Flapping dynamics of a flexible propulsor near ground

Transcription

Acta Mech. Sin. (2016) 32(6):991–1000
DOI 10.1007/s10409-016-0571-5
RESEARCH PAPER
Jaeha Ryu1 · Sung Goon Park1 · Boyoung Kim1 · Hyung Jin Sung1
Received: 3 February 2016 / Revised: 21 March 2016 / Accepted: 11 April 2016 / Published online: 1 June 2016
© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin
Heidelberg 2016
Abstract The flapping motion of a flexible propulsor near
the ground was simulated using the immersed boundary
method. The hydrodynamic benefits of the propulsor near
the ground were explored by varying the heaving frequency
(St) of the leading edge of the flexible propulsor. Propulsion near the ground had some advantages in generating
thrust and propelling faster than propulsion away from the
ground. The mode analysis and flapping amplitude along the
Lagrangian coordinate were examined to analyze the kinematics as a function of the ground proximity (d) and St.
The trailing edge amplitude (atail ) and the net thrust (F x )
were influenced by St of the flexible propulsor. The vortical
structures in the wake were analyzed for different flapping
conditions.
Keywords Flexible propulsor · Ground effect · Immersed
boundary method · Mode analysis
1 Introduction
Many biological propulsors, such as birds, fish, and microflies, exhibit a lower cost of transport [1] or require less power
to achieve a certain propulsion speed [2] near the ground.
Certain advantages achieved by propelling near the ground
are called ground effects. Well-known examples of ground
effects are gliding birds, sediment particles, and benthic fish.
To reveal the hydrodynamic benefits of a flexible propulsor
B
1
Hyung Jin Sung
[email protected]
Department of Mechanical Engineering, Korea Advanced
Institute of Science and Technology, 291 Daehak-ro,
Yuseong-gu, Daejeon 34141, Korea
near the ground, some questions are posed: How does the
ground offer hydrodynamic benefits to flexible propulsors?
Which parameters of flexible propulsors influence the ground
effect? What is the relation between the dynamics and kinematics of flexible propulsors near the ground?
The ground effects are classified into steady and unsteady
ground effects. A steady ground effect is achieved by a static lifting object moving parallel to the ground, while an
unsteady ground effect is created when an object oscillates
perpendicular to the ground. The hydrodynamic benefits of
the steady ground effect are a large lift force induced by slow
flow beneath a static lifting object and a smaller drag induced
by the vortices between a flexible propulsor and the ground.
These benefits of the steady ground effect are widely utilized by gliding birds and fish in schools [3,4]. The ground
effect acts to decrease the induced drag and increase the liftto-drag ratio of the wings for birds gliding horizontally near
the ground [5]. The ground proximity influences the gliding performance of birds as they glide near solid boundaries.
Hainsworth [6] showed a ground effect related to various
flight formations. Wing tip spacing, which is the distance perpendicular to the flight path between the wing tips of adjacent
birds at a maximum span, is the major parameter. The propelling performance of the benthic fish Agonopsis vulsa was
examined by Nowroozi et al. [7]. The fish overcomes substantial negative buoyancy while generating forward thrust.
The wing performance associated with its morphometry for
identifying the characteristics of flight was determined by
Park and Choi [8], who explored the hydrodynamic benefits, reduced drag, and fast propelling speed achieved by
means of the ground effect in various configurations of flying fish. In contrast, the unsteady ground effect has been less
explored than the steady ground effect. A few theoretical
[9] and experimental [10,11] studies have been performed.
123
992
Iosilevskii [12] addressed unsteady aerodynamic forces acting on a wing oscillating in a uniform flow in the presence
of a distant flag ground. An asymptotic theory was applied
to obtain hydrodynamic loads acting on a hydrofoil oscillating below the water. Quinn et al. [13] investigated the
unsteady ground effect by measuring net thrust and selfpropelled swimming speed. Many hydrodynamic benefits
were obtained for a flexible propulsor propelling near the
ground. Except for Quinn et al. [13], most earlier studies
were conducted on rigid objects.
In the present study, the immersed boundary (IB) method
was adopted to simulate a flexible propulsor near the ground.
Using the IB method, the structure was separated into the
massive part and the massless part. The fluid governing equations and the solid governing equations were independently
solved using the interaction momentum forcing term [14–
16]. The momentum forcing was added to induce the fluid
velocity equivalent to the velocity of the IB and to enforce
the no-slip boundary condition along the IB [17,18]. The IB
method can resolve the dynamics of a flexible body without
grid regeneration. Many studies have been conducted using
the IB method for applications. The dynamics of an oscillating cascade in turbomachinery was explored using the IB
method [19]. Kim et al. [20] exhibited the interaction modes
between two tandem flexible flags and examined the optimal
modes for drag reduction. Biomimetic studies using the IB
method are numerous. Uddin et al. [21,22] investigated the
efficiency of fish schooling for drag reduction. Fish schooling
behaviors are not merely a social behavior but a smart way
to improve movement efficiency. Zhu et al. [23] simulated
a self-propelled plunging foil using the IB method. The foil
moved forward with a heaving motion of the leading edge in
a quiescent flow. Zhu et al. [24] fleshed out a rationale for
the hydrodynamic benefits experienced by the downstream
flag in a system comprising two self-propelling flags. The
IB method was extended to heat transfer problems [25–27].
Liao and Lin [25] discussed the difference between natural
and forced convection flows with a moving rigid embedded
object.
The objective of the present study was to explore the
ground effect of a flexible propulsor using the IB method.
The kinematics of a flexible propulsor near the ground were
analyzed by varying the trailing edge amplitude (atail ), the
modal contributions (Θi ), the ground proximity (d), and
the flapping amplitude (a) along the Lagrangian coordinate. We explored dynamics such as the net thrust (F x ),
the propulsive efficiency (η), and the self-propelled swimming speed (u SPS ) by varying the heaving frequency (St) of
the leading edge. The flow field around a flexible propulsor near the ground was visualized. The hydrodynamic
benefits obtained from the unsteady ground effect were
examined.
123
J. Ryu et al.
U∞
L
ahead
d
y
O
x
Fig. 1 Schematic diagram of flexible propulsor near ground
2 Problem formulation
A schematic diagram of the flexible propulsor and the coordinate system is shown in Fig. 1. A flexible flag with a
heaving motion was subjected to the near-wall region. Noslip boundary conditions were applied at the wall, and a
Poiseuille flow with a maximum velocity U∞ was applied
to the upstream flow [13]. The interaction between the flexible propulsor and the surrounding fluid was modeled using
the IB method, in which the momentum forcing was added to
the Navier–Stokes equations. The fluid was governed by the
continuity equation and the incompressible Navier–Stokes
equations
∂u
+ u · ∇u = −∇ p + μ∇ 2 u + f ,
ρ0
∂t
∇ · u = 0,
(1)
(2)
where u is the velocity vector, p is the pressure, ρ0 is the flow
density, μ is the dynamic viscosity, and f is the momentum
forcing used to enforce the no-slip boundary conditions along
the immersed boundary.
The position of the flexible propulsor is denoted by X (s)
and governed by the equation
∂ 2X
∂
ρ1 2 =
∂t
∂s
∂X
T
∂s
∂2
− 2
∂s
∂ 2X
γ 2
∂s
− F,
(3)
where s is the arc length, ρ1 is the density difference between
fluid and flag, T is the tension force along the flag axis,
γ is the bending rigidity, and F is the Lagrangian forcing
exerted on the flag by the flow. Equations (1)–(3) were nondimensionalized using various characteristic scales. The flow
density ρ0 was used for normalizing the density, the flexible flag length L for the length, the maximum velocity of
a Poiseuille flow U∞ for the velocity, L/U∞ for the time,
2 for the pressure p, ρ U 2 for the tension force T ,
ρ0 U∞
1 ∞
2 /L for the momentum exerted on the fluid f , ρ U 2 /L
ρ0 U∞
1 ∞
2 L 2 for the bendfor the Lagrangian momentum F, and ρ1 U∞
993
ing rigidity γ . The nondimensionalized forms of Eqs. (1) and
(3) are
f (x, t) = ρ
(5)
where the density ratio ρ = ρ1 /(ρ0 L) comes from nondimensionalization. We set ρ = 1, indicating that the density
of the flexible propulsor is two times larger than that of
the fluid density (ρ0 ). Equations (8) and (10) show that the
IB was forced to move along with the local fluid. In this
study, the fluid equation was solved using the fractional step
method [29], and the structure equation was solved based on
the IB method [17]). Details regarding the numerical procedure can be found in Ref. [17]. The Eulerian grid was
uniformly distributed in the streamwise direction. In the
spanwise direction, the Eulerian grid was uniformly distributed for 0 y/L 8 and was stretched otherwise. The
physical and numerical parameters for the present simulation are summarized in Table 1. Several trial calculations
were repeated to reveal the sensitivity of the results to the
domain and grid sizes. A domain size of 8L ×8L was chosen
to show the vortical structures behind the flexible propulsor.
Three grids were tested for the displacement of the trailing edge (Fig. 2): N x × N y = 257 × 257, 513 × 513, and
where Re indicates the Reynolds number normalized by
ρ0 U∞ L/μ, set to 200 in the present simulation. The nondimensional continuity equation took the form of Eq. (2). The
flexible propulsor had a clamped leading edge and a free
trailing edge. A heaving motion was applied at the clamped
leading edge (s = 0),
2πU∞ St
∂X
ahead
sin
t ,
= (1, 0) ,
X = X 0 + 0,
2
L
∂s
(6)
where ahead is the heaving amplitude of the leading edge,
and St is the frequency normalized by 2πU∞ /L. We set
ahead /L = 0.1 in the present simulation. As shown in Fig. 1,
X 0 is (0, d), where d is the ground proximity. At the free
trailing edge (s = L),
T = 0,
∂ 3X
∂ 2X
=
0)
,
= (0, 0) .
(0,
∂s 2
∂s 3
t
F=α
(U IB − U) dt + β (U IB − U) ,
Γ
(8)
0
where α and β are large negative free constants (α =
−1.74 × 107 , β = −2.60 × 102 ) defined by Shin et al. [28],
U IB is the fluid velocity on the IB obtained by interpolation,
and U is the velocity of a flexible propulsor expressed by
U = dX/dt. In Eq. (8), the fluid on the IB and the flexible
propulsor are connected by a virtual stiff spring with damping. The IB and the flexible propulsor moved together owing
to a stiff connection applied between the IB and the flexible
propulsor. For this reason, the velocities of the fluid on the
IB and the flexible propulsor were equal, leading to no-slip
boundary conditions on the surface of the flexible propulsor.
The interaction force between the surrounding fluid and
the flexible propulsor was obtained using the Dirac delta
function. The velocity on the IB was interpolated according
to
Table 1 Physical and numerical parameters
Flag length
L
1
Flapping amplitude
ahead
0.1
Ground proximity
d/ahead
2–10
Reynolds number
Re
200–400
Bending rigidity
γ
0.005–0.025
Flapping frequency
St
0.01–1.0
Grid resolution
h
0.01563
Computational time step
t
0.00005
γ = 0.005, St = 0.2, d ahead = 2, Re = 200
0.3
513 × 513
0.2
257 × 257
1025 ×1025
0.1
0
1
Ω
2
3
4
t/T
U IB (s, t) =
(10)
(7)
The momentum forcing term, F, which was added to the
solid governing equation, was derived using the feedback
law,
F (s, t) δ (x − X (s, t))ds,
(4)
y L
1 2
∂u
+ u · ∇u = −∇ p +
∇ u + f,
∂t
Re
∂ 2X
∂ 2X
∂X
∂2
∂
T
− 2 γ 2 − F,
=
∂t 2
∂s
∂s
∂s
∂s
The Lagrangian forcing was distributed to the nearby grid
points,
u (x, t) δ (X (s, t) − x)dx.
(9)
Fig. 2 Grid convergence test
123
994
J. Ryu et al.
a
b
γ = 0.005
3
γ = 0.025
2.5
d ahead = 10
2.5
d ahead = 10
2
d a head = 5
1.5
d ahead = 5
1.5
1
1
0.5
atail ahead
atail ahead
2
d ahead = 2
d ahead = 2
0
0.2
0.4
0.5
0.6
0
0.2
0.4
St
0.6
0.8
St
Fig. 3 atail for various d/ahead
1025 × 1025. The outcome with the 513 × 513 grid was
satisfactory in this simulation.
0.8
γ = 0.005, d ahead = 2, Re = 200
1st mode
3rd mode
0.6
2nd mode
3 Results and discussion
The hydrodynamic benefits for many biological swimmers
cruising near the ground have recently attracted research
interests, and those benefits are related to their swimming
kinematics. The trailing edge amplitude (atail ) is an important
factor when it comes to analyzing the kinematics of propulsors. atail is a function of the Strouhal number (St) normalized
by 2π U∞ /L, where L is the flexible flag length, and U∞ is
the maximum velocity of the flow. atail has a local maximum at which the net thrust (F x ) reaches a plateau [13,30].
Figure 3 shows the trailing edge amplitudes with various
ground proximities (d) and bending rigidities (γ ). When St
is very low, atail shows a lack of consistency by varying the
ground proximity (d/ahead ). When St is high enough, atail
has a local maximum (high peak) and a local minimum (low
peak). In Fig. 3a, b, the values of St are almost the same for
different ground proximities when the local high peaks and
low peaks change with γ . atail is not large when the propulsor
is close to the ground since the flow between the propulsor
and the ground restrains flapping behavior. For γ = 0.005,
the low peaks appear at St of 0.3 and the high peaks at St of
0.15 and 0.4. For γ = 0.025, the low peaks are present at St
of 0.6 and the high-peaks are at St of 0.3.
Quinn et al. [30] showed that flexible propulsors near the
ground experienced several dynamical modes as St varied.
The orthonormal eigenfunctions (ξi ) were defined from the
Euler–Bernoulli beam equation [30]. The modal contribution
(Θi ) was calculated by taking an inner product of the flag
123
Θ 0.4
0.2
4th mode
0
0
0.2
0.4
0.6
St
Fig. 4 Time-averaged modal contributions
1.4
γ = 0.005, d ahead = 2
St = 0.4
St = 0.35
1
St = 0.15
a ahead
3.1 Kinematics of flexible propulsor
St = 0.2
0.6
St = 0.3
0.2
0
0.2
0.4
s L
0.6
Re = 200
0.8
1
Fig. 5 Time-averaged flapping amplitudes along Lagrangian coordinate
position with each eigenfunction (0 < ξi < 1). The modal
contribution (Θi ) was taken into account only when the leading edge passed through y = d and the time averaging was
995
b
a
γ = 0.005
0.4
γ = 0.025
1
d a head = 2
d a head = 2
0.8
0.3
d a head = 10
Fx
d a head = 10
Fx
0.6
0.2
0.4
0.1
d ahead = 5
d a head = 5
0.2
0
Re = 200
-0.1
0
0.2
0.4
0.6
St
Re = 200
0
0
0.2
0.4
0.6
0.8
St
Fig. 6 Time-averaged net thrust for St
was conducted over the course of more than 20 heaving
periods. Figure 4 shows the time-averaged modal contributions of each orthonormal eigenfunction for γ = 0.005 and
d/ahead = 2. For the low St region (St < 0.1), the flag flaps
very slowly and is deflected very little during the flapping
sequence. The first mode is dominant at St < 0.1 (Fig. 3).
When atail reaches the high peaks, i.e., St = 0.15 and 0.4
in Fig. 3, the second and third modes are dominant (Fig. 4).
As St increases, the modes are excited sequentially. For St =
0.3, several orthonormal eigenfunctions are overlapped and
mutually correlated. While multiple orthonormal eigenfunctions are interrupted, the kinematics of the flag is disturbed
and atail decreases. The overlapped mode restrains the flapping motion of the flag, leading to a decreased thrust (Fig. 6).
The modal contributions were generally lower than those in
Quinn et al. [30] regardless of the value of St. Since the
present simulation was conducted at relatively low Re, the
flapping motion of the flexible propulsor was interrupted by
the viscous incoming flow.
The time-averaged flapping amplitude along the flexible
propulsor is one of the key factors in representing the flapping
kinematics. Figure 5 illustrates the time-averaged flapping
amplitudes along the Lagrangian coordinate (a) with various
St. As illustrated in Fig. 3, the trailing edge of the propulsor
shows high peaks at St = 0.15 and 0.4, and both atail and a are
high. The a measured at St = 0.2, 0.3, and 0.35 are relatively
low compared to those at St = 0.15 and 0.4. Regardless of the
value of St, a local minimum is present at the first extreme
value from the trailing edge. As St increases, every local
minimum or local maximum is shifted to the trailing edge.
The first local minimum point is shifted from s/L = 0.8
to s/L = 0.9 as St increases. When the dominant dynamical mode is transitioned from the second mode (St = 0.15)
to the third mode (St = 0.4), an additional local maximum
point appears near the leading edge (s/L = 0.3). As the
dynamical mode is shifted to the next mode, the additional
local maximum point or minimum point appears alternately
near the leading edge. The superimposed dynamical modes
emerge at the low peaks, while one dominant dynamical
mode is observed at the high peak. The flapping amplitude
along the Lagrangian coordinate is low at the low peak since
the dynamical modes are significantly superimposed. At the
lower peak (St = 0.3), the flapping amplitude is minimized
( a/ahead |s/L=0.45 ≈ 0.3) owing to the superimposition of
the dynamical modes.
3.2 Hydrodynamic benefits due to ground effect
In experiments, thrust was measured along the leading edge
spar. The measured force at the spar was considered to be
the net propulsive force of the propulsor. Unlike the experiment, thrust is calculated through the view point of energy
conservation in the present study. Assuming that the internal energy and other reaction energy are neglected during
flapping, the energy
isconverted
to the
thrust along
2 directly
2
the x-direction Fx = dx/ dx
dE and thelateral
+ dy
force along the y-direction Fy = dy/ dx 2 + dy 2 dE .
The dissipated energy of the fluid is small enough that it can
be neglected. Figure 6 shows time-averaged net thrust (F x )
against various ground proximities (d). The net thrust near the
ground is generally high in most regions of St. For St = 0.35
with γ = 0.005, the net thrust at d/ahead of 2 is lower than
that at d/ahead of 5. This is related to the kinematics of the
propulsor, which will be discussed in connection with Fig. 10.
The propulsor in a slow oncoming flow can generate more
thrust than that in a fast flow. As the flow velocity increases,
the net thrust decreases consistently owing to the increased
drag force exerted on the immersed body. When the propulsor
123
996
J. Ryu et al.
a
0.5
γ = 0.005
0.4
b
d ahead = 2
0.4
γ = 0.025
0.3
d ahead = 2
η
η
0.3
0.2
0.2
d ahead = 5
d ahead = 5
0.1
0.1
d ahead = 10
d ahead = 10
0
0.2
0.4
0.6
St
0
0.2
0.4
0.6
0.8
St
Fig. 7 Propulsive efficiency
moves close to the wall, the oncoming flow is slower than the
flow in the center region, which reduces drag. Accordingly,
the propulsor can generate thrust easily and effectively near
the ground. The time-averaged net thrust reaches a plateau
at the high peak (St = 0.4), which is consistent with previous results [13,30]. The time-averaged net thrust at the low
peak is influenced by the ground proximity. When the propulsor is close to the ground (d/ahead = 2), the flow between
the propulsor and the ground interrupts the kinematics and
disturbs the thrust production. When the propulsors are positioned far enough from the ground (d/ahead = 10), the thrust
increases around the low peak (St = 0.6) in Fig.6b.
The propulsive efficiency (η = F x U∞ /P) is derived from
the time-averaged net thrust, far-field velocity (U∞ ), and the
input heaving power (P). In the present study, the heaving
motion of the leading edge was implemented by the prescribed motion. The input heaving power was determined
by multiplying the lateral force exerted on the flag by the
negative velocity at the leading edge [13]. The propulsive
efficiency was divided into the input heaving power. For
low St (0 < St < 0.25), the input heaving power is very
small, resulting in a high propulsive efficiency. Although the
propulsive efficiency was high, the net thrust was very low,
which is not very attractive to flexible propulsors. At St =
0.4, the propulsive efficiency showed a local maximum at
d/ahead = 2, a plateau at d/ahead = 5, and a descent at
d/ahead = 10 (Fig. 7a). The propulsive efficiency showed
no trend for γ = 0.005 while showing a consistent trend
when the bending rigidity was high enough (γ = 0.025)
(Fig. 7b). The propulsive efficiency had a local maximum
at the high peak (St = 0.3) and a plateau between the high
peak and the low peak, and a rapid descent at the low peak
(St = 0.6). After the local maximum, a rapid drop in the
efficiency was present in the plateau region of the net thrust.
The local maximum of the efficiency at the high peak was
123
shown experimentally [13]. As St increased, the propulsive
efficiency decreased, but the net thrust increased. A flexible
propulsor could not achieve the efficient propulsion and the
net thrust generation simultaneously.
As discussed in Fig. 6, a slow oncoming flow is beneficial
for a propulsor to generate a thrust since the drag force exerted
on the propulsor by the surrounding fluid is reduced. Quinn
et al. [30] showed that the flow velocity has an inversely proportional relation with the net thrust. In the present study, the
net thrust is reduced as the flow velocity increases. Figure 8
demonstrates time-averaged net thrust with respect to various flow velocities at the high peaks. The flow velocity (u) at
X 0 (0, d) was chosen as one of the governing parameters and
was increased as Re increased. The high flow velocity disturbed the flapping motion of the flag and led to a decrease in
the thrust. The drag increased as the flow velocity increased.
For those reasons, the net thrust generation was disturbed by a
high flow velocity (Fig. 8). When the flow velocity increased,
the net thrust decreased gradually and finally reached a negative net thrust region. The decrease in the net thrust was
mainly affected by the flow velocity, not the ground proximity. The flow velocity with zero net thrust was defined as
the self-propelled swimming speed (u SPS ) [13]. When the
propulsor moved at the self-propelled swimming speed, it
experienced no external forces from the upstream flow.
As mentioned earlier, the self-propelled swimming speed
is the flow velocity when the thrust generated by the propulsors is equivalent to the drag exerted on the propulsors by
the upstream flow. A high self-propelled swimming speed
indicates that the flag propels quickly for a given input force.
Figure 9 shows self-propelled swimming speeds as a function
of the ground proximity at the high peak. The self-propelled
swimming speed was normalized by the flow velocity at Re of
200. As depicted in Fig. 8, the self-propelled swimming speed
was high when the flag flapped near the ground for the high
a
997
γ = 0.005, St = 0.4
0.3
b
γ = 0.025, St = 0.3
Fx
0.2
d ahead = 2
0.1
0
d ahead = 2
d ahead = 10
d ahead = 5
1
1.5
2
d ahead = 5
d ahead = 10
1
1.5
u u Re = 200
2
u u Re = 200
Fig. 8 Time-averaged net thrust at high peaks
2.5
uSPS uRe = 200
peaks. The self-propelled swimming speed decreased linearly as the ground proximity increased. The self-propelled
swimming speed near the ground (d/ahead = 2) increased by
about 58 % of the self-propelled swimming speed far from
the ground (d/ahead = 10) for different bending rigidities.
The propulsors can propel about 58 % faster near the ground
when the equivalent heaving power is given. The propulsors propelling near the ground can take advantages of the
increased propulsion velocity for a given input force or the
reduced input force to achieve the same propulsion velocity.
Only the high-peak case was considered to measure the
self-propelled swimming speed [13]. The net thrust generated at the low-peak was meaningful as the flow velocity
increased and related to the trailing edge amplitude. Figure 9
depicts time-averaged net thrust and trailing edge amplitude
with respect to various flow velocities at the low peaks. As
shown in Fig. 10a, b, the trailing edge amplitude increased
continuously and plateaud as the flow velocity increased.
All the plateaus appeared when the trailing edge amplitude
was equivalent to the heaving amplitude (atail /ahead = 1.0).
When the trailing edge amplitude was lower than the heaving
amplitude, the transverse fluttering behavior of the trailing
edge was confined to the wake region affected by the heaving leading edge. The flow velocity in the wake region was
lower than that of open flow owing to the reduced drag exerted
on the trailing edge. For this region (atail /ahead < 1.0), the
net thrust and the trailing edge amplitude increased as the
flow velocity increased. Since the kinematics of the trailing edge was interrupted by upstream flow, the trailing edge
amplitude reached a plateau (atail /ahead = 1.0) as the flow
velocity increased. The fast flow velocity outside of the
wake region increased drag and disturbed the kinematics of
the trailing edge. The net thrust was maximized, while the
2
γ = 0.025, St = 0.3
1.5
1
γ = 0.005, St = 0.4
2
4
6
d ahead
8
10
Fig. 9 Self-propelled swimming speed against d/ahead
trailing edge amplitude reached a plateau. When the trailing edge amplitude was higher than the heaving amplitude
(atail /ahead > 1.0), the net thrust decreased continuously
because of the increased drag. After the plateau, the net thrust
also decreased gradually (Fig. 10). This maximized net trust
was present at the low peak. As shown in Fig. 10a, the net
thrust at d/ahead = 5 was larger than that at d/ahead = 2.
This is consistent with that of St = 0.35 (low-peak) in Fig. 5.
The net thrust at d/ahead = 5 was maximized in the region
u/u Re=200 = 1, but the net thrust at d/ahead = 2 was not in
the region.
3.3 Vortical structure interaction
Figure 11 displays vortical structures around a flexible
propulsor with various ground proximities when the leading
edge of the propulsor crossed the line, y = d. The vortic-
123
998
J. Ryu et al.
γ = 0.005, St = 0.3
a
γ = 0.025, St = 0.6
b
0.2
d ahead = 2
Fx
d ahead = 2
0.1
d ahead = 5
d ahead = 5
0
d ahead = 10
d ahead = 10
2.5
d ahead = 10
atail ahead
2
d ahead = 10
d ahead = 5
d ahead = 5
1.5
d ahead = 2
1
d ahead = 2
0.5
1
1.5
2
1
2.5
1.5
u u Re = 200
2
2.5
u u Re = 200
Fig. 10 Time-averaged net thrust and trailing edge amplitude at low peaks
123
y L
a
b
y L
ity contour is identified when the absolute vorticity is higher
than 10 % of the maximum vortex strength. Poiseuille flow
at the inlet leads to negative vortical structures (red contour)
being generated near the bottom channel wall and positive
vortical structures (green contour) formed underneath the
propulsor (Fig. 11b). When the propulsor was positioned at
d/ahead = 2, a back flow was generated near the ground
and the flow velocity at the space between the propulsor and
the ground became slow. This back flow generation induced
the drag reduction. This gave rise to high net thrust near the
ground. The back flow separated the positive vortical structure on the underside of the propulsor. The negative vortical
structure was induced underneath the propulsor by the back
flow generated near the ground, unlike the positive vortical
structures that formed only on the underside of the propulsor
at d/ahead = 10. When the propulsor was far from the wall
(d/ahead = 10), the upstream flow easily penetrated into the
gap between the propulsor and the ground. A counter vortical
structure was formed only while the propulsor was positioned
near the ground.
Figure 12 displays sequential vortical structures at the low
peak (St = 0.3) and high peak (St = 0.4) during one flapping period. For both cases, negative votical structures were
generated at the leading edge when the leading edge moved
downward (t ∗ = 0 ∼ 21 T ), and positive vortical structures
were generated at the upward movement of the leading edge
(t ∗ = 21 T ∼ T ). As shown in Fig. 5, the flapping amplitude of the propulsor at the low peak was low for all flag
1.5
d a head = 2
γ = 0.005, St = 0.3, Re = 200
1
0.5
0
1.5
d a head = 10
1
0.5
0
-0.5
0
0.5
1
1.5
2
2.5
x L
Fig. 11 (Color online) Vortical structures around flexible propulsor
and ground
positions and reduced the strength of the vortical structures.
The back flow was accordingly weakened and the hydrodynamic benefit was not obtained by the back flow for the low
peak (Fig. 12a). For the high peak (Fig. 12b), the back flow
was intensified and the advantage was greatest. Only for the
high peak did the positive vortical structures detached at the
trailing edge move upward since the positive vortical structure was intensified [31]. The negative vortical structures on
the ground were absorbed into the positive vortical structures
when those vortical structures escaped the space between the
propulsor and the ground. The high net thrust was generated
when its leading edge moved downward [30]. The back flow
was induced only when the leading edge moved downward.
b
y L
t* = 0
1
t* = T
6
1
t* = T
6
y L
St = 0.4, d ahead = 2
1
t* = T
3
1
t* = T
3
y L
t* = 0
1
t* = T
2
1
t* = T
2
y L
St = 0.3, d ahead = 2
2
t* = T
3
2
t* = T
3
y L
y L
a
999
5
t* = T
6
5
t* = T
6
x L
x L
Fig. 12 (Color online) Sequential vortical structure formation process around flexible propulsor and ground. a St = 0.3 (low peak). b St = 0.4
(high peak)
4 Conclusions
We simulated the dynamics of a flexible propulsor with a
clamped leading edge and a free trailing edge. The simulation was conducted using the IB method, in which the
solid motion equation and the fluid motion equation were
solved separately by adding the interaction momentum forcing term. The hydrodynamic benefits of the flexible propulsor
by the ground effect were discussed within the framework of
dynamics and kinematics. The trailing edge amplitude presented the high and low peaks. Each peak showed distinct
differences in terms of both the dynamics and the kinematics. The high peak of the trailing edge amplitude showed the
domination of the orthonormal eigenfunctions. The orthonormal eigenfunctions were defined from the Euler–Bernoulli
beam theory, and the modal contributions were calculated
by taking an inner product of the flag position with each
eigenfunction. As the mode changed, the kinematics of the
propulsor were affected accordingly. The time-averaged net
thrust reached a plateau at the high peak; otherwise the
time-averaged net thrust was influenced dramatically by the
ground proximity. When the propulsor was close to the
ground (d/ahead = 2), the flow between the propulsor and
the ground interrupted the kinematics and disturbed the thrust
production. The disturbance of the thrust production did not
show when the propulsors were positioned far enough from
the ground (d/ahead = 10). The self-propelled swimming
speed near the ground (d/ahead = 2) increased by approximately 58 % of the self-propelled swimming speed far from
the ground (d/ahead = 10) for both the different bending
rigidity cases. The propulsors could propel approximately
58 % faster near the ground with the equivalent heaving
power. All the plateaus for the net thrust generated at the low
peak appeared when the trailing edge amplitude was equivalent to the heaving amplitude (atail /ahead = 1.0). When the
propulsor was positioned at d/ahead = 2, a back flow was
generated near the ground and the flow velocity at the space
between the propulsor and the ground became slow. The back
flow was accordingly weakened and the hydrodynamic benefit was not obtained by the back flow for the low peak. For the
high peak, the back flow was intensified and the advantage
was the greatest.
Acknowledgments This work was supported by the Creative Research Initiatives (Grant 2016-004749) program of the National Research
Foundation of Korea (MSIP).
123
1000
References
1. Webb, P.W.: The effect of solid and porous channel walls on steady
swimming of steelhead trout Oncorhynchus mykiss. J. Exp. Biol.
178, 97–108 (1993)
2. Blake, R.W.: The energetics of hovering in the mandarin fish (synchropus picturatus). J. Exp. Biol. 82, 25–33 (1979)
3. Baudinette, R.V., Schmidt-Nielsen, K.: Energy cost of gliding flight
in herring gulls. Nature 248, 83–84 (1974)
4. Rayner, J.M.V.: On the aerodynamics of animal flight in ground
effect. Philos. Trans. R. Soc. B 334, 119–128 (1991)
5. Blake, R.W.: Mechanics of gliding birds with special reference to
the influence of ground effect. J. Biomech. 16, 649–654 (1983)
6. Hainsworth, F.R.: Induced drag savings from ground effect and
formation flight in brown pelicans. J. Exp. Biol. 135, 431–444
(1988)
7. Nowroozi, B.N., Strother, J.A., Horton, J.M., et al.: Whole-body
lift and ground effect during pectoral fin locomotion in the northern
spearnose poacher ( agonopsis vulsa). J. Zool. 112, 393–402 (2009)
8. Park, H., Choi, H.: Aerodynamic characteristics of flying fish in
gliding flight. J. Exp. Biol. 213, 3269–3279 (2010)
9. Tanida, Y.: Ground effect in flight. JSME Int. J. Ser. B. 44, 481–486
(2001)
10. Truong, T.V., Byun, D., Kim, M.J., et al.: Aerodynamic forces and
flow structures of the leading edge vortex on a flapping wing considering ground effect. Bioinspiration Biomim. 8, 036007 (2013)
11. Blevins, E., Lauder, G.V.: Swimming near the substrate: a simple
robotic model of stingray locomotion. Bioinspiration Biomim. 8,
016005 (2013)
12. Iosilevskii, G.: Asymptotic theory of an oscillating wing section in
weak ground effect. Eur. J. Mech. B/Fluids 27, 477–490 (2008)
13. Quinn, D.B., Lauder, G.V., Smits, A.J.: Flexible propulsors in
ground effect. Bioinspiration Biomim. 9, 036008 (2014)
14. Peskin, C.S.: The immersed boundary method. Acta Numer. 11,
479–517 (2002)
15. Zhu, L., Peskin, C.S.: Simulation of a flapping flexible filament in
a flowing soap film by the immersed boundary method. J. Comput.
Phys. 179, 452–468 (2002)
16. Kim, Y., Peskin, C.S.: Penalty immersed boundary method for an
elastic boundary with mass. Phys. Fluids 19, 053103 (2007)
17. Huang, W.-X., Shin, S.J., Sung, H.J.: Simulation of flexible filaments in a uniform flow by the immersed boundary method. J.
Comput. Phys. 226, 2206–2228 (2007)
123
J. Ryu et al.
18. Huang, W.-X., Sung, H.J.: An immersed boundary method for
fluid-flexible structure interaction. Comput. Methods Appl. Mech.
Eng. 198, 2650–2661 (2009)
19. Zhong, G., Sun, X.: New simulation strategy for an oscillating
cascade in turbomachinery using immersed-boundary method. J.
Propul. Power 25, 312–321 (2009)
20. Kim, S., Huang, W.-X., Sung, H.J.: Constructive and destructive
interaction modes between two tandem flexible flags in viscous
flow. J. Fluid Mech. 661, 511–521 (2010)
21. Uddin, E., Huang, W.-X., Sung, H.J.: Interaction modes of multiple
flexible flags in a uniform flow. J. Fluid Mech. 729, 563–583 (2013)
22. Uddin, E., Huang, W.-X., Sung, H.J.: Actively flapping tandem
flexible flags in a viscous flow. J. Fluid Mech. 780, 120–142 (2015)
23. Zhu, X., He, G., Zhang, X.: Numerical study on hydrodynamic
effect of flexibility in a self-propelled plunging foil. Comput. Fluids
97, 1–20 (2014)
24. Zhu, X., He, G., Zhang, X.: Flow-mediated interactions between
two self-propelled flapping filaments in tandem configuration.
Phys. Rev. Lett. 113, 238105 (2014)
25. Liao, C.-C., Lin, C.-A.: Simulation of natural and forced convection flows with moving embedded object using immersed boundary
method. Comput. Methods Appl. Mech. Eng. 213–216, 58–70
(2012)
26. Liu, Y.Z., Kang, W., Sung, H.J.: Assessment of the organization
of a turbulent separated and reattaching flow by measuring wall
pressure fluctuations. Exp. Fluids 38, 485–493 (2005)
27. Nagendra, K., Tafti, D.K., Viswanath, K.: A new approach for conjugate heat transfer problems using immersed boundary method
for curvilinear grid based solvers. J. Comput. Phys. 267, 225–246
(2014)
28. Shin, S.J., Huang, W.-X., Sung, H.J.: Assessment of regularized
delta functions and feedback forcing schemes for an immersed
boundary method. Int. J. Numer. Methods Fluids 58, 263–286
(2008)
29. Kim, K., Baek, S.-J., Sung, H.J.: An implicit velocity decoupling procedure for incompressible Navier-Stokes equations. Int.
J. Numer. Methods Fluids 38, 125–138 (2002)
30. Quinn, D.B., Lauder, G.V., Smits, A.J.: Scaling the propulsive performance of heaving flexible panels. J. Fluid Mech. 738, 250–267
(2014)
31. Park, T.S., Sung, H.J.: Development of a near-wall turbulence
model and application to jet impingement heat transfer. Int. J. Heat
Fluid Fl. 22, 10–18 (2001)

Flapping dynamics of a flexible propulsor near ground

Transcription

Similar documents