Ocean Engineering 27 (2000) 455–472

www.elsevier.com/locate/oceaneng

A hydrodynamic model of a two-part

underwater towed system
Jiaming Wu, Allen T. Chwang*
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

Received 10 September 1998; accepted 25 November 1998

Abstract

A three-dimensional model of a two-part underwater towed system is studied. In the model,

the governing equations of cables are established based on the Ablow and Schechter method.
The boundary conditions for the two-part underwater towed system are derived. The six-
degrees-of-freedom equations of motion for submarine simulations are adopted to predict the
hydrodynamic performance of a towed vehicle. The established governing equations for the
system are then solved using a central finite difference method. In this paper several algorithms
are used to solve this special form of finite difference equations. The results in this paper
indicate that the two-part underwater towed system improves the dynamic behavior of the
towed vehicle and is an easy way to decouple the towing ship motion from the towed vehicle.
Because the model uses an implicit time integration, it is stable for large time steps and is an
effective algorithm for simulation of a large-scale underwater towed system.  1999 Elsevier
Science Ltd. All rights reserved.

Keywords: Two-part tow; Underwater; Towed system; Numerical simulation; Cable

1. Introduction

Underwater towed systems are fundamental tools for many marine applications
which include naval defense, seabed mapping and ocean environmental measure-
ments. 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.

* Corresponding author.

0029-8018/00/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 9 - 8 0 1 8 ( 9 9 ) 0 0 0 0 6 - 2
456 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

In simulating the hydrodynamic performance of an underwater towed system, the

coupling effect between the towing ship and the towed system is generally neglected,
and the hydrodynamic model is usually composed of two parts: the cable and the
towed vehicle. It is well known that the equations of motion for the cable and the
towed vehicle are nonlinear 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 numerical methods
are usually employed.
The most prevalent approaches used nowadays in determining the hydrodynamic
performance of a cable in an underwater towed system are the lumped mass method
(Walton and Polachech, 1960) and the finite difference method (Ablow and
Schechter, 1983). However, according to Howell (1991) 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 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, Hearn and Thomas (1991) 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, 1989; Hearn and
Thomas, 1991).
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 for-
mulated in a local tangential-normal coordinate frame which has the unstretched
distance along the cable. The differential equations are then approximated by finite
difference equations centered in time and in 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 an 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 describing the hydrodynamic behavior of an underwater vehicle, the six-
degrees-of-freedom equations of motion for submarine simulation can be adopted.
In recent years, several hydrodynamic models of underwater towed systems have
been developed by various authors. Chapman (1984) proposed a model to describe
the dynamic behavior of an underwater towed system. The model is mainly used to
simulate the behavior when the towing ship travels in a circular trajectory and is
 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 457

not fully three-dimensional. Koterayama et al. (1988) developed a three-dimensional

model for a single cable towed system, in the model the behavior of the cable is
determined by the lumped mass method.
The research on hydrodynamics of an underwater towed system at present is
mainly concentrated on a single cable one. Owing to the demand of towed vehicle
operations, there is a need for these systems to have a capability of moving with a
stable attitude in ocean environments such as wave actions, and the towed vehicle
generally requires that the perturbations from the wave induced motion of a towing
ship are minimized. One easy method is to use a two-part underwater towed system
to decouple the surface vessel motion from the towed vehicle. Ranmuthugala and
Gottschalk (1993) proposed a two-dimensional hydrodynamic model of a two-part
underwater towed system. In the model the kinematic properties of the towed vehicle
are not discussed.
In this paper a three-dimensional hydrodynamic model to simulate a two-part
underwater towed system is proposed. In the model, the governing equations of
cables 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 hydrodynamic performance of a towed vehicle. The established governing
equations are then solved using a central finite difference method. In this paper
several algorithms are used to solve this special form of finite difference equations.
The numerical results indicate that a two-part underwater towed system improves
the dynamic behavior of the towed vehicle. Because the model uses an implicit time
integration, it is stable for large time steps. It 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. The model presented in this paper
can easily be extended to simulate multiple towed cables and towed bodies.

2. Governing equations and boundary conditions

The system described in this paper consists of primary, secondary and depressor
cables, and the towed vehicle and the depressor are attached to the end of secondary
and depressor cables respectively (Fig. 1). The primary cable and depressor cable
are negative buoyant, while the towed vehicle and the secondary cable are neu-
trally buoyant.
Three different coordinate systems are used in the derivation of equations, i.e. the
fixed inertial coordinate system (X,Y,Z) and local coordinate systems for cables (t,n,b)
and for the towed vehicle (x,y,z) as shown in Fig. 1. The governing equations for
cables are given first, then three cables are modeled separately and interfaced
dynamically at the conjunction. The speed of the towing ship on the water surface
and dynamic equations of the towed vehicle and the depressor are taken as boundary
conditions for governing equations of cables. For results presented in this paper,
computations start from a steady state solution which is taken as the initial condition
for the whole system.
458 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

Fig. 1. A two-part underwater towed system.

2.1. Cable modeling

The cable is treated as a long, thin, flexible circular cylinder. It is assumed that
the dynamics of a cable are determined by gravity, hydrodynamic loading and inertial
forces, no bending or torsional stiffness is taken into account in this study.
The equations are written in orthogonal coordinates (t, n, b) local to each point
of the cable. The orthogonal coordinates (i, j, k) represent the fixed inertial frame,
i, j being in the horizontal plane and k pointing downward. The origins of systems
(i, j, k) and (t, n, b) are coincident. The orientation of the local frame is so chosen
that t is tangent to the cable in the direction of increasing unstretched cable length
coordinate s, b is in the plane of i and j, t and n lie in a vertical plane. At any point
of a cable the two frames (i, j, k) and (t, n, b) are related by
(t,n,b) ⫽ (i,j,k)[D], (1)
where

冤 冥
cos␽cos␸ ⫺ cos␽sin␸ sin␽
[D] ⫽ sin␽cos␸ ⫺ sin␽sin␸ ⫺ cos␽ .
sin␸ cos␸ 0
Eq. (1) represents a rotation through angle ␽ about the k axis to bring i axis into
the plane of t and n, rotation about the new i axis through ␲/2 to bring k into
coincidence with b, and rotation about b through ␸ to bring i and j into coincidence
with t and n. The relative position of two coordinate systems is shown in Fig. 2.
Similar to Ablow and Schechter (1983), at any point of cables, the governing
equation can be written as
MY⬘ ⫽ NẎ ⫹ Q,
 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 459

Fig. 2. The relative position of two coordinate systems.

∂Y ∂Y
Y ⫽ (T,␯t,␯n,␯b,␽,␸)T, Y⬘ ⫽ , Ẏ ⫽ ,
∂s ∂t
1 0 0 0 0 0 

M⫽
 0 1 0 0
0 0 1 0
␯bcos␸
⫺ ␯bsin␸
⫺ ␯n
␯t 
, (2)

 
0 0 0 1 ⫺ ␯tcos␸ ⫹ ␯nsin 0
0 0 0 0 ⫺ Tcos␸ 0
0 0 0 0 0 T 
⫺ mevt
 m 0 0 m1␯bcos␸ ⫺ m1␯n 

 
1 ⫹ eT
e 0 0 0 0 0
0 0 0 0 0 1 ⫹ eT
N⫽ 0 0 0 0 ⫺ (1 ⫹ eT)cos␸ 0 ,

 
em1␯b
⫺ 0 0 m1 m1␯nsin␸ ⫺ m␯tcos␸ 0
1 ⫹ eT
em1␯n
 ⫺ 0 m1 0 ⫺ m1␯bsin␸ m␯ t 
1 ⫹ eT
 ⫺ w sin␸ ⫹ (1 ⫹ eT)1/2 21 ␳␲dCt␯t兩␯t兩 

Q⫽
 0
0 
,

 
0
1
2 ␳dCn(␯2n ⫹ ␯2b)1/2␯b(1 ⫹ eT)1/2
 ⫺ w cos␸ ⫹ 1
2 ␳dCn(1 ⫹ eT)1/2␯n(␯2n ⫹ ␯2b)1/2 
460 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

where T denotes the tension force of the cable, ␯t, ␯n and ␯b are the velocity compo-
nents along the t, n and b directions respectively, m is the mass per unit length of
cable, m1 is equal to m ⫹ ␳A (␳ being the density of water and A the cross-sectional
area of the unstretched cable), e is equal to 1/EA (E being the Young’s modulus of
cable), w is the submerged weight per unit length of a cable, and d is the cable diam-
eter.

2.2. Boundary and initial conditions for the system

The velocity components of three cables at the conjunction point must be identical
and resultant forces of three cables at that point must be zero. Therefore
VPN ⫽ VD0 ⫽ VS0, (3a)
⌺T ⫽ 0, (4a)
or

冤冥 冤冥 冤冥
␯t ␯t ␯t
[D]PN ␯n ⫽ [D]D0 ␯n ⫽ [D]S0 ␯n , (3b)
␯b PN ␯b D0 ␯b S0

冤冥 冤冥 冤冥
Tt Tt Tt
[D]PN 0 ⫹ [D]D0 0 ⫹ [D]S0 0 ⫽ 0, (4b)
0 PN 0 D0 0 S0

where subscripts P, D, and S denote the primary, depressor and secondary cables
respectively, and subscripts 0 and N denote the first and last node, the directions of
the cables are shown in Fig. 1.
The relation between the towing ship velocity and the towing point velocity of
the primary cable is
[␯t,␯n,␯b]P0 ⫽ [Sx,Sy,Sz][D]P0, (5)
where Sx, Sy and Sz are the velocity components of the towing ship in inertial coordi-
nates.
In this paper the depressor is assumed to be a weighted ball. Therefore, at the end
of the depressor cable (node DN) the boundary condition is
TDN ⫽ ⫺ 1
2 ␳CDDSD兩VDN兩VDN ⫺ (C1␳⌬ ⫹ m0)V̇DN ⫹ w0k, (6)
where CDD is the drag coefficient of depressor, SD the projected area of depressor,
C1 the added-mass coefficient of depressor, ⌬ the volume of depressor, m0 the mass
of depressor and w0 the submerged weight of depressor. The values of T, ␽ and ␸
at node DN are determined from Eq. (6).
The velocity coupling relation between the end of the secondary cable (node SN)
and the towing point of a towed vehicle is
 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 461

[VV ⫹ ␻ ⫻ rT] ⫽ [E][D]SNVSN, (7)

where VV ⫽ (u,v,w) and ␻ ⫽ (p, q, r) are translational and angular velocities of
T

the towed vehicle in the vehicle-fixed coordinate, rT ⫽ (xT, yT, zT) is the towing point
coordinates in the vehicle-fixed frame, VSN is expressed in local coordinates of
cable and
[E]

冤 冥
cos␪cos␺ cos␪sin␺ ⫺ sin␪
⫽ ⫺ sin␺cos␾ ⫹ sin␾sin␪cos␺ cos␾cos␺ ⫹ sin␾sin␪sin␺ sin␾cos␪ ,
sin␾sin␺ ⫹ cos␾cos␺sin␪ ⫺ sin␾cos␺ ⫹ cos␾sin␪sin␺ cos␾cos␪
where ␪, ␾ and ␺ are the pitch, roll and yaw angles of the towed vehicle.
The six-degrees-of-freedom equations of motion for a towed vehicle in surge,
sway, heave, roll, pitch, and yaw can be written as (Gertler and Hagen, 1967)
m[u̇ ⫺ ␯r ⫹ wq ⫺ xG(q2 ⫹ r2) ⫹ yG(pq ⫺ ṙ) ⫹ zG(pr ⫹ q̇)] ⫽ XH (8)
⫹ XW ⫹ XC ⫹ XT,
m[␯˙ ⫹ ur ⫺ wp ⫹ xG(pq ⫹ ṙ) ⫺ yG(p2 ⫹ r2) ⫹ zG(qr ⫺ ṗ)] ⫽ YH (9)
⫹ YW ⫹ YC ⫹ YT,
m[ẇ ⫺ uq ⫹ vp ⫹ xG(pr ⫺ q̇) ⫹ yG(qr ⫹ ṗ) ⫺ zG(p2 ⫹ q2)] ⫽ ZH (10)
⫹ ZW ⫹ ZC ⫹ ZT,
Ixṗ ⫹ (Iz ⫺ Iy)qr ⫹ Ixy(pr ⫺ q̇) ⫺ Iyz(q2 ⫺ r2) ⫺ Ixz(pq ⫹ ṙ) (11)
⫹ m[yG(ẇ ⫺ uq ⫹ vp) ⫺ zG(v̇ ⫹ ur ⫺ wp)] ⫽ KH ⫹ KW ⫹ KC ⫹ KT,
Iyq̇ ⫹ (Ix ⫺ Iz)pr ⫺ Ixy(qr ⫹ ṗ) ⫹ Iyz(pq ⫺ ṙ) ⫹ Ixz(p2 ⫺ r2)
⫺ m[xG(ẇ ⫺ uq ⫹ vp) ⫺ zG(u̇ ⫺ vr ⫹ wq)] ⫽ MH ⫹ MW ⫹ MC (12)
⫹ MT,
Izṙ ⫹ (Iy ⫺ Ix)pq ⫺ Ixy(p2 ⫺ q2) ⫺ Iyz(pr ⫹ q̇) ⫹ Ixz(qr ⫺ ṗ) (13)
⫹ m[xG(␯˙ ⫹ ur ⫺ wp) ⫺ yG(u̇ ⫺ vr ⫹ wq)] ⫽ NH ⫹ NW ⫹ NC ⫹ NT,
where the left-hand sides represent inertial forces and moments and the right-hand
sides denote external forces on the towed vehicle. Subscript H reflects hydrodynamic
contributions, W buoyant and weight effects, C forces arising from control surfaces
(control wing, flap etc.) and T towing forces. The symbols in the equations are based
on the standard notation.
The towed vehicle being investigated in this paper is assumed to be a circular
cylinder with symmetry about its center of uniform mass located at the mid point
of the vehicle. There is no control surfaces in such a type of towed vehicle. Thus
control forces do not exist. The hydrodynamic forces acting on the towed vehicle
are given by
462 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

XH ⫽ Xu̇u̇ ⫺ Xuuu兩u兩, (14)

(␯ ⫹ xr)
YH ⫽ Y␯˙ ␯˙ ⫺ 1
2 ␳CD␯D␯兰[(␯ ⫹ xr)2 ⫹ (w ⫺ xq)2] dx, (15)
Ucf(x)
(w ⫺ xq)
ZH ⫽ Zẇẇ ⫺ 1
2 ␳CD␯D␯兰[(␯ ⫹ xr)2 ⫹ (w ⫺ xq)2] dx, (16)
Ucf(x)
KH ⫽ Kṗṗ, (17)
(w ⫺ xq)x
MH ⫽ Mq̇q̇ ⫹ 1
2 ␳CD␯D␯兰[(␯ ⫹ xr)2 ⫹ (w ⫺ xq)2] dx, (18)
Ucf(x)
(␯ ⫹ xr)x
NH ⫽ Nṙṙ ⫺ 1
2 ␳CD␯D␯兰[(␯ ⫹ xr)2 ⫹ (w ⫺ xq)2] dx, (19)
Ucf(x)
where CDv is the drag coefficient of the towed vehicle in sway and heave, D␯ the
diameter of the towed vehicle, and
Ucf ⫽ [(␯ ⫹ xr)2 ⫹ (w ⫺ xq)2]1/2. (20)
The integrals in the above equations are over the length of the towed vehicle.
Hydrostatic restoring forces and moments due to vehicle weight W and buoyancy
B are given by
XW ⫽ ⫺ (W ⫺ B)sin␪, (21)
YW ⫽ (W ⫺ B)cos␪sin␾, (22)
ZW ⫽ (W ⫺ B)cos␪cos␾, (23)
KW ⫽ (yGW ⫺ yBB)cos␪cos␾ ⫺ (zGW ⫺ zBB)cos␪sin␾, (24)
MW ⫽ ⫺ (xGW ⫺ xBB)cos␪cos␾ ⫺ (zGW ⫺ zBB)sin␪, (25)
NW ⫽ (xGW ⫺ xBB)cos␪sin␾ ⫹ (yGW ⫺ yBB)sin␪. (26)

The towing forces and moments on the towed vehicle in local coordinates of the
vehicle are

冤冥
XT
FT ⫽ YT ⫽ ⫺ [E][D]SNTSN, (27)
ZT
[KT,MT,NT] ⫽ rT ⫻ FT. (28)
To complete the hydrodynamic model of the towed system, the expression for the
rates of change of Euler angles for the towed vehicle is
␾˙ ⫽ p ⫹ q sin␾tan␪ ⫹ r cos␾tan␪, (29)
␪˙ ⫽ q cos␾ ⫺ r sin␾, (30)
 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 463

sin␾ cos␾
␺˙ ⫽ q ⫹r . (31)
cos␪ cos␪

For results presented in this paper, constant velocity of the towing ship Vts is
assumed to be the initial state, that is, Vts ⫽ ⫺ V0i and
∂
␯t ⫽ ⫺ V0cos␸, ␯n ⫽ V0sin␸, ␯b ⫽ 0, ␽ ⫽ 0, ⫽ 0. (32)
∂t

3. Finite difference approximation

The governing equation of cables (Eq. (2)) with the conditions described in Section
2.2 determines the towed system parameters completely. In this study the partial
differential Eq. (2) is approximated by a finite difference equation centered in time
and in space. Three cables are divided into a series of segments of length ⌬Sj and
time is divided into a series of time steps ⌬t. The node numbers in three cables are
0 苲 NP, 0 苲 ND, 0 苲 NS respectively, 6 node variables Yj ⫽ (T, ␯t, ␯n, ␯b, ␽,␸)Tj
(j⫽0 苲 N, N ⫽ NP, ND or NS) are included at each node of cables. The finite differ-
ence approximation of Eq. (2) is written at mid point of each cable segment
(Burgess, 1991),
⫹ 1 ⫺ Yj
Yij ⫹ Yij ⫹ 1 ⫺ Yij
1 i⫹1

⫹ 1 ⫹ Mj
[Mij ⫹ ⫹ [Mij ⫹ 1 ⫹ Mij ] ⫽ [Nij ⫹
1 i⫹1 1
] ⫹1
⌬Sj ⌬Sj
⫹ 1 ⫺ Yj ⫹ 1
Yij ⫹ Yij ⫹ 1 ⫺ Yij
1 i
⫹ Nij ⫹ 1] ⫹ [Nij ⫹ 1 ⫹ Nij ] ⫹ 1 ⫹ Qj
⫹ Qij ⫹ 1 i⫹1
(33)
⌬t ⌬t
⫹ Qij ⫹ 1 ⫹ Qij ,
where superscript i denotes i-th time step.
The finite difference equations include all node variables in three cables, there are
6(NP ⫹ ND ⫹ NS ⫹ 3) unknown variables Yj of all three cables with 6(NP ⫹ ND ⫹
NS) finite difference equations in Eq. (33). The rest 18 scalar equations are provided
by Eq. (3a), (3b), (4a), (4b)–(7) in finite difference form. However, Eq. (7) introduces
9 new variables, that is, 3 translational velocity components (u, v, w), 3 angular
velocity components (p, q, r) and 3 Euler angles (␪, ␾, ␺) of the towed vehicle.
Therefore, 9 more equations are needed to solve the problem. These 9 equations are
given by Eqs. (8)–(13) and Eqs. (29)–(31).
Let g(Y) ⫽ 0 be the [6(NP ⫹ ND ⫹ NS ⫹ 3) ⫹ 9] nonlinear algebraic equations
as established above, and variables Y are ordered as
Y ⫽ 兵YPj,YDj,YSj,(u,v,w,p,q,r),(␾,␪,␺)其T, (34)
P D S
where Y , Y and Y are all node variables in the primary, depressor and secondary
j j j
cables ordered from the first to the last node. The components of g are arranged
as follows:
464 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

1. leading point boundary conditions (Eq. (5)),

2. 6 NP finite difference equations for the primary cable (Eq. (33)),
3. matching relations at the conjunction point (Eq. (3a), (3b), (4a) and (4b)),
4. 6 ND finite difference equations for the depressor cable (Eq. (33)),
5. boundary conditions at the end of the depressor cable (Eq. (6)),
6. 6 NS finite difference equations for the secondary cable (Eq. (33)),
7. boundary conditions at the end of the secondary cable (Eq. (7)), and
8. equations describing the behaviors of the towed vehicle (Eqs. (8)–(13) and Eqs.
(29)–(31)).
The nonlinear algebraic equations are solved by iteration in the time domain using
Newton’s method. Denote Y(m) an approximate value in the m-th iteration for the
solution of the unknown variables Y, and
Y(m ⫹ 1) ⫽ Y(m) ⫹ ⌬Y(m), (35)
where ⌬Y(m) is found by solving the linear equations
J(m)⌬Y(m) ⫹ g(m) ⫽ 0, (36)
∂g
and J ⫽ is the Jacobian matrix of equations g(Y) ⫽ 0. The structure of J is
∂Y
shown in Fig. 3. It can be seen from the figure that most of nonzero submatrices of
J are in block tridiagonal form except submatrix S01 which is introduced by the

Fig. 3. The block structure of the Jacobian. The shaded submatrices indicate potential fill-in outside
block tridiagonal in the same columns as S01.
 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 465

derivatives of Eq. (3a), (3b), (4a) and (4b) with respect to the variables at node S0
of the secondary cable.
The Gaussian elimination with partial pivoting is applied to solve Eq. (36). In this
work, the Jacobian J is stored in the matrices as shown in Fig. 4. The block tridia-
gonal submatrices are stored in M and in EN, and submatrix S01 together with out-
side block tridiagonal potential fill-in submatrices (shaded submatrices as shown in
Fig. 3) which are located in the same columns as S01 are stored in EX. Numerical
manipulation to solve Eq. (36) is carried out in these matrices, only nonzero submatr-
ices of the Jacobian are dealt with in this algorithm. The choice of pivotal element
and elimination of nonzero elements below the main diagonal are conducted in the
first column of matrix M or EN. When the first element is eliminated in a row, all
remaining elements in that row move up one entry and set the last entry (column
12 in M or column 15 in EN) in that row equal to zero. Same row exchange and
row elimination are conducted in the matrix EX when the numerical operations deal
with the outside block tridiagonal submatrices. Backsubstitution is carried out after
the elimination process, and finally Eq. (36) can be solved.

4. Numerical simulations and discussions

The computer program based on the above established model gives the three-
dimensional shape of the system, the tension and velocity components along the

Fig. 4. The storage of the Jacobian. The dashed line submatrix in EN indicates the transfer matrix
between M and EN.
466 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

three cable nodes and also the velocities and Euler angles of the towed vehicle at
each time step. The direction of towing path and the towing point speed may be
changed at any time. Thus the model can easily be used to predict the system
response during different maneuvers. In this section numerical simulations are given
to discuss the hydrodynamic characteristics of this kind of underwater towed system.
The particulars of cables, depressor and the towed vehicle used in the investi-
gations are
Depressor: diameter ⫽ 0.85 m, weight in water ⫽ 2000 kg,
Towed vehicle: D␯ ⫽ 0.3 m, L ⫽ 2.0 m, yT ⫽ zT ⫽ 0, xT ⫽ ⫺ L/2,
Cable: d ⫽ 0.02 m, Ct ⫽ 0.02, Cn ⫽ 2.0, SP ⫽ 300 m, and mP ⫽ 2.7 kg/m, SD
⫽ 6 m, and mD ⫽ 2.7 kg/m, SS ⫽ 20 m, and mS ⫽ ␳A kg/m,
where L is the length of towed vehicle, Ct and Cn are tangential and normal drag
coefficients of cables and SP, SD and SS are the lengths of three cables.

4.1. 3-D numerical simulations

For demonstration of the model on describing three-dimensional hydrodynamic

performance of the system, a numerical example of a towing ship traveling in wave
fields is presented. The incident wave propagates in the X direction with amplitude
␩0 and period T0. The values of towing ship response functions for heave and surge
are rh and rs. The towing ship travels sinusoidally on the horizontal plane along the
path ⫺ V0ti ⫹ ␩ycos(2␲t/Ty)j. The encountering frequency for the towing ship heave
and surge is ␻e ⫽ (kV0 ⫹ 2␲/T0). With the wave effects considered, the trajectory
of the towing point can be written as

rTS ⫽ 再 ⫺ V0t ⫹ ␩0rscos kV0 ⫹ 冋冉 2␲

T0 冊 册冎
t ⫹ ␸s i ⫹ ␩ycos
2␲
Ty
t j 冉 冊 (37)

冋冉
⫹ ␩0rhcos kV0 ⫹
2␲
T0 冊 册
t ⫹ ␸h k,

where k is the wave number, ␸s and ␸h are phase differences of surge and heave,
the values used in Eq. (37) are given in Table 1.
Fig. 5 shows the time histories of vertical displacements of the towed vehicle and
the depressor. It can be seen that the amplitude of the towed vehicle is about one
sixth of that of the depressor, which indicates that the two-part tow method does
reduce the heaving response of the towed vehicle. In Fig. 5 the low frequency oscil-

Table 1
The values used in the numerical simulation

V0 (m/s) ␩0 (m) T0 (s) ␩0 (m) Ty (s) rs rh

6.28 1.0 6.0 8.0 150.0 0.1 0.9

 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 467

Fig. 5. Time histories of vertical displacements of the towed vehicle and the depressor.

lations of the towed vehicle and the depressor are caused by the sinusoidal movement
of the towing ship in the horizontal plane.
The two-part tow method basically has no influence on the sway or surge behavior
of the towed vehicle. The time histories of the towed vehicle and the depressor are
similar in surge or sway (Figs. 6 and 7). In Fig. 8 the roll, pitch and yaw angles of
a towed vehicle during the above maneuver are shown. It can be seen from Fig. 8
that even though there is no other control measure, the pitching angle of the towed
vehicle remains very small (兩␽兩 ⱕ 1.0°), which is within the permitted limit of most
towed vehicle’s requirements. Fig. 9 shows time histories of cable tensions at three
cable ends (nodes P0, DN and SN). It is obvious that the maximum tension occurs
at node P0. The amplitudes of tension vary with different cable ends, the greater the
mean value of tension, the greater the amplitude.

Fig. 6. Surge movements of the towed vehicle and the depressor.

468 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

Fig. 7. Time histories of the towed vehicle and the depressor in sway.

Fig. 8. The roll, pitch and yaw angles of the towed vehicle.

In the above numerical simulation, the nodes on the cables are equally spaced.
The discrete segment numbers of the primary, depressor and secondary cables are
30, 3 and 5 respectively. The time steps are held fixed at 0.3 sec in the simulation
and the number of iterations needed for convergence at each time step is about 4.
The selection of the time step in the example is based on the nature of the high
frequency oscillation of the towing point. For low frequency maneuvers such as a
towing ship traveling in still water plane, the time step can be taken as large as 20
to 30 sec with no numerical instability.
 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 469

Fig. 9. Cable tensions at three cable ends.

4.2. Hydrodynamic characteristics of the system

To maintain the attitude of a towed vehicle as stable as possible under different

towing operations is one of the major concerns by the users of an underwater towed
system. It is evident that the hydrodynamic response of the towed vehicle in a two-
part underwater towed system varies with different towing conditions and different
system’s parameters. According to the results of our numerical simulation, for the
two-part underwater towed system, the heaving ratio between the towed vehicle and
the depressor depends on the encountering frequency of the towing ship in waves
and the length of the secondary cable etc. The higher the encountering frequency,
the smaller the ratio (Fig. 10); the longer the secondary cable, the smaller the ratio

Fig. 10. Heaving ratio between the towed vehicle and the depressor versus the encountering frequency.
470 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

Fig. 11. Heaving ratio between the towed vehicle and the depressor versus the secondary cable length.

(Fig. 11). In order to maintain a towed vehicle at a constant water depth, the length
of the primary cable in a two-part underwater towed system has to be changed under
different towing speeds. Fig. 12 shows the relationship between the towing speed
and the length of primary cable for the towed system as described in this section
when the towed vehicle is kept at 86.3 m water depth. Fig. 13 demonstrates the
heaving responses of the towed vehicle and the depressor versus the towing speed
on the condition that the towed vehicle is kept at about 86.3 m mean value of water
depth under the towing conditions as in Section 4.1. It is found that although the
heaving response of the depressor varies greatly with different towing speed, the
heaving amplitude of the towed vehicle remains very stable and its value is about
0.1 m. The lower the towing speed, the smaller the heaving ratio.

Fig. 12. Relationship between the primary cable length and the towing speed (the depth of the towed
vehicle is fixed at 86.3 m).
 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472 471

Fig. 13. Heaving responses of the towed vehicle and the depressor versus the towing speed.

Fig. 14 illustrates the time histories of the towed vehicle and the depressor in
heaving under the same towing conditions as in Fig. 5 except that the incident wave
propagates in the negative X direction. Comparing Fig. 14 with Fig. 5 it can be found
that the heaving patterns in two towing conditions are different. The heaving ampli-
tude of the towed vehicle in Fig. 14 is about 2.5 times than that in Fig. 5. The reason
for the difference is that the encountering frequencies of the towing ship in two
towing conditions are different, the former is 0.35 sec (following sea towing) and
the latter is 1.75 sec (head sea towing). According to the observation of Fig. 10, it
is expected that the heaving response of the towed vehicle in a head sea towing will
be smaller than that in a following sea towing.

Fig. 14. Time histories of the vertical displacements of the towed vehicle and the depressor (following
sea towing).
472 J. Wu, A.T. Chwang / Ocean Engineering 27 (2000) 455–472

5. Conclusions

A fully three-dimensional model of a two-part underwater towed system is pro-

posed. The computer program based on the model can give a three-dimensional shape
of the system, the dynamic properties of cables and the towed vehicle. The results
of numerical simulations indicate that the two-part tow method improves the hydro-
dynamic behavior of the towed vehicle in vertical plane motions (heave and pitch),
but no great difference is observed between the horizontal movements (surge and
sway) of the towed vehicle and those of the depressor. The encountering frequency
of a towing ship and the length of the secondary cable have notable influences on
the hydrodynamic behavior of the towed vehicle. In order to maintain a towed vehicle
as stable as possible during towing operations it is preferable to select a sufficiently
long secondary cable and choose a favorable manner of towing operation such as a
head sea towing. Because of the finite difference method and other algorithms
adopted in the model, more freedom is given to determine the time step and the
node number distributed along cables. The algorithms adopted in this paper guarantee
that the choice of the node number on cables does not constitute a substantial obstacle
on the calculation speed and computer memory or cause the numerical calculation
unstable. The model provides a good algorithm for simulation of the large scale
motion of an underwater towed system. The model presented in this paper can easily
be extended to simulate multiple towed cables and towed bodies.

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