Part B | 11.2
286 Part B Autonomous Ocean Vehicles, Subsystems and Control
Table 11.1 (continued)
Cruising, hovering, or yawing
Fin kinematics: straight
(Â Bias D 0), or
yaw (Â Bias ¤ 0)
swim
Fin optimization
parameters
based on thrust
or efficiency
Salient features of w .t/ on fin
surface (blue: low, red: high
wall-shear stress); both pressure
and suction surfaces are shown;
sinusoidal line: stagnation point
trajectory; vertical axis: %
chord, horizontal axis: time
Yawing while cruising or hovering
Hovering: U D 0
U wing =U t ! 1
0 ı < Â Bias < 45 ı
0 D 30 ı D
90 ı Â t0 D 0 ı
C x D fn.f 2 /
 0 D 45 ı
With increasing positive pitch
bias angles, regions of low w
increase; one pair of brief bursts
of low w extend over chord per
cycle of fin oscillation; oscillating
stagnation line is a band of low w
as when U D 0 and  Bias D 0 ı
0
1.6
1.2
0.8
0.4
Time (s)
S/c (% chord)
0
10
20
30
40
Hovering: U D 0
U wing =U t ! 1
0 ı < Â Bias <
45 ı 0 D 30 ı
D 90 ı Â t0 D
0 ı
C x D fn.f 2 /
 0 D 45 ı
With decreasing pitch bias angles
below 0 ı , the chordwise bands
of low w partly become compact
and shift mostly down chord
0
1.6
1.2
0.8
0.4
Time (s)
S/c (% chord)
0
10
20
30
40
Note: When w is very low, the local fin boundary layer approaches separation. When w is very high, the boundary layer is attached
and may be accelerating. The topology of the near-wall flow over the fin can be defined by critical points such as saddles and foci;
in the flapping fins, these flow bifurcation points can be identified: spatiotemporal stagnation point, separation and reattachment
points [11.15]
11.2 Theoretical Foundation of Animal-Inspired Hydrodynamics
and Control
Flapping fins are nonlinear oscillators. In the case
of a drag-producing cylinder in a stream, the lowest order solution of the Navier–Stokes equation of
the wake instability reduces to a van der Pol equation [11.31, Appendix]. The latter equation applies
also when the cylinder is freely suspended and is galloping (at the natural frequency) [11.32]. Obviously,
the cylinder can be replaced by a low-drag foil and
the van der Pol form of the Navier–Stokes equation
would still describe the lowest order solution of the
wake instability. It has been shown [11.15] that the
van der Pol solution can be extended to a flapping
fin (i. e., if the fin wake is perturbed by a flapping
fin motion) oscillating at the natural frequency when
the drag wake is replaced by a thrust wake after symmetry breaking [11.29]. In other words, in the transitional range of Reynolds number, vortex-propelled
animals can be described by nonlinear oscillators as
indeed was implied earlier by Karman and Burgers [11.1] and Lighthill [11.2] and Bandyopadhyay
et al. [11.15].
11.2.1 Hydrodynamics
As depicted in Fig. 11.2a,b, in the swimming animal, the oscillating surface and the vortex street couple [11.28]. The wake circulation alters the geometric
fin angle of attack given by the forward velocity and the
fin oscillation velocity. This mechanism allows a freely
suspended flexible fin oscillating at the natural frequency and its wake vortex train to adjust to each other
(lock-in) in terms of phase and energy, iterating to
a low-energy system (minimal losses).
For a single fin hinged at one end that is also twisting and flapping in the optimal manner (Fig. 11.2d,e)
near the natural shedding frequency of the wake instability of the nonflapping fin, the instantaneous fluctuating force F
0
x produced agrees well – to a leading order
particularly at high Reynolds numbers (fin chord and
cruise speed-based Re c 70 000) – with the van der Pol
formulation given below,
R
F
0
x ! s G.F
02
xo 4F
02
x / P
F
0
x C !
2
s F
0
x D ! s TR avg P
; (11.1)
286 Part B Autonomous Ocean Vehicles, Subsystems and Control
Table 11.1 (continued)
Cruising, hovering, or yawing
Fin kinematics: straight
(Â Bias D 0), or
yaw (Â Bias ¤ 0)
swim
Fin optimization
parameters
based on thrust
or efficiency
Salient features of w .t/ on fin
surface (blue: low, red: high
wall-shear stress); both pressure
and suction surfaces are shown;
sinusoidal line: stagnation point
trajectory; vertical axis: %
chord, horizontal axis: time
Yawing while cruising or hovering
Hovering: U D 0
U wing =U t ! 1
0 ı < Â Bias < 45 ı
0 D 30 ı D
90 ı Â t0 D 0 ı
C x D fn.f 2 /
 0 D 45 ı
With increasing positive pitch
bias angles, regions of low w
increase; one pair of brief bursts
of low w extend over chord per
cycle of fin oscillation; oscillating
stagnation line is a band of low w
as when U D 0 and  Bias D 0 ı
0
1.6
1.2
0.8
0.4
Time (s)
S/c (% chord)
0
10
20
30
40
Hovering: U D 0
U wing =U t ! 1
0 ı < Â Bias <
45 ı 0 D 30 ı
D 90 ı Â t0 D
0 ı
C x D fn.f 2 /
 0 D 45 ı
With decreasing pitch bias angles
below 0 ı , the chordwise bands
of low w partly become compact
and shift mostly down chord
0
1.6
1.2
0.8
0.4
Time (s)
S/c (% chord)
0
10
20
30
40
Note: When w is very low, the local fin boundary layer approaches separation. When w is very high, the boundary layer is attached
and may be accelerating. The topology of the near-wall flow over the fin can be defined by critical points such as saddles and foci;
in the flapping fins, these flow bifurcation points can be identified: spatiotemporal stagnation point, separation and reattachment
points [11.15]
11.2 Theoretical Foundation of Animal-Inspired Hydrodynamics
and Control
Flapping fins are nonlinear oscillators. In the case
of a drag-producing cylinder in a stream, the lowest order solution of the Navier–Stokes equation of
the wake instability reduces to a van der Pol equation [11.31, Appendix]. The latter equation applies
also when the cylinder is freely suspended and is galloping (at the natural frequency) [11.32]. Obviously,
the cylinder can be replaced by a low-drag foil and
the van der Pol form of the Navier–Stokes equation
would still describe the lowest order solution of the
wake instability. It has been shown [11.15] that the
van der Pol solution can be extended to a flapping
fin (i. e., if the fin wake is perturbed by a flapping
fin motion) oscillating at the natural frequency when
the drag wake is replaced by a thrust wake after symmetry breaking [11.29]. In other words, in the transitional range of Reynolds number, vortex-propelled
animals can be described by nonlinear oscillators as
indeed was implied earlier by Karman and Burgers [11.1] and Lighthill [11.2] and Bandyopadhyay
et al. [11.15].
11.2.1 Hydrodynamics
As depicted in Fig. 11.2a,b, in the swimming animal, the oscillating surface and the vortex street couple [11.28]. The wake circulation alters the geometric
fin angle of attack given by the forward velocity and the
fin oscillation velocity. This mechanism allows a freely
suspended flexible fin oscillating at the natural frequency and its wake vortex train to adjust to each other
(lock-in) in terms of phase and energy, iterating to
a low-energy system (minimal losses).
For a single fin hinged at one end that is also twisting and flapping in the optimal manner (Fig. 11.2d,e)
near the natural shedding frequency of the wake instability of the nonflapping fin, the instantaneous fluctuating force F
0
x produced agrees well – to a leading order
particularly at high Reynolds numbers (fin chord and
cruise speed-based Re c 70 000) – with the van der Pol
formulation given below,
R
F
0
x ! s G.F
02
xo 4F
02
x / P
F
0
x C !
2
s F
0
x D ! s TR avg P
; (11.1)