Slam induced loads on bow-flared sections with various roll angles
Introduction
Impulse loads with high pressure peaks occur during the impact between a body and water. This type of ‘slamming’ happens when a ship bottom hits the water with 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 sections corresponding to the ships sections. An example of such an approach is the one adopted by Guedes Soares (1989) who used a method to evaluate the vertical 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 simulation methods since von Kármàn (1929) who simplified the slamming problem of a ship bottom as a typical two-dimensional wedge impacting with water, neglecting the local uprise of the water. His idealised theory based on momentum conservation underestimates the impact load for wedges with small deadrise angle. Based on his work, Wagner (1932) proposed an asymptotic solution for water entry of two-dimensional bodies with small local deadrise angles, accounting for piled-up water on the wedge by simply introducing a constant surface wetting factor Cw, which results in overestimation on the impact load. The flow was divided into two fluid domains. The inner flow domain contains a jet 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 jet flow in the intersection region between the body and free surface using asymptotic matching. When the wedge impacts vertically with 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 work 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 with a jet flow approximation. As a further development, a fully nonlinear numerical simulation method which includes flow separation from 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 with a matching jet solution near the intersection between water and the body. This solution is valid for a wedge with arbitrary deadrise angles, however, the velocity of the wedge was assumed constant. Yettou et al., (2007) presented an analytical solution to symmetrical water impact problems of a two-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 with 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 variation of the total slamming load by analysing lots of test results. Drop tests for a wedge with 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 with the method used by Ramos and Guedes Soares (1998). These test results have also been used recently to validate CFD calculations 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.
With 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. Stenius et al., 2006, Stenius et al., 2007, Alexandru et al. (2007), Wang et al. (2012), Wang and Guedes Soares (2012).
The vertical slamming force, pressure distributions and pressure histories on a bow-flare section predicted by LS-DYNA, was presented by Wang et al. (2012), showing good agreement with published experimental results. In the present work, the water impact of a symmetric bow-flare section with 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.
Section snippets
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 with
Description of the bow-flare section and modelling
The geometry of the ship bow-flare section used in this work is shown in Fig. 1. It corresponds to the section used in the drop tests of Aarsnes (1996), who aimed at investigating the pressure distribution and the impact force for the section with 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
Impact force
During the water impact, vertical impact on the bottom surface of the two-dimensional section is given by:where, m is the measured mass of the section, a is the acceleration of the moving body, Fv 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 with the experimental ones in Fig. 4, together with the numerical results by using a BEM solution. The impact velocity
Conclusions
The slam induced loads on a two-dimensional bow-flare section with roll angle are evaluated by the explicit finite element method, when it impacts on water with a vertical velocity.
The simulated vertical impact force, impact velocity and pressure histories on the positions of P1–P4 are compared with experimental and numerical results. The predicted impact forces from present work and the calculations from BEM are in very good agreement, especially for the small roll angles. For the pressure
Acknowledgements
The work has been performed in the scope of the project EXTREME SEAS - Design for Ship Safety in Extreme Seas, (www.mar.ist.utl.pt/extremeseas), which has been partially financed by 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 with the numerical modelling.
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