Study on the steady state position of ocean towing cables

A study is conducted on the static equilibrium posture of the towing cable in water under a constant speed state in the towing system. Firstly, the underwater force state of the towing body is calculated and then the posture and tension of the towing cable are analyzed and calculated. As the cable is divided into two states which are above water and below water, a differential equation system is established as the boundary. The initial value of the equation can be transformed by the drag and gravity values of the towing body. The program solves the differential equations using the fourth-order Runge Kutta method and draws the attitude curve. Finally, it verifies the theoretical calculation results through experiments.


Introduction
At present, underwater acoustic or magnetic signal transmission or reception equipment in the ocean needs to be used in motion and often uses a combination of towing cables and towing bodies to achieve this goal.To capture the position and status of the towed body and its internal equipment in real-time, it often requires multiple sensors to be loaded and transmitted to the dry end through loadbearing cables.However, for some application scenarios, it does not require real-time capture of information such as the depth of the towed body.The equipment and power supply can be placed inside the towed body.The sensor information can be read and analyzed after recycling.At this time, the load-bearing cable or photoelectric composite cable can be directly replaced with steel cable or Kevlar cable.However, this method may expose the problem of unpredictable towing positions.The only variable that can be controlled on a ship during uniform motion is the length of the mooring line, so it is important to find a method that can accurately calculate the relationship between the length of the mooring line and the position of the towing body.
The research on the steady state of the streamer and the towed body has been conducted for a relatively long time.Assuming that there is no external force, the streamer will exhibit a stable curve.D Zoysa establishes a three-dimensional steady-state equilibrium equation for the streamer, analyzes in detail the resistance of the seawater fluid on the streamer in the equation and lists various calculation methods [1].Usually, the boundary conditions at the tail end of an underwater towing cable are determined and the steady-state problem of this towing system can be transformed into an initial value problem to solve [2].Analysis of the two-dimensional steady-state of a towing cable without considering the tangential force of the fluid [3] and classical methods for solving boundary value problems have high errors and low accuracy [4].Zhao developed a full-coupled threedimensional dynamic model of a towed cable body system.The generalized alpha algorithm is applied as the time integration method.The experimental sea trial data showed the proposed approach matches the sea trial data well [5].Wang studied the sharp turns, gradual turns and their transient states of towed cable dynamics for different course directions through the ratio of total length to turning radius R/L, the ratio of cable mass to vehicle mass and the ratio of mass unit length to hydrodynamic force w/r [6].Another article proposes a new formalism for the dynamic modeling of a cable towing system, in which both the tugboat and the towed vessel are subject to forces from waves on the sea surface [7].Tao conducted research on the design and control strategy of a streamer controller for the dynamic characteristics of streamers [8].Li and Yuan conducted on the effects of unit length streamer mass, streamer body mass, streamer resistance coefficient, streamer resistance coefficient and towing speed on the equilibrium configuration of streamers.It has been found that the equilibrium configuration of streamers can be divided into three types of internal mechanisms: "convex", "concave" and "straight" equilibrium configurations [9].
However, in practical engineering, the only variable that can be obtained after the towing cable enters the water is the length of the cable being laid out, while the theoretical calculation part often only focuses on the underwater part.The exposed part of the towing cable between the aft side of the ship and the water surface is often not included.For non-metallic towing cables with low density, this length cannot be ignored.For a fixed cable length, the calculation requires assuming the length of the water surface and iterating multiple times to solve the equation.This method is often not accurate and convenient.In this study, the water surface and underwater towing cables are expressed as a set of functional differential equations.The depth of the towing body and the posture of the towing cable are directly solved through programming.

Materials and methods
The dynamic equation for underwater towing cables is [10]: T is the tension of the towing cable, w is the net underwater gravity per unit length of the towing cable and D is the fluid resistance of the towing cable.The local coordinate system of the towing cable along its tangential direction is expanded as follows [10].
The streamer is discreated into microsegments, assuming that the cross-sectional area of the streamer is circular, the fluid resistance of the streamer can be obtained through the Morison equation as follows.
 is the density of seawater and d is the diameter of the towing cable.EA is Young's modulus of the towing cable multiplied by the cross-sectional area, T C is the tangential resistance coefficient and N C is the normal resistance coefficient.T V , n V and b V are the relative velocity in the three directions. Due The towing cable on the water surface ignores air resistance, which can be simplified as follows.
1 w is the gravitational force exerted on a unit length of towing cable in the air and  is the angle between the towing cable and the horizontal plane.Ignoring the change in wave height, the vertical distance from the towing cable fixing point on the stern deck of the mother ship to the water surface is known to be 1 z .
l is the length of the cable and a is the length of the cable below the water surface.( 4) and ( 5) are expressed as a segmented function with the cable laying amount x l as the independent variable.
T and 0  are the boundary conditions of the towed body, which can be calculated by programming the drag resistance and underwater gravity of the towed body.The coordinate positions of each point of the towing cable are plotted into an attitude map and the specific calculation formula for the coordinates is: T into Equation (4) and solve the above differential equation using the fourth-order Runge Kutta method.Assuming that the towing cable is completely in the water, the starting point of the aft deck of the mother ship is assumed to be the constant angle above the water surface.So is the upper limit value of a .Generally, the lower limit value can be taken as no more than 0.8 a , And starting the iterative calculation in the order is the height of the towing point on the stern deck of the mother ship to the water surface, the ship is traveling in the opposite direction, the flow velocity is 1.5 knots, the ship speed is 4 knots and the relative speed is 5.5 knots.
The Towing body CFD model and actual object are shown in Figure 1.Towing body parameters: the total length of towing body is 800 mm.The drag force of the drag body is interpolated through CFD software simulation results to obtain   

Results & discussion
By programming the calculation formula, the step size of Runge Kuta is chosen as 0.01 m and the value step size is 0.1 m.It can be calculated through iteration.At the same time, the posture diagrams of the 5.5 and 4.5 knots speed towing cable are plotted, as shown in Figure 2   The position of the towing point in the figure is the origin and the upper right end is the towing point of the mother ship.It can be seen that the slope of the towing cable is divided by the water surface, and there is a slight change in the slope.The turning point is the value.Although the slope change is small, there is still a significant error when it comes to lightweight cables and the height between the water surface and the towing point of the mother ship is relatively high.At the same time, the underwater streamer presents an upward convex shape, which is consistent with the description of the shape of the lightweight streamer for rapid towing.Comparing the two curves, it can be seen that their depth parameters are very sensitive as the speed changes, while the horizontal parameters change less.However, the depth value is often one of the most important parameters for towing bodies.
Through physical sea trial verification, a depth sensor was placed inside the towed body.During the data collection process, the influence of wind, waves and propeller propellers will generate irregular forces on the towing system with peaks of fluctuations ranging from about 18.0 to 19.0 meters, as shown in Figure 3.The data collected by the depth sensor at a speed of 5.5 knots was analyzed by using the least squares method for regression analysis, resulting in 18.50 m with a calculated value of 18.48 m.The data was consistent with 0.11% relative error and the reliability of the calculation results was high.Although we strive to create an ideal environment for comparative analysis with small errors, factors such as fluid flow direction, water resistance coefficient values and measurement errors can interfere with simulation and experimental data results.Compared with previous research results, this study uses segmented functions to fit the posture of the towing cable, which can quickly and effectively capture the posture of the towing cable through theoretical calculations.This method has a guiding significance for engineering practice.

Conclusion
By writing a program and using the fourth-order Runge Kutta method to solve the differential equations of the towing cable surface and underwater segmented functions, the attitude map of the towing cable can be quickly and accurately obtained.For cases where the vertical distance between the towing point of the mother ship and the water surface is large, the towing cable is light and the towing body is light, the part above the water surface has a significant impact on the calculation results.This method is of great significance for situations where the towing body cannot display depth in real time.However, there are few experimental data samples and the value of the resistance coefficient needs to be further corrected.In the future, more sensors will be installed in the towing system for further research.
writing a program, the logic of the program is first to determine the range of values.The method is to substitute the initial conditions l , 0  and 0 .1088/1742-6596/2756/1/012032 4 of small steps from large to small, when the final calculation is completed at statement ends and outputs the current value a and other related parameters.This study verified the correctness of the theoretical calculation conclusions through experiments.The experimental conditions are:

Figure 1 .
Figure 1.Towing body models and physical objects.Towing cable parameters: m l 100  , , through the coordinates of each point with experimental data corresponding to 5.5 knots.At this time,

Figure 2 .
Figure 2. Towing cables position of the computation model.