Hydrodynamic Interaction of Two Ships in Tandem under Current Jess´ e R. de Souza Junior Helio M. Morishita Dept. of Naval Arch and Ocean Eng, University of S˜ ao Paulo S˜ ao Paulo, Brazil Clodoaldo G. Ragazzo Dept of Applied Mathematics Inst of Mathematics and Statistics, University of S˜ ao Paulo S˜ ao Paulo, Brazil ABSTRACT There are various situations in the operation of ships that re- quire them to remain positioned close to another vessel at sea under the action of environmental agents. This scenario is com- mon in the offshore oil exploitation industry, where shuttle ves- sels attach themselves to FPSOs for offloading operations that may last for several hours. The dynamic behavior of the vessels during their exposure to the often-changeable weather is of vital importance for the safety of these operations. Having focused on this problem in previous works employing mathematical models that disregard the hydrodynamic interaction between the ships, we now take on the task of incorporating these potentially im- portant effects into the analyses. Literature on this problem is relatively scarce. As a point of departure, we assume that the ex- isting, time-proven single-ship models should not be discarded, but improved upon. We thus attempt to model the interaction between vessels through the modification of the current velocity field, and thus develop a model that supplies a velocity correc- tion term, intended for use in any of the existing models. In this first, heuristic approach to the issue, we show that a momentum- conservation argument can reduce the problem to that of deter- mining the geometry of the wake of the upstream vessel. Exper- imental results found in the literature were used to calibrate the parameters of the wake. The issue of the spatial averaging of the velocity field involved in calculating a unique correction term is also discussed. NOMENCLATURE a coefficient in the geometric parametrization of the wake B breadth of the ship C D drag force coefficient D drag force  F total hydrodynamic force L Length of the vessel  N hydrodynamic moment Copyright 2006 The International Society of Offshore and Polar Engineers. All rights reserved. r norm of position vector in spatial integration R flat disk radius Re Reynolds number T draft of the vessel  U free current velocity u x-component of the fluid velocity  u  u local fluid velocity  U eff effective current velocity v y-component of the fluid velocity  u w z-component of the velocity  u x longitudinal coordinate x 0 coefficient in the geometric parametrization of the wake y transversal coordinate z vertical coordinate δ wake width ∆U wake-produced velocity correction η non-dimensional norm of position vector in spatial integra- tion θ momentum thickness μ viscosity of the fluid ρ density of the fluid ψ angle between the current and the ship’s longitudinal axis INTRODUCTION Whenever two or more bodies are near each other in a flow there is some amount of fluid interaction between them. Situa- tions in physics and engineering in which this occurs are many and varied. They range from the study of the hydrodynamic in- terference between bridge caissons (Chakrabarti, 2005), to the investigation of the motion of bubbles (Zhang and Shoji, 2001). In the offshore oil industry there are several situations where two or more structures may be positioned in relative proximity. In the present work we shall consider, as a guiding example, the case of the offloading operations involving a VLCC-type tanker op- erating as a Floating Production Storage and Offloading (FPSO) unit and another tanker (the shuttle vessel). The latter has to re- main stationed near the FPSO to allow a hose to be connected between the two vessels. The hose itself is not designed to bear 1