Analyzing Parabolic Profile Path for Underwater Towed-Cable

Vineet K. Srivastava

Analyzing Parabolic Profile Path for Underwater Towed-Cable

Vineet K. Srivastava＊

ISRO Telemetry, Tracking and Command Network (ISTRAC), Bangalore 560058, India

This article discusses the dynamic state analysis of underwater towed-cable when tow-ship changes its speed in a direction making parabolic profile path. A three-dimensional model of underwater towed system is studied. The established governing equations for the system have been solved using the central implicit finite-difference method. The obtained difference non-linear coupled equations are solved by Newton’s method and satisfactory results were achieved. The solution of this problem has practical importance in the estimation of dynamic loading and motion, and hence it is directly applicable to the enhancement of safety and the effectiveness of the offshore activities.

underwater towed-cable; underwater towed system; parabolic profile; central implicit finite-difference method; Newton’s method; offshore activities

1 Introduction1

Underwater towed systems are fundamental tools for many marine applications including naval defense, oceanographic and geophysical measurements, e.g. in geophysical applications they are used for oil-prospecting whereas in naval applications they are used for acoustic detection of submerged targets. These systems can be as simple as a single cable with its towed vehicle or may be composed of multiple towed cables and multiple towed bodies. 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 performances of a cable in an underwater towed system are the lumped mass method (Walton and Polacheck, 1960), inverse methods (Duncan et al., 2007; Todd et al., 2013) and the finite difference method (Ablow and Schechter, 1983; Milinazzo et al., 1987; Thomas et al., 1988; Grosenbaugh et al., 1993; Burgess, 1991; Thomas and Hearn, 1994; Vaz and Patel, 1995 and 1997; Gobat and Grosenbaug, 2006). However, according to Grosenbaugh et al. (1993), the explicit time domain integration scheme used in the lumped mass method made the method conditionally stable. Burgess (1991) 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 (1994) believed that the collapse of the numerical 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, 1991; Thomas and Heam, 1994). 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 (1983) 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 un-stretched distance along the cable. The differential equations are then approximated by finite difference equations centered in time and space. By solving the equations, the motion of underwater cable can be determined 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.

This paper presents a three-dimensional hydrodynamic model to simulate an underwater towed system. In the model, the governing equations of cable are established based on the method of Ablow and Schechter (1983). The six degrees-of-freedom equations of motion for submarine simulations are adopted to predict the hydrodynamicperformance of a towed vehicle. The established governing equations are 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 (Tamsir and Srivastava, 2011 and 2012; Srivastava et al., 2013 and 2014). 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 maneuvering problems, and is an effective algorithm for the simulation of a large-scale towed system.

2 Mathematical formulation of the problem

A mathematical model of maneuvering of underwater towed cable array system (Choo and Casarella, 1973) 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.

Let (,)θφ be the Euler angles defining the position of local reference frame relative to the Newtonian reference frame (,,)ijk, where t is tangential to the cable-array system and b is in the plane of i and j.

The dynamic problem formulation is obtained by applying Newton’s second law of motion to the cable element of infinitesimally stretched length dS.

where B is the momentum per unit length, Tis the tension,wFis the weight minus buoyancy per unit length anddF is the force exerted by the fluid on the cable-array system per unit length and is taken to be the sum of independently operating normal drag and tangential. A system of three scale equations is obtained by separating the three components of vector equation in the independent directions (,,)tnb. The compatibility relations in terms of velocity can be expressed as

where r is a position vector from the origin of a fixed coordinate system (i,j,k) to a point on the cable-array system. The position vector r is a function of un-stretched cable-array system length coordinate s and the time t. By separating various components of the Eq. (2) in independent directions (t,n,b), a system of three scalar equations of compatibility is obtained. Three equations of motion and three equations of compatibility together present six scalar dynamic differential equations of first order in space variable s and time variable t. The six governing equations of motion in matrix form can be given as

where,

where A is the cross section area of un-stretched cable, Cnand Ctare normal and tangential drag coefficients, d is the diameter of cable, ρ is fluid density; dSis infinitesimal stretched cable length,e=, E is Young’s modulus, mis mass per unit length of cable, m1=m+ρA is virtual mass per unit length, w=(m?ρA)g is immersed weight per unit length, g is gravitational acceleration, T is cable tension magnitude; v is velocity of tow-ship, J=(Jt,Jn,Jb)is current velocity given in local frame (t,n,b), J˙=(J˙t,J˙n,J˙b)is the partial derivative of J with respect to time t holding s fixed, U=(Ut,Un,Ub)is tangential, normal and bi-normalcomponents of cable structural velocity relative to current velocity (V?J); x,y,z are trail, lateral shift and depth of a point on cable with respect to tow-point in the inertial frame, θ(s,t), φ(s,t ) Euler’s angles defining the position of local reference frame (t,n,b) relative to the inertial frame (i,j,k).

3 Boundary and Initial conditions

Three boundary conditions at the tow-point of the cable are provided by known velocity components of the tow-ship at any time, i.e.

In terms of y, we have

At the free end the three boundary conditions can be given as

where,

At t=0, it is assumed that the initial condition is known i.e. y(s, 0)is a known function of s(0≤s≤S). This condition along with six boundary conditions provides the complete solution of the governing equations. Computations start from a steady state solution (more precisely, the tow-ship is assumed to move with constant velocity), which is taken as the initial condition for the whole system. The variables T and φ are determined from equations

As ()0TS=, the critical angle ()Sφ satisfies Eq. (8). Where T′ is the partial derivative of T with respect to s, holding t fixed; φ′partial derivative of φ with respect to s, holding t fixed. The position (,,)xyz of the cable, in the inertial frame, can be obtained from the relations

Integrations in sdetermine (,,)xyz provided eT and the angles θ and φ are known.

If we take

ThentV,nV,bV can be obtained from the expressions

where

The angle θ is computed from the equation

Jt,Jn,Jbare computed from the expressions:

where J1, J2and J3are the current velocities in the inertial frame. Similarly, Ut,Un,Ubcan be computed from the relations

Three components of the tow-ship velocity, free end zero tension along with subsequent two more free end boundary conditions provide a total of six requisite boundary conditions. The solutions of the six governing equations along with six boundary conditions provide dynamic response of the cable array system.

4 Method of Solution

Second order central finite difference method is applied to the governing differential Eq. (3) to convert them into the algebraic difference approximations. The total cable-array length S is divided into N segments of arbitrary length

The time is divided into a number of intervals and various parameters are evaluated at all spatial grid points sjand temporal grid points ti. The discrete approximation to y(sj,ti)is taken to be Y with Yji≈y(sj,ti), forconvenience we use the following notations

Discretizing the governing equations of motion (3) at the half-grid pointsand dropping second and higher order terms, we get

Denoting left hand side of the Eq. (20) by, we have

Similarly, the boundary conditions can be approximated as

Eqs. (21), (22) and (23) can be together written as

where

The system (24) is an implicit, centered, second order approximation to the system of hyperbolic partial differential equations. GiveniY at timeit, the system of equations (24) determine1i+Y at1it+. Further we assume that the initial state of the cable0Y is known. The non-linear algebraic equations (24) are solved using Newton’s iterative method. The precise algorithm of Newton’s method is given below.

(1) Obtain an estimate for Yi+1by extrapolating Yi?1and Yi, i.e.

(2) Compute a correction to1i+Y by solving the linear system

(3) Yi+1=Yi+ΔY gives the improved value of Yi+1.

(4) If the absolute value of maximum relative change in any component of the solution Yi+1is less than 10?3, increment the time and go to step (a), otherwise repeat (b) and (c) using new value of Yi+1.

In step (a), using Yias the initial guess for Yi+1is sufficient to achieve (quadratic) convergence. Gauss elimination with partial pivoting has been used to solve the linear system.

5 Numerical results and discussions

The developed Newton’s iterative scheme is implemented on the underwater towed cable-array model. The underwater towed cable-array system is discussed under a six segment model. In the towed cable-array model, the steady ocean current (0.5 m/s) was taken.

5.1 Six segment cable model

Here we discuss dynamic behavior of the cable during the ship maneuvering for three different oceanic current conditions using the developed code. Fig. 1 illustrates the towed array system while Table 1 gives the physical characteristics of each segment of six segment cable model.

Fig. 1 Six Segment Towed Array System

Table 1 Tow cable system physical properties

5.2 Dynamic state analysis: parabolic profile path

In this section, we discuss the dynamic state analysis of underwater towed-cable when tow-ship changes its speed making a parabolic profile path. Linear profile case is discussed by Srivastava et al. (2011). The parabolic profile can be given as

where0v andfv are the tow-ship’s initial and final speeds, respectively and T is the elapsed time taken to reach the final speed. The tow-ship profile path is considered under three different conditions namely when its speed increases, decreases, and also with both situations, say 4 m/s to 12 m/s and 12 m/s to 4 m/s.

5.2.1 Tow-ship accelerates from 4 m/s to 12 m/s in parabolic profile

Fig. 2(a), (b) and (c) shows the graph between the trail and the cable depth when tow-ship accelerates from 4 m/s to 12 m/s in a parabolic path, when there is no current, against and along the current directions, respectively. The cable depth increases slightly, when the tow-ship accelerates, when there is no current, against and along the current directions, accordingly as shown in Table 2. It can be observed that maximum cable depth occurs when the tow-ship accelerates along the current direction and minimum cable depth occurs when the tow-ship is accelerating against the current direction. Fig. 2(d), (e) and (f) shows variation between the cable length and tension when tow-ship accelerates, when there is no current, against and along the current directions, respectively. The tow-point tension varies from 17.0 kN to 96.3 kN, 21.5 kN to 103.9 kN and 13.1 kN to 89.1 kN (refer Table 3), for three different current situations respectively. It can be seen that maximum tow-point tension occurs when tow-ship accelerates against the current direction and minimum occurs when tow-ship accelerates along the current direction.

Fig. 2 Tow-ship accelerates from 4 to 12 m/s in parabolic path

Table 2 Cable depth range (m) in parabolic profile

Table 3 Tow-point tension range in parabolic profile

5.2.2 Tow-ship decelerates from 12 m/s to 4 m/s in parabolic profile

Fig. 3(a), (b) and (c) shows the graph between trail and cable depth when the tow-ship decelerates from 12 m/s to 4 m/s in a parabolic profile, with no current, against and along the current directions, respectively. It can be observed that maximum cable depth occurs when the tow-ship decelerates along the current direction and minimum cable depth occurs when the tow-ship decelerates against the current direction (refer Table 2).

Fig. 3 Tow-ship decelerates from 12 to 4 m/s in parabolic profile

Fig. 3(d), (e) and (f) shows the graph between the cable length and tension when the tow-ship decelerates from 12 m/s to 4 m/s, under no current, against and along the current directions. From Table 3, it could be observed that maximum tow-point tension occurs when the tow-ship decelerates against the current direction and minimum tow-point tension occurs when the tow-ship decelerates along the current direction.

5.2.3 Tow-ship accelerates from 4 m/s to 12 m/s thereafter decelerates from 12 m/s to 4 m/s

Fig. 4(a), (b) and (c) shows the graph between the trail and the cable depth when the tow-ship accelerates from 4 m/s to 12 m/s thereafter decelerates from 12 m/s to 4 m/s in a parabolic profile path, in case of no current, against and along the current directions, accordingly. It is observed that maximum cable depth occurs when the tow-ship accelerates and decelerates along the current direction and minimum cable depth occurs when the tow-ship is accelerating and decelerating against the current direction (refer Table 2). Fig. 4(d), (e) and (f) shows the graph between the cable length and tension when tow-ship accelerates from 4 m/s to 12 m/s thereafter decelerates from 12 m/s to 4 m/s, when there is no current, against and along the current direction. It is observed that maximum tow-point tension occurs when the tow-ship accelerates and decelerates against the current direction and minimum tow-point tension occurs when the tow-ship accelerates and decelerates along the current direction (refer Table 3).

Fig. 4 Tow-ship accelerates from 4 m/s to 12 m/s then decelerates from 12 m/s to 4 m/s in parabolic curve

From Tables 2 and 3 we observe the same variation in the cable depth but more variations in tow-point tension in comparison with the linear profile given by Srivastava et al. (2011).

6 Conclusions

A three-dimensional numerical program had been carried out for the analysis of the underwater towed cable-array system when tow-ship makes parabolic profile during manuoeuring. An implicit central finite difference method had been employed for solving the three dimensional cable equations. In order to solve the non-linear and coupled problems, Newton’s iteration scheme had been used, and satisfactory results were obtained. The developed numerical program can be applied to towed array systems for detecting a moving object or submarine.

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Author’s biography

1671-9433(2014)02-0185-08

Srivastava

his M. Sc. (Mathematics) degree from University of Allahabad in year 2007 and M. Tech. (Industrial Mathematics and Scientific Computing) in 2010 from Indian Institute of Technology Madras, Chennai, India. His research interest are numerical PDE, mathematical modeling, ocean engineering, computational biology, computational physics, flight dynamics, digital signal processing, orbital and celestial mechanics. Presently, he is working as a Scientist/Engineer in Indian Space Research Organization (ISRO), India.

Received date: 2013-05-18.

Accepted date: 2014-01-02.

＊Corresponding author Email: vineetsriiitm@gmail.com

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014