Three-dimensional dynamics of long pipes towed underwater. Part 2 Linear dynamics

In this paper, a method of solution based on a finite difference scheme is developed, via which the partial differential equations of motion and boundary conditions, presented in Part 1, are converted into a set of first-order ODEs which are then solved numerically. The mathematical model is validated by considering some simplifications which enable us to compare the numerical results with the results of short pipes simply supported at both ends (pinned-pinned) and subjected to axial flow. A typical Argand diagram is then presented for a long pipe ((L) over cap = 2000 m) which shows the evolution of lowest three eigenfrequencies of the system as a function of nondimensional flow velocity (towing speed), For the same pipe, the deformation and time-trace diagrams at different values of flow velocity are also given. The results show clearly that a long pipe towed underwater may lose stability by divergence and at higher flow velocities by flutter; the deformation is confined to a small segment of the pipe, close to the downstream end. Some numerical comparisons are also presented in which the effects of cable stiffness and the skin friction coefficient on the onset of instabilities are studied. (C) 2013 Elsevier Ltd. All rights reserved.