The second-order diffraction-radiation for a piston-like arrangement

Transcription

1 The second-order diraction-radiation or a piston-like arrangement I.K. Chatjigeorgiou & S.A. Mavrakos Laboratory o Floating Structures and Mooring Systems, School o Naval Architecture and Marine Engineering, National Technical University o Athens, 9 Heroon Polytechniou Ave. Zograos Campus, 5773 Athens, Greece ABSTACT: The second-order double requency diraction-radiation hydrodynamic problem or a piston-like arrangement is considered. The structural model consists o two concentric cylinders, the outer being the liner and the inner being the piston. The geometry o the arrangement creates a cylindrical moonpool between the bodies that is potentially capable to stimulate resonances o hydrodynamic nature which accordingly will be relected on both the ree surace elevation o the inner luid domain and the hydrodynamic loading. INTODUCTION Moonpools are known to stimulate peaks in the transer unctions o the exciting orces and the hydrodynamic parameters (added mass and hydrodynamic damping). The location o maxima depends on the geometrical properties o the assembly and obviously, the period o the incoming waves. These artiicially conined luid regions may have consequences both rom the theoretical and the practical point o view. In the last years an increasing interest is reported especially in connection with their use as heaving wave energy converters or as oscillating water column (OWC) devices or the extraction o energy rom waves. Examples on theoretical approaches to the solution o the linear hydrodynamic problem related to structures with moonpools may be ound in the works reported by Mavrakos (985), (4) and (5) and Liu et al. (993). The structural assembly that is considered herein is the model used by Mavrakos (4) and (5). The present authors have extended the work o Mavrakos (4) and (5) in order to solve the second-order double requency diraction problem or the speciic piston-like arrangement assuming that is restrained in waves, and results have been presented in Mavrakos & Chatjigeorgiou (6b). The method they developed was used recently by Mavrakos et al. (8) or calculating the perormance characteristics o a tightly moored piston-like wave energy converter under irst- and second-order wave loads. Again, the second-order loading was derived assuming that both bodies are motionless. Here, the already developed solution tool, based on the semi-analytical ormulation o the velocity potentials, is properly extended in order to account or the radiation problem or the inner body in heave. The assumption that only the piston is allowed to perorm motions, relies on the piston-liner concept o the inner combustion engines where mechanical energy is created by the piston, moving along the liner in one direction. In the context o the combined diraction-radiation problem, the proper treatment o the ree surace boundary condition is o major importance. This is due to the extremely complicated orm o this condition due to the addition o more quadratic components arising rom the radiation potentials. The successul analytical approach o the solution requires the proper decomposition o the second-order potentials into a rational number o components which diers or each ield. In addition, second-order radiation potentials must be deined, while the

2 solution o the associated problem eventually results in the derivation o the double requency hydrodynamic parameters. POBLEM S DESCIPTION AND BIEF DISCUSSION ON THE SOLUTION The second-order diraction-radiation problem by an arrangement o two concentric cylinders is considered. eerence is made to Figure which depicts the layout o the arrangement. The cylinders are exposed to the action o a monochromatic incident wave o requency ω and linear amplitude H/ in constant water depth h. The draughts o the cylinders are h-h and h-h, respectively. A cylindrical coordinate system (r,θ,z) is chosen, with its origin on the sea bed and its Oz axis pointing vertically upwards. Viscous eects are neglected and the assumption is made that the luid is incompressible. Figure. Main dimensions and luid regions o the piston-like arrangement The time-dependent irst- and second-order potentials in all luid regions deined in the above igure can be expressed as: n { n iω e t } ( r, θ, z; t) = e ϕ ( r, θ, z) Φ or n=, () n The most diicult and challenging part related to the hydrodynamic problem o the speciic arrangement is the proper elaboration o the second-order ree surace boundary condition in the outer luid domain (ield A) and in the moonpool (ield C). The general orm o the associated expression is given by iω ν 4νϕ + ϕz = ϕ ϕ + ϕϕz ϕϕzz () g where ν=ω /g and the index z denotes dierentiation with respect to the vertical coordinate. The successul implementation o the semi-analytical ormulation or the velocity potentials requires the proper decomposition o the total velocity potentials into a number o components that accordingly will assist to the satisaction o all boundary conditions. In this context, the secondorder velocity potentials in ields A and C are split according to I ID DD A = ϕ + ϕ A + ϕ A ϕ A (3) ϕ + ID DD C = ϕ C + ϕ C ϕ C (4) ϕ +

3 The superscripts ID, DD and denote respectively the ree, the locked and the radiated components, while ϕ I represents the second-order double requency incident wave (Mei, 983). Using the decomposition principle, the potentials in the lower ields B and D will be given by D B = ϕ B ϕ B (5) ϕ + D D = ϕ D ϕ D (6) ϕ + Apparently, all velocity components deined previously should satisy the Laplace equation and the zero velocity condition on the sea bottom. In order to save valuable space we avoid giving the complete list o the mathematical ormulations or the boundary conditions. It is simply mentioned that the locked wave components satisy the inhomogeneous orm o the ree surace condition in the associated ield and the zero velocity condition or the hypothetical bottom seated cylinders or z h. Thus the normal velocities due to the locked wave DD component ϕ C r must be zero at r=b and r=b while ϕ DD A r must be zero at r=b. As a result the ree wave components should satisy the homogeneous orm o the ree surace condition while in addition they are used or satisying the zero velocity condition on the immersed part o the cylinders as well as or matching the pressures and the velocities along the common boundaries o the adjacent luid annuluses. For calculating the ree wave components, both constituent structural parts are considered completely stationary. In order to use the matching relations or the stationary cylinders, the total velocity potentials in the lower ields B and D should contain an assisting diraction term and a radiation term. The diraction terms, D D D D namely ϕ B and ϕ D are obtained assuming that ϕ B z and ϕ D z are zero at z=h and z=h respectively. According to the description that was just preceded, the second-order radiation components ϕ A and ϕ C must be in compliance with the homogeneous orm o the ree surace condition in the outer ield and in the moonpool respectively. Furthermore, as the toroidal body is stationary and the piston perorms only heaving motions the ollowing should apply: B = z=h ϕ z (7) D z = z= h ϕ iωξ (),() 3 (8) (),() where ξ3 is the double requency heaving motion o the piston (body ). Under these conditions, the irst- and the second-order radiation problems become identical. Below we provide the semi-analytical expressions or the various second-order radiation components that satisy the Laplace equation as well as the boundary conditions which are involved in the mathematical ormulation. ~ (),() (A),() () ϕ A iω( H ) ξ3 = F3 j Z j ( z) K ( κ jr) K ( κ jb) (9) j=

4 ~ (),() ( ) ( B) (B),() ~ ( B) ~ (B),() ϕ B = iω H ξ3 ε n [ n ( r) F3n + n ( r) F3n ] cos( nπz / h ) () n= ~ (),() (C) (C),() ~ (C) ~ (C),() ( H ) ξ [ ( r) F + ( r) F ] () C = iω 3 j 3 j j 3 j Z j ( z) j= ϕ () ϕ ~ z r / = iω 3 hh n= (),() (D),() ( H ) ξ + ε F cos( nπz / h ) I ( nπr / h ) I ( nπb h ) D n 3n / () where Z () j (z) are the orthonoral unctions in vertical direction that satisy the boundary condition on the bottom and the homogeneous boundary condition on the ree surace, κ j is used to denote the ininite solutions o the second-order transcendental equation including the ~ (),() (),() imaginary solution iκ or j=, ξ 3 = ξ3 h /( H / ), ε =, ε n =, n and I, K are the zeroth order modiied Bessel unctions o the irst- and the second kind respectively. For more details regarding the derivation o the above semi-analytical ormulations as well as or the ( B) ( ) complete mathematical expansion o the radial unctions n, ~ B n, (C) ~ ( ) j and C j the reader is reerenced to the works reported by Mavrakos (4) and (5). The symbols F in Eqs. (9)-() denote the unknown Fourier coeicients in the various regions which must be calculated by applying the zero velocity conditions on the walls o the bodies and the matching relations o the luid velocities and pressures along the common boundaries o the luid regions (see Mavrakos (985), (988), (4) and (5)). For the structural assembly under consideration and or the conditions o motions speciied (),() previously, the hydrodynamic coeicients that interest are the added mass A and the (),() damping coeicient B, namely the hydrodynamic parameters or body due to the motions o the same body in heave. These are obtained by integrating the pressure due to the radiation potential ϕ D on the bottom part o the piston. Following Mavrakos (4) F ( p),( s) ( p),( s) iωt ( s) ( p) e = p3 n3 ds = i ( p) S ( p),( s) ( p),( s) iωt ( ω A + ωb ) e = (3) (p) Herein n 3 denotes the vertical component o the outward unit normal vector to the mean wetted surace S (p) o the body p. Apparently in our case both p and s should be replaced by. The calculation o the ree and the locked wave components in regions A and C and the diraction components in the lower ields is perormed separately with the basic assumption that both bodies are immovable. The relevant procedure is extremely complicated and cannot be outlined in the present short text. The method was briely presented by Mavrakos & Chatjigeorgiou (6b) and its details were included in a recently submitted work by the same authors (Mavrakos & Chatjigeorgiou, 8). However it is enorceable to provide inormation at least or the expansion o the inhomogeneous term o the ree surace boundary condition due to the irst-order radiation components. Let s irst denote the right hand side term o Eq. () by Q(r, θ). Then, this term will be composed by quadratic terms due to i) incident wave and diraction components Q ID, ii) diraction component only Q DD, iii) incident wave and radiation components Q I, iv) diraction and radiation components Q D and v) radiation component only Q. Especially or the moonpool, Q(r, θ) will contain also the quadratic terms due to the incident wave component Q II. There is adequate discussion in the literature regarding the

5 quadratic terms Q ID and Q DD, or both ininite (e.g. Kim & Yue, 989; Huang & Eatock Taylor, 996; Malenica et al., 999) and conined luid domains (e.g. Chatjigeorgiou & Mavrakos, 5, 6, 7; Mavrakos & Chatjigeorgiou 6a). Here, we restrict our attention to the additional quadratic terms due to the irst-order radiation components. These are I Q D iω I ν I I ( r, θ ) + Q ( r, θ ) + Q ( r, θ ) = ϕ ϕ + ( ϕ ϕ z + ϕ ϕ z ) g I D ν D D ( ϕ ϕ + ϕ ϕ ) + ϕ ϕ + ( ϕ ϕ + ϕ ϕ ) ( ϕ ϕ + ϕ ϕ ) I zz zz ν + ϕ ϕ + ϕ ϕ z ϕ ϕ zz z Here, the index denotes irst-order components. Eq. (4) should be applied together with Q ID and Q DD to both ields A and C that extend up to the ree surace. It is evident that or each ield the relevant irst-order potentials should be used. Especially or the moonpool, the above should be supplemented by the quadratic components due to the irst-order incident waves D z I I ( ν k ) ϕ iω iω = (5) g r g II I I I I Q ( r, θ ) ϕ r ϕ r + ϕ θ ϕθ + 3 ϕ with k being the wave number. The complete analytical representation o Eq. (4) using the governing semi-analytical ormulations o the irst-order components (Mavrakos, 5) creates an enormously lengthy expression and its details are omitted. 3 NUMEICAL ESULTS AND DISCUSSION The structural model that was used in the present, is the piston-like arrangement employed and investigated by Mavrakos et al. (8). With reerence to the notation adopted in Figure, the dimensions o the arrangement in normalized orm are: b/h=., b /b=.5, b /b=.75, h /h=.7, h /h=.6. The numerical results shown in sequel have been plotted against the nondimensional requency o the incoming waves. All governing components which are derived rom the solution o the hydrodynamic problem have been considered, i.e. the hydrodynamic parameters (Figures and 3), the heaving motions o the piston (Figure 4) and the hydrodynamic loading (Figures 5 and 6). The calculation o the piston s displacements in heave is required due to their involvement in the inhomogeneous term o the ree surace boundary condition through the irst-order radiation potential. The motions are determined by examining the piston as a single degree o reedom oscillator and taking into account the hydrostatic restoring coeicient in heave ρga WL, where A WL stands or the waterplane area o the piston. Ater short mathematical manipulations, it can be shown that the irst-order heaving motions o the piston normalized by the amplitude o the incoming waves is given by zz D zz (4) (),() (),() (),() ξ (),() 3 Fz m + A B b = νb + 3 iνb π H / 3 ρgb H / b ρb ωρb (6) (),() where F z is the irst-order vertical diraction orce acting on body and m is the mass o the body. As no particular symbol is used to distinguish between irst- and second-order hydrodynamic parameters, it should be mentioned that the above reer to irst-order while those contained in Eq. (3) correspond to second-order. The added mass and the hydrodynamic damping coeicients are given in Figures and 3 respectively. The variation o the double requency hydrodynamic parameters is governed by the same idiomorphic behavior maniested

6 in the irst-order counterparts. In particular, the added mass coeicient exhibits an increase which is ollowed by a sadden drop down, and eventually, the speciic parameter tends asymptotically to a constant or increasing requencies. The resonance is also demonstrated in the hydrodynamic damping coeicients (Figure 3) and occurs at exactly the same requency. By inspecting Figures and 3 one may easily conclude that besides the apparent dierence in magnitudes, the resonance on the second-order components is more sharp and proound. In addition the second-order resonance is stimulated in a requency that is signiicantly smaller that the irst-order resonance requency..5 st order nd order a (),() Figure. First- and second-order hydrodynamic mass coeicients 3.5 a (),() (),() 3 = A ρb st order nd order b (),() Figure 3. First- and second-order hydrodynamic damping coeicients b (),() (),() 3 = B ωρb The calculation o the heaving motions o the piston through Eq. (6) requires that the mass m o the body should be known. As the mass may very depending on the application, dierent values or the mass were assumed and set as a percentage o the normalized added mass. The cases that were considered are listed in the caption o Figure 4. It is immediately apparent that

7 the maxima o the motion shit towards to lower requencies or increasing mass values. It can be saely concluded that this eature is associated with the location o the eigenrequency o the dynamic system as the peaks o irst-order hydrodynamic parameters occur at a requency which is apparently higher. However it is important to observe the eect o the hydrodynamic resonance to the motions o the piston, which is maniested through a small increase at approximately =.8. 8 ξ (),() 3 H / Figure 4. Vertical motions o the piston. Symbols indicate: m =, (),() m = a, * (),() m = 4a (),() m = a, () x 4 () x () x Figure 5. First- and second-order horizontal exciting orces on the piston First-order orce, Second-order orce due to the irst-order potential or the ixed body, Second-order orce due to the second-order potential or the ixed body, * Second-order orce due to the second-order potential obtained by combined diraction-radiation theory b/h=., b /b=.5, b /b=.75, h /h=.7, h /h=.6.

8 4 3.5 () z () z () z Figure 6. First- and second-order vertical exciting orces on the piston First-order orce, Second-order orce due to the irst-order potential or the ixed body, Second-order orce due to the second-order potential or the ixed body, * Second-order orce due to the second-order potential obtained by combined diraction-radiation theory b/h=., b /b=.5, b /b=.75, h /h=.7, h /h=.6. The hydrodynamic loading exerted on the piston is investigated with the aid o Figures 5 and 6 which depict the surge and heave orces, respectively. The orces are given in the ollowing nondimensional orm: k () =F k () /ρgb (H/), k () =F k () /ρgb(h/) and k () =F k () /ρgb(h/), where k=x and z. According to Figures 5 and 6 the second-order exciting orces due to the second-order potential exhibit two local maxima in the range o interest whereas the irst-order orces exhibit only one, which, in addition, is located at the high requency portion. That eature implies the signiicance o the second-order eects or the particular geometric arrangement as they can stimulate resonances in lower wave requencies which are more possible to be encountered in reality. The lower requency peak occurs at exactly the same requency as the second-order hydrodynamic coeicients (Figures and 3), while the second peak appears to be driven by the irst-order contributions. The symbols in Figures 5 and 6 denote the numerical results which were obtained by applying the present combined diraction radiation solution method and assuming that the mass o the piston is equal to the added mass in heave. The symbols are compared with the chained lines which assume that both bodies are ixed. It is evident that the results are comparable, which in turn implies that the energy oered to the piston is not signiicantly reduced due to the act that the piston is allowed to perorm heaving motions. The encouraging aspect is that the magnitude o the surge orce, which occurs at the irst resonance (Figure 5), exhibits a large scale reduction when the inner body is ree to heave. The last observation signiies that, at least according to the present calculations, the piston is less possible to be strongly impelled towards the inner surace o the hollow toroidal cylinder. 4 CONCLUSIONS The second-order diraction-radiation problem or a piston-like arrangement was considered. The solution was achieved by extending the solution method developed by the present authors or the diraction problem o the ixed arrangement. This was carried out by properly treating the ree surace boundary condition ater including the irst-order radiation potential as well.

9 The method resulted in the deinition and the calculation o the second-order hydrodynamic coeicients that govern the double requency motions o the piston. ACKNOWLEDGMENTS The results presented herein are part o a project supported by NTUA's undamental research program K. KAATHEODOI. 5 EFEENCES Chatjigeorgiou, Ι.Κ. & Mavrakos, S.A. 5. Second-order diraction by a surace piercing truncated compound cylinder, Proc th Int Congress o the Int Maritime Association o the Mediterranean, IMAM 5, Lisbon, Portugal, Vol. : Chatjigeorgiou, I.K. & Mavrakos, S.A. 6, Semi-analytical ormulation o the second-order wave diraction by a truncated-compound surace-piercing cylinder, Journal o Ship Technology esearch, Schistechnik, 53(): 6-38 Chatjigeorgiou, I.K. & Mavrakos, S.A. 7. Second-order hydrodynamic resonances on a compound truncated cylinder loating near the ree surace, Proc 6th Int Con on Oshore Mechanics and Arctic Eng, OMAE 7, Sam Diego, Caliornia, USA, Paper No 936 Huang, J.B. & Eatock Taylor, Semi-analytical solution or second-order wave diraction by a truncated circular cylinder in monochromatic waves. Journal o Fluid Mechanics, 39: Kim, M-H, & Yue, D.K.P The complete second-order diraction or an axisymmetric body. Part. Monochromatic incident waves. Journal o Fluid Mechanics, : Liu, Y., Yue, D.K.P. & Kim, M-H First- and second-order responses o a loating toroidal structure in long-crested irregular seas, Applied Ocean esearch, 5(3): Malenica, S., Eatock Taylor,. & Huang, J.B Second-order water wave diraction by an array o vertical cylinders. Journal o Fluid Mechanics, 39: Mavrakos, S.A Wave loads on a stationary loating bottomless cylindrical body with inite wall thickness, Applied Ocean esearch, 7(4): 3-4. Mavrakos, S.A Hydrodynamic coeicients or a thick-walled bottomless cylindrical body loating in water o inite depth. Ocean Engineering, 5(3): 3-9. Mavrakos, S.A. 4. Hydrodynamic coeicients in heave o two concentric surace-piercing truncated circular cylinders, Applied Ocean esearch, 6(3-4): Mavrakos, S.A. 5. Hydrodynamic characteristics o two concentric surace piercing loating circular cylinders, Proc th Int Congress o the Int Maritime Association o the Mediterranean, IMAM 5, Lisbon, Portugal, Vol. : -8. Mavrakos, S.A. & Chatjigeorgiou, I.K. 6a. Second-order diraction by a bottom seated compound cylinder, Journal o Fluids and Structures, (4): Mavrakos, S.A. & Chatjigeorgiou, I.K. 6b. Second-order diraction by two concentric truncated cylinders. Proc st Int Workshop on Water Waves and Floating Bodies, Loughborough, UK, 7-. Mavrakos, S.A., Katsaounis, G. & Chatjigeorgiou, I.K. 8. Perormance characteristics o a tightly moored piston-like wave energy converters under irst- and second-order wave loads, Proc 7th Int Con on Oshore Mechanics and Arctic Eng, OMAE 8, Estoril, Portugal, Paper No Mavrakos, S.A. & Chatjigeorgiou, I.K. 8. Second-order hydrodynamic eects on an arrangement o two concentric truncated vertical cylinders. Marine Structures (submitted) Mei, C.C The Applied Dynamics o Ocean Surace Waves, Wiley, New York.