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An axisymmetric underwater vehicle-free surface interaction: A numerical study Ali Nematollahi a, Abdolrahman Dadvand b,n, Mazyar Dawoodian b a b

Department of Mechanical Engineering, University of Manitoba, Winnipeg, Canada MB R3T 5V6 Department of Mechanical Engineering, Urmia University of Technology, Urmia 419-57155, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 30 March 2014 Accepted 20 December 2014

Underwater vehicles (UWVs) have been widely used in oceanographic applications. Investigation of their interaction with free surface when they are moving near the free surface is of great importance. In the present work, the hydrodynamic characteristics of a standard UWV (Afterbody-1) moving in water and its interaction with free surface are studied numerically using CFD software ANSYSTM CFX. The total drag coefﬁcient including the viscous and wave-making resistances acting over the UWV for its operating speeds ranging from 0:4 m=s (Re ¼ 1:05 105 ) to 1:4 m=s (Re ¼ 3:67 105 ) at different depths of submergence ranging from 0:75 to 4:0, is obtained. Also the wake formed behind the UWV is characterized to better understand the hydrodynamic behavior of the UWV motion in water at different submergence depths and vehicle speeds. The results were compared with available measured data and good agreements were observed. It was found that, for all submergence depths as the Reynolds number was increased the drag coefﬁcient was decreased. Besides, for a ﬁxed Reynolds number as the submergence depth was decreased the drag coefﬁcient was increased. Finally, for small submergence depths the effect of UWV motion on the free surface became more appreciable if the Reynolds number was increased. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Viscous resistance Wave-making resistance Underwater vehicle Free surface Numerical simulation

1. Introduction Autonomous Underwater Vehicles (AUVs) or simply Underwater Vehicles (UWVs) have increasingly found many ocean applications such as ocean surveillance and measurements, exploration and exploitation of seaﬂoor minerals, environmental monitoring and protection and deep-sea exploration of hydrocarbons (Jagadeesh and Murali, 2010; Saout and Ananthakrishnan, 2011). AUVs can be subdivided into three broad types based on operating regime: bottom layer, interior and surface layer (Curtin et al., 2005). Bottom layer AUVs would operate as a transport, tanker and relative navigation reference. Interior class AUVs would serve as a mobilesensor and are capable of navigating and communicating throughout the water column. Surface layer AUVs would act as communication links between the acoustic transmissions of the interior and bottom layer AUVs and the above surface radio transmissions. The interaction between the free-surface and an AUV could be signiﬁcant in littoral applications and when the AUV traverses near the free surface. In open ocean applications, an AUV would experience the free-surface effects periodically, particularly when it has to surface to get GPS ﬁxes for navigation (Saout

n

Corresponding author. Tel.: þ 98 4413980264; fax: þ98 4413554184. E-mail address: [email protected] (A. Dadvand).

http://dx.doi.org/10.1016/j.oceaneng.2014.12.028 0029-8018/& 2014 Elsevier Ltd. All rights reserved.

and Ananthakrishnan, 2011). This emphasizes the need for better understanding of the hydrodynamic forces acting on the underwater bodies under various conditions. Such understanding would help designing more efﬁcient powering systems for the AUV. There have been a number of experimental investigations on axisymmetric underwater bodies (Gertler, 1950; Granville, 1953; Nakayama and Patel, 1974; Patel and Lee, 1977; Huang et al., 1978; Roddy, 1990; Dash et al., 1996; Hackett, 2000; Jagadeesh et al., 2009; Jagadeesh and Murali, 2010). Most of these studies have been conducted in a wind tunnel. Zedan and Dalton (1979) made a critical comparison between the drag characteristics based on volume, surface area and frontal area for different axisymmetric bodies. Sayer (1996) conducted experimental tests to measure the drag and added mass coefﬁcients on a remotely operated vehicle and a solid box for various submergence depths in a towing tank. Although experimental tests are very useful for determining resistance and power requirements of UWVs, computational ﬂuid dynamics (CFD) can be used efﬁciently for the same purpose. The CFD procedure is shown to be capable of reproducing the experimental investigations quite well. Several researchers (Patel and Chen, 1986; Choi and Ching, 1991; Sung et al., 1993, 1995; Sarkar et al., 1997a, b; Ananthakrishnan and Zhang, 1998; Mulvany et al., 2004; Jagadeesh and Murali, 2006) investigated various issues related to the application of CFD to underwater hydrodynamics. The inﬂuences of turbulence models, grid generation, boundary resolution techniques and

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boundary conditions on CFD solutions, etc. were investigated over axisymmetric bodies. The studies suggest that the force coefﬁcients vary signiﬁcantly as a function of depth of submergence. The total resistance (drag) against the motion of a body in water comprises two components: viscous and pressure resistances, which are known respectively as viscous and wave-making resistances for vehicles traveling on or near the free surface. Although there are several investigations on the hydrodynamics of two- and threedimensional bodies submerged in water and on their interaction with the free surface, most of these studies considered the hydrofoil geometry and merely the wave-making resistance and were generally based on the potential theory (Sahin et al., 1997; Ahmed and Soares, 2009; Zhang et al., 2009; Chen and Zhu, 2010; Saout and Ananthakrishnan, 2011). It may be noted that potential theory does not consider the effects of viscosity or separation of the ﬂow around the body although empirical knowledge can be included to account for these effects. In the present work, the hydrodynamic characteristics of a standard UWV (Afterbody-1) motion in water and its interaction with free surface are studied numerically. The total drag coefﬁcient acting over the UWV for its operating speeds ranging from 0:4 m=s (Re ¼ 1:05 105 ) to 1:4 m=s (Re ¼ 3:67 105 ) at different depths of submergence H ranging from 0:75 to 4:0, is obtained. In addition, the wake behind the UWV is characterized to better understand the hydrodynamic behavior of the UWV motion in water at different submergence depths and vehicle speeds. It is worth mentioning that, in CFD simulation of ﬂow over UWVs, proper selection of turbulence model is a key issue that can directly affect the simulation accuracy and efﬁciency. The simulation of underwater hydrodynamics continues to be based on the solution of the Reynolds-averaged Navier Stokes (RANS) equations. The RANS based simulations are much more dependent on the selection of suitable turbulence model. In the current investigation, numerical studies are conducted using standard k ε turbulence model in the commercial ﬂow solver CFX (CFX, 2012). The computed drag force coefﬁcients are compared with the measured data of Jagadeesh et al. (2009), and Jagadeesh and Murali (2010).

2. Governing equations and numerical implementation 2.1. Problem description The UWV (Afterbody-1) conﬁguration and dimensions, which are chosen based on the work by Jagadeesh and Murali (2010), are shown in Fig. 1. The UWV is submerged at a distance h below the initially undisturbed free surface of water. The maximum diameter of the UWV is D. Due to the presence of free surface, the acceleration due to gravity g exerts an inﬂuence on the UWV motion and must be considered. The important physical parameters characterizing the ﬂow state are the Reynolds number Re ¼ ρU∇1=3 =μ, the Froude number Fr, and the initial UWV-free surface distance, H ¼ h=D,

where ρ is density of water (1000 kg/m3), U is the inlet velocity, and ∇ is volume of the body (0.018 m3), and m is the viscosity of water (0.001 kg/m s). 2.2. Governing equations The steady, incompressible, viscous and turbulent ﬂow ﬁeld around the UWV can be simulated by solving the RANS equations where the Reynolds stress ui uj is replaced with the standard k ε turbulence model. The governing equations of the ﬂow ﬁeld and mathematical expression of the standard k ε turbulence model are described below (Liu and Guo, 2013): (i) The RANS equations ∂U j ¼0 ∂xj

Continuity : Momentum :

∂U i ∂U ∂ui uj ∂p 1 þ Uj i þ ¼ þ ∇2 U i ∂xi Re ∂t ∂xj ∂xj

(ii) The k ε equations ∂k 1 ∂νt ∂k 1 þ Uj ¼ ∇2 k G þ ε ∂t σ k ∂xj ∂xj Rk ∂ε 1 ∂νt ∂ε 1 ε þ Uj ¼ ∇2 ε þ ðC ε1 G C ε2 εÞ ∂t σ ε ∂xj ∂xj Rε k

ð1Þ ð2Þ

ð3Þ ð4Þ

where U i denotes the mean velocity components, p is the pressure, and G is the turbulence production term, which can be expressed as: G ¼ νt

2 ∂U i ∂U j þ ∂xj ∂xi

ð5Þ

where νt ¼ C μ k =ε denotes eddy viscosity. The relationship between Reynolds stress ui uj , turbulence kinetic energy k and turbulence dissipation ε and eddy viscosity is written as: ∂U i ∂U j 2 ui uj ¼ νt þ ð6Þ δij k 3 ∂xj ∂xi 2

The values of constant coefﬁcients C μ , C ε1 , C ε2 , σ k and σ ε are set equal to 0.09, 1.44, 1.92, 1.0 and 1.3, respectively. The effective viscosities Rk and Rε are taken as 1=Rk ¼ ð1=ReÞ þ ðνt =σ k Þ and 1=Rε ¼ ð1=ReÞ þ ðνt =σ ε Þ, respectively. The main computational difﬁculty in the present work is related to the deformable free-surface. There are various ways to treat this situation numerically. A potential constraint in this case is that the surface may form breaking waves at high Froude numbers, which means that computational methods that track the surface directly as a computational boundary may fail or have difﬁculties. In order to tackle the problem, the interface capturing volume-of-ﬂuid (VOF) method was used. Here, both the liquid phase (water) and the much

Fig. 1. UWV conﬁguration and dimensions.

A. Nematollahi et al. / Ocean Engineering 96 (2015) 205–214

207

Fig. 2. Computational domain and grid used in the present study.

lighter gas phase (air) above it are treated explicitly by introducing a (liquid) volume fraction α1 , and gas volume fraction α2 . The combined volume fraction of both phases must satisfy the conservation P property, 2k ¼ 1 αk ¼ 1. A conservation equation is solved to transport the volume fraction of one of the phases. The viscosity μ and density ρ at any point are obtained by volume phase averaging,

ρ ¼ aρw þ ð1 aÞρa

ð7Þ

m ¼ amw þ ð1 aÞma

The subscripts w and a stand for water and air, respectively. A single momentum equation is solved for the whole domain resulting in a shared velocity ﬁeld for both phases. VOF deﬁnes a step function α equal to unity at any point occupied by water and zero elsewhere such that for volume fraction of α, three conditions are possible: 8 1 if cell is f ull of water > <

α¼

0 > : 0o α o 1

if cell is f ull of air

if cell contains the interf ace between the f luids ð8Þ

Tracking of interface between ﬂuids is accomplished by solution of a volume fraction continuity equation: ∂α ∂α þ Uj ¼0 ∂t ∂xj

ð9Þ

A description of the VOF method used to treat two-phase ﬂows is given in Hirt and Nichols (1981).

2.3. Numerical procedure The problem under consideration is three dimensional and the ﬂow regime is turbulent. The simulations were carried out using the computational ﬂuid dynamics software package CFX. Only a brief description of points, having direct relevance to the computations will be provided here. Further details of the implementation can be found in the CFX manuals. To solve the governing ﬂuid equations in CFX, the ﬂuid domain is subdivided into a ﬁnite number of cells and the equations are changed into algebraic form via a discretization process in which the Finite Volume Method (FVM) is used. Versteeg and Malalasekera (2007) provided an excellent description of the FVM on which the package is based. A collocated grid is used, i.e., the values of all variables are calculated at the cell centers. However, this approach can cause decoupling of the velocity and pressure ﬁelds. This is overcome by using the Rhie and Chow (1983) interpolation algorithm. The SIMPLE (semiimplicit method for pressure-linked equations) algorithm is used to couple the pressure and velocity. A high-resolution numerical scheme (Barth and Jespersen, 1989) was used for discretizing the advection terms. A linear interpolation scheme was used for interpolating the pressure, while the velocity was interpolated using a trilinear numerical scheme. The root-mean-square (RMS) criterion with a residual target value of 10 5 was used for checking the convergence of the solutions.

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2.4. Computational domain and grid generation

3. Results

The computational domain and the grid used in this study are given in Fig. 2. A tetrahedral unstructured body-ﬁtted grid was used for meshing the computational domain. It may be noted that the UWV was completely submerged in water. In order to achieve both the computational domain- and grid-independent solutions, domain size and grid convergence tests were conducted by computing the drag coefﬁcient (C D ¼ F D =0:5 ρU 2 ∇2=3 Þ, where FD is drag force, for the smallest submergence depth H ¼ 0:75 and the highest Reynolds number (Re ¼ 3:67 105 ). The tests indicated that the optimal computational domain size and grid size are 39D 10D 5D and 509,123 cells, respectively (see Tables 1 and 2). These domain and grid sizes were employed in all the simulations with H r 4. It may be noted that ﬁner grids were used in close proximity to the UWV body and around the air-water interface in order to accurately resolve the ﬂow properties having steep gradients in these regions. For cases with H 4 4, where the interaction of the UWV with the free surface is negligibly small, a smaller number of grid cells of 176221 was used as the optimal one (see Table 3).

The hydrodynamic characteristics of a standard UWV motion in water and its interaction with free surface are studied numerically. The drag coefﬁcient acting over the UWV for its operating speeds ranging from 0:4 m=s (Re ¼ 1:05 105 ) to 1:4 m=s (Re ¼ 3:67 105 ) at different depths of submergence H (0:75 4:0), is obtained. The results are compared with the experimental results of Jagadeesh et al. (2009) and Jagadeesh and Murali (2010). In the present study, both the viscous and wave-making resistances are investigated. It is worth mentioning that the results are given in terms of two different deﬁnitions of Froude number: pﬃﬃﬃﬃﬃ the Froude number based on the UWV's velocity Fr v ¼ U= gL ranging from 0:11 (U ¼ 0:4 m=s) to 0:38 (U ¼ 1:4 m=s), which is consistent with the deﬁnition given by Jagadeesh and Murali (2010) and pﬃﬃﬃﬃﬃﬃthe Froude number based on the submergence depth Fr H ¼ U= gh ranging from 0:223 (H ¼ 4) to 0:516 (H ¼ 0.75). The former deﬁnition is used merely for ease of comparison of the current numerical results with experimental ones reported by Jagadeesh and Murali (2010).

3.1. Effect of Reynolds number on drag coefﬁcient C D Table 1 Computational domain Fr H ¼ 0:516 ðH ¼ 0:75Þ

independent

test

for and

Re ¼ 3:67 105 ðU ¼ 1:4 m=sÞ. Domain (x y z)

Drag Coefﬁcient (C D )

25D 6D 3D 31D 8D 4D 39D 10D 5D 46D 12D 6D

0.0675 0.0531 0.0462 0.0476

Fig. 3 depicts the variation of drag coefﬁcient C D with the Re obtained for fully submerged UWV (i.e., H 4 4:0), and also the comparison between the experimental data obtained by Jagadeesh et al. (2009) and the current numerical results. It is observed from this ﬁgure that the total drag coefﬁcient C D decreases as Re increases. It may be noted that, as Re increases both the pressure drag and skin friction drag forces increase (not shown). However, both the drag coefﬁcients decrease (see Table 4). Hence, the total drag force increases as Re increases; however the total drag coefﬁcient C D decreases. In addition, the numerical predictions 0.06 Present Study

Table 2 Grid independent test for Fr H ¼ 0:516 ðH ¼ 0:75Þ

0.05

Drag coefﬁcient (C D )

263,664 367,002 509,123 838,090

0.0478 0.0460 0.0455 0.0456

CD

and Re ¼ 3:67 105 ðU ¼ 1:4 m=sÞ. Grid size

Exp.

0.04

0.03

0.02 0.5

Table 3 Grid independent test for fully submerged UWV for Re ¼ 3:67 105 ðU ¼ 1:4 m=sÞ.

1

1.5

2

2.5

3

3.5

4

Re×105

Grid size

Drag coefﬁcient (CD)

Fig. 3. Inﬂuence of Re on drag coefﬁcient C D for a fully submerged UWV (i.e., H 4 4:0).

84,861 120,117 176,221 279,492

0.0395 0.0384 0.0378 0.0377

Table 4 Pressure drag C p and frictional (viscous) drag C f coefﬁcients for fully submerged Afterbody-1 at different Re.

2.5. Boundary conditions The boundary conditions considered at various boundaries are as follows (see Fig. 2A). For inlet, the velocity inlet condition with velocity that varies from 0:4 to 1:4 m=s with an increment of 0:2 m=s; for outlet, the zero relative pressure, for UWV wall, the no-slip condition and for the far ﬁeld, the wall with zero speciﬁed shear.

Reynolds number Re

Pressure drag coefﬁcient Cp

Friction drag coefﬁcient Cf

Numerical Drag Coefﬁcient CD

Experimental Drag Coefﬁcient CD (Jagadeesh et al., 2009)

Deviation (%)

3.67 105 3.15 105 2.62 105 2.10 105 1.57 105 1.05 105

0.0060 0.0061 0.0062 0.0063 0.0065 0.0068

0.0318 0.0326 0.0337 0.0352 0.0370 0.0397

0.0378 0.0387 0.0399 0.0415 0.0435 0.0465

0.0389 0.0407 0.0419 0.0434 0.0451 0.0489

2.91 5.16 5.01 4.57 3.67 5.16

A. Nematollahi et al. / Ocean Engineering 96 (2015) 205–214

are found to be in good agreement with the experimental values with a maximum deviation of 5:16%. 3.2. Effect of submergence depth on drag coefﬁcient C D Fig. 4 illustrates the variation of drag coefﬁcient C D with submergence depth H for various UWV's velocity-based Froude number Fr v . The velocity-based Fr v is used in this section merely for the purpose of comparison of the current results with those of Jagadeesh and Murali (2010). It may be noted that in the present work, Fr v can be regarded to be equivalent in some sense to Re because in both only the velocity varies and the other parameters are kept ﬁxed. It can be seen that at all submergence depths, the total drag coefﬁcient decreases as the Fr v (or equivalently the Re) is increased (see Table 4). It is also observed that for a given Fr v (or equivalently the Re) as the submergence depth H is increased, the drag coefﬁcient C D decrease. This may be attributed to the fact that for small submergence depths, both the viscous and wave-making resistances become signiﬁcant. While at large submergence depths, the wave-making resistance becomes negligibly small. This can be deduced from the fact that the total drag coefﬁcient does not experience signiﬁcant changes from H ¼ 3 to H ¼ 4. Then we can conclude that for H Z 3, the effect of free surface on the UWV hydrodynamics is very small and can be neglected. This can also be explained by linear wave theory. According to this theory, the non-dimensional wave length λ=L is equal to 2π Fr 2v and the wave effect will be felt up to one-half the wave length (i.e., the depth of wave effect is H w ¼ 0:5λ=L ¼ π Fr 2v ). With L ¼ 1:4m and U ¼ 1:4m=s, we obtain H w ¼ π =g ¼ 0:32 where H w is nondimensional with respect to L. Since L ¼ 10D in the present work, H w in terms of D becomes is 10 (0.32) ¼3.2. This is in good agreement with our numerical result. To better understand the effect of free surface on drag coefﬁcient of the UWV, the total drag coefﬁcient is divided into two

components: the pressure drag (form drag) C p and the viscous drag (skin friction) C f . The individual components of the drag coefﬁcient are given in Table 5 for U ¼ 1:4 m=s pertaining to Re ¼ 3:67 105 . It can be seen that both the pressure drag coefﬁcient and viscous drag coefﬁcient decrease as the submergence depth H is increased. In addition, although both of the two drag components remain almost constant for H Z3, the rate of decrease of pressure drag is greater than that of the friction drag. This may be attributed to the fact that, the friction drag is a function of Re, while the pressure drag is a function of submergence depth H, so that for depths of submergence H smaller than 3:0 the effect of free surface waves on the UWV hydrodynamics becomes appreciable. 3.3. Effect of submergence depth H on free surface waves Thus far in the present work, the effect of submergence depth H on the UWV was investigated. In this section, the effect of UWV motion on the free surface is evaluated. Fig. 5A depicts the wave proﬁle due to ﬂow past the UWV (Afterbody-1) attained at U ¼ 1:4 m=s (Re ¼ 3:67 105 ) for different submergence depths ranging from 0:75 to 4:0. It can be observed that, the wave height decreases with increase in submergence depth H from 0:75 to 3:0 and remains constant afterwards. This implies that the interaction between the free surface and the UWV is negligible for H Z 3. Therefore, it is possible and computationally economical to simulate the problem in the absence of free surface (inﬁnite ﬂuid domain). In such situations, there is no need to use two-phase ﬂow (and accordingly the VOF model) to capture the air-water interface. The non-dimensional maximum free surface wave height H F ¼ hF =D as a function of submergence depth H corresponding to the cases considered in Fig. 5A is depicted in Fig. 5B, where hF is maximum free surface wave height. It can be observed that H F decreases as H is increased from 0:75 to 3:0 and remains almost unchanged afterwards. 3.4. Effect of Froude number Fr H on drag coefﬁcient C D for different Re

0.075 0.07 0.065 0.06

C

209

0.055 0.05 0.045 0.04 0.035 0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fr Fig. 4. Effect of submergence depth H on the drag coefﬁcient C D for different Fr v .

It is fundamentally known that for ﬂows with free surface effects, the Froude number deﬁned based on the submergence depth Fr H is the crucial criterion for evaluation of these effects. For this reason, the total drag coefﬁcient acting over the UWV for different values of Fr H and different values of Re ranging from Re ¼ 1:05 105 to Re ¼ 3:67 105 is depicted in Fig. 6. It is clearly evident that for all the Re considered, the drag coefﬁcient is increased as Fr H is increased. It is obvious that for a given Re, Fr H p 1=H. Therefore, for all the Re considered, the drag coefﬁcient is increased as H is decreased (see Fig. 7). This can be explained by the fact that the pressure around the UWV is strongly inﬂuenced by waves formed by the moving body and hence there is a net increase in the pressure drag as Fr H is increased. However, for a ﬁxed Re, the viscous drag is independent of the free surface waves

Table 5 Pressure drag C p and frictional (viscous) drag C f coefﬁcients for Afterbody-1 obtained for U ¼ 1:4 m=s ðRe ¼ 3:67 105 Þ. Submergence depth H

Pressure drag coefﬁcient Friction drag coefﬁcient Numerical drag coefﬁcient Experimental drag coefﬁcient C D (Jagadeesh and Cf CD Murali, 2010) Cp

Deviation (%)

0.75 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0161 0.0136 0.0103 0.0072 0.0064 0.0061 0.0059 0.0059

3.3 4.4 6.2 2.5 0.2 1.8 2.6 2.9

0.0334 0.0332 0.0329 0.0324 0.0322 0.0321 0.0320 0.0319

0.0495 0.0468 0.0459 0.0396 0.0386 0.0382 0.0379 0.0378

0.0479 0.0448 0.0432 0.0406 0.0387 0.0389 0.0389 0.0389

210

A. Nematollahi et al. / Ocean Engineering 96 (2015) 205–214

2.5 Re = 3.67 × 105

HF

2 1.5 1 0.5 0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

H Fig. 5. (A) Free surface deformation as a function of submergence depth H attained at Re ¼ 3:67 105 and Fr v ¼ 0:38 for different values of Fr H : (a) 0.516 (H¼ 0.75), (b) 0.447 (H¼1.0), (c) 0.365 (H¼ 1.5), (d) 0.316 (H¼ 2.0), (e) 0.283 (H¼2.5), (f) 0.258 (H¼ 3.0), (g) 0.239 (H¼ 3.5), and (h) 0.223 (H¼4.0). (B) Maximum free surface wave height H F as a function of submergence depth H corresponding to the cases considered in Fig. 5A.

and is almost the same for any submergence depth H (see Table 5). It is also observed that at small Fr H pertaining to the large submergence depths, say for H Z 3:0, the drag coefﬁcient remains unchanged. This is due to the fact that for small Fr H , the free surface effects on the UWV hydrodynamics are negligible. It is noteworthy that due to superposition effects, wave resistance (pressure drag) tends to show oscillations with respect to Fr H (see Lewis, 1988). However, as we have not covered a large range of Fr H , we would not observe the oscillations in the wave resistance values.

mainly to the fact that at depths H Z 3:0 the free surface does not have noticeable effect on the ﬂow around as well as behind the UWV. However, the wake loses its symmetry as the distance between the vehicle and the free surface H is decreased and there is some diffusion of vorticity so that the upper vortex disappears and the lower one becomes weaker. This would, in turn, conﬁrms that for H o3:0 the effect of free surface becomes more pronounced.

3.5. Effect of submergence depths H on the wake structure behind the UWV

In this section, the effect of velocity-based Froude number Fr v on the free surface waves is studied. The submergence depth H is ﬁxed at 0:75. However, the UWV speed is varied from 0:4 to 1:0 m=s giving rise to Fr v of 0:11 to 0:27, respectively. Fig. 9 depicts the wave proﬁle due to ﬂow past the UWV. It is observed as Fr v is increased (UWV speed is increased), the height of free surface waves is increased. However, for small Fr v the free surface is weakly inﬂuenced by the motion of the UWV. This would suggest

Fig. 8 shows streamlines behind the UWV at a ﬁxed Re, i.e., Re ¼ 3:67 105 for different submergence depths H (Fr H values). It can be observed that the wake has reasonable symmetry about the centerline for H Z 3:0 and two counter rotating vortices are formed behind the UWV. The symmetry of the wake is due

3.6. Effect of Froude number Fr v on free surface waves

A. Nematollahi et al. / Ocean Engineering 96 (2015) 205–214

0.07

0.07

Re = 1.05×105

CD

CD

Re = 1.57×105

0.06

0.06 0.05

0.05 0.04

0.04 0.03 0.05

0.07

0.09

0.11

0.13

0.03 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23

0.15

FrH

FrH

0.06

0.06

Re = 2.10×105

0.055 0.05

0.05

CD

CD

Re = 2.62×105

0.055

0.045

0.045

0.04

0.04

0.035

0.035

0.03 0.1

0.14

0.18

0.22

0.26

0.03 0.12

0.3

0.16

0.2

0.24

FrH

0.055 Re = 3.15×105

0.32

0.36

0.4

Re = 3.67×105

0.05

CD

0.05

CD

0.28

FrH

0.06 0.055

211

0.045

0.045 0.04

0.04

0.035

0.035 0.03 0.15

0.2

0.25

0.3

0.35

0.4

0.45

FrH

0.03 0.18

0.23

0.28

0.33

0.38

0.43

0.48

0.53

FrH

Fig. 6. Effect of Froude number Fr H on drag coefﬁcient C D for different Re.

centerline with two counter rotating vortices formed behind the UWV. The symmetry of the wake is due mainly to the fact that at this small Fr v the motion of UWV does not have noticeable effect on the free surface behavior and vice versa even for the small submergence depth H considered. However, the wake loses its symmetry as Fr v is increased.

0.07 Re = 1.05e05

0.065

Re = 1.57e05

0.06

Re = 2.10e05 Re = 2.62e05

CD

0.055

Re = 3.15e05

0.05

Re = 3.67e05

0.045 0.04

4. Discussion and conclusions

0.035 0.03 0

1

2

3

4

5

H Fig. 7. Effect of submergence depth H on drag coefﬁcient C D for different Re.

that for comprehensive investigation of the free surface-UWV interaction, the effect of both Fr v and Fr H are equally important and should be well taken care of. 3.7. Effect of Froude number Fr v on the wake structure behind the UWV Fig. 10 shows streamlines formed behind the UWV at a ﬁxed submergence depth H ¼ 0:75, and for various Fr v ranging from 0:11 to 0:27. It can be seen that at very small Froude number, say for Fr v ¼ 0:11, the wake is reasonably symmetric about the

The hydrodynamic characteristics of a standard UWV (Afterbody1) moving in water and its interaction with free surface are studied numerically using CFD software ANSYSTM CFX. The total drag coefﬁcient including the viscous and wave-making resistances acting over the UWV for its operating speeds ranging from 0:4 m=s (Re ¼ 1:05 105 ) to 1:4 m=s (Re ¼ 3:67 105 ) at different depths of submergence ranging from 0:75 to 4:0, is obtained. The standard k ε turbulence model and VOF interface capturing method are employed. Also the wake formed behind the UWV is characterized to better understand the hydrodynamic behavior of the UWV motion in water at different submergence depths and vehicle speeds. The results were compared with available measured data and good agreements were observed. It was found that, for all submergence depths as the Reynolds number was increased the drag coefﬁcient was decreased. Besides, for a ﬁxed Reynolds number as the submergence depth was decreased the drag coefﬁcient was increased due to the dominance of the wave-making resistance. For small submergence

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Fig. 8. Effect of submergence depths H on the wake structure behind the UWV at Re ¼ 3:67 105 , and FrH (a) 0.516 (H¼ 0.75), (b) 0.447 (H¼ 1.0), (c) 0.365 (H¼1.5), (d) 0.316 (H¼2.0), (e) 0.283 (H¼ 2.5), (f) 0.258 (H¼ 3.0), (g) 0.239 (H¼ 3.5), and (h) 0.223 (H¼4.0).

Fig. 9. Effect of Froude number Fr v on free surface waves, for Frv (a) 0.11 (Re¼1.05 105), (b) 0.16 (Re¼ 1.57 105), (c) 0.21 (Re¼ 2.10 105), and (d) 0.27 (Re¼2.62 105).

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Fig. 10. Effect of Froude number Frv on the wake structure behind the UWV for Frv (a) 0.11 (Re¼1.05 105), (b) 0.16 (Re¼ 1.57 105), (c) 0.21 (Re¼2.10 105), and (d) 0.27 (Re¼ 2.62 105).

depths the effect of UWV motion on the free surface became more appreciable if the Reynolds number was increased. Finally, it is observed that the drag coefﬁcients does not experience signiﬁcant changes from H ¼ 3 to H ¼ 4, implying that the free surface-UWV interaction is negligible for HZ3, which is obtained for the largest Frv tested in the present work. At larger Frv , the depth of effect would become greater.

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