G.X. Wu

FINITE ELEMENT ANALYSIS OF TWO-DIMENSIONAL NON-LINEAR TRANSIENT WATER WAVES

The two-dimensional nonlinear time domain free surface flow problem is analyzed by the finite element method. Two approaches are used. One is based on the velocity potential which is approximated by means of shape functions. The solution is obtained through use of a variational statement, and the velocity is obtained subsequently by the Galerkin method. The other approach is to write both potential and velocity in terms of the shape functions at the same time. Their solutions are derived from the same equation by using another variational statement. Numerical results are given for the vertical wave maker problem and for a transient wave in a rectangular container. They are compared with analytical solutions, and very good agreement is found.

The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies

A coupled finite element (FEM) and boundary element (BEM) method is developed to analyse the nonlinear interaction between bodies and water waves. The former is used away from the body while the latter is used in a region near body. The combination is based on consideration of the efficiency of FEM and BEM in computation and mesh generation, respectively. Results for wave/body interactions are obtained by using auxiliary functions to decouple the mutual dependence of the body acceleration and the wave force.

TIME STEPPING SOLUTIONS OF THE TWO-DIMENSIONAL NONLINEAR WAVE RADIATION PROBLEM

The two-dimensional nonlinear time domain free surface flow problem is analysed using potential flow theory. The problem is solved by a time marching method. At each time step two numerical approaches are used. One is based on the boundary element method in the complex plane. The complex potential is assumed to vary linearly within each element and the solution is obtained by imposing the boundary conditions at the nodes of the elements. The other approach is based on the finite element formulation. Triangular elements and linear shape functions are used. The solution is obtained by the Galerkin method. Numerical results are obtained for the wave elevation generated by a vertical wave maker. Results are also provided for a circular cylinder oscillating below the free surface. For these cases the finite element method is found to provide substantially more efficient computations than the boundary element method using equivalent discretizations.

Air cushioning with a lubrication/inviscid balance

The air cushioning effect in the gap between an almost inviscid body of water and a nearby solid wall (or another body of water) is studied theoretically and is found to depend on predominantly lubricating forces in the air, in certain applications. The situation in which the density and viscosity in air are taken as small compared with those in water is investigated. In this situation potential-flow dynamics in the water couples with lubrication behaviour in the air, leading to a nonlinear integro-differential system for the evolution of the interface. The numerical values of the main parameters are investigated and indicate a wide range of practical applications. Specifically, the lubrication/inviscid balance holds for typical global Reynolds numbers below the order of the viscosity ratio divided by the cube of the density ratio, i.e. below about 10

The effect of viscosity on the transient free-surface waves in a two-dimensional tank

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The hydrodynamic force on an oscillating ship with low forward speed

in the case of air and water; for Reynolds numbers of that order the lubrication behaviour is replaced by an unsteady boundary-layer response, whereas above that order formally the response is totally inviscid. A variety of spatio-temporal flow solutions are presented for the lubrication/inviscid system and these all indicate a relatively rapid closure of the gap, in a common form which is analysed.

Radiation and diffraction of water waves by a submerged sphere at forward speed

The paper attempts to develop some understanding of the interaction between viscous flow and a free surface by analysing the unsteady flow in an idealised two-dimensional rectangular tank. The mathematical model used is based on the linearized Navier-Stokes equations which are solved by use of the Laplace transform. Various results are provided to show the effect of viscosity on the free surface waves.

Hydrodynamic forces on a submerged circular cylinder undergoing large-amplitude motion

The mathematical formulation of the linearized potential theory for a slowly translating body undergoing oscillations in infinitely deep water is derived based on a perturbation series in terms of forward speed. It is shown that the hydrodynamic force on the oscillating body can be obtained from the solution of the velocity potential without forward speed, if second-order terms in forward speed are neglected. An example of a submerged circular cylinder is discussed. The numerical results are compared with the general solution of the linearized potential theory by a coupled finite-element method (Wu & Eatock Taylor 1987) which is not restricted to low forward speeds. Very good agreement is found. The nonlinear effect of the steady potential on the hydrodynamic forces is also discussed and is illustrated for a floating semicircular cylinder.

The exciting force on a submerged spheroid in regular waves

A submerged sphere advancing in regular deep-water waves at constant forward speed is analysed by linearized potential theory. A distribution of sources over the surface of the sphere is expanded into a series of Legendre functions, by extension of the method used by Farell (J. Ship Res. 17, 1 (1973)) in analysing the wave resistance on a submerged spheroid. The equations governing the velocity potential are satisfied by use of the appropriate Green function and by choosing the coefficients in the series of Legendre functions such that the body surface condition is satisfied. Numerical results are obtained for the wave resistance, hydrodynamic coefficients and exciting forces on the sphere. Some theoretical aspects of a body advancing in waves are also discussed. The far-field equation of Newman (J. Ship Res. 5, 44 (1961)) for calculation of the damping coefficients is extended, and a similar equation for the exciting forces is derived.

On singular and highly-oscillatory properties of the Green function for ship motions

The hydrodynamic problem of a circular cylinder submerged below a free surface and undergoing large-amplitude oscillation is investigated based on the velocity potential theory. The body-surface boundary condition is satisfied on its instantaneous position while the free-surface condition is linearized. The solution is obtained by writing the potential in terms of the multipole expansion. Various interesting results associated with the circular cylinder are obtained.