Experimental and numerical study of a containership under parametric rolling conditions in waves

Ocean Engineering 124 (2016) 385–403

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Experimental and numerical study of a containership under parametric rolling conditions in waves Arndt Schumacher a, S. Ribeiro e Silva b, C. Guedes Soares b,n a

HSVA, Hamburgische Schiffbau-Versuchsanstalt GmbH, Hamburg, Germany Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, Lisboa, 1049-001 Portugal b

art ic l e i nf o

a b s t r a c t

Article history: Received 23 October 2015 Received in revised form 17 May 2016 Accepted 15 July 2016 Available online 18 August 2016

A set of scaled model experiments have been conducted at HSVA towing tank to study the occurrence of parametric rolling on a containership in regular and irregular waves. Herein the performed model tests are described and the experimental results are presented. Forced rolling tests were also performed with the ship model in order to investigate its roll damping characteristics. By means of these tests the roll damping coefficients were determined for different maximum roll amplitudes and advance speeds. The experimental data of this containership is compared with results from a non-linear time domain model, where the prediction of maximum roll amplitude under parametric rolling conditions associated with different viscous roll damping models is thoroughly discussed. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Parametric rolling Seakeeping Ship stability in waves Roll damping model tests

1. Introduction In this work the effect of parametric resonance on a containership sailing in following or head seas is dealt with to illustrate the capabilities and limitations of both the experimental model testing and numerical simulations. An extensive series of tests on a containership were carried at “Hamburgische Schiffbau-Versuchsanstalt” (HSVA) and some of these are used to obtain accurate estimates of the roll damping coefficients at different speeds, and to validate the numerical model developed by Ribeiro e Silva and Guedes Soares (2013) to study the phenomenon of parametric rolling in waves. One main focus of the model tests was the investigation of conditions that lead to parametrically excited rolling in both regular waves and irregular seas. Notice that this phenomenon occurs only at certain combination of wave/seaway condition, load case, encounter angle and ship speed. In order to carry out effective model tests, it is very helpful to perform numerical calculations prior to the model tests to identify the combination of parameters at which parametric rolling might occur. Thus, in advance of the model tests HSVA extensive nonlinear time domain simulations of parametrically excited roll motions in waves were performed. For the container vessel simulations were mainly conducted in n

Corresponding author. E-mail address: [email protected] (C. Guedes Soares).

http://dx.doi.org/10.1016/j.oceaneng.2016.07.034 0029-8018/& 2016 Elsevier Ltd. All rights reserved.

regular bow and following waves at constant ship speed and encounter angle. The values of wavelength, ship speed, metacentric height and encounter angle were varied. The calculation results identify the parameter combinations where the vessel would properly execute parametrically excited roll motions in regular bow and following waves caused by low-cycle (1:2 ratio of wave to ship natural roll periods) resonance conditions. By means of these pre-determined parameter combinations it was also possible to generate irregular seaways which would be likely to lead to the occurrence of parametric rolling. The obtained information was used to choose the most promising test conditions and to compile a rough test program. Specifically, based on the computations in regular waves some additional simulations were conducted in irregular head and stern seas to identify wave sequences which would lead to parametrically excited roll motion. The identified wave sequences were then isolated and delivered to the University of Berlin, which generated the corresponding control signals for the wave generator in order to reproduce these sequences in the towing tank at HSVA during the test campaign.

2. Parametric rolling prediction models Theoretical works on parametric roll resonance can be found in the 1950s, where linear and non-linear roll damping models were taken into account by Grim (1952), Kerwin (1955) and Paulling and Rosenberg (1959). These studies enabled the discussion of

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parametric roll resonance with the Mathieu equation. Then, to investigate capsizing, non-linearity of restoring moment in still water was taken into account. At this stage, non-linear dynamical system approach including geometrical and analytical studies is required to identify all potential danger among co-existing states. Such examples can be found in Sanchez and Nayfeh (1990), Soliman and Thompson (1992) for uncoupled roll models and Oh et al. (2000) for a coupled pitch-roll model. These theoretical studies focused on understanding fundamental mechanism of the phenomena with rather simplified mathematical modeling. For example, the amplitude of restoring arm is often provided a priori without any relationship with wave steepness or exciting moment. On the other hand, several six degree-of-freedom (DOF) models such as Munif and Umeda (2000) and Matusiak (2003) have been developed for quantitative numerical prediction in the time domain. Here the relationship between wave steepness and restoring moment is fully taken into account. However, the works using these detailed models only show simulated results with limited number of initial condition sets. Because of the non-linearity of the system, there is a possibility to either overestimate or underestimate the roll responses and, therefore, to overlook some potential danger, as has been raised by some authors. Longitudinal waves i.e. head or following waves, cause the largest variations in stability and, therefore, create maximum parametric excitation. Whilst the physical basis for parametric rolling is the same in head and following waves, parametric rolling in head waves is more likely to be influenced by and coupled with heave and pitch motions of the ship, since these motions are typically more pronounced in head waves (Shin et al., 2004). Treatment of the coupling between the vertical motions of heave, pitch and roll varies in the numerical methods used. For example, Neves and Rodriguez (2005) used a two-dimensional analysis for a set of coupled heave, pitch and roll equations of motion with 2nd and 3rd order non-linearities describing the restoring actions. Levadou and van't Veer (2006) used coupled non-linear equations of motion in the time domain with 3 (heave, roll and pitch) and 5 (sway, heave, roll, pitch and yaw) DOF. Nonlinear excitations are incorporated by pressure integration over the actual wetted surface while diffraction forces are considered linear. Hydrodynamics are calculated in the frequency domain by a 3D panel code and are incorporated in the time domain by adopting the impulse response functions method. France et al. (2003) and Shin et al. (2004) adopted a similar approach but with a hybrid singularity based on the Rankine source in the near field and transient Green's function in the far field. On the other hand, Neves et al. (1999) used a system with 3 DOF, with the coupled heave and pitch motions providing input to the parametric excitation simulated using a one DOF non-linear roll equation of motion. The heave and pitch motions are solved simultaneously and independently of the roll motion, an assumption that has been shown to be adequate in simulating parametric roll and has been justified experimentally (Oh et al., 2000). More recently, Ahmed et al. (2010) used coupled non-linear equations of motion in the time domain with 4 (sway, heave, roll and pitch) DOF, where the non-linear incident wave and hydrostatic restoring forces/moments are evaluated considering the instantaneous wetted surface whereas the hydrodynamic forces and moments, including diffraction, are expressed in terms of convolution integrals based on the mean wetted surface. Numerical simulations and experimental measurements in regular waves are an effective procedure to observe and understand the physics of the parametric rolling phenomenon as well to validate numerical methods. Parametric rolling in realistic irregular seas, however, is of greater practical interest to shipmasters. The numerical work conducted by Bulian and Francescutto (2006) is an example of investigations in this field. This investigation is

conducted in long-crested head seas and makes use of the concept of Grim's effective wave amplitude within a one DOF equation of motion in roll. Ribeiro e Silva and Guedes Soares (2000), demonstrated that both linearized and non-linear theories could be used to predict parametric rolling in regular head waves. On the linear model (in the form of Mathieu's equation) stability variations were evaluated from the linearized righting arm curves with the wave crest varying longitudinally along the ship hull. However, this model was not adequate to predict ship's roll response magnitude under wave-induced parametric resonance conditions, since deck submergence effect on restoring characteristics of the vessel and nonlinear damping terms could not be included and therefore the limit cycle behavior could not be obtained. A non-linear numerical model of parametric resonance taking into consideration deck submergence and other non-linearities on restoring moment of ships in regular waves was also proposed by Ribeiro e Silva and Guedes Soares (2000). In that model a quasistatic approach was adopted to study the roll motion, where only the variations on transverse stability in regular waves were considered. For that purpose, an uncoupled roll equation, which included the effects of heave and pitch responses in regular waves, and immersed hull variations due to wave passage on roll restoring term, was used to describe the parametrically excited roll motions. While good agreement in terms of limited response behavior was found between the time domain simulation of roll motion in longitudinal regular waves and the existing experimental data, simulations of parametric rolling in irregular waves as presented in the literature by Francescutto and Bulian (2002), Belenky et al. (2003), and Pereira (2003), could not be performed using that single DOF model. To overcome these shortcomings a non-linear model, coupled in the five (sway, heave, roll, pitch and yaw) DOF was then developed and proposed Ribeiro e Silva et al. (2005) to simulate the time domain responses of a ship in long-crested irregular waves. The model is now extended to six DOF making use of a semiempirical formulation for surge motion. Notice should be given to the fact that accounting for nonpotential roll damping is of utmost importance for an accurate simulation of parametric rolling. Particularly, the magnitude of the roll response during parametric rolling is dictated in large part by the amount of viscous damping in the roll DOF, where damping improves the stability character of the system by reducing the instability regions in size. Moreover, from SAFEDOR results it has been commented by Spanos and Papanikoloau (2009) that accuracy of numerical simulations could be significantly improved by adopting a linear plus quadratic roll damping model. Various models are available in the literature for estimating a total roll damping coefficients of which, the most well-known and widely used model is the original method due to Ikeda et al. (1978). For example, Levadou and van't Veer (2006) used a time domain implementation of Ikeda's method with specific assessment of the fluid velocities at the bilge keel for evaluating the bilge keel damping. Neves and Rodriguez (2005) used Ikeda's method represented in a quadratic form. In this study the availability of an extensive series of free-decay and forced rolling experimental results on a scaled model of a containership, allowed the measurement of several roll time traces and maximum amplitudes of roll at different advance speeds to be utilized in the numerical simulations represented in a quadratic form. France et al. (2003) and Shin et al. (2004) used an empirical model for their C11 containership investigation, where the numerical roll decay response was tuned to match the experimental roll decay tests by specifying an equation with up to cubic order terms of roll angle and roll velocity. Ribeiro e Silva et al. (2005) have adopted Ikeda's method, while

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it was found by Ribeiro e Silva and Guedes Soares (2009) that for slender hull forms Miller's (1974) roll-damping method could also be used to accurately calculate the total roll-damping coefficient at different advance speeds. The approach adopted in this application of the non-linear time domain simulation is to estimate the roll damping from the experimental results, taking advantage of the availability of an extensive series of free-decay and forced rolling experimental results. 2.1. Prediction of occurrence of the parametric rolling in waves

(1)

These static and dynamic forces ⎡⎣ F ⎤⎦ and motions { ξ} are represented on an inertial coordinate system (see Fig. 1) fixed with respect to the mean position of the ship, X = (x, y, z ), with z in the vertical upward direction and passing through the center of gravity of the ship, x along the longitudinal direction of the ship and directed to the bow, and y perpendicular to the latter and in the port direction. The origin is in the plane of the undisturbed free surface. According to Fig. 2, the translatory displacements in the x , y , and z directions are respectively the surge ξ1, the sway ξ2, and the heave ξ3. The rotational displacements about the x , y , and z axes are respectively the roll ξ4 , the pitch ξ5, and the yaw ξ6 . This non-linear model attempts to extend the standard strip theory limitations associated with a fixed mean waterline and small displacements by considering the movement of the body induced by the incident wave and coupled the equations of motion in the five significant DOF. Using a time stepping scheme the wave forces on the actual underwater part of the hull are calculated at each time step. These forces together with the geometric, hydrostatic and most important hydrodynamic properties of the current wetted hull are used to derive the consequent translational and z0

O0

z

y0 x0

β

y x

O U.t

U

z Ω

η

y

O x

r

rotational motions. By using this approach, the coupling of the anti-symmetric and symmetric equations of motion may be retained via large angles Euler transformations followed by an intersection plane technique with free-surface for each strip (made up of several segments). Assuming that, apart from the restoring forces (which are time dependent and non-linear), all hydrodynamic forces are linear, and combining these with the mass forces one obtains six linear coupled differential equations of motion, given by: 6

If the susceptibility to parametric rolling has been determined by any simplified analytical or numerical model available, the application of a full numerical procedure with different possible analytical methods has to be applied to assess the severity of the phenomenon, as discussed by Umeda et al. (2004), Neves et al. (2003), Ribeiro e Silva and Guedes Soares (2008, 2009) and ABS (2004). Moreover, the numerical simulation system should be based on formulations capable of solving as well the incident wave forces (or Froude–Krylov forces) over an instantaneous submerged body. In this work unrestrained rigid body motions of a vessel with advancing speed are considered. The dynamics of oscillatory ship motions is governed by Newton's second law, which represents the equilibrium between the internal forces due to inertia, gravity, and the external forces acting on the ship, given by:

⎡⎣ M ⎤⎦{ ξ¨} = ⎡⎣ F Dyn⎤⎦ + ⎡⎣ F Sta⎤⎦

387

SW n Fig. 1. Definition of the earth-fixed coordinate system ( X0 ) and the inertial coordinate system ( X ), and mathematical relation between these two coordinate systems.

∑ { ( Mkj + Akj )ξ¨j + Bkjξj̇ + Ckj(t )ξj} = Fkeiωet j=1

(2)

If the ship travels along a prescribed heading β at an initial steady velocity U (see Figs. 1 and 2), she will encounter the regular wave crests with a frequency of encounter, given by:

ωe = ω − kU cos β

(3)

where the surface elevation of a regular wave is given by:

ηw = ηwa cos k⎡⎣ x cos β − ( c − U cos β )t ⎤⎦

(4)

An irregular wave profile can be described by a sum of harmonic plane progressive wave components given by: N

ηw =

⎡ ω2

∑ ηwa n cos⎢⎢

n= 1

n

⎣ g

⎤ ⎛ ⎞ ω2 x cos β − ⎜ ω n − n U cos β⎟t + εn⎥ ⎥⎦ g ⎝ ⎠

(5)

where N is the number of component waves, ωn is the circular frequency, εn is the random phase angle, and ηwan is the amplitude of the n-th component waves which are given by the wave spectrum S (ω). In general, the forces acting on the ship's hull consist of control forces from rudder, and active fins, environmental forces from wind and waves and reaction forces due to ship motions. When the problem of wave-induced parametric rolling onboard an unstabilised vessel is concerned, only the forces due to wave excitation and reaction forces due to wave-induced ship motions are taken into account in this model. Therefore, other forces, such as propulsive and control forces are assumed to be canceled by each other, which means, the ship and relative course of the ship to the wave direction is kept constant during the time domain simulation for a given loading condition. Hence, the wave excitation forces consist of incident wave forces (Froude–Krylov forces), diffraction forces, and the reaction forces of restoring and radiation. The surge motion has been described by the ellipsoid method and the Hutchison and Bringloe (1978) approach, which are semi-empirical methods to estimate the added mass and the Froude–Krylov wave excitation forces. The investigation of the transverse stability performance under parametric rolling on longitudinal waves represents a complex problem. In an approximate way the radiation and wave excitation forces are calculated at the equilibrium waterline (see Fig. 3) using a standard strip theory, where the two-dimensional frequencydependent coefficients of added mass and damping are computed by Frank's close fit method, and the sectional diffraction forces are evaluated using the Haskind–Newman relations as presented by Salvesen et al. (1970). For a time domain simulation of ship motion instabilities in waves, it is necessary to predict the body motions and forces at discrete time steps. The process by which this is done in the present method is now outlined. At each time step the underwater part of the hull is calculated, together with its geometric, hydrostatic and hydrodynamic properties. The forces and moments, calculated using these instantaneous properties, are used to derive consequential translational and rotational motions. These motions are applied to the

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Fig. 2. The coordinate system and six modes of motion, and definition of the ship heading angle.

Fig. 3. Definition of hydrostatic and hydrodynamic properties by integration along the length of the hull (strip theory).

hull and the time step incremented. This process is cyclic, the previous time step providing conditions for the current time step. As shown in Eq. (2), a time-dependency is adopted to calculate the non-linear restoring coefficients in heave, roll, and pitch motions in waves, in which hydrostatic plus Froude–Krylov pressure components are utilized for calculation of significant variations on these restoring coefficients over the instantaneous waterline. As illustrated in Fig. 4, the two-dimensional hydrostatic forces and moments calculations are made using the pressure integration technique along each segment ( Cx ) of each transverse section of the ship hull, rather than using area and volume integration of the ship offsets. The original theoretical approach to the pressure

Fig. 4. Definition of geometric properties of a transverse section of the underwater part of the hull.

integration technique (considering Gauss and Green's theorems), as outlined by Witz and Patel (1984), has been adopted in conjunction with a practical method to generate the segments required to calculate the hydrostatic pressure distribution under either a regular or irregular wave profile. In this methodology, the ship geometry is modified at each time step using the motions calculated at previous time step, and it becomes necessary to redefine the hullform through out the time simulation. The successive redefinition of the hull utilizes an intersection plane technique, whereby the hullform having undergone motions in five DOF is intersected with the planes of each transverse section (predefined in the input geometry file). At each time step of the simulation, these planes are always kept perpendicular to the x-axis of the inertial coordinate system ( X ) while the ship is, in general, displaced in five modes. Hence, the intersection of the planes and the hullform will differ from that at the beginning of the simulation, and the generated strips will have an arbitrary shape possessing, eventually, no symmetry. The intersection plane technique provides the total definition of each strip below and the above undisturbed free surface. The underwater part of the hull is extracted by further taking into account either the regular or irregular wave elevation at each strip, and is defined as the part of each strip which is below the wave surface elevation. Finally, as illustrated in Fig. 4 the evaluation of sectional wave forces and moments for modes j¼ 2, 3 and implicitly 4 are calculated at each strip by contour integration of the pressure force due to the incident wave along the segments placed around the wetted section perimeter. At this point it should be mentioned that a more sophisticated wave-induced parametric rolling model could be adopted, where added masses and damping coefficients and diffraction forces/ moments would be also calculated with consideration to the instantaneous submerged hull body under the wave surface. However, the hydrodynamic component of the parametric excitation is insignificant in comparison with quasi-hydrostatic excitation caused by the incident wave potential and the wave-induced heave and pitch motion. The hydrodynamic component depends upon the overall submerged hull form, while the quasi-static hydrostatic component is strongly dependent upon wave passage and hull-shape (i.e. variations about the still waterline). For this reason and others associated with larger computational efforts only the quasi-hydrostatic component of the parametric excitation is taken into account in here. In particular the roll added inertia and radiation coefficients are taken to be linearly proportional to roll acceleration and velocity, respectively. Hence, hydrodynamic memory effect due to roll motion and consequently its effect, expressed as roll damping, is practically negligible at frequencies lower than 0.5 [rad/s] (see Fig. 5). The development of the computer code to calculate the

A. Schumacher et al. / Ocean Engineering 124 (2016) 385–403

389

Fig. 5. Added mass and damping coefficients of roll motion.

radiation, diffraction and restoring forces/moments under parametric rolling conditions in all heading waves, represented a major step. However, in order to use this numerical model for the assessment of parametric rolling with adequate levels of confidence, it is necessary also to properly address the viscous roll damping term. 2.2. Susceptibility of occurrence of parametric rolling in regular waves Despite a fully analytical approach to non-linear dynamics of ship motion and capsizing in a longitudinal seaway due to parametric rolling being highly complex, considering some simplifications such as decoupling of motions and hull symmetry one can obtain a relatively simple second order differential equation of roll:

( M44 + A 44 )ξ¨4 + B44( ξ4, ξ4̇ )ξ4̇ + C44( ξ4, t )ξ4 = F4 cos( ωet − γ4)

(6)

Fig. 6. The righting arm ( GZ ) curves.

metacentric height can be determined on the basis of the linearized righting arm up to an appropriate roll angle as follows. The linearization procedure of the metacentric heights (equivalent area under the righting arm curves) is described in detail by Ribeiro e Silva et al. (2003), and this is mathematically given by:

2 ξ42

∫0

ξ4

in which ( M44 + A44 ) are the inertia terms (structural plus added mass) with respect to the x axis; B44 ξ4, ξ4̇ is the damping term,

GM(cond) =

considering viscous and wave-making components; C44( ξ4, t ) is the restoring term, including wave-induced time dependent variations; and F4 is the roll exciting moment. The heave and pitch motions may be obtained as the solutions of the set of coupled linear equations defined as:

Since the metacentric heights vary with the relative position of the ship in waves, further considerations are required to specify this variation on the basis of a reasonable method leading to a really equivalent solution. According to Kerwin (1955), to solve this problem, another assumption is made here that the variation of the metacentric height relative to the position of the wave along the ship hull GM(wave) is approximated by a cosine function, and numerically given by Eq. (6).

(

)

( M33 + A33)ξ¨3 + B33ξ3̇ + C33ξ3 + A35ξ¨5 + B35ξ5̇ + C35ξ5 = F3 cos( ωet − γ3)

(7)

(8)

In the particular case of longitudinal waves and for reasons associated with port and starboard symmetry of the hull, there is no external excitation ( F4 = 0). Therefore, the mathematical model employed to describe non-linear roll motion is of the form:

( M44 + A 44 )ξ¨4 + B44( ξ4, ξ4̇ )ξ4̇ + ΔgGZ (ξ3, ξ4, ξ5, ηwa, λw, ξG ) = 0

(10)

GM(wave) = GM(still)⎡⎣ 1 + a 0 + a1 cos( ωet )⎤⎦

A53ξ¨3 + B53ξ 3̇ + C53ξ 3 + ( M55 + A55)ξ¨5 + B55ξ5̇ + C55ξ5 = F5 cos( ωet − γ5)

GZ (cond)dξ4

(9)

where Δg is the ship's weight, GZ (ξ3, ξ4, ξ5, ηwa , λw , ξG ) is the instantaneous righting arm lever, ηwa is the wave amplitude, λw is the wavelength, and ξG is the relative position between waves and ship's hull. Let us consider the righting arm variation with respect to the relative position of ship in waves GZ(wave) on longitudinal waves. For monohulls the GZ(wave) decreases with the wave crest amidships and increases with the wave trough amidships, in comparison with the still water righting arm GZ(still ). For example, Fig. 6 shows the righting arm curves referring to the ship on the wave crest GZ(crest ) , wave trough GZ(trough), and in still water GZ(still ). According to Hamamoto and Panjaitan (1996), when the ship is rolling in head seas at large inclination angles, the equivalent

where a0 =

GM(crest ) + GM(trough) 2GM(still )

− 1, and a1 =

(11) GM(trough) − GM(crest ) 2GM(still )

.

Therefore, the equivalent linearized righting arm curve is GZ ( ξ4, t ) = GM(wave)ξ4 . Adopting the linearisation procedure for the restoring term above and an equivalent damping term (as shown in Section 2.3), Eq. (6) can be re-written as: 2 ⎡ ξ¨4 + b44 eqξ4̇ + ω 44 1 + a 0 + a1 cos( ωet )⎤⎦ξ4 = 0 n⎣

(12)

where 2 ω 44 = n

Δ. g . GM(still) M44 + A 44

is the undamped natural roll frequency (squared), and

b44 eq =

B44 eq( ξ4, ξ4̇ )

( M44 + A 44 )

is the equivalent linearized damping coefficient. Considering a0 relatively small, and introducing the change of variable τ = ωet into Eq. (12) the resulting equation is known as the linearly damped Mathieu equation and has solutions that have a property of considerable importance in ship rolling problems. For

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A. Schumacher et al. / Ocean Engineering 124 (2016) 385–403

certain values of the encounter frequency ωe , the solution is unstable, which implies that if the roll motion described by Eq. (12) is taking place in an unstable region, the amplitude will grow until limited by a physical constraint not included in this equation.

d2ξ4 dτ 2

+ b44 eq

dξ4 + ( δ + ε cos τ )ξ4 = 0 dτ 2

(13) 2

where δ = ( ω 44n /ωe) , and ε = a1( ω 44n /ωe) . To summarize, the most comprehensive mathematical method to assess susceptibility to parametric rolling is the solution of the Mathieu equation. Obviously, some computer codes of different sophistication levels can be also used in order to check if parametric rolling occurs in numerical simulations. For each time step the roll angle can be compared to a threshold level and if two subsequent roll amplitudes (negative or positive) are higher than that threshold, the group of roll angles is marked as parametric rolling. To conclude, parametric rolling occurs when the following requirements are satisfied: – Natural period of roll is equal to approximately twice the wave encounter period. – Wave length is on the order of the ship length (between 0.8 and 2 times LPP). – Wave height exceeds a critical level. – Roll damping is low.

value was measured, i.e., either a forced-rolling test or a free-decay experiment. 2.3.1. Non-linear roll damping model The equation of roll motion may be expressed as three DOF form, including sway and yaw motions simultaneously. On the other hand, the non-linearity in the roll restoring term is clearly also important in large roll amplitude situations, and possibly for ships of unusual hull forms or with very low freeboards. However, in order to limit the discussion here to the problem of non-linear roll damping, Eq. (6) will be considered. Hence, the damping term B44 ξ4, ξ4̇ can be expressed as a series expansion of ξ4̇ and ξ4̇ and therefore the governing equation of a free decay experiment can be presented in the form:

(

)

( M44 + A 44 )ξ¨4 + B44 1ξ4̇ + B44 2ξ4̇ ξ4̇

+ C44ξ4 = 0

(14)

where B441 is the linear damping term, and B44 2 is the quadratic damping term. Then using appropriate parametric identification techniques, the terms B44 1 and B44 2, can be obtained by fitting Eq. (14) to the recorded free-decay experiment data. Most of the times because of the limited number of cycles of roll decay traces, the energy balance method is adopted. This method is based on the concept that the rate of change of the total energy in roll motion equals to the rate of energy dissipated by the roll damping due to radiated waves and mainly by non-linear viscous effects.

2.3. Roll damping models A ship that oscillates transversally on the free surface of the sea is subjected to the resistance of both the water and air. For motor vessels the air resistance is negligible with respect to that caused by the liquid medium and, therefore, only the later is considered. In general, measurements of the damping moment from experimental tests on models or actual ships to introduce in the rolling equation of the ship, is the most accurate way to proceed. However, model testing is expensive and might not be readily available at the preliminary ship design stage, so that usage of adequate numerical predictions is sometimes the only way to proceed. Usually, major difficulties in predicting roll damping of ships arise from its non-linear characteristics (due to the effect of fluid viscosity) as well as from its strong dependence on forward speed of the ship. In terms of numerical prediction techniques, Ikeda et al.'s (1978) original method and subsequent revisions for predicting the roll damping of ships (Ikeda, 2004) are the most well known and widely used. In Ikeda's method, ship roll damping is divided into five components: friction, eddy, lift, wave and bilgekeels. Then the total damping is obtained by summing up these components predicted separately. However, other methods such as Miller's (1974) method can also be applied for predicting the roll damping of ships fitted or not with bilge-keels. In this case a linear component which depends on the fineness and on the shape of the displacement volume and on the transverse metacentric height subjected to the influence of forward speed is summed up to a non-linear component dependent on the maximum amplitude of roll and which is also subjected to the presence of bilge-keels and the fineness of the displaced volume. As mentioned before, in this study the availability of an extensive series of free-decay and forced rolling experimental results on a scaled model of a containership, allowed the measurement of several roll time traces and maximum amplitudes of roll at different advance speeds. To process the data of these tests many ways of representing roll damping coefficients have been used in the past, depending on whether roll damping is expressed as a linear or non-linear form, and by what experimental method its

2.3.2. Equivalent linear roll damping model Since it is difficult to analyze strictly the non-linear Eq. (14) in the preceding section, the non-linear damping is usually replaced by a certain kind of linearized damping. According to Lloyd (1989), equivalent linearized roll damping coefficient is therefore related to the dissipated energy and is given by:

b44 eq = b44 1 +

8 b44 2ξ4 aω 44 n 3π

(15)

The most common way to obtain these non-linear damping coefficients through forced oscillation tests is, first, to find the equivalent linear coefficient b44 eq in Eq. (6) by assuming that the forced-rolling test system is subject to a linear equation, and, second, to fit Eq. (15) to the b44 eq values obtained by several test sequences with the amplitude ξ4a varied. Then values of damping coefficients b44 1 and b44 2, which are independent of the amplitude of roll oscillation, can be easily obtained. 2.3.2.1. p–q analysis of roll damping. As shown by Ikeda et al. (1977) the p–q analysis of free roll experiments can be conducted to derive a non-linear roll damping coefficient. In this case nonlinearity means the roll damping coefficient that is obtained will be dependent on maximum roll amplitude. On a p–q analysis, several data points of a time record of a certain free roll experiment are first analyzed with respect to their logarithmic decrement ( Λ), so that a large number of damping factors ( ζ ) are obtained for different roll amplitudes. As illustrated in Fig. 7, the ratio of two successive positive peaks can be used to calculate a roll damping factor, given by:

⎛ ξ4a j ⎞ ⎟⎟ = ζω44nT44d Λ = ln⎜⎜ a ⎝ ξ4 j + 2 ⎠

(16)

As shown in Fig. 8, with increasing roll amplitudes the logarithmic decrement increases and thus the roll damping coefficient also increases. For practical purposes, a regression line can be fitted through the data points, resulting in the p–q definition of the logarithmic decrement ( Λ) as a function of the roll amplitude ( ξ4a ), given by:

A. Schumacher et al. / Ocean Engineering 124 (2016) 385–403

ξ 4amax, j

391

ξ 4amax, j +1

ξ 42,aj

ξ 42,aj +1

ξ 4amin, j

ξ 4amin, j +1 Fig. 7. Typical time history of a roll decay tests.

δξ4a j + 1 =

) = qξ

Logarithmic decrement Λ [-]

Λ (ξ

a 4

a 4

+p

Log. decrement from roll maxima Log. decrement from roll minima Log. decrement from double amplitudes Linear regression line (p-q values)

p Roll amplitude ξ [°]

Fig. 8. Typical plot of decrement vs. maximum roll amplitude, and its p–q regression line.

Λ( ξ4a) = qξ4a + p

(17)

Introducing Eq. (17) into (16) and using the definition of damping factor, the following linearized roll damping term is obtained:

B44( ξ4a) =

2( qξ4a + p)( M44 + A 44 ) T44d

(18)

B441

( M44 + A 44 )

=

2ω44n B442 3 a1p2 = = a π ( M44 + A 44 ) 4 2

δξ 4a j +1 = f (ξ 4a m )

(19)

6.00E-03

If roll maxima and minima are extracted from each half cycle of a free-decay test, as shown in Fig. 7, then mean roll amplitude decrease and roll amplitude decrement per half cycle can be calculated from:

5.00E-03

ξ4a m =

ξ4a j + ξ4a j + 1 2

δξ4a j + 1 = ξ4a j − ξ4a j + 1

(22)

As shown by Mathisen and Price (1984), the undamped natural frequency ( ω44n ) may be taken as the mean frequency of the decay record. Then in order to determine the three unknowns in Eqs. (20)–(22), ξ4a m , B44 1 and B44 2, at least three roll maxima or minima must be available from a decay test to provide estimates for these three parameters. A much larger number of maxima and minima will usually be available in practice, and some form of curve fitting technique is appropriate to minimize the effects of random error. Some care may be necessary to ensure that effects of experimental errors are avoided. For example, a least-squares technique of the full decay record has been tried initially, but occasionally resulted in a physically unrealistic value of the linear damping coefficient. In such cases, the largest deviating data points are eliminated and then a new least-square analysis of the truncated record is conducted. Notice should be given to the fact that this regressive analysis of the roll energy as illustrated in Fig. 9, allows the inclusion of the theoretical point (0,0), which has not be measured during the freedecay experiments. The value of the inertia ( M44 + A44 ) is also required to convert the damping terms to the normal form of the damping coefficients b44 1 and b44 2. The inertia may be estimated from the natural

2.3.2.2. Roll damping energy methods. Dividing the roll Eq. (14) by the inertia term ( M44 + A44 ) it results in: 2 ξ¨4 + p1ξ4̇ + p2 ξ4̇ ξ4̇ + ω44 nξ4 = 0

(21)

where the Krylov–Bogoliubov's solution of the linear and quadratic coefficients damping coefficients p1 and p2, see for example Flower and Sabati Aljaff (1980), are given by:

p1 =

q

4 π p ξ4a m + p2 ξ4a m2 = a1ξ4a m + a2ξ4a m2 2ω44n 1 3

y = 0.3535x 2 + 0.0289x R² = 0.9152

4.00E-03 3.00E-03 2.00E-03

(20)

By integrating the roll equation over half period and equating the energy dissipated by damping to the work done by the restoring moment, the following expression for roll decrement as a function of the mean roll amplitude is obtained:

1.00E-03 0.00E+00 0

0.02

0.04

(

0.06

)

0.08

Fig. 9. Typical plot of δξ4a j + 1 = f ξ4a m during a roll decay test.

ξ4am

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frequency and the restoring term ( C44), which is determined from the transverse metacentric height GMT and displacement weight ( Δ. g ) of the vessel. From previous studies, including Ribeiro e Silva and Guedes Soares (2000), it is known that when roll damping is tuned to model test results, a very good correlation of the roll motion can be achieved between model tests and the numerical analyses. Particularly, the magnitude of the roll response during parametric rolling is dictated in large part by the amount of viscous damping in the roll DOF, where damping improves the stability character of the system by reducing the instability regions in size. To account for these damping effects in this study, it was decided to establish an empirical formula directly through the statistical analysis of the energy method results on free roll decay model tests made on particular ship form of a containership.

(

)

3. Experimental model study The model tests were carried out in HSVA's large towing tank, which measures 300  18  5.6 [m] in length, width and water depth. The towing tank is equipped with a double flap wave generator at one end of the tank. A beach at the other end damps the waves. 3.1. Model tests for the Investigation of parametric rolling A completely computer controlled seakeeping model test technique which is combined with deterministic wave generation techniques was used for the model tests. By means of this technique the model's pitch, roll, yaw and heave motion, the model position in the tank, the propeller rpm, the ruder angle, the global ship loads and the undisturbed water level elevation at the position of the model could be determined. A scaled model of the container vessel was used here, and her principal particulars and one investigated load case as well as body lines are presented in Table 1 and Fig. 10, respectively (Schumacher, 2008a, 2010). The hull is vertical sided amidships above the waterline and presents large beam variations at aft and fore ends. The ship model of wooden construction was manufactured on a scale 1:24. The free-running radio controlled ship model was propelled by an electric motor with a constant revolution control system. As shown in Fig. 11, the superstructures and container stacks are simplified modeled and the hull is appended with three pairs of bilge-keels, and a rudder which is controlled by an auto-pilot. Surge, sway, heave, roll, pitch, and yaw were measured by an optical tracking system fixed to the moving carriage. The main pieces of equipment of the test technique are the master computer, the wave generator computer, the optical system measuring the model motion in six degrees of freedom, the measuring system recording the forces at midships section as well as the radio Table 1 Main particulars of containership. Principal particular

Value

Length between perpendiculars (LPP), in [m] Breadth, design waterline (BDWL), in [m] Depth at main deck (D), in [m] Draught at amidships (T), in [m] Displacement, design waterline (Δ), in [ton] Block coefficient (CB) Transverse metacentric height, still water (GMT), in [m] Natural roll period, linearized in waves (T44), in [s] Cruise speed (U), in [knots]

117.6 20.2 10.9 8.1 12811 0.6496 0.5 21.4 16.5

controlled ship model with propelling motor and steering engine. All computer systems and measurement systems are synchronized with each other. Prior to the model tests extensive numerical computations with adequate methods were performed in order to predict conditions (load case, ship speed, wave height, wavelength, etc.) in regular and irregular waves at which parametric rolling might occur. The obtained results were used to select the most promising test conditions and to compile a suitable test program. A test campaign of four days was performed with the container vessel model. As shown in Table 2, altogether 68 test runs in regular waves and irregular waves were carried out providing useful data for validation purpose. All test runs were recorded on video. In advance and during the model tests a staff member of the Technical University of Berlin provided the control signals for the generation of the required regular waves, irregular seaways and the user-defined wave sequences. Members of the IST attended the tests. First tests at different load conditions and advance speeds in head and following regular waves with varying wavelengths and wave heights were conducted in order to identify the conditions leading to large roll motions caused by oscillation of the righting levers in waves. Thereafter the model was tested in wave sequences of long-crested irregular head and following seas, which should be critical regarding parametric rolling. During the model tests the container vessel was excited to large roll motion (up to 30° roll amplitude), but no capsizing occurred. Notice should be given to the fact that no phenomena such as loss of stability at wave crest or broaching could be clearly observed, but the occurrence of parametrically excited roll motion was extensively investigated. Therefore, a significant amount of data for the validation of the numerical model was successfully obtained. After the model tests the measured data were analyzed and adequate time series and spectra data were prepared for all test runs. These data were delivered to IST for further analysis and direct comparison with numerical predictions. 3.2. Model tests for the Investigation of roll damping Prior to parametric rolling tests, forced rolling and free-decay model tests were conducted at different speeds in calm water to assess the ship's roll damping. 3.2.1. Forced roll motion tests in calm water The forced roll motion tests (Schumacher, 2008b) were performed with the free-running self-propelled model of the container vessel at a GMT = 0.65 [m]. Ship speeds of 0, 7, 12 and 16 knots were investigated. The ship model was equipped with a special apparatus which excites the model to harmonic roll motions by two contra-rotating arms without or with fitted weights. The total exciting moment amplitude is applied if both arms are situated one upon the other at the ship's sides. By varying the exciting frequency and the exciting moment nondimensional coefficients of the effective roll damping for different resonance roll angle amplitudes were obtained (Blume, 1979). The outcome of the performed test series plots shows the nondimensional roll damping coefficients depending on ship speed (0–16 knots) and resonance roll angle amplitude (between 5° and 20°). 3.2.2. Roll decay tests in calm water Roll decay tests at 0, 7, 12 and 16 knots with initial heeling angles of 5°, 10° and 15° each were conducted with free-running self-propelled model of the container vessel (Schumacher, 2010).

A. Schumacher et al. / Ocean Engineering 124 (2016) 385–403

393

Fig. 10. Body lines of the containership.

3.2.3. Analysis of roll damping results The free-decay experiments at 0, 7, 12 and 16 knots (Schumacher, 2010) imposing the largest heeling angle of 15° were first analyzed according to methodology described in Section 2.3.1. The obtained regression lines and damping ratios as a function of roll amplitudes are presented in Fig. 12. As expected, Fig. 12 shows that the vessel's roll damping increases with growing roll angles and ship's advance speed. Apart from the p–q analysis to estimate the total roll-damping coefficient, the energy method analysis of free-decay tests have been conducted as well to estimate linear and quadratic roll damping coefficients at different maximum amplitudes of roll and advance speeds. Fig. 13 shows the regression analysis of the linear and quadratic roll damping terms from roll decay tests at 0, 7, 12 and 16 [knots] with the largest heeling angle of 15°. Finally, from the statistical analysis of all the free decay tests at each advance speed, it was found that a first order polynomial equation can also be adopted to determine the quadratic and linear roll-damping coefficients as function of maximum amplitude of roll. Table 3 shows a comparison between roll damping terms ( B44 ) results obtained from free-decay and forced oscillation tests on the scaled model of containership. These results are presented for different ship's advance speeds and maximum roll amplitudes (each row), where: a) B44 for free-decay tests (3rd column) is obtained from Eq. (18) on the p–q analysis method. b) B44 for forced-rolling tests (4th column) is obtained from the amplitude of the resonant, the static and the dynamic heel angles, where, the resonant heel angle were taken from the measured resonance curves; the static heel angle is the heel

Table 2 Number of the usable test runs of the containership campaign stated for the different waves and sea conditions. Wave/Sea conditions

Tested conditions

Number of tests

Regular head waves

2 Load cases, 7 wave conditions, different speeds Regular following 2 Load cases, 6 wave conditions, waves different speeds Irregular head seas 1 Load case, 10 seaway conditions, different speeds Irregular following seas 1 Load case, 3 seaway conditions, different speeds

22 23 13 10

Containership - Roll Decay Analysis / GM = 0.65m / Tnatural roll period = 20.2s 1.4 1.3 1.2

Logarithmic decrement Λ [-]

The ship model was studied at nearly the same load condition as already investigated during the forced roll motion tests. For the adjustment of the desired initial heeling angles different weights were placed on the ship's side. During the tests the ship model was speeded up to the desired ship speed and the weight was quickly lifted from the side. Then the model's roll motions could freely decay. The measured roll motions were evaluated according to the p–q analysis as described in Section 2.3.1 and the non-linear roll damping coefficients were derived.

Λ = 0.0513ξ

1.1

+ 0.5916

1.0 0.9 0.8 0.7 0.6

Λ = 0.0221ξ

+ 0.4031

0.5 0.4 0.3

Λ = 0.0143ξ

+ 0.243

Λ = 0.0081ξ

+ 0.0532

0.2 0.1 0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

Roll amplitude ξ [°] 0 kts

7 kts

12 kts

16 kts

Fig. 12. Containership results of total roll-damping coefficient from p–q regression analysis.

angle of the model due to the amplitude of the exciting moment applied; and the dynamic heel angle is the static heel angle plus the angle resulting from the centrifugal forces acting on the model due to the rotating masses if the roll axis and the vertical center of the rotating masses are not at the same height; c) B44 1, B44 2 and B44 eq for free-decay tests (6–8th columns) were obtained from Eq. (22) on energy method and Eq. (15), respectively.

Fig. 11. A photograph of the scaled model of containership.

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2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8

1st order

Fig. 13. Containership results of regression analysis on linear and quadratic roll damping coefficients obtained from energy method analysis of free-decay experiments.

4th order

0.6

Containership (ABS) Containership (linearised)

As can be observed in Table 3, a large difference up to 48% on equivalent roll damping occurs between free-decay and forced oscillation test results, being the free-decay larger than the forced oscillation roll damping in almost all the tests performed. Aiming to investigate the possible causes of such large differences, it was concluded that no further checks were possible, since only resonant curves were available and, therefore, a Fourier analysis of time records of forced excitation tests could not be performed to determine the linear and quadratic roll damping terms (see Mathisen and Price (1984) for further details). Hence, linear and linear plus quadratic roll damping terms, as explained in c), were later utilized to perform two distinct types of numerical simulations shown next.

4. Comparison of numerical simulations with experimental results 4.1. Regions of instability For a specific loading condition, low cycle parametric rolling occurs when the four requirements presented at the end of Section 2.1 are satisfied. In respect to stability variations, as shown in Fig. 14 these were calculated based on either still water or linearized values of GMT (according to the linearization procedure defined by Ribeiro e Silva et al., 2005). For the containership, the requirements for low cycle parametric rolling to occur lead to a wavelength of 163.5 [m] and a ship's roll natural period of 21 [s], which then results on a wave frequency of 0.614 [rad/s] and the forward speed of 2.4 [knots]. From the Intact Stability Booklet of

0.4 0.2 0

-1.6 -1.4 -1.2

-1

-0.8 -0.6 -0.4 -0.2

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6

Fig. 14. Instability zone of the Mathieu equation (Ince–Strutt chart) for the in2 vestigated containership model, where according to Eq. (13) δ = ( ω 44n/ωe) and 2 ε = a1( ω 44n/ωe) . Solid and dashed lines indicate the damped boundaries of stability for the 1st and 4th order Taylor's expansion cases.

this ship the loading condition referred above for the containership has been confirmed as viable and the forward speed of 2.4 knots fits into the operational range of speeds. As shown in Fig. 14, these operational conditions also fall into a Mathieu type zone of instability so that critical wave-induced parametric rolling conditions could be identified prior to model tests. Fig. 14 also illustrates how much different can be the assessment of stability variations in longitudinal waves depending on the linearization procedure that has been adopted by different authors. Note that the circumference corresponds to ABS (2004) while the black dashed circle corresponds to a more conservative procedure described in Section 2.2. 4.2. Regular waves tests As referred before, the computer code is a non-linear time domain ship motion simulation program. Since this computer code is a full six DOF model it includes most of the couplings between the individual modes of motion. Both non-linearity of the excitation forces and coupling between the six motions are required to

Table 3 Comparison between roll damping coefficients ( B44 ) results, in [N m s], obtained from free-decay and forced oscillation tests on a scaled model of containership. Speed (U)

Roll (ξ4a)

[knots]

[deg]

Free-decay

6

Forced-oscill.

6

Var.

Energy method at free-decay tests

[%]

B44eq

B441

B442

7

7

0

5 10 15

8.04  10 1.19  107 1.51  107

5.27  10 1.16  107 1.99  107

34.53 2.69  31.91

6.07  10 2.94  107 3.84  107

6.07  10 2.93  107 3.82  107

1.85  107 2.06  107 1.43  107

7

5 10 15

2.79  107 3.38  107 3.91  107

1.45  107 2.28  107 3.21  107

48.01 32.54 17.89

4.75  107 4.96  107 2.98  107

4.73  107 4.92  107 2.91  107

6.55  107 6.59  107 7.46  107

12

5 10 15

4.65  107 5.23  107 6.23  107

2.61  107 3.65  107 4.06  107

43.82 30.20 34.82

1.84  108 1.26  108 8.42  107

1.84  108 1.25  108 8.32  107

8.41  107 9.51  107 1.01  108

16

5 10 15

7.52  107 9.75  107 1.23  108

5.43  107 5.92  107 6.60  107

27.77 39.34 46.16

3.55  108 2.18  108 1.23  108

3.55  108 2.17  108 1.22  108

1.29  108 1.31  108 1.44  108

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be able to predict parametric rolling. The code models a ship as a free running vessel in waves, comparable only to some extent to a free sailing model in the seakeeping basin. In this section, only the loading condition presented in Table 1, which corresponds to the roll natural period of 21 [s] and a GMT = 0.5 [m] is analyzed and the numerical predictions thoroughly compared and contrasted against experimental results. Notice same predictions as experimental results (no parametric rolling if roll angle is lower than 10°) in regular following and head waves were obtained when other ship's loading conditions corresponding to GMT = 0.82 and 1.25 [m], respectively, were considered, and therefore these results are not further discussed in here. Plotted side-by-side in the figures shown below are time traces of the wave surface elevation (at the origin), surge, sway, heave, roll, pitch, and yaw motions during low-cycle parametric resonance. The same scales for abscissas and ordinates axis have been adopted as much as possible. In respect to surge, sway, and

395

yaw modes, these should not be directly compared to experimental results, since the numerical model has no maneuverability capabilities embedded yet, and therefore, forces and moments exerted by the rudder and propulsion system of the model in order to maintain course cannot be taken into account during simulations. 4.2.1. Head waves For comparisons, the same critical speed, and wave conditions were used in the simulations as in the model tests (see Schumacher (2009)). Experimental and numerical results for a linearized and a linear plus quadratic roll damping models, assuming the same encountering conditions as those of Test nr. 5 (regular head waves, Hw ¼6 [m], Tw ¼10.23 [s], β ¼180° and U¼ 2.4 [knots]) are presented on Figs. 15 and 16, respectively. In Fig. 17 the numerical results of containership are presented for the same waves encountering scenario with a fully automatic procedure to

Fig. 15. Parametric rolling in regular head waves (Hw ¼6 [m] and Tw ¼ 10.23 [s]) on containership – time history of the Test nr. 5 obtained from experimental measurements (left side) and numerical simulations with linearized damping model (right side).

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Fig. 16. Parametric rolling in regular head waves (Hw ¼ 6 [m] and Tw ¼ 10.23 [s]) on containership – time history of the Test nr. 5 obtained from numerical simulations with a linear damping model (left side) and a linear plus quadratic damping model (right side).

Fig. 17. Parametric rolling in regular head waves (Hw ¼6 [m] and Tw ¼ 10.23 [s]) on containership – time history of the Test nr. 5 obtained from experimental measurements (left side) and iterative numerical simulations with a linear plus quadratic damping model (right side).

evaluate initial sinkage and trim on a wave; and a fully automatic iterative calculation of roll damping coefficients. In this case, initial sinkage and trim are obtained from heave and pitch amplitude and phase frequency domain transfer functions for a certain wave amplitude at the starting time instant (t¼0). Roll damping coefficients are obtained from a fully automatic iterative procedure, which takes into account the exact value of the maximum amplitude of roll given by the simulation until convergence is attained, while before maximum roll angle parameter was manually introduced in the input file. In these comparisons between experiments and simulations, due to practical reasons, the initial time instant of the experimental records has been truncated up to the instant (238 [s]) where the first regular wave with the predefined wave amplitude of 3 [m] arrived at the origin. As can be seen in Fig. 15, both measured and simulated wave amplitudes are practically constant. Therefore, as a result of this uniformity onto the excitation wave amplitudes, the vertical plane responses on surge, heave and pitch modes are also constant during the stationary phase. Measured and predicted heave motion is about 1.3 and 1.4 [m], and pitch is about 5 and 4°. On the transverse plane, although constant wave amplitude condition holds true, the time traces show increasing sway, roll and yaw

amplitudes up to their maximum amplitude at the end of the transient phase. Hence, the containership sailing at an advance speed of 2.4 knots encountering head waves of about 1.4 times her own length and a frequency twice her natural roll frequency will experience a wave-induced parametric rolling situation. As it can be observed, maximum measured and predicted steady roll amplitude is 29.8 and 24°, respectively. As can be seen in Fig. 16, there are no significant changes in the heave, roll, and pitch predicted responses between a linear damping and a linear plus quadratic damping model. However, as shown in Fig. 17, if an automatic iterative simulation is now adopted for calculation of initial sinkage and trim and roll damping coefficients, then both time traces of heave and pitch become closer to corresponding experimental traces, and the maximum roll numerical responses of 26° will be also closer to experimental result. At this stage let us focus our attention at the transient phase where measured and simulated wave amplitudes shown in Fig. 15 are practically constant from 238 to 588 and from 0 to 350 s, respectively, and, despite this uniformity onto the excitation wave amplitudes, the vertical plane responses on heave and pitch modes are showing some fluctuations due to strong coupling effects. Actually, heave and pitch motions for wave periods of 10.23 s

A. Schumacher et al. / Ocean Engineering 124 (2016) 385–403

397

Num. Lw/Lpp = 1.00

1.4

Exp. Lw/Lpp = 1.00

35.0

Num. Lw/Lpp = 1.39

1.2

Exp. Lw/Lpp = 1.39

30.0

Num. LwoLpp=1.57

Roll angle amplitude

25.0

Exp. LwoLpp=1.57

1

20.0

0.8 15.0 10.0

0.6 1st order

5.0

4th order

0.4

Num. Lambda/Lpp = 1.39

0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Exp. Lambda/Lpp = 1.39

16.0

Num. Lambda/Lpp = 1.0

Ship's advance speed

0.2

Exp. Lambda/Lpp = 1.0

Fig. 18. Comparison between numerical and experimental maximum roll amplitudes of the containership under parametric rolling conditions in regular head waves (Hw ¼ 6 [m]) for wavelength to ship's length ratios of 1.0, 1.39 and 1.57.

are strongly coupled so that heave motions due to wave passage are influenced by pitch and vice versa. On the transverse plane, although constant wave amplitude condition holds true, both experimental and numerical time traces show increasing roll amplitudes up to their maximum steady amplitude of 29.8 and 26° between time instants 238 and 388 and 0 and 150 s, respectively. As it can be observed from Fig. 17, this particular behavior during the transient phase is in fact due to significant transfers of energy from heave and pitch modes to roll motion at resonance, which are more easily noticed in the experiments. Fig. 18, presents a comparison between maximum roll numerical responses and experimental results of the containership sailing at different advance speeds ranging from 0 to 14 knots in regular head waves with a constant wave height of 6 m. As shown in Fig. 18, there are relatively small deviations between the experimental and numerical curves of steady roll amplitude vs. wave encounter frequency for wavelength to ship's length ratios 1.39 and 1.57. For the wavelength to ship's length ratio of 1.0, the deviations between numerical and experimental roll amplitudes are larger but again the numerical predictions of the wave encounter frequencies at which parametric rolling occurs are in reasonable good agreement with experimental results. In order to check the validity of utilizing the instability zones of the Mathieu equation for the preliminary investigation ship parametric rolling in regular head waves both the experimental results and numerical predictions were plotted in the Ince–Strutt diagram. Fig. 19, shows the solid and dashed lines indicate the damped boundaries of stability for the 1st and 4th order Taylor's expansion cases. Solid markers correspond to experiments where parametric rolling was observed and hollow markers correspond to numerical predictions of parametric rolling in head waves for wavelength to ship's length ratios 1.0 and 1.39. As it can be seen in Fig. 19, both the experimental results and numerical predictions are well contained within the boundaries of instability zones of the Mathieu equation so that it can be concluded that utilization of the Ince–Strutt diagram is a rather conservative approach to the assessment of susceptibility of occurrence of parametric rolling in regular head waves. 4.2.2. Following waves Experimental and numerical results for a linearized and a linear plus quadratic roll damping models, assuming the same encountering conditions as those of Test nr. 50 (regular following waves, Hw ¼ 4.7 [m], Tw ¼9.51 [s], β ¼0° and U¼4.7 [knots]) are presented on Figs. 20 and 21, respectively. From Fig. 19 it can be seen that once a stationary phase is reached, maximum measured

0 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

Fig. 19. Ince–Strutt chart for the investigated containership model. Solid markers correspond to experiments where parametric rolling was observed and hollow markers correspond to numerical predictions of parametric rolling in head waves for wavelength to ship's length ratios 1.0 and 1.39.

and predicted roll motion is 30.2 and 27.8°, respectively. Measured and predicted heave motion is about 1.0 and 0.8 [m], and pitch is about 3.0 and 2.9°. Hence, the containership sailing at an advance speed of 4.7 knots encountering following waves of about 1.2 times her own length and a frequency twice her natural roll frequency will experience a wave-induced parametric rolling situation. Similarly to head waves, roll numerical responses in following waves are practically the same either for a linear damping or a linear plus quadratic damping model. However, as shown Fig. 21, if more appropriate amplitudes and phase lags of heave and pitch, are now adopted for calculation of initial sinkage and trim, and a more precise iterative procedure is implemented for roll damping coefficients, the same trend holds true as in the head waves simulations, i.e., the maximum steady roll prediction will be closer to experimental result. As it can be observed from Fig. 21, during the transient phase, significant transfers of energy from heave and pitch modes to roll motion occurs at resonance, which again are more easily noticed in the experiments. Fig. 22, presents a comparison between maximum roll numerical responses and experimental results of the containership sailing at different advance speeds ranging from 0 to 10 knots in regular following waves with a wave height of 4.4, 4.5 and 4.7 [m] for wavelength to ship's length ratios of 1.0, 0.8 and 1.2, respectively. As shown in Fig. 22, although there are relatively large deviations between the experimental and numerical curves of steady roll amplitude vs. wave encounter frequency for wavelength to ship's length ratios 0.8 and 1.0, the numerical predictions of the wave encounter frequencies at which parametric rolling occurs are in a reasonable good agreement with experimental results. Contrarily, for the wavelength to ship's length ratio of 1.2, although the deviations between numerical and experimental roll amplitudes are smaller, the numerical predictions of the wave encounter frequencies at which parametric rolling occurs are in less agreement with experimental results. In particular, it can be observed that experimental onset of parametric rolling is terminated at an advance speed of about 7.3 knots while numerical predictions of parametric rolling are extended to advance speeds of about 11 knots. Finally, increasing of the level of severity of parametric rolling phenomenon on wave height is more easily detected in the following waves scenario where larger roll amplitudes of 19.1, 26.8 and 30.2° have been measured for the containership sailing at an

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A. Schumacher et al. / Ocean Engineering 124 (2016) 385–403

Fig. 20. Parametric rolling in regular following waves (Hw ¼ 4.7 [m] and Tw ¼9.51 [s]) on containership – time history of the Test nr. 50 obtained from experimental measurements (left side) and numerical simulations with linearized damping model (right side).

advance speed of 4.6 knots in wave heights of 4.4, 4.5 and 4.7 [m]. As it can be seen in Fig. 22, numerical predictions of the level of severity of parametric rolling phenomenon on wave height are also in a good agreement with these experimental results. Nevertheless, Neves et al. (2009) and Spanos and Papanikoloau (2009) highlighted the fact that parametric rolling may entirely disappear under certain initial conditions if the wave height exceeds a certain limit, which could not be experimentally confirmed this time due to limited amount data points available (i.e., large enough waves to cause significant change of the underwater part of the hull and the corresponding instantaneous roll restoring coefficients).

4.3. Irregular waves tests For the simulations of induced ship motions in irregular waves, distributed amplitude and phase components should be defined at the fixed coordinate system ( X0 ) to generate an incident wave realization that can be utilized at the computer code. However, the experimental wave components of the seaway were not readily available. Instead, only the wave components of the encounter spectrum, referred to the inertial coordinate system ( X ), were available. Hence, by applying a Direct Fourier Transform (DFT) analysis of the time record of encountered wave elevation it was possible to identify the largest wave harmonic components. As shown in Fig. 23, DFT analysis of data from Test nr. 33, revealed some small differences between measured and estimated space and time realizations of the wave elevation. However,

A. Schumacher et al. / Ocean Engineering 124 (2016) 385–403

399

Fig. 21. Parametric rolling in regular following waves (Hw ¼ 4.7 [m] and Tw ¼ 9.51 [s]) on containership – time history of the Test nr. 50 obtained from experimental measurements (left side) and iterative numerical simulations with a linear plus quadratic damping model (right side). Num. Lw/Lpp = 0.80 Exp. Lw/Lpp = 0.80

40.0

Num. Lw/Lpp = 1.00 35.0

Exp. Lw/Lpp = 1.00 Num. LwoLpp=1.20

30.0

Roll angle amplitude

Exp. LwoLpp=1.20 25.0

20.0 15.0 10.0 5.0 0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

Ship's advance speed

Fig. 22. Comparison between numerical and experimental maximum roll amplitudes of the containership under parametric rolling conditions in regular following waves (Hw ¼ 4.4, 4.5 and 4.7 [m] for wavelength to ship's length ratios of 0.8, 1.0 and 1.2, respectively).

despite these differences on the time histories their envelopes are in fact quite similar, and therefore the prediction is considered effective for 40 largest harmonic wave components referred to the inertial coordinate system ( X ). Then assuming a constant advance speed of 5.3 [knots] and a linear wave propagation model the measured and the estimated wave elevation time histories should be translated to the fixed coordinate system ( X0 ) so that numerical simulation of wave ζ (m

)

5.0

induced ship motions can be performed. As a result of differences shown in Fig. 24 for the incident wave conditions, some discrepancies between predictions and experimental results of ship motions induced by irregular head waves would be in principle anticipated. Since, despite simulation of the ship motions is performed for a ship heading of 180° and a constant advance speed condition where added mass and damping coefficients and diffraction forces are calculated for the corresponding values of peak spectral frequency and significant wave height, the Froude–Krylov forces are obtained from instantaneous surface elevation profile. Hence, the presence of group waves in an ensemble of irregular waves should be captured by the numerical model, and, therefore, it was decided to check on the occurrence of parametric rolling onboard containership in irregular head waves using the code. Experimental and numerical results for a linear plus quadratic roll damping model assuming the same encountering conditions as those of Test nr. 33 (Hs ¼6 m, Tp ¼10.23 [s], β ¼180° and U¼5.3 [knots]) are shown in Fig. 25. Here parametric rolling condition leads to the maximum roll amplitude of 12.2°, which is much smaller than that in regular waves. However, it should be mentioned that ship's advance speeds are quite different on these two tests. Similarly to head waves, also in following seas the same discrepancies have been detected between measured and estimated realizations of waves wave elevation, which aggravated again a directly comparison of the experimental and numerical results of Test nr. 56. Nevertheless, predicted and experimental results

Water surface elevation during test HSVA nr. 33 (exp.) Water surface elevation during test HSVA nr. 33 (num.)

4.0 3.0 2.0 1.0 0.0 -1.0

400

450

500

550

600

650

700

750

800

-2.0 -3.0 -4.0

t (s )

Fig. 23. Comparison between the measured and the estimated wave elevation time histories on Test nr. 33, referred to the inertial coordinate system ( X ).

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ζ (m )

5.0

Water surface elevation during test HSVA nr. 33 (exp.)

4.0

Water surface elevation during test HSVA nr. 33 (num.)

3.0 2.0 1.0 0.0 -1.0

400

450

500

550

600

650

700

750

800

-2.0 -3.0 -4.0

t (s )

Fig. 24. Comparison between the measured and the estimated wave elevation time histories on Test nr. 33, referred to the fixed coordinate system ( X0 ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

showed that the vessel may become subjected as well to heavy parametric rolling in irregular following waves, where maximum

steady roll amplitude of about 30° is attained. In this case, the maximum roll amplitude is practically the same as that obtained

Fig. 25. Parametric rolling in irregular head waves (Hs ¼6 [m] and Tp ¼ 10.23 [s]) on containership – time history of the Test nr. 33 obtained from experimental measurements (left side) and numerical simulations with linear plus quadratic damping model (right side).

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in regular waves for an identical speed. It is interesting to point out that a reasonable match was achieved between the predicted and experimental parametric rolling results in irregular waves despite some initial concerns about small discrepancies that have been detected between measured and identified realizations of waves elevation. In fact, it should be recalled that in previous studies (Ribeiro e Silva et al., 2005) where irregular waves were randomly generated by a computer seed number, the required synchronisation between waves and roll response period could not be sustained long enough due to variations of the wave realization in time and space. Moreover, since parametric rolling is a resonant non-linear phenomenon, it has been found that contrary to the regular wave's scenario, in computer generated irregular waves ships might not be prone to regularly exhibit the parametric rolling due to waves groupiness effect. Therefore, to evaluate the occurrence of parametric rolling in randomly generated irregular waves using a computer code, a new approach that goes beyond the standard linear seakeeping is required, where it is desirable to obtain not only a temporal average from a single run but also an ensemble average from multiple runs to alleviate the lack of randomness in the numerically simulated process for long durations.

5. Systematic predictions of parametric rolling The previous section has presented a limited sample of comparisons of numerical and experimental results, which were used to validate the present code. Having a validated code, it can then be used to predict the various conditions in which a vessel may be subjected to parametric rolling, which is a valuable information that can be used by ship masters to know when they will be close to situations liable of parametric roll. One useful way to show the results is to represent the likelihood of parametric roll in a polar plot depending on ship speed and heading. Such a polar plot can be calculated and simply integrated in decision support systems for fishing vessels (Rodrigues et al., 2012) or for general transportation vessels (Perera et al., 2012). A global user interface (GUI) has been developed by Uzunoglu (2011) to automatically prepare the required input files (with adequate initial condition sets) and iteratively run several simulations in order to produce systematic calculations leading, for instance, to polar diagrams on the risk of occurrence of parametric rolling. This GUI automatically calls the computer program and then saves all the output files on a well defined and ordered format. To check parametric rolling responses of the containership, simulations were made for the ship operating at various speeds from 0 to 16 knots and headings between 0–45° and 135–180° encountering regular waves with wave length to ship's length ratios of 1.4 and wave height of 6 m. Fig. 24 shows an operational polar diagram of the containership, where roll motions exceeding 20° do have shaded regions colored in “hot” and regions below threshold have “cold” colors. The diagram is created from a series of computer simulations from 0 to 16 knots in 0.5-knot increments at 2° heading increments. All speed/heading combinations inside the “hot” shaded regions exceed a 20° maximum roll angle during the 1200-s simulation. The regions of speeds between 0 and 9 knots and pure head waves (β ¼180°) correspond to a critical condition where the encounter frequency are strictly low-cycle parametric roll-induced motions. The maximum roll angle for this series of simulations is about 27.5° in the head waves 4.5-knot case. In following waves (β ¼000°), the maximum roll angle for this series of simulations is about 19.3° on 0-knot case, and also correspond to a critical condition where the encounter frequency is strictly low-cycle

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parametric roll-induced motions. While no experiments were performed in oblique waves, it should be noted that according to predictions, parametric rolling is still possible at these headings. This type of diagram can be very useful in helping the shipmaster avoid the occurrence of parametric rolling while the ship is operating in severe sea conditions. From Fig. 25 it can be seen that also in irregular waves the computer code is able to reproduce experimental results on parametric rolling providing the largest wave harmonic components of the encounter spectrum are identified applying a DFT analysis. The program gives also time traces of roll motion which can be used to determine if the vessel suffered from parametric rolling in that particular condition. The vessel could indeed be in parametric rolling conditions only for a small period of time, which nevertheless might be sufficient to cause large damage to the vessel. Numerical simulations with non-linear time domain computer codes can be used as an instrument of parametric rolling risk assessment. In this case a probability of exceed a given values could be determined. Then for parametric rolling avoidance, written information such as polar diagrams as the one shown in Fig. 26, could be generated for sea state defined with significant wave height and peak period, which nevertheless are significant only for a specific design point. Hence, this methodology has two major disadvantages which would result on a very large number of simulations to be carried out in advance, and several polar plots to be utilized by ship's operators onboard: – Specificity and variability of environmental factors, and ship's loading conditions. – Practical nonstationarity of non-linear parametric rolling, which means that several statistically independent wave records have to be used for numerical simulations assuming different initial phases.

Fig. 26. Polar diagram of predicted roll motion. containership sailing at speeds 0– 16 [knots] (¼ radial distance 0–100%) while encountering regular waves with wave length to ship's length ratios of 1.4 and wave height of 6 m with headings of stern to quartering (angles 0–45°) and following to bow (135–180°) waves. Vertical bar on the right hand side indicates level of severity of parametric rolling, where roll motions exceeding 20° have “hot” colors and regions below this threshold have “cold” colors. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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6. Conclusion In this study a non-linear numerical model in six DOF is presented and utilized to simulate parametric resonance in both regular and irregular waves encountered at arbitrary headings. With this time-domain numerical model, coupled heave, roll, and pitch responses in long-crested irregular waves can be simulated. In general, good agreement is found between the time domain simulation of ship's responses in a longitudinal seaway and existing experimental data. In addition, special attention has been given to the usefulness of model experiments conducted under conditions as realistic as possible to validate the theoretical approach proposed. It has been established that a statically stable ship encountering waves of about her own length and a frequency twice her natural roll frequency will experience a wave-induced parametric rolling situation. Therefore, under these conditions, roll angles exceeding 30° to each side can rapidly be produced, resulting sometimes in cargo loss, ship damage, and eventually in capsize. When roll damping is tuned to model test results, a very good correlation of the roll motion can be achieved between model tests and the numerical analyses. Moreover, in this study it was observed that linear and quadratic damping moments do lead in general to more accurate predictions than the linearized damping moment. Also important to improve accuracy of the numerical predictions, is to select appropriate initial values of sinkage and trim in waves and adopt an iterative procedure to estimate roll damping. In irregular waves the occurrence of parametric rolling can be highly dependent on the time and space realization of the water elevation profile and a large number of simulations can be required to obtain a parametric resonance condition with roll angles exceeding 20°. In this investigation and other studies it has been demonstrated that parametric rolling in head or following seas can occur when large transverse stability variations (driven by wave characteristics, coupled heave and pitch responses, and hull form parameters such as hull flare, end sections shape, and main deck position) are combined with low damping (directly dependent on ship's speed). Therefore, in addition to special checks on hull forms at a preliminary design stage, the installation of a decision support system for ship operation in rough weather is highly favored. The system can make use of polar diagrams for the ship operating at various speeds and headings for different sea states and ship's loading conditions, where this type of diagram can be very useful in helping the shipmaster avoid the occurrence of parametric rolling while the ship is operating in severe sea conditions.

Acknowledgments This work was carried out within the European project “HANDLING WAVES – Decision Support System for Ship Operation in Rough Weather” (www.mar.ist.utl.pt/handlingwaves/), which was partly funded by the European Union through the Growth program under contract TST5-CT-2006-031489.

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