Numerical Simulation of Flexible Flapping Airfoil ...

AIAA 2008-654

46th AIAA Aerospace Sciences Meeting and Exhibit 7 - 10 January 2008, Reno, Nevada

Numerical Simulation of Flexible Flapping Airfoil Propulsion using Dynamic Mesh at Low Reynolds Numbers

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Naveed Durrani* and Ning Qin† Department of Mechanical Engineering, University of Sheffield, Sheffield, S1 3JD, UK

Fast dynamic grid deformation based on Delaunay graph mapping is used to simulate flapping airfoil propulsion at a low Reynolds numbers of 9,000. A flexible plate with a given stiffness is attached to the trailing edge of a rigid NACA0033 airfoil. The ability to use a single Delaunay graph for moving mesh to simulate the complete flapping cycle considerably reduced the computational cost of dynamic grid deformation in the simulation. An ALE based unsteady Navier-Stoke solution with dual time stepping is used to determine the effects of various factors on thrust coefficient and efficiency due to flapping motion. Initial simulation has been carried out at a flapping frequency of 0.98 Hz with constant leading edge amplitude of 17.5mm according to the experimental work of Heathcote and Gursul, (AIAA J., vol. 45, No. 5, 2007, pp. 1066-1079).

Nomenclature ale ate b c Cp CT

= = = = = =

leading edge amplitude trailing edge amplitude plate thickness chord length time-averaged power-input coefficient time-averaged thrust coefficient

F Q

S ⎛ ⎞ = ratio of differential area of the element with element face area ⎜ ei = i , i = 1, 2,3 for a triangle ⎟ S ⎝ ⎠ = flux vector = primitive variable matrix

Re

= Reynolds number ⎜⎜

S SLE STE

= area = instantaneous leading edge distance along y-axis = instantaneous trailing edge distance along y-axis

Sr

= Strouhal number ⎜

t T U0 v W x X y

= = = = = = = =

ei

* †

⎛ ρU o c ⎞ ⎟⎟ ⎝ µ ⎠

⎛ 2 f ale ⎞ ⎟ ⎝ U0 ⎠

physical time step time period; Thrust free stream velocity airfoil leading edge velocity conservative variable matrix streamwise position position vector cross-stream position

Research Student, AIAA Member Professor of Aerodynamics, AIAA Associate Fellow

Copyright © 2008 by N Durrani and N Qin. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

η µ θ

= viscosity

τ ν

= = = =

Γ

= Jacobian Matrix ⎜⎜ Γ =

Ω

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⎛ CT ⎞ ⎟ ⎝ CP ⎠

= efficiency ⎜

pitch angle vorticity pseudo time step kinematics viscosity

⎛ ⎝

∂W ∂Q

⎞ ⎟⎟ ⎠

I. Introduction

O

BSERVING natural evolution of various mechanisms in living species has been very intuitive for the human kind. Spanning over hundreds of millions of years, these mechanisms for propulsion, stability and maneuverability are arguably thought to be the most refined ones for a particular environment and operating conditions for particular specie. Owing to prevailing geo-strategic circumstances, there is a surge in desire to build micro-air vehicles (MAVs) with various roles and capabilities. One recent example of this is “MOD Grand Challenge” by MoD, UK. Various applications of such MAVs range from military usage including reconnaissance and surveillance to urban environment usage including reconnoiter large buildings or region of interest, identification of a potential threat, and assist in rescue operations. Unlike conventional propulsion methods using separate thrust, lift and control devices for stability and maneuvering, MAV’s have various constraints including size, weight, stability and maneuverability prompting for non-conventional approach. The degree of suppleness, stability and maneuverability portrayed by various species in nature including insects, birds and fishes to name a few, have revitalized interest of large section of aerodynamic community looking for nonconventional MAV’s design. One distinct feature in above mentioned locomotion and propulsion is the use of single surface for lift and propulsion through flapping wing. Reynolds number is a very import aerodynamic parameter for the relative study of flapping motion of birds etc. Owing to their small size, typically in centimeters, and slow speeds, the Reynolds number range is generally from 103-105. Ref. [1] provides a good starting point about basics of MAV’s and some related research. The ability of oscillating wing to produce thrust was first described by Knoller[2] and Betz[3] in 1909 and 1912 respectively in their independent studies. They observed that a wing oscillating sinusoidally in a cross-stream of flow creates an effective angle of attack leading to a force vector normal to the relative flow direction. This force vector gives both lift and thrust. This effect known as the Knoller-Betz effect was demonstrated experimentally by Katzmayr[4]. In 1922, Birnbaum[5,6] , suggested oscillating wing as a possible ‘alternate to propeller’ mechanism for propulsion. The mainstream of the aerodynamic community, however, concentrated on lifting forces instead of drag or propulsive forces due to its application in flutter analysis, a topic of prime interest during that period. In 1935, Theodorsen[7] derived expressions for the unsteady lift and moment on a flat plate undergoing combined sinusoidal pitching and plunging motion. It was based on the inviscid, incompressible, oscillatory flat plate theory. Garrik[8] further theoretically determined the thrust force and predicted that the plunging airfoils generate thrust over the whole frequency range while pitching airfoils do so only beyond a critical threshold frequency as a function of location of the pivoting point. In 1939, Von Karman and Burgers [9] theoretically related the negative drag (thrust) production in conjunction with the shape and position of a double row of counter rotating wake vortices generated by bluff bodies. In 1939, Silverstein and Joyner[11] experimentally verified Garrick’s prediction while Bratt[12] did visualization experiments to bolster what we call now von Karman vortex street generation for different wake flows. Freymuth[13], Koochesfahani[13], Jones[14], Dohring and Platzer[15] and Lai and Platzer[16] showed experimentally the different wake structures generation based on plunge frequency and amplitude. They typically characterized wake structures as thrust producing, neutral and drag producing. The majority of the experiments carried out in different experimental studies in the literature have used rigid bodies. However, In view of the propulsion mechanism in the natural world, all the flapping, hovering or plunging motions are done by flexible surfaces. However,,as compared with rigid body experimentation, little work has been

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done on flexible bodies. It has been reported through numerical simulations that higher efficiencies are achieved with flexible surfaces as compared with rigid bodies[17-18]. Isaac et al.[19] have carried out water tunnel experiments on a flapping-and-pitching thin flat plate wing of semi elliptic planform apart from studying flapping wing based MAV’s[27,28]. More recent water tunnel experiments by Heathcote and Gursul[20] at low Reynolds numbers have shown that a degree of flexibility is beneficial for thrust supplement and better efficiency. The present study presents a numerical method for the solution of flapping wing problems based on the ALE dynamic mesh technique. The simulation revolves around the experiments by Heathcote and Gursul[20] to get further insight of the flow physics of flapping wing unsteady aerodynamics. With detailed unsteady flow field information, we aim to understand the mechanism for thrust production in relation to vortex shedding, plate flexibility, and the role that the leading edge induced vortices play in the process.

II. Governing Equations and Solution Methodology The flow governing equations are in the integral form for an arbitrary control volume V with differential surface area dA. ∂ WdV + ∫∫ [ F − G ] .dA = 0 ∂t ∫∫∫ V ∂A

(1)

A dual time stepping method as propounded by Jameson[30] is used in the numerical scheme. ∂ ∂ WdV + Γ QdV + ∂t ∫∫∫ ∂ τ ∫∫∫ V V 14243 14 4244 3 Physical time term

∫∫ [F − G ].dA = 0

(2)

A

Pseudo time term

where, ⎛ p⎞ ⎧ ρ ⎫ ⎜ ⎟ ⎪ ρu ⎪ ⎡ Fx ⎤ ⎡G x ⎤ u ⎪ ⎪ W = ⎨ ⎬ , Q = ⎜ ⎟ , F = ⎢ ⎥ and G = ⎢ ⎥ ⎜v⎟ ⎣G y ⎦ ⎣ Fy ⎦ ⎪ ρv ⎪ ⎜ ⎟ ⎪⎩ ρ E ⎪⎭ ⎝T ⎠ 0 ⎧ ⎫ ⎧ ρu ⎫ ⎪ ⎪ ⎪ ρ uu + p ⎪ τ xx ⎪ ⎪ ⎪ ⎪ Gx = ⎨ Fx = ⎨ ⎬ ⎬ ; τ xy ⎪ ⎪ ⎪ ρ uv ⎪ ⎪uτ xx + vτ xy − qx ⎪ ⎪⎩ ρ uE + pu ⎪⎭ ⎩ ⎭ ⎧ ρv ⎫ ⎪ ρ vu ⎪ ⎪ ⎪ Fy = ⎨ ⎬ ; ⎪ ρ vv + p ⎪ ⎪⎩ ρ vE + pv ⎪⎭

(3a)

0 ⎧ ⎫ ⎪ ⎪ τ yx ⎪ ⎪ Gy = ⎨ ⎬ τ yy ⎪ ⎪ ⎪uτ yx + vτ yy − q y ⎪ ⎩ ⎭

(3b)

Geometric Conservation Law (GCL) Basic idea of geometric conversation law (GCL)[21,22,35] is to preserve the freestream flow solution, unaffected by the movement of the mesh elements. ∂V (v − vg ).n dA = 0 + ∂t ∂∫∫ A(t )

(4)

3

where , ∂A(t ) is the time dependant boundary of the control surface. Arbitrary Lagrangian-Eulerian (ALE) Formulation Arbitrary Lagrangian-Eulerian (ALE)[37,38] formulation is widely used for the spatial discretization of problems in fluid and structural dynamics. In particular, ALE is frequently employed to simulate large distortion/deformation response of material structures, fluids and fluid-structure systems. The word “arbitrary” in ALE indicates that it could be both “Lagrangian” and ”Eulerian” or anywhere in between them as our control volume V (t ) and control area ∂A(t ) are function of time with moving meshes for unsteady flow

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problems involving geometrical deformation. If we take the velocity of moving control surface

∂A(t ) as v g

⎡u g ⎤ defined as v g = ⎢ ⎥ and put the grid velocity in equation 2, the resulting ALE formulation fluxes become: ⎣ vg ⎦ ⎧1 ⎫ ⎧0 ⎫ ⎪u ⎪ ⎪1 ⎪ ⎪ ⎪ ⎪ ⎪ Fx = ρ (u − u g ) ⎨ ⎬ + p ⎨ ⎬ v ⎪ ⎪ ⎪0 ⎪ ⎪⎩ E ⎪⎭ ⎪⎩u ⎪⎭ 144244 3 12 3 Convective terms

Pressure terms

⎧1 ⎫ ⎧0 ⎫ ⎪u ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪ ⎪ Fy = ρ (v − vg ) ⎨ ⎬ + p ⎨ ⎬ v ⎪ ⎪ ⎪1 ⎪ ⎪⎩ E ⎪⎭ ⎪⎩v ⎪⎭ 144244 3 123 Convective terms

(5a)

(5b)

Pressure terms

Replacing the flux terms in the NS equations with these flux terms will give the complete set of NS equations for the ALE formulation. It is evident that, for the grid speed v g = 0 , the system recovers to the Eulerian formulation. Solution methodology A cell centered, density based finite volume solver is used for all the simulations. It has dual time stepping capability with outer time loop advancing in real time and inner ‘pseudo time’ using 4 step Runge Kutta[29] scheme. Low speed preconditioning is used similar to Ref.[ 31]. Arbitrary Lagrangian Eulerian (ALE)[33] formulation is used for moving mesh solution.

III. Mesh Deformation Technique for Flapping Wing A schematic representation of the airfoil element is shown in figure1. The flapping airfoil element consists of a rigid airfoil section (NACA0033) and a flexible plate tail section. Different parameters used in this study are also presented in figure 1 for their clarification.

Fig. 1 Schematic diagram of flapping airfoil with flexible tail heaving sinusoidally in vertical (cross-flow) direction

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Due to the geometric deformation occurring at each time step including relatively large tail movement as shown in Fig. 2, selection of the dynamic grid technique to be used is of significant importance. Two options available are either to “re-mesh” at each time step or to “deform the mesh” by keeping the same topology. “Re-meshing”, though used by a number of researchers, is not an efficient option in terms of computational time and cost due to the very large number of time steps required in the simulation to resolve the unsteady flow. Therefore mesh deformation, which maintains the mesh topology, is believed to be a better choice here, involving large but periodical geometrical deformation. However, there are, of course, situations where re-meshing, mesh refinement and coarsening are unavoidable, such as the cases for store release. Most mesh deformation methods based on the spring analogy were iterative in nature such as that proposed by Batina et al.[21,22]. The basic idea of this methodology was to treat the edge between two nodes as a spring with certain stiffness. Therefore, the edge can be compressed and elongated similar to springs with certain governing rules to avoid mesh crossing over. Following the same principle, various improvements were proposed in the literature [23-26]. However, these iterative methods can drastically increase the computational cost of the grid deformation, particularly for highly stretched viscous meshes with large deformations, such as the situation in our case during the maximum deflection of tail in the plunge up and down.

Fig. 2 Flapping airfoil snapshots at different time during airfoil plunging motion showing flexible tail plate A. Mesh Details Although different meshes including structured and unstructured type can be generated, the baseline grid presented here is of hybrid nature as shown in figure 3. It has 27803 node points with 36651 triangular and quadrilateral elements. The wake size is about 3.5 times the length of the model (airfoil + the plate). The domain length is 5 times upstream and 8 times downstream with a height of 10 times of the length of the model. The mesh was generated through a commercial software “Gambit” in its generic mode. The grid near the airfoil element and in the wake is structured with clustering towards the wall to maintain the numerical accuracy of the boundary layer and the wake resolution. To facilitate the meshing, the trailing edge of the plate was made pointed by creating a small notch at the end. Since the plate is relatively thin, this should not alter the flow significantly. The Delaunay graph was generated using selected nodes of the model surface and the outer domain as shown in figure 4. The most critical grid deformation stage is when the tail is at the peak trailing edge amplitude in upward and downward plunging motion. Good control of near wall mesh was achieved using Delaunay graph mesh deformation as shown in Figures 5 and 6. Also, the fact that a single Delaunay graph was sufficient to simulate the complete flapping motion considerably reduced the computational cost of the whole mesh deformation. From our own experience, this large deformation and clustered viscous near wall mesh can cause huge problems for the spring analogy based methods in robustness and mesh quality. Also, with the increase in flexibility of the a flapping or moving surface, more number of iterations are required to ensure smooth mesh in the near wall region, which is most susceptible for grid crossing over. This puts a further computational cost and time constraint for iterative methods using large size meshes with regards to mesh quality. This situation gets worse when even a large number of stiffness smoothing iterations fail to give a valid grid. The Delaunay graph mapping method made ease the moving mesh process for the present problem when grid deformations from the mean starting position with straight flapping tail to the peak upper and lower amplitude locations with a fully bent tail in a single step.

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Fig.3 Mesh details of the flapping airfoil comprising of airfoil section NACA0033 and flexible tail section of attached plate: a) complete domain mesh; b) outline of the flapping airfoil element; c) mesh around flapping airfoil element; d)close-up view of airfoil section; e) close-up view of tail section B. Tail Motion Specification The time dependant displacement of the leading and trailing edge is known from the experimental data. For the flapping tail, the flexible plate curve is approximated by a quadratic function to give a close representation to the experiments. The displacement variation at the trailing edge of the plate is matched to the experimental data. The total length of the tail is kept constant by keeping the edge length of all the meshed elements of tail section constant during the motion. During the flapping motion, the leading and trailing edge location trajectories are followed in the modeling.. However, by keeping the length of the tail constant, the lengths of the upper and lower surfaces of tail will be very slightly different due to curvature at the trailing edge as shown in figure 3a. A small routine was written to extend the shorter surface to match the x-axis location by keeping the initial angle to the sharp trailing edge notch constant.

Fig. 4 Due to finite thickness of the flapping tail, the curvature effects will cause the trailing edge to have different x-axis location for upper and lower surface with fixed length

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C. Fast Grid Deformation using Delaunay Graph Mapping

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A simple, efficient and non-iterative grid deformation technique based on Delaunay graph mapping[32] is used. This method gives an order of magnitude improvement in CPU time over traditional spring analogy methods[32]. This method is divided into four main steps as follows: a) generate a Delaunay graph based on selected mesh points on the flapping model and the outer boundary to encompass all the interior mesh nodes as shown in figure 6. b) locate all the mesh points in the domain with respect to the generated Delaunay graph Locating the Graph elements for all the mesh points is done by checking the area ratios of the mesh nodes with Delaunay Graph nodes. e.g., a mesh node P will be in a Delaunay Graph element ∆ABC only if none of ∆ABP, ∆BCP and ∆CAP are negative. This logic is shown in Figure 5. xA 1 ∆ABC or S = xB 2 xC

yA 1

xA 1 1 ; ∆CAP or S3 = xP 2 1 xC

yA 1

yB yC

yP 1 yC 1

(6) xA

∆ABP or S1 =

1 xB 2 xP

yA 1

xP

1 yB 1 ; ∆BCP or S 2 = xB 2 yP 1 xC

yP 1 yB 1 yC 1

c) move the Delaunay graph according to the specified flapping airfoil motion d) relocate the interior nodes to adjust the whole domain mesh as per flapping airfoil motion Delaunay Graph element Location for interior mesh

Schematic Description Moving Mesh node Delaunay Graph Element Node

A

S1 or ∆ABP > 0 S2 or ∆BCP > 0 S3 or ∆CAP > 0 Conclusion: Point P is located in Delaunay Graph element ∆ABC

P C

B

A

S1 or ∆ABP = 0 S2 or ∆BCP > 0 S3 or ∆CAP > 0 P

Conclusion: Point P is located in Delaunay Graph element ∆ABC

C

B

A

S1 or ∆ABP = 0 S2 or ∆BCP = 0 S3 or ∆CAP > 0 Conclusion: Point P is located in Delaunay Graph element ∆ABC

P

C

B

A

S1 or ∆ABP < 0 S2 or ∆BCP > 0 S3 or ∆CAP > 0

P

Conclusion: Point P is not located in Delaunay Graph element ∆ABC

B

C

Figure 5 Various possible situations of a mesh point P in relation with a Delaunay element

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Note that we have used the area ratios calculated before the deformation to get the new position of moving mesh point P. Further indepth details are given in Ref.[32]. Due to the flapping motion, the airfoil element is simultaneously becoming convex at one side and concave on the other side. The Delaunay graph is obtained by creating Delaunay triangulation of the selected points as described in step ‘a’ above. To improve the robustness, a modification to the graph formation has been made so that a single graph is required without repeated graph generation during the flapping motion. To achieve this, we ensure that all the graph elements have at least one node from the outer domain or airfoil element to minimize any localized degradation effect due to convex/concave region swapping between the flap and the airfoil junction. By doing so, some of the graph elements are no longer Delaunay but this does not affect the purpose of serving as a mesh transformation map. Figures 6-7 show the Delaunay graph generation for the present problem, while, Figures 8-9 illustrate the moved meshes at two of the extreme displacement positions. The quality and the topology of the original mesh is maintained during the movement. For this problem, only a single Delaunay graph, shown in Figure 6(a), is required with the modification proposed above.

Fig. 6 Delaunay Graph generation over the flapping airfoil: a) complete domain showing only Delaunay triangles; b) close up view of Delaunay triangles around flapping airfoil element; c) close-up view of airfoil section; d) close-up view of tail section

Fig. 7 Location all interior mesh nodes (represented by ‘+’ symbol) through Delaunay graph

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Fig. 8 Deformed mesh over the flapping airfoil element during maximum downward tail deflection: a) complete domain mesh; b) close up view of flapping airfoil element; c) close up view of airfoil section; d) close up view of tail section

Fig. 9 Deformed mesh over the flapping airfoil element during maximum upward tail deflection: a) complete domain mesh; b) close up view of flapping airfoil element; c) close up view of airfoil section; d) close up view of tail section

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IV. Numerical Simulation of Flapping Airfoil Experimental Data The experimental data available[20] is for a water tunnel experiment. The Strouhal number and the Reynolds number are matched between the experiment and the CFD simulation input using air as working fluid, for dynamic similarity. The experimental and the corresponding CFD parameters are tabulated in Table 1, for the initial case run.

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Table 1 Comparison of CFD and Experimental Variables for the Case Parameter Description

Units

Experimental Data[20]

CFD Simulation Air

Water Density (ρ )

Kg/m3

Viscosity (µ)

1000

1.176

Pa.s

1.00E-03

1.846E-05

Incoming Flow Velocity

m/s

0.1

1.57

Frequency

Hz

0.98

15.3

Reynolds number

9000

Strouhal number

0.342

Chord length incl. plate tail

m

0.09

Laeding edge amplitude

m

0.0175 0.56 x 10-3

Plate thickness ratio

The motion of the flapping wing is shown in Figure 10(a) for the case studied here, which is extracted from the experimental work, Figure 10(b).

. Fig. 10-a Leading-edge and trailing-edge displacement as a function of time; Re=9,000, b/c = 0.56 x 10-3, Sr = 0.34

Fig. 10-b [Ref. 20] Leading-edge displacement, trailing-edge displacement and their difference as a function of time; Re=9,000, b/c = 0.56 x 10-3, Sr = 0.34 10

Results

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The initial results are presented for the flapping motion case. Figure 11 shows the instantaneous vorticity snapshot of the experiment for a similar case at the same Reynolds number. Due to flapping motion, counter rotating vortices are shed from the trailing edge downstream of the airfoil section. These vortices are similar in magnitude but opposite in direction as shown with different colors. There is some elongation in the vortices seen in the simulation, which may be due to the small differences in the Strouhal number and flexibility of plates. Now that the methodology is tested to work and a closer match to the experimental cases is under way.

a)

b)

Fig. 11 Comparison of experimental and numerical instantaneous vorticity a) Experiment Re= 9000, Sr=0.27, b/c= 0.85x 10-3 b) Numerical simulation Re=9000, Sr=0.34, b/c=0.56 x 10-3

Z vorticity plots from the flow field obtained during the simulation are shown in Figure 12 at various time during the flapping cycle.The phenomenon is consistent with the previous observations in literature. Firstly the upward and downward plunging motion produces counter-rotating vortices in the wake. This generates the thrust producing wake as described in Ref. 14. Secondly, as observed in the simulation of Lewin and Haj Hariri[34], leading edge vortices is also clearly shed,. The coefficient of thrust CT obtained for the baseline case of b/c=0.56x10-3 and the Strouhal number of 0.34 was 0.57 against the experimental value of 0.814. The flapping frequency of the teardrop element is 15.3 for this case with leading edge amplitude of 0.0175 m. The axial force plot in Figure 13 shows a convergence to a periodic distribution with some high frequency oscillations. The forces are predominantly in the negative domain, producing the trust to the flapping wing. We note a similar recent study was reported by Tang et al.[40] using a different methodology, which produced a rather regular periodical force variations. Although our initial calculation does show some qualitative agreement with the experiments, further verification of the calculations are underway to closely match the experimental condition and to study such factors as mesh sensitivity, time step sensitivity and the effects of numerical boundary conditions. Detailed validation will then follow against the experimental data.

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a)

b)

d)

c)

f)

e)

Fig. 12 Z-vorticity of flapping airfoil during one cycle; Re=9,000, b/c = 0.56 x 10-3, Sr = 0.34 a-b Mean position while stroking up and down c-d Tail in near horizontal position during downwards and upwards strokes e-f Flapping of tail during upward stroke near y=0

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Variation of Fx with time 0.1 0.05 0

Fx

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-0.05 0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

-0.1 Fx -0.15 -0.2 -0.25 -0.3 -0.35 Time (sec)

Fig. 13 Variation of FX with time

V. Conclusion A flapping wing simulation method has been proposed using a modified Delaunay graph mapping. A single initial Delaunay graph was found to work for complete flapping motion of the thinnest and most flexible flat plate (b/c = 0.56x10-3), the most challenging case with regard to moving meshes, from the experimental data. Due to the noniterative nature of Delaunay graph mapping, it is found to be an attractive tool for continuously deforming domains such as flapping surfaces of aerodynamic or marine interest. Average time it took for a single time step movement for this baseline case was around 0.1 second for mesh size of over 36500 elements. Vortex shedding in pairs from leading edge of the airfoil section is observed with one dominant in strength. The coefficient of thrust (CT) calculated with experiment for Sr=0.34 and b/c=0.56 x 10-3 is computed to be 0.57 against the experimental value of 0.814. Work is in progress to conduct a systematic comparison with the experiments for various Reynolds numbers and Strouhal numbers using the proposed method.

Acknowledgements The authors gratefully thank Dr. S. Heathcote and Prof I. Gursul for their help in providing the experimental data.

References [1] Mueller, T. J., “Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications”, Progress in Astronautics and Aeronautics Series, P. Zarchan, AIAA. [2] Knoller, R., “Die Gesetze des Luftwiderstandes,” Flug und Motortechnik (Wien), Vol. 3, No. 21, 1909, pp. 1–7. [3] Betz, A., “Ein Beitrag zur Erklaerung des Segefluges,” Zeitschrift fuer Flugtechnik und Motorluftschiffahrt, Vol. 3, Jan. 1912, pp. 269–272. [4] Katzmayr, R., “Effect of Periodic Changes of Angle of Attack on Behavior of Airfoils,” NACA Rept. 147, Oct. 1922 (translated from Zeitschrift fuer Flugtechnik und Motorluftschiffahrt, March 31, 1922, pp. 80–82, and April 13, 1922, pp. 95– 101). [5]Birnbaum,W., “Das ebene Problem des schlagenden Flugels,”Zeitschrift f¨ur Angewandte Mathematik und Mechanik,Vol. 4,No. 4, 1924,pp.277–292. [6] Birnbaum,W., “Der Schlag ugelpropeller und die kleinen Schwingungen elastisch befestigter Tragugel,” Zeitschrift fur Flugtechnik und Motorluftschiffahrt, Vol. 15, Nov.–Dec. 1924, pp. 128–134.

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