. We are appreciative of the Ecole Centrale de Nantes and the University of California at Berkeley for
promoting the collaborative environments.
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DISCUSSION
A.Clement
Laboratoire Mècanique des Fluides, Ecole
Centrale de Lyon, France
I may be wrong, but it seems to me, looking at moment curve results and at the animated video, that the global
quantitive like forces and moments are periodically established long before the vortical flow around the keel is itself
periodic. Moreover, one may wonder if the vortical flow ever reaches a periodical steady state? Could you comment
about that?
AUTHORS' REPLY
The amount of time that a particular solution takes to attain a periodical steady-state behavior depends on the
frequency of excitation. Lower frequency excitation generally takes more time to achieve steady state. Figure 10 of the
paper indicates that it took about 4 to 5 periods to attain a steady behavior for the frequency in question.
It is worthwhile to note that while the roll moment curve might not appear to vary significantly in amplitude during
the “transient state”, the phasing of the moment relative the to roll angle continue to adjust itself as more vortices are shed
off the keels. This phasing naturally determines the damping and added moment of inertia.
DISCUSSION
M.Kashiwagi
Kyushu University, Japan
In the classical potential theory, the conservation principles can be used to check the validity of numerical results
without resorting to comparison with experiments. I wonder if the same kind of principles can be used to validate the
numerical results in this paper.
What is the relation between the coupled coefficient of roll into sway and the coefficient of sway into roll?
AUTHORS' REPLY
Unlike classical potential-flow theory, the flows being considered in this paper are governed by the fully nonlinear
Navier-Stokes equations. Thus, none of the normal conservation or reciprocity relations, such as the wave-amplitude to
damping relation, symmetric of hydrodynamic coefficients, would be applicable. The FSRVM method contains no
artificial or numerical viscosity in the fluid domain, aside from the sponge layer used to damp out reflected waves on the
free surface. Its accuracy was earlier tested out by a number of experiments and numerical solutions reported in the
references. The nonlinear nature of the solution, particularly the importance of convective effects in the flow, renders the
sway-roll coupling coefficients to be non-symmetrical.
About a decade ago, we developed a viscous-flow formulation (Yeung & Wu, Ref. [35]) under a small motion-
amplitude assumption. The linearization process was applied to both the Navier-Stokes equations and the free-surface
boundary conditions It is conceivable that some kind of conservation and symmetry conditions can be derived for such a
linear model. The short-coming of such an approach is that if a substantial amount of flow separation takes place, as in the
present bilge-keel problem, the convective effects in the field equations cannot be neglected. The linearized model has
been pursued by a number of workers since our initial work.
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