Roll stabilization by vertical thrust-producing flapping wings using active pitch control

Transcription

1 Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 1 Roll stabilization by vertical thrust-producing flapping wings using active pitch control Kostas A. Belibassakis, School of Naval Architecture and Marine Engineering, National Technical University of Athens. kbel@fluid.mech.ntua.gr Gerassimos K. Politis, School of Naval Architecture and Marine Engineering, National Technical University of Athens. polit@central.ntua.gr ABSTRACT The analysis of vertical oscillating wing located beneath the ship s hull is investigated as an unsteady thrust production mechanism, augmenting the overall propulsion system of the ship. The wing undergoes a combined flapping and pitching oscillatory motion, in a uniform inflow and in the presence of waves. The flapping motion is induced by the motion of the ship in waves, essentially ship rolling and swaying. The pitching motion of the wing about its pivot axis, is properly selected as a function of the ship rolling motion in order to produce thrust, with simultaneous generation of useful antirolling moment for ship stabilization. Ship flow hydrodynamics are modeled using a Rankine source-sink formulation, and ship responses are calculated taking into account the additional forces and moments due to the above biomimetic propulsion system. Extending previous approach by the authors, a 3-D panel method is applied to model the unsteady lifting flow around the system. 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. Keywords: biomimetic ship propulsion, flapping wings, energy saving devices 1. INTRODUCTION Biomimetic propulsors are ideally suited for converting environmental (sea wave) energy to useful thrust. Recent research results concerning flapping foils and wings, supported also by extensive experimental evidence and theoretical analysis, have shown that such systems at proper conditions could achieve high thrust levels; see, e.g., Triantafyllou et al (, 4), Rozhdestvensky & Ryzhov (3), Taylor et al (1). In addition, response to the demand of making sea transport more environmentally friendly has been recognized to be an important factor concerning global warming and climatic change. The contribution of cargo ships in world pollution has been recognised as one of the most important factors (e.g. Colvile 1, Flannery 5), taking also into account the bad fuel quality of seagoing vessels in relation to other modes of transport. Indeed, images and data available from satellites reveal that large areas around the main sea-ocean shipping lines are almost permanently covered by clouds with large concentrations of pollutants from ships' engines. In real sea conditions the ship undergoes moderate or high amplitude oscillatory motions due to waves, the most significant being the heaving-pitching and rollingswaying combinations. A biomimetic propulsor, contrary to a conventional propeller, absorbs the required power by two independent motions. For a fish tail these two motions are heaving and pitching, while for a bird-flight flapping and pitching (or twisting) of bird wings, respectively (see Politis & Tsarsitalidis 1). The former biomimetic motion has already been considered by the authors, (Politis & Politis 1, Belibassakis & Politis 1), in connection with the problem of transforming the ship heavingpitching energy to useful thrust with simultaneous reduction of the corresponding ship motions. The latter biomimetic motion

2 Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, Figure 1. Ship hull equipped with a vertical flapping wing located below the keel, at the midship section. Geometrical details of the flapping wing (with NACA1 sections) are included in the upper subplot where the main flow direction relative to the flapping wings is indicated by using an arrow. is the subject of the present investigation. We shall show that such type of wing motion can be used to reduce the ship rolling-swaying motions with simultaneous production of useful propelling thrust. More specifically, in the present work the analysis of a randomly flapping-pitching wing, located beneath the ship s hull, is examined as an unsteady thrust production mechanism, augmenting the overall propulsion system of the ship. The above arrangement increases the draft of the ship, however the wing could be hinged to the hull permitting its folding and storage when not needed or when the ship is docking. The wing undergoes the combined flapping-pitching oscillatory motions, in a uniform inflow (due to ships speed) and in the presence of waves. The flapping motion is induced by the rolling motion of ship in waves. The pitching motion of the wing (about its pivot axis) is actively controlled in terms of the ship rolling-swaying motions (Politis & Politis 1) in order to (always) produce thrust and generate antirolling moment for ship stabilization. Ship flow hydrodynamics are modelled using a Rankine source-sink formulation and ship motions are calculated taking into account the additional forces and moments due to the above biomemetic wing propulsion system. For the needs of the linearized theory the wing is modelled using a combination of unsteady lifting line theory with D unsteady theory of Theodorsen. For an assessment of this assumption we present comparisons of the simplified model with a 3D panel time-stepping formulation of the unsteady problem (Politis 9, 11), which includes the complex unsteady trailing vortex rollup in the modelling of biomimetic wing performance. From this comparison, experience is obtained regarding the required corrections of the simplified model in the case of larger flapping angles of the wing exceeding the range of the linearized theory. It is shown that the above is a good approximation to our problem, due to the relatively small Strouhal number in which such biomimetic wings operate. Numerical results are presented for the developed thrust and the effect on ship motions of a biomimetic system. The present method can serve as a useful tool for assessment and the preliminary design and control of such systems for enhancing the overall ship performance in a wavy environment. The biomimetic propulsor considered in the present study consists of a single wing, as shown in Fig.1. However the whole performance of the system could be further enhanced by consideration of multiple wings in various arrangements (see also Belibassakis & Politis 1).. BIOMIMETIC WING THRUSTERS Kinematics: For the description of the kinematical characteristics of the flapping-pitching

3 wing and of the induced flow dynamics two reference systems are considered such as: the motionless inertial system and the ship-fixed coordinate system, which is steadily translated with velocity U with respect to the former and oscillating with respect to the fundamental degrees of freedom of the floating ship due to waves. This same system is also used to describe the pitching motion of the flapping wing. In the case of simple periodic ship oscillations the relative ship frequency ω = π f due to waves is the same as the flapping frequency of the wing. This is due to the fact that the vertical wing is attached to the hull, allowing only pitching motions relative to the ship. Furthermore, the active pitch selection algorithm (see next section) enforces pitch motion to follow the flapping motion with the proper phase angle: ψ = 9, which is necessary for the production of thrust. Consequently, pitching motion evolves at the same frequency ω as the ship rolling or wing flapping motions. Expressing the previous discussion quantitatively, the wing pitching axis performs a combined translation (sway) and rotation (roll) given by h( t) = h sin( ωt), where h is a spanwise variable amplitude h = ξ zξ4 (z denotes the vertical distance of a point of the pitching axis of the wing it is assumed that the origin of the moving x,y,z system coincides with the centre of flotation of the hull). Simultaneously, the wing undergoes a pitching motion around the pivot axis θ t = θ sin ω t + ψ. described by Dynamics: With the pivot point for the angular motion of the wing located around the 1/3 chord length from the leading edge, a minimization of the required torque for pitching is achieved (Anderson et al 1998, Schouveiler et al 5, Politis & Politis 1). For flapping systems steadily advancing in unbounded fluid the main flow parameter controlling the unsteady lift production mechanism is the Strouhal number St = f h / U, while, in the absence of flow separation, the Reynolds number has a secondary role affecting viscous drag. For the case of a flapping-pitching wing, the instantaneous angle of attack at the position z is given by: 1 1 ( z,t) ( z,t) ( t) tan ( U dh / dt ) ( t) α = ϑ θ = θ, (1) Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 3 where ( t) θ is the instantaneous pitch angle. For relatively low roll amplitude, purely harmonic motion, and the pitch angle selected according to the discussion of the previous section, relation (1) α z, t = ϑ( z) θ cos ω t, which can becomes be achieved by setting the pitch angle: ( t) ( t) w z ( t) θ = θ cos ω = ϑ cos ω, () 1 where ϑ ( z) U h ω and w acts as the pitch control parameter (Politis & Politis 1), usually taking values in <w<1, which is amenable to optimization. Decreasing the value of w, the maximum angle of attack is reduced and the wing operates at lighter loads. On the contrary, by increasing the above parameter the wing loading becomes higher and so is the danger of leading edge separation that would lead to significant dynamic stall effects. For a flapping wing the amplitude ϑ ( z) is variable spanwise. Thus determination of pitch requires selection of a reference z -position. We select the midsection of the wing ( z mid ) in order to control the wing pitch θ ( t). As a result, the instantaneous angle of attack is given by ( mid ) α z,t = tan U dh / dt wtan U dh / dt,(3) where h = ξ z ξ4. mid mid Free-surface effects: In the case of the biomimetic system under the calm or wavy free surface, additional parameters are important, as the Froude number F U / ( gl) 1/ =, where L denotes the characteristic (ship) length and g is gravitational acceleration, as well as various frequency parameter(s) associated with the incoming wave, like µ = ω L / g and τ = ω U / g, the latter being τ < from used to distinguish subcritical ( 1 / 4) supercritical ( τ > 1 / 4) condition. Geometrical parameters: As far as a standalone flapping wing is concerned, the selection of planform area, in conjunction with horizontal/vertical sweep and twist angles, and generating shapes ranging from simple orthogonal or trapezoidal-like wings to fish-tail like forms, constitutes the set of the most important 3

4 geometrical parameters (see Politis & Tsarsitalidis 9). Other significant 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, important additional geometrical parameters associated with its arrangement are the longitudinal position, and the clearance(s) with respect to the hull surface. As an example, which is used in the present work for demonstration purposes, we consider a variant of the series 6 - Cb=.6 ship hull form, shown in Fig.1. The main ship data are listed below: principal dimensions: L=5m, B=6.7m, T=.8m block coefficient: Cb=.533 immersed volume: = 5m 3, Δ=51tn (salt water) wetted/waterplane area: Sw=38m, A WL =5m center of flotation: x f = -1.15m (LCF aft midship) vertical center of buoyancy: KB=1.55m (from BL) longitudinal position: LCB=-.66m (from midship) metacentric radius: BM=1.15m. We also consider the long-center of gravity to coincide with the center of buoyancy, X G =-.66m (aft midship), Y G =, and KG=1.8m (from BL), and thus, the metacentric height in the above condition are estimated to be GM=.9m. Finally, the radii of gyration about the x-axis and y-axis, respectively, are taken R xx =.3B, R yy =.3L. Estimated 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. Table 1. Resistance data of ship hull of Fig.1 U F = U / gl C W R EHP (kn) (kp) (PS) The flapping wing propulsor, shown in Fig.1, is considered to be located at the midship section. The upper end (root) of the wing is assumed at a depth d=.8m and the span of the vertical wing is s=6m. The wing planform shape is trapezoidal, the root and tip chords of the wing have lengths c r =1m, c t =.5m, respectively, and the leading edge sweep angle is Λ=1.6deg (see Fig.1). On the basis of the Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 4 above, the wing planform area is S W =4.5m, and thus its aspect ratio is AR=8. The wetted area of the wing is the double of its planform area. The wing sections are symmetrical NACA1, and thus the local max thickness-to-chord ratio is kept constant over the span and equal to 1%. 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 that coincides with the vertical z- axis when the ship is at the upright position (see Fig.1). The same logic could be also applied to active anti-roll fins to generate propulsion. 3. SHIP AND WING HYDRODYNAMICS Methods for the calculation of dynamic ship responses in a given sea state have been developed up to a satisfactory level of accuracy; see, e.g., Ohkusu (1996). Various 3D hydrodynamic models based on BEM, both in the frequency and the time domain, are readily available nowadays, see, e.g., Beck et al (1996), Sclavounos et al (1997). Furthermore, simplified semi-3d theories are also useful in practice (e.g., Arribas 7). Methods based on numerical simulation of the hydrodynamic analysis of conventional and high-speed ships are valuable tools for use in ship design, and various commercial software packages & tools for the analysis of the dynamic behaviour of ships in waves are available today. In this category belong Rankine panel methods, based on B-spline and NURBS approximations, exhibiting low damping and numerical dispersion; see Sclavounos & Huang (1997), Kring & Sclavounos (1997), Huang & Sclavounos (1998), Kim & Shin (3). Comprehensive survey of theoretical and computational methods is presented in Beck & Reed (1). Moreover, lifting appendages are attached to ship hull in order to improve their calm water performance and reduce responses in waves; see, e.g. Sclavounos & Huang (1997). For example, passive and active systems are frequently used as anti-rolling stabilizers (e.g., Naito & Isshiki 5). In the context of 3D BEM applications a Rankine panel method developed by Sclavounos & Borgen (4) has been applied to study the seakeeping performance of a foil-assisted high-speed monohull. 4

5 In the latter work, the formulation of the seakeeping of ships equipped with lifting appendages is developed, and the mechanisms responsible for the reduction of the vertical motions of high-speed vessels equipped with hydrofoils is studied. The sensitivity of the heave and pitch motions on the longitudinal position of the hydrofoil are studied, finding that the most efficient location for the hydrofoil is at the ship bow. 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 & Sclavounos (6). Developments and applications of control theory for marine vessels are extensively discussed in Fossen (). 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., Thomas & Sclavounos (7). Standard seakeeping analysis in the frequency domain (see, e.g., Sclavounos & Borgen 4) is used in the present work to obtain the motions and responses of the examined system (ship and flapping wing). The coupled equations of ship s i t ξ Re ξ e ω i t ξ = Re ξ e ω sway = ( ) and roll 4 ( 4 ) motion (with corresponding complex amplitudes ξ and ξ 4 ) read as follows: ( ( m a ) i b ) ( a I ) i b ω + + ω ξ + ( ω ω 4 ) ξ + where ω a + I + iωb ξ = F + X (4a) , ω a + I + iωb + c ξ = F + X (4b) jk , a and b, j,k =, 4, are the (symmetric) jk added mass and damping coefficients, and m is the total mass of the ship and wing ( m = ). The involved hydrostatic coefficient is c44 = m g GM and the inertia coefficients are I = mr and Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 5 44 xx I4 = I4 = m ZG (here Z G =T-KG). The damping terms used in the above system are calculated on the basis of ideal flow hydrodynamics, and are used to provide a first demonstration of the present active anti-rolling thruster. However, this is expected to lead to erroneously large responses around the resonance condition. Enhanced expressions involving viscous damping effects are available and could be used in future extensions for more correct predictions. Moreover, the terms F j, j =, 4 appearing in the right-hand side of Eqs.(4) are the Froude-Krylov and diffraction vertical forces and rolling moment (about the x- axis) amplitudes, respectively. Finally, the terms X j, j =, 4, denote additional force and moment amplitudes due to the operation of flapping wing as an unsteady thruster. The latter are dependent on the ship s responses, as well as to the incoming wave field. In the present work we employ a simplified lifting-line model to derive analytic expressions of these forces in terms of the oscillatory ship amplitudes and include the effects of flapping wing in the system coefficients. 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 (, ) ξ ξ of the ship. However, taking into 1 6 account the large mass of ship, in conjunction with installation of energy storage (flywheel-type) systems and the effect of the rudder, at first level of approximation, the above motions are considered to be very small and are neglected. 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 forward speed. In both cases, the four-point, upwind finite difference scheme by Dawson (1977) has been used to approximating the horizontal derivatives involved in the (linearized) free surface boundary condition. Details concerning 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 (11). We mention here that a minimum number of 15- 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 also Sclavounos & Nakos 1988 and Janson 1997). An example concerning the calculated wave field exciting by rolling ship hull motion, for Froude number F=.5 and reduced frequency τ = ωu / g =.4, as predicted 5

6 by the present model, is shown in Fig.. In this case, the half-hull surface is discretized by using a mesh of (in the long direction) by 1 (sectional) panels, and the half symmetric part of the free surface by using a mesh 4 (in the transverse direction) by 118 (in the long direction) panels. Thus the total number of elements is 5. Although no special treatment is necessary for enforcing the radiation condition in the supercritical (τ>1/4) 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 Nakos & Sclavounos (1988), Nakos et al (1994), Sclavounos & Borgen (4). The thickness of the absorbing layer is of the order of 1- characteristic wavelengths and its coefficient is quadratically increasing. 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 steady flow around the same ship hull, also at Froude number F=.5, 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 calmwater resistance have been obtained from numerical analysis based on RANSE and experimental measurements (see, e.g., Tzabiras 4). In the present work, bow-quartering and beam waves (β=15 o,9 o ) have been considered as excitation of the hull oscillatory motion. In the examined case the working angle of attack of the flapping wing is given by: ξ 1 dξ t d t U dt dt Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 6 4 a t = W + z δ t INC INC where = y +, (5) ϕ ϕ W n nz is the incident wave y z velocity in the normal direction of the wing pivot n,n cos ξ, sin ξ δ t axis, ( y z ) ( ( 4 ) ( 4 ) ) =, and denotes the flapping wing pitch angle (controlled variable) with respect to the pivot axis. The above formula is obtained by linearizing for small angles. The first term in parenthesis in the right-hand side is due to the contribution of ship s oscillatory motion and the incoming waves, and this part, considered together, is denoted by: Figure. Calculated wave field exciting by ship rolling motion for F=.5 and τ=.4, as predicted by the present model. Horizontal distances are scaled with respect to ship length. Figure 3. Steady wave pattern for F=.5 as predicted by the present model. ξ 1 dξ t d 4 t ϑ ( z,t) = W + z U dt dt, (6) In the above formulas ϕ INC is the incoming wave potential, for unit amplitude (A) of free-surface elevation, which is given by: ig ϕinc ( x,z;t) = Re exp ( kz ) exp ( i( kx + ωt )), (7) ω where = kx cos ( β ) + ky sin( β ) kx, k = ω / g, is the wavenumber of the incident waves and ω is the absolute (angular) frequency. The corresponding relative frequency (frequency of encounter) is given by ω = ω ku cos β = ω ω U cos β / g. (8) 6

7 Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 7 Figure 4. Vortical wake behind the flapping wing in harmonic motion, as calculated by the 3D panel time stepping algorithm. Mean wing speed U=5.5m/s, flapping frequency ω=.68rad/s, rolling amplitude ξ 4 =deg, wing pitch control parameter w=.5. Figure 6. Sway response ( ξ ) / A of the ship against non-dimensional wavelength, for β=15 ο. The effect of flapping wing propulsor is shown by using a thick solid line, and the reduction of maximum response by an arrow. Figure 5. (a) Rolling motion and angle of attack of the flapping wing of Fig. 4. (b) Moment and (c) thrust as calculated by the simple model (thin lines) and the 3D panel time-stepping method (thick lines). As it was previously discussed in the case of harmonic flapping wing motion, the controlled δ t = wϑ z,t and the pitch pitch variable is set control parameter w ranges from to 1. Thus, the time variation of the angle of attack becomes ϑ ϑ a z,t = z,t w z,t, (9) mid with ϑ ( z,t) defined by Eq. (6). Unsteady lifting line models based on the integration of D sectional lift along the span, as the ones developed by Sclavounos (1987) and Guermond & Sellier (1991), mid Figure 7. Roll response ( ξ ) 4 / ka of the ship against non-dimensional wavelength for (a) β=15 ο and (b) β=9 ο. The effect of flapping wing propulsor is shown by using a thick line, and reduction of responses is indicated by an arrow. 7

8 can be used to obtain the wing lift and moment (about the pivot axis) coefficients. In the case of flapping wings operating at relatively low reduced frequencies of oscillation, the previous spanwise integration could be simplified (see also DeLaurier 1993), leading to approximate expressions as follows AR 1 D C = c( z) C ( z) dz +, (1) 3D L L AR Sw D where CL of the wing in the spanwise direction, and c( z ) is the chord distribution. z denotes the sectional lift coefficient The sectional lift can be estimated by means of unsteady hydrofoil theory (see, e.g., Newman 1977) in terms of the dynamic angle of attack: ( t) = Re ( A + A + A ) e iωt α S R W, (11) iωξ iωξ AS w R = mid, U U = 1 w W / U, the latter term being associated 4 where = ( 1 ), A ( z wz ) AW with incoming wave velocity at the flapping wing. 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 backsweep angle (as in the present example) is approximately expressed as follows (( C) S ) π k ( h / c) C = π k A + A + k A + D L S R W Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 8 +, (1) involving the Theodorsen C( k ) and the Sears functions S ( k ). The latter are defined in terms of the Hankel functions of the second kind as follows: ( = ) 1 Π = ( π Π) ( ) ( H ( k) ih ) ( k) C k H k /, S k i / k, Π = +, (13) 1 with k ω = c y / U the local spanwise value of the reduced frequency of the wing section. The last term in Eq. (1), with h = ξ zξ4, is the added mass effect due to roll-induced transverse motion of each wing section (see, e.g. Newman 1977, Sec.5.16), which is considered to be most significant in comparison to the corresponding one due to the rotational wing motion about the pivot axis. For low frequencies of oscillation, and wing reduced frequency k near zero, the functions C ( k ) S ( k ) 1., and this result permits us to obtain convenient expressions of the force and moment due to the operation of vertical flapping wing. Moreover, by considering the contribution of A << A, A, due to wave velocity very small W S R the fact that the wing operates submerged at considerable depth below the free surface, the above formulae result in the following approximate expressions for the amplitudes of hydrodynamic transverse force and rolling moment of the flapping wing X = iω G ξ + G ξ, (14a) S R 4 X = iω M ξ + M ξ, (14b) 4 S R 4 mid, where GS = γ ( 1 w) SW, GR = γ ( z wz ) c( z) dz M = γ ( 1 w) zc( z) dz, M = γ z( z wz ) c( z) dz, S AR γ = χ πρu, and χ is a correlation coefficient taking values near unity that will be explained AR + below. Using Eqs. (14) in the system (4), we obtain the following modifications of the system (hull and flapping wing) damping coefficients: = S δ 44 = R, δ 4 R δ 4 δb G, b M R mid b = G, b = M,(15) Also, the flapping wing added mass produce corresponding modifications in the system coefficients δ a, j,k =, 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. The thrust produced by the flapping wing is approximately calculated as follows ( G G ) sin ϑ ( z,t) T = +. (16) S R mid Enhanced expressions could be obtained by keeping more terms in the expansion of Theodorsen and Sears function about k =, which will result into additional modifications in the hydrodynamic coefficients and excitation forces. More complete unsteady lifting line models, as e.g., the one S 8

9 developed by Guermond and Sellier (1991), could be employed to overcome such limitations, and this task is left to be examined in future work. To illustrate the usefulness of the above approximate model, in the simple case of flapping wing in infinite fluid, we present in Figs. 4 and 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 (9, 11). In this case the wing undergoes simple harmonic transverse and rotational about its pivot axis motions. In particular, the wake rollup behind the wing as calculated by the 3D panel timestepping algorithm for the case: U=5.5m/s, ω=1.56rad/s and ξ 4 =deg, is illustrated in Fig.4. Predictions for the rolling moment and thrust as calculated by the above quasi-steady model and the 3D panel method in the example considered are presented in Fig.5, against the variation of the angle of attack, and are found in relatively good agreement. Moreover, from data shown in Fig.5 and systematic analysis, we can calculate frequencyand amplitude-dependent correction factor χ that could be used to better correlate the results of the above quasi-steady model to the 3D panel timestepping methodology (cf. Eqs. 14). The preceding analysis permits us to calculate, at a first order of approximation, the ship responses including the effect of the flapping wing as a thruster, and compare with corresponding seakeeping results without the wing. As an example, the normalized sway response of the ship with respect to the incident wave amplitude ( ξ / A) is plotted in Fig.6, as calculated by the present method, for various values of the non- λ / L, wave incidence dimensional wavelength β=15 ο and Froude number F=.5. In the same figure the corresponding result obtained with the consideration of the flapping wing is shown by using a thick solid line. 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. Furthermore, in Fig. 7 the same effect concerning the calculated ship-roll response ξ is presented. In this case head-quartering 4 / ka Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 9 (β=15 ο ) and beam (β=9 ο ) incident waves have been considered. Again we observe that the operation of the flapping wing (results shown by using thick solid lines) leads to significant reduction of roll responses, for all frequencies. This permits safe routing of the ship even in beam seas and rough sea-conditions that would be undesirable and avoided otherwise. The noticeable reduction of ship responses is evidence of significant wave energy absorption by the present flapping thruster, and also leads to reduction of the added resistance. To provide an indication, for the same as above Froude number F=.5 (U=1.6kn), the added wave resistance without the flapping wing has been estimated by the energy method (see, e.g., Arribas 8) and for sea state 6 to be of the order of calm-water ship resistance, and is reduced by 5% by the operation of the flapping wing. 4. FLAPPING WINGS AUGMENTING SHIP PROPULSION Having obtained an estimation of the time history of the ship oscillatory motion from Eqs.(4), including the effects of the oscillating wing operating as a thruster, the instantaneous angle of attack can be determined by Eq. (5) and unsteady 3D non-linear panel methods could be applied to obtain more accurate prediction of the flapping wing characteristics, as well as detailed information concerning local flow velocity and pressure distribution. For this purpose, in the present work we employ the non-linear 3D panel method developed by Politis (11), which is based on free wake analysis and Morino formulation based on Green s theorem. We note here that the above method includes also viscous drag effects. The thrust augmentation by means of the operation of the present flapping wing propulsor, as calculated by the present method, is illustrated in Figs. 8 and 9. In particular, we consider the ship of Fig.1 to travel at constant speed U=1.6kn, in head-quartering waves at sea state 6 (Beaufort scale 7), represented by a sea spectrum corresponding to significant wave height H s =5m, and peak period T p =11s, that is shown in Fig.8 by using dotted line (and the Bretschneider model spectrum). The same spectrum in the moving ship frame of reference is 9

10 also plotted in Fig.8 by using dashed line, as well as the spectrum of the transverse motion response at the wing midspan position, by using a thick solid line. Next, in Fig.9, time series of various quantities including the ship rolling ( ξ 4 ) motion, the the angle of attack (α ) and the thrust T (kp) production by the flapping wing, based on setting the control pitch variable w=.5, are shown, 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.9, the thrust oscillations are in the interval -75kp and have average value of T av =15kp. Similar analysis for same ship speed U=1.9kn and beam waves (β=9 ο ) provides higher value of average thrust T av =75kp, due to the increase of oscillatory frequency in this case. The last result concerning thrust production, as calculated by the present simplified model, is compared in Fig. 1 against corresponding prediction by the unsteady 3D panel method, showing quite good agreement. Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 1 Figure 8. Sea spectrum (H s =5m, T p =11s) and transverse motion spectrum at wing midspan, for head-quartering (β=15 ο ) seas and ship speed U=1.6kn (F=.5). In conclusion, in the example considered and discussed here 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 1% to 5% of the corresponding calm-water resistance of the ship at the same speed. Similar devices consisting of heaving horizontal wings, studied by the authors (Belibassakis & Politis 1), which convert oscillatory motion due to ship heave and pitch motion to thrust, are shown to be well capable to overcome the added resistance due to waves at these ship speeds. Finally, we remark that the torque required for wing pitching according to the control rule is very small and thus, the present system enhances the overall performance of the ship by producing thrust and providing significant stabilization. 5. CONCLUSIONS The analysis of a vertical oscillating wing, located beneath the ship s hull, in harmonic and multichromatic motion, is examined as an unsteady thrust production mechanism, augmenting the overall propulsion system of the ship. Figure 9. Stochastic responses of ship and flapping wing operating in head-beam (β=15 ο ) seas at ship speed U=1.6kn (F=.5). (a) Rolling motion with the flapping wing. (b) Normal velocity at the midspan position due to ship rolling. (c) Calculated angle of attack at the mid-span position of the flapping wing, using w=.5. (d) Thrust production by the flapping wing (time history). The time average is calculated to be 16kp and is indicated by using dashed line. The wing undergoes a combined flapping and pitching oscillatory motion, in a uniform inflow and in the presence of waves. The transverse flapping motion is induced by the rolling and swaying motion of ship in waves. The pitching motion of the 1

11 Figure 1. Thrust production by the vertical flapping wing (detail). Results by the simplified model (thin line) in comparison with the unsteady 3D panel method (thick line). wing about its pivot axis is properly controlled in terms of the ship rolling-swaying motion in order to result in thrust force production and generate useful antirolling moment for ship stabilization. Ship flow hydrodynamics are modelled 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. Flapping wing hydrodynamics are modelled through simplified lifting model and more accurate unsteady 3D panel method. Our results indicate that significant ship motion stabilization occurs with additional propulsive thrust production. The presentmethod can serve as a useful tool for assessment and the preliminary design and control of such systems for augmenting ship propulsion and stability in rough seas and enhancing overall ship operability. ACKNOWLEDGEMENTS This work has been supported by the ARISTEIA Programme of National Strategic Reference Framework of Greece (NSRF 7-13), project BIO- PROPSHIP entitled: Augmenting ship propulsion in rough sea by biomimetic-wing system. 6. REFERENCES Anderson, J. M., Streitlien, K., Barrett, D.S., Triantafyllou, M.S.,1998, Oscillating foils of Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 11 high propulsive efficiency, J. Fluid Mechanics, Vol. 36, Arribas, P.F., 7, Some methods to obtain the added resistance of a ship advancing in waves, Ocean Engineering Vol. 34, Beck, R.F., Reed, A.M, Rood, E.P., 1996, Application of modern numerical methods in marine hydrodynamics, Trans.SNAME, Vol.14, Beck, R.F., Reed, A.M, 1, Modern computational methods for ships in a seaway, Trans. SNAME, Vol.19, Belibassakis, K.A., Politis, G.K., 1, Hydrodynamic analysis of biomimetic wing systems for augmenting ship propulsion, to be presented at the nd Intern. Ocean and Polar Engineering Conf. (ISOPE 1). Belibassakis, K., 11, A panel method based on vorticity distribution for the calculation of free surface flows around ship hull configurations with lifting bodies. Proc. IMAM 11, Genoa, Italy. Chatzakis I. I., Sclavounos P. D., 6, Active motion control of high-speed vessels by statespace methods, J. Ship Res. Vol. 5(1). Colvile R.N et al., 1, The transport sector as a source of air pollution, Atmospheric Environment Vol. 35, Dawson, C.W.,1977, A practical computer method for solving ship-wave problems, Proc. nd Int. Conf. on Numerical Ship Hydrodynamics, Berkeley, USA. DeLaurier, J.D., 1993, An aerodynamic model for flapping wing flight. The Aeronautical Journal paper no Flannery,T., 5, The weather makers, Penguin. Fossen T. I.,, Marine Control Systems. Guidance, Navigation and Control of Ships Rigs and Underwater Vehicles. Marine Cybernetics AS, Trondheim, Norway. Guermond, J.-L., Sellier, A., 1991, A unified unsteady lifting line theory, J.Fluid Mechanics Vol., Huang, Y.F., Sclavounos P.D., 1998, Nonlinear ship motions, Journ Ship Research Vol. 4(), Janson, C.E., 1977, Potential flow panel methods for the calculation of free-surface flows with lift, Ph.D thesis, Chalmers Univ of Technology, Goeteborg

12 Kim, B., Shin, Y., 3, A NURBS panel method for three-dimensional radiation and diffraction problems, J Ship Research, 47(). Kring, D.C., Sclavounos, P.D., 1995, Numerical stability analysis for time-domain ship motion simulations, Journ Ship Res. 39, Naito S., Isshiki S., 5, Effect of Bow Wings on Ship Propulsion and Motions, Appl. Mech. Rev. Vol. 58(4), 53. Nakos D., Sclavounos P. (199), On steady and unsteady ship wave patterns, Journ. Fluid Mech. Vol. 15, Nakos D., Kring, D., Sclavounos P., 1994, Rankine panel methods for transint free surface flows, Proc. 6 th Int. Conf. Numerical Ship Hydrodynamics, Iowa City. Newman, N., 1977, Marine Hydrodynamics, MIT Press. Ochi, M.K., 1998, Ocean Waves. The Stochastic Approach, Cambridge University Press. Ohkusu, M. (Ed.), 1996, Advances in Marine Hydrodynamics, Computational Mechanics Publications. Politis, G.K., 9, A BEM code for the calculation of flow around systems of independently moving bodies including free shear layer dynamics, Proc. Advances in Boundary Element Techniques X. Athens, Greece. Politis, G.K., Tsarsitalidis, V.T, 9, Simulating Biomimetic (Flapping Foil) Flows for Comprehension, Reverse Engineering and Design, Proc. SMP 9, Trondheim, Norway. Politis, G. K. and V. T. Tsarsitalidis (1). Understanding Birds's Flight, using a 3-D BEM method and a time stepping algorithm. 4th IC- SCCE 1. Athens, Greece. Politis, G.K., 11, Application of a BEM time stepping algorithm in understanding complex unsteady propulsion hydrodynamic phenomena, Ocean Engineering 38, Politis, G., Politis, K., 1, Biomimetic propulsion under random heaving conditions, using active pitch control, Journal of Fluids & Sctructures (in press). Ships and Ocean Vehicles, 3-8 September 1, Athens, Greece, 1 Rozhdestvensky K.V., Ryzhov V.A. (3), Aerohydrodynamics of flapping-wing propulsors, Progress Aerospace Sci. Vol Sclavounos P. (1987), An unsteady lifting line theory, Journal of Eng. Mathematics Vol. 1, 1. Sclavounos P., Nakos, D., 1988, Stability analysis of panel methods for free surface flows with forward speed, Proc. 17th Symp. Naval Hydrodynamics, The Hague, Netherlands. Sclavounos, P., Kring, D., Huang, Y., Mantzaris, D., Kim, S., Kim, Y., 1997, A computational method as an advanced tool of ship hydrodynamic design, Trans. SNAME Vol. 15, Sclavounos P.D., Huang, Y.F. 1997, Rudder winglets on sailing yachts, Marine Technology Vol. 34(3), Sclavounos P., Borgen H., 4, Seakeeping analysis of a high-speed monohull with a motion control bow hydrofoil, J. Ship Research Vol. 48(), Schouveiler L., Hover F.S., Triantafyllou M.S. (5), Performance of flapping foil propulsion, J. Fluids Struct Vol., Taylor, G.K, Triantafyllou, M.S, Tropea, C., 1, Animal Locomotion, Springer Verlag. Thomas B.S., Sclavounos P.D., 7, Optimal control theory applied to ship manoeuvring in restricted waters, J. of Engineering Mathematics Vol. 58, Triantafyllou, M.S., Triantafyllou, G.S., Yue, D.,, Hydro-dynamics of fishlike swimming, An. Rev. Fluid Mech. Vol. 3. Triantafyllou, M. S., Techet, A. H., and Hover, F.S., 4, Review of experimental work in biomimetic foils, IEEE J. Ocean Eng. Vol. 9, Tzabiras G.D., 4, Resistance and Selfpropulsion simulations for a Series-6, Cb=.6 hull at model and full scale, Ship Technology Research, Vol. 51,