Part B | 12.4
310 Part B Autonomous Ocean Vehicles, Subsystems and Control
Here, C L and C D are the quadratic lift and drag coefficients, respectively. The drag coefficient is made up
of two terms, a profile drag term C D0 , that depends on
Reynolds number R e , and an induced drag term C Di ,
that increases with increasing lift coefficient and decreases with wing aspect ratio, N R [12.17]
C D D C D0 C C Di D K 0 R
e N A C K i
C
2
L
N R
: (12.10)
In (12.10), K 0 is the profile shape factor, N A is the ratio of the total wetted surface area to the wing area
N A D A t =A 0 , and K i is the wing plan form factor. The
aspect ratio is defined as N R D S=N c D S
2
=A 0 where S is
the wingspan and N
c D A 0 =S is the mean aerodynamic
chord of the glider’s wing. The power-law dependence
of profile drag is D 1=2 for completely laminar, unseparated boundary layer flow over the vehicle, and D
1=5 for fully turbulent un-separated boundary layers.
In (12.10), the Reynolds number is the size-dependent
scale factor R e Á UN c==, where is the kinematic viscosity of the fluid. Since the glider is considered
a streamlined body, the Reynolds number dependence
of the profile drag term in (12.10) is an approximation
of friction acting on the total wetted area, A t . Specific
energy consumption, E c , can be explicitly examined
for efficient gliders that achieve small glide angle during coordinated wings-level flight (v D v
0
D 0; ˇ D 0).
With the small glide angle assumption, the lift-to-drag
ratio can be written after [12.18] as
L
D
D
C L
C D
D
u
2
K 3 u 4 C K 4
D
1
E e
;
(12.11)
where
K 3 D
A 0
2gV b
K 0 R
e N A ;
K 4 D
2gV b
A 0
K i
N R
:
Taking @.L=D/=@u D 0, the speed at which the liftto-drag ratio is maximized becomes U D .K 4 =K 3 /
1=4
giving
 L
D
Ã
max
D
1
2
.K 3 K 4 /
1=2
D
1
.E e / min
:
(12.12)
It can be readily shown that the small glide angle approximation (L B D gV b ) also reduces the Reynolds
number to
R e Á
UN c
Š
1
 2gV b
C L N R
à 1=2
:
Using this approximation to eliminate the Reynolds
number in (12.10) and (12.11) gives the minimum specific energy consumption for nonturning, steady-state
flight in (12.12) as [12.18]
.E e / min D 2
"
K i K 0 C
=2
L N A
.2gV b / =2 N
1=2
R
# 1=2
D
 L
D
à 1
max
:
(12.13)
12.4.2 Size Factors
The minimum specific energy consumption from
(12.13) decreases as V
=4
b
, i. e., it decreases with increasing net buoyancy. Because net buoyancy is some
fraction n b of the total vehicle volume V 0 , where V b D
n b V 0 , bigger buoyancy-driven vehicles generally are
more transport efficient. Actually, surveys of natural
and man-made fliers by [12.12] and [12.14] demonstrate that specific energy consumption monotonically
decreases across 12 orders of magnitude of size increase. This size advantage is accentuated in an underwater glider because the buoyancy volume factor,
n b , increases with increasing vehicle volume approximately as n b 1:2 10
5 V
7=6
0 , due to economies of
scale in packing efficiency [12.7]. Larger n b permits
higher glide speeds (speed increases as the square root
of the increase in net buoyancy) and higher wing section Reynolds numbers, which results in higher wing
section lift-to-drag ratios (Fig. 12.9).
Equation (12.13) also indicates that specific energy consumption decreases with increasing aspect
ratio of the wing, as N
.1=2/=2
R
, favoring long tapered wingspans with relatively small wing chords.
The aspect ratio of the profiling gliders varies from
a maximum of N R D 9:75 for Spray to a minimum
of 4:4 for Seaglider. Specific energy consumption decreases as the lift to drag ratio (L=D) increases, but
L=D suffers a precipitous decline if the wing section
chord is made too small in an effort to achieve a high
aspect ratio. Figure 12.9 shows that this L=D crisis occurs when the wing section Reynolds number drops
into the mid 10
4 regime. This phenomenon arises because of laminar separation on the suction side of the
wing section, which destroys a large percentage of
the lift. Profiling gliders with their present 1020 cm
wing chords and 30 cm=s cruise speeds are operating within the laminar separation regime where their
wings will not be able to realize a higher L=D and
lower E e by simply flying at higher angles of attack
(Fig. 12.9). This laminar separation phenomenon suggests a need to go to bigger wing chords to get above
the mid 10
4 Reynolds number regime in order to improve flight efficiency. A primary design philosophy of
the larger cross-country gliders (Fig. 12.4) is to achieve
310 Part B Autonomous Ocean Vehicles, Subsystems and Control
Here, C L and C D are the quadratic lift and drag coefficients, respectively. The drag coefficient is made up
of two terms, a profile drag term C D0 , that depends on
Reynolds number R e , and an induced drag term C Di ,
that increases with increasing lift coefficient and decreases with wing aspect ratio, N R [12.17]
C D D C D0 C C Di D K 0 R
e N A C K i
C
2
L
N R
: (12.10)
In (12.10), K 0 is the profile shape factor, N A is the ratio of the total wetted surface area to the wing area
N A D A t =A 0 , and K i is the wing plan form factor. The
aspect ratio is defined as N R D S=N c D S
2
=A 0 where S is
the wingspan and N
c D A 0 =S is the mean aerodynamic
chord of the glider’s wing. The power-law dependence
of profile drag is D 1=2 for completely laminar, unseparated boundary layer flow over the vehicle, and D
1=5 for fully turbulent un-separated boundary layers.
In (12.10), the Reynolds number is the size-dependent
scale factor R e Á UN c==, where is the kinematic viscosity of the fluid. Since the glider is considered
a streamlined body, the Reynolds number dependence
of the profile drag term in (12.10) is an approximation
of friction acting on the total wetted area, A t . Specific
energy consumption, E c , can be explicitly examined
for efficient gliders that achieve small glide angle during coordinated wings-level flight (v D v
0
D 0; ˇ D 0).
With the small glide angle assumption, the lift-to-drag
ratio can be written after [12.18] as
L
D
D
C L
C D
D
u
2
K 3 u 4 C K 4
D
1
E e
;
(12.11)
where
K 3 D
A 0
2gV b
K 0 R
e N A ;
K 4 D
2gV b
A 0
K i
N R
:
Taking @.L=D/=@u D 0, the speed at which the liftto-drag ratio is maximized becomes U D .K 4 =K 3 /
1=4
giving
 L
D
Ã
max
D
1
2
.K 3 K 4 /
1=2
D
1
.E e / min
:
(12.12)
It can be readily shown that the small glide angle approximation (L B D gV b ) also reduces the Reynolds
number to
R e Á
UN c
Š
1
 2gV b
C L N R
à 1=2
:
Using this approximation to eliminate the Reynolds
number in (12.10) and (12.11) gives the minimum specific energy consumption for nonturning, steady-state
flight in (12.12) as [12.18]
.E e / min D 2
"
K i K 0 C
=2
L N A
.2gV b / =2 N
1=2
R
# 1=2
D
 L
D
à 1
max
:
(12.13)
12.4.2 Size Factors
The minimum specific energy consumption from
(12.13) decreases as V
=4
b
, i. e., it decreases with increasing net buoyancy. Because net buoyancy is some
fraction n b of the total vehicle volume V 0 , where V b D
n b V 0 , bigger buoyancy-driven vehicles generally are
more transport efficient. Actually, surveys of natural
and man-made fliers by [12.12] and [12.14] demonstrate that specific energy consumption monotonically
decreases across 12 orders of magnitude of size increase. This size advantage is accentuated in an underwater glider because the buoyancy volume factor,
n b , increases with increasing vehicle volume approximately as n b 1:2 10
5 V
7=6
0 , due to economies of
scale in packing efficiency [12.7]. Larger n b permits
higher glide speeds (speed increases as the square root
of the increase in net buoyancy) and higher wing section Reynolds numbers, which results in higher wing
section lift-to-drag ratios (Fig. 12.9).
Equation (12.13) also indicates that specific energy consumption decreases with increasing aspect
ratio of the wing, as N
.1=2/=2
R
, favoring long tapered wingspans with relatively small wing chords.
The aspect ratio of the profiling gliders varies from
a maximum of N R D 9:75 for Spray to a minimum
of 4:4 for Seaglider. Specific energy consumption decreases as the lift to drag ratio (L=D) increases, but
L=D suffers a precipitous decline if the wing section
chord is made too small in an effort to achieve a high
aspect ratio. Figure 12.9 shows that this L=D crisis occurs when the wing section Reynolds number drops
into the mid 10
4 regime. This phenomenon arises because of laminar separation on the suction side of the
wing section, which destroys a large percentage of
the lift. Profiling gliders with their present 1020 cm
wing chords and 30 cm=s cruise speeds are operating within the laminar separation regime where their
wings will not be able to realize a higher L=D and
lower E e by simply flying at higher angles of attack
(Fig. 12.9). This laminar separation phenomenon suggests a need to go to bigger wing chords to get above
the mid 10
4 Reynolds number regime in order to improve flight efficiency. A primary design philosophy of
the larger cross-country gliders (Fig. 12.4) is to achieve