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3.1
Shape Parameterization
The hull shapes are constrained to the most common AUV hull shape, namely, the
torpedo shape, i.e., a three section axisymmetric body with a nose, a cylindrical mid-
section, and a tail. Figure 2(a) shows a typical torpedo shaped hull with a nose of
length
a
, midsection of length
b
, overall length
L
, and maximum diameter of
D
. The
nose and the tail are parameterized using Bézier curves defined as [23]
m
n
n
!
==
n
−
i
i
B
(
t
)
=
(
−
t
(
k
))
t
(
k
)
P
(
i
)
,
(2)
i
!
n
−
i
)
!
k
10
i
where
P
i
,
i
= 0,1,…
n
, are the control points, and
t
is an 1
m
array from 0 to 1. We
use five control points for the nose and four for the tail, as shown in Figs. 2(b) and
2(c). Control points number three and eight are free (x- and y-coordinates), while the
other points are fixed. We, therefore, have two design variables for the nose and tail
curves, a total of four design variables, aside from the hull dimensions
a
,
b
,
L
, and
D
.
×
3.2
High-Fidelity CFD Model
The flow past the hull is considered to be steady and incompressible. The Reynolds-
Averaged Navier-Stokes (RANS) equations are assumed as the governing flow equa-
tions with the two-equation
k
-
ε
turbulence model with standard wall functions (see,
e.g., [24]).
The solution domain is axisymmetric around the hull centreline axis and extends
two body lengths in front of the hull, five body lengths behind it, and two body
lengths above it (Fig. 3). At the inlet, there is a velocity boundary condition where the
velocity is set parallel to the hull axis, i.e., zero angle of attack. Pressure is prescribed
at the outlet (zero gauge pressure).
The CFD computer code FLUENT [25] is used for numerical simulations of the
fluid flow. Asymptotic convergence to a steady state solution is obtained for each
case. The iterative convergence of each solution is examined by monitoring the over-
all residual, which is the sum (over all the cells in the computational domain) of the
L
2
norm of all the governing equations solved in each cell. In addition to this, the drag
force (defined in Sec. 3.4) is monitored for convergence. A solution is considered
converged if a residual value of 10
-6
has been reached for all equations, or the number
of iterations reaches 1000.
The computational grid is structured with quadrilateral elements. The elements are
clustered around the body and grow in size with distance from the body. The grids are
generated using ICEM CFD [26]. A grid convergence study was performed to deter-
mine the necessary grid density (Fig. 4). A torpedo shaped body with
L
/
D
= 5 was
used in the study. The inlet speed was 2
m
/
s
and the Reynolds number was 2 million.
Clearly, the drag coefficient value has converged at the finest grids (number 1 and 2)
(Fig. 4(a)). There is, however, a large difference in the simulation time between the
two finest grids (Fig. 4(b)). Therefore, we selected grid number 2, with 42,763 ele-
ments, to use for the high-fidelity CFD model in the optimization process. The veloci-
ty contours, and the pressure and skin friction distributions are shown in Fig. 5 for
illustration purposes.
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