The effects of the circulating water tunnel wall and support struts on
hydrodynamic coef
ficients estimation for autonomous underwater
vehicles
Hai Huang, Zexing Zhou, Hongwei Li
*, Hao Zhou, Yang Xu
National Key Laboratory of Science and Technology of Underwater Vehicle, Harbin Engineering University, No.145 Nantong Street, Harbin, 150001, China
a r t i c l e i n f o
Article history:Received 26 August 2018 Received in revised form 25 March 2019 Accepted 29 April 2019 Available online 6 August 2019
Keywords: CFD
Hydrodynamic coefficients Circulating water channel Side wall effect Struts effect
a b s t r a c t
This paper investigates the influence of the Circulating Water Channel (CWC) side wall and support struts on the hydrodynamic coefficient prediction for Autonomous Underwater Vehicles (AUVs) experiments. Computational Fluid Dynamics (CFD) method has been used to model the CWC tests. The hydrodynamic coefficients estimated by CFD are compared with the prediction of experiments to verify the accuracy of simulations. In order to study the effect of side wall on the hydrodynamic characteristics of the AUV in full scale captive model tests, this paper uses the CWC non-dimensional width parameters to quantify the correlation between the CWC width and hydrodynamic coefficients of the chosen model. The result shows that the hydrodynamic coefficients tend to be constant with the CWC width parameters increasing. Moreover, the side wall has a greater effect than the struts.
© 2019 Society of Naval Architects of Korea. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Accurate hydrodynamic coefficients in many marine engineer-ing applications are important for establishengineer-ing dynamic equations for single and multi-body submersibles. Generally, the inertial hy-drodynamic coefficient is solved by using the surface panel method, and the viscous hydrodynamic coefficient is obtained through the tank experiments or CFD (Humphreys, 1981; Prestero, 2001a;
Triantafyllou, 2004;Renilson, 2015). CFD has been widely used due to accuracy and low cost. Moreover, since CFD can simulate the experiment process with acceptable accuracy in engineering and obtain detailed flow field information, it is also used to replace some time-consuming and difficult-to-operate experiments, such as propeller-hull interaction, AUV underwater docking, the motion of a lifeboat falling into water, etc. (Shen et al., 2015;B€ohm and Graf,
2014;Leong et al., 2015a;Wu et al., 2014;Maximiano et al., 2017). One common method of obtaining viscous hydrodynamic co-efficients is to conduct captive model tests in CWC by using Planar Motion Mechanism (PMM). There have been many achievements in predicting hydrodynamic coefficients of the submersibles through
model test or CFD simulations (Yang et al., 2015;Leong et al., 2015b;
Kim et al., 2018,2002;Nouri et al., 2016). However, the free surface, side wall, and support struts of the towing tank or CWC may cause influences on the fluid field and the prediction accuracy, for example, the low-pressure zone formed by the high speed flow between the side wall and the surface of the AUV. An empirical response for the wall effect is to reduce the width of the ship to one tenth of the tank width (Kumar and Subramanian, 2007). This approach ensures that the wall influence is small, but too conser-vative, and excessive reduction of the model scale may increase the influence of the scale effect. The wall influence on the resistance of the surface ships in the towing tank test also catches the attention of some researchers.Kumar and Subramanian (2007)used the VOF method to simulate the free surface and studied the wall influence of the towing tank on the barge resistance estimation experiments. They maintained the size of the barge with increasing the tank width in the numerical simulations, and the experiment setup is contrary. The study shows that residuary resistance obtained at W¼ B ¼ 5:0 (tank width W, model width B) is free from tank wall influence for the chosen model.Liu et al. (2017)compared the re-sults of static drift towing tests with numerical simulations of a KVLCC2 model in CWC. The study indicated that there is remark-able difference of the pressurefield around the model ship between the tests in CWC and wideflow field.
* Corresponding author.
E-mail address:[email protected](H. Li).
Peer review under responsibility of Society of Naval Architects of Korea.
Contents lists available atScienceDirect
International Journal of Naval Architecture and Ocean Engineering
j o u r n a l h o m e p a g e : h t t p : / / w w w . j o u r n a l s . e l s e v i e r . c o m / i n t e r n a t i o n a l - j o u r n a l - o f - n a v a l - a r c h i t e c t u r e - a n d - o c e a n - e n g i n e e r i n g /
https://doi.org/10.1016/j.ijnaoe.2019.04.008
2092-6782/© 2019 Society of Naval Architects of Korea. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativeco mmons.org/licenses/by-nc-nd/4.0/).
In addition, the near-surface motion of the AUV generates a surface wave propagating along the free surface, which creates additional wave-making resistance acting on the AUV. Many re-searchers have noticed this and have achieved many results.
Mansoorzadeh and Javanmard (2014)studied the free surface effect on the AUV drag coefficient. The towing test is carried out at different depths. The results present that the AUV with depth greater than 2.6D (AUV height D) at an AUV speed of 1.5 m/s is not affected by the free surface, while the depth needs to be greater than 4.35D at a speed of 2.4 m/s.Polis et al. (2013)studied the effect of the free surface on the vertical hydrodynamic coefficients of SUBOFF model through CFD and tested the model in Australian Maritime College (AMC) towing tank.
Regarding the study of the influence of boundary conditions of the tank or channel in the experiments, many papers have studied the effect of the free surface on the total drag or wave making resistance. In this paper, the side wall and struts are modeled in CFD simulations to quantify their effects on the prediction of the total resistance and hydrodynamic coefficients, including X0ujuj, Y
0
v and
Y0vjvj. Moreover, the estimated values of CFD simulations have been compared with the experiment results.
This paper is organized intofive parts: Section2describes the experimental setup and the numerical methodology of investi-gating the effect of the CWC wall and struts on the AUV. In Section
3, the mesh sensitivity study has been made to optimize the mesh, and the accuracy of numerical simulations is discussed by comparing CFD and the model test. In Section4, the CWC wall in-fluence on hydrodynamic coefficients is presented. Conclusion is drawn in Section5.
2. Investigation methodology
To analyze the motion of AUVs in three-dimensional space, the earth-fixed frame and a body-fixed frame are defined asFig. 1. Obtaining accurate hydrodynamic coefficients is important to improve the vehicles' maneuvering performance. Generally, the hydrodynamic coefficients of the dynamics equations are predicted by conducting captive model tests. But the effects of tank side walls and struts on AUV may cause the decline of prediction accuracy in full scale model tests. The combination of the experiments and numerical simulations would provide an efficient method to investigate the effect of the CWC wall and struts on the AUV hori-zontal hydrodynamics characteristics in this paper. The fully-driven AUV horizontal three Degree of Freedom (DOF) motion equations are illustrated as follows (Gertler and Hagen, 1967;Feldman, 1979;
Prestero, 2001b;Fossen, 2011):
ðm X_uÞ _u myg_r ¼ XHSþ Xpropþ Xujujþ ðXvrþ mÞvr þXrrþ mxg
r2 (1)
ðmX_uÞ _umyg_r¼XHSþXpropþXujujþðXvrþmÞvrþXrrþmxg
r2 þmygr2þðYrmÞurþYvuv
(2) myg_uþmxgN_v_vþðIzzN_rÞ _r¼NHSþNpropþNvjvjvjvjþNrjrjrjrj
þNrmxg
urþmygvrþNvuv
(3)
where XHS, YHS, NHS are hydrostatic forces and moments, Xprop, Yprop, Nprop are propulsion forces and moments. m is mass of the AUV. Izz is mass moment of inertia about the x-axis.
2.1. Experiment setup
The experimental data is based on the work carried out in Harbin Engineering University Circulating Water Channel, which is equipped with a manned variable speed carriage and Vertical Planar Motion Mechanism (VPMM). As shown inFig. 2, a full scale AUV model is used to investigate the effects of the wall and support struts on the prediction of hydrodynamic coefficients, including Nomenclature
La Characteristic length of the AUV Wa Width of the AUV
Wc Width of the circulating water channel Rt Total resistance
Ct Total resistance coefficient; Rt=0:5
r
AfV2Rr Viscous pressure resistance
Cr Viscous pressure resistance coefficient Rr=0:5
r
AfV2Rf Frictional resistance
Cf Frictional resistance coefficient Rf=0:5
r
AfV2Af Hull frontal area
j
The drift angle of the AUV;j
¼ 10 10,j
max ¼10
V Forward velocity of vessel relative toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fluid V ¼ u2þ v2þ w2
p
u; v; w Velocities in the x, y and z directions respectively _u; _v; _w Accelerations in the x, y and z directions respectively r Angular velocities about the z-axis respectively
_r Angular accelerations about the z-axis respectively X; Y; Z Forces in the x-axis, y-axis, z-axis respectively X_u; Y_v; Y_r;N_v; N_r; Added Mass coefficients
X0ujuj Axial drag coefficient X
0
ujuj ¼ Xujuj=0:5
r
AfL2aXvr; Xrr Added mass cross-term
Yv; Yr;Nv; Nr; First order coefficients of force as functions of
velocities (v; r)
Yvjvj; Yrjrj; Nvjvj; Nrjrj; Cross-flow drag coefficient
r
Density offluidm
Viscosity offluidX0ujuj, Y0vand Y0vjvj. The scale of the AUV and CWC is shown inTable 1. The AUV model is mounted to the VPMM by the support struts presented inFig. 3. The center of gravity of the AUV model is 0.85 m below the free surface, and the support struts is modeled as two variable section poles located in the longitudinal section of the AUV and symmetrical about the center of gravity.
In order to obtain the hydrodynamic coefficients corresponding with varied CWC widths, the AUV scale is kept constant in the numerical simulations with various the CWC width to realize the variety of the non-dimensional width parameters Wc=Wa and
Wc=ðLa$
j
maxÞ. Since the flow speed is under 1.5 m/s in the CWCexperiment, the AUV would not interact with the free surface at a relative depth of about 2.6D (Liu et al., 2017), which is about 0.85 m for the chosen AUV model. Thus, the free surface has not significant influence on the hydrodynamic characteristics of the AUV in model tests.
The experiments consists of resistance test and static drift test. In resistance test, the full scale AUV is mounted under the VPMM, the current speed increases from 0.3 m/s to 1.5 m/s. The 3 DOF load
cells inside the model record the force data of X, Y and Z direction of the body-fixed frame. In static drift test, the sway velocity is generated by adjusting the AUV heading angle from10to 10,
and the current speed is 0.6 m/s. Thus, the parameter Wc=ðLa$
j
maxÞ can be replaced by 5.73 Wc=Wa. The recorded forcesacting on the AUV are represented by Eq.(4)and Eq.(5)(Renilson, 2015). X¼1 2
r
L 2 aX 0 ujuj$V2 (4) Y¼1 2r
L 2 a Y0*u2þ Y0vuv þ Y0vjvjvjvj (5) where Y0* ¼ Y*=12r
L 2a, Y*is sway force when the vehicle is travelling
at steady state with v¼ 0 and no appendage deflection angles.
2.2. Numerical simulation setup
The simulations are performed with Star-CCMþ12.02, a com-mercial CFD software. In order to simulate the two-phase flow, euler multiphase flow model and VOF are used, and Reynolds Averaged Naiver-Stokes equations (RANS) based on Shear Stress Transport (SST) k-
u
turbulence model have been solved. The SST k-u
is a classical turbulence model. Although it is not as good as Rey-nolds Stress Model (RSM) on anisotropic turbulence, SST k-u
is more stable in complexflow simulations (Kim et al., 2013;Gerber, 2006;Guilmineau et al., 2018;O'Brien et al., 2018). The RANS mean mass and momentum transport equations can be written as ( CD-adapco, 2015): vr
vtþ V$r
vv vg¼ 0 (6) v vtðr
vÞ þ V$r
vv vg¼ V$pI þ V$ðT þ TtÞ þ fb (7)where
r
is the fluid density, v and p are the mean velocity and pressure respectively, v is the reference frame velocity relative to the laboratory frame. I is the identity tensor, T is the viscous stress tensor, Tt is the Reynolds stress tensor, fb is the resultant of the body forces. In VOF method, the location of the twofluids around the interface is specified using a volume fraction function,a
q. Fortwo-phaseflow, the equations are shown as follows (Hirt and Nichols, 1981):
Fig. 2. Experiments in circulating water channel.
Table 1
Dimensions of the AUV and CWC.
parameter value description
AUV La 2.38 m Length
Wa 0.54 m Width
Ha 0.38 m Height
Af 0.1612 m2 Hull frontal area
Ap 1.0094 m2 Hull projected area (xz plane)
CWC Lc 7 m Length
Wc 1.5 m Width
Hc 1.7 m Height
X
a
q¼ 1 (8)r
¼a
1r
1þ ð1a
1Þr
2 (9)where
a
1is the volume fraction inside onefluid,a
2is the volumefraction inside the other. The volume fraction of water and air in simulations is shown inFig. 7.
Fig. 4shows the CWC computationalfluid domain. In order to assist with validation of the CFD results against the experimental data, the computationalfluid domain is given the same dimensions as the CWC (referred toTable 1). The struts used to support the vehicle are modeled in order to account for their effects on the forces and moments acting on the vehicle. The free surface is included in the CFD simulation as the influence of wave making increases the resistance acting on the vehicle. The surfaces of the AUV, the domain side-walls and bottom are prescribed as no-slip walls. The inlet is set as the desired vehicle speed while the outlet is set as an opening with zero relative pressure. Theflow at top of the domain is prescribed to zero velocity.
Fig. 5describes the“CWC computational domain” without free surface. This domain is created to probe the wave-making resis-tance of the AUV by comparing the“CWC computational domain” and the one without free surface. The boundaries of this domain are similar to the “CWC computational domain”. Moreover, the
underwater part of the struts is included in this domain.
Fig. 6shows the“no blockage” computational fluid domain in a fixed frame of reference. The far field boundaries are kept three body lengths away from the vehicle, with the exception of the outlet which is kept four body lengths away, to ensure that boundaries have no blockage effect onflow around the vehicle. The flow at the inlet is prescribed to match the desired vehicle speed while the outlet is set as an opening with zero relative pressure. The surfaces of the vehicle are prescribed as no-slip walls, while the remaining boundaries are set as symmetry.
The computation is performed with 28 processors (Xeon E5 2690, 2.60 GHz) at a workstation. The total wall clock time per run is approximately 8.5 h. The time step is
D
t¼ 5 103s. Twentyinner iterations arefixed per time step. Moreover, the minimum residuals are maintained for both continuity and momentum equations (in the range of 104), which match the usual conver-gence criteria.
3. CFD verification and validation 3.1. Grid dependency study
Due to numerical simulations are iteratively solved by discrete mesh, there would be discrete errors. As the mesh is refined, the discrete error of the calculation results and the grid sensitivity of the simulations will also decrease (STERN, 1999).
Fig. 4. Numerical simulations modeling the CWC test: A is the“CWC computational domain”, B is the free surface, C is the grid.
Fig. 5.“CWC computational domain” without free surface.
In this paper, a trimmed grid has been applied throughout the simulations. An anisotropic trimmed grid is used for the free sur-face region to model the sursur-face wave. Moreover, the scale ratio of the free surface mesh in the x, y and z direction is 4:4:1. A prism layer grid is placed near the AUV surface to model the boundary layerflow. The non-dimensional distance (yþ) of the first inflation layer around the vehicle for the various simulation runs is main-tained below one in order to adequately resolve the viscous layer and accurately predict the forces and moments on the vehicle. The boundary layer thickness can be computed as follows (M.White, 2005;Wu et al., 2014): Re¼
r
VLam
(10) y¼ Layþpffiffiffiffiffiffi80Re1314 (11)s
¼0:382La Re0:2 (12)where
s
is the boundary layer thickness, Re is the Reynolds number. Since the accuracy changes with the grid numbers,five different grid levels are used where the number of cells ranges from 0.5 to 3.5 million. An initial mesh model is created based on the“CWC computational domain” and the grid number is about 0.5 million. Based on the initial mesh, a more refined mesh is obtained by multiplying the number of mesh nodes in each direction bypffiffiffi2. The design of the grids includes refinement of the free surface and geometry surface. The grid topology is calculated to simulate the AUV in straight-ahead motion. The following discusses the mesh sensitivity study conducted at a speed of Re¼ 4:01 106.Fig. 8shows the dependency of the viscous pressure coefficient Cr, friction coefficient Cf, total drag coefficient Ct ¼ Cr þ Cf on the
Fig. 7. Volume fraction.
Fig. 8. Grid dependency study for Cr, Cf, Ct using a trimmed grid.
Fig. 9. Comparison of CFD and resistance test for estimating the axial resistance.
Table 2
Comparison of CFD and the resistance test. (V¼ 1:5 m/s).
Term Resistance(N)
Experiment 50.6
CFD CWC domain 51.83
CFD CWC domain without struts 48.31
CFD CWC domain without free surface 46.74
CFD no blockage domain 38.77
grid numbers. In thisfigure, Cr, Cf and Ct has converged after the mesh number exceeds 2.5 million. With the mesh cells increasing, Cr changes more than Cf. Since the meshfineness and the ability of modeling thefluid field are correlated, the finer mesh scale may be more accuracy for predicting the viscous pressure coefficient. To obtain more reliable results, a grid of 2.5 million cells is used for all numerical simulations in this paper.
3.2. Validation against experiments data
In order to compare the CFD with the model test and verify the accuracy of numerical simulations, CWC has been modeled in CFD. Comparing the simulation with struts and one without struts, the effect of the struts on the AUV is obtained.
As shown inFig. 9, the comparison data includes the experi-ments, the CWC domain, the CWC domain without struts, the CWC domain without free surface and the no“blockage domain” without struts. The above four groups have all predicted the axial drag of the AUV with an current speed from 0.3 m/s to 1.5 m/s. InTable 2, at 1.5 m/s current speed, the result of the resistance test is 50.6N, the result of the CWC domain is 51.83N, the result of the CWC domain without struts is 48.31N, the result of the CWC domain without free surface is 46.74N and the result of the “no blockage” domain is
38.77N. There is a little difference between the result of CWC domain simulations and the result of resistance test. The compar-ison of CWC domain and resistance test presents that there is a 2.83% difference in predicting X0ujuj. And the percentage difference in estimating X0ujuj is 4.53% between CWC domain without struts
and the resistance test. Thus, the numerical simulation of CWC domain with struts and free surface has higher accuracy. The struts should be included in simulations. Moreover, there is a 9.82% dif-ference between the CWC domain and the one without free surface in estimating resistance. Thus, the wave-making drag is about one tenth of the total resistance of the AUV in this paper.
In addition, there is a 3.45% difference between the result of the CWC domain and static drift test in predicting Y0vin Fig. 10. The
Fig. 11. Pressurefield. (A, CFD CWC domain; B, CFD CWC domain without struts; C, CFD CWC domain without free surface).
Table 3
Convergence of X0ujujfor increasing Wc=Wa.
No. Wc=Wa X0ujuj % deviation of X
0 ujuj 1 1.5 0.01384 129.68 2 2 0.00929 54.15 3 3.24 0.0083 37.71 4 5 0.00664 10.18 5 7 0.00610 1.21 6 10 0.00603 \
simulation of the CWC domain with struts and free surface can obtain an accurate result relative to experiments.
Fig. 11shows that there is some difference between the pressure field of the simulations with struts and without struts. In the simulation with struts, there is a small range of low pressure region behind the struts.
4. Numerical results and discussions
In order to quantify the effect of the CWC wall on the resistance test, this paper investigates the correlation between X0ujujand
non-dimensional width parameter Wc=Wa by adjusting the width of
computational domain modeling the CWC. Then, the same method is applied to model a series of static drift tests with varying CWC width to obtain the correlation between Y0v, Y0vjvj and non-dimensional width parameter 5.73 Wc=La.
Fig. 12. Pressure and velocityfield for the simulations modeling resistance tests. (V ¼ 1.5 m/s).
Fig. 13. Axial resistance vs Re with varying Wc=Wa.
Fig. 14. Viscous pressure coefficient Cr, friction coefficient Cf and total drag coefficient Ct vs Wc=Wa.
Table 4
Convergence of the crossflow drag coefficient Y0 vand Y 0 vjvjfor increasing 5.67 Wc=La. No. 5:73 Wc=La Y0v % deviation of Y 0 v Y0vjvj deviation of Y 0 vjvj 1 3.25 0.0719 62.67 0.0844 58.27 2 3.9 0.0631 42.76 0.3178 57.2 3 5.2 0.0571 29.19 0.176 12.97 4 6.5 0.0456 3.17 0.2141 5.94 5 9 0.0464 4.98 0.2138 5.81 6 11 0.0454 2.78 0.19508 3.47 7 13 0.0442 \ 0.2021 \
4.1. The influence of CWC side walls on resistance estimation of the AUV
Table 3shows that six group simulations with varying Wc=Wa
have been compared with the last group respectively, which is used as the benchmark. The range of Wc=Wa is from 1.5 to 10
(corre-sponding to numerical CWC width 0.81, 1.08, 1.75, 2.7, 3.78 and 5.4 m). In resistance tests, the current speed is varied between 0.3 and 1.5 m/s with the heading angle of the AUV keeping 0.
The axial drag coefficient X0ujuj gradually tends to a constant,
while Wc=Waincreases and the percentage difference of X
0
ujujdrops
to 1.21% for Wc=Wa ¼ 7. The influence of the CWC wall on the
pressure field in resistance test causes the resistance increasing with the reduce of CWC width.
The pressure and velocityfield at V ¼ 1.5 m/s with the compu-tational domain width varying from 0.81 to 5.4 m is shown in
Fig. 12. For the small CWC width, a narrow channel is generated between the AUV surface and the CWC sides wall. The high current speed in the narrow channel generates a negative pressure region, which is symmetrical to the longitudinal section of the AUV. As Wc=Wa increases, the value and range of the negative pressure
region continue to decrease. The influence of CWC side walls on the pressurefield vanishes for Wc=Wa¼ 7 and higher.
The axial drag corresponding to Re is plotted inFig. 13, which presents the effect of the numerical CWC wall on the AUV resis-tance. With increasing Wc=Wa, the axial drag curves trend to
overlap the curve Wc=Wa¼ 10. The resistance coefficients
corre-sponding to numerical CWC width are plotted inFig. 14. The values of Cr, Cf and Ct have attained constancy for Wc=Wa¼ 7 and higher.
Moreover, the effect of the CWC wall on the viscous pressure resistance is greater than the friction resistance.
4.2. The influence of CWC side walls on static drift test results
Table 4describes that seven group simulations with varying 5.73 Wc=Lahave been compared with the last group respectively,
which is used as the benchmark. The non-dimensional CWC width parameter 5.73 Wc=Lais in the range of 3.25e13 (corresponding to
the CWC width 1.35, 1.62, 2.16, 2.7, 3.74, 4.57 and 5.4 m). The sway force Y is shown in Fig. 16. In simulations, the current speed is 0.6 m/s in CWC, and the sway velocity is generated by adjusting the heading angle, which is varied from10to 10.
Fig. 15presents the pressure and velocity field at V ¼ 0.6 m/s with increasing CWC width. With the increase of the CWC width, the range and value of the negative pressurefield around the AUV
stern and bow are tending to reduce. The cross-flow coefficients Y0
v,
Y0vjvjcorresponding to 5.73 Wc=Laare plotted inFig. 17.
Both Y0v and Y0vjvj converge to a constant for CWC width increasing to 11 and higher, and the influence of the CWC wall on the second-order coefficient Y0vjvj is greater than thefirst-order
coefficient Y0v. The changes of the pressurefield may have greater
effect on the high order hydrodynamic coefficients. With the in-crease of the CWC width, the percentage difference of the Y0vand Y0vjvj respectively drop to 2.78% and 3.47%. From the above, more accurate results may be obtained in the model test by taking full account of the effect with the CWC side walls.
5. Conclusion
In this study, the steady-state two-phase ow numerical simu-lations have been carried to understand and quantify the effects of the CWC wall and the support struts on the prediction of hydro-dynamic coefficients such as axial drag coefficient X0ujujand
cross-ow drag coefficients Y0v, Y
0
vjvj. The numerical simulation results are
compared with those obtained from the experiments conducted in the CWC to verify the accuracy of numerical simulations. The re-sults show that the influence of struts needs to be considered in simulations. Moreover, the wave-making drag is less than one tenth of the total resistance of the AUV in this paper. The wave reflection from the boundaries in simulations has a marginal influence on the underwater vehicles.
The influence of the CWC wall on the hydrodynamic charac-teristics is presented with Wc=Lavarying from 1.5 to 10 in
resis-tance numerical simulations, while 5.73 Wc=Lavarying from 3.25
to 13 in the static drift numerical simulations. The axial drag co-efficient X0ujuj drops to a constant with the Wc=Laincreasing to 7
and higher for the current speed varying from 0.3 m/s to 1.5 m/s. Analogously, Y0vand Y0vjvjtend to be constant with the 5.73 Wc=La
increasing to 11 and higher for the current speed of 0.6 m/s. The present results show that the prediction values of the hydrody-namic coefficient may be larger than the true values for the model tests carried out in a comparatively narrow CWC.
Acknowledgments
Grateful acknowledgment is given to National Natural Science Foundation of China (No. 61633009, 51579053, 51209050, 51779052, 51779059), and the Key Basic Research Project of “Shanghai Science and Technology Innovation Plan” (No.15JC1403300). This work was also partially supported by the Field Fund of the 13th Five-Year Plan for the Equipment Pre-research Fund (No.61403120301).
Fig. 16. Lateral force Y vs the sway velocity with varying 5.73 Wc=La.
Fig. 17. Cross-flow drag coefficient Y0 vand Y
0
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