Hydrodynamic Coefficients for a 3D Uniform Flexible Barge Using Weakly Compressible Smoothed Particle Hydrodynamics
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Abstract
The numerical modelling of the interactions between water waves and floating structures is significant for different areas of the marine sector, especially seakeeping and prediction of waveinduced loads. Seakeeping analysis involving severe flow fluctuations is still quite challenging even for the conventional RANS method. Particle method has been viewed as alternative for such analysis especially those involving deformable boundary, wave breaking and fluid fragmentation around hull shapes. In this paper, the weakly compressible smoothed particle hydrodynamics (WCSPH), a fully Lagrangian particle method, is applied to simulate the symmetric radiation problem for a stationary barge treated as a flexible body. This is carried out by imposing prescribed forced simple harmonic oscillations in heave, pitch and the two and threenode distortion modes. The resultant, radiation force predictions, namely added mass and fluid damping coefficients, are compared with results from 3D potential flow boundary element method and 3D RANS CFD predictions, in order to verify the adopted modelling techniques for WCSPH. WCSPH were found to be in agreement with most results and could predict the fluid actions equally well in most cases.
Keywords
Weakly compressible Fluid structure interaction Smoothed particle hydrodynamics Seakeeping Hydroelasticity Radiation1 Introduction
During fluidstructure interaction (FSI), fluid forces acting on the structure result in the structure moving and deforming, which in turn affects the flow boundary conditions, hence fluid motion. This interaction, also known as twoway coupling, has been studied in marine, offshore, civil and coastal engineering. The majority of approaches used are based on potential flow where nonlinear effects are either ignored or allowed for through various assumptions on the body and free surface boundary conditions, leading to a range of partly to fully nonlinear methods (ISSC 2012). However, numerical models based on potential flow theory are unable to deal with extreme free surface deformations, as well as FSI problems where the effects of viscosity, turbulence and compressibility are significant. In order to address such shortcomings, there have been several attempts to solve nonlinear rigid body FSI and benefitting from rapid progress in RANS code development, using either the finite difference method or the finite volume method (FVM) (Weymouth et al. 2005; Wilson et al. 2006; Castiglione et al. 2011; Hochkirch and Mallol 2013; Tezdogan et al. 2015). However, most of these methods are Eulerian and, by and large, ineffective in the case of extreme events of wave breaking and water spray. Therefore, Lagrangian meshless methods are viewed as alternatives in providing accurate numerical solutions to improve inadequacy of meshbased discretisation.
Smoothed particle hydrodynamics (SPH) is a meshfree, Lagrangian method whereby the computational domain is represented by a set of interpolation points called particles where the fluid medium is discretised by the interaction between particles rather than grid cells (Shadloo et al. 2012; Chen et al., 2013). Each particle carries an individual mass, velocity, position and any other requisite physical characteristics, which evolve over time through the governing equations. All particles have a kernel function to define their range of interaction, while the hydrodynamic variables are defined by integral approximations.
There are limited number of studies performing seakeeping analysis using particle methods, mainly focusing on extreme events such as slamming and green water. For example, Shibata et al. (2009) used the moving particle semiimplicit (MPS) method to simulate shipping water on a moving ship and validate the impact force on the deck. Slamming events were also modelled where accurate slamming pressures can be estimated using the SPH algorithm (Veen 2010; Veen and Gourlay 2012). More recently, Kawamura et al. (2016) predicts 6DoF ship motions in severe conditions using a GPUaccelerated SPH simulation.
There have been quite a few recent investigations accounting for structural deformations and twoway coupling using conventional meshbased RANS methods. For example, El Moctar et al. performed twoway coupling between RANS code and Timoshenko beam model and investigated the effects of springing and whipping (El Moctar et al. 2011); Lakshmynarayanana investigated a containership in regular head waves coupling STARCCM+ with the ABAQUS finite element software (Lakshmynarayanana et al. 2015).
The work presented in this paper is the first step in extending the application of WCSPH (weakly compressible SPH) to simulate twoway coupling in FSI. Forced oscillation tests are performed on a uniform flexible barge using WCSPH for the 3D radiation problem. Hydrodynamic coefficients, namely added mass and damping coefficients, are obtained for the rigid body motions of heave and pitch and the twonode (2VB) and threenode (3VB) symmetric distortion mode shapes, including coupling terms. These are compared with potential flow (using 3D hydroelasticity) and RANS (using STARCCM+) predictions. Domain size, particle numbers and damping zones are modified based on different frequencies of oscillation, allowing the free surface to be well captured by WCSPH.
2 Methodology
2.1 SPH Interpolation
The parameter B is a constant related to the bulk modulus of elasticity of the fluid, ρ_{0} is the reference density, usually taken as the density of the fluid at the free surface, and γ is the polytrophic constant, usually between 1 and 7. The second term in Eq. (8), the minus one term, is in order to obtain zero pressure at a surface (Chen et al., 2013). In this paper, γ = 7, \( B={c}_0^2{\rho}_0/\gamma \), with c_{0} being the speed of sound at the reference density and \( {c}_0=c\left({\rho}_0\right)=\sqrt{\partial P/\partial \rho } \).
2.2 Implementation
2.3 Force Around Fixed and Floating Bodies
In the simulations, the repulsive force (DBC), f_{ak}, exerted by the boundary particle k on fluid particle a is the only force computed, and from Eq. (16), the force exerted by fluid particle a on the moving body can be calculated. By integrating Eq. (16) in time, the position of each boundary particle can be determined and moved accordingly. It can be shown that this technique conserves both linear and angular momentum (Monaghan et al. 2003; Tafuni 2016).
2.4 Computational Setup
Main particulars of the barge, m
Main particulars  Barge 

Length, L  120 
Breadth, B  14 
Depth, D  11.15 
Draft  5.575 
Simulation conditions at dx = 1.0 m
ω/(rad·s^{−1})  λ/m  z_{a}/m  Wave zone at long. and ath. direction/m  Total domain, a × b/m^{2}  Domain depth, d/m  Particle number 

0.2  1540.9  1.0  132.1  504.2 × 398.2  61.64  7.8 M 
0.4  384.24  1.0  132.1  504.2 × 398.2  61.64  7.8 M 
0.6  171.22  1.0  123.3  486.6 × 380.6  61.64  7.8 M 
0.8  96.309  1.0  123.3  486.6 × 380.6  30.69  3.9 M 
1.0  61.638  1.0  123.3  486.6 × 380.6  30.69  3.9 M 
1.2  42.804  0.2  123.3  486.6 × 380.6  30.69  3.9 M 
1.4  31.448  0.2  123.3  486.6 × 380.6  30.69  3.9 M 
1.6  24.077  0.2  123.3  486.6 × 380.6  30.69  3.9 M 
1.8  19.024  0.2  123.3  486.6 × 380.6  30.69  3.9 M 
2.0  15.410  0.2  123.3  486.6 × 380.6  30.69  3.9 M 
The NWT is extended either side of the barge in x and y directions that includes two zones, namely wave zone and damping zone. Wave zone allows time for the radiated wave from the oscillating barge to travel before being slowly damped by the beach at the end of the domain. Ideally, the lengths of both these zones should be set equal to the wavelength of the radiated wave. However, considering the number of particles used, wave zone and damping zone for all frequencies are first determined based on the characteristics of the wave with a frequency of 1.0 rad/s. In this case, the wave zone is set double the corresponding wavelength and damping zone is nearly equal to the wavelength. The water depth is kept large enough to avoid shallow water influence to the radiated wave for each individual frequency. However, for the case ω ≤ 1.0 rad/s, the dimension in the longitudinal and athwartship directions could not extend to double the length of the corresponding wavelengths due to the large particle numbers involved. Therefore, the domain sizes are reduced for ω = 0.2 rad/s and ω = 0.4 rad/s. The reductions are based on some preliminary simulations, and these domain sizes are believed to be adequate in providing accurate results and to avoid precision problems (Ramli et al. 2015). The excitation amplitude, z_{a}, is set to H/2 = 1 m for frequencies ω ≤ 1.0 rad/s and 0.2 m for frequencies ω > 1.0 rad/s. This large value of amplitude is assigned to lower frequency case to ensure that the dynamic force components are not too small compared to the static components.
2.5 Computational Parameters

First step: \( \dot{z}=\omega {z}_{\mathrm{a}}\cos \omega t\cdot \mathrm{eigenvector} \)

Second step: \( \dot{z}=\left(\begin{array}{c}u\\ {}v\\ {}w\end{array}\right)\cdot \left(\begin{array}{c}{R}_y\\ {}0\\ {}{T}_z\end{array}\right) \)
Time step size for all simulations is kept to 1.0e^{−4} s, following the highest frequencies tested in Table 2. Time duration of simulation varies depending on individual wavelengths, i.e. 15 s for 2 rad/s and 60 s for 0.2 rad/s with similar output files of every 0.05 s.
2.6 Hydrodynamic Coefficients
The computation of the hydrodynamic force, F(t), is carried out using the recorded information on the particle properties at each time step. The force is determined by first computing the acceleration vector of each fluid particle in the vicinity of the oscillating barge and, making use of Newton’s second law, by multiplying this value with the mass of each fluid particle. Referring back to Eq. (16), the resulting vector is assumed as the force exerted by the fluid on the barge with opposite sign. Since changes in x and y directions are negligible, only the z component of the acceleration is considered in obtaining the total hydrodynamic force, F(t).
Instantaneous values of the hydrodynamic coefficient are obtained by Fourier analysis of time history of the generalised force F_{rs}(t) using discrete window approach for one period, T = 2π/ω, of oscillation, namely
3 Results and Discussion
This section presents results for the stationary uniform barge oscillating on the free surface at different mode shapes, namely heave, pitch and twonode and threenode vertical bending modes. The predictions from WCSPH are compared to those obtained from 3D predictions using STARCCM+ and 3D (hydroelasticity) potential flow using the Green’s function for pulsating source (Bishop et al. 1986). Kim et al. (2014), in STARCCM+, imposed a boundary that oscillates simply harmonically in the shape of the selected modes. Both STARCCM+ and WCSPH simulations use inviscid flow, but account for nonlinearities. Each eigenvector relevant to the motion of the barge is determined by approximating the Euler beam solution by a polynomial. It is worthwhile mentioning that results from STARCCM+ used for the comparisons can be categorised into two sets. The first set is the initial setup of the numerical model, with mesh size ranging between 1.1 and 4.4 M (lowest frequency), while the second set is obtained by refinement of the mesh around the body and the free surface, resulting in quadrupling the size of the mesh. It should be noted that these refinements were only used for a few low frequencies of oscillation.
4 Conclusive Remarks
The accuracy of WCSPH in predicting the hydrodynamic coefficients of a stationary barge harmonically oscillating in still water is carried out and verified against predictions using 3D potential flow and 3D RANS CFD predictions. The WCSPH model has been successfully applied to the symmetric rigid body motions of heave and pitch and twonode and threenode distortion modes. The comparisons of the added mass and fluid damping coefficients show that WCSPH predictions agree well with STARCCM+ and BEM predictions for the vast majority of the cases investigated. It was observed that both STARCCM+ and WCSPH display similar features in the predicted hydrodynamic coefficients, including few discrepancies in the range of relatively low frequencies of oscillation. The most likely cause of the aforementioned discrepancies for WCSPH is the inadequacy of the domain size in the longitudinal directions, which requires to be larger (relative to the length of the radiated wave), as well as larger damping zones in preventing any wave reflections. Adopting such large domains may, however, be constrained by normally available computational power. Nevertheless, obtaining accurate predictions at such low frequencies is of academic interest, as they correspond to large wavelengths by comparison to the shipwave matching region of practical interest for motions and waveinduced loads. Extracting the dynamic force from the total generalised force in such large time window in the case of low frequencies is likely to increase the errors within the calculation of added mass and damping.
Moreover, poor performance of WCSPH particularly for the crosscoupling coefficients of pitch3VB modes and vice versa similarly may be related to the insufficient particle refinement of the total fluid domain, although it has been shown when using STARCCM+ that extending the domain does not improve significantly the quality of the results at all frequencies. It has been noted that the STARCCM+ predictions show similar, sometimes worse, trends in the frequency variation of these hydrodynamic coefficients, e.g. in the vicinity of 0.8 rad/s. The pressure distributions and wave contours of the radiated waves obtained from WCSPH are comparable to the refined results by STARCCM+.
Simulation of fluidstructure interaction involving a flexible body using the WCSPH method employed in this paper has shown that it is a reliable numerical tool in predicting added mass and fluid damping coefficients. Although this investigation is limited to the radiation problem, the conclusions drawn for this work show that it is applicable to modelling the behaviour of the 3D ship in waves. Future work should include double precision to increase accuracy of neighbouring particle tracking and variable resolution technique to improve particle refinement at large domains (Ramli et al. 2015; Dominuque et al. 2014).
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