Calculation of hydrodynamics coefficients for rolling in two dimension by particle hydrodynamics method

Many different analytical, experimental and numerical works have been done for the prediction of hydrodynamic forces and moments exerted on the oscillating body on the water surface. In this thesis, added mass of inertia and damping moment coefficient, vorticity around the body and surface waves generated due to roll motion of the body are investigated for 2-D rectangular bodies which are subjected to forced roll motion by means of Smoothed Particle Hydrodynamics Method (SPH) which is one of the meshless numerical methods. Obtained results have been compared with those of the numerical and experimental studies, by taking a rectangular cross-section into account. Moreover, sensitivity analysis have been performed to determine whether obtained results and generated SPH algorithm is robust or not. Various different numerical algorithms are still in development in order to tackle the problems of SPH method by researchers and engineers. Fundamentally, Continuity Equation and Euler's Equation of motion including Artificial Viscosity Terms are used as governing equations within the frame of Weakly Compressible SPH (WCSPH) approximation scheme. SPH method was first simultaneously introduced by Monaghan and Gingold (1977) and Lucy (1977)to find a numerical solution to the astrophysical problems. Then the method is applied to the fluid and solid mechanics problems. In 1990's SPH's usage is extended to the underwater explosion, shock, bio-mechanical, crash and variety of fluid and solid mechanics problems. Nowadays, different algorithms to find a sound and effective solution to the different kind of problems are still investigated. SPH method is one of meshless methods which has a Lagrangian nature. Concisely in SPH method, problem domain is discretized to the particles which represent fluid and fluid kinematic and dynamic properties such as velocity, force and etc. Mentioned properties can be taken from each particle throughout in analysis time. There shall be two indispensable steps in SPH numerical solution algorithm. First step is named as integral respresentation or kernel approximation. The second one is the particle approximation. • First step, the kernel approximation, is based on an interpolation process where the values of any field variable (such as velocity, pressure etc.) are calculated by using the integral representation of SPH approach. Kernel function, smoothing kernel function or smoothing kernel plays important role in the application of kernel approximation and it is explained in detail in this study. • Second step, particle approximation provides the discretization of these integral equations by using the relation through a summation procedure over all the particles in the problem domain. With the application of the kernel and particle approximations to the Continuity and Euler's Equation; WCSPH approximation is utilized. Artificial viscosity term is added to the pressure terms inside the Euler's Equation to obtain better stability for the generated İPH algorithm. With the addition of artificial viscosity term to the Euler's momentum equation, slight diffusion that contributes the numerical stability is acquired. This term is firstly used by Von Neumann and Richtmeyer (1950) in FDM analysis and then it is used in many SPH algoritms (Delorme et. al., 2005). Optimization of the artificial viscosity term is required.Artificial viscosity effects in results has to be taken into account while acquiring numerical stability. With the decrease of smoothing length artificial viscosity effects converges to zero and used governing equation for the momentum degraded to Euler's equation (Antuono et. al., 2011). WCSPH assumes that the fluid is incompressible, however it allows %1 variation from the intial value of the density distribution and it employs Equation of State (EOS) to couple density and pressure values of particles. After application of SPH discretization, Density Correction (Shephard filtering), Velocity Updated XSPH (VXSPH) and Particle Packing Algorithms (PPA) are employed. As the pressure is coupled with density in WCSPH approximation, precise calculation of density has utmost importance. When one uses the density evolution following from directly the continuity equation, the pressure field may fluctuate rapidly. Therefore, SPH computations which use WCSPH method usually employ density correction as introduced by Colagrossi and Landrini (2003), Dalrymple and Rogers (2006) and Özbulut et. al. (2014). It is also called as Shephard filtering. VXSPH is developed by Özbulut (2013) and it is a hybrid combination of Monaghan's (1994) Velocity Variant Algorithm (XSPH) and Shadloo et. al.'s (2011) Artificial Particle Displacement (APD) algorithms. In VXSPH, particle velocities and positions are updated on water surface by using XSPH algorithm to have better prediction and particle positions are shifted for the rest of the particles by using APD algorithm to obstruct particle accumulation. Additionally, Colagrossi et. al.'s (2012) Particle Packing Algorithm (PPA), which assures more homogenous particle distribution, is employed when the problem domain have oblique boundaries. Five different validations are executed to test the SPH algorithm presently developed. In the first case of validation, pressure values of the particles are observed for a satisfactory time interval while the body is at rest. Moreover in the first validation case two different tank geometries (one has vertical/horizontal boundaries and another one has oblique/horizontal boundaries) are tested. At the beginging hydrostatic pressure conditions are assigned to the fluid particles and variation of pressure values are observed. PPA algorithm success is sought for the tank which have oblique boundaries. In the second validation, moments are investigated in both when body is stable and it is harmonically oscillating. Obtained moment when the body is subjected to the harmonical oscillation is compared with that of experimental and numerical studies in the literature. Additionally, another test is executed for the oscillating with a harmonic charasteristics. Roll angle and moment obtained are taken into account and phase difference between these two parameters is also investigated. In the third validation, added mass of inertia and damping coefficients of the body are investigated for different angular frequencies and various roll amplitudes. Roll motion added mass of inertia and damping coefficient are compared with those of experimental and numerical studies in the literature. In the fourth validation, the number and location of the vortexes that occur around the body because of the motion of the body and the fluid particle velocities around the body are compared with the other studies in the literature. Fifth and the last; validation covers the analysis of the free waves that occur on free surface that is caused by the motion of the body. Validation activities are followed by the analysis of the impacts of the differences of the distance between the particles (dx), time step (dt) and problem geometry over the results to control whether the developed algorithm is robust. Nondimensional added mass of inertia (a66) and nondimensional damping moment coefficient (b66) are selected as the comparison criteria in all three sensitivity analyses. Through multiple validation and sensitivity analysis, the generated SPH algorithm is put to various tests. The algorithm, responded in satisfactory level of correctness to the experimental studies in the literature in all the tests implemented, and it is observed to yield more promising results to FSRVM which is another numerical method in the literature when the experimental data in the literature is taken as a basis. Even though successful results are obtained, the developed SPH Algorithm has a long computational time which needs to be shortened. During the literature review usage of IPH method, which was previously used for solution/verification of various problems, to the two dimensional body which is subjected to roll motion on the surface is not obtained. Therefore results that are achieved in the thesis through using IPH method to the two dimensional rolling body on the free surface are determined to contribute to the literature. Results that are examined in the scope of thesis regarding moment that is exerted on the rolling body, added mass of inertia moment, damping moment coefficient for the rolling body and surface waves generated due to roll motion of the body, contribute to the literature quantitatively. In addition vorticity and velocity vectors occur on the particles around the body that are examined in the scope of thesis contribute literature in a qualitative way.