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PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 206
Practical CFD Applications to Design of a Wave Cancellation
Multihull Ship
Chi Yang1, Francis Noblesse2, Rainald Löhner1, Dane Hendrix2
1 Institute for Computational Sciences and Informatics
George Mason University, Fairfax VA 22030–4444, USA
2 David Taylor Model Basin, CD-NSWC
9500 MacArthur Blvd, West Bethesda MD 20817–5700, USA
ABSTRACT
Four methods of analysis—a nonlinear method based on Euler's equations and three linear potential flow methods—
are used to determine the optimal location of the outer hulls for a wave cancellation multihull ship that consists of a main
center hull and two outer hulls. The three potential flow methods correspond to a hierarchy of simple approximations based
on the Fourier-Kochin representation of ship waves and the slender-ship approximation.
INTRODUCTION
This study considers an illustrative practical application of CFD tools to a simple ship design problem. This simple
design case is the wave cancellation multihull ship concept examined in [1], where experimental measurements and
theoretical calculations based on Michell's thin-ship approximation are given. The wave cancellation multihull ship that is
considered consists of a main center hull and two identical outer hulls centered at with respect to
the center of the waterplane of the main hull. The main center hull of the multihull ship considered in [1] and the present
study has a length The main hull and the outer hulls are defined in the Appendix.
The study considers the elementary design problem of determining the optimal location of the outer hulls with respect
to the main center hull, i.e. the optimal values of the two parameters LX and LY, for the purpose of minimizing the wave
drag of the ship. Four methods of analysis are used and compared to one another and to experimental data. One of the
methods is the nearfield flow calculation method presented in [2] and [3]. This method is based on the Euler equations and
the nonlinear free-surface boundary condition. The other three methods are linear potential flow methods that correspond to a
hierarchy of simple approximations based on the Fourier-Kochin representation of ship waves [4] and the slender-ship
approximation [5,6].
HAVELOCK AND FOURIER-KOCHIN REPRESENTATION OF WAVE DRAG
Consider a ship advancing along a straight path, with constant speed U, in calm water of effectively infinite depth and
lateral extent. The flow is observed from a Cartesian system of coordinates moving with the ship. The X axis is taken along
the path of the ship and points toward the ship bow; i.e., the ship advances in the direction of the positive X axis. The Z axis
is vertical and points upward, and the mean free surface is the plane Z=0. The flow appears steady in the translating system
of coordinates, and consists of the disturbance flow due to the ship superimposed on a uniform stream opposing the ship's
forward speed. The components of the disturbance velocity along the (X, Y, Z) axes are (U, V, W). Thus, the total velocity is
given by (U−U, V, W). Nondimensional coordinates and velocities are defined in terms of a characteristic length L, taken
and the ship speed U as
here as
Define the Froude number F and v as
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Here, g is the acceleration of gravity.
The drag DW=ρU2L2CW associated with the wave energy transported by the waves trailing the ship can be determined
from the Havelock formula
(1)
Here, the wavenumber k: is defined in terms of the Fourier variable β by
(2a)

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PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 207
Furthermore, Sr and Si are the real and imaginary parts of the spectrum function S=S( α, β). Here, α is defined in terms
of the Fourier variable β by
(2b)
F2α2=k:
The relations (2) follow from the dispersion relation with
The velocity representation of free-surface flows and the related Fourier-Kochin representation of waves given in [4]
define the spectrum function S in terms of the velocity distribution at the ship hull surface (or more generally at a boundary
surface that surrounds the ship). Specifically, [4] and [7] define the spectrum function S in terms of a distribution of
elementary waves exp[kz−i(αx−βy)]:
(3a)
with
(3b)
(3c)
Here, Σ stands for the mean wetted ship hull surface (or a boundary surface that surrounds the ship) and is the
intersection curve between the surface Γ and the mean free-surface plane z=0. Furthermore, and
respectively stand for the differential elements of area and arc length of Σ and Γ at the integration point
Finally, the amplitude functions A and A Γ are defined in terms of the boundary velocity distribution
by
(3d)
(3e)
is the unit vector normal to the ship hull surface Σ ; points inside the mean flow domain
Here,
(i.e. outside the ship). The unit vectors and are tangent and normal, respectively,
to the ship waterline Γ; is oriented clockwise (looking down) and points outside the ship, like Finally,
are the x and y components of the vector
Expressions (3) define the spectrum function S in the Havelock integral (1) in terms of the normal components
of the velocity at the ship hull surface Σ ∪ Γ .
and and the tangential components and
APPLICATION TO MULTIHULL SHIPS
∪ Γ consists of N component surfaces, i.e
If the ship hull surface Σ
(4a)
the spectrum function S can be expressed as
centered at
(4b)
with
(4c)
In the particular case of a ship consisting of three hulls, a center hull centered at (0, 0, 0) and two identical outer hulls
centered at (4b) becomes
If the outer-hull spectrum functions are assumed to be identical, we have
Here, and stand for the spectrum functions associated with the center hull and an outer
hull, respectively.
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Define
(5a)
are defined by (3b) and (3c) in which Σ , Γ are taken as
where the spectrum functions and
or Σo, Γo. The real and imaginary parts of the spectrum function S are then given by
These expressions yield

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PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 208
We thus have
with
Here, A, AR and Al are defined as (5b)
(6a)
The wave drag CW can then be expressed as
(6b)
Here, and are given by
(6c)
and represent the wave drags of the center hull and of an outer hull, respectively. The component accounts for
interference effects and is defined as
(6d)
with A, AR and Al given by (5b). The wave-interference coefficient Ci given by
(6e)
can be defined in terms of the wave drag coefficients and
FOURIER-KOCHIN REPRESENTATION OF NEAR-FIELD STEADY SHIP WAVES
The velocity representation of free-surface flows and the related Fourier-Kochin representation of waves expounded in [4]
and [8] show that, within the framework of potential-flow theory, the velocity field generated by a ship can be
(7a)
decomposed as
Here, and respectively represent a wave component and a local component that are associated with the
(7b)
decomposition
of the Green function G associated with the free-surface boundary condition w−F x 2u =0. The wave component is
given by a single Fourier integral.
Specifically, the ship hull Σ (which here stands for the center hull Σc or the outer hull Σ o) is divided into a set of
patches Σ p associated with reference points (xp, yp, zp≤0) located in the vicinity of Σ p. The patch reference points (xp, yp,
zp≤0) attached to the patches Σp need not lie on p. The size of the patches in the x direction is O(σF2) as required by the
function Θ defined by (9).
The wave component at a field point is defined in [7] as
(8)
Here, βc is a large positive real constant, and the Fourier variable α and the wavenumber are the
functions of the Fourier variable β given by (2). Furthermore, the function Θp is defined as
(9)
with σ a positive real constant.
Summation in (8) is performed over all the N patches Σ p ∪ Γ p that represent the surface Σ ∪ Γ , i.e.
(9) shows that Θp → 0 as (xp ξ)/( σF 2) → ∞ . Thus, the contribution of a patch Σp to the wave component is
located at a distance O(σF2) ahead of the reference point attached to Σ p.
negligible at a point
The functions in (8) are defined as
(10a)
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(10b)
(10c)
(10c)
with
Furthermore, and in (10) are defined as
(11a)
associated with the patch Σ p.
where and are the real and imaginary parts of the spectrum function

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PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 209
The spectrum function Sp is given by distributions of elementary waves over Σp ∪ Γp . Specifically, (3a)–(3c) yield
(11b)
(11c)
(11d)
The amplitude functions A Σ and A Γ are given by (3d) and (3e).
At a distance O(σF2) behind the ship, (9) yields p ≈ 2 and the foregoing representation of the wave component
can be simplified as in [7]. This simplified representation, which does not require subdivision of the ship hull (or more
generally boundary surface) Σ ∪ Γ into patches Σp ∪ Γ p, may be used to extend a nearfield flow past a ship into a farfield
region entirely located behind the ship [7]. The more general representation of given here, on the other hand, is valid
in the entire flow domain, and thus can be used to compute nearfield waves.
SLENDER-SHIP APPROXIMATION
The slender-ship approximation expounded in [5] and [6] defines the velocity field generated by a ship explicitly
in terms of the speed and shape of the ship:
(12)
The slender-ship approximation (12) may be regarded as a generalization of the Michell thin-ship approximation.
Specifically, the Michell approximation differs from (12) in that it defines in terms of a distribution of sources, with
strength 2nx, over the ship centerplane y=0 instead of a distribution of sources of strength nx over the port and starboard
sides of the actual ship surface Σ . In addition, there is no distribution of sources around the waterline Γ in Michell's
approximation.
The slender-ship approximation to the wave component in the decomposition (7a) is given by the Fourier-Kochin
representation defined by (8)–(11) with the amplitude functions AΣ and A Γ in (11c) and (11d) taken as
(13)
can be effectively evaluated from (12) with the Green function G taken as the simple
The local component
analytical approximation to the local component GL given in [9] or the simpler approximation
(14)
Here, r, r′ , r″ are defined as
The approximation (14) yields
as dictated by the free-surface boundary condition w− F2ux=0.
At the ship hull surface Σ , the velocity field given by the slender-ship approximation (12) can be modified using the
transformation
(15)
This transformation yields and thus ensures that the hull boundary condition is satisfied exactly by the
velocity distribution at the hull surface. The hull-condition transformation (15) can be applied to any velocity
distribution computed at a ship hull surface. In particular, (15) shows that the velocity distribution associated with the
trivial velocity distribution is given by
This velocity distribution is normal to the ship surface Σ (the tangential velocity component is null), and
evidently satisfies the hull boundary condition The velocity distribution corresponds to the
slender-ship approximation (12) and (13).
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FOUR METHODS OF ANALYSIS
CW
The wave drag coefficient defined by (6a)–(6d) and (5) account for interference effects of the farfield waves
generated by the center hull and the two outer hulls. The spectrum functions Sc and So in (5) are defined by (3) in terms of
the normal and tangential components of the velocity distributions at the center hull Σ c and the outer hulls Σo. The velocity
distributions at Σ c and Σo are affected by nearfield flow interactions between the center hull and the outer hulls. Thus, the
wave drag C W defined by (6a)–(6d), (5), and (3) account for both farfield wave-interference effects and nearfield flow
interactions. Evaluation of the wave drag CW requires evaluation of the nearfield velocity distribution at Σc and Σo for a
range of values of the Froude number F and of the parameters a and b

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PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 210
that define the location of the outer-hull. This computational task is daunting using typical calculation methods but can
actually be easily performed using the slendership approximation. Specifically, a single-loop set of computations of the
wave drag C W associated with the slender-ship approximation to the nearfield velocity distribution at Σ c and Σo for 4 values
of the Froude number, 61 values of a and 26 values of b, corresponding to 4×61×26=6,344 nearfield flow calculations
(using 11,525 panels to approximate the three hulls) requires 18 hours of CPU time using an SGI origin 2000 with 4
processors. This approach is identified as method 3 hereafter.
Considerable simplifications are obtained if nearfield flow interactions are ignored, i.e. if the velocity distribution at
the center hull Σc is evaluated for the center hull alone (i.e. without the two outer hulls) and the velocity distribution at an
outer hull is similarly evaluated for the outer hull alone (i.e. without the center hull and the other outer hull). This
approximation, which accounts for farfield wave interference effects but neglects nearfield flow interactions, only requires
two nearfield flow evaluations (one for the center hull, and one for an outer hull) per Froude number, a task that can easily
be performed using existing calculation methods. The spectrum functions Sc and So defined by (3) and the related functions
A, AR and A l given by (5) similarly need be evaluated only once per Froude number. These spectrum functions, like the
velocity distributions at the three hulls, are independent of the parameters a and b, which only appear in (6d), within the
“negligible nearfield interaction” approximation. Thus, this approximation effectively uncouples the outer-hull location
parameters (a, b) and nearfield flow calculations. For the purpose of estimating the importance of nearfield flow
interactions upon the wave drag CW, the nearfield velocity distribution is evaluated here using the slender-ship
approximation already used in method 3. This approach is identified as method 2 hereafter. Thus, comparison of methods 2
and 3 provides insight into the importance of nearfield flow interaction effects. Method 2 corresponds to the first-order
slender-ship approximation defined in [5].
Insight into the importance of using a sophisticated nearfield flow calculation method can be gained by comparing
method 2 and method 1, which corresponds to the zeroth-order slender-ship approximation in [5] and to the trivial
approximation
in (3d) and (3e). Thus, no nearfield flow calculation is required in this simplest approximation. Indeed, the zeroth-
order slender-ship approximation is associated with the trivial approximations and for the
tangential components of the velocity at the ship hull Σ and water line Γ , as previously explained.
Methods 1, 2, and 3 are based on the Fourier-Kochin representation of ship waves, i.e. on linear potential flow, and the
further simplification associated with the slender-ship approximation. Thus, even method 3 only accounts for effects of
nearfield flow interactions in an approximate manner. However, nearfield flow interactions are fully taken into account in
the nearfield flow calculation method presented in [2] and [3], which is based on the Euler equations and the nonlinear
free-surface boundary condition. This method is identified as method 4.
RESULTS OF ANALYSIS
Figure 1 depicts the experimental values of the residuary drag coefficient CR given in [1] and the corresponding
predictions of the wave drag coefficient C W given by methods 1, 2, 3, and 4 for four arrangements of the outer hulls. These
four arrangements of the outer hulls correspond to a= − 0.128, −0.205, −0.256, − 0.385 and b=0.136 (the same value for all
four cases). The left column in Fig. 1 compares CR and CW predicted by methods 1 and 2. The right column shows CR and C W
given by methods 2 and 3, and by method 4 at F=0.25, 0.3, 0.4, 0.5. Differences between the values of CW predicted by
methods 1 and 2 (left column) and between methods 2 and 3 (right column) are fairly small, and these 3 methods yield
values of CW that are in fair agreement with the experimental values of CR. In particular, the variation of C R with respect to
the Froude number F is well captured by the theory. The values of C W given by method 4 at F=0.25, 0.3, 0.4, 0.5 are in
even better agreement with CR on the whole, and are in fairly good agreement with CW predicted by methods 2 and 3.
Figs 2a,b,c,d compare the values of C W given by methods 1, 2, 3 for F=0.5, 0.4, 0.3, 0.25 and outer-hull arrangements
within the region
(16)
For a=0.75, the sterns of the outer hulls are aligned with the bow of the main center hull; similarly, the bows of the
outer hulls are aligned with the stern of the center hull if a= −0.75. The contour plots of CW depicted in Figs 2a–d are based
on calculations for increments in the values of the outer-hull location parameters a and b equal to ∆a=0.025 and ∆b=0.01.
Thus, the contour plots of CW in Figs 2 correspond to evaluations of CW using methods 1, 2, 3 for 4 values of the Froude
number F, 61 values of a and 26 values of b, i.e. 3×4×61×26=19,032 evaluations. Lowest values of C W, corresponding to
best outer-hull arrangements, are indicated in blue, and highest values of CW, i.e. worst
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PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 211
outer-hull arrangements, are indicated in red. Figs 2 indicate that methods 1, 2, and 3 predict best (blue regions) and worst
(red regions) outer-hull arrangements that are in fairly good agreement.
Fig. 3 shows the values of the wave drag coefficient CW given by methods 1, 2, 3, 4 at the three best distinct outer-hull
arrangements, i.e. for (ak, bk) with k=1, 2, 3, predicted by method 2. This figure shows that, although the values of C W
predicted by methods 1, 2, 3, 4 are not identical, these four methods yield The results depicted in
Figs 1, 2, and 3 indicate that methods 1 and 2, which are computationally more efficient than methods 3 and 4, may be used
for the purpose of determining optimal arrangements of the outer hulls. Method 2 is then used to further study the best and
worst outer-hull arrangements.
Fig. 4 depicts the variation of the wave drag coefficient CW , for F=0.3 and F=0.5, within the region
Consideration of this large region, which greatly extends the region (16) of practical interest for the design of a wave
cancellation multihull ship, provides a broader view of the variation of the wave drag coefficient CW within the smaller
region (16) examined in Figs 5a,b.
Figs 5a,b show the variation of the wave drag coefficient CW predicted by method 2 within the region (16) with
∆a=0.025 and ∆b=0.01, as in Figs 2a–d, for 20 values of the Froude number F in the range 0.2147≤F≤0.5426. Thus, Figs
5a,b present the result of 61×26×20=31,720 evaluations of CW . As in Figs 2, lowest and highest values of CW , i.e. best and
worst outer-hull arrangements, correspond to blue and red regions in Figs 5.
The left side of Fig. 6 depicts the variation, with respect to the Froude number F, of the wave drag coefficient
and of the wave drag coefficients and corresponding to the best and worst outer-hull
arrangements found (using method 2) within the region
The right side of Fig. 6 depicts the interference coefficients and defined by (6e) corresponding to the
best and worst outer-hull arrangements considered in the left side of Fig. 6. The large differences between and
and between and shown in Fig. 6 demonstrate the importance of selecting a favorable outer-hull
arrangement and the benefits that can derived from an optimal arrangement. Indeed, the left side of Fig. 6 shows that the
ratio approximately varies between 2 and 6.
The lower part of Fig. 7 depicts the experimental values of CR given in [1] for four arrangements of the outer hulls and
the wave drag coefficient already shown in the left side of Fig. 6, corresponding to the best outer-hull arrangement
(which varies with respect to the Froude number) given by method 2. As one expects, the wave drag coefficient is
smaller than the experimental values of CR (which do not correspond to optimal arrangements of the outer hulls over the
entire range of Froude numbers). The lower part of Fig. 7 also shows the values of the wave drag coefficient C W that
correspond to near-best arrangements of the outer hulls. These near-best arrangements are defined here as those outer-hull
arrangements for which CW does not exceed by more than 10%. The values of the outer-hull location parameters a
and b corresponding to the best and near-best arrangements considered in the lower part of Fig. 7 are depicted in the upper
part of Fig. 7. The upper part of Fig. 7 also shows the values of a and b for which experiments are reported in [1] and in the
lower part of Fig. 7.
CONCLUSION
Illustrative practical applications of CFD tools to the wave cancellation multihull ship concept examined in [1] have
been summarized. The wave cancellation multihull ship considered in [1] and here consists of a main center hull and two
identical outer hulls centered at with respect to the center of the waterplane of
the center hull. The elementary design problem of determining the optimal arrangement of the outer hulls with respect to
the main center hull, i.e. the optimal values of the two parameters a and b, for the purpose of minimizing the wave drag of
the ship has been considered using four methods of analysis.
One of the four methods is the nearfield flow calculation method presented in [2] and [3]. This method is based on the
Euler equations and the nonlinear free-surface boundary condition. The other three methods are linear potential flow
methods that correspond to a hierarchy of simple approximations based on the Fourier-Kochin representation of ship waves [4]
and the slendership approximation [5,6]. These four methods of analysis have been compared to one another and to the
experimental data given in [1] for the purpose of establishing and validating a practical methodology that can be used for
more complex hull-form design problems involving minimization of wave drag.
In addition to the problem of selecting optimal outer-hull arrangements considered here, a realistic multihull ship
design problem involves optimal selections of the lengths, beams, drafts, and shapes of the main center hull and the outer
hulls, which have been taken as in [1] for the purpose (of main interest for
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APPENDIX 212
the present study) of comparing and validating alternative methods of analysis of main interest in the present study.
Furthermore, constraints associated with mission requirements, structural considerations, seakeeping, and course keeping
must evidently be considered.
The three methods based on the Fourier-Kochin representation of ship waves and the slender-ship approximation,
especially the methods (called methods 1 and 2) corresponding to the zeroth-order and first-order slender-ship
approximations given in [5], provide simple and highly efficient tools. These practical tools have been shown to be
adequate for the purpose of determining optimal locations of the outer hulls. Method 4, based on a more refined flow
analysis, can then be used effectively to further evaluate the flow at the optimal outer-hull arrangement.
The use of a pragmatic approach that relies on a combination of simple and more refined tools evidently is a well-
established practice. In particular, the zeroth-order slender-ship approximation (i.e. method 1) had previously been used
with success for hull-form optimization in [10] and [11]. Thus, the practical usefulness of this remarkably simple
approximation is confirmed in the present study.
ACKNOWLEDGEMENTS
The work of Yang and Löhner was partially funded by AFOSR (Dr. Leonidas Sakell technical monitor) and by NRL
LCP&FD (Dr. William Sandberg technical monitor). The work of Noblesse and Hendrix was supported by the ILIR
program at NSWC-CD. All computer runs were performed on a 128-Processor R10000 SGI Origin 2000 at the Naval
Research Laboratory.
REFERENCES
[1] M.B.Wilson, C.C.Hsu & D.S.Jenkins (1993) Experiments and predictions of the resistance characteristics of a wave cancellation multihull ship
concept, 23rd American Towing Tank Conf., 103–112
[2] R.Löhner, C.Yang, E.Oñate & S.Idelssohn (1999) An unstructured grid-based, parallel free-surface solver, Applied Numerical Mathematics 31: 271–
293
[3] C.Yang & R.Löhner (1998) Fully nonlinear ship wave calculation using unstructured grid and parallel computing, 3rd Osaka Coll. Advanced CFD
Applications to Ship Flow and Hull Form Design, Osaka, 125–150
[4] F.Noblesse (2000) Velocity representation of free-surface flows and Fourier-Kochin representation of waves, s ubmitted
[5] F.Noblesse (1983) A slender-ship theory of wave resistance, Jl Ship Research 27:13–33
[6] F.Noblesse & G.Triantafyllou (1983) Explicit approximations for calculating potential flow about a body, Jl Ship Research 27:1–12
[7] C.Yang, R.Löhner & F.Noblesse (2000) Farfield extension of nearfield steady ship waves, Ship Technology Research 47:22–34
[8] F.Noblesse, X.B.Chen & C.Yang (1999) Generic super Green functions, Ship Technology Research 46:81–92
[9] F.Noblesse, C.Yang & D.Hendrix (2000) Steady free-surface potential flow due to a point source, 15th Intl Workshop on Water Waves & Floating
Bodies, Israel
[10] J.S.Letcher Jr., J.K.Marshall, J.C.Oliver III & Nils Salvesen (1987) Story & Stripes, Scientific American, 257:34–40
[11] D.C.Wyatt & P.A.Chang (1994) Development and assessment of a total resistance optimized bow for the AE 36, Marine Technology 31:149–160
APPENDIX
The outer hulls are considered first. Let 2Lo, 2Bo, and Do stand for the length, beam, and draft of an outer strut, which
consists of a parallel midbody and parabolic sharp-ended nose and tail regions. Define a system of coordinates (Xo, Y o, Zo)
with origin at the center of the outer strut. The upper part of the strut is given by with defined as
These equations hold for The lower part of the strut is given by
Thus, the outer hulls are defined by the five parameters Lo, Bo, Do,
with and
The centers of the water planes of the two outer struts are located at X=−LX and with
respect to the center of the water plane of the main center hull. Define the system of coordinates (X, Y, Z) with origin at the
the authoritative version for attribution.
center of the water plane of the center hull. Thus, the coordinates (X, Y, Z) corresponding to the outer-strut coordinates (Xo, Yo, Zo)
are given by
for the outer struts centered at . The outer-strut surfaces can be defined in terms of the

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APPENDIX 213
center-hull coordinates via the foregoing coordinate transformations.
The main center hull is now considered. Let and stand for the length, beam, and draft of the center hull.
The local beam (widest waterline) of the center hull is given by with defined as
with B0 defined as
The top waterline of the center hull is given by
Thus, the local beam and the top waterline of the main center hull are defined by the six parameters
and
Every frameline consists of a straight horizontal bottom and a straight (but not vertical) side connected by a portion of
circle of radius equal to σ is the local beam and σ=1/2. Thus, the hull bottom is defined by
, where
The hull side is given by
where C′ and Dg are the local slope and depth of the hull side. The circular bilge connecting the hull side and bottom is
defined by
Here, τ=tanθ* is defined by the condition of smooth contact between the hull side and bilge
for
circle, as shown below.
Consider the portion of a frameline located in the half plane Y≥0. The vectors
are normal to the side frameline and tangent to the bilge, respectively. Thus, we have
at the contact point θ=θ* between the side and the bilge. This condition defines C ′ as
It follows that the local depth Dg of the hull side is given by
Thus, the hull side is defined by
The foregoing equations fully define the main center hull except for the contact parameter τ =tanθ*. This parameter is
defined below.
At the contact point θ=θ*, the additional condition
holds. This condition yields
i.e.
where τ=tanθ*. This equation yields
We then have
with
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Thus, the contact parameter τ is determined by the 3 parameters and σ. If we have A0=0
and τ=0 as expected.

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APPENDIX 214
Framelines are vertical at some distance above the waterplane Z=0. The straight hull side below the waterplane,
inclined at an angle θ*=tan−1 τ with respect to the vertical, and the vertical upper hull are connected by a circular arc of
radius R. Consider the portion of a frameline located in the half plane Y≥0. The frameline intersects the waterplane at the
point (Y=B0, Z=0). The center of the circular arc that joins the lower and upper framelines is located at the point
The contact point between the upper vertical frameline and the circular transitional portion of the frameline is
Thus, the center hull above the waterplane Z=0 is defined by
The circular transition region between the upper vertical frameline and the hull side below the waterplane is defined by
were used above. The radius R may be taken
The relations
proportional to the beam of the top waterline, i.e.
with λ=1.
The six parameters Lc, Dc, B c, that define the main center hull and the five parameters Lo, Bo, Do,
and
and that define the outer hulls are given by
for the wave cancellation multihull ship considered in [1] and the present study. The experimental results given in [1]
and reported in Fig. 1 of this study correspond to four locations of the outer hulls given by (−LX,±Ly) with
Wave cancellation multihull ship model
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APPENDIX
Fig. 1. Calculated wave drag and experimental residuary drag
215

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APPENDIX
Fig. 2a. Wave drag coefficient predicted
by methods 1,2,3 for F=0.5
Fig. 2b. Wave drag coefficient predicted by methods 1,2,3 for F=0.4
216

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APPENDIX
Fig. 2c. Wave drag coefficient predicted by methods 1,2,3 for F=0.3
Fig. 2d. Wave drag coefficient predicted by methods 1,2,3 for F=0.25
217

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APPENDIX
Fig. 4. Wave drag coefficient for −3≤a≤3 and 0≤b ≤1
Fig. 3. Wave drag coefficient at three best arrangements of outer hulls
218

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APPENDIX
Fig. 5a. Wave drag coefficient for 10 Froude numbers
219

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APPENDIX
Fig. 5b. Wave drag coefficient for 10 Froude numbers
220

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APPENDIX
Fig. 6. Wave drag for best and worst arrangements of outer hulls
Fig. 7. Wave drag and parameters a and b for best and near-best arrangements of outer hulls
221

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APPENDIX 222
DISCUSSION
L.J.Doctors
University of New South Whales
Australia
I would like to express my appreciation to the four authors for a most interesting paper on the subject of multihulls, a
matter of interest to many researchers who are aiming to reduce either the overall resistance or the wave resistance (as in
the present work).
The contour plots of Figure 2, Figure 4, and Figure 5 are an excellent way of presenting the considerable quantity of
data showing the influence of longitudinal stagger and lateral offset on the wave resistance.
Could the authors kindly explain the background behind the breakdown of the Green function into the three terms in
Equation (14)? The first two terms are well known and the (new) third term provides the correct limiting behavior for low
and high Froude numbers. However, there seems to be no evidence of wave-like behavior in the third term. Indeed, there
would be a family of similar versions of the third term that would exhibit the appropriate limiting behavior.
Finally, can the authors indicate whether the effects of sinkage and trim are included in their work and whether they
feel these would affect the final predictions for the wave resistance?
AUTHOR'S REPLY
Thank you for your interest in our work and your questions. The flow field and the Green function are expressed as
sum of a wave component and a local component, as is indicated in equations (7a) and (7b). Equation (14) provides an
extremely simple approximation to the local component alone, not the wave component. This component is defined by
equations (8) through (11a–d) with expressions (3a–e), or expression (13) in the slender-ship approximation.
Effects of sinkage and trim have not been included in our calculations, although this could be done without any great
difficulty. Our guess is that sinkage and trim would not significantly modify the optimal hull arrangement, but it would be
interesting to verify that this is indeed the case.
As noted in our reply to the discussion by Profs. Nakatake and Ando, lifting effects are not included in methods 1–3
(thus, only sources are used in these 3 potential-flow methods) but are taken into account in method 4.
DISCUSSION
K.Nakatake and J.Ando
Kyushu University, Japan
We congratulate you on your paper to predict the optimal location of the outer hulls for the main hull by applying the
rather simple wave drag formulas. We also applied our Rankine source method to a trimaran and confirmed by some
experiments that the total wave drag fairly changes according to the location of the outer hulls for the main hull. From Fig. 4,
it is interesting to note that the wave drag coefficient changes like a wave contour.
Among your methods, method 4 seems to be most exact. In the strict sense, the main hull can be treated as a nonlifting
body by method 4, but the outer hull should be treated as a lifting body because the flow around it is not symmetrical with
respect to the center plane. Therefore the additional vortex (or doublet) distribution is needed to satisfy the Kutta's condition
at the trailing edge of the outer hull. This effect may become larger when the transverse distance between the main hull and
the outer hull is small. What do you think of this point?
AUTHOR'S REPLY
Thank you for your interest in our paper and for providing information about your own work on the effect of hull
arrangement upon wave drag. We agree that the outer hulls should in principle be treated as lifting surfaces, and that lifting
effects can be expected to be larger if the outer hulls are closer to the center hull. The 3 simple potential-flow methods we
have used (methods 1, 2, 3) do not include lifting effects. Method 4, based on the Euler equations, accounts for lifting
effects. The relatively good agreement between experimental residuary drag and wave drag predicted by all 4 methods
shown in Fig. 1 suggests that lifting effects may not be very important in the present case.
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