Fundamental viscous solutions or 'transient oseenlets’ associated with a body manoeuvring in a viscous fluid

Abstract

A viscous boundary element method involving a convolution-integral formulation is developed to determine directly the fluid actions and velocity flow field associated with a body manoeuvring in a viscous fluid. The proposed general approach is analogous to a potential flow singularity distribution panel method with the singularity replaced by a fundamental viscous solution. From the developed nonlinear mathematical model based on an integral identity relation, it is seen that the Oseen equation, its variant or modified form is central to the theoretical development of the fundamental viscous solution. Analytical expressions are presented for the fundamental solutions (or transient oseenlets) appropriate to two-dimensional and three-dimensional steady, and unsteady (transient) manoeuvring problems involving prescribed translational, rotational and their combined motions. The examples considered, and hence the fundamental viscous solutions, relate to model towing tank experiments (i.e. steady state, rotating arm, planar motion mechanism oscillatory tests, etc.) used extensively to derive fluid action data on manoeuvring ships and submersibles. Examples of combined translational and rotational motions are investigated and these relate to an elementary idealization of a propeller rotating in an axial flow and an idealization of a body moving ahead with parasitic oscillatory roll motion. The predicted fluid flow patterns associated with the various fundamental solutions are presented and these clearly illustrate the generation, shedding and decay of vortices in the wakes.