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Research Papers: Ocean Engineering

Numerical Investigation of Air Cavity Formation During the High-Speed Vertical Water Entry of Wedges

[+] Author and Article Information
Jingbo Wang

Centre for Ships and Ocean Structures,
Norwegian University of Science
and Technology,
Otto Nielsens v 10,
7491, Trondheim, Norway
e-mail: jingbo.wang@ntnu.no

Odd M. Faltinsen

Centre for Ships and Ocean Structures &
Department of Marine Technology,
Norwegian University of Science
and Technology,
Otto Nielsens v 10,
7491, Trondheim, Norway
e-mail: odd.faltinsen@ntnu.no

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 29, 2010; final manuscript received February 26, 2012; published online February 22, 2013. Assoc. Editor: Daniel T. Valentine.

J. Offshore Mech. Arct. Eng 135(1), 011101 (Feb 22, 2013) (10 pages) Paper No: OMAE-10-1106; doi: 10.1115/1.4006760 History: Received October 29, 2010; Revised February 26, 2012

In this paper, a nonlinear boundary element method (BEM) is developed for investigating air cavity formation during the high-speed water entry of wedges. A technique is proposed for dynamic re-gridding of free surface boundaries. This technique applies to both equally and nonequally spaced grids, and it is able to suppress the numerical instabilities encountered using a BEM for simulating free surface flows. The authors also develop a purely numerical method to simulate nonviscous flow separation, which occurs when the flow reaches the knuckle of the wedge. The present nonlinear BEM has been verified by comparisons with similarity solutions. We also compare numerical results with experimental results. Finally, we give a numerical prediction of the evolution of the cavity until the closure of the cavity, and the influence of the initial entry velocity, wedge mass, and deadrise angle on the characteristics of the transient cavities is investigated.

Copyright © 2013 by ASME
Topics: Cavities , Water , Wedges
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References

Von Karman, T., 1929, “The Impact of Seaplane Floats During Landing,” National Advisory Committee for Aeronautics, Washington, DC, Report No. NACA-TN-321.
Wagner, H., 1932, “Über Stoss- und Gleitvogänge an der Oberfläche von Flüssigkeiten,” Z. Angew. Math. Mech., 12(4), pp. 193–235. [CrossRef]
Cointe, R., and Armand, J. L., 1987, “Hydrodynamic Impact Analysis of a Cylinder,” ASME J. Offshore Mech. Arct. Eng., 109, pp. 237–243. [CrossRef]
Dobrovol’skaya, Z. N., 1969, “On Some Problems of Similarity Flow of Fluid with a Free Surface,” J. Fluid Mech., 36, pp. 805–829. [CrossRef]
Zhao, R., and Faltinsen, O. M., 1993, “Water Entry of Two-Dimensional Bodies,” J. Fluid Mech., 246, pp. 593–612. [CrossRef]
Vinje, T., and Brevig, P., 1981, “Nonlinear Two Dimensional Ship Motions,” Proceedings of the 3rd International Conference on Numerical Ship Hydrodynamics, Paris, pp. 257–266.
Greenhow, M., and Lin, W. M., 1985, “Numerical Simulation of Nonlinear Free Surface Flows Generated by Wedge Entry and Wave Maker Motions,” Proceedings of the 4th Numerical Conference on Ship Hydrodynamics, Washington, D.C.
Yim, B., 1985, “Numerical Solution of Two Dimensional Wedge Slamming With a Nonlinear Free-Surface Condition,” Proceedings of the 4th Numerical Conference on Ship Hydrodynamics, Washington, D.C.
Greenhow, M., 1987, “Wedge Entry into Initially Calm Water,” Appl. Ocean Res., 9, pp. 214–223. [CrossRef]
Kihara, H., 2004, “Numerical Modeling of Flow in Water Entry of a Wedge,” Proceedings of the 19th International Workshop on Water Waves and Floating Bodies, Cortona, Italy.
Longuet-Higgins, M. S., and Cokelet, E. D., 1976, “The Deformation of Steep Waves on Water I. A Numerical Method of Computation,” Proc. R. Soc. London, Ser. A, 350, pp. 1–26. [CrossRef]
Zhao, R., Faltinsen, O. M., and Aarsnes, J. V., 1996, “Water Entry of Arbitrary Two-Dimensional Sections With and Without Flow Separation,” Proceedings of the 21st Symposium on Naval Hydrodynamics, Trondheim, Norway.
Yettou, E., Desrochers, A., and Champoux, Y., 2006, “Experimental Study on the Water Impact of a Symmetrical Wedge,” Fluid Dyn. Res., 38, pp. 47–66. [CrossRef]
Blagoveshchensky, S. N., 1962, Theory of Ship Motions, Dover Publications, Inc., New York.

Figures

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Fig. 1

Coordinate system and symbol definitions used in numerical study of vertical water entry of a 2D wedge

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Fig. 2

Numerical simulation of nonviscous flow separation. The red line represents the free surface, the dashed line the artificial wall, and the green line the element on the artificial wall. The green element is completely beyond the separation point, and its boundary condition should be changed from the body boundary condition to the free surface condition.

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Fig. 3

Pressure distributions on the wedge. The wedge has a deadrise angle of 25 deg; it vertically enters the water at a constant velocity in a zero gravity environment.

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Fig. 4

The free surface shape during the vertical water entry of the wedge. The wedge has a 25 deg deadrise angle; it enters the water at a constant velocity in a zero gravity environment.

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Fig. 5

Conservation check of fluid mass, momentum and energy for the vertical water entry of the wedge with a 25 deg deadrise angle at a constant velocity in a zero gravity environment. T0 is the time when the tenth cutting-off of the jet on the impact side is performed.

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Fig. 6

Comparison of body velocity as a free-falling wedge cylinder enters the water. The wedge cylinder has a square 1.2 m × 1.2 m top section. The deadrise angle of the cross section of the wedge cylinder is 25 deg, and the mass of the wedge cylinder is 94 kg. The wedge cylinder contacts the water at a speed of V0 = 5.05 m/s.

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Fig. 7

Comparison of peak pressure on the impact side as a free-falling wedge cylinder enters the water. The wedge has a square 1.2 m × 1.2 m top section. The deadrise angle of the cross section of the wedge cylinder is 25 deg, and the mass of the wedge cylinder is 94 kg. The wedge cylinder contacts the water at a speed of V0 = 5.05 m/s.

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Fig. 8

The evolution of the free surface for the water entry of a wedge. The red lines represent the numerical results based on the present method.

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Fig. 9

Illustration of the jet shape after flow separation

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Fig. 10

The influence of the initial entry velocity on the free-surface shape at the initial closure of the cavity. In the numerical simulations, the mass, half beam c, and the deadrise angle of the wedge are 25 kg/m, 0.087 m, and 30 deg, respectively. The red, blue, and green lines correspond to initial entry velocities of 1 m/s, 5 m/s, and 9 m/s, respectively.

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Fig. 11

Nondimensional cavity closure period versus Froude number, V/gc. In the numerical simulations, the mass, half beam (c), and deadrise angle of the wedge are 25 kg/m, 0.087 m, and 30 deg, respectively.

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Fig. 12

Nondimensional submergence depth versus Froude number. In the numerical simulations, the mass, half beam c, and deadrise angle of the wedge are 25 kg/m, 0.087 m, and 30 deg, respectively.

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Fig. 13

Illustration of water entry when a cavity does not occur. At a later time, the free surface above the wedge will touch the top of the wedge.

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Fig. 14

The free-surface shapes for the different wedge masses at the closure of the cavity. In the numerical simulations, the initial entry velocity, half beam c, and deadrise angle of the wedge are 3 m/s, 0.087 m, and 30 deg, respectively. The red, blue, and green lines correspond to wedge masses of 25 kg/m, 50 kg/m, and 100 kg/m, respectively.

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Fig. 15

Nondimensional cavity closure period versus nondimensional mass. In the numerical simulations, the initial entry velocity, half beam c, and deadrise angle of the wedge are 3 m/s, 0.087 m, and 30 deg, respectively.

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Fig. 16

Nondimensional submergence depth versus nondimensional mass. In the numerical simulations, the initial entry velocity, half beam c, and deadrise angle of the wedge are 3 m/s, 0.087 m, and 30 deg, respectively.

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Fig. 17

The free-surface shapes for different deadrise angles at the closure of the cavity. In the numerical simulations, the mass, half beam c, and initial entry velocity of the wedge are set to 25 kg/m, 0.087 m, and 3 m/s, respectively. The red, blue, and green lines correspond to deadrise angles of 10 deg, 30 deg, and 50 deg, respectively.

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Fig. 18

Nondimensional cavity closure period versus deadrise angle. In the numerical simulations, the mass, half beam c, and initial entry velocity of the wedge are set to 25 kg/m, 0.087 m, and 3 m/s, respectively.

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Fig. 19

Nondimensional submergence depth versus deadrise angle. In the numerical simulations, the mass, half beam c, and initial entry velocity of the wedge are set to 25 kg/m, 0.087 m, and 3 m/s, respectively.

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