Author : Vineet Kumar Srivastava, YVSS Sanyasiraju, Mohammad Tamsir

International Journal of Scientific & Engineering Research Volume 2, Issue 6, June-2011

ISSN 2229-5518

Download Full Paper : PDF**Abstract—** In this paper, a numerical approach is presented which is capable of predicting dynamic behavior of underwater towed-cable structures when tow-ship changes its speed in a fixed direction making linear profile. A three-dimensional model of underwater towed system is studied. The governing equations for the system are solved by using a central finite-difference method. The solution of the finite-difference form of the assembled of non-linear algebraic equations is obtained by Newton’s method. Since the underwater towed cable model uses implicit time integration, it is stable for large time steps and is an effective algorithm for simulation of large-scale underwater towed systems. The solution of this problem is of practical importance in the estimation of dynamic loading and motion, and thus has direct application to the enhancement of safety and the effectiveness of the offshore activities.

**Keywords — **Underwater towed cable array; cable dynamics; towed systems; towing manoeuvres; cable tension; numerical simulation; linear profile

**1 INTRODUCTION**UNDERWATER towed systems are widely used for many marine applications (naval defense, oceanographic and geophysical measurements etc.). In naval applications, it is used for acoustic detection of submerged targets. In geophysical applications, it is used for oil-prospecting. These systems can be as simple as a single cable with its towed vehicle, or they may be composed of multiple towed cables and multiple towed bodies. A typical component of a towing system is shown in Fig 1. It is well known that the equations of motion for the cable and towed vehicle are non-linear and their dynamic behaviors during various operations are mutually dependent. As a result, these equations are strongly coupled. In order to study the complete problem, they must be solved simultaneously as a whole. It is not easy to solve such a complicated problem analytically and hence numerical methods are usually employed. The most prevalent approaches used in determining the hydrodynamic performance of a cable in an underwater towed system are the lumped mass method [1] and the finite difference method [2]-[10]. However, according to [5] the explicit time domain integration scheme used in the lumped mass method made the method conditionally stable. Burgess [6] pointed out that the time integration used in this algorithm requires the time step to be chosen so that the Courant-Friedrichs-Levy wave condition is satisfied for the highest natural frequency of the lumped mass system. This restricts the use of very small time steps. However, Thomas and Hearn [7] believed that the collapse of thenumerical procedure at large time steps in the method is not due to the instability of the numerical scheme, but is caused by the failure of the Newton- Raphson iterative procedure adopted to determine the correct tension levels to solve the nonlinear equations of motion.

The reason for the collapse of the numerical procedure in the lumped mass method may not be clear, however it is true that time steps in this method must be chosen very small in order to avoid the failure in numerical procedure on the basis of experiences (Burgess [6]; Thomas and Hearn [7]). In the finite difference method, the governing equations for the underwater cable are derived from the balance of forces at a point of cable. Among various finite difference methods, the model developed by Ablow and Schechter [2] is worthy to note. In this model, the cable is treated as a long thin flexible circular cylinder in arbitrary motion. It is assumed that the dynamics of cable are determined by gravity, hydrodynamic loading and inertial forces. The governing equations are formulated in a local tangential-normal coordinate frame which has the un-stretched distance along the cable. The differential equa-tions are then approximated by finite difference equations centered in time and in space. By solving the equations, the motion of underwater cable can be de-termined in the time domain. The principal advantage of this method is that it uses implicit time integration and is stable for large time step sizes. It is a good algorithm for simulation of large-scale underwater cable motion.

In this paper a three-dimensional hydrodynamic model to simulate an underwater towed system is presented. In the model, the governing equations of cable are estab-lished based on the method of Ablow and Schechter [2]. The six degrees-of-freedom equations of motion for sub-marine simulations are adopted to predict the hydrody-namic performance of a towed vehicle. The established governing equations are then solved using a central finite difference method. The solution of finite-difference form of the assembly of non-linear algebraic equations is obtained by the Newton’s method. Gauss elimination with partial pivoting is applied to solve the linear system obtained by Newton’s method. Since, the model uses implicit time integration; it is stable for large time steps. It also gives more flexibility in choosing different time steps for different manoeuvering problems, and is an effective algorithm for the simulation of a large-scale towed system.

**2 MATHEMATICAL MODEL**A mathematical model of manoeuvring of underwater towed cable array system [11] is used to find out the location and tension at any point on the cable as a function of time. The system is treated to be moving under the action of gravity, tow-ship, hydrodynamic loading and inertia forces. The loading function is taken to be the sum of independently operating normal and tangential drags.

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