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Research Papers: Ocean Engineering

A Meshless Boundary Element Method for Simulating Slamming in Context of Generalized Wagner

[+] Author and Article Information
Jens B. Helmers

Ship Hydrodynamics and Stability,
Det Norske Veritas,
Høvik 1363, Norway
e-mail: Jens.Bloch.Helmers@dnvgl.com

Geir Skeie

Riser Technology,
Det Norske Veritas,
Høvik 1363, Norway
e-mail: Geir.Skeie@dnvgl.com

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received October 22, 2013; final manuscript received November 23, 2014; published online February 20, 2015. Assoc. Editor: Wei Qiu.

J. Offshore Mech. Arct. Eng 137(2), 021103 (Apr 01, 2015) (10 pages) Paper No: OMAE-13-1100; doi: 10.1115/1.4029252 History: Received October 22, 2013; Revised November 23, 2014; Online February 20, 2015

A boundary element method (BEM) designed for solving the symmetric generalized Wagner formulation is presented. The flow field is parameterized with analytical functions and can describe the kinematics at any free surface or body location using a small set of parameters obtained from a collocation scheme. The method is fast and robust for all deadrise angles, even for flat plate impacts where classical BEMs usually fail. The method is easy to implement and is easy to apply. Given a smooth body contour the only additional input is the requested accuracy. There is no mesh involved. When solving the temporal problem, we exploit the analytical distribution of free surface velocities and apply an integral equation formalism consistent with the Wagner formulation. The output of the spatial and temporal scheme is a set of functions and parameters suitable for fast computation of the complete kinematics for any impact trajectory given the position of the keel and the body velocity. The method is developed to be combined with seakeeping programs for statistical impact and whipping assessment.

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References

Faltinsen, O. M., 2005, Hydrodynamics of High-Speed Marine Vehicles, Cambridge University Press, New York.
Tuitman, J. T., 2010, “Hydro-Elastic Response of Ship Structures to Slamming Induced Whipping,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
Hermundstad, O., and Moan, T., 2007, “Efficient Calculation of Slamming Pressures on Ships in Irregular Seas,” Mar. Sci. Technol., 12(3), pp. 160–182. [CrossRef]
Helmers, J. B., Sun, H., Landet, T., and Driveklepp, T., 2012, “Stochastic Analysis of the Extreme Impact Loads on Marine Structures,” 31st International Conference on Ocean, Offshore and Arctic Engineering (OMAE), ASME Paper No. OMAE2012-83849. [CrossRef]
Zhao, R., Faltinsen, O. M., and Aarsnes, J. V., 1996, “Water Entry of Arbitrary Two-Dimensional Sections With and Without Flow Separation,” 21st Symposium on Naval Hydrodynamics, NTNU, Trondheim, Norway.
Wagner, H., 1932, “Über stoß- und gleitvorgänge and der oberfläche von flüssigkeiten,” Z. Angew. Math. Mech., 12(4), pp. 193–235. [CrossRef]
Skeie, G., Kvålsvold, J., and Nestegård, A., 1996, “Computer Program 2DBEM for Prediction of Slamming Pressures,” Technical Report No. 96-2039, Det Norske Veritas.
Mei, X., Liu, Y., and Yue, D., 1999, “On the Water Impact of General Two Dimensional Sections,” Appl. Ocean Res., 21, pp. 1–15. [CrossRef]
Korobkin, A. A., 2010, “Generalized Wagner Model,” Technical Report, Lavrentyev Institute of Hydrodynamics.
Malleron, N., Scolan, Y. M., and Korobkin, A. A., 2007, “Some Aspects of a Generalized Wagner Model,” Proceedings of 22nd International Workshop on Water Waves and Floating Bodies (IWWWFB), pp. 137–140.
Korobkin, A. A., 2011, “Semi-Analytical Approach in Generalized Wagner Model,” Proceedings of 26th International Workshop on Water Waves and Floating Bodies (IWWWFB), pp. 85–88.
Khabakhpasheva, T. I., Kim, Y., and Korobkin, A. A., 2014. “Generalised Wagner Model of Water Impact by Numerical Conformal Mapping,” Appl. Ocean Res., 44, pp. 29–38. [CrossRef]
Korobkin, A. A., 2004, “Analytical Models of Water Impact,” Eur. J. Appl. Math., 15(6), pp. 821–838. [CrossRef]
Sun, H., 2007, “A Boundary Element Method Applied to Strongly Nonlinear Wave-Body Interaction Problems,” Ph.D. thesis, Norwegian Institute of Technology, Trondheim, Norway.
Calvetti, D., Golub, G. H., Gragg, W. B., and Reichel, L., 2000, “Computation of Gauss–Kronrod Quadrature Rules,” Math. Comput., 69(231), pp. 1035–1052. [CrossRef]
Helmers, S., and Skeie, G., 1999, “Slamming in Swan—Using 2DBEM as a Preprocessor,” Technical Report No. 99-2008, Det Norske Veritas.
Driveklepp, T., 2001, “On the Water Impact of Two-Dimensional Wedge-Shaped Sections,” (to be published).

Figures

Grahic Jump Location
Fig. 2

Exploiting symmetry the control volume applied in the BEM formulation is the enclosure of the body contour Sb, the vertical line (y = 0, z < 0) denoted Sz, the horizontal line (y > c, z = h) denoted Sy and the circular sector S connecting Sy and Sz at infinity. Given a collocation point xf on a smooth part of the enclosure, we modify the surface by making an infinitesimal semicircle Sϵ centered at xf pointing into the fluid domain.

Grahic Jump Location
Fig. 1

At time t, the intersection point between the body and free surface is located at (±c(t), h(t)) relative to a Cartesian coordinate system (y, z) with origin at the keel. The z-axis is pointing upward. The body is moving with the velocity Vb(t). The vertical distance from the keel to the undisturbed free surface is denoted H(t). The smooth symmetric body surface is defined by the coordinates (±y, ηb(y)). The deadrise angle of the body at the intersection point is denoted β and the unit normal vector n is pointing into the body.

Grahic Jump Location
Fig. 3

For wedges, the function QB(q) is smooth for all deadrise angles β. Within graphical accuracy, these results are indistinguishable from QB(q) derived from the analytical results presented in Appendix A. These observations are also valid for deadrise angles close to 90 deg but has been excluded from the plot due to differences in scale.

Grahic Jump Location
Fig. 4

The relative tangential velocity ∂φ/∂s along the hull is presented for various wedges. Within graphical accuracy, the results are indistinguishable from the analytical result presented in Appendix A in both logarithmic and linear scales.

Grahic Jump Location
Fig. 8

Distribution of QB(q) along a circular body contour for various penetrations of the cylinder. Within graphical accuracy, the results are indistinguishable from the analytical result presented in Appendix B.

Grahic Jump Location
Fig. 9

Distribution of QY(q) on the free surface given a circular body contour for various penetrations of the cylinder. Within graphical accuracy, the results are indistinguishable from the analytical result presented in Appendix B.

Grahic Jump Location
Fig. 5

The vertical velocity distribution on the free surface is here represented by QY(q) for a set of wedges. Within graphical accuracy, the results are indistinguishable from the analytical result presented in Appendix A for all deadrise angles. Due to differences in scaling in the far field, we have omitted the graph for β = 75 deg.

Grahic Jump Location
Fig. 6

Convergence of QB(q) as a function of nb and ny for a wedge with deadrise β = 30 deg

Grahic Jump Location
Fig. 7

Convergence of QY(q) as a function of nb and ny for a wedge with deadrise β = 30 deg

Grahic Jump Location
Fig. 10

Asymptotic values of k(h) for wedges of various deadrise angles solved from equation system (48). Within graphical accuracy, the results are indistinguishable from the analytical results represented by Eq. (A14) in Appendix B.

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