Journal of Fluids and Structures kinematics effects on flapping airfoil propulsion
Journal of Fluids and Structures 42 (2013) 166 – 186
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Journal of Fluids and Structures
journal homepage: www.elsevier.com/locate/jfs
High fidelity numerical simulation of airfoil thickness and kinematics effects on flapping airfoil propulsion
Meilin Yu
, Z.J. Wang
, Hui Hu
a
Department of Aerospace Engineering, The University of Kansas, Lawrence, KS 66045, USA b
Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA a r t i c l e i n f o
Article history:
Received 8 October 2012
Accepted 2 June 2013
Available online 17 July 2013
Keywords:
Flapping airfoil
Airfoil thickness
Kinematics
High order Navier – Stokes simulations
Spectral difference a b s t r a c t
High-fidelity numerical simulations with the spectral difference (SD) method are carried out to investigate the unsteady flow over a series of oscillating NACA 4-digit airfoils. Airfoil thickness and kinematics effects on the flapping airfoil propulsion are highlighted. It is confirmed that the aerodynamic performance of airfoils with different thickness can be very different under the same kinematics. Distinct evolutionary patterns of vortical structures are analyzed to unveil the underlying flow physics behind the diverse flow phenomena associated with different airfoil thickness and kinematics and reveal the synthetic effects of airfoil thickness and kinematics on the propulsive performance.
Thickness effects at various reduced frequencies and Strouhal numbers for the same chord length based Reynolds number ( ¼ 1200) are then discussed in detail. It is found that at relatively small Strouhal number ( ¼ 0.3), for all types of airfoils with the combined pitching and plunging motion (pitch angle 20 1 , the pitch axis located at one third of chord length from the leading edge, pitch leading plunge by 75 1 ), low reduced frequency ( ¼ 1) is conducive for both the thrust production and propulsive efficiency. Moreover, relatively thin airfoils (e.g. NACA0006) can generate larger thrust and maintain higher propulsive efficiency than thick airfoils (e.g. NACA0030). However, with the same kinematics but at relatively large Strouhal number ( ¼ 0.45), it is found that airfoils with different thickness exhibit diverse trend on thrust production and propulsive efficiency, especially at large reduced frequency ( ¼ 3.5). Results on effects of airfoil thickness based Reynolds numbers indicate that relative thin airfoils show superior propulsion performance in the tested
Reynolds number range. The evolution of leading edge vortices and the interaction between the leading and trailing edge vortices play key roles in flapping airfoil propulsive performance.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Unsteady flapping-wing aerodynamics has witnessed great prosperity in the last three decades. This prosperity is not only driven by the power of consistent curiosity about the effective propulsion exhibited by natural flyers, but also due to the increasing interest in the design of Micro Air Vehicles (MAVs), which have been considered to have great potential to open up new opportunities for surveillance-like missions in the near future. A large amount of experimental and computational research has been carried out to reveal flapping-wing aerodynamics and many comprehensive reviews n
Corresponding author. Tel.: + 1 7858644267.
E-mail address: mlyu@ku.edu (M. Yu) .
0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jfluidstructs.2013.06.001
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Nomenclature
Re
T
St
Δ P
α
η pow
θ
0 ϕ
, θ
C chord length, m
C
P
⟨ C
C
T pow
⟨ C
T
⟩
⟨ C
⟨ C
T _ P
T _ v
⟩
⟩ pressure coefficient, ð p − p
∞
Þ = ð 0 : 5 ρ U 2
∞
Þ
⟩ time averaged power coefficient, − ð 1 = T Þ thrust coefficient, Thrust = ð 0 : 5 ρ U
2
∞
C Þ time averaged thrust coefficient, ð 1 = T Þ
R
R t t
0
0
þ
T t t
0
0
þ T
C
½
T
Lift
ð t Þ
ð t dt
Þ _ ð t Þ þ Moment time averaged thrust coefficient calculated from the pressure force time averaged thrust coefficient calculated from the viscous force
ð t Þ _ ð t Þ dt = ð 0 : 5 ρ U
3
∞
C Þ f oscillation frequency, Hz
H k
0
Re
,
Ma
C y plunge amplitude and plunge position of airfoil, m reduced frequency, 2 π fC / U
∞
Mach number
Reynolds number based on the chord length, ρ U
∞
C / μ
Reynolds number based on the airfoil thickness, ρ U
∞
α C / μ
Strouhal number, 2 f H
0
= U
∞ averaged pressure differences in the streamwise direction,
R
∂ Ω pdS /FrontalArea thickness ratio (i.e. the ratio between the airfoil thickness and the chord length) propulsive efficiency, ⟨ C
T
⟩ = ⟨ C pow
⟩ pitch amplitude and pitch angle of airfoil, deg phase lag between pitching and plunging motions, deg
167
and book chapters ( Ho et al., 2003 ;
Platzer et al., 2008 ; Rozhdestvensky and Ryzhov, 2003
;
,
;
Triantaflyllou et al., 2004 ; Wang, 2005
) have systematically summarized the progress from different aspects. However, due to the intrinsic complexity of the unsteady aerodynamics, there are still many open questions to be answered. One such issue is the synthetic effects of the wing thickness and kinematics on the thrust production and propulsive efficiency during the flapping flight. As is widely recognized, natural flyers, e.g. birds and insects, maneuver their thin wings (i.e., wing thickness is only a few percent of the chord length) via the “ pitching and rolling ” motion. However, previous studies seldom investigated the synergy of thin airfoils and sophisticated flapping kinematics but were limited to thick airfoils (airfoil thickness larger than 10% of chord length) and relatively simple motions (i.e. pitching or plunging). These previous efforts are briefly reviewed as follows.
The first explanation for the thrust generation with oscillating airfoils was given by
and
independently based on the inviscid assumption and the effective angle of attack (AOA) concept. This was experimentally confirmed by
through mounting a stationary wing in an oscillating flow. After that,
showed a thought-provoking way to explain the thrust or drag production by checking momentum surfeit or deficit in the wake based on the wake vortices orientation and location. This has gradually become a qualitative principle to judge whether an oscillating mechanism generates thrust or drag. As the aerodynamic performance of the oscillating foils/ wings is closely tied to the vortical structures, the vortical structure analysis becomes an effective tool to unveil the key mechanism in flapping-wing aerodynamics. With different flow visualization techniques,
,
,
,
,
,
and
or pitching foils. In parallel, by using various numerical simulation or theoretical analysis methods, the evolution of vortex fields around plunging or pitching foils and the corresponding propulsion features (i.e. thrust production and propulsive efficiency) have been studied by many researchers (
;
;
simple airfoil kinematics, namely pure plunging or pitching motions, and relatively large airfoil thickness.
As reported by many researchers, a combined pitching and plunging motion is a more elaborate model of the flapping wing kinematics. Also, there is a growing awareness in this community that the airfoil thickness can affect thrust generation and propulsive efficiency. Based on extensive numerical study,
argued that the combined pitching and plunging motion could adjust the effective AOA of the airfoil, enhancing the thrust generation or power extraction performance.
experimentally examined the propulsive performance of a NACA0012 airfoil with a combined pitching and plunging motion, and gave the parameters for optimal thrust production.
,
Read et al. (2003) , Schouveiler et al. (2005)
and
confirmed that better propulsive performance was achieved when the combined pitching and plunging motion with suitable phase lag was adopted.
The studies in airfoil thickness have shown large discrepancy in its effects on aerodynamic performance.
compared the performances of several plunging airfoils with different thickness at Reynolds number 150, and concluded that airfoils with larger thickness could generate greater thrust than the airfoils with small thickness. Based on the inviscid simulation from an unsteady panel code,
found that thickness had a negligible effect on the propulsive efficiency.
found that thickness ratio was a crucial parameter for thrust production and their results indicated that
168 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 there existed an optimal thickness ratio for the thrust generation at Reynolds number 185.
systematically studied the thickness effects on the propulsive performances of flapping airfoils at different Reynolds numbers ranging from 200 to 2 10
6
. They found that at low Reynolds numbers, thin airfoils outperformed thick airfoils, while at high Reynolds numbers there existed an optimal thickness for the thrust production and propulsive efficiency. Recently,
found that at low
Reynolds numbers, viscous force could even generate thrust on plunging thin airfoils (e.g. NACA0004 and NACA0006) due to the unsteady circumnavigation of leading edge vortices (LEVs).
Based on the literature review, it is clear that studies on the synergy of the airfoil kinematics and thickness on flappingwing propulsion are limited, although this can be crucial for the MAV design. As is known, flapping-wing propulsion is closely related to the unsteady vortex dynamics. High order computational fluid dynamics (CFD) methods appear to be more accurate than their low order counterparts for simulations of vortex-dominated flow (
Wang, 2007 ). Some recent work on
the numerical simulations of bio-inspired flow with high-order methods (e.g.
Visbal, 2009 ; Visbal et al., 2009
;
;
the potential of the high-order numerical simulation in unsteady flapping-wing aerodynamics. The simulations in the present paper are carried out using the high-order accurate dynamic grid Navier – Stokes flow solver.
Although all nature flyers use complex 3D wing kinematics in the flapping flight, some intrinsic principles of unsteady aerodynamics do fall into the 2D frameworks and understanding these basic physics can help make progress on the study of the 3D flapping wing aerodynamics. Therefore, this study is only restricted to 2D flapping airfoil simulations. The remainder of this paper is organized as follows. In
Section 2 , the dynamic unstructured grid based spectral difference (SD) method is
first introduced. The simulation parameters are presented at the end of this section. Numerical results are displayed and discussed in
detail, and the aerodynamic performance dependency on the reduced frequency and Strouhal number is also discussed there.
briefly concludes this work.
2. Numerical method and simulation parameters
2.1. Governing equations
Numerical simulations are carried out with an unsteady compressible Navier – Stokes solver using dynamic unstructured grid based high-order SD method developed in
and
). The 2D unsteady compressible Navier –
Stokes equation in conservation form reads,
∂ Q
∂ t
þ
∂ F
∂ x
þ
∂ G
∂ y
¼ 0 : ð 1
Herein, Q ¼ ( ρ , ρ u , ρ v , E ) T flux vectors, i.e., F ¼ F i are the conservative variables, and F ; G are the total fluxes including both the inviscid and viscous
− F v
8
>
ρ u
9
>
F i ¼
>
: p þ ρ u 2
ρ u E uv
þ p Þ
>
;
; G i and G ¼ G i
8
>
ρ v
¼
>
: p
ρ uv
þ v E
ρ
þ
− G v
, which takes the following form:
9 v 2 p Þ
>
=
;
>
;
8
>
0
9
>
8
>
0
9
>
F v ¼
>
: u τ xx
þ v τ
τ xx
τ yx yx
þ
μ
C p
Pr
T x
>
;
; G v ¼
>
: u τ xy
þ v τ
τ xy
τ yy yy
þ
μ
C p
Pr
T y
>
;
ð 2 Þ
Þ
In Eq.
, ρ is the fluid density, u and v are the Cartesian velocity components, p is the pressure, and E is the total internal energy, μ is dynamic viscosity, C p is the specific heat at constant pressure, Pr is the Prandtl number, and T is the temperature. The stress tensors in Eq.
take the following form
τ xx
¼ 2 μ u x
− u x
þ v y
3
; τ yy
¼ 2 μ v y
− u x
þ v
3 y ; τ xy
¼ τ yx
¼ μ ð v x
þ u y
Þ ð 3 Þ
On assuming that the perfect gas law is obeyed, the pressure is related to the total internal energy by E ¼ ð p = ð γ − 1 ÞÞ þ
ð 1 = 2 Þ ρ ð u 2 þ v 2 Þ with constant γ , which closes the solution system.
To achieve an efficient implementation, a time-dependent coordinate transformation from the physical domain ( t , x , y ) to the computational domain( τ , ξ , η ), as shown in
(a), is applied to Eq.
(1) . The coordinate transformation is a mapping
between computational domain and the physical domain, which takes the following form, t ¼ τ ; x ¼ x ð τ ; ξ ; η Þ ; y ¼ y ð τ ; ξ ; η Þ : ð 4 Þ
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 169
Fig. 1.
(a) Transformation from a moving physical domain to a fixed computational domain; (b) Distribution of solution points (as denoted by circles) and flux points (as denoted by squares) in a standard quadrilateral element for a third-order SD scheme.
herein, ( ξ , η ) ∈ [ − 1,1] 2 , are the local coordinates in the computational domain. Readers are referred to the book by
for more details of the coordinate transformation.
Then the transformed equation of Eq.
takes the following form
∂
~
∂ τ
þ
∂
~
∂ x
þ
∂
~
∂ y
¼ 0 ; ð 5 Þ where
8
~
¼ j J j Q
~
¼ j J jð Q ξ t
~ ¼ j J jð Q η t
þ F ξ x
þ F η x
þ G ξ y
Þ
þ G η y
Þ
: ð 6 Þ
In the transformation shown above, the Jacobian matrix J takes the following form:
J ¼
∂
∂ ð
ð x
ξ ;
; y
η ;
; t
τ Þ
Þ
¼
0 x y
0
ξ
ξ x y
0
η
η x y
1
τ
τ
1
C
:
Note that the grid velocity v g x
τ
; y
τ
Þ is related with ( ξ t
, η t
) by
ξ t
: η t
¼ − v g
⋅∇ ξ
¼ − v g
⋅∇ η
: ð
ð 7
8 Þ
Þ
2.2. Space discretization
The SD method is used for the space discretization. In the SD method, two sets of points are given, namely the solution and flux points, as shown in
(b) for a 2D quadrilateral element. Unknown solutions or degrees of freedom (DOFs) are defined at the solution points (SPs), and fluxes are calculated on flux points (FPs). In the present study, the solution points are chosen as the Chebyshev – Gauss quadrature points. For a P N − 1 reconstruction, N solution points are needed in 1D and are specified as
ξ s
¼ − cos
2 s − 1
2 N
π ; s ¼ 1 ; 2 ; ⋯ ; N : ð 9 Þ
It has been proved in
) that the adoption of the Legendre – Gauss quadrature points as the flux points can ensure the stability of the SD method. Therefore, the flux points are selected to be the Legendre – Gauss points with end points − 1 and 1. These points are denoted as ξ f
; f ¼ 0 ; 1 ; … ; N .
Two sets of Lagrange polynomials based on the SPs and FPs respectively can be specified as follows.
SPs based Lagrange polynomial:
L s ; i
N
ð ξ Þ ¼ ∏ s ¼ 1 ; s ≠ i
ξ − ξ s
ξ i
− ξ s
; i ¼ 1 ; 2 ; … ; N : ð 10 Þ
FPs based Lagrange polynomial:
L f
; i
N
ð ξ Þ ¼ ∏ f
¼
0
; f
≠ i
ξ − ξ f
ξ i
− ξ f
; i ¼ 0 ; 1 ; … ; N : ð 11 Þ
170 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
The reconstruction of the SD method is stated briefly as follows. First of all, the inviscid fluxes are reconstructed. Note that the fluxes related to the grid movement are incorporated into the inviscid fluxes, e.g., F i
¼ j J jð Q ξ
τ
þ F i ξ x
þ G i ξ y
Þ . The conservative variables Q f on the flux points are interpolated from the conservative variables Q s on the solution points via a tensor production of the 1D Lagrange polynomials Eq.
(11) , which takes the following form
Q f
N
ð ξ ; η Þ ¼ ∑ j ¼ 1
N
∑ i ¼ 1
Q s
ð ξ i
; η j
Þ L s
; i
ð ξ Þ L s
; j
ð η Þ : ð 12 Þ
Then the fluxes can be reconstructed at the flux points using Q f
. Note that this reconstruction is continuous within a standard element, but discontinuous on the cell interfaces. Therefore, a Riemann flux or common flux needs to be specified on the interface to ensure conservation. Since the flow regime for flapping flight is almost incompressible and the present governing equations are compressible N-S equations, the Riemann solver should retain good performance at low Mach numbers. The AUSM
+
to behave well at low Mach numbers. The procedures to reconstruct the common fluxes from the AUSM
+
-up Riemann solver are stated as follows.
Denote the face normal of arbitrary interface by n m
1 = 2
¼ a
1 = 2
M
1 = 2
(
ρ
ρ
L
R if M
1
=
2
4 0 otherwise
;
, then the interface mass flow rate
_
1 = 2 reads
ð 13 Þ where the subscript ‘ 1/2 ’ stands for the interface, a and M are speed of sound and the Mach number respectively. Note that the grid velocity has been included in the interface Mach number M . The numerical normal fluxes F i and i can then be specified as
>
< i
¼ _
1 = 2
>
: i
¼ _
1 = 2
(
(
ψ
L
ψ
R
ψ
ψ
L
R if
_
1 = 2
4 otherwise if
_
1 = 2
4 otherwise
0
0
þ P
þ P
1 = 2
1 = 2
!
!
j J jj ∇ ξ j sign ð n j J jj ∇ η j sign ð n
U ∇ ξ Þ ;
U ∇ η Þ ;
ð 14 Þ where ψ ¼ ð 1 ; u ; v ; ð E þ p Þ = ρ Þ T ; P ¼ 0 ; pn x
; pn y
; pv g ; n
T
, with n x
, n y and v g , n y direction and the grid velocity in the face normal direction, respectively.
specifying the face normal components in x and
After this, the derivatives of the inviscid fluxes are calculated on the solution points using the following formulas:
∂ i
∂ ξ
N
ð ξ ; η Þ ¼ ∑ j ¼ 1
N
∑ i ¼ 0 i
∂
∂ η
N
ð ξ ; η Þ ¼ ∑ j ¼ 0
N
∑ i ¼ 1 i i
ð ξ i
ð ξ i
; η j
; η j
Þ L
0 f ; i
Þ L s
; i
ð ξ Þ L
ð ξ Þ L s
; j
0 f ; j
ð
ð
η
η
Þ
Þ
;
: ð 15
Since the viscous fluxes are functions of both the conservative variables Q and their derivatives ∇ Q , slightly more involved reconstruction procedures are needed. In the present study, the approach proposed in
known as ‘ BR1 ’ , is adopted. The implementation of this approach in SD is briefly introduced as follows.
Let R ¼ ∇ Q components of
. On transforming this formula from the physical domain to the computational domain, we obtain the
R in the conservation form as
R x ¼
R y ¼
1 j J j
1 j J j
∂ j J j Q ξ x
∂ ξ
∂ j J j Q ξ y
þ
∂ ξ
þ
∂ j J j Q η x
∂ η
∂ j J j Q η y
∂ η
;
: ð 16 Þ
Þ then the derivatives in Eq.
on the solution points can be calculated from the conservative variables Q f following the procedure as shown in Eq.
(15) . Note that the common conservative variables
Q com on the flux points on element interfaces are used in the derivative calculation. In BR1, Q com is the average of the left and right solutions on the interface,
Q com ¼
Q
L þ Q
R
2
: ð 17
After this, the gradient of Q is then interpolated back to flux points following the procedure as shown in Eq.
and the viscous fluxes can then be calculated on flux points. Again, the gradient of Q from the aforementioned reconstruction is generally discontinuous on the element interface, and BR1 is used to provide a common gradient ∇ Q com interface as follows, on the element
Þ
∇ Q com ¼
∇ Q
L þ ∇ Q
R
2
: ð 18
Thus the viscous fluxes v and v on flux points are uniquely specified in a local cell, and the flux derivatives on solution points can then be calculated via the approach as shown in Eq.
.
Þ
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Fig. 2.
Sketch of the kinematics of the NACA0012 airfoil.
171
Fig. 3.
(a) Convergence history of the energy residual for the steady solution of the viscous flow over a stationary NACA0012 airfoil with implicit (LU-SGS) time integration at Re ¼ 5000, Ma
∞
¼ 0.05 and zero AOA; (b) pressure coefficient contours for the converged steady flow; (c) Mach number contours for the converged steady flow.
Once all flux derivatives are available, the DOFs can be updated with either explicit or implicit time integrations.
2.3. Simulation parameters
In this study, the kinematics of the airfoil is specified as follows.
Plunging motion : y ¼ H
0 sin ð 2 π f t Þ ;
Pitching motion : θ ¼ θ
0 sin ð 2 π f t þ ϕ Þ : ð 19 Þ
Herein, y is the plunge position of the airfoil, H
0 is the plunge amplitude, f is the oscillation frequency, θ
0 is the pitch amplitude, and ϕ is the phase lag between pitching and plunging motions. In view of the optimal thrust generation conditions suggested by
, the pivot point for the pitching motion is located one-third chord length away from the leading edge, θ
0 is set as 20 1 and ϕ is fixed at 75 1 . Sketch of the airfoil kinematics is displayed in
. Grid deformation strategy associated with the kinematics of the airfoil can be found in the paper by
.
The aerodynamic parameters are specified as follows. The Reynolds number (Re
C
) based on the airfoil chord length C and the free stream velocity U
∞ is set as 1200 for most of simulations. Meanwhile, the airfoil thickness based Reynolds number
(Re
T
) is also used to characterize different flow features. All these Reynolds numbers fall well within the insect flight regime.
According to the optimal Strouhal number suggested by
Triantaflyllou et al. (1993) , the Strouhal number (St
¼ 2 fH based on the plunge amplitude H
0 is chosen as 0.3 or 0.45. The reduced frequency ( k ¼ 2 π fC / U
∞
0
/ U
∞
)
) based on the airfoil chord length C
∞ is crucial for the aerodynamic performance of a flapping wing if a compressible solver is adopted for the simulation. In the present study, Ma
∞ is set to be 0.05, which is in accordance with the value specified in
).
3. Results and discussion
The performance of the developed solver for low Mach number flow is first tested by computing the steady viscous flow over a NACA0012 airfoil at Re ¼ 5000, Ma
∞
¼ 0.05 and zero AOA with a 3rd order accurate SD scheme and an implicit LU-SGS time integration (
) on a coarse mesh. The residual convergence history, pressure coefficient ( C
P
) contour, and the Mach number contour are displayed in
(a) – (c) respectively. It is obvious that the Navier – Stokes solver works well at low Mach number as no oscillation of the flow field is observed.
Then h -refinement (grid refinement) and p -enrichment (polynomial degree enrichment) studies are carried out to determine the suitable grid and scheme accuracy, which ensure that the aerodynamic force and vortical structures around the
172 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 airfoil are grid and scheme accuracy independent for the simulations of the oscillating airfoils. The case of an oscillating
NACA0006 airfoil with the combined pitching and plunging motions at k ¼ 3.5, St ¼ 0.3 is used for the hp -refinement study.
Two sets of grids in the vicinity of the airfoil are displayed in
Fig. 4 (a) and (b). For the coarse mesh, 100 grid points are placed
on the airfoil surface and 35 grid points are assigned in the radial direction from the airfoil. The number of grid points for the fine mesh in both directions is 1.5 times that for the coarse mesh. An overview of the computational grid is shown in
(c) with the far field boundary imposed at about 10 chord length away from the airfoil surface. Time histories of the thrust coefficients ( C
T
) from the hp -refinement study are shown in
(d). It is found that time histories of thrust coefficients for the
3rd order scheme with both coarse and fine grids and the 4th order scheme with the coarse grid agree well with each other, while that for the 2nd order scheme with the coarse mesh shows marked deviation. Based on these results, the 3rd order accurate SD scheme with a coarse mesh is chosen to conduct the simulations. Moreover, the non-dimensional time step
( Δ t ¼ Δ tU
∞
= C ) used in the simulations is set as 1.2
10
− 5 , which is determined from a time refinement study on the case with the chosen accuracy and grid configuration. More validation of the present flow solver can be found in
).
Next it is firstly confirmed that the aerodynamic performance of airfoils with different thickness can vary dramatically under the same kinematics, and that the combined pitching and plunging motion with suitable phase lag is conducive for enhancing the aerodynamic performance. Then we examine the airfoil thickness effects on flapping airfoil propulsion with the combined motion at different reduced frequencies, the Strouhal numbers and the Reynolds numbers and unveil the hidden flow physics via vortex dynamics analysis.
3.1. Synthetic effects of airfoil thickness and kinematics
Three kinds of oscillation motions, namely the pure plunging motion, the pure pitching motion and the combined pitching and plunging motion, are studied in this section. The time histories of the thrust coefficients for NACA0006 and
NACA0030 with different oscillation motions at k ¼ 1 ; St ¼ 0.3 are displayed in
.
By comparing the time histories of the thrust coefficients in
and time averaged thrust coefficients in
observations can be summarized as follows.
Fig. 4.
hp refinement study for the flow over the NACA0006 airfoil with the combined pitching and plunging motion at k ¼ 3 : 5 ; St ¼ 0.3. (a) The coarse grid and (b) fine grid used in the grid refinement study. (c) Overview of the computational grid. (d) Time histories of the thrust coefficients from the hp refinement study.
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 173
Fig. 5.
Time histories of the thrust coefficients for (a) the NACA0006 and (b) NACA0030 airfoils with different motions at k ¼ 1, St ¼ 0.3.
Table 1
Time averaged thrust coefficients for the NACA0006 and NACA0030 airfoils with different motions at k ¼ 1 ; St ¼ 0.3.
⟨ C
T
⟩ denotes the total time averaged thrust coefficient; ⟨ C
T _ P
⟩ denotes the time averaged thrust coefficient calculated from the pressure force; ⟨ C
T _ v
⟩ denotes the time averaged thrust coefficient calculated from the viscous force.
⟨
⟨ C
⟨ C
C
T
⟩
T _ P
⟩
T _ v
⟩
NACA0006
Plu.
0.0085
0.032
− 0.0235
Pit.
− 0.240
− 0.198
− 0.042
Com.
0.418
0.453
− 0.035
NACA0030
Plu.
−
0.020
0.056
0.036
Pit.
− 0.399
− 0.327
− 0.075
Com.
0.105
0.166
− 0.061
First, it is found that for both airfoils, (1) although the combined motion is a linear superposition of the plunging and pitching motion, the aerodynamic force generated from the combined motion is not the linear superposition of the aerodynamic force generated from the plunging and pitching motion; (2) only drag is generated when pure pitching motion is used; (3) the combined motion outperforms pure plunging motion.
Second, NACA0006 can generate larger thrust than NACA0030 for the combined pitching and plunging motion. However, for pure plunging motion, NACA0030 outperforms NACA0006. The pitching motion did not help much in enhancing the aerodynamic performance of the oscillating NACA0030 airfoil, but it significantly increases the thrust generation of
NACA0006 as when pure plunging motion is conducted, the NACA0006 airfoil almost does not generate thrust.
The explanation for the common features displayed by both airfoils is given as follows. The nonlinear superposition of aerodynamic forces is attributed to the nonlinear effects of the unsteady vortical flow. A visual proof of this nonlinearity is that the vortical structures from the combined motion are not linear superposition of those from pure plunging and pitching motion as clearly displayed in
and
. According to the definition of the Strouhal number for the pitching motion as shown in
Yu et al. (2012 ), its value in the present case is St
pit
¼ 2 f (2 C sin θ
0
/3)/ U
∞
¼ 0.07. This value is well below the critical
Strouhal number for thrust production and it indicates that pure pitching motion is not powerful enough to generate thrust.
Instead, the plunging motion plays a major role in thrust generation as the plunging amplitude based Strouhal number is 0.3
here. Note that thrust generation via the pitching motion has been previously studied by
,
Platzer (1997) , Ramamurti and Sandberg (2001)
and
, just to name a few. There, similar propulsion performance from the pitching motion has been reported. Before explaining why more thrust is generated from the combined motion than pure plunging one, we recognize that the superior propulsion performance of the combined motion have been studied previously from different aspects, such as the effects of the phase lag between plunge and pitch, the effects of AOA and the effects of reduced frequency and Strouhal number by many researchers (e.g.
;
;
Ramamurti and Sandberg, 2001 ; Read et al., 2003
;
). It is reported from these work that aerodynamic parameters leading to good performance of the combined motion are generally associated with the favorable dynamic vortex configuration around the airfoils. Here, a comprehensive explanation for the superior performance of the combined motion is given by relating the aerodynamic force with kinematics and vorticity fields. First, we introduce the concept of ‘ frontal area ’ , which is defined as the projection of the airfoil in the free stream direction as shown in
Fig. 2 . Obviously, when the airfoil is parallel to the free stream, the frontal
area is actually the airfoil thickness. From
we find that the drastic change in the thrust coefficients between the combined motion and pure plunging motion is mainly due to the change of the pressure force contribution on the thrust.
Recall that this contribution is determined both by pressure difference in the free-stream direction and frontal area. From the vorticity and pressure fields with different motions at different phases as displayed in
and
(NACA0006) and
174 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Fig. 6.
Vorticity fields for the NACA0006 airfoil with different motions at k ¼ 1 ; St ¼ 0.3. (a) – (d) combined pitching and plunging motion at phases
0 ; π = 2 ; π ; 3 π = 2 respectively; (e) – (h) pitching motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively; (i) – (l) plunging motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
and
(NACA0030), it is clear that when a suitable pitching motion is added to the plunging motion, the frontal area can be effectively adjusted to enhance the thrust generation from pressure force. For example, at phases 0 and π , the low pressure region induced by LEVs is attached on the windward side of both airfoils. At the same phases, the frontal area of both airfoils almost reaches its maximum value. Thus, the performance in terms of thrust generation is significantly improved with the combined flapping motion.
Now we explain how airfoil thickness affects the aerodynamic performance with different kinematics. The inferior performance of the plunging NACA0006 airfoil is attributed to the small frontal area, which is also the airfoil thickness in this case. Specifically, it is observed from
– (l) and
(i) – (l) that at phases 0 and π , which are close to the thrust generation peaks, blunt leading edge of NACA0030 takes full advantage of the low pressure region induced by the leading edge vortices. From
it is clear that the thrust peak for the plunging NACA0030 airfoil is about four times of that for
NACA0006. It is also found from
that the drag peak for the plunging NACA0030 airfoil is only about twice of that for
NACA0006. This explains why NACA0030 outperforms NACA0006 for pure plunging motion. These results also agree with the observations by
that leading-edge effects are vital in determining aerodynamic forces.
The reason that the pitching motion almost does not enhance the thrust generation of the oscillating NACA0030 airfoil is given as follows. On examining
Fig. 8 (a) and (c), it is observed that although the pitching motion helps enlarge the frontal
area of the NACA0030 airfoil at phases 0 and π , i.e.
t / T ¼ 3 and 3.5 in
(b), the LEVs is still located in the rearward part of the airfoil. This vortex configuration does not contribute much to the thrust generation. Moreover, at phases π /2 and 3 π /2, i.e.
t / T ¼ 3.25 and 3.75 in
(b), the NACA30 airfoil with the combined motion experiences larger drag peak than that with the plunging motion. This is due to the interaction of the propagating LEVs and the shed trailing edge vortices (TEVs) during the oscillation stroke. Specifically, for the combined motion, parts of the TEVs are trapped on the trailing edge as shown in
(b) and (d), inducing a low pressure region there as shown in
Fig. 9 (b) and (d). However, for the plunging motion, the TEVs
are not trapped on the trailing edge, as displayed in
(j) and (l). Thus there does not exist a low pressure region in the rearward part of the airfoil.
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 175
Fig. 7.
Pressure fields for the NACA0006 airfoil with different motions at k ¼ 1 ; St ¼ 0.3. (a) – (d) combined pitching and plunging motion at phases
0 ; π = 2 ; π ; 3 π = 2 respectively; (e) – (h) pitching motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively; (i) – (l) plunging motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
The better performance of the NACA0006 airfoil with the combined motion is attributed to the LEVs dynamics as well.
From
and
that the low pressure region is located on the forward part of the airfoil at phases 0 and π , and no low pressure region is located on the rearward part of the airfoil at phases π /2 and 3 π /2. From
Fig. 5 (a), it is found that in contrast to NACA0030, the
NACA0006 airfoil with the combined motion does not experience larger drag peak than that with the plunging motion. This is due to that no TEVs are trapped on the trailing edge for both the combined and plunging motions.
3.2. Thickness effects at different reduced frequencies and Strouhal numbers
As aforementioned, a suitable combined motion can enhance the aerodynamic performances of the oscillating airfoil. In this section, the kinematics of the airfoils is chosen as a combined motion with the pitching motion leading the plunging motion by 75 1 . Thickness effects at different reduced frequencies for a given Strouhal number are compared and discussed at first. The evolution of LEVs and the interaction between LEVs and TEVs is confirmed to be vital for different performance of airfoils with different thickness. The physics behind these phenomena is unveiled by analyzing the vorticity fields and the corresponding induced pressure distribution. Then similar analysis strategy is adopted to reveal thickness effects at different
Strouhal numbers for the same reduced frequency.
3.2.1. Thickness effects at different reduced frequencies
The flow fields, namely the spanwise vorticity fields and the corresponding pressure fields of NACA0020 at k ¼ 1 and
St ¼ 0.3 are displayed in
. Note that the flow fields of NACA0012 are similar to those of NACA0006, and are not shown here. By comparing the vorticity fields of NACA0006 (
(a) –
(a) – (d)) and NACA0030 (
(a) – (d)), it is observed that as the airfoil becomes thicker, at the end of the oscillation strokes, i.e. at phases π /2 and 3 π /2, larger parts
176 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Fig. 8.
Vorticity fields for the NACA0030 airfoil with different motions at k ¼ 1 ; St ¼ 0.3. (a) – (d) combined pitching and plunging motion at phases
0 ; π = 2 ; π ; 3 π = 2 respectively; (e) – (h) pitching motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively; (i) – (l) plunging motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
of the shed TEVs will be trapped on the trailing edge of the airfoil, inducing an low pressure region on the rearward part of the airfoil. This process results in larger drag peaks at phases π /2 and 3 π /2 on thicker airfoils as shown in
(a). Moreover, it is also observed from the vorticity fields that as the airfoil thickness increases, the LEVs will be pushed further towards the trailing edge of the airfoil. This degrades the positive contribution of the pitching motion on thrust generation of the combined motion.
The pressure coefficient distribution on the airfoil surfaces at different phases are displayed in
Fig. 12 . From this figure it is clear
that the pressure distributions on the NACA0006 and NACA0012 airfoils are similar, while NACA0020 and NAC0030 show different trends. At phases 0 and π , it is found that a low pressure region with large suction peaks spans the forward part of the NACA0006 or
NACA0012 airfoil, while for thick airfoils the low pressure regions with small suction pressure spread out almost on the entire surface. As aforementioned, at phases 0 and π , the pressure distribution features of NACA0006 and NACA0012 are conducive for the thrust production, while NACA0020 and NACA0030 cannot take full advantage of the low pressure region. At phases π /2 and 3 π /2, it is observed that on the suction surfaces, thicker airfoils experience larger suction pressure, while on the pressure surfaces, thick airfoils experience excessive pressure on the forward part and wide-spread suction pressure on the rearward part. All these features dramatically increase the drag on thick airfoils at phases π /2 and 3 π /2.
Similar trends can be concluded for the oscillating airfoils with the combined motion at k ¼ 3.5 and St ¼ 0.3. However, there exist some distinct vortical features for these cases compared with the cases at k ¼ 1 and St ¼ 0.3. From the flow fields shown in
and
, it is found that for thin airfoils (e.g. NACA0006), the wake vortices formed during the upstroke or downstroke consist of two vortices of the same sign. One of the vortices is from LEVs while the other is from TEVs. For thick airfoils (e.g. NACA0030), only one vortex is shed during the upstroke or downstroke. However at k ¼ 1 and St ¼ 0.3, for both thin and thick airfoils, the wake vortices shed during the upstroke or downstroke consist of a chain of broken vortices. It is also observed that at k ¼ 3.5 and St ¼ 0.3, the LEVs formed at the beginning of a certain cycle are shed off from the airfoil at the end of the cycle. However at k ¼ 1 and St ¼ 0.3, the LEVs formed at the beginning of a certain stroke are shed off from the
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 177
Fig. 9.
Pressure fields for the NACA0030 airfoil with different motions at k ¼ 1 ; St ¼ 0.3. (a) – (d) combined pitching and plunging motion at phases
0 ; π = 2 ; π ; 3 π = 2 respectively; (e) – (h) pitching motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively; (i) – (l) plunging motion at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
airfoil at the end of the stroke. Note that one stroke equals to half the oscillating cycle. This indicates that at k ¼ 3.5 and
St ¼ 0.3, two LEVs on different sides of the airfoil coexist during the oscillation cycle. The configuration of the coexisting LEVs is not good for thrust production as the LEVs on the rearward part of the airfoil can induce a low pressure region there, which will hinder the thrust production as shown in
and
(e) and (g). Furthermore, it is observed from
and
pressure distribution further degrades the thrust production of the oscillating NACA0030 airfoil.
All these observations based on vortical structure analysis can be confirmed by the aerodynamic force results. From the surface pressure distribution at k ¼ 3.5 and St ¼ 0.3 as displayed in
Fig. 15 , it is found that there exist distinct differences in
pressure distribution on the forward part of airfoils with different thicknesses at phases 0 and π . But at phases π /2 and 3 π /2, the pressure distribution is similar for all airfoils. This suggests that larger thrust peak will appear on thin airfoils while the drag peaks will not be of much discrepancy, which is clearly shown from the time histories of thrust coefficients as shown in
(b).
Based on the above discussions, it can be concluded that the differences in aerodynamic features between the cases at k ¼ 1 and
3.5 are inherently related to the dynamics of LEVs.
The propulsive efficiency ( η pow
) and time averaged thrust coefficients ( ⟨ C
T
⟩ ) for several NACA 4-digit airfoils at k ¼ 1 and
3.5, St ¼ 0.3 are displayed in
. Following the work in
), the propulsive efficiency is defined as
η pow
¼
⟨ C
T
⟩
⟨ C pow
⟩
; ð 20 Þ where ⟨ C pow
⟩ is the time averaged power coefficient. From
Fig. 16 , it is found that the variation trends of the time-averaged
thrust coefficient and propulsive efficiency versus the airfoil thickness are similar at k ¼ 1 and 3.5. Thin airfoils perform well on thrust generation and can achieve relatively high propulsive efficiency at both k ¼ 1 and 3.5, St ¼ 0.3. Thick airfoils are not beneficial for either thrust production or propulsive efficiency with these parameters. It is also found that for the same
178 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Fig. 10.
Vorticity and pressure fields for the NACA0020 airfoil with the combined motion at k ¼ 1 ; St ¼ 0.3. (a) – (d) vorticity fields at phases 0 ; π = 2 ; π ; 3 π = 2 respectively; (e) – (h) pressure fields at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
Fig. 11.
Time histories of the thrust coefficients for a series of NACA 4-digit airfoils with the combined pitching and plunging motion at (a) k ¼ 1 ;
St ¼ 0 : 3 and (b) k ¼ 3 : 5 ; St ¼ 0.3.
airfoil, both the time-averaged thrust coefficient and propulsive efficiency are larger at lower reduced frequency than that at higher reduced frequency.
3.2.2. Thickness effects at different Strouhal numbers
In this section, the Strouhal number is increased from 0.3 to 0.45. The vorticity fields around the NACA0006 airfoil at k ¼ 1 and 3.5, St ¼ 0.45 are displayed in
and
18 , respectively. Comparing the vorticity fields with those shown in
and
– (d), it is found that as the Strouhal number increases, the vortical structures of the flow field do not change much.
However, the vorticity strength at St ¼ 0.45 greatly increases compared with that at St ¼ 0.3. Similar conclusions hold true for other airfoils with different thickness. These observations can be explained as follows. First, the configuration of vortices around the airfoil depends on two factors, namely the generation of LEVs and the relative position between LEVs and TEVs.
From the vorticity fields (e.g.,
and
), LEVs are generated during the oscillation cycle due to the change of the effective AOA. Its definition for the combined pitching and plunging motion is given as
α ef f
¼ arctan
U
∞
− θ ; ð 21 Þ
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 179
Fig. 12.
Surface pressure coefficienct distribution for a series of NACA 4-digit airfoils with the combined motion at k ¼ 1 ; St ¼ 0.3. (a) phases: 0 and π ;
(b) phases: π /2 and 3 π /2.
Fig. 13.
Vorticity and pressure fields for the NACA0006 airfoil with the combined motion at k ¼ 3 : 5 ; k ¼ 3 : 5 ; St ¼ 0.3. (a) – (d) vorticity fields at phases
0 ; π = 2 ; π ; 3 π = 2 respectively; (e) – (h) pressure fields at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
assuming that the pitch speed is negligible. The configuration of the vortices (including both LEVs and TEVs) is determined by the propagation of LEVs and the roll-up of TEVs. The temporal evolution of TEVs is mainly controlled by the oscillation frequency f . Specifically, TEVs shed from the airfoil trailing edge at the end of each oscillation stroke. One the other hand, the propagation of LEVs is closely related to the free-stream advection speed with characteristic time scale proportional to C / U
∞
Hence it can be inferred that the configuration of the vortices is mainly controlled by the reduced frequency ( ¼ 2 π fC / U
∞
).
.
This explains the fact that vortical structures of the flow field for a certain airfoil do not change much with Strouhal numbers at the same reduced frequency. The strength of LEVs is closely tied to the effective AOA under the same reduced frequency.
According to the definition of the plunge amplitude based Strouhal number and Eq.
can be rewritten as
α eff
¼ arctan ð π St cos ð 2 π f t ÞÞ − θ
0 sin ð 2 π f t þ ϕ Þ : ð 22 Þ
Considering the given parameters as stated in
, the maximum effective AOA at Strouhal number 0.3 and 0.45
are 25.2
1 and 36.7
1 respectively. This indicates that with the same reduced frequency, the airfoil at larger Strouhal number experiences more rapid AOA change during one oscillation cycle. Then because of the larger shear stress experienced by the airfoil, the vortex strength at St ¼ 0.45 is larger than that at St ¼ 0.3. Note that due to the nonlinearity of vortex dynamics, the interaction between stronger LEVs and TEVs can be different from that between weaker ones. This explains the mild difference of vortical structures discussed above at different Strouhal numbers for the same reduced frequency.
180 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Fig. 14.
Vorticity and pressure fields for the NACA0030 airfoil with the combined motion at k ¼ 3 : 5 ; St ¼ 0.3. (a) – (d) vorticity fields at phases
0 ; π = 2 ; π ; 3 π = 2 respectively; (e) – (h) pressure fields at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
Fig. 15.
Surface pressure coefficienct C
P
(b) phases: π /2 and 3 π /2.
distribution for a series of NACA 4-digit airfoils with the combined motion at k ¼ 3 : 5 ; St ¼ 0.3. (a) phases: 0 and π ;
Time histories of the thrust coefficients for the two sets of cases at k ¼ 1 and 3.5, St ¼ 0.45 are shown in
observations are made from the comparison between
and
. Firstly, at both k ¼ 1 and 3.5, all airfoils experience larger thrust peak at St ¼ 0.45 than at St ¼ 0.3. At k ¼ 3.5, the value of the thrust peak for thick airfoils (e. g. NACA0020 and
NACA0030) with St ¼ 0.45 is about four times of that with St ¼ 0.3; while the value of the thrust peak for thin airfoils (e. g.
NACA0006 and NACA0012) with St ¼ 0.45 is about three times of that with St ¼ 0.3. At k ¼ 1, the value of the thrust peak for all airfoils with St ¼ 0.45 is only about twice of that with St ¼ 0.3. Secondly, at k ¼ 3.5, drag peaks for all airfoils almost keep the same as the Strouhal number increases from 0.3 to 0.45. But at k ¼ 1, the drag peak noticeably increases when the
Strouhal number goes from 0.3 to 0.45. Thirdly, the flow field around the NACA0030 airfoil becomes unsteady at k ¼ 1,
St ¼ 0.45.
These aerodynamic features are closely tied to the evolution of LEVs on the airfoils. Herein, the surface pressure coefficient distribution for the NACA0006 and NACA0030 airfoils at the thrust peak (i.e. phases 0 and π ) as displayed in
are used to explain the diverse thrust production features of airfoils with different thickness at various Strouhal numbers and reduced frequencies. From
(a), it is clear that for NACA0006, as the Strouhal number increases, the
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 181
Fig. 16.
Time averaged thrust coefficients and propulsive efficiencies for a series of NACA 4-digit airfoils with the combined pitching and plunging motion at k ¼ 1 and 3.5, St ¼ 0.3.
Fig. 17.
Vorticity fields for the NACA0006 airfoil with the combined motion at k ¼ 1, St ¼ 0.45. (a) – (d) vorticity fields at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
Fig. 18.
Vorticity fields for the NACA0006 airfoil with the combined motion at k ¼ 3.5, St ¼ 0.45. (a) – (d) vorticity fields at phases 0 ; π = 2 ; π ; 3 π = 2 respectively.
strength of LEVs increases. As a result, low pressure region is induced on the airfoil, which is conducive for the thrust production. It is also found that at k ¼ 3.5, the pressure suction peak drops much near the leading edge of the airfoil at
St ¼ 0.45 compared with that at St ¼ 0.3, while at k ¼ 1, the reduction of the pressure suction peak due to the increase of the
Strouhal number is much smaller than that at k ¼ 3.5. On the other hand, it is obvious that at St ¼ 0.3, the pressure suction peaks at k ¼ 1 and 3.5 are almost the same, while at St ¼ 0.45, the pressure suction peaks at k ¼ 1 and 3.5 show large discrepancy. Similar conclusions can be drawn from
Fig. 20 (b) for the NACA0030 airfoil. These aerodynamic features indicate
that among all studied cases, the kinematics with k ¼ 3.5 and St ¼ 0.45 is more conducive for thrust production.
The propulsive efficiency and time averaged thrust coefficients for several NACA 4-digit airfoils at k ¼ 1 and 3.5, St ¼ 0.45
are displayed in
. Several features are observed by comparing the aerodynamic performance shown in
and
182 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Fig. 19.
Time histories of the thrust coefficients for a series of NACA 4-digit airfoils with the combined pitching and plunging motion at (a) k ¼ 1, St ¼ 0.45
and (b) k ¼ 3.5, St ¼ 0.45.
Fig. 20.
Surface pressure coefficienct distribution for (a) NACA0006 and (b) NACA0030 airfoils with the combined pitching and plunging motion at phases
0 and π and different reduced frequencies and Strouhal numbers.
Fig. 21.
Time averaged thrust coefficients and propulsive efficiencies for a series of NACA 4-digit airfoils with the combined pitching and plunging motion at k ¼ 1 and 3.5, St ¼ 0.45.
First, the time averaged thrust coefficients increase at St ¼ 0.45 for all cases. And the time averaged thrust coefficient at
St ¼ 0 : 45, k ¼ 3.5 exceeds that at St ¼ 0.45, k ¼ 1. This is different from the cases at St ¼ 0.3. There it is found that relatively small reduced frequency is conducive for thrust production. As a matter of fact, the drastic performance change at St ¼ 0.45,
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 183
Fig. 22.
Vorticity fields for the NACA0006 airfoil with the combined motion at Re
T
0 ; π = 2 ; π ; 3 π = 2 respectively.
¼ 144 ; k ¼ 1 and St ¼ 0.3. (a) – (d) vorticity fields at phases k ¼ 3.5 is due to the presence of strong pressure suction from the attached LEVs near the leading edge of the airfoil as shown in
and
. In terms of propulsive efficiency at St ¼ 0 : 45, it is found that at k ¼ 1, the propulsive efficiency for all airfoils drops dramatically compared with that at S ¼ 0.3 t , while at k ¼ 3.5, the NACA0020 airfoil exhibits the maximum propulsive efficiency and the propulsive efficiency of NACA0030 increases noticeably compared with that at St ¼ 0.3, k ¼ 3.5.
As aforementioned, all these aerodynamic features are closely related to the nonlinear vortex dynamics, and will not be further extended here.
According to the results shown in
, it is found that for the present kinematics and dynamic parameters, thin
airfoils perform well on both thrust production and propulsive efficiency at relatively small Strouhal number and reduced frequency. At relatively large Strouhal number and reduced frequency, there exists an optimal thickness range to maintain reasonable thrust production and propulsive efficiency.
3.3. Effects of the Reynolds number
In this section, the effects of the airfoil thickness based Reynolds number (Re
T
) on aerodynamic performance are discussed. It is clear that the relationship between the airfoil thickness based Reynolds number and the chord length based
Reynolds number can be expressed as Re
T
¼ α Re
C
, where α is the thickness ratio. Tests are done with three airfoils of different thickness, namely NACA0006, NACA0012 and NACA0030, at Re
T
¼ 72 ; 144 and 360 respectively. For all simulations presented in this section, the reduced frequency and Strouhal number are fixed at k ¼ 1 and St ¼ 0.3.
The time averaged thrust coefficients and propulsive efficiencies are tabulated in
. There, the corresponding chord length based Reynolds number (Re
C
) is also presented for reference. It is clear that both the time average thrust coefficient and propulsive efficiency increase as Re
T increases for all three types of airfoils. At the same Re
T
, relatively thin airfoils (e.g.
NACA0006 and NACA0012) show good propulsion performance. The vorticity fields for the NACA0006 airfoil at Re
T
¼ 144 and 360 are displayed in
and
23 . Comparing the results with those from
(a) – (d), it is found that vorticity fields do not change much when Re
T varies from 72 to 360. However, by comparing the vorticity fields of the oscillating NACA0030 airfoil at Re
T
¼ 72 and 144, as shown in
and
(a) – (d), dramatic change of the wake type is observed. It is clear that as Re
T increases, the wake gradually transforms from a drag-indicative type to a thrust-indicative
). This change can also be observed from the time histories of the thrust coefficient for the oscillating
NACA0030 at three different airfoil thickness based Reynolds numbers as depicted in
(c). Time histories of the thrust coefficient for the NACA0006 and NACA0012 airfoils are also shown in
greatly affect the thrust production of these two airfoils in the cases studied.
. It is clear that the variation of Re
T does not
4. Conclusions
In summary, a dynamic unstructured grid based high-order spectral difference Navier – Stokes solver is implemented in the present study to carry out high-fidelity simulations on the flow fields over several oscillating NACA 4-digit airfoils. It is demonstrated that the airfoil thickness has large impact on the performance of flapping airfoil propulsion at low Reynolds numbers. Moreover, by analyzing the flow fields with different kinematics, it is clearly shown that the synthetic effects of airfoil thickness and kinematics on aerodynamic performance are closely tied to the evolution of LEVs and the interaction between LEVs and TEVs. The key role of LEVs in thrust production is emphasized here. By adopting suitable kinematics, the low pressure region induced by LEVs can be well adjusted to enhance the thrust production.
By analyzing in detail the vortex dynamics and the corresponding aerodynamic force and propulsive efficiency, the airfoil thickness effects with the combined pitching and plunging motion at different reduced frequencies and Strouhal numbers are revealed. It is found that under suitable effective AOA, the vortical structures are dominated by the reduced frequency.
Since the vorticity strength increases as Strouhal number increases, the evolution of LEVs and the interaction between LEVs and TEVs can be changed. This results in alteration in propulsive performance at different Strouhal numbers. According to
184 M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186
Table 2
Time averaged thrust coefficients and propulsive coefficients for the NACA0006, NACA0012 and NACA0030 airfoils with the combined motion at different
Reynolds numbers.
⟨ C
T
⟩ denotes the total time averaged thrust coefficient, and negative value indicates drag.
η denotes the propulsive efficiency, and is not applicable in the drag production case.
Re
T
NACA0006 NACA0012 NACA0030
72
144
360
Re
C
1200
2400
6000
⟨ C
T
⟩
0.418
0.436
0.442
η (%)
34.2
35.7
36.6
Re
C
600
1200
3000
⟨ C
T
⟩
0.343
0.406
0.444
η (%)
29.0
34.6
38.4
Re
C
240
480
1200
⟨ C
T
⟩
− 0.213
− 0.0976
0.105
η (%)
–
–
10.1%
Fig. 23.
Vorticity fields for the NACA0006 airfoil with the combined motion at Re
T
0 ; π = 2 ; π ; 3 π = 2 respectively.
¼ 360 ; k ¼ 1 and St ¼ 0.3. (a) – (d) vorticity fields at phases
Fig. 24.
Vorticity fields for the NACA0030 airfoil with the combined motion at Re
T respectively.
¼ 72 ; k ¼ 1 and St ¼ 0.3. (a) – (d) vorticity fields at phases 0 ; π = 2 ; π ; 3 π = 2
Fig. 25.
Vorticity fields for the NACA0030 airfoil with the combined motion at Re
T
0 ; π = 2 ; π ; 3 π = 2 respectively.
¼ 144 ; k ¼ 1 and St ¼ 0.3. (a) – (d) vorticity fields at phases the present results, at relatively small Strouhal number ( ¼ 0.3), thin airfoils are conducive for thrust production and propulsive efficiency preserving no matter with a large or small reduced frequency; while at relatively large Strouhal number ( ¼ 0.45), the aerodynamic performance of different airfoils is closely related to the reduced frequency. The effects of
M. Yu et al. / Journal of Fluids and Structures 42 (2013) 166 – 186 185
Fig. 26.
Time histories of the thrust coefficients for a series of osicllating NACA 4-digit airfoils at Re
T
¼ 72, 144 and 360. (a) NACA0006, (b) NACA0012 and
(c) NACA0030.
airfoil thickness based Reynolds number on propulsion performance are also investigated. It is found that variation of the airfoil thickness based Reynolds number does not greatly affect the thrust production of thin airfoils and at the same
Reynolds number, thin airfoils exhibit superior aerodynamic performance.
Note that the maximum propulsive efficiency obtained from the present simulations is about 35%, which might not be optimal. Also, there might exist different optimal kinematic and dynamic ranges for flapping airfoil propulsion with different airfoil thickness. Therefore, future work will be devoted to the optimization of airfoil thickness and kinematics, which can be helpful to further elucidate the underlying flow physics.
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