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Prediction of added resistance of ships in waves
Ocean Engineering 38 (2011) 641–650
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Prediction of added resistance of ships in waves Shukui Liu, Apostolos Papanikolaou n, George Zaraphonitis Ship Design Laboratory, National Technical University of Athens, Greece
a r t i c l e in f o
abstract
Article history: Received 23 November 2009 Accepted 7 December 2010 Editor-in-Chief: A.I. Incecik Available online 3 January 2011
The prediction of the added resistance of ships in waves is a demanding, quasi-second-order seakeeping problem of high practical interest. In the present paper, a well established frequency domain 3D panel method and a new hybrid time domain Rankine source-Green function method of NTUA-SDL are used to solve the basic seakeeping problem and to calculate first order velocity potentials and the Kochin functions, as necessary for the calculation of the added resistance by Maruo’s far-field method. A wide range of case studies for different hull forms (slender and bulky) was used to validate the applicability and accuracy of the implemented methods in practice and important conclusions regarding the efficiency of the investigated methods are drawn. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Added resistance Drift forces Far-field method Near-field method 3D panel method Kochin function
1. Introduction When sailing in a seaway, the calm water resistance and powering of a ship is increased and this is accounted for in traditional ship design by some rough proportional increase of calm water resistance by about 20–40%. However, the accurate prediction of ship’s resistance in waves is nowadays of increased importance, both for the designer and the shipowner/operator, since it greatly affects the selection of ship’s engine/propulsion system and ship’s performance in terms of sustainable service speed and fuel consumption in realistic sea conditions. Also, accurate and efficient predictions of the added resistance in natural seaways are necessary for the implementation of modern onboard ship routing systems. The added resistance is a steady force of second-order with respect to the incident wave’s amplitude and acting opposite to ship’s forward speed in longitudinal direction. When the ship has zero forward speed, then the added resistance is trivially identical to the longitudinal drift force (Papanikolaou and Zaraphonitis, 1987). There are several alternative approaches to the added resistance problem; they can be generally classified into two main categories, namely far-field and near-field methods. The far-field methods are based on considerations of the diffracted and radiated wave energy and momentum flux at infinity, leading to the steady added resistance force by the total rate of momentum change. The nearfield method, on the other side, leads to the added resistance as the steady second-order force obtained by direct integration of the hydrodynamic, steady second-order pressure acting on the wetted
n
Corresponding author. E-mail address: [email protected] (A. Papanikolaou).
0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.12.007
ship surface. The latter can be calculated exactly from first order potential functions, and their derivatives. It is evident that though both above methods should lead to the same numerical results for the added resistance, this is not so in practice, because both methods request equally accurate solutions of the basic potential, seakeeping theory problem in the near and far field, which cannot be ensured in practice. This is a well-known observation in the similarly posed drift force problem. Thus, even if the same potential theory and related numerical methods are used for the solution of the linear seakeeping problem and the calculation of the governing potential function (and its derivatives and of related hydrodynamic quantities, including ship motions, as necessary) in the near- and far-field, the numerical results by the near- and far-field method for the added resistance are in general different, thus a basic question arises, namely which method is more efficient for practical applications. In this respect, several theoretical approaches of varying complexity and accuracy have been introduced in the past and numerically implemented/verified. The first far-field approach was introduced by Maruo (1957) in the late 50s; it was further elaborated in the following years by Maruo (1960, 1963) and Joosen (1966). In the early 70s, the first near field, direct pressure integration methods appeared; however, the considered hydrodynamic pressure distribution was highly simplified (e.g. Boese, 1970). At about the same time, the radiated energy approach of Gerritsma and Beukelman (1972) was introduced, which is basically following the far-field approach of Maruo. Strøm-Tejsen et al. (1973) did a thorough evaluation of all the above approaches to find large discrepancies between the numerical results form different theoretical approaches and relevant model experimental data. Later on, Salvesen (1974) investigated the added resistance problem by applying Gerritsma and Beukelman’s method, but using
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basic results of the STF seakeeping strip theory (Salvesen et al. 1970) and found quite satisfactory results for the investigated ship hull forms. This was not surprising, considering the superiority of the STF strip theory method for the prediction of ship motions over other methods at that time. Also, the significance of accurate ship motions predictions for a reliable estimation of the added resistance was acknowledged and one may have concluded that if a more accurate calculation of the ship motions could be used (or even ship motions from model experiments), all above alternative approaches to the added resistance may have led to better results. An exact in the sense of potential theory near-field, direct pressure integration approach to the added resistance problem was introduced by Faltinsen et al. (1980), with good validation results. The observed deficiency of the approach in short waves was addressed by the introduction of a simplified added resistance estimation formula, which models remarkably well the complicate interaction between the diffraction of waves and the steady flow around the ship. The same problem appears in exaggerated form with low speed, full hull forms with blunt bows (bulkcarriers and tankers), for which Ohkusu (1985) proposed an improved approach. This method was further elaborated by Iwashita and Ohkusu (1992), who applied Maruo’s (1963) improved far-field formulation and the concept of Kochin functions, to obtain very good results for the added resistance of a fully submerged spheroid by use of a 3D pulsating and traveling source, Green function method. In his most recent state of the art review, Naito (2008) concluded that y ‘‘good agreement can be obtained by applying this (latter) calculation method to all types of ship. However this method has not been widely utilized due to the complicated calculation technique and large computing time required’’. More recently, using an enhanced unified theory, Kashiwagi (2009) calculated the added resistance by a modified version of Maruo’s approach, obtaining also quite satisfactory results. In the present study, the well established frequency domain 3D panel method and computer code NEWDRIFT (Papanikolaou– Zaraphonitis, 1987, Papanikolaou–Schellin, 1992 and Papanikolaou et al. 2000), a new time domain Green function method (Liu et al. 2007) and a hybrid time domain the Rankine source-Green function method (Liu and Papanikolaou, 2009, 2010) of NTUA-SDL are used to solve the basic seakeeping problem and to calculate the first order potential and the linear ship responses, as necessary for the added resistance calculations; then Maruo’s far-field theory and the Kochin functions approach is adopted for the calculation of the added resistance. For comparison, the near-field, direct pressure integration method has been also partly investigated. Finally, for the short waves range, the asymptotic method of Faltinsen et al. (1980) and an improved derivative of it introduced recently by Kuroda et al. (2008) and Tsujimoto et al. (2008) are employed. A wide range of case studies of different hull forms (slender and bulky) was used to validate the applicability and the accuracy of the developed and implemented methods in practice.
2. Basic formulation A coordinate system is introduced, as shown in Fig. 1. The x–y plane coincides with the calm water free surface with the x-axis pointing towards the bow and the z-axis pointing upwards. The incident wave potential is given by h i gz ð1Þ F0 ¼ Re i a ekzikðx cos w þ y sin wÞ eioe t ¼ Re ðf0c þ f0s Þeioe t
o
where w is the incident wave heading (w ¼ p corresponds to head waves), oe ¼ okVcosw is the encounter frequency, k¼ o2/g ¼2p/l is the wave number, V is ship’s advance velocity and f0c, f0s are
Fig. 1. Coordinate system.
the symmetric and anti-symmetric parts of the wave potential, respectively,
f0c ¼ i f0s ¼
g za
o
g za
o
ekzikx cos w cosðkysin wÞ
ekzikx cos w sinðky sin wÞ
ð2Þ
ð3Þ
The diffraction potential may be also decomposed as follows:
F7 ¼ Re½f7 eioe t ¼ Re½ðf7c þ f7s Þeioe t
ð4Þ
with: @f7c @f @f7s @f ¼ 0c , ¼ 0s @n @n @n @n
ð5Þ
The diffraction potentials f7c and f7s share the same symmetry chrematistics with the symmetric and anti-symmetric parts of the incident wave potential, respectively. The complex function H(kj,y), known as Kochin function, describing the elementary waves radiated from the ships is given by ZZ @ @f Gj ðyÞds Hðkj , yÞ ¼ f ð6Þ @n @n S where: Gj ðyÞ ¼ exp½kj ðyÞz þ ikj ðyÞðx cos y þ ysin yÞ and kj(y), j ¼1, 2 are the unsteady wave numbers ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ :j ¼ 1 K0 12O cos y 7 14O cos y kj ðyÞ ¼ :j ¼ 2 2 cos2 y
ð7Þ
ð8Þ
In the above equations, y is the angle of elementary waves generated by the body, O ¼ oeV/g is the Hanaoka parameter, and K0 ¼g/V2 is the steady wave number. From Eq. (8) after some algebraic manipulation the following expressions may be derived for k1(y) and k2(y): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 17 14O cos y 2 ð9Þ k1 ðyÞ ¼ K0 sec y 4 k2 ðyÞ ¼ k
2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 14O cos y
ð10Þ
Following Maruo (1963), the added resistance may be expressed by the above Kochin function as (Z Z p=2 Z 3p=2 ) a0 r 2 k1 ½k1 cos ykcos w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy þ RAW ¼ 9Hðk1 , yÞ9 8p 14O cos y p=2 a0 p=2 þ
r 8p
Z 2pa0 a0
2
9Hðk2 , yÞ9
k2 ½k2 cos ykcos w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy 14O cos y
ð11Þ
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where r is the density of sea water and a0 is the critical angle (a0 ¼arcos(1/(4O)) for O 41/4 and a0 ¼0 for O r1/4). When V ¼0 then O ¼0, k1(y)-N and k2(y)¼k. Thus the wave systems are reduced to the ring wave only. For the zero speed case, the drift force may be expressed as follows (Maruo, 1960): Z rk2 2p 2 RAW ¼ HðyÞ9 ðcos ycos wÞdy ð12Þ 8p 0
1+ au for the effect of forward speed. The two integrals in Eq. (17) are calculated over the A–F section (integral I) and B–F (integral II) section of the non-shaded part of the waterline (see Fig. 2). I1 is the first order modified Bessel function of the first kind and K1 is the first order modified Bessel function of the second kind. Note that in the original proposal, the determination of CU was more complicated and required the use of experimental data.
where:
2.1. Computation of Kochin function
HðyÞ ¼
ZZ
@f ds exp½kz þikðx cos y þy sin yÞ fkðnz þ i cos ynx þ i sin yny Þ @n S
ð13Þ For the short waves range, Faltinsen et al. (1980) derived the following asymptotic formula for the added resistance, assuming that the incident waves are perfectly reflected from the non-shaded part of the ship surface that is exposed to the waves Z ð14Þ F1 ¼ F n sin jdl L
where Fn ¼
1 2o V rg z2a sin2 ðjwÞ þ 0 1 þcos jcosðjwÞ 2 g
ð15Þ
The integration in Eq. (14) is performed over the non-shaded part (A-F-B) of the waterline (Fig. 2). This expression yields good results for relatively full bodies; however, some poor results were obtained for fine hull forms like those of containerships. In order to improve this drawback, Kuroda et al. (2008) further investigated Fujii and Takahashi’s (1975) semiempirical original method and proposed an improved expression for the added resistance in short waves, which takes the following form: 1 rg z2a ad ð1 þ aU ÞBBf ðwÞ 2
RAW ¼
ð16Þ
where Bf ðwÞ ¼
ad ¼
1 B
Z
sin2 ðjwÞsin jdl þ I
2 2 I1 ðke dÞ 2 I2 ðk dÞ þK 2 ðk dÞ 1 e 1 e
p
Z
sin2 ðj þ wÞsin jdl
ð17Þ
II
The expression for the calculation of the Kochin functions involves the fluid velocity on the body surface. The latter may be calculated via the body boundary condition. For the symmetric potentials f1, f3, f5 and the symmetric part of the diffraction potential f7c, the Kochin function takes the following form: ZZ ekj ðyÞz þ ikj ðyÞx cos y cosðkj ðyÞy sin yÞ Hm ðkj , yÞ ¼ 2 S=2
@f kj ðyÞðfmc nz fms cos ynx Þ mc @n @fms ds þ ikj ðyÞðfms nz þ fmc nx cos yÞi @n ZZ 2 ekj ðyÞz þ ikj ðyÞx cos y sinðkj ðyÞy sin yÞ S=2
ðfmc þ ifms Þkj ðyÞsin yny ds
ð22Þ
where, m¼1, 3, 5, 7c. For the anti-symmetric potentials, i.e. f2, f4, f6 and f7s, the Kochin function takes the following form: ZZ Hm ðkj , yÞ ¼ 2i ekj z þ ikj x cos y sinðkj ðyÞy sin yÞ S=2 @f kj ðyÞðfmc nz fms cos ynx Þ mc @n @fms ds þ ikj ðyÞðfms nz þ fmc nx cos yÞi @n ZZ þ2 ekj ðyÞz þ ikj ðyÞx cos y cosðkj ðyÞy sin yÞ S=2
ðfmc þ ifms Þkj ðyÞisin yny ds
ð23Þ
ð18Þ
where, m ¼2, 4, 6, 7s.
1 þ aU ¼ 1 þ CU Fn
ð19Þ
2.2. Behavior of the wave systems
ke ¼ kð1 þ O cos wÞ2
ð20Þ
CU ¼ max½10:0,310Bf ðwÞ þ 68
ð21Þ
In the following, we investigate the behavior of the wave systems for the case of a submerged spheroid with a length to breadth ratio L/B ¼5 and draught to breadth ratio d/B¼0.75 where the draught d is measured from the free surface to the body center with Fn ¼0.2, k¼2.5 m 1, oe E7.14 s 1 in head waves. The corresponding values of the Hanaoka parameter, the steady wave number and the critical angle are: O E0.64, K0 E 12.7 m 1, a0 E671. The unsteady wave numbers of the two wave systems are plotted in Fig. 3. From this graph it may be observed that the wave number k1 approaches infinite when y-p/2, while the wave number k2 grows larger (although still remains bounded) as the wave direction approaches the critical angle. It has been generally found that the wave number k1 is much larger than k2, indicating that the k1 waves are very small and can be neglected in the calculation of added resistance in head waves. Fig. 4 shows the weighting functions in the added resistance formulation. Since the k1 wave system is neglected, only the weighting functions for the k2 wave system have been shown. The weighting functions are defined by the following expressions:
p
B and d are the beam and draught of the ship, Bf is the bluntness coefficient, ad accounts for the effect of draught and frequency and
Fig. 2. Coordinate system for the short waves range added resistance calculation methods.
k2 ðyÞ2 cos y WF2ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14O cos y
ð24Þ
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Fig. 3. Unsteady wave numbers of the two wave systems. Fig. 5. Kochin amplitude function of a submerged spheroid.
Fig. 4. Weighting function of the k2 wave system.
k2 ðyÞkcos w WF2ð2Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14O cos y
Fig. 6. Integrand in Maruo’s formula for a submerged spheroid.
ð25Þ
Fig. 5 shows the non-dimensional Kochin function amplitude for a spheroid in the range a0 o y o p and Fig. 6 shows the integrand of the second term in expression (11). A more detailed analysis can be found in Naito et al. (1988).
Generated data are validated in a series of test cases in comparison to numerical results of other authors, partly numerical results of a near-field, direct pressure integration method (Papanikolaou and Zaraphonitis, 1987) and experimental data, to the extent available. Typical results from these validation studies are presented and discussed in the following.
3. Numerical validations and discussion 3.1. Submerged spheroid The far-field method for the calculation of the added resistance, presented in Section 2, has been numerically implemented in a computer code, using velocity potential and other seakeeping data resulting from application of three different seakeeping assessment codes of NTUA-SDL, namely 1. The 3D frequency domain panel code NEWDRIFT (Papanikolaou– Schellin, 1992); relevant results are denoted in the graphs by ND. 2. The 3D time domain Green function code (Liu et al, 2007); relevant results denoted by LIU. 3. The 3D time domain hybrid Rankine source (near field)/Green function (far field) code HYBRID (Liu and Papanikolaou, 2010); relevant results denoted by HYBRID.
Iwashita and Ohkusu (1992) studied a shallowly submerged spheroid with a length to breadth ratio L/B¼5 and draught to breadth ratio d/B¼0.75, where the draught d is measured from the free surface to the body center. This study case has been addressed with two different approaches: The first one is using a hybrid time domain Green function method presented by Liu et al. (2007) for the calculation of the first-order potential. The second approach is based on a frequency domain 3D panel (i.e. NEWDRIFT, Papanikolaou and Zaraphonitis, 1987) for the solution of the first order problem and the calculation of the potential and the body motions. The deduced first-order results either from the time-domain or the frequency domain approach are introduced in Eqs. (22) and (23) for the
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calculation of the Kochin functions, from which the added resistance is calculated according to Eq. (11). In addition to the above, the drift forces and added resistance are calculated also applying the nearfield, direct pressure integration approach, implemented in the NEWDRIFT code (denoted as NDnear in the graphs). Results from the first approach (denoted as LIUfar in the graphs) for the drift force at zero speed, and for the added resistance when the body is moving at forward speed in incident head waves with suppressed responses (fixed body case) are presented in Figs. 7–9. Good agreement may be observed between the present results and the results of Iwashita and Ohkusu (1992), denoted as IWASHITA in the graphs, both for the zero speed and nonzero speed calculations (Figs. 7–9). The same spheroid has been also tested at a deeper submergence of d/B ¼1.25. Comparisons have been herein made only between the two NTUA codes, namely the time- and frequency domain potential solvers (LIU and NEWDRIFT) at zero and at a forward speed corresponding to Fn ¼0.2. Good agreement between both solvers was obtained using the far-field method in the zero speed case. For the forward speed case, though the trend and maximum values are in general in good agreement, the herein predicted peak value is actually slightly shifted, which might be credited to the fact that the employed 3D panel method corresponding to the NEWDRIFT code is essentially based on a
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Fig. 9. Added resistance on a submerged spheroid, Fn ¼0.3, d ¼ 0.75B.
Fig. 10. Horizontal drift force on a submerged spheroid at Fn ¼0, d ¼1.25B.
Fig. 7. Horizontal drift force on a submerged spheroid at Fn ¼ 0, d¼ 0.75B.
zero speed Green function and forward speed effects are taken into account in an approximate way by exploiting slender body theory assumptions. These results are shown in Figs. 10 and 11. 3.2. Floating spheroid
Fig. 8. Added resistance on a submerged spheroid, Fn ¼ 0.2, d ¼ 0.75B.
The added resistance of a half-immersed (floating) spheroid with a length to beam ratio L/B ¼5 has been studied by Kashiwagi (1997). Fig. 12 shows the drift force (zero speed) results for the freely moving spheroid, computed by NEWDRIFT (NDnear) and by the far field method (NDfar) using the first-order potentials from NEWDRIFT. A very good agreement between the near field and far field methods may be observed, along with a reasonable agreement with Kashiwagi’s enhanced unified theory results. Fig. 13 shows the added resistance of the half-immersed spheroid with suppressed motions at a forward speed corresponding to Fn ¼0.2. The results denoted by LIN are reproduced from Lin et al. (1993). Three different panelizations have been tested with NEWDRIFT, namely 360, 728 and 960 panels on half body. It may be observed that in some cases 360 panels are not enough for the accurate calculation of the added resistance, while with 728 panels, convergence has been generally achieved. Considering that 200 panels would lead in general to good motion results for such a simple body by the same code, we may conclude that added resistance is much more
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sensitive to the size of panelization, than results of first order. A similar observation was made by Kashiwagi (2009). Nevertheless, a significant deviation between a variety of numerical results and experimental values is observed for this case, which is theoretically and practically a difficult benchmark case, because of the complicated above still water level wetting of the half-immersed spheroid. Kashiwagi presented good results for this case with his enhanced unified slender body theory, when using correct n1 term which better accounts for the wave diffraction near the bow. 3.3. Wigley hull A series of Wigley hulls were tested and reported by Journee´ (1992). In this section the validation of Wigley-III is presented. The hull surface is defined by the following expression: ! 2 ! y x 2 z x 2 ¼ 1 1 þ 0:2 1 ð26Þ b L=2 D L=2 Fig. 11. Added resistance on a submerged spheroid, Fn ¼ 0.2, d ¼1.25B.
Fig. 12. Drift force on a freely moving spheroid at Fn ¼0.
Fig. 13. Added resistance on a fixed half-immersed spheroid at Fn ¼ 0.2.
where 2b/L¼0.1 and D/L¼0.0625. The hull surface was discretized by 800 panels (port side only). For the body at zero speed, the drift force for the fixed body case and the free floating case are predicted by both the near field and the far field method, as shown in Fig. 14. A very good agreement between the near field and far field drift force results may be observed. Calculations have been performed with non-zero forward speed as well. Fig. 15 presents results for the added resistance by the far-field method. Despite some scatter in the experimental data, clearly the numerical values NDfar are over predicting the added resistance and lead to a slight shift of the peak values towards smaller wavelengths. This tendency is partly attributed to inaccuracies in the consideration of forward speed effects by NEWDRIFT, but mainly attributed to a common shortcoming of linear seakeeping methods, which are based on potential theory, namely the under-estimation of damping coefficients, leading to predictions of increased first-order motions in the vicinity of the resonance. Another reason for the numerically predicted increased motion amplitudes in the resonance region (thus also of added resistance) are nonlinear effects due to the above still water level hull form, which is not considered in linear frequency domain methods; these effects become very significant when the motions are of large amplitude; Papanikolaou et al. (2000) tried to overcome this problem by including additional viscous damping to the potential theory terms, derived with the help of ‘‘empirical’’ viscous lift and cross-flow drag coefficients. The introduction of cross-flow viscous damping in the first-order calculations for the Wigley hull
Fig. 14. Drift force on a Wigley-hull in head seas at Fn ¼ 0.0.
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
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available. The obtained results for Fn ¼0.266 and 0.283 are presented in Figs. 17 and 18, respectively. Once again, as in the Wigley hull case, Fig. 15, numerical results without any viscous correction strongly over-predict the experimental measurements. The introduction of cross-flow viscous damping in the first-order calculations (Cd ¼0.1, za/L¼1/50) resulted in a considerable reduction of the peak values of the added resistance (and motion amplitudes), much closer to the experimental measurements. Note here the very good prediction by the new HYBRID time domain method of NTUA-SDL (Liu and Papanikolaou, 2010), which, compared to NEWDRIFT, takes accurately into account forward speed effects and in addition accounts for the above water hull form nonlinearities. The calculation by HYBRID method has been herein done only for the range of l/L¼0.8–1.6, where the resonance takes place and the nonlinearity plays a significant role. In the short waves range, added resistance calculations according to the asymptotic formulae of Faltinsen et al. (1980) and also of Kuroda et al. (2008), denoted as SW1 and SW2, respectively, are also presented. 3.5. S-175 container ship Fig. 15. Added resistance on a Wigley-hull in head seas at Fn ¼ 0.3.
Fig. 16. Drift force on a S60 hull in head seas at Fn ¼0.0.
Calculations were performed also for the S175 Container ship, a hull form that has been thoroughly investigated by many
Fig. 17. Added resistance on a Series 60 hull at Fn ¼0.266.
resulted in a reduction of the peak values for the added resistance (denoted as NDfarCD), much closer to the experimental measurements, as shown in Fig. 15. In this calculation, the cross-flow viscous damping coefficient is Cd ¼0.1, whereas the incident wave amplitude is assumed as za/L¼1/60. 3.4. Series 60 hull with Cb ¼0.6 Calculations were made for a Series 60 hull with block coefficient of 0.60, which has been studied extensively by many researchers (e.g. Strøm-Tejsen et al. (1973)). The hull surface was discretized by 809 panels (port side). Calculations were performed at first for the zero speed case. Fig. 16 presents the drift force result. For the fixed body case, there is a very good agreement between the near field and far field results. For the freefloating case however, a small discrepancy may be observed, although calculations with both methods were based on the same first-order potential and motion data. Calculations have been also carried out by the far field method for two forward speeds, for which experimental data were
Fig. 18. Added resistance on a Series 60 hull at Fn ¼0.283.
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S. Liu et al. / Ocean Engineering 38 (2011) 641–650
researchers (15th ITTC, 1978). A panelization with 885 panels was used in the calculations. Added resistance results are presented in Figs. 19–21. A quite satisfactory agreement between the numerical predictions and the experimental results (reproduced from Takahashi, 1988 and Seakeeping Committee report of ITTC 1984) is observed. Pure potential flow calculations by the panel code NEWDRDIFT resulted again in a small over-prediction of the added resistance at the peak, which is eliminated when the cross-flow drag viscous correction is introduced (Cd ¼0.1, za/L¼1/55). The results of the new HYBRID method of NTUA-SDL are in good agreement with the experimental data. In the short waves range, added resistance calculations using the asymptotic formulae of Faltinsen et al. and also of Kuroda and Tsujimoto are presented for this hull also, denoted as SW1 and SW2, respectively. In the short waves range, the agreement of added resistance calculations according to Kuroda and Tsujimoto with the experiments is very satisfactory. 3.6. Bulk-carrier Finally, results of drift force and added resistance calculations for a Bulk-Carrier, also studied by Kadomatsu (1988), are presented.
Fig. 19. Added resistance on the S-175 container ship at Fn ¼0.25.
Fig. 21. Added resistance on the S-175 container ship at Fn ¼ 0.30.
Fig. 22. Drift Force on a Bulk-Carrier in head seas at Fn ¼ 0.0.
This is a quite full hull form, with Lpp ¼285 m, B¼ 50.0 m, D¼18.5 m, Cb ¼0.829 and LCB ¼3.83% Lpp forward of the midship section. A panelization consisting of 819 panels on the half body was used for the calculations. The drift force and added resistance results are shown in Figs. 22–25, compared against experimental data. The numerical results agree well with the experiments, with the exception of the short waves range at the higher speeds corresponding to Fn ¼ 0.10 and 0.15, where the added resistance is under-predicted. However, in this range of wavelengths, calculations based on the asymptotic formulae of Faltinsen et al. and also of Kuroda and Tsujimoto resulted in a very good agreement with the experimental data. Note that in this case of full bodies at low speed, no viscous correction for motion damping proves necessary, because of the resulting small amplitude motion responses and of related nonlinearities, along with limited forward speed effects.
4. Summary and conclusions
Fig. 20. Added resistance on the S-175 container ship at Fn ¼0.275.
The added resistance problem of ships in waves has been solved following Maruo’s far-field theory and numerically implemented
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
Fig. 23. Added resistance on a Bulk-Carrier at Fn ¼0.05.
649
using velocity potentials calculated by a Time Domain Numerical Simulation method for submerged bodies (Liu et al. 2007), a 3D Frequency Domain Panel method (Papanikolaou and Zaraphonitis, 1987) and a 3D time domain HYBRID method (Liu and Papanikolaou, 2010) for bodies floating on the free surface. Results of a near-field, direct pressure integration method were used for comparison in the zero speed case. For the short waves range, the asymptotic formulae introduced by Faltinsen et al. (1980) and also by Kuroda et al. (2008) were employed for comparison. A systematic validation for different hull forms (submerged and floating, slender and bulky) and varying wavelengths and speeds were conducted. The obtained results were compared with those of other well established authors and experimental data and a reasonable agreement was observed, suggesting that the implemented procedure appears a reliable and robust method for the routine prediction of the added resistance of a ship in waves. Regarding computational time and efficiency of the implemented 3D panel code method, with a panelization in the order of 800 panels for the half the body, after obtaining the potential and motion data, which takes approximately 5 s with the NEWDRIFT code, the added resistance computation for one frequency takes approximately 1.5 s on a PC with Intel Core 2 QUAD CPU Q8200 2.33 GHz, which proves actually very efficient from the point of view of practical applications. Concluding, the original far-field method of Maruo, further elaborated by use of a 3D panel method and the asymptotic formulae of Faltinsen, Kuroda and Tsujimoto for the short wavelengths prove fully satisfactory for the prediction of the added resistance of ships (of slender and bulky hull form) in waves and may be practically implemented within reasonable computational times.
Acknowledgements The presented work is part of the Ph.D. work of the first author. The continuous guidance by the second and third author, as well as by Professor Wenyang Duan (Harbin Engineering University) is acknowledged. Finally, the financial support by NTUA-SDL and partly by DNV in the framework of the NTUA-DNV GIFT agreement (2007–2010) is acknowledged. References Fig. 24. Added resistance on a Bulk-Carrier at Fn ¼0.10.
Fig. 25. Added resistance on a Bulk-Carrier at Fn ¼ 0.15.
¨ Boese, P., 1970. Eine einfache Methode zur Berechnung der Widerstandserhohung eines Schiffes im Seegang. Journal Schiffstechnik-Ship Technology Research 17 (86). Faltinsen, O.M., Minsaas, K.J., Liapis, N., Skjørdal, S.O., 1980. Prediction of resistance and propulsion of a ship in a seaway. In: Proceedings of the 13th Symposium on Naval Hydrodynamics, Tokyo, pp. 505–529. Fujii H., Takahashi, T., 1975. Experimental study on the resistance increase of a ship in regular oblique waves. In: Proceeding of the 14th ITTC, pp. 351–360. Gerritsma, J., Beukelman, W., 1972. Analysis of the resistance increase in waves of a fast cargo ship. International Shipbuilding Progress 19 (217). ITTC Seakeeping Committee, 1978. Comparison of results obtained with compute programs to predict ship motions in six-degrees-of-freedom and associated responses. In: Proceedings of the 15th ITTC, pp. 79–92. ITTC Seakeeping Committee, 1984. Prediction of added resistance. In: Proceedings of the 17th ITTC, pp. 500–503. Iwashita, H., Ohkusu, M., 1992. The Green function method for ship motions at forward speed. Ship Technology Research 39 (2), 3–21. Joosen, W.P.A., 1966. Added resistance of ships in waves. In: Proceedings of the 6th Symposium on Naval Hydrodynamics, Washington, D.C. Journee´, J.M.J., 1992. Experiments and calculations on 4 Wigley hull forms in head waves, May 1992. DUT-SHL Report 0909. Kadomatsu, K., 1988. Study on the required minimum output of main propulsion engine considering maneuverability in rough sea, Ph.D. Thesis, Ship Design Lab., Yokohama Nat. Univ., Japan. Kashiwagi, M., 1997. Numerical seakeeping calculations based on the slender ship theory. Ship Technology Research 44, 167–192. Kashiwagi, M., 2009. Impact of hull design on added resistance in waves— application of the enhanced unified theory. In: Proceedings of the 10th International Marine Design Conference, Trondheim, Norway, pp. 521–535.
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Kuroda, M., Tsujimoto, M., Fujiwara, T., 2008. Investigation on components of added resistance in short waves. Journal of the Japan Society of Naval Architects and Ocean Engineers 8, 171–176. Lin, X., Takaki, M., Iwashita, H., 1993. Effect of the steady disturbance on the unsteady flow around a ship in waves. Journal Society of Naval Architects of Japan 174, 151–162. Liu, S., Papanikolaou, A., Duan, W., 2007. A time domain numerical simulation method for nonlinear ship motions, AsiaLink-EAMARNET. In: Proceedings of the International Conference on Ship design, Production & Operation, Harbin, China, pp. 177–185. Liu, S., Papanikolaou, A., 2009. A time-domain hybrid method for calculating hydrodynamic forces on ships in waves. In: Proceedings of the 13th International Congress of the International Maritime Association of the Mediterranean, Instanbul, Turkey. Liu, S., Papanikolaou, A., 2010. Time-domain hybrid method for simulating large amplitude motions of ships advancing in waves. In: Proceedings of the ITTC Workshop on Seakeeping, Seoul, Korea. Maruo, H., 1957. The excess resistance of a ship in rough seas. International Shipbuilding Progress 4 (35). Maruo, H., 1960. The drift of a body floating on waves. Journal of Ship Research 4 (3), 1–10. Maruo, H., 1963. Resistance in waves, 60th anniversary Series. The Society of Naval Architects of Japan 8, 67–102. Naito, S., Yamanoto, O., Takahashi, T., 1988. Effect of ship hull forms on resistance increase in waves. In: Proceedings of the Fifth Marine Dynamics Symposium, pp. 45–79. Naito, S., 2008. Propulsive performance of ships in actual seas. In: Proceedings of Seventh Osaka Colloquium on Seakeeping and Stability, Osaka, Japan.
Ohkusu, M., 1985. Added resistance in waves of hull forms with blunt bow. In: Proceedings of the 15th Symposium on Naval Hydrodynamics, pp. 135–147. Papanikolaou, A., Zaraphonitis, G., 1987. On an improved method for the evaluation of second-order motions and loads on 3D floating bodies in waves. Journal of Schiffstechnik 34, 170–211. Papanikolaou, A., Schellin, Th., 1992. A three dimensional panel method for motions and loads of ships with forward speed. Journal of Schiffstechnik—Ship Technology Research 39 (4), 147–156. Papanikolaou, A., Zaraphonitis, G., Maron, A., Karayannis, T., 2000. Nonlinear effects of vertical plane motions of ships with forward speed in waves. In: Proceedings of the Fourth Osaka Colloquium on Seakeeping Performance of Ships, Osaka, Japan, pp. 459–468. Salvesen, N., Tuck, E.O., Faltinsen, O., 1970. Ship motions and sea loads. Transactions of the Society of Naval Architects and Marine Engineers 78, 250–287. Salvesen, N., 1974. Second-order steady state forces and moments on surface ships in oblique regular waves. In: Proceedings of the International Symposium on Dynamics of Marine Vehicles and Structures in Waves, University College, London, pp.212–226. Strøm-Tejsen, J., Yeh, H.Y.H., Moran, D.D., 1973. Added resistance in waves. Transactions of the Society of Naval Architects and Marine Engineers 81, 109–143. Takahashi, T., 1988. A practical prediction method for added resistance of a ship in waves and the direction of its application to hull form design (in Japanese). Transactions on the West-Japan Society of Naval Architects 75, 75–95. Tsujimoto, M., Shibata, K., Kuroda, M., 2008. A practical correction method for added resistance in waves. Journal of the Japan Society of Naval Architects and Ocean Engineers 8, 177–184.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Prediction of added resistance of ships in waves Shukui Liu, Apostolos Papanikolaou n, George Zaraphonitis Ship Design Laboratory, National Technical University of Athens, Greece
a r t i c l e in f o
abstract
Article history: Received 23 November 2009 Accepted 7 December 2010 Editor-in-Chief: A.I. Incecik Available online 3 January 2011
The prediction of the added resistance of ships in waves is a demanding, quasi-second-order seakeeping problem of high practical interest. In the present paper, a well established frequency domain 3D panel method and a new hybrid time domain Rankine source-Green function method of NTUA-SDL are used to solve the basic seakeeping problem and to calculate first order velocity potentials and the Kochin functions, as necessary for the calculation of the added resistance by Maruo’s far-field method. A wide range of case studies for different hull forms (slender and bulky) was used to validate the applicability and accuracy of the implemented methods in practice and important conclusions regarding the efficiency of the investigated methods are drawn. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Added resistance Drift forces Far-field method Near-field method 3D panel method Kochin function
1. Introduction When sailing in a seaway, the calm water resistance and powering of a ship is increased and this is accounted for in traditional ship design by some rough proportional increase of calm water resistance by about 20–40%. However, the accurate prediction of ship’s resistance in waves is nowadays of increased importance, both for the designer and the shipowner/operator, since it greatly affects the selection of ship’s engine/propulsion system and ship’s performance in terms of sustainable service speed and fuel consumption in realistic sea conditions. Also, accurate and efficient predictions of the added resistance in natural seaways are necessary for the implementation of modern onboard ship routing systems. The added resistance is a steady force of second-order with respect to the incident wave’s amplitude and acting opposite to ship’s forward speed in longitudinal direction. When the ship has zero forward speed, then the added resistance is trivially identical to the longitudinal drift force (Papanikolaou and Zaraphonitis, 1987). There are several alternative approaches to the added resistance problem; they can be generally classified into two main categories, namely far-field and near-field methods. The far-field methods are based on considerations of the diffracted and radiated wave energy and momentum flux at infinity, leading to the steady added resistance force by the total rate of momentum change. The nearfield method, on the other side, leads to the added resistance as the steady second-order force obtained by direct integration of the hydrodynamic, steady second-order pressure acting on the wetted
n
Corresponding author. E-mail address: [email protected] (A. Papanikolaou).
0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.12.007
ship surface. The latter can be calculated exactly from first order potential functions, and their derivatives. It is evident that though both above methods should lead to the same numerical results for the added resistance, this is not so in practice, because both methods request equally accurate solutions of the basic potential, seakeeping theory problem in the near and far field, which cannot be ensured in practice. This is a well-known observation in the similarly posed drift force problem. Thus, even if the same potential theory and related numerical methods are used for the solution of the linear seakeeping problem and the calculation of the governing potential function (and its derivatives and of related hydrodynamic quantities, including ship motions, as necessary) in the near- and far-field, the numerical results by the near- and far-field method for the added resistance are in general different, thus a basic question arises, namely which method is more efficient for practical applications. In this respect, several theoretical approaches of varying complexity and accuracy have been introduced in the past and numerically implemented/verified. The first far-field approach was introduced by Maruo (1957) in the late 50s; it was further elaborated in the following years by Maruo (1960, 1963) and Joosen (1966). In the early 70s, the first near field, direct pressure integration methods appeared; however, the considered hydrodynamic pressure distribution was highly simplified (e.g. Boese, 1970). At about the same time, the radiated energy approach of Gerritsma and Beukelman (1972) was introduced, which is basically following the far-field approach of Maruo. Strøm-Tejsen et al. (1973) did a thorough evaluation of all the above approaches to find large discrepancies between the numerical results form different theoretical approaches and relevant model experimental data. Later on, Salvesen (1974) investigated the added resistance problem by applying Gerritsma and Beukelman’s method, but using
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basic results of the STF seakeeping strip theory (Salvesen et al. 1970) and found quite satisfactory results for the investigated ship hull forms. This was not surprising, considering the superiority of the STF strip theory method for the prediction of ship motions over other methods at that time. Also, the significance of accurate ship motions predictions for a reliable estimation of the added resistance was acknowledged and one may have concluded that if a more accurate calculation of the ship motions could be used (or even ship motions from model experiments), all above alternative approaches to the added resistance may have led to better results. An exact in the sense of potential theory near-field, direct pressure integration approach to the added resistance problem was introduced by Faltinsen et al. (1980), with good validation results. The observed deficiency of the approach in short waves was addressed by the introduction of a simplified added resistance estimation formula, which models remarkably well the complicate interaction between the diffraction of waves and the steady flow around the ship. The same problem appears in exaggerated form with low speed, full hull forms with blunt bows (bulkcarriers and tankers), for which Ohkusu (1985) proposed an improved approach. This method was further elaborated by Iwashita and Ohkusu (1992), who applied Maruo’s (1963) improved far-field formulation and the concept of Kochin functions, to obtain very good results for the added resistance of a fully submerged spheroid by use of a 3D pulsating and traveling source, Green function method. In his most recent state of the art review, Naito (2008) concluded that y ‘‘good agreement can be obtained by applying this (latter) calculation method to all types of ship. However this method has not been widely utilized due to the complicated calculation technique and large computing time required’’. More recently, using an enhanced unified theory, Kashiwagi (2009) calculated the added resistance by a modified version of Maruo’s approach, obtaining also quite satisfactory results. In the present study, the well established frequency domain 3D panel method and computer code NEWDRIFT (Papanikolaou– Zaraphonitis, 1987, Papanikolaou–Schellin, 1992 and Papanikolaou et al. 2000), a new time domain Green function method (Liu et al. 2007) and a hybrid time domain the Rankine source-Green function method (Liu and Papanikolaou, 2009, 2010) of NTUA-SDL are used to solve the basic seakeeping problem and to calculate the first order potential and the linear ship responses, as necessary for the added resistance calculations; then Maruo’s far-field theory and the Kochin functions approach is adopted for the calculation of the added resistance. For comparison, the near-field, direct pressure integration method has been also partly investigated. Finally, for the short waves range, the asymptotic method of Faltinsen et al. (1980) and an improved derivative of it introduced recently by Kuroda et al. (2008) and Tsujimoto et al. (2008) are employed. A wide range of case studies of different hull forms (slender and bulky) was used to validate the applicability and the accuracy of the developed and implemented methods in practice.
2. Basic formulation A coordinate system is introduced, as shown in Fig. 1. The x–y plane coincides with the calm water free surface with the x-axis pointing towards the bow and the z-axis pointing upwards. The incident wave potential is given by h i gz ð1Þ F0 ¼ Re i a ekzikðx cos w þ y sin wÞ eioe t ¼ Re ðf0c þ f0s Þeioe t
o
where w is the incident wave heading (w ¼ p corresponds to head waves), oe ¼ okVcosw is the encounter frequency, k¼ o2/g ¼2p/l is the wave number, V is ship’s advance velocity and f0c, f0s are
Fig. 1. Coordinate system.
the symmetric and anti-symmetric parts of the wave potential, respectively,
f0c ¼ i f0s ¼
g za
o
g za
o
ekzikx cos w cosðkysin wÞ
ekzikx cos w sinðky sin wÞ
ð2Þ
ð3Þ
The diffraction potential may be also decomposed as follows:
F7 ¼ Re½f7 eioe t ¼ Re½ðf7c þ f7s Þeioe t
ð4Þ
with: @f7c @f @f7s @f ¼ 0c , ¼ 0s @n @n @n @n
ð5Þ
The diffraction potentials f7c and f7s share the same symmetry chrematistics with the symmetric and anti-symmetric parts of the incident wave potential, respectively. The complex function H(kj,y), known as Kochin function, describing the elementary waves radiated from the ships is given by ZZ @ @f Gj ðyÞds Hðkj , yÞ ¼ f ð6Þ @n @n S where: Gj ðyÞ ¼ exp½kj ðyÞz þ ikj ðyÞðx cos y þ ysin yÞ and kj(y), j ¼1, 2 are the unsteady wave numbers ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ :j ¼ 1 K0 12O cos y 7 14O cos y kj ðyÞ ¼ :j ¼ 2 2 cos2 y
ð7Þ
ð8Þ
In the above equations, y is the angle of elementary waves generated by the body, O ¼ oeV/g is the Hanaoka parameter, and K0 ¼g/V2 is the steady wave number. From Eq. (8) after some algebraic manipulation the following expressions may be derived for k1(y) and k2(y): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 17 14O cos y 2 ð9Þ k1 ðyÞ ¼ K0 sec y 4 k2 ðyÞ ¼ k
2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 14O cos y
ð10Þ
Following Maruo (1963), the added resistance may be expressed by the above Kochin function as (Z Z p=2 Z 3p=2 ) a0 r 2 k1 ½k1 cos ykcos w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy þ RAW ¼ 9Hðk1 , yÞ9 8p 14O cos y p=2 a0 p=2 þ
r 8p
Z 2pa0 a0
2
9Hðk2 , yÞ9
k2 ½k2 cos ykcos w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy 14O cos y
ð11Þ
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
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where r is the density of sea water and a0 is the critical angle (a0 ¼arcos(1/(4O)) for O 41/4 and a0 ¼0 for O r1/4). When V ¼0 then O ¼0, k1(y)-N and k2(y)¼k. Thus the wave systems are reduced to the ring wave only. For the zero speed case, the drift force may be expressed as follows (Maruo, 1960): Z rk2 2p 2 RAW ¼ HðyÞ9 ðcos ycos wÞdy ð12Þ 8p 0
1+ au for the effect of forward speed. The two integrals in Eq. (17) are calculated over the A–F section (integral I) and B–F (integral II) section of the non-shaded part of the waterline (see Fig. 2). I1 is the first order modified Bessel function of the first kind and K1 is the first order modified Bessel function of the second kind. Note that in the original proposal, the determination of CU was more complicated and required the use of experimental data.
where:
2.1. Computation of Kochin function
HðyÞ ¼
ZZ
@f ds exp½kz þikðx cos y þy sin yÞ fkðnz þ i cos ynx þ i sin yny Þ @n S
ð13Þ For the short waves range, Faltinsen et al. (1980) derived the following asymptotic formula for the added resistance, assuming that the incident waves are perfectly reflected from the non-shaded part of the ship surface that is exposed to the waves Z ð14Þ F1 ¼ F n sin jdl L
where Fn ¼
1 2o V rg z2a sin2 ðjwÞ þ 0 1 þcos jcosðjwÞ 2 g
ð15Þ
The integration in Eq. (14) is performed over the non-shaded part (A-F-B) of the waterline (Fig. 2). This expression yields good results for relatively full bodies; however, some poor results were obtained for fine hull forms like those of containerships. In order to improve this drawback, Kuroda et al. (2008) further investigated Fujii and Takahashi’s (1975) semiempirical original method and proposed an improved expression for the added resistance in short waves, which takes the following form: 1 rg z2a ad ð1 þ aU ÞBBf ðwÞ 2
RAW ¼
ð16Þ
where Bf ðwÞ ¼
ad ¼
1 B
Z
sin2 ðjwÞsin jdl þ I
2 2 I1 ðke dÞ 2 I2 ðk dÞ þK 2 ðk dÞ 1 e 1 e
p
Z
sin2 ðj þ wÞsin jdl
ð17Þ
II
The expression for the calculation of the Kochin functions involves the fluid velocity on the body surface. The latter may be calculated via the body boundary condition. For the symmetric potentials f1, f3, f5 and the symmetric part of the diffraction potential f7c, the Kochin function takes the following form: ZZ ekj ðyÞz þ ikj ðyÞx cos y cosðkj ðyÞy sin yÞ Hm ðkj , yÞ ¼ 2 S=2
@f kj ðyÞðfmc nz fms cos ynx Þ mc @n @fms ds þ ikj ðyÞðfms nz þ fmc nx cos yÞi @n ZZ 2 ekj ðyÞz þ ikj ðyÞx cos y sinðkj ðyÞy sin yÞ S=2
ðfmc þ ifms Þkj ðyÞsin yny ds
ð22Þ
where, m¼1, 3, 5, 7c. For the anti-symmetric potentials, i.e. f2, f4, f6 and f7s, the Kochin function takes the following form: ZZ Hm ðkj , yÞ ¼ 2i ekj z þ ikj x cos y sinðkj ðyÞy sin yÞ S=2 @f kj ðyÞðfmc nz fms cos ynx Þ mc @n @fms ds þ ikj ðyÞðfms nz þ fmc nx cos yÞi @n ZZ þ2 ekj ðyÞz þ ikj ðyÞx cos y cosðkj ðyÞy sin yÞ S=2
ðfmc þ ifms Þkj ðyÞisin yny ds
ð23Þ
ð18Þ
where, m ¼2, 4, 6, 7s.
1 þ aU ¼ 1 þ CU Fn
ð19Þ
2.2. Behavior of the wave systems
ke ¼ kð1 þ O cos wÞ2
ð20Þ
CU ¼ max½10:0,310Bf ðwÞ þ 68
ð21Þ
In the following, we investigate the behavior of the wave systems for the case of a submerged spheroid with a length to breadth ratio L/B ¼5 and draught to breadth ratio d/B¼0.75 where the draught d is measured from the free surface to the body center with Fn ¼0.2, k¼2.5 m 1, oe E7.14 s 1 in head waves. The corresponding values of the Hanaoka parameter, the steady wave number and the critical angle are: O E0.64, K0 E 12.7 m 1, a0 E671. The unsteady wave numbers of the two wave systems are plotted in Fig. 3. From this graph it may be observed that the wave number k1 approaches infinite when y-p/2, while the wave number k2 grows larger (although still remains bounded) as the wave direction approaches the critical angle. It has been generally found that the wave number k1 is much larger than k2, indicating that the k1 waves are very small and can be neglected in the calculation of added resistance in head waves. Fig. 4 shows the weighting functions in the added resistance formulation. Since the k1 wave system is neglected, only the weighting functions for the k2 wave system have been shown. The weighting functions are defined by the following expressions:
p
B and d are the beam and draught of the ship, Bf is the bluntness coefficient, ad accounts for the effect of draught and frequency and
Fig. 2. Coordinate system for the short waves range added resistance calculation methods.
k2 ðyÞ2 cos y WF2ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14O cos y
ð24Þ
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Fig. 3. Unsteady wave numbers of the two wave systems. Fig. 5. Kochin amplitude function of a submerged spheroid.
Fig. 4. Weighting function of the k2 wave system.
k2 ðyÞkcos w WF2ð2Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14O cos y
Fig. 6. Integrand in Maruo’s formula for a submerged spheroid.
ð25Þ
Fig. 5 shows the non-dimensional Kochin function amplitude for a spheroid in the range a0 o y o p and Fig. 6 shows the integrand of the second term in expression (11). A more detailed analysis can be found in Naito et al. (1988).
Generated data are validated in a series of test cases in comparison to numerical results of other authors, partly numerical results of a near-field, direct pressure integration method (Papanikolaou and Zaraphonitis, 1987) and experimental data, to the extent available. Typical results from these validation studies are presented and discussed in the following.
3. Numerical validations and discussion 3.1. Submerged spheroid The far-field method for the calculation of the added resistance, presented in Section 2, has been numerically implemented in a computer code, using velocity potential and other seakeeping data resulting from application of three different seakeeping assessment codes of NTUA-SDL, namely 1. The 3D frequency domain panel code NEWDRIFT (Papanikolaou– Schellin, 1992); relevant results are denoted in the graphs by ND. 2. The 3D time domain Green function code (Liu et al, 2007); relevant results denoted by LIU. 3. The 3D time domain hybrid Rankine source (near field)/Green function (far field) code HYBRID (Liu and Papanikolaou, 2010); relevant results denoted by HYBRID.
Iwashita and Ohkusu (1992) studied a shallowly submerged spheroid with a length to breadth ratio L/B¼5 and draught to breadth ratio d/B¼0.75, where the draught d is measured from the free surface to the body center. This study case has been addressed with two different approaches: The first one is using a hybrid time domain Green function method presented by Liu et al. (2007) for the calculation of the first-order potential. The second approach is based on a frequency domain 3D panel (i.e. NEWDRIFT, Papanikolaou and Zaraphonitis, 1987) for the solution of the first order problem and the calculation of the potential and the body motions. The deduced first-order results either from the time-domain or the frequency domain approach are introduced in Eqs. (22) and (23) for the
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
calculation of the Kochin functions, from which the added resistance is calculated according to Eq. (11). In addition to the above, the drift forces and added resistance are calculated also applying the nearfield, direct pressure integration approach, implemented in the NEWDRIFT code (denoted as NDnear in the graphs). Results from the first approach (denoted as LIUfar in the graphs) for the drift force at zero speed, and for the added resistance when the body is moving at forward speed in incident head waves with suppressed responses (fixed body case) are presented in Figs. 7–9. Good agreement may be observed between the present results and the results of Iwashita and Ohkusu (1992), denoted as IWASHITA in the graphs, both for the zero speed and nonzero speed calculations (Figs. 7–9). The same spheroid has been also tested at a deeper submergence of d/B ¼1.25. Comparisons have been herein made only between the two NTUA codes, namely the time- and frequency domain potential solvers (LIU and NEWDRIFT) at zero and at a forward speed corresponding to Fn ¼0.2. Good agreement between both solvers was obtained using the far-field method in the zero speed case. For the forward speed case, though the trend and maximum values are in general in good agreement, the herein predicted peak value is actually slightly shifted, which might be credited to the fact that the employed 3D panel method corresponding to the NEWDRIFT code is essentially based on a
645
Fig. 9. Added resistance on a submerged spheroid, Fn ¼0.3, d ¼ 0.75B.
Fig. 10. Horizontal drift force on a submerged spheroid at Fn ¼0, d ¼1.25B.
Fig. 7. Horizontal drift force on a submerged spheroid at Fn ¼ 0, d¼ 0.75B.
zero speed Green function and forward speed effects are taken into account in an approximate way by exploiting slender body theory assumptions. These results are shown in Figs. 10 and 11. 3.2. Floating spheroid
Fig. 8. Added resistance on a submerged spheroid, Fn ¼ 0.2, d ¼ 0.75B.
The added resistance of a half-immersed (floating) spheroid with a length to beam ratio L/B ¼5 has been studied by Kashiwagi (1997). Fig. 12 shows the drift force (zero speed) results for the freely moving spheroid, computed by NEWDRIFT (NDnear) and by the far field method (NDfar) using the first-order potentials from NEWDRIFT. A very good agreement between the near field and far field methods may be observed, along with a reasonable agreement with Kashiwagi’s enhanced unified theory results. Fig. 13 shows the added resistance of the half-immersed spheroid with suppressed motions at a forward speed corresponding to Fn ¼0.2. The results denoted by LIN are reproduced from Lin et al. (1993). Three different panelizations have been tested with NEWDRIFT, namely 360, 728 and 960 panels on half body. It may be observed that in some cases 360 panels are not enough for the accurate calculation of the added resistance, while with 728 panels, convergence has been generally achieved. Considering that 200 panels would lead in general to good motion results for such a simple body by the same code, we may conclude that added resistance is much more
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S. Liu et al. / Ocean Engineering 38 (2011) 641–650
sensitive to the size of panelization, than results of first order. A similar observation was made by Kashiwagi (2009). Nevertheless, a significant deviation between a variety of numerical results and experimental values is observed for this case, which is theoretically and practically a difficult benchmark case, because of the complicated above still water level wetting of the half-immersed spheroid. Kashiwagi presented good results for this case with his enhanced unified slender body theory, when using correct n1 term which better accounts for the wave diffraction near the bow. 3.3. Wigley hull A series of Wigley hulls were tested and reported by Journee´ (1992). In this section the validation of Wigley-III is presented. The hull surface is defined by the following expression: ! 2 ! y x 2 z x 2 ¼ 1 1 þ 0:2 1 ð26Þ b L=2 D L=2 Fig. 11. Added resistance on a submerged spheroid, Fn ¼ 0.2, d ¼1.25B.
Fig. 12. Drift force on a freely moving spheroid at Fn ¼0.
Fig. 13. Added resistance on a fixed half-immersed spheroid at Fn ¼ 0.2.
where 2b/L¼0.1 and D/L¼0.0625. The hull surface was discretized by 800 panels (port side only). For the body at zero speed, the drift force for the fixed body case and the free floating case are predicted by both the near field and the far field method, as shown in Fig. 14. A very good agreement between the near field and far field drift force results may be observed. Calculations have been performed with non-zero forward speed as well. Fig. 15 presents results for the added resistance by the far-field method. Despite some scatter in the experimental data, clearly the numerical values NDfar are over predicting the added resistance and lead to a slight shift of the peak values towards smaller wavelengths. This tendency is partly attributed to inaccuracies in the consideration of forward speed effects by NEWDRIFT, but mainly attributed to a common shortcoming of linear seakeeping methods, which are based on potential theory, namely the under-estimation of damping coefficients, leading to predictions of increased first-order motions in the vicinity of the resonance. Another reason for the numerically predicted increased motion amplitudes in the resonance region (thus also of added resistance) are nonlinear effects due to the above still water level hull form, which is not considered in linear frequency domain methods; these effects become very significant when the motions are of large amplitude; Papanikolaou et al. (2000) tried to overcome this problem by including additional viscous damping to the potential theory terms, derived with the help of ‘‘empirical’’ viscous lift and cross-flow drag coefficients. The introduction of cross-flow viscous damping in the first-order calculations for the Wigley hull
Fig. 14. Drift force on a Wigley-hull in head seas at Fn ¼ 0.0.
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available. The obtained results for Fn ¼0.266 and 0.283 are presented in Figs. 17 and 18, respectively. Once again, as in the Wigley hull case, Fig. 15, numerical results without any viscous correction strongly over-predict the experimental measurements. The introduction of cross-flow viscous damping in the first-order calculations (Cd ¼0.1, za/L¼1/50) resulted in a considerable reduction of the peak values of the added resistance (and motion amplitudes), much closer to the experimental measurements. Note here the very good prediction by the new HYBRID time domain method of NTUA-SDL (Liu and Papanikolaou, 2010), which, compared to NEWDRIFT, takes accurately into account forward speed effects and in addition accounts for the above water hull form nonlinearities. The calculation by HYBRID method has been herein done only for the range of l/L¼0.8–1.6, where the resonance takes place and the nonlinearity plays a significant role. In the short waves range, added resistance calculations according to the asymptotic formulae of Faltinsen et al. (1980) and also of Kuroda et al. (2008), denoted as SW1 and SW2, respectively, are also presented. 3.5. S-175 container ship Fig. 15. Added resistance on a Wigley-hull in head seas at Fn ¼ 0.3.
Fig. 16. Drift force on a S60 hull in head seas at Fn ¼0.0.
Calculations were performed also for the S175 Container ship, a hull form that has been thoroughly investigated by many
Fig. 17. Added resistance on a Series 60 hull at Fn ¼0.266.
resulted in a reduction of the peak values for the added resistance (denoted as NDfarCD), much closer to the experimental measurements, as shown in Fig. 15. In this calculation, the cross-flow viscous damping coefficient is Cd ¼0.1, whereas the incident wave amplitude is assumed as za/L¼1/60. 3.4. Series 60 hull with Cb ¼0.6 Calculations were made for a Series 60 hull with block coefficient of 0.60, which has been studied extensively by many researchers (e.g. Strøm-Tejsen et al. (1973)). The hull surface was discretized by 809 panels (port side). Calculations were performed at first for the zero speed case. Fig. 16 presents the drift force result. For the fixed body case, there is a very good agreement between the near field and far field results. For the freefloating case however, a small discrepancy may be observed, although calculations with both methods were based on the same first-order potential and motion data. Calculations have been also carried out by the far field method for two forward speeds, for which experimental data were
Fig. 18. Added resistance on a Series 60 hull at Fn ¼0.283.
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researchers (15th ITTC, 1978). A panelization with 885 panels was used in the calculations. Added resistance results are presented in Figs. 19–21. A quite satisfactory agreement between the numerical predictions and the experimental results (reproduced from Takahashi, 1988 and Seakeeping Committee report of ITTC 1984) is observed. Pure potential flow calculations by the panel code NEWDRDIFT resulted again in a small over-prediction of the added resistance at the peak, which is eliminated when the cross-flow drag viscous correction is introduced (Cd ¼0.1, za/L¼1/55). The results of the new HYBRID method of NTUA-SDL are in good agreement with the experimental data. In the short waves range, added resistance calculations using the asymptotic formulae of Faltinsen et al. and also of Kuroda and Tsujimoto are presented for this hull also, denoted as SW1 and SW2, respectively. In the short waves range, the agreement of added resistance calculations according to Kuroda and Tsujimoto with the experiments is very satisfactory. 3.6. Bulk-carrier Finally, results of drift force and added resistance calculations for a Bulk-Carrier, also studied by Kadomatsu (1988), are presented.
Fig. 19. Added resistance on the S-175 container ship at Fn ¼0.25.
Fig. 21. Added resistance on the S-175 container ship at Fn ¼ 0.30.
Fig. 22. Drift Force on a Bulk-Carrier in head seas at Fn ¼ 0.0.
This is a quite full hull form, with Lpp ¼285 m, B¼ 50.0 m, D¼18.5 m, Cb ¼0.829 and LCB ¼3.83% Lpp forward of the midship section. A panelization consisting of 819 panels on the half body was used for the calculations. The drift force and added resistance results are shown in Figs. 22–25, compared against experimental data. The numerical results agree well with the experiments, with the exception of the short waves range at the higher speeds corresponding to Fn ¼ 0.10 and 0.15, where the added resistance is under-predicted. However, in this range of wavelengths, calculations based on the asymptotic formulae of Faltinsen et al. and also of Kuroda and Tsujimoto resulted in a very good agreement with the experimental data. Note that in this case of full bodies at low speed, no viscous correction for motion damping proves necessary, because of the resulting small amplitude motion responses and of related nonlinearities, along with limited forward speed effects.
4. Summary and conclusions
Fig. 20. Added resistance on the S-175 container ship at Fn ¼0.275.
The added resistance problem of ships in waves has been solved following Maruo’s far-field theory and numerically implemented
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Fig. 23. Added resistance on a Bulk-Carrier at Fn ¼0.05.
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using velocity potentials calculated by a Time Domain Numerical Simulation method for submerged bodies (Liu et al. 2007), a 3D Frequency Domain Panel method (Papanikolaou and Zaraphonitis, 1987) and a 3D time domain HYBRID method (Liu and Papanikolaou, 2010) for bodies floating on the free surface. Results of a near-field, direct pressure integration method were used for comparison in the zero speed case. For the short waves range, the asymptotic formulae introduced by Faltinsen et al. (1980) and also by Kuroda et al. (2008) were employed for comparison. A systematic validation for different hull forms (submerged and floating, slender and bulky) and varying wavelengths and speeds were conducted. The obtained results were compared with those of other well established authors and experimental data and a reasonable agreement was observed, suggesting that the implemented procedure appears a reliable and robust method for the routine prediction of the added resistance of a ship in waves. Regarding computational time and efficiency of the implemented 3D panel code method, with a panelization in the order of 800 panels for the half the body, after obtaining the potential and motion data, which takes approximately 5 s with the NEWDRIFT code, the added resistance computation for one frequency takes approximately 1.5 s on a PC with Intel Core 2 QUAD CPU Q8200 2.33 GHz, which proves actually very efficient from the point of view of practical applications. Concluding, the original far-field method of Maruo, further elaborated by use of a 3D panel method and the asymptotic formulae of Faltinsen, Kuroda and Tsujimoto for the short wavelengths prove fully satisfactory for the prediction of the added resistance of ships (of slender and bulky hull form) in waves and may be practically implemented within reasonable computational times.
Acknowledgements The presented work is part of the Ph.D. work of the first author. The continuous guidance by the second and third author, as well as by Professor Wenyang Duan (Harbin Engineering University) is acknowledged. Finally, the financial support by NTUA-SDL and partly by DNV in the framework of the NTUA-DNV GIFT agreement (2007–2010) is acknowledged. References Fig. 24. Added resistance on a Bulk-Carrier at Fn ¼0.10.
Fig. 25. Added resistance on a Bulk-Carrier at Fn ¼ 0.15.
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