1 Copyright © 2014 by ASME

HYDRODYNAMIC ANALYSIS OF FLAPPING-FOIL THRUSTER

OPERATING IN RANDOM WAVES

E.S. Filippas School of Naval Architecture

and Marine Engineering National Technical University of Athens

Zografos 15773, Athens, GREECE. Email: evfilip@central.ntua.gr

K.A. Belibassakis School of Naval Architecture

and Marine Engineering National Technical University of Athens

Zografos 15773, Athens, GREECE. Email: kbel@fluid.mech.ntua.gr

Abstract

Oscillating foils located beneath the ship’s hull are investigated an unsteady thrusters, augmenting the overall propulsion of the ship in rough seas and offering dynamic roll stabilization. The foil undergoes a combined oscillatory motion in the presence of waves. For the system in the horizontal arrangement the vertical heaving motion of the hydrofoil is induced by the motion of the ship in waves, essentially ship heave and pitch, while the rotational pitching motion of the foil about its pivot axis is set by an active control mechanism. In previous works, a potential-based panel method has been developed for the detailed investigation of the effects of free surface in harmonic waves, and the results are found to be in good agreement with numerical predictions from other methods and experimental data. Also, it has been demonstrated that significant energy can be extracted from the waves. In the present work we examine further the possibility of energy extraction under random wave conditions using active pitch control. More specifically, we consider operation of the foil in head waves characterized by a given frequency spectrum, corresponding to specific sea states. The effects of the wavy free surface are taken into account through the satisfaction of the corresponding boundary conditions. Numerical results concerning thrust coefficient are shown, indicating that significant efficiency can be obtained under optimal operating conditions. Thus, the present method can serve as a useful tool for the preliminary design, assessment and optimum control of such systems extracting energy from sea waves and augmenting marine propulsion. 1. INTRODUCTION

In the last period, requirements and intergovernmental regulations related to vehicle technology for reduced pollution and environmental impact have become strict, and response to the demand of greening transport has been recognized to be an important factor. In this connection, biomimetic systems are also examined. Actually, recent research and development

concerning flapping foils and wings, supported also by extensive experimental evidence and theoretical analysis, has shown that such systems at optimum conditions could achieve high thrust levels; see, e.g., Triantafyllou et al [1],[2], Taylor et al [3], Ellenrieder et al [4].

In real sea conditions, the ship undergoes moderate or higher-amplitude oscillatory motions, due to waves, and the vertical ship motion could be exploited for providing one of the modes of combined/complex oscillatory motion of a biomimetic propulsion system free of cost. The initial attempts in this direction were focused on the application of passively flapping wings underneath the ship hull, to transform energy stored in ship's motions to useful propulsive thrust, with simultaneous reduction of unwanted responses. In such cases pitching motion is induced by a spring, loaded by the unsteady wing pressure distribution, see for example Rozhdestvensky & Ryzhov [5] for an extensive review, Naito & Isshiki [6] for a review in flapping-bow wings on ship propulsion, and Bøckmann & Steen [7] and Terao & Sakagami [8] for experimental research and full-scale applications, respectively.

Working in the above direction we proceed some further steps, as follows: (i) Substituting the passive pitching setup by an active pitching control, using a proper control mechanism and a pitch setup algorithm, on the basis of the (known) random motion history of the wing, as described in Politis & Politis [9]. (ii) Developing theoretical/numerical tools capable of analysing/designing ships equipped with such biomimetic wing propulsors, in horizontal and vertical arrangement producing thrust by extracting energy from waves, see Belibassakis & Politis [10]. (iii) Developing numerical methods and tools for the detailed hydrodynamic analysis of the free-surface and incident-wave effects on the operation of flapping-foil thrusters Filippas & Belibassakis [11,12].

In particular, in Politis & Politis [9] a 3D nonlinear boundary element method (BEM) is used and a wing of finite aspect ratio is studied in random heaving motion and actively controlled pitching motion in infinite domain. The proposed

Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering OMAE2014

June 8-13, 2014, San Francisco, California, USA

OMAE2014-23314

2 Copyright © 2014 by ASME

control law is based on the signal of the angle of attack caused by heaving motion and a proportional controller is used. It has been shown that considerable thrust can be produced with the appropriate selection of the control parameter. Also, it has been mentioned that the power necessary for the active pitch motion is very small in comparison with the gained propulsion power.

In Belibassakis & Politis [10] the hydrodynamic analysis of flapping wings located beneath the ship’s hull, operating in random motion is examined. The wing undergoes a combined transverse and rotational oscillatory motion, while the ship is steadily advanced in the presence of waves, modeled by directional spectrum. The system is investigated as an unsteady thrust production mechanism, augmenting the overall propulsion of the ship. In the horizontal arrangement the wing performs combined vertical and angular (pitching) oscillatory motion, while travelling at constant forward speed. The vertical motion is induced by the random motion of the ship in waves, essentially due to ship heave and pitch, at the station where the flapping wing is located. Wing pitching motion is controlled as a proper function of wing vertical motion and it is imposed by an external mechanism. A second arrangement consisted of a vertical oscillating wing-keel located beneath the ship’s hull has also been considered (Belibassakis & Politis [10]). In that case, the transverse motion is induced by ship rolling and swaying motion in waves. The angular motion of the wing about its pivot axis is again properly controlled on the basis of ship motion in order to produce thrust, with simultaneous generation of significant antirolling moment for stabilization. Calculations are based on coupling the seakeeping operators associated with the longitudinal and transverse ship's motions with the hydrodynamic forces and moments produced by the flapping lifting surfaces, using unsteady lifting line theory and non-linear 3D panel methods (Politis [13]) in infinite domain. First numerical results presented in Belibassakis & Politis [10] indicate that significant levels of thrust and antirolling moment are produced in sea conditions of moderate and higher severity, under optimal control settings. Similar two-dimensional problem concerning wave energy extracting systems has been studied by Isshiki [14]. In the latter work a model, treating the problem of an oscillating hydrofoil in water waves, is developed, extending Wu's [15] theory by introducing a free surface effect, and applied to the investigation of the possibility of wave devouring propulsion. The latter work has been further extended both theoretically and experimentally by Isshiki and Murakami [16] where the basic concept of passive type wave devouring capability of an oscillating hydrofoil is studied. Furthermore, Grue et al [17] developed a theory for a two-dimensional flat plate near the free surface using frequency-domain integral equation approach, where unsteady foil motions and wave devouring capabilities are illustrated. Predictions from the above theoretical model are found to be in good agreement with the experimental measurements by Isshiki and Murakami [16] for both head and following waves. However, at lower wave numbers there were systematic discrepancies between theory and experimental results attributed to nonlinear and free surface effects which were not fully modelled. More recently, De Silva & Yamaguchi [18] examined in detail the possibility of extracting energy from gravity waves for marine propulsion, by numerical simulations of a two-dimensional

oscillating hydrofoil, using commercially available CFD software. The overall results suggest that actively oscillating-foil systems in waves, under suitable conditions, have the possibility to recover the wave energy, rendering these systems applicable to marine unsteady thrusters.

For the detailed investigation of the effects of the free surface on the oscillating body hydrodynamics, in authors' previous works Filippas & Belibassakis [11,12], a potential-based panel method has been developed and applied to the hydrodynamic analysis of a 2D hydrofoil operating beneath the free surface, undergoing flapping motion in monochromatic waves. Numerical results are presented concerning various hydrodynamic quantities over a range of parameters compared against other results from the literature and calculations in infinite domain. The results are found to be in good agreement with numerical predictions from other methods and experimental data. Also, it has been demonstrated that the free-surface effects are important and cannot be neglected. We have also seen that, for the appropriate conditions, significant thrust can be produced in the presence of incoming harmonic waves.

In the present paper we examine further the possibility of energy extraction under more realistic, multichromatic wave- conditions, using active pitch control. More specifically, we consider operation of the foil in head waves characterized by a given frequency spectrum, corresponding to specific sea states. The developed potential method is capable to take into account the effects of the wavy free surface through the satisfaction of the corresponding boundary conditions. Numerical results concerning thrust coefficient are presented, indicating that significant efficiency can be obtained under optimal operating conditions. Thus, the present method can serve as a useful tool for the assessment and optimum control of such systems extracting energy from sea waves for augmenting marine propulsion.

2. EFFECTS OF RANDOM UNIDIRECTIONAL WAVES In Politis & Politis [9] a 3D flapping wing in random heaving motion characterized by an energy spectrum that corresponds to a specific sea state is studied. It is assumed that the system encounters head unidirectional waves, and that the transfer function from sea wave spectra to heave energy spectra equals unity, i.e., there is no interaction between the wing and the hull and the forces generated by the lifting surface do not affect ship motions. Also, the wing does not interact with the free surface and the incident waves. Furthermore, Belibassakis & Politis [10] developed a method based on coupling the seakeeping operators associated with the longitudinal and transverse ship's motions with the hydrodynamic forces and moments produced by the flapping lifting surfaces, using unsteady lifting line theory and non-linear 3D panel methods in infinite domain. In this way, all the possible incident wave directions could be studied. The transfer function from sea wave spectra to heave energy spectra is based on the Response Amplitude Operator (RAO) of ship vertical motion at the exact location of the wing. The wing operation affects the system RAO, as calculated by a simplified coupled model of ship and flapping wing dynamics based on lifting line theory. At a next level the 3D non-linear BEM by Politis [13] has been used in order to enhance the predictions, based on kinematical information from the simplified model. However,

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in the aforementioned works, the calculation of forces by the lifting bodies does not take into account the effects from the free surface and the incident waves, as well as the effects of finite water depth. In the present work, the free-surface and the random incident waves are directly taken into account to the solution. For the formulation of the incident waves the standard random phase model is used; see Pierson [19] and Longuet-Higgins [20]. Details can also be found in Ochi [21]. A panel method is developed to numerically model the flapping foil operating in finite water depth. The solution is compatible with unsteady linearised theory at low amplitudes and frequencies, but also gives satisfactory comparisons with experiments at higher amplitudes and frequencies, as well as with RANS solvers for the cases of low and medium angles of attack and large Reynolds numbers (Filippas & Belibassakis, [11,12]). However our method is two dimensional and at the moment is applicable to head or following waves, leaving the extension to three dimensions to be examined in future work. As a first approximation we assume that the slope of the incident water waves is small, permitting as to use the linear free-surface gravity wave theory. The linear random incident wave field is represented by frequency spectra that correspond to different sea states. The realization of the random incident wave elevation is obtained using the random phase model

( ) ( ) ( )1

, cosRN

i i i ii

x t F t k x tη α ω ξ=

= − +∑ , (2.1)

where ( )( )2

0( ) 1 exp /F t f t T= − − is a filter function

permitting smooth transition from rest to the fully developed state of wave excitation andRN is the number of discrete frequencies. In this work the Bretschneider model spectrum is used

( )4 4

25 4

1.251.25 , 1,...,

4m m

i s Ri i

S H epx i Nω ω

ωω ω

= − =

, (2.2)

where

( ) max minmin0.5 , 1,..., ,i R

R

i i Nω ω

ω ω ω ω−

= ∆ − + = ∆ =Ν

, (2.3)

and mω denotes the spectrum modal circular frequency.

2m

m

Tπω

= is the corresponding modal (peak) period and sH

denotes the significant wave height. The angular-frequency range [ ]min max,ω ω is the essential support of the spectral

density. The wave numbers ik and the corresponding wave

lengths 2 /i ikλ π= are connected with the frequencies through the dispersion relation

2 tanh( )i i ik g k Hω = , (2.4) where g is the acceleration of gravity and H is the constant (finite) water depth. The modal wavelength mλ and wave

number mk are also be obtained from the dispersion relation at

the modal frequency mω . The independent random variables

, 1,...,i Ri Nξ = are uniformly distributed in [0,2 )π , and the amplitudes are given by

( )2i ia S ω ω= ∆ . (2.5)

Finally, the multichromatic incident wave potential is modeled as follows

( ) ( )( )( )( )

( )1

cosh, ; sin

cosh

RNi

I i i i ii i

k y Hx y t F t k x t

k Hϕ ω ξ

=

+Φ = − +∑ , (2.6)

where /i i igϕ α ω= are the corresponding potential amplitudes.

3. HYDROFOIL KINEMATICS AND ACTIVE CONTROL We consider a flapping hydrofoil in random wave environment (Fig.1). The hydrofoil starts from the rest and accelerates until a constant forward speed, that corresponds to

a Froude number defined as /Fn U gc= , where U is the speed of the foil (with positive values at the direction of the negative x-axis) and c its chord length. Also, in the present work we consider head waves, therefore the modal frequency of encounter is

0 m mk Uω ω= + . (3.1) We assume that there is no interaction between the wing and the hull and therefore, the transfer function from sea wave spectra to heave energy spectra equals unity and the random heaving motion has modal frequency the encounter frequency and its characteristic amplitude is the half of the significant wave height sH . Hydrofoil's rotational (pitching) motion (which is positive counterclockwise) is defined with respect to a rotation axis located at distance / 3RX c= from the leading edge for minimization of the required torque, see Belibassakis & Politis [10]. That motion is properly controlled as a function of the known random heave history. The control law we apply here had been proposed by Politis & Politis [9] and were applied by Belibassakis and Politis [10] and is based at the observation that a flapping hydrofoil in harmonic motion produces maximum thrust when pitching motion has the same frequency as heaving motion and the phase difference between the two motions is 90ψ = ° , see also Anderson et al [22] or Rozhdestvensky & Ryzhov [5]. As a result of the simultaneous heaving and pitching motions the instantaneous angle of attack is

1 ( ) /( ) tan ( )

( ) /

dh t dta t t

ds t dtθ− −

= − − , (3.2)

where ( )h t is the heaving motion, ( )s t is the forward motion

and ( )tθ the rotational. The first term denotes the angle of attack due to heave and the second is the influence of pitch angle. We mention here that at the above formula the effects of the wake and the incident wave field are omitted. For pure harmonic motion, relative low amplitudes and optimum phase

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difference 90ψ = ° , the corresponding optimum angle of attack can be achieved when pitch is proportional to the angle of attack due to heave

1 ( ) /( ) tan .

( ) /

dh t dtt w

ds t dtθ − −

= − (3.3)

The above relation is a proportional controller of pitch defined in terms of the known random heave signal. Coefficient w is called pitch control coefficient and usually takes the values 0 1w< < (see also [9,10]). The angle of attack can now be written

1 ( ) /( ) (1 ) tan .

( ) /

dh t dta t w

ds t dt− −

= − − (3.4)

As we can observe from the above formula when w decreases, the wing operates at higher load and that maximum angle of attack increases. The value where the maximum angle of attack reaches the static stall, where the thrust coefficient approximately reaches a saturation point, could be used as a limitation for minimum w (for more details see Politis & Politis [9]). For a better approximation of the effective angle of attack we will use the following formula that takes into account the influence of head waves

1 ( ) / ( , ; ) /( ) tan ( )

( ) / ( , ; ) /

dh t dt x y t ya t t

ds t dt x y t xθ− Ι

Ι

− + ∂Φ ∂= −

− + ∂Φ ∂ , (3.5)

where the derivatives of the incident wave potential can be calculated at the point of the rotation axis, which is a reasonable approximation when the modal wavelength is larger enough than the hydrofoil's chord length. We can notice that the effect of the incident head waves is to decrease the angle of attack and therefore smaller values of w, in comparison with the infinite domain case, are permitted. 4. MATHEMATICAL FORMULATION The studied configuration is depicted in Fig.1. The problem is time dependent and the oscillating body is represented by a moving boundary ( )BD t∂ with respect to the earth-fixed

frame of reference. The amplitude of the free surface waves is assumed to be small in comparison with the wave length permitting as a first approximation, linearization of the free-surface boundary conditions on the mean free surface level (shown in Fig.1 by using a dashed line). However, the present analysis could be extended to treat the non-linear problem and this task is left to be subject of future work. A Cartesian coordinate system is introduced with y-axis pointing upwards and x-axis on the mean free surface. The total wave potential

( ), ;T x y tΦ consists of the random incident wave potential

( ), ;I x y tΦ which is known from Eq.(2.6) and the disturbance

potential ( ), ;x y tΦ which satisfies the 2D Laplace equation

2 ( , ; ) 0 , ( , ) ,x y t x y D∇ Φ = ∈ (4.1)

Figure 1. Flapping hydrofoil (of chord c) moving under the free surface (at submergence d). supplemented by the body boundary condition

( , ; ), ( , ) ,B

BB

x y tb x y D

n

∂Φ= ∈∂

∂ (4.2)

where

( , ; )IB B

B

x y tb

n

∂Φ= − ⋅

∂n+V , (4.3)

for ( , ) Bx y D∈∂ , and the bottom no-entrance boundary condition

( , ; )0 ,H x y t

y Hy

∂Φ= = −

∂ . (4.4)

In the above relations n is the unit normal vector pointing into the interior of D . The linearized dynamic and kinematic boundary conditions are satisfied on the mean free surface

( , ; )( ; ) , on 0,F x y t

g x t yt

η∂Φ

= − =∂

(4.5)

( , ; ) ( ; ), 0,F x y t x t

yn t

η∂Φ ∂= − =

∂ ∂ (4.6)

where for the case of linearised horizontal free-surface boundary

( , ; ) ( , ; )F Fx y t x y t

n y

∂Φ ∂Φ= −

∂ ∂. (4.7)

We treat the above as an initial value problem and we assume that the potential and its derivatives vanish at large distance from the body. In the above equations ( );x tη denotes the

free-surface elevation associated with the disturbance field and g the acceleration of gravity. Furthermore, H is the constant (finite) depth, d is the mean submergence of the body and BV denotes the instantaneous velocity of the body (due to forward motion and oscillations) at each point on the boundary.

We note here that in the case of lifting flow around a hydrofoil, the problem is supplemented by the kinematic and dynamic conditions on the trailing vortex sheet

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( , ; ) ( , ; ) , ( , )u lW W Wp x y t p x y t x y D= ∈∂ , (4.8)

( , ; ) ( , ; ), ( , )

u lW W

Ww w

x y t x y tx y D

n n

∂Φ ∂Φ= ∈∂

∂ ∂, (4.9)

stating that the pressure Wp and the normal velocity W

n

∂Φ

∂ are

continuous through WD∂ . The superscripts { },u l are used to

denote wake's upper and lower side respectively, while the indices { }, , ,B H F W denote values of the disturbance field

( , ; )x y tΦ on the body surface, the seabed, the free surface and the wake of the hydrofoil, respectively. Using Eqs.(4.8)& (4.9) in conjunction with Bernoulli's equation

( )210, ( , ) ,

2ap p

gy x y DtρΤ

Τ

− ∂Φ+ + ∇Φ + = ∈∂

(4.10)

we obtain

( ), ;0, ( , ) ,W

W

D x y tx y D

Dt

µ= ∈∂ (4.11)

where u lW W Wµ = Φ −Φ denotes the potential jump (the dipole

intensity) on the wake and m

D

Dt t

∂= + ⋅∇∂

V is the material

derivative based on the mean total velocity

( )0.5 u lm T T= ∇Φ +∇ΦV on the trailing vortex sheet. The above

relation states that WD∂ evolves in time, as a material curve and its exact motion is part of the solution. In the present work a simplified wake model is used, based on the assumption that the generated vortex curve emanates parallel to the bisector of the trailing edge and has the shape of trailing edge's path. In this way, the free wake dynamics are linearized, simplifying the problem and providing satisfactory predictions in the case of low and moderate unsteadiness. A similar wake model, assuming that the generated components of the wake remain where shed, has also been described and used by La Mantia & Dabnichki [23-26]. Extension of the present approach in the direction of modelling the non-linear free-wake dynamics and the interaction between the vortex wake and the free surface is left to be examined in future work.

In lifting flow problems, also, enforcement of the Kutta condition is required, in order to fix the circulation at each time instant. A possible choice is a Morino-type version of the Kutta condition [27], which permits direct calculation of vorticity transport to the wake, expressed in terms of the potential's jump at the trailing edge. The latter is compatible with Kutta-Joukowski hypothesis that no vortex filament exists at the trailing edge, see also Mohammadi-Amin et al [28] and Morino et al [29]. That approach was followed in our previous work, Filippas & Belibassakis [12]. However, this approximation would lead to an overestimation of the thrust at high Strouhal numbers, while a significant pressure jump is calculated as the trailing edge is approached; similar findings has been also reported by other authors; see, e.g., Bose [30]. On the other hand, application of the pressure-type Kutta condition has been recently criticized by La Mantia and Dabnichki, stating that there is no experimental evidence supporting the notion that the pressure difference at the

trailing edge, especially for unsteady motion of high frequency and large amplitude, ought to be equal to zero; see the detailed discussion and justification of this approach in Ref. [23,24]. Leaving this interesting subject to be investigated in detail in future research, in the present work, a non-linear, pressure-type Kutta condition, requiring zero pressure difference at the trailing edge, is imposed as follows:

( ) ( )( )10

2

u l

u l u l

tΤ Τ

Τ Τ Τ Τ

∂ Φ −Φ+ ∇Φ −∇Φ ∇Φ +∇Φ =

∂. (4.12)

Furthermore, we exploit Kelvin’s theorem, in conjunction with the (linearized) wake model of the hydrofoil in order to calculate the transport of trailing vorticity in the wake. The latter, with respect to the body-fixed reference frame, is

( ) ( ); ; , ( , )W W TE Wt t t t x y Dµ µ= + ∆ − ∆ = ∈∂x x V x , (4.13)

where TEV denotes the velocity determined from the motion derivative of the foil trailing edge. 4.1 Boundary integral formulation

Applying the representation theorem of the potential (Green's formula, see e.g. Kress [31]) to our problem Eqs.(4.1-4.13), for points at the body contour and the free surface, respectively, we obtain a system of two equations, one for

BΦ and another for FΦ , written compactly as follows

0/ 0 0

( | )1( ; ) ( ; ) ( | ) ( ; ) ( )

2B

B F B

D

Gt b t G t dS

n∂Φ = −Φ

∂∫∫x x

x x x x x x

00

( | )( ; )( | ) ( ; ) ( )

F

FF

D

GtG t dS

n n∂

∂∂Φ+ −Φ

∂ ∂∫∫x xx

x x x x

( ) 0( | ); ( )

W

W

D

Gt dS

nµ

∂

∂−

∂∫∫x x

x x . (4.14)

In the above relation we have used the Green's function consisted of the fundamental solution of 2D Laplace equation corresponding to a Rankine source and a regular part corresponding to a mirror source with respect to the bottom surface (y H= − )

0 00

ln ( | ) ln ( | )( | ) ,

2 2mr r

Gπ π

= +x x x x

x x

0 0( | ) ,r = −x x x x (4.15)

where 0 0 0( , )x y=x is the field point, ( , )x y=x is the

integration point and ( , 2 )m x y H= − −x its image with respect

to HD∂ . The normal derivative on the boundary is easily

derived by differentiating, 0 0( | ) / ( | )G n G∂ ∂ = ⋅∇x x x xn . In this way the bottom boundary condition is identically satisfied and the corresponding boundary term of the integral representation is dropped.

The representation theorem (4.14) includes the contribution from the wake surface WD∂ in the last integral that introduces a memory effect. Additional memory effects are present due to diffraction and radiation caused by the moving body.

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Moreover, the contribution of incoming wave potential is included at the right-hand side of the body boundary condition Eq.(4.2). Relation (4.14) will be used to set-up a Dirichlet-to-Neumann map (DtN) of the boundary values concerning FΦ

and BΦ and its normal derivatives and also the value of the

potential jump or the dipole intensity on WD∂ in the vicinity of the trailing edge. The latter, after discretization, will be applied to the numerical integration of the free surface boundary conditions (4.5), (4.6) and the pressure Kutta condition (4.12), treated as a dynamical system, with dynamic variables FΦ , η , and Wµ , providing us with the evolution of the unknown free surface at the specified level of approximation.

4.2 Construction of the discrete DtN map Following a low-order panel method the body is replaced with a closed polygonal line (with BN denoting the number of panels), while the free surface and the wake are approximated by FN and ( )WN t straight-line panels, respectively. The

potential and its normal derivative, the potential jump on the wake, the potential and the elevation of the free surface, at each time step, are approximated by piecewise constant distributions. Applying a collocation scheme using the center of each panel as collocation point, where Eq. (4.14) is satisfied, the following discretization equations are obtained, written in matrix form

1W K WFF

n

µΒ

= + +∂ ∂

Φb

A S Wµ WΦΦ

, (4.16)

where

{ }

{ } { }2 ,in

in im

jn jm

B A

B A

δ + −

= −

A

{ } { }

{ },

2

in im

jmjn jm

A B

A Bδ

−

= − −

S

{ }{ }

{ }{ }

1

1

, ,ik i

K

jk j

B B

B B

− − = = − −

W W

{ } { } { }, 1,..., , , 1,..., , 2,..., ( )B F Wi n N j m N k N t∈ ∈ ∈ . (4.17)

In the above equations ijδ is Kronecker’s delta, { },B B i= ΦΦ ,

{ },F F j= ΦΦ , ,B i

n

∂Φ =

∂ b , { },

FF jn

∂= Φ

∂

Φ, { },W W kµ=µ . In

the sequel, we will denote with bold, quantities containing the values of piecewise constant hydrodynamic functions on the panels, at various parts of the boundary. Also, ijA and ijB are

induced factors representing the potential at the midpoint of the panel i due to a unit source or dipole distribution at panel j, defined as

( , | , )ij i i

panel j

A G x y x y dS= ∫ ,( , | , )i i

ij

panel j

G x y x yB dS

n

∂=

∂∫ .

(4.18)

The integrals ijA and ijB in the case of low order boundary

element methods can be calculated analytically (see e.g. Moran [32], Katz & Plotkin [33]). Multiplying equation (4.16) with 1−A we obtain a formula in the form

1( )W WFF

n

µΒ

= ⋅ + +∂ ∂

Φb

D P µ ZΦΦ

, (4.19)

with 1−=D A S ,

( )1 1

2

( )W Wµ − = =

PP A Wµ

Pand 1 1

2K

− = =

ZZ A W

Z, (4.20)

the above mapping is the discrete DtN operator which connects the potential with its normal derivative at each collocation point on the boundaries BD∂ and FD∂ , but also

involves the unknown value of the dipole intensity 1Wµ of the

first panel in the trailing vortex sheet WD∂ . We observe in the above equations that the influence of the wake, that introduces memory effects, is taken into account through ( )WµP , and the

effect of the incident wave through the component /I Bn∂ ∂Φ

included in b , as described by the body boundary condition Eq.(4.2). The nonlinear pressure-type Kutta condition can be put in discrete form as described in more detail in Ref.[11]. Using the appropriate part of the extended DtN map, Eq.(4.19), in the discrete form of the free-surface boundary conditions

,F gt

∂= −

∂

Φη (4.21)

,F

t n

∂∂= −

∂ ∂

Φη (4.22)

and the discrete pressure-type Kutta condition, we finally obtain the following extended system of ODEs, that approximately describe the dynamics of the system

( ),d

dt=

Uf U where [ ]1 ,

T

F Wµ=U Φ η (4.23)

where the vector function f is defined as follows

� �

( )� � � �

� �( )2 1

12 12

( ) ( )

1 1

g

d g d

dt dt

= − ⋅ + Ζ− − Ζ Ζ Ζ

22

0 - I 0

f U -D 0 Z U f U

D D� � � �� �� � � �� �

,

(4.24a) where

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� �

� �( ) ( ) ( )� �

1 2

11

( )

1 d P

dt

= + + ⋅ +− Ζ Ζ

21

0 0

f U -D b -P

U UD b� �� �� �

N LN LN LN L

, (4.24b)

and � �Ψ denotes the difference of a function Ψ at the

trailing edge. Equation (4.23) can be numerically integrated in order to calculate the evolution of the dynamic variables

FΦ η and 1Wµ , on the basis of information concerning the

functions 1

, ,F WµΦ η at previous time steps, in conjunction

with the history of the wake dipole intensity Wµ , included in

P , and the data IB Bn

∂= − ⋅

∂

Φb +V n , known at every time step

from the incident wave field Eq.(2.6), and the body boundary condition Eq.(4.2), respectively. Subsequently, the extended DtN map Eq.(4.19) is used to calculate the remaining

unknown field values ΒΦ and F

n

∂

∂

Φ.

After the solution has been obtained (at each time step) the pressure can be calculated through the unsteady Bernoulli’s theorem Eq.(4.10). Furthermore, forces and the moment are directly calculated through the integration of the instantaneous pressure acting on the solid boundary, as follows

( ), ,B B

ref y ref x

D D

p dS M p x n y n dS∂ ∂

= = − +∫ ∫F n (4.25)

where ( , )x yn n=n , and refx , refy are the horizontal and

vertical distances between the integration point and the reference point for the calculation of the moment.

4.3 Numerical time integration and PML model

Starting from a prescribed initial condition, e.g. from rest, a time-stepping method is applied to obtain the solution. After evaluation of different methods we found that the higher-order Adams-Bashford-Moulton predictor-corrector method provides the required accuracy, stability and efficiency; see also Longuet-Higgins-Cokelet [34] and Johannessen & Swan [35]. The used scheme requires calculation of only two derivative formulas at each time, and the error is of order (∆t5), where ∆t is the time step, ensuring that good convergence is achieved. Stability and convergence of the present scheme has been studied by extensive numerical investigation and relative discussion will be presented at the next section.

An important task concerning the present time-domain scheme deals with the treatment of the horizontally infinite domain and the implementation of appropriate radiation-type conditions at infinity. Although in the case of purely linear waves, conditions at infinity could be treated by using the appropriate time-dependent Green function, the present work is based on the truncation of the domain and on the implementation of Perfectly Matched Layer (PML) model, as e.g. described by Berenger [36] and Collino and Monk [37].

The latter model permits the numerical absorption of the waves reaching the left ( )x α= and right ( )x β= termination

ends of the truncated domain, with minimum reflection. In order to apply the PML-type absorbing layer, the time-derivative operator in the left hand-side of (4.23) is substituted by the mixed-type operator:

( ) ( ),d

xdt

σ+ =U

U f U (4.26)

where the PML-parameter ( )xσ is a positive absorption coefficient with support extended over several wavelengths from the artificial end-type boundaries used to truncate the computational domain. In the present work we use selection of the PML coefficients, as described by Collino and Monk [37] and applied by Belibassakis et al [38] and Belibassakis and Athanassoulis [39] to water wave problems. In accordance with the previous works the optimum distribution of the absorption coefficient is of the form:

00( )

px x

xσ σ−

=ℓ

, (4.27)

inside the absorbing layer and zero outside. In Eq.(4.26) 0σ is

a positive parameter controlling the magnitude of the ( )xσ

distribution, ℓ adjusts the support of ( )xσ and is comparable to the characteristic wave length of the generated wave and p is positive parameter that controls the rate of absorption in the PML. More details about the time-integration method and the PML absorbing model can be found in Filippas & Belibassakis [11,12].

5. NUMERICAL RESULTS AND DISCUSSION In this section numerical results are presented and discussed concerning the performance of a NACA0012 flapping hydrofoil in random wave conditions, at various sea states and a range of the following parameters: (i) mean submergence, (ii) Froude number and (iii) pitch control parameter. The random incident wave field is represented by frequency spectra corresponding to different sea states. We consider various sea conditions labeled by an index ranging from 1 to 5. The correspondence 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 modal period (Tm) is given in Table 1. As mentioned before, the Bretschneider model spectrum is used for calculations (see Fig. 2).

Sea conditions BF Sea State Hs (m) Tm (sec) 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

Table 1. Sea conditions and spectral parameters.

Before we proceed to the presentation of the results, we begin, with a short discussion concerning the numerical characteristics of the present numerical scheme. Extensive numerical investigation has been performed for cases of flapping motion in great submergence and near the free surface, as well as in cases of operation in incident waves. In

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all cases numerical stability is achieved when a condition of CFL-type criterion is satisfied, as follows

(1),U t

Ox

δ∆≤ =

∆ (5.1)

where U is the forward velocity of the hydrofoil and x∆ is the panel size on the free surface. Concerning the convergence characteristics of the numerical scheme, it is found that there exists an optimum relation between the size Ws∆ of the neighbouring to the trailing edge panels of the body and those that represent the nearby wake Bs∆ , see also Ref. [11]. We

found that a ratio / 9W Bs s∆ ∆ ≃ provides very satisfactory rate of convergence at all the cases of random motion examined here. In the case of oscillating hydrofoil under the free surface selection of numerical parameters for convergence also ensures stability of the numerical scheme. This is due to the fact that the wavelength of the wake is usually significantly smaller than the wavelength of the generated waves on the free surface. Thus, a small enough time-step is required in order to capture the wake dynamics, which in most cases suffices for satisfaction of the above criterion, Eq. (5.1).

As a first example we present in Fig.3 the time series of some important quantities relative to the problem of thrust production using flapping hydrofoils in random wave conditions. We present the signals after the third modal period, when the transition from the rest has been completed. The present example concerns the hydrofoil at mean submergence

/ 10d c = , Froude number 1Fn = , in random head waves that corresponds to sea state 4 (see Table 1) using pitch control parameter 0.3w = and pitching motion with respect to a rotation axis, located at distance / 3RX c= from the leading edge. In subplot 3(a) the evolution of the effective angle of attack (α ) is presented. With red line we denote the angle of attack without the influence of the incident wave as calculated by Eq.(3.2), while with blue line we denote calculation including the wave effects using formula (3.5). As we can see the effect of the waves is to reduce the effective angle of attack. In subplot 3(b) the time history of heaving motion and the calculated lift coefficient 2/ 0.5L YC F U cρ= are shown together. We observe that significant amplitudes of the vertical force can be produced. Moreover the phase lag between heaving motion and lift is approximately 180° and therefore lift acts as a restoring force that could possibly reduce the unwanted responses and enhance the stability of a ship equipped with flapping hydrofoils. Finally in subplot 3(c) we present the evolution of thrust coefficient

2/ 0.5T XC F U cρ= − . We observe that the thrust oscillations

are in the interval 0 2.1TC = ÷ , having an average value of

0.17TC = which is indicated by using bold red line.

Moreover, in Fig.4 we examine the effect of sea state on the augmentation of the overall propulsion of the ship in random waves. The same parameters as in the previous example are considered, and the significant wave height and the modal period take the values listed in Table 1. We notice that as the wave height and the available environmental energy increases, the value of mean thrust coefficient also increases, reaching a maximum value 0.17TC = at sea condition 4, corresponding to

Figure 2. Different sea spectra considered in this study.

Figure 3. Evolution of kinematic parameters and integrated hydrodynamic quantities for a NACA0012 hydrofoil in random head waves corresponding to sea state 4 at mean submergence / 10d c = , for time duration of 11 modal periods. 1Fn = and control parameter 0.3w = .

(a)

(b)

(c)

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Figure 4. Mean thrust coefficient by the NACA0012 hydrofoil, as a function of sea conditions (parameters as in Fig.3).

Figure 5. Mean thrust coefficient as function of (a) the mean submergence to modal wavelength ratio, and (b) Froude number, for the parameters of Fig.3.

4sH = m and 10mT = sec, for the specific speed submergence and pitch control parameter. We also observe that although at sea condition 5 the available environmental energy is greater than at sea condition 4, with the specific pitch control set up, the extracted energy and the thrust coefficient remains at the same level.

For the same hydrofoil at sea condition 4, and the other parameters as in Fig.4, we examine in Fig.5(a) the effects of free surface and of the incident waves. In particular, the mean thrust coefficient is shown as a function of the mean submergence to modal wavelength ratio. The mean submergence to chord ratio in this figure varies in the interval 6 / 50d c< < . We can see that the effect of free surface is to reduce the thrust coefficient due to the developed wave resistance, and it is quite important in a wide range of values of the submergence ratio. Moreover, in Fig.5(b) the mean thrust coefficient as a function of Froude number is presented. We observe that as the speed U of the flapping foil increases the mean thrust coefficient decreases and this is attributed partially to the reduction of mean and maximum angle of attack (see also Ref.[10]), as well as to the increase of the wave resistance.

Figure 6. Mean thrust coefficient (bold line) as a function of pitch control parameter, for the parameters of Fig.4. With circle and triangles we denote the RMS and maximum value of the effective angle of attack.

Finally, in Fig.6 the effect of the pitch control parameter w is demonstrated for the NACA0012 hydrofoil in the case examined in Fig.3. With bold line the mean thrust coefficient is plotted as a function of w. We clearly observe that as the pitch control parameter decreases, the calculated thrust coefficient increases, as expected due to the increase of the effective angle of attack. However, the present method provides an underestimation of the thrust coefficient at large angles of attack (see also Ref. [11]), due to leading edge separation and dynamic stall effects that are not presently modelled. For this reason the root mean square and the maximum value of the angle of attack are also plotted in Fig.6, using circles and triangles, respectively, supporting the selection of the pitch control parameter in order to avoid leading edge separation. More details concerning the efficiency of the examined system under various operating conditions can be found in Ref.[10].

CONCLUSIONS

An unsteady BEM is developed and applied to the hydro-dynamic analysis of flapping hydrofoils operating beneath the free surface, in the presence of incident waves. The possibility of energy extraction under random wave operating conditions using active pitch control is demonstrated. More specifically, we consider operation of the foil in head waves characterized by a given frequency spectrum, corresponding to specific sea states. The effects of the wavy free surface are taken into account through the satisfaction of the corresponding boundary conditions. Numerical results concerning the thrust coefficient are shown, indicating that significant efficiency can be obtained under optimal operating conditions. Thus, the present method can serve as a useful tool for the preliminary design, assessment and optimum control of such biomimetic systems extracting energy from sea waves and augmenting marine propulsion. Future extensions include the introduction and modelling of various non-linearities associated with the evolution of the free surface and the trailing vortex sheets, and dynamic stall effects. Finally, another important direction for future study is the 3D problem of flapping-wing thrusters in waves and their interaction with the hull of the ship.

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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». Support of the first author by Alexander S. Onassis Public Benefit Foundation scholarship is also acknowledged.

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