DYNAMIC BEHAVIOR OF SEMI-DISPLACEMENT AND PLANING VESSELS IN CALM WATER AND WAVES O.M. Faltinsen Cesos & Department of Marine Technology, NTNU, Trondheim, Norway

H. Sun Cesos, NTNU, Trondheim, Norway ABSTRACT: Main topics are dynamic instabilities, slamming and global wave-induced response of semi-displacement and planing vessels. Spinout and porpoising are discussed in detail. A nonlinear porpoising analysis shows that initial instabilities can lead to steady-state heave and pitch oscillations. A 2D+t nonlinear method is applied in the analysis of porpoising and wave-induced motions and accelerations of a planing vessel in head sea. The lack of correctly predicting the 3D flow at the transom stern influences, in particular, pitch accelerations. However, a fully 3D method can also lead to numerical difficulties in describing nonlinear wave effects on high-speed vessel. The latter occurred with a recently developed computer code in predicting second-order springing excitation in incident bichromatic waves. Wetdeck slamming is given special attention. 1 INTRODUCTION There is a broad variety of high-speed marine vehicles with challenging and diversified hydrodynamic problems (Faltinsen 2005). We will limit ourselves to semi-displacement and planing vessels. The Froude number plays an important role and affects the dynamic behaviour during manoeuvring and seakeeping. Heel is often neglected in manoeuvring analysis of displacement vessels but ought to be considered for semi-displacement and planing vessels. The importance of dynamic instabilities and nonlinear wave-vessel interaction increases with the Froude number. Since the wetted area of planing vessels changes easily in waves, a linear seakeeping analysis has limited applicability. A consequence is that stochastic simulations of planing vessels in a seaway become demanding. Many realizations of a sea state are needed to obtain reliable predictions of extreme response values. Cavitation and ventilation are limiting factors for propulsion and steering units and may contribute to dynamic instabilities. The latter fact implies that the cavitation number must be correct in model tests. A depressurized model tank with wave maker is needed.

Results by a nonlinear 2D+t method in calm water and waves will be presented. Slamming and added resistance are integral parts of the analysis. The method is applicable to semi-displacement and planing vessels of monohull and catamaran types. T-foils and antirolling fins must be considered as appendages in a complete seakeeping analysis. It depends on the detailed design of a wetdeck if it can be included as an integral part of the analysis. Simulations of bow-dive and green water require further studies. Slamming represents an important structural loading. One must distinguish between local and global slamming effects. Hydroelasticity may play an important role. Rules for wetdeck slamming needs to be improved and should consider details such as the influence of a bow ramp and vessel dynamics. If the local angle between the impacting free surface and the vessel surface is small, the impact pressures may be very high, sensitive to details of the inflow and unimportant for resulting local maximum stresses (Faltinsen 2005). Slamming is sometimes analysed separately from the vessel dynamics but ought to be an integral part of numerical predictions of wave-induced response.

IX HSMV Naples 25 - 27 May 2011: Keynote 1

Global wave-induced loads matter for larger vessels. Both springing and whipping may occur. Springing is steady-state resonant hydroelastic vessel oscillations while whipping represents transient resonant oscillations associated with slamming. Both the water entry (slamming) and exit (diminishing wetted surface) must be considered in a whipping analysis. Our focus is on dynamic instability and wave-induced motions and loads in deep water. The discussion involves predictions of spinout, porpoising, nonlinear wave-induced vertical motions and accelerations in head sea, slamming and global hydroelastic effects associated with whipping and springing. 2 DYNAMIC INSTABILITIES The importance of dynamic instabilities increases with the forward speed. The dynamic stability of high-speed vessels both in calm water and in waves is, in general, poorly understood. Examples of dynamic instability of monohulls are • Roll instability causing nonzero and non-

oscillatory heel • Chine walking associated with dynamic roll

oscillations • Trim instability with non-oscillatory bow drop • Dynamic pitch-heave oscillations (porpoising) • Non-oscillatory broaching in calm water • Broaching in waves • Corkscrew pitch-yaw-roll instabilities • Mathieu instability in waves • Capsizing in steep beam-sea waves • Spinout

Spinout is described by Pike (2004) while the other mentioned instability phenomena are discussed in Faltinsen (2005). Instabilities of high-speed vessels are influenced by the increased importance of the hydrodynamic pressure relative to the hydrostatic pressure and the occurrence of cavitation, e.g. in propeller tunnels. Ventilation may also be a contributing factor. Since the hydrodynamic pressure is partly associated with wave generation, shallow water effects may play an important role. However, we are not aware of quantitative information on how much shallow water effects can matter for high-speed vessels. We will elaborate more on spinout and porpoising. Spinout is a serious consequence of bad fast-boat driving and can occur when the boat is altering course and slows down. The bow will as a consequence drop. A scenario can be rounding a

mark in a race. Other scenarios involving waves are described by Pike (2004). We will explain the instability phenomena by generalizing Newman (1977)’s linear slender-body theory. The theory is developed for Froude numbers less than approximately 0.2 when free-surface waves do not matter. The method can be modified to very high speeds by changing the free-surface condition from a rigid-wall condition to a dynamic free-surface condition expressing zero velocity potential. The latter fact implies use of infinite-frequency added mass at high speeds. The effect of heel is not accounted for. The modified Newman analysis leads to that the boat is directional stable and can consequently make a stable turning if

( ) 2222

22T T

MAx a xM A

∞∞

∞>+

(1)

There is no explicit speed dependence. Tx is the longitudinal distance from the centre of gravity to the stern and ( )22 Ta x∞ is 2D infinite-frequency sway added mass at the stern. Further, M and 22A∞ are the boat’s mass and infinite-frequency sway added mass, respectively. Calculated values of ( )22 Ta x∞ for wedges of different deadrise angles β are presented in Table 1. The non-dimensional added mass values

( )222 /a Dρ∞ where ρ is the water density and D is

the draught, has a very small influence of the deadrise angle β in the considered range 5º ≤ β ≤ 45º. Since a bow drop causes decreased draught at the stern, eq. (1) together with the results in Table 1 shows that directional instability may occur. Porpoising has been extensively studied in the literature for prismatic planing hulls. A numerical tool can provide insight to how the hull form influences the heave-pitch instability in calm water. A possibility is to use a 2D+t theory as described by Sun & Faltinsen (2011a). The principle of a 2D+t theory is that 2D time-dependent hydrodynamic problems are solved in Earth-fixed planes that are parallel with cross-planes of the vessel. An illustration is given in Figure 1.When the vessel has passed the aft cross-plane, a new cross-plane is introduced at the bow. The solution starts from the stem and is stepped downstream until the stern by using the free-surface conditions. The solution does not realize that the flow separates at the stern which causes a 3D flow effect that is neglected in the analysis. Sun & Faltinsen (2011a) based their solution on nonlinear potential flow which was solved by a boundary element method (BEM) for infinite depth.

IX HSMV Naples 25 - 27 May 2011: Keynote 2

Table 1. Calculated non-dimensional 2D infinite-frequency sway added mass of wedges with different deadrise angles (Skejic, personal communication 2011).

( )β o   5  10  15  20  25  30  35  40  45 

( )222 /a Dρ∞   0.6501  0.6579  0.6650  0.6709  0.6757  0.6794  0.6820  0.6836  0.6841 

Figure 1. Examples on Earth-fixed cross-sectional planes for a 2D+t calculation. However, it is also possible to combine the 2D+t strategy with a CFD method such as the SPH (smoothed particle hydrodynamics) method. The fact that the applied free-surface conditions cause 3D flow implies that a 2D+t method is not a strip theory. The method neglects transverse wave systems, which means that the method is questionable for vertical motions of displacement vessels. We do not know to what extent the method is applicable for stepped hulls which would involve a separate analysis of how the flow reattaches to the hull after a step. Figure 2 exemplifies 2D+t simulations of heave and pitch for a prismatic planing hull with deadrise angle β = 20.5º, load coefficient C∆ = M/(ρB3)=0.36, beam Froude number FnB = U/(gB)1/2, normal distance vcg = 0.586 B from the centre of gravity (COG) to the keel and pitch radius of gyration k55=B with respect to COG. Here B denotes the beam. Two cases that involve flow separation from a hard chine are shown. Case 1 is with longitudinal position lcg=1.2B of COG relative to the stern and steady trim angle τ=5.06º. Case 2 is with lcg=1.3B and τ=4.7º. An initial disturbance is given in the simulations in order to investigate the stability. We see that case 2 is stable while the heave and pitch oscillations are initially unstable in case 1 before they reach a limit cycle with steady-state oscillations. A linear stability analysis says that case 2 is unstable. An operational limit for the vertical accelerations is needed to state that the nonlinear steady-state oscillations are acceptable.

0 1 2 3 4 5 6

-0.04

-0.02

0.00

0.02

0.04

0.06

Hea

ve η

3/B

Time (s)

C∆=0.36, FnB=3.0

lcg/B=1.2, τ =5.06o

lcg/B=1.3, τ =4.7o

0 1 2 3 4 5 6-3

-2

-1

0

1

2

3

Pitc

h η 5 (

degr

ee)

Time (s)

C∆=0.36, FnB=3.0

lcg/B=1.2, τ =5.06o

lcg/B=1.3, τ =4.7o

Figure 2. Time histories of heave and pitch for stable and unstable cases. Simulated by the numerical method for β = 20.5º, C∆ = 0.36, FnB=3.0 with vcg/B=0.586 and k55/B=1.0. A 2D+t theory together with a BEM is applied by Sun & Faltinsen (2010) to simulate the steady flow of a semi-displacement ship with round bilge at high speed. Figure 3 shows the free-surface elevations around the ship sections from station 0 to station 19. The Froude number Fn = 1.14. The ship model is the one used in Keuning (1988)’s model tests. The length of the waterline is 2.0m. The breadth of waterline is 0.25m. The draft is 0.0624m. The displacement is 0.01248m3 and the block coefficient is 0.396. The model is running with trim angle 1.62º. The non-viscous flow separation is simulated. As the jet is running fast around the bilge with large curvature, the speed of the running jet can be so high that the pressure on the jet-hull interface relative to the atmospheric pressure outside the jet becomes negative. When the area with low pressure is large enough, the air will ventilate the jet-hull interface and make the jet leave the body surface, which means the flow separation occurs. This flow

1 2 3 4 5 6 7 8 

U 

IX HSMV Naples 25 - 27 May 2011: Keynote 3

separation is numerically simulated. First, a threshold for the area with negative pressure is prescribed. When the area is larger than the threshold, the water-hull interface is made to detach from the hull surface. Details of the flow separation model are given in Sun &Faltinsen (2006). The method can also be used to study dynamic problems, e. g. porpoising.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

z (m

)

y(m)

Flow separation from the hull surface Section surfaces Free surfaces

Figure 3. Free-surface elevations with flow separation from the round bilge. Here we can see that the breaking waves caused by the vessel are accounted for. The resulting spray development is not predicted. However, it is believed to be secondary. It is the pressure gradient on the hull causing the spray that is important to correctly predict. The latter effect is adequately described. Since a potential flow method is used, water has to be artificially removed from the tip of a plunging breaking wave to avoid it impacting with the underlying free surface. A SPH method would not have such a deficiency. The described 2D+t method can be generalized to consider chine walking and “corkscrew” pitch-yaw-roll oscillations. It may also be attempted to consider calm-water broaching of round-bilge monohulls. Since instabilities may be associated with cavitation, a procedure that considers this effect combined with the propulsion and steering unit needs to be developed. It is harder to predict ventilation that often is initiated by cavitation. The fact that the method neglects viscous effects is not a serious draw-back for dynamic problems. What has to be improved is a more proper handling of 3D flow effects, in particular at the transom stern. However, there are also neglected 3D flow effects at the stem and where the flow separation from a chine occurs. A catamaran can be considered. However, the aerodynamics of the deck is important for

catamarans used in offshore racing. Studies of the steady waves caused by a trimaran, have shown that the 2D+t method is not well suited for trimarans. The same is believed to be true for pentamarans. 3 WAVE-INDUCED VERTICAL MOTIONS

AND ACCELERATIONS IN HEAD SEA

State-of-the-art numerical methods do not always give satisfactory predictions of wave-induced heave and pitch motions of semi-displacement vessels. The catamaran model presented in Figure 4 is used in the example. The pitch radius of gyration is 0.26 times the ship length L. A linear 3D Rankine Panel method (RPM) is used. Since the transom stern is assumed wet, there is no effect of hull-lift damping. The interaction between the unsteady and steady flow is handled in two different ways. One approach assumes that the steady flow can be calculated by a rigid free-surface condition saying that the free surface acts as a wall. This provides interaction between the local steady flow and the unsteady flow. However, this steady free-surface condition is only appropriate for low Froude numbers (Fn), let us say

0 2Fn .< . Since the steady flow can be calculated by considering a double body where the submerged hull surface is mirrored about the mean free surface, the model is referred to as a Double-body Model (DM). The second method assumes that the steady flow can be approximated as a uniform flow with a velocity equal to the ship speed. It is in the further discussion referred to as the Neumann- Kelvin (NK) method.

  

Figure 4. Body plan of the demi-hull of a catamaran and main particulars. Dimensions given in full-scale values. ΛL is the ratio between the full-scale and model-scale length. LCG is relative to station 10. Station 20 is at the transom stern. D = draught. B = beam. 2p is the distance between the centrelines of the catamaran demi-hulls (Lugni et al. 2004).

IX HSMV Naples 25 - 27 May 2011: Keynote 4

 

 

   Figure 5. Heave (η3) and pitch (η5) of a catamaran in head sea. Fn = 0.5. ζa and 2k /π λ= are the regular incoming wave amplitude and wave number, respectively. λ is the incoming wavelength. The ship model is presented in Figure 4 (Lugni et al. 2004). Head sea waves are considered and Response Amplitude Operators (RAO) of heave and pitch are experimentally and numerically predicted. The RAO can experimentally be obtained either by considering a transient test technique (Colagrossi et al. 2001) or by simply considering incident regular waves. The following results are for Fn = 0.5 and have been reported by Lugni et al. (2004). Tests in regular incoming waves with different wave amplitudes were performed in the resonant frequency range. The RAO experimental data are presented in Figure 5 together with the predictions by the 3D linear RPM code. The standard deviation (σ ) connected with the transient test technique is also given in the plots showing a good repeatability of the experiments. For the numerical results both the NK and DM approximations are considered. The

numerical results overestimate the resonant pitch motion. The DM linearization shows the best agreement with the experiments. A strong amplification of the motions is generally observed near the resonance due to a small damping level. Since each demi-hull has small beam-to-draught ratio, i.e. B/D = 1.14, we should expect small wave-radiation damping. The predictions at resonant pitch motions may have been improved by including hull-lift damping by accounting for flow separation at the transom. Further, an improved description of the interaction with the local steady flow should be investigated. The experiments show clear nonlinear effects. The regular wave results do not converge to the transient test results as the wave amplitude reduces. One possible error source is a variation of the wave amplitude along the track of the model. This aspect was not investigated. The RAO for the pitch motion shows double peak behavior, typical for the multi-hull vessels. The results for pitch showed that it is important to account for the hull interaction. Nonlinear effects are typically accounted for in a pragmatic way in numerical predictions by combining a linear 3D Rankine method with nonlinear Froude-Kriloff and restoring loads and 2D slamming theory. The importance of nonlinear effects increases with the Froude number and a linear theory has limited applicability for planing vessels. The description of the wave-induced response of a planing vessel will in the following text be based on a nonlinear 2D+t theory. Sun & Faltinsen (2011b) applied the previously described nonlinear 2D+t method to calculate heave and pitch of a monohull in regular head sea waves. The results were compared with the experiments by Fridsma (1969) who examined four configurations. The beam B is 9 inches, i.e. 0.2286m. The length of the model is L = 5B =1.143m. The deadrise angle is β = 20 degrees. The vertical distance of the COG to the keel is vcg = 0.294B. In the forthcoming numerical results, the attitude of the planing hull in calm water is given from those experimental data. The tests for regular incident waves are considered with amplitude ζa=0.0555B and wavelength given by λ/L = 1.0 ~ 6.0. The main part of the model, i.e. the part with a length equal to 4B measured from the transom, is prismatic. The bow of the hull has the same V-shaped cross-section but the keel in the bow does not follow a straight line extending from the main hull. The keel line in the bow is an elliptical curve. This special bow shape is not considered in the present numerical study. The keel is assumed to be straight in the calculations, but the maximum length of the keel equals the length of the model.

IX HSMV Naples 25 - 27 May 2011: Keynote 5

This simplification gives rise to numerical errors. The effect is more important when the bow gets wetted during the unsteady motions. 3D corrections at the transom stern were applied to see the influence of the 3D effect at the stern to the numerical results. The motivation for the corrections is that the pressure has to be atmospheric at the transom while the pressure predicted by the method depends only on upstream effects. The 3D corrections were done by setting the sectional force equal to zero either over a distance 0.25B or 0.5B from the transom. Since this is a crude approximation, it is only indicative of 3D effects. An example of comparisons of heave, pitch and vertical accelerations at the COG and 10%L from the stem is shown in Figure 6 for configuration B. The load coefficient C∆=M/(ρB3) is 0.608. The pitch radius of gyration is 0.255 times the vessel length L and the longitudinal position (L-lcg)/L of COG is 0.62. The beam Froude number is 3.99. The mean wetted length-to-beam ratio is 2.80 and trim angle is 4º ìn calm water. The ship motions are affected by the 3D corrections, especially near the resonance frequency, while the phase angles are slightly affected and the acceleration peaks at the bow near the resonance frequency are sensitive to the 3D corrections.The phase angle for heave/pitch is defined as the phase difference between the two time instants when the maximum heave/pitch appears and when the crest of the incident wave passes the COG. Nonlinearities are clearly present in the studied case which can be seen from the numerical time history presented in Figure 7 where we, for instance, see periodic sharp peaks. The incident wavelength is chosen as λ/L=3.0, because the maximum non-dimensional heave and pitch amplitudes appear at about this wavelength. No 3D correction is applied here. The non-dimensional hydrodynamic vertical force F3

* and pitch moment F5* are normalized by

ρU2B2 and ρU2B3, respectively. The accelerations at the COG and at the bow, with a distance of 10%L from the stem, are also presented. The accelerations are made non-dimensional by g, the acceleration of gravity. The direction of the acceleration at the COG is normal to the calm water surface and the acceleration at the bow is normal to the keel. The reason of the sharp peaks in the time histories of the pitch moment and the acceleration at the bow is analyzed. From the numerical results we can see the following phenomena. When the bow meets the front wave slope, the wetted length of the keel rapidly increases. At the same time, the bow is going downwards, so the bow impacts on the wave surface and causes a rapid increase in the vertical force on the bow. Although the vertical force on the stern also increases, the contribution from the bow to the

1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

Configuration B

Lλ

Heave

3 a

a

ηζ

Exp. Cal. no 3D correction Cal. 3D corr. -0.25B Cal. 3D corr. -0.5B

1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0

2.5 Configuration B

Lλ

5a

akηζ

Pitch

Exp. Cal. no 3D correction Cal. 3D corr. -0.25B Cal. 3D corr. -0.5B

 

1 2 3 4 5 6-150

-100

-50

0

50

100

Configuration B

Lλ

Phase angle (degree)

Heave Pitch Exp. Cal. no 3D correction Cal. 3D Corr. -0.25B Cal. 3D Corr. -0.5B

  

   1 2 3 4 5 6

0

2

4

6

8

Configuration B

Lλ

Acceleration (g)

At COG At bow Exp. Cal. no 3D correction Cal. 3D Corr. -0.25B Cal. 3D Corr. -0.5B

 

  

Figure 6. Heave motion, pitch motion, phase angles and accelerations of configuration B with Fn = 1.78, C∆=0.608 and τ=4º. Incident wave amplitude ζa=0.0555B. Exp. means the experimental results by Fridsma (1969); Cal. means the results by numerical calculations. Two 3D corrections (3D Corr.) are applied by neglecting the sectional force on the hull with length of 0.25B and 0.5B in front of the transom stern, denoted by ‘-0.25B’ and ‘-0.5B’ respectively.

IX HSMV Naples 25 - 27 May 2011: Keynote 6

pitch moment is greater. This results in a fast increase in the positive pitch moment. Afterwards, the downward speed of the bow is quickly decelerated, so the force in the bow decreases, however, the force in the stern is still increasing. Therefore, the pitch moment rapidly decreases but the total vertical force does not change much. The sharp peak in the pitch moment will then influence the pitch acceleration and therefore cause a sharp peak in the acceleration at the bow. It can be seen in Figure 7 that there are small sharp peaks in the total vertical force and the acceleration at the COG at the same time instants, but the peaks are not as prominent as for the pitch moment and the acceleration at bow.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.020.000.02

Pitch (degree)

Heave (m)

Time (s)

Acc. atbow(g)

Acc. atCOG(g)

F5*

F3*0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-202

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.05

0.10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-0.020.000.02

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.60.00.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

2

Configuration B

Figure 7. Time histories of the loads, motions and accelerations at λ/L = 3.0 and ζa=0.0555B for configuration B with Fn =1.78 and τ=4º. Acc. means the acceleration. 4 SLAMMING Slamming can both have a local and global effect. Local slamming analysis is typically based on 2D flow by using that the velocity potential is initially zero on the free surface which requires the section to be out of the water at start. For example, Sun & Faltinsen (2009) studied the water entry of a heeled ship section by a nonlinear 2D BEM with gravity. Figure 8 shows the free-surface elevations around the ship section and the pressure distribution on the ship section surface at different time instants. The ship section has a constant heel angle of θ = 28.3º. It is dropped from a height of h = 0.120m above the calm water surface. Non-viscous flow separation occurs at the knuckle of the right side and occurs also from the bottom of the section on the left side. An analysis like this does not account for the fact that the forward speed and three-dimensional flow may have an important effect on the slamming loads. A simple way to illustrate the forward-speed dependence on slamming is to follow a quasi-steady

approach and note that the steady wave elevation is influenced by the submergence of the vessel. We will use Ogilvie’s (1972) theory to illustrate this fact. Ogilvie (1972) derived a simple 2.5D solution for the wave elevation ζ in the bow region along the surface of a symmetrical wedge with draught D and wedge half angle α . The body-boundary condition was transferred to the centreplane and the linearised free-surface conditions were used. The maximum value

21 59maxU D.

gαζπ

=       (2)

of the wave elevation occurs at 0 91x . U D / g= . Here x is a longitudinal coordinate in the downstream direction with x = 0 corresponding to the edge of the bow. D is influenced by the heave motion at the bow in a quasi-steady approach. The consequence is that the bow wetted area and hence the slamming force is affected by U. This discussion does not imply that we suggest using a quasi-steady approach. The problem has to be solved dynamically.

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4

t = 0.03 st = 0.04 s

t = 0.05 st = 0.06 s

t = 0.07 s

y (m)  

-0.20.00.2

-0.20.00.2

0.0

0.8

-0.20.00.2

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.20.00.2

Pres

sure

(Bar

)

s (m)

t = 0.06 s

t = 0.05 s

t = 0.04 s

t = 0.03 s

t = 0.07 s

  Figure 8. Water entry of heeled ship section with non-viscous flow separation for θ = 28.3º and h = 0.120m.

IX HSMV Naples 25 - 27 May 2011: Keynote 7

Chezhian (2003) demonstrated by experimental drop tests and a 3D generalized Wagner theory at zero forward speed that 3D effects in the bow region was significant. However, the considered body has a blunter bow than a high-speed vessel. An important 3D effect for slender bows is through the speed-dependence of the unsteady free-surface conditions as accounted for by a 2D+t theory. The nonlinear analysis described in the last section incorporates slamming, i.e. it does not separate out the slamming analysis. The procedure thereby accounts directly for the effect of the vessel dynamics and the forward speed on the slamming loads. However, the procedure does not consider the fact that local hydroelasticity may have an important effect on the local slamming-induced stresses when the angle between the impacting free surface and the body surface is small (Faltinsen 2005). The latter fact would require a separate analysis. Water entry (slamming) and water exit (diminishing wetted surface) may be equally important in a global slamming analysis. The latter fact was evident in the numerical and experimental studies of global effects due to wetdeck slamming by Ge et al. (2005). They showed that the wetdeck-load analysis could be significantly simplified by using a von Karman method both for the water entry and exit and by adding the effect of nonlinear Froude-Kriloff and hydrostatic restoring loads. A Wagner method cannot be applied during water exit. The fine details of slamming occur on a very small time scale relative to the time scale of global response and are therefore not important for global response. Ge et al. considered the effect of global hydroelasticity and based the loads on the side hulls by using a frequency-domain strip theory with no hydrodynamic interaction between the side hulls. Good agreement with experimental results of water entry and exit forces on the wetdeck as well as global vertical bending moments and shear forces in transverse sections was demonstrated. Such a procedure becomes questionable at high Froude number. Since the response includes oscillations with several frequencies, e.g. the frequency of encounter and important wet natural structural frequencies, one ought to solve the problem in the time domain instead of in the frequency domain. Ge et al. showed that global hydroelasticity mattered. If the wetdeck is flat as it was in the Ge et al.’s case, an alternative would be to apply the previously described 2D+t theory to catamarans and thereby provide an inflow to the wetdeck. The reason that the water entry and exit loads cannot be directly included in the 2D+t analysis is that the flow caused by the wetdeck is mainly associated with longitudinal variations of the wetted deck area. The

water entry and exit forces and moments acting on the wetdeck have to be added to the forces and moments acting on the side hulls obtained by the 2D+t analysis. Further, global hydroelasticity must be incorporated. If the wetdeck is as shown for the “wave-piercing” catamaran in Figure 9, we do not need to separate the slamming analysis. When the wetdeck first touches the waves, the 2D+t analysis can be applied. Wetdeck slamming was the reason to a recent accident with MS “Sollifjell” in Norway. A drawing of a longitudinal cut of the wetdeck is shown in Figure 10. The wetdeck is flat in the cross-sectional plane. The design of the front panel of the wetdeck was special. It had an 45o angle relative to the rest of the wetdeck. The consequence is that the forward speed can contribute significantly to the relative impact velocity on the front panel and thereby to high loading on the front panel. The loading on the front panel may have initiated the damage on the rest of the wetdeck.

Figure 9. “Wave-piercing” catamaran.

Figure 10. Illustration of the wetdeck of MS “Sollifjell” containing a 45o front panel.

IX HSMV Naples 25 - 27 May 2011: Keynote 8

Figure 11. Calculated position of slamming on the wetdeck of a catamaran in regular head sea waves as a function of wavelength λ . The figure shows a longitudinal cross-section at the centre plane of the catamaran. The bow ramp is seen in the fore part. Fn = 0.5, ζa=ζslam=lowest incident wave amplitude when slamming occurs. L=Lpp = length between perpendiculars (Zhao & Faltinsen 1992). Design procedures of wetdecks considering the slamming load effects have to be improved. The details of the vessel dynamics at operating speeds in specified sea states as well as the slamming load effects have to be accounted for. The wetdeck geometry (bow ramp angle, deck height, deck flatness etc.) and material ought to be reflected in the rules. Let us illustrate the importance of the vessel dynamics. The example considers a wetdeck that is flat and horizontal in the cross-sectional planes. The longitudinal cut of the front part of the wetdeck has an angle in the front which influences where the water impacts initially. The latter fact depends on the wave conditions and the phasing of the wave-induced motions of the catamaran. It matters how the water hits the wetdeck. Figure 11 shows how the impact position depends on the wave period in regular head sea waves for a given catamaran and Froude number. The water always hits in the forward part of the wetdeck. This follows from that the relative vertical motions of a ship are always largest in the forward part for a ship at forward speed in head sea. We cannot in general for all speeds and wave headings say that the water will always impact in the forward part of the wetdeck. Figure 11 shows that the longer the wavelengths are,

the closer to the bow the initial impact occurs. The figure also presents the minimum wave amplitude

aζ for slamming to occur for a given incident wavelength λ . This minimum wave amplitude is smallest for 1 26/ L .λ = for the presented cases in Figure 11. The smaller the minimum wave amplitude is, the larger the amplitude of the relative vertical motion divided by aζ is. When the water does not initially hit at the end of the forward deck, the water surface has to be initially tangential to the wetdeck surface at the impact position. The steady trim angle influences the occurrences of wetdeck slamming and depends on the Froude number. Ge et al. (2005) showed that uncertainties in the trim angle could have a large effect on global wetdeck slamming-induced stresses. If the transverse cross-section of the wetdeck is not flat, let us say the deadrise angle of the wetdeck cross-section is larger than10o , local hydroelastic effect is not dominant (Faltinsen 1999). It means that slamming pressures obtained experimentally for a rigid model can be applied in a quasi-steady structural analysis. When local hydroelasticity matters, measured slamming pressures can be very high and sensitive to the inflow condition. Because the duration of high pressures are typically very small relative to important local structural natural periods, it is the force impulse that matters. The analysis by Faltinsen (1997) shows that the pressure is not needed to find the resulting maximum structural stresses in longitudinal stiffeners between transverse frames of aluminium and steel structures. The relative impact velocity is the important parameter. 5 SPRINGING Springing is steady-state global hydroelastic vibrations that may matter for large semi-displacement vessels. It is a resonance oscillation that can be excited by both linear and nonlinear wave-body interaction effects. The important structural natural periods are low relative to important mean wave periods. Figure 12 illustrates how the natural period nT for two-node vertical bending may vary as a function of ship length between 50m and 170m. The latter values are estimated by Faltinsen (2005) using class society rules for vertical bending moment midships. nT has a kink at L=100m and varies nearly linearly for L below 100m and for L above 100m. nT for L=100m is about 0.9s. For shorter ships, the stiffness is commonly higher than required by the rules, hence the period would be less.

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Figure 12. Estimated natural period nT for two-node vertical vibration as a function of ship length for a high-speed monohull vessel, based on uniform beam approximation (Faltinsen 2005). Normally the resonance frequency occurs in the high-frequency tail of the sea spectra. It is therefore of concern how the tail is represented. The Pierson-Moskowitz wave spectrum decays as ω −5 , while other spectra may have ω −4 . Here ω is the wave frequency. The wave energy in the high-frequency tail is an important uncertainty for prediction of linear springing response. Nonlinear springing has been observed in regular waves in model tests (Miyake et al. 2008; Slocum & Troesch 1982) when the encounter frequency is equal to 1/n of the structural natural frequencies where n is an integer, i.e. we can talk about second-order, third-order and so on nonlinearly excited springing. If we consider an irregular sea, second-order springing is associated with sum-frequency effects. An analysis can in moderate sea conditions be based on a perturbation scheme starting with a linear analysis. It is hard from a numerical point of view to go beyond second-order nonlinearly excited springing. Further, a perturbation scheme becomes increasingly tedious with increasing order of nonlinearly excited springing.  Shao & Faltinsen (2011 a & b) have studied numerically the second-order weakly-nonlinear hydrodynamic problem of a displacement ship moving with constant forward speed based on a consistent second-order theory. Potential flow theory is assumed. All 3D flow effects and interactions between the local steady and unsteady flow are accounted for. The boundary value problem is formulated in a body-fixed coordinate system. The formulation does not include any derivatives of the velocity potential on the right-hand side of the body boundary conditions, and thus avoid the difficulties associated with terms similar to the so-called mj-

terms and their derivatives. A time-domain higher-order BEM based on cubic shape functions is used as a numerical tool. A forward difference scheme is applied in the free-surface conditions in order to better numerically stabilize the solution. Both monochromatic and bichromatic head-sea waves are considered with different Froude numbers. The results have been validated by comparing with experimental results by Journee (1992) for added mass, damping, heave, pitch and added resistance in head sea of a Wigley hull at Froude number 0.3. For the same Wigley hull, it is found that the second-order velocity potential gives dominant contribution to the second-order wave excitation of ship springing in the wave frequency region where sum-frequency springing occurs. We are not aware of other studies solving the second-order problem consistently at forward speed. However, it is commonly done for large-volume offshore structures at zero speed. The numerical results also demonstrate strong dependency of the second-order wave excitation of ship springing on the Froude numbers for small wavelengths. A limited number of calculations indicate that the second-order transfer functions for springing excitation may be obtained from monochromatic waves which significantly simplify the estimation of the second-order transfer functions. An important issue is how to estimate the damping. A major hydrodynamic contribution for high-speed vessels is the hull-lift damping associated with the transom stern (Faltinsen 2005). We do not see any major obstacles in applying the numerical method to linear seakeeping and added resistance of semi-displacement vessels. One must of course include the important effect of the transom stern associated with the fact that the flow separates from the transom stern and leaves a hollow in the water behind. However, so far numerical spatial differentiation of the free-surface conditions causes numerical instability problems for the second-order problem for Froude numbers larger than 0.3. The method needs to consider oblique sea and springing of non-vertical eigenmodes of monohull and multi-hull semi-displacement vessels.  6 CONCLUSIONS The importance of dynamic instabilities and nonlinear wave-vessel interaction effects increases with the Froude number. Spinout which may occur when a fast boat turns is explained by stability analysis. A nonlinear 2D+t theory has been applied to study porpoising and the wave-induced motions and accelerations of a planing vessel in head sea. It is illustrated that linear heave and pitch instabilities (porpoising) in calm water may lead to steady-state

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finite amplitude motion due to nonlinear effects. 3D end effects at the transom stern that are neglected in a 2D+t formulation influence, in particular, the pitch acceleration predictions. The nonlinear 2D+t method incorporates water entry (slamming) and water exit loads and thereby includes the forward speed effect on slamming in a rational way. Slamming on a flat wetdeck must be done by a separate analysis. When the local angle between the impacting free surface and the vessel surface is small, the local slamming analysis must include the effect of local hydroelasticity. Springing predictions by a fully 3D perturbation method that considers the interaction between the local steady and unsteady flow is discussed. When an inertial system is used in the formulation, problems may occur with higher-order derivative terms appearing in the body-boundary conditions. A body-fixed coordinate system is therefore used. It is emphasized that the second-order potential plays an important role in second-order nonlinearly excited springing. 7 REFERENCES Chezhian, M. (2003) Three-dimensional analysis of slamming, Dr.ing thesis, Dept. of Marine Technology, NTNU, Trondheim, Norway. Colagrossi, A., Lugni, C., Landrini, M., Graziani, G. (2001) Numerical and experimental transient tests for ship seakeeping, Int. Journal Off. and Ocean Struct., 11, 67-73. Faltinsen, O.M. (1997) The effect of hydroelasticity on slamming, Phil. Trans. R. Soc. Lond.A, 355, 575-91 Faltinsen, O.M. (1999) Water entry of a wedge by hydroelastic orthotropic plate theory, J. Ship Res., 43, 3, 180-193. Faltinsen, O.M. (2005) Hydrodynamics of High-Speed Marine Vehicles. New York: Cambridge University Press. Fridsma, G. (1969) A systematic study of the rough-water performance of planing boats. Davidson Laboratory Report R-1275. Ge, C., Faltinsen, O.M., Moan, T. (2005) Global hydroelastic response of catamarans due to wetdeck slamming, J. Ship Res., 48, 1.

Journee, J.M.J. (1992) Experiments and calculations on 4 Wigley hull forms in head sea. Technical Report 0909, Delft University of Technology, Mekelweg 2, 2628 Delft. Lugni, C., Colagrossi, A., Landrini, M., Faltinsen, O.M. (2004) Experimental and numerical study of semi-displacement monohull and catamaran in calm water and incident waves, In Proc. 25th Symposium on Naval Hydrodynamics, Washington D.C.: Dept. of the Navy-Office of Naval Research Miyake, R., Matsumoto, T., Zhu, T., Abe, N. (2008) Experimental studies on the hydroelastic response due to springing using a flexible mega-container ship model. 8th International Conference on Hydrodynamics, Nantes. Newman, J.N. (1977) Marine Hydrodynamics, Cambridge: The MIT Press. Ogilvie, T.F. (1972) The wave generated by a fine ship bow, In Ninth Symp. Naval Hydrodynamics, ed. R.Brard and A.Castaro, vol. 2, pp. 1483-525, Washington, D.C.: National Academy Press. Pike, D. (2004) Fast Powerboat Seamanship. The complete Guide to Boat Handling, Navigation, and Safety, International Marine/ McGraw-Hill. Shao, Y., Faltinsen, O. M. (2011a) Numerical study of the second-order wave loads on a ship with forward speed, 26th Int. Workshop on Water Waves and Floating Bodies, April, Athens, Greece. Shao, Y., Faltinsen, O. M. (2011b) A numerical study of the second-order wave excitation of ship springing with infinite water depth, to be published in Journal of Engineering for the Marine Environment. Slocum, S., Troesch, A.W. (1983) Nonlinear ship springing experiments report no.266, The University of Michigan, Department of Naval Architecture and Marine Engineering, Ann Arbor, Mich. Sun, H., Faltinsen, O. M. (2006) Water impact of horizontal circular cylinders and cylindrical shells, App. Ocean Res.,28, 299-311. Sun, H., Faltinsen, O. M. (2009) Water entry of a bow-flare ship section with roll angle, J. Mar. Sci. Technol., 14, 69-79. Sun, H., Faltinsen, O. M. (2010) Numerical study of a semi-displacement ship at high speed, In Proc. 29th Int. Conf. on Ocean, Offshore and Arctic Eng. (OMAE2010-20565), Shanghai, China. Sun, H., Faltinsen, O. M. (2011a) Predictions of porpoising inception for planing vessels, Submitted for publication. Sun, H., Faltinsen, O. M. (2011b) Dynamic motions of planing vessels in head sea, Submitted for publication. Zhao, R., Faltinsen, O.M. (1992) Slamming loads on high-speed vessel, In Proc. Nineteenth Symp. on Naval Hydrodynamics, Washington, D.C.: National Academy Press.

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