Hydrodynamic performance of flapping wings for augmenting ship

propulsion in waves

Kostas A. Belibassakis n, Gerasimos K. Politis

School of Naval Architecture and Marine Engineering, National Technical University of Athens, Zografos 15773, Athens, Greece

a r t i c l e i n f o

Article history:

Received 8 November 2012

Accepted 24 June 2013

Keywords:

Biomimetic ship propulsion

Flapping wings

Energy from waves

a b s t r a c t

The present work deals with the hydrodynamic analysis of flapping wings located beneath the hull of the

ship and operating in random waves, while travelling at constant forward speed. The system is

investigated as an unsteady thrust production mechanism, augmenting the overall ship propulsion.

The main arrangement consists of a horizontal wing in vertical motion induced by ship heave and pitch,

while pitching about its own pivot axis that is actively set. A vertical oscillating wing-keel is also

considered in transverse oscillatory motion, which is induced by ship rolling and swaying. Ship flow

hydrodynamics are modeled in the framework of linear theory and ship responses are calculated taking

into account the additional forces and moments due to the above unsteady propulsion systems.

Subsequently, a non-linear 3D panel method including free wake analysis is applied to obtain the

detailed characteristics of the unsteady flow around the flapping wing. Results presented illustrate

significant thrust production, reduction of ship responses and generation of anti-rolling moment for ship

stabilization, over a range of motion parameters. Present method can serve as a useful tool for

assessment, preliminary design and control of the examined thrust-augmenting devices, enhancing

the overall performance of a ship in waves.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Biomimetic propulsors is a subject of intensive investigation,

since they are ideally suited for converting environmental (sea

wave) energy to useful thrust. Research and development results

concerning the operation of flapping foils and wings, supported

also by extensive experimental evidence and theoretical analysis,

have shown that such systems at optimum conditions could

achieve high thrust levels and efficiency; see, e.g., Triantafyllou

et al. (2000, 2004), Taylor et al. (2010). Exploitation of the above

systems for marine propulsion is thus an interesting subject,

taking also into account that ship energy efficiency management

and reduction of pollution is currently recognized to be an

important factor in sea transport. In particular, previous studies

by various authors, as the ones by Scherer (1968) and Yamaguchi

and Bose (1994), have illustrated the application of oscillating

wings to marine propulsion with considerable efficiency. In addi-

tion, significant progress has been reported concerning the possi-

bility of such systems to extract energy from waves. In this

direction, a two-dimensional oscillating hydrofoil has been exam-

ined by De Silva and Yamaguchi (2012) for wave devouring

propulsion, and the propulsive performance of 3D flapping wing

in unbounded fluid and random heaving conditions has been

studied by Politis and Politis (in press) using active pitch control.

A main difference between a biomimetic propulsor and a

conventional propeller is that the former absorbs its energy by

two independent motions: the transverse to the mean incoming

flow motion and the angular with respect to its pivot axis motion,

while for the propeller there is only rotational power feeding. In

real sea conditions, the ship undergoes a moderate or higher-

amplitude oscillatory motion due to waves, and the vertical and/or

transverse ship motions could be exploited for providing one of

the modes of combined/complex oscillatory motion of a biomi-

metic propulsion system free of cost; see Rozhdestvensky and

Ryzhov (2003). At the same time, due to waves, wind and

other reasons, ship propulsion energy demand in rough sea is

usually increased well above the corresponding value in calm

water for the same speed, especially in the case of bow and

quartering seas.

In the present work we consider the operation of randomly

oscillating wings, located beneath the hull of the ship, as unsteady

thrust-production mechanism, augmenting the overall propulsion

system of the ship. The case of a single biomimetic propulsor

consisting of a uni-block wing, as shown in Fig. 1, will serve as the

basis of our study. The main arrangement is shown in Fig. 1(a) and

consists of a horizontal wing undergoing combined vertical and

angular oscillatory motion. The vertical motion is induced by ship

heave and pitch, while the wing pitching motion about its pivot

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Ocean Engineering

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.oceaneng.2013.06.028

n Corresponding author. Tel.: +30 210 7721138; fax: +30 210 7721397.

E-mail address: kbel@fluid.mech.ntua.gr (K.A. Belibassakis).

Ocean Engineering 72 (2013) 227–240

axis is actively set in terms of the vertical motion. A second

arrangement is also considered, as shown in Fig. 1(b), consisted of

a vertical oscillating wing beneath the hull of the ship. In this case,

the transverse oscillatory motion is induced by ship rolling and

swaying, and the pitching motion of the wing about its pivot axis

is properly selected in order to produce thrust with significant

generation of anti-rolling moment for ship stabilization.

Ship flow hydrodynamics are modelled in the framework of

linear theory using a Rankine source-sink formulation and ship

motions are calculated taking into account the additional forces

and moments due to the above flapping propulsion system (see,

e.g., Sclavounos and Borgen, 2004). The latter are formulated using

a combination of unsteady lifting line theory, in conjunction with

unsteady hydrofoil theory (e.g., Newman, 1977, Section 5). At a

second stage, a fully 3D non-linear panel method developed by

Politis (2009, 2011) is applied to obtain the detailed characteristics

of the lifting flow around the flapping wing. Free-wake analysis is

incorporated to account for the effects of non-linear wing wake

dynamics at high translation velocities and amplitudes of the

oscillatory motion. Predictions of the performance of the present

system obtained by the above simplified model are compared

against the 3D panel time-stepping method which includes the

complex unsteady trailing vortex rollup in the modelling of

biomimetic wing flow dynamics. From this comparison experience

is gained regarding tuning corrections of the simplified model to

better approximate larger amplitudes of oscillatory wing motions.

It is shown that the linearized theory offers a generally good

approximation to our problem, due to the relatively small Strouhal

number (and reduced frequency) in which the flapping wings

operate. Finally, numerical results are presented concerning the

thrust produced by the examined biomimetic system as well as

the resulting reduction in ship responses over a range of motion

parameters, showing that the present method can serve as a useful

tool for the assessment and the preliminary design and control of

such thrust-augmenting devices, enhancing the overall propulsive

performance of a ship in a wavy environment.

The present paper is structured as follows: In Section 2 the

main kinematics and dynamics of flapping foils and wings in

infinite flow domain are reviewed and the geometrical details of

the examined configurations are presented. Subsequently, the

mathematical models applied to treat the ship and flapping wing

hydrodynamics are presented in Sections 3 and 4, respectively,

where also additional references to previous research are pro-

vided. Specific examples of the examined system responses in

harmonic waves are presented and discussed in Section 5, illus-

trating the thrust generation in conjunction with reduction of

responses by the operation of the flapping wing as first evidence

concerning energy extraction from waves. Finally, in Section 6

behaviour of the above system is examined in random waves,

represented by frequency spectra, and numerical results are

provided concerning the generation of thrust and anti-rolling

moment by both arrangements considered.

midship

wingx

yz

x

midship

yz

xLE

6m

TE

Fig. 1. (a) Ship hull equipped with a horizontal flapping wing located below the keel, forward the midship section. (b) Same hull with a vertical flapping wing located below

the keel, at midship. Geometrical details of the flapping wings are included in the upper subplots, where the main flow direction is indicated by using a blue arrow and the

oscillatory motions by using dashed black (ship induced oscillation) and red lines (controlled oscillation), respectively. (For interpretation of the references to color in this

figure legend, the reader is referred to the web version of this article.)

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240228

2. Biomimetic flapping-wing thrusters

2.1. Kinematics of flapping wings

For the description of the kinematical characteristics of the

flapping wing and of the induced flow dynamics various reference

systems are considered, as the motionless inertial system, the

ship-fixed coordinate system, which is moving forward with

velocity U and oscillating with respect to the fundamental degrees

of freedom (sway, heave, roll, pitch) due to waves ξ2;3;4;5, and the

body-fixed coordinate system attached to the flapping wing, that

undergoes a complex translational and oscillatory motion. In the

case of simple periodic oscillations, two distinct frequencies enter

into play, the relative frequency ω1 ¼ 2π f 1 due to waves and the

oscillatory wing frequency with respect to its own pivot axis

ω2 ¼ 2π f 2. In the case of horizontal arrangement (Fig. 1a), the

vertical motion is ZðtÞ ¼ Z0 þ hðtÞ, where hðtÞ ¼ h0 sin ðω1tÞ, Z0

denotes the average submergence of the wing, and h0 is the

amplitude of vertical oscillation of the ship (h0 ¼ ξ30$xwingξ50) at

the horizontal position xwing along the ship where the flapping

thruster is located. Simultaneously, the wing undergoes an

angular oscillatory motion with respect to its pivot axis θðtÞ ¼

θ0 sin ðω2 t þ ψÞ, at a possibly different frequency (ω2). In the case

of the vertical arrangement (Fig. 1b), a transverse oscillation is

induced by combined sway and roll of the ship hðz; tÞ ¼

h0ðzÞ sin ðω tÞ, where now h0ðzÞ ¼ ξ20$zξ40 is a spanwise variable

amplitude with z denoting the vertical distance along the pivot

axis of the wing. At the same time the vertical flapping wing

undergoes a controlled pitching motion abound its pivot axis

described by θ ðtÞ ¼ θ0 sin ðω2 t þ ψÞ.

2.2. Time-dependent angle of attack and active wing-pitch control

In the framework of potential flow applications the main

parameter controlling the unsteady thrust production of flapping

systems, advancing at forward speed U in unbounded liquid, is the

Strouhal number St ¼ 2f h0=U, while the Reynolds number is used

to calculate viscous drag corrections. Also, the phase difference ψ

between the two oscillatory motions is very important as far as

the efficiency of the thrust development by the flapping system

qis concerned. In the single harmonic thrust producing case (ω1 ¼

ω2 ¼ω) it usually takes the value ψ¼901 (see, e.g., Anderson et al.,

1998; Schouveiler et al., 2005), in which case the required torque for

wing pitching is found to be minimum when the pivot axis for the

angular motion of the wing is located around 1=3$1=4 chord length

from the leading edge. As a result of the simultaneous heaving and

pitching motions of the biomimetic wing, in the case of horizontal

arrangement (Fig. 1a), the instantaneous angle of attack is

αðtÞ ¼ θ HðtÞ$θ ðtÞ ¼ tan$1ðU$1dh=dtÞ$θ ðtÞ ð1Þ

For relatively low amplitudes of harmonic motion and opti-

mum phase difference ψ¼901, the angle of attack becomes

αðtÞ ¼ ðU$1h0ω$θ0Þ cos ðωtÞ, which can be equivalently achieved

by setting the pitch angle θ ðtÞ proportional to θ HðtÞ as follows:

θðtÞ ¼w tan$1ðU$1dh=dtÞ; and thus θ0 ¼w h0ω=U ð2Þ

In contrast to passively controlled flapping-wing thrusters (see,

e.g., Murray and Howle, 2003; Bøckmann and Steen, 2013) in the

present work an active system is studied based on the above

parameter w, as introduced and discussed by Politis and Politis (in

press). The latter is termed the pitch control parameter and usually

takes values in 0owo1, which is furthermore amenable to

optimization. Decreasing the value of w the maximum angle of

attack is reduced and the wing operates at lighter load. On the

contrary, by increasing the above parameter the wing loading

becomes strong and could lead to leading edge separation and

dynamic stall effects. In the present work we exploit the above

result as an active pitch control rule of the flapping-wing thruster,

not only for purely harmonic oscillations, but also in the general

multichromatic case, applied to the time history of vertical

oscillatory motion hðtÞ of the wing. Thus, the instantaneous angle

of attack, Eq. (1), takes the form

αðtÞ ¼ ð1$wÞ tan $1ðU$1dh=dtÞ ð3Þ

In the case of the vertical arrangement (Fig. 1b), the instanta-

neous angle of attack at any spanwise position z of the wing is

given by

αðz; tÞ ¼ ϑðz; tÞ$θ ðtÞ ¼ tan$1ðU$1dh=dtÞ$θ ðtÞ; ð4Þ

where ϑðz; tÞ ¼ tan $1ðhðz; tÞ=UÞ and θðtÞ is the wing pitch angle

about its pivot axis. For relatively low sway-roll amplitudes, purely

harmonic motion, and the pitch angle selected according to the

previous rule, Eq. (4) becomes αðz; tÞ ¼ ðϑðzÞ$θ0Þ cos ðω tÞ, and can

be equivalently achieved by setting the wing pitch angle as follows

θðtÞ ¼ θ0 cos ðω tÞ ¼wϑðzÞ cos ðω tÞ; ð5Þ

where ϑðzÞ≈h0ðzÞω=U and w as before acts as the wing-pitch

control parameter. We note that for the vertical oscillating wing

the amplitude ϑðzÞ is variable spanwise, and the setting of the wing

pitch about its pivot axis requires a reference spanwise position. To

this aim we select the midspan section of the wing ðzmidÞ and as a

result, the instantaneous angle of attack is defined by

αðz; tÞ ¼ tan$1ðU$1dh=dtÞ$w tan $1ðU$1dhmid=dtÞ;

where hmid ¼ ξ20$Zmidξ40 ð6Þ

2.3. Free-surface effects

In the case of the biomimetic system under the calm or wavy

free surface, additional parameters enter into play, as Froude

number(s) F ¼ U=ðgℓÞ1=2, with ℓ denoting the characteristic

length(s) and g is gravitational acceleration, as well as frequency

parameter(s) associated with the incoming wave, as μ¼ω2ℓ=g and

τ¼ ω U=g, the latter being used to distinguish subcritical ðτo1=4Þ

from supercritical ðτ41=4Þ conditions; see Nakos and Sclavounos

(1990).

2.4. Geometrical parameters of the examined configurations

Planform area parameters of the flapping wing including sweep

and twist angles and generating shapes, ranging from simple

orthogonal or trapezoidal-like wings to fish-tail like forms (see,

e.g., Politis and Tsarsitalidis, 2009), constitutes the set of the most

important geometrical parameters. Other important parameters

are the wing aspect ratio, spanwise distribution of chord, thickness

and possibly camber of wing sections, as well as the specific wing-

sectional form(s). In the case of the wing operating under the hull

of the ship, additional geometrical parameters are the longitudinal

position xwing , strongly connected with the amplitude of vertical

oscillations due to waves and thus, very important for optimiza-

tion, as well as the clearance(s) with respect to the hull surface.

As an example, which is used in the present work for demon-

stration, we consider a variant of the series 60–Cb¼0.60 ship hull

form, shown in Fig. 1. The ship has main dimensions length

L¼50 m, breadth B¼6.70 m and floats at a draft T¼2.80 m. The

exact value of the block coefficient at the above draft is Cb¼0.533.

From hydrostatic analysis, the immersed volume in the mean

position is 500 m3, and the corresponding displacement in salt

water, which equals the mass of the ship, is estimated Δ¼512 t.

Moreover, in the above draft, the wetted area of the hull is

calculated Swet¼380 m2, the waterplane area AWL¼225 m2, the

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240 229

center of flotation xf ¼!1.15 m (LCF aft midship), the long-

moment of inertia of the waterplane is IL¼28,800 m4, and the

corresponding metacentric radius BML¼57.6 m. The vertical center

of buoyancy is KB¼1.55 m (from BL), and the longitudinal position

is LCB¼!0.266 m. For simplicity, we consider the long-center of

gravity to coincide with the center of buoyancy, i.e., XG¼!0.266 m

(aft midship), YG¼0, and KG¼1.80 m (from BL), and thus, the

metacentric height in the above condition are estimated to be

GM¼0.9 m. Also, the longitudinal metacentric height in the above

condition is estimated to be GML≈BML. Finally, the radii of gyration

about the x-axis and y-axis, respectively, are taken Rxx¼0.32B,

Ryy¼0.23L. The definition of the above quantities can be found in

standard texts of Naval Architecture; see, e.g., Lewis, 1989. Esti-

mated data concerning wave and total resistance of the above hull,

for two representative values of the ship speed in calm water, are

given in Table 1.

In the case of horizontal arrangement (Fig. 1a) the flapping

wing propulsor is located at a distance 15 m fore the midship

section (station 8 of the ship), at a depth d¼3.6 m below WL, and

thus, there is a clearance of 0.8 m between the wing (at horizontal

position) and the keel of the ship. The half-wing planform shape is

trapezoidal and its span is s¼6 m. Moreover, the root and tip

chords of the wing have lengths cR¼1 m, cT¼0.5 m, respectively,

and the leading edge sweep angle is Λ¼9.41 (see subplot of

Fig. 1a). On the basis of the above, the wing planform area is

SW¼4.5 m2, and its aspect ratio AR¼8. The wetted area of the

wing is at a good approximation double of its planform area. The

wing sections are symmetrical NACA0012, and thus the local max

thickness-to-chord ratio is kept constant over the span and equal

to 12%. The flapping wing rotates about a transverse axis passing

through the hydrodynamic center of the root chord (located

approximately 1/4 chordlength distance from the leading edge).

The same wing conditions have been considered in a recent work

by Politis and Politis (in press), operating under random heaving

condition in unbounded fluid, showing that it can produce high

propulsive thrust values with minimum requirements for power-

ing the controlled pitch motion.

In the case of the vertical arrangement (Fig. 1b) the flapping

wing is located at the midship section. The upper end (root) of

the wing is assumed at a depth d¼2.8 m and the span of the

vertical wing is s¼6 m. As before, the wing planform shape is

trapezoidal, the root and tip chords of the wing have lengths

cR¼1 m, cT¼0.5 m, respectively, the sections are NACA0012 and

the leading edge sweep angle is Λ¼1.61 (see Fig. 1). Thus, the wing

planform area is again SW¼4.5 m2, and its aspect ratio is AR¼8.

The wetted area of the wing is the double of its planform area. The

flapping wing oscillates due to the ship swaying and rolling

motions and simultaneously it rotates about a pivot axis passing

through 1/3 chordlength distance from the leading edge coincid-

ing with the vertical z-axis when the ship is at the upright position

(as shown in the subplot of Fig. 1b).

3. Ship hydrodynamic analysis

Methods for the calculation of dynamic ship responses in a

given sea state have been developed up to a satisfactory level of

accuracy (Ohkusu, 1996) and various 3D hydrodynamic models

based on BEM, both in the frequency and the time domain, are

available, see, e.g., Beck et al. (1996), Bertram and Yasukawa

(1996), Sclavounos et al. (1997). Also high-order methods, based

on B-spline and NURBS approximations have been developed; see

e.g., Kring and Sclavounos (1995), Huang and Sclavounos (1998),

Kim and Shin (2003). Comprehensive survey of theoretical and

computational methods has been presented by Beck and Reed

(2001).

3.1. Ship vertical motions

Standard linearized seakeeping analysis in the frequency

domain (Sclavounos and Borgen, 2004) is used in the present

work to obtain the motions and responses of the examined system

(ship and flapping wing). The coupled equation of heave

ξ3 ¼ Reðξ30eiωtÞ and pitch ξ5 ¼ Reðξ50e

iωtÞ motion of the ship (with

corresponding complex amplitudes ξ30 and ξ50) is as follows

(taking the origin at the center of gravity):

ð!ω2ðmþ a33Þ þ iωb33 þ c33Þξ30

þ ð!ω2ða35 þ I35Þ þ iωb35 þ c35 þ pÞξ50 ¼ F30 þ X30; ð7aÞ

ð!ω2ða53 þ I53Þ þ iωb53 þ c53Þξ30

þ ð!ω2ða55 þ I55Þ þ iωb55 þ c55Þξ50 ¼ F50 þ X50; ð7bÞ

where ajk and bjk; j; k¼ 3;5, are added mass and damping coeffi-

cients, m is the total mass of the ship and wing and p¼!iωUm.

The involved hydrostatic coefficients are c33 ¼ ρgAWL, c35 ¼ c53 ¼

!ρgðxfAWLÞ and c55 ¼m g GML. The inertia coefficients involved in

the above system (7) are I55 ¼mR2yy and I35 ¼ I53 ¼!m XG.

The terms F j0; j¼ 3;5; appearing in the right-hand side of

Eq. (7a) is the Froude–Krylov and diffraction vertical forces and

pitching moment (about the ship y-axis) amplitudes, respectively.

Furthermore, the terms Xj0; j¼ 3;5; denote additional force and

moment amplitudes due to the operation of the horizontal

flapping wing as an unsteady thruster. The latter are dependent

on heave ðξ3Þ and pitch ðξ5Þ responses of the ship, as well as to the

incoming wave field. We note here that due to oscillatory thrust

developed by the flapping wing its responses are also coupled

with the surge motion ðξ1Þ of the ship. However, taking into

account the large mass of ship, in conjunction with installation

of energy storage and power feed smoothing (flywheel-type)

systems, at first level of approximation in the present work the

above effect is neglected and is left to be investigated in future

extensions.

3.2. Ship transverse motions

The coupled equations of sway ξ2 ¼ Reðξ20eiωtÞ and roll

ξ4 ¼ Reðξ40eiωtÞ motion of the ship (with corresponding complex

amplitudes ξ20 and ξ40) are

ð!ω2ðmþ a22Þ þ iωb22Þξ20

þð!ω2ða24 þ I24Þ þ iωb24Þξ40 ¼ F20 þ X20ðξÞ; ð8aÞ

ð!ω2ða42 þ I42Þ þ iωb42Þξ20

þ ð!ω2ða44 þ I44Þ þ iωb44 þ c44Þξ40 ¼ F40 þ X40ðξÞ; ð8bÞ

where ajk and bjk; j; k¼ 2;4, are the corresponding added mass

and damping coefficients. The involved hydrostatic coefficient in

Eq. (8) is c44 ¼m g GM and the inertia coefficients are I44 ¼mR2xx

and I24 ¼ I42 ¼!m ZG (ZG¼T!KG). Similarly, the terms F j0; j¼ 2;4

appearing in the right-hand side of the above system are the

Froude–Krylov and diffraction vertical forces and rolling (about the

x-axis) moment amplitudes, and the terms Xj0; j¼ 2;4; denote

additional force and moment amplitudes due to the operation of

Table 1

Bare-hull resistance of the examined ship.

U (kn) U (m/s) F ¼U=ffiffiffiffiffi

gLp

CW % 103 CT % 103 R (kp) EHP (PS)

10.6 5.5 0.25 0.5 2.5 1504 110

12.9 6.6 0.30 1.8 4.0 3465 305

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240230

vertical flapping wing as an unsteady thruster. The latter are again

dependent on the responses of the ship, as well as to the incoming

wave field.

In Section 4 of the present work we shall employ an unsteady

lifting-line model to derive analytic expressions of these forces

Xj ¼ ReðXj0eiωtÞ; j¼ 3;5 or j¼ 2;4; in terms of the oscillatory ship

amplitudes and include the effects of flapping wing in the system

(7) and (8) coefficients, respectively. We note here that due to ship

hydrodynamics and oscillatory thrust developed by the flapping

wing its responses are also coupled with the surge and yaw

motions ðξ1; ξ6Þ of the ship. However, at first level of approxima-

tion, the above motions are considered to be small and are

neglected.

3.3. Ship hydrodynamics boundary element model

A low-order panel method, based on simple Rankine source-

sink distributions and quadrilateral 4-node elements, is used to

obtain the hydrodynamic analysis of the oscillating ship-hull in

waves, in the frequency domain, and to treat the steady problem of

the ship advancing with mean forward speed. In both cases, the

four-point, upwind finite difference scheme by Dawson (1977) has

been used to approximate the horizontal derivatives involved in

the (linearized) free surface boundary condition. Details concern-

ing the application of the above method, in the case of the steady

problem and in the presence of additional effects from lifting

appendages, can be found in Belibassakis (2011). We mention here

that a minimum number of 15–20 elements per wavelength is

used in discretizing the free surface, in order to eliminate errors

due to damping and dispersion associated with the above discrete

scheme (see Sclavounos and Nakos, 1988 and Janson, 1977).

An example concerning the calculated wave field exciting by

heaving and pitching ship hull motions, for Froude number F¼0.25

and reduced frequency τ¼ ω U=g¼0.42, as predicted by the present

model, is presented in Fig. 2(a) and (b), respectively. Corresponding

results concerning the calculated wave field exciting by enforced

rolling motion is presented in Fig. 2(c). The half-hull surface is

discretized by using a mesh of 22 (in the long direction) by 12

(sectional) panels, and the half symmetric part of the free surface by

using a mesh 42 (in the transverse direction) by 118 (in the

longitudinal direction) panels. Thus the total number of boundary

elements is 5220. In order to enforce the radiation condition,

Fig. 2. Calculated wave field exciting by (a) heaving and (b) pitching ship motion for F¼0.25 and τ¼0.42, as predicted by the present model. (c) Corresponding results

concerning ship rolling motion. Horizontal distances are scaled with respect to the length of the ship.

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240 231

which is particularly important for the subcritical (τ41/4) case, an

absorbing layer technique is used, based on a matched layer all

around the fore and side borders of the computational domain on

the free surface; see Sclavounos (Chap. 4 in Ohkusu, 1996). The

thickness of the absorbing layer is of the order of 1–2 characteristic

wavelengths and its coefficient is taken quadratically increasing

within the layer. The efficiency of this technique to damp the

outgoing waves with minimal reflection is dependent on the

thickness of the layer.

A similar result concerning the calculated steady wave pattern

around the same ship hull, also at Froude number F¼0.25, is

shown in Fig. 3. Based on the above analysis the values of the wave

resistance coefficient listed in Table 1 have been calculated. The

corresponding ones concerning the total calm-water resistance

have been obtained by adding also the residual coefficient from

numerical analysis based on RANSE and experimental measure-

ments (see, e.g., Tzabiras, 2004).

4. Hydrodynamic analysis of flapping wings in waves

Lifting appendages attached to ship hull are used to improve

the calm water performance and reduce responses in waves; see,

e.g., Sclavounos and Huang (1997). For example, trim tabs attached

to transom stern of high-speed vessel have shown to reduce the

calm water resistance (Cusanelli and Karafiath, 1997) and passive

and active systems are frequently used as anti-rolling stabilizers

(e.g., Naito and Isshiki, 2005). In the context of 3D BEM applica-

tions a Rankine panel method developed by Sclavounos and

Borgen (2004) has been applied to study the seakeeping perfor-

mance of a foil-assisted high-speed monohull. Furthermore, active

lifting system and devices used for the motion control of ships and

high-speed vessels in a sea state as well as dynamic positioning of

offshore vessels have also been extensively studied by Chatzakis

and Sclavounos (2006). Developments and applications of control

theory for marine vessels are extensively discussed in Fossen

(2002). The complexity of the selected model depends upon the

underlying physics, the properties of the controller and the desired

performance of the controlled system; see, e.g., Sclavounos (2006),

Thomas and Sclavounos (2007).

In the present work we employ a simplified lifting-line model

to derive expressions of the flapping wing forces XðξÞ in Eqs. (7)

and (8) in terms of the oscillatory ship amplitudes and include the

effects of flapping wing in the system coefficients. These forces are

separated into two parts: XðξÞ ¼XAðξÞ þ XBðφINCÞ, one part

dependent on the oscillatory ship amplitudes and one dependent

on the incoming wave potential. The first part produces modifica-

tions of the hydrodynamic coefficients of the system and the other

part adds on the Froude–Krylov and diffraction forces in the right-

hand side (Sclavounos and Borgen, 2004).

4.1. Horizontal arrangement of the flapping wing propulsor

For simplicity only head waves (β¼1801) are considered here as

excitation of the hull oscillatory motion concerning the horizontal

arrangement, however the present analysis is easily extended to

treat the more general case of waves of relative direction β with

respect to the ship-axis. In the examined case the working angle of

attack of the flapping wing is given by:

aðtÞ ¼ $ξ5ðtÞ þ1

U

∂φINC

∂z$dξ3ðtÞ

dtþ xwing

dξ5ðtÞ

dt

!

$δðtÞ; ð9Þ

where δðtÞ denotes the flapping wing pitch angle (the controlled

variable). The above formula is obtained by linearizing the tangent

for small angles. The second term in the right-hand side is due to

the contribution of oscillatory motion(s) of the ship and the incoming

waves, and this part, considered all together, is denoted by

εðtÞ ¼1

U

∂φINC

∂z$dξ3ðtÞ

dtþ xwing

dξ5ðtÞ

dt

!

ð10Þ

In the above formulas φINC is the incoming wave potential, for

unit amplitude (A) of free-surface elevation, which is given by:

φINCðx; z; tÞ ¼ Reig

ω0expðkzÞexpðiðkx þ ωtÞÞ

" #

; ð11Þ

where k¼ω20=g is the wavenumber of the incident waves and ω0

the absolute (angular) frequency. The frequency of encounter

(relative frequency) for head seas (β¼1801) is then given by

ω¼ω0 þ kU ¼ω0 þ ω20U=g ð12Þ

Finally, ∂φINC=∂z denotes the wave vertical velocity evaluated at

points on the flapping foil (xwing ,zwing). As it was previously

discussed in Section 2.2, the controlled pitch variable is set

proportional to the oscillatory angle δðtÞ ¼w γðtÞ, with the pitch

control parameter w ranging from 0 to 1, and thus, the angle of

attack finally becomes

aðtÞ ¼ $ξ5ðtÞ þ ð1$wÞ εðtÞ ð13Þ

Unsteady lifting line models based on the integration of 2D

sectional lift along the span can be used to obtain the wing lift and

moment (about the pivot axis at x0) coefficients. In the case of

flapping wings operating at relatively low reduced frequencies of

oscillation, the spanwise integration could be simplified (see also

De Laurier, 1993), leading to following approximate expressions

C3DL ¼

AR

AR þ 2

1

Sw

Z

cðyÞC2DL ðyÞ dy≈

AR

AR þ 2C2DL ; ð14aÞ

C3DM ¼

AR

AR þ 2

1

cRSw

Z

ðxCðyÞ$x0ÞcðyÞC2DL ðyÞ dy; ð14bÞ

where C2DL ðyÞ denotes the unsteady sectional lift coefficient of the

wing in the spanwise (y) direction, cðyÞ the chord distribution and

xCðyÞ the corresponding hydrodynamic center. The sectional lift is

estimated by means of unsteady hydrofoil theory (see, e.g.,

Newman, 1977) in terms of the dynamic angle of attack, which is

αðtÞ ¼ Reðα0eiωtÞ ¼ Refð1$wÞðA1 þ A2 þ A3ÞexpðiωtÞg; ð15aÞ

where

A1 ¼ $iω

Uðξ30$xwingξ50Þ;A2 ¼ $ξ50; and A3 ¼ U$1ð∂φINC=∂zÞ ð15bÞ

Fig. 3. Steady wave pattern for F¼0.25 as calculated by the present hydrodynamic

model. Horizontal distances are scaled with respect to the length of the ship.

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240232

The last term is associated with incoming wave velocity at the

position of the flapping wing. The sectional lift is obtained as a

superposition of the solutions corresponding to the oscillating

heave-pitch and sinusoidal gust problems, and in the case of

flapping wings of small back-sweep angle (as in the present

example) is approximately expressed as follows

C2DL ¼ 2πð ℂðkÞð1#wÞA1 þ A2ð Þ þ SðkÞð1#wÞA3Þ þ 2πk

2 h0c; ð16Þ

involving the Theodorsen ℂðkÞ and the Sears functions SðkÞ. The

latter are defined in terms of the Hankel functions of the second

kind as follows

ℂðkÞ ¼Hð2Þ1 ðkÞ=Π; SðkÞ ¼ 2i=ðπkΠÞ; Π ¼Hð2Þ1 ðkÞ þ iH 2ð Þ0 ðkÞ ð17Þ

where k¼ω cðyÞ=2U the local spanwise reduced frequency of each

wing section. The last term in Eq. (16), with h0 ¼ ξ30#xwingξ50, is the

added mass effect due to the heaving motion of each section (see, e.

g., Newman, 1977, Sec. 5.16). This particular contribution is con-

sidered to be more significant in comparison to the corresponding

one due to the pitching wing motion around its pivot axis, which is

neglected. For low frequencies of oscillation, and wing reduced

frequency k near zero, the functions ℂðkÞ≈SðkÞ≈1:0, and this result

permits us to obtain the following approximations of the vertical

force and moment due to the operation of flapping wing

X30 ¼1

2ρU2Sw C3D

L ≈1

2ρU2Sw

AR

ARþ 2C2DL ¼ Gα0 þ

GcRh0

4U2ω2; ð18aÞ

X50 ¼1

2ρU2cRSw C3D

M #1

2ρU2SwC

3DL xwing≈#xwingX30; ð18bÞ

where G¼ χ πρU2Sw AR=ðARþ 2Þ with χ a correction factor taking

values near unity that will be explained below. Using the above in

Eq. (7a), we obtain the following modifications of the system (hull

and flapping wing) damping and restoring-force coefficients due to

the operation of the horizontal flapping wing (from the term XΑðξÞ):

δb33 ¼ Gð1#wÞ=U; δb35 ¼ δb53 ¼#Gð1#wÞxwing=U ð19aÞ

δb55 ¼ Gð1#wÞx2wing=U; ð19bÞ

δc35 ¼ G; δc55 ¼#G xwing ð20Þ

Also, the flapping wing added mass leads to corresponding

changes in the system coefficients as follows

δa33 ¼#GcR=ð4U2Þ; δa35 ¼ δa53 ¼ xwingGcR=ð4U

2Þ;

δa55 ¼#x2wingGcR=ð4U

2Þ ð21Þ

In accordance to the above formulas, the added mass of the

flapping wing in vertical (heaving) oscillation is calculated to be

jδa33j≈2898 kg. The last result is obtained using standard density

of salt water ρ¼1025 kgm3 and χ¼1, and is found to be compar-

able to corresponding prediction based on detailed analysis by La

Mantia and Dabnichki (2013), table 2, for k¼2, corresponding to

oscillation in the lift direction). However, due to the large dimen-

sions of the ship, the effect of the flapping-wing added mass in

heaving motion, which is the most significant, is only 0.5–1% of the

total added mass of the system (ship and flapping wing) and can

be neglected. In addition, the following expressions are derived

concerning the part XBðφINCÞ of the hydrodynamic vertical force

and moment due to the flapping wing which is not dependent on

the ship-motion amplitudes:

X30 ¼ Gð1#wÞ ðiω0=UÞ expð#kdþ ikxÞ ð22aÞ

X50 ¼#Gð1#wÞ xwing ðiω0=UÞ expð#kdþ ikxÞ ð22bÞ

where z¼#d denotes the mean position below the free surface of

the flapping wing (mean submergenge of the flapping wing). The

linearized thrust coefficient is estimated from the corresponding

lift coefficient as follows

CT ¼T

0:5ρU2Sw¼ C3D

L sin ðθnÞ ð23Þ

where θn ¼ θ þ α if the sectional lift is assumed normal to the

instantaneous inflow or θn ¼ θ if it is assumed approximately

normal to the chord line.

Enhanced expressions could be obtained by keeping more

terms in the expansion of Theodorsen and Sears function about

k¼ 0, which will result into additional modifications in the

hydrodynamic coefficients and excitation forces. The above analy-

sis has been based on using Prandl's lifting-line theory, (Eq. (14)),

developed for an elliptically loaded wing at stationary angle of

attack, which is approximately correct for slowly varying flow

conditions. More complete unsteady lifting line models, as the

ones developed by Sclavounos (1987) and Guermond and Sellier

(1991), or lifting surface models for flapping wings (e.g.,

Belibassakis et al., 1997), could be employed to enhance predic-

tions, and this task is left to be examined in future work.

To illustrate the usefulness of the present simplified model, we

consider the horizontal flapping wing alone in infinite fluid

(without the presence of the ship and the free surface), and we

present in Fig. 4(a) and Fig. 5 results and comparisons of the

calculated lift and thrust by means of the above equations and a

3D panel time stepping algorithm developed by Politis (2009,

2011). The latter model has been validated through comparisons

against predictions by other numerical models and experimental

data in Politis and Tsarsitalidis (2009,2011). In the examined case

the wing undergoes harmonic heaving and pitching motions:

hðtÞ ¼ h0 sin ðωtÞ, θðtÞ ¼ θ0 cos ðω tÞ. In particular, the wake rollup

pattern behind the oscillating wing, in the case U¼1 m/s,

ω¼1 rad/s, h0/cR¼0.75 and θ0¼221, is shown in Fig. 4(a) as

calculated by the nonlinear 3D unsteady panel method. The latter

method is based on Green's theorem and Morino formulation, in

conjunction with free wake analysis; for more details see Politis

(2011). For this example, predictions concerning the lift and thrust

coefficients of the flapping wing as calculated by the above

simplified model and the fully nonlinear 3D panel method are

comparatively presented in Fig. 5, against the variation of the wing

pitching angle about its pivot axis θðtÞ and the dynamic angle of

attack αðtÞ. Calculations by the simplified model are shown by

using thin lines and are found in relatively good agreement with

panel method results shown in the same figure by using thick

lines. We also observe in Fig. 5 a time delay between the force

history of the linear model (thin line) and the panel method (thick

line). The same effect has also recently reported by Politis and

Tsarsitalidis, 2013. Finally, from data shown in examples like the

one in Fig. 5 and similar, we can estimate frequency- and

amplitude-dependent correction factor χðω; α0Þ that could be used

to better correlate the results of the above simplified model to the

3D panel time-stepping methodology.

4.2. Vertical flapping wing in quartering and beam waves

Concerning the operation of the vertical flapping wing (Fig. 1b)

as unsteady propulsor, in this section we consider head-quartering

and beamwaves (β¼1501 and 901) as examples of excitation of the

hull oscillatory motion. The working angle of attack of the flapping

wing at each section (corresponding to submergence z) is given

by:

aðtÞ ¼1

UW#

dξ2ðtÞ

dtþ z

dξ4ðtÞ

dt

!

#δðtÞ; ð24Þ

where W ¼ ð∂φINC=∂yÞny þ ð∂φINC=∂zÞnz is the incident wave velo-

city in the normal direction ðny;nzÞ ¼ ð cos ðξ4Þ; sin ðξ4ÞÞ of the

flapping wing (at the position of the pivot axis), and δðtÞ denotes

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240 233

the flapping wing pitch angle (controlled variable) with respect to

the pivot axis. The above formula is again obtained by linearizing

for small angles. The first term in parenthesis in the right-hand side

of Eq. (24) due to the contribution of ship oscillatory motion(s) and

the incoming waves considered together, is denoted by

ϑðz; tÞ ¼1

UW#

dξ2ðtÞ

dtþ z

dξ4ðtÞ

dt

!

; ð25Þ

which is now dependent on the submergence (z) of each wing

section. In the above formulas φINC stands for the incoming wave

potential corresponding to unit amplitude of free-surface elevation,

which in the present case is

φINCðx; z; tÞ ¼ Reig

ω0expðkzÞexpðiðkx þ ωtÞÞ

" #

; ð26Þ

where kx ¼ kx cos ðβÞ þ ky sin ðβÞ, k¼ ω20=g, is the (vector) wave

number of the incident waves and ω0 is the absolute (angular)

frequency. The frequency of encounter is

ω ¼ ω0#kU cos ðβÞ ¼ ω0#ω20U cos ðβÞ=g ð27Þ

As it was previously discussed, the controlled pitch variable is

set δðtÞ ¼w ϑðzmid; tÞ, with 0owo1. Thus, the angle of attack

becomes

aðz; tÞ ¼ ϑðz; tÞ#w ϑðzmid; tÞ; ð28Þ

with ϑðz; tÞ given by Eq. (25). Working similarly as before, by

integrating the 2D sectional lift along the spanwize (z) direction

we obtain the following expression for the lift

C3DL ¼

AR

AR þ 2

1

Sw

Z

cðzÞC2DL ðzÞ dz; ð29Þ

where C2DL ðzÞ denotes the sectional lift coefficient and cðzÞ the

corresponding chord of the vertical flapping wing. A similar

expression is derived concerning moment about the pivot axis of

the vertical flapping wing. Again, the sectional lift can be esti-

mated by means of unsteady hydrofoil theory in terms of the

dynamic angle of attack, which is now

αðtÞ ¼ Re½ðAS þ AR þ AW Þeiωt &; ð30Þ

where AS ¼ #ð1#wÞðiωξ20=UÞ, AR ¼ ðz#wzmidÞðiωξ40=UÞ, AW ¼

ð1#wÞW=U, the last term being associated with incoming wave

velocity. The sectional lift is obtained as a superposition of the

solutions corresponding to the oscillating transverse-rotational

and sinusoidal gust problems, and in the case of flapping wings

of small back-sweep angle (as the ones considered here) is

approximately expressed as follows

C2DL ¼ 2πððℂðkÞðAS þ ARÞÞ þ SðkÞAW Þ þ 2πk

2ðh0=cÞ; ð31Þ

involving as before the Theodorsen ℂðkÞ and the Sears functions

SðkÞ, with k¼ ω cðzÞ=2U the local spanwise value of the reduced

frequency of each wing section. Again, the last term in Eq. (31),

with h0 ¼ ξ20#zξ40, represents the added mass effect due to sway

and roll induced transverse motions at each wing section, which is

most significant in comparison to the corresponding one due to

the rotational wing motion about its (vertical) pivot axis. Taking

into account the fact that the most energetic wing sections operate

submerged at increased depth below the free surface, we further

simplify the above equation by considering the contribution of

wave velocity very small ðAW≪AS;ARÞ. On this basis, the following

approximate expressions are derived concerning the amplitudes of

angle of

attack ( )tα

pitching angle θ(t)

Fig. 5. (a) Pitching motion and angle of attack of the horizontal flapping wing of

Fig. 3. (b) Lift ðCL ¼ L=0:5ρU2SwÞ and (c) thrust ðCT ¼T=0:5ρU2SwÞ coefficients as

predicted by the simple unsteady model and as calculated the 3D panel time-

stepping algorithm.

Fig. 4. Vortical wake development behind the flapping wing in harmonic motion, as calculated by the present nonlinear 3D time stepping panel method. (a) Horizontal

arrangement for mean wing speed U¼1 m/s, flapping frequency ω¼1 rad/s, heaving amplitude h0/c¼0.75, pitching amplitude θ0¼221, phase difference ψ¼901. (b) Vertical

arrangement for mean wing speed U¼5.5 m/s, flapping frequency ω¼0.628 rad/s, rolling amplitude ξ40¼201. In both cases the wing pitch control parameter is set w¼0.5.

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240234

hydrodynamic transverse force and anti-rolling moment of the

vertical flapping wing

X20 ¼ iωðGSξ20 þ GRξ40Þ; X40 ¼ iωðMSξ20 þMRξ40Þ ð32Þ

where the involved coefficients

GS ¼$γð1$wÞ SW ; GR ¼ γ

Zðz$wzmidÞ cðzÞdz ð33aÞ

MS ¼ γð1$wÞ

ZzcðzÞdz

MR ¼$γ

Zz z$wzmidð Þ cðzÞdz;

γ ¼ χ πρUAR

ARþ 2ð33bÞ

the z-integrals are considered along the span of the wing, and χ is

similarly as in the previous subsection a correction coefficient.

Using the above equations, we obtain the following modifications

of the system (hull and flapping wing) damping coefficients:

δb22 ¼!GS; δb44 ¼!MR; δb24 ¼!GR; δb24 ¼!MS ð34Þ

Also, the flapping wing added mass produce corresponding

modifications in the system coefficients δa22, j; k¼ 2;4, which

however, due to the large dimensions of the ship, are found to

be negligible in comparison with the corresponding ones of the

oscillating hull. Finally, the thrust produced by the vertical flap-

ping wing is approximately calculated as follows

T¼ ðGS þ GRÞ sin ðϑðzmid; tÞÞ ð35Þ

Again, enhanced expressions could be obtained by keeping

more terms in the expansion of Theodorsen and Sears function

about k¼ 0, and better using more complete unsteady lifting line

models.

To illustrate the usefulness of the simplified model, results and

comparisons are shown in Fig. 4(b) and Fig. 6 concerning the

calculated lift and thrust by means of the above equations and the

unsteady panel method (Politis, 2011), in infinite fluid. In this case

the wing undergoes simple harmonic transverse and rotational

about its pivot axis motions. More specifically, in Fig. 4 the wake

rollup behind the wing as calculated by the 3D panel time-

stepping algorithm in the case U¼5.5 m/s, ω¼1.256 rad/s and

ξ40¼201 is illustrated. Predictions for the anti-rolling moment

and thrust as calculated by the above simplified model (thin lines)

and the 3D panel method (thick lines) are compared in Fig. 6.

Numerical results are plotted against the variation of the rolling

angle ξ4ðtÞ and angle of attack αðtÞ at midspan, and are found to be

in relatively good agreement. Similarly as before, the same effect

concerning the time delay between the moment history predicted

by the linear model (thin line) and the panel method (thick line) is

observed. Also, from data shown in Fig. 6 and systematic analysis,

we can calculate frequency- and amplitude-dependent correction

factor χ (appearing in Eq. (33)) that could be used to better

correlate results from the above simplified model with the pre-

dictions of the unsteady nonlinear 3D panel method, which would

further enhance the applicability of the simplified model, particu-

larly to moderate frequencies of operation.

5. Ship responses in harmonic incident waves with the effects

of flapping wing

The preceding analysis, in conjunction with Eqs. (7) and (8) for

the horizontal and vertical flapping wing, respectively, permits us

to calculate the ship responses including the effect of the flapping

wing operating as an unsteady thruster, and compare with the

corresponding seakeeping responses concerning the bare hull

without the wing.

In the case of the horizontal flapping wing (Fig. 1a) in head

waves, the normalized heave response of the ship with respect to

the incident wave amplitude ðξ30=AÞ is plotted in Fig. 7(a), as

calculated by the present method, for various values of the non-

dimensional wavelength ðλ=LÞ and Froude number F¼0.25

(U¼5.5 m/s). Also, in the same figure the modification of the

response obtained due to the operation of the horizontal flapping

wing is shown by using a thick solid line. We observe a significant

reduction of the ship heaving motion, especially in the vicinity of

the resonant condition (indicated by using an arrow). This result is

due to the damping effect from the operation of the harmonically

oscillating wing in flapping mode of operation, using w¼0.5 to

control the wing pitching motion with respect to its own pivot axis

(cf. Eq. (13)).

Furthermore, in Fig. 7(b) the same effect concerning the

calculated ship-pitch response ðξ50=kAÞ is presented. We can

clearly observe in this figure that the operation of the flapping

wing propulsor, shown again by using a thick solid line, leads to

significant reduction of pitch ship response, and especially for all

frequencies corresponding to wavelengths longer than the ship

length. The noticeable reduction of the above ship responses in the

case of horizontal arrangement is also evidence of significant

energy extraction from the waves by the present unsteady

thruster. An extra effect, strongly connected with the reduction

of ship responses due to the flapping wing, is the expected drop of

the added wave resistance of the ship. Indicative results concern-

ing the latter additional benefit will be given in the next

subsection.

In the case of the vertical flapping wing (Fig. 1b) the normalized

sway response of the ship with respect to the incident wave

amplitude ðξ20=AÞ is plotted in Fig. 8(a), as calculated by the

present method, for various values of the non-dimensional wave-

length ðλ=LÞ, wave incidence β¼1501 and Froude number F¼0.25

(U¼5.5 m/s). In the same figure the corresponding result obtained

with the operation of the flapping wing thruster is shown by using

a thick solid line. Similarly as before, we observe a significant

reduction of the swaying motion, especially around the resonant

condition (indicated by using an arrow), which is due to the

damping effect from the operation of the harmonically oscillating

vertical wing.

( )tα

ξ4 (t)

Fig. 6. (a) Rolling motion ξ4ðtÞ and angle of attack αðtÞ of the vertical flapping wing

of Fig. 4. (b) Anti-rolling moment ðCM ¼M=0:5ρU2zmidSwÞ and (c) thrust coefficient

ðCT ¼T=0:5ρU2SwÞ as predicted by the simple unsteady model and as calculated by

the biblinear 3D panel time-stepping algorithm.

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240 235

Although the present potential-flow model is not very accurate

as concerns the prediction of the ship roll responses, and especially

near the resonant conditions when no viscous damping terms and/

or bilge keel effects are included, we still use it in order to

demonstrate in Fig. 7(b) the effect of flapping vertical thruster on

the ship-roll response ðξ40=kAÞ as calculated by the present method.

We consider the same as before conditions, i.e., U¼5.5 m/s

(F¼0.25) and head-quartering (β¼1501) incident harmonic waves

of various frequencies corresponding to wavelengths, ranging from

very short to very long waves with respect to the ship length. We

clearly observe in this figure that the operation of the flapping wing

(results shown by using thick solid lines) leads to significant

reduction of roll responses, especially around resonant frequencies.

At the same time significant anti-rolling moment is generated by

the vertical flapping wing, which could found useful for ship

stabilization, permitting safe routing of the ship even in beam seas

and rough sea-conditions that would be undesirable and avoided

otherwise. More detailed calculations concerning the latter addi-

tional benefit will be presented in the next subsection.

6. Flapping wings augmenting ship propulsion in rough seas

In the context of applications of linear theory, numerical results

are presented and discussed in this section concerning the

performance of the system consisted of the ship together with

the (horizontal/vertical) flapping wing thruster, operating in ran-

dom wave conditions, as represented by frequency spectra. We

consider the responses of the system operating at various sea

conditions labelled by an index ranging from 1 to 5. The corre-

spondence of sea conditions with Beaufort scale (BF), the sea state

and the main spectral wave parameters, i.e., the significant wave

height (Hs) and the peak period (Tp) is given in Table 2. Also, the

Bretschneider model spectrum (see, e.g., Ochi, 1998) is used for

calculations.

As a first example, the thrust augmentation achieved by the

operation of the horizontal flapping wing propulsor in head waves,

is illustrated in Figs. 9 and 10, as calculated by the present method.

In particular, we consider the ship and flapping wing of Fig. 1(a) to

travel at constant speed U¼10.6 kn, in head seas (β¼1801)

corresponding to significant wave height Hs¼5 m, and peak period

Fig. 8. (a) Sway response ðξ20=AÞ and (b) roll response of the examined ship against

non-dimensional wavelength λ/L, for incident wave direction β¼1501 and ship

speed U¼5.5 m/s. The effect of the vertical flapping wing propulsor is shown by

using a thick solid line, and the reduction of maximum responses is indicated by

using arrows.

Fig. 7. (a) Heave ðξ30=AÞ and (b) pitch response ðξ50=kAÞ of the examined ship

against non-dimensional wavelength λ/L (where A is the harmonic wave amplitude

and k the corresponding wavenumber), for head seas (β¼1801) and ship speed

U¼5.5 m/s. The effect of the horizontal flapping wing propulsor is shown by using

a thick solid line, and the reduction of maximum responses is indicated by using

arrows.

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240236

Tp¼11 s, i.e., at sea condition 5 (see Table 2). The frequency

spectrum associated with the incident wave is plotted in Fig. 9

by using dotted line and the same spectrum in the moving ship

frame of reference is also shown by using thick solid line.

Furthermore, the response operator (RAO) concerning the vertical

ship motion response of the ship ðξ3"xwingξ5Þ at Station 8, where

the flapping wing is assumed to be located (xwing ¼ 15 m forward

of midship section) is shown in the same figure by using a thin

solid line. Finally, the calculated frequency spectrum of the vertical

ship motion at Station 8 is plotted by using a thick dashed line.

On the basis of the previous results, in Fig. 10, realization of

various quantities obtained by means of the random-phase model

(see, e.g., Ochi, 1998) in the form of time series are presented as

calculated by the present method and the spectra of Fig. 9. The

results shown in the first subplot of Fig. 10(a) concerns the vertical

motion ðξ3"xwingξ5Þ at St. 8, where the horizontal flapping wing is

arranged. The second subplot (Fig. 10b) shows the incident wave

velocity at the same location, and in the next subplot (Fig. 10c) the

calculated time series of ship pitching angle (ξ5) together the angle

of attack (α) at wing sections are plotted. In the final subplot the

thrust T(kp) produced by the horizontal flapping foil is shown, as

calculated by setting the control pitch variable w¼0.5. We observe

in the last subplot of Fig. 10(d) that the thrust oscillations

produced by the horizontal flapping wing in the above sea

condition are in the interval 0–20,000 kp, having an average value

of Tav¼2900 kp which is indicated by using dashed line. Similar

analysis for the ship with horizontal flapping thruster travelling in

the same sea state with increased speed U¼12.9 kn leads to lower

value of mean thrust Tav¼2000 kp, a drop which is attributed to

the reduction of mean and maximum angle of attack in this case.

The above results, including the predictions of thrust produc-

tion by the horizontal flapping wing, are based on the present

simplified model as described in Section 4.1. On the basis of

calculated system responses, complete simulation results are

produced concerning the detailed motion of the flapping wing in

waves including the rotational oscillatory controlled motion about

its pivot axis. The latter information is then used to obtain

enhanced predictions by the nonlinear unsteady 3D panel method.

These results, in the interval 10 soto24 s, are presented in Fig. 11

by using thick solid line and compared against the predictions by

the simplified model shown in the same plot by using thin lines.

The same effect concerning the time delay between the predic-

tions by the linear model (thin line) and the panel method (thick

line) is also observed in the multichromatic case. Although 3D

panel method results indicate somewhat smaller values of max-

imum and mean thrust level(s), in general, the predictions

Sea spectrum for Hs=5m, Tp=11s

(Bretschneider model spectrum)

RAO vert.motion @ St.8

Vertical motion

spectrum @ St.8

moving

frame

spectrum

Fig. 9. Sea spectrum (Hs¼5 m and Tp¼11 s) and vertical motion spectrum at

station 8 where the horizontal flapping wing is located, for head seas (β¼1801) and

ship speed U¼10.6 kn (F¼0.25).

Fig. 10. Stochastic responses of ship with horizontal flapping wing operating in

head seas (represented by the spectrum of Fig. 9) at ship speed U¼10.6 kn

(F¼0.25). (a) Vertical ship motion at station 8 without (solid line) and with

(dashed line) the effect of the flapping wing. (b) Vertical wave velocity at the

flapping wing. (c) Calculated ship pitch (dashed line) and angle of attack (solid line)

at the flapping wing, using w¼0.5. (d) Thrust production by the flapping wing

(time history). The time average is calculated to be 2900 kp and is indicated by

using dashed line.

Fig. 11. Thrust production by the flapping wing (detail of Fig. 10d in the time

interval 10 soto24 s). Comparison of predictions by the simplified unsteady

model (thin line) against calculations by the nonlinear 3D panel time stepping

method (thick line).

Table 2

Sea conditions and spectral parameters.

Sea condition BF Sea state Hs (m) Tp (s)

1 3 3 1 7

2 4–5 4 2 8

3 5–6 4–5 3 9

4 6 5 4 10

5 7 6 5 11

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240 237

provided by the simplified model are in quite good agreement,

concerning both modulus and phase of thrust, and could be

further improved by iterations.

We would like to remark here that the pitching torque

necessary for rotating the wing according to the control rule used

is found to be negligible (see also Politis and Politis, in press), and

thus, the present system enhances the overall propulsion effi-

ciency of the ship by extracting energy from the waves, while at

the same time it provides very useful reduction of motions

increasing the seakeeping performance. An important additional

effect that should be mentioned is the reduction of the added

wave resistance. To provide an indication, for the same as above

condition corresponsing to Froude number (F¼0.25, U¼10.6 kn),

the added wave resistance without the flapping wing has been

estimated by the energy method (see, e.g., Arribas, 2007) and sea

state 4–5 corresponding to HS ¼ 3 m; Tp ¼ 9 s, to be of the order

of calm-water ship resistance (RA ¼ 1300 kp) and almost doubles

at sea condition 5 (HS ¼ 5 m; Tp ¼ 11 s). The above values are

expected to be reduced by approximately 25% by the operation of

the flapping wing. In concluding, in the examples considered and

discussed above the flapping wing mean thrust production ranges

from 190% to 60% of the calm-water resistance of the ship at the

corresponding ship speeds (U¼10.6 and 12.9 kn, respectively), and

is found to be capable to overcome the added resistance due to

waves at these conditions.

To proceed further, the thrust and ship stability augmentation

by means of the operation of the vertical flapping wing propulsor

is illustrated in Figs. 12 and 13, as calculated by the present

method. In particular, we consider the ship of Fig. 1(b) to travel

at constant speed U¼10.6 kn, in head-quartering waves (β¼1501)

at sea state 5 (Beaufort scale 7). Sea conditions are represented

again by the Bretschneider model spectrum corresponding to

significant wave height Hs¼5 m and peak period Tp¼11 s, which

is plotted in Fig. 12 by using dotted line. The corresponsing

frequency spectrum in the moving ship frame of reference is also

shown in Fig. 12 by using dashed line. Moreover, in the same figure

the RAO and the frequency spectrum of the transverse motion

response at the wing midspan position (zmid) are plotted by using

thin and thick solid lines, respectively, as calculated by the present

method and described in Section 4.2.

Fig. 12. Sea spectrum (Hs¼5 m, Tp¼11 s) and spectrum of the combined transverse

oscillatory motion (due to sway and roll) at the wing midspan position (zmid), for

head-quartering (β¼1501) seas and ship speed U¼10.6 kn (F¼0.25).

Fig. 13. Stochastic responses of ship and flapping wing operating in head-beam (β¼1501) seas, represented by the spectrum of Fig. 12, for ship speed U¼10.6 kn (F¼0.25).

(a) Rolling motion with the effect from the flapping wing. (b) Normal velocity at the mid-span position of the wing due to ship rolling and swaying. (c) Calculated angle of

attack at the mid-span position of the flapping wing, using w¼0.5. (d) Thrust production by the flapping wing (time history). The time average is calculated to be 160 kp and

is indicated in the last subplot by using dashed line.

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240238

On the basis of the above analysis we present in Fig. 13 a

stochastic simulation of the vertical flapping wing operating in the

above conditions while travelling at U¼10.6 kn. In particular, time

series of various quantities are presented, including the ship

rolling (ξ4) motion in Fig. 13(a), the normal velocity at wing

midspan in Fig. 13(b), the angle of attack (α) in Fig. 13(c) and,

finally, in Fig. 13(d) the thrust T(kp) production by the flapping

wing, based on setting the control pitch variable w¼0.5, as

obtained from the corresponding spectra by applying the random

phase model (see, e.g., Ochi, 1998). In this case, as illustrated in the

last subplot of Fig. 13, the thrust oscillations are in the interval

0-750 kp and have average value of T¼150 kp. Similar analysis for

same ship speed and beam waves (β¼901) provides higher value

of average thrust T¼750 kp, which is attributed to the increase of

oscillatory frequency in this case. The corresponding anti-rolling

moments are estimated to be 8220 kpm and 22360 kpm, respec-

tively. Again, detailed comparisons between the present simplified

model and fully nolinear 3D panel method indicate quite good

agreement, as previously reported for the horizontal flapping wing

(see Fig. 11).

In the example considered and discussed above the flapping

wing is shown to produce significant anti-rolling moment fully

capable of ship stabilization, while at the same time it provides

useful thrust production ranging from 10% to 50% of the calm-

water resistance of the ship at the same speed. As in the case of

the horizontal flapping wing, the torque required for wing pitch-

ing according to the control rule is small and thus, the present

system enhances the propulsive performance of the ship while

providing significant dynamic stabilization.

Finally, in Fig. 14 systematic results obtained by the simplified

model are presented for both horizontal and vertical arrange-

ments. In particular, in Fig. 14(a) the thrust production by the

horizontal flapping wing operating in head waves is shown for

various ship speeds and sea conditions, ranging from almost calm

conditions to rough seas (see also Table 2). To give a more clear

idea concerning the level of thrust augmentation achieved by the

operation of the examined system, predictions are expressed as

percentage of the corresponding values of the calm water resis-

tance at the ship speeds considered (see Table 1). Additionally, in

Fig. 14(b) the thrust (solid lines) and the anti-rolling moment

(dashed lines) are plotted, as produced by the vertical flapping

wing in head-quartering (β¼1501) and beam (β¼901) waves, for

ship speed U¼10.6 kn (5.5 m/s) and various sea conditions.

Numerical results concerning thrust are as before expressed with

reference to the calm-water resistance and the anti-rolling

moment as percentage of the static righting moment of the ship

at 101 heel angle (m GM sin101).

Our calculations indicate that the examined biomimetic system

could offer a suitable mechanism for converting hull kinetic

energy due to waves to useful thrust, augmenting ship propulsion

in rough seas with simultaneous reduction in ship motions. The

whole performance of the system could be further enhanced by

consideration of multi-block or multiple wings, in vertical (one

under the other) or in horizontal (fore/aft) arrangement, as well as

by the examination of more complex (as e.g., fishtail-like) wing

forms. Systematic applications to other ship hull and wing forms

and operation conditions will be subject of future work.

7. Conclusions

The analysis of horizontal and vertical flapping wings located

beneath the hull of the ship is presented and discussed. The

system is examined in harmonic and multichromatic motion as an

unsteady thruster, not in a stand-alone mode of operation but in a

mode augmenting the propulsion system of the ship. The wing

undergoes a combined flapping and pitching oscillatory motion,

while travelling at constant speed and in the presence of waves. In

the horizontal arrangement, the vertical wing motion is induced

by ship heave and pitch. A second arrangement is also considered

consisted of a vertical oscillating wing-keel. In this case, the

transverse oscillatory motion is induced by ship rolling and

swaying. The pitching motion of the wing about its pivot axis is

selected in order to produce thrust, with significant reduction of

responses and generation of anti-rolling moment by the vertical

wing, useful for ship stabilization. Ship flow hydrodynamics are

modeled in the framework of linear theory using Rankine source-

sink formulation, and ship responses are calculated taking into

account the additional forces and moments due to the above

unsteady propulsion systems. Also, a fully 3D non-linear panel

method is applied to obtain the detailed characteristics of the

unsteady flow around the flapping wing, incorporating free-wake

analysis to account for the effects of non-linear wake dynamics, at

high translation velocities and amplitudes of the oscillatory

motion. Numerical results are presented indicating significant

Fig. 14. (a) Thrust production by the horizontal flapping wing in head waves for

various sea conditions and ship speeds. (b) Thrust production (solid lines) and anti-

rolling moment (dashed lines) by the vertical flapping wing in head-quartering

(β¼1501) and beam (β¼901) waves for various sea conditions. Numerical results

concerning thrust are expressed as percentage of the corresponding calm-water

resistance (at the same speed from Table 1), and concerning anti-rolling moment as

percentage of the static righting moment of the ship at 101 heel angle (m GM

sin101).

K.A. Belibassakis, G.K. Politis / Ocean Engineering 72 (2013) 227–240 239

thrust produced by the examined biomimetic system and reduc-

tion of ship responses over a range of motion parameters. Thus,

the present method, after experimental verification, can serve as a

useful tool for the assessment, preliminary design and control of

the examined thrust-augmenting devices, enhancing the overall

performance of a ship in waves. In the same direction, application

of more elaborate control methods would be beneficial for

optimization of performance. Finally, the described model permits

extension to various directions, as the development and applica-

tion of more elaborate unsteady lifting line and lifting surface

models for enhancing the predictions concerning the operational

characteristics of the flapping system in waves, the inclusion of

various non-linear effects (waves and ship hydrodynamics,

dynamic stall) and consideration of hydroelasticity effects due to

wing(s) flexibility.

Acknowledgements

This research has been co-financed by the European Union

(European Social Fund – ESF) and Greek national funds through

the Operational Program “Education and Lifelong Learning” of the

National Strategic Reference Framework (NSRF) 2007–2013:

Research Funding Program ARISTEIA - project BIO-PROPSHIP:

«Augmenting ship propulsion in rough sea by biomimetic-wing

system».

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