Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering Conference
Shanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

www.isope.org

Fundamental investigation on the applicability of higher-order consistent ISPH method
Yuma Shimizu, Hitoshi Gotoh, Abbas Khayyer, Kazuki Kita
Department of Civil and Earth Resources Engineering, Kyoto University
Kyoto, Kyoto, JapanABSTRACT

In the context of ISPH method, first-order consistency correction on thepressure gradient model has been widely used and its enhanced effects
have been presented in the literature (e.g. Lind et al., 2012; Khayyer etal., 2017a; Zheng et al., 2017). On the other hand, there have been few
ISPH studies for development of second-order operator schemes orhigh-order Laplacian ones, although these topics have been attracting a
lot of interest in other branches of projection-based particle methods,
MPS or CPM (Consistent Particle Method) (e.g. Gao et al., 2021; Luoet al., 2022).

The paper presents a high-order consistent ISPH fluid model foraccurate simulation of ocean engineering problems. The high-orderconsistent discretization schemes on differential operators are derived
through consideration of Taylor-series expansion until second-orderdifferential terms. The derived consistent discretization schemes areapplied to the calculations of Laplacian term of pressure in PPE andpressure gradient term in the momentum equation. To enhance andensure the accuracy, stability and conservation property of the model,
the enhanced schemes developed by our research team, namely HS,ECS and DS schemes, are also incorporated. The proposed high-orderconsistent ISPH fluid model is validated through reproduction of a setof numerical examples.

In light of these backgrounds, this work aims at development of a highorder consistent ISPH fluid model for reliable simulation ofincompressible free-surface fluid flow. The SPH-based derivation on
high-order consistent discretization schemes for differential operators isperformed with consideration of Taylor-series expansion. The obtained
consistent discretization schemes are adopted to the approximations of
Laplacian term of pressure in PPE and pressure gradient term in themomentum equation. The ISPH fluid model is further enhanced byincorporating several existing refined schemes. The performance of the
proposed ISPH fluid model is validated through reproduction of oceanengineering numerical examples.

KEY WORDS: particle method; Smoothed Particle Hydrodynamics;
projection method; consistency; high-order model; accuracy; energy
conservation.

INTRODUCTION
Lagrangian meshfree computational methods, or the so-called particle
methods such as SPH (Smoothed Particle Hydrodynamics; Gingold andMonaghan, 1977) or ISPH (Incompressible SPH; Shao and Lo, 2003),
have recently been attracting a lot of interest in various engineering
fields. Thanks to their Lagrangian meshfree description of motion, theparticle methods possess great advantages, e.g. being free from
calculation of advection term and stable/natural tracking of complex
moving boundaries. In the field of ocean engineering, one can easilyfind a number of existing applications of particle methods towards
violent free surface fluid flow and its interaction with rigid/deformablestructures as comprehensively reviewed in Luo et al. (2021) and Gotoh
et al. (2021).

NUMERICAL METHODEnhanced Incompressible SPH method
In the ISPH method, the motion of fluid particle is governed by thecontinuity and Navier-Stokes equations, which is written as follows:

In the context of SPH method, consistency has been considered as oneof the key challenges (SPHERIC Grand Challenges; Vacondio et al.,2020) and much effort has been devoted to enhancement of consistencyof particle-based differential operator discretization schemes (e.g.
Khayyer et al., 2011; Fatehi and Manzari, 2011; Duan et al., 2021). The
application of high-order consistent discretization schemes fordifferential operators would reduce the so-called “discretization error”(Quinlan et al., 2006) and result in enhancement of accuracy.

D   u  0Dt(1)Dup 2 u  gDt(2)where D/Dt indicates Lagrangian time derivative, u represents velocity
vector, t is time, p refers to pressure, ρ signifies fluid density, ν means
kinematic viscosity and g denotes gravitational acceleration vector.
These principal equations (Eqs. 1 and 2) lead to the Poisson Pressure
Equation (PPE) based on the Chorin’s projection method (Chorin,1968):

1967