L. Seif. Síntesis Tecnológica. V.2 N° 2 (2005) 79-83
DOI:10.4206/sint.tecnol.2005.v2n2-06

 

Numerical Modeling of 2-D Water Impact in One Degree of Freedom

 

MOHAMMED S. SEIF
SEYED M. MOUSAVIRAAD, SEYED H. SADDATHOSSEINI
VOLKER BERTRAM

Sharif University of Technology, professor, seif@sharif.edu, P.O.Box 11365-9567, Tehran, Iran.
Sharif University of Technology, Ph.D. students, seif@sharif.edu, P.O.Box 11365-9567, Tehran, Iran.
ENSIETA, professor, Volker.Bertram@ensieta.fr, 2 rue F. Verny, F-29806 Brest.

Abstract

The Reynolds-averaged Navier-Stokes equations are solved in a commercial flow solver to compute the water impact of two-dimensional simple geometries, entering the water in vertical motion of constant speed. The simulation shows better agreement with measurements than classical theories and potential flow solutions. Surface tension and viscosity are shown to be negligible in this application, but gravity is important. A tilt of wedges introduces an asymmetry in the flow increasing maximum pressures as compared to the perfectly symmetric case.

Keywords: water entry, Navier-Stokes, free surface.

 

1. INTRODUCCIÓN

In rough seas with large relative ship motion, slamming may occur with large water impact loads. Usually, slamming loads are much larger than other wave loads. Sometimes ships suffer local damage from the impact load or large-scale buckling on the deck. The first well documented accident of such kind was reported by [1]. For high-speed ships, even if each impact load is small, frequent impact loads accelerate fatigue failures of hulls. Thus, slamming loads may threaten the safety of ships. The expansion of ship size and new concepts in fast ships have decreased relative rigidity causing in some cases serious wrecks.

We will here briefly review the most relevant theories. Further recommended literature include [2] for a short review for practice, [3] for numerical and stochastic slamming theories, [4] for theories with strong mathematical focus, and [5] for a comprehensive compilation (more than 1000 references) of slamming literature.

Classical theories approximate the fluid as ideal (inviscid, irrotational, incompressible) and free of surface tension. In addition, it is assumed that gravity effects are negligible. This allows a (predominantly) analytical treatment of the problem in the framework of potential theory. For bodies with small deadrise angle, the problem can be linearized. Von Karman [6] was the first to study theoretically water impact (slamming). He idealized the impact as 2-d wedge entry problem on the calm water surface to estimate the water impact load on a seaplane during landing. Since the impact is so rapid, von Karman assumed very small water surface elevation during impact and negligible gravity effects. Since von Karman's impact model neglects the water surface elevation, the added mass and impact load are underestimated, particularly for small deadrise angle.

Wagner [7] derived a more realistic water impact theory. Although he assumed still small deadrise angles β in his derivation, the theory was found to be not suitable for β < 3º, since then air trapping and compressibility of water plays an increasingly important role. If β is assumed small and gravity neglected, the flow under the wedge can be approximated by the flow around an expanding flat plate in uniform flow with velocity V. Using this model, the velocity potential and its derivative with respect to y on the plate can be analytically given. The impact pressure on the wedge is determined from Bernoulli's equation. Wagner's theory can be applied to arbitrary shaped two-dimensional bodies as long as the deadrise angle is small, but no air trapping occurs. Wagner's theory is simple and useful, giving conservative estimates for peak pressures. Subsequently, Wagner’s theory has been expanded by various scientists over decades, but results are hardly changed despite considerably higher effort, [4].

All slamming theories treated so far were two-dimensional, i.e. they were limited to cross sections (of infinite cylinders). For most practical impact problems, the body shape is complex, the effect of gravity is considerable, or the body is elastic. In such cases, analytical solutions are very difficult or even impossible. This leaves CFD (computational fluid dynamics) as a tool. The three-dimensional treatment of slamming phenomena is still subject to research. It is reasonable to test and develop first numerical methods for two-dimensional slamming, before one progresses to computationally more challenging three-dimensional simulations. We present here such initial two-dimensional studies with a method that can be in principle easily extended to three dimensions and incorporate less model simplifications than the classical approaches. Previous CFD applications to slamming include [8] to [14].

2. THEORY

2.1. Analytical Method

For the water entry of a 2-d symmetric wedge with a small dead-rise angle β, Wagner’s theory gives good result in predicting the slamming pressure distribution. With a constant drop velocity V, the slamming pressure distribution is, [15]:

with ζ = x/c. ρ is the water density. Toyama [16] modified Wagner’s theory to calculate the transient pressure distribution on a 2-d asymmetric wedge in water entry (still neglecting spray effects). The pressure distribution along the wetted line is then determined by:

with ζ = (x-μc)/c . µ is a parameter describing the asymmetry:

with T = tg β2 / tg β1 , Fig.1, and the function f(T) defined by:

ċ is the time derivative of the half wetted length:

 

 
 
Fig.1: Asymmetric water entry of an edge body

 

2.2. CFD Method

Our CFD (computational fluid dynamics) results were obtained using the commercial RANSE (Reynolds averaged Navier-Stokes equations) solver Fluent version 5.2.3. Computations were reproduced in Fluent 6.0, but version 6.1.18 showed unphysical numerical instabilities with a water rising from the water surface well before the body would enter.

Fluent is based on the finite volume method, solving the flow simultaneously for water and air. The treatment for the two-phase flow uses an interface capturing method in combination with a volume of fluid (VOF) approach. In this method, an additional transport equation is solved for the volume percentage of air in each cell. The two-phase fluid is treated as one effective fluid with effective viscosity and density determined by a weighted sum of air and liquid in each cell. Turbulence is modeled using a Reynolds stress model (RSM).

3. APPLICATION: SYMMETRIC CASE

To evaluate the accuracy of the numerical model, a circular cylinder of radius R=5.5 m and constant drop velocity of 10 m/s is considered. The computational domain considered was a semi-circular lower region with radius 8 times larger than the body, and a upper rectangular region extending 3 times the radius of the body. A structured grid with finer cells near the body was used, Fig.2, using approximately 90000 cells in total. Grid independence studies verified that resolution and lateral extension of the grid were sufficient to make grid errors insignificant.

 

 
 
Fig.2: Detail of grid around semi-cylinder in water entry

 

The side walls of the domain were treated as planes of symmetry. At the lower inlet, vertical velocities were specified giving constant flux, making the water rise at the water entry velocity. At the upper outlet, zero pressure was enforced. At the body surface, a wall-condition enforced the no-slip condition. The PISO (pressure implicit with splitting of operator) scheme for pressure-velocity coupling and the QUICK (quadratic upstream interpolation of convective kinematics) scheme for the discretization of momentum and turbulence parameters. A time step size of 0.01 s was used to catch all detail information of solution process.

Numerical results compare the slamming force in the form of the non-dimensional coefficient Cs:

Fs is the slamming force. Fig.3 compares our CFD results with other data. Karman’s theory predicts Cs=π at initial contact, while the Wagner’s theory predicts Cs=2π. Experimental data [17] vary between Cs=5.5 and Cs=6.5. Potential flow results based on a finite difference method, [8], lie closer to the experimental results, but the best agreement is for the CFD method.

The time history of the computed slamming coefficient, Fig.4, and the computed pressure at the lowest point of the section, Fig.5, show some unphysical fluctuations due to numerical instabilities in the beginning, but these remain relatively small. Fig.5 shows the typical rapid increase of pressure (up to 10 times the hydrostatic pressure). The pressure peak is important for structural dimensioning, but the simulations will over-predict this pressure peak as hydro-elastic deflections in reality reduce the maximum pressures. This is subject to current research, [18].

 

 
Fig.3: Slamming coefficient for different h/R values

 

 
 
Fig.4: Slamming coefficient, semi-cylinder water entry

 

 
 
Fig.5: Pressure at lowest point, semi-cylinder water entry

 

As the sphere immerses, the pressure peak gradually decreases and its effective area increases. Fig.6 shows that the maximum pressure gradient and maximum pressure appear at the spray root. This remains so until the spray detaches. The problem was solved including surface tension, gravity and turbulence. Then these three features were individually switched off and computations repeated. Only gravity was found to have a significant effect in this case, with results for the slamming force coefficient being approximately 23.2% lower without gravity.

 

 
 
Fig.6: Pressure contours at various time steps

 

4. APPLICATION: ASYMMETRIC CASE

The often studied symmetric impact in slamming is actually not representative of real conditions. In reality, irregular seaways and ship motions (notably roll) introduce an asymmetry in the water entry leading to higher hydrodynamic pressures for chined hulls, [19]. [15] pointed out the importance of asymmetric water impact in the assessment of design loads for high-speed vessels.

Our numerical test case was a two-dimensional wedge with a dead-rise angle of 30° and drop velocity 5 m/s. The simulation was performed for symmetric condition and a heel angle of 10°. A square structured grid is used with approximately 62000 total cells. The domain was extended 7 times the body width to the side and 4 times to the inlet and outlet boundaries. The boundary and inlet conditions are as for the semi-circle test case described above. Fig.7 compares analytical and numerical results for the symmetric water entry. The pressure is made non-dimensional by stagnation pressure based on entry velocity. Wagner’s theory gives a maximum hydrodynamic pressure 38% higher than our CFD simulation. The difference will be largely due to assorted simplifications in Wagner’s theory. Since the difference is much more in larger values of x, where the spray takes place, the effect of spray is the most important one.

Fig.8 compares analytical and numerical results for asymmetric water entry. Toyama’s method, [16] also overestimates the hydrodynamic pressure and it is possible to get more exact values from numerical modeling. The asymmetric water entry increased in this case the maximum pressure to almost twice its value for symmetric water entry. The pressure level increases on the side with the lower dead-rise angle. Asymmetric loads will also induce global torsion moments in the ship.

 

 
 
Fig.7: Analytical and numerical results of symmetric impact

 

The slamming force increases until the spray root separates from the body, Fig.9. Then the force decreases to reach the hydrostatic value. The hydrodynamic pressure increases with time in both sides and the pressure value is much higher on the side of the body with the lower dead-rise angle. The surface capturing method employed is capable of reproducing the fine spray formation.

 

 
 
 
Fig.8: Analytical and numerical results of asymmetric impact

 
 
Fig.9: Non-dimensional slamming force coefficient for asymmetric wedge entry

 

5. CONCLUSION

In usual design procedures, approximate formulas are used for slamming load prediction. Our study showed that considerable differences exist between these results and CFD results even for simple geometries. Asymmetric impact has a considerable effect on value and location of maximum pressure. The possibility to extend the numerical approach to complex three-dimensional geometries and hydro-elastic coupling makes it very promising, but more and better validation data are needed to support the research in this direction.

REFERENCES

[1] Y. Yamamoto, K. Iida, T. Fukasawa, T., T. Muramami, M. Arai, and M. Ando, (1985), "Structural damage analysis of a fast ship due to bow flare slamming," Int. Shipb. Progress 32, pp.124-136, 1985.

[2] V. Bertram, Practical Ship Hydrodynamics, Butterworth+Heinemann, Oxford, 2000.

[3] S. Mizoguchi and K. Tanizawa, "Impact wave loads due to slamming - A review," Ship Technology Research 43, pp.139-154, 1996.

[4] A. Korobkin, "Water impact problems in ship hydrodynamics," in Advances in Marine Hydrodynamics, Comp. Mech. Publ., pp.323-371, 1996.

[5] N.N., "Hydrodynamic impact on displacement ship hulls," Ship Structure Committee, SSC-385, SSC-385A, 1995.

[6] T. von Karman, "The impact on seaplane float during landing," NACA TN321, 1929.

[7] H. Wagner (1932), "Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten," ZAMM 12, pp.193-215, 1932.

[8] M. Arai, L.Y. Cheng and Y. Inoue, "A computing method for the analysis of water impact of arbitrary shaped bodies," J. Society Naval Architects Japan 176, 1994

[9] M. Arai, L.Y. Cheng, Y. Inoue and T. Miyauchi, "Numerical study of water impact loads on catamarans with asymmetric hulls," in Proc. FAST’95, Travemünde, 1995.

[10] M. Arai, L.Y. Cheng, Y. Inoue, T. Miyauchi and T. Ishikawa, "A study on slamming characteristics and optimization of bow forms of ships," in Proc. PRADS’95, Seoul, 1995.

[11] C. Schumann, "Volume of fluid computations of water entry of bow sections, " in Proc. Euromech Coll. 374: Recent Comp. Developments in Steady and Unsteady Naval Hydrodyn., Poitiers, 1998.

[12] S. Muzaferija and M. Peric, Computation of free-surface flows using interface tracking and interface-capturing methods, in Proc. Nonlinear Water Wave Interaction, Advances in Fluid Mech., Comp. Mech. Publ., 1998.

[13] S. Muzaferija, M. Peric, P. Sames and T. Schellin, "A two-fluid Navier-Stokes solver to simulate water entry," in Proc. 22nd Symp. Naval Hydrodyn., Washington, 1998.

[14] P. Sames, T. Schellin, S. Muzaferija and M. Peric, "Application of two-fluid finite volume method to ship slamming," in Proc. OMAE’98, Lisbon, 1998.

[15] J. Hua, J. Wu, and W. Wang, "Effect of asymmetric hydrodynamic impact on the dynamic response of a plate structure," J. Marine Science and Technology 8/2, pp.71-77, 2000.

[16] Y. Toyama (1993), "Two-dimensional water impact of unsymmetrical bodies," J. Soc. Naval Arch. Japan 173, 1993.

[17] I.M.C. Campbell and P.A. Weinberg, "Measurement of parameters affecting slamming," Wolfson Unit, University of Southampton, Report No. 440, 1980.

[18] A. Constantinescu, A. Neme, V. Bertram, Y. Doutreleau, and B. Peseux, "Finite element simulations of dihedral and conical shell structures in slamming," FIV Conf., Paris, 2004.

[19] A. Rosen and O. Rutgersson, "Full-scale trials on a small high speed naval craft with focus on slamming," in Proc. OMAE’98, Lisbon, 1998.