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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

Abstract

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: jobasic@fesb.hr (Josip Bašić), bblag@fesb.hr (Branko Blagojević),

mandru00@fesb.hr (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

2

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

3

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.

4

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) = <

−π/2

ˆ

π/2

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:

k=k0λ2,(2)

where k0is the basic wave number, k0=g/U2,Uis the ship speed, and λis the secant of

wave propagation angle, λ= sec θ.

5

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

−π/2

ˆ

π/2

|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.,

2014).

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.

6

1. The lowest-level integral obtains the contribution of a ship section for a wave propaga-

tion angle along the draught:

F(x, λ) =

Zmax

ˆ

Zmin

∂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

0

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

7

Figure 1: An example of the tangency correction applied on a bluﬀ body, and the resulting change in

streamlines.

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?:

∂sδ?≡∂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

8

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

9

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, k0→ˆ

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:

ˆ

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).

10

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:

CT=RT

(1/2) ρ U2SW

,(14)

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

2"1−2x

L2#·1−z

T2,(15)

11

Figure 3: Total resistance coeﬃcient (top graph) and force (bottom graph) of the parabolic Wigley hull

model.

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

12

0.0 0.2 0.4 0.6 0.8 1.0

x/L

0.00

0.02

0.04

0.06

0.08

0.10

0.12

δ?/B

Experiment

XFOIL

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

13

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

14

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

15

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

16

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

17

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

18

0.0

0.5

01234567

0.0

0.5

−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. 3–8, 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. 3–8show 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

19

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.

20

Further numerical experiments are necessary in order to examine the validity of the present

approaches for multihulls and hulls with immersed transom sterns.

Acknowledgements

The authors wish to thank Leo V. Lazauskas for useful and interesting discussions on the

subject.

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