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For the prediction of ship resistance in the preliminary stages of the design process, naval architects often use methods that are less complex and expensive than CFD simulations and experiments. The linear wave-making theory can be used to quickly evaluate wave resistance, although, the theory gives poor estimates for conventional hull forms due to neglecting viscosity. This paper introduces improvements to the original theory by including boundary layer effects through the tangency correction that can handle flow separation. The improvements that account for viscous and nonlinear effects are implemented within Michell’s thin-ship theory to extend its applicability to non-slender hulls, which is validated by numerical simulations of five profoundly different hull forms: the Wigley hull, Series 60, Delft 372, a yacht hull, and the KRISO containership hull. The modified theory yielded more accurate resistance curves compared to the original theory and Holtrop–Mennen’s method, and gave new insight into ship wave-making.
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Improved estimation of ship wave-making resistance
Josip Bašića,, Branko Blagojevića, Martina Andruna
aFaculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split,
Rudera Boskovica 32, 21000 Split, Croatia
For the prediction of ship resistance in the preliminary stages of the design process, naval
architects often use methods that are less complex and expensive than CFD simulations
and experiments. The linear wave-making theory can be used to quickly evaluate wave
resistance, although, the theory gives poor estimates for conventional hull forms due to
neglecting viscosity. This paper introduces improvements to the original theory by including
boundary layer effects through the tangency correction that can handle flow separation.
The improvements that account for viscous and nonlinear effects are implemented within
Michell’s thin–ship theory to extend its applicability to non-slender hulls, which is validated
by numerical simulations of five profoundly different hull forms: the Wigley hull, Series 60,
Delft 372, a yacht hull, and the KRISO containership hull. The modified theory yielded more
accurate resistance curves compared to the original theory and Holtrop-Mennen’s method,
and gave new insight into ship wave-making.
Keywords: Wave-making; ship resistance; wave resistance; boundary layer; far-field waves
1. Introduction
The problems of predicting wave patterns and wave resistance of ships are one of the most
important subjects in ship theory. Wave-making resistance at large Reynolds numbers is
determined by the potential component in the far–field without the rotational component.
The paper published by Michell (1898) marks the beginning of the theory of wave resistance
?Published in Ocean Engineering, 200, pp. 107079. doi:10.1016/j.oceaneng.2020.107079
Email addresses: (Josip Bašić), (Branko Blagojević), (Martina Andrun)
Preprint submitted to Elsevier 26th February 2020
of floating bodies, which was overlooked for years until Havelock and Wigley discovered it
and continued to work on it (Gotman,2002). Utilising the potential flow theory, Michell
derived the linear-wave theory by representing the ship hull by sources distributed along the
centreplane with strengths proportional to the longitudinal slope of the hull, assuming the
slope is small. Therefore, it is often referred to as the thin–ship theory. The theory is also
applicable on submerged bodies near free surface (Tuck et al.,2001), multi-hulls (Yu et al.,
2015), shallow waters (Gourlay,2008), etc. Computational Fluid Dynamics (CFD) nowadays
provides more precise tools for ship resistance prediction (Bašić et al.,2017b). In spite of their
limitations, potential–flow based tools are compelling for early stages of the design process due
to their simplicity, low cost and small amount of engineer’s time consumption (Chen et al.,
2018). Wave-making simulated by the higher–order Rankine Boundary Element Method
(BEM) can yield adequate results (He,2013), and therefore, can be used for seakeeping
analyses in the framework of linear potential theory (He and Kashiwagi,2014).
On the other hand, it may be argued that the thin–ship theory does not yield sufficiently
accurate results, especially not for conventional ship forms that have low length–breadth
ratio, i.e. L/B < 8(Gotman,2002). Wigley, Havelock, Tuck (1974) and other researchers
argued that the humps and hollows on a resistance curve obtained by the potential-flow
theory supervene due to neglecting the viscosity. They argued that the boundary layer (BL)
and the wake affect the wave system, which should smooth out the humps and hollows.
Besides neglecting of the viscous effects, most of the remainder of the discrepancies accounts
for neglecting of the nonlinear terms that do not affect the wave pattern amplitude, but
modify the phase of the far field wave.
To avoid directly solving these issues within the theory, viscous–inviscid domain decom-
position became a popular approach. Viscous effects are taken into account through the
Navier-Stokes equations in the neighbourhood of the hull and in the wake, and the inviscid
assumption far from the hull allows a description by the potential flow (Sahoo et al.,2007;
Raven,2010). Gotman (2002) gives a good review of published papers in which the wave
resistance is obtained by linear theories variously incorporating the viscosity effects on both
wave generation and wave propagation. To name a few, Milgram (1969) was one of the first
to prove that the fluid viscosity affects the resistance of a thin ship obtained by modifying
the beam of the Wigley hull based on the assumed separation of the streamlines. Wang
(1985) made a modification for viscous flow to thin– and slender–ship theories by adding the
BL displacement thickness to the actual ship-hull geometry. The displacement thickness was
calculated by solving the integral BL (IBL) equations on a double–body model of the hull.
Ikehata and Tahara (1987) combined the Rankine-source method with the IBL method to
investigate the influence of the BL and wake on the free surface flow around a ship model.
Shahshahan and Landweber (1986,1990) calculated the wave resistance of the Wigley hull
using the thin–ship theory, and found that including the viscosity effects modifies the theor-
etical resistance in the right direction. Tuck (1974) added a modification to the free-surface
condition to account for wave dissipation due to viscosity and concluded that a ship is a less
efficient generator of waves in the presence of surface viscosity, and that the skin friction
effect on the wave resistance should also be investigated. Moreno et al. (1975) thus con-
ducted experiments on smooth and rough ship model to investigate the skin friction effect
on the resistance, and found a reduction of the wave-making when the BL and wake are
thickened. Doctors (2003) and Doctors and Zilman (2004) included the surface tension and
eddy viscosity damping factor in the computation of the wave pattern around a catamaran,
and concluded that the inclusion of the surface tension desirably reduces the amplitude of
waves at lower Froude numbers, but without determining appropriate damping values for
general ship forms.
Generally, a potential-flow solver may incorporate the effects of BL by physically shifting
boundaries outward in their normal direction using the calculated displacement thickness,
by “blowing” specified velocity from geometry to displace the surface streamlines, or by
tangency correction introduced by Bašić et al. (2017a). Similar to Wang (1985) and Milgram
(1969), Peng et al. (2014) excluded a part of the stern region from the computational domain,
while modifying the boundary conditions. In this case, the BL displacement thickness was
incorporated by physically shifting the geometry, but without extending the applicability
to arbitrary ship forms. Furthermore, iteratively solving the problem using the IBL theory
(Drela,2013) with shifting or blowing techniques, the convergence cannot be reached due to
stern effects, since the methods are not suitable for the calculation of wake with vortices, e.g.
see (Tanaka,1988). In addition, stern surface areas with normals pointing back should be
immensely shifted away from the initial positions or should blow infinite velocity to represent
the wake.
Imposing the displacement thickness using the displacing or blowing techniques implicitly
modifies the tangency of the flow near the body, while the tangency correction technique
makes the same modifications to flow without changes in geometry and boundary conditions.
Consequently, the tangency correction technique does not suffer from the listed problems.
It directly incorporates the BL displacement thickness along streamlines on the surface, and
works with separated flows with wide wakes (Bašić et al.,2017a). This study deals with
a possibility of enhancing the linear wave-making theory by introducing approximate, but
physically-correct enhancements. The linear wave-making theory is examined and enhance-
ments are introduced based on the following:
1. The BL and wake significantly affect the wave-making: the humps and hollows in the
wave resistance curve are a consequence of the interaction of the inviscid bow and stern
wave systems.
2. Any potential flow solver can yield more accurate pressure field by incorporating the
BL displacement thickness information.
3. From the listed coupling techniques, the tangency correction technique has a direct
physical connection and intrinsically handles separated flows.
4. Using the tangency correction, the BL displacement thickness may be incorporated
in the linear wave–making theory without adding complexity to obtain more accurate
wave–resistance results.
The above points are explained throughout the paper, and the resulting method is validated
by calculating the resistance of five various hull forms and comparing the results to known
experimental values. The remainder of the paper is structured as follows. Section 2describes
the linear theory and corresponding numerical procedure for predicting wave resistance of
ships. Section 3analyses how to improve the theory by including viscous and phase-shifting
effects. The validation of the extended theory is given in Section 4. Finally, Section 5lists
the drawn conclusions.
2. Wave-making theory
2.1. Far–field wave resistance
Wave-making is determined by the potential component in the far field. Zhu et al. (2017)
showed that Hogner’s modification of Michell’s theory may benefit prediction of wave-making
for high Froude numbers. Since the area where the wave resistance begins to increase is of
interest to the naval architect, the problem of wave-making is examined through somewhat
simpler foundations of Michell’s thin–ship theory. The problem of wave-making is nonlinear
due to the quadratic nature of the dynamic free-surface condition, while the free surface elev-
ation ζ(x, y)is not known a priori. Michell (1898) approached the problem by linearisation
of the boundary conditions, which are imposed only on the centreplane (y= 0) and still
waterline (z= 0). He used his Fourier–integral theorem to obtain the velocity potential and
pressure on the centreplane, i.e. on the projection of the hull. Havelock later replaced the
Fourier–integral theorem with a distribution of sources and sinks on the centreplane. The
thin– and slender–ship approximations satisfy the Laplace equation, the radiation condition,
and the Kelvin–Michell linearised free–surface boundary condition.
A steadily moving body near the free surface produces free waves with a steady surface
wave pattern. Free–wave elevation far away from the ship equals to the sum of amplitudes of
waves that travel at various angles of propagation, θ, relative to the vessel direction. The hull
and flow around the hull are assumed symmetrical about the centreplane, so that only the
starboard is considered, θ[π/2, π/2]. In ship–fixed reference system, the wave elevation
is defined as:
ζ(x, y) = <
A(θ) exp {−i k (θ) [xcos θ+ysin θ]}dθ, (1)
where A(θ)is the complex amplitude function of the free wave for an angle of propagation,
and kis the wave number that is defined as:
where k0is the basic wave number, k0=g/U2,Uis the ship speed, and λis the secant of
wave propagation angle, λ= sec θ.
Because the total energy in the far–field wave pattern arises from the moving ship, it is directly
related to the wave–pattern resistance, RW P . The value of the wave pattern resistance force
is given for infinite water depth by the following expression:
RW P =π
2ρ U2
|A(θ)|2cos3θdθ, (3)
where ρis the fluid density. Therefore, the wave resistance depends quadratically on the
wave amplitude, with a cubic weighting factor related to the wave propagation angle.
The complex amplitude function, A(θ), depends on the hull shape, angle of propagation
of waves, and the ship speed, U. It can be computed numerically, e.g. by the thin–ship
theory, CFD (Amini-Afshar and Bingham,2018), or it can be obtained by experimental
measurements (Degiuli et al.,2003). The complex amplitude function in infinite depth is
generally defined as:
A(θ) = 2
πi k (θ)2[P(θ) + i Q (θ)] .(4)
For water of infinite depth and a thin body whose geometry is described with a half-breadth
function, Y(x, z), Michell’s theory indicates:
P(θ) + i Q (θ) = ¨S
xY(x, z) exp {k z +i x k0λ}dxdz, (5)
where sources are described with the longitudinal slope of the geometry, xY(x, z)Y (x, z)/∂x.
Besides the wave–pattern component, wave–making resistance includes nonlinear compon-
ents, which are primarily arising from wave breaking at the bow and stern (Miyata et al.,
2.2. Numerical solution
If the ship hull geometry is described with a half-breadth function Y(x, z), then each station
of the hull can be described with a curve Y(X, z)from its lowest point of the section Zmin 0
to its uppermost point Zmax 0. Rather than evaluating Michell’s triple integral, Eq. (3),
integration by parts is used to separate the integral, which solves the problem with the
following steps.
1. The lowest-level integral obtains the contribution of a ship section for a wave propaga-
tion angle along the draught:
F(x, λ) =
xY(x, z) exp {z k (λ)}dz. (6)
2. The higher-level integral gathers contributions of ship sections for a wave propagation
angle along the ship (from bow to stern):
P(λ) + iQ (λ) = ˆL
F(x, λ) exp (i x k0λ) dx. (7)
In order to avoid dealing with complex–number numerics, P(λ)and Q(λ)from Eq.
(7) are described separately by two complementary integrals that replace exp (i . . .)
by cos (. . .)and sin (. . .)components. Those oscillatory integrals are then computed by
Filon’s integration formula (Tuck et al.,2001).
3. Finally, the following integral gathers the amplitudes of the free-wave spectrum for all
possible wave propagation angles:
RW P =4
πρ U2k2ˆ
0P2(λ) + Q2(λ)1 + t2dt, (8)
where the substitution λ=1 + t2is introduced to avoid integration difficulties due
to the integral singularity of Eq. (3) (Noblesse et al.,2008).
In conclusion, the wave resistance of a ship hull is obtained by evaluation of three separate
integrals, described by Eqs. (6)–(8). A single-threaded evaluation of the wave–resistance
numerical procedure takes on average 50 ms on a modern 3.6 GHz processor, using 250
transverse sections that describe the frame section by an array of 50 points distributed along
the vertical axis. This 250×50 grid was shown to offer adequate precision when evaluating
Eqs. (6)–(8) for various hull forms tested in Section 4.
3. Theory modifications
The linearised wave–making resistance theory is usually valid for ships with large L/B ratio.
For conventional ship forms which have the L/B ratio from 6 to 8, the thin–ship theory gives
Figure 1: An example of the tangency correction applied on a bluff body, and the resulting change in
poor estimate of the wave–making resistance. The modifications of the thin–ship theory
described below expand the limitations of the theory so that it can be applied to common
hull forms.
3.1. Boundary layer
Bašić et al. (2017a) introduced a BL correction for potential flows by letting the flow follow
the outer BL frontier and the mean wake direction after the separation occurs. They showed
that the tangency correction technique may be applicable not only to streamlined forms,
but to bluff bodies as well. The treatment, called the tangency correction, is implemented
through virtual rotation of surface normals, and correspondingly tangents as shown in Fig. 1.
It was found that the correction angle, β, has a direct connection to the BL displacement
thickness, δ?:
∂s = tan {β(s)},(9)
where sis the direction of the streamline path. The distribution of BL displacement thick-
ness along the streamline, δ?(s), obtained by rotating normals, β(s), is responsible for
reproducing viscous pressure fields.
The question that remains is how to assume or calculate the distributions of βand δ?for
general ship hull forms. δ?should be calculated by marching downstream the streamlines from
the stagnation point, which are not known a priori. The problem can be solved by obtaining
streamlines on the double–body and then solving the IBL equations along the streamlines
Von Kerczek (1973) with empirical closure relations Drela (2013). To avoid the cumbersome
three–dimensional calculation of a double–body problem, in this study δ?is approximately
predicted by marching downstream the waterlines. Actual streamlines are three-dimensional,
and therefore, this cannot be reliable for areas where streamlines abruptly change their depth,
which is usually near the baseline. On the other hand, the tangency correction applied on
the bottom of the hull does not affect the wave pattern significantly, since the influence of
depth changes exponentially, which is visible in Eq. (6).
Based on the above assumptions, it can be concluded that if actual streamlines near the free
surface roughly follow the waterlines of the hull, then the inclusion of the waterline tangency
correction should yield more realistic wave patterns than the original thin–ship theory. In this
study, the distribution of δ?is calculated by solving the IBL equations along hull waterlines
waterline using XFOIL software (Drela,1989). The tangency correction angle is calculated
according to Eq. (9) as:
β= arctan {xδ?},(10)
where xδ?is the longitudinal rate of change of the displacement thickness. Following the
introduced assumptions, Eq. (6) is evaluated for modified slopes, not the original hull geo-
metry. The definition of the tangency correction defines the modified slope, or the tangency
of flow as:
xY|BL = tan {arctan {xY}+ ∆β}.(11)
Eq. (11) modifies surface tangents inside Eq. (6) to include the BL displacement effect by
rotating the surface normal around the vertical axis, depicted in Fig. 1, due to approximating
streamlines confined in their waterplane. Besides the thin–ship theory, the correction is
applicable to other variants of linear wave–making theory, e.g. (Noblesse et al.,2013;Zhu
et al.,2018).
3.2. Phase shifting
Difference of the pressure and velocity between the thin–ship and real geometry results in
difference of the phase and amplitude of the generated wave pattern. Besides the BL correc-
tion explained in the former section that modifies the generated wave amplitude, a correction
of the phase of the wave pattern around the thin–ship is needed (Tsubogo,2014). Even if
the dynamic pressure on the hull near free surface is similar between the thin and real ship,
the difference in the phase will arise nonetheless. The higher order influence is known to
be the major reason for the phase shift of the regular waves (Han et al.,2003). Total wave
phase advances considerably compared to the first order wave, while the amplitude of the
total wave height does not differ too much from that of the first order wave.
The second–order effect is strong near the bow and shoulder of the hull (Miyata et al.,2014).
Therefore, the hull shape forward from the fore shoulder (FS) is the main factor for the phase
shift. Since the influence of wave-making decays exponentially with depth, the shape of the
waterline may be taken as the main variable. In the spirit of simplicity of Michell’s theory,
approximate resolution for the problem is to modify the basic wave number within numerical
calculations, k0ˆ
k0, based on the waterline shape forward from the fore shoulder. It was
empirically found that there is a strong linear dependence between the change in the basic
wave number and the fore waterline “arc to projection length” ratio. The following expression
enhances the phase due to nonlinear effects for most common ship hulls:
k0=k01+3.7SF S
LF S 1,(12)
where ˆ
k0is the modified basic wave number, SF S is the length of the waterline curve measured
from the forward perpendicular (FP) to the FS, and LF S is the longitudinal distance measured
from the FP to the FS.
3.3. Oscillations
The resulting wave–resistance curve is known to oscillate, mostly at low Froude numbers.
Ananthakrishnan (1999) showed that the wave dampening by viscosity is stronger on short
waves. Although, Tuck et al. (2001) concluded that there is little dependence of the vis-
cosity on the free–surface elevation, unless the viscosity is increased over several orders of
magnitude to be even larger than the oceanographically relevant eddy viscosity. Without a
physical connection confirmed, any kind of damping in mathematical sense contributes the
stabilisation of the resistance curve for low Froude numbers. In this work, waves with very
small wavelength compared to the length of ship are simply discarded from the numerical
evaluation of the complex amplitude function, Eq. (7).
Figure 2: Sections of tested hull forms, which are sized relative to their model sizes.
4. Numerical experiments
This section presents the validation of the introduced enhancements for the thin–ship theory.
Numerical simulations predicted the total resistance of both slender and full hull types, shown
in Fig. 2, and the results are compared to experimental data. The hulls have different block
coefficients and L/B ratios from 3.6 to 12.5. The transoms were kept dry to remove their
dependency on the wave-making.
The total resistance force, RT, comprises of the wave and viscous resistance:
RT=RW+RF(1 + k),(13)
where RWis the wave resistance obtained by the numerical procedure explained in Section
2.2,RFis the frictional resistance obtained by the ITTC-1957 frictional correlation line, and
(1 + k)is the form factor obtained from the experimental measurements. In addition to Eq.
(13), the total resistance coefficient, CT, is also used as a relevant measure:
(1/2) ρ U2SW
where SWis the at-rest wetted surface area of the hull.
4.1. Wigley hull
Due to its simple geometry, the parabolic Wigley hull is often used as a benchmark for
methods that handle free surface (Andrun et al.,2018). The mathematical representation of
the Wigley hull is defined by:
y(x, z) = B
Figure 3: Total resistance coefficient (top graph) and force (bottom graph) of the parabolic Wigley hull
where Bis the breadth, Lis the length, and Tis the draught of the hull. Main dimensions
of the Wigley hull considered in this study are taken as L= 4.0 m, B= 0.40 m, and T=
0.25 m, which correspond to those of the model made by Ship Research Institute (Kajitani
et al.,1983).
Fig. 3compares the total resistance curves of the original and modified thin–ship theory to
the experimental measurements. Both resistance curves in graphs include the same amount
of viscous resistance force, RF(1 + k). The form factor 1 + k= 1.095 reported by Kajitani
et al. (1983) was used in the numerical simulations. The hollows of the numerical curve
follow the measurements, indicating that there is no need for shifting of the phase, since
0.0 0.2 0.4 0.6 0.8 1.0
Figure 4: Distribution of the boundary layer displacement thickness along the Wigley hull at depth of 0.2T.
SF S /LF S 1. The curve obtained by the original theory has exaggerated humps due to the
stern being too strong of a wave generator.
Fig. 4shows a distribution of the boundary layer displacement thickness at depth of 0.2Tpre-
dicted by XFOIL, which is compared to the experimental data of Shahshahan and Landweber
(1986). The value of δ?reaches 10% of the hull beam. More importantly, slope of the curve
defines the tangency correction, Eq. (10). It is noticeable that the wave pattern and viscous
pressure resistance strongly depends on the ship after–body, where the value of δ?rises ab-
ruptly and induces flow separation. The adverse pressure at stern leads to the rapid increase
of the BL thickness and viscous stresses, which is even the case for the slender Wigley hull.
Normals of aft regions in slender hull are only slightly rotated backwards, but still the BL
displacement thickness is significant and βcorrections must be incorporated. This is an
important understanding, which verifies the need for including the BL effects even for slender
forms with ratio L/B 10. Consequently, results of the modified theory in Fig. 3oscillate
to smaller extent, and are in better agreement with the experimental measurements than the
original theory.
4.2. Series 60 with CB= 0.6
Series 60 is a general-cargo hull form without a bulb, having U–shaped sections. It is often
used as a CFD validation test in combination to the Wigley hull. A model defined with LP P
= 3.0 m, B= 0.40 m, T= 0.163 m and CB= 0.6 in fixed model condition is simulated and
Figure 5: Total resistance coefficient (top graph) and force (bottom graph) of the Series 60 hull model with
CB= 0.6.
compared to the experimental data of Toda et al. (1992). The form factor 1 + k= 1.135,
calculated by Prohaska’s method, is used for obtaining the viscous resistance.
The comparison of the total–resistance force and coefficient obtained numerically and exper-
imentally is shown in Fig. 5. The original theory produces a local peak in the resistance
curve, which is unrealistically high and starts prematurely. The local maximum is reached
at F n = 0.28, while the experiment shows it should be reached around F n = 0.30. Inviscid
flow around the stern generates waves that are excessively superposed to waves generated by
the fore. Compared to the original–theory curve, the local peak and hollow of the modified
curve only slightly oscillates around the experimental points. As in the case of the Wigley
hull, the ship resistance curves benefit from the introduced modifications.
4.3. Delft 372
Broglia et al. (2014) conducted an experimental investigation on models of Delft 372 monohull
and catamarans. The model has typical lines of a high-speed displacement hull shown in Fig.
2, and principal dimensions of LP P = 3.0 m, B= 0.24 m, T= 0.15 m and CB= 0.403. The
experiments were conducted on a free model, while the simulations were done for a fixed model
in the displacement regime. This simplifies the validation without introducing additional
effects of the immersed transom during the semi-displacement regime. It is worth noting
that the relatively high ratio L/B = 12.5characterises the hull as slender, and Prohaska’s
method yields low form factor 1 + k= 1.05. Nevertheless, the original thin–ship theory
over-predicts the wave resistance of the hull for all simulated Froude numbers, while the
inclusion of the BL displacement thickness through the tangency correction suppresses the
stern wave system, and therefore, yields lower wave-making resistance. The measurements
by Broglia et al. (2014) show that for F n > 0.5both trim and sinkage change significantly.
Consequently, numerically obtained resistance curve for the fixed model start to deviate from
the measured values on a dynamic model. The resistance at high-speed displacement Froude
numbers is under-predicted without taking the trim and sinkage into account. The validation
of the modified theory for semi-displacement regimes is left for future work.
4.4. Wide-Light yacht
Sailing Yacht Research Foundation (SYRF) published the tank test data from their “Wide-
Light Yacht Project” for the hydrodynamics of a high performance yacht (Claughton,2015).
A comprehensive set of data is collected for the canoe body with and without appendages in
upright, heeled and yawed conditions. The canoe has a displacement of 197 kg, the waterline
length LW L = 4.60 m, and the maximum beam of B= 1.28 m. The hull ratio L/B = 3.6is
far from usual limits that characterise the ship as slender. Compared to the Delft 372 model,
the upright canoe model had experienced significant sinkage even at lower Froude numbers.
Therefore, this numerical experiment applies the change in sinkage for each simulated Froude
number, which is obtained from the experimental data by Claughton (2015). The trim is
Figure 6: Total resistance coefficient (top graph) and force (bottom graph) of the Delft 372 hull model.
ignored as it slowly changes with the canoe speed. The wave resistance of the upright canoe
body is computed using the introduced method, and the used value of the form factor 1+ k=
1.18 was obtained by Prohaska’s method. The results are compared against the measured
data in Fig. 7. Very good agreement between the measured and computed data is obtained.
4.5. KRISO containership
Besides a hull with a very small L/B ratio that was reported in the previous section, a difficult
test for the introduced corrections is to predict the resistance of a full hull form. KRISO
containership (KCS) has a typical containership form with a bulbous bow. The model is
Figure 7: Total resistance coefficient (top graph) and force (bottom graph) of the Wide-Light yacht canoe.
defined with LP P = 7.279 m, B= 1.019 m, T= 0.342 m and CB= 0.651, which is the
largest of the tested models as shown in Fig. 2. It has a conventional value of L/B = 7.1
and an extremely high value of the midship section coefficient, CM= 0.985.
Fig. 8shows the resistance curves of the KCS model obtained by the modified theory, which
is compared to the experimental data from (Chen et al.,2016) and Holtrop-Mennen’s method.
The value of the form factor 1+k=1.165 was calculated from the experimental data using the
method of Prohaska, which was used in the simulation. Very good agreement is found between
the modified thin–ship theory and the experiment. The simulation properly predicts rising of
the wave resistance with the speed, unlike the Holtrop-Mennen’s method that approximates
the wave resistance rise and misses its gradient from F n = 0.25 to F n = 0.3. The original
Figure 8: Total resistance coefficient (top graph) and force (bottom graph) of the KCS model.
theory frantically over-predicts the wave resistance. Fig. 9shows the wave-making pressure
distribution, obtained as described by Noblesse et al. (2008), which renders the difference of
the original and modified thin-ship theories. The hull lines rendered in Fig. 2show that the
stern region has steep geometry changes within the relatively small stern length. Changes
in geometry induce adverse pressure gradients that cause sudden development of the BL
displacement thickness and probable flow separation with vortices (Duy et al.,2017). The
previous numerical experiments had validated that the inclusion of BL displacement thickness
indeed damps far-field waves, but the KCS experiment shows that the tangency correction
can damp far-field waves irrelevant of the adverse–pressure gradient strength. This indicates
that the modified method can be potentially applied to full ship forms with realistic L/B
0.800 0.533 0.267 0.000 0.267 0.533 0.800
Pressure coefficient
Figure 9: Wave-making pressure distribution on the centreplane, obtained by the original thin-ship theory
(top image) and the modified theory (bottom image).
ratios and CBvalues.
4.6. Discussion
As it is noted in literature, the BL and wake affect the wave-making of a steadily moving
vessel. The resistance curves shown in Figs. 38, which are obtained using the unmodified
theory, have humps and hollows that are a consequence of the interaction of the inviscid bow
and stern wave systems. Since the tangency correction has proven to correctly model the
rapid increase of the BL displacement thickness for bluff bodies, the technique was applied
to ship hulls with streamlined forms.
In order to avoid difficulties of implementing an IBL solver, a simple empirical approximation
for βalong a general streamline was introduced, with the intent to validate the possibility
of improving the wave–making prediction. Although the approximation has corrected the
total resistance of the five introduced hull forms, the more general method using the IBL
should be implemented for general forms. Figs. 38show that the approximate modifications
enhance sources of far–field waves irrelevant of the adverse–pressure gradient strength, and
validate that the inclusion of the BL displacement thickness improves the interaction of the
bow and stern wave systems.
Generally, if the trim and sinkage are not taken into account for high-speed vessels, then
numerical methods frequently under-predict the resistance. In these cases, the original thin–
ship theory may purely accidentally yield accurate results, since the theory regularly over-
predicts the wave resistance at higher Froude numbers. Naval architects should avoid relying
on absolute results of the uncorrected original linear theories.
5. Conclusions
Commercially viable ships sail in the displacement regime with speeds merely smaller where
the wave-making resistance begins to increase toward its global maximum. Therefore, this
area is of interest to the naval architect. Michell’s integral has been analysed for over 120
years in order to understand why it yields errors to the order of magnitude within the area
of interest, especially for hulls that experience strong adverse pressure gradients. The aim of
this study was to give new insight into why the errors and oscillations in the resistance curve
occur, and to offer a simple approximate solution to the problem. The following conclusions
are drawn:
The inviscid wave-making theory generally over-predicts the wave resistance, due to
the stern being too strong of a wave generator as the inviscid potential-flow theory
restores (too) high pressure at the stern. Unrealistic stern–generated waves are hence
superposed to those generated by the ship fore part.
Inclusion of the displacement thickness of the boundary layer yields realistic pressure
gradients along the wetted surface. Consequently, including it in a potential–flow wave-
making method, such as the thin–ship method, generates more accurate far-field waves.
The numerical experiments conducted on five hull forms have validated that the tan-
gency correction properly includes the displacement thickness that damps far-field
waves, and therefore, makes resistance prediction more accurate.
It was found that the resistance curve of a thin ship starts to grow prematurely com-
pared to experimental measurements. The phase shift can be made by virtually scaling
the wave number, based on the waterline shape up to the fore shoulder.
Implementing the enhancements within the thin–ship method obtains more accurate
results compared to the original thin–ship theory and Holtrop-Mennen’s method. The
trend of the total resistance curve is properly estimated for the tested hull forms,
meaning that the thin–ship theory may be applied for arbitrary monohulls.
Further numerical experiments are necessary in order to examine the validity of the present
approaches for multihulls and hulls with immersed transom sterns.
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The wave-resistance coefficients of a Wigley ship form have been calculated by the Michell integral, using the Michell thin-ship centerplane distribution, a slender-body hull-surface distribution, and a slender-body centerplane distribution. These calculations have been carried out for two different cases:wave-resistance calculation without the effects of the boundary layer and wake, andcalculations including these effects. Results have been compared with the residuary resistance and the "measured" wavemaking resistance, obtained as the difference between the measured total and wake-survey resistances.
A turbulent boundary layer calculation scheme is presented which combines the Cumpsty-HeadSmith three-dimensional turbulent boundary layer calculation method, under the small cross flow approximation, with the Tuck-von Kerczek slender body potential flow at zero Froude number. A sample calculation for the Lucy Ashton double model conforming to the experiments of Joubert and Matheson is presented.
The far-field waves, and the related far-field wave drag, predicted by the Michell or Hogner theories are considered. These two classical theories are found to predict nearly identical transverse waves for all practical cases. However, the Michell and Hogner models predict divergent waves—dominant at high Froude numbers—that can differ significantly. Differences between the far-field waves predicted by the Michell and Hogner models are much smaller for catamarans, for which lateral interferences between the twin hulls overwhelm lateral interferences between the port and starboard sides of the hulls, than for monohull ships. Moreover, differences between the divergent waves predicted by the Michell and Hogner theories are larger for higher Froude numbers, bigger beam/length ratios, smaller draft/length ratios, or larger separation distances between the twin hulls of a catamaran.
Kelvin's classical 1-point ship model, the basic 2-point models of a monohull ship and a catamaran previously studied via elementary geometrical considerations, optimal versions of these two basic 2-point models, and a 4-point model of a catamaran (not considered previously) are analyzed within a common theoretical approach based on potential flow theory, the Green function associated with the Kelvin–Michell linear boundary condition at the free surface, the related Fourier representation of farfield ship waves, and the farfield stationary-phase approximation. The analysis shows that the apparent wake angle associated with the highest waves found inside the cusps of the Kelvin wake as a result of interferences among the divergent waves created by a ship at a high Froude number can be realistically predicted via elementary ship models. In particular, the apparent wake angles for monohull ships and catamarans can be predicted very well via a 2-point model or a 4-point model. Moreover, these models provide basic insight into main features of the farfield waves created by fast ships. Notably, differences between the amplitudes of the bow and stern waves are shown to have no influence on the occurrence of constructive or destructive interferences, although they affect the intensity of wave-interference effects (strong if the amplitudes of the bow and stern waves are commensurate, weak otherwise). Another notable conclusion is that lateral interference effects become more important as the Froude number increases. Indeed, lateral interferences between the twin hulls of a catamaran dominate longitudinal interferences between the fore and aft of the ship at Froude numbers greater than about 1. However, longitudinal interferences between the bow and the stern waves remain dominant for common fast monohull ships.
A Modification to the gravity-wave free-surface condition is suggested, to account for dissipative properties of thin surface layers of material such as broken-up ice, slush, or oil spills. A corresponding correction to michell's wave-resistance integral is established, and computations are carried out for a parabolic strut.
One of the reasons making the optimal design for a stern shape of a ship hull still difficult even nowadays is the complexity of flow field behind a transom stern. Study on effects of transom configurations on flow fields as well as on the ship resistances is therefore necessary and useful for the transom shape optimization. In this paper, the simulated results of flow fields behind transom stern of a container ship hull with various transom configurations using a CFD method are presented. Transom stern of an original container ship hull is modified in both breadth and height to get new models with various stern shapes. A flow solver which solves the Reynolds Averaged Navier-Stokes (RANS) equations for incompressible flows is employed in the computations of flow fields around ship hulls. The computational results are compared with each other to investigate correlations of the transom shapes with flow fields behind the ship stern and ship resistances. It is considered that the achieved results provide useful information for the practical optimization design of a stern shape of a ship hull.