Effects of surface and internal waves on Unmanned Underwater Vehicles (UUVs)

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Effects of surface and internal waves on Unmanned Underwater Vehicles (UUVs)

Zaman, Hasanat; Millan, Jim

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Surface wave kinematics, UUV, internal wave loading i, 23 16

OCRE-TR-2013-010 _{A1-002609 UNCLASSIFIED UNLIMITED}

Effects of Surface and Internal Waves on Unmanned Underwater Vehicles (UUVs)

Hasanat Zaman and Jim Millan

NRC Ocean, Coastal and River Engineering

DRDC

--- --- Research

Marine Vehicles St-John's: P.O. Box 12093, Arctic Ave, St. John's, NL A1B 3T5

In the present work a vertically 2D numerical model is utilized to study the free surface wave kinematics in the absence and presence of a submarine. Description or derivation of formulae is provided that we have used to compute the internal wave amplitude along the interface of fluids of two different densities when free surface wave conditions are known. Finally another numerical model is utilized to compute the second order drift force on the underwater vehicles due to the presence of internal waves along an interfacial boundary.

Ocean, Coastal and River Génie océanique, côtier et fluvial Engineering

Effects of Surface and Internal Waves on

Unmanned Underwater Vehicles (UUVs)

Technical Report UNCLASSIFIED OCRE-TR-2013-010 Hasanat Zaman Jim Millan March 2013

Table

of Contents

Abstract ... 1 

1. Introduction ... 1 

2. Surface wave model ... 1 

2.1 Computational conditions ... 2 

2.2 Computation of wave kinematics ... 3 

2.3 Computation of the wave height and mean water level (MWL) ... 9 

3. Model for stratified fluid ... 12 

3.1 Model for internal wave computation ... 13 

3.2 Application of the model ... 16 

4. Drift forces on the submerged body ... 18 

4.1 Definition of different drift forces ... 18 

4.2 Drift forces on ellipsoidal body ... 19 

4.3 Examples of the drift forces on the underwater vehicles due to internal wave ... 21 

5. Conclusions ... 23 

6 Acknowledgement ... 23 

Effects of Surface and Internal Waves on

Unmanned Underwater Vehicles (UUVs)

Abstract

In the present work a vertically integrated 2D numerical model is utilized to study the free surface wave kinematics in the absence and presence of a submarine. Description or derivation of a numerical model is provided which was used to compute the internal wave amplitude along the interface of fluids of two different densities when free surface wave conditions are known. Finally a numerical model is used to compute the second order drift force on the underwater vehicles due to the presence of internal waves along an interfacial boundary.

1. Introduction

UUV docking to a mother-ship (submarine) is a very sensitive task as it requires quite a lot of parameters to be taken care of simultaneously during the docking operation. Any one of the parameters can significantly influence the whole docking process and might result in wrong information about the location coordinates, velocities, etc of the vehicles involved. If the docking activity is carried out at a relatively deeper water (deep-vehicle condition) where the ratio of the (deep-vehicle-depth from the free surface to the surface wavelength is equal to or more than 0.5, then the effects of the surface waves on UUVs should not be important provided no internal waves due to density stratification are available in the vicinity to influence the motion of the UUVs. However, in shallower water when the ratio of the vehicle-depth from the free surface to the surface wavelength is less than 0.5 then the scenario gets very much different and problematic as the velocity field of the surface waves is extended up to the locations of the vehicles in the water. In such a situation, if the area is exposed to high wave conditions then the local wave kinematics would be very complicated. As a result, there will be huge changes in the particle velocities, mean water levels, wave heights and energy distribution, etc. over the vehicles. The presence of internal waves and their interaction with the velocity parameters of the surface waves can again contribute extra loads on the UUVs. Second order drift force is another factor that also could influence the motion of the UUVs and mislead the docking operation. Fig. 1 shows the definition sketch of a submarine and UUVs in a docking process. In this work a vertically 2D numerical model is used to study the free surface wave kinematics. Derivation of a numerical model is provided that can compute the internal wave amplitude along the interface of fluids of two different densities when the free surface wave conditions are known. Finally a numerical model is used to compute the second order drift force on the underwater vehicles due to the presence of internal waves.

2. Surface wave model

Following continuity equation and equation of motions are utilized in a vertically 2D numerical model with the assumption of inviscid and incompressible fluid:

0       z w x u (1) 0 1 2             x p z uw x u t u  (2) 0 1 2              g z p z w x uw t w  (3)

where u and w are the velocity components in the x and z directions, respectively ;  _is the water density, g the acceleration due to gravity and t is the time. The total pressure p is the summation of the hydrostatic pressure p_s below the mean water level and p is _d

the dynamic pressure due to wave motion on the free surface.

Vertical integration of the above equations are carried out from bottom (-h) to instantaneous free surface () to find the governing equations. The obtained equations are discretized following a semi-implicit finite difference numerical technique. This model is then utilized to compute wave kinematics in a homogeneous fluid domain. More detail of the model, numerical scheme and computational procedure can be found in Zaman et al (2007).

2.1 Computational conditions

In this work it is assumed that submarine (mother ship) is located in the middle of the water column (h 2) as shown in Fig. 1.

2 ) ( 2 q x h h_e   _d

Fig. 1 Definition sketch of surface wave model

As a test case a regular wave of sea state-6 with wave period of 11s and wave height of 6m is used for the simulation here. The dimension of the submarine (or, under water

z x h2=h/2 h1=h/2 L qd (x)

vehicle) is assumed to be 70m long (VL) and height qd(x) varies along the length to

maintain an elliptical shape. The height at the center of the submarine is 8m. In this simulation UUVs are not included due to their smallness in comparison to the submarine size. The still water depth over the flat bottom of the domain is 30m but the effective still water depth (he) at the center of the submarine reduces to 11m [half of the water depth

minus half of the vertical height of the submarine (h_e h 2q_d(x) 2)]. Additionally, drop of the mean water level will also reduce the effective water depth over the submarine. As a result, particle velocity and acceleration will increase over this region.

2.2 Computation of wave kinematics

The particle trajectory in the wave field is given by the following elliptical expression at any arbitrary point xo and zo. The trajectory in X-Z coordinates is then:

1 ) ( sinh sinh ) ( cosh sinh 2 2 2 2        _        _ z h k kh a Z z h k kh a X (4)

where a is the wave amplitude and k is the wave number.

It is assumed that a water particle moves from its old position (xo, zo) to a newer position

(xo+X, zo+Z) as shown in Fig. 2.

Fig. 2 Axes orientation for particle trajectory computation in 2D flow field

Fig. 3 shows a typical particle trajectories view over submarine and its surrounding area. For deep water



h L0.5



particle orbital motion is circular and slowly decreases towards the bottom and does not extend up to the bottom of the ocean, for intermediate water



0.05 Lh 0.5



the orbital motion is almost elliptical and slowly decreases towards bottom and usually reaches the bottom of the ocean and for shallow water



h L0.05



the orbital motion is significantly elliptical or orbit can be transformed into line depending on the local water depth and horizontal particle velocity and, usually vertical velocity component is uniform along the water depth.

x, X

xo, zo

xo+X, zo+Z

Fig. 3 General view of particle trajectory on the flat bottom and on the submarine

In these computations we have used a wave with period T = 11s, incident wave height H = 6.0m and water depth h = 30m. So we have an intermediate water depth



h L0.191



condition. It is assumed that the submarine is located at the middle of the water depth and is stationary at its location. Figs 4a to 4b show the particle trajectory at the free surface (z=0) on the flat bottom and on the submarine. Figs. 4a and 4b represent the water particle path for wave along xoyo-XZ vertical plane. Figs. 5a and 5b describe the

comparison of the horizontal and vertical velocity components on the flat bottom and on the submarine, respectively. Figs. 6a and 6b show the comparisons of the acceleration on flat bottom and on the submarine. Figs. 7a and 7b show the horizontal x-direction unit wave force on the structure on the above two locations in the domain.

15m 15m

Fig. 4a Particle trajectory over flat bottom [he=30m, T=11s and H=6m, see Fig. 2]

Fig. 4b Particle trajectory at submarine center [he=11m, T=11s and H=6m, see Fig. 2]

z

x z

Fig. 5a Velocity at the surface for flat bottom [he=30m, T=11s and H=6m]

Fig. 5b Velocity at the surface due to presence of submarine [he=11m, T=11s and H=6m]

Velocity (m/s)

time (s)

Velocity (m/s)

Fig. 6a Acceleration due to flat bottom [he=30m, T=11s and H=6m]

Fig. 6b Acceleration due to presence of submarine [he=11m, T=11s and H=6m]

Acceleration (m/s2)

time (s)

Acceleration (m/s2)

Fig. 7a x-direction unit force time series due to flat bottom [he=30m, T=11s and H=6m]

Fig. 7b x-direction unit force due to presence of submarine [he=11m, T=11s and H=6m]

Force (Newton)

time (s) Force (Newton)

2.3 Computation of the wave height and mean water level (MWL)

Root mean square method is utilized to compute the wave height at any spatial location using respective time series of the free surface elevation over one wave period. The following equations are deployed here for the computation of the wave heights over

x-spatial distance: N T H_s 2 2_s2( )/ (5) ) ( . )... 2 ( ) ( ) ( 2 2 2 2 t N t t T _{s s s} s        (6)

Where T is the wave period, t the time increment, Hs the wave height at any location on

the spatial grid (s= 1, 2,…..), )2(_T

s

 the algebraic summation of the square of the surface elevations at any location on the spatial grid over one wave period and N is the number of data points (T/t) in one wave period.

Location of the mean water level (MWL) changes due to wave propagation can be computed using following equations. This tilt of the MWL is also known as setup or setdown: N T s( )/    (7) ) ( . )... 2 ( ) ( ) (T _s t _s t _s N t s        (8)

In this computation it is assumed that the submarine is stationary and no relative motion of the submarine, waves or UUVs is considered here. Fig. 8 compares the wave height distribution in the presence and in the absence of a submarine. Calculated wave height (H) is normalized by the incident wave height (Ho). We can see from this figure

that the wave height increases significantly over the submarine as its presence reduces the effective water depth (he) for the propagating wave over it. Due to change of water depth,

the wave over the submarine will undergo significant shoaling effects. Consequently, its length will be decreased and height will be increased to conserve the energy. Fig. 9 compares the distribution of the MWL with and without the presence of the submarine. The drop of the MWL is a second order effect and considerably depends on the local velocity square of the flow. From Fig. 9 one can observe that the MWL drops significantly due to shallower water depth and stronger wave condition over the submarine. It is found that the peak drop of the MWL is about 46.7cm for the case we have used here. That is to say that the effective water depth (he) at a certain point on the

submarine would also be reduced by 46.7cm. In this study UUVs are not taken into consideration while computing resulted MWL but it is always good to include everything together. Fig. 10 shows the distribution of the instantaneous wavelength (L) normalized by incident wavelength (Li) after two wave periods. Instantaneous wavelengths are

normalized by the incident wavelength. On the other hand, Fig. 11 shows the concentration of energies over the submarine after two wave periods. In this figure instantaneous energies are normalized by the incident wave energy. All related formulation can be obtained in Zaman et al (2007).

0 0.5 1 1.5 2 0 70 140 210 280 350 420 H/H o x (X 0.5 m)

Wave height with SUB Wave height without SUB

Fig. 8 Wave height distribution in the presence and absence of submarine

[he=30m and 11m (at center), T=11s and H=6m; SL=70m and qd(x)=8m at center]

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0 70 140 210 280 350 420  /Ho x (X 0.5 m)

Mean water level with SUB Mean water level without SUB

Fig. 9 Mean water level distribution in the presence and absence of submarine

0 0.5 1 1.5 2 0 70 140 210 280 350 420 L/L i x (X 0.5 m)

Fig. 10 Instantaneous wavelength distribution after two wave periods

[he=30m and 11m (at center), T=11s and H=6m; SL=70m and qd(x)=8m at center]

-75 -50 -25 0 25 50 75 0 70 140 210 280 350 420 E/E i x (X 0.5 m)

Fig. 11 Instantaneous energy distribution after two wave periods

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -30 -25 -20 -15 -10 -5 0 5 u/ Co Water depth, h (m) At incident Grid

At the center of the domain (with SUB)

Fig. 12 Structure of the vertical velocity distribution under wave trough

[he=30m and 11m (at center), T=11s and H=6m; SL=70m and qd(x)=8m at center]

Fig. 12 shows the vertical structure of the velocity distribution at incident grid and at the center of the domain and on the submarine. It can be observed from Fig. 12 that the velocity at the submarine top is higher and more extended than that on the flat bottom at the entrance. Velocity component is normalized by the incident wave celerity (Co).

3. Model for stratified fluid

In this work it is assumed that the submarine (mother ship) is located on the interface of two fluids of different densities.

Fig. 13 Definition sketch of two layered fluids

h2 h1 L 2 1 2 1

The upper layer of the fluid has water depthh₂, density ₂ and surface elevation₂. On the other hand, lower layer has water depthh₁, density  and surface elevation₁  . ₁ The water depth of both layers is equal h1 h2 and the still water depth over flat bottom

ish(h₁h₂).

In the upper layer the pressure variation is solely due to the gradient of the free surface, however, in the lower layer the pressure variation depends not only on the gradient of the interface between the two fluids, but also on the gradient of the free surface.

3.1 Model for internal wave computation

The linearized shallow water equations in a two layer fluid are described in many literatures (such as, Gill, A.E., 1982). The upper layer momentum and continuity equations in the x-horizontal direction are as follows:

x g t u       ₂ ₂ (9) x u h t t          ₂ 2 1 2   (10)

The bottom layer momentum and continuity equations in the x-horizontal direction can be shown in the following way:

x g x g t u           _{2 1} 1 2 1     (11) x u h t       1 1  (12) g g _         1 2 1    (13)

The quantity g in Eq. 11 is known as reduced acceleration. It should be mentioned here that when ₁ ₂ theng g.

Now differentiating Eq. 9 and Eq. 10 with respect to x and t respectively one can find the following equations: 2 2 2 2 2 x g t x u         (14)

t x u h t t           2 2 2 2 1 2 2 2 2  (15)

Substituting Eq. 14 into Eq. 15 we can find the following equation for upper layer fluid which is independent of velocity component:

2 2 2 2 2 2 2 2 1 2 x gh t t            (16)

In the similar fashion, for bottom layer fluid, we can derive Eq. 17 which is again independent of velocity component from Eq. 11 and Eq. 12 in the following form:

2 2 2 1 2 2 2 1 2 1 2 1 2 x gh x h g t               (17)

Let’s assume that the surface undulations at the free surface and at the interface between two layers are as follows:

kx t i Be   _  2 (18) kx t i Ae   _  1 (19)

Substituting Eqs. 18 and 19 into Eqs. 16 and 17 we can derive following relations between amplitudes B and A:



2 ₂ 2



0 2    B k gh A   (20)







2 2



1 2 0 2 1 2     B k gh A k h g    (21)

The ratio of the undulation amplitudes at the free surface B and at the interface A, are found from Eq. 20 as follows:

k c gh c c A B _    : 2 2 2 (22)

Eliminating A and B from Eqs. 20 and 21, we can find the following forth order equations for angular frequency :

0 4 2 1 2 2 4     k h h g g k gh  (23) 0 2 1 2 4 _{ }  _ h h g g ghc c (24)

Now expression for celerity square (c2) of the wave can be obtained from Eq. 24 as follows: 2 4 _{1 2} 2 2 2 gh g h gghh c     (25)             1₂ 2 2 4 1 2 1 2 1 gh h h g gh c (26)

If the magnitudes of the fluid densities  and ₁  differ insignificantly then we can ₂ write from Eq. 13:

1 1 2 1              g g (27)

so that we can write:

1 4 2 2 1   gh h h g (28)

Using Eq. 28, a Taylor series expansion of term



14gh₁h₂ gh2



in Eq. 26 becomes



12g h₂h₂ gh2



. We can then rewrite Eq. 26 as follows:

                1₂ 2 2 2 1 2 1 2 1 gh h h g gh c (29)

If we take positive root in Eq. 29, then we get the following equation, which is identical to the celerity of wave in a homogeneous fluid:

gh

c2  (30)

Now Eq. 22 becomes as follow:

1

h h A

B  (31)

In the case of positive root, free surface wave and internal wave are always in phase. On the other hand, for negative root, free surface wave and internal wave are always in anti-phase (180o out of phase). For negative root following expressions can be obtained:

h h h g c2   1 2 (32) 1 1 1 2 1 2 1 2 1         gh h g gh h g h gh h h g h h g A B (33) Since gh₁ gh1 then gh gh gh h g gh h g 1 1 1 1     h h g g A B  ₁   (34)

So if we know the amplitude of the surface wave (B) at any instance then we can find the amplitude of the internal wave (A) propagating along the interface of the fluids of two different densities (see Fig. 12) by Eq. 31 or by Eq. 34.

3.2 Application of the model

With the above computational domain and incident wave conditions when the surface wave height is 6.0m then the internal wave height would be calculated as 3.0m (Hin) by Eq. 31. Figs. 14a, 14b, 14c and 14d respectively, show the particle trajectory in

the interface of two different fluids, time series of velocity components, time series of accelerations and x-directional loading component on the submarine.

Fig. 14a Particle trajectory over interface of two fluids [h1=15m, T=11s and Hin=3m]

z

Fig. 14b Velocity along the interface of two fluids [h1=15m, T=11s and Hin=3m]

Fig. 14c Acceleration along the interface of two fluids [h1=15m, T=11s and Hin=3m]

Velocity (m/s)

time (s)

Acceleration (m/s2)

Fig. 14 d x-direction unit force due to two fluids [h1=15m, T=11s and Hin=3m]

4. Drift forces on the submerged body

The second order drift force might be an important issue in the process of the docking system where multiple underwater vehicles of different sizes are involved. This is because drift force varies with the size of the submerged vehicles. Fig. 13 shows the computational setup and domain considered here. The shapes of the submerged vehicles (submarines, UUVs) are assumed to be of elliptical and dimensions are shown in Fig. 15.

4.1 Definition of different drift forces 1

F is the mean drift force due to free surface effect which is obtained by integrating the free surface elevation over one wavelength. It can be shown as:



 Wa bndl g F   2 1 1 (35)

where _bis wave surface elevation at the body surface, n is the normal to the surface, and

Wa is the wetted line of the body in the still water.

2

F is the second kind of drift force results from the velocity squared term in the Bernoulli’s equation and can be expressed as

Force (Newton)

ndA u F A N i



 0 1 2 2 2  (36)

where u_i represents the water particle velocity component along the N body coordinates,

dA is the incremental area and Ao is submerged body surface in still water.

The third component of the drift force, due to motion term is

ndA t X F A    



 0 3 (37)

where again the integration is done over the wetted body surface and X’ represents the translational body motion of inertia like, surge, sway or heave.

The fourth component of the mean drift force is due to rotation of body and can be as

3

4 A F

F   (38)

where A is the radius of the gyration corresponding to the surge or sway motion of the body. For additional discussion see, Chakrabarti (1987, pages 260-262).

4.2 Drift forces on ellipsoidal body

In the figure l is the length of major axis, q is the vertical minor axis and r is the horizontal minor axis.

Fig. 15 Dimension of the submerged vehicle

Assuming that the wave is traveling in the x positive direction so the velocity potential can be expressed in the following form:

) ( cosh ) ( cosh 2 ) , , , ( eikx t kh z h k igH t z y x    _   (39)

Ellipsoidal coordinates (X, Y, Z) are connected to Cartesian coordinates (x, y, z) in the following way (Dechambre and Scheeres, 2002):

l q r VL  

KS XYZ x (40) ) ( ) )( )( ( 1 2 2 2 2 2 2 2 2 K S S Z S Y S X S y      (41) ) ( ) )( )( ( 1 2 2 2 2 2 2 2 2 S K K Z K Y K X K z      (42) in which, 2 2 _r l K   (43) 2 2 _q l S  (44)

Using the chain rule of the differentiation and with further simplification the X component of the velocity along the major axis (l) u, Y component of horizontal minor axis (q) v and Z component of vertical minor axis (r) w can be expressed in normalized form as follows (Chakrabarti and Gupta, 2007; Gupta, 2008):

X component of the velocity along the major axis (p):

X z z X y y X x x u                    (45)             kh h h k z K S K Z K K Y X t kx gka cosh ) ( sinh ) ( ) )( ( 2 ) sin( ₂ 2 2 2 2 2 2 2                       ) sin( ; cosh ) ( sinh ) ( ) )( ( 2 2 2 2 2 2 2 2 2 _{gka kx t} u u kh h h k z K S K Z K K Y X u (46)

Y component of the velocity along the major axis (q):

Y z z Y y y Y x x v                    (47)             kh h h k z K S K Z K K X Y t kx gka cosh ) ( sinh ) ( ) )( ( 2 ) sin( ₂ 2 2 2 2 2 2 2                      ) sin( ; cosh ) ( sinh ) ( ) )( ( 2 ₂ 2 2 2 2 2 2 2 _{gka kx t} v v kh h h k z K S K Z K K X Y v (48)

Z component of the velocity along the major axis (r): Z z z Y y y X x x w                    (49)             kh h h k z K S K X K K Y Z t kx gka cosh ) ( sinh ) ( ) )( ( 2 ) sin( ₂ 2 2 2 2 2 2 2                      ) sin( ; cosh ) ( sinh ) ( ) )( ( 2 ₂ 2 2 2 2 2 2 2 _{gka kx t} w w kh h h k z K S K X K K Y Z w (50)

The normalized velocity squared force is then obtained by integrating the summation of the squares over the whole surface.

The force components are integrated for X from –l/2 to l/2, for Y from –q/2 to q/2 and for

Z from –r/2 to r/2 that covers the whole surface of the body. In the process of integration

it is observed due to symmetry of cross section that X and Y components of the forces have the even functions and reduce to zero in any given X-Y plane. So only Z component of the force contributes to the total velocity squared term.

There is no contribution of the drift forces due to the free surface waves on the ellipsoid as it is submerged in the water. No translation and rotation motion are considered in this computation. So only velocity squared terms are contributing to the second order drift force.

The normalized drift forces on the whole underwater body can be computed utilizing the equations below (see also Eq. 36):

  

      /2 2 / 2 / 2 / 2 / 2 / 2 2 2 sin cos ) ( l l q q r r X u v w dXdYdZ F   (51)

  

      /2 2 / 2 / 2 / 2 / 2 / 2 2 2 _{)cos cos} ( l l q q r r Y u v w dXdYdZ F   (52)

  

      /2 2 / 2 / 2 / 2 / 2 / 2 2 2 _)sin ( l l q q r r Z u v w dXdYdZ F  (53) where                    Y X Y X Z 1 2 2 1 _{; tan} tan   (54)

4.3 Examples of the drift forces on the underwater vehicles due to internal wave

A numerical model is developed using Eq. (36) and Eqs. (39) to (54) to compute the drift forces on the underwater vehicles. In this computation a sea state #6 wave conditions is adopted. We have used a wave with period T is 11s, wave height H is 6.0m and water depth h is 30m. It is also assumed that the submarine is located at the middle of

the water depth and the vehicle is assumed to be stationary at its location. Translation and rotation motions are not considered here.

Using Eq. 31 we can approximate the incident internal wave height Hin to be 3m. All

the normalized drift forces are plotted here with respect to ka. Fig. 15 shows the normalized second order drift forces for submarine (mother-ship). On the other hand, Fig. 16 shows the normalized drift forces on the UUVs of different sizes. It is observed from Figs. 15 and 16 that drift forces increases with the increasing size of the underwater vehicles. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 k a F (N orm a liz ed)

Submarine : Length = 70m and Diameter = 8m

Fig. 16 Normalized drift force on the submarine

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 k a F ( N o rm a liz e d )

UUV : Length = 10m and Diameter = 1.0m UUV : Length = 8m and Diameter = 0.8m UUV : Length = 5m and Diameter = 0.5m UUV : Length = 4m and Diameter = 0.4m UUV : Length = 3m and Diameter = 0.3m UUV : Length = 2m and Diameter = 0.2m

Fig. 17 Normalized drift force on the UUVs of different sizes

[h2=15m and Hin=3m; different UUV lengths and diameters]

5. Conclusions

Two numerical models are used here: (1) a vertically integrated numerical model to study the wave kinematics at the free surface in the presence and absence of submarine in the water and, (2) a numerical model to compute the drift forces on the underwater vehicles. A simplified derivation is provided to compute the internal wave amplitude along the interface of fluids of two different densities due to surface waves. For the above computational domain and incident wave conditions when the surface wave height is 6.0m then the internal wave height would be calculated as 3.0m by Eq. 31. This wave height can produce as much as the half of the loadings that shown in Figs. 7a and 7b. In the computed results wave particle trajectories, particle velocities, wave heights, MWL, change in wavelengths, energy distribution, vertical velocity distribution, x-direction forces and second order drift forces are shown. All the above information are necessary to predict the relative motions of the submarine (mother-ship) and UUVs for a smooth docking process.

6 Acknowledgement

NRC and authors are grateful to Dr. George Watt of DRDC-Atlantic for his financial support and for providing NRC with a very interesting project to work on.

7. References

Chakrabarti, S.K. (1987): Hydrodynamics of Offshore Structures. Computational Mechanics Publications, Springer-Verlag, London, 1987. pages 420.

Dechambre, D. and Scheeres, D. J. (2002): Transformation of spherical harmonic coefficients to ellipsoidal harmonic coefficients, Astronomy and Astrophysics 387, pages 1114-1122.

Gill, A. E. (1982): Atmosphere-Ocean Dynamic, International Geophysics Series, Academic press, pp 662.

Gupta, A. (2008): Steady wave drift force on basic objects of symmetry, a thesis submitted to the Office of Graduate Studies of Texas A&M University, for M.Sc.

Chakrabarti, S. and Gupta, A. (2007): Steady wave drift force on basic objects of symmetry. In: 4th International Fluid-Structure Interaction Conference 92, pages 1-11. Zaman, M. H., Togashi, H. and Baddour, E. (2007): Propagation of monochromatic water wave trains, Ocean Engineering, 34 (13), pages 1850-1862.