The aim of this paper is to describe the development of the method of fundamental solutions (MFS) and related methods over the last three decades. Several applications of MFS-type methods are presented. Techniques by which such methods are extended to certain classes of non-trivial problems and adapted for the solution of inhomogeneous problems are also outlined.

https://msvlab.hre.ntou.edu.tw/grades/meshless/MFS98.pdf

The method of fundamental solutions for elliptic boundary value problems

Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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Liquid Sloshing Dynamics

Publisher Summary This chapter discusses the varied physical circumstances in which interactions among water waves and currents occur. Different mathematical approaches, relevant observations, and experiments that are applicable to all or some of these physical circumstances are described. The emphasis is on waves and their interaction with preexisting currents rather than on wave-generated currents. Common simplifying assumption is that the waves are of sufficiently small amplitude for the free-surface boundary conditions to be linearized and evaluated at, or close to, the mean free surface. Most progress can be made in this subject with such a constraint, but wherever possible, finite-amplitude effects are discussed. Unlike some other common forms of wave motion, water waves involve water motion varying with direction perpendicular to the space in which they propagate. The chapter concludes on the interaction of waves generated by a ship with the flow around it.

Interaction of Water Waves and Currents

The lubrication of external liquid flow with a bubbly mixture or gas layer has been the goal of engineers for many years, and this article presents the underlying principles and recent advances of this technology. It reviews the use of partial and supercavities for drag reduction of axisymmetric objects moving within a liquid. Partial cavity flows can also be used to reduce the friction drag on the nominally two-dimensional portions of a horizontal surface, and the basic flow features of two-dimensional cavities are presented. Injection of gas can lead to the creation of a bubbly mixture near the flow surface that can significantly modify the flow within the turbulent boundary layer, and there have been significant advances in the understanding of the underlying physical process of drag reduction. Moreover, with sufficient gas flux, the bubbles flowing beneath a solid surface can coalesce to form a thin drag-reducing air layer. The current applications of these techniques to underwater vehicles and surface ships are discussed.

Friction Drag Reduction of External Flows with Bubble and Gas Injection

Introduction Overview of problems and approaches Model test and similarity laws Full scale tests Numerical approaches (Computational Fluid Dynamics) Basic equations, Basic techniques Applications. Propeller Flows: Propeller geometry and other basics, Propeller curves Numerical methods for propeller design Lifting line theory Lifting surface theory BEM for propellers Field methods Cavitation Experimental approach Propeller design procedure. Resistance and propulsion: Resistance and propulsion concepts Interaction between ship and propeller Decomposition of resistance Experimental approach Towing tanks and experimental set up Resistance test Method ITTC 1957 Method of Hughes-Prohaska Propulsion test Additional resistance under service conditions Simple design approaches CFD approaches for steady flow Wave resistance computations Viscous flow computations Problems for fast and unconventional ships. Ship Seakeeping: Introduction to seakeeping Experimental approaches (model and full-scale) Waves and seaway Airy waves (harmonic waves of small amplitude) Natural seaway Wind and seaway Wave climate Numerical prediction of ship seakeeping Overview of computational methods Strip method Rankine panel methods Problems for fast and unconventional ships Further quantities in regular waves Ship responses in stationary seaway Simulation methods Long-term distributions Slamming. Manoeuvring: Simulation of manoeuvring with known coefficients Coordinate systems and definitions Body forces and manoeuvring motions Linear motion equations CFD for manoeuvring Experimental approaches Manoeuvring tests for full-scale ships in sea trials Model tests Rudders Computation of body forces Slender-body theory Influence of heel Shallow-water effect Jet thrusters Stop manoeuvres. Boundary element methods: Green function formulation Integral equations Source elements Point source Regular first-order panel Jensen panel Higher-order panel Vortex elements Dipole elements Point dipole. Numerical examples for BEM: Two-dimensional body in infinite flow Theory Numerical implementation.

Practical Ship Hydrodynamics

SUMMARY The concept of desingularization in three-dimensional boundary integral computations is re-examined. The boundary integral equation is desingularized by moving the singular points away from the boundary and outside the problem domain. We show that the desingularization gives better solutions to several problems. As a result of desingularization, the surface integrals can be evaluated by simpler techniques, speeding up the computation. The effects of the desingularization distance on the solution and the condition of the resulting system of algebraic equations are studied for both direct and indirect versions of the boundary integral method. Computations show that a broad range of desingularization distances gives accurate solutions with significant savings in the computation time. The desingularization distance must be carefully linked to the mesh size to avoid problems with uniqueness and ill-conditioning. As an example, the desingularized indirect approach is tested on unsteady non-linear three-dimensional gravity waves generated by a moving submerged disturbance; minimal computational difficulties are encountered at the truncated boundary. Boundary integral methods provide a powerful technique for the solution of linear, homogeneous boundary value problems. The method employs a fundamental solution, which satisfies the differential equation (and possibly part of the boundary conditions), to reformulate the problem as an integral equation on the boundary. In conventional boundary integral formulations, singularities of the fundamental solution are placed on the domain boundary. This requires special evaluation of singular integrands, which can result in costly numerical calculations. In time-dependent non-linear free surface problems' - a boundary integral problem is solved at each time step. Since most of the computation time is devoted to the boundary integral problem, an effective solution method is critical in the time-marching procedure. When the singularity of the fundamental solution is placed away from the boundary and outside the domain of the problem, a desingularized boundary integral equation is obtained. We will show two advantages to this desingularization: a more accurate solution can be obtained for a given truncation, and a numerical quadrature can be used to reduce the computational time to obtain the algebraic system representing the discretized boundary integral problem. There are two types of non-singular boundary integral formulations: direct and indirect. In the direct method, Green's second identity is used to derive the boundary integral equation, and the solution of the problem is obtained directly by solving the boundary integral equation. In the

/pdf/three-dimensional-desingularized-boundary-integral-methods-4pzrtgq8fg.pdf

Three-dimensional desingularized boundary integral methods for potential problems

Transient and steady-state nonlinear free surface waves and wave-body interactions are investigated in a 3-dimensional Numerical Wave Tank (NWT) using an indirect Desingularized Boundary Integral Equation Method (DBIEM) and a Mixed Eulerian-Lagrangian (MEL) time marching scheme. The Laplace equation is solved at each time step by using Rankine sources distributed outside the solution domain, and the fully nonlinear free surface boundary conditions are integrated with time to update its position and boundary values. A regridding algorithm is devised to eliminate the possible instabilities without using artificial smoothing. The incident waves are generated either by a piston type wavemaker or by feeding analytic forms on the input boundary. The outgoing waves are sufficiently dissipated by using spatially varying artificial damping on the free surface of the damping zone before they reach the downstream wall boundary. In some cases, side beaches are also implemented to minimize the sidewall reflection. Computations are first performed for nonlinear long-crested regular waves in a 3-dimensional numerical wave tank without a body. Subsequently, the nonlinear diffractions by a bottom-mounted vertical cylinder and truncated vertical cylinders are investigated and the present NWT results are compared with available experimental results and second-order diffraction computations.

Fully nonlinear 3-d Numerical Wave Tank simulation

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/31913/0000866.pdf?sequence=1

Time-domain computations for floating bodies

Linear, time-domain analysis is used to solve the radiation problem for the forced motion of a floating body at zero forward speed. The velocity potential due to an impulsive velocity (a step change in displacement) is obtained by the solution of a pair of integral equations. The integral equations are solved numerically for bodies of arbitrary shape using discrete segments on the body surface. One of the equations must be solved by time stepping, but the kernel matrix is identical at each step and need only be inverted once. The Fourier transform of the impulse-response function gives the more conventional added-mass and damping in the frequency domain. The results for arbitrary motions may be found as a convolution of the impulse response function and the time derivatives of the motion. Comparisons are shown between the time-domain computations and published results for a sphere in heave, a sphere in sway, and a right circular cylinder in heave. Theoretical predictions and experimental results for the heave motion of a sphere released from an initial displacement are also given.

Transient motions of floating bodies at zero forward speed

The problem to be treated here is that of steady longitudinal motion of a ship in a dredged channel, with specific attention to the predicton of sinkage and trim, and wave resistance. The dredged channel is idealized to be of rectangular cross section, and linearized shallow- water solutions are developed for the dredged or deep section, and for the two shallow regions alongside the channel. The ship is assumed slender, moving along the centerline of the channel, and is replaced by an equivalent source distribution, as in the earlier theories of Tuck for shallow water of constant depth. (Author)

/pdf/hydrodynamic-forces-on-ships-in-dredged-channels-1qvkvdpxby.pdf

Robert F. Beck

Papers

Three-dimensional desingularized boundary integral methods for potential problems

Fully nonlinear 3-d Numerical Wave Tank simulation

Time-domain computations for floating bodies

Transient motions of floating bodies at zero forward speed

Hydrodynamic Forces on Ships in Dredged Channels