Effects of wind gusts on a rigid flapping NACA 0012 airfoil at Re = 3000
Marcus Lundberg [email protected]
Degree Project in Engineering Physics, First Cycle, SA104X at KTH Mechanics Supervisors: Luca Brandt & Walter Fornari Examiner: M˚artenOlsson Stockholm, Sweden 2015 Abstract
Insects and small flyers operate at Reynolds numbers ranging from ap- proximately 10 − 105, where viscous forces are important. Due to their small size and weight, they are sensitive to small changes in the free stream during flight, such as wind gusts. First, the aerodynamics of flapping flight is briefly explained. Then the lift, drag and power us- age for a flapping NACA 0012 airfoil is simulated in ANSYS Fluent for different oncoming wind directions. The aim of the report is to un- derstand how the pitching amplitude, the flapping frequency and the plunging amplitude can be adjusted to compensate for oncoming wind gusts. The simulation is modelled as quasi-static since the time-scale of the flapping wings of the insect is much shorter than the time-scale of the wind gusts.
2 Contents
1 Introduction 4 1.1 Objectives ...... 4
2 Theory of flapping wing aerodynamics 5 2.1 Definitions ...... 5 2.1.1 Wing nomenclature ...... 5 2.1.2 Reynolds number, Re ...... 5 2.1.3 Reduced frequency, k ...... 5 2.1.4 Strouhal number, St ...... 6 2.2 Aerodynamic forces ...... 6 2.2.1 Lift ...... 6 2.2.2 Drag ...... 7 2.2.3 Thrust ...... 7 2.2.4 Leading edge vortices (LEVs) ...... 8 2.2.5 Dynamic stall ...... 8 2.3 Wing model ...... 8 2.3.1 Equations for wing motions in 2D ...... 8 2.4 Power considerations ...... 9
3 Numerical simulations 10 3.1 Governing equations and turbulence model ...... 10 3.2 Geometry and mesh parameters ...... 10 3.2.1 Grid characteristics ...... 11 3.2.2 Validation ...... 12 3.3 Simulation model ...... 13 3.3.1 Boundary conditions ...... 15 3.3.2 Calculation of power usage ...... 16 3.3.3 Moving airfoil setup ...... 16 3.4 Results ...... 17 3.4.1 Reference simulation ...... 17 3.4.2 Varying the reduced frequency, k ...... 19 3.4.3 Varying the plunging amplitude, ha ...... 21 3.4.4 Varying the pitching amplitude, αa ...... 23 3.5 Conclusion ...... 25 3.5.1 Wind from below, β > 0◦ ...... 25 3.5.2 Wind from above, β < 0◦ ...... 25 3.5.3 Final remarks ...... 25
4 Appendix A 27 4.1 User Defined Functions ...... 27
3 1 Introduction
The flight of birds, bats and insects has intrigued humans for many centuries. There exists almost a million species of flying insects and over 13000 warm-blooded vertebrate species have the ability to roam the sky. Although great progress has been made in aeronautical technology over the past 100 years, birds’, bats’ and insects’ ability to manoeuvre a body efficiently through air is still very impressive. Studying their flight mechanisms is important in the development of Micro Air Vehicles with flapping wings. By studying the flapping wing flight in animals, we can improve the designs and build smaller, more efficient and more manoeuvrable MAVs. However, a consequence of the small size of insects is that they are very sensitive to disturbances in the environment, such as wind gusts.
1.1 Objectives It is the aim of this report to first explain the basics of flapping wing flight and then study the effects of wind gusts on lift, drag and thrust on insect-sized bodies, and to give a recommendation for how the wings should be aligned to counteract the gust and have optimal performance. The frequencies of typical wind gusts, around O(1) Hz, are low compared to the flapping frequencies of the insects, which are O(101) Hz to O(102) Hz. Therefore, in the time-scale of the flapping wings, the wind gusts can often be modelled as quasi-steady [1].
4 2 Theory of flapping wing aerodynamics
In this section a short theoretical background on flapping wing aerodynamics at low Reynolds numbers is introduced.
2.1 Definitions 2.1.1 Wing nomenclature Figure (1) shows the different parts of an asymmetric airfoil.
Figure 1: The terminology for lift producing airfoils.
The chord line is the imaginary straight line between the leading edge and the trailing edge of the airfoil. The camber line is defined as the curve that lies halfway between the upper and lower surfaces of the airfoil. The chord line and camber line coincide for symmetric airfoils, such as the NACA 0012 airfoil, which will be used in the simulations. The maximum distance between the camber line and the chord line is the maximum camber, which is a measure of the curvature of the airfoil.
2.1.2 Reynolds number, Re The Reynolds number is defined as the ratio between the inertial and viscous forces. It is defined as
U L Re = ref ref (1) ν where Uref is a reference velocity, Lref is a reference length and ν is the kinematic viscosity of the fluid. In flapping wing flight, the reference length is generally taken to be either a mean chord length or the wing length R. The mean chord length is defined as the average chord length in the spanwise direction. In forward flight, the reference velocity Uref is the forward velocity [1].
2.1.3 Reduced frequency, k The ratio between the forward velocity and the flapping velocity is another impor- tant dimensionless parameter used to analyse the aerodynamic performance of a natural flyer [2]. It is expressed using the reduced frequency k,
5 πfc k = , (2) U where f is the flapping frequency, c is the mean chord length and U is the forward velocity U.
2.1.4 Strouhal number, St The Strouhal number St is a dimensionless parameter that describes the wing kine- matics for flying animals [2]. It is well known to govern the vortex dynamics and shedding behaviour for airfoils undergoing pitching and plunging motions. It is defined as the stroke frequency f times a reference length Lref = 2ha, where ha is the plunging amplitude, divided by the forward speed U.
2fh 2kh St = a = a , (3) U πc where St is expressed in the reduced frequency using equation (2). The Strouhal number for natural flyers and swimmers in cruising conditions has been found to be in the range 0.2 < St < 0.4 [2].
2.2 Aerodynamic forces 2.2.1 Lift When an airfoil moves forward through air, the air is deflected and exerts an aero- dynamic force on the airfoil. Lift is defined as the component of this force perpen- dicular to the direction of the oncoming flow, as shown in Figure (2). The airfoil exerts a downward force on the air when it flows past. According to Newton’s third law, the air exerts an equal but opposite force on the airfoil, the lift. The direction of the air flow changes as it passes over the airfoil and curves downward. This change of direction in the air results in a reaction force opposite to the directional change. Reynolds numbers for insects ranges between approximately 10 to 105 [3]. Elling- ton [4] observed, in a comprehensive investigation, that classical, steady state the- ories predicted insufficient forces in flapping flight, to explain the characteristics of insect and bird flight [5]. It has been shown in experiments that the reduced frequency increases as the mass and size of the flying animal decreases, which in- dicates that unsteady effects are important in lift and thrust generation for small flyers [5].
Figure 2: Directions of forces. Lift L, drag D, total aerodynamic force R and angle of attack α
6 2.2.2 Drag Drag is a mechanical force generated by the contact and interaction of a solid body and a fluid. It is a non-conservative force which depends on the speed of the body relative to the speed of the fluid. If the relative speed is zero, there is no drag. The drag force is the component of the aerodynamic force that is opposed to the motion of the object, as shown in Figure (2). It is a type of friction and refers to forces acting in the opposite direction of a body’s motion through a fluid. A drag force can exist between a body and a fluid or between two fluid layers. It acts to reduce the relative speed between the body and the fluid [6].
2.2.3 Thrust Generally, bird and insect flight can be separated into two types; unpowered (gliding and soaring) and powered (flapping wings). The thrust is the component of the total aerodynamic force parallel to the direction of the flying animal and depends on the power output of its flight muscles. Consider an airfoil with a sinusoidal plunging motion, with no pitching, flying forward through a stationary fluid. During the motion of the airfoil, the effective angle of attack changes, as can be seen in Figure (3). Positive lift is generated during the downstroke since the airfoil is exposed to a flow with positive angle of attack, and vice versa on the upstroke.
Figure 3: The effective angle of attack varies during the plunging cycle [1].
Cross-sections of the wings further from the body will move faster than parts closer to the body during upstrokes and downstrokes. Consequently, the flow di- rection hitting the wing during forward flight will vary along the wing span. Bird wings are flexible and can twist to make sure the angle of attack is correct along the entire wing span. Since the vertical speed of the wings is highest at the tips, they are angled more forward during down-stroke than the inner parts of the wing. This lets the bird generate a forward propulsive force without any loss of altitude [7]. Birds and insects can pitch their wings to get the optimal angle of attack. Lift and thrust is generated during the down-stroke. During the upstroke, the wing is pitched so that the effective angle of attack is zero, which results in the smallest drag force possible and as little negative lift as possible. Birds often partially fold their wings during the upstroke to reduce the surface area, which reduces the drag even more [7].
7 2.2.4 Leading edge vortices (LEVs) When the angle of attack or speed of an airfoil is changed, a corresponding amount of vorticity is deposited in the wake. It takes time for the bound vortex to reach its steady state strength when an airfoil is accelerated quickly [1]. The LEV is trapped by the airflow and remains trapped to the upper surface of the wing for several chord-lengths of forward flight, shown in Figure (4). When air flows around the leading edge, it flows over the trapped vortex and is pulled in by the lower pressure generated by the vortex, which in turn generates lift. This mechanism was first discovered by Ellington et al. [4], when they studied the mechanics of forward flight in bumblebees. The lift enhancing LEV is a main feature during the plunging motion of the stroke.
Figure 4: Leading edge vortex formation in flapping flight.
2.2.5 Dynamic stall The rapid change in angle of attack when the airfoil switches stroke direction sheds a strong vortex from the leading edge. This vortex then moves along the surface of the airfoil, causing a reduction in pressure and increase in lift. However, when the vortex passes the trailing edge, the lift is dramatically reduced. This non-linear unsteady aerodynamic effect is called dynamic stall [1].
2.3 Wing model 2.3.1 Equations for wing motions in 2D Free flight wing kinematics measurements of many insects using high speed video showed that the translational velocity and pitching of the wings varies approxi- mately as simple harmonic functions [8]. The airfoil’s instantaneous location and
8 incidence can be uniquely defined using its translational and rotational coordi- nates [1] [5],
h(t) = ha sin(2πft + φ) (4)
α(t) = α0 + αa sin(2πft), (5) where h(t) is the instantaneous plunging amplitude, ha is the maximum plunging amplitude, normalized by the chord length, f is the plunging frequency, α(t) is the instantaneous angle of attack, α0 is the initial pitching angle, αa is the pitching amplitude and φ is the phase difference between plunging and pitching motion.
Figure 5: Snapshots of wing profile motion during the upstroke for advanced, syn- chronized and delayed rotation.
The phase difference between the pitching and plunging motion has a big impact on the generation of thrust and lift. It was found by Kramer et al. that an advanced wing rotation produces a mean thrust coefficient that is 72% higher than the mean thrust coefficient of an airfoil with delayed rotation [1]. A symmetric rotation, used in the simulations in this report, produced a 65% higher mean thrust coefficient compared to the delayed rotation.
2.4 Power considerations Most birds and other flying animals have well developed flight muscles to be able to generate the power required for flight. The strength of these muscles imposes limits on the flight modes of the animal. In birds, the flight muscles constitute around 17% of the total weight of the animals [1]. The maximum flapping frequency is a physical limitation for flapping animals and birds heavier than around 12 to 15 kg are not able to maintain a flapping frequency high enough to sustain horizontal powered flight [1].
9 3 Numerical simulations
The symmetric NACA 0012 airfoil, undergoing plunging and pitching motion, is studied numerically for different values of plunging amplitude h0, reduced fre- quency k, pitching amplitude αa and oncoming freestream angle β.
3.1 Governing equations and turbulence model The fluid flow around an insect wing is adequately described by the incompress- ible two-dimensional Navier-Stokes equation (without gravity) together with the continuity equation for incompressible flow [3].
∂u 1 + u · ∇u = − ∇p + ν∇2u, (6) ∂t ρ ∇ · u = 0. (7)
ANSYS Fluent (version 15.0) is used to solve these non-linear equations numer- ically using the Spalart-Allmaras turbulence model, which is based on the Reynolds Averaged Navier-Stokes (RANS) model available in Fluent. It models the extra dissipation produced by the turbulent fluctuations and was used in all the simu- lations. It is a one equation model that solves a transport equation for kinematic eddy (turbulent) viscosity without calculating the length scale related to the shear layer thickness [9].
3.2 Geometry and mesh parameters The airfoil geometry is modelled in ANSYS DesignModeler using coordinates down- loaded from the on-line database Airfoil Tools1. It is centered in a circular domain with a diameter of d = 20c, where c = 1 is the chord length. The airfoil is translated so that the origin is located a distance c/4 to the right of the leading edge. The ◦ airfoil is then rotated to give it a positive mean angle of attack α0 = 5 . The origin’s location inside the airfoil will be the center of rotation for the pitching motion.
1http://airfoiltools.com/airfoil/details?airfoil=n0012-il, Retrieved 2015-03-29
10 Figure 6: Simulation geometry
The simulation grid is created using the meshing software included in ANSYS Workbench. The grid density is higher near the airfoil to capture more detail there.
◦ Figure 7: Simulation grid with a mean angle of attack α0 = 5 .
3.2.1 Grid characteristics Mesh Size Level Cells Faces Nodes Partitions 0 24736 37380 12644 1 1 cell zone, 4 face zones.
Domain Extents: x-coordinate: min (m) = -1.000000e+01, max (m) = 1.000000e+01 y-coordinate: min (m) = -1.000000e+01, max (m) = 1.000000e+01 Volume statistics: minimum volume (m3): 2.224370e-06 maximum volume (m3): 4.570552e-02 total volume (m3): 3.140571e+02 Face area statistics: minimum face area (m2): 1.351579e-03
11 maximum face area (m2): 3.907141e-01
3.2.2 Validation A validation is performed on the simulation mesh to verify that is produces accurate results. Figure (8) is a plot of the lift and drag coefficients as functions of the angle of attack for a stationary NACA 0012 airfoil.
1.2 c L c 1 d
0.8
0.6
0.4 Coefficient values [ − ] 0.2
0
−0.2 0 5 10 15 20 25 Angle of attack α [degrees]
Figure 8: Plot of lift and drag coefficients versus angle of attack.
In Figure (8) it can be seen that around α ≈ 20◦ the airfoil reaches the stall angle where the flow separates from the top surface of the airfoil, which results in reduced lift. The drag is increasing even when the stall angle has been passed. This is the expected behaviour of the lift and drag coefficients [10].
12 3.3 Simulation model
L
D (t)
^y
U Upstroke motion ^z ^x
Figure 9: Definition of angles and coordinate system used in the simulation.
The free stream angle β is defined as in Figure (9), positive when the free stream is hitting the airfoil from below. In this simulation Re = 3000 will be used, the value in forward flight of bum- blebees (Bombus terrestris) [1]. Dynamic scaling, conserving the dimensionless parameters Re and k, ensures that the underlying fluid phenomena are conserved [3]. Since the simulation depends only on the Reynolds number and the reduced frequency, the airfoil chord is scaled to be unit length, c = 1, and the freestream ve- locity is set to U = 1. The viscosity is calculated using the formula for the Reynolds number, equation (1).
Uc 1 ν = = . (8) Re 3000 Bumblebees have a flap frequency of around 150 Hz during forward flight, mean chord length c ≈ 0.002 m and average forward flight speed of about 3 m/s [1][4]. Inserting these parameters into equation (2) results in a reduced frequency k ≈ 0.3. Substituting the reduced frequency k from equation (2) into the equations for plunge height, (4), and angle of rotation, (5), yields
h(t) = ha sin(2kt + φ) (9)
α(t) = α0 + αa sin(2kt). (10)
The phase difference φ = 90◦ is chosen to be constant. Although changes in φ results in variation in performance, it has been found that φ = 90◦ is a good choice for high thrust and efficiency [11]. The airfoil’s position and pitch during upstroke is shown in Figure (10).
13 ◦ ◦ Figure 10: Plunging and pitching with phase difference φ = 90 and α0 = 0 . The figure shows snapshots of the airfoil during upstroke. For this φ and α0, the airfoil is horizontal in the min and max positions.
The thrust coefficient CT is the negative part of the drag coefficient CD, which is calculated in Fluent by integrating the total pressure and viscous forces along the boundary of the airfoil. These coefficients are convenient for comparing the efficiency of different parameters for the oscillating airfoil.
L
Φ
Figure 11: Front view of flapping wing with stroke amplitude Φ and length L. The maximum plunge amplitude is reached when the wing has a stroke amplitude Φmax = π.
The amplitude parameter span in the simulation is calculated using the stroke amplitude for bumblebees, which is Φ = 2.1 rad [1]. The following relations can be seen in Figure (11),
14 ( h = L sin Φ ref 2 (11) hmax = L. which gives a maximum amplitude increase
h 1 max = ≈ 1.15. (12) h Φ ref sin 2
The amplitude span used in the simulation is therefore 0.7 ≤ ha/href ≤ 1.15. The oncoming wind angle parameter is confined to the interval −20◦ < β < 20◦ since this is approximately the stall angle found in Figure (8).
3.3.1 Boundary conditions The boundary conditions for the different parts of the domain are as follows; The left side of the circular outer boundary is the inlet where the inlet velocity is defined with magnitude and direction. The direction of the inlet flow is the wind direction parameter. The right side of the circular boundary is set to pressure-outlet. There is a no-slip condition on the wing profile.
3.3.2 Calculation of power usage The instantaneous power exerted by the fluid on the wing profile during steady forward flapping flight is
Pfluid = F · v + MA · ω, (13) where F = FLift + FDrag is the total aerodynamic force, v is the velocity of the airfoil, Ma is the moment with respect to the center of rotation of the airfoil and ω is the angular velocity of the airfoil. The power exerted by the insect on the fluid is obtained by introducing a negative sign,
Pinsect = −Pfluid. (14) The moment is calculated in Fluent using a moment coefficient monitor which gives the moment with respect to the origin for each time step. Since the moment needed in equation (13) is with respect to the center of rotation of the airfoil, the moment is moved using the formula
MA = MO + RAO × F, (15) where O is the origin, the center of the domain, and RAO is the vector from the origin to the center of the airfoil. The simulation starts with the airfoil in its top position which means that the position of the airfoil is given by h π i R = −h 1 − sin 2kt + y.ˆ (16) AO a 2 The velocity v and angular velocity ω are given by the time derivatives of equations (9) and (10).
15 π v = 2kh sin 2kt + y,ˆ (17) a 2 ω = 2kαa sin(2kt)ˆz. (18)
The lift force, drag force and moment are defined as
FL = qACL(− sin βxˆ + cos βyˆ), (19)
FD = qACD(cos βxˆ + sin βyˆ), (20)
MO = qAc · CM z,ˆ (21)
1 2 where β is the freestream angle, q = 2 ρU is the dynamic pressure, A is the planform area of the wing, c is the chord length and CM is the moment coefficient.
3.3.3 Moving airfoil setup The motion of the airfoil is defined using a User Defined Function (UDF) in a separate file, which can be found in appendix A. However, since Fluent uses the UDF to set the translational and angular velocity of a boundary, the time derivatives of equations (9) and (10) are used.
3.4 Results 3.4.1 Reference simulation The reference simulation is done with freestream velocity coming in with zero an- gle, which means that the insect is flying in direct headwind. The wind velocity including the wind gust is normalized to 1 in the simulation.
Parameter Value k 0.3 [rad/s] Re 3000 [ - ] ha 1 [-] ◦ αa 15 [ ] ◦ α0 5 [ ] β 0 [◦] St 0.19 [ - ]
Table 1: Parameters used in the reference simulation. β is the oncoming freestream angle. A timestep dt = 0.1 s is used. These parameters generate positive thrust and lift.
16 Figure 12: Plot of velocity magnitude for the reference simulation which shows a periodic shedding of vortices.
The simulation is run for 10 periods to remove transients from the initial values. The instantaneous lift and thrust coefficients during the tenth period is shown in Figure (13).
Thrust coefficient Lift coefficient 0.6 3
0.4 2
0.2 1 T L C C 0 0
0.2 1
0.4 2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time (t/T) Time (t/T)
Figure 13: Thrust and lift coefficients over one period, T . The airfoil starts at the top position and begins the downstroke at t/T = 0. Thrust and lift generation is higher during the downstroke than during the upstroke.
Parameter Value
CT 0.125 CL 0.572 Table 2: Lift and thrust coefficients averaged over four periods. The values indicate that the airfoil produces positive lift and thrust during a reference case flap cycle.
It is informative to look at the instantaneous power during a flap cycle for different β. There are physical limitations to how much power an insect can produce in its muscles.
17 1.5 Reference case 1 Airfoil position refMax
0.5
0
−0.5 Relative power P(t/T) / P
−1 0 0.2 0.4 0.6 0.8 1 Time t/T
Figure 14: Power usage during one flapping cycle with period T in the reference case, normalized with the maximum instantaneous power. The dashed red line represents the position of the airfoil and shows that the airfoil starts off from its topmost position at t/T = 0. The position is not to scale in this plot.
Figure (14) indicates that the insect needs to spend more energy during the downstroke, where the effective angle of attack of the airfoil is higher. The angle of attack is lower during the upstroke and the insect needs less energy to push the wing through the air.
3.4.2 Varying the reduced frequency, k In this section, the effect on mean lift and thrust coefficients resulting from varying the reduced frequency is studied when keeping amplitude and pitch constant. The lift and drag coefficients are calculated in the directions of the lift and drag forces as defined in Figure (9) and CT = −CD.
18 0.8
0.6
0.4
0.2 β = 20o β o 0 = 10 β = 0o [ − ] T −0.2 β o C = −10 o −0.4 β = −20 Reference −0.6
−0.8
−1
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Reduced frequency k [rad/s]
Figure 15: Thrust coefficients for different β and reduced frequencies k.
It can be seen in Figure (15) that the thrust generally increases with increasing k for β ≤ 0◦. However, for β > 0◦ the thrust coefficient appears to have a minimum around k = 0.4 rad.
3
2.5
2
1.5 β = 20o 1 β = 10o o 0.5 β = 0 [ − ] L β o C 0 = −10 β = −20o −0.5 Reference −1
−1.5
−2
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Reduced frequency k [rad/s]
Figure 16: Lift coefficient as a function of the reduced frequency k for different β.
As expected, Figure (16) shows that the lift coefficient depends strongly on the oncoming freestream angle since CL increases for larger β. The mean angle of ◦ ◦ attack α0 = 5 results in a positive lift coefficient for β = 0 . With β ≥ 0, CL generally increases. However, for β = 20◦, the lift coefficient reaches a maximum at k = 0.4 rad.
19 6 k = 0.2 5 k = 0.3 (Reference case) 4 k = 0.4
refMax k = 0.5 3 k = 0.6 2 Airfoil position
1
0
−1 Relative power P(t/T) / P −2
−3 0 0.2 0.4 0.6 0.8 1 Time t/T
Figure 17: Instantaneous power needed to push the airfoil through the air during one flapping cycle. The airfoil position curve is only for visualization purposes and is not to scale.
5
4.5
4 refMax 3.5 β = 20o β = 10o 3 β = 0o 2.5 β = −10o o 2 β = −20 Reference 1.5 Relative power P(t/T) / P 1
0.5
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Reduced frequency k [rad]
Figure 18: Maximum power usage as a function of k. The plot shows the max- imum instantaneous power needed during one flap cycle. It is a measure of how much power the insect maximally needs to exert during a flap cycle compared to the reference case. These peaks occur during the downstrokes, which can be seen in Figure (17).
It is clear in Figure (18) that the maximum power the insect needs to exert during a flapping cycle increases rapidly with increasing frequency. If the insect were to double its flapping frequency when facing direct headwind (β = 0◦), it
20 would require a 5 times increase in maximum power during the downstroke. Since the insect will not be able to exert the necessary extra power for extended periods of time, the results indicate that the frequency change must be small.
3.4.3 Varying the plunging amplitude, ha
In this set of simulations the plunging amplitude ha is varied while keeping the reduced frequency and pitching amplitude constant.
0.2
0.1
0 β o −0.1 = 20 β = 10o −0.2 β = 0o [ − ] T β o C −0.3 = −10 β = −20o −0.4 Reference −0.5
−0.6
−0.7
0.7 0.8 0.9 1 1.1 Plunging amplitude h /h [ − ] a ref
Figure 19: Thrust coefficient as a function of plunging amplitude for different β.
The thrust coefficient is strongly influenced by the freestream angle and varies approximately linearly with the plunging amplitude, as can be seen in Figure (19). |CT | increases with increasing plunging amplitude. However, positive average thrust is only generated for β = −10◦ and β = 0◦.
21
2
1.5
1 β = 20o β = 10o 0.5 β = 0o [ − ] L β o C 0 = −10 β = −20o −0.5 Reference
−1
−1.5
0.7 0.8 0.9 1 1.1 Plunging amplitude h /h [ − ] a ref
Figure 20: Lift coefficient as function of plunging amplitude for different β.
Figure (20) shows that there is an approximately linear relation between the plunging amplitude and CL. Wind coming from above, corresponding to negative β, pushes the airfoil downwards, reducing the lift. The positive lift for β = 0◦ is a result of the positive mean angle of attack α0.
2.4
2.2
2
1.8 refMax β = 20o 1.6 β = 10o 1.4 β = 0o β o 1.2 = −10 β = −20o 1 Reference 0.8 Relative power P(t/T) / P 0.6
0.4
0.7 0.8 0.9 1 1.1 Plunging amplitude h /h [ − ] a ref
Figure 21: Power exerted by the insect as a function of plunging amplitude.
The power usage in Figure (21) increases with increasing amplitude. The slope of the curves increase slightly with increasing β.
22 3.4.4 Varying the pitching amplitude, αa
In this section the effect of the pitch amplitude αa on lift, thrust and power is studied while other parameters are kept constant.
0.2
0.1
0
−0.1 β = 20o −0.2 β = 10o o −0.3 β = 0 [ − ] T β o C = −10 −0.4 β = −20o −0.5 Reference −0.6
−0.7
−0.8
10 12 14 16 18 20 Pitching amplitude α [degrees] a
Figure 22: Thrust coefficient as a function of pitching amplitude for different β.
Although the changes in thrust coefficient are small when varying the pitching amplitude, shown in Figure (22), a general trend is that the magnitude of the thrust force decreases with increasing αa. An increased pitching amplitude results in a lower effective angle of attack on both upstroke and downstroke.
2
1.5
1 β = 20o β = 10o 0.5 β = 0o [ − ] L β o C 0 = −10 β = −20o −0.5 Reference
−1
−1.5
10 12 14 16 18 20 Pitching amplitude α [degrees] a
Figure 23: Lift coefficient as a function of pitch amplitude for different β.
23 As for the thrust coefficient, an increase of pitching amplitude leads to a decrease in |CL|, as shown in Figure (23).
2
1.8
refMax o 1.6 β = 20 β = 10o 1.4 β = 0o β = −10o 1.2 β = −20o Reference 1 Relative power P(t/T) / P 0.8
0.6
10 12 14 16 18 20 Pitching amplitude α [degrees] a
Figure 24: Power exerted by the insect as a function of pitching amplitude.
The effects of varying the pitching amplitude can be seen in Figure (24). When the pitching amplitude is increased, the maximum power during a flap cycle tends to decrease slightly for negative β and increase slightly for positive β.
3.5 Conclusion According to the results of the simulations, the ideal situation for the insect is to ◦ fly in a horizontal freestream, β = 0 , since CT and CL always are positive when varying the parameters. If this is not the case, the following qualitative rules can be used to adjust the reduced frequency, plunge amplitude and pitch amplitude to compensate for the oncoming wind gust while keeping the maximum power needed as low as possible for different β.
3.5.1 Wind from below, β > 0◦ The thrust coefficient was found to have a minimum around k = 0.4 for wind gusts coming from below. It was also found that CL increases with increasing k. However, increasing k leads to a rapid increase in the maximum power usage during the downstroke. Since the wind comes from below, it pushes the insect upwards, which is a big contribution to the lift force. Therefore, if k is decreased, the thrust is increased and lift is reduced. Reducing k will reduce the maximum power usage of the insect. An increased plunging amplitude was found to decrease the thrust, increase the lift and increase the power usage. This indicates that the plunging amplitude should be decreased as well, to counteract the extra lift from the wind coming from below, generate extra thrust and further reduce the power usage.
24 The thrust increases, the lift decreases and the power usage is marginally in- creased when the pitching amplitude is increased. It is therefore a good choice to increase the pitching amplitude.
3.5.2 Wind from above, β < 0◦ Wind coming from above pushes the insect downwards. To counteract this, the lift and thrust force should be increased to improve the manoeuvrability of the flyer. This could be accomplished by increasing the frequency and increasing the pitching amplitude. The increase in k will result in an increase in CT , a small decrease in CL and an increase in power usage. For β ≈ −10◦, an increase in plunging amplitude and decrease in pitching amplitude will result in a marginally increased thrust, unchanged lift and increased power usage. When β ≈ −20◦, a decrease in plunging amplitude and increase in pitching amplitude will result in a marginally increased thrust and lift.
3.5.3 Final remarks There are more ways for a flyer, natural or artificial, to compensate for incoming wind gusts than was studied in this report. The flyer can for example turn in the air and change its direction to counteract the wind gust, instead of being locked in one direction, as is assumed in the simulations. This would be particularly helpful when the wind comes from above, since changing the flapping parameters only marginally increases the thrust and lift. The results may differ for finite 3D wings where 3D effects are accounted for, such as tip vortices and the clap-and-fling mechanism [1].
25 References
[1] Wei Shyy, Hikaru Aono, Chang kwon Kang, and Hao Liu. An Introduction to Flapping Wing Aerodynamics. Cambridge University Press (CUP), 2013.
[2] Graham K. Taylor, Robert L. Nudds, and Adrian L. R. Thomas. Flying and swimming animals cruise at a strouhal number tuned for high power efficiency. Nature, 425(6959):707–711, oct 2003.
[3] S. P. Sane. The aerodynamics of insect flight. Journal of Experimental Biology, 206(23):4191–4208, dec 2003.
[4] R. Dudley and C. P. Ellington. Mechanics of forward flight in bumblebees. Journal of Experimental Biology, 148:19–52, jun 1990.
[5] J. Tang, D. Viieru, and W. Shyy. Effects of reynolds number and flapping kinematics on hovering aerodynamics. AIAA Journal, 46(4):967–976, apr 2008.
[6] Steven Vogel. Life in Moving Fluids. Princeton University Press, 1994.
[7] Joel Guerrero. Numerical simulation of the unsteady aerodynamics of flapping flight. Unpublished doctoral thesis, University of Genoa, 2009.
[8] Y. Sudhakar and S. Vengadesan. Flight force production by flapping insect wings in inclined stroke plane kinematics. Computers & Fluids, 39(4):683–695, apr 2010.
[9] Nadeem Akbar Najar, D. Dandotiya, and Farooq Ahmad Najar. Comparative analysis of k-ε and spalart-allmaras turbulence models for compressible flow through a convergent-divergent nozzle. The International Journal Of Engi- neering And Science (IJES), 2:8–17, 2013.
[10] NASA turbulence modeling resource. http://turbmodels.larc.nasa.gov/ naca0012_val.html. Accessed: 2015-04-27.
[11] F.S. Hover, Ø. Haugsdal, and M.S. Triantafyllou. Effect of angle of attack profiles in flapping foil propulsion. Journal of Fluids and Structures, 19(1):37– 47, jan 2004.
26 4 Appendix A
4.1 User Defined Functions Below is the UDF c code file which defines the vertical translational velocity and the angular velocity of the airfoil. The parameters are passed in from Fluent, where they are defined for each simulation case using a journal file. The journal file, generated by a Python script, contains all text commands needed to set up and run all simulations automatically.
#include "udf.h"
DEFINE_CG_MOTION(profilemotion, dt, vel, omega, time, dtime) { real t = time; real k = RP_Get_Real("frequency"); real ampl = RP_Get_Real("amplitude"); real pitch = RP_Get_Real("pitchrange")*3.141592/180.0;
vel[1] = ampl*2*k*cos(2*k*t+3.141592/2); omega[2] = pitch*2*k*cos(2*k*t); }
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