BIOMIMETIC SYSTEMS OPERATING AS MARINE ENERGY DEVICES IN WAVES AND SHEARED CURRENTS

Transcription

1 th HSTAM International Congress on Mechanics Athens, 7 3 May, 6, Greece BIOMIMETIC SYSTEMS OPERATING AS MARINE ENERGY DEVICES IN WAVES AND SHEARED CURRENTS K.A. Belibassakis, E.S. Filippas and Th.P. Gerostathis School of Naval Architecture & Marine Engineering National Technical Univ. of Athens, Athens, GR- 578, Greece kbel@fluid.mech.ntua.gr, evfilip@central.ntua.gr Dept. of Naval Architecture, Faculty of Technological Applications, Technological Educational Institute of Athens, Athens, GR-43, Greece tgero@teiath.gr Keywords: Marine renewable energy, Waves and currents, Biomimetic systems, BEM Abstract. The present work is focused on the investigation of oscillating hydrofoils in the presence of waves and currents studied as new systems for the exploitation of this kind of combined renewable marine energy sources. We examine the possibility of energy extraction by oscillating hydrofoils operating as biomimetic systems in harmonic waves and currents in coastal regions using active pitch control. Except currents of uniform structure in depth, special attention is paid to the case of waves and vertically sheared currents. The present method takes into account the effect of the wavy free surface through the satisfaction of the corresponding boundary conditions, the velocity component due to waves and currents on the formation of the incident flow, as well as the effects of variable bathymetry and sheared currents. Numerical results concerning the extracted power and the operability characteristics of the system over variable bathymetry are presented, indicating that significant output can be obtained under general operating conditions. Our method after further enhancements and verification can be applied for the design and optimum control of such biomimetic systems operating in the nearshore/coastal region and extracting energy from waves in the presence of ambient currents. INTRODUCTION Understanding the interaction of water waves with varying currents in nearshore and coastal areas is important for a variety of engineering applications, including interaction of waves with structures, coastal management and harbor maintenance, as well as design and development of systems for exploitation of marine renewable energy resources. In particular, the effects of inhomogeneous currents on wave transformation in the nearshore and coastal environment are significant, since they are responsible for Doppler shifting and additional wave refraction, reflection, and breaking, completely changing the wave energy pattern. In particular, the characteristics of surface waves present significant variation as they propagate through non-homogeneous ambient currents, in the presence of depth inhomogeneities in variable bathymetry regions. Thus, large amplitude waves can be produced when obliquely propagating waves interact with opposing currents []. This phenomenon could be further enhanced by inshore effects due to sloping seabeds, and has been reported to be connected with the appearance of giant waves []. Extensive analysis concerning the subject of wave-current interaction in the nearshore region have been presented by various authors [3-5]. Recent information can also be found in the corresponding sections of reviewing articles [6,7]. For the prediction of wave-seabed-current interaction quite complete and accurate coupled-mode models have been recently developed, with application to the propagation/scattering of water waves over variable bathymetry regions, in the presence of strong spatially varying currents, extending previous simplified mildslope/mild shear models that are applicable to cases of slowly varying bathymetry and current [8]. Also, the problem of transformation of the directional spectrum of an incident wave system over a region of strongly varying three-dimensional bottom topography is further studied in Belibassakis et al (4) [9], where also the accuracy and efficiency of the coupled-mode method is tested, comparing numerical predictions against experimental data and calculations by phase-averaged model SWAN []. Results are shown for various representative test cases, demonstrating the importance of the first evanescent modes and the additional slopingbottom mode when the bottom slope is not negligible.

2 Fig. (a) Interaction of waves with vertically sheared currents over variable seabed topography. (b) Flapping hydrofoil biomimetic system (BS) operating as energy device in waves and sheared currents. The mutual existence of waves with strong following, oblique or opposing currents at various nearshore places, which otherwise is characterized by quite low wave potential, offers a motivation for comprehensive investigation of such resources, and the development of hybrid technological devices, based on novel ideas such as biomimetic systems (see, e.g. BioPower Systems: The latter are appropriate for the efficient energy extraction and exploitation of this complex type of renewable energy resources. In fact, recent research and development results, concerning flapping-wing systems, supported also by extensive experimental evidence and theoretical analysis, have shown that such systems, operating under conditions of optimal wake formation, could achieve high levels of efficiency [,]. On the other hand, the complexity of kinematics of flapping wings necessitates the development of sophisticated power transmission mechanisms and control devices, as compared to the standard hydrodynamic systems, such as water turbines and marine propellers. Novel biomimetic systems based on oscillating hydrofoils operating in the presence of waves and currents could offer an alternative for extraction and exploitation of this kind of marine renewable energy sources. It has been demonstrated by previous and ongoing research that flapping foil thrusters operating in waves, while travelling at constant forward speed, are very efficient, and could be exploited for augmenting the overall ship propulsion in waves by directly converting kinetic energy from ship motions to thrust [3-7]. In previous works potential based panel methods and CFD tools have been developed for the hydrodynamic analysis of these systems, including the effects of the free surface. Predictions are found to be in agreement with other methods and experimental data [4,5]. Also, it has been demonstrated that significant energy can be extracted by the system [3,6,7]. In the above cases, the hydrofoil is controlled in order to develop positive thrust operating as a wave device. An important aspect of the present work is to examine the possibility of energy extraction by oscillating hydrofoils in waves and currents, also by using active pitch control. In this case, the pitching of the foil is appropriately tuned in order to develop heaving forces due to alternating lift and drive the hydraulic generator (HD); see Fig.. The present work is focused on the investigation of oscillating hydrofoils in the presence of waves and currents, examined as biomimetic systems for extraction and exploitation of this kind of marine renewables. The hydrodynamic analysis is based on the coupled-mode model in conjunction with time domain Boundary Element Methods (BEM). Except currents of uniform structure in depth studied [8], special attention is paid to waves and vertically sheared currents, that is the case of tidal currents and nearshore flows strongly interacting with the seabed [9]. The present method takes into account the effect of the wavy free surface through the satisfaction of the corresponding boundary conditions, as well as the effects of variable bathymetry and sheared currents, for determination of the incident flow on the oscillating hydrofoil. Numerical results concerning the extracted power and the operability characteristics of the system in variable bathymetry are presented, indicating that significant output can be obtained under general operating conditions. The present method, after validation, could be found useful for the design and optimum control of such biomimetic systems operating in the nearshore/coastal region and extracting energy from waves and ambient currents. WAVE PROPAGATION IN THE PRESENCE OF SHEARED CURRENTS AND DEPTH VARIATIONS We consider wave propagation in the presence of horizontal, non-homogeneous current, in a variable bathymetry region. The present analysis follows the work by Belibassakis et al () [8] with the basic difference that the current is now considered to be vertically sheared; see Fig.. The current flow is assumed to vary very slowly with respect to the wave period, as it is in the case of tidal and low frequency environmental flows, so that it may be considered steady at the scale of wave evolution. Furthermore, the current is characterized by a vertical variation, possibly stronger that the horizontal one, so that the vorticity associated with the background

3 flow is essentially horizontal like the current itself. Moreover, the current flow velocity is assumed to be small and, and thus, the associated mean free-surface elevation (set-down) is also small. This will constitute the background flow component. On the other hand, the wave flow perturbing the background current flow, is generated by an incident wave system coming from the far up-wave region (see Fig.), and is assumed to be irrotational. A Cartesian coordinate system is used, having its origin at some point on the unperturbed free-surface ( w u u, u, has been assumed to be steady satisfying the z ); see Fig.. The current q u,, with q where continuity equation 3 conditions H H w u u w H f f x x h h w u u w h x x where x, x f denotes the divergence operator in 3D, and the boundary 3, z u, u, z h z H f x, () x. () x the horizontal coordinates, h is the depth and H f denotes the mean set-down associated with the background current flow. Using the fact that the current velocity has been assumed to be small, the mean set-down is a higher-order quantity and the kinematic free-surface boundary condition can be linearised. Also the current is assumed to be nearly horizontal w represented as follows, z z u x U x S x, (3) where Ux, Sx, are horizontal vector fields. Obviously, the first term in the above expansion Ux corresponds to the horizontally varying, but vertically constant current component, which is also the value of the current on the free surface, and the second term zs x corresponds to the component associated with linear vertical shear []. It must be notted that in two-dimensional vertical plane flow fields, with linear sheared currents, the vorticity production terms reduce to zero and the wave motion remains irrotational []. Using this fact that the total velocity field, in the presence of waves, is decomposed as v u 3 x, zt ;, (4) we obtain the following approximate version of Bernoulli-type equation for the pressure p pa u3 gz t z ΨS, (5) where Ψx, z;t is the stream function corresponding to the wave potential, z;t x, z;t Ψx, z;t x : Ψx, z;t, x, z;t. (6) z z Restricting ourselves tο the case of harmonic propagating waves of absolute frequency ω, Re, exp x z;t x z i t, (7) and assuming mild-bottom slope and current variations, we obtain the following expression for the complex wave potential approximated by the propagating mode x; z f xz x, z, where Z, z cosh k( ) z h( ) / cosh k( ) h( ) x x x x x. (8) In the case of vertical sheared current with linear structure in depth, the wavenumber is obtained as root of the following modified version of the dispersion relation [] : kg tanh kh, (9)

4 formulated with respect to the local depth and an appropriate geometric mean of the intrinsic-frequency, defined as follows u, where x is the phase of the wave, () which is calculated by using information concerning the values of the current on the free surface ux, z Ux, and at a characteristic depth u x ux, z d e. In accordance with Eqs. (6) and (8), and the mild-slope assumption, the associated vector stream function is calculated as follows h z sinh k z Ψ x, z, z dz f x. () k cosh kh zh Using the above on the variational principle of the wave-current flow problem based on the pressure functional, we eventually arrive at the following Extended Mild-Slope equation (EMS): a c U u b, () where the coefficients are recently derived and studied (see [] for more details) and are given by,, CC i a CC g b U u, c k CC i g U. (3) g In the above equations, C / k and C / k denote the local wave phase and group velocities relative g to the current, respectively, calculated at the mean (geometric) intrinsic frequency, where, are defined by Eq.() using current speed at z= and z d e. The latter depth is also parametrically dependent on x, defined as follows d tanh kh / k. (4) e In the present work we will restrict ourselves to the D problem formulated on the vertical plane, x x, x x,, as shown Fig.(b). As an example, we consider a variable bathymetry region of the form of smooth shoal characterized by continuously decreasing depth, from h =5m to h 3 =3m; see Fig.. The depth function, presenting monotonic variation, is defined by.5 tanh, where the mean depth is h.5h h h x h h h x m 3 m, (5) and the coefficient is selected so that the mean bottom slope is 6% and the max bottom slope %. Normally incident waves of nondimensional frequency St h / g.4 are propagating in the variable bathymetry in 3 the presence of sheared current. The current speed at the entrance of the domain corresponds to bathymetric Froude number U / gh.5 and the current distribution in the domain is calculated so the mass flow rate is constant U x U x Sh x, (6) where Ux denotes the mean over depth current speed. In this example the shear is selected to be distributed following a Gaussian shape function becoming maximum ( S.5s ), at the centre of the domain (x=) and zero at the ends ( x 5 m), and the horizontal distributions of the current speed at the free surface and at the bottom surface of the domain are shown in the top subplot of Fig. by using solid and dashed lines, respectively. In the middle of the domain (x=) the current speed ranges from.m/s near the free surface and drops linearly to.8m/s at the bottom. In this area the depth is h =9m and the corresponding mean Froude number U / ghm.. Results concerning the calculated wave field in the presence of the above sheared current and bottom topography, corresponding to incident wave height H=.m, are obtained by the present EMS Eq.(), using second-order finite differences to discretize the equations, and are presentd in Fig. by using contour lines. / / are also shown by using solid In the same plots the values of the normalized free-surface elevation lines. It must be noted that the current significantly changes the wave characteristics, and thus the consideration m

5 (a) (b) (c) Figure. Spatial wave evolution over a shoal. Incident waves of nondimensional frequency h / g.4 propagating in the presence of sheared current. (a) Current velocity at free surface and bottom surface. (b) Real part and (c) Imaginary part of the calculated wave field. of shear in the current further modifies the wave field, especially in the vicinity of the sloping bottom subregion. 3 FLAPPING HYDROFOIL AS A BIOMIMETIC ENERGY DEVICE IN WAVES AND CURRENTS We consider waves and currents propagating in a variable bathymetry region, lying between two regions of constant, but possibly different depth, as the shoaling environment shown in Fig.. In this area a biomimetic system (BS) is operating as a combined wave and current energy extraction device, which for simplicity is modelled as an oscillating hydrofoil of chord c, performing simultaneously vertical heaving and rotational pitching oscillations, while being elastically connected to the energy generator subsystem. Restricting ourselves for simplicity in a D problem in the vertical plane, under the assumption of small wave amplitude the disturbance flow due to oscillating foil is formulated as a time-dependent problem characterized by a moving foil boundary D t with respect to the earth-fixed frame of reference. The total velocity field B V ( x, z ; t ) consists of the incident field velocity v ( x, z ; t ) and the disturbance potential ( x, z ; t ). Moreover, the T incident velocity also decomposes to the sheared current velocity u U(x) Sz, and the propagating wave velocity ( x, z ; t ), as given by Eq.(7). The disturbance potential ( x, z ; t ) satisfies the D Laplace equation ( x, z; t), ( x, z) D, (7) supplemented by the body boundary condition ( x, z; t) v n B +VB nb b, ( x, z) DB, (8) n where the right hand side of the Neumann boundary condition is denoted by b and no-entrance boundary condition is also imposed, z h x, n v u Φ. The bottom (9) as well as the linearized dynamic and kinematic boundary conditions, including the effect of the sheared current, are satisfied on the mean free-surface level ( x, z; t) ( x, z; t) U x g ( x; t), on z, () t x ( x ; t ) ( x, z ; t ) ( x ; t U x ), z. () t z x

6 In lifting flows around a hydrofoil, the problem is supplemented by the Kutta condition, necessitating equal pressure at the trailing edge and the kinematic and dynamic conditions on the trailing vortex sheet stating that the pressure and the normal velocity are continuous through the trailing vortex wake u l p ( x, z; t) p ( x, z; t), u l ( x, z; t) ( x, z; t), ( x, z ) DW, () n n where the indices ul, are used to denote the upper and the lower sides of the wake. We treat the above as an initial value problem and we assume that the disturbance field due to the presence of the body vanishes at large distance from the body. In the above equations ( xt ; ) denotes the free surface elevation associated with the disturbance field and g the acceleration of gravity. Furthermore, d is the mean submergence of the hydrofoil, V B denotes the instantaneous velocity of the body at each point on the boundary and n is the unit normal vector pointing into the interior of D. The unsteady pressure field, including the wave component and the disturbance due to the oscillating hydrofoil, in the domain is calculated by the Bernoulli's equation including the effect of the sheared current as follows p U U U u SU gz, ( x, z) D, (3) t where U, U is the corresponding stream function, and p denotes the pressure of the unsteady wave field. In the present work we have assumed that the disturbance stream function is small and thus, we have used the approximation U. Using Eqs. () in conjunction with Bernoulli's theorem Eq. (3) we obtain the following formula D x, z; t W Dt, ( x, z) D, (4) W u l D where W denotes the potential jump (the dipole intensity) on the free wake and V m is Dt t the material derivative based on the mean velocity V on the trailing vortex sheet. The above relation states that m D W evolves in time as a material curve moving with the mean velocity. The continuous generation of the sheet is also associated with the trailing edge motion and a simplified wake model is derived by approximating V u, resulting in sinusoidal shape of the trailing vortex sheet emanating at the trailing edge of the m oscillating hydrofoil. This wake model provides satisfying predictions in the case of low and moderate unsteadiness [4]. Concerning now the motion of the foil, it consists of vertical oscillations Z(t) simultaneously with pitching oscillations Θ(t) with respect to a pivot point selected near the pressure center of the hydrofoil (usually at a distance c/3 from the leading edge). In the present case the pitching motion is enforced (and thus the angle of attack can be controlled) and the response of the examined system is the power extracted by the vertical oscillations, P b Z, where mz bz kz L, t cos t ψ, (5) o Z where,, ψ are the frequency, amplitude and phase of the pitching motion, m is the mass of the hydrofoil and bk, are characteristics of the power take off and stiffness of the generator and the elastic connector, respectively. Also LZ t denotes the vertical lift forces obtained through hydrodynamic pressure integration on the foil. In the present work we consider incident waves of angular frequency ω equal to the frequency of the controlled pitching motion. 4 THE BOUNDARY ELEMENT METHOD Applying the Green's representation theorem [4] to our problem for points on the free surface, the body and the bottom, we obtain a system of three equations for the unknown boundary distributions of the potential and its normal to the boundary derivative, written compactly as follows

7 G( x x) ( x; t) G( x x) ( x ; t) ( ; t) G( ) ( ; t) ds( ) G( ) ( ; t) ds( ) b x x x x x x x x x F F / B/ H B F D n t D n n B H ( x; t) G( x x) G( x x) G( x x) ds( x) H( ; t) ds( ) W ; t ds( ) D n x x D n x x. (6) D n H H W In the above relation we have used the fundamental solution of D Laplace equation corresponding to a Rankine source G( x x) ln r( x x ), where x ( x,z ) is the field point and x ( xz, ) is the integration point. The normal derivative on the boundary is defined by differentiating G( x x) / n n G( x x). Eq.(6) is used in conjunction with low-order panel method [4,] to derive a discrete version of the Dirichlet-to-Neumann map (DtN). Using the latter in the free-surface boundary conditions and the pressure-type Kutta condition a set of ODEs is derived governing the dynamics of the system, as follows F Φ t F η ΦF gη D U, xφf, D U, xη, (7) t n where the vectors ΦF, η denote the discrete values of the potential on each panel on the free surface, and the values of the free surface elevation. The horizontal derivatives appearing in Eq.(7) (stemming from Bernoulli s equation) are approximated by the following finite difference scheme on the free surface U 3 4 D X, x Ui, where F, F x X X Φ η. (8) U U N 4 3 f Moreover, the nonlinear pressure-type Kutta condition is put in discrete form, permitting to obtain an extra equation involving the dipole distribution in the hydrofoil wake in the vicinity of the trailing edge [4]. Using the latter, in conjunction with the above equations, we finally obtain the following system of ODEs dy T fy ( ), where Y ΦF η W. (9) dt Starting from a prescribed initial condition, e.g. from rest, a time-stepping method is used to obtain the numerical solution of the problem concerning the disturbance potential ( x, z; t) due to hydrofoil oscillation in the incoming waves and currents field v ( x, z; t), that is previously calculated by means of the present EMS, Eq.(). After evaluation of different methods, we found that the higher-order Adams-Bashford-Moulton predictor-corrector method provides the required accuracy, stability and efficiency. The used scheme requires calculation of only two derivative formulas at each time, and the error is of order (Δt 5 ), where Δt is the time step, ensuring that good convergence is achieved. 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. The present work is based on the truncation of the domain and on the implementation of a Perfectly Matched Layer (PML) model. The latter model permits the numerical absorption of the waves reaching the left and right termination ends of the truncated domain, with minimum reflection. More details about the time-integration method, stability and convergence of the present scheme as well as the PML absorbing model can be found in our previous work [4]. After the solution has been obtained (at each time step) the free surface elevation and the pressure distribution are calculated through the unsteady Bernoulli s theorem. Furthermore, forces and the moment are directly calculated through the integration of the instantaneous pressure on the solid boundary. 5 NUMERICAL RESULTS AND DISCUSSION In this section numerical results are presented and discussed indicating at a first level of approximation the performance of oscillating hydrofoils operating in incoming waves and currents in variable depth regions and in the presence of sheared currents. As an example, we consider a NACA hydrofoil of chord c=m operating

8 (a) t/t=4. K.A. Belibassakis, E.S. Filippas, and Th.P. Gerostathis. C P x/c (b) C P t/t=4.5 x/c C P t/t=4.5 x/c C P t/t=4.75 x/c Fig. 3 (a) Simulation of the oscillating hydrofoil operating as biomimetic energy system in waves and sheared currents over variable bathymetry, at various time instants in a period. (b) Output angle of attack, pitching angle, heaving motion and lift and moment coefficients. (a) (b) Fig.4 (a) Output power coefficients in the case of Fig. 3. (b) Increase of output power coefficient with incident wave amplitude. (a) (b) Fig.5 Output power coefficient in the case of Fig. 3 for (a) various values of St, (b) various values of ψ.

9 in coastal region in the presence of propagating waves and vertically sheared currents with parameters described in the last paragraph of Section. The hydrofoil, arranged at mean submergence d=h/, is performing simultaneous heaving and pitching oscillations, and the corresponding reduced frequency is kr U / c.3. In this case, the enforced pitching motion has amplitude Θ =deg and its phase lag with respect to the incoming wave is deg while the resulting heaving motion is part of the solution of Eq. 5. In Fig. 3(a) we plot at various time instants of the fifth period of oscillation the foil and the trailing vortex curve modelling the foil's wake, including the calculated dipole intensity (potential jump) on the vortex sheet, which is illustrated by using arrows normal to the wake curve with length proportional to the local dipole strength. The latter result is associated with the memory effect of the generated lifting flow around the hydrofoil. Moreover, in the right subplots of Fig. 3(a) the instantaneous distribution of the pressure coefficient c p p /.5 U p a is shown, at the same time instants, as calculated by the present method. From the calculated pressure distributions, lift and thrust components are obtained at each time step by surface integration. In this case, the effective angle of attack tan Z u z x, against the pitching motion is shown in Fig. 3(b) and the resulting heaving oscillation is plotted, as obtained by the present system, using b / ( m R).5 to model the constants of the power take off (PTO), corresponding to resonant frequency k/ m, equal to R.7 rad / s. The output of the considered system operating in the above conditions is plotted in Fig. 4(a) using the power coefficient 3 C P / (.5 U c). We observe in this case that the extracted energy with mean Pb value C Pb =.95 is significant, especially as compared to the power needed for controlling the pitching foil motion, indicated in Fig. 4(α) through the corresponding pitching power coefficient and shown by using dashed lines. Next results from parametric investigation of the examined system are presented. More specifically, in Fig. 4(b) the increase of the average output power with incident wave amplitude is shown, keeping the rest of parameters as in Fig.3. Also, is presented in Fig. 5(a) the effect of Strouhal number is demonstrated. Finally, an examination of the pitching phase effect on the performance of the system operating in waves and current is presented in Fig. 5(b). The values of average power coefficient, due to energy extraction from waves and currents, is presented as a portion of its value C P. In the same figure, the case where there is no propagating wave is shown with horizontal bold line at %. This result, in the case of the examined example, indicates that the output power due to current energy can be significantly increased up to almost 5% by suitable tuning of the phase of the pitching oscillation with respect to the phase of the incoming wave. 6 CONCLUSIONS Oscillating hydrofoils in waves and currents are investigated as novel biomimetic systems for extraction and exploitation of this kind of marine renewable energy. The present method takes into account the effect of the wavy free surface through the satisfaction of the corresponding boundary conditions, as well as the velocity component due to waves and vertically sheared currents on the formation of the incident flow. Numerical results concerning the extracted power and the operability characteristics of the system are presented, indicating that significant output can be obtained under general operating conditions. The present method, after validation, could be found useful for the design and optimum control of such biomimetic systems operating in the nearshore/coastal region and extracting energy from waves and ambient currents. Future work is planned towards the detailed investigation and systematic examination of the effects of various parameters of the environment as well as the controlled pitching motion. Also, very important is the further validation and examination of the range of applicability of the present method through comparison with other numerical methods. ACKNOWLEDGEMENTS This research is funded by the Special Account for Research Grants of the TEI of Athens, in the framework of the Internal Programme for the Support of the TEI of Athens Researchers, for 5. R REFERENCES [] Mei, C.C. (996), The applied dynamics of ocean surface waves, nd ed., Singapore, World Scientific. [] Lavrenov, I.V., Porubov, A.V. (6), Three reasons for freak wave generation in the non-uniform current, European Journal of Mechanics - B/Fluids, 5, pp [3] Peregrine, D.H. (976), Interaction of waves and currects,, Adv. Appl. Mech. 6, pp.9 7. [4] Jonsson, I.G. (99), Wave-current interactions, in B. LeMehaute & D.M. Hanes Eds, The Sea, 9(a), pp.

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