Finding the best combination of numerical schemes for 2-D SPH simulation of wedge water entry for a wide range of deadrise angles

International Journal of Naval Architecture and Ocean Engineering.
2014.
Sep,
6(3):
638-651

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Published : September 30, 2014

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INTRODUCTION

Water impact pressure arising during the impact and especially during the initial stages of water entry when the maximum pressure occurs, is extremely important in the design of marine structures.
Abrate (2011)
has presented an in-depth review of the state of the art on the hull slamming or water entry problem. Based on this article and other recent papers, research conducted in the field of water entry is hereby surveyed.
Initial study of water entry problem for determination of water entry impact forces and pressures was done by
Von Karman (1929)
. He used simple principles such as conservation of momentum and the concept of added mass.
Wagner (1932)
continued the theoretical solution of water impact problem. He analyzed the vertical water entry of a 2D wedge of small deadrise angle.
Sedov (1934)
, on the other hand, extended Wagner’s work to study the impact of 2D wedge of larger deadrise angle.
In order to simulate the water entry problem, many numerical methods have been implemented by different researchers (
Greenhow, 1988
;
Zhao et al., 1997
;
Yan and Ma, 2007
;
Zhang et al., 2010
;
Ghadimi et al., 2012
;
Ghadimi et al., 2013
). Common methods of simulation have been grid-based. These schemes have encountered some computational difficulties when phenomena such as flow surface piercing, separation and large movement are involved in the modeling. As a result of this, it has become difficult to capture the movement of fluid and free surface flow due to a moving body. To overcome this difficulty, Lagrangian computational method such as Smoothed Particle Hydrodynamics (SPH) was adopted which is a mesh free method. SPH method was first developed for simulation of the particle motion in astrophysics by
Gringold and Monaghan (1977)
and
Lucy (1977)
. Later, SPH was employed for simulating flow through porous media, large deformation of free surface flows, impact problem, Fluid Structure Interaction (FSI) and water wave generation (
Shao, 2010
;
Dalrymple and Herault, 2009
;
Ferrari, 2010
;
Gómez-Gesteira et al., 2005
;
Gómez-Gesteira and Dalrymple, 2004
;
Gómez-Gesteira et al., 2010
;
Monaghan, 1994
;
Rogers et al., 2008
).
Application of the SPH method has also been extended to simulation of water impact problem.
Oger et al. (2006)
modeled symmetric wedge water impact of 30
Summary of wedge water entry studies by SPH method.

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SPH FORMULATION

- Integral presentation of a function

SPH interpolation of a quantity
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- Particle approximation

To apply SPH interpolation for a fluid flow, computational domain is divided into a set of particles, as shown in
Fig. 1
. Every element has a mass
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GOVERNING EQUATIONS

Fundamental physical laws of conservation are the basic governing equations of fluid dynamics which are conservation of mass, momentum and energy. For this problem, conservation of mass and momentum should be used.
In the SPH method, derivative of the density for particle
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NUMERICAL FORMULATION OF SPH METHOD

- Viscosity treatment

To consider the diffusion term in the momentum equation, three different approaches including (1) artificial viscosity, (2) laminar viscosity and (3) laminar viscosity plus sub-particle scale turbulence, are investigated.
- Artificial viscosity

Monaghan (1992)
introduced the artificial viscosity. Based on this concept, the momentum conservation equation can be written as
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- Laminar viscosity

Momentum equation in the form of laminar viscous stresses is given by
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- Laminar viscosity and sub-particle scale (SPS) turbulence

Gotoh et al. (2004)
introduced Sub-Particle Scale approach for modeling the turbulent effects. Momentum conservation equation can be presented as
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- Time stepping scheme

SPH algorithm reduces the original partial differential equations to a set of ordinary differential equations. Then, any time stepping scheme for ordinary differential equations can be used.
In this study, Predictor-Corrector algorithm is used. The momentum, density and position equations can be rewritten in the following form:
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- Density filter

In the SPH simulation, pressure oscillations may be observed in the obtained results. To overcome this difficulty,
Colagrossi and Landrini (2003)
applied a filter over the density of the particles and re-assigned a density to each particle. In the current paper, zero order filter and first order filter are considered as compiling options.
The following procedure is known as zero order filter and is applied every 100 time steps:
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- Pressure calculation

In the studies conducted by
Monaghan (1994)
and
Batchelor (1974)
, the relationship between pressure and density is assumed to be
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RESULTS AND DISCUSSIONS

- Computational setup

The geometry of the considered problem is illustrated in
Fig. 2
. Width and height of the domain are equal to 1
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Compiling options.

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- Results

In the present work, pressure distribution of the wedge water entry is studied. Nine tests including entry of constant velocity of wedge of 45
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Comparison of SPH results and BEM solutions.

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CONCLUSIONS

In this paper, wedge water entry for a wide range of deadrise angles is simulated by SPH method. In order to find the best numerical configuration, nine different test cases are considered. These cases describe the water entry of wedge of 45
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Citing 'Finding the best combination of numerical schemes for 2-D SPH simulation of wedge water entry for a wide range of deadrise angles
'

@article{ E1JSE6_2014_v6n3_638}
,title={Finding the best combination of numerical schemes for 2-D SPH simulation of wedge water entry for a wide range of deadrise angles}
,volume={3}
, number= {3}
, journal={International Journal of Naval Architecture and Ocean Engineering}
, publisher={The Society of Naval Architects of Korea}
, author={Farsi, Mohammad
and
Ghadimi, Parviz}
, year={2014}
, month={Sep}