Journal of

Marine Science
and Engineering

Article
Six-DOF CFD Simulations of Underwater Vehicle Operating
Underwater Turning Maneuvers
Kunyu Han 1,2 , Xide Cheng 1,2, *, Zuyuan Liu 1,2, *, Chenran Huang 3 , Haichao Chang 1,2 , Jianxi Yao 1,2
and Kangli Tan 3

1 Key Laboratory of High Performance Ship Technology, Wuhan University of Technology,

Ministry of Education, Wuhan 430063, China; kyhan@whut.edu.cn (K.H.);
changhaichao@whut.edu.cn (H.C.); yao@whut.edu.cn (J.Y.)
2 School of Naval Architecture, Ocean and Energy Power Engineering Wuhan University of Technology,
Wuhan 430063, China
3 China Ship Development and Design Center, Wuhan 430064, China; hubcp520@163.com (C.H.);
tankangli7012020@126.com (K.T.)
* Correspondence: xdcheng@whut.edu.cn (X.C.); wtulzy@whut.edu.cn (Z.L.)

Abstract: Maneuverability, which is closely related to operational performance and safety, is one of
the important hydrodynamic properties of an underwater vehicle (UV), and its accurate prediction is
essential for preliminary design. The purpose of this study is to analyze the turning ability of a UV
while rising or submerging; the computational fluid dynamics (CFD) method was used to numerically
predict the six-DOF self-propelled maneuvers of submarine model BB2, including steady turning
maneuvers and space spiral maneuvers. In this study, the overset mesh method was used to deal
with multi-body motion, the body force method was used to describe the thrust distribution of the



propeller at the model scale, and the numerical prediction also included the dynamic deflection of the


control planes, where the command was issued by the autopilot. Then, this study used the published
Citation: Han, K.; Cheng, X.; Liu, Z.;
model test results of the tank to verify the effectiveness of the CFD prediction of steady turning
Huang, C.; Chang, H.; Yao, J.; Tan, K.
maneuvers, and the prediction of space spiral maneuvers was carried out on this basis. The numerical
Six-DOF CFD Simulations of
results show that the turning motion has a great influence on the depth and pitch attitude of the
Underwater Vehicle Operating
Underwater Turning Maneuvers. J.
submarine, and a “stern heavier” phenomenon occurs to a submarine after steering. The underwater
Mar. Sci. Eng. 2021, 9, 1451. https:// turning of a submarine can not only reduce the speed to brake but also limit the dangerous depth.
doi.org/10.3390/jmse9121451 The conclusion is of certain reference significance for submarine emergency maneuvers.

Academic Editor: Alessandro Ridolfi Keywords: 6-DOF; CFD; self-propelled; steady turning maneuver; space spiral maneuver; autopilot

Received: 25 November 2021

Accepted: 15 December 2021
Published: 18 December 2021 1. Introduction
Maneuverability, an important hydrodynamic property of underwater vehicles (UVs),
Publisher’s Note: MDPI stays neutral
is closely related to the safety and combat capability of UV functions. Conventionally, a UV
with regard to jurisdictional claims in
usually adjusts the control unit by transmitting signals from the control system and changes
published maps and institutional affil-
or maintains the established course according to the requirements of the mission. The
iations.
signals transmitted to different control units are affected by the maneuverability of UVs.
Therefore, the accurate estimation of the maneuver performance of the UV plays a crucial
role in the design of the control system and the ability of the UV to achieve the desired
trajectory during the tasks. Maneuverability can be studied by means of a numerical
Copyright: © 2021 by the authors.
technique and test procedures or a combination of the two. The former includes the
Licensee MDPI, Basel, Switzerland.
method based on the hydrodynamic coefficients and self-propulsion model test prediction.
This article is an open access article
The hydrodynamic coefficients are brought into the motion equations to simulate the
distributed under the terms and
maneuvers of UVs (Gertler and Hagen [1], 1967; Feldman [2], 1979). It has been widely
conditions of the Creative Commons
used as it quickly predicts the ability of the setting coefficients and simulation time in
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
maneuver predictions; however, the coupling effects of various factors in maneuvers are
4.0/).
ignored due to several simplifications (Huang [3], 2018). More importantly, the prediction

J. Mar. Sci. Eng. 2021, 9, 1451. https://doi.org/10.3390/jmse9121451 https://www.mdpi.com/journal/jmse

J. Mar. Sci. Eng. 2021, 9, 1451 2 of 24

was limited due to the fact this method cannot capture the changes of flow and vorticity
in the maneuvers (Bettle [4], 2013); therefore, it can only be used as a tool and method
to predict the maneuvering trajectory and attitude of the UV in early research. Hence,
this study proposes a numerical prediction method based on a self-propulsion model that
involves a body force propeller and the response of an autopilot controller to improve the
accuracy of the simulations.
The UV model test was used to verify the simulation prediction results based on
coefficients (Itard [5], Issac et al. [6], Jun et al. [7], Toxopeus et al. [8], Quick and Woody-
att [9]); however, this test method generally has limitations due to its cost and the need
for specialized equipment and facilities. With the improvement of computational fluid
dynamics (CFD) and high-performance computing (HPC) capabilities, CFD numerical
simulation based on self-propulsion model tests provides a new direction for maneuver-
ability prediction and research and is well suited as a complement to experimental studies,
although validation may require experimental results.
Chase [10] designed a one-DOF (degree of freedom) self-propelled CFD simulation
for a full appendage general submarine SUBOFF equipped with a fixed control plane and
rotating propeller (Groves [11]). The CFD results are compared with the self-propelled
model tank test results to verify that the propulsion performance of the two methods is
in good agreement at the speed of 1.75 m/s (thrust coefficient KT , torque coefficient KQ ,
propeller efficiency η). However, since the model test and numerical simulation of the
propeller require significant computational costs, the study was limited to calculating
the trajectory velocity and acceleration of the UV at one-DOF only. Meanwhile, Chase
suggested that the real propeller model in the numerical simulation of the steering motion
could be replaced with a body force propeller model to greatly reduce the computational
time and cost. Chase [12] carried out a three-DOF zig-zag maneuver simulation of SUBOFF
in the horizontal plane. As there were no test results of free-sailing self-propulsion, he
adopted two methods (direct simulation of the propeller and body force to replace the
propeller) to compare the accuracy. The results show that the body force method can replace
the propeller effect well. At the same time, the research pointed out that the potential of
this method and it can be adapted to simulate a rising maneuver.
In an earlier study, the UV maneuver simulations mainly aimed at simple planar
motions that were three-DOF motions (Broglia et al. [13]; Dubbioso et al. [14]; Feng et al. [15];
Yasemin et al. [16]). With the enhancement of computing performance and the development
of dynamic grid and other technologies, the simulation of UV maneuvers took the ship,
propeller and controlling planes into consideration at the same time. Meanwhile, more
attention was paid to the attitude of a UV during navigation. The full six degrees of the self-
propelled maneuvering motion appendage control algorithm and extensible performance
are the basis of the implementation of free navigation maneuvering simulation.
Carrica [17] carried out a series of six-DOF numerical simulations of the general
submarine model Joubert BB2 (designed by MARIN) based on self-propulsion and self-
sailing tests, including self-propulsion near the surface and at depth, turning circles, vertical
and horizontal zigzag maneuvers at depth, and rise to the surface maneuvers with stops
by crash-back. The calculation was modeled after the principle of the autopilot in the
test model. In all conditions, the autopilot controlled the propeller and control plane and
used the vertical command to control pitch and depth and the horizontal command to
control the yaw and sway of BB2. The results show that the CFD method can predict the
self-propulsion maneuver distinctions well within 5%, and the motion and speed can be
predicted under free-sailing conditions well. The study also pointed out that the attitude of
the submarine controlled by autopilot is the most difficult part to predict, and the command
correlation of the controller in the test is difficult to replicate completely.
Kim [18] explored the ability of the CFD method to predict the six-DOF free-sailing
maneuver of a fully appendaged UV based on the commercial software STAR-CCM+ ac-
cording to research by Bettle [4] and Coe [19]. The study adopted movable control planes
and a body force propeller represented by an actuator disk incorporating predetermined
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 3 of 25

J. Mar. Sci. Eng. 2021, 9, 1451 3 of 24

and a body force propeller represented by an actuator disk incorporating predetermined

propulsion properties. The aft control planes were X-shaped and consisted of four inde-
pendent planes;
propulsion the horizontal
properties. The aft and vertical
control motion
planes wereofX-shaped
the UV was andcontrolled
consisted of byfour
autopilot
indepen-
using a proportional-differential (PD) controller that has proportional
dent planes; the horizontal and vertical motion of the UV was controlled by autopilot using and differential
coupling control parameters. The
a proportional-differential (PD)validation
controllerof thehas
that experimental
proportional data
andprovided by Over-
differential coupling
peltcontrol
[20] established
parameters. the credibility
The validation of theofCFD free-running simulation
the experimental data provided results.
by Overpelt [20]
A CFD study
established theof a UV is surely
credibility of themoreCFD complicated
free-runningand difficultresults.
simulation compared with surface
ships due to the increase in the vertical degrees of freedom
A CFD study of a UV is surely more complicated and difficult compared (pitch and heave). Sincewithverti-
surface
cal control is related to the safe navigation of a UV, its prediction should
ships due to the increase in the vertical degrees of freedom (pitch and heave). Since vertical be important as
well. Zhou [21] simulated a submarine’s rising maneuvers in still water
control is related to the safe navigation of a UV, its prediction should be important as well. and waves and
analyzed the feasibility
Zhou [21] simulated aand potential of
submarine’s a submarine’s
rising maneuversemergency
in still waterbuoyancy
and waves maneuvera-
and analyzed
bility
theinfeasibility
a harsh and environment
potential through direct numerical
of a submarine’s emergency simulation.
buoyancy Wu [22] used thein a
maneuverability
multi-block hybrid gridthrough
harsh environment and removable region method
direct numerical to simulate
simulation. Wu [22] a UV-forced self-pro-
used the multi-block
pelled diving
hybrid gridmaneuver,
and removable summarized and analyzed
region method the maneuverability
to simulate of a UV diving
a UV-forced self-propelled diving
motion qualitatively. Carrica [23] used dynamic grid technology
maneuver, summarized and analyzed the maneuverability of a UV diving motion qualita- to numerically predict a
submarine’s vertical
tively. Carrica zigzag
[23] used maneuverability
dynamic grid technology and verified the feasibilitypredict
to numerically of the CFD calcu-
a submarine’s
vertical
lation zigzag maneuverability
with experimental results. and verified the feasibility of the CFD calculation with
experimental
This study aimed results.to comprehensively analyze the turning ability as well as the rising
This studyabilities
and submergence aimed to in comprehensively
the vertical direction. analyze
First,the turning
the 6-DOFability as well asturning
self-propelled the rising
and submergence
maneuver of a general abilities
submarinein theisvertical direction.
simulated, First, thecontrol
the movable 6-DOF planes
self-propelled turning
and a body
force propeller for free-sailing are adopted, and the deflections of the control planes aare
maneuver of a general submarine is simulated, the movable control planes and body
force propeller for free-sailing are adopted, and the deflections
determined, by autopilot, that the settings are based on the test; the results showed that of the control planes
the are determined,
6-DOF by autopilot,
CFD maneuvering that the method
simulation settings can
are based
predictona the test; the
vehicle’s results
speed andshowed
ma-
that thecharacteristics
neuvering 6-DOF CFD maneuvering
well. Then, this simulation method
study presents thecan predict a vehicle’s
maneuvering speedofand
performance
the maneuvering
submarine under characteristics
spiral rising well.andThen, this study presents
submergence conditions, the maneuvering
and the spaceperformance
turning
of the submarine under spiral rising and submergence
performance of the submarine is verified as well. The operational performance conditions, and the space and turning
safe
performance
navigation of the submarine
performance is verified
of the submarine areasdiscussed
well. Thebased operational
on the performance
trajectory andand atti-safe
tudenavigation performance
of the submarine under oftwo
the submarine are discussed based on the trajectory and attitude
operating conditions.
of the submarine under two operating conditions.
2. Materials and Methods
2. Materials and Methods
2.1.2.1.
Model and and
Model Coordinate
Coordinate
TheThe target underwater
target underwater vehicle in the
vehicle present
in the study
present is the
study generic
is the submarine
generic submarine Joubert
Joubert
BB2,BB2,
which is introduced as an international benchmark for submarines. This
which is introduced as an international benchmark for submarines. This submarine is submarine
is ininthe
the modern generic SSK-class;
modern generic SSK-class;ititwas
wasdesigned
designedbyby Professor
Professor Joubert
Joubert [24,25]
[24,25] from from
DSTO
DSTO in Australia, and MARIN modified the original geometry later;
in Australia, and MARIN modified the original geometry later; the geometry is available the geometry is
available from MARIN in several solid body formats, as shown by Watt
from MARIN in several solid body formats, as shown by Watt [26]. The full-scale length [26]. The full-
scaleis length
70.2 m,isand 70.2the
m,appendages
and the appendages
include a include a sail, x-configuration
sail, x-configuration stern
stern control control
planes and a
planes and a casing on top. The model scale BB2 geometry with a scale
casing on top. The model scale BB2 geometry with a scale factor (λ) of 18.348 was factor (λ) of 18.348
utilized
wasinutilized
the study in the study according
according to Froudetoscaling
Froudelaws,
scalingas laws, as theismodel
the model 3.826 is
m 3.826
long. mThere
long.is a
There is a six-bladed
six-bladed stock propeller
stock propeller (MARIN
(MARIN 7371R) 7371R)for
attached attached for propulsion.
propulsion. Table 1 shows Table 1
the main
shows the main parameters of the BB2 model, and the entire model (including
parameters of the BB2 model, and the entire model (including all appendages) is reported all append-
ages)in is reported
Figure 1. in Figure 1.

Figure 1. BB2 geometry (hull, propeller, rudder).

Figure 1. BB2 geometry (hull, propeller, rudder).
 Figure 2 shows that the coordinate system applied in this study includes the follow-
ing two right-handed coordinate systems: the space coordinates system and body-fixed
J. Mar. Sci. Eng. 2021, 9, 1451 coordinate system, with its origin at the center of gravity on the hull’s centerline. The 4 of 24
translational and rotational motions in the body-fixed coordinate system are defined by
an inertial reference frame in the space coordinates system. Additionally, the x-axis is pos-
itive pointing upstream. The y-axis is positive pointing starboard, and the z-axis is positive
Table 1. Main parameters of BB2 (model scale 1:18.348). Mass properties of the scaled BB2 submarine
pointing downward. In the space coordinates system, the coordinates X, Y and Z are used
by a factor of 18.348; the longitudinal and vertical CG was measured from the front nose tip and the
to express the position of the UV coordinate system. Additionally, the orientation of the
keel, and the moments of inertia (i.e., r x , ry and rz ) are about the CG.
body-fixed coordinate system is described by the Euler angles ψ (yaw), ϕ (pitch) and θ
(roll) as describedModel
by Pan [27].
Parameters Symbol Scale (Full) Scale (Model)
Length L0 ( m ) 70.2 3.8260
Table 1. Main parametersBeam of BB2 (model scale 1:18.348). Mass B(m)properties of 9.6the scaled BB20.5232
subma-
rine by a factor of 18.348; the longitudinal
Draft to Deck and vertical CG was
Dd ( m ) measured from
10.6 the front0.5777 tip
nose
and the keel, and theDraft
moments inertia (i.e., 𝑟 , 𝑟 and D
to SailofTop 𝑟 s)(are
m) about the16.2
CG. 0.8829
Propeller Diameter D p (m) 5 0.273
Displacement ∆ (tonnes) 4440 Scale
0.7012
Model Parameters Symbol Scale (Full)
Longitudinal Center of Gravity (from nose) XCG (m) 32.31 (Model)
1.761
Vertical Center of Gravity (from keel)
Length ZLCG
0 ( m)
(m) 4.844
70.2 0.2856
3.8260
Vertical Center of Buoyancy (from keel) ZCB (m) 5.644 0.3076
Beam B(m) 9.6 0.5232
Roll Radius of Gyration r x (m) 3.433 0.1871
Draft to Deck
Pitch Radius of Gyration Drdy ((mm)) 10.6
17.600 0.5777
0.9592
Yaw Radius
Draft to SailofTop
Gyration Dz (m))
r
s
( m 17.522
16.2 0.9550
0.8829
Propeller Diameter D p (m ) 5 0.273
Figure 2Displacement
shows that the coordinate system Δ (tonnes
applied ) in this4440
study includes the follow-
0.7012
ing two right-handed coordinate systems: theXspace coordinates system and body-fixed
Longitudinal Center of Gravity (from nose) CG ( m) 32.31 1.761
coordinate system, with its origin at the center of gravity on the hull’s centerline. The
Vertical Center ZCG (m)
translational andof Gravity (from
rotational motionskeel)
in the body-fixed 4.844
coordinate system are 0.2856
defined by an
Vertical Center offrame
inertial reference Buoyancy
in the(from coordinatesZsystem.
spacekeel) CB ( m ) 5.644 the x-axis
Additionally, 0.3076
is positive
pointingRoll
upstream.
Radius ofThe
Gyration r
y-axis is positive pointing
x ( m )starboard, and the z-axis
3.433 is positive
0.1871
pointing downward. In the space
Pitch Radius of Gyration coordinates r (
system,
y m) the coordinates
17.600 X, Y and Z are used
0.9592
to express the position of the UV coordinate system. rz (m) Additionally, the orientation of the
Yaw Radius of Gyration 17.522 0.9550
body-fixed coordinate system is described by the Euler angles ψ (yaw), φ (pitch) and θ (roll)
as described by Pan [27].

Figure 2. Body fixed coordinate with the origin (O) located at the center of gravity (CG).
Figure 2. Body fixed coordinate with the origin (O) located at the center of gravity (CG).
2.2. Experimental Details
The experiment was carried out by The Australian Defense Science and Technology
Group (DSTG) and the Dutch Defense Materiel Organization (DMO) in 2014 to work
together on background research (R&D) on the hydrodynamic behavior of submarines
(Overpelt, 2015). The free sailing maneuvering tests were conducted in the Seakeeping
and Maneuverings Basin (SMB) in June 2014. The tests included roll decay and kinds of
maneuvers in the horizontal plane or the vertical plane, but the downloadable data set does
not contain all the maneuvers conducted, and only the roll decay at 0 kn, the horizontal
zigzag and turning circle and vertical zigzag are available.
The BB2 model is equipped with X-planes, and all of these four rudders can rotate on
their own axis; hence, the deflection of four rudders results in a combined horizontal and
 maneuvers in the horizontal plane or the vertical plane, but the downloadable data set
does not contain all the maneuvers conducted, and only the roll decay at 0 kn, the hori-
zontal zigzag and turning circle and vertical zigzag are available.
The BB2 model is equipped with X–planes, and all of these four rudders can rotate
on their own axis; hence, the deflection of four rudders results in a combined horizontal
J. Mar. Sci. Eng. 2021, 9, 1451 5 of 24
and vertical motion. Figure 3 shows the arrangement and the norm direction of the rota-
tion of the X-planes. Throughout the model tests, the submarine was controlled by an
autopilot that kept the submarine on course and at depth. The autopilot commanded ef-
fective rudder
vertical motion. (𝛿 )Figure
and effective
3 showsstern plane (𝛿 ) angles,
the arrangement and thewithnormindividual
direction of plane angles of
the rotation
(based on the right-hand
the X-planes. rulethe
Throughout with
model the thumb
tests, the pointing
submarine awaywas
from the body)
controlled bycalculated
an autopilot
using
thatthe following
kept equations.
the submarine onAscourse
there are andeffectively
at depth. only Thetwo autopilots
autopilot (horizontal
commanded and
effective
vertical)
rudder some
(δr ) formulas werestern
and effective usedplane
to arrive(δs ) at four individual
angles, plane angles:
with individual plane angles (based on
the right-hand rule with the thumb1pointing away from the body) calculated using the
following equations. As there are δ s =effectively
(−δ1 + δ 2only− δ 3 two
+ δ 4 )autopilots (horizontal and vertical)
4
some formulas were used to arrive at 1 four individual plane angles:
δ r = (δ1 + δ 2 + δ 3 + δ 4 )
δs = 414 (−δ1 + δ2 − δ3 + δ4 )
δδ1 ==δ r1 (−δδ s+ δ + δ + δ ), (1)
r 4 1 2 3 4
δδ21 ==δδr r+−δ sδs
(1)
δδ32 ==δδr r−+δ sδs
δδ43 ==δδr r+−δ sδs
δ4 = δr + δs
where
(1) 𝛿 > 0, diving rudder, model goes diving and pitch down; 𝛿 < 0, rising rudder,
where
model goes s > 0, diving
(1) δfloating up. model goes diving and pitch down; δs < 0, rising rudder,
rudder,
and pitch
(2) 𝛿 > 0, starboard rudder,
model goes floating and pitch up.model turns starboard; 𝛿 < 0, port rudder, model
(2)
turns port. r δ > 0, starboard rudder, model turns starboard; δr < 0, port rudder, model
turns port.maximum steering angle for each of the control surfaces is 30 degrees, and
(3) The
the planes(3) go
Thetomaximum steering angle
their commanded angle for with each of maximum
their the control plane
surfaces is 30 degrees,
velocity and the
of 7.11 deg/s
planes
at full go to their commanded angle with their maximum plane velocity of 7.11 deg/s at
scale.
full scale.

δ3>0 δ4>0

δ2>0 δ1>0
Figure 3. Norm direction of the X-planes.
Figure 3. Norm direction of the X-planes.
The attitude control of the model was based on the principle of horizontal and vertical
autopilot. A PD (proportional derivative) controller was used to adjust the translation and
rotation of the model according to its characteristics during sailing. In practice, for large
depth deviation, the translation part was temporarily ignored to prevent the excessive
pitch of the model, and the pitch value of vertical motion commands by the PD controller
was set to zero degrees.
The parameters of the autopilot used in the control simulation can be obtained accord-
ing to the scale of the model by a factor of 18.348 (see in Table 2), and the equations for the
preset plane angles can be obtained as follows (Kim et al., 2018):

de(t) de(t)
δs (t) = Pz e(t) + Dz dt + Pθ e(t) + Dθ dt
de(t) de(t) (2)
δr (t) = Py e(t) + Dy dt + Pϕ e(t) + D ϕ dt

Additionally, input offset error can be calculated as follows:

e(t) = edesire − ecurrent

de(t) e(t)current −e(t) previous (3)
dt = tcurrent −t previous
J. Mar. Sci. Eng. 2021, 9, 1451 6 of 24

where
the subscript “desire” represents the default value of the parameter, the subscript “cur-
rent” represents the actual value of the parameter, and the subscript “previous” represents
the value of the previous time step of the parameter.
For the simulation in this study, it was necessary to control each stern rudder through
the steering rate; therefore, the steering rate was set to associate with the planes angle
as follows:
δ − δcurrent
rudderanglerate[deg/s] = desire , (4)
∆t
where
δdesire is the preset plane angles based on the PD controller, δcurrent is the current plane
angles, and ∆t is the time step.

Table 2. Autopilot PD parameters for the scaled BB2 submarine by a factor of 18.348.

Proportional Parameter (P) Derivative Parameter (D)

Description
Symbol Value Symbol Value
Translation in y direction (y) Py 18.3 [deg/m] Dy 0 [deg/(m/s)]
Translation in z direction (z) Pz −55.04 [deg/m] Dz −12.85 [deg/(m/s)]
Rotation about y axis (θ) Pθ 3 [deg/deg] Dθ 0.7 [deg/(deg/s)]
Rotation about z axis (ϕ) Pϕ 3 [deg/deg] Dϕ 2.85 [deg/(deg/s)]

In this study, the straight-line case was chosen and the vehicle speed was 1.2 m/s
(10 kn for full scale) (equivalent to a Reynolds number of 5.2 × 106 ), and the results of
the CFD and EFD of the steady turning maneuvers (20 deg to port and starboard) were
compared to verify the feasibility and accuracy of the CFD method. In every CFD case,
movable control planes in conjunction with a body force propeller using an actuator disk
were adopted, and the sailplanes kept no deflection in all CFD maneuvers.

2.3. CFD Method

In this study, the commercial CFD software STAR-CCM+, based on incompressible
RANS (the Reynolds averaged Navier–Stokes) simulations, was used to model the flow
around the UV and the following RANS equations:

∂ui ∂u 1 ∂p 1 ∂ ∂u j
+ uj i = fi − + (µ − ρui0 u0j ), (5)
∂t ∂x j ρ ∂xi ρ ∂x j ∂x j

∂ui
= 0, (6)
∂xi
where ui is the averaged velocity components in the Cartesian coordinates at the meantime,
subscript i is the direction in Cartesian coordinates, p is the time-averaged pressure, µ is
the viscous coefficient, ρui0 u0j are the Reynolds stresses, and f i is the force source term at
which a momentum source can be added to simulate flow field. To allow the closure of
the time-averaged Navier–Stokes equations, the Reynolds stresses were estimated using
various turbulence models. Here, the SST k-ω high Reynolds turbulence model was chosen
because of its accuracy and the reliability of the viscous flow around the wall and far-field.
STAR CCM+ is software based on the finite volume method, in which the discretization
of the governing equation is carried out on a series of control volumes constituting the
computational domain. In this study, the separation flow calculation model was used to
separate the velocity term from the pressure term, and the SIMPLE algorithm of prediction–
correction was used to solve the flow field. For temporal discretization, the transient term
was separated by the second order. The convection term was discretized by the second
order upwind, the diffusion term by the central difference scheme, and the gradient using
the mixed Gaussian least square method.
J. Mar. Sci. Eng. 2021, 9, 1451 7 of 24

2.3.1. Gird
For the direct free sailing maneuvers simulation of UV, the relative motion of each
appendage is an inevitable difficulty. For the complex movement of the ship maneuvering
system, the rudder planes provide steering force to the hull because of their deflection, and
the propeller provides thrust pushing hull sailing due to its rotation. In the meantime, the
planes and the propeller move together with the ship’s six degrees of freedom. Overset
gird can generate meshes of different regions independently and can deal with the relative
movement of multiple bodies very flexibly. In the design process, if it is necessary to
modify the grid details or add or subtract or replace parts, the overset grid can reduce its
difficulty and, for the free sailing of UVs, it has significant advantages; therefore, in this
study, the overset grid was adopted to deal with the rudder rotation.
Figure 4 shows the background and the overset regions of the self-propulsion model,
with the local refinement on the hull body, appendages and wake region. The wall function
was used for the near-wall treatment, and the all-wall y+ wall treatment was carried out for
the simulations. The wall spacing was designed to satisfy the condition that the distance
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW
to
8 of 25
the wall of the first point lies within y+ = 1 for the designed speed, as required by the SST
k-ω turbulence mode (Kim et al., 2015; Huang et al., 2017).

(a)

(b) (c)
Figure
Figure4.4.The
Thebackground
background andand
thethe
overset regions
overset containing
regions the control
containing planes:
the control (a) Boundary
planes: con-
(a) Boundary
ditions of the computational domain, (b) view from the aft and (c) magnified view of the stern
conditions of the computational domain, (b) view from the aft and (c) magnified view of the stern region
showing the positions
region showing of the inflow
the positions of theplane.
inflow plane.

To
Toensure
ensurethe theaccuracy
accuracyand andquality
qualityof ofinterpolation,
interpolation,the themesh
meshsize
sizeshould
shouldbe bekept
keptasas
consistent as possible around overset grids; if it is not, interpolation may
consistent as possible around overset grids; if it is not, interpolation may be impossible be impossible to
achieve. Figure
to achieve. 5 shows
Figure 5 showsthe background
the background gridgrid
encrypted
encryptedat the
at rudder plane’s
the rudder rotation
plane’s re-
rotation
gions; thethe
regions; sizesize
of the overset
of the region
overset is the
region same
is the same as that of encryption
as that of encryption region.
region.
We
Wenoticed
noticed that
that there
there should be a gap gap between
betweenthe therudder
rudderplanes
planesand
andthe
thehull
hullbody
bodyto
toensure
ensurethe thedeflection
deflection ofof
thethe planes.
planes. The
The gap
gap between
between thethe planes
planes and
and thethe adjacent
adjacent surface
surface on
on the hull was around 1 mm in the physical model. However, the
the hull was around 1 mm in the physical model. However, the overset interface requires overset interface re-
quires
at leastat three
least three
to fivetolayers
five layers
in theingap
the(CD-adapco
gap (CD-adapco [28], 2021);
[28], 2021); therefore,
therefore, the gaps
the gaps were
were encrypted
encrypted to ensure
to ensure that itthat
hadit enough
had enough
mesh. mesh.
 achieve. Figure 5 shows the background grid encrypted at the rudder plane’s rotation re-
gions; the size of the overset region is the same as that of encryption region.
We noticed that there should be a gap between the rudder planes and the hull body
to ensure the deflection of the planes. The gap between the planes and the adjacent surface
J. Mar. Sci. Eng. 2021, 9, 1451 on the hull was around 1 mm in the physical model. However, the overset interface 8 ofre-
24
quires at least three to five layers in the gap (CD-adapco [28], 2021); therefore, the gaps
were encrypted to ensure that it had enough mesh.

Figure 5. Free-running
Figure 5. Free-running model
model grid adopted on
grid adopted y=
on y = 00 symmetry
symmetry plane
plane around
around the
the hull
hull body
body (left)
(left) and
and overset
overset grid
grid area of
area of
rudder planes area (right).
rudder planes area (right).

To ensure the convergence of the computational grid, the mesh was encrypted accord-
ing to the fineness ratio in the
√ITTC recommendation rules, that is, the mesh was encrypted
in three directions by rG = 2, while other parameters remained unchanged (Stern [29];
Zhang [30]). A series of grid levels from coarse to fine are shown in Table 3; the drag force
in each case of the towing test is also exhibited.

Table 3. Grid dependence study of the discretized hull and control planes, showing the percentage
difference to the fine grid level.

Gird Level Cells (in Millions) Drag Force (N) Difference to Fine (%)
Fine 15.83 24.02 -
Medium 6.58 24.13 0.46
Coarse 2.92 24.50 2.00

The differences of drag forces calculated by medium–fine and coarse–medium is

represented by ε drag , then:
ε drag1 = Fmedium − Ff ine , (7)
ε drag2 = Fcoarse − Fmedium , (8)
The changes of ε drag are used to define the convergence ratio Rdrag :

Rdrag = ε drag1 /ε drag2 , (9)

The result of the convergence ratio Rdrag = 0.23, which is in the range of 0 < Rdrag < 1.
Therefore, the three sets of gird levels are monotonically convergent. For comprehensive
consideration, the medium grid configuration (6.58 M cells) was adopted for the self-
propulsion simulations in subsequent studies.

2.3.2. Body Force Propeller Model

Normally, viscous numerical methods, such as RANS CFD, detached eddy simulation,
and large eddy simulation, have the most potential to capture viscous effects accurately
around the propeller. However, when these methods are carried out, the propeller time
scale is smaller than the ship time scale, which results in even more costly solutions because
the ship must be analyzed with the propeller time scale as Bradford’s [31] work. In other
words, using the CFD method to simulate the real propeller directly requires a significant
amount of time and computing resources; under the circumstances, the body force approach
came up (Oda [32]) and was quickly applied to study the disturbance between the propeller
and ship body (Kawamura and Miyata [33]; Nakatake [34]; Stern [35]). In research on
underwater vehicles, this method has been widely used according to its eligibility in a
J. Mar. Sci. Eng. 2021, 9, 1451 9 of 24

study on the ship propeller issue (Phillips [36]; Broglie [37]). Recently, UV’s free-sailing and
self-propulsion study has adopted it as well (Dubbioso [38]; Sezen [39]; Li [40]). Therefore,
in this study, an actuator disk to model the force of the propeller was used to analyze the
turning motion ability of a self-propelled submarine. For an overview of the state of the art,
the interested reader can be referred to the review conducted in previous work (Huang [3];
Han [41]).
The force source model was used to simulate the influence that the propeller inflicts
on the flow, and the body force adopted in this study was uniformly distributed along
the axis direction. The propeller used in the simulation was the MARIN 7371R propeller.
To obtain the same effect as the real propeller, the characteristics of the propeller should
be added into the code. The body force depends on the thrust coefficient Kt and torque
coefficient Kq , which can be obtained from the open-water test, and these two coefficients
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW
are related to the advance coefficient (J): 10 of 25

J = V/nD, (10)
where 𝑉 is the fluid velocity at the propeller location, 𝑛 is the rotation velocity of the
where V is the fluid velocity at the propeller location, n is the rotation velocity of the
propeller, and 𝐷 is the diameter of the propeller dish. The open-water curves were ob-
propeller, and D is the diameter of the propeller dish. The open-water curves were
tained
obtainedfrom the experimental
from the experimentalresults in a study
results conducted
in a study by Kim
conducted by[18],
Kimas[18],
shown in Figure
as shown in
6.Figure
Illustratively, propeller performance properties were obtained under captive
6. Illustratively, propeller performance properties were obtained under captive self-pro-
self-
pulsion
propulsioncondition, andand
condition, thethe
advance
advancecoefficients (𝐽)(J)
coefficients were
werecomputed
computedbased
basedononthe
theaverage
average
velocities measured at a plane placed 0.136 m in front of the propeller origin;
velocities measured at a plane placed 0.136 m in front of the propeller origin; therefore, therefore,
the
the propeller
propeller coefficients
coefficients included
included thethe
wakewake field
field influenced
influenced by the
by the submarine
submarine bodybody
andandthe
the controller
controller planes,
planes, andand there
there waswas no need
no need to calculate
to calculate the the wake
wake fraction
fraction additionally.
additionally.

Theopen-water
Figure6.6.The
Figure open-watercurves
curvesof
ofthe
theMARIN
MARIN7371R
7371Rpropeller
propellerconducted
conductedby
byKim.
Kim.

3.3.Results
Resultsand
andDiscussion
Discussion
All the working conditions of the simulation were set at the same scale as those used in
All the working conditions of the simulation were set at the same scale as those used
the tank tests, including the hull body, control planes and sail, to ensure that the model and
in the tank tests, including the hull body, control planes and sail, to ensure that the model
real submarine had similar Froude numbers. Additionally, all the maneuver parameters
and real submarine had similar Froude numbers. Additionally, all the maneuver param-
should be in accordance with certain proportion scaling; the results are shown in Table 4.
eters should be in accordance with certain proportion scaling; the results are shown in
Pay attention to the fact the test was conducted in fresh water; all the results, ultimately,
Table 4. Pay attention to the fact the test was conducted in fresh water; all the results,
need to be converted to sea water density.
ultimately, need to be converted to sea water density.
To ensure the validation, the simulation results need to be translated into a real
To ensure the validation, the simulation results need to be translated into a real scale.
scale. The results and discussion are divided into two parts, including steady turning
The results and discussion are divided into two parts, including steady turning maneu-
maneuvers in deep water and space spiral maneuvers. In the end, this study analyzed the
vers in deep water and space spiral maneuvers. In the end, this study analyzed the ma-
maneuverability and the safety performance comprehensively.
neuverability and the safety performance comprehensively.

Table 4. Scaling laws (with λ = 18.348; λ ρ = ρ sea ρ ba sin ).

Parameters Coefficients
Time 𝜆.
Speed 𝜆.
Distance 𝜆
Displacement 𝜆
J. Mar. Sci. Eng. 2021, 9, 1451 10 of 24

Table 4. Scaling laws (with λ = 18.348; λρ = ρsea /ρba sin .

Parameters Coefficients
Time λ0.5
Speed λ0.5
Distance λ
Displacement λ3
Moments of inertia λ4
Position of CoG λ
Angular velocity λ−0.5
Angular acceleration λ −1
Force λ3 λ p
Moment λ4 λ p
Power λ3.5 λ p

3.1. Straight-Line Maneuver

The turning motion is the most common form of underwater vehicle navigation
and is closely related to safety ability. The self-propelled model test is one of the most
suitable ways to evaluate the availability and space required for turning maneuvers; it
can be the precondition to obtaining the credible results of the six-DOF simulations. The
maneuvering simulations usually start from the state of self-propulsion and keep a constant
propeller rotation speed after reaching a situation when the propeller thrust is equal to the
hull resistance.
In this part, the six-DOF free maneuvering motion of the submarine model in straight-
line navigation was carried out based on the numerical simulation results of the early
towing tank test. The effects of the propeller are described by the body force model to
achieve the thrust and torque on the submarine, and finally, the model reached the target
speed of 1.2 m/s, equivalent to about 10 knots in the real-scale submarine. In the process
of straight-line sailing, the course is maintained through the autopilot system, and the
depth and pitch are kept by vertical control commands; the commands in this study were
0. A noteworthy point is that the experimental data employed for validation used stern
and sailplanes for vertical control, while the CFD simulations only used stern planes for
vertical control because of the limited availability of experimental data. We speculate that
the experimental data were still deemed to be acceptable for the validation of the CFD
prediction because both the sail and stern planes are mainly used to maintain the vertical
position for a straight-line course, and one of their effects may be enough. The results show
that the vertical (pitching angle) control is very good. The results of CFD simulations and
tests are shown in Table 5.

Table 5. Result of CFD simulations and tests (all results are at full scale).

Parameters CFD EFD Error (%)

Vehicle Speed (kn) 10.2 10 2.00
Propeller Revolution (rpm) 272 266 2.26
Thrust Force (N·105 ) 1.61 1.63 1.23
Pitch Angle (deg) 0.89 0.92 −3.26
Vertical Position (/LPP ) 0.02 0 -

Figure 7 shows the vortex structure diagram of the submarine model. The wake area
of the propeller is shown as a circle, which is also the part where the vortex structure of the
body force model is different from that of the discrete propeller. Meanwhile, the horseshoe
vortices caused by the shell, tip vortex and hub vortex can be seen in the figure as well.
 Parameters CFD EFD Error (%)
Vehicle Speed (kn) 10.2 10 2.00
Propeller Revolution (rpm) 272 266 2.26
Thrust Force (N∙105) 1.61 1.63 1.23
Pitch Angle (deg) 0.89 0.92 −3.26
J. Mar. Sci. Eng. 2021, 9, 1451 11 of 24
Vertical Position (/LPP) 0.02 0 -

Figure
Figure 7. Vortical
7. Vortical structures
structures shown
shown as isosurfaces.
as isosurfaces.
3.2. Steady Turning Maneuver
3.2. Steady Turning Maneuver
Steady turning maneuvers include turning to the portside and the starboard side on
Steady
the turning
horizontal maneuvers
plane at a rudder include
angleturning to theThe
of ±20 deg. portside and the
submarine starboard
model side on
starts direct flight
the at
horizontal plane at a rudder angle of ±20 deg. The submarine model starts
a target speed of 1.2 m/s (10 kn for full scale). When it reaches this, it keeps the propeller direct flight
at aspeed
target constant
speed of 1.2
andm/s sets(10
theknturning
for full rudder
scale). When
angleittoreaches
±20 deg. this,After
it keeps
that,the
thepropeller
submarine
model starts to turn. The heave and pitch of the submarine model are controlled by the
autopilot, which noticed that the shell planes maintain 0 deg in the whole process. The
related data and icons are converted into real-scale data through scale ratio and compared
with the results of the tank test.
Table 6 shows the difference between the test and CFD turning motion parameter
results, and the results are dimensionless according to the requirements of ITTC [42].
Figure 8 shows the comparison of the left and right rudder trajectories predicted by CFD
with the test results and Carrica’s results shown in the graph as well. The comparison
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW only
12 of 25
includes the tactical diameter and longitudinal distance in the test and CFD prediction,
since the test did not perform a complete turning operation. The results show that the
trajectory obtained by the CFD numerical prediction is in good agreement with the test
speed constant and sets the turning rudder angle to ±20 deg. After that, the submarine
results. While the error between the 180◦ turning time of the left turn is 10.19%, the other
model
turning motion parameters can be within 10%.the
starts to turn. The heave and pitch of Thesubmarine
CFD simulationmodelinare controlled
this study canby the
better
autopilot, which noticed that the shell planes maintain 0 deg in the
predict the free sailing maneuver characteristics of the submarine and provides an effective whole process. The
related data andmethod
pre-evaluation icons are forconverted
evaluating into
thereal-scale data through
maneuverability of thescale ratio and compared
submarine.
with the results of the tank test.
Table 6. Percentage difference between Table CFD
6 shows the difference
and experiment (frombetween
Overpelt’s the test and CFD
reports)/study turning
results motion
for length parameter
(L)-based
results, and the results are dimensionless according to the requirements
non-dimensional maneuvering characteristics (ITTC [42], 2002) for effective rudder angles 20 deg and −20 deg steady of ITTC [42]. Fig-
turning maneuvers. ure 8 shows the comparison of the left and right rudder trajectories predicted by CFD with
the test results and Carrica’s results shown in the graph as well. The comparison only
Turning Diam
includes the tactical diameter Tactical
and Diam
longitudinal distance 90◦ Turning 180◦ Turning
Effective Rudder Angle Transfer (/L) Advace (/L) in the test and CFD prediction,
(/L) (/L) Time (s) Time
since the test did not perform a complete turning operation. The results show that the
CFD 2.89 1.45 3.13 2.53 47.8 93.4
−20 deg PT EFD
trajectory
-
obtained by the CFD numerical
1.55 3.24
prediction
2.65
is in good agreement
52.6
with the test
104.0
Difference (%) results. -While the error −6.45 between the−3.39 180° turning time −4.91 of the left − turn
9.12 is 10.19%,−the 10.19other
CFD turning2.91motion parameters
1.50 can be within
3.15 10%. The CFD 2.50 simulation in this study can
49.8 101.9better
20 deg SB EFD predict -the free sailing 1.63maneuver characteristics
3.33 of2.61
the submarine54.1 and provides106.2 an effec-
−7.97 −5.41 −4.21
Difference (%) -
tive pre-evaluation method for evaluating the maneuverability of−the
7.95
submarine. −9.25

Figure 8. X-Y trajectories for steady turning maneuvers at a forward speed of 1.2 m/s.
Figure 8. X-Y trajectories for steady turning maneuvers at a forward speed of 1.2 m/s.

Table 6. Percentage difference between CFD and experiment (from Overpelt’s reports)/study results for length (L)-based
non-dimensional maneuvering characteristics (ITTC [42], 2002) for effective rudder angles 20 deg and −20 deg steady
turning maneuvers.
 J. Mar. Sci. Eng. 2021, 9, 1451 12 of 24

Turning diameter: diameters of the circular arc traveled by the CG at a vehicle’s

heading angle of 180◦ .
Transfer: perpendicular distance traveled by the CG at a vehicle’s heading angle
of 90◦ .
Tactical diameter: perpendicular distance traveled by the CG at a vehicle’s heading
angle of 180◦ .
Advance: distance traveled by the center of gravity (CG) in a direction parallel to the
original course at a vehicle’s heading angle of 90◦ .
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW
Figure 9 shows the time history curve of the pitch angle and roll angle. The results
13 of 25
show that the submarine presents a slightly bow-down posture during the straight-line
sailing. During the turning maneuver’s progress, the submarine is affected by the hy-
drodynamic force and affected by a bow-up moment, and the steady-turning submarine
reaches
reachesandandfinally
finallymaintains
maintains a stable
a stable bow-up
bow-up angle.
angle.This may
This maybe be
due to the
due external
to the externalforce
force
generated by the control plane surfaces, resulting in a downward
generated by the control plane surfaces, resulting in a downward force around the tail force around the tail of of
thethe
submarine (Leong [43] et al., 2016). The prediction results also reflect
submarine (Leong et al., 2016 [43]). The prediction results also reflect that the results of that the results
of the
the left
left and
andright
rightrudder
rudderrotation
rotation pitchpitch angles
angles predicted
predicted by CFD
by CFD are about
are about 0.7 deg0.7 greater
deg
greater
than than theresults,
the test test results, but because
but because the CFDthe CFD
resultsresults are more
are more inclined
inclined to thetobowthe during
bow
during the straight
the straight flightflight
stage, stage, the results
the test test results are based
are based on theon net
the change
net change in pitch
in the the pitch
angle,
angle,
whichwhich
shows shows
goodgood agreement
agreement withwith the CFD
the CFD forecast
forecast results.
results. In theInprocess
the process
of theof the
turning
turning
motion,motion, the model
the model tends to tends to which
fall in, fall in,iswhich is consistent
consistent with theregular
with the general general regular
submarine’s
submarine’s
rolling rule. rolling rule. The rate
The changing changing
of therateroll of the predicted
angle roll angle by predicted
CFD at by theCFD at the of
beginning be-the
ginning
turningof motion
the turning motionthe
is basically is basically
same as the same as the gradually
test results; test results; gradually
and in the end,andthe
in the
value
end, the value
seems seems than
to be larger to bethat
larger than
of the thatSince
test. of thethetest.
CFD Since the CFDissimulation
simulation based on the is based
six-DOF
on maneuvering
the six-DOF maneuvering
motion, theremotion, there
is a certain is a certain
coupling couplingbetween
relationship relationship between
the pitch angletheand
pitch
theangle and the
roll angle; roll angle;
therefore, thetherefore,
prediction the prediction
error errorangle
of the pitch of thealso
pitch anglethe
affects alsoprediction
affects
theresult
prediction
of theresult of the roll angle.
roll angle.

Figure
Figure 9. Time
9. Time histories
histories of pitch
of pitch (top)
(top) andand
rollroll (bottom)
(bottom) forfor
thethe
20 20
degdeg controlled
controlled turn
turn maneuvers.
maneuvers.

Figure 10 shows the time history curve of the depth change. The results show that at
the beginning of the turning motion, the depth predicted by CFD is above the initial posi-
tion, and there is a certain error, which is caused by the slight difference between the cen-
ter of gravity position of the CFD model and the test model. Meanwhile, there is a gap
between the control plane and the hull body when using overset grids to achieve the de-
J. Mar. Sci. Eng. 2021, 9, 1451 13 of 24

Figure 10 shows the time history curve of the depth change. The results show that
at the beginning of the turning motion, the depth predicted by CFD is above the initial
position, and there is a certain error, which is caused by the slight difference between the
center of gravity position of the CFD model and the test model. Meanwhile, there is a
gap between the control plane and the hull body when using overset grids to achieve the
deflection; we chose to cut part of the model volume, which, to a certain extent, caused an
error in the submarine model weight and the test value, and finally may have caused the
submarine model to be generated. The rising force causes the submarine to be higher than
its initial depth during straight-line sailing. However, during steady turning, the depth
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW
changes of the CFD prediction results and the test results gradually decrease and14finally
of 25
converge to a fixed value, showing good consistency.

Figure10.
Figure Timehistories
10.Time historiesofofdepth
depthchanges
changesfor
forthe
the20
20deg
degcontrolled
controlledturn
turnmaneuvers.
maneuvers.

Thechange
The change curve
curve of of the
thesubmarine
submarinespeed speedisisshown
shown in in
Figure
Figure11.11.
TheThe
results show
results that
show
CFD can predict the speed reduction in the model in good agreement
that CFD can predict the speed reduction in the model in good agreement with the exper- with the experimental
results. results.
imental It can be seenbe
It can from
seenthe figure
from the that thethat
figure righttheturn speed
right turn of the CFD
speed of thehas
CFDfurther
has
decreased, from about 10 knots to less than 6 knots. This phenomenon
further decreased, from about 10 knots to less than 6 knots. This phenomenon indicates indicates that
the submarine received more resistance when turning to starboard.
that the submarine received more resistance when turning to starboard. Figure 915shows Figure 9 shows that
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW of 25
the submarine presents a greater bow lift when turning right, which also causes greater
that the submarine presents a greater bow lift when turning right, which also causes
resistance in the sailing direction and make the model slower.
greater resistance in the sailing direction and make the model slower.
Figure 12 shows the evolution of the yaw rate. The results show that the changing
trend of the yaw rate is slightly over-predicted by CFD but totally consistent with the
experimental results during the steady turning process. The yaw rate gradually decreases
and converges to a stable result and is larger than those of the test, both in portside and
starboard side turning, which leads to a better turning performance. It is interesting that
CFD predicts the speed drop in the turning process well. Figure 11 shows that the speed
drop is basically the same as the test results, which means that at the same speed, the
turning trajectory predicted by CFD will be more compact, that is, the turning circle will
be smaller. This is also consistent with the results shown in Figure 8.
The propeller thrust results are shown in Figure 13; when the model starts turning,
the thrust of the propeller drops significantly. The test thrust drops by up to one-third,
while the CFD-predicted drop is only about 12%, which may be because of the heavy at-
tack on the stern’s surface. The body force propeller used in the software uses an approx-
imate model to replace the real propeller, ignoring the effects of blades and gaps; thus, the
captured incoming flow cannot completely simulate the real flow field. When the speed
isFigure
reduced from 10 to
Evolution 6speed
knots, that is, after entering stable turning, the thrust is increased
Figure 11.Evolution
11. ofofspeed forthe
for theturn
turn maneuvers.
maneuvers.
by about 14% compared with the direct flight state. The CFD prediction results are in good
agreement
Figurewith the testthe
12 shows results. At the
evolution same
of the time,
yaw theThe
rate. propeller
results thrust is well
show that thepredicted
changing
when
trendturning to therate
of the yaw starboard side.
is slightly At the beginning
over-predicted of thebut
by CFD turning,
totallythe yaw rate with
consistent and thethe
experimental
transverse results
velocity duringrapidly,
increase the steady turninginprocess.
as shown Figure 12;Thethus,
yawtherateinflow
gradually decreases
in front of the
propeller plane increase as well and may be larger than the velocity of the vehicle, and the
influence of the yaw rate and transverse velocity makes the freestream velocity increase
while the vehicle speed decreases. Thus, the axial velocity of the propeller increases, and
u propeller > uvehicle as a result. Additionally, with the turning motion continuing, the influ-
J. Mar. Sci. Eng. 2021, 9, 1451 14 of 24

and converges to a stable result and is larger than those of the test, both in portside and
starboard side turning, which leads to a better turning performance. It is interesting that
CFD predicts the speed drop in the turning process well. Figure 11 shows that the speed
drop is basically the same as the test results, which means that at the same speed, the
turning trajectory predicted by CFD will be more compact, that is, the turning circle will be
smaller.
Figure 11. This is also
Evolution of consistent with
speed for the turnthe results shown in Figure 8.
maneuvers.

Figure12.
Figure Evolutionofofyaw
12.Evolution yawrate
ratefor
forthe
theturn
turnmaneuvers.
maneuvers.

The propeller thrust results are shown in Figure 13; when the model starts turning, the
thrust of the propeller drops significantly. The test thrust drops by up to one-third, while
the CFD-predicted drop is only about 12%, which may be because of the heavy attack on the
stern’s surface. The body force propeller used in the software uses an approximate model
to replace the real propeller, ignoring the effects of blades and gaps; thus, the captured
incoming flow cannot completely simulate the real flow field. When the speed is reduced
from 10 to 6 knots, that is, after entering stable turning, the thrust is increased by about 14%
compared with the direct flight state. The CFD prediction results are in good agreement
with the test results. At the same time, the propeller thrust is well predicted when turning to
the starboard side. At the beginning of the turning, the yaw rate and the transverse velocity
increase rapidly, as shown in Figure 12; thus, the inflow in front of the propeller plane
increase as well and may be larger than the velocity of the vehicle, and the influence of the
yaw rate and transverse velocity makes the freestream velocity increase while the vehicle
speed decreases. Thus, the axial velocity of the propeller increases, and u propeller > uvehicle
as a result. Additionally, with the turning motion continuing, the influence of the decrease
of vehicle speed dominates, and the thrust begins to increase, which can be proved by the
curves of yaw rate, since the yaw rates are basically constant after t = 10 s.
FigureFigure
13. Evolution of propeller
14 shows thrust forcescharacteristics
some interesting for the turn maneuvers.
of the flow field. The tip vortices
generated by the right plane of the shell are captured by the tip vortices generated because
of the separation of the shell tip, and only the tip vortices of the shell are left afterward.
The horseshoe vortices generated at the roots area of the stern planes are all clearly visible.
The secondary vortices in the hull area intersect with the horseshoe vortex generated by
the right downside plane, cause a low-pressure area above the entire right side of the
model and work together with the other three planes to make the model drift and turn.
The separation vortex is then captured by the propeller wake, fused with the unique wake
vortex of the body force propeller and deformed.
J. Mar. Sci. Eng. 2021, 9, 1451 15 of 24

Figure 12. Evolution of yaw rate for the turn maneuvers.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 16 of 25

Figure 14 shows some interesting characteristics of the flow field. The tip vortices
generated by the right plane of the shell are captured by the tip vortices generated because
of the separation of the shell tip, and only the tip vortices of the shell are left afterward.
The horseshoe vortices generated at the roots area of the stern planes are all clearly visible.
The secondary vortices in the hull area intersect with the horseshoe vortex generated by
the right downside plane, cause a low-pressure area above the entire right side of the
model and work together with the other three planes to make the model drift and turn.
The separation vortex is then captured by the propeller wake, fused with the unique wake
vortex of the body force propeller and deformed.
Figure13.
Figure Evolutionofofpropeller
13.Evolution propellerthrust
thrustforces
forcesfor
forthe
theturn
turnmaneuvers.
maneuvers.

Figure14.
Figure Vortexview
14.Vortex viewofofthe
the-20
-20deg
degcontrolled
controlledturn
turnmaneuver
maneuverat
attt==50
50s.s.

InInsummary,
summary,thethesix-DOF
six-DOF CFD
CFD prediction
prediction is
is consistent
consistent with
with the
the test
test results,
results, the
the simu-
sim-
lation of turning motion has good accuracy and the control effect of the autopilot
ulation of turning motion has good accuracy and the control effect of the autopilot on onthe
the
depth and posture of the submarine is also good. Ideally, the same numerical
depth and posture of the submarine is also good. Ideally, the same numerical method will method will
beused
be usedtotopredict
predictthe
thespace
spacespiral
spiralmaneuvering
maneuveringof ofthe
thesubmarine
submarinein insubsequent
subsequentstudies.
studies.

3.3. Space Spiral Maneuver

3.3. Space Spiral Maneuver
When the submarine is diving and floating underwater to achieve the tactical goal of
When the submarine is diving and floating underwater to achieve the tactical goal of
changing depth, it may maneuver with a turning motion to avoid attacks or just to ensure
changing depth, it may maneuver with a turning motion to avoid attacks or just to ensure
the comfort of the crew. The space spiral maneuver is the most common, and normally the
the comfort of the crew. The space spiral maneuver is the most common, and normally
submarine deflects its rudders and stern-planes to a predetermined degree—sometimes it
the submarine deflects its rudders and stern-planes to a predetermined degree—some-
only needs its rudders—and the submarine gradually moves into space and spirals. In the
times it onlythe
simulation, needs its rudders—and
vertical command is sent the submarine gradually
by the autopilot, moves
and at into spaceofand
the beginning spi-
the CFD
rals. In the simulation,
prediction, the submarinethe vertical command
self-propelled andis sent bydirectly
sailed the autopilot, and at
into deep the beginning
water at a speed
ofofthe CFD prediction, the submarine self-propelled and sailed directly
1.2 m/s (10 kn for full scale). At t = 0 s, the effective rudders rotate ±20 into deep
deg,water
andatat
athe
speed of 1.2 m/s (10 kn for full scale). At t = 0 s, the effective rudders rotate
same time, the vertical command is set to ±8 deg for the autopilot. The 3D trajectory ±20 deg, and
atprediction
the same time, the vertical
diagram is shown command
in Figureis15.
setIttois±8worth
deg for the autopilot.
noting The 3Dshow
that the results trajectory
some
prediction diagram is shown in Figure 15. It is worth noting that the
interesting phenomena, which indicates that the submarine is in an underwater space. results show someIt
interesting phenomena,
has more complex which indicates
maneuverability and that
combat the performance
submarine is during
in an underwater
movement.space. It
has more complex maneuverability and combat performance during movement.
Figure 16 shows the X–Y projections of the spiral trajectory under the four cases. The
figure shows some unexpected results: the upward and downward spiral circles show
larger differences. The trajectory parameters are shown in Table 7. The trends of the left
and turn trajectory curves are relatively close in combination with Figure 16. However, it
is worth noting that when the submarine reaches the starboard steady spiral maneuver, it
deviates further from the original line, with a first heading change of 180 deg. From this
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 17 of 25

J. Mar. Sci. Eng. 2021, 9, 1451 16 of 24

ask for more space to complete the spiral rising maneuvers, which indicates that the sub-
marine has a better flowing performance when submerged.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 17 of 25

ask for more space to complete the spiral rising maneuvers, which indicates that the sub-
marine has a better flowing performance when submerged.

Figure 15. 3D trajectories for the space spiral maneuvers.

Figure 15. 3D trajectories for the space spiral maneuvers.
Figure 16 shows the X–Y projections of the spiral trajectory under the four cases. The
figure shows some unexpected results: the upward and downward spiral circles show
larger differences. The trajectory parameters are shown in Table 7. The trends of the left
and turn trajectory curves are relatively close in combination with Figure 16. However, it is
worth noting that when the submarine reaches the starboard steady spiral maneuver, it
deviates further from the original line, with a first heading change of 180 deg. From this
point of view, the port spiral maneuver shows a better turning ability. Another noteworthy
phenomenon is that the predicted longitudinal distance is larger when rising, which causes
the trajectory of the rising maneuver to overpass the initial position. Thus, it must ask for
more space to complete the spiral rising maneuvers, which indicates that the submarine
Figure
has a 15. 3D trajectories
better for the spacewhen
flowing performance spiral submerged.
maneuvers.

Figure 16. X–Y trajectories for the space spiral maneuvers.

Table 7. Parameters for effective rudder angles 20 deg and −20 deg space spiral maneuvers.

−20 deg Turn to Port 20 deg Turn to Starboard

Parameters
Rising Submergence Rising Submergence
Turning diam/Lpp 3.42 2.64 3.41 2.69
Tactical diam/Lpp 3.39 3.00 3.36 3.03
Longitudinal distance/Lpp 3.00 2.85 2.99 2.81
Positive constant distance/Lpp 1.78 1.65 1.75 1.61

Figure
Figure16.
16.X–Y
X–Ytrajectories
trajectoriesfor
forthe
thespace
spacespiral
spiralmaneuvers.
maneuvers.

Table 7. Parameters for effective rudder angles 20 deg and −20 deg space spiral maneuvers.

−20 deg Turn to Port 20 deg Turn to Starboard

Parameters
Rising Submergence Rising Submergence
Turning diam/Lpp 3.42 2.64 3.41 2.69
Tactical diam/Lpp 3.39 3.00 3.36 3.03
Longitudinal distance/Lpp 3.00 2.85 2.99 2.81
Positive constant distance/Lpp 1.78 1.65 1.75 1.61
J. Mar. Sci. Eng. 2021, 9, 1451 17 of 24

Table 7. Parameters for effective rudder angles 20 deg and −20 deg space spiral maneuvers.

−20 deg Turn to Port 20 deg Turn to Starboard

Parameters
Rising Submergence Rising Submergence
J. Mar. Sci. Turning
Eng. 2021,diam/Lpp
9, x FOR PEER REVIEW 3.42 2.64 3.41 2.69 18 of 25
Tactical diam/Lpp 3.39 3.00 3.36 3.03
Longitudinal distance/Lpp 3.00 2.85 2.99 2.81
Positive constant distance/Lpp 1.78 1.65 1.75 1.61
Figure 17 shows the time history curve of the pitch and speed. The speed of the sub-
marine decreases and gradually converges to a stable value. According to the CFD pre-
Figure 17 shows the time history curve of the pitch and speed. The speed of the
diction, the speed drops by about 30% when rising, while it drops by 45% when sub-
submarine decreases and gradually converges to a stable value. According to the CFD
merged. The turning diameter of a submarine is related to the speed and the efficiency of
prediction, the speed drops by about 30% when rising, while it drops by 45% when
the rudder angle. The diameter of the rising maneuver will surely be larger because of the
submerged. The turning diameter of a submarine is related to the speed and the efficiency
higher speed with the efficient rudder deflection angle of 20 deg, which is consistent with
of the rudder angle. The diameter of the rising maneuver will surely be larger because of the
the X–Yspeed
higher planewith
projection of therudder
the efficient CFD prediction.
deflection Theangle yaw rates
of 20 deg,are also is
which shown in Figure
consistent with
17, which experienced an increased peak, and then decreased and
the X–Y plane projection of the CFD prediction. The yaw rates are also shown in Figure converged to a stable
17,
result. The results of the portside and starboard side maneuvers show
which experienced an increased peak, and then decreased and converged to a stable result. good consistency,
while the rising
The results results
of the are aand
portside little bit larger
starboard than
side those while
maneuvers submerged.
show In general,
good consistency, the
while
turning abilities of the space spiral maneuvers show little difference while
the rising results are a little bit larger than those while submerged. In general, the turning rising and sub-
merged;
abilities in
of other words,
the space themaneuvers
spiral effect of an showefficient rudder
little is on while
difference the same level,
rising andand thus the
submerged;
ability to follow is hardly influenced by vertical control or the deflection
in other words, the effect of an efficient rudder is on the same level, and thus the ability of efficient stern
planes. Based on the CFD prediction in Figure 16, a larger speed means
to follow is hardly influenced by vertical control or the deflection of efficient stern planes.a larger turning
diameter,
Based on while
the CFDa similar yaw in
prediction rate, that 16,
Figure is, the turning
a larger speed ability
meansof submergence,
a larger turning appears to
diameter,
be better.
while a similar yaw rate, that is, the turning ability of submergence, appears to be better.

(a) (b)
Figure
Figure17.
17.Evolution
Evolutionof
ofspeed
speed(a)
(a)and
andyaw
yawrate
rate(b)
(b)for
forthe
thespace
spacespiral
spiralmaneuvers.
maneuvers.

Figure18
Figure 18shows
showsthe theevolution
evolutionof ofthe
thepitch
pitchangle
angle and
and roll
roll angle.
angle. TheTheroll
roll angle
angle first
first
quicklyincreases
quickly increasesto toaapeak
peakand andthen
thengradually
graduallyconverges
converges to to aastable
stableresult.
result. The
Thepeak
peakvalue
value
ofrising
of rising(about
(about33deg) deg)isissmaller
smallerthan
thansubmergence
submergence(about(about44deg).
deg).From
Fromwhen whenthetheplanes
planes
finish rotating (about 1 s) until the model is under the steady spiral
finish rotating (about 1 s) until the model is under the steady spiral maneuvers (about 30 maneuvers (about
30that
s), s), that is, the
is, the moment
moment when
when thethe parameters
parameters just
just stop
stop changing,the
changing, themodel
modelisisaffected
affectedby by
resistance, and the speed is significantly reduced. Additionally, at the same time, there isa
resistance, and the speed is significantly reduced. Additionally, at the same time, there is
alateral
lateralmoment
momentthat thatacts
acts on
on the
the center
center of
of gravity and makes
gravity and makes the the model
modelrotate
rotateand
andfall
fallin.
in.
The results show a larger roll angle of submergence, which indicates
The results show a larger roll angle of submergence, which indicates worse safety. When worse safety. When
thedrag
the dragtorque,
torque,lateral
lateralmoment
momentand andcontrol
controlplane
planetorque
torquearearebalanced,
balanced,the themodel
modelachieves
achieves
steady spiral motion, and the results suggest a stable roll angle with little
steady spiral motion, and the results suggest a stable roll angle with little difference in difference inthe
the
four cases.
four cases.
J.J. Mar.
Mar. Sci. Eng.
Eng. 2021,
2021, 9,
9, x1451
FOR PEER REVIEW 1918of
of 25
24

(a) (b)
Figure 18. Evolution of roll (a) and pitch (b) for the space spiral maneuvers.
Figure 18. Evolution of roll (a) and pitch (b) for the space spiral maneuvers.
The result of the pitch angle presents an interesting phenomenon; vertical commands
The result of the pitch angle presents an interesting phenomenon; vertical commands
of 8 deg and −8 deg set in the autopilot at t = 0 s make the model maneuver the bow up
of 8 deg and −8 deg set in the autopilot at t = 0 s make the model maneuver the bow up
and bow down. The pitch angle prediction of rising fluctuates several times and finally
and bow down. The pitch angle prediction of rising fluctuates several times and finally
converges, maintaining slightly less than 8 deg. However, the prediction of submergence
converges,
seemed to maintaining slightly less
realize the command than
was 8 deg. However,
impossible; the resultthe shows
prediction
that of
thesubmergence
pitch angle
seemed
reaches the peak (about −7 deg) quickly and then, as the speed decreases, itpitch
to realize the command was impossible; the result shows that the angle
gradually
reaches the peak (about −7 deg) quickly and then, as the speed
converges, maintaining about −1 deg for spiral submerging. When the submarine is decreases, it gradually
converges, maintaining
turning underwater, theabout −1 deg
pressure for spiralcaused
difference submerging.
by theWhen speedthe submarine
drop works with is turn-
the
ing underwater, the pressure difference caused by the speed
resistance t caused by the internal roll to cause an objective sinking force behind drop works with the the
re-
sistance t caused
shell. From by theofinternal
the effect rolldeflection
force, the to cause an of objective
the modelsinking showsforce
therebehind
must be thea shell.
pitch
From
moment that causes the body to bow up to oppose the moment of stern planesmoment
the effect of force, the deflection of the model shows there must be a pitch during
that causes
turning the body to bow up to oppose the moment of stern planes during turning
underwater.
underwater.
Figure 19 shows the projection of trajectory on the X–Z plane and the evolution of
FigureThe
the depth. 19 shows
curvesthe projection of seem
of submergence trajectory
to beon the X–Z
sharper than plane and the
the rising evolution
curves, of
and the
the depth.
results Thetwo
of the curves
depth of changes
submergence seembetween
are closer to be sharper
portsidethanand thestarboard
rising curves, and the
side turning,
results
while theof the two trend
change depthof changes are closer between
the submergence maneuver portside andGenerally,
is gentler. starboard thesidechange
turning,
of
while
depththe change
while trend of
the model the submergence
heading is 360 deg ismaneuver
defined asisthe gentler. Generally,
lift distance, ∆ζ. the change of
According to
depth while the model heading is 360 deg is defined as the lift distance,
the CFD prediction, the lift distance for the portside and the starboard side turning is 0.66 Δ ζ . According
and
to the0.65
CFDwhile rising, and
prediction, −0.36
the lift and −for
distance 0.33theduring submergence,
portside respectively.
and the starboard From the
side turning is
results,
0.66 andwhen the model
0.65 while rising,submerges,
and −0.36 part of theduring
and −0.33 bow-down moment caused
submergence, by the From
respectively. stern
planes
the is balanced
results, when the by model
the bow-up hydrodynamic
submerges, part of the moment
bow-down that reduces
momentthe submarine’s
caused by the
pitch angle and slows the tendency to submerge.
stern planes is balanced by the bow-up hydrodynamic moment that reduces the subma-
rine’s pitch angle and slows the tendency to submerge.
J.J.Mar.
Mar.Sci.
Sci.Eng.
Eng.2021,
2021,9,9,1451
x FOR PEER REVIEW 20 of
19 of2425

(a) (b)
Figure 19.X–Z
Figure19. X–Ztrajectories
trajectories(a)
(a)and
andevolution
evolutionofofdepth
depth(b)
(b)for
forthe
thespace
spacespiral
spiralmaneuvers.
maneuvers.

The evolution of the controller plane deflections, the forces at the Z–axis and the
The evolution of the controller plane deflections, the forces at the Z–axis and the pitch
pitch moments of the portside turning spiral maneuver are shown in Figure 20. The (a)
moments of the portside turning spiral maneuver are shown in Figure 20. The (a) and (b)
and (b) are actually the defections of the effective rudder and effective stern plane. The
are actually the defections of the effective rudder and effective stern plane. The defections
defections follow the commands of autopilots (PD controllers), and if the controller wants
follow the commands of autopilots (PD controllers), and if the controller wants the vehicle
the vehicle to sail with pitch angle while turning, two commands are given: the horizontal
to sail with pitch angle while turning, two commands are given: the horizontal angle is
angle is maintained by effective rudders, and the vertical motion (pitch angle) relies on
maintained by effective rudders, and the vertical motion (pitch angle) relies on the auto-
the autopilot commands (transferred to effective stern planes). When the model rises, the
pilot commands
autopilot (transferred
input commands tomodel
the effective
to stern
bow upplanes). When
to 8 deg andthe model
the rises, the
stern-plane autopilot
deflection
input commands the model to bow up to 8 deg and the stern-plane
is more than 25 deg at the beginning. After the command is completed, the deflection deflection is more than
25 deg at rapidly
decreases the beginning. After approximately
and is finally the command isequal completed, the deflection
to 0. However, decreases
the autopilot rapidly
command
and is finally approximately equal to 0. However, the autopilot command
does not seem to be well satisfied when the model submerges. The stern plane deflection does not seem
to be well satisfied when the model submerges. The stern plane
is almost −20 deg during the whole spiral maneuver, which forces the model to pitchdeflection is almost −20
deg during the whole spiral maneuver, which forces the model to
to −2 deg, rather than −8 deg, according to Figure 18. The comparisons of force and pitch to −2 deg, rather
than −8 show
moment deg, according to Figure has
that the submarine 18. The comparisons of
the characteristic of “stern
force and moment
heavier” show
during thethat the
spiral
submarinethe
maneuver; hasmodel’s
the characteristic of “stern
body is subjected heavier”moment
to a bow-up during while
the spiral
risingmaneuver;
but is almost the
0model’s body is subjected
while submerging, to a bow-up
which means moment
the moment while
of the risingforce
sinking but is almost the
balances 0 while
momentsub-
merging,
of the sternwhich
planes,means thewhen
that is, momentthe of the sinking
model force
is rising, thesebalances the moment
two moments are inofthethesame
stern
direction and work together to make the model bow-up, and the autopilot only needsand
planes, that is, when the model is rising, these two moments are in the same direction a
workvertical
small together to make to
command themake
modelthebow-up, and the
stern planes autopilot only needs a small vertical
deflect.
command
Figure to21 make
showsthethestern planes
surface deflect.
pressure of the model body at t = 90 s. At this moment, the
hull shows bow-up (a) and bow-down (b) motions. In general, during the turning motion,
the flow field around the body changes, and the phenomenon “sidewash” shows up, which
creates a pressure difference between the top and bottom of the model. The high-pressure
areas are located around the shell, while the low-pressure areas are located at the tail zone
and the forepart of the control planes at the top of the body; the pressure difference between
these regions forms the sinking force. A distinct low-pressure area appears at the rear of the
bottom, producing a bow-up moment, which is balanced with the stern planes moment and
the sinking moment when turning. At the same time, an obvious pressure gradient appears
from the starboard side to the port side of the hull, forming a lateral force pointing to the
left side of the model, providing a turning moment for portside turning. A comprehensive
comparison shows that the peaks in the high-pressure zone and low-pressure zone are
higher during the rising maneuver, which also shows that the submarine receives a larger
pitching moment and results in a larger pitch angle when rising.
J.J.Mar.
Mar.Sci.
Sci.Eng.
Eng.2021,
2021,9,9,1451
x FOR PEER REVIEW 21 25
20 of 24

(a) (b)

(c) (d)
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 22 of 25
Figure 20.Evolution
Figure20. Evolutionof
ofthe
thecontroller
controllerplane
planedeflections
deflections(a,b),
(a,b),forces atZ-axis
forcesat Z-axis(c)
(c)and
andpitch
pitchmoments
moments(d)
(d)for
forthe
theleft
leftturning
turning
space spiral maneuvers.
space spiral maneuvers.

Figure 21 shows the surface pressure of the model body at t = 90 s. At this moment,
the hull shows bow-up (a) and bow-down (b) motions. In general, during the turning mo-
tion, the flow field around the body changes, and the phenomenon “sidewash” shows up,
which creates a pressure difference between the top and bottom of the model. The high-
pressure areas are located around the shell, while the low-pressure areas are located at the
tail zone and the forepart of the control planes at the top of the body; the pressure differ-
ence between these regions forms the sinking force. A distinct low-pressure area appears
at the rear of the bottom, producing a bow-up moment, which is balanced with the stern
planes moment and the sinking moment when turning. At the same time, an obvious pres-
(a) (b)
sure gradient appears from the starboard side to the port side of the hull, forming a lateral
Figure 21.
Figure 21. Top
Top and
and bottom
bottom surface
surface pressureof
force pointing
pressure of rising
to rising
the left(a)side
(a) and submergence
and submergence
of the model,(b)(b) for the
providing
for the left–turning
left–turning space
a turningspace
momentspiral
for
spiral maneuvers.
portside turn-
maneuvers.
ing. A comprehensive comparison shows that the peaks in the high-pressure zone and
Figure
Figure 22zone
low-pressure shows area higher
vortex during
near thethe
model’s
rising body during
maneuver, the steady
which spiralthat
also shows maneuvers.
the sub-
The
The vortex
vortex structure
structure on
on the
the leeward
leeward side
side is
is obvious
obvious as
as well
well as
as the separation
separation
marine receives a larger pitching moment and results in a larger pitch angle when rising. phenomenon
phenomenon
around
around the the body’s
body’s surface
surface whenwhen thethe hull
hull is
is in
in the
the side
side wash.
wash. There
There are
are several
several tip
tip vortices
vortices
formed
formed by the shell and its upper tip, and at the bottom of the shell, the horseshoe vortex
by the shell and its upper tip, and at the bottom of the shell, the horseshoe vortex
extension
extension merges
merges with
with the
the hull
hull vortex
vortex and
and isis transported
transportedto to the
the propeller
propellerarea.
area. At
At the
the same
same
time,
time, the
the chain
chain vortex
vortex formed
formed by by the
the control
control planes
planes is is also
also merged
merged with
with the
the body
body vortex
vortex
at
at the
the stern,
stern, as
as well
well asas the
the unique
unique circular
circular vortex
vortex that
that belongs
belongs toto the
the body
body force
force propeller
propeller
and the secondary
and the secondaryvortex,
vortex,whichwhichmakes
makes thetheflowflowmoremore complicated
complicated at the
at the sternstern
zone.zone.
The
The phenomenon in the figure also shows that the vortex near
phenomenon in the figure also shows that the vortex near the stern of submergence sepa-the stern of submergence
rates more thoroughly; however, the speed of flow seems to be smaller than it was during
the rising maneuver. It can be seen also in Figure 17 that the speed drop is larger during
submergence, as the separation surely interferes with the inflow of the propeller and has
a negative effect on the maneuverability of the submarine.
 around the body’s surface when the hull is in the side wash. There are several tip vortices
formed by the shell and its upper tip, and at the bottom of the shell, the horseshoe vortex
extension merges with the hull vortex and is transported to the propeller area. At the same
time, the chain vortex formed by the control planes is also merged with the body vortex
J. Mar. Sci. Eng. 2021, 9, 1451 at the stern, as well as the unique circular vortex that belongs to the body force propeller
21 of 24
and the secondary vortex, which makes the flow more complicated at the stern zone. The
phenomenon in the figure also shows that the vortex near the stern of submergence sepa-
rates more thoroughly; however, the speed of flow seems to be smaller than it was during
separates more thoroughly; however, the speed of flow seems to be smaller than it was
the rising maneuver. It can be seen also in Figure 17 that the speed drop is larger during
during the rising maneuver. It can be seen also in Figure 17 that the speed drop is larger
submergence, as the separation surely interferes with the inflow of the propeller and has
during submergence, as the separation surely interferes with the inflow of the propeller
a negative effect on the maneuverability of the submarine.
and has a negative effect on the maneuverability of the submarine.

(a) (b)
Figure 22.
Figure 22. Vortex
Vortexof
ofrising
rising(a)
(a)and
andsubmergence
submergence(b)
(b)for
forthe
theleft–turning
left–turningspace
spacespiral
spiralmaneuvers.
maneuvers.

3.4.
3.4. Results
Results Discussion
Discussion
The
The simulations of of our
ourwork
workarearedivided
divided into
into three
three parts,
parts, including
including straight-line
straight-line ma-
maneuvers,
neuvers, steadysteady turning
turning maneuvers
maneuvers in deep
in deep water
water andand space
space spiral
spiral maneuvers.
maneuvers. TheThe
set-
settings
tings werewere based
based ononthe
theexperiments,
experiments,and andaabody
bodyforce
force model
model was used to to simulate
simulate the
the
effects of the propeller. All the planes can rotate within their own axis, and their
effects of the propeller. All the planes can rotate within their own axis, and their deflec- deflections
were
tionscommanded
were commanded by horizontal and vertical
by horizontal autopilots,
and vertical which which
autopilots, were normal PD controllers
were normal PD con-
with a combined proportional and differential control parameter for translations
trollers with a combined proportional and differential control parameter for translations and
rotations. The comparison
and rotations. The comparisonabout straight-line
about maneuvers
straight-line and and
maneuvers steady turning
steady maneuvers
turning maneu-
of CFD
vers and experiments
of CFD showed
and experiments the vertical
showed (pitch
the vertical angle)
(pitch control
angle) is very
control good,
is very and and
good, the
submarine could reach the target speed of 1.2 m/s, equivalent to about 10
the submarine could reach the target speed of 1.2 m/s, equivalent to about 10 knots in the knots in the
real-scale
real-scale submarine.
submarine.
In the simulations of space spiral maneuvers, we conducted scenarios for diving
and floating as well as the turn to portside and starboard. The vertical commands in this
manuscript are 8 deg and −8 deg. The results are very interesting, and the submarine
showed the phenomenon of “stern heavier” when turning underwater. The flow fields
around the sail and the flow fields between the top and bottom all changed when the
vehicle turned. The side wash appeared as a result; thus, the speed difference appeared in
these zones. Based on Bernoulli’s equations, the difference in speed caused a difference in
pressure between the top and bottom—the bow and stern—which was also the reason the
vehicle rolled to the inside. The vertical component of the resistance acting on the bow was
larger than that acting on the stern, so there would be a considerable vertical force point at
the bottom, behind the sail, and so the vehicle appeared to bow up.

4. Conclusions
In this study, the CFD method was used to predict the underwater turning ability of
the general submarine model BB2, including steady turning maneuvers and space spiral
maneuvers. The overset mesh was carried out to deal with the relative multibody motion,
and the variations in the free sailing trajectories and hydrodynamic loads were analyzed in a
Reynolds-averaged Navier–Stokes (RANS) simulation with the assumption of a body force
propeller model to ensure it was self-propelled. The numerical prediction also included the
dynamic deflection of the control planes, where the deflection angle command is issued
by the autopilot. The numerical prediction of the characteristic parameters of the turning
maneuvers agreed well with the results of a tank test, and the CFD method used in this
study can accurately simulate self-propelled tests and is therefore is a cost-effective tool that
J. Mar. Sci. Eng. 2021, 9, 1451 22 of 24

can replace more expensive self-propelled tests. The prediction of the six-DOF maneuvers
enables designers to determine the maneuverability and safety of the submarine and help
with the research on and design of the underwater vehicle.
Space spiral maneuvers are predicted based on the horizontal turning, and the attitude
is controlled by the autopilot. The predicted results of rising maneuvers are in line with
expectations, and the pitch angle is within 3% of the preset value of the final steady
spiral maneuvers. However, the submergence maneuvers have not performed as well as
expected, and the vertical and horizontal motions of the submarine under six-DOF show
strong mutual interference effects. Through the analysis, the turning motion was found
to have a greater impact on the depth and pitch, and the effective rudders’ deflection
makes the submarine appear to be “stern heavier”. This prediction shows that, even if the
vertical command is over 20 deg, the pitch angle eventually remains below 2 deg of the
submergence maneuvers. The “stern heavier” and motion characteristics when turning
underwater might help to save the submarine in a situation of dangerous submergence,
as the turning motion can be an effective way not only to reduce the speed but also to
limit the dangerous depth, which plays a role in correlational research between safety
and maneuverability.
The work in this article made certain reference to the CFD prediction of the maneu-
verability of an underwater vehicle and evaluated the research on a dangerous situation.
Since the current CFD simulation only considered the space turning performance and the
stability of roll, further design and research on repeated steering is needed.

Author Contributions: Conceptualization, K.H., X.C., Z.L. and C.H.; methodology, K.H., X.C., Z.L.
and C.H.; software: K.H. and K.T.; validation: K.H., H.C. and J.Y.; formal analysis: K.H. and X.C.;
data curation: K.H., H.C. and J.Y. All authors have read and agreed to the published version of
the manuscript.
Funding: This research was funded by the National Natural Science Foundation of China [grant num-
bers 551720105011, 51979211], Key Research and Development Plan of Hubei Province(2021BID008),
Research on the Intelligentized Design Technology for Hull Form. Green Intelligent Inland Ship
Innovation Programme. High-tech ship research project (2019[357]).
Data Availability Statement: The data presented in this study are available in this article (Tables
and Figures).
Conflicts of Interest: The authors declare no conflict of interest.

References
1. Gertler, M.; Hagen, G.R. Standard Equations of Motion for Submarine Simulation. (NSRDC-2510). Engineering 1967. [CrossRef]
2. Feldman, J. DTNSRDC Revised Standard Submarine Equations of Motion. 1979. Available online: https://ntrl.ntis.gov/NTRL/
dashboard/searchResults/titleDetail/ADA071804.xhtml (accessed on 1 November 2021).
3. Huang, C. Direct Simulation of Maneuver of the Underwater Vehicle Based on Overset Grid. Master’s Thesis, Wuhan University
of Technology, Wuhan, China, 2018.
4. Bettle, M. Unsteady Computational Fluid Dynamics Simulations of Six Degrees-of-Freedom Submarine Manoeuvres. Ph.D. Thesis,
University of New Brunswick, Fredericton, NB, Canada, 2013.
5. Itard, X. Recovery procedure in case of flooding. In Proceedings of the Warship International Symposium Conference, Brisbane,
Australia, 22–25 August 1999.
6. Issac, M.T.; Adams, S.; He, M.; Bose, N.; Williams, C.D.; Bachmayer, R.; Crees, T. Manoeuvring experiments using the MUN
Explorer AUV. In Proceedings of the 2007 Symposium on Underwater Technology and Workshop on Scientific Use of Submarine
Cables and Related Technologies, Tokyo, Japan, 17–20 April 2007.
7. Jun, B.; Park, J.; Lee, F.; Lee, P.; Lee, C.; Kim, K.; Oh, J. Development of the AUV ‘ISiMI’ and a free running test in an ocean
engineering basin. Ocean Eng. 2009, 36, 2–14. [CrossRef]
8. Toxopeus, S.; Atsavapranee, P.; Wolf, E.; Daum, S.; Pattenden, R.; Widjaja, R.; Zhang, J.T.; Gerber, A. Collaborative CFD exercise
for a submarine in a steady turn. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering,
Rio de Janeiro, Brazil, 1–6 July 2012; pp. 761–772.
9. Quick, H.; Woodyatt, B. Experimental Testing of a Generic Submarine Model in the DSTO Low Speed Wind Tunnel. Phase
2. Defence Science and Technology Organisation Fishermans Bend (Australia) Aerospace Div. 2014. Available online: https:
//citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.915.6230&rep=rep1&type=pdf (accessed on 1 November 2021).
J. Mar. Sci. Eng. 2021, 9, 1451 23 of 24

10. Chase, N. Simulations of the DARPA Suboff Submarine Including Self-propulsion with the E1619 Propeller. Master’s Thesis,
University of Iowa, Iowa, IA, USA, 2012.
11. Groves, N.C.; Huang, T.T.; Chang, M.S. Geometric characteristics of DARPA (Defense Advanced Research Projects Agency)
SUBOFF Models (DTRC Model Numbers 5470 and 5471). David Taylor Research Center Bethesda MD Ship Hydromechanics
Dept. 1989. Available online: https://www.researchgate.net/publication/235092809_Geometric_Characteristics_of_DARPA_
Defense_Advanced_Research_Projects_Agency_SUBOFF_Models_DTRC_Model_Numbers_5470_and_5471 (accessed on
1 November 2021).
12. Chase, N.; Michael, T.; Carrica, P.M. Overset simulation of a submarine and propeller in towed, self-propelled and maneuvering
conditions. Int. Shipbuild. Prog. 2013, 60, 171–205.
13. Broglia, R.; Dubbioso, G.; Durante, D.; Di Mascio, A. Turning ability analysis of a fully appended twin screw vessel by CFD. Part
I: Single rudder configuration. Ocean Eng. 2015, 105, 275–286. [CrossRef]
14. Dubbioso, G.; Broglia, R.; Zaghi, S. CFD analysis of turning abilities of a submarine model. Ocean Eng. 2017, 129, 459–479.
[CrossRef]
15. Feng, D.; Chen, X.; Liu, H.; Zhang, Z.; Wang, X. Comparisons of Turning Abilities of Submarine with Different Rudder
Configurations. In Proceedings of the ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering,
Madrid, Spain, 17–22 June 2018.
16. Yasemin, A.O.; Munir, C.O.; Ersin, D.; Sertaç, K. Experimental and Numerical Investigation of DARPA Suboff Submarine
Propelled with INSEAN E1619 Propeller for Self-Propulsion. J. Ship Res. 2019, 63, 235–250.
17. Carrica, P.M.; Kerkvliet, M.; Quadvlieg, F.; Pontarelli, M.; Martin, J.E. CFD Simulations and Experiments of a Maneuvering
Generic Submarine and Prognosis for Simulation of Near Surface Operation. In Proceedings of the 31st Symposium on Naval
Hydrodynamics, Monterey, CA, USA, 11–16 September 2016.
18. Kim, H.; Ranmuthugala, D.; Leong, Z.Q.; Chin, C. Six-DOF simulations of an underwater vehicle undergoing straight line and
steady turning manoeuvres. Ocean Eng. 2018, 150, 102–112. [CrossRef]
19. Coe, R.G. Improved Underwater Vehicle Control and Maneuvering Analysis with Computational Fluid Dynamics Simulations.
Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, 2013.
20. Overpelt, B.B.N.; Anderson, B. Free running manoeuvring model tests on a modern generic SSK class submarine (BB2). In
Proceedings of the Pacific International Maritime Conference, Sydney, Australia, 6–8 October 2015.
21. Zhou, G.; Ou, Y.; Gao, X.; Liu, R.J. Overset Simulations of Submarine’s Emergency Surfacing Maneuvering in Calm Water and
Regular Waves. J. Ship Mech. 2018, 22, 1471–1482.
22. Wu, L.; Feng, X.; Ye, Z.; Li, Y. Physics-Based Simulation of AUV Forced Diving by Self—Propulsion. J. Shanghai Jiao Tong Univ.
2021, 55, 290–296.
23. Carrica, P.M.; Kim, Y.; Martin, J.E. Vertical zigzag maneuver of a generic submarine. Ocean Eng. 2021, 219, 108386. [CrossRef]
24. Joubert, P.N. Some Aspects of Submarine Design, Part 1. Hydrodynamics, Defence Science and Technology Organisation,
DSTO-TR-1622, October 2004. Available online: https://primoa.library.unsw.edu.au/primo-explore/fulldisplay?vid=UNSWC&
tab=default_tab&docid=UNSW_ALMA21140087510001731&lang=en_US&context=L (accessed on 1 November 2021).
25. Joubert, P.N. Some Aspects of Submarine Design, Part 2. Shape of a Submarine 2026, Defence Science and Technology Organisation,
DSTO-TR-1920, December 2006. Available online: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.508.5665&rep=
rep1&type=pdf (accessed on 1 November 2021).
26. Watt, G.D. DRDC-Atlantic Research Centre. BB3: A Generic BB2 Based Submarine Design Using Sternplanes, Rudder, and
Bowplanes. 2019. Available online: https://cradpdf.drdc-rddc.gc.ca/PDFS/unc341/p811112_A1b.pdf (accessed on 1 Novem-
ber 2021).
27. Pan, Y.; Zhang, H.; Zhou, Q. Numerical Prediction of Submarine Hydrodynamic Coefficients Using CFD Simulation. J. Hydrodyn.
2012, 24, 840–847. [CrossRef]
28. CD-Adapco. 2020 STAR CCM+ User’s Guide Version 15.04. Available online: https://www.plm.automation.siemens.com/
global/zh/products/simcenter/STAR-CCM.html (accessed on 1 November 2021).
29. Stern, F.; Wilson, R.; Shao, J. Quantitative V&V of CFD simulations and certification of CFD codes. Int. J. Numer. Methods Fluids
2006, 50, 1335–1355.
30. Zhang, N.; Shen, H.; Yao, H. Uncertainty analysis in CFD for resistance and flow field. J. Ship Mech. 2008, 12, 211.
31. Bradford, G.K.; Kevin, J.M. A semi-empirical multi-degree of freedom body force propeller model. Ocean Eng. 2019, 178, 270–282.
32. Oda, J.; Yamazaki, K. Technique to Obtain Optimum Strength Shape by the Finite Element Method: On Body Force Problem.
Trans. Jpn. Soc. Mech. Eng. 1978, 44, 1141–1150. [CrossRef]
33. Kawamura, T.; Miyata, H.; Mashimo, K. Numerical simulation of the flow about self-propelling tanker models. J. Mar. Sci. Technol.
1997, 2, 245–256. [CrossRef]
34. Nakatake, K.; Sekiguchi, T.; Ando, J. Prediction of Hydrodynamic Forces Acting on Ship Hull in Oblique and Turning Motions
by a Simple Surface Panel Method. In Practical Design of Ships and Other Floating Structures; Elsevier Science Ltd.: Amsterdam,
The Netherlands, 2001; Volume 1, pp. 645–650.
35. Stern, F.; Yang, J.; Wang, Z.; Sadat-Hosseini, H.; Mousaviraad, M.; Bhushan, S.; Xing, T. Computational ship hydrodynamics:
Nowadays and way forward. Int. Shipbuild. Prog. 2013, 60, 3–105.
J. Mar. Sci. Eng. 2021, 9, 1451 24 of 24

36. Phillips, A.B.; Turnock, S.R.; Furlong, M. Comparisons of CFD simulations and in-service data for the self-propelled performance
of an Autonomous Underwater Vehicle. In Proceedings of the 27th Symposium of Naval Hydrodynamics, Seoul, Korea,
5–10 October 2008.
37. Broglia, R.; Dubbioso, G.; Durante, D.; Di Mascio, A. Simulation of turning circle by CFD: Analysis of different propeller models
and their effect on manoeuvring prediction. Appl. Ocean Res. 2013, 39, 1–10. [CrossRef]
38. Dubbioso, G.; Durante, D.; Di Mascio, A.; Broglia, R. Turning ability analysis of a fully appended twin screw vessel by CFD. Part
II: Single vs. twin rudder configuration. Ocean Eng. 2016, 117, 259–271. [CrossRef]
39. Sezen, S.; Dogrul, A.; Delen, C.; Bal, S. Investigation of self-propulsion of DARPA Suboff by RANS method. Ocean Eng. 2018, 150,
258–271. [CrossRef]
40. Li, S.; Ye, J.; Zhang, H. Analysis of underwater fixed depth turning characteristic of X-rudder and C-rudder submarine. J. Ship
Mech. 2020, 24, 1433–1442.
41. Han, K.; Cheng, X.; Huang, C.; Kangli, T. Impacts of Rudder Profiles on UV Maneuverability Based on Numerical Simulation. In
Proceedings of the 30th International Ocean and Polar Engineering Conference, Virtual, 11 October 2020; OnePetro: Shanghai,
China, 2020.
42. Uncertainty Analysis for Free Running Model Tests[S]. ITTC Recommended Procedures and Guidelines. 7.5-02-06-05:2014
Revision 00. Available online: https://www.ittc.info/media/8091/75-02-06-05.pdf (accessed on 1 November 2021).
43. Leong, Z.Q.; Piccolin, S.; Desjuzeur, M.; Ranmuthugala, D.; Renilson, M. Evaluation of the Out-of-Plane loads on a submarine
undergoing a steady turn. In Proceedings of the 20th Australasian Fluid Mechanics Conference, Perth, Australia, 5–8 Decem-
ber 2016.