Numerical Validation of Flapping Airfoil Experiments P. Bilgi1 and J. Soria 2 Department of Mechanical and Aerospace Aerospace Engineering Monash University (Clayton (Clayton Campus), Melbourne, VIC, AUSTRALIA The advantages offered by flapping wing aerodynamic design are readily seen in the flight of insects and to a lesser extent, birds. Tight maneuvering and hovering are examples of flight behavior that fixed wing aircraft are incapable of which makes study in flapping wing aerodynamics attractive to Micro Air Vehicle designers. Due to the complexity of flapping wing kinematics, CFD has become an effective method of investigation into this problem. A limitation in this method however is accurately assessing the reliability of results obtained under different conditions. This paper presents the results of comparisons made between 2D flow simulations and PIV experiments of a pitching quasi-elliptical wing section in forward motion. Numerical solutions were obtained using the OpenFOAM CFD toolbox. Specifically, a Finite Volume solver of the Incompressible Navier-Stokes equations using the PISO algorithm was used, with a General Grid Interface used to handle the rotation and an inertial reference frame to handle translation. Comparisons are made for Reynolds numbers of 500 to 1500 thus negating the necessity for turbulence models. The solutions were compared to experimental results of planar Digital PIV flow measurements in the Monash University Water channel for a solely pitching case. Good agreement is observed in all cases, some more so than others. Further study using this numerical method is certainly warranted.
Nomenclature
Frequency (Hz) Reference length (m) Strouhal number Reynolds number Pitch angle amplitude (deg) Chord length (m) Period of pitching motion (s) Pitch angle function (rad) Surface area ( )
ℎℎ master patch face slave patch face Number of master patch faces Number of slave patch faces factor of master patch patch Weight face to slave patch face ∩
Intersection areas of master patch face and slave patch face
I. Introduction
I
NSECT flight has stifled stifled the efforts of conventional aerodynamic aerodynamic wing theory as developed by Lanchester (1907) and Prandtl (1914-1918) for quite some time now. This is because according to this theory the wing of the insect cannot generate enough lifting force to allow an insect of any size to stay aloft, let alone perform complex maneuvers like hovering hovering and rapid trajectory alteration. alteration. However, recent efforts efforts in visualizing flow flow over insect wings have revealed a complex interaction of different aerodynamic flow phenomena which, if understood more thoroughly will reveal the principles of insect flight – these can be used in engineering engineering applications such such as Micro Air Vehicles Vehicles (MAVs) (Sane, 2003). The mystery of insect flight and flapping wings in particular persisted until the discovery of the unsteady vortical flow field and especially the generation of the leading edge vortex. The potential benefit of vortices attached the wing has been discussed previously (Maxworthy, 1979) (Dickinson & Gotz, 1993). The significance of the presence
[email protected], 1 Student, School of Aerospace, Mechanical Engineering, Monash University Clayton campus,
[email protected], StudAIAA 2 Professor, School of Aerospace, Mechanical Engineering, Monash University Clayton campus,
[email protected],
[email protected], SMAIAA
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of the leading-edge vortex (LEV) which is generated over the top surface of the moving wing was only noted relatively recently, however (Ellington, van den Berg, Willmott, & Thomas, 1996). It is this vortex that affects the pressure distribution over the the surface of the wing, thus increasing the lift force force to values much higher than predicted by wing theory. This vortex is known to be three-dimensional however and its stability is not yet fully understood and does appear to heavily depend on the movement characteristics of the wing and the Reynolds number. Still more recent studies have revealed that flapping foil aerodynamics involves vortical shedding that can form either a periodic or chaotic wake pattern depending on the kinematics of the movement. The most significant parameters have been shown to be the Reynolds number and the flapping amplitude (Thaweewat & Bos, 2009). It has been seen that the origin of the vortex developed on the leading edge is the roll-up of shear layers created in highly viscous flows. Indeed these flows are found in the low Reynolds number flight regime in which insects (and hence MAVs) operate in, due to the scale of their wings. It is hypothesized that the wing kinematics influences the development of the shear layer direction and flow accelerations which in turn will influence the evolution of the leading edge vortex. Expanding this area of study is part of this study’s objective.
Figure 1. Aircraft mass vs. Flight regime
The problem of flow over insect wings has been investigated both numerically and experimentally through various methods and techniques (Sane, 2003). On the experimental side there is the ubiquitous Particle Image Velocimetry (Ching, Lim, & Soria, 2004), hotwire anemometry (Chang & Eun, 2003) and dye flow visualizations. On the numerical side, various techniques have been employed in a large body of literature such as DNS, RANS, LES, Vortex method and so on in various flow regimes (Wang, 2008). Additionally, while differences have been observed to exist between three and two dimensional studies (Wang Z. J., 2000), two dimensional simulations simulations still garner interest from researchers due to the vast array of similarities they share with their three dimensional counterparts. On this basis and the relatively large computational expense of 3D simulations, the present study focuses on 2D simulations. This paper presents the results of a Finite Volume solver of the incompressible Navier-Stokes equations found in an open-source software known as OpenFOAM (Weller, Greenshields, & Janssens, 2004). The degree of accuracy or relevance that the results of any numerical method have to offer is always a question that appears at the fore of any such study. Thus, this study aims to address this issue by presenting the consistency between the numerical results and experimental results. OpenFOAM is an open-source developed suite of finite volume solvers and utilities which for obvious reasons is an attractive alternative to commercial codes. Furthermore it provides one with an object oriented programming environment allowing the user far more flexibility than commercial codes do. OpenFOAM has been validated in several PhD and MSc theses around the world and has been adapted for purposes such as free surface flow, multi-phase flow, Direct Numerical Simulation, Large Eddy Simulation, turbulence modeling and fluid-structure interaction (Jasak, 2010). Thus by demonstrating the capability of OpenFOAM in the capacity of simulating moving boundary problems like flapping wings the present study aims to open the doors to further work in this area using this tool. 2 American Institute of Aeronautics and Astronautics
II. Flow equations and Numerical Method The equations to be solved in the present study are a simplification of the Navier-Stokes equations (Anderson, 1991). The basis of simplification is the assumption of incompressibility of the flow which is the case for the flight regime of insect flight (Williamson, 1995). This is when the velocity of the flow is nowhere greater than 0.3 times the speed of sound and we can neglect thermal expansion effects. These equations then become,
⋅ ∗ = 0 1. ∗ + ⋅ ∗∗ = −∗ + 1 ∗ Momentum 2. Cons. Here, the, the main variables, ,, , and and have been scaled with their reference values as follows: ∗ = ∕ ∞, ∗ = ⋅ , ∗ = ⁄, ∗ = ⁄∞ 3. Furthermore the variables and in equations (1) and (2) are two extra dimensionless variables known as the Strouhal and Reynolds numbers defined as, = ⁄ and = ∞ ⁄. These two variables represent the ratio of time scales from that of the convective transport to the motion of the body ( , and that of the viscous transport to the convective transport respectively (). Mass cons.
As stated earlier, a solver using the second order Finite Volume discretisation method (FVM) was used. The advantage of this method is that it may be used effectively on an arbitrary arrangement of polyhedral cells and it is the basis of many widely used CFD packages. Once the discretisation is performed on equations (1) and (2), an iterative solving algorithm was implemented known as the PISO algorithm (Ferziger & Peric, 2002) which is purportedly much more efficient than the SIMPLE algorithm which is still widely used in many solvers. Furthermore, since the flow is assumed laminar for the Reynolds numbers considered (on the order of 200 to 2000) thus negating the necessity for turbulence models, the method may be considered as a Direct Numerical Simulation where all scales of motion of the fluid are considered to be resolved by the solution.
III. Problem definition In the summer of 2005, two dimensional Digital PIV (DPIV) visualizations were carried out on the flow over a symmetric aerofoil pitching about the mid-chord in the context of insect aerodynamics (Green, Parker, & Soria, 2005). This was performed in a water tunnel at the Laboratory of Turbulence Research for Aerospace and Combustion (LTRA&C). The working section of the water tunnel measures 1m long with a 0.25m 2 cross section. Furthermore, turbulence turbulence intensity levels in the core region are purportedly less than 0.35%. Measurements were taken of flow visualized in a single 2D plane at the mid-span of the wing. While the flow is intrinsically 3D, for reasons outlined in section 1.2 these flow features were not pursued.Thus, just one Pixelfly CCD camera with an array size of 1280 by 1024 pixels was required for the experiment with 11 m hollow glass spheres to be used as seeding particles. A Nd-Yag laser was responsible for illumination, providing a 3mm thick laser sheet shone on the mid-span section of the wing.
Figure 2. Schematic of problem setup
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Since the present study aims to corroborate these experimental results, 2D simulations of the aforementioned problem were performed. performed. A schematic schematic is shown in figure 2 while the pitching displacement displacement profile may be described described by equation 4 in terms of the relevant non-dimensional parameters and constants. Three aspects of the kinematic parameters were were varied, namely namely the Reynolds number, number, Strouhal number and pitch angle amplitude, amplitude, .
⋅⋅ ⋅ ⋅ ⋅) radian = sin(2 sin (2 ⋅ tan
4.
IV. Dynamic Mesh method Since the problem is one of a moving boundary, this presents challenges in the way of discretising (meshing) the solution domain since this must evolve in time. Various methods have been adopted in the past and the key criteria to a dynamic mesh method are accuracy and efficiency, both equally important. In this study rotation of the airfoil is treated using a sliding interface method known as the General Grid Interface (GGI) in OpenFOAM. The GGI algorithm is very efficient and fully parallelized meaning the 3D extension of the problem is certainly permissible using this method (Beaudoin & Jasak, 2000).
Figure 5. Example GGI configuration
In the GGI method, it is the interface between two meshes which is treated at the matrix level in order to balance the mass flow through the interface. A GGI is represented by a meeting of a master and shadow patch. The master patch is composed of faces while the shadow patch is composed of faces. During rotation, these patches do not meet seamlessly and may intersect each other. Thus the field variables on a particular patch are to be calculated using the values on the corresponding patch as necessary.
= ∑ ⋅
5.
= ∑ ⋅
6.
Equations 5 and 6 dictate that a field variable value on a particular face of a patch is calculated as a weighted sum of the values on neighboring faces. These weighting factors are calculated through a geometric consideration of 4 American Institute of Aeronautics and Astronautics
the ratio of intersection of areas of a patch face with each of its neighboring patches. Using the GGI method, a circular interface was required at the meeting between the rotating part of the mesh and the outer static part. A conventional OH type grid is adopted with a high density structured conformal grid used in the vicinity of the wing and wake region and tetrahedral mesh elements to inflate the grid to the outer boundaries. Figure 6 shows a schematic of the mesh used to solve the problem.
Figure 6. Solution domain (left) and inner rotating mesh around the wing (right)
V. Results and Discussion 7. Qualitative comparison
Two representative cases from the experiment were taken for simulation. This section gives a comparative analysis of the results. Figure 7 shows the comparison of in-plane vorticity distributions for computations and experiment for representative test case 1 where ∞ and seconds ( Hz). The comparisons are made in intervals of 10 degrees of pitching motion starting from . The spatial dimensions are scaled by the reciprocal of the chord length, , the velocity is scaled by the reciprocal of the free-stream velocity, ∞ and the vorticity is scaled by the chord length divided by the velocity scale, . ∞
= 0.16667 / 1⁄
= 2.18 = 0.46 = 0° ⁄
1 ⁄
The comparisons indicate good accuracy for the numerical simulations and most of the features of the flow present in the PIV visualizations are also present in the simulations. In figure 7 (a) we can clearly see the roll up of the LEV in the early stages of the pitching motion and at or around -10 degrees displacement the LEV detaches from the leading edge. Shortly after this, the TEV also detaches from the trailing edge and begins to convect downstream. Meanwhile the shear layer created at the trailing edge during the birth of the TEV begins to move up the surface of the wing as indicated in figure 7 (d). Figure 7 (e) then shows the LEV being absorbed into the advancing shear layer as the wing then begins to pitch upward and move into the LEV. Since the shear layer and the LEV are of the same sign vorticity, they are able to merge constructively. Subsequently, as seen in figure 7 (g) at 20 degrees positive displacement of the wing, the shear layer has reached the leading edge of the wing which is about to pitch down. This shear layer contributes to the roll-up development of the next LEV (with vorticity of direction into the page) which explains why the LEV is created stronger than the TEV.
=0
It should be noted that since we are dealing with symmetrical motion, the flow is symmetrical about the line. Thus although the flow on the bottom of the wing is not captured in the experimental visualizations due to the laser shadow cast by the wing, this region of the flow is simply the reflection of the top side about with a phase lag of half a period of motion, or .
/2
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=0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 7. Vorticity contour contour solution comparisons for
= , = .. , = ° at ° intervals of motion with computation above and PIV below.
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 8. Vorticity contour contour solution comparisons for
= , = .. , = ° at ° intervals of motion with numerics above and PIV below.
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Figure 8 depicts comparisons of vorticity distributions for representative test case 2. The velocity in this case was
∞ = 0.05 m/s and the period, = 1.09 seconds which corresponds to a pitch frequency of = 0.92 Hz. Scaling was done as before and the contour colors range from red to blue and the vorticity from −15 1/s to 15 1/s. Once again the comparisons show that the simulations are quite agreeable to the experimental PIV results with all the flow features observed in the experimental results being present in the simulations.
We can see in figure 8 (a) that the LEV has already been created in the first half of the pitch stroke while the TEV is only beginning to form. Then, in figure 8 (c) we see that at the TEV has formed and detached while the LEV is proceeding down-stream across the wing surface. As before in test number 1, we see that the shear layer is advancing across the top surface of the wing – however however we also see that while it comes into contact with the LEV, they do not merge and the LEV proceeds down-stream and absorbs the rear of the shear layer. Also worth noting is that in figure 8 (e) we can see that the LEV on the upper surface of the wing is paired with a counter rotating vortex of much smaller proportions which is evident in the experimental results also.
= −20°
= ⁄2
Thus, since the LEV is not absorbed, it is released at the trailing edge to join the TEV formed at time from when the LEV was formed. This produces an oblique wake in which counter rotating vortex pairs travel downstream. The bottom part of the wake consists of pairs where a clockwise rotating vortex leads the pair while the top part of the wake consists of pairs where the anti-clockwise cortex leads the pairs. These pairs produce jets of fluid moving away from the wing in either direction (up and down). Some differences arise between the simulations and experiment however, most notably in the distribution of
Figure 8 depicts comparisons of vorticity distributions for representative test case 2. The velocity in this case was
∞ = 0.05 m/s and the period, = 1.09 seconds which corresponds to a pitch frequency of = 0.92 Hz. Scaling was done as before and the contour colors range from red to blue and the vorticity from −15 1/s to 15 1/s. Once again the comparisons show that the simulations are quite agreeable to the experimental PIV results with all the flow features observed in the experimental results being present in the simulations.
We can see in figure 8 (a) that the LEV has already been created in the first half of the pitch stroke while the TEV is only beginning to form. Then, in figure 8 (c) we see that at the TEV has formed and detached while the LEV is proceeding down-stream across the wing surface. As before in test number 1, we see that the shear layer is advancing across the top surface of the wing – however however we also see that while it comes into contact with the LEV, they do not merge and the LEV proceeds down-stream and absorbs the rear of the shear layer. Also worth noting is that in figure 8 (e) we can see that the LEV on the upper surface of the wing is paired with a counter rotating vortex of much smaller proportions which is evident in the experimental results also.
= −20°
= ⁄2
Thus, since the LEV is not absorbed, it is released at the trailing edge to join the TEV formed at time from when the LEV was formed. This produces an oblique wake in which counter rotating vortex pairs travel downstream. The bottom part of the wake consists of pairs where a clockwise rotating vortex leads the pair while the top part of the wake consists of pairs where the anti-clockwise cortex leads the pairs. These pairs produce jets of fluid moving away from the wing in either direction (up and down). Some differences arise between the simulations and experiment however, most notably in the distribution of vorticity around each vortex. It seems that the vortices generated in simulation are markedly more coherent and diffuse less when convecting down-stream. This is evident in figure 8 (a) where the right most TEV is very diffuse in the experiment while still retaining coherence in the simulation. These differences are better seen in the quantitative comparisons in section 3.2.2. 8. Quantitative comparison
We first consider the position of the LEV and TEV over time from the time time of birth for representative test 1 (the flow solution is pictured in figure 7). Figure 9 shows a favorable comparison of the movement of the LEV and TEV over time – time – the the simulations accurately predict the time of detachment of the vortices and their speed as they travel downstream. Furthermore, in figure 9, the variation of the peak vorticity of the LEV and TEV are shown as compared to that of the experiment. Again, the comparison is quite favorable with the initial rapid decay of both vortices being correctly predicted and a slowing slowing of the decay being observed later on. The The quicker decay of the LEV can be understood understood to be due to the interaction with the shear layer on the surface of the wing which is of oppositely signed vorticity as seen in figure 7 (c).
Vortex position vs Time
2.5 ) n o i t i s o p ( e t a n i
2
1.5
d r o o c x
LEV TEV ExpLEV ExpTEV
1
0.5
Vortex intensity vs Position
35 ) s / 1 ( y t i c i t r o v k a e P
30 LEV TEV ExpLEV ExpTEV
25 20 15 10 5 0
0 0
1
2
3
0
Time (s)
Figure 9. Comparison of vortex advection (left) and decay (right) for
1
2
3
x-coordinate
= , = .. , =
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Figure 10 shows the peak vorticity comparisons for the second test case. Naturally since a greater amount of activity is present in these test conditions (due to a much higher Reynolds number) the comparisons are not as favorable as for test number 1. Nonetheless patent agreement is found between simulation and experiment in the behavior of the vortices. Figure 10 clearly shows the accurate prediction of the vortex speed over time. The speed-up of the TEV is evident in both simulation and experiment and is due to the separation of the vortex from the shear layer. This can also be observed in visualizations of figure 8 (g) where the TEV is about to detach from the trailing edge. The LEV on the other hand can be seen to be speeding up at around 0.5 seconds which corresponds approximately to figure 8 (e) where the LEV is halfway across the upper surface of the wing. This is where the LEV splits the advancing shear layer into two parts and consumes the rear of the shear layer while proceeding down the remainder of the wing. Finally, when the LEV departs from the trailing edge the speed of the vortex begins to drop again. All of the above mentioned phenomena are observed in the computations and the experimental results are a good indication.
Vortex position vs Time
Vortex intensity vs position
) e t a n i
d r o o c x ( n o i t i s o P
LEV TEV ExpLEV
LEV
) s / 1 ( y t i c i t r o v k a e P
TEV ExpLEV ExpTEV
ExpTEV Time (s)
Vortex position (x-coordinate)
Figure 10. Comparison of vortex advection (left) and decay (right) for
= , = ., =
As before in figure 9, figure 10 shows the variation of vortex intensity with vortex position for the second test. We can see that while the large changes in the rate of decay of the vortex are correctly modeled by the simulations, smaller changes effected by complex phenomena are seemingly not represented. The LEV begins a strong rate of decay from birth but then slows almost simultaneously with the increase in velocity observed in figure 10 (left). This indicates a possible correlation between the vortex velocity and the rate of decay of the strength of the vortex. According to the experiment however, the TEV is still building up strength where the simulations indicate that it is decaying. Thus, differences arise concerning the formation and detaching of the TEV but agreement resumes once the TEV is released from the wing at around 1.6 seconds corresponding to figure 8 (e).
VI. Conclusions and Further work The preliminary results presented in this paper demonstrate the applicability of the incompressible fluid solvers in OpenFOAM and the GGI dynamic mesh method to low Reynolds number flapping wing flows. This has been done by comparing pitching wing simulations with DPIV experimental results. While only airfoil pitching has been considered thus far, full insect wing kinematics can easily be simulated by introducing translation via the application of a non-inertial reference frame (which simply involves the introduction of an extra term into the momentum equation). With these satisfactory validation test results, one may also confidently extend the study into considering three-dimensional flow as well as three-dimensional wing kinematics kinematics which are closer to that of insect flight. This is due to the efficiency and robustness robustness of the dynamic mesh method method used.
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Acknowledgments The author would like to acknowledge the guidance and supervision of Prof. Julio Soria and Brendon Anderson and the ongoing advice and support from fellow students and friends. In particular, the efforts of Dr. Melissa Green (currently working at the Department of Energy in the USA) in recovering and explaining the experimental data of the companion PIV experiments are to be recognized as a major contribution to the present work. Thanks are also extended to the various Monash Mechanical Engineering postgraduates who provided useful insight into understanding CFD and the Finite Volume Method. Finally the author also acknowledges the computational facility provided by the National National Computing Computing Infrastructure without which the present work work would not have been possible.
References 2D DPIV of a Pitching Aerofoil. Green, M, Parker, K and Soria, J. 2005. 2005, Fourth Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion. Aerodynamic efficiency of flapping flight: analysis of a two-stroke model. Wang, Z. J. 2008. Journal of Experimental Biology, pp. 234-238.
Anderson, J.D. 1991. Fundamentals of Aerodynamics, second edn. s.l. : McGraw-Hill Inc., 1991. Beaudoin, Martin and Jasak, Hrvoje. 2000. Turbomachinery section. University of Zagreb, Power Engineering Department. [Online] 2000. http://powerlab.fsb.hr/ped/kturb http://powerlab.fsb.hr/ped/kturbo/OpenFOAM/B o/OpenFOAM/Berlin2008/Sessio erlin2008/SessionIV/. nIV/.
Springer-Verlag, Ferziger, J.H. and Peric, M. 2002. Computational Methods for Fluid Dynamics, 3rd ed. Berlin : Springer-Verlag, 2002. Flow past an impulsively started oscillating Elliptical Cylinder. Ching, T.L., Lim, T.T. and Soria, J. 2004. 2004, 15th Australasian Fluid Mechanics Conference, pp. 13-17 December.
Jasak, Hrvoje. 2010. List of Papers: Hrvoje Jasak. Hrv's Homepage. [Online] 2010. [Cited: July 1st, 2010.] http://www.h.jasak.dial.pipex.com/. Leading-edge vortices in insect insect flight. Ellington, C. P., et al. 1996. 1996, Nature, pp. 626-630. Numerical study of vortex-wake interactions and performance of a two-dimensional flapping foil. Thaweewat, N. and Bos, F.M. 2009. 2009, 47th AIAA Aerospace Sciences Meeting, Orlando. reduced Frequency Effects on the Near-Wake of an Oscillating Elliptic Airfoil. Chang, Jo Won and Eun, HeeBong. 2003. 2003, KSME International Journal, pp. 1234-1245. The aerodynamics of insect flight. Sane, Sanjay P. 2003. 2003, The Journal of Experimental Biology, pp. 41914208. Vortex shedding and frequency selection in flapping flight. Wang, Z. Jane. 2000. 2000, Journal of Fluid Mech., pp. 323-341.
Weller, H, Greenshields, C and Janssens, A. 2004. OpenFOAM: The Open Source CFD Toolbox. OpenCFD. [Online] 2004. www. opencfd. co. uk/openfoam. Williamson, C.H.K. 1995. Fluid Vortices. s.l. : Kluwer Academic Publishers, 1995. Vol. 30.
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