Ocean Engineering 67 (2013) 45-57
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Slam induced loads on bow-flared sections with various roll angles
( | f ) c r o s s M a x kShan Wang, C. Guedes Soares*
Centre for Marine Teciinology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Lisboa, Portugal
A R T I C L E I N F O
Article history:
Received 18 August 2012 Accepted 6 April 2013 Available online 18 May 2013
Keywords:
Slamming load Bow-flare section Multi-material formulation
Arbitrary Lagrangian-Eulerian method Penalty coupling
Roll angle
A B S T R A C T
The two-dimensional water entry of a bow-flared section w i t h different roll angles is studied by using an explicit finite element code. The modelling technique o f the fluidstructure interaction adopts a m u l t i -material Arbitrary Lagrangian-Eulerian formulation and a penalty coupling method. The simulated vertical slamming force and pressure histories are compared w i t h experiments and other numerical calculations. The effects of the roll angle on the slamming load are studied through simulations for a bow-flare section w i t h different roil angles.
© 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Impulse loads w i t h high pressure peal<s occur during the impact between a body and water. This type of 'slamming' happens when a ship bottom hits the water w i t h a high velocity in rough sea. This slam induced loads can cause local damage to ship hulls and induce global whipping responses. Ship motions and wave induced loads are often calculated by strip theory programs, in which case sectional forces are required and the slamming loads need to be assessed for two dimensional secdons corresponding to the ships secdons. An example of such an approach is the one adopted by Guedes Soares (1989) who used a method to evaluate the verrical transient load on the ship hull when the forward bottom impacts in water, and later checked it experimentally (Ramos et al., 2000).
There is a considerable amount of research conducted on slamming by experimental, analytical, and numerical simuladon methods since von Karman (1929) who simplified the slamming problem of a ship bottom as a typical two-dimensional wedge impacting w i t h water, neglecting the local uprise of the water. His idealised theory based on momentum conservadon underesti-mates the impact load for wedges w i t h small deadrise angle. Based on his work, Wagner (1932) proposed an asymptotic solution for water entry of two-dimensional bodies w i t h small local deadrise angles, accounting for piled-up water on the wedge by simply introducing a constant surface wetting factor C^, which results in overestimation on the impact load. The flow was divided into two
'Corresponding author. Tel.: +351 218417957.
E-mail address: guedessffimar.ist.utl.pt (C. Guedes Soares).
0029-8018/$-see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j,oceaneng.2013.04.009
fluid domains. The inner flow domain contains a j e t flow at the intersection between the body and the free surface, while the body boundary condition and the dynamic free surface condition were transformed to a horizontal line in the outer flow domain. For his theory, the pressure is singular on the edge of the expanding plate when the deadrise angle is small. Much work has been done by other researchers to further develop this theory. Armand and Cointe (1987) and Howison et al. (1991) developed this work by accounting for the effect of nonlinear j e t flow in the intersection region between the body and free surface using asymptotic matching. When the wedge impacts vertically w i t h water at a constant velocity, Dobrovol'skaya (1969) derived an analytical solution by transferring the potential flow problem for the constant water entry into a self-similar flow problem in complex plane, which took advantage of the simplicity of the body geometry and is valid for any deadrise angle.
Zhao and Faltinsen (1993) generalised the w o r k of Wagner (1932) and presented a numerical method for studying the water entry of a two-dimensional body of arbitrary cross-section which is a nonlinear element method w i t h a j e t flow approximation. As a further development, a fully nonlinear numerical simulation method which includes flow separation f r o m knuckles of a body and an approximation solution which does not include flow separation were presented by Zhao et al. (1996) to predict slamming loads on two-dimensional sections.
Motivated by that work, Mei et al. (1999) developed an analytical solution for the general impact problem by adopting the conformal mapping technique. For numerical confirmation, they developed a fully nonlinear simulation by using a Cauchy-integral method w i t h a matching j e t solution near the intersection between water and the body. This solution is valid for a wedge
46 S. Wang, C. Cuedes Soares / Ocean Engineering 67 (2013) 45-57 w i t h a r b i t r a r y deadrise angles, however, the velocity o f the wedge
was assumed constant. Yettou et al., (2007) presented an analytical solution to symmetrical water impact problems of a t w o -dimensional wedge by taking into account the effect of velocity reduction of the solid body upon impact.
In the field of experimental research, Stavovy and Chuang (1976) obtained the coefficients of peak pressure for the wedges w i t h different deadrise angles according to experiments results. Ochi and Motter (1973) obtained the slamming loads in terms of slamming pressure, the pressure distribution and the time varia-t i o n of varia-the varia-tovaria-tal slamming load by analysing lovaria-ts of varia-tesvaria-t resulvaria-ts. Drop tests for a wedge w i t h deadrise angle 30° and a bow-flare section were carried out by Zhao et al. (1996). Ramos et al. (2000) conducted an experimental programme assessing the slam induced loads on a segmented ship model, which was analysed w i t h the method used by Ramos and Guedes Soares (1998). These test results have also been used recently to validate CFD calcula-tions by Paik et al. (2009).
Most investigations of water entry problems, including the Studies mentioned above have been focused on the symmetric impact of wedge, while less attention has been given to bow-flared sections or oblique cases. For bow-flared sections, the water entry often involves complicated phenomena, such as different free surface geometries, vortices and secondary impacts.
Aarsnes (1996) performed free drop tests of two ship sections, i.e., one wedge section and one bow-flared section which are the same ones used in the tests of Zhao et al. (1996), aiming at investigating the pressure distributions and the impact forces for different roll angles. Zhu et al. (2005) analysed the effects of the roll angle and discussed the problems related to large roll angles with a constrained interpolation profile method based on the Navier-Stokes equations. Sun and Faltinsen (2009) studied the two-dimensional water entry of a bow-flared section with a constant roll angle, or heel angle by using a boundary element method and compared their results with experiments. They found that for the ship studied, the vertical slamming force did not change much with the roll angle when the roll angle is small, whereas the horizontal force increased with the increasing roll angle, and a high localised pressure occurred in the flare area for the larger roll angle.
W i t h the development of computing technology and capability, much work about the two-dimensional ship section impacting a calm water surface were investigated by using explicit finite element methods, e.g. Senius and Rosen (2006, 2007), Alexandru et al. (2007), Wang et al. (2012), Wang and Guedes Soares (2012). The vertical slamming force, pressure distributions and pres-sure histories on a bow-flare secrion predicted by LS-DYNA, was presented by Wang et al. (2012), showing good agreement w i t h published experimental results. In the present work, the water impact of a symmetric bow-flare section w i t h a constant roll angle and vertical velocity is studied by using the same explicit finite element method. The effects of the roll angle and impact velocity are discussed based on the predicted results and the comparisons between the numerical predictions and the available experimental values. Furthermore, the secondary impact on the opposite side of the section is observed in the profiles of the water surface elevation. The induced loads on the sections by the secondary impact are also studied. Furthermore, as a result of the oblique water entry air ventilation and air pocket, which might happen on one side of the section, are discussed as well.
2. Explicit finite element method
The explicit finite element code LS-DYNA is applied to simulate the water impact of a bow-flare section In present work. This finite element analysis is based on a multi-material Eulerian formulation
and a penalty coupling method, of which the former describes the fluid domain, and the latter enables the interaction between the body and the fluid. Luo et al. (2011) and Wang et al. (2012) verified this approach by comparing the simulated slamming loads on the wetted surface of a rigid wedge w i t h corresponding experimental and analytical results. Luo et al. (2012) have even studied the hydroelastic response of the stiffened panels that were part of a wedge used in a drop weight test
The multi-material Eulerian formulation is a part of the Arbitrary Lagrangian-Eulerian (ALE) solver w i t h LS-DYNA. The ALE solver involves a Lagrangian step, where the mesh is allowed to move and a second step that moves the element state variables back onto the reference mesh. The remap step i n the ALE algorithm needs an advection algorithm to update fluid velocity and history variables. LS-DYNA incorporates both first and second order accurate advection algorithms. Here, the governing equa-tions of the fluids domain are presented.
In Arbitrary Lagrangian-Eulerian (ALE) formulation, a reference coordinate which is not the Lagrangian coordinate and Eulerian coordinate is Induced. The differential quotient for material w i t h respect to the reference coordinate is described as following equation:
a / ( X , t )
dt dt --h W ' dx (1)
where, X is Lagrangian coordinate, x is Eulerian coordinate, and
w is relative velocity between the particle velocity 1? and the
velocity of the reference coordinate 1Ï. Therefore, the Arbitrary Lagrangian-Eulerian formulation can be derived f r o m the relation between the time derivative of material and that of reference geometry configuration.
Let n^sR^ represent the fluid domain, and di/ denote its boundary. The equation of mass, momentum and energy conser-vation for a Newtonian fluid in ALE formulation i n the reference domain, are given by:
dp dt = -pdiv(w)-w -gradip) dir "-df dE = ^di - " d!V(cr) -I- ƒ -pw-gradi v ) :r : grad('v')+ f •iJ'-pw•grad(E) (2) (3) (4) where p is the fluid density, ƒ is the body force and a is the total Cauchy stress given by:
a = -p-ld + p(grad(v) + (grad{v))'^) (5) where p is the pressure and p is the dynamic viscosity. The part of the boundary at which the velocity is assumed to be specified is denoted by dO^,, the inflow boundary condition is:
(6) 1? = ~g(t) on dd^
The traction boundary condition associated w i t h Eq. (3) is the conditions on stress components. These conditions are assumed to be imposed on the remaining part of the boundary:
^ . T f = (t) on dni, (7) The multi-material Eulerian formulation is a specific ALE case
where the reference mesh velocity is zero, which means:
I f = 0 (8)
3. Description of the bow-flare section and modelling The geometry of the ship bow-flare section used in this work is shown i n Fig. 1. It corresponds to the section used in the drop tests
S, Wang, C. Cuedes Soares / Ocean Engineering 67 (2013) 45-57 47 of Aarsnes (1996), who aimed at investigating the pressure
distribution and the impact force for the section w i t h different roll angles. The total weight of the falling equipment including the test section was 261 kg, and the total length of the section is 1 m. In order to eliminate the three-dimensional effects, the length of the measuring section is only 0.1 m in the middle of the total section. The pressures on the locations of P1-P4 were measured by pressure gauges. The test sections are mounted directly to a trolley which is connected to a free-falling rig. After the test section has
Force transducer
P l
Fig. 1. The geometry of the bow-flare secdon and die locations of the pressure sensors.
Syintnetiic line n •' Air
!
v i
Water X Non-renccliiig boundaiies 3 ^t t t
Fig. 2. Model setup of the asymmetric water entry of a two-dimensional bow-flare
section with constant roll angle ff.
hit the water surface, the trolley is stopped by two elastic ropes. The time history of the impact velocity was obtained by using the measured value by a sensor in combination w i t h the measured vertical acceleration of the drop rig. The friction between the test rigs may cause decrease of the velocity. Some vibrations in the test rig may also produce oscillations in the measurements.
In the drop tests, different roll angles, including 0°, 4.8°, 9.8°, 14.7°, 20.3° and 28.3°, are used for this bow-flare section. The initial vertical impact velocity is determined by the drop height. The main parameters for some cases were listed in Table 1 of Sun and Faltinsen (2008), who compared the numerical calculations w i t h the experimental results for different roll angles. During the water entry, the section has only a vertical velocity. At least two drops were performed for each test condition.
The calculated impact force and pressure distributions are later compared w i t h the measured values and the BEM solution as well. Unless otherwise specified, t = 0 means the keel point touches the calm water for all the plotted results below. The finite element model setup of the asymmetric water entry of a two-dimensional bow-flare section is shown in Fig. 2. The section drops vertically into the calm water as a constant roll angle ö. The x-axis is located at the calm water surface, and the y-axis is placed in the vertical line which includes the lowest point of the section. Obviously, z-axis means the three-dimensional direction of the model. Since only two-dimensional water impact is investigated, the size of the model in z-axis is set as one element length and the displacements of all nodes i n the z-direction are fixed.
Correspondingly, the meshed model is shown in Fig. 3. Only part of the mesh is illustrated and the mesh of the air is not included, in order to illustrate the detailed mesh around the impact domain clearly. Taking the computational time into consideration, the size of the water domain is limited to 700 mm's width and 500 mm's height, while that of air domain is 700 mm's width and 300 mm's height. As seen in Fig. 3, the Impact domain is uniformly meshed, and except for this domain, the mesh of other parts is moderately expanded towards the boundaries. The mesh sizes of the uniformly meshed area and the bow-flare section are all 2.5 mm, which means that the model has an extension of 2.5 m m in z-axis.
The fluid that is subjected to large deformations at the free surface, is modelled as multi-material Eulerian, while the structure is modelled as Lagrangian. For the Lagrangian solid part, the element type can be a shell element, a solid element or a beam element, while only solid elements can be applied for the Eulerian mesh.
In this work, the fluid, water and air, are modelled w i t h solidl64 which is a 3D structural element, and they are defined
48 S. Wang, C. Guedes Soares / Ocean Engineering 67 (2013) 45-57 as void materials wliich have no yield strength and behave in a
fluid-like manner. This material allows equations of state to be considered computing deviatoric stress. Optionally, a viscosity can be deflned, however, in this work, the fluids are considered as inviscid. In the definition of the material model of fluids in LSDYNA, the cut-off pressure must be defined to allow for a material to 'numerically' cavitate. In other words, when a material undergoes dilatation above certain magnitude, it should no longer be able to resist this dilatation. Since dilatation stress or pressure is negative, setting the cut-off pressure limit to a very small negative number or zero would allow for the material to cavitate once the pressure in the material goes below this value. In this work, the cut off pressure is set as 0. Therefore, the negative pressure is filtered out f r o m the results. On the other hand, the fluids are compressible in ALE simulations, while theyare assumed incompressible in Sun and Faltinsen (2009)'s boundary element method. Kaushik and Romesh (2011) examined the effects of the compressibility of fluids on the pressure distribution acting on the rigid panels entering calm water for the simulations of LS-DYNA, and the results show that the effects are very limit. The bow-flare section is modelled w i t h Lagrangian sheII163 elements and rigid body material, but the method is not restricted to rigid structures. The boundaries of fluids are defined as non-reflecting which are used on the exterior boundaries of an analysis model of an infinite domain, such as a half-space to prevent artificial stress wave reflec-tions generated at the model boundaries. Non-reflecting boundary conditions simulate the effect of a surrounding semi inflnite solid by imposing a traction along the designated boundary based upon impedance matching. Furthermore, all nodes are fixed in z-axis considering the two-dimensionality. The initial impact velocity of the body is deflned as the estimated value in the drop tests of Aarsnes (1996).
Although a comparison of first order (donor cell) and second order (Van Leer MUSCL) accurate advection algorithms resulted in almost identical results (Tutt and Taylor, 2004), was found by Wang et al. (2012) that the fluid leakage could be prevented by using the donor cell advection algorithm (IVIETH = 1). Therefore, the donor cell advection algorithm is used in this work.
The parameters study on the simulation of the same bow-flared section entering calm water was conducted by Wang et al. (2012), including the mesh size, penalty factor, time step factor, and the number of coupling points. Considering the computational time and the accuracy of the prediction, the mesh size on the impact region is deflned as 2.5 mm as mentioned above, the penalty factor is 0.1, and the number of coupling points is 3. To obtain accurate predictions, it is critical that the time step is sufficiently small to provide solution stability. The time step used in the simulation can be defined by one scale factor based on the critical one which is the minimum time value that the sound travels. In this work, the scale factor for the symmetric cases is set as 0.2 and that for the asymmetric ones is set as 0.1, which are both much smaller than the default one, thus double-precision solver is to be used in the LS-DYNA calculations.
4. Results and discussion 4.J. Impact force
During the water impact, vertical impact on the bottom surface of the two-dimensional section is given by:
ma = Fy-mg (9)
where, m is the measured mass of the section, a is the acceleration of the moving body, Fy is the vertical impact force, and g is the acceleration of gravity.
The simulated vertical impact forces for various roll angles, which are obtained by Eq. (9), are compared w i t h the experi-mental ones in Fig. 4, together w i t h the numerical results by using a BEM solution. The impact velocity and roll angle for each case are illustrated in these figures.
As seen in Fig. 4, the predictions from LS-DYNA are in good agreement w i t h the BEM's calculations. The experimental results do not agree w i t h the numerical predictions well, In particular, the measured peak forces occur later In general, the vertical impact force is higher for the section w i t h a larger roll angle. A large roll angle corresponds to a small deadrise angle, and a ship-like section w i t h a small deadrise angle causes a high impact force as presented in Wang et al. (2012). Actually, the peak force does not increase much until the roll angle is around 20°, thus, more attention must be paid to the Impact force when the roll angle is large. As illustrated in Fig. 4(a, c, d and f), these cases have the same initial impact velocity but different roll angle. It can be found that the impact force comes up to the maximum value earlier for a larger roll angle.
For the maximum vertical force, the simulated results In present work are in good agreement w i t h the BEM calculations, except when the roll angle is 28.3°. The drop tests gave lower peak values than the calculations from both of the numerical solutions, except for the case when the roll angle is 4.8°. A probable reason for the exceptional case is the uncertainty of the test rig at the late stage of the impact Besides, the measured peak forces occur later than the predicted ones from present work and BEM. The differences between the tests and the numerical methods are mainly due to the experimental errors which are discussed by Sun and Faltinsen (2008), and other effects caused by three-dimensionalities and hydroelasticity during the tests. Because the same differences are observed for the simuladon of 2D wedges In Wang et al. (2012). For a three-dimensional case, the water pile up evolves slower than a two-dimensional case, thus the peak force occurs later Furthermore, after the free surface separation at the knuckle, LS-DYNA predicts a stronger oscillation in the vertical forces. It is because the mesh size of the water jet becomes larger when it goes beyond the impact region. The oscillations seen in the force measurement may due to some vibrations in the test rig.
As seen In Eq. (9), the vertical slamming force is related to the gravity and the acceleration of the section, while the mass of the structure is constant during the entire water impact This means that If the acceleration of the structure is larger, the ratio of the gravity to the total force is lower Correspondingly, the accelera-tion of the secaccelera-tion is larger for a larger roll angle. Therefore, the gravity effects become more apparent for a bow-flare section w i t h a smaller roll angle which corresponds to a larger deadrise angle. This is consistent w i t h the fact proposed by Faidie-Clarke and Tveitnes (2008) that the gravity effects are greater for a wedge w i t h a larger deadrise angle.
Furthermore, as seen i n Fig. 4, the impact forces keep a value between 100 N and 200 N for all cases in the later stage of the impact It can be concluded that the gravity plays an important role during this period.
Fig. 5 shows the comparison of the impact force for different impact velocities. The bow-flared section w i t h roll angles of 0°, 9.8°, 14.7° and 28.3° are analysed. For each case, three different drop velocities of 0.5 m/s, 1.5 m/s and 2.43 m/s, are chosen. For all the cases, the impact force increases from the beginning and reaches the peak value, then reduces quickly after flow separation, and then decays slowly.
As expected, the maximum value of the vertical impact force Is larger for a higher velocity when the roll angle is constant as happens earlier for a larger roll angle. As seen i n Fig. 5(d), the peak value increases faster compared to the other two cases. It shows
S, Wang, C. Cuedes Soares / Ocean Engineering 67 (2013) 45-57 49 200 150 100 50 0 - Aarsnes(1996) -BEM - LS-DYNA e=9.8°,v=-0.61m/s 0.05 0.1 Time (s) 0.15 200 150 100 50 0 -Aarsnes(1996) -BEM - LS-DYNA =4.8°,v=-0.57m/s 0.05
d
„ 200 z ^ 150 ro 100I ro
0 - Aarsnes{1996) -BEM - LS-DYNA =14.7'',v=-0.61m/s TTv 0.1 Time (s) 0.15 0.1 Time (s)Fig. 4. Vertical impact force for various roll angles.
0.05 0.1 Time (s) 0.15 500 g 400
1 300
I
200 E ra Ö5 100 0 0 0.61 m/s 1.50m/s 2.43m/sb
500 z 400 •ara
o 300 O) c e 200 Era
W 100 0d
1000 800 ra o 600 D) C E 400 Era
w 200 0.61m/s 1.50m/s • 2.43m/s 0.05 0.1 0.15 0.2 Time (s) 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 Time (s) Time (s)Fig. 5. Vertical impact force for different impact velocity, (a) (7=0°; (b) fl=9.8°,(c) 6i=14.7°; (d) 0=28.3°
0.2
that the impact force is more sensitive to the impact velocity when the roll angle is large.
On the other hand, as seen in Fig. 5(b and c), there are several impulses in the curves after the first peak value. It seems that
the secondary impact happens between the leeward side of the section and the elevated water When the roll angle is 9.8°, the impulse caused by the secondary impact of the section is higher than the first peak value for the cases w i t h an impact velocity of
50 S. Wang, C. Cuedes Soares / Ocean Engineering 67 (2013) 45-57
1.5 m/s and 2.43 m/s. The effects of the secondary impact on the vertical force become more apparent for a higher impact velocity. Furthermore, the secondary impact occurs earlier for a higher impact velocity at a given roll angle. When the roll angle is 28.3°, there is no obvious secondary impulse in the curves, but it cannot be easily shown that a secondary Impact does not happen. It is not apparent probably just because the effects are not important in this case. This w i l l be discussed later related to the profiles of the elevated water.
4.2. Impact velocity
The impact velocities of the bow-flare section w i t h roll angle 9.8° and 20.3° are presented i n Fig. 6. The predictions of this work are compared w i t h the experimental and numerical results, respectively published by Aarsnes (1996), Sun and Faltinsen (2008) and Zhu et al. (2005). Although the curves of the impact force have many high frequencies, the ones of the impact velocity are smooth. At the first stage of the water impact, the four methods give very similar results, but very small differences are still observed between the experimental ones and the numerical values. It is probably due to the error of manually measurement from the published paper or the experimental error. Brizzolaraa et al. (2008) stated the accurate impact velocity is very important to obtain actual pressure distribution during the impact. Though the differences are very small, it affects the pressure estimates. So, this is one reason for the differences between the numerically predicted pressure and the measured values. The experimental velocities become much larger than the values pf other methods in the later stage. It is believed that this is mainly due to the elastic rope that is used to stop the water entry in the tests.
4.3. Pressure history
4.3.1. Points on the downward side of the section
The predicted time histories of the measured points P1-P4 which are illustrated in Fig. 1, are plotted in Fig. 7, together w i t h the experimental and numerical results. For the maximum value of the pressure, the predicted values by LS-DYNA agree well w i t h the measured ones and the numerical calculations on the locations of P2 and P3, but they are usually smaller than the published ones on the locations of Pl and P4. Since Pl is at the lowest part of the section, the under-estimated values on Pl may be due to the mesh size of the model which was discussed in Wang et al. (2012). Corresponding to the vertical force the predicted peak pressure occurs eariier than the results of other two methods.
When the roll angles are 20.3° and 28.3°, the pressure on Pl becomes negative in the later stage as shown. The possible reasons for this, including the high velocity around the bottom and the elastic ropes used i n the test, were discussed by Sun and Faltinsen
(2008). As known, negative pressure may lead to ventilation. Zhu et al. (2005) mentioned that ventilation actually occurred in the case for 0=28.3°, but Sun and Faltinsen (2008) suggested that it did not happen in the cases above, according to their numerical results. Since that LS-DYNA gets rid of the negative pressure from the results it is not possible in this work to determine if ventilation has happened through the pressure values, but this will be discussed later related to the profiles of the elevated water
4.3.2. Points on the leeward side of the section
Due to the symmetric geometry of the section, the points P1-P4 on the leeward side are defined as the symmetric ones w i t h respect to the symmetric line. The predicted pressure histories are presented In Fig. 8 together w i t h the numerical and experimental results, for 5 = 9 . 8 ° and 51=20.3°.
As shown, the pressures of Pl are similar to those on the downward side of the section, however, when the roll angle is 9.8°, an obvious increase happens in the late stage. The small impulse, which can also be observed f r o m the pressure histories of P2 and P3, is probably due to tbe secondary impact on the leeward side. When the roll angle is 20.3°, the pressures on the leeward side are smaller than those on the downward side. In particular for P2, the pressure predicted by LS-DYNA is nearly zero. This is because of the 'air pocket' that is created after the secondary impact on the leeward side.
4.4. Pressure distribution
The predicted pressure distributions on the wetted surface of the bow-flare section w i t h different roll angles are plotted in Fig. 9, together w i t h the calculations f r o m the BEM. The initial drop velocity and the roll angles are respectively illustrated in these figures. The time corresponds to the instant when the simulated vertical force comes up to the maximum value. In agreement w i t h the variable of Fig. 6 in Sun and Faltinsen (2009), the horizontal axis is a curvilinear coordinate along the section surface, where s = Ois at the keel of the section, and the coordinates for s > 0 and s < 0 respectively correspond to the body surfaces on the right and the left of the keel.
When the roll angle is larger, the pressures on the downward side are higher, while those on the leeward side are lower Compared to the BEM's calculations, the predictions f r o m LS-DYNA are lower, especially for the maximum value at the down-ward side for a large roll angle. As seen in Fig. 7, the simulated pressures of Pl are lower than the experimental and BEM's results for all the cases, which are consistent w i t h the differences of the pressure at the keel ( s = 0 ) between the results from LS-DYNA and BEM.
S. Wang, C. Cuedes Soares / Ocean Engmeering 67 (2013) 45-57 51
On the other hand, the simulated pressures on the leeward side are very low for the bow-flared section w i t h a roll angle. It indicates that the free surface of water does not reach the flared part of the section at this time instant for a roll angle larger than 4.8° in this work.
4.5. Pressure variation
Fig. 10 shows the predicted pressure distributions at different time instants for the cases of a roll angle 14.7° and 20.3°. Both the right and the leeward side of the section are investigated here.
0.1 e 0.05 0.1 Si 0.05 0.05 0.1 Time (s) 0.15 0.2 - Aarsnes(1996) -BEM - LS-DYNA P3 0.05 0.1 Time (s) 0.15 0.2 0.1 S 0.05 Aarsnes(1996) BEM LS-DYNA P2 0.1 S! 0.05 0.05 0.1 Time (s) - Aarsnes(1996) -BEM - LS-DYNA 0.15 0.2 P4 2 0.05 0.1 0.15 Time (s) 0.2 0.1 a? 0.05 0.1 2; 0.05 - Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time(s) Aarsnes(1996) -BEM - LS-DYNA PI 0.15 0.2 0.05 0.1 Time (s) 0.15 0.2 0.1 S 0.05 0.1 2! 0.05 Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time(s) 0.15 0.2 Aarsnes(1996) BEM • LS-DYNA 0.05 0.1 Time (s) 0.15 0.2 0.1 & 0.05 0.1 0.05 - Aarsnes(1996) -BEM - LS-DYNA P1 0.1 e 0.05 0.1 e 0.05 Aarsnes(1996) BEM - LS-DYNA 0.05 0.1 Time (s) 0.15 0.2 Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) 0.15 0.2
Fig. 7. Pressure histories on the points P1-P4 for different roil angles (1 bar=10^Nm ). (a) 0=0°, v
0=14.7°. i/=-0.61 m/s, (e) 0=20.3°, v=-0.75 m/s and (f) 0=28.3°, i;=-0.61m/s.
0.05 0.1 Time (s) = -0.61 m/s, (b) 0=4.8°,
0.15 0.2
52 S. Wang, C. Cuedes Soares / Ocean Engineering 67 (2013) 45-57 0.2 0.15 0.1 0.05 0 0.2 0.15 0.1 0.05 0 -Aarsnes{1996) - B E M - LS-DYNA 0.05 0.1 Time (s) - Aarsnes(1996) - B E M - LS-DYNA P I 0.15 0.2 P3 0.05 0.1 Time (s) 0.15 0.2 0.2 0.15 2 0.1 3 W I 0.05 ai 0.2 0.15 0.1 0.05 0 Aarsnes(1996) -BEM - LS-DYNA P2 0 0.05 0.1 0.15 0.2 Tme (s) - Aarsnes(1996) - B E M • LS-DYNA 0 0.05 0.1 0.15 0.2 Time (s) 0.2 0.15 0.1 0.05 0 0.2 0.15 0.1 0.05 0 - Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) - Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) P I 0.15 0.2 P3 0.15 0.2 0.2 0.15 0.1 0.05 0 0.25 0.2 0.15 0.1 0.05 0 -Aarsnes(1996) -BEM - LS-DYNA - Aaisnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) P2 0 0.05 0.1 0.15 0.2 Time (s) P4 0.15 0.2 0.1 0.05 0 -0.05 0.2 0.15 0.1 0.05 0 - Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Tme (s) -Aarsnes(199B) -BEM - LS-DYNA P1 0.15 0.05 0.1 Time (s) 0.15 0.2 P3 0.2 0.15 0.1 0.05 0 •c- 0.4
I
0.3 a 0.2 li; 0.1 0 0.2 - Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) - Aarsnes(1996) - B E M - LS-DYNA 0.05 0.1 Time (s) P2 0.15 0.15 0.2 P4 0.2 Fig. 7. Continued.The variable in the y-axis is given by iY~Yk)/Yd, where y means the vertical coordinate on the body surface, yk is the vertical coordinate of the keel and yd Is the draft of the section when then roll angle is 0°.ln other words, this variable is related o the position on the secdon. Obviously 0 means the keel, 1 represents the
highest point in the downward side and - 1 means the highest part in the leeward side of the section.
For the evolution of the pressure distribution on the downward side, when the roll angle is 14.7°, the maximum pressure on the wetted section is located in the keel of the body at the time instant
S. Wang, C. Guedes Soares / Ocean Engmeering 67 (2013) 45-57 53 0.1 2 0.05 0.1 0.05 Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) - Aarsnes(1996) -BEM • LS-DYNA PI 0.1 2 0.05 0.05 0.15 0.2 0.1 g 0.05 - Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) Aarsnes(1996) - B E M - LS-DYNA 0.15 0.2 P4 0.05 0.1 Time (s) 0.15 0,2 0,2 0.1 0,1 0,05 - Aarsnes(1996) -BEM • LS-DYNA 0.05 0.1 Time (s) - Aarsnes(1996) -BEM - LS-DYNA P1 0.05 0,1 Time (s) 0,15 0.2 0.15 0,2 0.06 0.04 0.02 0.1 0.05 Q-- Aarsnes(1996) -BEM - LS-DYNA 0.05 0.1 Time (s) 0.15 0.2 Aarsnes(1996) BEM • LS-DYNA P4 0.05 0.1 Time (s) 0.15 0,2
Fig. 8. Pressure liistories on the points P1-P4 on the leeward side of the bow-flare secdon for different roll angles (1 bar=10^ N m"^). (a) 0=9.8', i/=-0.61 m/s and (b) 0=20.3°, v=-0.75 m/s.
of 0.08 s after the impact, as plotted in Fig. 10(a), and then it moves to near the spray root of the water jet. Before 0.11 s, the pressure on each position of the section increases w i t h time. It seems that the f l o w separation occurs after the time instant of 0.11 s, because after that the pressure decreases fast, as seen the results of 0.12 s. At the later stage of the impact, the pressure distribution does not change too much w i t h time and is almost uniformly distributed along the wetted bottom.
When the roll angle is 20.3°, the pressure distributions are similar w i t h the results of the model w i t h a roll angle of 14.7° before flow separation occurs. However, at the later stage, the maximum pressure is located near the keel of the section, and i t increases w i t h time. Perhaps, i t is due to the section impact at the leeward side of the body.
For the evolution of the pressure distribution on the leeward side, the pressures on the leeward side are very small for these two cases in the early stage of the water impact When the roll angle is 14.7°, the pressures are increasing w i t h time in the late stage, and the maximum value is located i n the middle part o f t h e section. Even at t=0.14 s the maximum pressure is larger than the value on the downward side. This is because f l o w separation occurs on the downward side but does not happen on the leeward side at that time instant When the roll angle is 20.3°, the maximum pressure on the leeward side is also located i n the
middle part of the section in the late stage, and the pressures at f=0.16 s are smaller than those at t=0.15 s, which means that the flow separates from the section at t = 0.16 s. Furthermore, the pressures on the lower part of the section are close to zero i n the late stage. The possible reason is that the air pocket is created during the water entry.
4.6. Water surface elevation and pressure contour
Corresponding to the calculated pressure distributions at the different time instants mentioned above, the pressure contours at those time instants are plotted, together w i t h the free surface elevations, as seen in Fig. 11.
Since the simulated free surface elevation greatly depends on the mesh size, and is also affected by the fluids domain and even material model of the fluids, i t is difficult to predict i t precisely. It can only be qualitatively estimated for its evolution during the water entry. As plotted, the pressure contours at different time instants in Fig. 11 are in good agreement w i t h the predicted pressure variations mentioned above. For both of the cases, the pressures on leeward sides are merely zero at the initial stage, and the peak values on downward sides move toward the knuckles. After the flow separation from the knuckles, the pressures on the downward sides drop, and the values on the leeward sides
S. Wang, C. Guedes Soares / Ocean Engineering 67 (2013) 45-57 55
Fig. 11. Free surface elevation and pressure contour for different roll angles, (a) 0=14.7°, i/=-0.61 m/s. (b) 0=20.3°,'i'=-0.75 m/s,
increase. The secondary impact between the water and the leeward side of the body is observed as well as the air pocket, as seen i n the last two figures o f both cases.
4.7. Discussions about the flow on the leeward side of the section 4.7.1. The secondary impact
Fig. 12 shows the pressure contours and the free surface elevations at the instant when the elevated water impacts w i t h the leeward side of the section for different roll angles. The results for the cases w i t h roll angles of 4.8°, 9.8°, 14.7° and 20.3° are presented, which indicate that the secondary impact happens for any roll angle.
As mentioned before, the effects of the secondary impact are large when the roll angle is 9.8° and 14.7°, while there is no obvious change of the vertical force for the 28.3° roll angle, as plotted in Fig. 5. The time Instants included in these figures show that, as the roll angle increases, the secondary impact happens later When the roll angle is large, the secondary impact happens in the later stage of the water impact, for which the relative velocity between the body and the water is small; therefore, the force caused by the secondary impact is small as well. When the roll angle is 4.8°, the effects of the secondary impact are also limited as seen in Fig. 4(b). This is because the force caused by the
secondary impact is small compared to the total vertical force on the body.
4.7.2. Ventilation and air pocket
On studying the asymmetric impact of wedges, Xu. et al. (1998) described two types of impact. Type A flow occurred when there was small asymmetry and the water moved out towards the chine on the both sides of the keel. Type B flow happened when there was large asymmetry and the flow separated f r o m the hull at the keel on one side. As mentioned in Judge et al. (2004), 'Type B
impact' and 'ventilation' were used interchangeably, and both of them implied a flow detachment at the keel of the wedge that may or may npt eventually produce reattachment to the body w i t h an air pocket.
From this point of view, ventilation does not happen i n present work, because the pressures of Pl in Fig. 7 do not go to zero. As seen f r o m the pressure distribution i n Fig. 9, although the pressures on the keel of the sections w i t h a roll angle are very small, they are not zero.
On the other hand, although there is no ventilation, the secondary impact on the leeward side happens as mentioned before. As plotted in Fig. 12, the air pocket is created In the later stage of the Impact or after the secondary impact occurs. This
56 S. Wang, C. Cuedes Soares / Ocean Engineering 67 (2013) 45-57
Time = 0.1188 Time = 0.1198
Fig. 12. Pressure contours at'the time instant when the secondary impact at the leeward side happens for different roll angles, (a) 0 = 4.8', v = Cb) 0 = 9.8', v =-0.61 m/s; (c) 0 = 14.7", v =-0.61 m/s; (d) 0 = 20.3", v =-0.75 m/s.
-0.57 m/s;
means that the flow separates from the lower part of the left surface, but not from the keel, and then touches the section again. Furthermore, as the roll angle increases, the air pocket becomes larger and moves towards the keel. Meanwhile, i t is also the reason that the pressure value of P2 plotted in Fig. 8(b) is close to zero. It shows that P2 is located i n the air pocket when the roll angle is 20.3°.
As mentioned before, the two-dimensional simulations are studied hy fixing the displacement of all the nodes in z-direction. For these cases, the air pocket cannot escape easily during the impact, thus follows the secondary impact. A three dimensional simulation may produce a different flow field around the struc-ture. As mentioned above, the predicted impact force drops quickly after the flow separation f r o m the knuckle. We can see that this numerical model can simulate the flow separation well. For the free surface of water i n the late stage, the water jet moves to a far region, so the mesh size of this region is concerned. Though some oscillations i n the vertical forces after the free surface separation at the knuckle, they do not matter the results much. It is considered that the mesh size of the water jet is fine enough.
5. Conclusions
The slam induced loads on a two-dimensional bow-flare sec-tion w i t h roll angle are evaluated by the explicit finite element method, when it impacts on water w i t h a vertical velocity.
The simulated vertical impact force, impact velocity and pres-sure histones on the positions of P1-P4 are compared w i t h experimental and numerical results. The predicted impact forces f r o m present work and the calculations from BEM are in very good agreement, especially for the small roll angles. For the pressure distribution, LS-DYNA gives small values than the ones from BEM generally. The experimental measurements do not agree very well w i t h the numerical results. It is mainly because the differences betiA/een the time histories of impact velocity from these methods. The effects of the roll angle on the slamming load are investigated through the calculations for different roll angles
which include 0°. 4.8°, 9.8°, 14.7°, 20.3° and 28.3°. The results show that the higher the roll angle, the larger the maximum vertical force is, and the gravity effects become more apparent for a larger roll angle. For a constant roll angle, the maximum force is more sensitive to the impact velocity when the roll angle is large. The predicted pressure distributions on the wetted surface of the bow-flare section w i t h different roll angles are plotted at the time instants when the vertical impact force comes up to the peak value. It is found that, when the roll angle is larger, the pressures on the downward side are higher, while those on the leeward side are lower. Meanwhile, the predicted pressure distributions are not in good agreement w i t h the BEM's calculations, especially that on the leeward side of the section. The evolutions of pressure distribution and free surface elevation are presented as well. In the initial stage of the water entry, the pressure distributions on the downward side are similar for any case and that on the leeward side are very small. In the later stage, the maximum pressure on the downward side is located at the keel of the section while that on the leeward side is located in the middle part of the section when the roll angle is 20.3°, and as the time increases, the peak value on the downward side becomes larger, while that on the leeward side becomes smaller
Furthermore, the free surface elevations indicate that the secondary impact happens at the leeward side of the body for all the cases. When the roll angle is smaller, i t occurs earlier. Based on the predicted impact forces, it can be concluded that the effects of the secondary impact become more apparent for a higher velocity. In present work, the effects on the vertical force are most obvious when the roll angle is 9.8°, while they are limited for a small or a larger angle. This means that the effects are related to the occurrence time of the secondary impact, and the m a x i m u m value of the impact force.
Acknowledgements
The work has been performed in the scope of the project EXTREME SEAS - Design for Ship Safety in Extreme Seas, ( w w w . manisLuti.pt/extremeseas), which has been partially financed by
S. Wang, C. Cuedes Soares / Ocean Engineering 67 (2013) 45-57 57 the EU under contract SCP8-GA-2009-234175. Dr Hanbing Luo has
been involved in the earlier phases of this research and has contributed to the present understanding on how to deal w i t h the numerical modelling.
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