Mathematical and Computational Techniques for Predicting the Squat of Ships

Abstract

This thesis deals with the squat of a moving ship; that is, the downward displacement and angle of trim caused by its forward motion. The thesis is divided into two parts, in which the ship is considered to be moving in water of constant depth and non-constant depth respectively. In both parts, results are given for ships in channels and in open water.

Since squat is essentially a Bernoulli effect, viscosity is neglected throughout most of the work, which results in a boundary value problem involving Laplace's equation. Only qualitative statements about the effect of viscosity are made.

For a ship moving in water of constant depth, we first consider a one-dimensional theory for narrow channels. This is described for both linearized flow, where the disturbance due to the ship is small, and nonlinear flow, where the disturbance due to the ship is large. For nonlinear flow we develop an iterative method for determining the nonlinear sinkage and trim. Conditions for the existence of steady flow are determined, which take into account the squat of the ship.

We then turn to the problem of ships moving in open water, where one-dimensional theory is no longer applicable. A well-known slender-body shallow-water theory is modified to remove the singularity which occurs when the ship's speed is equal to the shallow-water wave speed. This is done by including the effect of dispersion, in a manner similar to the derivation of the Korteweg-deVries equation. A finite-depth theory is also used to model the flow near the critical speed.

For a ship moving in water of non-uniform depth, a linearized one-dimensional theory is derived which is applicable to unsteady flow. This is applied to simple bottom topographies, using analytic as well as numerical methods. A corresponding slender-body shallow-water theory for variable depth is also developed, which is valid for ships in channels or open water. Numerical results are given for a step depth change, and an analytic solution to the problem is discussed.