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
Abrate S.
2011
Hull slamming
Applied Mechanics Reviews
64
(6)
1 -
35

Batchelor G.K.
1974
Introduction to fluid dynamics
Cambridge University Press
Cambridge

Colagrossi A.
,
Landrini M.
2003
Numerical simulation of interfacial flows by smoothed particle hydrodynamics
Journal of Computational Physics
191
(2)
448 -
475

Dalrymple R.A.
,
Rogers B.D.
2006
Numerical modeling of water waves with the SPH method
Coastal Engineering
53
(2-3)
141 -
147

Dalrymple R.A.
,
Herault A.
2009
Levee breaching with GPU-SPHysics code
Fourth International SPHERIC Workshop
Nantes, France
May 2009
27 -
29

Dilts G.A.
1999
Moving-least-squares-particle hydrodynamics, I. consistency and stability
International Journal for NumericalMethods in Engineering
44
(8)
1115 -
1155

Dobrovol’skaya Z.N.
1969
On some problems of similarity flow of fluid with a free surface
Journal of Fluid Mechanics
36
(4)
805 -
829

Ferrari A.
2010
SPH simulation of a free surface flow over a sharp crested weir
Advanced in Water Resources
33
(3)
270 -
276

Ghadimi P.
,
Dashtimanesh A.
,
Djeddi S.R.
2012
Study of water entry of circular cylinder by using analytical andnumerical solutions
Journal of the Brazilian Society of Mechanical Sciences and Engineering
34
(3)
225 -
232

Ghadimi P.
,
Farsi M.
,
Dashtimanesh A.
2012
Study of various numerical aspects of 3D-SPH for simulation of thedam break problem
Journal of the Brazilian Society of Mechanical Sciences and Engineering
34
(4)
486 -
491

Ghadimi P.
,
Dashtimanesh A.
,
Farsi M.
,
Najafi S.
2012
Investigation of free surface flow generated by a planing flatplate using smoothed particle hydrodynamics method and FLOW3D simulations
Journal of Engineering for the MaritimeEnvironment
227
(2)
125 -
135

Ghadimi P.
,
Feizi Chekab M.A.
,
Dashtimanesh A.
2013
A numerical investigation of the water impact of an arbitrarybow section
ISH Journal of Hydraulic Engineering
9
(3)
186 -
195

Gómez-Gesteira M.
,
Cerqueiro D.
,
Crespo C.
,
Dalrymple R.A.
2005
Green water overtopping analyzed with a SPHmodel
Ocean Engineering
32
(2)
223 -
238

Gómez-Gesteira M.
,
Dalrymple R.A.
2004
Using a Three-Dimensional Smoothed Particle Hydrodynamics Methodfor Wave Impact on a Tall Structure
Journal of Waterway Port, Coastal and Ocean Devision
130
(2)
63 -
69

Gómez-Gesteira M.
,
Rogers B.D.
,
Dalrymple R.A.
,
Crespo A.J.C.
2010
State-of-the-art of classical SPH for freesurface flows
Journal of the Hydraulic Research
48
6 -
27

Kai G.
,
Liu H.
,
Wang B.L.
2009
Water entry of a wedge based on SPH model with an improved boundary treatment
Journal of Hydrodynamics
21
(6)
750 -
757

Gotoh H.
,
Shao S.
,
Memita T.
2004
SPH-LES model for numerical investigation of wave interaction with partiallyimmersed breakwater
Coastal Engineering Journal
46
(1)
39 -
63

Greenhow M.
1988
Water entry and exit of horizontal circular cylinders
Applied Ocean Research
10
(4)
191 -
198

Gringold R.
,
Monaghan J.J.
1977
Smoothed particle hydrodynamics: theory and application to non-spherical stars
Monthly Notices of the Royal Astronomical Society
181
375 -
388

Issa R.
2004
Numerical assessment of the Smoothed Particle Hydrodynamics grid-less method for incompressibleflows and its extension to turbulent flows. Ph.D. Thesis
University of Manchester Institute of Science and Technology(UMIST)

Lo E.Y.M.
,
Shao S.
2002
Simulation of near-shore solitary wave mechanics by an incompressible SPH method
Applied Ocean Research
24
(5)
275 -
286

Lucy L.
1977
A numerical approach to testing of the fusion hypothesis
Astronomical Journal
88
1013 -
1024

Monaghan J.J.
1992
Smoothed particle hydrodynamics
Annual Review of Astronomy and Astrophysics
30
543 -
574

Monaghan J.J.
1994
Simulating free surface flows with SPH
Journal of Computational Physics
110
(2)
399 -
406

Muzaferija S.
,
Perie M.
,
Sames P.
,
Sehellin T.
1999
A two-fluid navier-stokes solver to simulate water entry
Twenty-Second Symposium on Naval Hydrodynamics
Washington, DC
638 -
651

Oger G.
,
Doring M.
,
Alessandrini B.
,
Ferrant P.
2006
Two-dimensional SPH simulations of wedge water entries
Journal of Computational Physics
213
(2)
803 -
822

Oger G.
,
Touzé D.L.
,
Alessandrini B.
,
Maruzewski P.
2008
A new parallelized 3D SPH model: resolution of waterentry problems and scalability study
ERCOFTAC Bulletin
76
35 -
38

Panizzo A.
2004
Physicfal and numerical modeling of sub-aerial landslide generated waves. Ph.D thesis
Universitadegli Studi di L'Aquila

Pope S.B.
2000
Turbulent flows
Cambridge University Press
United Kingdom

Rogers B.D.
,
Dalrymple R.A.
,
Stansby P.K.
2008
SPH modeling of floating bodies in the surf zone
Proceeding of31st Conference on Coastal Engineering
Hamburg, Germany
204 -
215

Sedov L.
1934
The impact of a solid body floating on the surface of an incompressible fluid, CAHI Report 187
CAHI
Moscow

Shao S.
2010
Incompressible SPH flow model for wave interactions with porous media
Coastal Engineering
57
304 -
316

Tveitnes T.
,
Fairlie-Clarke A.C.
,
Varyani K.
2008
An experimental investigation into the constant velocity water entryof wedge-shaped sections
Ocean Engineering
35
(14-15)
463 -
1478

Vandamme J.
,
Zou Q.
,
Reeve D.E.
2011
Modeling floating object entry and exit using smoothed particle hydrodynamics
Journal of Waterway, Port, Coastal, Ocean Engineering
137
(5)
213 -
224

Veen D.
,
Gourlay T.
2012
A combined strip theory and smoothed particle hydrodynamics approach for estimatingslamming loads on a ship in head seas
Ocean Engineering
43
64 -
71

Vepa K.S.
,
Van Nuffel D.
,
Van Paepegem W.
2011
Pressure predictions during water entry of a 2D rigid cylinderusing SPH method
26th International Workshop on Water Waves and Floating Bodies (IWWWFB)
Athens
197 -
200

Von Karman T.
1929
The impact of seaplane floats during landing. NACA TN 321
National Advisory Committee for Aeronautics
Washington, USA
1 -
16

Wagner H.
1932
Phenomena associated with impacts and sliding on liquid surfaces
Mathematik und Mechanik
12
(4)
193 -
215

Yan S.
,
Ma Q.W.
2007
Numerical simulation of fully nonlinear interaction between steep waves and 2D floatingbodies using the QALE-FEM method
Journal of Computational Physics
221
(2)
666 -
692

Zhang Y.
,
Zou Q.
,
Greaves D.
,
Reeve D.
,
Hunt-Raby A.
,
Graham D.
,
James P.
,
Lv X.
2010
A level set immersedboundary method for water entry and exit
Communications in Computational Physics
8
(2)
265 -
288

Zhao R.
,
Faltinsen O.M.
1993
Water entry of two-dimensional bodies
Journal of Fluid Mechanics
246
593 -
612

Zhao R.
,
Faltinsen O.M.
,
Aarsnes J.
1997
Water entry of arbitrary two-dimensional sections with and without flowseparation
21st Symposium on Naval Hydrodynamics
Washington DC
408 -
423

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}