Elsevier

Ocean Engineering

Volume 209, 1 August 2020, 107533
Ocean Engineering

Lagrangian finite-difference method for predicting green water loadings

https://doi.org/10.1016/j.oceaneng.2020.107533Get rights and content

Highlights

  • Meshless weighted-least squares operators are stable and efficient for solving the pressure Poisson equation.

  • Fluid points are projected to static or moving boundary geometry, where the Neumann boundary condition is imposed.

  • Mass conservation remains stable by imposing equidistant neighbouring points during the Lagrangian advection.

  • The resulting method is validated for simulating violent flows with impacts and free-surface fragmentation, i.e. green water events.

Abstract

A meshless CFD method is proposed for numerical simulation of incompressible flows and estimation of hydrodynamic loads during green water events. In this paper, governing equations are described in Finite Differences context, where spatial operators are derived using Weighted-Least Squares. Volume-conservative Lagrangian advection is handled by solving a set of geometrical constraints, while adjusting to moving boundaries represented by triangulated meshes. The validation of the method is performed by simulating four green-water experiments on fixed and moving structures, and comparing the kinematics of flow and hydrodynamic loads during impacts. It is shown that the method accurately reproduces patterns of waves that are wetting the deck, as well as the pressure distribution along the deck. The resulting method intrinsically handles violent fluid–structure interaction with free surface fragmentation, while providing second-order accurate pressure field.

Introduction

A green water event is characterised by some amount of rapidly moving water, which interacts with a moving ship. These events include violent fluid–structure interaction, and therefore need to be examined with the physics of the structure and fluid flow considered in a coupled manner to accurately estimate hydrodynamic loads during the interaction. There are hundreds of scientific papers and research reports that deal with green water effects, i.e. the physics of water on deck and possible prediction methods. Temarel et al. (2015) in their review conclude that the green water problem has been investigated experimentally and numerically, but with very limited success. The issues arise due to the fact that green water is very difficult to measure either under laboratory conditions or in the field, and very difficult to simulate numerically. The problem, due its fast moving, multiphase and highly turbulent nature, is not very amenable for accurate numerical simulation, although three-dimensional (3D) models help to understand and simulate the mechanisms of green water impinging on deck. Computations of green-water loads with various wave headings are still a challenge.

Yamasaki et al. (2005) developed a Finite Difference Method (FDM) to predict the water impact pressure caused by green water phenomena, where the density function method was employed in the framework of a locally refined overlapping grid system. Colicchio et al. (2010) analysed the water shipping caused by head sea waves for a Floating Production Storage and Offloading (FPSO) vessel at rest with a 3D domain-decomposition strategy, where a linear potential flow seakeeping analysis of the vessel is coupled with a local nonlinear rotational-flow investigation for the prediction of water-on-deck phenomena. Greco and Lugni (2012) and Greco et al. (2012) conducted a synergic 3D experimental and numerical investigation for wave–ship interactions involving the water-on-deck and slamming phenomena. A weakly nonlinear external solution for the wave–ship interactions was combined with a two-dimensional (2D) shallow-water approximation. Lee et al. (2017) studied the behaviour of green water impacting a fixed rectangular structure, and investigated the flow kinematics of a series of experiments and CFD simulations based on the Finite Volume Method (FVM). Gatin et al. (2018) validated a FVM solver for green water simulations by employing Volume of Fluid (VOF) and Ghost Fluid Method (GFM) to discretise the free-surface boundary conditions at the interface. Mesh-based methods for solving Partial Differential Equations (PDEs) have achieved adequate success and they have been applied in computational ship hydrodynamics. During the ship–wave interaction, simulation requires large mesh deformations by cautious node movements or re-meshing of deforming areas to avoid mesh tangling and loss of its regularity. Alternatively, the overset grid technique can be used for rigid body motion in free floating-body simulations (Östman et al., 2014).

Lagrangian meshless or mesh-free methods have emerged with the goal of avoiding difficulties encountered in conventional mesh-based methods. Mesh-free methods adapt to domains with complex and changing geometries and moving phase boundaries without the complexity that is usually required within methods that rely on topological data structures. Gotoh and Khayyer (2018) made a review of state-of-the-art meshless methods that are applicable on ocean engineering problems, taking into account the problem of green water. Dalrymple and Rogers (2006) analysed the potential of the SPH method for wave–structure interactions by reproducing breaking waves that overtop the deck. Shibata et al. (2009) have proposed a variant of the Moving Particle Semi-implicit (MPS) method, and compared the estimated pressure with the experiments for a ship in head waves. The investigation has shown that there is still some lack of agreement in terms of both pressure and forces acting on the deck due to relevant oscillations in time. Shibata et al. (2012) extended the MPS method for a 3D ship motion under high wave-height conditions where shipping of water occurs. The nonlinear effect of shipping water was successfully simulated by the MPS method, although quantitative differences between the calculated and experimental results still remained. Bellezi et al., 2013a, Bellezi et al., 2013b analysed the green water phenomenon on 3D models using the MPS method. They focused on the validation of the method comparing the numerical results with experimental results for green water on reduced scale models. Gómez-Gesteira et al. (2005) analysed the phenomena within the framework of the Smoothed Particle Hydrodynamics (SPH) method, and noted how the flow in the wave crest is split into two, showing a different behaviour above and below the deck. Le Touzé et al. (2010) applied the SPH method to predict the fluid behaviour for two different dynamic flooding scenarios: the interaction between a ship and travelling waves, and the transient flooding behaviour that occurs on a side collision between two ships. Pakozdi et al. (2012) showed the feasibility of the SPH method to give more detailed forecast of the hydrodynamics on the deck than the simplified water-on-deck estimation.

The green-water problem arising due to complex physics is a significant problem for ships and offshore structures, and the listed numerical methods give limited insight in determining the occurrence and magnitude of green water loads. Consequently, classification societies seek for new procedures that may approve and improve designs of new buildings. Commonly used meshless methods have different advantages and disadvantages (Gotoh and Khayyer, 2018). A promising class of meshless methods concerning the solution accuracy are Generalised Finite Difference Methods (GFDM), which are strong form methods that usually rely on Moving-Least Squares (MLS), as they can be extended to reach arbitrary order of accuracy. Since GFDMs do not require basis functions or quadrature, boundary and interface conditions can be implemented in a straightforward manner. Tiwari and Kuhnert (2003) introduced a Lagrangian GFDM, called Finite Pointset Method (FPM), for simulating incompressible flows based on Chorin’s projection method. The FPM was recently generalised to work on dynamically changing interfaces (Suchde and Kuhnert, 2019), it was successfully applied to FSI problems (Kuhnert et al., 2019), and for simulating free surface flow about ships (Lu et al., 2016). Finite Point Method, not to be confused with FPM, is a mesh-free Eulerian GFDM that stabilises the discrete differential equations by balancing momentum in the finite control domain (Oñate et al., 1996), which was extended to Lagrangian formulation for solving dam-break problems in 2D (Rodolfo Idelsohn et al., 2001). Fu et al. (2020) applied a GFDM to simulate interaction of waves and a cylinder array, which was reduced to a 2D formulation. Wang et al. (2018) proposed a Lagrangian Mesh-free Finite Difference Particle Method (FDPM) with variable stencil sizes for solving wave equations. Suchde and Kuhnert (2017) analysed volume conservation errors when performing Lagrangian advection in GFDMs. One of keys problems with MLS-based GFDMs, besides volume conservation, is that they express the vector space of polynomials up to an arbitrary order, which are approximated by solving a linear system for each point at each time step. Size of the system depends on number of neighbours and basis vector, which may be costly for 3D time-dependent problems. In addition, GFDMs may have difficulties when describing linear system of equations, i.e. the system may be hardly solvable since the matrix may not be an M-matrix. Seibold (2008) investigated least-squares approaches that fail to yield positive stencils, and introduced a technique of selecting neighbours that ensure an M-matrix structure for Poisson problems.

This paper introduces a new Lagrangian GFDM, built as a compromise between fast but oscillatory SPH/MPS methods, and slow but accurate MLS-based methods. The method is based on the Weighted-Least Squares (WLS) operators, which require slight computational cost each time step in a unsteady simulation, but still yield second-order convergence for Poisson problems on irregular point clouds as MLS-based methods do (Basic et al., 2018). Moreover, the stencils produce an M-matrix for the system of Navier–Stokes equations (NSE), which are represented in the Poisson pressure equation (PPE) form instead of the commonly used projection method. Finally, it is worth noting that the Lagrangian advection is volume-conservative. These are important ingredients for obtaining accurate flow kinematics and pressure fields for problems in marine hydrodynamics. The method is based on the following hypotheses:

  • The Lagrangian description of flow naturally describes the advection of fluid with the free surface, as compared to the Eulerian description of flow.

  • It is possible to solve the nonlinear problem of green water with high-fidelity by solving NSE in PPE form, based on the Lagrangian description of flow and WLS.

  • The global domain can be decomposed to improve computational efficiency, by coupling the Lagrangian nonlinear region with some simpler solver for the outside region.

The remainder of the paper is organised as follows. Section 2 describes the governing equations, their numerical counterparts, and the solution algorithm. The method is validated by 2D numerical experiments described in Section 3, and by 3D simulations of wave–ship interactions of a fixed and a heaving FPSO model in Section 4. The results of the simulations are further discussed in Section 5, and the conclusions are drawn in Section 6.

Section snippets

Governing equations

For an incompressible fluid, the NSE form the time-dependent initial–boundary-value problem: DuDt=1ρp+ν2u+g,xΩ,u=0,xΩΓ,u=U,xΓ,ut=0=u0,xΩ, where DDt is the Lagrangian or material derivative, u is the velocity, ρ is the fluid density, p is the pressure, ν is the kinematic viscosity of the fluid, g is the external acceleration acting on the fluid, U is the boundary velocity, and u0 is the initial velocity. The time and spatial dependency from Eqs. (1)–(4) is implied, and the fluid is

Wave patterns

Hernández-Fontes et al. (2018) experimentally investigated the generation of isolated events of green water on a fixed structure using the wet dam-break approach. The authors verified that it is possible to reproduce different types of green water resembling those obtained with unbroken regular waves reported in literature. The experimental setup is shown in Fig. 7, where H=450 mm, W=335 mm, H1=150 mm, L1=195 mm, L2=505 mm, and L3=300 mm. The testing matrix is presented in Table 1. The authors

Fixed model

In this section, a validation of the numerical method for 3D green water loads on a fixed structure is conducted. Several experimental results on green water have been published, but there is small amount of data available to be useful for CFD validation. In this study, the experiments conducted by Lee et al. (2012) are numerically reproduced in order to understand the physics of green water and to quantify the pressure distributions due to green water on deck. The model geometry and the

Discussion of the results

A green-water event can comprise of complex phenomenons, i.e. a breaking wave on the moving deck impulsively impacts a moving wall while generating a jet, falling pile of water, spray, etc. The validation cases of isolated events of wetting the deck, and periodic green water events in regular seas, were successfully simulated in a short amount of time. The study showed that wave impact pressures and pressure impulses can be predicted with very good accuracy. Furthermore, a test with moving ship

Conclusions

Predicting the effects of green water on the stability and structure of the vessel is a challenging problem, which needs to be assessed in the design process. By analysing the limitations of currently used methods, a rationale was developed and the research resulted in a novel design tool, i.e. a methodology for incompressible flows suitable for simulating green water events. The introduced methodology can be described as: meshless, Lagrangian, volume-conservative, FD-based, second-order

CRediT authorship contribution statement

Josip Bašić: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Writing - original draft, Writing - review & editing, Visualization. Nastia Degiuli: Investigation, Methodology, Writing - original draft, Writing - review & editing, Supervision, Resources. Šime Malenica: Investigation, Methodology, Writing - original draft, Writing - review & editing. Dario Ban: Methodology, Investigation, Writing - review & editing, Supervision, Resources.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors thank Inno Gatin for interesting discussions and for sharing details on their numerical experiments, and prof. Shin Hyung Rhee and Jeonghwa Seo for sharing their experimental data-set.

Third author acknowledges the support from a National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) through GCRC-SOP (Grant No. 2011-0030013).

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