A Nonlinear Simulation Method of 3-D Body Motions in Waves (1st Report)

Abstract

A full nonlinear method to simulate three dimensional body motions in waves is presented. This is a time domain method to simulate Euler's equation of ideal fluid motion coupled with the equation of solid body motions.Introducing Prandtl's nonlinear acceleration potential, whose gradient gives acceleration of the fluid, Euler's differential equation of the ideal fluid motion is converted to the integral equation of the acceleration potential. The boundary condition of the acceleration potential on the body surface is systematically derived from the kinematic relation between the acceleration of the solid body and the acceleration of the fluid on the body surface. Since this kinematic boundary condition is a function of the body acceleration, the boundary values on the floating body can not be evaluated explicitly. To overcome this point, the unknown acceleration of the free floating body is eliminated by substituting the equation of body motion into kinematic condition, then implicit body surface boundary condition is derived. This is the kinematic and dynamic condition which guarantees dynamic equilibrium of forces between ideal fluid and the solid body at any instance. With the free-surface boundary condition of the acceleration potential, the formulation of the boundary value problem for the acceleration field is completed.Although this formulation of the acceleration field is mathematically correct, this is not appropriate to numerical computation, because Prandtl's nonlinear acceleration potential does not satisfy Laplace's equation. Therefore, the nonlinear part is shifted from the governing equation to the boundary condition, then the alternative formulation for the numerical computation is derived. The computational flow of the nonlinear simulation method based on this alternative formulation is also given. In order to show the accuracy of this new method, two dimensional numerical results are presented. They show that the conservation of mass, momentum and energy are satisfied excellently.